A Pedagogical Note on Multitier Pricing Scheme

Abstract

This note derives a new formula for determining a monopolist’s optimal multitier pricing scheme for any given number of tiers. It further characterizes Gabor’s (Review of Economic Studies) two-tier pari passu marginal revenue function to the $n$ -tier case. By introducing the individual tier’s marginal revenue and the pari passu marginal revenue in a linear demand case, this note provides a perceptive graphical representation of the optimal pricing scheme, revealing that all tiers’ outputs are equal, the last tier’s price is always higher than the marginal cost, and an increase in the number of tiers increases social welfare. In a class of nonlinear demand functions, it shows that starting from the first tier, the tiers’ outputs are monotonically increasing (decreasing) if the demand function is strictly convex (concave). It also shows that the equal-tier-output property preserves in the linear demand case with the total output fixed as a constraint.

JEL Classification : D01, D21, D42, L12, L21

Keywords

monopoly second-degree price discrimination multitier pricing public utility pricing social welfare

Introduction

The theory of imperfect price discrimination has developed into various nonlinear pricing schemes such as quantity discounts, two-part tariffs, block pricing, bundling, tie-in-sales, and so on. In practice, multitier pricing and quantity-discount pricing such as “buy more, save more” or “buy one and get second one 50% off” are ubiquitous in the business world; for example, Häagen-Dazs prices its 3.6 oz. cup vanilla-flavored ice cream at US$2.19, one-pint size at US$6.99, and one-half gallon size at US$9.99, respectively. This is equivalent to the first 3.6 oz. at US$0.608 per oz., the next 12.4 oz. at US$0.387 per oz., and the last 48 oz. at US$0.062 per oz. In this pricing structure, one can interpret it as buying the first tier of 3.6 oz., the second tier of 12.4 oz., and the third tier of 48 oz. at three different, declining, prices per oz. Unlike two-part tariffs, it charges consumers declining tier prices for the same good as the quantity purchased increases without charging a fixed or entry fee. Enke (1964) classified this as a sort of discrimination among units. By doing so, a firm can extract more, but not all, of the consumer surplus, compared with uniform pricing.

Earlier attempts in the economic literature to derive a monopolist’s optimal tier-pricing structure and its implications for social welfare include Paine (1937); Buchanan (1952); Gabor (1955); Spence (1977); Willig (1978); Varian (1989); Yano (2001); Allen, Doherty, Weigelt, and Mansfield (2005); Wetzstein (2005); Baye (2010); Salvatore (2011); and Chang and Chen (2017). Whenever a seller has some market power, a multitier pricing scheme can generate a higher profit, output, and social welfare than a uniform pricing scheme. Ideally, a firm would like to engage in first-degree price discrimination to extract all surplus from consumers and thus earns the highest possible profits. In situations where the firm does not know the maximum price that each consumer will pay for a good or when it is not practical to post a continuous schedule of prices for each incremental unit purchased, a firm may resort to second-degree price discrimination, which posts a discrete schedule of declining prices for different blocs of quantities. If demand is linear, it is well known that all tiers will have the same quantity. This has been popularly illustrated, for example, in Baye (2010, pp. 402-406) for a two-tier model, which is reproduced in Figure 1.

Figure 1.

Baye’s illustration of second-degree price discrimination in a two-tier case.

He showed that each tier has a quantity of two, with the first tier’s price being US$7.60 and the second being US$5.20. The shaded region between the prices and the marginal cost line is the producer surplus, which is profit plus fixed cost (if any). As the MC is drawn as linear, the total cost function is quadratic as shown in Appendix A.¹

Given that the monopolist knows the demand and cost functions, how can we provide a precise graphical representation of the optimal tier-pricing structure when demand is linear and the number of tiers is given? One of the main purposes of this note is to explore this topic. The model to be studied, though frequently illustrated in textbooks, is subject to some restrictive assumptions. We assume that the firm has relevant information on the aggregate demand function without having to know each consumer’s preferences and income. Moreover, to buy a higher numbered tier, a consumer must have purchased all quantities offered in the preceding tiers. Either goods are quickly perishable and must be consumed on the spot, or resale is not feasible or prohibited. These assumptions bypass the thorny problems such as incentive compatibility and participation constraints—which may arise from consumers having different preferences and incomes—have been examined in various models in the literature.² We also assume that the number of tiers is predetermined. This can be the result of the firm having considered the costs and benefits of the optimum number of tiers. It can also be the result of regulation, which is often the case in the public utility industry. But the fixing of the number of tiers in the public utility industry is frequently the result of ensuring equitable consumption of basic need among all households so that the first tier of consumption can be provided at a low price. The model we present in this note is for a monopoly whose pure goal is profit maximization.³

