3. THE EXCHANGE ECONOMY
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- Godwin Cameron
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1 Essential Microeconomics THE EXCHNGE ECONOMY Pareto efficient allocations 2 Edgewort box analysis 5 Market clearing prices 13 Walrasian Equilibrium 16 Equilibrium and Efficiency 22 First welfare teorem 24 Second welfare teorem (convex, differentiable economy) 28 Te omotetic preference 2 2 economy 41 Second welfare teorem (convex economy) 49 Jon Riley
2 Essential Microeconomics -2- Private goods excange economy Consumer (or ouseold), = 1,..., H as strictly increasing preferences over n X + =. We assume tat te basic preference axioms are satisfied so tat tese are represented by a continuous utility function U (). 1 Were it is elpful we will assume tat U is continuously differentiable ( U () ). Endowments: Te initial allocation of commodities is { ω } = 1. H Feasible allocations: Te final allocation { x } H is feasible if = 1 x = 1 = 1 H H ω * Jon Riley
3 Essential Microeconomics -3- Private goods excange economy Consumer (or ouseold), = 1,..., H as strictly increasing preferences over n X + =. We assume tat te basic preference axioms are satisfied so tat tese are represented by a continuous utility function U (). 1 Were it is elpful we will assume tat U is continuously differentiable ( U () ). Endowments: Te initial allocation of commodities is { ω } H = 1. Feasible allocations: Te final allocation { x } H is feasible if = 1 x = 1 = 1 Pareto-efficient allocations H H n allocation { x ˆ } H = 1 is Pareto efficient if tere is no oter feasible allocation in wic at least one consumer is strictly better off and no consumer is worse off. Consider an alternative allocation { x } H in wic consumers 2,,H = 1 are all at least as well off. Tat is, U ( x ) U ( xˆ ) Uˆ ω Ten U ( x ) U ( xˆ ) and so { x ˆ } H = 1 solves te following maximization problem. H H ˆ arg { ( ) ( ) ˆ x = Max U x U x U, = 2,..., H, } x ω { x } = 1 = 1 Jon Riley
4 Essential Microeconomics -4- Two commodity 2 consumer case (lex and ev) For te special 2 2 case, lex and ev must sare te aggregate endowment ω = ( ω1, ω2). Let x ˆ be te allocation to ev and let ˆ be te set of allocations tat ev prefers over x ˆ. Tis is depicted in Figure For any te allocation to lex is x x ˆ, = ω x. Tus te best possible allocation to lex tat leaves ev no worse off is lex s utility maximizing allocation in ˆ. Figure 3.1-2: ev s upper contour set Jon Riley
5 Essential Microeconomics -5- Edgewort box diagram Since preferences are strictly increasing a PE allocation uses all te endowment x + x = ω = ω + ω In te diagram te sum of te two consumption vectors is te vector ( ω1, ω 2), tat is, te rigt and corner of te Edgewort box. Figure 3.1-3: Edgewort box Diagram Jon Riley
6 Essential Microeconomics -6- For Pareto-efficiency, tere can be no mutually preferred alternative. One PE allocation is depicted in Figure s long as an allocation { xˆ, xˆ = ω xˆ } is in te interior of te Edgewort box, a necessary condition for te allocation to be PE is tat te slopes of te two indifference curves must be equal. Tus te grap of te PE allocations is te set of allocations to lex (and ence ev) satisfying Figure 3.1-4: PE allocations wit identical CES preferences MRS U U ( xˆ ) ( xˆ ) 1 1 ( xˆ ) = = U U ( xˆ ) ( xˆ ) 2 2, were xˆ + xˆ = ω. Jon Riley
