Abraham Wald s equilibrium existence proof reconsidered 1

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1 Abraham Wald s equilibrium existence proof reconsidered 1 Reinhard John Wirtschaftstheoretische Abteilung II, Universität Bonn, Adenauerallee 24-42, D Bonn, GERMANY ( rjohn@econ2.uni-bonn.de) Economic Theory 13 (1999), Summary. For his proof of the existence of a general competitive equilibrium Abraham Wald assumed a strictly pseudomonotone inverse market demand function or, equivalently, that market demand satisfies the Weak Axiom of Revealed Preference. It is well known that more recent existence theorems do not need this assumption. In order to clarify its role in Wald s proof, the question of existence of an equilibrium for a modified version of the Walras-Cassel model is reduced to the solvability of a related variational inequality problem. In general, the existence of a solution to such a problem can only be proved by advanced mathematical methods. We provide an elementary induction proof which demonstrates the essence of Abraham Wald s famous contribution. Keywords and Phrases: Abraham Wald - Existence of equilibrium - Pseudomonotonicity - Variational inequality problem JEL Classification Numbers: B 21, C For their helpful comments I would like to thank Juan Enrique Martínez-Legaz, Matthias Raith and, in particular, Werner Hildenbrand who also suggested the idea for this paper. Financial support by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303 at the University of Bonn is gratefully acknowledged.

2 1 Introduction One of the most distinguished achievements in mathematical economics is the first rigorous proof of the existence of a competitive equilibrium of a production economy by Abraham Wald (1935, 1936a,b). Nevertheless, the essence of his contribution has not been fully recognized in the existing literature. It is well known that one of Wald s assumptions concerning the demand side of the economy corresponds to the Weak Axiom of Revealed Preference which implies partial uniqueness of the equilibrium. Although more recent existence theorems, e.g. by Arrow and Debreu (1954), do not need this assumption, they do employ an advanced fixed point argument (or equivalent mathematical results), while Wald proceeded in a more elementary way by induction on the number of commodities. Consequently, Hildenbrand (1987) assumes that many readers would like to know why in Wald s proof his famous assumption (6) (i.e. a version of the weak axiom of revealed preferences for the market demand function) is needed for the existence part of the proof. Dorfman, Samuelson, and Solow (1958) implicitly disagree: But the only use made of this overly strong assumption is to show that the competitive equilibrium is unique. The purpose of this paper is to clarify this issue. In particular, it turns out that, contrary to the opinion of Dorfman, Samuelson, and Solow, Wald s assumption 6 is crucial for the existence part of his induction proof. Its role is to guarantee the partial uniqueness of equilibrium for all lower dimensional subsystems. It is exactly this property which enables the induction step. Moreover, we will demonstrate that only convexity instead of uniqueness is required for this kind of proof. As a consequence, Wald s assumption 6 which, in mathematical terminology, means strict pseudomonotonicity of inverse demand can be weakened to pseudomonotonicity. There is still another question, also raised by Hildenbrand (1987): What role does the assumption play that the market demand function is invertible? Is this assumption only made since Schlesinger s modification of Cassel s equation is formulated in this (unsatisfactory) way? We are not able to answer the latter question. However, we show that Schlesinger s inverse demand formulation is not required for an elementary existence proof. It is well known that existence results, which in general are equivalent to Brouwer s fixed point theorem, can be proved by more elementary methods (e.g. separation of convex sets) under special circumstances. For example, this is true for the KKM-Lemma if the closed sets under consideration are also convex (see Valentine (1964)) or for monotone generalized variational inequality problems as shown by Oettli (1988). Elementary proofs of the existence of a competitive 1

