Barbolla, R.

Corchón, Luis C.

The Quarterly Journal of Economics

English

Name not available

An Elementary Proof of the Existence of a Competitive Equilibrium in a Special Case

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AN ELEMENTARY PROOF OF THE EXISTENCE OF A COMPETITIVE EQUILIBRIUM IN A SPECIAL CASE* ROSA BARBOLLA AND Luis C. CORCH6N I. INTRODUCTION The proof of the existence of a competitive equilibrium has played an important role in the development of modern microeco- nomics. Such a proof usually involves the use of an advanced fixed point theorem: either Brouwer's or Kakutani's. Since competitive excess demand functions do not possess any structural properties beyond continuity and Walras' Law, it is sometimes argued that the proof of the existence of a competitive equilibrium and the proof of the existence of a fixed point are actually equivalent. Once such equivalence has been established, the following questions appears to be quite natural: how robust is the previous result to changes in the admissible space of economies? Or in other words: is it possible to introduce any additional assumption on preferences, technology, etc., such that the proof of the existence of an equilibrium is drastically simplified? In 1977 Joseph Greenberg proved that the introduction of an additional assumption, namely that goods are gross substitutes, simplified the proof of the existence of a competi- tive equilibrium. The present paper aims to contribute to such a line of research. We shall show that if the gross substitutes assump- tion is replaced by the assumption that good n is a weak gross substitute with the rest of goods and the assumption that the determinant of a Jacobian of excess demand functions is every- where nonvanishing, we still have a simple proof of the existence of an equilibrium. A motivation for our work can be found in the fact that equilibrium is a totally meaningful concept in the case in which uniqueness and stability are taken as granted. But precisely our condition on the Jacobian implies uniqueness (see Varian [1984, p. 244]). Moreover, a theorem by Hands [1981] on the stability of equilibrium requires a special case of our condition on the Jacobian (see Hands [1981, p. 209]). Therefore, the methods developed in this paper work for the class of economies for which the concept of equilibrium may be considered as totally satisfactory. *We would like to express our gratitude to Andreu Mas-Colell and J. Greenberg for their helpful suggestions and to Ana Ant6n-Pacheco for her linguistic advice. Referees supplied enormously useful criticisms. The usual caveat applies. ? 1989 by the President and Fellows of Harvard College and the Massachusetts Institute of Technology. The Quarterly Journal of Economics, May 1989 386 QUARTERLY JOURNAL OF ECONOMICS II. THE MODEL The are n goods. The price of good i is denoted by pi. Let S = {p E Rn I y~,_lp, = 11 be the price simplex. Rn is the nonnegative orthant of Rn. Let R = RU{?+jo; i.e., the real line extended at +?c. Let z: S - Rk be an excess demand function. If b E int S,z(p) E Rn. We shall assume the following: (1) z(p) is continuously differentiable Vp int S. zn(P) is continuous VPn >- 0; (2) z(p) is homogeneous of degree zero; (3) p. z(p) = 0 (Walras' Law); (4) If p - with some zero component (but not the nth) then n-i Zi(p)2 +ms i=1 p p Assumptions 1-4 are more or less standard. The continuity of z(P-) must be understood in an extended sense; i.e., if z (-) = +?c, and p- zn(p)+ Notice that the boundary condition is usually stated in terms of all goods. Let us denote by zi; = azi( )/Opj. L(p) is an (n - 1) x (n - 1) matrix with typical component zi;. (5) DeterminantL(p) = 0 Vp E S,pi > 0, i = 1, 2, . . ., n - 1; (6) Z2yn-1 piZi( ) is a nondecreasing function of Pn for pi > 0, i = 1, 2,...,n - 1. A possible interpretation of (5) is the nonexistence of large income effects (see Varian [1984, p. 244]). This assumption is a generalization of the so-called Gale property (see Arrow-Hahn [1971, p. 208]) which in turn is a generalization of gross substitutes, dominant diagonal, and other properties that are used in the areas of uniqueness, comparative statics, and stability in general equilib- rium. As we know from the work on the properties of excess demand functions [Sonnenschein, Debreu, Mas-Colell, and others], such an assumption is not necessarily true (see Shafer and Sonnenschein [1982]). A sufficient condition