PS1 solution

# PS1 solution - Rice University Professor Diamond Economics...

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Rice University Economics 370 Professor Diamond Microeconomic Theory SOLUTION TO PROBLEM SET 1: INTRODUCTION AND APARTMENT SUPPLY AND DEMAND I. Suppose the production function for the output produced by a firm is f ( L , K ) = 40 L 3 / 4 K 1 / 4 , where L is labor and K is capital. 1. What is the marginal product of labor? What is the marginal product of capital? The marginal product of labor is 1 f ( L , K ) (40 L 3/ 4 K 1/ 4 = ) = 40 · 3 L 3/ 4 1 K 1/ 4 = 30( K ) 1/ 4 . L L 4 L The marginal product of capital is f ( L , K ) (40 L 3/ 4 K 1/ 4 = ) = 40 · 1 L K 1/ 4 1 = 10( L ) . K K 4 K 2. Suppose the rental rate of capital in the market is r =2, the wage of labor is w =10, the price of output is p =1/3 and the amount of capital is fixed at 16 units (in the short run). Assuming that the firm acts to maximize profit, what is the optimal level of labor? Show that both the first and second order conditions for a maximum are satisfied. What is the associated optimal output level? Since profit is pf ( L , K ) ( wL + rK ) and the capital stock K is fixed, the optimization problem can be expressed as a function of a single variable, labor L: Max 1 · 40 L 16 1/ 4 10 L 2 · 16 L 3 Max 1 · 40 L 16 1/ 4 10 L L 3 F . O . C . 2 · 40 · 3 L 1 10 = 0 3 4 20 L 1/ 4 = 10 2 = L 1/ 4 L * = 16 S . O . C . 20( 1 ) L 1 / 4 1 = 5 L 5 / 4 < 0 , so L* =16 maximizes profit.

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PS1 solution - Rice University Professor Diamond Economics...

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