Under the restrictive assumptions discussed above, the model to be examined clearly holds if all consumers are identical. This in effect is a special case of tie-in sales in which the basic good and the complementary goods are the same goods. It is also a special case of a quantity discount in that only the largest quantity package is made available to the consumers. In practice, the sales scheme in which the two units are tied-in, with the first unit sold at full price and the second one at half price, with the package price being an average of the two, is such an example. In the case where consumers are heterogeneous, the model still holds if the monopolist has perfect information on the consumers, as long as resale is prohibited and the monopolist has a fixed number of tiers to offer.⁴

This note also intends to show that the marginal cost-pricing scheme will not be optimal for a pure profit-maximizing monopolist unless it is regulated by the government to use such a rule. The marginal cost-pricing rule was used by Wetzstein (2005) in a two-tier model. He first determined the second-tier price and the total output from the intersection point of the demand and marginal cost curves and then used that to derive the first-tier prices (and output) under profit maximization.

Recently, Chang and Chen (2017) characterized in detail the optimal quantity-discount pricing scheme with general demand and cost structures. They showed that each tier’s marginal cost (other than the last one) is irrelevant in determining the optimum, an increase in the equilibrium total output increases each tier’s output as well as its cumulative output, and the amount of increase in each tier’s cumulative output rises monotonically from the first tier.

Under linear demand, this note derives a new formula to determine the optimal total output. It further characterizes Gabor’s (1955) two-tier pari passu marginal revenue function to the $n$ -tier case. The pari passu marginal revenue measures the whole scheme’s marginal revenue as a result of a change in total output when all tiers’ outputs are simultaneously optimally adjusted. By introducing the individual tier’s marginal revenue curves and the pari passu marginal revenue curve in the linear demand case, this note provides a perceptive graphical representation of the optimal pricing scheme, yielding the result that the last tier’s price is always higher than the marginal cost. Our graphical approach also readily implies the well-known results that all tiers’ outputs are equal, and the larger is the number of tiers, the larger are the total output, profits, and social welfare but the lower is the consumer surplus.

In this note, we also consider a class of nonlinear demand functions and show that starting from the first tier, if the demand function is strictly convex (concave), the individual tiers’ outputs are monotonically increasing (decreasing). In addition, we also consider the $n$ -tier linear demand model with a constraint on total output. We show that the equal-tier-output property in the no constraint output case carries over to the constraint case, but the last tier’s marginal revenue may not be equal to the marginal cost.

The remainder of this note is organized as follows. “A Simple Illustrative Model: Linear Demand and Three Pricing Tiers” section presents a simple illustration with a linear demand and three pricing tiers. “Nonlinear Demand and Optimal Pricing Scheme” section characterizes an optimal pricing scheme in a class of nonlinear demand functions. “Output Constraint and Optimal Pricing Scheme” section identifies the equal-tier-output property in the constraint case. In the final section of the note, we offer concluding remarks on our findings.

A Simple Illustrative Model: Linear Demand and Three Pricing Tiers

For simplicity of exposition, we consider a monopolist offering a three-tier pricing schedule to consumers.⁵ Let the inverse demand function facing the monopolist be $p = f (Q) = a - b Q$ , where price $p$ is a function of the cumulative output $Q$ . Let tier $i$ ’s output be $q_{i}$ , and $Q_{i}$ be its cumulative output $\sum_{k = 1}^{3} q_{k}$ . Thus, $Q_{1} = q_{1}$ , $Q_{i} = Q_{i - 1} + q_{i}, i = 2, 3$ . For each tier’s price, we define $p_{i} = f (Q_{i})$ and assume $f^{'} (Q_{i}) \equiv \partial f (Q_{i}) / \partial q_{i} < 0$ . It is convenient to use the tier outputs as our variables. Let tier $i$ ’s revenue be

r^{i} = q_{i} f (Q_{i}) = q_{i} f (\sum_{k = 1}^{3} q_{k}) = r^{i} (q_{1}, q_{2}, q_{3}), i = 1, 2, 3,

and its marginal revenue with respect to $q_{i}$ be

r_{i}^{i} = \frac{\partial r^{i}}{\partial q_{i}} = p_{i} + q_{i} f^{'} (Q_{i}) = p_{i} - b q_{i}, i = 1, 2, 3 .

