Network Effect and Multi-Network Sellers’ Dynamic Pricing:

Evidence from the US Smartphone Market

Rong Luo ∗

Department of Economics, University of Georgia

October 23, 2016

Abstract

The utility a consumer realizes from owning a smartphone increases with its oper-ating system (OS) network size. The OS network effect influences the dynamic pricingstrategy of multi-OS telecom carriers, who can internalize the competition across OSs.In this paper, I first study the carriers’ pricing strategy in a two-period, two-OS the-oretical model. I then estimate a structural model of consumer demand and telecomcarriers’ dynamic pricing game for two-year contract smartphones, using product leveldata from August 2011 to July 2013 in the US. From both the theoretical model andthe estimates, I find that multi-OS carriers choose lower prices for large OSs than smallOSs, while single-OS sellers would do the opposite in equilibrium. The estimate of theOS network effect in consumer utility is positive and significant. Counterfactual anal-yses show that, if the carriers specialized in different OSs or if the two-year contractswere eliminated, consumer surplus and smartphone penetration would decrease, andthe big carriers’ profits would increase. This paper is the first to study multi-networksellers’ dynamic pricing problem and to estimate a model of dynamic continuous choicegame among asymmetric multi-product firms.

Keywords: OS Network Effect, Carrier Dynamic Pricing Game, Two-Year Contract,Asymmetric Multi-OS Sellers, Value Function Approximation, MPEC.

∗University of Georgia Terry School of Business, 310 Herty Drive, Athens, GA 30602, USA. Email:rluo@uga.edu. This paper is previously circulated as The Operating System Network Effect and the Carri-ers’ Dynamic Pricing of Smartphones in US. I am very grateful to Mark Roberts and Paul Grieco for theirconstant support and very helpful comments. I would also like to thank Ying Fan, Gautam Gowrisankaran,Martin Hackmann, Hiroyuki Kasahara, Kenneth Judd, Robin Lee, Huihui Li, Yue Liu, Charles Murry, PeterNewberry, Kathleen Nosal, Joris Pinkse, Marc Rysman, Philipp Schmidt-Dengler, John Turner, Daniel Xu,and participants in PSU IO workshop, 13th Annual International Industrial Organization Conference, 2015CEPR-JIE School and Conference, and 2016 SHUFE IO summer school for their thoughtful comments. Allerrors are my own responsibility.

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1 Introduction

In markets with network effects, such as the computer and smartphone markets, consumers

value the size of the installed consumer base. Due to the network effect, the price of a

product affects its network size and thus future demand. This leads the sellers to make

dynamic pricing decisions. However, while the literature has studied the pricing strategy

of single-network sellers, little attention has been paid to multi-network sellers’ dynamic

pricing strategy, who can price differently across networks and internalize the network

competition.

I study the multi-network sellers’ pricing game in the US smartphone industry. Smart-

phones are subject to a network effect that arises through the operating systems (OS) for

two reasons. First, application stores of the OSs generate an indirect network effect. Ap-

plication developers choose to launch apps on large OSs. In return, more consumers adopt

large OSs because of more variety and better quality of applications.1 Second, a direct

OS network effect exists. Friends and family members prefer adopting the same OS. The

benefits of doing so include convenient communication, the ease of sharing files and app

purchases, and lower learning costs.

Telecom carriers act like multi-network sellers due to two facts. First, all four of the

biggest carriers — Verizon, AT&T, Sprint, and T-Mobile — have been selling smartphone

models with iOS, Android, Blackberry, and Windows Phone operating systems. Second,

the two-year contract model of selling smartphones provides the carriers a channel to price

differently across OSs. Both the availability of smartphones with different OSs from a

carrier and the two-year contract policy have important impacts on consumer surplus, the

carriers’ profits, and the smartphone industry growth.

To provide the intuition of the multi-OS carriers’ pricing strategy, I first study a two-

period, two-OS theoretical model. I find that a multi-OS seller chooses a lower price for

the large OS than the small OS in the first period, while the opposite happens in the

equilibrium of the single-OS sellers’ price competition. Single-OS sellers can’t internalize

the OS competition. The large OS seller has stronger market power and chooses a higher

price. In contrast, a multi-OS seller is able to internalize the OS competition, and it prefers

OS concentration to close OS competition, because a large OS can increase both future

demand and prices due to strong network effect. Thus, the multi-OS seller chooses a lower

1There is great variation in the numbers of applications across OSs. The correlation between the numberof applications and the number of users for an OS is 0.85, according to monthly data from 2011 to 2013.

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price for the initially larger OS to achieve concentration.

Data show that the carriers’ pricing of smartphones is consistent with the multi-OS

seller’s differentiated pricing strategy. To measure the impacts of the carriers’ pricing on

their profits and consumer surplus, I develop a structural model of consumers’ demand

and carriers’ dynamic pricing game over smartphones bundled with two-year contracts. I

assume that consumers make static discrete purchase decisions, where the OS network sizes

affect consumers’ utility, and the carriers choose smartphone prices to maximize long-run

profits. I estimate the model using product level data from 2011 to 2013.

It is a challenge to estimate the model of the carriers’ dynamic pricing game because

it features continuous choices, asymmetric multi-product firms, and high dimension state

space. Existing estimation methods for dynamic games can not be used in this paper.

I deal with this challenge by combining the value function approximation approach and

mathematical programming with equilibrium constraints (MPEC) to find equilibria of the

game in the estimation. Multiple equilibria of the game may exist and present issues to the

estimation. I explain the assumptions needed for identification, the continuity of the GMM

objective function of the estimation, and the implications on the counterfactual results in

the corresponding sections.

I use instrumental variables to identify the OS network effect and the price coefficient.

The estimation results show that the OS network effect is positive and significant, and

adding 19.41 million users for an OS increases a consumer’s utility by the same amount

as cutting the price by $100. Consistent with the results in the theoretical model, I find

that the carriers’ estimated markups on the large OSs (Android and iOS) are lower than

on small OSs (Blackberry and Windows Phone). Nonetheless, 38.25% of the four carriers’

profits are from Android model and 55.71% from iOS models due to their high volume of

sales.

With the estimates, I study two counterfactual cases. In the first case, I find that if each

carrier specialized in one OS and no two carriers specialized in the same OS, the prices of

the Android and iOS models would increase; the profits of Verizon (specialize in Android)

and AT&T (iOS) would increase by 48.31% and 74.45%, while Spring (Blackberry) would

and T-Mobile (Windows Phone) would each lose profits by more than 60%. The large

profit changes are due to OS specialization and price adjustments. The OS market would

be less concentrated, and the smartphone penetration rate by October 2013 would drop

from 78.36% in the data to 62.53%. Consumer surplus would decrease by $0.14 billion

during the sample period.

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In the second case, I find that if the two-year contracts were eliminated, AT&T’s profit

would increase by 18.25% because it would no longer offer high discounts on iPhones but

still have strong iPhone demand; Verizon’s profit would drop by 8.69% because of lower

demand of Android models, and more than half of Verizon’s customers bought Android

models in the data; and the profits of Sprint and T-Mobile would decrease by 44.32%

and 40.18%. The smartphone penetration rate would decrease from 78.36% in the data to

58.31%. Consumer surplus would decrease by $0.86 billion.

This paper contributes to the network effect literature by studying the multi-network

sellers’ dynamic pricing strategy, which hasn’t been analyzed either theoretically or em-

pirically. Theoretical research on network effects has focused on the competition between

single-network sellers, but not the prices of multi-network sellers. Katz and Shapiro (1985),

Farrell and Saloner (1986), Katz and Shapiro (1992), Katz and Shapiro (1994), Shapiro and

Varian (1999), Rochet and Tirole (2003), Armstrong (2006), Rochet and Tirole (2006) Zhu

and Iansiti (2007), Rysman (2009), and Weyl (2010) all study the impact of the network

effect on the prices of either monopolistic or oligopolistic single-network sellers. In this

paper, I find that the dynamic pricing strategy of multi-network sellers is opposite to the

equilibrium strategies of single-network sellers.

The empirical literature on network effects has studied markets in which single-network

manufacturers set retail prices, including papers that study the network effect in the yellow

page industry (Rysman (2004)), the VCR industry (Park (2004)), the PDA industry (Nair,

Chintagunta, and Dube (2004)), the ACH banking industry(Ackerberg and Gowrisankaran

(2006)), the video game industry (Dube, Hitsch, and Chintagunta (2010), Lee (2013)), and

the DVD player industry (Gowrisankaran, Park, and Rysman (2014)). In these markets,

manufacturers can’t internalize competition across networks. However, in this paper, I am

interested in the multi-OS carriers’ dynamic pricing strategy.

This paper also makes a contribution to the literature on the estimation of dynamic

games. Aguirregabiria and Mira (2007), Pakes, Ostrovsky, and Berry (2007), Pesendor-

fer and Schmidt-Dengler (2008), Kasahara and Shimotsu (2012) have proposed estimation

approaches for dynamic discrete choice games, which are based on conditional choice proba-

bilities. Since the pricing game in this paper is a continuous choice game, these approaches

can not be used. Bajari, Benkard, and Levin (2007) propose a two-step method that

can estimate dynamic games with continuous choices. The first step estimates the policy

functions, and the second step estimates the full model using the method of simulated min-

imum distance. If I use this two-step method in this paper, the first stage results would

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be very inaccurate, because there aren’t enough observations for each product due to the

high-dimension state space and the large number of products. In addition, this method

requires monotonicity of the policy functions on shocks, and linearity of the value functions

in parameters is assumed to reduce computation time. It is hard to prove monotonicity

of prices in shocks under the differentiated pricing strategy, and the value functions are

nonlinear in model parameters. Therefore, the two-step method can not be applied to this

paper either.

Several papers have studied the dynamic pricing problems of single-product firms.

Benkard (2004) use static estimates to solve a dynamic oligopoly model with four single-

product aircraft firms. Liu (2010) and Dube, Hitsch, and Chintagunta (2010) analyze the

dynamic decisions of two oligopolistic video game console manufacturers. They assume

parametric forms of the policy functions instead of solving for equilibrium. Goettler and

Gordon (2011) analyze the dynamic investment and pricing problem of two single-product

microprocessor firms. Sweeting (2012) tests how accurately dynamic pricing models de-

scribe sellers’ behavior in secondary markets for event tickets. Since the telecom carriers in

this paper are asymmetric multi-product firms, value function iteration or policy function

iteration would be difficult to implement and inaccurate.

Instead of using the two-step methods or policy function iteration, I solve the multi-

network carriers’ dynamic pricing game in the estimation. To do so, I approximate the

carriers’ value functions with basis functions to derive the carriers’ first order conditions

and develop an efficient algorithm to find equilibria.2 I estimate the model using Gener-

alized Methods of Moments (GMM) with MPEC, which was introduced by Su and Judd

(2012). The moment conditions are based on the orthogonality between unobserved shocks

and exogeneous variables and instrumental variables. The carriers’ Bellman equations are

imposed as constraints on the value function approximation. Ai and Chen (2003) and

Chen (2007) propose a minimum distance estimator with sieve approximation and show

efficiency. Barwick and Pathak (2015) use sieve approximation and MPEC method to

estimate a model of dynamic optimization problem.