7 Essential Microeconomics -7- Example: Identical CES Preferences If preferences are CES wit elasticity of substitution σ, lex and ev ave a 1/ σ x 2 MRS ( x ) = k x 1 For a PE allocation in te interior of te Edgewort box, te indifference curves of te two consumers must ave te same slope, tat is,. 1/ σ 1/ σ 2 x2 = 1 x1 x x x2 x2 ence =. x x 1 1 Figure 3.1-4: PE allocations wit identical CES preferences * Jon Riley
8 Essential Microeconomics -8- Example: Identical CES Preferences If preferences are CES wit elasticity of substitution σ, lex and ev ave a 1/ σ x 2 MRS ( x ) = k x 1 For a PE allocation in te interior of te Edgewort box, te indifference curves of te two consumers must ave te same slope, tat is,. 1/ σ 1/ σ 2 x2 = 1 x1 x x x2 x2 ence =. x x 1 1 Figure 3.1-4: PE allocations wit identical CES preferences a1 b1 a1+ b1 Ratio Rule: = = a b a + b a1 b1 Proof: If = = k ten a1 = ka2 and b 1 = kb 2 and so a1+ b1 = k( a2 + b2). a b a1+ b1 Hence a + b = k. Jon Riley
9 Essential Microeconomics -9- ppealing to te Ratio Rule and ten setting demand equal to supply, x x x + x x x x x ω. ω = = = 1 1 a Tus, in a PE allocation eac consumer is allocated a fraction of te aggregate endowment. It follows tat for eac consumer te marginal rate of substitution is ω 2 MRS ( xˆ ) = k ω1 1/ σ. (3.1-1) Te PE allocations are depicted in Figure Figure 3.1-4: PE allocations wit identical CES preferences Jon Riley
10 Essential Microeconomics -10- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange economy. In a WE eac consumer is a price taker. We write te set of consumers as H= {1,..., H}. *** Jon Riley
11 Essential Microeconomics -11- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange 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 convex so consumer as is a unique most preferred consumption vector, x ( p, ω ). x ( p, ω ) = arg Max{ U ( x) p x p ω }. x ** Jon Riley
12 Essential Microeconomics -12- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange 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 convex so consumer as is a unique most preferred consumption vector, x ( p, ω ). x ( p, ω ) = arg Max{ U ( x) p x p ω }. Total endowment vector: x ω = H ω Total or market demand: ( ) x p = x ( p, ω ) H Excess demand: z( p) = x( p) ω. * Jon Riley
13 Essential Microeconomics -13- Walrasian Equilibrium for an Excange Economy Let p 0 be a price vector of tis excange 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 convex so consumer as is a unique most preferred consumption vector, x ( p, ω ). x ( p, ω ) = arg Max{ U ( x) p x p ω }. Total endowment vector: x ω = H ω Total or market demand: ( ) x p = x ( p, ω ) H Excess demand: z( p) = x( p) ω. Definition: Market Clearing Prices Let zj ( p ) be te excess demand for commodity j at te price vector p 0. Te market for commodity j clears if zj ( p) 0 and pz j j( p ) = 0. Jon Riley
14 Essential Microeconomics -14- Walras Law: If preferences satisfy local non-satiation and all markets but one clear ten te remaining market must also clear. If preferences satisfy local non-satiation, ten a consumer must spend all of is income. Wy is tis? ** Jon Riley
15 Essential Microeconomics -15- Walras Law: If preferences satisfy local non-satiation and all markets but one clear ten te remaining market must also clear. If preferences satisfy local non-satiation, ten a consumer must spend all of is income. Wy is tis? Ten for any price vector p te market value of excess demands must be zero. p z( p) = p ( x ω) = p ( ( x ω )) = ( p x p ω ). H ecause all consumers spend teir entire wealt te rigt and expression is zero. Hence n p z( p) = p z ( p) + p z ( p) = 0. i i j j j= 1 j i H * Jon Riley
