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Note on the Economic Equilibrium for Nonlinear ModelsAuthor(s): Nicholas Georgescu-RoegenSource: Econometrica, Vol. 22, No. 1 (Jan., 1954), pp. 54-57Published by: The Econometric SocietyStable URL: http://www.jstor.org/stable/1909832 .

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7/27/2019 Note on the Economic Equilibrium for Nonlinear Models

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NOTE ON THE ECONOMICEQUILIBRIUMFORNONLINEAR MODELS

BY NICHOLAS GEo1RGESCU-RoEGEN

IT ISTHE purposeof this note to present urtherresultsconcerning he existenceof economicequilibrium n the case of a model more generalthan that con-sidered by either Mr. Nikaido in the note published in this issue or by thiswriterin ActivityAnalysisof Production nd Allocation.!

1. The results reached by Nikaido can be obtained as an application ofTheorem2 (and its Corollaries) f Bohnenblust,Karlin, andShapley.2Nikaido'snote offers, however, a much more elementary proof for the particular casetreated by him. Sincethis particularcase is of specialimportancen the discus-

sion of economicequilibrium ny elementarization f its treatment deservesourattention.3In this respect, one may point out that Nikaido's results, concerningthe

existenceof saddlepointsfor both K(A, j) andO(A,Y), canbeprovedbyfollow-ing step by step the procedureadopted by this writer in the paper alreadymentioned.The resultsof Theorems - V are valid, mutatismutandis or eitherK(A, j) or 4(A, Y) in case of concavity. In particular,TheoremIII offers asharpeningof Nikaido'sresults.

Finally, it should be remarked hat Nikaido's model does not constitute a

real generalizationof von Neumann's model since parallel to dropping theassumptionof linearity,an important restrictionof the latter, it introducesanew restriction:limitationality. ndeed, in Nikaido's model, to each input Athere corresponds nly one output f(A). A reexaminationf my argumentn thevolume on ActivityAnalysis, in the light of Nikaido'sresult, leads to the con-clusion that the limitationality restrictioncan be eliminatedwithout affectingthe existenceof economicequilibrium.These new results are presented n thenext sections.

2. Referringto the technologicalhorizonH as the set of all feasible trans-formationsT = (A, B),4we shall introduce he followingpostulates,I - III.I. The technologicalorizon orms a closed convexset in the positiveorthant

R+2 f thespaceof coordinatesal, a2, * , an, bi, b2, *, bn).The problemof justifyingthis postulateis rather delicate.Someimplications

of (I) representwell-known nterpretationsof "decreasing eturns."Thus:

1 "The Aggregate Linear Production Function and its Application to von Neumann'sEconomic Model," loc. cit., pp. 98-115.

2 "Games with Continuous, Convex Pay-Off," Contributions to the Theory of Games,

Vol. 1, Annalsof Mathematics Studies, No. 24, pp. 181-192. (Their main Theorem 1 (p. 185)requires the additional condition that "inf, supa (pa(y)< + co . Indeed if c = + oo theapplication of Lemma 1.5 is incorrect.)

3E.g., J. von Neumann and 0. Morgenstern, Theory of Games and Economic Behavior,2nd edition, p. 154n.

4 Georgescu-Roegen, op. cit., pp. 99-102. The present paper will use the same notations,with the exception of that for a transformation, introduced above.

54

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NOTE ON GENERAL EQUILIBRIUM 55

I.1. The convexitypropertyapplied to the transformationsA, B1), (A, B2),yields a nondecreasingmarginal rate of productsubbstitution.

I.2. The same property appliedto (A1, B), (A2, B) yields a nonincreasing

marginal rate of factor substitution.I.3. Finally, the convexityproperty applied to (A, B), (XA,MB)yields non-increasing returns to scale.

Obviously, (I) involves more than (I.1) - (I.3). One may, however,consider(I) as the most complete condition or nonincreasing eturns.

II. For any T e H,

n

Ea i ) a- > 0; ai + bi > hyi> O.

Any closed subset of R+2 satisfyingthese conditionswill be referred o as an61set.

Definition. The normalizedshadow t of any point (A, B) e 6R s the point(XA, XB), whereX2(ai + bi) = 1. The normalized hadowof a set is the setof normalized hadowsof its elements.

III. The normalizedshadow r of H is closed.5

3. From the above postulates,the results (1) - (9), presentedbelow, can

easily be derivedby followingthe main line of the argumentpresented n Ac-tivity Analysis.(1). If we write k(T) = min (T, Y), then k(T)is continuous overany 61.(2). 4(X1T1 X2T2) min kb(T1), b(T2)]forany T1, T2belongingoa convexR,

(0 X= 1 - X2K 1).This follows immediately from relations (40) - (42), Activity Analysis,

p. 110.(3). The subset of a convexat such that + (T) > K is a closed convex set 6(R.

