Short Run and Long Run Average Cost Curve

In this article we will discuss about the relationship between the short run and long run average cost curve.

Any discussion of a firm’s long-run cost behaviour starts with the proposition that in general firms plan in the long run and operate in the short run. In other words, the long run is the firm’s planning period and the short run its production period. Indeed, we call the long run the firm’s planning horizon. The long-run cost function gives the most efficient (the least-cost) method of producing any given level of output, because all inputs are variable.

But, once a particular firm size is chosen and the firm starts producing, the firm is in the short run. Plant and equipment have already been constructed. Now, if the firm wishes to change its level of output it cannot vary the usage of all inputs. Some inputs, the plant and so forth, are fixed to the firm. Thus, the firm cannot vary all inputs optimally and therefore cannot produce this new level of output at the lowest possi­ble cost.

Such a situation is shown in Fig. 13 — LRTC is the firm’s long- run total cost curve. Suppose that, when making its plans, the firm had decided that it wanted to pro­duce Q₀ units of output at the low­est possible cost, which is shown as TC₀ in the figure. Since it would not wish to vary any of its inputs so long as it continues to produce Q₀, the short-run cost of producing Q₀ is the same as the long-run cost.

Thus, the short-run total cost curve, SRTC, is tangent to LRTC at Q₀. But, .since some inputs are fixed in the short run, it cannot produce the new output at the lowest possible cost. At any output other than Q₀, the short-run input combination is less efficient—would result in a higher cost—than the combination that would be chosen if all inputs were variable.

Suppose, for example, that the firm wants to increase its output from Q₀ to Q₁. If all inputs were variable, it could produce this new output at TC_L. But its plant and some other inputs are fixed; so SRTC gives the cost of producing Q₁. This cost is TC_S, which is clearly higher than TC_L. The fixed cost is F and the variable cost is TC_S—F. Clearly, from Fig.13, SRTC is higher than LRTC at any output other than Q₀. Only at output Q₀ are the two curves equal.

The relation between LRTC and SRTC will, of course, determine the relation between the long-run and short-run average cost curves. Given the total cost curves in Figure 13, short-run average cost will be equal to long-run average cost only at an output of Q₀. (Since LRAC=LRTC is equal to SRTC). At any other level of output, short-run average cost is higher than long-run average cost, because SRTC is greater than LRTC.

Fig. 14 shows the typical relation between short and long-run average and marginal cost curves. In this figure, LRAC and LRMC are the long-run average and marginal cost curves.

Three short-run situ­ations are indicated by the three sets of curves:

1. SRAC₁, MC₁;

2. SRAC₂, MC₂ and

3. SRAC₃, MC₃.

We may now look at SRAC₁ and MC₁. These are the short- run curves for the plant size designed to produce output Q_S optimally. Since the short-run total cost curve would be tangent to the long-run total cost curve at this output, the two average cost curves are also tangent at this output.

Since marginal cost is given by the slope of the total cost curve, long-run marginal cost equals short-run marginal cost at the output given by the point of tangency, Q_S. Finally, short-run marginal cost crosses short-run average cost at the latter’s minimum point. We may note that since Q_S is on the decreasing portion of LRAC, SRAC₁ must also be decreasing at the point of tangency.

SRAC₃ and MC₃ show another short-run situation—a different plant size. Here tangency occurs at Q_L on the increasing part of LRAC. Thus, SRAC₃ is increasing at this point also. Again the two marginal curves are equal at Q_L, and MC₃ crosses SRAC₃ at the minimum point of the latter.

Finally, SRAC₂ is the short-run average cost curve corresponding to the output level — plant size — at which long-run average cost is at its minimum. At output level Q_m the two average cost curves are tangent. The two marginal cost curves, MC₂ and LRMC₂, are also equal at this output. And, since both average cost curves attain their minimum at Q_m, the two marginal cost curves must intersect the two average cost curves. Thus, all four curves must intersect the two average cost curves, i.e., all four curves are equal at output Q_m. That is, at Q_m, LRAC=SRAC₂= MC₂= LRMC.

If the firm is limited to producing only with one of the three short-run cost structures shown in Fig.2, it would choose the cost structure—plant size—given by SRAC₁ to produce outputs from zero to Q₁, because the average cost, and hence the total cost of producing each output over this range, is lower under this cost specification.

If it wishes to produce any output between Q₁ and Q₂, it would choose the plant size given by SRAC₂. This average cost curve lies below either of the other two for any output over this range. It would choose the cost structure shown by SRAC₃ for any level of output greater than Q₂.

But, typically the firm is not limited to three sizes — large, medium or small. In the long run it can build the plant whose size leads to lowest average cost. The long-run average cost curve is a planning device, because this curve shows the least cost of producing each possible output. Managers therefore are normally faced with a choice among a wide variety of plant sizes.

Conclusion:

The long-run planning curve, LRAC, is a locus of points representing the lowest possible unit cost of producing the corresponding output. The manager determines the desired size of plant by reference to this curve, selecting the short-run plant that yields the lowest unit cost of producing the volume of output desired.