3 equilibrium can be also found in Greenberg (1977) or Barbolla and Corchón (1989). Our main concern here, however, is the particular proof by Abraham Wald. The paper is organized as follows. In Section 2 we consider a simple twocommodity excess demand system which serves as an example to illustrate the essential idea of an existence proof by induction. We emphasize that this is the sole objective of this section since the model is quite different from Wald s which is presented in Section 3. There we reduce the problem of existence of an equilibrium for Schlesinger s version of the famous Walras-Cassel model to the solvability of a related variational inequality problem. Section 4 provides an elementary induction proof for the existence of a solution of a general variational inequality problem which resembles Abraham Wald s proof. Finally, in Section 5 this result is applied in order to obtain a similar proof for Cassel s original formulation. 2 An Example Let E : IR 2 ++ IR 2 be a continuous function. For p = (p 1, p 2 ) IR 2 ++ the value E(p) = (E 1 (p), E 2 (p)) is interpreted as the excess demand for two commodities with market prices p 1 and p 2. We assume that E statisfies the following boundary conditions: (2.1) For every sequence (p n ) in IR 2 ++ with I = {i p n i 0} there exists k I such that E k (p n ) > 0 for large n. (2.2) For every sequence (p n ) in IR 2 ++ with J = {j p n j } there exists k J such that E k (p n ) < 0 for large n. It is known that under these assumptions there exists an equilibrium for the excess demand system E, i.e. there is p IR 2 ++ such that E(p ) = 0. The result generalizes to any number of commodities as was shown by Schulz (1985). Its proof is not elementary, even for the case of only two commodities. However, if we assume that any partial equilibrium analysis of one market for a given price on the other market leads to at most one partial equilibrium then we obtain existence of a global equilibrium by employing only the intermediate value theorem. Proposition 2.1. Let E : IR 2 ++ IR 2 be a continuous function which satisfies the conditions (2.1), (2.2), and (2.3) For any p 2 > 0 there is at most one p 1 > 0 such that E 1 (p 1, p 2 ) = 0. Then there exists p = (p 1, p 2) such that E(p ) = 0. 2

4 Proof: Consider an arbitrary fixed p 2 > 0. By (2.1) there exists p 1 with E 1 (p 1, p 2 ) > 0 and by (2.2) there exists p 1 with E 1 (p 1, p 2 ) < 0. Since E is continuous, the intermediate value theorem implies the existence of ˆp 1 such that E 1 (ˆp 1, p 2 ) = 0. By (2.3) ˆp 1 is uniquely determined and, thus, can be denoted by ˆp 1 (p 2 ). The function which associates to every p 2 the partial equilibrium price ˆp 1 (p 2 ) of the first market is continuous. In order to prove this, consider an arbitrary p 2 > 0 and a sequence (p n 2) converging to p 2. E 1 (ˆp 1 (p n 2), p n 2) = 0 for all n implies that the sequence (ˆp 1 (p n 2)) is bounded from above and away from zero (otherwise one would get a contradiction to (2.2) or (2.1)). Hence, there is a subsequence of (ˆp 1 (p n 2)) which converges to some p 1 > 0. By continuity of E, it follows that E 1 ( p 1, p 2 ) = 0. Consequently, by (2.3), we obtain p 1 = ˆp 1 ( p 2 ). Since any convergent subsequence of (ˆp 1 (p n 2)) has the same limit ˆp 1 ( p 2 ), the sequence ˆp 1 (p n 2) converges to (ˆp 1 ( p 2 )). Thus, ˆp 1 (p 2 ) varies continuously in p 2. This implies that the function ϕ : IR ++ IR, defined by ϕ(p 2 ) = E 2 (ˆp 1 (p 2 ), p 2 ), is also continuous. Consider a sequence (p n 2) converging to 0. It defines a sequence p n = (ˆp 1 (p n 2), p n 2) with E 1 (p n ) = 0. Consequently, by (2.1), ϕ(p n 2) = E 2 (p n ) > 0 for large n. Analogously, (2.2) implies ϕ(p 2 ) < 0 for some large p 2. Applying the intermediate value theorem to the function ϕ yields the existence of p 2 such that ϕ(p 2) = 0. Thus, for p = (p 1, p 2), where p 1 = ˆp 1 (p 2), we obtain E(p ) = 0. Q.E.D. Let us conclude this section with two remarks. First, it is easy to see that Proposition 2.1 can be generalized to any number of commoditites by requiring all multicommodity generalized partial excess demand systems to have at most one equilibrium. Second, it is not the boundary behaviour which is essentially responsible for the main idea of the proof. This will be made clear in Section 4 where we formulate a version of an existence result without it. Since Wald also employed a boundary condition we have chosen to present an example including it in order to indicate the unimportance of this point. 3 The Walras-Cassel Model In this section we consider a general equilibrium model with a production sector characterized by fixed input coefficients. Its origin is the simplification of Walras production equations by Cassel (1899, 1918). 3

