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2
$ ( / I , I > >
as t 0. Finally, Jb,&P, 1))= ( P , o h 4 ( p , 1 ) ) ) = < P , (vi) For any x in R+” satisfying the budget constraint --f
4(p, 1 )> =I.
( 4 , -y> 5 J ( q ,4,
we let .r(t) = t s . Then, sinceJ(y, a ) > 0, we find (y, ~ ( t )<) J(q, n) ( I > t which rules out x ( t ) 2 a. Hence
>= 0),
o(q, a) 2 a > x ( t ) (1 > t 2 01,
which becomes by continuity 4 % a) 2 a 2 x
as t + I . (vii) Similarly to the argument for (iii), we find by definition (9,4Y (9 +
&?
+ Ac73 0 ) ) 2 < 4 , 4 Y >a)), 4 % a ) ) 2 ( 4 + Aq,CT(q + Aq, 0 ) ) .
Whence ( 4 , A o ) 2 0 2 ( q + Aq, ACT).
Hence, by subtracting the leftniost term from the rightmost one, we arrive at ( A q , ACT)5 0, Q.E.D. From Lemma 18.I , the economic significance of o ( q , a ) is now clear. For an arbitrarily fixed a, J(q, a ) is the income level that guarantees under the price vector q the same preference level as that of the commodity bundle a, the corresponding (unique) optimum commodity bundle being n(q, a). Using o(q, a ) and J ( y , a), we can decompose the change of demand A4 = 4(P
+ AP>I ) - 4 ( P >I ) ,
302
VI. GROSS SUBSTITUTABILITY
responding t o a change Ap in price to two terms in the following way. Performing the decomposition
A4
= 4(P
+ AP, I ) - 4(P, I )
+ A/]> I ) - 5(P + AP, 4(P>1 ) ) + 4 P + AP>4(P, 1))- 4 ( P >I ) ?
= 4(P
and noting
4 ( P + AP, I ) = 4(P + 4 1 , J(P> 4@,I ) ) ) , + AP, 4(/& 1 ) ) = 4@ + A/& J ( P + AP>4(P> I))), +(/& 1 ) = 4 / 1 7 4 h 111,
4 P
ensured by Lemma 18.1, we finally find
A4 = 4 ( P + AP? 4 4 =
-(P7
1)
+ A/& J ( Y >4(P>1 ) ) )- 4 ( P + AP>J(P + AP, 4(P, 1 ) ) ) + 0+ AP? 4 h 1 ) )- 4 P >4(P, 1)).
(14)
4 P + AP, 4(LJ+ AP, J ( P , 4(/& 1 ) ) )- 4(P + AP? J(P + AP, 4 h 1 ) ) ) and 4(p,I )) - o(p, 4(p, I ) ) are called the inconie eflect term and the suhstitutioii
ejjGcr term, respectively. Equation (14) is the Slutskj. equation in the finite increment version. A change Ap in price affects the behavior of a consumer in two ways. It entails a change of price ratios that induces substitution among goods i n the optimum men~iof consumption, and it causes a change i n his real income, or his purchasing power, although the nominal level of income remains unchanged. Therefore, the substitution effect represents the shift of the optimum menu of consumption within the indifference class to which the original + ( / I , I ) belongs. J ( p Ap, $(p, I)) is the corresponding (virtual) level of nominal income that keeps the consumer within the same indifference class as before the price change, even in the new price situation p + Ap. Thus, the income effect represents the shift of the optimum menu of consumption, in the price situation 1) Ap, responding to the change of income level, from the virtual J ( p Ap, (p(p, I ) ) t o the actual J ( y , 4(p, I ) ) .
+
+
+
THEOREM 18.1 (Slutsky, 19 15). Assume fhar tlie deemand function $ ( p , I ) ( p > 0, I > 0 ) //as partial c/erii.atii.es (!4;/?pj, (i,j = I , . . . , t i ) atid continuous partial dcrii.atii.e.s &bi/?I ( i = I , . . . , n). Then, we kaiv the Slutsky equutio~iin tlie d(f~erentia1i-evsioii
4 18.
GROSS SUBSTITUTABILITY AND EXTREMIZATION
303
where
A j p = ( 0 , . . . , O , A p j , 0 , . . . ,O)'
By the mean-value theorem in calculus, and Lemma lS.l(iii), the income effect term can be rewritten as
where
#i,
is by assumption continuous, whereas aj is also continuous by Lemma
18. I(ii). Hence, in view of (IS), we have
because of Lemma lS.l(v). On the other hand, we have by definition
304
V1. GROSS SUBSTlTUTABlLlTY
Therefore the limit o i j of substitution effect term
as Apj 3 0 also exists and equals
This proves Eq. ( 1 5). In the differential version of the Slutsky equation (15), as in its finite increment version, we also call - $ j d q 5 J d l and crij the income effect term and the substitution effect term. We observe, in the light of Lemma lS.l(iii), that if oi(q,a ) have partial derivatives (i,j = I,
. . . ,n )
that are continuous with respect to q for each fixed a, we find
( i , j = 1 ,..., n ) .
In such a case, we have the symmetry of the substitution terms in Theorem 18.1 0 i.J.
= 0.J 1
=
'5
" ' 3
">
from (17). On the other hand, we can say nothing definite about the signs of uij except for crii 5 0 ( i = I , . . . , I?), which is immediately implied by Lemma lS.l(vii) for q = y , Aq = A i p and (17). Based on the symmetry of the substitution terms, Hicks (1939) put forward definition of substitutability and complementarity as follows. The ith and j t h goods are substitutes or are complementary according to o i j > 0 or oij < 0. As a matter of fact, the intervention of the income effect makes the general determination of the sign of ?4J?pjdifficult, even if the sign of oij is known. In this connection, the concept of gross substitutes, which was advanced by Hicks (1939) and Mosak (1944), is of interest. The ith good is termed a gross substitute for the jth good if ?4J?pj > 0. Here the word gross means the actual total effect, including the income effect, as opposed to the (net) substitution effect along an indifference surface (class).
6 18. GROSS SUBSTITUTABILITY AND EXTREMIZATION
305
It should be noted that substitutability and complementarity, as well as gross substitutability, are concepts whose definitions depend on the individual consumer and the price-income situation ( y , I ) . However, we can also define counterparts of these three concepts for the aggregate demand function by means of the signs of the sum of the individual substitution effects and the sum of the individual total effects, respectively. I n particular, gross substitutability is conceivable for the aggregate excess demand function, even if supplies of goods by production intervene, because this concept does not premise a decomposition to the income effect and the substitution effect of the change of the aggregate excess demand but is based on the total change itself. Moreover, this concept may be intensified to its overall prevalence throughout a range of price-income situations. Usually the individual incomes enter an aggregate excess demand function as functions of prices rather than as independent variables. More recently, a weakened version of gross substitutability, called weak gross substitutability, requiring only d4JdyJ 2 0 (i # j ) has received much attention by several authors, particularly Hahn (1958), Negishi (1958, 1962), Arrow er al. (1959), Nikaido (1959c, 1964c), Arrow and Hurwicz (1960, 1962), McKenzie (1960a, b), Morishima (1960a, 1964), Uzawa (1961) and Kuga, 1964, 1965). Their results seem to suggest that it might be better to modify the terminology in such a way that we refer to weak gross substitutability simply as gross substitutability and the original version of gross substitutability as strong gross substitutability. With the foregoing as background, we now have DEFINITION 18.1 (Gross Substitutability in the Finite Increment Version). Giiwi ti excess deematid functions E l ( p ) ( i = I , . . . , ti)
(19)
ti-ciiiiieiisional price rector p = ( p , ) , dejned oii a conitiioti doimiti P, that gross substitutabi/itji prerails iii P if for any y , q E P satisfying p 2 q we Iiaw of the
1i.e SUJ~
E L p ) 2 E,(q) (i # N P> 4 ) = { iI PI > q , } ) .
(20)
DEFINITION 18.2 (Gross Substitutability in the Differential Version). (19) hare partial dericatires dE,/c?p, (i,j = 1, . . . , t i ) in P, tie SUJ~ that gross substitutability precails iti P if tile haue
If E,(p) in
everywhere in P.
306
VI. GROSS SUBSTITUTABILITY
DEFINITION 18.3 (Generalized Gross Substitutability). If there exist partial tieo.iiwtir.es i E l / 2 p , ( i ,j = I , . . . , n ) in P as in Dejnition 18.2, tt.e say that yeneralizetl gross suhstitutabilitj>prevails i i i P i j itse hai3e (22) ererj'ii l7rre in
P. Remark 1. Gross substitutability in the finite increment version is nothing but an alternative nomenclature of weak monotonicity (6,) in Chapter 111, $10.1. But the former seenis to fit the present economic background better.
Remark 2. I t is easy to verify that in the case of the existence of the partial derivatives ?El,'?/), , Definitions 18.1 and 18.2 are equivalent. Verification will be left to the reader. Remark 3. Generalized gross substitutability includes as special cases gross siibstitutability i n the differential version as well as sh e~~,-sj,i?uiietricifj., which means
Nikaido (I964c) seems to be the only source dealing with generalized gross substitutability. It is difficult, however, to conceive a workable finite increment version of this concept. Remark 4. I n view of Remarks I and 2, we may reasonably carry over the concept of indecomposability (Chapter I I I , Definition 10.2) to gross substitutes systems. Explicitly, a gross substitutes system (a system satisfying Definition 18. I ) is said to be ii~tI~~c~oiiipo.vcihle if for any 17, 4 E P satisfying p 2 (1 but / I 3 q, at least one inequality of (20) is a strict inequality for some i. When strict inequality holds for all i i n (20), the system is called a strong gross substitutes system. For differentiable cases, a gross substitutes system (a system satisfying Definition 18.2) and a generalized gross substitutes system (a system satisfying Definition 18.3) are said to be indecomposable if the matrices (?E,/?pj)and (?Ei/pj + ? E j / ? p t are ) each everywhere indecomposable i n P. The corresponding strong versions of gross substitutability are defined as the positivity of all off-diagonal entries of the above matrices.
EXAMPLE 1. A simple nontrivial gross substitutes system satisfying(]), >= O), ( I l l ) i n $18.2, and continuous differentiability can be given on the set of a11 positive price vectors p = (11;) by
( 1 1 ) ( 1 > nz
I
1
PiJ=l
where 1 > 117 2 0 and
n
"
Ei(p) = -
N,,
t il .J . p Jl . + " l-
1
LljiPi))l
( j = I , .. . , n),
j = 1
( i , j = I , . . . , n) are nonnegative constants.
(24)
$1 8. GROSS
307
SUBSTITUTABILITY A N D EXTREM!ZATION
If there are / consumers s = I , . . . , indicator
I, each of whom has
a preference
on the set of all nonnegative commodity bundles and a nonnegative initial holding vector a" = (a,"), the (aggregate) excess demand functions in the corresponding pure exchange situation are of the above type, with m = 0 and
The reader may readily verify these facts.
EXAMPLE 2. If n? = I , we can have nontrivial generalized gross substitutes systems satisfying (I), (11) (n7 = I ) , (111) in $18.2, and continuous differentiability on P = R+"\{O). The simplest example is a skew-symmetric system with
c aijpj n
E;(p) =
( i = 1, . . . , 17);
j= 1
where ( a j j )is a nonzero skew-symmetric matrix. An equilibrium price vector (see Definition 18.4, below) of this system is nothing but a synonym for an optimal mixed strategy of the two-person game having ( a i j )as its payoff matrix. Every game having a general payoff matrix reduces to a game with a skew-symmetric payoff matrix [cf. Gale et al. (1 950)]. Gross substitutes systems have various interesting properties of economic and mathematical significance, which are not shared by general competitive systems such as those considered in Chapter V. These properties have been brought to light by the results of At-row, Hurwicz. and the other authors mentioned above. It is difficult to present all of them in full detail here, however, because of limited space. We shall therefore confine ourselves to describing two lines of results. One is the establishment of both the existence of a competitive equilibrium and the prevalence of Samuelson's weak axiom of revealed preference between an equilibrium price vector and a disequilibrium price vector, by means of further intensified utilization of extremization methods such as those given in Nikaido (1964~).The other is the presentation of a few representative results in 519 from among the numerous results on global stability in gross substitutes systems by several authors mentioned i n the paragraph preceding Definition 18.I . 918.2. Extremization and the Existence of Equilibrium
In this section, as well as in all the following ones of the present chapter, we are concerned with a system where there are given 17 single-valued numerical functions E j ( p ) (i = I , . . . , 17) of an n-dimensional argument vector
308
V1. GROSS SUBSTITUTABILITY
p = (I,<), defined in a common domain P. The function E,(p) indicates the amount of the excess demand for the ith good, while the ith component p L of the argument vector p stands for the price of the ith good. It is understood that the behavior of these functions represents the state of a n economy involving 17 items of goods i = I , . . . , 17, so long as the aggregate excess demands for them are concerned. We list below three basic assumptions on E , ( p ) that will either be more completely specified o r will be reinforced by additional assumptions later, depending on the proposition in question. ( I ) The common domain P of E l @ ) (i = 1, . . . , n ) is a nonvoid subset of the nonnegative orthant R+" of R" such that (1.3)
(I./?) (I.?) (11) 111,
P does not contain the origin 0; 1.1, E P for a n y 1) E P and any positive number i. > 0: P is open in R," in the relative topology of K,". The excess demand functions are homogeneous of a common order
1 2 171 2 0. That is, Ei(ip) = i,"'E,(p) for p (111)
E
P , 1, > 0 (i
=
I,
. . . , 17).
The Walras law in the narrow sense holds. That is. n
2 piE,(p)= 0
everywhere in
P.
i= I
Remark 1. The simplest example of P in assumption (1) is the set of all semipositive vectors. The second simplest example may be the set of all positive vectors. In general, howeker, P can be of more complicated structure than is usually premised. P need not be convex and may even have holes in its intersection with the normalizing hyperplane p , = 1 . As will be shown later, if the excess demand functions are continuous as well a s homogeneous of order 0, there is no gross substitutes system in the whole set of all semipositive price vectors other than the trivial one consisting of the identically vanishing excess demand functions. On the other hand, under gross substitutability or generalized gross substitutability, we can obtain interesting results even for P of more general structure. These facts motivate a drabtic relaxation of the conventional restriction on the structure of P in ( I ) . Remark 2. I n traditional economic situations, of course we have i n assumption (11) (see Chapter V, $16.4).
iu =
0
Remark 3. Proposition ( 1 1 1 ) is one of the most important features of a Walrasian competitive economy, as was explained i n Chapter V. As a counterpart of Definition 16.6 in Chapter V i n the present setting of the problem, we have
$1 8. GROSS SUBSTITUTABILITY
A N D EXTREMIZATION
309
DEFINITION 18.4. A price rector 8 = ( p i ) is an equilibrium price uertor of a system of’ excess demand functions E i ( p ) on P if j? E P a i d
E , ( j ) 5 0 (i = 1, . . . , M), E , ( j ) = 0 provided p i > 0. It should be noted that (2) is implied by ( I ) on the basic assumption (111). Recall Theorem 16.6 in Chapter V, which proves, by means of Kakutani’s fixed-point theorem (a generalized version of that of Brouwer), the sufficiency of certain conditions of broad applicability for the existence of an equilibrium, and which is equivalent to Brouwer’s fixed-point theorem by virtue of Chapter V, Theorem 16.7. There seems t o be little need to reconsider the existence of equilibrium here. But the situation is a bit different. Note that though the mere imposition of the Walras law and the closedness (continuity in a general sense) o n excess demand functions in Chapter V, Theorem 16.6 is indeed hardly restrictive, the theorem does restrict the domain of thc excess demand functions t o an extremely special set, namely, the whole standard simplex, or-what amounts to the same thing under the homogeneity assumption-the set of all semipositive price vectors As was already noted in Remark 1 and will be explicitly shown later, there is no other gross substitutes system of continuous excess demand functions homogeneous of order 0 apart from the trivial system. Nontrivial gross substitutes systems can be conceived only on its proper subsets. This necessitates a reconsideration of the existence of equilibrium i n situations in which fixed-point methods of proof such as that for Chapter V, Theorem 16.6 w e not applicable because of the general structure of the domain of the excess demand functions. I n this connection, it is of interest that gross substitutability enables us to prove the existence of equilibrium in a domain of general structure as formulated in the basic assumption ( I ) . Results along this line, which are due to Nikaido ( 1 9 6 4 ~ )and Kuga (1965) will be given in more general forms in this section. Nikaido ( 1 964c) dealt with generalized gross substitutes systems. Later, Kuga (1965) gave an alternative proof for gross substitutes systems under 0th order homogeneity, relaxing the differentiability assumption in Nikaido ( 1 9 6 4 ~ )to continuity. Though Kuga’s result is interesting, it premises an extremely stringent boundary condition that is vital to his method of proof but that is invalidated even in some familiar traditional pure exchange situations (see Remark 6, below). Therefore we shall treat all interesting cases, including a generalization of his result, by the extremization methods of Nikaido ( 1 9 6 4 ~ ) .For most cases, the principal extremand will be the sum of squared positive excess demands, namely,
c O,(P)*, I,
@(P) =
(3)
, = I
O , ( p ) = max(E,(p), 0) ( I = I , . . . , 17).
(4)
310
VI. GROSS SUBSTITUTABILITY
Clearly we have, by definition, @(p)>= 0. Therefore, if I", is an equilibriuni price vector, @(@) = 0, and @ ( p )takes on a minimum 0 at p. It is also obvious that satisfies ( I ) if and only if @(p) = 0. Our method of proof is to see, by taking advantage of gross substitutability, that the extremum of @(p) (Theorems 18.2, 18.4, and 18.5) or an alternative extremand (Theorem 18.3), if any, is zero. As a matter of course, in proving the existence of an equilibrium in P, we need a sort of well-behavedness of the excess demand functions in P , such as their continuity or differentiability. In addition to this requirement, we need a further requirement for their behavior. We realize that P is a nonvoid open subset of R+"\(O), and, in view of the connectedness of R+"\(O), that P is not closed in R+"\{O) unless P = R+"\{O). Whence P has in general boundary points in R+"\{O}, which are not included in P, and on whose vicinities the behavior of the excess demand functions are further required to be nonviolent. These sorts of well-behavedness, which may be referred t o as boundary conditions, will be given below. In what follows, we mean by a boundary point of P that in the relative topology of R+"\{O],so that it cannot be the origin, but is a semipositive vector. Boundary Condition A . For atiy boundary point p of P and any sequence { p ' } in P that coniwges to p , there is an element i j in P , dependent possibly ON p atid { p " ) , ~ c that h p, = p, and
C:=
lim sup @(p") >= @(p). Y
+
(5)
+a
Boundary Condition B. For any boundary point p of P and afiy sequence { p ' ) in P which coiiiwges to p , 1t'e hare lim sup E,(p") =
+ co
(,for at least one
i).
v - + r
(6)
Boundary Condition C. In the case of a special P none of whose boundary points hare all con7potieiits positire, ,for any boundary point p , let K = {iI p i = 0). Then, ,for any sequence { p " ) in P contierging to p , we haw lim sup E i ( p v ) = +co
(,for at least one
i E K).
(7)
Y - + L
Boundary Condition D. I n the same setting of Boundary Condition C , ( 7 ) is replaced bj*
lirn sup E,(p") > 0 (,for at least one i E K ) .
(8)
Y'+T.
Remark 4. I n the above alternative boundary conditions, C is obviously a special case of B. One may readily observe that (6) amounts to the requirement that lim sup @(p")= +GO. Whence B is a special case of A. O n the
$18. GROSS SUBSTITUTABILITY AND EXTREMIZATION
31 1
other hand, C is also a special case of D. it is of vital importance to note that the i in (6), (7), and (8) may depend not only on the boundary point but also on the sequence { p " } (cf. Remark 6 , below). Remark 5. If the excess demand functions are bounded from below, as is often the case, Eq. (6) occurs only when the corresponding ith component p i of the boundary point in question is zero, and hence B is equivalent to C. In fact, if Ei (p) 2 y ( p E P, i = 1, . . . , n ) by lower boundedness, the Walras law implies pkvEk(pv)
5
-
c
1piv. lim sup Ek(p")= + co. We
i#k
pivEi(p") 5 -7
ifk
Suppose that limp" = p = ( p i ) , may assume lim E,(p") = +m, since this can be achieved by choosing some sub-sequence. > 0 for large 11,the above inequality reduces to Now, since
which becomes in the h i i t 0 5 pk 0. Hence p k = 0 if lim sup &(p") = +a, limp" = p . Tn proving the main results in $18.2, we need only Boundary Condition A for Theorems 18.2 and 18.4, whereas Theorem 18.3 premises Boundary Condition D. Before stating alternative existence theorems for price domains of general structure under gross substitutability, we note several simple facts, given in the following lemmas.
LEMMA 18.2. Let f ( x ) be a convex,fimction diflerentiable on a convex = f i ( x ) dxi . subset X of R", with the corresponding total diflerential being Then we haue
cy
Proof. We first observe that if x ( t ) is a differentiable function o f t on an interval of real numbers, with all its values in ,'A and if , f ( x ) has a total f i ( x )dxi in A', then 4 ( t ) =f(x(t)) is also differentiable in the differential interval, with the derivative
Applying this result to the given convex functionf(x) and x ( t ) = ( I - t)x + fy (1 >= t 2 0), we have
312
VI. GROSS SUBSTITUTABILITY
Clearly d ( t ) is convex on the unit interval. Thus by Theorem 3.15(iii) in Chapter 1 as applied to u = 0, v = 1, we have (9), Q.E.D.
LEMMA 18.3. The function
has the following properties (i), (ii): (i) f(x) is a convex function and has continuous partial derivatives
af
axi = 2 max(x,, 0) -
(i = 1 , .
. . , n).
(10)
(ii)
Proof. (i) It suffices to show that the function a ( t ) = (max(t, 0))2 has a continuous derivative 2 max(t, 0) in (- co, + co). First, o ( t ) as a function having a nondecreasing derivative is convex by Corollary l(i) to Theorem 3.15 in Chapter I. This implies that each (max(x,, 0))’ is convex on R”, whence the convexity off’(x) follows. Second, we find af/dx, = o’(xi) = 2 max(x,. 0). Now, since a(t) = 0 for t < 0, we have ~ ’ ( t=) 0 = 2 max(t, 0) in ( - co,0). On the other hand, a(t) = t 2 for t > 0, so that a’(t) = 2t = 2 max(t, 0) in (0, + a).At t = 0, o(t)has a left-hand derivative and a right-hand derivative, both of which are zero and therefore equal 2 max(0, 0). Hence o’(t)= 2 max(t, O), which is continuous. (ii) Since continuous differentiability implies differentiability, and continuous partial derivatives can serve as coefficients of a total differential, (ii) is implied by (i) and Lemma 18.2, Q.E.D.
We now proceed to the main results. The following theorem weakens the assumptions of Kuga (1965) in several respects, except for a slight strengthening by assumption (6) below.
THEOREM 18.2. There is an equilibrium price rector fi in P on the basic assumptions (I), (11) (in = 0), and (111) if ( a) Boundary condition A is assumed. (p) The excess demand,functions E,(p) (i = 1 . . . , t i ) are continuous in P. ( y ) Gross substitutability in the Jinite increment version (Definition 18.1) holds in P . ( 6 ) For arij. i, j = 1, . . . , n we haw
$18.
GROSS SUBSTITUTABILITY AND EXTREMIZATION
313
-
at a n y p in P, where
i
A j p = (0,. . . , 0, A p j , 0 , . . .,O)'. ProoJ
For the function @(p)in (3), let
6
= inf
@(p) over all p
E
P
satisfying
C p i= 1,
(1 1)
i= 1
and choose a sequence {p"} in P which fulfills
c piy n
=
1 (v
= 1,
2,
. . .),
(12)
i=l
lim @(p')
= 6.
"++a0
Clearly 6 2 0 by the definition of @. From ( 1 2) and the nonnegativity of p", (p'f is a sequence of the standard simplex P,, . In view of the compactness of Pn, we may therefore assume, without loss of generality, that { p " ] converges to a limit p in P,. This p may or may not belong to P . If p q! P, then p is a boundary point of P. Whence there is p , by boundary condition A, such that p E P n P,, and ( 5 ) holds. But, in the present case the left-hand side of (5) equals 6 by (13). Thus we end up with 6 2 @(p), p E P n P,, , which, combined with the definition of 6, implies @(p) = 6. On the other hand, if the limit p belongs to P , we must also have @ ( p ) = 6 from (13) and the continuity of @. We can therefore conclude that in both cases @ ( p ) actually takes on a minimum over P n P , . It should be noted that the minimum of @ ( p ) over P n P, is even the minimum over P because of the 0th order homogeneity (11) ( n 7 = 0). We have thereby shown that @(p) is minimized at some j? over all p E P. It will be seen that fi is an equilibrium price vector. To this end, it suffices to show the fulfillment of ( I ) by j?, since (2) follows from (1) by the Walras law (111). Suppose that ( I ) were violated by j?. Then, if we let K ( j ) = {i I E,(j?)> 0}, we have K ( j ) # @. Since P is open in R,", we find j? + A,p E P for m a l l Apj > 0. If we make Ap] > 0 sufficiently small, we still have by continuity
E,(j? + A j p ) > 0 (i E K ( $ ) , j = 1, . . . , n ) . Let us now consider the Walras law (Ill), which ensures
2 fii E,($ + A j p ) + A p j E j ( f i + A, P)
i= I
= 0.
(14)
314
VI. GROSS SUBSTITUTABILITY
Thus, upon subtraction, we find t l j j ( A p j ) l j j = E j ( B + A j ~ ) + C(-dij(Apj))$i i#j
( j = 1, . ., .
where dij(Apj1 =
Ei(lj) - Ei(B + Aj P> ( i , j = 1, AP j
..., n ) .
