1. business and industrial
2. company
3. earnings

Private contracts in two-sided markets
Gast´on Llanes
Francisco Ruiz-Aliseda
Preliminary and incomplete
March 16, 2015
Abstract
We develop a model of a two-sided platform that unites sellers and
buyers, and study the effect of private contracts between sellers and the
platform provider on the structure of prices, platform adoption, profits,
and welfare. We study a game in which the platform provider charges perunit royalties to sellers, and in which sellers cannot observe the royalties
imposed on other sellers, and compare the resulting equilibrium with that
of a model in which sellers can observe all royalties. We show that the
secrecy of contracts lead to lower profits for the platform provider, and
may lead to higher or lower profits for sellers, depending on the degree of
substitution between their products. In particular, seller profits are lower
when contracts are secret if and only if sellers’ products are substitutes,
and the degree of substitution between sellers’ products is large enough.
We also find that secrecy may increase the area of parameters for which the
platform provider chooses to subsidize consumers, and may lead to lower
or higher platform prices for sellers and consumers.
Keywords: Two-sided markets, platforms, private contracts.
1
Introduction
Private contracts are common in many two-sided platforms. For example, Amazon signs private contracts with publishers, Netflix with movie studios, Sony with
game developers, HMOs with health-care providers, Google with phone manufacturers, and Apple with cellphone carriers.
An extensive literature has studied the effects of private contracts on market
structure and competition in one-sided markets, but the effects of private contracts
in multi-sided markets are largely unexplored.
1
In this paper, we develop a model of a two-sided platform that unites sellers and
buyers, and study the effect of private contracts between sellers and the platform
provider on the structure of prices, platform adoption, profits, and welfare.
Sellers and buyers contract with the platform provider to access the platform.
Sellers then sell platform-specific products to buyers. As an example, consider
console-specific games sold to consumers. The contract terms offered to buyers
are publicly observed, but the contract terms (fixed fees or per-unit royalties)
offered to each seller are not observed by other sellers and consumers.
We first study the case in which the platform provider charges royalty fees to
sellers. We focus on the equilibrium in which consumers form “passive” beliefs
and sellers form “wary” beliefs (McAfee and Schwartz, 1994; Rey and Verg´e,
2004) about the royalties they cannot observe. We compare the equilibrium of
this game with that of two additional games: (i) a game in which firms can observe
all royalties, but royalties are still unobservable for consumers, and (ii) a game in
which royalties are observable for all agents.
We show that the secrecy of contracts lead to lower profits for the platform
provider, and may lead to higher or lower profits for sellers, depending on the
degree of substitution between their products. In particular, seller profits are
lower when contracts are secret if and only if sellers’ products are substitutes, and
the degree of substitution between sellers’ products is large enough. We also find
that secrecy may increase the area of parameters for which the platform provider
chooses to subsidize consumers, and may lead to lower or higher equilibrium
royalties for sellers.
2
The model
There exist n + 1 firms labeled 0, 1, . . . , n. Firm 0 produces platforms at no cost
that enable the other firms to sell their respective products to consumers who
purchase firm 0’s platform. Leaving firm 0 aside, all other firms are symmetric
and each of them produces a certain product at no cost as well.
A consumer finds the product sold by firm i ∈ {1, . . . , n} valuable only if she
has acquired the platform good from firm 0. Consumers are uniformly spread on
the positive real line and firm 0 is located at the left end.
Given a consumer at distance x ∈ [0, ∞) from firm 0, consider her utility if she
purchases one unit of the product sold by such a firm at price p0 and purchases
2
qi ≥ 0 units of the product sold by firm i ∈ {1, . . . , n} at price pi each. We assume
(see e.g. Vives 1999) that such utility equals
Ux (q1 , . . . , qn ) = u(q1 , . . . , qn ) − x − p0 ,
where
u(q1 , . . . , qn ) =
n
X
i=1
1
qi −
2
n
X
qi2 + θ
i=1
n
n
X
X
!
qi qj
i=1 j=1;j6=i
−
n
X
pi qi .
i=1
Parameter θ ∈ [−1, 1] captures the degree of complementarity/substitutability
between the products of firms i and j. Products are complements, substitutes, or
independent if θ is negative, positive or zero, respectively. As θ grows, the degree
of complementarity between the products decreases, and the degree of substitution
increases.
