Simple Pricing for Information Services: Optimal

Three-Part Tariff

June 22, 2015

Abstract

Mobile and smartphone apps, and other information services, are transforming business andsociety. Most of these goods are offered as pay for service rather than pay to own, and manyuse the simplest form of pricing, namely per-unit (“Pay as you Go”) or buffet (“All you can eat”)pricing. It is known that a single three-part tariff (3PT) has better revenue performance, but theoptimal 3PT is hard to design. Because of the mathematical complexity in optimizing a 3PT,analytical results exist only for the simplest cases. This paper provides optimal solutions for mostwidely used demand functions, and develops fundamental properties of the optimal solution forthe general case. Additionally, it develops insights regarding the optimal 3PT design under variousforms of consumer heterogeneity. We also compare the optimal 3PT against per-unit and buffetpricing under variety of market conditions. The 3PT dominates on profit when usage heterogeneityis low and value heterogeneity is high, but per-unit pricing achieves maximum market coveragewhile buffet pricing generates highest consumer surplus. This suggests the potential dominance ofa menu comprising a per-unit pricing and buffet pricing. Yet, surprisingly, a single 3PT produceshigher profit than this menu under most conditions.

1 Introduction

Many products, especially in information, computing and service industries, feature multi-unit consump-

tion. Each buyer consumes a variable number of units, making it important for the firm to design an

appropriate non-linear pricing mechanism that implements multiple objectives, including price simplicity,

fairness in pricing, profit, market share, and consumer surplus (Jain and Kannan, 2002; Lehmann and

Buxmann, 2009). Several firms in these industries practice “simple” pricing: essentially a single pricing

scheme offered to all consumers, even when the product caters to a very large market (Essegaier et al.,

2002). One implementation of this is an “all you can eat” buffet plan, which provides unlimited usage for

a fixed price. For example, Netflix employs an all-you-can-watch buffet price for video content; Rhapsody

and Spotify have similar pricing for music. Another implementation is a per-unit usage fee, under which

total price is a linear multiplier of quantity consumed. For instance, Amazon Web Services has per-unit

prices for numerous computing, analytics, networking, storage, applications, deployment and manage-

ment, databases, and mobile services. A few firms combine these two methods into a two-part tariff (a

fixed membership or access fee, plus a per-unit fee for actual usage). Some firms (e.g., EMR hosting

services) use three-part tariffs where users pay a fixed monthly fee and receive some “free” allowance,

then pay a usage fee for above-allowance consumption.

The striking advantage of per-unit pricing and buffet pricing is their utter simplicity: the firm can

communicate its pricing plan with just a single parameter and can effectively differentiate themselves by

offering different pricing schemes (Choudhary, 2010). The two pricing techniques differ in an important

way. With buffet pricing, all users pay the same regardless of consumption, hence light users face a

substantial entry barrier while heavy users enjoy substantial quantity discount, at a cost to light users

(Schlereth et al., 2010). In contrast, heavy users pay proportionately higher prices under per-unit pricing

and light users face no entry barrier, but per-unit pricing is unable to implement quantity discounts,

which are fundamental to improving profit when the firm faces heterogeneous consumers.

A three-part tariff (3PT, represented as (F,Q, s)—a fixed fee F , allowance Q, and overage fee s)

generalizes the remaining schemes, and will naturally have highest profit. A per-unit plan corresponds

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to F=Q=0, a buffet plan to s = 0, and a two-part tariff to Q = 0. Conceptually, a three-part

tariff emulates both a per-unit plan’s ability to monetize heterogeneity in usage levels, and a buffet

pricing’s ability to collect the extra consumer surplus that results from consumers’ diminishing marginal

valuations. Moreover, while the theoretically optimal menu under classical demand conditions might be

an infinite menu of quantity price bundles, a single 3PT can capture most of the profits while maintaining

price simplicity and fairness (Bagh and Bhargava, 2013). Three-part tariffs remain advantageous under

demand variations such as network effects (Oren et al., 1982), usage uncertainty (Lambrecht, Seim,

and Skiera, 2007), consumer overconfidence (Grubb, 2009), etc. Several firms find a small sacrifice in

revenue worthwhile, in exchange for the incredible simplicity and sense of fairness offered by a single

plan. Hence three-part tariffs play a critical role for multi-unit consumption products. But, how should

a three-part tariff be designed?

