Voting, spatial monopoly, and spatial price regulation.

Meng-Hua Ye ; Yezer, Anthony M.J.

VOTING, SPATIAL MONOPOLY, AND SPATIAL PRICE REGULATION

Regulations often require that local public utilities engage in high rates of freight

absorption. These regulations, often mandating uniform pricing, are shown to arise

logically as a consequence of self-interested voting behavior. We specifically consider

the case of a single-plant spatial monopoly which is regulated by consumers distributed

around the plant. Consumers may change their delivered price by voting to require a

rate of freight absorption which differs from the profit-maximizing rate. Voting

outcomes under a median voter model predict the high rate of freight absorption often

observed in practice.

I. INTRODUCTION

While the literature on spatial monopoly is large, and research on regulated firms larger still, very little attention has been given to the study of regulations affecting the spatial monopolist.(1) Regulated spatial monopolies could include either public or private production of telephone service, natural gas, cable television, waste removal, water, sewage service, etc. Indeed the analysis could be adapted to mass transportation systems providing access to an employment center. It is common to find that regulations governing the spatial monopolist require uniform delivered pricing in which the firm must absorb most or all transportation costs.(2) It appears that regulated freight absorption by the spatial monopolist is one of those cases that everyone knows about but which no one has bothered to analyze in detail.

An economic analysis of regulated freight absorption requires that the reaction of a spatial monopolist to the full range of requirements for freight absorption, from zero to full absorption, be available. But there is no such treatment in the literature. The first objective of this paper is to fill this gap by analyzing the effects of continuous variation in freight absorption on prices, output, profits, consumers' surplus and total welfare. The results which we obtain are rather surprising. For example, the unregulated profit-maximizing spatial monopolist is found to be a knife-edge case in which second-order conditions are not satisfied. In addition, welfare maximization is achieved by regulating the rate of freight absorption so that it lies between zero and the rate practiced by an unregulated firm.

Having determined the effects of alternative freight absorption regulations on the firm, we then construct a voting model in which spatially distributed consumer-voters choose the rate of regulated freight absorption. In this spatial voting model, the spatial distribution of consumer surplus is used to generate voter preferences regarding the regulated level of freight absorption. This analytical approach, which we call spatial surplus surface analysis, shows that the median voter prefers a high level of freight absorption. This explains the popularity of regulations requiring high rates of freight absorption, the regulatory outcome most often observed. These results have significant welfare implications. Consumer voters choose freight absorption regulations with higher required rates of freight absorption than practiced by an unregulated spatial monopolist engaged in spatial price discrimination. The voting outcome lowers both total welfare and total consumers' surplus below the levels obtained under unregulated spatial monopoly.

II. MODEL OF THE SPATIAL MONOPOLIST: ASSUMPTIONS

Following the conventional stylized assumptions for single-plant spatial monopoly, there is a given linear (one-dimensional) market characterized by location along X, a subset of (- [infinity], [infinity]), with a single plant located at point 0.(3) There is a single firm producing and distributing a homogenous product, whose quantity is denoted by q, to consumers distributed at uniform density across the market. Resale or transportation of the product by consumers is not possible, so that demand at any distance, r, is independent of the price charged at alternative distances. Total quantity demanded at distance r is given by(4) (1) q(P(r)) = [Beta] [P.sub.2] - [Beta] P(r), q([Beta]) = 0, where P(r) is the delivered price and [Beta] is a parameter equal to the choke price.(5) Subsequent measures of consumer surplus will be based on the area under this demand function. Use of such a measure implicitly assumes the marginal utility of income is constant and, with a homogenous population, distributional questions are not considered.

The spatial price function, P(r), takes the particular form P(r) = P(0) + etr where P(0) is price at r = 0, t is the transportation cost per unit of output per unit of distance and e is the fraction of freight cost passed on to consumers, (1 - e) is the fraction of freight cost absorbed by the firm. In the case of mill or Free on Board (FOB) pricing, e = 1 and there is no freight absorption. Uniform pricing occurs when e = 0 and there is complete freight absorption.

