Microeconomic Teory -1- Equilibrium and efficiency Equilibrium and Pareto Efficiency in an excange economy 1. Efficient economies 2 2. Gains from excange 6 3. Edgewort-ox analysis 15 4. Properties of a consumer s coice 20 5. Walrasian equilibria are Pareto Efficient 24 Jon Riley October 15, 2018
Microeconomic Teory -2- Equilibrium and efficiency Efficient economies Definition: Pareto preferred allocation Te allocation { xˆ } H is Pareto preferred to { x } H if all consumers weakly prefer { xˆ } H over { x } H and at least one consumer strictly prefers{ xˆ } H. Jon Riley October 15, 2018
Microeconomic Teory -3- Equilibrium and efficiency Definition: Pareto preferred allocation Te allocation { x } H { xˆ } H is Pareto preferred to { } and at least one consumer strictly prefers x H { xˆ } if all consumers weakly prefer H. { xˆ } H over Definition: Pareto efficient allocation { xˆ } H is Pareto efficient if tere is no feasible Pareto preferred allocation. Jon Riley October 15, 2018
Microeconomic Teory -4- Equilibrium and efficiency Definition: Pareto preferred allocation Te allocation { x } H { xˆ } H is Pareto preferred to { } and at least one consumer strictly prefers x H { xˆ } if all consumers weakly prefer H. { xˆ } H over Definition: Pareto efficient allocation { xˆ } H is Pareto efficient if tere is no feasible Pareto preferred allocation. First welfare teorem for an excange economy If U ( x ), H= {1,..., H} satisfies te non-satiation property and { x } H is a Walrasian Equilibrium allocation, ten { x } H is Pareto Efficient. Jon Riley October 15, 2018
Microeconomic Teory -5- Equilibrium and efficiency 2. Gains from excange Preliminary observation Consider te standard utility maximization problem wit two commodities. If te solution x 0 ten te marginal utility per dollar must be te same for eac commodity slope = 1 U 1 U ( x) ( x) p x p x 1 1 2 2. ** Jon Riley October 15, 2018
Microeconomic Teory -6- Equilibrium and efficiency 2. Gains from excange Preliminary observation Consider te standard utility maximization problem wit two commodities. If te solution x 0 ten te marginal utility per dollar must be te same for eac commodity slope = 1 U 1 U ( x) ( x) p x p x 1 1 2 2 Equivalently te marginal rate of substitution satisfies. MRS ( x, x ) 1 2 U x U x 1 1 2 p p 2 * Jon Riley October 15, 2018
Microeconomic Teory -7- Equilibrium and efficiency 2. Gains from excange Preliminary observation Consider te standard utility maximization problem wit two commodities. If te solution x 0 ten te marginal utility per dollar must be te same for eac commodity slope = 1 U 1 U ( x) ( x) p x p x 1 1 2 2 Equivalently te marginal rate of substitution satisfies MRS ( x, x ) 1 2 U x U x 1 1 2 2. p p 1 In te figure te slope of te budget line is. t te maximum tis slope is te same as te slope of te indifference curve. Terefore MRS ( x1, x2) is te slope of te indifference curve. p p 2 Jon Riley October 15, 2018
Microeconomic Teory -8- Equilibrium and efficiency Pareto Efficient allocation in a 2 person 2 commodity economy n allocation xˆ and xˆ is not a PE allocation if tere is an excange of commodities e ( e1, e2) suc tat U ( xˆ e) U ( xˆ ) and U ( xˆ e) U ( xˆ ) ** Jon Riley October 15, 2018
Microeconomic Teory -9- Equilibrium and efficiency Pareto Efficient allocation in a 2 person 2 commodity economy n allocation xˆ and xˆ is not a PE allocation if tere Is an excange of commodities e ( e1, e2) suc tat U ( xˆ e) U ( xˆ ) and U ( xˆ e) U ( xˆ ) Proposition: If xˆ 0 and xˆ 0 ten a necessary condition for an allocation to be a PE allocation is tat marginal rates of substitution are equal. * Jon Riley October 15, 2018
