The Economics of Product-Line Restrictions
With an Application to the Network Neutrality Debate
Benjamin E. Hermalin and Michael L. Katz∗
Revised February 9, 2007

Abstract
We examine the welfare eﬀects of product-line restrictions, such as thosecalled for by some proponents of network neutrality regulation. We consider aplatform that brings together households and application providers. We ﬁndthat restricting a monopoly platform to a single product has the following
eﬀects: (a) application providers that would otherwise have purchased a lowquality variant are excluded from the market; (b) applications “in the middle”
of the market utilize a higher and more eﬃcient quality; and (c) applications
at the top utilize a lower and less eﬃcient quality than otherwise. Total
surplus may rise or fall, although the analysis suggests to us that harm towelfare is likely. We also examine a duopoly model and ﬁnd that the welfare
eﬀects are similar.

∗

The authors would like to thank Kathryn Graddy, Yossi Spiegel, John Tirole, John
Vickers, two referees, and seminar participants at IDEI and Oxford University. Katz has
consulted on telecommunications policy issues for a number of private and governmental
clients. None of that consulting concerned network neutrality, and the views expressed in
the present paper should not be attributed to any of those organizations.

1Introduction

Firms often oﬀer product lines—several variants of the same product. Examples include diﬀerent programming packages oﬀered by cable televisioncompanies and diﬀerent versions of software (e.g., standard or professional)
oﬀered by software manufacturers. Typically, a ﬁrm’s decision to oﬀer aproduct line is viewed as unexceptional. In some circumstances, however,there are calls for public policy to limit the range of products oﬀered. At thetime we write this paper, for example, there is an intense debate taking placein the halls of Congress regarding “network neutrality” regulation. One of
the central issues in this debate is whether providers of “last mile” Internet
access services (typically a local telephone company oﬀering dsl service, acable company oﬀering cable modem service, or a wireless service such as
3G) should be allowed to oﬀer more than one grade of service.1 Proponentsof regulation argue that oﬀering multiple grades is unfair and results in someconsumers’ being provided unduly low-quality service.2 For instance, Senator Olympia Snowe warned that, absent regulation, “Consumers will have all
the selections of a former Soviet Union supermarket. We are going to createa two-tier Internet, for the haves who can pay the price, and the have nots
who will be relegated to the Internet dirt road.”3
In this paper, we examine the eﬀects of product-line restrictions in markets where, absent any restrictions, any given supplier could oﬀer a continuumof vertically diﬀerentiated variants. Our results suggest that product-line re1

The term “network neutrality” means many things to many people. Other policies
referred to by this term include prohibition of vertical foreclosure, a ban on third-degree
price discrimination (e.g., setting Internet access prices that vary with the identity of theapplication provider), and prohibition of two-sided pricing of platform services.
2
This type of concern arises in other sectors of the economy as well. For example, asomewhat similar set of issues arise with respect to supermarkets and the quality of shelf
space that they provide to diﬀerent vendors; the sale of preferential product locations has
been controversial. Turning to publicly provided goods, there is typically strong resistanceto having premium lanes on toll bridges or highways that allow travel in a less congested
lane in return for payment of a fee. Ironically, Air France has complained about plans by
the Marseilles airport to oﬀer lower-price, lower-quality terminal services to other airlines.
(John Lichﬁeld and Jen Wainwright, “France takes cheap ﬂights to the next level withlaunch of budget airport in Marseilles,” The Independent, 26 October 2006, at 18.)
3

Ira Teinowitz, “Senate Panel Kills ‘Net Neutrality’ Proposal: Web Access Providers
Free to Charge More for Better Service,” TVWeek.com, June 28, 2006, available at
http://www.tvweek.com/news.cms?newsId=10287, site visited July 24, 2006.

Hermalin & Katz

Introduction

2strictions aﬀect welfare through several mechanisms and that the eﬀects areoften negative.
Most of our analysis concerns a monopoly supplier. Other authors havepreviously examined a monopolist’s choice of product quality. Spence (1975)
showed that a single-product monopolist may choose a quality level higher
or lower than the eﬃcient level, depending on certain properties of consumer
demand. Mussa and Rosen (1978) conducted a classic analysis of a multiproduct monopolist showing that the monopolist generally distorts the setof products oﬀered to consumers. The present model extends these earlier
analyses by comparing the single-product with the multi-product equilibrium
and by examining the qualities of the oﬀerings of a platform service thatbrings two sides of a market together.
We introduce the model in Section 2. In order to shed light on the network
neutrality debate, we examine situations corresponding both to traditional
markets and to what have come to be known as two-sided markets.4 Speciﬁcally, we examine a platform provider that oﬀers services that connect applications providers with households. For example, the platform provider may
be an Internet service provider (isp) selling services that connect householdswith Internet application providers, such as Google or a newspaper’s proprietary web site. We explicitly model the platform provider’s simultaneous
choices of how to price to households and to application providers.
In Section 3, we compare the levels of proﬁts, consumer surplus, and total
surplus when a monopoly platform provider can oﬀer a full range of service
qualities with the corresponding levels when the monopolist can oﬀer only
one product. We examine situations in which applications providers haveheterogeneous demands for the qualities of their connections to households.
We ﬁnd that, as a result of the single-product restriction: (a) application
providers who would otherwise have purchased low-quality connections are
excluded from the market; (b) application providers “in the middle” of themarket purchase connection of higher and more eﬃcient qualities; and (c) application providers at the top of the market purchase connections of lower andless eﬃcient qualities. Eﬀects (a) and (c) reduce total surplus, while eﬀect(b) raises it. Although we ﬁnd that the negative eﬀects frequently dominate,
there are situations in which restricting a monopolist raises welfare. Thatsaid, it should be observed that application providers at the bottom of themarket—the ones that single-product restrictions typically are intended to4

In our formal model below, σ = 0 corresponds to a one-sided market.

Hermalin & Katz

The Model

3

aid—are almost always harmed by the restriction.5 Moreover, the restriction
reduces the set of applications available to households.
In Section 4, we consider several extensions of our baseline model. The
most important is to allow for platform competition. The network neutrality
debate, for example, concerns the regulation of cable and telephone companies that often are local duopolists in the provision of broadband, wireline
Internet access services. We examine the eﬀects of product-line restrictions
in a Hotelling duopoly. Our principal technical ﬁnding is that the equilibrium
quality schedules oﬀered by the duopolists are analogous to the schedule offered by a platform monopoly. The central policy implication of this ﬁnding isthat the welfare results obtained in the monopoly case carry over to this one.
Speciﬁcally, a single-product restriction excludes application providers at thebottom of the market, has mixed eﬀects on application providers higher up
in the quality spectrum, and typically—but not necessarily—lowers overall
welfare.
The paper closes with a brief conclusion. Proofs not given in the text
may be found in Appendix A.

2

The Model

We begin by considering a monopoly platform that provides an intermediation service that facilitates exchange between application providers andhouseholds. We consider a monopoly platform for two reasons. First, thelack of competition simpliﬁes the analysis and allows us to identify forcesthat are also at work in settings with imperfect competition. Second, some
proponents of network neutrality regulation argue that market power lies at
the heart of the problem. Thus, this is a useful setting to examine in its own
right.
The platform charges a hookup fee of h to households and p(q) to an
application provider purchasing connection quality q. Under some public
policies, the platform can oﬀer diﬀerent connection qualities (e.g., an isp canoﬀer diﬀerent combinations of bandwidth, latency, and packet loss rate) to
diﬀerent application providers. As a convention, we use q = 0 to indicatethat an application provider has not connected to the platform.
5Interestingly, this point is implicit in Walras’s (1897) [1980] discussion of the diﬀerent
classes of travel on the French railway system in the middle of the 19th century (page 94).

Hermalin & Katzp(q)Application

4

The ModelhPlatform

Householdt

Figure 1: Fee-for-service model. Solid lines indicate service flows (e.g., Internet
connections). Dotted curves indicate payments.

Figures 1 and 2 illustrate two relevant market structures, which diﬀer interms of the economic relationship between application providers and households. In the ﬁrst market structure—the fee-for-service model—a householdpays a transaction fee t to the application provider for each use. In the second market structure—the advertiser-supported-content model—application
providers earn revenues solely from the sale of advertising and oﬀer theirapplications to households at no charge.
There is a continuum of potential application providers providers, which
we normalize to have unit mass. Each application provider has type, θ ∈[0, θ̄] ≡ Θ, where households value the content from high-θ providers morethan from low-θ providers ceteris paribus. Let F (·) denote the cumulative
distribution function for types, which has a continuous density f (·) such that0 < f (θ) < ∞ for all θ ∈ [0, θ̄].6 The last assumptions ensure that the inverse
hazard rate exists for all θ and is bounded away from 0 for all θ < θ̄. Eachapplication provider knows its type, while the platform provider knows only
the population distribution of application types.
Technology is such that the platform cannot provide quality of service
less than some minimum level q , where q > 0.7 Denote the platform’s choice¯¯
space for quality by
Q ≡ {0} ∪ {q|q ≥ q } .
¯The platform incurs a cost of c(q) in providing quality q. We assume c(·) is
at least twice diﬀerentiable, with c (·) > 0 and c (·) > 0. We also assume
that q c (q ) > c(q ). Given the convexity of c(·), this last assumption implies¯is increasing¯
that ¯c(q)in q (i.e., the average cost of quality rises with theq
67

In Appendix B, we discuss how our results can be extended to discrete type spaces.

Alternatively, the platform could provide lower-quality service but no application
provider would be willing to use that quality even if it were priced at cost.

