3.2 THE FUNDAMENTAL WELFARE THEOREMS

Transcription

1 Essential Microeconomics THE FUNDMENTL WELFRE THEOREMS Walrasian Equilibrium 2 First welfare teorem 3 Second welfare teorem (conve, differentiable economy) 12 Te omotetic preference 2 2 economy 21 Second welfare teorem (conve economy) 24 Jon Riley October 4, 2013

2 Essential Microeconomics -2- Review: Walrasian Equilibrium for an Ecange Economy Let p 0 be a price vector of tis ecange economy. In a WE eac consumer is a price taker. We write te set of consumers as H= {1,..., H}. We assume tat preferences are strictly conve so consumer as is a unique most preferred consumption vector, ( p, ). ( p, ) arg Ma{ U ( ) p p }. Total endowment vector: H Total or market demand: ( p) ( p, ) H Ecess demand: z( p) ( p). Definition: Market Clearing Prices Let zj ( p ) be te ecess demand for commodity j at te price vector p 0. Te market for commodity j clears if zj ( p) 0 and p jz ( p) 0. j Definition: Walrasian Equilibrium Te price vector p 0 is a WE price vector if all markets clear. Jon Riley October 4, 2013

3 Essential Microeconomics -3- First welfare teorem Proposition 3.1-2: First welfare teorem for an ecange economy If preferences satisfy local non-satiation, a WE allocation in an ecange economy is PE. Tis teorem formalizes dam Smit s idea tat consumers are better off if cartels are blocked from setting prices and instead, via competition, te invisible and of te marketplace acieves a better outcome. Note tat te statement of te proposition appears to be almost assumption free. However it depends critically on te assumption tat tere eists a WE allocation. Te proof appeals to te Duality Lemma (Section 2.2). Tat is, if te local non-satiation property olds, ten te utility-maimizing bundle is cost minimizing among all preferred consumption bundles. Duality Lemma arg Ma{ U ( ) p p } p Min{ p U ( ) U ( )}. Jon Riley October 4, 2013

4 Essential Microeconomics -4- Proposition 3.1-2: First welfare teorem for an ecange economy If preferences satisfy local non-satiation, a WE allocation in an ecange economy is PE. Proof: Let { } H p 0 be te WE price vector. Consider any allocation { } be a WE allocation for te ecange economy wit endowments { } H. Let H tat is Pareto-preferred to { } H. ecause none of te consumers can be worse off in te Pareto-preferred allocation, it follows from te Duality Lemma tat p p 0, H. ** Jon Riley October 4, 2013

5 Essential Microeconomics -5- Proposition 3.1-2: First welfare teorem for an ecange economy If preferences satisfy local non-satiation, a WE allocation in an ecange economy is PE. Proof: Let { } H p 0 be te WE price vector. Consider any allocation { } be a WE allocation for te ecange economy wit endowments { } H. Let H tat is Pareto-preferred to { } H. ecause none of te consumers can be worse off in te Pareto-preferred allocation, it follows from te Duality Lemma tat p p 0, H. Moreover at least one consumer must be strictly better off. Since te budget set, it follows tat is te most preferred allocation in p p 0, for some. * Jon Riley October 4, 2013

6 Essential Microeconomics -6- Proposition 3.1-2: First welfare teorem for an ecange economy If preferences satisfy local non-satiation, a WE allocation in an ecange economy is PE. Proof: Let { } H p 0 be te WE price vector. Consider any allocation { } be a WE allocation for te ecange economy wit endowments { } H. Let H tat is Pareto-preferred to { } H. ecause none of te consumers can be worse off in te Pareto-preferred allocation, it follows from te Duality Lemma tat p p 0, H. Moreover at least one consumer must be strictly better off. Since te budget set, it follows tat is te most preferred allocation in p p 0, for some. Summing over consumers, p ( ) 0. H H lso all markets clear in a Walrasian equilibrium. Terefore p ( p ) 0. H H Jon Riley October 4, 2013

7 Essential Microeconomics -7- Summarizing tese results, p ( ) 0 H H and p ( p ) 0. H H Terefore p 0 H H ecause p 0, it follows tat tere must be some commodity j suc tat 0. Tus all Pareto-preferred allocations are infeasible. Q.E.D. H j H j Jon Riley October 4, 2013