Microeconomic Teory -10- Equilibrium and efficiency Pareto Efficient allocation in a 2 person 2 commodity economy n allocation xˆ and xˆ is not a PE allocation if tere Is an excange of commodities e ( e1, e2) suc tat slope = U ( xˆ e) U ( xˆ ) and U ( xˆ e) U ( xˆ ) Proposition: If xˆ 0 and xˆ 0 ten a necessary condition for an allocation to be a PE allocation is tat marginal rates of substitution are equal. Suppose instead tat, as depicted, MRS ( xˆ ) MRS ( xˆ ) Consider a proposal by lex of e ( e1, e2) were e 0 e 1 2 slope = and te excange rate lies between te two marginal rates of substitution Jon Riley October 15, 2018
Microeconomic Teory -11- Equilibrium and efficiency Suc an excange is depicted. On te margin, lex is willing to give up more of commodity 2 In excange for commodity 1. Terefore lex offers ev some of commodity 2 In excange for commodity 1. * Jon Riley October 15, 2018
Microeconomic Teory -12- Equilibrium and efficiency Suc an excange is depicted. On te margin, lex is willing to give up more of commodity 2 In excange for commodity 1. Terefore lex offers ev some of commodity 2 In excange for commodity 1. If te proposed trade is too large it may not be better for bot consumers due to te curvature of te level sets. ut for all sufficiently small, te proposed trade e must raise te utility of bot consumers. So te initial allocation xˆ allocation., x ˆ is not a Pareto efficient Jon Riley October 15, 2018
Microeconomic Teory -13- Equilibrium and efficiency Wat if tere are more tan two commodities? For all possible allocations we can, in principle compute te utilities and ence te set of feasible utilities. Pareto preferred allocation For any point in te interior of tis set tere is anoter allocation suc tat ev is no worse off and lex is strictly better off. * Jon Riley October 15, 2018
Microeconomic Teory -14- Equilibrium and efficiency Wat if tere are more tan two commodities? For all possible allocations we can, in principle compute te utilities and ence te set of feasible utilities. For any point in te interior of tis set tere is Pareto preferred allocation Pareto efficient allocation anoter allocation suc tat ev is no worse off and lex is strictly better off. Consider te following maximization problem. Max { U ( xˆ e) U ( xˆ e) U ( xˆ )} e Class Exercise Wat excange xˆ, x ˆ is Pareto efficient? * e solves tis problem if te allocation Jon Riley October 15, 2018
Microeconomic Teory -15- Equilibrium and efficiency 3. Efficiency in an Edgewort-ox diagram Consider lex and ev wit endowments and. MRS ( ) MRS ( ) so tere are gains from excange. Jon Riley October 15, 2018
Microeconomic Teory -16- Equilibrium and efficiency Efficiency in an Edgewort-ox diagram If te endowments are and, te set of feasible allocations for ev is te set of allocation in te rectangle or box Te set of allocations preferred by ev Is te dotted region in te lower box. On te next slide we rotate te box 180. Jon Riley October 15, 2018
Microeconomic Teory -17- Equilibrium and efficiency ox rotated 180 Jon Riley October 15, 2018
Microeconomic Teory -18- Equilibrium and efficiency We also add te level set for lex troug te endowment. ecause MRS ( ) MRS ( ) tere is a vertically lined region of Pareto preferred allocations Jon Riley October 15, 2018
Microeconomic Teory -19- Equilibrium and efficiency Te allocation xˆ and ˆ is Pareto- efficient since te x xˆ marginal rates of substitution are equal. Jon Riley October 15, 2018
Microeconomic Teory -20- Equilibrium and efficiency Group exercise Suppose tat U ( x ) 2( x ) 3( x ) and U ( x ) 2( x ) 3( x ) 3/4 3/4 1 2 3/4 3/4 1 2 Te aggregate endowment is (100,200). (a) Sow tat for all allocation to be a PE allocation, bot consumers are allocated twice as muc of commodity 2. (b) Wat is te MRS if an allocation is Pareto Efficient? Jon Riley October 15, 2018
