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l, d>2, 4>3» ) of the "). «(*))A<*J( > «P. u)ds + JDi/is, xs(c, cp, u)), u{s)ds . a a u")€Mtn)-
= \y(s), R(s), L(s), K(s), p(s), E(s)], 0 Now set X J^-i(/ -s)x(s)ds
= g-\(t),
-00
becomes d_ dt
x(t)-XJA-i(.t-s)y(s)ds-g^(t)
0 X \A\(t-CO
SE(-CO,0]
s)x(s)ds =g)(t)> system (1.82)
Economic Dynamic Model
41 I
= A0x(t)+X
)A](t-s)x(s)ds+g](t)-o(l) + q(t). (1.83) 0 We have rewritten this linear neutral Volterra integrodifferential equation. Hereditary systems as dynamics of economic variables have been studied extensively. (See [2, 3, 7, 8, 9].) With -a(/) as a control variable available to the representative firm and with q{t) as government control instrument, the system (1.83) is a differential game of pursuit in the spirit of Hajek [15]. Government is the quarry and the firm is the pursuer. Government acts first with , T\, money supply, tariffs, investment, exchange rate, and preferential trade agreement policies. The firm reacts with autonomous consumption, autonomous investment, autonomous net export, autonomous real money demand, productivity, wage rates, income consumption intercept, income export intercept, and price intercept. It is expected that public investment is self-financing. The system of six key economic variables has the following dynamics, d_ x(t) - X dt
JA.i(t-s)y(s)ds-g-i{t) 0
t = A(,y(t) + X $A\(t -s)y(s)ds+ 0
g\(t) - p{t) + q{t).
Let x(t) be a solution of (1.83) defined for / > 0 with initial value g(t) = <(>(/), where § is a continuous function which is bounded and defined on (-oo, 0]. By the theorem of Wu [25] the solution is given by
x(t) = U(t)[x(0)-g_l(0)]
t + g-\ + 0
+ jU(t-s)[q(s)-p(s)+g](S)]ch,
jO(t-s)g_l(s)ds
(1.84)
where U is an n x n continuously differentiate matrix function which is the solution of the resolvent equation
42
Differential Models and Neutral Systems
dt
£/(/)-
JA-i(t-s)U(s)ds = A0U(t)+ JA\{t-s)U(s)ds.
(1.85)
U(0) = I, identity matrix. In the sequence we treat the system, d_ dt
c(t)-\JA-\(t-s)x(s)ds
(1.86)
Aox(t) + X jA\ (t - s)x(s)ds, -p(t) + q(t)
p(t)eP
qit)eQczEn,
which is associated with (1.83). Dual to the linear game (1.83) is the control system d_ x(t)-k dt
t
f/Li(f - s)x(s)ds
■ AQx(t) + X \Al(t-s)x(s)ds-u(t)
(1.87)
where u(t) e W c E" and W is the Pontryagin difference of sets, defined by W = (P + kerU(ti -s))*Q.
(1.88)
U with the matrix solution of (1.85). The Pontryagin difference of sets is used by Hajek [15, p. 109]. The computation of £/can be done as explained in Chukwu [7, p. 46] by the method of steps. A numerical approach which extends the work of Hirsch [6] is the focus of recent effort by the author. Definition 1.1. We say that the system (1.87) is Euclidean controllable on the interval [0,t\] if for every initial function §:(-«>, 0]-> En which is continuous and bounded with
Economic Dynamic Model
43
(i) For each t z[0,t\] the value pit) depends on q(t) only (and of course on xu = ¥$\ 1, At /'= -1,0,1 and A.). (ii) The pair of controls p, q so obtained is such that the solution of (1.83) satisfies *('1>>A<7)=*1. *(0, *o. A ) = *o • In the next chapter we state and prove our main results.
2.
Main Results
First we shall state a controllability result for (1.87) with - u(t) = Bu(t), B an n x m matrix. Define the controllability matrix
//(0,n)=
\u(t\-s)BB*U*(t\-s)ds
(2.1)
0
where the star (*) denotes the matrix transpose. Theorem 2.1. Assume that (i)
A_\(t), A\(t) are n x n continuously differentiable matrix valued functions. (ii) AQ is n x n constant matrix, and B, n x m constant matrix. (iii) Suppose the symmetric matrix H(0, t\) is nonsingular for some t\ > 0. Then system (1.87) is Euclidean controllable. Proof. Define the control function u* by
u*(f) = B*U*(ti-t)W-1
(0,tx) xx
-U(t{X*(0)-g_i(0))-g_j(/,)
1
- Jr(/! -s)g_x(s)ds0
Jt/(/i -s)gl(s)ds
(2.2)
0
System (1.87) has the solution,
45
46
Differential Models and Neutral Systems t x(t) = U(t)[y(0) - g_x (0)] + g_x (0 + JY(t - s)g_x (s)ds 0 t
+ \u(t-s)[Bv(s)+gx(s)]ds.
(2.3)
0
Therefore if $ and y\ are given and we use the control u in (2.2), then the solution which begins at ty(0) - yo e En with all the past history <j>, shall have 'i
*('!) = W\ )W0) - g-\ (0)] + g-i (/i) + \Ht\ - s)g-\ (s)ds 0 'i
+ jt/(r, - s)BB * U * (tx -1) x W~X (0, tx) ■ [xx - U(xx - U(tx X*(0) - g_x (0)) 0 'i
'i
'i
- g - l ( ' l ) - J ^ ' l - s)g-\(*)ds - ju(tx - s)gx(s)ds) + ju(tx - s)gx(s)ds 0 0 0 'i
= U(tx)[x(0) - gX(0)} + g_,(/,) + j>(r, - s)g_x(s)ds + xx- U(tx)[x(0) - g_!(0)] 0 'i
'i
'i
- S-l('l) ~ JHti - s)g-\(s)ds - p(tx - s)gx(s)ds + ju(ti - s)gx(s)ds = xx. 0
0
0
Hence system (1.87) is Euclidean controllable. It will be interesting to obtain from the non-singularity of //(0, tx) in (2.1) computational conditions in terms of the systems coefficients^/ for Euclidean controllability. If (ii) is invalid and (1.87) not controllable then the "solidarity function" q{f) =
Main Results
47
B2(t)v(i) can be brought to ensure the controllability of (1.87) with - u(t) = Bv{i) = (B\ + B2)v(t). An adequate amount of public government intervention q{t) is needed. We now treat the pursuit game (1.83) or equivalently. t 0 t
+ XJMI-
s)x(s)ds + g,(0 - p{t) + q(t),
(2.4)
0
with o(t)€PaEn,
q(t)*Q.
Theorem 2.2 (Duality Theorem of Complete Capture). Assume 0 e Q and P compact. There is a complete Euclidean space capture everywhere at time t\ for game (2.4) if, and only if the associated linear neutral Volterra integrodifferential control system (1.87) with control (1.88) is Euclidean space controllable at time t\. Furthermore a(q, t) = u(t) + q modulo ker (U(t\ -1) for all q e Q, t e [0, /]]) can be used to determine a suitable strategy from u e Z,oo([0, t\], U) in (1.87), and vice versa. Here u{t) + q = o(q, t) + ker U(t\ - f)see 2.11 and Hajek [15, p. 102]. Proof. Assume that there is complete Euclidean space capture everywhere at time /] for (1.83). It now follows from the definitions that there is a mapping a : Q x [0, t\] -> P such that for any quarry control q e Ioo([0, t\], Q) the map p, pit) = o(q(t), t) is a pursuer control; and p and q steer any arbitrary if with <|>(0) = %o to a n y *1 6 En m ^me '1 : x(tu*,p,q)=x(tl) 'i
= xx =U(trtx(0)-g-\(0)] 'i
+ jU(li -s)g.i(s)ds+jU0\ 0
0
+ g-i(f\)
(2-5)
'i
-s)g](s)ds-jU(t] 0
-s)[p(s)-q(s)]ds,
48
Differential Models and Neutral Systems
or i
(r,,
(2.6)
where i
JC(/1,4>,0,
0)= l/(ri)[x(0)- g-i(0)] +g_!(/,) + Jl/(/, -
s)g-i(s)ds
0 '1
+ J*/('i -*)*i(*)*. o
(2.7)
For the quarry control 0, (0 e 0 , 'i
x(t], (j., 0,0) = Jt/(/, - j)«(s)& + x, ,
(2.8)
0
where n(s) = a(0, .$)• If we take any point q e Q, and time / e [0, t\] and consider the piecewise constant quarry control with 0 € [0, /] and q e [x, i\\, then we deduce from (2.6) that <
JC(/1,4>,0,0)=
JU(1{ -s)u(s)ds+xl 0
'l +
Jl/(n -s)[a(q,s)-q]ds. f
If we subtract (2.9) from (2.8), then 'i
]*£/(/, - s)[u(s) + q- a(q, s)]ds = 0,
for all t e [0, t\}. Since the integrand is independent of / we have
(2.9)
Main Results
49
U(ti-sMs)+g-o(q,s)] = 0,
(2.10)
almost all s e [0, l\]. By the definition of kernel reinterpret this as u(s) + qe o(q, s) + ker U(tx - s).
(2.11)
Since a has values in P, u(s) + q e P + ker U{t\ - s). From Hajek's lemma [15, p. 59] u(s) + Q c P + ker £/(/, - 5) «.
x(/l,M,0)= J^('I -*)«(*)*+.V1, 0
or '1
*1=.K'l,,0)- JUOI -s)u(s)ds. 0
Since *(0,*,0) = {/(0X*(0)-g_i(0)]+g_ 1 (0)=*(0), and equation (2.11) proves the assertion on the control set, then one direction of the proof is complete. Conversely assume that (1.87) is Euclidean controllable at time t\, and let <>| G C((-oo, 0], En), ($(0) = x(Q)), x\ e En be given. Suppose u e Ioo([0, t\], W) is the admissible control which steers (<j)(0) = x(Q)), to x\:
50
Differential Models and Neutral Systems
jr(0,fcii) = x(0),
x(/i,<M)=*l-
Then, 'i
x, =*(/,,♦, 0 , 0 ) - Jt/ft
-s)u(s)ds,
0
where n(s) e W(s) yields u(s) + q eP + ker U(t\ - s). Apply Filippov's Lemma in the form stated in [15, p. 119] and construct a pursuer strategy: There exist measurable mappings, a:0x[O,fi]->/ , ,u:gx[O,/i]->kert/(/i,-), such that u(s)+ q=a(q,s)
+ u(q,s).
Since the values of a lie on the compact set P, a e Loo[(0, t\), P]. Clearly a is admissible for any quarry control q, since for any q e £oo([0, t\], Q), a - q = u - u. Thus the solution of (2.4) at t\ with the admissible pair a and q and initial § e C([-oo, 0], En), (<|)(0) =.yo)> is x(t\, <> |,
'V x(t\A,°,q)=x(nA,0,0)-
)U(t\ ~s)[c(q{s)s)-
q(s)]ds,
0
(wherex{t\, <(), 0, 0) is as in (2.7)). The proof is complete. If we assume a nonlinear A.\ the adaptation of the proof is slightly complicated but straightforward. One uses an adaptation of nonlinear variation of parameter formula such as is contained in [7, p. 102]. In (1.25) the control u(t) takes values in the control set
Main Results
51
W(t) = (/> + ker U{t\ - t))*Q = {u:u + Qc P + kert/(/! - / ) , 0
0
(2.12)
is necessary and sufficient for the domain of null controllability (
52
Differential Models and Neutral Systems
\u(li-s)u{s)ds:u(s)eW(s)
is the reachable set of (1.87). Thus zero is in the domain of null controllability of (1.87), which by the duality theorem coincides with that of (1.83).
3.
Economic Interpretation and Fundamental Economic Principles
From our discussion in Chapter 1 it is reasonable to postulate that d_ x(t)-\ dt
\A_\(t - s)x(s)ds = AQX(t) + X JA{(t - s)x(s)ds - pit) + q{t), -QO
(3.1) describes the dynamics of income (GNP), employment, value of capital stock, price, and cumulative balance of payment. The function q(i) in (1.75) describes government intervention, i.e., policy instrument. The function pit), is private initiative, describes the firms' strategy. We assume instantaneous stroboscopic firms' response - the "snap decision rule", p. Income can grow from a past path (j) and an initial point <|>(0) = xo to the prescribed target, using p and q strategies, if and only if, the control system, d_ t(t) dt
\A_\(t - s)x(s)ds = AQX(t) + X \A\(t - s)x(s)ds - u(t),
(3.2)
can be controlled from <>| to the target at time t\. Though zero is the target treated, any arbitrary fixed point jq e En can be a target (See [7, p. 399]). The firms' control strategy which reacts to any government action q, is J(q, f) = v(t) + q modulo ker U(t\, f) where v is an admissible control for (3.2). If Q is the totality of government power, and if the algebraic sum of government action may be zero, (0 e Q), and if the firms' economic capacity P is limited and small (compact) the optimal control strategy of the firm lies in W, which is defined by W(t) = (P + ker U(t\, 0 * 0 . We now 53
54
Differential Models and Neutral Systems
have the following universal principles on the limitation of government economic power. QcInt(P + kaU(ti-t)).
(3.3)
Principle 3.1. No initial value of income and past path of income can be controlled to the target unless the firms initiative (consumption, investment, export, real money demand) contains solidarity, i.e., government intervention (investment, taxation, money supply) as a subset. Here the totality of all possible government control strategies in 1.75, Q, is defined as solidarity. Principle 3.2. To ensure income growth to a target it is necessary that the firm's capacity or initiative and its internal power for waste (ker U{t\, t)) dominate whatever government can do. Principle 3.3. If the growth of income is not controllable to a target when government intervention is effectively zero the introduction of public investment, taxation and manipulation of money supply and public investment can enforce controllability to the desired target. Only a proper amount of intervention is needed, since the intervention can make matters worse. It is obvious that these principles remain valid as they were for systems of neutral type treated in [7, 8, 9, 10]. Remark 3.1: The application of these principles for specific nations rests on the calculations of some key coefficients in the dynamics, by the socalled method of system identification of hereditary systems. The argument of Robert E. Lucas [19] that the coefficients do not describe the real structure of the economy is mitigated in our case. As Lucas suggested we have taken into account the dependence of private decision rule (pit)) on the government choice of a policy rule (q(t)). An objection to our paper may be made: p(t) = fq(t)), whereas "reaction to changing rules" requires p(t) = fq(0), (s < t). The "snap decision rule is too much." This objection is demolished by the following famous argument of Hajek [15, p. 56]. Suppose the firm is allowed to observe and retain the past history of its own control p, and of the state variable x governed by the equation t
m(t) = Lx(0 - pit) + q(t), where Mf) = x(t) -\J A_x(t - s)x(s)ds -
Economic Interpretation
55
t
Lx(t)+ AQx(t) - X J A_\(t - s)x(s)ds - q\(t). Then at each t the rate of change 0
is available ™,-c\ \\ m(s-h)-m{s) , m m(s)= urn h-+0+ -h for all s
4(0=lim+-
' \q(s)ds, t-h
almost everywhere, by Lebesgue Theorem. Therefore at almost all times t the current value q(t) of government control choice for setting up its own policy is available. If one is unhappy with this conclusion then there must be realistic time delays. In this case Hajek's treatment of the case with delays in the pursuer's response [15,p. 96] may be appropriate, for example:
p(t)=f(q(t-h),t) the dynamical system becomes
dt
y(t)-X
\A-i(t-s)x(s)ds
= Aoy(t) + K JAi(t- s)y(s)ds - p(t-h) + q(t).
Differential Models and Neutral Systems
56
Thus the control set for the dual system can be defined as in [15, p. 97, eq. 5], where x = Ax- u(t), ueU = p*eA^~^Q. We now address an objection to Principle 3.1 and Principle 3.2 which are sometimes brought up in connection with the inclusion relation between government control Q and private strategy set P. Since government could cooperate it is claimed that (2.12) is false. But Q is symmetric (i.e., q e Q implies - q e Q), therefore government and the firms can {sometimes) cooperate when W = (P + kerU(ti-s))*Q, i.e.,
Q c (P + ker U(tx -s))
0<s
and (3.3) holds. But if government and the firms always cooperate then, of course W = P + Q, can be taken as the control set for (1.21) with u(t) = q{t) - p{t). The condition for controllability is then the usual one. But is it realistic to assume that government and the firms will always cooperate? In our analysis we have assumed instantaneous stroboscopic firms response to government control strategy [15, p. 55]. The response is nonanticipatory in the sense of Hajek [15, p. 39]. More general firms' response will presumably yield a wide result. With our assumption our result (3.3) is valid. The duality theorem gives us a control strategy which can be optimal feedback, allowing polices to change as the state of the income changes, [39, pp. 301-304]. Note that our system is allowed to pursue maximization of profit [10, p. 280] while both are committed to time optimal growth with perhaps minimum investment. The most interesting argument against the result in this chapter is the criticism that it is "not even scientific", "only mathematical", and it has no established relation to the real world. One could argue that Newton's magnum opus, the Principia, and the greatest scientific work of all times was "not yet scientific" when it was published: He had not performed the experiments and validated/deduced his three laws of motion with data from the "real world" but later the laws of motion were put to work to solve many problems both on earth and in heavens. It was able to predict the path of a
Economic Interpretation
57
comet and when it would return. Edmond Haley made an independent check for Newton: Haley's comet returned as it should because it obeyed the mathematical universal laws of motions of Newton. A similar validation may be undertaken for our principles of income growth, using the computer and the past trajectories. This philosophy is exactly that of most recent note worthy books by Taylor [24]. Since our mathematics is valid, validation from experience- the usual economics variables time series can now begin. This is the philosophical position of Taylor [24, p. 4]. If by parameter estimation of the coefficients and the empirical evaluation of the controls the dynamics are true and realistic, then the principles we have enunciated are epoch making and have profound consequences for the organization of society. We now explore the simple case models of Canada and India.
4.
Economic Hereditary Model of Canada
We postulate a simplified version of our key equation to avoid collection of data from infinite past. We use our MATLAB program and data from International Statistic Yearbook 1994 to deduce the linear functional differential equation of neutral type as the economic model of Canada, India, and USA. The system is given by: i(t) - A-\Xt -h) = AQXQ) + Axx(t -h) + Bxq{t) + BlP(t). Let
A(\) =XI- A_ike~X -AQ- A\e~X B = [Bh
Bj],
a = [A(X),
B],
cc = [KI,
-A_\\,
c = [cc,
B].
Because in appendix 1, rank (a) = 6, rank (c) = 6, the model is controllable. See Salamon [23]. One needs to test constrained controllability to a given target. This will be explored elsewhere, after we have studied stability.
59
5. Soft Landing of Key Economic Indicators with Private and Government Controls under Scarcity We now consider the controllability with control constraints of the system,
A
x(t)-~k
\A\(t
-s)x{s)ds = Atfc(t)+\
JA+\(t-s)x(s)ds-u(t),
(5.1)
u(t) e W(t) = {P + ker U(t\ - s))*Q
or d_ x(t)-JA-X(t-s)x(s)ds-g-l{t) dt
= AQx(t) + JAx(t-s)x(s)ds-u(t)
+ gl(t),
0
(5.2) where g _ l ( 0 = J AA(t -s)x(s)ds, -oo
fl(/)=
JAi{t-s)x{s)ds.
(5.3)
-oo
The solution of (5.2) is t x(0 = U(t)[x(0) - g_, (0)] + g_i (/) + J0(f - s)g_i (s)ds
61
62
Differential Models and Neutral Systems t +
\u(t-s)[gl(s)-u(s)]ds. 0
(5.4)
See [25], and [32]. Let
//(/,♦) = f/(/)[*(0)-g_i(0)] + g_,(/)+ J0(t-s)g_l(s)ds.
(5.5)
0
Then / *(/) = #(/,♦)+ Jl/(/-j)[ f f l (j)-«(j)]*. 0
(5.6)
Since ty(s) = x(s) se(-oo,0], //(/,0)=0. Definition 5.1 The Euclidean reachable set to XQ = y(0) at time s > 0, Rs(i|/(0)) is the set of all Euclidean states from which a given point \\i can be reached at time /. The system (5.2) is constrained controllable to *o = v|/(0) at time t if all points in same neighborhood of y can be steered to v|/(0) in time t by some control. Definition 5.2 The reachable set to \\i at time s > 0, J^(vj/) is the set of all states from which a given point \\i can be reached at time /. The system (5.2) is said to be constrained controllable to \\t at time t if all points in some neighborhood of \\i can be steered to \\i in time / by some admissible control. The reachable set to v|/(0) is
R(t) = U(t)[x(0) - g_x (0)] + g_, (/) + jt/(t - 5)g_, {s)ds
I
0
Soft Landing
63 + ju(t - s)[g-\(s) - u(s)]ds
(5.7)
u e W(t)
0
The reachable set to \\i is KKHO = {*XG> W> u):
u e
^(0)-
Theorem 5.1 The point JCI e Int R(t, \y) for some t > 0, if and only if 0 e Int K(t), where R{i) is the reachable set of the auxiliary control system
dt
t(0 - X \A-\(f - s)x(s)ds = Aox(t) + X J/Li(/ - s)x(s)ds - v(t), (5.8)
where V(t,s)zV
+ Ax«-s)(x0-gi(0))
+
gl(s)+U*(t-sXA-i(s)Ws)+A0g-l(s)). (5.9)
Proof. Let S denote the unit ball centered at the origin in En. Then §(0) = XQ e Rs(\\)(0)) implies that there exists a 8 > 0 such that
<|>(0) + 5Sc ^(V(0)) = H(t, <)>)+ \u{t - s)[®(s) - W]ds = H(t, if)
+ ju(t-s)gl(s)-lu(t-s)Wds
= H(t^) + ^U(.t-s)gl(s)+^t
where
lit
=-ju(t-s)Wds0
This is the reachable set at time t with controls in W. It follows that
(5.10)
Differential Models and Neutral Systems
64
bS c //(/,$) -
ds
[H(t,
It is true that d_ U(t)-^A-\{t-s)U(s)ds dt
= A0U(t)+ JA\(t -s)U(s)ds.
Therefore d dt
[H(t, <> | ) - «0)]= A-\(0)U(t) + AoU(t) +$A(t - s)U(s)ds WO)-g-i(O)) Lo (5.11)
+ g-l(0 + O(0)g-\(0
A-\(0)U{t) + A0U(t)+
JA(t-s)U(s)ds (x(0)-g_i(0))
+ g-i(0 + t/(0)g-i(0We can therefore deduce that
8Sc |^-1(0) + 4) t/(*) + J / * I ( J - w)U(w)dw (*(0)-g_|(0))
+ g-l (0 + £/(0)*_i (0 + t/(r - s)[gl (s) - W(s)]ds.
(5.12)
Soft Landing
65
We now use 0(0) = A_x (0) + AQ ,
£_, (s) = A_x (-5)X(0)
to conclude that ft
55 c J-4_l(0) + AQ U(S) + JA](s - w)U(w)dw(x(0)V0
+ M_i(0) + ^o]g_i(j) + t/(r-*)[gi(j)- »f]ds,
g_i(0)) + ^_,(-s)x(0)
o<*<*.
Proceeding we note that if t
v(s) = U+(t -*)[yLi(0)+ A0]U(s)+ p , ( 5 - w)U(w)dw(x(0)- g_,(0)) 0 + /*_l(-*)*(0) +(/L, (0) + /
0<5
(5.13)
5 bScjU(t-s)v{s), 0 where L^ is the generalized inverse of U, S v(5) ei/-(/ - J ) M _ I ( 0 ) + AQ]U(S) + JAi(s - w)U(w)dw(x(0))-8-\ (0)) 0 + A-i(-s)x{0) +{A-i (0) + A0)g-i(s) + gi(5) - W(s).
(5.14)
Thus zero is in the interior of Euclidean reachable set at time / of the auxiliary system
dt
(l)-K JA-\(t - s)x(s)ds =Aox(t) + xJA\(t-s)x(s)ds+v(t).
(5.15)
66
Differential Models and Neutral Systems
This is equivalent to the Euclidean Controllability of (5.15) if v(/)=Cr(/). Theorem 2.1 implies that if rank C-n then (5.15) is Euclidean controllable. Thus any initial data can be driven to zero in the dynamics (5.15) by v in (5.13) if d_
x(t)-\
$A-\(t -s)x(s)ds = AQX(I) +
dt
\A\(t-s)x(s)ds
is globally exponentially stable. This ends the proof of the Theorem. This means that any initial data can be driven to x\ in finite time in the system (3.1) by the action of both government and private firms. From our definition it is clear that controllability implies the ability to control national income, GNP, y, interest rate R, employment, L, value of capital stock, k, prices, p (inflation) and cumulative balance of payment, E, simultaneously. In particular national growth rate, employment and inflation can be controlled simultaneously to a desired vector value. This seems to be happening in the USA under the Clinton-Gore presidency. See New York Times June 7, 1997 (p. 1, p. 22). To test whether a given country's economy is currently controllable we follow the following steps which are implied by the remarks above. Test for Controllability of National Economic State 1. Modify if possible the system (3.1) and identify the data X, A_\, AQ , A\, B\, B2 where -p(t) = B\r(t), q(t) = B2g(t). This is done by using the economic time series and regression by MATLAB. See the Appendix and Chapter 9. With the MATLAB program and the definition of the economic variables the best fitting formula is identified, using regression. The coefficients of the economic variables are used with Ndu.m to iden tify the dynamics. It is possible to identify a better fitting model by combining the methods of Kwon, etc [42] and of Chukwu [37]. A modi fied simple version of (3.1) is given by ■j [x(t) - M_,g(f - h)} = A0x{t)+ at
Aig(t)-
B\r(t) + B2g(t).
2. Select a feasible target i|/= [v)/i,v(/2,v(/3>M/4.M/5.V6]-
Soft Landing
67
3. Compute the constraint sets P, Q. 4. Compute the constraint set W{t) = (P + ker U(t\ - t))* Q, where U is the fundamental matrix of the system without controls. 5. Compute the control set V= W- A_\^(-h) + /*0V(0) + A\\y(-h). Test: OelnfV d 6. Examine the stability of (3.2), — [x(/)-/l_i(r -s)x(s)ds] = AQX(J) I
+ k \Ai(t - s)x(s)ds, or its simplified version above (see Theorem 5.2, -00
Theorem 5.4 below). d 7. Examine the controllability of ~ x(i)-X
JA-\(f-s)x(s)ds
A0x(t)
+ X \A^(t - s)x(s)ds - u(t) where u(t) = Bw(t), by computing the rank -00
of B which ensures controllability. 8. Conclude by the stability of 6 and the controllability of 7 the Euclidean constrained controllability for (3.1). Remark. Suppose system (3.2) is not stable we can select a feedback control strategy with the constraints in W: u(t) = k_{x(t - h) + %r(0 + *!*(/ - h) t + f*oi (s)x(t- s)ds which stabilizes (3.2), i.e., the system
d_ t(t)-X dt
§A-}(t -s)x{s)ds
~k\x{t-h) = (Ao+ko)x(t)
+ X 1 A\(t- s)x(s)ds + \k0\(s)x(t - s)ds + k\x(t - h), -00
-00
is uniformly asymptotically stable. For example, if the modified simple
Differential Models and Neutral Systems
68
version — [x{t)-lA-\{t-h)] at then,
= AQx(t) + A\(t -s)-u(t)
is considered and
4 WO - (4-i - k-l)*(t ~ h)] = (AQ- Ik0)x(t) + (A\ - /*,)*(/ - h) at
- Bw(t). We choose k.\, % k\ so that this system is stable, i.e. d. [x(t)-(A-\ dt
-Ik-i)x(t-h)]
= (Ao -lko)x(t)+(A
-lki)x(t-h)
is uniformly asymptotically stable. We continue with 7. Stability We need the global stability of the system d_ dt
s{x)-
^i(t-s)x(s)ds-g-i(t)
JAi(t-s)x(s)ds+gi(t),
= AQXO)+
-oo
(5.16) which is associated with the system t
t
d_ x(i)dt
JA^(t-s)x(s)ds-g-i(t) = Aox(t)+ JAi(f-s)x(s)ds+gl(t)
(5.17)
0
via the resolvent equation d_ (/(/)dt
JA^(t-s)U(s)ds = A0U(t) + U(t ~ s) U(s)ds . 0
£/(0) = /.
(5.18)
We assume /*_i(/),^i(') are n x n matrices continuous for t > 0 g.\(t), g\(t) are n vectors continuous for / e E with g\(t + T) =gi(t),g-\(t + T)= g-\(t) for constant T > 0. With these assumptions the solution can be represented by
69
Soft Landing
x(r) = l/(OW0)-g_ 1 (0)] + g_1(O+ Jl/(r-*)g_,(i)A+ Jt/(/-*)g,(5)
0
The next result states when all solutions of (5.20) converge to a ^-periodic solution. Theorem 5.2 If A_h AhU, UeLl[0,oo), then
dt
x(t)-
JA_i(t-s)x(s)ds-g-\(t) = Aox(t)+
JA](t-s)x(s)ds+g](t), -oo
(5.20) has a /"-periodic solution t
t
g_j(0+ Jl)(f-*)g_i(j)
ju(t-s)gl(s)ds,
(5.21)
and all solutions of (5.16) defined for t > 0 with bounded continuous functions on (- co, 0] as their initial values tend to this ^-periodic solution as t —> co. Conditions for stability are stated in the next Theorem. Theorem 5.3 Suppose that AQ is stable, B is a positive n x n matrix with ATB + BA = - /, and a, p are positive constants with aP-x^x <xT Bx< ^P-x^x. If \BA0A\ (t)+ BAi(t)dt
j\A^(t)\dt<\,
2(3'
a
jM-i(0l dt
Differential Models and Neutral Systems
70
then U,U,Del)[0,<»), with D=U(t)-\/Li(t-s)U(s)ds. Also Km D(t)=0 0 '-*00 and lim U(t) = 0. Hence *(/)-> g-i(t) as / -> <x>. As a consequence every <->QO
solution *(f) of (5.16) satisfies x(t) -> 0 as / -»• oo. The main global constrained null controllability result can now be stated. Theorem 5.4 Consider the auxiliary system d_ x(t)-XJA_i(t-s)x(s)ds dt
= AQx(t) + p ! (/ - s)x{s)ds + v(/),
(5.22)
where v(s) elf(t
-i)^-i(O)
+ AQ]U(S) + J ^ ( s - w)U(w)dw(x(0) -g_!(0))
+ A-X(-s)x(0)+(AA(0)+A0)g-l(s)
+ gi(s)-W(s),
(5.23)
and W = (P + ker U(t - s))*Q.
(5.24)
Suppose (5.22) is Euclidean controllable to the target 0 e E". Assume all the conditions of stability in Theorem 5.2 are valid for (5.22). Then (5.22) is globally null controllable with constraints. Hence system (3.1) can be driven from <> | to an arbitrary point vj/ in finite time using admissible controls, u or p + q. This Theorem is used in a computer test for controllability of economic systems. Consider the system
Soft Landing
7] t jC2(t-s)i(s)ds
x(t)-B2x(t-h)-
t = Ax(t) + B]X(t-h)+
-00
jc^l - s)x(s)ds, -00
(5.25) where A, B, Bj are constant n x n matrices C\, Cj are continuous matrix functions defined on [0, +oo] and +oo
ji|Q(e)||< + oo,
/=1)2
(5.26)
D
Theorem 5.5. Suppose A is stable, and there exist a positive definite matrix B and positive constants X\ X2 such that (i)
ATB +
BA=-I
X\\x\2<xTBx<X2\x\2 + 00
(")
l|/kll+ jiic2(0ll«*
(iii)
1-211BH-2 ll«i+Mfcll + J|iq , (/)||df+*i J||c}(/)||df >o A.,
+00
where
MII + ||Blll+ \\\Ci(t)\\dt k=+00
B2\\+ /l|C2(0ll
then the zero solution of (5.25) is asymptotically stable. Proofs for these Theorems are contained in Ref. 37 and Ref. 26 of Chapter 0.
5
-^^r!•„ram;wo-
^
C •
a
^
9
i
r
C
RR
C
`
•
R
^ Y Z Z
Ecw yR s
y
R R =
R IL LL ^ n Y
R Y ^
_ R s Tt w OI r G S ^ Y 2
N
C D
^ ^= Z^ Z G
-_,Z ='' ^ ^^ N
^^ R^ p h^~ N r3
^^
Z
=
.~. ^
r i
?
24 us
Pv
70 Sweden
O
21 Swltterland 22 Turkey 23 UK
h
N
2 Ainlrla 3 Belgium 4 Canada 5 Denmark 5 Finland 7 Franca " W r * t Germany • Greece 10 Iceland 11 Ireland 12 Italy 1 3 Japan M Lmembotiry 15 Netherlands 16 New Zealand I f Norway IS Portugal 19 Spain
1 A U5lrilia
n
30
Government Consumption for OECD Countries % of GDP at market prices
72 Differential Models and Neutral Systems
C
0'
0
N
C
Soft Landing
73
6.
Economic Systems With Delay in Control
In our earlier treatment of controllability we accepted instantaneous stroboscopic nature of pursuer response to government action, the snap decision rule. We now assume realistic time delays in the pursuer's response: delays in the transmission of information: a time delay 8 > 0 in the pursuer's observation of quarry's action and a time lag X > 0 in the system's reaction to pursuers controls. As Hajek [33] suggested for ordinary differential games the two lags are identified by the dynamics
dt
x{i)- fat - s)x(s)ds - g-i(t) = Ax(t) + \G(t -s)x(s)ds + g\(t)
- p(t -X) + q(t)
(6.1)
where we can set, />(0 = B(0/n(0 q(t) = C(t)qx{t). The corresponding dual control system is of the form, d_ x(t) - jc(t - s)x(s)ds -g_j(0 = Ax(t) + JG(t - s)x(s)ds - M(0 dt 0
0
u(t)eW(t).
(6.2)
See Proof of Theorem 2.2, and Hajek [39, pp. 95-98]. The solution of (6.1) is given by
75
Differential Models and Neutral Systems
76
t x(t) = U(t)[x(0) - g_i (0)] + g_, (0 + J0(t - s)g_i (s)ds 0
- JU(( - s)[p(s -X)-
q(s) + g](s)]ds,
(6.3)
0
where U(t) solves (1.85). The pursuer has no influence on the component t
X+5
f U(t-s)q(s)ds= /-X-8
[U(s)q(t-s)ds. 0
Define X+8 A=
\lJ(s)Qds.
(6.4)
0
Theorem 6.1. Consider the system (6.1), including the data A. > 0, 8 > 0, and (6.4). Assume that Q, P are compact and 0 e Q, 0 e P. Pursuer can force an initial state <> t (with <|>(0) = *(0)) to target Q. + A at time t > 0, if within the control system (6.2), «(/) e W where JFis defined by W{t) = (P + kert/(/! - /)) * V(k + b)Q,
(6.5)
! : Pontryagin Difference and A is defined in (6.4), § can be steered to Q at time t by the admissible control. Furthermore the time of steering can be taken as the termination time in the game (6.1); if u is a control steering <j) to Q c E" at time / within (6.2), then a winning pursuer strategy 0 with delay 8 is determined by a(q, s + 8) = u(s + X + 8) + qU(X + 8) + ker({/(/i -1)).
(6.6)
for q e Q, 0 < s + X + 8 < /. The "winning strategy" is a control defined by (6.6). It steers the initial state $ to Q c En at time t within the system (6.1) once government has declared its policy.
Economic Systems
77
Proof. Because w steers 9 to Q at time t within the control system (6.2), / t U(t)[x(0) - g_, (0)] + g_j (0 + jl)(t - s)g_1 {s)ds + jU(t - s)gi (s)ds 0
0
t -
\u(t-s)u(s)dseQ. 0
Let X + 8 = 8. Then for any control q() use (6.6) a(q, s + S) = u(s + l + b) + U(k + b)q mod k e ^ t / ^ - /)) to define a pursuer response/?, p(s) = c(q(s - 8), s) = u(s + X) + U(Q)q(s - 6) mod ke^f/ft - 0 ) (we interpret q(s) = 0 for s < 0). Then the solution with this/? in (6.3) is t *(/,♦, A q) = U(t)[x(0) -*_i(0)]+g_i(f)+ \u(t-s)[g^(s) 0 - Jf/(f- s)/?(.s- X.)A 0
+ gi(s)]ds
$ U(t - s)q(s)ds 0
$U(t-s)q(s)ds t-B
t
= u(t)[x(0) - g_i(0)]+ g_i(0+ \u{t - *)[g_i(5) + a(5)]& 0 r e - J l / ( f - s)u(s)ds + fu(s)q(t 0
-s)dseQ
+ A.
0
The constraint set W is non-empty if and only if U(X + S)QcP
+ ker U(tx -1),
Differential Models and Neutral Systems
78
provided P, Q are non-void, convex and symmetric. Thus the effective influence of government
is dominated by private initiative and its internal power for waste, ker
mti-Q. Recall that P is the aggregate control set and instrument of the private firms. Also Q is the control strategy instrument set of government. We showed in (3.3), and (6.5) that if U(k + &)Q is the effective government control set, then controllability is possible if the private control set and the systems possibility for waste dominate effective government control set. Government acts, and after some delay in information 8 and some lag X in the system's reaction to the private firms (pursuers control) the dynamics is established. Each firm reacts with its own strategy. The collection of all possible control strategies of all the firms, the aggregate control, is P. That of the government is Q. The author does not suggest that private sector has the ability to counteract any government intervention. Rather it is suggesting that private sector reacts to government and together in aggregate, a control strategy u is formed which is contained in the Pontryagin difference of sets: W(t) = (/> + ker £/(/, - t))*Q,
0<s
(6.7)
fV(l)=(P+ker U(t\ -O)!t/(\ + 8)0.
(6.8)
or
These sets are non-empty if P+kerU(t\-t)z>Q
(6.9)
or P + ker U(t}-t)ziU(X
+ b)Q.
(6.10)
This is the exact meaning of "dominate" in proposition 3.2. Crudely if the inputs are measured in money, the aggregate effective total government outlay is smaller than the private outlay together with its (allowed monetary) capacity to waste. There is nothing in our analysis which implies that the
Economic Systems
79
private sector is overly co-ordinated: each firm reacts to government intervention, and the statistician can find the aggregate possibilities P. Remark 3.1 comments on the plausibility of the information pattern on which the firms base their decisions. If we allow realistic delays, the second set relation (6.8) is valid.
7. The Nonlinear Theory of Controllability of Volterra Neutral Integrodifferential Dynamics In (6.1) we considered the system i
d_ x(t)-jC(t-s)x(s)dS-g.i(t) dt
= Ax(t) + fat -s)x(s)ds
0
+ gl(t)-p{t-X)
+ q(t),
(7.1)
where there is delay in private firms observation of government action and in the systems reaction to firms' controls. It seems more appropriate to consider distributed time lags with lower time limit of almost zero in the model for firms' control:
jdQB(t,Q)p(t + Q), -h and a nonlinear function f(t,x(t),q(t)) = v(t) as a model for the government controls. Thus we study the nonlinear system, d_ x{i)-\c{t,s)x(s)ds-g-x(i) dt
= A(t)x(t)+ JG(t,s)x(s)ds
81
82
Differential Models and Neutral Systems 0
+ 81(0+ ldQB(t,6)p(t + Q) + f(t,x(t),q(t))
(7.2)
-h
and its linear version d_ dt
c(/)-Jc(/,*)x(5>fc-*_,(/)
= A(t)x(t)+
JG(t,s)x(s)ds
0
+ gl(0+ $ddB(t,&)p(t + B)+q(t).
(7.3)
x(t) = <(>(/) on (- oo, 0] where p(t) e P, q(t), Q, and where 0
g_l(/)=
gl(/)=
\c(t-sMs)ds,
(7.4)
JG(t-sMs)ds.
(7.5)
Here *(/) e E" and /? e C([- h, t\] ), and 5(/, 9) is an n x m matrix continuous in / and of bounded variation in 9 on [- h, 0], in each t e [0, t\] = J. The n x n matrices A(t), C(t, s) and G(t, s) are continuous in their arguments. The w-vector functions / and gi (i = - 1, 1) are respectively continuous and absolutely continuous. In (7.2),
dt
x(t)-jc(t-s)x(s)ds-g^(t)
= Ax(0+ $G(t -s)x(s)ds
+ g\(0+ J < W , Q)p(t + 8) + /(/,x(t% q(t))
Nonlinear Theory
83
where p(t) e P, q(t) e Q. The effective control set of the government is F{t,x)={f(t,x,q):qeQ},
(7.6)
so that V(t)
=
f(t,x(t),q(t))eF(t,x).
(7.7)
The solution of (7.3) can be written as t
x(r,(|)(A9) = t/0.0)[x(0)-5 1 (0)] + g_ 1 (0- j | £/(f,s)g_,(s>fc
t + ju(t,s)
(0 jdQB(s,0)p(s + Q) + gi(s) + q(s)ds
(7.8)
\~h
where U(t, s) and — U(t, s) are continuous matrices satisfying
[jt J t/(/, 5) - j " | t/(/, T)C(T, 5) A + C(f, 5) = - £/(f, s)A(S)- JU(I, T)G(T, 5)rft, 0
0
(7.9) where U{t, t) = /. The solution of (7.2) is
x(t,b,p,q) = U(t,0)[x(P)-g-i(0)] + g-{(t)-
^£jU(l,S)g_x(s)ds 0
0
/
ju«,s)
\dQB(s, Q)p(s + 0) + gi (s)+f(s,
x(s),q(s)) ds.
.-h
Let
S(t,s)= Jt/(/,5-9>%fli(s, 9)
(7.10)
84
Differential Models and Neutral Systems
where \B(s, 9) Bt(s,Q) = { [ 0
s
Define
W(0,t)= ls(t,s)S*(t,s)ds.
(7.11)
0
Then Dauer [35] has shown that W is non-singular if, and only if, (7.3) is Euclidean controllable with q = 0. In [35], Dauer proves some results on this without government controls (i.e., q = 0). His results can be modified. Theorem 7.1. Suppose there are measurable functions <J>/ : £ " + w and L\ functions OLJ:J-> E+, / = 1,2,..., y such that y
l/(/,*,9)|£2>,-(')<>/(*,/0,
(7.12)
i=l
for every (/, x, q) e Jx E and every p. Then (7.2) is Euclidean controllable provided (
q
\
Iim sup r - £ c , sup{
V
i=l
(7.13)
J
Remark 7.1. The proof is a slight modification of Dauer's and Chukwu [36]. The nonlinearity of / is natural and essential in the economic application. In addition to the firms' initiative
»(0= \dQB{t,Q)p{t + Q).
(7.14)
-h
The control of government v(/) =J(t, x(t), q(t)) is not linear in q = [T\, go, e, x,d,mhfo]. (see [37]) The growth conditions mean that the firms' control set (initiative)
85
Nonlinear Theory
should dominate the governments. It is related to the result in Chukwu [36, p. 296]. It settles a basic problem: how much (in comparison with private effort) should government intervention be in the economy. This debate occupied a great part of human endeavor in the 20th century. (Example: capitalism, communism.) In my model it is now settled. Definition Function g is uniformly nonatomic at zero on ExC if, for any (t, §) e ExC, there exists SQ > 0,u0 > 0 independent of (t, <j>), and a scalar function p(t, <J>, H, s), defined and continuous for (t,
|g(',v)-sM*polk-4||. for t e E, \\i e Q(t, <> | , \i, s) and all s < s0,0 < u < ]i0. Here Q is defined as follows: Q(t, (J>, u, s) = {y e C: (t, v) e ExC, ||v - ^| < u, y(9) = +(8), 9 < - s, 9 € [- h, 0]}. Theorem 7.2. Consider the system 4 D(f, *,) = /(/, *, pit)) + B(t)p(t),
(7.15)
at
and assume that (i) (ii)
rank B(t) = n on [/i -h, t\] 5(r) is continuous on [a, f i], with continuous derivative;
(iii) The function g in DO,
(716> ExCxEm
86
Differential Models and Neutral Systems
-> En is continuous; (iv)
There are continuous functions Fj C"' x Em' -> E* and i) functions ccy : E -» E* j = 1,..., s such that s \f(t,^p)\<^daj(t)FJ^,p)
V(t,bp)eE*CxEm>,
(7.17)
>1
where
lim sup r -£cysup{/y(*,/>):||ft,/0|| < r} = + oo. r . 7=1
(7.18)
Then (7.15) is controllable on [a, /j], with /j > a + A. The proof is contained in Chukwu [36, p. 295]. The result gives conditions for function space controllability when government strategy is effectively zero, i.e., q = 0 and v(t) = k(t,
+ k(<,x,,q(t)),
(7.19)
and assume all the (i) (ii)
conditions of Theorem 7.2; (i) - (iv) for eachy = 1,..., £ there are continuous functions Fj : CxEm -> En and Z,1 functions fy : E -> E+j = 1,..., q such that e \\k(t,$,qy\<^Pj(t)FjW,p), y=i
for all (t, <)>,/>), Pi < a/, where
Nonlinear Theory
87
>
lim sup
-Y,CjSup{Fj(^p):m,p)\\
= +00.
► 00
Then (7.3) is controllable on [a, t\], t\ > a + h. The proof is contained in Chukwu [36, p. 296]. Remark The non-linearity of/and the solidarity function k in (7.19) are natural and essential in economic applications. In addition to the firms initiative w(0= B(t)p(t), the control action of government, the "solidarity" function k(t, x\,q(t)) is (realistically) not linear in q = [T, gQ, e, co, /, Ml, / 0 ] The basic contribution of Lucas theory demands a game theoretic formulation of the dynamics [19]. This idea is the thrust of Milliners argument [21, p. 91] and the role of p and q. Condition (ii), p, < a, points to the requirement that the control set of private firms contain that of the government. In the linear situation Int P z> Q the firms control set contains the government control set as a proper subset. See Hajek [15, p. 61] for the genesis of this idea. It settles a basic problem which dominated much of this century: How much (in comparison with private effort) should government intervention be in the economy? Figure 1A displays government consumption for OECD countries as percentage of GDP at market prices. In Salamon [23], the following theorem on function space controllability is stated. Theorem [23, p 157]. The system 4 MO - M*<J ~ h)] = 4)x(0 + Axx{t -h) + Bui}) at is exact controllable in the state space W^,
1 < p < oo if and only if rank
[A(X), B] = n VA.(complex) and rank[M- A_iB] = n for all X(complex) where A(X) = W - te-Xh A_! - Ao - A j e - ^ , and X is complex. This result is needed to test the function space controllability of some economic models of nations.
8.
Economic Models of U.S.A., Canada, U.K., Germany, and India
In Chapter 4, we postulate a simplified version of our key equations (1.1)(1.7), (1.10), (1.23), (1.33), (1.37), (1.56), (1.66), (1.82), and avoid collection of data from the infinite past. Using the rational expectations principle of Luigi Amoroso (1886-1995) and of Lucas [19] in the formulation of Ray C. Fair [29] that the expected values of macroeconomic variables are functions of the current and past values of the variables we derive the hereditary system i ( ' ) - <4-\x(t -h) = AQXQ) + Axx{t -h) + B,g(r) +
Bjpif).
The coefficients A.i, Ao, Aj, Bi, B2 are determined by regression on the economic variables using our MATLAB program and data from International Statistic Yearbook 1994. Details are given in the India case. The computation of other cases and the resulting dynamics are summarized. Function Space Controllability Compute A(X) = )J-A-Xk!~X-AQ-
A\e~X.
Let a = [A(X), B],
c = [cc, B].
For the U. S. A., Canada, U.K., Germany, and India, rank(a) = 6, rank(c) = 6 because rank(B) = 6. The more appropriate B is obtained from the 89
90
Differential Models and Neutral Systems
Pontryagin difference of sets and u in (3.2). In summary, we derived the dynamics of x=[y,R,L,k,P,E], (y = GNP, R = interest rate, L = employment, k = value of capital stock, p = prices, and E = cumulative balance of payment as equation (1.82) or (1.83)). This model assumes the use of data from the infinite past. We avoid collection of data from infinite past by postulating a simplified version of our key equations given by x(t) - A_]X(t - h) AQX(I) + Axx(t -h) + Blg(t) +B2p(t). We implement our dynamics with microeconomic data from the U.S.A., Canada, Germany, and India. Using MATLAB program formulated in 14.8 of "System Identification Toolbox" [18], we identify the coefficients A.], Ao, A], Bi, B2. Salamon's Theory [23] helps us to decide that the systems are controllable. With Lp 1
Economic Models
91
REFERENCES 1. 2. 3. 4. 5. 6.
7. 8.
9.
10. 11.
A. B. Abel and B. S. Bemanke, Macroeconomics, Addison-Wesley Publishing Co., 1991, Reading, Massachusetts 1991. R. D. G. Allen, Mathematical Economics, McMillian and Co. Ltd., London, 1960. Luigi Amoroso, Economia di Mecato, Bologna, Zuggi, 1949. K. Balachandran and P. Balasubramaniam, "Controllability of Nonlinear Neu tral Volterra Integrodifferential Initial Systems," pending. W. H. Branson, Macroeconomic Theory and Policy, Harper and Row Publish ers, New York, 1989. J. A. Burns and P. D. Hirsh, A Difference Equation Approach to Parameter Es timation for Differential-Delay Equations, Applied Math. Computation, 7 (1980), 281-311. E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, Academic Press, Boston, 1992. E. N. Chukwu, "Mathematical Controllability Theory of Growth of Wealth of Nations," Proceedings of the 1st World Congress of Nonlinear Analysts, Tampa, Florida, August 1992. E. N. Chukwu, Control of Global Economic Growth: "Will the center hold" In ordinary delay differential equations, Ed., J. Wiener and J. K. Hale, Pitman Re search Notes in Mathematics Series 272 Longman group, UK 1993. E. N. Chukwu, Optimal Control of Growth of Income of Nations, Applied Math. Computation, 62 (1994), 279-309. E. N. Chukwu, Interconnected Nonlinear Delay Differential Equations in W\ ', Proc. Indian Acad. Sci. (Math. Sci.) 105(1) (February 1995), 73-98.
12. E. N. Chukwu, Control in of Nonlinear Interconnected Systems of Neu tral Type, J. Austral. Math. Soc, Ser B 36 (1994), 286-312. 13. G. Gandolfo and P. C. Padoan, The Italian Continuous Time Model Theory and Empirical Results, Economic Modeling, Butterworth and Co., Publishers Ltd., April 1990, pp. 91-132. 14. R. M. Goodwin, "The Nonlinear Accelerator and Persistence of Business Cy cle," Econometrica, 19(1951), 1-17. 15. O. Hajek, Pursuit Games, Academic Press, New York, 1975. 16. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1997. 17. M. Kalecki, "A Macrodynamic Theory of Business Cycles Econometrica," 3 (1935), 327-344. 18. Lennart Ljung, System Identification Toolbox for Use with Matlaba, Users Guide, The Math Works Inc., July 1992. 19. R. E. Lucas, Econometric policy evaluation: A critique, The Philips' Curve and Labor Market (K. Brunner and A. Meltzer, eds.), Vol. I of Carnegie-
Differential Models and Neutral Systems
92
20. 21. 22. 23. 24. 25. 26.
Rochester Conference in Public Policy, a supplementary to series to the Journal of Monetary Economics, Amsterdam, North-Holland, 1976. M. McElroy, Macroeconomy; Private Actions, Public Choices and Aggregate Outcomes, Prentice Hall, New Jersey. A. W. Mullineux, The Business Cycle after Keynes: A Contemporary Analysis, Barnes and Noble Books, New Jersey, 1984. Jeffrey Sachs, and Lecipe Larrain, Macroeconomics in the Global Economy, Prentice Hall, 1993. D. Salamon, Control and Observation of Neutral Systems, Pitman Advanced Publishing Program, Boston, 1984. J. B. Taylor, Macroeconomic Policy in a World Economy, from Econometric Design to Practical Operation, W. W. Norton and Company 1993. J. Wu, Globally Stable Periodic Solutions of Linear Neutral Volterra Integrodifferential Equations, J. Math. Anal. Application, 130 (1988), 474-483. IMF International Financial Statistics.
27. E. N. Chukwu, Control in W~
of Nonlinear Intercorporated Systems of
Neutral Type, J. Austral Math. Soc, Ser. B 36(2) (1994), 286-312. 28. UN: National Accounts Statistics, The Economists Book of Vital World Statis tics, Editor Miles Smith-Morris, Butler and Tanner, Frome, England, 1990. 29. G. Gandolfo, Economic Dynamics, Third Edition, Springer-Verlag, Berlin, Heidelberg, 1996. 30. Raye C. Fair, Specification, Estimation, and Analysis of Macroeconometric Models, Harvard University Press, Cambridge Massachusetts, 1984. 31. Luigi Amoroso, Economia di Mercato, Bologna, Zuggi, 1949. 32. Wu Jianhong, Stability of Neutral Functional Differential Equations with Infi nite Delay, Funkcialag 29 (1986), 131-139. 33. O. Hajek Pursuit Games with Delay, Funkcalaj Ekvacioj, 18 (1975), 93-98. 34. K. Balachandran, Controllability of Neutral Volterra Integrodifferential Sys tems, J. Australian Math. Society, Ser. B, 34 (1992), 19-25. 35. K. Balachandran and J. P. Dauer, Relative Controllability of Nonlinear Neutral Volterra Integrodifferential Systems, J. Austral. Math. Soc, Ser. B, 37 (1996) 346-353. 36. E. N. Chukwu, Control of Interconnected Functional Differential Equations of Neutral Type in W^ , J. Austral. Math. Soc, Ser. B, 36 (1994), 286-312. 37. E. N. Chukwu, On the Controllability of Nonlinear Economic Systems with Delay: The Italian Example, Applied Math. Computation, 95 (1998), 245-274. 38. J. Stiglitz, The Journal of Economic Perspectives, 11(1) written 1997, pp. 3-10. 39. O. Hajek, Pursuit Games, Academic Press, 1975. 40. E. N. Chukwu, Control Under Scarcity of the Growth of Wealth of Nations: With Austria and U.S.A. Examples, (preprint).
Economic Models
93
41. E. N. Chukwu, "Optimal Control for the Growth of Wealth of Nations," in Fall 1999 classnotes, N.C.S.U., Raleigh, NC. 42. W. H. Kwon, S. H. Han, J. W. Kang, A. Kim, V. Pimenov, A. Lozsnikov, and O. Onegova, Time-Delay Systems Toolbox Beta Version for use with Matlab User's Guide, Engineering Research Center for Advanced Control and Instru mentation, Seoul National University, 151 -742, Korea.
9.
Model Programs and Graphs
Volterra Integrodifferential Neutral Dynamics and Differential Neutral Systems for the Growth of Wealth of Nations: An Optimal Control Theory Models MATLAB Programs and Graphs Ethelbert Nwakucke Chukwu
U.S.A. Program US2.M Figure US1.
U. S. - Private Consumption, Exports, p. 479
Figure US2
U. S. - Investment, Government Consumption, p. 479
Figure US3
U. S. - Consumer Prices, Income, p. 480
Figure US4
U. S. - Money Demand, Aggregate Demand, p. 480
Figure US5
U. S. - Income/Consumption, Income/Investment p. 481
Figure US6
U. S. - Income/Government, Income/Export, p. 481
Figure US7
U.S. - Balance of Payment, log(income) p. 482
Figure US8
U. S. - Increase in Stock, p. 482
95
Differential Models and Neutral Systems
96
Program US2.M This chapter is a sign post which identifies the economic variables, terms and programs which are used to build the economic dynamics of nations. It is helpful to use the electrocial disc. The programs, US 2.m, Canada 2.m, India 2.m, Australia 2.m, Japan 2.m, are contained in the computer discs. The symbols are identified in both the numbered equations and the appendix. For example z is aggregate demand of equations (1.8), (1.10) and (1.11). The reading of the program is made easy by this identification. 1.
2.
3.
Definition of Economic Variables and Terms, pp. 367-368. In this section the economic variables extracted from the Inter national Financial Statistic Yearbook are defined by the symbols. The USA data pp. 366-368. Displayed here are data extracted from the International Finan cial Statistic Yearbook 1994, UN Financial 1974. UN National Accounts Statistics. Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. 367-370.
ML = L-M L / C X T G Z R(t)
p. 367 p. 366 (1.2) (1.4) p. 369 (1.l)p. 366 (1.9) p. 366 p. 369 (1.3) (1.7) p. 367p.369 (1.8),(1.10),(1.11) (1.20), (1.21), (1.22)
p(t)
(1.32), inflation
B = Balance of payment E = Cumulative balance of payment y = National income D = deliveries of new equipment
(1.38) p. 370 (1.39) (1.45) pp. 367-369 (1.49) p. 370
^1= flow of capital dt L = employment
(1.53), (1.57) (1.59) p. 366
Model Programs
97
y = income Gy = income government Xy = income export 4.
5. 6.
(1.52) p. 370 (1.50) p. 369 (1.50) p. 370
MATLAB Regression Programs for Economic Variables pp. 368370. See "System Identification Toolbox" for use with MATLAB, Lenhart Ljung, The Math Works, Inc. I.30-I.32, 14.8, (1.79), (2.16). MATLAB Plot, Subplot programs pp. 370-371. Identification of economic dynamics x(t) - A_xx{t -h) = AQx(t) + Axx(t -h) + B* u(t)
7. 8.
with coefficients A-\, AQ, A\,B, B\, Bi,B=[B\, B2], and given in pp. 372-377 and identified in pp. 370-372 using (Ndu.M, US3.M) program. Diagrams and plots: Fig. U.S.I -> Fig. U.S. 7, pp. 479-482 We use the rank condition of Salamon and the full rank of B to deduce the controllability in Wy of the economic state variables of the dynamics x(t) - A_\x(t -h) = AQX(0 + Axx{t -h)+
9.
10.
Bu(t)
If X is any complex number rank[A(k), B] = n = 6, and ranktyJ A.\, B] = 6, is the required condition (p. 157 D. Salamon. Control and Observation of Neutral Systems, Pitman Advanced Publishing Program, Boston). This rank condition is satisfied by our linear model p. 378. Diagrams: Figures U.S. 1 - U.S. 7, pp. 479-482 We use the same definitions as in US2.M to study Canada, Germany, India, Japan, Australia, and UK. Program Canada Program Germany Program India Economic Models for Australia and Japan Program Australia Program Japan
98
Differential Models and Neutral Systems
CANADA 1.
2.
3.
Definition of Economic Variables and Terms, p. 3 82. In this section the economic variables extracted from the inter national Financial Statistic Yearbook are defined by the symbols. The data pp. 381-382. Displayed here are data extracted from the International Finan cial Statistic Yearbook 1994, UN Financial 1974, UN National Accounts Statistics. Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. 382-385.
ML = L-M L / X T G Z *(') />(') B = Balance of payment E = Cumulative balance of payment y = National income
p. 382 p. 381 (1.2) (1.4) p. 381 (1.1) p. 382 (1.6) p. 381 (1.3) p. 381 (1.7) p. 381 (1.8), (1.10), (1.11), (1.2) (1.20), (1.21), (1.22) (1.32), (138) p. 384 (1.39) (1.45) p. 384
D = deliveries of new equipment
(1.49) p. 3 85
— f ^ = flow of capital
(1.53), (1.57)
L = employment y = income Gy = income government Xy = income export
(1.60) (1.52) p. 384 (l .50) p. 384 (1.50) p. 384
c
4. 5. 6. 7.
MATLAB Regression Programs and plots pp. 383-385. Model Dynamics and Coefficient Identification pp. 386-394. Rank condition for controllability p. 393. Graphs, Diagrams for Canada Economic Indicators pp. 483-486.
Model Programs
op
Figure CAl Canada Investment, Government Consumption, p. 483 Figure CA2 Income/Consumption, Income/Investment, p. 483 Figure CA3 Canada - Income/Government, Canada - Income/Export, p. 484 Figure CA4 Canada - Balance of Payment/Canada-log(Income), p. 484 Figure CA5 Canada-Increase in Stock, p. 485 Figure CA6 Canada-Private Consumption, Export, p. 485 Figure CA7 Canada - Money demand, Aggregate demand, p. 486 Figure CA8 Canada - Consumer Prices, Income, p. 486
GERMANY 1.
2.
3.
Definition of Economic Variables and Terms, pp. 394-397. In this section the economic variables extracted from the inter national Financial Statistic Yearbook are defined by Symbols. The data pp. 394-397. Displayed here are data extracted from the International Finan cial Statistic Yearbook 1994, UN Financial 1974, UN National Accounts Statistics. Formulae postulated for economic variables in the body of the book are now identified, pp. 398-400.
ML = L-M L I C X T G Z R(t)
(1.2X1.4) (1.1) (1.9)4 (1.3) (1.7) (1.8), (1.10), (1.11) (1.20), (1.21), (1.22)
Differential Models and Neutral Systems
100
Pit) B = Balance of payment E = Cumulative balance of payment y = National income D = deliveries of new equipment
(1.32), inflation (1.38) (1.39) (1.45) (1.49)
—— = flow of capital dt L = employment y = income Gy = income government Xy = income export
(1.53), (1.57)
4. 5. 6. 7.
(1.60) (1.52) (1.50) (1.50)
MATLAB Regression Programs and plots pp. 398-401. Model Dynamics and Coefficient identification pp.403-408. Rank condition for controllability p. 402. Graphs, Diagrams for German Economic Indicators Figures pp. 487-489.
Figure G1
Germany - Private Consumption, Export, p. 487
Figure G2
Germany - Investment, Government Consumption, p. 487
Figure G3
Consumer Price, Income, p. 488
Figure G4
Germany - Income/Consumption, Income/Investment, p. 488
Figure G5
Germany - Increase in Stock, p. 489
Figure G6
Germany - Money demand, Aggregate demand, p. 489
GERMD.M
Generated data pp. 407-408
INDIA Program India2.M 1.
Definition of Economic Variables and Terms, pp. 409-411. In this section the economic variables extracted from the inter national Financial Statistic Yearbook are defined by the symbols.
101
Model Programs
2.
3.
The data pp. 409-410. Displayed here are data extracted from the International Finan cial Statistic Yearbook 1994, UN Financial 1974, UN National Accounts Statistics. Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. 409-411.
ML = L - M L I C X T G Z R(t) Pit) B = Balance of payment E = Cumulative balance of payment y = National income D = deliveries of new equipment —— = flow of capital dt L = employment y = income Gy = income government Xy = income export
(1.2) (1.4) (1.14 (1.9) (1.3) (1.7) (1.8),(1.10),(1.11) (1.20), (1.21), (1.22) (1.32), inflation (1.38) (1.39) (1.45) (1.49) (1.53), (1.57) (1.60) (1.52) (1.50) (1.50)
4. 5. 6.
MATLAB Regression Programs and plots pp. 411-414. Model Dynamics and Coefficient identification pp. 415-425. Rank condition for controllability pp.415-416.
7.
Graphs, Diagrams for India Economic Indicators
Figure IN 1
India - Private Consumption, Exports, p. 490
Figure IN2 Figure IN3
India - Investment, Government Consumption, p. 490 India Money demand, Aggregate demand, p. 491
102
Differential Models and Neutral Systems
Figure IN4
India -■ Consumer prices/Income, p. 491
Figure IN5
India ■■ Income/Consumption, Income/investment, p. 492
Figure IN6
India -• Income/Government, Income/Export, p. 492
Figure IN7
India -• Balance of Payment, log (income), p. 493
Figure IN8 India ■■ Increase in stock, p. 493
AUSTRALIA Program Australia2.M 1.
2.
3.
Definition of Economic Variables and Terms, pp. 450-455. In this section the economic variables extracted from the inter national Financial Statistic Yearbook are defined by the symbols. The Australian data pp. 450-452. Displayed here are data extracted from the International Finan cial Statistic Yearbook 1994, UN Financial 1974, UN National Accounts Statistics. Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. 452-460.
ML = L-M L I C X T G Z
ho B = Balance of payment E = Cumulative Balance of payment y = National Income D = deliveries of new equipment
(1.2) (1.1) (1.9) (1.3) (1.7) (1.8),(1.10),(1.11) (1.20), (!.21), (1.22) (1.32), inflation (1.38) (1.39) (1.45) (1.49)
Model Programs
103
^ - = flow of capital dt L = employment y = income Gy = income government Xy = income export 4. 5. 6.
(1.53), (1.57) (1.60) (1.52) (1.50) (1.50)
MATLAB Regression Programs for Economic Variables pp. 450460. MATLAB Plot, Subplot programs pp. 453-454. Identification of economic dynamics x(t)- /L.xx{t -h)= AQX{I) + Axx(t -h) + B*«(r)
7.
with coefficients A.\, AQ, [B\, B2]=B, and given in pp. 461-464. Diagrams and plots pp. 505-508. Controllability pp.461-464.
8.
Standard Deviation p. 461.
Figure AU1 - Private Consumption, Exports, p. 505 Figure AU2 - Investment, Government Consumption, p. 505 Figure AU 3 - Consumer Prices, Income, p. 506 Figure AU4 - Money Demand, Aggregate demand, p. 506 Figure AU5 - Income/Consumption, Income/Investment, p. 507 Figure AU 6 - Income/Government, Income/Export, p. 507 Figure AU7 - Balance of Payment, log(income), p. 508 Figure AU8 - Increase in Stock, Australian Growth rate, p. 508
104
Differential Models and Neutral Systems
JAPAN Program Japan2.M
2.
Definition of Economic Variables and Terms, pp. 465-470. In this section the economic variables extracted from the inter national Financial Statistic Yearbook are defined by the symbols. The Japan data pp. 465-467. Displaced here are data extracted from the International Finan cial Statistic Yearbook 1994, UN Financial 1974, UN National Accounts Statistics. Equations and Formulae postulated for economic variables in the body of the book are now identified, pp. 467-470.
ML = L-M L I C X T G Z R(t) p(t) B = Balance of payment E = Cumulative balance of payment y = National income D = deliveries of new equipment dk(t) - flow of capital dt L = employment y = income Gy = income government Xy = income export
5. 6.
1.2) 1.1) 1.9) 1.3) 1.7) 1.8), (1.10), (1.11) 1.20), (1.21), (1.22) 1.32), inflation 1.38) 1.39) 1.45) 1.49) 1.53), (1.57) 1.60) 1.52) 1.50) 1.50)
MATLAB Regression Programs for Economic Variables pp. 468470. MATLAB Plot, Subplot programs pp. 470-471. Identification of economic dynamics
Model Programs
105 x(0 - A_\x(t -h) = AQXQ) + A\x(t -h) + Bu(t)
7.
with coefficients A. \, AQ, A \, B = [B\, B2], given in pp. 471 -472. Diagrams and plots Fig. Japan. 1 (Japan.8).
8.
Controllability p. 472.
Figure JA1 Japan - Private Consumption, Exports, p. 501 Figure JA2 Japan - Investment, Government Consumption, p. 501 Figure JA3 Japan - Money demand, Aggregate demand, p. 502 Figure JA4 Japan - Consumer prices, Income, p. 502 Figure JA5 Japan - Income, Consumption; Income/Investment, p. 503 Figure JA6 Japan - Income/Government, Income/Export, p. 503 Figure JA7 Japan - Balance of Payment, log (income), p. 504 Figure JA8 Japan - Increase in Stock, p. 504
10. Optimal Control of Volterra Integral Neutral Equations and of Linear Neutral Equations
10.1 Introduction In this chapter we use a maximum principle developed by Carlson [12] and [13] generalized by Burnp and Kazemi [10] to deduce optimal control strategies of linear Volterra integrodifferential dynamics of neutral type. First we consider a variation of parameter version of
dt
x(t)-X
\A(t-s)x(s)ds = A0x(t) + X JAx(t-s)x(s)-Bu(t)
(lo.l)
—°°
namely, x(t) = £/(/)[*(0) - g_!(0)] + g_!(r) + p(t
- s)g_xds + jU(t - s)[gl(s) - Bu(s)]ds
0
0 (10.2)
where 0 I JA_\(t - s)x(s)ds s= g_i(r), -oo
(10.3) 0 X
JAi(t-s)x(s)ds^gi(t)
107
108
Differential Models and Neutral Systems
and U is an n x n continuous differentiable matrix function which is the solution of the resolvent equation. d_ dt
t
U(t)-lA-.\(t-s)U(s)ds
= AQU(1)+ \AIO-
0
s)U(s)ds,
(10.4)
0
with U(0) = I, the identity matrix. A more general version of (10.2) can be identified as follows: Let g(0 = t/(0M0)-g-i(0)]+g_i(0, h(t, s, x(s)) = U(t - s)g_l(s) + U(t -
s)gi(s)
k(t, s) U(t -s) = G(t, s, x(t)) f(s,x(s),x(o(s)),u(s))
= - Bu(s).
(10.5)
Hence, t
*(0 = S(0 + \W, s, x(s)) + k{t, s)f(s, x(s), x(a(s), u(s)))]ds
(10.6)
0
tel *(0 =
/e(-oo,0],
[a(0),0]
y = a + l,...,p.
Here u = u(f) is control and x = x(u) is the corresponding state of the system and the control constraint. u(t)eWa,e, 0
(10.7)
We assume that W is compact and convex with 0 e Int W. In particular, W is an n-dimensional cube
Optimal Control
109
W = {ueEn :\UJ \
(10.8)
Assumptions Let g, G,f, Q>j, 1 <j < q, satisfy the general hypotheses: fory = 0, 1,2, ..., q the functions g En -» EX are continuously differentiable; g:[0,T\ ->• En is continuous; /J:[0, T] x [0, 7] x £" -> f72 is continuous with continuous partial derivatives with respect to x e £ " ; G : [ 0 J ] x [ 0 J ] x £ " - > £ n x ' is a continuous n x / matrix where entries (Gy) have first partial derivatives with respect tox; and f:[0, T]x E" xU-+ E1 is continuous, with continuous first partial derivatives with respect to x, the second variable. Also for every compact set H c En there exist kf, m/ such that s < t, \f(t,s,u)\<mf \Mt,s,u)\
(/,jt,M)e[0, T]xHxW
,
where fx denotes the matrix of the first partial derivatives with respect to /, x. Theorem 10.1. Let the general assumptions 10.1 prevail. Let f° : M^>EX be a given continuous function which satisfies the same conditions a s / , if {xo, HO} is an optimal solution of the optimal control problem which consists of minimizing T jf°(t,x(t),u(t))dt,
(10.9)
0
overall pairs of functions {x, u) satisfying
x(t)=g(t)+ j[h(t,s,x(s))ds + G(t,s,x(s))f{s,x(slu(s))ds 0
on[0,7L
(10.10)
110
Differential Models and Neutral Systems
with terminal constraints
j=l,2,...,p
j = p + l,p + 2,...,q,
(10.11)
and the control constraint u(t)eW a.e. 0
(10.12)
There exist constants X,, 0 <j: < q and a function p:[Q, 7] -> En, such that: P (i) rf(X) = - 2 > ; < D y x(x0(T)), j=0
(k0,Xh...,Xp)*0
=
(0,0,0,...,0)
Kj < 0, 0 < j < a, Xjj(xo(T)) = 0
l<j
The functionp:[0, T]-> En \s such that (ii) Pi0 = -d(\)»
Dx[HT, t,x0(0) + G(T, t, x0(t))f(t, x0(t), «<)(/))]
T + IpMD^K^tiXQW+G&^xoitW^xonuoWys / (iii) For almost all t e [0, 7], T d(\)G(T, t, x(t)) - J(G(s, t, x0(t)))p(S)ds • / ( / , x0(t), uQ(0)
.
(10.13)
Optimal Control
HI
= max d(X)G{T, t, *o(0) - \(G(s, t, xQ(t)))p(s)dsf(t, x 0 (0, v)
(10.14)
Thus the optimal control is of the form v°(t)=s&i[U(t-s)p(s)B(t)l
t<,s
(10.15)
Remark on (10.15) The theorem states that the required optimal control is bang-bang. Of course, if u e W,as defined in (10.8), the components of the optimal control are± a , j = 1, ..., m: That is withdraw ay(-a^) from the system or invest (a,-) into the system. If the component is taxation, impose ay taxes or subsidize by ay, etc. If V (/) is a positive scalar, then V° = 1. This implies that if U(t - s)p(s)B(t) > 0, and u = p - q, and p - q < 0 effective private initiative is weaker than public strategy. On the other hand, if p - q > 0, effective private initiative is stronger than solidarity (government intervention). Optimal Control of Linear Neutral Systems We now consider a related neutral system and state its optimal control policy both in Euclidean and function spaces [22]. Consider the linear control system N
i(0-i4_ii(r-A) = 4o*(0 + 5]*(/-*y) + ««(0 t
x0=*.
(10.16)
7=1
where - w(t) = Bu(t) e W(t). We could consider N
m-A-lx(t-h)
= x(t) + ^Ajx(t-hj) >1
+ q(t)-p(t),
(10.17)
112
Differential Models and Neutral Systems
where q(i) e Q, p(t) e P and W(t) = P + ker U(th t) *Q, ( / i s the fundamental matrix solution of N x(t) - A_xx(t -h) = x(t) = £ Ajx(t -hj)
+ q(t) - p(t) .
(10.18)
We assume Q = {qeET :\qj,\Sq0, P={peEn:\pj\Zl,
j=\,...,n}, j=l,...,n}.
Then W(t) =P+ kerU(tu t)*Q = {w:\wj \£l+ko-q0}3
(10.19)
where ko is the bound of elements of ker U{t\, t). If det A-\ * 0, then ker U(ti,t) = 0, so that W = {w:\Wj\Zl-q0\.
(10.20)
If we also assume that |^Q| < 1, then there is a "winning strategy", a control defined by J(q, t) = w(t) + ^ where w is a control which steers (10.16) to the target within the control system x(t) - A_xx(t) =Lc, + (1 + A0 - qo)p(0,
(10.21)
with p(t) e />. We rewrite (10.21) as x(t) - A_xx{t) = Lx,+ Bu(t), and consider the problem of optimality when the target is Euclidean.
(10.22)
Optimal Control
113
Minimum "Investment" Control Consider N
Mf) - A_xx{t) = AQxit) + £ x ( f ~ hi) + Bu(t),
(10.23)
t=\
where 0 < h\ < hi <... < h^ = h, A{ / = 0,..., N are n x n constant matrices and B is an n x m matrix. The controls are bounded measurable functions. Find a u (subject to some constraint, i.e., u <= W) that minimizes an "investment" function E(u(t\)) subject to (10.23), where % =§,x(t\,a,§,u)=x\
e£".
(10.24)
Here x = x(-,a, <> | , w) is the solution of (10.23) with XQ = <j>. We identify the following investment functions: t
£0(«(1))= \uT(t)R(t)u(t)dt,
(10.25)
0
where R(f) is a continuous positive definite m x m matrix. When R(f) = I, the identity matrix, then t
Eo(u(n))=$\u(t)\2dt. 0
Another effort function is defined by £l("('l))= max sup | My (01, \<j<m
(10.26)
Q
where ueW = {ueEn,u
measurable] Uj(t)\
j = \,...,m
a
114
Differential Models and Neutral Systems
In (10.26), E(u(t\)) describes the maximum thrust available to the system. We define investment energy function,
£2(«('I)) =
]Y)uj{t)\p dt
K||„,
(10.28)
,0 7=1
where p> 1, and ueU = {ueE?,u measurable || u \\p = E2 (u(t\)) < 1} .
(10.29)
If/7 = 2 in (10.28), then £2(M('l))=ll Mll2> represents the investment energy or power of the system, and this is to be minimized. Another investment energy is £3 hM
E3(u(ti))= J S M / ' ) | P * = I«|, •
(10.30)
o;=i This is cumulative economic stimulus. assume the constraint,
In (10.30), we may sometimes
U<EW = {U measurable, u e Em, | uj (t) | < 1}.
(10.31)
For the solution of the minimum investment problem with investment defined by E\(t) in (10.26), set yit) = x\-x{t,o,if,Q),
(10.32)
and call it the reachable state. Let githc) = cTY(tht) = cTU(tht)B,
(10.33)
where c is an n-dimensional vector. Here U(t\,t) is the fundamental matrix solution of
Optimal Control
// j N x(t) - A_xx{t -h) = AQX(t) + £ Ajjxit -hj).
(10.34)
jt=\
Note that g is an m-vector. Suppose m t\
fl(c,n)=Zjl«/0.c)|df.
(10.35)
y=oo
Let
ejk = Z ^ 0 * - i ( * - ^ ) + ^ i e 4 ( * - A ) , *=1,2,...,ii-i,
te[o.n],
[B
s = hj
n
,.
00=
[0
otherwise,
Theorem 10.2. System (10.23) is Euclidean controllable if rankQn(tO = n,
(10.36)
where Qn={QoW>-,Qn-i(s)
s
e[(Wi]}.
(10.37)
Theorem 10.3. In (10.23) assume that: (i) Equation (10.36) holds. (ii) The system (10.34) is uniformly asymptotically stable. (iii) System (10.34) is complete; i.e., the map T{t, o):C^En defined by r(r,o)$ = x(/,a,t,0), where x(t, a, <|>,0), is a solution of (10.34) and satisfies
116
Differential Models and Neutral Systems
T(t,ayc = En.
(10.38)
(iv) The system (10.23) is normal, and this holds if for eachy = 1, ..., m, the matrix Qnj = {Q0j(s),.-,Qn-ij(s),
se(o,
(10.39)
has rank n, where Qy is defined by N Qkj(s) = S AiQk-\j(s ~ h) + A-lQg(s ~h),
k = 1, 2,..., n -1,
ss[o,/i],
i=0
\bi for
and B =
s = h,
(bi,...,bj,...,bm).
Then there exists a minimum investment control u*(t) such that £!(«*(/!))££!(«(/!)),
Vwe^,
with W defined in (10.27) subject to (10.23) and (10.24) with *i=0. Furthermore, if y(t\)*0, the minimum investment £imin =E\(u*(t\)) is given by l
— = minF1(c,
£] min
c6
p
ft,
(10.41)
T
where P is the plane c y(t\) = 1. The optimal control w*(0 is unique almost everywhere and is given by u*(i) = Ex min sgn g(t, c*)
(10.42)
where c* is any vector in P for which the minimum in (10.41) is attained. If X l ) = 0, the control u*(t) = 0 is the minimum-effort control. Theorem 10.4. In (10.23) assume that:
Optimal Control
117
(i) Equation (10.23) is Euclidean controllable. (ii) Equation (10.23) is uniformly asymptotically stable. (iii) Equation (10.23) is normal. (iv) System (10.23) is complete. If W^ =Z.oo([0,fi], Cm), t* is the minimum time, and it* is the time-optimal control, then w* is uniquely determined by uj (t, c) = sgfllc1 U(Tt\, t)bj (0],
a < t< f
(10.43)
where gLt,c) =
sBnc1U(f*,tW),
(10.44)
^-/p
f m
(10.45)
K= V0y=l
Theorem 10.5. Let the prevailing assumptions ((i)-(iv)) hold for system (10.23), and let system (10.34) be complete. The optimal control u* that steers <j) e C to 0 in minimum time t* while minimizing E\{u{t*)) is given by «*(0=sgn[g(f,c*)],
a
(10.46)
where c* is the vector in P that minimizes a = min F(t*, c), ceP
(10.47)
t* m
J\«*,c)= jY,\S(t,c)\dt
(10.48)
07=1
and g(f,c) = c'a(fV)fl(0.
(10.49)
118
Differential Models and Neutral Systems
The optimal strategy which is time-optimal and minimizes Ei(u(t*)) is u* with component K\glit,<*)\«/PsBtigj(t,c*),
Uj(t) =
(10.50)
where -i/p K=
(10.51)
fEl*/(',C*)l*«ft
and where c* is the minimizing vector in a = min /*2(/ * c)
(10.52)
ceP
with -\lq
F2(t*,c) =
(10.53)
The following fundamental principle is now obvious. In our dynamics (10.23), the time-optimal control is also the control that minimizes the investment function. The question posed by this report is now answered. Subject to the hypotheses of Theorem 10.5, system (10.23) can be steered from any initial function to zero as fast as possible while minimizing E. Our strings of theorems are proved in the graduate text [22] for the delay case and below for neutral system. The solution of (10.16) in the state space En is given by the variation of parameter x(t, a, 4, u) = x(t, a, <j>, 0) + ju(t, s)B(s)u(s)ds , where U(t, s) is the fundamental matrix solution of
(10.54)
Optimal Control
119
N x(t) - A.xx(f -h) = ^ Q ( / ) * ( 0 + ]►>(;)*(/ - h), 7=1
Xa =
U(t, s)=0
s > t.
In (10.54), set Y(t,s) = U(t,s)B(s). This is an n x m matrix function which, because B is analytic, is piecewise analytic in /, s, [54] and at least measurable in /, s [33, p. 145], and continuous in / for / > s for each fixed s. We now state the solution of the minimum-effort problems for the various efforts. Theorem 10.6. Assume that (i)
System x(t) - A_\x{t -h) = AQx(t) + Ai(t)x(t -h)+Bu is Euclidean controllable on [a, t\], and this holds if rankQn(tx) = n,
(10.55)
where fi„(
sB[o,tx%
(10.56)
and Qk(s) is defined by the n x n matrix function defined by
120
Differential Models and Neutral Systems
Qk(s) = 4)fit-l(*) + AQk-l(s -h) + A-\Qk(* -h)
00(0) = /» (ii)
Qfc(O = 0i/
k = 0,1, 2
i
(10.57)
Let h W = ]Y(ths)R-\s)[Y(ths)f
ds .
(10.58)
a The control u* is defined by u*(t) = R-\t)(Y(tht)TWq~l,
te[a,H],
(10.59)
where g = [*i(0-*(4,*4»,0)]
(10.60)
is the optimal control that minimizes Eo(u(t)), i.e., £0("*(>i))<£o("('l))> Proof. Since (10.55) holds, W u*(i) in (10.52), we obtain
VM.
exists and u*{t) is well defined. If we use
n x(fi,a, b «*) = *(>1, a, <(>, 0) + fan, s)R-\s)(Y(tz, s))T W~Xqdt a = x(t\, a,$,Q) + q = x\(t\). Thus, indeed u* transfers to x\ in t\. That w* minimizes EQ follows the standard arguments. Indeed, let u be any other control that transfers § to * 1 , we have the equality
Optimal Control
121 'l
'1
JUQI,S)B(S)U*
a
(s)ds= ju(th s)B(s)Ii(s)ds . a
Using the inner product on both sides of this equality, we obtain
JY(tx,s)(U(S)-u*(s))ds, Wq~X
= 0.
Using (10.59) and the properties of the inner product, we obtain
\(u(s)-u*(s),
u*(s))ds = 0
We now use this equality to derive h h £o(" * 0l)) = fM *(s)R(s)u *(s)ds < {u(s)R(s)U(s)ds = Efjifa)).
This completes the proof. Remark. We can easily show that - 11. E0(u*(tl)) = (q,Wq)
Observe that there are no constraints on u except that it is measurable and integrable. The controllability assumption enables one to infer that § is steered to x\ in time t\. The rank condition is lucidly proved by Ukwu [56, Chapter 6]. If the controls are constrained to lie on a bounded set U, then some stability conditions on (10.23) are required. For the solution of the minimum-effort problem with effort defined by E\(t) in (10.26), set ><0 = x(0-x(f,cj,(|),0) J
(10.61)
122
Differential Models and Neutral Systems
and call it the reachable state. Let g(thc) = cTY(fh t) = cTU(thtW)
,
(10.62)
where c is an w-dimensional vector. Note that g is an m-vector. Suppose m t\
^ > ' l ) = Z \\sit,c)\dt.
(10.63)
j=\a
Theorem 10.7. In (10.53) assume: (i) (ii)
That (10.55) holds, The systems (10.53) is uniformly asymptotically stable.
(iv)
Also (10.53) is complete, i.e., the map T(t, CJ)C = En defined by T(t,o)b = x(t,a,
(iv)
(10.64)
The system (10.53) is normal and this holds if for eachy = 1,..., m the matrix Qnj(t\) = {QoMt\)~Qn-\j{s,t\),
se[o,tx}}
(10.65a)
has rank n where Qy is defined by Qkj(s) = A-\Qkj(s ~h)+ AoQk-lj(s) + 4Qk-lj(s - h), £ = 0, 1,2, ...,s = 0,h,2h Q0j(s) = bj B
s = hi=0
= (*1> *2> - . bj,...,
otherwise. b„).
