Download Nonlinear Pricing In Oligopoly: An Application to the US Mobile

Nonlinear Pricing In Oligopoly:
An Application to the US Mobile Phone Industry
Zsolt Macskási¤y
Northwestern University
October 7, 2003
Abstract
I develop a structural oligopoly model with nonlinear pricing. I estimate the model using data
from the US wireless phone service industry. The structural parameter estimates are used to forecast
the e¤ects of policy alternatives. First, I measure the welfare consequences of nonlinear pricing by
comparing the actual outcome to a regime where …rms are required to post uniform prices. Second, I
assess the potential anti-competitive e¤ects of mergers.
1
Introduction
In this paper, I develop an oligopoly model with non-linear prices. On the demand side, consumers
simultaneously make a discrete choice and a continuous choice. In particular, they choose which …rm to
buy from (if any), and a quantity level of the product o¤ered by the chosen …rm. On the supply side,
…rms simultaneously post tari¤ schedules. I assume that observed tari¤s correspond to an equilibrium
outcome of this game. I take a non-parametric approach, in that I do not restrict tari¤ schedules to be
of any speci…c functional form. My supply-side model does not yield closed-form solutions, therefore I
calculate equilibrium tari¤s numerically. The advantage of this strategy is that I do not have to make
restrictive assumptions neither on the distribution of consumer types nor on the functional form of the
tari¤s. In addition, this numerical approach can handle an arbitrary number of …rms. The disadvantage is
that nonparametric methods require more parameters to estimate, hence computational time and e¢ciency
might be an issue.
¤I
would like to thank Robert Porter and Shane Greenstein for their help. Financial support of Northwestern University’s
Dissertation Year Fellowship and of the Center for the Study of Industrial Organization is gratefully acknowledged.
y [email protected]
1
Generally speaking, there are two major welfare questions that theoretical and empirical papers on
price discrimination explore. First, given industry structure (holding the number of …rms …xed), what is
the e¤ect of price discrimination, compared to a benchmark case of uniform pricing? Second, given that
price discrimination is allowed, what is the e¤ect of imperfect competition, compared to the monopoly
benchmark? In line with this tradition, after having estimated the structural parameters, I simulate two
hypothetical scenarios. First, I am interested in the welfare e¤ects of nonlinear pricing. I simulate what
would happen if a regulatory agency restricted the form of the tari¤s that the …rms can o¤er to the public.
In particular, instead of fully nonlinear tari¤s, I consider a) two-part, b) linear or c) ‡at tari¤ rates. There
is no theoretical evidence about this question yet. It is not known whether in oligopoly, allowing …rms to
post fully nonlinear prices would increase or decrease pro…ts, consumer surplus and total welfare. Second,
I calculate the welfare e¤ects of hypothetical mergers. Nonlinear pricing poses new questions in merger
simulations. Theoretical and empirical evidence tells that changing the competitiveness of an industry
can have di¤erential consequences for consumers. Small and large users can be a¤ected di¤erently. There
can be welfare redistributions from one group of consumers to the other when the number of …rms in the
industry changes.
The application chosen is the US wireless phone industry in the late 1990’s. In this era, there was
already a large number of carriers, six of which had nationwide presence. By contrast, almost all previous
studies on the wireless phone industry (Marciano (1995), Parker and Röller (1997), Busse (2000), Miravete
and Röller (2003)) have analyzed data from an earlier period, when the industry structure was at most
duopolistic. The wireless service industry is characterized by highly nonlinear pricing, some amount of
product di¤erentiation, and almost no price regulation.
Leslie (2001) studies the welfare e¤ects of nonlinear pricing of theater tickets, where the theater is
assumed to be a local monopolist. Not surprisingly, he …nds that price discrimination increases pro…ts.
The e¤ect on aggregate consumer surplus is close to zero. There is a redistributional e¤ect, though. Low
quality buyers bene…t to the detriment of high quality buyers. This …nding is not surprising, since buyers
have unit demands and under uniform pricing, all tickets are sold at the same price. By contrast, in models
of quantity (quality) discrimination, uniform pricing means that tari¤s are linear in quantity (quality). It
is not obvious, which way the redistributional e¤ect would go then.
Crawford and Shum (2001) get similar results in a monopoly model of cable TV service. The counterfactual in that paper is not uniform pricing, but restricting the …rm to o¤er the e¢cient set of qualities,
while letting it to choose prices. The …rm is better o¤ by being able to price discriminate, while the e¤ect
on consumers is ambiguous.
2
1.1
Theoretical literature
Price discrimination is said to exist when the same product is sold to di¤erent consumers at di¤erent prices.
Under perfect competition, of course, this cannot happen, unless marginal costs di¤er across consumers.
The monopoly case is well understood since the seminal works of Mussa and Rosen (1978) and Maskin and
Riley (1984). Later, explorations were made into imperfectly competitive environments. It was recognized
by Katz (1984), Borenstein (1985) and Holmes (1989) that price discrimination can persist even if …rms
have some market power. Extensive surveys about nonmonopoly price discrimination theories can be
found in Varian (1989), Wilson (1993) and Stole (2001). Among many possible discriminatory techniques,
second-degree price discrimination (also called indirect discrimination or nonlinear pricing) represents a
very important case. Major contributions to this area include Spulber (1989), Champsaur and Rochet
(1989), Stole (1995), Rochet and Stole (2001, 2002) and Armstrong and Vickers (2001).
For a monopolist, nonlinear pricing is better than uniform pricing. Trivially, the former yields at least
as high pro…ts than the latter. It can be shown that pro…ts are indeed strictly higher. Under oligopoly,
to my knowledge, there are hardly any unambiguous theoretical results. As Stole (2001) remarks: “While
banning price discrimination unambiguously harms the monopolist’s bottom line, the e¤ect may be to
increase [oligopoly] industry pro…ts when all …rms must use uniform prices.” He writes also: “When
competition is introduced, then the size of the cross-…rm elasticities relative to the industry elasticities
will be relevant for determining output, prices and welfare under price discrimination.” The trade-o¤ that
…rms face is whether the rent-extraction or the business-stealing e¤ect dominates once they are allowed to
depart from uniform pricing. Which way the welfare e¤ects go, depends on the underlying cost and demand
structure. In addition, in many models on nonlinear pricing, the vertical dimension is quality instead of
quantity. Since quality is typically unobserved (although one can use proxies, of course), it is di¢cult to
conceptualize what uniform pricing means. As a result, the welfare e¤ects of price discrimination with
nonlinear pricing did not receive much theoretical attention.1 In any case, it is safe to say that there are
no general results.
1 Models
of third-degree price discrimination also do not have clear predictions. Thisse and Vives (1988) note that competi-
tion with price discrimination may be more intense than with uniform pricing. Corts (1998) suggests that price discrimination
may lead to “all-out-competition.” In these situations, …rms would be better o¤ if they committed not to price discriminate.
By contrast, Armstrong and Vickers (2001) show in a Hotelling-type locational model that price discrimination increases
duopoly pro…ts.
3
1.2
Empirical literature
To the best of my knowledge, there is no concise survey on empirical works on price discrimination.
McManus (2001) o¤ers an incomplete review. Without trying to be complete, I name a few frequently cited
papers. Reduced form techniques were used by Borenstein (1989, 1991), Shepard (1991), Borenstein and
Rose (1994), Busse and Rysman (2001), Clerides (2002). Structural models include Ivaldi and Martimort
(1994), Cohen (2000), Leslie (2001), McManus (2001), Miravete (1996, 2002, 2003), Miravete and Röller
(2003). These papers span a large range of industries: retail gas, airlines, electricity, local and wireless
phone, books, paper towel, theater, co¤ee shops, and Yellow Pages directories.
