Supplemental Materials to “The Operating System Network

Effect and Carriers’ Dynamic Pricing of Smartphones”

Rong Luo

October 5, 2016

A Proofs in the Two-OS, Two-Period Model

Lemma 1. Given that the two models have the same unit cost, the multi-network seller

chooses the same price for A and B in the second period: p∗cA2 = p∗cB2.

Proof. Let the unit cost of the two models be c. The seller’s profit in the 2nd period is:

π2(pA1, pB1, pA2, pB2) =∑j=A,B

(pj2 − c)M2sj2

=∑j=A,B

(pj2 − c)M2e(δj+γnj2−αpj2)

1 +∑

k=A,B e(δk+γnk1−αpk2).

The FOC with respect to pA2 is:

sA2 − α(p∗A2 − c)sA2(1− sA2) + α(p∗B2 − c)sA2sB2 = 0, (A.1)

which is equivalent to:

1− α(p∗A2 − c) + α(p∗A2 − c)sA2 + α(p∗B2 − c)sB2 = 0. (A.2)

Similarly for B, we have:

1− α(p∗B2 − c) + α(p∗A2 − c)sA2 + α(p∗B2 − c)sB2 = 0. (A.3)

Compare equations (A.2) and (A.3), we get p∗A2 = p∗B2.

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A.1 Proof of Proposition 1

Proof. I solve the two-period game backwards. Let the price in the 2nd period of the two

models be p∗2. Then plug p∗2 and c = 0 into equation (A.1). We get:

1− αp∗2(1− sA2 − sB2) = 0.

Plug in the sales market share equations and rearrange the terms. We get:

αp∗2 − 1 = e(δA+γnA2−αp∗2) + e(δB+γnB2−αp∗2). (A.4)

The total differentiation of (A.4) is:

α∂p∗2∂nA2

= e(δA+γnA2−αp∗2)(γ − α ∂p∗2∂nA2

)− αe(δB+γnB2−αp∗2) ∂p∗2

∂nA2.

Then we can solve for∂p∗2∂nA2

:

∂p∗2∂nA2

=γ

αsA2. (A.5)

Similarly for∂p∗2∂nB2

, we have:

∂p∗2∂nB2

=γ

αsB2. (A.6)

Then the profit in the 2nd period is:

π2 =∑j=A,B

pj2M2sj2

= p∗2M2(sA2 + sB2)

= p∗2M2(1− 1

1 + e(δA+γnA2−αp∗2) + e(δB+γnB2−αp∗2))

= p∗2M2(1− 1

αp∗2)

=M2

α(αp∗2 − 1).

(A.7)

2

Then the maximization problem in the first period is:

maxpA1,pB1

π1(pA1, pB1) + βπ2(pA1, pB1, p∗2(pA1, pB1))

= maxpA1,pB1

∑j=A,B

pj1M1sj1 +β

α(αp∗2(p∗A1, p

∗B1)− 1)M2

= maxpA1,pB1

∑j=A,B

pj1M1sj1 + β(p∗2(pA1, pB1)− 1

α)M1s01.

(A.8)

in which s01 = 1− sA1 − sB1 (0 means the outside option) and those who didn’t buy any

smartphone in the first period enter the second period, M2 = M1s01. Then the FOC with

respect to pA1 is:

0 =sA1 − αp∗A1sA1(1− sA1) + αp∗B1sA1sB1

+ αβ(p∗2(pA1, pB1)− 1

α)sA1s01 + β

∂p∗2∂pA1

s01,(A.9)

in which the partial derivative of price p∗2 with respect to 1st period price pA1 is:

∂p∗2∂pA1

=∂p∗2∂nA2

∂nA2

∂pA1+

∂p∗2∂nB2

∂nB2

∂pA1

=∂p∗2∂nA2

M1(−α)sA1(1− sA1) +∂p∗2∂nB2

M1αsA1sB1.