16 Essential Microeconomics -16- Walras Law: If preferences satisfy local non-satiation and all markets but one clear ten te remaining market must also clear. If preferences satisfy local non-satiation, ten a consumer must spend all of is income. Wy is tis? Ten for any price vector p te market value of excess demands must be zero. p z( p) = p ( x ω) = p ( ( x ω )) = ( p x p ω ). H ecause all consumers spend teir entire wealt te rigt and expression is zero. Hence n p z( p) = p z ( p) + p z ( p) = 0. i i j j j= 1 j i Terefore if all markets but market i clear ten market i must clear as well. H Definition: Walrasian Equilibrium price vector Te price vector p 0is a WE price vector if all markets clear. Jon Riley
17 Essential Microeconomics -17- Edgewort box example In a Walrasian equilibrium consumers coose te best point in teir budget sest given a price vector p = ( p1, p2). In te figure tese are te ligtly saded orange and green triangles. N Figure 3.1-5: Excess supply of commodity 1 ** Jon Riley
18 Essential Microeconomics -18- Edgewort box example In a Walrasian equilibrium consumers coose te best point in teir budget set given a price vector p = ( p1, p2). In te figure tese are te ligtly saded orange and green triangles. It is very important to note tat consumers consider only teir budget sets. In te case depicted, bot of tese budget sets extend beyond te boundary of te N Edgewort box (te set of feasible allocations). Figure 3.1-5: Excess supply of commodity 1 * Jon Riley
19 Essential Microeconomics -19- Edgewort box example In a Walrasian equilibrium consumers coose te best point in teir budget set given a price vector p = ( p1, p2). In te figure tese are te ligtly saded orange and green triangles. It is very important to note tat consumers consider only teir budget sets. In te case depicted, bot of tese budget sets extend beyond te boundary of te N Edgewort box (te set of feasible allocations). Te eavily saded triangles indicate te desired trades of te two consumers. s depicted, lex wants to trade from te endowment point N to is most preferred desired consumption x, wereas ev wises to trade from N to x. Tus, tere is excess supply of commodity 1. Figure 3.1-5: Excess supply of commodity 1 Jon Riley
20 Essential Microeconomics -20- y lowering te price of commodity 1 (relative to commodity 2) te budget line becomes less steep until eventually supply equals demand. Te Walrasian equilibrium E is depicted in Figure N Figure 3.1-6: Walrasian equilibrium Jon Riley
21 Essential Microeconomics -21- Class Exercise: Wic (if any) of tese figures depicts a Walrasian equilibrium? In te left figure te budget line is tangential to ev s indifference curve at x ˆ. In te rigt-and figure te budget line is tangential to lex s indifference curve. Jon Riley
22 Essential Microeconomics -22- Equilibrium and Efficiency In Figure te WE allocation is in te interior of te Edgewort box. Tus te marginal rates of substitution must bot be equal to te price ratio: U U ( x ) ( x ) p MRS x MRS x ( x ) ( x ) ( ) = = = = ( ) U p2 U 2 2 N Since te MRS are equal, it follows tat te WE allocation must be PE. Figure 3.1-6: Walrasian equilibrium * Jon Riley