Let $C (Q_{3}) = C (Σ_{i = 1}^{3} q_{i})$ be the total cost function and the marginal cost be $C^{'} (Q_{3})$ . Assume $C^{'} (Q_{3}) \geq 0$ and $C^{″} (Q_{3}) \geq 0$ . The profit function can be expressed as

π = \sum_{i = 1}^{3} r^{i} (q_{1}, q_{2}, q_{3}) - C^{'} (\sum_{i = 1}^{3} q_{i}) = π (q_{1}, q_{2}, q_{3}) .

To determine the optimal size of each tier, the monopolist maximizes profit with respect to $q_{i}$ . The first-order conditions for profit maximization are

\frac{\partial π}{\partial q_{1}} = p_{1} - b \sum_{k = 1}^{3} q_{k} - C^{'} (Q_{3}) = r_{1}^{1} - b q_{2} - b q_{3} - C^{'} (Q_{3}) = 0,

\frac{\partial π}{\partial q_{2}} = p_{2} - b \sum_{k = 2}^{3} q_{k} - C^{'} (Q_{3}) = r_{2}^{2} - b q_{3} - C^{'} (Q_{3}) = 0,

\frac{\partial π}{\partial q_{3}} = p_{3} - b q_{3} - C^{'} (Q_{3}) = r_{3}^{3} - C^{'} (Q_{3}) = 0,

which can be solved for

q_{i} = \frac{1}{4 b} (a - C^{'} (Q_{3})) = \frac{1}{b} (p_{3} - C^{'} (Q_{3})), i = 1, 2, 3,

Q_{3} = \frac{3}{4 b} (a - C^{'} (Q_{3})) .

All $q_{i}$ are seen to be equal and are denoted by $q$ . The optimal pricing structure, therefore, is

p_{i} = a - i b q = a - \frac{i}{4} (a - C^{'} (Q_{3})), i = 1, 2, 3,

and

p_{i} = p_{i - 1} - b q, i = 2, 3 .

The gap between any two adjacent prices is simply $b q$ . In addition, from the first-order conditions, we obtain the ranking conditions⁶:

p_{1} > r_{1}^{1} = p_{2} > r_{2}^{2} = p_{3} > r_{3}^{3} = C^{'} (Q_{3}) .

Figure 2 illustrates the optimal pricing structure under profit maximization. We will introduce two alternative ways to view the equilibrium configuration. The first one is the use of three $r_{i}^{i}$ ’s and $C^{'} (Q_{3})$ . The second one relies on the pari passu marginal revenue which was termed by Gabor (1955) and we denote it by $γ (Q)$ . It measures the whole scheme’s marginal revenue as a result of a change in $Q$ (the total output) when all tiers’ outputs are simultaneously optimally adjusted. In Appendix C, we explore the pari passu marginal revenue in some detail for the $n$ -tier case.

Figure 2.

Equilibrium configuration: Optimal pricing structure in the three-tier case.

