Uniform two-part tariffs and below marginal cost prices: Disneyland revisited.

Cassou, Steven P. ; Hause, John C.

I. INTRODUCTION

Since Cournot's [1838] classic analysis of a hard-nosed sole proprietor of a spring pricing water to maximize his profit, the simple theory of monopoly has concluded that a monopolist will produce the level of output at which marginal revenue is equal to marginal cost, and sell this output at the market clearing price, (Hicks [1935]). With a negatively sloped demand curve, this results in the monopoly equilibrium price exceeding marginal cost, and in less output than would be demanded if price is equal to marginal cost. In a seminal contribution to the analysis of twopart tariffs, Oi [1971] elicited some surprise when he pointed out that a profit maximizing monopolist might, under some conditions, choose a (marginal) price that is less than the marginal cost.(1) Later work by Schmalensee [1981] described a specific set of conditions in which pricing below marginal cost would be observed. Despite these demonstrations, the subsequent literature on uniform two-part tariffs for the most part considers below marginal cost pricing a thoroughly atypical outcome.(2)

However, as an empirical matter, pricing below marginal cost is easily found. Disneyland-like recreation, ski resorts, buffet service of food, and the ubiquitous presence of open bowls of peanuts in bars are but a few examples. Srinagesh [1991] has argued convincingly that "the practice of setting marginal prices below marginal cost is so common in telecommunications offerings that it can justifiably be labeled a stylized fact." This paper fills the theoretical void by answering the following questions about pricing below marginal cost, using two-part tariffs in Oi's model with demand heterogeneity. Is the set of demand and cost parameters that induce such pricing small or large? When it occurs, will the price typically be slightly or substantially less than marginal cost? How large of an effect can such pricing have on profits, and does it typically have a small or large effect?

We consider the class of all linear demand models with two types of consumers and a monopolist with constant marginal costs who charges a profit-maximizing uniform two-part tariff. We first show that two conditions are necessary for below marginal cost pricing to occur: 1) The consumer demand curves intersect at a strictly positive price and quantity; 2) The marginal cost is less than the demand curve intersection price. It is shown that pricing below marginal cost arises when consumers have sufficiently different demand elasticities so that the revenue losses arising from the high usage of the elastic consumer are more than offset by revenue gains obtained from the large entrance fee charged to the inelastic consumer.3 Next, we determine how the monopolist sets prices to maximize profits, given parameter values of the model. We then describe the behavior of p*/c the ratio of the optimal (marginal) price to marginal cost, over the parameter space. This provides the basis for determining regions in the parameter space in which p* [less than] c, i.e., where p*/c [less than] 1. To indicate quantitatively the potential significance of pricing below marginal cost, we find that sampling uniformly over the parameter space satisfying the two necessary conditions induces below marginal cost pricing 40% of the time. Furthermore, conditional on price being less than marginal cost, the mean ratio of p*/c with uniform sampling is .55.

What relevance do these calculations with such a simple and austere model have for understanding nonlinear pricing in the real world? In an important recent paper, Sibley and Srinagesh [1997] study nonlinear pricing in multiproduct markets. They show that a critical assumption for below marginal cost pricing to occur is a "uniform ordering of demand curves" condition which is the generic equivalent of the non-crossing of demands condition in the analysis of nonlinear pricing of a single product. They demonstrate that violation of the uniform ordering condition is necessary for below marginal cost pricing to occur (when demand cross-elasticities are zero), that such pricing easily arises when the condition is violated, and indicate how extremely restrictive the condition is. Our analysis takes the first steps down the path to understanding the more empirically realistic structures where the uniform ordering condition is relaxed by thoroughly exploring the simplest case of one product and two types of consumers.

The paper is organized as follows. In section II we describe the model. Section III derives pricing formulas for an optimizing monopolist. In section IV we show that pricing below marginal cost occurs often and the incentives for doing so can be large. Section V concludes.

II. THE MODEL

Our model is a parameterized version of Oi's [1971] two-consumer example. It consists of a market in which there are three agents, a single firm and two consumers of the firm's output. The firm is interpreted as a monopolist for the market's output and is able to charge a two-part tariff to those who want to consume its goods. The first part is an entrance fee, which we denote by a, and the second part is a marginal price, which we denote by p. The monopolist has a constant marginal cost, c. In our analysis, we assume that consumers have linear demand curves for the monopolized commodity with no income effects.(4)

Our analysis is carried out in a parameter space which consists of parameter combinations that characterize the consumers' preferences and the firm's marginal cost. We wish to determine regions in this parameter space where the optimal price [p.sup.*] is less than the marginal cost c. Some of the parameter space can be immediately excluded from the analysis since in this excluded region, pricing below marginal cost will never occur. Two necessary conditions for pricing below marginal cost are: 1) The consumer demand curves intersect at a strictly positive price and quantity; 2) if [p.sub.1] is the price coordinate at which such a pair of demand curves intersect, the marginal cost c lies in the interval (0, [p.sub.1]).(5) Condition (2) actually has two parts: (i) When c [element of] (0, [p.sub.1]), it is possible for p* [less than] c; and (ii) when c [not an element of] (0, [p.sub.1]), it is not possible for p* [less than] c. We now show why (2) (i) is true. The demonstrations of (1) and (2) (ii) are similar.

