Download Simple Pricing for Information Services: Optimal Three

Simple Pricing for Information Services: Optimal
Three-Part Tariff
June 22, 2015
Abstract
Mobile and smartphone apps, and other information services, are transforming business and
society. Most of these goods are offered as pay for service rather than pay to own, and many
use 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 the
optimal 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 most
widely used demand functions, and develops fundamental properties of the optimal solution for
the general case. Additionally, it develops insights regarding the optimal 3PT design under various
forms of consumer heterogeneity. We also compare the optimal 3PT against per-unit and buffet
pricing under variety of market conditions. The 3PT dominates on profit when usage heterogeneity
is low and value heterogeneity is high, but per-unit pricing achieves maximum market coverage
while buffet pricing generates highest consumer surplus. This suggests the potential dominance of
a menu comprising a per-unit pricing and buffet pricing. Yet, surprisingly, a single 3PT produces
higher profit than this menu under most conditions.
1
Introduction
Many products, especially in information, computing and service industries, feature multi-unit consumption. 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 management, 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
1
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 perunit 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; Köhler, 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
2
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 q th 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.
3
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
Rq
into the total valuation function V (x, q) = 0 v(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 x̂ in two ways: i) x̂ achieves indifference at the free allowance level Q (i.e., consumes
exactly Q units), or ii) x̂ 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 x̂ be the marginal buyer for the optimal three-part tariff. x̂
consumes exactly Q units, equivalently v(x̂, Q) ≤ s. The indifference equation is V (x̂, Q) = F .
Applying Lemma 1, the firm earns V (x̂, Q)(1 − G(x̂)) from fixed fees, and overage revenue from
x > x̂ whose consumption exceeds Q. Given Q and x̂, this yields a stand-alone optimization problem in
RX
which overage fee is set to optimize residual revenue, x̂ (q ∗ (x̂, s)−Q)+ g(x)dx. The optimal fee s∗ may
lead to overage consumption by all x > x̂ (implying that v(x̂, Q) = s), or only for x ∈ (y, X), where
y ≥ x̂ is the highest x who consumes exactly Q units at rate s (i.e., v(y, Q) = s; y =
2.2
s+Q
).
1+Q ln(δ)
Profit Function with 3PT
The firm’s total profit function is:
Z
Π = max F (1−G(x̂))+s
F,Q,s
where F = V (x̂, Q);
X
(q ∗ (x, s)−Q)g(x)dx
(1a)
y
y = v-1 (s, Q);
4
and q ∗ = v-1 (x, s)for x ≥ y
(1b)
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.]
5
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 (x̂, Q, s) space using the indifference equation
F = V (x̂, Q). Given x̂ 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 x̂. This reduces the problem into a
single-variable optimization with no constraints (barring that x̂ 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,
bQ2
s(a − (s+bQ))3 (a + (s+bQ))
3x̂2 2x̂3
.
Qx̂ −
+
Π= 1− 2 + 3
a
a
2
2a3 b
(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) x̂ = 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 x̂ = s + bQ. Third, combining this identity with the
property a2 − 2as = (s + bQ)(5s + bQ) yields s and Q uniquely as functions of x̂, namely
s=
a2 − x̂2
;
2(a + 2x̂)
Q=
2ax̂ − a2 + 5x̂2
.
2b(a + 2x̂)
(3)
Substituting these values into the original profit function converts the problem to an order-7 polynomial
optimization problem in a single variable x̂. Differentiating with x̂ 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
6
at that point (it is a local minimum). This isolates the unique optimal solution,
a
x̂ =
17
∗
!
√
1
21
151) 3 − 2 ,
√
1 + (230 − 17
(230 − 17 151) 3
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=
2x̂ x̂2
+ 2
1−
a
a
bQ2
Qx̂ −
2
+
s(a − (s+bQ))3
.
3a2 b
From first-order conditions, after rejecting s + bQ = a as a solution we derive Q =
(4)
4x̂−a
3b
and s =
a−x̂
.
3
Substituting this into the profit function yields a single-variable optimization problem in x̂, and x̂ as
the solution of the following cubic equation, (2a − 5x̂)(a − 4x̂)(a − x) = 0. Of the three solutions,
the optimal solution x̂ =
2a
5
is identified because it is adjacent to the solution x = a which is a local
minimum. Hence the full solution is
2a
x̂ = ;
5
2.3.3
a
Q= ;
5b
a
s= ;
5
3a2
F =
;
50b
9a2
Π=
.
250b
(5)
Uniform Distribution
Now, g(x) =
1
a
∀x ∈ [0, a]. After substituting for F , y and q ∗ the profit function in this case is
Π = (a − x̂)(x̂Q − b
Q2
s(a − (s + bQ))2
)+
.
2
2ab
(6)
First-order conditions are obtained by differentiating with s, Q and x̂, respectively. These conditions
yield i) s =
a−bQ
,
3
ii) Q = (x̂ − s)/b, and iii) x̂ =
a
2
7
+ Q4 , after eliminating a boundary solution Q = 0.
Solving this system of equations yield the optimal 3PT:
x̂ =
2.4
3a
;
5
Q=
2a
;
5b
a
s= ;
5
4a2
;
25b
F =
Π=
2a2
.
25b
(7)
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(x̂, Q)(1 − G(x̂)) − s(1 − G(y)) − 0,
yields the optimality condition for Q.
Proposition 1 (Optimal Allowance). The optimal allowance Q is the level at which x̂’s adjusted marginal
value equals the weighted overage fee s.
