Murray Manufacturing Company produces pantyhose. The firm's production function is given as: Q = 5LK, where Q = pairs of pantyhose, L = labor measured in person hours, and in machine hours. Murray's labor cost, including fringe benefits, is $20 per hour, while the firm uses $80 per hour as an implicit machine rental charge per hour. Murray's current budget is $64,000 per month to pay labor and capital.

a. Given the information above, determine Murray's optimal capital/labor ratio.

b. Using the Lagrangian technique, determine the quantities of labor and capital that will allow the firm to maximize output given their budgeted input expenditure. What is the firm's output?

c. Again using the Lagrangian technique, demonstrate the duality in production and cost theory