A Graphical Exposition of the Profit Maximizing Two-Part Tariff

Abstract

This note presents a simple graphical approach for deriving the profit maximizing two-part tariff when there are two types of consumers in the market. The exposition covers both the case in which a monopoly can offer only one two-part tariff plan and the case in which the monopoly can offer two different plans.

JEL Classification: D4

Keywords

two-part tariff price discrimination

Introduction

Price discrimination as a topic of study is one of the pillars of intermediate level microeconomics, and two-part tariff is probably the most commonly taught form of second degree price discrimination.

In the case where the market under consideration has only one type of consumer, the exposition of the profit maximizing two-part tariff is straightforward—one that most of my intermediate level students are able to grasp fairly easily. Things do not move as smoothly, however, once I start teaching about finding the optimal two-part tariff pricing scheme when there are two types of buyers in the market. Several years of teaching experience have led me to the conclusion that many students have a harder time grasping how price discrimination with multiple types of buyers works because they tend to get too “caught up” in the mechanics or mathematics of the problem, and they neglect to focus on the underlying intuition for the solution. Too often, students view and treat a problem involving two-part tariffs as a mechanical, algorithmic exercise in arithmetic (finding areas that correspond to producer surplus) and calculus (taking derivatives to find the first-order condition for a maximum). A consequence of this type of “mechanical learning” is that many of them fail to understand the fundamental principles of the issue and hence struggle to solve similar problems with twists or variations that they have not seen before.

To get students to pay more attention to the concepts and ideas involved in price discrimination—and rely less on a robotic, algorithmic way of solving problems—I utilize a graphical approach—which I present below—to derive the profit maximizing two-part tariff scheme when buyers are heterogeneous in their preferences. This approach better illustrates—no pun intended—the trade-offs involved for a monopoly in devising a two-part tariff plan, and allows students to see—without using any calculus—the importance of the second-order condition for a maximum. Moreover, in a market with two types of consumers, the method described here gives a short-cut for deriving the profit maximizing two-part tariff scheme, both for the case in which the monopoly offers only one plan and the case in which the monopoly offers two plans.

Deriving the Profit Maximizing Two-Part Tariff

Consider a market with a monopoly that has a constant marginal cost of production $c \geq 0$ . There are two types of consumers: the low-value consumers, whose demand curve is given by $P = a_{L} - b Q$ , where P and Q, respectively, denote price and quantity; and the high-value consumers, whose demand curve is given by $P = a_{H} - b Q$ . Assume $a_{H} > a_{L} > c$ and $b > 0$ .

Two-Part Tariff: One Plan

A two-part tariff is a pricing scheme where each unit of the good is sold at price $p \geq 0$ , and a consumer has to pay a fixed fee $F \geq 0$ for the right to purchase the good. Let us assume for now that the monopoly offers the same two-part tariff plan to all consumers. With two types of consumers, the monopoly has two options to consider.

High-fee option: Set the fixed fee F at a relatively high level so that only the high-value consumers would purchase the good.

Low-fee option: Set the fixed fee F at a relatively low level so that both types of consumers would purchase the good.

High-fee option

For this plan, we know that the profit maximizing scheme is to set $p = c$ and to choose F so as to take away all of the consumer surplus of the high-value buyer. With such a pricing scheme, the fixed fee would be too large to induce the low-value buyers to consume; for the high-value buyers, the consumer surplus from buying under this plan would be zero so that they could not be better off by not consuming. This scheme yields a producer surplus of $\frac{{(a_{H} - c)}^{2}}{2 b} N_{H}$ , where $N_{H}$ is the number of high-value buyers in the market.

Low-fee option

For this plan, we know from intermediate microeconomics textbooks (see, for example, Perloff [2017]) that the profit maximizing scheme is to set $p > c$ and to choose F so as to take away all of the consumer surplus of the low-value buyer. Such a plan leaves the low-value buyer indifferent between consumption and not buying anything—hence, such a buyer would not be able to do better by staying out of the market; and the high-value consumers would accept such a pricing plan since it gives them a positive consumer surplus from consumption.

To find the profit maximizing price p, consider Figure 1.

Figure 1.

The shaded areas indicate the producer surplus the monopoly can earn from the two types of buyers when the monopoly charges a per unit price of $p$ and a fixed fee that takes away all of the consumer surplus of the low-value buyer.

