TSE‐691
September2016
“CompatibilityChoicesunderSwitchingCosts”
Doh‐ShinJeon,DomenicoMenicucciandNikroozNasr
Compatibility Choices under Switching Costs∗
Doh-Shin Jeon†
Domenico Menicucci‡
Nikrooz Nasr§
August 31, 2016
Abstract
We study firms’ compatibility choices in the presence of consumers’ switching
costs. We analyze both a model of once-and-for-all compatibility choices and that
of dynamic choices. Contrary to what happens in a static setting in which firms
embrace compatibility to soften the current competition (Matutes and Régibeau,
1988), when consumer lock-in arises due to a significant switching cost, firms make
their products incompatible in order to soften future competition, regardless of the
model we consider. This reduces consumer surplus and social welfare.
Key words: Compatibility, Incompatibility, Switching Cost, Lock-in
JEL Codes: D43, L13, L41.
∗
We thank Paul Hünermund, Joshua Gans, Lukasz Grzybowski, Jorge Padilla, Patrick Rey, Yaron
Yehezkel, and the audience who attended our presentations at the Net Institute Conference 2015, Ninth
IDEI-TSE-IAST conference on the Economics of Intellectual Property, Software and the Internet, Workshop on the Economics of Network Industries 2016, 14th ZEW Conference on The Economics of Information and Communication Technologies 2016, and 31st Annual Congress of The European Economic
Association. We gratefully acknowledge financial support from the NET Institute. Jeon acknowledges
financial support from the Jean-Jacques Laffont Digital Chair.
†
Toulouse School of Economics,
University of Toulouse Capitole (IDEI), E-mail:
[email protected].
‡
Dipartimento di Scienze per l’Economia e l’impresa, Università degli Studi di Firenze. Via delle
Pandette 9, I-50127 Firenze (FI), Italy. E-mail: [email protected].
§
Toulouse School of Economics. Manufacture de Tabacs, 21 allees de Brienne - 31000 Toulouse, France.
E-mail: [email protected]
1
Introduction
Will the future of the Internet be dominated by incompatible platforms? Back in the 90s’
when the Internet was at its dawn, openness and compatibility seemed to be the future.
For instance, during the times, Microsoft was the dominant player in the personal computer market but decided to bring two of its most successful software, Internet Explorer
and Microsoft Office, to Macs. However, after the turn of the 21st century, we seem to
enter a new era in which platforms try to lock-in customers by making it hard to move
data across platforms or by providing some benefits exclusively to those who use all from
the same ecosystem. According to Larry Page, a co-founder of Google and the CEO of
its parent company, Alphabet Inc.,:
"The Internet was made in universities and it was designed to interoperate.
And as we’ve commercialized it, we’ve added more of an island-like approach
to it, which I think is a somewhat a shame for users."1
If a consumer purchased applications from iTunes (Google Play), she is more likely to
choose an iPhone (an Android phone) instead of an Android phone (an iPhone). The same
logic applies to services like: iMessage and iCloud versus Hangouts and Google account,
One Drive versus Google Drive, iCloud Keychain versus Chrome password syncing. In
cloud computing, from the beginning, a major concern was vendor lock-in where due to
incompatible technologies, customers would incur a huge cost if they wish to transfer their
data from one vendor to another2 . This kind of phenomenon is known as "walled garden".
As cloud computing technology becomes ubiquitous, the risk of lock-in in walled gardens
of data generalizes.
Furthermore, Apple and Google are expanding their businesses outside of the phone
industry by leveraging their established smartphone platforms: smart car, smart TV,
Google glasses, Apple watch, Apple pay etc. These new hardware and software bring the
ultimate lock-in experience. For instance, Android Wear only works with Android phones
and tablets. Apple Watch supports iPhones and iPads exclusively. We enter an era where
working across multiple ecosystems deprive consumers of real benefit. This is why Drew
Houston, CEO of Dropbox, says that consumers are "trapped" in each platform.3
1
http://fortune.com/2012/12/11/fortune-exclusive-larry-page-on-google/
http://fortune.com/2015/10/08/aws-lock-in-worry/
3
http://vimeo.com/70089044
2
1
In this paper, we attempt to offer a novel insight to understand platforms’ compatibility choices from a dynamic perspective, which is relevant because hardware must be
upgraded and subscription to content/application often must be renewed. As the main
result, we show that when consumer lock-in arises due to a significant switching cost,
firms make their products incompatible in order to soften future competition, which hurts
consumers and reduces welfare. More generally, we find a strong conflict between the
compatibility regime chosen by the firms and the one maximizing consumer surplus. Our
result is opposite to what happens in a static model in which symmetric firms make their
products compatible to soften current competition (Matutes and Régibeau, 1988).
We extend the model of Matutes and Régibeau (1988) to two periods. They study
compatibility choices made by two symmetric firms (A and B) who compete to sell a system of complementary products (x and y). Therefore, under compatibility, four systems
are available ((A, A), (A, B), (B, A), (B, B)) while under incompatibility, only two pure
systems, (A, A) and (B, B), are available. They study a two-stage game in which the first
stage of non-cooperative choice between compatibility and incompatibility is followed by
the second stage of price competition. We consider two different dynamic extensions of
their setup. In our model 1 of once-and-for-all compatibility choices, firms make their
choices only in period one and the compatibility regime determined in period one is maintained in period two. In model 2 of dynamic compatibility choices, firms make their
choices every period. Both models are identical to that of Matutes and Régibeau if there
were no second period. In the second period of our models, each platform competes to
poach consumers by offering prices dependent on past purchase behavior (Chen, 1997 and
Fudenberg and Tirole, 2000): there is price discrimination depending on which product
or system a consumer bought in the first-period.4
Consider first model 1 of once-and-for-all compatibility choices. When products are
compatible, there are four submarkets in period two: the market ij refers to the market
composed of the consumers who bought product j = x, y from firm i = A, B in period
one. We assume that each consumer discovers the value that she obtains from a product,
which is a random draw from a uniform distribution, only after consuming it. We assume
for simplicity that all consumers incur the same switching cost s > 0. For instance, if a
consumer wants to switch from x of A to x of B, she should incur s and therefore firm A is
4
For instance, apple launched Android device trade-in program to encourage iPhone upgrade:
http://9to5mac.com/2015/03/16/applewillofferandroidswitchersgiftcardstotradeinrivalsmartphonesforiphones/
2
dominant and firm B is dominated in market Ax . When products are incompatible, there
are two submarkets (A, A) and (B, B) in period two where market (A, A) is composed of
the consumers who bought system (A, A) in period one. If a consumer wants to switch
from (A, A) to (B, B), she should incur a switching cost of 2s.
When we study how incompatibility affects the firms’ second-period profits from the
consumers who bought say (A, A) in period one, we find three thresholds of switching
costs (s1 , s2 , s3 ) with s1 < s2 < s3 such that the dominant firm’s profit is higher under
incompatibility than under compatibility for s > s1 , the industry profit is higher under
incompatibility for s > s2 and the dominated firm’s profit is higher under incompatibility for s > s3 . In other words, when the switching cost is high enough, incompatibility
softens second-period competition relative to compatibility. This result can be understood from Hahn and Kim (2012) and Hurkens, Jeon and Menicucci (2016), who extend
Matutes and Régibeau (1988) to asymmetric platforms (one is dominant and the other is
dominated).5 They consider a static model and find that when the level of dominance is
large enough, both the dominant firm and the dominated firm prefer incompatibility since
incompatibility softens competition. This mechanism is in place in our model as the level
of dominance increases with s. In Section 3.3, following Hurkens, Jeon and Menicucci
(2016), we decompose the effect of incompatibility into demand size effect and demand
elasticity effect and explain how incompatibility softens second-period competition when
switching cost is high.
When we study first-period competition for a given regime of compatibility, each firm
charges a price smaller than the one in Matutes and Régibeau (1988) by the difference
between the dominant firm’s second-period profit and the dominated firm’s one. In other
words, each firm dissipates the rent from a locked-in consumer by competing more aggressively in period one. This implies that each firm’s total profit is the sum of the profit
in Matutes and Régibeau (1988) and the dominated firm’s (second-period) profit. Therefore, for s > s3 , if the relative weight of the second-period payoff is large enough, the
firms choose incompatibility in period one in order to soften future competition even if it
5
Hahn and Kim (2012) consider uniform distribution of consumers on a Hotelling line as Matues and
Régibeau do. Hurkens, Jeon and Menicucci (2016) consider the family of log-concave distributions which
include the uniform distribution and offer a general intuition based on the property that the distribution of
average location is more peaked than the original distribution for this family. In addition, they decompose
the effect of pure bundling into demand size effect and the demand elasticity effect and use the insight
to build a theory of credible leverage of market dominance.
3
intensifies current competition; for all other parameter values, they choose compatibility.
As the firms make compatibility choices mainly to soften competition, we find a strong
conflict between the compatibility regime chosen by the firms and the one maximizing
consumer surplus, regardless of the model we consider. In particular, in model 1 of onceand-for-all compatibility choices, whenever the firms choose incompatibility, it generates a
lower consumer surplus than compatibility. We also find a similar conflict between firm’s
choices and welfare. In particular, whenever firms choose incompatibility, it generates a
smaller welfare than compatibility, regardless of the model we consider. Welfare is lower
under incompatibility as consumers lose the ability to mix and match the products.
The analysis of model 2 of dynamic compatibility choices qualitatively confirms the
findings from model 1. The key difference between the two models is that in model 2, firms
make compatibility choices in period two as well, which implies that when considering
a deviation in period one, a firm needs to think about how the deviation will affect the
compatibility regime in period two. In a few cases, this generates more powerful deviations
than the ones under model 1 such that no pure strategy equilibrium exists in model 2. The
analysis of model 2 uncovers some novel mechanisms regarding dynamics of compatibility
as follows.
First, given first-period market shares, the dynamics of compatibility is asymmetric
in that firms are more likely to embrace compatibility tomorrow if products are compatible today but today’s incompatibility does not necessarily imply that the firms are
more likely to embrace incompatibility tomorrow. For instance, if the first-period market
shares are symmetric, first-period compatibility leads to second-period compatibility no
matter what the level of switching cost. In contrast, first-period incompatibility leads
to second-period incompatibility if and only if the level of switching cost is high enough.
To understand the results, notice that if first-period compatibility is followed by secondperiod incompatibility, there are four submarkets (A, A), (A, B), (B, A), (B, B) in period
two. But if incompatibility is maintained in both periods, there are only two submarkets
(A, A), (B, B). Hence, first-period compatibility choices dramatically affects the distribution of switching cost in that all consumers switching to (B, B) must incur a total
switching cost of 2s under the first-period incompatibility whereas their total switching
cost can be 2s or s under the first-period compatibility. As competition in market (A, B)
and (B, A) is much stronger under incompatibility than under compatibility, firms choose
compatibility in period two if compatibility was chosen in period one.
4
Second, when first-period market shares are endogenous and the relative weight of the
second-period payoff is large, we find that the asymmetry in the dynamics of compatibility
disappears. For switching costs low (high) enough, firms end up choosing compatibility
(incompatibility) in period two no matter what the first-period compatibility regime. In
particular, when switching cost is high enough, given first-period compatibility, there
exists no symmetric equilibrium in period two and instead there exists a cornering equilibrium in which a firm corners both markets and chooses incompatibility which softens
second-period competition compared to compatibility.
Finally, we find that there exist two kinds of pure strategy equilibria in model 2: either
compatibility is chosen in both periods or incompatibility is chosen in both periods. We
also find that if the incompatibility equilibrium arises in model 1, it arises also in model
2. But there are parameter values under which the incompatibility equilibrium arises in
the latter while it does not in the former. Symmetrically, if the compatibility equilibrium
arises in model 2, it arises also in model 1. But there are parameter values under which
the compatibility equilibrium arises in the latter while it does not arise in the former.
In summary, when consumer lock-in arises due to high switching cost, firms are likely
to make their products incompatible in order to soften the future competition that occurs
after consumer lock-in. We also provide an extension to show that incompatibility is more
likely to be chosen in a mature market than in a growing market.
We merge two different strands of literature, the one on compatibility and the one on
poaching. First, as incompatibility is equivalent to pure bundling, our paper is related
to literature on bundling or incompatibility. The bundling (or tying) literature can be
divided into three categories. The first one includes the papers that view bundling as a
price discrimination device for a monopolist (Stigler, 1968, Schmalensee, 1984, McAfee
et al. 1989, Salinger 1995, Armstrong 1996, Bakos and Brynjolfsson, 1999, Fang and
Norman, 2006, Chen and Riordan, 2013, Menicucci, Hurkens, and Jeon, 2015). The
second is about competitive bundling where entry and exit is not an issue (Matutes and
Régibeau 1988, 1992, Economides, 1989, Carbajo, De Meza and Seidmann, 1990, Chen
1997, Denicolo 2000, Nalebuff, 2000, Armstrong and Vickers, 2010, Carlton, Gans, and
Waldman, 2010, Thanassoulis 2011). The last is about leverage theory of bundling in
which the main motive of bundling is to deter entry or induce the exit of rival firms
in the competitive segment of the market (Whinston, 1990, Choi and Stefanadis 2001,
Carlton and Waldman, 2002, Nalebuff, 2004, Peitz 2008, Jeon and Menicucci, 2006, 2012).
5
Our paper is closely related to the second literature, in particular, Matutes and Régibeau
(1988), Economides, (1989), the extension of Matutes and Régibeau (1988) to asymmetric
firms by Hahn and Kim (2012) and Hurkens, Jeon and Menicucci (2016) and the recent
papers on pure bundling when n(> 2) symmetric firms compete (Kim and Choi (2015) and
Zhou (2015)). Economides (1989) extends the finding of Matutes and Régibeau (1988) to
n > 2 firms. However, Kim and Choi (2015) and Zhou (2015) show that when there more
than two firms compete, incompatibility can generate higher profits than compatibility,
which contradicts the finding of Economides (1989).6
Second, our two-period game is similar to the games considered in poaching literature
(Chen, 1997, Fudenberg and Tirole, 2000). The main difference is that we consider multiproduct firms and their compatibility choices. This generates an interesting dynamics as
the firms’ first-period compatibility choices change the distribution of switching costs and
thereby affects the second-period compatibility choices and the second-period competition
to poach consumers
The paper is organized as follows. Section 2 describes the model. Section 3 analyzes
second-period price competition given a compatibility regime and introduces demand size
effect and demand elasticity effect of incompatibility. Section 4 analyzes model 1 of
once-and-for-all compatibility choices. Section 5 analyzes model 2 of dynamic compatibility choices. Section 6 presents the analysis of forward-looking consumers and Section
7 provides the analysis of consumer surplus and welfare. Section 8 presents an extension
studying the relationship between compatibility choices and market growth. Section 9
provides the conclusion. All the proofs are gathered in Appendix.
6
Kim and Choi (2015) extend Matutes and Régibeau (1988) by allowing for n ≥ 3 symmetric firms,
and assuming that competition occurs on a torus (i.e., the Cartesian product of two circumferences)
rather than on Hotelling segments. They prove that if n ≥ 4, then there exists a way to symmetrically
locate the firms on the torus which results in pure bundling generating higher profits than independent
pricing. Zhou (2015) studies a model with m ≥ 2 symmetric products and n ≥ 2 symmetric firms, but
instead of adopting a spatial competition model he uses the random utility framework of Perloff and
Salop (1985). He proves that if the number of firms is sufficiently large, then pure bundling increases the
firms’ profits with respect to no bundling. For both papers, the results are a consequence of a demand
elasticity effect, which makes the demand under bundling less elastic and thus softens competition, as it
occurs in the second-period of our model when switching cost is large (see Section 3.3).
