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Lecture 25

Lecture 25 - Economics 101A(Lecture 25 Stefano DellaVigna...

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Economics 101A (Lecture 25) Stefano DellaVigna December 1, 2009

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Outline 1. Example of General Equilibrium 2. Existence and Welfare Theorems 3. Asymmetric Information: Introduction 4. Hidden Action (Moral Hazard)
1E x a m p l e Consumer 1 has Leontie f preferences: u ( x 1 , x 2 )=m in ³ x 1 1 ,x 1 2 ´ Bund ledemandedbyconsumer1 : x 1 1 = x 1 2 = x 1 = p 1 ω 1 1 + p 2 ω 1 2 p 1 + p 2 = = ω 1 1 +( p 2 /p 1 ) ω 1 2 1+( p 2 /p 1 )

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Comparative statics: increase in ω increase in p 2 /p 1 : dx 1 1 dp 2 /p 1 = ω 1 2 (1 + ( p 2 /p 1 )) ³ ω 1 1 +( p 2 /p 1 ) ω 1 2 ´ (1 + ( p 2 /p 1 )) 2 = = ω 1 2 ω 1 1 (1 + ( p 2 /p 1 )) 2 E f ect depends on income e f ect through endow- ments: A lot of good 2 > increase in price of good 2 makes richer Little good 2 > increase in price of good 2 makes poorer Notice: Only ratio of prices matters (general feature)
Consumer 2 has Cobb-Douglas preferences: u ( x 1 , x 2 )= ³ x 2 1 ´ . 5 ³ x 2 2 ´ . 5 Demands of consumer 2: x 2 1 = . 5 ³ p 1 ω 1 1 + p 2 ω 1 2 ´ p 1 = . 5 Ã ω 1 1 + p 2 p 1 ω 1 2 ! and x 2 2 = . 5 ³ p 1 ω 1 1 + p 2 ω 1 2 ´ p 2 = . 5 Ã p 1 p 2 ω 1 1 + ω 1 2 !

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Comparative statics: increase in ω > Increase in f nal consumption increase in p 2 /p 1 > Unambiguous increase in x 2 1 and decrease in x 2 2
Impose Walrasian equilibrium in market 1: x 1 1 + x 2 1 = ω 1 1 + ω 2 1 This implies ω 1 1 +( p 2 /p 1 ) ω 1 2 1+( p 2 /p 1 ) + . 5 Ã ω 1 1 + p 2 p 1 ω 1 2 ! = ω 1 1 + ω 2 1 or . 5 . 5( p 2 /p 1 ) 1+( p 2 /p 1 ) ω 1 1 + . 5( p 2 /p 1 )+ . 5( p 2 /p 1 ) 2 1 1+( p 2 /p 1 ) ω 1 2 =0 or ³ ω 1 1 2 ω 1 2 ´ + ³ ω 1 1 + ω 1 2 ´ ( p 2 /p 1 )+ ω 1 2 ( p 2 /p 1 ) 2 =0

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Solution for p 2 /p 1 : p 2 p 1 = ³ ω 1 1 2 ω 1 2 ´ + v u u u t ³ ω 1 1 + ω 1 2 ´ 2 4 ³ ω 1 1 2 ω 1 2 ´ ω 1 2 2 ³ ω 1 1 2 ω 1 2 ´ Some complicated solution! Problem set has solution that is easier to compute (and interpret)
2 Existence and Welfare Theorems Does Walrasian Equilibrium always exist? In general, yes, as long as preference convex Is Walrasian Equilibrium always unique? Not necessarily Is Walrasian Equilibrium e cient? Yes.

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First Fundamental Welfare Theorem. All Wal- rasian Equilibria are on Contract Curve (and there- fore are Pareto E cient). Figure
Second Fundamental Welfare theorem. Given convex preferences, for every Pareto e cient alloca- tion ³ ( x 1 1 ,x 1 1 ) , ( x 2 1 ,x 2 2 ) ´ there exists some endow- ment ( ω 1 2 ) such that ³ ( x 1 1 ,x 1 1 ) , ( x 2 1 ,x 2 2 ) ´ is a Walrasian Equilibrium for endowment ( ω 1 2 ). Figure

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Lecture 25 - Economics 101A(Lecture 25 Stefano DellaVigna...

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