Mathematical approach

Mathematically, the producer surplus function is

π (p) = [\frac{{(a_{L} - p)}^{2}}{2 b} + \frac{(p - c) (a_{H} - p)}{b}] N_{H} + [\frac{{(a_{L} - p)}^{2}}{2 b} + \frac{(p - c) (a_{L} - p)}{b}] N_{L},

where $N_{L}$ denotes the number of low-value buyers. It is straightforward to show that $π (p)$ is strictly concave and that it is maximized at price $p^{*}$ such that $π ˈ (p^{*}) = 0$ . Now, taking derivatives and simplifying, we get

π ˈ (p) = (\frac{a_{H} - a_{L}}{b}) N_{H} - (\frac{p - c}{b}) (N_{H} + N_{L}),

(1)

which yields $p^{*} = c + (\frac{N_{H}}{N_{H} + N_{L}}) (a_{H} - a_{L})$ . It can be easily verified that the low-fee option beats the high-fee option in terms of producer surplus if $a_{H} - a_{L}$ is not too large, for example, $a_{H} - a_{L} < (\sqrt{6} - 2) (a_{L} - c)$ if $N_{H} = N_{L}$ . Let us henceforth assume that $a_{H} - a_{L}$ is small enough so that the low-fee option is better than the high-fee option.

Graphical approach

It is possible to derive $p^{*}$ purely graphically without using any calculus. In fact, we can easily use a graphical approach to verify that the monopoly’s second-order condition for a maximum—that is, the difference between the marginal gain and the marginal loss from increasing the price p is diminishing—must be satisfied without having to take any derivatives.

To see this, let us take an arbitrary price $p \in (c, a_{L})$ ; change it by a small amount $Δ > 0$ ; and consider how the monopoly’s producer surplus from the low-fee option is affected. Figure 2 shows all the components of the resulting change in producer surplus.

Figure 2.

When the unit price $p$ is raised, there is a decrease in the producer surplus coming from the low-value buyer shown by the red area in the left panel.

When the unit price p is increased slightly, the producer surplus generated from the low-value consumer unambiguously decreases. There are, however, two opposing effects on the producer surplus generated from the high-value consumer. On the one hand, there is a negative effect on producer surplus because the monopoly sells fewer units to the high-value buyer when the unit price increases. On the other hand, there is a positive effect on producer surplus because the higher per unit price means that each unit sold yields more revenue.

The two negative effects on total producer surplus—one coming from the low-value consumer side (the red area in the left panel of Figure 2, multiplied by the number of low-value buyers); the other one coming from the high-value consumer side (the red area in the right panel of Figure 2, multiplied by the number of high-value buyers)—can be thought of as the monopoly’s marginal loss from raising p. The positive effect on total producer surplus—which occurs on the high-value consumer side (the blue area in the right panel of Figure 2, multiplied by the number of high-value buyers)—is the monopoly’s marginal gain of raising p.

When p is close to c—and the number of low-value buyers relative to the number of high-value buyers is not too large—the marginal benefit of raising p exceeds the marginal loss from raising p. Hence, the profit maximizing p cannot be too close to c. Notice, when the demand curves of the two types of consumers have the same slope, the marginal benefit of raising p does not depend on the value of p. However, the marginal loss from raising p increases with p. Therefore, as long as $a_{H} - a_{L}$ is not too large, the marginal loss from raising p would exceed its marginal benefit when p is too high (see Figure 3), which means that the profit maximizing unit price p cannot be too far from c. The optimal unit price, $p^{*}$ , must be such that the marginal benefit of raising p equals the marginal loss from raising p.

Figure 3.

When p is too high relative to c, the marginal loss from raising p (the two red areas, each multiplied by their respective number of buyers) exceeds the marginal gain of raising p (the blue area multiplied by the number of high-value buyers).

With the aid of Figure 2 or 3, the marginal benefit of raising p is

[(\frac{a_{H} - a_{L}}{b}) Δ - \frac{Δ^{2}}{2 b}] N_{H},

which corresponds to the positive term in (1), and the marginal loss from raising p is

[(\frac{p - c}{b}) Δ + \frac{Δ^{2}}{2 b}] N_{L} + [(\frac{p - c}{b}) Δ] N_{H},

which corresponds to the negative term in (1). Setting them equal to each other, taking the limit as $Δ \to 0$ , and solving for p yields the profit maximizing unit price: $p^{*} = c + (\frac{N_{H}}{N_{H} + N_{L}}) (a_{H} - a_{L})$ .

