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A note on nonlinear prices under rate-of-return regulation.

I. Introduction

Since the 1962 publication of the seminal work of Averch and Johnson (A-J), many studies have analyzed the effects of linear price under rate-of-return regulation, notably, Baumol and Klevorick [1970], Bailey [1973], and Callen, Mathewson and Mohring [1976]. In contrast, sherman and Visscher [1982], examining the interaction between rate-of-return regulation and nonlinear prices, argued that nonlinear prices are very common in regulated industries not because they contribute to economic efficiency or to equity, but rather because such price structures can augment the profits of rate-of-return regulated firms. (1) Srinagesh [1986, p. 52] argued that Sherman-Visscher's argument may not be true because in their model, "the production structure is of the constant returns, fixed coefficients type (which) rules out any A-J input bias." Using a Cobb-Douglas example, Srinagesh [1986, p. 68] further claimed that "the move from linear to nonlinear prices can lead to a reduction in input bias and an unambiguous increase in welfare."

This paper argues that Srinagesh's claim is doubtful because it actually comes from an example where the rate-of-return regulation plays no role at all (see Appendix). It shows that when a rate-of-return regulated monopolist moves from linear to nonlinear prices, the A-J input bias will decrease (increase) if the move increases (decreases) the marginal revenue product of capital. Since the general arguments of the changes of social welfare (the sum of producer's profits and consumer's surplus) are very difficult (if not impossible) to derive, numerical examples of moving from linear price to perfect price discrimination and to uniform two-part tarriffs were used. 2 These examples demonstrate that when the regulated firm moves from a linear price to either of the two nonlinear prices, the social welfare decreases but the percentage of the welfare loss will be smaller if flatter demand function and lower scale economies are present. Also, when the regulated firm moves from linear price to uniform two-part tariffs, the smaller users lose more of their consumer surplus than the larger users.

II. The A-J Model and Numerical Examples

The A-J rate-of-return regulation model concerns a monopolist who produces a single output, Q, with two inputs, capital, K, and labor, L, and faces constant input prices, r and w. The firm strives to maximize its profit, [pi], under the constraint of earning on its capital some allowed rate of return, s, that is greater than the rental cost of capital, r, i.e.,

Max.[pi] = R - rK - wL

subject to R - wL [is less than or equal] sK, s [is greater than] r, (1)

where R = P(Q) Q is the revenue, and P(Q) is the inverse demand function. Using the Lagrangian function, L, with a multiplier, [lambda], the first order conditions are:

[unkeyable]L/[unkeyable]K = [R.sub.k] - r - [[lambda]R.sub.k] + + [lambda]s [unkeyable] O, (2)

[unkeyable]L/[unkeyable]L = (1 - [lambda]) [R.sub.L] - (1 - [lambda]) w [unkeyabla] 0, (3)

[unkeyable]L/[unkeyable][lambda]=-(R - wL - sK) [unkeyable] 0, (4),

where [R.sub.k] [unkeyable] [unkeyable]R/[unkeyable]K [unkeyable] ([unkeyable]R/[unkeyable]Q) ([unkeyable]Q/[unkeyable]K) [unkeyable] [R.sub.Q][Q.sub.K] is the marginal revenue product of capital, and [R.sub.L) [unkeyable] [R.sub.Q][Q.sub.L] is the marginal revenue product of labor. From (2) and (3),

[Q.sub.K./Q.sub.L] = r/w + [lambda] (1 - [lambda] [r - s)/w]. (5)

The second order conditions require that 0 < [lambda] < 1, and the A-J input bias exists [Baumol and Klevorick, 1970].

In (2), [lambda] = (r - [R.sup.K])/(s - [R.sup.K], where [R.sup.K] < r <s. Because [Q.sub.L] and w are positive, (3) shows that the marginal revenue [R.sub.Q] is positive. Therefore, if the firm's move from linear to nonlinear prices increases (decreases) the marginal revenue product of capital [R.sub.K], then [lambda] will decrease (increase) and the A-J input bias will decrease (increase). A numerical example can be used to show the changes of the A-J input bias and social welfare when a rate-of-return regulated monopolist moves from linear price to perfect price discrimination.

