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Rubinstein2005-page145 - 12:18 master Sheet number 143 Page...

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Unformatted text preview: October 21, 2005 12:18 master Sheet number 143 Page number 127 Review Problems 127 good 1 for f ( x ) units of good 2, or he can exchange y units of good 2 for g ( y ) units of good 1. Assume the consumer can only make one exchange. 1. Show that if the exchange functions are continuous and the consumer’s preference relation satisfies monotonicity and con- tinuity, then a solution to the consumer problem exists. 2. Explain why strong convexity of the preference relation is not sufficient to guarantee a unique solution if the functions f and g are increasing and convex. 3. What does the statement “the function f is increasing and con- vex” mean? 4. Suppose both functions f and g are differentiable and concave and that the product of their derivatives at point 0 is 1. Sup- pose also that the preference relation is strongly convex. Show that under these conditions, the agent will not find two dif- ferent exchanges, one exchanging good 1 for good 2, and one exchanging good 2 for good 1, optimal.exchanging good 2 for good 1, optimal....
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