23 Essential Microeconomics -23- Equilibrium and Efficiency In Figure te WE allocation is in te interior of te Edgewort box. Tus te marginal rates of substitution must bot be equal to te price ratio: U U ( x ) ( x ) p MRS x MRS x ( x ) ( x ) ( ) = = = = ( ) U p2 U 2 2 N Since te MRS are equal, it follows tat te WE allocation must be PE. Figure 3.1-6: Walrasian equilibrium To prove tat tis result olds very generally, we will appeal to te Duality Lemma (Section 2.2). Tat is, if te local non-satiation property olds, ten te utility-maximizing bundle is cost minimizing among all preferred consumption bundles. Duality Lemma x = arg Max{ U ( x ) p x p ω } p x = Min{ p x U ( x ) U ( x )}. x x Jon Riley
24 Essential Microeconomics -24- Proposition 3.1-2: First welfare teorem for an excange economy If preferences satisfy local non-satiation, a WE allocation in an excange economy is PE. Proof: Let { } x H p 0 be te WE price vector. Consider any allocation { } be a WE allocation for te excange economy wit endowments { ω } H. Let x H tat is Pareto-preferred to { x }. H ecause none of te consumers can be worse off in te Pareto-preferred allocation, it follows from te Duality Lemma tat p x p x 0, H. ** Jon Riley
25 Essential Microeconomics -25- Proposition 3.1-2: First welfare teorem for an excange economy If preferences satisfy local non-satiation, a WE allocation in an excange economy is PE. Proof: Let { } x H p 0 be te WE price vector. Consider any allocation { } be a WE allocation for te excange economy wit endowments { ω } H. Let x H tat is Pareto-preferred to { x }. H ecause none of te consumers can be worse off in te Pareto-preferred allocation, it follows from te Duality Lemma tat p x p x 0, H. Moreover at least one consumer must be strictly better off. Since te budget set, it follows tat p x p x > 0, for some. x is te most preferred allocation in * Jon Riley
26 Essential Microeconomics -26- Proposition 3.1-2: First welfare teorem for an excange economy If preferences satisfy local non-satiation, a WE allocation in an excange economy is PE. Proof: Let { } x H p 0 be te WE price vector. Consider any allocation { } be a WE allocation for te excange economy wit endowments { ω } H. Let x H tat is Pareto-preferred to { x }. H ecause none of te consumers can be worse off in te Pareto-preferred allocation, it follows from te Duality Lemma tat p x p x 0, H. Moreover at least one consumer must be strictly better off. Since te budget set, it follows tat p x p x > 0, for some. x is te most preferred allocation in Summing over consumers, p ( x x ) > 0. H H lso all markets clear in a Walrasian equilibrium. Terefore p ( x p ω ) = 0. H H Jon Riley
27 Essential Microeconomics -27- Combining tese results, p x ω > 0 H H ecause p 0, it follows tat tere must be some commodity j suc tat x ω > 0. Tus all Pareto-preferred allocations are infeasible. Q.E.D. H j H j Jon Riley
28 Essential Microeconomics -28- Second Welfare Teorem We now argue tat, as long as preferences are convex, any PE allocation is also a WE allocation for some redistribution of resources. Consider te PE allocation ˆ x, x ˆ were xˆ + xˆ = ω in Figure Te saded regions are te allocations were eiter lex or ev is better off. Figure 3.1-7: PE allocation ** Jon Riley
29 Essential Microeconomics -29- Second Welfare Teorem We now argue tat, as long as preferences are convex, any PE allocation is also a WE allocation for some redistribution of resources. Consider te PE allocation ˆ x, x ˆ were xˆ + xˆ = ω in Figure Te saded regions are te allocations were eiter lex or ev is better off. If preferences are convex, eac of tese sets is convex so, by te Separating Hyperplane Teorem, we can draw a line p x = p xˆ troug x ˆ separating te two sets. Figure 3.1-7: PE allocation * Jon Riley
