Walrasian Equilibrium in an exchange economy
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1 Microeconomic Teory -1- Walrasian equilibrium Walrasian Equilibrium in an ecange economy 1. Homotetic preferences 2 2. Walrasian equilibrium in an ecange economy Te market value of attributes Efficient economies 23 Jon Riley October 10, 2017
2 Microeconomic Teory -2- Walrasian equilibrium A. Homotetic preferences Analysis of markets is greatly simplified if we are willing to make two strong assumptions 1. Identical strictly increasing utility functions 2. Utility is omotetic Definition: Homotetic preferences Homotetic preferences Preferences are omotetic if for any consumption bundle preferred to 1, for all 0. 1 and 2 preferred to 1, 2 is (Scaling up te consumption bundles does not cange te preference ranking). * Jon Riley October 10, 2017
3 Microeconomic Teory -3- Walrasian equilibrium A. Homotetic preferences Analysis of markets is greatly simplified if we are willing to make two strong assumptions 1. Identical strictly increasing utility functions 2. Utility is omotetic Definition: Homotetic preferences Homotetic preferences Preferences are omotetic if for any consumption bundle 1 preferred to, for all 0. 1 and 2 preferred to 1, 2 is (Scaling up te consumption bundles does not cange te preference ranking). Homotetic utility function A utility function is omotetic if for any pair of consumption bundles 1 and 2, U 2 1 ( ) U ( ) implies tat U U for all ( ) ( ) Proposition: Wit identical omotetic preferences, market demand is te same as te demand of a single representative consumer wit all of te income. Jon Riley October 10, 2017
4 Microeconomic Teory -4- Walrasian equilibrium Let ( p, I) be te coice of a consumer wit income I. i.e. for all I ( p, I) solves Ma{ U( ) p I} 0 In particular, p (,1) solves Ma{ U( ) p 1} 0 We will sow tat ( p, I) I( p,1) Since I( p,1) costs I it is a feasible consumption bundle for a consumer wit tis income. Hence U( I( p,1)) U( ( p, I)). (1.1) *** Jon Riley October 10, 2017
5 Microeconomic Teory -5- Walrasian equilibrium Let ( p, I) be te coice of a consumer wit income I. i.e. for any income, ( p, I) solves Ma{ U( ) p I}. 0 In particular, p (,1) solves Ma{ U( ) p 1} 0 Since I( p,1) costs I it is a feasible consumption bundle for a consumer wit tis income. Hence U( I( p,1)) U( ( p, I)). (1.1) Since 1 ( p, I) I costs 1, it is a feasible consumption bundle for a consumer wit income 1. Hence 1 U ( ( p, I)) U ( ( p,1)) I ** Jon Riley October 10, 2017
6 Microeconomic Teory -6- Walrasian equilibrium Let ( p, I) be te coice of a consumer wit income I. i.e. for any income, ( p, I) solves Ma{ U( ) p I}. 0 In particular, p (,1) solves Ma{ U( ) p 1} 0 Since I( p,1) costs I it is a feasible consumption bundle for a consumer wit tis income. Hence U( I( p,1)) U( ( p, I)). (1.1) Since 1 ( p, I) I costs 1 it is a feasible consumption bundle for a consumer wit income 1. Hence 1 U ( ( p, I)) U ( ( p,1)) I Appealing to omoteticity, U ( ( p, I)) U ( ( p,1)) for all I * Jon Riley October 10, 2017
7 Microeconomic Teory -7- Walrasian equilibrium Let ( p, I ) be te unique coice of a consumer wit income I. i.e. for any income, ( p, I) solves Ma{ U( ) p I}. 0 In particular, p (,1) solves Ma{ U( ) p 1} 0 Since I( p,1) costs I it is a feasible consumption bundle for a consumer wit tis income. Hence U( I( p,1)) U( ( p, I)). (1.1) Since 1 ( p, I) I costs 1 it is a feasible consumption bundle for a consumer wit income 1. Hence 1 U ( ( p, I)) U ( ( p,1)) I Appealing to omoteticity, U ( ( p, I)) U ( ( p,1)) for all I Coose I U( ( p, I)) U( I( p,1)) (1.2) Togeter (1.1) and (1.2) imply tat ( p, I) I( p,1) Jon Riley October 10, 2017
8 Microeconomic Teory -8- Walrasian equilibrium Homotetic preferences For any Wy? 0 and any MRS MRS 0 ( ) ( ) Jon Riley October 10, 2017