The first objective of this paper is to specify optimal solutions for the 3PT design problem. For per-

unit pricing and buffet pricing, deriving the optimal solution is relatively straightforward. But, optimizing

a 3PT is non-trivial because of the non-linearity in the profit function created by a non-concave payment

function. Hence, despite the demonstrated advantages of three-part tariffs, no paper has stated analytical

results for 3PT design except for simple examples in Wilson (1993), Grubb (2009), and Fibich et al.

(2015). Discussion of the optimal 3PT is absent in recent surveys of non-linear pricing (Iyengar and

Gupta, 2009; Lambrecht, Seim, Vilcassim, et al., 2012; Kohler, 2012). This paper contributes by

specifying the properties and economic interpretations of the optimal solution under very general demand

formulation, and by deriving closed-form expressions for commonly used demand functions. Neither has,

to our knowledge, been published in the literature previously.

The second objective is to describe the pros and cons of a 3PT relative to using the more extreme and

simpler per-unit pricing or buffet pricing schemes. While several papers have compared per-unit pricing,

buffet pricing, and two-part tariffs (e.g,, Fishburn et al., 2000; Altmann and Chu, 2001; Essegaier et al.,

2002; Sundararajan, 2004; Choudhary, 2010), this paper is the first one to extend the comparison to

three-part tariffs. Crucially, our comparative evaluation covers variations in both consumers’ valuation

heterogeneity and their usage or appetite heterogeneity. Additionally, it covers multiple performance

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metrics, including the firm’s profit, market coverage, and consumer surplus. This multi-metric evaluation

is crucial because the alternative schemes operate in fundamentally different ways with respect to the

seller’s conflicting desires for profitability, market coverage (which may help create economies of scale

and network benefits), and sufficient consumer surplus (e.g., in order to manage customer retention).

2 Modeling Framework and Optimal Three-Part Tariff

Let v(x, q) ≥ 0 be type x consumer’s marginal valuation for the qth unit. Let G represent the distribution

of the type variable x over set Θ (with maximum element X, and ordered such that v(x, q) is increasing

in x). Marginal valuations and the distribution function are subject to the following assumption that

guarantee the Spence-Mirrlees single-crossing property and non-decreasing demand elasticity (Lariviere,

2006).

Assumption 1 (Demand). (i) v(x, q) ≥ 0, (ii) ∂v∂q< 0, (iii) ∂2v

∂x2≤ 0, (iv) G is log concave.

The choice of pricing scheme depends on the levels of marginal costs and transaction or monitoring

costs. Intuitively, buffet pricing dominates when monitoring costs are high because it can avoid these

costs where usage-based (or metered) pricing cannot (Sundararajan, 2004; Levinson and Odlyzko, 2008).

Conversely, buffet pricing is less efficient as marginal costs rise because it promotes more consumption

and leads to higher costs. However, modern information technology has made both these arguments less

relevant, with an end-to-end digital infrastructure that makes marginal costs negligible, lowers monitoring

and transaction costs, and eliminates resale. Moreover, the difference in monitoring costs between per-use

and buffet plans has nearly been eliminated due to other reasons. Today, firms monitor and collect data

not for determining the customer’s payment level, but to enable data analytics and customer relationship

management, as well as to satisfy legal and security requirements. Hence, they would incur these costs

even under buffet pricing. In line with these arguments, our analytical framework is set up to exclude a

decisive role for both marginal costs and transaction costs.

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2.1 Consumer Choice

We use the notation v-1 to denote both inverses of the two-argument function v(x, q), depending on

the omitted variable. Specifically, v-1(x, s) maps into a quantity q, while v-1(s, q) identifies a customer

x; in both cases v(x, q) = s. For customer x, the optimal consumption subject to per-unit rate s is

q∗(x, s) = v-1(x, s), i.e., v(x, q∗(x, s)) = s. Conditional on purchasing the 3PT, q∗ is the higher of Q

and q∗(x, s). Formally, q∗ = max{v-1(x, s), Q}. The optimal consumption quantity can be plugged

into the total valuation function V (x, q) =∫ q

0v(x, q)dq to get the maximum surplus customer x would

obtain conditional on purchasing the plan. The 3PT plan is purchased by all consumers whose gross

valuation, at their q∗ level, is no less than the total payment for q∗ units. In principle, this can identify the

marginal consumer x in two ways: i) x achieves indifference at the free allowance level Q (i.e., consumes

exactly Q units), or ii) x has negative surplus at Q but achieves indifference by consuming overage units

at a cost s less than the marginal value of these units. Lemma 1 substantially prunes the search space

without sacrificing optimality, by eliminating the second case.

