5. Suppose that a firm produces park benches and the production function is Q=2KL. The firm's MPL=LK and the firm's MPK=KL. Suppose that the firm's goal is to produce Q=200 in the cost minimizing way. The cost of a unit of capital (K) is $40(r=$40) and the cost of a unit of labor is $40(w=$40). a. Write an equation for the firm's MRTS with K and L in it. Simplify it as much as possible. b. If the firm is cost minimizing, the marginal benefit of one more unit of L in terms of K. (MRTS) will be equal to the marginal cost of one more unit of L in terms of K(rw). Write an equation that shows the MRTS =rw for this park bench producing firm. c. Write an equation for the isoquant when the firm is producing Q=200 (just put Q=200 into the production function: Q=2KL). d. Use the information from parts b and c to find the cost minimizing combination of K and L for producing Q=200. Show your work: Two equations 2 unknowns: -K=-MRTS=L=rw= e. At these levels of K and L, what are the (show your work): - TC= - ATC= f. Suppose the firm producing benches wants to decrease their output to 60 . However, in the short run capital is fixed at 100:K=100. - How much L is needed to produce Q=60 in the short run? Show your work. - What is the total cost of producing Q=60 in the short run? Show your work. - What is the average total cost of producing Q=60 in the short run? Show your work. - MPL= - MPK= - MRTS (use the equation from part a) = - rw= g. Suppose the firm producing benches wants to decrease their output to 60 . In the long run when the firm is cost minimizing, what are the following (show your work) - K= - L= - MRTS= - rw= - TC= - ATC= When producing Q=60, are the firm's ATC higher in the long run or the short run (when K=100 )? Explain. Compare the ATC when the firm is cost minimizing at Q=200 and Q=60. Do they differ? If so, why? If not, why not? Use returns to scale in your response. 5. Suppose that a firm produces park benches and the production function is Q=2KL. The firm's MPL=LK and the firm's MPK=KL. Suppose that the firm's goal is to produce Q=200 in the cost minimizing way. The cost of a unit of capital (K) is $40(r=$40) and the cost of a unit of labor is $40(w=$40). a. Write an equation for the firm's MRTS with K and L in it. Simplify it as much as possible. b. If the firm is cost minimizing, the marginal benefit of one more unit of L in terms of K. (MRTS) will be equal to the marginal cost of one more unit of L in terms of K(rw). Write an equation that shows the MRTS =rw for this park bench producing firm. c. Write an equation for the isoquant when the firm is producing Q=200 (just put Q=200 into the production function: Q=2KL). d. Use the information from parts b and c to find the cost minimizing combination of K and L for producing Q=200. Show your work: Two equations 2 unknowns: -K=-MRTS=L=rw= e. At these levels of K and L, what are the (show your work): - TC= - ATC= f. Suppose the firm producing benches wants to decrease their output to 60 . However, in the short run capital is fixed at 100:K=100. - How much L is needed to produce Q=60 in the short run? Show your work. - What is the total cost of producing Q=60 in the short run? Show your work. - What is the average total cost of producing Q=60 in the short run? Show your work. - MPL= - MPK= - MRTS (use the equation from part a) = - rw= g. Suppose the firm producing benches wants to decrease their output to 60 . In the long run when the firm is cost minimizing, what are the following (show your work) - K= - L= - MRTS= - rw= - TC= - ATC= When producing Q=60, are the firm's ATC higher in the long run or the short run (when K=100 )? Explain. Compare the ATC when the firm is cost minimizing at Q=200 and Q=60. Do they differ? If so, why? If not, why not? Use returns to scale in your response