Contracts between sellers and the platform provider may include a per-unit
royalty fee wi or a fixed fee fi . In Section 3, we study the case of per-unit royalty
fees. In Section 4, we study the case of fixed fees. In Section 5, we study the case
in which both types of fees may be used.
We will seek for symmetric Perfect Bayesian Equilibria (PBE) given some standard constraints on how off-the-equilibrium-path beliefs are formed. In particular,
we assume that consumers and firms have either passive or wary beliefs (McAfee
and Schwartz, 1994; Rey and Verg´e, 2004). We discuss each type of belief, and
the reasons for assuming agents have such beliefs, below.
3
Private contracts with royalty fees
Consider the following two-period game. In the first period, firm 0 sets p0 and
offers contracts with per-unit royalty fees wi to firms i ∈ {1, . . . , n}, consumers
decide whether to purchase firm 0’s platform, and firms i ∈ {1, . . . , n} decide
whether to accept the contracts offered by firm 0. Before accepting firm 0’s offers,
consumers observe p0 , and each firm i ∈ {1, . . . , n} observes p0 and wi . Consumers
do not observe any royalty fee, and firm i does not observe any royalty fee wj for
j 6= i. In the second period, firms 1, . . . , n form beliefs about the royalty fees
generated by firm 0 given their available information, and then set the price for
their respective product to those consumers who acquired firm 0’s platform.
3
Our timing reflects the fact that consumers use the platform for many periods
(i.e., the platform is a durable good), during which platform-specific products are
continuously being launched. For instance, buyers of a video console often buy it
without observing the prices charged for the games that will be marketed during
the lifetime of the console.
Let p∗0 and denote the price charged to consumers, and let w∗ denote the
royalty fee offered to firm i ∈ {1, . . . , n} in a symmetric PBE. We assume that
consumers form so-called “passive” beliefs upon observing some price p0 6= p∗0 :
they still believe that firm i ∈ {1, . . . , n} has been offered w∗ , and therefore
expect price competition between all these firms to be unaffected. Consumers are
not sophisticated upon observing a deviation from equilibrium behavior, and this
is known by firm i ∈ {1, . . . , n}, so such a firm believes that p0 6= p∗0 conveys no
further information and ignores it, as do consumers. Firms therefore form passive
beliefs with respect to to price deviations by firm 0 and believe that no signaling
is taking place by firm 0.
On the other hand, firms interpret an unexpected royalty fee as an intended
deviation by firm 0 and form “wary beliefs.” Thus, if firm i ∈ {1, . . . , n} receives
a private offer of wi 6= w∗ , it believes that firm 0 must have made an offer to
j ∈ {1, . . . , n} (j 6= i) that maximizes firm 0’s total profit given the price that
firm 0 charges to consumers and the royalty fee demanded from firm i ∈ {1, . . . , n}.
We also assume that a firm that forms “wary beliefs” conjectures that its rivals
also do, and it also conjectures that firm 0 does not want to drive any firm out of
the market.
In this section, we assume that the number of sellers is fixed. Note, however,
that consumers’ utilities depend on (expected) royalty fees, whereas firms’ profits
depend on the price charged by the platform to consumers. Therefore, the model
allows us to study how the interactions between the two sides of the market affect
the equilibrium. In Sections 4 and 5 we allow for the endogenous determination
of the number of sellers, and discuss additional issues that arise in this case.
3.1
Solution of the model
We begin by studying the equilibrium of the second stage after consumption and
membership decisions by consumers and sellers have been made.
After observing pi , consumers who have purchased firm 0’s platform, choose
{qi }ni=1 to maximize u(q1 , . . . , qn ). Looking at interior solutions yields (see Vives
4
1999, pp. 146-147) the following demand for the product sold by firm i ∈ {1, . . . , n}:
1 − θ − (1 + (n − 2) θ) pi + θ
qi (p1 , . . . , pn ) =
(1 − θ) (1 + (n − 1) θ)
P
j6=i
pj
.
Suppose that there are x(p0 ) consumers who have purchased firm 0’s platform
at price p0 . Because their consumption of {qi }ni=1 does not depend on their distance
from firm 0, it holds that the overall demand for firm i’s product equals
Qi (p1 , . . . , pn ) = x(p0 ) qi (p1 , . . . , pn ).