Assume that the firm's production function is Q = [(KL).sup.1/[alpha]],where [alpha] = 1 represents increasing returns to scale (IRS); [alpha] = 2, constant returns to scale (CRS); and [alpha] = 3, decreasing returns to scale (DRS). Substituting for K for the production function into (4) (where positive root implies under-capitalization and negative root, over-capitalization), the regulated firm's input demand functions can be derived as

L = (1/2w) [R - [([R.sup.2. - 4ws[Q.sup.[alpha]]1/2, (6)

K = 2q[Q.sup.[alpha][R-(R - R2,4ws[Q.sup.[alpha]]1/2]-1 (7)

Assume that r = w = 1, s = 1.05, and that two linear market demand functions with different elasticities exist: P(Q) = 4 - (0.1)Q and P'(Q) = 4 - Q. By substituting (6) into (3), the firm's output Q can be calculated. The result recorded in Table 1 show that when the firm moves from linear price to perfect price discrimination, its profits, outputs, and capital usages increase, and the marginal revenue product of capital may decrease (under IRS), remain constant (under CRS), or increase (under DRS). Hence, the A-J input bias, indicated by (1 - (L/K)], may increase (under IRS), remain constant (under CRS), or decrease (under DRS). Significantly, unlike the case with no regulation, where perfect price discrimation (without considered equity) always outperforms profit-maximized linear price in terms of given higher social welfare (the sum of the firm's profits and the consumer's surplus), the rate-of-return regulated perfect price discrimination (at least in this example) always produces smaller social welfare because of deadweight loss. However, with a flatter demand function (i.e., P(Q) = 4 - 0.1(Q)) and lower scale economics(i.e., [alpha] = 3), the percentage of the social welfare loss will be smaller. (3) Also, under perfect price discrimination, marginal prices are always less than unregulated marginal costs, but under linear price, marginal prices may be greater (when [alpha] = 1 or 2) or lower (when [alpha] = 3) than unregulated marginal costs. (4)

The above example runs counter to the fact that antitrust laws usually prohibit unequal treatment of customers. The second example assumes that a rate-of-return regulated monopolist is allowed to move from linear price to uniform two-part tariffs, i.e., the customers must pay a uniform fixed entrance fee T for the right to buy a product at a unit price P. The firm's revenue is R = P(Q)Q + nT, where n is the number of customers. This equation implies that the firm's profit hill will be raised by nT, which may cause its output to increase and its marginal revenue to decrease. Hence, if the firm's marginal product of capital decreases, its marginal revenue product of capital will also decrease and the A-J input bias will increase. The input prices and the firm's production function are assumed to be the same as the preceding example and the allowed rate of return s = 1.01. (5) Two sets of three individual demand functions are assumed: [P.sub.1](Q) = 4 - Q,(P.sub.2)(Q) = 4 - 0.5Q, and [P.sub.3](Q) = 4 - 0.25Q;[P'.sub.1](Q) = 12 - 12Q, [P'.sub.2](Q) = 12 - 10Q, and [P'.sub.3](Q) = 12 8Q. Hence, the market demand functions are P(Q) = 4 - (1/7)Q and P'(Q) = 12 - (120/37)Q, respectively.

[TABULAR DATA OMITTED]

The results recorded in Table 2 show that when the firm moves from linear price to uniform two-part tariffs, (1) its profits, outputs, and capital usages increase, (2) the A-J input bias increases (because the marginal product of capital decreases), (3) all the customers suffer a welfare loss and the smaller uses lose more of their consumer surplus than the larger users, and (4) as in Table 1, the social welfare decreases and the percentage of the welfare loss will be smaller if

[TABULAR DATA OMITTED]

flatter demand function and lower scale economies exist. (6)

III. Concluding Remarks

As is well-known in the A-J literature, under linear price conditions, regulated firms overcapitalize, and social welfare may be greater or lower than the social welfare under no regulation [see Callen et al.]. Sherman-Visscher and Srinagesh have analyzed the A-J effects under nonlinear prices. However, since they assume either fixed coefficients production technology or a model with no regulation, their results are inconclusive.