9 Microeconomic Teory -9- Walrasian equilibrium Eamples of omotetic utility functions (i) ( ) U a a a, a (ii) U( ), (iii) (iv) U ( ) ( ) U( ) 1/2 1/ (v) U ( ) Jon Riley October 10, 2017
10 Microeconomic Teory -10- Walrasian equilibrium Definition: Market demand If ( p, I ), 1,..., H uniquely solves consumers wit incomes 1 H I,..., I is Ma{ U ( ) p I }, ten te market demand for H 0 H ( p) ( p, I ) 1 ** Jon Riley October 10, 2017
11 Microeconomic Teory -11- Walrasian equilibrium Definition: Market demand If ( p, I ), 1,..., H uniquely solves consumers wit incomes 1 H I,..., I is Ma{ U ( ) p I }, te market demand for H 0 H ( p) ( p, I ) 1 Consider a 2 consumer economy wit incomes 1 I and 2 I. Proposition: Market demand in a 2 person economy wit identical omotetic preferences. If ( p, I ) is te demand for a consumer wit income I ten p I p I p I I (, ) (, ) (, ) is te market demand. Proof: Group Eercise Jon Riley October 10, 2017
12 Microeconomic Teory -12- Walrasian equilibrium Corollary: Representative consumer Suppose tat consumers ave identical strictly increasing omotetic preferences and tat solves H Ma{ U( ) p I I } 0 1 Ten is a market demand. Proof: Follows almost immediately from te proposition Jon Riley October 10, 2017
13 Microeconomic Teory -13- Walrasian equilibrium B. Walrasian equilibrium (WE) in an ecange economy In a WE consumer knows is own endowment and preferences but knows noting about te economy ecept te vector of prices. Consumer ten solves for te set of Walrasian (utility maimizing) demands ( p, ). Te price vector is a WE price vector if tere is some WE demand ( p, ), 1,..., H suc tat te sum of tese demands (te market demand) is equal to te total endowment. Jon Riley October 10, 2017
14 Microeconomic Teory -14- Walrasian equilibrium Walrasian equilibrium (WE) in an ecange economy wit identical omotetic preferences Consider te representative consumer wit endowment. We assume H 1 0. Let be a demand of te representative consumer. Ten solves Ma{ u( ) p p I} *** Jon Riley October 10, 2017
15 Microeconomic Teory -15- Walrasian equilibrium Walrasian equilibrium (WE) in an ecange economy wit identical omotetic preferences Consider te representative consumer wit endowment. We assume H 1 0. Let be a demand of te representative consumer. Ten solves Ma{ u( ) p p I} FOC for a maimum. 1 U 1 U ( )... ( ) p p 1 1 n n For p to be a WE price vector markets must clear. Wit only one consumer,. Terefore te WE prices satisfy 1 U 1 U ( )... ( ) p p 1 1 n n. Note tat tis only determines relative prices (i.e. price ratios.) * Jon Riley October 10, 2017
16 Microeconomic Teory -16- Walrasian equilibrium Walrasian equilibrium (WE) in an ecange economy wit identical omotetic preferences Consider te representative consumer wit endowment. We assume H 1 0. Let be a demand of te representative consumer. Ten solves Ma{ u( ) p p I} FOC for a maimum. 1 U 1 U ( )... ( ) p p 1 1 n n For p to be a WE price vector markets must clear. Wit only one consumer,. Terefore te WE prices satisfy 1 U 1 U ( )... ( ) p p 1 1 n n. Note tat tis only determines relative prices (i.e. price ratios.) Above we argued tat if consumer as an endowment of value p I ten I I I I, were I is te sum of all te incomes 1 H I I... I is a WE demand. Jon Riley October 10, 2017
17 Microeconomic Teory -17- Walrasian equilibrium Terefore in te WE of te omotetic economy, consumer aggregate endowment. consumes a fraction I I of te Eample: U ( ) ln 1 2ln 2 1 (36,6) 2 (12,42) 1 2 Eercise: Use te representative consumer to sow tat p (, ) is te unique WE price vector normalized so tat te sum of te prices is Jon Riley October 10, 2017