30 Essential Microeconomics -30- Second Welfare Teorem We now argue tat, as long as preferences are convex, any PE allocation is also a WE allocation for some redistribution of resources. Consider te PE allocation ˆ x, x ˆ were xˆ + xˆ = ω in Figure Te saded regions are te allocations were eiter lex or ev is better off. If preferences are convex, eac of tese sets is convex so, by te Separating Hyperplane Teorem, we can draw a line p x = p xˆ troug x ˆ separating te two sets. Figure 3.1-7: PE allocation If te endowments are ˆ ω = xˆ, H eac individual maximizes 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
31 Essential Microeconomics -31- Define te transfer payment T = p ( xˆ ω ), H. H ecause xˆ = ω te sum of tese transfers is zero so tis is a feasible redistribution of wealt. H Te budget constraint p x p xˆ can be rewritten as follows: p x p T ω +. Ten given transfers T, H, te price vector p is a WE price vector. Jon Riley
32 Essential Microeconomics -32- Proposition 3.1-3: Second welfare teorem for an excange economy In an excange economy wit endowments { ω }, H, suppose tat U ( x ), is continuously differentiable, quasi concave on n U + and tat ( ) 0 x >> were xˆ 0, H, can be supported by a price vector p > 0., H. Ten any PE allocation { xˆ } H For expositional 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 maximization problem and ten sow tat te associated sadow prices are no-trade WE prices. * Jon Riley
33 Essential Microeconomics -33- Proposition 3.1-3: Second welfare teorem for an excange economy In an excange economy wit endowments { ω }, H, suppose tat U ( x ), is continuously differentiable, quasi concave on n U + and tat ( ) 0 x >> were xˆ 0, H, can be supported by a price vector p > 0., H. Ten any PE allocation { xˆ } H For expositional 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 maximization problem and ten sow tat te associated sadow prices are no-trade WE prices. Proof: If xˆ, x ˆ is a PE allocation ten xˆ = arg Max{ U ( x ) x + x ω + ω, U ( x ) U ( xˆ )}. (3.1-2) x, x Class exercise: Explain wy te assumptions imply tat te Kun-Tucker conditions are necessary conditions. Jon Riley
34 Essential Microeconomics -34- Te Lagrangian for te optimization problem (3.1-2) is L = U ( x ) + νω ( + ω x x ) + μ( U ( x ) U ( xˆ )). Kun-Tucker conditions. L U = ( ˆ ) 0 x ν U, were xˆ ( ( xˆ ) ν ) = 0. (3.1-3) L U = μ ( ˆ ) 0 x ν U, were xˆ ( μ ( xˆ ) ν) = 0. (3.1-4) L = ω + ω xˆ xˆ 0, were ν ( ω + ω xˆ xˆ ) = 0. (3.1-5) ν *** Jon Riley
35 Essential Microeconomics -35- Te Lagrangian for te optimization problem (3.1-2) is L = U ( x ) + νω ( + ω x x ) + μ( U ( x ) U ( xˆ )). Kun-Tucker conditions. L U = ( ˆ ) 0 x ν U, were xˆ ( ( xˆ ) ν ) = 0. (3.1-3) L U = μ ( ˆ ) 0 x ν U, were xˆ ( μ ( xˆ ) ν) = 0. (3.1-4) L = ω + ω xˆ xˆ 0, were ν ( ω + ω xˆ xˆ ) = 0. (3.1-5) ν U ecause >> 0 it follows from (3.1-3) tat ν >> 0. From (3.1-5) it ten follows tat ω + ω xˆ xˆ = 0. (3.1-6) ** Jon Riley
36 Essential Microeconomics -36- Te Lagrangian for te optimization problem (3.1-2) is L = U ( x ) + νω ( + ω x x ) + μ( U ( x ) U ( xˆ )). Kun-Tucker conditions. L U = ( ˆ ) 0 x ν U, were xˆ ( ( xˆ ) ν ) = 0. (3.1-3) L U = μ ( ˆ ) 0 x ν U, were xˆ ( μ ( xˆ ) ν) = 0. (3.1-4) L = ω + ω xˆ xˆ 0, were ν ( ω + ω xˆ xˆ ) = 0. (3.1-5) ν U ecause >> 0 it follows from (3.1-3) tat ν >> 0. From (3.1-5) it ten follows tat ω + ω xˆ xˆ = 0. (3.1-6) U ecause x ˆ > 0 and >> 0 it follows from (3.1-4) tat μ > 0. * Jon Riley
