Positive Theory of Equilibrium: Existence, Uniqueness, and Stability

Chapter 7 Nathan Smooha Positive Theory of Equilibrium: Existence, Uniqueness, and Stability 7.1 Introduction Brouwer s Fixed Point Theorem. Let X be a non-empty, compact, and convex subset of R m. If a function f : X X is continuous on X, then f has a fixed point, i.e., there is a point x X such that f (x ) = x. Kakutani s Fixed Point Theorem. Let X be a non-empty, compact, and convex subset of R m. If a correspondence F : X 2 X is an upper hemi-continuous correspondence with non-empty compact and convex values on X, then F has a fixed point, i.e., there is a point x X such that x F(x ). FKKM Theorem. Let Y be a convex set and /0 X Y. Suppose F : X 2 Y is a correspondence such that a. F(x) is closed for all x X; b. F(x 0 ) is compact for some x 0 X; c. F is FS-convex, i.e., for any x 1,...,x m X and its convex combination x λ = m i=1 λ ix i, we have x λ m i=1 F(x i). Then, x X F(x) /0. 7.2 The Structure of the General Equilibrium Model A general equilibrium is defined as a state where the aggregate demand does not excess the aggregate supply for all markets. Thus, equilibrium prices are endogenously determined. The general equilibrium approach has to central features: 1

It views the economy as a closed and inter-related system in which we must simultaneously determine the equilibrium values of all variables of interest (i.e., consider all markets together). It aims at reducing the set of variables taken as exogenous to a small number of physical realities. We use the general equilibrium approach to predict the final consumption and production in the market mechansim. 7.2.1 Economic Environments The fundamentals of the economy are economic institutional environments that are exogenously given and characterized by the following terms: n : the number of consumers N = {1,...,n} : the set of consumers J : the number of producers (firms) L : the number of (private) goods X i R L : the consumption space of consumer i = 1,...,n, which specifies the boundary of consumptions, collection of all individually feasible consumption of consumer i. Some components of an element may be negative such as labor supply i : Preference ordering (or u i if a utility function exists) of consumer i = 1,...,n. Remark: i is a preference ordering if it is reflexive (x i i x i ), transitive (x i i x i and x i i x i implies x i i x i ), and complete (for any pair x i and x i, either x i i x i or x i i x i or both). It can be shown that it can be represented by a continuous utility funciton if i is continuous. The existence of general equilibrium can be obtained even when preferences are weakened to be non-complete or non-transitive. w i X i : initial endowment vector of consumer i. e i = (X i, i,w i ) : the characteristics of consumer i. Y j : production possibility set of firm j = 1,2,...,J, which is the characteristic of producer j. y j Y j : a production plan, y jl > 0 means y jl is output and y jl < 0 means y jl is input. Most elements of y j for a firm are zero. Remark: Recall that there can be three types of returns about production scales: non-increasing returns to scale (i.e., y j Y j implies that αy j Y j for all α [0,1]), non-decreasing returns to scale (i.e., y j Y j implies that αy j Y j for all α 1), and constant returns to scale (i.e., y j Y j implies that αy j Y j for all α 0). In other words, non-increasing returns to scale implies any feasible input-output vector can be scaled down; non-decreasing returns to scale implies any feasible inputoutput vector can be scaled up, and constant returns to scale implies the production set is the conjuntion of non-increasing and non-decreasing returns. With non-decreasing returns to scale, firms profits are infinite, so we will avoid this case in the model. Page 2 of 27

e = ( {X i, i,w i }, { Y j }) : an economy, or also called an economic environment. X = X 1 X n : consumption space Y = Y 1 Y J : production space 7.2.2 Institutional Arrangement: Private Market Mechanism p = (p 1,..., p L ) R L + : a price vector p x i : the expenditure of consumer i for i = 1,...,n p y j : the profit of firm j for j = 1,...,J p w i : the value of endowments of consumer i for i = 1,...,n θ i j R + : the profit share of consumer i from firm j, which specifies ownership (property rights) structures, so that n i=1 θ i j = 1 for j = 1,...,J, and i = 1,...,n. For i = 1,...,n, consumer i s budget constraint is given by p x i p w i + J j=1 and the budget set is given by { B i (p) = θ i j p y j (1) x i X i : p x i p w i + J j=1 θ i j p y j } A private ownership economy then is referred to ( e = e 1,...,e n, { } J Y j j=1,{ } ) θ i j (3) The set of all such private ownership economies are denoted by E. (2) 7.2.3 Individual Behavior Assumptions a. Perfect competitive markets: Every player is a price-taker. b. Utility maximization: Every consumer maximizes his preferences subject to B i (p). That is, subject to max x i u i (x i ) p x i p w i + J j=1 θ i j p y j (5) (4) c. Profit maximization: Every firm maximizes its profit in Y j. That is, max y j Y j p y j for j = 1,...,J (6) Page 3 of 27

7.2.4 Competitive Equilibrium Before defining the notion of competitive equilibrium, we first give some information on allocations, which identify the set of possible outcomes in economy e. For notational convenience, â will be used throughout the notes to denote the sum of vectors a i, i.e., â = a i. An allocation (x,y) is a specification of a consumption vector x = (x 1,...,x n ) X and production vector y = (y 1,...y J ) Y. An allocation is individually feasible if x i X i for all i N and y j Y j for all j = 1,...,J. An allocation is weakly balanced if ˆx ŷ + ŵ (7) or specifically n i=1 x i j j=1 y j + n w i i=1 (8) When inequality holds with equality, the allocation is called a balanced or attainable allocation, i.e., aggregate demand equals aggregate supply. An allocation (x, y) is feasible if it is both individually feasible and (weakly) balanced. Thus, an economic allocation is feasible if the total amount of each good consumed does not exceed the total amount available from both the initial endowment and production. Denote the set of all feasible allocations by A = {(x,y) X Y : ˆx ŷ + ŵ}. We now define the notion of competitive equilibrium. Definition. 7.2.1 (Competitive Equilibrium or also called Walrasian Equilibrium). Given a private ownership economy, e = (e 1,...,e n,{y j },{θ i j }), an allocation (x,y) X Y and a price vector p R L + constitutes a competitive equilibrium if the following conditions are satisfied a. Utility maximization: i N, x i B i (p), x i B i(p) : x i x i b. Profit maximization: j, y j Y j, y j Y j : p y j p y j c. Market clearing condition: ˆx ŵ + ŷ Denote x i (p) = {x i B i (p) : x i B i(p), x i i x i } : the demand correspondence of consumer i under utility maximization; it is called the demand function of consumer i if it is a single-valued funciton. { } y j (p) = y j Y j : y j Y j, p y j p y j : the supply correspondence of the firm j; it is called the supply function of firm j if it is a single-valued function. So, we can define the following ˆx(p) = n i=1 x i(p) : the aggregate demand correpondence Page 4 of 27