18 Microeconomic Teory -18- Walrasian equilibrium We know tat if income Goes up by a factor of Ten so does consumption. Te value of consumer 1 s endowment is 48 and te value of consumer 2 s endowment is 96 so tey consume respectively 1/3 and 2/3 of te aggregate Endowment. Te trade triangles are depicted in te figure. Jon Riley October 10, 2017
19 Microeconomic Teory -19- Walrasian equilibrium C. Te market value of attributes In studying industries like te airline industry economist often try to determine te implicit value of different attributes (for eample, air travel: leg-room, percentage on-time arrival etc.) We now consider a simple eample to illustrate. Eac unit of commodity 1 and commodity 2 (fligts on different airlines) ave different amounts of two attributes (attribute A and B) commodity 1 commodity 2 Attribute A 2 1 Attribute B 1 3 Total endowment * Jon Riley October 10, 2017
20 Microeconomic Teory -20- Walrasian equilibrium C. Te market value of attributes In studying industries like te airline industry economist often try to determine te implicit value of different attributes (for eample, air travel: leg-room, percentage on-time arrival etc.) We now consider a simple eample to illustrate. Eac unit of commodity 1 and commodity 2 (fligts on different airlines) ave different amounts of two attributes (attribute A and B) commodity 1 commodity 2 Attribute A 2 1 Attribute B 1 3 Total endowment A consumer cares about te quantity of eac attribute consumed. Let ( 1, 2, 3,...) be te consumption coice z z A , B U U ( z, z,,..., ) ln z ln z ln... A B 3 n A b 3 3 ln(2 ) ln( 3 ) ln Jon Riley October 10, 2017
21 Microeconomic Teory -21- Walrasian equilibrium To keep te model simple we assume tat every consumer as te same log utility function. Eercise: Is te log utility function omotetic? p2 4 Eercise: Sow tat p 3. 1 An alternative approac Imagine a market for attributes. Wat would be te market clearing prices of eac attribute? commodity 1 commodity 2 Total endowment of eac attribute Attribute A Attribute B Total commodity endowment Let ( 1, 2) be te sadow (implicit) prices of eac attribute. 2 Eercise: Sow tat 1. 1 Eercise: Using tese attribute prices, wat is te value of eac commodity? Jon Riley October 10, 2017
22 Microeconomic Teory -22- Walrasian equilibrium Group Eercise Eac unit of commodity 1, 2 and 3 (fligts on different airlines) ave different amounts of two attributes (attribute A and B) commodity 1 commodity 2 commodity 3 Attribute A Attribute B Total endowment A consumer cares about te quantity of eac attribute consumed. z z A , B U U ( z, z,,..., ) ln z ln z... A B 3 n A b ln(2 5 ) ln( 3 5 ) ln Odd numbered groups: Solve for te equilibrium prices directly Even numbered groups: Solve for te sadow prices of eac attribute. Jon Riley October 10, 2017
23 Microeconomic Teory -23- Walrasian equilibrium D. Efficient economies Definition: Pareto preferred allocation Te allocation { } H { ˆ } H is Pareto preferred to { } H and at least one consumer strictly prefers { ˆ }. H if all consumers weakly prefer { ˆ } H over Jon Riley October 10, 2017
24 Microeconomic Teory -24- Walrasian equilibrium Definition: Pareto preferred allocation Te allocation { ˆ } { } H H is Pareto preferred to { } H and at least one consumer strictly prefers { ˆ } if all consumers weakly prefer { ˆ } over H H. Definition: Pareto efficient allocation { ˆ } H is Pareto efficient if tere is no feasible Pareto preferred allocation. Jon Riley October 10, 2017
25 Microeconomic Teory -25- Walrasian equilibrium Definition: Pareto preferred allocation Te allocation { ˆ } { } H H is Pareto preferred to { } H and at least one consumer strictly prefers { ˆ } if all consumers weakly prefer { ˆ } over H H. Definition: Pareto efficient allocation { ˆ } H is Pareto efficient if tere is no feasible Pareto preferred allocation. First welfare teorem for an ecange economy If U ( ), H = {1,..., H} satisfies te non-satiation property and { } Equilibrium allocation, ten { } H is Pareto Efficient. H is a Walrasian Jon Riley October 10, 2017
