Two views of applied welfare analysis: comment.

Mayo, John W.

I. Introduction

In a recent article in this Journal, Professor John Wenders provides an exposition of applied welfare analysis, with an application to the pricing of local telephone service |4~. Such inquiries offer the dual hope of both improving our basic knowledge of the principles of applied welfare analysis and of bringing such principles directly to bear on practical economic matters. Accordingly, Professor Wenders is to be commended for undertaking this task. At the same time, however, fundamental aspects of his analysis are considerably less general than Professor Wenders's conclusions suggest. Accordingly, the purpose of this paper is two-fold. First, we point out the special assumptions upon which Professor Wenders's analysis is based and show how modest (and plausible) variations in the model yield substantially different conclusions. Second, we extend Professor Wenders's analysis to the case of self-selecting two-part tariffs. In this case, we argue that Wenders's alleged discrepancy between the Harbeger and voluntary exchange standards of welfare maximization disappears.

II. Background

In his paper, Professor Wenders considers a firm providing local telephone service. The firm can offer either a flat-rate price, where consumers pay a fixed access charge and a zero marginal price per call, or local measured service (LMS), where the subscriber pays a reduced fixed access charge and a positive marginal price for usage.

The firm's costs are incurred according to

C(q) = {N|C.sub.A~ + cq if the tariff is flat rate, or {N|C.sub.A~ + (c + m)q if the tariff is LMS,

where q is firm output, N is the number of customers served (assumed constant for all prices considered), |C.sub.A~ is the customer-specific fixed cost incurred by the company, c is constant marginal cost, and m is the constant marginal cost associated with measuring usage.

Prices in Wenders's model may be regarded as two-part tariffs (p, E), where p is the marginal price and E is the access charge. The incumbent firm is assumed to break even, so tariffs under the two alternative pricing regimes are

|Mathematical Expression Omitted~

where |q.sub.i~ is the consumption of individual i.

Professor Wenders analyzes the sustainability of these prices under the alternative assumptions of homogeneous versus heterogeneous consumers, and welfare dominant LMS versus welfare dominant flat-rate pricing, where "welfare dominant" means that the surplus of an average consumer is larger under the dominant tariff structure. Either tariff can dominate in this sense because aggregate measurement costs may either be larger or smaller than the deadweight loss associated with the (non-marginal cost price) flat-rate tariff. When consumers are homogeneous, Wenders shows that if LMS welfare dominates flat-rate pricing then the viability of equally efficient competitors will force the incumbent firm to adopt LMS. Similarly, with homogeneous consumers, if flat-rate pricing welfare dominates LMS, competition will force the implementation of flat-rate pricing.

In the more provocative (and realistic) case, however, consumers are allowed to be heterogeneous in their usage characteristics. In this case, Wenders argues that even if flat-rate pricing welfare dominates LMS, it will nonetheless prove unsustainable from competition. The basic reason for this unsustainability is that flat-rate prices must be set to recover the costs of the "average" consumer. But when some customers are smaller than average, the "prices" these customers face exceed their stand-alone costs and, therefore, the small customers will prove susceptible to competitive attacks from entrants who offer LMS. Thus, Professor Wenders draws "the immediate practical conclusion . . . that flat-rate telephone service is not sustainable, even if it is optimal in a Harberger benefit/cost sense"(1) |4, 347~. In the following section, however, we show from a theoretical perspective that this principal conclusion from Professor Wenders's analysis is not in any sense general, and that given a plausible alternative structure of industry costs, conclusions directly counter to those of Wenders may be realized.