To establish (2) (i) we use Figure 1 which depicts a pair of consumer demands that cross at a price level above the marginal cost. Consider two prices, [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where [Mathematical Expression Omitted] is depicted by the horizontal line beginning at point A and [Mathematical Expression Omitted] is depicted by the horizontal line beginning at point G. If the monopolist is to serve both consumers, then the entrance fee a will equal the smaller of the two consumer surpluses, evaluated at the marginal price. As Figure 1 indicates, the lower marginal price results in an increase in the entrance price equal to the area of the trapezoid ACEG and an increase in costs equal to [Mathematical Expression Omitted] times ([q.sub.1] + [q.sub.2]).(6) It follows that the net revenue received by the firm is lower from the elastic consumer but higher from the inelastic consumer.(7) Under some circumstances, illustrated by Figure 1, it is possible for the net loss from the elastic consumer (area of triangle CDE) to be more than offset by the net gain from the other consumer (area of trapezoid BCEF), increasing the firm's total return.

Figure 1 suggests that there are three dimensionless parameters that essentially describe when the necessary conditions for pricing below marginal cost are present. These are the demand elasticities of the two consumers where their demands intersect, and the relative distance of the marginal cost below the intersection price. We achieve the desired parameterization in the following way. It is convenient to normalize the demand functions so that they intersect at the price and quantity ([p.sub.I], [q.sub.I]) = (1,1). There is no loss of generality from this normalization since both the unit of measure for the quantity of the monopolized product and the numeraire of the price system are arbitrary.

With this specification, the inverse demand of consumer i, for i = 1,2, is given by

(1) [p.sub.i] = (1 + [k.sub.1]) - [k.sub.i] [q.sub.i], for 0 [less than] [k.sub.i],

where [p.sub.i] is consumer i's marginal valuation of the commodity when he is consuming [q.sub.i], units, and [k.sub.i] is the value of the consumer's demand parameter. (Since all consumers purchasing the commodity face the same marginal price, p, chosen by the monopolist, all purchasing consumers in equilibrium have their [p.sub.i] = p.) The most useful interpretation of [k.sub.i] is that -1/[k.sub.i] is the demand elasticity of i at the common demand curve intersection point ([p.sub.I], [q.sub.I]) = (1,1). Hence, at this point, consumer i's demand is relatively elastic (exceeds 1) if [k.sub.i] [less than] 1 and is relatively inelastic (is less than 1) if [k.sub.i] [greater than] 1. Because the price coordinate of the demand curve intersection point is [p.sub.I] = 1, the marginal cost c (0 [less than] c [less than] 1) can be interpreted as the relative distance of marginal cost below the intersection price.

With this specification of the consumer demands, our objective is to describe the location in the parameter space {[k.sub.1], [k.sub.2], c [where] 0 [less than] [k.sub.1], 0 [less than] [k.sub.2], 0 [less than] c [less than] 1} where a profit maximizing monopolist chooses a two-part tariff [a.sup.*] and [p.sup.*] so that [p.sup.*]/c [less than] 1.(8) We describe the monopolist's optimization problem as the optimum of four separate optimization subproblems.

SUBPROBLEM 1: Serve both consumers and charge fixed fee equal to consumer 1's surplus.

[Mathematical Expression Omitted],

subject to (1),

(2) a = .5[q.sub.1](1 + [k.sub.1] - p),

(3) a [less than or equal to] .5[q.sub.2] (1 + [k.sub.2] - p),

and

(4) 0 [less than or equal to] p.

SUBPROBLEM 2: Serve both consumers and charge fixed fee equal to consumer 2's surplus. This constrained optimization problem is identical to subproblem 1, with subscripts 1 and 2 interchanged.

SUBPROBLEM 3: Serve only consumer 1.

[Mathematical Expression Omitted]

subject to (1), (2) and (4).

SUBPROBLEM 4: Serve only consumer 2. This constrained optimization problem is identical to subproblem 3 with subscripts 3 and 1 replaced by subscripts 4 and 2.