We may observe, from gross substitutability,
which, combined with assumption (d), implies the local boundedness of the left-hand side of ( 1 8), i.e., the existence of some positive constant M satisfying Iu‘ij<Apj>I
5 M (i # j )
(19)
for sufficiently small Apj > 0. Resting on (19), for each j we may choose a sequence ( A p y } of positive numbers satisfying lim A p j v = 0
(20)
V-t+CC
and ensuring the existence of n - 1 limits (211
lim d i j ( A p Y )= d i j ( i # j ) . V-+m
If we let J = { j l p j > O}, then (16), (20), and (21) together also ensure the existence of limits lim djj(Apj”)= d j j ( j E J) v+
+
(22)
rT
by virtue of the continuity of Ej at lj. Hence, substituting Ajp”, Ap: for A j p , Apj, respectively, in ( I 6), and letting 11+ + co, we find, after rearrangement,
c L l i j pi
=
Ej@) ( j E J).
(23)
ieJ
Clearly, the coefficients d i j i n (23) fulfill dij 5 0 ( i # j , i , j E J )
(24)
from (18) and (21). Next, recalling that @(jl) is the minimum of @(p) over all p the light of Lemma l8.3(ii), we have
,,
0 5 @ ( j + A j p ) - @(fi) S 2 C Oi(j
E
P , and in
+ A j p ) ( E i ( l j + A j p ) - Ei(lj))
, = I
( j = 1, . . . , n ) .
(25)
$18. GROSS SUBSTITUTABILITY AND EXTREMIZATION
315
Note that (16) and E j ( j + A j p ) > 0 ensure rij > 0 because of the nonnegativity of Ci+j( -dij(Apj))fii( j = 1, . . . , n) implied by pi >= 0 and (18). Whence we realize J
= K ( j ) # @.
(26)
With these remarks in mind and using ( 1 7) and (25), as well as the positivity of Ap:, we find n
The above relations become in the limit
from (21), (22), continuity, and the fact that Oi(fi)
A.
= max
Qi($>
- pi
over all
= 0 (i 4
J ) by (26). If we let
iEJ,
A. must be positive by (26), and Oi(j3) 5 A.ji (i E J ) , with equality holding for some i, say k . Then, for this k , we have
=
/lo,($) = A.2$,
>0
from (23), (24), and (27), which is a contradiction. This proves that K ( $ ) must be empty and $ is an equilibrium price vector, Q.E.D. As a matter o f course, assumption (6) in Theorem 18.2 by no means requires the existence o f finite-valued derivatives, but merely rules out the existence of vertical tangents to the graph of Ei(p) as a function of t h e j t h component pi o f p . It says nothing about the existence of nonvertical tangents. This assumption can be dispensed with, however, for price domains of a special and familiar structure under an alternative boundary condition, as is the case in the following theorem, which gives a direct generalization of Kuga's result (1965).
THEOREM 18.3. There is an equilibrium price vector 3 in P if we preniise the saiiie assicniptions as in Theorem 18.2, except for (6) i i o ~ 'dropped and ( a ) replaced by (a') :
(d) Boundarj>condition D holds. Proof. Let
Q ={PIP
~f',
P > 01.
316
VI. GROSS SUBSTITUTABILITY
Since P satisfies the basic assumption (I), so does Q, in which the original set of excess demand functions Ei(p) (i = 1, . . . , n ) fulfills the basic assumptions (11) ( m = 0) and (111). Taking n2 constants aij having the properties
( i , j = i ,..., n ) ,
aij>O n
1a , . = i= I
IJ
1 ( j = 1, ..., n ) ,
arbitrarily once for all, we let 11
1
F,(p) =
aij P j
j = I
, n). - 1 ( p ~ Q , i = l ...,
~
Pi
(28)
As a matter of course, (28) is a system of excess demand functions of the types (24) in $18.1 considered in Example I , which fulfills the basic assumptions (I), (11) ( m = O), ( I l l ) , continuity, and strong gross substitutability in Q. One can readily verify that this system has a very gentle boundary behavior. ( I ) Any boundary point p ( = p i ) of Q has at least one zero component because of the assumed same property of P , and (2) we have lirn Fi(q) =
+ co
(for any i E K = {i ( p i = 0 ) ) .
(29)
4-P
With these facts in mind, for any
E,“(q) = Ei(q)
E
> 0, we define the functions
+ EFi(q)
(i = 1, . . . ,n),
(30)
which obviously form a strong gross substitutes system of excess demand functions on Q satisfying the basic assumptions (I), (11) (m = 0), ( I l l ) , and continuity. Moreover, this system fulfills boundary condition C. In fact, let K = {i ( p i = O} for a boundary point p of Q. We have K # 0, as noted above. Let { 4 ” ) be any sequence converging to p . p may or may not belong to the original P . We consider two cases separately. Case (i) p E P . In this case Ei(q) are well-defined and continuous at p , so that lim Ei(q”)= E i ( p ) (i = 1, . . . , n ) . This, combined with (29), implies lim E,”(q’) = + co for any i E K. Case (ii) p $ P . In this case, we find Jim sup Ei(q”)> 0 for some i E K, say k , from boundary condition D for the original E i . Then, in view of (29) again, we have lim sup Eke(q’)= + co.We have thereby shown that boundary condition C holds on Q for system (30). The remaining part of the proof consists of the minimization of a suitable function over Q. In doing this, we have several useful minimands at hand, e.g., the sum of the squared positive excess demands (3), the sum of the positive excess demands, and the maximum of the excess demands $(q) = max E:(q)
over all i = 1 , . . . , n.
$18. GROSS SUBSTITUTABILITY A N D EXTREMIZATION
317
Although we shall work out the remaining part of the proof in terms of the latter, it should be noted that essentially the same argument can be made in terms of any of the three alternative minimands. We first observe that at every point 4 E Q we have
from the Walras law (III), which rules out the possibility of having negative excess demands for all goods. Since boundary condition C, which is a very stringent version of boundary condition A , holds in Q, we can show, in the same way as in the minimization of @ in the proof of Theorem 18.2, that @(q)is minimized at some 4 = (4,) having unit component sum over Q. We now assert that
q"4)
= 0.
T o this end, suppose that ~'((4) > 0. Then, we must have E,"(4) < 0 for some i, say k , again from the Walras law (111) and 4 > 0. If for such a fixed k , we make Aq, > 0 sufficiently small, we find
4 -'Ah 4 E Q, E,"(4 - Ah 4 ) < from the positivity of $, the openness of Q in R,", and the continuity of E,"(q). Moreover, the strong gross substitutability of E,'((q) (i = 1, . . . , n) implies
E,'(Lj - A k q ) < E,'(Q) (for all
i # k).
Whence a contradiction @(@) > q'(4 - A k 4 )follows. This proves
~ ' ( 4 )= 0.
Next, choose a sequence ( E , } of positive numbers converging to zero and a corresponding sequence (4') in Q n P,, satisfying q"u(Lj') = 0
(v
=
1, 2, . . .)
This is possible by the result just proved for any E > 0. Moreover, as usual, we may assume by virtue of the compactness of P,,that (4") converges to 3 in P,. It will be shown that fi E P and is a desired equilibrium price vector of the original E,(p). In fact, noting that (28) satisfies F,(p) 2 - 1 ( p E Q, i = 1 , , . . , n), we obtain for any i, v , 0
= q"(4')
L E:"(4")
+ CvF,(qV)2 E,(Q')
= E,(Q")
which entails E,
2 E,(4") ( i = I , . . . , n ; v
=
I , 2 , . . .).
- E,,
318
VI. GROSS SUBSTITUTABILITY
Since lim E , = 0, the above result rules out the possibility of any i having lim sup Ei(q^")> 0, so that fi cannot be a boundary point of P but must belong to P by virtue of boundary condition D. Whence the above relations become in the limit
0 = lim E,, ,,- + x
>= E,(q^)
(i
=
1, _ .., n )
because of the continuity of E,(p) in P, showing that fi is an equilibrium Q.E.D. price vector. Incidentally, fi need not belong to Q, while P 3 /I, Next we shall establish a similar result for generalized gross substitutes systems, for which n i need not be zero. T o this end, we recall a well-known result due to Euler.
LEMMA 18.4 (Euler's Theorem on Homogeneous Functions). Let,f(p) be a cliffircntiahlc, firtictioii, liotiiogetic~ousof order t n , oti P satisjjitig (I). Then, i f C ,fi(p) chi is a total ciijierentiaI qf',f, we liai-e
Proof. For each p E P , we find tp E P (0 < t < + m). Then, by the differentiability of,/; the function 4 ( t )= f ' ( t p ) is differentiable, and its derivative is given by
On the other hand, since
4(t)= t'';f(p) by
homogeneity, we find
f$'(t) = i71tn1- If ( p).
Hence the above alternative representations of d ' ( j ) yield the desired result, particularly for t = 1. Note that./(/)) has right-hand partial derivatives everywhere in P and partial derivatives at any positive p of P, all of which coincide with,fi(p), Q.E.D.
THEOREM 18.4. Tlicrc i s (I), ( I I ), at1d ( I I I ) if
aii eqirilibriittii
price rector
fi
iii
P
011
tlie basic
n.ssLll1lptioll.s
(2)
(/I)
Boirtitlur~,coiiditioti A is ass~tt~ictl. Tlic e.\-c'c.rsik~~iiiutic/,Jiciic.ti~ll.(. E i ( p )( i = I , . . . , t i ) arc rJ(fuwitiable n,itli total cii/firoitials gii.cti I ~ J -
dE, =
Eij(p) tlp,.
iii
P,
0 18.
GROSS SUBSTITUTABILITY AND EXTREMIZATION
319
Proof: We note that the differentiability of Ei implies its continuity and right-hand partial differentiability in P, with E i j being its right-hand partial derivative with respect to p i . Then, it can be seen, in a similar way as in the first part of the proof for Theorem 18.2, that @ ( p ) is minimized at j? over P n Pn. It is not readily shown, however, that @ ( j is ) also the minimum of @ ( p )over P because this time m need not be zero. Thus we will proceed somewhat differently. In what follows, it should be understood that all Api > 0 are small enough to ensure j + A j p E P. Since j E P,,the component sum of fi + A j p equals 1 + A p j , so that
320
VI. GROSS SUBSTlTUTABlLlTY
from the definitions of the respective derivatives, and lim oi(P(APj)) = Oi(lj), Ap,-
+O
litn @(p(Apj)) = @ ( l j ) Ap,-+O
from the continuity of E,(p). Therefore, (33) becomes in the limit
C Oi(lj)Eij(P)2 in@(/?) ( j
I , . . . , n).
=
r = I
Multiplying the ,jth relation of (34) by U j ( l j )
>= 0
(34)
and summing over all
j = I , . . . , n, we find It
n
i.j= 1
j = 1
We have thereby obtained (35) as a necessary condition for @(B) t o be a minimum of @ ( p ) over P n P,. At this stage of the proof, we need digress to the following lemma.
LEMMA 18.5. Under the same assuniptions as in Theorem 18.4, except that ( a ) is noit' dropped, i1.e have (i) ff 1 > n z 2 0, /et K(p) = {i 1 E,(p) > O} ,for a p E P . Then, i f K ( p ) # @, the quadratic,forni
C
Eij(P)Si<j i.jaK(p)
is negatire definite.
(ii) I/ m
=
1, the quadratic form
is tiegatii.e semidefinite. Pvooj: By the Walras law ( I I I ) , we have p i E,(p) = 0, which, upon right-hand partial differentiation with respect to p j , yields
C;=
,I
C p j E , j ( p ) + E j ( p )= 0 (,j= 1, i= I
. . . , n).
On the other hand, Euler's theorem (Lemma 18.4) gives n
1 EJi(P)Pi = rllEj(p) i= I
( j = 1, . . . , n ) .
(37)
9 18. GROSS SUBSTITUTABILITY AND EXTREMIZATION
32 1
from generalized gross substitutability. Then, adding (36) to (37) for each j we end up with
Henceforth we will consider (i) and (ii) separately. (i) Rearranging (39) in a similar way as in the proof of Theorem 18.2, and using (38) as well as the nonnegativity of p i , we find
Since 1 > M and Ej(p) > 0 for j E K ( p ) , the above relations reduce to
Pi
2 0 (i E K(P)).
(41)
By Chapter 11, Theorem 6.2, we deduce from (38), (40), and (41) that the matrix ('ij(P>)
('7
j
E
K(P))
must have all its principal minors positive. Moreover, the matrix is symmetric by construction. These results imply that the quadratic form
is positive definite. Hence
is negative definite. (ii) Since rn = I , (39) becomes n
dij(p)pi= 0 ( j = I,
..., n ) .
i= I
Using Kronecker's deltas rewrite (42) as n
(E&j
i= I
and an arbitrary positive number
+ d i j ( p ) ) p i=
with the new coefficients still satisfying
"j
(j
=
1,
. .., n),
8,
we can
(43)
322
V I . GROSS SUBSTITUTABILITY
Hence, if p j > 0 ( j = I , . . . , i?) in (43), the matrix (cSij + dij(p))has all principal minors positive by Chapter I I , Theorem 6.2, so that the matrix (dij(p)) has all principal minors nonnegative at any positive p E P because of the arbitrariness of c > 0. Since P is an open nonvoid subset of R+",P obviously contains positive vectors that f i l l out a dense subset of P . Thus, the continuity of Eij(p) at any point of P having at least one zero component ensures that the matrix (dij(p))has all principal minors nonnegative everywhere in P . Since the matrix i s symmetric, this implies the negative semidefiniteness of
2
i. j = I E i j ( p ) t i t j =
-;
i,z,dij(~)titj.
The proof of the lemma is thereby complete. We now return to the proof of Theorem 18.4. To see that j satisfying (35) must also fulfill ( I ) , let K ( b ) = { i 1 E i ( j ) > 0), as in Lemma 18.5. It will be shown that the supposition K ( j ) # 0 will lead to a contradiction. (a) The case 1 > nz 2 0. Since Ui(fi) = 0 for i $ K ( j ) , the left-hand side of (35) equals i, j
1 E
Eij(B)oi(fl>oj(P),
K(P)
a.
which must be negative by Lemma 18.5(i) if K ( j ) # On the other hand, the right-hand side is nonnegative. Whence K ( B ) = 0. (b) The case m = 1. By Lemma 18.5(ii), the left-hand side of (35) is nonpositive. On the other hand, if K ( j ) # we have the right-hand side positive in (35) becauseti? = I , @ ( $ ) > 0,cOi(b) > 0,yieldingacontradiction. Therefore K(fi) = 0. Thus, in both cases (a) and (b), j satisfies Eq. ( I ) . Equation (2) follows from ( I ) and the Walras law ( I l l ) , as already noted. Incidentally, the extremization method of proof stated here can apply mutatis titutaiidis to a general minimax theorem on zero-sum two-person games played on compact convex strategy spaces which need not be finite-dimensional [cf. Nikaido ( I 959a)l. The treatment of further generalized types of games seems to be possible, however, only by means o f fixed-point theorems and related concepts [cf Nikaido (1954b, 1959b). Sion (1958)], Q.E.D. Under differentiability, we can readily obtain an interesting result o n the maximization of cP(p).
a,
THEOREM 18.5. Uiider the saiiie assimiptiotis as iti Theoreni 18.4, ercept that ( x ) is dropped. n'e hare (i), (ii) ,for the case 1 > m 2 0: ( i ) I f ' a riia.\-iiiiiiiii of' @(pi i s attaiiietl at sortie point of P n P,, , tfie excess tkwiaiirl jirrtctioiis r.aiiish itleiiticall~iii P. (ii) There is iio s j ~ s t e r i i of e.ycess ~~et?iatid,fiitictiotis 011 P = R+"\{O) othcr thnir the tririal S ) ~ . S ~ C I I IOf icletitically i~anisliitigoties.
$1 8. GROSS SUBSTITUTABILITYA X D
EXTREMIZATION
323
Proof. (i) It will be shown that if 8 maximizes @ ( p )over all p in P n P,, we must find @(fi) = 0. Once this is seen, @ ( p )= 0 everywhere, which proves that Ei (p) 5 0 ( i = 1, . . . , / I ) . This ensures E,(p) = 0 at any positive p in P by (111). Since any p in P can be approximated arbitrarily closely by positive vectors in P , we finally have E i( p) = 0 (i = 1, . . . , n ) everywhere in P. To this end, we note that as a counterpart of (31) we find, for such a j,
0 L t ( @ ( p ( A p j ) ) - WP)) n
L
1 Oi(fi)(Ei(p(Apj))- E;(B))
(j=
1 9
.. . n ) .
i= 1
Then, using an argument similar to the one by which (34) is derived from (31), we obtain
c Oj(j?)Eij(fi)5 m @ ( j ) n
( j = I , . . .,n ) .
i= I
(44)
Since we have n
1~
~ ~ = ( 1j 7 >7 ~ j~ (~j ()i = I , . . . , n )
j = I
from the nith order homogeneity of E , and Euler’s theorem, and O;(j)E,(fi)= 0 , ( $ ) 2
( i = I , . . . ,n )
from the definition of O,(p), multiplying the,jth inequality in (44) by f i j 2 0 and summing over all j will give rise to the same result n7@(fi) on both the left-hand and the right-hand side. Whence equality holds in (44) if j j > 0. Now, let K ( B ) = {i 1 E,(fi) > 0) as before and suppose K(fi) # 0. We recall (40) and (41), together with (38), which ensure p i > 0 for i E K(p). If we apply this result to K(fi), we find that Oj(fi) > 0 ensures jj > 0 and hence equality in thejth relation of (44). From these results it follows that
Then the left-hand side of (45) is negative by Lemma l8.5(i), whereas its right-hand side is nonnegative. This is a contradiction. Therefore K(fi) = and @(fi) = 0. (ii) In this special case, we have P n P,, = P,, which is compact. Since @ ( p ) is continuous, it must take on a maximum on P,. Therefore the assertion follows from (i), completing the proof. Remark 6. Economic intuition seems to claim the plausibility of the hypothesis that the convergence of the price of a good to zero will result in
a
324
VI. GROSS SUBSTITUTABILITY
either a finite o r an infinite excess demand for the good. Explicitly, it presumes that if p converges to a boundary point p, we may. have a well-defined limit, finite or infinite, lirn E,(p). (46) P+B
B u t intuition is deceptive in this case. There is often a possibility that for two sequences { p ” ) , 14’) converging to b, we have a finite lirn E j ( p v )but an infinite We may even have a n undamped oscillation of E,(r”) lirn Ei(4“)as I’ + +a. for a third sequence { r ’ } converging to j?. Consider (24) in Example 1 in $18.1 for I I = 3, a I 3= 0, and a i j > 0 for all other pairs ( i , j ) . When a sequence { p ” } of positive price vectors converges t o the boundary point fi = (0, 0, l)’, we find, for example, lirn El(/?”)= a, I if limp,”/p,” = 0, whereas lirn E,(p”) = + co if Iimp2’/p,’ = +a.The first situation occurs for the sequence { p ” = ( I / \ * , ijr2, I)’), and the second for { p ”= I/$,, I)‘). Diversity in the mode of convergence to fi makes this point a maligrant singular point of E l @ ) . This suggests that a boundary condition requiring (7) for all i E K , as adopted by K U ~ (1963, I is too stringent. Remark 7. I n Theorem 18.2, boundary condition A is indispensable. The following example, which satisfies all the assumptions except that boundary condition A is violated a t B = ( I , O)’, has n o equilibrium price bector i n the set of all positive vectors:
E z ( p ) = - 1. The boundary condition is violated because @ ( p )> 0 for p > 0 but Q ( p ) -+ 0 ( p + 9). Nonetheless, if this system is considered in the set of all p = (pi), pl > 0, p2 2 0, it satisfies all the assumptions of Theorem 18.2 in the extended price domain and has 3 as an equilibrium price vector. El(/]) = PZ/l’,>
$18.3. Extremization and a Revealed Preference Relation
I n this section, we are concerned exclusively with gross substitutes systems homogeneous of order 111 = 0, a typical situation i n economic theory. Arrow ct ul. (ISSS), and Arrow and Hurwicz (1960) have proved that if E,(p) (i = I , . . . , 1 1 ) are the excess demand functions of a gross substitutes system, we find, for any equilibrium price vector fi = (8,) and any disequilibrium price vector p,
This relation is very important because it enables LIS to establish the convexity of the set of all equilibrium price vectors (Corollary to Theorem 18.6)a fact observed for the first time by McKenzie (1960~1)-as well as global stability (519.2).
$1 8.
GROSS SUBSTITUTABILITY AND EXTREMIZATION
325
Samuelson’s weak axiom of revealed preference (Samuelson, 1948a, Chapter 5 ) means, if stated in terms of E i ( p ) , that for any two semipositive vectors p = ( p , ) ,q = (qi),the relations n
and Ei(p ) # Ei(9) (for some i)
(3)
entail
If the Walras law is explicitly premised, (2) is obviously true for q = j?, an equilibrium price vector, because in (2) we then have zero on the left-hand side and a nonpositive number on the right-hand side. Furthermore, if p is a disequilibrium price vector, we also have (3) for 9 = jj, since Ei(p) > 0 for some i but Ej(q) 5 0 for all i. Then (4) for 9 = 6,whose right-hand side vanishes by the Walras law, is nothing but ( I ) . For this reason, ( I ) will henceforth be referred to as a revealed preference relation. The purpose of the present subsection is to prove this revealed preference relation ( I ) between any equilibrium price vector and any disequilibrium price vector in a novel way by a method that differs completely from that of Arrow and Hurwicz. The principal idea of our method of proof is as follows. We regard the left-hand side of (1) as a function n
of two argument vectors j?, p . To establish the revealed preference relation ( I ) between any equilibrium price vector j? and any disequilibrium price vector p , it suffices to prove the following: (i) When j?, p range over all equilibrium price vectors and over all price vectors, respectively, the function ~ ( j ? , p ) takes on a minimum; (ii) The corresponding minimum must be zero and is attained only if p is an equilibrium price vector. Throughout this subsection, the basic assumptions (I), (11) (117 = 0), and (111) stated in $18.2 are premised, with (I) specialized to (1’) P = { p J p=(pi) 2 0, p i > 0 (i E M ) ] ,where M is a given fixed subset of { I , 2 , . .., 1 7 ) . M may be empty, nonempty, proper, or improper.
The additional assumptions are as follows: (IV) The excess demand functions E i ( p ) (i = 1, . . . , 1 7 ) are continuous in P .
326
VI. CROSS SUBSTITUTABILITY
(V) Gross substitutability in the finite increment version (Definition 18.1) prevails in P. (V1) The excess demand functions are bounded from below, i.e., there is a lower bound y such that E,(p) 2 y ( p E P, i
=
I , . . . , 11).
(VII) There is at least one equilibrium price vector in P.
We shall begin with preliminary results, which may also be of independent interest. Lemma 18.6, whose proof does not rest on (VI) and (VII), will be frequently used, not only in this section but also in $19.
LEMMA 18.6. The ,/ollowiiig itenis hokd: (i) For each Ic # M ,we hare €,(p) 5 0 eivrywliere iii P. 1ii
particrrlar, if' M
= @, ire
hare
reP Ei(p) = 0 r w r ~ ~ i c h e it7 (ii)
For ecicIi.fised k # M , let P ( k ) = { p IpL = 0, p E,(p)
(iii)
(i = 1 , . . . , 11).
Fwflieriiiore,
= constaiit
if' E6(y)= 0 at
soiiic
q
E
P I . Tlieii,
in P ( k ) . = (qi)E
P
iii tlie
,foregoing (ii),
theii
E,(p)
=0
ercrjmhcrc~iii
P(k).
ProoJ (i) Let p = ( p i )E P. Two cases (i.x), (i./l) will be considered sc pa r a t e I y . (i.2) p i = 0 ( i # k ) . (This case can occur only when M = 0.) Then, / i n > 0 and p6 EL(/>) = p i € i ( p ) = 0 by the Walras law (111). Whence E6(p) = 0. (i./l) pj > 0 (for some ,j # 1,). Denote by p ( t ) = ( p i ( t ) )the vector whose components are given by p,(t) = p i ( i # k ) , pi(!) = t . Clearly p ( t ) E P for any t >= 0. For any t > 0, we lind p , ( t )= p i (0 ) = p i ( i # k ) , p r ( t ) > pA(0),so that by gross sii bs t it 11t a hi I it y
E , ( p ( t ) )2 Ei(P(0)) (i # k ) .