If ?b(T) = K, T belongs to B(6(K), the boundary of (Rc .

This follows from (2) and from the fact that ?(T) = K requires bi = Kai,bj > Kaj, (i $ j) for at least one i = 1, 2, ***, n.

(4). The return to the dollar, O(T, Y), admits a saddle point (TO, Y?) overH

and S.6

(5). If q(T0, Y?) = M, then b? = Ma?, i , o-;b? > Mai, yi = 0, i>(6). The set of all To is a convexset Ho n H, whereHo is a conewhosenormalized

shadow ro C B((r). The set of all Y? is a convexSo C S.

(7). rF, so (and henceHO) are closed.We need now the

Definition.H is saidto be strictlyB-convexf for T1, T2, e B(H) and A1 - A2,the transformationX1Tj X2T2,where0 < XI = 1 - X2< 1, doesnot belongto B(H).

(8). If H is strictlyB-convex, the AOof all To is unique.

6 Ibid., pp. 103n, 109. This postulate is a consequence of (I), if H is a cone.6Ibid., p. 110, (39).

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56 NICHOLAS GEORGESCU-ROEGEN

This follows immediately from the fact that To e Ho n H c B(H), result

(6). One may go even a little further:(9). If H is strictlyB-convex, and the rank of To is n,7 then To is unique.

4. The essential difference between a linear model and a nonlinear modelis that the equilibrium of the former leads to a set of linear processes, that of

the latter, to a set of transformations.Since scale is the element that differentiates

transformation from linear process, it is only natural that the nonlinear modelshould raise additional problems related to size. One of these problems concerns

the availability of AO. The technologicalequilibrium To may not be achievablebecause the resources at the disposal of the community do not suffice in all

respects. However, this new aspect of the problem does not interfere with the

results of the preceding section.

In general, the available amounts of resourceslimit the ai's either individuallyor by groups. The latter is the case when a certain type of labor, for instance,

may have alternative input uses. On the other hand, the employment of some

factors may have to be maintained above a certain minimum level. All these

restrictions lead to

EaE ai <, L, ,,#2 ai >, M, ((x' 0i).

Therefore, one may now introduce the additional postulate:

IV. The available nputs form a closed convexset, a, havinga closed normalizedshadow.8

The set of all technologically feasible equilibria which are at the same time

economically achievable, i.e., for which A E a, forms the economic horizon,30 c H. Because of IV, the economic horizon satisfies postulates (I) - (III).

Therefore, the results regarding H, presented in the preceding section, are

valid for SC oo. In particular

(10). A nonlinear model satisfying postulates (I) - (IV) has a technologicalequilibrium To as well as an achievableequilibriumgo.'

5. The distinction between the technological and the achievable equilibria

raises a series of problems. The following preliminary remarks are intended

to offer a sample of the problems that could be pursued further.

(a) If N is the maximum value of the integer n such that ng0 Ea, the achieva-

ble equilibrium of the entire economy consists of N units go.(b) If two economies, E1, E2, have the same technological horizon, but

k(go1)> k(go2), 2 is underdeveloped relative to E1. To # <J0is a reasonable

essential criterion for absolute economic underdevelopment. Economic under-

development is not necessarily caused by lack of know-how alone. Any economymay be underdeveloped in both the absolute and relative sense.

7Ibid., p. 110.8 Cf. Nikaido, op. cit., p. 49. The assumption of boundedness is not necessary.9 The role of postulate IV appearsnow in its true light. It does not replace any character-

istic of the technological horizon, in spite of the fact that from the analytical point ofview one may regard all transformations which are not in JCasif not technically achievable.

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NOTE ON GENERAL EQUILIBRIUM 57

(c) If a1 C a2 and 92 # To then p(9?, Y?) < 0(J?2 Y?) < 4(T, Y?).Thisexplains why, in an underdeveloped economy, additional amounts of any short

factors can earn increasing returns.

(d) If To 5 Jo, then \jo + (1 - X)T0, (0 < X < 1) cannot belong to a.This follows from (c) and (2). Obviously, this shows that there is a kind of

optimum utilization of a. It also shows that by introducing some "rationing"

of the short factors, one can increase the employment of the other factors.

(e) Finally, the question of development, i.e., the structure of the sequence

N19?, N2c92, * , where the output of NJ influences the available inputs for

Ni+1j+j1, is mentioned here only as an illustration of the rather interesting

problems related to the dynamic aspects of the model.

Vanderbilt University