26 Microeconomic Teory -26- Walrasian equilibrium Non satiation property For every, tere is at a commodity j suc tat for all sufficiently small U (,...,,,..., ) U (,...,,,..., ) ** 1 j 1 j j 1 n 1 j 1 j j 1 n 0, Jon Riley October 10, 2017
27 Microeconomic Teory -27- Walrasian equilibrium Non satiation property For every, tere is at a commodity j suc tat for all sufficiently small U (,...,,,..., ) U (,...,,,..., ) 1 j 1 j j 1 n 1 j 1 j j 1 n 0, Properties of a consumer s coice Consider a consumer wit a utility function wose utility satisfies tis very weak property. Let be te coice of consumer. (i) If U ( ˆ ) U ( ) ten p ˆ p and (ii) if U ( ˆ ) U ( ) ten p ˆ p. * Jon Riley October 10, 2017
28 Microeconomic Teory -28- Walrasian equilibrium Non satiation property For every, tere is at a commodity j suc tat for all sufficiently small U (,...,,,..., ) U (,...,,,..., ) 1 j 1 j j 1 n 1 j 1 j j 1 n 0, Properties of a consumer s coice Consider a consumer wit a utility function wose utility satisfies tis very weak property. Let be te coice of consumer. (ii) If U ( ˆ ) U ( ) ten p ˆ p and (ii) if U ( ˆ ) U ( ) ten p ˆ p. Proof of (i): If U ( ˆ ) U ( ) ten p ˆ p Suppose instead tat U ( ˆ ) U ( ) and p ˆ p. Ten is not te coice of te consumer since it does not maimize utility among commodity bundles in te budget set p I. Jon Riley October 10, 2017
29 Microeconomic Teory -29- Walrasian equilibrium Proof of (ii): If U ( ˆ ) U ( ) ten p ˆ p. Suppose instead tat U ( ˆ ) U ( ) and p ˆ p. Ten for some j and all small 0 U ( ˆ,... ˆ, ˆ, ˆ,..., ˆ ) U ( ˆ,... ˆ, ˆ, ˆ,..., ˆ ). 1 j 1 j j 1 n 1 j 1 j j 1 n Define ˆ ( ˆ ˆ ˆ ˆ ˆ 1,... j 1, j, j 1,..., n). * Jon Riley October 10, 2017
30 Microeconomic Teory -30- Walrasian equilibrium Proof of (ii): If U ( ˆ ) U ( ) ten p ˆ p. Suppose instead tat U ( ˆ ) U ( ) and p ˆ p. Ten for some j and all small 0 U ( ˆ,... ˆ, ˆ, ˆ,..., ˆ ) U ( ˆ,... ˆ, ˆ, ˆ,..., ˆ ). 1 j 1 j j 1 n 1 j 1 j j 1 n Define ˆ ( ˆ ˆ ˆ ˆ ˆ 1,... j 1, j, j 1,..., n). Since p ˆ p we can coose 0 suc tat ˆ is in te budget set and U ( ˆ) U ( ˆ ). Ten again is not te coice of te consumer since it does not maimize utility among commodity bundles in te budget set p I. Jon Riley October 10, 2017
31 Microeconomic Teory -31- Walrasian equilibrium First welfare teorem for an ecange economy If U ( ), H = {1,..., H} satisfies te non-satiation property and { } Equilibrium allocation, ten { } H is Pareto Efficient. H is a Walrasian Proof: Remember tat { } H is an equilibrium allocation. Consider any Pareto preferred allocation { ˆ } Step 1: H For some, U ( ˆ ) U( ). By te non-satiation property (i) * p ˆ p. Jon Riley October 10, 2017
32 Microeconomic Teory -32- Walrasian equilibrium First welfare teorem for an ecange economy If U ( ), H = {1,..., H} satisfies te non-satiation property and { } Equilibrium allocation, ten { } H is Pareto Efficient. H is a Walrasian Proof: Remember tat { } H is an equilibrium allocation. Consider any Pareto preferred allocation { ˆ } Step 1 H. For some, U ( ˆ ) U( ). By te non-satiation property (i) p ˆ p. Step 2 For all, U ( ˆ ) U( ). By te non-satiation properties (i) and (ii) p ˆ p Jon Riley October 10, 2017
33 Microeconomic Teory -33- Walrasian equilibrium Summarizing, (a) ˆ p p for some H (b) ˆ p p for all H Summing over consumers, Step 3: ˆ ˆ p p p p H H For any feasible allocation { } H. H te total consumption vector must satisfy Since p 0 it follows tat for any feasible allocation p p. Since p ˆ p it follows tat ˆ is not a feasible allocation. Jon Riley October 10, 2017
34 Microeconomic Teory -34- Walrasian equilibrium Remark: An 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 10, 2017
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