ŷ(p) = J j=1 y j(p) : the aggregate supply correspondence ẑ(p) = ˆx(p) ŵ ŷ(p) : the aggregate excess demand correspondence If we have correspondences, we just add the sets together, i.e., X +Y = {x + y : x X y Y }. An equivalent definition of a competitive equilibrium is that a price vector p R L + is a competitive equilibrium price vector if there exists ẑ ẑ(p ) such that ẑ 0. If ẑ(p) is single-valued, then ẑ(p ) 0 is a comeptitive equilibrium. Note, competitive equilibria can still exist if ẑ(p) is a correspondence instead of a function. If we do not assume the production possibility sets satisfy free disposal, then ˆx = ŷ + ŵ ẑ(p) = 0 and the budget constraint holds with equality for all i. 7.3 Examples of GE Models: Graphical Treatment In most economies, there are three types of economic activities: produciton, consumption, and exchanges. Before formally stating the existence results on competitive equilibrium, we will first give two simple examples of general equilibrium models: exchange economies and a production economy with only one consumer and one firm. These examples introduce some of the questions, concepts, and common techniques that will occupy us for the rest of this part. 7.3.1 Pure Exchange Economies A pure exchange economy is an economy in which there is no production. This is a special case of a general economy. In this case, economic activies only consist of trading and consumption. The aggregate excess demand correspondence becomes ẑ(p) = ˆx(p) ŵ so that we can define the individual excess demand by z i (p) = x i (p) w i for this special case. The simplest exchange economy with the possibility of mutual benefit exchange is the two-good and two-consumer exchange economy. Note that the point denoted by w in the Edgeworth Box can be used to represent the initial endowments of two people. Edgeworth Box Consider an excahnge economy with two goods (x 1,x 2 ) and two persons. The total endowment is ŵ = w 1 + w 2. For example, if w 1 = (1,2) and w 2 = (3,1), then the total endowment is ŵ = (4,3). Note that the point, denoted by w in the Edgeworth Box, can be used to represent the initial endowments of two persons. The advantage of the Edgeworth Box is that it gives all the possible (balanced) trading points. That is, x 1 + x 2 = w 1 + w 2 for all points x = (x 1,x 2 ) in the box, where x 1 = (x 11,x 12 ) and x 2 = (x 21,x 22 ). Thus, every point in the Edgeworth Box stands for an attainable allocation so that x 1 + x 2 = w 1 + w 2. Page 5 of 27

The shaded lens (portion) of the box represents all the trading points that make both persons better off. Beyond the box, any point is not feasible. In the box, one person s budget line is also the budget line of the other person. They share the same budget line in the box. In the interior equilibrium case, two indifference curves are tangent to each other at a point that is on the budget line so that the marginal rates of substitutions for the two persons are the same and equal to the price ratio, and aggregate demand equals aggregate supply. A competitive equilibrium may still exist even if two persons indifference curves are not tangent. In this case, there is no tangent point as long as the slopes of the indifference curves of the two persons are not the same. Even so, there sill exists a competitive equilibrium. Offer curve: The locus of the optimal consumptions for the two goods when price varies. Note that it consists of tangent points of the indifference curves and budget lines when price varies. The intersection of the two offer curves of the consumers can be used to check if there is a competitive equilibrium. One intersection of two consumers offer curves is always given at the endowment w. If there is another intersection of two consumers offer curves rather than the one at the endowment w, the intersection point must be the competitive equilibrium. Note that a preference relation i is convex if x i x implies tx i + (1 t)x i x i for all t (0,1) and for all x i,x i X i. A preference relation i has a satiation point x i if x i x i for all x i X i. Cases in which there may be no Walrasian Equiliria Case 1. Case 2. Indifference curves are not convex. If the two offer curves are not intersected except for at the endowment points, then there may not exist a competitive equilibrium. This may be true when preferences are not convex. The initial endowment may not be an interior point of the consumption space. Consider an exchange economy in which one person s indifference curves put no values on one commodity, like when agent 2 s indifference curves are vertical lines so that u 2 (x 21,x 22 ) = x 21. Agent 1 s utility function is a regular one, which is given by a quasilinear utility function, u 1 = (x 11,x 12 ) = x 11 + x 12. The initial endowments are given by w 1 = (1,0) and w 2 = (0,1). There is no competitive equilibrium. Why? There are two cases to be considered. a. p 1 = 0 or p 2 = 0. Agent 1 would demand infinite amount of the good whose price is zero. Thus, there is no competitive equilibrium. b. p 1 > 0 and p 2 > 0. Agent 2 s demand for good 1 is x 21 = 1, and agent 1 would demand a positive quantity of good 1 because of the quasi-linear utility function. MU 1 p 1 = MU 2 p 2 1 2p 1 x1 = 1 p 2 If x 1 0, then the left-hand side approaches infinity, and the left-hand side is greater than the right-hand side. So, to obtain equality, we must have x 1 > 0. As such, the aggregate demand for good 1 is greater than 1, i.e., x 11 (p) + x 21 (p) = ˆx 1 (p) > ŵ 1, which violates the feasibility conditions. Thus, no competitive equilibrium exists. Page 6 of 27

