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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 ) = min { x, y } (weakly monotonic) ω 2 = (1 , 1) For any p 0, D 2 ( p ) = (1 , 1) = ω 2 D 1 ( p ) 1 > ω 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 , 0) is a Walrasian Quasi-Equilibrium Price. 1

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Even without local nonsatiation, p Δ , x i Q i ( p ) I 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 i =1 x i ¯ ω If, in addition, all agents exhibit local nonsatiation, I i =1 x i = ¯ ω If ω i 0 for all i , p · ω i > 0 x i D i ( p ) I 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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