1.2.1
Identifying price discrimination
At …rst, the problem was the mere identi…cation of the presence of price discrimination. Discrepancies from
uniform pricing can partly be explained by di¤erences in marginal costs.2 Consumers might pay di¤erent
unit costs, simply because serving them involves di¤erent marginal costs. Size discounts can be given,
because larger sizes are associated with lower marginal costs, due to a less than proportional increase in
storage, inventory or other types of costs that do not vary with size. By contrast, prices might vary due
to the fact that di¤erent types of consumers have di¤erent willingness to pay. Only this second source of
price dispersion should be called price discrimination. So, initially, the problem was to empirically separate
cost-based and demand-based e¤ects.3
Borenstein (1991) investigates the retail market for leaded and unleaded gasoline. He …nds that margins
were initially higher for unleaded gas, but this tendency has reversed after many stations discontinued selling
leaded gas. The explanation is that buyers of unleaded gas are, for some reason, more loyal to the station.
More market power associated with unleaded gas allows stations to charge higher margins. Likewise, when
many stations discontinued o¤ering leaded gas, stations that kept selling it, gained a lot of market power.
Shepard (1991) investigates self-service and full-service margins at gas stations. She …nds that …rms
o¤ering both types of service, charge a larger margin for full-service and a lower margin for self-service
than stations that o¤er only a single type of service. Multi-product stations can charge a high margin for
their high quality good, because their marginal consumers switch to the low quality good. By contrast,
marginal consumers of single-service …rms would switch to a di¤erent station. Similarly, multi-product
stations charge lower price for their low-quality good, because they have fewer inframarginal customers
that buy this good, due to the existence of the other good.
2 This
is a standard criticism against any empirical paper about price discrimination. See, for example, Lott and Roberts
(1991).
3 When cost data is not available, as it is often the case, price-cost discrepancies should be inferred from price di¤erences,
for example.
4
Cohen (2000, 2001) examines the paper towel market. He applies an idea that follows from Shepard’s
model. In particular, a single-product …rm that introduces a second, higher quality good, should decrease
its price for the low quality good. Accordingly, Cohen …nds a negative e¤ect on price associated with being
the lowest quality in the product line. He also estimates that 55-65% of the variation in margins is due to
cost reasons, but the rest is due to price discrimination.
Clerides (2002) investigates price-cost margins of a book publisher. Unlike the above studies, he …nds
that almost all variations are due to cost-shifters. Demand shifters in‡uence quantities sold, but not prices.
Finally, it is an open question, whether it is the absolute di¤erence (margin, equal to price minus cost)
or the relative di¤erence (markup, equal to price divided by cost) that matters. There seems to be no
consensus about which one is more appropriate. Clerides (2003) shows that the two can sometimes lead
to opposite conclusions. In other words, for the same prices and costs, margin and markup can rank two
products di¤erently.
1.2.2
E¤ects of increased competition
The second question tackled by empirical papers was how the degree of competitiveness a¤ects pricing.
Theory gives a clear prediction about monopolistic distortions of prices and allocations relative to the …rstbest (e¢cient) outcome. The monopolist enlarges the quantity (quality) spectrum towards the lower end
of the range. It assigns too low quantities (qualities) and too high marginal prices to low type consumers.
On the other hand, it assigns the e¢cient quantity and charges marginal cost to the highest type consumer
(no distortion on the top). How the presence of a second …rm changes the nature of these distortions, is
a relatively recent area of research. There are two qualitatively di¤erent theoretical predictions, both of
which are obtained only for duopoly. According to the …rst, if the market is fully covered (full competition)
then price is equal to cost plus …xed fee.4 Hence, the outcome is e¢cient, because marginal utilities are
equal to marginal costs. On the other hand, if the market is not fully covered, then …rms behave as local
monopolists. In this case, the outcome lies between the monopoly and the e¢cient one. McManus (2001)
analyzes product lines of competing co¤ee shops. He …nds that observed price-cost margins are indeed
nonuniform, and estimates di¤erences between marginal utilities and marginal costs to be nonuniform.
It is disturbing, however, that distortions are increasing with size, a …nding, which contradicts to both
theoretical predictions.
As expected, all theoretical models agree that duopoly prices are lower than in monopoly. However,
theory is not clear about the relative gains of the presence of a second …rm. In particular, it is not
4 The
exact conditions, under which this result is true, are more stringent. See Rochet and Stole (2002) and Armstrong
and Vickers (2001).
5
clear whether the price decrease is relatively more important for small or for large users. On one hand,
the articles of Spulber (1989) and Stole (1995) imply that small users gain more. This happens because
they assume that low types (who consume less) have weaker brand loyalties, hence they have a higher
elasticity of substitution between …rms. Therefore, price competition is more intense in the low end of
the spectrum. It is important to notice that this result hinges on the assumption that brand preference
and vertical preference are positively correlated. Borenstein (1989) has …rst documented that increased
competition a¤ects margins associated with goods of di¤erent qualities in an uneven way. Borenstein and
Rose (1994) note that price dispersion of airline tickets increases with competition. The explanation is
best illustrated by the following stylized story. Suppose there are two kinds of buyers: low (tourist) types
and high (business) types. A monopolist would o¤er an expensive ticket for high types and a cheap one for
low types. Suppose that a second …rm enters. Suppose that tourists have a higher cross-elasticity between
the two airlines. When a second …rm enters, both prices will decrease. However, the price for the low type
will drop by more, due to the fact that tourists are more eager to react to a price cut o¤ered by the rival
airline.
On the other hand, Rochet and Stole (2002) predict the opposite, i.e. that competition causes a greater
price decrease for high types than for low types. In their model, brand preference is independent from
vertical preference. Busse and Rysman (2001) provides empirical support to this result. Their intuition
is the following. Under monopoly, high types get more surplus. Hence, in duopoly, they would be able to
expend some travel cost, buy their less preferred product, and still receive positive surplus. For low types,
there may be only one …rm that can pro…tably serve consumers. In other words, at the low end of the
spectrum, markets do not join, and each …rm behaves as a local monopolist. Therefore, competition does
not a¤ect the prices charged to them.
Miravete and Röller (2003) …nd, surprisingly, that markups increase when the industry structure changes
from monopoly to duopoly. Accordingly, pro…ts increase and consumer surplus decreases. The …rst e¤ect
dominates, so overall welfare increases, too. This …nding is due to fact that the comparison is not made
on a ceteris paribus basis. Instead, they compare two di¤erent time periods. Between the monopoly and
the duopoly era, costs have decreased substantially.
1.2.3
Wireless service pricing
A few empirical papers have already been written about pricing in the US wireless phone industry. Parker
and Röller (1997) estimate the conduct parameter of …rms’ supply function. They …nd evidence that …rms
collude somewhat but not as much as they potentially could. Busse (2000) …nds that when the same
two …rms compete against each other at multiple markets, they tend to charge higher prices. This result
6
supports the theory about multimarket contact. Marciano (1995) …nds a positive relationship between
price discrimination (measured by the degree of concavity of the posted tari¤) and product di¤erentiation.
1.2.4
Endogeneity of product menus
A major drawback of almost all empirical papers on price discrimination is the following. In real world,
…rms o¤er a number of di¤erent product sizes. The determination of what sizes to o¤er is an important
choice that …rms make. By contrast, almost all papers treat product menus as exogenous. In other words
product menu choice is not modeled. As Cohen (2000) puts it: “The exogeneity of product characteristics
is taken as a matter of course by almost every author who has estimated structural models of supply
and demand.” In reality, product menus are unlikely to be exogenous. In particular, it is reasonable
to believe that …rms with stronger market presence tend to o¤er a broader product spectrum. Treating
them as exogenous, could lead to erroneous results. The traditional solution to control for endogeneity
of product menus is to include …rm-speci…c …xed e¤ects. However, this might not always be adequate.