(10.65b)
Optimal Control
123
Then there exists a minimum-effort control u*{t) such that £i(«*(/l))££l(ii(/l)),
V«e(/,
with £/ defined in (10.27) subject to (10.53) and (10.52) with ;q=0. Furthermore, if >
(10.66)
ceP
where /> is the plane c y(t\) = 1. The optimal control u*(t) is unique almost everywhere and is given by «*(0 = £iminsgn«(f,c*),
(10.67)
where c* is any vector in P for which the minimum in (10.66) is attained. If y(t\) = 0, the control u*(t) = 0 is the minimum effort control. Theorem 10.8. If the effort function E2(u(ti)) is defined as in (10.28), define
^('1.^) =
X
j\8j0,c)fdt
(10.68)
where — +— = 1. For (10.23) assume conditions (i)-(iv) °f Theorem 10.6. P <7 Then for each <> ) eC, 0=x\ eEn, and some t\, there exists an optimal control u*(f) that minimizes E2(u(t\)), i.e., £ 2 ( " * ( ' I ) ) : £ £ 2 ( K ( ' I ) ) for all M e [/defined in (10.29) subject (10.23) and (10.24). Furthermore, if )in)*0, the minimum-effort £2min = £i(" * CO) is given by 1 £
lmin
min F2(t\,c), ceP
(10.69)
124
Differential Models and Neutral Systems
where P= {ce.En \c X'l) = l}everywhere and is given by
The optimal control u* is unique almost
u*
sgn gj(t,c*),
(10.70)
where
»= ElminWh
c*)Tq/p,
and c* eEn is any vector in P where the minimum is attained. If yit\) = 0 the minimum effort control is u*(t) s 0. The solution of the minimum effort problem when E2(u(t\)) is as defined in (10.30) (and it is not constrained) does not exist among integrable functions. If impulsive functions, the so-called Dirac functions, are admissible, then an optimal solution exists. These observations are contained in the next theorem. Theorem 10.9. Consider the minimum effort problem with effort function E3(u(n)) defined in (10.30), where u eU = {u:\\u\\\ <1} is defined in (10.31). Assume that for (10.23), the following conditions exist: (i)
(i)-(iii) of Theorem 10.5 Assume Metanormality for (10.23), i.e., rankQn+\j=n, j = 1, ..., m where Qn+ij ={QoJis,tl)...Qnj(s,ti)
for each
se[a,ti]}
with
Qy defined as in (10.65). Then there is no optimal solution with w an integrable function that steers <|) (unless x(t\, cy, <j>, 0) = 0) to 0 in some time t\, while minimizing £3 (u(t)). But if impulsive controls are admissible and F3(thc)=
max sup \gj(t,c)\,
(10.71)
\<j<m Q
then there exists a minimum effort control u*(t) if X'l) *0. The minimum effort is given by £3 (u * (t\)) = £3 m i n and
Optimal Control
125
min F3 (^, c), ii . J
(10.72)
ceP
min
where p= {c&En :c y(t\) = 1}. The optimal control given by u*, where N,
(Nj Emin
uAt)=
•C;=S<
C,
J=l
1 <j < m.
(10.73)
Here 8(r, - ijj) is the so-called Dirac delta function. The suprema (10.71) may occur at multipley and at multiple instances of time TJJ, i= 1,2, ..., Nj where Nj equals zero if gj does not contain this suprema. Thus, Zjh e [a, t\] are the finite number of times at which \gj(t\, c)\ = F^{t\, c*). In Theorem 10.9, the components of the controls are not bounded. If U={u measurable u(t)eEn\uj(t)\<\,
j = \,...,m
||u||i
then the optimal controls are "bang-off-bang". Theorem 10.10. Assume in (10.23) the following: (i)
Conditions (i) and (ii) of Theorem 10.9. Suppose the problem is to minimize ||u||i subject to u e Uand (10.23) and (10.24). Then there exists a unique minimum fuel control u (t). This control is "bang-off-bang" in the sense of having only the values ±1, and 0, with no switches +1 and -1 or back (- 1 and 1) (however, 1, 0, 1 is possible) unless a = mt\ andy(t\) e dG&t\) where i * . ( ' ! )
= ■
jU(t,s)B(s)u(s)ds: H|co
Optimal control is given by u where
(10.74)
/ 26
Differential Models and Neutral Systems
«Un=\Tgj(f,c) J
if 8j(t c)Ur]> u
[ :
\Q
-
cio.75)
otherwise
where r\ > 0 is some constant. To prove the above theorems on minimization of effort, we recall the variation of parameter in (10.54) and Y in (10.54), and define the function t
S,(u)=lY(t,s)u(s)ds,
(10.76)
which maps the control space L into the state space En: Sf. L -> En. This map is continuous and linear with S(£L) = 0. If U c L is control constraint set, then the reachable set is
{Sl(u):ueU}=' JY(t,s)u(s)as:
u&U
(10.77)
(a
Thus if y(t) is a reachable state defined in (10.61) i.e.,y(t) = x\ - x(t, a , ((», 0), the coincidence y(t) = xl(t)-x(t,<j,hu)
= Sl(u),
(10.78)
for some u' e U, implies that y(t) e
(10.79)
The time-optimal control u* e f/ is such that j>(/*) = S,*(u*). The minimum effort control is admissible control u e (/such that St*(u*) =y(t\) with £(«*(/!)) <£ 2 («('l)).
Vuell.
In what follows assume L is either L^ orLp,l<,p< oo. In this case, the map S,: L->E" as defined in (10.76), its adjoint as S{ :E" -» L, represented by
Optimal Control
127
STV) =
0, nt,xfcTt
a.e. te[(,d], a.,.e[0,<].
< 10 - 8 °)
If L = Lao, S* maps En into L\([0, ( i ] , F ) c A » so that
|s*(c r )| = j | | n ' i , * ) r c7]! ds,
(10.81)
c e En, where ||-||i is the L\ norm in Em. If L = Lp, then 5T is still given by (10.80) with
\i \py\- J K ^ T * Va
<7
(10.82)
J
for c G En. We now impose some conditions on S,: L -> E" and then show what conditions on the system's coefficients ensure that the assumptions hold. Prevailing Assumptions: I.
The reachable set *(/) = {S,(u):ueU}
II.
is closed. With Uin (10.27) or (10.29), the closure of
III.
transformations from L into En. The function y.[o,tl]-*EH
128
Differential Models and Neutral Systems
defined by y(t)=x\{t)-x(t,o,$,0), in (10.61) is continuous. Here x\(t) is continuous point target. In the next assumption we assume that it is constant. IV. y:[o, t\] —> EP is constant and not zero: y(t) =y\ for all / e [a, V.
VI.
*ll For each c e En, c * 0, the function t -> | \S (c )| | is strictly in creasing. This condition is guaranteed by the following condition: For each x\, x2 e [a, t\], cT U(t\, t)B(t) = 0, V r e [xi, T 2 ], if and only ifc = 0. This condition (10.83) is equivalent to Euclidean controllabil ity. See Manitius [44, pp. 77-86]. In terms of the system's coeffi cients, Euclidean controllability is assured by the rank condition (10.55). The system (10.23) is normal in the following sense S( (c ) is not identically zero for / e [a, t\], and c * 0.
In view of (10.80), we have that system (10.23) is normal on [a, /]] if for each c e En, and eachy = 1,..., m the set {t>a,cTU{t\,t)bj{t)mQ,
/e[a,/i]}
has measure zero. In this case, US, (x )|| > 0 in (10.80) or (10.81), for each c # 0 , / > a . Conditions on the systems' coefficients for normality are given in (10.65). We state and prove this for the autonomous simple system i(t) + /!_!*(/ - h) = AQXU) + AYx(t -h) + Bu(t),
(10.83)
where the "determining equations" are given by Qkj(s) = 4)Qk-l(s) + 4Qk-l(s - h) + A-iQkis- h), k= 1,2,3,...,5 6(-oo,oo),
Optimal Control
129 \bi,
s=0
(1084)
a^'Ho, ,*o. Set Qnj(ti) = {Qoj(s),Qij(s),...,Qn^j(s),
se[0,tfl.
(10.85)
Theorem 10.11 The system (10.83) is normal on [0, t\] if and only if for each/' = 1,..., m, rank Qnj(tl) = n. Proof: We note that s -> U(t, s) is the fundamental solution to the adjoint equation: | - £/(«i, s) = | - U(th s + h)A.x - U(ti, s)Ao - U(tx, s + h)A\ OS
OS
o<s
y=o, i, ...
subject to \t
s=t
[0
s>t.
By Tadmore [54, p. 80] s -> U(t\, s) is analytic on (t-(j+ \)h, t -jh),j 1,... and 5 —► U(t\, s) is C^ on this interval. Hence the function
= 0,
gj(s,c) = cTU(ths)bj the j-th component of index of the control system, has the following properties: s -> gj{s, c) is defined on [0, oo); it vanishes on (t, oo) and is piecewise analytic in (0, oo). The isolated exceptional points are at points s = t\, t\ -h,t\- 2h...t\ - hi where t\ -hi>0 (See Tadmore [54]). Define Agj(f,c) = g * ( f - 0 , c ) - gj(t+0,c)
fe(0,oo),
and we have designated gj(t-0, c), gj(t + 0, c) as the left- (respectively
130
Differential Models and Neutral Systems
right) hand side limit of the k-th derivative of gj at s = t k = 0, 1, 2, ... . Clearly Ag*(*Xc, n - ih) = (-if cTQkJ(ih)
i = /, - ih > 0,
where Ag°{c,ti-ih) kg{j\c,t\-ih)=-cTQl(jh)bj
= cTA'_]bj, forall
k=\,j:t\-ih>Q.
Thus Ag*(*XCl -ih) = (-l)kCTQk(jh)bi
j:tx -jh>0
A = 0,1,... /=1
n,
on [0, fi]. Now *(/I,T)S0
(10.86)
cTX(tx, i)bi= 0 for all t e[0, oo).
(10.87)
Jt^^itxJx-jhyji^Q,
(10.88)
for all x e (t\, oo). Therefore,
Consequently,
for k = 0, 1, 2, ... and j : t\ - jh > 0. Now cTAS^kXt\, t\ - jh)bi =(-l)k cTQk{jh)bi, for k = 0, 1, ...;y: /j -yV? > 0. From (10.87), we deduce that cTQk<Jh)Bi = Q, i = \,...,m,
(10.89)
for some c e £ " , c * 0 and for all j:t\ - jh>0; k =0,1,2,... (10.89) implies that the nonzero vector c is orthogonal to all columns of Qay(ft) and consequently orthogonal to all columns of Qnj(t\). Thus Qnj(t\) does not
Optimal Control
131
have full rank. This proves that Qnj(t\) < n, which in turn proves the contrapositive statement. Therefore, if rank Qnj(t\) = n, then (10.83) is normal on [0, t\]. Assume that rank Qnj(t\) = n , then (10.83) is not normal on 0, t\]. Then there is a c *■ 0 c e En such that for somey = 0, I, ...; m gj (s, c) = 0 on [0, t\]. Because of this gj(s, c) = 0 on [0, oo), so that on differentiating 0 = cTAgk (f, - in), tx-ih>0 = (-1)* c r ^ (//>), for k= 0,1,2,..., * = 0,1,2,..., /, -ih>0. We deduce that c e En is orthogonal to all the vectors of Qnj(t\), which is a contradiction. We have proved normality. We now prove the necessity for this. Necessity. Suppose that ^ybo('i)
jh > 0. / = \,...,m .
(10.90)
But (-l)kcTQk(Jh)bi =Ag{k\c,n-jh)
= ^k\th(t{
-jh)~)
-V*(/i,('l-yA) + )=0
(10.91)
g
(10.92)
by (10.90). Hence
for * = 0, 1, ... andy':/i -jh > 0. In particular, if/ = 0 then (10.92) yields i|/(*)(c,/J")=vi/(*)(c,r*). Now
Differential Models and Neutral Systems
132
(10.93) TT->'l
t\<1
cTU{k){c,
x)bi
XT—>t\
t\<x
lim
U{k\t\,i)bi
TT-W]
t\
(10.94) because U(k)(thx) = 0 for T > / I .
(10.95)
Therefore gik)(c,t-)
= g(k\c,tf)
=0
(10.96)
for k = 0, 1, 2, ... . Now £/(/i,fj~) = £/(/],/j) because £/(/], x) is left continuous at x = t\. So g(c, tf) = g(c, t^) = g(c,t\) = 0. By virtue of the fact that x -> g(c, x) is analytic for x e(t\-(j+ \)h,t\-jh);j such that : t\ - (j + \)h > 0, the theorem below is needed for the rest of the proof. Taylor's Theorem [55, pp. 214-220] and [30 pp. 686-688]. Let/be a realvalued function on [a - h, a] such thatj^x) is analytic for every x e ((a - h), a). ThenXx) = £ J-—K-—i- (T - af for T e ((a - h), a). Now set a = t\, a k=0
k\
- h = t\-h. Then each component of the m-vector function \\i(c, t) satisfies the hypotheses of Taylor's Theorem on [t\ - h, t\] because x -> g(c, x) is analytic on (t\ - (j' + \)h, t\ -jh). Let g> '(c, T) denote the /-th component of g(*)(c, x), / = 1,2,..., m. Then,
Optimal Control
133
, T, V
~n
for t e ((/i - h), t\). Therefore gl(c,
T)= 0 by (3.3.13), / = 1, 2,.... m .
(10.97)
But g(c, ?i) = g(c, rj") = 0. Therefore, gl(c,x)=0 for all
xe(fi-h,ti].
Now, set a = t\ - h, a - h = t\ - 2h and apply Taylor's Theorem on the interval (t\ - 2h, t\ - h) to get
Butg\k\c,(tl-h)-) 0 by (10.97).
= g(ik\cy(ti-h)+) by (10.91). Also g\k\c, (/, - h)+) = Therefore gi(c,x) = 0, i = 1,2,..., m. Now gt(c,t\-h)
= gi(c, fa - h)~) = 0.
Hence g\(c, x) = 0 for all T e (/1; - 2/i, fj - A) • The
process is continued until g\(c, x) = 0, x e(0, /?]. But g\{c, 0) = gi(c, 0 + ) = 0. Hence g(c, x)=0 for all T €[0, * I ] . Thus, there exists c e En, c * 0 such that for i = 1,..., m, U{t\, i)bj = 0 on [0, t\]. We conclude that (10.83) is not normal on [0, t\\. This proves that if (10.83) is normal on [0, t\], then fory = 1, ..., m rankQcaj{t\) = n. But rank QaaQi) = rank Qn(t\) as can be proved by the method of Ukwu [56]. This completes the proof. The next prevailing assumption on (10.23) is that of metanormality. The notion was introduced by Hajek [35, p. 416].
/ 34
Differential Models and Neutral Systems
VII.
The system (10.23) is metanormal on [a, t\] if and only if every index g(t,c) = cTU{tx,t)B{t) with c * 0 has each component gj{t, c) constant only on sets of measure zero. The set {t>o:cTU(tut)bjit)
= a}
has measure zero for each column bj of B, constant a e E c ^ O in En. Though metanormality is a condition on S( , we characterize it as the full rank of some of the system parameters. We restrict our discussion to (10.83). Lemma 10.1. The system (10.83) is metanormal if and only if eachy = \,...,m rank{Q\j(s),...,Qnj(s),s e[0,t\)} =n where Q/y is as defined in (10.65b). Proof. Assumey'-th co-ordinate gj(t,c) = cTU(t],t)bj=a a a constant on a set of positive measure. Then t -> gj(t, c)-a = M(t) = 0 on a set of positive measure. Since it is piecewise analytic except at the points s = t\, t\ - 2h,...,t\ - ih where / is such that t\ - ih > 0, the function M(t) = 0 on [0, oo). Hence it follows that 0=AMk(t\-ih), = (-\)k cTQ/g(ih)
t]-ih>0 (10.98)
k = 1,2, ..., n i = 0, 1, ... t\ - ih > 0. Hence c is orthogonal to all vectors Qlj(s),...Qnj(s) s e[0,/i J and the assertion rankQ^j = n means that there are n linearly independent columns in the sequence
Optimal Control
135 {Qkj(s),
se[0,ti),
k=\,2,...}.
Suppose this is false. Then there is a c * 0 c = En such that cTQk/{s) = 0, k = \,2,... Since A.\,AQ,A\,
se[0,ti).
(10.99)
B are constant, the function t->gj(t,c)
= cTU(ti,t)bj
is piecewise analytic except at isolated points of [0, t\], which are seen to be t\,t\-
h, t\ - 2h, etc. But gj(t, c) vanishes for t> t\, hence M(t) = gjit, c) -
a s - a for t > t\. It follows that h^k\t\
+ 0 - Mk\t
- 0) - AK% * 0), / e
(0, oo), we deduce that A#)(/i - 0) = gy (t\ - 0, c) = 0, k = 1,2. Because gj(t, c) is piecewise analytic on [0, t\\. M(t) = 0 on [ri - 2h, t\ - h], this contradicts metanormality. To complete the proof it can be shown that rankQoj = n implies rankQ„ = n where Qnj=[Qlj(s),...,Qnj(s)
i6[0,(il.
10.2 Proof of Minimum-Effort Theorems With the six general assumptions I - VI stated, and condition for their validity in terms of the system's coefficients deduced, we are now prepared to state three preliminary results on which the solution of the problem of the minimum effort control strategies are based. Theorem 10.12. In (10.23), let t e [a, t\], and consider W in (10.29) or (10.31). Then there exists an admissible control u e fFsuch that St(u) = )it)=xl(t)-x(t,
a, 4», 0),
(10.100)
if and only if c r J <0<||S*(c r )||,
Vce£\
(10.101)
136
Differential Models and Neutral Systems *
nT
where St is the adjoint of St, a map of E to L. Proof. It is assumed that there is a u e W such that sfcu) = yii). It follows that cTy(t) = cT(St(u)) = S,V(w) < || S*(cT) || || u | | S , V ) ) || . Therefore (10.101) is valid. To prove the converse, we recall that the Euclidean reachable set 4&t) is closed. It is also convex, being a linear image of the convex set W. If there is no u e W such that S((u) = yii), then yif) € 3{(0- Because 4&t) is a closed and convex subset of En, the Separation theorem [60, p. 33] asserts that there is a hyper-plane that separates 4t£t) and yit): This means there exists a c e En, such that cr><0>sup{cr(S,(K)):w eW) = sup{5*(cT)(u):u
eW}.
This invalidates (10.101). The assumption that there exists u e Wha statement on the constrained controllability of (10.101) at some t. The optimal (minimum) time t* for hitting the target is defined as t* = M{t e [a,/! ]:$ (u)=y(t) for some ueW).
(10.102)
The admissible control u* e W that ensures the coincidence St*(u*) = y(t*), is the time-optimal control. For the minimum effort problem, the optimal strategy u* that ensures that Stx (u*)= y(t\), while minimizing E(u(t\)) is the (minimum) optimal control. Inspired by the ideas of Hajek and Krabs [38], we propose the following. Theorem 10.13. In (10.100) assume that point target x\(t) e En is continuous. Then there exists a c e En with ||c|| = 1 such that cTy{t*) = \\S*,(cT)\\,
(10.103)
S*(cT(u*)) = cT(S,*(u*)) = cTy(t*),
(10.104)
so that
Optimal Control
137
where w* is the time-optimal control, and II «* 11 = 1,
(10.105)
if the system is normal. Remark 10.1. Becausex\(t) is continuous, K/)=XI(/)-*(/,CT,*,0)
is continuous. Since the next results link the time-optimal control and the minimum fuel strategy, it is described by Hajek and Krabs [38] in another setting [38] as the Duality Theorem. It is simple and fundamental. Theorem 10.14. Assume null controllability with constraints in (10.16). This is satisfied if (10.34) is uniformly asymptotically stable and (10.16) Euclidean controllable. If/* is the minimum time, then / * = max{/ e(0, /i]such that cTy(t) =\\S*(cT^\ for some czEn: \\c\\ = 1}. (10.106) Proof: Since (10.16) is null controllable with constraints, we set x\(t) = 0 for each non-trivial <> ) eC, )(t)= x\(t)-x(t, a, <J>, 0) becomes XO = - x(t, a, <j>, 0) = y[, V/>a then (IV) is satisfied. Because of Euclidean controllability, assumption (V) is satisfied. Since Theorem 10.13 is valid, the minimum time /* is a point over which the maximum in (10.106) is assumed. Suppose there is / > t* such that cTyi=\\S*(cT)\\ for some c eEn with ||c||=l. Then c r ^^||5*(/)i|<||S'*(c r )||. The first inequality follows from Theorem 10.11, the second from assump tion V. The obvious contradiction proves our assertion. We now designate U to be as in (10.29) or (10.31), and derive from Theorems 10.12-10.14 optimal control for minimizing effort and time.
Differential Models and Neutral Systems
138
Theorem 10.15. In (10.16), assume: (i) (ii) (iii) (iv)
(10.23) is Euclidean controllable. (10.34) is uniformly asymptotically stable. (10.23) is normal. System (10.34) is complete. If U^ = A*>([0, t\lCm), minimum time, and u* is uniquely determined by uj(/,c)
= sgn[c U(n,t)bj(t)},
t* is the
c
(10.107)
1 1 for some c * 0 and eachy = \,...,m. If U„ c L„([a, t\],E ), p > 1, — + - = 1 p
q
such that u e U implies \\u\\p < 1, then the time-optimal controls are uniquely determined by u*(t) =
k\gj(t,c)fPs&>(gj(t,c)),
g(t,c) = ft* I* M
k=
c'U(t*,t)B{t)
c)|9
(10.108) (10.109)
%
{IXC- *
(10.110)
Proof: From Theorem 10.14, if u* is a time-optimal control and /* the minimum time, then S*,(cT)(u*) = cT(St*(u*)) = cTyit*) = | | £ ( c r ) | | . But then u* e Uoo is such that cTyit*) = |c r [/(/*, t)B(t)u*{t)dt = cT Jt/(/*, t)B(t)u*(t)dt
Optimal Control
139 \\\cTU(t*,t)B(t)\\\dt-
This implies that u* is of the form a.e. uj {t)s&[cTU{t*, 06/(0],
o
T
when c U(t*, t)bj(t) * 0, for c * 0 , andy = 1, ..., m, which is true since the system is normal and (10.34) complete. For Up, we obtain from (10.103) in Theorem 10.13 that cT)
= \\\cTU(t*,t)B(t)\\q dt)XI1
1
1
(10.111)
T
where - + - = 1. Since g(t, c) = c' £/(/♦, t)B(t), P <7 cTy(t*)= \Y.gj{.t,c)Uj{t)dt< a J'=l (
i
m
\Y)gj{t,c)uj(t)dt\ a 7=1
> \lq (
)
t* ( m
\ \lq (
I *Z\gj(t,ctf U=i
\ 1/?
dt
cr V / = l
a
m
J
Vv'=l
J ^l\IP
m
^UjWfdt
U=i
J
Differential Models and Neutral Systems
140 (
t*
^'q
n,
YY}zj^c^q
dt
V a 7=1 But then (10.111) is valid, i.e., \*4
T
q
c *n = j£\gj(t,c)\ dt and so we have equality everywhere in the above estimate. The control u* that gives the equality is (by inspection) j l ' 1« Sgn(gy(/,C)), j = 1, ..., i Uj(t) = k\gj(t, Cf" ,m
VA/ where k = !I.\gjO,cyiqds Voy=l
(10.112)
^," . This is the time-optimal control, and as j
observed before, it is uniquely determined when the system is normal. The proof is complete. Proof of Theorem 10.13: Because of the Euclidean controllability and the stability assumption, there is indeed a t\ such that the solution x of (10.23) satisfies %(-,CT,
x(t\,G,if,u) = Q.
We observe that in this case yit) defined in (10.61), y(t) = x\(t) - x(t, a, <j>, 0) = -x(tho,$,0) = yi 5*0 and ^(r1)e4J.(/1). Observe that u* in (10.107) is a boundary control in the sense that if i
z(/l,c)=
ju((i,t)B(t)u*(t,c)dt,
then z(t\, c) is on the boundary of the reachable set <$£t), so that
Optimal Control
141
cTz(t\,c)
m h = £ ]\gj(t, c)\dt = F(c, /l) = | s * ( c r ) | | ,
and cT z(thc)>
JyVy
&QL(.t\),
y±z(thc).
With y(t\ ) = y\ eQL(n), w e c a n extend this to reach the boundary as follows. Let a = max{p: ( 3 ^ ) 6 «{.(/!)}. (10.113) Since y(t\) * 0, a can be assumed to be positive and obviously ay(t\) is a boundary point of 4&t). This means that ay(t\) = z{t\,c) for some c, where cTy{tx)=\=-cTx{tx,a,$,Q).
(10.114)
It is easy to verify that the control jj(/)=«!iL£) a
(10.115)
steers <j) to 0 in time t\ while minimizing E\(u(t\)). Also 1 a
1 = min £I(H(?I)), M(tx)
(10.116)
where a = min{Fi(t,c): ce P= {ce En: cTyx = I}}.
(10.117)
Details of the verification are the same as in the case of ordinary linear differential equations (see Neustadt [38]). We conclude that
142
Differential Models and Neutral Systems
-, x u*(t,c)
u*(t)
(10.118)
is the minimum energy control, where c* e E" is the vector P on which the minimum in (10.117) is attained. Because «*(/) = sgn(g(/, c)), we deduce that g(/)=
'8n^.^)}>
(10.119)
where c* is the minimizing vector (10.117). Normality ensures that u is uniquely determined. If y{t\) = y\ = 0, then the choice u s 0 is appropriate. This completes the proof of Theorem 10.13. Proof of Theorem 10.14: In this case Up in (10.29) is used to define the reachable set i
*(fl)=i \u(n,t)B(t)u(t)df.
u:\\u\\p
which, as we have noted earlier, is closed. Just as in the proof of Theorem 7.5.2, [22] the time-optimal control «*(/, c) in (10.118) is a boundary control that is uniquely defined. Thus if i
z(thc)= ju(tut)B(t)u*(t,c)dt,
z(thc)ed^(ti).
With a as defined in (10.113), it is easy to verify that the minimum effort control is given by -/
_*x
U*(t,C*)
u(t,c*) = — ^ - ^ ,
(10.120)
where 1 a
1 = min£ 2 ("('l)), A/(/,)
(10.121)
Optimal Control
143
and a=
min
ceP={ceEnc'yi
(10.122)
F(t, c), = \}
and ^ ( f j , c) is as given in (10.62): Mq
m M f2(t\,c)
=
X ]\8jif,cf dt
The vector c* is the one that minimizes the function in (10.122). Since «*(/, c*) in (10.120) is given by (10.108), the minimum effort strategy is given by *g/qc*)' ? / p sgn(g / (r,c>*(/,c) «/(',**) = a
(10.123)
where N-l/p
'<* M
k=
c)! 9
Through Theorem 10.13 and the general ideas of Hajek and Krabs [38], we can link up the time-optimal controls and the minimum effort controls. In some situations we see that they are the same. We observe that \\S( (c )|| is given by (10.111) and it is to be minimized in (10.117) and (10.122) over the hyperplane P. We are led to the following definition. For each / e [0,t\] let a = inf{||5*(cr)| \:ceP} where P= {ce En:cTyi
= l}.
(10.124)
Theorem 10.16. Let the prevailing assumptions I-V hold for (10.23). Then for each t e [0,t\], we have that if/* is the minimum time, then
144
Differential Models and Neutral Systems
> 1
^^~~ ' = •
«(0
< 1<-W =
<
(10.125)
/*.
>
Remark 10.2. Note that 1 is the norm bound of the control set W, and t\ is defined in the minimum effort problem in (10.102). Proof. Immediately from (10.124), one has that crH^||$V)ll,
Vce£".
(10.126)
From an elementary approximation theory arguments we can verify that for each / e [0, t\] there exists some c(t) e P such that
HS,V(0)ll = a(0>0,
(10.127)
Also there exists a »/ e I such that <S"/(w;) = y\ and ||«/|| =
. From <x(0
(10.127) we argue that there exists a c(i) e En such that c(0^i = l = ^l|S*(c r (r))||. Invoke Theorems 10.12 and 10.13 to validate (10.125). Theorem 10.17. Let the prevailing assumptions I-VI hold for system (10.23) in which (10.34) is complete. The optimal control u* that steers <j) e C to 0 in minimum time t* while minimizing E(u(t*)) is u* with component «*(0 = sgn[g(f,c*)],
a
(10.128)
where c* is a vector in P that minimizes (10.117), i.e., oc = min F(t*, ceP
and
c*),
(10.129)
Optimal Control
145
F\(t*,c)=Ydj\gj(t,c)[dt,
(10.130)
7 = la g(t,c) = c U(t*, »*j ( 0 = k\gj(t, c)f/p
t)B(t), sgn(g/(r, c*)).
(10.131) (10.132)
Here -\iP in
k =
j*Z\gj(',c*yiq dt
(10.133)
with c* the minimizing vector in a = min F\(t*, c),
(10.134)
ceP
and ft*
\i/g
(10.135)
F(t*,c) = a 7=1
Proof. By assumption, the feasible t\ of (10.24) in the minimum effort problem is the minimum time t*. But then a ( / * ) = i n f {||S,,(c')||} = l , ceP
by (10.115) in Theorem 10.16. Since the minimum fuel controls are the functions
« ' ) - ^ - ( ' . ^
Differential Models and Neutral Systems
146
by (10.118) or (10.120), the corresponding expressions in (10.107) and (10.30) for the time-optimal controls of Theorem 10.10 prove that (10.107) and (10.108) are correct. The following fundamental principle is now obvious. In our dynamics (10.23), the time-optimal control is also the control that minimizes the effort function.
10.3 Optimal Absolute Fuel Function The solutions of the minimum time/minimum fuel problems contained in Theorems 10.6, 10.7, 10.8, and 10.15 depend on the closure of the reachable sets. Since the reachable sets are also convex, optimal controls are boundary controls in the sense that they generate points on the boundary of ^(/]). But if the effort function is defined by E^(u(t\)) in (10.30), as the absolute fuel function '1 m
£3(«('i)) = INIi = J]£l«/(OI
(10.136)
0 j=\
and if W is defined by W = {umeasurable\\u\\\<\},
(10.137)
then the reachable set
*(/l) = j \U(tht)B(t)u(t)dt:u e\v\
(10.138)
is not closed, but open. Since the reachable set is open, its boundary points can only be "reached" by convex combination of delta or Dirac functions, which are impulsive in nature. If we admit such functions as controls in absolute fuel minimization problems, optimal control exist. If we rule them out and
Optimal Control
147
Mf) + A_{i(t -h) = AQX(1) + A]X(t -h) + Bu(t).
(10.139)
Lemma 10.2. Suppose (10.139) is metanormal, i.e., fory = 1,..., m, rank[Qf,...Q^,
s e[0, /,]]=»,
where Qkj is defined in (10.84)
yi = \lJ{t\,t)Bm(t)dt a
(10.140)
is a boundary point. Let c * 0 be an outer normal to (the closure of) 4&t) at y\. With this c, define the index of (10.139) g(/, c) = cTU(t\, t)B. Obviously, «o is a control that maximizes 'l
'i
jg(t, c)u(t)dt < jg(t, c)uo(s)ds whenever 11«| |i < 1. a a
(10.141)
'1
But the mapping M( ) -» $g(t,c)u(t)dt is a linear functional on L\([c,ty]) a
with norm ll£lloo= max
max |g,-(/,c)|.
\£j<,m te[a,t\]
The inequality on (10.141) implies that the value UgJIoo is attained at the element MQ of the unit ball L\. Therefore
148
Differential Models and Neutral Systems
llglloo = jg(f,c)uo(t)dt < )\gj\\uoj\
1
nt
= llgllooll«Olll^ll«lloo.
This shows that we have equality throughout, so that 'l M
JXllslUH^C^IlKOyCOI^O. aj=\
It follows that fllslloo -\gj(t,c)\y\uojit)\
= 0 a.e.foreach j = 1, ...m,
t e [a, q ] .
Because the system is metanormal, \gj\ is constant (= ||g||oo) only on a set of measure zero. Hence, MO = 0 a.e. on [a, t\]. But with wo = 0, our boundary point isjvi = 0. This contradicts the following containment: (l)
«.a('l)c: Atf*p(/i), M l /0X.0C1) =1 ■« I * p .
(10.142)
whenever 0 < a < p < mfi, which is an easy consequence of metanormality and completeness as can be proved by the methods of Hajek [35, p. 432]. Recall that (in Hajek's notation) 'l m
I Nil = \y]\uj(t)\dt, CT
y=l
INloo = max esssup|wy(0|,
Optimal Control
149 i
*a(0 = \u{t\,t)Bu{t)dt: |M]|OO < 1, ||«l||
(10.143)
Thus the reachable set
*.('!)= U"*-« = ju(tht)Bu{t)dt: INIooSl, |Mli0 where — u = v is an open set. a We conclude that if the controls u are measurable with ||w|| < k, then the reachable set is k<Sit\) for k > 0. Since this open, the minimal k can never be attained unless y\ = -x(t\, a, <j>, 0) = 0. Continuing, if g is the index, i.e., g(/) = g(t, c) = cTU(tu t)B, then 'l
cTy\ =cTy(t\)= jgit,c)u(t)dt.
(10.145)
a
The control that realizes a boundary point of 3{(/i) definitely maximizes 'l
'l
\g(s)u(s)< jg{s)u0(s), a
(10.146)
a
whenever ||w||i < 1, i.e., over u e W. We have that '1 m
'1 m
T
c M)= jX«/(0«,-(0<*£ |5j«/ll«/l
max |g/(c,/)| = |gll<x>
(10.147)
150
Differential Models and Neutral Systems
If t* is a minimum time for the time-optimal control problem with u e U, then c r X'*) = ll£(c r )lloo,
(10.148)
where \\sUcT)\\o0=
max max \gj(c,t)\.
(10.149)
\<,j<mo
If equality holds in (10.147) then impulse functions applied at the points where gj is the largest that maximize (10.147). The maximum values in (10.149) occur at multipley and at multiple instances Ty/, / = 1,2, ...Nj, where Nj is taken to be zero if gj does not contain the maximum. Because of these, the time-optimal controls (which are "boundary" controls) are u* NJ * uj {f, c) = Xsgntey(V/ ,c)n<-
m
I Vi) / £ Nj,
\<j<m.
(10.150)
For the minimum-fuel problem we deduce as before that the optimal control is u*(t,c*)^ a
(10.151)
where
- = -J---min£ 3 («(/i)), a
(10.152)
M(/])
and a(/i) = minf(/i,c), ceP F(/,,c)= max
/»= {c eEn: CTyitx) = 1}, sup |^/CI =11 ^lloo •
(10.153) (10.154)
Optimal Control
151
In (10.151) c* is the minimizing vector in (10.152). The proof is complete. Remark 10.3. If W in (10.137) is replaced by '1
n W = u measurable«(/) e E : \\ Uj(t) \dt<\,
j = 1,..., m
(10.155)
then h m
m
cTy(tO=\j^gj(t,c)uj(t)dt
a
7=1
The maximum in (10.156) occurs as before at multiple instances of time iy, / = 1, ..., Mi, where M, > 1. The maximizing impulsive controls that approximate reachable points on the boundary of R(t) are Mj
»y-(',c)=^-|-sgn(gy(r(/,cMr-r(,-),
i<j<m.
(10.157)
Therefore, with M
F«hc)=Y
mzx \gj(t,c)\,
(10.158)
a(/i) = minF(ri,c)= F(c*), ceP the optimal control that minimizes the absolute fuel is ■»(',c»)
j7(/)="*^), a(/,) ' where c* and a are determined by (10.158). Proof of Theorem 10.11. Recall the following definitions:
(10.159)
152
Differential Models and Neutral Systems i
*.(
«.a = *.('! ) = \u(ti,t)Bu(t)dt:\\u\\co
11M||I < a
(10.160)
(10.161)
where a > 0. These sets are non-void, convex, and symmetric about 0. Clearly 3{.a
y= ju(tht)Bu(t)dt,
MMIOO
<1-
But then '1 m
\\u\\x =
'1 m
\Yt\uj(t\)dt\<\Yd»>h,
a 7=1
a 7=1
this shows that y eQLmt] c gfo. We also observe that 0 < a < P, *.ac*.p,
a'^Dp
The first is obvious from the definition. following containment:
1
^.
(10.162)
The second follows from the
A*.y + ( l - X K 5 c ^ y + ( 1 _ X ) g , for all y, 8 > 0, 0 < A. < 1. If 5 > 0, y = P, and X = cc/p, then the second containment follows. From (10.162) we deduce that <*.a=>(a//n/i)<*.(fi),
0
Optimal Control
153
If (10.139) is controllable, then 9^t[) has a non-void interior, and this forces QLa to have non-void interior for each a > 0. From the usual weak-star compactness argument we can prove that Q(£t\) is compact. Other properties ofQ&t\) are contained in the next lemma. Lemma 10.3. For each 0: x(f\, a,
(10.163)
where
( 10164 )
*«l)=LKa£0
It is clear that there is a sequence of admissible controls «( ), each steering <> j + to 0 at time t\ with ||w||i -> 0 . The usual weak compactness argument pp. 164 and 165 yields an optimal control u with 0 = \\u \\\ = £3(u(/i)) and £3(7i(/1))<£3(U(f1)),
V«.
To see this, interpret admissible controls as points in l2-space, i.e., L2([o, t\),En).
Thus the conditions INIooSi,
||«||,<e+e
determine a convex set that is bounded in L2 norm. It is weakly closed since, if a sequence converges weakly, then a sequence of convex combination converges strongly in L2, and as a result a subsequence converges poinrwise a.e. Thus x(t\, a,
154
Differential Models and Neutral Systems
x(t\, a, <)), 0) e Int G(Q. This implies that for sufficiently small A. - 1 > 0, kx(t\, a, <)>, 0) e 9Q. But if 0 < a < 9, then by what was proved before
«tec(e/a)*fo). Let a = Q/X, then \x(t\,a,$,0)e\9La
x(/i,a,<|>,0)6<*a
even though a < 9 and 9 is minimal. If 9 = 0, 3{.= 0. The proof is complete. The next lemma is the maximum protoprinciple of Hajek, which corresponds to our system (10.4). Lemma 10.4. If <) e C is such that x(t\, a, <}>, 0) e d$Q, and c is an exterior normal to <SQ &tx(t\, a, <> | , 0), and if UQ() is any admissible control steering <> f toO at time t\, then 'i
'i
\git)u(t)dt< \g(t)u0{t)dt, a 0
(10.165)
Vw(-) with |M|OO < 1, ||«||i < a, where g is the index of the control system: g(t) = JU(tUt)B. It is not necessary that |w||l < a. Conversely if c ? 0 and ]|w|oo < 1, ||u||i < a, and x(t\, a, <}>, 0) e d%Q for some <}> e C, at which c is an external normal. Proof. Let <J) € C be point for which x(t\, a, <j), 0) is on the boundary of 3fo with c an external normal. If c i- 0, then this is equivalent to c^y(t\) < cTy(t\, a,
x(tl,c,$,0)=
\g(tu a
and
c)u0(t)dt,
Optimal Control
155 l\
X'l)= jg(t, c)u(t)dt. a
The proof is complete. Corollary 10.1. Assume that the boundary control MQ satisfies ll«olll SoThen each co-ordinate UQJ of MO satisfies gj(t,c)uoj(t)>0a.e
[a,/,].
(10.166)
Proof. Define a control u with co-ordinates Uj =(Sffigj)-\UQj
|.
This control satisfies the assumptions. Since (10.165) is valid, 'I
m
'l
'l
Jgwo = j Xsy'"0 - j s " a
a 7=1
a
'i =
Jl«/I"l"07la
Because of this, '1 m
f£(W/-IS/«0;l)*0, a 7=1
so that gjUQj =
\gjUoj\>Q,
a.e. since the summands are positive. We know from the lemma that optimal controls exist as boundary control of QQ. We obtain more information by applying further extension of Pontryagin's maximum principle reported by Manitius [37, p. 98], Define the Hamiltonian
156
Differential Models and Neutral Systems N
H = y{t) AOx(t) + Y, AJx(t ~hj) + Bu{t) - T] X 7=1
uj
(10.167)
7=1
for some constant r| > 0, where the adjoint equation may be written as N
y(t) - y(t + /0/L, = - y(t)Aox(t) - ^y(t
- hj)AjX(t - h),
7=1
a.e. in [a, n], >t\.
(10.168)
The optimal control that maximizes H is the u(f), which maximizes the expression M
y{t)Bu{t)-r\
I
(10.169)
7=1
ButXO = cTV{t\, t) is the solution of the adjoint equation [33, pp. 147-149] w i t h ^ i ) = cT. Thus with the index of the control system g(t,c)^y(t)B
= cTU(tut)B,
we desire that u that maximizes M
Y,^j(t,c)uj-r]\uj\)
(10.170)
7=1
overall m-vector u whose co-ordinates satisfy \u/\ < 1. If r\ = 0, then the optimal control would coincide with that for the time-optimal problem and we should therefore have to assume r\ > 0. In this case we maximize each summand in (10.170) separately:
Optimal Control
157 Uj =
Sfflgj(t,c), 0
if I g y (/, C ) I T)<1 otherwise,
(ioi?i)
provides that gj ■£ r\. The co-ordinates of u have values ± 1 and 0 only. As t varies, there is no switch from 1 to - 1 or vice versa. Just as in ordinary differential systems studied by Hajek, optimal controls either always use all the available capacity or are completely dormant. The expression (10.171) uniquely determines optimal controls provided gj ± r\ on set of positive measures, i.e., provided (10.139) is metanormal, which is assumed in the theorem. The expression for optimal control depends on a careful choice of r\. The following construction follows very closely those of Hajek for ordinary linear systems. First, choose any n > 0, such that (10.171) holds and hunt for some u: 11«||oo ^ 1- But h m
m
INIl = { 2 > ; l = 2>eas{b,|>Ti}. We select some suitable r| for which ||u||i = a. Indeed, since (10.139) is metanormal, gj{t, c) is non-constant a.e. and the mapping m(r\): n-> meas{t e [o\/i } \gj(t, c)\ > n.} is continuous, strictly decreasing, 0 < m(r\) < t\. Because 0 < a < mt\, there exists a unique value of n > 0 such that m ^meas{\gj\>r)}=a. 7=1
With this r\ set Ej = {te[a, t\]:\gj(t, c^>^}, so that
Fj = [a,/i ]/£y,
Differential Models and Neutral Systems
158 m
£measEj =a\gj(t,
c)\>r\
onEj,
\ gj(t, c)\>r\
on Fj
With this, define the control «() as follows
ujity
sgngj(t,c),
on Ej ,
0
on Fj.
This u has the following properties: IMhc
for all controls «(•) with N|oo
-s)x(s) = ^0^(0+ )A\(t-s)x(s)+
Bu(t),
can be obtained as in the treatment above. The variation of parameter solution is given by i
x(t,ty,o,u)= g(t)+
$U(t-s)Bu(s)ds,
Optimal Control
159
where g(t) = U(t)[x(0) - g_,(0)] + g_,(/) + Jt/(/ - s)g_x(s)ds. 0
We consider the following problem: Find u (subject to some constraints, i.e., u e W) that minimizes an effort function E(u(t\)) subject to (8.1) in place of (10.139) where xa = <(>, x(t, a, <J>, u) e E". Set Y(t,s)=U(t,s)B(s). The solution of the minimum effort problems for the various efforts are parallel to the investigations above for (10.139) where x{t, a, <(>, 0) = g(t). The controllability, normality, and metanormality criteria can be formulated and proved. Details will be supplied elsewhere. We note that if the optimal control strategy of government is q = - qop, then the optimal strategy of firm is p(i). It is reasonable to assume that government has to decide on a policy before the firm makes a decision, i.e., government chooses q(t) in (1.82) before firms choose pit) in (1.82). The best results are achieved when government and the firm co-operate with a common objective of "full employment" as soon as possible or attainment of some growth target as soon as possible with minimal investment. It is conjectured that the best value of ^o = 0.618. We have seen that the optimal control game is equivalent to the optimal control system. The latter problem is solved by adapting a very recent theory of Angell and Kirsh [4] and Chukwu [22]. It is more appropriate to work in function space. For example in July 1992, China proposed to move the national GNP from current growth trend V (0 = —- t + c to a target growth trend of \\i (t) = — t + c. The initial state and the target are both functions. The following problem is appropriate. Problem 10.1. Minimize t\ subject to the constraints: x(t) - A_\x(t -h) = LXi + Bu(t) on [0, /j]
160
Differential Models and Neutral Systems
=v,-jt(Dy)eC\[-h,0],En),
xh
u(t)eWae
on [0,t\],
(10.172)
where fV = {ueEn:\uj\
j=\,...,m).
Theorem 10.18. [22, p. 468]. Assume that the system x(t)-A_xx{t-h)
= LXt +Bu(t) onfO,/!],
x0=0
(10.173)
with attainable set ji={xtl(u)eC([-h,0],En),
Dx,v
eC\[-h,OlEn):ueU},
where W = {ueLo0([0, h),En):,
ut], eC([-h,0],
En)}
is such that A = f ^ C ( [ - A , 0], En), Dy, ec\[-h,
- ] , En)}.
(This condition implies that (10.173) is a function space controllable with controls in W, i.e., given ^ , f € w^ \[-h, 0],En), there exists a e JFsuch that the solution x of (10.173) satisfies XQ = $,xt] = ij/.) Then there exists a triple (a, v, q): azLoo({0,t\\,En),vsNBV([-h,txlEn) {a, v, q) * (0, 0, 0) such that
and
q*EH,
Optimal Control
161 1
'1
a(t) = - q - a(t + h)A^ - jr\(s)a(s)ds + jr)(s)dv(s -t\), t
(10.174)
t
for all f e [0, t\], and 'i
'i
fat)Bu*Q)dt-
]dv(t - t\)Bu(t)
t
l\-h
'l
'I
3 JO(O0H*(/)4/
\dv(t-t\)Bu*{t)
(10.175)
t\-h
for all M e Uad, where Uad = {ueU:u(t)€W
on [0,/,]}.
Remark 10.5. The theorem asserts that optimal control is «* = v(f), / e [ 0 , n = w(i),
-h],
te[t\ -h,t\],
(10.176)
v(/) = sgn[a(/)S].
(10.177)
where
The control w(l) on [t\ - h, t\] is computed from i(t) - A_xx(t) = LXi +Bu(t) on [<,_/,,/,], xn-h If 5 is invertible, then
= x{t,v),xn
=v
on [/i_fc,fi].
(10.178)
Differential Models and Neutral Systems
162 lr.:
w(t) = B '[v(/)-i<-i\j/(/)-Ii|/]
= B~
^(t)-A_MO- jdwQMQ)- p9Ti(eMe,<(.,v)
(10.179)
t-h
The proof of Theorem 6.1 is contained in [22, p. 469]. The function a is computed from (10.174) by method of steps. The construction of optimal control u on [0, t\ - h] is similar to that in Euclidean space. The delay model is treated in Chukwu [22, Chapter 7]. Example. Let y be national income, R interest rate, D aggregate demand, Ml money supply, S1 = y aggregate supply, C consumption, and / investment. Assume that C = cy I = -aR L(y) = ky
(0
money demand for transaction purposes. Let s= 1 - c . Assuming the market principle dy dt
= k](D -y) = -sy-aR
(Aj = 1, s, d > 0)
^-=k2(L(y)-Ml) dt = ky-M\
(k2 = 1 , A > 0 ) .
Differentiating the first equation and substituting the second into it, we have d y dy ... —i- + s — + aky = aM\. dt dt'
Optimal Control
163
This equation is a very naive and simple description of the "growth" of gross national product. It will illustrate the principle of economic stimulus. Let y(P) = yo,
y(0) = yo,
yo is the initial value and yo is the initial velocity imparted by the external impact of Mj by the central bank. Thus the dynamics of the gross national product GNP, y is given by dly 1
dt
dy
+ s — + aky = aM\, at
y(0) = y0,
y(0) = y0.
(10.180)
Our aim is to damp out the oscillatory behavior of y(t) by using money supply to bring it to rest (0,0) = (y(t*), y(t*)) in minimum time /* or minimizes J->,u(T)
J2u(T)= ji«(0|
where the set of admissible controls is given by W = \u: u(t)e E, ju(t)dt < 1 Assume the solution is such that g(t, c) = De^' sin(P/ + 5) where
D
c\
I P
a + c2 -
P
(10.181)
164
Differential Models and Neutral Systems
- q x C20.
F(c*,T)=
•^min
o?P
l
8= tan
max 0
g{c,T)
=minF(c,r). ceP
The maximum occurs at 11
x
—- 71 - 8 T = '
2 P
(approximately) (see Redmond and Silverberg [61]) F(c, T) = DeaT I sin(pt + q) \ F(c, T) is minimized over H by letting C2 = 0 so J_ •^min
_ ! _ ax VTOC
0 e
,
in which 1 = — . Optimal strategy is u*(t) = fix^e"0* sgn(sinpx)8(/ - T). Figures 5.1A-5.1D allow the following observation to be made. Impulse control-or money supply economic stimulus allows for a period of free oscillation during which the natural damping present in the system removes energy from the system. Then at last instant, when the growth is identically zero, an impulse of magnitude equal to the system's momentum is applied. This instantaneous change in velocity abruptly transfers the system to the origin at time r. The timing of the stimulus is extremely important. The stimulus is applied when the potential energy is minimum and the kinetic energy is a local maximum. This observation is of importance in the development of optimal control laws of a full-blown economic system. If the cumulative money supply in propulsive (stimulative) system is defined by
Optimal Control
165 T 0
where u(t) = Ml(t) represents the Central Bank thrust and Isp the specific impulsive. This can be set to 1. There is a large saving in the use of money as a control (and therefore of inflation) if money stimuli (impulse controls) are used. It is illustrated in Table 5.1. Remark 10.6. We recall that the control u in our analysis belongs to W, the Pontryagin difference of sets. Thus define absolute economic fuel T fuel = Eu = \\u(t)\ dt. 0 Absolute economic private fuel, T E(p)=\\p{t)\dt 0 and absolute economic solidarity or government fuel to be T £(<7)= |k(0l<* • 0
This remark on when to apply economic stimulus (impulsive control) is now clear. The U.S.A. situation in 1990 is displayed on the next page. / economic private fuel = private fuel = E(u)=
11 p(t )\ dt, 0
and economic solidarity or government fuel
166
Differential Models and Neutral Systems
Optimal Control
167
£(
Fu(tx)=\\u(0\dt
ueP*Q.
0
The following optimization problem which is a consequence of Theorem 10.12 may have a solution which throws some light on how efficient and effective economic policies are. It can also be used to compare different administrations and their policies. Optimization problem. Let (yo, RQ, LQ, KQ, PQ, EQ) = XQ be the optimal (ideal) economic path and
% = Flo* So, eo> xo, ^o. M\o> ^io> /]>
the optimal government and private policies. Our aim is to minimize 2: u*P!Q ]{\AO-yo]l + (*(')-«o(0)L+W/)-eo(0)L +(*(/)-*o(0) p
y
L
+ ( £ ( 0 - EoO))lE + («(')- " o ( ' ) ) V ,
where ueW = {u:u + Qci P} = P1Q subject to
dt
x(t)-
\A-i(t-s)x(s)ds = AQX(() + X JAi (/ - s)x(s) - u(t)
Differential Models and Neutral Systems
168
when QczInt(P + terU(t\-t)) and U is the fundamental matrix. When the system is linear without delay, x(t) = A0x(t) + Bu(t) and the controls are u=a +g g = g(G, T, m, e) a = o(ZQJ, T 0 ),
Gandolfo and Petit presented a solution with Italian data. This can be tested using least square optimization MATLAB technique [64]. We now describe some result from the Italian solution. We want to see whedier the old government policy and the firms reaction could have been improved upon to yield better outcomes. Gandolfo and Petit performed some control exercises to evaluate the dynamic properties of their model and to analyze the optimal policy responses to different targets and different instruments. They consider the rate of inflation, the rate of growth output and the rate of growth of international reserves. The ideal target is EEC average rate of inflation, EEC average rate of growth output and EEC average rate of growth of reserves. Gandolfo and Petit choose public expenditure, g, taxes, T; and money supply, M; as control instruments. They discover that the "best" set of tools is (G, m). See the table. The three targets (higher output growth, lower inflation and slightly growing reserve) are met entirely through optimal control policies. It seems therefore that the poor behavior of the Italian economy as described by the solution of the model (base run) in that period, could have been improved upon by adequate policies. They write: "Our optimal control exercises show in fact that with an adequate (optimal) policy mix, the rate of inflation could have been lowered to a yearly average rate of 8.4%, while the growth output could have been pushed up to 8%, at the same time, reserves would have been growing at more than 22% per "annum". This conclusion is consistent with other researchers, (Perkins [1985]). Despite this conclusion Gandolfo and Petit also do not believe that policy makers should determine economic
Optimal Control
169
policy on the basis of optimal control results only. But the experiments do test the effectiveness of different optimal policies. See Figures 1-11.
Figure 1: Output
Figure 4: Public expenditure
-0.100
Figure 2: Price level
Figure 5: Rate of Growth of Money Supply
10 60
Figure 3: Reserves
Figure 6: Output
170
Differential Models and Neutral Systems
-0.076
Figure 10: Rate of growth of money supply
H7
Figure 7: Price level 10.76
Figure 11: Exchange rate
Figure 8: Reserves
Note: The symbols with star indicate the optimal trajectories. The non-starred symbols indicate the base run. Figure 9: Public expenditure
We consider another example - the Rigid Body Maneuver. mx(t) = u(t). F(c, T) = sup 0
Xntn
m
Optimal Control
171
= nun sup •^min
XQ/M
c2 Oit
m
Considering all three possible ranges of argument (positive, negative, and sign change) the minimum is attained at
2x M
/jmin
0
This minimum occurs both at x i = 0, and 12 = T. From (10.181) with m = 2, the open-loop optimal control law is XnM
«*(/) = - ^ r sgn(f2 - am
+ Ht - T)}.
This control consists of one initial impulse to impart a velocity to the system. Then the system drifts until the desired time T, when a final impulse is applied to terminate the motion at rest. See Figures 5.1 A, 5.IB, 5.1 C, and 5.1D. Redmond and Silverberg have determined that if fuel consumption in propulsive systems is defined by
Fu=-^ sp
T \\u(t)\dt,
o
where u(t) represents the thrust and Isp the specific impulsive, which is set to 1, there are large savings in fuel if impulsive controls are used. The result is illustrated in the table below. Continuous controls are those of Theorem 5.4.1; bang-bang controls represent Theorem 5.2.1 in [22]. Table 5.1: Fuel consumption Bang-bang Oscillator 1.168 Maneuver 4.0
Continuous 0.921 3.0
Impulse 0.595 2.0
172
Differential Models and Neutral Systems I 1t
I
"I
1
1
1
—
* - t
-* W)
A
7 !I -0.2
20
1
L
0
0.2
25
1 .... 1 0.4
Q&
1 OJ
L_ 1
U
OJ
1
U
Tim*
Figure 12c Figure 12a
-15
-1
-0.5
0 X{t)
Figure 12b
5
1
li
-02
0
0.2
0.4 0.S X(t)
Figure 12d
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11.
Nonlinear Neutral Systems
11.1 Introduction This chapter introduces the problem of reaching a continuously moving target z by a trajectory of the control system described by the nonlinear functional differential equations of neutral type, Jt[D(t)xt] = f{t,x„u(t)),
in minimum time. The state space is either E", the Euclidean ^-dimensional vector space; the space C = C([-h, 0], En) of continuous functions from [-h, 0] into E" with the sup norm, or the Sobolev space W^> = W^\[-h, 0], E") of functions [-h, 0] -> En whose derivatives are Lp integrable.
The
m
admissible controls are measurable functions u:[t, <x>] -» E whose values u(t) e U, form a compact convex set. The existence theory in C is established in 11.3; the corresponding constrained controllability questions are answered in Chapter 12. The necessary condition for time-optimal control, the maximum principle, is stated in 11.4. It is very desirable for nations to have a high level of growing gross national product, low interest rate, very low unemployment, or full employment, good value of capital stock, low inflation and low cumulative balance of payment. The problem of immense practical importance is to determine whether the economic state can be driven to this state of "paradise" using the private firms control instruments and government controls. In our dynamics government intervenes, the firms react and the resultant control u is the control in our nonlinear systems. One is interested 177
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178
in driving fluctuations of economic state to the state of 'paradise' as articulated by the country as rapidly as possible. This is the time-optimal problem. It is old, very important, and continues to be interesting even for linear ordinary differential systems (see [22]). Our aim here is to present a comprehensive theory that will be needed for the construction of an optimal feedback control similar to the development in [23]. We consider the problem of reaching, in minimum time, a continuously moving target z/, in function space, by a trajectory of the control system described by the nonlinear functional differential equation of neutral type, ^([D(t)xt} = At,x„u(t)\
t>0,
(11.1)
In (11.1), D is a continuous function D():[x, oo) x C -> E" given as D(t)xt=x«)-g(t,xt),
(11.2)
and f.[x, oo) x C x Em -» En is a continuous function. Here x is a real number. The target z may be taken as a point of En, or of C or of wy \ 1 < p < oo, where Wr-*' is the Sobolev space of all absolutely continuous functions dx x:[-h, 0] -> E" whose derivative x(t) = — belongs to the/?-integrable space Lp([-h, 0], En). The admissible controls are measurable functions u:[x, oo] -> E" with w(0 e U(t, x{), where U:[x, oo) x C -> 2E" is a set-valued map that is nonempty and compact. We can also consider the set of admissible controls that is a closed and bounded subset of Lp([a, oo), En) with zero in its interior, when the state space is WT '. The time-optimal problem is to determine an admissible control u* such that the solution x(v, q>, u*) of (11.1) hits a continuously moving target z (in the appropriate space) in minimum time /* > a. Such a control is called time-optimal, and /* is the optimal time. This problem can be divided into three parts: (i)
Does there exist an admissible control u such that
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179
xT(a, cp, u) = zT ;
for some x?
(11.3)
This is the problem of controllability. With the optimal time TO defined by T 0 =inf{x:x such that (11.3) holds}, (ii)
(11.4)
does there exist an optimal control u such that XT0(CT,(P,M) = Z T ( ) .
(U.5)
Finally: (iii) what are the form, uniqueness, and general properties of the opti mal control if it exists. Part three of the problem is inferred from the necessary condition for an optimal control, the so-called maximum principle. Though a very large body of research is available on the control of neutral systems, most are concerned with optimal control of systems with quadratic cost functions, [1-2]. The time-optimal problem was implicitly touched upon in [1], as a consequence of a result on minimizing a general cost function. Here a necessary condition was formulated. For linear systems with unrestrained controls, controllability questions were investigated by Gabasov and Kirillova [3], Rodas and Langenhop [4], and Salamon [5] and more recently in [7] for linear systems. The present treatment gives a comprehensive theory for the nonlinear system (11.1). In Section 11.2, we establish the existence and uniqueness and some regularity properties of solutions of the differential equations (11.1). We introduce some conditions that will prevent finite escape times of trajectories. This will enable us to restrict our attention to a closed and bounded set of the (f, ;t/)-space. If x(t, a, u) is a solution of (11.1) and ^lAv>") i s the Frechet derivative o f / o n E x C x Em with respect to rth variable, then the linear approximation of (11.1) about x{t, a, w) is Jt[D(t)yt]=L(t,y!)
+
B{tHt),
(11.6)
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Differential Models and Neutral Systems
where L(t, yt) = D2f(t, xt, u)yt, B(() = 1^/(1, x„u). Some of the properties of the solution map of 11.6 is outlined in 11.2. In Section 11.3, we prove the closure of the attainable sets of (11.1). Under the constrained controllability assumption, we prove the existence of a time-optimal control. Sufficient condition for controllability of (11.1) is discussed in Chapter 12. For completeness we include the necessary condition of an optimal control, a maximum principle in Section 11.4. There the theory is applied to linear systems.
11.2 Existence, Uniqueness, and Continuity of Solutions of Neutral Systems In (11.2), assume that g{t, ■) is a bounded linear operator from C into En for each / e (x, oo), g(t, cp) is continuous for (t,
I g(t, cp) | <; k(t) ||q>||,
(11.7)
for some non-negative function k e C((x, °o),is), and \i(t, •) is an n x n matrix function of bounded variation on [-h, 0]. We assume that g is uniformly nonatomic at zero; that is, there exists a continuous, non-negative, nondecreasing function L(s) for 5 e [0, h] such that L(0) = 0,
for all t e
(T, OO),
J [ ^ ( r , e)](p(6)
(11.8)
We require that D in (11.2) be uniformly stable in the following sense: Definition 11.1: The operator D is uniformly stable if there are constants, a, P > 0 such that for every a e [a, oo),
Nonlinear Neutral Systems
181
homogeneous difference equation D(t)xt=0
t>a, (11.9)
%=cp,
£>(a) = 0 ,
satisfies ||x,(a,
'>0.
(11.10)
In (11.1) we assume that /:[T,oo)xCx£ m ->£" is continuously differentiable and is C 1 . We now establish the existence of a solution of (11.1) for each initial data (a, (p) e E x C([-h, 0], O), where O c £ " is an open convex set containing the origin, and where / : E x C([- A, 0], O) x £ m -> £",
g: E x C([- /i, 0], 0} -> £" ,
are continuous and g satisfies (11.7) and (11.8). The existence result of Cruz and Hale [8] does not quite cover the situation here. One can use the ideas of Melvin [9] to prove existence and uniqueness with initial data (a, cp) e Ex W^\ and this is done by Chukwu and Simpson [24]. Theorem 11.1 Suppose in (11.1), (i) /L;;-):E x C([-h, 0], 0)xEm-+ (ii)
En is continuously differentiable.
For each compact convex set K c O there exists an integrable function M\ :E -» [0, °o) and integrable functions M\:E -» [0, oo), i = 2, 3, such that II D2f(t, cp, w) || < Mx{t) + M2(t) || w ||,
|| D^f{t, cp, w) || < M 3 (0 (11.11)
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Differential Models and Neutral Systems
for all /, w, and q> C([-h, 0], K) where D,/(-, •, •) is the Freshet de rivative of/with respect to the /7-h variable. (iii) / ( / , 0 , 0 ) = 0,
V/.
(iv) If L(t, •) = D2f(t, 0, 0), 5(0 = D?f(t, 0, 0) where 0
Z(/,cp)= \[dQr\(t, e)]cp(9),
|/;(/,q>)l = M(OII
(H.12)
-h (t, cp) e (x, 00) x C
for some M e LJ 0C ((T, 00), £), and some n x n matrix r|(x, 6) of bounded variation in 6 e [-//, 0]. (v)
The function g in (11.2) and (11.7) satisfies (11.7) and (11.8). Then there exists an open neighborhood OQ, of the origin in C, and an open neighborhood OJJ of the origin in Z,oo([o% 00), Em) such that for each ( p e O f and each u e OJJ, (11.1.1) has a unique solution that is continuously differentiable.
Proof: Let Cf = Cji[a-h, T], En), and define the function G:E by f0,
if Q -h
[gti, xt)-g{o,y),
if a
G{x){t)=\
Define H.CT*
XCT^CT
(11.13)
Ioo([a, T], Em) -> Cfby 0,
H(x,u)(t) = \'e jf(s, xs> y(s))ds,
if a - h < t < a, (11.14) if
a<(
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Nonlinear Neutral Systems
Finally, define r.Cf-> Cj f(p(/-a),
if a -h
M0 = '
[(p(0),
if
a
Written in integral form, Eq. (11.1) is equivalent to xa
(11.16)
=
x(t) = g(t,x,)- g(a, (p) +
t2c.
Using this fact and the operators G, H, and /, we see that x satisfies (11.1) if and only if (11.17)
x = Gx + H(x, u) + Iy
for x e CjIt is easy to see that (11.16) is well defined, since/-,-,) is assumed to be continuous and by the mean value theorem (we use the convexity of K), || /(/, q>, w) || < | /(/, 0,0) | £"+W1(0l|q»|| + A/3(0l|w||,
(11.18)
for all / € [a, t\], (p e C([-h, 0], K), w e LooNote condition (ii). We need two lemmas. Lemma 11.1 Let H:C([o-h, T], 0) x L w -> Cj be as defined in (11.14), G be as defined in (11.13). Then H is continuously differentiable and 0, ( A t f (0, 0)x)(0 = o JD2/(5, 0, 0)xsds,
a - h
Differential Models and Neutral Systems
184
CT-h
0,
(£»3//(0,0)«XO =
(11.20) \D2f{s, 0, 0)xsds,
a
for all x<=CT,u<= ^([a, T\,Em), t e [a-h, T\. G is also continuously differential and for any x e Cj([a-h, t\], En), (DGx)x = Gx.
(11.21)
Proof: By [12, p. 177],
{L^Hix, u)x){t) = \D2f(s,
xs, U(s))xsds,
(11.22)
a t
(D2H(x, «)w)(0 = JDsfis,, xs, u(s))u(s)ds,
(11.23)
where t e [a-h, T\, x e C([a-h, T],0)xeCr= C([o-h, 7], En), u, u e £»([<*, 7], Em). The results follow quickly from these. Also, because of the linearity of g(t, 9) in (11.7), (11.21) follows at once. Lemma 11.2. Define K: C([CT - h, T], En) -► C([o - h, T], En) by Kx = DG(0)x+DlH(0,0)x.
(11.24)
Then (1 - K)'^ exists as a bounded linear map. Let q be any function in C([x, 00), En). Consider
Nonlinear Neutral Systems
185
Jt WO - gff, xt) - q(t)] = Dif{t, 0, 0)x, where *a=(P = <7oo-
(11.25)
4 [£>(')*,-<7(')1 = £(',*,)•
(11.26)
This is the same as
at
By assumptions (iv) and (v) and Theorem 2.1 of Hale and Cruz [10, p. 333], there exists a function x(a, (p, q)(t) with initial value (p at a uniquely defined and continuous on [a-h, oo) and satisfying (11.6). Furthermore, for any fixed a and / < a, x(a, •, 0X0 is a continuous linear operator from C into £". The integrated form of (11.26) is
x(t) = g(t,xt)-
g((J,(p) + q(t) + cp(0) + J i ( s , X S ) A . a
But this is *(/) = DG(0)x(t) +
186
Differential Models and Neutral Systems
that its solution £(cp, u) e Cj is necessarily unique for any pair of cp, w such that the solution exists. Define M(x, u,
Cj([o,h,T\,En)xLn([o,T},Em) such that M(£(cp, u), u, (p) = 0 for all (cp, u) e N. Furthermore, x = £,(cp, u) satisfies (11.17) or (11.1). We now choose 0QT ,Oy so that OQ * Oy czN to complete the proof. It is clear from Theorem 11.1 that the solution x(a, cp, u) of (11.1) is continuously differentiable for each (a, cp) e E x C and each u s Zoo([cr, 00), Em). Corresponding to the point (/, a, 9, u) e E x E x C x Lao, the mapping x:E xExCxL^-^C defined by x(a, cp, «)(/) = xfa, (p) represents a point in C. We now give some very useful properties of this mapping. Lemma 11.3. Let (/, a, q>, u) e E x E x C x L^. Assume all the conditions of Theorem 11.1. Let Dx(t, a, cp, u) denote the partial derivative of x(t, a,
Nonlinear Neutral Systems
187
d - [D(t)yt] = D2f(l, xt(o, cp,«), u(t))yt + D}f(t, x,(o, cp, u), w)v
*,=¥•
(11.27)
Also for v G Z,oo([cr, oo), Em) we have D^x{t, a, (p, «XV) = X'.CT>9' "> v ) where the mapping / -> >>(/, a, cp, M, V) of E into £M is the unique solution of (11.27) satisfying ya(o, cp, w, v) = 0. Proof: For each (a, ( ( i ) e £ x C , the unique solution of (11.1) with initial data (a, cp) is given in (11.16) by xa = cp in [- h, 0], x(a, cp, w)(0 = x(/, a, cp, M) = g(t, x, (a, cp, u)) - g(a, cp) + cp(0)
+ jf(s,xs(a,
t>a.
Let £>4Jt(/, a,
D4x(/, a, cp, w) = JZ^/O. *5(a, cp, u)), u(s)Duxs(a, cp, w)
+ JD3/(s,xs(a,
+ £>2g('> *f(o\cp, "))"^J (°> 'P' ")• Thus, by the same reasoning in (11.21), we have dx, ( dx, Dig{t, x,iq, cp, «)—*- (a,cp, u) = gl / — (a, cp, u)
188
Differential Models and Neutral Systems
Thus, D4x(t, a, cp, u) - D2g\ t, xt(p,
t CT
= p V C * . **,(<*>
On differentiating with respect to /, d — [Dux(t, a, (p, u)v - Digit, xt(o,
f
^ic
+ / + Difis, xs (a, cp, w), u(s)) —^ (a, cp,«) ds J da* d
189
Nonlinear Neutral Systems
D4jc(r, CT, cp, u) = jD2/(s, XS(O, cp, u), u(s))Duxs{a, cp, u)ds
+ Jz>3/(s, xs,(c, cp, u), u(s))ds + g(/, D u x,(o, ip,«)).
Thus £>jr(f, CT,
1
V I " «(<"»») Djf C ■••)
' dx + y + JD2f{s, xs(o, cp,«), u(s)) —*- (a, cp, u)yds
+ JDif(s, x,(a, (p, «), if(5))Duxf(<J, cp, u)vds
+
[DIAS,
xs(a, cp, u), u(s))vds + g(t, Duxt(a, cp, u)v)
o
Now take the t derivative to obtain
dt
dx(t,a, cp, «X
We have used the linearity of g. Note that
Ip + D u X.s(0, Cp, u)v
190
Differential Models and Neutral Systems Dx a (a,
This concludes the proof. We require conditions that will prevent finite escape times of trajectories of (11.1). It is given in the next lemma. Lemma 11.4. In (11.1), assume: (i)
D is uniformly stable.
(ii) (D(t)xt,Mxt,u))
(11.28)
for some constant k > 0, for all / >CT,x,eC, u{t)e U(t, x,). Then there exist an M> 0 and anN> 0 such that every solution x(a, cp, u) of (11.1) satisfies \\xt(<3,iv,u)\\<MeN(t-°\ so that in any compact interval [CT, T], ||jf/(a,(p,«)||sA/eJV<'-a).
Proof: Let x be a solution of (11.1). Let V{x(t)) = (Dx,,Dxt) = \Dxt\2. Then y V(x{Q) = 2(Dxt,f(t, x„ «(/))) <2K(1 +1 Dx, | 2 ) , at by condition (ii). Thus ^\Dxt\2<2K{\+\Dxt\2). at It follows that
Nonlinear Neutral Systems
191
| Dxt | 2 < (1 + | Dtp|2)exp 2kt = g(t). Since D is uniformly stable [13, Lemma 3.4], there exist some constants a, b, c, d such that | x , ( a , ( p , u ) | | < e ' -a(T-cr) A||cp|| + c sup \g(u) + d sup a
\g(u)\.
Since g(t) = (1 + |D(p|2)exp 2ki, we obtain || xt(a, (p, K)|| < e'a{'~a)[b\\
3rt? ao , we have \\xt(a,ip,u)\\<MeNl.
Remark: Once a value T > a is chosen T < co, then x(a, cp, M) is bounded. We work in the region that is a closed and bounded subset of E x C([-h, 0],
NT-,
M = {(/,x / ):o
(11.29)
Remark 11.1: If q>, the initial point, lies on a certain compact initial subset of C and u(t) e U(t, xf), then the bound MeN^\s independent of the controls. In the region M, when this uniform bound is in force, we shall examine the behavior of the attainable set Ji of (11.1). This is done in the next section.
Differential Models and Neutral Systems
192
11.3 Existence of Optimal Controls of Neutral Systems Let M be the closed and bounded subset of E x C that was introduced in (11.29). In order to study the existence of a time-optimal control of -[Dit)x,} = f(t, x„ «(/)), xa=y,(t,x,)eM
t>c,
,
we introduce the associated contingent equation,
{im>,UF«,*,).
(1]3])
xa=(f>,(t,x,)eM , where the set-valued map
F:ExC-^2t is given by F(t,
U:ExC^2L is said to be upper semicontinuous with respect to inclusion in t, cp if for any e > 0 there is a 8 > 0 such that
Nonlinear Neutral Systems
193
£/(/],
Dx( is absolutely continuous and has a continuous derivative on [o\7],
(ii)
(11.32)
—(Dxt) satisfies Eq. (11.31), dt (iii) xa = cp, (/, xt) G M.
Clearly, any solution of (11.31) gives rise to a solution of the orientor problem (11.32). By standard measurable selection arguments of [14], we know that every solution of the relation (11.32) can be viewed as a trajectory of the control system (11.31). It is summarized in the next lemma. Lemma 11.5: Let F(t, cp) be convex for each (/, q>) G M. Then a function x G C([a-h, 7], En) is a solution of (11.32) if and only if x is a solution of (11.31) for some admissible control u, «(/) e U(t, xt). Definition 11.4: The attainable set JUJ) of (11.31) at time / > a is a subset of C = C([-h, 0], E") defined by Ji{t) = {x, €C ,x is a solution of (11.31) for some u(t) <=U(t,xt),{t,xt)eM). By Lemma 11.5, this is the same as: A(0 = {x, e C, x is a solution of (11.3.2), i.e., — dt xo=
[D(t)xt]sF(t,xt),
(t,xt)eM).
We assume the conditions of Theorem 11.1, which ensure the existence and uniqueness of solutions. Subject to these conditions, Ji(t) is nonempty. We now prove that the attainable set is closed as well.
/ 94
Differential Models and Neutral Systems
Theorem 11.2. In (11.31) assume that: (i) f.E x C([-h, 0], En) x Em) -> En is continuously differentiable. (ii) D is uniformly stable and satisfies (11.7) and (11.8). (iii) For any measurable yif) satisfying y(t) e F(t, X(), we have \y(t)\ < m(l\ a.e., m e L\([o, T\, E). (iv) F(t, (p) is compact and convex for each /, (p. (v) U(t, q>) is upper semicontinuous with respect to inclusion. Then the attainable seM(f) of (11.31) is closed. Proof: To prove the closure of the attainable set, let x" e Mt) n = 1, 2, ... . Let x" -> X( as n -> oo. We prove JC/ e Ji(t). Since x" e j?(/) for each n, ^-[D(t)x?]eF(t,x?), at
tel.
By condition (iii),
\l,^
<m(t),
where m e L\(I, E). From the above inequality, we deduce that / / | D(t, x?) - D(a, cp) | < jm(s)ds, | D(t, x?) - D{t)x? | < jm(s)ds, 0 / so that [D(s,x")ds->0,
/-»<»,
uniformly with respect to n for each decreasing sequence {£/}, £/ c / with void intersection. Therefore, by [15, p. 292] there is a sequence (we retain
Nonlinear Neutral
Systems
195
the same notation) weakly convergent in L\(I, E") to a function ^ e L\(I, En)). Then for each t e I, t t D(t)xt = lim [£>(/)*"] = "m 0(a>p+ \D(s)(x?)ds = D(a>p+ fe(.s)
/i->oo
J
J
CT
O
It now follows from p. 422 of [15] that there is a sequence {^} of convex combinations of the functions {D(t,xf)D(t,xf+')...}
converging in L\(I,
n
E ) norm to £. From this sequence (£jfc) select a subsequence that converges to £ a.e. Thus, almost everywhere on / 00
£(0ef)co *=1
\
00
sflco
\jb{t)x? \n=k
J
*=1
f l F ^ x " ) cF{t,xt), \n=k
(11.33)
J
where co(M) is the closed convex hull of M e E". Hence
Hence J?(/) is closed, since also (/, */) e M from the closure of M. This concludes the proof. Note that we have used the upper semicontinuity of F in (11.33). The so-called property Q of Cesari, a much milder condition, could have replaced upper semicontinuity [see 16, p. 7]. Condition (iv) could be replaced by the closure and convexity of F{t, (p). The existence of an optimal control for (11.31) requires a controllability assumption. This concept is defined as follows: Definition 11.5: Let zt e C([-h, 0], En) = C be a target point function that is time varying. Suppose z-p e MX) for some T > o\ then the system is controllable to the target. Thus for each y e C there exists a ! > o and an admissible control u{t) e U(t,xt), t e [a, T], such that the solution of (11.31) satisfies xda, (p, u) = (p, xj(o, cp, u) = zj. In this case we also say that system (11.31) is function space z-controllable with constraints.
196
Differential Models and Neutral Systems
Definition 11.6: System (11.31) is null controllable with constraints if for each (p G C, there exists a t\ < <x> an admissible control u(t) G U(t, xt), t G [cr, t\ ] such that the solution of (11.31) satisfies %(CJ, cp, u) = cp,
xtx (a, (p,u) = 0 .
The main result of this section is now stated. Theorem 11.3. In (11.31), assume that: (i) f.E x C([-h, 0], E")xEm^> E" is continuously differentiate. (ii) D is uniformly stable and satisfies (11.7) and (11.8). (iii) For each measurable yif) satisfying y(f) G F(t, xt), we have \y(t)\ < m(f) a.e, m G L\([a, 7], E\. (iv) F(t, cp) is closed and convex for each /, cp. (v) U(t, cp) is upper semicontinuous with respect to inclusion. (vi) System (11.31) is function space z-controllable with constraints. Then there exists time-optimal control for (11.31). Proof: By assumption, there exists some t\ < 7/such that z,x
=xheA(tx)
where Mt\) is the attainable set of (11.31) at time t\. Let t* = inf{t:zt<=A(t)}. It follows thatCT< t* < t\ < T. There is a sequence of nonincreasing times /„ G [a, fi] converging to t* and a sequence un of admissible controls, un(t) e U(t, x"), such that z =x tn tn(a>
Let jc(a, cp, M") = x". Then
Nonlinear Neutral Systems
Ar dt
n
197
[x Vn) -gUtn,x,
)]eF{tn, x, )*a = 9, (tn, x't' ) e M .
Since Zf is continuous and tn -> t* as n -> oo, we have Z; —» ztif. Also x" —> xtt as « -> oo. By condition (iii), k [*"('«) - g('n, *? )1 * «(4i) a.e. where w e ^([o, 7] £). Let
y(n0 = *n(0-g(',*f)The set H = {yn.yi{t) = JCW(0 - g(t, *")} is an equicontinuous family. Indeed, for every measurable subset E of [a, 7] we have
Let e > 0. By integrability of TM we conclude that there is a 5 > 0 such that if meas(.E) < 8, then \Em{t)dt < e. For this E we have ! ' " « > dt
198
Differential Models and Neutral Systems
6 C([CT, 7], En), that is absolutely continuous on [a, 7]. By condition (iii), we deduce | yn(t) - yn(a) | < £m(s)ds ,\yn{t)-y{t)[<
]\m(s)ds, so that
f d n I ~7 [y (s)]ds -> 0 as n -> <x> uniformly with respect to each decreasing sequence {Ej}, E\ c [o% 7] = 7 with void intersection. Therefore [see 15, p. 292] there is a sequence >'" (we retain the same notation) weakly convergent in L\(J, E") to a function 4 e Zi(7, £"). Since the functions y* converge uniformly on [a, 7] to y, certainly they converge pointwise almost everywhere to y and hence by absolute continuity of these functions we have y(t) = 4(0 a.e. Thus, for tn
eI,t„-> t*, *('•) - *('. .**,)= I™ [*"(/„) - g(fB, x" )]
lim [xn(o) - g(a, 4)] + \^-[yn(s)]ds yt—L
J dt at <7
=
We note x£ = q> for each w, so that lim„^.oo *a
=
*P- Because M is closed,
(tn, x" ) -> (/*, xf) e Mas « -> oo. It now follows from [15, p. 422] that there is a sequence {£,„} of convex combinations of {yn,yn ,■■■} that converges in L\(I, En) norm to 4- From this sequence {£,„}, select a subsequence which converges a.e. to 4Thus, almost everywhere on I,
Nonlinear Neutral Systems
199
n=\
\k=n
cf)c0 Q^.*i) fi=l
£F((„i,t)
(11.34)
U=*
where co(N) is the closed convex hull of N in E". Hence ^[*0*)-g('«,*/,)]efO*,*/,). We have already seen that xa =
This completes the proof. We observe that we used the upper semicontinuity of F in (11.34). It could be replaced by using the weakened version of Cesari's property Q. SeeAngell[16,p. 7].
11.4 Optimal Control of Neutral Systems in Function Space In this section we present a maximum principle for optimal control of nonlinear systems of neutral type with function space initial and terminal conditions. In our setting we guarantee the existence of regular nontrivial multipliers when the controls are constrained to lie on a unit m-dimensional cube. This problem was broached in [2] and [26], but not completely resolved. In [26] the nontriviality of the multipliers could not be guaranteed and pointwise control constraints were ruled out. This section continues the exciting recent work of Angell and Kirsch [25] on delay equations. We maintain the basic definitions and notation of the corresponding treatment in [25], and consider the following problem: Find a control u that minimizes
200
Differential Models and Neutral Systems T J(x,u)=jf°(t,x„u{0)dt,
(11.35)
0
subject to the constraints N
i(t) - £ A-U(')*(' - hi) = /(/, xt, u(t)),
*0 =
a.e. on
(11.36)
(11.37) [0,7],
(11.38)
j=\,...,m),
(11.39)
where Cm={ueEr":\uJ\
i=\,...,p,
te[0,T\,
(11.40)
and N
xzC((-h,T),
En),
^0-£^l/(0i(<-^)eAx,([0)r])£"), i=l
u e M t O , 7], C"). We assume further that the initial data § and the terminal function VJ/ satisfy <>| G C([-A, 0], E"),
M/ G C([-A, 0], E"),
with D(7)v G C ! ( R , 0], £").
(11.41)
Nonlinear Neutral Systems
201
We observe that if the admissible control u is such that uj is continuous, N
then by Eq. (11.4.2) * ( 0 - £ ^y (')■*('-*/) is continuous on (T-h, T), since 7=1
we have assumed the following properties of/ °,/,/*: The mappings /°:[0, 7] x C x Em -> £, /:[0,
T\xCxEm-+E»,
/HO,T\*C-+EP, are continuous; / / " are Frechet differentiable with respect to the second argument and continuously differentiable with respect to its third argument, the derivatives being assumed continuous with respect to all arguments. Also,/*:[0, 7] x C-> EP is continuously Frechet differentiable with respect to its second argument. We associate with (11.36) the linear equation N
x(f)-^ji(t-hj) ;=i
= Lit,xt) + B{t)u(f),
*0=4>,
(11.42)
where we U,x eX\; and these subspaces are defined as follows: X\ = {x e C([-h, 7], EP):D{T)XT e C\[0, 7], # " ) } ,
(11.43)
t/= {u € Zoo([0, 7], £ W ) : „ r 6 C([-h, 0], £ " %
(11.44)
17=C([0, 7],£P),
(11.45)
K2 = {(z, u, (3) e Zoo([0, 7], £") x C([-fc, 0], E") x E":ZT e C([-h, 0], £»)}. In (11.42),
(11.46)
202
Differential Models and Neutral Systems
L(t,xt) =
D2Kt,x;,u\t))xt, (11.47)
B(t)v =
D?fit,x;,u*(t))v,
where Djf (t, x, , u ) is the Frechet derivative with respect to the /th argument at (/, x{ , u (t)) and x , u (/) are the optimal trajectory and optimal control respectively. We introduce the following maps: g°X\ x U->E, gl-JC\xU-+Y\, g2:X\
xU->Y2,
which are defined as follows:
g°(x,u)=lf°(t,xtMO)dt,
(11.48)
g 1 (*,")=/(•, *(•)).
(11.49)
N
N
gz(x,u) = *(•) - Z AJ <•)*<• - hj) - /(•• *(•)> «(•» - (v(0) - X ^y (H v(-Ay-)) 7=1
7=1 N
^
+ x(T) - X Aj(T)x(T - hj), x(T) - y(T) 7=1
Observe that if g*(x, u) = 0, then
(11.50)
Nonlinear Neutral Systems
203
N
* ( ) - X ^ ( ) * 0 - A y ) = /(•-*(•)>"(•)), 7=1 N
N
x{T)-YJAj(T)x{T-hj)
= y(fi)-Y,AjV)V<<-hj)>
7=1
x
(11.51)
7=1
Because of our previous assumptions o n / / 0 , / l , the functions g°, g1, g 2 are continuously differentiable at each (x ,u ) e X\ x U. Indeed, T
Dig°(x*, «*) = J/>2/°(/, **, « * ('))*A
* e *,,
(11.52)
0
r D 2 gV.«*)«=J^/V**,«*(0«(0'*,
fllgW^/W.*,*)* D2g\x\u*)
ueU,
^C,
= 0.
(11.53)
(11.54) (11.55)
Also TV
£>1 g V , u*)x = i(.) - £ ^ ()i(- - hj) - D2f(; *(.), » * (■))*(.) , 7
N x(T) - £ Aj(T)x(T - Ay),
x(7),
where XB Xu
(11.56)
7=1
Z)^**, M*M/) = W , JC/, «(/))v(0, v € [/. (11.57) Due to the validity of the Riesz Representation Theorem, there exist
204
Differential Models and Neutral Systems
r, i(/, ■) € NBVpxril-h,
0]),
no(/, 0 e NBVnxn([-h, 0]),
y](t,-)eNBVnxn([-h,0]), such that o
Io(0* = D2f°(t, x*,u* (0) = jdmB(t, 6)<|)(e), -h
(11.58)
0
4(0*=D2f\t, x*, u * (/))♦ = periiC e)<j>(e),
(l 1.59)
-h
o
I(/)4> = Ihf(t, x*, u * (/))<(. = p9Ti(/, 0>K6).
(11.60)
-A
In all these representations, r\,r\i, i = 0, 1 are measurable. We extend these functions by defining r|(/,6) = 0
for
t>T.
To match up our notation in the statement of the multiplier rule in [28, Theorem 7.11.2], we set X=X\ x U,
W= {(*, u) e X:xo = $},
V={(x,u)e
W:U<E
£/ad},
U ad = {u e C/:w(0 e CW a.e. on [0, 7]}, where C«={« e JE w »:|«y|^l,
y=l,...,«}.
Note that £/ad has a nonempty interior relative to U.