6
2
Model
There are two firms, i = A, B, who produce two products, j = x, y, at constant marginal
cost c. In our model, the two products can be perfect complements or can be independently
consumed but our results are the same as long as each consumer obtains a high enough
value from each product, which we assume. Since each firm produces each product,
four systems are possible: (x, y) = (A, A), (A, B), (B, A), (B, B). We consider two-period
models in which consumers have switching costs in the second period.
Each firm’s marginal cost of producing a product is c > 0. c is large enough such that
if a consumer bought product j = x, y from a firm, he or she has no incentive to buy the
same product from the competing firm. Thus, at each period t = 1, 2, every consumer
has a unit demand for each product. Given this assumption, we normalize c to zero, and
interpret prices as the mark up above marginal cost without loss of generality.
Regarding compatibility choices, we consider two different models. In model 1, we
consider once-and-for-all compatibility choices and assume that in the beginning of period
one, each firm simultaneously and non-cooperatively chooses between compatibility and
incompatibility. Compatibility regime prevails only if both firms choose compatibility;
otherwise, incompatibility regime prevails. In model 1, the regime determined in period
one is maintained in period two. In model 2, we consider dynamic compatibility choices
in which each firm makes a compatibility choice at the beginning of every period.
In the first period, consumers have heterogeneous costs of learning to use each product
as in Klemperer (1995). Precisely, a consumer’ learning cost for product j = x, y is
uniformly distributed along a line segment [0, 1]. It is independently distributed across
different products and different consumers. For instance, a consumer located at (θx , θy ) ∈
[0, 1]2 incurs a total learning cost of tθx + tθy to use a system (A, A) or tθx + t(1 − θy ) to
use system (A, B). In the beginning of period one, every consumer has the same expected
valuation for each product, v e . Therefore, depending on the compatibility regime, the
first-period utility of a consumer located at (θx , θy ) from purchasing (A, A) is given as
follows; under compatibility, it is
A
U1 (A, A) = 2v e − pA
1,x − p1,y − tθx − tθy ;
7
(1)
under incompatibility, it is
U1 (A, A) = 2v e − P1A − tθx − tθy ,
where pi1,j 7 is the price for product j of firm i in period one under compatibility and P1i
is the price of the system (i, i) in period one under incompatibility. If there is no second
period, our models are identical to that of Matutes and Régibeau (1988).
Once a consumer uses firm i’s product j in period one, she discovers her own true
value vji for the product, which is ex ante a random draw from a uniform distribution over
[v, v], with v > 0. Hence, (v + v) /2 = v e . We assume that the distribution is independent
across different products and different consumers.
Now let us describe what happens in the second period. Consider for instance a
consumer who bought product x from A in period one. Then she has learnt her value vxA .
Her choice is either to consume the same product and obtain vxA , or to switch to product
x of B. In the latter case, her expected value is v e and in addition she has to incur a
switching cost of s > 0. The switching cost includes not only learning cost but also other
costs due to a consumer’s product-specific investment. For simplicity, we assume that
the switching cost is the same for all consumers and products. In addition, we assume
that each firm can engage in behaviour-based price discrimination to poach consumers:
the price a firm charges to a consumer in period two can depend on the history of the
consumer’s purchase in period one. Suppose now that a consumer who bought (A, A) in
the first period, switches to (A, B) in the second period (which is possible only under the
second-period compatibility). Then, her second-period utility is given by:
B
U2 (A, B)|(A,A) = vxA + v e − pA
2,x (A, A) − p2,y (A, A) − s,
(2)
where pi2,j (i, h) is the second-period price charged by firm i for product j under compatibility to the consumers who bought (i, h) in the first period with i, h ∈ {A, B}. Similarly,
if a consumer who bought (A, A) in the first period switches to (B, B) in the second
period, her second-period utility under incompatibility is given by:
U2 (B, B)|(A,A) = 2v e − P2B (A, A) − 2s,
7
We use lower-case letters for compatibility case and upper-case letters for incompatibility.
8
where P2i (i, h) is the second-period price charged by firm i for its system under incompatibility to the consumers who bought (i, h) in the first period with i, h ∈ {A, B}.
All players have a common discount factor δ > 0: δ can be larger than one since
it represents the weight assigned to the second-period payoff. All firms have rational
expectations. We consider myopic consumers for our exposition and consider forwardlooking consumers in Section 6 and show that if consumers were forward-looking, our
results remain unaffected in model 1 and remain qualitatively similar in model 2.
We introduce the following assumption to guarantee that a positive measure of consumers switch in each sub-market in period two:
Assumption 1: s < 23 ∆v where ∆v ≡ v − v.
If the assumption is not satisfied, no switching occurs under second-period compatibility.
In model 2 of dynamic compatibility choices, the timing in each period is given by:
• Stage 1: Each firm simultaneously and non-cooperatively chooses between compatibility and incompatibility.
• Stage 2: After observing the outcome of the compatibility choices, each firm simultaneously and non-cooperatively chooses its prices.
• Stage 3: Consumers make purchase decisions.
In model 1 of once-and-for-all compatibility choices, Stage 1 occurs only in period one.
Notice that there always exist an equilibrium in which both firms choose incompatibility.
But we assume that each firm plays its weakly dominant action and therefore compatibility
arises if and only if both firms prefer compatibility.
In order to solve this two-period model, we first solve for firms’ second-period equilibrium behavior for any given first-period compatibility choice and market shares. We find
that equilibrium prices and profits are linearly homogenous in ∆v regardless of compatibility regime. Therefore sometimes it is useful to normalize ∆v to 1; the model of (∆v, s)
is equivalent to that of (1, s/∆v).
9
3
Second-period price competition given a regime of
compatibility
In this section, we first study the second-period price competition to poach consumers for
a given regime of once-and-for-all compatibility and show how incompatibility affects the
second-period profits.
3.1
Given compatibility
Suppose that the firms chose compatibility in the first period. Consider the market
composed of the consumers who have bought product j from firm i in the first period.
We call it market ij . Although there are four different markets Ax , Ay , Bx , By in period
2, all four markets are alike. Hence, it is enough to analyze just one market ij 8 . We
normalize the total mass of consumers in market ij to one. In this market, the firms offer
prices pi2,j (ij ) and ph2,j (ij ) with i 6= h and i, h = A, B. To simplify notation, let us define:
i
−
h
+
i
−
h
+
i
−
h
p+
2 ≡ p2,j (ij ), p2 ≡ p2,j (ij ), d2 ≡ d2,j (ij ) , d2 ≡ d2,j (ij ) , π2 = π2,j (ij ), π2 = π2,j (ij ),
where + (respectively, −) refers to the advantage of firm i (respectively, the disadvantage
of firm h) due to the switching cost and di2,j (ij ) is the demand for product j of firm i and
i
(ij ) is firm i’s profit in market ij . A consumer with valuation vji for product j of firm
π2,j
i is indifferent between staying with firm i or switching to firm h (6= i) if and only if
e
−
vji − p+
2 = v − p2 − s.
−∗
+ +
− −
The equilibrium prices, p+∗
2 and p2 , maximize p2 d2 and p2 d2 , respectively.
We have:
Lemma 1. Suppose that compatibility was chosen in both periods. Consider the secondperiod competition in market ij composed of the consumers who bought product j from
firm i in period one. We normalize the total mass of consumers in market ij to one.
There exists a unique equilibrium, which is characterized as follows:
8
Without loss of generality, we can consider that the poaching price for product x (say) depends only
on whether a given consumer bought x from A or B but not on whether he bought y from A or B.
10
(i) The equilibrium prices in period 2 are
p+∗
2 =
∆v s
+ ,
2
3
p−∗
2 =
∆v s
− .
2
3
(3)
(ii) The firms’ second period profits from this market are given by:
π2+∗
1
=
∆v
∆v s
+
2
3
2
,
π2−∗
1
=
∆v
∆v s
−
2
3
2
Corollary 1. π2+∗ (s) + π2−∗ (s) strictly increases with s for s ∈ 0, 3∆v
2
.
(4)
−∗
+∗
−∗
As socially optimal switching requires p+∗
2 = p2 but we have p2 > p2 , there always
exists excessive switching from a social point of view. The corollary says that the industry
profit in a given product market strictly increases with s.
We note that in model 2 of dynamic compatibility choices in which each firm makes
a compatibility choice every period, Lemma 1 applies to market ij as long as there is
compatibility in the second period, independently of the first-period compatibility regime.
3.2
Given incompatibility
Suppose that the firms chose incompatibility in the first period. Consider market (i, i)
composed of the consumers who purchased both products from firm i in the first period.
We normalize the total mass of consumers in this market to one. In this market, firms
offer prices P2i (i, i) and P2h (i, i) with i 6= h and i, h = A, B. To simplify notation, let us
define:
i
−
h
P2+ ≡ P2i (i, i), P2− ≡ P2h (i, i), D2+ ≡ D2i (i, i), D2− ≡ D2h (i, i), Π+
2 = Π2 (i, i), Π2 = Π2 (i, i).
A consumer with valuations vxi , vyi is indifferent between buying (i, i) and (h, h) in period
two if and only if
vxi + vyi − P2+ = 2v e − P2− − 2s.
We have:
Lemma 2. Suppose that incompatibility was chosen in both periods. Consider the secondperiod competition in the market composed of the consumers who bought a pure system
11
(i, i) in period one. We normalize the total mass of consumers in market (i, i) to one.
There exists a unique equilibrium, which is characterized as follows:
(i) The equilibrium prices in period 2 are
P2+∗
i
i
1 hp
1h p
−∗
2
2
2
2
3 (2s − ∆v) + 8∆v + 5(2s − ∆v) , P2 =
(2s − ∆v) + 8∆v − (2s − ∆v)
=
8
8
(ii) The firms’ second period profits from this market are given by:
Π+∗
2 =
2 P2−∗
1−
∆v 2
2 !
P2+∗ , Π−∗
2
2 P2−∗
=
∆v 2
3
−∗
3∆v
.
Corollary 2. Π+∗
2 (s) + Π2 (s) strictly increases with s for s ∈ 0, 2
As the socially optimal switching in market (i, i) requires P2+∗ = P2−∗ but we have
P2+∗ > P2−∗ , there is always a socially excessive switching. In addition, for some consumers the socially optimal switching involves switching only one product. Therefore, the
incompatibility generates inefficiency as it forces a consumer either not to switch at all
or to switch both products. The corollary shows that the industry profit in market (i, i)
strictly increases with s.
We note that in the case of dynamic compatibility choices in which each firm makes
a compatibility choice every period, Lemma 2 applies to market (i, i) as long as there is
second period incompatibility, independently of the first-period compatibility regime.
3.3
The effects of incompatibility: the demand size effect and the
demand elasticity effect
Before we compare second-period profits, we here summarize the findings of Hurkens, Jeon
and Menicucci (2016), which are useful to understand our main results. They consider
competition between a dominant firm and a dominated firm in a static model and decompose the effects of incompatibility (i.e., pure bundling in their paper) into the demand
size effect and the demand elasticity effect. We here apply their analysis to the second
period in the current model by considering the market composed of the consumers who
bought product j (= x, y) from firm A (for instance) in period one. Then, firm A is the
12
dominant firm and firm B is the dominated firm in this market given s > 0. We normalize
the mass of consumers in this market to one.
Demand size effect. Suppose that under compatibility firm A charges p+
2 and firm B
−
A
charges p2 . Let vej be the valuation of the consumer who is indifferent between switching
and no switching and hence the demand for A’s product is 1 − F (e
vjA ) where F (·) is the
distribution function of vjA over [v, v], which has a symmetric log-concave density f (·)
(including the uniform distribution). In any equilibrium with s > 0, we expect vejA < v e
and hence 1 − F (e
vjA ) > 1/2. Suppose now that there is incompatibility and that each firm
sells its system at a price equal to the sum of the previous prices, that is P2+ = 2p+
2 and
−
−
i
P2 = 2p2 . Then the indifferent consumer has the average valuation equal to vej . Letting
F̂ denote the distribution function for the average valuation, we infer that the demand for
A’s system is 1 − Fb(e
vjA ). Since the distribution of the average valuation is more-peaked
around v e than the distribution of the individual location for any symmetric log-concave
vjA ) for any vejA < v e . Hence, for given prices,
density f (·), we have 1 − Fb(e
vjA ) > 1 − F (e
incompatibility increases the demand for the dominant firm A and decreases the demand
+
for the dominated firm B. In particular, these arguments apply when p+
A and pB are equal,
respectively, to their equilibrium values under compatibility for any s ∈ (0, 3∆v/2).
Demand elasticity effect. After the regime changes from compatibility to incompati−
−
bility, firms will have incentives to choose prices different from P2+ = 2p+
2 and P2 = 2p2 .
Whether they want to charge higher or lower prices depends on how incompatibility affects
demand elasticity, which in turn depends on the valuation of the indifferent consumer and
hence on the level of the switching cost. For low levels of switching cost (that is, when
vejA is not much smaller than v e ), incompatibility makes the demand more elastic: a given
decrease in the average price of a system under incompatibility generates a higher boost
in demand than the same decrease in the prices of single products under compatibility
because the distribution of the average location is more-peaked around v e than the distribution of individual locations. On the other hand, for high levels of switching cost
(that is, when vejA is close to v), incompatibility makes the demand less elastic: because
fb(v)/f (v) converges to zero as v tends to v, where fb(·) is the density of the average valuation, for vejA close enough to v, we find that a given decrease in the average price of a
system generates a smaller boost in demand than the same decrease in the prices of single
products. In summary, incompatibility changes the elasticity of demand such that firms
compete more aggressively for low levels of switching costs but less aggressively for high
13
levels of switching costs.
Second-period profit comparison. As a result of these two effects, we find the following
profit comparison:
Corollary 3. Let ∆v = 1 without loss of generality. Then, there are three threshold
values of switching cost, s1 , s2 , s3 with s1 < s2 < s3 , such that
1
2π2+∗ (s) R Π+∗
2 (s) if and only if s Q s ;
−∗
2
2π2+∗ (s) + 2π2−∗ (s) R Π+∗
2 (s) + Π2 (s) if and only if s Q s ;
3
2π2−∗ (s) R Π−∗
2 (s) if and only if s Q s .
More precisely, s1 = 0.701, s2 = 0.825 and s3 = 1.187.
2.5
Π+2 ∗
2π2+ ∗
2
1.5
1
0.5
2π2− ∗
Π−2 ∗
0
0
0.5
0.701
1
1.187
1.5
s
Figure 1: Second-period profits with or without compatibility
−∗
Even if each of π2+∗ , π2−∗ , Π+∗
2 , Π2 depends on s, in what follows we do not highlight
−∗
this dependence unless it is necessary. The next graph shows 2π2+∗ , 2π2−∗ , Π+∗
2 and Π2
14
as a function of s ∈ (0, 3/2) where we set ∆v = 1. There are three thresholds such that
+∗
+∗
−∗
+∗
−∗
for s > 0.701 we have Π+∗
2 > 2π2 , for s > 0.825 we have Π2 + Π2 > 2π2 + 2π2 , and
−∗
for s > 1.187 we have Π−∗
2 > 2π2 .
This result can be understood in view of the two effects introduced above. For low
level of switching costs (for instance, s close to zero), the demand size effect is negligible
relative to the demand elasticity effect and the latter intensifies competition compared to
compatibility. Therefore, incompatibility reduces both firms’ profits. In particular, when
s = 0, there is only the demand elasticity effect, which explains the result of Matutes
and Régibeau (1988) that incompatibility intensifies competition for symmetric firms. For
high level of switching costs (for instance, s close to 3/2), the demand size effect is again
negligible relative to the demand elasticity effect but now the latter softens competition
compared to compatibility. Therefore, incompatibility increases both firms’ profits. For
intermediate level of switching costs, the demand elasticity effect might be neutral but the
demand size effect is positive for firm A and negative for firm B. Therefore, incompatibility
increases A’s profit but reduces B’s profit. These kinds of results were obtained by Hahn
and Kim (2012) and Hurkens, Jeon and Menicucci (2016).9
4
Model 1: once-and-for-all compatibility choices
In this section, we consider model 1 of once-and-for-all compatibility choices. As the
second-period price competition is analyzed in Section 3, we start by analyzing the firstperiod price competition.