8 Essential Microeconomics -8- Second Welfare Teorem We now argue tat, as long as preferences are conve, any PE allocation is also a WE allocation for some redistribution of resources. Consider te PE allocation ˆ, ˆ were ˆ ˆ in Figure Te saded regions are te allocations were eiter le or ev is better off. Figure 3.1-7: PE allocation ** Jon Riley October 4, 2013

9 Essential Microeconomics -9- Second Welfare Teorem We now argue tat, as long as preferences are conve, any PE allocation is also a WE allocation for some redistribution of resources. Consider te PE allocation ˆ, ˆ were ˆ ˆ in Figure Te saded regions are te allocations were eiter le or ev is better off. If preferences are conve, eac of tese sets is conve so, by te Supporting Hyperplane Teorem, te tangent line Figure 3.1-7: PE allocation troug ˆ is a supporting line for bot X and X. * Jon Riley October 4, 2013

10 Essential Microeconomics -10- Second Welfare Teorem We now argue tat, as long as preferences are conve, any PE allocation is also a WE allocation for some redistribution of resources. Consider te PE allocation ˆ, ˆ were ˆ ˆ in Figure Te saded regions are te allocations were eiter le or ev is better off. If preferences are conve, eac of tese sets is conve so, by te Supporting Hyperplane Teorem, te tangent line Figure 3.1-7: PE allocation troug ˆ is a supporting line for bot X and X. If te endowments are ˆ ˆ, H eac individual maimizes by coosing is or er endowment. ecause demand equals supply for eac individual, all markets clear. Tus te price vector p is a WE price vector. Jon Riley October 4, 2013

11 Essential Microeconomics -11- Define te transfer payment T p ( ˆ ), H. H ecause ˆ te sum of tese transfers is zero so tis is a feasible redistribution of wealt. H Te budget constraint p p ˆ can be rewritten as follows: p p T. Ten given transfers T, H, te price vector p is a WE price vector. Jon Riley October 4, 2013

12 Essential Microeconomics -12- Proposition 3.1-3: Second welfare teorem for an ecange economy In an ecange economy wit endowments { }, H, suppose tat U ( ), is continuously differentiable, quasi-concave on n U and tat ( ) 0 were ˆ 0, H, can be supported by a price vector p 0., H. Ten any PE allocation { ˆ } H For epositional simplicity, consider a two person economy. Te generalization is direct. Te idea on te proof is to argue tat a PE allocation must be te solution to a maimization problem and ten sow tat te associated sadow prices are no-trade WE prices. * Jon Riley October 4, 2013

13 Essential Microeconomics -13- Proposition 3.1-3: Second welfare teorem for an ecange economy In an ecange economy wit endowments { }, H, suppose tat U ( ), is continuously differentiable, quasi-concave on n U and tat ( ) 0 were ˆ 0, H, can be supported by a price vector p 0., H. Ten any PE allocation { ˆ } H For epositional simplicity, consider a two person economy. Te generalization is direct. Te idea on te proof is to argue tat a PE allocation must be te solution to a maimization problem and ten sow tat te associated sadow prices are no-trade WE prices. Proof: If ˆ, ˆ is a PE allocation ten ˆ arg Ma{ U ( ), U ( ) U ( ˆ )}. (3.1-1), Class eercise: Eplain wy te assumptions imply tat te Kun-Tucker conditions are necessary conditions. Jon Riley October 4, 2013

14 Essential Microeconomics -14- Te Lagrangian for te optimization problem (3.1-1) is L U ( ) ( ) ( U ( ) U ( ˆ )). Kun-Tucker conditions. L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-2) L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-3) L ˆ ˆ 0, were ( ˆ ˆ ) 0. (3.1-4) *** Jon Riley October 4, 2013

15 Essential Microeconomics -15- Te Lagrangian for te optimization problem (3.1-1) is L U ( ) ( ) ( U ( ) U ( ˆ )). Kun-Tucker conditions. L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-2) L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-3) L ˆ ˆ 0, were ( ˆ ˆ ) 0. (3.1-4) U ecause 0 it follows from (3.1-2) tat 0. From (3.1-4) it ten follows tat ˆ ˆ 0. (3.1-5) ** Jon Riley October 4, 2013