(11.61)
Nonlinear Neutral Systems
205
We now state a local maximum principle. Theorem 11.4: Let u be the optimal solution of (11.35) - (11.41) and x* 0 < t < T the corresponding optimal solution. Assume that Dg2(x , u ) = D\g\x*,u*) + &2Sp-{x , u*) has a closed range in Yi- Then there exists (X, a, p,v,q)zEx
Loo([0, T\) xNBVp([0, T\) x NBV„([0, T\) x E»,
(X, a, p, v, q) * (0, 0, 0, 0, 0) such that X £ 0,
p = (p 1,..., pp), pj is nondecreasing in [0, 7]
(11.62)
and pj is constant on every interval where /j-(/,x* ) < 0 ;
(11.63)
r r
N
T+
- £ a ( f + AfcM_U(/ + />*) + JTI_I(6, / - 9)
t
-
fn(6, / - 6)a(Q)dQ + fn(9, / - 9)^(9 - T) for all / e [0, T\. t
t
(11.64) Also T T T - X \l30(t)u(t)dt + \a(t)B(t)u(t)dt - jdv(t - h)B(t)u{t) 0
t
T-h
206
Differential Models and Neutral Systems T
<-X JB0(t)u*(t)dt + \a{t)B(t)u*(t)dt-
jdv(t-h)B(t)u*(t),
(11.65)
T-h
for all u e CadCorollary 11.1: In addition to all the conditions of Theorem 11.4, assume that the mapping Dgix*,u*):(W-(x*,u*))^Y2 is surjective. Then we can take the multiplier A, = 1. Proof: Carefully note that the functions g°, g 1 , g 2 of the Multiplier Rule of Theorem 7.11.2 in [28] satisfy the same assumptions of the mappings of (11.48) - (11.50) and the spaces (11.43) - (11.46). We are guaranteed the existence of the multipliers. Thus if we denote the space L by L = {x e Zoo([0, T\, E"):xT e C([-h, 0], E")}, and its dual by £*, we have the existence of I e £*, v e NBVn([-h, 0]), q e E", p e NBVp([0, 7]), and X > 0, (X, e, v, q, p) * (0, 0, 0, 0, 0) such that p is non-decreasing and
jdp(t)f\t,x*(t))
T X \LQ(t)xtdt + |dp(t)Li(t)x t
= 0,
(11.66)
N +1
*(•)-Z ^y 0*0-£()*(•)
N
J dv(t~T)) m-Y,Ajit)X*(t) T-h
for all x e X\,
XQ
y'=i
= 0; and
+ qx(T) = 0,
(11.67)
Nonlinear Neutral Systems
207
T
X Jflb(0«(/)df + ([- B(-X«0 - « * (•))] * 0,
(11.68)
0
for all u e f/ad- Recall the definition of r|, r|j and their extension outside their intervals of definition, and extend the definition of v by constancy and continuity outside its original interval of definition. We need an expression for the linear functional £. It is contained in the next lemma. Lemma 11.6: The linear functional £ has the form
£(z)=^a(t)z(t)dt-
\a\{t-T)z{t),
(11.69)
T-h
0
for all z e £ and a e £«,([(), T\, En). Proof: Consider (11.67). Let z e £ and consider the equation N
m-Y,A_Xj(t)x(t-hj)-L(t)xt=z(t)
in [0,7-],
x0=0.
(11.70)
The variation of the constant formula of Hale and Meyer [27] gives the solution as
jc(/,z)=j>(/,s)z(s)*,
'e[0,/],
0
where X{t, s) e Loo([0, 7], En2) is the fundamental matrix defined by *u, s) = — r — 95
(a) 0W-, *) = <).
a.e in 5.
(11.71)
208
Differential Models and Neutral Systems N
t
(b) W(t,s) = YJA(tW(t-s-hj)+ 7=1
[L(X,Wk(;s))dk-Q-s)I,0<s
With this expression, (11.67) takes the form T
T
X JLQ(t)xt(;
0
Z)dt + jdpiOmt)^;
T z) + I(Z) +
0
jdv(t
~
T)z(t)
T-h
T + jdv(t-T)L(t)tl(;z) T-h
+ gx(T,z) = 0
(11.72)
for all z e L. But then Z,n, L\, L, are as given in (11.58) - (11.60), so that (11.67) yields
t(z) = -XJ
> jdsx]Q(t,s)x(t + s,z) dt
T 0 - p p ( 0 Jd s m(t, s)x(t + s,z)- jdv(t - T)z(t) 0 -h T-h
-
| dv(t- T) ysn(t, T-h -h
s)x(t + s,z)- qx(T) .
(11.73)
We now substitute the expression x(t; z) in (11.71) into (11.73), and change the order of integration to deduce (11.69), where a e L depends on X, q, X, v, "Ho, T|i, T|. With (11.69) established, we rewrite (11.72) as follows:
x Jio(/)V+pP(OA(0*,
Nonlinear Neutral Systems
209
T
N
+ ja(t)[x(t) - YjA-ijiW 0
-
- hj) - L(t)xt]dt
7=1
T N | dr{t- T)[x(t) - £ A_Xj (t)x(t -hj)T-h
7=1
T
N
+ j dr(i -TXxiO-^A-XjiOUt T-h
L(t)xt ]
- hj)] + qx(T) = 0,
;=i
for all x\ e X\. Once again we use the expressions for r|o, t|i, r| to deduce that T
t
T t
^ { J dQVoit, 0 - t)x(Q)dt + | 0 t-h
jdp(t)dQm(t, 0 - t)x(6)
0 t-h T
N
t
+ Ja(/)[x(0 - £ X_,,(t)x(t -hj)0
7=1
T t = f \dv(t-T)dQ^t,Q-t)x(Q) T-h t-h
Jd0Ti(/, 0 - t)x(0)]dt t-h
+ qx(T) = 0,
for all x e X\ with *o = 0. We consider each of the first four terms above, beginning with r t
x\
peT1o(/,0-o*(eM,
0 t-h
and ending with
210
Differential Models and Neutral Systems T
t
T-h
t-h
and integrate by parts: T
T
t
- X JTIOC*. - h)(t - h)dt - X | JTI0(/, S - t)x(s)dsdt 0 t-h
0
- Jrfp(/)Tii(/, - A)*(r -h)- jdp(f)iii(/ *,-/)*(*)«&
W
/ * ) i ( / ) - £ / L I y ( r ) i ( / - A y ) <* + Ja(0n(>, - h)xQt - h)dt 7=1 N
- \
a(t)
m-^A^ijiOxit-hj)
T + qx(T)-
dt+
J T-h
jdv(t-T)T](t,-h)x(t-h) T-h
7=1 t jdv(t-T)T](t,s-t)x(s)ds
= 0,
t-h
for all x e X\ with XQ = 0. As in Angell and Kirsch [25], use the following abbreviation:
- fa(s)ds, u(0 =
0 <, t < T - h,
t - v(t -T)-
T \a(s)ds,
T
-h
Nonlinear Neutral Systems
211
For any y e L that is extended by zero, set
x(t) = \y(s)as
for
te E .
0
Use this above and change the order of integration:
T T-h T s+h - * { f ilo(e + h, - h)ddy(s)ds = X J JTI 0 (8, S - Q)dQy(s)ds 0
S
0
5
T T-s T s+h - J pP(6)Tli(e + h, - h)y(s)ds - J pp(e)n.i(e, s - Q)d0y(x)ds 0
s
0 /
T T s+h jd\i(eM
0
5
T T-h + J U i ( 8 + h)T](8 + h,- h)y(s)ds 0
5
T + q^y(s)ds+ 0 for all y e
T-hk N J a(t)^A_lk(t)x(t -hk
+ hk)dt = 0
*=•
L. Note that
N T-hk N ]T J a{t)A_xk(t)x{t -hk)dt-JT
T h
~k ]a(s + hk)Ak(s + hk)y(s)ds.
Since the integrand above must vanish pointwise, we have for s e [0, T],
212
Differential Models and Neutral Systems T-s
-X
s+h
f r)0(Q+ h,-h)dQ-\ s
{r\0{0,s-B)dd 0
T-h
s+h
J rfp(9)ru(9+/*,-/*)- jdP(Q)m(e,s-e) 0 s+h
T-h
+ a(s) + f 4I(9)TI(9, s - 9) + JdWG + /J)TI(9 + h,-h) + q 0
jfc=l
which yields T
T
a(s) = A JTI 0 (9, * - 9)d9 + pp(9)TH(9, j - 9) s
s
T
T
- Ja(9)Ti(9, s-Q)dQ-
pv(9)ti(9, s - 9)
AT
- <7 - £
a
* C*+ A* )y4-!* C*+ A* ) •
k=l
We have proved that (11.64) holds. To prove (11.65), we insert £(z) in (11.69) into (11.68):
X JB0(t)[u(t)
- u * (t)]dt + ja(t)[-
B(t)(u(t) - u * (t))dt]
Nonlinear Neutral Systems
213 T
- jdv(t - T)[- B(t)(u(t) - u * (f))] > 0 0
for all u e £/ac|. The result follows at once, since p is monotonic. Note that (X, a, p, v, q) do not vanish simultaneously, since otherwise (A., (., v, q, p) = (0, 0, 0, 0, 0). The proof of Corollary 11.1 and Theorem 11.2 follows as in [25]. Remark: The properties of the range of Dg^ are conjectured as follows: Closure: Proposition 1.1 of Section 12.1 of [28]. Surjectivity: Controllability of the linear system N
i{t) - X A} (0*0 -hj) = L(t, xt) + S(0"(0
in W<J). See for example Theorem 10.8.4 of [28]. We now formulate the necessary conditions of the time-optimal control problem. Theorem 11.5: Consider the following problem: Minimize T subject to the constraints N
m - £ A_u (/)*(/ -*,-) = /(/, x,,«(/)),
x0 = *elfM{[-h,0\,En), *r=V,
DxTyeCl&-h,
uiOeCf
(11.36)
(11.37) 01 En),
a.e. on [0, T],
(11.38)
where Cm={ueEm:\uj\<\,
j=\,m}.
(11.39)
214
Differential Models and Neutral Systems
(i)
We assume that/is continuous, and is Frechet differentiable with respect to its second argument and continuously differentiable with respect to its third argument, the derivative being continuous with respect to all arguments. Consider the linear approximation (11.40) of (11.36), and assume that: This system
(ii)
N
x(t)-^Ajx(t-hj)
= L(t,x,) + B(t)u(t),
XQ=0,
7=1
with attainable set Jl defined by A= {xj{u) e C([-h, 0], EP) DXTe C([-h, 0], E»):u e V}, where V={ue
Zoo([0, 7], £M):uT e C([-h, 0], £")}
is such that A= {V e C([-h, 0]):Z)H/ e C([-h, 0], E")}. Then there exists (a, v, q) e Zoo([0, 7], E») x NBVn([0, 7]) x EP (a,v,g)
* (0,0,0)
such that N a(0 = -g-^a(t k=l
T + hk)A_ik(t + hk)-
jr\(s,t-s)a(s)ds
,
(11.40)
Nonlinear Neutral Systems
215
T jr\(s,l-s)dv{s-T), t
forall/e[0, T], and T T \a{t)B(t)u(t)dt - \dv{t - T)B{t)u(t) 0 T-h T T < ja(t)B(t)u * (/) - jdv(t - T)B(t)u * (/) 0 T-h for all u € £/adProof: We need to prove that the attainable set Jl is all of
= Y2.
Suppose that A= W e C([-fc, 0]):Dv e C{[-h, 0])}, and fix a point (z, v,/») e J^- We shall prove that there exists a pair (x, w) e A^ x 1Jsuch that if x =x-x* and u =u-u*, then A'
JKO -
Z ^ - i ; ( W - hj) - !(/, *,) - Bu(t) = 2{l),
DxT =v
and
x(T) = p.
Now define y e C([-/J, 0], £") with Dy e C'([r - h, 7], £") by vj/(/) = p - J v(s)ds T- h < t < T. By assumption, there exists a control Z such that
216
Differential Models and Neutral Systems T x(t;0,u) = \|/(/)- [U(t,s)z(s)ds, 0
T-h<>t
It follows from the variation of constant formula in Proposition 2.3.3 that t x(t; z, u) = x(t, 0, u) + \U(t, s)z(s)dt. 0 The required choice is the triplet x = (x; z, u), u(t)=u(t). To see that the converse is also valid, we note that if \y e C([-h, 0], En), with Ehy e C'([Th, 7], En), then the fact that Dg is surjective implies there exists a pair (x, u) that is the preimage of the triple (0, \\>, v}/(7)). We conclude that (ii) holds if and only if the required map Dig is surjective. But this is equivalent to the controllability of the linear system (11.42) with controls in 1). Theorem 11.4 and its corollary can therefore be invoked to conclude the proof. Remark 11.2: For autonomous systems, necessary and sufficient conditions for exact controllability on the interval [0, 7] are given by Salamon [5, p. 155]. There we can identify our interval to be [0, T - h] with controls u e Zoo([0, T- h], Em) and state space W^([-h, 0], En). On the interval [T- h, T\ one can use a control that is continuous. Thus it can easily be proved that in the autonomous case Salamon's conditions suffice for the surjectivity conditions.
REFERENCES 1. G. A. Kent, Optimal Control of Functional Differential Equations of Neutral Type, Ph.D. Thesis, Brown University, 1971. 2. H. T. Banks and G. A. Kent, "Control of Functional Differential Equations to Target Sets in Function Space," SI AM J. Control 10 (1972), pp. 567-593. 3. R. Gabasov and F. Kirillova, The Qualitative Theory of Optimal Processes, Marcel Dekker, New York, 1976. 4. H. R. Rodas and C. E. Langenhop, "A Sufficient Condition for Function Space Controllability of a Linear Neutral System," SI AM J. Control Optimization 16 (1978), pp. 429-435.
Nonlinear Neutral Systems
217
5. D. Salamon, Control and Observation of Neutral Systems, Pitman Advanced Publishing Program, Boston, 1984. 6. E. N. Chukwu, "The Time Optimal Control Problem of Linear Neutral Functional Systems," J. of Nigerian Mathematics Society 1 (1982), pp. 39-55. 7. E. N. Chukwu, The Time Optimal Control Theory of Linear Differential Equations of Neutral Type, Proceedings, Second Bellman Continuum, Georgia Institute of Technology, Atlanta, Georgia, June 24, 1986. 8. M. Cruz and J. K. Hale, "Existence, Uniqueness and Continuous Dependence for Hereditary Systems," Ann. Math. Pura Appl. 85 (1970), pp. 63-82. 9. W. R. Melvin, "A Class of Neutral Functional Differential Equations," J. Differential Equations 12 (1972), pp. 524-534. 10. M. A. Cruz and J. K. Hale, "Asymptotic Behavior of Neutral Functional Differential Equations," Arch. Rational Mech. and Anal. 34 (1969), pp. 331353. 11. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1974. 12. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1969. 13. M. A. Cruz and J. H. Hale, "Stability of Functional Differential Equations of Neutral Type," J. Differential Equations 7 (1970), pp. 334-355. 14. K. Kuratowski and C. Ryll-Nardzewski, "A General Theorem on Selectors," Bull. Acad. Polon. Sci. 12 (1965), pp. 397-403. 15. N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Interscience, New York, 1958. 16. T. S. Angell, "Existence Theorem for Hereditary Lagrange and Mayer Problems of Optimal Control," SIAM J. Control Optimization 14 (1976), pp. 1-18. 17. S. Lang, Analysis II, Addison-Wesley, Reading, MA, 1969. 18. E. N. Chukwu, "An Estimate for the Solutions of a Certain Functional Differential Equation of Neutral Type," Proceedings of the International Conference on Nonlinear Phenomena in Mathematical Sciences, Academic Press, 1982, edited by V. Lakshmikantham. 19. E. N. Chukwu, "Global Asymptotic Behavior of Functional Differential Equations of the Neutral Type," J. Nonlinear Anal. Theory, Method and Applications 5 (1981), pp. 853-872. 20. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 21. O. Lopes, "Forced Oscillation in Nonlinear Neutral Differential Equations," SIAM J. Appl. Math. 29 (1975), pp. 196-207. 22. H. J. Sussmann, "Small-Time Local Controllability and Continuity of the Optimal Time Function for Linear Systems," J. Optimization Theory and Applications 53 (1987), pp. 281-296. 23. E. N. Chukwu and O. Hajek, "Disconjugacy and Optimal Control," J. Optimization Theory and Applications 27 (1979). 24. E. N. Chukwu and H. C. Simpson, "Perturbations of Nonlinear Systems of Neutral Type," J. Differential Equations 82 (1989), pp. 28-59.
218
Differential Models and Neutral Systems
25. T. S. Angell and A. Kirsch, "On the Necessary Conditions for Optimal Control of Retarded Systems," Appl. Math. Optimization 22 (1990), pp. 117-145. 26. H. T. Banks and M. Q. Jacobs, "An Attainable Sets Approach to Optimal Control of Functional Differential Equations with Function Space Terminal Conditions," J. Differential Equations 13 (1973), pp. 129-149. 27. J. K. Hale and R. R. Meyer, "A Class of Functional Equations of Neutral Type," Mem. Amer. Math. Soc, No. 76, 1967. 28. E. N. Chukwu, Stability and Time Optimal Control of Hereditary Systems, Academic Press, 1992, 2nd Edition, World Paradise Press, Raleigh, North Carolina (in preparation). Obtain from Copytron, Inc. Academic Publishing Division, 100 West Franklin Street, Chapel Hill, NC 27516. Fax: 919-9332680.
12. Controllable Nonlinear Neutral Systems
12.1 Introduction Criteria for linear systems controllability and constrained controllability can be found in 10.2 and 10.3. There Euclidean and function-space targets are considered. In this chapter we treat the general nonlinear situation in fir• '.
12.2 General Nonlinear Systems We now consider the general system (12.1) where f.E x C x Em -» En is continuously differentiable in the second and third arguments, and is continuous, and where (12.1) satisfies the basic assumptions on D and/in Section 11.2. In particular, we assume that there exists an m e Z,2([cr, oo), E) such that \Wt,
(j>, M)H + U A ^ , $, «)|| < m{t),
and
D(t,xt) = x(t)-g(t,xt) Where |g(/,4»|£A(0ll
Differential Models and Neutral Systems
220
for some k continuous, g(t, <> | ) is linear in (|>. We need some preliminary definitions from analysis. We work in the space W^pDefinition 12.1: Let X and Y be real Banach spaces and F a mapping from an open set S of X into Y. If for each fixed point XQ e S and every h e X, the limit lim [F(XQ + th) - F(xo)]/1 = 5F(XQH)
t->0
exists in the topology of Y, then the operator 8 .F(;co, h) is called the Gateaux differential of F at JCO in the direction of h. If for each fixed XQ e X the Gateaux differential 8 F(XQ, ■) is a bounded linear operator mapping X into Y, we write 8F(^o, h) = F(xo)h, and F(XQ) is called the Gateaux derivative of F at JCO- If F has a Gateaux derivative at XQ, we say F is G-differentiable at XQ, and F(XQ) e L(X, Y), the space of bounded linear operators from X into Y. Definition 12.2: FX -> Y is weakly G-differentiable at JCO if there exists a bounded linear map -F(JCO) e L(X, Y) such that ([F(x0+th)-F(x0)]/t-F'(x0)h,y*)^0, as t -> 0 for al\ h e X,y* e Y*. As a consequence, if F is G-differentiable, it is weakly G-differentiable. Consider •jtm,xt)]=f(t,xt,u(t)),xa=$.
(12.1)
Corresponding to the point (/, a, <|), u) e E x E x fr^ ^ x Lp, the mapping x:ExEx
VfW
xLp-^W^,
is defined by x(o,<|),MX0 = ^(cT,<|),M)e^ 1 ) .
We now give some useful properties of this mapping.
We assume
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221
conditions on (12.1) are valid. Also/is x C x Em -» E" is continuous and continuously differentiable in the second and third argument. Lemma 12.1: Let {t, a, §, u) e E x E x W^ x Lp L et v e I p ([a, oo), £7") and F(u) be the G-derivative of x(t, a, <|), w) with respect to w. Assume all the conditions of existence and uniqueness of solutions of (12.1) in [5] or [11]. Then we have F(H)(V) =
Dux(t,a,ty,u)(v)= y(t,a,ty,u, v),
where the mapping t -» y(t, a, <> | , u, v) of E into En is the unique solution of -\IKi)yt\=D2f{t,xt{o,^u),u)yt at
+ Dif{t,xt{o,^u),u)V(
ya=$> (12.2)
with <>| = 0. Also, for every (\j/, v) e ^ *' x Z p we have that if F(u, v) denotes the G-derivative of x(t, a, §, u) with respect to its last two variables at the point (t, a, <> | , u), then Dx{t, a, §,u)(w,v) = X(t, a, <(>, u,v), where t ->~k{t,a, <j>, w, v) is a mapping of E into £", which is the unique solution of (12.2) with^ 0 = \|/. Proof of Lemma: For each (a, (|)) e E x fF^ % the unique solution of (12.1) with initial data (a, (|)) is given by xa = § in [-/*, 0], x(a, <|>,wX0 = *(',<*, <M) = g(t,xt (a, <)),«)) -g(a,(|>) + <|>(0)
+ J/(x, jft(a, <|i, «X a(j))d[s.
(12.3)
We claim that x(a, <|), M) is Frechet differentiable with respect to u(t) e Em. Indeed, if Du denotes this partial derivative, then by Dieudonne [10], Duxa = 0 in [-h, 0],
222
Differential Models and Neutral Systems t \D2/(S,XS(O, a
Dux(t,aA,u)=
<(>,«), u(s))Duxs(o, <|>, (u))ds
t + Jz>3/[(5, xs(q,
ax, + D2g(t, x,(a,
Due to the linearity of g, Dnit, x,(a, <(>, u))—£ (a, <(>, w)= g[t, —£ (a, f , « ) | .
Thus Z^JT(/,
a,<M) - D2g0,*/(c, ((),
M))—
(a,<(), u)
= j j V f o ^(
Furthermore, on differentiating with respect to / we obtain — [Dux(t, a,
A,X,(CT, <> |,
w)v]
$,«), n ( 0 ) A i ' ( ( ^ ♦. «) v + *>}/('> *<(<*.
or equivalently — [Dux(t,o, <(», u)v - g(/, D K ^(a, <j>, u))v] at = £>2/(r, *,(CT- <|>, u), u(t)Dux,(a, f, u))v
Neutral Systems
223 +
Dif(t,xt(a,hu)MO)v.
We now prove that F{u)v = Dux(t, a, *, «)(v) to complete the first assertion. Consider w-»F(w)=x(f, a, <> | , u), which is a mapping
Let UQ, h G Z,p, F(«o + th) - F(MO)= g(f, xt(a,<|>, HO, + T/J)) - g(f, xt{c, <> | , «0))
a
- f(s, xs(a,$,uQ), MO (s))ds. But f(s, xs(a,
224
Differential Models and Neutral Systems git,xt(u,^uo+rh))-g{t,xt(a,^UQ))/r^5giuo,h)
as
r->0,
and from linearity g(t, (xt(o, <j>, UQ + rh) - xt(a, <> | , u0))/r) -> g(t, 5X,(CT, UQ, h)) as r -> 0 so that 8g("0. h ) = «('> A/ *f (o, <|>, «o VO •
We now have bF(uQ, h) = lim (F(UQ + rh) - F(UQ))Ir
r->0
t = gO. DuXt{a,^,UQ)h) + ]Duf(s,xs{c, <(>, UQ), uQ(s))h(s)ds. a We now prove that 8F is continuous at (wo, 0) e 1^ x Lp, so that by Theorem [9, p. 336] 5F(u0, h) = F'(u0)h, F'(« 0 eL(Lp,WJP): F'(«o): Lp ->
^
is a bounded linear map. Suppose («*, A*) -> (MQ, 0). Then lim*-**) llg('> A«*r(<*. ♦,«*))** - * ( / , A i * / f o ♦, «oX 0) II
a - Duf(s,xs(c,$,uo),u0(s)),0\\ds
=0
since 4//C*, **(<7,
Neutral Systems
225 DuX,(a,
Duf{s, xs(a,
are continuous in u, and k(t) (in 12.1) is continuous on [a, oo), for every finite r > 0 and u e B^UQ) = {u:\\u - UQ\\P < r] there is some L < oo such that iff e [a, fj], fi <0, then ||F'(«)-r(«o)||sL
(12.4)
We note that | P(« 0 ) - F'(« 0 ) I < I g(/, Dux,(a, 4, «)) - g(t, Du (xt(a, *, u0))) | + Jl Duf{s, xs(a,
+ jl A*/(*> xs(a,
+ JlDuRs,x s (o, <(>, w),u(s)) - Duf(s, xs(a, <j>, K 0 ), "o(s))\ds.
If x(o", <j), M) is a solution of (12.1), the associated variational control system along the response / -> x£a, <j>, «) is the system (12.2). In particular, if M 0,0) = 0
(12.5)
so that the trivial solution x(a, 0, 0) = 0 is a solution of (12.1), the variational control system along the trivial solution is J [D(t)y, ]= Dif(t, 0,0)y, + £>3/0, 0, 0)v.
(12.6)
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Differential Models and Neutral Systems
The following theorem is reproduced from [9, p. 155]. Theorem (Open Mapping Theorem): Let X, Y be Banach spaces and B,{XQ) an open ball of radius r about Jto e X. Suppose F:B,{XQ) a X -> Y is weakly G differentiable such that R(F(x0)) = Y and \F(x) - F(x0)\
=
Utyzt)+B(t)y(t),
is controllable on [a, t\] where t\ > a + h and where D2f(t,0,0)z, = lit, z,),
D}f(t, 0, 0)v = fi(/)v.
Assume that /(/,0,0)=0,
V/>CT.
Then 0 tint
r\(t\,a)
where AOi, or) = {xt^o, 0,u):u eLp([a, trf, Em),\\u\\Lp 51, x(o ( 0, u) is a solution of (12.1) with xa = 0} is the attainable set associated with (12.1). Proof of Theorem: Let <>| e JFJJ\ w e Lp([o, t\\, Em), t\> a + h. Let u -> x,,(a, ()>, M) be the mapping F:Lp([a, tx], E") -► ^\[-h,
0], E") given
by Fu = xtl(a, <j>, u), where jc(a, <J>, w) is a solution of (12.1). Then by Lemma 12.1 ^(»)v = Duxh (o-,<j>,«XV) =yt(a,
Neutral Systems
227
where y is the solution of the variational equation (12.2) and F(u):Lp([o, t\], Em) -~* Hp • Evidently F is a bounded linear surjection if and only if the control system (12.2) is controllable on [a, t\\. As a consequence, F(Lp[a, t\], Em) = W*p'; and Lemma 12.1 guarantees that all the requirements of the open mapping theorem are met for the map F:Lp(l<j,tl],Em)-+Wpi\ Thus for w0 = 0 e Lp([a, t\)), F(u0) = F(0) = 0 e W^\ for every r > 0 and open ball B(0, r) c Lp([o, t\], Em) of center 0 e Lp and radius r there is an open ball B(0, p) c W^' center 0 of radius p such that ), p) c B(F(u0), p) F(B(U0, r)) = F(B(0, r)). Thus 0 e B(0, p) c F(B(0, /•)) c F(Lp([c, t\], £"). It follows from this that 0 e Int, A(t\, a ) ^ ^ where Mtx,a) = {xtl(G,0,u):ueLp([a,taEm),
\\u\\Lp
is a solution of (12.1)}. We have proved that for any r > 0, 0eIntA(/i,o)
( 12 - 7 )
(12.8)
where Ji(t\, a) is defined in (12.7). Since r is arbitrary and the set of admissible controls f/a(j with constraints is Uad = {ueLp(l<5,tl]):\\u\\p<\}, (12.3) is valid.
(12.9)
228
Differential Models and Neutral Systems
Corollary 12.1: Assume all the conditions of Theorem 12.1. Then System (12.1) is locally null controllable with constraints. That is, there is a neighborhood O of zero in W^p such that every initial point of O can be driven to zero in some finite time t\ using some u e U^. Theorem 12.1 will prove not only local null controllability but constrained null controllability as well, because (12.8) implies that OeMV
(12.10)
where
Indeed, suppose (12.10) is false. Then there is a sequence {(Mi", §n'-[-h, 0] -> En, §„-> 0 as n -> oo and no <|)„ is in
Neutral Systems
229
-J [*(/) -A-ix(t-
A)] = /(x,,«(0).
(12.11)
Let A>/( 0 0)« = 5 " ,
D\f(0,0)yt = /<
rank
[A(X), B] = «, V ^ e
(ii)
rank [XI-AA,B]
= n,
C
wAere A(X) = A - A0 - A\ e~h
VUC.
Then (12.11) is locally null controllable and locally null controllable with constraints. Proof: Conditions (i), (ii) are the necessary and sufficient conditions for the controllability of the linear variational equation 4 [x(Q - A-\x(t - h)] = Aox(t)+ AvUt -h) + Bu(t). at
(12.12)
Theorem 12.1 yields the required assertion. In Theorem 12.1, the controllability of the linear system (12.12) is of fundamental importance. Can this be weakened? The next result asserts that the closure of the attainable set of the linear system (12.12) can replace the controllability requirement. The result is stated for the autonomous system x(t)-A_xx(t-h)
= f{xt,u)
(12.13)
and the associated linear system x(t) - A_lX(t -h) = AQx{t) + A{x(t -h) + Bu(t), where A/(0,0)r/ = Aox(t) + A\x{t - h),
DifQ), 0)v - Bu.
(12.14)
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Differential Models and Neutral Systems
Following [1] we introduce the following notation: Let s be fixed. Define the n x n-matrices A0_r/, J
7 = 0,
(12.15)
[o, h = l,...,s,
&j = 4), AJJ[
j = 0, j-l,...,s,
(A-IAQ + AI), j
j Z
>
=
Z
A
^ - T ^ i = 0,...,n(s + \)-l, j = 0,...,s,
Z'^tzi.ZJ,,,...,^], r}=Z*j-AoZ}-\
(12.16)
i = l,...,n(s + l)-l,
j = 0,...,s
r'' = [ri,...,ri].
(12.17)
The matrices Kt are defined as follows:
i-\
Kt =T'' - J V ^ W ' + Kj -
i = l,...,n(5+l)-l,
(12.18)
where (Z°)+ is afixedright inverse of Z°, i.e., Z9(Z9)+ = /. Bartosiewicz [1] has proved the following: Proposition 12.1: Let
Neutral Systems
231
A(t) = {xt eW^\[-h,
0], En): x is a solution of (12.14)
for some u e Lp([0, t\], Em)}. The setA(t\), t\ = (s + \)h, is closed in Wy if and only if A:,(kerZ°)cImZ 0 ,
/= 1,. ..,n(s + 1)-1.
(12.19)
If/1! > n/r, then put s = n in (12.19). As an immediate consequence of Proposition 12.1, the following is valid: Corollary 12.3: Mf\) is closed in W^' whenever imT1 elm Z°, i = 1, ...,«(s + l) - 1 ,
(12.20)
rankZ° = n,
(12.21)
rankB = n.
(12.22)
rank Z° = rank[AixB,..., B).
(12.23)
or
or
Note that
Thus (12.23) is equivalent to closedness and finite codimensionality of Mt\) inwy>[l,p.l81]. Theorem 12.1 can now be weakened. Theorem 12.2: In (12.13), assume that all the conditions of Proposition 12.1 are fulfilled. Then there is a closed subspace Y of and a neighborhood O of zero in Y such that every initial point of O can be driven to zero in some time t\ using some u e f/a(j. Proof The proof follows the same pattern as that of Theorem 12.1, with the new hypothesis replacing the controllability condition. The mapping
232
Differential Models and Neutral Systems
F:Lp([a,n],Em)-*wP, Fu = xtx (a, 0, u) h = g(xtl (a, 0, w)) - g(o, 0) + jf(s, xs (CT, U), u(s))ds, a
t>a
where x(o, 0, u) is a solution of (12.13), does not have a surjective Gateaux derivative F at 0 e Xp([a, /j], £"). Instead its range is closed: F(0)(Z p ([a, /i], £ m )) s y is closed. Therefore it is a closed subspace of W^\[-h, 0], En). We can therefore study the mapping F:Lp -> 7 of the Banach space Lp into the Banach space Y, whose Gateaux derivative F(0):Lp([a, t\], Em) -> Y is a surjection. It satisfies the conditions of Corollary 15.2 of [7, p. 155]. The proof ends in the same way as before. Global controllability can be proved by imposing a stability condition on
m-A-ii(t-h)
= f(x„0).
Theorems 12.1 and 12.2 are local controllability results. We are now ready for the global constrained null controllability result. Theorem 12.3: In (12.1) assume that (i)
/ i s smooth enough for uniqueness, existence, and continuous de pendence on initial data of solutions of (12.1). (ii) / ( / , 0 , 0 ) = 0 f o r f > c . (iii) System (12.2) is controllable. (iv) The system Jt[D{t)xt] = f(t,xt,0)
(12.24)
is uniformly exponentially stable, that is, every solution of (12.24) satisfies ||*,(^9)il^ll
t>o,
(12.25)
for some k > 0, a > 0. Then (12.1) is null controllable with constraints, i.e., with controls in t/a(j of (12.24) for some t>a + h.
Neutral Systems
233
Remark 12.1: Conditions that guarantee the behavior (12.25) for System (12.24) are contained in [24] [5]. We now consider a special case of System (12.1), namely (12.26). The results obtained are sharp. We now study the nonlinear in the state difference-differential control system ■j [x(t) -A-ix(tat
h)\ = /(*(/), x(t - h)) +Bu(t),
(12.26)
where A.\ is an n x n constant matrix,/is continuously differentiable, and B is an n x m constant matrix. Let H\ - D\J{x(t), 0), HI -D2J[x(t), x(t - h)) where Dif is the rth partial derivative of/ Let AQ = Di/(0, 0), A\ = Dif{Q, 0), and denote by Z)a the symmetric n x n matrices X
-{Ha + Hl)±
Da,
a = \,2.
Define Ja as follows: Ja = ADa + Da A, where A is a positive definite symmetric n x n matrix. We have the following special result for (12.26): Theorem 12.4: In (12.26) assume: (0 (ii)
f(0,0) = 0. rank [A(X), B] = n, for all X e C where A(X) = f-XA-\e - A\e~Xh, rank [XI - A^B] = n for all X e C. (iii) All the roots of the equation
Xh
- AQ
det(/-/L 1 r _ / '] = 0, have moduli less than 1; and for some positive definite n x n matrix A we have
234
Differential Models and Neutral Systems
J\ < - 8/
where, for some q > 1, S-9ll^+^ill«H>0. Then (12.26) is null controllable with constraints. Proof: Conditions (ii) and (iii) are the necessary and sufficient conditions for the linear approximation of (12.26): ■j[z(t)-A-lz(t-h)]=AQz(t) at
+ Alz(t-h) + Bu(t)
to be controllable. Condition (iii) is the required condition for the system jt[x(t)-A_ix(t-h)]
=
f(x(t),x(t-h))
to have all its solution x(a, (p) satisfy ll*/(<*>
12.3 Nonlinear Interconnected Systems We now investigate the rth interconnected subsystem described by ■7[A-(/)x/] = /■(/, *{,«''(0)+*i(',*/,v''), at
i=l,...J,
(12.27)
where £, describes the action of the whole system - CXt,xt) =f(t,xt,u(t)) at
+ k(t, xt, v(/))
on its interconnected system (12.27), which when "free" and "disconnected" is given by
Neutral Systems
235 ■^lDt(t>i] = fi(t,xi,ui(t)).
In (12.27), «'' eL2(lo,tilEmi\ and v''s L2([cs,tx], Em). sometimes restrain the control w/ to lie in the set P,- ={«':«' eL2do,tx],Em
(12.28) We shall
), ||«'|| 2 < 1},
and v' e Qi where Qi={visL2([c,tl],Emi):\\vi\\2<\}. Let f/(/, ♦,') = l/i-(/, ♦,«'): a1'e P/
We now assume that 7Q(t,xt)'zMFi(t,xlt). The following result is valid. Theorem 12.5: In (12.27) assume that: (i) f,{t, 0,0) = 0, *,, 0) = 0. (ii) fi, kj, and Dj satisfy all the smoothness conditions of existence, uniqueness, and continuous dependence on initial data. (iii) Assume that the linear variational system j t [D,'(r)z,'(r)]= £,-(/,*j) + ^ ' ( / M / ) , of (12.28), where ^/,-(/,0,0)zj = L/(r,z|),
/>}>;•(/, 0,0)w= rf'Ww,
(12.29)
236
Differential Models and Neutral Systems
is controllable on [o~, t\], t\ > a + h. (iv) *)) c Int
Condition (i) - (iii) of Theorem 12.5. The system | [A(')*!l = /K'> *;,<>),
i = l, ...,<,
(12.30)
is globally exponentially stable. Then the interconnected subsystem is (globally) null controllable with constraints, that is, with controls M'eP
= { M 'GZ,2([a,n],£ m '):||«'ll2
v'' eQ = {vivi €£ 2 ([<M 1 ],£ m ):||v i '|| 2
(12.31)
where J?,
solves (12.28) with xa = 0}.
This is also the solution set of (12.27) with «' as the control of the interconnected subsystem and with v = 0 as the control from the large-scale system. Thus x'(a, 0, u', 0) = x'(a, 0, «')• Suppose x* is a solution of (12.27) with x'G = 0, then
*'(
Neutral Systems
237
for t > a. On defining the set Hi(tho) = ixi(a,0,ui,vi)ewV\[0,tx],Eni):ui
e P,, v< e Q,}
we deduce (on account of the hypothesis (i) and the fact that *'(/, 0, 0, 0) = 0 is a solution of (12.27), that 0 € fl,{t\, a) c Ht{t\, a). Since (iv) is valid and (12.31) holds, Oelnt A,(n,CT)c//,(n,a)
(12.32)
0eInt//,(/i,CT)
(12.33)
and
as well. If
proving local null controllability with constraints. Proof of Corollary 12.4: One uses the control «' = 0 e £/,-, / = 1, ..., i to glide along System (12.30) and approach an arbitrary neighborhood of the origin. Since (i) guarantees that all initial states in this neighborhood can be driven to zero in finite time t\, we are done. Remark: Conditions for uniform asymptotic stability are available in [3]. Remark 12.2: It is very important to study the condition (12.32) from which (12.33) was deduced. Suppose Oelnt A,(/j,CT)
(12.34)
fails, and the isolated system is not even controllable locally and is "not well behaved". Condition (12.33) can still be salvaged and the interconnected system made controllable. What may be needed is a criteria that ensures that 0eIntG,-(>i,a),
(12.35)
Differential Models and Neutral Systems
238
where
Gj«ua)
= - jki(s,xsy(s))ds:v>
eQj
In other words we require a sufficient amount of control impact from external sources ((ki), which is monitored locally at the /-subsystem) to be "forced" on (12.28). The source of power senses the control u' available to (12.28) and sends a sufficient controlling signal kt to the large-scale system, which is a function of the power's state variable x and the control vi. With this the right behavior is enforced. Policy implications of this were pursued elsewhere in [7]. There we observe that the interaction does not ignore the subsystems' (initiatives) v> e Em\ it utilizes them. For example, were they to be ignored, and ki = ki(t,xt)
still be operative and nontrivial, the condition 0 € H,{t, a) and subsequently (12.33) will fail. The interconnected system will not be null controllable. Failure to acknowledge individual subsystems' initiatives (v>) in highly centralized interconnected organizations monitored by our dynamics leads to lack of growth from initial endowment <> j to desired economic target \\i. On the other hand, failure to enforce regulations or use subsidy, bringing £,• to bear when same subsystem misbehaves or is depressed, and is uncontrollable, in "individualistic" systems may make the interconnected system uncontrollable and trigger an economic depression. In place of (12.27), consider the system . t - [A(')*|] = fi(U */,«'(/)) + £ % « , xj, «'(/)).
(12.36)
The following can be proved: Theorem 12.6. For (12.36) assume (i) - (iii) of Theorem 12.5. But in (12.28), £,(/, zj) and Bl(t) are defined as
Neutral Systems
239
£b/,-(/,0,0)rJ = I/(/,z{), PiifiO, 0,0)+ with
kj(t,xt,u')=
£*£•(/,*/,«'(/)).
ki(t,x,,0))=Bi(t) Then
(12.36)
is locally
null
7 =1
controllable with constraints. We note that condition (iii) of Theorem 12.6 requires the controllability of jt[D>(t)zit) = Li(t,zit)+(Bli{t)+B2i(t))ui(t),
(12.37)
where %(') = 03/, 0,0,0), B2i(t) = Dikj(t,xt,0). Suppose (12.31) is invalid for the isolated system (12.29), and this can happen when (12.29) is not controllable with the control matrix B\,{t), the "solidarity function" £, can be brought to bear to force the controllability of (12.37) with B' = B\t + #2, to ensure (12.31). If the system is autonomous, the requirement would be (i) and (ii) of Corollary 12.2. Even if (i) and (ii) hold with B\i and (12.31) is assured, the composite system need not be locally null controllable unless (12.37) is controllable wkh B' = B\i + .#2/. An adequate proper amount of intervention in the form of a solidarity function £, is needed. The linear case was explored in Chapter 11. It is quite possible to extend this treatment to the situation ^ [ ^ ' ( ^ j ] =//(/, * / ) - w ( 0 + *,(0,
(12.38)
where /,(/, xl() is independent of control pt, which enters the dynamics in an additive fashion. One can then deduce the four universal principles for the control of an interconnected organization as contained in Chapter 3.
240
Differential Models and Neutral Systems
REFERENCES 1. Z. Bartosiewicz, "Closedness of the Attainable Det of the Linear Neutral Control System," Control and Cybernetics 8 (1979), pp. 179-189. 2. E. N. Chukwu, "On the Euclidean Controllability of a Neutral System with Nonlinear Base," Nonlinear Analysis, Theory, Methods and Applications 11(1) (1987), pp. 115-123. 3. E. N. Chukwu, "Uniform Asymptotic Stability of Large Scale Systems of Neutral Type," IEEE Proceedings of the Eighteenth Southeastern Symposium on Systems Theory, April 7-8, Knoxville, TN, Computer Society Press, 1986. 4. E. N. Chukwu, "The Time Optimal Control Theory of Linear Differential Equations of Neutral Type," Comput. Math. Applic. 6(10/11) (1988), pp. 851866. 5. E. N. Chukwu and H. C. Simpson, "Perturbations of Nonlinear Systems of Neutral Type," J. Differential Equations 82 (1989), pp. 28-59. 6. E. N. Chukwu, "Some Controllability Questions of Linear and Nonlinear Neutral Functional Differential Equations in ^ '" Preprint. 7. E. N. Chukwu, "Global Economic Growth: Will the Center Hold?" In Ordinary and Delay Differential Equations, Ed. J. Wiener and J. K. Hale Longman, UK, 1992. 8. J. P. Dauer, "Nonlinear Perturbations of Quasilinear Control Systems," J. Math. Anal. Appl. 54(3) (1976). 9. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. 10. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1960. 11. J. Hale and M. A. Cruz, "Existence, Uniqueness and Continuous Dependence for Hereditary Systems," Ann. Math. Pura. Appl. 85 (1970), pp. 63-82. 12. J. Hale, "Forward and Backward Continuation of Neutral Functional Differential Equations," J. Differential Equations 9 (1971), pp. 168-181. 13. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 14. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, Macmillan Company, New York, 1964. 15. G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Academic Press, New York, 1972. 16. A. N. Michel and R. K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, Academic Press, New York, 1977. 17. M. Slemrod, "The Flip-Flop Circuit as a Neutral Equation," in Delay and Functional Differential Equations and Their Applications, edited by K. Schmitt, Academic Press, 1972. 18. V. N. Do, "Controllability of Semilinear Systems," J. Optimization Theory and Applications 65 (1990), pp. 41-52.
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241
19. D. Salamon, Control and Observations of Neutral Systems, Pitman Advanced Publishing Program, Boston, 1984. 20. H. T. Banks, M. Q. Jacobs, and C. E. Langenhop, "Characterization of the Controlled States in of Linear Hereditary Systems," S1AM J. Control 13 (1975), pp. 611-649. 21. M. Kalecki, "A Macrodynamic Theory of Business Cycles," Econometrica 3 (1935), pp. 327-344. 22. R. G. D. Allen, Mathematical Economics, Macmillan, London, 1960. 23. K. J. Arrow, Production and Capital, Collected Papers of Kenneth J. Arrow, Belknap Press of Harvard University Press, Cambridge, Massachusetts, 1985. 24. E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, Academic Press, 1st Edition 1992, Second printing, World Paradise Press (in preparation). Distributaed by Copytron, Inc., Academic Publishing Division, 100 West Franklin Street, Chapel Hill, NC 27516. Email: [email protected].
13. Function Space Control of Nonlinear Interconnected Economic Systems of Neutral Type In this section we tackle the problem of controlling economic variables in nonlinear interconnected systems of neutral type. Two types of systems are considered, a general one, and one in which some control appears linearly. We insist that each system unit is controlled by its own internal variables (e.g., by autonomous investment, autonomous consumption, net export and autonomous money demand) while taking into account the effect of solidar ity. To prove controllability, we assume some controllability criteria of each isolated system and some growth conditions of the interconnecting function. The effort is mathematical and it was reported in Chukwu [34].
13.1 Introduction Kalecki [24] and [1] argued that the growth of capital stock x(t) of a single firm is given by x(t) = I(t) = a0x(t) + aix(kt-hy
(l3A)
where a/ are constants and the delay h represents the time lag between the decision to invest and the deliveries of capital equipments. The crucial assumption for (13.1) is that the net capital formation x{t) is given by /(/). To obtain (13.1), Kalecki assumes that the decision to invest/? is given by B(t) = a(\ - c)y(t) - kx(t)+ e, a, c, k are constants. The "windfalls", s may be time varying, y is income (output), x(i) denotes the stock capital assets at time /. Later as an exercise 243
Differential Models and Neutral Systems
244
[1, 8.4, no. 3, p. 254], Kalecki suggests that the decision to invest should be 5(0 = o ( l - c M O - * * ( ' ) + e + v - ^ . The outcome of this analysis is the functional differential equation of the neutral type — (x(t) - a_,x(/ - h))= aox(t) - a[x{t -h) + a2.
(13.2)
This is a functional differential equation of neutral type which describes the growth of capital stock of a single firm. We can introduce b(t)u(t) at the right hand side of this system to obtain x{t) + a_xx{t -h) = a0x(t) + axx(t -h) + b(t)u(t).
(13.3)
We interpret b(t)u(t) as follows. If 0 < u(t) < 1, then u(t) is the fraction of "available capital assets, £(/)" at a time / that is allocated to investment. If - 1 <, u(t) < 0 then u(t) is the fraction allocated to consumption or for payment of taxes. Thus -1 < u{t) <, 1 and u(t) is an investment consumption strategy which is appropriate as a control to drive an initial capital endowment § to a target while minimizing a cost function. To motivate a nonlinear system of the form (13.3) which is interconnected by the so-called "solidarity" function we argue as follows. Let Z denote aggregate demand consisting of consumption ( Q , investment (7), net exports (X) and government outlay (G). These differentiable functions are related as follows Z=C + I + X+G,
(13.4)
where C = C0 + c(y-T),
0
(13.5)
and y - T is the current after-tax income; T = T0+fx(y),
(13.6)
Function Space Control
245
TQ > 0 is the level of non-income taxes, and f\(y) is the income taxes. X = X0-f2(y)-eR,
0<m
e,>0,
(13.7)
f2(y) the part of income that is spent on other countries' products, XQ is autonomous net exports, and R is the real rate of interest.
G = /30#-*))+*/)
(13.8)
where fy is public consumption which is dependent on the previous high income, and v(t) public investment:
/ ( ' W o O + gW'-*)).
(13.9)
Thus Z(t) = (Z 0 (/) + f3(y(t - h)) + v(t) + cy{t) - cT0(f) - cfx(y) + g(y(t - h)) - f2{y)) - e^R,
(13.10)
where ZQ(/) = IQ + XQ + CQ. From the model equations on money demand and supply [27] we deduce that R=
ky
1 Mo
r
r
since LIP=MIP, LIP = j +
hy-rR,
(13.11)
246
Differential Models and Neutral Systems
Hence M is the nominal value of money supply which is controlled by the Central Bank, P is the prize level. The real demand for money is denoted by LIP. The symbol j is autonomous real money demand. With this R we deduce that Z(t) = Z0(t) + y(t)[c - & I + f3(y(t - A)) - (c/i (y(t)) + f2(y(t)))
e\
-j \-cT0(t) + g(Jit-h)) + v(t).
Following Allen [1] we postulate that dy(t) / dt = ~ X(y(t) - Z(t)) where X is a constant. Thus KO - W f l - h)) = - X\ 1 + & - c \y(t) - \(cfx (y(t)) + f2(y(t)))
+ \f3(y(t-/,))
+ %■Ml -J
+ \v(t) + XZ0(t)-\cT0(t).
(13.12)
Denote as "solidarity function" , 9(0 = M(ei//-)(M 0 // o)-c7b(0 + v(0];and "private initiative" by p(t) = X[ZQ(I) - (e\ lr)j]. Then the dynamics of income is }
+ &-c
\y(t) - UcMy(t)) + f2(y(t)))
+ ¥3(yH-h)) + p(t) + qit).
(13.13)
It is an interconnected nonlinear system whose controllability is investigated for all admissible values ofp and q. It is proper to consider x'(t) = G\(t),.... u'n(tu to be the value of n capital stocks with strategy «' =L\, ...,u'nJ-\
which are located
in an isolated region (Si). They are linked to t other such regional systems
Function Space Control
247
in the country and the "interconnection" or "solidarity functions", or government intervention given by qi =
Here qi describes the action of the whole system in its rth interconnected subsystem (5/) *'(/) - A_\i\t -h) = AQx\t) + V f c - *) +to*'(?)■
(Si)
Thus xi(t)-/L^i(t-h)
= A0xi(t) + AixiO-h) + Bui(t) + qi.
(13.14)
Thus formulated we are interested in using the firm's strategy M' and government interventions qi to control the growth of capital stock on which the wealth of a nation depends. Theorems 13.1 and 13.2 can be stated loosely as follows. If a regional economy is well behaved, carefully weighted government interventions qi can maintain the country's economic growth. Even if a regional economy is not controllable the intervention of solidarity function can render the system controllable. See Remark 13.1 and Theorem 13.7. Implications of controllability questions of neutral and other systems for the control of global economy are pursued elsewhere in [13], [14]. The issue of optimality is investigated in Chapter 10. Because of its apparent importance the author takes the rather general system of neutral type whose qualitative properties have been established by Hale and his school.
13.2 Preliminaries The notations and definitions of [10] will be maintained as follows: Let E be the real line (-oo, oo) and E" the Euclidean ^-dimensional vector space. The symbol Ljiia, b], Em) denotes the usual Hilbert space of "square integrable" functions from [a, b] into Em. The Sobolev space
Differential Models and Neutral Systems
248
W$'([a,b],En) = Wjj' is the space of all absolutely continuous functions x:[a, b] -> En with the property that t -*■ x(t) = (dx/dt)(t) belongs to L2([a,b],En). Let h > 0. Let x be a function from [a - h, t\] -» En. Let f e [a, t\] c is. We use xt to denote the function on [-h, 0] defined by x((Q) = x(t + 9), -A < 9 < 0. The symbol C = C ([-h, 0], is") is the space of continuous functions from [-h, 0] -> En, with sup norm. We first consider systems of nonlinear functional differential equations of neutral type with the form |
EKf, xt) = /(/, xt, «(0) + B(t)u(t),
(13.15)
where the operator DJ): E x C -> En is atomic at 0 and uniformly atomic at 0 in the sense of Hale [21, pages 170-173], and where f.E x C x F ^ F i s continuous and uniformly Lipschitzian in the last two arguments. We assume that B(t) is a continuous nxm matrix function which is continuously differentiable, and the controls are 2-2 functions. In (13.15), instead of the atomicity assumption on D, we may assume that D is of the form
0fc4O = «o)-s(/,4O, where g:E x C -» En is continuous and is uniformly nonatomic at zero on E x C in the following sense. Definition 13.1 For any (t, §) e Ex C, and u > 0, s > 0 let (VeC:(/,\|/)e£xC;||\|/-$||£|i,]
lv(e) = «e), e<-4 e e [-M} J We say that a continuous function g:£ x C -> £ " is uniformly nonatomic at zero on E x C if for any (t,ty)eExC there exist SQ > 0, UQ > 0 independent of (t, <(>), and a scalar function p(t, §, u, s), defined and continuous for (/, §), 0<,s<so,0<\i<,no, nondecreasing in u, s such that
Function Space Control
249
PO = p ( £ x C , n o . *o) =
SU P P('. ^ ^0> *o) < £xC
!
and I g{t, \|/) - *(f,«))) | < Po II V -
for t € £, \|/ e g(/, (J), u, 5) and all 0 < 5 < *0> 0 ^ H ^ HoIt is known ([15], [21] and [23]) that under these assumptions on D,f, B and u for each f e C there is a unique solution of (13.15) with initial value <>| at a. The solution is continuous with respect to initial data and parameter u. For definition of solutions see [23]. In the sequel t\ is fixed and is in the interval of definition [a, x], of solutions of (13.15). Definition 13.2 The system (13.15) is on [a, t\] if for any function § eW^' and any vector x\ e En, there exists a control u e Z,2([°% 'l]> Em) such that the solution x(t) = x(t, a, <> | , u) of (13.15) satisfies •%(•,<*, <M) = »
xOi,o\ <> | , M) = * 1 -
It is Euclidean null controllable on [a, /1] if xj = 0 in the above definition. The system (13.15) is controllable on the interval [a, t\], t\ > a + h if for each <|», vy € there is a controller w e = L2([a, t\], Em) such that xCT(-, a, <)), w) = <j>, x/, (•, a, (J>, u) = vjy. It is null controllable on [a, fj] if \\i = 0 in the preceding definition. In Wip, w e drop ^ e qualifying phrase "on tile interval [a, t\]" if controllability obtains on every interval [a, t\], with t\ > a + h. In E" we drop "on the interval [a, t\]" if Euclidean controllability holds on every interval [a, t\], t\ > a. To motivate the problem, we consider a simple linear state equation of the i-th subsystem of an interconnected control system,
250
Differential Models and Neutral Systems
x(t) - A_xix\t
-h) = AQix\t) + aux(t -h) + £,-«''(/) + ^ . / ( O + ^
0
~
h
)
(13.16) where xl{t)&Efi the w/-dimensional Euclidean state vector of the /th subsystem; w'(/)e£ m ' is the control vector and A.\it AOJ, A\i, A2i, A^i, Bj are time invariant matrices of appropriate dimensions. Furthermore y'(t) is the supplementary variable of the /-th subsystem and it is a function of its own Euclidean state vector x\t) and other subsystems' state vector x)(t)J = 1,..., I. This is expressed as follows: e /'(/) = Mux^it) + 2>yx>'(0 ,
(13.17)
7=1
where My, Myij = 1, 2,..., I) are constant matrices. On substituting (13.17) into (13.16) we deduce that the interconnected system has the equation x\t) - A_ux(t -h) = Hmxl(t) + HXix\t -h)+ BiU\t) + ht(t) + e,-(/ - h), (13.18) where #0; = 4)i + ^ M i >
#1/ = AU +
e
A M
3i U
e j
hi(t)=^M2iMiJx (t), 7=1
ej(t-h)=
Yj^i^ijxht-h). 7=1
In (13.18), the function git, t-h) = h,{t) + e,{t - h) describes the interaction, the effects of the whole system on the /th subsystem. This can be measured locally. We may view the decomposed system (13.18) as an interconnection of (. isolated subsystems, *'(/) - A.^(t - h) = H0ix\t) + HXix\t - A) + V ( 0 ,
(13.19)
Function Space Control
251
with an interconnection structure described by Ki(?,t-h) = h,ii) + ei(t-h),
(13.20)
a function which does not depend on the state variable *'(/)• With this linear system as a guide, we consider the more general interconnected nonlinear system ,
I
- Di(t,x't) + fi(t,x'hu'(t))+
3(0"'(0+ £ % ( ' , * / V ( 0 ) ,
(13.21)
7=1 j*i
where
A(',*/W(0-aM)
(13.22)
and where x't e C"' ([-h, 0], E"') = Cn' is the space of continuous functions from [-h, 0] into Em . We write Cn = C([-A, 0], £") = C. Equation (13.21) may be viewed as an interconnection of i isolated free subsystems described by the equations j t Di(t,xil) = Mt,xit,ui(t))+ BjitW®,
(13.23)
with interconnecting structure
e Ki{t, x,, v') = £Kij(t, xi v' (0) • y=i
We assume v' is in a different control set from u'. Here, fi-.ExCfJ
x £"i -+ £ *
Ka-.ExCfJ x £ m ' ->£"'".
(13.24)
252
Differential Models and Neutral Systems
t t We let « = ]Trt / m = £ O T / .,x 7 '=[(;r') 7 ',...,(/) 7 '] e £'', i=l
/=l
I/O, */. u)f = [fi(t, x],ul)f,...,
(fe(t, x{,
[Kit, x„ v)]7" = [K^t, X), v1))7",.... (K((t, xf,
u')f\ vi)ff
B(t) = diag[B](t),...,B((t)]> Z)(/ ) x,) = [D 1 (/,x / 1 ) 7 '
Di{t,x\)T\.
With this notation, the decomposed system (13.21) can be represented by j ( D(f, xt) =/(/, x„ «(/)) + B(t)u(t) + K(t,x„ v(/)).
(13.25)
The free system (13.23) is given as in (13.15) by j t D(t,xt)=Mt, x,,u(t))+ B(t)u(t).
(13.26)
We assume in the sequel that/ K, B and D have the basic properties for the existence and uniqueness of solutions as well as the continuous dependence of solution on the initial data and parameter u.
13.3 Main Results In this section we maintain the notations of Section 13.2 and consider the free system Jt W,xt) = Mt,xi,,ui(t))+Bi(t)ui(t), and the decomposed interconnected system
(13.27)
Function Space Control
253
7 Di{t,x\) = fi(t,xi,,u\t)) at
+ £K0«''(0+ Kb x/,v\t)),
(13.28)
where £)/, £/ are identified in (13.22) and (13.24) respectively. As observed, (13.28) can be represented by (13.25), namely ■j D(t,Xt)=f(t,xt.u(t))+
B(t)u(t)+ K(t,xt,v(t)).
(13.29)
The first theorem is on the free subsystem (13.27). It extends the earlier paper [10] on Euclidean controllability to a result on function space controllability; it generalizes the more recent paper [7] by covering cases on which fi is not necessarily bounded. Theorem 13.1 In (13.27) assume that (i)
rank B,{t) = n,on [t\ - h, t\];
(ii)
Bj(t) is continuous on [a, t\], with continuous derivative;
(iii) the function gf.E x C -> E"1 is uniformly nonatomic at zero on E x C* , and f\\ ExCx
Em -> E" is continuous;
(iv) the continuous functions Fy :C"' x Em' -> E+ and L\ functions ctj.E -» E*,j = 1,..., q be such that
| jf(', if, « ' ' ) | < ^ a ; ( 0 ^ ( * , « ' ) ,
V(/, *,«') e £ x Cx £ f
7=1
where
lim sup r - £ c ; s u p { ^ ( < M ' ) : | ( < l > V ) l ^ } r-XJO
Then (13.27) is controllable on [a, t\] with t\>a + h.
= + ao.
254
Differential Models and Neutral Systems
In Theorem 3.1 Euclidean controllability of (13.27) is proved if we assume the condition f rank Hj(ti-o)=
/,-c ^B(s)B*(s)ds = n
t\>a,
instead of rank Bj(t) = n
on [t\
-h,t\],
of Theorem 13.1, condition (i). The hypotheses (i)-(iv) are conditions on each subsystem which ensure controllability. We now state a result on the interconnected system. Remark 13.1: Conditions (ii) and (iii) are essentially the required criteria for the existence and uniqueness of solutions of (13.27). Condition (iv) is a growth condition which should be compared to a uniform bound imposed on / b y Mirza and Womack [29, Theorem lc] when treating delay equations. Such growth conditions have a long history: see Dauer [16], Chukwu [8], Balachandra and J. P. Dauer [3], and Do [19]. In Do [19] one sees the consequences of the growth condition. Theorem 13.2 In (13.29) assume (i) (ii)
all the conditions (i)-(iv) of Theorem 13.1. For each ij= \,..., t,i ±j there are continuous functions E+ and L\ functions fy.E -> E+J = 1,..., q such Fy.E* x£m< that
!/:,(/, ♦,',i/)|sj]p_/(0Fjr-(V,",«/) 7=1
for all (/, \|/', u>), P/ < a/ where,
Function Space Control
lim sup
255
- £cySupJFy(4/V)|(H',"i)}
= + 00 .
Then (13.28) is controllable on [a, t\],t\ >a + h. Remark 13.2: Theorem 13.2 (ii) reminds one of the growth conditions in Michel and Miller [17, Theorem 5.8.4 (ii), Theorem 3.3.5 (iii), Theorem 3.3.2 (iii), Theorem 2.4.20 (iii)]. The condition 3.2 (ii) says that the external intervention ky to the system is dominated by some "power" "P/^"//'" of the system. Remark 13.3: The nonlinearity and interconnectedness which are introduced in (13.24) and (13.25) are natural and essential in the economic application. In addition to the firm's initiative pit) = B(t)u(t) in (13.13) the control action of government q(t) = l{e\lr){MQl PQ) - cT0(t) + v(t)], is (realistically) not linear in MQ, PQ, TQ, etc. The basic contribution of Lucas's critique [31] is that pit) must be incorporated. (See [31, Macroeconomics in the Global Economy, Chapter 19]). A deep reflection on Lucas's contribution demands a game theoretic formulation. This is the thrust of MuUinex's argument [30, p. 91] and the reason for the insertion of a non-trivial Kj. It is also reasonable to have a nonlinear/. Investment and savings functions may be nonlinear, [30, p. 87] (see, (13.6)-(13.9) and (13.2)). Of course results for the linear situation are already available in Chukwu [13]. Mathematicians often object to the full rank of B, but the economic insight of Tinbergen in [31, pp. 5, 90] shows how essential this "classical" non-degeneracy assumption is in executing monetary and fiscal policies. That Ki is assumed to be sufficiently "small" compared to the linearly acting control is nothing but a non-linear generalization of the requirement in the linear pursuit game HO ~ a-\Kt ~h) = <>oyO) + a\y(t ~h) + PiO +
Pit) e P, qit) e Q ,
that Int P^>Q. The firm's control set (or initiative) should dominate the government's.
256
Differential Models and Neutral Systems
This is a necessary and sufficient condition (on the control sets) for controllability. See Hajek [20, p. 61] for the genesis of this idea. It settles a basic problem: how much (in comparison with private effort) should government intervention be in the economy? Proof: Under the conditions of the Theorem, the system (13.27) is Euclidean controllable. Indeed, let xj eE"' ,<}>' eC"', then the solution x' of (13.27) is given by jr''(/ + o) = ♦''(/), / i
i
i
x (t) = Di(a^ )+gi(t,x t)+
/e[-M] I
i
i
JMs,x s,u (s))ds+JBi(s)ui(s)ds, a a
t>o. (13.30)
From (i) with fl(. the algebraic adjoint of Bj, Bfo) Bt (t) has rank «/ on [t\ h, t\] so that Hi(t\)= \Bi(s)Bi (s)ds has rank «/ for t\ > a + h, since
trk
//,(/))=
j Bi(s)B*(S)ds+ a
]Bi{s)B*(S)ds tx-h
and the last term is positive definite. As a consequence of this, r,-o Hiif\-a)= JB(s + o)B*(s + o)dsy
is nonsingular for a carefully chosen t\ (e.g., t\ > 2a + h). The solution of (13.27) with UJ as a control is given by (13.30). Let xl e C([CT - h, x], E*1); and define %(t)=xi(t),
te[o-h,a],
Function Space Control
257
%(t)= xl (s),
te[a,a + j],
and define $' \-h, °°] -> En', by «/) = ♦''(/),
/e[-A,0],
♦M = *''(0),
/e[0,oo].
It follows from these definitions and from (13.30) that x' is a solution of (13.27) on [a, a + x], 0 < x < a>, if and only if x'(a +/) = ♦'(/)+ *'(/),
-h<,t£x,
where z'(t) satisfies / z\t) = g,(t + o, ♦; + z|) - f t ( o , ♦'') + J/,-(s + a , ^ + zj, «'"(* + a))ds or
(13.31) / + \Bj(s + o)u'(s + o)ds, 0
2Q = 0 .
Note that if t\ e [a, x] then x'(t\)= x\ if and only if z/(ri-o) = *{-♦''(/!-a). The corresponding «' which steers 9' to xj in time /j is given by «'(/)= S*(OW- , ('l-crX^-?'('l-cT) + g,(a,
-ft('|.^-
p I ^ + o^i+zi.B'^ + a))*). 0
(13.32)
258
Differential Models and Neutral Systems
We need the following lemmas. Lemma 13.1 (Hale and Cruz, [23, p. 70]) Let U be an open set in IxC"' xC2([o,t\],Em') 1
and let V be its
n
projection on/xCJ ' where C^ = C\([-h,0], E >). Let W= {(CT, XO, u(a))}. Let Uz{W) be an s neighborhood of PTc U. Let fi:U-> E"< be continuous, |/)i <Mon f/E(^0 and gi'-V-> E"1 nonatomic at zero. There are positive real numbers YQ, 5Q, 8, v and t\, 0 < t\ < 8Q, 0 < 5 < 8Q/2, such that (i) 00 (Hi) (iv)
(a) Mt\ < vS/2 or (b) Mt\ < (v(l - v)8/2); \xc\-xo\<80/2,ts[a,G + t\]; |g(f,x 0 ,)-g(a,x C T )|
(v)
+
b0]>\\xV-xa\\<-80};
F(Y0)8O)CK
Lemma 13.2 Let B:I -> Enxm be a continuous n x m matrix function. Then the constants t\, v in Lemma 13.1 can be chosen such that (l-vXl + | | 5 * ( 0 / r , ( a , / i - a ) | | ) < l . We now prove that such a u' in (13.32) exists. Now set
r(zVX0 = 0
if/e[-M];
5(Z/,M'X/)=0
if/e[-A,0];
r(z/,«,'xo = (y(0,iKO)
if 'e[o,t] ;
259
Function Space Control 5(z f ,« , XO = (A(0,v(0)
if'e[0,T],
where X 0 = Si* + a, <)>,' + z\) -gi(o,+'); _.
f
'
w(t)= B. (t)H~\ti -aXx[ - ♦ ' ( / ! - a) H-gKa,*')-«,(
v(/) = - ^(OtffV'l - a) Jy-(* + o,ti+ 4>"'(* + <*))* •
Recall that /] e [a, <»). From the earlier remarks it is clear that if an operator T+S given by ( r + SX*\«'XO=(z'('X«(0), has a fixed point, so that
/,-a /I-o f ft(s + c,$i + zis, u'(s + a))ds+ fas
+ c)B*(s + o)ds
x[ - <(.'(?! - a) + ft(o, (j)') - ft-ft, ♦ ^ + z; t _ 0 ) xHi -1
-
Jy;(5 + a,(|)j,Zj,M'(s + CT))£fa 0
'x|-♦'(*!-a).
260
Differential Models and Neutral Systems
then (13.27) is Euclidean controllable. In this case z'(j) is given by (13.30) and u *(t) by (13.31) so that
z(n-a)=x[-V(ti-o). That is
x[ = zi(ti-a)+V(ti-'o)
= xi(fi).
Our immediate aim now is to establish the existence of a fixed point of T+ S in an appropriate function space. Introduce the following space: X= C([-h,
k*.«oM*l+l«l2. where |<|>| = supse[^h,t\]!(■* )|, and \u\2 is the L2 norm of w. We show that there is a positive constant /-Q and a subset A of X such that ^ = A\(th /Q)X ^2(4, >&), where A\{thrQ) = K:[-A,4]->£"' continuous, ; 0 = 0,||
^t\
^2(
Function Space Control
261
Since the growth condition in (iv) is valid, there exists a constant TQ > 0 such that
or
7=1
see [19, p. 44]. By direct calculations and estimates contained in [10, pp. 121-122] we easily
show that
if
(z^,u))eA,j
= \,2,
T{z[,u\)(t)
+ A(z2, 4 ) ( 0 = 0, / e [-/?, 0]. Also
where b(0l s l«(' + a . * /
+
2i')-g(' + o.*»)l + ls(' + a . ^ ) - « ( a . * ) l
< p(/+CT,i/_ a ,-Mi) 8 + vS / 2 < ( 1 - V ) 8 + V5/2 = 8 - V 8 / 2
and we have used Lemma 1.2 of Hale [23] where v, 8 are identified in Lemma 13.1. Also I K')\ * | fi(s + a, $s + z2s, u2(s + a)) \ds + J| B,(S + a)u2{s + cs)\ds
y=i o
Differential Models and Neutral Systems
262
<M>? + £||a y ||/-,(r 0 ),
where N - maxse^Qt^|Bj(s
+ o)||and a: = J| a O + o) \ds. Hence
| y{t) | +1 h(t) | < [ 1 K
V
\ + Nvq + £ | |
J
aj
\\
F^)
j=\
>1 where
d\ = max < 1
6 + JVv^,||B*(/W- I ||(|jei-*(/.
c l -=««fl|o i / ||,||B»7/- 1 ||||ay||}. Hence lbll + l|A||
1
|l(l^l-
5||B*(/)// _ 1 ||a*i-*(/i-a)|) + p(/i,^ l - 0 ,6,/i)8 + -Y S ( I - v ) 8 + - y + ||fi*// ' l l d x i - ^ / i - o ) ! ) .
263
Function Space Control
Also ti-a | v(01 <|| B *{t)H~X || J|//(s + a,
J
0 ftl-G
g
<\\B*(t)H0 7=1
<||B//-1||J||ay||fi/(Al)), 7=1
where
j|ay| = ||a 7 ||. Thus 0
|W(0I +| v(OI <(1 - v)5+ — + \\B*lTX\\(\xi -WX - o)|)
|5//-1||£;i|(xy||/V/(^) y=i
£
We have proved that J + S:^ -» A. It is clear that 5 is continuous and S:A^A. That S is compact is due to the following arguments. |S(r'",«,'XO-5(z,',«,'X/1)l = |A(0-A(' 1 )l + mO-v(OI < M]r - f1! + 7Vr0|f - / ! | + |5* (0 - 73*(r')}G
for
t,tX &J,
264
Differential Models and Neutral Systems
where G = ||//r 1 ||A/« 1 - a ) , M= max |/,-(*+a, <j>' + z' u'(s + a)X se[OA] N=
max \\Bi(s + ayi\, se[0,/,]
p = M + NM + mnc, db] , c=max max —-*■, (b-) = B*. ij OitZT dt V We now prove that Tis a contraction. Let (z'.,u'j) eA,j = 1,2. Then W S|7X*J, «j)C) - 7X4'«2W* " ^ ( / ) - ^ 2 ( 0 1 + |wi(0 - w2(0lBut
Sp(/ + a , ^ + z ^ / , 8 , r i ) | | z ^ - z J / | | <(l-v)||z^-z(,||, where || • || is the norm in A\ and we have used the nonatomicity of g as was done in Chukwu [10, p. 120]. Also |M'i(0-M'2(r)|<||S*(OW"1|l(lg('.Va+4/-o)-^*'-o+zl/-o)l) <\\B*(t)H-]\\{(l
-v)\\z^_a
Z\\B*{t)H-l\\iQ-Vyi\z^_a-zit^\\}.
-(z{t_a)\\}
Function Space Control
265
We have obtained |H'l(0-H'2(^
where H={(l-vXl+||B*(0/r 1 ||)}. Because v can be chosen so that the sum 7t is less than 1, proving that T is a contraction. We use Lemma 1.1 of [ 10] to assert that T + S has a fixed point which proves that Jc/(/l) = x ( = z ' ' ( n - a ) + * ' ' ( 1 - a ) ; and any <j>' sfVy can be transferred to x{ eEn in time t\. Thus (13.27) is Euclidean controllable on [a, t\\. The same argument proves that (13.27) is Euclidean controllable on [a, t\ - h], since B(t) is continuous. Indeed, since Bi is continuous and Bi(t\ - h)B*(t - h) has rank n, there is an e > 0 such that 5,<'l - / » - e)B?(t\ - h - e ) has rank« and B,{t-h-s)Bj*(t-h-s) has rank n for all s e [0, e]. Therefore 'i-A
H(t\-h)= JBi(s)B*(s)ds o fj-/l-E
=
J a
/|-E
Bi(s)B*(s)ds+
JBi(s)B*(s)ds, /-/i-e
is positive definite. As argued before (13.27) is Euclidean controllable on [a, t\ - h]. Because of this given <)>', \\i' e W^\[-h, 0], En') there exists a
266
Differential Models and Neutral Systems
«eZ,2([cj, t\],Em>) such that 4=<J> and x'(fi -h, a, $', u) = \V'(-h). l
extend this u and x (•, a, <J>', u) to the interval [v,t\],t\>
We
a +h,so that
J (£>,(/, */)) = /,-(/, x\Mt)) + Bi(t)u(t),
(13.33)
on [t\ - h, t\] where x(t) = \]i(t -1\), t\ - h < t < t\, on the right-hand side of (13.34). In view of hypothesis (i) we define a control u as follows,
"(0= < ( 0 ^ ( A ( ^ - / / C *>(')))),
(13.34)
for t\ - h < t < t\, where B+ is the generalized inverse of Bj. With this u, x satisfies (13.33). Indeed insert u defined by (13.34) in (13.33): j t Di{t,x\) = fi(t, x/,a(O) + fl«(O<(0(-^ A C * / ) - / « ( ' , * ! , "(0)
with x(t) = y(t -1\), t\ - h < t < t\. Evidently u is square integrable on [t\ h, t\] because of hypothesis (ii) and the continuity off, x and \\i. We now prove that u is well-defined as a solution of the functional equation (13.34). We now define the following set: u €L2([t\ -K n], E ):\u(t)\
=
J|v(/ + s)-v(/)| dt->0
a.e. andv eA implies
as 5 ->0 uniformly with v eA.
l'i-*
By [13, p. 2971], ^ is compact. Define an operator 1\u)(t) = v(t) where
v(o=BT (t)(jt at, Wt_tx) -f(t, x\,«(/))) • We now show that there is a constant r$ and a Banach space A = A{TQ) such
Function Space Control
267
that T:A -»• A where T is continuous. Thus we are guaranteed a fixed point T\u) = («) e A which proves that (13.34) holds. The existence of a fixed point follows from [25, pp. 297 and 645] because A is compact convex subset of the Banach space Z.2- To demonstrate the validity of our assertions we observe that by (i) and (ii), the generalized inverse B+ (t) is continuous [6, p. 225], and is therefore uniformly bounded on [t\ -h,t\\. Also £>/(/, xi) is differentiable with an L2 derivative. Let («) e A. Since the growth condition (iv) is valid there exists r§ > 0 such that <7
Y*cJFiJ^
+ d r
-b
for some d. With this r$, define A = A{r^). Now introduce the following notations p = max{||B(/,
X=max{p+||//-1||,||fi*W71||},
a,|| = max' f t
a,- = 3U||a,||, 6,- = 3110/11, q =max{ai,bi},
268
Differential Models and Neutral Systems
d2 =max{3R|D(v|/,_/,)|2,sup|Z>(\)/,_?1)|}, d = max{d\, d2}Since w e ^ w e have \1\u){t)\ < |v(/)| where |v(OI ^ \B+(t,x")\(\j
via/
D(t,Wt-tl )-fi(t,xt-t,,
K(0)
sp+fliKMi'^l+l Mt, */,«' (0)1)
< p+1 tit, M/t_h) i+Y/tjWiM»»' w) ;=i
^P4 7=1
a13*
* a ■ , ^2A
3*
3
y
U
i ^2 + 2L,cjFii(.ni) ' 3*
>i
Hence |7T(«)(V)| < A"O- We have proved that T:A -> ^4 provided the second condition is satisfied. To verify the second condition we note that
j \T(uXt + s)-T(uXt)i2
dt=
$\Bt(t + s)Ht+s)-BfX(t)\2
dt
u-h (13.35)
where
Function Space Control
269
Ht)=~
dt
(D(t, yt-h )) - fi{t, xt-t{, «(/))
is integrable. Consider the function k(t) = B? (t)X(t),, which is measurable in / and is L2. Therefore there is a sequence {kn(t)} of continuous functions such that '1
J|*(0-M')| 2 dr->0
as n -xa.
ti-h
Hence
tf
1
J|A(r + s)-*(0| 2 \-n
1
1
j|A„(/+ S )-*„(/)| 2 ^
J|A(/+i)-A„(/+5)|2^
i j|*„(/)-*(OI2 0 by selecting n very large. The second integral can be made less than e if 5 is selected small enough. This proves that
j\B+(t + s)Ut + s)- B+0)H'f H-h
Therefore,
dt^O
as s -> 0.
2 70
Differential Models and Neutral Systems
h J| 7XC u\t +s)- 7XC «X0| 2 dt ->• 0
as s -> 0.
We now verify continuity. Note that u -> x(-, u) is continuous. Also t -> £(/), (/, <|), w) ->./X'>
(v'(t)) = T{w).
Then I v(/) - v'(r)| = |5, + (0(«;" - Cw) + if At, x*h , w(t)) - / , ( ' , JCJ^ . «(')))}
=Pi+(oii(ic"' - c ' i+/,('. *,"!,,,«'«»- /,•(/, *,%, w(o)i). (13.36) Since ul -» f(t,xf
, u\t))u is continuous, given any e > 0, there is an r\ > 0
such that if \u(t) - w(t)\ < r\ then |/,(f, xf,«''(/)) - //(/, *"', w'(0)i < e / for all r e [a, t\]. Divide [a, /1] into two sets e\ and 22; and P u t the points at which \u\i) - w'(r)| < r\ to be ej and the remainder e2- If we write ||« -w ||2 =y then '• . . y 2 = J l i A O - w'(0| 2 (//> f|M f (0- w\tf e2
so that 2 2 mes e2 ^ Y / r\ ■
Consider
dt>r\2 mes ej
Function Space Control
-27/
Obviously
/= J + j|/(',*£,,, "'«) -fi(t,x?Jtl, "(0)| * e,
e2
2 4v 2 < e mes e\ + —hr- {sup / ( )} Tl
2
4Y
2
< e /wes e\+~k R , for some /?i. Because u -> (dldt)D(tjct) is continuous, given e > 0, there exists an «2 > 0 such that if \u{t) - w(t)\ < T|2 then \{d I dt)D(t, v | / ^ ) -(d/dt)D(t,
v | / ^ )Us for all f 6 [a, fi]. Consider as defined
above, and
Clearly
M+J||0<<^>-i**S->,,>|2''' ej
ej
2 4y 2 < e mes e\ + 2" {sup{D(t, •)}} Y 2 „2 < e mes ej + —5- A , Tl2
2
4
for some /?2- We now insert these estimates of / and J in (13.36). We deduce that if \u - w\ < r\, then |(v(0 - v'(0)l2 < II #11 I?"' ~C \
Differential Models and Neutral Systems
272
+ P +2 (e 2 + e 2 ). Thus i
ll(v-v')||2=
Jl(v(/)-v'(0)| 2 ^
/,-/. r ^+p
2
2
+
Because y = \\u - w\\ , the left-hand side of this inequality can be made as small as possible if u and w are sufficiently close. This proves that T is a continuous map, as claimed T:A -> A is a continuous mapping of a compact convex subset of I2 m t o itself. By Schauder's fixed point theorem [25, p. 645], Thas a fixed point. With u, (13.34) is satisfied since x(t) = VJ/(/ - 1 \ ) on [t\-h,t\], and /(r,*;, M (0) + B(Ofi + (o("| £>(', V / - r , ) - / ( ' , */,«('))) =j( D{t,
yt_h).
The history of the growth condition (iv) of Theorem 13.1 is outlined in the recent paper, [19, pp. 46-51]. Inspired by these ideas we have the following corollaries. Corollary 13.1 In the system in (13.27) assume (i) (ii)
condition (i)-(»0 of Theorem 6.6.1; Urn |/i(',*.«0|/|l(*.«OII = 0 IK4>,")lh>°°
uniformly in t{t e [0, t\]). Then (13.27) is controllable on [a, t\] = /with t\ > a + h. Remark: (13.37) is used in [16] and generalizes [29]. Proof: Let /7(
(13.37)
Function Space Control
273
Then <7
)
lim r-^CiSvpiFy^u'y^u'^r) /•-+00
y=i
= +00
(13.38)
J
if lim inf{-) r—HO
sup{Fi($, u):U, «)| < / • } < - .
K r /
(13.39)
c
--
\
But condition (13.37) implies (13.39) by a modification of an argument of Do [19, p. 47]. The required modification is the proof that if the corresponding sequence {(<|>j, uj)} is bounded we can assume it compact (see the definition of the set A = A\ x Aj). Therefore (13.38) is valid and Theorem 13.1 can be concluded. Recall that//, <J>, M) is said to be locally bounded in u if for any M> 0 there is an L > 0 such that \\/{t, <> | , u)\\ < L for all ( / J ) e £ x C and for all ||u|| < M The next corollary is valid. Corollary 13.2 For the system (13.27) assume conditions (i)-(iii) of Theorem 13.1. Ifff.E x C x Em is locally bounded in u and lim ||/(f,
(13.40)
||B||->CO
uniformly in (t,§) e E x C then (13.27) is controllable on [a, t\], t\>a + h. Proof: LetF,<<(>, u) = sup{\\f,{t,$, u):tel}. Then ||/,- (t,buy\
(<)),«)
V((,^,u)6/xCx£m
Due to (13.40) the following is valid: lim!-)
sup{Fi(bu):m,uy\
= 0.
274
Differential Models and Neutral Systems
As a consequence (13.39) holds, and the result follows. Proof of Theorem 13.2: The proof is similar to that of Theorem 13.1. The integral equation of (13.28) corresponding to (13.30) is *''(/) = Di(o,tf) + gi(t, x\) + fffa, x's, u\s))ds
tt
f
+ \Kj(s,xs,v'(s))ds+
\Bi(s)ul(s)ds,
t>a.
The control function is defined by ui(t)=B*(t)H-\t]-aXx\-V(n-o)
-
jfi(s
+ gi(o^)-gi(thVt].a
+
^ra)
+ a,Vs +rj,«''(5 + o)) + Ki(s + a , ^ +z^, v'(s + a ) ) * ) . ( 1 3 . 4 1 )
This control steers §l to xj in time t\. t
The additional term J£,-($, xs, v'(s))ds is taken care of in the various o estimates by using condition (ii), and noting that P/ < a/. Just as in the proof of Theorem 13.1, under the conditions of Theorem 13.2, system (13.28) is Euclidean controllable. The corresponding «' which steers <J>' to xj in time t\ is given by (6.5.41). This is comparable to (13.32), the difference being the "Interconnection integral" i
-
in
\Kj(s + a, <j£ + z's, v'(s + a))ds. The operators T and S are defined as i
the proof of Theorem 13.1 with the modification that S(zi,uiXt)=(K0M0) where
if fG[o,/]
Function Space Control
275
t
+ o,ti+zi,ui(s
Kt)=jfi(s
+ a))ds
0 t
/
+ lBj(s + a)u'(s + a)ds+ JK,(.s + a, <J4 + 4> v'(s + o))ds ; 0 0
+ \Bj(s + cs)u'(s+o)ds+ \Ki(s + a,^'s + z's, vl(s + o))ds . o 0 Just as before we prove that the operator T + S defined by ( r + SX*\ «'X0= (KO + h(t\ w(t) + v(/)), has a fixed point:
(r + sxz''y'xO=(z''(0,"''(0) so that (13.28) is Euclidean controllable. We proceed as before to prove this fixed point. Suppose /V/-(r)=5i(p{F&-(♦,«):! | (♦,«)! 15 r} so that by the growth condition there is some /-Q such that
7=1 and Kij(r) =
sup{Kij®i,ui):m,ui)\\
2 76
Differential Models and Neutral Systems
so that
We use Lemma 1.2 of Hale [23] to estimate that
| h{t) | <; Y)\ «; II Fyiti) + "vg + j j l Py II ^(t)) 7=1
7=1
'1
'1
where ||ccy|| = l\aj(s + o)\ds,\\Vj 11 = JIPy(* + a)|&. Thus 0
0
I A(f)|<£||a,-1|(fjy(/&) + Fy(n)) + /Vv
since Py < ay by condition (ii). This and the previous estimates yield
lyiOl +
lKO^di+^WajWFy^)^, 7=1
where d\ is as before and q = /Wax{2||a;||,||p*//-1||||ay||}.
In the estimate for v(/) we replace ||ay|| by 2||ay||. Thus T+ SA -» A. The previous argument shows that SA -> S is continuous and compact. In the earlier argument replace Mby M= M\ + Mi, where M\= max \fj(s + o,$'s+z's,u'(s + o))\, /6[0,/,]
Function Space Control M2=
277 max \Ki(s + a,^'s+z'
te[0,t{]
v'(s + a ) ) | . *1
The argument that T is a contraction is unchanged. Lemma 1.1 of [10] asserts that T+ S has a fixed point, so that (13.28) is Euclidean controllable on [a, t\]. It is also Euclidean controllable on [a, t\ - h], given ty, i]/ e Jf 2 (,) ([- h, 0], E"t), there is a « e Z,2([^ 'i - h], Em>) such that x'a = <>j and *'(/! - A, o, $', M) = vj/(?i - A). This control u and the solution x' (•, a, ^ , u) are extended on the interval [o,t\],t\ |
> a + h so that
A(/,xj) = /,-(/, *;,«(/)) + £,(',*;, v(/)) + $ « « ( ' )
(13.42)
on [t\ - h, t\] where
Define a control
«(o= <(o(^Aa,*!)-/ia,*;,«(0)-Ai-(r,*;',v(/))), (13.43) for /j - h
yields f
|v(/)|
+
1
\iKt,Vt-ti)\ +
'Z2\aj(')\Fij{r0)
which
2 78
Differential Models and Neutral Systems ( < — dl + 3k V
q
\
HCJ Fij(ro) ) 7=1
With minor modification the rest of the proof is complete. Remark: It may be said that p;- is a measure of the action of the whole system S on its rth interconnected subsystem, while a/ is a measure of the subsystem's internal power. To ensure controllability a/ > P,. Broad policy implications of this are pursued in [9].