This paper also contributes to the literature on the smartphone industry. Parker and

Van Alstyne (2010) analyze innovation and the platform openness control. Zhu, Liu, and

Chintagunta (2011) and Sinkinson (2014) study the incentives and the effects of the exclu-

sive contracts between Apple and AT&T. Kim (2012) focuses on variations in consumers’

2There is a multi-equilibria issue when solving for equilibria. I will discuss the issue whenever it is aconcern in the following sections.

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adoption of mobile apps across different OSs. Boudreau (2012) discusses app developers’

innovation patterns. Bjorkegren (2014) estimates consumers’ dynamic demand for mobile

phones. Bresnahan, Orsini, and Yin (2014) model and estimate mobile app developers’

OS platform choice. In this paper, I study the impact of the carriers’ two-year contract

discounts.

The paper proceeds as follows. In Section 2, I study a two-period, two-OS model to

compare prices in two different supply settings. Section 3 sets up the consumer demand

and the carriers’ supply model for smartphones with two-year contracts. The industry

background and data used in this paper is described in Section 4. Section 5 discusses

identification and estimation details. The estimated results are presented in Section 6.

Section 7 shows the two counterfactual analysis results. Section 8 concludes the paper.

2 A Two-Period, Two-OS Model

In this section, I study a multi-network seller’s pricing strategy in a two-period, two-OS

model and compare it with single-network sellers’ strategies. This two-period model helps

to understand the telecom carriers’ pricing problem of smartphones in Section 3 and the

observed pricing strategy in data in Section 4.

2.1 Demand Model

There are two smartphone models, A and B, in a two-period economy. The two models have

different OSs, which are also denoted by A,B to simplify notation. Let the total mass of

consumers be 1 and the market shares of the two OSs be nt = (nAt, nBt) at the beginning

of period t. Assume that nA1 + nB1 < 1, which implies that not all consumers have

smartphones in the beginning of the first period. In each period, only consumers who do

not own smartphones enter the market. So the market size in period t is Mt = 1−nAt−nBt.Consumer i’s utility of buying smartphone j ∈ A,B in period t is:

uijt = δj + γnjt − αpjt + εijt,

in which δj , j ∈ A,B, is the consumer utility from the characteristics of model j; pjt is

the carrier’s smartphone price in period t; εijt is the idiosyncratic utility shock of option

j ∈ A,B in period t; and γ and α are parameters for OS network size and price.

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Assumption 1. A positive network effect exists, γ > 0. Without loss of generality, assume

that network A has a higher market share than B in period 1: nA1 > nB1.3

Assumption 2. The two models have the same characteristics, δA = δB, and the same

unit cost, c = 0.4

Assumption 2 means that the only difference between A and B lies in their network

sizes. This assumption allow me to focus on the network size effect in this model.

An outside option, not to buy any smartphone, exists in each period. Normalize the

mean utility of the outside option to be zero. Consumer i’s utility of the outside option is

ui0t = εi0t. Assume that the utility shock εijt follows a Type-I extreme value distribution

and is i.i.d. across consumers, models, and periods. The sales market share of model j in

period t is

sjt(pcAt, p

cBt, nt) =

e(δj+γnjt−αpjt)

1 +∑

k=A,B e(δk+γnkt−αpkt),∀j ∈ 0, A,B. (1)

An OS network grows due to the new sales of smartphones. At the beginning of the second

period, the market share of OS j ∈ A,B is

nj2 = nj1 +M1sj1(pcA1, pcB1, n1). (2)

Therefore, the market size in the second period is M2 = 1−nA2−nB2 = M1s01, where s01

is the market share of the outside option in the first period.

2.2 Two Single-Network Sellers’ Pricing Game

Consider two single-network sellers who play a dynamic pricing game in the two periods.

Denote the two sellers by A,B too for notation simplification. Seller j ∈ A,B only

sells smartphone j. In this subsection, I use superscript m to denote the single-network

sellers (manufacturers who sell directly to consumers) in this subsection. For seller j, the

3There are many reasons that the networks are asymmetric in a particular point of time. Different OSsmay enter the market in different years. Their companies may have different demand and supply shocks.Different operating systems may have different openness towards smartphone manufacturers.

4By normalizing costs to be zero, the prices in this section can be interpreted as markups that the carrierearns. When the costs are not zero, choosing prices is equivalent to choosing markups.

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profit in each period t is

πmjt (pmAt, p

mBt) = pmjtsjt(p

mAt, p

mBt)Mt.

The dynamic problem of seller j ∈ A,B is

maxpmj1πmj1(pmA1, p

mB1) + βdmax

pmj2πmj2(pmA2, p

mB2, n2|pmA1, p

mB1).

subject to the OS share transition rule in equation (2). Let (p∗mA1 , p∗mB1 , p

∗mA2 , p

∗mB2) be the

equilibrium prices.

Proposition 1. (1) The optimal price of A is higher than that of B in both periods:

p∗mAt > p∗mBt , for t = 1, 2.

(2) Network A keeps its network advantage in the second period.

Proof. See Section A.1 of the Supplemental Material.

The first statement of Proposition 2 says that, seller A chooses higher prices than B

does in both periods because seller A has initial OS network size advantage. Suppose that

both models have the same price in the first period, then A’s marginal profit is greater

than B’s, which is zero. So seller A would increase price. In equilibrium, A’s price is higher

than B’s price in the first period. The second statement says that A can keep its network

advantage in the second period.

2.3 A Monopoly Multi-Network Seller’s Pricing Problem

I now consider one seller who sells both models and chooses prices in the two periods to

maximize the total discounted profits. The OS market shares at the beginning of period t

are nt = (nAt, nBt). The seller’s profit in period t is:

πt(pcAt, p

cBt|nt) = [pcAtsAt(p

cAt, p

cBt, nt) + pcBtsBt(p

cAt, p

cBt, nt)]Mt.

where the superscript c means a multi-network seller (carrier). In the first period, the

seller’s profit maximization problem is:

maxpcA1,p

cB1

π1(pcA1, pcB1|n1) + βd max

pcA2,pcB2

π2(pcA2, pcB2, n2|pcA1, p

cB1) (3)

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subject to the OS share transition rule in equation (2). βd is the discount factor. The first

period prices affect the seller’s profits in both periods because of the network effect and

market size transition. The multi-network seller can internalize the competition between

A and B since it chooses the prices of both products. Optimal prices require that the

marginal profits from pcAt and pcBt are zero. The dynamic optimization problem can be

solved backwards. Let the optimal prices of the multi-network seller be (p∗cA1, p∗cB1, p

∗cA2, p

∗cB2).

Proposition 2. (1) The price of A is lower than that of B in the first period: p∗cA1 < p∗cB1.

(2) The price difference between the two models |pc∗A1 − pc∗B1| increases as the OS network

effect, γ, increases.

(3) The OS market share difference at the beginning of the second period (nA2 − nB2)

increases with the OS network effect, γ.

Proof. See Section A.2 of the Supplemental Material.

The first statement in Proposition 1 says that, when the OS network effect exists,

the multi-network seller chooses a lower price for the larger network A than for B in the

first period. It can be shown that the multi-OS seller’s second-period profit is convex in

(nA2, nB2). The seller prefers OS concentration because the strong network effect of A can

attract more future consumers, and it can get more profits in the second period.5 The sec-

ond statement means that the seller chooses more differentiated prices as the network effect

become stronger. The third statement says that the market becomes more concentrated in

the large operating system as the network effect increases.

The discount factor also makes an impact on the first period prices. Without the

dynamic effect (βd = 0), the optimization problem is static. In a static pricing problem

with Logit demand model, the multi-OS seller will choose the same price for the two

products, because the cross-derivatives of prices (αsAtsBt) are the same, and so are the

carrier’s two first-order conditions, as shown in Supplemental Material A.1. However, when

βd > 0, the large network has a lower price in the first period, pcA1 < pcB1, though the prices

of A and B in the second period are the same, pcA2 = pcB2.

The OS network effect affects optimal prices similarly as the discount factor. If there is

no network effect (γ = 0), A and B will have the same price in both periods since they are

symmetric products. However, if the network effect exists (γ > 0), the larger network A

5The reason for selling products with different OSs by the carriers is because that consumers are het-erogeneous. Some consumers might prefer B no matter how large network A is. So the carrier still sells B,which can increase its profits.

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will have a lower price than B in the first period because the seller’s second-period profit

increases with the OS concentration. Thus, both βd and γ need to be positive for the seller

to choose different prices for A and B.

The Logit demand model is an ideal choice to analyze the impact of network effect

on the multi-network seller’s prices. With the Logit demand model, the seller chooses the

same price for both OSs in each period when γ = 0 and βd > 0. This provides a clear

benchmark for studying the seller’s prices when γ > 0. Any price differentiation across the

two OSs will reflect only the impact of the network effect.

Though I consider a monopoly multi-network seller, its dynamic pricing strategy carries

over to the case with multiple multi-network sellers.6 Because the competition among the

sellers doesn’t change their incentive to take advantage of the initially larger network, and

using prices to grow different OSs can not be an equilibrium.

2.4 Comparing the Two Models

Single-OS sellers and the multi-OS seller choose the opposite pricing strategies in the first

period. This is because of the difference in their ability to internalize competition effect

across OS and the network effect. The multi-network seller prefers network concentration

and uses price differentiation to achieve that. Single-network sellers set prices based on

their network sizes.

In Section 4, I show that the carriers’ pricing strategy in data is consistent with the

multi-network seller’s dynamic pricing strategy. Next, I set up an empirical model of the

consumers’ demand and the carriers’ pricing of smartphones in the US, which is more

general than the two-period, two-OS model.

3 An Empirical Model of Demand and Supply of Smart-

phones

In this section, I design a structural model of consumer demand and carriers’ dynamic

pricing game for two-year contract smartphones. There are four leading wireless carriers,

four operating systems, and hundreds of smartphone models. I assume consumers make

static purchase decisions of smartphones and use the random coefficient demand model

as in Berry, Levinsohn, and Pakes (1995). I discuss more about the assumption of static

6I discuss more about multiple-carrier competition in Section A.3 in the Supplemental Material.

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decisions at the end of Section 3.1. The carriers play an infinite horizon dynamic pricing

game.7 Each carrier sold phones with all the four OSs, but the sets of phone models vary

by carrier and over time.

One challenge in modeling consumers’ demand for smartphones is that the sales market

shares are only available at the carrier-OS level, not the smartphone model level. Mean-

while, characteristics and prices data are at the model level in my dataset. To use as much

information on prices and characteristics as possible, I introduce a carrier-OS specific un-

observed demand shock. I explain this in detail next.

3.1 Consumer Demand

Each period, consumers who don’t own any smartphone or have ended previous two-year

contracts enter the market. Each consumer chooses one option from the choice set to

maximize utility. The choice set in period t is Ωt = (j, s, c, t)jsc ∪ (0, t), where j ∈1, 2, ..., Jt is a smartphone model, s ∈ 1, 2, ..., S is the operating system of model, and

c ∈ 1, 2, ..., C is a carrier. An outside option (0, t) exists, and it means not buying any

smartphone. Jt is the total number of models in Ωt.