For example, Cohen (2001) shows that this empirical strategy could falsely reject the existence of price
discrimination. He proposes a test for the presence of price discrimination. The test requires to have panel
data, where changes to actual product menus are frequent enough. The test exploits the fact that changes
in the product menu will have a di¤erent e¤ect on prices depending whether variations in unit price are
cost-based or discrimination-based.
The exogeneity of product menus is often justi…ed by arguing that these choices are less ‡exible than
pricing choices, because they require some adjustments in the production procedure. However, this claim
is untenable in many service industries, where contracts typically involve pricing schedules de…ned over a
range of possible quantity levels.5
Miravete, in a series of papers, (1996, 2002, 2003) makes an important departure from treating products
as exogenous. He estimates structural models, where a local phone company is assumed to set optimal
nonlinear prices. In these papers, the …rm is assumed to be a monopolist. Crawford and Shum (2001)
is another study on monopolistic cable TV providers. To my knowledge, there are only two duopoly
papers: Ivaldi and Martimort (1994) and Miravete and Röller (2003). These authors take a common
agency approach. Goods are assumed to be imperfect substitutes, hence consumers always choose buying
both of them. By contrast, I work with an exclusive agency framework, in which consumers buy either one
of many competing products or none.
The remainder of the paper is structured as follows. In Section 2, I outline a behavioral model of
5 It
is true, of course, that …rms in service industries typically o¤er a menu of tari¤s. Since introducing an additional tari¤
probably involves some kind of …xed costs, the optimal number of tari¤s is …nite. In this paper, I abstract from this question,
by assuming that consumers choose a tari¤ and a quantity level simultaneously.
7
supply and demand in an oligopolistic industry, where …rms post nonlinear prices. In Section 3, I describe
the estimation strategy. Section 4 presents some facts about the wireless phone service industry. Section
5 describes the data. Section 6 describes the results of the estimation. Section 7 and 8 presents two
simulation exercises. First, I calculate the welfare consequences of nonlinear pricing, then I address the
e¤ects of mergers. Section 9 concludes.
2
The economic model
A realistic model of the mobile phone industry would account for the sequential nature of consumer
decisions. In reality, tari¤ choice and quantity choice are not simultaneous decisions. In treating them
as simultaneous, I make an important simplifying assumption. Otherwise, I would face a complicated
problem. The set of tari¤s o¤ered by one …rm has to be a best response against the sets of tari¤s o¤ered
by other …rms. A …rm has to choose optimally both the number of tari¤s o¤ered, and the shape of each
tari¤ o¤ered. By contrast, if consumer choices are simultaneous, then consumers would only choose a point
on the lower envelope of the set of tari¤s. Hence, we can assume without loss of generality that each …rm
o¤ers a single nonlinear tari¤.
In this section, I outline a model of nonlinear pricing in oligopoly. The model is a mixture of the
basic logit oligopoly model and the canonical monopoly model of nonlinear pricing. Therefore it is best to
describe the two components separately. Subsection 2.1 follows Sections 7.2, 7.4 and 7.5 of Anderson et.
al (1992), whereas Subsection 2.2 reiterates the results of Section 3.5.1.2 of Tirole (1988).
A common feature among all subsections below that tastes are concave, costs are linear in quantity
levels. Both …rms and consumers are risk neutral.
2.1
The basic logit oligopoly model
Consider J …rms that sell a di¤erentiated product. Firms have constant marginal costs (cj ) and …xed
costs are zero. Firms set prices (pj ) simultaneously. Consumers have unit demand, with the possibility of
choosing the “outside good.” In other words, each consumer buys either one good from one of the …rms,
or buys nothing. Formally, when purchasing from …rm j, buyer i has utility:
uij = v ¡ pj + » j + "ij
v is the reservation value of the product, assumed to be identical across buyers and …rms. » j is a
…rm speci…c component that represents possible quality di¤erences across …rms. "ij is an individual and
…rm speci…c taste component that represents horizontal heterogeneity across buyers. v and » j are …xed
8
components that …rms know. However, …rms only know the distribution of "ij in the population. Assume
that "ij has an i.i.d. extreme value distribution. The cdf is then (because of the i.i.d. assumption, subscripts
can be omitted):
µ
µ
¶¶
"
G (") = exp ¡ exp ¡ ¡ °
½
where ½ is a scale parameter, and ° is Euler’s constant (approximately 0:5772). In what follows, I
characterize the Bertrand-Nash equilibrium of this game. It is well-known that extreme-value disturbance
terms give rise to logit choice probabilities. Market share of …rm j is given by:
sj (pj ; p¡j ) =
Firm j’s problem is then
e(v¡pj +»j )=½
PJ
1 + k=1 e(v¡pk +»k )=½
max ¦j = (pj ¡ cj ) sj (pj ; p¡j )
pj
Di¤erentiating w.r.t. pj , after simple manipulations yields:
pj = cj + ½
1
1 ¡ sj (pj ; p¡j )
(1)
As it is apparent from (1), equilibrium margin is increasing in own market share. In other words, a
…rm with higher market share, has more market power. Also, the equilibrium margin is increasing in ½.
Remember that ½ is the scale parameter of the distribution of ", thus it is proportional to its variance. In
words, a larger degree of horizontal di¤erentiation leads to larger margins. To see the e¤ects of competition
on margins, assume that …rms are symmetric. Absent cost and quality di¤erences (when cj = c and » j = »
for all j), a symmetric equilibrium is given by the solution to the following …xed point problem:
p=c+½
1 + Je(v¡p+»)=½
1 + (J ¡ 1) e(v¡p+»)=½
It is straightforward to show that the equilibrium margin is decreasing in the number of …rms (J), and
lim p = c + ½.
J!1
2.2
The canonical nonlinear pricing model in monopoly
Consider a single …rm that sells a good in di¤erent quantities. The …rm has constant marginal cost (c) and
…xed costs are zero. The monopolist sets a tari¤ P (q). Consumers di¤er in their preferences for quantity.
This vertical taste component is denoted by µ. The monopolist only knows its distribution. Assume that
9
the support of this distribution is [0; 1), with pdf f and cdf F . Assume the increasing hazard rate property,
i.e. f = (1 ¡ F ) is increasing in µ. Consumers’ indirect utility is given by:
v (µ) = max µV (q) ¡ P (q)
q
(2)
where V is a nonnegative, strictly increasing and strictly concave function. For the moment, there is
no restriction on P (:). Instead, it should be checked at the end that it has such form that gives rise to a
globally concave problem. In what follows, I characterize the optimal tari¤ set by the monopolist. Denote
the optimal quantity level by q ¤ . Formally,
q ¤ (µ) = arg max µV (q) ¡ P (q)
q
By the envelope theorem:
@
v (µ) = V (q ¤ (µ))
@µ
(3)
Therefore, it is possible to express v (µ) by integrating V (q ¤ (µ)):
v (µ) = v (0) +
Z
µ
V (q ¤ (µ)) dµ
(4)
0
Incentive compatibility with full separation is guaranteed by either (3) or (4), provided that q ¤ (µ) is
nondecreasing. This last condition means that higher types are assigned higher quantities. Individual
rationality is ful…lled for every type, if it is ful…lled for the lowest type. Assuming that the reservation
utility is zero, amounts to v (0) = 0. From (2), it follows that P (q ¤ (µ)) = µV (q ¤ (µ)) ¡ v (µ). Hence, we
can write the monopolist’s problem as:
max¤ ¦ =
vj (µ);qj (µ)
s.t. v (µ) =
R1
Rµ
@ ¤
@µ q (µ)
0
0
fµV (q ¤ (µ)) ¡ v (µ) ¡ cq ¤ (µ)g f (µ) dµ
(5)
V (q ¤ (µ)) dµ
¸0
One could substitute the equality constraint into the objective function. Under fairly mild conditions,
the …rst-order conditions are su¢cient. Pointwise maximization yields
µV 0 (q ¤ (µ)) = c +
1 ¡ F (µ) 0 ¤
V (q (µ))
f (µ)
(6)
The expression on the left-hand-side of (6) is the marginal utility of consumer of type µ. In equilibrium,
this coincides with the marginal price paid by type µ. Hence, the monopolist charges a positive price-cost
margin for almost all consumer types. Marginal utilities and marginal prices are distorted upward from
10
marginal cost, except for the highest type. Since consumers pay higher marginal price than e¢cient, they
also consume smaller than e¢cient quantities. To put it di¤erently, the monopolist enlarges the quantity
spectrum relative to the socially e¢cient allocations. Also, the price-cost markup: (p ¡ c) =p = (1 ¡ F ) =µf
is decreasing in µ. In other words, the distortion is relatively lower for higher types, and it completely
disappears at the limit, when µ ! 1.