7.3.2 The One-Consumer and One-Produce Economy Now, we introduce the possibility of production. We analyze the simplest possible setting in which there are only two price-taking economic agents. The two agents are one producer so that J = 1 and one consumer so that n = 1 and θ = 1. There are also two goods: labor (leisure) and the consumption good produced by the firm. w = ( L, 0) is the consumer s endowment, where L denotes the total units of leisure time. We also have f (z) : the production funciton is strictly increasing, concave, and differentiable, where z is labor input. To have an interior solution, we assume f satisfies the Inada conditions lim z 0 f (z) = + and lim z + f (z) = 0. (p,ω) : the price vector of the consumption good and labor. u(x 1,x 2 ) : is the utility function which is quasi-concave, increasing, and differentiable. To have an interior solution, we assume u satisfies the Inada conditions lim xi 0 u x i (x i ) = + and lim xi + u x i (x i ) = 0. q(p,ω) : the profit maximizing output for the consumption good z(p,ω) : the profit maximizing input for the labor π(p,ω) : the profit maximizing funciton The firm s problem is to choose the labor z so as to solve max z 0 p f (z) ωz (9) The first-order condition is given by p f (z) = ω = f (z) = ω p (10) which means the marginal rate of technical substitution of labor for the consumption good equals the price ratio of the labor input and the consumption good output (i.e., MRT S z,q = ω/p). The consumer s problem is to choose the leisure time and the consumption for the good so as to solve subject to max u(x 1,x 2 ) x 1,x 2 px 2 ω( L x 1 ) + π(p,ω) where x 1 is leisure and x 2 is the consumption good. The first-order condition is given by u x 1 u x 2 = ω p (11) Page 7 of 27

which means the marginal rate of substitution of leisure for the consumption good q equals the price ratio of the leisure and consumption good (i.e., MRS x1,x 2 = ω/p). By (10) and (11) MRS x1,x 2 = ω p = MRT S z,q (12) A competitive equilibrium for this economy involves a price vector (p,ω ) at which x 2 (p,ω ) = q(p,ω ) x 1 (p,ω ) + z(p,ω ) = L Thus, the aggregate demand for the two goods equals the aggregate supply for the two goods. 7.4 The Existence of Competitive Equilibrium The proof of the existence of a competitive equilibrium is generally considered one of the most important and robust results of economic theory. In this section, we will examine the existence of competitive equilibrium for the three cases: (1) the single-valued aggregate excess demand function; (2) the aggregate excess demand correspondence; (3) a general class of private ownership production economies. In this section we will examine the existence of competitive equilibrium for the three cases: (1) the single-valued aggregate excess demand function; (2) the aggregate excess demand correspondence; (3) a general class of private ownership production economies. The first two cases are based on excess deman instead of underlying preference orderings and consumption and produciton sets. There are many ways to prove the existence of general equilibrium. For instance, one can use the Brouwer fixed point theorem approach, KKM lemma approach, and abstract economy approach to show the existence of competitive equilibrium for these three cases. 7.4.1 The Existence of CE for Aggregate Excess Demand Functions The simplest case for the existence of a competitive equilibrium is the one when the aggregate excess demand correspondence is a single-valued function. Note that, when preference orderings and production sets are both strictly convex, we obtain excess demand functions rather than correspondences. Also, if preferences are strictly monotone, then ẑ(p) is not well-defined when p = 0. A very important property of the aggregate excess demand function ẑ(p) is Walras law, which can take one of the following three forms: a. The strong form of Walras law is given by p R L + : p ẑ(p) = 0 Page 8 of 27

b. The weak form of Walras law is given by p R L + : p ẑ(p) 0 c. The interior form of Walras Law is given by p R L ++ : p ẑ(p) = 0 Another important property of the aggregate excess demand funciton is homogeneity of ẑ(p). It is homogeneous of degree zero in price, i.e., λ > 0 : ẑ(λp) = ẑ(p). Important to note, if i is a preference ordering, locally non-satiated, and continuous, then x i (p) is homogenous of degree zero. Also, if Y j is non-empty, closed, and satisfies free disposal, then y j (p) is homogenous of degree zero. Normally, we assume ẑ(p) is always homogenous of degree zero. From this property, we can normalize prices. For example, we can normalize a price vector as follows: a. p l = p l/p 1, l = 1,...,L. Here, good 1 is known as the numeraire, i.e., its price has been normalized to 1. b. p l = p l/ L p l. This normalization is often used in proofs of existence of CE. Thus, without loss of generality, we can restrict our attention to the unit simplex: { } S L 1 = p R L + : L p l = 1 R L + which is a non-empty and compact set. Note, the second normalization can be used in the construction of this set, but not the first. Theorem. 7.4.1 (Existence Theorem I). For a private ownership economy e = ({X i,w i, i },{Y j },{θ i j }), if ẑ(p) is a homogeneous of degree zero and continuous function, and satisfies the strong Walras Law, then there exists a competitive equilibrium, i.e., there is p R L + such that ẑ(p) 0. Proof. Define a continuous function g : S L 1 S L 1 by p l = g l (p) = p l + max{0,ẑ l (p)} L k=1 [p k + max{0,ẑ k (p)}] = p l + max{0,ẑ l (p)} 1 + L k=1 max{0,ẑ k(p)} (13) for l = 1,...,L. First, we have constructed a mechanism to increase the price of good l when there is a shortage of the respective good (i.e., ẑ l (p) > 0). Second, note that g is a continuous function since max{ f (x),h(x)} is a continuous function when f (x) and h(x) are continuous. By Brouwer s Fixed Point Theorem, there exists a price vector p such that g(p ) = p, i.e., p l = p l + max{0,ẑ l(p )} 1 + L k=1 max{0,ẑ k(p, l = 1,...,L (14) )} We want to show that p is in fact a competitive equilibrium price vector. Page 9 of 27

Cross multiplying 1 + L k=1 max{0,ẑ k(p )} on both sides of (14), we have p l L max{0,ẑ k (p )} = max{0,ẑ l (p )} (15) k=1 Then, multiplying equation (15) by ẑ l (p ) and taking summation, we have [ L ][ L ] p l ẑl(p ) max{0,ẑ l (p L )} = ẑ l (p )max{0,ẑ l (p )} (16) Then, by the strong Walras Law, we have L ẑ l (p )max{0,ẑ l (p )} = 0 (17) Therefore, each term of the summation is either zero or (ẑ l (p )) 2 > 0. Thus, to have the summation be equal to zero, we must have each term be zero. That is, ẑ l (p ) 0 for l = 1,...,L. Remark 7.4.1. Do not confuse competitive equilibrium of an aggregate excess demand function with the strong Walras Law. Even though Walras Law holds, we may not have ẑ(p) 0 for all p. Also, if ẑ(p ) 0 for some p, i.e., p is a comeptitive equilibrium price vector, the strong Walras Law may not hold unless some types of monotonicty are imposed, such as local non-satiation. Fact 1. (Free goods). Under the strong Walras Law, if p is a competitive equilibrium price vector and ẑ l (p ) < 0, then p l = 0. Proof. Suppose not. Then, p l > 0. So, p l ẑ l (p ) < 0, and thus, p ẑ(p ) < 0. However, this contradicts strong Walras Law. Note. (Desirable goods). If p l = 0, then ẑ l (p ) > 0. Fact 2. (Equality of demand and supply). If all goods are desirable and p is a competitive equilibrium price vector, then ẑ(p ) = 0. Proof. Suppose not. Then, we have ẑ l (p ) < 0 for some l. By the free goods condition, p l = 0. However, since all goods are desirable, then ẑ l (p ) > 0, which contradicts ẑ l (p ) < 0. Remark 7.4.2. By the strong Walras Law, if p > 0 and L 1 markets are in equilibrium, then the L-th market is also in the equilibrium. Thus, because of the strong Walras Law, to verify that a price vector p > 0 clears all markets, it suffices to check that it clears all markets but one. Theorem. 7.4.2 (Existence Theorem I ). For a private ownership economy e = ({X i,w i, i },{Y j },{θ i j }), if the aggregate excess demand function ẑ(p) is a lower semi-continuous function and satisfies strong or weak Walras Law, then there exists a competitive equilibrium, i.e., there is p S L 1 such that ẑ(p ) 0. Page 10 of 27