REFERENCES 1. R. G. D. Allen, Mathematical Economics, MacMillan & Co., Ltd., London, 1973. 2. K. J. Arrow, Production and Capital, Collector Papers of Kenneth J. Arrow, The Belenap Press of Harvard University Press, Cambridge, Massachusetts, 1985. 3. K. Balachandra and J. P. Dauer, "Relative Controllability of Perturbations of Nonlinear Systems", J. ofOptim. Theory Appl. 63(1989). 4. H. T. Banks, M. Q. Jacobs, and C. E. Langenhop, "Characterization of the Controlled States in of Linear Hereditary Systems", SIAMJ. Control 13 (1975), pp. 61-649. 5. Z. Bartosiewicz, "Closedness of the Attainable set of the Linear Neutral Control System", Control and Cybernetics 8 (1979), pp. 179-189. 6. S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Pitman, London, 1979. 7. E. N. Chukwu, "Some Controllability Questions of Linear and Nonlinear Neutral Functional Differential Equations in fV^ "preprint. 8. E. N. Chukwu, "Controllability of Delay Systems with Restrained Controls", J. of Optim. Theory Appl. 29(1979). 9. E. N. Chukwu, "Uniform Asymptotic Stability of Large Scale Systems of Neutral Type", in IEEE Proceedings of the Eighteenth Southeastern Symposium on Systems Theory, Computer Society Press, Knoxville, TN, April 7-8, 1986. 10. E. N. Chukwu, "On the Euclidean Controllability of a Neutral System with Nonlinear Base", Nonlinear Analysis, Theory, Methods and Applications 11 (1987), pp. 115-123. 11. E. N. Chukwu, "The Time Optimal Control Theory of Linear Differential Equations of Neutral Type", Comput. Math. Applic. 16 (1988), pp. 851-866.
Function Space Control
279
12. E. N. Chukwu, "Global Economic Growth: Will the Center Hold?", in Ordinary and Delay Differential Equations, eds. J. Wiener and J. K. Hale, Longman, New York (1992), pp. 19-23. 13. E. N. Chukwu, "Mathematical Controllability Theory of the Growth of Wealth of Nations", in First World Congress of Nonlinear Analysis, Tampa, Florida, ed. V. Lakshmikantham, 1992. 14. E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, Academic Press, New York, 1992. 15. E. N. Chukwu and H. C. Simpson, "Perturbations of Nonlinear Systems of Neutral Type" J. of Differential Equations 82 (1989), pp. 28-59. 16. P. Dauer, "Nonlinear Perturbations of Quasilinear Control Systems", J. of Math Anal. Appl. 54 (1976), pp. 717-725. 17. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New York, 1985. 18. J. Dieudonne, Foundations of Modern Analysis, Academic Press, New York, 1960. 19. V. N. Do, "Controllability of Semilinear Systems", J. of Optim. Theory Appl. 65 (1990), pp. 41-52. 20. 0. Hajek, Pursuit Games, Academic Press, 1975. 21. J. Hale, "Forward and Backward Continuation of Neutral Functional Differential Equations", J. of Differential Equations 9 (1971), pp. 168-181. 22. J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. 23. J. Hale and M. A. Cruz, "Existence, Uniqueness and Continuous Dependence for Hereditary Systems", Ann. Math. Pura. Appl. 85 (1970), pp. 63-82. 24. M. Kalecki, "A Macrodynamic Theory of Business Cycles", Econometrica 3 (1935), pp. 327-244. 25. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, MacMillan, New York, 1964. 26. G. E. Ladas and V. Lakshmikantham, Differential Equations in Abstract Spaces, Academic Press, New York, 1972. 27. M. McElroy, The Macroeconomy; Private Actions; Public Choices and Aggregate Outcome, Prentice-Hall, New Jersey, 1996. 28. A. N. Michel and R. K. Miller, Qualitative Analysis of Large Scale Dynamical Systems, Academic Press, New York, 1977. 29. K. B. Mirza and B. F. Womack, "On the Controllability of Nonlinear TimeDelay Systems", IEEE Trans, on Automatic Control (December 1972), pp. 312814. 30. A. W. Mullinex, The Business Cycle after Keynes: A Contemporary Analysis, Barnes and Noble, New Jersey, 1984. 31. Jeffrey Sachs and Felipe Larrain, Macroeconomics in the Global Economy, Prentice-Hall, 1993. 32. D. Salamon, Control and Observations of Neutral Systems, Pitman, Boston, 1984.
280
Differential Models and Neutral Systems
33. M. Slemrod, "The Flip-Flop Circuit as a Neutral Equation", in Delay and Functional Differential Equations and Their Applications, Academic Press, 1972 34. E. N. Chukwu, Control in of Nonlinear Interconnected Systems of Neutral Type, J. Austral. Math. Soc, Ser B 36 (1994), pp. 286-312.
14. Nonlinear Mathematical Controllability Theory of the Growth of Wealth of Nations
14.1 Introduction Earlier we accepted instantaneous stroboscopic nature of pursuer firms response to government action. We can assume realistic time delays in the firms' response-delays in the transmission of information: a time delay 8 > 0 in the firms observation of government action and a time lag in the systems' reaction to the firms controls. We can now also allow the treatment of distributed lags.
14.2 The Main Results Let 0 L (t, if, \\i)xt = jdsA(t, s,ty(s),y(s))x(t + s), -h where A{t, s, §(s), \\i(s)) is measurable in (t, s) e E x E, for all § e C, \\i e Lao([-h, 0], En) normalized so that A{t, s, §, \\i) = 0 for * > 0 V<|>, v|/ A(t, s, ty(s), \\i(s)) = A(t, -h, \\>(-h), \\i(-h)), s < -h. There is an integrable m such that [L(t, <)>, \\i)xt] < »i(/)||jcr||, for all t e (-oo, oo), <j>, \\i, xt e C. We also assume that the function (/, \\i, v) -» B(t, ()), v) is a continuous n x m matrix function which is given B(t, xt, ut)ut = )_h dsH(t, x(t + s), u(t + s)) ■ u(t + s), where H is an n x m matrix function. Also \B{t, <J>, i|/)w,| < *(0llM/ll where k is integrable. The solidarity function q : EI Q x El -> En where C/ 281
282
Differential Models and Neutral Systems
= C([-h, 0], E1) is assumed continuous. Enough smoothness conditions are imposed on (14.2) and (14.1) below to ensure the existence of solutions and the continuous dependence of solution on initial data, (see [17]), and to prevent finite escape time: If the controls are bounded, the solutions are uniformly bounded by some M\. Consider the isolated system *(t)-AA
x(t-h)=L(t,xt,pt)xt
+B{t,xt,pt)pt ,
(14.1)
together with its composite or interconnected system *(0-^.1 k(t-h) = L{t,xt,pt)xt +B(t,xt,pt)pt +q(t,xt,v(t)) (14.2) where v(t) is in L^([a, t\], En). Thus we assume the "solidarity" function: q(t) = q(t, <|), v(/)) belong to Q. Its values lie on some set Q, En 3 Q. As the control strategies v(/) vary, the values of q(t) lie on Q. The following assumption is made on q. It is called the solidarity assumption. Let measurable functions q"Cy.Em - > £ + and Z1 functions a ; : £ -» E*
\q(t, <> | , v)| < ^
i = \,...,q , be such that
aj(t) qj®, u) for all (/,
7=1 where
lim s u p ( r - >
c, sup{q{$,p) :||(<)>,/?)||
7=1 Remark 14.1: The solidarity assumption can be translated loosely as saying that if q(§, p) is a measure of the "power" of firms and q(t, (|>, v) the
Nonlinear Mathematical Controllability
283
corresponding measure of the power of solidarity, this should later be dominated by the "power" or "initiative" of the firms. Remark 14.2: As a consequence of the solidarity assumption if qj(r)
= sup {, «):||(
then there exists r 0 > 0 such that
r
0
2^
Cj qfa)
>d
or
7=1
2_,
c
j ?/ r u)
+ d
*> r0
7=1
We tackle the problem of Euclidean controllability of (14.1). First consider the simpler system, * (t) - AA *(t-h)
= L(t, z, v)x, + B{t, z, v)«, ,
(14.3)
where the arguments xt and u( of L and B has been replaced by specified functions zeC([-/i,0],£")=C,« 6 Lao([-h,0],Em).
Here C is the space of
n
continuous E -valued functions on [-/?, 0], and L^ is the space of essentially bounded measurable Em -valued functions on [-h, 0]. For each (z, u)e C x Las, the system (14.3) is linear and its solution in the form of variation of parameter can be deduced by the ideas of Klamka [25] which are utilized by Sinha [30] and Balachadran and Dauer [4]. Let X(t, s) = X(t, s, z, v) be the transition matrix of the system *(0-/*_, i(t-h)
= L(t,z,v)xl
(14.4)
so that dX(t,s) . dXU-h,s) ..,, . T, — ^ - A-\ \ = L(t, z, v)Xt{; s) , dt ot ' where
,, . ,. (14.5)
Differential Models and Neutral Systems
284
X(t,s) =
0 1
s-h
and where Xt(; s)(Q) =X(t + 9, s),
-h
The solution of equation (14.3) is "0
*(/) = *(/, a, 4>,0)+ JAU-O ldQH(s,z(Q),u(Q))p(s + Q)ds ,
(14.6)
a<,t
/>(/) = C(0, / e [a - A, a],
(14.7)
where x(t, a, 9, 0) is the solution of (14.4) with xa(-, a, 9, 0) = 9. The last term in (14.6) contains values of u(s) for s < a. We like to separate them from the other values by changing the order of integration by means of unsymmetric Fubini theorem. These important ideas are familiar. See [4], [25], [30]. We have
x(t, a, 9, u) = x(t, a, 9, 0) + g(t, z, u) + js(t, s)p(s)ds ,
*(/) =
te[c-h,a],
u(t) = rj(0
/ e [a-h, a ] ,
(14.8)
where 0
g{t, z,u)= J
a
\{X{t, s - Q)dB[H(s - 6, z(9), p(Q))l T](s)} ds,
-h o+e
(14.9)
Nonlinear Mathematical Controllability
285
wherepa(s) = r\(s), s e [~h, 0]. (Note that g{t, z, v) = 0 if T| = 0.) Also 0
S(t,s)=S{t,s,z,u)
= jX(t,s-Q)dQ[H(s-Q,z(d),u(Q))]. -h
(14.10)
The controllability map of (14.4) at time t is defined by t W(t) = W(t, o, z, u) = Js(/, s)S*(t, s)ds , a where *, denotes matrix transpose. In what follows we write W(t. ) = W. For Euclidean controllability we use the control set U= {p measurable,/K/) e Em , \pj(t) | < 1 ,
j= l,...,w}
and define the reachable set R(t) of (14.3) as follows:
R(t) = jS(t,s)p(s)ds:peU
(See 1.21.) Since x(t, a, <(>, 0) = 0 and g = 0 if $ = 0 and r| = 0, the reachable set is the set of all solutions *(-, p) of (14.3) with <J» = 0, r) = 0 with p € ^oo ([<*, 'i ], Cm), where Cm is the unit/w-dimensional cube. The Euclidean attainable set of (14.1) is the subset of En defined by A as follows: A{t) = {x{p\t)\ pe U, x{p) is a solution of (14.1) with xa(p) = 0, T| = 0} . With T| s 0 fixed, the domain D of null controllability is the set of initial <J> such that there exists a time f, and a p e ^ ( [ a , /]],C m )such that the solution of AT of (14.1) satisfies xa(p) = <j>, x(p)(t\) = 0. following Lemma. Lemma 14.1: In (14.1) we assume as basic
We need the
286
Differential Models and Neutral Systems
(i) (ii)
the conditions on L and B of section 3. there exists a constant c* > 0 such that det W(a, t\, z, u) > c* > 0 for all (z, u) e C x Lx .
Then Oelnt^/j);
(14.11)
OelntZ).
(14.12)
and consequently
Proof: The argument is standard that the condition (ii) implies the Euclidean controllability of (14.3) see Manitius [27, pp. 77 and 78]. By the standard arguments (see [19]) this implies that OeInt/?(/,z, v),
(14.13)
where we have displayed the dependence of R on z, v. Since (ii) is uniform for all (z, v) e C x Z,ro, 0 e Int R(t, z, u), V(z, u) e C x L^. Note that if (ii) is valid the map T: ^ ( [ o , /]], Em)-+ En , defined by
(7>X0 = JS(1, s, z, u)p{s)ds
(14.14)
o
is a continuous linear subjection for each (z, u) e C x L^, and is therefore an open map, and maps open balls into open sets. Since there is an open ball O, Zoo([°, 'l ], Cm) => O with O e O , R(t, z, v) = r(4o([a, *iL C"))(/,)z> 71[0). This proves that (14.13) is valid. With Euclidean controllability assumed the transfer of any <>| e C to any x, e En is achieved by the control
Nonlinear Mathematical Controllability
p{t) = r\(t)
287
/ e [a-h, a]
p{t) = S\tx ) , 0 FT1 [x, - *(/, , <> | , 0) - g<7(>, )1,
/ e [a, /, ] .
If this p is inserted in the variation of parameter formula (14.8) then x(t) = x(t,o,0) + g(t,z,u)
(14.15)
+ jS(t, s)S*(th s)W-l[Xl - x(th a, <(>, 0) - g(tx, z, «)],
/ e [a, / , ] .
a
With initial data w(a) = (x(a),
with norm defined
by II (x,p) || = || x || + ||p IL, where || x || = sup | x(s) |, s e [a - A, /, ] ||p IU = maxi
- js(r,, s, zh, ptl)q(s,
xs,vs)ds,
a
p(t) = T|(0 and
/ e [ a - A , a]
t e [a, tx ] .
(14.16)
288
Differential Models and Neutral Systems t
x(t, a, <►, p, q) = x(t, a, <(», 0, 0) + g(t, *,,/>,)+ js(t, s, xt, p, )p(s)ds
+ js(t,s,x„p,)q(s,xs,v(s))ds,
x(t) = W),
t>a,
(14.17)
te[a-h,a].
Then p e Lo0([cs-h,t\],Em), and it can be shown by differentiation that x solves (14.2). With u so defined and with initial data (x(a), <> | , T|) where Pa = r), we have JC(/J ,CT,<)», /?, B by 71[x, ) = (y, iX where
u
S*(tht,xtl,utlW~l[x\-x(tha,hO,0)-g(thxtvptl)]
y(t) =
- jS(th s, xt], p,x )q(s, xs, ps)ds,
V(0 = TI(0,
t>a,.
te[c-h,o);
(14.18)
y(t) = x(/, a, <j>, 0, 0) + g(t, xt, pt) t
t
+ ]S(t, s, xt> p,)v(s)ds + jS(t, s, x„ pt)q(s, xs, v(s))ds , a
t>a
a
J<0 =
te[o-h,o].
(14.19)
Let a, = sup||S*(>, ,',*,, ,/>,,)!!
te [ a , / , ]
Nonlinear Mathematical Controllability
289
(x,p)eB,\\p\\
+ \g(t,x!,pl)\
+ \x{ \
t e [a, tx ]]
(x,p)eB,\\p\\
|| (t, s) e [0, f, ] x [a, f, ]]
(x,p)eB,\\p\\
=taax[CljtCy],
rfj = foj a 2 a 3 b,
d2 = 6a3
fc = max{(/j -
290
Differential Models and Neutral Systems
| v(/) | < a\ ca
d+
6b
JXV
^
b '
'
6b
Z^y^o)
6b
u
6
r o
Also
| >
jJTa,( S )<7 y (/- 0 ) «y=i
< « 3 + X « 4 II cxy || ^ ( r 0 ) + A |j v || < ^ || v || + i S % ^ ^ 0) 6 ^ 6 >1 r
r
o
o
< b\\v\\ + -7; < -T + 6 " 6 6
r
o
r
o 3
Since
|v(0|<-^,
WOl^y
II v ||oo< -g-
and
|| ^ | | <
if we let H(r0) =
{(x,p)eC([o-h,t],En)xLOD([a-h,ti],Em)
s5:||(jr,/>)|| = ||x|| + ||/>|loo£'b>'
j
Nonlinear Mathematical Controllability
291
we have proved that / / ( r 0 ) 3 7r//(r 0 )). We have tacitly assumed that r 0 is large enough for || rj || < rQ , || <> j || < rQ . It is clear that //(rg) is closed bounded and convex. Also T is continuous because q is assumed continuous and (l,ty,v) -> B(t, <(>, v) is continuous. For more general q where / -> q(t, <j>, «(/)) is only measurable, an argument similar to that contained in [24; pp. 654-656] can be used to prove continuity. To prove that the set of MO} defined in (14.19) is equicontinuous is routine. The set of {v} defined by (14.18) is relatively compact since all such v are uniformly bounded in L^ norm, we need only verify [24, p. 296] that max ess sup|v-(/ + s) - v{t) \ = || av - vH^ -> 0 as s -> 0 , J J
\<.j<m
uniformly. Since v(r) is measurable in / and is L^ there exists a sequence {vn(t)} of continuous functions such that l|v-v„ m - * 0
as
n->oo.
Hence II v, - v |L < || v5 -
Vns
IL + || v„ - v„ IU + || vn - v IL.
The last and first terms can be made arbitrarily less than e by selecting n very large. The second term can be made less than e if s is selected small enough. This proves \\vs - v ||oo-> 0 as s -> 0. Therefore r(//(r 0 ))is relatively compact. Invoke Schauder's Theorem to conclude that there is a fixed point (x, p) e Hfo) such that T\x, p) = (x, p). We can deduce several conditions from this existence of a solution pair for (14.16) and (14.17). Assume q = 0, i.e., consider (14.8) which is a solution of (14.1). Since T defined by (14.14) is an open map for each (r, v) e C x L^ and in particular for (z, v) = (x, p) we have that solution of (14.1). Since T defined by (14.14)
292
Differential Models and Neutral Systems
is an open map for each (z, v) e Xx LK and in particular for (z, v) = (x, p) we have that 'i
Tu= jS(ths,xtl,pt])p(s)ds;
T:
L00([o,tl],Em)-->En
a
tx], Cm)) = A(tx ). This proves that
is open. It follows that 0 e Int T^fto,
Oelnt^/,).
(14.11)
With (14.11) we also prove that OelntD.
(14.12)
Indeed suppose 0 is not contained in the interior of D, and aim at a contradiction. With our assumption there is a sequence {„}*, $n : [-h, 0] -> En with
-*(/,CT,<J>„, 0, 0) = %„ * p(t, s, x„ pt)p(s)ds
,
a
for any n, any / £ a + h, and any/? e Z,00([CT,r],C'"). Therefore \n is not contained in A(t) for any f > a + A. From the uniqueness and continuity properties of x(t, a, <)>„, 0, 0) we have produced a sequence {^w}^=1 such that %„ -> 0 as « -> oo £„ £ ^(/) for any f > a + A, ^„ * 0. Note that - x(t, a, «))„, 0, 0) -> 0 as <j>„-» 0. Hence 0 £ Int ,4(0 for any / > a + h. This contradicts (14.11). Lemma 1.1 and its proof can be used to prove controllability results. Theorem 14.1: First (14.1) assume in (14.3) that
Nonlinear Mathematical Controllability
(i) (ii)
293
The smoothness conditions on L and B of section 3 hold. There exists a constant c* > 0 such that det W(a, t\, z, v) > c* > 0 for all (z, v) e C x /,<„. Then (14.1) is locally Euclidean null con trollable with controls in U.
This result is the content of Lemma 14.1 and it has been proved. Theorem 14.2: Assume all the conditions of Theorem 14.1. Furthermore impose the solidarity assumption; then (14.2) is Euclidean controllable on [a, t\] with ^ ([<*» t\]> Em) controls. Proof: The proof is an easy deduction from the existence of a fixed point of T in the lemma. This theorem can be interpreted as follows. If the isolated system (14.1) is controllable then so is the "perturbed system", obtained from government and other interventions, provided the resulting solidarity function q is "rice" and dominated by the firms power or initiative. The following fundamental principle is now obvious. Fundamental Principle 1: If the isolated system is "proper", which means that it is locally controllable with constraints and if the "Solidarity" functions is nice and dominated by the firms power or initiative, then the composite system is also proper and therefore locally null-controllable. Theorem 14.3: In (14.2) assume that (i) (ii)
q(t, (|), 0) = 0, q(t, 0, 0) = 0 for all t > 0. All the conditions of Theorem 14.1 are valid. Then (14.2) is lo cally Euclidean null-controllable on some interval [a, t j ]. If in
addition the system (iii) x (f) = L{t, xt ,ut)xt
(14.20)
is globally exponentially stable then (3.2) is globally nullcontrollable with controls in U. Proof: Let x(p, q) be a solution of (14.2) with xa(p, q) = 0 , p0 = 0 = r|. Then x(a, Q,p,q){tx) =
x{p,q\tx)
Differential Models and Neutral Systems
294 '1
'1
js(t, s, xs, ps)p(s) + jS(t, s, xs, ps)q(s, xs, v(s))ds.
Define the set A(tl)={x(j>,q)(ti):peU,qe
V),
which is the attainable set associated with (14.2). Since q{t, <> | , 0) = 0 and because *(0, 0)(/) = 0 is a solution of (14.2) we have that 0 e A(t\). But by Lemma 14.1, 0 e \n\A(t\), we easily deduce that 0 e Intv4(/i), from which one easily deduces 0 € Int D, where D is the domain of null controllability of (14.2). We prove the second assertion as follows. Utilize the control u eUp = 0, q = 0 6 Fand describe the motion of (14.2). Thus consider (14.20), since in (iii) we assume exponential stability there exists IQ < oo such that the solution x(§; •) of (14.20) is such that v|/ = */.(<)>, 0,0) belongs to O where D z> O, and O is an open ball containing zero in C. With \]i as initial state there is a time /j > a and strategies p e U, q e V such the solution of (14.2) satisfies *t0 0> V» P>
* ( ' i » V. P>
Hence the control 0
on [oy 0 ]
Nonlinear Mathematical Controllability
295
is admissible and does the transfer of <>| to 0 in time t\. The theorem is verified. In all these theorems we assume conditions which ensure that the isolated system (14.1) is locally controllable. If (14.1) is not well behaved, and the controllability assumptions are not valid, it is still possible to ensure controllability. It is then necessary to rely on some hypothesis on the solidarity function q. One such condition is the local controllability of x{t)-A_x
x {t-h) = L(t, xt, 0)x, +q(t, x,, u(t)) ,
(14.21)
which is the system (14.2) with the subsystems control set to zero. Thus with the solidarity function q{t) as a control and q{t) e Q, where Q is a subset of £" with 0 e Q. The controllability of the linear version of (14.21) will yield the desired result. In other words we require a sufficient amount of control impact or "solidarity intervention" q to be brought to bear on the system. Suppose condition (ii) of Lemma 3.1 fails so that the system x{t)-A_x
x(t-h)
= L(t,z,v)xt+B(t,z,v)ut
,
(14.3)
is not Euclidean controllable for each (z, v) e C x L^. Then (14.11) fails and (14.12) will not follow. The intervention of q(t) - q{t, xt, v(0) can help. For example if q(i) = C(t, z, u( )pt where C is an n x m matrix function, the system is perturbed, and becomes x (0 - A_x x(t-h) = L{t, z, v)xt + H(/, z, v)pt ,
(14.22)
where H(t,z,v) = B(t,z,v) + C(t,z,v). To ensure growth from an initial endowment to the target we assume (14.22) is Euclidean controllable. Hence matters should be so arranged that the solidarity function q brought to bear on the isolated system ensures controllability. These observations are formalized in the following principle. Principle 2: If an isolated system is not proper and is not locally controllable with constraints, the composite system may be made "proper "
296
Differential Models and Neutral Systems
and locally controllable provided there is an external source of power or initiative q available to enforce controllability and proper behavior. It is possible that the intervention of solidarity can make matters worse. Only a "proper" amount is needed. There is no theorem at the present time which states that the interconnected system is well behaved when it is misbehaving in isolation and there is no compensating external solidarity q. We observe that the intervention of q can make matters worse. If (14.11) holds and we consider (14.2) with q(t) = q(t, xt) = qQ * 0. Then 0 need not be in A(t\) and Theorem 1.3 may fail. In this case solidarity function is not flexible. This seems to be the case in centralized economies. Principle 3: If the solidarity function is inflexible and rigid, or if the isolated systems control initiatives are ignored in the construction of a nontrivial solidarity function the interconnected system will fail to be proper and locally null controllable. Failure to acknowledge individual subsystem's control initiative in highly centralized economic systems which are described by our dynamics may lead to lack of growth from initial endowment <>| to the desired economic target (which is an equilibrium in this case) in centralized systems. On the other hand, failure to enforce regulation bringing to bear the right force q when some isolated system misbehaves and is locally uncontrollable in "individualistic" systems may make the composite system locally uncontrollable and may trigger an economic depression. These observations seem to have a bearing as an explanation of what is happening in the East and what may happen in the West as a consequence of the 198089 U.S.A. dismantling of "regulations" on the economy. In the East economic target cannot be reached because of too much centralization: to make firm controllable, market forces are being advocated. In the third world, the popular policy prescription of dismantling of "solidarity function", which rides the wave of deregulations of the 1980's in the West, is at work with fury. Aside from its hardships, it does not seem to have a solid theoretical foundation. A certain amount of "solidarity function" is effective for economic growth. This can be provided by central governments which cherish individual initiatives. As this book is being written, the economies of Thailand and South Korea are threatened to collapse. It is now being recognized that the string of large corporate bankruptcies and bank defaults are due to weak solidarity functions - bad government, under-regulated banking systems, too much "private initiative": insider dealings and bookkeeping practices. It is being recognized that the problem can be solved perhaps internally by reforms enforced by "solidarity function" -
Nonlinear Mathematical Controllability
297
more effective government intervention with perhaps some cushion from other countries. This is the insight the principles provided here. We have isolated "null-controllability" in En as our objective in the analysis of the growth of capital stock. Though zero target at the final time seems artificial, the theory incorporates nontrivial targets. Indeed if x\ is a nontrivial arbitrary target and y(t) =y{t, <> | , w°) is any solution of i
(t)~A_l Ht~h)=AUxnut)
(14.23)
with j^ CT = 4> and w° admissible such thai y{t^ ) = Xj , one can equivalently study the null controllability of the system * (t) -A_x
Ht-h) =fft, zt+y,,ut)
-Myt , w,),
where zff = 0, so that xa = ya = <)> and x(t) = z(t) + y{t). If we can show that there is a neighborhood O of the origin in z space such that all initial <J> e 0 can be brought to z(t\, u, (j), ifl) = 0 by some admissible control at time f, , then z(th u, <j>, u°) = x(th <J>, u) -y(ti) = 0 , so t h a t x ^ , <> | , M) = X ' i ) = *i •
REFERENCES 1. R. G. D. Allen, Mathematical Economics, MacMillan and Co., Ltd, London (1960). 2. K. J. Arrow, Production and Capitol, Collected papers of Kenneth J. Arrow, the Beknop Prec. of Harvard University Press, Cambridge, Mass. (1985). 3. J. A. Burns, Finite Difference Method for Identification of Hereditary Control Systems. Nonlinear Phenomena in Mathematical Science (ed. V. Lakshmikantham), Academic Press, New York (1982). 4. K. Balachadran and J. P. Dauer, "Relative Controllability of Perturbations of Nonlinear Systems," J. Optim. Theory Appl., 63(1) (1989), pp. 51-56. 5. W. H. Branson, Macroeconomic Theory and Policy, Harper and Row, Publishers, New York (1989). 6. E. N. Chukwu, Control of Global Economic Growth: Will the Center Hold. Ordinary and Delay Differential Equations, eds. J. Wiener and J. K. Haole, Longman Scientific and Technical (1992), pp. 19-23.
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I.
Differential Models and Neutral Systems
E. N. Chukwu, "Universal Laws for the Control of Global Economic Growth with Nonlinear Hereditary Dynamics," Appl. Math. Comput., 78 (1996), pp. 1981. 8. E. N. Chukwu, "The Time Optimal Control Theory of Linear Differential Equations of Neutral Type," Comput. Math. Appl. 16 (1988), pp. 851-866. 9. E. N. Chukwu, The Time Optimal Control Problem of Linear Neutral Functional Systems, J. Nigerian Math. Soc. 1 (1982), pp. 39-55. 10. E. N. Chukwu, Symmetries of Linear Control Systems, SIAMJ. Control 12(3) (1974), pp. 436-488. II. E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, Academic Press, New York (1992). 12. K. L. Cooke and J. A. Yorke, "Equations Modelling Population Growth, Economic Growth and Gonorrhea Epidemiology in Ordinary Differential Equations," ed. by L. Weiss, Academic Press, New York (1972). 13. R. M. Goodwin, "The Nonlinear Accelerator an Persistence of Business Cycle," Econometrica 19 (1951), pp. 1-17. 14. K. Gopalsamy, Staility and Oscillations in Delay Differential Equatins of Population Dynamics, Kluwer Academic Publishers, Boston (1992). 15a. J. R. Graef, M. K. Grammatikopoulos and P. W. Spikes, "Asymptotic and Oscillatory Behavior of Solutions of First Order Nonlinear Neutral Delay Differential Equations," J. Math. Anal. Appl. 155 (1991), pp. 562-571. 15b. L. GyOri and G. Ladas, Oscillation Theory of Delay Differential Equations with Application, Clarendon Press, Oxford (1991). 16. O. Hajek, Pursuit Games, Academic Press, New York (1975). 17. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York (1977). 18. Pierre-Yues H£nin, Macrodynamics Fluctuation and Growth, Routledge and Kegan Paul, New York (1986). 19. H. Hermes and J. P. LaSalle, Functional Analysis and Time Optimal Control, Academic Press, New York (1969). 20. R. Isaacs, Differential Games, Wiley, New York (1965). 21. N. Kaldor, Capital Accumulation and Economic Growth, The Theory of Capital New York, eds. F. A. Lutz and D. C. Hague, St. Martin's Press (1961), pp. 177222. 22. N. Kaldor, "A Model of Trade Cycle," Economic J. 50 (1940), pp. 78-92. 23. M. Kalecki, "A Macrodynamic Theory of Business Cycles," Econometrica, 3, (1935), pp. 324-344. 24. L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, MacMillan Co., New York (1964). 25. J. Klamka, "Controllability of Nonlinear Systems with Distributed Delays in Control," Int. J. Control, 31 (1980), pp. 811-819. 26. G. Knowles, An Introduction to Applied Optimal Control, Academic Press (1981).
Nonlinear Mathematical Controllability
299
27. A. Manitius, "Optimal Control of Hereditary Systems", Control Theory and Topics in Functional Analysis International Center for Theoretical Physics Trieste International Atomic Energy, Vienna (1976). 28. M. McElroy, The Macroeconomy; Private Actions, Public Choices and Aggregate Outcomes, Macmillan Publishing Company, New York, forthcoming. 29. A. W. Mullineux, The Business Cycle after Keynes: A Contemporary Analysis, Barnes and Noble, New Jersey (1984). 30. A. S. C. Sinha, "Controllability of Large Scale Systems with Distributed Delays in Control and State," Proceedings of the 1988 International Conference on Advances in Communication and Control Systems, eds. A. Porter and Sabhash C. Kah, Baton Rouge, LA, II (Oct. 19-21, 1988), pp. 817-828. 31. R. H. Strotz, J. C. McAnulty and J. B. Naines, "Goodwin's Nonlinear Theory of the Business Cycle: An electro-Analog Solution," Econometrica 21 (1953), pp. 390-411. 32. C. Tadmor, "Functional Differential Equations of Retarded and Neutral Type: Analytic Solutions and Piecewise Continuous Controls", J. Differential Equations 51(1984), pp. 151 -181. 33. A. Takayama, Mathematical Economics, 2nd Edition, Cambridge University Press, Cambridge (1984). 34. R. G. Underwood and D. F. Young, "Null Controllability of Nonlinear Functional Differential Equations," SIAMJ. Control 17 (1979), pp. 753-772.
IS. Oscillation
15.1 Introduction In this section we derive a scalar equation of gross national product, y and of Labor, L. Each is a linear functional differential equation of neutral type. For these scalar equations conditions are stated for oscillation and for nonoscillation. The basic research is contained in [23] and [24]. These conditions are applied here to the economic model and the implications explored. Also treated is the (M) six-dimensional economic state. Necessary and sufficient conditions for oscillation are stated. Definition 15.1
A real valued function x defined on a half-line (0, oo) is said to be oscillatory, if there exists a sequence {tm} -> oo as m -> oo such that x(tm) = 0 (m = 1, 2, 3,...) and x is said to be nonoscillatory, if there exists a T e [to, oo) such that \x(t)\ > 0 for t > T. Definition 15.2 Solution x(t) = (x\(t), ..., Xr^t))T is said to oscillate if every component x^t) of the solution has arbitrarily large zeros, otherwise the solution is called non-oscillatory. Thus a solution oscillates if it oscillates componentwise and is non-oscillatory if at least one component is eventually positive or eventually negative. The basic results on oscillation are introduced.
301
302
Differential Models and Neutral Systems
15.2 Basic Results Theorem 15.1 Consider the scalar equation
x(t)-d(t-h)
=
(15.1)
-^PiX(t-h).
Assume that the real numbers c, h, h\, ..., hm, p\, ..., pm satisfy 0 < c < 1; h>0, 0
>0, /= 1, 2,...,/wand \
m e > 1 - c\ 1 + h /
\i=\
J
(15.2)
pj
i=l
Then all nontrivial solutions of (15.1) are oscillatory, see Gopalsamy [12]. Theorem 15.2 [5, p. 124]: Consider the system of neutral delay differential equations
dt
xM + j^PjxQ-hj) y=i
(15.3) <=i
where Pj, Qi are m x m matrices and the hj and a, are non-negative real numbers. Associated with (15.3) is its characteristic equation -lh,
det cp=l
where / is the mx m identity matrix. i= 1,2,..., n Pj,QeEmxm
-Xu. :0
hj e(0,oo)
Then the following are equivalent
'
(15.4)
i=l
Assume that for j = 1, ..., £,
and
CJ, e(0,oo).
(15.5)
Oscillation
303
(a)
Every solution of (15.3) oscillates componentwise.
(b)
The characteristic equation (15.4) has no real roots.
Theorem 15.3 Assume thaty = 1, 2, ...(. and for i = 1, 2, ..., n Pj, Qi eE
, hj, at e £ .
Let u 0 be real. Then the following are equivalent. (a)
Every solution x(t) of equation (15.3) with Lyapunov exponent .■ ln||jc(0|| r . \x[x] = hm sup —" w " < |LI0, <->oo
(b)
t
oscillates. The characteristic equation (15.4) has no roots in (-oo, uo].
Corollary 15.1: Assume (15.5). Every bounded solution of equation (15.3) oscillates if and only if the characteristic equation (15.4) has no roots in ( - » , 0]. Theorem 15.4 Consider the scalar neutral delay differential equation Jt[)it) + py(t-h)} = -qy(t-a),
(15.6)
where peE,
h,q e(0,oo),
oe[0,oo).
Lety(t) be a non-oscillatory solution of (15.6). The following are true
(15.7)
304
Differential Models and Neutral Systems
(a)
p<-\
if and only if lin | y(t) | = oo.
(b)
I f / ? * - l and —?— (a -h)<~. 1+ p e
(15.8)
Then every solution of equation (15.6) is oscillatory.
15.3 Economic Model We use the formula for aggregate demand in [23] and [24]: z=C + I+X+G= (z0(t) + Go(t) - cT0(t) + u><0 + g(y(t - h))
r
\Po
where the formulae for C, I, X and G are simple and can be estimated by regression. Then we deduce the dynamics of GNP, y as ^ P - kg(y(t - h)) = - a0y(t) + axy(t -h) + u(t) + q(t).
(15.9)
Here X, QQ, a\ are constants: ao=X(l + - 9 ^ - | i ) , v
(15.10)
«1 = *P,
(15.11)
a_l =<**,!.
(15.12)
The following formulae are used: C = C0+c(y-T) T=T0+tiy,
0
(15.13) (15.14)
Oscillation
305 X = Xo - my - e\R, G = GQ+G\
0 <m<\,
=$y(t-h)
e[ > 0
+ G02
(15.15) (15.16)
/=/o(0 + /iO),
(15.17)
I\ = a-xy(t - h),
(15.18)
and u=c(l-/i)-m,
0
0
0<m<\.
m = nominal value for money p - price level L
=j +
ky-vR,
P R =--
dt
(15.19)
interest rate
= -X.(3<0-z(0),
(15.20)
A, = constant, the speed of response when output is lagging behind the demand z. Fluctuations of income (GNP), y can occur when y satisfies (15.9). Indeed, Gyori and Ladas [5, p. 154] have proved: Theorem 15.5: If (15.9) is considered and if we assume
(i)
«(/) + g(0 = o
(ii)
a_\, QQ, - a\,h are non-negative real numbers such that
(iii) a_ 1 £l,fl_ 1 (fl t o -2a,)-fl 1 >—. eh
Then every solution of (15.9) oscillates.
(15.21)
Differential Models and Neutral Systems
306
(See Theorem 15.1.) The following arguments can be made, by decreasing the value of - a\ = - xp (and this can be done by decreasing the numerical value of P, i.e., decreasing public consumption), the second inequality in (iii) can be destroyed. This will remove oscillation. Oscillation is guaranteed by (15.21). We now examine policies that will negate this inequality and can terminate oscillation and render (15.9) nonoscillatory. 1. a.\: Because a smaller a.\ can render (15.21) invalid and because a.\ #
= a,X
*
where o_j is the acceleration coefficient of the private investment *
*
function g(y(t - h)) = a_\y(t - h) a smaller a_j can make the system nonoscillatory. 2. The speed of response X, when the output y is lagging behind the demand z,
can be decreased to terminate oscillation. In this case, the time constant of the lag 1/A. is increased. 3. OQ: Since a0 = ^ l + - ^ - + (« + c(/i-l))J, 0 < c < 1, 0 < t\ < 1, 0 < e\, 0 < m < 1, we can decrease «o by increasing net export X = XQ -my- e\R, which can be achieved by decreasing real rate of interest, (e\), and decreasing purchase of foreign goods, m. Clearly «Q c a n be made smaller by increasing the numerical value of the negative number c(t\ - 1), i.e., increasing c the marginal prosperity to consume after taxes income. 4. k: Since the real demand for money (L/P) varies directly with income and k as the constant of proportionality, a$ can be reduced by decreasing k, the proportion of money held. 5. r. Because (L/P) (in (15.19) is our "demand for money," our desire to hold some average level of money (currency and checking account)
Oscillation
307
declines with the rate r of interest R. A careful balance (between e\, k, and r) has to be maintained in decreasing e\klr. The last important government instrument is contained in a\: 6. a| = Xp. In (15.16) government outlay, GQ\ = $y(t - h), can be decreased by decreasing (3. Thus, a net government outlay - decrease in government spending on consumption - can destroy oscillation or fluctuating income stream. Of course, this has to be done in a very responsible way, by either spending cash at hand or borrowing within limits defined by future income levels. If B is borrowed, then the capital account deficit Bk = B - GQ2 = GQ\ - TQ< 0. In our model, GQJ - v(t) presupposes public investment which can be self-financing. The policies now outlined in (1-6) can destroy the oscillation in the system y(t) - a_xy{t -h) = - a0y(t) + axy(j - h). When j{t) = u(t) + g(t) is introduced, one can avoid oscillation while maintaining growth by assuming properties on/, which will ensure that the system is nonoscillatory (see Theorems 3.2-3.4). The strategy can be constructed optimally. A theory of optimal control of the growth of income and of employment has been constructed. It opens up the possibility of using minimum investment to steer growth to the desired target as fast as possible. From some basic postulates of economics and by the methods of control theory and pursuit games, some laws have been deduced. With Professor Paul Samuelson [1] we have the following sentiments: "whether or not the result is of some interest is an empirical question that cannot be decided by logical manipulations ... a priori constructs, definitions ... ." In Chapter 9, we presented how close to the real world our growth model of income and employment is for each nation of the world. We used data available from the United Nations and from each nation studied.
15.4 Employment We now consider employment oscillation where labor dynamics is described by L(t) + a_{L(t-h) = a0L(t) + alL(t-h) + p(t) + g(t),
(15.22)
308
Differential Models and Neutral Systems
where a.\, an, a\ are derived in [24, 1.39b] with
v_Yl
v (. °~X
Hoh V
HohJ
-i^-X-s)"' ao=
{^m)v-'th} '
-i^-)('-^r' _>(i-g), *3
l
a4=
M
flbA JV
//oA
afl-cfrV
l":i^1)(1
v V
ft^
1
^feX -"^) ' -1
1
^"(rtfoJl '^) «7=«l»-^r] • ^OA; where B(/) = a(l - c M O - *0*(>) + vv + e ,
O5-23)
Oscillation
309
is the rate of deliveries of new equipment and e(l - c)y(t) is the rate of savings. All these constants can be identified by regression. We state conditions which will guarantee that employment is nonoscillatory and will grow without bound. Theorem 15.6 [6, p. 379]: Assume that (0 (ii)
/>(0 + g(0 = 0 for all / > 0 ; (15.24) a_], OQ, ax are real numbers such that
Then every nonoscillatory solution of (5.22) tends to + oo or - oo, as t-> oo. Remark: Since it is reasonable to assume that L(i) > 0 for all t, Wudu's theorem gives condition for unbounded growth for nonoscillatory solutions. It is an easy consequence of a Theorem 15.6 of Lalli [7, p. 510] that if 0<
and
tf0=0.ai>0,
(15.25)
then (15.22) always has an unbounded positive solution, and every bounded solution of (15.22), either oscillates or tends to zero as / tends to infinity, provided that (15.24) holds. The condition (15.24) seems too restrictive. We assume that it is not valid. Then the following is true. Theorem 15.7 [7]: Consider (15.22) and assume that there exists a function /such that F(t)= p(t)+git) = /'(/) and lim sup/(/) = + oo, lim inf/(0 = -°°r->oo t-+co Suppose a-\ >0,/i>0, CIQ =0, and F is continuous on [0, oo). Then every bounded solution of (15.22) is oscillatory.
Differential Models and Neutral Systems
310
It is always assumed in some schools that government will always "mess up," causing possible oscillation in the economy. The following remarks provide a partial elucidation to the problem so raised. Suppose the homogeneous equation L(t) + a_! L(t -h) = a0 L(t) + axL(t-h),
(15.26)
is nonoscillatory. (We discuss shortly policies that will bring about this.) The introduction of fit) = q(t) + pit) can provoke oscillation to the system L(t) + a_lL(t-h) = a0L(t) + alL(t-h) + f(t).
(15.27)
Indeed, we have; Theorem 15.8 [8]. Let (i)
L(t) + a.\L(j-h)
be stable,
(15.28)
(i.e.,|a-i|
jy(t)f(t)dt
= 0,
(15.29)
0
for all w-periodic solutions^ of the adjoint equation y(t) - y(t + h)a_\ = - y(t)aQ - y(t + h)ax.
(15.30)
Hale's theorem gives conditions on/which will initiate government inspired oscillations in a nonoscillatory economic system. In section 1 we deduced that the dynamics of x = [y, R, L, k,p, E] is modeled by x(0 - A_xx{t - h) = A0x(t) + AlX(t -h) + p{t) + q(t).
Oscillation
311
Theorem 15.1 gives condition for oscillation. It will be interesting to give these conditions in terms of A.\, AQ, A\. This will help state economic conditions for oscillation or nonoscillations. See Theorem 15.3.
15.5 Policy Implications of Damping Out Employment Oscillation The dynamics of employment is given in 15.22 where the coefficients are given (15.23). From Theorem 15.5 consider the required condition a_l
a _ i ( a 0 - 2 a i ) - a i >—. en
It is possible to break the second inequality and perhaps stamp out oscillation if a.\ is decreased. Recall that u = c(l - fi) - m Ho = \\+-^-
+ m+ c{t\
-\\\,
and v ( 1 Hoh U - v / Hoh v Hoh-v' Here v is the coefficient of the growth rate of GNP which is a component of the rate of deliveries of new equipments. The derivative of a.\ with respect to v is Hoh
2 >0. {Hoh - v)
312
Differential Models and Neutral Systems
Thus a.\ is increasing as v increases. Smaller v and higher growth rate of GNP makes businessmen want to invest more which increases productivity. Faster GNP growth rate bolsters wages and coaxes more people into the labor market. Thus the innovation of new equipment, etc. will increase productivity and employment and damp out labor oscillation. This may contribute to higher rates of sustainable growth. Oscillation can also be broken by decreasing
This can be achieved by decreasing the rate of savings a{\ - c) or by increasing Hoh - v. Increase Hoh - v:
(i)
Decrease v the coefficient of speed of rate of national income growth. (ii) Increase Ho: decrease c{\ - t\), i.e., decrease c the propensity to consume after tax income. (iii) Increase tax rate t\. (iv) Increase m the fraction of income spent on foreign goods. Increase
—-: r
k = proportion of money held, ej, the real rate of interest, r rate of interest Recall that //o = [ l + — ^ +
m-c(\-t\)).
a(l - c) + kftHoh X
~" ~
Hoh-v
Oscillation
313
Decrease - a\ by decreasing k$ - the fraction of new stock of capital assets equipment which is decided to invest. If h is very small, oscillation can be broken; i.e., if h, the gestation period before deliveries are made, are smaller, oscillation may be prevented. The analysis we have carried out here are rudimentary, and linear. Perhaps a nonlinear analysis may be more fruitful, but the method above can be imitated.
REFERENCES 1. P. A. Samuelson, A Universal Cycle? Oper. Res. Verefahren 3 (1967), pp. 307320. 2. W. H. Branson, Macroeconomic Theory and Policy, Harper and Row, New York, 1989. 3. M. McElroy, The Macroeconomy: Private Actions, Public Choices and Aggregate Outcomes, Prentice Hall, Upper Saddle River, NJ, 1996. 4. E. N. Chukwu, Mathematical Controllability Theory of the Growth of Wealth of Nations, Proceedings of the 1st World Congress of Nonlinear Analysts, Tampa, FL, August 1992. 5. I. Gyflri and G. Ladas, Oscillation Theory of Delay Differential Equations with Application, Clarendon Press, Oxford, 1991. 6. Lu Wudu, The Asymptotic and Oscillatory Behavior of the Solutions of Higher Order Neutral Equations, J. Math. Anal. Appl. 148 (1990), pp. 378-389. 7. B. S. Lalli and B. G. Zhang, Oscillation and Non-Oscillation of Some Neutral Differential Equations of Odd Order, Internal. J. Math. Math. Sci. 15 (1992), pp. 509-515. 8. J. K. Hale, Oscillation in Neutral Functional Differential Equations, in Nonlinear Mechanics C.I.M.E., June 1972, Edizioni Cremonese, Rome, (1973), pp. 99-111. 9. J. K. Hale, Theory of Functional Differential Equations, Springer- Verlag, New York, 1977. 10. J. R. Graef, M. K. Grammatikopoulos, and P. W. Spikes, Asymptotic and Oscillatory Behavior of Solutions of First Order Nonlinear Neutral Delay Differential Equations, J. Math. Anal. Appl. 155(2) (1991), pp. 562-571. 11. E. N. Chukwu, Stability and Time-Optimal Control of Hereditary Systems, Academic Press, New York, 1992. 12. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Boston, 1992. 13. Hajek, Pursuit Games, Academic Press, New York, 1975. 14. A. W. Mullineux, The Business Cycle After Keynes: A Contemporary Analysis, Barnes and Noble Books, New Jersey, 1984.
314
Differential Models and Neutral Systems
15. E. N. Chukwu, The Time Optimal Control Problem of Linear Neutral Functional System, J. Nigerian Math. Soc. 1 (1982), pp. 39-55. 16. E. N. Chukwu, The Time Optimal Control Theory of Linear Differential Equations of Neutral Type, Comput. Matah. Appl. 16(10/11) (1988), pp. 851866. 17. J. A. Burns, Finite Difference Methods for Identification of Hereditary Control Systems, in Nonlinear Phenomena in Mathematical Sciences (V. Lakshmikanntham, Ed.), Academic Press, New York, (1992), pp. 169-178. 18. E. N. Chukwu, Optimal Investment Strategy for Time Optimal Income Growth of Nations, Preprint. 19. T. S. Angell and A. Kirsch, On the Necessary Conditions for Optimal Control of Retarded Systems, Appl. Math. Optim. 22 (1990), pp. 117-145. 20. R. M. Goodwin, The Nonlinear Accelerator and Persistence of Business Cycle, Econometrica 19 (1951), pp. 1-17. 21. R. G. D. Allen, Mathematical Economics, MacMillan, London, 1960. 22. N. Kaldor, Capital Accumulation and Economic Growth, in The Theory of Capital (F. A. Lutz, Eds.), St. Martin's Press, New York, (1961), pp. 177-222. 23. E. N. Chukwu, Mathematical Controllability Theory of the Growth of Wealth of Nations, Japan J. Indust. Appl. Math. 11(1) (Feb. 1994), pp. 87-111. 24. E. N. Chukwu, "Optimal Control of the Growth of Income of Nations," Appl. Math. Comput. 62 (1994), pp. 279-309.
16. Construction of Econometric Meter and Reform of Global Economic Systems Structure In "Recent Advances in Modeling and Optimal Control of the Growth of Wealth of Nations, with Austria, Australia and Russian Examples", we postulate various formulae for consumption, C, Investment, /, net export, X, and Government, G, as components of aggregate demand, z. The past history of income and the rate of change of income, y, and delays in the values of other economic variables enter the formulae. From the familiar economic principle of rational expectations and the principle that the rate of change of income, y, is proportional to z{t)-y(t) we derive a hereditary dynamic economic model, a neutral dynamic game of pursuit with the state vector consisting of GDP, Interest rate, Employment, Value of Capital Stock, and Cumulative balance of payment. Different dynamical systems are obtained depending on the postulated formulae. As a dynamic game of pursuit the quarry control is government investment, taxes, exchange-rate, tariff, preferential trade arrangement and money supply and its flows. The pursuer consists of labor productivity, wage rate, autonomous consumption, autonomous investment, autonomous net export, and autonomous real money demand. Conditions are given for the state economic variables to be controllable from one initial function to a terminal function. In particular, employment, national income, prices (and therefore inflation) and balance of payment can be controlled from one arbitrary vector to another when the controls of the systems are subject to scarcity, i.e., to constraints. Conditions for constrained controllability are presented. We show that high growth of GNP, and full employment are compatible with low inflation (or even deflation) and zero (or surplus) balance of payment, provided government control instruments and the private firms' controls are appropriate, and are fully effectively used. The economic model is confronted and tested with data from Austria, Australia, and Russia. The parameters are estimated witJi the state at equilibrium, the coefficients of the components of aggregate demand are then known. This serves as a "first guess" in a dynamic model. We 315
316
Differential Models and Neutral Systems
estimate parameters in a dynamic model, and then study the optimal control of the economic model. With the constructed model as a tool, we propose a way to now redesign national economic systems to prevent unwanted crises, by using the model to serve as 'national surveillance' for needed structural reforms and effective fiscal measures which will guarantee optimal growth of the wealth of nations. This is accomplished as follows: In recent papers and books [3, 12-16] the author derives hereditary differential games of pursuit as realistic models of the economic state of nations. Their numerical solutions (with MATLAB) have been proposed [17] for such systems with identified coefficients. The models use regression methods and assume Fair's type of "rational expectation", and market principles. Inspired by the spirit of much earlier works of Goodwin and Kalecki, Smith and Erdley [19] Tustin [21] and Phillips [18, 19] we construct electronic circuits for electronic analog computers or simulators. The models of the economic state are themselves analogs of the current and voltage equations in lossless transmission lines, tunnel diode circuit, equations derived from Kirchoffs current law with phase shifting mechanisms or even simpler circuits. The solutions of the hereditary systems (both linear and nonlinear) for the economic state x = [y, R, L, K, p, E] which are measured as current and voltage in the electronic models are solved by MATLAB programs and displayed in a monitor. The control strategies of government are devices for impressed voltage designed to mirror accurately the vector policy g = [T, go, e, x, d, Ml, M\',fd\ which include monetary and fiscal policies. The aggregate private reactions to government intervention is a stroboscopic strategy a which include wages and productivity: a = [Co, Io, Xo, Mo, n, w xo, yo, po]. An impressed voltage is designed to capture this. The combined resultant control strategy u lies in the so-called Pontryagin Difference of sets, a control device represented by E(t) = u{t). With u and a formal agreement of similarity between the equation of economic state and the current in the electronic meter, one can simulate and try out quickly many alternative hypothesis, and examine their consequences. The prototype device is being assembled. In the models we have constructed for various countries (e.g., Austria, Australia, Japan, U.K., USA, Canada, Germany, Nigeria, South Africa, etc.), good and complete financial data are needed and should be available. With this one constructs a linear or nonlinear model,
Construction of Econometric Meter
317
dldt[x{t) - g(t, x(t - h)} = / ( / , x(t), x(t -h) + p(t) + q(t), where x = \y, R, L, K, p, E], a state vector of GNP, Interest rate, Employment, Value of capital stock, Prices and Cumulative balance of payment. The control strategies of government consists of government spending (or investment) taxes, exchange rate, tarriff, preferential trade arrangement and money supply and its flows. The private firms controls are labor productivity, wage rate, autonomous real money demand, autonomous investment, autonomous net export. Various control strategies are incorporated in the dynamics which are solved and displayed in a graphical form. Computer programs are available in MATLAB for computing such numerical solutions. The cited book [17] contains such methods. The "predict" command can be used to display future economic trends. Looming crises can be detected and controlled. Thus the device can serve as a "surveillance" machine for the economic state, a warning for invading oscillation, depression or good growth. The econometric meter is the 'computer' that executes this. To build the model and have the meter performed the collection of reliable economic data and the disclosure to a world institute that builds the model is critical. The computation is best carried out in an independent world institute where the analysis and policy implications are highlighted. Thus a nation's economic reporting must be improved. Equally important is the openness of the institute that builds the model, openness to the world. Should the body recommend a policy to a nation, it should display this model and its dynamics, the crises it is about to correct, and how the model corrects it with the control policies recommended. Preferably all these should be displayed to the leaders of the nation in computer graphical forms which are checkable in various computer languages. Suppose depression is predicted by the model, or the economic system is seen to be uncontrollable. Suppose it is necessary to "reform", i.e., to modify the economic system's structure, (A_i, AQ, A\,B\, B2) in d/dt[x(t) - A_ix(t - h)] = AQXU) + A]X(t -h) + Bxp(t) + B2q(t). This can be done by a juducious choice of «(/) = Bp(t) + Cq{t)
318
Differential Models and Neutral Systems
u(p, q, x) = A_x xdldt{x(t) + 4o,*(r) + Ax {x(t -h) + Bx lP(t) + C, #(/)), so that the resulting system
i(0-(^_i+^-ii)^-^)=(^o+^oiWO+(^+AiW/-A)+Sp(0+c?(0 is controllable and nonoscillatory. The economic meaning of this strategy can be deciphered and unraveled from the derivations. The assumption that p and q can be chosen effectively rests on the moral values implicit in the ethics of cooperation both national and international. The mathematical basis is already outlined in Chapter 1 of the book, "Optimal Control of the Growth of Wealth of Nations". There it is argued that the social environment for economic activity is based on a mathematical theory of competition and cooperation. The dynamics include ordinary differential equations, function differential equations, or parabolic partial differential equation. From existing mathematical research, competitive systems have either (at best) bounded dynamics or tends to extinction in finite time. For cooperative systems, every solution converges to a positive equilibrium or leaves any closed and bounded set. Thus solutions of cooperative systems can either grow unbounded or at (worst) if bounded will converse to a positive equilibrium. There is no extinction; there is the possibility of "stupendous orgy of
R<< gM
\
—i—^
E —
1
FIGURE 1.
».
319
Construction of Econometric Meter lossless transmission line
1 i » 9(v)
C t © l(t)
x=/
x=0 FIGURE 2.
lossless transmission line »»•
i = g(v)M
C,© KO
0E(t) x=/
x=0
FIGURE 3.
320
Differential Models and Neutral Systems
mutuality," an unbounded growth of all concerned. Person x gives himself in service to y and y devotes his effort and his service to x. As a result both grow unbounded. With this sort of ethics, private investors will bear the good of all in mind, and if they make bad decisions effecting the good of all, they should not expect a big brother to bail them out to the ruin of the common good: they will fully bear the consequences of their actions. They will put the "life of the people before debt". They will help "put these countries back on their feet without sacrificing the health and education needs of a whole generation" (Jerry Fitleau, NC Catholic, Vol. 52, No. 16, April 26, 1998). They will support to link debt cancellation with investment in human development, health, education, and food production. In this way and in finite time investors will reap a sustainable rich harvest of profit and an unbounded growth for themselves and families and the country of residence. What is needed is greater openness in statistics, in central and commercial bank dealings, for "corruption free - government practices," and for growth built on sound policies that enjoy public confidence". All these are achieved in a cooperative system. Technical details of the construction of the models, the simulation of the consequences of various economic control strategies and the design and role of an economeric meter which can be used by individuals and nations to obtain better policies can now be explained.
16.1 Design of an Econometric Meter or Econometric Simulator In Chapter 1, we derived an economic model for an isolated nation. But a nation is interconnected with others in a well-defined but complicated fashion whose model is an interconnected composite system. The dynamics is an isolated functional differential equation of neutral type, or an interconnected large scale system of neutral type. The models can be simulated by the flip-flop circuits in Figs. 1 and 2. The equations are obtained by the conversion of the partial differential equation satisfied by the current, I, and the voltage, v, into a functional differential equation. The dynamics can be displayed on T.V. monitor when the private and government strategies are supplied. The interconnection, the so-called "solidarity function," is the external economic intervention from outside the country, the I.M.F., the world bank,
Construction of Econometric Meter
321
external aid, etc. They are denoted by [g\\ --- gnol = k(t, xt> v(t))x=[y,R,k,L,p,E] v = v(p, q). The intervention can be viewed as functions of GNP, Investment, I, net export, X, and Government, and Consumption. With K. fed into the system, some effect is produced in the dynamics, hopefully to produce what is required. This can be T.V. monitored. In this way the design can be viewed as an econometric simulation which are operated by one or more computers. Given the past history of the key economic variables, and the initial conditions, the government strategies and the firms reactions, the computer can determine the solutions of the model and what the economy will do under the given conditions including the action of external solidarity. The dynamic characteristics of the economy must be accurately modeled, and the differential equations solved during the operation of the simulator. Various scenario's can be explored and analyzed. Remark. It was pointed out by Slemrod [7, p. 388] that the flip-flop circuit in Fig. 1 is the basic element in a digital computer which can also operate as a memory device provided the single equilibrium (v*, I*) of (16.11) and (16.12) are globally asymptotical stable. A simple model of an econometric meter or economic simulator can now be described and constructed. It consists of the lossless transmission line connected as in Fig. 1, p. 264. Here g(v) is a nonlinear function of v and gives the current in the indicated box in the direction displayed. It is well known (see [2, p. 5]) and described by Hale and S. M. V. Lunel [2, p. 5] that the following system of partial differential equations can be deduced from physical principles: . di dv dv di L—= , C—= , dt dx dt dx The boundary conditions are E - v(0, 0 - Ri(0, t) = 0,
0<x
Q^ ±
r>0.
, u n (16.1)
= /(l, /) - g(v(l, /)). (16.2)
at
Presumably E is an electric generator (battery or device having an emf of E
Differential Models and Neutral Systems
322
volts). It will serve as a control device. The resistance is a resistor of R ohms. In the indicated box the capacitance is given by C\ farads, etc. Set 1/2
s = (LCy"\
zJ±
The general solution of the partial differential equation is seen to be v(x, t) = <|>(x - st) + VJ/(JC + st),
i(x, t) = - [<|>(* - st) - \\i(x + st)]; z
or 2$(x - st) = v(x, t) + zi(x, t),
2v|/(x + st) = v(x, t) - zi(x, t).
Let i
k=
z — R
-, z +R
a =
IE
-, z+ R
«(0 = v(l,0,
and observe, with the aid of the first and second boundary conditions, that the partial differential equation can be converted to the equation,
4[«(0-A»('-2/*)] = ^ - ( C / I ) , / 2 « ( 0 dt
R +z
<£zJ2].(£V'V ««*«,. z+R J
(16.3)
yL)
Let the delay equal h = 2(Z,C)1/2. Then a linear differential neutral system is deduced:
±Ht)-m-h))=2^ dt
R+z
rc,U2
1 1 »"»-
( . -R T.\
z + R)
(Q^11
u(t-h), L)
(16.4)
Construction of Econometric Meter
323
or
4 [«(/) - A-\u(t - h)] = 4o«(0 + 4«(> - A) + g(0
(16.5)
where, A_x=k
z-R z+R
= ((I/C) 1 / 2 -R)/((L/C)l/2
+ R)
AQ=-(C/L)X/2,
(L/C) 1 ' 2
fCNl/2
((Z,/C) l / 2 + /?)
«M-
" %
((L/Cy,l
■
(16.6)
+ R)
The values of A_\, AQ, A\ can be fixed, constructed and inserted in the circuit by engineers. A carefully constructed generator, or source of power can be made to generate E{t) and therefore a g(t). With the personal help and communication of Michael Tom, we give the details of the derivation from di
dv
—= dt
_, dv
, dx
di
C —= dt
,
0<x
dx
with boundary conditions E - v(0, /) - Ri(0, 0 = 0 e , ^ ( l , / ) = /(l,0-g(v(l,0) at
f>0
324
Differential Models and Neutral Systems
for (16.5). Differentiate the second equations with respect to t and use the first
dt1
dxdt
v
tt = *2vxx
w
L fa* s2 = — .
'th
This is the wave equation. Using the canonical variables h,-
X - St,
T) = X - St
one can rewrite (16.7) into v^ = 0
(16.8)
which has the general solution vfo Tl) = v|/(4) + \|/(il)
or V(JC, /) = <}>(x - ^ ) + \\i(x + st).
(16.9)
From the first equation 9/_ dt
dv dx
or
L — = -[y'(x-st) 9/ We now integrate with respect to /
+ \v'(x + st)].
(16.10)
Construction of Econometric Meter Li(x, t) ■■
325
- y(x - st) — i|/(> + st) s s
or i(x, t) = -—[lyO - st) - M/(x + st)]. Ls But Ls = L-
4-
(16.11)
so that
i(x, t) = ~[\\i(x - st) - \\/(x + st)].
(16.12)
z
We obtain \\>(x - st) + \\i(x + st) = v(x, t),
\]i(x - st) - \\i(x + st) = zi{x, t) .
Adding this and subtracting we deduce - st) = v(x, t) + zi(x, t),
(16.13)
2y(x + st) = v{x, t) - zi{x, t).
(16.14)
2VJ/(X
In (16.13) take JC = I and replace / by / + l/s to obtain 1
2\j/(-s/) = v| 1,/ + s) + Zl]l,t + -
In (16.14) take x = 1 and replace / by t - Ms 2V|/(J/)
= v(l, t-\ls)-
zi(\,
t-\ls).
Note that
v(o,o = -[v(-rf)-v(*0].
Differential Models and Neutral Systems
326
i(0, r) = - [\|/(- st) z
y(st)].
Thus the first boundary condition becomes E - [ y ( - st) + y{st)} - - [v|/(- st) - Wist)] = 0, z or E — 2
[,/ + -
+ Zl
V
sj
+ v| 1 - / + - | - zi \,t — V
R_ 1,/ + - + z 1,/ + - -v|l,/--° 2z V sj Sj ^ s) /
E — v(l,t) + zi(l,t) + v 1,' —A -zi V s) 2 2_ v{\,t) + 2z
zi(\,t)-v
f + Zl
f
2"| 1-" L s)
t - - \ + zi \,t —s
Rewrite the last equation to obtain i(l,0-** l , / - - | = a - ^ v ( l , / ) - - v l l , f ^ s) 2 z v where , z-R 2E k= , a = z+R z+R Finally letting
sj
^
\,t--
\\
=0
Construction of Econometric Meter
327
«(0 = v(i,o, ti(0 = - v ( U ) dt and using the second boundary condition, (
u{t) - ku t
2^
= _v(l,0-*Tv dt dt
U
/
-^['(U)-g(v(U))]-^
c,
c,
J_ a
u—
-g
V 1,/
—
2 / 1 -g(v(l,/)) + *g /-v^1 , / — v(l, /) — v , z z vU - -
1
^ * r 2i
c, a - - u ( / ) - - w / - z
z ^
1
/x * f
z
z ^
f
■g(u(t)) + kg \u
2
^
2^
s)
2
^ -g(w(/)) + *g rt-~ 2V
s,
(16.15)
/(«(/),«(*-2/J)).
Equation (16.3) is deduced. We now relate this to an economic dynamics of a national economy. Our aim is to monitor the dynamics of the gross national product under the action of private and government controls. To do this we appropriate equations (4.15-4.19) of [1] which are derived from the definition of aggregate demand: Z = l + C + X + G.
(16.16)
We now consider this to be our economic dynamics of a nation whose aggregate demand is postulated to be
328
Differential Models and Neutral Systems Z = I +C + X + G,
which is given by z = z0 + zxy{t) + z2y(t - h) + z3y{t) + z4y(t - h) + IX3M - CxT(t) - C2T(t -h)~
C3t(t) - C4t(t -h) + Xx6x(t) + XX5e(t)
+Xl7d(t), (16.17)
as derived in (4.15-4.19) of Italy.2 [1] where we have ignored, interest rate value of capital stock, prices, employment and balance of payment. This simplification is still meaningful see Allen [11, pp. 252-254] and Chukwu [8, p. 1180]. The simpler version is good for ease of presentation here. Thus Z = I + C + X + G, where we define ZQ = h + c0 + C-JMQ + g0 + XQ,
ZX = IX+CXZ2=I2+C2-
IX3M0 + C 7 M, hlM2
+gx+Xx,
+ X2 + S2C7M2,
Z 3 = - 7 3 + C 3 + g3 + X3 , Z4 = I4+C4
+ g4 + X4<
and Z = z0 + zxy(t) + z2y(t - h) + z3y(t) + z4y(t -h) + IX3M- CxT(t) - C2T(t -h)Thus
C3T(t) - C4t{t -h) + XX6x(t) + XX5e(t) +
Xxld{t).
Construction of Econometric Meter
329
MO = A.,(2 -
y(t)),
dt
or ^p-
- X^yit
-h) = z,(X, - l)y(t) + Xxz2y{t - h) + V 3 X O
+
*0*1
+ X,/13A/ - (C,7X0 + C 2 r(r - A) + C37-(/) + C4t(t - h)) + Xl6i(t) +
Xl5e(t) + X^d(t)
(1 - X&) ^ - V ^ O - A) = M(z, - l)y(t) + z2y(t -h) + z0 + InM at + Tl(t) + Xl6x(t) + X]5e(t) dy±t) _ _ ^ 4 _ ^ dt
_h)=
1 - A,]Z3
Xi
[(z(
_{
X^d(.t)}, ) m
+ z^((
_h)+ZQ
+
IuM
(1 - /-iz 3 )
+ go + 7i(0 + Xl6z(t) + Z 1 5 e(0 + * 17
(16.18)
The equation of the gross national product is given by Ml
-a_ny(t _ h)
= fll0>,(,)
+ fl,,^ _ /,) + fl(/) + 9 l ( / ) , (16.19)
at where «-ll =
°01
A.,z4 1 - X,z 3
*l(*l ~ 1) 1-V3
330
Differential Models and Neutral Systems \\z2
«11
l
Pl(t) = zokia
=X\a
1 - X,z 3
1
[C0 + I0 +
X0_M0Vi34C7)],
91(0 = r-^— {in** + 7i(0 + *ie*(0 + *is«<0 + *i7<*(0), 1 - A.1Z3
with z
0 = 70 + C 0 + C 7 w 0 +
x
0»
Z! = /1 + Q - / ! 3 W i + C7A/1 + A", + g , , Z2 = /2 + C2 - /13A/2 + Xj + C 7 A/ 2 + g2 > z 3 = - 73 + C 3 + JT3 + g 3 , Z4 = /4 + C4 + 54 + X 4 .
Also 7*0 = - ^ - , I-X1Z3
9(0 = — T — (7i3A/ + r,(0 + X16x(0 + g0 + *15*(0 + Xxid{t)), (16.20) 1 - A.(Z3
where T2 is defined by 7i(0 = cxT(t) - c2T(t -h)-
c2t(t) - c4t( - h).
(16.21)
Suppose our aim is to monitor the flow of the growth or decay fluctuations of GDP under the action of private (p(t)) and government controls g(t). Suppose we have accurately identified the coefficients of equation (4.15) by the method of [1]. Then analogously, in the circuit, we can identify the constants of the economic dynamics as follows:
Construction of Econometric Meter a
-\ 1
331
hz4
_
Xl
1 — X\Z-$
=0-'
1 — A,jZ3
^-l z l 1 - X,z 3
-1
A-iz2 -l *11 = " " = o *2, 1 - A,)Z3
1- ^ 3
+ 7j(/) + Xl6z(t)
- *\ \y(t-h)
+ X 1 5 e(/) + jr, 7 < /(/))X,ar
= al0y(t) + any{t-h)
+ />,(/) + qx(t) .
Thus, . = /L a_l! = ^ a ,- 1 z^ n
=k
(L/C)l/2
-R
12
+R
(L/C) '
%\ = "
ij
= A.1a1
a,,=
= *•!<*! (zi - 1 -Z13A/1)
(z! - 1 - Z 1 3 J W , ) .
(L/cyu -R {(UC)U2 ' Z2^1CT1 •
+
R)
(16.22)
332
Differential Models and Neutral Systems
9l(o
= W t s o - *unt) + *\9nt -v- zoht) + z2lt(t - h) + zl5e(t) + z 16 x(0 + zn<*(0
+/
13 M 1
A ( 0 = W [ ( Q ) + A) + *o) - MoV\3 + C 7 )].
(16.23)
Comparing equations (16.5), (16.3) with (16.18) or (16.19) we deduce that z-R «-ll =
z + tf
= ((L/C) 1 / 2 - R)/((L/C)U2
-r R),
1/2
«,.--(£] (L/C)l/2-R
fC>
1/2
((Z,/C),/2 + /?) U J 0 = 9i(0 + i(0 = -
2£(/ )
(( L/C)
17 2
+fl)
We assume "stroboscopic strategy" in the sense of Hajek [4] and now compute Q, the feasible government control set and the firms' reaction defined in (16.1.23): r\{t) =
qeP.
Define ve(P*Q) Then
= {v:v + QczP}.
Construction of Econometric Meter
^
at
333
- fl-i \)>(t -h) = a0ly(t) + ax xy(.t - h) + v(/),
(16.24)
where v(0-
< 16 - 25 )
^ 1 • l/2 ((Z,/C) + fl)
We view £(f) as a control device which introduces "Impressed Voltage" for the fluctuations of GDP, yij). By manipulating E{t), i.e., government autonomous outlay q\(i) - taxation and its flows and its hereditary values, exchange rate, tarriff and distance between trading partners, and preferential trading laws all measured by impressed voltage, E{t) carefully crafted, we program the solution of (16.19) and graphically display the solution on the dynamics of y on an attached T.V. monitor. Thus using, go, T\i), 1\t - h), t(t - h), e(t), x(t) and d(i) for q and Co, IQ, XQ, MQ(I\I + C7), one constructs v(/) and displays the solution of (16.19) and then select the best outcome. For example various taxation strategies can be constructed and the effects of GDP observed. Of course, a little more sophisticated six-dimensional meter can be designed and constructed to monitor x = [y,R,L,k,p,E], by a carefully designed circuit with several "loops", in Figures 3 and 4. See for example Fig. 4. Indeed consider equation (16.19), which is equation (4.19) of [1], which is simplified as suggested before: dv - f - o_j \y{t -h)at
a_uL(t -h) = a0iy(t) + a, xy{t -h) + anR(t) + anR(t - h)
+ ai4L(t) + a]5L(t-h)
+ a{6k(t)-a]Sp(t)
+ qx(t) + r](t).
(16.27)
Suppose this equation has been identified with definite coefficients, using the program of Italy.2 of [3]. For ease of presentation we neglect (for the moment) the terms, L, R, k and p. This is still meaningful. See Allen [11, pp. 252-254] and Chukwu [8, p. 1180]. Thus
334
Differential Models and Neutral
dv -ya-ny(t-f') at
= a0lyit) + anyit-h)
+ ql(O + rl(O
where
= ^ i a j " 1 ( 5 4 + 7 4 + C 4 + ^ 4 + M4(C-j +
In)
n(0 = W [ ( C 0 + A)+*o)-AWi3+c 7 )],
y
1]
ft
'
L
I
K
r
'
E FIGURE 4.
Systems
(16.28)
Construction of Econometric Meter
335
qi(t) = Xid, ' ( g 0 - z\AT(t) + 2X9T(t -h)-
+ zX5e{t) + z16x(/) +
z0f(t) - z2\t(t - h)
zl7d(t).
For the equation, ^ 7 ^ fl-1 \y(! -h)-a\iUt at
-h) = a0ly(t) + a, xy(t -h) + anR(t) + anR(t-h)
+ aHL(t) +al5L(t -h) + al6k(t) - aiiP(t) + qx{t) + i\{t),
(16.29)
define fn(L) = anUt-h)
+ aHL(t) + al5L(l).
This is the output from a device from L into the first circuit, a "labor integrator" or current added into the first output. We define the function f\2(R) = a\2R(t) + a\2R(t-h), measured from the /^-circuit and added the first circuit. /13(A) = oi6*(0, is a fraction of stock output, integrator, added into the first circuit. Finally, f\4(P) = -al$P(<) is an electric measure of price added to the circuit of y{t), the GNP. All these yield the equation for y{f), the electronic equivalent or measure of GDP. In the same way the dynamics (2.23), (2.35), (2.43) of [1] are modeled by the other five circuits which are similarly constructed. It is easy to see that some boundary conditions, similar to the one given will yield the full equation. Full details of the electronic device which yields the complete model of economic system generated by Italy.2m of [3] is reported elsewhere, announced and with copyright and provisional patent application on the way.
336
Differential Models and Neutral Systems
16.2 Salamon's Circuit We now consider a more complicated system in which the gross national product, interest rate, employment and the value of capital stock form the economic state in a coupled linear fashion. The circuit for a lossless transmission line which was designed by Salamon can be appropriated and used to design the econometric meter The lossless transmission l i n e Ilt.XI
U, I
Uo; Uir.xj
L,
c,
I„t,
..T ^7777 I— FIGURE 5.
Consider the lossless transmission line described by the hyperbolic partial differential equation du__ dx
dl_ dt'
dl___ du_ dx dt
(16.30)
with the following boundary conditions «(r,0) = «o(0-fy/(/,0) + £b(0;
«(/,!) = Ml(0 + /?i/(r,l)
(16.31)
Construction of Econometric Meter
337
«o(0 = -Vo(<).«i(0 = £ i t e .
(16.32)
/(/,0)-/o(0 = -c0«o(0.
(16.33)
/(M) = /,(0 = q«i(0-
(16.34)
If
h = y[CL, x4(t) = 2-JZ /, (/)
(16.35)
x3(t) = 2y[Ll0 a«rf
(16.36)
»(/) = 2 Vc£ 0 (')
a solution of (16.31)-( 16.36) leads to a neutral functional differential equation of the form - M/) - A_xx{t -h)- B_xu(t)] = AQXU) + A]X(t -h) + B0u(t) at
where x = [x\ *2, x^ X4], and -a0
0
<*0
0
0
-a,
0
-ai
-o2
0
0
0
0
«3
0
0
AQ =
0
A=
»
a0
0
0"
0
0
0
0 0 B
0 a^a<
a2a4
0
0
0
0
0
o a2Po 0
(16.37)
338
Differential Models and Neutral Systems
a
/L,=
0
a4
5
0
0 0" 0
"Po"
0
0
0
0
0
0
0
0
0
0 B-0 = 0 0
where a0 =
_/?oVc-VZ RQyfC+yfl'
04
a
l
i /?nVc + VZ h Vc
1 Vc" C0 RQifC+yfl'
Po =
1 VC C\ /?,Vc + VZ'
VI fyVc + VZ' I /?.Vc + VZ
^i
Vc
_R\4c-4I <*5 =
(16.38)
^ V c + VZ'
We identify x with [y, /?, Z,, A:] ,
(16.39)
the economic state of a country A. Then
m
-A4
0 0
>(/-/»)
5
0
0 0
R(t-h)
L{t)
0
02^4
0 0
L{f-h)
k\t)
a3a5
0
0 0
k(t-h)
Mo
0 a
Construction of Econometric Meter
339
0
-%
0
0" 'y(t-h)
«I
0
0
0
R{t-h)
0
a2a4
0
0
L{t-h)
o3a5
0
0
0
W-h)
0
0
0
a
\
+
"0
0
Ol
~m
0
0
0
R(t)
(16.40)
«(0 + 012P0
0
a2a4
0
0
L(t)
0
- a3a5
0
o
oj _*(<)
Equivalently, y(t) - aAR(t -h)-
P 0 «(0 = - a0R(' ~h) + «0#C)
or y(t) - a4R(t -h) = a0R(t) - a0R(t ~h) + P0ii(0 /?(/) - a5y(t -h) = a]y(t) + cqX' - h) L(J) - atfuRfj
-h) + a 2 a 4 Z,(/) + a 2 Po"(0
*(/) = - a3a5><(/) + a 3 a 5 >'(/ - h)
(16.41)
or — WO " *-\x0 ~h)- 5_,w(/)] - A0x(t) + Ax(t ~h) + B0u(t), dt
(16.42)
Differential Models and Neutral Systems
340
for the matrices A_\,B_\,AQ,A\ and BQ. C\,L\,RQ, R\ are nonzero and finite. Definition 16.1. The system
dt
We assume C, L,
[x{t) - A_]X(t - h)] = AQXU) +AlX{t -h)+ B0u(t)
CQ,LQ,
(16.43)
is approximately controllable if for all <|>e WJ,' and e > 0 there exist a time / > 0 and an input u e Lp([0, t],Em), such that the corresponding forced motion x(t) of (16.43) with initial condition § satisfies ->\/p
0 |
j\ks)-x('
+
s)\Pds
<e.
-h
Suppose country A has the dynamics (16.41) for>>, R, L, k, then the circuit equations Salamon [5, p. 151] Figure 5 "models", mimics and "measures" the economic state, x = [y, R, L, k]. The e.m.f. EQ at the boundary will provide the control u{t) = 2Vc£o(0- Conditions for the economic state to be called "F-controllable", i.e., (approximately) F-controllable, are given by Salamon as rank[A(X), BQ + \B_\] = n,
(16.44)
where A(X) = IX- XA_xe~Xh -AQ-
A{e~Xh ,
This can be expressed in terms of the coefficients of the economic state. To explore this concept further we give the following definitions. Theorem 16.1. The system (16.41) is approximately controllable in W^' if and only if rank[A(X), BQ ] = n, VX e <2,
Construction of Econometric Meter
341
max rank[A\ + "KA_\, BQ]- n, Xe
AQ-\I
max rank XsC A\+hL\,
A\+ XA_\,
BQ
0,
0
= n + max rank[A\ + A*4_j]. XeC
This concept of (approximate) F-controllability is weaker than approximate controllability. The system derived for x = [y, R, L, k] which is equivalent to that of Salamon's circuit is F-controllable if and only if C0L0=ClLi=C
L/k2n2
keN
(16.45)
does not hold. Note that as can never be allowed to be equal to one. The economic meaning of this condition which allows initial states to attain arbitrarily close to the target can be interpreted from the derived economic system. The design of an econometric meter for the full Italian model of [3] requires several coupled partial differential equations derived from the lossless transmission lines. It will be presented elsewhere. The outcome is a system of differential equation of neutral type.
16.3 Simulation of a Nonlinear Economic State of Gross Domestic Product and Interest Rate There is an analogy between the electric circuit problem and the derivation of the economic state variable of gross domestic product y, and the interest rate. This is an old basic idea in the field of ANALOG Computer. Consider the following: Let z be aggregate demand, C consumption, / aggregate investment. Assume that
342
Differential Models and Neutral Systems z = C + I, C = CQ+ c\y I = IQ-
0 < c\ < 1
I\R
I[ > 0
Aggregate supply s=y, y = gross national product, R = interest rate, L = Liquidity (i.e., real money demand)
M(f) = money supply. Assuming the principle of supply and demand
at ^ = X2(L-M(t)). at Then ^- = Xl(C0+(Cl-l)y at ay — = Xl(sy-IlR at
+ I0 -IiR)
+ C0 + / 0 )
s = 1 - C\ (savings)
(16.46)
343
Construction of Econometric Meter
*£ = \2(J + ky-M(t)), at — = X2(£y +
j-M).
The system of equations ^ = ^ - V i * + (^i(Co + 'o)) at — = X2ky + K2U ~ at
M
)
is calculated from above and made into a second order equation d2y -f + \lSy(t) + XxI{k2ky + V i M / - M),
(16
dr since CQ, IQ are zero. Thus the equation fory is given by d
y
dt2
+ a ^ + by = g(t) + p(t), dt
(16
where a = Xxs = A,j(l — C\), b — 'k\I\k2k, g{t) = XxI{k2M s cM .
p(t) = -hhhJ
= Po.
(16
344
Differential Models and Neutral Systems
For a specific nation we can use MATLAB program to identify a, b, c, po as follows: Program 3.M C = C0+C{y
temp = [C, ones(size(y)), y] the = arx(temp, [0,1 1 0 0]) cp = predict(C, ones (size(y)), y the, 4); cc = th(l 1:2); cc(l) = C0
cc(2) = Ci.
7=7 0 -I\R. temp = [/, ones (size(7?)), /?]; thl = arx(temp [ 0 1 1 0 0]); Ip = predict (7, ones (size(y)), .y, //?/, 4); 77 =th( 1,1:2); 7/(1) = / 0 ,
7/(2) = -/,.
Also L=j + ky temp = [L, ones (size(y)), y] thl = arx(temp, [0, 1 1 0 0]); Lp = predict (L, onesize(y), y, thL, 4);
Construction of Econometric Meter
LL(l)=j
345
LL(2) = k.
Then a = (\-cc(2)), b = 11(2) LL(2), g(t) = II(2)M=cM, pit) = -11(2) LL(\)=p0. The equation ±J. dt1
+ a±
+
by = g(t)
+ P0 =
Ei + E0
(16.50)
*
can be modelled as the voltage/current in the electric circuit L
+R
^T
+
f
2
dt
-Q = W, c
dt
or dt2
L dt
cL
where R
CL
L '
(16.51)
Differential Models and Neutral Systems
346
H i-
R
V
fwmr\ L FIGURE 6.
e(t)
(16.52)
Consider the circuit depicted in Figure 7. It contains a linear resistor R, and a nonlinear resistor characterized by the function J[v), (for example f{v) = v3 /3 - 3v212 + 2v in tunnel diode circuit). If v is the voltage and / the current through the inductor then by the KirchhofPs Voltage Law , di L — + iR + v - E\, dt ' L — + iR + v = E\, dt '
C^- + [f(y)-i] at
= E2.
(16.53)
Construction of Econometric Meter
347
R
L
HM^iO-L/ A E
T
w
^
\ /
J
FIGURE 7.
Equivalently di _ It' dv dt
l
=
. R iR L ~T+
/ c
/(v) c
*'=Xv),
E{ L |
E2 c
(16.54)
348
Differential Models and Neutral Systems
dt dv dt
i = c —.
di
v = Li — dt We allow the following postulates z = aggregate demand C = consumption / = investment z= C+I C = Co + C\y I =
IQ-I\R
Aggregate supply y = gross national product GNP R = interest rate L = real money demand (liquidity)
=j + ky-M)
*>o
where J[R) is a nonlinear function of R; for example /(/?) = aR3 + bR2 + cR, (a, b, c to be estimated). By the familiar principle of supply and demand,
Construction of Econometric Meter
349
dy
^ = - M y - V l * + * - i ( C o + 4>><* = l - C i ) . at dR — at
=\2ky-X2f(R)+X2U-M).
Using MATLAB program one identifies the coefficients c 0 , cx, j , t\ i, c\ j, and therefore a\, a2, b\, in dy — = axy + a2R + at
k,an,
p\+q\,
— = bly + f(R) + p2 + q2. at
(16.55)
In analogy, we compare it with the nonlinear circuit (16.53). We have R
!
i
w
Pi +91 s £ , / L = ^.1(C0 + / 0 ) 6 l = l / c = X 2 *.
E2/c = X2(j-M).
(16.56)
Thus after system (16.55) has been identified, the coefficients are used to construct the monitoring electric circuit whose current and voltage are measured, and represent the GNP and the interest rate. The impressed voltage Ex = U ( C 0 + /„), is constant and represents private firms strategy. The impressed voltage
Differential Models and Neutral Systems
350
E2 = l2CU ~
M
)
is considered government strategy which includes money supply, and autonomous money demand. It is possible to use E\,E2 to induce some motion on y and R and extract the best outcome for the state. It is more appropriate to designate (C0 + I0)M
0
(16.57)
where/? is private firm's control strategy and q is the government strategy, a multiple of money supply.