Microeconomic Teory -21- Equilibrium and efficiency 4. Properties of a consumer s coice Non satiation property For every x, tere is at a commodity j suc tat for all sufficiently small 0, U ( x,... x, x, x,..., x ) U ( x,... x, x, x,..., x ) 1 j 1 j j 1 n 1 j 1 j j 1 n Consider a consumer wit a utility function wose utility satisfies tis very weak property. Let x be te coice of consumer. (i) If U ( xˆ ) U ( x ) ten p xˆ p x and (ii) if U ( xˆ ) U ( x ) ten p xˆ p x. * Jon Riley October 15, 2018
Microeconomic Teory -22- Equilibrium and efficiency 2. Properties of a consumer s coice Non satiation property For every x, tere is at a commodity j suc tat for all sufficiently small U ( x,... x, x, x,..., x ) U ( x,... x, x, x,..., x ) 1 j 1 j j 1 n 1 j 1 j j 1 n 0, Consider a consumer wit a utility function wose utility satisfies tis very weak property. Let x be te coice of consumer. (i) If U ( xˆ ) U ( x ) ten p xˆ p x (ii) If U ( xˆ ) U ( x ) ten p xˆ p x Proof of (i): If U ( xˆ ) U ( x ) ten p xˆ p x Suppose instead tat U ( xˆ ) U ( x ) and p xˆ p x. Ten x is not te coice of te consumer since it does not maximize utility among commodity bundles in te budget set p x I. Jon Riley October 15, 2018
Microeconomic Teory -23- Equilibrium and efficiency (ii) If U ( xˆ ) U ( x ) ten p xˆ p x Proof: If U ( xˆ ) U ( x ) tis follows from (i) Suppose instead tat U ( xˆ ) U ( x ) and p xˆ p x. Define xˆ ( xˆ 1,... xˆ 1, xˆ, xˆ 1,..., xˆ ). j j j n Ten for some j and all small 0 U ( xˆ ) U ( xˆ,... xˆ, xˆ, xˆ,..., xˆ ) U ( xˆ,... xˆ, xˆ, xˆ,..., xˆ ). * 1 j 1 j j 1 n 1 j 1 j j 1 n Jon Riley October 15, 2018
Microeconomic Teory -24- Equilibrium and efficiency (ii) If U ( xˆ ) U ( x ) ten p xˆ p x Proof: If U ( xˆ ) U ( x ) tis follows from (i) Suppose instead tat U ( xˆ ) U ( x ) and p xˆ p x. Define xˆ ( xˆ 1,... xˆ 1, xˆ, xˆ 1,..., xˆ ). j j j n Ten for some j and all small 0 U ( xˆ ) U ( xˆ,... xˆ, xˆ, xˆ,..., xˆ ) U ( xˆ,... xˆ, xˆ, xˆ,..., xˆ ) 1 j 1 j j 1 n 1 j 1 j j 1 n Since p xˆ p x we can coose 0 suc tat xˆ is in te budget set and U ( xˆ) U ( xˆ ). Ten again x is not te coice of te consumer since it does not maximize utility among commodity bundles in te budget set p x I. Jon Riley October 15, 2018
Microeconomic Teory -25- Equilibrium and efficiency 3. Walrasian equilibria are Pareto Efficient First welfare teorem for an excange economy If U ( x ), H= {1,..., H} satisfies te non-satiation property and { } Equilibrium allocation, ten { x } H is Pareto Efficient. x H is a Walrasian Proof: Remember tat { x } H is an equilibrium allocation. Consider any Pareto preferred allocation { xˆ } Step 1: For some, U ( xˆ ) U( x ). y te non-satiation property (i) * p x ˆ. p H Jon Riley October 15, 2018
Microeconomic Teory -26- Equilibrium and efficiency First welfare teorem for an excange economy If U ( x ), H= {1,..., H} satisfies te non-satiation property and { } Equilibrium allocation, ten { } x H is Pareto Efficient. x H is a Walrasian Proof: Remember tat { x } H is an equilibrium allocation. Consider any Pareto preferred allocation { ˆ } Step 1 For some, U ( xˆ ) U( x ). y te non-satiation property (i) p x ˆ. p Step 2 For all, U ( xˆ ) U( x ). y te non-satiation properties (i) and (ii) p x ˆ p x H. Jon Riley October 15, 2018
Microeconomic Teory -27- Equilibrium and efficiency Summarizing, p x p for some H (a) ˆ p x p for all H (b) ˆ Summing over consumers, Step 3: ˆ ˆ p x p x p p H H For any feasible allocation { x } H te total consumption vector must satisfy x x. H Since p 0 it follows tat for any feasible allocation p x p. Since p xˆ p it follows tat ˆx is not a feasible allocation. Remark: n almost identical argument can be used to sow tat a Walrasian Equilibrium allocation for an economy wit production is also Pareto Efficient Jon Riley October 15, 2018