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Unformatted text preview: 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, ) = py rk wl + ( k a l b y ) . b. According to the KuhnTucker theorem, the values y * , k * , and l * that solve the firms problem, together with the associated value * for the multiplier, must satisfy the firstorder conditions L 1 ( y * ,k * ,l * , * ) = p * = 0 , L 2 ( y * ,k * ,l * , * ) = r + a * ( k * ) a 1 ( l * ) b = 0 , and L 3 ( y * ,k * ,l * , * ) = w + b * ( k * ) a ( l * ) b 1 = 0 , the constraint L 4 ( y * ,k * ,l * , * ) = ( k * ) a ( l * ) b y * , the nonnegativity condition * , and the complementary slackness condition * [( k * ) a ( l * ) b y * ] = 0 . c. The firstorder condition for y * reveals that * > 0 whenever p > 0. Assuming that this is the case, the complementary slackness condition requires the constraint to hold as an equality. Hence, the firstorder conditions can be used together with the bindingan equality....
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 Fall '09
 IRELAND
 Economics

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