16.4 Delay Systems A simple delay equation can be constructed from a circuit with phaseshifting network A. If curent x(t) at time / seconds is measured in ampers and resistance R is measured in ohms while voltage, inductance and capacitance are measured respectively in volts, henry and farads, then by Kirchoffs law, L — + (J? + fli)x(0 + - = £ . dt c The sum of voltage drop across the inductance, the resistances and the capacitances is equal to the Applied Voltage. See Figure 3.3 [1, p. 9]. We now introduce a phase-shifting network p to account for the natural lag between the input and the output. The voltage drop across R in series with P is given by e«) = qg(x(t - h)),
Construction of Econometric Meter
351
j
imm^
FIGURE 8.
q is the gain of the amplifier. Thus
L
2 i ^ l + Rjc{t) + qgk(t - h) + - X(t) = E . dt c
We now convert this into
m = y(0,
W) = - f AO - \ g(y« - m - -^ x(t) + E . E can be converted to a control device E = {b\u\, bjui), which introduces an impressed voltage b]u\ and a device that creates an impulse of/. Thus
Differential Models and Neutral Systems
352
z(t) = A&(f) + Axgz(t -h) + Bu(t) X
where z =
"i
«= _y_
."2.
o j_
4>
R
TV 0
0
0
b
b=™.
bi
0
0
bj
B=
A delay economic system x(t) = AQX(0 + A\x{t -h) + Bu(t)
where x = \y, R] CT
a!_,A.l(zi-l)
l z
I
h5
AQCT
o~2~ A-2^1
2
CT
1 A-lz3
CT
i
^-2^3
^lz6
4 0 2 A.2A/1
o, %
CT9
0
A2A/3
0
B= 0
CT2
A, 2
Q"2 A, 2
Construction of Econometric Meter
353
a,
\\ZQ
u=
is derived in [12]. The coefficients can be identified by MATLAB regression. An electronic econometric meter of such a system can also be designed using the phase-shifting method and circuit in loops. Details will be reported below
16.5 Tunnel Diode Circuit with Delay: Analogue, Econometric Meter with Delay Consider the circuit:
^ijyyu—$# s —pMq
>K-j
** QZ: R3
1?
FIGURE 9.
Then I ^
at
C ^pat
+ /?!/(/) + * 2 i(/ - h) + v(t) + vc(t -h) = £,(/),
(16.58)
+ [/(v(/), v(/ - h) - /(/)] + *,/(/ - A) = £2(0,
(16.59)
354
Differential Models and Neutral Systems
f(v(t), v(t - h)) = ^ p
- | v 2 (0 + 2v(/) + v(t - h).
(16.60)
It follows that di(t) dt
Rxi(t) L
=
dv(t) _ /(/) dt C
R2i(t-h) L
v(/) L
R3i(t - h) C
v(t-h) L
Ex{t) L
=
/ ( v ( 0 , v(t - h)) _ E2(t) C
or x(t) = Aox(t) + Axx(t -h) + g + Bu(t),
(16.61)
where x = [i, v ] ,
1 L
*1
4) =
z' 1
g(v) =
"l i
(16.62) *3
. c'
c
»
0 " E\ ' (16.63)
-
Zfa(0 = E2(t)/C
I
Z l
[o. v3(" - 2 * 0
~Ex(t)IL
Thus
4=
>
2 C .
c
/?2 Z
0
1
c.
M*\
Construction of Econometric Meter
355
j ( 0 = 4>x(0 + A{x(t -h) + Bu(t) + g(x(t))
(16.64)
Note that rank B = 2 since — * fj. IC Observe that v
c=i>
1 charge,
c = capacitance, vc - voltage drop across the capacitance, (in farads) while charge, q is in coulombs. If/ =XV) = -?v then/is linear and g(v(/)) = 0. This is the Modified tunnel diode circuit. We examine the system for controllability and the use of the circuit as an econometric meter. From Chapter 18 of "Control of Nonlinear Delay Differential Economic Systems in W2
in [12]. The aggregate demand
z = z0 + zxy(t) + z2y{t) + z3R(t) + z5y(t - h) + z6R(t -h), (16.65) so that
^P-Mx-XOl. at — = l2[L-M\], yields x(t) = AQx(t) + Axx{t -h) + Bu(t),
(16.66)
356
Differential Models and Neutral Systems
where o"j A.j(zj — 1)
cjj A,jZ5
a<2 A,2zj
a2 ^2 O T 3.
AQ =
aj
A,JZJ
CTJ
A.jZg
4= <*2 ^2 OT 2
B=
CT
2 ^2W4
a, A-! 0
C2 A-2
O2 ^-2
356 M =
^^(Wo-Wl)
CT =
< = CI + g ,
CT2^1»»0.
(16.67)
S = G21X.2M1
z
O
U =
mQ
(16.68)
Ml
JC(0 = Aox(t) + Axx(t -h) + Bu(t).
(16.69)
This is a delay differential equation. We identify coefficients as follows:
Construction of Econometric Meter
In
357
AQ:
| - = ar%(z,-l), ^ = ci%(\-zl), 1 T
1
= CT
1
-l, A z l 5>
-h
— = a 2 A2ZI »
— = a 2 A2/w3 • In A, *2
-1,
1
-K
7 J = CT2 ^2W2 »
a =
- = ° 2 ^2OT4 •
( Q +'o + *o)CTi a21A.,/w0
h
= A,
= E2(t).
Differentia! Models and Neutral Systems
358
Note that it is possible to design E
\ =[gll>e12l>
E
=[e2he2l\-
2
If
B
a, X{
0
0
a 2 X2
then 1
CT
1 *1
l
a 2 X2
11,(0 =
«2(0 = E2(')
(16.70)
Also
flw =
of'X,,
0
0
cr 2 A.2
z
0
fflQ - M l
It is now clear what method to adopt in constructing a delay econometric meter.
Construction of Econometric Meter
359
16.6 Perspective There is an intense desire for prosperity by many people in all nations in a world of harmony. World leaders of thought have postulated that the global view, collaboration, cooperation and diversity are the key values that can yield meaningful prosperity. Brown University accepted this as its philosophy and ethos. In this year's convocation ceremony, Wake Forest University President, Thomas K. Hearn, Jr. argued eloquently this global view and diversity: Love is and should be the moral basis of the pursuit of prosperity, the role of a university, as an intellectual enterprise is to propose and evaluate new and existing policies that lead to prosperity. Peace College, Raleigh, NC and University of Mississippi have within the last month established national institutes of policy analysis - a pointer to the need to fulfill the current aspirations, and further the training of students in leadership. Yes, society needs to create and sustain prosperity. The survey outlined here was a long intellectual journey. We now give a brief historical survey of the theory of aggregate economic dynamics which we have studied. It begins with Say's law of the market which can simply be stated that "Demand is equal to supply" or "Supply creates its own demand". Hence there is no involuntary unemployment because of overproduction. The next dominant idea is the quantity theory of money traceable to Jean Bodin and refined, deepened and popularized by Milton Friedman. It states that the general level of prices, p, is a function of money supply: p = j(M). Since western economy collapsed in the 1930's with national output and prolonged under-employment, Say's law of the markets aggregate and the quantity theory of money were discarded as too simple to explain current actual behavior of the economy. In fact, supply is not always equal to demand. It was John Maynard Keynes who constructed a rich and better theory. His aggregate demand and supply framework and its subsequent refinement with this framework are the core of modern microeconomics. In our work we postulate formulae for aggregate demand, Z and its components and aggregate supply, y. The components of z and y utilize the new classical economic theory of rational expectations in the spirit of Fair. Accepting also the market principle that dyldt = k(z-y), (k constant) for example, the book derives the dynamics of income, interest rate, employment, value of capital stock, prices, and cumulative balance of payment. It is an integral, differential and neutral system, a game of pursuit. The quarry control is government intervention in the form of taxation,
360
Differential Models and Neutral Systems
control of money supply, tariffs, foreign credit, interest equalization tax, preferential trade agreement (which reduces trade barriers and enhance trade flows between nations), transportation and distance between trading partners. The pursuer is the firms' private autonomous investment, consumption, net export and real money demand, productivity, wage rates, income, export, intercepts and price intercept. We derive conditions for controllability and then deduce how government intervention (compared with private firms' contribution) should be to ensure the possibility of growth. The universal principles deduced are fundamental and consistent with earlier ones derived by the author for hereditary systems. They are solutions to this century's burning problem. To avoid using data from the infinite past, we simplify our dynamics and then illustrate our method with a study of the USA, Canada and India. Optimal control strategies are explored. The book opens up hope for pleasant possibilities for the world. The current world economic situation raises the hope of mankind for better things and for possibility that it will conquer scarcity at a global level with big profit for all. This hope is inspired by the recent USA accomplishment: the growth of gross national product GNP, prices (and therefore inflation), value of capital stock, employment, and cumulative balance of payment can all be controlled simultaneously. Historically, it was unthinkable to many because for example, low unemployment is associated with high inflation. Ideas seem to be changing: controllability from an arbitrary economic state to another seems to be an emerging new phenomenon. For example, it was recently reported: "As for inflation, Greenspan said that in the past the current low level of joblessness was associated with worsening inflation. But for a variety of reasons, he continued, "we have yet to see early signs in prices themselves of intensifying pressures" that point to accelerating inflation. In fact, by some measures, the inflation rate is falling rather than worsening. Rennert of the News & Obsever, Washington Bureau (Saturday, June 19, 1996) reported: President Clinton won a vote of confidence from major allies for his economic growth policies. In a 29-page economic statement, leaders of the world's seven biggest industrial democracies praised US success in achieving sustained non-inflationary growth. With unemployment in Western Europe stuck above 10 percent, several summit leaders spoke enviously of American economic performance in creating nearly 10 million jobs since Clinton took office, while they still struggle to cope with 20 million idled workers. This book shows that the USA achievement is not by chance. It is
Construction of Econometric Meter
361
because of some general principles of controllability of the national economic state: any initial aggregate state of GNP, interest rate, employment, value of capital stock, prices (and therefore inflation) and balance of payment can be driven simultaneously to some fixed and arbitrary target selected by the people subject to scarcity. The success in the USA can be made permanent and universal under the same type of controllability conditions. Once it is planted in other nations, the sustained growth of wealth of the USA is assured, and the possibility of world peace increases. Though the economic ideas of Supply and Demand and of Rational Expectations are the basis of our research, they are not the old ideas which are flawed. This is articulated forcefully by Pope John Paul II, (see Appendix I) and by world leaders. The G-8 group of industrialized nations has recently focused as a priority "Building lasting growth in our economies", and "achieving sustainable economic growth and development" in the world, a view shared by Asia and Africa. The old theories cannot do it. Witness the economic crises in Asia, Russia, Latin America and Africa, and their worldwide impact. The global economic turnoil include economic disaster zones from Japan to Brazil, South Korea, Thailand, and Russia. Because exports make up about 11% of US Gross Domestic Product, the resulting decline in USA exports has hit US farmers, many manufacturers and USA banks. Consider two North Carolina Industries-The Nucor Cooperation led by John Correnti whose wage income was cut by 30%, with a $30 million loss of community revenue, and Burlington Industries, managed by G. W. Henderson, whose industrial capacity was reduced by 25% with 2,900 jobs eliminated-all a result of the overseas economic crises. It is important that the international situation not worsen. As observed by senator John D. Rockefeller IV "modern 'experts' are so confused. There is no Economics 101 which carries us through what is going on today. We are probably writing the text books, but we don't have them to read yet". Meeting the Challenges of the Global Economy, March 30, 1999. If economic growth slows in Europe, if Brazil falls, then Argentina and Mexico can be speculated to tumble with bad effects in Hong Kong and China. It is important to realize that the continued sustained economic prosperity of the USA depends in a significant way on the global growth of the world economy. It is of fundamental importance that the economic advice and assistance
362
Differential Models and Neutral Systems
given to the world does not make matters worse. For example, there is a broad debate among economists and policy makers about whether the IMF decision last year to push Asian nations to send interest rate soaring and prices of fuel higher is a crucial blunder. The World Bank believes that this is a misjudgment which worsened the crises. While it stabilized national currencies in Thailand and South Korea, the bank argues that it plunged the countries into a recession with substantial unemployment, i.e., it lowered the GDP and decreased employment so much that civil order broke down. This book draws attention to the study of the Optimal Control for the Growth of Wealth of Nations in order to obtain new and better and more accurate economic models for all nations and use them to obtain very good economic outcomes, high growth rate, low interest rate, low inflation, and full employment, soaring value of capital stock, and surplus balance of payment. The models constructed using the principle of supply and demand and Fair's version of the principle of rational expectation yield a dynamic model, that is, a differential game of pursuit which is equivalent to a control system with delays in the state variables and perhaps in the controls. Our ultimate aim is to find out the best way to drive the current economic state of nations using the available strategies to attain the chosen economic target as soon as possible and minimize the cost. To do this we have in this chapter shown how to construct national Economic Meter or Econometric Simulator which will serve as "surveillance" device for the economic state, a warning for invading oscillation, depression or growth in a country. Circuits are identified which are analogous to the economic state. Once assembled through simulation and the use of the machine and the computer, various strategies and their consequences can now be explored. Our effort will succeed more rapidly if we can secure the assistance of industry, private philanthropies, government and colleagues. The assistance will eventually be focused on an established institute, the United Nations Institute for The Study of The Optimal Control of the Growth of Wealth of Nations. There, our studies will flourish. Economic policies will be evaluated, simulated and obseved before implementation. The center will use United Nations data and data directly from countries to construct realistic dynamic models of national economy. This will be useful for forecasting for ensuring sustainable growth and prevention of depression. The studies will be useful for policy making. Because of the use of the economic simulator, the recommended policy can be studied for its effectiveness and impact in the institute before implementation. The data, the model, and the policy recommendation will be transparent to all
Construction of Econometric Meter
363
countries in question for scrutiny and verification. What is needed is a new and fresh look, a new study of growth and development in a new place which is committed to build on and sustain globalization, an institute that believes in the importance of sound economic policy, transparency and good governance. Such a place of such research does not exist in the world. This book highlights the need, and points to a possible realistic solution to the most important problem of all times: the conquest of scarcity and a return to the garden of Eden. As President Molly Broad of North Carolina University System said: "If we have learned any lesson from witnessing the economic storms raging around the globe, it should be this: none of our organization is selfsufficient. We must forge alliances beyond our borders." This is a challenge to the nations! We have built national economic models, and confronted them with data. The economic state of some nations are shown mathematically to be controllable, and in some cases optimally controllable. The models can be enlarged to include nonlinear systems whose optimal controllability is explored. The construction of optimal feedback control is open. The refinement of a workable econometric meter needs to be undertaken beyond the prototype level to handle real national economy. Hopefully, the required good first rate scholarship will be available in all societies for this endeavor.
REFERENCES 1. E. N. Chukwu and H. C. Simpson, The Solution Operator in W^ ' for Systems of Neutral Type, Nonlinear Analysis, Theory Methods and Application, Vol. 29, No. 10, pp. 1173-1195, 1997. 2. J. Hale and Sjoerd M. Veiduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. 3. Ethelbert N. Chukwu, On the Controllability of Nonlinear Economic System with Delay: The Italian Example Applied Math, and Computation, Vol. 95, pp. 245-274, 1998. 4. O. Hajek, Pursuit Games, Academic Press, New York, 1975. 5. D. Salamon, Control and Observation of Neutral Systems, Pitman Advanced Publishing Program. Research Notes in Mathematics, Boston, 1984. 6. R. K. Miller, Introduction to Differential Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1987.
364
Differential Models and Neutral Systems
7. M. Slemrod, "The Flip-Flip Circuit as a Neutral Equation" in Delay and Functional Differential Equatins and Their Applications, pp. 387-389, Academic Press, New York, 1972. 8. E. N. Chukwu, Control in W^ ' of Nonlineaar Interconnected Systems of Neutral Type, J. Austral. Math. Soc, Ser B 36, pp. 286-312, 1994. 9. E. N. Chukwu, Nonlinear Delay Differential Economic Systems in Wj ', Chapter 18, pp. 157-170: Stability and Control: Theory Methods and Applications, Vol. 5, Advances in Nonlinear Dynamics (ed), S. Sivasundam and A. A. Martynyuk, 1997. 10. E. N. Chukwu, "Stability and Time Optimal Control of Herditary Systems," Academic Press, New York, 1991. 11. R. G. D. Allen, Mathematical Economics, McMillan & Co., Ltd, London, 1973. 12. E. N. Chukwu, Optimal Control of the Growth of Wealth of Nations: With Austria and USA Examples. Nonlinear Analysis, Theory Methods and Applications, Vol. 30, pp. 1029-1041, 1997, Proc. 2nd World Congress of Nonlinear Analysis. 13. E. N. Chukwu, "Stability of Time Optimal Control of Hereditary Systems," 2nd Edition 1999, World Paradise Press (Distributed by Copytron, 100 West Franklin Street, Chapel Hill, NC 27516. 14. E. N. Chukwu, Optimal Control of the Growth of Wealth of Nations with Austria and U.S.A. Examples, Proceedings of the Second World Congress of Nonlinear Analysis, Athens, Greece, July 10-17, Edited by V. Lakshmikanthan. 15. E. N. Chukwu, Modeling and Optimal Control of the Growth of Wealth of Nations, with Austria, Australia and Chinese Examples, 1999 IFAC 14 Triennial World Congress, Beijing, P.R. China, July 5-9. 16. E. N. Chukwu, Optimal Control of the Growth of Wealth of Nations, 500 pages (with the Publisher). 17. Wook Hyun Kwon and Arkadii Kim and Co., "Time Delay Systems Toolbox," Engineering Research Center for Advanced Control and Instrumentation, Seoul National University, Seoul, Korea, 1999. 18. A. W. Phillips, "Mechanical Models in Economic Dynamics," Economica, 17, 283-305, 1950. 19. A. W. Phillips, "Some notes on the Estimation of Time-forms of Reaction in Interdepenent Dynamic Systems," Economica, 23, 99-113, 1956. 20. O. J. Smith and H. F. Erdley, "An Electronic Analog of an Economic System," Electrical Engineering, 71, 362-366, 1952. 21. A. Tustin, "A Method of Analyzing, the Effect of Certain Kinds of Nonlinearity in Closed Cycle Control Systems," J. Inst. Electrical Eng., 94, part-2A, 152160, 1947.
17. Appendix
365
366
Differential Models and Neutral Systems Appendix: U.S.A
% USA's data is in billions of US dollars % adopted from International Financial Statistic Yearbook 1994, except where noted. % Use ndu.in to obtain the matrix entries YEAR-[1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]'; X X =[39.0 41.4 45.3 49.3 57 59.3 66 2 91.8 124.3 136 3 148.9 158.8 186.2 228.9 279.2 303 282.6 276.7 302.4 302.1 319.2 364.0 444.2 508 557 601.5 640.5]'; IMP -[-37.1 39 9 -46.6 -50.5 -55.8 -62.4 -74.2 -91.2 -127.5 -122.7 -151.2 -182 5 -212.3 -252.7 -293.9 -317.7 303.2 -328.1 -405.1 -417.6 -451.7 -507.1 -552.2 -587.7 -625.9 -621.1 -670.1]'; X -XX-IMP; V.TX "[1967-70.1972-90 233.9 269.3 305.0 258.5 292.7 291743 354182 399776 443494 458256 523873 587945 668257 757114 757114 826725 949668 970493 1029670 1142340 1239250 1311950 1424120 1513520 1630440]'; T X -[190000 233900 254200 293196 299387 317571 361635 406617 453017 467002 536907 602083 668257 757114 828888 850000 974757 1036008 1148734 1248179 1322479 1435529 1526427 1657800 1749200 1829800 1877200]': d =[)': % =[112 113 117 117 100 102 106 108 104 105 108 1977-84 4.04.7 5 3 3.9 1.3 1.9 1.84.5]'; (a =[ 1900 2000 2300 2400 2584 2767 3124 3620 3771 3780 4675 5485 7162 7202 7535 9993 8688 9430 12042 13067 13312 13923 15054 16096 16339 16197 17164]';
E =[]'; F -[-0.65-0.88-0.84-0.94-1.10-1.11-1.II-1.25-1.01 -0.92-0.91 -0.82-0.86-0.91 -1.03-4.51 -8.73-9.069.75 -9 56 -10.12 -10.55 -11.96 -12.32-12.39-14.04 -14.46]'; D -[I3.S 10.5 9.1 9.7 2.3 8.1 9.9 17 7 14.3 -5.7 16.7 24.7 27.9 12.8 -9 5 25.5 -16.0 -5.5 71.1 24.6 8.7 26.3 16.3 33.3 6.3-0.1 7.3]',
K =[116.6 117 6 130.8 145.5 147.7 167.2 195.3 224.8 230.8 230.7 268 3 333 5 406.1467.5 477.1 532.6 519.4 552.2 647.8 690 0 709.0 723.0 777.4 798.9 793.2 736.9 789.1 ]'; klc -[116.6 117.6 130.8 145.5 147.7 167.2 195.3 224.8 230.8 230.7 268 3 333.5 406.1 467.5 477.1 532.6 519.4 552.2 647.8 690 0 709.0 723.0 777.4 798.9 793.2 736.9 789.1 876.1]': 1 =[130.4 128.1 139.9 155.2 150.0 175.3 205.2 242.5 245.1 225.0 285.0 358.2 434.0 480.3 467 6 558.1 503.4 546.7 718.9 714 6 717 7 749.3 793.7 832.2 799.5 736.8 796.4]'; B =(2.83 1.16 4.12 8.25-7.26-9.88 1.25-1.28-3.47-12.89-10.39-4.73-17.53 14.75 30.53 35.99 -42.27 15.66 19 33.3 22 I 46.5 16.07 11.69 17.8-6.38 41.22]'; BD =(0472.83 1.164 12 8.25 -7.26 -9.88 1.25 -1.28 -3.47 -12.89 -10.39 -4.73 -17.53 14.75 30.53 35.9942.27 15.66 19 33.3 22.1 46.5 16.07 11.69 17.8 -6.38]'; T -[X-B-F]'; e -[I I I 1 1 1.0857 1.0857 1.2064 1.2244 1.1707 1.1618 1.2147 1.3028 1.3173 1.2754 1.1640 1.1031 1.0470 0.9802 1.0984 1.2232 1.4187 1.3457 1.3142 1.4227 1.4304 1.3750]': w =[25.1 26.1 27.8 29.4 31.0 33.0 35.3 37.7 40.9 44.6 48.2 52.4 57.0 61.8 67.2 73 8 78.5 81.5 84.8 88.1 89.9 91.5 94.1 96.8 100 103.3 105.8]'; G -[622.4 667 9 686.8 682.0 665.8 652 4 653.0 644.2 655.4 663.5 659.2 664.1 677.0 689.3 704.2 713.2 723.6 743.8 766.9 813.4 855.4 881.5 886.8 904.4 929.9 941.1 943.0]'; L -[58.4 60.1 62.1 64.3 64.8 65.1 67.3 70.2 71.5 70.3 72.5 75.4 79.2 82.1 82.6 83.3 81.8 82.4 86.3 89.0 90.8 93.2 96.2 98.6 100 98.9 99.2]'; LD =[55.5 58.4 60.1 62.1 64.3 64.8 65.1 67.3 70.2 71.5 70.3 72.5 75.4 79.2 82.1 82.6 83.3 81.8 82.4 86.3 89.0 90.8 93.2 96.2 98.6 100 98.9]'; LP =[1.70 2.00 2.20 0.50 0.30 2.20 2.90 1.30-1.20 2.20 2.90 3.80 2.90 0.50 0.70-1.50 0.60 3.90 2.70 1.80 2.40 3.00 2.40 1.40-1.10 0.30 1.50 ]'; LPD -[ 2.90 1.70 2.00 2.20 0.50 0.30 2.20 2.90 1.30-1.20 2.20 2.90 3.80 2.90 0.50 0.70 -1.50 0.60 3.90 2.70 1.80 2.40 3.00 2.40 1.40-1.10 0.30]'; C -[481.6 509.3 559.1603.7 646.5 700.3 767.8 848.2 927.7 I024.9II43.1 1271.5 1421.3 1583.7 I748.I 1926.3 2059.2 2257.6 2460.3 2667.4 2850.6 3052.2 3296.1 3523.1 3748.4 3906.4 4139.9]',
Appendix
367
M -[180.4 194 0 209 6 216.6 225 9 240.7 262.2 277.3 286.9 302.0 318.7 345.7 375 2 408.2 432 2 460.3 494.2 540 3 585 0 655 0 776 1 786 2 820 I 830.9 870.5 935.7 1065.8]'; VIP =[13 6 15 6 7 0 9 . 3 14.8 21.5 15.1 9.6 15.1 16 7 27.0 29.5 33.0 24.0 28 I 33.9 46,1 44.7 70.0 121.1 10.1 33 9 10.8 39.6 65.2 130.1 164.5]'; MPD=[4 2 13.6 15 6 7.0 9 3 14.8 21.5 1 5 ! 9.6 15.1 16.7 27.0 29 5 33 0 24.0 28.1 33 9 46 1 44.7 70.0 121.1 10.1 33 9 10 8 3 9 6 6 5 2 130.1 ]'; ML =(26.2792 23 6938 11.7627 9.7887 12.6905 14 2568 12.8253 3.9740 2 1125 14.3533 33.3342 38.6344 28 0482 7.6254 11 3427 22.2640 65.5101 64.9009 63 8482 56.3244 -13.8977 26.3147 31.4402 72.6962 91 7929 87.9106 14.1891)'; MD =[176.2 180.4 194.0 209.6216.6 225.9 240.7 262.2 277.3 286 9 302.0318.7 345.7 375.2 408.2 432.2 460 3 494.2 540.3 585.0 655.0 776.1 786.2 820.1 830.9 870.5 935.7]'; P =[24.9 25.5 26.6 28 I 29 7 51.0 32.0 34.0 37.8 41.2 43.6 46.4 49.9 55.6 63.1 69 9 73 9 76 2 79.5 82.4 83.9 87.0 90.5 94 9 100.0 104 2 107.4]', Pf =P.'e; Pfe =Pf.*e; R -[4.5 4.5 5.5 6 5.5 4.5 4.5 7.5 7.75 6 5.25 6 9.5 12 13 12 8.5 8.5 8 7.5 5.5 6 6.5 7 6.5 3.5 3 ]'; RD -[4.5 4.5 4.5 5.5 6 5.5 4.5 4 5 7.5 7.75 6 5.25 6 9.5 12 13 12 8.5 8.5 8 7.5 5.5 6 6.5 7 6.5 3.5]'; RP -[0 1.0 0.5 -0.5-1.0 0 3.0 0.2-1.75 -0.75 0.75-0.75 0.75 3.5 2.5 1.0-1.0-3.5 0-0.5-0.5-2.0 0.5 0.5 0.5-0.5 -3.0]'. RPD=[00 1 0 0 5 -05-1.0 03.00.2-1.75 -0.75 0.75-0.75 0.75 3.5 2.5 1.0-1.0-3.5 0-0.5-0 5-2.0 0 5 0.5 0 5-0 5]'; GDP'[769 8 814.3 889.3 959.5 1010.4 1096.8 1206.5 1349 1 1458.8 1584.8 1767.1 1974.1 2232.7 2488.7 2708 1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9 6038.5]'; y =[769.8 814.3 889.3 959.5 1010.4 1096.8 1206.5 1349 I 1458.8 1584.8 1767 1 1974.1 2232.7 2488 7 2708.1 3030.6 3149.6 3405.1 3777.2 4038 7 4268.6 4539.9 4900.4 5250 8 5522.2 5722.9 6038 5]'; yD =[702 7 769.8 814.3 889.3 959.5 1010.4 1096.8 1206.5 1349.1 1458.8 1584.8 1767.1 1974.1 2232.7 2488.7 2708.1 3030.6 3149.6 3405.1 3777.2 4038.7 4268.6 4539.9 4900.4 5250.8 5522.2 5722.9]'; yP =(67.10 44 50 75.0 70.2 50.90 86.40 109.70 142.60 109.70 126.0 182.30 207.0 258.60 256.0 219.40 322.50 119 0 255 5 372.10 2615 229.9 2713360.5 350.4 271.40 200.70 315.60]'; yPD =[54 60 67.10 44.50 75.0 70.2 50.90 86.40 109.70 142.60 109 70 126.0 182.30 207 0 258.60 256.0 219.40 322.50 119.0 255.5 372.10 261.5 229.9 271.3 360.5 350.4 271.40 200.7]'; YT =y-TX; YTD-(516760 579800 580400 635100 666304 711013 779229 844865 942483 1005783 1117798 1230193 1372017 1564443 1731586 1879212 2180600 2174843 2369092 2628466 2790521 2946121 3104371 3373973 3593000 3773000 3893100]'; YTP-[600 5470031204 44709 6821665636 9761863300 112015 112395 141824 192426 167143 147626 301388-5757 194249 259374 162055 155600 158250 269602 219027 180000 120100 268200 316600]'; YTPD=[63040 600 54700 31204 44709 68216 65636 97618 63300 112015 112395 141824 192426 167143 147626 301388-5757 194249 259374 162055 155600 158250 269602 219027 180000 120100 268200)'; N = y.>.L; price-[23.7 24.1 24.9 25.5 26.6 28.1 29 7 31.0 32.0 34 0 37.8 41.2 43.646.4 49 9 55 6 63.1 69.9 73.9 76.2 79.5 82 4 83 9 87.0 90.5 94.9 100.0 104 2 107.4]'; for i= 1:27 pp(i)=price(i+1 >price(i): kpnm(i)=kk(i+l)-kk
368 %
Differential Models and Neutral Systems =From Yearbook of labor statistic^ international labor office)
% % K -capital stock(Gross fixed capital formation) % I -investment(Gross capital formation) % D =increase in stock % The UN(National accounts statistics) defines Gross capital "/■ formation as the sum of the increase in stocks and gross fixed % capital formation
"/. % % % % % % % % % % % */o % "/• "/• % % % % % % % % % %
y -GDP(Gross domestic product) yD-y(t-l) yP - y W yPD=y'(H) X "Net export(export-impoft of goods and services) d -preferential trade agreement transportation Pf'import price level in foreign currency ta =tariffs(from: statistical abstract of the united states-1990) ta for 1966-69 comes from UN financial statstics 1974 E -cumulative balance of payment F -Net private outflow of capital C -private consumption M,MI =money supply Md -money demand(liquidity), where, ML = md-M P =consumer prices R -interest rate(discount rate) RD-R(t-l) RP=R'(t) RPD=R'(t-l) pie=inflation L -nonagri. employment B =Balance of payment(deposit money banks) T -Net government transfer of capital to foreigers and firms G -Government consumption DP =D*log(P) = p(t)Vp(I)
%format long; %Bert I -(X,G,I.C,GDP,R.M, YEAR] [years,n]=size(YEAR); XH-21.8;X(l:years-l)J; Gl-[475.3;G(l:years-l)]; II-[296.4;l(l:years-I)]; Cl=(1178.9;C(l:years-l)]; Ml=[140;M(l:years-l)]; R]-[3.36:R(l:years-l)]; GDPl=[494.2;GDP(l:years-l)], DGDP-GDP-GDP1; Z-X+G+I+C; ZI-X1+GI+I1+C; %temp=[YEAR XI Gl II Cl Zl yl yD Ml 1 RDI yPDl RP RPD Tl T P P-PF PI P-PFI G L LD LP LPD LI LD1 LP1 LPD1 t t l e e l d d l ] ; %save Bert I .asc temp -ascii -double -tabs VoclearBertl temp; %ML - L-M - ml-mO - ml«y(t) - m2*y(t-h) - m3*R(t) - nvJ^ROh) - m6«P(t) - m7*R'(t-h)
Appendix
369
% L(t) =mO +ml*y(t)-r m2«y(t-h) +m3'R(l) +m4*R
3 70
Differential Models and Neutral Systems
ggy=thgy(5,l:6); %Xy(t) = xO +xl'y(t) <-x2*y(t-h) +xJ*R(t) +x8'L(t) +xlO'L'(t) +xll'e(0 +xl2*ta(t) +xlj'd(t)*(0) temp=[Xones(size(y)),y,yD,R,LXP,e,ta]; thxy-arx(temp,[0,l I 1 1 I I 1 I 0 0 I O 0 0 0 0]); xyp=predict([X,ones(size(y)).y,yD,R,L,LP,e,ta],thxy,4); xxy»nhxy(7.1:8): %B(t) = bO +bl*y(t) +b2»y(t-h) +b3*P(t) +b4»y'(t-h) -rbS'R(t) +b6*R(t-h) +b7»e(t) +b8*R'(t-h) +b9»L(t) ->-blO«L(t-h) +bl2'f(t-h) +bl3«ta(t) --b15*d(t) +b!7«B(t-h) temp-{Bones(size(y)),y.yD.P,yPD,R,RD,e.RPD,L,LD,LPD,ta,BD]; lhb=arx(temp,[0.1 1 I 1 I I 1 I 1 1 1 1 I 1 0 0 1 0 I 0 1 0 1 0 1 1 0 I]); Bp=predicl([G,ones(si2e(y)),yIyD,P.yPD,R,RD,e,RPD,L,LD,LPD,ta,BD])thb,4); BB-thb(13,l:14); %y(t) = A'K-WXI-a) =>ln(y(t)) - a'ln(K) + (l-a)*ln(L) temp=[!og(y)ones(size(y)),tog(K).log(l)]; thly=arx(temp.IO,I I I 0 0 0]); lyp -predict([log(y) ones(size(y)),log(K),log(L)J,thly,4); lyy-thly(2,l:3); %D(t) - a(l-c)*y(t) -kO*K(t) +kl3'K'(t) +L4'R(t) +L5'L(t) -L6*P(t) -v»y'(t) temp-[Dones(size(y)),y,K,KP,R,L,P,yP]; thD=arx(temp,[0,l 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 ] ) ; Dp=predict([D,ones(size(y)),y.K,KP,R.L.P.yP],thD,4); DD-thD(7.1:8); %DP(t) = pO -pl*Pfe(t) +p2'w(t) -p3'n(t)-p4»P(t) +p5*M'(t) +p6'ML(t) temp=(DPones(size(w)),Pfe.w,N,P,MP.ML]; thp=arx(temp,{0.1 1 1 1 1 ! I 0 0 0 0 0 0 0 ] ) ; dlp=predict([P,ones(size(w)),Pfe.w,N,P,MP,ML]>thp,4); dp=thp(6,l:7);
subplot (2,1,1 ),p!ot(YEAR.C. YEAR.Cp, V) title('US--PRl VATE CON SUMPTION-'); ylabel('C); xlabel('YEAR'); subplot (2.1 %2).plot(YEAR.X,YEAR,Xp,'+') titkCUS-EXPORTS--'); ylabel('X'); xlabelCYEAR); pause subplot (2,1.l)>plot(YEAR.I,YEAR,Ip,•+•) titlefUS-INVESTMENT-'); ylabel('l'); xlabel('YEAR'); subplot(2,1,2),plot(YEAR,G,YEAR.Gp,'+') titleCUS-GOVERNMENT CONSUMPTION-'); ylabelCG1); xlabelCyear'); pause clg subplot(2,1,1 ),plot(YEAR,M.YEAR,Lp,V) title('US-MONEY DEMAND-1); ylabelCMT); xlabelCYEAR');
Appendix
371 ,
subptoa?. I J).pl«(YEAIU. YEAR,{Xp+lp+<}p+Cp), +0 (itkCUS-AOOREGATE DEMAND- 1 ); yiabeKT);
xlabtlCYEAR'): MtISC
subplot (2.1,1 ),plot(YEAR,DP,YEAJt1dlp.,+') titleCUS-CONSUMER PRICES-"); ylabeKT); xlabelCYEAR'); subplot (2.! .^.plotCYEAR.y.YEAR.yp.v) tWeCUS-INCOME--1); ytabelfGDPO; xlabelCYEAR"); pause subplot (2,1 ,l)4»lot(YEAR,C,YEAR,cyp:+') tilkfUS-INCOMEAIONSUMPTION-'); ylabelCcy1); xlabelCYEAR"); subpk«(2. l,2),plo«YEAR,l,YEARlIyp.•+,) tJUefUS-INCCH^E/INVESTMENT-'); vUbeKlyp'); xlabetCyear*); pause cl»
subplc<2,l,llplot(YEAR,0,YEAR^yp,'+') iMeCUS—INCOME/OOVERNMENT-'); ylabelCgyp*); xlabelCYEAR"); subplot(2,U)*Jo«(YEAR,X,YEAR^p. , +0 titkrUS-INCOME^XPORT--); yiabeK'xyp"); xlabelCYEAR'); pause subplot (2,l.l),plot(YEAR£,YEAR3p.'+') UUefUS-BALANCE OF PAYMENT-1); ytabelCBT. xlabelCYEAR"); subplot (2.ll2).ploKYEAR,logy1YEAR,lyp,•+,) tMeCUS-togCincoine)-*); ylabelClny"); xlabclCYEAR'); pause
subplot (2,1,1 ),plot(YEAR.D,YEARJ)p,V) DtlefUS-lncitase in stock-*); ylabeK'DO; xlabelCYEAR"); H The economic dynamics can bt put in matrix form % as follows: x'(t) - A-l 'x'(t-W - AO,x(t>*Al»x(t-h)*B'Mt Hx-brRLKPEr.
3 72
Differential Models and Neutral Systems
V.A-1 =[aml I -aml4 aml3 0 0 0:0 am22 0 0 0 0; Lm03 0 -LmOl 0 0 0; a3 0 a6 -ami 0 0; 0 -M7*p6'P(t) 0 0 0 0;b4b8bl2 0 0 0] "/.AO -(aOl a!2 al4 al6 a!8 0; a2l a23 0 0 a25 0; 0 0a33 0 a35 0; 0 0 0 a44 a45 0; a51*p(t) a52*p(t) 0 0 a55*p(i) 0: a61 a62 a63 0 0 0] %A1 -[all al3al5O0 0:al21 al22 0 0 0 0; al31 al32 al33 0 0 0; aI41 aI42 al43 0 0 0; aISI*p(') al52*p(t)0 0 0 0;al61 al62 al63 0 0 al66] %B1 =[-ssl ssl sslzl5sslzl8sslzl7 0 0 0;0 0 00 0-L2 0 0;-mwzsl4mwmwzsl5 mwzsl6 0 mwzsl3 0 0;ZJ|4 1 zsl5 zsl6 0zsl3 0 0;0 0pl*pfl;t)0 0-p6p5 0;0 0b7b8bl5 0 0-l] %B2 -[ssl ssl ssl ssl(II3+c7)00 00 0;0 00-L20 00 0 0;OmwOOOOmwmwO;0 1 0 0 0 0 1 I 0;00 0p6-p3p2 0 0 l;000-l 00 I 00] %u - (q S]; %q -(Tl gOetadMl Ml'(Of, %q -[Tl -9.6126etadMl Ml'-347.6812]'; %S =[C0 10 xO MO n w xO y 10 pO]'; %S =[-0.0002-211.6960-291.5135-1.5231 nw-291.5135-0.1225-55.5034]'; %B=[B1 B2] %AB=A0»B % A2B-
% G-[B AB A2B A3B A4B A5B] %rank(B10>=6 %rank(G,0)-6 % For spectral controllability % Deltaj-eye(size(AI)).'j -Aml.'O'2.7814^)) - AO % -Al.*(2.78l4*(-j)) %a=[DeltajB) %forj =0.4312, 4, 10,23.100,... % rank(a)-6 % For nonzeroeigenvalues of A-1 to be controllable via matrix B, */. cc^y^sizeCAml)).*] -Ami % c-[cc B] %forj -0.4312,4, 10.23, 100,... % rank(c)=6 •/. Since rank(B) is 6, where rank(a) and rank(c) are % also 6, the system is function space controllable LI -I;L2-0.9;L5-DD(5) 1.6 -DD(6): h -1
Appendix •ifory(t)«A , K A a*L A (l-a) a=- 0.9 % a -alpha aa-0.1 % aa-(l-a)=beta W=sum(w)/27 mw -(aa/W).*a'a) % cobb Douglas states: aa+a=l "constant return to scale. % aa-"-a> 1 "increasing return to scale. % aafa=l ->decreasing return to scale. % Increasing return to scale if doubling L&K more than doubles y. % constant return to scale if doubling L&K exactly doubles y. % decreasing return to scale if doubling L&K less y.
% % In cobb.m program log(A) is found to be 2 to give the best ?/o aproximation for y(t). zO-OG(l )*IT( I )+MM( 1)«(CC(8)+1I( 11 ))+CC( 1 )+XX( 1) zl»GU(2)+II(2)+MM(2)'(CC(8)+ll(ll))-K:C(2)+XX(2) z2"GG(3)+l!(3)+CC(3HXX(3)+MM(3)'(CC(8>+ll(ll)) z3-GG(4>H(4)+CC(4)+XX(4) z4-GG(5Ml(5)+CC(5)+XX(5) z5-GG(6)+II(6)+CC(6)+XX(6>+MM(4r(CC(8)+II(ll)) z6»lI(7)+CC(7)-KCC(8HI(H))'MM(5) z8=GG(7)+U(8)+XX(7) z9~ll(9) zlO=XX(8) zll=-l!(10) Zl3=dl(ll)+CC(8))*MM(6) zl4=-CC(2) zl5=XX(ll) zl6-XX(l0) zl7=0 zl8=XX(9)+(CC(8)+zl3)*MM(7) zl9--CC(3) z20-CC(4) z2l=CC(5) %z22--(CC<8)Hl(ll))'Ml zsO - ccy( I )+1 ly( I)-ggy( I )+xxy( 1); '/.constant term zsl = ccy(2)+ggy(2)+xxy(2) ; %y(t) term zs2 xxy(3) ; %y(t-h) term zs4 - ccy(3)+ggy(3) ; %y"(t) term zs5 = ccy(4)+xxy(4)+gsy(4) ; %R(t) term zs8 = xxy(5)+ggy(5) ;%L(t)term zsl0-xxy(6) ; %L'(t) term zsl3=ccy(5) ;%Ml(t)term zs!4=ccy(6) ; %T(t) term zsl5=xxy(7) ;%e(t)term zs!6=xxy(8) ;%ta(t)term si -1-Ll*z3 % assumption yyl ■ zsO; yy2 = zsl; yy3 = zs2; y y 4 - zs4;
373
374 yy5 - zs5; yy6 " zs8; yy7=»zsl0; yy8 = zsl3; yy9 = zsH: yylO=zsl5: yyll -z$16; zsO= GG( 1 )*XX( 1 )+CC( 1 )+H( I) aO = DD(lV(h*(l-yy(2))) al = -DD(l),yy3/((l-yy2)«h> a2»-al'h a3 - aO*h'yy4 a4 - DD{! )*yy5/(( 1 -yy2) +DD(4)) a5 - DCKD'yy6/(l-yy2) +DD(5) a6 ^aO'h'yy? a8 = DD(6)
%A-I starts here: ami I = a-11 amll =Ll*z4/sl am22 -L2'MM(6) aml3-Ll*2lO/sl aml4=Ll,2lO/sl Lm03 = DD( I )'yy4'mw/( I -yy2) LmOl - DD(3),DD(l)/(l-yy2) -mw*DD(l)«yy7/(l-yy2) a3 =DD(l),yy4/(l-yy2) a6 =DD(i)'yy7/(i-yy2) ami -DD{3),DD(l)/(l-yy2) M7p6P = MM(7)*dp(6)»sum(PV27 b4 = -BB(5) b8 - -BB(9) bl2--BB(l2) •/oAO starts here! aOI =Ll*(zl-l)/sl all -Li*z2/sl al2=Ll'z5/sI al3=LI'z6/sl al4=Ll*z8/sl al5'LI , r9/sl al6-Ll"zll/sl al7-Ll*zl9/sl al8=LI'zl8/sl
a2l =L2*MM(2) a23 =L2*MM(4) a25 -L2 4 MM(7)
a33 ~a0; a35 =mw*a8 a44 -aO a45 -a8 a5 l=»-dp(7),MM(2),sum(P)/27 % includes P(t) a52=-dp(7)»MM(4)»sum(P)/27 % includes P(t) aJ5 --(MM(7)+dp(5))*sum(P)/27 % includes P(t) a61 =BB(2) a62 ~BB(6)
Differential Models and Neutral Systems
375
Appendix (63 -BB(IO) KAl lUro here! •11 -Ll*z2/sl
al3-LI*z67sl ili=LI»z9/il 122 -L2'MM(3) i24 -L2»MM(5)
al--DD(l)»yy3/((l-yy2rh) i2--«l*h 11 --al 12 --mw»DD{l)»yyV(l-yy2) 14 -mw'DD(l)»yy5/(l-}y2+DD(4)) b2 -BB(3) b6 -BB(7) blO - B B ( I I )
bl7 -BB(O) il 11 -LI*(GG(3)+ll(3>*XX(3)-MM(3)»(ll(l l>+CC(8))yjl 1112-113 H13-115 «121-i22 H22-124
H31 «mw*DD(l )'XX(3y(l-<XX(2)+tty(2>+ccy(2))) a 132-mw'DD( I )*yy5/( I -yy2) -DD(4) H33-i3-ll»h tl41 -DD(l)*yy3/(l-yy2) H42 -DD(l)*yy5/(l-yy2) + DD(4) il4J -DO(l)»yy6/
blzl4--BB{2),zl4; bl -BB(1) MzlJ-BBflJ'zIS blzl6-BB(2)*zl6 blzl7«BB(2)*zl7 blzl3-BB(2)»zl3 mwzil4-mw*yy9 nwolS =mw*yylO nwzil6-mw*yyM row»17 -0 mwz»13 -mw*yyl plpf -dpO'wmtPfcyZ? P« -MS)
Hdp(2)»Pfl[t)
376
Differential Models and Neutral Systems
p5 -dp(6) b7 -BB(8) b8 -BB(9) b!5 -0 %B2 starts here! ssl =Ll/sl
sslil3c7=-ssl,(II(ll)+CC(8)) 12 -mw*DD(l)»yy3/(l-yy2) p6 -dp(7) p3 -dp(4) p2 - dp(3) al7=LI*zl9/sl a45 -a8 cO-CC(l) 10-11(1) xO-XX(l) pO = dp(l) MO-MM(I) yl0 = ccy(l) mO =MM(1) gO -GG(1) K)=BB(I)-XX(I) g2 --L2*MM(1)
%
%
The following data were the output results for the program us2.m Ll-0.1870, L2- 1 4462e-004. L5 - 0.0525,h - I,a - 0.9000,aa - 0.1000 W = 63.1704,mw = 7.7325e-O04,zO = -512.6880,zl - -0.75l9,z2 = 0.6830.Z3 - 0.9443 z4 - -0 2400 z5 = 15.4108
z6 - 10.329l,z8 - 12.6083.z9 - -8.8065,zl0 -45.3681.zl I - 0.3089^13 - 0.0192 zl4 = -1.4798e-008,zl5 - 7.8295^16 = -0.0429,zl7 - 0,zl8 -15.0963,zl9 - -6.9638e-009 z20 - 6.5288e-009,z21 - 6.5699e-009,sl - 0.0557,zs0 = -512.8223,a0 = 1.5554,al - -0.1870 a2 = 0.1870,a3 = -0.0022,a4 = 20.9668,a5 = -16.136S.a6 = 10.1608 a8 = -O.0243,aml 1 - -4.3105,am22 - -0.1961,aml3 - 8l4.8353.aml4 = 814.8353 Lm03 - -1.5478e-006,LmO 1 - -0.0O47,a3 - -O.002O,a6 » 9.4257,am 1 - 0.0026 M7p6P - 0.2685,64 - 0.1990,b8 = -5.9898,b 12 - 4.9703,a01 = -31.4655 all-l2.2672,al2--276.7857,al3- I85.5l72.al4 -226.45l9,al5 --I58.I694 al6-5.5489.al7--l.2507e-007.al8-271.1380,a21 --0.008l,a23 =0.2186 a25 =O.I076.a35 =-l.8772e-O05,a44 = l.5554,a45 = -0.0243,a51 =0.0081 aS2--0.2195.aS5- 132.4471.a61 --0.070S.a62--2.4137.a63-34.3642 a l l - 12.2672,al3 = l8S.5172.al5 - -158.1694.e22 - 0.0071.a24 - -0.1522 al - -0.1870,a2 - 0.1870,11 - 0.1870.12 --1.4462e-004,l4 - O.0I62,b2 = 0.3164 b6-32.1284,bl0--20.5989bl7--O.0267.alll - 12.2672,al 1 2 - 185.5172 al 13 = -158.l694.al2l -0.007l,al22--O.I522,al3l -4.l324e-004 al32 = O.OII5,al33--l6.3235,al4l - 0.1870.a 142-21.0641 al43--l6.l365.al51 --O.9037.al52-0.1528,al6! -0.3164 al62-32.l284,al63--20.5989,al66--0.2439,ssl - 17.9606 sslzlJ -0.3449,sslzl5 - I40.6221,sslzl6 - -0.7698.sslzl7 ■= 0 sslzl8 - 271.1380,bl=-639.1947,blzl5 =-0.5518 blzl6 - 0.0030,blzl7 - 0,blzl3 - -0.0014,mwzsl4 - 5.4905e-009,mwzsl5 - 0 0956 mwzsl6- 1.0145e-O0S,mwzsl7-0,mwzsl3-4.7S87e-O07,p1pf-7.6410
Appendix
377
p4 = -2.3281.p5 = 0 0374.b7 - -I54.5053.M - 5.9898 bl5 - O.ssl - 17.9606.sslil3c7 - -1.5833,12 - l.4462e-004 p 6 - O O I 5 l , p 3 - 6 4 2 1701,p2 - 2.5444,al7 =-I 25O7e-O07.a45 »-0.0243 cO - -1.6295e-004.IO = -21I.6960.x0 - -291.5135,pO --53.5O34.M0- -I 5231 y l O - O t225.m0- -1.523! gO - - 9 6126.(0--347 6812,g2 = 1 3708
The matrices for economic dynamics are: Ami - K 3 1 0 5 -814.8353 814.8353 0 0 0 -0.1961 0 0 0 0 00000 0 00047 0 0 0 -O.0020 0 9 4257 -0.0026 0 0 0 -02685 0 0 0 0 0.1990 5.9898 4.9703 0 0 0]
0
A0[-31.4655-276.7857 226.4519 5.5489 271.1380 0 -0.0081 0.2186 0 0 0.1076 0 0 0 1.5554 0 0.000O 0 0 0 0 1.5554 -0.0243 0 0.0081 -0.2195 0 0 132.4471 0 -OO705
-2 4137
34.3642
0
0
0)
Al -
(12.2672 0.0071 0.0004 0.1870 -0.9037 0 3164
185.5172-158 1694 0 0 0 -0.1522 0 0 0 0 0.0115 -16 3235 0 0 0 21.0641 -16 1365 0 0 0 0.1528 0 0 0 O 32.1284 -20.5989 0 0 -0.2439]
Bl =■ [-17.9606 17.9606 140.6221 271.1380 0 0 0 0 0 0 0 0 -0.9000 0 0 0.0000 0.0008 0.0956 0.0OO0 0 O.OOOO 0 0 0 0000 1.0000 123 6930 0.0131 0 0.0006 0 0 0 76410 0 0 -0.0151 0.0374 0 0 0 -154.5053 5.9898 0 0 0 B2(17.9606 0 0 0 0 0
17.9606 0 0.0008 1.0000 0 0
0
0 -1.0]
17.9606 -1.5833 0 0 0 0 0 0 -0.9000 0 0 0 0 0 0 0 0 0 O.0008 0.0008 0 0 0 0 0 1.0000 1.0 0 0 -O.0151 -642.1701 2.5444 0 0 1.0 0 -1.0000 0 0 10000 0 0]
q -{Tl -9.6126 e n d Ml Ml* -347 6812]-; S -[-0.0002 -211.6960-291.5135 -I 5231 n w -291.5135 -0 1225 -55.5034]';
% Here we experiment with the values given by cobb % Douglu(alpha>=0.9 and beia-0.1) to obtain the best
378
Differential Models and Netural
Systems
% possible constant term A in the equation: y =A'K A a'L'
% % In the final analysis for United States data log(A) = 2. logk =log(K)'(.9); iogL-iog(L)«(.l): logy -log(y); llpy- 2 + logL + logk; subplot (2,1,1 ),plot( YEAR.Iogy, YEAR,llpy,'+') title('cobb douglas on United States-log(income)-'); ylabel('llpy'); xlabe!('validity of the use of cobb douglass alpha and beta');
The polynomial coefficients and their standard deviations for US3 m are: % C(t) -cO +cl*YT(t) +c2,YT(t-h) + tl'YT{l)' +c4*YT(t-h) +c5*R(t) + c6'R(t-h) +c7*(17.2335 +O.8380.*y(0-0 3061.-yD(t-h)-0.3175.'R(t)+3.6165.*RD(t)-3.3413.*P(t)-l.77l4.«RPD(t)) - M(0); c0=-25.2258. cI =-0.0017, c2 =0.0002736. c3 =-0.0001802, c4 =-0.0001135. c5 =0 6036, c6= 3.9353, c7=0.4041 16.5075 0.0004 0.1777 0.0808 0.1668 2 6985 4.1023 0.2254 % l(t) -iO +il*y(t) + i2'y(t-h) -i3'y'(0 + J4»y'(«-h) +i5'R(t) +i6'R(t-h) +i8'L(t) +i9»L(t-h)-il l*K(t) -i!3'ML(t) iO =-53.5756, il =-0.0326, i2- 0.0477, i3- 0.2187, i4 --0.I292, i5 - -26440, i6-5.7933, 95.7551 0.06310.0615 0.0482 0.0465 1.2571 3.2116 iS = 7.5730, i9- -7.1687, il 1 - 0.8737. i 13 =0.0439 3.8055 2.8628 0 1146 0.0800 % X(t) =x0 + xl*y(f) + x2«y(t-h) +x3»/(0 +x4,y'('-h) +x5*R(t)+ x8*L(t) +xl0'L'(t-h) *xl2*P(t) +xl6'ta(t) +xl5'e(t)+xl7*d(t) xO =-89.1936. xl= 0.6169. X2= -0.1565, x3= -0.3130, x 4 - -0.1824, x5 = 19.6237, x8= -2.3689, xl0= -16.9896 298.4460 0.2162 0.1735 0.1939 0.1311 4.4968 6.5091 6.7356 x 12 -8.7879. x 16 --0.0226, x 15- 122.4798, x 17 - 0 3.3356 0.0142 98.0708 •/. G(t) =g0 - gl'y(t) - g2*y(t-h) +g3'y'(t) +g4'y-(l-h) +g5'R(t) +g8*L(t); gO- 675.2828, gl - 0.0436, g2 -0.0347, g3 --0.1267, g4--0.0527. g5--3.9923, x6 --0.9466 181.3267 0.1076 0.1028 0.1143 0.1084 3.3706 3.2503 %L(t) =m0 -ml*y(t)+ m2*y(t-h) +m3*R(t) +m4»R(t-h) +m5'R,(t-h) +m6,p(t); mO -108.1063, ml = -0.0195, m2 -0.1594, m3 --3.3462, m4 --7 2497, m5 --0.7915. x6 = 1.9784 35.1699 0.1250 0.0890 3.6606 6.7577 3.9036 4.9497 % The economic dynamics can be put in matrix form '/. as follows: x'(t) - A-l*x(t-h) - AO'xOHAl'xft-hJ+B'uft) %x-[y R L K P E]'; %u-[qS];
Luxembourg
24 US
23 UK
22 Turkey
21 Switzerland
70 S w e r l r n
19 S p a i n
1« P o r l u g i l
17 Norway
1 > N e m Zealand
IS N e l l m l a n d s
H
13 J a p a n
12 Italy
I I Ireland
10 Iceland
9 Greece
8 W e a l Germany
7 Franca
C Finland
5 Denmark
4 Canada
3 Belgium
2 Austria
I Australia
25
30
Government Consumption for OECD Countries % of GDP at market prices
I
579
380
Differential Models and Netural Systems
Appendix
381
% This program was written for chapter 6 at 7:16PM 2/23/97 % Canadian's data is in Billions of Canadian dollars % adopted from International Financial Statistic Yearbook 1994&1995 except where noted % Use ndu.m to obtain the matrix entries YEAR '[1966 1957 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 199! 1992]'; EX -[12.56 14.16 16 17 17.84 20.08 21.17 23.74 29.77 37.81 38.95 44.25 51 18 61.15 75.07 87.58 96.88 9665 103 44 12604 134.92 138.12 145.42 159.31 163.9 168.92 164.85 181 19)'; IMP -[-12.58 -13 46 -15 19 -17.71 -17.83 -19.53 -22 78 -28 02 -37 37 -41 36 -45 28 -51.25 -60.05 -73.28 81.93 -93 -82.6 -89.83 -110.63 -123.39 -133.37 -140.5 -156.38 -166.09 -171.13 -172 81 -187.25)'; X -EX-IMP; TX -[9.64 10.57 11.82 16.41 17.23 19.1122.04 25.85 31.62 34.89 37.72 38.24 41.64 48.22 57.49 72.28 73 34 77.33 85.3 91.199.21 111.55119.31 132.54 138.79 142.4 146.01]'; K -[15.92 16.25 16.49 18.14 19.01 21.57 23.88 28.86 35.78 41.85 46.71 50.23 54.58 63 44 72.29 86 12 81.33 81.23 84 7 94 2 101 56 116.72 132.79 146.08 141.38 132.01 128 921'; KP =|0 33 0.24 1 65 0 87 2.56 2.31 4.98 6.92 6.07 4.86 3.52 4.35 8.86 8.85 13 83 -4.79 -0.10 3 47 9 50 7.36 15 16 16 07 13 29 -4 70 -9.37 -3.09 -0.04]': D - [ 1 28.21 74 1 48 24.37.78 I 86 3.59 1 37 2 33 1.86 1 05 4.99 34 1.19 -9 75 -2.85 4.76 2 28 2 56 3 07 3 8 3.62 -2.84 -3.68 -3.28]'. DD = [ I . J 128 21 74 148 24 37.78 1.86 3.59 1.37 2.33 1.86 1.05 4 99 .34 1.19-9.75-2 85 4.76 2.28 2.56 3.07 3.8 3.62-2 84-3 68]': I =[16 586 15 888 16 499 18.699 18.244 20.728 22.809 29 436 37.771 39 805 47.02149.262 52.872 64.801 69.137 84.100 70 195 76.423 89 460 96.479 104.117 119 788 136 585 149 682 138.554 132.641 127.390]'; e -[108 3 108.1 108.3 108.3 1118 1155 117.9 116.7 119 3 1)4 7 118.3 109.8 102.3 99.6 99.8 97.3 94 6 94 7 90 I 85 5 84 88 94.8 98.5 100 101 8 96 6]': P -[22 3 23 1 24 25.1 25.9 26.7 28 30.1 33.4 37 39 7 42 9 46.7 51.0 56.2 63.2 70 74.1 77.3 80.4 83.7 87.4 90 9 95 5 100 105 6 107.2)': Pfe - P ' e ' e : ta =[8 030 8 852 9.662 10 722 11.302 12.320 13.711 15.598 18.257 17.584 21.520 23.907 25.563 27.728 28 733 37 737 40.356 41 417 42.714 47.212 53.827 59.719 67.790 76.214 76.596 80.772 84.8)'. F =[6 24 92 38 80 173 177 240 428 375 431327 240 369 53 153 25! 27 24 7 41394 663 763 766 710 869]'; B -[-438 327-318-440-238 1335 673-252-1001 1016-715 1147 1206 5164 491 15413-J12S 488-348 307 -2033 2806 3166 -951 2517 3738 -4239]'; BD =[424-438-327-318-440-238 1335 673-252-1001 1016-715 1147 12065164 491 15413 -4125 488348 307 -2033 2806 3166 -951 2517 3738)': T = |X-B-F]; L =[48.7 50 5 53 9 57.5 58.4 61.5 66.2 72 8 75.3 70.8 74 6 76 3 79 4 84.1 81.3 83 74.8 79.7 89.3 94.3 94.2 98.8 103 3 103.4 100 95.6 95.9]'; LD - [45 8 48.7 50.5 53.9 57.5 58.4 61.5 66.2 72.8 75.3 70.8 74.6 76.3 79.4 84.1 81.3 83 74.8 79.7 89.3 94.3 94.2 98 8 103.3 103.4 100 95.6]'; LP -(1.80 3 40 3.60 0.90 3.10 4.70 6.60 2.50 -4.50 3.80 1.70 3.10 4.70 -2.80 1.70 -8.20 4 90 9.60 5.0 -0 10 4.60 4 50 0.10 -3.40 -4.40 0 30 5.90]'; LPD -[2.9 1.80 3.40 3 60 0 90 3.10 4.70 6.60 2.50 -4.50 3.80 1.70 3.10 4.70 -2 80 1.70 -8.20 4.90 9.60 5.0 0.10 4.60 4.50 0.10 -3.40 -4.40 0.30]'; w -[15.1 16.2 17.3 18.8 20.3 22.1 23.8 25.929.4 34 38.7 42.9 45.9 50 5561.6 68.9 73.7 78 81 83.6 857 89.7 94.6 100 105.5 108.2]'; y -[64.39 69.06 75.42 83.03 89.12 97.29 108.63 127.37152.11 171.54 197.92 217.88 241.6 276.1 309.89 355 99 374 44 4O5.72 444.74 477.99 505.67 551.6 605.91 650.75 669.51 676.48 690.12]'; vD -[57.52 64.39 69.06 75.42 83.03 89.12 97.29 108.63 127.37 152.11 171.54 197.92 217.88 241.6 276.1 509 89 355 99 374.44 405.72 444.74 477.99 505.67 551.6 605.91 650.75 669.51 676.48]'; GDP «v: \? -[4 67 6.36 7.616.09 8.17 1134 18.74 24.74 19.43 26.38 19.96 23.72 34.50 33.79 46.10 18.45 31.28 39.02 33.25 27.68 45.93 54.3144.84 18.76 6.97 13.64 22.74]'; vPD -[6.87 4 67 6.36 7 61 6.09 8.17 11.34 18.74 24.74 19.43 26.38 19.96 23 72 34.50 33.7946.10 18.45 31.28 39.02 33 25 27 68 45.93 54 31 44.84 18.76 6.97 13.64]'; G -[9.64 11.09 1269 14.19 16.45 18.23 20.1422.85 27.48 33.27 38.27 43.4147.3952.29 59.25 68.79 78.66 84.57 89.0995 52 100.13 10579 114.47 124.11 135 15 144.89 150.39)';
382
Differential Models and Netural Systems
C *(37.95 41.07 44.84 49.09 51.85 56.27 63.02 72.07 84.23 97.57 111.5 123.56 137.43 153.39 172.42 196.19 210 51 231 45 251.65 274.5 297.48 322.77 349.94 378.94 394.32 411.96 422.52]'; n =y.\L: M =[10.13 16.19 15.97 15.30 15.57 17.61 19.77 21.5 21.83 25.97 26.36 29.1 31.12 31.56 34.75 36.75 39.82 44.09 52.44 68.91 80.22 85.20 90.12 95.45 96.22 100.45 108.02]'; MD -[9.43 10.13 16.19 15.97 15.30 15.57 17.61 19.77 2J.S 21.83 25.97 26 36 29.1 31.12 31.56 34.75 36.75 39.82 44.09 52.44 68.91 80.22 85.20 90.12 95.45 96.22 100.45]'; MP =(6 06 -0.22 -0.67 0.27 2.04 2.16 1.73 0.33 4.14 0.39 2.74 2.02 0.44 3.19 2.00 3.07 4.27 8.35 16.47 11.31 4.98 4 92 5.33 0.77 4.23 7.57 6.38]', MPD -[0.70 6 06 -0.22 -0.67 0.27 2.04 2.16 1.73 0.33 4.14 0.39 2.74 2.02 0.44 3.19 2.00 3.07 4.27 8 35 16.47 11.31 4 98 4.92 5.33 0.77 4.23 7.57]'; ML -[-9.2127 -15.2220 -14.9552 -14.2137 -14.4200 -16.4242-18.5578 -20.1980 -20.3830 -24.3417 -24.6128 -27.2154 -29.1072 -29.3618 -32.3007 -33.9659 -36.7074 -40.8828 -49.1377 -65.4246 -76 6375 -81.4946 86.1959 -91.2781 -91.8415 -95.8893 -103.4429]'; R =[5.69 5.94 6.75 7.58 7.91 6.95 7.23 7.56 8.9 9.04 9.18 8.7 9.27 10.21 12.48 15.22 14 26 11.79 12.75 11.04 9 52 9 95 10.22 9.92 10.85 9.76 8.77]'; RP =[0.25 0 810 83 0.33-0.96 0.28 0.33 1.34 0.14 0.14-0.48 0.57 0.94 2.27 2.74-0.96-2.47 0.96-1.71 -1.52 0.43 0 27 -0.30 0.93 -1.09 -0.99 -0.92]'; RPD -[0.48 0.25 0.81 0.83 0.33 -0 96 0.28 0.33 1.34 0.14 0.14 -0.48 0.57 0.94 2.27 2.74 -0.96 -2.47 0.96 1.71-1.52 0.43 0.27 -0.30 0.93 -1.09 -0.99]'; RD -[5.21 5.69 5.94 6 75 7.58 7.916.95 7.23 7.56 8.9 9.04 9.18 8.7 9.27 10.21 12.48 15.22 14 26 11.79 12.75 11 04 9.52 9 95 10.22 9.92 10 85 9.76]'; YT =y-TX, YTD =[48.66 54 75 58.49 63.60 66.62 71.89 78 18 86.59 101.52 120 49 136.65 160 20 179.64 199.96 227.88 252.40 283 71 301.20 328.39 359.44 386.89 406.46 440.05 486.60 518.21 530.72 534.08]'; YTP =[3 74 5 11 3.02 5.27 6.29 8.41 14.93 18.97 16.16 23.55 19 44 20.32 27.92 24.52 31 31 17.49 27.19 31.05 27.45 19 57 33.59 46.55 31.61 12.51 3.36 10.03 19.13]'. YTPD-(6.09 3.74 5.11 3.02 5.27 6.29 8.41 14.93 18.97 16.16 23.55 19 44 20.32 27.92 24.52 31.31 17.49 27.19 31.05 27.45 19.57 33 59 46.55 31.61 12.51 3.36 10.03]'; price-[ 21.5 22.3 23.1 24 25.1 25.9 26.7 28 30.1 33.4 37 39.7 42.9 46.7 51.0 56.2 63.2 70 74.1 77.3 80.4 83.7 87-190.9 95 5 100 105 6 107.2]'; for i=1:27 pp(i)-price(i-M)-price(i); end pie-[pp]': DP- pie P: logy-log(y): % TX -Revenue.fbillions Canadian dollars) %n »laborproductivity(A/l - GNP/employment). % -From Yearbook of labor statistics(intemational labor office) % K -capiial stock(Gross fixed capital formation) % I -investmenHGross capital formation) % D -increase in stock
% The UN(National accounts statistics) defines Gross capital % formation as the sum of the increase in stocks and gross fixed •/• capital formation % y -GDP(Gross domestic product)
%yD-y(t-l) %yP-y(t) •/.yPD-y'(t-l) % X -Net export export-import of goods and services) % d -preferential trade agreement transportation •/• Pfe -import price level in foreign currency (P'e'e) % ta -tariffs (indirect taxes net subsides: % by. UN National acounts statistics)
Appendix
383
% E -cumulative balance of payment % F =Net private outflow of capital % C =private consumption % M.M1 "money supply % Md =money demand(liquidity). Where, ML- md-M % P -consumer prices % R -interest rate(Govemment bond yeild) %RD-R(t-1) % RP -R'(t) %RPD-R(t-1) % pie^inflation % L =labor(industrial production) % B -Balance of payment(deposit money banks) % T -Net government transfer of capital to foreigers and firms % G -Government consumption % DP -D'log(P) -p(t)'/p(t)
% %Bert=|X.G.I.C.GDP.PGDP.R,M.QM.YEARJ [years.v]=sue(YEAR); XI=[2I.55.X(I .-years-1)]; GI=[8.27:G(l:years-t)]. 11 =[ 14.423;I( 1 :years-1 >]: C:=[34.7|:C(I years-!)]: Mil -[9.43:M(i.years-l)]: Rl=[5.21:R(l:years-l)). GDPl-|57.52;GDP(l:years-l)l. DGDP-GDP-GDPI: Z=X-G+I-C:
ZI=XI~GI-II+C. %temp-[YEARXI Gl II CI Zl y l y D M U RDl yPDl RPRPDTI TPP-PFPl P-PFl G L L D L P L P D L l LD1 LPI LPD1 t t l e e l d d l ] , °'osav« Bert I asc temp -ascii -double -tabs %clear Bertl temp; %ML - L-M - ml-mO - ml*y(t) - m2'y(t-h) - mS'RO) - m4'R(t-h) - m6'P(t) - ntf'R'O-h) % L(t) =m0 -^ml >(()+ m2*y(t-h) +ird,R(t) +m4*R(t-h) +m5*R'(t-h) +m6,p(t); temp-(Mones(si2e(y)),y,yD.R,RD,RPD.PJ; thl=arx(temp.[0.1 1 1 1 1 1 1 0 0 1 0 1 1 0]); Lp-predict([M.ones(site(y)),y,yD.R,RD,RPD,P],thl,4); MM-lhl(6.1:7); % C(t) -cO -cl 'YT(t) -c2«YT(t-h) + c3*YT(t)' +c4*YT(t-h) *c5'R(0 * c6*R(t-h) +c7*(17_2335 +0.8380 ♦>(!) -0.3061.'yD(t-h) -0.3175.'R(t) +3 6165 •RD(t>3.3413',P(t) -1.7714.'RPD(i)) - M(t)); temp=[Cones(size(R)),YT,YTD,YTP>YTPD>R,RD,Ml.], thc=arx(temp,I0.1 1 1 1 1 1 1 1 0 0 1 0 1 0 10]}; Cp=predict([C.ones{sUe(R)),YT,YTD,YTP,YTPD,R,RD,ML],thc,4); CC-thc(7,l:8): % l(t) ' i 0 ^il'y(t) - i2'y(t-h) - i 3 V ( 0 + i4,y'(Mi) +i5'R(t) +i6'R(t-h) +i8'L(t) +i9'Ut-h)-il l*K(t) -il3'ML(t) temp-[Iones(si2e(y)),y,yD.yP,yPD.R.RD,L,LD,K,ML]; thi-arx(temp.[0,l l l i l l l l l l l O O I O I O l O l O 0]); Ip=predict([I.ones(size{y)).y,yD,yP,yPD,R,RD,L,LDJCML],thi,4); Il=thi(IO.I:ll); •/. X(t) -xO + xl'y(t) - x2'y(t-h) +x3'y'(<) ♦x4«y'(t-h) -x5'R(t)+ x8'L(t) ^xlO»L*(t-h) +xl2*P(t) -x!6*ta(t) +xl5'e
384
Differential Models and Neturat Systems
temp-[Xonej(size(y)),y,yD,yP,yPD,R,L,LPD,P,ta,e]; thx-anc(temp,[0.1 I 1 1 I I I I 1 I 1 0 0 1 0 I 0 0 I 0 0 0]); Xp-prcdict([X,ones(size(y)),y,yD,yP,yPD,R,L,LPD,P,ta,e],thx,4); XX-thx( 11.1:12); % G(t) "gO + gl*y(t) + g2'y(t-h) +g3*y(t) +g4»y(t-h) +g5»R(t) -g8'L(t); temp-[Gones(size(y)),y,yD,yP,yPD,R.L]; thg-arx(temp,[0,l 1 1 1 I 1 1 0 0 1 0 1 0 0)); Gp=predict([G,ones(size(y)),y,yD,yP,yPD,R,L],thg,4); GG=lhg(6.l:7); %y(t) - zsO +zsl 'y(t) +zs2»y(t-h) +zs4«y'(t) +zs5'R(t) +»8*L(t) +zslO»L'(t) +zsl3*Ml(t) -zsI4«TX(t) +zsl5'e(t) -zsl6»ta(t) +zsl7»d(t)*(0) +l/h,K,(t); h-l" temp-[yones(size(y)),y,yD,yP,R.L.LP,M.TX,e,ta,KP]; thy=arx(temp.[O.I 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0]); yp-predict([y,ones(size(y)),y.yD,yP.R,L,LP.M,TX,e,ta,KP].thy,4); yy=thy(l 1.1:12); %cy(i) = ylO -yl l'y(t) - yl2*y'(t) + yl3*R(t) + yl5»ML<«) +yl8'YT(t) temp-[Cones(size(y)),y.yP,R,ML.YT]; thcy«arx(temp,[0,l 1 1 I 1 1 0 0 0 0 0 0 ] ) ; cyp-predict([C.ones(size(y)),y,yP.R,ML,YT],thcy.4); ccy=thcy(5.l:6); «cly(t) = 10 H.VKP(t): dK(t)/dt = D
%w- real wage rate per unit time •/.m(w) - [Myy(2V(w)Hl/lyy(l))
Appendix •/.D(t) - a(l-c)'y(t) -kO'K(t) +kl3 # K(t) +U*R(t) +L5*L(1) -L6»P(t) +v»y(0 temp-[Dones(size
subplot (2.1. l),plot(YEAR.C,YEAR.Cp,'-) uileCCanada-PRlVATE CONSUMPTION-'); ylabel('C); xlabel('YEAR-): subplot (2.1.2),plol(YEAR,X.YEAR,Xp.-+') title('Canada-EXPORTS-'): ylabel('X'); xlabel('YEAR'); pause subplot(2.l.l),plot(YEAR.I.YEAR.Ip.V) tiIle(■Canada-I^VESTMENT--,): vlabel('l'): xlabel('YEAR'): subplotC.1.2).plot(YEAR.G.YEAR.Gp>') litle('Canada--GOVERNMENT CONSUMPTION-); ylabel('G): xlabelCyear'); pause clg subplotC 1. l).plot(YEAR,M.YEAR,Lp.V) title('Canada--MONEY DEMAND--1): ylabel('Mr); xbbeK'YEAR): subplot(l 1.2).plot( YEAR^,YEAR,[Xp+Ip-Gp-Cp)>) titleCCanada-AGGREGATE DEMAND-1); ytabeK'Z'); xlabel(YEAR'); pause
% subplot (2.1,1 ),plot(YEAR,DP,YEAR,dlp,'+') titleCCanada-CONSUMER PRICES-'); ylabel('DF); xlabelCYEAR); subplot (2.1.2),plot(YEAR,y,YEAR,yp,'+') title('Canada-INCOME-'); ylabel(GDP'): xlabel('YEAR'): pause subplot (2.1.1 ),plot(YEAR,C. YEAR,cyp,'+') litM'Canada-rNCOME'CONSUMPTlON--'); ylabel('cy'). xlabel('YEAR'). subplot(2.1,2).pk)t( YEAR.I.YEAR.lyp,1*') titlefCanada-lNCOME/INVESTMENT-');
385
386
Differential Models and Netural Systems
yUbtlOypI; xlibtlfyniO; piuse dS
wbplotfl. 1.1 ).pl«(YEAR.G.YEAR,»p,*') tit^'Caaadt-irttOME/CXWERNMENT-'); ytob«K'»pl: xlabtlfYEAtf); rafeplot(2.1 J)4*)«YEARJ(,YEA JUtyp.'+T titkfCuKh-INCOME/EXPORT-'); ytabelCxyp'); xhbeirYEAR); pause % subplot (2.l.l),plot(YEAR,B,YEAR3p,'-') dtkCCmada-BALANCE OF PAYMENT-'); ytabeKB-); xJkbeVYEAR); subplot (2.1.2),plor(YEAR,l©gy,YEAR,lyp.,+1 ritleCCanadi-logl income)-"): ylabelCmy): xlabelCYEAR); pause subplot (2.1.1 ),plo« YEAR.D. YEAR,Dp>) titld'Canada-lncrcasc in stock-'): yltbeieD'). xlabclfYEAR): H The economic dynamics can be pot in matrix form % asfollow*xV) • A-1 'xV-W " A0»x(t)+A I *x(t-h>«-B»u(0 %x-[y R L K P EJ'; HA-1 -(amll -aml4aml3 0 00;Oam22 0000;Lni03 0-Lm010 0 0;a3 0t6-amlOO;0-M7*p6*P
HS-t00939-0.9052-3.22230.0S64nw-37223 0.6207-29.19731; HB-fBlB2)
Appendix % % % % %
AB=A0«B A2B=(A0*A0)«B A3B=(A0*AO*AO)*B A4B=(A0,A0,AO'AO)'B ASB-fAO'AO'AO'AO'AO'AO^B
% G=(B AB A2B A3B A4B A5B] %rank(B)=6 % rank(G)= 6 % For spectral controllability % Deltaj=eye(size(Al))."j -Aml.*(j,2.7814A(-j)) - AO % -Al.*(2 78l4"<-j)) % a-[Deltaj B) •-bforj-0.4312. 4, 10.23. 100.... % rank(a)-6 % For nonzeroeigenvalues of A-1 to be controllable via matrix B. °'b cc=eye(size(Aml)).*j -Ami % c=[cc B] •-bforj =0.4312. 4. 10.23, 100,... % rank(c)*6 % Since rank(B) is 6, where rank(a) andrank(c)are °'b also 6. the system is functional controllable LI =I:L2=0.9:L5=DD(5) L6=DD(6);h =1 •/ofory(t)-A"K'a*L\!-a) a= 0.9 % a =alpha aa=0.1 % aa=(l-a)=beta W=sum(w)/27 mw -(aa/W)."(l.-'a) % cobb Douglas states: aa+a=l =>constant return to scale. % aa+a>l =>increasing return to scale. % aa+a=l ->decreasing return to scale. % Increasing return to scale if doubling L&K more than doubles y. % constant return to scale if doubling L&K exactly doubles y. % decreasing return to scale if doubling L&K less y.
