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Homework 4 l. A perfectly competitive rm has a Cobb-Douglas production function f(x1, x2) = 3612353. Suppose that output and input prices are p = 6, \"'1 = 3, and W2 = 1 , respectively. a. Suppose that in the short run the quantity of input factor 2 is xed at x2 = 64. Solve the rm's short-run prot maximization problem to derive the optimal input quantity xi'. (1 p2\") Use the information in part a. to derive the optimal short-run output level 3!\" and the corresponding prots fr\". (2 pt) Now consider the long run, in which both input factors are variable. Solve the rm's long-run prot-maximization problem to derive the optimal input quantities x; and x; (2 pt) Use the information in part c. to derive the optimal longrun output level y'\" and the corresponding prots fr\". (2 pt) 2. A perfectly competitive rm has a Cobb-Douglas production function f(x1, x2) = xfxz. Suppose that input prices are W1 = 4 and wz = 1. The rm wants to nd the cheapest way of producing y = 32. a. Suppose that in the short run the quantity of input factor 2 is xed at x2 = 32. Solve the rm's short-run cost minimization problem to derive the optimal input quantity xi. (1 pt) Derive the corresponding costs Cs of producing y = 32 in the short run. (1 pt) Now consider the long run, in which y = 32 and both input factors are variable. Set up the Lagrangian function for this firm's long-run cost minimization problem. (1 pr) Derive the rst-order conditions of the above long-run cost minimization problem. (3 pt) Solve the above rst-order conditions to derive the optimal input quantities x; and xi\". (2 pt) Derive the corresponding costs 6* of producing y = 32 in the long run. (1 pr)