When a consumer purchases model (j, s, c, t), s/he signs a two-year wireless service

contract with carrier c. Assume that consumer i’s utility of buying the model is:

uijsct = x′jscβi − αi(pcjsct + fct) + γNst + ηsvst + ψsc + ξsct + εijsct. (4)

xjsc is a K × 1 vector of observed smartphone characteristics. pcjsct is the price of the

two-year contract price of the smartphone. fct is carrier c’s price for wireless service for

two years.8 Nst is the number of users of OS s at the beginning of period t. It captures

both the indirect network effect and the direct network effect of an OS, as described in the

introduction.9 vst is the number of new versions of operating system s since the beginning

of data. ηs measures the improvement of a new version of OS s. ψsc is a dummy for the

carrier-OS pair (s, c). It captures the carrier-OS quality that is constant across periods.

Hence, (ηsvst +ψsc) represents the time-varying quality of the carrier-OS group. ξsct is the

7In this paper, I don’t model the manufacturers’ choices of prices in a Stackelberg leader-and-followerframework. I focus on the price decisions of the carriers after the manufacturers set their wholesale prices.

8I assume that the consumer takes the two years’ wireless service costs into account when signing thecontract.

9I can not separate the two kinds of network effect in this paper, and each type of network effect can bethe main force in the network effect in this paper.

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carrier-OS level unobserved demand shock in period t. εijsct is a consumer idiosyncratic

utility shock.

The unobserved quality shocks ξsct is assumed to be carrier-OS specific because the

market share data are at the carrier-OS level. The implication of the assumption is that the

market share variation across different models in the same carrier-OS group is determined

by their observed characteristics and prices, not by their individual unobserved quality

shocks.

Consumers have different income levels and tastes for the smartphone characteristics.

The parameters θi = (βi, αi) are consumer specific. Consumer i is described by vi =

(yi, vi1, ..., viK), where yi is income and viks are independent standard normal taste shocks.

Assume that vi is independent of the unobserved quality shock ξsct. Thus,(βi

αi

)=

(β

α

)+ Φvi,

in which β is the mean of βi over all consumers; α is the part of αi that is the same for all

consumers; and Φ is a diagonal matrix that measures the impact of vi on (βi;αi). Rewrite

the utility function (4) as:

uijsct = δjsct + µijsct + εijsct,

whereδjsct = x′jscβ − α(pcjsct + fct) + γNst + ψsc + ξsct,

µijsct = [x′jsc; (pcjsct + fct)]′ ∗ Φvi.

The mean utility of the outside option is normalized to zero. Consumer i’s utility of

the outside option is:

ui0t = εi0t.

Assume that the εijsct follows a Type-I extreme value distribution and is i.i.d. across

(i, j, s, c, t). Then consumer i’s probability of choosing product (j, s, c) in period t is:

sijsct(vi) =e(δjsct+µijsct)

1 +∑

j′s′c′∈Ωte(δj′s′c′t+µij′s′c′t)

.

Let Ajsct be the set of consumer characteristics such that j has the highest utility for

consumers in this set. That is, Ajsct = vi|uijsct(vi) ≥ uij′s′c′t(vi), for all (j′s′c′t) ∈ Ωt.

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Then the market share of product (j, s, c) in period t is:

sjsct =

∫Ajsct

sijsct(vi)dF (vi). (5)

I aggregate the smartphone model shares to carrier-OS levels, which are observed in

the data. Let Ωsct be the set of models with OS s by carrier c in period t. The market

share of the carrier-OS group (s, c) in period t is:

ssct =∑j∈Ωsct

sjsct. (6)

Consumers are assumed to make static decisions because it is very challenging to esti-

mate a model with dynamic demand and dynamic continuous game of firms for two reasons.

First, it’s rare to have explicit forms of the market share derivatives in a dynamic demand

model because consumers’ value functions are unknown. But the derivatives would be

required to analyze the FOCs in the firms’ dynamic problems. Second, with a dynamic de-

mand model, the carriers’ state variables would have to include the smartphone ownership

distribution, which would be a high dimension vector (shares of different models and the

contract status distribution) in this paper.

The main concern with using a static demand model is that it ignores the consumers’

expectation of future prices and OS network sizes. However, the static demand model

still gives the carriers the incentive to make dynamic price decisions, because as long as

consumers value the OS network sizes, current prices will always affect the future OS

network sizes.

The static model may lead to biases in the network effect estimate. The directions

of the biases depend on which of the following two dynamic factors dominates the other.

On one hand, forward-looking consumers may postpone purchasing a large OS model if

anticipating a decrease in price. A static model treats this low current demand for a

large OS as if that consumers do not value the OS network effect enough, which results

in underestimation. On the other hand, consumers may adopt a small OS because they

believe this OS would eventually dominate, which could lead to overestimation.

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3.2 The Carriers’ Dynamic Pricing Game

The carriers play a dynamic pricing game of smartphones due to the evolving OS network

sizes and market size. In this subsection, I model the carriers’ costs, the market size, and

the carriers’ dynamic pricing game over smartphones. Each period, the carriers observe

the OS market shares and choose the two-year contracts prices of smartphones.10

3.2.1 The Carriers’ Unit Costs

A carrier pays a wholesale cost, a service cost, and an unobserved cost shock on each smart-

phone model. Carrier c’s wholesale cost of model (j, s, c, t) is the product of a manufacturer

wholesale-retail price ratio ωj and the manufacturer retail price pmjsct. ωj is assumed to

be manufacturer specific.11 It captures the manufacturer’s bargaining power against the

carriers.

In addition to the wholesale cost, carrier c also pays a carrier-OS specific monthly

service cost κsc. It includes the costs of selling a phone, maintaining wireless coverage, and

providing customer service. There is also an unobserved cost shock at model level, λjsct.

Hence, carrier c’s unit cost of selling model (j, s, c) in period t is:

kjsct = ωjpmjsct + 24κsc + λjsct. (7)

3.2.2 The Market Size and the Transition of OS Market Shares

Let the cumulative OS market shares be nt = (n1t, ..., nSt) at the beginning of period

t. Let M be the population of potential smartphone users. The OS cumulative share of

OS s is nst = Nst/M . The sum∑S

s=1 nst is less than 1 because not all consumers have

smartphones at the beginning of period t.

In each period, two types of consumers enter the market, those who do not own any

smartphone yet and those who have finished their previous contracts. The share of the first

type consumers is 1−∑S

s=1 nst. To get the share of the second type consumers, I assume

that each user has the same probability to end his/her current contract every period. This

10The manufacturers may also affect the two-year contract prices of smartphones by negotiating withcarriers. I assume that a carrier would only reach an agreement with a manufacturer if the negotiatedprices can maximize its long run profit. Then the carriers indirectly choose the two-year contract priceseven if the agreements between manufacturers and carriers specify the two-year contract prices.

11 The manufacturers’ wholesale prices could change with the OS network sizes. However, this effectcould be already reflected in the manufacturer retail prices.

14

probability is fixed to be one eigth because a contract is 24 months long and one period is

three months in this paper. Hence, the market size in period t is:

Mt = 1−∑s

nstM +1

8

∑s

nstM = 1− 7

8

∑s

nstM. (8)

There are two limitations of assuming the same probability of ending contracts. First,

in reality, the consumers who end their contracts in a period should be the customers who

bought new smartphones two years ago. Tracking the distribution of the smartphone users’

contract status (shares of customers on each OS with different numbers of periods left) is

feasible. However, this would make the distribution vector a state variable for the carriers.

This would increase the dimension of the state space by at least four, which would be very

challenging to estimate. Second, some consumers may use a smartphone for more than two

years, in which case the carriers would face smaller markets in the data than in the model.

This implies that the carriers’ observed differentiated prices are for smaller markets than

in the model. Thus, I may underestimate the network effect parameter using the larger

markets in the model to match the same degree of price differentiation.

The market size decreases with the sum of OS shares. By assumption, the market

size cannot be zero, because 1/8 of the previous users end their contracts and re-enter the

market each period. At the beginning of period t+1, an OS has two types of users, previous

users from period t− 1 and new users in period t. Let Ωst be the set of smartphones with

OS s in period t. The transition of the cumulative market share of OS s is:

nst+1(nt, pct) =

7

8nst + (1− 7

8

S∑s′=1

ns′t)∑

(j,c)∈Ωst

sjsct(pct , ξt). (9)

3.2.3 Timing of the Pricing Game and the Carriers’ Bellman Equations

At the beginning of period t, the carriers observe the OS market shares, nt. Then the

demand and cost shocks (ξt, λt) are realized. Each carrier is assumed to have perfect infor-

mation of all carriers’ demand and cost shocks. The carriers choose prices simultaneously,

then consumers make purchase decisions. At the end of period t, the state variables update

to nt+1.

Denote the set of carrier c’s smartphones in period t by Ωct. Let λt be the vector of all

cost shocks in period t and pct be the vector of all carriers’ prices in period t. Carrier c’s

15

profit in period t at price pct is:

πct(pct , ξt, λt) =

∑(j,s)∈Ωct

(pcjsct + fct − kjsct)sjsct(pct , ξt)Mt, (10)

where the demand shocks affect the market shares, and the cost shocks affect the unit costs

as in equation (7).

The carriers play dynamic pricing game because the prices in period t affect the OS

market shares in the next period. The demand shocks and cost shocks affect their prices,

so the price of each product is a function of these shocks, pjsct(ξt, λt). Carrier c’s Bellman

equation is:

Vc(nt) = Eξ,λ

[max

pcjsct(ξt,λt),(j,s)∈Ωct

πct(p

ct , ξt, λt) + βdVc(nt+1(nt, p

ct(ξt, λt)))

], (11)

subject to equation (9). βd is the discount factor across periods. The expectation is over

the unobserved shocks (ξ, λ), and the carriers choose prices after they observe the shocks.

The value functions are carrier specific because the carriers have different smartphone sets,

wireless service costs, and wireless plan prices.

I assume that the OS market shares are the only state variables in the carrier’s dynamic

game. Ideally, the evolving sets of smartphone models and their characteristics should also

be state variables. The number of smartphone models by each carrier is not a concern,

since it is relatively stable over time as shown in Section D.1 in the Supplemental Materials.

But the characteristics of the smartphones have been improving over time, so there should

be an increasing trend in the carriers’ period profits since the outside option always has

zero mean utility in any period. However, to keep the problem tractable, I don’t add these

variables to the state space. The value functions in this paper can be interpreted as the

averages of the discounted profits across all periods in the data.12

Assumption 3. (1) ξsct and λjsct follow the normal distributions N(0, σ2ξ ) and N(0, σ2

λ),

respectively.

12I also considered whether time, t, should be a state variable to count for the evolving set of smartphones.However, it’s not reasonable to assume that the carriers make decisions based on the time periods assigned inthis paper. An alternative is to add an inclusive value that represent the overall utility of all smartphonesin each period. See Gowrisankaran, Park, and Rysman (2014) for more details about inclusive value indiscrete choice demand models. However, I don’t use this method either because the inclusive value is anunobserved variable and must be calculated for each set of structural parameters, and it also increases thedimension of the state space.

