Lecture 8: Market Equilibria

Transcription

1 Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecturer: Bruno Codenotti Lecture 8: Market Equilibria The market setting transcends the scenario of games. The decentralizing effect of equilibrium prices mediates among the economic agents thus making their strategic decisions independent. However there is a natural interplay between market equilibria and Nash equilibria for games: one of the very first proofs of existence of a market equilibrium is built upon the existence of a Nash equilibrium in an associated game. In this lecture we first describe a general model (the competitive economy), but then only consider exchange economies, a market model which does not include the production of goods, and analyze the market equilibrium problem in such a restricted context. 8.1 The Competitive Economy We assume that there is a finite number l of commodities. These include goods, which may be produced and traded, as well as labor (which can be supplied by individuals, but cannot be produced.) The decision makers in our model are n production units and m consumption units. Production units correspond to companies. These take some commodities as input (labor, raw materials), and produce others as output (finished goods). Consumption units correspond to individuals, who generally supply labor but consume goods. At a high level, our goal is to set prices for each commodity so that each producer will decide its production strategy based only on the prices and the total amount of raw materials (and independently of what each other company does); each consumer will decide a consumption strategy based only on the prices and his/her budget, and moreover there will be no excess of supply or demand. By excess of supply, we mean a commodity for which the price is nonzero but there is supply left over. By excess of demand, we mean a commodity which is sold out, but of which some producer or consumer wanted more. Each producer j has a set Y j of production plans, which are vectors in R l, with the coordinates representing the various commodities. Inputs (raw materials, labor) are represented by negative coordinates. The set Y j is assumed to be a closed convex subset of R l containing 0. Similarly, each consumer i has a set X i of consumption plans, which are vectors in R l. Here labor has a negative coordinate, while bought goods have positive coordinates. Consumers begin the day with an initial endowment, ζ i, of commodities, which is another vector in R l. These may be sold to other consumers or to producers, or kept. We make the following basic assumptions about the sets X i and Y j : Labor cannot be produced, i. e. there exist commodities h such that for all j, for all production plans y j Y j the hth coordinate of y j is non-positive No consumer works more than 24 hours per day. In other words, there is an absolute bound on the labor coordinate for all consumption plans x i X i, 1 i m. All goods coordinates of x i are non-negative (these goods are consumed). Thus there is an absolute lower bound on all coordinates of each x i X i, 1 i m. 8-1

2 8-2 Lecture 8: Market Equilibria Suppose each consumer i has chosen a consumption plan x i X i and each producer j has chosen a production plan y j Y j. Let x = m i=1 x i, y = n i=1 y j, and ζ = m i=1 ζ i. Then x ζ +y. In other words, consumption cannot exceed initial supply plus production. We will call such an overall selection feasible. We further restrict the sets X i and Y j to only contain plans which belong to some feasible overall selection. This restriction, together with closure and convexity, forces the sets to be bounded, so that they are compact. (Note that the restriction does not force every selection in i X i j Y j to be feasible in the above sense.) A price vector, p R l must satisfy p h 0 for all commodities h, and l h=1 p h = 1. Let P be the space of all such price vectors. Note that labor has its price as well as goods. Each company j aims to maximize the dot product y j p, which is its net profit (sales minus costs). Note that since the zero strategy is always available, a company s profit at equilibrium must be non-negative. Each consumer i has a utility function u i : X i R, which is continuous and concave. The consumer s goal is to maximize this utility function, subject to her/his budget, which is defined as B i = p ζ i + j α i,j p y j. Here the α i,j s represent the dividends paid to consumer i by company j. We require α i,j 0, and m i=1 α i,j = 1, for all j s. A consumer s budget represents the total assets available for her/him to spend. The consumption vector x i must satisfy the budget constraint x i p B i. A competitive equilibrium in this setup is an (m + n + 1)-tuple (x 1,..., x m, y 1,..., y n, p ) of vectors in R l such that y j maximizes p y j over y j Y j. x i maximizes u i(x i ) over x i X i, and satisfies the budget constraint, given (y, p ). x y ζ 0, and (x y ζ) p = 0. The first condition is a resource constraint, while the second condition is an example of complementary slackness, and models the classical law of supply and demand. We can now state the fundamental theorem by Arrow and Debreu [2]. Theorem 8.1 (Arrow-Debreu) In the model described above, a competitive equilibrium exists. 8.2 The Exchange Economy We describe now the so-called exchange economy, an important special case of the model described above. The exchange economy does not include the production of goods. Let us consider m economic agents who represent traders of n goods. Let R n + denote the subset of R n with all nonnegative coordinates. The j-th coordinate in R n will represent good j. Each trader i has a concave utility function u i : R n + R +, which represents her preferences for the different bundles of goods, and an initial endowment of goods w i = (w i1,..., w in ) R n +. At given prices π R n +, trader i will sell her

