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Special Appendix to Chapter 6 Short-Run and Long-Run Costs 279

and SMC dC/dQ Q/(2K). It is easy to check that SAC is U-shaped. For

instance, by setting Kequal to 9 thousand square feet, we obtain the SAC function:

This is the equation of the first SAC curve graphed in Figure 6.3. By setting

K18 thousand and K27 thousand, we trace out the other SAC curves in

the figure.

LONG-RUN COSTS We can now confirm that the firm’s LAC and LMC curves

are constant, as shown in Figure 6.3. One way of doing so is to note that the

level of LAC is given by the minimum point of each SAC curve. Returning to

the SAC expression in Equation 6A.3, we can show that the point of minimum

average cost occurs at output

[6A.4]

To see this, remember that the SMC curve intersects the SAC curve at its min-

imum point. Equating the preceding expressions for SAC and SMC, we find

16K/Q Q/4KQ/2K. The solution is Q^2 64(K)^2 , or Q 8K. After

substituting Q 8Kinto Equation 6A.3, we find the firm’s minimum average

cost to be min SAC 16K/8K8K/4K4/unit. In turn, substituting

Q 8Kinto Equation 6A.2 implies

[6.A5]

This equation specifies the necessary amount of labor to be used in conjunc-

tion with a plant of size K.

In summary, Equation 6A.3 describes the short-run average cost of pro-

ducing Q units of output using Kunits of fixed capital, whereas Equation 6A.2

specifies the requisite amount of labor. For instance, the short-run average cost

of producing 54 thousand units of output in a 9-thousand-square-foot plant is

SAC 144/54 54/36 $4.17. The necessary amount of labor is L 

(54)^2 /[(144)(9)] 2.25 thousand labor-hours. In turn, the LAC of producing

Q units is $4 and is achieved using the amounts of capital and labor given by

Equations 6A.4 and 6A.5. Thus, to produce 54 thousand units in the long run,

the firm should use a 54,000/8  6.75-thousand-square-foot plant and

(4/9)(6.75) 3 thousand labor-hours.

Questions and Problems

1. a. Using Equation 6A.3, find the short-run average cost of producing
   108 thousand units in a 9-thousand-square-foot plant. Determine the
   necessary quantity of labor.
   b. Would the same output be less expensive to produce using an
   18-thousand-square-foot plant?

L(4/9)K.

Q8K.

SAC144/QQ/36.

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