16

(2) The cost shocks are independent across models and periods, and they are independent

with demand shocks.

The assumption of normal distribution and independence will be used in the estimation.

The cost shocks are assumed to be independent across periods so that they wouldn’t be state

variables. The independence of cost shocks across models and that between demand shocks

and cost shocks are not essential to identification and would not add much computation

burden if relaxed. The unobserved demand shocks are allowed to be correlated over time,

even though I control the time-invariant carrier-OS quality and the OS quality improvement

over time in the utility function. In case that they are correlated, I use instrumental

variables to deal with the possible endogeneity problem, which will be explained in Section

5.1, but the demand shocks are not added as state variables for tractability.

For any (ξt, λt), carrier c’s first-order conditions (FOC) with respect to price pjsct is

Mtsjsct(pct , ξt) +Mt

∑(j′,s′)∈Ωct

mj′s′ct∂sj′s′ct∂pcjsct

+ βd∂Vc(nt+1(nt, p

ct))

∂pcjsct= 0, (12)

where mjsct denotes the carrier markup:

mjsct = pcjsct + fct − kjsct. (13)

The price pcjsct affects not only the carrier’s current profit, but also its future profit through

the OS network size. The FOC implies that the marginal profit should be zero at the

equilibrium prices.

The carriers’ choices of service prices, fcts, are taken as exogenous for several reasons.

First, the wireless service plan price of each carrier rarely changed in the data. Second,

since each carrier sets the same service price for all models and OSs, the choice of optimal

service price is not key to the price differentiation across OSs. Lastly, the few service price

adjustments by the carriers in the data don’t appear periodically, which makes it hard to

define a period. Thus, I take the carriers’ service prices as given.

3.3 Equilibrium

The equilibrium concept used in the carriers’ dynamic pricing game is Markov Perfect Nash

equilibrium (MPNE). In this paper, an MPNE is a subgame perfect equilibrium where the

strategies depend on the past only through the state variables. An equilibrium consists of

17

a vector price function, pc(n, ξ, λ), and value functions, Vc(n)s, such that (1) Vc(n)s are

the expected discounted long run profits given pc(n, ξ, λ); and (2) pc(n, ξ, λ) maximizes the

long run profit for each carrier, given the rival’s prices in pc(n, ξ, λ) and Vc(n)s.

The dynamic game may have multiple equilibria for a given set of model parameters.

De Paula (2013) reviews the methods to deal with issues in identification, estimation, and

post-estimation analyses raised by the existence of multiple equilibria. I assume that the

data are from the same equilibrium and there is a unique vector of parameters that can

generate this equilibrium. Then the existence of multiple equilibria will not be an issue

for identifying the parameters. Since I use panel data from one single market (the US) as

opposed to multiple markets, assuming that the data are from the same equilibrium is not

as strong as when the data are from multiple markets. In this paper, multiple equilibria

issue will not cause discontinuity in the objective function in the estimation. I will explain

this at the end of Section 5.2.2. But it may affect the evaluation of the Bellman equation

and counterfactual computation, which will be discussed in Section 5.3 and Section 7.1.

4 Industry Background and Data

4.1 Background of the U.S. Smartphone Industry

A consumer has to choose both the smartphone model and the service provider when buying

a smartphone. The top service providers (carriers) in the U.S. are Verizon Wireless, AT&T

Mobility, Sprint Corporation, and T-Mobile US. According to Kantar World Panel data,

the average combined share of smartphone sales for the four carriers is 88.72% from October

2011 to November 2013. The top four OSs in the U.S. smartphone industry are Android,

iOS, Blackberry, and Windows Phone. Their combined market share increased from 94%

in 2011 to 99% 2011 in 2014.13 Therefore, I consider the top four OSs and top four carriers

in this paper.

Verizon, AT&T, Sprint, and T-Mobile all have been selling smartphone models with

Android, Blackberry, and Windows Phone since these OSs were first launched. AT&T

had been selling iOS exclusively until January 2011 when iPhones became available from

Verizon. Sprint started selling iPhones in October 2011, and T-Mobile started in April

2013. Since then, all carriers have been selling models with all the four OSs.

The two-year contract policy has been used by all carriers from the beginning of the

13Cromar (2010) has a thorough description of the US smartphone industry.

18

smartphone industry. Consumers get discounts on smartphones off the manufacturer retail

prices if they sign two-year service contracts. For example, Apple’s retail price of the

iPhone 5 was $649 in Oct 2012. Consumers could pay $199 to get an iPhone 5 if they sign

contracts with AT&T. If consumers need to end the contract early, they have to pay early

termination fees. Depending on the number of months left in an contract, the termination

fee could be between $150 and $350. According to the Statista.com, the average monthly

churn rate for the four carriers is 2% on average.14 Since 2013, the carriers gradually went

away from the two-year contract model, which has lead to a lot of discussions about its

impacts on the carriers and consumers.

I focus on the consumers’ demand for smartphones with two-year contracts in this

paper. According to the US Wireless Industry Overview 2011, more than 78% of mobile

phone users were on two-year contracts. The percentage is expected to be even higher for

smartphones alone because they are much more expensive without contracts.

4.2 Data

The data used in this paper come from several sources, and the sample period is from

Aug 2011 to Oct 2013. comScore.com reports the U.S. cumulative smartphone subscriber

market shares every month. Kantar World Panel has been publishing the sales market

shares by carrier. Carriers’ two-year contract prices and manufacturers’ retail prices for

smartphones are collected via the web archive website. The smartphone characteristics

data are from phonearena.com. I get household income distribution data from the yearly

Current Population Survey (CPS).

I exclude the population younger than twelve years old and older than 70 years old as

smartphone consumers. This assumption makes the potential market size of smartphones

to be 75% of the population, according to the 2010 US population distribution by age. I

calculate the market size Mt and state variables nt using market size and the cumulative OS

shares. The average household income increased from $69,677 to $72,641 and the standard

deviation increased from $368 to $499.

Each month, the sales market shares by carrier by OS are for the past three months.

For example, the sales market shares in Jan 2012 are for the three months ending in Jan

2012. comScore.com also reports the OS sales shares conditional on the sales shares of

14Churn rate is the proportion of contractual customers or subscribers who leave a supplier during agiven time period. Data source is from this link: http://www.statista.com/statistics/ .

19

AT&T and Verizon. Combining the carriers’ sales shares and the OS shares within each

carrier, I get the sales market share for each carrier-OS group. One missing piece of the

sales data is the conditional OS shares for Sprint or T-Mobile. Since only the joint OS

shares for the two carriers are observed, I assume that their OS share ratio is equal to their

number of models ratio. In the end, there are sales market shares for 16 carrier-OS groups

for 8 non-overlapping 3-months periods (out of 26 consecutive months).

The web archive website has been keeping records of the carriers’ webpages every month

sicne 2008.15 The carriers’ two-year contract prices and the listed manufacturer retail price

of each model can be collected by month. In the sample period, the data has 2283 model-

month observations. The highest two-year contract price is $399 for the 64 GB iPhones

from multiple carriers.

The monthly wireless plan prices are also from the web archive website. Each carrier

offers multiple wireless plans each month. I use the single line price for medium amount of

data and minutes.16 Verizon’s wireless plan price was the highest at $70 and T-Mobile’s

was the lowest at $50. The average across carriers is $60 per month during 2011 to 2013.17

To match the sales shares data, a period is three months in the structural model.

But the smartphone prices are observed every month. Thus, to use as much information

as possible, I construct consumers’ choice set every period in the following way. If a

smartphone model is observed in multiple months in a three-months period, I treat them

as different choice options in that period. The data are seasonally adjusted when used in

estimation.18

The smartphone characteristics include camera pixels, built-in storage, 4G dummy,

weight, screen size, resolution, processor speed, memory, and battery capacity. All the

characteristics are scaled so that their values are in similar range to compare their coeffi-

cients in the utility function.

Table 1 shows the summary statistics of the number of models, characteristics, average

smartphone two-year contract price, and manufacturer retail price by carrier-OS group

by month. All four carriers have more than 10 Android models each month on average.

Windows Phone has the fewest number of models, with an average lower than two. iOS

15 The web archive website link is: http://archive.org/web/16I use the prices for the following minutes and data bundles for the 4 carriers: Verizon (unlimited minutes,

2GB), AT&T (450 minutes, 300MB), Sprint (unlimited minutes, 1GB), T-Mobile (unlimited minutes anddata).

17This matches the $61 average reported by New Street Research company for 2013.18See Supplemental Material Section D.2 for details on removing seasonality from the data.

20

Tab

le1:

Des

crip

tive

Sta

tist

ics

by

Car

rier

-OS

grou

p(A

ver

age

over

Mon

ths)

Car

rier

-OS

Gro

up

No.

ofC

arri

erP

rice

Manu

f.P

rice

Batt

ery

Cam

era

Scr

een

Pix

elP

roce

ssor

Sale

sM

od

els

100$

100$

1000mAh

megapixels

inches

100/inch

2Ghz

Sh

are

%

Ver

izon

-iO

S4.

692.

266.7

41.4

37.1

43.6

53.2

61.7

217.8

0V

eriz

on-A

nd

16.1

51.

225.0

41.9

26.8

64.2

42.6

82.6

018.9

4V

eriz

on-B

la3.

001.

454.6

61.3

15.0

33.0

12.6

01.3

40.4

7V

eriz

on-W

in1.

811.

034.3

21.4

95.6

83.9

42.4

51.6

10.0

8A

T&

T-i

OS

5.07

1.98

6.4

71.4

16.7

53.6

63.1

11.6

620.7

5A

T&

T-A

nd

13.8

80.

984.6

91.9

07.0

44.3

12.6

62.9

89.5

8A

T&

T-B

la3.

810.

864.4

81.2

55.3

03.0

72.5

91.2

51.4

AT

&T

-Win

4.81

0.75

4.3

61.6

77.8

24.2

62.4

71.8

11.3

6S

pri

nt-

iOS

4.91

2.04

6.7

11.4

37.3

63.6

63.2

81.7

93.0

7S

pri

nt-

An

d11

.38

0.90

4.4

11.8

05.7

73.9

72.4

52.0

311.9

8S

pri

nt-

Bla

1.65

1.30

4.5

31.1

85.0

62.6

82.7

61.1

10.3

8S

pri

nt-

Win

1.13

0.79

4.4

21.5

75.6

03.7

72.5

71.3

60.4

5T

-Mob

ile-

iOS

4.00

—6.9

61.4

47.9

43.9

83.2

62.5

50.3

6T

-Mob

ile-

An

d10

.62

1.58

4.0

51.7

76.1

05.1

92.4

32.6

211.4

4T

-Mob

ile-

Bla

3.69

2.01

4.4

61.3

95.1

92.9

42.6

11.1

90.8

3T

-Mob

ile-

Win

1.31

1.15

3.4

91.4

65.8

03.9

12.4

81.8

90.8

0

Not

es:

Th

efi

rst

thre

ele

tter

sof

the

OS

sar

eu

sed

inth

eta

ble

.F

or

exam

ple

,“A

nd

”is

for

An

dro

id.