Let us start with the first way. Recall that $r_{i}^{i}$ is tier $i$ ’s own marginal revenue. The first-order conditions show that all three $r_{i}^{i}$ s and $C^{'} (Q_{3})$ play roles in determining the optimal tier outputs and corresponding prices. From Equation 2, the starting points of all three $r_{i}^{i}$ lines are, respectively, $r_{1}^{1} |_{q_{1} = 0} = a$ , $r_{2}^{2} |_{q_{2} = 0} = p_{1}$ , and $r_{3}^{3} |_{q_{3} = 0} = p_{2}$ , and the vertical gaps between two adjacent starting points are $r_{1}^{1} |_{q_{1} = 0} - r_{2}^{2} |_{q_{2} = 0} = b q_{1}$ and $r_{2}^{2} |_{q_{2} = 0} - r_{3}^{3} |_{q_{3} = 0} = b q_{2}$ , respectively. We then draw the three parallel $r_{i}^{i}$ lines with the slope of $- 2 b$ and utilize $r_{1}^{1} = p_{2}$ and $r_{2}^{2} = p_{3}$ in the above ranking conditions to obtain points $G_{1}$ and $G_{2}$ . $G_{1}$ is the optimal point that satisfies $p_{2} = r_{1}^{1}$ where $r_{1}^{1}$ is evaluated at $q_{1} = Q_{1}$ . Similarly, $G_{2}$ is determined by $p_{3} = r_{2}^{2}$ where $r_{2}^{2}$ is the second tier’s own marginal revenue evaluated at $Q_{2} = q_{1} + q_{2}$ . Point $E_{3}$ satisfies $r_{3}^{3} = C^{'} (Q_{3})$ , where $C^{'} (Q_{3})$ is the marginal cost evaluated at the total output $Q_{3}$ . Clearly, all three $r_{i}^{i}$ lines span each $q_{i}$ equally, as indicated by Equation 4. It follows that $p_{1} = a - b q$ , $p_{2} = a - 2 b q$ , and $p_{3} = a - 3 b q$ . In this three-tier case, the line $\bar{I J}$ is divided into four equal segments, each with a length of $q$ . Thus, the vertical line segment $\bar{a I}$ is divided equally into three parts, and each tier’s pricing step equals $b q$ , as proved in Equation 6. It is instructive to see that $Q_{3}$ is determined by point $E_{3}$ , and the latter is found by lengthening or shortening the three equal-length, and parallel, $r_{i}^{i}$ lines in tandem until $r_{3}^{3}$ just hits the $C^{'} (Q)$ curve.

The second way to view the optimal pricing scheme is to make use of the pari passu marginal revenue curve. From Equation 5, we notice that it contains only $Q_{3}$ as a variable. Clearly, $Q_{3}$ is its fixed point. Rearranging Equation 5, we obtain,

C^{'} (Q_{3}) = a - \frac{4}{3} b Q_{3} .

This gives us the idea that $a - (4 / 3) b Q_{3}$ must be the marginal revenue of the whole pricing scheme when the total output $Q$ is evaluated at $Q_{3}$ ; namely, it must be the pari passu marginal revenue curve $γ (Q)$ . In the present three-tier linear demand case, we show in Appendix C that $Q_{1} = Q_{2} / 2$ , $Q_{2} = 2 Q_{3} / 3$ , and $f (Q_{3}) - (1 / 3) b Q_{3} = C^{'} (Q_{3})$ . It follows that

γ (Q_{3}) = f (Q_{3}) - \frac{1}{3} b Q_{3} = a - \frac{4}{3} b Q_{3} .

The $γ (Q)$ line is drawn as the blue dotted line in Figure 2 when we treat $Q_{3}$ as a variable $Q$ . Thus, point $E_{3}$ is the intersection point of $γ (Q)$ and $C^{'} (Q)$ . The pari passu marginal revenue function is conceptually totally different from the usual uniform-pricing marginal revenue function. With a finite number of tiers, the $γ (Q)$ curve is below the demand curve for positive $Q$ . This helps clarify why the last tier’s price must be higher than the marginal cost. It readily helps us find the point $E_{3}$ in Figure 2 that nails down the total output $Q_{3}$ and hence the associated optimal $p_{3}$ . As we already know that $q_{1} = q_{2} = q_{3} = q$ , we know that each tier’s output is $Q_{3} / 3$ and their associated prices are readily implied.

To sum up, the optimal pricing structure satisfies the following:

q_{1} = q_{2} = q_{3} = q = \frac{Q_{3}}{3} = \frac{1}{b} (p_{3} - C^{'} (Q_{3})),

p_{i} = a - \frac{i}{3} b Q_{3}, i = 1, 2, 3 .

As the number of tiers increases, the value of $γ (Q)$ increases at a given $Q$ ; namely, it swings counterclockwise toward the demand curve. Thus, the optimal total output will increase.⁷ In the limit, when the number of tiers approaches infinity, it will result in perfect price discrimination and the last tier’s price would be equal to the marginal cost. Clearly, the total output will also increase if the market size, $a,$ is larger or if $b$ is lower.

