Suppose that a firm's production function is . The cost of a unit of labor is $20 and the cost of a unit of capital is $80.
a. The firm is currently producing 100 units of output, and has determined that the cost-minimizing quantities of labor and capital are 20 and 5 respectively. Graphically illustrate this situation on a graph using isoquants and isocost lines.
b. The firm now wants to increase output to 140 units. If capital is fixed in the short run, how much labor will the firm require? Illustrate this point on your graph and find the new cost.
c. Graphically identify the cost-minimizing level of capital and labor in the long run if the firm wants to produce 140 units.
d. If the marginal rate of technical substitution is , find the optimal level of capital and labor required to produce the 140 units of output.