Proof. Let S S L 1. Define a correspondence F : S 2 S by p S : F(p) = {q S : p ẑ(q) 0} First, note that for each p S, F(p) is nonempty since p F(p) by strong or weak Walras Law. Since p 0 and ẑ( ) is lower semi-continuous, then the function defined by φ(p,q) p ẑ(q) = L p lẑ l (q) is lower semi-continuous in q. Hence, the set F(p) is closed for all p S. Also, since F(p) S and S is bounded, then F(p) is compact for all p S. We now prove F is FS-convex. Suppose for contradiction that there are some {p 1,..., p m } S and some convex combination p λ = m t=1 λ t p t such that p λ / m t=1 F(p t). Then, p λ / F(p t ) for all t = 1,...,m. This implies that p t ẑ(p λ ) > 0 for all t = 1,...,m, so λ t p t ẑ(p λ ) > 0 for all t = 1,...,m. Consequently, m t=1 λ t p t ẑ(p λ ) = p λ ẑ(p λ ) > 0, which contradicts the fact that ẑ satisfies strong or weak Walras Law. Thus, F must be FS-convex. Therefore, by the KKM Lemma, we have p S F(p) /0 Then, there exists a p S such that p p S F(p), i.e., p F(p) for all p S. Thus, p ẑ(p ) 0 for all p S. Now, let p 1 = (1,0,...,0), p 2 = (0,1,0,...,0),..., and p L = (0,...,0,1). Then, p l S, and thus, p l ẑ(p ) = ẑ l (p ) 0 for all l = 1,...,L. Thus, we have ẑ(p ) 0, which means p is a competitive equilibrium price vector. The two theorems above assume that the excess demand function is well defined for all prices in the closed unit simplex S L 1, including zero prices. When preferences are stricly monotone, excess demand functions are not well defined on the boundary of S L 1 so that the above existence theorems cannot be applied. Then, we need an existence theorem for strictly positive prices, which is given below. Theorem. 7.4.4 (Existence Theorem I ). For a prive ownership economy e = ({X i,w i, i },{Y j },{θ i j }), suppose the aggregate excess demand function ẑ(p) is defined for all strictly positive price vectors p R L ++, and satisfies the following conditions a. ẑ( ) is continuous; b. ẑ( ) is homogeneous of degree zero; c. p R L ++ : p ẑ(p) = 0 (interior Walras Law); d. There is an s > 0 such that ẑ l (p) > s for every commodity l and all p R L ++; e. If p k p, where p 0 and p l = 0 for some l, then max{ẑ 1 (p k ),...,ẑ L (p k )} Then there is p R L ++ such that z(p ) = 0, and thus p is a competitive equilibrium. Page 11 of 27

Proof. Because of homogeneity of degree zero, we can restrict our search for an equilibrium in the unit simplex S S L 1. Denote its interior by int(s). We want to construct a correspondence F from S to S such that any fixed point p of F is a competitive equilibrium, i.e., p F(p ) implies ẑ(p ) = 0. To start, define a correspondence F : S 2 S by { {q S : q S, q ẑ(p) q ẑ(p)} if p int(s) F(p) = {q S : q p = 0} if p is on the boundary Note, for p int(s), F( ) means that, given the current proposal p int(s), the counterproposal assigned by the correspondence F( ) is any price vector q that maximizes the values of the aggregate excess demand vector among the permissible price vectors in S. Here, F( ) can be thought of as a rule that adjusts current prices in a direction that eliminates the largest, but not all of the excess demand. The correspondence F( ) assigns the highest prices to the commodities that are most in excess demand. In particular, we have F(p) = {q S : q l = 0 if ẑ l (p k ) < max{ẑ 1 (p k ),...,ẑ L (p k )}} Observe that if ẑ(p) 0 for p int(s), then because of the interior Walras law we have ẑ l (p) < 0 for some l and ẑ l (p) > 0 for some l l. Thus, for such a p, any q F(p) has q l = 0 for some l (to maximize the values of excess demand vectors). Therefore, if ẑ(p) 0, then F(p) Boundary(S) = S\Int(S). In contrast, if ẑ(p) = 0, then F(p) = S. Next, we want to show that the correpondence F is an upper hemi-continuous correspondence with non-empty, convex, and compact values. First, note that S is a compact set and q ẑ(p) is a continuous function on S. By Weierstrass s Theorem, each continuous function on a compact set has a maximizer. When p Int(S), F(p) is non-empty since any price vector q that maximizes the value of ẑ(p) is in F(p). When p is on the boundary of S, p l = 0 for at least some good l, so there exists a q S such that q p = 0, which also implies F(p) is also non-empty. Moreover, since q ẑ(p) is a linear function and F(p) is defined to be the set of maximals of q ẑ(p), then F(p) is compact and convex-valued. Now, we show the correspondence F is upper hemi-continuous, or equivalently, it has closed graphs, i.e., for any sequences p t p and q t q with q t F(p t ) for all t, we have q F(p). There are two cases to consider. Case 1. Case 2. p int(s). Then, p k int(s) for k sufficiently large. From q k ẑ(p k ) q ẑ(p k ) for all q S and the continuity of ẑ( ), we get q ẑ(p) q ẑ(p) for all q S, i.e., q F(p). p is a boundary point of S. Take any l with p l > 0. We want to show that for k sufficiently large, we have q lk = 0. Therefore, it must be that q l = 0; from this, q F(p) follows. Take a sequence p k such that p lk p l > 0 as k. Then, for all δ > 0, there exists N N such that for all k N, p lk B δ (p l ). This implies that there exists ε > 0 such that p lk > ε for k sufficiently large. Moreover, if p k is on the boundary of S, then q lk = 0 by the definition of F(p k ). Instead, if p k Int(S), then by conditions (iv) and (v), we must have ẑ l (p k ) < max{ẑ 1 (p k ),...,ẑ L (p k )} Page 12 of 27

