Price Theory of Monopoly Platforms in Two-Sided...

Price Theory of Monopoly Platforms in Two-Sided Markets

Yan Miao

Advisor: Prof. William Rogerson

Submitted for Honors in MMSS

Contents

1 Introduction 6

2 Review of Frameworks 9

2.1 Re�ned De�nition of Two-Sided Markets . . . . . . . . . . . . . . . . . . . . 9

2.2 Monopoly Quality Distortion in Pro�t-Maximizing Outcomes . . . . . . . . . 9

2.2.1 Quality Distortion in One-Sided Markets . . . . . . . . . . . . . . . . 9

2.2.2 Preview of Quality Distortion in Two-Sided Markets . . . . . . . . . . 11

3 Models of One-Dimensional Heterogeneity and a Natural Extension 11

3.1 General Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Special Cases with User Heterogeneity in One Dimension . . . . . . . . . . . 12

3.2.1 Armstrong (2006): Heterogeneous Membership Values and Homoge-

neous Interaction Value . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2.2 Rochet and Tirole (2003): Heterogeneous Interaction Values and Ho-

mogeneous Membership Value . . . . . . . . . . . . . . . . . . . . . . 16

3.3 A Natural Extension: Combining Two Dimensions of Heterogeneity (Model I) 20

3.4 Quality Distortion in the Pro�t-Maximizing Outcome . . . . . . . . . . . . . 26

4 Weyl (2010): Heterogeneity in Two Dimensions 29

4.1 Weyl's Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.1 Distribution Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1.3 Quality Distortion in the Pro�t-Maximizing Outcome . . . . . . . . . 32

4.2 Comparing Model I and Weyl's Model . . . . . . . . . . . . . . . . . . . . . 32

5 Generalized Model (Model II): Arbitrary Relationship between Two Di-

mensions 33

5.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2

5.2 Quality Distortion in the Pro�t Maximizing Outcome . . . . . . . . . . . . . 38

5.3 Special Case: Cobb-Douglas Interaction Utility Function . . . . . . . . . . . 39

5.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

5.3.2 Quality Distortion in the Pro�t Maximizing Outcome . . . . . . . . . 40

6 Conclusion 41

3

Acknowledgements

I would like to thank my advisor, Professor William Rogerson, for introducing me to the study

of industrial organization and generously o�ering his time, ideas, and support throughout

the thesis process. I would also like to thank the thesis seminar organizer, Prof. Joseph

Ferrie, for providing guidance and keeping me on track.

4

Abstract

A two-sided market involves two groups (sides) of users who interact on platforms that

sell distinct goods to each group. One side's utility from participation depends not only

on the value of the good (membership value) itself, but also on the participation rate of

the other side (interaction value). This paper studies how monopoly platforms set prices

in such markets when users within each side di�er from one another on the membership

values and/or interaction values. Using the more succinct approach of viewing platform's

problem as choosing participation rates directly, I reformulate cases with single-dimensional

user heterogeneity studied in Rochet and Tirole (2003) and Armstrong (2006) and combine

these cases to form an extension with two-dimensional heterogeneity. I compare this natural

extension with the basic two-dimensional heterogeneity model in Weyl (2010) and �nally

provide a generalized model that allows arbitrary relationship between users' membership

and interaction utility. Analysis of the generalized model suggests that the common features

of monopoly prices in a two-sided market can be easily studied without assuming speci�c

relationship between the two sources of user heterogeneity.

5

1 Introduction

Two-sided markets are markets in which economic platforms enable and encourage interac-

tions between two sides of users by appropriately charging each side. Cross-side externalities

are present, where the utility of a user on one side of the platform depends on the number

of users on the other side that the platform attracts.

Examples of two-sided markets are fairly easy to come by. As a classic one, shopping

malls are platforms where stores and shoppers interact. Merchants would pay more to open

stores in malls with lots of shoppers, and shoppers like to go to malls with a variety of

stores. Heterosexual dating clubs appeal to both female and male members. Customers of

each gender would like to join the clubs that have plenty of members of the opposite gender.

Credit card providers facilitate transactions between cardholders and merchants. Cardhold-

ers prefer credit cards accepted by most merchants, and a�liated merchants bene�t from

cards used by many customers. More recently, PC and smartphone operating systems (Win-

dows, Mac OS X, Linux, Google Android, etc.) attract both device users and third-party

application providers. Device users enjoy using operating systems that support a spectrum

of applications, while application developers prefer launching applications compatible with

widely-used operating systems. In some cases, cross-side externalities are negative for one

side of the market. For instance, advertisers prefer to put ads on newspapers with large

reader bases, yet newspaper readers normally prefer papers with limited numbers of ads.

Rochet and Tirole (2003) provide an extensive list of existing and prospective two-sided

markets.

Within the context of two-sided markets, members of one side can commit to using only

one platform (single-home), or they can use multiple platforms at the same time (multi-

home). A shopping mall in a remote town is a platform where all merchants wishing to sell

and all consumers hoping to purchase are con�ned (both sides single-home). In the case

of PC operating systems, PC users mostly single-home (using only one operating system),

while application developers mostly multi-home (launching applications compatible with

6

multiple operating systems). In the newspaper example, however, a reader can read multiple

newspapers, and an advertiser can advertise on more than one newspapers as well (both sides

multi-home). Various studies have been done on price theory of competing platforms in two-

sided markets, including Caillaud and Jullien (2003), Rochet and Tirole (2003), Rochet and

Tirole (2006), Armstrong (2006), and Hagiu (2009). Focuses include how di�erent price

structures compare in extracting network surpluses and how markups vary in di�erent states

of platform competition. Nevertheless, the monopoly platform on its own renders much of

the unique characteristics of prices in a two-sided market that are worth studying. Therefore,

in this paper I will focus on the price theory of monopoly platforms alone.

Distinct special cases of monopoly platforms in two-sided markets are studied in Rochet

and Tirole (2003) and Armstrong (2006). According to Rochet and Tirole (2006) and Weyl

(2010), the crucial di�erence between these cases is the source of user heterogeneity. On

each side, users can have utility that di�ers in two distinct dimensions, membership and

interaction. For example, members of a dating club di�er mainly in the interaction utility

they derive from getting a date (as oppose to the membership utility of being in the club

alone). In contrast, newspaper readers di�er mostly in the membership utility they take

from reading content (as oppose to the interaction (dis)utility of encountering ads while

reading). In a given case, which dimension of heterogeneity is dominant has implications

on equilibrium prices. Weyl (2010) shows that both special cases with one-dimension het-

erogeneity presented in Rochet and Tirole (2003) and Armstrong (2006) can be �t into a

more general model that allows two-dimensional heterogeneity in certain forms. Weyl (2010)

transforms the platform's problem from choosing prices charged to the two sides to choosing

participation rates directly, which signi�cantly simpli�es the analysis of the general model.