III. Sustainability and Flat-Rate-Pricing

Consider an incumbent firm offering a price schedule (p, E) where p is the marginal usage price and E is the fixed access (or entry) charge. Let the cost function be C = 1000 + q, which differs from Wenders's structure in that the per-customer access cost is zero, but firm-level fixed cost is present.(2) Assume there are 100 consumers divided into two groups: large and small. Large consumers have demand |x.sub.2~ = 20 - p and small consumers have demand |x.sub.1~ = 10 - p. Consistent with Wenders's distribution, let two-thirds of the customers be small and one-third of the customers be large. Under these conditions, we show that it is not possible for an equally efficient entrant to profitably capture any part of this market by offering a two-part tariff when the incumbent offers a break-even flat-rate price. We demonstrate this result for the case of zero costs of measuring usage. Therefore, the results hold a fortiori for the case of positive measurement costs. Hence, Wenders's conclusion does not generalize to the case of positive firm-level fixed cost.

For a two-part tariff (p, E), the surpluses realized by consumers in each group are

|S.sub.1~(p, E) = 1/2|(10 - p).sup.2~ - E,

and

|S.sub.2~(p, E) = 1/2|(20 - p).sup.2~ - E. (1)

A firm offering (p, E) and serving both consumer types earns profit |Pi~(p, E),

|Pi~(p, E) = (p - 1)|2|x.sub.1~ + |x.sub.2~~(100/3) + 100E - 1000

= (p - 1)(40 - 3p)(100/3) + 100E - 1000. (2)

Within this context, a flat-rate tariff is defined by an entry fee |E.sub.FR~ that achieves zero profit when p = 0. Thus, from equation (2) we have |E.sub.FR~ = (70/3). Note that both consumer groups stay in the market when faced with (0, |E.sub.FR~) since |S.sub.1~(0, |E.sub.FR~) = (80/3) |is greater than~ 0 and |S.sub.2~(0, |E.sub.FR~) = (530/3) |is greater than~ 0.

Now, consider an equally efficient entrant offering any tariff (p, E) with p |is greater than~ 0.(3) To appeal to both consumer groups when (0, |E.sub.FR~) is available from the incumbent, any offering must satisfy |S.sub.1~(p, E) |is greater than or equal to~ 80/3 and |S.sub.2~(p, E) |is greater than or equal to~ 530/3, which from equation (1) implies

3|(10 - p).sup.2~ - (160 + 6E) |is greater than or equal to~ 0,

and

3|(20 - p).sup.2~ - (1060 + 6E) |is greater than or equal to~ 0. (3)

Rearranging these constraints shows that for p |is greater than or equal to~ 0, |S.sub.1~(p, E) - 80/3 |is greater than or equal to~ |S.sub.2~(p, E) - 530/3. Hence, the constraint for the small consumers is automatically satisfied whenever the constraint for the large consumers is satisfied, implying that an entrant seeking to serve both consumer types can ignore the constraint for small consumers and just ensure that large consumers find his offering attractive. Because profit is strictly increasing in E, and |S.sub.2~ is strictly decreasing in E, the constraint will bind at the optimum. Thus, the entrant's problem becomes

|Mathematical Expression Omitted~

Substituting E from the constraint |equation (3)~ into the objective function (equation (2)) yields

|Mathematical Expression Omitted~

Taking the derivative of equation (5) with respect to p yields

|Delta~|Pi~/|Delta~p = (100/3)| - 17 - 3p~ |is less than~ 0 |for all~p |is greater than or equal to~ 0. (6)

Since we know from the definition of |E.sub.FR~ that |Pi~(0, |E.sub.FR~) = 0 this derivative tells us that any tariff yielding larger surplus than (0, |E.sub.FR~) to both consumer groups and having a positive marginal price must yield negative profit. Thus, an entrant cannot profitably serve both customer groups.(4)

Since any tariff that appeals to large consumers will also appeal to small consumers, the only other possibility for an entrant is to try serving only the small consumers. Once again, the constraint will bind with equality (for small consumers), but the profit function is different since the entrant is only serving one consumer group. We have

|Mathematical Expression Omitted~

subject to |S.sub.1~(p, E) = (80/3). (7)

Substituting for E from the constraint |equation (3)~ yields

|Mathematical Expression Omitted~,

and the derivative is

|Delta~|Pi~/|Delta~|Pi~ = (200/3)|1 - p~. (9)

The objective function is concave, so the maximum occurs at p = mc = 1.

