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Pareto Optimality, the Core, and Competitive Equilibrium Econ 2100 Lecture 16, October 28 Outline 1 Pareto E¢ ciency 2 The Core 3 Planner’s Problem(s) 4 Competitive (Walrasian) Equilibrium Fall 2015 General Equilibrium Consumers i = 1; :::; I are described by a consumption set Xi RL , preferences J %i 0, endowments ! i 2 Xi , and shares i 2 [0; 1] of the pro…ts of each …rm. Firm j = 1; :::; J are described by a production set, Yj RL . P The aggregate endowment is ! = i ! i . P Firms are owned by someone: 1 = i ij with ij 2 [0; 1] for all j. De…nitions fXi ; %i gi =1 I An economy is a tuple An economy with private ownership is a tuple fXi ; %i ; ! i ; J fYj gj =1 I i gi =1 ! J fYj gj =1 An allocation: (x; y ) = (x1 ; :::; xI ; y1 ; :::; yJ ) 2 RL(I +J ) where xi 2 Xi for each i = 1; :::; I , and yj 2 Yj for each j = 1; :::; J. De…nition An allocation (x; y ) is feasible if I X i =1 xi !+ J X j =1 yj Pareto E¢ ciency Question How should we think about e¢ ciency? A minimal requirement is that there is no waste of resources. De…nition A feasible allocation (x; y ) is Pareto optimal if there is no other feasible allocation (x 0 ; y 0 ) such that xi0 %i xi for all i = 1; :::; I and xi0 i xi for some i Nothing is left on the table: an allocation is e¢ cient if we cannot …nd another one that harms no one and bene…ts someone. This only looks at preferences to de…ne e¢ ciency. Firms do not have preferences, but they matter via what they produce. Remarks Nobody chooses anything here. This is not about markets and/or prices. O2 x x’ O1 x and x’are not Pareto optimal O2 ω x O1 x is Pareto optimal O2 O1 The locus of all Pareto optimal allocations (contract curve) Individual Rationality Suppose we let the individuals exchange goods with each other. Observation A consumer should not want allocations that do not improve over her initial conditions. Therefore, she may not want outcomes that are worse than consuming her initial endowment since she can always refuse to trade. De…nition A feasible allocation (x; y ) is individually rational if xi %i ! i for all i: An individually rational allocation represents a trade that improves everyone position relative to their initial endowment. Pareto optimal allocations are not necessarily individually rational. O2 ω x x’ O1 x is individually rational, x’is not The Core A related concept requires that no group of consumers can gain by ‘seceding’ from the economy. This is easier to de…ne for an exchange economy: let J = 1, and YJ = RL+ . De…nition A coalition is a subset of f1; :::; I g. De…nition A coalition S f1; :::; I g blocks the allocation x if for each i 2 S there exist xi0 2 Xi such that X X xi0 i xi for all i 2 S and xi0 !i i 2S i 2S A blocking coalition can make all its members better o¤. One can think of a weaker de…nition where a coalition bene…ts at least one of its members strictly without hurting the others. De…nition A feasible allocation x is in the core of an economy if there is no coalition that blocks it. Pareto Optimality, Individual Rationality, and the Core Easy to Prove Results Any allocation in the core of an economy is also Pareto optimal. Obvious since the ‘whole’(sometimes called ‘grand coalition’S = f1; :::; I g) is not a blocking coalition. Not all Pareto optimal allocations are in the core. Slightly less obvious: xi = ! (consumer i gets everything) is Pareto optimal but not in the core. Any assumptions requred here? In an Edgeworth box (I = 2), the core is the set of all individually rational Pareto optimal allocations. This is an (easy) homework problem. With more consumers this result does not hold. As the number of consumers grows, there are more possible coalitions, and more allocations will be blocked. So the core is typically much smaller than the set of Pareto optimal allocations that are individually rational. De…nitions With Utility Functions Suppose individuals’preferences are represented by a utility function. Consumer i’s utility function is denoted ui (xi ). We can rewrite Pareto e¢ ciency and individual rationality as follows. De…nitions with utility functions A feasible allocation (x; y ) is Pareto optimal if there is no other feasible allocation (x 0 ; y 0 ) such that ui (xi0 ) ui (xi ) for all i and ui (xi0 ) > ui (xi ) for some i: A feasible allocation (x; y ) is individually