Plug∂p∗2∂pA1

into (A.9), we get:

1− αp∗A1 + αp∗A1sA1 + αp∗B1sB1 + β(p∗2 −1

α)αs01

+ βM1s01[−α ∂p∗2∂nA2

(1− sA1) + α∂p∗2∂nB2

sB1] = 0.(A.10)

Similarly for OS B, the first-order condition of profit with respect to price pB1 gives the

following equation:

1− αp∗B1 + αp∗B1sB1 + αp∗A1sA1 + β(p∗2 −1

α)αs01

+ βM1s01[−α ∂p∗2∂nB2

(1− sB1) + α∂p∗2∂nA2

sA1] = 0.(A.11)

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Then by comparing equations (A.10) and (A.11), we get:

− αp∗A1 + βM1s01[−α ∂p∗2∂nA2

(1− sA1) + α∂p∗2∂nB2

sB1]

=− αp∗B1 + βM1s01[−α ∂p∗2∂nB2

(1− sB1) + α∂p∗2∂nA2

sA1].

Plug in the partial derivatives in (A.5) and (A.6), we get:

p∗A1 − p∗B1 =βγ

αM1s01(sB2 − sA2). (A.12)

Next we need to prove p∗A1 < p∗B1 if nA1 > nB1. That is, the 1st period price for the

large OS is lower than that for the small OS. Notice that sB2 < sA2 ⇔ nB2 < nA2, given

p∗B2 = p∗A2 and δA = δB = δ. Next I discuss the three cases of possible relationship between

p∗A1 and p∗B1 to prove that p∗A1 < p∗B1 must hold to maximize profit.

First, suppose p∗A1 = p∗B1. Then nB2 < nA2. Because nj2 = nj1 + M1sj1, nA1 > nB1,

and sA1 = sB1. But this means (A.12) is violated because sB2 − sA2 < 0.

Second, suppose p∗A1 > p∗B1. If this is true, then (A.12) implies that sB2 > sA2, which

means that nB2 > nA2. That is, the initial OS network advantage of A is reversed in

the 2nd period due to high price of A. Since nj2 = nj1 + M1sj1, nB2 > nA2 implies that

sB1 > sA1. That is, the sales share of B is higher than that of A in the first period. Use

the equation of sales market shares, we get:

e(δ+γnB1−αp∗B1) > e(δ+γnA1−αp∗A1). (A.13)

Let (p∗A1, p∗B1, p

∗2) bet the profit maximization prices in the two periods. Then the

seller’s total profit is:

Π∗ = M1p∗A1e(δ+γnA1−αp∗A1) + p∗B1e(δ+γnB1−αp∗B1) + β/α(αp∗2 − 1)

1 + e(δ+γnA1−αp∗A1) + e(δ+γnB1−αp∗B1).

Now consider a different price plan for the two periods (p′A1, p′B1, p

∗2), in which:

γnA1 − αp′A1 = γnB1 − αp∗B1,

γnB1 − αp′B1 = γnA1 − αp∗A1.

Then p′A1 =γnA1−γnB1+αp∗B1

α and p′B1 =γnB1−γnA1+αp∗A1

α . The seller’s total profit with this

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price plan is:

Π′ = M1p′A1e(δ+γnA1−αp′A1) + p′B1e(δ+γnB1−αp′B1) + β/α(αp∗2 − 1)

1 + e(δ+γnA1−αp′A1) + e(δ+γnB1−αp′B1)

= M1p′B1e(δ+γnA1−αp∗A1) + p′A1e(δ+γnB1−αp∗B1) + β/α(αp∗2 − 1)

1 + e(δ+γnA1−αp∗A1) + e(δ+γnB1−αp∗B1).

Take the difference of the two profits with the two price plans, we have:

Π′ −Π∗ = M1γ/α(nA1 − nB1)(e(δ+γnB1−αp∗B1) − e(δ+γnA1−αp∗A1))

1 + e(δ+γnA1−αp∗A1) + e(δ+γnB1−αp∗B1).

