indirectutility

# indirectutility - Notes on Indirect Utility How do we show...

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Notes on Indirect Utility How do we show that the indirect utility function is quasi-convex? We want to show that if v ( p,m ) v ( p 0 ,m 0 ), then the indirect utility of the convex combination budget is worse than the indirect utility of the ( p,m ) budget. That is, for any λ such that 0 < λ > 1, v ( λp + (1 - λ ) p 0 ,λm + (1 - λ ) m 0 ) v ( p,m ) . The key is that anything in the convex combination budget can be aﬀorded with one or the other of the original budgets. Therefore at least one of these two budgets is at least a good as the convex combination budget. So the convex combination is worse than the better one. How do we show this? Where 0 < λ < 1, let p λ = λp + (1 - λ ) p 0 and m λ = λm + (1 - λ ) m 0 . Let x λ = x ( p λ ,m λ ). Then p λ x λ m λ (since x λ = x ( p λ ,m λ ) is something you can aﬀord with income m λ .) But this means that λpx λ + (1 - λ ) p 0 x λ λm + (1 - λ ) m 0 . This implies that λ ( m - px ) + (1 - λ )( m 0 - p 0 x ) 0, which implies that either m px λ or m 0 p 0 x λ , (or possibly both) which implies that at least one of the following two things are true: px λ m and hence v ( p λ ,m λ ) v ( p,m ) or p 0 x λ m 0 and hence v ( p λ ,m λ ) v ( p 0 ,m 0 ). So it must be that v ( p λ ,m λ ) max { v ( p,m ) ,v ( p 0 ,m 0 ) }

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indirectutility - Notes on Indirect Utility How do we show...

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