Chang and Chen (2017) compared the marginal revenue under uniform pricing and the pari passu marginal revenue when both are evaluated at the optimal uniform-pricing output $Q_{u}$ . They showed that if $Q_{u}$ were set at $Q_{n}$ , where $Q_{n} > Q_{u}$ , the marginal revenue under uniform pricing would fall short of the marginal cost at $Q_{n}$ . It is therefore not profitable to produce to the level $Q_{n}$ under uniform pricing. The larger the $n$ , the closer the pricing scheme approaches first-degree perfect price discrimination.

Some Special Cost Functions

In Appendix B, we further illustrate the effects of an increase in $n$ with two special cost functions: increasing marginal cost, $C (Q) = Q^{2} / 2$ , and constant marginal cost, $C (Q) = m Q$ . Specifically, we illustrate with the parameters $a = 10$ , $b = 1$ , and $m = 2$ . Table 1 shows that an increase in $n$ increases profit and social welfare ( $W$ ) but decreases all tiers’ outputs and consumer surplus ( $S$ ).

Table 1.

The Effects of an Increase in $n$ on $q$ , π $π$ , $S$ , and $W$ .

	$\frac{\partial q}{\partial n}$	$\frac{\partial π}{\partial n}$	$\frac{\partial S}{\partial n}$	$\frac{\partial W}{\partial n}$
$C = Q^{2} / 2$	$- \frac{20}{{(2 n + 1)}^{2}}$	$\frac{50}{{(2 n + 1)}^{2}}$	$- \frac{50 (2 n - 1)}{{(2 n + 1)}^{3}}$	$\frac{100}{{(2 n + 1)}^{3}}$
$C = 2 Q$	$- \frac{8}{{(n + 1)}^{2}}$	$\frac{32}{{(n + 1)}^{2}}$	$- \frac{32 (n - 1)}{{(n + 1)}^{3}}$	$\frac{64}{{(n + 1)}^{3}}$

Tables 2 and 3 show the key qualitative results for the general audience to easily grasp the findings in this note. The tables below clearly illustrate that an increase in $n$ increases the first-tier price and decreases the last-tier price. Recall that $π = \sum_{i = 1}^{n} r^{i} - C^{'} (\sum_{i = 1}^{n} q_{i}) = (p_{1} q_{1} + p_{2} q_{2} + \dots + p_{n} q_{n}) - C^{'} (Q_{n})$ and $W = S + π$ , where $Q_{n} = q_{1} + q_{2} + \dots + q_{n}$ . Recall also that $q_{1} = q_{2} = \dots = q_{n}$ at the optimum in the linear demand case. It follows that $Q_{n} = n q$ .

Table 2.

Some Numerical Results in the Case of $C = Q^{2} / 2$ (Increasing Marginal Cost).

		$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 8$
Tier output	$q$	$3.33$	$2$	$1.43$	$1.11$	$0.59$
Total output	$Q$	$3.33$	$4$	$4.29$	$4.44$	$4.71$
First-tier price	$p_{1}$	$6.67$	$8$	$8.57$	$8.89$	$9.41$
Last-tier price	$p_{n}$	$6.67$	$6$	$5.71$	$5.56$	$5.29$
Profit	$π$	$16.66$	$20$	$21.43$	$22.22$	$23.53$
Consumer surplus	$S$	$5.56$	$4$	$3.06$	$2.47$	$1.38$
Social welfare	$W$	$22.22$	$24$	$24.49$	$24.69$	$24.91$

Table 3.

Some Numerical Results in the Case of $C = 2 Q$ (Constant Marginal Cost).