5 The economic system is described by m pure factors of production i = 1,..., m which are available in fixed positive amounts r = (r 1,..., r m ) and which can be used to produce n final commodities j = 1,..., n. The production of one unit of commodity j requires a ij 0 units of the primary resource i, i.e. the production sector is characterized by the input coefficient matrix A = (a ij ) m n. Assume that price vectors p = (p 1,..., p n ) 0 for final commodities and q = (q 1,..., q m ) 0 for factors are given. Then the output vector x = (x 1,..., x n ) 0 maximizes profits if unit costs are greater or equal than output price, equality being implied by a nonzero output quantity, i.e. if 1 qa p and (qa p)x = 0. (3.1) All factor markets are in (free disposal) equilibrium if on each market demand is less than or equal to supply, equality being implied by a positive factor price, i.e. if Ax r and q(ax r) = 0. (3.2) By the duality theory of linear programming, it is well-known that (3.1) and (3.2) are equivalent to the statement that x and q solve the dual problems and max px subject to x 0 and Ax r min qr subject to q 0 and qa p, or, equivalently, that p, q and x satisfy the conditions Ax r, qa p and px = qr. What now remains is to close the model by an equilibrium condition for the product markets. If the resources of the economy are owned by consumers who demand the final commodities according to a homogeneous demand function F (p, q), equilibrium in these commodity markets is described by the condition x = F (p, q). (3.3) Thus, a general equilibrium is characterized by a vector (p, q, x ) such that (3.1), (3.2) and (3.3) hold. It follows immediately that, for all λ > 0, the vector (λp, λq, x ) is also an equilibrium. Therefore, prices can be normalized by fixing the total value of the resources, e.g. q r = C > 0. Originally, Cassel (1918) required a solution where unit costs are equal to output price, i.e. qa = p, (3.1 ) 1 Prices are represented by row vectors, quantities by column vectors. 4

6 and where demand equals supply on factor markets, i.e. Ax = r. Later, Zeuthen (1933) and Schlesinger (1935) proposed to replace the latter condition by (3.2). In general, demand depends on the income distribution derived from q and the distribution of ownership. To assume independence, as it was explicitly done by Wald (1936b), is a strong additional assumption (for example, it is satisfied if consumers own collinear individual resources). However, another interpretation of demand being independent of factor prices was already given in Cassel (1899). There, as well as in his provisional (short run) model (1918), he supposed consumers expenditures on final commodities to be determined by exogenously given amounts of money. In that case, aggregate demand depends only on p and the fixed distribution of total expenditure C. 2 Both views enable to replace (3.3) by the simpler equation x = F (p). (3.3 ) Finally, Schlesinger (1935) modified this demand condition by assuming the existence of an inverse market demand function f, i.e. by writing p = f(x). (3.3 ) It was essentially the system of conditions (3.1), (3.2) and (3.3 ) for which Wald proved the existence of a solution (p, q, x ). His proof is based on the following assumptions: 3 (W1) For every j = 1,..., n there exists i {1,..., m} such that a ij > 0. (W2) f : IR n + IR n ++ is continuous. (W3) For every x, x IR n + such that x x, the weak inequality f(x)(x x) 0 implies the strict inequality f(x )(x x) < 0. Assumption (W1) guarantees the boundedness of the production-possibility set X = {x 0 Ax r}. This implies for any p 0 the existence of an element x in this set which maximizes px, i.e. the primal linear programming problem mentioned above has the optimal solution x. By the (elementary) duality theorem of linear programming it follows that the dual problem can be solved by some q. Consequently, (p, q, x ) satisfies the conditions (3.1) and (3.2), but not necessarily p = f(x ). 2 Cassel had in mind a model which Malinvaud (1972) has called a distribution economy. 3 To be precise, we consider here the variant of Wald s model by Kuhn (1956). Wald looked for a solution satisfying (3.1 ) which is stronger than (3.1) and ensured by strictly positive outputs. These he enforced by an additional boundary condition on f. Our assumption (W3) is assumption 6 in Wald (1936a,b) and equivalent to the Weak Axiom of Revealed Preference for the direct demand function F. 5