2.3
Nonlinear pricing and logit oligopoly
Essentially, I extend the canonical nonlinear pricing model to an oligopolistic environment. Consider J
…rms with constant marginal costs cj , and zero …xed costs. Firms simultaneously set fully non-linear pricequantity schedules. Denote the pricing schedule of …rm j by Pj (q). Consumers (individuals) are indexed
by i. They choose both a …rm (j) and a quantity level (q). They are allowed to “opt out,” in which
case, they do not choose any quantity. Buyers are modeled as having two-dimensional types. The vertical
dimension (preference for quantity) is indexed by µ. (Although consumers have di¤erent µ’s, I will not use
the i subscripts to ease notations.) The horizontal dimension (brand preferences) is represented by a vector
f"ij gJj=1 . An important simplifying assumption is that the two type components are independent.6 As a
consequence, the optimal quality choice is independent from the brand choice. To …nish the description
of consumer preferences, denote utility by uij . More precisely, by choosing …rm j and the optimal quality
level, consumer i gets utility uij . The utility function is a sum of three terms. First, there is a vertical
component, vj , that stems from the optimal quantity choice. Similarly to (2), we have
vj (µ) = max µV (q) ¡ Pj (q)
q
(7)
I make the same assumptions about V than in the previous section. Denote the optimal quantity level
by qj¤ (µ). The second and the third term of the utility function are associated with brand preferences. I
¡ ¢
assume a …xed-e¤ect » j and an individual-speci…c e¤ect ("ij ). Hence, utility is:
uij (µ; "ij ) = vj (µ) + » j + "ij
Types are privately known to consumers. Firms know their distributions only. Assume that both
type components have a continuous distributions that are i.i.d across consumers. Denote the density of
µ by f, with support [0; 1). Assume that "ij has extreme-value distribution, independent across …rms
and consumers. The extreme-value distribution is useful for two reasons. First, it ensures that choice
probabilities have analytic forms. Second, it ensures that for any type µ, the market shares of each …rm
(as well as the share of the outside option) will be strictly positive.
6 This
is the same assumption as in Rochet and Stole (2001, 2002).
11
From the …rms’ point of view, the probability that a consumer with type µ chooses …rm j is given by:
Pr (type µ chooses j) ´ sj (vj (µ) ; v¡j (µ)) =
e(vj (µ)+»j )=½
PJ
1 + k=1 e(vk (µ)+»k )=½
(8)
¡
¢
¡
¢
From (7), it follows that Pj qj¤ (µ) = µV qj¤ (µ) ¡ vj (µ). Hence, we can write …rm j’s problem as:
max¤ ¦j =
vj (µ);qj (µ)
ª
R1© ¡ ¤ ¢
µV qj (µ) ¡ vj (µ) ¡ cj qj¤ (µ) sj (vj (µ) ; v¡j (µ)) f (µ) dµ
0
s.t. vj (µ) = vj (0) +
@ ¤
@µ qj (µ)
¸0
Rµ
0
¡
¢
V qj¤ (µ) dµ
(9)
Firm j has to choose two functions: vj (µ) and qj¤ (µ), to maximize the objective function subject to an
incentive compatibility constraint similar to (4), and that the assigned quantity is an increasing function
of type.7 Unlike in the monopoly case, here the market shares are functions of the control variable, vj (µ).
This makes the problem very complicated.8 Generally, one cannot get analytic solutions. See more on this
in Rochet and Stole (2002).
Although …rms behave as local monopolists, they price di¤erently than a monopolist would. In other
words, equilibrium pricing outcomes are di¤erent from what a single …rm would post when facing a similar
demand structure. For example, Rochet and Stole (2002) derive optimal price schedule of a monopolist
that faces consumers with random participation constraints. There, the randomness comes from an exogenous type distribution. By contrast, here, the participation constraints (embedded in the market share
expressions) are partly determined by the rivals’ prices.
2.4
Discrete type space
It is possible to circumvent analytical di¢culties with a numerical approach. To do numerical computations,
it is necessary to discretize the type-space.9 In particular, assume that µ can take on a grid of T +1 possible
values: µ0 < µ 1 < :: < µT , with respective probabilities f0 ; f1 ; :::; fT . Denote the assigned quantity and
© ¤ ªT
T
utility levels by qj;t
and fvj;t gt=0 , respectively. The continuous case can be approximated with …ner
t=0
and …ner grids. I will apply the …nest grid that is computationally manageable.
7 Note
that the individual rationality (participation) constraints are implicitly built into the …rms’ problem through the
market share formulae.
8 As Rochet and Stole (2002) point out, to solve for the appropriate …rst-order conditions, one needs to appeal to controltheoretic techniques. The resulting Euler equation is a second-order di¤erential equation with boundary conditions. In
general, this cannot be expressed in closed-form. The above authors present an example, where horizontal di¤erentiation
takes the form of a simple Hotelling-duopoly. Even with a simple form of utility and cost function, the model can be only
solved numerically.
9 Crawford and Shum (2001), in their monopoly model, also use discrete type-distributions.
12
With discrete types, we need to distinguish two kinds of incentive compatibility constraints. The
downward constraints ensure that no type wants to pretend to be a lower type. The upward constraints
ensure that no type wants to pretend to be a higher type. Formally:
DIC
U IC
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¸ µt V qj;t¡1
¡ Pj qj;t¡1
: µ t V qj;t
¡ Pj qj;t
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
: µ t¡1 V qj;t¡1
¡ Pj qj;t¡1
¸ µt¡1 V qj;t
¡ Pj qj;t
(10)
(11)
It can be shown that whenever these constraints hold between adjacent types (µt and µt¡1 ), they also
hold between any two types.10 Furthermore, it is standard to show that in the optimal solution, the
downward constraints hold with equality, whereas the upward constraints are slack. Assume therefore
equalities in (10), and apply recursive substitutions of the vj;t terms. In the end, we get the following:
vj;t = vj;0 +
t
X
s=1
¡ ¤
¢
(µs ¡ µs¡1 ) V qj;s¡1
(12)
Expression (12) is the discrete equivalent of the incentive compatibility constraint (4), under continuous
types. Similarly to (8), the market shares are given by:
Pr (type µt chooses j) ´ sj;t (vj;t ; v¡j;t ) =
e(vj;t +»j )=½
PJ
1 + k=1 e(vk;t +»k )=½
(13)
An important consequence of the logit assumption is that every …rm, as well as the outside option has
a strictly positive market share. For other modeling choices of horizontal di¤erentiation, this would not
necessarily be the case.11 We are now ready to state the oligopolists’ problem:
max T ¦j =
¤
fvj;t ;qj;t
gt=0
s.t. vj;t = vj;0 +
PT
t=0
Pt
©
¡ ¤ ¢
ª
¤
µt V qj;t
¡ vj;t ¡ cj qj;t
sj;t (vj;t ; v¡j;t ) ft
s=1 (µs
¡ ¤
¢
¡ µ s¡1 ) V qj;s¡1
(14)
¤
¤
¤
qj;0
· qj;1
· ::: · qj;T
© ¤ ªT
Firm j maximizes the above objective function with respect to T + 2 free unknowns: qj;t
and vj;0 .
t=0
In the end, I am looking for a Nash-Bertrand equilibrium. In other words, each …rm’s tari¤ has to be a
10 A
common way to state this fact is that “local” incentive compatibilty constraints are su¢cient.
example, Rochet and Stole (2002) extend the classic Hotelling-duopoly to nonlinear pricing. Suppose that the two
11 For
…rms are at the endpoints of the unit interval, consumers are uniformly distributed and the unit transportation cost is ¾.
