Thayer Watkins
Silicon Valley
& Tornado Alley

Marginal Cost and Social Efficiency

In Multi-Plant Industries
the Efficiency-Relevant Marginal Cost
is the Minimum Average Cost
of the Marginal Plant

Revision of an article published in Southern Economic Journal, July 1981, pp. 149-155.

I. Introduction

According to standard economic analysis social efficiency requires market prices equal marginal social cost [4]. This marginal cost-pricing principle, which has had a discernible influence on public policy, goes back overa century to Jules Dupuit [1]. Yet, despite the hallowed position of this principle in microeconomics , there are major ambiguities about the appropriate definition of marginal cost. Consider the conclusion of P.R.G. Layard and A.A. Walters in their book, Microeconomic Theory[5]. They say,

"The pricing rule is: Find that output which, for the existing plant, equates the demand price to the marginal cost; then charge the marginal cost (and demand price) for that output. The investment rule is: Do all investments whose benefits (given the prices that will be charged) exceed their costs. ...If there is a facility with fixed capacity and optimal output equals full capacity output, then the optimal price is the demand price for the full capacity output...The optimal price is the demand price for capacity output, though in a sense it can also be called the marginal cost since at [capacity] the marginal cost is anything between the [demand price] and infinity."

What Layard and Walters say is not so much wrong as it gets the matter of defining marginal cost mixed up with determining optimal output. Layard and Walters seem to be trying to define marginal cost as whatever will save the marginal cost-pricing rule. In particular the marginal cost at physical capacity in the sense of a derivative is undefined. With marginal cost undefined the choice of optimal output is not determined by their rules. Layard and Walters are blurring the distinction between social desirability and financial feasibility, two related but separate concepts. There is also a problem of priority: first the optimal number of plants should be determined and then the optimal price.

II. Physically Indivisible Units and Efficiency Pricing

Optimal output and efficiency pricing are determined from the total social benefit and total social cost as functions of output. The incremental private benefit of an increment in consumption is the area under the demand schedule over that range in consumption (and hence output) and, by extention, the total private benefit of any level of consumption is the area under the demand schedule up to that level of consumption.

Assume for now that there are no externalities so the private benefit of consumption is the same as the social benefit and that the prices of the inputs are independent of the industry output so that the private cost of production is the same as the social social cost of production. The case of interest is when the industry cost function is piecewise continuous and differentiable but has discontinuities. This type of cost function arises in many applications where the discontinuity is the fixed cost of an addition plant.

Consider, for example, the cost function of an airline (total cost versus passengers carried between two points) as shown in Figure 1.

Figure 1

There is a small increase in cost for each additional passenger and a big discontinuous increase when an additional plane has to be put into service. An incorrect interpretation of the marginal cost-pricing rule would suggest that for economic efficiency the passengers should be charged the negligible cost of carrying one more passenger on a partially filled plane or the enormous cost of putting another plane into service. The correct interpretation of the marginal cost-pricing principle is that for economic efficiency the passengers should be charged the average cost per passenger of another planeload of passengers. As will be demonstrated, the relevant marginal cost for economic efficiency is the minimum average cost of the marginal plant rather than the intra-plant marginal cost. This is equivalent to the condition that the marginal plant is earning no economic profit, a condition that prevails when there is freedom of entry and exit to and from the industry.

An appropriate algorithm seems to be that one uses the average cost for the marginal units to determine a close approximation to the optimal number of plants and then determines the price necessary to establish the optimum output.In the case of the airline example suppose the inverse demand function is:

p = p0 - bx
so the net social benefit is
NSB = p0x - (1/2)bx2.

Let C0 be the cost of flying a plane with 0 passengers and C1 the cost of flying a plane with a full load of q passengers. The average cost is then C1/q. If the cost function is taken to be the envelope (convex hull) then the optimal output is given by:

MB = p(x) = p0 - bx = MC* = C1/q
x = (p0 - MC*)/b = (p0 - C1/q)/b

where MB stands for marginal social benefit and MC* is the slope of the convex hull envelope.