Two-Part Tariff: Two Plans

Assuming the monopoly cannot distinguish between the two types of consumers, the monopoly can do better—in terms of producer surplus—than the two-part tariff plan described above by offering two two-part tariff plans—one designed for the high-value consumer and one designed for the low-value consumer. To see this graphically, suppose the monopoly offers two plans.

Plan 1: The unit price is $p_{1} > c$ and the fixed fee $F_{1}$ extracts all of the consumer surplus of the low-value consumer.

Plan 2: The unit price is $p_{2} = c$ and the fixed fee is $F_{2}$ .

Now, if both types of consumers choose Plan 1, then the producer surplus, consumer surplus, and deadweight loss from each type of buyers in this market are as shown in Figure 4.

Figure 4.

The producer surplus, consumer surplus, and deadweight loss from each type of buyers when all buyers choose Plan 1.

In order to induce the high-value consumer to choose Plan 2 over Plan 1, Plan 2 has to give the high-value consumer a consumer surplus at least equal to the consumer surplus that Plan 1 yields. Hence, by setting $F_{2}$ to be equal to the shaded area in Figure 5, the monopoly can raise producer surplus by inducing the high-value consumer to choose Plan 2—without changing the low-value consumer’s behavior. Note that the increase in producer surplus comes from converting the deadweight loss resulting from selling Plan 1 to the high-value consumer to surplus for the monopoly.

Figure 5.

The monopoly can increase producer surplus by offering Plan 2—designed for the high-value buyer—with a fixed fee equal to the shaded area.

To see graphically how the monopoly should set $p_{1}$ , the unit price for Plan 1, consider what happens to producer surplus when the monopoly raises this price by a small amount $Δ > 0$ (for a mathematical derivation of the optimal $p_{1}$ , see Gotlibovski and Kahana (2009)]. Figure 6 shows graphically the two effects on producer surplus.

Figure 6.

When the unit price $p_{1}$ for Plan 1 is raised, there is a decrease in the producer surplus coming from the low-value buyer shown by the red area.

Increasing $p_{1}$ reduces producer surplus generated from the low-value consumer—this is the marginal loss from raising $p_{1}$ . At the same time, raising $p_{1}$ increases the producer surplus generated from the high-value consumer—this is the marginal benefit of raising $p_{1}$ . Note from Figure 6 that the marginal benefit of increasing $p_{1}$ is a constant, while the marginal loss from raising $p_{1}$ changes with $p_{1}$ —more specifically, the marginal loss is increasing in $p_{1}$ . Assuming $a_{H} - a_{L}$ is not too large, the profit maximizing $p_{1}$ equates the marginal benefit and the marginal loss of raising $p_{1}$ . Now, with the aid of Figure 6, we can quantify these marginal effects. The area of marginal benefit is

(\frac{a_{H} - a_{L}}{b}) Δ N_{H},

and the area of marginal loss is

[(\frac{p_{1} - c}{b}) Δ + \frac{Δ^{2}}{2 b}] N_{L} .

Setting these equal to each other, letting $Δ \to 0$ , and solving for $p_{1}$ yields the profit maximizing price: $p_{1} * = c + \frac{N_{H}}{N_{L}} (a_{H} - a_{L})$ .

Conclusion

In 20 years of teaching intermediate microeconomics, I have found that some students struggle to understand the basic intuition and principles of two-part tariff pricing plans (as well as other types of price discrimination schemes) because they get lost in the mathematics of the problems. To avoid this pitfall and allow the students to focus on the broad, general ideas, I have presented a graphical illustration of how two-part tariff works when there are two types of consumers. While the exposition is restricted to linear demand curves, it accommodates any number of buyers of either type. In addition, it covers both the case in which the monopoly offers only one plan to all consumers and the case in which the monopoly offers two different plans. The graphical approach is not intended as a substitute for the mathematical approach; rather, it should be viewed as a vital complement of it.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Frederick Chen

Author Biography

Frederick Chen is a professor of Economics at Wake Forest University, where he teaches microeconomics, mathematical economics, and game theory. His primary research interests include the application of economic principles to issues in public health, population biology, and sustainability studies.

References

Gotlibovski

Kahana

(2009). Second-degree price discrimination: A graphical and mathematical approach. The Journal of Economic Education, 40(1), 68–79.

Perloff

(2017). Microeconomics (8th ed.). Pearson.