In previous posts we made a distinction between relevant and irrelevant costs. Any negative-implications phenomenon which is consequent upon the production process is a relevant cost, whether it is denominated in money terms or not. We also noted that there are some negative-implications phenomena which are not relevant to a particular decision context, but often are mistaken as being relevant to it.
One example lies in the distinction between direct costs and overhead costs, or to use the economist’s preferred terms, between variable and fixed costs. Variable- or direct costs are those which vary with the level (or rate) of productive output. Variable costs are always relevant to the rate-of-production decision. Fixed or overhead costs are associated with the existence of the manager, the plant, and the equipment. Examples are contractual salaries and insurance premiums. They continue at the same levels or rates irrespective of the rate of production, even if it is zero. Once the plant has been put in place, these fixed or overhead costs are, so to speak, “sunk” costs, and sunk costs are not relevant to any rate-of-production decisions.
The distinction between variable and fixed costs permitted us in the past posts to distinguish between the time frames of the short and long runs. In regard to costs, the short run is the period of time within which some contractual obligations associated with management, plant, and equipment are not alterable by changing the firm’s managerial capacity or its scale of operations. The duration of the short run of course varies from enterprise to enterprise and situation to situation, and thus cannot be specified in discrete terms.
In the long run all aspects of the enterprise’s operations can be adjusted, so all costs are variable in the long run. Yet, as we noted in previous posts, any long run consists of a sequence of short runs. All decisions affecting both the enterprise’s scale and rate of operation are made in short-run settings, even those decisions affecting the long runs. The distinction between the short and long runs may be more pertinent to academic analysis than to operational decision making. But once we have distinguished the concepts of the short and the long runs, we can assert that the costs which are relevant to short-run decisions (i.e., the rate of production) include no overhead or fixed costs (i.e., sunk costs are “gone” costs and hence irrelevant to short-run decision making). Fixed costs, though irrelevant to rate-of-production decisions in the short run, become relevant to the scale-of-operations decisions of the long run.
Before the making of short-run output and pricing decisions, overhead costs should be ignored as irrelevant costs; after the point of decision, they may be regarded purely as information to be considered in any forth-coming long-run decision set. Particularly, the decision maker should not attempt to set price to cover overhead costs or total costs (including overhead); the output decision should not be oriented specifically toward the “spreading of the overhead.”
After the fact of selecting the price and output upon appropriate criteria, the manager may observe in retrospect whether or not total costs were covered, and by how much the overhead was spread across the number of units produced. If the total costs were not covered, or the overhead costs were not met, then a long-run change may be warranted.
In the long run all aspects of the enterprise’s circumstances are changeable; all costs are variable. The impetus for long-run change may come from a variety of sources:
- recognition from short-run decision contexts that total and overhead costs are not being adequately covered
- increasing or decreasing demand which requires changes in the scale of operations
- the departure or acquisition of managerial ability and entrepreneurial capacity
- technological advances which alter production functions or result in new products
and yet other possibilities.
Possibilities (3) and (4) above will cause shifts and perhaps twists or warps in the production surface as described previously, with consequent alterations in the short-run cost curves which may be generated from any total product curve sliced through the production surface. How any such change in technology, management, or entrepreneurship likely affect costs must be analyzed on the merits of each change?
We can reach some general conclusions on how costs are affected when the long-run change is to adjust the scale of the enterprise’s operation by constructing a larger or smaller plant. A larger plant than the one represented by the K1 quantity of capital results in another slice through production function surface at, say K2. The total product curve drawn from K2 likely rises higher into the output dimension, and may reach its peak at a larger volume of labor utilization than does the K1 total product curve. Any number of successively larger plants could be represented by cutting total product slices through the production surface at quantities of capital greater than K2.
The total cost curves associated with the K2 plant likely lie farther to the right than those associated with plant K1, as illustrated by TC2 relative to TC1. Correspondingly, the average total cost curve for plant K2, illustrated by ATC2 likely lies slightly to the right of the position of ATC1 for plan K1, and ATC2 may reach minimum at a lower per-unit cost level than does ATC1.
An idealized representative set of such ATC curves might result from building a sequence of ever larger plants, given the same technology and managerial capacity. This is an admittedly heroic representation since most firms would have the occasion to build only one plant for any particular technology. However, if we may indulge in the presumption that the management might at least contemplate the construction of any of a range of possible plant sizes, we may note that the successively larger plants initially achieve ever higher and wider output ranges accompanied by falling per-unit costs. This occurs up to point A. Economists describe this phenomenon as the range of economies of scale and note that the decreasing costs correspond to the phenomenon of increasing returns to scale described in previous posts. Increasing returns to scale, with the consequent scale economies, are attributable to greater division and specialization in the use of labor, or (and perhaps to say the same thing in other terms) more efficient use of plant and equipment.