The monopolist produces output subject to increasing returns, which are traditionally represented by fixed cost for an essential input and constant marginal cost. Transportation cost is proportional to distance and quantity transported. The total cost function of the monopolist is given by (2) C = [C.sub.T] + F + [C.sub.V] = [C.sub.T] + F + mQ where C is total cost, [Mathematical Expression Omitted] is total transportation cost, F is total fixed cost, [Mathematical Expression Omitted] is the total variable cost, m is constant marginal cost, q(P(r)) is total output sold at distance r, and Q is total quantity sold. The market extends in two directions from the plant covering a total distance of 2R.

The welfare measure is the sum of monopoly profit and consumers' surplus. The voting behavior of consumers is motivated by an attempt to maximize their personal consumer surplus.

III. STANDARD RESULTS ON THE BEHAVIOR OF THE SPATIAL MONOPOLIST

First consider the production and pricing decision of the unregulated spatial monopolist maximizing profit in the environment described in section II. The monopolist chooses a market radius R, so that the actual market served [-R, R] is a subset of X, selects P(0) which determines the price at r = 0 and determines e, the fraction of transportation cost passed on to consumers. Pricing over space is determined by P(r) = P(0) + etr, r [Epsilon] [0, R], but the monopolist must choose P(R) [is less than or equal to] [Beta], the choke price. Thus the general pattern of spatial pricing is a function of P(0) and e, which vary independently of R except when the choke price is binding--i.e., in the case where P(R) = [Beta]. When P(R) < [Beta], P(0), e, and R are chosen independently. Recalling that the total cost at distance r is given by equation (2), and letting total revenue = Z, profit may be written as (3) [Pi] = Z - [C.sub.T] - [C.sub.V] - F [Mathematical Expression Omitted] As is well known, for linear demand the level of e which maximizes profit is 0.5, i.e., half of transportation costs are absorbed.(6)

Welfare maximization is based on the sum of [Pi] and consumers' surplus. Consumer's surplus for households located at r, s(r), is given by (4) [Mathematical Expression Omitted] Overall consumers' surplus is (5) [Mathematical Expression Omitted] Welfare maximization implies maximization of [Pi] + S, which has been shown to required e = 1,(7) or marginal cost pricing. Other well-understood results involve the case of e = 0, uniform pricing or complete transportation cost absorption, which is often prompted by regulation.

IV. REGULATING FREIGHT ABSORPTION BY THE SPATIAL MONOPOLIST

The major object of this study is the effect of regulations controlling the level of transportation cost absorption. Given that welfare is maximized at e = 1 (zero absorption or mill pricing) and firm profit is maximized at e = 0.5 (spatial price discrimination), it is curious that regulations commonly require that the monopolist set e = 0 (uniform delivered price). It is always possible that such regulations result from ignorance, cultural notions of fairness, or problems implementing distance-based pricing. Our analysis is designed to determine if voter self-interest and knowledge of the effects of e on consumer surplus can lead to regulations requiring low values of e.

To determine the effect of variation in e on the decision variables, we treat e in equation (3) as a parameter and use standard maximizing techniques. The first-order necessary conditions of the optimal P(0) and R for the maximization of the total profit [Pi] given by equation (3) are (6) P(0) = ([Beta] + m)/2 + (1 - 2e)tR/4, and (7) q(P(R))[P(0) - m - (1 - e)tR] = 0. Whether equation (7) implies q(P(R)) = 0 or [P(0) - m - (1 - e)tR] = 0 depends crucially on the concavity of [Pi] over R, i.e., on the second-order conditions for R: (8) [Mathematical Expression Omitted] We find, applying equations (6) and (7), that [Mathematical Expression Omitted] for q(P(R)) = 0 and e [Epsilon] [1/2, 1], and that ([Delta].sup.2] [Pi])/[[Delta]R.sup.2]) = (1 - e) [Delta] [t.sup.2] R(2e-1) [is less than or equal to] 0 for P(0) - m - (1-e)tR = 0 and e [Epsilon] [0, 1/2]. Consequently, we have the optimal solutions as follows. For e [Epsilon] [0, 1/2], (9) P(0) = [2(1 - e) [Beta] + m]/ (3 - 2e), and (10) R = 2([Beta] = m)/t(3 - 2e). For e [Epsilon] [1/2, 1], (11) P(0) = ([Beta] + 2em)/(1 + 2e), and (12) R = 2([Beta] - m)/t(1 + 2e). This results in the following series of propositions. For purposes of comparison with results already in the literature, outcomes obtained for specific values of e = 1/2, e = 1, and e = 0 will be noted using subscripts 1/2, 1 and 0 respectively and the subscript [Omega] will indicate a result that maximizes welfare. First we develop three propositions about the effects of varying e on the [0, 1] interval which are implicit in the standard literature on spatial monopoly.