4.1
First-period price competition given a regime of compatibility
We study the first-period price competition given a compatibility regime. As we mentioned
in Section 2, we consider myopic consumers and analyze forward-looking consumers in
Section 6.
9
The intuition based on the demand size effect and demand elasticity effect are provided only by
Hurkens, Jeon and Menicucci (2016). Both papers find two thresholds of dominance corresponding to s1
and s3 . This suggests that there should be another threshold between the two such that the industry
profit is higher under incompatibility if the dominance level is higher than it.
15
4.1.1
Given compatibility
Suppose first that compatibility was chosen in the beginning of the first period. For any
vector of the first-period prices pi1,x , pi1,y , ph1,x , ph1,y , firm i’s profit is given as follows:
Πi = di1,x (pi1,x + δπ2+∗ ) + (1 − di1,x )(δπ2−∗ ) + di1,y (pi1,y + δπ2+∗ ) + (1 − di1,y )(δπ2−∗ ),
(5)
where di1,j represents demand for product j of firm i in period one (for j = x, y) and is
given by
1
1
di1,j = + (ph1,j − pi1,j ).
(6)
2 2t
We have:
Proposition 1. Consider the model of once-and-for-all compatibility choices. Given compatibility, there exists a unique equilibrium, which is symmetric. The equilibrium prices
in the first period and the total equilibrium profits are given by
+∗
−∗
pi∗
=
t
−
δ
π
−
π
≡ p∗1 ,
1j
2
2
for i = A, B and j = x, y.
(7)
t
+ δπ2−∗ ≡ π ∗ for i = A, B.
(8)
2
Given compatibility, the pricing in (7) is quite intuitive. For δ = 0, each firm charges a
price per product equal to t as in a standard Hotelling model. For δ > 0, if firm i attracts
a consumer from the rival in the first period in a given product market, its expected profit
from the customer in the second period is π2+∗ . But if the customer stays with the rival,
its expected profit from that customer in the second period is π2−∗ . Therefore, each firm
is ready to pay δ π2+∗ − π2−∗ to attract a consumer from the rival. The profit in (8) can
be understood in a similar way. For δ = 0, each firm gets a profit of t/2 as in a standard
Hotelling model. For δ > 0, each firm’s profit from the second period is equal to π2−∗ as
π2+∗ − π2−∗ is dissipated away during the first-period competition.
π i∗ =
4.1.2
Given incompatibility
Suppose now that incompatibility was chosen in the beginning of the first period. For
any pair of the first-period prices P1i , P1h , the profit functions are as follows
i
−∗
Πi = D1i (P1i + δΠ+∗
2 ) + (1 − D1 )(δΠ2 ),
i
−∗
Πh = (1 − D1i )(P1h + δΠ+∗
2 ) + D1 (δΠ2 ), (9)
16
where D1i is firm i’s market share in period one and is given by (for P1h −2t < P1i < P1h +2t)
D1i
1
=
2
2
1 h
1
i
1 + (P1 − P1 ) − 2 (P1h − P1i ) max{0, P1h − P1i }.
2t
4t
(10)
We have:
Proposition 2. Consider the model of once-and-for-all compatibility choices. Given incompatibility, there exists a unique equilibrium, which is symmetric. The equilibrium
prices in the first period and the total equilibrium profits are given by
−∗
P1i∗ = t − δ Π+∗
≡ P1∗
2 − Π2
Πi∗ =
t
∗
+ δΠ−∗
2 ≡ Π ,
2
for i = A, B.
for i = A, B.
(11)
(12)
Under incompatibility, if δ = 0, it is well-known from Matutes and Régibeau (1988)
that each firm charges a price for its system equal to t as in a two-dimensional Hotelling
model. For δ > 0, if firm i attracts a consumer from the rival in the first period, its
expected profit from the customer in the second period is Π+∗
2 . But if the customer stays
with the rival, its expected profit from him in the second period is Π−∗
2 . Therefore, each
+∗
−∗
firm is ready to pay δ Π2 − Π2 to attract a consumer, which is dissipated away. For
this reason, each firm’s equilibrium profit is 2t + δΠ−∗
2 .
4.2
Compatibility choice
From Propositions 1 and 2, each firm chooses incompatibility if and only if 2t + δΠ−∗
2 >
−∗ ∗
−∗
−∗
t
t + δ2π2 π , which is equivalent to Π2 − 2π2 > 2δ . In the next proposition, we neglect
the incompatibility equilibrium when a compatibility equilibrium exists for the reasons
explained in Section 2. We have:
Proposition 3. Consider the model of once-and-for-all compatibility choice.
−∗
t
(i) If Π−∗
2 −2π2 < 2δ , there is an equilibrium in which both firms choose compatibility.
This equilibrium Pareto-dominates the equilibrium in which both firms choose incompatibility.
−∗
t
(ii) If Π−∗
2 − 2π2 > 2δ , there is a unique equilibrium and it is such that both firms
choose incompatibility.
17
Incompatibility intensifies competition between symmetric firms in a one-period game.
Precisely, when δ = 0, incompatibility reduces each firm’s total profit from t to t/2,
which is well-known from Matutes and Régibeau (1988). Proposition 3(i) generalizes their
result for δt small enough: as long as δt is small enough, the firms choose compatibility
because this softens competition in period one and the second-period profits are relatively
unimportant.
Proposition 3(ii) shows that the finding of Matutes and Régibeau (1988) is reversed if
both the switching cost and the weight of the second period are large enough. Precisely,
if s > s̄3 , then Π−∗
> 2π2−∗ . Then, the firms choose incompatibility for high δt since
2
incompatibility softens competition in period two. Even if part of the increased secondperiod profit is dissipated away during the first-period competition, each firm retains
−∗
δ(Π−∗
2 − 2π2 ) in terms of increased profit, which more than compensates the reduction
in the first-period profit of t/2 if δt is large. Recall that in a standard one-period Hotelling
model, t measures products differentiation between firms, and the higher is t, the higher
are equilibrium profits. Therefore, the ratio δt measures the importance of the second
period relative to the first-period profitability.
5
Model 2: dynamic compatibility choices
In this section, we consider model 2 in which each firm makes a compatibility choice in
each period. As usual, we start by analyzing the second period.
5.1
5.1.1
Second-period price competition and compatibility choice
Second-period price competition
Lemma 1 can be applied to the second-period price competition in each market ij as long
as there is second-period compatibility, independently of the first-period compatibility
regime. Similarly, Lemma 2 can be applied to each market (i, i) as long as there is secondperiod incompatibility, independently of the first-period compatibility regime. Therefore,
what remains to be studied is the competition under incompatibility in the market (i, h)
composed of the consumers who have bought a hybrid system (i, h) in the first period
with i 6= h and i, h = A, B. This competition arises only if the firms chose compatibility
in period one but incompatibility in period two.
18
Consider market (i, h). We normalize the total mass of consumers in this market to
one. A consumer with valuations vxi , vyh is indifferent between buying (i, i) and (h, h) if
and only if
(13)
vxi + v e − P2i (i, h) − s = v e + vyh − P2h (i, h) − s.
Note that s cancels out in (13). It turns out that the demand for firm i’s system is
identical to the demand for its system in the market composed of the consumers who
bought system (i, i) when s = 0 (see the proof of Lemma 3). Therefore, Lemma 3 below
is a special case of Lemma 2. As the two firms are symmetric in this market, we have a
symmetric equilibrium. We introduce simplifying notation:
i∗
h∗
P20∗ = P2i∗ (i, h) = P2h∗ (i, h), Π0∗
2 = Π2 (i, h) = Π2 (i, h),
where the superscript 0 refers to the fact that no firm has any advantage in market (i, h).
We have:
Lemma 3. Suppose that compatibility was chosen in period one while incompatibility was
chosen in period two. Consider the period-two competition in the market composed of the
consumers who bought the hybrid system (A, B) (or (B, A)) in period one. We normalize
to one the total mass of consumers in this market. There exists a unique equilibrium,
which is symmetric.
(i) The equilibrium price for each firm is
P20∗ =
∆v
.
2
(ii) Each firm’s second-period profit from this market is
Π0∗
2 =
∆v
.
4
In the market composed of the consumers who bought a hybrid system in period one,
neither firm has any advantage no matter what the level of switching cost. Therefore,
neither the equilibrium prices nor the profits depend on s.
Note that we have
+∗
−∗
Π0∗
2 = Π2 (0) = Π2 (0).
19
−∗
+∗
−∗
2
As 2π2+∗ (s) + 2π2−∗ (s) > Π+∗
2 (s) + Π2 (s) if and only if s < s and both 2π2 (s) + 2π2 (s)
−∗
and Π+∗
2 (s) + Π2 (s) are strictly increasing, the second-period competition in market
(A, B) or (B, A) under incompatibility is more intensive than competition in market
(A, A) or (B, B) under incompatibility and competition in any product market under
compatibility.
5.1.2
Second-period compatibility choice
Compatibility choice when compatibility was chosen in period one. Suppose
that both firms chose compatibility in the first period. Let ∆v = 1 without loss of
generality. If both firms choose compatibility in the second period, then firm i’s secondperiod profit is:
di1,x + di1,y π2+∗ + dh1,x + dh1,y π2−∗ ,
(14)
where di1,j is the first-period market share of firm i for product j, and di1,j + dh1,j = 1. If
any of the two firms chooses incompatibility in the second period, then i’s profit is
0∗
i
h
h
i
h
h
−∗
di1,x di1,y Π+∗
+
d
d
+
d
d
2
1,x 1,y
1,x 1,y Π2 + d1,x d1,y Π2 .
(15)
Therefore, firm i chooses compatibility in the second period as long as (14) is at least
as large as (15). Using Lemmas 1-3, we have the following result:
Lemma 4. Suppose that compatibility was chosen in period one (and let ∆v = 1 w.l.o.g.).
Then compatibility choices in the second period are as follows:
1
(i) If s is such that 2π2+∗ − Π+∗
2 ≥ 0 (i.e., s ≤ s ), then both firms choose compatibility
for any (di1,x , di1,y ) in [0, 1]2 .
(ii) If s is such that 2π2+∗ − Π+∗
< 0 (i.e., s > s1 ), then at least one firm chooses
2
incompatibility if and only if di1,x and di1,y are both close to 1 or both close to 0.
For s > s1 firm i prefers incompatibility if
di1,x >
−∗
+∗
−∗ i
0
2π2−∗ − Π−∗
π2+∗ + π2−∗ − Π02
2 − (Π2 − Π2 − π2 + π2 )d1,x
i
and
d
>
,
1,y
+∗
−∗
+∗
−∗
−∗
+∗
0 i
0
Π+∗
(Π02 − Π−∗
2 + π2 − π2 − Π2
2 − π2 + π2 ) + (Π2 + Π2 − 2Π2 )d1,x
20
that is if di1,x and di1,y are close enough to one. This is quite intuitive, as when di1,x and di1,y
are both close to 1, the comparison between (14) and (15) approximately reduces to 2π2+∗
+∗
+∗
1
vs Π+∗
2 , and we have 2π2 < Π2 given s > s . Likewise, firm h prefers incompatibility if
dh1,x and dh1,y are close to 1 (i.e., when di1,x and di1,y are close to zero). For instance, figure
2 shows firm i prefers incompatibility if (di1,x , di1,y ) is above the thin curve, and firm h
prefers incompatibility if (di1,x , di1,y ) is below the thick curve, when s = 1.1.
1
0.8
i
d1,y
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
i
d1,x
Figure 2: Second-period compatibility choice when compatibility was chosen in period
one for s = 1.1
As s increases, the set of (di1,x , di1,y ) such that at least one firm prefers incompatibility
becomes wider, but for no s it includes the point such that di1,x = di1,y = 12 , as the following
corollary states.
Corollary 4. Suppose that the firms chose compatibility in the first period (and let ∆v = 1
B
A
w.l.o.g.). In the case of symmetric first-period market shares, that is dA
1,x = d1,x = d1,y =
1
dB
1,y = 2 , both firms prefer compatibility to incompatibility for any s ∈ (0, 3/2).
21
Given symmetric first-period market shares, we find for any s ≥ 0
1
1 0∗
−∗
π2+∗ (s) + π2−∗ (s) > (Π+∗
2 (s) + Π2 (s)) + Π2 ,
4
2
where the L.H.S. (the R.H.S) is each firm’s second-period profit under compatibility (under incompatibility). As we wrote after Lemma 3, competition in market (i, h) under
incompatibility is more intensive than competition in (i, i) under incompatibility or under
+∗
+∗
−∗
−∗
1
(Π
(s)
+
Π
(s))
for any
compatibility. More precisely, Π0∗
<
min
π
(s)
+
π
(s),
2
2
2
2
2
2
s > 0. This force induces the firms to embrace compatibility no matter what the level of
−∗
switching cost even if π2+∗ (s) + π2−∗ (s) < Π+∗
(s)
+
Π
(s)
/2 holds for s > s2 .
2
2
Compatibility choice when incompatibility was chosen in period one. We now
consider the case in which incompatibility was chosen in the first period. This means
there is no hybrid system in the first period. Let D1i denote i’s first-period market share.
Suppose first incompatibility in both periods. The equilibrium of the competition
under incompatibility in each market (i, i) is given in Lemma 2. Hence firm i’s secondperiod profit is
h −∗
D1i Π+∗
(16)
2 + D1 Π2 .
Suppose now that the first-period incompatibility is followed by the second-period
compatibility. Then, as we wrote just after Lemma 1, the firms’ second-period profits in
each market ij do not depend on the first-period compatibility regime. Therefore, Lemma
1 applies and firm i’s second-period profit is
2 D1i π2+∗ + D1h π2−∗ .
(17)
In order to compare (16) and (17), we use D1h = 1 − D1i and see that firm i prefers
second-period compatibility if and only if
i
−∗
D1i 2π2+∗ + (1 − D1i )2π2−∗ > D1i Π+∗
2 + (1 − D1 )Π2 ,
which is equivalent to D1i <
10
2π2−∗ −Π−∗
2
+∗
+∗
−∗
Π2 −Π−∗
2 −(2π2 −2π2 )
≡ D.10 Likewise, firm h with h 6= i
−∗
+∗
−∗
3
Notice that the denominator Π+∗
2 − Π2 − (2π2 − 2π2 ) is positive for each s ∈ (0, 2 ).
22
prefers second-period compatibility if and only if
+∗
i
D1i 2π2−∗ + (1 − D1i )2π2+∗ > D1i Π−∗
2 + (1 − D1 )Π2 ,
which is equivalent to D1i >
immediate.
+∗
Π+∗
2 −2π2
−∗
+∗
−∗
−(2π
−Π
Π+∗
2 −2π2 )
2
2
≡ D. Then the following lemma is
Lemma 5. Suppose that the firms chose incompatibility in the first period (and let ∆v = 1
w.l.o.g.). Then compatibility choices in the second period are as follows:
+∗
(i) If Π+∗
(that is, if s < s1 ), then D > 1 and D < 0. Hence compatibility
2 < 2π2
emerges in the second period for any D1i ∈ [0, 1].
−∗
+∗
−∗
2
(ii) If Π+∗
2 + Π2 ≥ 2π2 + 2π2 (that is, if s ≥ s ), then D ≤ D. Hence incompatibility
emerges in the second period for any D1i ∈ [0, 1].
