Dynamical Economics

Vo,. 13, 1927

MA THEMA TICS: C. F. ROOS

14So

this point is difficult to determine since direct comparison with a series of control animals is always more or less of an arbitrary procedure. By the second method of administering the material in the drinking water, we have obtained a considerable number of cases of complete regression- a phenomenon we have never witnessed in the controls. We have also brought about the complete regression of the spontaneous mass by the injection of the solution in a greatly reduced concentration subcutaneously. By the combination of administering the material by mouth and by injection we have also witnessed a high percentage of complete regression of the spontaneous tumors. In fact, we have never had a mouse with a spontaneous tumor that did not respond to some extent at least to this double treatment. In all cases, if the treatment be not too severe, the animals are capable of living for several months free of any evidence of new growth after the treatment has been stopped. No tumors have been found in any part of the body on the post mortem examination of such animals. The details of these experiments will be published in the near future.

D YNAMICAL ECONOMICS By C. F. Roos* DZPARTMZNT OF MATHgMATICS, UNIVrRSITY OF CHICAGO

Communicated February 4, 1927

1. Introduction.-As far back as 1838, Augustin Cournot realized the need for a dynamical theory of economics but chose to avoid the mathematical difficulties by considering a so-called average price and average function of demand.' On page 104 of Irving Fisher's doctoral dissertation we find the following statement: "The dynamical side of economics has never yet received systematic treatment.. When it has it will reconcile much of the apparent contradiction."2 In 1901 we find Pareto introducing derivatives of price with respect to time in his analysis.3 The %rst American economist to clearly recognize the limitations of the static theory of Cournot and Walras and, to suggest a method of overcoming them was H. L. Moore.4 It was he who first applied the method of relative changes and the method of trend ratios to eliminate long time changes from the law of demand More recently Irving Fisher has shown that there is a correlation of 0.79 between the volume of trade and a curve representing rapidity of rise or fall of the price level with minor irregularities smoothed out.5 G. C. Evans has suggested that the demand for a commodity is a function of the price and the rate of change of price.6 When the rate of change of price is thus introduced into the demand function the problems of competition and monopoly become problems in the calculus of variations.7 The monopoly problem is of the classic Lagrange type in the calculus of variations. The competition problem, how-

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ever, is a problem dealing with partial maxima of integrals and can be said to be a generalization of the Lagrange problem.8 In this paper it is planned to discuss the phenomena of competition, monopoly and cooperation for general functions of demand and cost. We will make use of the following postulates of competition: 1. Each individual engaged in trade is motivated by an attempt to make his net profits as large as possible over a given interval of time. 2. The rate of production of producers 1, 2, ...., k-1, k + 1., n can have no influence on the rate of production of the kth producer except in so much as they cause a change in price which uill influence the rate of production of k through the demand function. 2. Equations of Demand.-Let us suppose there are n producers manufacturing subject to cost functions plO, p2, . . ., (P. amounts ul, U2, . . ., U respectively, of a like commodity in unit time and further that each article sells at a theoretical price p which at any time t is the same for each article produced. Let us suppose further that as many articles are consumed as are produced. As G. C. Evans has suggested the demand 2u is often a function of the rate of change of price, dp/dt, as well as of the price. It is conceivable that the demand might also depend upon the rate of production and upon the acceleration of production. In order to insure generality let us write UPSUl, ,2P . . . =,,, pt pI) t) g(ul, Z 8ui where primes denote derivatives with respect to the time. There is then a differential equation =

;

-

g(ul, . . .ux,u, * * .,u, p,op,t)

=O

which we shall call G(ui, u1, . . ., u,,, u', P p, t) = 0 relating the price of the commodity and the rates of production of the n producers. 3. Equations of Cost.-Inasmuch as the price and rates of production have been chosen as functions of the time the total net profit rk of each producer for the period of time to to ti is given by the integral 7rk=

(PUk -(Pk)dt

(k

=

1, 2, ..., n)

where (pk is a function representing the total cost for the kth producer of manufacturing and distributing per unit time. In general this function (Pk will be a function of the rate of production, the price and the acceleration of production. In order to insure generality let us write (Pk = (pk(ul, u, . .* u*, u,pU, P,Pt). The integrand of 7rk then becomes PUk - (pk(Ul, u1,II .. ., un, u, p,p',tI) which we will call Fk(ul, ul,. u1, . ., u, u', p P, t) in order to save notation.