General Profit maximization problem:

[Mathematical Expression Omitted]

A substantial simplification of the subsequent analysis of these subproblems and their relationships is achieved by recognizing the symmetry between the pair of subproblems 1 and 3 and the pair of subproblems 2 and 4. This implies that regions relevant for any of the subproblems are also realized by the corresponding symmetric subproblem in the symmetric regions that have been reflected about the 45 [degrees] line in the [k.sub.1] - [k.sub.2] plane.

III. DETERMINING THE PROFIT-MAXIMIZING MARGINAL PRICE

The optimization problem described above also indicates the solution algorithm. For given values of c, [k.sub.1], and [k.sub.2], each of the subproblems is solved and the largest profit selected. The three dimensional parameter space in c, [k.sub.1], and [k.sub.2], can be partitioned into regions such that within each region a single subproblem is always used. Such a partitioning allows one to focus on a specific type of monopolist behavior within a given region. In this section we describe the monopolist pricing behavior within these regions.

In all of the following analysis, c is restricted to the interval (0,1) except where explicitly noted. We begin by investigating subproblem 1. Because of the inequality constraints, subproblem 1 partitions its own region of the parameter space further. In particular, Kuhn-Tucker optimization implies that

(5) [Mathematical Expression Omitted],

and

[a.sup.*] = .5[(1 + [k.sub.1] - [p.sup.*])2/[k.sub.1]]

where

(6) p[prime] = (-[k.sub.2]/2) (1/[k.sub.1] - 1/[k.sub.2] - c/[k.sub.1] - c/[k.sub.2])

p[double prime] = 1 - [-square root of [k.sub.2][k.sub.1]]

and the regions in which each case holds are defined by various Kuhn-Tucker inequalities.(9) The pricing formula p[prime] occurs when there is slack in both the inequality constraints (3) and (4). p[double prime] occurs when the inequality constraint (3) holds with equality and there is slack in the inequality constraint (4). And p = 0 occurs when the inequality constraint (4) holds with equality. The optimal [a.sup.*] is the consumer surplus of consumer 1 facing a linear price of [p.sup.*]. As Oi [1971] and Schmalensee [1981] pointed out, pricing below marginal cost does occur. For instance, when [k.sub.2] [greater than] [k.sub.1] [greater than] 1 (so that both demands are relatively inelastic) and case 1 of subproblem 1 is used, equation (6) implies c [greater than] [p.sup.*]. In the following section we show that such pricing can occur often.

Proceeding with the other subproblems shows that the subproblem 2 results in pricing formulas that are symmetric analogues to those for subproblem 1. Subproblem 3 calculations result in [p.sup.*] = c (marginal cost pricing) and [a.sup.*] = .5[[(1 + [k.sub.1] - [p.sup.*]).sup.2]/[k.sub.1]], where [a.sup.*] is the consumer's surplus of consumer 1. And, subproblem 4 is symmetric to subproblem 3.

IV. HI HO! ON TO DISNEYLAND

In this section, two quantitative measures are used to summarize the monopolist's optimal solution over the parameter space. We first examine the ratio of the optimal marginal price to the marginal cost, [p.sup.*]/c, and how it is distributed over the parameter space. Next the interesting and extreme case when [p.sup.*]/c = 0 is investigated with particular attention devoted to showing how changes in c alter the size and location of the set of demand parameters for which this is the optimal condition. We then turn to the second quantitative measure, the ratio of unconstrained monopoly profit to monopoly profit when the monopolist is constrained to p [greater than or equal to] c. The ratio suggests how strong the incentives are to price below marginal cost over various regions of the parameter space.

Distribution of the [p.sup.*]/c ratio over the parameter space

The ratio [p.sup.*]/c quantifies the pricing decision in a way that facilitates the comparison of optimal prices under different cost conditions. In this section a brief geometric tour of the parameter space is provided, to show the regions in which [p.sup.*]/c [less than] 1, i.e., price is less than marginal cost. A plot of [p.sup.*]/c in [k.sub.1] - [k.sub.2] space with c = .5 is provided in Figures 2a and 2b. This relationship is shown from two viewpoints, to obtain good resolution over a large domain of the parameter space.(10) Consider first the various cases of subproblem 1. Figures 2a and 2b show a side view of the 3-dimensional relationship of [p.sup.*]/c as a function of [k.sub.1] and [k.sub.2]. If [k.sub.1] = [k.sub.2], the two consumers have identical demands. Here the monopolist sets [a.sup.*] to the common value of the consumer surpluses and [p.sup.*] = c. This corresponds to the ridge [p.sup.*]/c = 1 on the vertical axis beginning at the origin in the right corner and extending to the left corner of the figures. The curved, negatively sloped surface with the ridge as its back border shows the behavior of the ratio in part of the case 1 region of subproblem 1. In this part of the case 1 region, [p.sup.*]/c [less than] 1. The figures show clearly that a large set of parameters lead to this outcome. The small surface facing the viewer in the right corner, around the [k.sub.1] - [k.sub.2] origin, corresponds to the other part of the case 1 region of subproblem 1. In this part of the case 1 region [p.sup.*]/c [greater than] 1. The case 2 region of subproblem 1 can be found near both of the axes. This is the region along the [k.sub.1] axis where the surface falls as [k.sub.2] increases. Similarly, this region can also be found along the [k.sub.2] axis and is the portion of the surface which falls as [k.sub.1] increases.(11) The figure also shows that in much of the case 2 region, [p.sup.*]/c [less than] 1. The case 3 region for subproblem 1 is the low flat region where [p.sup.*]/c = 0 that includes the point ([k.sub.1], [k.sub.2]) = (.6,3) and is better viewed in Figure 2b.

