5First Welfare Theorem

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online