aa r X i v : . [ ec on . T H ] S e p Existence of Equilibrium Prices:A Pedagogical Proof ∗ Simone Tonin † September 25, 2018
Abstract
Under the same assumptions made by Mas-Colell et al. (1995), I develop a short, sim-ple, and complete proof of existence of equilibrium prices based on excess demand func-tions. The result is obtained by applying the Brouwer fixed point theorem to a trimmedsimplex which does not contain prices equal to zero. The mathematical techniques arebased on some results obtained in Neuefeind (1980) and Geanakoplos (2003).
This paper aims to provide a short, simple, and complete proof of existence of equilibriumprices under the same set of assumptions made in Mas-Colell et al. (1995). The existenceresult is derived by two lemmas which define, respectively, the non-empty compact convexset and the continuous function required by the Brouwer fixed point theorem. As usual, it isshown that the fixed point is an equilibrium price vector.It would be pretentious to say that the proof is new as it follows closely some results ofNeuefeind (1980) and Geanakoplos (2003). However, the approach developed here highlightsclearly the central role of fixed point theorems in proving the existence of equilibrium prices.This can be interesting from a pedagogical and historical point of view as the two results arestrongly linked. I further discuss this point in the last section.Let me briefly outline the results of Neuefeind (1980) and Geanakoplos (2003). Neuefeind(1980) showed that, under a proper boundary behaviour of the excess demand function, it ispossible to consider a trimmed simplex which does not contain prices equal to zero. In the firstlemma I prove the same result by considering a different boundary condition. Geanakoplos(2003) added a perturbation in a correspondence considered by Debreu (1959) to make itsingle-valued and to apply the Brouwer fixed point theorem. The second lemma states thiskey result and I report its proof for completeness. Furthermore, by using the result of Neuefeind(1980), the boundary condition of the excess demand function is used in a different way fromGeanakoplos (2003) (for a detailed analysis of the boundary condition see Ruscitti, 2012).
Consider an exchange economy with L commodities, l = 1 , . . . , L . Let z : R L ++ → R L bethe exchange economy’s excess demand function defined over the set of positive price vectors.I make the same assumptions used in Proposition 17.C.1 of Mas-Colell et al. (1995):(A1) z ( · ) is continuous;(A2) z ( · ) is homogeneous of degree zero; ∗ I would like to thank Giulio Codognato and Marialaura Pesce for their helpful suggestions. † Durham University Business School, Durham University, Durham, DH1 3LB, UK p · z ( p ) = 0 for all p ∈ R L ++ (Walras’ law);(A4) There is an s > z l ( p ) > − s for all l and for all p ∈ R L ++ .(A5) If p n → p , where p = 0 and p l = 0 for some l , thenmax { z ( p n ) , . . . , z L ( p n ) } → + ∞ . Exchange economies with a positive aggregate endowment of each commodity and with con-sumers having continuous, strongly monotone, and strictly convex preferences satisfy Assump-tions A1-A5 (see Proposition 17.B.2 in Mas-Colell et al., 1995).Let ∆ = { p ∈ R L + : P l p l = 1 } be the unit simplex. Since the excess demand function ishomogeneous of degree zero, its domain can be restricted to the interior of the unit simplexint∆. However, this is an open set and to apply the fixed point theorem it is required aclosed set which contains only positive prices. For this reason, I define a trimmed simplex∆ ǫ = { p ∈ ∆ : p l ≥ ǫ, for all l } , with ǫ ∈ (0 , L ]. In the paper it is always assumed that ǫ liesin (0 , L ]. Finally, in the proof I will repeatedly use the price vector ¯ q = ( L , . . . , L ), at whichall prices are equal, and the following straightforward result. Proposition.
The sets ∆ and ∆ ǫ are non-empty, compact, and convex.I finally state the theorem of existence of equilibrium prices. Theorem.
