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probset3 - Problem Set 3 EC720.01 Math for Economists...

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Problem Set 3 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, October 1 Many famous results from microeconomic theory are now understood to be special cases of the envelope theorem. This problem set will ask you to invoke the envelope theorem repeatedly to “prove” some of these results. 1. Hotelling’s Lemma Consider a firm that produces output y with capital k and labor l according to the technology described by f ( k, l ) y. The firm sells each unit of output at the price p , rents each unit of capital at the rate r , and hires each unit of labor at the wage w . Hence it chooses y , k , and l to maximize profits py - rk - wl subject to the technological constraint just shown above. a. Set up the Lagrangian for this problem, letting λ denote the multiplier on the constraint. b. Next, write down the conditions that, according to the Kuhn-Tucker theorem, must be satisfied by the values y * , k * , and l * that solve the firm’s problem, together with the associated value λ * for the multiplier. c. Assume that the output and input prices p , r , and w and the production function f are such that it is possible to solve uniquely for the values of y * , k * , l * , and λ * in terms of the parameters p , r , and w . Then the function y * ( p, r, w ) describing the optimal level of output represents the firm’s supply function, and functions k * ( p, r, w ) and l * ( p, r, w ) describing the optimal inputs are the firm’s factor demand curves. Along with these functions, define the firm’s profit function as π ( p, r, w ) = max y,k,l py - rk - wl subject to f ( k, l ) y.
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