Journal of Regulatory Economics; 4:197-204 (1992) �9 Kluwer Academic Publishers

Capital Waste in the Rate-of-Return Regulated Firm

ROGER SHERMAN University of Virginia

Department of Economics, Charlottesville, VA 22901

Abstract Here it is shown how capital waste can be motivated under rate-of-return regulation, even when the marginal product of capital is positive. This result means that the sign of the marginal product of capital is not adequate as a basis for defining waste. The wasteful use of capital is motivated to avoid an inelastic region of demand, and involves pricing as well as technical input decisions of the firm.

1. Introduction

The incentive of a rate-of-retarn regulated monopoly to waste capital in the static certainty case was demonstrated unambiguously by Fred Westfield (1965). 1 He showed that, if confined to operate on its production function, the rate-of-return regulated finn could want to use capital of negative marginal product rather than operate where demand was inelastic. He also showed that, in those circumstances, the firm would prefer to depart from the production function and use capital of zero marginal product (or, what was equivalent in Westfield's model, pay a higher price for its capital equipmen0 rather than of negative marginal product. Both Elizabeth Bailey (1973) and Edward Zajac (1970) later concluded that the firm would not waste capita/if capital had a positive marginal product, but their positive-marginal-product qualification ruled out the very circumstances revealed by Westfield to invite capital waste. Wesffield's perverse finding that the firm would use capital of negative, or preferably of zero, marginal product in an effort to avoid operating where demand is inelastic is the important conclusion. But the Bailey and Zajac positions bring out a difficulty in defining capital waste that deserves attention, because even the positive marginal product of capital they refer to will not ensure that capital is not being wasted.

We shall first sketch briefly in part 2 how the question whether capital use is wasteful cannot be answered based solely on the sign of the marginal product of capital. In doing so, we use an Averch-Johnson (1962) style model, with its assumption that the allowed return exceeds the cost of capital. First, we show that the marginal product of capital may be positive while capital is being wastefully employed. We shall claim that waste of capital arises whenever the firm uses it as a means of avoiding inelastic demand. Because Das

198 ROGER SHERMAN

(1980) argued that under uncertainty the rate-of-return regulated firm will not waste capital, and likened his result to that found by Bailey (1973) and Zajac (1970), we shall examine the case of uncertainty in part 3. Whether capital waste will occur under price-cap regulation is briefly examined in part 4, where limits of the Averch-Johnson framework are also noted. Part 5 contains a summary.

2. The Difficulty in Defining Capital Waste

Obviously, the use of capital with zero or negative marginal product qualifies as waste. But we want to show how capital can be wastefully used even when it has a positive marginal product. Generally, we shall be concerned with inefficient capital choices by the firm, from which more output could be obtained at lower cost. The more extreme possibility, where the word, "waste," seems appropriate, arises when a high level of capital is chosen by the firm to raise cost, and therefore price, as a means of avoiding a region where demand is inelastic. Empirical demand studies from the electric utility industry are at least consistent with such behavior, in that long run estimates nearly always find demands to be elastic (Taylor 1975).

A useful example of perverse incentive to use capital is found in the borderline case of fixed proportions for only two inputs, say labor and capital. By examining this simple case, Kennedy (1977) showed how a monopoly would respond if rate-of-return regulation was imposed on it. The finn would expand along the efficient path only to a point where the labor input was chosen independently to satisfy the monopoly condition. From there, labor and output would be held constant while capital was expanded, even though its marginal product was zero, until the rate-of-return constraint was just satisfied. In behaving in this clearly inefficient way, the firm would be avoiding a region where demand was inelastic.

That monopolies avoid regions of inelastic demand is well known. Raising price to move away from a region of inelastic demand has to improve profit, because it raises revenue while lowering output and thus total cost. That rate-of-return regulated monopolies also can be motivated to avoid inelastic demand is not as widely appreciated, even though Westfield (1965) and Kennedy (1977) have convincingly demonstrated it. If it is in a region where demand is inelastic, the firm can increase its capital, which under rate-of-return regulation allows it to earn more profit. Increased capital will justify a higher price due to the resulting higher cost per unit, which with inelastic demand will bring in more total revenue. Since output will be slightly lower at the higher price, that will tend to reduce total cost so profit can be raised toward the now higher allowed level. Thus allowed profit, which matters for the regulated firm, will increase as the ftrm has more capital. And profit at the higher allowed level can be obtained by a regulated monopoly when it avoids an inelastic region of demand, just as an unregulated monopolist would.