−∗
(iii) If Π2+∗ ≥ 2π2+∗ and Π+∗
< 2π2+∗ + 2π2−∗ (that is, if s1 ≤ s < s2 ), then
2 + Π2
0 ≤ D < 21 < D ≤ 1. Hence compatibility (incompatibility) emerges in the second
period if D < D1i < D (if D1i ≤ D, or D1i ≥ D).
We know from Corollary 3 that if the dominant firm prefers compatibility (i.e., if Π+∗
2 <
+∗
−∗
−∗
2π2 ), then also the dominated firm prefers compatibility (i.e., Π2 < 2π2 ). Therefore,
as long as the dominant firm prefers compatibility (i.e., if s < s1 ), we have that (16) is
greater than (17) for any D1i and both firms choose compatibility in the second period. As
−∗
+∗
the switching cost becomes larger than s2 , the inequality Π+∗
+ 2π2−∗ holds.
2 + Π2 > 2π
Then incompatibility emerges since this inequality means that the sum of the secondperiod profits is greater under incompatibility than under compatibility. Therefore at
least one firm prefers incompatibility. Lemma 4(i) and Lemma 5 show that if s < s1 , the
firms always end up choosing compatibility in period two regardless of the first-period
compatibility regime and market shares. But such strong prediction cannot be made for
s ≥ s2 .
The following corollary covers the case of equal market shares.
Corollary 5. Suppose that the firms chose incompatibility in the first period. In the
case of symmetric market shares, i.e., D1A = D1B = 21 , the firms prefer incompatibility to
−∗
+∗
−∗
2
compatibility in the second period if and only if Π+∗
2 + Π2 ≥ 2π2 + 2π2 (i.e., s ≥ s ).
23
5.2
First-period price competition
In this subsection, we study the first-period competition given a first-period compatibility
regime. As the case of first-period incompatibility is easier to study than that of the firstperiod compatibility, we start with the former.
5.2.1
Given incompatibility in the first period
If incompatibility was chosen in period one, then Lemma 5 applies. As a consequence, for
+∗
any s ∈ [0, s1 ) (i.e., when Π+∗
2 < 2π2 ) and for any pair of the first-period prices under
incompatibility P1i , P1h , the firms will choose compatibility in the second period. Hence,
we consider the following profit functions for any P1i , P1h :
Πi = D1i (P1i + δ2π2+∗ ) + (1 − D1i )(δ2π2−∗ ),
Πh = (1 − D1i )(P1h + δ2π2+∗ ) + D1i (δ2π2−∗ ),
(18)
where D1i is firm i’s market share in period one and is given by (10).
−∗
+∗
A similar argument can be applied to any s ∈ [s2 , 23 ) (i.e., when Π+∗
2 + Π2 ≥ 2π2 +
2π2−∗ ). In this case, for any pair of the first-period prices under incompatibility P1i , P1h ,
the second-period compatibility regime will be incompatibility. Therefore, for any s ∈
[s2 , 32 ), we consider the profit functions from (9).
Proposition 4. Suppose that incompatibility was chosen in the first period. Let ∆v = 1
without loss of generality.
+∗
1
(i) If Π+∗
2 < 2π2 (i.e., if s < s ), there exists a unique equilibrium, which is symmetric,
and such that
P1i∗ = t − δ 2π2+∗ − 2π2−∗
for i = A, B.
Both firms choose compatibility in the second period. Each firm’s total profit is
Πi∗ =
t
+ δ2π2−∗
2
for i = A, B.
−∗
(i.e., if s ≥ s2 ), there exists a unique equilibrium,
(ii) If 2π2+∗ + 2π2−∗ ≤ Π+∗
2 + Π2
which is symmetric. Both firms choose incompatibility in the second period. The
first-period price is P1i∗ = P1∗ and each firm’s total profit is Π∗ as described in
Proposition 2.
24
Proposition 4 covers s in [0, s̄1 ) and in [s̄2 , 23 ). Precisely, Proposition 4(ii) which considers s in [s̄2 , 23 ) is similar to Proposition 2, as both yield the same equilibrium prices and
profits. The difference is that Proposition 2 refers to a setting in which incompatibility
has been chosen once-and-for-all in period one while Proposition 4(ii) refers to a setting
in which firms can change the compatibility regime in period two.
Proposition 4(i) considers s in [0, s̄1 ) and describes a change in the compatibility regime
over the two periods, which is implied by Lemma 5(i). However, the equilibrium prices
and profits it identifies can be explained by arguments very similar to those provided after
Proposition 1 and 2.
Finally, as Lemma 5 states, if s is between s1 and s2 , then the compatibility/incompatibility
regime in period two depends on whether D1i belongs to the interval (D, D) or not: compatibility emerges if the first-period market shares are not too different. In Figure 3 below
we represent D (the thin curve) and D (the thick curve), for s ∈ [s1 , s2 ).
Figure 3: (D, D) under first-period incompatibility choice for s ∈ [s1 , s2 ).
In case there exists a symmetric pure-strategy equilibrium, then D1i = 21 and thus
the firms choose compatibility in the second period. However, in period one a firm may
deviate in a way to induce incompatibility in period two, that is in a way such that D1i ≤ D
or D1i ≥ D holds, and an equilibrium needs to be robust to this sort of deviations. Next
25
proposition identifies a symmetric equilibrium candidate
√ and shows that it is robust to
2(1−D̄)+2·(3D̄−2)
√
. Conversely,
such deviation if δ/t is not larger then δ̄(s) ≡ +∗ −∗
−∗
+∗
−Π
2(1−D̄)
Π
( 2
2 −(2π2 −2π2 ))
if δ/t is large then a firm can profitably deviate by choosing a small price in period one in
order to gain a large market share, which yields a large profit in period two.11 In such a
case no symmetric pure-strategy equilibrium exists, and the next proposition shows that
no asymmetric pure-strategy equilibrium exists either.
Proposition 5. Suppose that incompatibility was chosen in the first period. Let ∆v = 1
+∗
+∗
−∗
+∗
−∗
without loss of generality. Suppose that Π+∗
2 ≥ 2π2 and 2π2 + 2π2 > Π2 + Π2 (i.e.,
s is between s1 and s2 ).
(i) If a pure strategy equilibrium exists (symmetric or asymmetric), then P1A∗ = P1B∗ =
t − δ 2π2+∗ − 2π2−∗ , as in Proposition 4(i).
(ii) For each s ∈ (s1 , s2 ), the prices in (i) constitute an equilibrium if δ/t ≤ δ̄(s).
5.2.2
Given compatibility in the first period
If compatibility was chosen in period one, then Lemma 4 applies. Therefore, for s ≤ s1 ,
compatibility arises also in period two as no firm wants to choose incompatibility in period
two regardless of first-period market shares. Hence, we consider the profit function from
(5). Then there exists an equilibrium which is outcome-equivalent to the equilibrium
described by Proposition 1:
Proposition 6. Suppose that compatibility was chosen in the first period. Then, there
+∗
t
exists a unique equilibrium, which is symmetric, if and only if Π+∗
2 − 2π2 ≤ δ . In the
equilibrium, each firm chooses compatibility in the second period. The equilibrium prices in
∗
the first period are pi∗
1j and each firm’s total profit from market j = x, y is π as described
in Proposition 1.
However, for s greater than s1 each firm can induce incompatibility in the second period
by choosing low prices such that it corners both markets in the first period. Indeed, for δ
large, cornering is the best deviation as there are economies of scale under second-period
A
A
incompatibility. Suppose that firm A deviates by capturing dA
1,x = d1,y = d1 ∈ (1/2, 1]
11
Precisely, the unique candidate equilibrium is such that each firm charges the price P ∗ = t−δ(2π2+∗ −
but if firm i deviates with P i = P ∗ − 2t then firm i obtains full market share and profit −t −
−∗
t
− 2π2−∗ ) + δΠ+∗
if δt is large.
2 . This is greater than the candidate equilibrium profit 2 + δ2π2
2π2−∗ ),
δ(2π2+∗
26
share of consumers.12 Then, under the second-period incompatibility, firm A’s second2 +∗
A
0∗
A 2 −∗
period profit is dA
Π2 + 2dA
1
1 (1 − d1 )Π2 + (1 − d1 ) Π2 , which is strictly convex in
+∗
−∗
0∗
dA
1 as Π2 + Π2 − 2Π2 > 0 holds for s > 0. To corner both markets, the deviating firm
should charge p∗1 − t in period one and then obtains a profit of Π+∗
2 in period two, which
+∗
+∗
−∗
generates the total profit of δ Π2 − 2 π2 − π2
. Therefore, the cornering deviation
+∗
+∗
t
is not profitable if Π2 − 2π2 ≤ δ holds. This inequality is obviously satisfied for for
+∗
t
s ≤ s1 , but for s ∈ (s1 , s3 ) there exist t, δ satisfying Π+∗
2 − 2π2 > δ , and then we find no
pure strategy equilibrium.
However, there exists a cornering equilibrium when s > s3 and δ/t is large. In this
equilibrium, one firm charges a price below the marginal cost (i.e., p̄ < 0) in each market
and the other firm charges p̄ + t such that the first firm gains full market share in both
markets in period one and therefore chooses incompatibility in the second period, earning
a total profit of 2p̄ + δΠ+∗
2 . The other firm has zero market share in both markets in
period one and earns a total profit of δΠ−∗
2 . The value of p̄ must be small enough in
order to deter the cornered firm from trying to earn a positive market share in period
one, and high enough to deter the cornering firm from reducing its period-one market
share in order to reduce its period-one losses. Since the possible deviations are somewhat
intricate to describe, stating precisely the conditions under which this equilibrium exists
is complicated, except that there exists a function δ` such that for each s > s3 , a cornering
. It turns out that for each s ∈ (s̄3 , 23 ), the ratio
equilibrium exists if and only if δt ≥ δ` (s)
t
δ` (s)
is between Π+∗3/2
and Π+∗16/9
: see the proof in the appendix for the full details.
t
−2π +∗
−2π +∗
2
2
2
2
Proposition 7. Suppose that compatibility was chosen in the first period, and assume
−∗
Π−∗
2 − 2π2 > 0. Then there exists an asymmetric equilibrium like the one described above
if and only if δt ≥ δ` (s). Precisely,
−∗
(i) in the first period, one firm charges price p̄ = γ − 2δ (Π+∗
2 − Π2 ) in both markets
(for a suitable γ in (0, t)), the other firm charges price p̄ + t in both markets, and
the first firm corners both markets;
(ii) in the second period, incompatibility is chosen at least by the firm who cornered the
markets, which earns a total profit of 2γ + δΠ−∗
2 , whereas the other firm’s total profit
is δΠ−∗
2 .
12
Lemma 6 in the proof of Proposition 6 shows that the optimal deviation is characterized by symmetric
market shares if the deviating firm expects incompatibility in period two.
27
5.3
First-period compatibility choice
Up to now we analyzed the first-period price competition given a first-period compatibility
regime. We now study the firms’ first-period compatibility choices. Let us normalize ∆v
to one without loss of generality. If the switching cost is low enough (i.e., s < s1 ), then by
Lemmas 4 and 5 the firms choose compatibility in period two no matter their compatibility
choices in period one. As compatibility in period one generates a higher first-period profit
than incompatibility (see Propositions 6 and 4(i)), the firms will choose compatibility in
both periods.
Suppose now that the switching cost is high enough (i.e., s > s3 ). Then, incompatibility in period one leads to incompatibility in period two, with the profit of 2t + δΠ−∗
2 for
each firm (see Proposition 4(ii)). Regarding the scenario of compatibility in period one,
+∗
we have to distinguish δ/t small from δ/t large. If δ/t is small (i.e., Π+∗
2 − 2π2 ≤ t/δ),
compatibility in period one leads to compatibility in period two (see Proposition 6). In
this case, each firm’s profit is t + δ2π2−∗ . Therefore, the firms choose compatibility for δ/t
small. If δ/t is large enough (see the inequality in Proposition 7), compatibility in period
one leads to incompatibility in period two through the cornering equilibrium. In such an
−∗
t
equilibrium, the cornered firm earns δΠ−∗
2 , which is less then 2 + δΠ2 . Therefore such a
firm will choose incompatibility in period one.
Summarizing, we have
Proposition 8. (first-period compatibility choice) Consider the case in which each firm
makes a compatibility choice every period.
(i) If either the switching cost is low enough (i.e., s ≤ s1 ) to satisfy 2π2+∗ (s) ≥ Π+∗
2 (s),
1
or if δ/t is small enough to satisfy δ/t ≤ Π+∗ −2π+∗ (and, furthermore, δ/t ≤ δ̄(s)
2
2
for s ∈ (s̄1 , s̄2 )) then there is a unique equilibrium, which is symmetric. In the
equilibrium, the firms choose compatibility in both periods.
−∗
(ii) If both the switching cost and δ/t are high enough to satisfy Π−∗
2 (s) > 2π2 (s)
and δ/t ≥ δ` (s)
, then there is a unique equilibrium, which is symmetric. In the
t
equilibrium, the firms choose incompatibility in both periods.
28
5.4
Comparison: Model 1 vs. Model 2
We now compare the results we obtained in Proposition 3 from model 1 of once-and-for-allcompatibility choices with those in Proposition 8 from model 2 of dynamic compatibility
choices. The key difference between the two models is that in model 2, firms make compatibility choices in period two as well, which implies that when considering a deviation
in period one, a firm needs to think about how the deviation will affect the compatibility
regime in period two. In a few cases, this generates more powerful deviations than the
ones under model 1 such that no pure strategy equilibrium exists in model 2. For instance,
+∗
given first-period compatibility, if δt is large and Π+∗
2 > 2π2 , it is profitable for a firm to
deviate in such a way to gain a large market share in period one in order to induce incompatibility in period two. As a consequence, Proposition 8 covers values of parameters
such that a pure strategy equilibrium exists in model 2 given each compatibility regime
in period one.
With this caveat in mind, when we compare Proposition 3 with Proposition 8, we find
that the incompatibility equilibrium arises in model 2 more frequently than in model 1.
In order to see why, notice that in model 1 the incompatibility equilibrium emerges if and
only if each firm’s profit under incompatibility, 2t + δΠ−∗
2 , is larger than the one under
compatibility, t + δ2π2−∗ . In model 2, the incompatibility equilibrium arises as long as the
cornering equilibrium exists given compatibility in period one – see Proposition 8 – since
this induces at least one firm to choose incompatibility in period one. The profit under
compatibility is not relevant because when s > s̄3 and δt is large, the cornering equilibrium
is such that compatibility in period one is followed in period two not by compatibility
but by incompatibility. The existence condition for the cornering equilibrium is less
−∗
restrictive than the inequality 2t + δΠ−∗
2 > t + δ2π2 , and therefore the incompatibility
equilibrium is found more frequently in model 2 than in model 1. As a consequence, the
compatibility equilibrium arises less often in model 2 than in model 1. Precisely, in model
−∗
1 the compatibility equilibrium exists as long as 2t + δΠ−∗
2 < t + δ2π2 , but in model 2
this inequality does not suffice since the compatibility equilibrium must be robust also
to a period-one deviation which induces incompatibility in period two. Summarizing, we
have:
Proposition 9. (i) If the incompatibility equilibrium arises under once-and-for-all compatibility choices, it arises also under dynamic compatibility choices. But there are param29
eter values under which the incompatibility equilibrium arises in the latter while it does
not arises in the former.
(ii) If the compatibility equilibrium arises under dynamic compatibility choices, it arises
also under once-and-for-all compatibility choices. But there are parameter values under
which the compatibility equilibrium arises in the latter while it does not arise in the former.