The according to (A.13), we know that Π′ > Π∗. Hence, there exists another price plan

that leads to higher profit than (p∗A1, p∗B1, p

∗2), when p∗A1 > p∗B1. Therefore p∗A1 > p∗B1 can

not be the profit maximization solution.

Therefore, the seller’s profit maximization prices in the first period must satisfy p∗A1 <

p∗B1. This implies that nA2 > nB2 and thus sA2 > sB2. Then we have:

p∗A1 − p∗B1 =βγ

αM1s01(sosB2 − sosA2),

nA2 − nB2 = nA1 − nB1 +M1(sA1 − sB1).

Based on these two equations, we have the following conclusions:

(1) The optimal price of A is lower than that of B in the first period: p∗A1 < p∗B1.

(2) The price gap |p∗A1−p∗B1| between the two models increase as the OS network effect

becomes stronger (γ increases).

(3) The OS market share difference (nA2 − nB2) increases in the OS network effect γ.

A.2 Proof of Proposition 2

Proof. This proof of the single-network sellers’ dynamic pricing game has three steps. The

first step shows that the price of the larger OS network is higher in the second period. The

second step shows that the price of the larger OS network is higher in the first period. The

third step shows that the larger operating system keeps its advantage in the second period.

Combining the three steps, Proposition 2 is proved.

1. Step 1. This steps shows that, if nA2 > nB2 at the beginning of the second period,

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then pmA2 > pmB2. Seller j’s problem in the second period is:

maxpmj2πmj2(pmj2, p

m−j2) = pmj2sj2M2

= pmj2e(δj+γnj2−αpmj2)

1 +∑

k=A,B e(δk+γnk1−αpmk2)M2.

Then the FOC w.r.t. price is:

sj2 + p∗mj2 (−αsj2 + αs2j2) = 0,

which is equivalent to the following equation since sj2 > 0:

p∗mj2 =1

α(1− sj2). (A.14)

By comparing the equations (A.14) for model A and B, we have the following equa-

tion:p∗mA2

p∗mB2

=1− sB2

1− sA2. (A.15)

From equation (A.15) and the assumption that nA2 > nB2, the result p∗mA2 > p∗mB2

holds. The proof is as follows. Suppose that p∗mA2 ≤ p∗mB2 . This implies that model A

not only has larger OS network size (nA2 > nB2), but also has a lower price in the

second period. then sA2 > sB2. So the LHS (left hand side) of equation (A.15) is less

than 1 but the RHS (right hand side) is greater than 1. This contradiction shows

that p∗mA2 > p∗mB2 when nA2 > nB2. That is, the model with larger OS network size at

the beginning of the second period has higher price in the second period.

2. Step 2. This step is to show that if nA1 > nB1 and nA2 > nB2, then p∗mA1 > p∗mB1 .

That is, the optimal price of the larger OS model is higher in the first period.

From the Step 1, the maximum profit in the second period for j is:

π∗mj2 = p∗mj2 sj2M2

=sj2

α(1− sj2)(1− nA2 − nB2)

=sj2

α(1− sj2)M1s01,

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in which s01 is the market share of the outside option in the first period. Then seller

j’s profit maximization problem in the first period is:

maxpmj1πmj1(pmj1, p

m−j1) + βπ∗mj2

= pmj1sj1M1 +β

α

sj21− sj2

s01M1

= pmj2e(δj+γnj1−αpmj1)

1 +∑

k=A,B e(δk+γnk1−αpmk1)M1 +

β

α

sj21− sj2

s01M1.