(6)
Multiplying the ith ineqiiality in (6) by p i ( t )= p i ( 0 ) and summing over all i # k , we see
from the Walras law. Also from the Walras law,
$18. GROSS
SUBSTITUTABILITY A N D EXTKEMIZATION
327
Whence pk(t)Ek(p(t)) 5 0, that is, tE,(p(t)) 5 0 for any t > 0. This implies E,(p(t)) i0 for t > 0 and hence also for t 2 0 by continuity. Thus the desired result follows upon equating t to p k . Finally in the extreme case M = @, the above result implies E,(p) 5 0 everywhere in P (i = I , .. ., / I ) . Then, Ei(p) = 0 (i = I , . . . , n) for any positive p in P from the Walras law. Since positive vectors f i l l out a dense subset in P, we must have Ei(p) = 0 everywhere in P (i = I , . . . , n ) by continuity. (ii) Let p = ( p i ) , q = (4,) be in P(k), and p i > 0, qi > 0 ( i # k ) . Let . 9i cx=min-. i#A
Pi
Then, cx > 0, q 2 my, qk = upk = 0. Whence, by gross substitutability and homogeneity, we find Ekh) 2 Ek(V) = Ek(P),
and hence also Ek(p)2 Ek(q) from the symmetry between p and 4. Thus, E,(p) = Ek(q) for p , 4 in P(k) as long as their components are all positive except for the vanishing kth one. Since such p , q form a dense subset in P(k), we also have Ek(p)= Ek(q)for any p , q i n P ( k ) by continuity. (iii) Since Ek(p)equals a constant in P(k) by (ii), it suffices to see that Ek(p)= 0 at some p in P(k). If qk = 0, this q belongs to P ( k ) and can serve as a desired vector. The case qk > 0 will be divided in two subcases (iii.x) and (iii.p). (iiicx) If q has all components zero except for the kth one, M must be empty. Therefore, all E,(p) identically vanish by (i), so that we have, in particular, E k ( p ) = 0 in P(k). (iii.P) Suppose now that t h e j t h component of q is positive for some,j # k . Let the vector q ( t ) = ( q i ( f ) be ) defined by 4 i ( t )= qi (i # k ) , q k ( t )= t. Then, q ( t ) E P ( f 2 0), q(q,) = q. Clearly, if 4,, 2 t 2 0, we have q 2 q ( t ) , qi = qi(r) ( i # k), so that gross substitutability ensures Ei(q) 2 E M [ ) ) ( i # k , q k 2 t 2 0). Multiplying the ith inequality above by qi ( = q i ( t ) ) and summing over all i # k, we find
But the left-hand side of (7) must vanish because q, E,(q) = 0 by assumption, and the sum of all q , € , ( q ) ,including q, €,(q), is zero from the Walras law. Whence the right-hand side of (7) is nonpositive. Then, again by the Walras law, we must have q,(t)E,(q(f)) 2 0, that is, t€,(q(t)) 1 0 (4, 2 t 2 0). Whence it follows that E,(q(t)) 2 0 (4, 2 f 2 0, f > 0), and hence E,(q(t)) 2 0 (qr 2 t 2 0) by continuity. We have thereby found, in particular, E,(q(O)) 2 0. Note that q(0) E P(k) and therefore E,(q(O)) 5 0 by (i). These results combined imply E,(q(O)) = 0 at q(0) E P ( k ) , Q.E.D.
328
VI. GROSS SUBSTITUTABILITY
Remark 1. An indecomposable, homogeneous, gross substitutes system can exist only when P in ( I ) contains no vector having some components zero; hence A4 = { I , 2, . . . , / I ) in the case of (1’). Suppose that there were a vector p = ( p i ) , with pn = 0, in P. Then, ocp 2 p , and N(ocp,p) = { i I @pi> p i ) would be a nonempty proper subset of { I , 2, , . . , t i } provided M > I . Hence, by gross substitutability and indecomposability, we must have E , ( M ~2) E i ( p ) ( i 4 N ( x p , p ) ) , with a strict inequality for at least one i. On the other hand, E,(xp) = E i ( p ) ( i = I , . . . , 11) from homogeneity, arriving at a contradiction. Throughout the following, we shall frequently use the notation f ( p ) to denote the set defined for a price vector p by
I ( p ) = { i I p i > 01.
(8)
LEMMA 18.7. Let = ( B i ) he ail arbitrarj~$xedequilibrium price vector. Suppose tliat $(b,p ) iii ( S ) , n ~ l ~ e rieii.ed ti as a ,fuiictioti of a single argument rector p, is riiiiiitiiizctl o i w P at .such a price rector p tliat l ( P ) ={(it).
(9)
T / m , if’ I I V let w(p, p ) = min
Pf
i E f(B),
over all
(10)
Pi
sea, P ) = { i I P i ’W ( B >P ) B i ) ,
(11)
we liarv, for aiiy price rector q jiiljilliiig p 2 q 2 ~ ( p)P, j ,
k ( q ) 5 0 ( i !$ I ( l i ) ) ,
(12)
E;(q) = 0 (i E I(B)\S(B, PI).
(1 3)
Pr*oof. We first note that p i p
= q i = w(b, p ) b i
for i
4 S(B,p ) . Whence
2 q 2 w ( j ,p ) b , gross substitutability, and homogeneity entail E;(P)= E,(wB)
where w
= w(B, 17)
s Ei(9) 5 Ei(P)
( i 4 S(P>P))>
(14)
> 0 by (9). Relation (14) implies, in particular, 0 5 E i h ) 5 E;(P) ( i fzI(B)\S(B, P ) ) ,
(15)
since Ei(fi) = 0 for i E f ( b ) . Note that (12) holds trivially by Lemma 18.6(i), since f(j) 3 M . Equation ( 1 3) is also trivially true by ( I 5 ) if p is an equilibrium price vector. Therefore, it remains to consider ( 13) for a disequilibrium price vector p. I n what follows we use, as before, the notations K(cr) = { i I E , ( L I > ) 01, i
Aju
= (0,. . . ,0, A l l i ,
0 , . .. , 0 )
4 18. GROSS SUBSTITUTABILITY AND EXTREMIZATION
32 9
for a price vector u. Then, since p is a disequilibrium price vector, we must have K ( P ) f 0, and hence also, by homogeneity,
0-
S(B, P ) #
Moreover, for sufficiently small Apj > 0 ( j = I , . . . , t i ) , we find K(p
+ Ajp)
3
K ( p ) ( j = 1, . . . , / I )
(16)
from the continuity of E i . Then, setting
+
djj Ei(p) - E;(p Ajjp) ( i , j = 1 , . . . , n) and using the Walras law, we obtain as counterpart relations of (16) in $18.2 djjpj=Ej(p+Ajp)Apj- xdijpi (j= l,...,~). i+ j
Since$ is minimized over P at p , we must have $ ( @ , p + Ajp) that is, n
$(B,p),
n
1 Di Ei(P + Aj p ) 2k1= Bi E j h )
i= I
1
( j = I , . . ., n ) ,
which can be put in the form d J. J. *P J< = - Ci +dj i j p i
( j = J , ..., n).
Multiplying the ,jth relation in (IS) by w the j t h relation in ( 1 7), we find
d j j ( p j- o b j ) 2 E j ( p
= w@,p)
and subtracting it from
+ A j p ) A p j C d i j ( p i - obi) -
(18)
( j = I , . . . , n).
(19)
i#j
Since p i - wBi 2 0 ( i = 1, . . . ,n ) by definition, and dij 5 0 (i # , j ) from gross substitutability, (16) and (19) imply
which, in conjunction with (1 5), yields (1 3), Q.E.D. We are now ready to prove, with the vital aid of Lemmas 18.6 and 18.7,
THEOREM 18.6. (Arrow et al., 1959; Arrow and Hurwicz, 1960). W e haoe, for any equilibrium price rector j? and any diseyuilibriuni price rector p , the rerealed preference relation ( 1 ), namely,
330
VI. GROSS SUBSTITUTABILITY
Proof. The principal idea of our novel method of proof is the minimization of the function cc/(j,p) over all equilibrium price vectors /j and all price vectors p , as already noted. The proof will be worked out in steps 1-4. Step I . This step establishes that $ is actually minimized at some pair ( 8 , ~of) an equilibrium price vector j3 and a price vector p , and the corresponding minimum value is zero. We note that the minimization problem is well posed because of assumption (VII). To accomplish step I , we need only show that for each fixed equilibrium price vector fi, we always have cc/(jl, PI 2 0 (Y
E
(21)
PI,
which is part of the assertion of Theorem 18.6. For, if (21) holds for each equilibrium price vector 8, we find, in conjunction with the Walras law,
$ ( AP ) 2 0 = $ ( A li) for any pair (b,p ) of an arbitrary equilibrium price vector 8 and an arbitrary price vectory, so that cc/ is minimized at special pairs (b, j?), with the minimum value being zero. To prove (21) for any fixed equilibrium price vector 8 = (BJ, whose component sum may be assumed to be unity because of the first-order homogeneity of the relation (21) in /j, let
P ={pIp
E
P , pi > 0 (i E I@))),
where f(b) is defined by (8) f o r p = j?. Clearly, Z@) M , so that Q is a subset of P and satisfies in itself assumption (f'). Define auxiliary excess demand functions Fi(p) on Q by =J
n
This being done and taking an arbitrary Gic(p,)= E,(p)
+ cF,(p)
E
> 0, let
( p E Q , i = 1, . . . , 1 7 ) .
The system of excess demand functions C,'(p) (i = 1 . . . , 17) is readily verified to fulfill assumptions ( 1 ' ) and (11)-(VII) in Q , with f(b) as an M, y - E as a lower bound. and fi as a n equilibrium price vector. Next let n
6
=infx 1=
1
a, G , ' ( p )
over all
p
E
Q n P,,
$18.
GROSS SUBSTITUTABILITY A K D EXTREMlZATlON
33 1
where P, is the standard simplex as before. Since 6 2 'J - E , (5 is finite. Moreover, 6 5 0 = fiiCt(j3) from the Walras law. Let {p'] be a sequence in Q n Pn such that
c
'-+I,
1=1
We may assume, as usual, that { p ' ) converges to a vector p = ( p i ) in P,, . Suppose that this p were a boundary point of Q. Then clearly p, = O for some i E I(/?),say k. Note that from (22) we find litn F,(p') = +m V++CL
because k E I @ ) . Whence, using (VI) and F,(p') 2 = I , 2, . . .), we see
-
1 (i = I,
. . . ,n ;
n
This contradicts 6 5 0. Hencep cannot be a boundary point of Q but belongs to Q. It follows that
from the continuity of G,', and p is a minimizer. Then, by virtue of Lemma 18.7 as applied to GIE,b, q
G,"(p)= 0 (i E
r(B)\s(P,PI>.
= p-
we find (23)
We now assert that I($) n Sib, P> =
Izr.
(24)
Suppose (24) t o be false. Then, since by definition (10) we cannot have Z(B) c S(b,p), the sets /(B)\S(fi,p ) and I @n ) S(B, p ) would be nonempty proper subsets of /(,6). These results, combined with p 2 o($, p ) p , give rise to
'G,F(P)
G,W
=0
(i E
/iB)\S(B,/?)I
by gross substitutability, homogeneity, and definition (22) of F l . which contradicts (23). Whence (24) must hold, so that (23) turns out to be G,'(p) = 0
/(F,,.
(I E
Hence we find
s=
n
piG,'(p) = i= I
f i i G , " ( p )= 0. i
E
ICp)
(25)
332
VI. GROSS SUBSTITUTABILII~Y
If we take the 0 t h order homogeneity of G;' into account, the above result implies n
n
;= 1
;= I
2 B; E ; ( l / )+ 2 B; F ; ( 4 ) 2 0
(26)
( Y E Q)
for any c > 0 and therefore for E = 0. Finally, since Q is dense in P, we have (26) for c = 0 and 4 E P by continuity. SIP/)2. Let E be the set of all equilibrium price vectors. If we write A for the set of all pairs ( f i , p ) minimizing $ over E x P, we have seen that A # 0, and that the corresponding minimum value is zero, in step I . Therefore if it is shown that A contains no pair p ) for which y is a disequilibrium price vector, the theorem will have been proved. p ) i n E x P an e-d pair when is S / q 3. For convenience, we call an equilibrium price vector and p is a disequilibrium price vector. Denote by r the subset of A. formed by all e-d pairs i n A. The final goal of the proof is to see that assuming r # 0 will lead to ;I contradiction. To this end, we pay attention to special e-d pairs. called c ~ a i ~ o ~ ~ones. i c u l A pair p ) is said to be canonical if their components a;. p i satisfy I ( p ) = I(/?). We write C for the set of all canonical e-cl pairs i n r. The purpose of step 3 is to see that I- # 0 implies C # @. To this end. let (a, p ) E r be an e-d pair. We have
(a,
(a.
(a.
K ( p ) = ( i I E ; ( p )> 0; # @, because 1'
= ( p i ) i q ;I
disequilibriuni price vector. Note
I
=
/(B)
nI ( p ) 3 M.
E;(q)5 0 ( 4 E P, i
(27)
4 I),
(28)
from Lemma 18.6(i), so t h a t
I With ihe\e remarks in mind, let
= K ( p ) # @.
fi*
=
{/?TI,
/I* =
{ p : ] be defined by
The above definitions make sense because of (27) and. (29). It will be shown that (,V, p * ) E C and hence C # Clearly, it sufices to show that (fi:*. p * ) E r, since it is canonical by construction. I n fact, we find >= and l;; = i,: ( i E I ) so that Ei(l;") 5 E ; ( f i )5 0 ( i E I ) from gross substitutability. On the othcr hand. E;(fi")5 0 ( i 4 I ) by (28). Therefore a* is an equilibrium price vector. Regarding / I * . we first observe E;(fi)= 0 ( i E from the fact that fi is an eqiiilibrium price \'ector. Hence. by Lemma l8.6(iii),
a.
a*
/(a))
E ; ( / l )= E , ( / P )= 0 ( i E I ( / ? ) \ / ) .
(30)
5 18. GROSS SUBSTITUTABILITY Moreover, we find p 2 p* and p i
= p'
A N D EXTKtMIZATION
333
(i E I ) , whence
Ei(P) 2 Ei(P*) (i 1).
(31)
In view of (30) and (31) we see that n
1 B,Ei(P) = U P ) 2 1 l i m p * ) = $(B*, P*) 2 0,
0 = $(A P> =
Bi
i= I
is1
(32)
i t 1
whence equality turns out to hold in all the relations in (32). This also entails the validity of equality in (31) because p , = j?: > 0 (i E I ) . That is. E i ( / l )=
$(p. p ) =
.€;(p$:) (i E I ) ,
(33)
$(P*. p * ) = 0.
(34)
Equation (33) ensures that / I : % is a disequilibrium price kector. since Ei(p*)= E i ( p ) > 0 for i E K ( p ) c I . Then. (34), in cotijunction with the fact that B* E E, shows (p*, p * ) E P. We have thereby proved that (j?*,p*) is a canonical e-d pair in and hence C # Step 4. This step deduces a final contradiction from the conclusion C # fa in step 3. For any pair (fi, p ) in C, we can define the set S(b. p ) in Lemma 18.7 by (9), (10) and ( I I ) . Since S($,p ) c { I . 2, . . . , 1 7 ) for any (P, p ) E C, the number of elements i n S(p,p ) attains a minimum over C at some pair, say (b,p ) . We observe that Lemma 18.7 applies to this pair p). The canonicality of ( i , p ) , homogeneity, and the fact that ( b , p ) is an e-d pair ensure
a.
(a,
I
=
I ( b ) = /(PI = s(p,p ) #
a.
(35)
Now, letting
2
=
min
Pi
over all
i E SO, p)
Pi
(36)
and noting 1. > w = w(p, / I ) > 0,
we define a new price vector y^ A
qi =
=
(37)
(4;) by
\).fi,
( i E S(B, p ) ) , lop, ( i $ S(F, P ) ) .
Then / ( / I ) = f ( p ) = I ( y ^ ) , so that (4, p ) is canonical. Relations (36) and (37) imply p 2 4 2 o p , whence from ( 12) and ( 13) for q = 4 in Lemma 18.7, we find E,(y^)5 0 ( i $l
PI).
(38)
334
V I . GROSS SUHSTlTUTAHlLlTY
On the other hand, appealing to (37), we have 4iljb =
Bi
(i
E
s(fi,PI),
Lc)
gill. = fii 5 pi ( i !$ S(fi, p ) ) . l. Whence
Ei(4) = E i ( 4 / j . )5 E i ( j ) 5 0 ( i E S(fi,p ) )
(39)
from homogeneity, gross substitutability, and the fact that fi is a n equilibrium price vector. Therefore 4 is also an equilibrium price vector by (38) and (39). Finally, again from ( I 31, for 4 = / J in Letntna 18.7, we find Ei(/7)= 0 ( i E f\S(fi, I))).
(40)
Then, letting S = S(p. / I ) , for simplicity, and taking the definition of (35) into account, we have $( g, p ) =
14; Ei(p ) = i:c f i i Ei( p ) = I.$(
its
4
and
fi, p ) = 0.
iSS
Therefore, (4, / I ) E C. But S(4,p ) is ;I proper subset of S(fi,p ) , which contradicts the minimality of S(p, / I ) over C with respect to the number of its elements. This completes the proof, Q.E.D. Si/,i/i/ificu/io/iw d c r I/rc/c,c.oi,r/io.sLihi/ir.i.. I n special cases, certain steps in the proof of Theorem 18.6 can be dispensed with, so that the essence of the method of proof t-e\,eals itself more clearly. For example, if the price domain P consists exclusively of positive price vectors. we can dispense with Lemma 18.6, step 3, and all about canonicality i n step 4, since any e-d pair is canonical i n these cases. Moreover, it should be noted that further simplification of the proof is effected f o r indecomposable gross substitute systems. First, as was already noted in Remark I , such systems can be conceived on the basic assumptions (1’) and ( 1 1 ) ( n i = 0) only f o r A4 = [ I , 2, . . . , n ) , whence the general simplification of the proof noted above applies. Second, step 4 can be consiclet-ably simplified by dispensing with thc minimization of S(p, p ) with respect t o the tiumber of its elements. In fact, any e-d pair (fi, /I) satisfies
Ei(/i)= 0 ( i $ scp, /7)) f r o m ( l 3 ) in Lcmma 18.7 f o r q = / I . l ( f i )= i l , 2 , . . . , 1 1 ) . However this result clearly contradicts inciecomposability, since S(fi,/ I ) is ;I nonempty proper subset of { I , 2, . . . . / I ) from the hypothesis at the outset of the proof. Therefore, the argument i n terms of 4 i n step 4 is not needed under indecomposability. I t i general cases, the proof is made somewhat intricate by corn hi ti at or ia I jobs wit 11 c;i ti o ti ica I i t y and in i t i i ma I i t y .
$19.
STABILITY IN GROSS SUBSTITUTES SYSTEMS
335
The following corollary, which was first discovered by McKenzie (19604, is an important, though immediate, consequence of Theorem 18.6.
COROLLARY. On the sanie assumptions as in Tlieoreni 18.6, let E be the set of all equilibriuin price rectors. Then, E u ( 0 ) is a coiii~e.~ cone. Proof. We first note that the price domain P in (1’) is a convex set. Consider any convex linear combination ir = (ir,) = zfi + p$ (a 2 0, p >= 0 , LY + p = 1) for any fi = ( P I ) , (i = (4,) E E. If ir E, we would have
+
n
n
1
i= 1
i)i
E,(ir) > 0,
c
i= I
lji E,(ir) > 0 ,
by Theorem 18.6. Whence
i= I
i= I
i= I
which contradicts the Walras law. This proves that ir E E. Finally, the cone property of E u (0) is obvious from homogeneity, Q. E. D. Remark 2. Under indecomposability, the E in the above corollary is a ray generated by a positive price vector, that is, we have the uniqueness of equilibrium price vectors up to the multiplication by positive scalars. This can easily be seen as follows. Let fi, 4 be two arbitrary equilibrium price vectors. As noted in Remark I , fj, 4 are positive vectors. Whence E,(fi) = E,(Lj) = 0 ( i = I , . . . , n ) . Consider S(fi,Q)= { i 14, > toi),), where (u = min g,/i), for all i = I , . . . , 11, as in step 4 in the proof of Theorem 18.6. Suppose S(fi,Lj) # @. Then, S(fi,4) is a nonvoid proper subset of { I , 2, . . . , n } , so that homogeneity and indecomposability imply E,(lj) # E , ( o f i ) = E,(fj) = 0
(for some i
+ S(fi,$)),
contradicting E , ( $ )= 0 (for all i = I , . . . , 1 1 ) . Whence S(i),$) = @ and = wi), which proves uniqueness. In an indecomposable case, the revealed preference relation ( I ) therefore holds between an equilibrium price vector i ) and any p not proportional to fi.
4
619. STABILITY IN GROSS SUBSTITUTES SYSTEMS
$19.1. Dynamic Processes of Price Formation
I ti the foregoing sections, we have explored such fundamental static features of competitive equilibria i n a Walrasian decentralized economy as their existence (Chapter V, $ 1 6 and Chapter VI, $ 1 8) and their socioeconomic
336
VI. GROSS SUBSTITUTABILITY
implications (Chapter V, $1 7). Nevertheless these results say little about how an equilibrium price system is established in a competitive market. Walras, who was content with counting equations and unknowns to justify the existence of a competitive equilibrium, conceived of a competitive market as being a giant computer locating an equilibrium by virtue of the so-called law of supply and demand. He attached great importance to this practical solution of the corresponding huge system of equations of general equilibrium by the market mechanism. One typical process of competitive price formation that he and his followers had in mind is as follows, To a current quoted price system correspond aggregate supplies of, and aggregate demands for, goods which originate in the integration of the behaviors of individuals as price-takers. The classical law of supply and demand presumes that prices respond at the next instant to the current levels of supplies and demands in such a way that there will be a rise i n the price of every good with positive excess demand and a fall i n the price of every good with negative excess demand. Then the modified price system is met by new levels of excess demands for goods, which give rise to another price change, etc. Thus a trial and error process of price adjustment, called a tiitonnernent, is generated. It should be noted that no transaction takes place during the process until a general equilibrium situation is reached. Mathematically, a tiitonnen?ent process is formulated in terms of differential equations, difference equations, and other types of functional equations. This clear formulation of tiitonnenient processes from a dynamic viewpoint is due to Samuelson (1948a). For example, a familiar typical formulation of a tiitonnetnent process by a system of differential equations is d Pi
- = /ZiEi(p) (i = 1,. nt
..
where p = (11,) is the price vector; E,(p) is the excess demand function for the ith good ; 1, is a positive constant, called the coefficient of price adjustment of the ith good. Note that E,(p) > 0 entails a rise in the price p I , and E,(p) < 0 implies a fall in the price p , in ( I ) . The system of differential equations ( I ) may therefore be conceived as a representation of the classical law of supply and demand, and its solution p ( t ) = ( p r ( t ) )is expected to simulate the process of price change by tiitonnement. More recently, the forniulation of a ~iitonnernentprocess by ( I ) was generalized by Arrow et al. (1959) and others to the system
(!Pi -= H,(p) ( i ri t
=
1 , ...
$19.
STABILITY IN GROSS SUBSTITUTESSYSTEMS
337
where H,(p) (i = 1 , . . . , n) are functions defined on the price domain P and are of the same sign as the excess demand functions E , ( p ) , i.e., sgn H i ( p ) = sgn Ei(p) ( p E P , i = 1,
. . . , n),
(3) in the sense that for each i, H i ( p ) is positive, negative, or zero according as E i ( p ) is positive, negative, or zero. When there is a numhraire good, whose price is always kept at unity because of its special role as a unit of accounting and transaction, the requirements of the positivity of A i in ( I ) and the sign-preserving property of H , ( p ) are imposed on all goods other than the numhraire good, for which we set 2, = 0 and H , ( p ) = 0. In this section, we shall be concerned with properties of the solutions p ( t ) of ( I ) or ( 2 ) , particularly with their asymptotic properties as t tends to infinity. In view of the Walrasian concept of tbtonnement, economists are very interested in the convergence of a solution p ( t ) to an equilibrium price vector 1 satisfying ( I ) , (2) in 518.2 as t tends to infinity. Mathematically, this convergence property means the following: ( a ) Existence of Local Solutions. Given any initial price vector p o , there is a local solution p ( t ) of (1) (or (2)) defined on an interval [0, a), a > 0 and satisfying the initial condition p ( 0 ) = p o . (p) Possibility of Continuation of Local Solutions. Any local solution p ( t ) of ( I ) (or ( 2 ) ) on [0, a) can be continued to a global solution defined on [O, +a>. any global (y) Convergence to an Equilibrium Price Vector. As t --+ +a, solution p ( t ) of ( I ) (or (2)) on [0, +a)converges to an equilibrium price vector 1 characterized by (l), (2) in $18.2.
DEFINITION 19.1. Global stability is said to prwail in systen7 ( I ) (system ( 2 ) ) if(@), (fl) and (7) hold. Undoubtedly, (a) is a prerequisite to the possibility of tiitonnernent. If a local solution starting at some initial position cannot be continued beyond a moment t = z, the tbtonnenient breaks down at T and is not workable. (p) therefore guarantees the workability of tbtonnen7ent. Finally, (y) ensures that a tlitonnement eventually leads to an equilibrium. As a matter of fact, global stability is so special a dynamic property that contrary to the Walrasian view, one can hardly expect it to be shared by all competitive economies. As Scarf's interesting examples of instability (Scarf, 1960) suggest, instability seems to be a universal phenomenon in competitive economies, rather than an exceptional one, whereas global stability is expected to prevail only in very well-behaved systems. It is important as well as interesting, however, to single out special types of competitive economies that enjoy global stability. I n other words, it is worthwhile finding conditions of economic significance that ensure global stability. In this respect, recent
338
VI. GROSS SUBSTITUTABILITY
studies have concentrated on gross substitutes systems and revealed the prevalence of global stability in them. On the other hand, no drastically different kinds of competitive economies have been known to parallel gross substitutes systems with regard t o this global stability aspect. The purpose of $19.2 is to discuss global stability under gross substitutability. This will be done, however, by presenting only certain results by Arrow et a/. (1959), by Arrow and Hurwicz (1960), and by Uzawa (1961) as representative examples from among the bulky literature on the subject. This opening section lists a few important basic results i n the theory of ordinary differential equations and formulates a useful lemma on majorization.
THEOREM 19.1 (Special Form of the Cauchy-Peano Local Existence Theorem). Let tixi -
(it
=
G,(x)
(i
=
I , . . ., n i )
(4a)
he a .Y.I..Y t el 7 i 9f rh/ferei i t i d equations, 1I here (i) s = (xi)stands,for the in-ditiietrsiorial argrrnient cector. , n i ) are continuous functions defined on a coninion (ii) G,(x) ( i = I , open subset A of R". Then, ,for an?' .yo E A and an)' time poitit t o , there is a set of continuously dijerentiahle ,functions of' t '
.Yi(t)
( i = I , ...,nl),
(4b)
which are &fined in an interra1 It - t o / < E ( I : > 0 ) nYt/i x ( t ) = (.yi(t))E A, and satisfj, (4a) iti the i i i t c r r d as \ i d 1 as the initial condition ~ ( t , = ) xo.