37 Essential Microeconomics -37- Te Lagrangian for te optimization problem (3.1-2) is L = U ( x ) + νω ( + ω x x ) + μ( U ( x ) U ( xˆ )). Kun-Tucker conditions. L U = ( ˆ ) 0 x ν U, were xˆ ( ( xˆ ) ν ) = 0. (3.1-3) L U = μ ( ˆ ) 0 x ν U, were xˆ ( μ ( xˆ ) ν) = 0. (3.1-4) L = ω + ω xˆ xˆ 0, were ν ( ω + ω xˆ xˆ ) = 0. (3.1-5) ν U ecause >> 0 it follows from (3.1-3) tat ν >> 0. From (3.1-5) it ten follows tat ω + ω xˆ xˆ = 0. (3.1-6) U ecause x ˆ > 0 and >> 0 it follows from (3.1-4) tat μ > 0. Now consider an economy wit endowments ˆ ω = xˆ, H and consider te price vector p = ν. Jon Riley
38 Essential Microeconomics -38- Consumer cooses x = arg Max{ U ( x ) ν x ν xˆ }. x Te FOC for tis optimization problem are L U = ( ) 0 x λν U, were x ( ( x ) λν) = 0. Moreover, because U () is quasi-concave te FOC is also sufficient. Coose λ = 1 and λ = 1/ μ. Ten, appealing to (3.1-3) and (3.1-4), te FOC old at x = xˆ, H. ** Jon Riley
39 Essential Microeconomics -39- Consumer cooses x = arg Max{ U ( x ) ν x ν xˆ }. x Te FOC for tis optimization problem are L U = ( ) 0 x λν U, were x ( ( x ) λν) = 0. Moreover, because U () is quasi-concave te FOC is also sufficient. Coose λ = 1 and λ = 1/ μ. Ten, appealing to (3.1-3) and (3.1-4), te FOC old at x = xˆ, H. Tus at te price p vector is a WE price vector. = ν no consumer wises to trade. Terefore supply equals demand and so te price * Jon Riley
40 Essential Microeconomics -40- Consumer cooses x = arg Max{ U ( x ) ν x ν xˆ }. x Te FOC for tis optimization problem are L U = ( ) 0 x λν U, were x ( ( x ) λν) = 0. Moreover, because U () is quasi-concave te FOC is also sufficient. Coose λ = 1 and λ = 1/ μ. Ten, appealing to (3.1-3) and (3.1-4), te FOC old at x = xˆ, H. Tus at te price p vector is a WE price vector. = ν no consumer wises to trade. Terefore supply equals demand and so te price Finally define transfers T = ν ( xˆ ω ). ppealing to (3.1-2), te sum of tese transfers is zero. Consumer s budget constraint wit tese transfers is ν x ν ω + T = ν xˆ. Tus te PE allocation is acievable as a WE wit te appropriate transfer payments among consumers. Q.E.D. Jon Riley
41 Essential Microeconomics -41- Homotetic Preferences Suppose tat te two individuals in te economy (lex and ev) ave different convex and omotetic preferences. t te aggregate endowment, ( ω1, ω 2), lex as a stronger preference for commodity 1 tan ev. Tat is, lex is willing to give up more units of commodity 2 tan ev in excange for an additional unit of commodity 1. ssumption: Differing Intensity of preferences t te aggregate endowment, lex as a stronger preference for commodity 1 tan ev. MRS U U U U ( ω, ω ) = > = MRS ( ω, ω ) (3.1-7) / / Tis is depicted in Figure Figure 3.1-9: lex as a stronger preference for commodity 1 Jon Riley
42 Essential Microeconomics -42- We now explore te implications of tis assumption for te PE allocations. Consider te PE allocation C in te interior of te Edgewort box. Class exercises Commodity 2 D 1. Explain wy all PE allocation lie below te diagonal. F Commodity 1 2. Explain wy te allocations in te yellow and Fig : Pareto efficient allocations dark blue regions are not PE. Tus any oter PE allocation C preferred by lex must lie above te line OD. ecause lex s MRS is constant along tis line, te marginal rate of substitution at C will be iger, reflecting te greater influence of lex s stronger preference for commodity 1. Ten x x < x, H, and MRS ( C ) > MRS ( C), H. C C x1 Jon Riley