Hermalin & KatzaAdvertisers

5

The Modelp(q)ApplicationhPlatform

Household

Figure 2: Advertiser-supported-content model. Solid lines indicate service flows(e.g., Internet connections). Dotted curves indicate payments.

quality level). Lastly, in keeping with our notational convention that q = 0
corresponds to non-consumption, c(0) = 0.
x)There is a unit mass of households with identical preferences. Let u( θqthdenote the marginal utility derived from consumption of the x unit of a
type-θ application with connection quality q. We assume that u(·) is continuous and decreasing (i.e., there is diminishing marginal utility from consuming
the content of any speciﬁc provider) with limx→∞ u(x)x = 0.
A household’s utility from connecting to the platform and consuming theavailable applications is quasi-linear and given by  x(θ) zU≡uf (θ)dzdθ + y ,
θq(θ)
Θ 0
where q(θ) is the connection quality chosen by a type-θ application provider,x(θ) is the household’s consumption of each type-θ application, and y is thenumber of units of the numeraire good consumed.
The quantity a household purchases from a given application provider atprice t is the solution to

 x 
z
u
dz − tx .
max
x
θq(θ)
0
The ﬁrst-order condition, which is suﬃcient as well as necessary because u(·)
is decreasing, isxu− t = 0.
θq(θ)
Because u(·) is monotonic, it is invertible. Hence, an application provider
faces demandθqd(t)

Hermalin & Katz

The Model

6

from each household subscribed to the platform, where d(t) ≡ u−1 (t).
The marginal cost of providing an application is a constant, k ≥ 0, common to all application providers. To ensure that trade is socially desirable,
assume u(0) > k. In the fee-for-service case, we assume that any one application provider generates a suﬃciently small proportion of household beneﬁts
that it ignores the eﬀect of its pricing on households’ network connection
decisions.8 Conditional on its choice of quality, each application provider
chooses its price to solve
max θq(t − k)d(t) .
t(1)

Observe that θ and q are irrelevant to the solution of this maximization
problem. For expositional convenience, we assume that u(·) gives rise toa unique proﬁt-maximizing application price, t∗ .9 Observe that the proﬁtmaximizing price is common to all content providers.
In the advertising-supported-content model, let a > k be the amount paidby advertisers to the application provider per unit of demand.10 The proﬁts
of an application provider conditional on q areθq(a − k)d(0) .

(2)

Observe that, under either business model, an application provider’s profits (gross of fees paid the platform) have the form θqρ, whereρ = (t∗ − k)d(t∗ )under the fee-for-service business model andρ = (a − k)d(0)8Were this not the case, the platform would be able to engage in a price squeeze,
whereby application providers would be forced to price at cost in order to induce households to connect to the network.
An equivalent assumption to the one made in the text would be to have the content
providers set their prices before the platform; this timing, however, seems less realistic
than the one used in the text.
9

A suﬃcient condition for this would be that xu (x) be decreasing in x; this conditionentails that d(t) is log-concave, which is suﬃcient for (1) to have a unique solution.
10

One can also think of a as advertisers’ expected payment to the content provider given
the click-through rate.

Hermalin & Katz

The Model

7

under the advertiser-supported-content business model.11 Henceforth, wewill express the gross proﬁts of an application provider of type θ that haspurchased connection quality q as θqρ.
The consumer surplus that a household derives from consumption of an
application provider’s service isθqs(t) ,∞

where s(t) ≡ t d(z)dz. The surplus a household derives from consumptionof a type-θ application whose provider has purchased a connection of quality
q can compactly be expressed as θqσ, where σ = s(t∗ ) in the fee-for-service
model and σ = s(0) in the advertising model.
To rule out the uninteresting scenario in which q is the only eﬃcient¯ ). Given our earlier
level of quality to oﬀer, assume that (ρ + σ)θ̄ > c (qassumption that the marginal cost of quality exceeds¯ the average cost at
the lowest feasible quality level, this inequality also implies that some tradeacross the platform is always socially desirable (i.e., (ρ + σ)θ̄q > c(q )).
¯¯
Absent any ﬁxed costs, welfare (i.e., total surplus) is





(ρ + σ)θq(θ) − c q(θ) f (θ)dθ ,
(3)
W =
Θ

where q(θ) is the connection quality supplied to type-θ applications.12 Let
qw (·) denote the quality-consumption schedule that maximizes welfare.
Lemma 1 Welfare is maximized by: (i) excluding a type-θ application if
the cost of the lowest quality service exceeds the gross beneﬁt the application
generates (i.e., c(q ) > (ρ + σ)θq ); and (ii) otherwise providing access of¯ qw (θ) is the solution¯
toquality qw (θ), wheremax(ρ + σ)θq − c(q) .
q≥q¯11There can also be a hybrid business model in which an application collects revenues
from both households and advertisers. If one assumes that advertising does not aﬀect
households’ willingness to pay for an application, the hybrid model is equivalent to thefee-for-service model with a cost of k − a.
12

Observe that this expression does not account for the welfare of advertisers. One
can interpret the model as assuming that advertising is provided in a broader, perfectlycompetitive market, such that changes in the quantity of online advertising do not giverise to changes in advertiser welfare. Alternatively, some of the results below generalize tosituations in which increases in advertising correspond to increases in advertiser welfare.
See, for example, Proposition 4 below.

Hermalin & Katz

The Model

8

Under the ﬁrst-best outcome, a positive measure of application types are excluded and a positive measure of types are served.
The proof of Lemma 1 establishes the existence of a marginal type, θw ,¯such that this and all higher types of application provider operate and lower
types shut down. Because the marginal contribution of quality to welfare isincreasing in type, it follows that θ > θ implies qw (θ) ≥ qw (θ ), with theinequality being strict if qw (θ) > q . Recalling our notational convention, set¯ that do not operate.
qw (θ) = 0 for those application types
Now consider the equilibrium behavior of a proﬁt-maximizing platform
provider. The timing of play is as follows.
• The platform sets its price schedule to application providers, p(·), andhookup fee to households, h.
• Application providers simultaneously choose their connection qualities,q(θ), and, in the fee-for-service model, their prices to households.
• Households observe the hookup fee, connection qualities, and, in thefee-for-service model, application prices. Households then choose whether
to connect to the network and their consumption levels of the various
applications.
We assume that the platform quotes its prices to application providers on
a per-household basis. That is, the platform commits to a schedule p(q) thatis prorated by the number of end-users it attracts, so that an application
provider that selects quality q pays the platform provider λp(q), where λ isthe proportion of households that subscribe to the platform.
We make this assumption as a means of approximating the fact that,in reality, platforms attract new applications and households over time andthat a platform may adjust its price over time as well.13 In our one-shot(albeit multi-stage) model, having the platform set access prices for application providers that were independent of the eventual number of household
subscribers could lead to widely varying equilibrium outcomes as the resultof bootstrapping on diﬀerent expectations. For example, as is well known,
the two-sided nature of the market makes the analysis vulnerable to a degenerate equilibrium in which applications choose not to connect to the platform
13

For an insightful model of adoption dynamics in a two-sided market, see Caillaud andJullien (2003).

Hermalin & Katz

Effects of a Single-Quality Restriction

9

because they anticipate that households will not connect, and—given thatno applications are connected—households indeed choose not to connect.
This degenerate outcome is not an equilibrium in our model; because theplatform’s charge is prorated, it is a weakly dominant strategy for a type-θ
application provider to sign with the platform as long as
max ρθq − p(q) ≥ 0 .

q∈Q\{0}In equilibrium, the surplus realized by a household from all applications—and hence the household’s value of subscribing to the platform—isθq(θ)f (θ)dθ .
(4)σΘ

Because households are identical, the platform can capture their entire grosssurplus by setting the hookup fee, h, to equal (4). Recalling that there is aunit mass of households, h is also the platform’s revenues from householdsusers. The platform’s proﬁts are

 


π =h+
p(θ) − c q(θ) f (θ)dθ
Θ

 


σθq(θ) + p(θ) − c q(θ) f (θ)dθ .
(5)
=
Θ

3

The Eﬀects of a Single-Quality Restriction

In this section, we compare equilibrium with and without a single-quality
restriction.

3.1

The Unrestricted Monopoly Equilibrium

We ﬁrst characterize the equilibrium when a proﬁt-maximizing monopoly
platform is not subject to legal restrictions on the range of qualities oﬀered.
Standard analysis (e.g., Mussa and Rosen, 1978; Caillaud and Hermalin,
2000) demonstrates that the proﬁt-maximizing, incentive-compatible, individually rational price schedule to oﬀer is θρq(τ )dτ .
(6)p q(θ) = ρθq(θ) −0Hermalin & Katz

Effects of a Single-Quality Restriction

Consequently, the provider’s problem is θ θ̄ 
max(ρ + σ)θq(θ) −ρq(z)dz − c q(θ) f (θ)dθq(θ)∈Q

0

10

(7)

0

(8)
subject to q ≤ q(θ ) ≤ q(θ) ∀θ, θ such that θ < θ ,¯where (7) follows from (5) and (6), and (8) is necessary and suﬃcient for the
resulting quality schedule to be incentive compatible. Integration by parts
allows us to rewrite (7) as θ̄ 
max(ρ + σ)θq(θ) − m(θ)ρq(θ) − c q(θ) f (θ)dθ ,q(θ)∈Q

(7 )

0

where
m(θ) ≡

1 − F (θ)
f (θ)

is the inverse of the hazard rate.14 Let qu (·) denote the solution to (7).
As we will show shortly, there exists a lowest type of application provider,θu > θw , that connects (i.e., qu (θ) = 0 for all θ < θu ). Equation (6) implies¯p (θ )¯ = ρθ q (θ ); as is well known, the lowest type¯ served enjoys no surplus.
u u
u u u
¯The marginal¯¯ contribution to proﬁts of an increase in q(θ) is proportional
to


(9)
(ρ + σ)θ − ρm(θ) − c q(θ) .
The presence of m(θ) captures the fact that increasing the quality level oﬀered
to type-θ application providers increases the information rents that have to begiven to all higher types to keep them from purchasing the quality intended
for type-θ providers.
Because the quality allocated to the top type cannot aﬀect the information
rent received by any other type (m(θ̄) = 0), the monopolist has no incentive
to distort the quality allocated the top type away from its eﬃcient level; as
is well known, qu (θ̄) = qw (θ̄).
Next, consider application types at the bottom of the distribution. Consider the eﬀects of lowering the lowest application type served by an inﬁnitesimal amount. The revenues collected from the incremental application
providers and through the increase in the hookup fee charged to households
14

We use m as a mnemonic because the inverse hazard rate is also known as the Mills
ratio.