Then the FOC w.r.t. pmj1 is:

sj1 − αsj1(1− sj1) + βs01sj1sj2

1− sj2

+β

αs01

1

(1− sj2)2

∂sj2∂sj1

∂sj1∂pmj1

= 0 .(A.16)

Using the definition of sj2 and sj1, we get the following partial derivatives:

∂sj2∂sj1

= γsj2(1− sj2),

∂sj1∂pmj1

= −M1αsj1(1− sj1).(A.17)

Plug equations in (A.17) into equation (A.16) and rearrange the terms, we get the

following equation:

1 + βs01sj2

1− sj2= (1− sj1)(αp∗mj1 − βM1s01

γsj21− sj2

). (A.18)

Equation (A.18) can be applied to both model A and model B, then by comparing

the two sides for the two models, we get:

1 + βs01sA2

1−sA2

1 + βs01sB2

1−sB2

=1− sA1

1− sB1︸ ︷︷ ︸R1

∗αp∗mA1 − βM1s01

γsA21−sA2

αp∗mB1 − βM1s01γsB2

1−sB2︸ ︷︷ ︸R2

. (A.19)

Given the assumption that nA2 > nB2, it is shown in Step 1 that sA2 > sB2. So for

the equation (A.19), LHS > 1. Next I show that p∗mA1 > p∗mB1 .

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Suppose p∗mA1 ≤ p∗mB1 , then sA1 > sB1 because model A not only has the OS network

advantage but also lower or equal price than model B. So on the RHS of equation

(A.19), we have R1 < 1. In addition, since sA2 > sB2, then R2 < 1 in equation

(A.19). Thus, the RHS < 1 for equation (A.19), which contradicts the result that

LHS > 1.

Therefore, in the first period, the price of model A is higher than model B, p∗mA1 > p∗mB1 ,

when nA2 > nB2 and the nA1 > nB1. In the next step, I show that nA2 > nB2 holds

if nA1 > nB1.

3. Step 3. This step shows that the manufacturer A keeps its OS network advantage

to the second period: nA2 > nB2 if nA1 > nB1. Suppose on the contradictory that

nA2 ≤ nB2, then according to Step 1 result, sB2 > sA2. So LHS < 1 for equation

(A.19). Also nA2 ≤ nB2 implies that sA1 < sB1, so R1 > 1 for equation (A.19).

In addition p∗mA1 > p∗mB1 and that sB2 > sA2 imply that R2 > 1 for equation (A.19).

Hence, the RHS > 1, which contradicts that LHS < 1. Thus, nA2 ≤ nB2 can’t hold,

which means that nA2 > nB2 holds by contradiction.

The three steps above shows that when the two manufacturers choose prices, the one

with initial OS network advantage choose higher prices in both periods and keeps its

advantage in the second period. So I have proved that: (1), p∗mAt > p∗mBt , for t = 1, 2; and

(2), nA2 > nB2.

A.3 Competition among Multi-Network Sellers

In this subsection, I use two steps to argue that if there are multiple multi-network sellers

in the two-period model, the equilibrium prices will still be that the product with the

large network has a lower price than the small network in the first period. Without loss

of generality, suppose there are two symmetric multi-network sellers. Notice that, in the

second period, the two sellers choose same prices for both A and B and have same profit.

The following arguments are for the prices and profits in the first period.

First, there exists a symmetric equilibrium in which the two sellers choose the same

prices for the two models. In such a symmetric equilibrium, the large platform will have a

lower price than the small platform in the first period. Suppose instead, that both sellers

choose a higher price for the large platform in the first period. Then given the rival’s

8

prices, each seller has the incentive to deviate and choose a lower price for the product

with the large network. Because the deviating seller gains consumers from both the other

seller and the outside option. This makes the deviation profitable. Hence, in a symmetric

equilibrium, the price of the product with the large network is lower than that of the small

network in the first period.

Second, there doesn’t exist an asymmetric equilibrium. Suppose instead, that seller

1’s price of the large network is lower than the small network and seller 2’s price of the

large platform is higher than the small network. Then seller 2 could be better off by

choosing the same price as seller 1. Because to get the same profit as seller 1 in the first

period using the asymmetric strategy, seller 2 has to choose a very low price for the small

platform to compete for customers, when seller 1 can easily get consumers with a low price

on the initially large platform. So the sellers will not choose asymmetric pricing strategy

in equilibrium.