The corresponding cases for subproblem 2 can readily be observed in these figures by the symmetric locations of the subproblem 1 surface about the ridge line [k.sub.1] = [k.sub.2]. The high flat region with [p.sup.*]/c = 1 is a combination of subproblems 3 and 4, where only one customer is being served.

Figures 2a and 2b clearly demonstrate that for c = .5, a large set of demand parameter pairs induce the monopolist to price below marginal cost. We also explored other values of c [greater than] 0. Our main findings about the location of the parameter space that results in the optimal [p.sup.*] [less than] c are as follows. 1) The marginal cost must be strictly less than 1, i.e., strictly less than the price coordinate of the intersection of the consumer demand curves. 2) Although it cannot happen if the demand curves are identical, it always occurs if the demands differ by a small amount and the demands are sufficiently inelastic.(12) 3) It cannot occur if both demands are too elastic, i.e., [k.sub.1] and [k.sub.2] are both small. 4) It cannot occur if the demands are so different that the monopolist finds it most profitable to serve only one of the customers.

One way to quantify the extent to which typical parameter values induce pricing below marginal cost is to impose a distribution on the parameter space and sample, to determine the frequency of the parameter values that induce [p.sup.*]/c [less than] 1. We focus on models that satisfy the necessary conditions described earlier, thus the following probabilities are conditional on these assumptions. To this end, [x.sub.i] is sampled independently and uniformly over the unit interval, and transformed by letting [k.sub.i] = [x.sub.i]/(1 - [x.sub.i]). In Table I, the first three lines of column 3 report the estimated probability that independent pairs of [k.sub.i] lead to [p.sup.*]/c [less than] 1 for three constant levels of c (.25, .5, and .75). The fourth line of column 3 reports the estimated probability when c is uniformly distributed over the unit interval along with the demand parameters. The estimated probabilities are the percentage of 100,000 random draws [TABULAR DATA FOR TABLE I OMITTED] which resulted in below marginal cost pricing. Column 3 line 4 shows that the fraction of below marginal cost outcomes was .39 when marginal cost is uniformly sampled from c [element of] (0,1). Furthermore, holding c constant at widely different values led to relatively little change in the fraction, although there is some suggestion that the fraction increases modestly with c (at least from .25 to .5).(13) The last three lines of Table I partition the results of line 4 to reveal how the demand pair elasticities affect the outcome. These lines show the sampling results when both demands are relatively inelastic (both [k.sub.i]'s [greater than] 1), one demand is relatively elastic and the other is relatively inelastic, and both demands are relatively elastic (both [k.sub.i]'s [less than] 1), respectively. Column 3 shows that the fraction of cases with [p.sup.*]/c [less than] 1 decreases from .52 to .29 across these categories. Thus, when the necessary conditions are satisfied, below marginal cost pricing occurs more than half the time when both demands are inelastic. But even when both demands are elastic, it still occurs nearly a third of the time.

Another important quantitative question is the extent to which [p.sup.*]/c [less than] 1 for typical parameter values. We address this issue by estimating the conditional mean of this ratio (conditional on [p.sup.*]/c [less than] 1), using the same sampling procedure described in the preceding paragraph.(14) Column 4 presents estimates for the conditional means (and standard deviations). Column 4, line 4 shows the conditional mean is almost .55 when marginal cost is uniformly sampled from c [element of] (0,1). The conditional mean is a strongly increasing function of c over the unit interval, changing from .26 to .83 as c changes from .25 to .75. As indicated in the next section, this result is primarily a consequence of the fact that smaller c values imply a large increase in the demand parameter space where [p.sup.*]/c = 0. What role do the demand elasticities play in determining the conditional means of [p.sup.*]/c? The fourth column of the last three rows of the table show that the greatest relative deviation of [p.sup.*] below c among the conditional means occurs when one demand is inelastic and the other elastic and the highest conditional mean of[p.sup.*]/c (.91) occurs when both demands are inelastic. These differences in conditional means are strongly associated with the relative frequency of the optimal [p.sup.*] = 0 for the three different demand elasticity categories. It follows that for typical values in the parameter space, if [p.sup.*]/c [less than] 1, it should not be difficult to detect this outcome empirically.