Lemma 1 (Overage Consumption). Let x be the marginal buyer for the optimal three-part tariff. x

consumes exactly Q units, equivalently v(x, Q) ≤ s. The indifference equation is V (x, Q) = F .

Applying Lemma 1, the firm earns V (x, Q)(1 − G(x)) from fixed fees, and overage revenue from

x > x whose consumption exceeds Q. Given Q and x, this yields a stand-alone optimization problem in

which overage fee is set to optimize residual revenue,∫ Xx

(q∗(x, s)−Q)+g(x)dx. The optimal fee s∗ may

lead to overage consumption by all x > x (implying that v(x, Q) = s), or only for x ∈ (y,X), where

y ≥ x is the highest x who consumes exactly Q units at rate s (i.e., v(y,Q) = s; y = s+Q1+Q ln(δ)

).

2.2 Profit Function with 3PT

The firm’s total profit function is:

Π = maxF,Q,s

F (1−G(x))+s

∫ X

y

(q∗(x, s)−Q)g(x)dx (1a)

where F = V (x, Q); y = v-1(s,Q); and q∗ = v-1(x, s)for x ≥ y (1b)

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The profit function is non-concave, flat upto Q and then inclined with slope s. It is also non-linear,

with order greater than the order of V (≥ 2) plus the order of G. Analytical solutions to this problem

exist only for restricted cases of G and v. For instance, Fibich et al. (2015) derive the optimal 3PT when

G is discrete with the entire mass at just one or two points, and for a particular marginal value function

v(x, q) = x− bq. In Essegaier et al. (2002), each consumer’s demand curve is defined at just one point

V (x, q(x)) (where x takes only two values, representing heavy and light users), rather than downward

sloping. Similarly, consumers are ex ante homogeneous in Grubb (2009). In contrast, most mass-market

products face a richer distribution of consumers (Wedel et al., 1999), and empirical research in 3PT has

emphasized both diminishing marginal valuations and that consumer heterogeneity is best represented

with a continuous distribution.

2.3 Illustrations: Optimal 3PT and Solution Procedure

To illustrate the optimization process and solution, consider the case where v(x, q) = x − bq, the

valuation function commonly used in empirical research (Iyengar and Gupta, 2009; Lambrecht, Seim,

and Skiera, 2007). While Fibich et al. (2015) solve the optimal 3PT with this function, our illustrations

in this section generalize their two-point distribution with a continuous G distribution. Given the critical

role of consumer heterogeneity (Iyengar and Gupta, 2009; Lambrecht, Seim, Vilcassim, et al., 2012), we

pick three distributions that cover the space of real-world distributions. Depicted in Fig. 1, these are i)

a normal-like distribution where there is a greater density of moderate-valuation customers and density

decreases with extremeness in valuation, ii) a triangle distribution where the density of users is inversely

proportional to their valuation, so that customers are clustered towards the lower end of valuations, and

iii) the widely used uniform distribution, where every type is equally likely, so that there is no clustering of

users and low-value users have the same mass as higher-value users. These distributions can be generated

as special cases of a B(ν, ω) distribution with parameters (2,2), (1,2), (1,1) respectively.

[Figure 1 about here.]

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Our solution procedure for these illustrations and for the general problem is as follows. We reformulate

the 3-variable (F,Q, s) optimization problem into an (x, Q, s) space using the indifference equation

F = V (x, Q). Given x we identify properties that s and Q must satisfy in optimality, and employ these

properties to derive the optimal s and Q values as functions of x. This reduces the problem into a

single-variable optimization with no constraints (barring that x lies in [0, a]). While the profit function

is of high order, we have analytically isolated the unique optimal solution for each distribution.