The optimal strategy is to price at marginal cost in order to generate the maximal aggregate surplus, and then to recoup as much of the surplus as possible by raising the entry fee to the level at which the consumer is just indifferent to the incumbent's tariff.(5) Hence, E = 83/6 from the constraint with p = 1. But with (p, E) set optimally at (1, 83/6) the entrant still earns a negative profit since |Pi~(1, 83/6) = -700/9 from equation (7).

We conclude, then, that for the case considered here, the flat-rate tariff (0, |E.sub.FR~) is sustainable not only against the specific LMS analyzed by Wenders but against all other two-part tariffs. Thus, contrary to Professor Wenders's claim, flat-rate pricing may indeed be sustainable against alternative rate designs.

IV. Pricing Strategy and the Harberger Standard

At several points Wenders argues that, in the presence of heterogeneous consumers, there is a fundamental discrepancy between the two alternative welfare standards he considers: the Harberger |3~ standard of social welfare maximization and the competitive standard of voluntary exchange maximization. This discrepancy, however, is attributable to an implicit (and unnecessary) assumption that whichever pricing regime is optimal for the average consumer must be applied by the incumbent firm to all customers across the entire market. That is, Wenders's analysis does not permit an incumbent to offer a self-selecting tariff structure that offers consumers a choice of either a flat-rate price or local measured service. Hence, the only way multiple-part tariffs can be made available to consumers is through heterogeneous price offerings across firms. This, in turn, creates a role for an entrant when such heterogeneous prices are welfare-enhancing, thereby generating voluntary exchanges with the entrant. This is the entire reason for the discrepancy between the two standards. Moreover, for the particular case of telecommunications, the restriction on individual firms that only one tariff be offered is unwarranted since optional multi-tariff structures are commonly used in pricing local telephone services.(6)

Given the possibility of offering heterogeneous customers a self-selecting multi-tariff structure containing both pricing regimes so that some consumers prefer the measured service option while some prefer the flat-rate option, analysis based on the average consumer's net benefit only is a misleading application of the Harberger standard. Under these circumstances, this standard requires implementation of an optional tariff structure in order to fulfill the criterion of maximizing total net benefits.

Using Wenders's model, which is portrayed in Figure 1 of his paper, an individual consumer will choose local measured service if the following inequality is satisfied |4, 343~:

c |is greater than~ d + e. (10)

On the other hand, if this inequality is reversed, the flat-rate pricing option will be chosen.(7)

Assume we have two consumers--a small consumer, S, and a larger consumer, L. Further, assume that the inequality in expression (10) is satisfied for S and is reversed for L, so that S would voluntarily choose measured service, and L would voluntarily choose flat-rate pricing. The Harberger standard of welfare maximization then requires that the utility offer a self-selecting optional multi-tariff structure that contains both pricing alternatives. This is true because the net social benefits of such a pricing regime will be greater than the net social benefits of either of the two pricing structures imposed on both groups. That is,

|Mathematical Expression Omitted~

where NB is net social benefits, subscripts refer to pricing structures (MS for measured service and FR for flat rate), and superscripts refer to the individual customers.(8)

Within the context of modern nonlinear price theory, it is standard practice to maximize the sum of the surpluses received by individuals purchasing from a schedule of pricing options |2~. This is accomplished by choosing multiple pricing options tailored to heterogeneous customers and is consistent with Harberger's |3~ postulate (c), which states that ". . . the costs and benefits accruing to each member of the relevant group . . . should normally be added without regard to the individual(s) to whom they accrue."

The voluntary exchanges with the entrant that accompany the emergence of competition in Wenders's model is directly attributable to the elimination of the cross-subsides inherent in the flat-rate pricing regime that is uniformly applied to all customers. Indeed, once such cross-subsidization is eliminated by offering an optional multi-tariff pricing structure, the alleged discrepancy between the welfare standards disappears. Maximization of overall social welfare is entirely consistent with maximization of voluntary exchanges.