PROPOSITION 1. A single-plant spatial monopolist under transportation cost absorption regulation will charge the choke price at r = R if regulated e [Epsilon] [1/2, 1]; otherwise it will charge a price less than the choke price at r = R if regulated e [Epsilon] [0, 1/2).

Proof. Substituting equations (11) and (12) into the spatial pricing definition P(R) = P(0) + etR, we obtain that P(R) = [Beta] for all e [Epsilon] [1/2, 1]; and substituting equations (9) and (10) into the spatial pricing equation at r = R, we obtain that P(R) < [Beta] strictly for all e [Epsilon] [0, 1/2).

PROPOSITION 2. Regarding the maximum market radius of the single-plant spatial monopolist under transportation cost absorption regulation, [R.sub.W] [is greater than or equal to] [R.sub.1] = [R.sub.0]; and [Delta] R/ > 0, [[Delta].sup.2] R/[[Delta] e.sup.2] > 0 for all e [Epsilon] [0, 1/2), and [Delta] R/[Delta] e < 0, [[Delta].sup.2] R/[[Delta] e.sup.2] > 0, for all e [Epsilon] (1/2, 1].

Proof. The rank of R with respect to welfare maximization, and profit maximization with e = 0, 1/2, and 1 is determined simply by plugging the respective values into equations (10) and (12). The monotonicity results are obtained by taking derivatives of equations (10) and (12) with respect to e and the concavities are given by the second-order derivatives of R with respect to e. These results are illustrated below in Figure 1.

Note that the choice of market radius is continuous in e but not differentiable at the profit-maximizing radius [R.sub.1/2]. The first condition, derived from equation (7), is switched from one to another as the sign of the second-order conditions is flipped around this point. This switching arises for a variety of demand curves, convex, concave, and linear, but it does require that there be a choke price. In such cases, there are two factors which can produce a market boundary at R. First, if the rate of transportation cost absorption is high, marginal cost of production plus transportation may rise to equal price, or m + tR = P(R). In such cases, P(r) is below the choke price and q(P(r)) > 0 for r < R, and P(r) becomes infinite and q(P(r)) = 0 for r > R. Second, if transportation cost absorption is small (1 [is greater than or equal to] e > 1/2), delivered price reaches the choke price at the market boundary and q(P(R)) = 0, although the marginal cost of production and transportation is less than delivered price at the boundary, m + tR < P(R). This dual nature of the conditions characterizing the market boundary, not noted in previous studies, appears to have serious consequences for the shape of the welfare surface and consequent regulatory decisions.

PROPOSITION 3. Concerning total output, [Q.sub.w] > [Q.sub.1/2] > [Q.sub.1] = [Q.sub.o]; and [Delta] Q/[Delta] e [is greater than or equal to] 0, [[Delta].sup.2] Q/ [[Delta]e.sup.2] [is less than or equal to] 0 for all e [Epsilon] [0, 1/2], and [Delta] Q/ [Delta] Q/ [Delta] e [is less than or equal to] 0, [[Delta].sup.2] Q/ [[Delta]e.sup.2] [is less than or equal to] 0 for all e [Epsilon] [1/2, 1].(8)

These results are fairly obvious from the literature. The profit-maximizing spatial monopolist sets e = 1/2 and produces a large output because sales extend over the largest market radius excepting the welfare-maximizing case.

Now we come to a series of three new propositions, not implicit in previous work, which provide the basis for statements about welfare and for voting analysis conducted in the next section.

