201Lecture92006

201Lecture92006 - Economics 201BSecond Half Lecture 9...

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Economics 201B–Second Half Lecture 9 Existence of Walrasian Equilibrium (Wrap-Up) Still need to prove Boundary Condition, it’s on Problem Set 4. What happens if we weaken the strong monotonicity assump- tion? local nonsatiation implies Walras’ Law holds with equal- ity, but is not sufficient to give Walrasian Equilibrium with I i =1 x i ¯ ω . In Edgeworth Box Economy, let u 1 ( x, y )= y + x (strongly monotonic) ω 1 =( 0 , 1) u 2 ( x, y )=m in { x, y } (weakly monotonic) ω 2 =( 1 , 1) For any p ± 0, D 2 ( p )=(1 , 1) = ω 2 D 1 ( p ) 1 11 For p =(1 , 0) or p =(0 , 1), D 1 ( p )= But notice for p =(1 , 0) ω 1 Q 1 ( p ) ω 2 Q 2 ( p ) so (1
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Even without local nonsatiation, p Δ ,x i Q i ( p ) I X i =1 x i ¯ ω Walrasian Quasi-Equilibrium exists, some goods may be left over; local nonsatiation does not imply allocation is exact, since some prices may be zero. If one agent (WLOG agent 1) is strongly monotonic and ω 1 ± 0, then p ± 0, so x i D i ( p )( i =1 ,...,I ) I X i =1 x i ¯ ω If, in addition, all agents exhibit local nonsatiation, I X i =1 x i ω If ω i ± 0 for all i , p · ω i > 0 x i D i ( p ) I X i =1 x i ¯ ω Local nonsatiation need not imply allocation exact, since some prices may be zero. With nonconvex preferences or indivisibilities, see Lecture 12.
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201Lecture92006 - Economics 201BSecond Half Lecture 9...

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