Despite the differences explained above, the two models generate quite similar predic−∗
t
tions. If δt and the switching cost are large enough (i.e., Π−∗
2 − 2π2 > 2δ holds), then the
incompatibility equilibrium arises in both models. Similarly, if either the switching cost
1
is low enough
) to satisfy 2π2+∗ (s) ≥ Π+∗
2 (s) or δ/t is small enough to satisfy
n (i.e., s ≤ s o
1
1 2
δ/t ≤ min Π+∗ −2π+∗ , δ̄(s) for s ∈ (s̄ , s̄ ), then the compatibility equilibrium arises in
2
2
both models.
6
Forward-looking consumers
Up to now, we have assumed that consumers are myopic. We here show that if consumers
were forward-looking, our results remain unaffected in model 1 and remain qualitatively
similar in model 2.
First notice that our analysis of the second period continues to apply regardless of
whether consumers are myopic or forward-looking. Second, in model 2 of dynamic compatibility choices, each single consumer knows in period one that he cannot be pivotal
in influencing the firms’ second-period compatibility choices since he is an infinitesimal
fraction of the population of consumers. Then, taking as given the compatibility regimes
in period one and two, we inquire how forward-looking consumers make their purchases
in period one.
It is simple to see that if a consumer expects in period one compatibility to prevail
period two, then his expected second-period utility does not depend on which products
he buys in period one. Precisely, if he buys product j from firm i in period one, then his
−∗
e
utility in period two is the expectation of max{vji − p+∗
2 , v − p2 − s}; if he buys product j
−∗
e
from firm h in period one, then his utility is the expectation of max{vjh −p+∗
2 , v −p2 −s}.
Since vji and vjh are identically distributed, it follows that the two expectations are equal,
and the consumer will choose in period one on the basis of his first-period utility, as a
myopic consumer does. The same principle holds if incompatibility prevails in period
30
one, and a consumers expects that it will be maintained in period two as well. If a
consumer chooses the system of firm i in period one, then his payoff in period two is
equal to the expectation of max{vxi + vyi − P2+∗ , 2v e − P2−∗ − 2s}. If he buys the system
of firm h in period one, then his payoff in period two is equal to the expectation of
max{vxh + vyh − P2+∗ , 2v e − P2−∗ − 2s}. Since the distribution of vxi + vyi is the same as the
distribution of vxh + vyh , the two expectations are equal, and the consumer will choose in
period one on the basis of his period one utility, as a myopic consumer does. This proves
that in model 1 of once-and-for-all compatibility choices, a forward-looking consumer
behaves exactly the same as a myopic consumer does.
Proposition 10. (forward-looking consumers) Suppose that consumers are forward-looking.
In model 1 of once-and-for-all compatibility choices, forward-looking consumers behave in
the same way as myopic consumers do.
However, things are quite different if compatibility prevails in period one, and a consumer expects incompatibility to prevail in period two. In this case, if a consumer buys
both products of firm i in period one, then he obtains a payoff in period two equal to the expectation of max{vxi +vyi −P2+∗ , 2v e −P2−∗ −2s} ≡ uii2 , which is equal to uhh
2 . Conversely, if in
period one a consumer buys product x from firm i and product y from firm h, then his payoff in period two is equal to the expectation of max{vxi +v e − ∆v
−s, vyh +v e − ∆v
−s} ≡ uih
2 ,
2
2
hi
ii
ih
which is equal to u2 . The inequality u2 < u2 holds since (as we previously noted in
section 5.1.1) second-period competition under incompatibility is more intensive in the
market of the consumers who bought a hybrid system than in the market of the consumers who bought a pure system. This implies that given compatibility in period one,
a forward-looking consumer who expects incompatibility in period two makes more frequently hybrid purchases in period one than a myopic consumer does. However, given
incompatibility in period one, there is no difference in behavior between forward-looking
consumers and myopic consumers no matter the second-period compatibility regime which
they expect to prevail.
Consider now the compatibility equilibrium. In model 1, this equilibrium exists when
it is robust to the deviation starting with first-period incompatibility. In model 2 with
myopic consumers, the equilibrium exists if it is robust to, in addition to the aforementioned deviation, the deviation in which first-period compatibility is followed by secondperiod incompatibility. Now consider model 2 with forward-looking consumers. Relative
31
to when consumers are myopic, the deviation starting with first-period incompatibility
is unaffected but the deviation in which first-period compatibility is followed by secondperiod incompatibility is less profitable. These arguments imply (i) if compatibility arises
in model 2 with myopic consumers, it arises also in model 2 with forward-looking consumers; (ii) if compatibility arises in model 2 with forward-looking consumers, it arises
also in model 1.
Consider now the incompatibility equilibrium. In model 1, this equilibrium exists when
it generates a payoff larger than the one when both firms choose compatibility. In model
2 with myopic consumers, for relevant parameter values, first-period compatibility leads
to a cornering equilibrium inducing second-period incompatibility. The incompatibility
equilibrium exists when it generates a payoff larger than the one obtained by the cornered
firm in the cornering equilibrium. In model 2 with forward-looking consumers, relative to
when consumers are myopic, first-period compatibility is likely to lead less frequently to
the equilibrium in which one firm gains sufficient market shares to choose incompatibility
in period two because of uii2 < uih
2 . Therefore, there are two possibilities: the relevant
payoff conditional on the first-period compatibility is either the one from symmetric equilibrium leading to second-period compatibility or the one from asymmetric equilibrium
leading to second-period incompatibility. In the former case, the incompatibility equilibrium exists in model 2 with forward-looking consumers if and only if it exists in model 1.
In the latter case, the incompatibility equilibrium would exist in model 2 with forwardlooking consumers more frequently than in model 1 but less frequently than in model 2
with myopic consumers.
7
Consumer surplus and welfare
In both models, we have, as pure strategy equilibrium, either the equilibrium in which
compatibility is chosen in both periods or the equilibrium in which incompatibility is
chosen in both periods. In this section, we compare, in terms of consumer surplus and
welfare, the allocation that competition generates when compatibility is maintained in
both periods with the one when incompatibility is maintained in both periods. Without
loss of generality, we normalize ∆v to one.
32
7.1
Consumer surplus
In period one, consumer surplus is greater under incompatibility than under compatibility
for any s and δt for two reasons. The first reason is known from Matutes and Régibeau
(1988): for symmetric firms, incompatibility intensifies first-period competition. As we
explained in Section 3.3, this has to do with the demand elasticity effect of incompatibility.
Even if incompatibility increases transportation costs incurred by consumers, this effect
is dominated by the effect from more intensive competition. The second reason is that
the difference between the dominant firm’s second-period profit and the dominated one’s
profit greater under incompatibility than under compatibility, that is we have
Π+∗
2 (s)
−
Π−∗
2 (s)
>
2π2+∗ (s)
−
2π2−∗ (s)
3
.
for any s ∈ 0;
2
Hence, under incompatibility, the firms dissipate more the second-period profit from
locked-in consumers by competing more aggressively in period one.
Consumer surplus in period two depends on s. For instance, consider a consumer who
bought both goods from firm A in period one. Then, for large values of s, we find that
−∗
+∗
−∗
P2+∗ > 2p+∗
> 2p−∗
> p+∗
2 , P2
2 , P2
2 + p2 .since incompatibility softens competition in
period two with respect to compatibility. In particular, the first inequality implies that if
this consumer buys both goods from A in period two, then he is better off under compatibility than under incompatibility, but in fact we find that for any possible realization of
vxA , vyA , this consumer is better off under compatibility. As a consequence, the secondperiod consumer surplus is higher under compatibility for large s (i.e., for s > 0.876).
−∗
Conversely, for s close to zero the opposite inequalities hold: P2+∗ < 2p+∗
< 2p−∗
2 , P2
2 ,
+∗
+∗
−∗
P2 < p2 + p2 . The second-period consumer surplus is higher under incompatibility
for s < 0.876.
Let CS I (CS C ) denote the total consumer surplus under incompatibility (compatibility). The previous arguments seem to suggest that CS C > CS I when s > 0.876 and δt
is large. But in fact we need to take into account that an increase in δ reduces both 2p∗1
and (especially) P1∗ . Hence, we find CS C > CS I if s > 1.168 and δ/t is above a suitable
threshold which we denote `(s).
Regarding the effect of equilibrium compatibility choices on total consumer surplus,
we find a minor difference between model 1 and model 2. Precisely, Proposition 3 proves
33
that incompatibility arises in the equilibrium of model 1 if and only if s > s̄3 and
δ
1
> 2(Π−∗ −2π
−∗ , whereas Proposition 8 establishes that incompatibility arises in the equit
)
2
2
9/5
1
librium of model 2 if s > s̄3 and δt > Π+∗9/5
+∗ . We find that
+∗ < `(s) <
−∗ ,
Π+∗
2(Π−∗
2 −2π2
2 −2π2
2 −2π2 )
1
and therefore whenever incompatibility emerges in model 1, (i.e., if δt > 2(Π−∗ −2π
−∗ ), it
2 )
2
necessarily reduces consumer surplus. Conversely, in model 2, there exist parameter valδ
I
C
ues (i.e., if Π+∗9/5
+∗ < t < `(s)) such that firms choose incompatibility and CS > CS .
2 −2π2
Similar properties hold with respect to compatibility: in both models, when compatibility
arises in equilibrium, it is often the case that CS I > CS C .
Proposition 11. (consumer surplus) (i) Consumer surplus under compatibility is CS C =
1 −∗
2[v e − p∗1 − 41 t + δ(v e − p+∗
2 + 2 π2 )].
(ii) Consumer surplus under incompatibility is CS I = 2v e − P1∗ − 32 t + δ(2v e − P2+∗ +
2 −∗
Π )].
3 2
(iii) The inequality CS C > CS I holds if and only if s > 1.168 and δt > `(s) =
5
+∗
+∗ 1 −∗ . Whenever incompatibility arises in equilibrium, in model
6(2π2+∗ −π2−∗ −2p+∗
2 +P2 −Π2 + 3 Π2 )
1, it always implies CS C > CS I ; in model 2, it implies CS C > CS I if δt > `(s) and
CS C < CS I if δt < `(s). Whenever compatibility arises in equilibrium, CS I > CS C for
s < 1.168 regardless of the model.
What is remarkable is the conflict between the firms’ compatibility
choices and conn
−∗
1
sumer surplus. Except for a very small range of parameters ( s ∈ (1.168, 23 ) and `(s)
> δt > 2(Π−∗
2 − 2π2
whenever the firms choose incompatibility (compatibility), it reduces CS compared to
compatibility (incompatibility). This is because firms choose a regime of (in)compatibility
in order to soften competition, which hurts consumer surplus.
7.2
Social welfare
We start by describing the first-best allocation that maximizes social welfare. In period
one, the first–best requires a consumer of location θj for good j (j = x, y) buys good Aj
if θj < 21 and buys good Bj if θj > 12 . In period two, the first-best requires a consumer
who purchased good j of A (for instance) in period one and observes vjA to keep buying
good j from A if vjA > v e − s and to switch to B if vjA < v e − s. In particular, notice that
no switching should occur in period two if s ≥ 21 .
34
Under compatibility, the first-period allocation coincides with the welfare-maximizing
allocation, whereas the second period allocation is inefficient (as we noticed after Lemma
−∗
1) since p+∗
2 > p2 implies that too much switching occurs. However, this efficiency loss
is small if s is close to zero or if s is close to 23 . In the former case, we have p+∗
2 close
−∗
to p2 . In the latter case, maximizing the second-period welfare requires no switching
and the second-period allocation under compatibility is such that the market share of the
poaching firm (i.e., the proportion of switching consumers) tends to 0 as s tends to 23 .
Incompatibility generates an allocation which is inefficient in both periods. In period
one, consumers cannot mix and match, which increases the transportation costs incurred
by consumers. In period two, likewise, incompatibility forces consumers to make switching
decisions only at a system level, which prevents it from reaching the first-best. However,
for some intermediate values of s (i.e., for s ∈ (0.535, 1.152)), the second-period social
welfare is higher under incompatibility than under compatibility. This occurs because for
these values of s, no switching maximizes social welfare and the market share of the dominant firm is significantly larger under incompatibility than under compatibility precisely
because of the demand size effect of incompatibility explained in Section 3.3. As a consequence, total social welfare is greater under incompatibility than under compatibility for
s ∈ (0.535, 1.152) if δ/t is large since the superior performance of incompatibility in period
two dominates its inefficiency in period one. Conversely, if s < 0.535 or s > 1.152, then
social welfare is greater in both periods under compatibility than under incompatibility.
From Propositions 3 and 8, we see that s > s̄3 is a necessary condition for incompatibility to emerge in equilibrium regardless of the model. Since s̄3 > 1.152, we conclude
that whenever the firms choose incompatibility, this necessarily reduces social welfare.
When the firms choose compatibility, this choice generates a higher social welfare except
for the case in which s ∈ (0.535, 1.152) and δ/t is large. Summarizing, we have
Proposition 12. (welfare) (i) Social welfare under compatibility is SW C = 2[v e − p∗1 +
5 −∗
3
t + δ(v e − p+∗
2 + 2 π2 )].
4
(ii) Social welfare given under incompatibility is SW I = 2v e − P1∗ + 13 t + δ(2v e − P2+∗ +
8 −∗
Π )].
3 2
(iii) The inequality SW I > SW C holds if and only if s ∈ (0.535, 1.152) and δt >
1
. Whenever incompatibility arises in equilibrium, SW I <
+∗
+∗
−∗
+∗
5 −∗
6(Π+∗
2 + 3 Π2 −P2 −(2π2 +3π2 −2p2 ))
SW C holds regardless of the model. When compatibility arises in equilibrium, SW C >
35
SW I holds except when s ∈ (0.535, 1.152) and
8
δ
t
>
1
.
+∗
−∗
+∗
+∗
5 −∗
6(Π+∗
2 + 3 Π2 −P2 −(2π2 +3π2 −2p2 ))
Extension: growing vs mature market
We here consider a simple extension of model 1 (once-and-for-all compatibility choices)
in order to study how the compatibility choices are affected by whether the market is
growing or mature. For this purpose, we assume that in the second period, a measure
σ > 0 of new consumers appears. As the firms can apply behavior-based price discrimination, the firms can offer targeted prices to these consumers. Therefore, each firm’s
profit from the new consumers is σt under second-period compatibility, is σt/2 under
second-period incompatibility. Hence, the firms will choose compatibility in period one if
t + δ 2π2−∗ + σt > t/2 + δ Π−∗
+
σt/2
and choose incompatibility otherwise. Therefore,
2
we have:
Proposition 13. Suppose that the firms make once-and-for-all compatibility choices in
period one and that there is a measure σ > 0 of new consumers who arrive in the second
period. Then the firms choose compatibility if t + δ 2π2−∗ + σt > t/2 + δ Π−∗
2 + σt/2
holds and choose incompatibility otherwise. Therefore, compatibility is more likely to be
chosen in a growing market than in a mature market.
The above proposition generates an interesting prediction that platforms are more
likely to embrace compatibility in a growing market than in a mature market.
9
Conclusion
Our paper captures well Larry Page’s claim of an "island-like" Internet. Consumers’ data
are stocked in platforms’ data center and consumers access them through cloud. As platforms typically own such data and moving data across platforms is hard, consumer lock-in
arises very naturally.13 Then, our theory predicts that platforms embrace incompatibility
today in order to soften future competition after consumer lock-in even if this intensifies
today’s competition. Therefore, incompatibility hurts consumers. It also reduces welfare
by making switching at product level impossible.
13
See Schneier (2015) for a good introduction to "the hidden battles to collect your data and control
your world".
36
Our results are derived when any second-period rent from locked-in consumers is
completely dissipated away during the first-period competition. In reality, such full passthrough of rent to consumers seems unlikely. If pass-through is partial and hence some
rent is retained, this will further expand the regime of incompatibility.