(1 − G(y))
(1 − G(y))
-1
Q=v
x̂, s
.
(8)
;
equivalently, v(x̂, Q) = s
(1 − G(x̂))
(1 − G(x̂))
This equation defines a relationship between the optimal allowance, overage fee s, and the marginal
buyer x̂. 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,
Z
s= −
,
X
-1
(v (x, s) − Q)g(x)dx
y
d
ds
Z
X
(v-1 (x, s) − Q)g(x)dx.
(9)
y
RX
which is a differential equation of the form s = −J (s) dFds(s) , where J (s) = y (v-1 (x, s) − Q)g(x)dx
is the total over-allowance consumption computed over those whose consumption exceeds Q: the overallowance consumption of each buyer weighted by density of buyers. Rewriting the differential equation
as
−J (s)
s·J 0 (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
8
of overage demand equals 1. Formally,
−J (s)
= 1;
s · J 0 (s)
where J (s) =
RX
y
≡
d ln(J (s))
= −1 ,
d ln(s)
(10)
(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 x̂1 . 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 0 1 − G(x̂)
= 1.
(11)
F
g(x̂)
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
ṽ(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
9
heterogeneity decreases (conversely, increases) with x (equivalently, marginal value functions converge
vs. diverge on the horizontal axis).
[Figure 2 about here.]
For the ṽ function, y =
s+bQ
)
1+Q ln(δ)
and q ∗ (x, s) =
x−s
.
b−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)
1
(1−x̂) ln δ + (1−x̂ ln δ)(ln(1− ln δ)− ln(1−x̂ ln δ))
s =
ln δ (1−x̂) ln δ + 2(1−x̂ ln δ)(ln(1− ln δ)− ln(1−x̂ ln δ))
1 1
(1−x̂) ln δ + (1−2x̂ ln δ)(ln(1− ln δ)− ln(1−x̂ ln δ))
∗
Q =
b ln δ (1−x̂) ln δ + 2(1−x̂ ln δ)(ln(1− ln δ)− ln(1−x̂ ln δ))
∗
(13a)
(13b)
The optimal overage fee and allowance levels are expressed above as functions of only x̂. Hence the
original problem is reduced to a single-variable optimization problem in x̂. Optimizing the resulting profit
function yields x̂ as a solution to a 4th -order equation. Since this equation also has logarithmic terms
(in the variable x̂) 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 x̂∗ 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
10
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
11
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 heterogeneity. 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.
13
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, perunit 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 perunit 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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17
0
20
40
60
80
100
1.0
0.8
0.6
1.0
0.2
0.5
0.4
like−normal distribution
Beta(2,2)
0
20
40
δ
60
80
100
uniform B(1,1)
triangle B(1,2)
normal B(2,2)
0.0
0.0
1.5
1.0
triangle distribution
Beta(1,2)
0.0
0.5
1.5
2.0
1.4
1.2
1.0
0.6
0.8
uniform distribution
Beta(1,1)
0
20
40
δ
60
80
100
0
20
40
δ
60
80
100
δ
Figure 1: Three distributions that explore different extent and location of clustering of consumers. The
final right panel compares the cumulative density functions for the three distributions.
marginal value
δ=2
marginal value
δ=1
marginal value
δ = 0.02
quantity
quantity
quantity
fixed fee
0.6
0.6
0.5
1.0
1.5
δ
2.0
2.5
0.2
0.2
0.0
Uniform
Triangle
Normal
0.0
0.1
0.4
0.4
0.3
0.2
allowance
0.8
1.0
3PT
buffet
0.8
r
s
0.4
0.5
Figure 2: Level of usage heterogeneity increases with δ, moving from the left panel to the right.
0.5
1.0
1.5
δ
2.0
2.5
uniform
triangle
normal
0.5
1.0
1.5
2.0
δ
Figure 3: Optimal plan Design: Impact of Usage and Value Heterogeneity.
18
2.5
60
bucket plan profit: % increase over r plan
bucket plan profit: % increase over unlimited plan
60
uniform
triangle
normal
40
40
30
30
20
50
20
10
50
uniform
triangle
normal
10
δ
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0
0
0.5
δ
0.5
1.0
1.5
2.0
2.5
80
60
40
0.5
per−unit
3PT
buffet
0
per−unit
3PT
buffet
0
0
per−unit
3PT
buffet
% market covered
B(2,2): Normal
20
40
60
80
% market covered
B(1,2): Triangle
20
20
40
60
80
% market covered
B(1,1): Uniform
100
100
100
Figure 4: 3PT plan: Profit increase relative to per-unit and buffet pricing.
1.0
1.5
δ
2.0
2.5
0.5
1.0
δ
1.5
2.0
2.5
δ
Figure 5: Market share across pricing schemes, and impact of heterogeneity.
0.5
1.0
1.5
δ
2.0
2.5
r
3PT
buffet
0.5
1.0
1.5
δ
2.0
2.5
28
30
32
34
36
% CS captured
uniform distribution
r
3PT
buffet
26
26
r
3PT
buffet
28
30
32
34
36
% CS captured
triangle distribution
26
28
30
32
34
36
% CS captured
uniform distribution
0.5
1.0
1.5
2.0
2.5
δ
Figure 6: Percentage of Consumer surplus captured under different prices schemes and heterogeneity.
19
−2
−1
0
1
2
bucket plan profit: % increase over (r + unlimited) plan
uniform
triangle
normal
0.5
1.0
1.5
2.0
2.5
δ
Figure 7: 3PT plan vs. Menu of Per-Unit and Buffet Pricing.
20