Beyond point A in the ATC curves, though they reach higher output ranges, incur ever increasing per-unit costs. Economists describe this as the range of diseconomies of scale which are associated with the phenomenon of decreasing returns to scale. Scale diseconomies are most often attributed to limitations on the abilities of management to coordinate and control the ever more complex productive relationships involved in the larger-scale operations.
The curve in is tangent to the sequence of ATC curves is a long run average total cost curve, LATC. It is also referred to as an “envelope” curve because it envelopes the sequence of ATC curves.
The LATC curve is an idealized representation of production conditions in the long run because, given its U-shape, it contains ranges representing both scale economies (the down-sloping portion) and scale diseconomies (the up-sloping portion). Other shapes for the LATC curve are possible, and empirical data for a variety of industries suggests that some of the other shapes may be more common than the theoretical U-shape. There are three possible alternate shapes. This suggests possible early (small-plant) scale economies, but after a rather small least-cost plant size has been reached, diseconomies of scale if the company attempts to build larger-scale plants. These long-run cost conditions are conducive to the development of a market populated by a rather larger number of smaller firms (or firms with a multiplicity of small-capacity plants). Alternately, a market populated by a large number of small firms will likely be characterized by a long-run cost curves.
Depicting long-run cost conditions characterized by the progressive, though gradual, exploitation of economies of scale never seem to reach exhaustion. A market characterized by these long-run cost conditions will tend to be populated by a rather smaller number of larger-sized firms, and perhaps ever fewer firms as time passes and competition ensues. Economists classify such a market as oligopolistic. The ultimate end of competitive evolution in such a market might be a single, very large surviving firm, a so-called “natural monopoly.” Real-world industries with a small number of large firms are likely have reached such a state by exploiting scale economies while realizing no significant diseconomies of more growth.
The LATC may have involved a few minor scale economies as firms grew from very small size at start-up, and could eventually reach a range of diseconomies of large scale if existing firms grow much larger. But the salient characteristic of the LATC curve is that it is flat-bottomed, i.e., there are no significant scale economies to be exploited or diseconomies to be suffered over a very wide range of plant sizes. Another way to say this is that there appears to be no purely cost-related incentives or penalties to further growth.
The significant implication of a flat-bottomed LATC curve is that small-sized firms (or firms with small plants) may coexist with and thrive along side of much larger firms (or firms with very large plants). It is not possible to predict with confidence whether such a market will be populated by many or few firms with large or small plants, but many economists believe that the growth pressures felt by managerial personnel in such industries likely militate in favor of an evolution toward a smaller number of larger firms. These long-run cost conditions may be even more favorable to the oligopolistic evolution. Empirical data for a variety of industries suggest that many are in fact characterized by flat bottomed LATC curves.
We shall draw our discussion of costs in the long run to a close with a reconsideration of the implications of technological, managerial, and entrepreneurial change. It may be recalled from previous posts that changes in any of these factors likely shifts the entire production surface upward or downward from its previous position, and may have the effect of twisting or warping it with respect to the input axes. These may be exogenously sourced changes to which the management must adjust; for example, a key member of the management staff may leave the enterprise or die, or some natural disaster may destroy the plant or equipment. Other such changes may be sought and initiated by the management of the firm in response to changing market realities or scientific developments. An example of the former might consist in the falling market price of one input or the rising market price of another input which induces management to seek to replace existing technologies with those using the lower-priced inputs more intensively, possibly even displacing the higher-priced inputs to the extent that they are substitutable. Technological advances not only produce new products for which new production processes are developed and production functions must be formulated; they may also render some inputs so much more productive than they were formally that managers will find incentive to replace the old technologies with the new as occasion arises.
Fundamentally there are two ways for a managerial decision maker to regard the information about costs presented thus-far in this post: grasp the essential theoretical relationships and employ the understanding of them to make “seat-of-the-pants” (but informed) decisions (as suggested by Fritz Machlup, “Marginal Analysis and Empirical Research,” Essays in Economics and Semantics, W. W. Norton & Company, 1967); or, take the information provided thus-far as a platform for departure into the realm of attempting to model the enterprise’s cost functions by mathematical specification and statistical estimation. It is likely that the vast majority of all who study the behavior of costs take the former route, so we give the reader an opportunity at this point to skip to the last section of this post.
One who chooses to attempt to model the cost function may use an intuitive rather than a statistical approach to the specification process. This involves making an informed judgment about the likely shape of the cost function (e.g., linear, quadratic, cubic). The function is then tailored to the specific situation by guessing at the magnitudes and signs of the constant coefficients of the variables in the equation of the function which will make it simulate the actual behavior of costs.
