Economics of Business Decisions

1. Consider production in a two-input economy, with inputs L and K as usual, but without any substitution possibilities. Specifically, suppose that every unit of L requires exactly four units of K if output is to be increased. (If the firm hires another unit of L and less than four more units of K, then output does not increase. If the firm hires another unit of L and more than four more units of K, then output rises by the same as if just four more units of K were purchased with the new unit of L.) Suppose that L costs the firm $90 per unit employed and K costs the firm $180 per unit employed. Given cost minimizing employment of the inputs, output is given by: Q =3L½. A. (7 pts.) Find the equation for minimized total cost in the long run, C(Q). (Hint: The standard rule for long run cost minimization that inputs should be employed so that the ratios of each input’s marginal product to its cost are equalized is not applicable here, because that rule applies when inputs can be substituted. Cost minimization is simple though; think about it.)

Since the two inputs cannot be substituted, even if MPL/w does not equal MPK/r, the firm cannot shift to make it equal to minimize cost at the same output level.

Given:
1) the cost minimizing output equation Q = 3L1/2  L(Q) = Q2/3
2) the restriction of increasing the output: L/K = 1/4  K = 4L
3) W = $90, r = $180

We know that when cost is minimized, the total cost in the long run:
C(Q) = wL(Q) + rK = wL(Q) + 4rL(Q) = (w+4r)L(Q) = (90+720)Q2/3 = 270Q2

B. (6 pts.) Find the equations for long-run marginal cost (LMC(Q)) and long-run average total cost (LAC(Q)).

LMC(Q) = dC(Q)/dQ = 270*2*Q = 540Q,
LAC(Q) = 270Q2/Q = 270Q.

C. (4 pts.) Suppose that the firm is in the short run, stuck with K*= 200. Find the equation for marginal cost in the short run.

In the short run, K is fixed, and L is variable. L is a function of Q: L(Q).

Since:
1) Q= 3L1/2  L(Q) = Q2/3,
2) K* =