rational if ui (xi ) ui (! i ) for all i: Pareto E¢ ciency and Utility Possibility Set De…nition The utility possibility set is U= (u1 ; :::; uI ) 2 RI : there exists a feasible (x; y ) such that ui u(xi ) for i = 1; :::; I The utility possibility frontier is the boundary of U, and a Pareto optimal allocation must belong to the frontier. Draw a picture. De…nition The Pareto frontier is de…ned as UP = (u1 ; :::; uI ) 2 RI : there is no (u10 ; :::; uI0 ) 2 U such that ui0 ui for all i and ui0 > ui for some i Question 6, Problem Set 9 Show that a feasible allocation (x; y ) is Pareto optimal if and only if (u1 (x1 ); :::; uI (xI )) 2 UP. Pareto E¢ ciency and Social Welfare De…nitions A (linear) social welfare function is a weighted sum of the individuals’utilities: W = I X i ui = u with i =1 The social welfare maximization problem is max u2U P i i 0 i ui This maximization problem is sometimes called the “planner’s problem”. Proposition P If u 2 RI solves maxu2U i i ui with i > 0 for all i, then u 2 UP. Moreover, if U is convex then for any u~ 2 UP there exists 0 with that u~ u for all u 2 U. 6= 0 such An allocation that solves the social welfare maximization problem is Pareto optimal; morever, if the utility possibility set is convex, any Pareto optimal allocation solves the social welfare maximization problem. Convexity of U follows from convexity of production and consumption sets and concavity of utility functions. The proof uses the separating hyperplane theorem. Proof. First show that u 2 arg max u2U X i ui with i i > 0 for all i ) u 2 UP By contradiction. If u is not Pareto optimal, then there exists u 2 U with u u and u 6= u ; since i > 0 for all i . This contradicts that u solves the social welfare maximization problem ( u> u ). Now show that U convex ) for any u~ 2 UP there exists 0 with u~ 6= 0 for all u 2 U Since u~ belongs to the frontier of U, a convex set, by the separating hyperplane theorem there exists a vector 6= 0 such that u~ Verify that we also have u u such that for all u 2 U 0 to complete the proof. Pareto E¢ ciency and Constrained Planner Problem Another planner’s problem aims to maximizie the utility of one consumer subject to everyone else achieving a prespeci…ed utility level. Consider, for simplicity, an exchange economy. Proposition Suppose that, for each i = 1; :::; I , %i can be represented by a strictly increasing and continuous utility function ui (xi ). Then, an allocation x is Pareto optimal if and only if it solves the following ui (xi ) ui for all i 6= j, max uj (xj ) x subject to: I P xi = ! and i =1 xi 0 for all i for some choice of fui gi 6=j . An allocation is Pareto optimal if and only if it maximizes one consumer’s utility subject to all others obtaining some given utility level. Notice that this does not say for any fui gi 6=j (easy to …nd a counterexample). Proof is an homework problem. Pareto E¢ ciency: An Example Finding Pareto Optimal Allocations: An Example Consider an economy: where consumer 1 ! 1 = (! 11 ; ! 21 ) u1 (x1 ) = u1 (x11 ; x21 ) consumer 2 ! 2 = (! 12 ; ! 22 ) u2 (x2 ) = u2 (x12 ; x22 ) To …nd the set of Pareto optimal allocations for an Edgeworth box economy in which consumers’utility functions are u1 (x11 ; x21 ) = (x11 ) (x21 ) 1 and 1 u2 (x12 ; x22 ) = (x12 ) (x22 ) We must solve the following planner’s problem: 1 max x11 ;x21 ;x12 ;x22 (x11 ) (x21 ) 1 subject to (x12 ) (x22 ) u x11 + x12 = ! 11 + ! 12 x21 + x22 = ! 21 + ! 22 x11 ; x21 ; x12 ; x22 0 for some u. In this case, the utility functions are di¤erentiable, so we can write the Lagrangean, write the …rst order conditions, and then solve. Competitive (Walrasian) Equilibrium De…nition Given an economy fXi ; %i ; ! i ; i gi =1 ; fYj gj =1 , a competitive (Walrasian) equilibrium is formed by an allocation (x ; y ) 2 RL(I +J ) and a price vector p 2 RL such that: I yj p yj for all xi 2 fxi 2 Xi : p xi p 1 For each j = 1; :::; J: 2 For each i = 1; :::; I : xi %i xi 3 p I X i =1 1 2 3 J xi I X i =1 !i + J X for all yj 2 Yj !i + yj PJ j =1 ij p yj g j =1 Firms’choices maximize pro…ts given the equilibrium prices; Consumers’choices are optimal in the budget set at the equilibrium prices; and In all markets, demand cannot exceed supply . Remark Equilibrium prices make all optimization problems ‘mutually compatible’. Individual Demand Consumer’s demand depend on prices and initial endowment