Essential Microeconomics : EQUILIBRIUM AND EFFICIENCY WITH PRODUCTION Key ideas: Walrasian equilibrium, first and second welfare theorems

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1 Essential Microeconomics : EQUILIBRIUM AND EFFICIENCY WITH PRODUCTION Key ideas: Walrasian equilibrium, irst and second welare teorems A general model 2 First welare Teorem 7 Second welare teorem 9 Jon Riley October 9, 202

2 Essential Microeconomics -2- A general model Firms Firm, =,..., F as a set o easible production plans Y n and cooses a production vector y = ( y,..., y n ) Y. A production plan or te economy { y } F = is a plan or eac o te irms. Te aggregate production plan or te economy is te sum o all te individual plans y F = y. = Te set o all easible aggregate production plans, Y, is te aggregate production set. Jon Riley October 9, 202

3 Essential Microeconomics -3- Consumers Commodities are private, tat is, consumer as preerences over is own consumption vector = (,...,. n ) and not tose o oter consumers. Let X n be te consumption set o consumer, =,..., H. Tat is, preerences are deined over X. We assume tat consumer as an endowment vector ω X. A consumption allocation in tis economy { } H = is an allocation o consumption bundles X, =,..., H. Te aggregate consumption in te economy is te sum o te individual consumption vectors Sareoldings H =. Similarly te aggregate endowment is = ω H = ω. = Firms are owned by consumers. Consumer as an ownersip sare in irm o θ. Ownersip sares must sum to, tat is H θ =, =,..., F. = Jon Riley October 9, 202

4 Essential Microeconomics -4- Feasible allocation Given te aggregate demand, endowment ω and supply y, deine ecess demand z = ω y. An allocation is easible i aggregate ecess demand is negative z = ω y 0. Pareto eicient allocation A easible plan or te economy { ˆ } H, = { y ˆ } F = is Pareto eicient (PE) i tere is no oter easible plan { } H, = { y } F = tat is strictly preerred by at least one consumer and weakly preerred by all consumers. Jon Riley October 9, 202

5 Essential Microeconomics -5- Price taking Let p > 0 be te price vector. Consumers and irms are price takers. Tus i irm cooses te production plan y it as a proit o p y. Consumer receives er sare o te proit as a dividend payment. Te total dividend payment received by consumer is tereore θ p y. Adding te value o er endowment, consumer as a wealt W = p ω + θ p y. Se ten cooses a consumption bundle in er budget set { X p W }. Note tat er budget set is largest i wealt is maimized. Tus, as a sareolder, consumer s interests are best served by irm managers wo maimize proit. Jon Riley October 9, 202

6 Essential Microeconomics -6- Walrasian equilibrium Given te price vector p, let. Tat is y,,..., = F, be a production plan tat tat maimizes te proit o irm p y p y, or all y Y, =,..., F. (*) Also, let be a most preerred consumption plan in consumer s budget set. Tat is, U ( ) U ( ), or all suc tat p W (**) Te aggregate ecess demand vector is ten z = ω y. Deinition: Walrasian equilibrium prices Te price vector p 0 is a WE price vector i or some { y } satisying (*) and = { } H satisying = (**), te ecess demand vector is negative ( z 0). Moreover p = 0 or any market in wic ecess demand is strictly negative ( z j < 0). F j Jon Riley October 9, 202

7 Essential Microeconomics -7- Proposition 5.2-: First welare teorem I te preerences o eac consumer satisy te local non-satiation postulate, te Walrasian equilibrium allocation is Pareto eicient. Proo: We appeal to te Duality Lemma (se EM 2.3). Lemma 2.2-3: Duality Lemma I te local non-satiation assumption olds and * arg Ma{ U ( ) 0, p I}, ten * * U( ) U( ) p p and so * * arg Min{ p 0, U( ) U( )}. (5.2-) In addition, note tat since bundle maimizes utility over consumer s budget set, any strictly preerred must cost strictly more tan te equilibrium allocation. Tat is, F U ( ) > U ( ) p > p ω + p θ y. (5.2-2) = Consider any allocation { } H tat is Pareto preerred to te WE allocation = { } H = consumers, it ollows tat. Summing over H H F p > ( p ω + p θ y ). = = = Jon Riley October 9, 202

8 Essential Microeconomics -8- Since sares sum to, tis can be rewritten as ollows. p > p ω + p y. Tereore p ( ω y) > 0, Also y is proit maimizing over Y. Hence p y p y. Tereore p ( ω y) p ( ω y) > 0. (5.2-3) Suppose tat te allocation { } H, = { y } F = is easible. We will sow, by contradiction, tat no suc allocation eists. For easibility, ecess demands must be negative so z = ω y 0. Ten, since te Walrasian equilibrium price vector is positive, p ( ω y) 0. But tis contradicts (5.2-3). Tus tere is no Pareto preerred easible allocation. Q.E.D. Jon Riley October 9, 202