The parameter space where the price of rides is 0

This section discusses further the interesting and extreme case in which [p.sup.*] = 0 for c [greater than] 0. As Figure 2b demonstrates, the set of demand parameter pairs that generate the case 3 region where [p.sup.*] = 0 is sizable even for a marginal cost, c, as large as .5. This implies that the case 3 region is of interest not only because it represents the most extreme form of below marginal cost pricing, but also because the size of the case 3 region indicates that [p.sup.*] = 0 should be the most frequently observed below marginal cost price. Indeed, most ski resorts and amusement parks, and even Disneyland, currently price some activities at [p.sup.*] = 0.(15)

Further analysis of the case 3 region reveals that the marginal price [p.sup.*] is always greater than 0 when c [greater than] .685. At this critical value of c, the case 3 region consists of the two points ([k.sub.1], [k.sub.2]) = (.432, 2.315) and ([k.sub.1], [k.sub.2]) = (2.315, .432).(16) With these demand parameter values, the consumer's surplus at [p.sup.*] = 0 for the two consumers are equal. If the monopolist is restricted to p [greater than] c, the monopolist would only serve the consumer who has the less elastic demand.

As c decreases, the size of the case 3 region expands around these points and becomes very large for values of c near 0. The expansion about the point ([k.sub.1], [k.sub.2]) = (.432, 2.315) occurs in the region where [k.sub.2] [greater than or equal to] 1/[k.sub.1] and [k.sub.2] [greater than or equal to] (1 + c) [k.sub.1]/(1 - c). The expansion about the point ([k.sub.1], [k.sub.2]) = (2.315, .432) is symmetric.

Distribution of the ratio of unconstrained to constrained profits over the parameter space

This section studies a second quantitative summary of the monopolist's optimal choice: The ratio V/[V.sub.c], where V is the unconstrained monopoly profit and [V.sub.c] is monopoly profit if the monopolist is constrained to choose a marginal price at least as great as marginal cost. Because the profit function is concave, the optimal (constrained) price is [Mathematical Expression Omitted] if the constraint is binding. The ratio V/[V.sub.c] and its distribution over the parameter space is of some interest for several reasons. First, V/[V.sub.c] = 1 if [p.sup.*] [greater than or equal to] c, and V/[V.sub.c] [greater than] 1 if [p.sup.*] [less than] c. Hence a graph of this ratio makes it easy to see the parameter region for which [p.sup.*] [less than] c, and this gives an alternative to the Figure 2 graphs of [p.sup.*]/c for visualizing parameter regions where [p.sup.*] [less than] c. Second, the ratio V/[V.sub.c] suggests the relative strength of a monopolist's incentive to price below marginal cost if the exogenous parameter values make [p.sup.*] [less than] c optimal. This is relevant for assessing whether the possibility of pricing below marginal cost may provide a monopolist with significant incentives to engage in such pricing.

Figure 3 graphs the V/[V.sub.c] ratio as a function of [k.sub.1] and [k.sub.2], holding c = .5. As it must, this figure confirms the qualitative conclusion of Figures 2a and 2b that a large region in the parameter space generates below marginal cost pricing. The figure shows that there are some parameter values for which the ratio is almost 1.3, indicating that the monopolist can gain almost 30% in profits by being able to sell at less than the marginal cost. It can be shown analytically that the highest values of the ratio approach [-square root of 2] and that this occurs for parameters depicted in Figure 3.(17) Figure 3 does not indicate these values since this occurs under very special circumstances which the relatively crude grid used to plot Figure 3 is unable to capture. The highest values in the graph, as well as overall, occur where the demand curves of the consumers have very different elasticities, and have the property that the surplus of the consumers are equal at the optimal price [p.sup.*]. As Figure 3 indicates, the parameter regions where the big profit incentives are available are not very large. The vast majority of parameter values that lead to [p.sup.*] [less than] c yield only a small increase in profit beyond what could be attained with marginal cost pricing.

The last column of Table I quantitatively confirms this result. It shows the conditional mean of V/[V.sub.c] (conditional on [p.sup.*]/c [less than] 1) is 1.066 when c = .5, and that this conditional mean is insensitive to c. We conclude that for typical parameter values that make below marginal cost pricing profitable, profits are increased by an average factor of only .05. This result needs to be interpreted with care, since the analysis assumed the monopolist had constant marginal costs and no fixed costs. In important areas of the economy where nonlinear pricing is used, such as telecommunications and electric power, fixed costs are large. Everything else the same, the introduction of positive fixed costs will increase the size of the ratio if the constraint is binding.(18) Thus the existence of large fixed costs could significantly increase the V/[V.sub.c] ratio for given parameters that induce [p.sup.*] [less than] c.