10
References
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11
Appendix A
Proof of Lemma 1
The demand for product j of firm i in market ij is given by:


 1
if
+
−
+
1
1
d2 =
+ ∆v s + p2 − p2 if s −
2


0
if s +
∆v
2
∆v
2
+ p−
2
−
+ p2
p+
< s−
2
+
≤ p2 ≤ s +
< p+
2
∆v
2
∆v
2
+ p−
2
−
+ p2
(19)
+
and the demand for product j of firm h is d−
2 = 1 − d2 .
+ +
−∗
− −
(i) By definition of equilibrium, p+∗
2 maximizes p2 d2 and p2 maximizes p2 d2 . Hence,
+
+
− −
the following F.O.C. need to hold (they are sufficient as p+
2 d2 is concave in p2 , and p2 d2
is concave in p−
2 ):
p+
2
+ d+
= 0⇔
2
∆v
p−
− 2 + d−
= 0⇔
2
∆v
−
∆v
+
+ s + p−
2 − 2p2 = 0
2
∆v
+
− s − 2p−
2 + p2 = 0
2
39
−∗
s
∆v
s
∆v
This implies p+∗
2 = 2 + 3 , p2 = 2 − 3 .
(ii) Using (19), and the equilibrium prices, we have π2+∗ =
1
∆v
∆v
2
+
s 2
,
3
π2−∗ =
1
∆v
∆v
2
−
s 2
.
3
Proof of Lemma 2
The demand function for firm h’s system is given by (for −∆v < P2+ − P2− − 2s < ∆v)
D2− =
2
1
1
∆v − 2s − P2− + P2+ −
(P + −P2− −2s) max{0, P2+ −P2− −2s}, (20)
2
2(∆v)
(∆v)2 2
and D2+ = 1 − D2− is the demand for the system of firm i.
(i) The equilibrium prices maximize D2+ P2+ and D2− P2− , which implies the F.O.C.
2
P2+ = 2s − ∆v + 3P2− , and −8(P2− ) − 2(2s − ∆v)P2− + ∆v 2 = 0.14 Hence,
P2+∗ =
i
i
1 hp
1h p
(2s − ∆v)2 + 8∆v 2 − (2s − ∆v)
3 (2s − ∆v)2 + 8∆v 2 + 5(2s − ∆v) , P2−∗ =
8
8
(ii) Substituting the equilibrium prices in (20) yields the equilibrium demands and eventually the equilibrium second-period profits.
Proof of Proposition 1
Given compatibility, the total profit of firm i from product j is:
di1,j pi1,j + δ di1,j π2+∗ + (1 − di1,j )π2−∗
where di1,j = 21 +
order condition
1
(ph
2t 1,j
(21)
− pi1,j ) is the first-period demand. From (21) we obtain the first
di1,j −
1 i
(p + δπ2+∗ − δπ2−∗ ) = 0
2t 1,j
A similar first order condition,dh1,j − 2t1 (ph1,j + δπ2+∗ − δπ2−∗ ) = 0, needs to hold for firm
h.15 Solving then, we find the first-period equilibrium prices in Proposition 1, and the
equilibrium profits are obtained by replacing equilibrium prices in (21).
14
Since D2− is such that
∂D2− /∂P2−
D2−
is negative and decreasing in P2− (i.e., D2− is log-concave in P2− ),
satisfying the first order condition relative to P2− D2− suffices to maximize P2− D2− with respect to P2− . A
similar remark applies to the maximization of P2+ D2+ with respect to P2+ .
15
As in Lemma 1, these first order conditions are sufficient since the profit functions are concave.
40
Proof of Proposition 2
Assume without loss of generality that the equilibrium first-period prices, P1i∗ , P1h∗ ,
satisfy P1i∗ ≥ P1h∗ . Then given the profit functions in (9), with D1i in (10), P1i∗ , P1h∗ solve
the F.O.C. given by16
1 h
i
−∗
1 + (P1 − P1 ) P1i + δ Π+∗
− 2tD1i = 0
2 − Π2
2t
1 h
−∗
i
− 2tD1h = 0
1 + (P1 − P1 ) P1h + δ Π+∗
2 − Π2
2t
−∗ 17
The unique solution is P1i∗ = P1h∗ = t − δ Π+∗
−
Π
. From (9), the equilibrium profit
2
2
+∗
−∗
+∗
−∗
1
1
for each firm is 2 (t − δ Π2 − Π2 + δΠ2 ) + 2 (δΠ2 ) = 2t + δΠ−∗
2 .
Proof of Proposition 3
The proof is omitted as it is straightforward.
Proof of Lemma 3
The demand for firm i’s system is given by (for P h − ∆v < P i < P h + ∆v)
2
1
1 h
i
h
i
∆v
−
P
P
(i,
h)
+
P
−
(i,
h)
−
P
(i,
h)
(i,
h)
max{0, P2h (i, h)−P2i (i, h)}
2
2
2
2(∆v)2
(∆v)2 2
(22)
and D2h (i, h) = 1 − D2i (i, h) is the demand function for firm h’s system. Notice that (22)
coincides with (20) when s = 0. Hence, Lemma 3 is a corollary of Lemma 2 obtained by
setting s = 0 into Lemma 2.
D2i (i, h) =
Proof of Lemma 4
(i) Using (14) and (15), we see that firm i prefers compatibility if and only if
1
1
i
i
−∗
+∗
−∗
−∗
i
i
+∗
2π2−∗ − Π−∗
2 ≥ d1,x d1,y (Π2 + Π2 − ) + (d1,x + d1,y )( − Π2 − π2 + π2 )
2
4
(23)
16
Since D1i (D1h ) is log-concave in P1i (P1h ), the same given in footnote 14 applies here to establish that
F.O.C. are sufficient.
17
If we consider P1i > P1h , then D1i < D1h , which implies that not both equalities can be satisfied.
41
Likewise, firm h prefers compatibility if and only if
1
1
i
i
+∗
−∗
i
i
+∗
+∗
−∗
2π2+∗ − Π+∗
2 ≥ d1,x d1,y (Π2 + Π2 − ) + (d1,x + d1,y )( − Π2 + π2 − π2 )
2
4
(24)
If s ≤ s̄1 , then we prove that (23) holds for each (di1,x , di1,y ) ∈ [0, 1]2 by showing that
the maximum of the right hand side, considered as a function of (di1,x di1,y ) ∈ [0, 1]2 ,
is not larger than 2π2−∗ − Π−∗
2 , the left hand side. To this purpose, notice that the
Hessian matrix of the right hand side is indefinite, hence no maximum point exists in
the interior of [0, 1]2 , and the four edges of the square [0, 1]2 have to be examined. For
instance, if we consider the edge such that di1,x = 1, then the right hand side of (23) is
−∗
−∗
+∗
−∗
1
1
i
i
di1,y (Π+∗
2 + Π2 − 2 ) + (1 + d1,y )( 4 − Π2 − π2 + π2 ): this is a linear function of d1,y
i
i
which is smaller than 2π2−∗ − Π−∗
2 both at d1,y = 0 and at d1,y = 1 . The other three edges
of the square are dealt with similarly. The proof for (24) follows the same lines.
(ii) The proof follows from immediate manipulations of (23)-(24).
Proof of Lemma 5
The proof is omitted as it is straightforward.
Proof of Proposition 4
(i) Given s < s̄1 , we will have second period compatibility regardless of first period
market shares, by Lemma 5(i). Given the profit functions in (18), with D1i in (10), we
can argue as in the proof of Proposition 2 to find the first period equilibrium prices
P1i∗ = P1h∗ = t − δ 2π2+∗ − 2π2−∗ . Therefore the equilibrium profit for each firm is
t
+ δ2π2−∗ .
2
(ii) Given s > s̄2 , we will have second period incompatibility regardless of first period
market shares, by Lemma 5(ii). Then the proof of Proposition 2 applies.
Proof of Proposition 5
Suppose that s ∈ (s̄1 , s̄2 ). We prove three claims: (i) no equilibrium exists such that
incompatibility arises at stage two; (ii) the only candidate for equilibrium is such that
P1i∗ = P1h∗ = t − δ 2π2+∗ − 2π2−∗ ; (iii) no profitable deviation exists if and only if δt is not
too large.
42
Proof of claim (i) There exists no equilibrium such that D < Di < D̄ for
some firm i
We first prove that in no equilibrium we have Di ≥ D̄.
Suppose that P1i∗ , P1h∗ is an equilibrium with D1i = D. Then the profit of firm i if
P i < P̄ i is Πi in (9), and its derivative with respect to P i needs to be non-negative at
dDi
−∗
i
i
i
P i = P̄ i , that is dP 1i (P1i∗ + δ(Π+∗
2 − Π2 )) + D ≥ 0. The profit for firm i if P > P̄ is Π
1
in (18) and its derivative with respect to P i needs to be non-positive at P i = P̄ i , that is
dD1i
(P i∗ + δ(2π2+∗ − 2π2−∗ )) + D ≤ 0. However, these inequalities cannot both hold since
dP i
1
dD1i
dP1i
−∗
+∗
−∗
< 0 and Π+∗
2 − Π2 > 2π2 − 2π2 .
Suppose that P1i∗ , P1h∗ is an equilibrium such that D < D1i < 1, which implies P1i∗ <
P1h∗ . Then the profit functions are (9) in a neighborhood of (P1i∗ , P1h∗ ), and from these
profit functions. We obtain first order conditions as in the proof of Proposition 1(i), which
rule out P1i∗ < P1h∗ .
Suppose that P1i∗ , P1h∗ is an equilibrium with D1i = 1. Then the derivative of the profit
function of firm i with respect to P i is equal to one at P i = P i∗ . Therefore it is profitable
for firm i to increase slightly P i above P̄ i .
Now we rule out that Di ≤ D. If there exists an equilibrium such that Di ≤ D, then
Dh ≥ D̄ and we can rule out this case as we have ruled out above that Di ≥ D̄.
Proof of claim (ii) The only possible equilibrium is such that P1i∗ = P1h∗ =
t − δ 2π2+∗ − 2π2−∗
We know that if P ∗i , P ∗h is an equilibrium, then D < Di < D̄. Hence the profit
functions are given by (18) in a neighborhood of (P ∗i , P ∗h ), and from these profit functions
we obtain first order conditions as in the proof of Proposition 4(i), which imply P1i∗ =
P1h∗ = t − δ 2π2+∗ − 2π2−∗ .
Proof of claim (iii) The prices P1i∗ = P1h∗ = t − δ 2π2+∗ − 2π2−∗ constitute an
equilibrium if and only if δ/t is sufficiently close to 0
Given P1i∗ = P1h∗ = t − δ 2π2+∗ − 2π2−∗ , it is immediate that for no firm there exists
a profitable deviation which leads to compatibility in period two, hence we consider here
deviations which lead to incompatibility in period 2, that is such that Di ≤ D, or Di ≥ D̄.
Deviations with large P Given that firms are symmetric, it suffices to consider the
point of view of firm i at stage one, given incompatibility at stage one. Given P1h∗ = P1∗ =
t−δ(2π2+∗ −2π2−∗ ), let P be the price of firm i such that D1i = D (recall that for s ∈ (s̄1 , s̄2 ),
43
P ∗ −P
we have 0 < D < 21 ). Precisely, P belongs to (P1∗ , P1∗ +2t) and is such that 12 (1+ 12t )2 =
√
D, that is P = P1∗ + 2t − 2t 2D. Since P1i = P induces incompatibility in period two, the
√
profit of firm i from playing P1i = P is D t − δ(2π2+∗ − 2π2−∗ ) + 2t − 2t 2D + δΠ+∗
+
2
−∗
(1 − D)δΠ2 , that is
p
+∗
−∗
(3 − 2 2D)Dt + δ Π+∗
2 − 2π2 + Π2
Π+∗ −2π +∗
2
after using D = Π+∗ −Π−∗2 −(2π+∗
−∗ . The difference between the equilibrium profit
2 −2π2 )
2
2
−∗
δ2π2 and (25) is
(25)
t
2
+
p
1
−∗
− (3 − 2 2D)D t + δ(2π2+∗ + 2π2−∗ − Π+∗
2 − Π2 )
2
√ √
√
√
which is positive for each D ∈ (0, 12 ) since 21 −(3−2 2D)D = (1+2 2D)( 12 2− D)2 > 0
−∗
2
and 2π2+∗ + 2π2−∗ − Π+∗
2 − Π2 > 0 as s < s̄ .
In fact, firm i can achieve incompatibility in the second period by choosing any P i ∈
−∗
−∗
(P , P ∗ + 2t], and in such a case the profit of firm i is D1i (P i + δ(Π+∗
2 − Π2 )] + δΠ2 , with
derivative
dD1i i
−∗
i
(P + δ(Π+∗
(26)
2 − Π2 )) + D1
dP i
Now consider the profit under compatibility in the second period, which is D1i [P i +δ(2π2+∗ −
dDi
2π2−∗ )]+δ(2π2−∗ ), and its derivative with respect to P i , dP 1i [P i +δ(2π2+∗ −2π2−∗ )]+D1i , which
dD1i
+∗
−∗
−∗
+∗
−∗
i
i
is larger than (26) because Π+∗
2 −Π2 > 2π2 −2π2 . Since dP i [P +δ(2π2 −2π2 )]+D1 <
0 for each P i > P1∗ ,18 in particular for P i > P , it follows that (26) is negative.
Deviations with small P Given P1h∗ = P1∗ = t − δ(2π2+∗ − 2π2−∗ ), let P̄ be the price
of firm i such that D1i = D̄ (recall that for s ∈ (s̄1 , s̄2 ), we have 21 < D̄ < 1). Precisely,
∗
P̄ belongs to (P ∗ − 2t, P ∗ ) and is such that 1 − 12 (1 − P 2t−P̄ )2 = D̄, hence P̄ = P ∗ − 2t +
p
2t 2(1 − D̄). Since P1i = P̄ induces incompatibility in period two, the profit
of firm i from
p
playing P1i = P̄ is D̄ t − δ(2π2+∗ − 2π2−∗ ) − 2t + 2t 2(1 − D̄) + δΠ+∗
+ (1 − D̄)δΠ−∗
2
2 ,
that is
q
2 2(1 − D̄) − 1 D̄t + δ2π2−∗
(27)
18
This follows from the log-concavity of D1i : see footnote 16.
44
2π −∗ −Π−∗
2
after using D̄ = Π+∗ −Π−∗2−(2π+∗
−∗ . The difference between the equilibrium profit
2 −2π2 )
2
2
−∗
δ2π2 and (27) is
q
1
− 2D̄ 2(1 − D̄) + D̄ t
2
t
2
+
2
which is positive for each D̄ >
1
2
since
1
2
− 2D̄
p
2(1 − D̄) + D̄ =
8D̄+1)
(2D̄−1) (√
2(1+2D̄+4D̄
2(1−D̄))
> 0.
In fact, i can achieve incompatibility in the second period by choosing any P i ∈
−∗
−∗
[P ∗ − 2t, P̄ ), and in such a case the profit of firm i is D1i (P i + δ(Π+∗
2 − Π2 )] + δΠ2 , with
derivative (26) which is equal to
3 i 2
4(∆π)2 δ 2 − 7t2 + 2∆Πtδ + 4∆πtδ + 4∆Π∆πδ 2
i
− (P ) − (2t + ∆Πδ + 4∆πδ)P −
2
2
(28)
+∗
−∗
+∗
−∗
with ∆Π = Π2 − Π2 , ∆π = 2π2 − 2π2 . Notice that if (28) is positive or zero at
P i = P̄ , then we can conclude that the profit from deviation is increasing with respect
to P i for P i ∈ [P ∗ − 2t, P̄ ), hence no profitable deviation exists, since we have already
proved that deviating with P i = P̄ is unprofitable. Precisely, at P i = P̄ we find that
p
2(1 − D̄), which is non-negative if and only if
(28) is equal to −2 + 3D̄ + t+2∆πδ−∆Πδ
2t
δ/t ≤ δ̄(s).
