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# answer1 - Solutions to Problem Set 1 EC720.01 Math for...

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Solutions to Problem Set 1 EC720.01 - Math for Economists Peter Ireland Boston College, Department of Economics Fall 2009 Due Thursday, September 17 1. Profit Maximization Consider a firm that produces output y with capital k and labor l according to the technology described by k a l b y, (1) where 0 < a < 1, 0 < b < 1, and 0 < a + b < 1. 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 constraint just shown in (1). a. The Lagrangian for this problem is L ( y, k, l, λ ) = py - rk - wl + λ ( k a l b - y ) . b. According to the Kuhn-Tucker theorem, the values y * , k * , and l * that solve the firm’s problem, together with the associated value λ * for the multiplier, must satisfy the first-order conditions L 1 ( y * , k * , l * , λ * ) = p - λ * = 0 , L 2 ( y * , k * , l * , λ * ) = - r + * ( k * ) a - 1 ( l * ) b = 0 , and L 3 ( y * , k * , l * , λ * ) = - w + * ( k * ) a ( l * ) b - 1 = 0 , the constraint L 4 ( y * , k * , l * , λ * ) = ( k * ) a ( l * ) b - y * 0 , the nonnegativity condition λ * 0 , and the complementary slackness condition λ * [( k * ) a ( l * ) b - y * ] = 0 . c. The first-order condition for y * reveals that λ * > 0 whenever p > 0. Assuming that this is the case, the complementary slackness condition requires the constraint to hold as an equality. Hence, the first-order conditions can be used together with the binding 1

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constraint to solve for y * , k * , l * , and λ * in terms of the model’s parameters a , b , p , r , and w : y * = a
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