SRAC r
LRAC r
LRMC r
LRAC r
Figure 5.8 Relation between Short Run and Long Run Cost Curves for Different Rates of Return than output O1 are greater than long run total cost. Hence, the short run and long run total cost curves are tangent to each other at the output O1. It follows that the long run average cost curve for a particular rate of interest is the 'envelope' of all the short run average cost curves which 'yield' that particular return. It also follows that the short run and the long run marginal cost curves intersect at the output at which the short run and long run average cost curves are tangent to each other.
As was mentioned earlier, with a given plant there always exists a short run average cost curve that is tangent to a given demand curve. If the firm is operating as a short run profit maximizer, it will operate so as to produce the output which yields this short run tangency relation. If this solution is achieved with a given demand curve and a given plant, the long run average cost curve which is the envelope of the short run average cost curve that is tangent to the demand curve may not be tangent to the demand curve. This means that there is another plant size which could yield a higher rate of return than the given plant. This plant size is given by the tangency of a long run average cost curve with the demand curve. The tangency of a long run average cost curve with the demand curve represents the highest rate of return that can be earned with a given demand curve. At the output where the long run average cost curve and the demand curve are tangent, the long run marginal cost curve for this rate intersects the marginal revenue curve. By building the plant so determined, the firm will earn the highest attainable rate of return given the demand curve.
If the firm uses a planning rate lower than the highest attainable rate (rj < r,) in determining its expansion path, then the maximizing plant will
Costs L LRMCr;
LRMCr
LRMCr
Output
Figure 5.10 Profit Maximization: Planning Rate - Maximum Attainable Rate
Figure 5.10 Profit Maximization: Planning Rate - Maximum Attainable Rate be determined by the intersection of the long run marginal cost curve and the marginal revenue curve. At the output where they intersect SRMC = LRMC = MR and LRAC(r^) = SRAC(r^) The average cost curves for r, will be parallel to the demand curve. However, if the firm maximizes profit with the plant so determined, the rate of return earned will be given by SRAC(rk) where ri > rk > r, and the LRAC(rk) will intersect the demand curve. The argument is simple if we use total cost and total revenue relations. With a given total revenue curve, there is a rate of return, r, which for some output Oi will make total revenue just equal to total cost
and yield rt on the plant. If a lower rate, say r., is used in determining the size of plant, then O} will be greater than Oi and total revenue will be greater than total cost based upon r. However, rk will be the actual yield and rt > rk > r..
Our conclusion is that if a firm is attempting to earn the highest rate upon its total investment, the plant decision is based upon the tangency of a long run average cost curve and the demand curve. If a firm uses a rate of return either higher or lower than the highest attainable rate in order to determine its long run marginal cost curve, it will earn a lower rate of return upon its total investment than the maximum attainable.
The history of such long run cost curves has been rather peculiar. The original article by Viner has achieved as much renown for the controversy between Professor Viner and Dr Wang13 as for its content. However, after the clearing up of the interpretation of these curves,14 the material now appears in price theory textbooks under the heading of planning curves,15 and aside from its use as a classroom exercise, these curves, as far as I know, have not been used in further analysis. This neglect seems to me to be particularly unfortunate. However, for these curves to be truly useful, it is necessary to drop the assumption of a unique long run marginal cost curve for the firm, which we have done.
The distinction drawn in these curves between the short run and the long run refers to the scale of plant. 'The short run is taken to be a period which is long enough to permit any desired change of output technologically possible without altering the scale of plant, but which is not long enough to permit any adjustment in the scale of plant. It will arbitrarily be assumed that all of the factors can for the short run be sharply classified into two groups, those which are necessarily fixed in amount and those which are fully variable.'16 Viner went on to define the increases of the scale of plant by asserting that 'each scale will be qualitatively indicated by the amount of output which can be produced at the lowest average cost possible of that scale'.17 As our broadened concept of the average cost curve is inconsistent with the existence of a unique average cost curve for each scale of plant, we have to define the measure of the scale of plant in another way.