³
´
v
v ¡v
Then, the market share of …rm j would be equal to sj (vj ; vk ) = min ¾j ; 12 + j ¾ k . Then, depending on ¾, there are two
distinct cases: when sj + sk = 1 and when sj + sk < 1. The authors call the …rst as the fully competitive, the second as the
local monopoly regime. In this sense, in my model, …rms are in the local monopoly regime, for all µ.
13
best response to all other …rms’ tari¤s.12
The equilibrium schedules are conditional on the primitives of the model, which are the cost and demand
©
ªJ
parameters, and the latent type distribution. That is, for given values of cj ; » j j=1 , ½, and fµt ; ft gTt=0 ;
n©
ªJ oT
¡ ¤ ¢
¤
.
; sj;t j=1
we can calculate equilibrium quantities, prices and market shares:
qj;t
; Pj qj;t
t=0
As a remark, notice that my approach does not have a closed-form solution. In other words, I cannot
calculate the optimal pricing schedules analytically. Similarly to my approach, Leslie (2001) also does numerical calculations when he simulates the optimal pricing of a monopolist under hypothetical regulatory
constraints. By contrast, Ivaldi and Martimort (1994) and Miravete and Röller (2003) obtain exact formulae. However, they get them by restricting attention to duopoly, and to very special type-distributions.
3
The econometric model
The major question is how to take this model to data. The above theory predicts the joint distribution
© ¤
¡ ¤ ¢ª
of prices and quantities for each …rm. In other words, it assigns to each pair qj;t
; Pj qj;t
a relative
frequency: sj;t ft . It is the distribution of the latent type (µ) that ultimately generates this distribution.
Similar predictions were obtained in models by Ivaldi and Martimort (1994), Crawford and Shum (2001),
and Miravete and Röller (2003), all of them proceeded in slightly di¤erent ways.
One approach is to …t the predicted joint distribution with the actual price-quantity distribution. To
follow this strategy, one needs of course, to observe the actual distribution. In other words, one needs to
know how many people were at each price-quantity pair, for any given …rm. Then, by minimizing some
measure of distance between the predicted and actual distribution, one can recover the distribution of µ.
Lack of data often makes this strategy infeasible. For example, Miravete and Röller (2003) only observe
actual tari¤s used by competing …rms. Hence, their empirical strategy amounts to minimize the distance
between predicted and actual tari¤s. Crawford and Shum (2001) observe qualities, prices and aggregate
market shares for each quality-price pair. Nevertheless, they only exploit information in prices and market
shares. Ivaldi and Martimort (1994) observe individual transactions and choose to ignore actual tari¤s.
My dataset permits me to follow a similar approach. When individual purchases are observed, one can
take advantage of observed market shares. Using this information, one does not need to make restrictive
parametric assumptions to recover the type (µ) distribution.
12 I
should note that in general, there is no guarantee that an equilibrium exists or it is unique. Even when …rms compete in
a single price, the equilibrium is typically assumed to exist. See, for example, Berry, Levinsohn and Pakes (1995). A common
defense against this criticism is to try to …nd an equilibrium starting from di¤erent initial values. Getting the same solution
each time would give some reasons to believe that the equilibrium is unique. Caplin and Nalebu¤ (1991) provide su¢cient
conditions on the demand structure, under which existence is guaranteed for the single price problem.
14
My dataset contains provider choice, quantities consumed, and price paid by households. Formally,
there are I households, indexed by i. Let di;j = 1 if household i has chosen …rm j, and zero otherwise. If
no …rm is chosen then di;0 = 1. For the moment, assume that households can only choose one option, i.e.
PJ
13
In case where the household chooses a …rm, it also chooses a quantity level, qi;j and there
j=0 di;j = 1.
is a price pi;j associated with this level.
I use a pseudo-likelihood method, similar to Ivaldi and Martimort (1994). I assume that quantity levels
and prices are observed with measurement errors. If consumer i was of type µt then his “true” quantity
¡ ¤ ¢
¤
and price would be qj;t
and Pj qj;t
. The measurement error is the di¤erence between observed and “true”
values. This is the same “trick” as in Ivaldi and Martimort (1994).
qi;j
pi;j
¤
= qj;t
+ ºq
¡ ¤ ¢
= Pj qj;t + º p
I assume that the disturbance terms are joint normal, with (º q ; º p ) » N (0; §) Why do I allow for
correlation between the measurement errors?14 Notice that I have chosen not to include any aspects of
the calling plans other than quantity and price. However, these plans have other (both observed and
unobserved) product characteristics that are likely to a¤ect price. To name only three, plans might or
might not o¤er free long-distance calls or roaming and they might be prepaid cards or might require
a (one or two-year) subscription. Any “good” characteristic is likely to increase price, and/or decrease
quantity, whereas the opposite holds for “bad” characteristics. Hence it is likely that the two error terms
are negatively correlated due to the omitted product characteristics. Denote by Á the pdf of the standard
normal distribution. The pseudo-likelihood function is the following:
L=
I X
T
Y
i=1 t=0
(
(di;0 = 0) Á
µ
¤
qi;j ¡ qj;t
¾q
)
¡ ¤ ¢!
¶ Ã
pi;j ¡ Pj qj;t
Á
sj;t + (di;0 = 1) s0;t ft
¾p
(15)
Notice that I do not assign a direct likelihood-value to an observation-pair (qi;j ; pi;j ). If I did, I would
have a proper likelihood-function. Instead, I assume that these choices contain measurement errors. This
is the sense, in which I use a pseudo-likelihood.
13 Strictly
speaking, this is not true in my dataset. In the section, where I describe the data in detail, I will specify how I
deal with this problem.
14 Ivaldi and Martimort (1994) also do not assume independence of the two measurement errors. Their data is about
electricity purchase contracts of a few hundred dairy …rms. The worry is that the two electricity providers (EDF-GDF and
a cartel of oil companies) know their clients closely. Hence, the two providers can o¤er di¤erent tari¤ schedules to di¤erent
clients. As a consequence, observed quantity-price pairs, in reality, lie on di¤erent schedules. The problem is that the model
assumes one schedule per provider. This could generate correlation between price error and quantity error.
15
To summarize: the estimation involves a two-step procedure. In the inner loop, based on Section 2.4, I
determine equilibrium prices, quantities and market shares, given structural cost and demand parameters.
In the outer loop, I maximize the log of (15) over the structural cost and demand parameters.