7 We thus have to find a vector x that maximizes f(x )x on X, i.e. we have to find x X such that for all x X or, equivalently, f(x )x f(x )x, f(x )(x x ) 0. Such an x is called a solution to the variational inequality problem given by X and f. Assumption (W2) guarantees that there is at least one solution. This is implied by Kakutani s fixed point theorem as was shown by Kuhn (1956). It follows directly from an existence result of Hartman and Stampacchia (1966). But then, why do we need assumption (W3)? Of course, these theorems were not available to Abraham Wald. On the other hand, assumption (W3) immediately implies the uniqueness of a solution x. Since this does not imply the uniqueness of the factor prices q, its purpose cannot be to prove uniqueness of equilibrium. The crucial point is that assumption (W3) ensures a unique solution to the variational inequality problem with respect to f and every convex subset of X. It was this consequence which enabled Wald to obtain an existence proof by induction on the number n of final commodities without the use of an advanced result like Brouwer s fixed point theorem. In the next section we provide an elementary proof of the Hartman- Stampacchia theorem which resembles Wald s proof. In fact, we even weaken the uniqueness property implied by assumption (W3) by only assuming convex solution sets to the variational inequality problems. This shows again that it is not uniqueness per se which distinguishes Wald s result from the more recent equilibrium existence theorems. 4 An Elementary Existence Proof We have demonstrated in the previous section that an equilibrium of the Schlesinger system (3.1), (3.2), and (3.3 ) can be reduced to the following solution concept, which generalizes the notion of a solution to an equation with l variables. Definition 4.1. Let f : X IR l be a function defined on a nonempty subset X of IR l and let Y be a nonempty subset of X. 6

8 A solution to the variational inequality problem V IP (Y, f) associated with (Y, f) is a vector y Y such that for all y Y : f(y ) (y y ) 0. We denote the solution set of V IP (Y, f) by (Y, f). It is well known that the following general existence result due to Hartman and Stampacchia (1966) is equivalent to Brouwer s fixed point theorem (see e.g. Border (1985)). Theorem 4.1. If f : X IR l is a continuous function defined on a nonempty compact and convex subset X of IR l, then V IP (X, f) is solvable. In the sequel we shall give an elementary proof of this theorem under an additional assumption which slightly generalizes Wald s famous condition 6. As Wald s proof it proceeds by induction and uses only the intermediate value theorem. by First, notice that the solution set of V IP (X, f) can be equivalently described (X, f) = x X{y X f(y) (x y) 0}. Since f is continuous, it is an intersection of closed sets in the compact set X. Consequently, V IP (X, f) is solvable if every finite collection of these sets has a nonempty intersection, i.e. if for any x 1,..., x n X there is x X such that for i = 1,..., n f(x ) (x i x ) 0. Thus, it is sufficient to prove the existence of x co {x 1,..., x n } satisfying these inequalities which is equivalent to the solvability of V IP (X n, f) where X n := co {x 1,..., x n }. For this proof, we make use of the following generalized version of the intermediate value theorem which was also employed by Wald 4. Lemma 4.1. Let F : [a, b] IR be an upper hemi-continuous correspondence with compact and convex values defined on a real interval [a, b]. If F (a) IR + and F (b) IR are nonempty then there is x [a, b] such that 0 F (x ). Proof. Consider the sets A := {x [a, b] F (x) IR + } and B := {x [a, b] F (x) IR }. By assumption, A and B are nonempty and, trivially, 4 We assume that the reader is familiar with the notion of a correspondence and its continuity definitions. A good reference is Border (1985). 7