Each long run average cost curve has associated with it a long run marginal cost curve. The plant which a firm would build if it were maximizing the rate of return upon its total investment in plant is given by the tangency of a long run average cost curve and the demand curve. This is equivalent to the intersection of the long run marginal cost curve for that rate of return and the marginal revenue curve. The plant that will be built is given by this intersection and will determine a short run marginal cost curve. Each short run marginal cost curve therefore represents a scale of plant. If the firm is using a unique long run marginal cost curve in its planning, the optimum scale of plant for the firm can be defined in terms of this long run marginal cost curve's intersection with the marginal revenue curve. Inasmuch as the zero return on total investment has significance for survival (Chapter 6), we can define each scale of plant as the intersection of the short run marginal cost curve with the zero return long run marginal cost curve. The determination of the measure of the scale of plant is purely arbitrary: all we have to do is to be consistent in our usage.
'The long run is taken to be a period long enough to permit each producer to make such technologically possible changes in the scale of his plant as he desires, and thus to vary his output either by a more or less intensive utilization of existing plant, or by varying the scale of his plant by some combination of these methods.'18 The ability of an existing producer in the long run to change the scale of his plant as he desires leads us to consider the relation between an existing plant and alternative plants. If we are really interested in the static long run in our attempt to analyse the generating factors for the coefficients in macroeconomic models, then we can ignore the possible existence of variations in the production relations other than along a uniquely defined long run production function. If we can restrict ourselves to a unique production function we can unambiguously define investment as a movement from one point to another on a particular return long run marginal cost curve.
However, if the time period which is relevant to investment decisions is short enough so that existing plant is a relevant factor in the incremental outlay19 necessary to increase the scale of the plant, we cannot analyse investment in terms of a unique particular return long run marginal cost curve. We have to distinguish between the optimum manner of producing a given output at a given factor price ratio when no plant is in existence, and the optimum way of producing a given output at a given factor price ratio when the initial conditions include the existence of a plant. Of course, the second alternative, where the initial conditions include the existence of a plant, implicitly assumes that the relevant time period for our problem is short of that time period in which the plant can be reduced to a zero productive capacity, for in such a time period there is no difference between the choice with no existing plant and with an existing plant.
In an iso-product map, the long run marginal cost curve is the rate of change of total costs as you move along an expansion path. Short run marginal costs are the rate of change of total costs with plant factors held constant. If output is increased from O1 to O2, as shown in Figure 5.12, total costs (constant factor prices) are increased from L1 to L2 in the long run, from L1 to L'2 in the short run. However, there may be an inter-
mediate time period (let us label it an investment period) in which plant may be modified so that total costs fall from L'2 to L'2 and nevertheless total costs are higher than they would be if, starting from scratch, a plant for output O2 were built. The dotted iso-product curve O'2 is the relevant curve for plant expansion. It may be that in time a firm keeps on producing output O2, the O2 iso-product curve approaches the O2 iso-product curve; it may be that it never does. That is an element of the technique of production.
The cost curves as defined here are transformations of a production function. As such, once the factor price ratios are given (including the relevant earning rate), the long run marginal cost is given too. We can define the long run marginal cost curve as the marginal cost of a particular output starting from no plant as an initial condition. We can also draw a longer than short run marginal cost curve on the assumption that we are modifying an existing plant. If the existing plant is a relevant variable in determination of investment decisions (in business cycle analysis it is necessary to argue from where you are rather than in terms of long run timeless considerations), then the long run marginal cost curve using an existing plant as an initial condition is the relevant curve for the expansion of plant by a firm with an existing plant.
For a given firm, therefore, we can distinguish two types of long run marginal cost curves.20 One is a long run marginal cost curve in which the firm completely ignores the existing plant: that is, a 'new plant' long run cost curve. Another is a long run cost curve which considers the existing plant as a given and modifies the existing plant. The modification long run marginal cost curve never lies below the zero plant initial condition long run marginal cost curve for outputs greater than the optimum output with the original scale plant. For a given output larger than the optimum with a given plant (factor prices fixed), the amount of capital which the modified plant will use is less than the amount of capital which the optimum plant for that scale of output would use. That is, in Figure 5.12, the modified plant uses K of capital to produce the same output as a plant built to produce O2 but starting from scratch (zero plant) and using K2 of capital.
The relevant marginal cost curve for a firm with a given plant is different from, and it lies above (for outputs greater than the original output), the long run marginal cost curve drawn on the assumption that no plant exists. For the expansion of the plant, the curve labeled LRMC' is the relevant curve, and it always entails a smaller amount of investment for a given output larger than O1 than the long run marginal cost curve LRMC drawn on the assumption that no plant exists. However, for a firm with an existing plant, we have another long run marginal cost curve to consider - a long
122 Cost
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