Question

Consider the production function Q= 20K^(1/2)L^(1/2). The firm operates in the short run with 16 units of capital.

a. The firm's short-run production function is Q=?
b. The average product of labor function is AP=?
c. The marginal product of labor function is MP=?
d. Show that marginal product diminishes for all levels of labor usage.

For the short-run production function in exercise 1, let the wage be $20.

a. Derive AVC(Q).
b. When 160 units are produced, ______ units of labor are employed, and the average product is ____. Average variable cost is $______.
c. Derive SMC(Q).
d. Using the marginal product (MP) function derived in part c, the marginal product is _____ when 160 units are produced. SMC is $_______. Verify that SMC (Q) evaluated at Q=4 is identical to calculating SMC by using the ratio w/MP.

Please explain fully because I am really lost.

Answer 1

Minimum AVC occurs when MP=AP.
This occurs at 80 units of labor and 20000 units of output (AP x L = 20000). From this AVC = w / AP = 2 / 250 = .008
For part 4, at 100 units of labor, AP = 225 so Q = 22500 (225 x 100). You can calculate TVC = $2 x 100 units = 200, then calculate AVC = TVC / Q = 200 / 22500 = .0088