2.3.1 Normal-Like Distribution B(2, 2)

The density function is g(x) = 6x(a− x)/a3 over [0, a] (so, G(x) is order 3), and the profit function is

an order 5 expression in the decision variables,

Π =

(1− 3x2

a2+

2x3

a3

) (Qx− bQ2

2

)+s(a− (s+bQ))3(a+ (s+bQ))

2a3b. (2)

First, differentiating with s, the first-order condition admits multiple solutions, of which we reject two

that have s+bQ = a (because this leads to zero overage revenue). Hence this condition yields a2−2as =

(s + bQ)(5s + bQ). Second, differentiating with Q yields that either i) x = s + bQ or ii) a3 − 3aZ +

2xZ + 2s(s + bQ)2 = 0, where Z = sx + x2 + s(s + bQ). However, we can eliminate condition (ii)

because it requires that Z > 3a2, which is impossible because each component term is less than a.

Therefore, the optimal solution satisfies the identity x = s+ bQ. Third, combining this identity with the

property a2 − 2as = (s+ bQ)(5s+ bQ) yields s and Q uniquely as functions of x, namely

s =a2 − x2

2(a+ 2x); Q =

2ax− a2 + 5x2

2b(a+ 2x). (3)

Substituting these values into the original profit function converts the problem to an order-7 polynomial

optimization problem in a single variable x. Differentiating with x and solving the first-order condition

produces 6 solutions, of which 4 are easily rejected, and a fifth is rejected because the profit is increasing

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at that point (it is a local minimum). This isolates the unique optimal solution,

x∗ =a

17

(21

(230− 17√

151)13

+ (230− 17√

151)13 − 2

),

and s,Q are easily computed using Eq. 3, while F and optimal profit are obtained from Eq. 1.

2.3.2 Triangle Distribution

We follow the same approach when G(x) is a “triangle” distribution (g(x) = 2(a − x)/a2 linearly

decreasing density over [0, a]). The profit function is

Π =

(1− 2x

a+x2

a2

) (Qx− bQ2

2

)+s(a− (s+bQ))3

3a2b. (4)

From first-order conditions, after rejecting s + bQ = a as a solution we derive Q = 4x−a3b

and s = a−x3

.

Substituting this into the profit function yields a single-variable optimization problem in x, and x as

the solution of the following cubic equation, (2a − 5x)(a − 4x)(a − x) = 0. Of the three solutions,

the optimal solution x = 2a5

is identified because it is adjacent to the solution x = a which is a local

minimum. Hence the full solution is

x =2a

5; Q =

a

5b; s =

a

5; F =

3a2

50b; Π =

9a2

250b. (5)

2.3.3 Uniform Distribution

Now, g(x) = 1a∀x ∈ [0, a]. After substituting for F , y and q∗ the profit function in this case is

Π = (a− x)(xQ− bQ2

2) +

s(a− (s+ bQ))2

2ab. (6)

First-order conditions are obtained by differentiating with s, Q and x, respectively. These conditions

yield i) s = a−bQ3

, ii) Q = (x − s)/b, and iii) x = a2

+ Q4

, after eliminating a boundary solution Q = 0.

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Solving this system of equations yield the optimal 3PT:

x =3a

5; Q =

2a

5b; s =

a

5; F =

4a2

25b; Π =

2a2

25b. (7)

2.4 General Solution

Returning to the general problem (Eq. 1), we derive optimality conditions successively differentiating Π

with Q, s, F . Differentiating with Q, applying Leibniz Rule, ∂Π∂Q

= v(x, Q)(1−G(x))− s(1−G(y))−0,

yields the optimality condition for Q.

Proposition 1 (Optimal Allowance). The optimal allowance Q is the level at which x’s adjusted marginal

value equals the weighted overage fee s.

Q = v-1(x, s

(1−G(y))

(1−G(x))

); equivalently, v(x, Q) = s

(1−G(y))

(1−G(x)). (8)

This equation defines a relationship between the optimal allowance, overage fee s, and the marginal

buyer x. The optimal allowance Q is the quantity that the marginal buyer would demand (given rate s),

weighted by the fraction of 3PT-plan buyers who would exceed Q.

Next, we differentiate the profit function with overage fee s. Applying ∂F∂s

= 0 in this expression

yields the optimality condition,

s =

(−∫ X

y

(v-1(x, s)−Q)g(x)dx

)/d

ds

∫ X

y

(v-1(x, s)−Q)g(x)dx. (9)

which is a differential equation of the form s = −J (s)/dF (s)

ds, where J (s) =

∫ Xy

(v-1(x, s)−Q)g(x)dx

is the total over-allowance consumption computed over those whose consumption exceeds Q: the over-

allowance consumption of each buyer weighted by density of buyers. Rewriting the differential equation

as −J (s)s·J ′(s)

= 1 highlights that the optimal overage fee is set such that the inverse elasticity of overage

consumption equals 1. This parallels the classical optimal pricing rule, “inverse elasticity equals markup”

which (under zero variable cost) is 1−G(p)p·g(p) = 1.