16 Essential Microeconomics -16- Te Lagrangian for te optimization problem (3.1-1) is L U ( ) ( ) ( U ( ) U ( ˆ )). Kun-Tucker conditions. L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-2) L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-3) L ˆ ˆ 0, were ( ˆ ˆ ) 0. (3.1-4) U ecause 0 it follows from (3.1-2) tat 0. From (3.1-4) it ten follows tat ˆ ˆ 0. (3.1-5) U ecause ˆ 0 and 0 it follows from (3.1-3) tat 0. * Jon Riley October 4, 2013

17 Essential Microeconomics -17- Te Lagrangian for te optimization problem (3.1-1) is L U ( ) ( ) ( U ( ) U ( ˆ )). Kun-Tucker conditions. L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-2) L U ( ˆ ) 0 U, were ˆ ( ( ˆ ) ) 0. (3.1-3) L ˆ ˆ 0, were ( ˆ ˆ ) 0. (3.1-4) U ecause 0 it follows from (3.1-2) tat 0. From (3.1-4) it ten follows tat ˆ ˆ 0. (3.1-5) U ecause ˆ 0 and 0 it follows from (3.1-3) tat 0. Now consider an economy wit endowments ˆ ˆ, H and consider te price vector p. Jon Riley October 4, 2013

18 Essential Microeconomics -18- Consumer cooses arg Ma{ U ( ) ˆ }. Te FOC for tis optimization problem are L U ( ) 0 U, were ( ( ) ) 0. Moreover, because U () is quasi-concave te FOC is also sufficient. Coose 1 and 1/. Ten, appealing to (3.1-2) and (3.1-3), te FOC old at ˆ, H. ** Jon Riley October 4, 2013

19 Essential Microeconomics -19- Consumer cooses arg Ma{ U ( ) ˆ }. Te FOC for tis optimization problem are L U ( ) 0 U, were ( ( ) ) 0. Moreover, because U () is quasi-concave te FOC is also sufficient. Coose 1 and 1/. Ten, appealing to (3.1-2) and (3.1-3), te FOC old at ˆ, H. Tus at te price p vector is a WE price vector. no consumer wises to trade. Terefore supply equals demand and so te price * Jon Riley October 4, 2013

20 Essential Microeconomics -20- Consumer cooses arg Ma{ U ( ) ˆ }. Te FOC for tis optimization problem are L U ( ) 0 U, were ( ( ) ) 0. Moreover, because U () is quasi-concave te FOC is also sufficient. Coose 1 and 1/. Ten, appealing to (3.1-2) and (3.1-3), te FOC old at ˆ, H. Tus at te price p vector is a WE price vector. Finally define transfers no consumer wises to trade. Terefore supply equals demand and so te price T ( ˆ ). ppealing to (3.1-1), te sum of tese transfers is zero. Consumer s budget constraint wit tese transfers is T ˆ. Tus te PE allocation is acievable as a WE wit te appropriate transfer payments among consumers. Q.E.D. Jon Riley October 4, 2013

21 Essential Microeconomics -21- Homotetic Preferences Suppose tat te two individuals in te economy (le and ev) ave different conve and omotetic preferences. t te aggregate endowment, ( 1, 2), le as a stronger preference for commodity 1 tan ev. Tat is, le is willing to give up more units of commodity 2 tan ev in ecange for an additional unit of commodity 1. ssumption: Differing Intensity of preferences t te aggregate endowment, le as a stronger preference for commodity 1 tan ev. MRS U U U U (, ) MRS (, ) (3.1-6) / / Tis is depicted in Figure Figure 3.1-9: le as a stronger preference for commodity 1 Jon Riley October 4, 2013

22 Essential Microeconomics -22- We now eplore te implications of tis assumption for te PE allocations. Consider te PE allocation C in te interior of te Edgewort bo. Class eercises Commodity 2 D 1. Eplain wy all PE allocation lie below te diagonal. F Commodity 1 2. Eplain wy te allocations in te yellow and Fig : Pareto efficient allocations dark blue regions are not PE. Tus any oter PE allocation Cpreferred by le must lie above te line OD. ecause le s MRS is constant along tis line, te marginal rate of substitution at C will be iger, reflecting te greater influence of le s stronger preference for commodity 1. Ten C, C H, and MRS ( C) MRS ( C), H. Jon Riley October 4, 2013