T-M

ob

ile

elim

inate

dth

etw

o-y

ear

contr

act

pol

icy

sin

ceit

star

ted

tose

lliO

Sm

od

els.

Th

esa

les

share

sare

con

dit

ion

al

share

sam

on

gth

eli

sted

gro

up

san

dare

reco

rded

for

the

sale

sin

the

pas

tth

ree

mon

ths

ever

ym

onth

.

21

models have the highest two-year contract prices and manufacturer retail prices on average.

Windows Phone models have the lowest carrier prices and manufacturer prices.

The pattern of hardware characteristics across OSs is mixed. iOS models outperform

other models in camera pixels and screen pixels per square inch. Android models have

the best battery capacities, screen sizes, and processor speeds. From the last column, I

find that Android has the highest sales market shares, and most new iPhone users signed

contracts with Verizon and AT&T.

Figure 1 shows the cumulative monthly market shares of the four operating systems

from August 2011 to October 2013. The market shares of iOS and Android both increased

from below 6% to above 25%. Blackberry’s market share dropped from 10% to less than

5%. The Windows Phone market share was stable and small, at around 3%.

Figure 1: The Cumulative OS Market Shares Over Time (2011.08-2013.10)

Table 2 shows the results from regressing the carriers’ contract prices of smartphones

on past month OS shares, manufacturer retail prices, model characteristics, OS dummies,

manufacture dummies, and month dummies. The results show that the carriers choose dif-

ferentiated prices across OS, lower prices for smartphones with larger OSs. The coefficient

−1.0362 implies that for every 10% increase in an OS share, the carriers drop the two-year

contract price of a model with that OS by $10.36.

22

Table 2: OLS Regression of Carriers’ Contract Prices of Smartphone ($100)

Variables Estimates Std

OS Share (Lag 1 month) -1.0362*** (0.3708)

Manufacturer price 0.6425*** (0.0162)

(Monthly data from Aug 2011 to Oct 2013. Control for

phone characteristics and fixed effects of OSs, carriers,

manufacturers, and months.)

The regression results are consistent with Proposition 1 and 2 in Section 2. The car-

riers are multi-network sellers, so they use a differentiated pricing strategy to achieve OS

concentration and attract more future consumers.

5 Identification and Estimation

The structural model parameters are θd = (α;β; γ;ψ; η; Φ;σξ) in the carrier-OS share

equation (6) and θs = (ω;κ;σλ) in the carriers’ FOCs in equation (12). I use GMM to

estimate (θd, θs). The moment conditions are based on the carrier-OS level unobserved

quality shock ξsct in equation (6) and the model level cost shock λjsct in equation (12). In

the rest of this section, I discuss the identification of the parameters, describe the moment

conditions and objective function in the GMM, and explain how I calculate the demand

and cost shocks for a given set of parameters.

5.1 Identification

The identification of the network effect parameter, γ, and the price coefficient, α, is subject

to endogeneity issues. First, if the demand shock in the past period ξsct−1 affects the current

shock ξsct, then ξsct is correlated with the cumulative network size Nst, which leads to an

endogeneity problem when estimating γ. Second, due to the carriers’ endogenous pricing

decisions on the supply side, the prices are correlated with the demand shocks. To deal

with these issues, I use moment conditions that are based on the orthogonality between

ξscts and a set of instrumental variables (IV).

Identification of network effect is a challenge in the literature, and previous empirical

studies have used different instrumental variables for different industries.19 In this paper,

19 Rysman (2004) uses the population coverage by a yellow page directory as an instrument to identifythe indirect network effect of advertising on the number of uses of the directory. Lee (2013) uses the lagged

23

I use the total number of smartphone models for each OS since the beginning of the data

until period t as the IV for the network size Nst. That is, the instrumental variable for Nst

is∑t

t′=1

∑js′c I(OSjs′ct′ = s), where I(OSjs′ct′ = s) is equal to one if s′ = s and is equal

to zero otherwise. Their simple correlation is 0.67. The underlying assumption is that

the number of smartphone models for each OS in the US in each period is not correlated

with the unobserved demand shocks (which do not include the carrier-OS fixed effects

and the OS quality improvement). This assumption is reasonable for several reasons.

First, all the manufacturers sell smartphones to many countries, and their decisions of

releasing new smartphone models mainly depend on demand from the global market and

competition among manufacturers. Second, most manufacturers (e.g. Apple, Samsung,

and LG) release new models of their flagship smartphones every year. This pattern didn’t

change with demand shocks. Third, the number of smartphone models for each OS is

largely determined by how many manufacturers adopt the OS, which is further determined

by the OS openness that is exogenous to the model. Therefore, the unobserved demand

shocks in the US market should have little impact on the number of smartphone models

in the US market.

To identify the price coefficient, I follow Berry, Levinsohn, and Pakes (1995) and use IVs

including the average characteristics of models in the same carrier-OS group, Ωs,c,t, that of

other OSs by the same carrier, Ω−s,c,t, and that of the same OS by other carriers, Ωs,−c,t,

in which the Ωs are sets of smartphone models and the subscripts denote the OS/carrier

information of the models. The average characteristics in the same carrier-OS group are

cost shifters, which are assumed to be exogenous. The average characteristics of other OSs

or by other carriers are correlated with prices due to the competition among the carriers,

but are also uncorrelated with demand shocks, ξscts. Hence, the average characteristics are

valid IVs.

After dealing with the endogeneity issues, γ is identified by the variation of the sales

shares and the past OS network sizes, and α is identified by the variation of sales shares

and prices across carrier-OS groups.

The diagonal matrix Φ has the non-linear coefficients in the consumer specific coeffi-

cients. It is identified by the variation of market shares and consumer income distribution

across periods.20 In particular, if after controlling for everything else, the average con-

values of software utility as instruments to identify the indirect network effect of video game software titleson hardware adoption. Ackerberg and Gowrisankaran (2006) avoid the endogeneity issue by fully specifyan equilibrium model of the adoption decisions of the two sides of the ACH network.

20See Supplemental Material Section C.1 for the details of simulation and normalization of individual

24

sumer income increases over time and so do the sales shares, then this implies that the

price coefficient decreases as income goes up.

The unit costs then provide identification for service cost κsc and the wholesale pa-

rameters in ω as in equation (7). The differences between observed prices and predicted

markups are the unit costs kjsct. The structural model predicts product level markups for

any given (θd, θs). I don’t directly estimate the standard variances of the demand and cost

shocks, σξ and σλ. Instead, I calculate them using the demand and cost shocks backed out

from the market share equation and the FOCs. Following Goettler and Gordon (2011), I

fix βd to be 0.975 as the three-month discount rate for carriers’ profits.

5.2 Estimation

5.2.1 Moment Conditions and Objective Functions

The moment conditions are based on the orthogonality between the instrumental variables

and the random shocks, (νsct, λjsct). Let (θd0, θs0) be the vector of true parameters. The

moment conditions are:

E[ξsct(θd0)|Z1sct] = 0, (14)

E[λjsct(θd0, θs0)|Z2jsct] = 0. (15)

Z1sct is the vector of IVs as explained above. Z2jsct is a vector of the variables including

manufacturer dummies and carrier-OS dummies in the unit cost equation, the model char-

acteristics, xjsct, the numbers of models in (Ωsct,Ωst,Ωct), the average characteristics over

models in (Ωct,Ωst,Ωt). There are 44 moment conditions in (14) and 67 moment conditions

in (15). The number of structural parameters to be estimated is 62.

To get efficient estimates, I use the two-stage GMM. In the first stage, the weight matrix

for the demand side moment conditions is Wd1 = (Z ′1Z1)−1 and that for the supply side

moment conditions is Ws1 = (Z ′2Z2)−1. The second stage uses the optimal weight matrix

estimate W2, estimated using the first stage results. The second stage objective function

is:

Q2(θd, θs) = (1

Nλ

Nλ∑jsct

[Z1sctξsct(θd);Z2jsctλjsct(θs; θd)])′W2(

1

Nλ

Nλ∑jsct

[Z1sctξsct(θd);Z2jsctλjsct(θs; θd)]).

(16)

income levels.

25

where (Z1sctξsct(θd);Z2jsctλjsct(θs; θd)) is a vector of moment conditions from both the

demand shocks and cost shocks.

5.2.2 Solve for the Demand and Cost Shocks

To evaluate the GMM objective function for a (θd, θs), I first calculate the model level

shares in equation (5) using observed model level characteristics, prices, and the number of

OS users. The model level shares and their derivatives are then used to solve for demand

shocks and cost shocks for any given vector of (θd, θs).

To calculate demand shocks, ξsct(θd), I aggregate the model level shares to the carrier-

OS level, so that they match the observed carrier-OS shares. I show that there exists a

unique vector of demand shocks that make the model prediction match the data for any

θd. See Appendix A for the proof of the invertibility from the carrier-OS sales shares to

ξsct(θd).

The model market share in equation (5) is an integration of individual probabilities

over the distribution of consumer characteristics vi. Since vi is a nine-by-one vector, I

use numerical approximation to calculate the integration. I simulate Ns = 200 consumers

with different characteristics each period and use the averages of their individual choice

probabilities to approximate the sales market share of each model.21

sjsct(θd) =1

Ns

Ns∑i=1

sijsct(θd). (17)

The predicted carrier-OS sales share is the sum over all models in the group.

ssct(θd) =∑j∈Ωsct

sjsct(θd).

Following Berry, Levinsohn, and Pakes (1995) and Nevo (2001), I apply an iterative

procedure to solve for the unobserved shocks ξ(θd). Let the observed carrier-OS shares

be sosct. Given an initial guess of the vector of unobserved demand shocks, ξ0 = ξ0sct,

calculate the predicted market shares ssct(θd, ξ0). Then compare the predicted shares with

the observed shares. The updating rule is to increase ξsct if the predicted share is less than

21To check the accuracy of the numerical integration with 200 simulated individuals, I compare the inte-gration results for some known joint normal distributions by simulating 1,000, 5,000, and 10,000 individuals.I find that simulating 5,000 individuals only changes the integration by a magnitude of less than 10−4.

26

the observed market share for the group (s, c, t) and decrease it otherwise. Repeat this

updating process until the vector ξk converges. The iteration is summarized by:

ξk+1sct (θd) = ξksct(θd) + χ(sosct − ssct(θd, ξk(θd))),

where χ is a constant, set to be 0.9. The proof of unique fixed point in Appendix A

guarantees the convergence of this iteration.

Next, I describe the algorithm for calculating the cost shock λjsct for a given (θd, θs).

I first solve for the smartphone model level markup mjsct using data and FOCs. Then it’s

straightforward to calculate λjsct, using the definition of markup. As in equation (12), the

FOC w.r.t. price pcjsct is:

Mtsjsct(pct , ξt; θd) +Mt

∑(j′,s′)∈Ωct

mj′s′ct∂sj′s′ct∂pcjsct

+ βd∂Vc(nt+1(nt, p

ct(ξt)))

∂pcjsct= 0. (18)

An important feature of the FOCs is that they are linear in the markups mjscts, given

the observed prices. I take advantage of this linearity when solving for equilibrium prices.

