5First Welfare Theorem

5First Welfare Theorem - Welfare in the 2-good quasilinear...

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Unformatted text preview: Welfare in the 2-good quasilinear model (or: concave programming and the invisible hand) Let = (( x 1 ,m 1 ) ,..., ( x I ,m I ) ,q 1 ,...,q J ) X I R J + = A denote an allocation, where X = R + R is each consumers consumption set. The input allocation ( z 1 ,...,z J ) is given by the cost vector, ( c 1 ( q 1 ) ,...,c J ( q J )), where for j = 1 ,...,J , c j ( q j ) = f- 1 j ( q j ). Consider the following problem (maximize the sum of utilities over all feasible allocations) ( WM-0 ) max A ( i ( x i ) + m i ) s.t. x i q j m i + c j ( q j ) i By Lemma 1 in the Pareto Optimality notes, any solution to ( WM-0 ) is Pareto Optimal. Let each i and f j be C 1 , strictly increasing and concave . Claim : Under these hypotheses, if ( * ,p * ) is a Competitive Equilibrium, then * solves problem ( WM-0 ) . Since the objective is increasing in m i for every i , the second constraint holds as an equality (and since we are ignoring the boundary condition on the second good) we may substitute...
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