43 Essential Microeconomics -43- We summarize tese results below. Proposition 3.1-4: Pareto Efficient llocations Wit Homotetic Preferences In te 2 2 excange 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, x x x < x1 Moreover, along te locus of efficient allocations, as consumer s utility rises, te consumption ratio x2 / x 1 and marginal rate of substitution of 1 x for x 2 of bot consumers rises. Note tat if lex 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 lex as a stronger preference for commodity 1, te iger endowment, te more te relative price reflects is preferences. Jon Riley
44 Essential Microeconomics -44- closer look at te second welfare teorem Te economy Commodities are private: Consumer H = {1,..., H} as preferences over is own consumption vector x = ( x 1,...,. x n ) Consumption set: Preferences are defined over te convex set n X. Endowments: Consumer as an endowment vector ω X. Consumption allocation: { x } were, H x X H. ggregate consumption: x = H x. ggregate endowment is ω = H ω. Excess demand: z = x ω z Jon Riley
45 Essential Microeconomics -45- Feasible llocation: n allocation { x } satisfying H z = x ω 0. *** Jon Riley
46 Essential Microeconomics -46- Feasible llocation: n allocation { x } satisfying H z = x ω 0. Pareto-Efficient llocation feasible allocation { xˆ }, 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
47 Essential Microeconomics -47- Feasible llocation: n allocation { x } satisfying H z = x ω 0. Pareto-Efficient llocation feasible allocation { xˆ }, 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 ω. Se cooses a consumption bundle x X p x p ω. x in er budget set { } * Jon Riley
48 Essential Microeconomics -48- Feasible llocation: n allocation { x } satisfying H z = x ω 0. Pareto-Efficient llocation feasible allocation { xˆ }, 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 ω. Se cooses a consumption bundle x X p x p ω. x in er budget set { } Walrasian Equilibrium Eac consumer cooses te most preferred consumption plan in er budget set. Tat is, U ( x ) U ( x ), for all x suc tat p x p ω Let x =. = x be te total consumption of te consumers. Excess demand is ten z x ω Definition: Walrasian equilibrium prices Te price vector p 0 is a Walrasian equilibrium price vector if tere is no market in excess demand ( z 0) and p = 0 for any market in excess supply ( z < 0). j j Jon Riley
49 Essential Microeconomics -49- 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 { xˆ } H is PE it { xˆ } H must solve te following optimization problem. ˆ 1 1 n Max{ U ( x ) U ( x ) U ( x ), = 2,..., H, ( ω x ) 0, x } + { x } H H Consider te optimization problem wen te aggregate supply is x. PE( x: ) ˆ 1 1 n Max{ U ( x ) U ( x ) U ( x ), = 2,..., H, ( ω x ) 0, x } + { x } H H Define H V x Max U x U x U xˆ H x x H { x } = 1 = 1 ( ) = { ( ) ( ) ( ), = 2,...,, 0} (3.2-1) Note tat { ˆ } x H solves te optimization problem PE ( ω ). Jon Riley
50 Essential Microeconomics -50- Second welfare teorem Te earlier proof appealed to te Kun-Tucker conditions. s w ave seen, tese follow from te Supporting Hyperplane Teorem. We now dispense wit differentiability assumption and appeal directly to te Supporting Hyperplane Teorem. If { xˆ } H is PE it { xˆ } H must solve te following optimization problem. ˆ 1 1 n Max{ U ( x ) U ( x ) U ( x ), = 2,..., H, ( ω x ) 0, x } + { x } H H * Jon Riley