For the proof of Theorem 19.1 the reader may refer t o a standard textbook on ordinary differential equations (e.g., Coddington and Levinson (1953, Chapter I ) . Henceforth the system of functions (4b) formulated in Theorem 19.1 will be referred to as a local solution of (421). It should be notcd that Theorem 19. I says nothing about the uniqueness of local solutions o r their continuity with respect to the initial value .yo. All the proofs that will be stated i n $19.2 below need no reference to the continuity of solutions with respect to the initial value, as opposed to the proof in Arrow and Hurwicz (19621, so that Theorem J9.1 will suffice. Incidentally, Theorem 19.1, by not having G i depend on t , is distinguished from the general form of the Cauchy-Peano existence theorem in the cases where Gi are functions not only of s but also of t . of Continuation). Suppose that G,(x) are cotititiiious in a subset A o f R, , and that ,for each poitit o f A, s l ~ s t o i i(4a) has a local ~Y#lliti#l7starting at the point. I f anj' local solution p ( t )
LEMMA 19.1 (Possibility
(i = 1. . . .
.
111)
$19.
STABILITY IN GROSS SUBSTITUTES SYSTEMS
339
of (4a) in [ t o ,a) is contained in a compact subset r of A, which may depend not only on its initial position but also on the solution itself, any local solution of(4a) can be continued to [ t o , 03).
+
Proof. Take any local solution (4b) in an interval [ t o ,a) a > t o . To prove the possibility of its continuation, consider the set of numbers z such that T > to and the given local solution can be continued to the interval [ t o ,T). This set is nonempty, since it contains a. Now let w be the least upper bound of the set. It will be shown that Q = +a. Suppose w < fco. The solution can be continued to the interval [ t o ,w ) . Let x ( t ) = (x,(t)) be a continued solution to [ t o ,0). Then, x ( t ) E r (w > t 2 to)for a compact subset r of A by assumption. Thus there is a bound M from the compactness of r and the continuity of G,(x) ( u = I , . . . , m ) such that JG,(x)I 5 A4 (x E r, i = I , . . ., m). Now, by integrating (4a), we find
Whence for i = I , . . . , n7, S2
Ixi(s1)
j
- x i ( s z ) l ~ IGi(x(t))l dt 5 M IS^
- s21
51
(w > s 2 ,
s1
2 to),
so that xi(sI)- xi(s2)-+ 0 ( i = 1, . . . , m ) as sl, s2 w . In view of the Cauchy criterion of convergence, this implies the existence of --f
lim xi(t) = xiw ( i
= 1,
. . . , m).
r-w-0
As a matter of course, x"' = (x,"') E r by the compactness of r. Since x" E A, there is by assumption a local solution y ( t ) = (yi(t))of (4a) on an interval [w, 41, 4 > w satisfying the initial condition y(w) = x"). Define z ( t ) = (zi(t)) on [to, 4 ) by
i
z(t) = x(t) Jft)
t 2 to), t 2 0).
(0>
(V
It can be readily verified that this ~ ( t is) a continuation of the given solution beyond Q, which contradicts the definition of w . Whence w must be +03 Q.E.D. Next we proceed to a majorization lemma that will play a vital role in the proofs of stability.
340
VI. GROSS SUBSTITUTABILITY
LEMMA 19.2. Let f ( t ) be a real-calued ,function on a semiir.tterra1 [ t o , a), a > t o , irhere a niay or niay not be co. Then (i) I f a real nimiber (5 satisfies
+
we hare
(ii) l f a real number S satisJies lim sup f ( t '
+
h-0
I?
-
f
(
f
)
s- 6
(.for all t E [ t o , a)),
vve Iiaoe .f(t)
5 &t
- I,)+ f ( t , )
(a > f 2 to).
Proof. Clearly either of (i) o r (ii) follows from the other. It therefore suffices to prove only (i). Moreover, if it is seen that , f ( t ) 2 (6 - ~ ) (-t 1,) +,f(to) (a > t 2 r,) for any E > 0, the arbitrariness of E ensures the result of (i). Thus letting L
=
{ t I f ( t > I (6 - E ) ( t - to) + f ( t o ) , a > t
2
to>,
we shall show L = [ t o , a). To this end, let w be the least upper bound of all numbers T such that 7 2 to and [ t o ,T ] c L. w is well-defined because [ t o ,T] c L for T = t o . Then, clearly CJ 2 w 2 t o , and proposition (i) is proved if w = a. With this in mind, suppose a > o.Then, let us first show that f ( w ) 2 (6
-
1(
-
to)
+ .f(to).
Relation (5) is true in an equality form if w > o > t o . Since
= to.
Consider the
5
by assumption, we must have / ( o - 11) - f ( w ) <
for sufficiently small f(0)
- 11)
11
-(ii
- c)l1
> 0. Moreover,
>= ( ( j - c)((o - / I
- t o )+ f ( t O ) (I? > 0,
-
from the definition of w . Relations (6) and (7) combined entail
T(w) - (0 - & ) / I 2 (6 - &)(a- I?- t o )+ f ( r o )
2
to)
$19.
STABILITY I N GROSS SUBSTITUTES SYSTEMS
34 1
for sufficiently small h > 0. This becomes (5) in the limit, as 42 + +O. On the other hand, there must be a sequence { h v ] of positive numbers converging to 0 and satisfying
+
.f(o h,)
< (4 - E 1 ( 0 + h,
-
to) +. / U O ) ,
(8)
again from the definition of 0).Then, subtracting ( 5 ) from (8) and dividing the resulting relation by 11, > 0, we find
f ( w + k )- f ( w ) < h"
- E.
Whence
which contradicts the assumption on 6. This proves o = n, Q.E.D.
A special case of Lemma 19.2 deserves attention. This will be stated as follows.
LEMMA 19.3. Suppose that 6 (i) (ii)
f(t)
=0
in Lemma 19.2. Theri we hare
in Lemma 19.2(i) is nondecreasitig if7 [ t o , n). 19.2(ii) is nonincreasing in [ t o , 0).
f ( t ) in Lemma
Proof. (i) One can readily see that Lemma 19.2(i) ensures f(t)
2 6(t - s) +.f(s)
(a > t 2 s 2 to).
If we take the assumption S = 0 into account, the above inequality implies t 2 s 2 to), as was to be shown. The proof of (ii) is completely similar, Q.E.D.
f ( t ) z f ( s ) (n >
Remark. In view of the fact that in Chapters 11, 111, and IV we worked on the dynamic aspects of intermediately disaggregated systems by using formulations in terms of difference equations, the reader may wonder why we employ differential equation formulations here. Most of the results of dynamic nature in Chapters I I through IV can also be established i n terms of differential equations, so that the choice of types of equations there is mainly a matter of convenience. Here the situation is different. Almost all of the results achieved so far on stability i n Walrasian competitive economies are in terms of differential equations [except for that of Uzawa (1959-1960)], and it seems to be impossible to establish their difference equation versions. This is the main reason for employing differential equations in the present chapter.
342
VI. GROSS SUBSTITUTABILITY
$19.2. Stability under Gross Substitutability
I t i this section, global stability will be established under gross substitutability for system ( I ) or its more general version ( 2 ) in $19.1, namely, “Pi
-=AiEi(p) rlt
( i = 1,
or
P
2=
rlt
Hi(p) ( i = 1, . . . , I ? ) .
The basic assumptions premised in this section are (l’), (11) (rn = 0), ( I l l ) , (IV), and ( V ) given in $18.3. For easier reference they are restated below: (1’) P = { p 111 = ( j j l ) 2 0, p l > 0 ( I E M ) ] ,where M is a given subset of { I . 2. . . . , I t ) . (11) The excess demand functions E,(p) (i = I , . . . , 11) are homogeneous of order zero ( n i = 0). ( 1 1 1 ) The Walras law i n the narrow sense holds everywhere i n P. ( I V ) E , ( p ) ( i = I . . . .. 1 1 ) are continuous in P. ( V ) Gross substitutability in the finite increment version (Definition 18.1) prevails in P.
We also assume (Vll’), a stringent version of (VII) in 418.3: (VII‘)
There is a positive equilibrium price vector p
=(PI).
A < noted several times before, I, must satisfy
E,(p)=O ( i = l , . . . , / I )
(3)
from Definition 18.4 and the Walras law. Ac to system (2). we premise (a)?(b): (a) H , ( p ) (i = I , . . ., 1 7 ) are continuous in P . (b) For each i = I , . . . , 11. H , ( p ) and E , ( p ) are of the same sign,
sgn H , ( p ) = sgn E , ( p ) ( p E P, i
=
I , . . . , n),
(4)
with the understanding that in the case of a tdtoIitieInnit where a good, say the Ith one, is a I I I I I I I ~ ; Y C I ~ Ithe . ~ ’ , Ith relation i n (4) is removed and replaced by
H,(p)
=0
everywhere in P.
(5)
If system (2) is specialized to system ( I ) , assumption (a) is implied by assumption ( I V ) , whereas assumption (b) means
2, > 0 (i = I , . . ., I ? )
(6)
$19.
STABILITY I N GROSS SUBSTITUTES SYSTEMS
343
except for a numgraire good, if any, say the lth good for which /I1 = 0.
(7)
We begin with the following two lemmas. LEMMA 19.4. Any equilibrium price rector j?
=
(Fi) satiJfies
Ei(fl) = 0 (i = 1, ..., 11). Proof. l j I > 0 implies E,(P) = 0 by (2) in $18.2. Moreover, since E,(p) = 0 ( i = 1 , . . . , t i ) for the equilibrium price vector p in assumption (VII’), Lemma 18.6(iii) encures E,@) = 0 for i such that j3, = 0. This completes the proof. LEMMA 19.5. L e t ji = (p,) be the equilibriuni price w c t o r in assumption (VII‘) and I be an arbitrary nonroid subset of { I , 2, . . . ,n } such that I 3 M. Then the price rector p ( I ) = ( P I ( / ) ) in P, kchich is dejined bJ-
is again an eqirilibriuni price rector. PvooJ First we have E , ( p ( I ) )5 0 ( i ef I ) by Lemma 18.6(i). Second, p ( l ) 5 p , p i ( / ) = Pi (i E I ) imply E i ( p ( / ) )5 Ei(p) = 0 ( i E I ) from gross substitutability. Thus, p ( / ) satisfies ( I ) in $18.2. Hence (2) in $18.2 is also ensured by the Walras law. Therefore p ( I ) is a n equilibrium price vector, Q.E.D. THEOREM 19.2. On the basic assuniptions (l’), (11)-(V), (Vll’), (a) and (b), i1.e haw (i) rf’ asswiiptioii (VI) in $18.3, namely, the I o n w boundedness of the excess deniand functions, is also premised, global stability precuils in s-btstetn ( I ) (Arroic. and Hurnicz, 1960). (ii) rf’ gross substitutabilitji (V) is strengtllenetl to strong gross substitutability in the finite increnient rwsion (see Reniark 4 in $18. I), then global stahilitj. prerbils in system ( 2 ) (Arrow et al., 1959). (iii) Specialize assuriiptions (IV), (V), (a), and (b) to (IV’), (V’), (a’), and (b’), respectiivly: (IV’) E i ( p ) ( i = I , . . . , n ) are drflerentiable in P. (V‘) Gross substitutabilitj. in the cf(fcvential rersion prerails in P (see Defiriitioii 18.2). (a’) H i ( p ) are of’the ,form Hi(p) =.A.(Ei(/))) ( i = I , . . , I ] ) , ichere , f ; ( t ) (i = I ,
able
t.
. . . , t i ) are cli~i~rctitiahle,~irIlc.riolls o / a real rari-
344
VI. GROSS SUBSTITUTABILITY
(b')
f,'(t)
ij
> 0 for all t (i = 1, . . . , n), and each f , ( t ) canislies i f and onl,v f i ( t ) = 0 identically for a nuniPraire good I, if
t = 0, except j o r
any. Then, global stability preoails in system ( 2 ) (Uzawa, 1961).
Pvoof. Recalling Definition 19.1 of global stability, we shall have to prove (a), (p), and ( y ) in $19.1. In doing this, (a ) and (8) can be proved for system (2) on the basic assumptions, without appealing to the specific additional assumptions in (i)-(iii). Moreover, the respective proofs of ( y ) for (i)-(iii) commonly rest on a fundamental relation, derived from gross substitutability, appearing in the course of the proof of (p), while they proceed differently only in the remaining parts. The proof is also intended to be worked out in as unified a way as possible for tfitonnenient processes both with and without numPraire. Readers who are exclusively interested in processes without nuniPraire may skip the passages pertaining to a iiuniPraire good. (a ) Existence of Local Solutions. Let p o = (p,") be any price vector in P. It will be shown that there is a local solution of (2) starting at p a . Let Z ( po) = { i I p," > 0}, as before (ste (8) in $18.3). Since there is a positive equilibrium price vector p by (VII'), which satisfies (3), we find
from Lemma 18.6(iii) and assumption (b). Let
Consider the system of differential equations
If Z ( p o ) consists of ni integers, H i ( p ) (i E Z(po)) are continuous on A(po), which can obviously be regarded as an open subset of R". Since p o E A(po), there is, by virtue of Theorem 19.1, a local solution p ( t ) = ( p i ( t ) ) of (10) defined in an interval [0, a), o > 0. This solution satisfies At)E by definition, so that Pk(t) = 0 (k
# I ( p o ) ) in [O, 0), and hence automatically
9d3t = H , ( p ) = 0
( k # Z(p0))
from (8). Whence p ( t ) is a desired local solution of (2).
$19.
STABILITY IN GROSS SUBSTITUTES SYSTEMS
345
Incidentally, it should also be noted that any solution p ( t ) of (2) in an interval [0, a), a > 0 is of the above type, whether 0 < +co or 0 = + a.In fact, upon integration, we find Ph(') = Pko
+fHL(ds))
ds = ? X H k ( d s ) ) d s
( k $ I(pO>, a > t 2 0). (1 I )
The integrands f f k ( p ( t ) ) must be nonpositive by Lemma 18.6(i) and assumption (b), whereas p k ( t ) must be nonnegative by definition. The integrals in the rightmost of (1 1) therefore vanish and hence Pk(f) = 0
(k
# r ( p o ) , a > t 2 0).
(12)
(p) Possibility of Continuation. Let p ( t ) be any solution of (2) in [0, a), a > 0, local (a < +a),or global (a = 03). For any equilibrium price vector
+
B such that I(B)
=
4P0)?
(13)
define x(t,
j?)
=
min -T over all
Pi
i E f(pO).
Let CI) be the least upper bound of all 7 such that 0 > 7 > 0 and I ( p ( t ) ) 3 Z(po) (7 > t 2 0). w is well defined and positive, since i ( p ( t ) )=I /(PO) for a small 7 > 0 because ofp(0) = po and the continuity o f p ( r ) .Clearly, a >= o > 0 and P(r) 2 ~ ( t>) 0 in [0, w ) . Let J&,
B) = iiI P i t ) = d t , m,>.
(16)
Then, p , ( t ) 2 ~ ( f B)B, , (i E I ( p o ) ) , while p k ( t )= cc(t, fi)j$ = 0 ( k $ / ( P O ) ) by (I 2) and (13). Hence p ( t ) 2 cc(t, B)B in [0, a). Moreover, a(t, B ) > 0; hence cc(r, p)fi E P in [0, 0). Consequently, E,(p(t))2 E,(cc(r,j?)fi) = E , @ ) = 0 (i E J,(t, j), t E [0, w ) ) from homogeneity, gross substitutability, and Lemma 19.4. Therefore, in view of assumption (b), we find
H,(p(O) 1 0 (i E J,@,
B),
t E [O, a)).
(17)
It should be noted that (17) holds even if a nume'raive good happens to be listed in J J t , B) because H , ( p ( t ) )= 0 by definition when the ith good is a numkraire good.
346
V I . GROSS SUBSTITUTABILITY
We now assert
In fact, at each t E [0, w), there is a sequence {h,) of nonzero numbers converging to zero such that lirn inf
4 tf
h+O
h, B ) - 4 h
L B) =
lim v- + m
+ hv B ) - 44 B)
4 t
5
hV
.
(19)
By definition of a({, fi), we can choose integers k(v) satisfying 4t
+ h v ,ri) = ~ ! + ) ( t+ hvYBh(Y)( v
= 1,
2, . . .),
k ( v ) ~ , I ( p ’ ) n J , ( t + h , , f l ) ( v = l , 2 ,...).
(20) (21)
For all values of v = 1, 2, . . . , k(v) lies in { I , 2, . . . , n}, so that k(v) takes on a common value, say k , for an infinite number of We may therefore assume, without loss of generality, that for a fixed k we have $1.
cc(t+h,,B)=p,(t+h,)/B,, k
Moreover, as v
4
E
( v = 1,2,...),
+
I ( p o ) n ~ , ( t h v , 9) (v
=
1,2,
...).
(22) (23)
+ co, (22) becomes in the limit B ) =P k ( t ) / b k
(24)
from the continuity of p k ( f ) and u(t, P), and hence k
E
I(pO) n JA,B).
(25)
Taking (22), (24), and (25) into account, we find lim V ’ i o o
cr(t
+ h , , j3) - a(t, j3) = -I hV
Bk
lim Pk(t v*+m
+ hv) - P d t ) hV
from (2). The rightmost of (26) is nonnegative by (17) and (25). This, combined with (19), ensures
I n exactly the same way, we find
$19.
STABILITY IN GROSS SUBSTITUTES SYSTEMS
347
Whence a(t, #) is nondecreasing and P(t, fi) is nonincreasing in [0, o) by Lemma 19.3. This implies, in particular, p i ( t ) >= a(0, fi)fii > 0 (i E I@'), o > t 2 0). If o > o,we would have pi(w) 2 a(0, fi)fii > 0 ( i E Z(p')) and hence also pi(o + h) > 0 ( i E Z ( p o ) ) for any small h > 0 satisfying o + h E [0, o), both from continuity at o,thus contradicting the definition of w. Therefore o = 0 . In summary, we have shown that 0 < 40,B) 5 a(s,
3) 5
fi) 5 Pi(t)/'fii
I P(r, B) 5 B(s, B) 5 B(0, B) (i E I(po), o > r
1 s 2 0). (29)
This result, in conjunction with (12), is of vital importance in proving (p) as well as (y). If we denote by j j = (pi) the special p( I) in Lemma 19.5 for I = I ( p o ) , then (12) and (29) for # = p imply in particular that any solution p ( t ) of (2), whether it is local or global, remains in the cube r(po), TCP')
== {P
I pi
=0
(i $ Z(Po)>, a(0, B ) i i i 5 P i
5 P(0, P ) B i (i E I(po)>>>.
(30)
T(po) is a compact subset of P. Moreover, for each point of P, system (2) has a local solution, starting at the point, by (a). Hence any local solution of (2) can be continued to [0, + a)by Lemma 19.1 as applied to system (2). ( y ) Convergence to an Equilibrium Price Vector. The basic idea common to all the methods of proving (i), (ii), and (iii) is as follows. We first note that the convergence of a global solution p ( t ) of (2) to some vector p ( a ) occurs if and only if p , ( t ) + p i ( a ) ( t -+a)for i € I ( p ' ) and p k ( a ) = 0 for k 4 I ( p o ) , since the convergence of p k ( t ) ( k I ( p o ) ) to zero is always ensured by (12). On the other hand, for any equilibrium price vector fi satisfying I @ ) = I ( p o ) and any global solutionp(t) of (2) starting at p o , we have by (29), f$
( I ) a(t, fi) is nondecreasing and bounded from above in [0, + 0 0 ) ; (2) P (t, fi) is nonincreasing and bounded from below in [0, + a). Hence lim a(t, fi) = a(#), (31) t++m
lim P(4 t++m
#I = P(#>
(32)
exist and P(P) >= a(P) > 0. As a matter of fact, .(B) need not equal /I@). But, if there is an equilibrium price vector fi satisfying = I(PO)>
a@) = P(#),
(33)
(34)
348
VI. GROSS SUBSTITUTABILITY
we find lim pi(t) = u(fi)ji = p ( j ) j i ( i E I(p'))
(35)
f'+cc
from (29), and
+
pi(t) = fii = o ( i $ ~ ( p ' ) , co > t 2 0)
(36)
from (1 2). That is,
as was to be proved. Therefore, it suffices to have (33) and (34) for some equilibrium price vector j. To d o this, however, we have only to show that there is a sub-sequence {tvf,"=l of time points t L , t , , ..., t , , . . ., tending to +a,for which the sequence { p ( t ' ) } converges to an equilibrium price vector p ( w ) . First, we find Z(p(00)) =I@') because p ( t ) E T(p'), and hence p(co) = lim p ( t , ) E T(p'). Second, for j = p(co) we have lim a ( t , , p ( c c ) ) =
lim P ( t , , p ( c o ) ) = l , v-+m
V-'+a,
so that lim
cc(t,
p ( c o ) ) = lim P ( t , p ( c o ) ) = 1
t++m
f'+m
from their monotonic convergence ensured by ( I ) and (2) for j = p ( c o ) . The proof will henceforth proceed differently for (i), (ii), and (iii). (y.i) Case of (i). We recall Theorem 18.6, which, under assumptions (1') and (II)-(VI I), ensures, in particular, n
for a positive equilibrium price vector P = (pi) whose existence is ensured by (VI1') and any price vector p E P, with strict inequality holding if p is a disequilibrium price vector. With these remarks in mind, we first consider the non-nurnr'raire case. Let " lim inf C piE i ( p ( f ) = ) 6. f++o,
Then 8 2 0 by (38). If 6 which
= 0,
i=l
there is a sub-sequence
Iim x p i E i ( p ( t v ) ) = 6 Y+
+m
= 0.
{I,}
tending to
+ co for (39)
$19. 5-1- W I L I T \
Since p ( t , ) E r ( p o ) ( v
=
1, 2,
GKC)SS SUBSTITUTESSYSTEMS
I\
349
. . .), which is compact, we may assume that
lim P(f,> = p ( . o ) ,
p(m>E Up").
, - + 3
Then we find n
1
i= 1
Pi
Ei(p(co)) = 0
from (39) and continuity. Whence p ( c o ) must be an equilibrium price vector because otherwise we would have n
1
Pi
i=1
E i ( p ( a > >> 0.
In the light of the conclusion of the general remark for (y), this proves (i). Thus, it remains to be shown that the supposition 6 > 0 would lead to a contradiction. Suppose 6 > 0. Then
for some t , > 0. Hence we find for f 2
I,,
from ( I ) , the Walras law, and (40). Then, by Lemma l9.2(ii), we must have for t 2 t o ,
whose right-hand side tends t o - co as t n
This contradicts p ( t ) E r(po) ( t 2 0).
-+
+ co. Whence
350
VI. GROSS SUBSTITUTABILITY
I n the case where the Ith good is a numr‘raire, by using a normalized positive equilibrium price vector p with its fth component PI = 1 and noting p l ( t ) = PL = I over time, we find
n
=
-2
C Pi E i ( P ( 0 )
i= 1
from ( I ) and the Walras law. Thus, the counterpart of (41) is
so that the proof proceeds in the same way. (y.ii) Case of (ii). Since a strong gross substitutes system is a further stringent version of an indecomposable gross substitutes system, we must have M = { I , 2, . . . , 1 1 ) by Remark I in $18.3, and the uniqueness of equilibrium price vectors up to multiplication by positive scalars by Remark 2 in $18.3. The case without /i/mi&aire will be discussed first. Taking a positive equilibrium price vector p stated in ( V l l ’ ) , let
where J,(t, p ) is defined by ( I 6) for substitutability and (b), and hence lim inf r-+
fi = P.
((2)
Then we find
= 20
t(t)>= 0 from gross
2 0.
7
Then, supposing ii = 0, and taking, as in (y.i), a sub-sequence {t,,) tending to +cc and fulfilling liin ( ( t , ) = \-t
26
= 0,
liin p(t,) = p(c0)
,-+
(44)
I
r
in
qpO),
(45)
$19.
STABlLlTY IN GROSS SUBSTITUTES SYSTEMS
35 1
we shall show that p(m) is a positive multiple of P . In fact, as in the argument leading to (22), (23) from (20), (21), we may further assume that for a common k,
k E J,(t,, I j )
(V =
1, 2,
. . .).
Relation (47) implies P k ( t v ) = a(t,, p)&, p ( t v ) 2 a(t,, pk(co) = n P k , p(co) >= np by (45) and continuity, 72 = or@)
=
(47)
P)p (v
=
1, 2,
. . .), so that
lim a ( t v , Is). V++cO
Hence k E J = {i I p i ( a ) = npi},
(48)
P(W>2
(49)
nP.
Moreover, from (44), (45), and (46), and continuity, Hk(p(oo))
=
lim H k ( p ( t v ) )
=pk
v++m
lim ((t,)
= 0,
V’+m
which entails, by (b), & ( P ( ~ ) >= 0.