Also introduced is the notion of quality choice that is linked to Spence (1975): for side I

on the platform, the monopoly chooses quantity (participation rates on side I) as well as

quality (participation rate of side J) provided, both of which a�ect side I's gross utility.

Like the one-sided monopoly in Spence (1975) that chooses a single quality for all purchas-

7

ing consumers, here a platform can only internalize valuations of incremental quality by the

marginal users, which in turn leads to nonoptimal quality provision in pro�t-maximizing out-

come. Relying on the Weyl (2010) notion of choosing participation rates, this paper o�ers a

more succinct representation of special cases with one-dimensional heterogeneity, suggests a

natural combination of the one-dimensional cases with utility functions di�erent from those

Weyl's model, and propose a generalized model. Analysis of the generalized model sug-

gests that the study of prices in monopoly two-sided markets can be done without assuming

special relations between the two sources of heterogeneity. I will also explore whether the

generalized model sheds light on interesting special cases that were di�cult to study using

earlier models.

The rest of the paper is organized as follows: In Section 2, I review a more re�ned

de�nition of two-sided market and the Spence model of monopoly quality choice. In Section

3, I formulate special cases with one-dimensional user heterogeneity studied in Rochet and

Tirole (2003) and Armstrong (2006) viewing the platform as choosing participation rates and

provide a natural combination of the special cases into a model with two-dimensional user

heterogeneity (Model I). In Section 4, I review the Weyl (2010) model with two-dimensional

user heterogeneity and show that Weyl's Model permits utility functions that di�er from

what Model I allows. In Section 5, I propose a generalized model (Model II) that does not

assume special relations between the two sources of user heterogeneity and introduce special

utility functions that can be easily studied in the context of Model II. Analyses of monopoly

quality distortions in each model are presented in sections 3 to 5. Section 6 concludes.

8

2 Review of Frameworks

2.1 Re�ned De�nition of Two-Sided Markets

Although the concept of a two-sided market is quite intuitive, the exact de�nition is more

debatable. Here I adopt the de�nition summarized by Weyl (2010) with three features.

1. Multi-product �rm: A platform provides distinct services to two sides of the market,

which can be explicitly charged di�erent prices.

2. Cross network e�ects: Users' bene�ts from participation depend on the extent of user

participation on the other side of the market, which varies with market conditions.

3. Price-setting monopoly: A platform is a price setter (monopolistic or oligopolistic) on

both sides of the market and typically sets a uniform price on each side.

2.2 Monopoly Quality Distortion in Pro�t-Maximizing Outcomes

2.2.1 Quality Distortion in One-Sided Markets

Spence (1975) addresses market ine�ciency that arises when a monopoly chooses the quality

of production as well as the price. He shows that this ine�ciency stems from the �rm

responding to the marginal instead of the average consumer's valuation of quality increments.

As a result, a monopoly often fails to set quality optimally, adding a second dimension of

deviaton from the social optimum (in addition to maintaining a price markup above marginal

cost).

More speci�cally, consider a monopoly that sells a single product. Let P be the price

charged by the monopoly, and q the quality of the product. Let x be the quantity sold at

these corresponding P and q. Thus, given two out of the three variables the third can be

9

pinned down uniquely. The demand for the product is D (p, q) and the inverse demand is

P (x, q). The cost of producing x units of quality q is c (x, q). Consider the choices of q in

di�erent outcomes for given x and P .

Consumer surplus S can be written as

S =

ˆ x

0

P (v, q) dv − xP (x, q) .

Pro�t, denoted by π is

π = xP (x, q)− c (x, q) .

Therefore, total welfare W is

W = S + π =

ˆ x

0

P (v, q) dv − c (x, q) .

Social welfare is maximized when

0 =∂W

∂q=

ˆ x

0

Pqdv − cq.

But pro�t is maximized when

0 =∂π

∂q= xPq − cq.

Therefore, in the pro�t-maximizing equilibrium,

∂W

∂q=∂π

∂q+

(ˆ x

0

Pqdv − xPq)

= 0 +

(ˆ x

0

Pqdv − xPq)

= x

(1

x

ˆ x

0

Pqdv − Pq). (1)

10

In the pro�t maximizing outcome, equilibrium quality is underprovided if 1x

´ x0Pqdv −

Pq > 0 and overprovided if 1x

´ x0Pqdv−Pq < 0. Notice that Pq is the valuation of incremental

quality by the marginal consumer, and 1x

´ x0Pqdv is the average valuation of incremental

quality over all the consumers who make purchases. In words, since the monopoly only

internalizes marginal consumer's valuation of incremental quality, quality is underprovided

(overprovided) when the marginal consumer values incremental quality less (more) than all

consumers on average.

2.2.2 Preview of Quality Distortion in Two-Sided Markets

As stated in Weyl (2010), one can see a monopoly platform in a two-sided market as choosing

both quantity and quality for each side of users. For each side, the quantity is this side's own

participation rate and the quality (or negative quality, if this side dislikes interaction with

the other side) is the other side's participation rate. Similar to the case in one-sided markets,

here the monopoly platform can only internalize the marginal user's valuation of incremental

quality on each side, leading to potential quality distortions in the pro�t-maximizing out-

come. This paper adopts Weyl's notion of quality and analyzes pro�t-maximizing quality

distortion in each model.

3 Models of One-Dimensional Heterogeneity and a Nat-

ural Extension

Armstrong (2006) and Rochet and Tirole (2003) study special cases of monopoly platforms

where users on each side have utility that di�ers in only one dimension. In Armstrong

(2006), users on the same side can have di�erent membership values, but their interaction

value, captured by utility per interaction with a user from the other side, is the same. In

Rochet and Tirole (2003), users assume no membership value, but their interaction values

11

vary within each side. In these two papers, equilibria are calculated viewing the platforms

as choosing prices, which leads to various di�culties in solving the models. Here I present

both cases viewing platforms as choosing participation rates on two sides, which simpli�es

derivations as suggested in Weyl (2010).

3.1 General Settings

Consider a monopoly platform serving two sides of the market, side A and side B. Without

loss of generality, normalize number of users on each side to 1. If a user from side I ∈ (A, B)

decides to be on the platform under given prices and participation information, she will

purchase one unit of good I. N I denotes the quantity of good I purchased by the entire

side I, which equals the rate of participation on side I. For each side, the platform charges

a single price for each unit of good purchased: P I is the price of one unit of good I.