that guarantees (6) is that '( ) 0, i 1 2, .... n -1 49pn PROOF OF A COMPETITIVE EQUILIBRIUM 387 i.e., good n is a weak gross substitute of goods 1, 2, ..., n - 1. Alternatively, if the elasticity of demand for good n with respect to p, is negative and greater than or equal to one, (6) holds. Comparing our assumptions with those of Greenberg [1977], we notice that our boundary condition is slightly stronger. However, our assumptions (5) and (6) can be regarded as generalizations of his gross substitute assumption. Finally, we shall define our equilib- rium notion. DEFINITION. A vector p* = (p, ... , n*) is said to be an equilibrium price vector if (a) p* E S; (b) zi(p*) - O for every i = 1, . . ., n; (c) if zi(p*) < 0, then pi* = 0. And notice that pi* > 0, i = 1, 2, . . . , n - 1 (from (4) above). III. THE THEOREM In this section we shall state and prove our main theorem. THEOREM. Under assumptions (1)-(6) above, there exists an equi- librium price vector. Proof. Let p- int S. Let us define n-i w =ZEzi(m)2. i=1 Now let n-i B = {p E SIZzi(p)2 _ 2w}. i=1 The set B has the following properties: it is nonempty (since p E B), bounded (since p Q S), and closed (by the continuity of zi( ) and assumption (4)). Also V b E B, the first n - 1 components of j- must be strictly positive (by (4)). Therefore, En L1zj(p)2 is continu- ous on B. Now consider the following problem: n-i To min Mz(p)2 p Q B. i.ll A solution exists. Call it p*'. Also notice that the above 388 QUARTERLY JOURNAL OF ECONOMICS minimization can be written in the following form: n-i To min E z(p) i=1 subject to [Pi < ? i = 1, 2, .. ., n (1) n E Pi = i (2) i=1 n-i z z1(p)2 _ 2w (3). First, if zi(T-) = 0, i = 1, 2, .. ., n - 1, the argument is done. Now notice that, at any solution, (3) holds with strict inequality, as well as (1) for i = 1, 2, ..., n - 1. Moreover, the Lagrange multiplier associated with (2) is zero because of our assumption (2). Therefore, the first n - 1 first-order conditions read L(p*). z = 0 [Z = Z1 ... Zn-1] and this implies that zi(p*) = 0 i = 1, 2, . . . , n - 1. If pn* 0, then zn(p*) = 0; i.e., p is an equilibrium. So let us assume that pAn = 0. In this case if zn(p*) < 0, an equilibrium exists, so we only have to consider the case in which either zn(P*) > 0 or zn(P*) = + o0 Let us denote by P-n = (P1,P2 ... Pn-l)- If we take pn#+ 0 but sufficiently close to zero, we have, n-i 0 > -Pn'Zn n = E P*Zi (P*_Pns) -? 0 (the first inequality is due to the continuity of zn( ) and the last to condition (6)) which is a contradiction. Assumptions (5) and (6) can be replaced by the following assumption: let J(p) be the n x n matrix with typical component zij. Then det I # O Vp E S. p o The difficulty with this approach is the economic interpreta- tion of such a condition. It is true that when z = 0, this condition is equivalent to our assumption (5). (See Mas-Colell [1985, pp. 178- PROOF OF A COMPETITIVE EQUILIBRIUM 389 80]). However, we have been unable to interpret such a condition when z # 0. U. COMPLUTENSE DE MADRID REFERENCES Arrow, K., and F. Hahn, General Competitive Analysis (San Francisco: Holden Day, 1971). Greenberg, J., "An Elementary Proof of the Existence of a Competitive Equilibrium with weak Gross Substitutes," this Journal, XCI (1977), 513-16. Hands, D. W., "A Note on Global Hicksian Stability Conditions," Economics Letters, VIII (1981), 221-26. Mas-Colell, A., The Theory of General Economic Equilibrium: A Differentiable Approach (Cambridge: Cambridge University Pres, 1985). Shafer, W., and H. Sonnenschein, "Market Demand and Excess Demand Function," in K. Arrow and M. Intriligator, eds., Handbook of Mathematical Economics (Amsterdam: North-Holland, 1982). Varian, H., Microeconomic Analysis, 2d ed. (New York, London: W. W. Norton and Company, 1984). 

A Note on Global Hicksian Stability Conditions,&quot; Economics Letters,

An Elementary Proof of the Existence of a Competitive Equilibrium with weak Gross Substitutes,&quot; this Journal, XCI

General Competitive Analysis

Market Demand and Excess Demand Function,&quot;

Microeconomic Analysis, 2d ed.

The Theory of General Economic Equilibrium: A Differentiable Approach (Cambridge: Cambridge University Pres,

http://e-archivo.uc3m.es/bitstream/handle/10016/3819/CORCHON_QJE_1989_104_2_CL.pdf?sequence=1

An Elementary Proof of the Existence of a Competitive Equilibrium in a Special Case

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