Hermalin & Katz

Effects of a Single-Quality Restriction

11

are just equal to the social beneﬁts. However, the information rents of higher
application types rise. Hence, the platform’s incremental proﬁts are less than
the social beneﬁts. Combined with Lemma 1, this logic establishes:
Lemma 2 The unrestricted platform excludes more application provider
types than is socially optimal. Speciﬁcally, there exists an application type,θu , such that θw < θu < θ̄ and qu (θ) = 0 for all θ < θu .
¯¯¯¯Total surplus under the unrestricted equilibrium is θ̄ (ρ + σ)θqu (θ) − c qu (θ) f (θ)dθ .
Wu ≡

(10)

θu

¯

3.2

The Restricted Monopoly Equilibrium

Now suppose the platform is restricted to oﬀering only a single level of quality,
and let qr denote the monopolist’s choice.
Clearly, it would be suboptimal for the monopolist to charge a price, pr ,such that pr > ρθ̄qr or pr ≤ 0. It also could never maximize proﬁts to oﬀer aprice and quality such that all types of application providers earned positive
surplus. Instead, the monopolist chooses a price and quality such that there
is a marginal application provider type just indiﬀerent between connecting
and not. Rather than view the monopolist’s problem as one of choosing
an optimal quality and price, we can view it as one of choosing an optimal
cutoﬀ type and quality. Note the cutoﬀ type satisﬁes θ = pρqr . Writing theplatform’s problem in this manner, we have
 θ̄
max

{θ∈Θ,q∈Q}θρθq + στ q − c(q) f (τ )dτ .

Maximization with respect to q is equivalent to
max ρθ + σΛ(θ) q − c(q) ,
q∈Q

(11)

(12)

where Λ(θ) is the expected value of a content provider’s type conditionalon that type not being less than θ. Let θr be the marginal type (i.e., the¯
solution to (11)).

Hermalin & Katz

Effects of a Single-Quality Restriction

12

Proposition 1 A proﬁt-maximizing platform restricted to oﬀering a single
product quality excludes more application providers than would a welfare maximizing platform (i.e., θw < θr ). Moreover, the proﬁt-maximizing platform¯ is¯lower than the one that would maximize totalchooses a quality level thatsurplus conditional on the types of application providers that connect to theplatform.
For the case of traditional, one-sided markets, Spence (1975) showed thatthe fundamental source of distortions in a single-product ﬁrm’s choice ofquality is that prices and proﬁts are driven by the marginal buyer’s valuationof quality, while welfare depends on the average buyer’s valuation of quality.
In a two-sided market, one must consider the eﬀects of an incremental quality
increase on the willingness of both sides to pay. Because all households are
identical in our model, the marginal and average valuations of quality are thesame. The distortion arises on the application provider side of the market—
the marginal application provider values increased connection quality by lessthan do the inframarginal application providers.

3.3

Comparison of the Equilibria

We now compare the equilibrium in which the monopolist is restricted toa single product to the equilibrium in which the platform is free to oﬀer afull product line. One consequence of imposing a single-product restrictionis that low-value application providers can get priced out of the market.
Lemma 3 Suppose that at least one of the following conditions holds:(i) There is a unique solution to the restricted platform’s problem;(ii) There is a unique solution to the unrestricted platform’s problem;(iii) If there are multiple solutions to the unrestricted platform’s problem,
then the ﬁrm chooses the one that maximizes the set of application types
served (i.e., that minimizes θu );¯(iv) It is not a solution to the restricted platform’s problem to set quality
equal to the minimum feasible quality (i.e., qr > q );¯(v) The hazard rate associated with the distribution over application types
is everywhere non-decreasing.

Hermalin & Katz

Effects of a Single-Quality Restriction

13

Then a single-product restriction weakly reduces the set of application types
oﬀered to households in equilibrium (i.e., θr ≥ θu ).15¯¯
Contrary to the wishes of network neutrality proponents, regulation may
move the equilibrium outcome closer to “a former Soviet Union supermarket”
rather than away from it.
Lemma 3 leaves open the possibility that, if there are multiple equilibria
absent the single-product restriction and there are multiple equilibria withthe single-product restriction, including one in which the platform provider
oﬀers only the lowest technically feasible quality, and the hazard rate isdecreasing on at least one interval, then the lowest application type served insome of the restricted equilibria may be lower than the lowest type served insome—but not all—of the unrestricted equilibria. Given the highly restrictive
set of conditions this case would have to satisfy, we strongly doubt that thispossibility is of empirical importance.
Lemma 3 also leaves open the possibility that the lowest application typeserved would be unaﬀected by the single-product restriction. As we now
show, this can happen only if the restricted platform would maximize its
proﬁts by oﬀering the lowest technologically feasible quality. This seems anunlikely scenario in many markets and is certainly not the scenario envisioned
by proponents of network neutrality, who worry that applications with low
types will receive worse service (“the Internet dirt road”) absent network
neutrality.
Proposition 2 If the restricted platform does not oﬀer the product withthe lowest technically feasible quality level, then a single-product restriction
strictly reduces the set of applications oﬀered to households in comparison
with the unrestricted equilibrium.
The intuition underlying this result is the following. When the unrestricted platform connects additional application types at the bottom of themarket, it must lower the prices charged to higher-type application providers,
but it does not have to distort higher types’ quality levels further. Whenthe restricted platform serves additional application types at the bottom of
the market, it does so by lowering both the price and the quality level consumed by all types connecting to the platform. The change in quality is a15This proposition also holds for a discrete type space.

Hermalin & Katz

Effects of a Single-Quality Restriction

14

costly distortion that aﬀects the revenues collected from both applications
providers and households.
Restricting the platform to oﬀering a single quality level results in some
applications’ purchasing lower quality connections than they would have purchased if platform were unrestricted. One response could be to require theplatform to oﬀer only the highest level oﬀered prior to imposition of regulation.16 Recall that the highest quality product oﬀered by the unrestricted
platform is the quality that is eﬃcient for the most valuable application,
qw (θ̄).
Proposition 3 Consider a policy that requires the platform to oﬀer only thehighest quality variant that would be oﬀered absent regulation. In equilibrium,
strictly fewer application providers connect to the platform than under eitherof the following policies: (a) the platform is allowed to oﬀer at most one
quality level but can choose any technologically feasible quality level, and (b)the platform is free to oﬀer a full range of qualities.
In other words, a policy that mandates the highest service quality reduces
the set of applications available to households.
Now, return to consideration of a single-product restriction that grants
the platform the freedom to choose the quality. As Proposition 2 shows, alikely consequence of this restriction would be greater exclusion of low-value
applications in comparison with the unrestricted equilibrium. Consider theapplications that are excluded under the single-product restriction, but thatwould have been provided under the unrestricted equilibrium. These applications, with the possible exception of the lowest type, would generateprovider proﬁts plus household surplus strictly greater than the platform’s
costs of transmitting the content. Hence, the increased exclusion is a reduction in total surplus ceteris paribus. We refer to this as the exclusion eﬀectof a single-product restriction.
There are two other eﬀects of a single-product restriction. One is thatthe connection quality for the highest-type application falls from qw (θ̄) to qr .
It follows that there is a positive measure of application types that enjoy lesseﬃcient quality in the sense that(ρ + σ)θqr − c(qr ) < (ρ + σ)θqu (θ) − c qu (θ) .
16

We are assuming here that regulation is unanticipated, so that the service provider did
not strategically reduce the highest quality oﬀered prior to the imposition of regulation.

Hermalin & Katz

Effects of a Single-Quality Restriction

15

We refer to this reduction in welfare as the reduced-quality eﬀect of a singleproduct restriction.
On the other hand, the connection quality for some application types
is greater under the single-product restriction than it would be in the unrestricted equilibrium. In Appendix A (Lemma A.1), we show that there
is one type, θ̂ ∈ (θr , θ̄), such that qr = qw (θ̂). Because the unrestrictedequilibrium entails ¯a downward distortion in quality for all types except the
top one, qu (θ̂) < qw (θ̂). By continuity, there must be a positive measure oftypes for which quality is more eﬃcient under network neutrality than in theunrestricted equilibrium. That is,(ρ + σ)θqr − c(qr ) > (ρ + σ)θqu (θ) − c qu (θ)
for a positive measure of types. We refer to this welfare beneﬁt as the
improved-quality eﬀect of a single-product restriction.
In Appendix B, we show by example that the net welfare eﬀects of asingle-product restriction can be positive or negative.17 Despite the overall
ambiguity of the welfare consequence of imposing a product-line restriction,
we can derive a condition that must necessarily hold if the restriction is toincrease welfare (correspondingly, the negation of that condition is a suﬃcientcondition for the unrestricted equilibrium to yield greater welfare). Recall
that qu (θ) = 0 for θ < θu . Deﬁne qr (θ) = qr if θ ≥ θr and = 0 if θ < θr .
¯¯¯Proposition 4 Suppose the connection quality oﬀered when the platform issubject to a single-product restriction exceeds the lowest technically feasible
quality level (i.e., qr > q). If qr (·) dominates qu (·) in the sense of second-orderstochastic dominance, then the single-product restriction lowers welfare.18,19
The condition identiﬁed in the previous result provides the basis of intuition. Because the single-product restriction results in greater exclusion of17
Some of these examples assume a discrete type space. This is done to simplify the
exposition. A more general analysis of the discrete-type case is presented in Appendix B.
18
 xRecall that qrx(·) dominates qu (·) in the sense of second-order stochastic dominance
if 0 qr (τ )dτ ≤ 0 qu (τ )dτ for all x in the common domain. Because neither qr (·) norqu (·) is a distribution, it would arguably be more appropriate to describe the condition interms of weak majorization (Pečarić et al., 1992, §12.1). We use the language of stochastic
dominance because it is more common in economics.
19

This result can readily be extended to allow for advertiser welfare. Speciﬁcally, as longas an advertiser’s welfare is increasing in the volume of advertising, Propositions 2 and 4
imply welfare, including advertisers’ welfare, is reduced by a single-product restriction.

Hermalin & Katz

Extensions

16

applications, it follows that θ θqu (τ )dτ >qr (τ )dτ = 0
00for θ ∈ (θu , θr ]. Because qu (θ̄) = qw (θ̄) > qr , it follows that, if it is not to be¯
the case ¯that


θ

0

qu (τ )dτ ≥θ

0

qr (τ )dτfor all θ, then qr must be considerably larger than qu (θr ). In other words, for¯
the single-product restriction to raise welfare, the marginal
application typeserved under the restricted equilibrium must obtain a connection of muchhigher quality than the the connection it would obtain absent the restriction.
We next make an observation about the political economy of singleproduct restrictions, such as those advocated by some proponents of network
neutrality regulation. We show by example in Appendix B (Example 2) that
there exist situations in which application providers favor network neutrality, the platform (e.g., an ISP) opposes it, and imposition of the restriction
would reduce overall welfare. It should be emphasized, however, that there
are also examples in which a single-product restriction reduces application
providers’ aggregate surplus (Appendix B, Example 3).