V. CONCLUSION

The introduction noted the ubiquity of below marginal cost pricing in multiproduct pricing in the telecommunications industry. Sibley and Srinagesh [1997] show that this outcome requires violation of the uniform ordering of demands condition (when product demands are independent). Thus, a key to a better understanding of such pricing requires a more detailed quantitative picture of what happens when this condition is violated. Since the non-crossing of demands condition in the analysis of a single product is the generic equivalent of this condition, our paper provides an initial attack on this issue. We thoroughly explore a very simple single product model with two types of consumers, limiting attention to a parameter space where the necessary conditions for below marginal cost pricing hold (where the demands intersect, and marginal cost is less than the intersection price). These results should provide a useful benchmark for quantitative studies of the more complex multiproduct model. We think further work on the multiproduct extension is an important direction for future work.

Sampling experiments with our model reveal that pricing below marginal cost occurs much of the time; over 50% when both demands are relatively inelastic at the intersection price. Conditional on parameter values that lead to below marginal cost pricing, the mean ratio of [p.sup.*]/c is considerably less than 1, .56 when both demands are relatively inelastic, and .40 when one is relatively inelastic and the other is not. Although it is possible that the unconstrained to constrained profit ratio V/[V.sub.c] from pricing below marginal cost can approach 1.41, the sampled mean of the ratio is much smaller at 1.05.

APPENDIX

Kuhn-Tucker Conditions

The subproblem 1 optimization problem can be written as

[Mathematical Expression Omitted],

subject to

[(1 + [k.sub.1] - p).sup.2]/[k.sub.1] + [less than or equal to] [(1 + [k.sub.2] - p).sup.2]/[k.sub.2] and 0 [less than or equal to] p.

The Lagrangian for this problem is

L([center dot]) = [(1 + [k.sub.1] - p).sup.2]/[k.sub.1] + (p - c) [(1 + [k.sub.1])/[k.sub.1]

+ (1 + [k.sub.2])/[k.sub.2] - p (1/[k.sub.1] + 1/[k.sub.2])]

+ [Lambda][[(1 + [k.sub.2] - p).sup.2]/[k.sub.2] - [(1 + [k.sub.1] - p).sup.2]/[k.sub.1]].

Kuhn-Tucker conditions are:

[Delta]L([center dot])/[Delta]p = -2(1 + [k.sub.1] - p)/[k.sub.1] + (1 + [k.sub.1])/[k.sub.1]

+ (1 + [k.sub.2])/[k.sub.2] - (2p - c) (1/[k.sub.1] + 1/[k.sub.2])

+ [Lambda][-2(1 + [k.sub.2] - p)/[k.sub.2] + 2(1 + [k.sub.1] - p)/[k.sub.1]] [less than or equal to] 0,

0 [less than or equal to] p, and [Delta] L([center dot])/[Delta]p [multiplied by] p = 0.

[Delta]L([center dot])/[Delta][Lambda] = [[(1 + [k.sub.2] - p).sup.2]/[k.sub.2]

- [(1 + [k.sub.1] - p).sup.2]/[k.sub.1]] [greater than or equal to] 0,

0 [less than or equal to] [Lambda] and [Delta]L([center dot])/[Delta][Lambda] [multiplied by] [Lambda] = 0.

These conditions imply three possible cases:

Case 1: [(1 + [k.sub.2] - p).sup.2]/[k.sub.2] - [(1 + [k.sub.1] - p).sup.2]/[k.sub.1] [greater than] 0 and p[greater than]0.

This implies [Lambda] = 0 and p = (-[k.sub.2]/2)(1/[k.sub.1]-1/[k.sub.2] -c/[k.sub.1] - c/[k.sub.2]). The case 1 region of subproblem 1 consists of parameter values for c, [k.sub.1], and [k.sub.2] such that, with the value of p given by this expression, p [greater than] 0 and [(1 + [k.sub.2] - p).sup.2]/[k.sub.2] - [(1 + [k.sub.1] + p).sup.2]/[k.sub.1] [greater than] 0.

Case 2: [(1 + [k.sub.2] - p).sup.2]/[k.sub.2] - [(1 + [k.sub.1] - p).sup.2]/[k.sub.1] = 0 and p[greater than]0.