3 Lecture 8: Market Equilibria 8-3 endowment, and get the bundle of goods x i = (x i1,..., x in ) R n + which maximizes u i (x) subject to the budget constraint 1 π x π w i. An equilibrium for the exchange economy is a vector of prices π = (π 1,..., π n ) R n + at which, for each trader i, there is a bundle x i = ( x i1,..., x in ) R n + of goods such that the following two conditions hold: 1. For each trader i, the vector x i maximizes u i (x) subject to the constraints π x π w i and x R n For each good j, i x ij i w ij. The Fisher model. A special case of the general exchange model defined above occurs when the initial endowments are proportional, i.e., when w i = δ i w, δ i > 0, so that the relative incomes of the traders are independent of the prices. This special case is equivalent to the Fisher model, which is a market of n goods desired by m utility maximizing buyers with fixed incomes. We assume that each buyer has a concave utility function u i : R n + R + and an endowment e i > 0 of money. There is a seller with an amount q j > 0 of good j. An equilibrium in the Fisher setting is a nonnegative vector of prices π = (π 1,..., π n ) R n + at which there is a bundle x i = ( x i1,..., x in ) R n + of goods for each buyer i such that the following two conditions hold: 1. The vector x i maximizes u i (x) subject to the constraints π x e i and x R n For each good j, i x ij = q j. Demand and excess demand. For any price vector π, the vector x i (π) that maximizes u i (x) subject to the constraints π x π w i and x R n + is called the demand of trader i at prices π. 2 Then X k (π) = i x ik(π) denotes the market (or aggregate) demand of good k at prices π, and Z k (π) = X k (π) i w ik the market excess demand of good k at prices π. The vectors X(π) = (X 1 (π),..., X n (π)) and Z(π) = (Z 1 (π),..., Z n (π)) are called market demand (or aggregate demand) and market excess demand, respectively. The market is said to satisfy positive homogeneity if for any price vector π and any λ > 0, we have Z(π) = Z(λπ). It is said to satisfy Walras Law if for any price π, we have π Z(π) = 0. Both positive homogeneity and Walras law are considered standard and extremely mild assumptions in the theory of equilibrium. Positive homogenity follows simply from the agent s utility maximizing behavior whereas Walras Law follows from a very weak assumption on agent preferences called local nonsatiation 3. In terms of the aggregate excess demand function, the equilibrium is a vector of prices π = (π 1,..., π n ) R n + such that Z(π) is well-defined and Z j (π) 0, for each j. GS and WARP. Two properties play a significant role in the theory of equilibrium and in related computational results: gross substitutability (GS) and the weak axiom of revealed preferences (WARP). The market excess demand is said to satisfy gross substitutability (resp., weak gross substitutability - WGS) if for any two sets of prices π and π such that 0 < π j π j, for each j, and π j < π j for some j, we have that π k = π k for any good k implies Zk (π) < Z k (π ) (resp., Z k (π) Z k (π )). That is, increasing the price of 1 Given two vectors x and y, x y denotes their inner product. 2 Unless otherwise stated, we assume that there is at most one such vector, i.e., the demand is a single-valued function. This is the case for utility functions that satisfy strict quasi-concavity. In general, the demand is a set-valued function. 3 Local nonsatiation means that for any bundle of goods there is always another bundle of goods arbitrarily close to it that is preferred to it.