Since no statistical curve-fitting procedures are employed, the validation process involves trying the model equation as data become available, and then making adjustments to the magnitudes and a sign of the coefficients until the function behaves like the actual cost phenomenon. The analyst may go through several iterations of adjustment before getting the function to behave closely enough to the real situation. If one is attempting to model the cost relationships for a new product or a new production technology for which no historic data exist or can be generated, the intuitive approach may be the only reasonable way to proceed.
The statistical estimation of a cost function is an arduous task at best, and one which has a high risk of yielding an unsatisfactory result. Why should one go to the effort? If one can adequately model the enterprise’s cost conditions, the cost model can be used to simulate (or “what-if”) the effects upon his costs of output and other changes. And if appropriate data are at hand, the statistical procedures may be able to do a better job in tailoring the equation to the specific situation than can the intuitive approach.
The functional-notation representation of a cost function can be given as
(1) C = f(X1, X2, … , Xn ),
where C is cost, Q is output, and X1 through Xn are possible independent variables which the analyst feels may be significant to the explanation of the behavior of costs. One obvious candidate for inclusion as an independent variable in time series data is t, a time period counter to indicate trend if the productive capacity of the capital equipment is thought to diminish due to weathering or obsolescence. Other independent variables to be included may be dummy variables (values of 0 or 1) to indicate the presence or absence of some condition, for example, whether or not the work force in each plant is unionized.
The analyst may attempt to model short- or long-run cost conditions. If the objective is analysis of the short-run, then the analyst may choose to model a total variable cost, an average variable cost, or a marginal cost. It really does not matter which form is chosen to model since it is possible by algebraic manipulation to derive the other two forms. For short-run analysis, if the enterprise produces only one product, the analyst could just as well model the TC relationship as the TVC relationship since TC and TVC have the same shapes (i.e., they are parallel along verticals) and share a common MC. However, if (as is usually the case) the enterprise produces a multiplicity of products, some of which may be jointly produced, there is a problem of how to allocate the overhead costs to the various products. This problem can be avoided by choosing to analyze only direct costs, i.e., the TVC relationship, and this is entirely appropriate since overhead costs are not relevant to short-run decision making anyway. If the object is to model the long-run cost relationship, then the appropriate focus of attention should be upon TC (i.e., including both direct and overhead costs) since in the long run all costs are variable and there is no difference between TC and TVC.
The usual statistical-estimation procedure is to subject the available data to regression analysis assuming a higher-order relationship than is expected. Then, if the highest-ordered terms are not statistically significant, the next lower-ordered regression equation may be tested. The highest-ordered total cost equation tested usually is cubic (the conceptual sense of an equation of higher-order than 3 is obscure) of form
TVC = b0 + b1Q + b2Q2 + b3Q3,
from which may be derived
MC = dTC/dQ = b1 + 2 b2Q + 3 b3Q2
AVC = TVC/Q = b0/Q + b1 + b2Q + b3Q2.
If b3 turns out not to be statistically significant but both b1 and b2 are significant, the model may be respecified as a quadratic of form
TVC = b0 + b1Q + b2Q2,
from which may be derived
MC = dTVC/dQ = b1 + 2 b2Q
AVC = TVC/Q = b0/Q + b1 + b2Q.
And if upon regression analysis of the quadratic form the coefficient b2 appears not to be statistically significant, the equation may be respecified as a linear form,
TVC = b0 + b1Q,
from which may be derived
MC = dTVC/dQ = b1,
AVC = TVC/Q = b0/Q + b1.
If none of the coefficients, b1, or b2, or b3 are statistically significant, then the effort to model the cost relationship by statistical means has failed.
The theoretically expected shapes of the TVC, AVC, and MC functions for a cubic (3rd order) relationship can be derived, illustrating the expected shapes of the quadratic (second order) and linear (first order) cost functions, respectively. Since so many real total cost functions turn out to be approximately linear over the range of data that are available, we should point out the fact that if TVC is linear, MC is constant (a horizontal line), and AVC falls to converge upon MC. The significance of this fact is that it may be possible to base output decisions upon the readily available average cost data (rather than the difficult-to-compute marginal costs) since average and marginal costs are approximately equal beyond some level of output.
The methods for specifying and statistically estimating a cost function are clear enough; the real difficulty is in getting enough useable data to permit specification of a reliable cost function. The problems fall into three categories: ensuring that the cost data include all relevant costs and only relevant costs; identification of output where joint-products are produced; and selecting the time period for data collection.