The implies [Lambda] = {-2(1 + [k.sub.1] - p)/[k.sub.1] + [(1 + [k.sub.1])/[k.sub.1] + (1 + [k.sub.2])/[k.sub.2] - (2p - c)(1/[k.sub.1] + 1/[k.sub.2])]}/{[2(1 + [k.sub.2] - p)/[k.sub.2]] - [2(1 + [k.sub.1] - p)/[k.sub.1]]} [greater than or equal to] 0 and p = 1 - [-square root of [k.sub.2][k.sub.1]]

The case 2 region of subproblem 1 consists of parameter values for c, [k.sub.1] and [k.sub.2] such that, with the values for [Lambda] and p given by these expressions, p[greater than]0 and [Lambda] [greater than or equal to] 0.

Case 3: p = 0.

For this case, [Lambda] is not uniquely determined. The case 3 region of subproblem 1 consists of parameter values for c, [k.sub.1] and [k.sub.2] such that two conditions hold. First [(1+ [k.sub.2]).sup.2]/[k.sub.2] - [(1+ [k.sub.1]).sup.2]/[k.sub.1] [greater than or equal to] 0. Second, either -2(1 + [k.sub.1])/[k.sub.1] + (1 + [k.sub.1])/[k.sub.1] + (1 + [k.sub.2])/[k.sub.2] + c (1/[k.sub.1] + 1/[k.sub.2]) [less than or equal to] 0 or -2(1 + [k.sub.2])/[k.sub.2] + 2(1 + [k.sub.1])/[k.sub.1] [less than or equal to] 0.

The authors thank Min Liu, Bridger Mitchell, Walter Oi, Marius Schwartz, David Sibley, Padmanabhan Srinagesh, Lester Taylor, and the anonymous referees of this journal for comments that have materially improved the final version of this paper. Neither they nor Cournot can be blamed for any errors in the following.

1. A uniform two-part tariff is a pricing schedule of the form a + pq which requires each consumer to pay a fixed fee, a, for the right to purchase any amount of the good and to pay a constant amount, p, for each of the q units of the good purchased. In his paper, Oi interpreted a as the entrance fee to an amusement park and p as the price of a ride. A two-part tariff is uniform if all consumers face the same price structure coefficients, a and p, when purchasing the good.

2. See, e.g., Ng and Weisser [1974], Tirole [1988], Varian [1989], and Carlton and Perloff [1993]. Furthermore, the magisterial study of nonlinear pricing by Wilson [1993] contains 244 references with only the prepublication title of Srinagesh [1991] suggesting a focus on below marginal cost pricing.

3. Our analysis maintains the standard assumption of two-part tariff models, that consumers are price takers who can buy as much as they want at the marginal price p, but are unable to resell the good. If one assumed the monopolist could impose quantity constraints onto the tariff, one might relax the assumption that p [greater than or equal to] 0.

4. The assumption of no income effect enables one to use the consumer's surplus as an exact measure of the excess amount a consumer would be willing to pay over the variable payment, pq, which he makes under the twopart tariff.

5. Intersection of the demand curves is incompatible with the assumption of "strong monotonicity" which is often imposed in theoretical discussions of nonlinear pricing, (Brown and Sibley [1986]). It is also incompatible with the "quasi-supermodularity" assumption recently developed by Milgrom and Shannon [1994] as a broad regularity condition for monotone comparative statics.

6. The possibility of collecting the indicated increase in the entrance price from the inelastic consumer requires the latter's consumer surplus to be at least as large as the surplus of the elastic consumer at p.

7. For two linear demand curves with different intercepts on the price axis, we refer to the demand curve with the larger intercept as the "inelastic" demand and the curve with the smaller intercept as the "elastic" demand. This terminology is natural since for any price at which both consumers purchase positive quantities, the demand curve with the larger intercept is less elastic.

8. In the theoretical literature on two-part tariffs and more general forms of nonuniform pricing, it is common to consider a one-parameter family of demand functions q(p,t). Most of this literature assumes that the consumer preferences are "weakly monotonic" in the parameter t, which means that for all p, as t increases, a consumer gets a greater surplus from consumption of the commodity (Brown and Sibley [1986]). Our one-parameter family of linear demand functions is not weakly monotonic. If two linear demands intersect, the steeper one necessarily has greater surplus for prices above the intersection price [p.sub.I]. But the less steep demand may have greater surplus for sufficiently low prices that are substantially less than [p.sub.I]. The lack of weak monotonicity for the demand system substantially complicates the geometry of the parameter space for the monopolist's optimization problem. The text shows that four subproblems must be considered with the linear system. There would only be two subproblems with a weakly monotonic system: Whether to serve both consumers or just the high surplus consumer.

9. The Kuhn-Tucker inequalities and how they define the different regions are formally described in the appendix.

10. The three dimensional graphs in Figures 2a, 2b and 3 are based on a gridding procedure. Some of the abrupt discontinuities in the graphs are a consequence of this procedure. The actual behavior of the ratios as a function of [k.sub.1] and [k.sub.2] is much smoother.