4 8-4 Lecture 8: Market Equilibria some of the goods while keeping some others fixed can only cause an increase in the demand for the goods whose price is fixed. The market excess demand is said to satisfy WARP if for any two sets of prices π and π such that Z(π) Z(π ) either π Z(π ) > 0 or (π ) Z(π) > 0. This means that if the demands at prices π and π are different, then either Z(π ) is not within budget when the price is π or Z(π) is not within budget when the price is π. It is well known that GS implies that the equilibrium prices are unique up to scaling ([8], p. 395) and that WARP implies that the set of equilibrium prices is convex ([6], p. 608). It is also well known that the market excess demand satisfies weak GS when each individual excess demand does. In contrast, WARP is satisfied by the individual excess demand of a trader with a concave utility function, but it is not in general satisfied by the aggregate excess demand. Commonly used utility functions. u(αx) = αu(x), for all α > 0. A utility function u( ) is homogeneous (of degree one) if it satisfies A linear utility function has the form u i (x) = j a ijx ij. The Cobb-Douglas utility has the form u i (x) = j (x ij) aij, where j a ij = 1. The Leontief utility function has the form u i (x) = min j a ij x ij. A CES (constant elasticity of substitution) utility function has the form u(x i ) = ( j (a ijx ij ) ρ ) 1/ρ, where < ρ < 1 but ρ 0. In all of these definitions, a ij 0. As ρ tends to 1 (resp. 0, ), the CES utility function tends to a linear (resp. Cobb-Douglas, Leontief) utility function ([1], page 231). The CES class of functions are popular among economists for their ability to express a wide variety of consumer preferences as well as their mathematical tractability which allows for explicit computation of the associated demand function. Example 8.2 (Cobb-Douglas Utilities) Let consider utility functions of the form u(x) = x a 1 1 xa 2 2 xan n, a i 0, a i = 1. These utility functions are known as Cobb-Douglas utilities. It is easy to see that a trader with this utility function will spend the same fraction a j of her income M on the jth good, independently of relative prices. In other words, the demand function will be x j (π) = M a j π j. i=1 Implications of Gross Substitutability denote the market excess demand function. Let us consider prices on the unit simplex S n, and let Z( ) We show that, if all the goods are gross substitutes at all prices, then, if p is an equilibrium price vector, we have p Z(p) > 0, for all p p S n. In other words, gross substitutability implies that the weak axiom of the revealed preferences holds with respect to the equilibrium price. This result is the basis of several computational results for economies where gross substitutability holds. We will actually show that p Z(p) is minimized at p = p, and that p is the unique minimum. Note that the function p Z(p) is bounded below on the simplex, so that the minimum does exist. Furthermore, the minimum can not be attained on the boundary of the simplex, because of gross substitutability. Therefore we can find the minimum by taking the partial derivatives and setting them to zero:

5 Lecture 8: Market Equilibria 8-5 j=1 p Z j (p) j = 0, k = 1,..., n. p k We first show that the equality above is satisfied for p = p, and then we will prove the uniqueness. By Walras law, we have 0 = (p Z(p)) p k = Since Z k (p ) = 0, we have n j=1 p Z j(p ) j p k = 0. j=1 p j Z j (p) p k + Z k (p). We now show that no other price vector satisfies the condition above. By contradiction, assume that there exists p p such that n j=1 p Z j(p) j p k = 0. Suppose now that h = p r p r Now we have p = max j j p. Since GS implies that Z r (p ) = 0, we have that Z r (p) < 0. j 0 = (p Z(p)) = Since Z r (p) < 0, we must have n j=1 p j Z j(p) > 0. Z j(p) We have 0 = h n j=1 p j This last term is strictly greater than n Z. Now note that the right hand side can be written as p r(p) r +h n j r p Z j(p) j., which is positive. Therefore we have reached a contradiction. j=1 p j Z j (p) + Z r (p). j=1 p j Zj(p) 8.3 Existence of Equilibria Recall that the market equilibrium is a vector of prices π = (π 1,..., π n ) R n + such that the market excess demand Z(π) is well-defined and Z j (π) 0, for each j. Since in general the market excess demand is a set-valued function, the proofs of existence of economic equilibria are based on Kakutani s fixed point theorem. This theorem provides a generalization of Brouwer s fixed point theorem to set-valued functions. However under mild assumptions which avoid boundary complications, the excess demand becomes a singlevalued function, and in this case it is possible to prove the existence of equilibria by means of Brouwer s fixed point theorem. Theorem 8.3 (Equilibrium Theorem) Let Z(π) be a function defined for all nonzero nonnegative price vectors π R m +, and satisfying continuity, homogeneity of degree zero and Walras law. Then there is price vector π such that Z(π ) 0. Proof: The proof is very similar to the proof of Nash Theorem of lecture 4. Consider the functions g j (π) = max{0, Z j (π)}. Define the function f mapping the unit simplex into itself, whose j-th component