A geometric understanding of this result comes from considering how a monopoly's revenue is related to its input choices when there are both elastic and inelastic regions of demand, as illustrated in figure 1. Consider a familiar set of production isoquants in a standard two dimensional diagram relating labor and capital inputs to identified levels of output. Figure 1 converts such a diagram into a three dimensional one, with the third dimension representing not output but instead revenue. The dashed lines in the figure are isorevenue curves. Although output will increase as both inputs increase, revenue increases

CAPITAL WASTE IN THE RATE-OF-RETURN REGULATED FIRM 199

L $

" . . ' ,

K

Figure 1

only up to the output represented by isorevenue curve AA, where demand is unit elastic. Beyond curve AA, further increases in output will cause revenue to fall, since greater output brings lower revenue when demand is inelastic. Thus, in moving out from the origin, the isorevenue curves form a rising, side hill surface as inputs are increased, until a highest revenue level is reached at AA. Then the surface declines into a revenue valley. The maximum revenue level, corresponding to the output where demand is unit elastic, forms a ridge (in the shape of curve AA) at the top of this revenue hill. All positions on the higher output, or valley, side of the ridge are positions where demand is inelastic.

The point of this three-dimensional imagery is to show the profit advantage of operating where demand is elastic. Higher elevations in the revenue surface naturally offer advantages to the firm. The profit advantage is greater in the elastic region because total cost rises in moving out from the origin and so tends to be lower in the elastic region, allowing profit to be higher. Even a rate-of-return regulated firm can clearly earn more profit on the elastic side of the revenue hill. But such a firm wants also to be allowed to keep its profit. Since allowed profit depends on capital used, the firm can be motivated to move more deeply into the capital dimension than is efficient. Returning to figure 1, suppose the efficient point lies in the inelastic region of demand, say at point E. If the rate-of-return regulated firm ventures into the revenue valley that corresponds to inelastic demand,its profit will fall. Since the firm's allowed profit grows with capital, K (as (s - r)K), the firm will prefer to expand its use of capital to remain near the ridge along the top of the revenue surface as at point R, where more profit can be earned while more allowed profit also can be justified.

Recall that, if the firm produces under a Cobb-Douglas production function, it will always have a positive marginal product. Nevertheless, it can still be motivated to expand its capital perversely, as an alternative to expanding output efficiently into an inelastic region of

200 ROGER SHERMAN

demand. A rate-of-return regulated fn'm operating according to a Cobb-Douglas production function, with always positive marginal products, may operate at R, far from an efficient point such as E that lies in the inelastic region of its demand, as shown in figure 1. This firm is operating at a point of very high capital intensity to earn, and to be allowed to keep, more profit where demand is elastic and revenue and allowed profit are both higher. Yet at R, first-order conditions for a constrained profit maximum can look perfectly normal, despite serious capital waste. In this case, capital can be seen as being wastefully employed even though its marginal product is positive.

Of course, it is possible for the extreme capital waste that is motivated in a rate-of-return regulated firm avoiding inelastic demand to be revealed by first-order conditions. For this to happen, a region of negative marginal product of capital must exist, and, in expanding capital to raise allowed profit, the firm must be driven to it. It is then possible to see whether the firm will want to depart from its production function to use purely wasteful capital, which is the method Wesffield originally employed to demonstrate waste. We shall use this device to examine the claim by Das that under uncertainty rate-of-return regulated firms will not waste capital.

3. Uncertainty

In the first model examined by Das, a quanfity-settng firm chooses capital and labor, and hence output quantity, before observing the current level of demand. The inverse demand is p(q,u) = p~(K,L) ,u ) , where p is price, q is quantity,f is the production function, K and L are capital and labor, and u is a random term. Quasi rents are p f - wL, where w is the price of labor, as long as u < u0, with uo defined so that p f - w L = s K when s is the allowed rate of return. The firm's profit is then

~1 = P q - w L - r K for ot < u < u 0

and

rc 2 = (s - r )K for u 0 < u < 9,

where ot and 13 are lower and upper limits of u and r is the cost of capital.2 The problem of maximizing the expected utility of profit is then set out as

M a x G(K,L) = ~ u~ V(rCl)(P(u)du + U(x 2) j0 qo(u)du, a u o

where q~(-) is the density function of u. The first-order conditions derived by Das are

fK ~ u~ U'(Xl) (P + fpq)qI(u)du Ot

= r ~ ~ U'(nl)q~(u)du - (s - r)U'(~2) J~ (p(u)au (1)

u 0

and

CAPITAL WASTE IN THE RATE-OF-RETURN REGULATED FIRM 201

Oo r (2) 0 l

and from these conditions the Averch-Johnson conclusion thatfK/fL < r/w was straightfor- wardly derived.