% % In cobb.m program log(A) is found to be l.S to give the best % aproximation for y(t). z0=GG( 1 )+H( 1 )+MM( 1 )*(CC{8)+1I( 11 ))+CC(l >+XX( 1) z I -GG(2M[(2)+MM(2)'(CC(8)+II( 11 ))+CC(2)+XX(2) z2=GG(3)+ll(3h-CC(3)+XX(3)+MM(3)*(CC(8)+II(ll)) z3=GG(4)-H(4)+CC{4)+XX(4) z4=GG(5)+ll(5)-CC(5)+XX(5) zS=GG(6)+Il(6)+CC(6)+XX(6>+MM(4)«(CC(8)+Il(ll)) z6=II(7)i-CC(7)+(CC(8)-ll(ll))'MM(5) z8=GG(7)+II(8)+XX(7) z9-ll(9)
387
388 zlO=XX(8) zii--il(io) zl3=(Il(ll)-CC(8))*MM(6) zl4=-CC(2) zl5-XX(ll) z!6=XX(10) zl7=0 zl 8-XX(9)+(CC(8)-zl 3)'MM(7) zI9=-CC(3) z20-CC(4) z21=CC(5) %z22=-(CC(8)+II{]l))*Ml zsO - ccy( 1 )+IIy( 1 )+ggy( 1 )+xxy(l); %constant term zs I - ccy(2)+ggy(2) i xxy(2) ; %y(t) tenn zs2 = xxy(3) ; %y(t-h) term zs4 - ccy(3)+ggy(3) : %y'(t) term zs5 = ccy(4)+xxy(4)+ggy(4) ; %R(t) term zs8 = xxy(5)-ggy(5) ; %L(t) term zs 10 -xxy(6) ; %L'(t) term zsl 3 =ccy(5) ; %Ml(t) term zsl4=ccy(6) ;%T(t)term zs 15 =xxy(7) ; %e(t) term zs 16 ^xxy( 8) ; %ta(t) term si =I-Ll*z3 "o assumption yyl = zsO: y>2 = zsl: \>3 = zs2: yv4 = zs4; y>5 = zs5: yy6 - zs8: yy7»zsl0: yy8 = zsl3: yy9 = zsl4: yyl0=zs!5; yyll =zs!6: zsO= GG( 1 )-XX( I )+CC( 1 )+U( I) aO = DD(iy(h«(!-yy(2))) al=-DD(l)*y}'3/((l-yy2)«h) a2 = -al*h a3 - a0*h*yy4 a4 - DDU)'yy5/((l-yy2) +DD(4)) a5 - DD( 1 )'yy6/( 1 -yy2) +DD(5) a6 = a0*h*yy7 a8-DD(6)' %A-I stans here; am 11 - a-11 am 11 =Ll*z4/sl am22 =L2*MM(6) aml3 -Ll'zlO'sl aml4=LI*zlO/sl Lm03 - DD(l)'y)4'mw/(l-yy2) LmOl - DD(3)*DD(l)/(l-yy2)-mw'DD(l)'yy7/(l-yy2)
Differential Models and Netural Systems
Appendix a3 = DD( 1 )*yy4/( 1 -yy2) a6 = DD(l) , y>7;(l-yy2) ami - DD(3)*DD
389
Differential Models and Neutral Systems
390 ,
al43 -DD{l) yy6/(l-yy2) - DD(5) al51 =-dp(7)»sum(P)/27 %includes P(i) al52 --MM(5)'dp(7)*sum(Py27 Voincludes P(t) al6l =BB(3) al62=BB(7) al63-BB(ll) al66=BB(H) %B1 stanshere! ssl -Ll/sl %ss!=bl s s l z l j -ssl*zl3 sslzl5-ssl'zI5 sslzl6=ssl'zl6 sslz!7=0 sslzlS =ssl'zl8 blzl4-.BB(2) , zl4; bl -BB(I) blzI5-BB(2)'zl5 blzl6=BB(21'z16 blzl7-BB(2)*zl7 blzl3=BB(2)'z!3 mwzsU -mw'yy9 muzslj -mw*yylO muzsl6 =mw*yyl I mwzsl7 - 0 m\vzsl3 =mw , yy8 plpf p4 p5 b7 b8 bl5
=dp(:)*sum(PfeV27 -dpi 5) 'dpi.6) =BB(8) =BB(9) =0
%dp(2)«Pf(t)
°'oB2 siarts here1 ssl =11 si
sslil3c7=ssl , (ll(ll)+CC(8)) 12 =m w DD( I )*yy3/( 1 -yy2) p6 =dp(7) p3 -dp(4) p2 - dp(3) al7-LI*zl9-sl a45 =a8 cO = CC(l) 10 = 11(1) xO-XX(l) pO = dp(l) MO-MM(I) ylO = ccy(l) mO 'M.VI(I) gO-GG(l) R)-BB(I)-XXU) g2 --L2'MM(I)
Appendix L5 = -0.0182;h =1 a - 0.9000 aa = 0.1000 W = 55.0333 mw = 9.0128e-004 zO --4.1312.zl = -0.0059 z2 - -0.0012 z3 - 0.0139 z4 - -0.0385 z5 = 0.1310 z6 = 0.0946 z8 - -0.0372 z9 = -0.0174 zlO -0.1220 zll = -0.0442 zl3 = -1.0741e-004 zl4 = 0.0026 zl5 = 0.0413 zl6 = -0.0079 zl7 ■ 0
z!8 = 0.0222 zl9 = -0.0033 z20 = 0.0044 z21 - 0.0043 si - 0.9861 zsO = -4.1299 aO--0.6187 al - 0.0153 a2 = -0.0153 a3 = -0O052 a4 - -0.3843 a5 = -0.0889 a6 - -0.0547 a8 - 3.7853e-O04 ami I = -0.0391 am22 - 0.0066 aml3 - 0.1237 aml4 - 0.1237 Lm03 - -4.5567e-006 LmOl = -0.0144 a3 = -0.0051 a6 - -0.0532 ami = -0.0145;M7p6P - 0.7540 b4 - -5.4877e+004 b8-6.5475e+004bl2 = 1.1014c+O05 aOl = -J.0201 all--0.0012 al2 = 0.1328 al3 = 0.0960 al4 - -0.0377 al5 - -O.0176 al6 - -0.0448 al7 = -0.0033; al8 = 0.0225; a21 = 3.9513e-004; a23 = 0.0143; a25 = -0.0191; a35 = 3.4116e-007; a44 = -0.6187: a45 = 3.7853e-004 ai 1 - -0.0047; a52 - -0.1684; a55 - 206.1996; a61 - -4.3044e+O04 a62 = -3.3688e-003; a63 = 9.7945e->-004; al I = -0.0012; al3 - 0.0960 al5 = -0.0176; a22 - 0.0021; a24 - 0.0191: al = 0.0153; a2 = -0.0153 11 =-0.0153:12 = 1.3832e-005;14 = -3.4637e-O04;b2= 1.9831e+004 b6 = -1.9580e*005; blO = -6.6309e+004; bl7 - 2.8985e-004; al 11 = -0.0045 al 12 - 0.0960; al 13 - -0.0176; al21 - 0.0021; a!22 - 0.0191; al31 = 1.0676e-0O5 al32 = -0.0407; al33 = -0 0736; al4l - -0.0153; al42 - -0.3590 al43= -0.0889 al51 =-10.6013; al52 --0.2249; al61 = 1.983 le+004 al62 - -1 9580e-005 al63 = -6.6309e-K)04 al66 » 8.8335 ssl = 1.0141 sslzl3 = -l.0893e-004 ssIz15 = 0.0419 sslzl6 = -0.0080 ss 1 z17 - 0 sslz!8 = 0 0225: bl = -1.9445e-006; blzl5 = -l.7772e+O03; blz!6 = 339.6066 blzl7 - 0: blzlj = 4.6235; mwzsl4 = -4.8708e-005; mwzsIS = -3.9703e-005 mwzsl6 - 3.09O2e-004: mwzsl7 - 0; mwzs!3 - 2.5492e-005; plpf- -6.1604 p4 = -3.5767; p5 - -0.6212; b7 = -5.314le+O03; b8 - -6.5475e+004 bl5 = 0: ssl = 1.0141: sslil3c7 = -O.0150; 12 - -1.3832e-005: p6 = 0.1850 p3 = 71.3819; p2 = 4.0858: al7 = -0.0033; a45 - 3.7853e-004; cO = 0.0939 10 = -0.9052: .\0 = -3 2223: pO = -29.8973: MO » 0.0864; ylO - 0.6207 mO = 0.0864: gO = -0.0964; fO = -1 9445e+006; g2 = -0.0777
391
Differential Models and Neutral Systems
392 > ndu Ami 1.0*+005 • OOOOO 0.0000 OOOOO 0 0 0 0 00000 0 0 0 0 0.0000 0 OOOOO 0 0 0 OOOOO 0 0.0000 OOOOO 0 0
o o.oooo -0.5488 -0 6547
o
o
1 1014
o
o
0
0
0
A0« 1.0e*004* -0 0001 OOOOO 0 0000 0.0000 0.0000 OOOOO 0.0000 0 0 0.0000 0 0 0 -00001 0 0.0000 0 0 0 0 -0.0001 0.0000 0 OOOOO 0.0000 0 0 0.0208 0 -4 3044 -0 3369 9.7945 0 0 0
0
A11.0»*005* 00000 0.0000 0.0000 0.0000 OOOOO 0 0.0000 0.0000 00000 00000 OOOOO 00000 -0.0001 0 0000 0 01983 -1.9580 -06831
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B1* IOe-0041 Columns 1 through ^ -0 0001 0 0001 OOOOO 0.0000 0 0 0 O O O O O -0.0001 0 00000 OOOOO 00000 0.0000 0 0.0000 0 OOOOO 0 0001 OOOOO 0.0000 0 0.0000 0 0 0 -00006 0 0 00000 -0.0001 0 0 -05314 -65475 0 0 0 Column 8 0 0 0 0 0 -0.0001
B2Cohimns 1 through 7 1.0141 1.0141 1.0141 -0 0150 0 0 0 0 0 0 -0 9000 0 0 0 0 00009 0 0 0 0 0.0009 0 10000 0 0 0 0 1.0000 0 0 0 -0.1850 -71.3819 4 0858 0 0 0 0 -1.0000 0 0 1.0000 Columns 8 through 9'
393
Appendix 0 0 0 0 00009 0 1.0000 0 0 V0000 0 0
1 0e*006* 00000 -1.9445
S00939 -0.9052 -3.2223 » rank(B.O) ans =
6 » rank(G.O) an» = 6 »rank(a) ans = 6 »rank(c) ans =
6
0.0864 -3.2223
0.8207 -29.8973
394
Differential Models and Neutral Systems GERMANY2.M
%10:05AM
8/17/95
% Germany's data is in billions of deutsche mark % and it was adopted from International Financial Statistic Yearbo ok 1994 YEAR =[1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]'; EX -[98.5 106.3 120.2 137.4 152.9 167.0 181.9 214.1 275.9 269.6 306.5 324.5 341.2 374.6 420.7 481.7 517.7 522.2 585.8 644.7 637.0 637.5 687.9 788.3 879.7 1015.3 1069.4]'; IMP -[-90.9 -89.0 -100.4 -119.8 -138.8 -152.9 -163.7 -185.6 -232. 4 -238.4 -277.8 -295.6 -303.0 -359.9 -422.2 -464.8 -477.3 -481.6 530.1 -569.5 -526.3 -525.0 -566.1 -643.5 -713.8 -827.9 -870.2]'; X -EX-IMP; %TX =[1972-91 200.165 231.465 247.992 263.210 293.542 315.21 33 6.77 362.16 407.23 430.04 449.03 462.00 485.36 508.96 530.79 552.3 7 573.78 617.65 654.88 756.40 ] ' ; TX =[67300 68700 73000 86200 258900 288900 327570 387490 420780 437470 492200 538350 573330 617030 661200 690430 724200 754320 795 200 834300 866290 890300 923590 998710 1043290 1171120 1267090]'; Tx =[150850 162280 67300 68700 73000 86200 258900 288900 327570 387490 420780 437470 492200 538350 573330 617030 661200 690430 724 200 754320 795200 834300 866290 890300 923590 998710 1043290 11711 20 1267090]'; %K = (NaN NaN NaN 139590 173690 199600 214340 225440 216390 2145 40 232540 250210 266790 304820 337980 338220 327890 340810 350670 355810 373480 385780 409900 448520 507780 564900 593590]'; i -[NaN NaN NaN 154990 189090 203400 216940 232640 221890 21124 0 246240 261410 274690 331420 356880 338120 335090 339310 355990 3 57060 376400 385220 420200 464530 519590 571500 588970]'; d =[]'; e «[40.3 40.4 40.4 41.1 44.2 83.5 50.5 60.8 62.3 65.7 64.1 69.5 80.4 88.0 88.8 71.6 66.5 63.3 56.8 55.2 74.5 89.8 92.0 85.8 100.0 97.5 103.5]'; ta -[9900 11500 13000 14350 18600 19700 21270 23510 26910 28180 32800 34970 38560 45150 13480 14000 14810 16210 16480 16440 17020 19150 19530 23380 24980 29260 30580]'; E = (]'; epl - [ 1 1 6 . 9 1 0 9 . 3 1 1 0 . 1 1 1 5 . 8 1 1 9 . 9 1 1 9 . 1 1 1 6 . 4 1 1 6 . 9 1 1 3 . 8 106 .4 1 0 3 . 9 1 0 3 . 0 1 0 2 . 3 1 0 2 . 6 1 0 3 . 3 1 0 1 . 0 9 7 . 5 9 3 . 5 9 2 . 5 9 3 . 7 9 5 . 3 95 .2 9 5 . 0 9 7 . 3 1 0 0 . 0 1 0 1 . 4 9 9 . 0 ] ' ; w =[17.1 18.1 18.9 20.8 24.5 27.7 30.6 34.6 39.6 44.1 46.7 50.7 54.1 57.9 62.8 67.4 70.9 73.8 76.2 79.3 83.5 87.6 91.0 94.9 100.0 107.2 1 1 4 . 8 ] ' ; B = [ 0 . 0 7 - 1 . 4 3 - 0 . 2 3 - 1 . 2 9 2.32 1.51 0.20 - 0 . 6 8 - 4 . 3 7 - 5 . 9 7 0.6 6 - 0 . 0 3 4.26 5.24 - 6 . 6 1 -1.56 3.55 -0.36 - 0 . 8 6 - 9 . 7 6 - 2 6 . 4 3 - 8 . 2 7 - 2 . 8 7 - 2 5 . 0 0 - 1 5 . 1 2 6.56 4 0 . 9 1 ] ' ; BD - ( 0 . 0 3 0.07 - 1 . 4 3 - 0 . 2 3 - 1 . 2 9 2.32 1.51 0.20 - 0 . 6 8 - 4 . 3 7 - 5 . 9 7 0.66 - 0 . 0 3 4.26 5.24 - 6 . 6 1 -1.56 3.55 - 0 . 3 6 - 0 . 8 6 - 9 . 7 6 - 2 6 . 4 3 -
Appendix
395 GERMANY2.M
8.27 - 2 . 8 7 - 2 5 . 0 0 - 1 5 . 1 2 6 . 5 6 ] ' ; F - [ - 0 . 8 4 -0.76 -0.79 -1.11 -1.65 -2.04 -2.39 -3.21 -3.57 -3.66 -3.74 -3.82 -4.93 -5.43 -5.91 -4.88 -4.56 -4.41 -4.09 -3.73 -4.90 -5.72 -6.39 -5.71 -7.15 -7.03 - 8 . 6 1 ] ' ; T « [X-B-F]'; GDP = [ 4 8 8 . 2 4 9 4 . 4 5 3 3 . 3 5 9 7 , 0 6 7 5 . 3 7 4 9 . 8 8 2 3 . 1 9 1 7 . 3 9 8 3 . 9 1 0 2 6 . 6 1120.5 1195.3 1283.6 1388.4 1472.0 1535.0 1588.1 1668.5 1750.9 1 823.2 1925.3 1990.5 2096.0 2224.4 2425.2 2635.0 2 7 9 4 . 2 ] ' ; y - [ 4 8 8 . 2 494.4 533.3 597.0 675.3 749.8 823.1 917.3 983.9 1026. 6 1120.5 1195.3 1283.6 1388.4 1472.0 1535.0 1588.1 1668.5 1750.9 1 823.2 1925.3 1990.5 2096.0 2224.4 2425.2 2635.0 2 7 9 4 . 2 ] ' ; Y - [ 4 5 9 . 2 488.2 494.4 533.3 597.0 675.3 749.8 823.1 917.3 983.9 1 0 2 6 . 6 1 1 2 0 . 5 1 1 9 5 . 3 1 2 8 3 . 6 1 3 8 8 . 4 1 4 7 2 . 0 1 5 3 5 . 0 1 5 8 8 . 1 1 6 6 8 . 5 17 50.9 1823.2 1925.3 1990.5 2096.0 2224.4 2425.2 2635.0 2794.2 2832. oj'; yD - [ 4 5 9 . 2 4 8 8 . 2 494.4 533.3 597.0 675.3 749.8 B23.1 9 1 7 . 3 9 8 3 . 9 1 0 2 6 . 6 1 1 2 0 . 5 1 1 9 5 . 3 1 2 8 3 . 6 1 3 8 8 . 4 1 4 7 2 . 0 1 5 3 5 . 0 1 5 8 8 . 1 1 6 6 8 . 5 17 50.9 1823.2 1925.3 1990.5 2096.0 2224.4 2425.2 2 6 3 5 . 0 ] ' ; yP - [ 2 9 . 0 0 6.20 38.90 63.70 78.30 74.50 73.30 94.20 66.60 42.70 93.90 74.80 88.30 104.80 83.60 63.00 53.10 80.40 82.40 72.30 102. 10 6 5 . 2 0 0 0 105.50 128.40 200.80 209.80 1 5 9 . 2 ] ' ; yPD - [ 3 9 . 0 0 2 9 . 0 0 6 . 2 0 3 8 . 9 0 6 3 . 7 0 7 8 . 3 0 7 4 . 5 0 7 3 . 3 0 9 4 . 2 0 66.60 42.70 93.90 74.80 88.30 104.80 83.60 63.00 53.10 8 0 . 4 0 82.40 72.3 0 102.10 65.2000 105.50 128.40 200.80 2 0 9 . 8 0 ] ' ; G - [ 7 5 . 4 8 0 . 1 8 2 . 7 9 3 . 0 1 0 6 . 5 1 2 6 . 8 1 4 1 . 0 1 6 3 . 1 1 9 0 . 1 2 1 0 . 1 221 .9 2 3 5 . 1 2 5 3 . 1 2 7 3 . 5 2 9 8 . 0 3 1 8 . 4 3 2 6 . 4 3 3 6 . 4 3 5 0 . 4 3 6 5 . 7 3 8 2 . 6 397 .3 412.4 418.8 444.1 466.5 5 0 0 . 6 ] ' ; C - [ 2 7 5 . 1 282.6 300.7 330.9 368.9 409.0 452.0 495.6 533.6 583.5 6 3 1 . 9 6 8 2 . 1 7 2 5 . 9 7 8 1 . 3 8 3 7 . 0 8 8 3 . 5 9 1 6 . 1 9 5 9 . 3 1 0 0 1 . 2 1 0 3 6 . 5 106 6.4 1108.0 1153.7 1221.0 1318.7 1428.3 1 5 1 0 . 0 ] ' ; K - [ 1 2 4 . 1 114.2 119.3 138.8 172.1 196.1 209.2 2 1 9 . 3 212.7 2 0 9 . 4 225.7 242.4 264.9 301.3 332.1 331.3 323.5 340.8 350.7 355.8 373.5 385.8 409.9 448.5 507.8 564.9 5 9 3 . 6 ] ' ; KP = [ 4 . 2 - 9 . 9 5.1 19.5 33.3 24.0 13.1 10.1 - 6 . 6 - 3 . 3 16.3 16.7 2 2.5 36.4 30.8 - 0 . 8 - 7 . 8 17.3 9.9 5.1 17.7 12.3 24.1 38.6 5 9 . 3 57.1 28.7]'; D - [ 5 . 2 - 0 . 9 11.1 17.3 14.2 4.5 4.3 12.4 3.7 - 6 . 4 15.7 7.0 7.2 23.0 11.8 - 1 0 . 6 - 1 6 . 0 - 1 . 5 5 . 3 1.3 2 . 9 - 0 . 6 1 0 . 3 1 6 . 0 1 1 . 8 6.6 - 4 . 6]'; 1 =(129.3 113.3 130.4 156.1 186.3 200.6 213.5 231.7 216.4 203.0 241.4 249.4 2 7 2 . 1 324.3 343.9 320.7 3 0 7 . 5 339.3 356.0 3 5 7 . 1 3 7 6 . 4 385.2 420.2 464.5 519.6 571.5 5 8 9 . 0 ] ' ; YT =y-TX; yt -Y-Tx; M = [ 7 4 . 2 8 1 . 6 9 0 . 6 9 5 . 4 1 0 3 . 7 1 1 6 . 9 1 3 3 . 4 1 3 5 . 7 1 5 0 . 2 1 7 1 . 7 177 . 3 1 9 8 . 6 2 2 7 . 5 2 3 4 . 1 2 4 3 . 4 2 3 9 . 6 2 5 6 . 7 2 7 8 . 2 2 9 4 . 8 3 1 4 . 5 3 4 0 . 2 365 .7 4 0 8 . 3 4 3 1 . 6 5 5 1 . 4 5 7 5 . 0 6 4 0 . 7 ] ' ; ML - [ 6 1 3 . 3 614.9 611.7 638.1 686.9 711.0 723.0 764.8 832.9 838.5 868.2 884.2 8 9 3 . 0 944.9 1011.6 1079.2 1106.8 1113.6 1131.5 1149.7 1150.2 1136.9 1132.1 1178.5 1160.7 1227.8 1 2 4 8 . 7 ) ' ; L =M;
396
Differential Models and Neutral Systems GERKANY2.M
1 - [ 7 2 . 8 74.2 81.6 90.6 95.4 103.7 7 177.3 198.6 227.5 234.1 243.4 239.6 2 365.7 408.3 431.6 551.4 575.0 640.7 LD - [ 7 2 . 8 74.2 81.6 90.6 95.4 103.7 7 177.3 198.6 227.5 234.1 243.4 239.6 2 365.7 408.3 431.6 551.4 5 7 5 . 0 ] ' ; P =[43.3 44.0 44.7 45.5 47.1 49.6
116.9 133.4 256.7 278.2 697.6]'; 116.9 133.4 256.7 278.2
135.7 150.2 294.8 314.5
171. 340.
135.7 150.2 294.8 314.5
171. 340.
52.3 55.9 59.8 63.4 66.1
68.5
70.4 73.3 77.3 82.1 86.5 89.3 91.5 93,5 93.3 93.6 94.8 97.4 100.0 103.5 107.6]•; Pf -P.*e; Pfe -Pf.*e; r -[3.0 4.0 5.0 3.0 3.0 6.0 6.0 4.0 4.5 7.0 6.0 3.5 3.5 3.0 3.0 6.0 7.5 7.5 5.0 4.0 4.5 4.0 3.5 2.5 3.5 6.0 6.0 8.0 8.3 5.8]'; R -[5.0 3.0 3.0 6.0 6.0 4.0 4.5 7.0 6.0 3.5 3.5 3.0 3.0 6.0 7.5 7.5 5.0 4.0 4.5 4.0 3.5 2.5 3.5 6.0 6.0 8.0 8.3]'; RD -{4.0 5.0 3.0 3.0 6.0 6.0 4.0 4.5 7.0 6.0 3.5 3.5 3.0 3.0 6.0 7.5 7.5 5.0 4.0 4.5 4.0 3.5 2.5 3.5 6.0 6.0 8.0]'; RP -[-2.0 0 3.0 0 -2.0 0.50 2.50 -1.0 -2.50 0 -0.50 0 3. 0 1.50 0 -2.50 -1.0 0.50 -0.50 -0.50 -1.0 1.0 2.50 0 2.0 0.30 -2.50]'; RPD =[1.0 -2.0 0 3.0 0 -2.0 0.50 2.50 -1.0 -2.50 0 -0.50 0 3.0 1.50 0 -2.50 -1.0 0.50 -0.50 -0.50 -1.0 1.0 2.50 0 2.0 0.30]'; n -y.\epl; YTD -[-150400 -161800 -66800 -68200 -72400 -85500 -258200 -288100 -326700 -386500 -419800 -436300 -491000 -537100 -571900 -615600 659700 -688800 -722500 -752600 -793400 -832400 -864300 -888200 -92 1400 -996300 -1040700]'; YTP -(-94990 -1360 -04240 -13120 -172630 -29930 -38580 -59850 -33 250 -16600 -54660 -46060 -34880 -43620 -44110 -29180 -33690 -30040 -40810 -39000 -31920 -23900 -33160 -74920 -44370 -127670 -95930]' YTPD -[-11400 94990 -1360 -04240 -13120 -172630 -29930 -38580 -598 50 -33250 -16600 -54660 -46060 -34880 -43620 -44110 -29180 -33690 -30040 -40810 -39000 -31920 -23900 -33160 -74920 -44370 -127670]'; LP -[7.4 9.0 4.8 8.3 13.2 16.5 2.3 14.5 21.5 5.6 21.3 28.9 6.6 9 .3 -3.8 17.1 21.5 16.6 19.7 25.7 25.5 42.6 23.3 119.8 23.6 65.7 56 • 9]'; HP *LP' LPD -fl.4 7.4 9.0 4.8 8.3 13.2 16.5 2.3 14.5 21.5 5.6 21.3 28.9 6 .6 9.3 -3.8 17.1 21.5 16.6 19.7 25.7 25.5 42.6 23.3 119.8 23.6 65. 7]'; price-[43.3 44.0 44.7 45.5 47.1 49.6 52.3 55.9 59.8 63.4 66.1 68.5 70.4 73.3 77.3 82.1 86.5 89.3 91.5 93.5 93.3 93.6 94.8 97.4 100.0 103.5 107.6 112.0] '; for i-l:27 pp(i)-price(i+1)-price(i); end pie«[pp]'; DP-pie.\P;
Appendix
397 GERMANY2.M
% TX -tax revenue % Tx -Total current Receipts of general Governinent (million deutsch e marks) % n -labor productivity(A/1). 1966-69 data is based on (1963-100) i % % % % % % % % •d % % % % % % % % % % » % % % % % % % % % % t % % %
1970-76 data is based on (1970-100). apparent labour productivity data was used for (1984-92) which iB based on percentage change from preceding year K -capital stock(Gross fixed capital formation) I -investment(Gross capital formation) D -increase in stock The UN(National accounts statistics) defines Gross capital formation as the sum of the increase in stocks and gross fix capital formation y -GDP(Gross domestic product) yD -y(t-l) yP =y'(t) yPD-y'(t-l) X -Net export d -preferential trade agreement transportation n -labor productivity Pf -import price level in foreign currency t -tariffs E -cumulative balance of payment F -Net private outflow of capital C =private consumption M -money demand P -consumer prices R -interest rate(discount rate) RD =R(t-l) RP -R'(t) RPD-R'(t-l) pie-inflation L -employment B -Balance of payment(deposit money banks) T "Net government transfer of capital to foreigers and firms G ^Government consumption
format long; %keitcubl«(X,G,I,C,GDP,PGDP,R,M,QM,YEAR] [years,nn]=size(YEAR); Xl=[153.6;X(l:years-l)],• Gl»[62.1;G(l:years-l)]; Il«(lll.B;I(l:years-l)]; Cl«[233.6;C(l:years-1)]; Ml-[67.6;M(l:years-l)]; Rl«[3.0:R(l:yeara-l)];
398
Differential Models and Neutral Systems
GERHAMT2.M
GDPl-[420.2;GDP(l:years-l)]; DGDP-GDP-GDP1; Z-X+G+I+C; Z1-X1+G1+I1+C; %terap»[YEAR XI Gl II Cl Zl yl yD Mil RDl yPDl RP RPD Tl T P P-PF P 1 P-PF1 G L LD LP LPD LI LD1 LP1 LPD1 t tl e el d dl); taave Bertl.asc tamp -ascii -double -tabs tclaar Bertl temp; %ML « L-M = ml-mO - nl*y(t> - m2*y(t-h) - m3*R(t) - m4*R(t-h) - m6 *P(t) - ra7*R'(t-h) % L(t) -mO +ml*y(t)+ i«2*y(t-h) +n3*R(t) +m4«R(t-h) +m5*R'(t-h) +m6 *P(t); temp-[M o n e s ( s i z e ( y ) ) , y , y D , R , R D , R P D , P ] ; thl«arx(temp,[0,1 1 1 1 1 1 1 0 0 1 0 1 1 0 ] ) ; Lp«predict([M,ones(size(y)),y,yD,R,RD,RPD,P],thl,4); MM=thl(6,1:7); »L(t) =17.2335 + 0 . 8 3 8 0 . * y ( t ) -0.3061.»yD(t-h) - 0 . 3 1 7 5 . * R ( t ) +3.616 5.*RD(t-l)-3.3413.*P(t) -1.7714.*RPD(t-l); % C ( t ) »c0 + c l * Y T ( t ) + c 2 » Y T ( t - h ) + c 3 * Y T ( t ) ' + c 4 * Y T ' ( t - h ) + c 5 * R ( t ) + c6*R(t-h) +c7*(17.2335 +0.8380.*y(t) -0.3061.*yD(t-h) - 0 . 3 1 7 5 . * R(t) +3.6165.*RD(t)-3.3413.*P(t) -1.7714.*RPD(t)) - M(t)); temp-(C ones(size(R)),YT,YTD,YTP,YTPD,R,RD,ML]; thc»arx(temp,[0,1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 ] ) ; Cp-predict([C,ones(size(R)),YT,YTD,YTP,YTPD,R/RD,ML],thc,4); CC=thc(7,1:8),• % l ( t ) =»i0 + i l * y ( t ) + i 2 * y ( t - h ) - i 3 * y ( t ) + i 4 * y ( t - h ) + i S * R ( t ) 6*R(t-h) +i8*L(t) + i 9 * L ( t - h ) - i l l * K ( t ) -il3»ML(t) terap«[I o n e s ( s i z e ( y ) ) , y , y D , y p , y P D , R , R D , L , L D , K , M L ] ; thi-arx(temp,[0,l 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 0]); Ip=predict([I,ones(size(y)),y,yD,yP,yPD,R,RD,L,LD,K,MI,],thi,4); II«thi(10,l:ll);
+i
% X ( t ) -xO + x l * y ( t ) + x 2 * y ( t - h ) + x 3 * y ( t ) + x 4 * y ( t - h ) + x 5 * R ( t ) + x 8*L(t) +xlO*L'(t-h) +xl2*P(t) + x l 6 * t a ( t ) + x l 5 * e ( t ) + x l 7 * d ( t ) temp«[X o n e s ( s i z e ( y ) ) , y , y D , y P , y P D , R , L , L P D , P , t a , e ] ; thx»arx{temp,[0,1 1 1 1 1 1 X 1 1 1 1 0 0 1 0 1 O 0 1 0 O 0]>; Xp=predict([X,ones(size(y)),y,yD,yP,yPD,R,L,LPD,P,ta,e],thx,4); XX=thx(ll,l:12); % G ( t ) -gO + g l « y ( t ) + g 2 * y ( t - h ) + g 3 * y ' ( t ) + g 4 « y ( t - h ) 8«L(t); temp»[G o n e s ( s i z e ( y ) ) , y , y D , y P , y P D , R , L ] , • thg-a^x( temp,[0,1 1 1 1 1 1 1 0 0 1 0 1 0 0]); Gp^predictaG^nesfsizetyn^yD^P/yPD.R^J.thg^); GG=thg(6,l:7);
+g5*R(t)
+g
%y(t) = zsO + z s l * y ( t )
+zs8*L(t)
+
+zs2*y(t-h)
+zs4*y'(t)
+zs5*R(t)
Appendix
399 GERMANY2.M
z s l O * L ( t ) +zsl3*Ml(t) -zsl4«YT(t) +zsl5*e(t) + z s l 6 * t a ( t ) +zsl7*d( t ) * ( 0 ) + l / h * K ' ( t ) ; h=l temp»[y o n e s ( s i z e ( y ) ) , y , y D , y P , R , L , L P , M , T X , e , t a , K P ] ; thy=arx(temp,[0,1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 ] ) ; yp-predict((y,ones(size(y)),y,yD,yP,R,L,LP,M,TX,e,ta,KP],thy,4); yy=thy(ll,l:12); %cy(t) » ylO + y l l * y ( t ) + y l 2 * y ' ( t ) + y l 3 * R ( t ) + y l 5 * M L ( t ) t) temp»[C o n e s ( s i z e ( y ) ) , y , y P , R , M L , Y T ] ; thcy=arx(temp, [ 0 , 1 1 1 1 1 1 0 0 0 0 0 0 ] ) ; cyp»predict([C,ones(size(y)),y,yP,R,ML,YT],thcy,4); ccy»thcy(5,1:6);
+yl8*YT(
%Iy(t) - 10 + l / h * K P ( t ) temp«(I o n e s ( s i z e ( K P ) ) , K P ) ; thly»arx(temp,[0,1 1 0 0 ] ) ; Iyp-predict([I,ones(size(y)),KP],thly,4); Ily-thly(l,l:2); » G y ( t ) - gsO + g s l * y ( t ) + g s 4 * y ' ( t ) + y s 5 « R ( t ) + g s 8 * L ( t ) temp»[G o n e s ( s i z e ( y ) ) , y , y P , R , L ] ; thgy»arx(temp,[0,1 1 1 1 1 0 0 0 0 0 ] ) ; gyp-predict([G,ones(size(y)),y,yP,R,L],thgy,4) ; ggy=thgy(5,1:6); %Xy(t) = xO + x l * y ( t ) + x 2 * y ( t - h ) + x 5 * R ( t ) + x 8 * L ( t ) + x l O * L ' ( t ) e(t) +xl2*ta(t) +xl3*d(t)»(0) terap=[X o n e s ( s i z e ( y ) ) , y , y D , R , L , L P , e , t a ] ; thxy-arx(temp,[0,1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 ] ) ; xyp=predict([X,ones(size(y)),y,yD,R,L,LP,e,ta],thxy,4); xxy=thxy(7,l:8);
+xll»
%B{t) - bO + b l * y ( t ) + b 2 * y ( t - h ) + b 3 * P ( t ) + b 4 * y ( t - h ) + b 5 * R ( t ) +b6*R ( t - h ) + b 7 * e ( t ) + b 8 » R ' ( t - h ) +b9»L(t) + b l 0 * L ( t - h ) + b l 2 * L ' ( t - h ) + b l 3 * t a ( t ) +bl5*d(t) +bl7*B(t-h) temp=[B o n e s ( s i z e ( y ) ) , y , y D , P , y P D , R , R D , e , R P D , L , L D , L P D , t a , B D ] ; thb-arx(temp,[0,1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1]); Bp»predict( (G,ones(size(y)) (yjyD.PjyPDjRjRD^RPD/LjLD/LPDjta.BD], thb,4); BB=thb(13,l:14); % y(t) - A»K*a*L"(l-a) =■->ln(y(t)) - a*ln(K) + (l-a)*ln(L) temp=[log(y) ones(size(y)),log(K),log(L)]; thly«arx(temp,[0,1 1 1 0 0 0]); lyp -predict([log(y) ones(size(y)),log(K),log(L)],thly,4); lyy =thly(2,1:3),• %w« real wage rate per unit time %m(w) - [l-lyy(2)/(w)p(l/lyy(l))
400
Differential Models and Neutral Systems GERMANY2.M
%D(t) - a ( l - c ) * y ( t ) -kO*K(t) + k l 3 * K ' ( t ) +L4*R(t) + L 5 * L ( t ) +v*y'(t) terop=(D o n e s ( s i z e ( y ) ) , y , K , K P , R , L , P , y P ] ; thD-arx(temp,(0,1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0]); Dp-predict<[D,ona8<8ize{y)),y,K,KP,R,L,P/yP],thD/4); DD-thD(7,l:8);
+L6*P(t)
%DP(t) » pO + p l * P f e ( t ) + p 2 * w ( t ) - p 3 * n ( t ) - p 4 * P ( t ) + p 5 * M ' ( t ) +p6*ML
(2,l,l),plot(YEAR,C,YEAR,Cp,'O')
title{ 'GERMANY- -PRIVATE CONSUMPTION—' ); ylabel('C'); xlabel('TEAR'); subplot (2,l,2),plot{YEAR,X,YEAR,Xp,'O') title(-GERMANY—EXPORTS--') ; ylabel('X'); xlabel('YEAR'); pause subplot (2,1,1),plot(YEAR,I,YEAR,Ip,'—') title( 'GERMANY—INVESTMENT—'); ylabel(•I•); xlabel( YEAR'); 8ubplot(2,l,2),plot(YEAR,G,YEAR,Gp,'—') title('GERMANY—GOVERNMENT CONSUMPTION--'); ylabel('G'); xlabel ('year'); pause clg subplot(2,1,1),plot(YEAR,L,YEAR,Lp,'—') title( ' GERMANY—MONEY DEMAND— ' ) ; ylabel('Ml'); xlabel('YEAR'); subplot(2,1,2),plot(YEAR,Z,YEAR,[Xp+Ip+Gp+Cp],'—•) title( "GERMANY--AGGREGATE DEMAND—'); ylabel('Z'); xlabel('YEAR'); pause % subplot (2,1,1),plot(YEAR,P,YEAR,dip,'—') title( 'GERMANY—CONSUMER PRICES— ' ) ; ylabel('P'); xlabel('YEAR'); subplot (2,1,2),plot(YEAR,y,YEAR,yp,'—•)
Appendix
401
title( 'GERMANY—INCOME— ') ; ylabelfGDF'); xlabel(-YEAR'); pause •ttbplot (2,1,l),plot(TEAR,C,YEAR,Cyp, ' — •) title('GERMANY—INCOME/CONSUMPTION-- 1 ); ylabel('cy'); xlabel<'TEAR'); •ubplot(2,l,2),plot(TSAR,I,YEAR,Iyp,' — ' ) title( •GERMANY—INCOME/INVESTMENT--'); ylabel('lyp'); xlabel('year'); pass* clg •ubolot<2,l,l),olot(TEAR,G.YEAR,gyp,' — ' ) tit)g('G«inMy-INCOMEXX)VERNMENT-');
ylabai('gyp); xlabal('TCAR'); •ubplot(2,l,2)/plot(YIAR,X,TEAR/xyp,'—') title( 'GERMANY—INCOME/EXPORT—'); ylabel('xyp); Xlab«l('YEAR'); pause % subplot (2,l,l),plot(TEAR,B,TEAR,Bp,'—') title('GERMANY—BALANCE OF PAYMENT--'); ylabal('B'); xlabel('YEAR'); subplot (2,l,2),plot
-[a01 a!2 al4 al6 al8 0; a21 a23 0 0 U S 0; 0 0 a33 0 a35 0;
402
Differential Models and Neutral Systems GERMANY2.M
0 0 0 a44 a45 0; a51 a52 0 0 955 0; a61 a62 a63 0 0 0]; %A0 -[-1.0392 0.61 0.0147 -9.7792e-6 0;-0.0056 0.0856 0 0 -0.0614 0;0 0 -0.028 0 8.4419e-7 0;0 0 0 0 5.9005e-4 0;0.0366 609.8401 0 0 -394.4422 0;-0.0034 0.0544 0.0086 0 0 0] %A1 -[alll all2 all3 0 0 all6; al21 al22 0 0 0 0; al31 al32 al33 0 0 0; al41 al42 a!43 0 0 0; al51 al52 al53 alS4 al55 al56; al61 a 162 al63 al64 al65 al66]; %A1 -[0.0212 0.3597 0.0059 0 0 0;0.0083 .0082 0 0 0 0;-1.4135e-6 -0.0234 0 0 0;-0.028 -0.7954 0.0046 0 0 0;-5.880S -0.0539 0 0 0 0; 1.8299e-4 -0.0329 -0.0037 0 0 -0.0053 ] %q %q
-[T gO e ta d Ml Ml' f0]'; = [T -0.0422 e ta d Ml Ml' 11.4319]
%S %S
-[CO 10 xO (pO-p6M0) n w xO ylO]'; -[.0029 3.8926 -12.8977 9.4705e3 n w -12.8977 -0.0968]
\ w- real wage rate per unit time % m(w) - [lyy(2)/w]-(l/lyy(l)( - 5.4718e-4 % where w ■ average of w(t) - 17.1 %B1 -[-blzl4 bl -blzl5 blzl6 blzl7 blzl3 0 0; 0 0 0 0 0 -11 0 0; -m(w)z5l4 m(w) n(w)zslS n(w)zsl6 m(w)zsl7 m(w)2sl3 0 0; -2«14 1 zs 15 zsl6 zsl7 zsl3 0 0; 0 0 plpf(t) 0 0 p6 p5 0; 0 0 b7 b8 bl5 0 0 i]; %B1 -[1.389e-8 -1.4658 -2.154e-4 7.106e-8 0 -3.081e-6 0 0; 0 0 0 0 0 0.028 0 0;9.3162e-10 0.0014 3.2584e-4 3.8212e-6 0 5.07224e-8 0 0;6.5116e-7 1 0.2277 0.0027 0 3.5454e-5 0 0;0 0 -0.0051 0 0 -5.88 08 -14.5701 0;0 0 0.0544 -0.0329 -1.6607e-6 0 0 1] %B2 - [ s s l s s l s s l - s s l ( I 1 3 + c 7 ) 0 0 0 0 0; 0 0 0 - 1 2 0 0 0 0 0 ; 0 m (w) 0 0 0 0 m m 0; 0 1 0 0 0 0 1 1 0 ; 0 0 0 - p 6 - p 3 p2 0 0 l ; 0 0 0 - 1 0 0 1 0 0] %B2 - [ 0 . 9 9 0 . 9 9 0 . 9 9 0 . 0 1 3 1 0 0 0 0 0 ; 0 0 0 4 . 0 0 6 3 e - 5 0 0 0 0 0 ; 0
0.0014 0 0 0 0 0.0014 0.0014 0;0 1 0 0 0 0 1 1 0,-0 0 0 -5.8808 3. 0917e3 286.76 0 0 1;0 0 0 -1 0 0 1 0 0] %B - [Bl B2]; %u - [q S]; %Rank(Al) *Rank(B2) %Rank(Bl) %Rank(B)
= » =
4 5 6 6
format short; LI -1 L2 -0.9 L5 -DD{5) L6 -DD(6)
Appendix
403 GERMANY2.M
h -1 zO=CC(l)+II(l)-MM(l)*(CC(8)+II( 11))+CC(1)+XX(1) zl-GG(2)+II(2)-MM(2)*(CC(8)+II( 11))+CC(2)+XX(2) z2-GG(3)+II(3)+CC(3)+XX(3)+MM(3 )*(CC(8)+II(U)) z3=GG(4)-II(4)+CC(4)+XX(4) z4-GG(5)+II(5)+CC(5)+XX{5)+MM(5 )*(CC(8)+II(11)) z5-GG(6)+II(6)+CC(6)+XX(6)-CC(8 )*MM(6) Z6-II(7)+CC(7)-CC(8)*MM(7) z8-GG(7)+II(8)+XX(7) z9-II(9) zlO«XX(8) zll--II(lO) zl3=(II(ll)+CC(8))*MM(7) zH—CC(8) zl5»=XX(ll) Zl6=XX(10) Z17-0 zl8-(CC(8)-t-zl3)*MM(7) zl9=CC(3) z20-CC(4) z21=CC(5) zsO - ccy(l)+IIy(l)+ggy(l)+xxy( 1) %constant term %y(t) term zsl - ccy(2)+ggy(2)+xxy(2) %y(t-h) terra zs2 » xxy(3) %y'(t) terra zs4 » ccy(3)+ggy(3) %R(t) term zs5 - ccy(4)+xxy(4)+ggy(4) %L(t) term zs8 = xxy<5)+ggy(5) %L'(t) terra zslO «xxy(6) %Ml(t) term zsl3 -ccy(5) %T(t) term zsl4 «ccy(6) %e(t) term zsl5 -xxy(7) %to(t) term zsl6 =xxy(8) si =1-L1*(GG(4)-II(4)+CC(4)+XX(4)) % assumption yyl = zsO; yy2 - zsl; yy3 ■« zs2; yy4 = zs4; yy5 = zs5; yy6 = zs8; yy7 - zslO; yy8 = zsl3; yy9 - zsl4; yylO =zsl5; yyll -zsl6; W - 17.1 % wage rate for the year 1966 a » 0.5582 % By Cobb Douglas aa - 0.4418 % aa « (1-a)
404
Differential Models and Neutral Systems GERMANT2.M
raw
»(aa/W).*(l/a)
zsOaO al a2 a3 ■ a4 a5 « a6 « a8 -
GG(1)+XX(1)>CC(1)+II(1) DD(l)/(h*(l-yy(2))) -DD(l)*yy3/((l-yy2)*h) -al*h a0*h*yy4 DD(l)*yy5/((l-yy2) +DD(4)) DD(l)*yy6/(l-yy2) +DD(5) a0«h*yy7 DD(6)
tA-1 starts here; anil - a-11 amll ara22 ami3 Lm03 LmOl a3 a6 ami m7p6 b4 » b8 = bl2-
«Ll*(GG(5)-II(5)+CC(5)+XX(5)+MM(5)*(CC(8)+II(ll)))/sl -L2*MM(6) =Ll*zlO/sl - DD(l)*yy4*mw/(l-yy2) « -DD(3)*DD(l)/(l-yy2) -mw*DD(l)*yy7/(l-yy2) = DD(l)*yy4/(l-yy2) - DD(l)*yy7/(l-yy2) - -DD<3)*DD(l)/(l-yy2) » HM(7)*dp(6) BB(5) BB(9) BB{12)
%A0 starts here I aOl =(L1*(GG(2) +11(2) -MM(2))*(II(11)+CC(8))+ CC(2)+XX(2)+CC(8)1-(II(11)+CC(8))*MM(7)*MM(2) )/sl al2 «L1*((GG(6)+II(6)+CC(6)+XX(6)-CC(8)*MM(7))-(II(11)+CC(8))-MM( 3)*MM(7))/6l al4 -Ll»(GG(7)+II(8)+XX(8))/sl al6 «-Ll*(II(10))/sl al8 —Ll*(MM(4)*MM(6)*(II(ll)+CC*(8)))/sl a21 -L2*MM(2) a23 «L2*MM(4) a25 -L2«MM(7) a33 »a0; a3S »ai*r*a8 a44 -aO a45 -a8 a51--dp(7)*MM(2) % a51*P(t) a52«-dp(7)*MM(4) % a52*P(t) a55 »(-MM(7)+dp(5)) t a55*P(t) a61 -BB(2) a62 -BB(6) a63 -BB(10) %A1 starts here!
Appendix
405 GERMANY2.M
all -L1*(GG(3) +11(3) +CC(3) +XX(3) +MM( 3)*(II(11)+CC(8)))/8l al3 -Ll*(II(7)+CC(7)-CC(B)*MM(7))/sl al5 =Ll*(II(9))/ol a22 -L2*MM(3) a24 =L2«MM(5) al « -DD(l)*yy3/<(l-yy2)*h) a2 - -al*h 11 *al 12 —mw*DD(l)*yy3/(l-yy2) 14 *=mw*DD(l)*yy5/
406
Differential Models and Neutral Systems GBRMANT2.H
blzl5«BB(2)»zlS blzl6-BB(2)*zl6 blzl7«BB(2)*zl7 blzl3«BB(2)»zl3 mw2sl4 »raw«yy9 IDVZS15 »niw*yylO mwzal6 »mw«yyll mwzsl7 »0 mv2Sl3 »m*f*yy8 plpf -dp(2) P4 -dp(5) P5 -dp(6) b7 -BB(6) b8 »BB(7) bl5 «BB(13)
%dp(2)*Pf(t
%B2 starts here! ss2 L2 P6 P3 p2
«-ssl*(II(ll)+CC(8)) -0.9 - -dp(7) - -dp(4) » dp(3)
% ln(y) =a*ln(K) + (1-a)*ln(L); % Xyy(l) - a; * lyy(2) - l-a; al7 =Ll*CC(2)/sl a45 -a8 cO «= CC(1) 10 - 11(1) xO - XX(1) pO - dp(l) MO » MM(1) ylO » ccy(l) pOp6mO»dp(1)-dp(7)*MM(1) gO -CG(1) fO -BB(1)-XX(1) g2 --L2*MM(1) a25 -L2*MM(6) all5-0; al23-0; al24-0; al25-0; al25»0; al26-0; al34-0;
Appendix
407 GERMD.H
The following data for germany was generated using the program germany2.m. LI - 1, L2 = 0.9, L5 - 0.0192, L6 - 5.9005e-004, h - 1, 20 = -8.9741, 2l - -0.0408, 22 - 0.0215, 23 - -0.0101, 24 =0.0057, 25 -0.6023, Z6 -0.3633, z8 -0.0342, 29 - 0.0059 210 - 0.1289, zll - 0.0148, zl3 - 9.0208e-004, 214 = 4.0661e-006, zl5 « 0.0631, zl6 * -2.08076-005, Zl7 «0, zl8 - -6.1283e-005, zl9 - 1.6255e-008, z20 « 4.8616e-008, 221 = 1.5561e-008, si ■ 1.0101, W = 59.0667, zsO - -9.0444, aO - 0, al = 0.0280, a2 - -0.0280, a3 = 0, o4 - -0.8011, a5 - 0.0046, a6 = 0, a8 - 5.9005e-004, amll - 0.0231, am22 = -0.0071, aml3 - 0.1276, Lm03 = 3.9474e-009, LmOl - -3.2318e-005, a3 - 2.7591e-006, a6 = -5.1656e-004, ami = -3.3057e-005, m7p6 =0.9943, b4 - -0.0012, b8 = 0.0199, bl2 « 3.1523e-004, aOl = -1.0392, al2 = 0.61, al4 = 0.1057, al6 » 0.0147, al8 = -9.7792e-006, a21 = -0.0056, a23 - 0.0856 a25 - -0.0614, a35 = 8.4419e-007, a44 - 0, a45 -5.9005e-004, a51 = 0.0366, a52 = -0.5590, a55 = -394.4422, a61 - -0.0034, a62 => 0.0544, a63 - 0.0086, all - 0.0212, al3 - 0.3597, al5 = 0.0059, a22 - 0.0083, a24 - 0.0082, al - 0.0280, a2 = -0.0280, 11 - -0.0280, 12 - 4.0063e-005, 14 - -0.0011, a51 = -5.8808, a52 - 609.8401, b2 - 1.8299e-004, a33 » 0.028 b6 = -0.0329, blO - -0.0037, bl7 - -1.6607e-006, alll - 0.0212 all2 - 0.3597, all3 - 0.0059, all6 = 0, al21 = 0.0083, al22 - 0.0082, al31 = -1.4135e-006, al32 - -0.0099, al33 = 0.0326, al41 = -0.0280, ai42 - -0.7954, 8143 =0.0046,
408
Differential Models and Neutral Systems GERMD.M
al44 - 0, al45 = 0, al46 - 0, 6151 - -5.8808, al52 --0.0539, al61 -1.8299e-004, al62 - -0.0329, al63 - -0.0037, al64 - 0, al65 - 0, al66 - -0.0053, ssl - 0.99, sslzl3 = -0.0131, aslzl5 - -0.0558, sslzl6 - 0.0624, sslzl7 - 0.1276, blzl4 - 1.38876-008, bl - 1.4658, 881I13C8 - -0.0131, blzl5 - 2.15416-004, blzl6 « 7.1064e-008, blzl7 » 0, blzl3 - -3.08106-006, mw -1.55286-004, rawzsl4 - 1.0111*-10, mw2Bl5 - 3.53656-005 mwzsl6 - 4.1473e-007, mwzsl7 - 0, mwzsl3 - 5.5053e-009, zl3 - 3.5454e-005, zl4 - 6.5116e-007, 215 -0.2277, zl6 - 0.0027, plpf - -0.0051, P4 - -394.5105, P5 - -14.5701, b7 -0.0544, b8 - -0.0329, bl5 = -1.66076-006, 8S2 - 0.0131, L2 - 0.9, P6 - -5.8808, P3 - 3.09176+003, p2 - 286.7589, al7 - -1.85276-008, a45 * 5.9005e-004, cO = 0.0029, 10 - 3.8926, xO - -12.8977, pO - 9.4817e+003, MO =1.9039, ylO - -0.0968, p0p6m0 -9.4705e+3, gO --0.0422 fO - 11.4319, g2 - -1.7135, a25 - -0.0071
Appendix IND1A2M V. This program was written at 8:26PM 4/11/96 % India's data is in billions of Rupees % adoptedfromInternational Financial Statistic Yearbook 1994. except where noted. •/o Use ndu m to obtain the matrix entries YEAR-(1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989]'; EX -[13.3 15.1 16 16.3 16.4 17.9 21.5 26.6 36.9 49.5 61.3 64 71.2 83.4 90.3 102.6 115.6 131.4 158.5 149.5 165 4 202.8 259 1346.1]'; IMP -[-21.2 -22 -19 -17.5-18.2 -21.8 -20.5 -31.8 -47.8 -56.6 -56.1 -65.2 -74.2 -100 9 -136 -148.1 -157.4 176 8 -194.8 -217 5 -223.6 -252.6 -320.1 -402.11; X -EX-IMP; K =[46 50.8 53.8 59 63.1 70.8 81.3 90.7 110 133.3 153 172.2 188.8 213.1 262.8 314.6 357.7 399.9 455.7 542.6 620 5 721.9 856.7 1027.8]'; KP -(4.8 3.0 5.2 4.1 7.7 10.5 9.4 19.3 233 19.7 19.2 16.6 24.3 49.7 51.8 43.1 42.2 55.8 86.9 77.9 101.4 134.8 171.1 212.2]';
TX -(25 3 25.5 27 5 30.5 33.3 40.6 45.7 49.7 75.3 91.7 102.6 113.5 129.6 144 161.1 195.9 225 1 255.2 296 I 361 2 420.7 480.8 557.4 675.4]'; D - ( 7 2 6 2 I 65.8 107 14 6 5 9 22.6 34.7 30 8 24.7 18 41 I 48 4 21 8 85 6 50.2 38 34.4 91.9 58.5 26.9 113 9 80.2]'; I -[40 8 46 3 51 8 57 3 62 7 68.2 73.7 113 92 145.82 165 177.05 186.21229.84 261.4 313.44 360.99 404.22 473.77 487 88 626 21 714 43 790.25 1013.28 1170.6]'; ta -[20 7 24 27.3 30.6 35.2 38.5 41.8 51.68 63.32 77.14 85.33 89.17 105.34 121 84 139 05 169.14 191.83 223 19 228 1 284.44 329.19 383.5 430.76 481.59]'; F - | 1 9 8 135 160 105 77 110 104 142 222 414 634 926 1147 1424 2743 2281 2599 2650 2278 2456 2223 2636 2295 2567]': B =(-318 17-5-9 1 I -5-80-1 78 24 160-7-207 129-1 56 100 72-88-44-57 29)'; BD -[2-31 8 17-5-9 1 I -5-80-1 78 24 160-7-207 129-1 56 100 72-88-44-57]'; T -(X-B-F): e -17.576 7.547 7.628 7.559 7.576 7.903 8.773 9 896 9 978 10.462 10.3189.971 10668 10416 10 114 10 591 10627 10.986 12.205 13.363 16 051 18.268 20.117 22.387]'; L - (26.9 26.6 28 3 30.3 31.9 33.3 35.2 35.8 36.J 38.2 42 44.2 47.2 47.7 48.1 52.4 53.5 56.2 61.5 67 71.5 79.5 85 4 90 I]': LD - [27.1 26.9 26.6 28.3 30.3 31.9 33.3 35.2 35.8 36.5 38 2 42 44.2 47.2 47.7 48.1 52.4 53.5 56.2 61.5 67 715 79.5 85.4]': LP -(-0.3 17 2 0 16 14 1.9 0.6 0.7 1 7 3.8 2.2 3.0 0.5 0.4 4.3 1.1 2.7 5.3 5.5 4.5 8.0 5.9 4.7 9.9]'; LPD -1-0.2-0.3 1.7 2 0 1.6 1.4 1.9 0.6 0.7 1.7 3.8 2.2 3 0 0 5 0.4 4.3 1.1 2.7 5.3 5.5 4.5 8.0 5.94.7]'; w - [229.4 257.6 285.8 313.9 342 2 370.4 398.6 438.65 511.49 528.54 551.07 630.96 672.08 726.3 881.85 986.86 1082.05 129.02 149.26 740.2 890.7 862.9 913.2 886.1]'; y -[295 7 346.1 366.7 403 9 431.6 462.6 510.1 620.1 732.4 787 6 848.9 960.7 10419 1143.61360.1 1597.6 1781.3 2075.9 2313.4 2622.4 2929.5 3332 3957.8 4568.2]': GDP - y;
>D -[261.5 295.7 346.1 366.7 403.9 431.6 462.6 510.1 620.1 732.4 787.6 848 9 960.7 1041.9 1143.6 1360.1 1597.6 1781 3 2075.9 2313.4 2622.4 2929.5 3332 3957.8]'; yP -[50.4 20.6 37.2 27.7 31.0 47.5 110.0 112.3 55.2 61 3 111.8 81.2 101.70 216.5 237.5 183.7 294.6 237.5 309.0 307.1 402.5 625.8 610.40 787.0]'; yPD -[34.2 50 4 20 6 37.2 27.7 31.0 47.5 110.0 112.3 55.2 61.3 111.8 81.2 101.70 216.5 237.5 183.7 294.6 237.5 309.0 307.1 402.5 625.8 610.40]'; G -[25 27.9 30.6 34.4 38.4 44.6 47.3 51.6 62.4 73 82.3 87.6 97.2 111.7 130.8 153.6 182.7 211.4 243.5 291.7 346.3 408.4 473.3 542]'; C -[237 9 283.4 283.9 307.8 325.5 351 386.9 434.3 565 1 578.2 600.8 691.8 752.4 817 992.9 1137.7 1251.7 1469 5 16209 1777.6 1986 2225.5 2574.2 2882.4]'; n -y.\L: M -[46.8 51 53 9 60.4 67.6 76.5 86.2 101 111.3 122.3 152 8 178 5 157 6 176.9 204 6 232.5 273.7 308.6 365.6 412.4 478.7 543 2 632.8 746.9]'; M D =(43 46.8 5153.9 60.4 67.6 76 5 86 2 101 111.3 122.3 152.8 178.5 157 6 176.9 204.6 232.5 273.7 308.6 365.6 412 4 478.7 543 2 632.8]';
409
410
Differential Models and Neutral Systems
MP -[4.2 2.9 6.5 7.2 8.9 9.7 14.8 10.3 11.0 30.5 25.7-20.9 19.3 27.727.9412 34.9 57.0 46.8 66.3 64S 89.6 114.1 106.7]', MPD =[3.8 4.2 2.9 6.5 7.2 8.9 9.7 14.8 10.3 11.0 30.5 25.7-20.9 19.3 27.7 27.941.2 34.9 57.0 46.8 66.3 64.5 89.6 114.If, ML - [ -46.4 -50.4 -53.4 -59.8 -66.8 -75.8 -85.6 -100.2 -110.0 -121.3 -151.6 -177.4 -156.4 -175.5 -203.4 230.9 -271.9 -306.6 -363.1 -409.9 -475.8 -540.2 -629.4 -743.1]'; P -[16.4 18.6 19 1 19.2 20.2 20.8 22.2 25.9 33.3 35.3 32.6 35.3 36.2 38.4 42.8 48.8 5 2 2 58.4 63.3 66.8 72.6 79 86.4 91.8J-; Pf -P.'e; Pfe - P f e : R =[4.37 5.36 3 9 3.92 5 68 6.3 4.69 6.64 13.52 10.4 11.28 10.18 8.05 8.47 7.24 8.61 7.27 8.3 9.95 10 9.97 9.83 15.5 15.5]": RP =[0.99-1.46 0.02 1.760.62-1.61 1.95 6.88-3.12 0.88-1.1 -2.13 0.42-1.23 1J7-1.34 1.03 1.65 0.050.03-0.14 5.67 0 0.07)': RPD =[-1.910.99-1.46 0.02 1.76 0.62-1.61 1.95 6.88-3.12 0.88-I.I -2.13 0.42-1.23 1.37-1.34 1.03 1.65 0.05-0.03-0.14 5.67 0]'; RD =[6.28 4.37 5.36 3.9 3.92 5.68 6.3 4.69 6.64 13.52 10.4 11.28 10.18 8.05 8.47 7.24 8.61 7.27 8.3 9.95 10 9.97 9.83 15.5]'; YT =y-TX. YTD =[237.7 270.4 320.6 339.2 373.4 398.3 422.0 464.4 570.4 657.1 695.9 746.3 847.2 912.3 999.6 1199.0 1401.7 1556.2 1820.7 2O17.3 2261.2 2508.8 2851.2 3400.4]'; YTP-[32.7829 101.5 135.7 160.6 184.3 226.7332.7 419.4458.2 508.6 609.5 674.6 7619 961.3 1164.0 1318.5 1383.0 1779.6 2023.5 2271.1 2613.5 3162.7 3655.lf; YTPD -(82.9 1015 135.7 160.6 184.3 226.7 332.7 419.4 458.2 508.6 609.5 674.6 761.9 961.3 1164.0 1318.5 1583.0 1779.6 2023.5 2271.1 2613.5 3162.7 3655.1 4393.9]'; price=[l4 8 16.4 18.6 19.1 19.2 20.2 20 8 22.2 25.9 33 3 35.3 32 6 35 3 36.2 38.4 42.8 48.8 52 2 58.4 63.3 66.8 72.6 79 86.4 9! 8]'; for 1=1:24 pp(i)=price< i-1 Vpricefi): end pie-[ppj'; DP- pieAP; logy-log(y);
% TX -Total current receipts of general govemment(From UN financial % statisticsXthousand million kalian lire) %n -labor productivity(A/l - GNP/employment). % -From Yearbook of labor jtatistics(intemational labor office)
% % K =capital stockfGross fixed capital formation) % I =investmem(Gross capital formation) % D -increase in stock % The UN(National accounts statistics) defines Gross capital % formation as the sum of the increase in stocks and gross fixed % capital formation
% V, w - compesation of employees(UN financial accounts statistics. 1965-84 % plus (wages) in manufacturing and yearbook of labor statistics 1985-89) % y -GDP(Gross domestic product) %yD-y{t-l) •/,yP-y'(t) VoyPD-y'll-l) % X =Net export(export-import of goods and services) % d -preferential trade agreement transportation % Pfe -import price level in foreign currency (P'e'e) */.u -tariffs •/. E ^cumulative balance of payment
Appendix
411
%F ^Net private outflow of capital V. C -private consumption % M,MI -money supply % Md =money demind(liquidity); Where, ML" md-M % P -consumer prices */■ R -interest rate(money market rate, except l988&1989:narional rate) •/.RD-RO-I) % RP -R'(t) V.RPD-R'(t-1) •/• pie-inflation % L - industrial production % B -Balance of payraent(deposit money banks) % T -Net government transfer of capital to foreigers and firms %G -Government consumption •/.DP-D'logtPl-pW/pO)
% %Bert-[X.G.I.C.GDP,PGDP.RM.QM,YEAR] Iyears.v]*size
412
Differential Models and Neutral Systems
thx-arx(lemp,|O.I I 1 I I I M I 1 I 0 0 I 0 100IOOO]); XpT)redict((X.ones(size(y)).y.yD,yP,yPD.R,L,LPD.P.ta.e],!hx,4); XX=thx(11.1:12); •/. G(t) -gO ♦ gl *y(t) ♦ g2*y(t-h) ~gi*y(.i) +g*,y'(t-h) -g5*R(t) *g8«L(t); temp-[Gones(size(y)),y,yD,yP,yPD,R,L|; thg-arx(temp.|0.1 1 1 1 1 1 1 0 0 ) 0 1 0 0]); Gp-predict([G.on«(size(y)).y.yD.yP,yPD.R,L].thg,4), GG-thg(6.l 7); */.y(t) - ziO <-ul*y(t) *i$2*y(l-h) *u4 , y , (t) -2s5'R(t) »zsS*L(t) ♦zslO,L'(t) *zs13*Ml(t) -zsl4*TX(t) ♦alS'eCt) ♦z»l6*ti(t) «-zsl7»d(t)'(0) +l/h«K'(t);h-l temp-[yones(size(y)).y,yD,yP,R,L,LP,M,TX1t,u,KP]; thyarx(temp.|0.l 1 I I I 1 I 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 ] ) ; yp-predici((y,ones(size(y)).y,yD,yP.R.L,LP.M.TX,e.ta,lG,],thy.4); yy-thy(ll.ll2). •/^y(t)-ylO'yll*y(t)*-yl2'y'(t) + y13 , R(t)*yl5 , MUO*yl8 , YT(t) temp-[Cones(size(y)).y.yP.R.ML.YT), thc>-anc(ttmp.(0.1 I I I I 10 0 0 0 0 0 ] ) ; cyp-pfedia{(C,ones(size(y)).y,yP,R.ML.YT],thcy14); ecy=thcy(S.I:6); •'.Iv(t)-lO*l/h»K.P(l) tenp-II ones(size
Appendix temp-[D ones(siie(y)).y.K,KP,R,L.P.yP]; thD-arx(temp.[0,l I I 1 1 I I I 0 0 0 0 0 0 0 0]). Dp-predict([D.onej<siK
subplot (2.1,1 ).plot(YEAR,C,YEAR,Cp,'■'■') titleClndia-PRJVATE CONSUMPTION-1); ylibeKO: xlabel(YEAR); subplot (2.1,2).pk>t(YEAR.X.YEAR,Xp.'+') titleCIndia-EXPORTS--'), ylabel('X');
xlabelCYEAR). pause
subplot (2.1.1 ),plot(YEAR.I,YEAR.Ip>') titleClndn-lNVESTMENT--); ylabel(T); xlabel(YEAR). jubplot(2.1 ^.plotfYEAR.G.YEAR.Gp:*) utleClndia-GOVERNMENT CONSUMPTION--); ylabeK'G"); xlabel('year'); pause clg subplm(2.1,1 ).plot(YEAR,M,YEAR,Lp.v) titleCIndia-MONEY DEMAND-'); ylabelCMT): xlabetCYEAR'); subplot(2.l.2).plot(YEAR.Z.YEAR.[Xp+lp+Gp+Cp],*') titleClndia-AGGREGATE DEMAND-); ylabelCZ): xlabelCYEAR); pause
subplot (2,1.1 ),plot(YEAR.DP.YEAR,dlp,+') titleCIndia-CONSUMER PRICES-"); ylabelCDP1); xlabel('YEAR'); subplot P.UXploUYEAK.y.YEAR.yp.1*') titleClndia-INCOME-'); ylabelfGDP-); xlabelCYEAR'); pause subplot (2,1,1 ),plot(YEAR,C,YEAR,cyp/+T tilleClrdia-INCOMBCONSUMPTION-'); ylabelCc/); xlabelCYEAR). subplot(2.1.2 ).plot( YEAR.l,YEAR,lyp> •) titleClndia-INCOME'INVESTMENT-); ylabel('lyp);
413
414
Differential Models and Neutral Systems
xlabelCyetO; pauic cl» Mbptot(2.1.1XploKYEAIt.C.YEAIl4yp>') titleflnduH-INCOME/GbVERNMEOT--); ylabalfjyp-); xJaMfYEART; wbpJottt1,2Xplo«(YEAIWC YEAR^yp,*") tJOcTtodia-INCOME^XPORT--); yhbelCxypI: xlabeJCYEAR-); p-tusc
% subplot P , 1.1 )4>lct( YEAR.B.YEAR3p.'+-) ntfeftodil-BALANCE OF PAYMENT-"); ylabdCBO; xlabelfYEAR"); subplot (2.1.2).pl«K(YEAR.logy.YEAR.Iyp.'-) titleClndla-logXincoine)-'); ylabeKTii/): xlabdCYEARO; pause
subplot (2.1,1 ).ploKYEAR.D.YEARJ>p,V) titkf India-Increaw in stock-*); vhbeK'DT. xlabelCYEAR-): % The economic dynamic! can be put in matrix form % asfollows'.x(t) '• A-lMt-h)" A0*x(t)+Al,x(t-h)+B,'u(t) Sx-[>RLKPEr: HA-1 -[amH Hml4iinl3 000:Oam22OO00;LmO3 0-UnOI00O;a3 0*6-aml0O;0-M7,p6*P(«)0OO 0;MMbi:000] HAO -[a0lal2a|4al«al«0;a21 a2300a25 0;00i330a35 0;000a44a45 0;i51,p(t)aJ2*p(t)OO a55*p(l)0:e6la62i63 000] HA1 -{allal3al5 0QD;il2lil22 0000;il31i:32*I33000;at41aU2al43000;al5! , p
Appendix
415
pause subplot (2.1.1 ),plot(YEAR.D,YEAR,Dp.v) titleflndia~lncrease in slock--'): ylabel('D'): xlabelCYEAR'); % The economic dynamics can be put in matrix form % as follows: \'(t) - A-l 'x'(t-h) - A0*x((h AI ♦x(t-h)+B'u(t) •-.x=[y R L K P E]\ %A-1 =[aml 1 -am 14 am 13 0 0 0;0 am22 0 0 0 0; Lm03 0-LmOI 0 0 0; a3 0 a6 -ami 0 0; 0-M7*p6 , P(I) 0 0 0 0;b4b8bl2 0 0 0] %A0 «{a0l al2 al4 al6 al8 0; a21 a23 0 0 a25 0; 0 0 a33 0 a35 0; 0 0 0 a44 a45 0; a51*p(t) a52*p(t) 0 0 a55'p(t)0:a61 a62a63O0O] %A1 - [ a l l a l 3 a l 5 0 0 0;al2l al22 0 0 0 0;al31 al32 al33 0 0 0; aNl al42 a!43 0 0 0; a!51*p(t) al52'p(') 0 0 0 0; al6l al62 al63 0 0 a 166) %BI -[-ssl ssl S s l z l 5 s s l z l 8 s s l z l 7 0 0 0 ; 0 0 0 0 0 - l 2 0 0:-m\vzsl4mwmwzsl5mwzsl60mwzsl3 0 0;isl4 I zsl5 zs!6 0 zsl3 0 0: 0 0 pl*pf(t) 0 0 -p6 p5 0; 0 0 b7 b8 blJ 0 0 -1] %B2 =[ssl ssl ssl s s l ( H 3 + c 7 ) 0 0 0 0 0 ; 0 0 0 - L 2 0 0 0 0 0 ; O m w O O O O m w m w O ; 0 1 0 0 0 0 1 1 0 ; 0 0 0 p6-p3p2 0 0 1,00 0-1 0 0 I 001 %u = [q S]; °/«.\ =(TI g O e l a d M l MI'fD]1; %S =[C0 10 ,\0 M0 n w xO y 10 p0]'; %B-[BI B2] %AB-A0*B %A2B-(A0'A0)'B K A3B-(A0'A0«A0)*B % A4B-(A0'A0*A0'A0) , B "..ASB-IAO'AO'AO'AO'AO'AOJ-B %G-[B AB A2B A3B A4B A5BJ %rank(B)-6 % rank(G)= 6
% Where, % Ami =(0 0300 -7.3492 7.3492 0 % 0 -00019 0 0 0 % 0.0000 0 0.6852 0 0 % 0.1343 0 12.5592 0.6844 % 0 -0 0001 0 0 0 % -0.2010 10.0167 52.7368 0
0
0
0 0 0 0 0
0 0]
% AO -[-0.9448 -1.2805-1.8774 0.1262 17977 0 % -0.0007 0.0127 0 0 -0 0050 0 % 0 0 -4.2991 0 0.0000 0
416 % '/. %
Differential Models and Neutral Systems 0 0 0 -4.2991 0.2026 0 00000 -00005 0 0 -1.0846 0 0.4589 7 1118 86.6667 0 0 01
% A1 =[-0.0650 2.2146 -1.1848 0 0 0 % 0.0010 -0.0123 0 0 0 0 % 0.0000 -0.3253 169422 0 0 0 % -05834 -363063 16.3588 0 0 0 % -00375 0.0005 0 0 0 0 % -1 1138 -6.5845 -56.0216 0 0 0.1897] % Bl =[-1.0152 1.0152 0 3898 1.7977 0 0 0 0 % 0 0 0 0 0 -0.9000 0 0 % 0.0000 0.0001 -0.0003 0.0000 0 0.0000 0 0 •/• 0.5183 1.0000 -3.8873 0.3012 0 0.3444 0 0 •/. 0 0 -0.2031 0 0 -0.0009 -0.0005 0 % 0 0 -40.4471 10.0167 0 0 0 -1.0) •/. B2-[1.0152 1.0152 1.01520.1334 0 0 0 0 0 % 0 0 0 -0.9000 0 0 0 0 0 % 0 0.0001 0 0 0 0 0.0001 00001 0 % 0 1.0000 0 0 0 0 1.0000 1.0 0 % 0 0 0 -0.0009 -25.0144 0.0013 0 0 1.0 % 0 0 0 -1.0000 0 0 1.0000 0 0] %q -[Tl 0.4078 etad Ml Ml'-278.0597]'; %S -f14471 25.1406 15.7077 0.0400 n w 15.7077 13.9320 -2.8120]; % For spectral controllability
%Deltaj-eye(size(Al))'j -Ami.»(j*2.78I4'<-j)) - A0 % -Al. '(2.7814*<-j)) % a-[Deliaj B) %forj -0.4312.4. 10.23, 100....
% rank(a)=6 % For nonzerocigenvalues of A-1 to be controllable via matrix B, % cc-eye(size(Aml)).,j -Ami % c-[cc B] %forj'0.43l2,4, 10,23, 100,... '/. rank(c)-6 % Since rank(B) is 6. where rank(a) and rank(c) are % also 6. the system is function space controllable. LI~I;L2-0.9:L5-DD(5) L6=DD(6);h -1 % for y(t) - A'lOa'lAl-a) a- 0.9 % a -alpha aa-0.1 % aa^O-a^beta W-sum(w)^4 mw »(aa/W).*(!/a)
Appendix zll--ll(lO) zlS-OKIlHCCWrMMW zl4--CC(2) zl5=XX(ll) zl6-XX(IO) zl7=0 zl 8-XX(9)+(CC(8)->-zl 3)'MM(7) ZI9--CCO) z20-CC(4) z21=CC(5) %z22--(CC(8)+II(ll))«Ml zsO = ccy(IH'y(l)+ggy(l) + My
417
418 a6 -DD(])'yy7/(l-yy2) ami -DD(3)*DD(iy(l-yy2) M7p6P - MM(7)'dp(6),sum(PV24 b4--BB(5) b8 - -BB(9) bl2--BB(l2) 4
/4A0 starts here!