6 8-6 Lecture 8: Market Equilibria is f j (π) = πj+gj(π) 1+ k g. It is easy to check that the conditions needed to apply Brouwer s fixed point theorem k(π) hold, so that there is a point π such that f j (π ) = πj, for all j. We want to show that Z j(π ) 0, for all j. Note that there must exist an index t such that Z t (π ) 0. For such a t, we have πt π = t 1+ k g k(π ). But this equality holds only if k g k(π ) = 0, i.e., g k (π ) = 0, for all k. It is interesting to show that from the Equilibrium Theorem it is possible to derive Brouwer s theorem. Theorem 8.4 (Uzawa) The Equilibrium Theorem implies Brouwer s Theorem. Proof: Let f be a continuous function mapping the unit m-simplex into itself. Define a function g mapping the unit m-simplex into R m+1 as follows: g(x) = f(x) x f(x) x x x. It is easy to check that g satisfies the conditions needed to apply the Equilibrium theorem (continuity, homogeneity of degree zero and Walras law). Therefore there exists x such that g(x ) 0. Then some simple algebraic derivations show that the set of points x for which g(x) 0 coincides with the set of points for which f(x) = x. We now sketch the idea of another proof of the existence of equilibria for exchange economies which is based on a transformation of the market scenario into a non-cooperative game. Let us assume that each trader has some positive amount of each good. There are n + 1 players, i.e., the n traders, and the market player. The strategy set of the i-th trader is the positive orthant corresponding to all possible bundles. The strategy set of the market player is the set of points in the price simplex. Given a price π from the simplex, the best response of the i-th player is just her demand at price π. In other words, she picks the bundle x i from the positive orthant maximizing her utility function u i (x) subject to π T x π T w i. Note that the best response of the i-th player depends only on the choice made by the market player. The payoffs are defined in the natural way: a strategy profile (x 1,..., x m, π) yields a payoff of 1 for the i-th player if x i is her demand at price π. Otherwise her payoff is 0. Given a strategy profile (x 1,..., x m, π), the payoff for the market player is π T z, where z = i x i i w i. That is, given (x 1,..., x m ), the best response of the market player is to pick any price from the price simplex that is nonzero only for those components of z that are maximal. Now look at a Nash equilibrium (x 1,..., x m, π) of the game. Since x i is a best response given π, we must have π T x i = π T w i if the function u i ( ) is nonsatiable. It follows that π T z = 0. But then for π to be the best response given (x 1,..., x m ), it must be that each component of z is non-positive: otherwise, the market player can choose π π to get a strictly positive payoff. Thus, both utility maximization and market clearance hold, and we have an equilibrium for the market. Bibliographic notes In 1874, Walras published the famous Elements of Pure Economics, where he describes a model for the state of an economic system in terms of demand and supply, and expresses the supply equal demand equilibrium

7 Lecture 8: Market Equilibria 8-7 conditions [10]. In 1936, Wald gave the first proof of the existence of an equilibrium for the Walrasian system, under severe restrictions [9]. In 1954, Arrow and Debreu proved the existence of an equilibrium under milder assumptions [2]. The main general reference for the market equilibrium problem is [6]; in particular Chapters 16 and 17 present existence proofs at various levels of generality. The version we have shown (the Equilibrium Theorem above) corresponds to Proposition 17.C.2 in [6]. The reverse implication, i.e., the proof of Brouwer s theorem via the Equilibrium Theorem is due to Uzawa [7]. For the definitions, notation, etc., of this lecture we have mainly followed [4]. References [1] K.J. Arrow, H.B. Chenery, B.S. Minhas, R.M. Solow, Capital-Labor Substitution and Economic Efficiency, The Review of Economics and Statistics, 43(3), (1961). [2] K.J. Arrow and G. Debreu, Existence of an Equilibrium for a Competitive Economy, Econometrica 22 (3), pp (1954). [3] K.C. Border, Fixed point Theorems with Applications to Economics and Game Theory, Cambridge University Press (1985). [4] B.Codenotti, S. V. Pemmaraju, K. R. Varadarajan, The computation of market equilibria. SIGACT News 35(4): (2004) [5] E. Eisenberg, Aggregation of Utility Functions. Management Sciences, Vol. 7 (4), (1961). [6] A. Mas-Colell, M.D. Whinston, and J.R. Green, Microeconomic Theory, Oxford University Press (1995). [7] H. Uzawa, Walras s Existence Theorem and Brouwer s Fixed Point Theorem, Econ. Stud. Quart. 13, (1962). [8] H. Varian, Microeconomic Analysis, New York: W.W. Norton (1992). [9] A. Wald, On Some Systems of Equations of Mathematical Economics, Zeitschrift für Nationalökonomie, Vol.7 (1936). Translated: Econometrica, Vol.19 (4), p (1951). [10] L. Walras, Éléments d économie politique pure, ou théorie de la richesse sociale (Elements of Pure Economics, or the theory of social wealth), (1899, 4th ed.; 1926, rev ed., 1954, Engl. transl. Homewood, Ill.: Richard Irwin.)