The first two terms in equation (18) can be calculated using the demand side market

share functions. To calculate the last term in equation (18), I approximate each carrier’s

value function with a linear combination of basis functions, so that that there is explicit

functional form for ∂Vc(nt+1)∂pcjsct

.

Each carrier’s value function is a multivariate function of the four operating systems’

market shares nt. I use the second-order complete polynomials as basis functions. I also

use higher order polynomial approximation to check the robustness of the estimates and

find the results don’t change much.22 Hence, there are 15 basis functions for the four state

variables. Denote the basis functions by Bf(n) = (1, bf1(n), ..., bf14(n)). Let θvc be the

coefficient vector for carrier c ∈ 1, 2, 3, 4.

Vc(nt) = Bf(nt) ∗ θvc .

Let θv = (θv1 ; ...; θv4) be the vector of approximation coefficients. With the approximated

value functions, I solve for the model level markups using the FOCs. In Supplemental

Materials Section B, I derive the function forms for the derivatives in equation (18). Let

22Barwick and Pathak (2015) use the Multivariate Adaptive Regression Spline (MARS) method to ap-proximate multivariate functions.

27

mjsct(θd, θs) be the solution for the given (θd, θs). Then the unobserved cost shock is:

λjsct(θd, θs) = pcjsct + fct −mjsct(θd, θs)− ωjpmjsct − 24κsc. (19)

Therefore, given an alternative of the parameters (θd, θs), the shocks ξsct(θd) and

λjsct(θd, θs) can be solved using observed data and approximation of the value functions.23

The next subsection describes the restrictions on the approximation of value functions,

which will be the constraints in the MPEC estimation method.

The possibility of having multiple equilibria of the dynamic pricing game will not lead

to discontinuity in the λjsct(θd, θs) or the GMM objective function in this paper. The

discontinuity problem rises when the equilibrium found by solving (18) for an updated

(θd, θs) is far from the equilibrium found for the previous (θd, θs). This may cause jumps

in the solved cost shocks, thus the GMM objective function. However, in this paper, the

solution of mjsct(θd, θs|data) is unique and continuous in (θd, θs). There is a unique solution

of the markup vector because the coefficient matrix for the markup vector has full rank

due to diagonal dominance of∂sj′s′ct∂pcjsct

. This markup vector is continuous in (θd, θs) because

all other terms in the FOCs are continuous in (θd, θs). The uniqueness of mjsct(θd, θs|data)

doesn’t imply that there is a unique equilibrium for a (θd, θs), but that the prices observed

in data imply a unique vector of markups for the given (θd, θs).

5.3 Equilibrium Constraints

The approximation parameters are identified by the Bellman equations at the true struc-

tural parameters. Instead of using a nested loop to solve for θv(θd, θs), I impose the carriers’

Bellman equations as constraints in the MPEC algorithm to reduce computation time. In

the estimation, both sides of the carriers’ Bellman equations need to be evaluated for any

given (θd, θs). The right hand sides of the Bellman equations are integrations of the car-

riers’ discounted profits over (ξ, λ), which requires solving for the equilibrium prices for

any possible (ξ, λ). Since direct integration over the distributions of (ξ, λ) is not feasible, I

23When calculating the GMM objective function, I exclude observations with $0 carrier price and T-Mobile models in May 2013-July 2013. Since $0 is the lower bound of the carrier prices, they wouldbe conner solutions in the equilibrium prices. The corresponding cost shocks could be far from the actualshocks if included in the GMM objective function. T-Mobile completely dropped the two-year contract sinceMay 2013. There are no carrier discounts for those models. So the cost shocks calculated in equation(19)for these observations would have a different meaning if I include them in the GMM objective function.Therefore, I exclude the cost shocks of these observations in the GMM objective function.

28

simulate R vectors of the demand and costs shocks.24 For each simulated (ξr, λr), I solve

for the equilibrium prices pc,rjsct(ξr, λr). I design a Newton-Raphson iteration algorithm that

takes the advantage of linearity of markups in the FOCs to get the equilibrium prices.25

There is potential multi-equilibria issue when solving for the equilibrium prices for the

simulated shocks. I take the markups implied by the observed prices and a (θd, θs, θv) as

the starting point when solving for an equilibrium for each vector of shocks. In the cases

where multiple equilibria exist, my algorithm always selects the equilibrium that is closest

to the observed one.26 Let pc,rjsct be the equilibrium price of (j, s, c, t) for (ξr, λr). Then I

use the average long run profit across simulations to approximate the integration on right

hand side of the Bellman equations:

∼RHSct(θd, θs, θ

v) =1

R

R∑r=1

[∑(j,s)

(pc,rjsct − ωjpmjsct − 24κsc − λrjsct)Mtsjsct(p

c,rt , ξrt ; θd)

+ βdVc(nt+1(nt, pc,rt ), θv)]

In the estimation, I allow an approximation error between the two sides of the Bellman

equations, a percentage difference smaller than τ between the two sides of the Bellman

equation at each observed state and for each carrier. This flexibility is added due to the

finite basis functions used in the approximation. Imposing strict equality of the Bellman

equations may lead to algorithm to fail to find an approximation vector or taking too long a

time to find the approximation coefficients. Barwick and Pathak (2015) also approximate

the value functions and impose the FOCs of the distance between the two sides of the

Bellman equations with respect to approximation parameters as equilibrium constraints.

Their method also allows errors in the value function approximation.

|Vc(nt; θd, θs, θv)−∼

RHSct(θd, θs, θv)|/ |Vc(nt; θv)| < τ, ∀c, t. (20)

The constraint (20) implies that the relative difference between the two sides of each

carrier’s Bellman equation at any state nt should be smaller than the tolerance τ . The

24Due to computation time issue, I use R = 50 in the estimation. After I get the parameter estimates, Icheck the impact of the number of Monte Carlo simulation on the value functions and find that increasingR doesn’t affect the integration much. See Supplemental Materials Section C.2 for details.

25See Section C.3 of the Supplemental Materials for details on the iteration algorithm.26See Supplemental Materials Section C.3 for details on using iterations to solve for prices.

29

smaller is τ , the better the value functions are approximated. In the estimation, I set the

value of τ be 5% and impose the constraints at the observed states. At the converged

parameter estimates, the average error in Bellman equations is 3%.

6 Estimation Results

Table 3 shows the estimates of demand parameters. Model (1) shows the estimates using

only the demand model. The potential endogeneity issues are dealt with. Model (2) is the

full model, and the demand model and supply model are estimated jointly. The coefficient

estimate of the OS network effect, γ, is 0.0814 and significant in the full model. This means

that a consumer’s utility of buying a smartphone increases with the number of existing users

of the smartphone OS. The network effect is more important in the full model because the

carriers’ differentiating pricing strategies in Table 2 imply strong OS network effect. The

price coefficient estimate α is −1.5793. Comparing the parameters for OS network size and

price, I find that adding 19.41 million users for an OS increases a consumer’s utility by the

same amount as cutting the price by $100. To see the scale of the OS network effect, iOS

and Android had 55.47 and 86.69 million users at the beginning of 2013, respectively. The

interaction between price and income has a positive coefficient, meaning that high income

consumers are less price sensitive. The income effect implies that the price of smartphones

will not affect the utility of an individual who has an annual income of $201,000.

In model (1), the estimates of β imply that consumer utility increases with camera

pixels and CPU speed. But the coefficients for storage, battery, screen size, and pixels

are negative. This is probably because that the best selling models do not have the most

advanced characteristics. For example, iPhone 5’s battery is 1540mAh, while most other

top models have capacities more than 2000mAh. Similarly, the average screen size for

carrier-iOS groups is 3.66 inches, while that for carrier-Android and carrier-Windows Phone

groups are 4.24 and 3.94 inches, respectively. The estimates become insignificant in the

full model, where the important factors in determining consumers’ purchase decisions are

only price, OS network size, and OS quality.

30

Table 3: Demand Model Parameter Estimates

(1) (2) (1) (2)

Variable RDC-IV Full Fixed Effects RDC-IV Full

OS Subscribers (Million), γ 0.0107*** 0.0814*** Verizon-iOS, ψiv 1.3285 1.7664(0.0038) (0.0171) (0.9082) (5.4545)

Carrier Price ($100), −α -1.5516*** -1.5793* Verizon-Android, ψav -0.2821 -0.4445(0.6481) (0.9097) (0.8698) (5.8247)

Price*Income, −φ1 1.2705*** 1.2537 Verizon-Blackberry, ψbv -3.3741*** -2.3277(0.5117) (0.8270) (0.7780) (4.7689)

Storage (GB), β1 -0.0863*** -0.3082 Verizon-Windows, ψwv -0.1147 -0.0770(0.0161) (0.2248) (0.8756) (5.3396)

Battery (1000mAh), β2 -0.6356*** -1.2989 AT&T-iOS, ψia 1.5404 1.2312(0.4659) (3.7447) (0.9245) (5.4172)

Camera(100MP ), β3 0.2930*** -0.3792 AT&T-Android, ψaa -0.9411 -1.7506(0.0327) (0.4563) (0.8182) (5.3095)

Screen Size (inch), β4 -2.000*** -2.8462 AT&T-Blackberry, ψba -2.8566*** -2.3800(0.1921) (2.9428) (0.8211) (4.9948)

Dummy 4G, β5 0.3127 -0.1253 AT&T-Windows, ψwa -0.7067 0.5833(0.4799) (1.8099) (0.9702) (5.3578)

Pixels (100/inch2), β6 0.2885 0.4169 Sprint-iOS, ψis -0.2821 -0.8214(0.1715) (1.5726) (0.9270) (5.4625)

RAM (GB), β7 -0.0261 0.2039 Sprint-Android, ψas -0.2071 -0.7632(0.2131) (1.3457) (0.8482) (5.8705)

CPU (Ghz), β8 0.7518*** -0.5389 Sprint-Blackberry, ψbs -4.0000*** -3.1028(0.1138) (0.9647) (0.8039) (4.9485)

ηios 0.1468 3.8196*** Sprint-Windows, ψws 0.4296 0.3924(0.0951) 0.6049 (0.8663) (5.3910)

ηand 0.0000 1.4887*** T-Mobile-iOS, ψit -0.4043 0.6467(0.0581) 0.3524 (0.8421) (4.9588)

ηbla 0.0000 4.4243*** T-Mobile-Android, ψat -0.7020 -1.3600(0.0830) 0.6845 (0.8666) (5.3040)

ηwin 0.2537*** 4.5885*** T-Mobile-Blackberry, ψbt -4.000*** -3.1594(0.0676) 0.6541 (0.7988) (5.1802)

T-Mobile-Windows, ψwt -0.3009 0.0052(0.9221) (4.7559)

The number of OS version updates has positive impacts on consumer utility in model

(2). Android’s improvement in each new version is the smallest, increasing a consumer’s

utility by 1.4887, while Windows Phone’s improvement is the highest. The carrier-OS fixed

effects, ψs, measure the unobserved characteristics of these groups that are constant over

time. The results of model (1) show that Blackberry has the lowest time-invariant quality

among all OSs, and iOS has the highest quality. Though the estimates become insignificant

in the full model, the rank of the OS quality stays the same.