23 Essential Microeconomics -23- We summarize tese results below. Proposition 3.1-4: Pareto Efficient llocations Wit Homotetic Preferences In te 2 2 ecange economy, suppose eac consumer as omotetic preferences. Suppose also tat at te aggregate endowment, consumer as a stronger preference for commodity 1. Ten at any interior efficient allocation, Moreover, along te locus of efficient allocations, as consumer s utility rises, te consumption ratio 2 / 1 and marginal rate of substitution of 1 for 2 of bot consumers rises. Note tat if le become relatively more wealty so tat te WE moves from C to C, te equilibrium MRS rises. Tus p1/ p 2, te equilibrium relative price of commodity 1 rises. Intuitively, since le as a stronger preference for commodity 1, te iger endowment, te more te relative price reflects is preferences. Jon Riley October 4, 2013

24 Essential Microeconomics -24- closer look at te second welfare teorem Te economy Commodities are private: Consumer H {1,..., H} as preferences over is own consumption vector ( 1,...,. n ) Consumption set: Preferences are defined over te conve set n X. Endowments: Consumer as an endowment vector. X Consumption allocation: { } were, H X H. ggregate consumption: H. ggregate endowment is H. Ecess demand: z z Jon Riley October 4, 2013

25 Essential Microeconomics -25- Feasible llocation: n allocation { } satisfying z 0. H *** Jon Riley October 4, 2013

26 Essential Microeconomics -26- Feasible llocation: n allocation { } satisfying z 0. H Pareto-Efficient llocation feasible allocation { ˆ } H, is Pareto-efficient if tere is no oter feasible plan tat is strictly preferred by at least one consumer and weakly preferred by all consumers. ** Jon Riley October 4, 2013

27 Essential Microeconomics -27- Feasible llocation: n allocation { } satisfying z 0. H Pareto-Efficient llocation feasible allocation { ˆ } H, is Pareto-efficient if tere is no oter feasible plan tat is strictly preferred by at least one consumer and weakly preferred by all consumers. Price-Taking Let p 0 be te price vector. Consumers are price takers. Consumer as an endowment. Se cooses a consumption bundle in er budget set { X p p }. * Jon Riley October 4, 2013

28 Essential Microeconomics -28- Feasible llocation: n allocation { } satisfying z 0. H Pareto-Efficient llocation feasible allocation { ˆ } H, is Pareto-efficient if tere is no oter feasible plan tat is strictly preferred by at least one consumer and weakly preferred by all consumers. Price-Taking Let p 0 be te price vector. Consumers are price takers. Consumer as an endowment. Se cooses a consumption bundle in er budget set { X p p }. Walrasian Equilibrium Eac consumer cooses te most preferred consumption plan in er budget set. Tat is, U ( ) U ( ), for all suc tat p p Let. be te total consumption of te consumers. Ecess demand is ten z Definition: Walrasian equilibrium prices Te price vector p 0 is a Walrasian equilibrium price vector if tere is no market in ecess demand ( z 0) and p 0 for any market in ecess supply ( z 0). j j Jon Riley October 4, 2013

29 Essential Microeconomics -29- Second welfare teorem Te earlier proof appealed to te Kun-Tucker conditions. s we ave seen, tese follow from te Supporting Hyperplane Teorem. We now dispense wit differentiability assumption and appeal directly to te Supporting Hyperplane Teorem. If { ˆ } H is PE it must solve te following optimization problem. Ma U U U ˆ H 1 1 n { ( ) ( ) ( ), 2,...,, ( ) 0, } { } H H Consider te optimization problem wen te aggregate supply is. PE( ): Ma U U U ˆ H 1 1 n { ( ) ( ) ( ), 2,...,, ( ) 0, } { } H H Define H V Ma U U U ˆ H H { } 1 1 ( ) { ( ) ( ) ( ), 2,...,, 0} (3.2-1) Note tat { ˆ } H solves te optimization problem PE ( ). Jon Riley October 4, 2013