3.2 Special Cases with User Heterogeneity in One Dimension

3.2.1 Armstrong (2006): Heterogeneous Membership Values and Homogeneous

Interaction Value

Denote U I(N I)the total membership utility on side I when N I ∈ [0, 1] is the participation

rate. Suppose U I(N I)is increasing, twice di�erentiable, and strictly concave, and U I (0) =

0. Let αI be the interaction value that each user on side I derives from having one side J

user on board, thus the total interaction utility on side I for participation rates(N I , NJ

)is αIN INJ . Notice that αI is the same for all users on side I. Let CI be the cost that the

platform incurs from supplying each unit of good I.

Denote side I's gross utility T I(N I , NJ

)when participation rates are

(N I , NJ

)T I(N I , NJ

)= U I

(N I)

+ αIN INJ . (2)

12

When facing a price of P I and a side J participation rate of NJ , side I chooses a

participation rate N I to maximize its consumer surplus SI(N I , NJ

), which equals the gross

utility T I(N I , NJ

)minus the amount paid to the platform P IN I .

maxNI

SI(N I , NJ

)= T I

(N I , NJ

)− P IN I ,

which is equivalent to

maxNI

U I(N I)

+ αIN INJ − P IN I .

First order condition with respect to N I gives

0 = U I1

(N I)

+ αINJ − P I ,

where

U I1

(N I)

=dU I

(N I)

dN I.

Thus, given(N I , NJ

), price on side I is

P I(N I , NJ

)= U I

1

(N I)

+ αINJ , (3)

which is the inverse demand function on side I.

First Best Outcome

Suppose(N I , NJ

)are chosen to maximize social welfare W

(N I , NJ

)maxNI ,NJ

W(N I , NJ

),

where

13

W(N I , NJ

)= T I

(N I , NJ

)+ T J

(NJ , N I

)−(CIN I + CJNJ

).


0 = U I1

(N I)

+ αINJ + αJNJ − CI .

From (3), U I1

(N I)

+ αINJ = P I(N I , NJ

). Then given

(N I , NJ


P I(N I , NJ

)= CI − αJNJ︸︷︷︸

2−sided factor

. (4)

Therefore, the welfare maximizing unit price for good I equals production cost minus

cross-side externality that one side I user exerts on all participating side J users. Intuitively,

if side I exerts a positive externality on side J (i.e. αJ > 0), then participation of side I is

especially encouraged since it tend to bring up participation of side J . Such encouragement

is re�ected in a reduction of unit price P I to below the platform's production cost. In other

words, for each unit of good I, side I is charged the social (instead of platform) cost.

Pro�t Maximizing Outcome

Suppose(N I , NJ

)are chosen to maximize platform pro�t π

(N I , NJ

)maxNI ,NJ

π(N I , NJ

),


maxNI ,NJ

[P I(N I , NJ

)− CI

]N I +

[P J(NJ , N I

)− CJ

]NJ .

14


0 = P I1

(N I , NJ

)N I + P I

(N I , NJ

)− CI + P J

2

(NJ , N I

)NJ ,

where subscript k indicates taking partial derivative with respect to the kth argument. De�ne

µI ≡ −P I1N

I = P I/εI , where εI is the elasticity of demand. This together with (3) gives

P I(N I , NJ

)= CI − αJNJ︸︷︷︸

2−sided factor

+ µI︸︷︷︸market power

. (5)

Compared with the price in the �rst best outcome in (4), the price in the pro�t maximizing

outcome is higher by a market power term µI , which is identical to what happens in a one-

sided market. Thus, in the pro�t maximizing outcome, for each unit of good I, side I is

charged the social cost plus a monopoly market power factor.

Second Best Outcome

Notice that by pricing below production costs, the platform makes a negative pro�t when

social welfare is maximized. Therefore, the �rst best outcome is hard to achieve in reality.

As a result, a second best outcome, in which social welfare is maximized with the constraint

that the platform makes nonnegative pro�t is more plausible.

maxNI ,NJ

W(N I , NJ

),

s.t. π(N I , NJ

)≥ 0.

Lagrangian of this problem is

L = W(N I , NJ

)+ λπ

(N I , NJ

).

15


0 = U I1

(N I)

+ αINJ︸︷︷︸P I(NI ,NJ )

+αJNJ − CI

+λ

P I1

(N I , NJ

)N I︸︷︷︸

−µI

+P I(N I , NJ

)− CI + P J

2

(NJ , N I

)NJ︸︷︷︸

αJNJ

.This simpli�es to

P I(N I , NJ

)= CI − αJNJ︸︷︷︸

2−sided factor

+λ

1 + λµI︸︷︷︸

market power

. (6)

Compare (6) with (4) and (5), we see that the monopoly market power component in the

second best price is a fraction of that in the pro�t maximizing price. Now we move on to

the special case in Rochet and Tirole (2003).

3.2.2 Rochet and Tirole (2003): Heterogeneous Interaction Values and Homo-

geneous Membership Value

Suppose that conditions in 3.1 General Settings still hold. Assume that neither side derives

membership utility from being on the platform. However, now the interaction utility that

users derive from having one side J user on board is allowed to vary within side I. More

speci�cally, let V I(N I)denote the interaction utility that the entire side I on board enjoys

from having one participating side J user, where V I(N I)is increasing, twice di�erentiable,

and strictly concave for N I ∈ [0, 1], and V I (0) = 0. Then V I(N I)NJ denotes the total

interaction utility for side I. On the platform costs, suppose instead of the per unit cost of

CI , the platform incurs a per interaction cost of c. Thus the total cost for the platform is

cN INJ .

16

Side I's gross utility when participation rates are(N I , NJ

)is

T I(N I , NJ

)= V I

(N I)NJ . (7)

When facing a price of P I , side I chooses a participation rateN I to maximize its consumer

surplus SI(N I , NJ

), which equals the gross utility T I

(N I , NJ

)minus the amount paid to

the platform.

maxNI

SI(N I , NJ

)= T I

(N I , NJ

)− P IN I ,


maxNI

V I(N I)NJ − P IN I .


0 = V I1

(N I)NJ − P I .



P I(N I , NJ

)= V I

1

(N I)NJ , (8)

which is the inverse demand function on side I. I now consider the �rst best, pro�t maxi-

mizing and second best outcomes for the Rochet and Tirole (2003) special case.

First Best Outcome

Suppose(N I , NJ

)are chosen to maximize social welfare

maxNI ,NJ

W(N I , NJ

),

where

17

W(N I , NJ

)= T I

(N I , NJ

)+ T J

(NJ , N I

)− cN INJ .

First order condition with respect to N Igives

0 = V I1

(N I)NJ + V J

(NJ)− cNJ .