Thus, all that Wenders has really shown is that markets with free entry are prone to not tolerate cross-subsidies.(9) He has not shown any fundamental discrepancy between the two welfare standards he considers. To allow the average consumer's preferences to determine the pricing structure faced by all consumers represents a misapplication of the Harberger welfare criterion.

V. Conclusion

Professor Wenders has shown that if the only fixed costs a company incurs are customer specific, flat-rate pricing is not sustainable if it is applied throughout the market even when it dominates LMS on an "average consumer" welfare criterion. We have shown, however, that in the presence of a cost structure more typically attributed to public utilities that includes firm-level fixed cost, and with sufficiently diverse consumers, flat-rate pricing may be sustainable. Our results hold whether the fixed costs result from measurement or not. The basic reason is that when consumers are sufficiently diverse it is hard for an entrant to attract large consumers who only pay an "average" flat rate. Attempts to target small customers may similarly be foreclosed if fixed costs are sufficiently large. Thus, the sustainability of flat-rate pricing is an empirical question that depends on the size of fixed costs and the dispersion of consumers, and is not a question that is answered by theory as suggested by Professor Wenders.

We have also shown that the apparent discrepancy between the two alternative welfare standards analyzed by Professor Wenders is attributable to his implicit assumption that a single tariff structure (either flat rate or LMS) must apply to all customers of the regulated firm. If the firm is, instead, allowed to offer a self-selecting optional tariff structure (as most do), this discrepancy is eliminated.

David L. Kaserman Auburn University Auburn, Alabama

David M. Mandy University of Tennessee Knoxville, Tennessee

John W. Mayo University of Tennessee Knoxville, Tennessee

1. Professor Wenders also argues |4, 342~ ". . . any debate about Harberger benefits and costs of LMS is irrelevant. Beneficial or not on these grounds, strict flat-rate pricing is unsustainable in a competitive environment."

2. It should be noted, however, that the subsequent analysis is not dependent on our exclusion of a per-customer cost component in the cost function. All that is required is that some fixed costs exist at the firm (as opposed to the customer) level.

3. The specific two-part tariff considered by Wenders (referred to as LMS) dictates that price be set equal to marginal cost and E be set to break even. Our analysis considers the more general case where the entrant is allowed to chose among any two-part tariff such that p |is greater than~ 0 and E |is greater than or equal to~ 0.

4. We assume here that prices must be positive.

5. This result is not surprising since an entrant attempting to serve only the small consumers is formally equivalent to the problem dealt with in (1). In both scenarios the optimal two-part tariff maximizes surplus by marginal cost pricing (together with an entry fee).

6. In fact, local measured service has been offered on an optional basis in nearly every jurisdiction in which it has appeared.

7. For linear demand with parallel shifts, areas c and e will remain constant. Area d, however, increases with increasing demand. Thus, inequality (10) will tend to be satisfied for relatively small consumers, and it will tend to be reversed for relatively large consumers.

8. Proof of this inequality involves some fairly tedious algebra and is, therefore, not presented here. Moreover, the logic behind this expression is straightforward, and a formal proof is probably not necessary. Such proof is, nonetheless, available from the authors upon request.

9. Though as we have shown, such markets may tolerate considerable price discrimination, depending on the industry cost structure and the distribution of consumer demands.

References

1. Coase, Ronald H. "The Marginal Cost Controversy." Economica, 1946, 169-82.

2. Goldman, M. B., H. E. Leland and D. S. Sibley, "Optimal Nonuniform Pricing." Review of Economic Studies, April 1984, 305-19.

3. Harberger, Arnold C. "Three Basic Postulates for Applied Welfare Economics: An Interpretive Essay." Journal of Economic Literature, September 1971, 785-97.

4. Wenders, John T. "Two Views of Applied Welfare Analysis: The Case of Local Telephone Service Pricing." Southern Economic Journal, October 1990, 340-48.