11. It is useful to note that the case 2 regions for subproblem 1 and subproblem 2 are identical. This follows because these are the regions where the inequality constraint (3) and its symmetric pair for subproblem 2 hold with equality.

12. To be sufficiently inelastic, [k.sub.1] and [k.sub.2] must be sufficiently large. The minimum value for [k.sub.1] and [k.sub.2] depends on c. However, a sufficient condition which holds for all cis, [k.sub.1] [greater than] 1 and [k.sub.2] [greater than] 1.

13. Note that the uniform sampling of [x.sub.i] from the unit interval implies that [k.sub.i] and 1/[k.sub.i] have the same distribution. Since 1/[k.sub.i] is the demand elasticity (at the demand intersection point), it follows that the median elasticity magnitude of the sample is 1. Furthermore, the probability that a random pair of elasticities are relatively elastic ([greater than] 1) is .25, that a random pair of elasticities are relatively inelastic ([less than] 1) is .25, and that a random pair of elasticities has one relatively elastic and the other relatively inelastic is .5. While we recognize the arbitrariness of any sampling distribution (including ours), these sampling properties seem reasonable to us. We do not think that other reasonable sampling of the parameter space will alter the important qualitative conclusion that a large measure of the sample space leads to [p.sup.*]/c [less than] 1, given that the necessary conditions are satisfied.

14. The conditional ratio obviously lies between 0 and 1.

15. The abundance of marginal pricing at 0 could also be due to high costs of collecting the variable tariff relative to the marginal costs of production.

16. The first point is the solution to three equations: [k.sub.2] = (1 + c)[k.sub.1]/(1 - c), [k.sub.2] = 1/[k.sub.1], and the implicit function giving the boundary between subproblems 1 and 4. The intersection of these three lines corresponds to the situation in which the case 2 region of subproblem 1 consists of the single point ([k.sub.1], [k.sub.2]) = (.4320, 2.3146) and thus accounts for why it is the maximum marginal cost at which a zero price is observed. The second point is determined by the symmetric versions of these equations.

17. The least upper bound for the ratio V/[V.sub.c] of [-square root of 2] = 1.41 ... occurs when the unconstrained problem results in prices such that the two consumers have equal surplus while the constrained problem is at the threshold between serving both customers and serving only one customer. To verify this, one can set formulas for the constrained problem where both customers are served equal to the constrained problem where only one customer is served and solve for c to get c = 1 + ([k.sub.1] [-square root of 2 [k.sub.2]] - [k.sub.2] [[-square root of [k.sub.1]]) / ([-square root of 2[k.sub.2]] - [-square root of [k.sub.1]])]. This value of c can then be substituted back into the formula for V/[V.sub.c], and after some algebra get V/[V.sub.c] = {[-square root of 2] + (2 [-square root of 2] - 3)[[-square root of [k.sub.1][k.sub.2]] / ([k.sub.2] + [k.sub.1] - 2 [-square root of[k.sub.1][k.sub.2]])]}. Next, noting that the equal surplus price p[double prime] requires [[-square root of [k.sub.1][k.sub.2]] [element of] (0, 1), we see that the least upper bound is [-square root of 2]. It is also interesting to note that this maximal ratio is approached for any c [element of] (1 - 1/[-square root of 2], 1). To see this, note that a smaller c implies a smaller [p.sup.*]. Since [p.sup.*] [greater than or equal to] 0, p[double prime] implies that the smallest price where surpluses are equal occurs when [k.sub.1] = 1/[k.sub.2]. Plugging this into the expression for c and using L'Hopital's rule gives the lower limit for c.

18. Denote fixed costs by F. If the constraint is binding, [V.sub.c] (F)[less than or equal to] V(F), where V(F) = max {[S.sub.1] - F, [S.sub.2] - F, [S.sub.3] - F, [S.sub.4] - F, 0} and [V.sub.c](F) is analogously defined. Assuming F [less than] [V.sub.c](0), then V/[V.sub.c] [equivalent to] V(F)/[V.sub.c](F) = (V(0) - F)/([V.sub.c] (0) - F) is an increasing function of F, where V(0) and [V.sub.c] (0) are profits in the zero fixed cost economy.

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Steven P. Cassou: Associate Professor, Department of Economics, Kansas State University, Manhattan Phone 1-785-532-6342, Fax 1-785-532-6919 E-mail scassou@ksu.edu

John C. Hause: Professor, Department of Economics, SUNY-Stony Brook, New York, Phone 1-516-632-7547 Fax 1-516-632-7516 E-mail jhause@datalab2.sbs.sunysb.edu