Suppose that a firm has production function F(L, K) = L^2/3 K^1/3 for producing widgets, the wage rate for labor is w = $400, and the rental rate of capital is r = $25. a) Suppose that the firm has received an order for Q = 120 units of output. Neatly specify this firm’s cost minimization problem, using the particulars associated with this problem. b) Give two equations that an interior solution satisfies, tailoring your equations to the particulars of this problem. c) Solve the two equations for the firm’s optimal choice. Show your work. d) Determine this firm’s minimum cost of producing 120 units. e) Now suppose that the firm’s production goal is left as the variable Q. Come up with the firm’s cost function C(Q). Show your work.

Suppose that a firm's production function is: 112 q= 10L05K0.5 98 Its marginal product functions are: 84 MP =5KL05 = 0.5q/L and 70 MP 5L05K5 =0.5g/K. The cost of a unit of labor, w, is $20 and the cost of a unit of capital, r, is $80 Isoquants for output of 140 and 280 are illustrated in the figure to the right 42 20 Initially, the firm is producing 140 units of output and has dotermined that the cost-minimizing quantities of labor and capital are 28 and 7, respectively. 14+ Suppose now that the firm wants to increase output to 280 units. If capital is fixed in the short run, how much labor will the firm require? 9. 14 25 42 04 LUnits of labor The firm will now require units of labor. (Enter your response rounded to two decimal places. ) K Unts of capta

The production function for a product is given by q = 100KL. Its marginal product functions are MPL = 100K and MPK = 100L. If the price of capital, r, is $120 per day and the price of labor, w, is $30 per day, what is the minimum cost of producing 20,000 units of output? The minimum cost of producing 20,000 units of output is $. (Round all intermediate results to 3 decimal places and enter your response rounded to two decimal places.)

Miguel and Jake run a paper company. Each week they need to produce 1,000 reams of paper to ship to their customers. The paper plant's long-run production function is: q= 4K^L+ where q is the number of reams produced, K is the quantity of capital rented, and L is the quantity of labor hired. K For this production, MP, =| L and MP, = 34 K K The weekly cost function for the paper plant is C=10K+2L where C is the total weekly cost How much capital and labor will Miguel and Jake need to rent and hire in order to produce 1,000 reams of paper each week, and how much will hiring these inputs cost them?