aOI -Ll«(zl-l)/s) all -Ll'22/sl al2-Ll'z5/sl al3-LI*z6/sl al4-Ll*z8>sl al5=Lrz9.'sl ilrj-LI'zll.'sl al7-LI*z!9/sl al8-ll'zl8/sl a2l *L2*MM(2) a23 =L2*MM(4) a25 -L2*MM(7) a33 ^aO; a35 -mw*a8 a44-aO a45 -a8 a51 *-dp(7)»MM(2)"sum(P)/24 % includes P(l) a52--dp(7)»MM(4),sum(P)/24 •/. includes P(t) 155 --(MM(7)-dp(5))«sum(PV24 % includes P(l) a61 «BB(2) a62 -BB(6) a63 =BB(10) VoA I slarts here1 all -Ll*z2'sl ilj-Ll'zfrsl al5 =Ll'z9.'sl a22 -L2'MM(3) a24 -L2*MM(5) al=-DD(l)*yy3'((l-yy2)*h) a2--al'h 11 - - a l 12 --mw*DD
Differential Models and Neutral Systems
419
Appendix a 151 --dp(7)«sum(PV27
%includes P(t)
1152 --MM(5)*dp(7)*sum(P)/24 %includes P(t) • 161 -BB(3) a 162 -BB(7) 1I63-BB01) a!66=BB(l4) •/•BI starts here! ssl =LI'sl%ssl=bl sslzl3-ssl*z]3 sslzl5=ssl , zl5 sslzl6-ssl«zl6 sslzl7 -0 sslzlS-ssl'zlS blzl4=-BB(2) , zl4: bl =BB(1) blzl5-BB(2)*zl5 b!zl6-BB(2)*zl6 blzl7-BB(2)*zl7 blzl3=BB(2)-zl3 mwzs 14 =rmv"yy9 mwzsl5 =mw*y\IO mwzs 16 -mw'yv 11 mwzs 17 -0 mwzs 13 -mw*yy8 plpf =-dp(2)*sum(Pfey24
p4 pi b7 b8 bl5
=dp<5) -d(X6) «BB(8) -BB(9) -0
%B2 starts here! ssl -LI.'si
sslil3c7-ssl'(lI(ll>+CC(8)) 12 -mw«DD(l)'>y3/(l-yy2) p6 = dp(7) p3 - dp(4) p2 - dp(3) a17-Ll # zl9/sl i45 -a8 cO-CC(l) 10 = 11(1) xO-XX(l) pO-dp(t) MO-MM(l) ylO-ccy(l) mO -MM(1) gO-GC(l) fl)=BB(l)-XX(l) g2 --L2*MM(1)
°/idp(2),Pfl;t)
Differential Models and Neutral Systems
420
Project Reflections on the above models inspire one to wonder whether such models can be attempted for Russia, China, and Brazil. The data is given for the reader to build the model for each of these nations. An ordinary differential model should be attempted first. Is the system stable, oscillatory, controllable? Appendix Russiacu.m % Russia's data is in millions of rubbles % taken from the international financial statistics yearbook 1997. *QM - Quasi money %EXGS = Exports of goods and services *IMGS « Imports of goods and services kl =■ changes in inventories %YD=Y(t-l) YEAR=[941 942 943 944 951 952 953 954 961 962 963 964]'; EXGS-[23.227 35.504 44.417 66.387 90.387 114.271 105.217 118.241 105.1 119.739 128.600 145.716]'; IMGS=[-21.406 -29.743 -34.367 -56.419 -68.394 -92.807 -95.196 -106.238 5.410 -97.479 -101.389 -94.658]'; X=EXGS - IMGS; G=(20.744 28.814 65.860 51.265 47.967 68.504 78.602 110.554 94.032 117 4 103.117 138.489]•; I-[-2.069 11.598 13.543 -0.310 -6.617 12.094 41.143 6.781 16.200 18.58 6.631 -20.849]'; C-I47.873 57.403 78.990 101.104 136.690 179.343 228.006 252.474 245.32 70.161 290.907 300.717]'; GDP=[87.441 130.156 168.257 224.738 255.969 359.867 473.147 541.096 46 23 512.642 593.064 685.491]'; PGDP-[75.593 42.715 38.101 56.481 31.231 103.898 113.280 67.949 -76.17 7.719 80.422 92.427]'; QM=[24.340 29.675 44.027 61.187 89.825 107.739 114.056 124.514 138.888 9.663 160.420 169.606]'; M-[29.547 42.404 54.142 68.544 68.056 106.311 121.302 151.267 155.905 .106 180.083 197.449] '; R=[190.S 1 8 5 . 5 1 8 1 . 3 1 2 5 . 2 1 4 2 . 9 1 2 2 . 4 7 2 . 9 6 9 . 6 6 1 . 7 5 5 . 4 6 0 . 1 4 3 . 0 ] Y-GDP; YD-[75.500 87.441 130.156 168.257 224.738 255.969 359.867 473.147 541. 464.923 512.642 593.064]'; Yp-Y-YD; DGDP = YD; russicul=tX,G,I,C,GDP,PGDP,R,M,QM,YEAR] [years,n]=aize(YEAR); Xl=[28.179;X(l:years-D] ; Gl-[2.634;G(l:years-D] ; Il=[2.031;I(l:years-l>] ;
Appendix Cl-[6.S50;C(l:years-l)] ; Ml-[23.881;M(l:years-l)] ; Rl-[175:R(l:years-l)]; GDPl«[19.006;GDP(l:years-l)J; DGDP-GDP-GDP1; Z-X+G+I+C; Z1-X1+G1+I1+C; temp.[YEAR XI Gl II Cl Zl GDP1 DGDP M l ] ; save russicul.asc temp -ascii -double -tabs clear russicul temp; temp-[C ones(size(GDP)),GDP,DGDP,R]; thc-arxltemp, [0,1 1 1 1 0 0 0 0] ) ; Cp=predict([C,ones(size(GDP)),GDP,DGDP,R],thc,l); CC-thc(3,l:4) ; terap-[I ones(size(GDP)),GDP,DGDP,R); thi-arx!temp, [0,1 1 1 1 0 0 0 0]); Ip-predict([I,ones(size(GDP)),GDP,DGDP,R],thi,l); II-thi(3,l:4); temp-(X ones(size(Z)),GDP,DGDP,R]; thx-arx(temp,[0,1 1 1 1 0 0 0 0]); Xp»predict([X,ones(size(Z)),GDP,DGDP,R],thx,1); XX.thx(3,1:4); temp.[G ones(size(Z)),GDP,DGDP,R]; thg=arx(temp,[0,1 1 1 1 0 0 0 0 ] ) ; Gp-predict ( [G, ones (size (Z) ! , GDP, DGDP, R] ,thg,l) ; GG-thg(3,l:4); temp= [M ones(size(G)) ,GDP,R] ,thm.arxftemp,[0,1 1 1 0 0 0]); Mp.predict([M ones(size(G)),GDP,R],thm,l); MM-thm(3,l:3) ; subplot (2,1,1),plot(YEAR,C,YEAR,Cp, '--•) title('PRIVATE CONSUMPTION'); ylabel{•€•);
xlabel('YEAR'); subplot (2,1,2),plot(YEAR,X,YEAR,Xp, '--' ) title('EXPORTS');
421
422
Differential Models and Neutral Systems Chinacu.m
% china's data is in billions of yuan % taken from the international financial statistics yearbook 1994 &199 %QM - Quasi money *X - Export(net) *I - Investment(Gross capital formation) %YD-Y(t-1) YEAR-[1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 93 1994 1995 1996 1997]•; X-L-1.5 1.1 9.1 5.1 0.1 -36.7 -25.5 1.1 -15.1 -18.6 51.0 61.8 27.6 -68 63.4 99.9 145.9 274.5]'; G-[65.9 70.5 77.0 83.8 102.0 118.4 136.7 149.0 172.7 203.3 225.2 283.0 9.2 449.7 598.6 669.1 785.2 8 6 5 . 0 ] ' ; I-[131.8 125.3 149.3 170.9 212.6 264.1 309.8 374.2 462.4 433.9 473.2 5 0 8 1 3 . 7 1298.0 1685.6 2030.1 2333.6 2 5 6 9 . 8 ] ' ; C-[231.7 260.4 286.4 318.3 367.5 458.9 517.5 596.1 763.3 852.4 911.3 1 .6 1 2 4 6 . 0 1 5 6 8 . 2 2 1 2 3 . 0 2 7 8 3 . 9 3 3 1 8 . 8 3 6 1 1 . 8 ] ' ; GDP-[455.1 4 9 0 . 1 548.9 607.6 716.4 879.2 1013.3 1178.4 1470.4 1646.6 1 .0 2 1 2 8 . 0 2 5 8 6 . 4 3 4 5 0 . 1 4 7 1 1 . 1 5 9 4 0 . 5 6 9 3 6 . 6 7 6 0 7 . 7 ] * ; PGDP-[47.7 3 5 . 0 5 8 . 8 8 . 7 1 0 8 . 8 1 6 2 . 8 1 3 4 . 1 1 6 5 . 1 2 9 2 . 0 1 7 6 . 2 1 8 5 . 4 296 458.4 863.7 1261.0 1229.4 996.1 6 7 1 . 1 ] ' ; QM-C52.23 6 3 . 2 5 7 8 . 9 6 9 6 . 3 9 1 1 4 . 9 1 1 8 5 . 7 6 2 4 8 . 9 6 3 3 8 . 3 4 4 1 1 . 4 7 5 5 5 . 8 9 .24 961.11 1261.30 1797.88 2538.04 3514.67 4 5 4 3 . 2 7 5 3 5 2 . 4 5 ] ' ; M-C114.88 134.52 1 4 8 . 9 1 8 4 . 8 9 2 4 4 . 9 4 3 0 1 . 7 3 3 8 5 . 9 4 5 7 . 4 5 4 8 . 7 4 5 8 3 . 4 2 .95 898.78 1171.43 1676.11 2153.99 2559.68 3066.26 3 8 3 4 . 3 3 ] ' ; R-[5.4 5.4 5.20 5.76 5.76 7.20 7.20 7.20 8.64 11.34 8.64 7.56 7.56 10. 10.98 10.98 7.47 5 . 6 7 ] ' ; Y-GDP; YD-[407.4 455.1 490.1 548.9 607.6 716.4 879.2 1013.3 1178.4 1470.4 164 1832.0 2128.0 2586.4 3450.1 4711.1 5940.5 6936.6]'; Yp-Y-YD; DGDP - YD;
Appendix
423 Appendix Brazil2.m
Money demand may be taken as quasi-money. t Braxil-taachftr
(ordinary differential aquation)
YEWt-[197i 1972 1973 1574 1975 1976 1977 1975 1979 1960 1991 1992 1983 84 198S 1984 199? 1988 1989 1590]'j XX -[6162 9091 14S45 20727 37 43 66 88 157 408 840 1399 4870 19020 614 117 397 3427 38004 8 S 3 P ; IMP-(-7434 -11273 -14727 -3600 -42 -54 -72 -104 -202 4S27 887C 18065 4 5 134219 504364 1336 4209 31491 400000 10900]'; X -XX-IKPj TX -(17455 34000 34192 50162 71 114 230 301 493 1009 2183 4604 11267 3 0 133883 297 1441 «J1« 102291 3639]'; P -[3 3 3 4 5 7 10 14 31 39 81 160 387 0 0 0 0 0 0 0]',B -(727 1091 1819 4000 7 13 22 23 50 106 252 530 1351 4153 17997 0 0 9845 397)•; BD -[700 727 1091 1819 4000 7 13 22 23 SO 106 252 530 1351 4153 17997 0 663 8845]'; L -[6069 6676 7614 8803 9477 38038 40179 41564 42707 44086 4546S 4792 8466 62100 60633 5*729 57410 5S436 53861 5O209]'; LD -[6876 7614 6803 9477 38039 40179 41564 42707 44086 4S465 47926 484 62100 60622 58729 S7410 55436 53661 50209 50000]'; LP -abs(L-LD); D -[2545 3273 7636 19273 35 10 18 28 -13 20 13 -43 -617 -1600 11273 0 0 00 00 0 0 ] ' ; DP -[3273 7636 19273 25 10 18 28 -13 20 13 -63 -617 -1600 11273 0 0 0 OP; K -[33638 31636 4S4S7 73004 69 133 193 293 507 1031 2046 3940 7731 22 96364 225 936 7151 114496 2500)'; KD -[31636 45457 72004 69 133 193 293 S07 1031 2044 3940 7731 22182 96 225 936 7151 114496 2500 112Q0P j KP -abs(K-XD); 1 -[26163 34911 53093 91277 114 143 311 321 494 1051 2059 3677 7114 2 2 65091 22S 936 71S1 114496 25001'; G -[1054S 13455 1654S 35455 39 63 85 137 315 414 831 1139 4119 11632 55 142 510 3951 6S948 1B39P ; C -[61455 81091 111273 1(9728 243 407 626 904 1532 313J 4064 12765 30 98771 330572 902 2412 18727 200000 6 4 Q 0 P j R .(20 20 19 18 16 28 30 33 35 93 121 174 194 272 380 69 401 2262 363 1063] ' ,RD -[20 20 20 18 16 16 29 30 33 35 93 121 174 194 372 390 89 401 3383 41] ' ; HP -[0 0 2 0 0 10 2 3 2 59 28 S3 29 53 20 79 109 291 312 1991 36059 37 H
-[15636 214SS 31273 42909 61 84 116 156 276 466 655 1441 2922 6887
00 166 524 2623 37697 917]•j HD -[21455 15636 214SS 31273 42909 61 84 116 158 276 466 855 1441 2923 87 38600 166 524 3623 37697]'; HP -aba(M-MD); Ml -[-0.0333 -0.0S90 -0.1053 -0.1341 0.0331 0.0155 0.0209 0.0204 0.030 0.0133 0.0080 -0.0044 0.0157 0.0454 0.1427 0.0189 -0.0197 -0.1226 0.10 1.3517]'; HL -ml./2; w -(670 849 049 0370 1936 2644 31796.7S 31796.75 21796.75 21796.75 21 .75 21796.75 21796.75 423.6 1607.2 3S07.0 10949.0 75579.0 26076.0 1366 01 '; t* -[15 18 27 32 30 38 42 43 4 6 7 7 6 9 9 9 10 11 13 12]'; • -[2.33 3.46 2.73 3.33 3. BC 5.32 7.09 9.91 30.37 30.39 54.09 101.35 .63 1.13 4.19 6.43 37.27 374.5 5.43 91.4]*; Pf*-P.•«.*«; y -( 94125 124269 174077 255120 363 581 899 1337 2235 4353 8501 17112 948 126183 477455 1376 4050 30230 448503 10600]'; yO -(70503 94125 124269 174077 255120 362 S61 699 1337 2235 4353 9501 12 36946 126183 477455 1376 4050 30230 4465031'; yP -[33622 30164 49768 81043 245758 219 306 448 898 2118 4148 6611 216 67334 351273 476779 3774 36190 14620 4379031'; YT -y-TX; YTD-[100269 13989S 204938 291 445 569 1034 1742 3344 6311 12506 27661 72 344S72 979 2609 23912 346212 6972 39947]'; YTPO- (29604 65043 204647 174 194 377 704 1602 2974 6190 15173 64S91 25 0 343593 1630 21303 322300 339240 32575 33100]'; TTP-[33419 39606 6S043 304447 174 194 377 706 1603 2974 6190 15173 645 251400 3435)3 1630 31)03 J22300 339240 32975P; logy-log(y); * -y./L; Z-X*Q*I*-C;
424
Differential Models and Neutral Systems
UK2.M * This program was written for chapter 6 at 9:55PM 4/9/96 % The United Kingdom's data is in billions of pounds '« and it was adopted from International Financial Statistic Yearbook 1994 % The interest rate used here is the short term Government bond yield i Use ndu.m to obtain the matrix entries YEAR =(1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1968 1989 1990 1991 1992)'; EX -[7.17 7.39 8.98 10.09 11.51 12.92 13.62 17.07 22.8 26.86 35.04 43.3 47. 48 S4.9 62.62 67.43 72.69 79.88 91.63 102.04 97.89 106.4 107.27 121.49 133.28 134.15 140.25]'; iiH> -1-7.26 -7.85 -9.38 -9.93 -11.10 -12.16 -13.74 -18.95 -27.15 -28.8 -36.6 4 -42.38 -45.37 -54.35 -57.61 -60.37 -67.76 -77.59 -92.76 -98.99 -106.22 -111 .74 -124.8 -142.61 -148.29 -148.78 -149.46]'; X -EX-imp; TX -(12.957 14.397 16.249 18.306 20.748 21.940 23.394 26.497 33.256 42.610 49.632 56.559 62.839 75.044 92.331 105.946 119.048 127.857 136.605 149.706 15 7.199 170.396 187.992 20.2863 217.959 223.299 225.693]'; Tx -(10.409 11.786 12.957 14.397 16.249 18.306 20.748 21.940 23.394 26.497 33.256 42.610 49.632 56.559 62.839 75.044 92.331 105.946 119.048 127.857 136. 605 149.706 157.199 170.396 187.992 20.2863 217.959 223.299 225.693]'; n -(108 111 116 98 100 103 106 109 108 106 10 1977-85 3.7 6.7 6.3 4.4 1. 4 2.0 5.4]'; ta -(4700 5017 5680 6686 7265 7598 7836 6254 5444 6497 8635 11912 13685 166 28 18132 22869 25326 25831 25473 27281 31843 35381 39064 40749 36938 38295 38 229]'; B -[-809 -414 -823 -838 1474 2462 -105 922 -761 1367 -994 2593 -2725 -1153 1 2929 4121 15844 4772 221 11241 25865 -19900 19424 22620 13634 26326 187]'; BD =(112 -809 -414 -823 -838 1474 2462 -105 922 -761 1367 -994 2593 -2725 -11531 2929 4121 15844 4772 221 11241 25865 -19900 19424 22620 13634 26326]'; F =(-65 -77 -106 -70 -31 -3 -129 -244 -280 -307 -29 -81 -244 -413 -473 115 82 510 493 405 112 -205 -480 -491 -535 -531 -485]'; T - [X-B-F]'; D »(0.29 0.29 0.45 0.54 0.38 0.11 0.03 1.53 1.05 -1.35 0.90 1.62 1.60 2.16 - 2 . 5 7 - 2 . 7 7 - 1 . 1 9 1.47 1 . 3 0 0 . 8 2 0 . 6 8 1 . 3 3 4 . 7 5 3 . 5 9 - 1 . 1 2 - 5 . 0 7 - 1 . 9 9 ] ' ; K - [ 7 . 0 6 7.71 8.51 8.83 9.74 10.89 11.94 14.73 17.5 21.04 24.5 27.07 31.06 36.93 41.56 4 1 . 3 44.82 48.62 55.18 60.72 65.03 75.02 91.12 104.58 106.78 96. 53 9 2 . 5 9 ) ' ; kk - [ 6 . 6 3 7.06 7.71 8.51 8.83 9.74 10.89 11.94 14.73 17.5 21.04 2 4 . 5 27.07 31.06 36.93 41.56 41.3 44.82 48.62 55.18 60.72 65.03 75.02 91.12 104.58 106.7 8 96.53 9 2 . S 9 ] ' ; I =( 7 . 3 5 8 . 0 8 . 9 6 9 . 3 7 1 0 . 1 2 1 1 . 0 1 1 . 9 7 1 6 . 2 6 1 8 . 5 5 1 9 . 6 9 2 5 . 4 0 2 8 . 8 9 3 2 . 86 3 9 . 0 9 3 8 . 9 9 3 8 . 5 3 4 3 . 6 3 5 0 . 0 9 5 6 . 4 8 6 1 . 5 4 6 5 . 7 1 7 6 . 3 S 9 5 . 8 7 1 0 8 . 1 7 1 0 5 . 6 6 91.46 9 0 . 6 ] ' ; e - ( 1 5 6 . 5 153.9 134.1 133.9 134.3 136.9 140.2 137.4 131.1 124.5 101.2 97.8 107.6 118.9 130.3 113.6 96.1 85.0 74.9 72.6 82.2 91.8 99.8 91.9 100.0 9 9 . 1 9 8.9] ' ; L - [ 1 0 0 . 4 9 8 . 4 9 7 . 7 9 7 . 6 9 6 . 9 9 7 . 4 9 7 . 4 9 9 . 8 1 0 0 . 4 1 0 0 . 0 9 9 . 3 9 7 . 4 1 0 0 . 3 10 1.9 1 0 1 . 2 9 6 . 3 9 4 . 2 9 2 . 7 9 3 . S 9 4 . 3 9 4 . 2 9 5 . 1 9 8 . 1 9 9 . 9 1 0 0 . 0 9 6 . 8 9 4 . 1 ] ' ; LD = ( 9 9 . 6 1 0 0 . 4 9 8 . 4 9 7 . 7 9 7 . 6 9 6 . 9 9 7 . 4 9 7 . 4 9 9 . 8 1 0 0 . 4 1 0 0 . 0 9 9 . 3 9 7 . 4 100 .3 1 0 1 . 9 1 0 1 . 2 9 6 . 3 9 4 . 2 9 2 . 7 9 3 . 5 9 4 . 3 9 4 . 2 9 5 . 1 9 8 . 1 9 9 . 9 1 0 0 . 0 9 6 . 8 ] ' ; LP - [ - 2 . 0 0 0 0 - 0 . 7 0 0 0 - 0 . 1 0 0 0 - 0 . 7 0 0 0 0 . 5 0 0 0 0 2 . 4 0 0 0 0 . 6 0 0 0 - 0 . 4 0 0 0 - 0 . 7 0 0 0 -1.9000 2.9000 1.6000 - 0 . 7 0 0 0 -4.9000 -2.1000 -1.5000 0.8000 0.6000 - 0 . 1 0 0 0 0 .9000 3 . 0 0 0 0 1 . 8 0 0 0 0 . 1 0 0 0 - 3 . 2 0 0 0 - 2 . 7 0 0 0 0 . 1 0 0 0 ] ' ; LPD - ( 0 . 8 0 0 0 - 2 . 0 0 0 0 - 0 . 7 0 0 0 - 0 . 1 0 0 0 - 0 . 7 0 0 0 0 . 5 0 0 0 0 2 . 4 0 0 0 0 . 6 0 0 0 - 0 . 4 0 0 0 0.7000 - 1 . 9 0 0 0 2.9000 1.6000 - 0 . 7 0 0 0 - 4 . 9 0 0 0 -2.1000 - 1 . 5 0 0 0 0.8000 0.8000 -0 .1000 0 . 9 0 0 0 3 . 0 0 0 0 1.8000 0 . 1 0 0 0 - 3 . 2 0 0 0 - 2 . 7 0 0 0 ] ' ;
Appendix
425
UK2M w ^(9.0 9.3 10.0 10.8 12.1 13.5 15.2 17.3 20.3 25.7 29.7 32.7 37.5 43.3 51 .4 56.0 63.4 68.8 73.0 79.0 65.3 92.0 63.5 91.1 100.0 108.0 114.6]'; GDP =[38.37 40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.85 125.25 148 .98 168.53 198.22 231.77 254.93 279.04 304.46 325.85 357.34 384.84 423.38 471 .43 515.96 551.12 573.65 595.26]',y =[38.37 40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.8S 125.25 148 .98 168.53 198.22 231.77 254.93 279.04 304.46 32S.85 357.34 384.84 423.38 471 .43 515.96 551.12 573.65 595.26]'; yD -[36.04 3B.37 40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.85 125. 25 148.98 168.53 198.22 231.77 254.93 279.04 304.46 325.85 357.34 384.84 423. 38 471.43 515.96 551.12 573.65]'; yP -(2.33 2.03 3.41 3.34 4.62 5.98 6.91 9.60 9.60 21.99 19.40 23.73 19.55 2 9.69 33.55 23.16 24.11 25.42 21.39 31.49 27.50 38.54 48.05 44.53 3S.16 22.53 21.61 ] • ; yPD -(2.60 2.33 2.03 3.41 3.34 4.62 5.98 6.91 9.60 9.60 21.99 19.40 23.73 19 .55 29.69 33.55 23.16 24.11 25.42 21.39 31.49 27.50 36.54 48.05 44.53 35.16 2 2.53 ] • ; G .-(6.67 7.73 7.48 8.19 9.25 10.55 12.06 13.77 17.15 23.65 27.7 30.18 34.1 3 34.61 49.98 56.51 61.64 67.2 71.2 75.27 80.91 87.05 93.64 101.6 112.93 124. 21 132.66]'; C -[24.37 25.62 27.59 29.29 31.91 35.76 40.44 46.21 53.26 65.59 76.23 87.1 7 100.52 119.21 138.56 154.27 169.37 185.61 198.82 217.49 241.55 265.29 299.4 5 327.36 347.53 365.06 382.36]'; M -[7.84 8.44 8.78 8.81 9.64 11.09 12.66 13.30 14.74 17.48 19.47 23.52 27. 36 29.86 31.04 34.59 40.66 42.46 48.05 56.67 69.27 154.12 170.67 195.31 214.9 4 229.22 238.73]'; ML -[-5.1962 -4.0476 1.7207 14.2848 8.1216 -6.4183 -1.4609 23.7577 39.2476 16.3852 14.2173 0.4014 11.7645 20.3134 22.3611 27.4556 11.8899 15.2224 19.570 2 27.5928 43.8390 -6.6041 34.8675 51.7866 40.7535 17.8065 -0.3106)'; MB -[7.85 7.84 6.44 8.78 a.81 9.64 11.09 12.66 13.30 14.74 17.48 19.47 23.5 2 27.36 29.86 31.04 34.59 40.66 42.46 48.05 S6.67 69.27 154.12 170.67 195.31 214.94 2 2 9 . 2 2 ] ' ; MP = ( 0 . 6 0 . 3 4 . 0 3 0 . 8 3 1.45 1.57 0.S4 1.44 2 . 7 4 1.99 4 . 0 5 3 . 8 4 2 . 5 1.16 3 . 5 5 6.07 1.80 5 . 5 9 8 . 6 2 1 2 . 6 0 B4.85 1 6 . 5 5 2 4 . 6 4 1 9 . 6 3 1 4 . 2 8 9 . 5 1 1 3 . 4 3 ] ' ; MPD - ( - 0 . 0 1 0 . 6 0 . 3 4 . 0 3 0 . 8 3 1.45 1.57 0 . 6 4 1.44 2 . 7 4 1.99 4 . 0 5 3 . 8 4 2 . 5 1. 18 3 . 5 5 6 . 0 7 1.80 5 . 5 9 8 . 6 2 1 2 . 6 0 8 4 . 8 5 1 6 . 5 5 2 4 . 6 4 1 9 . 6 3 1 4 . 2 8 9 . 5 1 ] ' ; P - [ 1 2 . 2 12.S 1 3 . 1 13.8 14.7 16.1 17.2 18.8 21.8 2 7 . 1 31.6 3 6 . 6 39.6 4 4 . 9 53 5 9 . 3 6 4 . 4 6 7 . 4 7 0 . 7 75 7 7 . 6 8 0 . 8 8 4 . 7 9 1 . 3 100 1 0 5 . 9 1 0 9 . 8 ] ' ; Pf -P.*e; Pfe -Pf.*e; r =[6.77 6.66 7.59 8.81 7.26 6.69 7.55 10.41 12.51 10.57 12.06 10.08 11.32 11.73 13.84 14.65 12.79 11.19 11.29 11.13 10.01 9.36 9.66 10.73 12.08 10.18 8.96]'; R - [ 6 . 7 7 6.66 7.59 8.81 7.26 6.69 7.55 10.41 12.51 10.57 12.06 10.08 11.32 11.73 13.84 14.65 12.79 11.19 11.29 11.13 10.01 9.36 9.66 10.73 12.08 10.18 8.96)'; RD = [ 6 . 5 7 6.77 6 . 6 6 7 . 5 9 8 . 8 1 7 . 2 6 6 . 6 9 7 . 5 5 1 0 . 4 1 1 2 . 5 1 1 0 . 5 7 1 2 . 0 6 1 0 . 0 8 11.32 11.73 13.84 14.6S 12.79 11.19 11.29 11.13 10.01 9.36 9.66 10.73 12.08 1 0.16]'; RP =[ - 0 . 1 1 0 . 9 3 1 . 2 2 - 1 . 5 5 - 0 . 5 7 0 . 8 6 2 . 6 6 2 . 1 0 - 1 . 9 4 1.49 - 1 . 9 6 1.24 0 . 4 1 2 . 1 1 0 . 8 1 - 1 . 8 6 - 1 . 6 0 0 . 1 0 - 0 . 1 6 - 1 . 1 2 - 0 . 6 5 0 . 3 0 1.07 1.35 - 1 . 9 0 - 1 . 2 2 - 2 . 3 1]'; RPD - [ 0 . 2 0 - 0 . 1 1 0 . 9 3 1.22 - 1 . 5 5 - 0 . 5 7 0 . 8 6 2 . 8 6 2 . 1 0 - 1 . 9 4 1 . 4 9 - 1 . 9 8 1.24 0 . 4 1 2 . 1 1 0 . 8 1 - 1 . 8 6 - 1 . 6 0 0 . 1 0 - 0 . 1 6 - 1 . 1 2 - 0 . 6 5 0 . 3 0 1.07 1.35 - 1 . 9 0 - 1 . 2 2 ] ' ;
YT YTD
-y-TX; -(25.413 26.003 27.561 28.844 31.022 35.81 41.266 47.763 50.604 63.24 75
426
Differential Models and Neutral Systems
UK2.M .618 92.421 105.691 123.176 139.439 148.984 159.992 176.603 189.245 207.634 2 27.641 252.984 283.438 495.6737 333.161 350.351 369.S67]'; YTP = [ 1 1 5 9 0.59 1.558 1.283 2.178 4.788 5.456 6.497 2.841 12.636 12.37B 16. 803 13.27 17.485 16.263 9.545 11.008 16.611 12.642 18.389 20.007 25.343 30.45 40 212.2357 -162.5127 17.19 19.2160J'; VTPD -(1.223 1.159 0.59 1.558 1.283 2.178 4.788 5.456 6.497 2.841 12.636 12.3 78 16.803 13.27 17.485 16.263 9.543 11.008 16.611 12.642 18.389 20.007 25.343 30.4540 212.2357 -162.5127 17.19]"; N -y.\L; prlce-[11.2 11.8 12.2 12.S 13.1 13.6 14.7 16.1 17.2 18.8 21.6 27.1 31.6 36.6 39.6 44.9 53 59.3 64.4 67.4 70.7 75 77.6 80.8 84.7 91.3 100 105.9 109.8]'; for 1-1:27 ppli)=price(i+l)-price(i); kprim(i) -kk(i+l) -kk(i|; end pie-lppj'; KP -kprira'; DP -pie.NP; logy -log(y); % TX -total current receipts of general government(millions pounds sterling) % TX -Total current receipts of general government(From UN financial » statistics) * n "labor productivity(A/1 = GNP/employment). » »From Yearbook of labor statistics(international labor office)
% t K "capital stock(Gross fixed capital formation) % I -investment(Gross capital formation) * D 'increase in stock % The UN(National accounts statistics) defines Gross capital % formation as the sum of the increase in stocks and gross fixed % capital formation
% i y -GDP(Gross domestic product) % yD -y(t-l) » yP =y'(t) % yPD-y' (t-1) * X -Net export(export-import of goods and services) * d -preferential trade agreement transportation i Pfe =import price level in foreign currency (P*e*e) % ta -tariffs i E -cumulative balance of payment * F -Net private outflow of capital * C -private consumption * M,M1 -money supply % Md -money demand(liquidity); Where, ML= rad-M % P -consumer prices * R "interest rate(discount rate) % RD "R(t-l) * RP =R'(t) * RPD-R'(t-1) * pie=inflation i L -employment % 8 -Balance of payment(deposit money banks) % T -Net government transfer of capital to foreigers and firms % G "Government consumption * DP -DMog(P) =p(t)'/p(t)
Appendix
427
UK2M %format long; Bertl«[X,G, I,C,GDP,R,M,YEAR] ; (years,n]-size(YEAR); Xl-[13.02;X(l:years-l)); Gl-[S.57;G(l:years-ll]; Il«[6.12;I(l:years-l)]; Cl=(21.64;C(l:years-l)]; Ml-(7.S6;M(l:years-ll]; Rl*(S.S4:RU:years-l) ] ; GDPl=»[33.44;GDPU:y«ars-l)I; DGDP-GDP-GDP1; Z-X+G+I+C; Z1-X1+G1+I1+C; *temp-[YEAR XI Gl II Cl Zl yl yD Mil RDl yPDl RP RPD Tl T P P-PF PI P-PFl G L LD LP LPD LI LD1 LP1 LPD1 t tl e el d dlj; «save Bertl.asc temp -ascii -double -tabs >clear Bertl temp; iML ■= L-M = ml-mO - ml*y(t) - m2*ytt-h) - m3*Rlt) - m4*R(t-h) - m6*P|t) - m7* R'(t-h) * L(tl -mO +ml*y(t)+ m2*y(t-h| +m3*R(t) »-m4»R(t-h) ■HnS'R" (t-h) +m6*p|t); temp"[M onea (size(y)) , y,yD,R,RD,RPD,P] ; thl=arx(temp,[0,1 1 1 1 1 1 1 0 0 1 0 1 1 0 ] ) ; L p - p r e d i c t ([M, o n e s ( s i z e ( y ) ) , y, yD,R, RD,RPD,P] , t h l , 4 l ; MM=thll6,l:7); »L|t) -17.2335 +0.8380.*y(t) -0.3061.*yD(t-h) -0.317S.*R(t) +3.6165.*RD
+i
» X(t) =x0 + xl»y(t) + x2*y(t-h) +x3*y'(t) +x4*y'(t-h) +x5*R|t)+ x8«L(t) +xl0 *L'(t-h) +xl2*P(tl +xl6*ta(t) +xl5*e(t) txl7*dlt) temp-[ X ones (sizely) ) ,y,yD,yP,yPD, R,L,LPD,P, ta,e] ; thx-arxlteap,(0,1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0]); Xp-predictI[X,ones(sizely)),y,yD,yP,yPD,R,L,LPD,P,ta,e],thx,4); XX=thx(ll,l:12); » G i t ) -gO + g l * y ( t ) + g 2 * y ( t - h ) + g 3 * y ' ( t ) + g 4 * y - ( t - h ) temp=(G o n e s ( s i z e l y ) ) , y , y D , y P , yPD,R, L] ; thg=arx(temp,(0,1 1 1 1 1 1 1 0 0 1 0 1 0 0]);
+g5*Rlt)
+g8*L(t);
428
Differential Models and Neutral Systems
UK2.M G p - p r e d i c t ( [ G , o n e s ( s i z e ( y ) ) , y , y D , y P , y P D , R, L ] , t h g , 4 ) ; GG=thg(6,l:7); %y(t) - zsO + z s l * y ( t ) + z s 2 » y ( t - h ) + z s 4 * y ' ( t ) + z s 5 * R ( t ) + z s 8 * L ( t ) + z s l O * L ' l t ) + z s l 3 * M l ( t ) - z s l 4 * T X ( t ) + z s l S * e | t ) + z i l 6 * t a l t ) + z s l 7 * d ( t ) * ( 0 ) + l / h * K ' ( t ) ; h-1 t e i n p - l y o n e s ( s i z e (y) ) , y , y D , yP,R, L,LP,M,TX,e, t a , K P ) ; thy-arx(temp,(0,1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0]); y p = p r e d i c t ( [ y , o n e s ( s i z e ( y ) ) , y , y D , y P , R , L , LP,M,TX,e, t a , K P ] , t h y , 4 ) ; yy-thy(ll,l:12); %cy(t) - ylO -fyll*y(t) + yl2*yMt) + yl3*R(t) + yl5*ML(t) +yl8*YT(t) temp-[C ones(size(y)) ,y,yP, R,ML, YT]; thcy-arx(temp,[0,1 1 1 1 1 1 0 0 0 0 0 0)); cyp«predict([C,ones(size(yI),y,yP,R,ML,YT],they,4); ccy-thcy(5,1:6); »Iy(t) - 10 +l/h*KP(t) temp=[I ones(size(KP)),KP]; thly=arx|temp,[0,1 1 0 0]); lyp»predict((I,ones(size(y)),KP],thly,4); Ily«thly(l,l:2); %Gylt) - gsO + g s l * y ( t ) + g s 4 * y ' ( t ) + y s 5 « R ( t l + g s 8 * L ( t ) temp«(G o n e s ( s i z e ( y ) ) , y , y P , R , L ) ; thgy-arx(temp,(0,1 1 1 1 1 0 0 0 0 0]); g y p - p r e d i c t ( [G, o n e s ( s i z e ( y ) ) , y , y P , R , L ] , t h g y , 4 I ; ggy»thgy(5,l:6); »Xy|t) - xO +xl*y(t) +x2*y(t-h) +x5»R(t) +x8»L(t) +xlO»L'(t) +xll»e(t) +xl2»t alt) +xl3*d(t)*<0) temp-[X ones (size (y)) , y,yD,R, L,LP,e,ta] ; thxy-arxftemp,(0,1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0]); x y p - p r e d i c t ( [X, o n e s ( s i z e (y) ) , y , y D , R , L , L P , e , t a ] , t h x y , 4) ; xxy"thxy(7,l:8); » B ( t ) = bO + b l * y ( t ) + b 2 * y ( t - h ) + b 3 * P ( t ) + b 4 * y ' ( t - h ) + b 5 * R ( t ) + b 6 * R ( t - h ) +b7*e ( t l + b 8 * R ' ( t - h ) +b9*L(t) + b l O * L ( t - h ) + b l 2 * L ' ( t - h ) + b l 3 * t a ( t ) + b l 5 * d ( t ) +bl7*B (t-h) teinp-(B o n e s ( s i z e ( y ) ) , y, yD, P,yPD, R,RD,e, RPD, L,LD,LPD, t a , B D ] ; thb-arx(temp,(0,1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1]); B p - p r e d i c t ( (G, o n e s ( s i z e (y) ) , y, yD, P, yPD, R, RD, e , RPD, L, LD, LPD, t a , BD] , t h b , 4 ) ; BB-thb(13,1:14); » y ( t ) - A*KAa*LA(l-a) — > l n ( y ( t ) ) - a * l n ( K ) + ( l - a ) » l n ( L ) temp-[log(y) ones!size(y)),log(K),log(L)]; thly-arx(temp,[0,1 1 1 0 0 0]); lyp =predict([log(y) ones(size(y)),log(K),log(L)],thly,4); lyy - t h l y ( 2 , l : 3 ) ; «D(t) - a ( l - c ) * y ( t ) - k 0 * K ( t ) + k l 3 * K ' ( t ) +L4*R(t) +L5*L(t) temp-[D o n e s ( s i z e (y) ) , y , K , K P , R , L , P , y P ] ; thD-arx(temp,[0,1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0]); D p - p r e d i c t ( [ D , o n e s ( s i z e ( y ) ) , y, K,KP, R, L, P, y P ] , thD, 4 ) ; DD-thD(7,l:8|;
+L6*P(t)
+v*y'(t)
«DP(t) = pO +pl*Pfe(t) +p2*w(t) -p3*n(t) -p4*P(t) +p5»M'(t) +p6*ML(t) temp-[DP ones(size(w)),Pfe,w,N,P,MP,ML);
Appendix UK2.M thp - a r x ( t e a p , [ 0 , 1 1 1 1 1 1 1 0 0 0 0 0 0 0 ] ) ; dip-predict([P,ones!size(w|),e,w,N,P,KP,KL),thp,4); dp-thp(6,1:7); subplot (2,1,1) ,plot (YEAR, C, YEAR,Cp, '*■') title ('UK—PRIVATE CONSUMPTION— ') ; ylabel('C'|; xlabel('YEAR'); subplot (2,1,21,plot(YEAR,X,YEAR,Xp, ' + ') title (' UK—EXPORTS— • ) ; ylabelI'X'I; xlabel('YEAR'); pause subplot (2,1,1),plot(YEAR,I,YEAR,Ip, ' + ') title ('UK— INVESTMENT— ■); ylabelCI') ; xlabel('YEAR'); subplot(2,1,2),plot)YEAR,G,YEAR,Gp, • + ') title ('UK—GOVERNMENT CONSUMPTION— •) ; ylabel ("GM; xl*bel('year'); pause clg subplot(2,1,1),plot(YEAR,M, YEAR, Lp, ' + ') title ('UK—MONEY DEMAND—'); ylabel('Ml'); xlabel('YEAR'); subplot(2,l,2),plot(YEAR,Z,YEAR,[Xp+Ip+Gp+Cp], ' + ') title ('UK—AGGREGATE DEMAND—'); ylabel('Z'); xlabel('YEAR'); pause « subplot (2,l,l),plot(YEAR,DP,YEAR,dlp, ' + ') title (• UK—CONSUMER PRICES— ■); ylabel('P'l; xlabel('YEAR'); subplot (2,l,2l,plot(YEAR,y,YEAR,yp, ■ + ') title('UK—INCOME—') ; ylabel('GDP'); xlabel('YEAR'); pause subplot (2,1,1),plot(YEAR,C, YEAR, cyp, * + ') title (' UK—INCOME/CONSUMPTION—'); ylabel1'cy'); xlabel('YEAR'); subplot(2,1,2), plot(YEAR,I,YEAR,Iyp,* +') title ('UK—INCOME/INVESTMENT—'); ylabel('Iyp'); xlabel('year')i pause clg subplot(2.1.1),plot(YEAR,G, YEAR, gyp,' + ') titM'UK-&rcOME/GOVraNMENT-'); ylabel('gyp');
429
Differential Models and Neutral Systems
430 UK2M xlabeli'YEAR'); subplot(2,1,2),plot!YEAR,X,YEAR,xyp,'+') title!'UK--INC0ME/EXPORT--'); ylabell'xyp'); xlabel('YEAR'); pause ■j
subplot (2,1,1),plot!YEAR,B,YEAR,Bp,•♦') title('UK--BALANCE OF PAYMENT--'); ylabel('B'); xlabel('YEAR'); subplot (2,1,2),plot(YEAR,logy,YEAR,lyp, ' + ') title('UK--log(income)--') ; ylabell'lny'); xlabel('YEAR') ; pause subplot (2,1,1),plot I YEAR,D,YEAR,Dp, '♦') title('UK—Increase in s t o c k — ' ) ; ylabel ('DM; xlabel('YEAR'); 4 The economic dynamics can be put In matrix form * as follows: x'(t)- A-l*x'(t-h) =» A0*x(t|+Al*x(t-h)+B*u (t) 4x-|y R L K P E] '; % The economic dynamics can be put in matrix form % as follows: x'(t)- A-l*x'(t-h) - A0*x(t)+Al«x(t-h)+8*uIt) »x«(y R L K P E ) ' ; »A-1 -[anil -ami4 ami3 0 0 0; 0 am22 0 0 0 0; Lm03 0 -LmOl 0 0 0; a3 0 a6 -an 1 0 0; 0 -M7»p6*P(t) 0 0 0 0; b4 b8 bl2 0 0 0) «A0 -|a01 al2 al4 ale al8 0; a21 a23 0 0 a25 0; 0 0 a33 0 a35 0; 0 0 0 a44 a 45 0; aSl*p(t) a52*p(t) 0 0 a55»p(t) 0; a61 a62 a63 0 0 0] IA1 -[all al3 alS 0 0 0; al21 al22 0 0 0 0; al31 al32 al33 0 0 0; al41 al42 al43 0 0 0; al51*p(t) al52*p(t) 0 0 0 0; al61 al62 al63 0 0 al66] »B1 "[-ssl ssl sslzl5 sslzl8 sslzl7 0 0 0; 0 0 0 0 0 -L2 0 0; -mwzsl4 mw rawz slS mwzsl6 0 mwzsl3 0 0;-zsl4 1 zsl5 zsl6 0 zsl3 0 0; 0 0 pl*pf(tl 0 0 -p6 p5 0; 0 0 b7 bB blS 0 0 -1] fB2 -[ssl ssl ssl 331(113+07) 0 0 0 0 0; 0 0 0 -L2 0 0 0 0 0,-0 mw 0 0 0 0 nw mw 0; 0 1 0 0 0 0 1 1 0; 0 0 0 -p6 -p3 p2 0 0 1;0 0 0 -1 0 0 1 0 0] «u - [q S ) ; iq »q
-[Tl gO e ta d Ml Ml' £0]'; »[T1 -0.3683 e ta d Ml Ml' -5.5844e+8]';
IS -[CO 10 xO M0 n w xO ylO pO]'; »S -(-16.7625 -1.7795 6.3646 -1.5714 n w % B=[B1 B2)
6.3646
0.0047 0.1518]';
Appendix UK2.M % AB=A0*B * A2B-(AO*A0)*B * A3B-(AO*A0*AO)*B % A4B-(A0*A0*A0*A0)*B % A5 B= (AO * AO * AO * AO * AO *AO) * B » G=[B AB A2B A3B A4B A5B] % rank(B)= 6 % rankG » 6 * For spectral controllability % Deltal-eye(sizelAl)I.*j -Ami.*(j*2.7814'(-j)) - AO * + Al.M2.7814-(-j)) i a=[Deltal B) * for J =0.4312, 4, 10, 23, 100, ... i rank(a)=6 i For nonzeroeigenvalues of A-1 to be controllable via matrix B, * cc-eye(size(Ami)).*j -Ami % c=[cc B] » for j "0.4312, 4, 10, 23, 100, ... % rank(c)=6 % Since rank(B) is 6, where rank(a) and rank(c) are % also 6, the system is function controllable LI =1; L2 =0.9; L5 =DD(5) L6 =DD (6) ; h -1 % for y(t) - A*K A a*L A (l-a) a= 0.9 % a =alpha aa=0.1 % aa=(l-a)-=beta W«sura(w)/27 mw =(aa/W).*(l/a) % cobb Douglas states: aa+a=l »>constant return to scale. % aa+a>l =>increasing return to scale. % aa+a="l =>decreasing return to scale. % Increasing return to scale if doubling LSK more than doubles y. % constant return to scale if doubling L*K exactly doubles y. % decreasing return to scale if doubling LSK less y.
* % In cobb.m program log(A) is found to be 1.6 to give the best % aproximation for y(t). Z0=GG(1)+11(1)+MM(1)* ICC(8)+11(111)+CC(1)+XX(1) zl=GG(2)+II(2)+MM(2)MCC(8)+II(ll) )+CC(2)+XX(2) z2-GG(3)+II(3)+CC(3)+XX(3)+MM(3)MCC(8)+II(ll) ) z3=GG(4)-II(4)+CC(4)+XX(4) z4=GG(5)+II(5)+CC(5)+XX(5) Z5»GG(6)+II(6)+CC(6)+XX(6)+MM(4)*(CC(8)+11(11)) Z6=II(7)+CC(7) + (CCI8)+II(11) )*MM(5) Z8=GG(7)+II<8)+XX(7)
431
Differential
432 UK2.M z 9 = I I (9) Z10-XXI8) zll=-II<10) zl3-(II(ll)+CC(8))*MM(6) zl4—CC(2) zl5«XX(U> Z16»=XX(10) Z17-0 z l 8 - X X [ 9 ) + (CC(8)i-zl3)*MM(7) zl9=-CC(3) Z20-CCI4) z21-CC(5) % z22=-(CC(8)+II(ll))*M1
zsO - c c y ( l ) + I I y ( l ) + g g y ( l ) +xxy(l); »constant term zsl - ccy(2)+ggy(2)+xxy(2) ; »y(t) term zs2 ■= xxy(3) ; %y(t-h) term zs4 - c c y ( 3 ) + g g y ( 3 ) ; %y'(t) term z«5 » c c y ( 4 ) + x x y ( 4 ) + g g y ( 4 ) ; %R(t) term za8 - x x y ( 5 | + g g y ( 5 ) ; %L(t) term zslO »xxy(6l ; %L'(t) term z s l 3 =ccy(S) ; *Ml(t) term zsl4 »ccy(6) ; »T(t) term zslS " x x y ( 7 ) ; *e(t) term zsl6 «xxy(8) ; %ta(t) term si
-1-Ll*z3
% assumption yyl = zsO; yy2 « z s l ; yy3 = z s 2 ; yy4 = z s 4 ; yyS = z s 5 ; yy6 ■ z s 8 ; yy7 » z s l O ; yy8 » z s l 3 ; yy9 - z s l 4 ; yylO » z s l 5 ; yyll »zsl6; zsO= G G | 1 ) + X X ( 1 ) + C C ( 1 ) + I I ( 1 ) aO - D D ( l ) / ( h * ( l - y y ( 2 ) ) ) a l =» - D D ( l ) * y y 3 / ( ( l - y y 2 ) * h ) a2 = - a l * h a 3 - aO*h*yy4 a4 =■ D D ( l ) * y y 5 / ( ( l - y y 2 ) +DD(4)) a5 - D D ( l ) * y y 6 / ( l - y y 2 ) +DD(5) a6 - a 0 * h * y y 7 a8 = DD(6) %A-1 starts here; amll » a-11 amll =Ll*z4/sl am22 =L2*MM|6) aml3 =Ll*zlO/sl am!4 -Ll*2l0/sl
Models
and Neutral
Systems
Appendix UK2.M Lm03 =■ DD(l) * y y 4 * m w / < l - y y 2 | LmOl - D D ( 3 ) * D D ( l ) / ( l - y y 2 ) -mw*DD(l)*yy7/Il-yy2I a3 - DDil)*yy4/(l-yy2) a6 ' DD|l)*yy7/(l-yy2) ami - D D ( 3 ) * D D ( l l / ( l - y y 2 ) M7p6P - M M ( 7 ) * d p ( 6 ) * s u m ( P ) / 2 7 b4 = - B B ( 5 ) b8 ■= - B B O I bl2= -BBU2)
%AO starts here! aOl -Ll*(zl-l)/sl all -Ll*z2/sl al2 -Ll*z5/sl al3 -Ll*z6/sl al4 =Ll*z8/sl al5 =Ll*z9/sl al6 =Ll»zll/sl al7 =.Ll»zl9/sl alB «=Ll*zl8/sl a21 = L2*MM(2) a23 =L2*MM(4) a25 =L2*MM(7) a33 =aO; a35 -mw*a8 a44 =aO a45 -a8 a51—dp(7)»MM(2)*sum(P)/27 % includes Pit) a52—dp(7)*MM(4)*sum(P)/27 * includes Pit) aS5 —(MM(7)+dp<5))*sum(P)/27 % includes P(t) a61 «BB(2) a62 =BB(6) a63 -BB(IO) %A1 starts here! all =Ll»z2/sl al3 =Ll*z6/sl al5 =Ll*z9/sl a22 =L2*MM(3) a24 »Ii2»MM(5) al = -DD(l)*yy3/((l-yy2)*h) a2 » -al*h 11 = -al 12 —mw*DD(l)*yy3/(l-yy2) 14 -mw*DD(l)*yyS/(l-yy2 +DDI4)) b2 =BB(3) b6 »BB(7) blO =»BB(11| bl7 =>BB(13) alll -I,1MGG<3)+II<3)+XX(3)+MM(3)MI1(11)+CC18) ) )/sl all2 -al3 all3 =al5 al21-a22 al22-a24 al31»mw*DD(l)*XX(3)/(l-(XX(2)+ggy(2)+ccy(2))) al32-mw*DD(l)*yyS/(l-yy2) -DD(4)
433
Differential Models and Neutral Systems
434
UK2.M al33=a5-ll*h al41 -DD(l)*yy3/(l-yy2) al42 =DD(l)*yy5/(l-yy2) + DD(4) al43 =DD(l)*yy6/(l-yy2) + DD(5) alSl =-dp|7)*sum(P)/27 ^includes Pit) al52 =-MM(5)*dp(7)*sum(P)/27 %includes Pit) al61 =BB(3] al62 =BB(7) al63 -BB(ll) al66 =BB(14) % Bl starts here! ssl =Ll/sl « ssl=bl sslzl3 =ssl*zl3 sslzlS =ssl*zl5 sslzl6 =ssl*zl6 sslzl7 «0 sslzl8 =ssl*zl8 blzl4—BB(2)*zl4; bl -BB(l) blzlS=BB(2)*zl5 blzl6-BB(2)*zl6 blzl7=BB(2)*zl7 blzl3=BB(2)*zl3 mwzsl4 =mw*yy9 mwzslS =mw*yylO mwzsl6 =mw*yyll mwzsl7 =0 rawzsl3 «"mw*yy8 plpf p4 p5 b7 b8 bl5
=dp(2)*suin
»dp(2)*Pf(t)
«B2 starts here! ssl -Ll/sl sslil3c7 -ssl*(II(ll)+CC(8)) 12 =ntw*DD(l)*yy3/(l-yy2) p6 = dp(7) p3 =» dp (4) p2 = dp(3) al7 =Ll*zl9/sl a45 -a8 cO =» CC(1) 10 = 11(1) xO = XX(1) pO - dp(3) MO » MM(1) ylO - ccy(l) mO -MM(1) gO «=GG(1)
fO»B(l)-XX(l) g2-L2*MM(l)
Appendix
435
We now use the MATLAB program India?..M which we present again for ease of reference to display the polynomial coefficients of the economic variable and their standard deviations and plots INDIA2.M •/« This program was written at 8:26PM 4/11/96 "A India's data is in billions of Rupees % adoptedfromInternational Financial Statistic Yearbook 1994. except where noted. % Use ndu m to obtain the matrix entries YEAR-[1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989]'; EX -[13.3 15.1 16 16.3 16.4 17.9 21.5 26.6 36.9 49.5 613 64 71.2 83.4 90.3 102.6 113.6 131.4 158.5 149.5 165 4 202.8 259.1 346.1)'; IMP -[-21.2 -22 -19 -17.5 -18.2 -21.8 -20.5 -31.8 -47.8 -56.6-56.1 -65 2 -74.2-100.9 -136 -148.1 -157.4 176.8 -194.8 -217.5 -223.6 -252.6 -320.1 -402l]\ X -EX-IMP; K =[46 50 8 53.8 59 63.1 70.8 81.3 90.7 110 133.3 153 172.2 188.8 213.1 262.8 314.6 357.7 399.9455.7 5426620.5 721 9856.7 1027.8]'; KP -[4.8 3.0 5.2 4.1 7.7 10.5 9.4 19.3 23.3 19.7 19 2 16.6 24.3 49.7 51.8 43.1 42.2 55.8 86.9 77.9 1014 1348 171 1 2122]'. TX =[25 3 25.5 27 5 30.5 33 3 40.6 45 7 49 7 75 3 91 7 102.6 113.5 129.6 144 161.1 195 9 225 1 255 2 296 1 361.2 420.7 480.8 557.4 675.4)': D - [ 7 2 6 2 1 6 5 8 10 7 14 6 5 9 22.6 34.7 30 8 24.7 18 41 148.4 21 8 85.6 50 2 38 34.4 91.9 58.5 26.9 113.9 80.2)'; I =[40.8463 jl 8 5 7 3 6 2 7 6 8 2 7 3 7 11392 14582 165 177 05 186.21 229.84 261.4 313.44 360.99 404.22 473.77 487 88 626 21 714.43 79025 1013.28 1170.6)'; ta -(20 7 24 27.3 30.6 35.2 38.5 41.8 51.68 63.32 77.14 85.33 89 17 105 34 12184 139.05 169.14 191.83 223.19 228 1 284.44 329.19 383.5 430 76 481.59]'; F -[198 135 160 105 77 110 104 142 222 414 634 926 1147 1424 2743 2281 2599 26S0 2278 2456 2223 2636 2295 2567]': B - [-318 17-5-9 1 1 -5-80-1 78 24 160-7-207 129-1 56 100 72-88-44-57 29)'; BD -[2-31 8 17-5-9 1 1 -5-80-1 78 24 160-7-207 129-1 56 100 72-88-44-57]', T = [X-B-FJ: e -*[7 576 7 547 7.628 7 559 7.576 7.903 8.773 9 896 9.978 10 462 10.318 9.971 10.668 10 416 10 114 10.591 10 627 10.986 12.205 13.363 16.051 18.268 20.117 22.387]'; L - [26.9 26.6 28.3 30.3 31.9 33.3 35.2 35.8 36.5 38.2 42 44.2 47.2 47.7 48.1 52.4 53.5 56.2 61.5 67 71.5 79.5 85 4 90 1]': LD - [27.1 26.9 26.6 28.3 30.3 319 33.3 35.2 35.8 36.5 38 242 44 2 47.2 47.7 48.1 52.4 53.5 56.2 61.5 67 71.5 79.5 85.4]'; LP -[-0.3 1.7 2.0 16 14 1.9 0.6 0 7 1.7 3.8 2.2 3.0 0.5 0 4 4.3 1.12.7 5.3 5.5 4.5 8.0 5.9 4.7 9.9]'; LPD -1-0.2-0 3 1.7 2.0 1 6 1.4 1 9 0.6 0.7 1.7 3.8 2.2 3.0 0.5 0.4 4.3 1.1 2.7 5.3 5.5 4.5 8.0 5.9 4 7]'; w - [229 4 257.6 285.8 313 9 342 2 370.4 398.6 438.65 511.49 528.54 551.07 630.96 672.08 726.3 881.85 986 86 1082 05 129 02 149.26 740.2 890.7 862.9 913 2 886.il'; y -[295 7 346 1 366.7 403.9 431.6 462.6 510.1 620 1 732 4 787 6 848.9 960.7 1041 9 1143 6 1360.1 1597.6 1781.3 2075.92313.4 2622.4 2929.5 3332 3957.8 4568.2]'; GDP = y: yD -[261 5 295.7 346.1 366.7 403.9 431.6 462.6 510.1620.1 732.4 787.6 848.9 960 7 104! 9 1143 6 1360.1 1597.6 1781 3 2075.9 2313.4 2622 4 2929.5 3332 3957.8)', yP -[50.4 20 6 37.2 27.7 31.0 47.5 110.0 112.3 55.2 61 3 111.8 81.2 101.70 216.5 237.5 183.7 294.6 237.5 309.0 307.1 402.5 625.8 610.40 787.0)'; yPD =[34 2 50.4 20 6 37.2 27.7 31.0 47.5 110.0 112.3 55.2 61.3 111 881.2 101 70216.5 237.5 183.7294.6 2375 309.0 307.1 402.5 625.8 610.40)'; G -[25 27.9 30.6 34.4 38.4 44.6 47.3 51.6 62.4 73 82.3 87.6 97.2 111.7 130.8 153.6 182.7 211.4 243.5 291.7 346.3 408.4 473.3 542]'; C -[237 9 283.4 283.9 307.8 325.5 351 386.9 434.3 565.1 578.2 600 8 691.8 752.4 817 992.9 1137 7 1251.7 1469.5 1620 9 1777.6 1986 2225.5 2574.2 2882.4)'; n =yAL: M =[46 8 5153 9 60.4 67.6 76.5 86.2 101 1113 122 3 152.8 178 5 157 6 176.9 204.6 232 5 273 7 308.6 365.6 412.4 478.7 543.2 632.8 746.9]'; MD =(43 46.851 53.9 60.4 67.6 76.5 86.2 101 111.3 122.3 152.8 178.5 157.6176.9 204.6232.5 273.7 308.6 365 6 412 4 478.7 543 2 632.8)';
436
Differential Models and Neutral Systems
MP -{4.2 2.9 6.5 7.2 8.9 9 7 14.8 10.3 11.030.525.7-20.9 19.327.727.941.234.957.046.866.364.589.6 114.1 106.7]'; MPD -[3.8 4.2 2.9 6.5 7.2 8.9 9.7 14.8 10.3 11.030.525.7-20.9 19.3 27.727.941.2 34.9 57.046.8 66.3 64.5 89.6 114.1]'; ML - [ -46.4 -50.4 -53.4 -59.8 -«6.8 -75.8 -85.6-100.2 -110.0 -121.3-151.6 -177.4 -156.4 -175.5 -203.4 230.9 -271.9 -306.6 -363.1 -409.9 -475.8 -540.2 -629.4 -743.1]'; P -[16.4 18.6 19.1 19.2 20.2 20.8 22.2 25.9 33.3 35.3 32.6 35.3 36.2 38.4 42.8 48.8 52.2 58.4 63.3 66.8 72.6 79 86.4 91.8]\ Pf -P.'e; Pfe - P f e : R "[4 37 5 36 3 9 3.92 5.68 6.3 4.69 6.64 13.52 10.4 1128 10 18 8 05 8.47 7.24 8.61 7.27 8.3 9.95 10 9.97 9.83 15.5 15.5]'; RP =[0.99-1.46 0.02 1.76 0.62-1.61 1.95 6.88-3.12 0.88-1.1 -2.13 0.42-1.23 1.37-1.34 1.03 1.65 0.050 03-0.14 5.67 0 0.07]'; RPD -[-1.91 0.99-1.460.02 1.76 0.62-1.61 1.95 6.88-3.12 0.88-I.I -2.13 0.42 -121 137-134 1.03 1.65 0.05 -0.03 -0.14 5.67 0]'; RD =[6.28 4.37 5.36 3.9 3.92 5.68 6.3 4.69 6.64 13.52 10.4 11.28 10.18 8.05 8.47 7.24 8.61 7.27 8.39.95 10 9.97 9.83 15.5]'; YT -y-TX. YTD -[237.7 270.4 320.6 339.2 373.4 398.3 422.0464.4 570.4 657.1 695.9 746.3 847.2 912.3 999.6 1199.0 1401.7 1556.2 1820.7 2017.3 2261.2 2508.8 2851.2 3400.4]'; YTP =[ 32.7 82 9 101.5 135.7 160.6 184.3 226.7 332.7 419.4 458.2 508.6 609.5 674.6 761.9 961.3 1164.0 1318.5 1583.0 1779.6 20235 2271.1 2613.5 3162.7 3655.1]'; YTPD -[82.9 101.5 135.7 160.6 184.3 226.7 332.7419.4 458.2 508.6 609.5 674.6 761.9 961.3 1164.0 1318.5 1583 0 1779.6 2023.5 2271.1 2613.5 3162.7 3655.1 4393.9]'; price=[ 14.8 16.4 18.6 19.1 19.2 20.2 20.8 22.2 25.9 33.3 35.3 32.6 35 3 36.2 38.4 42.8 48.8 52.2 58.4 63.3 66.8 72.6 79 86.4 91 81'; for i-1 24 pp(i)=price(i-l}-price(i); end pie-[pp]'; DP« pie.'P; logy=log(y):
% TX -Total current receipts of general govemment(From UN financial % statisticsXthousand million Italian lire) •/« n "labor productivity A/1 - GNP/employment). % -From Yearbook of labor staturics(uuemational labor office)
% % K =capital stock(Gross fixed capital formation) % I =investment(Gros5 capital formation) % D =increase in stock % The UNfNational accounts statistics) defines Gross capital % formation as the sum of the increase in stocks and gross fixed % capital formation
% % w » compesation of employees(UN financial accounts statistics. 1965-84 % plus (wages) in manufacturing and yearbook of labor statistics 1985-89) %y -GDP(Gross domestic product) %yD-y(t-l) V.yP=y'(t) VoyPD-y'd-l) %X -Net expon(export-tarpori of goods and services) % d ^preferential trade agreement transportation % Pfe =import price level in foreign currency (P*e"e) •/•ta -tariffs % E -cumulative balance of payment
Appendix % F -Net private outflow of capital •/» C 'private consumption '/• M.MI "money supply % Md -money demand(liquidity); Where, ML= md-M •/• P "consumer prices V. R -interest raie(money market rate, except 1988&l989:national rate) 0 /.RD-Wt-0 % RP =R'(t) V. RPD-R'(t-l) % pie-inflation % L - industrial production 'A B -Balance of payment(deposit money banks) % T -Net government transfer of capital to foreigers and firms % G ^Government consumption V. DP -D # log(P) - p W W
% ■V.Bert-[X,G.I.C.GDP,PGDP.R.M.QM,YEAR] [yean.vj-sizefYEAR); Xl-[23.9;X(l:years-l)]; Gl-[22.9:G(lyears-l)j; ll-[35 3;I(l:years-l)]: Cl-[204.2:C(l:years-l)], Mil -l43.M(l:years-l)]: RI-|6:R(l:years-l)]; GDPI-[261.5:GDP(l:years-l)]; DGDP-GDP-GDP1; Z-X-K}-l+C. Zl-XI+GI-ll-fC; %temp-[YEAR XI Gl II Cl Zl yl yD Ml 1 RDl yPDl RP RPD Tl T P P-PF PI P-PFl G L LD LP LPD LI LD1 LP1 L P D I t t l e e l d d l J ; %save Bert 1 asc temp -ascii -double -tabs Sclear Bert 1 temp; •/.ML = L-M - ml-mO - ml*y(t) - ro2\v(t-h) - m3«R(t) - m4«R(t-h) - m6*P(t) - m7»R'(i-h) '/, L(t) -mO *ml*y(t) + m2'y(t-h) -m3'R(t) +m4«R(t-h) +m5«R'(t-h) -mi6'p(t); temp=[Mones(size(y)),y,yD,R.RDJU>D,P]; t h l - a r x < t e m p , [ 0 . 1 I I I I 1 1 0 0 I 0 1 I 0]); Lp-predict([M,ones(si2e(y)),y,yD,R,RD,RPD,P],thl,4); MM-thl(6.1:7); % C(t)-cO *cl*YT(t) +c2«YT(t-h) + c3'YT(0' +c4«YT(t-h) +c5'R(t) + c6'R(t-h) +c7'(17.2335 +O.8380.«y(t)-0.3061.*yD(t-h) -0.3 l75.»R(t) +3.6l65.«RD(t)-3.3413.'P(t)-1.7714.*RPD(t)) - M(t)); temp-[Cones(size
437
438
Differential Models and Neutral Systems
thx-arx(temp,[O.I l l l l l I I I 1 1 0 0 1 0 I 0 0 1 0 0 0 ] ) ; Xp^redict(ix,ones<size
Appendix temp-[Dones(size(y)),y,ICKP.R,L.P.yP]; thD=arx
subplot (2.1,1 ),plot(YEAR,C,YEAR,Cp,+') title(india--PRJVATE CONSUMPTION--'); ylabeK'C); xlabeK'YEAR'); subplot (2. l,2),pk>t(YEAR,X.YEAR,Xp,'V) title('India-EXPORTS-'); ylabel('X'); xlabel('YEAR'); pause subplot (2.1.1 ),plot(YEAR,l,YEAR.Ip,'+') titleCIndia-lNVESTMENT--); ylabel(T); xlabel(YEAR'); subplot(2,1.2),plot(YEAR,G.YEAR,Gp,'+) tttleClndia-GOVERNMENTCONSUMPTION-); ylabel(G'); xlabel('year'); pause clg subplot(2.1,1 ).plot(YEAR,M,YEAR,Lp.'+) title('lndia-MONEY DEMAND-1); ylabelCMl'): xlabel(YEAR'); subplot(2.1.2),plot(YEAR,Z.YEAR.[Xp+Ip+Gp+Cpl,+') titleClndia-AGGREGATE DEMAND-); ylabel('Z); xlabel(YEAR'); pause V subplot (2,1,1 ),plot(YEAR.DP.YEAR,dlp,'+') titleCIndia-CONSUMER PRICES-'); ylabel('DF); xlabel('YEAR'); subplot (2.1,2),plot(YEAR.y,YEAR,yp,'+') title('lndia-INCOME-); ylabelOGDP'); xlabel('YEAR'): pause subplot (2,1,1 ),plot(YEAR,C,YEAR,cyp,'+') title('lndia-INCOME/CONSUMPTION--'); ylabelCcy1); xlabel(YEAR'); subplot(2.1,2 ).plot(YEARJ,YEAR,lyp,'+') titleCIndia-INCOME/INVESTMENT-'); ylabel('Iyp');
439
440
Differential Models and Neutral Systems
YT -[270.4 320.6 339.2 373.4 398.3 422.0 464.4 570.4 657.1 695.9 746.3 847.2 912 3 999.6 1199.0 1401.7 1556.2 1820.7 2017.3 2261.2 2508.8 2851.2 3400.4 3892.8]'; YTD =[237.7 270.4 320.6 339.2 373.4 398.3 422.0 464.4 570.4 657.1 695.9 746.3 847.2 912.3 999.6 1199.0 1401.7 1556.2 1820.7 2017.3 2261.2 2508.8 2851.2 3400.4]'; YTPD-(32.7 82.9 101.5 135 7 160.6 184.3 226.7 332.7 419.4 458.2 508.6 609.5 674.6 761.9 961.3 1164 1318.5 15S3 1779.6 2023.5 2271.1 2613.5 3162.7 3655.1]': YTP=[82.9 101.5 135.7 160.6 184.3 226.7332.7419.4458.2 508.6 609.5 674.6 761.9961.3 1164 1318.5 1583 1779 6 2023.5 2271.1 2613.5 3162.7 3655.1 4393.9]'; price-[l4.S 16.4 18.6 19.1 19.2 20.2 20.8 22.2 25.9 33.3 35.3 32.6 35.3 36.2 38.4 42.8 48.8 52.2 58.4 63.3 66 8 72.6 79 86.4 91.8]'; for i=l:24 pp(i)=price(i+1 )-price(i); end pie-(pp]': DP- pie \P; logy-"logo); [years,v]-size(YEAR); Xl=[23 9;X(l.years-l)]; GI-[22 9:G(l:years-l)]; ll«[35.3:I(l:years-l)]; Cl-[204.2:C(l:years-l)J, Mil «[43:M(lyears-l)]; Rt=[6;R(l:years-l)J; GDP I -[261 5;GDP( 1 :years-1)]; DGDP-GDP-GDP1; Z=X+G+I+C; Zl-Xl+Gl+II+C; %temp-(YEARXIGI II Cl Zl y l y D M I l RDl yPDI RP RPDTl T P P-PF PI P-PFl G L LD LP LPD LI L.D1 LPI LPD1 ttl e e l d d l ] , •/isave Bertl.asc temp -ascii -double -tabs %clear Bert I temp: %ML = LL-M - ml-mO - ml'yft) - m2*y(t-h) - m3*R(t) - m4*R(t-h) - m6#P(t) - m7*R'(t-h) % where I.L =■ money demand. % L(t) -mO +m 1 *y(t)+ m2'y(t-h) -m3*R(t) +m4*R(t-h) +m5 , R(t-h) -m6*p(t); temp-[Mones(size(y))y.yD,R.RD.RPD,P]; thl-arx(temp,[O.I I I I I 1 I 0 0 I 0 1 I OJ); Lp»predicl([M.ortes(size(y)),y,yD.R,RD,RPD,P],thl,4); MM=thl(6.1:7); % C(t) -CO -cl»YT(t) +c2«YT(t-h) + c3«YT(t)' +c4*Yr(t-h) -c5'R(l) + c6*R(t-h) +c7'( 7.8783 +0 1041.♦>•(!) +0 1283 «yD(t.h) + 2.5354.^(1) +1.086l.*RD(t)-2.4128.*P(t)+ 1.3031.*RPD(t))- M(t)); temp=[Cones(size(R)),YT,YTD,YTP,YTPD.R,RD,ML]; thc-arx(temp.[O.I 1 I I I I I 1 0 0 1 0 I 0 1 0]); Cp-predict(|C.ones(size(R)),YT,YTD,YTP,YTPD,R,RD.ML].thc,4); CC-thc(7.l:8); % I(t)-i0 +il*>(0 + i2«y(t-h)-i3V(t) + i4*y'(t-h)+i5*R(t)*i6*R(t-h)+i8*L(0+i9*L(t h)-iU*K(t) -il3'ML(r) temp-[lones(size(y)),y,yD,yP,yPD,R,RD,L,LD,K,ML]; thi-arx(temp.[O.I I 1 I M 1 I 1 1 I 0 0 1 0 1 0 1 0 I 0 01); lp-predict([l,ones(size(y)),y,yD,yP,yPD£,RD,L.LD,K,ML],!hi>4); ll-*i(IO.I:ll);
Appendix % X(t) =xO + xl 'y(i) + x2«y(l-h) +x3'y'(0 +x4V(t-h) +x5«R(t)+ x8*L(l) +xl0*L'(t-h) +xl2*P(t) +xl6* "xl5*e(l)"xl7'd(t) temp=lXoiies(size(y)).v,yD,yP,yPD,R,L.LPD,P,ta,e]; thx-arx(iemp.[O.I 1 I M 1 1 I I M 0 O I 0 I 0 O I 0 0 0]); Xp-predict([X,on«s(size(>0).y.yD.yP,yPD.R,L.LPD.P.ta,e],thx.4); XX-thx(l 1.1:12); •/, G(t) =g0 - gl'y(t) - g2*y(t-h) +g3'y(t) *e4«/(i-h) +g5*R(t) -g8*Ul); temp=-[Gones(size(y)),y,yD,yP,yPD,R,L]; thg-arx(temp.[0.1 1 1 1 f l 1 0 0 1 0 1 0 0]). Gp»predici([G.ones(size(y)),y,yD,yP.yPD,R,L],thg.4);
GG=lhg(6,l:7); %y(t) " zsO +zsl'y(t) *zs2*y(t-h) *zs4*y,(t) +zs5*R(t) -zs8*Ut) tzslO'L'Ct) +zsl3*Ml(l) -zsl4'TX(i) *zsl5'e(t) -rzsl6'ta(t) +zsl7 , d(t) , (0) + l.VK-(t); h-l temp-[yones(size(y)),y.yD,yP,R,L,LP.M,TX,e,la,KP), thy-arx(temp,[0,l 1 1 1 1 M 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0]); yp-predici([>.ones(size(y)),y,yD,yP,R,L,LP,M,TX,e,ta,KP],thy,4); yy=thy(ll.l:12): •■ocy(i) - ylO +y 11 'y(t) + y 12*y(0 + yl3*R(l) + yl5*ML(t) *y l8'YT(t) temp»[C ones(size(y)),y.yP,R.ML,YT]; thcy-arx(temp.[0.1 I 1 1 1 I 0 0 0 000]): cyp=predict([C,ones(size(y)),y,yP,R.ML.YT],ihcy,4); ccy-thcy(5,l:6): %ly(t) - 10 + I.VKP(t). dK(t>'dt - D(i-h) iemp=[l ones(size(K.)),K], thly-arx(temp,[0.l I 0 0]); lyp-predict([l.ones<size(K)),K],lhly.4); llythlyt 1,1:2): %Gy(t) = gsO +gsl •>■(!) +gs4V(0 +ys5*R(t)*g58*L(t) temp^IG ones(size(y)).y,yP.R,L]; thgy«arx(temp,[0.1 I 1 1 1 0 0 0 0 0 ] ) ; gyp^predict([G,ones(size(y)),y,yP,R,L],lhgy,4); ggy-thgy(5,l:6); %Xy(t) =■ xO +xl«y(0 +x2*y(t-h) +x5*R(t) +x8*L(t) +xl0*L'(t) +xll'e(t) +xI2*ta(t) +xl3M(t) , (0)' temp-[Xones(sizt(y)),y.yD,R,L,LP,e.ia]; thxy-arx(temp.[0,l I I 1 1 1 I 1 0 0 1 0 0 0 00]); xyp-predict(|X.oncs(sizc(y)),y yD.R,L.LP.e,ta],rhxy.4); xxvMhxy(7.]:8);
*(>B(t) = bO *bl *y(i) +b2'y(t-h) +b3'P(t) +b4'y'(t-h) +b5*R(t) +b6'R(t-h) +b7'e(t) +b8'R'(t-h) +b9*L(l) +blO*L(t-h)+bl2 , L'(<-h)+bl3'ta(t)+bl5M(t)+bl7 , B(t-h) temp=lBones(sizc(y)),y,yD,P,yPD,R,RD,e,RPD.L,LD,LPD,ia,BD]; dib=arx(temp.[0.1 1 I I 11 I I I I t I 1 1 0 0 1 0 1 0 1 0 1 0 1 I 0 1]); Bp=predict([G,ones(size(y)).y.yD.P,yPD,R,RD,e.RPD,L,LD,LPD,la,BD),thb,4); BB-ihb(l3.l:14): % y(l) - A'K^'L^l-a) —>ln(y(t)) - a*ln(K) - (l-a)'ln(L) temp-[log(y)ones(size(y)),log(K.),log(L)]: thly-arx(iemp,[O.I 1 10 00]); lyp -predict([log(y) ones(sizc(y)),los(K),log(L)J,thly,4); lyy-thly(2,l:3); •/•»= real wage raie per unit time
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Differential Models and Neutral Systems
%m(w) = [ I -lyy(2)/(w)]*( I /lyy( I)) %D(t) - a(l-c)*y(t) -kO'K(t) -kl3*K'(l) ^L4«R(t) +L5#L(t) -L6'P(t) +v'y'(t) temp=(Dones(size(y)).y.K.KP,R.L.P.yP]; thD=arx(temp.[0,l I M I 1 1 I 0 0 0 O 0 0OO]); Dp-prediu(ID,oncs(size(y)).y.K.KP,R,l.,P,yP],thD.4); DD-thD(7,l:8); %DP(t) - pO +p]*Pfc(t) +p2*w(t) -p3*n(0 -p4*P(i) +p5'M'(t) +p6*ML(t) temp=[DPones(5ize(w)),Pfc,w,n,P,MP,ML]; thp-arx(tcmp,[0,l 1 I I I 1 I 0 0 0 0 0 0 0]); dlp-predict(|P,ones(size(w)).Pfe,w.n,P.MP,ML],lhp,4); dp-thp(6.l:7); '/«The polynomial coefficients and their standard deviations % for India2.m arc: % C(t) -cO +cl*YT(t) +c2*YT(l-h) T c3*YT(t)' ^c4'YT(t-h) -*c5*R(t) + c6*R(l-h) c7*( 7 8783 +0 104! *y(t) +0 1283.'yD(t-h) - 2.5354.*R(t) +1.0861.»RD(t)-2.4128.*P{t) + l.303l.'RPD(t))- M
Appendix
443 1
citleandia-tNVESTMENT- ); ylttwK'l'); xlabel('YEAR'); subplot(2.1.2).plot( YEAR,G,YEAR,Gp.'+') title(india--GOVERNMENTCONSUMPTION~'); vlabeK'G'): xlabeK'yeaO: pause clg jubptot(2. l,l),plot(YEAR,M,YEAR,Lp.--) utleflndia-MONEY DEMAND-"); ylabelCMI): xlabeK'YEAR'). subptot{2.1,2),p!ot(YEAR^.YEAR.[Xp+lp+Gp+Cpi;+') wleClndia-AGGREGATE DEMAND-1); ylabeK'Z"); xlabeK'YEAR'); pause
subplot (2,l,l).ptot(YEAR,DP.YEAR,dlp>') titleClwlia-CONSUMER PRICES-'); ylibelCDP): xlabelCYEAR); subplot (2,1 ^.plotfYEAR.y.YEAR.yp.V) titleClndia-INCOME-1); ylabel('GDP); xlabelCYEAR); pause subplot (11.1 ),p!o«YEAR,C,YEAR,cyp,V) titleflndia-lNCOME/CONSUMPTlON-); ylabe'Ccy'); xlabelCYEAR'); subplot(2.1,2),plot(YEAR,l,YEAR,Iyp,' ♦') titleClndia-INCOME/IN VESTMENT-1); ylibel(lyp'); JclabelCyear'); pause clg »ubplet(2.1,1 XpW YEAR.G, YEAR^yp.-1) AK'todii-lNCOME^GOVERKMENT-'V. ylabelCjyp'); xlabeK'YEAR"); subplot(2,1.2).plot(YE AR.X,YEAR,xyp.'-') titleOndia-lNCOME/EXPORT-'); ylabelCxyp"); xlabelCYEAR'); pause
subplot (2,1,1 ),plot(YEAR.B, YEAR.Bp.V) titleClndia-BALANCE OF PAYMENT-*); ylabeK'B'); xlabeK'YEAR'); subplot (2, l.2),ploi(YEAR,|piy,YEAR,lyp,,V) dtleCIndia-log(incoine)-'); ylabeK'Iny*): xlabeK'YEAR');
444
Differential Models and Neutral Systems
pause subplot (2,1,1 ),plot( YE AR.D, YEAR.Dp. V ) lilleCIndia-lncrcase in stock-'); ylabeU'D'); xlabel(YF.AR'); % The economic dynamics can be put in matrix form % as follows: x\l) - A-l V
% Where, «/. Ami -[0.0300 -7.3492 7.3492 0 0 0 % 0 -0.0019 0 0 0 0 % 0.0000 0 0.6852 0 0 0 % 0.1343 0 12.5592 0.6844 0 0 % 0 -0.0001 0 0 0 0 % -0.2010 10.0167 52.7368 0 0 0]
% A0-[-0.9448 -1.2805 -1.8774 0.1262 17977 0 % -O.0O07 0.0127 0 0 -0.0050 0 % 0 0 -4.2991 0 0 0000 0
Appendix 0 0 0 -4 2991 0.2026 0 0.0000 -0.0005 0 0 - 1 0846 0 0.4589 7.111$ 86.6667 0 0 0]
% Al =[-00650 2.2146 -1.1848 0 0 0 % 0 0010 -0.0123 0 0 0 0 % 0.0000 -0.3253 16.9422 0 0 0 % -0.5834 -36.3063 1635SS 0 0 0 % -00375 00005 0 0 0 0 •i -1.1138 -6 5845-56.0216 0 0 0 1897'
% Bl =[-1.0152 1.0152 0.3898 1.7977 0 0 0 0 % 0 0 0 0 0 -0.9000 0 0 % 0 0000 0.0001 -0O003 0.0000 0 O.00O0 0 0 % 05183 1.0000 -3 8873 0.3012 0 0 3444 0 0 % 0 0 -0.2031 0 0 -0.0009 -0.0005 0 % 0 0 -404471 10.0167 0 0 0 -1.0] % B2=[l 0152 1.0152 1.0(52 0 1334 0 0 0 0 0 % 0 0 0 -0.9000 0 0 0 0 0 % 0 00001 0 0 0 0 0.000! OOOOI 0 % 0 1 0000 0 0 0 0 1.0000 1.0 0 % 0 0 0 43.0009 -25.0144 0.0013 0 0 1.0 % 0 0 0 -1.0000 0 0 I.OO00 0 0] %q =[TI 0.4078 e lad Ml MI' -278.0597]'; %S-[1.4471 25 1406 15.7077 0.0400 n w 15.7077 13.9320 -2.8120]; % For spectral controllability % Deltaj-eye(size(AI)).*j -Aml.'(j'2.78l4' > <-j))- AO % -AI. , (2.78l4'X-j)) %a=[DeltajB] %forj =04312,4. 10,23, 100,... "/o rank;a)=6 % For nonzeroeigenvalues of A-1 to be controllable via matrix B, % cc-eye(ii7e(Am I )).*j -Am 1 % c=[cc B] "/oforj -0.4312. 4, 10.23, 100.... % rank(c)=6 "/• Since rank(B) is 6. where rank(a) and rank(c) arc "i also 6. the system is function space controllable LI =1;L2'0.9;L5=DD(5) L6-DD(6):h - I %fory(i) = A*K'a*LA(l-a) a- 0.9 % a =alpha aa=0.1 % aa=(l-a)-beta W=sum(w)/24 mv. -(aa/W).A(l.'a)
445
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Differential Models and Neutral Systems
% cobb Douglas states: aa+a-l =>constam return to scale. % aa+a>l -increasing return to scale. % aa+a=! ^decreasing return to scale. % Increasing return to scale if doubling L&K more than doubles y % consiam return to scale if doubling L&K exactly doubles y. % decreasing return to scale if doubling L&K less y.