From (8), V I1

(N I)NJ = P I

(N I , NJ

). Then given

(N I , NJ


P I(N I , NJ

)= cNJ − V J

(NJ)︸︷︷︸

2−sided factor

. (9)

Notice that (9) has a very similar interpretation as (4). The welfare maximizing unit

price for good I equals average production cost minus cross-side externality that one side I

user exerts on all participating side J users. Intuitively, if side I exerts a positive externality

on side J (i.e. V J(NJ)> 0), then participation of side I is especially encouraged since it

could bring up participation of side J . Such encouragement is re�ected in a reduction of

unit price P I below the platform's production cost. Same as in the Armstrong (2006) case,

for each unit of good I, side I is charged the social (instead of platform) cost.


Suppose(N I , NJ

)are chosen to maximize platform pro�t

maxNI ,NJ

π(N I , NJ

),


maxNI ,NJ

P I(N I , NJ

)N I + P J

(NJ , N I

)NJ − cN INJ .

18


0 = P I1

(N I , NJ

)N I + P I

(N I , NJ

)+ P J

2

(NJ , N I

)NJ − cNJ ,

This together with (8) gives

P I(N I , NJ

)= cNJ − V J

1

(NJ)NJ + µI ,

which equals

P I(N I , NJ

)= cNJ − V J

(NJ)︸︷︷︸

2−sided factor

+ µI︸︷︷︸mkt power

+[V J(NJ)/NJ − V J

1

(NJ)]NJ︸︷︷︸

Spence distortion

. (10)

Compared with the price in the �rst best outcome (9), the price in the pro�t maximizing

outcome is adjusted upward by two factors. The �rst is the monopoly market power factor

identical in one-sided monopoly price and the Armstrong (2006) case. The second is a new

factor not presented in the one-sided or the Armstrong (2006) model. It is proportional to the

di�erence between per interaction utility enjoyed by the average and the marginal side J user

on board. Since V I(N I)is twice di�erentiable and concave for N I ∈ [0, 1], this last factor

is positive. Adopting Weyl's terminology I name this factor Spence distortion, as it will be

shown that this factor is directly linked to monopoly quality distortion in pro�t-maximizing

outcomes.

Second Best Outcome

Notice that in this model the platform still makes a negative pro�t if social welfare is maxi-

mized. As a result, the second best outcome is more plausible for policy makers.

maxNI ,NJ

W(N I , NJ

),

19

s.t. π(N I , NJ

)≥ 0.


L = W(N I , NJ

)+ λπ

(N I , NJ

).


0 = V I1

(N I)NJ︸︷︷︸

P I(NI ,NJ )

+V J(NJ)− cNJ

+λ

P I1

(N I , NJ

)N I︸︷︷︸

−µI

+P I(N I , NJ

)+ P J

2

(NJ , N I

)NJ︸︷︷︸

V J1 (NJ )NJ

−cNJ

.Then simplifying to get

P I(N I , NJ

)= cNJ − V J

(NJ)︸︷︷︸

2−sided factor

+λ

1 + λµI︸︷︷︸

mkt power

+λ

1 + λ

[V J(NJ)/NJ − V J

1

(NJ)]NJ︸︷︷︸

Spence distortion

. (11)

Compare (11) with (9) and (10) we see that both the monopoly market power factor and

the Spence distortion factor in the second best price are fractions of those in the pro�t

maximizing price.

3.3 A Natural Extension: Combining Two Dimensions of Hetero-

geneity (Model I)

From the analysis above we observe that special cases studied in Armstrong (2006) and

Rochet and Tirole (2003), mainly di�er in the sources of user heterogeneity, have di�erent

20

pricing implications in the pro�t maximizing (and second best) outcomes. More speci�cally,

when user heterogeneity is present in the interaction value dimension, pro�t maximizing

price is further distorted up by the Spence distortion factor.

A natural extension of the special models combines the two dimensions of heterogeneity.

Assume that conditions in 3.1 General Settings still hold. For a side I participation rate

N I , denote U I(N I)the total membership utility for all participating side I users, and

V I(N I)the total interaction utility that all participating side I users enjoy from having

one participating side J user. Then V I(N I)NJ gives the total interaction utility for side

I. These formulations of membership and interaction utility make sure that both values

are allowed to vary within a side, which creates user heterogeneity in two dimensions as

expected. Then the gross utility for side I when participation rates are(N I , NJ

)is

T I(N I , NJ

)= U I

(N I)

+ V I(N I)NJ , (12)

and the side I net surplus is

SI(N I , NJ

)= T I

(N I , NJ

)− P I

(N I , NJ

)N I .

On the cost side, the platform incurs a cost CI for each unit of good I produced, and a cost

c for each interaction between one side I and one side J user. Thus the total cost for the

platform when participation rates are(N I , NJ

)is given by CIN I + CJNJ + cN INJ .

To allow for a wider set of interesting cases, assume that one of the two dimensions of

utility can take nonpositive values. More speci�cally, assume that T I(N I , NJ

)is increasing,

strictly concave and twice di�erentiable in N I . Also assume that both U I(N I)and V I

(N I)

are twice di�erentiable. U I (0) = 0 and V I (0) = 0. Unlike the models before, here V I(N I)

can be either strictly concave or strictly convex. V I(N I)is also allowed to be a negative

function as long as the above conditions are met.

The above assumptions are realistic in many cases. Take the example of a newspaper. The

21

readers (side A) have positive membership values (from reading the content) and negative

interaction values (from reading the ads placed by advertisers (side B)). In this special case,

it is reasonable to assume that UA(NA)is positive, increasing and concave, and V A

(NA)is

negative, decreasing and convex. In words, the average reader derives higher (more positive)

membership value and lower (more negative) interaction value than the marginal reader.

Side B's participation rate NB is interpreted as �negative quality� of good A.

It can be shown that the two special cases in 3.2.1 and 3.2.2 both �t in this combined

model. For Armstrong (2006), set V I(N I)

= αIN I to eliminate interaction value hetero-

geneity, and c = 0 to eliminate platform interaction costs. For Rochet and Tirole (2003), set

U I(N I)≡ 0 to eliminate membership value heterogeneity, and CI = 0 to eliminate platform

unit costs.

Side I's Decision Problem


surplus SI(N I , NJ

).

maxNI

SI(N I , NJ

)= T I

(N I , NJ

)− P IN I ,


maxNI

U I(N I)

+ V I(N I)NJ − P IN I .


0 = U I1

(N I)

+ V I1

(N I)NJ − P I .


),

P I(N I , NJ

)= U I

1

(N I)

+ V I1

(N I)NJ , (13)

22


For Armstrong (2006), P I(N I , NJ

)= U I

1

(N I)

+ αINJ , which is (3). For Rochet and

Tirole (2003), P I(N I , NJ

)= V I

1

(N I)NJ , which is (8).