Having obtained demands, we can solve for the second-period subgames. Recalling that we are examining symmetric equilibria, let B(w)
b denote the belief
formed by firm i ∈ {1, . . . , n} about the royalty fee paid by firm j ∈ {1, . . . , n}
(j 6= i) to firm 0.1 We begin by studying the case of n = 2, and leave the case of
n ≥ 2 for Section 3.4.
Let pi (wi ) denote the strategy of firm i ∈ {1, 2} in the second-period subgame
if it has observed an offer of wi . Having observed w1 , firm 1 chooses p1 to maximize
π1 (w1 ) = (p1 − w1 ) Q1 (p1 , p2 (B(w1 ))).
The first-order condition is
1 − θ + w1 − 2p1 (w1 ) + θp2 (B(w1 )) = 0.
Similarly, if firm 2 observes w2 , then its first-order condition is
1 − θ + w2 − 2p2 (w2 ) + θp1 (B(w2 )) = 0.
(1)
Let us now turn to analyzing the first period of play. Regardless of the price
p0 that consumers observe, they believe that both sellers face a royalty fee of w∗ .
1
Because we are looking at symmetric equilibria, the belief function B(·) does not depend on
the label of the firm receiving the unexpected offer. Note that, in general, B(·) is an unrestricted
function except for the constraint that B(w∗ ) = w∗ (i.e., conjectured beliefs are fulfilled along
the equilibrium path). In our case, we will restrict the function so that beliefs are ”wary”. If
firm 1 observes a deviation involving w1 6= w∗ and thinks it was intended, it may use a forwardinduction argument and form a conjecture based on the idea that firm 0 must be making an offer
to firm j ∈ {2, . . . , n} so as to maximize its overall profit given w1 . The ”rational” beliefs that
result from solving this fixed-point problem were coined ”wary beliefs” by McAfee and Schwartz
(1994), but we use the modification advocated by Rey and Verg´e (2004).
5
Given the above second-stage first-order conditions, consumers believe they will
face a price
1 − θ + w∗
pC
i =
2−θ
for each unit they purchase from firm i ∈ {1, 2} in the second period. The overall
utility expected by a consumer located at distance x from firm 0 equals
Ux∗ (p0 ) =
(1 − w∗ )2
− x − p0 .
(1 + θ) (2 − θ)2
As a result, firm 0 attracts
(1 − w∗ )2
x(p0 ) =
− p0
(1 + θ) (2 − θ)2
consumers when charging p0 .
Firm 0’s total profit if it charges p0 and makes a private offer of w1 and w2 to
firms 1 and 2 is
π0 (w1 , w2 , p0 ) = x(p0 ) (p0 + w1 q1 (p1 (w1 ), p2 (w2 )) + w2 q2 (p1 (w1 ), p2 (w2 ))) .
Wary beliefs imply that the inference made by firm 2 about firm 1’s royalty
fee upon observing a price of p0 and an offer of w2 is
B(w2 ) = argmax π0 (w, w2 , p0 ).
w
The first-order condition for this problem is
∂q2 (p1 (w), p2 (w2 )) dp1 (w)
∂π0 (w, w2 , p0 )
= x(p0 ) q1 (p1 (w), p2 (w2 )) + w2
∂w
∂p1
dw
∂q1 (p1 (w), p2 (w2 )) dp1 (w)
+w
∂p1
dw
dp1 (w)
x(p0 )
1 − θ − p1 (w) + θp2 (w2 ) + (θw2 − w)
. (2)
=
1 − θ2
dw
Thus, beliefs B(w2 ) must satisfy the following condition:
1 − θ − p1 (B(w2 )) + θp2 (w2 ) + (θw2 − B(w2 ))
dp1 (B(w2 ))
= 0.
dw
(3)
We will now seek for a linear symmetric PBE sustained by wary beliefs. Let
6
us conjecture that pi (w) = Θ + Σ w for i ∈ {1, 2}, and B(w) = Γ + Φ w for
parameters Θ, Σ, Γ and Φ to be determined. Conditions (1) and (3) yield the
following conditions:
(1 − θ)(1 − Θ) − 2Σ Γ + 2 Σ (θ − Φ) w2 = 0,
1 − θ + θ Σ Γ − (2 − θ) Θ + (1 − 2 Σ + θ Σ Φ) w2 = 0.