3.1
Further functional speci…cations
Assume the following functional form for V :
V (q) = q ® , with 0 < ® < 1
Assume that µ has exponential distribution, with scale parameter ¯ > 0. Then the cdf is
F (µ) = 1 ¡ e¡¯µ
Next, I formulate a discrete distribution that approximates the continuous exponential distribution:
ft = F (µ t ) ¡ F (µt¡1 ) = e¡¯µt¡1 ¡ e¡¯µt
Also, suppose that the grid fµt gTt=0 is equally spaced, that is µt = t¢. Hence, the problem is to
©
ªJ
maximize (15) with respect to the following set of unknowns: » j ; cj j=1 ; ½; ¾ q ; ¾ p ; ®; ¯; T; ¢.
In describing the economic model, I have implicitly assumed that there is only one market, at which J
…rms are present. The reality of the US mobile phone industry is more complex, of course. Consider M
di¤erent geographic markets. Each market m has Jm …rms present. I assume that …rms post the same
tari¤ schedules at all markets, which they are present. Let zjm be an indicator variable, taking a value
of 1 if …rm j is present at market m, zero otherwise. Then the modi…ed market share equations are the
following:
Pr (type µt chooses j at market m) ´
sm
j;t (vj;t ; v¡j;t )
Hence, …rm j’s objective function is the following:
¦j =
M
X
m=1
zjm
=
8
>
<
e(vj;t +»j )=½
P Jm (vk;t +»k )=½
1+ k=1 e
>
: 0
!
à T
X©
ª m
¡ ¤ ¢
¤
µ t V qj;t ¡ vj;t ¡ cj qj;t sj;t (vj;t ; v¡j;t ) ft
t=0
16
if zjm = 1
if zjm = 0
3.2
Identi…cation
Without being formal, I discuss how variation in the data is able to identify my model. First, there is
variation across households regarding carrier choices as well as choosing the outside option. This variation
© ª
identi…es the …rm-speci…c …xed e¤ects » j (with the exception of one) and the parameter on the inclusive
value of inside options ½. Second, there is variation in minutes consumed and prices paid among households
that choose the same carrier. This variation identi…es parameters that characterizes vertical heterogeneity
(preferences for quantity): ®; ¯; T; ¢; ¾q and ¾ p . Third, from the equilibrium conditions (stemming from
pro…t maximization) of the …rms’ problem, one can recover marginal costs: fcj g.
Notice that I do not use information on household demographics. I am currently thinking about how
to incorporate this. Ideas welcome.
4
Facts about the wireless phone industry
To be written.
5
Data
I have got the Convergence Audit household survey data from Claritas, Inc. The data was collected in
April-May of 2000. It contains extensive information about energy, television, computer, Internet and
telephone use of 36,741 American households, as well as demographic variables.15
To maintain tractability, I only keep households that live at the 25 most populous markets. These
markets are the following cities: Atlanta, Boston, Chicago, Cincinnati, Cleveland, Dallas, Denver, Detroit,
Houston, Kansas City, Los Angeles, Miami, Minneapolis, New York, Philadelphia, Phoenix, Pittsburgh,
Portland, Sacramento, St. Louis, San Diego, San Francisco, Seattle, Tampa, Washington DC. This elimination has produced 10,652 households. Also, I have selected the 9 most important providers of that time
period: AT&T, AirTouch, Ameritech, Bell Atlantic, BellSouth, Cellular One, GTE, Southwestern Bell,
and Sprint. The table below describes the carrier choices of 5,384 households, all of those who reported to
have a wireless phone.
15 These
kinds of surveys are typically not representative, because they tend to select too many wealthy households and too
few poor households. The reason is that these surveys aim to assist marketing and sales strategy of rather upscale products. I
haven’t made any check on this, nevertheless the data contains a weight variable, which can help to restore representativeness,
if it is needed.
17
Carrier
# of clients
AT&T
824
AirTouch
671
Ameritech
293
Bell Atlantic
639
BellSouth
143
Cellular One
707
GTE
228
Southwestern Bell
125
Sprint
505
Other provider
1249
Total
5384
Households reported data about their primary wireless phone account. What I intend to use is their
typical monthly consumption (in minutes) and their typical monthly bill (in dollars).16
Some of the
households have chosen other providers than the 9 most important ones. Also, some households reported
more than one primary wireless accounts. Instead of disregarding them, I have chosen to assign average
quantity and price values to these households. The average was taken across the 9 most important carriers.
5.1
Price variations
In this section, I take a …rst look at price variations across markets and …rms. I want to see a) how large
is the price variation of a given …rm across di¤erent markets, and b) how large is the price dispersion at a
given market across …rms. Remember that for computational reasons, I need to assume in my model that
each …rm posts the same price at all the markets it is present. This assumption would be justi…ed if the
price variation was largely of inter…rm nature.
In this section, I present some evidence on pricing schemes. To this end, I do not use the survey data
but a di¤erent dataset that provides direct information on actual posted prices. Details about carriers’
calling plans are freely available over the Internet. My work was facilitated by having access to the May
2000 Wireless Survey of EconOne, Inc, a consulting company. This survey keeps monthly records about
the available calling plans in the 25 most important MSA’s. These cities are: Atlanta, Boston, Chicago,
Cincinnati, Cleveland, Dallas, Denver, Detroit, Houston, Kansas City, Los Angeles, Miami, Minneapolis,
New York, Philadelphia, Phoenix, Pittsburgh, Portland, Sacramento, St. Louis, San Diego, San Francisco,
16 The
reported number is likely to include taxes, hence it does not exactly correspond to …rms’ revenues. I abstract from
this issue.
18
Seattle, Tampa, Washington DC. The surveyed companies are AT&T, Aerial, Ameritech, Bell South,
Cellular One, Cincinnati Bell, GTE, Houston Cellular, OmniPoint, Paci…c Bell, Powertel, PrimeCo, RCS
Wireless, Southwestern Bell, Sprint, US West, Verizon, VoiceStream. At each market, there are 4 to 7
carriers present. Companies o¤er a large number of di¤erent plans at each market. The survey contains
2248 di¤erent plans. Data about the plans include all pricing and non-pricing features.
To facilitate comparisons between …rms and markets, I needed to cut the data substantially. First, I
selected plans that speci…ed a one-year contract. Second, I only kept plans that o¤ered digital service.
Third, I have dropped four relatively small companies: Cincinnati Bell, Houston Cellular, Powertel and
RCS Wireless, because each of them operate at a single market. Finally, I was forced to drop a large
company, Cellular One as well. Apparently, this company assigns di¤erent names to the same plan o¤ered
at di¤erent markets, and this makes product classi…cation very cumbersome. I ended up having 1071
di¤erent plans. Next, I have divided plans according to the usual classi…cation: local, regional, and national
ones. Unfortunately, the actual name of the plan does not always tell which category does it belong to.
In addition, some companies de…ne their regional and local plans di¤erently from the common practice.
Therefore, I used the following criteria. I considered a local plan where roaming inside the network area
was not free of charge. Regional plans were the ones where roaming inside the network area was free but
the consumer had to pay for either roaming outside the network area or for long-distance calls. Finally,
any plan that o¤ered all the above three features free of charge, was considered as a national plan.
My …nal dataset covers 13 …rms, 25 markets and 3 plan types (local, regional or national). Some of
the …rm-market-type combinations had no entries. The non-empty cells contained 3 to 10 di¤erent plans.
I took a grid of 1 to 3000 corresponding to the hypothetical number of minutes consumed in a month. I
have assigned to each gridpoint the cheapest option. Essentially, I have calculated the lower envelope of
the tari¤s. Denote this fee by pm
j;t , where j refers to …rm, t to number of minutes, m to the market. Denote
the number of markets that …rm j is present at by Mj . Denote the number of …rms at market m by Jm .