31

Table 4 shows the demand elasticities across eight smartphone models by Verizon and

AT&T in November 2012. The own elasticities are stronger than cross elasticities. The

own elasticities of Android models (Samsung Galaxy S3 and LG Lucid) are the highest.

Since there are a lot of Android models available, a price increase of an Android model will

make consumers to choose other Android models. Consumers have quite robust demand

for iPhone 4s(16GB), less elastic than for iPhone 5(16GB). This is due to lower price of

iPhone 4s than iPhone 5. There is high cross demand elasticities among iPhone models.

The eighth column shows that the price change of Blackberry 9900 has small impacts on

other models’ sales, similar for the Nokia 920 with Windows Phone. Overall, Android and

iOS models are closer substitutes for each other than for Blackberry and Windows Phone

models.

Table 4: Demand Elasticities w.r.t. Prices at Model Level

V-iPhone4s V-iPhone5 V-S3 V-Lucid A-iPhone4s A-iPhone5 A-B9900 A-Nokia

V-iPhone4s -1.6116 0.1866 0.0243 0.0247 0.7768 0.1842 0.0047 0.0186V-iPhone5 0.5874 -2.0203 0.0246 0.0395 0.5798 0.2029 0.0056 0.0161V-S3 0.2004 0.0645 -2.5905 0.0413 0.1978 0.0637 0.0007 0.0098V-Lucid 0.1701 0.0862 0.0345 -2.6651 0.1679 0.0851 0.0028 0.0060A-iPhone4s 0.7869 0.1866 0.0243 0.0247 -1.6118 0.1842 0.0047 0.0186A-iPhone5 0.5874 0.2055 0.0246 0.0395 0.5798 -2.0204 0.0056 0.0161A-B9900 0.3272 0.1230 0.0058 0.0283 0.3230 0.1214 -2.0048 0.0062A-Nokia 0.6854 0.1890 0.0439 0.0322 0.6766 0.1865 0.0033 -1.7542

Notes: V: Verizon. A: AT&T. The (g, g′)th element is the demand elasticity of product g when prices of productg′ increase by 1%. S3 is Samsung Galaxy S3 with Android OS. Lucid is an Android smartphone by LG. B9900 isa Blackberry model. Nokia is the Nokia 920 model with Windows Phone OS.

The estimates of the supply side parameters are in Table 5. The wholesale cost co-

efficients ω are significant. Apple’s wholesale-retail price ratio is the highest at 89.63%.

This implies that the carriers pay $581 to Apple for a $649 iPhone on average. This ex-

actly matches the $581 estimate from Apple’s balance sheet according to a report in 2013

by Digital Trends.27 Samsung’s wholesale-retail price ratio is 85.89%, implying that the

carriers pay $514 for a Samsung S3 whose manufacturer retail price is $599. Motorola’s

wholesale discount is the highest among all manufacturers, at 16.55%.

The monthly service cost estimates, κs, are at the carrier-OS level. The four carriers’

average monthly service cost estimate is $22.14. Sprint has the highest estimated service

cost, $26.73 per month per user. T-Mobile has the lowest service cost with an average of

27See the report here: http://www.digitaltrends.com/mobile/iphone-cost-what-apple-is-paying/

32

$19.60 per month. The average monthly service costs for Verizon and AT&T are $20.31

and $21.91. The results don’t show that one OS costs more than others for the carriers

to provide services. Comparing the estimated unit costs with the service price data, the

monthly markups per customer on wireless services for Verizon, AT&T, Sprint, and T-

Mobile are $44.13, $36.44, $34.59, and $32.80, respectively.

Table 5: Supply Model Estimates

Variable Estimate Variable Estimate

Wholesale Apple, ωa 0.8963*** AT&T-iOS, κai 0.2360price (0.1971) (0.2932)ratio Samsung, ωs 0.8589*** AT&T-Android, κaa 0.2671ω : (0.2819) (0.2313)

Motorola, ωm 0.8345*** AT&T-Blackberry, κab 0.1842(0.2819) (2.7595)

LG, ωl 0.8675*** AT&T-Windows, κaw 0.1891(0.3052) (2.5295)

HTC, ωh 0.8744*** Sprint-iOS, κsi 0.3047(0.2481) (0.4797)

Blackberry, ωb 0.8399*** Sprint-Android, κsa 0.2324(0.2625) (0.2477)

Nokia, ωn 0.8354*** Sprint-Blackberry, κsb 0.2335(0.2627) (2.7546)

Monthly Verizon-iOS, κvi 0.2244 Sprint-Windows, κsw 0.2987service (0.0207) (5.2813)cost (100$) Verizon-Android, κvi 0.1885 T-Mobile-iOS, κti 0.2244κ : (0.0538) (0.0000)

Verizon-Blackberry, κvb 0.1936 T-Mobile-Android, κta 0.1680(0.2898) (0.4144)

Verizon-Windows, κvw 0.2058 T-Mobiel-Blackberry, κtb 0.1711(0.1253) (2.2132)

T-Mobiel-Windows, κtw 0.2206(0.5362)

Given the estimates, I calculate the carriers’ markups per two-year contract customer.

Take AT&T and iPhone 5 in December 2012 for example. AT&T sells the iPhone 5 at

$199 while paying $581 to the manufacturers, which implies a $382 net cost on the phone.

AT&T earns a net margin of $36.44 each month on wireless service, which makes a net

profit of $874.56 on the two-year service. So AT&T earns a net profit of $492.56 from an

iPhone 5 two-year contract customer. Table 6 shows the carriers’ average estimated profits

per two-year contract customer by carrier-OS groups.

33

Table 6: Carriers’ Markups per Two-Year Contract by Carrier-OS ($100)

Verizon AT&T Sprint T-Mobile

iOS 5.95 5.01 3.18 -Android 6.99 5.13 5.93 5.24Blackberry 7.95 7.38 6.59 5.30Windows Phone 7.31 7.33 4.78 4.26

The carriers get lower markups on iOS and Android models than on Blackberry and

Windows Phone models, which is consistent with the results in the two-period theoretical

model in Section 2. Their markups on iOS models are the lowest since they get the small

wholesale discounts from Apple and offer high discounts to consumers. This finding is

in line with the news about the carrier margin drops in 2011 because of iPhones.28 The

carriers’ markups on Blackberry and Windows Phone models are higher than on iOS and

Android models. This is because that the carriers give lower discounts on Blackberry and

Windows Phone models and that they pay lower wholesale prices to the manufacturers.

Among the carriers, Verizon has the highest overall markups due to its high margin on the

wireless service. AT&T has the second highest average markups. T-Mobile’s markups are

the lowest due to their low margins on wireless service.

The value function approximation results show that Verizon’s value function is the

highest among the carriers. AT&T has the second highest value function. T-Mobile’s

value function is the lowest. See the Supplemental Materials for plots of the carriers’ value

functions.29

7 Counterfactuals

In this section, I study two counterfactual cases to measure the impacts of the carriers’ abil-

ity to internalize across operating systems and the carriers’ two-year contract discounts on

smartphone penetration, OS market concentration, carriers’ profits, and consumer surplus.

7.1 Carriers Specialize in Operating Systems

As shown in Section 2, multi-OS carriers’ ability to internalize prices across OSs leads to

price differentiation across OSs. In this counterfactual analysis, I assume the four carriers

28Data source: http://money.cnn.com/2012/02/08/technology/iphone carrier subsidy/ .29The value functions are plotted in Supplemental Material D.2.

34

specialize in the four OSs to measure the impact of internalization of OS competition. I

assume that Verizon, AT&T, Sprint, and T-Mobile specialize in Android, iOS, Blackberry,

and Windows Phone, respectively.

To keep consumers’ choice sets the same as in data, I keep all the Android smartphones

that were sold by AT&T, Sprint, and T-Mobile and assume that they are now only available

at Verizon, same for the other OSs. In addition, I still use the estimated carrier-OS

fixed effects from original carriers and estimated demand shocks from data, so that only

price and OS network size in a consumer’s utility from a smartphone would change in the

counterfactual case. On the supply side, I also use the cost shocks backed out from data.

This scenario is designed in this way so that only the supply side OS ownership structure

changes from multi-OS carriers to single-OS carriers.

The four carriers still play a dynamic pricing game after they specialize. I first solve

for the value function approximation parameters if they specialize, by minimizing the

differences between the two sides of Bellman equations. I then calculate the counterfactual

equilibrium prices by iterating guesses of prices (markups) using new value functions. There

could be multiple equilibria when solving for prices. The counterfactual equilibrium shown

below is the one that is closest to the equilibrium in data, since I use the vector of observed

prices as the starting point in the iterative algorithm.

Table 7 shows the changes in smartphone prices and the carriers’ profits if they spe-

cialized in OSs. The average two-year contract prices of Android and iOS smartphones

would increase by $286 and $26, respectively, due to their OS network size advantage and

OS specialization. The average prices of Blackberry and Windows Phone models would

drop $131 and $72. This is consistent with the results in Section 2. If the carriers become

single-OS sellers, the carriers on larger networks will choose higher prices than carriers

on small networks. Comparing the second row and the last row, I find that the profits

of the carriers who specialize in Android (Verizon) and iOS (AT&T) would increase by

48.31% ($14.68 billion) and 74.45% ($15.28 billion), while Spring (Blackberry) would and

T-Mobile (Windows Phone) would lose profits by more than 60%. This is because that

most consumers would still buy either Android models or iPhones, and both would have

higher prices. The third row shows the profits by OS before specialization. Carriers get

most of their profits from Android and iOS models in the data.

35

Table 7: Price ($100) and Profit ($billion) Comparison

Verizon AT&T Sprint T-Mobile

(And) (iOS) (Bla) (Win)

∆Price by OS +2.86 +0.26 −1.31 −.72Profits by Carrier (Multi-OS) 30.40 20.52 10.99 7.91Profits by OS (Multi-OS) 26.46 38.54 2.11 2.07Profits by OS (Single-OS) 45.87 35.80 4.07 3.03

Figure 2: OS Growth if the Carriers Specialize in OSs

Figure 2 show the OS market shares over time. The dashed curves show the counter-

factual OS shares, and the solid curves are the OS shares in the data. The two curves in

the same color represent the same OS. The OS market would become less concentrated,

and the smartphone penetration rate would drop from 78.36% in the data to 62.53% if

the carriers specialized. The market share of Android would decrease from 41% to 26%,

while iOS would drop from 31% to 29%. Android’s market share would be below iOS

from January 2013 on, due to its increased prices and the growing number of iOS users.

However, Verizon still has the highest profits among all carriers, because of the large price

increases of Android models as in Table 7. The market share of Blackberry and Windows

Phone would increase slightly due to lower prices. Their joint share would increase from

36

7% to 9%.

If the carriers specialized in OS, consumers’ surplus would decrease by $137.40 million

during the sample period.30 Consumers would get higher utility from Blackberry and

Windows Phone models because of lower prices and larger OS networks, but their utility

from Android and iOS models would decrease. Since most consumers are Android and iOS

users in the data, the aggregate consumers surplus would drop.