for k sufficiently large. Therefore, q lk = 0 by the definition of F(p k ). Before proving the inequality above, note that some prices go to zero as k because p is a boundary point of S. Consequently, the sequence of maximums on the right-hand side of the expression approaches infinity as k by condition (v). The left-hand side hand, on the other hand, is bounded from above. Now, since p lk > ε for k sufficiently large, then if the left-hand side is positive, we have εẑ l ( ˆp k ) p lk ẑ l (p k ) ẑ l (p k ) 1 ε p lkẑ l (p k ) By the interior Walras Law, p k ẑ(p k ) = 0, and so L p lkẑ l (p k ) = 0. This implies that p lk ẑ l (p k ) = l l p l kẑ l (p k ). As a result, 1 ε p lkẑ l (p k ) = 1 ε p l kẑ l (p k ) l l Finally, by condition (iv), there exists s > 0 such that ẑ l (p) > s ẑ l (p) < s for all l and for all p S. Hence, 1 ε p l kẑ l (p k ) < s ε p l k < s ε l l l l since l l p l k < 1. Thus, the sequence of ẑ l (p k ) is bounded from above. In summary, for p k close enough to the boundary of S, the maximal demand corresponds to some of the commoditites whose price is close to zero. Therefore, we conclude that, for large k, any q k F(p k ) will put nonzero weight only on commodities whose prices approach zero and zero weight on all commodities whose prices don t approach zero. This guarantees that q p = 0 and so q F(p). Hence, F must be upper hemi-continuous. Thus, the correspondence F is an upper hemi-continuous correspondence with nonempty, convex, and compact values. Therefore, by Kakutani s Fixed Point Theorem, we conclude that there is p S such that p F(p ). Finally, we show that any fixed point p of F is a competitive equilibrium. Suppose that p F(p ). Then, p cannot be a boundary point of S because p p > 0 and p q = 0 for all q F(p) cannot occur simultaneously, so p int(s). If ẑ(p ) 0, then ẑ l (p ) < 0 for l and ẑ k (p ) > 0 for k by the interior Walras Law. Thus, for such a p, any q F(p ) must have q l = 0 because q is the maximum of the function q ẑ(p ), which means F(p ) is a subset of boundary points of S. But, p F(p ) and p is not in the boundary of S. Therefore, if p F(p ), we must have ẑ(p ) = 0. Conditions for Walras Law to be true From the above theorems, Walras Law is important to prove the existence of a competitive equilibrium. Under which conditions is Walras Law held? Page 13 of 27

When each consumer s budget constraint holds with equality: for all i we have p x i (p) = p w i + J j=1 θ i j p y j (p) n i=1 p x i (p) = = n i=1 n i=1 p w i + p w i + n J i=1 j=1 θ i j p y j (p) J p y j (p) j=1 which implies that so that p [ n i=1 x i (p) n i=1 w i + J j=1 y j (p) ] = 0 (18) p ẑ(p) = 0 (19) Thus, as long as the individual constraints hold with equality, Walras Law must hold. The above existence theorems on competitive equilibrium are based on the assumptions that the aggregate excess demand correspondence is single-valued and satisfies Walras Law. The questions are under what conditions on economic environments a budget constraint holds with equality, and the aggregate excess demand correspondence is single-valued or convex-valued? The following various types of monotonicities and convexities of preferences with the first one strongest and the last one weakest may be used to answer these questions. Types of monotonicity conditions a. Strict monotonicity: For any two consumption market bundles (x x ) with x x implies x i x. b. Monotonicity: x > x implies that x i x. c. Local non-satiation: For any point x and any neighborhood, N(x), there is x N(x) such that x i x. d. Non-satiation: For any x, there exists x such that x i x. Remark 7.4.4. Monotonicity of preferences can be interpreted as individuals desires for goods: the more, the better. Local non-satiation means individuals desires are unlimited. Page 14 of 27

Types of convexities a. Strict convexity: For any x and x with x i x and x x, x λ λx + (1 λ)x i x for λ (0,1). b. Convexity: If x i x, then x λ λx + (1 λ)x i x for λ (0,1). c. Weak convexity: If x i x, then x λ i x The above conditions are ordered by increasing weakness, i.e., each of the above condition implies the next one, the converse may not be true by examining thick indifference curves and linear indifference curves, strictly convex indifference curves as showed in figures. Remark 7.4.5. The convexity of preferences implies that people want to diversify their consumptions and thus, convexity can be viewed as the formal expression of a basic measure of economic markets for diversification. Note that the strict convexity of i is equivalent to the strict quasiconcavity of utility functions u i. It also implies the convential diminishing marginal rates of substitution (MRS). Weak convexity of i is equivalent to the quasi-concavity of utility funcitons u i. Also, notice that the continuity of i is a sufficient condition for the continuous utility representations, i.e., it guarantees the existence of a continuous utility function u i ( ). Remark 7.4.6. Under the convexity of preferences i, non-satiation implies local non-satiation. Proof. Assume i satisfies convexity and non-satiation. Take x X i. Take ε R ++. By nonsatiation, x X i : x i x. Let x λ λx +(1 λ)x, λ (0,1). By convexity, λ (0,1) : x λ i x. Also, λ (0,1) : x λ B ε (x). Therefore, i satisfies local non-satiation. Proposition. 7.4.1. Under local non-satiation, we have the budget constraint holds with equality, and thus Walras Law holds. Proof. Assume i satisfies local non-satiation. Let x i (p) be agent i s demand correspondence under utility maximization. Let y j (p) be firm j s supply correspondence under profit maximization. First, we claim that the budget constraint holds with equality. Suppose for contradiction that it does not, then x x i (p) : x Int(B i (p)). Since Int(B i (p)) is an open set, then ε > 0 : B ε (x) Int(B i (p)). By local non-satiation, x B ε (x) : x i x. However, this contradicts the fact that x x i (p). Thus, for all i, the budget constraint holds with equality. Now, summing over i we have N i=1 [ N p i=1 p x i (p) = x i (p) p ẑ(p) = 0 N i=1 N i=1 Therefore, Walras Law holds. N J p w i + θ i j p y j (p) i=1 j=1 ] J w i y j (p) = 0 j=1 Page 15 of 27