Proposition 2 (Optimal Overage Fee). The optimal overage fee is the level at which inverse elasticity

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of overage demand equals 1. Formally,

−J (s)

s · J ′(s)= 1; ≡

(d ln(J (s))

d ln(s)= −1

), (10)

where J (s) =∫ Xy

(v-1(x, s)−Q)g(x)dx is total overage demand at s, given F,Q.

To compute the third optimality condition, differentiate Π with x1. Interestingly, just like for the

optimal overage fee, the optimality condition for fixed fee reduces to the classic relationship between

price elasticity and percentage markup (which, due to zero fixed costs in this case, is 1).

Proposition 3 (Optimal Fixed Fee). The optimal fixed fee is such that inverse elasticity of demand for

the bucket plan equals the percentage markup for that plan.

F ′

F

(1−G(x)

g(x)

)= 1. (11)

2.5 Optimal 3PT and Usage Heterogeneity

The previous sections developed economic properties of the optimal solution for the general case, as well

as detailed optimal 3PT solutions for the commonly used marginal value function v(x, q) = (x − bq).

This formulation accounts for marginal diminishing utility from increasing consumption. However, it

does so by restricting marginal valuation (or demand) curves of different consumers to be parallel (i.e.,

marginal value diminishes at the same rate for all consumers), and hence satiation levels (which measure

usage heterogeneity) vary the same way as valuation for the first unit. For a deeper investigation, we

employ the more flexible marginal valuation function to capture more facets of consumer hetrogenity

v(x, q) = x−b(1−x ln(δ))q ...(δ ∈ [0, e]), (12)

where δ measures the degree of usage heterogeneity in the market. This formulation allows marginal

valuations to decline at different rates (either increasing or decreasing with x), covering the major

scenarios of interest with respect to how usage heterogeneity interacts with per-unit valuation (see

Fig. 2). Setting δ = 1 simplifies this function to v(x, q) = x − bq, for which satiation levels vary the

same way as valuation for the first unit. Values of δ < 1 (conversely, δ > 1) cover the cases where usage

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heterogeneity decreases (conversely, increases) with x (equivalently, marginal value functions converge

vs. diverge on the horizontal axis).

[Figure 2 about here.]

For the v function, y = s+bQ1+Q ln(δ)

) and q∗(x, s) = x−sb−x ln(δ)

. We first illustrate the solution for the

case where G is uniform over [0, 1]. Differentiate the overage revenue function with s and with Q, and

solving the first-order conditions yields (for δ 6= 1)

s∗ =1

ln δ

((1−x) ln δ + (1−x ln δ)(ln(1− ln δ)− ln(1−x ln δ))

(1−x) ln δ + 2(1−x ln δ)(ln(1− ln δ)− ln(1−x ln δ))

)(13a)

Q∗ =1

b

1

ln δ

((1−x) ln δ + (1−2x ln δ)(ln(1− ln δ)− ln(1−x ln δ))

(1−x) ln δ + 2(1−x ln δ)(ln(1− ln δ)− ln(1−x ln δ))

)(13b)

The optimal overage fee and allowance levels are expressed above as functions of only x. Hence the

original problem is reduced to a single-variable optimization problem in x. Optimizing the resulting profit

function yields x as a solution to a 4th-order equation. Since this equation also has logarithmic terms

(in the variable x) it must be solved numerically. This is straightforward and yields a unique solution for

each value of δ. The complete solution can be obtained by plugging in x∗ into Eq. 13a and Eq. 1b.

Solutions for the Triangle and Normal-Like distributions can be obtained using the same procedure,

but are suppressed due to space limitations.

3 Comparison of Alternative Pricing Schemes

Having solved the optimal 3PT problem under a wide variety of market conditions enables us to compare

the 3PT design against alternative simple pricing schemes, namely per-unit pricing and buffet pricing.