Therefore, when there are multiple sellers, they would coordinate on choosing the same

low price for the large network than the small network. Intuitively, the network effect is

not seller specific, so the sellers can’t exclude others from benefiting from the growing OS

networks. As a result, no seller would like to grow the small platform by sacrificing the

first period profit.

B Solve for the Model Markups

The first-order conditions with respect to carrier prices are:

Mtsjsct+Mt

∑(k,s′)∈Ωct

(pcjsct−ωjpMjsct+ppct−24κsc−λjsct)∂sks′ct∂pcjsct

+βd∂Vc(nt+1)

∂pcjsct= 0. (B.1)

Next the partial derivatives in equation (B.1) will be explicitly derived. First, given the

model market share equation in the paper, the partial derivatives of shares with respect to

prices can be derived.

Consumer i’s price coefficient can be written as:

αi = α+ φαyi.

9

If (j′, s′, c′) = (j, s, c), then

∂sj′s′c′t∂pcjsct

= − 1

Ns

Ns∑i=1

αisijsct(1− sijsct). (B.2)

If (j′, s′, c′) 6= (j, s, c), then the partial derivative is:

∂sj′s′c′t∂pcjsct

=1

Ns

Ns∑i=1

αisij′s′c′tsijsct, (B.3)

where Ns is the number of simulated consumers.

Second, for ∂Vc(nt+1)∂pcjsct

, we have:

∂Vc(nt+1)

∂pcjsct=

S∑l=1

∂Vc(nt+1)

∂nlt+1

∂nlt+1

∂pcjsct. (B.4)

Given an approximate of the value function form Vc(nt), then ∂Vc(nt+1)∂ns′t+1

can be derived.∂ns′t+1

∂pcjsctcan be derived from the network size transition rule. It also depends on whether

s = s′ or not. The network size transition rule is:

ns′t+1 =7

8ns′t +

Mt

M

∑(j′,c′)∈Ωs′t

sj′s′c′t(pct).

If s = s′, then

∂ns′t+1

∂pcjsct=Mt

M

∑(j′,c′)∈Ωs′t

∂sj′s′c′t∂pcjsct

=Mt

M

∑(j′,c′)∈Ωs′t

[− 1

Ns

Ns∑i=1

αisijsct +1

Ns

Ns∑i=1

αisijsctsj′s′c′t].

If s 6= s′, then the partial derivative of the OS shares with respect to carrier price is:

∂ns′t+1

∂pcjsct=Mt

M

1

Ns

∑(j′,c′)∈Ωs′t

αisijsctsij′s′c′t.

10

Then we have:

∂Vc(nt+1)

∂pcjsct=

S∑s′=1

∂Vc(nt+1)

∂ns′t+1

∂ns′t+1

∂pcjsct

=∂Vc(nt+1)

∂nst+1

Mt

pop

∑(j′,c′)∈Ωs′t

[− 1

Ns

Ns∑i=1

αisijsct +1

Ns

Ns∑i=1

αisijsctsj′s′c′t]

+∑l 6=s

∂Vc(nt+1)

∂ns′t+1

Mt

pop

1

Ns

∑(j′,c′)∈Ωs′t

αisijsctsij′s′c′t

=Mt

pop[

S∑s′=1

∂Vc(nt+1)

∂ns′t+1(∑

j′,c′∈Ωs′t

1

Ns

Ns∑i

αisijsctsij′s′c′t)−∂Vc(nt+1)

∂nst+1

1

Ns

Ns∑i

αisijsct].

(B.5)

Then the equations (B.2)-(B.5) can be plugged back into equation (B.1). Define the

markup as mjsct = pcjsct + fct − ωjpmjsct − κsc − λjsct. Then plug the derivatives into the

FOC, we have:

sjsct +∑

(j′,s′)∈Ωct

mj′s′ct1

Ns

Ns∑i=1

αisijsctdij′s′ct −1

Ns

Ns∑i=1

αisijsctmjsct +βd

Mt

∂Vc(nt+1)

∂pcjsct= 0.