First Best Outcome

Suppose(N I , NJ


maxNI ,NJ

W(N I , NJ

),

where

W(N I , NJ

)= T I

(N I , NJ

)+ T J

(NJ , N I

)−(CIN I + CJNJ + cN INJ

)= U I

(N I)

+ V I(N I)NJ + UJ

(NJ)

+ V J(NJ)N I

−(CIN I + CJNJ + cN INJ

).

First order condition with respect to N Igives

0 = U I1

(N I)

+ V I1

(N I)NJ + V J

(NJ)− CI − cNJ .

From (13), U I1

(N I)

+ V I1

(N I)NJ = P I

(N I , NJ

). Then given

(N I , NJ


P I(N I , NJ

)= CI + cNJ − V J

(NJ)︸︷︷︸

2−sided factor

. (14)

Therefore, the welfare maximizing unit price for good I equals production cost (unit plus

interaction costs) minus cross-side externality that one side I user exerts on all participating

users of side J . In other words, for each unit of good I, side I is charged the social (instead

of platform) cost.


)= CI − αJNJ , which is (4). For Rochet and Tirole

23

(2003), P I(N I , NJ

)= cNJ − V J

(NJ), which is (9).


Suppose(N I , NJ


(N I , NJ

)maxNI ,NJ

π(N I , NJ

),


maxNI ,NJ

P I(N I , NJ

)N I + P J

(NJ , N I

)NJ −

(CIN I + CJNJ + cN INJ

).

First order condition with respect to N I together with (9) gives

0 = P I1

(N I , NJ

)N I︸︷︷︸

−µI

+P I(N I , NJ

)+ P J

2

(NJ , N I

)NJ︸︷︷︸

V J1 (NJ )NJ

−CI − cNJ ;

P I(N I , NJ

)= CI + cNJ − V J

1

(NJ)NJ + µI

= CI + cNJ − V J(NJ)︸︷︷︸

2−sided factor


+[V J(NJ)/NJ − V J

1

(NJ)]NJ︸︷︷︸

Spence distortion

. (15)


outcome is adjusted by two factors. The �rst is the upward monopoly market power factor

identical in one-sided monopoly pricing. The second factor, the Spence distortion, equals

the side J participation rate times the di�erence between per interaction utility enjoyed

by the average and the marginal side J users on board. To contrast with the Rochet and

Tirole (2003) special case, now depending on whether V J is strictly concave or convex, the

Spence distortion can be positive or negative. A detailed analysis of the Spence distortion

24

and nonoptimal quality provision follows in 3.4.


)= CI − αJNJ + µI , which is (5).

For Rochet and Tirole (2003),

P I(N I , NJ

)= cNJ − V J

(NJ)

+ µI +[V J(NJ)/NJ − V J

1

(NJ)]NJ ,

which is (10).

Second Best Outcome

Notice that in this model the platform still makes a negative pro�t on side I in the

�rst best outcome when V J(NJ)is positive. As a result, the second best outcome is more

plausible for policy makers in these cases.

maxNI ,NJ

W(N I , NJ

),

s.t. π(N I , NJ

)≥ 0.


L = W(N I , NJ

)+ λπ

(N I , NJ

)= U I

(N I)

+ V I(N I)NJ + UJ

(NJ)

+ V J(NJ)N I


)+ λ

[P I(N I , NJ

)N I + P J

(NJ , N I

)NJ −


)]= U I

(N I)

+ V I(N I)NJ + UJ

(NJ)

+ V J(NJ)N I

+ λ[P I(N I , NJ

)N I + P J

(NJ , N I

)NJ]

− (λ+ 1)(CIN I + CJNJ + cN INJ

).

25


0 = U I1

(N I)

+ V I1

(N I)NJ︸︷︷︸

P I(NI ,NJ )

+V J(NJ)

+ λ

P I1

(N I , NJ

)N I︸︷︷︸

−µI

+P I(N I , NJ

)+ P J

2

(NJ , N I

)NJ︸︷︷︸

V J1 (NJ )NJ

− (λ+ 1)

(CI + cNJ

).

This simpli�es to

P I(N I , NJ

)= CI + cNJ − V J

(NJ)︸︷︷︸

2−sided factor

+λ

1 + λµI︸︷︷︸

market power

+λ

1 + λ

[V J(NJ)/NJ − V J

1

(NJ)]NJ︸︷︷︸

Spence distortion

. (16)



maximizing price.


)= CI − αJNJ + λ

1+λµI , which is (6).

For Rochet and Tirole (2003),

P I(N I , NJ

)= cNJ − V J

(NJ)

+λ

1 + λµI +

λ

1 + λ

[V J(NJ)/NJ − V J

1

(NJ)]NJ ,

which is (11).

3.4 Quality Distortion in the Pro�t-Maximizing Outcome

As suggested in 2.2.2, for a monopoly platform in a two-sided market, it is natural to regard

the cross-side participation rate as an expression of product quality. Consider the combined

model in 3.3, for users on side I, consumer surplus is determined by N I(own participation

26

rate, or quantity), P I(own price), and NJ (cross-side participation rate); thus NJ �ts nat-

urally as the quality variable. This notion works �ne as long as side I enjoys interaction,

i.e. as long as the interaction value function V I(N I) is positive. When V I(N I) is negative,

however, an increase in NJ leads to a decrease side I's utility, so NJ needs to be interpreted

as the �negative quality.� As a result, I will start with the Armstrong (2006) and the Rochet

and Tirole (2003) special cases in which V I(N I) is assumed nonnegative and then proceed

to combined model in which V I(N I) is allowed to have either sign.

From (13), the marginal side I user's valuation of incremental quality Pq is

P I2

(N I , NJ

)= V I

1 (N I). (17)

Therefore, in Armstrong (2006), P I2

(N I , NJ

)= αI ; and in Rochet and Tirole (2003),

P I2

(N I , NJ

)= V I

1 (N I).

On the other hand, the average side I user's valuation of incremental quality 1x

´ x0Pqdv

is

1

N I

ˆ NI

0

P I2

(x,NJ

)dx =

1

N I

ˆ NI

0

V I1 (x)dx

=1

N IV I(N I). (18)

For Armstrong (2006), 1NI

´ NI

0P I

2

(x,NJ

)dx = 1

NIαIN I = αI . For Rochet and Tirole

(2003), 1NI

´ NI

0P I

2

(x,NJ

)dx = 1

NI VI(N I).

Compare (17) and (18), when V I(N I) > 0, quality for side I as measured by side J

participation is underprovided if 1NI V

I(N I) − V I1 (N I) > 0, overprovided if 1

NI VI(N I) −

V I1 (N I) < 0, and optimally provided if 1

NI VI(N I)− V I

1 (N I) = 0. Now recall from (15) that

N I[

1NI V

I(N I)− V I1 (N I)

]is just the Spence distortion term in side J 's price P J , which now

makes perfect sense: side J 's participation rate NJ is less than optimal when P J is distorted

upward, more than optimal when P J is distorted downward, and optimal when P J is not

27

distorted.

