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Unformatted text preview: 2004 Spring 1. Homothetic Preferences (a) Homothetic utility function is a utility function u that satisfies u ( x ) ≥ u ( y ) ⇔ u ( kx ) ≥ u ( ky ) for all k > Under these preferences, the income expansion path will be a ray from the origin. To see this, let x * M be the optimal consumption under income M . Since a budget constraint is linear, both x * M and 1 k x * kM are feasible under M . By definition of x * , u ( x * M ) ≥ u 1 k x * kM ¶ By homotheticity, u ( kx * M ) ≥ u ( x * kM ) Also both x * kM and kx * M are feasible under kM . Again by definition of x * , u ( x * kM ) ≥ u ( kx * M ) So we have u ( x * kM ) = u ( kx * M ) It follows that 1 x * kM = kx * M Differentiating this with respect to k , ∂x * kM ∂M · M = x * M Letting k = 1, we have E ( x * ,M ) = ∂x * ∂M · M x * = 1 (b) If a utility function is identical as well as homothetic, both consumers will consume a fraction of the aggregate endowment, so the equilibrium price p will satisfy p = ∂U 1 ∂x fl fl fl fl x * 1 = ∂U 2 ∂x fl fl fl fl x * 2 = ∂U ∂x fl fl fl fl...
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This note was uploaded on 02/01/2011 for the course ECONOMY 6 taught by Professor Fallahi during the Spring '10 term at Cambridge.
- Spring '10