# Analyzing production function

The manager of a national retailing outlet recently hired an economist to estimate the firm's production function. Based on the economist's report, the manager now knows that the firm's production function is given by Q = K^(1/2)*L^(1/2) and that capital is fixed at 1 unit.

a. Calculate the average product of labor when 9 units of labor are utilized.

b. Calculate the marginal product of labor when 9 units of labor are utilized.

c. Suppose the firm can hire labor at a wage of $10/hr and output can be sold at a price of $100 per unit. Determine the profit maximizing levels of labor and output.

d. What is the maximum price of capital at which the firm will still make non-negative profits?

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#### Solution Preview

a. Calculate the average product of labor when 9 units of labor are utilized.

Average Product of labor, APL =Q/L=(K^1/2*L^1/2)/L=K^(1/2)*L^(-1/2)

Put K=1 and L=9

APL=1^(1/2)*9^(-1/2)= 0.33333

b. Calculate the marginal product of labor when 9 units of labor are utilized.

Marginal Product of labor, MPL=dQ/dL=K^(1/2)*(1/2)*L^(1/2-1)

MPL=(1/2)*K^(1/2)*L^(-1/2)

Put K=1 and L=9

MPL=(1/2)*1^(1/2)*9^(-1/2)= ...

#### Solution Summary

Solution describes the steps to calculate average product, marginal product and optimal combination of inputs.