For Armstrong (2006), 1NI V

I(N I) − V I1 (N I) = αI − αI = 0, so quality for side I is

optimal (i.e. side J participation rate NJ is optimal) and there is no Spence distortion in

the pro�t-maximizing P J . Intuitively, to maximize pro�t the platform rewards side J for

interaction based on side J 's contribution to the marginal side I user. Since the interaction

value is homogeneous on side I, this is the same as rewarding side J based on its contribution

to the average of all participating side I users, which is the socially optimal pricing strategy.

Therefore, side J is fully compensated for its positive externality on side I, leading to optimal

side J participation.

For Rochet and Tirole (2003), 1NI V

I(N I)−V I1 (N I) > 0 since V I(N I) is twice-di�erentiable

and strictly concave and V I(0) = 0. Therefore for side I quality as NJ is underprovided

in the pro�t-maximizing outcome, and the pro�t-maximizing price P J includes a positive

Spence distortion component. Intuitively, to maximize pro�t the platform rewards side J for

interaction based on side J 's contribution to the marginal side I user. The socially optimal

pricing strategy, however, rewards side J according to its contribution to the average of all

participating side I users. When heterogeneity is present in interaction utility, the marginal

side I user values interaction less than the average of all participating side I users. Therefore

side J is not fully compensated in P J for its positive externality on side I, leading to less

than optimal side J participation.

For the combined model with user heterogeneity in two dimensions, however, the sign

of 1NI V

I(N I) − V I1 (N I) is no longer �xed: it is positive when V I

(N I)is strictly concave

and negative when V I(N I)is strictly convex. Therefore, side J is overcharged and NJ

underprovided when side I's interaction utility function V I(N I)is strictly concave; side

J is undercharged and NJ overprovided when V I(N I)is strictly convex. When V I () is

a positive function, NJ is interpreted as quality for side I. In contrast, when V I () is a

negative function, NJ is interpreted as negative quality for side I. Notice that the scenario

of overprovision of NJ is not presented in either special model with single-dimensional het-

28

erogeneity. For a concrete example of this, consider again the platform of a newspaper.

Recall that for readers (side A), UA(NA)is positive, increasing and strictly concave, and

V A(NA)is negative, decreasing and convex. For advertisers (side B), both UB

(NB)and

V B(NB)are positive, increasing and concave. Therefore, in the pro�t-maximizing outcome

1NAV

A(NA) − V A1 (NA) < 0 and 1

NBVB(NB) − V B

1 (NB) > 0. Thus NB, in this case the

�negative quality� for readers, is overprovided. On the other hand, NA, the quality for

advertisers, is underprovided. In words, the pro�t-maximizing outcome involves more ads

and fewer readers than optimal. In terms of prices charged to each side, the readers are

overcharged and the advertisers are undercharged compared to what is optimal.

4 Weyl (2010): Heterogeneity in Two Dimensions

4.1 Weyl's Model

4.1.1 Distribution Formulation

Let I, J ∈ (A,B) be the two sides of a market with a monopoly platform. Consider an

arbitrary agent i from side I. Let BIi be i's membership utility from being on the platform.

Let bIi be i's utility from interacting with one agent on side J . Then i′s total utility when

participation rate on side J is NJ is given by BIi + bIiN

J . For a pair(P I , NJ

), agent i would

stay on board if and only if her gross utility is no less than the price of good I, i.e.

BIi + bIiN

J ≥ P I .

De�ne

S ′I ≡

{(BI , bI) : BI + bINJ ≥ P I

}S ′I ≡

{(BI , bI) : BI + bINJ = P I

}

29

Notice that for side I users who chooses to participate, it is not necessary that both BI

and bI are positive. Similar to what is assumed in the combined model (Model I), users

can have negative (positive) membership values together with positive (negative) interaction

values.

For given price P I and side J participation NJ , de�ne participation rate on side I as

N I(P I , NJ

)=

¨S′If I(BI , bI

)dBIdbI ,

for some well-behaved joint density function f I () .

By multivariable Leibniz Integral Rule,

N I1

(P I , NJ

)= 0 +

¨S′I

∂(BI + bINJ − P I

)∂P I

f I(BI , bI

)dBIdbI

= −¨S′If I(BI , bI

)dBIdbI < 0.

Therefore invertingN I(P I , NJ

)with respect to P I gives a well-de�ned function P I(N I , NJ).

For a �xed pair of participation rates (N I , NJ), there is a unique pair of prices and respective

utility. Thus, one can treat the monopoly platform as choosing participation rates directly.

De�ne

SI ≡{

(BI , bI) : BI + bINJ ≥ P I(N I , NJ

)}SI ≡

{(BI , bI) : BI + bINJ = P I

(N I , NJ

)}Gross utility for side I users on the platform for a participation rate pair

(N I , NJ

)is

τ I(N I , NJ

)=

¨SI

(BI + bINJ

)f I(BI , bI

)dBIdbI . (19)

In addition, assume the same cost structure as in Model I: the platform incurs a cost CI

for each unit of good I produced, and a cost c for each interaction between one side I and

30

one side J user. Thus the total cost for the platform when participation rates are(N I , NJ

)is again given by CIN I + CJNJ + cN INJ .

Total welfare is

ω(N I , NJ

)= τ I

(N I , NJ

)+ τJ

(NJ , N I


)=

¨SI

(BI + bINJ

)f I(BI , bI

)dBIdbI

+

¨SJ

(BJ + bJN I

)fJ(BJ , bJ

)dBJdbJ


). (20)

Platform's pro�t is the same as in the Combined Model

π(N I , NJ

)= P I

(N I , NJ

)N I + P J

(NJ , N I

)NJ


). (21)

4.1.2 Results

Solve for the �rst best and pro�t-maximizing outcomes.

First Best Outcome

To maximize total welfare, price charged to side I is

P I(N I , NJ

)= CI + cNJ − bJNJ︸︷︷︸

2−sided factor

, (22)

where

bJ ≡˜SJ b

JfJ(BJ , bJ

)dBJdbJ˜

SJ fJ (BJ , bJ) dBJdbJ

31

is the average per interaction value of all participating users on side J .


To maximize platform's pro�t, price charged to side I is

P I(N I , NJ

)= CI + cNJ − bJNJ︸︷︷︸

2−sided factor


+(bJ − bJ

)NJ︸︷︷︸

Spence distortion

, (23)

where

bJ ≡ P J2 = −N

J2

NJ1

=

˜SJ b

JfJ(BJ , bJ

)dBJdbJ˜

SJ fJ (BJ , bJ) dBJdbJ

is the per interaction value of marginal users on side J .

