Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand


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1 Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such as existence, uniqueness, or stability of equilibria. The notes are based on MWG, chapter Walrasian Equilibrium and Excess Demand I > 0, J > 0, L > 0, all of them are finite. ({X i, i } I i=1, {Y j } J j=1, {(ω i, θ i1,..., θ ij )} I i=1). An economy is defined by Definition 1 A Walrasian (or pricetaking) equilibrium is an allocation (x, y ) and a price vector p = (p 1,..., p L ) if (i) For every j, y j Y j maximizes profits in Y j : p y j p y j for all y j Y j. (ii) For every i, x i X i is maximal for i in the respective budget set {x i X i : p x i p ω i + j θ ijp y j }. (iii) i x i = i ω i + j y j. For a while, let us consider exchange economies. Most results obtained from this exercise easily carry over to economies with production. Definition 2 An exchange economy is defined by E ({X i, i } I i=1, Y 1 = R L +, {ω i } I i=1). We assume that preferences are continuous, strictly convex, and locally nonsatiated (before long: strictly monotone). Moreover, we assume: i ω i 0. Notice also that we assume free disposal, which is taken into account by allowing for one firm whose only available technology is that of free disposal: Y 1 = R L + y l1 0 for all l = 1,...L. Query V.1 What is the relationship between the assumption of free disposal and prices? In the setting of an exchange economy, an allocation (x, y ) and a price vector p constitute a Walrasian equilibrium if and only if (i) y 1 0, p y 1 = 0, p 0, (ii) x i = x i (p, p ω i ) for all i, and (iii) i x i = i ω i + y 1. Ronald Wendner V1 v1.1
2 Notice that (i) we will prove in class. However, by (i) and (iii) aggregate demand cannot exceed aggregate supply of a commodity (as y 1 0). Thus, by (i), a price p l is zero if and only if aggregate demand is smaller than aggregate supply (i.e., only free goods can be in excess supply). Proposition 1 Suppose, preferences in E are strictly convex and locally nonsatiated. Then, p is a Walrasian equilibrium price vector if and only if: i (x i(p, p ω i ) ω i ) 0. Query V.2 Prove Proposition 1. Definition 3 Consumer i s excess demand function is z i (p) = x i (p, p ω i ) ω i, where x i (p, p ω i ) is her Walrasian demand function. The aggregate excess demand function is z(p) = i z i(p). From here on, we ll state most results in terms of the excess demand (rather than Walrasian demand). Definition 4 E + defines an exchange economy where i are strictly monotone, continuous, and strictly convex. In E +, p constitutes a Walrasian price vector if and only if z(p) 0. If, moreover, preferences are strictly monotone this we ll assume from here on a Walrasian price vector has the property that p 0. Query V.3 If i are strongly monotone for all i, why must a Walrasian price vector be strictly positive: p 0? Under strong monotonicity of preferences, p is a Walrasian equilibrium price vector if and only if z l (p) = 0 for every l = 1,..., L. I.e., z(p) = 0. Proposition 2 Consider an economy E + where i ω i 0. Then z(p) is defined on p 0 and satisfies: (i) z(p) is continuous. (ii) z(p) is HD 0. (iii) Walras law: p z(p) = 0. (iv) z(p) is bounded below. I.e., there is some number s : z l (p) > s for every l = 1,..., L and all p. (v) If p n p, where p 0 and p l = 0 for some l, then: {max{z 1 (p n ),..., z L (p n )}} n=1. Ronald Wendner V2 v1.1
3 Property (i) follows from the fact that x i (p, p ω i ) is continuous. Query V.4 From which property about i does it follow that x i (p, p ω i ) is continuous? Property (ii) follows from the fact that x i (p, p ω i ) is HD 0, (iii) comes from strong monotonicity of i, (iv) stems from the fact that demand cannot be negative. Finally, (v) we ll show in class. 2 Some Mathematical Prerequisites for Existence Proofs Correspondence. A correspondence is a multivalued function. Suppose our domain is A R N. A (real valued) function f : A R is a rule that assigns to every x A a single value f(x) R (a singleton). In contrast, a (real valued) correspondence ϕ(x) : A R K is a rule that assigns to every x A a set ϕ(x) R K (which is not necessarily a singleton). Obviously, every function is a correspondence. But a correspondence is a function if and only if for every x A we have that ϕ(x) is a singleton. ConvexValuedness of a Correspondence. Suppose, a correspondence ϕ(x) : A R K assigns to every x A a set ϕ(x) R K. This correspondence is convex valued at x if ϕ(x) is a convex set. This correspondence is convex valued if ϕ(x) is a convex set for all x A. Upper Hemicontinuity (uhc) of a Correspondence. Let ϕ : A Y be a correspondence, where A R N, Y R K, both A and Y are closed, and Y is bounded. Consider any two converging sequences {x n } and {y n } such that for all n, y n ϕ(x n ), where x n x and x, x n A, and y n y and y, y n Y for n = 1, 2,... The correspondence ϕ : A Y is said to be uhc at x if y ϕ(x). The correspondence ϕ : A Y is said to be uhc if it is uhc at all x A. Brouwer s FixedPoint Theorem. Suppose that A R N is nonempty, compact, and convex. If f : A A is a continuous function from A to itself, then f(.) has a fixed point; i.e., there is an x A such that: x = f(x). Kakutani s FixedPoint Theorem. Suppose that A R N is nonempty, compact, and convex. If ϕ : A A is an upper hemicontinuous correspondence Ronald Wendner V3 v1.1
