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© John Riley 28 July 2011 ANSWERS TO ODD NUMBERED EXERCISES IN APPENDIX C SECTION C.1: TWO VARIABLES Exercise C.1-1: Consumer Choice (a) Since U is strictly increasing, for any x not on the boundary of the budget set there exists a 0 δ > such that x + is in the budget set and () ( ) Ux + > . Therefore x is not optimal. It follows that 11 2 2 p xp xI += an so 21 1 2 / x Ip x p = . Then the consumer chooses [0, / ] x to maximize 1 1 2 () (, ) / ) ux ux I px p =− . If 1 is increasing at 1 0 x = the optimal 1 x is strictly positive. Thus a necessary condition for a corner solution at 1 0 x = is 1 12 1 12 2 0 dx p UU xx d x p ∂∂ =+ at 1 0 x = . Rearranging this inequality, 2 2 (, ) U x p MRS x x U p x =≤ at 2 (0, / ) x = . Alternatively, x x p p at 2 (0, / ) x = . The expression on the left hand side of this inequality is the marginal utility of a dollar spent on commodity 1. If at the corner this is less than the marginal utility of a dollar spent on commodity 2, the consumer is better off at the corner than moving slightly away from the corner. Similarly, there is a corner solution at 2 0 x = and so / x = if

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© John Riley 28 July 2011 21 1 12 1 12 2 () 0 dx p UU ux xx d x p ∂∂ =+ =− at / x Ip = . Rearranging this inequality, x x p p at 1 (/ , 0 ) xI p = . (b) These conditions cannot hold if the marginal utility of each commodity increases without bound as each commodity approaches zero. Thus * x is strictly positive and so x x p p = at * x .
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