4.1.3 Quality Distortion in the Pro�t-Maximizing Outcome

From (23), in the pro�t-maximizing outcome with positive interaction values on side J , qual-

ity for side J users as measured by side I's participation rate is underprovided (overprovided)

when side I is overcharged (undercharged). This happens when the average interaction value

of participating users on side J is higher (lower) than the average interaction value of the

marginal user on side J . When interaction values on side J is negative, interpret side I's

participation rate as the �negative quality� and the same arguments follows.

4.2 Comparing Model I and Weyl's Model

Both Model I and Weyl's model in section 4 allow users' utility within one side to di�er

in both the membership and the interaction dimensions: model I speci�es utility in each

32

dimension on an aggregate level, while Weyl's model starts on an individual level with

distribution functions. However, a brief comparison will show that the two models cover

di�erent forms of utility functions.

Starting with Weyl's model, let us examine (19) to see how side I's gross utility changes

when side J 's participation rate NJ increases by a small amount. Twice di�erentiate τ I

with respect to NJ using multivariable Leibniz Integral Rule gives a function containing NJ .

Therefore, side I's gross utility is non-linear in NJ . This contrast with that in Model I, where

T I is apparently linear in NJ .

Having examined two models of single-dimensional heterogeneity and two models of two-

dimensional heterogeneity, one can observe that equilibrium outcomes in all these models

share similar characteristics despite di�erent assumptions on the two dimensions of utility.

More speci�cally, comparison of equilibrium prices in two-sided and one-sided markets, com-

parison of �rst best and pro�t-maximizing outcomes within a two-sided market, as well as

analysis of quality distortion in a two-sided market yield similar results with di�erent user

utility assumptions. As a result, in the next section I will remove speci�c assumptions on

the relationship between membership and interaction utility and study the two-dimensional

heterogeneity model in a generalized form.

5 Generalized Model (Model II): Arbitrary Relationship

between Two Dimensions

Both Model I and Weyl's model are still rather restrictive in at least one important way:

they assume special relations between the two sources of heterogeneity. In Model I, the side

I aggregate utility is additively separable in total membership utility and total interaction

utility. Moreover, there is constant return to quality on the aggregate level (i.e. side I as a

whole derives the same utility from interacting with each additional side J user). In Weyl's

model, each user i's utility is additively separable in her individual membership utility and

33

interaction utility. In addition, there is constant return to quality for each user (i.e. user i

from side I derives the same utility from interacting with each additional side J user). This

section introduces a more generalized monopoly model that allows arbitrary relationship

between the membership and interaction dimensions. One result of the relaxation is that

the new model allows variations in marginal utility of quality. It will also be shown that

distortions to the equilibrium pro�t maximizing prices can be explained using the same

notions as in the previous models (namely the classic monopoly market power, the two-sided

factor, and the Spence distortion). I will end this section with a simple but potentially useful

special case that can be easily studied under this generalized formulation.

5.1 Formulation

Assume that 3.1 General Settings still holds. Denote T I(N I , NJ

)the total utility for

side I users on board if(N I , NJ

)are the respective participation rates. Suppose also that

T I is twice di�erentiable in each argument and strictly concave in N I , and T I(0, NJ

)= 0.

Thus the side I net surplus is

SI(N I , NJ

)= T I

(N I , NJ

)− P I

(N I , NJ

)N I .

The cost structure is assumed to be the same as Model I and Weyl's model, in which

the platform incurs a cost CI for each unit of good I produced, and a cost c for each

interaction between one side I and one side J user. Thus the total cost for the platform

when participation rates are(N I , NJ

)is still given by CIN I + CJNJ + cN INJ .

Side I's Decision Problem


surplus SI(N I , NJ

).

maxNI

SI(N I , NJ

)= T I

(N I , NJ

)− P IN I ,

34


0 = T I1(N I , NJ

)− P I .


),

P I(N I , NJ

)= T I1

(N I , NJ

), (24)


First Best Outcome

Suppose(N I , NJ


maxNI ,NJ

W(N I , NJ

),

where

W(N I , NJ

)= T I

(N I , NJ

)+ T J

(NJ , N I


)First order condition with respect to N Igives

0 = T I1(N I , NJ

)+ T J2

(NJ , N I

)− CI − cNJ .

From (24), T I1(N I , NJ

)= P I

(N I , NJ

). Then given

(N I , NJ


P I(N I , NJ

)= CI + cNJ − T J2

(NJ , N I

)︸︷︷︸2−sided factor

. (25)

Therefore, the welfare maximizing unit price for good I equals production cost (unit plus

interaction costs) minus the total cross-side externality that the marginal side I user exerts

on side J . Notice that the two-sided factor in the �rst best price is present without any

special relationship between the two dimensions of utility assumed in Model I and Weyl's

35

model.


Suppose(N I , NJ


(N I , NJ

)maxNI ,NJ

π(N I , NJ

),


maxNI ,NJ

P I(N I , NJ

)N I + P J

(NJ , N I

)NJ −


).


0 = P I1

(N I , NJ

)N I︸︷︷︸

−µI

+P I(N I , NJ

)+ P J

2

(NJ , N I

)NJ︸︷︷︸

TJ12(NJ ,NI)NJ

−CI − cNJ ;

P I(N I , NJ

)= CI + cNJ − T J12

(NJ , N I

)NJ + µI

= CI + cNJ − T J2(NJ , N I



+[T J2(NJ , N I

)/NJ − T J12

(NJ , N I

)]NJ︸︷︷︸

Spence distortion

. (26)


outcome is again adjusted by the one-sided monopoly market power and the Spence distor-

tion. Details on the Spence distortion and the quality provision argument are presented in

5.2.

Second Best Outcome

Notice that in this model the platform still makes a negative pro�t on side I in the �rst

best outcome when T J2(NJ , N I

)is positive. As a result, the second best outcome is more

36

plausible for policy makers in these cases.

maxNI ,NJ

W(N I , NJ

),

s.t. π(N I , NJ

)≥ 0.


L = W(N I , NJ

)+ λπ

(N I , NJ

)= T I

(N I , NJ

)+ T J

(NJ , N I


)+ λ

[P I(N I , NJ

)N I + P J

(NJ , N I

)NJ −


)]= T I

(N I , NJ

)+ T J

(NJ , N I

)+ λ

[P I(N I , NJ

)N I + P J

(NJ , N I

)NJ]

− (λ+ 1)(CIN I + CJNJ + cN INJ

).