4 from A to itself, with ϕ(x) A being nonempty and convex for every x A (i.e., convexvalued ), then ϕ(.) has a fixed point; i.e., there is an x A such that: x ϕ(x). 3 Existence of Equilibrium This is the first (positive) question. We cannot use our GEframework unless there is an equilibrium. The conditions for which an equilibrium exists are clarified in this section. By HD 0 of an excess function, we are allowed to normalize the price vector (e.g., set one price equal to unity, or normalize prices to the unit simplex in R L +). The unit simplex is defined by {p R L + : l p l = 1}. Moreover, denote the interior of by i, and the boundary of the simplex by. Before going to the propositions, please be sure you understand the following concepts: convexvaluedness of a correspondence, (upper) hemicontinuity of a correspondence, Brouwer s FixedPoint Theorem, and Kakutani s FixedPoint Theorem. I start with the general result first, and give a simplified (more special, but probably more instructive) version thereafter. However, for the real existence proof which is also applicable for production economies I ask you to read my Notes VI. Proposition 3 Consider an exchange economy E + with ω 0. There exists a Walrasian equilibrium, i.e., there exists an allocation (x, y ) and a price vector p that constitute a solution to the system of equations z(p) = 0. Proof. First, construct a correspondence f(p) from all p into. Step (i) considers f(p) : i, step (ii) considers f(p) :. (i) Construct a correspondence for all p i : f(p) = {q : z(p) q z(p) q for all q }, which assigns an element (a set) of to every p i. Observe that if z(p) = 0 (i.e., we are having a Walrasian equilibrium), f(p) =. However, if z(p) 0, then f(p). In particular, q l = 0 if z l (p) < max{z 1 (p),..., z L (p)}. (ii) Construct a correspondence for all p : f(p) = {q : p q = 0} = {q : q l = 0 if p l > 0}. As for any p : p p > 0, no fixed point can be represented by a price vector p. (iii) Certainly, a fixed point of f(p) is a Walrasian equilibrium. Notice that Ronald Wendner V4 v1.1
5 a fixed point means p f(p ). In this case, p. Thus, p 0. But if z(p ) 0, then p. Hence, a fixed point represents a Walrasian equilibrium. (iv) The fixed point correspondence is convexvalued and upper hemicontinuous (as will be shown in class). (v) Now we can apply Kakutani s FixedPoint Theorem to establish that there is a fixed point. By (iii), then, there is a Walrasian equilibrium.. W.H.O.W. All right, this was pretty tough. The difficulty in the preceding proof arose from boundary complications, i.e., excess demand is not well defined when p, as the maximum z l (p) is going to infinity. For purely instructive reasons, we proceed as follows. Assume properties (i) to (iii) from Proposition 2, and z(p) is well defined for all nonzero p R L +. 1 Remember that in equilibrium we have z(p) 0. Corollary 1 Consider an exchange economy E with ω 0 and z(p) being well defined for all p R L +. Then there exists a Walrasian equilibrium, i.e., there exists an allocation (x, y ) and a price vector p 0 that constitute a solution to the system of equations z(p) 0. (i) As z(p) are HD0 in prices, we can restrict our attention to the price simplex: = {p R L + l p l = 1}. (ii) Define the function z + l (p) = max {z l (p), 0}. The function z + (p) is continuous, and z + (p) z(p) = 0 implies z(p) 0. (iii) Define α(p) = l (p l + z + l (p)) 1. (iv) f(p) = (p + z + (p))/α(p) is a continuous function from the price simplex to itself. (v) By Brouwer s FixedPoint Theorem there exists a p such that p = f(p ). (vi) By Walras law: 0 = p z(p ) = f(p ) z(p ) = (1/α(p )) (p + z + (p )) z(p ) = (1/α(p )) z + (p ) z(p ). But then, z + (p ) z(p ) = 0, which implies, by (i), that z(p ) 0. W.H.O.W. Query V.5 Show that f(p) :, as claimed in step (iv). 1 Such excess demand functions are not possible with monotone preferences, yet they exist with locally nonsatiated preferences. Ronald Wendner V5 v1.1
