CONSTRAINED OPTIMIZATION

Capital and Labor are never free and budget is never anything one wants. The production function is the same. The market price of capital is $1, that is, P_k = 1 and the price of labor is $2, that is, P_L = 2. The firm has a budget of $5 to spend on capital labor.

a. Calculate and solve the first-order conditions for the optimum capital and labor levels and the Lagrangean multiplier.

b. If I spend an additional dollar on labor, how much additional output would I get? (remember it really is not a full dollar but as budget increases by a small amount but as the derivative is a ratio I explain it in per unit terms). How about capital?

c. Calculate the second-order condition.

d. If the firm allocated more budget, at what rate would output rise?

e. Now suppose the firm increases it budget to $16. Solve the problem as you did in part above. What is the value of lamda (the Langangean multiplier)? Why did it take on the value you calculated. How constraining is the budget?