7.2 The Impact of the Two-Year Contract Discounts

In this counterfactual case, I assume that the carriers sell smartphones at the manufacturer

retail prices and cut the service prices by $15 each month, in order to measure the impacts

of the two-year contract discounts. This is motivated by the fact that the carriers started to

sell smartphones at the manufacturer prices and cut monthly service prices by at least $15

for non two-year contract subscribers from late 2013. By January 2016, all the four leading

carriers stopped the two-year contract option for subscribers.31 To make this counterfactual

results comparable with data, I assume that each consumer uses a smartphone for two years.

By using the manufacturer retail prices, I eliminate the carriers’ discounts on smart-

phones and their price differentiation across OSs. Without the differentiated discounts,

iOS and Android models become relatively more expensive than Blackberry and Win-

dows Phone. Table 8 compares the carrier profits with and without the carrier discounts.

AT&T could get 18.25%($3.51 billion) more profits than in the data if it didn’t offer high

discounts on iPhones, because 63% of its new service subscribers during the sample period

bought iPhones on which AT&T had low markups, while the iPhone sales shares for Ver-

izon, Sprint, and T-Mobile were 47.33%, 19.56%, and 3.24%. Verizon’s would lose profits

by 8.69% ($2.97 billion), Sprint by 44.32%($4.91 billion), and T-Mobile by 40.18% ($2.92

billion).

30I calculate the consumer surplus following the definition in Chapter 3 of Train (2009) and use thepotential population of smartphone users as the market size in each period. If I use the market sizedefinition in Section 3.2.2, then the consumer surplus would increase if carrier specialized due to largermarket sizes from lower OS shares, though an individual consumer’s surplus would decrease.

31Sprint brought the two-year contract option back in March 2016.

37

Table 8: Profit Comparison by Carrier ($billion)

Verizon AT&T Sprint T-Mobile

Profits by Carrier (Estimated) 30.40 20.52 10.99 7.91Profits by Carrier (No Contract) 27.43 24.03 6.08 4.99

Figure 3 compares the OS growth paths in this counterfactual case with data. The

dashed curves show the counterfactual OS shares, and the solid curves are the OS shares

in the data. The two curves in the same color represent shares of the same OS. Without

the discounts, the overall smartphone penetration rate by the end of May 2013 would be

58.31% (102.33 million), compared with 78.36% (137.51 million) in data. Because less

consumers would be willing to buy iOS and Android models, which have a lot of users in

the data.

Figure 3: OS Growth without two-year Contract

The OS concentration would decrease without the carrier discounts because the larger

OSs no longer have high carrier discounts. Blackberry and Windows Phone would share

10.08% of the market, more than the 6.39% in the data. The lead of Android to Windows

Phone would decrease to 14.30%, compared with 28.57% in the data. iOS and Android

would still dominate the market for several reasons. First, they still have the initial OS

38

network advantages, which would result in higher demand in all periods. Second, iOS has

high fixed effects from the demand estimates and Android has many models in the market,

so consumers’ demand for the two OSs would still be stronger.

If the carriers didn’t offer the two-year contracts, consumers’ surplus would decrease

by $855.37 million during the sample period. Consumers would get higher utility from

Blackberry and Windows Phone models, because the service fee deduction would be more

than the carriers’ contract discounts on smartphones and OS network sizes would increase.

Their utility from Android and iOS models would decrease, because the service fee deduc-

tion would be less than the carriers’ discounts on smartphones and their OS network sizes

would be smaller. Again, since most consumers are Android and iOS users in the data,

consumers surplus would drop without the two-year contract discounts.

8 Conclusion

The literature on network effects focused on pricing decisions of single-network firms, but

not multi-network sellers. In this paper, I analyze the impact of a network effect on a

multi-network seller’s dynamic prices in a two-period, two-OS theoretical model. I find

that, multi-network sellers choose lower prices for the products with larger networks in

early periods, while single-network sellers choose higher prices for products with larger

networks, . This is due to the difference in the two types of sellers’ ability to internalize the

competition effect across networks. A multi-network seller’s profit increases with network

concentration, so they use differentiated prices across networks to achieve concentration.

The multi-network seller’s pricing strategy is present in the smartphone industry, where

the smartphone operating system network effect exists and telecom carriers act like multi-

network sellers. To measure the impact of the network effect and the carriers’ pricing

strategy, I set up a structural model of consumers’ demand and carriers’ dynamic pricing

game for smartphones. I use smartphone model level data from 2011 to 2013 to estimate

the structural model. The results show that the OS network sizes affect consumers’ utility,

and thus carriers should make dynamic pricing strategies.

The dynamic oligopolistic game in this paper features continuous choice variables, con-

tinuous state variables, multi-product firms, and asymmetric firms. The existing estima-

tion algorithms for dynamic discrete choice games and dynamic continuous choice games

among single-product firms can’t be used in this paper. I solve the dynamic pricing game

by approximating the firms’ value functions with linear combinations of basis functions. I

39

develop an iterative procedure to efficiently solve for the equilibrium prices. As a result,

the oligopolistic multiple-product firms’ asymmetric pricing game can be solved efficiently

in the estimation.

With the estimates, I study two counterfactual cases, eliminating carriers’ ability to in-

ternalize competition across OSs and the two-year contracts, respectively. I find that both

the carriers’ ability to internalize OS competition and the two-year contract discounts sig-

nificantly accelerated the smartphone industry’s growth and the concentration of operating

systems and increased consumer surplus.

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Appendix

A The Inversion of Carrier-OS Shares to Carrier-OS Unob-served Quality

In this section, I prove that the observed vector carrier-OS sales market shares sscts uniquelydetermine a vector of the carrier-OS unobserved quality ξsct when the utility function isat smartphone model level. The proof is based on that in Berry (1994). For notationsimplicity, I use a logit model version of the demand model with carrier-OS unobservedquality.

Consider the following consumer utility function:

uijsct = δjsct + ξsct + εijsct,

where δjsct is the mean observed utility and ξsct is carrier-OS specific unobserved quality.The utility of the outside option is assumed to be ui0t = εi0t. Given the assumption thatεijscts follow Type-I extreme value distribution and are i.i.d. across consumers, models,and periods, the market share of carrier-OS group (c, s) in period t is:

ssct = Ssct(ξt) =∑j∈Ωsct

e(δjsct+ξsct)

1 +∑

(j′,s′,c′)∈Ωte(δj′s′c′t+ξs′c′t)

. (A.1)

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I need to prove that there is a unique ξt = S −1(sdatat ) ∈ RG for any fixed finite δvector, where sdatat is the observed vector of carrier-OS market shares in period t givenother unobserved shocks ξ−sct. G is the number of carrier-OS groups. The equation (A.1)has the following properties. (1), ∂Ssct/∂ξsct > 0; (2) ∂Ssct/∂ξs′c′t < 0, if (s, c) 6= (s′, c′);and (3) Ssct approaches to zero as ξsct goes to −∞ and it approaches to 1 as ξsct goes to∞.

Define the element-by-element inverse, rsc(ξt, sdatasct ), as the unobserved quality value

for group (s, c, t) such that the model predicted carrier-OS share Ssct equals the observedshare sdatasct . That is:

sdatasct = Ssct(rsc((ξt, sdatasct )), ξ−sct; δt). (A.2)

Since the market share function S is continuously differentiable and satisfy the threeproperties above, the function rsc(ξt, s

datasct ) is well defined and differentiable. In particular,

rsc is strictly increasing in ξs′c′t for any (s′, c′) 6= (s, c) and doesn’t depend on ξsct. So avector ξt solves equations in (A.1) if and only if it is a fixed point of the element-by-elementinverse: ξt = r(ξt, d

data). Next, I first show the existence of a fixed point of r(ξt, ddata),

then show the uniqueness of the fixed point.First, r(ξt, d

data) has a lower bound ξ. The lower bound for the (s, c)th element is the

value of rsc(ξ′t, d

data), with ξ′s′c′t = −∞, for all (s′, c′) 6= (s, c). Define ξ as the smallestvalue across the products of these lower bounds. Note that there is no upper bound forrsc, but a slight variant of the element-by-element inverse has.

Lemma 1. There is a value ξ, with the property that if one element of ξt, say ξsct is greaterthan ξ, then there is another carrier-OS pair (s′, c′) such that rs′c′t(ξt, s

datat ) < ξs′c′t.

Proof. To construct ξ, set ξs′c′t = −∞, for all (s′, c′) 6= (s, c). Then define ξsct as thevalue of ξsct that set the outside option market share S0t(ξsct, ξ−sct) = sdata01 . Define ξas any value greater than the maximum of the ξsct. Then, if for the vector ξt, there isan element (s, c) such that ξsct > ξsct, then S0t(ξsct, ξ−sct) < sdata01 , which implies that∑

s′c′ Ss′c′t(ξt; δt) >∑

s′c′ sdatas′c′t, so there is at least one carrier-OS pair (s′, c′) such that

Ss′c′t(ξt; δt) > sdatas′c′t. Then for this pair (s′, c′), rs′c′t(ξt, sdatat ) < ξs′c′t.

Now define a new function which is a truncated version of rsc: rsc(ξt, sdatat ) =

minrsc(ξt, sdatat ), ξ. Then r is a continuous function which maps [ξ, ξ]G into itself. Thenby Brouwer’s fixed-point theorem, r is has a fixed point ξ∗. By the definition of ξ and ξ,

ξ∗ can’t have a value at the upper bound, so ξ∗ is in the interior of [ξ, ξ]G. This implies

that ξ∗ is also a fixed point of the function r(ξt, sdatat ). So there exists a fixed point for the

element-by-element inverse function.Next I show the uniqueness of the fixed point. One sufficient condition for unique-

ness is the diagonal dominance of the Jacobian matrix of the inverse functions. That is:∑(s′,c′)6=(s,c) |∂rsc/∂ξs′c′t| < |∂rsc/∂ξsct|. By the implicit function theorem on equation

(A.2), we have:∂rsc/∂ξs′c′t = −[∂Ssct/∂ξs′c′t]/[∂Ssct/∂ξsct],

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which implies that ∂rsc/∂ξsct = 1. Then the sum is:∑(s′,c′)6=(s,c)

|∂rsc/∂ξs′c′t| =1

|∂Ssct/∂ξsct|∑

(s′,c′)6=(s,c)

|∂Ssct/∂ξs′c′t|. (A.3)

Note that increasing all the unobserved quality levels (including the outside option ξ0t)by the same amount wouldn’t change any market share. That is:

K∑s′c′=0

∂Ssct/∂ξs′c′t = 0

Then it implies that:

∂Ssct/∂ξsct = −[∂Ssct/∂ξ0t +K∑

s′c′ 6=(sc)

∂Ssct/∂ξs′c′t].

Since all terms on the right hand side are strictly negative, so

|∂Ssct/∂ξsct| > |K∑

s′c′ 6=(sc)

∂Ssct/∂ξs′c′t|.

Then the sum in equation (A.3) is:∑(s′,c′)6=(s,c)

|∂rsc/∂ξs′c′t| < 1 = |∂rsc/∂ξsct|.

Hence the sufficient condition for uniqueness is satisfied. Therefore, the element-by-element inverse function has unique fixed point ξ∗, which is the solution of the marketshare inversion function.

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