9 A B = [a, b]. Since F is compact-valued and upper hemi-continuous, A and B are closed. Hence, connectedness of [a, b] implies that A B, i.e. there is x [a, b] and ȳ, y F (x ) such that ȳ 0 y. Convexity of F (x ) implies 0 F (x ). Q.E.D. Proposition 4.1. Let f : X IR l be a continuous function defined on a nonempty convex subset X of IR l such that for all convex sets Y X the set (Y, f) is convex. Then for all x 1,..., x n X there exists a solution to V IP (X n, f) where X n = co {x 1,..., x n }. Proof. For n = 1 the claim is trivially true. Assuming the existence for n 1 it remains to show the existence for any x 1,..., x n+1 X. by First, notice that the elements of co {x 1,..., x n+1 } = X n+1 can be denoted x(λ, µ) := µ x n+1 + (1 µ) n λ i x i = i=1 n λ i (µx n+1 + (1 µ)x i ), where λ n 1 and µ [0, 1], i.e. X n+1 = 0 µ 1 X n(µ) with X n (µ) = co {µx n+1 + (1 µ)x i i = 1,..., n}. By the induction hypothesis, V IP (X n (µ), f) is solvable. Since X n (µ) is a convex subset of X, the assumption of the proposition implies that (X n (µ), f) is a nonempty convex set for each µ. Consequently, the correspondence i=1 (X n ( ), f) : [0, 1] X n+1 is convex-valued. In addition, it is compact-valued and u.h.c. if it is closed. In order to prove the closedness of this correspondence, consider sequences µ k µ and x k x such that x k (X n(µ k ), f) for all k. For any λ n 1, this implies f(x k ) (x(λ, µ k) x k ) 0 for all k and, consequently, f(x ) (x(λ, µ) x ) 0. Hence, x (X n (µ), f). Now consider the continuous function ψ : X n+1 IR, defined by ψ(x) = f(x) (x n+1 x). The composition F := ψ (X n ( ), f) from [0, 1] into IR is again a convex- and compact-valued u.h.c. correspondence. Assume µ [0, 1[ and x(λ, µ) (X n ( µ), f). We claim that f(x(λ, µ)) (x(λ, µ) x(λ, µ)) 0 (4.1) 8

10 for every µ such that (µ µ) ψ(x(λ, µ)) 0, and every λ n 1. In order to show this, it is straightforward to check that x(λ, µ) x(λ, µ) = µ µ 1 µ (xn+1 x(λ, µ)) + 1 µ 1 µ (x(λ, µ) x(λ, µ)). Since x(λ, µ) (X n ( µ), f), it follows that f(x(λ, µ)) 1 µ 1 µ (x(λ, µ) x(λ, µ)) 0. Moreover, by definition and by the assumption, we obtain f(x(λ, µ)) µ µ 1 µ (xn+1 x(λ, µ)) = µ µ 1 µ ψ(x(λ, µ)) 0. Adding both inequalities leads to the desired result (4.1). We now distinguish between three mutually exclusive and exhaustive cases. a) F (0) IR + =, i.e. ψ(x(λ, 0)) < 0 for some x(λ, 0) (X n (0), f) : By (4.1) we obtain f(x(λ, 0) (x(λ, µ) x(λ, 0)) 0 for all λ and µ. Hence, x(λ, 0) solves V IP (X n+1, f). b) F ( µ) IR = for all µ < 1: Consider a sequence µ k 1 and a sequence x(λ k, µ k) (X n ( µ k ), f). Then ψ(x(λ k, µ k)) > 0 which implies (µ µ k ) ψ(x(λ k, µ k)) < 0 for µ < µ k. Thus, by (4.1), for any λ and any µ < 1 we obtain f(x(λ k, µ k )) (x(λ, µ) x(λ k, µ k )) 0 for all k such that µ < µ k. Since x(λ k, µ k) x n+1, continuity of f implies f(x n+1 ) (x(λ, µ) x n+1 ) 0. Hence, x n+1 solves V IP (X n+1, f). c) F (0) IR + and F (µ) IR for some µ < 1: By Lemma 4.1 there exists µ [0, µ] such that 0 F (µ ). Consequently, there is x(λ, µ ) (X n (µ ), f) with ψ(x(λ, µ )) = 0. Hence, by (4.1), f(x(λ, µ )) (x(λ, µ) x(λ, µ )) 0 for all λ and µ, i.e. x(λ, µ ) solves V IP (X n+1, f). Q.E.D. The assumption of a convex solution set (Y, f) for any convex Y X is the key to the proof of Proposition 4.1. It remains to relate this property to condition (W3). 9