		$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 8$
Tier output	$q$	$4$	$2.67$	$2$	$1.6$	$0.89$
Total output	$Q$	$4$	$5.34$	$6$	$6.4$	$7.12$
First-tier price	$p_{1}$	$6$	$7.33$	$8$	$8.4$	$9.11$
Last-tier price	$p_{n}$	$6$	$4.66$	$4$	$3.6$	$2.88$
Profit	$π$	$16$	$21.33$	$24$	$25.6$	$28.45$
Consumer surplus	$S$	$8$	$7.11$	$6$	$5.12$	$3.16$
Social welfare	$W$	$24$	$28.44$	$30$	$30.72$	$31.61$

In the case of increasing marginal cost, $C = Q^{2} / 2$ :

In the case of constant marginal cost, $C = 2 Q$ :

We summarize the preceding results in the following proposition:

Proposition 1: Under linear demand $p = a - b Q$ , a profit-maximizing monopolist considers a multitier pricing scheme.

i. In the three-tier pricing scheme, all individual tiers’ outputs are equal and the price gaps between any two adjacent tiers are also equal. The optimal tiers’ quantity and price schedules are $q_{i} = Q_{3} / 3$ and $p_{i} = a - i b Q_{3} / 3$ , $i = 1, 2, 3$ , as shown in Equation 9. The last tier’s price is higher than the marginal cost $C^{'} (Q_{3})$ .

ii. In the $n$ -tier scheme ( $n \geq 2$ ), the total output will increase if $a$ or $n$ increases, or $b$ decreases.

iii. In the $n$ -tier scheme ( $n \geq 2$ ), an increase in $n$ increases the first-tier price and decreases the last-tier price. Furthermore, if $n$ increases, both profit and social welfare will increase, but the tier output and consumer surplus will decrease.

Nonlinear Demand and Optimal Pricing Scheme

In the case of nonlinear demand, the tier-output allocation scheme clearly depends on the shape of the demand curve. It is natural to ask the following: If demand is nonlinear, what are the sizes of tiers’ outputs? We answer this question below by simulating a special nonlinear demand case.

Consider the inverse demand function:

p = f (Q) = α + \frac{e^{- β Q}}{β}, β \neq 0 .

$f (Q)$ is strictly convex or concave if $β$ is positive or negative. Denote $p_{i} = f (Q_{i})$ . We have $p_{i}^{'} = f^{'} (Q_{i}) = - e^{- β Q_{i}} = - β (p_{i} - α)$ . From the first-order conditions of profit maximization, we obtain,

q_{i} = \frac{p_{i} - p_{i + 1}}{β (p_{i} - α)} = \frac{1 - e^{- β q_{i + 1}}}{β}, i = 1, \dots, n - 1,

q_{n} = \frac{p_{n} - C^{'} (Q_{n})}{β (p_{n} - α)},

and

(q_{i} - q_{i + 1}) β = 1 - e^{- β q_{i + 1}} - β q_{i + 1} = e^{- β q_{i + 2}} - e^{- β q_{i + 1}}, i = 1, 2, \dots, n - 1 .

It follows that

q_{i} ≶ q_{i + 1} if β ≷ 0, i = 1, 2, \dots, n - 1 .

To illustrate further, we consider the following examples: $α = 10$ , $n = 3$ , $C^{'} (Q) = Q$ with two values of $β = 1$ and $- 1 .$

Case 1: $β = 1$ (strictly convex demand)

From Equation 11, we obtain $q_{1} = 1 - e^{- q_{2}}$ , $q_{2} = 1 - e^{- q_{3}}$ , and $10 + e^{- Q_{3}} - q_{3} e^{- Q_{3}} - Q_{3} = 0$ . These three equations yield $(q_{1}, q_{2}, q_{3}) ≅ (0.63, 1.00, 8.37)$ , which confirms Equation 12.

Case 2: $β = - 1$ (strictly concave demand)

In this case, $q_{1} = e^{q_{2}} - 1$ , $q_{2} = e^{q_{3}} - 1$ , and $10 - e^{Q_{3}} - q_{3} e^{Q_{3}} - Q_{3} = 0$ . The approximate solutions are $(q_{1}, q_{2}, q_{3}) ≅ (0.75, 0.56, 0.44)$ . $q_{i}$ is seen to be decreasing in $i$ , and thus Equation 12 is confirmed. This case is seen to be relevant to Häagen-Dazs example we provided in the “Introduction” section. The three-tier pricing structure of ice cream is at three different, but declining, prices per oz.

Proposition 2: Consider an $n$ -tier pricing scheme under the inverse demand function $p = α + e^{- β Q} / β$ . If the demand function is strictly convex (concave), the individual tiers’ outputs are monotonically increasing (decreasing), starting from the first tier.