O2 ω21 Supply of good 2 x121 x1 x221 x321 O1 x2 ω11 x1 11 Demand of good 1 x211 x3 x311 Excess Demand The sum of consumers’s demands does not equal the initial endowment ω1j ω ω2i x 1j Oj Supply of good 1 ω2j Demand of good 2 x2j Supply of good 2 x2i Demand of good 1 Oi ω1i x 1i There is excess demand of good 1 (and excess supply of good 2) Equilibrium ω12 Supply of good 1 x*12 O2 ω ω21 ω22 Supply of good 2 Demand of good 2 x* x*21 Demand of good 1 O1 ω11 x*11 Excess demands of good 1 and good 2 are ZERO x*22 Competitive Equilibrium in an Edgeworth box with Cobb-Douglas utilities An equilibrium is given by a price vector p and an allocation x such that: xi = arg max x 2fx 2R2 :p x p !1 g (x1i ) (x2i ) 1 xj = arg the Walrasian demands are: p ! +p ! x1i = 1 1ip 2 2i 1 p ! +p ! x2i = (1 ) 1 1ip 2 2i 2 an equilibirum must also solve x1 i + x1 j = ! 1 i + ! 1 j max x 2fx 2R2 :p x1j = and x2j = (1 x p !2 g (x1j ) (x2j ) 1 p 1 ! 1j +p 2 ! 2j p1 p ! +p ! ) 1 1jp 2 2j 2 x2 i + x2 j = ! 2 i + ! 2 j We can …nd an equilibrium solving these six equations in six unknowns, but... ... Walrasian demand is homogeneous of degree zero, so if one multiplies all prices by some constant demand does not change. Thus, we can only pin down the relative prices p1 p2 . So we have six equations but …ve unknowns: are we in trouble? Walras’law to the rescue: if the market for one good clears (supply equals demand for that good) then the market for the other good does also clear (we will prove this later). One of the equations linearly depends on the others (it is redundant). Econ 2100 Fall 2015 Problem Set 9 Due 3 November, Monday, at the beginning of class 1. Suppose there are two goods and the consumer has a vNM utility index over consumption bundles of v(x) = f (x1 + x2 ) where f : R ! R is a strictly increasing function. Let p = (1; 3) and p0 = (3; 1). Let q = :5p + :5p0 = (2; 2). In Regime 1, there is a 1=2 probability that the realized price will be p and a 1=2 probability that the realized price will be p0 , so her ex-ante utility is :5 max f (x1 + x2 ) + :5 max f (x1 + x2 ): x2Bp;w x2Bp0 ;w In Regime 2, the price is q with certainty, so her ex-ante utility is max f (x1 + x2 ): x2Bq;w Prove that, for any strictly increasing f , the consumer is ex-ante better o¤ in Regime 1. 2. Kreps 6.4 3. Kreps 6.14. 4. Suppose a consumer’s utility for CDFs is U (F ) = R where V ar(F ) = (x 2 F ) dF F V ar(F ) is the variance of F . (a) Prove that if > 0 then the preference corresponding to corresponding to 0 . is more risk averse than the one (b) Prove that U violates expected utility. 5. Prove that in an exchange economy with I = 2, the core is the set of all Pareto optimal allocations that are individually rational. Hint: enumerate all possible coalitions. 6. Show that a feasible allocation (x; y) is Pareto optimal if and only if (u1 (x1 ); :::; uI (xI )) 2 U P . 7. Show that in an exchange economy where all consumption sets are convex and each consumer’s utility function is concave then the utility possibility set is convex. 8. Consider an exchange economy. Suppose that %i can be represented by a strictly increasing and continuous utility function ui (xi ) for each i = 1; :::; I. Prove that an allocation x is Pareto optimal if and only if it solves the following X max uj (xj ) subject to: ui (xi ) ui for all i 6= j, xi = ! and xi 0 for all i i for some choice of fui gi6=j . Hint: …rst prove the case I = 2 …rst, and then think about how to extend it. 1 9. For each Edgeworth Box economy below do the following: (i) …nd the Pareto optimal allocations, (ii) …nd the Pareto optimal allocation that gives consumers equal utility (ui = uj ), (iii) …nd the competitive equilibria, (iv) draw the Edgeworth box and clearly show the initial endowment, the equilibrium allocation and equilibrium prices, and the allocation you found in (ii). HINT: You can solve most of these without calculus (just draw pictures). (a) ! i = (1; 0) and ui (xi ) = p x1i x2i ; ! j = (0; 1) and uj (xj ) = p x1j x2j . (b) ! i = (0; 1) and ui (xi ) = 2x1i + x2i ; ! j = (1; 0) and uj (xj ) = x1j + 2x2j . (c) ! i = (2; 0) and ui (xi ) = x1i + x2i ; ! j = (0; 1) and uj (xj ) = x1j + x2j . (d) ! i = (1; 0) and ui (xi ) = min f2x1i ; x2i g; ! j = (0; 1) and uj (xj ) = min fx1j ; 2x2j g. (e) ! i = (1; 0) and ui (xi ) = min fx1i ; x2i g; ! j = (0; 2) and uj (xj ) = min fx1j ; x2j g. 2