Proposition. 7.4.2. Under the strict convexity of i, x i (p) becomes a (single-valued) function. Proof. Assume i satisfies strict convexity. We want to show that x i (p) is single-valued. Suppose for contradiction that it is not. Then, {x,x } x i (p) : x x. Note that x x, which is equivalent to x i x and x i x. Without loss of generality, consider x i x. Let x λ λx+(1 λ)x, λ (0,1). Now, x λ B i (p) because [ ] [ ] p λx + p (1 λ)x p λ = p λw i + λ j J = p w i + w i + θ i j y j (p) j J + p (1 λ) θ i j p y j (p) + p w i + j J J θ i j p y j (p) j=1 w i + θ i j y j (p) j J θ i j p y j (p) p λw i λ θ i j p y j (p) j J Finally, by strict convexity, λ (0,1) : x λ i x. However, this contradicts the fact that x x i (p). Therefore, x i (p) is single-valued. Proposition. 7.4.3. Under the weak convexity of preferences, the demand correspondence x i (p) is convex-valued. Proof. Assume i satisfies weak convexity. We want to show that x i (p) is convex-valued. Take {x,x } x i (p). Then, by definition, x i x and x i x. Let us consider x i x. Define x λ λx+(1 λ)x, λ (0,1). Now, because p λx+ p (1 λ)x p w i + p j J θ i j y j (p), x λ B i (p). By weak convexity, x λ i x. Therefore, x λ x i (p), and thus, x i (p) is convex-valued. Proposition. 7.4.4. If Y j is compact (i.e., closed and bounded) and strictly covex, then the supply correspondence y j (p) is a well defined single-valued and continuous function. Proof. Assume Y j is compact. By the maximum theorem, we know that y j (p) is a non-empty valued upper hemi-continuous correspondence by the compactness of Y j (by noting that 0 Y j ) for all p R L +. Now, we show it is single-valued. Suppose that it is not. Then, y j1 and y j2 are two profit maxizing production plans for p R L +, and thus p y j1 = p y j2. By the strict convexity of Y j, we have that λ (0,1) : λy j1 + (1 λ)y j2 Int(Y j ). So, t > 1 : t [ λy j1 + (1 λ)y j2 ] Int(Yj ). However, t [ λp y j1 + (1 λ)p y j2 ] = t p y j1 > p y j1, which contradicts the fact that y j1 is a profit maximizing production plan. Thus, y j (p) is a single-valued function. Therefore, by the upper hemi-continuity of y j (p), we know it is a single-valued and continuous funciton. Proposition. 7.4.5. If i is continuous, strictly convex, locally non-satiated, and w i > 0, then x i (p) is a continuous single-valued function and satisfies the budget constraint with equality for all p R L ++. Consequently, Walras Law is satisfied for all p R L ++. Page 16 of 27

Proof. Assume i is continuous, strictly convex, and locally non-satiated. First, note that since w i > 0, then one can show that the budget constrained set B i (p) is a continuous correspondence with non-empty and compact values and i is continuous. Then, by the maximum theorem, we know the demand correspondence x i (p) is upper hemi-continuous. Futhermore, by the strict convexity of preferences, it is single-valued and continuous. Finally, by local non-satiation, we know the budget constraint holds with equality, and thus, Walras Law is satisfied. Important to note, there is no utility function representation when preferences are lexicographic. Also, i are continuous if the upper contour set U w (x i ) {x i X i : x i x i} and the lower contour set L w (x i ) {x i X i : x i x i } are closed. From the above propositions, we can have the following existence theorem that provides sufficient conditions directly based on the fundamentals of the economy by applying the Existence Theorem I above. Theorem. 7.4.6. For a private ownership economy e = ({X i,w i, i },{Y j },{θ i j }). there exists a competitive equilibrium if the following conditions hold a. X i R L +; b. i are continuous, strictly convex, and strictly monotonic; c. ŵ > 0 d. Y j are compact, strictly convex, 0 Y j ; j = 1,...J. Proof. Important to note, we assume strict monotonicity, instead of a weaker condition, to guarantee the aggregate excess demand function is continuous. Then, by the assumptions imposed, we know that x i (p) and y j (p) are continuous, single-valued, and homogenous of degree zero. Thus, the aggregate excess demand function is continuous, single-valued, and homogenous of degree zero. Also, by the strict monotonicity of preferences, ẑ(p) satisfies interior Walras Law, i.e., p R L ++ : p ẑ(p) = 0. Now, we only need to show that conditions (iv) and (v) of Theorem I are satisfied. The bound in (iv) follows from the non-negativity of demand (i.e., the fact that X i = R L +) and bounded production sets. These conditions imply that a consumer s total net supply to the market of any good l can be no greater than the sum of his initial endowment and upper bound of production sets. Finally, we show that condition (v) is satisfied. As some prices go to zero, a consumer whose wealth tends to a strictly positive limit (note that, because p ŵ > 0, there must be at least one such consumer) and who has strict monotonicity of preferences will demand an increasingly large amount of some of the commodities whose prices go to zero. Therefore, by Theorem I, there is a competitive equilibrium. Examples of Computing CE Example. 7.4.1. Consider an exchange economy with two consumers and two goods with u 1 (x 1 ) = x11 a x1 a 12, 0 < a < 1; w 1 = (1,0) (20) u 2 (x 2 ) = x21 a x1 a 22, 0 < b < 1; w 2 = (1,0) (21) Page 17 of 27

