5. A recent Duke graduate has opened a factory for production of textbooks in Durham. The production function is as follows f«(K;L) = K L1/3 Currently, she uses K = 2000 and L = 500 and the rental rates of capital and labor are w=r=$1. Now she is considering expanding her business and opening another factory in Chapel Hill. Using her microeconomic skills, she estimates that the cost function for the factory in Chapel Hill will be C(q) = q2 The firm is facing a competitive market and the price at which it can sell its output is p = $30. a. Suppose that in the short-run, both inputs in the Durham factory are fixed. What would be the profit maximizing output in the Chapel Hill factory? What would be the total output QsR of the firm? Calculate the profit SRthat the firm will make in the short run. b. In the long run, both capital and labor are adjustable. Find the long-run cost function C(q) for the factory in Durham. c. Suppose that the firm wants to produce Q units in total from the two factories in the long-run. How should production be allocated between the two factories in order to minimize the cost of production? Find the total cost CLR(Q) associated with this allocation.

Does the production function y=0.8x_1x_2 exhibit constant returns to scale, increasing returns to scale, decreasing returns to scale, or homothetic returns to scale

RBW produces light sticks for one of their legendary girl groups, Kara. Suppose that RBWuses capital, K, and labor, L, to produce the light sticks. The production function is given as: Q(K,L)=2K0.75L0.51. Identify if RBW’s production function is increasing, constant, or decreasing returns to scale with a brief explanation. Also, solve for the company’s cost minimization problem by providing conditional demand functions of labor and capital, with respect to their input prices. Assume that w is the wage of labor and r is the rent cost of capital.2. Suppose that RBW is preparing for a mini-fan meet for the group’s “When I move” mini-album launch and that they have to provide 200 light sticks. The wage of labor, w, is $6.00 and the cost of capital, r, is $2.00. Determine how much labor and capital they have to use and the total cost of producing the light sticks.

22. Suppose a firm has a production technology given by the Cobb-Douglas production function: Q = f(L,K) = AL¤ KB with a + ß 0, where L and K represent labour and capital respectively and A is technology. (a) Does this production function exhibit constant, increasing or decreasing returns to scale? Explain how you would be able to see this on an isoquant graph. (b) Set out the unconstrained profit maximisation problem (assuming input prices are w for labour and r for capital and the output price is p) and then solve for the unconditional input demand functions for capital and labour and find the firm's supply function (c) Now set up the cost minimisation problem and solve for the conditional labour and capital demands. What is the difference between the unconditional input demand functions from part (b) and the conditional input demand functions you have just derived? (d) Give an expression for the total cost function (no need to simplify) and explain how the average and marginal cost…