PROPOSITION 4. Total consumer surplus generated by the single-plant spatial monopolist under alternative transportation cost regulations follows [S.sub.w] > [S.sub.1] > [S.sub.1/2] > [S.sub.0]; and [Delta] S/ [Delta] e > 0, [[Delta].sup.2] S/ [[Delta]e.sup.2] > 0 for all e [Epsilon] [10, 1/2], and [Delta] S/ [Delta] e > 0, [[Delta].sup.2] S/[[Delta] e.sup.2] < 0, fall all e [Epsilon] [1/2, 1].(9)

PROPOSITION 5. Profit of the single-plant spatial monopolist under alternative transportation cost regulations ranges from [[Pi].sub.1/2] > [[Pi].sub.1] = [[Pi].sub.0] > [[Pi].sub.w] = 0; and [Delta] [Pi]/[Delta] e [is greater than or equal to] 0, [[Delta].sup.2] [Pi]/ [[Delta]e.sup.2] [is less than or equal to] 0, for all e [Epsilon] [0, 1/2] and [Delta] [Pi]/[Delta] e [is less than or equal to] 0, [[Delta].sup.2] [Pi]/[[Delta]e.sup.2] [is less than or equal to] 0, for all e [Epsilon] [1/2, 1]. (As expected, profit rises with e until it is maximized at e = 1/2 and then it falls steadily.)(10)

PROPOSITION 6. Welfare associated with the regulated single-plant spatial monopolist varies as [W.sub.w] > [W.sub.0.683] > [W.sub.1/2] > [W.sub.1] > [W.sub.0]; and [Delta] W/[Delta] e > 0, [[Delta].sup.2] W/[[Delta]e.sup.2] [is greater than or equal to] 0 for all e [Epsilon] [0, 1/2], [Delta] W/[Delta] e = 0 for e = 0.683, and [[Delta].sup.2] W/[[Delta]e.sup.2] [is less than or equal to] 0 for all e [Epsilon] [1/2, 1].(11)

The relation between e and W is displayed in Figure 3. There is the familiar result that a global maximum of welfare is obtained at an extreme point through strict marginal cost pricing at [W.sub.w]. Of course, this special case assumes some non-distorting scheme which can be devised to compensate the spatial monopolist for operating where profit is negative (specifically [Pi] = -F). The most surprising result, to our knowledge not obtained elsewhere, is that, ignoring the extreme point maximum under marginal cost pricing, welfare is maximized at e = 0.683. It is useful to distinguish this as the "internal welfare maximum" which will be noted as [W.sub.m] hereafter.

Overall these results appear substantially inconsistent with regulatory decisions to require high rates of transportation cost absorption by the regulated spatial monopolist. Both consumer surplus or total welfare are minimized under uniform pricing. It appears that consumers should instruct regulators to require that transportation cost be fully reflected in price.

V. CONSUMER VOTING AND REGULATED TRANSPORTATION COST ABSORPTION

Consumer-voters base decisions on the regulated rate of transportation cost absorption based on the consumer surplus generated by the spatial monopolist. Following Mueller [1989], if consumer welfare can be reduced to a single metric, voting outcomes on regulation are determined by the median voter.

The spatial distribution of consumer's surplus for different values of transportation cost absorption is given by the s(r, e) function displayed in Figure 4 below.(12) Total surplus, which was discussed in Proposition 4 and displayed as a function of e in Figure 2, is the integral of s(r, e) on the interval r [Epsilon] [0, R].

The spatial surplus surface formed by s(r, e) is quite irregular. Voting behavior of consumers located at a particular r is motivated by an attempt to maximize surplus generated by different rates of transportation cost absorption. Finding the value of transportation cost absorption, e*(r), which maximizes consumer surplus for voters at a given r involves rather long and cumbersome manipulation. Conceptually, we are slicing the spatial surplus surface along a given r and determining the level of e, noted e*(r), which maximizes the function thus obtained. It is necessary to find e*(r) separately on the interval [10, 1/2] and on [1/2, 1]. Comparison of the maxima generated on each of these intervals allows one to compute the e*(r) associated with a maximum of surplus on the entire interval e [Epsilon] [0, 1]. Figure 5 shows the relation between location and e*, the surplus-maximizing level of transportation cost absorption.(13)

Location is expressed in terms of the maximum market radius, [R.sub.1/2], shown in Figure 1. There are two significant groups of consumer voters, one with e* = 1 and another with e* = 0. The median voter preference is for e = 0.33, as illustrated in Figure 5.(14) The voting outcome which prefers e [is greater than or equal to] 0.33 consists of consumers located beyond [0.357R.sub.1/2] and extending nearly to [R.sub.1/2]. While voters consider only their own self interest, the final outcome appears to have an obvious element of cross-subsidy. Because regulated e is below 1/2, the market will end with P(R) < [Beta], the choke price, and consumers living near [R.sub.1/2] will be cut off from service. Voters living from point 0 to R = [0.357R.sub.1/2] are forced to subsidize those living past [0.357R.sub.1/2].