1
4t2
Proof of Proposition 6
Consider an equilibrium such that for each firm i, the profit in (14) is strictly greater
than the profit in (15) at the equilibrium values for di1,x , di1,y . Then compatibility arises
in both period and the profit functions are given by (5) in a neighborhood of first period
equilibrium prices. Then we can argue as in the proof of Proposition 1 and find first
period equilibrium prices and profits p∗1 , π ∗ in (7)-(8). Moreover, thanks to Corollary 4
we know that given di1,x = di1,y = 12 , both firm choose compatibility in the second period.
Given the equilibrium candidate we mentioned above, now we show that no profitable
+∗
t
deviation exists if and only if Π+∗
2 − 2π2 ≤ δ .
(a) First, by the virtue of Proposition 1, we know that there exists no profitable
deviation which induces compatibility in the second period. Therefore, no profitable
+∗
1
deviation exists if Π+∗
2 ≤ 2π2 (that is, if s ≤ s̄ ) because then Lemma 4(i) reveals that
in any case compatibility emerges in the second period.
> 2π2+∗ . Then, by Lemma 4(ii), there exist
(b) Now consider the case in which Π+∗
2
45
deviations of firm i which lead to incompatibility in the second period: They are such
that di1,x , di1,y are both close to 1, or both close to 0. We show that no profitable deviation
+∗
t
exists for firm i if and only if Π+∗
2 − 2π2 ≤ δ . Given incompatibility in the second period,
the total profit of firm i is
Πi = di1,x pi1,x + di1,y pi1,y
i
h
0∗
i
h
0∗
h
h
−∗
+ δ di1,x di1,y Π+∗
2 + d1,x d1,y Π2 + d1,y d1,x Π2 + d1,x d1,y Π2
i
−∗
= di1,x pi1,x + di1,y pi1,y + δBdi1,x di1,y + δ Π0∗
−
Π
(d1,x + di1,y ) + δΠ−∗
2
2
2
(29)
−∗
0∗
0∗
where di1,j = 21 + 2t1 (p∗1 − pi1,j ) for j = x, y, and B = Π+∗
2 − Π2 − Π2 + Π2 > 0. We
now prove that to maximize (29) with respect to pi1,x , pi1,y , we can restrict to considering
pi1,x = pi1,y :
Lemma 6. Suppose that the firms chose compatibility in the first period and that a deviating firm expects that the incompatibility would prevail in the second period. Then,
without loss of generality, we can restrict attention to deviations in the first period such
that pi1,x = pi1,y .
Proof of the lemma. Suppose that firm i deviates with pi1,x 6= pi1,y . Then consider
p̂i1,x , p̂i1,y such that p̂i1,x = p̂i1,y = 21 (pi1,x + pi1,y ) ≡ p̂i1 . Then the demand for the product of
firm i is dˆi1 = 21 (di1,x + di1,y ) in each market and i’s profit is19
i
−∗
2dˆi1 p̂i1 + δB(dˆi1 )2 + δ Π0∗
2dˆ1 + δΠ−∗
2 − Π2
2
which is larger than (29) because (i) 2dˆi1 p̂i1 = 2 12 (di1,x + di1,y ) 12 (pi1,x + pi1,y ) = di1,x pi1,x +
di1,y pi1,y + 12 (pi1,x − pi1,y )(di1,y − di1,x ) = di1,x pi1,x + di1,y pi1,y + 4t1 (pi1,x − pi1,y )2 > di1,x pi1,x + di1,y pi1,y ;
(ii) δB(dˆi1 )2 > δBdi1,x di1,y because δB > 0 and (dˆi1 )2 > di1,x di1,y , given that dˆi1 = 12 (di1,x +di1,y )
i
i
−∗
−∗
; (iii) δ Π0∗
2dˆ1 = δ Π0∗
(d1,x + di1,y ). 2 − Π2
2 − Π2
Given a symmetric deviation such that pi1,x = pi1,y ≡ p, we have di1,x = di1,y = 1 −
Notice that if incompatibility emerges in the second period given di1,x , di1,y , then it also emerges
given the market shares dˆi1 , dˆi1 in each market, given the shape of the set described in Lemma 4(ii), and
immediately after the lemma.
19
46
π2+∗ − π2−∗ −
δ
2t
1
p
2t
≡ d and
−∗
+ δΠ−∗
Πi = 2dp + δd2 B + 2δd Π0∗
2 − Π2
2
δ
2δ
= −4t 1 − B d2 + 4t 1 − A d + δΠ−∗
2 ,
4t
4t
(30)
−∗
+∗
−∗
where A = −Π0∗
2 + Π2 + π2 − π2 . Then we need to consider a few cases, as a function
of A and B.
Suppose that 1 − 4tδ B ≤ 0, which implies that Πi is convex in d. Then Πi is maximized
at d = 0 or (and) at d = 1. Precisely, the profit at d = 0 is δΠ−∗
2 , which is smaller than
+∗
+∗
−∗
the profit at d = 1, equal to δ Π2 − 2π2 + 2π2 . The latter is not larger than the
+∗
t
candidate equilibrium profit t + 2δπ2−∗ if and only if Π+∗
2 − 2π2 ≤ δ .
Now suppose that 1 − 4tδ B > 0, which implies that Πi is concave in d Then the
maximum point for Πi is either d = 0, or d = 1, or d =
1
δ
− 4t
A
2
δ
1− 4t B
if 0 <
1
2
We have already dealt with the cases of d = 0 and d = 1. When 0 <
the deviation profit is:
2
1
δ
−
A
Πi = 4t 2 4tδ
+ δΠ−∗
2 .
1 − 4t B
− 4tδ A < 1 − 4tδ B.20
1
2
−
δ
A
4t
<1−
δ
B,
4t
The difference between the candidate equilibrium profit t + 2δπ2−∗ and the deviation profit
is
"
2 #
δ
δ
1
δ
1
t(1 − B) + δ(1 − B)C − 4t
− A
t + 2δπ2−∗ − Πi =
4t
4t
2 4t
1 − 4tδ B
δ
δ 2
B
=
A − + C − (A + BC) ,
4
4t
1 − 4tδ B
∆v
where C = 2π2−∗ − Π−∗
2 . In case s = 0 we have A = B = 0 and C = 4 , hence the
deviation is not profitable. Otherwise, the inequalities 12 − 4tδ A ≤ 1 − 4tδ B, 0 < B − A
20
verify whether d =
d=
1
δ
2 − 4t A
is the point where the derivative of Πi in (30) vanishes.
δ
1− 4t
B
1
δ
2 − 4t A
induces incompatibility in period two, but we prove below
δ
1− 4t
B
Notice that d =
1
δ
2 − 4t A
δ
1− 4t
B
is smaller than the candidate equilibrium profit.
47
In fact, we should
that Πi in (30) at
imply
δ
4t
≤
1
,
2(B−A)
and since BC + A2 > 0 we find that
A−
B
δ
B
A2 + BC
+ C − (A2 + BC) ≥ A − + C −
4
4t
4
2(B − A)
(31)
Since the right hand side in (31) is positive for each s ∈ (0, 32 ), it follows that t + 2δπ2−∗ −
Πi > 0.
Proof of Proposition 7
We are considering the equilibrium candidate in which firm i corners the other firm,
h, in the first period in both markets. Precisely, firm i plays price p̄ in each market, and
firm h plays price p̄ + t in each market, with p̄ to be determined. As a consequence, the
−∗
profit of firm i is 2p̄ + δΠ+∗
2 , and the profit of firm h is δΠ2 .
Step 1
First, we focus on deviations given second-period incompatibility: this requires that
the market share of the deviating firm is either close to 1 in both markets, or close to
zero in both markets. Precisely, consider a deviation of firm i such that it plays price
pi in both markets, which implies that the quantity sold by firm i in each market is
di = 12 + 2t1 (p̄+t−pi ), and hence the profit of firm i is f i (di ) = 2di pi +δB(di )2 +2δγdi +δΠ−∗
2 ,
+∗
−∗
−∗
0
0
i
i
with B = Π2 − 2Π2 + Π2 and C = Π2 − Π2 . Using p = 2t + p̄ − 2d t, we obtain
+∗
i
f i (di ) = (Bδ − 4t) d2 + 2 (2t + p̄ + Cδ) d + δΠ−∗
2 ; of course, f (1) = 2p̄ + δΠ2 . Since we
are considering δt > B4 , it follows that f i is convex. Hence no deviation of firm i which
induces incompatibility in the second period is profitable for i if and only if f i (1) ≥ f i (0),
+∗
−∗
−∗
δ
that is if 2p̄ + δΠ+∗
2 ≥ δΠ2 , or − 2 (Π2 − Π2 ) ≤ p̄.
Now consider firm h and a deviation such that firm h plays price ph in both markets,
which implies that the quantity sold by the firm in each market is dh = 21 + 2t1 (p̄ − pi ),
h
h
and hence its profit is f h (dh ) = 2dh ph + δB(dh )2 + 2δCdh + δΠ−∗
2 . Using p = t + p̄ − 2d t
−∗
h
we obtain fh (dh ) = (Bδ − 4t) (dh )2 + 2 (t + p̄ + Cδ) dh + δΠ−∗
2 (of course, f (0) = δΠ2 ),
and since f h is convex (as δt > B4 ), it follows no deviation of firm h which induces
incompatibility in the second period is profitable for h if and only if f h (0) ≥ f h (1) that
−∗
is if p̄ ≤ t − 2δ (Π+∗
2 − Π2 ).
Therefore, no deviation which induces incompatibility is profitable for any firm if and
only if
δ
δ +∗
−∗
−∗
− (Π+∗
(32)
2 − Π2 ) ≤ p̄ ≤ t − (Π2 − Π2 )
2
2
48
Step 2
−∗
In view of (32), consider p̄ = γ − 2δ (Π+∗
2 − Π2 ), with γ ∈ [0, t]. We need some
−∗
1
preliminaries in order to state the result precisely. In addition to B = Π+∗
2 + Π2 − 2 ,
+∗
−∗
+∗
−∗
+∗
+∗
−∗
1
1
+
define E = Π+∗
2 + π2 − 4 − π2 > 0, F = Π2 + π2 − 4 − π2 > 0, ∆ = Π2 − 2π2 > 0,
−∗
±
+
−
∆− = Π−∗
2 − 2π2 > 0, ∆ = ∆ − ∆ > 0.
Compatibility in period two arises if and only if the inequalities (23)-(24) in the proof
of Lemma 4 are satisfied, and notice that they can be indifferently written as a function
of dix , diy , or of dhx , dhy . Here we use the latter variables:
F (dhx + dhy ) − Bdhx dhy ≥ ∆−
E(dhx + dhy ) − Bdhx dhy ≥ ∆+
(33)
The set of dhx , dhy which satisfies (33) is represented in figure 2 (after changing the labels
of the axes to dhx , dhy ). Notice that
• in case that firm h plays the same price in both markets, we have dhx = dhy ≡ dh ,
and then both inequalities in (33) are satisfied if and only if dh is included between a suitable lower bound d` and a suitable upper bound du , that is d` ≤
√
√
√
2
−
2
+
2
+
dh ≤ du with d` = max{ F − F B−B·∆ , E− EB−B·∆ } = E− EB−B·∆ and du =
√
√
√
2
−
2
+
2
−
min{ F + F B−B·∆ , E+ EB−B·∆ } = F + F B−B·∆ ;
• the southwest border of the feasible set is the graph of the curve
dhy = q(dhx ), with q(dhx ) =
∆+ − Edhx
E − Bdhx
for dhx ∈ [0,
∆+
] and q(d` ) = d` (34)
E
We now define several values of δ:
(4t + 2γ)d` − 2γ − 4td2`
δ̃(s, γ) ≡
,
∆− + ∆± d`
δi (s, γ) = max{δ̃(s, γ), δ̄(s, γ)}
p
2 (2t + γ) ∆± + 8t∆− − 4 2t (γ∆± + 2t∆− ) ∆+
δ̄(s, γ) ≡
(35)
(∆± )2
2d` + 2 γt d` − 4d2`
δh (s, γ) ≡ t
(36)
∆− + ∆± d`
Finally, for each s ∈ (s̄3 , 23 ), let γm (s) denote the unique γ ∈ (0, t) such that δi (s, γ) =
δh (s, γ) (existence and uniqueness are proved in Step 1.3 of the proof below). The function
δ` we have mentioned in Proposition 7 is such that δ` (s) = δh (s, γm (s)). Now we can state
Proposition 7 in detail:
49
Proposition 7 Suppose that compatibility was chosen in the first period, and suppose
that s ∈ (s̄3 , 32 ). Then a cornering equilibrium exists if and only if δt ≥ δh (s,γtm (s)) . Precisely,
−∗
in the first period, one firm charges price p̄ = γm (s) − 2δ (Π+∗
2 − Π2 ) in both markets, the
other firm charges price p̄ + t in both markets, and the first firm corners both markets.
In the second period, incompatibility is chosen at least by the firm who cornered the
markets, which earns a total profit of 2γm (s) + δΠ−∗
2 , whereas the other firm’s total profit
δ
(s,γ
(s))
−∗
3
m
and Π+∗16/9
.
is between Π+∗3/2
is δΠ2 . For each s ∈ (s̄3 , 2 ), the value of h t
−2π +∗
−2π +∗
2
2
2
2
In the proof we give below we consider first (in Step 2.1) deviations such that each firm
plays the same price in both markets. In Step 2.2 we show that no profitable deviation
exists even if the deviating firm can charge different prices in different markets.
Step 2.1: For each s ∈ (s̄3 , 32 ), for no firm there exists a profitable deviation
which induces compatibility in period two and such that the firm chooses the
same price in both markets if and only if δ ≥ δh (s, γm (s)) Step 2.1.1: No profitable
deviation exists for firm i if and only if δ ≥ δi (s, γ); moreover, δi = δ̄(s, γ) if γt ≥ 0.05921
Given the price pi played by firm i in both markets, the demand for product ij is
di = 21 + 2t1 (p̄ +t − pi ) (for j = x, y), and inverting this relation we obtain pi = 2t + p̄
−2tdi . Hence, the profit of firm i is g i (di ) = 2di (pi + δz) + 2δπ2−∗ = −4t(di )2 + (4t +
2γ − δ∆± )di + 2δπ2−∗ for di ∈ [d` , du ]. Notice that a necessary condition for no profitable
−∗
deviation to exist for firm i is g i (d` ) ≤ 2γ + δΠ−∗
2 (2γ + δΠ2 is the equilibrium profit of
firm i), which is equivalent to δ ≥ δ̃(s, γ). . We need to distinguish two cases, depending
on the value of γt .
• Case A: 0.05921 < γt ≤ 1. The derivative of g i is g i0 (di ) = −8tdi +4t+2γ −δ∆± ,.and
`
g i0 (d` ) ≤ 0 if and only if δ ≥ 4t+2γ−8td
≡ δ̂(s, γ) . It turns out that δ̃(s, γ) < δ̂(s, γ)
∆±
γ
for each s ∈ (s̄3 , 23 ) since t > 0.05921.21 Therefore, when δ ≥ δ̂(s, γ), no profitable
deviation exists for firm i.