Output Constraint and Optimal Pricing Scheme

Here, we consider the case in which the monopolist must produce and sell a predetermined amount of output ${\bar{Q}}_{n} = Σ_{i = 1}^{n} q_{i}$ .⁸ As the total output becomes a fixed number, some of the results obtained in “A Simple Illustrative Model: Linear Demand and Three Pricing Tiers” section will not hold. This section tries to answer questions like “How are the pricing schemes differ between them?” and “Will the tier outputs still be all equal?”

Using our notations in Appendix B, let $r^{i} = q_{i} f (Q_{i}) = r^{i} (q_{1}, q_{2}, \dots, q_{i})$ , $r = Σ_{i = 1}^{n} r^{i} = r (q^{1}, q^{2}, \dots, q^{n})$ , and $π = r (q_{1}, q_{2}, \dots, q_{n}) - C (\sum_{i = 1}^{n} q_{i}) = π (q_{1}, q_{2}, \dots, q_{n})$ . We let the Lagrange function be

ℒ = π (q_{1}, q_{2}, \dots, q_{n}) + λ ({\bar{Q}}_{n} - \sum_{i = 1}^{n} q_{i}),

where $λ$ is the Lagrange multiplier. Differentiating Equation A2 with respect to $q_{i}$ and $λ$ , we obtain the first-order conditions:

\frac{\partial ℒ}{\partial q_{i}} = \frac{\partial π}{\partial q_{i}} - λ = r_{i}^{i} - b \sum_{k = i + 1}^{n} q_{k} - C^{'} (\sum_{i = 1}^{n} q_{i}) - λ = 0, i = 1, \dots, n - 1,

\frac{\partial ℒ}{\partial q_{n}} = \frac{\partial π}{\partial q_{n}} - λ = r_{n}^{n} - C^{'} (\sum_{i = 1}^{n} q_{i}) - λ = 0,

\frac{\partial ℒ}{\partial λ} = {\bar{Q}}_{n} - \sum_{i = 1}^{n} q_{i} = 0 .

Comparing Equation A1a and Equation A1b in Appendix B with Equation 14a and Equation 14b, we readily obtain the counterparts of Equation A2, Equation A3, and Equation A4 in Appendix B with some minor modifications:

\begin{matrix} q_{i} = \frac{1}{(1 + n) b} (a - C^{'} (Q_{n}) - λ) = q = \frac{1}{n} {\bar{Q}}_{n} = \frac{1}{b} (p_{n} - C^{'} (Q_{n}) - λ), i = 1, \dots, n, \\ Q_{n} = {\bar{Q}}_{n}, \end{matrix}

\begin{array}{l} p_{i} = a - i b q = a - \frac{i}{(1 + n)} (a - C^{'} (Q_{n}) - λ), i = 1, \dots, n, \\ p_{i} = p_{i - 1} - b q, i = 2, \dots, n, \\ p_{n} = p_{n} ({\bar{Q}}_{n}), \end{array}

and

p_{1} > r_{1}^{1} = p_{2} > r_{2}^{2} = p_{3} > \dots > r_{n - 1}^{n - 1} = p_{n} ({\bar{Q}}_{n}) > r_{n}^{n} = C^{'} (Q_{n}) + λ .

Clearly, all individual tiers’ outputs remain equal as in the no constraint case.

From the envelope theorem, we know that $λ$ is the marginal profit. As we require that the total output must be $\bar{Q}$ and all must be sold, the constraint is always binding. Therefore, there is no sign constraint on $λ .$ In the case where $λ > 0$ , we have $r_{n}^{n} > C^{'} (Q_{n})$ . This implies that the output constraint is too strict in the sense that by relaxing it would have increased the firm’s profit. Likewise, if $λ < 0$ , the output target is larger than desired as we have $r_{n}^{n} < C^{'} (Q_{n})$ —reducing the target level would have increased the firm’s profit. It is now clear that the no-output constraint case corresponds to $λ = 0$ ; namely, the total output is at the optimal level where the marginal profit is zero. In Appendix C, we further show that $r_{n}^{n} ({\bar{Q}}_{n} / n) = γ ({\bar{Q}}_{n})$ is satisfied in the constraint case.

It is clear that unless $n$ is allowed to change, the presence of output constraint, in general, will not produce the amount of profit attained by the case of no constraint.

