Now that you understand how long-run production decisions determine the structure of long-run costs, we can demonstrate more clearly the important relations between short-run and long-run costs. As we explained at the beginning of Chapter 8, the long run or planning horizon is the collection of all possible short-run situations, one for every amount of fixed input that may be chosen in the long-run planning period. For example, in Table 8.2 in Chapter 8, the columns associated with the 10 levels of capital employment each represent a different short-run production function, and, as a group of short-run situations, they com- prise the firm’s planning horizon. In the first part of this section we will show you how to construct a firm’s long-run planning horizon—in the form of its long-run average cost curve (LAC)—from the short-run average total cost (ATC) curves as- sociated with each possible level of capital the firm might choose. Then, in the next part of this section, we will explain how managers can exploit the flexibil- ity of input choice available in long-run decision making to alter the structure of short-run costs in order to reduce production costs (and increase profit).
Long-Run Average Cost as the Planning Horizon
To keep matters simple, we will continue to discuss a firm that employs only two inputs, labor and capital, and capital is the plant size that becomes fixed in the short run (labor is the variable input in the short run). Since the long run is the set of all possible short-run situations, you can think of the long run as a catalog, and each page of the catalog shows a set of short-run cost curves for one of the possible plant sizes. For example, suppose a manager can choose from only three plant sizes, say plants with 10, 30, and 60 units of capital. In this case, the firm’s long-run plan- ning horizon is a catalog with three pages: page 1 shows the short-run cost curves when 10 units of capital are employed, page 2 shows the short-run cost curves when 30 units of capital are employed, and page 3 the cost curves for 60 units of capital.
The long-run planning horizon can be constructed by overlaying the cost curves from the three pages of the catalog to form a “group shot” showing all three short- run cost structures in one figure. Figure 9.16 shows the three short-run average total cost (ATC) curves for the three plant sizes that make up the planning horizon in this example: ATCK510, ATCK530, and AT C __K 560 . Note that we have omitted the associated AVC and SMC curves to keep the figure as simple as possible.
When the firm wishes to produce any output from 0 to 4,000 units, the manager will choose the small plant size with the cost structure given by ATCK510 because the average cost, and hence the total cost, of producing each output over this range is lower in a plant with 10 units of capital than in a plant with either 30 units or 60 units of capital. For example, when 3,000 units are produced in the plant with 10 units of capital, average cost is $0.50 and total cost is $1,500, which is better than spending $2,250 (5 $0.75 3 3,000) to produce 3,000 units in the medium plant with 30 units of capital. (Note that if the ATC curve for the large plant in Figure 9.16 is extended leftward to 3,000 units of production, the average and total cost of producing 3,000 units in a plant with 60 units of capital is higher than both of the other two plant sizes.)
When the firm wishes to produce output levels between 4,000 and 7,500 units, the manager would choose the medium plant size (30 units of capital) because ATC __K 530 lies below both of the other two ATC curves for all outputs over this range. Following this same reasoning, the manager would choose the large plant size (60 units of capital) with the cost structure shown by ATC __K 560 for any output greater than 7,500 units of production. In this example, the plan- ning horizon, which is precisely the firm’s long-run average cost (LAC) curve, is formed by the light-colored, solid portions of the three ATC curves shown in Figure 9.16.
Firms can generally choose from many more than three plant sizes. When a very large number of plant sizes can be chosen, the LAC curve smoothes out and typically
ATCK=10 r
m
g e
f s
Output 0
0.80 0.75 0.72
0.50 0.30
2,000 3,000 4,000 5,000 7,500 10,000 12,000
Average cost (dollars)
ATCK=30
ATCK=60 LAC F I G U R E 9.16
Long-Run Average Cost (LAC) as the Planning Horizon
takes a ứ-shape as shown by the dark-colored LAC curve in Figure 9.16. The set of all tangency points, such as r, m, and e in Figure 9.16, form a lower envelope of average costs. For this reason, long-run average cost is called an “envelope” curve.
While we chose to present the firm’s planning horizon as the envelope of short- run average cost curves, the same relation holds between the short-run and long- run total or marginal cost curves: Long-run cost curves are always comprised of all possible short-run curves (i.e., they are the envelope curves of their short-run counterparts). Now that we have established the relation between short- and long-run costs, we can demonstrate why short-run costs are generally higher than long-run costs.