Under Assumptions A1-A5, there exists a p ∗ ∈ R L ++ such that z ( p ∗ ) = 0.It is immediate to see that z ( p ) = 0 corresponds to the system of L equations in L unknownsintroduced by Walras (1874-7) and the solution p ∗ is the equilibrium price vector at which allmarkets clear. The first step consists in defining the non-empty, compact, and convex set required by thefixed point theorem. To this end, the result of the next lemma implies that if an equilibriumprice vector p ∗ exists, then there is an ǫ such that p ∗ belongs to the interior of the set ∆ ǫ . Lemma 1.
Let Q = { p ∈ int∆ : P l z l ( p ) ≤ } . Under Assumptions A1-A5, there exists an ǫ ∈ (0 , L ] such that Q ⊆ int∆ ǫ . Proof.
First, the set Q is non-empty as the price vector ¯ q belongs to Q by Walras’ law, i.e., L P l z l (¯ q ) = 0. Next, I show, by contradiction, that there exists an ǫ ∈ (0 , L ] such that Q ⊆ int∆ ǫ . Suppose that for all ǫ ∈ (0 , L ] there exists a price vector p ∈ Q such that p / ∈ int∆ ǫ . Consider a sequence of { ǫ n } with ǫ n = nL . Then, there exists a sequence of pricevectors { p n } such that p n ∈ Q and p n / ∈ int∆ ǫ n for all n . Since the sequence { p n } belongs tothe compact set ∆, there is a subsequence p k n → p with p ∈ ∆. As p k n / ∈ int∆ ǫ kn for all n ,it follows that, for all n , p k n l ≤ ǫ k n for some l . Then, I can conclude that p l = 0 for some l because ǫ k n →
0. Hence, max { z ( p k n ) , . . . , z L ( p k n ) } → ∞ by A5. Moreover, since z ( · ) has alower bound by A4, the inequality max { z ( p k n ) , . . . , z L ( p k n ) } − s ( L − < P l z l ( p k n ) holdsfor all n . As the left hand side converges to infinity by the previous result, there exists an m such that max { z ( p k n ) , . . . , z L ( p k n ) } − s ( L − > n > m . But then, it follows that P l z l ( p k n ) > n > m . As p k n ∈ Q for all n , it also follows that P z l ( p k n ) ≤ n , a contradiction. Therefore, there exists an ǫ ∈ (0 , L ] such that Q ⊆ int∆ ǫ .The next step consists in defining the continuous function on ∆ ǫ to itself required by theBrouwer fixed point theorem. A possible approach would be to consider the correspondence µ ( p ) = arg max q ∈ ∆ ǫ { q · z ( p ) } , The symbol 0 denotes the origin in R L as well as the real number zero. −k q − p k , to make the correspondenceabove single-valued. The next lemma deals with the continuous function φ ( · ) on which I willapply the Brouwer fixed point theorem. Lemma 2.
Let φ : ∆ ǫ → ∆ ǫ be such that φ ( p ) = arg max q ∈ ∆ ǫ (cid:8) q · z ( p ) − k q − p k (cid:9) . Under Assumptions A1-A5, φ ( · ) is a continuous function. Proof.
Define the function g ( q, p ) = q · z ( p ) − k q − p k . Since I consider only prices belongingto ∆ ǫ , g ( · , · ) is continuous as it is a sum of continuous function. Let p be a constant. Then, g ( · , p ) has a maximum point on the non-empty compact set ∆ ǫ by the Weierstrass Theorem. Itis immediate to verify that the square of the Euclidean distance is strictly convex. In fact, itsHessian matrix is a diagonal matrix with positive entries and it is then positive definite. Then, g ( · , p ) is strictly concave because it is a sum of a linear function and a strictly concave function.But then, the maximum is unique and φ ( · ) is a function. Finally, I prove that φ ( · ) is continuous.Let q ∗ = φ ( p ) be the unique maximum point of g ( · , p ). Since ∆ ǫ is compact, I can consider asequence p n → p and a corresponding subsequence { p k n } such that φ ( p k n ) → r with r ∈ ∆ ǫ .By the definition of φ ( · ), the following inequality holds g ( q ∗ , p k n ) ≤ g ( φ ( p k n ) , p k n ). As thefunction g ( · , · ) is continuous, it follows that g ( q ∗ , p k n ) → g ( q ∗ , p ) and g ( φ ( p k n ) , p k n ) → g ( r, p ).Then, g ( q ∗ , p ) ≤ g ( r, p ) by the properties of sequences. Since the maximum point is unique, itfollows that r = q ∗ . But then, if the subsequence p k n → p , it follows that φ ( p k n ) → q ∗ = φ ( p ).Hence, φ ( · ) is continuous.I finally prove the existence theorem Proof.