(50)
Now, if we had p ( c o ) 2 np in (49), we would find E,(p(m)) > 0 from strong gross substitutability, homogeneity, and (48), thus contradicting (50). Hence p ( m ) = np, the desired situation, as expected in the concluding part of the general remark for (y). Finally, to deduce a contradiction from the supposition 6 > 0, we use (19), (25), and (26) for f i = p and rest on Lemma 19.2(i), to obtain for some I, > 0, a((, F )
2 6(t - to) + a(to,
( t 2 to),
as in (7.i). This results in a contradiction a(/, p ) -+ + 00 ( t -+ + co). Next it will be observed that the above argument for the nonnormalized case applies directly to the case where the Ith good is a numhraire, provided the argument is worked out in terms of a normalized positive equilibrium price vector with its fth component p1 = I . Then, p l ( / ) = p1 = 1 over time, so that
@, PI 5 1 5 bv? Is). With the monotonicity of a(t, p ) , P(t, p ) in mind, consider the case where ~ ( tp ,) < 1 over time. Since the nunihraire good is never listed in J,(t, p ) , the
352
VI. GROSS SUBSTITUTABILITY
argument in terms of eft) in (43) for the non-numkraire case is directly applicable. The case where /I(?,p) > 1 over time can be handled by the corresponding argument in terms of /I(t, p) and q ( t ) = max H k ( p ( t ) ) over all ~
k
E J,(t,
p)
Pk
I Pi(t) = P ( t , F > P i > . Finally, if u(t,, p ) = /I(t,, p) = 1 at some moments ={i
4 4 PI
= B(t, P ) =
t, , r, , we find
1 (t 2 max(t,, t,))
by monotonicity, whence the convergence of p ( t ) to p takes place in a finite period. (y.iii) Case of (iii). Recall that any solution p ( t ) of ( 2 ) satisfies Z ( p ( t ) ) = Z(po). With this in mind, for any global solution p ( t ) of (2) starting at po, let
Z(p(t)) = Z(po) justifies the definition of p ( t ) by (51). At each t , we find E i ( p ( t ) )2 0 for at least one i in I ( p o ) , from (8), (12), and the Walras law, so that p ( t ) 2 0 ( t 2 0) (52)
from assumptions (a’) and (b’). Equation (51) implies H,(p(t)) 5 p ( t ) p i ( t ) ( i E I ( p o ) ) .On the other hand, Hk(p(t))= p ( f ) p k ( r )= 0 ( k $ Z(po)) by (8), and Z(p(t)) = Z(po). Hence Hi(P(t)) I P(tlPi(t) (i
=
1 , . . . , 4.
(53)
Then, in a way similar to the derivation of (26), if we let J ( t ) = ( i I H i ( A t ) ) = p(tlpi(t>>,
we find
Taking assumption (a’) into account, and performing differentiation, we have
”
where p i ( t ) = d p i / d t , Eij(p) = d E i / d p j ( i , j = 1, . . . , n).
$19.
353
STABILITY I N GROSS SUBSTITUTES SYSTEMS
Note fk’(E,(p(f)))2 0 by assumption (b’). Moreover,
n
2 A t ) C E,j(p(t))pj(t) j = 1
from pj(t) = Hj(p(t)), gross substitutability Ekj(p(t))2 0 (k Z j ) , (53), and H,(p(t)) = p(t)p,(t). The rightmost of (56) vanishes by homogeneity and Lemma 18.4. Since pk(t) > 0, the first term on the right-hand side of (55) is nonpositive by the above remarks. On the other hand, its second term reduces to -p(t)’ because d,(t) = H,(p(t)) and k E J ( t ) . These results, combined with (541, imply
By (57) and Lemma 19.3, the function p ( t ) is nonincreasing. This, together with the lower boundedness by (52), ensures the monotonic convergence of p ( t ) to a nonnegative limit lim p ( t )
=
t++m
inf p ( t ) = 6 2 0. t20
If 6 > 0, and hence p ( t ) 2 S > 0, we would find, by (57) and Lemma 19.2(ii),
+ p(0)
p ( t ) 5 -6’t
which results in a contradiction: p ( t )
--f
( t 2 O),
- 00 ( t --t
+ co). Hence
lim p ( t ) = 0 (monotonically). r++m
In view of the boundedness p ( t ) 5 p(0, ,7)@ by (29) for n
lim sup t++m
6 = ,7,
we find
n
C ffi(p(t)) = E 5 p(O, j )iC= l pi lim i=l
p(t) = 0
t++m
from (53) and (58). Suppose imply
(59)
< 0. Then, (59), combined with (2), would
E
C hi(2) 5 4 2 < 0
i= 1
(t
2 to)
for some t o , which yields, by Lemma 19.2@), a contradiction n
lim t++m
C pi(t) = - 00. i=l
Hence E = 0, so that there is a sub-sequence (t,,} tending to lim v++m
1 H.i ( p ( t v ) )= E = 0 ,
i=l
+ co and satisfying
354
VI. GROSS SUBSTITUTABILITY
where {p(t,)} may be assumed t o converge to p ( m ) in r(po) from the fact that p(t,) E l-(po) (v = I , 2, . . .), a compact set. It will be shown that p ( m ) is an equilibrium price vector fulfilling I ( p ( c 0 ) ) = Z(po), and we therefore have a situation such as was desired in the conclusion of the general remark for (7). In fact, f ( p ( o 0 ) ) = f(po) is obvious from p(m) E Tfp'). On the other hand, substituting t , for t in (53) and letting v + + m , we find
H i ( p ( c o ) ) 5 p i ( . o ) . lini p ( t , ) v- +
=0
(i
=
1,
..., n )
(61)
rl)
from continuity and (58), whereas n
from continuity and (60). Thus (61) and (62) entail
H , ( p ( m ) ) = 0 (i = I , . . . , 17).
(63)
. . . , n)
(64)
Equation (63) implies E ; ( p ( m ) )= 0 (i = I ,
by assumptions (a') and (b'), except possibly for a numr'raire, if any, say the Ith good, whose excess demand E,(p(co))must also vanish following (64) for i # I and the Walras law, as was to be shown. This completes the proof. The general idea of the method of proof stated above is due t o Arrow et al. ( I 959) and Uzawa. Of its particular parts, (y.i) is due t o Arrow et al. (1959) and to Arrow and Hurwicz (1960); (y.ii) is due to Arrow et al. (1959);
and (y.iii) is due to Uzawa (1961). An alternative result by Arrow and Hurwicz (1962), which is not presented here, can claim extreme generality among numerous results on global stability i n gross substitutes systems. Unlike the results given here, it takes explicit advantage of the continuity of solutions of differential equations with respect to the initial position.
Chapter VII
THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
I n special reference to factor price equalization and the uniqueness of competitive equilibrium, conditions Lor the global univalence of a mapping on its Jacobian matrix are stated. 920 expounds univalence theorems. 921 explains their applicability to special Jacobian matrices.
520. CONDITIONS O N T H E JACOBIAN M A T R I X FOR GLOBAL UNIVALENCE 020.1. Underlying Motivations
Consider a system of n equations in n unknowns , f i ( x l ,x2,. . . , x,) = p i (i = 1,
. . . , n),
(1)
where ( a ) J;(x,, x2, . . . , x,) (i = 1, . . . , n ) are real-valued differentiable functions on a region X of R” (as to differentiability, cf. Definition 5.2 in Chapter I) with total differentials
dfi =
C fij(x,, x 2 , . . ., x,)
dxj
j= 1
( i = I , . . ., n ) ,
(2)
(p) p i (i = I , . . . , n) are given constants. The Jacobian matrix (3) fnl
fn2
355
356
VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
provides us with some information about the properties of solutions of (I). I t is a well-known classical result that the nonvanishingness of the determinant of J at a solution of ( I ) rules out the existence of a second solution of ( I ) in the vicinity of the former (see Chapter I , Theorem 5.6(i)). More specifically, the mapping ( x i )+ (fi(xl, x ~ . ., . , x,)) : X + R" defined by the functions on the left-hand side of ( I ) is univalent in the vicinity o f a point if the elements f i j of J are continuous and the determinant of J does not vanish at the point (see Chapter I, Theorem 5.6(ii)). These classical results are all local and ensure only the local uniqueness of solutions or the local univalence of the mapping. There is often a demand i n economic theory and related fields, however, for conditions on J that ensure this global uniqueness of solutions of ( I ) or the global univalence of the mapping in a given whole region; and little related material seems to be available in the orthodox mathematical literature. The present chapter is designed to fill this gap to some extent by giving certain results ensuring global univalence and their applications on the basis of recent work by Gale and Nikaido (1965) and lnada (1966). Of various motivations for the study of global univalence in terms of the Jacobian matrix, a typical and most stimulating one is that from the proposition of factor price equalization in the theory of international trade. A controversial theme concerns whether or not rewards to production factors equalize by trade, without their international movements, among countries having the same technological possibilities but different factor endowments. The equalization thesis has both advocates and antagonists. Samuelson (1948b, 1949, 1953-1954), an advocate of the thesis, bases his argument on a serious consideration of the important, though special, case where the number of goods equals the number of production factors. This case is formulated mathematically as follows. Suppose that there are / I goods and 17 factors of production, and that all the countries involved possess the same technological possibilities in which constant returns to scale prevail, but from which both joint production and external diseconomies (within each sector producing a single good in each country) are assumed to be absent. Recalling the formulation of production possibilities in terms of technology sets (cf. Chapter IV, especially i n reference to (A.I)-(A.5) in sI3.l), for each i = I , Ti: the technology set of the sector producing the ith good, where ( A . l ) Tiisaclosedconvexconelyinginthe Cartesian product R," x R + ' . I f (xi, 1.J E T i ,then xi = (xji)and R i represent the input vector and the corresponding output of the ith good, respectively, with x i i standing for the required amount of thejth factor of production. (A.2) (0, 1.;) E Tiimplies 1.; = 0 (the impossibility of the land of Cockaigne). (.4.3') There is a process in Tithat produces a positive output.
$20.
CONDITIONS FOR UNIVALENCE
357
For any given factor price vector w 2 0, the value of the (unit) cost function ci(w) of the ith good is defined to be the minimum of the inner product ( w , x i ) over all processes
(.xi,1) E Ti.
The reader can readily verify that the cost function ci(w) is well defined for w 2 0 under the above assumptions. Now let us further assume that (A.5) If ( x i , Li) E Ti,(ui, pi) E T i , and if x i and u i are linearly independent, then for any positive numbers a > 0, p > 0, a + /3 = 1, there is a level of output of the ith good q i such that
aLi + ppi < q i (strict convexity).
(axi + /hi, qi) E Ti,
Then, in essentially the same way as in the treatment of J(q, a) = ( q , o(q, a ) ) in Chapter VI, $18.1, the reader can also verify the following facts: (i) For any positive w > 0, the unit input vector minimizing the unit cost of each good i is unique and can therefore be regarded as a single-valued function ai(w) = (aji(w))of w > 0. Hence 0
Ci(W)
=
.
1Ujt(W)Wj
(w > 0).
(4)
j = 1
(ii) aji(iv) ( i , j = 1, . . . , n) are continuous and nonnegative for w > 0. (iii) ci(w*)(i = 1, . , n) have continuous partial derivatives for M' > 0 and % ,
meaning the equality of the partial derivatives of the (unit) cost functions to the (unit) factor intensities. As a matter of course, the cost functions are common to all countries from the hypothesis of their common production possibilities. Therefore, if goods are traded in the competitive world market under a price system p = (pi),the world price p iof the ith good must equal its unit cost ci(ii,) in equilibrium, thus resulting in a system of 17 equations in the single unknown factor price vector it-, = p ; ( i = 1, . . . , n).
(6) It should be noted that (6) must hold in equilibrium, however unevenly endowed countries may be with factors of production. Samuelson argues that if (6) admits only a single solution i t ' , then the equalization of factor prices results without international factor movements. System (6) is undoubtedly a typical example of (1). The elements of the Jacobian matrix of system (6) are factor intensities. Whence conditions on the Jacobian matrix of ( I ) for the global uniqueness of solutions of (1) can be translated into conditions on factor intensities for that of (6) over the whole region of positive factor price vectors it', ensuring the factor price equalization in Samuelson's sense. Ci(ll.)
358
VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
Another motivation for the study of global uniqueness in system (1) in terms of its Jacobian matrix comes from the system of equations in general equilibrium. Let E,(p) = 0 (i = 1, . . . , n)
(7)
be a system of n equations, where p = (pi)is the positive unknown n-dimensional price vector, p i standing for the price of the jth good, and E , ( p ) (i = I , . . . , n) are excess demand functions for the ith good. Suppose that the Walras law in the narrow sense holds, and E,(p) are homogeneous of order zero. Then, if the nth good is taken as a numeraire, the determination of equilibrium price ratios p 1 : p 2 : . . . : p , through the solution of (7) reduces to the study of the system of n - I equations in the positive unknowns p l , p z ,. . . , Pn-1:
E i ( p , ,p 2 , . . . ,p n - l , 1) = 0 ( i = I , . . . , n - 1).
(8)
If the excess demand functions are differentiable in the region of positive p i - l), conditions on its Jacobian matrix ( E i i ) of (8) for the global uniqueness of solutions, if any, will be of great importance in traditional economic theory. System (8) is obviously of type (1). Remark 1. The cost functions ci(w) have the following interesting properties : (i) Homogeneity of the First Order. That is (i = 1, . . . , n
ci(cut') = ctc,(~v) (a 2 0,
11'
20, i
. . . , n).
= 1,
This is obvious by definition. (ii) Nonnegativity of Function Values. That is Ci(lt.)
20
(11%
2 0, i = I , . . . ,n),
and they are strictly positive if 11' > 0. Their general nonnegativity is obvious by definition. Their strict positivity for I(' > 0 is implied by (A.2) and iv > 0. it** 2 0, I 2 r 2 0, we have (iii) Concavity. That is, for 111,
ci(( 1
- t ) n , + m*)2 ( I
- r)ci(rr.)
+ rc,(iv*)
(i = I , . . . , n).
Let (x, I ) , (x+,I), and ( y , I ) be unit cost minimizing processes in Ti associated with II'* and ( I - t ) ~+, t i t , * . Then, one finds, for each i, 111,
Ci((
1
proving concavity.
- t)lV
+ m*) = (( 1 - t)ll, + t w * , y ) = (1 - t)(Il., y ) + f ( W * , 2 ( 1 - t)(W, x) + /(It'*, = (1 - t)Ci(ll.) + fC,(N'*),
y) x*)
520.
CONDITIONS FOR UNIVALENCE
359
(iv) Monotonicity. That is, w* 2 u’ 2 0 implies Ci(W*) 2 Ci(u’) (i = 1, . . . , n), which is obvious by definition and from the fact that x i 2 0 for ( x i , I ) E Ti by (A.1). For positive w > 0, this monotonicity is represented as the nonnegativity of the partial derivatives c i j ( w ) . Remark 2. The classical result for n = 2 by Samuelson on the factor price equalization, in his sense as global uniqueness in system ( 6 ) , is decisive and suggestive, although it is simple and easy to prove. Assume that the Jacobian determinant is nonvanishing in the region of all positive two-dimensional factor price vectors u’. This assumption ensures by virtue of the continuity of cij(w) that the determinant is either everywhere identically positive or everywhere identically negative. Whence the factor proportions of both sectors never reverse. In the light of the homogeneity and positivity of c,(\ti1, w2) (i = I , 2) noted in Remark 1, the uniqueness of positive solutions wl, 1 1 ’ ~ of (6) (n = 2) reduces to the uniqueness of positive solutions of the equation
in the single unknown w (= w2/wl), the factor price ratio. Using (4) and (5), we can compute the derivative of the function on the right-hand side of (9), obtaining
which is of the same sign for all o > 0. Whence the function c2(1, o)/c,( I , w ) is either strictly increasing or strictly decreasing, so that it can be equated to p 2 / p 1only once at most. Remark 3. The classical result for M = 2 in Remark 2 takes considerable advantage of certain of the properties of cost functions. The same conclusion of global uniqueness can also be reached by merely relying on the nonnegativity of the elements of the nonsingular Jacobian matrix, without appealing to properties (i) and (ii) in Remark I (see $20.5). The first higher dimensional determinantal condition on the Jacobian matrix for global uniqueness was proposed by Samuelson (1953-1954). This condition is as follows: If its equations and unknowns are suitably renumbered, the Jacobian matrix of system ( I ) has all its upper-left-hand-corner principal minors nonvanishing everywhere in the region X , that is,
lfkl
.’‘
fkk
I
360
VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
The condition has not yet been proved to ensure global uniqueness for system (6), which has cost functions on its left-hand side and is therefore endowed with certain special properties not shared by ( I ) in general. But it has been shown (Gale and Nikaido (1965) that condition (10) alone is not sufficient for global uniqueness in system (1) in general, even in a rectangular region. This is verified by the following example. The system of equations
+ 3 = 0,
f l ( x , , x2)= exp(2x,) - x 2 2
f2(x,, x2) = 4 exp(2x,)x2 - x23 = 0 has two solutions x, = 0, x2 = k2, although fl ,(XI> x2)
=2
exP(2x*)> 0,
the whole plane R2. Remark 4. Let J ( w ) = (cJN~))be the Jacobian matrix of system (6). If J ( w ) is a constant matrix independent of MI,(6) is in effect a system of linear equations, so that the nonsingularity of J ( w ) entails the uniqueness of solutions of (6) in this extremely special case. McKenzie (1955) observed that if one takes the concavity of cost functions ((iii) in Remark I ) into account, the uniqueness of solutions by the constancy of J ( w ) can be extended to a somewhat general nonlinear situation whereJ(1v) need not be constant, but is nearly constant in the following sense. For a square matrix A of order n, let us denote by K ( A ) and K(A)* the convex cone generated by the n column vectors of A and its interior in R". Then, J ( w ) may be said to be nearly constant if for any H!' > 0, i t 2 > 0, we have in
a.
~ ( ~ ( r t , ' ) ' ) ' n K ( J ( M , ~ ) '# )"
(1 1)
It can be readily seen that if J ( w ) is nearly constant, the uniqueness of solutions obtains in system (6). Let IV', w2 be two solutions of (6). Using the vectors c(w) = (cl(it-)), p = ( p , ) , we can put (6) in the form ~ ( 1 1 3 ) = p , so that c(d)
= c(w2) = p .
(12)
By Lemma 18.2 in Chapter VI, we find J(\tJ)(td
- I?)
5 C(lV1)
J(IP)(lV' - w') 2
C(W')
- c(,v') = 0,
- C(M'2)
=0
from (12) and the concavity of cl(it>). Note that (1 1) implies in particular the nonsingularity of J(it1). Hence, if either J(wl)(wl- w 2 ) = 0 or J(w2)
$20.
CONDITIONS FOR UNIVALENCE
36 1
w2)= 0, we have w1 = w2 from nonsingularity. It therefore remains to consider the case where
(wj’ -
J(W’)(W’
- d)5 0,
(13)
J(w2)(w’
-
2 0.
(14)
By (1 I), there are two positive vectors
WJ) X I
> 0, x2 > 0, for which
J(W”)’X’ = J(WJ)’X2 = u.
(15)
Forming inner products between x i and (13) and between x2 and (14), we obtain (u, w1 - Wl’) < 0 < (u, w1 - UJ), yielding a contradiction. Despite motivations from many disciplines, however, there are only a few works in the orthodox mathematical literature that deal with conditions on the Jacobian matrix for global univalence; and even these are quite recent. The joint research of Gale and Nikaido (1965), which considers the case of the Jacobian matrix having positive principal minors as well as other cases, seems to be the first general work along this line. Later, an interesting related result on the case of the Jacobian matrix having negative principal minors was obtained by Inada (1966), who applied a similar method of proof. The main purpose of the present chapter is to present these results (420) with applications (421). 020.2. P-Matrices and Related Concepts
For a real square matrix A of order n, we present a few definitions.
DEFINITION 20.1 (Gale and Nikaido, 1965). A is termed a P-matrix if all the principal minors of A are positive. DEFINITION 20.2 (Inada, 1966). A is termed an N-matrix if all the principal minors of A are negative. Moreover, N-matrices are classiJied according to the fof/owYng cutegories: (i) A n N-matrix A is said to be of the Jirst category if A has at least one positive element. (ii) A n N-matrix A is said to be of the second category if all the elements of A are nonpositive. DEFINITION 20.3. A is termed an N-P-matrix if it has all the principal minors of odd orders negative and those of even orders positive. DEFINITION 20.4. A is termed a P-N-matrix if it has aN the princ@al minors of odd orders positive and those of even orders negatice.
362
VIJ. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
We begin by stating several lemmas and theorems that are of fundamental importance in what follows. As will be made clear, the role played by P-matrices is so decisive that the handling of N-matrices not only depends on the results on P-matrices, but is also very similar in treatment. Unless otherwise stated, in what follows, we mean by a matrix a real square matrix of order n. LEMMA 20.1. The follon.ing (i), (ii) hold: (i) A is an N- P-matrix if and only if - A is a P-matrix. (ii) A is a P-N-matrix ij' and only if - A is an N-matrix. Proof. Propositions (i) and (ii) are clear from the basic properties of the determinant, Q.E.D. LEMMA 20.2. A simultaneous and identical reordering of rows and columns converts P-matrices, N-matrices, N-P-matrices, and P-N-matrices again to P-matrices, N-matrices N-P-matrices, and P-N-matrices, respectively. Proof. By such a reordering, the values of the principal minors of a matrix do not change, whence follows the assertion, Q.E.D. LEMMA 20.3. Let L be a (proper, nonproper, or empty) subset of N = (1, 2, . . . , n}, and D be the matrix obtained by replacing the j t h column ej of the identity matrix by -e' f o r j E L. Then, D - ' A D remains to be a P-matrix, an N-matrix, an N-P-matrix, or a P-N-matrix, respectively, accordittg as A is a P-matrix, an N-matrix, an N-P-matrix, or a P-N-matrix. Proof. Every principal minor of D - ' A D is derived from the correspond ing original principal minor of A by multiplying the latter by an even power of - 1 ; therefore, they must be equal. This proves the lemma. Also note that D-' = D, an obvious but important fact, Q.E.D. Remark. It should be noted that if A is an N-matrix i n Lemma 20.3, A and D - I A D need not be of the same category. LEMMA 20.4. Every principal submatrix of a matrix A is a P-matrix, an N-matrix, an N-P-matrix, or a P-N-matrix, according as A is a P-matrix, an N-matrix, an N- P-matrix, or a P- N-matrix. Proof. This is clear by definition, Q.E.D. THEOREM 20.1. For a given matrix A , the system of linear inequalities
A x S 0,
x>,o has only the triuial solution x = 0 if either (i) or (ii) holds: (i) A is a P-matrix (Gale and Nikaido, 1965); (ii) A is an N-matrix o j thejirst category (Inada, 1966).
$20.
CONDITIONS FOR UNIVALENCE
363
Proof. (i) We proceed by induction on n. The result for n = 1 is obvious. Assuming the result for n - 1, consider the case for n. Since A as a P-matrix has all its diagonal elements positive, we can increase some components of x, if necessary, until we obtain AY 5 0, (3)
with equality to zero for at least one component of A y , say the kth one, and
(4)
y?x_20.
Clearly, (i) will have been proved if it can be seen that y = 0. Now, in the light of Lemma 20.2, we may assume, without loss of generality, that k = 1. Explicitly, we find ally1 + a , 2 ~ 2 + . . . + a , n ~ n = O
+
~ 2 1 ~ ~1 2
2
+~ * 2. * + a2nyn 5 0 (5)
only1
+ a n 2 ~ 2+
+ Q n n Y n 5 0.
We note a,, > 0 and perform the following elimination: the ith relationai,/al times the first relation in ( 5 ) (i = 2, . , . , n) as in the proof of Chapter 11, Theorem 6.1. Thus we obtain
,
ally,
+ ... + Q l " Y " = 0 4 2 y 2 + ... + az*yy,5 0
-I-a , , y ,
a:, y2
+ . . . + a:n y , 5 0.
Let A* = (a:) ( i , j = 2, .. ., n) be the coefficients matrix of the system, consisting of all the relations in (6) except the.first one. If C* = (a,*,)( i , j E K ) , where K is a nonempty subset of ( 2 , . . ., n), is an arbitrary principal submatrix of A*, then C = (f . j ) (i, j E { 1) w K ) is a principal submatrix of A and
det C =
from the method of elimination. Moreover, a, > 0, det C > 0 because A is a P-matrix. Whence (7) implies det C* > 0, showing that A* is a P-matrix of order n - 1. Then the fulfillment of the last n - 1 relations in (6) by y i 2 0 (i = 2, . . . , n) entails y i = O ( i = 2 , ..., n) (8)
364
VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
from the induction hypothesis. Finally, we find y , = 0 from the first relation in (6) and (8) in conjunction with a, > 0. Hence x = y = 0, Q.E.D. (ii) Since the diagonal elements of an N-matrix are negative, the order of an N-matrix of the first category must be at least two. With this in mind, consider (I), (2) for an N-matrix A of the first category. Since A has positive elements by assumption, we can have the same situation (3), (4) as in (i). Moreover, the argument for (i) leading from (5) to (7) applies directly to the present case, since a,, # 0 ( a l l < 0 ) and the order n of A is at least two, as already noted. The only difference is that we have a , , < 0, det C < 0 in (7) from the property of A being an N-matrix. Nonetheless (7) obviously ensures det C* > 0, whence A* is a P-matrix of order n - 1. Hence (i), as applied to the last n - 1 relations in (6) and y i 2 0 (i = 2, . . . ,n), entails y i = 0 (i = 2, . . . , n), which also implies y , = 0 by the first relation in (6) and a, # 0. Hence x = y = 0, Q.E.D.
,
,
COROLLARY 1. For a given matrix A , the system of linear inequalities A x > 0,
(9)
x>O
(10)
has a solution x i f A is either a P-matrix or an N-matrix of the first category. Proof. The transpose A’ of A is clearly either a P-matrix or an N-matrix of the first category, since A and A’ have the same principal minors in common. Therefore, Theorem 20.1 as applied to A’ implies that A’p 5 0, p 2 0 entail p = 0. Expressed in terms of the n x 2n matrix (Z, - A ) , where I is the identity matrix of order n, this means that the system of linear inequalities p’(Z, - A ) 2 0
has only the trivial solution p’ = 0, - p ’ A = 0. This is nothing but the Stiemke condition (Chapter I , Theorem 3.7(i)) for the existence of positive n-dimensional solution vectors u, x of the system of linear equations A)(:) = 0.