9 Essential Microeconomics -9- Decentralization Teorem Net we consider te second welare teorem wit production. We ollow te same line o argument as in te proo or te ecange economy. However we no longer assume tat consumption bundles are necessarily positive. Tus consumers may supply commodities (e.g. labor services) and production vectors ave bot positive components (outputs) and negative components (inputs.) Proposition 5.2-2: Second welare teorem wit production Let { ˆ H }, = { y ˆ } F = be a Pareto eicient allocation. Suppose (a) consumption vectors are private, (b) consumption sets X, =,..., Hare conve (c) utility unctions are continuous, quasi-concave and satisy te local non-satiation property (d) or eac tere is some X suc tat < ˆ, and (e) production sets Y, =,..., Fare conve and satisy te ree disposal property. Ten tere eists a price vector p > 0 suc tat ˆ p > p ˆ y Y p y p yˆ., =,,H Jon Riley October 9, 202

10 Essential Microeconomics -0- Proo: Let { ˆ }, = { y ˆ } F = H be a PE allocation. As in te pure ecange economy we introduce te indirect utility unction H V = Ma U U U ˆ H { } = = = H ( ) { ( ), ( ) ( ), 2,..., }. As argued or te ecange economy (see Capter 3), V( ) is quasi-concave. Also since local non-satiation property, so does V. U as te We also deine te set o easible aggregate consumption 2 vectors Z = { ω} +Y. Tis is depicted or a two commodity V ( ) V ( ˆ ) = U ( ˆ ) eample in wic some o te initial endowment o ˆ commodity is transormed into commodity 2. Z = { ω+y } +Y p = p ˆ Note tat since production sets are conve, so is te aggregate production set. ω Figure : Supporting yperplane Jon Riley October 9, 202

11 Essential Microeconomics -- Consider te ollowing optimization problem: Ma V Z. (5.2-4) { ( ) } { },{ y } Since { ˆ } H, = { y ˆ } F is Pareto eicient, ˆ = U ( ) is te igest easible utility or consumer, given tat no oter consumer is made worse o tan in te PE allocation. Tus { ˆ } H, = { y ˆ } F = solves tis maimization problem 2 ˆ Z = { ω} +Y V ( ) V ( ˆ) = U ( ˆ ) p = p ˆ ω Figure : Supporting yperplane and so V ( ˆ) = U ( ˆ ). Te solution is depicted in te igure. Given local non-satiation, te maimizing aggregate consumption vector ˆ = ω + yˆmust lie on te boundary o Z. Jon Riley October 9, 202

12 Essential Microeconomics -2- Step : Appeal to te Supporting Hyperplane Teorem Deine ˆ X { V ( ) V ( ˆ )}. Ten int Xˆ Z = so tat 2 ˆ = ω + yˆ is also on te boundary o ˆX. Since bot Z and V ( ) V ( ˆ) = U ( ˆ ) ˆX are conve it ollows tat ˆX Z is conve. Moreover ˆ ˆ ω yˆ = 0 lies on te boundary o ˆX Z. It ollows rom te Supporting Hyperplane Teorem tat tere Z = { ω} +Y p = p ˆ eists a supporting vector p 0 suc tat p ( ω y) p ( ˆ ω yˆ) = 0 or all Xˆ and y Y ω Figure : Supporting yperplane Setting = ˆ, (i) p y p yˆ or all y Y. Setting y = yˆ, (ii) p p ˆ or all Xˆ. See EM Proposition.- and Appendi B. Jon Riley October 9, 202

13 Essential Microeconomics -3- Step 2: Establis tat te vector p must be positive We suppose tat some components o p are negative and sow tat tis yields to a contradiction o (i). Deine δ = ( δ,..., δ n ) > 0 suc tat δ > 0 i and only i p < 0. Consider te vector y = yˆ δ. Given ree disposal ŷ δ Y. Also p y = p yˆ p δ > p yˆ since p δ < 0. But tis violates (i). Tus p > 0 ater all. j j Jon Riley October 9, 202

14 Essential Microeconomics -4- Step 3: Sow tat y ˆ is proit-maimizing. Net note tat we can rewrite conditions (i) and (ii) as ollows: (i) F y Y p y p yˆ and (ii) F = = H H V V ˆ p p ˆ ( ) ( ). = = j j Setting y = yˆ, j in (i) it ollows tat (i) y Y p y p yˆ, =,..., F. Tus y ˆ is proit maimizing or irm. Jon Riley October 9, 202

15 Essential Microeconomics -5- Step 4: Sow tat ˆ is utility-maimizing. i i Similarly, setting = ˆ, i in (ii) on te previous slide, it ollows tat 2 (ii) U ( ) U ( ˆ ) p p ˆ. Te inal step is to sow tat i U ( ) > U ( ˆ ) ten p > p ˆ. Suppose instead tat U ( ) > U ( ˆ ) and p = p ˆ. Consider te consumption vector = ( λ) + λ X is conve, X. Since < ˆ, p = ( λ) p + λp < ( λ) p + λp ˆ = p ˆ, were λ (0,). Since. X and Also, given te continuity o preerences, U ( ) > U ( ˆ ) or λ suiciently close to. But tis is impossible since condition (ii) is violated. QED 2 From te deinition o te indirect utility unction ( ) ( ˆ U U ), = 2,..., H. Also U ( ) = V ( ) V ( ˆ) = U ( ˆ ). Jon Riley October 9, 202

3.2 THE FUNDAMENTAL WELFARE THEOREMS

3.2 THE FUNDAMENTAL WELFARE THEOREMS Essential Microeconomics -1-3.2 THE FUNDMENTL WELFRE THEOREMS Walrasian Equilibrium 2 First welfare teorem 3 Second welfare teorem (conve, differentiable economy) 12 Te omotetic preference 2 2 economy