4

Extensions

We next extend the basic model to consider alternative cost assumptions,
technological restrictions on quality choice, and competitive platforms.

4.1

Alternative Cost Assumptions

In this subsection, we brieﬂy discuss three extensions with respect to ourmodeling of costs. First, our model assumes that there are no productspeciﬁc ﬁxed costs. In the presence of ﬁxed costs, variety is costly as wellas potentially beneﬁcial. For one-sided markets, Katz (1980) has shown
that, in the presence of such ﬁxed costs, a proﬁt-maximizing monopolist may
oﬀer more or fewer than the total-surplus-maximizing number of products(quality levels). Depending on the structure of household and application
provider preferences, similar eﬀects arise in a two-sided market. Thus, inHermalin & Katz

Extensions

17

some cases, restricting the number of products oﬀered by the monopolist
would exacerbate the distortion, and in other cases, it would ameliorate it.
Second, we have assumed that there are no economies of scale in production. Hence, marginal-cost pricing would allow a platform to cover its costs.
The presence of economies of scale can give rise to an additional social beneﬁtof allowing the platform to oﬀer a variety of qualities: a multi-product ﬁrm
has a greater ability to cover its costs. Moreover, in a model that requireda large, ﬁxed investment in facilities, an unrestricted platform would havegreater investment incentives than would a restricted platform.
Third, we have assumed that unit costs rise with quality. Some participants in the network neutrality debate have argued that increased quality
is essentially costless, at least up to some point. We doubt the empiricalvalidity of this claim, but it is nonetheless of interest to examine the case inwhich c(q) ≡ c for all q not exceeding some maximum possible quantity, q̄.
Observe that, when c(q) ≡ c, there is no social beneﬁt of variety inthe following sense. Under the ﬁrst-best outcome, all content that is worthtransmitting (i.e., such that (ρ + σ)θ ≥ c) is transmitted at the highest
possible quality, q̄. Now, consider the proﬁt-maximizing outcomes. Recall
that the marginal contribution to the unrestricted platform’s
proﬁts of anincrease in q(θ) is proportional to (ρ+σ)θ−ρm(θ)−c q(θ) (see equation (9)above). It can be seen by inspection that the platform’s problem of picking
a quality has a bang-bang solution when c (θ) = 0: a type-θ application iseither connected at quality q̄ (if (ρ + σ)θ > ρm(θ)) or is excluded from themarket (if (ρ + σ)θ < ρm(θ)). Given that the platform oﬀers only a single
quality level absent any restriction, imposing a net-neutrality restriction has
no eﬀect on the equilibrium outcome.
Summarizing this analysis,
Proposition 5 Suppose c(q) ≡ c for all q ∈ [q , q̄]. Then a single-productrestriction has no eﬀect on the set of equilibrium¯ outcomes.
It should be noted that this result depends on the functional form we haveassumed for user beneﬁts. Speciﬁcally, let v(q, θ) denote the gross beneﬁts
enjoyed by a type-θ application provider that connects at quality q. The
standard screening condition requires that vqθ (q, θ) > 0 (where subscripts
here indicate partial derivatives). Our functional form imposes the stronger
condition that vqθ (q, θ) is a positive constant. When this cross partial canvary with q, there are cases in which the unrestricted platform would oﬀerHermalin & Katz

Extensions

18

quality variety even though there is no marginal cost of higher quality. Although there is no social value to variety, the platform would oﬀer a productline as a screening device (i.e., as a form of second-degree price discrimination). Relative to the ﬁrst best, this is a privately proﬁtable action that issocially wasteful (some content is transmitted via an ineﬃciently low quality
connection).20
Even with a more general v(q, θ) function, a platform restricted to oﬀeringa single quality would oﬀer the eﬃcient quality level, q̄. It does not follow,
however, that the restriction would raise welfare. One would also have tocheck whether the restricted platform would serve more or fewer application
providers than would the unrestricted platform. In the context of a one-sided
market with two possible quality levels, for example, Deneckere and McAfee(1996) show that this type of price discrimination can result in a Pareto
improvement in consumer welfare.
The welfare eﬀects in a general model of a two-sided market remain aquestion for future research. Observe that—because the connection quality
provided under a single-product restriction is eﬃcient, but those utilized
when a variety of qualities are oﬀered are not—a suﬃcient condition for the
restriction to raise welfare is that it not reduce the range of applications
oﬀered to households in equilibrium. The contrapositive of this result is thata necessary condition for the platform’s oﬀering a variety of qualities to raise
total surplus is that it lead to more applications’ being oﬀered.
This ﬁnding parallels the well-known result that a necessary condition
for third-degree price discrimination to raise welfare is that total output
rise under discrimination.21 There are, however, important diﬀerences between the eﬀects of quality variety (or second-degree price discrimination)and third-degree discrimination. First, a simple measure of total applications lacks economic meaning when applications are heterogeneous and aredelivered via connections of varying qualities. Second, the distributions of
the welfare eﬀects across market segments are somewhat diﬀerent. In thecase of third-degree price discrimination, discrimination lowers eﬃciency inhigh-value markets (prices rise), raises eﬃciency in low-value markets (pricesfall), and has ambiguous eﬀects in middle-value markets. In contrast, “discrimination” in the form of a product line increases eﬃciency at both the20

In the presence of economies of scale, however, second-best (Ramsey) pricing could
entail the use of a product line even when quality were otherwise free.
21

See, e.g., Varian (1985) and references therein.

Hermalin & Katz

Extensions

19

high and low ends of the market, and eﬃciency losses occur in the middle.

4.2

Technologically Restricted Quality levels

Heretofore, we have allowed the producer to choose the quality level from acontinuum of possibilities. In this section, we assume that there are only two
technologically feasible quality levels, high (h) and low ( ). This assumption is motivated by the fact that often quality levels (e.g., bandwidth andlatency) are limited and, in part, determined by forces beyond a platform’s
control (e.g., capabilities of routers and computers or the installed base ofcomplementary products).22
Let c > 0 be the cost of providing a unit of the high-quality good. For
convenience, we set the cost of providing a unit of the low-quality good to 0.
So that there is a welfare beneﬁt to high quality, we assume (ρ + σ)θ̄h − c >(ρ + σ)θ̄ . Observe that there is also a welfare beneﬁt from oﬀering the lowquality product (i.e., (ρ + σ)θh − c < (ρ + σ)θ for θ suﬃciently close tozero).
At prices ph and p , a type-θ content provider is indiﬀerent between thetwo qualities if and only ifρhθ − ph = ρ θ − p .
Solving this expression for θ and noting that the left-hand side increases
faster in θ than the right-hand side, we can conclude that all types such thatθ≥ph − pρ(h − )

prefer high to low quality. If the ratio on the right-hand side exceeds θ̄, thenthere is no demand for the high-quality product.
An application provider prefers low quality to not connecting at all if andonly ifρ θ − p ≥ 0 .
Provided22ph − fp>,ρ(h − )
ρ(13)

In practice, a continuum of quality levels may be feasible, but it may be eﬃcient tooﬀer only a ﬁnite, discrete set.

Hermalin & Katz

we have

20

Extensions⎧⎪
no connection , if θ < pρ⎪⎨
ph −p
quality , if pρ ≤ θ < ρ(h−).
demand of type θ =⎪⎪⎩ h quality , if ph −p ≤ θ(14)ρ(h−)

We ﬁrst characterize the unrestricted equilibrium.
Lemma 4 Suppose only two quality levels are technologically feasible andthe distribution of application types has a non-decreasing hazard rate. An
unregulated platform will oﬀer both qualities and set prices to application
providers deﬁned by
 
ρ
p
p =
ρ m
ρ+σ
ρ
and

ρ
ρ
c+
ρ(h − )mph = p +
ρ+σ
ρ+σ



ph − pρ(h − )

.

(15)

It follows from this result that the marginal application types purchasing
low- and high-quality connections satisfy
θ =
and

ρ
θh =
ρ+σ



ρ
m(θ )
ρ+σ

(16)


c
+ m(θh ) ,ρ(h − )

(17)

respectively.
Welfare absent regulation is
 θh
θ

 θ̄
(ρ + σ) θf (θ)dθ +θh(ρ + σ)θh − c f (θ)dθ .

(18)

The derivative of this expression with respect to θh isc − (ρ + σ)θh (h − ) f (θh ) .
By (17), this last expression is equal to −ρm(θh )(h− )f (θh ) < 0. Hence, welfare would rise if the platform lowered θh while holding θ constant. In otherHermalin & Katz

21

Extensions

words, conditional on the set of active application providers, the platform
transmits too little content via the high-quality connection.
Now, suppose that regulation forces the platform to oﬀer at most oneof the two possible qualities. A restricted platform chooses price, pr , tomaximize θ̄ pr  
pr − c(qr ) +σqr θf (θ)dθ ,
(19)
1−F
pr
ρqr
ρq
r

where c(h) = c and c( ) = 0.
Lemma 5 Suppose that only two quality levels are technologically feasible
and the distribution of application types has a non-decreasing hazard rate.
If the monopolist oﬀers a single product of quality qr , then it maximizes its
proﬁts by charging a price, pr , that satisﬁes pr 
ρ
.
(20)
pr =
c(qr ) + ρqr m
ρ+σρqr
The proof is similar to that of Lemma 4 and is omitted.
It follows from this result that
ρc(qr )+ m(θr ) .
θr =ρ + σ ρqr¯¯
When qr = , this expression is identical to (16). Thus, if regulation induces
the platform to provide only low-quality service, then the restriction has noeﬀect on the set of applications served in equilibrium. However, the restriction reduces the amount of content transmitted via high-quality connections
to zero from a level that itself would have been too low absent the restriction.
We can thus conclude:
Proposition 6 Suppose that only two quality levels are technologically feasible and the distribution of application types has a non-decreasing hazard rate.
If the platform chooses low quality service under a single-product restriction,then that restriction lowers total surplus.
Lastly, suppose that the restricted platform oﬀers only high-quality service. We have found numerous examples (e.g., application types are uniformly distributed) in which this restriction lowers total surplus relative tothe unrestricted equilibrium, but we have been unable to construct an example in which it raises total surplus. That said, we have also been unable
to construct a proof that such an example does not exist.

