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201Lecture152006 - Economics 201BSecond Half Lecture 15 GEI...

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Economics 201B–Second Half Lecture 15 GEI Model (Continued) Saw in Lecture 14 that Equilibrium need not be Pareto Opti- mal. Model V(b): Either 2 periods, L > 1, securities pay off in goods; or more than 2 periods with L = 1. Two situations work much the same way. Natural Conjecture : In the GEI model, Walrasian Equi- librium is constrained Pareto Optimal, i.e. there is no allo- cation achievable by the given markets that Pareto domi- nates the Equilibrium allocation. Note that the conjecture is ambiguous, since the allocations achievable by the given markets depend on the prices. Under any reasonable interpretation, the conjecture is false . Hart Example (19.F.2 in MWG, but with different spin). · I = 2, L = 2, S = 2. There are no securities, so you can’t trade consumption across states. · States s = 1 and s = 2 have identical Edgeworth boxes. · Each Edgeworth box has three Equilibrium prices; let p and p be two of them. · Within each Edgeworth box, agent 1 prefers p to p , agent 2 prefers p to p 1
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· Fix utility functions v 1 , v 2 realizing the preferences of agents 1,2 in the common Edgeworth box. · x si denotes the consumption of good by agent i in state s . Let u 1 ( x 111 , x 211 , x 121 , x 221 ) = 100 v 1 ( x 111 , x 211 ) + v 1 ( x 121 , x 221 ) u 2 ( x 112 , x 212 , x 122 , x 222 ) = v 2 ( x 112 , x 212 ) + 100 v 2 ( x 122 , x 222 ) Agent 1 cares much more about state 1, agent 2 cares much more about state 2.
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