Together with a constant marginal cost $C^{'} (Q) = m$ , Table 4 illustrates some numerical results in the output constrained case with the parameters $a = 10$ , $b = 1$ , $m = 2$ , and $\bar{Q} = 4$ .

Table 4.

Some Numerical Results Under $a = 10$ , $b = 1$ , $m = 2$ , and $\bar{Q} = 4$ .

		$n = 1$	$n = 2$	$n = 3$	$n = 4$	$n = 8$
Tier output	$q$	$4$	$2$	$1.33$	$1$	$0.5$
Total output constraint	$\bar{Q}$	$4$	$4$	$4$	$4$	$4$
First-tier price	$p_{1}$	$6$	$8$	$8.67$	$9$	$9.5$
Last-tier price	$p_{n}$	$6$	$6$	$6$	$6$	$6$
Profit	$π$	$16$	$20$	$21.27$	$22$	$23$
Consumer surplus	$S$	$8$	$4$	$2.75$	$2$	$1$
Social welfare	$W$	$24$	$24$	$24$	$24$	$24$

Proposition 3: Consider a profit-maximizing monopolist facing linear demand and offering an $n$ -tier pricing scheme with a total output constraint.

i. The equal-tier-output property carries over the present output constraint case.

ii. The last tier’s marginal revenue $r_{n}^{n}$ may or may not be equal to the marginal cost. If the output constraint is too tight, the last tier’s marginal revenue will be higher than marginal cost, signaling that profit can still be increased if the output constraint is relaxed. It is even possible that if the output target level is set too high, it will cause the marginal profit to be negative at the required output point.

iii. Other properties such as $p_{i + 1} = r_{i}^{i}, i = 1, 2, \dots, n - 1$ and $p_{n} > r_{n}^{n}$ carry over from the no-output constraint case to the constraint case.

Concluding Remarks

This note examined the optimal multitier pricing scheme for a profit-maximization monopolist. Its main purposes are to provide a simple graphical illustration of the optimal pricing scheme in the linear demand case and to characterize some special pricing structure in the nonlinear case. We characterized in Appendix C the pari passu marginal revenue function in the $n$ -tier case under general demand.

In the linear case, we provided two ways of determining the tier-output structure. One is by the use of individual marginal revenue curves and the other by the pari passu marginal revenue curve. A new formula is derived that readily determines the optimal total output for any given number of tiers. We illustrated that all tiers’ outputs are equal, the last tier’s price is always higher than the marginal cost, and an increase in the number of tiers increases social welfare but decreases consumer surplus. We further illustrated two specific cost functions (one with increasing marginal cost and the other with constant marginal cost) and showed that each tier’s output becomes smaller as the number of tiers is increased. We also compared our results to those presented in some textbooks.

In the nonlinear case, we examined a class of demand functions and showed that if the function is strictly convex (concave), then the individual tiers’ outputs are monotonically increasing (decreasing), starting from the first tier.

Finally, we examined the case of a constraint on total output in the $n$ -tier linear demand case. We showed that the equal-tier-output property remains true in this case, but the last tier’s marginal revenue may be higher or lower than the marginal cost, depending on whether the output constraint is too tight or too loose. The no-output constraint case is the knife-edge case in which at the optimal output, the marginal profit is zero so that the last tier’s marginal revenue is exactly equal to the marginal cost.

Our approach can be extended to the case where the targeted output that is produced may or may not have to be completely sold. When unsold inventory is allowed, this will involve the consideration of carrying costs, which will have to be included in the total cost consideration. This should affect its implied marginal cost, which is crucial in our illustration.

Footnotes

Appendix A

Appendix B

Appendix C

Acknowledgements

The authors are indebted to the editor-in-chief, the associate editor, and a referee for constructive comments and helpful suggestions. Also, authors are indebted to Zheng Han, Nicole Hunter, and Michael Wetzstein for helpful comments and suggestions.

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

Tai-Liang Chen

Notes

Author Biographies

Winston W. Chang is a professor of Economics at the State University of New York at Buffalo. His personal website is at , and his recent publications can be found at SSRN and ResearchGate.

Tai-Liang Chen is an associate professor in the Wenlan School of Business at the Zhongnan University of Economics and Law, Wuhan, China.

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