Let p = p 2/p 1. Consumer 1 s problem is to solve max x 1 u 1 (x 1 ) (22) subject to x 11 + px 12 = 1 (23) Since utility functions are Cobb-Douglas types of functions, the solution is then given by x 11 (p) = a x 12 (p) = 1 a p (24) (25) Consumer 2 s problem is to solve max x 2 u 2 (x 2 ) (26) subject to x 21 + px 22 = p (27) The solution is given by x 21 (p) = bp x 22 (p) = (1 b) By the market clearing condition, x 11 (p) + x 21 (p) = 1 a + bp = 1 (28) and thus the competitive equilibrium is given by p = p 2 = 1 a p 1 b This is true because, by Walras Law, for L = 2, it is enough to show only one market clearing. Remark 7.4.7. Since the Cobb-Douglas utility function is widely used as an example of utility funcitons that have nice properties such as strict monotonicity on R L ++, continuity, and strict quasiconcavity, it is useful to remember the functional form of the dmean function derived from the Cobb-Douglas utility functions. It may be remarked that we can easily derive the demand function for the general function: u i (x i ) = xi1 α xβ i2 ;α > 0,β > 0 Page 18 of 27

by the suitable monotonic transformation. Indeed, since utility functions are invariant to monotonic transformations, we can rewrite the utility funciton as so that we have and [ ] xi1 α 1 xβ α+β i2 = x x i1 (p) = x i2 (p) = α α+β i1 when the budget line is given by β α+β xi2 (29) α α+β I p 1 (30) β α+β I p 2 (31) p 1 x i1 + p 2 x i2 = I (32) Example. 7.4.2. Consider an exchange economy with two consumers and two goods with u 1 (x 1 ) = x11 a x1 a 12, 0 < a < 1; w 1 = (1,0) (33) u 2 (x 2 ) = min{x 21,bx 22 }, b > 0; w 2 = (0,1) (34) Let p = p 2/p 1. Important to note, if w il = 0 for some i and l, then a competitive equilibrium may not exist. However, in this example, a competitive equilibrium still exists under certain conditions. For consumer 1, we have already obtained and x 11 (p) = a x 12 (p) = 1 a p For consumer 2, his problem is to solve max x 2 u 2 (x 2 ) (35) (36) (37) subject to x 21 + px 22 = p (38) At the optimal consumption, we have x 21 = bx 22 (39) By substituting the solution into the budget equation, we have bx 22 + px 22 = p (40) Page 19 of 27

and thus and x 22 (p) = x 21 (p) = p b + p bp b + p (41) (42) Then, by x 11 (p) + x 21 (p) = 1, we have or a + bp b + p = 1 (43) (1 a)(b + p) = bp (44) so that (a + b 1)p = b(1 a) (45) Thus, p = b(1 a) a + b 1 To make p be a competitive equilibrium price, we need to assume a + b > 1. 7.4.2 The Existence of CE for Aggregate Excess Demand Correspondences When preferences and/or production sets are not strictly convex, the demand correspondences and/or supply correspondences may not be single-valued, and consequently, the aggregate excess demand correspondence may not be single-valued. As a result, one cannot use the above existence results to argue the existence of competitive equilibrium. Nevertheless, by using the KKM lemma, we can still prove the existence of competitive equilibrim when the aggregate excess demand correspondence satisfies certain conditions. Theorem. 7.4.7 (Existence Theorem II). For a private ownership economy e = ({X i,w i, i },{Y j },{θ i j }), if ẑ(p) is a non-empty, convex, and compact-valued upper hemi-continuous correspondence and satisfies Walras Law, then there exists a competitive equilibrium, i.e., there is a price vector p S such that ẑ(p ) { R L +} = /0. Proof. Let S S L 1. Define a correspondence F : S 2 S by, F(p) = {q S : p ẑ 0 for some ẑ ẑ(q)} In this setup, F(p) is the set of new prices q such that at least one consumption bundle chosen at q is still affordable under the old prices p. Page 20 of 27

Now, F(p) c = {q S : p ẑ(q) R ++ }. Take q F(p) c, then p ẑ(q) R ++. Since R ++ is open, then there is an open set O such that p ẑ(q) O R ++. By the upper hemi-continuity of p ẑ(q), there is an open set N(q) such that for all x N(q), p ẑ(x) O R ++. So, for all x N(q), x F(p) c, and hence, N(q) F(p) c. Thus, for all q F(p) c, there is an open set N(q) such that N(q) F(p) c. Therefore, F(p) c is open, and equivalently, F(p) is closed. Also, since F(p) S and S is bounded, then F(p) is compact for all p S. Next, we prove F is FS-covex. Suppose for contradiction that there are some {p 1,..., p m } S and some convex combination p λ = m t=1 λ t p t such that p λ / m t=1 F(p t). Then, p λ / F(p t ) for all t = 1,...,m. Thus, for all ẑ ẑ(p λ ), we have p t ẑ > 0 for t = 1,...,m. Thus, m t=1 λ t p t ẑ = p λ ẑ > 0, which contradicts the fact that ẑ satisfies Walras Law. So, F must be FS-convex. Therefore, by the KKM Lemma, we have p S F(p) /0 So, there exists a p S such that p p S F(p), i.e., p F(p) for all p S. Thus, for each p S, there is ẑ p ẑ(p ) such that p ẑ p 0 Finally, we prove ẑ(p ) { R L +} /0. Suppose for contradiction that this is not true. Since ẑ(p ) is convex and compact and R L + is convex and closed, then by the Separating Hyperplane Theorem, there is p R L with p 0 and a value c R such that p ( R L +) < c < p ẑ(p ) Since ( R L +) is a cone, we must have c > 0 and p ( R L + ) 0. So, we have 0 < c < p ẑ(p ), ẑ(p ) { R L +} = /0, and p 0. Thus, p R L + and p ẑ(p ) > 0 for some p, which contradicts Walras Law. Similarly, we have the following existence theorem that provides sufficient conditions directly based on economic environments by apply the Existence Theorem II above. Theorem. 7.4.8. For a private ownership economy e = ({X i,w i, i },{Y j },{θ i j }), there exists a competitive equilibrium if the following conditions hold a. X i R L +; b. i are continuous, weakly convex, and strictly monotone; c. ŵ > 0 d. Y j are closed, covex, and 0 Y j ; j = 1,2,...,J. Page 21 of 27