Much as Westfield pointed out for the certainty case, a positive marginal product of labor will require in (2) that marginal revenue, p +fpq, be positive because all other terms are positive. Of course, this means that demand must always be elastic. Should regulators set s in an effort to force the firm into a region of inelastic demand, the firm can escape to the elastic demand region if it keeps output low, price high, and capital high, by wasting capital.

171

Use of inefficient capital will keep price high as long as it causes jv tp(u)du to be large; then uo

rc 2 will determine profit much of the time and that could cause 3

r)U'(rc2) : tp(u)du > r ~ u~ U'(rCl)CP(u)du. (3)

(s u0

This inequality implies that the R.H.S. of (1) will be negative, in which case one of two candidates on the L.H.S.,fK or (p +fpq), must be negative. We have already noted that a positive marginal product of labor in (2) requires p +fpq > 0, so the only available possibility isfK < 0, which would mean the marginal product of capital would be negative.

It is possible to replace the K variable with K + K*, where K* is "useless" capital having zero marginal product. Allowing for clearly wasteful use of capital in this way will reveal whether K* > 0 may be chosen by the fLrrn over the use of more (negatively) productive capital, K. A new necessary condition is obtained if K* > 0, because then we must have OG/OK* = 0, which implies

(s - r)U'(rc2) : tp(u)du = r ~ u~ U'(nl)tP(u)du. (4)

u o ct

Satisfying (4) rather than (3) allowsfK = 0 rather thanfK < 0 in (1), and so, if it can be used, useless capital K* (capital of zero marginal product) would be preferred over capital of negative marginal product.

The second model examined by Das involves a price-and-quantity-setting firm, in which the quantity that can be produced need not match the quantity demanded. Here not only quantity, through capital and labor decisions, but also price, is set before demand is known. More than an efficient level of capital can now be employed in production without forcing reductions in price, yet in this model Das proved that K* = 0 and there would be no capital waste. However, his argument overlooked the fact that when K* > O,fK = 0; i.e., capital of zero marginal product will be used only when the marginal product of productive capital K has been reduced to zero. 4 His proof rested on a contradiction that implicitly assumed fK ~ O. With fK = 0, there is no contradiction, and K* > 0 again is feasible. Thus, the rate-of-return regulated firm may waste capital under uncertainty as well as under certainty.

202 ROGER SHERMAN

4. Price-Cap Regulation

The motive that can arise in rate-of-rettLrn regulated fm'ns to use wasteful levels of capital is due entirely to the influence such firms can have on price. A price in the inelastic region of demand is never attractive to a monopoly that is free to propose its own prices. We have seen how the rate-of-return regulated monopoly will want to distort its capital-output ratio in an effort to avoid proposing a price that would take it into such a region. By removing price choice from the firm, and.also by breaking the connection between the level of capital employed and price, price-cap regulation can avoid this most serious flaw of rate-of-return regulation. It can force the firm to operate where demand is inelastic.

If it cannot influence its price, the firm confined to producing output that will be sold at an imposed price, if, will be motivated to produce efficiently. In the case of only labor and capital units, for example, the firm's profit would b e ~ q ( L , K ) - w L - rK. Maximizing profit with respect to the inputs immediately yields necessary conditions that are consistent with efficient input use (marginal products of inputs equal their prices). To maintain appropriate- ness, output price can be changed to match some general price index less any allowance to be made for improved productivity, such as the RPI - X formula used in England. 5

Some concern might arise about whether the firm will avoid inelastic demand if it offers a basket of services and therefore can propose a set of relative prices. Even if allowed some flexibility in setting re la t i ve prices within an overall constraint of the RPI - X form, however, the firm need not be motivated to operate only in elastic regions of the demands it faces. As long as firms are required to adjust prices according to a price index, holding the previous period's quantities fixed, the influence of marginal revenue on their price choice can be broken (B0s 1978). Indeed, their price adjustments can even lead to Ramsey prices (Vogelsang and Finsinger 1979). Price-cap adjustments take this form and can induce socially desirable pricing (BOs 1990; Vogelsang 1988) rather than the refusal to operate under inelastic demand that can be induced by rate-of-return regulation.