6 4 Bonus Stuff Uniqueness A few Results Suppose there exist Walrasian equilibria. The question then is: How many equilibria are there? If there is a (globally) unique equilibrium, we can perform meaningful comparative static analysis. However, if there is more than one equilibrium (i.e., multiplicity) the next best thing is to have a finite number of equilibria. In this case, we have local uniqueness, i.e., at every Walrasian equilibrium (x, y ), there exists an ɛ > 0 and an ɛ ball about (x, y ), B ɛ (x, y ), such that there is no other Walrasian equilibrium within B ɛ (x, y ). More precisely, a Walrasian equilibrium price vector p 0 is locally unique, if there is an ɛ > 0 such that if p p, and p p < ɛ then z(p ) 0. In contrast to local uniqueness, we might encounter indeterminate equilibria, in which case for every ɛ > 0 however small there is an infinite number of Walrasian equilibrium price vectors in p p < ɛ. Indeterminateness is not a desirable property. If the economy is regular, all equilibria are locally unique (determinate). Moreover, an economy is regular, if the Jacobian matrix of price effects Dẑ(p) has rank L 1 (is nonsingular). 2 Query. Suppose, L = 2. Under which condition is E + regular? Under which condition does E + face indeterminate equilibria? We now consider a condition that guarantees global uniqueness of equilibrium. Definition 5 (Gross Substitution) The excess demand function has the gross substitution (GS) property if whenever p and p are such that, for some l, p l > p l and p k = p k for all k l, we have z k (p ) > z k (p) for all k l. Notice that the gross substitution property (as defined above) implies: z l (p ) < z l (p)! In a differential version, GS implies: δ z k (p)/δ p l > 0, i.e., all the offdiagonal entries of Dz(p) are positive. Proposition 4 In E +, there is a globally unique equilibrium, if z(p) satisfies the gross substitution property. 2 Normalize the price vector such that the price of good L = 1: p = (p 1, p 2,..., p L 1, 1). The normalized excess demand function is then: ẑ(p) = (z 1 (p), z 2 (p),..., z L 1 (p)). Then, p 0 is a Walrasian equilibrium price vector if ẑ(p) = 0. Ronald Wendner V6 v1.1
7 Observe that the GS property is sufficient, not necessary! Excess Demand in Economies with Production Definition 6 An economy with production is defined by P ({X i, i } I i=1, {Y j } J j=1, {(ω i, θ i1,..., θ ij )} I i=1). Let P + be an economy with production, where all production sets are closed, strictly convex and bounded. Consider an economy P. The productioninclusive excess demand is given by: z(p) = i x i(p, p ω i + j θ ijπ j (p)) i ω i j y j(p). Proposition 5 Consider an economy P +. (i) to (v), as given by Proposition 2. See Exercise 17.B.4 (MWG, p.642). Local Nonsatiation and Positivity of Prices Then, z(p) satisfies properties Notice that local nonsatiation implies that there is at least one desirable good, otherwise 0 would be a global satiation point. Thus, p x i = p ω i. Proposition 6 Suppose, preferences in E are strictly convex and locally nonsatiated. Then, p is a Walrasian equilibrium price vector if and only if: z(p) i (x i(p, p ω i ) ω i ) 0. Proof (Sketch). It can easily be shown that z(p) 0 [(y 1 0, p y 1 = 0, p 0) & (x i = x i (p, p ω i ) for all i) & ( i x i = i ω i + y 1)]. Proposition 7 Let p be a Walrasian price vector in E. Then, no commodity has a negative price: p l 0 for all l = 1,..., L. Proof (direct). Because of the possibility of free disposal, there are no transactions with a negatively priced commodity (nobody is willing to sell). So there are is no trade with such commodities hence, no good has a negative price. Proposition 8 Let p be a Walrasian price vector in E. Then, p 0. Proof (direct). Suppose p = 0. By local nonsatiation there is a desirable commodity, say l. But then, as the budget set is unbounded, there exists no maximal element, x i, in the budget set. From HD0 of the excess demand functions, and from Proposition 3, we can normalize prices without loss of generality: l p l = 1. Ronald Wendner V7 v1.1
8 Corollary 2 Let p be a Walrasian price vector in E. every desirable commodity l is strictly positive: p l > 0. Then, the price of Corollary 3 Let p be a Walrasian price vector in E. If all commodities are desirable (strong monotonicity), p 0. Proposition 9 Let p be a Walrasian price vector in E. If some commodities are not desirable, i.e., z l (p) < 0, then, p l = 0 and the price vector is not strictly positive. Proof (direct). Suppose first, all goods are desirable. Then, p 0. As p z(p) = 0 we have z(p) = 0, i.e., z l (p) = 0 for all l = 1,..., L. Next, suppose that l is not desirable, i.e.: i z l i(p) < 0. Define z (p) = (z 1, z 2,...z l 1, z l +1,..., z L ), and p = (p 1,...p l 1, p l +1,..., p L ). Then, p l z l (p)+ p z (p) = 0. By Corollary 1, p 0. Moreover, z (p) = 0, as all those goods are desirable (and z (p) 0). Thus, p z (p) = 0. As z l (p) < 0, we must have p l = 0. The argument can easily be extended to the case with several bads. Ronald Wendner V8 v1.1
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