7.4.3 The Existence of CE for General Production Economies For a general private ownership production economy e = ({X i,w i, i },{Y j },{θ i j }) (46) recall that a competitive equilibrium consists of a feasible allocation (x,y ) and a price vector p R L + such that (i) x i D i (p ) x i (p ) (utility maximization) (ii) y j S j (p ) y j (p ) (profit maximization) We now state the following existence theorem for general production economies without proof since it is very complicated. Theorem. 7.4.9 (Existence Theorem III, Debreu (1959)). A competitive equilibrium for the private-ownership economy e exists if the following conditions are satisfied a. X i is closed, convex, and bounded from below; b. i are non-satiated; c. i are continuous; d. i are convex; e. w i Int(X i ); f. Y j are closed and convex (continuity and no IRS); g. 0 Y j (possibility of inaction); h. Y j { Y j } = {0} (Irreversibility); i. { R L +} Y j (free disposal). 7.5 The Uniqueness of Competitive Equilibria So, we now know that a Walrasian equilibrium will exist under some regularity conditions. We worry next about the other extreme possibility: for a given economy, there are many equilibria so that we do not know what is the outcome of the market process. We can easily give examples in which there are multiple competitive equilibrium price vectors. When is there only one normalized price vector that clears all markets? The free goods case is not of great interest here, so we will rule it out by means of the desirability assumption so that every equilibrium price of each good must by strictly positive. We want to also assume the continuous differentiability of the aggregate excess demand function. The reason is fairly clear; if indifference curves have kinks in them, we can find whole ranges of prices that are market equilibria. Not only are the equilibria not unique, they are not even locally unique. Thus, we answer this question while only considering the case of p > 0 and ẑ(p) differentiable. Page 22 of 27

Theorem. 7.5.1. Suppose all goods are desirable and gross substitutes for all prices (i.e., z h(p) p l > 0,l h). If p is a competitive equilibrium price vector and Walras Law holds, then it is the unique competitive equilibrium price vector. Proof. Since all goods are desirable, then p > 0. Suppose p is another competitive equilibrium price vector that is not proportional to p, i.e., α R + : p αp. Let m = max p l p = p k l p for k some k (at least one k satisfies this, but there can be more). In this setup, good k becomes the most relatively expensive good at prices p. Since all goods are desirable, then, by the definition of a competitive equilibrium, ẑ(p ) = ẑ(p) = 0. By homogeneity of degree zero, we know that ẑ(p ) = ẑ(mp ) = 0. Also, we know that m = p k p p l k p for all l and m > p h l p for some h. So, we h have mp l p l for all l and mp h > p h for some h. Now, if the price of good k stays the same when prices change from p to p, i.e., p k = p k, then m = 1. Thus, p l p l for all l and p h > p h for some h. This means that when the price of good k is fixed, the prices of the other goods are down. As a result, we must have the demand for good k go down by gross substitutes. Hence, ẑ k (p) < 0, but this contradicts ẑ(p) = 0. If the aggregate demand function satisfies the Weak Axiom of Revealed Preference (WARP), and Walras Law holds, then the competitive equilibrium is unique. The Weak Axiom of Revealed Preference (WARP of the aggregate excess demand function): {p, p } R L + : p ẑ(p) p ẑ(p ) = p ẑ(p) > p ẑ(p ). WARP implies that, if ẑ(p ) could have been bought at p where ẑ(p) was bought (so that ẑ(p) ẑ(p ) since ẑ(p) is the optimal choice), then at price p, ẑ(p) is outside the budget constraint (otherwise it contradicts the fact that ẑ(p) is the optimal choice). WARP is a weaker restriction than the continuous concave utility funciton. However, the restriction on the aggregate excess demand function is not as weak as it may seem. Even though two individuals may satisfy the individual WARP, the aggregate excess demand function may not satisfy the aggregate WARP. Lemma. 7.5.1. Under the assumptions of strong Walras Law and WARP, we have p ẑ(p) > 0 for all p kp where p is a competitive equilibrium. Proof. Assume p is a competitive equilibrium. Then, ẑ(p ) 0. By Walras Law, p ẑ(p) = 0. Because p R L +, p ẑ(p ) 0. So, we have p ẑ(p) p ẑ(p ). Thus, by WARP, p ẑ(p) > p ẑ(p ) = 0, which implies that p ẑ(p) > 0 for all p kp. Theorem. 7.5.2. Under the assumptions of Walras Law and WARP of the aggregate excess demand function, the competitive equilibrium is unique. Proof. By Lemma 7.5.1, p kp : p ẑ(p) > 0, which means at least for some l, ẑ l > 0. Page 23 of 27

7.6 Stability of Competitive Equilibrium The concept of competitive equilibrium is a stationary concept. However, it has given no guarantee that the economy will actually operate at the equilibrium point, or an economy may deviate from an equilibrium due to economic shocks. What forces exist that might tend to move prices to a market-clearing price? This is a topic about the stability on the price adjustment mechanism. A paradoxical relationship between the idea of competition and price adjustment is that: if all agents take prices as given, how can prices move? To solve this paradox, one introduces a Walrasian auctioneer whose sole funciton is to seek for the market clearing prices. The Walrasian auctioneer is supposed to call the prices and change the price mechanically responding to the aggregate excess demand until the market clears. Such a process is called a Tatonnement adjustment process. A Tatonnement Adjustment Process is defined, according to the laws of demand and supply, by d p l dt = G l (ẑ l (p)), l = 1,...,L (47) where G l is a sign-preserving function of ẑ(p), i.e., G l (x) > 0 if x > 0, G l (x) = 0 if x = 0, and G l (x) < 0 if x < 0. The above equation implies that when the aggregate excess demand is positive, we have a shortage and thus price should go up by the laws of demand and supply. As a special case of, G l can be an identical mapping such that ṗ l = ẑ l (p) ṗ = ẑ(p) (48) (49) Under Walras Law, [ d L dt (p p) = d p 2 l dt ] = 2 L p l d p l dt = 2p ṗ = p ẑ(p) = 0 which means that the sum of squares of the prices remain constrant as the price adjusts. So, the ratios of the prices change, but the length of the price vector does not. This is another price normalization. The path of the prices are resticted on the surface of a k-dimensional sphere. Definition. 7.6.1. An equilibrium price p is globally stable if a. p is the unique competitive equilibrium, b. for all p o there exits a unique price path p = φ(t, p o ) for 0 t < such that lim t φ(t, p o ) = p We also define what it means for p to be locally stable. Definition. 7.6.2. An equilibrium price p is locally stable if there is δ > 0 and a unique price path p = φ(t, p o ) such that lim t φ(t, p o ) = p whenever p p o < δ. Page 24 of 27