This paper showed that when a rate-of-return regulated monopolist moves from linear to nonlinear prices, the A-J input bias will decrease (increase) if the marginal revenue product of capital increases (decreases). The numerical examples showed that when the regulated firm moves from linear price to perfect price discrimination or to uniform two-part tariffs, the social welfare decreases but the percentage of the welfare loss will be smaller if flatter demand function and lower scale economics exist, and that when the firm moves from linear price to uniform two-part tariffs, the smaller users lose more of their consumer surplus than the larger users.

APPENDIX

Srinagesh's numerical example shows that a linear-price monopolist earns unregulated rate of return of 2.5 on capital with profits 1.5625. Under perfect price discrimination, its unregulated rate of return is 2.083 with profits 2.167. Srinagesh's point is that if a regulator allows a rate of return less than 2.5 but greater than 2.083 and permits multipart pricing, the monopolist will use multipart pricing, maximize profits, and satisfy the rate of return constraint with no A-J input bias. There are two reasons to argue that Srinagesh's example is inappropriate. First, the A-J literature assumes that a regulated firm earns no more than the allowed rate of return, which is less than the rate of return the firm can earn if it were allowed to maximize profits with no regulation. Srinagesh's example is somewhat inconsistent with the literature. Second, generally, Srinagesh's example is not applicable. For instance, assumed that with no regulation, a firm uses two inputs: capital (K) and labor (L), to produce one output (Q). The firm's production function is Q = [(LK).sup.1/[alpha], where [alpha] = 1, 2 or 3, the rental cost of capital (r) = the wage rate (w) = 1, and the market demand function is P(Q) = 4 - Q. Then, a shown in the following table, when [alpha] = 3 (i.e., decreasing returns to scale), the earned rate of return under perfect price discrimination is 2.5, which is less than the earned rate of return under linear price, i,e., 2.8508. However, when [alpha] = 1 increasing return to scale) or 2 (constant returns to scale), the earned rates of return under linear price are not greater than those under perfect price discrimination.

[TABULAR DATA OMITTED]

REFERENCES

H. Averch and L. Johnson "Behavior of the Firm under Regulatory Constraint, "American Economic Review, 52, 5, December 1962, pp. 1053-69.

E. Bailey, Economic Theory of Regulatory Constraint, Lexington, MA: Lexington Books, 1973.

W. Baumol and A. Klevorick, "Input Choice and Rate-of-Return Regulation: An Overview of the Discussion," Bell Journal of Economics, 1, 2, Autumn 1970, pp. 162-90.

J. Callen, G. Mathewson and H. Mohring, "The Benefits and Costs under Rate-of-Return Regulation, American Economic Review, 66, 3, June 1976, pp. 290-97.

R. Sherman and M. Visscher, "Rate-of-Return Regulation and Two-part Tariffs," Quarterly Journal of Economics, 97, 1, February 1982, pp. 27-42.

P.Srinagesh, "Nonlinear Prices and the Regulated Firms," Quarterly Journal of Economics, 101, 1, February 1986, pp. 51-68.

(1) In other words, regulated agencies are captured by the firms they regulate.

(2) Callen et al. also use numerical examples to show the changes of social welfare under rate-of-return regulated linear price.

(3) Callen et al.'s Cobb-Douglas example shows that in comparison with marginal cost pricing, a rate-of-return regulated linear price would be desirable when low demand elasticity, modest scale economies, and a small exponent on capital in the production exist.

(4) This result refutes another Srinagesh claim [1986, p. 64] that "in our (nonlinear pricing) model, rate-of-return regulation does lead to the sale of output at a marginal price below true (unregulated) marginal cost. This result has no counterpart in linear pricing models for a single-output monopolist."

(5) It was found that different values of s did not alter the basic results.

(6) A numerical method was employed to find the optimal solutions, i.e., with (3) and (6), increasing the entrance fee T gradually from zero and solving for Q until the maximum allowed profit was found. In the example, the profit-maximizing firm will not price T too high for fear of losing the lowest demand customers.
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Title Annotation:includes appendix
Author:Kuo-Ping Chang
Publication:Atlantic Economic Journal
Date:Jun 1, 1991
Words:2101
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