# LONG-RUN COST FUNCTIONS

In the long run, all inputs are variable, and a firm can have a number of alternative plant sizes and levels of output that it wants. There are no fixed cost functions (total or average) in the long run, since no inputs are fixed. A useful way of looking at the long run is to consider it a planning horizon. The long run cost curve is also called planning curve because it helps the firm in future decision making process.

The long run cost output relationship can be shown with the help of a long run cost curve. The long run average cost curve (LRAC) is derived from short run average cost curves (SRAC). Let us illustrate this with the help of a simple example. A firm faces a choice of production with three different plant sizes viz. plant size-1 (small size), plant size-2 (medium size), plant size-3 (large size), and plant size-4 (very large size). The short run average cost functions shown in Figure-4 (SRAC1, SRAC2, SRAC3, and SRAC4) are associated with each of these plants discrete scale of operation. The long run average cost function for this firm is defined by the minimum average cost of each level of output. For example, output rate Q1 could be produced by the plant size-1 at an average cost of C1 or by plant size-2 at a cost of C2. Clearly, the average cost is lower for plant size-1, and thus point a is one point on the long run average cost curve. By repeating this process for various rates of output, the long run average cost is determined. For output rates of zero to Q2 plant size-1is the most efficient and that part of SRAC1 is part of the long run cost function. For output rates of Q2 to Q3 plant size-2 is the most efficient, and for output rates Q3 to Q4, plant size-3 is the most efficient. The scallop-shaped curve shown in boldface in Figure 9.4 is the long run average cost curve for this firm. This boldfaced curve is called an envelope curve (as it envelopes short run average cost curves). Firms plan to be on this envelope curve in the long run. Consider a firm currently operating plant size-2 and producing Q1 units at a cost of C2 per unit. If output is expected to remain at Q1, the firm will plan to adjust to plant size-1, thus reducing average cost to C1.

Most firms will have many alternative plant sizes to choose from, and there is a short run average cost curve corresponding to each. A few of the short run average cost curves for these plants are shown in Figure-5, although many more may exist. Only one point of a very small arc of each short run cost curve will lie on the long run average cost function. Thus long run average cost curve can be shown as the smooth U-shaped curve. Corresponding to this long run average cost curve is a long run marginal cost (LRMC) curve, which intersects LRAC at its minimum point a, which is also the minimum point of short run average cost curve 4 (SRAC4). Thus, at a point a and only at a point a, the following unique result occurs:

SRAC = SRMC when LRAC = LRMC

The long run cost curve serves as a long run planning mechanism for the firm. It shows the least per unit cost at any output can be produced after the firm has had time to make all appropriate adjustments in its plant size. For example, suppose that the firm is operating on short run average cost curve SRAC3 as shown in Figure-5, and the firm is currently producing an output of Q*. By using SRAC3, it is seen that the firm’s average cost is C2. Clearly, if projections of future demand indicate that the firm could expect to continue selling Q* units per period at the market price, profit could be increased significantly by increasing the scale of plant to the size associated with short run average cost curve SRAC4. With this plant, average cost for an output rate of Q* would be C2 and the firm’s profit per unit would increase by C2 – C1. Thus, total profit would increase by (C2 – C1)* Q*.

The U-shape of the LRAC curve reflects the laws of returns to scale. According to these laws, the cost per unit of production decreases as plant size increases due to the economies of scale, which the larger plant sizes make possible. But the economies of scale exist only up to a certain size of plant, known as the optimum plant size where all possible economies of scale are fully exploited. Beyond the optimum plant size, diseconomies of scale arise due to managerial inefficiencies. As plant size increases beyond a limit, the control, the feedback of information at different levels and decision-making process becomes less  efficient. This makes the LRAC curve turn upwards. Given the LRAC in Figure-5, we can say that there are increasing returns to scale up to Q* and decreasing returns to scale beyond Q*. Therefore, the point Q* is the point of optimum output and the corresponding plant size-4 is the optimum plant size.

If you have long run average cost of producing a given output, you can readily derive the long run total cost (LRTC) of the output, since the long run total cost is simply the product of long run average cost and output. Thus, LRTC = LRAC * Q.

Figure-6 shows the relationship between long run total cost and output. Given the long run total cost function you can readily derive the long run marginal cost function, which shows the relationship between output and the cost resulting from the production of the last unit of output, if the firm has time to make the optimal changes in the quantities of all inputs used.

##### ECONOMIES AND DISECONOMIES OF SCALE

We have seen in the preceding section that larger plant will lead to lower  average cost in the long run. However, beyond some point, successively larger plants will mean higher average costs. Exactly, why is the long run average cost (LRAC) curve U-shaped? What determines the shape of LARC curve? This point needs further explanation.

It must be emphasized here that the law of diminishing returns is not applicable in the long run as all inputs are variable. Also, we assume that resource prices are constant. What then, is our explanation? The U-shaped LRAC curve is explainable in terms of what economists call economies of scale and diseconomies of scale.

Economies and diseconomies of scale are concerned with behaviour of average cost curve as the plant size is increased. If LRAC declines as output increases, then we say that the firm enjoys economies of scale. If, instead, the LRAC increases as output increases, then we have diseconomies of scale. Finally, if LRAC is constant as output increases, then we have constant returns to scale implying we have neither economies of scale nor diseconomies of scale.

Economies of scale explain the down sloping part of the LRAC curve. As the size of the plant increases, LRAC typically declines over some range of output for a number of reasons. The most important is that, as the scale of output is  expanded, there is greater potential for specialization of productive factors. This is most notable with regard to labour but may apply to other factors as well. Other factors contributing to declining LRAC include ability to use more advanced technologies and more efficient capital equipment; managerial specialization; opportunity to take advantage of lower costs (discounts) for some inputs by purchasing larger quantities; effective utilization of by products, etc.

But, after sometime, expansion of a firm’s output may give rise to diseconomies, and therefore, higher average costs. Further expansion of output beyond a reasonable level may lead to problems of overcrowding of labour, managerial inefficiencies, etc., pushing up the average costs.

In this section, we examined the shape of the LRAC curve. In other words, we have analysed the relationship between firm’s output and its long run average costs. The economies of scale and diseconomies of scale are sometimes called as internal economies of scale and internal diseconomies of scale respectively. This is because the changes in long run average costs result solely from the individual firm’s adjustment of its output. On the other hand, there may exist external economies of scale. The external economies also help in cutting down production costs. With the expansion of an industry, certain specialized firms also come up for working up the by-products and waste materials. Similarly, with the expansion of the industry, certain specialized units may come up for supplying raw material, tools, etc., to the firms in the industry. Moreover, they can combine together to undertake research etc., whose benefit will accrue to all firms in the industry. Thus, a firm benefits from expansion of the industry as a whole. These benefits are external to the firm, in the sense that these have arisen not because of any   effort on the part of the firm but have accrued to it due to expansion of industry as a whole. All these external economies help in reducing production costs.

Economies of scale are often measured in terms of cost-output elasticity, Ec. Ec is the percentage change in the average cost of production resulting from a one percent increase in output:

Ec = (WTC/TC) / (WQ/Q) = (WTC/ WQ) / (TC/Q) = MC/AC

Clearly, Ec is equal to one when marginal and average costs are equal. This means costs increase proportionately with output, and there are neither economies nor diseconomies of scale. When there are economies of scale MC will be less than AC (both are declining) and Ec is less than one. Finally, when there are diseconomies of scale, MC is greater than AC, and Ec is greater than one.