Hermalin & Katz

4.3

Extensions

22

Oligopoly

We now examine a market in which two platforms (e.g., a local telephonecompany oﬀering dsl service and a cable company oﬀering cable modem
service) compete to serve households and applications providers.
We model the platforms as competing à la Hotelling. Variants of theHotelling model have been used by several previous authors to examine competition between single-product platforms.23 Examples include Anderson and
Coate (2005), Armstrong (in press), Gabszewicz et al. (2002), and Rochet
and Tirole (2003).24
There are two platforms, located on the unit interval. Platform i is locatedat point i on the line, where i = 0, 1. The timing of play in the duopoly gameis as follows.
• The platforms simultaneously set their price schedules to application
providers, pi (·), and hookup fees to households, hi , i = 0, 1. As in
the monopoly model, pi (·) is the price per household that platform icharges an application provider for access to the platform’s household
subscriber base.
• Application providers simultaneously choose their connection qualities
on the two platforms, qi (θ), and, in the fee-for-service model, theirprices to households.
• Households observe the hookup fees, connection qualities, and, in thefee-for-service model, application prices. Households then choose, for
23

There have also been many insightful papers examining quality competition in onesided markets, with several focusing on the issue of whether suppliers will choose to compete head to head or oﬀer diﬀerent products than one another to avoid direct competition.
For single-product suppliers, Hotelling (1929) found that minimum diﬀerentiation would
result as each ﬁrm staked out the middle of the famous Hotelling line. Shaked and Sutton (1982), however, analyzed a multi-stage game in which single-product ﬁrms choose
to diﬀerentiate themselves in the ﬁrst stage (i.e., choose product locations that are notnear one another) in order to relax second-stage price competition. Brander and Eaton(1984), Champsaur and Rochet (1989), and others have extended this analysis to multiproduct ﬁrms. However, DeFraja (1996) ﬁnds that suppliers oﬀer identical product lines ina single-stage game with competition in quantities and vertically diﬀerentiated products.
As we will demonstrate below, the platforms in our model compete head to head in any
equilibrium in which households do not multi-home.
24

For a survey of research on two-sided markets, see Rochet and Tirole (in press).

Hermalin & Katz

23

Extensions

each platform, whether to connect to it and their consumption levelsof the various applications.
We assume households are uniformly distributed along the Hotelling line.
A household incurs “transportation costs” equal to T times the distance
between the household and a platform to which it subscribes. By analogy toequation (4), the surplus gross of transportation costs derived by a householdfrom consuming the applications available on platform i isθqi (θ)f (θ)dθ − hi .
Si ≡ σΘA household at location L prefers platform 0 to platform 1 if and only if
S0 − T L > S1 − T (1 − L) .

(21)

If the right-hand side of (21) is positive, then location L is a contested household. More generally if there are households for whom either platform would
provide positive net surplus (including transportation cost), we say there arecontested households. If each household can obtain positive surplus from atmost one platform, we say there are no contested households.25
Up to this point, we ignored the cost incurred by the platform to connect
a household because that cost was irrelevant provided that it was not sogreat that hooking-up a household to the network would yield negative surplus. The hookup cost, denoted K, is now relevant because it aﬀects whether
households wish to single-home (connect to at most one platform) or multihome (connect to both platforms). For the time-being, we will analyze themarket under the assumption that household multi-homing is infeasible. Below, however, we will establish a condition on K such that no household
multi-homes in equilibrium even when it is otherwise feasible to do so.
Recall that platforms’ prices to application providers are quoted on aprorated basis; hence, when households single-home, it is a weakly dominant strategy for a type-θ application provider to sign with any platform forwhich26
max ρθq − pi (q) ≥ 0 ,
q∈Qi

25

We use the contested/not contested distinction rather than the covered/uncovered
distinction common to the literature because, for a range of parameter values, a market
can be covered—all customers are served—without either ﬁrm’s setting a price that would
attract its rival’s customers if the rival were to raise its price.
26

We do not allow a platform to require applications to sign exclusive-dealing contracts.
To date, isps have not demanded exclusivity for most applications.

Hermalin & Katz

24

Extensions

where Qi is the range of qualities oﬀered by platform i.
Let Di (S0 , S1 ) denote the number of households that prefer platform iwhen platforms 0 and 1 oﬀer surplus levels S0 and S1 , respectively. Observe
Di (S0 , S1 ) =Si,Tif no contested households

S −S
1+ i2T j , j = i ,2

if there exist contested households.

Platform provider i’s problem is to choose hi and qi (·) to maximize proﬁts.
One can equivalently think of a platform having a choice of both quality
schedule and household surplus level. Platform i’s proﬁts are
 






σθqi (θ) + pi qi (θ) − c qi (θ) f (θ)dθ − Si − K Di (S0 , S1 ) . (22)πi =ΘObserve that, conditional on the value of Si , a duopoly platform’s optimization problem with respect to qi (·) is identical to the monopolist’s (i.e.,
expression (7) in the unrestricted case and expression (11) in the restrictedcase). We have shown,
Lemma 6 Consider a Hotelling market in which household multi-homing is
infeasible. The quality schedule oﬀered by each duopoly platform in equilibrium is the same as the schedule oﬀered by the platform in the monopoly
model of Section 3.
This result is a manifestation of what has come to be known as the “terminating access problem.”27 Although there are competing platforms, anapplication provider has only one way to connect with a given householdthat has chosen a particular platform. There is a sense in which, from theapplication providers’ perspective, that platform has a monopoly over accessto that household. The result is also a consequence of the fact that the degreeof market power a platform has with respect to single-homing households isirrelevant to determining the nature of qi (·) and aﬀects only hi .
Deﬁne

 




M
σθqz (θ) + pz qz (θ) − c qz (θ) f (θ)dθ ,
Π ≡
Θ

27

For a regulatory body’s expression of concern with this problem, see Federal Communications Commission (2001), paragraphs 13 and 14. For an early analysis of the problem,see Doyle and Smith (1998).

Hermalin & Katz

25

Extensions

where z = u or r depending on regulatory regime. We can rewrite (22) as
πi = (ΠM − K − Si )Di (S0 , S1 ) .

(23)

Observe, from (23), that the market exists only if the hookup cost is lessthan the monopoly proﬁt per customer (i.e., ΠM ≥ K), a condition we havebeen assuming implicitly and, now, assume explicitly.
We can now establish the following.
Lemma 7 The Hotelling duopoly model in which household multi-homing is
infeasible has a unique equilibrium, which is symmetric. Each platform oﬀers
the same quality schedule and price schedule to the application providers aswould the monopolist in the model of Section 3 when subject to the same
regulatory regime. Moreover, hookup fees are as follows:(i) If T > ΠM − K ≥ 0, then

 




1
σθqz (θ) + c qz (θ) − pz qz (θ) + K f (θ)dθ .
h=2 Θ(ii) If ΠM − K ≥ T > 23 (ΠM − K) ≥ 0, then1h=σθqz (θ)f (θ)dθ − T .
2Θ(iii) If 23 (ΠM − K) ≥ T ≥ 0, then

 




h=T +K −pz qz (θ) − c qz (θ) f (θ)dθ .

(24)

(25)

(26)

Θ

Observe that we can view the platform’s marginal cost of signing up
a household as the hookup cost minus the proﬁts the platform earns fromapplication providers wishing to serve the household:

 




pz qz (θ) − c qz (θ) f (θ)dθ .
K−ΘIn this light, (24) is just the usual result that a monopolist facing linear
demand with price intercept P̄ charges price 12 (P̄ + C), where C is marginalcost. Similarly, (26) is the standard result for Hotelling models with contested

Hermalin & Katz

26

Extensions

consumers in which the equilibrium price is transportation cost plus marginalcost. Observe, too, from (26) that, as T ↓ 0, the equilibrium approaches the
Bertrand equilibrium in which ﬁrms price at marginal cost. Absent platform
diﬀerentiation, Bertrand competition forces the platforms to bid away the
proﬁts they make from the application providers in their eﬀorts to entice
households to sign with their platform.
We now return to the question of multi-homing. SupposeK ≥ max θ̄ 

q(θ)

0






p q(θ) − c q(θ) f (θ)dθ ,(27)

where q(·) is required to be technically feasible, incentive compatible, and
p q(θ) = ρθq(θ) − θ0ρq(τ )dτ .

The right-hand side of expression (27) is the maximum proﬁt an unrestricted monopoly platform can earn from trade with application providers
on a per-household basis. If expression (27) didn’t hold, then it would bepossible for a platform to charge households negative hookup fees and stillearn a positive proﬁt. As we see no evidence of platforms engaging in such extreme subsidization of households, we view (27) as a reasonable assumption.
Given this assumption we can establish the following result.
Lemma 8 Suppose that the platforms’ choices of product lines are unrestricted, condition (27) holds, and application providers believe that no household will multi-home regardless of the platforms’ play. Then the equilibrium
identiﬁed in Lemma 7 is also an equilibrium when household multi-homing isfeasible.
Next, suppose the public policy limits each platform to oﬀering a single
product quality.
Lemma 9 Suppose that each platform is restricted to oﬀering a single quality, application providers believe that no household will multi-home regardlessof the platforms’ play, and the following conditions hold: θrρθq + σθq − c(q) f (τ )dτ < K ,(28)max{θ∈Θ,θ≤θr ,q∈Q,q≤qr }θHermalin & Katz

andConclusion

 θ̄
max

{θ∈Θ,θ≥θr ,q∈Q,q≥qr }θρθq + σθ{q − qr } − c(q) f (τ )dτ < K .

27

(29)

Then the equilibrium identiﬁed in Lemma 7 is also an equilibrium whenhousehold multi-homing is feasible.
The two inequalities stated in the lemma guarantee that there is no quality
level and set of application providers served such that the incremental surplus
created by multi-homing is greater than or equal to the cost of a household’s
second network connection. Hence, there is no proﬁtable deviation that will
induce multi-homing.
Because welfare is determined by the quality schedule, Lemmas 6 through9 establish conditions under which all of our earlier welfare results apply tothe case of competing platforms:
Proposition 7 Suppose that conditions (27) through (29) hold and application providers believe that no household will multi-home regardless of theplatforms’ play. Then the welfare consequences of imposing a single-productrestriction on Hotelling duopolists are identical to the consequences of applying such a restriction to a monopoly platform.
We close our discussion of platform competition and product-line restrictions by noting that, in some circumstances when the conditions of Lemma 9
are not satisﬁed, households may multi-home in equilibrium. Household
multi-homing can arise because a single-product restriction creates an artiﬁcial value of platform variety. It is artiﬁcial because—absent regulation—asingle platform would have the ability to oﬀer a full range of qualities. We
observe that, in addition to ineﬃciently doubling expenditures on network
connections when it induces household multi-homing, a public policy thatforces any given ﬁrm to oﬀer at most one quality level may not result in all
households or applications consuming the same quality of service—suppliers
can collectively oﬀer a range of products even if each platform oﬀers onlyone.