The optimal number of flights is then either [x/q] or [x/q]+1. To determin which, one would have to compute the net social benefits at the two outputs, q[x/q] and q([x/q]+1). Here is a calculator for inputing the parameters and obtaining the optimal output and price for the type of problem. Under some conditions the optimal number of plants is [x/q] and under other conditions it is [x/q]+1. And, of course, under special conditions the net social benefit at [x/q] and [x/q]+1 could be equal and thus both would be optimal.

In the case where the optimal number of flights is [x/q] the optimal price is above the relevant marginal cost MC* by whatever amount is necessary to reduce the quantity demanded to the number of trips that can be provided by the [x/q] flights. This would mean that the marginal unit was earning some economic profit but that an additional unit would lose. Thus this case would be consistent with a competitive market equilibrium.

For the case in which the optimal number of flights is [x/q]+1 the optimal price would be below the relevant marginal cost MC* and thus the marginal firm would be experiencing an economic loss. A competitive market equilibrium would result in one less plant than the optimum.

III. The Socially Optimal Number of Plants

In the preceding analysis there was a physical capacity which necessitated the introduction of additional plants, but in general the introduction of an additional plant and its operating level is a matter of choice. Suppose the cost functions for the different plants are known, say

ci = fi(xi)
for i=1,...,n.

Since the focus of interest is which plants should be brought into operation it will also be presumed that the fixed cost for a plant does not apply until that plant is in operation. Otherwise the fixed costs of all the plants would be sunk costs and would not be relevant in determining the optimal set of operating plants and their outputs. The exclusion of the fixed costs of the plants if they are not in production can be achieved by requiring that for all i,

fi(0) = 0,
even though
limxi→0+(fi(xi)) > 0.

Note also that this formulation which makes the cost of a plant a function only of its output and thus presumes that each plant's costs are unaffected by the level of output of the industry otherwise.

The industry cost function is defined as:

C(X) = minxi Σin fi(xi)
subject to the constraint
Σinxi = X

The first order conditions are easily derived as:

fi'(xi) = λ
for all i such that
xi >0.

This means that for the optimum operating plants the intra-plant marginal costs are all the same; i.e., λ.

For purposes of finding the optimal operation of the plants the concept of the envelope of the industry cost function is relevant. The envelope of the cost function could be defined in terms of the boundary of the convex closure of the cost function; i.e., the intersection of all the convex sets which enclose the feasible cost combinations. But geometrically the envelope is easily constructed by drawing tangent lines to the industry cost function. Suppose the envelope cost function is designated as C*(X). The algorithm proposed is to find the set of plants that should be in operation by finding the value of X which maximizes SB(X) - C*(X), say X*. The value of X* is then the basis for finding the value of X which maximizes SB(X) - C(X).

On the basis of a consideration of production runs the industry cost function can be reformulated as:

C(X) = minzi,ti Σin tifi(zi)
subject to the constraints
Σintizi = X
ti≥0, zi≥0,

where ti is the fraction of the time the i-th plant is running, zi is the rate of production of the i-th plant when it is operating and the annual rate of production xi is equal to tizi.

The first order conditions with respect to the production levels in the plants are:

tif'i(zi) = λti
for all i such that
zi > 0.

This reduces to the condition previously found; i.e.,

fi'(zi) = λ
for all i such that
zi > 0.

The interesting new element in the problem is the first order conditions with respect to the ti's:

f(zi) = λzi

Since λ = fi'(zi) from the first order conditions for zi and λ = f(zi)/zi from the first order condition for ti, it follows that an operating plant is operating at its economic capacity; i.e., the rate of production at which average cost is equal to marginal cost and therefore average cost is a minimum. This means that the industry cost function is constructed by putting the plant into operation that has the lowest minimum average variable cost. This plant's value for t increases from 0 to 1. At the level of industry output that has the first plant operating full time another plant is brought into production. That plant is the one with the lowest minimum average costs of those not yet in production. The value of t for that plant increases from 0 to 1. This process is repeated until all plants are in operation and the industry is producing at the highest level it can.

The method of construction of the industry cost function is equivalent to finding the convex hull of the cost functions. The relevant marginal cost is the minimum average cost of the marginal plant.

IV. Conclusions

The construction of the industry cost function involves minimizing the cost of producing each level of output. The first order conditions for this constrained minimization result in the condition that, at the levels of output at which additional plants are brought into production, the slope of the industry cost function (the marginal cost) is equal to the minimum average cost of that marginal plant.

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