Suppose that a firm's production function is q = 10L 1/2K1/2 . 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 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 graphically and find the firm's new total 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) Use the method of Lagrangian multipliers to find the cost-minimizing combination of capital and labor for an output level of 140.