5

Conclusion

We have formally modeled the eﬀects of product-line restrictions such as
those sought by some proponents of network neutrality regulation. We exam-

Hermalin & Katz

Conclusion

28

ined a two-sided market for platform services in which application providers
on one side of the market purchase platform services of varying qualities thatallow the applications to be provided to the households that subscribe to therelevant platform.
For both a monopoly platform and Hotelling duopolists, we found thata single-product restriction results in: (a) application providers that would
otherwise have purchased a low-quality variant being excluded from the market; (b) applications “in the middle” of the market purchasing a higher andmore eﬃcient quality; and (c) applications at the top of the market purchasing a lower and less eﬃcient quality. We ﬁnd that the net welfare eﬀects
can be positive or negative, although the analysis suggests to us that harmis the more likely outcome. Moreover, applications at the bottom of themarket—the ones that a single-product restriction is typically intended toaid—are almost always harmed by the restriction, and consumers have fewer
applications available to them as a consequence.28
In summary, although our speciﬁc ﬁndings must be regarded as tentative given the exploratory nature of our model, we believe that our analysis
urges caution. A rigorous case for network neutrality regulation entailingproduct-line restictions has not been made, and there are sound reasons to
expect such policies to harm consumers and economic eﬃciency. Moreover,there are additional eﬀects that would need to be considered in a full policy
assessment. For instance, a single-product restriction will generally reduce
the platform’s provider’s proﬁts and, consequently, might adversely aﬀect theplatform provider’s investment incentives.
We close by noting that this type of analysis could be extended to otherelements of proposed network neutrality regulation, including the proposalthat providers of last-mile Internet access services be allowed to charge households but not applications providers. It is readily shown that—because all28

In an earlier version of this paper, we examined a model of Cournot competition
between undiﬀerentiated duopolists in a one-sided market in which each ﬁrm could either
oﬀer two quality levels or was restricted to oﬀering only one. In that model, imposition ofa single-product restriction always reduces welfare. Absent the restriction, the two ﬁrms
engage in head-to-head competition across full product lines, a result ﬁrst obtained by
DeFraja (1996). In some circumstances, the single-product restriction induces the two
ﬁrms to oﬀer identical products. The resulting loss of variety reduces welfare. In othercircumstances, a restriction on the number of products that each ﬁrm is allowed to oﬀer
induces the ﬁrms oﬀer non-overlapping, or vertically diﬀerentiated, products. Here, theresulting loss of competition harms both consumers and economic eﬃciency.

Hermalin & Katz

Appendix A: Proofs

29

application providers would utilize the highest available quality level—theplatform would oﬀer application providers only one quality of service. Moreover, because application-provider surplus would be increasing in the service
quality, the monopolist in our model would oﬀer a quality level that would belower than either the socially eﬃcient quality level or the quality level thatwould be oﬀered if the platform were restricted to oﬀering a single quality
but allowed to charge both sides of the market for access services.29

Appendix A: Proofs
Proof of Lemma 1: Recall that c(q)is increasing. Hence, ifq(ρ + σ)θq − c(q ) < 0 ,
¯
¯

(30)

then
(ρ + σ)θq − c(q) < 0 ,for all q ≥ q . Thus, for all feasible q, welfare is greater if these types are¯ if they are served.
excluded thanIf(ρ + σ)θq − c(q ) ≥ 0 ,(31)¯¯
then it is welfare enhancing (at least weakly) to serve these types. Note thatthe minimum θ satisfying (31) is the least upper bound of θ satisfying (30).
It follows that there is a marginal type, θw , such that θ < θw should be¯ assumption¯θw > 0, and theexcluded and θ ≥ θw should operate. Clearly,¯¯ θ < θ̄. Hence, there is a positive
measure of types whothat θ̄q > c(q ) impliesw¯¯¯
should be excluded and a positive measure of types who should be served. If
a type is served, then its contribution to welfare is maximized by choosing q
to maximize(ρ + σ)θq − c(q)subject to q ≥ q .
¯29

The ﬁnding that the quality level would be less than socially optimal builds on thefact that, in our model, the monopolist is able to appropriate all of the marginal beneﬁts
of quality enjoyed by households.

Hermalin & Katz

Appendix A: Proofs

30

Proof of Proposition 1: Utilizing the envelope theorem, the ﬁrst-ordercondition for θr is given by
¯

 

ρq 1 − F (θr ) − (ρ + σ)θr q − c(q) f (θr ) = 0 .
(32)
¯
¯
¯
Suppose that 0 ≤ θr ≤ θw . Then¯¯(ρ + σ)θr q − c(q) ≤ (ρ + σ)θw q − c(q)¯¯(because c(q)/q is increasing in q)≤ (ρ + σ)θw q − c(q )¯ ¯¯=0(by deﬁnition of θw )¯
The ﬁrst term on the left-hand side of (32) is positive, and we have justshown that the second is non-negative, which contradicts (32). The resultthat θr > θw follows.
¯¯Conditionalon connecting only types θ ≥ θr , welfare maximization re¯quires maximizing(ρ + σ)Λ(θr )q − c(q) .
¯
Because Λ(θr ) > θr , the q that solves that problem must exceed the q that¯
solves (12). ¯
Proof of Lemma 3: Deﬁne quality q  such that q ≤ q  ≤ min{qu (θu ), qr }.
Suppose, counterfactually, that θr < θu . Consider¯an extension of ¯the un¯ oﬀerings in which q(θ) remains the
restricted platform’s equilibrium¯quality
same for θ ≥ θu , but now q = q  for θ ∈ [θr , θu ). By construction, this ex¯ the order restriction and, ¯thus,¯ it would have been feasible
tension satisﬁes
for the unrestricted platform to have oﬀered it. By (6), this extension is
incentive compatible if and only if the prices charged to all application typesθ ≥ θu are reduced by ρ(θu − θr )q  . Because this extended line was not the¯¯platform, the change in proﬁts from adopting¯
line chosen
by the unrestricted
this extension cannot be positive. Hence, recalling (7 ), θu¯ (ρ + σ)θq  − m(θ)ρq  − c(q  )f (θ)dθ
θr



¯
= (ρ + σ)θr q  − c(q  ) F (θu ) − F (θr )
¯
¯

¯
(33)
− 1 − F (θu ) (ρ + σ)(θu − θr )q ¯¯¯  θu¯

1 − F (θ) dθ ≤ 0 .
+ σq
θr

¯

Hermalin & Katz

Appendix A: Proofs

31

Note this inequality is strict if conditions (ii) or (iii) are satisﬁed.
By revealed preference under the restricted regime,


 

θr ρqr − c(qr ) F (θu ) − F (θr ) − 1 − F (θu ) (θu − θr )ρqr
¯ θu
¯
¯
¯
¯
¯
¯
+ σqr
θf (θ)dθ
θr
 


(34)
¯ 
= (ρ + σ)θr qr − c(qr ) F (θu )−F (θr ) − 1−F (θu ) (ρ + σ)(θu −θr )qr
 θu
¯
¯
¯
¯ ¯
¯


¯
1 − F (θ) dθ ≥ 0 .
+ σqr
θr

¯
Note this inequality is strict if condition (i) is satisﬁed.
Let Δ ≡ θu − θr and ΔF ≡ F (θu ) − F (θr ). Note both Δ and ΔF are¯(33) and (34) imply
positive given¯the ¯supposition that θ¯u > θr . Expressions
¯
¯
 θu




(35)
(ρ + σ)θr ΔF − 1−F (θu ) (ρ + σ)Δ + σ ¯ 1 − F (θ) dθ¯¯θr¯c(q  )c(qr )≤ ΔF  ≤ ΔF
(36)
q
qr
 θu




≤ (ρ+σ)θr ΔF − 1−F (θu ) (ρ+σ)Δ+σ ¯ 1−F (θ) dθ . (37)¯¯θr¯The ﬁrst inequality in (36) follows from (33) and the inequality in (37) follows
from (34). The second inequality in (36) follows because q  ≤ qr and c(q)/q is
an increasing function. Observe that (35) and (37) are the same expression,
so this chain of inequalities yields a contradiction if any of the inequalities
are strict. However, as noted, under condition (i), the inequality in (37) isstrict; and, under conditions (ii) or (iii), the ﬁrst inequality in (36) is strict.
Hence, by contradiction, we can conclude θr ≥ θu if any of conditions (i)–(iii)
¯
¯
hold.
If qr > q , then q  can be chosen so q  < qr . Hence, the second inequality¯ be strict. So, by contradiction, θr ≥ θu if condition (iv) holds.
in (36) would¯ (v).¯ Note the same revealed
We now turn to the suﬃciency of condition

Hermalin & Katz

Appendix A: Proofs

32

preference argument used to derive (33) applies to any θ ∈ [θr , θu ]. Hence,¯ ¯ θu¯ (ρ + σ)τ q  − m(τ )ρq  − c(q  )f (τ )dτθ= (ρ + σ)θq  − c(q  ) F (θu ) − F (θ)
¯


(38)
− 1 − F (θu ) (ρ + σ)(θu − θ)q ¯¯  θu¯

1 − F (τ ) dτ ≤ 0+ σqθ

(i.e., the unrestricted platform would not strictly prefer to extend down toθ). Similarly, the same reveal preference argument used to derive (34) applies
to any θ ∈ [θr , θu ]. Hence,
¯ ¯

 

θr ρqr − c(qr ) F (θ) − F (θr ) − 1 − F (θ) (θ − θr )ρqr
¯
¯
¯
 θ
+ σqr

τ f (τ )dτ

 

¯ 
= (ρ + σ)θr qr − c(qr ) F (θ)−F (θr ) − 1−F (θ) (ρ + σ)(θ−θr )qr
¯
¯
¯
 θ


1 − F (τ ) dτ ≥ 0
+ σqr
θr

θr

(39)