These comparisons enable us to examine whether implementing a three-part tariff confers significant

advantage in terms of profit, and whether it involves sacrifice on other dimensions such as market share

or consumer surplus. Our computations cover the optimal 3PT for the entire range of δ ∈ (−1, e) and

for the three distributions (uniform, triangle, like-normal). This setup enables us to account for both

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forms of heterogeneity (consumer usage heterogeneity and traditional market valuation heterogeneity).

Similarly, varying G affects the proportion of consumer types, nature of value heterogeneity. Together,

the marginal value function and the variations on G fully account for the various scenarios of relevance

to our research questions.

3.1 Impact of Consumer Valuation and Usage Heterogeneity

We present the results visually, to facilitate comparison along the two dimensions of heterogeneity. First,

consider the impact of δ (usage heterogeneity) and G (value heterogeneity) on the optimal design of

each pricing scheme (see Fig. 3). As δ increases, there is an increase in both r (in per-unit plan) and

U (buffet pricing). Intuitively, δ has greater impact on V of higher x customers, which tilts the margin

volume trade-off towards higher margin, higher fees. However, while s does increase in the 3PT, F falls,

indicating that as usage heterogeneity increases, the 3PT design places more prominence on the usage or

overage fee than on the fixed fee. For the impact of value heterogeneity, compare prices across the three

distributions. The per-unit price (r and s) is highest in the uniform distribution. It drops for the normal

and triangle distributions, because the shift in cumulative density makes it attractive to target customers

at the lower end. Fixed fees are also highest in the uniform distribution, even after normalizing the fees

in the other two distributions, accounting for lower total value.

[Figure 3 about here.]

Comparison of parameters in the different pricing plans reveals the versatility of the 3PT in responding

to different market conditions. For instance, increase in usage heterogeneity leads to a greater difference

in fixed fees increases between the 3PT and buffet plan. With buffet pricing, the firm has only one lever,

the flat fee, with which to address the changed demand environment. But with 3PT, it can choose to

either employ the fixed fee lever or, conversely, place less weight on fixed fees (and, correspondingly,

the allowance, see panel 3) and more on usage fees. As users become more diverse in demand quantity,

the firm deploys the usage fee lever to monetize this demand. Similarly, compare the usage rates under

per-unit pricing and 3PT: the latter has lower usage fee because it can also monetize through the fixed

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access fee. The use of this lever itself changes with an increase in δ, which causes the 3PT to mimic

the per-unit plan, hence shrinking the difference between s and r.

Result 1. The overage fee s increases and the implicit per-unit rate F/Q in the 3PT decreases, as

usage heterogeneity increases. The optimal 3PT plan begins resembling the per-unit plan as δ becomes

sufficiently high.

Next, compare the market outcomes under different pricing schemes. Note that the total value under

the demand curve changes with variation in value and usage heterogeneity. Hence the plan metrics must

be normalized to ensure a meaningful comparison. Specifically, profit and consumer surplus are divided

by the maximum trade surplus available in the market; fixed fee and allowance (in the 3PT plan) are

divided by the average of maximum consumption across all consumers; while the usage fee component

remains the same (between 0 and 1 by construction).

[Figure 4 about here.]

Being a more general pricing scheme than both per-unit and buffet pricing, a 3PT naturally produces

higher profit. However, a 3PT is marginally more complex than the other two plans, hence the magnitude

of profit increase matters. Fig. 4 presents the result with respect to the nature of value and usage het-

erogeneity. The profit advantage of a 3PT plan over per-unit pricing is highest when usage heterogeneity

is low, because it is then very efficient at re-collecting the excess consumer surplus that would be lost

under per-unit pricing. Conversely, the profit advantage reduces under high usage heterogeneity because

a usage fee is best able to respond to high variation in consumption. Indeed, when usage heterogeneity

is very high, a 3PT essentially mimics a per-unit plan. The comparison flips in the case of buffet pricing,

except that there is a much larger profit difference between the two plans. Surprisingly, the 3PT plan’s

advantage over buffet pricing is highest under high usage heterogeneity. Buffet pricing leaves firms with

a very stark contrast between volume (which requires low fixed fee) and margin (high fixed fee). In

contrast, the 3PT plan is able to add revenue through the overage fee. Conversely, when there is very

little usage heterogeneity, then the simpler buffet pricing is able to achieve most of the profit obtained

from the 3PT plan.