(B.6)

Since the individual choice probabilities from the demand model and the carriers’ value

functions, the only unknowns are the markups mjsct’s in equation (B.6), given a set of

parameters. There are Jt equations and Jt unknowns. The equations are linear in the

unknowns. So the markups (mjsct’s) can be solved using matrix inversion. Then we can

calculate the unobserved cost shock:

λjsct = pcjsct + fct − ωjpmjsct − κsc −mjsct. (B.7)

C Computation Details

C.1 Simulating the Income Levels of Consumers

In the estimation, I simulate the income levels Yit of 300 individuals for each period t

based on household income distributions using yearly Current Population Survey (CPS)

data. Each individual’s income level is assumed to independently drawn from a lognormal

distribution, with mean and standard deviation from the CPS data.

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I normalize the simulated individuals’ income levels by the log mean income in 2011,

such that the mean of normalized log income levels is zero in 2011. That is, in the esti-

mation, the log income level of consumer i in year is yit = log(Yit)− µ2011, where µ2011 is

the log of mean income in 2011 and Yit is i’s income level in dollars. The income levels in

2012 and 2013 are also normalized by µ2011. In this way, the normalized log-income data

keep the pattern of growing household income over time.

C.2 Number of Monte Carlo Draws to Approximate Integration over

Shocks

In the estimation, the MPEC constraints are the carriers’ Bellman equations. I impose

these constraints so that the value functions are well approximated.

Vc(nt) = Eξ,λ[ maxpcjsct(ξt,λt),(j,s)∈Ωct

πct(p

ct , ξt, λt) + βdVc(nt+1(nt, p

ct(ξt, λt)))

].

To calculate the expectation over (ξ, λ) on the right hand side, I simulate R vectors of

quality shocks and cost shocks (ξr, λr), r = 1, ..., R to approximate the integration of

discounted profits over (ξ, λ).

V Rc (nt) ≈

1

R

R∑r=1

[ maxpcjsct(ξ

r,λr),(j,s)∈Ωct

πct(p

ct , ξ

r, λr) + βdVc(nt+1(nt, pct(ξ

r, λt)))

].

Given a set of parameter values, the algorithm solves the equilibrium prices of the carriers’

dynamic pricing game for each (ξr, λr) and use the average over the draws to approximate

the value functions.

In the estimation, I use 50 draws (R=50) due to computation burden. But since the di-

mension of (ξ, λ) is more than 200, I need to check whether R = 50 is generating significant

errors in the value functions. To check this, I simulate more draws (R = 100, 300, 1000)

and use the estimation results to re-calculate the V Rc (nt)’s for all carriers in all periods.

I find that when R increases from 50 to 1000, V Rc changes by 1.35% on average with a

standard deviation of 1.61%.

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C.3 Iteration Algorithm to Solve the Carriers’ FOCs

For each simulated shock (ξr, λr), the algorithm for solving the equilibrium prices in period

t follows four steps.

1. Guess an initial price vector for all models, p0r. With p0r., I calculate the consumers’

choice probabilities s0rijsct for all models and the next period state variable nt+1(nt, p

0r)

using p0r and nt.

2. Calculate the markups m0rjscts using FOCs in (B.6), which are linear in the markups.

3. Update the price for all models simultaneously. The new price for model (j, s, c, t) is:

p1rjsct = m0r

jsct + cjsct(θs, λr).

4. Compare p1r with p0r. If the distance between the two prices, in L1 norm, is greater

than 10−3, then repeat step 2-4. If the distance is smaller than 10−3, then the prices

are solved.

D Tables and Graphs

D.1 Stationarity of the Carriers’ Dynamic Problem

I assume that the set of smartphone models are not state variables. Ideally, the number

of smartphone models of each OS by each carrier and its characteristics should be state

variables because they affect the carriers’ period profits. However, the dynamic problem

would quickly become intractable as the dimension of the state space increases. Table 1

shows the mean and the standard deviation of monthly number of smartphone models by

OS by carrier from August 2011 to October 2013. The standard deviations in the number

of smartphone models over time are relatively small compared with the mean number of

smartphone models, except for iPhones by T-Mobile. T-Mobile only started to sell iPhones

since early 2013.