Consider …rst intra…rm price dispersion, denoted by Sj;intra . Calculate average prices of …rm j as:
¢2
PMj m
PMj ¡ m
pj;t = M1j m=1
pj;t . Then price variances of …rm j are given as ¾2j;t = M1j m=1
pj;t ¡ pj;t . I normalize
the standard deviation by the average fee, then take the average across the gridpoints:
Sj;intra =
3000
1 X ¾ j;t
3000 t=1 pj;t
m
Next, consider inter…rm price dispersion, denoted by Sinter
. Calculate average prices of market m as:
¢
P
PJm ¡ m
J
m
1
1
m 2
m
m 2
pm
. I
t = Jm
j=1 pj;t . Then price variances of market m are given by (¾ t ) = Jm
j=1 pj;t ¡ pt
normalize the standard deviation by the average fee, and take the average across the gridpoints:
19
m
Sinter
3000
1 X ¾m
t
=
3000 t=1 pm
t
The results are reported in the Appendix. The …rst table displays intra…rm variations for each of the
13 di¤erent …rms. The second table displays inter…rm variations, for each of the 25 markets.
The picture that arises from the above two tables is the following. Price variations from any source
remains under 30%. As the …rst table shows, …rms do o¤er di¤erent prices at di¤erent markets. This e¤ect
is about 10-20% for local and regional calling plans. However, …rms tend to post the same prices at all
markets for their national plans. When considering inter…rm variations (second table) then the variation
is just as large among national plans than among the other two types. When comparing magnitudes, it
seems that di¤erences between …rms is somewhat more important, but the evidence is not strong at all.
5.2
Nonlinearities in pricing
It is a widely accepted claim that wireless pricing exhibits substantial nonlinearities. Calling plans that
include larger amounts of “free minutes” cost less on a per-minute basis. As a consequence, plotting
observed prices and quantities would produce a diagram that could be best described by some concave
function. In this section, I provide a preliminary statistical evidence about the concavity of price-quantity
schedules. To this end, I use again the Claritas database. I perform the analysis on the full dataset. I ran
the following two regressions:
price = ® + ¯ 1 min +¯ 2 min sq + ° 1 family + ° 2 longdist + ° 3 roa min g + ° 4 prepaid
ln price = ® + ¯ 1 ln min +° 1 family + ° 2 longdist + ° 3 roa min g + ° 4 prepaid
The dependent variables are the actual monthly price paid (…rst equation), and its log (second equation).
The main interest is in the parameters of the quantity variables: actual monthly minutes talked and its
square (…rst equation), and log of actual monthly minutes talked (second equation). Both equations
include additional explanatory variables. Family means that the plan allows unrestricted calling between
family members. Longdist and roaming means that the plan o¤ers long-distance calls and roaming without
additional charges. Finally prepaid means that the plan includes a so-called prepaid calling card.17
Detailed results are reported in the Appendix. Among the 36; 741 households, 16; 647 reported that
they had a wireless phone. The actual numbers of observations in the regressions were about 11; 000
17 A
prepaid card is a package of minutes. One has to pay upfront, and can use them until they expire (typically 6 or 12
months). This option is typically more expensive than subscription plans, because it o¤ers more ‡exibility. First, one does
not have to pay a …xed monthly service fee over a year or two. Second, minutes do not expire at the end of each month.
20
because of missing data. When estimating the log-form equation, I had slighthly less data points, because
when an exact zero for minutes or prices was reported, its logarithm was considered to be a missing value.
Almost all parameters were signi…cant, and they all had the expected sign. In the …rst equation, ¯ 2 was
negative and statistically signi…cant, indicating concavity. However, the coe¢cient was so small, that it
hardly seems to have any economic signi…cance. In the logarithmic speci…cation, an estimated value of 1
for ¯ 1 would mean linear schedules. I have obtained a very precise estimate of 0:22, which ocurred to be
signi…cantly di¤erent from 1. This indicates very strong concavity.
6
Results
To be written.
7
Simulation: Two-part tari¤s
In this section, I describe what the model predicts if …rms are forced to charge two-part tari¤s. Since it is
simpler, I consider …rst the continuous case. However, I can only estimate the nonlinear pricing model by
assuming a discrete type space. Ultimately, I want to compare the nonlinear regime with the hypothetical
two-part regime. Therefore, I will simulate the latter under the assumption of discrete type space.
With two part tari¤s, …rm j has two strategic variables, a …xed fee, aj and a variable fee, pj : Similarly
to (7), we can write
vj (µ) = max µV (q) ¡ pj q ¡ aj
q
Since V is concave, the …rst order condition is su¢cient, hence:
qj¤ (µ) = V 0¡1
Hence
³p ´
j
µ
³
³ p ´´
³p ´
j
j
vj (µ) = µV V 0¡1
¡ pj V 0¡1
¡ aj
µ
µ
Market shares are as before:
Pr (type µ chooses j) ´ sj (pj ; p¡j ) =
Then, pro…ts can be written as:
21
e(vj (µ)+»j )=½
PJ
1 + k=1 e(vk (µ)+»k )=½
¦j =
Z
1³
pj V 0¡1
0
³p ´
j
µ
´
+ aj sj (pj ; p¡j ) f (µ) dµ
Firm j maximizes this expression with respect to its choice variables, aj and pj .
Special cases arise when aj = 0 (uniform prices), or pj = 0 (‡at rate).
7.1
Discrete type space
In this case, the incentive compatibility constraints are a little more involved. Similarly to the nonlinear
regime, we have two kinds of constraints: an upward and a downward one.
DIC
UIC
¡ ¤ ¢
¡ ¤ ¢
¤
¤
: µt V qj;t
¡ pj qj;t
¸ µt V qj;t¡1
¡ pj qj;t¡1
¡ ¤ ¢
¡ ¤ ¢
¤
¤
: µt¡1 V qj;t¡1
¡ pj qj;t¡1
¸ µt¡1 V qj;t
¡ pj qj;t
After rearrangements, they can be written as:
¡ ¤ ¢
¡ ¤ ¢
V qj;t
¡ V qj;t¡1
pj
pj
·
·
¤ ¡ q¤
µt
qj;t
µ
t¡1
j;t¡1
As before, under common regularity conditions, only the downward constraints will be binding. Hence,
we have
¡ ¤ ¢
¡ ¤ ¢
V qj;t
¡ V qj;t¡1
pj
=
¤ ¡ q¤
µt
qj;t
j;t¡1
(16)
© ¤ ªT
¤
As a result, quantity levels qj;t
can be computed recursively, by using (16) and assuming qj;0
= 0.
t=1
¡ ¤ ¢
¤
Then vj;t = µt V qj;t ¡ pj qj;t
¡ aj .
Market shares are as before:
Pr (type µ t chooses j) ´ sj (pj ; p¡j ) =
Pro…ts can be written as
¦j =
e(vj;t +»j )=½
PJ
1 + k=1 e(vk;t +»k )=½
T
X
¡ ¤
¢
pj qj;t + aj sj (pj ; p¡j ) ft
t=0
Firm j maximizes this expression with respect to its choice variables, aj and pj .
22
8
Simulation: Merger analysis
Undoubtedly, the two most important consolidations in the US wireless phone industry were the creation of
Verizon Wireless and Cingular Wireless. Verizon was formed by the combination of the wireless businesses
of Bell Atlantic, GTE and Vodafone AirTouch plus the earlier acquisition of PrimeCo. Cingular was created
by the merger of the wireless businesses of Bell South and SBC. SBC itself has earlier acquired Ameritech,
Cellular One, Paci…c Bell, and Southwestern Bell, to name only the most important transactions.
My dataset predate these mergers. Calling plan contracts cover either a one-year or a two-year period.
Therefore, consumers could have signed up any time in a two year time span previous to the date of the
survey, which is April-May, 2000. It is apparent from the Claritas dataset that most consumers signed up
before the mergers happened.
These consolidations largely happened between …rms that operated at geographically di¤erent markets.