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Result 2. The 3PT plan’s dominance over the per-unit plan is highest when value heterogeneity is high

and usage heterogeneity is low. Conversely, its dominance over buffet pricing is highest when usage

heterogeneity is high and value heterogeneity is low.

[Figure 5 about here.]

Next, consider market share. Because use fees impose greater total costs on high usage customers

while fixed fees cause light users to subsidize heavy users, a buffet plan and 3PT plan favor high-usage

customers, working in a strikingly different way than a per-unit plan, which favors low-usage customers.

Hence, market coverage is highest under a per-unit plan (see Fig. 5). However, as usage heterogeneity

increases, the per-unit plan faces a starker tension between volume and margin, and market coverage

drops. Market coverage falls even more rapidly for the buffet plan, because besides this tension, the

buffet plan is already ill-suited to high levels of usage heterogeneity. The trade-off is milder with the

3PT, because it can leverage both the fixed and usage fee components. The gap in market coverage

between the 3PT and per-unit plan shrinks as δ increases.

[Figure 6 about here.]

Besides market share, another metric for evaluating impact on consumers is consumer surplus. This

metric paints a different picture, because now the buffet plan (which has lowest market coverage)

encourages all buyers to consume up to their satiation level, creating additional surplus for consumers.

Naturally, this effect is more pronounced for heavy users. Consequently, the buffet plan performs quite

well when usage heterogeneity is high. Conversely, the per-unit plan performance degrades as δ increases,

because the use price places a heavy tax on consumption and surplus creation. The 3PT, being able to

use both F and s improves modestly with usage heterogeneity. Fig. 6 visualizes these observations, and

the results on market coverage and surplus are summarized below.

Result 3. Per-unit pricing results in maximum market coverage, and buffet pricing the least. However,

buffet pricing leads to greater consumption, and creates more consumer surplus when usage heterogeneity

is high.

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3.2 3PT vs. Menu of Per-Unit and Buffet Pricing

Our results have demonstrated the overall dominance of 3PT over the alternative pricing schemes, per-

unit pricing and buffet pricing. While the profit from these simpler schemes can be close to the 3PT

profit under some market scenarios, it can be substantially lower in others. One implication of this is

that if a marketer decides to use a simple plan—per-unit or buffet—then, picking the wrong one, given

the market conditions, could cause a big sacrifice in profit. The results also demonstrated that the two

simpler pricing schemes work in starkly contrasting ways. One captures most of the potential profit

(relative to 3PT) when value heterogeneity is high, the other when usage heterogeneity is high. This

suggests that a marketer could get the best of both by simply combining the two schemes, offering a

menu with one per-unit price and one unlimited-use price. For instance, Apple offers two choices for

music: a per-unit price ($0.99/song) and a $9.99/month buffet plan. The per-unit plan can cater to the

low-value customers, while the buffet price can be used to lure high-usage customers. Intuitively, then,

this menu should produce both higher market coverage and higher profit than a three-part tariff.

[Figure 7 about here.]

We examine this conjecture, pitting a single 3PT against a menu of per-unit and buffet prices, and

varying both usage heterogeneity and value heterogeneity. The results are presented in Fig. 7, which

displays the percentage increase (or decrease) in profit by using a 3PT vs. the menu. Surprisingly, the

3PT beats the menu under most conditions. Specifically, the best relative performance of 3PT occurs

when value heterogeneity involves clustering towards mid-value consumers. It also performs well even

when clustering is towards low-value customers. When there is no clustering (uniform distribution), then

the 3PT performs well under low usage heterogeneity but not when usage heterogeneity is high.

Result 4. The single 3PT produces higher profit than the optimal menu of per-unit and buffet prices,

except when consumers are very heterogeneous on appetite, and highly scattered on per-unit valuation.

14

4 Conclusion

To the best of our knowledge this is the first paper that provides an analytical solution for designing an

optimal 3PT plan for most commonly employed value functions in literature. Furthermore, the paper

derives the properties of optimal 3PT for the general case and provides its economic interpretation. A

major contribution of this paper is to highlight, analyze and understand the implications of multi-facets

of consumer heterogeneity for goods with multi-unit consumption. We carefully distinguish valuation

heterogeneity (captured through distributional assumptions) from usage heterogeneity to understand

their impacts on optimal design. We found that usage heterogeneity has larger role to play in deciding

optimal plan rather than traditional value heterogeneity.