11 Definition 4.2. Let f : X IR l be a function defined on a nonempty set X IR l. f is called pseudomonotone (resp. strictly pseudomonotone) if for every x, y X such that x y, f(y) (x y) 0 implies f(x) (x y) 0 (resp. < 0) The following characterization has been shown in John (1998). Proposition 4.2. If f : X IR l is a continuous function defined on a nonempty convex set X IR l then the following conditions are equivalent: (1) f is (strictly) pseudomonotone. (2) For every convex subset Y of X, the set (Y, f) is convex (contains at most one element). By Theorem 4.1 and Proposition 4.1 we now obtain Corollary 4.1. Suppose the conditions (W1) and (W2) are satisfied. Then there exists a solution (p, q, x ) to (3.1), (3.2) and (3.3 ). If, in addition, f is pseudomonotone (in particular, if (W3) holds) then the same can be proved by elementary methods. Observe that strict pseudomonotonicity (W3) allows the application of the standard intermediate value theorem instead of its generalization to correspondences in the proof of Proposition 4.1. However, since Wald did not (as we have done) separate the existence question for factor prices from those for output quantities and prices he also needed Lemma 4.1 in his elementary existence proof. 5 Cassel Revisited We now return to Cassel s short run model as modified by Zeuthen, i.e. we look for a solution (p, q, x ) to the system (3.1 ), (3.2) and (3.3 ). Consider the following assumptions on the input coefficient matrix A, the aggregate demand function F, and total income C: (C1) F : IR n + IR n + is continuous. (C2) pf (p) C for every p, and pf (p) = C if AF (p) r. (C3) F is pseudomonotone, i.e. for every p, p IR n +: (p p)f (p) 0 implies (p p)f (p ) 0. Condition (C1) implies that demand is defined even if some or all prices 10

12 are zero, i.e. there is satiation. 5 This is a convenient assumption for purely technical reasons since it avoids tedious arguments which make use of boundary conditions. It does not remove the essential feature of factor scarcity as long as satiation only occurs for high levels of consumption. Indeed, this is guaranteed by the second part of condition (C2) which describes a sufficient desirability of the final commodities. (C3) is a weak version of the Weak Axiom of Revealed Preferences. Thus, it is even weaker than Wald s assumption. The essential idea for the following existence proof is also due to Cassel. It reduces the whole system to the excess demand function for the primary resources which is indirectly defined for given factor prices. Corollary 5.1. Suppose the conditions (C1) and (C2) are satisfied. Then there exists a solution (p, q, x ) to (3.1 ), (3.2), and (3.3 ). If, in addition, (C3) holds then the same can be proved by elementary methods. Proof. The set Q := {q IR m + qr = C} is nonempty, compact, and convex. Define the indirect factor excess demand function E : Q IR m by E(q) = AF (qa) r. Continuity of E follows from continuity of F which is assumed by (C1). By Theorem 4.1, there exists a solution q to V IP (Q, E). We first show that q yields a solution to (3.1 ), (3.2) and (3.3 ) provided that (C2) holds. Let p = q A and x = F (p ). Obviously, (p, q, x ) satisfies (3.1 ) and (3.3 ). It only remains to show (3.2). Since q solves V IP (Q, E) we know, by definition, that (q q )E(q ) 0 for every q Q. By the first part of (C2), this implies qe(q ) q E(q ) = (q A)F (q A) q r 0 for every q Q. In particular, for q i := (0,..., C r i,..., 0) we obtain q i E(q ) = C r i E i (q ) 0, i.e. E i (q ) 0. Since this holds for any i = 1,..., m we have proved that E(q ) 0 or, equivalently, Ax r. Moreover, by the second part of (C2), it follows that q (Ax r) = q (AF (q A) r) = (q A)F (q A) q r = 0, which completes the first part of the proof. To apply Proposition 4.1, i.e. to give an elementary proof, it remains to show that E is pseudomonotone provided that (C3) is satisfied. 5 If A contains no row with all entries equal to zero, F (0) need not be defined. 11