51 Essential Microeconomics -51- 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: Class exercise. Proposition 3.2-2: Second Welfare Teorem for an Excange Economy Consumer H as an endowment n ω +. Te consumption set for eac individual X is te positive ortant. Suppose also tat utility functions U (), n + H are continuous, quasi-concave and strictly increasing. If { ˆ } suc tat x H is PE suc tat xˆ 0, H ten tere exists a price vector p > 0 U ( x ) > U ( xˆ ) p x > p xˆ, H Jon Riley
52 Essential Microeconomics -52- Proof: ppealing to te Lemma, lso V 1 () is strictly increasing because V 1 ( x) is quasi-concave. U 1 () is strictly increasing and any increment in te aggregate supply can be allocated to te first consumer. Fig 3.2-1: Supporting yperplane ** Jon Riley
53 Essential Microeconomics -53- Proof: ppealing to te Lemma, lso V 1 () is strictly increasing because V 1 ( x) is quasi-concave. U 1 () is strictly increasing and any increment in te aggregate supply can be allocated to te first consumer. n indifference curve for V 1 () is depicted. s we ave noted tat { xˆ } solves H PE( x ) if x = ω. Fig 3.2-1: Supporting yperplane Moreover, because U 1 () is strictly increasing, H xˆ = ω. (3.2-2) = 1 * Jon Riley
54 Essential Microeconomics -54- Proof: ppealing to te Lemma, lso V 1 () is strictly increasing because V 1 ( x) is quasi-concave. U 1 () is strictly increasing and any increment in te aggregate supply can be allocated to te first consumer. n indifference curve for V 1 () is depicted. s we ave noted tat { xˆ } solves H PE( x ) if x = ω. Fig 3.2-1: Supporting yperplane Moreover, because U 1 () is strictly increasing, H xˆ = ω. (3.2-2) = 1 ecause ωis on te boundary of te upper contour set xv x V ω 1 1 { ( ) ( )}, it follows from te Supporting Hyperplane Teorem tat tere is a vector p 0, suc tat all te points in te upper contour set lie in te set { x p x p ω} Jon Riley
55 Essential Microeconomics -55- Formally, > ω > ω ω ω. (3.2-3) V ( x) V ( ) p x p and V ( x) V ( ) p x p We now argue tat te vector p must be positive. If not, define δ = ( δ1,..., δ n ) > 0 suc tat δ > 0 if and only if p < 0. j j Ten V ( ω + δ) > V ( ω) and p ( ω + δ) < p ω. 1 1 ut tis contradicts (3.2-3) so p must be positive after all. Fig 3.2-1: Supporting yperplane Jon Riley
56 Essential Microeconomics -56- To complete te proof we appeal to (3.2-1) - (3.2-3). H V x Max U x U x U xˆ H x x H { x } = 1 = 1 ( ) = { ( ) ( ) ( ), = 2,...,, 0} (3.2-1) H xˆ = ω (3.2-2) = V ( x) V ( ) p x p and V ( x) V ( ) p x p > ω > ω ω ω. (3.2-3) From (3.2-3) H U ( x ) U ( xˆ ), = 1,..., H p x= p x p ω. (3.2-4) = 1 * Jon Riley
57 Essential Microeconomics -57- To complete te proof we appeal to (3.2-1) - (3.2-3). H V x Max U x U x U xˆ H x x H { x } = 1 = 1 ( ) = { ( ) ( ) ( ), = 2,...,, 0} (3.2-1) H xˆ = ω (3.2-2) = V ( x) V ( ) p x p and V ( x) V ( ) p x p > ω > ω ω ω. (3.2-3) From (3.2-3) H U ( x ) U ( xˆ ), = 1,..., H p x= p x p ω. (3.2-5) = 1 Substituting for ω from (3.2-2) it follows tat H H. U ( x ) U ( xˆ ), = 1,..., H p x p xˆ = 1 = 1 k k Setting x = xˆ, k, we may ten conclude tat for consumer, U ( x ) U ( xˆ ) p x p xˆ. Jon Riley
58 Essential Microeconomics -58- It remains to sow tat any strictly preferred bundle costs strictly more. Suppose instead tat U ( x ) > U ( xˆ ) and p x = p xˆ. Ten for all λ (0,1), p λx < p xˆ. lso because U () is continuous, for all λ sufficiently close to 1, U ( λ x ) > U ( xˆ ). ut tis cannot be true since we ave just sown tat U ( x ) U ( xˆ ) p x p xˆ Hence U ( x ) > U ( xˆ ) p x > p xˆ. Q.E.D. Jon Riley
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