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4. Statement of the Problem of Competition.-The mathematical interpretation of postulate 2 is to regard ul, . . , u, as independent variables defining p by the equation of demand. A brief statement of the problem is as follows: Given the end times to and t1 and the end values of the prices and rates of production fixed at to, choose the theoretical price p and the rates of production u1, .. ., un defining a curve r in the space ul, ..., un, p, t satisfying an equation of demand of the type G(ui, uV,u2, U2, . . ., u, u P, p', t) = O, such that the profit

7rk =

r2

Fk(Ul,U,

. .

., un up un

p'p t)dt

(k =

1, ... ., n)

of each producer is a maximum when the rate of production Uk alone is allowed to vary with p. It is importanit to notice that there are two problems involved in the above ambiguous statement. If we know in advance the rates of production at the end of the interval of time to < t < t1 the problem is a generalized Lagrange problem with fixed end-points. If we wish to analyze data which is presented to us in an attempt to determine a suitable form of the law of demand it is this problem with fixed end-points which is of chief interest, for we know in advance the end values of the quantities Ulf . . ., Un, p, t. If, however, we wish to forecast probable rates of production and prices the values of the quantities ul, . . ., un, p at t1 are not known in advance and the problem is a generalized Lagrange problem with one end-point available. 5. First Necessary Conditions.-To obtain a solution of either of these problems it is convenient to introduce the mathematical concept of partial variations as I have done in another paper.9 If an n-parameter family of admissible arcs ui = uj(t, a,, . . ., a"), i = 1, . . ., n containing a particular admissible arc r for the parametric values a, = a2 = ... = an = 0 be given, the quantities

{j(X)

= ui (t, 0, . . . 0)/ai

are said to be the partial variations of the family along F. If we assume aG/ap' 0 the partial variations of p are determined by G = 0 except for an arbitrary constant as continuous functions of (t, a,, ... ., a.) with continuous derivatives of the first order except possibly at corners. These partial variations satisfy the following partial equations of variations a(_ ) + aG (x) + + of i = 1, 2, ..., n. (1)

auj

aiuj

apoai

a

a

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The first of these determines bp/ba1 in terpis of 4'_ and the partial derivatives of G with respect to ul and u', whereas the second determines bp/a12 in terms of 4/2 and the partial derivatives of G with respect to u2 and u2 and so on. Choosing each of the partial variations bp/1aj, bp/aZ bp/ba,. equal to zero at t = to implies that the total variation of p be zero at to, i.e., bp = bp/Jal8a, + bp/6a25a2 + . .. + bp/baj,a, = 0 at t = to. The equations of partial variations and the initial conditions, therefore, completely determine the variation§ of p. The partial variation of the integral 7rk with respect to ak reduces for a, = a2 = .. . a. = 0 to the expression .

)7rk/akSak =f

.

.

[(iFk/1uk)`k + (OFk/OUk)4k + (bFk/bP)(bP/1ak) + (aMOa) (OP'lbak) ]bakdt.

By solving the kth differential equation of (1) for iP/?ak, a first necessary condition analogous to the Euler-Lagrange condition may be obtained without the use of multipliers. Before performing the customary integration by parts it is convenient to apply Dirichlet's formula for changing the order of integration to the iterated integral resulting from the substitution of the value of bp/bak in the equation defining the partial variation of rk with respect to ak. 'It follows readily that the solutions must satisfy the n equations 6Fk + +bFk GP+ GWk d (Fk +Fk W + w )=o JUk dt kJk Uk ap IUk ap JUk ?kUk

(k =1,...,n) (2) where [aFk + eAx bG;/dp.dsa

Wk=

_

1

ap

= ?G/ap

ap, ap

as

0O and

WI =

? = ?G/luk WI_ bUk 6G16pt

aG/Iuk

auk aGlap'