From (1 l), combined with the positivity of u, x, we have Ax=u>O,
x>O;
hence x is a desired solution of (9) and ( l o ) , Q.E.D. COROLLARY 2. 0 1 1 the same assumptions as in Corollary I , it’e catifitid a positice number 6 > 0 such that the maximum of components of A x is as large as 6 f o r all x 2 0 with unit norm (Ix/I= 1.
$20.
CONDITIONS FOR UNIVALENCE
365
Pvoof. Write [ A x l i for the ith component of Ax as before, and let 6 ( x ) = max[Ax], over all i. 6 ( x ) is continuous in R" and therefore takes a minimum 6 on the compact set of all x 2 0 with unit norm j/x//= 1. This minimum 6 must be positive by Theorem 20.1. Clearly, 6 is a desired positive
number, Q.E.D. Geometric Characterization. From the geometric point of view, Theorem 20.1 means that a P-matrix A induces a linear mapping that does not send a point from the positive orthant R," into the negative orthant - R," except for the origin. But this property alone by no means characterizes P-matrices, since it is also shared by N-matrices of the first category, as shown i n Theorem 20.1 (ii). The following theorem provides us with a complete characterization in terms of the concept of sign-reversibility.
DEFINITION 20.5. For a matrix A and a vector x,denote by [ A x l i the ith component of A x , as before. Then A is said to reverse the sign of x if x i [ A x l i 5 0 ( i = 1, . . . , n).
(12)
THEOREM 20.2 (Gale and Nikaido, 1965). A is a P-matrix ifand only i f A does not reverse the sign of any vector except that of the zero .vector. Proof. (i) Necessity. Suppose that (12) holds for a P-matrix A and a vector x . Let L = { j I x j < O), where x = (xi), and D be the matrix in Lemma 20.3 associated with the present L. Then, D - I A D , which is again aP-matrix by Lemma 20.3, clearly reverses the sign of Dx 0. It therefore suffices to deduce the conclusion x = 0 from (12) for x 2 0, so that henceforth we may assume x 2 0. Next, let K = ( k 1 xk > o}. Assume K # 0. Then, let A be the principal submatrix of A obtained by deleting its ith rows and columns for i $ K , and R be the vector obtained from x by deleting its ith components for i 6K. Then A^ is again a P-matrix by Lemma 20.4 and reverses the sign of 2 . Moreover, we observe that 22 can have no positive component, since 2,[A^,t], 5 0, 2k > 0 ( k E K ) . That is, A^.? 5 0, 2 > 0, which contradicts Theorem 20. I(i). (ii) Sufficiency. Let A^ = ( a i j ) (i,,jE K ) be an arbitrary principal submatrix of A , where K is a nonempty subset of N = { 1, 2, . . . , n ). We shall have to show that det A^ > 0. In fact, if det A^ 5 0, A^ would have a real nonpositive eigenvalue 15 0 and an associated real eigenvector R # 0. This is obvious from the fact that det A^ equals the product of all the eigenvalues of A, with their multiplicities taken into account, and complex eigenvalues, if any, have their conjugate counterparts. We may conveniently number the components of 2 in a natural order by using integers in K. Then define an n-dimensional vector x = (xi) by x i = 2i (i E K ) , = 0 (i $ K ) . A reverses the sign of x # 0, since x i [ A x l i = 2i[A^2]i = Hi25 0 (i E K ) , xi[Axi] = 0 (i 4 K ) . This contradicts the assumption. Therefore det A^ > 0, and hence A is a P-matrix. This completes the proof.
366
VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
920.3. Univalence Theorems Let f : X - t R"
(1)
be a differentiable mapping, where (E)
X is a region of R";
(b) f ( x )= (f i ( x ) )( x E X , i = 1, . . . , n),f i ( x )being differentiable functions on X , with total differentials n
dfi
=
C fij(x)dxj
(i = 1,
. . ., n ) .
i =1
Coupling the mapping (1) with a constant vector p
f (4 = P?
= (pi) E
R", we obtain (3)
an equation in the unknown vector x in X , which is a vector form of (1) in $20.1. The uniqueness of solutions of (3) for the unspecified (i.e., indefinite)p implies the (global) univalence of the mapping (l), meaning that (1) is one-toone in X , and vice versa. We are concerned with conditions on the Jacobian matrix J ( x ) = CfijCX)) (4) of (1) that ensure the univalence of (1). Special attention will be paid to univalence theorems that are derived by extending Theorem 20.1 to nonlinear situations.
THEOREM 20.3. Assume that X is a rectangular region of R". Then, for any fixed a E X , the inequalities
f(4 5 f (a),
(5)
X Z U
(6)
have only the trivial solution x = a in X if either (i) or (ii) holds: (i) The Jacobian matrix J ( x ) of the mapping (1) is everyctihere a P-matrix in X(Gale and Nikaido, 1965); (ii) The Jacohian matrix is continuous and everyicyhere an N-matrix of the first category (Inada, 1966). Proof. (i) The proof will be worked out by induction on M, the common dimension of both the argument and the image vectors x, f ( x ) of ( I ) . The result for n = 1 is obvious. Assuming the result for M - 1, let us consider the case for 17. Denote by M the set of all solutions x of (5) and (6) in X . Since the result holds trivially in the case where s 2 a for no x E X , we have only to consider
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CONDITIONS FOR UNIVALENCE
the nontrivial case where the opposite situation prevails. With this premise in mind, it will be shown first that a is an isolated point of M . In fact, differentiability (see Definitions 5.2 and 5.3 in Chapter I) implies
Since J(a) is a P-matrix, there is a positive number S > 0, by Corollary 2 to Theorem 20.1, such that for any x in X fulfilling x 2 a, some component of J ( a )
x-a ~
IIX
- all
2s>o. -
From (7) and (8), it follows that in the vicinity of a, some component of f ( x ) - f ( a ) must be positive for x 2 a in X . This proves that a is an isolated point of M . What we shall have to prove is that A4 contains only the single point a. To this end, suppose that A4 contained a second point b # a, which will be fixed once for all in the following argument. Clearly b 2 u, and X includes the bounded closed cube {x I a 5 x 5 6). Consider the subset of X :
Q = {x I a S x S 6,
A x ) Sf(a)>.
We recall the remark immediately following Definition 5.2 in Chapter I that differentiability implies continuity; hence mapping ( I ) is continuous. Whence Q is a closed subset of the compact cube, and is compact by Theorem 1.2(ii) in Chapter I. Moreover, Q\{a) is a closed set in Q , since a is an isolated point of M . Hence Q\{a} is also compact, again by Chapter I, Theorem 1.2(ii), while it is nonempty because of the inclusion of b. The sum of the components of x is therefore minimized at some X over Q\{a}. Clearly, this X is minimal in Q\{a} in the sense that no other element x i n Q\{a} fulfills x X. Now, in view of the obvious fact X 2 a implied by X E Q\{a}, only two possible situations are conceivable: (i.1) X > a ; (i.2) X 3 a. It will be shown separately in what follows that either situation entails a contradiction, thereby proving A4 = { a } . (i.1) X > a. Corollary 1 to Theorem 20.1, as applied to J ( X ) , a P-matrix, ensures the existence of a vector u satisfying u
< 0,
J(X)u < 0.
Then, noting .? > a, u < 0, we can choose a positive number we can let the vector-valued function
x(t) = ,u
+ tu
of a real variable t satisfy
x > x(t) > a
(e > t > 0).
(9) E
so small that
368
VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
Since the rectangular region X contains X as well as a, we find X 3 x(t)
(E
> t > 0)
from (10). Moreover, by differentiability, we have
. f ( x ( t ) >= f G >+ j J ( +
+ 4Ilull)
(E
> t > o>,
so that
can be made as small as we like by letting t > 0 approach zero. This, in conjunction with the second inequality in (9), implies that for a sufficiently small positive I, f ( 4 t >I < f ( 32 f ( 4 . (12) It follows that x ( t ) E Q\{u} for this value o f t . In view of (lo), however, this contradicts the minimality of X. (i.2) X 3 a. In this case a t least one component of X = (Xi)equals the corresponding component of a = (ai). We may assume, renumbering the equations and variables if necessary, that X1 = a,, without invalidating the basic premise of a P-Jacobian matrix. With this understanding, define a new differentiable mapping?: 2 + R"-', where
(4) 8 = {(xz,xj , . . . , x,,)' I ( u l , x 2 , xj , . . . , x,,)' ion in R " - ' ; (/I> f(.x2,
-Yj,
. . . , .x,) = ( f i ( X 2 ,
Sj,
. . . , x,) =.f;.(u,, s 2 , x3,
x3,
E
x } ,a rectangular reg-
. . . , x,))
and fl(.Y2,
. . . , x,J (i = 2,
. . . ,n).
The Jacobian matrix of this new mapping is a principal submatrix of order I o f J(.x), evaluated for x 1 = ctI, which is a P-matrix by Lemma 20.4. Moreover,
17 -
- Y ~ ,. . . ,
x,> z f i ( u 2 , u 3 , . . . , a,) -Yi
(i = 2, . . . , n),
a , ( i = 2, . . .,1 7 ) .
Therefore we must have Xi = a , ( i = 2, . . . , n ) by the induction hypothesis. This contradicts S # a. (ii) The proof of (ii) almost parallels that of (i) except for the need to supplement the latter because of the classifications of N-matrices into the two categories. The supplementary remark is due to the fact that although a principal submatrix of an N-matrix is also an N-matrix, they need not be of the same category.
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CONDITIONS FOR UNIVALENCE
Now, in view of the indiscriminate validity of Theorem 20.1 and its two corollaries for N-matrices of the first category as well as for P-matrices, we can observe the direct applicability of the argument in the proof of (i) to the present case (ii), except for two points: (a) The induction must start at the case for n = 2, since N-matrices of the first category are at least of order 2; (b) The Jacobian matrix of the new mappingf : r? + R"-' should be shown not only to be an N-matrix but also to be of the first category. The proof of (ii) will therefore be complete if these two points are thoroughly discussed. (a) The Treatment of the Case for n = 2. We have the inequalities fi(x1, x 2 )
S f i ( a l , a21 ( i = 1, 21,
(13)
xi 2 a, ( i = I , 2)
(14)
in a rectangular region in R2 with the N-Jacobian matrix of the first category. We find
from theN-matrix property ofthe Jacobian matrix. Whencef,, f 2 , >fl f Z 2> 0, so thatf,, , f 2 1 are of the same sign. Note that the Jacobian matrix has positive elements by its first categoricity, while its diagonal elements are negative. Hence f 1 2 > 0, f 2 1 > 0. (15) Therefore, interchanging the numbering of equations, we can reduce (ii) for n = 2 to a case with a P-Jacobian matrix, while essentially preserving (13) and (14). (b) It should first be noted that under continuous differentiability, as is the case with (ii), the elements of an N-Jacobian matrix J ( x ) = ( f i j ( x ) ) do not change signs but are of definite signs throughout the whole region. In fact, the diagonal elements must be negative by the very definition of an N-matrix. Next, for i, j ( i # j ) we find
from the N-matrix property ofJ(x). Whencefij,fji >fii,fjj > 0, which implies that.fij ,.fji are not zero and that they are of the same sign. Since they are continuous by assumption,fij,fji ( i # , j ) are of the same and definite sign in the whole region (a connected set) for each pair (i, j ) , i # j , by Chapter I, Theorem 1.8. Let us now see that the new mapping f : 8 R"-' i n (i.2) has an NJacobian matrix of the first category in the present case (ii). As a matter of course, the Jacobian matrix j(x,, x3, . . . , x n ) o f f is the principal submatrix --f
'
370
VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
J,(al, x,, . . . , x,,)obtained by deleting the first row and column of the original Jacobian matrix J(x) and evaluated for x1 = a,. It therefore suffices to see that J,(x,, x2, . . . , xn)is an N-matrix of the first category in X . That J 1 ( x l ,x2,.. . , x,,)is an N-matrix is obvious by Lemma 20.4. Next, suppose .fi,(Xl,
Xz . . . , X") > 0 (.i= 2, . . ., n). 3
(16)
Then,
f,,(x) > 0 (for any x E X , j = 2, . . . , n),
(17)
by the preliminary remark in (b). But, since XI = a , , X i 2 ai (i = 2, . . . , n), with strict inequality for at least one i 2 2, (17) must entail fi(X) >fi(a), contradicting (5). Hence (16) is ruled out, so that the left-hand side of (16) must be nonpositive for at least o n e j 2 2, say k , obtaining fik(X1, 2 2
9
.. . > Xn) I 0.
Then, if J,(X,, X, , . . . , 2,) were of the second category and hence had only the nonpositive elements, for the N-matrix of the first category J(X), the vector u = (vi) defined by i l k = 1, v i = 0 (i # k ) would be a nontrivial solution of the inequalities J(X)v 5 0, v >= 0. This contradicts Theorem 20.1(ii). Thus J,(X,, X,, . . ., X,) has some positive elements, which must remain positive in the whole region, again by the preliminary remark on their definite signs. Hence, J,(x,, x, , . . . , x,) is everywhere of the first category. The supplementary remarks are thereby complete. The reduction of the proof of (ii) to that of (i), as well as the indispensable supplementary remarks, are essentially due to Inada (1966). It is now possible to derive useful univalence propositions from Theorem 20.3. THEOREM 20.4. The mapping ( I ) , it-here X is a rectangular region, is univalent if either (i) or (ii) holds: (i) The Jacohiari matrix J ( x ) o f tlie mapping is euery,ihere a P-matrix in X (Gale and Nikaido, 1965); (ii) The Jacobiaii matrix is coiitiiiuous and is everyic,here an N-niatrix in X (Inada, 1966). Pvoof. We shall have to prove that f(a)
= f ( b ) , a, b E X
(18)
imply a = 6. To this end, let L = { , j I a j > b j } and D be the matrix corresponding to this L in Lemma 20.3. We define a new mapping g : Y + R" by the following formulas: (a") Y = D - ' X is again a rectangular region in R"; (Po) g ( y ) = D - ' f ( D y ) , with its Jacobian matrix being
J&)
=
D-'J(Dy)D.
$20. For a*
=
D-'a, b*
=
CONDITIONS FOR UNIVALENCE
37 I
D - ' b in Y, clearly we find s(a*) = g(b*),
(20)
a* 5 b*.
(21)
Equations (20) and (21) imply a* = b*, and hence a = b, as will be shown separately for (i) and (ii), below. (i) J,(y) is everywhere a P-matrix in Y by (19) and Lemma 20.3. Hence, Theorem 20.3(i) as applied to (20) and (21)-a special situation of the inequalities g(a*) 2 g(b*), b* 2 a*-ensures a* = b*. (ii) J,(y) is everywhere an N-matrix in Y by (19) and Lemma 20.3. Jg(y)is continuous in Y from expression (19) and the assumed continuity of J(x). Therefore, J,(y) is of one and the same category throughout Y for the reason stated in the preliminary remark in (b) in the proof of Theorem 20.3(ii). The discussion henceforth will deal separately with (ii. I ) and (ii.2), depending on the category of Jg(y). (ii.1) The first category case. a* = b* is implied, as in (i), by Theorem 20.3(ii) as applied to g(a*) 2 g(b*), b* 2 a*. (ii.2) The second category case. All the elements of Jg(y)are everywhere negative by the preliminary remark in (b) in the proof of Theorem 20.3(ii). Thus, the supposition a* # b* in (20) and (21) would entail a contradiction g(a*) > g(b*). This proves a* = b*, Q.E.D.
COROLLARY. The mapping (I), where X is a rectangular region, is univalent i f either (i) or (ii) holds: (i) The Jacobian matrix J ( x ) of the mapping is everywhere an N-P matrix in X ; The Jacobian matrix is continuous and is everywhere a P-N matrix in (ii) X. Proof. This is immediately implied by Lemma 20.1 and Theorem 20.4. Univalence by a P-Jacobian matrix in Theorem 20.3 (i) can be put in a more general form, which is a nonlinear counterpart of the necessity part of Theorem 20.2.
THEOREM 20.5.
For the mapping (I), the inequalities
(xi- a,)(fi(x) -fi(a)) 5 0 (i = I , . . . ,n)
(22)
have only the trivial solution x = a if the region of dejnition X is rectangular and the Jacobian matrix J ( x ) is every\z>herea P-matrix in X . Proof. For a solution x of (22), letting L = { j I x j < a j } , we can define the corresponding coupled mapping g : Y + R" exactly as in the proof of Theorem 20.4. The Jacobian matrix J,(y) is again everywhere a P-matrix in Y.
312
VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
After the transition from the original f to the new g, (22) obviously takes the form (Xi*- a,*)(g,(x*)- g,(a*)) 5
0 ( i = 1, . . . , n),
(23)
with x* 2 a*,
(24)
where x* = D - ' x = (xi*), a* = D - ' a = (ai*). Next, let K = { j 1 xj* > a j * ) .The proof will be complete, if it is shown that K = @. With this in mind, suppose K # @. We may assume, by virtue of Lemma 20.2, that K = { I , 2, . . . , k } while (23), (24), and the P-matrix property of J&y) are still valid. Consider the mapping h : Z + Rk defined by (E) and (8):
(G) z = { z I z = ( z i ) , (zl, . . . ,z k , a,*+l,. . . , a,*)' tangular region in R k ; (8) h(z) = (hi(z>>, hi(z)=gi(ZI,
.. . ) z k , a,*,,, .. . ) a,*)
E
Y } is again a rec-
...
( i = 1,
9
k),
(25)
with the Jacobian matrix Jh(z)being the principal submatrix of Jg(zl, . - . >z k , a:+,,
. - . )a n * )
(26)
formed by its first k rows and k columns. Then, we find hi(xl*, . . . , xk*) 5 hi(al*, . . . , ak*) (i = 1, xi* > a,*
. . . ,k),
(i = 1, . . . , k )
(27) (28)
from (23), (24), and the definition of K . On the other hand, Jh(z)is everywhere a P-matrix in Z , by Lemma 20.4. These results contradict Theorem 20.3(i). hence x* = a*, which implies x = a, Q.E.D. This proves K = 0; COROLLARY. (Xi
For the mapping ( I ) , the inequalities - Ui)(fi(X)-.f,(a))
1 0 (i = I , . . . , n)
have only the trivial solution x = a if the region of definition X i s rectangular and the Jacohiun matrix J ( x ) is everywhere an N-P-matrix in X .
Proof. This is immediately implied by Lemma 20.1 and Theorem 20.5.
The remaining part of $20.3 will be devoted to special situations where the Jacobian matrix is a particular type of P-matrix.
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CONDITIONS FOR UNIVALENCE
THEOREM 20.6 (Gale and Nikaido, 1965). Assume that the Jacobian matrix J ( x ) = (f i j ( x ) )of the mapping (1) has nonpositive of-diagonal elements fij(x)
2 0 ( i Zj)
(29)
in the region X . Then (i) and (ii) hold: (i) I f X is rectangular and Jex) is everywhere a P-matrix in X , the mapping (1) is unicalent, and its inverse mapping f is monotonic increasing. That is,
f (a) 5 f (b)
for a
= (a,),
b
= (bi)E
X
(30)
implies a 5 b. (ii) Suppose that the mapping (1) is univalent in an arbitrary open region X , not necessarily rectangular. I f its inverse mappjng f is diferentiable and monotonic increasing, then the Jacobian matrix of the mapping (1) is everywhere a P-matrix in X . Proof. (i) Univalence is already proved in Theorem 20.4(i). Monotonicity remains to be proved. Noting obvious monotonicity for n = 1, we proceed by induction on n. Assume monotonicity for n - 1, and consider it for n. It should be noted first that (30) rules out a > b, by Theorem 20.3(i), and hence we must have ak 5 b, for some k. By Lemma 20.2 and the invariance of condition (29) under a simultaneous and identical renumbering of equations and unknowns, we may assume that k = 1. As was noted before (Chapter I, §5.3), differentiability implies partial differentiability, whenever partial differentiation can be performed. In reality, partial differentiation can be performed everywhere in a rectangular region provided one-sided partial differentiation is applied to boundary points, if any. Under this convention, we find fi(b,,a2, . . . , a,) 5 f i ( a l ,a,, . . . ,a,) s f i ( b , , b,, .. ., b,)
( i = 2, . . . , n),
(31)
the first inequality holding from-f,, 5 0 (i > 1) and a, S b,, and the second being implied by (30). We next define a rnapping.7: ??-+ R"-' by the rules:
- X- = {(x,,
(ct)
- -
x 3 , . . . , x,)' I ( b l ,x2, x3, . . . , x,)'
E
X } , a rectangular re-
gion in R"-' ;
(P)
f(X2
2
x3 2
' ',>
x,) =
(.hx3 , 3
fi(b,, x2 , x 3 , . . . , x,) ( i = 2, . . . , n).
In terms o f x (31) is written as
-
.f(az, . . . ,a,)
3
' . . , x,)>
-
and
5f@, , . . . ,b,,h
icy,
>
-y3,
. . . , x,) =
(32)
374
VII.
THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
The Jacobian matrix off^is again everywhere a P-matrix having nonpositive off-diagonal elements in r?, by Lemma 20.4 and (29). Then (32) implies ai5 bi (i = 2, . . . ,n) from the induction hypothesis. This, in conjunction with a1 <= b,, proves a 5 b, as was to be shown. (ii) We first note that f ( X ) , the image of X underf, is an open region in R", by the Corollary to Theorem 5.7 in Chapter I. Let a be an arbitrary point of A', and put a* =f ( a ) . Choose an arbitrary positive vector u* > 0 in R". Then, since a* is an interior point of f ( X ) , there is such a positive E > 0 that for a real variable t we find x*(t) =a*
+ tu* E f ( X )
(It1
< E).
(33)
Let us consider the vector-valued function x ( t ) = f - ' ( x * ( t ) ) in the interval E ) . Then, sincef-' is assumed to be differentiable as well as monotonic increasing, x ( t ) is differentiable and
(-E,
d dr
-x(t) = u(t) 2 0
(It1
< E).
Whence, differentiating x*(t) = f ( x ( t ) ) at t = 0, we find
J(a)u(O) = u* > 0, u(0) 2 0. (34) Since J ( a ) has nonpositive off-diagonal elements, (34) implies, by Theorem 6.1 in Chapter 11, that J(a) satisfies the Hawkins-Simon condition and hence is a P-matrix. This completes the proof. Another noteworthy special situation is the case with a quasidefinite Jacobian matrix. This enables us to establish global univalence even in convex regions, and in a simpler alternative way. DEFINITION 20.6. A matrix A is said to be positive (negative) quasidejinite if the quadratic f o r m ( A u , u ) is positive (negative) definite. Clearly, any principal submatrix C of a positive (negative) quasidefinite matrix A is again positive (negative) quasidefinite. A matrix A is positive quasidefinite if and only if its symmetric part +(A + A') has all principal minors positive. Let 2 be any eigenvalue of A , and p,,,, p M be the minimum and the maximum of the eigenvalues of * ( A + A ' ) . Then it is well known that Prn 5 R e ( 2 ) ~S M . If A is positive quasidefinite, and hence 11, > 0, the real parts of all the eigenvalues 2 of A are positive from the above inequality, which implies that det A > 0. Since this argument applies to any principal submatrix C of A , a positive quasidefinite matrix is a P-matrix. Hence global univalence is ensured in rectangular regions by Theorem 20.4(i) if the Jacobian matrix is everywhere positive quasidefinite. Nonetheless quasidefinite cases deserve a reconsideration.
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CONDITIONS FOR UNIVALENCE
375
THEOREM 20.7 (Gale and Nikaido, 1965). The mapping ( I ) is univalent is convex and if its Jacobian matrix J(x) = (fi,(x))is everywhere positive (negative) quasidejinite in X .
if X
Proof. For obvious reasons, it suffices to prove a positive quasidefinite case. For a, b E X with a # b, let x ( t ) = a + th, where h = (hi)= ( b - a) # 0. By convexity we have x ( t ) E X ( l >= t 2 0). Consider the function
Then, upon differentiation, we find
which is identically positive by positive quasidefiniteness. Since (D(0)= 0, we must have @(1) # 0 and hence ,fi(b)-fi(a) # 0 for some i. This proves univalence, Q.E.D. The following application of Theorem 20.7 to analytic functions is of some interest, although it has no special bearing on economic theory.
COROLLARY (Noshiro, 1934). Let f ( z ) be an analytic (complex) function of a complex variable z in a convex region X of the complex plane. Then, f ( z ) is univalent in X ij'the derivativef ' ( z )lies in a half-plane whose interior does not contain the origin. Proof. It will be noted first that it suffices to consider the case in which the real part of the derivative is positive in X . In fact, putf(z) = u(x, y ) + iv(x, y ) , where z = x iy, u(x, y ) = Re(f(z)), u(x, y ) = Im(f(z)). Hence f'(z) = &/ax + i(av/ax),and there are, by assumption, some real numbers a,p, not all of them zero, such that
+
all
av ax
a-+D->O
ax
(inx).
Then, h(z) = (IX- ip)f ( z ) is analytic and its derivative has a positive real part in X . Clearly, f ( z ) is univalent if and only if h(z) is univalent. Now, let us assume Re(f'(z))
aLl
= -> 0
ax
(in X ) ,
(35)
and consider the Jacobian matrix J(x, y ) of the mapping
( x , y)'
-+
Y ) , v(x, Y))' :
x
-+
R2.