% % In cobb.m program log(A) is found to be 1.5 to give the best % aproximation for y(t). zO=GG( I H i t 1)-MM( I )4(CC(S) Hl( 11))' CC( 1 )-XX( 1) zl-GC(2^ll(2)-MM(2)*(CC(8)-ll(ll))+CC(2)+XX(?) z2-GG(3)-ll(3)-CC(3)+XX(3)-MM(3> , (CC(8)-ll(ll)) z3-GG(4)-ll(4)-.-CC(4)'rXX(4) 74-GG(5)TII(5)I-CC(5)+XX(5)
z5-GG(6HI(6)+CC(6hXX(6)-MM(4)*(CC(8)+II(1l)) z6«ll(7hCC(7)+(CC(8)+ll(ll))*MM(5) z8=GG(7)-ll(8)-XX(7) z9=II(9) zin-xX(8) zll--II(lO) zl3=(II(llXC(8))'MM(6) zl4--CC(2) zl5=XX(ll> zl6'XX(10) zl7=0 118-XX(9)+(CC(8)+zl 3 )*MM(7) zl9=-CC(3) z20
Appendix zsO=GG(l)*XX(l)*CC(i)-ll(l) aO = DD(l>'(h'(l-yy(2))) nl --DD(l)\>.y3/(i.l-yy2)*h) a2 - -al"h a." = aO*h*vv4 a4 = DDO j\vy5/l(l-yy2) -DD(4)) a.< - DD(l)'w60-yy2)-OD(5) n6 - aO'h'yy? aS = DD(6) °'oA-1 starts here, am I I = a-11 ami 1 = LI*z4/sl am22 -L2'MM(6) amlj =L I -zlO/sl amU'LI'zlO/sl Lm03 - DD(l)'yvrmw/(l-yy2) LinOl " DD(3)*DD(lHI-y72)-mWDD(iryy7/(l-yy:) a? =■ Dl)lli*yy4/
447
Differential Models and Neutral Systems
448 al--DD(l)*yy3/((l-yy2)'h) a2=--al*h !l = -al 12 --mWDD(l)'yy3/(l-yy2) 14 =mWDD( I Tyy5/( 1 -yy2 *DD(4)) b2 =BB(3) b6 -BB(7) bit) =BB(11) bl7 -BB(I3) al 11 =Ll'(UO(3)+ll(3)^XX(3)*MM(3)'(ll(ll)+CC<8)))/sl all2=a!3 a!13-al5 a!21-a22 al22-a24 a 131 -mw'DCX 1 )'XX(3y( I -<XX(2hggy(2)+ccy(2») al32=mw«DD(i)'yy5.'(l-yy2) -DD(4) al33=a5-IKh aMl -DD(])*yy3/(l-yy2) a 142 -DD( I )*yyS/( 1 -yy2) + DD(4) a 143 «DD( 1 )'yy6/( I -yy2) + DD<5) al51 —dp(7),sum(PV24 ^includes P(t) a 152 --NfM(5),dp(7),suin(Py24 %inchides P(t) a 161 =BB(3) al62-BB<7) al«-BB(1l) al66-BB(l4) %BI starts here! ssl -Ll<'sl%ssl"=bl sslzlj=ssl*zl3 sslzlS'ssl'zlS sslzl6-ssl'zl6 ss I z 17 - 0 islzIS -ssl*zl8 blzl4"-BB(2) , zU; bl -BB(l) blzl5=BB(2)*zl5 blzl6-BB(2)*zl6 blzl7-BB(2)*zl7 blzl3-BB(2) , zl3 mwzsl4 »rnw'yy9 mwzsl5 =mw,yylO rrmzsl6 "n™*j)l 1 m\vzsl7 »0 mwzsl3 -mw*yy8 plpf p4 p5 b7 b8 bl5
-dp(2)"sum(PfeV24 -dp(5) -dp(6) -BB(8) =BB(9) =0
*«B2 starts here! ssl -Ll/sl ssltl3c7-55l*(lI(ll>-CC(8))
%dp(2),Pfl[t)
Appendix 12 p6 p3 P2
=mWDD(I)'>y3.'(l-yy2) - dp(7) = dp(4) - dp(3)
al7 =Ll*zl9/sl a45 =a8 cO-CC(l) 10=11(1) xO-XX(l) pO«dp(l) MO-MM(I) ylO'ccy(l) mO =-MM(l) gO=GG(l) fD-BB(l)-XX(l) g2 --L2*MM(1)
449
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Differential Models and Neutral Systems
AUSTRALIA2.M % 8:31PM 9/15/97 */• australia's data is in Billions of australian dollars % adopted from International Financial Statistic Yearbook(IFSY) 1996 except where noted. % Use ndu.m to obtain the matrix entries YEAR =[1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991]'; EX -[3.26 3.59 3.68 4.3 4.95 5.35 6.22 7.55 8.93 10.57 12.46 13.95 14.86 19.75 22.55 22.55 25 51 26.47 31.44 38.75 40.92 47.53 53.46 58.30 63.8S 68.41 J"; IMP -[-3.45 -3.89 -4.3 -4.45 -4.91 -5.33 -5.07 -6.31 -9.94 -10.02 -12.27 -14.79 -16.53 -19.25 -23.36 -26.73 30.19 -28 97 -35.66 -44.25 47.06 -49.86 -56.15 -66.78 -66.82 -66.01]'; X -EX-IMP; TX -[5.178 5.299 5.730 6.442 7.323 8.241 9.131 9.740 12.296 15.749 18.853 22.076 24.219 26.336 30.584 36.244 41.814 45.936 50.450 59.298 66.793 75.486 84.235 91.574 100.426 103.135]'; K =[6.03 6 38 7.1 7.85 8.63 9.84 10.36 11.95 13.88 16.71 19.88 21.74 24.24 27.32 31.97 38.67 42.06 41.41 47.30 56.53 61.31 68.05 77.66 90.38 85.68 78 111'; KP -[0.35 0 72 0.75 0.78 1.21 0.52 1.59 1.93 2.83 3.17 1.86 2.5 3.08 4.65 6.7 3.39 -0.65 5.89 9.23 4.78 6.74 9.61 12.72-4.70-7.57 1.18]'; D - [ 5 .35 .45 .29 .51 .33 -.16 .29 2.32 -.16 .42 .62 .29 2.15 .79 1.5 -.51 -1.42 2.62 .58 -1.33 .64 2.95 5.1 .17 -2.84]': DD =[.48 .5 .35 .45 .29 .51 .33 -.16 .29 2.32 -.16 .42 .62 .29 2.15 .79 1.5 -.51 -1.42 2.62 .58 -1.33 .64 2.95 5.1 .17]'; I -[6.317 6.627 7.872 8.336 7.872 7.306 6.949 12.985 15.314 17.223 20.398 20.386 24.672 25.931 31 882 38.899 35.078 45.677 52.034 60.628 62.879 71.086 89.261 94.083 80.347 96.021]'; e =[1.114 1.121 1.11 1.118 1.115 1.097 1.1743 1.2335 1.0838 1.0738 1.0738.9351 .9396.8831 .8392.9257 .969 .8889 .8616 .8445 .6199 .5435 .5093 .6357 .6032 .5436)'; P ={15 15.5 15.9 16.4 17 18.1 19.1 20.9 24.1 27.7 31.5 35.4 38.2 41.6 45.8 50 50.3 55.9 61.6 64 68.3 74.5 80.8 86.7 93.2 100]'; Pfe =P.V»e; ta =[.138 .169 194 .220 .270 .278 .374 .314 .333 .327 .320 .315 .516 .632 .815 .997 1.097 1.306 30.109 32 735 37.663 42.484 46.364 50.651 55.328 57 896]'; F - [62 67 94 80 80 56 66-11 -32 -82 -72 -21 24 104 275 298 300 412 451 686 817 1196 1706 2124 1939 2062]'; B -[8 19 27-8-15-22 138-9 81 -193 99 73 -24-24 60 230 449 293 612 1195 3107 1725 2882 4843 4412 3192]'; B D =(15 8 19 27-8-15-22 138-9 81 -193 99 73-24-24 60 230 449 293 612 1195 3107 1725 2882 4843 4412]'; T = [X-B-F]; M -[4.203 4.495 4.752 5.205 5 446 5.749 6.899 8.050 7.990 9.807 10.681 11.390 12.709 14.661 17.220 18.063 18.032 20.796 22.492 23.298 25.947 31.218 40.470 43.514 46.699 49 983]'; MD "[3.907 4 203 4.495 4.752 5 205 5.446 5.749 6.899 8.050 7.990 9.807 10.681 11.390 12.709 14.661 17.220 18.063 18.032 20.796 22.492 23.298 25.947 31.218 40.470 43.514 46.699]'; MP -[.257 .453 .241 .303 1.150 1.151 -0.060 1.817 .874 .709 1.319 1.952 2.559.843 -0.031 2.764 1.696 0.806 2.649 5.271 9.252 3 044 3.185 3.284 10.309 10.735]'; MPD -[.296.257.453.241 .303 1.150 I.IS1 -0.060 1.817.874.709 1.319 1.952 2.559.843-0.031 2.764 1.696 0.806 2.649 5.271 9.252 3.044 3.185 3.284 10.309]'; w -[10.4 11.0 11.7 12.7 13.8 15.5 16.8 18.9 23.2 27.4 31.3 34.7 37.7 40.9 45.9 51.0 51.1 S7.1 61.1 66.9 70.3 75.8 79.7 85.2 92.1 100]'; y -[23.1 24.77 27.13 30.15 33.29 37.42 41.82 49.32 59.17 70.54 82.88 91.99 101.19 116.63 132.4 150.26 166.22 181.8 207.88230250.35 281.83 318.27 356.45 377.30 379.95]'; yD -[21.21 23.1 24.77 27.13 30.15 33.29 37.42 41.82 49.32 59.17 70.54 82.88 91.99 101.19 116.63 132.4 150.26 166.22 181.8 207.88 230 250.35 281.83 318.27 356.45 377.30 ]'; GDP =y. yPD -[1.89 1.67 2.36 3.02 3.14 4.13 4.4 7.5 9.85 11.37 12.34 9.11 9.2 15.44 15.77 17.86 15.96 15 58 26.08 22.12 20.35 31.48 36.44 38.18 20.85 2.65]'; yP -[1.67 2.36 3.02 3.144.134.47.59.85 11.37 12.34 9.11 9.2 15.44 15.77 17.86 1596 15.58 26.08 22.12 20.35 31.48 36.44 38.18 20.85 2.65 14.82]';
Appendix G =[2.96 3 44 3 82 4 11 4.57 5.25 5.88 7 13 9.22 12.11 14.52 16.25 18.29 20.0123.26 26.79 30.48 34.08 38.08 42.55 47.4 50.39 54.75 59.13 65.1 69.65]', C =[13 38 15 16.34 17 98 19 83 22 05 24 41 28.23 33.9 41.22 48 35 53.98 60.52 68 12 77 86 88 7 100.54 111.39 12197 136 99 150 165.59 184.35 206.91225.06 235.57]'; L =[48.3 51 53.9 57.5 60.1 62.5 63.2 69.7 71.9 66.4 69 8 68.7 69.7 73.3 73.7 76 76 75 6 73.3 78.4 84.7 84.5 88.5 93 4 97.6 100]'; LD -[47 5 48 3 51 53.9 57.5 60.1 62.5 63.2 69.7 71.9 66.4 69.8 68.7 69.7 73.3 73.7 76 76 75.6 73 3 78.4 84.7 84.5 88.5 93 4 97.6]'; LP -[0.8 2 7 2.9 3.6 2.6 2.4 0.7 6.5 2 2 -5.5 3 4-1.1 1.0 3.6 0.4 2.3 0 -0.4 -2.3 5.1 6.3 -0.2 4.0 4.9 4 2 2.4]'; LPD -[2.7 2.9 3.6 2.6 2.4 0.7 6.5 2.2 -5.5 3.4 -1.1 1.0 3.6 0.4 2.3 0 -0.4 -2.3 5.1 6.3 -0.2 4.0 4.9 4.2 2.4 -3.6]'; ML =[-4.1116 -4 4051 -4.6608 -5.1040 -5.3316 -5.6361 -6.7937 -7.9160 -7.8256 -9.6332-10.4888 -11.1786 12 5007 -14 4209 -16.9375 -17.7429 -17.6974 -20.4594 -22.1330 -22.9039 -25.5338 -30.7764 -40.0063 42 9821 -46 1287-49.399]'. n -y.\L; R -[5 25 5 25 5 21 5 71 6 72 6 87 5.99 7.11 9.07 9.75 10.18 10.28 9.06 9.76 11.65 13.96 15.35 14 33 13 83 14.10 13 56 13 47 12.32 13.62 13.18 10.68]'; RP =[0 -0.04 0.50 1.010.15-0.88 1.12 1.96 0.68 0.43 0.10 -1.22 0.70 1.89 2.31 139-1.02-0 50 0.27-0.54 0.09-1.15 1.30 -0.44 -2.50 2.53]"; RPD =[0.530 -0.04 0.50 1.010.15-0.88 1 12 I 960.68 0.43 0 10-1.22 0.70 1.892 31 139-1.02-0 50 0.270.54-0.09-1.15 1.30-0.44-2.50]'; RD -[4 72 5.25 5.25 5 21 5.71 6.72 6.87 5.99 7.11 9.07 9.75 10 18 10.28 9.06 9.76 11.65 13.96 15.35 14.33 13.83 14.10 1356 13.47 12.32 13.62 13.18]'; YT -v-TX; YTD =[17.92 19.47 21.40 23.71 25.97 29.18 32.69 39.58 46 87 54 79 64.03 69.91 76.97 90 29 101.82 114.02 124.41 135.86 157.43 170.70 183.56 206.34 234 034 264.88 276.87 276.82]'; YTP =[1.549 1929 2.308 2.259 3.212 3.510 6.891 7.294 7.917 9.236 5.887 7.057 13.323 11.522 12.200 10.390 11.458 21 566 13.272 12.855 22.787 27.691 30.841 11.998 -0.059 13.74]'; YTPD =[1.544 1.549 1.929 2.308 2.259 3.212 3.510 6.891 7.294 7.917 9.236 5 887 7 057 13.323 11.522 12.200 10.390 11.458 21.566 13.272 12.855 22.787 27.691 30 841 11998-0 059]'; price-[15 15.5 15.9 16.4 17 18.1 19.1 20.924.1 27.7 31 5 35.4 38.241.645.8 50 50.3 55.9 61.6 64 68.3 74.5 80.8 86.7 93.2 100 107.2]', for i= 1:26 ppO)"-price(i-l)-price(i); end pie-lpp]'; DP= pie.\P; logy=log(y); unL= 100.'oncs(size(L))-L; '/. TX =Revenue,(billions auslralian dollars) % n -labor productivity(A/l = GNP/employment) % 'From Yearbook of labor statistics(intemational labor office) % w - wages, weekly earnings % K -capital stock(Gross fixed capital formation) V. I -investment(Gross capital formation), (UN National acounts statistics). '/• D ^increase in stock % The UN(National accounts statistics) defines Gross capital % formation as the sum of the increase in stocks and gross fixed % capital formation
% % y =GDP(Gross domestic product) %yD=y(t-l) %yP-y'(t) %yPD-y'(t-l) % X =Net export(export-impont of goods and services) % d -preferential trade agreement transportation % Pfe -import price level in foreign cunency (P*c*e) % a -tariffs (indirect taxes net subsides:
451
452
Differential Models and Neutral Systems
Ve by, U N National acounts statistics) % E -cumulative balance o f payment %? - N e t private outflow ofcapiul(private unrequited transfers),(IFSY, 1992). % C =private consumption V. M.M I -money supply '/• Md -money demand(liquidity); Where, ML- md-M % e » exchange rate(market rate, ac) V.P -consumer prices. (IFSY, 1992). •/• R interest rate(Goveniment bond yeild, long term),(IFSY. 1992) V. RD-R(M) '/. RP -R(t) •/.RPD-R'ft-l) '/• pie-inflation % L -labor(industriil production) '/. B ^Balance of payment(deposit money banks), (IFSY, 1992). % T -Net government transfer of capital to foreigers and firms % G -Government consumption %DP-D*log(P)-p(t)7j)(0
'/. •/•Bert-IX.G.1.C.GDP,PGDP,R,M,QM.YEAR| |years.v]-size(YEAR): Xl-[4.88;X(l:years-l)l; Gl=[2.69;G(l:yean-l)], U=[5.793;l(l:years-15]; CI=[12.89;C(l:years-t)]; Mil -[4.203;M(l:years-l)]; RI-[5 2IR(l:yean-l)]; GDPI-(2l.21;GDP(l:years-l)J. DGDP-GDP-GDP1; Z-X+G+l+C; Z1=XI+GMI+C, %temp=[YEAR XI GI II Cl ZI yl yD Ml 1 RD! yPDI RP RPD Tl T P P-PF PI P-PF) G L LD LP LPD L! LDI LPI LPD1 ttl e e l d d l ] , •/.save Ben I asc temp -ascii -double -tabs •/.clear Bert I temp; •/.ML - L-M - ml-mO - ml«y(t) - m2'y(t-h) - m3«R(i) - m4«R(t-h) - m6,P(t) - m7'R'(t-h) •/. L(t) =m0 +rnl,y(t>+ m2«y(t-h) +m3*R(t) +m4'R(t-h) +m5«R'(t-h) +m6*p(0; temp=[M ones(size
Appendix
453 ,
% X(t) =xO + xl*y(t) + x2*y(t-h) +x3*y (0 +x4*y'(t-h) +x5»R(t)+ x8»L(t) +xlO»L'(t-h) -xl2'P(t) +xl6'ta(t) +xl5*e(i)tx!7'd(t) temp«[XoiKS<size(y)).y.yD,yP,yPD,R,L.LPD.P,ta,e]; lhx=arx(temp.(0,l 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0)); Xp-predict([X.ones(sitt(y)),y,yD,yP,yPD,R,L,LPD.P.ta.e].lhx,4); XX-thx(ll.l:12); %G(t) =g0 + sl*y(0 + g2*y(t-h) ♦g3*/(0 +84*y'(t-h) +g5'R(0 +g8*L(t); temp=(G ones(size(y)).y,yD,yP1yPD,RXl; thg=arx(temp,[0,l 1 1 1 1 1 1 0 0 1 0 1 0 0]); Gp=predict([G.ones(size(y)).y,yD,yP,yPD,R,L],thg,4); GG=thg(6.!:7); %y(I) - zsO +zsl«y(t) +ra2»y(t-h) +zs4*y'(t) +zs5'R(t) +zs8*L(t) +zslO*L'(t) +zsl3 , MI(0-«14 , TX(t) +zsl5'e(t) +=16*13(0 -zsl7M(0'(0) +l/h,K'(t); h-l temp-[yon«(siz*(y)),y,yD,yP,R.L,LP.M,TX,e,ta,KPl; thy-arx(temp,[0,l 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0|); yp-predict([y,ones(size(y)),y,yD,yP,R,L.LP,M,TX,e,ta,KP],thy,4); yy=thy( 11.1:12); %cv(t) = y ID -yl 1 'y(l) + yl2*y'0) + y!3'R(t) + yl5*ML(t) +yl8'YT(t) tcm'p-[Cones(size(y)),y,yP,R,ML,YT]; thcy=arx(temp,[0.1 1 I 1 1 1 00 0 0 0 0]); cyp-predict<[C,ones(size(y)).y,yP,R,ML,YT]>thcy,4); ccy=lhcy(5,1.6); %ly(t) - 10 + l;VKP(0; dK(t)/dt = D(t-h) temp-[I ones(size(K)),K); lhly-arx(temp.[0,l 10 0]); lyp=predict((!,ones<size(K)),KJ,thly,4); lly-thly(l,l;2); %Gy(i) = gsO +gs! 'yd) +gs4'y'(l) +ys5'R(l) +«s8,L(t) temp-[G ones(size(y)),y,yP,R,L]; thgy-arx(temp,(0,l 1 1 1 1 OOOOO]); gyp-predict([G,ones(size(y)),y,yP,R,L],thgy,4); ggy-ihgy(5,l 6); %Xy(t) = xO +xl'y(t) -x2*y(t-h) +x5*R(t) +x8«L(t) +xlO"L'(t) >xl l'e(t) +xl2'ta(t) -xl3'
454
Differential Models and Neutral Systems
%m(w) - [l-lyy(2)/(w)]-(1/lyy(l)) %D(l) = a(l-c)*y(t) -k0*K(t) +kl3'K'(t) +L4'R(t) +L5'L(t) +L6'P(t) +v*y(0 temp-[Dones(size(.v)),y.K,K.P.R,L,P,yP]; thD=arx(lemp.[0.l I 11 1 1 1 1 0 0 0 0 0 0 0 0 1 ) ; Dp-predict([D.ones(size
% subplot (2.1,1 ),plot(YEAR,DP, YEAR,dlp,'+) title('Australia--CONSUMER PRICES-'); ylabel('DP); xlabelCYEAR'); subplot (2.1,2),plot(YEAR,y.YEAR,yp,'+') title('Australia-tNCOME--'); ylabelCGOP-): xlabelCYEAR'); pause subplot (2,1,1 ),plot( YEAR,C,YEAR,cyp,,+ ) title('Australia-INCOME/CONSUMPT10N-'); ylabelCcy"); xlabel('YEAR'); subplot(2,1,2),plot(YEAR,l, YEAR,Iyp,'+')
Appendix tftlefAustralia-INCOMEm^VESTMENT-"); ylabelClyp-); xbbelfyear'); pause elf fUbpkK(2.1,Upto((YEAR,0,YEAJUyp.v') tideCAMnlia-INCOME/GOVERNMENT-'); yhbeJCgyp-);
xlabelTYEAR*); subplot(2.1 J)4>lot(YEAIU(, YEAR,xyp>') thiefAustralia-INCOME/EXPORT--1); ylabelfxyp-); XlabekTYEAR'); pause
subplot (2.l,l),plot(YEAR,B,YEARfBp.,+') titlefAuttralia-BALANCE OF PAYMENT-1); ylabeaTB"); xlabelfYEAR1); subplot (2,1,2),otot(YEAR.Iogy.YEAR,Iyp,1-1) thiefAustraHa-logtincoroeK); ylabelflnyO; xlabelfYEAR1); pause subplot (2,l,l),ptot(YEARJ>,YEAR,Dp,'+1) titlefAustralia-Increase in stock-*); ytabetCDO; xlabelfYEAR1); subplot (2,1 J)jAx(YEAR4)ie,YEAR^:+',YEAR,unL,'o') titlefAustralian growth rare-'}, ytabelfInnalioa(-) ft GDP(+) ft unemptoymene'o)'); xlabelfYEAR') % The economic dynamics can be put in matrix form % asfellows:x"(t) - A-l'xXt-h)" AO'xltKAIVG-hHB'uO) Sx-[yRLKPET; SA-l-Iamll-aml4aml3 000;Oam22 0000;Lm03 0-Lm010 00;a3 0a6-amlOO;0-M7*p6'>P(0000 0;b4Mbl2000] HAO -Ca01al2al4al6alS0;a2la2300a25 0;00a330a3S0;000a44a45O;aSI*p(t)a52*p(t)0O aS5*p(t)0;a6la62a63 000] SAI -{allal3al5 000;al2lal22 0 00 0;al31al32al33 000;al41al42al43 0 0 0;al31*p(t) al $2«p(t)0 0 0 0; al61 al62 al63 0 0 al66] HBI -(-ssl ssl sslzl5 sslzlS sslzl7 0 0 0; 0 0 0 0 0 -L2 0 0; -mw»l4 mwmwzslS mwzsl6 0mwzsl3 0 0;zsl4lzslSzsl6Ozsl3O0;OOp1*pf(t)O0-p6pSO;00b7b8bl300-l] KB2 -(ssl ssl ssl ssl
455
456
Differential Models and Neutral Systems
%S -(CO 10 xO MO n w xO ylO pO]'; %S-(0.0057 -1.8803 0.0768 0.0028 nw 0.0768 -0.0800 -173.0705); %B-IB1B2| % AB=AO»B %A2B-(A0 , A0)*B % ASB-fAO'ACAO^B % A4B-
% % In cobb.m program log(A) is found to be 1.5 to give the best •/■ aproximation for y(t). z0-GG(0+[l(lHMM(l)'(CC(8>Hi(ll))+CC(l>tXX(l) z I -GG(2)+Il(2)+MM(2)'(CC(8)+ll( 11 )>K:C(2)+XX(2) z2-GG(3)-ll(3)+CC(3)+XX(3)4-MM(3),(CC(8)+IKH)) z3=GG<4)-II(4)+CC(4)+XX(4)
Appendix z4=GG(5Ml(5)+CC(5)+XX(5) z3=GG(6hll(6)-CC(6)+XX(6)+MM{4)*(CC(8)+[l(ll)) Z6=U(7)+CC(7)-(CC(8)+I1( 11 ))'MM(5) z8-GG(7HI(8)-XX(7) z9=U(9) zI0-XX(8) zll--II(lO) zl3=
457
Differential Models and Neutral Systems
458 am22 =L2'MM(6) amI3'Ll*zlO/sI aral4-Ll*zlO/sl Lm03 - DD(l)*yy'l , mw/(l-yy2) LmOl » DDQ)*DD(iy(l-yy2)-mWDD(l)*yy7/(l-yy2) a3 «DD(l)'yy4/(l-yy2) a6 -DD(l)'yy7/(l-yy2) ami =DCK3)'DD{iy(l-yy2) M7p6P =■ MM(7)»dp(6),!um(P).'26 b4 - -BB(5) b8 = -BB(9) bl2--BB(12) %AO starts here! aOI - L l ' f z l - i y s l all
=U*T2/S\
al2=Ll*z5/sl a 13 -L1 *z6/s 1 al4=LI'z8/sl al5=Ll'z9/sl al6-LI'zlUsl al7-LI«zl9/sl al8-LI»zl8/sl a2l -L2*MM(2) a23 =L2'MM(4) a25 -L2«MM(7) a33 =aO; a35 -mw'aS a44 =aO a45=a8 a5l»-dp(7),MM(2),sun>(P)/26 % includes P(t) a52—dp(7)*MM(4)*sum(Py26 % includes P(l) a55 --(MM(7)+dp(5)),sum(Py26 % includes P(t) a61 -BB(2) a62 -BB(6) a63 =BB(10) %A1 starts here! all -LI*z2/sl a)3 =Ll*z6/sl al5 =Ll«z9/sl a22 -L2'MM(3) a24 -L2*MM(5) al--DDd)*yy3/((l-yy2)'h) a2--al'h ll=-al 12 -mw , DD(i)*yy3'(i-yy2) 14 -mw'DD(l)'yy5/(l-yy2+DD(4)) b2 =BB(3) b6 -BB(7) blO - B B ( l l ) bl7 -BB(I3) al 11 =Lr(GG{3)+lK3)+XX(3)*MM(3)*{ll(l l)+CC(8))ys) all2-al3 all3=a!5 a!2!-a22 a!22-a24
459
Appendix Dl31-mw«DD(l)*XX(3)/(l-(XX(2hgS>(2) + ccy(2))) a 132«mw« DD( 1 )'yySI{ I -yy2) -DLX4) al33=a5-ll*h aI41 =DD(l)»yy3/(l-yy2) aI42 =DD(l)*yy5/(l-yy2) + DD(4) a 143 -DD( 1 )*yy6/( 1 -yy2) + DD(5) a! 51 =-dp(7)*sum(PV26 V.includes P(t) a 152 --MM(5)*dp(7),suni(P)/26 Voincludes P(t) a!6l -BB(3) al62-BB(7) al63 -BB(1I) al66-BB(14) °-'»BI starts here! ssl =LI.'sl%ssl=bl sslzl3=ssl*z!3
ssU15-ssl*zl5 sstzl6-ssl'zl6 sslzl7=0 sslzl8 =ssl*zl8 blzl4--BB(2)-zl4; bl =BB(I) bU15=BB(2)'z!5 blzl6=BB(2)'zl6 blz!7-BB{2)'zl7 blzl3=BB(2)*zl3 mwzsl4 =mw'yy9 mwzsl5-mw*yyl0 mwzsl6=tnw*yyll m*zsl7 =0 mwzsU =mw'yy8 plpf p4 p5 b7 b8 bl5
-dp(2)*sum(Pfey26 =dp(5) -dp(6) =BB(8) =BB(9) -0
%B2 starts here! ssl =Ll/sl sslil3c7=ssl'(l!(ll)+CC(8)) 12 =mw-DD(l) , yy3/(l-yy2) p6 = dp(7) p3 -dp(4) p2 - dp(3) al7=Ll*zl9/sl a45 -a8 cO = CC(l) 10-11(1) xO = XX(l) pO-dp(l) MO-MM(l) ylO = ccy(l) mO -MM(1)
%dp(2)«Pf{t)
460 gO=GG(l) fl)=BB(l)-XX(l) g2 =-L2*MM(l)
Differential Models and Neutral Systems
Appendix
461
The polynomial coefficients and their standard deviations for (he Australian regression are: % C(t) =c0 +cl'YT(t) +c2'YT(t-h) + c3*YT(t)' ^c4'YT(l-h) +c5'R(t) t c6*R(t-h) ■Ktf'f.-O.sm +0.3657 *y(t) -0.54 l7.«yD(t-h) +0.1068.*R(t) -0.1963.*RD(t) -0.1223.«P(t) +O.0O9I «RPD(0) - M(t)); cO =-0.8173, cl-0.3657, c2 =0 5417. c3=0.1068, C4--0.I963, c5-O.0091, c6=-0.l223, c7-0.2464 0.8819 0.0370 0 0307 0.0297 0.0506 0.1035 0.1388 0.1107 % l(t) »i0 +il*y(t) + i2'y(t-h) -i3V(0 + M*y'(l-h) +i5*R(t) +i6'R(t-h) +i8*L(t) +i9,L(t-h)-il l«K(t) -il3*ML(t) iO -8.5876, il "-0.0906. 12 =0 1185. i3-0.3873, i4=0.0535. i5 =-0.5383, i6 =0.1649 9.4900 0.2708 0.1770 0.1215 0.3493 0.5491 0 7597 i8=-0.2849, i9-0 1227, i l l - 0.6413, i 13 =-0.8098 0.2332 0.2898 0.4645 0.6609 % X(t) =x0 + xl»y(t) + x2«y(t-h) +x3*y(t) -x4'y,(t-h) +x5'R(t)- x8'L(t) +xl0*L'(t-h) +xl2*P(t) +x!6,ta(t) +xl5*e(t)+xl7'd(t) xO- 8.5437. x I =0.2148, x2-0 0946, X3--0.0458, x4«0 2038, x5= 0.2490, x8 =0.0152, 7.8870 0 0937 0.0844 0 0738 0.2019 0.3067 0.0853 xlO-0.3100. xl2=-00145. x 16-0.1367. X15--10.7265 0.1273 0.2114 0 0649 5.6659 '/. G(t) =g0 + gl»y(t) + g2»y(t-h) fg3".V(t) +g4,y'(t-h) +g5*R(t) +g8»L(t); gO = 1 3666, g I - -0.0345, g2=0 2235, g3 -0.1253, g4-O2024, g5=0.5116, g8--0.0971 1.7271 00313 00301 0.0235 0.0725 0.0670 0.0301 % Ut) - m O -ml «y(t)*- m2*y(t-h) +m3 , R(t) +m4«R(t-h) ♦tr^R'tr-h) +m6*p(t); raO-4.4513, ml-0.0976, m2=-0.0540. m3--0.8126. m4=-0.5765,m5-0.4l26,m6H)4351 10034 0 0259 0.0357 0.2881 0.2511 0.3807 0.1276 %D(t) - a( I -c)*y(t) -kO'K(t) +k 13,K'(t) +L4»R(t) +L54L(t) +L6*P(t) H-v'yW a- -3.0921. ac = -0.1346, kO - 0.5739, kl3 = 0.3398. L4 = -0 1694, L5 = 0.0710, L6= 0.0412, v-0.1976 2.9120 0.0502 0.1269 0.1455 0.1661 0.0524 0.1369 0.0821 %B(t) - bO +bl'y(t) +b2«y(t-h) +b3'P(t) +b4«y'(t-h) +b5'R(t) +b6«R(t-h) +b7*e(t) +b8*R'(t-h) +b9*L(t) +blO*L(t-h) +bl2,L'(t-h) +bl3*ta(t) +bl5M(t) +bl7«B(t-h) bO =30.0, bl = 32.342, b2--0.4733, b3 - -71.7905, b4=30.7391, b5=-289.5411, 2.242 17.7807 25.9831 76.5825 40.0749 145.9437 06-29.8877. b7 -282.7, b8-229 6072, b9=49.4826, blO --31.4879, bl2=26.909. 101.0495 1.6207 1909514 41.8284 48.8974 36.0804 bl3-7.0718.bl7--0.4923 13.1538 0.2219
%% The economic dynamics can be put in matrix form % as follows: x'(t) - A-l'x'(t-h) - AO'x(t><-AI»x(t-h>+B»u(t)
462
Differential Models and Neutral Systems
x-[y R L K P E]'; Where, Ami = l.Oe+003 ♦ 0.0000 0.0000 0.0000 0 0 0 0 0.0000 0 0 0 0 0.0000 0 0.0000 0 0 0 0.0000 0 0.0000 O.OOOO 0 0 0 -0.0016 0 0 0 0 0.2113 4.0439 0.9254 0 0 0
A0 = ].0e+O03 • -0.0011 0.0000 0.0001 -0.0001 0.0000 O.OOOO 0.0000 0 0 0.0000 0 0 0 -0.0001 0 0.0000 0 0 0 0 -0.0001 0.0000 0 -0.O005 0.0061 0 0 3.9122 0 0.1753 -3.3442 1.7496 0 0 0
l.Oe+003 * 0.0000 0.0000 0.0000 O.OOOO 0.8359 -0.6571
0.0006 0.0000 0.0000 0.0000 -0.0016 30650
0
-0.0001
0
0 0 0
0.0000 0.0000
0 -1.8690
0
0 0 0
0 0
0
0 0
0 0
0 0 0.0029
Bl ■= 1.0e+004' Columns 1 through 7 -0.0001 0 0.0000 0.0000 0 0 Column 8 0 0 0
0.0001 0.0000 0.0000 0 0 0 0 0 0 0 -0.0001 0 0.0000 O.OOOO 0.0000 0 0.0000 0 0.0001 0.0000 0.0000 0 0.0000 0 0 O.OOIl 0 0 0.0019 -0.0013 0 4.5766 0.4044 0 0 0
463
Appendix 0 0 -0.0001 B2Columns 1 through 7 1.0311 1.0311 1.0311 -03378 0 0 0 -0.9000 0 0 0.0012 0 0 0 0 1.0000 0 0 0 0 0 0 18.6146-254.0384 0 0 0 -1.0000 0
0 0 0 0 0 0.0012 0 1.0000 80.6258 0 1.0000
0
0
Columns 8 through 9 0 0 0.0012 1.0000
0 0
0 0
1.0000 0
0 0
q =[T1 0.01374 e ta d Ml Ml' -71188.348]'; S =[0.0057-1.8803 0.0768 0 0028 n w 0.0768 -O.0800 -173 0705]'; ranB » 6 ranG - 5 ranka = 6
nnikc"6
464
Differential Models and Neutral Systems
•/iNdu.m Ami -[ami I -aml4 am 13 0 0 0; 0 am22 0 0 0 0; Lm03 0 -LmOl 0 0 0; a3 0 a6 -ami 0 0; 0 -M7p6P 0 0 0 0; b4 bSbl2 0 0 01 AO -[aOl a l 2 a l 4 a l 6 a ! 8 D ; a 2 l a23 0Oa25O;O0 a33 0a35 0; 0 0 0 a 4 4 a 4 3 0; a51 aS2 0 0&S5 0, a61 562 a63 0 0 0) Al -[all a ! 3 a t 5 0 0 0 ; a l 2 1 a l 2 2 0 0 0 0; a 131 al32al33 0 0 0 ; al4l a!42 al43 0 0 0 ; al5l al52OOO0; al6l al62al63 0 0a!66] B1 =[-tsl s$l sslzlSsslzl* J S U 1 7 D 0 0 ; 0 D 0 0 0 - L 2 0 0; -mwzsl4 mwmwzsIS mwzs!6 0 mwzjl3 0 0;zsl4 I Z s l 5 z s l 6 0 n l 3 0 0 ; 0 0 p l p f 0 0 - p 6 p 5 0 ; 0 0 b 7 b 8 b l 5 0 0 - l ] B2 - | n l sil isl u l i l 3 c 7 0 0 0 0 0; 0 0 0 -L2 0 0 0 0 0;0 mw 0 0 0 0 mw mwO; 0 1 0 0 0 0 1 I 0; 0 0 0 -p6 -p3 p2 0 0 1 ; 0 0 0 - 1 0 0 1 0 0 ] q =[gOR>l S =[cOIOxOmOxOylOpO] B'(B1 B2); AB-AO'B; A2B=(A0*A0)*B; ASB-KAO'AO'AOJ'B, A4B-
Appendix
465 JAPAN2.M
* » % %
This program Japan's data adopted from Use ndu.m to
was written for chapter 6 at 10:34AM 4/11/96 is in Billions of yens International Financial Statistic Yearbook 1994 obtain the matrix entries
YEAR -11966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 19B6 1987 1988 1989 1990 1991 1992J'; EX -[4031 4311 5348 6558 7926 9452 9779 11291 18258 18982 22582 24308 22729 25627 32887 37977 39391 39275 45066 46307 38090 36210 37483 42352 45920 4681 0 47409)'; IMP -[-3434 -4211 -47S7 -S567 -6985 -7254 -7645 -11261 -19257 -18919 -21247 -21267 -19174 -27629 -35036 -35927 -37341 -34258 -36866 -35532 -24791 -25195 -29065 -36768 -42872 -38529 -36184)'; X -EX-IMP; Tx -(5650 6522 7358 B636 10317 12335 15166 17250 19855 24845 32752 35571 39 180 45686 49939 58243 66208 72835 79455 83505 90707 98764 103860 113723 12277 4 132240 147021 155023 157915)'; TX -[7358 B636 10317 12335 15166 17250 19855 24845 32752 35571 39180 456B6 49939 58243 66208 72835 79455 83505 90707 98764 103860 113723 122774 132240 1 47021 155023 157915]'; tkk -[NaN NaN NaN 21742 26043 27637 31524 41339 46960 47972 51499 56177 62 384 70248 75420 78399 79173 78881 83251 88040 91310 99160 111074 122766 13673 3 143429 142999]«; %i -(NaN NaN NaN 22986 28852 32823 43236 50482 48520 52592 57388 63421 720 65 77011 79752 80407 79067 84262 90198 92953 99850 113705 125855 139054 14670 0 144593]'; d -[)'; n -(124 138 155 91 100 107 116 125 124 127 134 1977-84 2.7 -1.2 5.0 5.8 4.8 1.7 -0.2 -6.7)'; P -[27.8 28.9 30.5 32.1 34.6 36.8 38.5 43.0 53.0 59.2 64.8 70.1 73.0 75.7 81.6 85.6 88 89.7 91.7 93.5 94.1 94.2 94.9 97 100 103.3 105.1)'; e -[39.6 39.8 40.0 40.2 40.2 67.9 47.S 53.1 49.4 48.6 48.6 53.8 69.1 66.0 63.8 65.5 58.1 60.7 60.7 60.9 86.1 99.9 112.6 104.7 100.0 107.2 113.9) 1 ; Pf =P.*e; Pfe -Pf.*e; ta -[185 205 295 432 498 478 510 647 639 549 711 777 759 946 1313 1300 1310 1187 1337 1353 1046 1166 1217 2252 2713 2872 2887)'; E -[)'; F -[-.05 -0.02 -0.03 -0.02 -0.03 -0.04 -0.13 -0.10 -0.10 -0.14 -0.06 -0.26 -0.23 -0.23 -0.24 -0.21 -0.09 -0.18 -0.13 -0.28 -0.59 -0.99 -1.12 -0.99 -1. 01 -0.66 -1.31)'; D -[815 1528 1910 1938 2573 1215 1299 1885 3396 476 1092 1280 1027 1813 16 13 1424 1187 187 1011 2159 1643 690 2630 3089 2322 3271 1594)'; K -[11562 14288 17567 21441 26043 27637 31527 40938 46695 48136 51945 5598 2 62147 70171 75821 78908 79735 78881 83251 88040 91310 99160 111074 122766 1 36733 143429 142999)'; kk -(11562 14288 17567 21441 26043 27637 31527 40938 46695 48136 51945 5598 2 62147 70171 75821 78908 79735 78B81 83251 88040 91310 99160 111074 122766 1 36733 143429 142999 141322)'; 1 -(12377 15816 19477 23379 28616 28852 32826 42823 50091 48612 53037 5726 2 63174 71984 77434 80332 80922 79068 84262 90199 92953 99850 113704 125855 139055 146700 144593]'; B -(-0.39 0.50 -0.25 -1.49 -0.39 2.46 -4.16 0.64 7.84 2.12 1.16 -0.09 -1.4 8 - 0 . 3 1 12.76 4.72 - 3 . 3 5 - 7 . 7 4 8.85 3.19 51.91 62.11 38.39 3.68 - 1 9 . 7 0 - 9 1 . 9 4 -70.45]'; BD - ( - 0 . 2 8 - 0 . 3 9 0.50 - 0 . 2 5 - 1 . 4 9 - 0 . 3 9 2.46 - 4 . 1 6 0.64 7.84 2.12 1.16 - 0 . 0 9 - 1 . 4 8 - 0 . 3 1 12.76 4.72 - 3 . 3 5 - 7 . 7 4 8.85 3.19 51.91 62.11 38.39 3.68 - 1 9 . 7 0
466
Differential Models and Neutral Systems JAPAN2.M
-91.94]'; w -(14.2 15.9 17.8 20.2 20.7 23.7 27.4 32.6 40.7 48.0 54.0 59.0 63.2 66.9 70.7 74.3 78.1 80.6 83.3 85.9 88.4 90.2 93.4 96.3 100.0 103.4 105.6]'; T - (X-B-F] '; GDP -[38170 44731 52975 62229 73345 80701 92394 112498 134244 148327 166573 185622 204404 221547 240176 257963 270601 281767 300543 320419 334609 348425 371429 396197 424537 451297 463850]'; y -(38170 44731 52975 62229 73345 80701 92394 112498 134244 148327 166573 185622 204404 221547 240176 257963 270601 281767 300543 320419 334609 348425 371429 396197 424537 451297 463850]'; yD -[32866 38170 44731 52975 62229 73345 80701 92394 112498 134244 148327 1 66573 185622 204404 221547 240176 257963 270601 281767 300543 320419 334609 3 48425 371429 396197 424537 451297)'; yP -[3325 5304 6561 8244 9254 11116 7356 11693 20104 21746 14083 1B246 1904 9 18782 17143 18629 17787 12638 11166 18776 19876 14190 13816 23004 24768 28340 26760]'; yPD -[6561 8244 9254 11116 7356 11693 20104 21746 14083 1B246 19049 18782 17 143 18629 17787 12638 11166 18776 19676 14190 13816 23004 24768 28340 2676 0 12553 4919)'; YT -y-TX; YTD -[26344 30812 36095 42658 49894 58179 63451 72539 87653 101492 112756 12 7393 139936 154465 163304 173966 185128 191146 198262 209636 221655 230749 23 4702 24B655 263957 277516 296274)'; YTP -[4468 5283 6563 7236 8285 5272 9088 15114 13839 11264 14637 12543 14529 8839 10664 11160 6018 7116 11574 11819 9094 3953 13953 15302 13559 18758 966 11*.• YTPD =[2653 4468 5283 6563 7236 8285 5272 9088 15114 13839 11264 14637 12543 14529 8839 10664 11160 6018 7116 11574 11819 9094 3953 13953 1S302 13559 1875 8]'; L -[86.4 89.2 92.9 96.3 100 100.6 98.7 99 98.6 93.4 91.5 90.7 88.7 88.3 89 .2 90.2 91.3 91.8 93.6 95.9 96.9 95.6 96.2 97.9 100.0 102.1 102.8)'; LD -[86 66.4 89.2 92.9 96.3 100 100.6 98.7 99 98.6 93.4 91.5 90.7 88.7 88.3 89.2 90.2 91.3 91.8 93.6 95.9 96.9 95.6 96.2 97.9 100.0 102.1]'; LP =[0.40 2.80 3.70 3.40 3.70 0.60 -1.900 0.30 -0.40 -5.20 -1.9 -0.8 -2.0 0.40 0.90 1.00 1.10 0.50 1.B0 2.30 1.00 -1.30 0.60 1.700 2.10 2.10 0.70]'; LPD -[2.80 3.70 3.40 3.70 0.60 -1.900 0.30 -0.40 -5.20 -1.9 -0,8 -2.0 -0.40 0.90 1.00 1.10 0.50 1.80 2.30 1.00 -1.30 0.60 1.700 2.10 2.10 0.70 -0.4]'; N -y.\L; G -(3054 3410 3934 4558 5455 6422 7537 9336 12240 14890 16417 18243 19753 21486 23568 25585 26796 27996 29449 30685 32388 32975 34184 36275 38807 41232 43254]'; C -[22142 25405 28974 33300 38333 43230 49901 60308 72912 84763 95784 1070 76 117923 130078 141324 149997 160834 169687 178631 188760 195969 2045B5 2151 22 228483 243628 255084 264778]'; M -[11720 13370 15160 18280 21360 27690 34530 40310 44950 49950 56180 6079 0 68930 71020 69570 76510 80900 80800 86380 88980 98210 102970 111840 114470 119630 131040 136140)'; ML =[-0.0304 -0.1087 0.0009 -0.0400 0.0444 -0.3947 -0.6660 -0.9854 -0.6586 -0.4291 -0.5654 -0.5329 -0.9102 -1.1997 -0.5459 -0.3663 -0.6485 -0.5067 -0.33 91 -0.0831 -0.6496 -0.8256 -0.9927 -0.7401 -0.4562 -0.5829 -1.0563]') MD =[10290 11720 13370 15160 18280 2136C 27690 34530 40310 44950 49950 561B 0 60790 6B930 71020 69570 76510 80900 80800 86380 8B980 98210 102970 111840 1 14470 119630 131040]'; MP -[1650 1790 3120 30BO 6330 6840 5780 4640 5000 6230 4610 8140 2090 -1450 6940 4390 -100 5580 2600 9230 4760 8870 2630 5160 11410 5100 9470]'; HPD "(1430 1650 1790 3120 3080 6330 6840 S780 4640 5000 6230 4610 8140 2090 -1450 6940 4390 -100 5580 2600 9230 4760 8870 2630 5160 11410 5100]';
Appendix
467 JAPAN2.M
x - [ 6 . 5 7 5.48 5.48 5.84 5.84 6.25 6 4 . 7 5 4.25 9 9 6.5 6.5 4.25 3 . 5 6.25 7. 25 5 . 5 5 . 5 5 . 0 5 . 0 5 . 0 3 . 0 2 . 5 2 . 5 4 . 2 5 6 . 0 4 . 5 3 . 2 5 1 . 7 5 ] ' ; R - [ 5 . 4 8 5.84 5.84 6.25 6 4.75 4.25 9 9 6.5 6.5 4.25 3.5 6.25 7.25 5.5 5.5 5.0 5.0 5.0 3.0 2.5 2.5 4.25 6.0 4.5 3 . 2 5 ] ' ; RD - [ 5 . 4 8 5.48 5.84 5.84 6.25 6 4.75 4.25 9 9 6.5 6.5 4.25 3.5 6.25 7.25 5. 5 5.5 5.0 5.0 5.0 3.0 2.5 2.5 4.25 6.0 4 . 5 ] ' ; RP -10 0.36 0 0.41 -0.25 -1.25 - 0 . 5 4.75 0 - 2 . 5 0 - 2 . 2 5 - 0 . 7 5 2 . 7 5 1.0 -1.75 0 - 0 . 5 0 0 - 2 . 0 - 0 . 5 0 1.75 1.75 - 1 . 5 - 1 . 2 5 ] ' ; RPD - [ - 1 . 0 9 0 0.36 0 0.41 -0.25 -1.25 - 0 . 5 4.75 0 - 2 . 5 0 - 2 . 2 5 -0.75 2.75 1.0 -1.75 0 - 0 . 5 0 0 - 2 . 0 - 0 . 5 0 1.75 1.75 - 1 . 5 ] ' ; p r i c e - [ 2 7 . 8 28.9 30.5 32.1 34.6 36.8 38.5 43.0 53.0 S9.2 64.8 70.1 73.0 8 1 . 6 8 5 . 6 88 8 9 . 7 9 1 . 7 9 3 . 5 9 4 . 1 9 4 . 2 9 4 . 9 97 100 1 0 3 . 3 1 0 5 . 1 1 0 6 , 4 ] ' ; for i - l : 2 7
pp(i)-price(1+1)-price|i); kprimfi) -kk(i+l)-kk(i); end pie=[pp]'; KP =kprim'; DP -pie./P; logy-log(y); % TX -Total current receipts of general government(From UN financial » statistics) » n -labor productivity(A/1 - GNP/employraent). % -From Yearbook of labor statistics(International labor office)
% % K -capital stock(Gross fixed capital formation) i I -investment(Gross capital formation) % D -increase in stock % The UN(National accounts statistics) defines Gross capital I formation as the sura of the increase in stocks and gross fixed % capital formation
% % y -GDP(Gross domestic product) % yD -y(t-l) % yP -y' (t) i yPD-y'(t-1) % X -Net export(export-import of goods and services) * d -preferential trade agreement transportation * Pfe -import price level in foreign currency (P*e*e) % ta -tariffs % E -cumulative balance of payment % F "Net private outflow of capital % C =private consumption » M,M1 -money supply « Hd -money demand(liquidity); Where, ML- md-M * P -consumer prices % R -interest rate(discount rate) I RD -R(t-l) » RP -R'(t) % RPD-R' (t-1) % pie-inflation
% % % 4
L -employment B -Balance of payment(deposit money banks) T -Net government transfer of capital to foreigers and firms G -Government consumption
75.7
Differential Models and Neutral Systems
468 JAPAN2.M % DP * D * l o g ( P )
=p|t)'/p(t)
«fonoat long; B r e t l = [ X , G , I,C,GOP,R,M,YEARi j (years,n]-size(YEAR); Xl-[5652;X(l:years-l)J; Gl-[2352;G(l:years-l)J; Il-[9362;I(l:y««rs-l)]i Cl-(17028;C(l:years-l)); Ml - [ 8 . 7 ; M ( l : y e a r s - l ) J ; Rl-[6.57:R(l:years-l)J; GDPl~{2SS41;GDP(l:years-l)]; DGDP-GDP-GDP1; Z-X+G+I+C; Z1-X1+G1+I1+C; *temp«[YEAR XI Gl II Cl Zl yl yD Mil RDl yPDl RP RPD Tl T P P-PF PI P-PFl G L LD LP LPD LI LD1 LP1 LPD1 t tl e el d dlj; (save Bertl.asc temp -ascii -double -tabs tclear Bectl temp; %ML - L-M - ml-mO - ml*y
-0.3175.*R(t)
+m6*p(t);
+3.6165.*RD(t-l)-
« C(t) =c0 + c l » Y T ( t ) +c2*YT(t-h) + c 3 * Y T ( t ) ' + c 4 , Y T ' ( t - h ) + c 5 » R ( t ) + c 6 * R ( t - h ) +c7*(17.2335 +0.8380.*y(t) -0.3061.'yD(t-h) -0.3175.♦R(t) +3.6165.*RD(t)-3. 3413.*P(t) -1.7714.»RPD(t)) - M ( t ) ) ; temp-[C o n e s ( s i z e (R)) ,YT,YTD, YTP,YTPD,R,RD, ML); thc=arx(temp,[0,1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 ) ) ; C p - p r e d i c t | [ C , o n e s ( s i z e ( R ) ) , Y T , Y T D , Y T P , Y T P D , R , R D , M L ] , t h c , 4),• CC=thc{7,l:8); % I ( t ) "i0 ♦ i l » y [ t ) + i 2 * y ( t - h ) - 1 3 * y ' I t ) + i 4 * y ' ( t - h ) +i5*R(t) 8*L
+i6»R(t-h)
% X ( t ) -xO + x l * y ( t ) + x 2 * y ( t - h ) + x 3 * y ' ( t ) + x 4 * y ' ( t - h ) + x 5 * R ( t ) + x 8 * L ( t ) *L'(t-h) +xl2*P(t) +xl6*ta(t) +xlS*elt| +xl7*d(t) temp=[X ones ( s i z e ( y ) ) , y , y D , y P , y P D , R , l , L P D , P , t a , e ) ; thx=arx(temp,[0,1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 0 1 0 0 0)); Xp-predict([X,ones(sizely)),y,yD,yP,yPD,R,L,LPD,P,ta,e),thx,4); XX=thx(ll,l:12); « G(t) =g0 + g l * y ( t ]
+ g2«y(t-h)
+ g 3 " y ( t ) +g4»y' ( t - h l
+g5*R(t)
+g8*L(t);
+i
+X10
Appendix
469 JAPAN2.M
temp-[G ones ( s i z e (y) 1, y, yD, yP, yPD, R, L) ; thg-arx(temp,(0,1 1 1 1 1 1 1 0 0 1 0 1 0 0)1; G p - p r e d i c t ( [ G , o n e s ( s i z e ( y ) ) , y , y D , y P , y P D , R,L) , t h g , 4 ) ; GG«thg(6,l:7); » y ( t ) - zsO + z s l * y ( t ) + z s 2 * y ( t - h ) + z s 4 * y ' ( t ) + z s 5 * R ( t ) +zsB*L(t) + z s l O » L ' ( t ) +zsl3»Ml(t) -zsl4*TX(t) + z s l 5 » e ( t ) + z s l 6 * t a | t ) + z s l 7 * d ( t ) * ( 0 ) + l / h * K ' ( t ) ; h-1 temp=[y o n e s ( s i z e ( y ) ) , y , y D , y P , R , L , L P , M , T X , e , t a , K P ] ; thy=arx(temp,[0,1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0]); yp=predict([y,ones(size(y)),y,yD,yP,R,L,LP,M,TX,e,ta,KP],thy,4); yy=thy(ll,l:12); » c y ( t ) - ylO ♦ y l l ' y l t ) + y l 2 » y ' ( t ) + y i 3 * R ( t ) + ylS*ML(t) t e a p - [ C 6 n e s ( s i z « ( y l ) ,y,yP,R,ML, YT]; thcy-arx(temp,[0,1 1 1 1 1 1 0 0 0 0 0 0]i; cyp-predict([C,ones(size(y)),y,yP,R,ML,YT],they,4); ccy-thcy(5,1:6);
+yl8*YT(tl
%ly(t) = 10 + l / h « K P ( t ) temp-[I ones(size(KP)t.KPJ; thly-arx(temp,[0,1 1 0 0)]; I y p - p r e d l c t ( [ I , o n e s ( s l z e ( y ) ) ,KP) , t h l y , 4 ) ; Ily»thly(l,l:2); %Gy(t! = gsO + g s l * y t t ) + g s 4 * y ' ( t ! +ys5*R(t) + g s 8 « L ( t ) temp-[G o n e s ( s i z e l y ) ) , y , y P , R , L ] ; t h g y - a r x ( t « m p , [ 0 , 1 1 1 1 1 0 0 0 0 OJ); gyp-predict([G,ones(size(y)) ,y,yP,R, L],thgy,4); ggy-thgy(S,1:6); *Xy(t) - xO + x l » y ( t ) + x 2 » y ( t - h ) +x5»R|t) + x 8 ' L ( t ) + X 1 0 * L ' ( t ) alt) +xl3*d(t)«(0) temp-[X ones ( s i z e ( y l ) , y , y D , R , L , L P , e , t a ] ; t h x y » a r x ( t e m p , [ 0 , 1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 01 I; x y p - p r e d i c t ( [ X , o n e s ( s i z e (y)) , y , y D , R , L , L P , e , t a ] , t h x y , 4) ; xxy-thxy(7,l:8);
+xll«e(t)
+xl2«t
«B(t) - bO + b l * y ( t l rt>2*y(t-h) +b3*P(t) + b 4 * y ' ( t - h ) +b5*R(t) +b6*R[t-h] +b7*e ( t ) + b 8 * R ' ( t - h ) +b9*L(t) + b l 0 * L ( t - h ) +bl2*L' ( t - h ) + b l 3 « t a | t ) + b l 5 * d l t ) +bl7*B (t-h) temp=[B ones ( s i z e ( y ) ) , y, yD, P,yPD,R, RD, e, RPD, L, LD, LPD, t a , BD] ; thb=arx(temp,[0,1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 0 1 0 1 1 0 1]); Bp=predict( [G, ones (size (y) ), y, yD, P, yPD, R, RD, e, RPD, L, LD, LPD, ta, BD) ,thb,4); BB«thb(13,l:14); % y(t) - A*K*a*L*(l-a> -=>ln(y(t)) = a*ln(K) * |l-a)*ln(L) temp-[log(y) ones(size(y)),log(K) ,log(L)); thlyarx (temp, [0,1 1 1 0 0 0]); lyp -predict![log(y) ones(size(y)),log(K),log(L)],thly,4); lyy -thly(2,l:3); %D(t) - a(l-c)*y(t> -k0*K(t) +kl3*K"(t) +L4»R(t) +L5*L(t) +L6*P(t) +v*y'(t) temp-[D ones(size(y)) ,y,K, KP, R,L,P,yP] ; thD»arx(temp,[0,1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0]); D p - p r e d i c t ! [D, o n e s ( s i z e ( y ! >, y, K, KP, R, L, 1, y P ] , thD, 4) j DD-thD(7,l:8);
470
Differential Models and Neutral Systems
JAPAN2.M «DP(t) = pO + p l * P f e ( t ) +p2*w(t) -p3*n(t> -p4*P(t) +pS*M'(t) +p6*ML(t) temp-[DP ones ( s i z e (w) ) , Pf e, w,N, P,MP,ML]; thp « a r x [ t e m p , 1 0 , 1 1 1 1 1 1 1 0 0 0 0 0 0 0 ] ) ; dlp=predict([P,ones(size(w)),Pfe,w,N,P,MP,ML],thp,4); dp=thp(6,l:7); subplot (2,l,l),plot(YEAR,C,YEAR,Cp, ' + ') title (' japan—PRIVATE CONSUMPTION—') ; ylabel C O ; xlabel('YEAR'); subplot (2,l,2),plot(YEAR,X,YEAR,Xp, ' + ') title (' japan—EXPORTS—'); ylabel('X'); xlabelCYEAR*); pause subplot (2,l,l),plot(YEAR,I,YEAR,Ip, ' + ■) title (' japan --INVESTMENT— •>; ylabel('I'); xlabelCYEAR'); subplot (2,1,2), plot! YEAR, G, YEAR, Gp, ' + ') title (' japan--GOVERNMENT CONSUMPTION— ') ; ylabel C G ' ) ; xlabel('year') ; pause clg subplot [2,1,1),plot (YEAR,M, YEAR, Lp, ' +') title C japan—MONEY DEMAND--'); ylabel('Ml'); xlabel('YEAR'); subplot(2,1,2),plot(YEAR,Z,YEAR,[Xp+Ip
% subplot (2,l,l),plot(YEAR,DP,YEAR,dlp, ' + ') title C japan—CONSUMER PRICES—'); ylabel CP') ; xlabelCYEAR'); subplot (2,l,2),plot(YEAR,y,YEAR,yp,'+') title (' japan—INCOME--' I ; ylabel('GDP'); xlabelCYEAR'); pause subplot (2,l,l),plOt(YEAR,C,YEAR,cyp, * + •) title (' japan—INCOME/CONSUMPTION— ') ; ylabel('cy'); xlabel('YEAR'); subplot(2,1,2),plot(YEAR,I,YEAR,Iyp, ' + ') title (' japan—INCOME/INVESTMENT—'); ylabelCIyp'); xlabel('year'); pause clg subplot(2,1,1),plot!YEAR,G,YEAR,gyp, ' + •)
Appendix
471
tkW^japaa-QCOME/GOVEKMENT-'); ylab*ll•gyp'11 aubplot(2,1,2).plot(YEAR,X,YEAR.xyp,•♦•) t i t l a (* J apan—IWCCHE/EXPORT--'); ylabelCxyp'); xlaball'YEARM; pauae
\ aubplot (2,1.1), plot (YEAR, B, YEAR, Bp, '♦•) titl.C japan—BALANCE OP PAYMENT--'); ylab*lCB');
xlabaM'YEAR'W aubplot (2,l,2),plot(YEAR,lo4y,YEAR,lyp, ' + ') t i t l * (• Japan—log (incom) ~ ' ) . ; ylabalflnyM) xlabalCYEAR'); pauia aubplot (2,l,l),plottYEAR,D,YEAR,I>p,'♦') titlaCJapan—Inctaaaa in a t o c k — ' ) ; ylabalCD'H xlabal('YEAR'))
I The economic dynaiaici can b* put in matrix foza t aa follow*: x ' ( t ) - A - l * x ' ( t - h ) - AO»x(t)*Al*x(t-h)+B*u(t) %x-[y R L K P E l ' l tA-1 - [ a n i l -aml4 aal3 0 0 0; 0 an22 0 0 0 0; Ua03 0 -ImOl 0 0 0; a3 0 a6 -an 1 0 0; 0 -MTp6*P(t) 0 0 0 0; b« be bl2 0 0 0] %A0 -laOl al2 al4 a l 6 alB 0; a21 a23 0 0 a2S Oi 0 0 «33 0 a3S 0; 0 0 0 «44 a 45 0; aSl*p(t) aS2*p(t) 0 0 a5S*p(tl 0; a61 a62 a63 0 0 0) (Al - [ a l l al3 alS 0 0 0; al21 al22 0 0 0 0; al31 al32 al33 0 0 0 ; al41 a!42 al43 0 0 0; a l S l - p l t ) al52*p(t) 0 0 0 0; «161 al62 al63 0 0 al66) tBl » [ - » » ! aal a a l z l 5 a a l z l t a i l x l 7 0 0 0) 0 0 0 0 0 -L2 0 0; -mwal4 mu nuz •15 zwz*16 0 *Mzal3 0 0;-zal4 1 if15 zal6 0 zal3 0 0; 0 0 pl*pf(t) 0 0 -p6 p5 0; 0 0 b7 bB blS 0 0 -1] IB2 - [ a a l aal aal aal(I13+c7) 0 0 0 0 0; 0 0 0 -L2 0 0 0 0 OlO m 0 0 0 0 m mw 0; 0 1 0 0 0 0 1 1 0; 0 0 0 -p6 -p3 p2 0 0 1;0 0 0 - 1 0 0 1 0 0) tu • [q S]i Iq tq
-IT1 flO « ta d Ml Ml" f0]'| -IT1 -634.1202 • ta d Ml HI' 1.9562e+006]■;
IS -[CO 10 X0 M0 n w xO ylO p0)'; %S -(-133.6142 -3.9738*4 -1.9561*6 ] ' t
» B-(B1 B2} % AB-A0*B I A2B-(A0*A0)*B
-5.6973
n M -3.9738*4
29.3647 1.9959* 4
Differential Models and Neutral Systems
472 % A3B-(A0*A0*A0)*B % A4B-(A0*A0*A0*A0)*B « A5B-(AO*A0*A0*A0*A0*AO)*B » G-[B AB A2B A3B A4B A5B] % rank(B)« 6 % rankG - 6 % For s p e c t r a l
controllability
% D e l t a l - e y e l s i z e ( A l ) ) . * j -Ami . * ( j * 2 . 7 8 1 4 " ( - j ) ) » + Al.*(2.7814*(-j)) % a«=[Deltal B] * for j - 0 . 4 3 1 2 , % rank(a)«6
4, 1 0 , 2 3 , 100,
- AO
...
% For nonzeroeigenvalues of A-1 to be controllable via matrix B, % cc-eye(size(Ami)). *j -Ami % c»[cc B] i for j =0.4312, 4, 10, 23, 100, ... % rankle)=6 % Since rank(B) is 6, where rank(a) and rank(c) are 4 also 6, the system is function.,-, controllable. Ll «=1, L2 -0.9; LS -DD(5) L6 »DD(6); h »1 » for y(t) - A*KAa*L*(l-a) a- 0.9 » a 'alpha aa-0.1 % aa-(l-a)-beta W-sum(w)/27 raw -
473
Appendix JAPAN2.M zll=-II(10) zl3-=(II(ll)+CC(8) )*MM(6) zl4—CC(2) zl5=XX(ll) 2l6-XX(10) zH-0 zl8-XX{9)+(CC(8)+zl3l*MM(7| zl9»-CC(3) z20-CC(4) z21-CC(5) » z22—(CC(B)+II(11))«M1 zsO = ccy(l)+IIy(l)+ggy(l)+xxy(l) zsl - ccy(2)+ggy(2)+xxy(2) zs2 xxy(3) zs4 » ccy(3)+ggy(3) zs5 - ccy(4)+xxy(4)+ggy(4) zs8 - xxy(5)+ggy(5) zslO =xxy(6) zsl3 =ccy(5) zsl4 »ccy(6) zslS «xxy(7) zsl6 «xxy(8)
%constant term %y(t) term %y(t-h) term »y'(t) term *R(t) term *L(t) term »L'(t) term %Ml(t) term %T(t) term »e(t) term *ta(t) term
si -1-Ll*z3 % assumption yyl - zsO; yy2 = 2sl; yy3 = zs2; yy4 = zs4; yy5 - zs5; yy6 = zs8; yy7 « zslO; yy8 - zsl3; yy9 » zsl4; yylO =zsl5; yyll =zsl6; zsO= aO al a2 » a3 •= a4 a5 ■> a6 a8 «=
GG(1)+XX(1)+CC(1)+II(1) DD(l)/(h*U-yy(2))) -DD(l)*yy3/((l-yy2)*h) -al*h a0*h*yy4 DD(l)*yy5/((l-yy2) +DD14)) DD(l)*yy6/(l-yy2) +DD(5) a0*h*yy7 DD(6)
%A-1 starts here; amll = a-11 amll am22 aml3 aml4 Un03 UnOl
=Ll*z4/sl -L2*MM(6) =Ll*ZlO/sl -Ll*zlO/sl - DD(l)*yy4*mw/(l-yy2) « DD(3)*DD(l)/(l-yy2) -mw*DD(l)*yy7/
474
Differential Models and Neutral Systems JAPAN2.M
a3 = DD(l)*yy4/(l-yy2) a6 - DD(l)*yy7/(l-yy2) ami = D D ( 3 ) * D D ( l ) / ( l - y y 2 ) M7p6P - MM(7)«dp(6)'sura(P)/27 b4 = -BB(5) b8 - -BB(9) b l 2 - -BB(12) %A0 s t a r t s h e r e ! aOl = L l * ( z l - l ) / s l a l l =Ll*z2/sl al2 -Ll*z5/sl al3 -Ll*z6/sl al4 - L l * z 8 / s l al5 - L l * z 9 / s l al6 - L l * z l l / s l al7 =Ll*zl9/sl al8 =Ll*zlB/sl a21 -L2*HM(2) a23 «L2*MM(4) «25 =L2*MM(7) a33 -aO; a35 «mw*a8 a44 =aO a4S =a8 aSl=-dp(7)*MM(2)*sum(P)/27 » includes Pit) a52"=-dp(7)*MM(4)"suni(P)/27 » includes Pit) a55 —(MM(7)+dp(5))*sum(P)/27 % i n c l u d e s P i t ) a61 -BB(2) a62 =BB(6) a63 =BB(10) %A1 starts here! all =Ll»z2/sl al3 =Ll*z6/sl al5 =Ll*z9/sl a22 »L2*MM(3) a24 -L2*MM(5) al - -DD(l)*yy3/lIl-yy2)*h) a2 = -al*h 11 - -al 12 —mw*DD(l)*yy3/(l-yy2) 14 -mw*DD(l)*yyS/(l-yy2 +DD(4)) b2 »BB(3) b6 -BB(7) blO -BB(ll) bl7 »BB(13) alll -L1*(GG(3)+II(3)+XX(3)+MM(3)*(II(11)+CC(8)) )/sl all2 -al3 all3 =a!5 al21=a22 al22-a24 al31»mw*DD(l)*XX(3)/(l-(XX(2)+ggy(2l+ccy(2))) al32»mw*DD(l)«yyS/(l-yy2) -DD(4) al33-a5-ll*h al41 =DD(l)*yy3/(l-yy2)
475
Appendix JAPAN2.M ■142 al43 al51 alS2 al61 al62 al63 al66
-DDIl)*yyb/(l-yy2) + DD14) =DD(l)«yy6/(l-yy2) + DDI5) —dp(7)*sum(P!/27 %includes Pit) — MM(5)*dp(7)*sum!P)/27 Sincludes Ptt) -BB(3) •BB|7) -BB(ll) -BBI14)
» Bl starts here! ssl -Ll/sl % ssl-bl sslzl3 "=B»l*2l3 sslzl5 =ssl*zl5 sslzie -ssl*zl6
ssizn -o SSlZlS -SSl'ZlB blzl4=-BB(2l*zl4; bl -BBUI blzl5-BB(2l«zlf> blzl6-BB(2)*zl6 blzl7-BB(2)»zl7 blzl3-BBl2)*zl3 mwzsl4 =mw*yy9 mwzsl5 "jnw'yylO mwzsl6 -mw'yyll mwzsl7 «0 mwzsl3 =»mw*yy8 plpf p4 p5 b7 b8
-dp(2)*sum(Pfe)/27 -dp(S> -dp(6) =»BB(8| «BB(9)
bib
-0
4B2 starts here! ssl -Ll/sl sslil3c7 -ssl«tII{11)+CC18)1 12 =mw«DD(l)*yy3/(l-yy2) p6 - dp(7) p3 - dp(41 p2 - dp(3) al7 -Ll«zl9/sl a45 -a8 CO - CC(1> 10 = 11(II XO - XX(II pO = dp(II MO - mill ylO » ccy!l) mO ->M! 1) <jO - G G U ) £0 =BBU)-XX(1) g2 —L2*MM(1)
*dp(2i*Pf(t)
Differential Models and Neutral Systems
476 JAPDM
The following data fox Japan was generated using the prograia Japan2.m. LI - 1,L2 -0.9.LS -1.2861,L6 - -0.1039,h -l,a =0.9,aa - 0.1,W -61.2778 mw - 7.9983e-004,z0 = -1-8273e+006,zl = 7.5207e+010,z2 - -7.5207e+010 23 - -7.5207«+01,z4 - -7.5207«+01,z5 - -5.6406e+5,z6 - 8.0241«+004 z8 - 1.3840e+004,z9 - -3.9753e+003,zlO - 1.3745e+005 zll —0.2269,213 - -1.132Se+005,zl4 - 0.0213.Z15 - 7.3516e+003 zl6 - -222.0336,217 = 0,zl8 - -4.0124e+0O4,zl9 - -0.0215,z20 -0.0393 Z21 - 0 . 0 0 3 5 , s i - 7 . 5 2 0 7 e + 0 1 0 , z s 0 - -1.9966e+006,a0 «= - 0 . 6 7 1 0 , a l = 0.7398 a2 - - 0 . 7 3 9 8 , a 3 - - 3 . 6 8 6 4 e - 0 0 4 , a 4 - -7,0901e+003,a5 - 145.4243 a6 -6.18S4e+4,a8 - - 0 . 1 0 3 9 , a m l l - -1.0,am22= 3.5514,80113 -1.8277e-O06 ami4 -1.8277e-6,Lm03 «= -3.0053e-9,Lm01 - - 0 . 5 0 4 3 , a 3 - - 3 . 7 5 7 5 e - 0 0 6 a6 - 630.4745,ami - 2.0183e-006,M7p6P - -1.6356e-008,b4 - -0.0041 bB - 9.4764,bl2 -18.4975,a01 -1.0000,all —1.000,al2 —7.5001e-6 al3 -1.0669e-006,«14 = 1.8403a-007,alS - -5.2859e-008,al6 - -3.0164e-012 al7 —2.8640e-013,«18 =-5.3351e-007,a21 »-0.0057,a23 -17.0002,»25 ' 0.4833 a35 —8.3105e-OO5,a44 =-0.6710,a45 —0.1039,aSl —5.8654e-007,a52 - 0.0018 a55 —37.5250,a61 - -0.0013,a62 - 8.4629,a63 » 10.3534,all - -1.0000 al3 • 1.0669e-006,al5 - -5.2859e-008,»22 = 0.0061,a24 - -2.2046 al - 0.739B,a2 - -0.7398,11 - -0.7398,12 - 5.9173«-004,14 - -5.6708 b2 -0.0015,b6 - 13.3059,bl0 - -13.2036,bl7 » -0.0045,alll - -1.0000 all2 -1.0669e-006,all3 —5.2859e-008,al21 = 0.0061,al22 • -2.2046 a!31 - -5.1906e-004,al32 —5.6720,«133 - 146.1641,al41 - -0.7398 al42 —7.0901e+003,al43 -145.4243,»151 -9.2705e-005,al52 —2.2709«-004 al61 -0.001S,al62 -,13.3059,al63 —13.2036,al66 - 0.0370,sal -1.3297e-011 sslzl3 - -1.5058e-006,sslzl5 «= 9.7752e-008,sslzl6 - -2.9523e-009 sslzl7 - 0,sslzl8 - -S.33Sl«-007,bl « 109.7219,blzl5 - -9.3074 blzl6 - 0.2811,blzl7 - 0,blzl3 - 143.3742,inwzsl4 - -2.4322e-006 rawzslS - 40.2234,mwzil6 = -0.8208 mwz«17 -0,mwzsl3 - -0.0141,plpf ■ -6.3388e-005,p4 - -1.5140e-005 p5 - -4.3587e-010,b7 - 0.6002,b8 - -9.4764,bl5 = 0,ssl - 1.3297e-011 salil3c7 - -3.8160e-007,12 = -5.9173e-004,p6 - -1.3267e-006,p3 « 0.0048 p2 -1.5011e-005,al7 - -2.8640e-013,a45 - -0.1039,c0 - -133.8142 10 - -3.9738«+004,x0 - -1.9561e+006,p0 -1.9959e-004,M0 - -5.8973 ylO - 29.3847,m0 —5.8973,g0 —634.1202,fO -, 1.9S62e+006,g2 - 5.3075
477
Appendix • ndu Ami ■ 0.0000 3.5514
0.0000
-0.0041
-9.4764
16.4975
O 0
1.0000 -0.0057
0.0000 17.0002
0.0000
0.0000
0
0 0
0 0
-0.6710
0 0
0.0000 -0.0013
0.0018 8.4629
-0.0010 0.0000 0.0000 -0.0007 0.0000 0.0000
0.0000 -0.0022 -0.0057 -7.0901 0.0000 0.0133
-1.0000
0 0.0000 0.0000
0
0 0 0.5043 0 630.4745 0.0000 0
0 0 10.3534
0.0000
0 0.1462 0.1454
0 -0.0132
0 0 0 0.0000
-0.6710
0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0.0000 0.4833 -0.0001 -0.1039 -37.5250
0 0 0 0 0 0
0
0 0 0 0 0 0
0 0 0 0 0 0.0000
Coluatns 1 t h r o u g h 7 0.0C00 0 0.0000 0.0000 0 0
0.00OD 0 0.0000 0.0001 0 0
0.0000 0 0.0040 5.0290 0.0000
o.oooi
0
0000 0 0001 1026 0 0009
0 0 0 0 0 0
-0.0001 0.0000 -0.0018 0.0000
0.0000
0
0
0.0000 -0.9000
0 0 0 0
0 0 0 0
0 0
-0.0048
0.0000
0
0
0
1.0000
0 0 0 0
Column 8 0 0 0 0 0 -0.0001 B2 Columns 1 through 7 0.0000
0 0 0 0 0
0.0000
0.0000
0
0 0 0 0 0
0.0008 1.0000
0 0
Columns 8 through 9
0 0 0.0008 1.0000
0 0
0 0 0 0 1.0000
0
0 0 0.0000 -1.0000
0.0008 1.0000
478
Differential Models and Neutral Systems
q 1.0e*006 • -0.0006
1.9562
S 1.0e*006 * -0.0001
-0.0397
-1.9561
0.0000
-1.9561
0.0000
0.0000
Appendix
479 US-PRIVATE CONSUMPTION-
1970
1975
1980 YEAR
1945
1990
1995
1990
1995
US- EXPORTS-
tsoo 1500 -
1000 •
500
1865
1470
1S75
19B0 YEAR
19S5
solid- real data
Figure US1 US-INVESTMENT-
0 1965
1970
1975
1980 YEAR
1995
1985
US- GOVERNMENT CONSUMPTION-
1990 solid- rtal data (*) modal
Figure US2
1995
480
Differential Models and Neutral Systems US-CONSUMER PRICES--
1965
1870
1975
1880 YEAR
1885
1990
1995
US-INCOME--
1865
1870
1875
1880 YEAR
1885
1980 1885 •olid- rial d a n (♦) modal
Figure US3 U S - M O N E Y DEMAND1500
1885
1SS5
1870
1875
1880 YEAR
Figure US4
1985
1880 1885 •olid- raal data («■) modal
481
Appendix Ut-INCOME/CONBUMPTION5000
ION
1
r—
■ < - —'
l
^
too (00
l
•
400
\^-
/ +
♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦
* ^ < 1F - O ^ *
♦
♦ ♦ ♦ ♦ , ♦ *
•
toe _
i
1
1170
1171
1N0 1*"
1MS
1IW u M - r a i l data (♦}«odt1
II
Figure US5 US-{NC0ME«OVERNMENT1000
I too
1175
IttO YEAR U8-MC0ME/EXPORT-
1MS
1M0
1000
too
IMS
1170
ItM 1HI •ati-raoldtu (♦) nodal
Figure US6
Differential Models and Neutral Systems
482
US-BALANCE OF PAYMENT100
Figure US7 US--lncre*s« in stock-
1895
solid- real data (•*■) model
Figure US8
Appendix
483 Canada-INVESTMENT-
0 1965
1970
-50 1965
1970
1975
19S0 1985 YEAR Canada-GOVERNMENT CONSUMPTION-
1975
1980
1985
1990
1995
1990
1995
year
Figure CA 1 Canada-INCOME/CONSUMPTION-
1975
1980 YEAR Canada-INCOME/INVESTMENT-
1995
1995
Figure CA 2
484
Differential Models and Neutral Caada-QCOME/QOVERNMENT-
1880 YEAR
1995
Canada-INCOME/EXPORT-
1985
1970
1975
1960 YEAR
1985
1990
1995
Figure CA 3 Canada-BALANCE OF PAYMENT-
-SOW
1965
1970
1975
1980 YEAR
1985
1990
1995
Canada-logOneoflw)-
1995
Figure CA 4
Systems
Appendix
485 Canada-Increase in stock-
1995
Figure CA 5
Canada-PRIVATE CONSUMPTION-
1980 YEAR Canada-EXPORTS-
0 1965
1970
1975
1980 YEAR
Figure CA 6
1995
1985
1990 1995 solid-real data (+) model
486
Differential Models and Neutral Systems Canada-MONEY DEMAND-
1965
1970
1975
1980 YEAR
1985
1995
Canads-AGGREGATE DEMAND1500
1990 1995 solid- real data M modal
Figure CA 7 Canada-CONSUMER PRICES-
19 1985 YEAR Canada-INCOME-
Figure CA 8
1990
1995
1990 solid- real data (♦) model
1995
Appendix
487 GERMANY-PRIVATE C0NSUMPTK3N-
1970
1980 1985 YEAR GERMANY-EXPORTS-
1975
1980 YEAR
1985
1990
1990 solld-raal data (♦) model
1995
1995
Figure G 1 GERMANY-INVESTMENT-
1980 YEAR GERMANY-GOVERNMENT CONSUMPT10N-
1980 yaar
Figure G 2
1985
1990 tolid-ma) data (♦) modal
1995
1995
488
Differential Models and Neutral Systems GERMANY-CONSUMER PRICES-
-600 1965
1970
1975
1960 1965 YEAR GERMANY-INCOME-
1990
1995
1990 *olid-raal data H model
1995
3000
2000Q O
1000
1980 YEAR
1985
Figure G 3 GERMANY-MCOME/CONSUMPTION-
1980 YEAR GERMANY-INCOME/INVESTMENT-
1995
1995
•olid-raal data (♦) modal
Figure G 4
Appendix
489 GERMANY-lncrease in stock-
1995 solkt-real data (+) model
Figure G 5 GERMANY-MONEY DEMAND-
1995
1980 YEAR GERMANY-AGGREGATE DEMAND-
0 1965
1970
1975
1960 YEAR
Figure G 6
1965
1990 •oUd-raal data (♦) model
1995
Differential Models and Neutral Systems
490
India-PRIVATE CONSUMPTtON3000
2000
1000
1965
1970
1975
1980 YEAR Indla-EXPORTS-
1965
1970
1975
1980 YEAR
1985
1985 solid-real data (+) model
1990
1990
Figure IN 1 India-INVESTMENT1500
1000
500
1965
1970
-200 1965
1970
1975
1980 1985 YEAR India-GOVERNMENT CONSUMPTION-
1975
1980 year
Figure IN 2
1990
1985 1990 solid- real data (♦) model
Appendix
491 Indla-MONEY DEMAND-
1965
1970
197S
1980
1985
1990
YEAR India-AGGREGATE DEMAND6000]
4000
2000
1975
1980 YEAR
1985 toid- real data (♦) model
1990
Figure IN 3 Indw-CONSUMER PRICES-
1975
1980
1985
1990
YEAR India-INCOME-
1975
1980 YEAR
Figure IN 4
1990 solid-real data (+) model
Differential Models and Neutral Systems
492
India-INCOME/CONSUMPTION3000
2000-
1000
1965
1970
1975
1980 YEAR India-INCOME/INVESTMENT-
1985
1990
1500
1000
500-
1965
1970
1975
1980 year
198S solid- real data (+) model
1990
Figure IN 5 Iodi»-INCOME/GOVERNMENT-
1965
1970
1985
1970
1076
1980 YEAR Indto-INCOME/EXPORT-
1079
1080 YEAR
Figure IN 6
1985
1990
108S 1990 toW-mat data («) modal
ADDendix
-500 1985
493
1970
1975
1980
1985
1990
YEAR lndla-tog(inoome}9 8
*7
^ t ^ * - * - ^ * " ^
6
* 1065
1970
—
-■
—
1975
i
1980 YEAR
—
'
1985 1990 ■old- m l data (♦) model
Figure IN 7 India-Increase in stock-
150 100
1965
1970
1975
1980 YEAR
Figure IN 8
1985 1990 solid- real data (+) model
Differential Models and Neutral Systems
494
UK-PRIVATE CONSUMPTION-
-100 1965
1970
1975
1980
1985
1990
1995
YEAR UK-EXPORTS-
300
1990 1995 solid-real data (+) model
Figure UK 1 UK-INVESTMENT-
1665
1970
1975
1980 1985 YEAR UK-GOVERNMENT CONSUMPTION-
1960
1995
1990 1995 •ofid- real data (♦) model
Figure UK 2
495
Appendix UK-MONEY DEMAND300 200 -
i ioo 1+
0-
—T—<
+ -100 1965
1070
1975
1980 1985 YEAR UK-AGGREGATE DEMAND-
1970
1975
1990
1995
1000
1985
1980 YEAR
1985
1990 solid-real data (+) model
1995
Figure UK 3 UK-CONSUMER PRICES-
1995
1965
1970
1975>
1980 YEAR
Figure UK 4
1985
1990 11995 •oBd-real data (♦) modal
Differential Models and Neutral Systems
496
UK-INCOME/CONSUMPT10N-
1965
1970
1980 19S5 YEAR UK-tNCOME/lNVESTMENT
1990
1975
1995
150
100-
50
1965
+ + + +
1970
1975
+
<-
+
+ ^#^*
1980 year
+
+
+
1985
+
1990 1995 solid-realdata (♦) model
Figure UK 5 UK-INCOME/GOVERNMENT-
1080
1970
1973
1980 1988 YEAR UK-INCOME/EXPORT-
1990
1995
300
1990 1995 solid-real data
Mr
Figure UK 6
Appendix
497 UK-BALANCE OF PAYMENT-
.x10
0 « -5 -10 1965
1970
1876
1980 YEAR UK-log(income)-
1985
1969
1970
1975
19B0 YEAR
1985
1990
1996
1990 1995 ■old-real data Wmo
Figure UK 7 UK-Increase in stock-
o(-
-10 1965
1970
1975
1980 YEAR
Figure UK 8
198S
1990 1995 solid- real data (+) model
498
Differential Models and Neutral Systems
Ami . 1.0e»006
o.ocoo
0.0000 0 0.0000
o.acoo 0 0.5997
0.0000 0 0.8378
0 0 0 0.0000 0 0
0.0000 0.0000 0 0 0.0000 -2.0273
0.0000 0 0.0000 0 0 5.4770
0.0000 0 0 0.0000 0 0
0.0000 0.0000 0.0000 0.0000 0.0000 2.6624
0.0000
0.0000 0
0.0000 0 0 3.0000 1.4942
o.oooo
1.0e*006 •
o.oooc 0.0000 0 0
a.oooo
-0.864}
0.0000 0.0000 0.0000 0.0000 0.0000 0
0 0 0 0 0 0
.0t>»005 •
o.oooc 0.0000 0.0000 o.ooco 0.0000 5.2099
0 0 0 0 0 0.0001
0
0.0000 0.0000 0
6.8529
1.0«»006 • Coluans 1 through 7
0.0000 0
0.0000 0.0000 0 0
0..0000 0 0,.0000 0..cooo 0 0
0 0 0 0 0 0
0 9.0000 0.0000 0.0000 0.0000 0
0 0 0 0 0.0000 0
0.0000 -0.9000
0 0 0 0
0 0 0 0
0.0010 1.0000
-0.0260 -1.0000
-0.2701
0.3038
0.0000
0.0000
0
0
0.0000 0.0000 0.0000 -0.3460
0.0000 0.0000
0.0000
0.0000
0
0 0 0 0 0
a
1.4942
Colvun 8 0 0 0 0 0
0.0000
Colunns I through 7
0.0000 0 0 0 0 0
0.0010 1.3000 0 0
0 0
ilunns 9 through 9 0 0
0.0010 1.0000
0
m
o 0 0
Fi g .
UK 9
0
0
0 0 0
1.0000
499
Appendix 0 0
1.0000 C
q 1.0«*008 • 00000
-5.5844
-16.1625
-1.7795
s 6.3646
-1.5714
6.3646
0.0047
0.1518
Figure UK 10
UKD.M The following data for United Kingdom was generated using the program uk2.m. LI" 1,L2 » 0.9L5 »5.9149e-4,L6 «0.0070,h »l,a =0.9,aa -0.1,H = 50.1667 mw = 9.9894«-4,z0 » -13.0515,zl - 4.2197e+13,z2 - -4.2197e+013 z3 - -4.2197e+013,z4 » 0.0193,z5 = -2.2968,z6 - 4.0961 z8 = -0.0460,z9 » -5.7224e-004,zl0 - 0.1404,zll - -0.0085 zl3 - -0.0019,zl4 —4.2197e+13, zl5 -8.4197e-4,zl6 —6.0196e-006,zl7 -0 zl8 » -0.0195,zl9 = 4.2197e+13,z20 » -4.2197e+13, z21 =» 0.0030 si - 4.2197e+013,zs0 - -12.5457,aO - -0.7016,al - 0.0238 a2 - -0.0238,a3 » 1.3B03e-4,a4 - -0.2580,a5 « -0.0291,a6 - -0.1483 a8 ■ 0.0070,anil « 4.5648e-16,am22 - -0.0055,aml3 -3.3261e-15 ami 4 = 3.3261«-15,Lm03 - 1.332Se-7,Lm01 »'0.0020,a3 = 1.3339e-4 a6 - -0.1433,ami - 0.0018,K7p6P - 0.1223,b4 » 5.9970e+005 b8 - -1.4942e+6,bl2 - 8.3778e+S,a01 =■ 1.0000,all - -1.0000 al2 = -5.4429e-014,al3 » 9.7071e-014 al4 » -1.0900e-015,al5 - -1.3561e-017,al6 - -2.0114e-016,al7 - 1.0000 al8 - -4.6206e-16, a21 • -0.0154,a23 ■= 0.1654,a25 =» -0.0724 a35 = 6.9681e-006,a44 - -0.7016,a45 = 0.0070,a51 - 0.0224 a52 - -0.2404,a55 - 20.6239,a61 - -8.6435e+5,a62 » -2.0273e+6 a63 - 5.4770e+6,all - -1.0,al3 - 9.7r>71«-14,al5 - -1.3561e-17 a22 - 0.0303,a24 - 0.1028,al «= 0.0238,a2 = -0.0238,11 = -0.0238 12 - 2.3740e-5,14 - -2.5776e-4,b2 - 5.2099e+5,b6 » 2.6624e+5 blO - 6.8529e+005,bl7 - 2.3991e+003,alll - 6.5368e-016 all2 - 9.7071e-14,all3 - -1.3561e-17,al21 - 0.0303,al22 - 0.1028 al31 - -2.1473e-6,al32 - -0.0079,al33 —0.0054,al41 "-0.0238 al42 - -0.2523,al43 —0.0291,al51 = -1.3080,al52 - -0.1494 al61 = 5.2099e+005,al62 = 2.6624e+005,al63 - 6.8529e+005,al66 - 10.0500 ssl - 2.3698e-14,sslzl3 - -4.6205e-17,3slzl5 » 1.9953e-17 sslzl6 — 1.4265e-19,sslzl7 -0',sslzl8 — 4.6206e-16,bl - -5.5844e+8 blzlS - -727.7578,blzl6 «= 5.2031,blzl7 =0,blzl3 =■ 1.6853e+3 B W Z S 1 4 - -1.3281e-007,nwzsl5 -1.3413e-5,mwzsl6 =,6.3573e-8 mwzsl7 -,0,mwz3l3 -, 1.3283»-007,plpf =■ 0.7891,p4 - -0.3290 pS = -0.0302,b7 « -3.4599e+005,b8 = 1.4942e+006,blS » 0 ssl - 2.3698e-014,sslil3c7 -7.6280e-015,12 = -2.3740e-005 p6 - 0.0260,p3 - 0.2701,p2 »0.3038,al7 - 1.0,a45 - 0.0070 CO - -16.7625,10 —1.7795.X0 - 6.3646,p0 - 0.1518,M0 - -1.5714 ylO « O.O047,m0 —1.5714,gO --0.3683,f0 » -5.5844e+008,g2 - 1.4142
Figure UK 11
500
Differential Models and Neutral Systems cobb douglas on United Kingdom-log(income)T
31
1965
I
1
1
1
I
1
1
1
1970 1975 1980 1985 1990 validity of the use of cobb douglass alpha and beta
1
1995
solid- real data (+) model
Figure UK 12
UKCOBB.M % Here we experiment with the values given by cobb « Douglas(alpha«0.9 and beta-0.1) to obtain the best 4 possible constant term A in the equation: y »A*K*a*L A (1-a) . i In the final analysis for United Kingdom data log(A)"1.6. YEAR -[1966 1967 196B 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992]'; L -[100.4 98.4 97.7 97.6 96.9 97.4 97.4 99.8 100.4 100.0 99.3 97.4 100.3 10 1.9 101.2 96.3 94.2 92.7 93.5 94.3 94.2 95.1 98.1 99.9 100.0 96.8 94.1]'; y =[38.37 40.4 43.81 47.15 51.77 57.75 64.66 74.26 83.86 105.85 125.25 148 .98 168.53 198.22 231.77 254.93 279.04 304.46 325.85 357.34 384.84 423.38 471 .43 515.96 551.12 573.65 595.26]'; K -[7.06 7.71 8.51 8.83 9.74 10.89 11.94 14.73 17.5 21.04 24.5 27.07 31.06 36.93 41.56 41.3 44.B2 48.62 55.18 60.72 65.03 75.02 91.12 104.58 106.78 96. S3 92.59]'; logic -log(K)*(.9) ; log! -log ID * 1.1); logy -log(y); llpy- 1.6 + logL ♦■ logic; subplot (2,1,1), plot (YEAR, logy, YEAR, llpy, '<•*) title('cobb douglas on United Kingdom—log(income)—'); ylabel('llpy');
xlabeK 'validity at the use of cobb douglass alpha and beta');
Appendix
501 japan-PRIVATE CONSUMPTION-
x10
1965
1970
1975
1970
1975
x10
1965
1980 1985 YEAR japan-EXPORTS-
1980 YEAR
1985
1990
1995
1990 1995 solid-real data (+) model
Figure JA1 japan-INVESTMENT-
X10
1965 x10 4
1970
1980 1985 YEAR japan-GOVERNMENT CONSUMPTION1975
1980 year
1985
1990
1995
1990
1995
solid- real data (+) model
Figure JA2
Differential Models and Neutral Systems
502
japan-MONEY DEMANO-
x10
1995
1990 1995 sold-real data (♦) model
Figure JA3 japan-CONSUMER PRICES-
1965 x10
1970
1975
1980 YEAR japan-INCOME-
1985
1990
1995
1990 1995 tow- real data (+) model
Figure JA4
Appendix
503 ."10"
i M i l MCOMCAIOMUM'IIOH-
Figure JA 5 .«»
1ST9
x10
mo
IOOB
IflM
1990
YEAR UpM-MQOMEEXPORT-
1978
1990 YEAR
1988
1990 1996 •olrf-Mfdtfa <♦)«
Figure JA6
Differential Models and Neutral Systems
504
japen-BALANCE OF PAYMENT-
-300 1965
1970
1975
1980 YEAR
1990
1985
1995
japan-4og(lncome)-
1985
1980 YEAR
1990 •olid-real data (♦) model
1995
Figure JA7 japan-Increase in stock4000 3000 O2000 1000
1965
1970
1975
1986
1980 YEAR
1990
1995
•olid-real data (+) model
eobb douglat on Jao»n-(oo(tncome>-
14
'
'
13
—
I—
I
^ r r t T T ^ ^ ^ i > ^ ^ »
1965
.
^i-**"^
*« 11
1
1
I
i
.
_. 1
1970 1975 1980 1985 1990 validity of the use of cobb douglasa alpha and beta solid- real data (♦) modal
Figure JA8
1995
Appendix
505 Australla-PRIVATE C O N S U M P T I O N -
1995
1990 1995 solid-- real data (+) model
Figure AU1 Australla-INVESTMENT-
19S5
1970
1975
1980 YEAR
1985
1990
1995
1990
1995
Australla-GOVERNMENT C O N S U M P T I O N -
Figure AU2
506
Differential Models and Neutral Systems Australia -CONSUMER PRICES -
1985
1975
1880 YEAR
1985
1990
1995
1990
1995
Autiralii-INCOME"
1965
1975
I960 YEAR
198S
•olid - real data (*) modal
Figure AU3 Aualrata -MONEY DEMANO-
1985
1970
1975
1980 YEAR
1985
1990
1995
1990
1995
Auilralla -AGGREGATE DEMAN0600
400
200 '
1965
1970
1975
1980 YEAR
Figure AU4
1985
solid- real data (♦) modal
507
Appendix Anmh-MCOMEffiONtUUPTION--
Figure AU5 kmtlm
WOOMBWOVBtWMBKT-
1IH
1M0 1tW MM-mldali Mnedd
Figure AU6
Differential Models and Neutral Systems
508
Amlralla-BALANCE OF PAYMENT-.
tS05
1870
1890 1995 • o l M - real data («■) modal
1880 YEAR
1878
Figure AU7 Au»tialla-lncr»a»» In d o c k -
1S65
1995 Auitrallin growth rale
E400 >»
a
+
| 300 c
+
1 ♦
+
3
+
*200|-
aT □
+
+
+
O100
. * ** + $ £ 5 * ® * J o O o o o o o o o o o
0 1S6S
1
1970
1
1975
1_
1980 YEAR
Figure AU8
Q o 0 n ^ 1 >■ ' " "
1985
t
II
1990 •olid- rial data (♦) modol
1995
Index A Absolutely continuous, 178,193,198 Admissible control, 50, 51, 53, 56 Adjoint equation, 128, 155 Aggregate demand, 2 distance, 16 economic, 2 supply, 2 Agreements, 1, 3 Amoroso Luigi, 95 Analog computer, 316, 344 Angell, 199 Angell and Kirsh, 159,200 Animal spirit, 17 Applied voltage, 353 Approximately controllable, 342, 343 Asymptotically stable, 68, 72 Atomicity, 248 Attainable set, 180, 192-194, 196, 214, 215,226,229,236 Australia, 5, 96,97, 102, 105, 315, 316 Austria, 315, 316 B Balachandra and Dauer, 254,283 Balance of payment, 25, 26, 39, 55, 185 Banach space, 220, 226, 232 Bang-bang, 170 Bang-off-bang, 126 Bartosiewicz, 230 Between trading partners, 1,3,9 Birmingham, viii Bonds, 12 Bounded solution, 303, 309 Bounded linear transformation, 127
Boundary, 140, 141, 145-150, 154 Brazil, 363, 364 Broad, Molly, 365 Bump and Kazemi, 107 Burton, T. A., 4, 14 C Canada, 1,4,5, 11,89,90,96,98 Capital stock, 1, 3-5, 9, 10, 18, 19, 27, 32, 39, 297 Capital stock, K, 3 Central Bank, 162, 164 Cesari's property Q, 195, 197 Characteristic equation, 302, 303 Chukwu, vii-x, 7, 14, 31, 32,41, 67, 85, 87,90,99,159,161 Clinton-Gore, 66 Closure, 180, 194, 195 Cobb-Douglas, 31 Collapse, 297 Compact, 76 Compactness, 127, 152 Complete Euclidean space capture, 47 Constrained controllable, 61 Consumption, 1, 3, 10, 11 Consumption, C, 3,7,9 Control instruments, 3, 10 autonomous investment, 1, 3, 10, 11 autonomous net export, X 0 , 3, 10 autonomous real money demand M 0 = j , 3,10 I 0 , 3,9,10 labor productivity, n, 3, 10 wage rate, w, 3, 10 Controllable with constraints, 71 Controllability, 1, 4, 5, 10-12, 14, 122 509
510
Differential Models and Neutral Systems
matrix, 45 theory, 4, Convex, 135, 145, 146, 151, 153,291 Convexity, 146 Corduneanu, C. C, 4 Covariance matrix, 12 Cruz and Hale, 181, 185 Cumulative balance of payment, E, 3, 9 Current, 316, 320, 321, 337, 347, 348, 351 D Dauer, 85 Delay, 75, 76, 78, 322 system, 352 Depression, 239 Deta macroeconomic, 8, 9, 12, 13 Determining equations, 128 Deterministic rational expectations, 9 Dieudonne, 221 Differential game, 9 Differential neutral systems, 95 Dirac functions, 124, 152 Distance between, 1,3,9 Domain of null controllability, 53 Duality theorem of complete capture, 47 Dynamics, 1, 2, 5, 7, 9, 12, 13, 66, 75, 78, 81,89,90,96 E Econometric meter, 315, 317, 320, 321, 338, 343, 355, 357, 362,363 Economic dynamic model, 315 indicators, 61 model, 320, 364, 365 reform, ix solidarity, 164, 166 stimulus, 115, 162-164 variables, 89, 243, 315, 321 EEC, 167 Eisner, Robert, 10 Stiglitz, Joseph, 10 Employment, x, 1, 3-5, 10, 11, 13, 17, 27, 31,39
Employment, L, 3, 9, 16,307 Employment oscillation, 307, 311 Estenson, P. S., 4 Euclidean controllable, 42,45,46, 51,66, 71,85, 119,256,260,293, 295 Europe, 363, 364 European 3, 11, 17, 40 Exchange rate, e, 3,9, 16, 24, 27 F F-controllable, 340, 341 Fair, Ray C, 3, 7, 13 Feedback control, 67 Filippov's Lemma, 51 Finite escape time, 179, 190 Foreign credit, 1 Foreign goods, 306, 312 Full-employment, 3 Function space controllable, 196 Fundamental matrix, 112, 115, 121,160, 207 G Gandolfo and Petit, 167, 168 Gateaux derivative, 220,232 Gateaux differential, 220, 223, 232 Generalized inverse, 65 Germany, 5 Global stability, 68 Government controls exchange rate, e, 11, 16, 19, 24, 27 money supply, Ml, 9 preferential trade, 1, 3,9, 11, 24, 40 tariff, T, 9, 16 taxes, T, 2,3,9 wage rate, w, 3, 10 transportation, 1, 3, 9 Government intervention, 1, 3, 5, 9,42, 112 Grace, Andrew, 4 Gronwall, 186 Gross domestic product, I, 9 national product, 1
Index Growth, viii-x, 1 H Hajek, 75 Haley, 59 Hereditary systems, 1, 4 Hilbert space, 247
511 K Kalecki, 7 Keynes, 1, 3 Keynes, John Maynard, 2 Keynesian dynamics, 22 KirchhofPs voltage law, 348 L Least square optimization, 167 Lenhart Ljung, 97 Linear regression, 31 Liquidity function, 24 Ljung, L., 4 Lossless transmission line, 316, 321, 338, 343 Lucas, Robert E., 56 Lyapunov exponent, 303
I Identification toolbox, 97 Implicit function theorem, 186 Impressed voltage, 316, 333, 351, 354 Impulse function, 149 Impulsive control, 150 Income export intercept, 42 India, viii, 1, 5, 59, 61, 89, 90 Individualistic, 239 Infinite past, 1 Inflation, viii, x, 5, 10, 11, 12, 315, 362364 Innovation, 311 Initial endowment, 11 Institute, 317, 361,365 Integrable, 122, 146 Integral coefficients, 39 Interconnected system, 234, 238, 239 Internal power, 280 International pressures, viii reserve, 167 statistic yearbook, 61 Interest equalization tax, 1 Interest rate, 1, 4,11, 12, 23, 39 Interest rate, R, 1, 3, 9, 17, 162 Investment, ix, 1-3, 11, 17, 18,40, 90, 111, 159, 161 energy, 114, 115 Investment (government), 12,45 Investment, I, 2, 3, 7, 9, 59, 244 Italian model, 343
M Macroeconomics, 255 Manitius, 127, 155 Market principle, 161 MATLAB, 3, 12,31,66,316,317 MATLAB program, 61, 99, 316, 344, 353 MATLAB technique, 137 Matrix form, 33 Maximization of profit, 32, 59 Maximum principle, 155 Measurable, 51,52,57 Melvin, 181 Metanormal, 133, 146, 147 Metanormality, 125, 133, 134, 148 Minimizing vector, 120, 150 Minimum-effort problem, 158 Minimum investment control, 113, 118 Money demand and supply, 245 Money supply, 1-3, 9, 10, 40, 56 Money supply, M, 3, 15, 16 Multipliers, 206
J Japan, 5,97, 104 Jobless rate, 12
N National income, 23, 27, 39, 66 Natural rate doctrine, 10 Ndu, 97
512 Neck and Karbuz, 12 Net export, 1, 11 Net export, X, 2, 7, 9, 15-17 Neustadt, 141 Neutral systems, 112,121 Newton, 59 New York Times, 66 Nonlinear model, 316 Nonlinear neutral system, 177 Nonlinear variation of parameter, 52 Non-oscillatory, 301, 303 Nonsingular, 45 Normal, 116, 123,153 Normality, 127,128,130,133,138, 141 Null controllability, 11, 136, 293 O Open mapping theorem, 186,226,227 Optimal control, 111, 112, 118, 120, 121, 123-127, 135-139, 141-143, 145, 146, 149, 151, 152, 179, 180, 192,195,196, 199 Optimal control strategies, 107 Optimal control problem, 109 Oscillation, 301, 306, 307, 309-311 Oscillatory, 301, 302, 305, 309 Oscillatory behavior, 162 P Paradise, 177,178 Perkins, 167 Perspective, 361 Peterson, Wallace C , 4 Piecewise analytic, 119 Pontryagin difference of sets, 3, 76, 78 Pope John Paul II, vii Preferential trade, 1, 3, 9, 11,40 President Robert Mugabe, viii Prices, 1-4,10,17,23,39 Prices, p, 2, 3, 9,66, 90 Price intercept, 1, 40 Prime Minister Inder Kumar Gujral, viii Private autonomous, 1 Private controls
Differential Models and Neutral Systems autonomous consumption, C 0 , 3, 10 autonomous investment, 1, 12 autonomous net export, XQ, 10 autonomous real, 3, 10 money demand, 1, 3,10 intercept income consumption, 3, 10 intercept income export, yjo, 10 intercept prices, 10 labor productivity, n, 3, 10 money demand, M 0 = j , 3 Private fuel, 164 Productivity, 1, 11 Proper, 240, 246 Propulsive system, 170 Public consumption, 306 Pursuer, 1, 40, 47 Pursuit game, 255 Q Quadratic cost, 179 R Rank, 66,67 Rational expectation, 3, 8, 9, 315, 316, 362-364 Reachable set, 53,127,136,141, 142, 146, 149 Real money demand, 1,315, 327, 342, 348 Redmond and Silverberg, 163, 170 Reform, 297, 315,317 Regression, 4, 9, 66 Resolvent equation, 68 Rigid body maneuver, 169 Russia, 315, 363 S Salamon, 87, 336, 341, 363 Samuelson, Paul, 307 Say's law, 2 Scarcity, 61 Separation theorem, 135 Snap decision rule, 55, 57 Sobolev space, 177, 178
Index Social exclusion, ix Soft landing, 61 Solidarity, viii, 243 "Solidarity function", 49, 52, 244, 246, 247, 282, 293, 295- 297 South Korea, 364 Speed of adjustment, 20 Speed of response, 305, 306 Stable, 66-68, 70-72 Standard deviation, 104 State equilibrium, 3 Stochastic, 12 Stocks, 12 Stroboscopic, 281 Subsystem, 239, 247, 249-254 Supply and demand, 342, 362, Surjectivity, 213,216 Sustained economic development, viii T Tariffs, 1,3, 11, 19,30 Tariffs, x, 16 Taxes, 3, 11, 17, 27 Taylor, 59 Taxation, 1, 56 Terminal constraints, 109 Termination time, 76 Thailand, 363, 364 The Globe and Mail, 90 Time delay, 57, 75 Time-optimal, 120, 121, 156, 179 Time-optimal control, 118, 121, 127, 130, 136-142, 145, 177, 180, 192, 196,213 Tinbergen, 255 Trading partners, 1, 3,9, 16 Transportation, 1, 3, 9, 16, 17, 27
513 U U. K., 5 U.S. A., 5, 11,89,99 Unbounded growth, 320 UN National accounts, 96, 98, 100, 101, 103 Unemployment, viii, 2, 3, 10-12 Uniform boundedness, 289 Uniformly asymptotically stable, 68, 122, Uniformly nonatomic, 180, 248, 253 Uniformly stable, 181, 190, 191 Unique, 123, 124, 126 Unit ball, 63 United Nations, viii, x, 307 United Nations Organization, vii United States National Science Foundation G-8 summit communique, viii Universal opinion, vii Universal principles, 240 Unsymmetric Fubini theorem, 284 Upper semicontinuous with respect to inclusion, 192, 193, 196 V Value of capital stock, K, 3, 9, 55, 66, 90 Variation of parameter, 107, 121, 126 Variational control system, 225, 226 Vice President Al Gore, viii W Wage rates, 1, 11,40 Weakly G-differentiable, 220, 226 World economy, 362, 364