aW/abt

The extremals of the problem with fixed end-points must, in addition, satisfy the initial conditions p(to) = po, uj(to) = uio, . . ., u.(to) = U.o

~~~~~~~~Ui(ti)

=

-

Uii,

. .

.1

Un(tl)

=

unl.

The extremals for the problem with one end-point variable must satisfy, the initial conditions p(to) = pop ul(to) = ulo, .. . u,,(to) = uno and the transversality conditions

[Fkdt + (aI64 +

I

k

b

0, Wk) (dUk - U;dt)] t+ (kk= ( (k=1...,n.(3)

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It will be noticed that equations (2) are functional equations for the determination of ul, u2, ... ., u,, and p in terms of the time. The problem. is possible inasmuch as we have given initial conditions and the n functional equations plus the one differential equation of demand to determine the n + 1 functions ul, u2, ... ., u, and p. 6. Further Generalizations.-Obviously, the first order differential equation suggested in paragraph 2 is not sufficiently general to be taken as the typical relation existing between price, rate of production and time, for very often the demand depends upon the history of prices as well as upon the present price and the rate of change of price. As Karl Karsten points out "purchasers everywhere watch not only the present level of prices, but also the direction in which these prices are changing and are expected to change." He uses the mean between a curve and its first integral as typical of the relation existing between price and demand.10 I treated the matter in a much more general manner when I wrote the demand as a Volterra integral equation

,

= ap(t) + b +

J'p(t-r)P((r)dr

where p(- co ) is assumed finite and p(t- r) is negligible if t- r is large and negative. Such a form of the equation of demand assumes that the demand depends not only on the present price but upon all previous prices as well.8 As a first approximation we might write (o(t-Tr) =Xea(t7) where X and a are constants. In the present paper we shall write the demand equation in the even more general form.

G(ui,

4, . . ., u,, uP, p', I) = J P(U, u'l

... .

u, UUn p, ply t, r)dr (4)

where G and P are assumed to be continuous with continuous derivatives of the second order with respect to all of their arguments. With such a general function of demand the dynamic problem of competition can be stated as follows: Given the end time to and the end values of the price and rates of production fixed at to, choose the price p and the rates of production ul, u2 . . ., uU defining a curve r in the space ul, ..., un, p, t satisfying an equation of demand of the type

G(ul, u', . . ., u,, u,, p, p , t) =

P(U1, uun

. .

.P un, un, ip, t, T)dT

such that the profit 7rk =

,Jto

Fk(ul, u1,

.

*

*, un, un, p, pI, t)dt

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of each producer is a maximum when the rate of production Uk alone is allowed to vary with p. Again there are two problems of interest involved in the above statement of the problem. If we are analyzing data at hand the problem is one with fixed end-points. If we wish to forecast probable rates of production and prices one end-point is variable. Let us for brevity consider the problem with fixed end-points. Let r be the curve defined by ui(x) and p(x) if such curves exist and write ui(t) = ui(t) + Oi(t, a,, . . ., ax); p(t) - p(t) + t(t, a,, . . ., ax) where 0s and t are functions, continuous with their second derivatives with respect to ul, ul,. . ., u, u,u, p, t and vanishing when a, = a2 = ... an = 0. If these functions be substituted in the equation (4) it becomes a function of these parameters and yields by a partial differentiation with respect to a, the equations of partial variation

a

i

+ a o$! + atG + aX bu ai

apt

50 ^ sGoj t, ~'[ u,, + bui + a at + aP5']dr.

If we integrate this expression with respect to t and then perform an integration by parts on the primed variations we obtain an expression which by the help of Dirichlet's formula can be written as f _[ aG d aG _P aP aG = _aG us Jt au ui dsau

Ep_;

ujM'

js

dt ?u,/ [ I.G

ddWbG _

(P _'s

d

aP

dt]aWs

ds apt apt -dt Zip' pi which is a Volterra integral equation for the determination of bt in terms of 50,.

J Las

*

NATIONALi RESEARCH FiLLow

b

Iw

MATElSxbATICS.

l A. Cournot, The Math. Prin. of the Theory of Wealth, pp. 52. 2 J. Fisher, Trans. Conn. Acad. Arts Sci., 9, part 1, 1892. 3 V. Pareto, Giorn. Econi., 283 (1901). ' H. L. Moore, Economic Cycles, 69-70, see also J. Amer. Stat. Ass. (1922), and Quar. J. Econ. (1926). I. Fisher, J. Amer. Stat. Ass., 8 (1923). *G. C. Evans, Amer. Math. Monthly (1924). 7 Evans, these PROCMDINGS, 11, 90. 8 Roos, Amer. J. Math., 163-175 (1925). 9 Roos, Trans. Amer. Math. Soc., not yet published. 10 K. Karsten, J. Amer. Stat. Ass. (1924).