13 Suppose that (q q)e(q) 0. Since q, q Q implies (q q)r = 0 we obtain by definition of E (q A qa)f (qa) = (q q)af (qa) (q q)r = (q q)e(q) 0. By (C3), it follows that (q A qa)f (q A) 0. Hence, (q q)e(q ) = (q q)af (q A) (q q)r = (q A qa)f (q A) 0, i.e. E is pseudomonotone. Q.E.D. Finally, we conclude with two remarks. Assumption (C3) only ensures that E is pseudomonotone, i.e. in Proposition 4.1 the application of the generalized intermediate value theorem is needed. To employ its standard version requires that E is strictly pseudomonotone. With minor changes in the foregoing proof this can be shown if F is strictly pseudomonotone and if the rank of A is equal to m. In that case, the equilibrium is of course unique. Thus, since invertible demand satisfying the Weak Axiom is strictly pseudomonotone, the direct demand approach also yields Wald s additional uniqueness result (1936b). Is there an elementary existence proof even for Cassel s long run system (3.1 ), (3.2), and (3.3)? The answer depends on the assumptions one is willing to accept. The proof of Corollary 5.1 can easily be adapted to a demand function F (p, q) where the second argument indicates the possible dependence on the income distribution. While the first part requires no change, the second part needs the assumption that the function AF (qa, q) is pseudomonotone in q. It is well known that such a condition requires a specific correlation between factor ownership, factor intensities in production, and marginal propensities to consume the final commodities (see e.g. Mas-Collel (1991)). Since it implies convexity of the set of equilibrium prices for the long run model it is a considerably stronger assumption than (C3). References Arrow, K.J., Debreu, G.: Existence of an Equilibrium for a Competitive Economy. Econometrica 22, (1954) Barbolla, R., Corchón, L.C.: An Elementary Proof of the Existence of a Competitive Equilibrium in a Special Case. Quarterly Journal of Economics 104, (1989) Border, K.C.: Fixed Point Theorems with Applications to Economics and Game Theory. New York: Cambridge University Press

14 Cassel, G.: Grundriß einer elementaren Preislehre. Zeitschrift für die gesamte Staatswissenschaft 55, (1899) Cassel, G.: Theoretische Sozialökonomie. Leipzig: Winter 1918 Dorfman, R., Samuelson, P., Solow, R.: Linear Programming and Economic Analysis. New York: McGraw-Hill 1958 Greenberg, J.: An Elementary Proof of the Existence of a Competitive Equilibrium with Weak Gross Substitutes. Quarterly Journal of Economics 91, (1977) Hartman, P., Stampacchia, G.: On Some Non-linear Elliptic Differential Functional Equations. Acta Mathematica 115, (1966) Hildenbrand, W.: Review on General Equilibrium Analysis: Studies in Appraisal by E.R. Weintraub. Journal of Economic Literature 25, (1987) John, R.: Variational Inequalities, and Pseudomonotone Functions: Some Characterizations. In: Crouzeix, J.-P., Martinez-Legaz, J.E., Volle, M. (eds.) Proceedings of the 5th Symposium on Generalized Convexity. Dordrecht: Kluwer 1998 Kuhn, H.W.: On a Theorem of Wald. In: Kuhn, H.W., Tucker, A.W. (eds.) Linear Inequalities and Related Systems. Annals of Mathematics Studies No. 38. Princeton: Princeton University Press 1956 Malinvaud, E.: Lectures on Microeconomic Theory. Amsterdam: North-Holland 1972 Mas-Colell, A.: On the Uniqueness of Equilibrium once again. In: Barnett, W., Cornet, B., D Aspremont, C., Gabszewicz, J., Mas-Colell A. (eds.) Equilibrium Theory and Applications. New York: Cambridge University Press 1991 Oettli, W.: Monoton-konvexe Funktionen Eine Bemerkung zum Satz von Browder-Minty. In: Guddat, J. et al. (eds.) Advances in Mathematical Optimization. Berlin: Akademie- Verlag 1988 Schlesinger, K.: Über die Produktionsgleichungen der ökonomischen Wertlehre. Ergebnisse eines mathematischen Kolloquiums 6 ( ), (1935) Eco- Schulz, N.: Existence of Equilibrium Based on Continuity and Boundary Behaviour. nomics Letters 19, (1985) Valentine, F.A.: Convex Sets. New York: McGraw-Hill 1964 Wald, A.: Über die eindeutige positive Lösbarkeit der neuen Produktionsgleichungen. Ergebnisse eines mathematischen Kolloquiums 6 ( ), (1935) Wald, A.: Über die Produktionsgleichungen der ökonomischen Wertlehre. Ergebnisse eines mathematischen Kolloquiums 7 ( ), 1-6 (1936a) Wald, A.: Über einige Gleichungssysteme der mathematischen Ökonomie. Zeitschrift für Nationalökonomie 7, (1936b) Zeuthen, F.: Das Prinzip der Knappheit, technische Kombination und ökonomische Qualität. Zeitschrift für Nationalökonomie 4, 1-24 (1933) 13