2004ps_solutions

# 2004ps_solutions - Problem Set 1 Econ 702 Spring 2004...

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Problem Set 1 Econ 702, Spring 2004 Prepared by Ahu Gemici February 7, 2004 Problem 1 Show that without the assumption of continuity of the price func- tion, the proof of First Basic Welfare Theorem doesn’t go through. Solution: I will write down the proof of FBWT with the assumption that the price function is continous and point out to the places we need this property for the proof to go through. Theorem 1 (First Basic Welfare Theorem) Suppose that for all i , all x X i there exists a sequence { x n } n =0 in X i converging to x with u ( x n ) u ( x ) for all n (local nonsatiation). If an allocation [( x * i ) j J , ( y * i ) i I ] and a conti- nous linear functional ν constitute a competitive equilibrium, then the allocation [( x * i ) j J , ( y * i ) i I ] is Pareto optimal. Proof. As you will remember from your 701, the ﬁrst thing before we prove FBWT is to show that if x * i solves agent i’s optimization problem, and if there is another allocation that gives him at least as much (or strictly more) utility, then it must be that that allocation is at least as (or strictly more) expensive as x * i . In other words, i , all x i X , u ( x i ) u ( x * i ) ν ( x i ) ν ( x * i ) (1) u ( x i ) > u ( x * i ) ν ( x i ) > ν ( x * i ) (2) First, showing (1): Suppose not, so that, u ( x i ) u ( x * i ) and ν ( x i ) < ν ( x * i ) By local nonsatiation of u , for every ± 0, there is x 0 i X i ∩{ ˜ x i : k ˜ x i - x i k < ± } with u ( x 0 i ) > u ( x i ) u ( x * i ) But then for suﬃciently small ± , ν ( x 0 ) < ν ( x * i ).(Note that without the continuity we would not be able to show the last point (i.e. that ν ( x 0 ) < ν ( x * i ))). This contradicts with x * i being the optimal solution to 1

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the agent’s problem. Showing (2): This follows from the fact that x * i solves the agent’s problem. Now consider an allocation [( x i ) , ( y i )] and suppose that each consumer prefers x i to x * i ,with strict preference for at least one consumer. From (1) and (2), it must be that, ν ( x i ) ν ( x * i ) i (3) ν ( x i ) > ν ( x * i ) for some i (4) Sum over i, X i ν ( x i ) > X i ν ( x * i ) (5) By the linearity of ν we can write ν ( x ) > ν ( x * ) where x = i x i andy = i y i Moreover, proﬁt maximization implies ν ( y ) ν ( y * ) ν ( x ) - ν ( y ) > ν ( x * ) - ν ( y * ) (6) ν ( x - y ) > ν ( x * - y * ) ( By linearity of ν ) (7) Feasibility implies x * - y * = 0. By (7), it must be that x - y 6 = x * - y * which in turn implies x - y 6 = 0. Hence, the allocation [( x i ) , ( y i )] is not feasible. Contradiction. You can see where we need the continuity of ν : Supposing that ( x * , y * ) is not PO, we take another feasible allocation that gives some of the households more utility. Then we arrive at the contradiction through showing that such an allocation is not feasible. To conclude that such an allocation is not feasible, we need to use (1) and (2) which holds under the assumption of continuity of the price functional. Hence with no continuity, we have no (1) and (2) and the proof doesn’t go through.
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2004ps_solutions - Problem Set 1 Econ 702 Spring 2004...

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