¯
(i.e., the restricted platform would not strictly prefer to cutoﬀ sales at θ).
Observe (38) implies θu 
1 − F (τ ) dτθu − θ
(ρ + σ)θq  − c(q  )θ¯−σ≤ (ρ + σ) 1 − F (θu )¯qF (θu ) − F (θ)¯ F (θu ) − F (θ)¯¯
for all θ ∈ [θr , θu ]. Hence, in the limit as θ → θu :¯ ¯¯(ρ + σ)θu q  − c(q  )1 − F (θu )1 − F (θu )≤ (ρ + σ)
¯ 
¯ −σ
¯
q
f (θu )f (θu )¯¯
= ρm(θu ) .
(40)¯
Similarly, we can write (39) asθ 
1 − F (τ ) dτ
θ − θr(ρ + σ)θr qr − c(qr )θr≥ (ρ + σ) 1 − F (θ)
−σ ¯
¯
¯
qr
F (θ) − F (θr )F (θ) − F (θr )¯¯for all θ ∈ [θr , θu ]. Hence, in the limit as θ → θr :¯ ¯¯(ρ + σ)θr qr − c(qr )≥ ρm(θr ) .
(41)
¯
qr
¯

Hermalin & Katz

33

Appendix A: Proofs

Recall that −c(q)/q is a decreasing function of q and q  ≤ qr . By supposition θu > θr , and hence the left-hand side of (40) is strictly greater than¯ side of (41). Thus, m(θ ) > m(θ ), which means the hazard¯
the left-handur¯ it is ¯evaluated at θ , a contradicrate evaluated at θu is strictly less thanr¯ (v) holds.
¯ Reductio ad absurdum, θ ≥ θ if condition
tion of condition (v).
ru¯¯Proof of Proposition 2: From Lemma 3, θr ≥ θu . Suppose, counterfactually, that θr = θu . Select a q  such that q ≤¯ q  <¯qr and q  ≤ qu (θu ). The¯¯¯ preference¯ establish that the unrestricted
same revealed
arguments used toplatform would not wish to extend downward (i.e., serve types worse thanθu ) and that the restricted platform would not wish to have a higher cutoﬀ¯than θ continue to apply, so expressions (40) and (41) remain valid. Byr¯
supposition
θr = θu . Making that substitution into (40) and utilitizing the¯
fact that c(q)/q is¯ a strictly increasing function of q, we can combine (40)and (41) asρm(θr ) ≥¯(ρ + σ)θr q  − c(q  )(ρ + σ)θr qr − c(qr )>≥ ρm(θr ) ,
¯ 
¯
q
qr
¯

which is impossible. Hence, the supposition θu = θr must be false reductio
ad absurdum. Given θu ≤ θr , we are left with¯ θu <¯ θr , as was to be shown.
¯¯¯¯
Proof of Proposition 3: First, we assemble some facts. Recall thatqw (θ̄) > qw (θ) for all θ < θ̄, including θr , where θr is the marginal type served
under the ﬁrst policy. By Proposition ¯1, qr < qw¯(θr ). Hence, qr < qw (θ̄) ≡ q̄w .
¯ policy. We can establish
Let θR denote the marginal type under the second
that θR¯ ≥ θr by mimicking the proof of Lemma 3. Speciﬁcally, suppose¯ θ play the role played by θ in the proof of Lemma 3 andθR <¯θr . LetRr¯¯let θ ¯play the ¯role played by θ in that proof.
Note, because qr < q̄w , theru¯¯
equivalent of condition (iv) holds.
To establish that θR > θr , suppose, counterfactually, that θR = θr . Then
¯ conditions for both of the following¯ programs,¯¯
θr must satisfy the ﬁrst-order¯max ρθqr + σΛ(θ)qr − c(qr ) 1 − F (θ)
θ



and



max ρθq̄w + σΛ(θ)q̄w − c q̄w
θ


 


1 − F (θ) ,Hermalin & Katz

34

Appendix A: Proofs

where, recall, Λ(θ) is the expected type conditional on type not being lessthan θ. Hence,








qr ρ + σΛ (θr ) 1 − F (θr ) − ρθr + σΛ(θr ) qr − c(qr ) f (θr ) = 0
¯
¯
¯
¯
¯
and






 
q̄w ρ + σΛ (θr ) 1 − F (θr ) − ρθr + σΛ(θr ) q̄w − c q̄w f (θr ) = 0 .
¯
¯
¯
¯
¯




Rearranging and combining, we have
 
 

c q̄w
c(qr ) = ρθr + σΛ(θr ) − ρ + σΛ (θr ) m(θr ) =
.
qr
q̄w
¯
¯
¯
¯

(42)

But c(q)/q is a strictly increasing function of q. Hence, (42) contradicts thefact that q̄w > qr . The result follows reductio ad absurdum.
The second part of the proposition follows from the ﬁrst coupled with
Lemma 3.
Lemma A.1 There exists an application type, θ̂ ∈ (θr , θ̄), such that the¯for that type
quality under a single-product restriction (i.e., qr ) is eﬃcient(i.e., qr = qw (θ̂)).
Proof:
Hence,

Because some trade is always proﬁtable for the platform, θr < θ̄.
¯

ρθr + σΛ(θr ) < ρθ̄ + σΛ(θ̄) = ρθ̄ + σ θ̄ ,¯¯which implies that qr < qw (θ̄). But, becauseρθr + σΛ(θr ) > ρθr + σθr ,¯¯¯¯qr > qw (θr ). The result follows because qw (·) is a continuous function.
¯
Proof of Proposition 4: By revealed preference, the platform’s proﬁtcannot be less absent regulation than under the single-product restriction.

Hermalin & Katz

35

Appendix A: Proofs

It follows that a suﬃcient condition for welfare to be greater in the unrestricted equilibrium is that application-provider surplus (A) be greater inthat equilibrium than under regulation.30 Observe θ̄   θ
 θ̄


Ar =
ρθqr − ρθr qr f (θ)dθ =ρqr (τ )dτ f (θ)dθ
(43)
¯
θr
0
0
¯
and


 θ̄  θ̄   θρθqu (θ)−p q(θ) f (θ)dθ =ρqu (τ )dτ f (θ)dθ, (44)
Au =
θu

0

0

¯
where the second equality in (43) follows from the deﬁnition of integrationand the fact that qr (t) ≡ 0 for t < θr , while the second equality in (44)¯ ≡ 0 for t < θ . If q (·) dominates
follows from (6) and the fact that qu (t)ur¯ then it follows fromqu (·) in the sense of second-order stochastic dominance,(43) and (44) and the fact that θu < θr by Proposition 2 that Au > Ar .
¯ condition is the contrapositive of¯
Suﬃciency follows. Because the necessary
the suﬃciency condition, it follows as well.
Proof of Lemma 4: Deﬁne Δf = fh − f and let Δq = h − . Consider
the platform’s maximization problem: θ̄ Δf  Δf + f − c + Δ σhθf (θ)dθ
max 1 − Ff{f ,Δf }
ρΔq
ρΔ
q


 Δf
ρΔq
 f 
 Δf 
−Fσ θf (θ)dθ . (45)
f +
+ F
f
ρΔq
ρ
ρ


For the moment, we ignore the constraints (i.e., that (13) hold and that both
fractions in (13) be between 0 and θ̄). As we will demonstrate, they are notbinding. Observe (45) is equivalent to the following program.
 θ̄ Δf  Δf − c + Δ σhθf (θ)dθ
max 1 − Ff{f ,Δp }
ρΔq
ρΔ
q


 Δf
ρΔq
 f 
σ θf (θ)dθ . (46)
f +
+ 1−F
f
ρ
ρ


30

Recall end-users’ surplus is fully captured by the platform so is accounted for in theplatform’s proﬁt.

Hermalin & Katz

Appendix A: Proofs

36

If the solution to the unconstrained problem satisﬁes the constraints, thenthat solution is also the solution to the constrained problem. The ﬁrst-orderconditions for (46) can be reëxpressed as
 
f
σ f
f
−
−
=0
(47)
m
ρ
ρ
ρ ρ
and

m

Δf
ρΔq


−

Δ f − c σ Δf
−
= 0.
ρΔq
ρ ρΔq

(48)

Because m(·) is non-increasing, it is readily seen that if solutions exist theymust be unique and they satisfy the second-order condition because the derivatives (the left-hand sides of (47) and (48)) are positive to the left of the
optimum and negative to the right of the optimum.
Given the continuity of m(·) and that m(0) > 0 = m(θ̄), (47) must havea solution such thatf
< θ̄ .
0<ρ
Because m(·) is non-increasing and c > 0, it follows that, if (48) has asolution, then it satisﬁes
Δf
f
>
;
ρΔq
ρ
that is, (13) is satisﬁed. Hence, we’re done if we can show that there exists
Δ
a Δf such that ρΔfq < θ̄. Suppose not. Thenσc − Δf
σ Δf−> 0 or, equivalently, c > Δf 1 +
ρΔq
ρ ρΔq
ρ
for any Δf ≥ ρθ̄Δq . In particular, this would require thatσc > θ̄ρΔq 1 += θ̄(h − )(ρ + σ) ,ρwhich violates the assumption that there is a welfare beneﬁt to high quality.
By contradiction, (48) has a solution.
We have shown that the solutions to (47) and (48) maximize (46) andsatisfy the relevant constraints. Algebra yields (15).

Hermalin & Katz

37

Appendix A: Proofs

Proof of Lemma 7: Regardless of which deﬁnition of Di (S0 , S1 ) applies,(23) is global concave in Si and, thus, has a unique solution. If, in equilibrium, there are no contested households, then Sj , j = i, is irrelevant to the
optimal Si . Given the symmetry between the platforms and the concavity of(23), it follows that any equilibrium without contested households must be
symmetric. Suppose there are contested households in equilibrium, then S0and S1 must simultaneously satisfy the ﬁrst-order conditions:


1
M
(49)
Π − K − 2S0 − T + S1 = 0
2T


1
M
Π − K − 2S1 − T + S0 = 0 .
2TandStraightforward algebra reveals
S0 = S1 = ΠM − K − T .

(50)

Hence, we’ve established uniqueness and symmetry.
In a no-contested-households equilibrium, a platform selects S to maximizeS(ΠM − K − S)
(51)
T
subject to the constraint that it not “intrude” into the other platform’s market. Given symmetry, we can express this constraint as
S≤

T
.
2

(52)

Maximizing (51), we ﬁnd that the proﬁt-maximizing (and, thus, equilibrium)
S satisﬁes
S=

ΠM −K
, if T > ΠM − K (i.e., (52) doesn’t bind)2T, if T ≤ ΠM − K (i.e., (52) binds)
2

.