0 = T I1(N I , NJ

)︸︷︷︸P I(NI ,NJ )

+T J2(NJ , N I

)

+ λ

P I1

(N I , NJ

)N I︸︷︷︸

−µI

+P I(N I , NJ

)+ P J

2

(NJ , N I

)NJ︸︷︷︸

TJ12(NJ ,NI)NJ

− (λ+ 1)

(CI + cNJ

).

37

This simpli�es to

P I(N I , NJ

)= CI + cNJ − T J2

(NJ , N I


+λ

1 + λµI︸︷︷︸

market power

+λ

1 + λ

[T J2(NJ , N I

)/NJ − T J12

(NJ , N I

)]NJ︸︷︷︸

Spence distortion

. (27)



maximizing price.

5.2 Quality Distortion in the Pro�t Maximizing Outcome

Same as in Model I and Weyl's model, here one can still treat NJ as the quality to side

I users. According to 2.2.1, whether quality is overprovided or underprovided in the pro�t

maximizing outcome depends on the sign of equation (1).

From (24), the marginal side I user's valuation of incremental quality Pq is

P I2

(N I , NJ

)= T I12(N I , NJ). (28)

On the other hand, the average side I user's valuation of incremental quality 1x

´ x0Pqdv

is

1

N I

ˆ NI

0

P I2

(x,NJ

)dx =

1

N I

ˆ NI

0

T I12(x,NJ)dx

=1

N IT I2 (N I , NJ). (29)

38

Combining (28) and (29), the quality distortion indicator in (1), x(

1x

´ x0Pqdv − Pq

), is

N I

[1

N I

ˆ NI

0

P I2

(x,NJ

)dx− P I

2

(N I , NJ

)]

= N I

[1

N IT I2 (N I , NJ)− T I12(N I , NJ)

]. (30)

Notice from (26) that N I[

1NI T

I2 (N I , NJ)− T I12(N I , NJ)

]is exactly the Spence distortion

on the pro�t maximizing price for side J . From the perspective of quality provision, qual-

ity as measured by side J participation rate NJ is underprovided (overprovided/optimally

provided) for side I when the average of participating side I users' valuation of incremental

quality is greater than (less than/equal to) the marginal side I user's valuation. From the

perspective of pro�t-maximizing price setting, side J is overcharged (undercharged/optimally

charged) when NJ is less than (more than/equal to) optimal. One important observation

from the above interpretation is that the existence of non-optimal quality provision and

its associated Spence distortion in pro�t-maximizing price are present without assuming

particular relationship between the two dimensions of utility.

5.3 Special Case: Cobb-Douglas Interaction Utility Function

5.3.1 Formulation

With the model in 5.1 that allows arbitrary relationship between two dimensions of hetero-

geneity, let us consider a special case with Cobb-Douglas total utility function. Speci�cally,

let T I(N I , NJ

)=(N I)α (

NJ)β

for some scalar α, β > 0 and I ∈ {A,B}.

From (24), inverse demand function on side I is

P I(N I , NJ

)= α

(N I)α−1 (

NJ)β. (31)

From (25), �rst best price on side I is

39

P I(N I , NJ

)= CI + cNJ − β

(NJ)α (

N I)β−1︸︷︷︸

2−sided factor

. (32)

Therefore, welfare maximizing price is less than cost of production if β > 0, equal to

cost of production if β = 0 (degenerated Armstrong special case), and more than cost of

production if β < 0.

From (26), pro�t maximizing price on side I is

P I(N I , NJ

)= CI + cNJ − β

(NJ)α (

N I)β−1

+ µI

+[β(NJ)α (

N I)β−1 − αβ

(NJ)α (

N I)β−1

]= CI + cNJ − β

(NJ)α (

N I)β−1︸︷︷︸

2−sided factor


+ (1− α) β(NJ)α (

N I)β−1︸︷︷︸

Spence distortion

. (33)

5.3.2 Quality Distortion in the Pro�t Maximizing Outcome

From (30), the quality distortion indicator for side I is

x

(1

x

ˆ x

0

Pqdv − Pq)

= N I

[1

N IT I2 (N I , NJ)− T I12(N I , NJ)

]= N I

[1

N Iβ(N I)α (

NJ)β−1 − αβ

(N I)α−1 (

NJ)β−1

]= (1− α) β

(NJ)α (

N I)β−1

. (34)

Notice from (33) that (1− α) β(NJ)α (

N I)β−1

is exactly the Spence distortion on the

pro�t maximizing price for side J . From the quality provision perspective, quality for side

I as measured by side J participation rate NJ is underprovided (overprovided/optimally

40

provided) when α is less than (larger than/equal to) 1. From the pro�t maximizing price

setting perspective, side J is overcharged (undercharged/optimally charged) when NJ is less

than (more than/equal to) optimal.

6 Conclusion

This paper studies price theory of monopoly platforms in two-sided markets where each side

of users derives both membership and interaction utility from market participation. Ap-

proaching the platform's problem as choosing participation rates instead of choosing prices,

I reformulate and solve special cases with single-dimensional user heterogeneity studied in

Rochet and Tirole (2003) and Armstrong (2006). I then present a natural combination of the

single-dimensional models that allows speci�c relations between membership and interaction

utility and compare it with the basic two-dimensional heterogeneity model in Weyl (2010).

From there, I construct a generalized model that allows arbitrary relationship between the

two sources of utility. With the generalized model, I perform comparison of equilibrium

prices in two-sided and one-sided markets, comparison of �rst best and pro�t-maximizing

outcomes within a two-sided market, as well as analysis of quality distortion in a two-sided

market. Compared to prices in one-sided markets, prices in two-sided markets are adjusted

by a two-sided factor that (partially) internalizes the own side's externality on the other

side. Within a two-sided market, the pro�t-maximizing price di�ers from the �rst best price

by a conventional, one-sided market power factor and a Spence distortion factor unique for

two-sided markets. The Spence distortion results from the monopoly platform's inability to

provide optimal quality (as measured by other side's participation rate) when heterogeneity

is presented in users' valuation of quality within the other side. Notice that both the two-

sided factor and the Spence distortion factor are associated with participation of the other

side.

Derivation and interpretation of equilibrium prices in the generalized model suggest that

41

all the factors mentioned above, which are among the most interesting attributes of monop-

olistic pricing in two-sided markets, can be studied without assuming special relationship

between the two dimensions of user heterogeneity. Special models like those in Armstrong

(2006), Rochet and Tirole (2003) apply well to certain markets with only one prominent

dimension of user heterogeneity. Similarly, distribution formulation like that in Weyl (2010)

might be needed for empirical testing. However, for the objective of understanding the fun-

damentals of pricing in monopoly two-sided markets, the generalized formulation presented

in this paper can serve as an easily accessible model.

42

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