DEPARTMENT OF ECONOMICS ECON 2750A H.W. #6 Prof. K.C. Tran Due Date: Wednesday April 14, 2009 1. Suppose that a monopolist produces an identical product in three plants and face an inverse demand function P = 4 - QT where total quantity in all three plants is QT = Q1 + Q2 + Q3. The output from the three plants is produced at the costs C1 = Q1 + Q12, C2 = 3Q2, C3 = 2Q32 – Q3 where Ci refers to the total cost required to produce Qi units from each facility. (a) Assuming that the total revenue TR = PQ and the total cost TC = C1 + C2 + C3, determine the profit function (b) Determine the levels of output, Q1, Q2, and Q3 that maximize this firm’s profits. 2. Are the following production functions homogeneous? If so, to what degree? (a) y = AL2K2 – BL3K3 with A, B >0 (b) y = (AL1+aK1+b)(BL + CK) A, B > 0 and 0 < a, b < 1 (c) y = [aL-1/2 + (1 – a)K-1/2] -2 3. Suppose you have the Indirect Utility function V ( p1 , p 2 , m) 1 mp11 / 2 p 21 / 2 2 (a) Use Roy’s Identities to determine the two uncompensated demand functions (b) Determine the expenditure function E( p1 , p2 , u ) (c) Use Shephard’s Lemma to determine the two compensated demand functions 4. Suppose a firm uses two inputs, labor, L and capital, K to produce the output Q according to the following production function: Q K 2L Let r be the price of a unit of capital and w be the price of a unit of labor, then the firm wishes to minimize the cost of producing some given level of output Q. That is, the firm minimizes: min C ( K , L) rK wL s.t. Q K 2 L (a) Solve this minimization problem to obtain the two input demand functions (also known as conditional demand functions), and the supply function. (b) Check the SOC to ensure that the solution is indeed minimum (c) Compute the minimal cost function and its marginal cost. What is the interpretation of the Lagrange multiplier here?