(53)

A no-contested-households equilibrium when T ≤ ΠM − K and, thus, S =T /2 can exist only if the best response to Sj = T /2 is not Si > T /2. Using(49), this requiresT
π M − K − 2S − ≤ 0(54)2Hermalin & Katz

Appendix A: Proofs

38

for all S > T /2. Expression (54) can hold for all such S if and only if
2
T > (π M − K) .
3

(55)

If (55) doesn’t hold, then the equilibrium results in the platforms contestingfor some households, so S is given by (50). Straightforward algebra on expressions (50) and (53) yield expressions (24)–(26).
Proof of Lemma 8: First, given their beliefs—which are correct if theLemma 7 equilibrium is played—application providers will sign with any
platform for whichmax ρθq − pi (q) ≥ 0 .
q∈Qi

Second, we verify that no household would multi-home given the equilibrium strategies set forth in Lemma 7. Observe h > 0 as a consequence of(27). It follows, therefore, given the symmetric oﬀerings of the two platforms,
that a household could only lose by multi-homing.
Third, we verify that neither platform would wish to deviate. Without
loss of generality, suppose that network 1 is oﬀering the candidate equilibrium
surplus level and quality schedule, and network 0 deviates. By deﬁnitionof equilibrium in the model when multi-homing is infeasible, there is nodeviation that is both proﬁtable and does not induce any multi-homing.
Hence, to complete the proof, we need only rule out deviations by a platformthat induces at least some multi-homing.
For a household at location L that multi-homes, the incremental surplus
oﬀered by platform 0 over signing exclusively with platform 1 isθ max q0 (θ) − qu (θ) , 0 f (θ)dθ − h0 − LT .
(56)
σ
Θ

A multi-homing household will sign with platform 0 only if that surplus is
non-negative. Observe that platform 0 will lose money on every household
it connects if

 




p0 q0 (θ) − c q0 (θ) f (θ)dθ .
(57)
h0 < K −ΘHermalin & Katz

Appendix B: Examples

Using (57), a necessary condition for (56) to be non-negative isθ max q0 (θ) − qu (θ) , 0 f (θ)dθ
σ
Θ

 




+
p0 q0 (θ) − c q0 (θ) f (θ)dθ − K > 0 .

39

(58)

Θ

Clearly, the left-hand side of (58) cannot exceed the value of that expression
maximized by the choice of q0 (·). That maximization program can be writtenas θ̄ +σ q(θ) − qu (θ) + ρθq(θ) − m(θ)ρq(θ) − c q(θ) f (θ)dθ (59)
max
q(θ)

0

subject to the constraint that q(·) be non-decreasing, where
x , if x > 0
+
(x) =
.
0 , if x ≤ 0
Suppose we maximized (59) ignoring the constraint that q(·) be non-decreasing.
That maximized value must be at least as great as the constrained maximumof (59). Observe that the ﬁrst-order condition for the unconstrained problem
if q(θ) ≥ qu (θ) is the same as that of the original unrestricted monopolist’s
problem, (9); so, in these cases, q(θ) = qu (θ). Hence, optimally q(θ) ≤ qu (θ)
almost everywhere. Consequently, the top line of (58) is zero. But the remainder of (58) cannot be positive by (27). By contradiction, we have shownthat Platform 0 can never oﬀer a household enough surplus to induce it tomulti-home.

Appendix B: Examples
It is useful to illustrate certain points using discrete examples. The following
analysis is a straightforward extension of the continuous case.
Assume that there is a ﬁnite number, N , of content types indexed sothat i < j implies θi < θj . Let fi denote the probability a randomly drawn
household is type θi . The distribution, correspondingly, can be denoted
Fi =

i

j=1

fj .

Hermalin & Katz

Appendix B: Examples

Deﬁne
mi ≡
Note mN = 0. Deﬁne

40

1 − Fi.
fi

Ri (q) ≡ ρθi+1 q − ρθi q .

As will become apparent, the fact that RN (·) is not deﬁned is not an issue.
The function Ri (·) is positive, strictly increasing, and convex on R+ . Asa consequence, Proposition 2 of Caillaud and Hermalin (1993) implies thatthe monopolist’s proﬁt-maximization problem with respect to the choice ofqualities reduces to the following:31
N



max
fi (ρ + σ)θi qi − c(qi ) − mi Ri (qi ){q1 ,...,qN }

(60)

i=1

subject to q ≤ q1 ≤ · · · ≤ qN .
(61)¯
Let {q̂1 , . . . , q̂N } denote the solution. Observe this solution must satisfy, foreach i, the following condition.
(ρ + σ)θi − mi (θi+1 − θi )ρ − c (q̂i )  0 ,(62)

where the expression is an equality if the relevant order restriction, condition(61), doesn’t bind, is greater than zero if the upward restriction binds, andis less than if the downward restriction binds.
If we impose the assumption that mi is non-increasing in i (i.e., a monotonehazard rate), then it can be shown that (61) is not binding except, possibly,at q . In that case, if q̂i = 0 or q , then the left-hand side of expression (62) is¯
less¯ than zero.
Welfare when the monopolist is unrestricted is
Wu =

N




(ρ + σ)θi q̂i − c(q̂i ) fi .

(63)

i=1
31

Caillaud and Hermalin’s Proposition 2 is a fairly straightforward extension of standard
results in mechanism design for two types or for a continuum of types to an arbitrary, but
ﬁnite, number of types.

Hermalin & Katz

Appendix B: Examples

41

Example 1: Consider a discrete type space with three elements. Let√√
1
15 + 2 2
53
θ1 =
θ2 =
+
(15 + 2 2)
θ3 = 310 000 381
6
7
13
40
f2 =
f3 =
f1 =
60
60
60
(the value of θ1 is set just above the value at which the unrestricted platform
would exclude that type). Suppose ρ = 100, σ = 50, and c(q) = q 2 /2, which
implies qw (θ) = 150θ, and assume q < 1/20.
Consider the unrestricted case ¯ﬁrst. Using expression (62) and the factthat c (q) = q, one ﬁnds that all types are served, with
127
≈ .09071400√
25
qu (θ2 ) = (345 + 238 2) ≈ 436.9
39

qu (θ1 ) =qu (θ3 ) = qw (3) = 450 .
Consequently, unrestricted welfare is

3 



θi qu (θi ) − c qu (θi ) fi
Wu =
i=1


61, 669, 559, 167, 397 35, 014, 567√=+
786, 240, 000
2340 2≈ 89, 016.8 .

Now consider the imposition of a single-product restriction. Calculations
reveal⎧√(12,525,381+1,190,000 2)2⎪≈ 77, 884 , if θr = θ1⎪2,592,000,000⎪¯⎨√
125(2505+238 2)2.
proﬁt =
≈ 88, 166 , if θr = θ2
11,448⎪⎪¯⎪⎩
67, 500 , if θr = θ3¯
Under the single-product restriction, the platform excludes the lowest type.

Hermalin & Katz

42

Appendix B: Examples

Nevertheless, welfare increases:
Wr =

3 



 
θi qr − c qr fi

i=2√
125(7, 438, 393 + 505, 020 2)=
11, 448≈ 89, 017.7 > Wu . 32

Example 2: Assume that c(q) = q 2 /2 and θ is distributed on [0, 1] according
to the distribution F (θ) = θ5 . Let ρ = 2, σ = 1, and q = 0. Calculations¯
reveal:33
θr ≈ .741
Wr ≈ 2.77
ASr ≈ .572¯

 1/5 1/5 232
31
5(221 − 2 5 17 5 25)
82
Wu = 5
ASu =
θu =
−
17
42 21 17357¯
≈ .652≈ 2.88≈ .566


Example 3: Suppose F (·) is the uniform distribution, θ̄ = 1, c(q) = q 2 /2,
ρ = 2, σ = 1, and, for convenience, set q = 0. Note that¯m(θ) = 1 − θ .
Hence, from (9),qu (θ) = max{0, 3θ − 2(1 − θ)} = max{0, 5θ − 2} .
32

The precise diﬀerence between Wr and Wu is√
115, 982, 179, 127, 959 − 81, 985, 604, 568, 000 2
Wr − Wu =.
41, 670, 720, 000

33

The Mathematica program used is available from the authors upon request.

Hermalin & Katz

43

Appendix B: Examples

Note qu (θ) ≥ 0 for all θ ≥ 2/5 = θu . Hence,
¯
1

 1 


1
5 3
63
2
2
3θ 5θ − 2) − 5θ − 2) dθ = θ + 2θ − 2θ =
Wu =
2
6
50
2/5
2/5
and

 θ

 1
Au =

2
2/5

 1 

=2
2/5

0

 1  θqu (τ )dτ5 2
2
θ − 2θ +25dθ = 2

2/5dθ = 22/55τ − 2 dτ


dθ

 1
2 
9
5 3
2
θ −θ + θ  =
.
6
5
25
2/5

Turning to the case of the single-product restriction, observe
1
2
1
τ dτ
− θ2
θ+1
θ
2
=
=
.
Λ(θ) =1−θ
1−θ
2Hence, solving (12),θr + 15θr + 1= ¯.
qr = 2θr + 1 ¯
2
2¯
The ﬁrst-order condition for (11), utilizing the Envelope Theorem, isρ + σΛ (θ)(1 − θ)qr − ρθqr + σΛ(θ)qr − c(qr ) =5θr + 1 θr + 1 5θr + 1 1  5θr + 1 2
1
5θr + 1= 0.
(2 + )(1 − θr ) ¯− 2θr ¯
+¯
−
¯
¯
2
2
2
2
2
2
2
¯
¯
Solving, θr = 3/5 and qr = 2. Hence,
¯
1

 1 

1 2
56
2
3θ × 2 − 2 dθ = 3θ − 2θ =
Wr =
2
50
3/5
3/5
and

 1


3 
2 × 2(θ − ) dθ
5
3/5
1
12 
8
= 2θ2 − θ =
.
5
25

Ar =

3/5

In this example, the improved-quality eﬀect is positive but is dominatedby the negative exclusion and reduced-quality eﬀects. A single-product restriction reduces welfare from Wu = 63/50 to Wr = 56/50.
Hermalin & Katz

44

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