Question

Using the production function Q = ( and output levels of Q=2, Q=4, Q=6

V(L *K)

A). Suppose the price of L and K is $3/hr. On a graph show isocost lines corresponding to total costs of $12, $24, and $36. Using isoquants and isocost lines, locate three points on the expansion path and draw the expansion path. Show your calculations.

B). Using the three points on the expansion path, calculate the firm's long run total and average costs at each of thos points. Summarize your clculations in a table and in a graph.

C). Suppose amount of capital the firm uses in the short run is fixed at 4 units. On your isoquant/isocost diagram illustrate the three points corresponding to output levels of 2, 4, and 6. Calculate the firm's short run total and average costs and draw them in the same graph with the long run costs.

Answer 1

Q = L^1/2 k^1/2 rightarrow isoquant c = 3L + 3k rightarrow insolent C_1 = 3L + 3K = 12 C_2 = 32 + 3k = 24 C_3 = 32 + 3k = 36 Q_1 = 2 Q = 4 Q = 6 Z = L^1/2 k^1/2 T lambda [12 - 3L - 3K] = 1/2 L^-1/2 k^1/2 - 3 lambda = 1/2 L^1/2 k^-1/2 - 3 lambda => k/L = 1, k = 2 At optimum, 3L + 3K = 12 => k = 2, L = 2 => LTC = 12 (2, 2) LAC = 6 At Q = 4, 3L + 3K = 24 K = L = 4 LTC = 24, LAC = 6 At Q = 6, 3L + 3K = 36 K = L = 6 LTC = 36, LAC = 6 At optimum L = k => some SAC = LAC STC = LTC

In the above the SAC is the short run average cost, the LAC is the long run average cost, the STC is the short run total cost and the LTC is the long run total cost. The long run average cost is an envelope of the short run average costs.