376
V11. T H E JACOBIAN MATRIX AND GLOBAL UNIVALENCE
J(x, y ) is positive quasidefinite in X because
from the Cauchy-Riemann equations
au ay
au -_ -a u _ ax
-=--
ay’
aU ax
and from (35). Whence univalence follows from Theorem 20.7, Q.E.D. Remark. The convexity assumption in Theorem 20.7 is indispensable. The following example shows that Theorem 20.7 is no longer true in a nonconvex region. In the light of the basic relation (36), the existence of a nonunivalent analytic function f ( z ) having the real part of its derivative positive in a nonconvex region will be shown. An example is given by the Joukowski mapping 1 f ( z )= z - ( 2 # O), z
+
when it is considered in suitable regions. Let R be the common exterior portion of two circles of radius having their centers at and -+, respectively, on the real axis. The real part of the derivative f ’ ( z ) = I - 1/z2 can be written as
+
+
which is positive in R. Butf(i) = f ( - i ) = 0 and i, - i E R. Moreover, one can easily construct a simply connected subregion X of Q containing i, - i .
$20.4. Weakening of Conditions The conditions on the Jacobian matrix for univalence in Theorems 20.4(i) and 20.7 can be relaxed to some extent. The purpose of $20.4 is to prove univalence under slightly weaker conditions with the aid of Kronecker’s existence theorem (Chapter I, Theorem 5.5). The results will be considered only for open regions, however, in order to avoid complications that might arise on the boundary.
DEFINITION 20.7. A matrix A is called a weak P-matrix if det A > 0 and all its proper principal submatrices have nonnegative determinants. A M9eak N-P-matrix is dejined likewise.
$20.
CONDITIONS FOR UNIVALENCE
377
DEFINITION 20.8. A matrix A is called weakly positive quasidefinite i f det A > 0 and the quadratic form ( A x , x ) is positive semidefinite. A weakly negative quasidefinite matrix is defined likewise. THEOREM 20.8 (Gale and Nikaido, 1965). The mapping ( 1 ) in 620.3 is univalent i f either (i) or (ii) holds: (i) X i s an open rectangular region of R" u*herethe Jucobian matrix J ( x ) is a weak P-matrix everywhere; (ii) X is an open convex region of R" where the Jucobian matrix J ( x ) is weakly positive quasidefinite everywhere. Proof. The method of proof is exactly the same for (i) and (ii). We first prove that for any fixed a in X there is no x in X such that
f ( x )-f(a)
=
-A(x - a),
x # a (for some A > 0).
(1)
Suppose that (1) holds for some b in X . Then, by rearrangement, we find
Aa + f ( a ) = Ab + f ( b ) . Using this A, define a new mapping f : X
+ R"
(2)
by the formula
f ( x ) = Ax + f ( x ) .
(3)
The mapping f is clearly differentiable, with its Jacobian matrix being J,(x)
= AZ
+ J(x),
(4)
where Z is the identity matrix. Regarding the properties of J,(x), we have (lo), (2') below: (lo) J,(x) is everywhere a P-matrix in X in case (i); (2') J,(x) is everywhere positive quasidefinite in X in case (ii).
To prove (1') for a weak P-Jacobian matrix, let K , be an arbitrary principal submatrix of J,(x) of order k . Then, K, = AZ, + K for the identity matrix Zk of order k and a principal submatrix K of J(x). Whence we find det K,
= det(A1,
+ K ) = Ak + &(A),
(5)
where &(A) is a polynomial of degree k - 1 at most, having coefficients equal to certain sums of the principal minors of K . Since these principal minors are assumed to be nonnegative for a weak P-Jacobian matrix, we have det K, > 0 from (9, proving (1'). On the other hand (2') is obvious from A > 0 and (4), since (J,(x)u, u ) = A(u, u ) (J(x)u,u ) > 0 for u # 0. From (1') and (2'), it follows by virtue of either Theorem 20.4(i) or 20.7 that the mappingfis univalent. But (2) impliesf(u) = f ( b ) , and hence a = b, contrary to the assumed fulfillment of (1) by b.
+
378
VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
Henceforth we will rely on the concepts and result in Chapter I, 55. For each a E X , let
X,= {x IfW # f (a)>.
(6)
Clearly Xu # 0 from the nonvanishingness of det J(x). Moreover, the mapping (1) in 520.3, if confined on X,,is homotopic to the mapping 9, : X ,
+
R"\{f(a)),
(7)
+x -a
(8)
g"(x) = f ( a )
by the homotopy bridge
H(x,t ) = (1 - t),f(x) + @(x)
(x E X , ,
1 2 t 2 0).
(9)
Note thatf(XJ $ f (a)from the definition (6). We also have g"(X,) $ f ( a ) , since x E Xu must imply x # a. To prove the asserted homotopy, it therefore suffices to see that (9) never coincides with ,f(a) for any x E X , and any t, 1 > t > 0. Suppose on the contrary that
H(x,t ) = f ( a ) (for some x E ,'A , t , 1 > t > 0).
(10)
Rearranging (lo), we would then find
x # a,
t > 0, 1-t
which contradicts the result in the foregoing step of the proof. This proves the homotopy. We are now ready t o prove univalence. To this end, if we suppose that
f(4=f@)
(1 1)
for some two distinct a, b in ,'A we shall have to deduce a contradiction. If we write 4 $ to mean that a mapping (b is homotopic to another $, from the above results on homotopy, we have the two homotopy relations
-
f - 9" : X,
-+
R"\{f(a)l,
f- gb : x b * R"\{ f ( b ) ) . But, since X ,
= Xb
by ( I l), we can reduce the above relations to
f
N
g" : M
-+
R"\(c),
f - g b : M - + R"\{c), where A4 = X , = Xb, c = f ( a ) = f ( b ) . Then, (12) and (13) imply, by the transitivity of homotopy, ga -gb : M+R"\{c}.
(14)
$20.
CONDITIONS FOR UNIVALENCE
379
In view of the openness of X , det J(a) # 0, and a # b, we can find, by Chapter I, Theorem 5.6(i), an n-ball C,,(a, 6) lying in X such that b 4 C,(a, 61, f ( x ) #f(4 ( x E C,(a, 61, x
(15)
+ 4.
(16)
Note that (16) ensures, for the boundary sphere &',,(a, S) of C,,(a, S), S,(a, 6) c M .
(17)
From (14) and (17), it follows that
-
gQ gb : S,(U,S ) + R"\{c).
(18)
Consider dymc : S,,(a, S) &(a, 6), the mapping associated with g" (see Chapter I, $5.2 and the remark preceding Lemma 5.3 in Chapter I). Clearly, bgacis the identity mapping and hence, by Theorems 4.3 and 5.2, in Chapter I, is not homotopic to a constant mapping. Therefore, applying Kronecker's existence theorem (Chapter I, Theorem 5.5) to g', go asf , g , we can conclude that --f
c E g"C,(a, 6)).
(19)
But (19) implies b E C,,(a, 6) because c = gb(x) only for x = b. This contradicts (15). We have thereby proved that f(a) = f ( b ) for a, b E X entails a = b, Q.E.D.
COROLLARY. The mapping (1) in $20.3 is univalent i f either (i) or (ii) holds: (i) X i s an open rectangular region of R" where the Jacobian matrix J(x) is a weak P-N-matrix everywhere; (ii) X is an open convex region of R" tchere the Jacobian matrix J(x) is weakly negative quasidejinite everywhere. Proof. This follows by applying Theorem 20.8 to - f : X - + R", Q.E.D. 620.5. Miscellaneous Results for n = 2
The univalence proposition for a special two-dimensional case in Remark 2 in $20.1 rests not only on the information about the Jacobian matrix but also on certain properties of the cost functions, namely, their homogeneity and the positivity of their values. While this proposition is undoubtedly of great relevance in so far as factor price equalization is concerned, the special lower dimensionality of two-dimensional cases enables us to establish global univalence without relying heavily on the specific properties of cost functions.
380
VlI.
THE JACOBIAN MATRIX AND GLOABL UNIVALENCE
The results for two-dimensional cases in this section, although apparently fragmentary, may be useful. Throughout this section, f : X+R2
(1)
is a differentiable mapping, where
X is a rectangular region in R 2 ; xZ) = (,fi(xi, ~ 2 1 fi(x1, 7 x2))’, fi(x1, able functions on X , with total differentials (LX)
(B)
f(x17
d f i =fi1(x,,
x2)
dx,
~
+ f i 2 ( x , , x 2 ) dx2
2
(i) = 1, 2) are differenti-
(i = 172).
We shall consider two kinds of conditions on the Jacobian matrix for global univalence: (I) the one-signedness of principal minors and (2) the onesignedness of a row. ( I ) One-Signed Principal Minors. In the two-dimensional case, global univalence by a P-Jacobian matrix (Theorem 20.4) essentially implies global univalence by a Jacobian matrix having one-signed principal minors, as will be stated in the following theorem.
THEOREM 20.9 (Gale and Nikaido, 1965). Suppose that the Jacobian matrix
of the mapping ( I ) is continuous in X . Then, the mapping (1) is univalent in X if either (i) or (ii) holds: (i) N o principal minors of J ( x , , x2) vanish in X ; (ii) det J ( x , , xz)# 0 and no diagonal elements of J(x,, xZ) change signs in
an open X . Proof. (i) By virtue of the continuity of J(x,,x2) on a connected set X , Theorem 1.8 (Chapter 1) ensures that each principal minor is everywhere positive or everywhere negative in X . Multiplying some or all of the functions fi(xl, .rz)(i = 1, 2) by - 1 if necessary, we may assume that all the diagonal elements f j j ( x l ,x2) ( i = 1 , 2) are everywhere positive in X . Note that det J(xl, x 2 ) is also everywhere positive or everywhere negative for the same reason. Then if detJ(x,, x2) is everywhere positive in X , the present (i) is a special case of Theorem 20.4(i). Next consider the case where det J(x,, x 2 ) is everywhere negative. In this case, we find h 2
from
f i l f Z 2> 0, ,fil f 2 2
SZl > 0
-fi2f21
(in X)
= detJ(x,, x2) < 0,
(3) as in the preliminary
$20.
CONDITIONS FOR UNIVALENCE
38 1
remark in the discussion of (b) in the proof of Theorem 20.3(ii). Inequality (3) implies, again from continuity and connectedness, that either
or In the case of (4), a new mapping f : X + R2,where f(xl, x2) = (f2(x1, x2), fi(x,, x2))', has a P-Jacobian matrix in X , and hence is univalent by Theorem 20.4(i). In the case of (5), another new mappingf : X + R2,wheref(x,, x2) = (-f2(x1, x2)> -fl(xl, x,))', has a P-Jacobian matrix in X , and hence is univalent by Theorem 20.4. (i) Clearly, in both cases the univalence o f f or implies that of the original mapping. (ii) If we proceed in the same way as in (i), we have three situations to consider: (m) the Jacobian matrix of the original mapping is a weak P-matrix; (fl)fhas a P-Jacobian matrix; (y)rhas a P-Jacobian matrix. Since Xis further assumed to be open this time, univalence obtains for any of the above three mappings by Theorem 20.8(i). This always ensures univalence for the original mapping, Q.E.D. ( 2 ) One-Signed Elements in a Row. We shall pay attention to the implication of the presence of a one-signed row in the Jacobian matrix, or, put in a seemingly more general form though amounting to the same thing, that of the existence of two real numbers n1,n 2 , not both zero, such that qfij+ n2 f i j ( j = 1,2) are one-signed.
THEOREM 20.10 (Gale and Nikaido, 1965). Suppose that det J(xl, x2) is one-signed and nonvanishing in X . Then the mapping (1) is univalent i f (i) There are two real numbers n,, 7 c 2 , not both of them zero, such that nlflj(x1,
~
2
+) 712 fij(x13 x2)
( j = 122)
(6)
are one-signed in X and at least one of the functions of ( 6 ) never vanishes in X . (ii) The nonvanishingness of one of the functions of ( 6 ) in (i) can be dispensed with, provided X is further assumed to be open. Proof. (i) Suppose n1 # 0. Then, the original mapping is obviously univalent if and only if the mappingf: X - t R2 defined by
f<x,,x 2 ) = (%fl(Xl, x 2 ) + 7(2 f i ( X 1 , X 2 ) > f 2 ( X l > X Z ) ) , is also univalent. The Jacobian matrix of,f, which is obtained by premultiplying that of the original mapping by the constant matrix
382
VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
has one-signed elements in its first row. Thus, we can assume that the original Jacobian has one-signed elements in its first row. Moreover, by tra'nsforming some or all of the variables x l , x 2 to - x l , - x 2 , respectively, if neeessary, we may also assume that fij(xl, x 2 ) 2 0 ( j = I , 2) in X . Note that one of flj ( j = I , 2) is by assumption everywhere positive. Thus, renumbering the variables if necessary, we may finally assume that f l I ( X 1 , x2)
> 0,
f i 2 ( X I , x2)
2 0 (in
w.
(7)
The case n2 # 0 can be also reduced to a situation fulfilling (7). It therefore suffices to prove univalence for a situation where (7) is satisfied. Let a = (a,), b = (bi)E X be two points that are mapped under (1) to the same point, say p = ( p , ) , so that fj(a1, a2)
=fi(bl, b2) = p i
(i
=
1, 2).
(8)
We shall deduce a contradiction from the supposition a # b. We first observe that a2 # b, . Otherwise we would also have a, = b, byf,, > 0 in ( 7 ) . Hence u2 # 6 , , and we may assume that a2 < b2 .
(9)
Next, it will be shown that there is a function 4(x2) (b, 2 x2 2 a21
(10)
fulfilling f 1 ( 4 ( x 2 ) , x 2 ) = p1
@2
L x2 2 a2).
(1 1)
Indeed, for each x, in the interval [ a z , b2] we find fl(%
x2)
f l ( b 1 , x2)
2fl(%a21 = P1, S f , @ , , b2) = P1
fromf,, 2 0 in ( 7 ) and 6 , 2 x 2 2 a 2 . Then, from continuity and connectedness, f l ( x , , x 2 ) can be equated to p , at some x1 in the interval [min(a,, bl), max(a,, b,)] (cf. Example 1 in Chapter I, 9 1.4 and Theorem 1 .8). Such a value of x1 is unique for each x 2 E [ a 2 ,b,] byf,, > 0 in (7), and can therefore be regarded as a single-valued function (10) satisfying ( I 1). As will be noted at the end of the proof, the function 4 ( x z )is differentiable in [a,, b 2 ] , with its derivative +'(x2) satisfying fll(4(X,>,
xz)4'(x,) + h 2 ( 4 ( x 2 ) , x2) = 0.
(12)
Then the function b+z)
=f2(4(x2),x2)
is also differentiable in [a,, b 2 ] , and its derivative $'(x2) is given by V(x2)
= det 4
4 W ?x2>/fi1(4(x2),x2)
(13)
@0.
383
CONDITIONS FOR UNIVALENCE
from (12) and .fil # 0 ensured by (7). Equation (13) implies that $'(x,) is nonvanishing and one-signed fromf,, > 0 in (7) and the nonvanishingness and one-signedness of det J(x,, x,). Whence $(x2) is monotonic. But this monotonicity contradicts $(a2)=f2(a,,a 2 )= p 2 =f,(b,, b,) = $(b2),a, f b, because a, = 4(a2),b, = 4(b2). The differentiability of 4(x2) and (12) remains to be shown. From (1 1) and the differentiability off,(x,, x,), we find
E(h) = fl l ( d ( X 2 1 , x2)A4
((A4)2
+fl2(4(XA
+ h2)1i2
x2)h
~
(h
+
O),
(14)
where
We first show
h+O
Suppose, contrary to (15), that there were a sequence {h,} such that
v- + ( r l 1 A4V
lim
where
A4v = 4(xz + h,)
- =
hv
+ 00,
- 4(x2) ( v = 1,2,
lim v-+
+
lim h,
= 0,
v++m
. . .). Equation (16) implies
hV
((A4,,),
+ hy2)l/, = O'
which, combined withf,, > 0 in (7) and (14), entails
But (17) and (18) contradict
This proves (15). Relation (15) ensures the existence of a constant M such that (for sufficiently small h).
384
V11. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
Then, using (14), (19), andf,, > 0 in (7), we have
whence
l i m Ad, -- - .fl2(d,(X,), 11-0 h fl I ( 4 ( X Z ) >
x2) x2)‘
This proves the differentiability of (P(x,) and (12). (ii) This case can also reduce t o the case where
.f;j ( d ~ l x2) , 2 0 (.i= 1 , 2) in X , as in (I). Suppose that there were two distinct points a = (a,), b = (b,) in X satisfying (8). Since X I S assumed to be open, we can find real numbers a t , p, ( i = I , 2) such that
a, b E R = {(xl,x,)’ 1 c[, < x,< B, (i = 1, 2)) c X . Then, choosing a sufficiently small positive number a2
- EM1
B2
E,
(21)
we can let
- EB1,
and the parallelogram subregion of R, R* = { ( X I , x,)’
1011
< x] < /II, c(,
< fi2 - E D 1 } ,
- &a, < x2 - E X ,
contain the two points a and h. Let us now define a new mapping g: Y+R2,
where
(4 (P)
y
= {(YI
1
YJ
I
< Y1 < P I
d Y , Y J = (gl(Yl3 Y * ) ,92(1’13 3
910‘1, Y 2 ) = fl(1’13
Note that (yl, c j ~ ,+ y,)’ 112) of 9 1s
J,Oj
E
c1’1
a2
7
-
< 4’2 < /I,
- &PI
1;
1’2))‘>
+ Y 2 h 92(1’13 Y,) =fz(Y,,
EYl
+ Y2).
R* for (yl, y 2 ) ’E Y. The Jacobian matrix
3
i Whence
/ I i(1’1,
~ J ’ I J 2)
/,I(.Yl?
CJ’l
(22)
+ cfi ~ ( J ’ IEYI + Y 2 ) 1
+ Y 2 ) + c/22(L’1.
EY1
+ 1’2)
fi
+ Yz) V l + 1’2)
~ ( Y I EYi ,
f22(1’1?
$21.
APPLICATIONS TO SPECIAL JACOBIAN MATRICES
385
from (23), and is therefore nonvanishing and one-signed in Y . On the other hand, the nonsingularity ofJ(x,, x2)rules out flj(xl,x2) = 0 ( j = 1, 2) in (20), so that fil(Yl7 EYl
+ Y2) + Efi2(Yl. Vl + Y2) > 0,
f12(Y1r
W l
+ Y2) 2 0
in (23). Thus Sll(Y1, Y 2 )
> 0,
912(Yl, Y2)
2 0 (in Y ) ?
(24)
which is a counterpart of(7). But, Ycontains two distinct points ( a l , a2 - &a,)’, ( b l , b, - ~ b , ) ’that are mapped under g to the same point p . This contradicts (i), Q.E.D. Remark. Theorems 20.9 and 20.10 include as special cases situations where the corresponding Jacobian matrices have all their elements nonnegative. These cases can be handled by Theorem 20.9 since the diagonal elements of a nonnegative Jacobian matrix are one-signed. They can also be handled by Theorem 20.10 since all the rows of a nonnegative Jacobian matrix are one-signed. Factor price equalization for n = 2 is therefore already achieved by the irreversibility and nonnegativity of factor intensities before other specific properties of cost functions take effect (cf. Remark 2 in $20.1).
021. APPLICATIONS TO SPECIAL JACOBIAN MATRICES 021 .I.Special P-Matrices
In applying the univalence theorems proved in $20, in most of which P-matrices play a central part, it may be useful to enumerate certain specific types of P-matrices. This will be done in the present section. All matrices that appear in the following are real square matrices unless otherwise stated. ( I ) Matrices Having a Positive Dominant Diagonal. Let us put forward
DEFINITION 21.1. A complex matrix A = ( a l j )qf order n is said to have a dominant diagonal, if‘ there is a set of n positive numbers n i ( i= 1, . , . , n ) such that (i = I, . . . , n). luiil ni > ]aij(TC, (1)
C
jSi
Remark 1. It is clear from Definition 21.1 that a complex matrix A = ( a l j ) has a dominant diagonal if and only if the matrix D = (d,.,) defined by dii = laii(( i = I , . . . , n), dij = - laijl (i #j) satisfies the Hawkins-Simon condition or an equivalent (cf. Chapter 11, Theorem 6. I). This implies in particular that a matrix has a dominant diagonal if and only if its transpose has a dominant diagonal.
386
V11. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
Remark 2. It is.also clear from Definition 21.1 that every principal submatrix of a complex matrix having a dominant diagonal has a dominant diagonal.
THEOREM 21.1 (Ostrowski, 3937-3938, 1956; McKenzie, 1960b). A matrix A = ( a i j )liming a positive dominant diagonal is a P-matrix.
Puoof; First, it will be shown that a (complex) matrix B = (bij)having a dominant diagonal is nonsingular. Let n
2b i j Z j = 0
(i
=
I , .. . , n )
j = 1
for (complex) numbers zi( i = 1, . . . , 17). To prove the nonsingularity of B, we have only to see that all the z i must be zero. There are a set of positive numbers x i ( i = 1, . . . , n ) satisfying
from the dominancy of the diagonal. Let
lzil 0 = inax - over all
i
=
1,
. . . , n.
(4)
xi
Thus 0 >= 0, and it clearly suffices to show 0 = 0. Suppose 0 > 0 and this maximum is attained for i = k . Then lzll = Ox,,
lzil
5 Oxi
( i # I<).
(5)
From (2) for i = k , we find
bkjzj,
Okkz,= j#k
which, combined with (3) for i = k , 0 > 0, and (5), implies
5 Ox IbkjI " j j#k
<
lbkkl xk
=z l b k k l
lzkl >
arriving at a contradiction. Whence 0 = 0, so that z i = 0 (i = 1, . . . , n). Now let us consider a (real) matrix A = ( a i j )having a positive dominant diagonal. Since every principal submatrix of A also has a positive dominant diagonal (cf. Remark 2), to verify the P-matrix property of A, we need only show that det A > 0. To this end, we observe that in the present case (1) becomes a i i n i > Iai,ilxj ( i = 1 , .. ., n )
C
.j # i
$2 1.
387
APPLICATIONS TO SPECIAL JACOBIAN MATRICES
by virtue of the positivity of the diagonal elements, and hence (p
+ aii)ni> C
laijl xi
(i
=
I, . . . , n ; p 2 0 ) .
(6)
j+i
Inequality (6) implies that p Z + A , where Z is the identity matrix of order has a dominant diagonal for any p 2 0. Therefore det(p1
+ A) # 0
(p
2 0)
ti,
(7)
from the nonsingularity established above. Recall that det(pZ + A ) is a polynomial of degree n, with the coefficient of p” being 1 ;hence det(pZ A ) + 00 ( p -+ + a).Then if det A 5 0, we could let det(pZ + A), a continuous function, vanish for some p 2 0 by Theorem 1.8 in Chapter I, and Example 1 in Chapter I, 91.4. This contradicts (7). Whence det A > 0, Q.E.D. Remark 3. Matrices having nonpositive off-diagonal elements and satisfying the Hawkins-Simon condition form a special subclass in the set of all matrices having dominant diagonals (cf. Chapter 11, $6, Theorem 20.6, and Remark 1). ( 2 ) Nonsingular Skew-Symmetric Matrices. It is well known that the determinant of a skew-symmetric matrix of order n (see Chapter I, Definition 3.5) vanishes for an odd n and is the square of a polynomial of the matrix elements for an even n. Hence a skew-symmetric matrix can be nonsingular only when it is of an even order.
+
--f
THEOREM 21.2. A nonsingirlar skewsymmetric matrix A is a weakly positive quasidefinite matrix (see Definition 20.8) as itsell as a iiseak P-matrix (see Definition 20.7). Proof. Note that principal submatrices of a skew-symmetric matrix are also skew-symmetric. From the foregoing, it is therefore obvious that a principal minor of order k of A vanishes for an odd k and is nonnegative for an even k , and positive for k = n, the highest even order. This proves that A is a weak P-matrix. On the other hand, the quadratic form ( A x , x) identically vanishes by virtue of skew-symmetricity and hence is positive semidefinite. Since det A > 0 as noted above, A is weakly positive quasidefinite, Q.E.D. (3) Hicksian Matrices. Hicks (1939) put forward a condition on the Jacobian matrix of a system of excess demand functions which was asserted to be necessary for a kind of stability of equilibrium. A Hicksian matrix is a matrix whose principal minors of odd orders are negative and whose principal minors of even orders are positive. A Hicksian matrix is therefore nothing but an N-P-matrix (see Definition 20.3). It should be noted that the results in $20 have brought to light the implication of the Hicksian condition for the uniqueness of equilibrium in a competitive price formation, rather than the stability implication originally presumed by Hicks himself. This will be explained below.
388
Vll. THE JACOHIAN MATRIX AND GLOBAL UNIVALENCE
Let us recall (8) in $20.1. This is a system of 11 - 1 equations of general equilibrium in the 11 - 1 unknown levels of equilibrium prices of goods apart from therith one, which is anumeruirr. If the Jacobian matrixJ(p,,p,, . . . ,p,-,) of order n - 1 is Hicksian i n a rectangular region of prices, the Corollary to Theorem 20.4 ensures the uniqueness of solutions of (8) in $20.1. What is more interesting is the fact that the Hicksian condition entails the uniqueness of equilibrium price vectors determined, as in Chapter VI, Definition 18.4, by a system of inequalities E,(P,, P 2 >
. . . , P,) 5 0
( i = 1, . . . > n),
(8)
. . ., p,) = 0 (if P , > O), (9) ( i = I , . . . , n) are excess demand functions. We shall
€,(PI,
p2
3
where E,(pl,p 2 , . . . , p,) now consider a situation in which the excess demand functions are homogeneous of order zero in the region of all semipositive price vectors and are differentiable for p , 2 0 ( i = I , . . . , 17 - I), p , > 0. Assume further the Walras law in the narrow sense n
in the price region. For the case where the nth good serves as a numgraire, the system of inequalities (8) and (9) is equivalent to the normalized system (1 1)
E i ( ~ I , ~ 2 , . .1.) 5,0~ n( i = 1l,, . . . , n - l ) , p i E i ( p l , p 2 ,. . . ,p n p l ,1) = 0 (i = 1, . . . , 17
-
I),
p i 2 0 ( i = 1, ..., n - 1).