Now consider δ between δ̃(s, γ) and δ̂(s, γ). Then the profit maximizing di is
±
4t+2γ−δ∆±
, which is larger than d` but smaller than du .22 We have that g i ( 4t+2γ−δ∆
)=
8t
8t
Precisely, δ̃ < δ̂ is equivalent to 2(1 − 2l)DY + γt DX ≥ 2(l)2 DZ (,).
22
For each s ∈ (s̄3 , 32 ), we have that 43 < du and g i0 ( 43 ) , 2γ − 2t − δDZ < 0.
21
50
1
16t
2
(4t + 2γ − δ∆± ) + 2δπ2−∗ , which is smaller than 2γ + δΠ−∗
2 if and only if
−(∆± )2 δ 2 + 4 (2t + γ)∆± + 4t∆− δ − 4(2t − γ)2 ≥ 0
(37)
At δ = δ̃(s, γ), the left hand side of (37) is negative since g i (d` ) = 2γ + δΠ−∗
2 and
32t
g i0 (d` ) > 0. At δ = δ̂(s, γ), the left hand side in (37) has value ∆± (γ∆+ + 2t∆− − 4td` ∆− − 2td2` ∆±
which has the same sign as δ̂(s, γ) − δ̃(s, γ), hence it is positive. As a consequence,
in case A no profitable deviation exists for firm i if and only if δ is at least as large as
the smallest solution to (37), which is δ̄(s, γ), a number between δ̃(s, γ) and δ̂(s, γ).
• Case B: 0 ≤ γt ≤ 0.05921. In this case δ̂(s, γ) ≤ δ̃(s, γ) for some s ∈ (s̄3 , 23 ), and
then no profitable deviation exists for firm i if and only if δ ≥ δ̃(s, γ) (in this case
δ̄(s, γ) < δ̃(s, γ)). If for some s ∈ (s̄3 , 23 ) the inequality δ̃(s, γ) < δ̂(s, γ) holds, then
we can argue as for case A above that no profitable deviation exists for firm i if and
only if δ ≥ δ̄(s, γ). Hence, in case B no profitable deviation exists for firm A if and
only if δ ≥ max{δ̃(s, γ), δ̄(s, γ)}.
Joining the two cases A and B, we conclude that no profitable deviation exists for firm
i if and only if δ ≥ max{δ̃(s, γ), δ̄(s, γ)}, which we have defined as δi (s, γ) in (36).
Step 2.1.2: No profitable deviation for firm h exists if and only if δ ≥ δh (s, γ)
Given the price ph played by firm h in both markets, the demand for product hj is
dh = 12 + 2t1 (p̄ − ph ) (for j = x, y), and inverting this relation we obtain ph = t + p̄ − 2tdh .
Hence the profit of firm h is g h (dh ) = −4t(dh )2 +(2t+2γ −δ∆± )dh +2δπ2−∗ for dh ∈ [d` , du ].
Notice that a necessary condition for no profitable deviation to exist for firm h is g h (d` ) ≤
−∗
δΠ−∗
2 (δΠ2 is the equilibrium profit of firm h), which is equivalent to δ ≥ δh (s, γ).
The derivative of g h is g h0 (dh ) = −8tdh + 2t + 2γ − δ∆± , and g h0 (d` ) ≤ 0 if and only
`
`
. It turns out that δh (s, γ) ≥ 2t+2γ−8td
, hence no profitable deviation for
if δ ≥ 2t+2γ−8td
∆±
∆±
firm h exists if and only if δ ≥ δh (s, γ).
Step 2.1.3: For each s ∈ (s̄3 , 23 ), there exists a unique γ which minimizes max{δi (s, γ), δh (s, γ)}
For given s and γ, from Steps 1.1 and 1.2 it follows that no profitable deviation exists
for firm i and firm h if and only if δ ≥ max{δi (s, γ), δh (s, γ)}. Then, for a given s, we
choose γ to minimize max{δi (s, γ), δh (s, γ)}. It turns out that (i) δi (s, 0) > δh (s, 0) and
δi (s, t) < δh (s, t) for each s ∈ (s̄3 , 23 ); (ii) δi (s, γ) is decreasing in γ and δh (s, γ) is increasing in γ. Hence, for each s ∈ (s̄3 , 23 ) we conclude that max{δi (s, γ), δh (s, γ)} is minimized
51
by the unique γ ∈ (0, t) such that δi (s, γ) = δh (s, γ). We denote this value with γm (s), and
precisely γm (s) is the smaller solution to the equation w1 γ 2 +w2 γ +w3 = 0, with w1 = (∆−
∆± )2 , w2 = 2∆− ∆± (td` (1 + 2d` )(∆± )2 + 2t(1 + 2d` )∆− ∆± + 4t(∆− )2 ) − 8t∆± ∆+ (∆−
2
+∆± d` )2 , w3 = (td` (1 + 2d` )(∆± )2 + 2t(1 + 2d` )∆− ∆± + 4t(∆− )2 ) − 16t∆− ∆+ (∆− +
3
t), which implies that δi (s, γm (s)) = max{δ̃(s, γm (s)), δ̄(s, γm (s))
∆± d` )2 . It turns out that γm (s) ∈ ( 92 t, 11
δ̄(s, γm (s)).
Step 2.2: For each s ∈ (s̄3 , 32 ), let γ be equal to γm (s). Then for no firm there
exists a profitable deviation which induces compatibility in period two, even
though each firm can charge different prices in different markets Step 2.2.1:
For each s ∈ (s̄3 , 23 ), let γ be equal to γm (s) and δ ≥ δi (s, γm (s)). Then for firm i there
exists no profitable deviation which induces compatibility in period two, even though firm
i can charge different prices in different markets
Let pix , piy denote the prices charged by firm i in period one and dix , diy the demands
for the products of firm i in period one. Arguing as in Step 1.1 we find pix = 2t + p̄ −2tdix
and piy = 2t + p̄ −2tdiy , hence the profit of firm i under compatibility is
δ
δ
Gi (dix , diy ) = −2t(dix )2 + (2t + γm (s) − ∆± )dix − 2t(diy )2 + (2t + γm (s) − ∆± )diy + 2δπ2−∗
2
2
Notice that Gi is a concave function to maximize in the feasible set we denote with F: see
i∗
i
figure 2 in section 5.1.2. Hence, if there exists a critical point (di∗
x , dy ) of G in F (a point
i
i∗
where both partial derivatives of Gi vanish), then (di∗
x , dy ) is a global max point for G .
±
±
4t+2γm (s)−δ∆
(s)−δ∆
i∗
The proof of Step 1.1 reveals that (di∗
, 4t+2γm8t
) is a critical
x , dy ) = (
8t
i∗
point for Gi in F if δ ∈ [δi (s, γm (s)), δ̂(s, γm (s))], but since di∗
x = dy , we know that it does
not constitute a profitable deviation for firm i.
However, for δ > δ̂(s, γm (s)) there exists no critical point for Gi in F. This suggests
to inspect the boundaries of F, but it is immediate that the north-east boundary does
not contain any maximum point, by the virtue of the remark in footnote 22. About the
southwest boundary, consider any point (dix , diy ) which belongs to this boundary for which
we know that G(dix , diy ) < 2γ + δΠ−∗
2 when δ = δ̂(s, γm (s)). For δ > δ̂(s, γm (s)), the same
inequality holds because the derivative of the right hand side with respect to δ, Π−∗
2 , is
greater than the derivative of the left hand side with respect to δ, 2π2−∗ − 12 (dix + diy )∆± .
Therefore, for each δ ≥ δi (s, γm (s)), no profitable deviation exists for firm i even though
52
firm i can charge different prices in different markets.
Step 2.2.2: For each s ∈ (s̄3 , 23 ), let γ be equal to γ(s). Then for firm h there exists
no profitable deviation which induces compatibility in period two, even though firm h can
charge different prices in different markets
Let phx , phy denote the prices charged by firm h in period 1 and dhx , dhy the demands for
the products of firm h in period one. Arguing as in Step 1.2 we find phx = t + p̄ − 2tdhx
and phy = t + p̄ − 2tdhy , hence the profit of firm h under compatibility is
δ
δ
Gh (dhx , dhy ) = −2t(dhx )2 + (t + γm (s) − ∆± )dhx − 2t(dhy )2 + (t + γm (s) − ∆± )dhy + 2δπ2−∗
2
2
From the proof of Step 1.2 we know that there exists no critical point of Gh in F, hence
each global maximum point of Gh belongs to the southwest boundary of F,23 which is the
B·∆+ −E 2
0
curve dhy = q(dhx ) described in (34) Notice that q 0 (dhx ) = (E−Bd
h )2 < 0 with q (d` ) = −1,
and q 00 (dhx ) =
by
2B(B·∆+ −E 2 )
3
(E−Bdh
x)
x
< 0. Hence firm h’s profit along the southwest border is given
δ
δ
gq (dhx ) = Gh (dhx , q(dhx )) = −2t(dhx )2 +(t+γm (s)− ∆± )dhx −2t(q(dhx ))2 +(t+γm (s)− ∆± )q(dhx )+2δπ2−∗
2
2
+
for dhx ∈ [0, ∆E ]. First, we prove below that no profitable deviation exists for firm h when
δ = δh (s, γm (s)).
When δ = δh (s, γm (s)), we prove that gq is maximized at dhx = d` since gq0 (d` ) = 0
and gq is concave. Therefore no profitable deviation exists for firm h since gq (d` ) = δΠ−∗
2 .
Precisely, we find that
δ
δ
gq0 (dhx ) = −4tdhx + t + γm (s) − ∆± − 4tq(dhx )q 0 (dhx ) + (t + γm (s) − ∆± )q 0 (dhx )
2
2
and gq0 (d` ) = 0 since q(d` ) = d` and q 0 (d` ) = −1. Moreover,
δ
gq00 (dhx ) = −4t − 4t(q 0 (dhx ))2 − 4tq(dhx )q 00 (dhx ) + (t + γm (s) − ∆± )q 00 (dhx )
2
(B · ∆+ − E 2 )2
B(∆+ − Ed)(B · ∆+ − E 2 )
δ ± 00 h
= −4t − 4t
−
8t
+
(t
+
γ
(s)
−
∆ )q (dx )
m
(E − Bd)4
(E − Bd)4
2
23
The northeast boundary cannot include any global optimum point because
δ
2 DZ,
which is negative for each dhx ≥ 12 . A similar property holds for
53
h
∂G
∂dh
y
.
∂Gh
∂dh
x
= −4tdhx +t+γm (s)−
and at δ = δh (s, γm (s)) we have (t + γm (s) − 2δ ∆± )q 00 (dhx ) < 0 because t + γm (s) −
− +2t(d )2 ∆±
δh (s,γm (s)) ±
`
∆ = (t+γm (s))∆
> 0 and q 00 (dhx ) < 0. The other terms in gq00 (dhx ) have
2
∆− +∆± d`
the same sign as −(E − Bdhx )4 − (B · ∆+ −E 2 )2 + 2B(E 2 − B · ∆+ )(∆+ −Edhx ). For any
3
1/3
+ ·E
E
fixed s, this is a concave function of dhx which is maximized dhx = B
− B1 E −B·∆
2
and the maximum value of the expression is negative. Therefore gq00 (dhx ) < 0.
+
We have established above that gq (dhx ) ≤ δΠ−∗
for each dhx ∈ [0, ∆E ] when δ =
2
holds for
δh (s, γm (s)). When instead δ > δh (s, γm (s)), the inequality gq (dhx ) < δΠ−∗
2
+
each dhx ∈ [0, ∆E ] because the derivative of the right hand side with respect to δ, Π−∗
2 ,
−∗
is greater than the derivative of the left hand side with respect to δ, which is 2π2 −
1 ± h
∆ dx − 12 ∆± q(dhx ).
2
Proof of Proposition 8
The proof is straightforward and is omitted.
Proof of Proposition 9
The proof is omitted as we explained the results in the main texts.
Proof of Proposition 11
(i) In the compatibility equilibrium, the second-period consumer surplus in market j
is given by
CSj
1
=
∆v
ˆ
v̄
(vji
v
−
i
p+∗
2 )dvj
1
−
∆v
ˆ
−∗
v e +p+∗
2 −p2 −s
v
(vji
−
i
p+∗
2 )dvj
−∗
ˆ ve +p+∗
2 −p2 −s
1
i
+
(v e − p−∗
2 − s)dvj
∆v v
e
1
−∗
+∗
−∗
v e + p+∗
2 − p2 − s − v (v − p2 − p2 − s + v)
2∆v
e
1
−∗
−∗
v e + p+∗
+
2 − p2 − s − v (v − p2 − s)
∆v
2
1
∆v
∆v −∗ 2
1 −∗ −∗
e
+∗
+∗
−∗
= v − p2 +
+ p2 − p2 − s = v e − p+∗
(d2 ) = v e − p+∗
2 +
2 + p 2 d2
2∆v
2
2
2
1 −∗
= v e − p+∗
2 + π2 .
2
= v e − p+∗
2 −
54
The first-peirod consumer surplus in market j is 2
Hence, the total consumer surplus in market j is
v e − p∗1 −
´1
2
0
(v e − p∗1 − tx)dx = v e − p∗1 − 4t .
t
1 −∗
+ δ(v e − p+∗
2 + π2 ).
4
2
(ii) In the incompatibility equilibrium, the second-period consumer surplus is given by
1
CS =
(∆v)2
1
−
(∆v)2
1
+
(∆v)2
e
= 2v −
−
1
(∆v)2
ˆ
ˆ
v̄
v̄
(vxi + vyi − P2+∗ )dvxi dvyi
v
v
ˆ
2v e −2s−P2−∗ +P2+∗ −v
ˆ
2v e −2s−P2−∗ +P2+∗ −v
ˆ
v
+∗
P2
ˆ 2ve −2s−P2−∗ +P2+∗ −v
v
(2v e − P2−∗ − 2s)
+
(∆v)2
e
(vxi + vyi − P2+∗ )dvxi dvyi
v
v
ˆ
2v e −2s−P2−∗ +P2+∗ −vxi
ˆ
2v e −2s−P2−∗ +P2+∗ −vxi
(2v e − P2−∗ − 2s)dvxi dvyi
v
(2v e − 2s − P2−∗ + P2+∗ − vxi − v)(2v e − 2s − P2−∗ − P2+∗ + vxi + v)dv
2v e −2s−P2−∗ +P2+∗ −v
(2v e − 2s − P2−∗ + P2+∗ − vxi − v)dvxi
v
P2+∗
= 2v −
(2v e − 2s − P2−∗ + P2+∗ − 2v)2
1 e
−∗
+∗
e
−∗
−
− (2v − 2s − P2 + P2 − 2v) + (2v − 2s − P2 )
2(∆v)2
3
−∗
+∗
2
e
(2v − 2s − P2 + P2 − 2v)
+
(2v e − 2s − P2−∗ )
2(∆v)2
= 2v e − P2+∗
1 (2v e − 2s − P2−∗ + P2+∗ − 2v)2
(2v e − 2s − P2−∗ + P2+∗ − 2v)
+
3
2(∆v)2
2
2
= 2v e − P2+∗ + D2−∗ P2−∗ = 2v e − P2+∗ + Π−∗
.
3
3 2
The first-period consumer surplus is
ˆ
1
ˆ
2
0
0
1−x
2
(2v e − P1∗ − tx − ty)dydx = 2v e − P1∗ − t.
3
55
Hence, the total consumer surplus is
e
2v −
P1∗
2 −∗
2
e
+∗
− t + δ 2v − P2 + Π2 .
3
3
(iii) As enough details are given in the main text, the proof is omitted.
Proof of Proposition 12
The welfare is simply obtained by adding consumer surplus to profits, which we presented in previous propositions. Regarding (iii), as enough details are given in the main
text, the proof is omitted.
Proof of Proposition 13
The proof is omitted as it is a minor adaptation of the proof of Proposition 3.
56