We have relied here on the static model of rate-of-return regulation introduced by Averch and Johnson (1962), even though it is known that delays in regulatory adjustments can bring stronger incentives for efficient operation (Bawa and Sibley 1980). During the period between rate decisions, a rate-of-return regulated firm is subject to something like price-cap regulation, in that its prices are fixed and it can earn more profit only by operating more efficiently. Of course the rate-of-return regulated firm can be expected to keep an eye on the next rate case, and how its record will influence the prices to be set then.

Note that the influence of the firm on p r i c e s is the source of inefficiency examined here under rate-of-return regulation. Even with regulatory lags, if they are not very long, it is possible for the firm to pay attention to its current actions for how they will influence future prices, and to prefer strongly to remain in elastic regions of demand. During the lag periods, prices remain distorted, so while they may invite the firm to produce efficiently, the lag periods will not affect the allocative consequences of inefficient prices. And the negative effect on profit of operating in inelastic demand regions may keep the firm from proposing prices that would cause it to operate there.

CAPITAL WASTE IN THE RATE-OF-RETURN REGULATED FIRM 203

5. Conclusion

On the possibility of capital waste due to rate-of-return regulation, the lessons of Westfield need reemphasis. How the rate-of-return regulated firm can avoid inelastic demand to earn more profit under its regulatory constraint has been presented here in simple terms, not only because it is a basic point that can be presented this way but also because first-order conditions will not reliably reveal it. Capital waste cannot be inferred solely from the sign of the marginal product of capital. Yet, it is motivated any time an efficient solution lies in an inelastic region of demand. Instead of concluding that there can be no wasteful use of capital, we find that, to avoid a region of inelastic demand, rate-of-return regulated firms can have incentive to waste capital. Moreover, wasteful input mixtures may exist even if the marginal product of capital is positive. And such perverse behavior extends to models that allow for uncertainty. Under uncertainty, firms may even be able to argue for excess capital more easily, because judging what is an appropriate level of capital is more difficult when demand is uncertain and excessive capital is not so obvious. 6

All of this analysis is carried out in models of the Averch and Johnson (1962) type, with or without uncertainty. In a dynamic model, where the firm must respond to exogenous prices such as under price-cap regulation, incentives are consistent with efficiency. The problem we describe that keeps the firm from an inelastic region of demand derives essentially from the firm's pricing incentives. Even with regulatory lags, a rate-of-return regulated firm may persist in behavior that will keep it from inelastic demand regions. Since the incentive of the firm to avoid inelastic demand can be so perverse in existing models of regulated firms, further study seems warranted that may hasten the adoption of alternative regulatory institutions.

Notes

I thank Satya Das, David Mills, and referees of this Journal for helpful comments. 1. The logic behind his conclusion is compelling. Suppose the allowed rate of remm is lowered in an effort to

push the regulated firm into an inelastic region of its demand curve. If the firm increases output, its revenue will fall while its costs will increase, so profit must fall. If, instead, the firm substitutes capital for labor, it can justify more profit while remaining in the elastic region of its demand curve, as an unregulated monopolist would want to do. A nice description of this behavior by the firm was made by Leonard (1971).

2. Not having u0 depend on K is criticized by Chang (1991) in this model, but the point about capital waste will hold also in Chang's model.

3. This inequality can be restated as

ff q)(u)du>r E[U'(x)] ~o s U'(~2) "

In this extreme case, the quantity demanded may virtually always exceed output, so j9 r and u0

E[U'(~)]/U'(x9 both are close to 1, and r < s causes the inequality to be satisfied. Inefficiently high capital (capital of negative marginal product) keeps output low and allowed profit high.

4. This point is shown in Sherman (1989, 221-2). 5. Regulation of British Telecom uses the Retail Price Index (RPI) less a required rate of productivity

improvement (X). 6. Crew and Kleindorfer (1979, 142-43) point out that whereas under certainty a firm's capacity will exactly

serve demand, under uncertainty the firm has an opportunity to choose reliability of service through its capacity

204 ROGER SHERMAN

choice. And with the added scope that the service reliability decision offers, the firm may be able to justify a greater level of capital.

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Economic Review 67:968-71. Leonard, R. 1971. "On the Optimal Use of Useless Capital." University of Virginia Working Paper. Sherman, R. 1989. The Regulation of Monopoly. New York: Cambridge University Press. Taylor, L.D. 1975. "The Demand for Electricity: A Survey." Bell Journal of Economics 6:74-110. Vogelsang, I., and J. Finsinger. 1979. "A Regulatory Adjustment Process for Optimal Pricing by

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