The high rate of freight absorption selected by the median voter is inconsistent with a maximum of profit, welfare, or consumers' surplus. Indeed consumers' surplus is lower at the e = 0.333 level selected by voters than when the unregulated profit-maximizing firm engages in spatial price discrimination and sets e = 0.5. This surprising voting outcome is generated in part because the median voter selects such high freight absorption that consumers near the maximum market radius, i.e., near [R.sub.1/2 [prime]] are cut off from service.

Thus far the one-dimensional representation of the market has been adequate. However, analysis of voting behavior should take place in a two-dimensional market where, under the assumption of uniform consumer density, the number of consumers increases with the square of r. If the results in Figure 5 are modified, essentially by "spinning" Figure 5 about the origin to sweep out a circle and hence consider the number of consumers a function of r in a circular market with uniform consumer density, the level of e* characterizing the median voter is e = 0.297.(15)

Thus a spatial model of consumer voting on the level of transportation cost absorption to be imposed on the regulated spatial monopolist suggests that high rates of absorption will be required. voting outcomes which set e = 0.297 (or e = 0.333 for the one-dimensional case) imply that regulators will be compelled to require that the spatial monopolist absorb transportation costs at rates which greatly exceed those associated with either profit maximization by the firm, surplus maximization for consumers, or overall welfare maximization.

V. SUMMARY AND CONCLUSIONS

It is quite common to find a regulatory requirement that spatial monopolists absorb most, if not all, transportation cost, i.e., that they engage in uniform delivered pricing. Given that this is a clear departure from marginal cost pricing, the conclusion that it is not consistent with welfare maximization follows easily. The results presented here extend this easy conclusion in several ways and, based on the spatial surplus surface, provide a voting model which explains the political pressure for uniform pricing schemes.

By merging theoretical models of the spatial firm with regulatory and voting analysis in a manner not previously reported in the literature, we obtain a variety of original and, we hope, interesting results. First, analysis of the effect of varying e over the entire choice set e [Epsilon] [0, 1], shows that uniform pricing, e = 0, minimizes both firm profits and total consumer surplus. Second, internal welfare maximization is achieved with fairly low levels of transportation cost absorption, i.e., with e = 0.683 for the specific case considered here. Third, the spatial distribution of consumer surplus generated by alternate levels of transportation cost absorption is such that the median voter prefers regulations requiring very high levels of absorption, with e = 0.297 for the case analyzed. Thus the spatial distribution of benefits from regulatory decisions may result in voting outcomes which are far removed from those which maximize either the profits of firms or the collective welfare of consumer-voters. Indeed, these voting outcomes produce levels of total welfare and even total consumers' surplus which are below those which would be obtained if the spatial monopolist were not regulated at all. Given the popular notion that uniform delivered pricing is "fair," politicians may find that both cultural pressures and self-interested voting outcomes force them to require high rates of freight absorption by firms subject to regulation.

APPENDIX

If we spin the linear market around the origin, a circular market is generated. The total profit for such a circular market is [Mathematical Expression Omitted] Following the same approach we developed to deal with the linear market, we have for e [Epsilon] [0, 1/2], tR = 3([Beta] - m)/2(2 - e) and P(0) = [3(1 - e) [Beta] + (1 + e)m]/2(2 - e); and for e [Epsilon] [1/2, 1], tR = 3([Beta] - m)/2(1 + e) and P(0) = [(2 - e) [Beta] + 3em]/2(1 + e). A graph similar to Figure 5 can then be drawn accordingly: from 0 to [0.375R.sub.1/2 [prime]] consumers prefer e = 1; from 0.375[R.sub.1/2] to 0.525R, consumers located at r prefer an e that satisfies 2 [(1 + e).sup 2] r = 3; from [0.525R.sub.1/2 [prime]] to [0.750R.sub.1/2 [prime]] consumers all prefer e = 0; and finally, from [0.750R.sub.1/2 [prime]] to [1.00R.sub.1/2 [prime]], consumers located at r prefer an e that satisfies tr = 3([Beta] - m)/2(2 - e).