By the results in Lemmas 1 and 2, I can apply the Brouwer fixed point theorem andthen there exists a fixed point p ∗ of the function φ ( · ), i.e., p ∗ = φ ( p ∗ ). Next, I show that p ∗ ∈ int∆ ǫ . Since p ∗ is the unique maximum point of g ( · , p ∗ ) on ∆ ǫ and g ( p ∗ , p ∗ ) = 0 byWalras’ law, it follows that g ( αp + (1 − α ) p ∗ , p ∗ ) = ( αp + (1 − α ) p ∗ ) · z ( p ∗ ) − k ( αp + (1 − α ) p ∗ ) − p ∗ k < , for each α ∈ (0 ,
1] and p ∈ ∆ ǫ with p = p ∗ . By applying the Walras Law the previous inequalitysimplifies in p · z ( p ∗ ) < α k p − p ∗ k , for each α ∈ (0 , p ∈ ∆ ǫ such that p · z ( p ∗ ) >
0. Then, thereexists a sufficiently small ¯ α ∈ (0 ,
1] such that p · z ( p ∗ ) > ¯ α k p − p ∗ k , a contradiction. Hence, p · z ( p ∗ ) ≤ p ∈ ∆ ǫ . Since ¯ q ∈ ∆ ǫ , it follows that ¯ q · z ( p ∗ ) = L P l z l ( p ∗ ) ≤ p ∗ ∈ Q . But then, p ∗ ∈ int∆ ǫ by Lemma 1. Furthermore, note that p ∗ maximises p · z ( p ∗ ) on ∆ ǫ because p ∗ · z ( p ∗ ) = 0, by Walras’ law, and p · z ( p ∗ ) ≤ p ∈ ∆ ǫ , bythe previous result. Finally, I show that the fixed point p ∗ is an equilibrium price vector, i.e., z ( p ∗ ) = 0. I proceed by contradiction and I suppose that there exists a commodity l suchthat z l ( p ∗ ) <
0. As p ∗ maximises p · z ( p ∗ ) on ∆ ǫ , it follows that p ∗ l = ǫ . But p ∗ ∈ int∆ ǫ , acontradiction. Hence, z l ( p ∗ ) ≥ l . By the fact that P l z l ( p ∗ ) ≤
0, I can conclude that z ( p ∗ ) = 0. The symbol k x − y k denotes the Euclidean distance between the vectors x and y . Fixed point theorems and equilibrium prices
In the literature there are alternative approaches to prove the existence of equilibriumprices which do not rely on fixed point theorems (see Greenberg (1977) and Quah (2008) amongothers). However they require stronger assumptions such as the weak gross substitutability orthe weak axiom of revealed preference. This is due to the fact there is an equivalence betweenfixed point theorems and the existence of equilibrium prices under the classical assumptionsas shown by Uzawa (1962). Debreu (1982) also pointed out that the proof of existence ofequilibrium prices, under the classical assumptions, requires mathematical tools of the samepower as fixed point theorems.As the Brouwer fixed point theorem was published in 1911, it is not surprising that theproblem of existence of equilibrium prices formulated by Walras (1874-7) was solved in ageneral way by McKenzie (1954) and Arrow and Debreu (1954) for the first time.Under more restrictive assumptions some rigorous existence results were also obtained byA. Wald and J. von Neumann in the 1930s (for modern explanation of Wald’s result see John,1999). It is also worth mentioning that A. Wald wrote another paper on the existence problemin 1935 which unfortunately went lost. Duppe and Weintraub (2016) gave a detailed historyof this lost proof and clarified that it applied a fixed-point theorem to show the existence ofequilibrium prices in exchange economies.
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