30 Essential Microeconomics -30- Lemma 3.2-1: Quasi-concavity of V 1 () If U, H is quasi-concave ten so is te indirect utility function V 1 (). Proof: Follows directly from te conveity of te upper contour sets (Class eercise.) Lemma: Suppose tat for ˆ in consumer s consumption set, X n te vector p 0 is supporting at ˆ for te upper contour set CU ( ˆ ) { U ( ) U ( ˆ ) were U () is strictly increasing, tat is, Ten U ( ) U ( ˆ ) p p ˆ (1) pˆ 0 and U ( ) U ( ˆ ) p p ˆ Proof: Suppose tat for some, U ( ) U ( ˆ ) and p p ˆ. (2) Since p ˆ 0, it follows tat for (0,1) p p ˆ. lso, since U () is continuous it follows tat U ( ) U ( ˆ ) for sufficiently close to 1. ut tis contradicts (1). Terefore (2) is false. It te follows from (1) tat U ( ) U ( ˆ ) p p ˆ. QED Jon Riley October 4, 2013

31 Essential Microeconomics -31- Second welfare teorem Te earlier proof appealed to te Kun-Tucker conditions. s we ave seen, tese follow from te Supporting Hyperplane Teorem. We now dispense wit differentiability assumption and appeal directly to te Supporting Hyperplane Teorem. Jon Riley October 4, 2013

32 Essential Microeconomics -32- For any aggregate supply and allocation { ˆ } H 2 to all consumers oter tan consumer 1, let V 1 () be te maimum utility of consumer 1. Tat is ˆ { } H H V ( ) Ma U ( ) U ( ) U ( ), 1, 0 If te utility functions are quasi-concave it follows tat tat V 1 () is quasi-concave as depicted below. lso V 1 () is strictly increasing as any increase in can be given to consumer 1. Te conveity of te upper contour sets of V 1 () plays te critical role in te proof of te second welfare teorem. Jon Riley October 4, 2013

33 Essential Microeconomics -33- Second Welfare Teorem for an Ecange Economy Consumer H as a consumption set, X n. Te aggregate endowment is strictly positive. Utility functions U( ), H are continuous, quasi-concave and strictly increasing. Let { ˆ } H be a PE allocation in wic ˆ 0, H. Ten tere eists a strictly positive price vector p suc tat U ( ) U ( ˆ ) p p ˆ Proof: Since { ˆ } H is a PE allocation, it solves te following optimization problem ˆ { } H H V ( ) Ma U ( ) U ( ) U ( ), 1, 0. Moreover, because U 1 () is strictly increasing, ˆ ˆ. H ecause is on te boundary of te set { V1( ) V1( )}, it follows from te Supporting Hyperplane Teorem tat tere eists a vector p 0, suc tat V1( ) V1( ) p p (3) Jon Riley October 4, 2013

34 Essential Microeconomics -34- We ave establised tat V1( ) V1( ) p p (3) We now argue tat te vector p must be positive. If not, for some j, p j 0. Define (0,.., j,0,...,0) 0 and ˆ. Since V 1 () is strictly increasing, V1( ) V1( ) and p p. ut tis contradicts (3), so p 0 after all. From (3) and te definition of te indirect utility function H U ( ) U ( ˆ ), H p p p. Since ˆ it follows tat pˆ p. Terefore U ( ) U ( ˆ ), H p p ˆ. H Setting k ˆ k, k, we may conclude tat for consumer, U ( ) U ( ˆ ) p p ˆ. (4) H Jon Riley October 4, 2013

35 Essential Microeconomics -35- We now sow tat p 0. Since ˆ 0 and p 0 it follows tat p ˆ p ˆ 0 p. H Ten for some, p ˆ 0. ppealing to te Lemma, for tis consumer U ( ) U ( ˆ ) p p ˆ If p j 0, define (0,.., j,0,...,0) 0. Since U () is increasing, U ( ˆ ) U ( ˆ ) and p( ˆ ) p ˆ. ut tis contradicts (4). Tus p must be strictly positive. Since ˆ 0 it follows tat for all H, p ˆ 0. gain appealing to te Lemma U ( ) U ( ˆ ) p p ˆ QED Jon Riley October 4, 2013