Therefore, we find that 50 percent of the consumers prefer e > 0.297 and 50 percent of the consumers prefer e < 0.297; 27.6 percent prefer e > 1/2 and 72.4 percent prefer e < 1/2; 14.1% consumers prefer e = 1 and 28.7% consumers prefer e = 0; and consumers at the middle of the market (i.e., r = [0.5R.sub.1/2]) prefer e just above 1/2. [Figures 1 to 5 Omitted]

(1)Analyses of spatial pricing and output decisions by firms have traditionally considered three cases: mill pricing with zero freight absorption, profit-maximizing spatial price discrimination, and uniform pricing with complete freight absorption. A number of significant results on the behavior of the spatial monopolist under these specific alternative freight absorption schemes have been obtained beginning with Singer [1937] and Hoover [1937] and extending to papers by Holahan [1975], Beckmann [1976], Heffley [1980], and Hsu [1983]. In addition, papers on spatial competition by Norman [1981], Villegas [1982] and Capozza and Van Order [1977] have considered the effects of mill pricing vs. spatial price discrimination on price, output, and welfare under spatial oligopoly. An excellent summary of results is found in Greenhut, Norman, and Hung [1987]. (2)Often the price of public services and utilities does not vary with location of the consumer. Such nominal uniform pricing does not automatically imply that all transportation costs are absorbed. Differences in maintenance, service frequency, repair service, emergency response time, etc. may result in higher total cost to consumers located further from point of production. As in other cases, economic analysis should go beyond statutory provision to actual performance measures in measuring the degree of transportation cost absorption. (3)We will "spin" the line around the origin to discuss cases for circular markets below in section V. (4)A general form of the demand function can be specified as q(P) = ([Alpha] - 1) [P.sup.2] - [Alpha] [Beta] P + [[Beta].sup.2], which is well behaved because the choke price is [Beta] and the maximal demand is [[Beta].sup.2] no matter what [Alpha] is, while [Alpha] controls the concavity of the demand function: [Alpha] > 1 implies that [Alpha] is convex, [Alpha] < 1 concave and [Alpha] = 1 for the linear case we are studying. (5)As price rises to the choke price, quantity demanded goes to zero. (6)From equation (3), the first-order condition, [Delta] [Pi]/[Delta] e = 0, gives P(0) = ([Beta] + m)/2 + (1 - 2e) [Iota] R/3. Combining this equation with the other first-order condition, equation (6), we obtain that e = 1/2. It also can be shown that, for the general demand function defined in footnote 3, we have the optimal e for profit maximization is greater than 1/2 if the demand function is convex ([Alpha] > 1) and is less than 1/2 if the demand is concave ([Alpha] < 1). (7)From the first-order conditions, [Delta] W/[Delta] e = 0 and [Delta] W/[Delta] P(0) = 0, we have, respectively, P(0)-m = 2(1-e)tR/e, and P(0)-m = (1-e)tR/2. Solving these equations simultaneously, we obtain e = 1. (8)These results can be derived by defining [Mathematical Expressions Omitted] and applying the optimal values of P(0) and R given by equations (9) to (12). (9)The total surplus is calculated according to equations (4) and (5). These results are illustrated in Figure 2. (10)These results are derived from equation (3). (11)These results are derived by adding [Pi] and S together. (12)A detailed derivation of this graph is available from the authors upon request. (13)A detailed derivation of this graph is available from the authors upon request. (14)From Figure 5 we can see that 50 percent of the consumers along the market prefer e > 0.333 and 50 percent prefer e < 0.333. About 33.3 percent of all consumers prefer e > 0.5 and 66.7 percent prefer e < 0.5. Finally, approximately 33.3 percent of the consumers prefer e = 0 and 22.2 percent prefer e = 1. Consumers located at the middle of the market (where r = 0.5R1/2) prefer e = 0. (15)See the appendix for a derivation.

REFERENCES

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MENG-HUA YE and ANTHONY M.J. YEZER, Department of Economics, George Washington University. Helpful comments were made by an editor, referee, and a number of our colleagues at department seminars.