(12) (13)
It will be shown that if J ( p , , p r , . . . ,pnp1)= ( E i j ( p l p, 2 , . . . , P , , - ~ ) ) ( i , , j = 1. . . . , 17 - I ) is Hicksian, the uniqueness of solutions obtains for the system of inequalities(ll)-(l3). L e t q i ( i = 1, . . . , n - l)alsosatisfy(ll),(12)and(13), so that E i ( 4 , . 4 2 , . . ., Yn-], 1) 5 0 (i = 1, . . . , f l - 11, (14) qiE,(ql,q, ,..., q n - l , I ) = O
( i = I , ..., n - l ) ,
y i 2 _ o ( i = I , ..., n - I).
(1 5 )
(16)
It is then immediately possible to reduce (1 I)-( 16) to (pi -qi)(Ei(Pl>
~2
7
. . . > P n - 1 , 1) - Ei(q1>42 . . . > 7
(;=I
1)) 2 0 , . . . , 17-1).
qn-1,
(17)
In the light of the Corollary to Theorem 20.5, the Hicksian Jacobian matrix of system (17) i n the region of all (ri - 1)-dimensional nonnegative vectors therefore implies p i = q i ( i = 1, . . . , n - l), as was to be shown.
$21.
APPLICATIONS TO SPECIAL JACOBIAN MATRICES
389
021-2. The Stolper-Samuelson Condition
In a special reference to the protection of domestic factor owners by tariff, Stolper and Samuelson (1941) studied the effect on factor rewards of a change in the world good prices. They consider a simple two-good two-factor situation. As in the consideration of factor price equalization, they rely on (6) in $20.1 for n = 2 as the fundamental relationship between the world good prices p i and domestic factor prices w j . Explicitly
where ci(wl, w2) is the cost function of the ith good (i = I , 2) defined for positive factor prices wl, w 2 . As was already noted in 520.1, the elements cij(wl, w2) of the Jacobian matrix of (I),
are factor intensities, which are assumed to be positive for simplicity. If the factor proportions satisfy CIl(W,,
w2)
> C2I(Wl,
C 1 2 ( W I , w2)
w2)
C 2 2 ( W I , w2)'
(2)
we say that the first good is intensive in the use of the first factor, and the second good is intensive in the use of the second factor. On the other hand, if
we say that the first good is intensive in the use of the second factor, and the second good is intensive in the use of the first factor. When the factor proportions never reverse in the region of all positive factor prices I I ' ~ , H ' ~one , can, in a unique way, associate with each good the factor in the use of which it is intensive. For easier reference, we mean by the associated factor of a good, the factor that is associated with the good in the above good-factor association. Then the noted Stolper-Samuelson proposition asserts that a rise in the price of a good entails a rise in the price of its associated factor and a fall in the price of the other factor. The proposition is verified as follows. Since case (3) can be put in form ( 2 ) by renumbering either goods or factors, it suffices to discuss case (2). From (2), we find detJ(ttr,,
1it2)
> 0,
(4)
390
VII. THE JACOBIAN MATRIX AND GLOBAL UNIVALENCE
which, in conjunction with the positivity of c i j , implies that J(w,, w2) is a P-matrix in the region W of all positive two-dimensional factor price vectors w = (wi). Denote by P the image of W under the mapping w + c ( w ) = (ci(w)) induced by the cost functions. Then, by the Corollary to Theorem 5.7 in Chapter I, P is an open region of RZ,from (4), whereas, by Theorem 20.4(i), the mapping is invertible from the P-matrix property of J ( w l , wz). Let the inverse mapping be p+w(p):P+
w.
(5)
By Lemma 5.4 in Chapter I, the mapping ( 5 ) is also differentiable because of (4), its Jacobian matrix being the inverse matrix J(kvI, ivJ1 of J(w,, N , ~ ) .The Stolper-Samuelson proposition will be established if it can be verified that .f(wl, w2)- has positive diagonal elements and negative off-diagonal elements. In fact,
which has the desired property from (4) and the positivity of c i j ( w , , w z ) . In the general n-good n-factor case, there seem to be no well-accepted factor intensities conditions that make possible an n-dimensional version of the above good-factor association. I n what follows, we will therefore only touch upon the implication of an n-good n-factor version of the StolperSamuelson proposition for factor price equalization. Let us reconsider the original Stolper-Samuelson proposition. We recall that the factor intensities condition (2) first ensured the P-matrix property of the Jacobian matrix and hence the global invertibility of the mapping i t ' + c(iis), before the signs of the Jacobian matrix of the inverse mapping were detected. It seems to be better, however, to reverse our procedure in the general n-good n-factor case. Consider the system of equations (6) in 520.1, which governs the fundamental relation between good prices p i and factor prices in the general case. We shall consider the mapping It' + c(w)
: W + P,
(6)
where W is the region of all positive factor price vectors; W under (6); ( y ) c(it.) = (ci(ii,)), with cj(ii.) ( i = I , . . . , n ) being the cost functions. (x)
( p ) P is the image o f
Instead of advancing a faithful n-dimensional version of the StolperSamuelson proposition, we put forth a condition on the Jacobian matrix J(It7)
= (Cjj(Ilj)
(7)
$21.
APPLICATIONS TO SPECIAL JACOBIAN MATRICES
39 1
of the mapping (6), which is suggested by the proposition, and we examine the implication of the condition for factor price equalization. The condition is formulated as follows :
DEFINITION 21.2. A matrix A is said to satisfy the Stolper-Samuelson condition if (i) A is nonnegative; (ii) A is nonsingular; (iii) its inverse A - ' = ( d i j ) has all its ofl-diagonal elements nonpositive, that is, dij 5 0 (i # j ) .
THEOREM 21.3 (Chipman, 1964). condition, A must be a P-matrix. Proof. Let D as A , we have
= A-'
= (dij).If
(8)
If A satisjies the Stolper-Samuelson
I is the identity matrix of the same order
D A = I.
(9)
Further let c be an arbitrary positive vector. Then, premultiplying c by (9), we find D ( A c ) = c > 0,
(10)
while Ac
>= 0,
(1 1)
from (i) in Definition 21.2. By virtue of Theorem 6.1, in Chapter 11, it follows from (8), (lo), and (11) that D satisfies the Hawkins-Simon condition. We shall now prove the assertion by induction on the order n of A . Equation (9) for n = l is d,,a,, = l . This implies a,, > 0 because d,, > 0 by the Hawkins-Simon condition. Next, assuming the theorem for 17 - I , consider the case for n 2 2. First, we find det D . det A = 1 from (9). This implies det A > 0,
(12)
since det D > 0 by the Hawkins-Simon condition. To prove the P-matrix property of A , it therefore suffices to show that every principal submatrix of order 17 - I of A is a P-matrix. Let A* be any of such principal submatrices. Performing an identical and simultaneous renumbering of the rows and columns of A , D, f, if necessary, we may assume that A* is located in the lower right-hand corner of A . I n view of d,, > 0 implied by the HawkinsSimon condition. define the matrix
392
VII. THE JACOBIAN MATRIX A N D GLOBAL UNIVALENCE
1
0
0
1
T=
:I
0 0
0
an 1
1
BIBLIOGRAPHY
I. Books for General Mathematical Reference ANALYSIS Courant, R. (1934). “Differential and Integral Calculus,” Vols. I, I1 [transl. from the original German ed.]. Wiley (Interscience), New York. ELEMENTARY TOPOLOGY Alexandroff, P., und Hopf, H. (1935). “Topologie,” Vol. I . Springer, Berlin. Lefschetz, S. (1949). “Introduction to Topology.” Princeton Univ. Press, Princeton, New Jersey. Pontryagin, L. S. (1952). “Foundations of Combinator-ial Topology” [translated from the first (1947) Russian ed.]. Graylock Press, Rochester, New York. Berge, C. (1959). “ EspacesTopologiques, Fonctions Multivoqucs.” Dunod, Paris [English transl. “Topological Spaces, including a Treatment of Multi-Valued Functions, Vector Spaces and Convexity.” Oliver & Boyd, Edinburgh and London, 19631. MATRIX THEORY Bellman, R . (1960). Introduction to “ Matrix Analysis.” McCraw-Hill, New York. Gantmacher, F . R. (1960). “The Theory of Matrices,” Vols. 1, 11. Chelsea, New York.
DIFFERENTIAL EQUATIONS Coddington,E. A.,and Levinson, N.( 1955). “ Theoryof Ordinary DifferentialEquations.” McGraw-Hill, New York. MATHEMATICAL ECONOMICS Gale, D. (1960). “The Theory of Linear Economic Models.” McCraw-Hill, New York. Karlin, S. (1959). “Mathematical Methods and Theory in Games, Programming and Econonomics.” Addison-Wesley, Reading, Massachusetts. 11. References
Only papers and bocks which are specifically 1-efer-redto in the text are listed hei-ein. Arrow, K . J . (1951). Alternative proof of the substitution thcoi.em for Leontief models in the general case. In “Activity Analysis of Production and Allocation .’ (T. C. Koopnians, ed.). Wiley, New York. Arrow, K . J . , Block, H . D., and Hurwicz, L. (1959). On the stability of the competitive equilibrium, 11. Ecotromtricn 27, No. I . Arrow, K . J., and Debreu, G. (1954). Existence ofan equilibrium for a competitiveecononiy. Eeotionrerricn 22, No. 3. Arrow, K. J., and Hurwicz, L. (1960). Competitive stability under weak gross substitutability: The ‘Euclidian distance’ approach. Interiint. €con. Reu., 1, No. I . Arrow, K . J., and Hurwicz, L. (1962). Competitive stability under weak gross substitutability: Nonlinear pi ice adjustment and adaptive expectations. Inrertrrrt. Eeoii. Rev. 3, No. 2.
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Atsumi, H. (1965a). Ncoclassical growth and the efficient program of capital accumulation. Reit. Econ. Stiitlies 32, No. 2. Atsumi, H. (1965b). The efficient capital accumulation i n a generalized Leontief model. Mimeographed (cited with the kind permission of the author). Aumann, R. J. (1964). Markcts with a continuum of traders. Eeonornetricn 32, No. 1. Aumnnn, R. J. (1966). Existence of competitive equilibria in markets with a continuum of traders. Eeonoiwtv;c~i34, No. I . Begle, E. G. (1950). A fixed point theorem. Ann. of Moth. 51, No. 3. Bourbaki, N. ( 1955). ’‘ Espaces Vectoriels Topologiques, Elements de Mathematique,” Vol. V. Hermann, Paris. Brouwer, L . E. J. (1909). On continuous vector distributions on surfaces. Amsterdam Proc. 11. Brouwer, L. E. J. (1910). Anisterrltiin Proc. 12; 13. Chipman, J. S. ( 1964). Factor price equalization and the Stolper-Samuelson theorem (mimeographed, 1963). Abstract. Eeonornetvicn 32, No. 3 (cited with the kind permission of the author). Coddington, E. A,, and Levinson, N. (1955). “Theory of Ordinary Differential Equations.” McGraw-Hill, New York. Dantzig, G. B. ( I 963). “Linear Programming and Extensions.” Princeton Univ. Press, Princeton, New Jersey. Debreu, G . (1954). Representation of a preference ordering by a numerical function. In *‘ Decision Processes” (R. M. Thrall, C. H. Coomb, and R. L. Davis, eds.). Wiley, New York. Debreu, G. (1959). “Theory of Value.” Wiley, New York. Debreu, G . (1962). New conccpts and techniques for equilibrium analysis. Internnt. Eeon. Rev. 3. Debreu, G. (1963). On a theorem of Scarf. Rev. €con. Stiidies 30, No. 3. Debreu, G., and Scarf, H. (1963). A limit theorem on the core of an economy. Ititernat. Econ. Rev. 4, No. 3. Dorfinan, R., Saniuclson, P. A., and Solow, R . M. (1958). “Linear Programming and Economic Analysis.” McGi-aw-Hill, New York. Edgcworth, F. Y. (1881). ‘‘ Mathematical Psychics.” Kegan P a d , London. Eilenberg. S., and Montgomery, D. (1946). Fixed point theorem for multivalued transformations. Aino.. J . Mtrth. 68, No. 2. Farkas, J. (1902). Uber die Theorie der einfachen Ungleichungen. J . Reine Angeii,. Math. 124. Frobcnius, Ci. ( 1908). Uber Mati-izen aus positiven Elementen I. Sitzher. Ktjnig/, Pre,
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Gale, D. (1955). The law of supply and demand. M ~ t h Scnnd. . 3. Gale, D. (1965). Optimal programs for a multi-sector economy with an infinite time horizon. Tech. Rcp. No. I . Dept. of Math., Brown Univ. Providence, Rhode Island (cited with the kind permission of the author). Gale, D. (1967). On optimal development in a multi-sector economy. Rev. Econ. Studies, 34, No. I . Gale, D., Kuhn, H . W., and Tucker, A. W. (1950). On symmetric ganies. In “Contributions to the Theory of Ganics” ( H . W. Kuhn and A. W. Tucker, eds.). Princeton Univ. Press, Princeton, New Jersey.
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Author Index Numbers in italic refer to pages where thc complete reference is listed.
F
A Alexandroff, P., 393 Arrow, K. J., 189, 191, 236, 249, 253, 254, 260, 305, 324, 329, 336, 338, 343, 354, 393 Atsumi, H., 220, 226, 229, 394 Aumann, R. J . , 295,394
B Begle, E. G . , 64, 394 Bellman, R., 393 Berge, C . , 393 Block, H. D., 305, 324, 329, 336, 338, 343, 354, 393 Bourbaki, N., 28, 31, 394 Brouwer, L. E. J., 63, 270, 394
C Chipman, J. S . , vii, 391, 394 Coddington, E. A., 338, 393, 394 Courant, R., 393
Farkas, J., 38, 394 Frobenius, G., 102, 394
G Gale, D., vii, 220, 229, 236, 249, 265, 307, 356, 360, 361, 362, 365, 366, 370, 373, 375, 377, 380, 381, 393, 394, 395 Gantmacher, F. R., 393 Georgescu-Roegen, N., 90, 189,395 Goldman, A. J., 139, 395 H
Hahn, F., 305, 395 Hawkins, D., 90, 395 Hicks, J . R., 304, 387, 395’ Hopf, H.. 393 Howe, C. W.. 145, 395 Hurwicz, I.., 305, 324, 329, 336, 338, 343, 354, 393 1
D Dantzig, G. B., 132, 394 Debreu, G., 236, 237, 246, 249, 253, 254, 260, 265, 281, 283, 284, 393, 394 Dorfman, R., 196, 207, 394
E Edgeworth, F. Y . , 237, 284, 394 Eilenberg, S., 64, 394
Inada, K., vii, 208, 356, 361, 362, 366, 370, 395
K Kakutani, S., 64, 67, 395 Karlin, S., 393 Kemeny, J . G., 145, 395 Knaster, B., 63, 395 Koopnians. T. C., 189, 220, 395 Krein, M., 39, 395 399
400
AUTHOR INDEX
Kronecker, L., 78, 79, 395 Kuga, K., 305, 309. 312, 315, 324, 395 Kuhn, H. W., 307, 394 Kuratowski, C . , 63, 395
L
R Rader, J. T., 284, 397 Radner, R., 208, 210, 21 1,397 Ramsey, F. P., 220, 397
5
Lefschetz, S., 393 Leontief, W. W., 88, 395 Levinson, N., 338, 393, 394
M McKenzie, L. W., 208, 236, 249, 305, 324, 335, 360, 386, 395, 396 Mazurkiewicz, S., 63, 395 Milman, D., 39, 395 Minkowski, H., 38, 396 Montgomery, D., 64, 394 Morgenstern. O., 145, 395, 398 Morishima, M., 150, 208, 249, 305, 396 Mosnk, I. L., 304, 396 Muth, J . F., 150, 396
Samuelson, P. A., 150, 189, 196, 207, 220, 325, 336, 356, 359, 389, 394, 397, 398 Scarf, H., 237, 284, 337, 394, 397 Shapley, L. S., 296, 397 Shubik, M., 296, 397 Simon, H. A., 90,395 Sion, M., 322, 397 Slutsky, E., 302,397 Smith, A,, 87, 397 Solow, R. M., 150. 196, 207, 394, 397 Sperner, E., 61, 397 Stiemke, E., 36, 38, 398 Stolper, W. F., 389, 398 Suits, D. B., 150, 398
T N Negishi. T.. 249, 305, 396 Nikaido. H., vii, 67, 132, 162, 236, 249, 265, 305, 306, 322, 356, 360, 361, 362, 370, 373, 375, 377, 380, 396 Noshiro, K., 375, 396
0 Ostrowski, A,, 386, 397 P Perron, O., 102, 397 Pontryagin, L.. S., 393
208, 307, 365, 381,
215, 309, 366, 395,
Thompson, G . L., 145,395, 398 Tsukui, J., 208, 220, 230, 398 Tucker, A . W., 36, 38, 39, 139, 307, . 394, 395, 398
U Uzawa, H., 52, 220, 249, 254, 255, 256, 168, 305, 338, 341, 344, 354, 398 V Vind, K., 295, 398 von Neumann, J., 64, 67, 69, 142, 398
W Wald, A., 236, 249, 398 Walras, L., 1, 235, 270, 398
Subject Index A Additivity, 214 Affine subspace, 16 spanned by convex set, 22 Affinely independent points, 53 Allocation, of production and consumption, 247 Arrow-Debreu model, 250 Average, of allocation, 290
B Balanced growth, 99, 199 of first kind, 163 of second kind, 164 Balanced-growth solution. 99, 100, 112, 126, 130, 150, 159, 171, 179 Barycenter, 55 Barycentric subdivision, 55 Brauer-Solow’s condition, 94 Budget constraint. 344
C Cardinalist. 245 Cartesian product of mappings, 70 of sets, 24 Cauchy-Peano local existence theorem, 338 Cauchy-Riemann equation, 376 Characteristic equation, 100 Closed interval, /I-dimensional. 9 Coalition, 286 blocked by, 287 Commodity bundle, 238 indifferent to, 221, 238 preferred to, 231, 238 Competitive equilibrium. 251
Composition, of set-valued mappings, 71 Concave function, 51 strictly, 51 Constant returns to scale, 88, 182 Consumer, positively endowed, 291 Consumption function, 162 set, 238 Continuation, possibility of, 338 Convex body, 2 1 Convex cone, 33 dual, 33 pointed, 43 polyhedral, 41 Convex function, 26. 44 strictly, 50 Convex hull of mapping, 72 of set, 17 Convex polyhedron, 41 Convex set, 16 dimension of, 23 Convexity, aggregative, 255 of function, 26, 44 of preference field, 241 of set, 16 Core, of economy, 287 Cube. 9, 10 Cyclical decomposition, 115
401
D Demand function, 244 individual, 261 Diameter, of \et, 55 Differenti tbility of function, 79 of mapping, 80 Dimension, of convex set, 23 Disposal hull, 213
402
SUBJECT INDEX
Dominant diagonal, 385
Dua I it y for closed convex cones, 34 in linear programming, 133 for polyhedral convex cones, 43
E c . 4 Pair, 332 canonical, 332 E-Ncighborhood, 12 €-Net, 12 Eipcnvalue problem, 100 nonlinear, 151 Equilibrium price vector, 251, 263, 309 Equiljbrium solution, 263 Euclidean distance, 7 Euclidean norm of matrix, 81 of vector, 6 Euler's theorem, on homogeneous functions. 315 Excew \upply function apprcpate, 261 modilied, 274 Extcrnal diseconomies, 183 Extreme point. 39
F Feasible path of finite horizon N , 196, 221 of infinite horizon, 196 Finite intersection property, I 1 Fixcd point of set-valued mapping, 67 of singlevalued mapping. 60 Fixed-point theorcm Rrouwcr's, 63 Kakutani's, 67 Free disposability, 184 Frobenilis-Perron's theorem, 102
G Golden rule, of accumulation, 224 Good desired. 153 productive, 253
Gross substitutability, 305 in differential version, 305 generalized, 306 strong, 305 Gross substitute, 304 Growth fast, 175 slow, 175 Growth factor. 199
H Hawkins-Simon condition, 90 Homogeneity. 150, 270, 358 Homotopy, 74, 378 bridge, 74, 378 Hyperplane separating, 32 supporting, 27
I Implicit function theorem, 85 Income effect term, 302, 304 Indecomposability, 156, 306 of rnattix. 105 for technology set, 204 Indifference class, 238 relation, 238 Individual supply function, 260 Inner product, 6 Input coefficients, 88 Insatiability, 240 Invariance of interior points, 83 11-revcrsihility of agpi-egate processes, 250 o f all processes, 184 of process, 183
J Jacobian, 81 Jacobian matrix, 81, 355 nearly constant, 360
K Kronccker's existence theorem, 78, 79
SUBJECT INDEX
L Lagrangian multiplier, 51, 140, 225 Land of Cockaigne, impossibility of, 183, 184, 250, 356 Leontief's system, 88 profitability of, 93 workability of, 89 Linear combination convex, 16 nonnegative, 18 of sets, 25 Local inverse, 85 Local univalence theorem, 81
M Majorization lemma, 339 Mapping differentiable, 80 point-to-set, 64 set-valued, 64 Marginal productivity theory, 141 Matrix decomposable, 105 Hicksian, 387 imprimitive, 115 indecomposable, 105 Jacobian, 81, 355 multiplier, 96 N-, 361 N-P-, 361 nonnegative, 9.5 P-,361 P-N-, 361 period of, 119 positive, 95 positive quasidefinite, 374 primitive, 1 15 semipositive, 95 skew-symmetric, 38 stable, 114 super-multiplier, 99 weak N-P-, 376 weak P-, 376 weakly positive quasidefinite, 377 Metric, 11 , Metric space, 11 Minkowski-Farkas lemma, 38
403
Monotonicity, 150 strict, 150 weak, 150 Moving equilibrium,99, 144 Multiplier, matrix, 96 numerical, 96 Multiplier process, dynamic, 98
N ri-Ball, 10 rz-Disk, 10 (rr-1)-Sphere, 10 Nonnegative invertibility, 95 Nonnegative orthant, 33 Nonsubstitution theorem, 190 Nrrniiraire, 337, 358, 388
0 Open covering, 11 Ordinalist, 245 Orthogonal complement, 3 1 , 37 of vector, 2 15
P Pareto optimum, 280 Path balanced-growth, 199, 223 maximal, 199 efficient, 197 feasible, 196, 221 golden, 224 optimal, 222 rr-optimal, 208 Period, of matrix, 119 Point nonnegative, 34 po$itive, 34 semipositive, 34 Preferable element least, 240 most, 240 Preference cardinal measurability of, 245 continuity of, 239 field, 238 indicator, 244
404
SUBJECT INDEX
Preference ( c o n ! . ) relation, 238 weak axiom of revealed, 325 Preference relation, 238 increasing, 243, 297 strong, 238 Primitivity, 157 of matrix, 115 Process, 181 aggregate, 246 disposal, 184 efficient, 185 flow, 182 of inaction, 184 irreversibility of, 183 production, 181 Production function classical, I88 neoclassical. 189 Productive technology, 147 Projection, 24 orthogonal, 31, 42, 215 Proximity consecutive, 212, 215, 230 weak, 229
Q Quadratic form, positive semidefinite, 50
R R;iy golden, 224 maximal balanced-growth. 203 von Neuninnn, 203 Region closed. 80 open. 80 rectangular, 80 Relation eq ti i va 1elice, 2 38 indifference, 238 preference, 238 Repetition, r-fold, 289 Replica, r-fold, 288 Resolvent equation. 165 positive. 165 set. 165 Reverie, \i:n of vector, 365
Root dominant, 103 Frobenius-Perron, 103 RouchC’s theorem, 82
S Satiation point, 240 Scalar multiplication, 6 Schwarz’s inequality, 7 Segment, 16 Separation theorems, 27 Set arcwise connected, 15 compact, 8, I 1 connected, 14 contractible, 76 convex, 16 resolvent, 165 Set-valued mapping closed, 65 graph of, 65 upper semicontinuous, 65 Simplex, 54 derived, 57 face of, 54 regular derived, 61 regular derived face, 61 standard, 10 Skew-symmetricity, 306 Slater condition, 5 2 Slutsky equation, 302 Sol LIt i on niinorant, 130, 131 optimal, 135 stationary, 158 Spending function, 162 Spel-ner’s lemma, 61 Stability global, 337 relative. 99, 112, 155, 171 Stiemke’s theorem, 36 Stolper-Samuelson condition, 391 Strict convexity of function, 50 of preference field, 242, 297 of technology set, 195, 221, 357 Sribstitution effect term, 302, 304 Super-multiplier, matrix, 99 Symnielry, of substitution terms, 304
405
SUBJECT INDEX
T TPtonnement, 268, 336 Technology set, 181 aggregate, 246 flow version of, 182 production, 181 stock version of, 182 Total differential, 79 Transformability, 195 Triplet golden, 224 productive, 223 Tucker’s theorem, 36 Turnpike theorem consumption, 228 Radner’s, 21 1 strong, 212 weak, 212
U Unique profitability, 205 Utility function, 221 indicator, 221
V von Neumann model, 142 von Neumann price vector, 205 von Neumann ray, 203
W Walras law in general sense, 263 in narrow sense, 263