We also compared three most widely used pricing plans for computing services, and the impact

of consumer heterogeneity on their relative performance over multiple metrics, namely profit, market

coverage and consumer surplus. Generally, both forms of heterogeneity favor a “Pay as you Go” plan

relative to an “All you can Eat” buffet plan. A three-part tariff (being more general) beats both on profit,

although per-unit pricing yields higher market coverage and buffet pricing yields maximum consumer

surplus. The 3PT profit beats the buffet plan handily in all conditions, while its advantage over a per-

unit plan is highest under high value heterogeneity and low usage uncertainty. Moreover, the 3PT is more

versatile: the overage fee serves as a device to meter high type consumers, but when market conditions

favor per-unit pricing, it essentially acts as one. We also examine a menu of per-unit pricing and buffet

pricing which, intuitively, should exhibit the best performance because it combines the positive, but

contrasting, qualities of both pricing schemes. Surprisingly, the 3PT exceeds the profit from this menu

under most conditions. This is consistent with (Bagh and Bhargava, 2013)’s finding on the efficiency of

price discrimination via three-part tariffs. We hope that the availability of optimal 3PT design will spur

more research that leverages 3PT (e.g., in pricing, salesforce compensation, and other contracts), and

also make 3PT pricing more accessible to practitioners.

15

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0 20 40 60 80 100

0.6

0.8

1.0

1.2

1.4

δ

uniform distribution Beta(1,1)

0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

δ

triangle distribution Beta(1,2)

0 20 40 60 80 100

0.0

0.5

1.0

1.5

δ

like−normal distribution Beta(2,2)

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

δ

uniform B(1,1)triangle B(1,2)normal B(2,2)

Figure 1: Three distributions that explore different extent and location of clustering of consumers. Thefinal right panel compares the cumulative density functions for the three distributions.

quantity

mar

gina

l val

ue

δ = 0.02

quantity

mar

gina

l val

ue

δ = 1

quantitym

argi

nal v

alue

δ = 2

Figure 2: Level of usage heterogeneity increases with δ, moving from the left panel to the right.

0.5 1.0 1.5 2.0 2.5

0.0

0.1

0.2

0.3

0.4

0.5

δ

rs

UniformTriangleNormal

0.5 1.0 1.5 2.0 2.5

0.0

0.2

0.4

0.6

0.8

1.0

δ

3PTbuffet

fixed fee

0.5 1.0 1.5 2.0 2.5

0.2

0.4

0.6

0.8

δ

uniformtrianglenormal

allowance

Figure 3: Optimal plan Design: Impact of Usage and Value Heterogeneity.

18

0.5 1.0 1.5 2.0 2.5

010

2030

4050

60

δ

uniformtrianglenormal

bucket plan profit: % increase over r plan

0.5

1.0

1.5

2.0

2.5

0

10

20

30

40

50

60

δ

uniformtrianglenormal

bucket plan profit: % increase over unlimited plan

Figure 4: 3PT plan: Profit increase relative to per-unit and buffet pricing.

0.5 1.0 1.5 2.0 2.5

020

4060

8010

0

δ

per−unit3PTbuffet

% market covered B(1,1): Uniform

0.5 1.0 1.5 2.0 2.5

020

4060

8010

0

δ

per−unit3PTbuffet

% market covered B(1,2): Triangle

0.5 1.0 1.5 2.0 2.5

020

4060

8010

0

δ

per−unit3PTbuffet

% market covered B(2,2): Normal

Figure 5: Market share across pricing schemes, and impact of heterogeneity.

0.5 1.0 1.5 2.0 2.5

2628

3032

3436

δ

r3PTbuffet

% CS captured uniform distribution

0.5 1.0 1.5 2.0 2.5

2628

3032

3436

δ

r3PTbuffet

% CS captured triangle distribution

0.5 1.0 1.5 2.0 2.5

2628

3032

3436

δ

r3PTbuffet

% CS captured uniform distribution

Figure 6: Percentage of Consumer surplus captured under different prices schemes and heterogeneity.

19

0.5 1.0 1.5 2.0 2.5

−2

−1

01

2

δ

uniformtrianglenormal

bucket plan profit: % increase over (r + unlimited) plan

Figure 7: 3PT plan vs. Menu of Per-Unit and Buffet Pricing.

20