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Table 1: Mean and Std Deviation of Monthly Number of Models

Verizon AT&T Sprint T-Mobile

iOS 4.69 5.08 4.54 0.92(1.32) (1.02) (1.88) (2.04)

Android 16.15 13.88 11.38 10.62(2.77) (2.97) (3.21) (1.79)

Blackberry 3.00 3.81 1.65 3.69(1.13) (1.02) (0.49) (0.74)

Windows Phone 1.81 4.81 0.65 1.12(1.06) (1.52) (0.63) (0.65)

D.2 Seasonality of Increases of Smartphone Users

The green curve in Figure 1 shows the adjusted monthly increases of smartphone sub-

scribers. The monthly increases are adjusted so that (1) the geometric mean of adjusted

sales is the same across months of the year and (2) the total increase of smartphone users

from Sep 2009 to Jan 2014 is equal to the original data.1

Figure 1: Increase of Smartphone Users by Month

1Following Gowrisankaran & Rysman(2009), I first regress the monthly log increases on the monthdummies. Then divide each monthly increase by the exponentiated dummy for its month of the year. Theadjusted increase is constructed by multiplying the divided increases by a constant such that the totalincrease during Sep 2009 to Jan 2014 is the same with that in the original data

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D.3 Plots of the Carriers’ Value Functions

Each carrier’s value function is a function of the market shares of the four OSs. Figure

2 and 3 are the plots of the value functions. The impact of market shares of iOS and

Android on the carriers’ value functions are in Figure 2. The market shares of Blackberry

and Windows Phone are fixed at the July 2013 shares. The plot of the value functions

against the market shares of Blackberry and Windows Phone are in Figure 3. The market

shares of iOS and Android are fixed at the July 2013 shares. The value function of Verizon

is the highest among the four carriers, and T-Mobile is the lowest in both figures.

(a) Verizon (b) AT&T

(c) Sprint (d) T-Mobile

Figure 2: Value Functions (iOS and Android Market Shares)

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(a) Verizon (b) AT&T

(c) Sprint (d) T-Mobile

Figure 3: Value Functions (Blackberry and Windows Phone Market Shares)

There are different effects of the OS market shares on the carriers’ profits. First, there is

a market size effect. As more consumers adopt smartphones, there are less non-smartphone

consumers in the market and only a small portion of existing users re-enter the market.

This effect would make the value functions decreasing with OS shares. Second, there is

OS network effect. As the market share of an OS increases, this OS can attract more

consumers in the future periods. This effect would make the value functions increasing

with OS shares. Third, there is price differentiation effect. Since the carriers give earn

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lower markups on larger OSs, the differentiating pricing strategy alone would drive down

a carrier’s profit as the market share of large OSs increase. Forth, the OS shares may

have different impacts on the value functions of the carriers, who are asymmetric in their

product sets. On one hand, carriers that sell more products with large OS networks would

benefit more from an increase in the market share of that OS. On the other hand, the

carriers selling less smartphones of large OSs don’t need to give high two-year contract

discounts to as many customers. Vice versa for an increase in the share of a small OS.

Therefore, it’s not clear to see the monotonicity of the value functions.

From Figure 2 The carriers’ value functions decrease with the market shares of Android

and iOS. This is because that the market size effect dominates the OS network effect.

though the increasing OS network sizes could increase sales to a certain degree. T-Mobile’s

value function increases with iOS market share. This is probably because that T-Mobile

didn’t sell iPhones for most of the time in the data, so it doesn’t need to offer high discounts

towards iOS as other carriers. When this effect dominates the market size effect, T-

Mobile’s value function would increase with iOS market share. The monotonicity of the

value functions against the market shares of Blackberry and Windows Phone become is

less clear as shown in Figure 3. This could be because of the different effects listed above

of Blackberry and Windows Phone work together and show concavity or convexity in some

OS shares.

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