There were only two exceptions. Both Ameritech and Cellular One had signi…cant market shares at the
Chicago market, and the same is true for Ameritech and Southwestern Bell in St. Louis. Thus, it was
tempting to say at the time of the mergers that competition would not be lessened because the merging
…rms were not competing directly with each other anyway. Remember, however, that I assume that each
…rm has to post the same price schedules across all its markets. Therefore, when two …rms merge, the new
…rm cannot replicate the old pricing schedules, unless they were identical. Due to this fact, mergers could
have welfare e¤ects.
I can analyze the e¤ect of these historic mergers, from an a priori standpoint. By estimating a structural
model of the industry based on this dataset, I only use information that was available prior to the mergers.
9
Conclusion
I have treated tari¤ choice and quantity choice as simultaneous decisions. This assumption enabled me to
assume w.l.o.g that each …rm posts a single, fully nonlinear tari¤. In reality, these decisions are separated
in time. One could argue, that consumers are not fully informed when they make their tari¤ choices.
In other words, they are not fully aware about all the circumstances that will determine their subsequent
quantity choice. As a consequence, they might select a tari¤ that will not be optimal from a cost-minimizing
perspective. A natural assumption would be, of course, that the tari¤ choice is nevertheless optimal, in
the sense, that it maximizes expected utility. It is an interesting question what are the welfare e¤ects of
the fact that consumers are only partially informed ex-ante, or that they cannot costlessly switch between
plans. One would be tempted to say that consumers would bene…t from being able to switch costlessly.
However, under these kinds of arrangements, …rms would post di¤erent prices, hence it is far from being
23
certain whether consumers would bene…t in the end. Miravete (1996, 2002, 2003) in a series of papers,
explores this question.18 In all the above, there is a single …rm only. It would be interesting to address
this question in oligopoly.
References
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for Books, International Journal of Industrial Organization, Vol. 20, pp. 1385-1408
18 In
a theoretical paper, Miravete (1996) gets ambigous results. Basically, he …nds that the e¤ects on pro…t and consumer
welfare could go either way, depending on the relative magnitude of the uncertainty in consumers’ tastes.
24
[13] Clerides S. (2003): Price Discrimination with Di¤erentiated Products: De…nition and Identi…cation,
Mimeo, University of Cyprus
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Commitment, RAND Journal of Economics, Vol. 29(2), pp. 306-323
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Television, Mimeo, Duke University and Johns Hopkins University
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Review, Vol. 79(1), pp. 244-250
[19] Ivaldi M., Martimort D. (1994): Competition under Non-Linear Pricing, Annales d’Economie et de
Statistique, Vol. 34, pp. 72-114
[20] Katz M. (1984): Price Discrimination and Monopolistic Competition, Econometrica, Vol. 52(6), pp.
1453-1471
[21] Leslie P. (2001): Price Discrimination in Broadway Theatre, Mimeo, UCLA
[22] Lott J., Roberts R. (1991): A Guide to the Pitfalls of Identifying Price Discrimination, Economic
Inquiry, Vol. 29(1), pp. 14-23
[23] Marciano S. (1995): The E¤ect of Product Heterogeneity on Duopoly Pricing: The Case of Cellular
Telephony, Mimeo, Kellogg School of Management, Northwestern University
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University of Virginia
[26] Miravete E. (1996): Screening Consumers Through Alternative Pricing Mechanisms, Journal of Regulatory Economics, Vol. 9, pp. 111-132
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25
[29] Miravete E., Röller L-H. (2003): Competitive Nonlinear Pricing in Duopoly Equilibrium: The Early
U.S. Cellular Telephone Industry, Working Paper
[30] Parker P.M., Röller L-H. (1997): Collusive Conduct in Duopolies: Multimarket Conduct and CrossOwnership in the Mobile Telephone Industry, RAND Journal of Economics, Vol. 28(2), pp. 304-322
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26
A
Appendix
The two types of incentive compatibility constraints are:
DIC
U IC
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¸ µt V qj;t¡1
¡ Pj qj;t¡1
: µ t V qj;t
¡ Pj qj;t
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¡ ¤ ¢
¸ µt¡1 V qj;t
¡ Pj qj;t
: µ t¡1 V qj;t¡1
¡ Pj qj;t¡1
Substituting vj;t and vj;t¡1 yields
vj;t
vj;t¡1
Rearranging yields
Hence,
¡ ¤ ¢
¡ ¤ ¢
¡ µt¡1 V qj;t¡1
¸ vj;t¡1 + µt V qj;t¡1
¡ ¤ ¢
¡ ¤ ¢
¸ vj;t + µt¡1 V qj;t
¡ µt V qj;t
¡ ¤ ¢
vj;t ¡ vj;t¡1 ¸ (µt ¡ µt¡1 ) V qj;t¡1
¡ ¤ ¢
(µt ¡ µt¡1 ) V qj;t
¸ vj;t ¡ vj;t¡1
¡ ¤ ¢ vj;t ¡ vj;t¡1
¡ ¤ ¢
V qj;t¡1
·
· V qj;t
µt ¡ µt¡1
Under typical regularity conditions, only the downward incentive compatibility constraints bind. In
other words, the upward constraints are automatically satis…ed if the downward constraints hold with
equality. Hence, we can write
or
¡ ¤ ¢ vj;t ¡ vj;t¡1
V qj;t¡1
=
µt ¡ µt¡1
¡ ¤ ¢
vj;t = vj;t¡1 + (µt ¡ µt¡1 ) V qj;t¡1
Hence, using recursive substitutions, we can express vj;t as:
vj;t = vj;0 +
t
X
s=1
¡ ¤
¢
(µs ¡ µs¡1 ) V qj;s¡1
27
B
Tables
Intra…rm variations
Local plans
Regional plans
Price dispersion
# of …rms
AT&T
22%
20
Aerial
0%
1
Ameritech
0%
2
Bell South
Price dispersion
3%
National plans
# of …rms
Price dispersion
# of …rms
0%
20
0%
2
4
0%
3
GTE
14%
7
18%
7
0%
7
OmniPoint
14%
5
14%
5
0%
5
12%
4
3%
5
17%
3
Sprint
0%
25
US West
0%
5
0%
20
16%
5
Paci…c Bell
PrimeCo
Southwestern Bell
Verizon
0%
20%
1
10
14%
30%
VoiceStream
28
2
14
Inter…rm variations
Local plans
Regional plans
National plans
Price dispersion
# of …rms
Price dispersion
# of …rms
Price dispersion
# of …rms
Atlanta
0%
1
4%
2
15%
3
Boston
10%
2
12%
2
26%
4
Chicago
18%
2
18%
4
Cincinnati
8%
2
10%
2
15%
3
Cleveland
12%
3
11%
2
14%
4
Dallas
0%
1
17%
5
Denver
0%
1
0%
1
17%
5
Detroit
12%
3
11%
2
26%
4
Houston
13%
3
0%
1
17%
4
14%
2
21%
2
Kansas City
Los Angeles
19%
2
0%
1
15%
3
Miami
12%
2
23%
2
29%
5
Minneapolis
0%
1
7%
2
18%
4
New York
23%
3
8%
2
26%
4
Philadelphia
11%
3
7%
2
26%
4
Phoenix
0%
1
0%
1
24%
5
Pittsburgh
17%
2
37%
2
15%
3
Portland
0%
1
0%
1
19%
5
Sacramento
20%
2
0%
1
15%
3
San Diego
22%
2
28%
2
15%
3
San Francisco
0%
1
11%
2
17%
2
Seattle
10%
2
14%
2
19%
5
St. Louis
17%
2
0%
1
20%
4
Tampa
10%
2
20%
3
18%
5
Washington DC
11%
2
0%
1
17%
4
29
30