Restructuring Short-Run Costs
In the long run, a manager can choose any input combination to produce the de- sired output level. As we demonstrated earlier in this chapter, the optimal amount of labor and capital for any specific output level is the combination that minimizes the long-run total cost of producing that amount of output. When the firm builds the optimal plant size and employs the optimal amount of labor, the total (and average) cost of producing the intended or planned output will be the same in both the long run and the short run. In other words, long-run and short-run costs are identical when the firm produces the output in the short run for which the fixed plant size (capital input) is optimal. However, if demand or cost conditions change and the manager decides to increase or decrease output in the short run, then the current plant size is no longer optimal. Now the manager will wish to restructure its short-run costs by adjusting plant size to the level that is optimal for the new output level, as soon as the next opportunity for a long-run adjustment arises.
We can demonstrate the gains from restructuring short-run costs by returning to the situation presented in Figure 9.4, which is shown again in Figure 9.17. Recall that the manager wishes to minimize the total cost of producing 10,000 units when the price of labor (w) is $40 per unit and the price of capital (r) is $60 per unit.
As explained previously, the manager finds the optimal (cost-minimizing) input combination at point E: L* 5 90 and K* 5 60. As you also know from our previous discussion, point E lies on the expansion path, which we will now refer to as the
“long-run” expansion path in this discussion.
We can most easily demonstrate the gains from adjusting plant size (or capital levels) by employing the concept of a short-run expansion path. A short-run expansion path gives the cost-minimizing (or output-maximizing) input combination for each level of output when capital is fixed at __K units in the short run. To avoid any confusion in terminology, we must emphasize that the term “expansion path” always refers to a long-run expansion path, while an expansion path for the short run, to distinguish it from its long-run counterpart, is always called a short-run expansion path.
Suppose the manager wishes to produce 10,000 units. From the planning hori- zon in Figure 9.16, the manager determines that a plant size of 60 units of capital is the optimal plant to build for short-run production. As explained previously, once the manager builds the production facility with 60 units of capital, the firm
Now try Technical Problem 11.
short-run expansion path Horizontal line show- ing the cost-minimizing input combinations for various output levels when capital is fixed in the short run.
operates with the short-run cost structure given by ATC __K 560 . This cost structure corresponds to the firm’s short-run expansion path in Figure 9.17, which is a hori- zontal line at 60 units of capital passing through point E on the long-run expansion path. As long as the firm produces 10,000 units in the short run, all of the firm’s inputs are optimally adjusted and its long- and short-run costs are identical: Total cost is $7,200 (5 $40 3 90 1 $60 3 60) and average cost is $0.72 (5 $7,200/10,000).
In general, when the firm is producing the output level in the short run using the long-run optimal plant size, ATC and LAC are tangent at that output level. For example, when the firm produces 10,000 units in the short run using 60 units of capital, ATC __K 560 is tangent to LAC at point e.
If the manager decides to increase or decrease output in the short run, short- run production costs will then exceed long-run production costs because input levels will not be at the optimal levels given by the long-run expansion path. For example, if the manager increases output to 12,000 units in the short run, the man- ager must employ the input combination at point S on the short-run expansion path in Figure 9.17. The short-run total cost of producing 12,000 units is $9,600 (5 $40 3 150 1 $60 3 60) and average total cost is $0.80 (5 $9,600/12,000) at point s in Figure 9.17. Of course, the manager realizes that point F is a less costly input combination for producing 12,000 units, because input combination F lies on a lower isocost line than S. In fact, with input combination F, the total cost of producing 12,000 units is $9,000 (5 $40 3 120 1 $60 3 70), and average cost is
$0.75 (5 $9,000/12,000), as shown at point f in Figure 9.16. Short-run costs exceed
Q1=10,000
Q2=12,000
Labor (L) 0
160150
120
70 60
90 120 150 180 225 240
Capital (K)
Short-run expansion path (K=70) Long-run
expansion path
Short-run expansion path (K=60)
E F S
F I G U R E 9.17
Gains from Restructuring Short-Run Costs
long-run costs for output levels below 10,000 units as well, because a plant size of 60 units of capital is too big (i.e., larger than the optimal plant size) for every out- put below point E on the long-run expansion path.
At the next opportunity to adjust plant size, the manager will increase plant size to 70 units, as long as the firm plans to continue producing 12,000 units. In- creasing capital to 70 units causes the short-run expansion path to shift upward as shown by the broken horizontal line in Figure 9.17. By restructuring short-run pro- duction, the manager reduces the short-run total costs of producing 12,000 units by $600 (5 $9,600 2 $9,000). As you will see in Part IV, firms can increase their profits—sometimes even convert losses to profits—by adjusting their fixed inputs to create a lower cost structure for short-run production operations. We can now summarize this discussion with the following principle.
Principle Because managers have the greatest flexibility to choose inputs in the long run, costs are lower in the long run than in the short run for all output levels except the output level for which the fixed input is at its optimal level. Thus the firm’s short-run costs can generally be reduced by adjusting the fixed inputs to their optimal long-run levels when the long-run opportunity to adjust fixed inputs arises.