Since these two conditions should be satisfied for all w2 , we must have
(1 − θ)(1 − Θ) − 2 Σ Γ = 0,
2 Σ (θ − Φ) = 0,
1 − θ + θ Σ Γ − (2 − θ) Θ = 0,
1 − 2 Σ + θ Σ Φ = 0,
which implies that parameters Θ, Σ, Γ and Φ are
(2 + θ)(1 − θ)
,
4 − θ(1 + θ)
1
Σ =
,
2 − θ2
(1 − θ)(2 − θ2 )
Γ =
,
4 − θ(1 + θ)
Φ = θ.
Θ =
Beliefs are fulfilled in equilibrium (w∗ = B(w∗ )), which implies that
w∗ =
2 − θ2
> 0.
4 − θ(1 + θ)
Also, firm 1 should find it optimal to choose p0 = p∗0 and w1 = w2 = w∗ , so
(w∗ , w∗ , p∗0 ) ∈ arg max{π0 (w1 , w2 , p0 )}. Note that the optimal choices of w1 and
w1 ,w2 ,p0
w2 do not depend on the choice of p0 , so firm 0 can maximize with respect to
w1 and w2 ignoring the value of p0 ; the analysis above leading to expression (3)
shows that private offers are chosen optimally, since second-order conditions are
satisfied.2 As for the optimal choice of p0 given that firm i ∈ {1, 2} receives an
2
From (2), we obtain
∂ 2 π0 (w1 , w2 , p0 )
∂ 2 π0 (w1 , w2 , p0 )
2Σ
=
=−
≤ 0,
2
2
∂w1
∂w2
1 − θ2
7
offer equal to w∗ , we need
dx(p0 )
x(p0 ) + p0 + 2 w∗ q1 (p1 (w∗ ), p2 (w∗ ))
= 0,
dp0
which implies
p∗0 =
(1 − w∗ )2
2w∗ (1 − Θ − Σw∗ )
.
−
2(1 + θ)(2 − θ)2
2(1 + θ)
It readily follows that
3 − 2θ2
< 0.
2(1 + θ)[4 − θ(1 + θ)]2
p∗0 = −
In the unique linear equilibrium, firm 0 makes sales of
x∗ ≡ x(p∗0 ) =
5 − 2 θ2
> 0,
2 (1 + θ) (4 − θ (1 + θ))2
whereas firm i ∈ {1, 2} sells
qi∗ =
1
>0
(1 + θ) (4 − θ (1 + θ))
because it charges
p∗i = 1 −
1
4 − θ (1 + θ)
so as to earn a markup equal to
p∗i − w∗ =
1−θ
> 0.
4 − θ (1 + θ)
Profits for firm 0 are
π0∗
=
5 − 2θ2
2 (1 + θ) (4 − θ(1 + θ))2
2
,
whereas firm i ∈ {1, 2} earns
πi∗
(1 − θ)(5 − 2θ2 )
=
,
2(1 + θ)2 [4 − θ(1 + θ)]4
∂ 2 π0 (w1 , w2 , p0 ) ∂ 2 π0 (w1 , w2 , p0 )
−
∂w12
∂w1 ∂w2
∂ 2 π0 (w1 , w2 , p0 ) ∂ 2 π0 (w1 , w2 , p0 )
+
∂w12
∂w1 ∂w2
8
=
4Σ2
≥ 0.
1 − θ2
so
π0∗ − 2 πi∗ =
3.2
(1 − 2 θ2 + 4 θ) (5 − 2 θ2 )
.
4 (1 + θ)2 (4 − θ (1 + θ))4
Royalties are observable for firms
To understand the foregoing results, we analyze the effect of removing the assumption that a firm cannot observe the other firm’s royalty fee, keeping the
assumption that consumers do not observe the royalty fees offered to firms.
In the second stage, firms set prices, given that x(p0 ) consumers have traded
with firm 0. The equilibrium prices for firms 1 and 2 are
(2 + θ)(1 − θ) + 2w1 + θw2
,
(2 − θ)(2 + θ)
(2 + θ)(1 − θ) + 2w2 + θw1
.
p2 (w1 , w2 ) =
(2 − θ)(2 + θ)
p1 (w1 , w2 ) =
if their respective royalty fees are equal to w1 and w2 . Consumers believe that
firms face a royalty fee of w∗ , so their demand equals
x(p0 ) =
(1 − w∗ )2
− p0
(1 + θ) (2 − θ)2
when observing p0 . Firm 0 chooses p0 , w1 and w2 to maximize
x(p0 ) (p0 + w1 q1 (w1 , w2 ) + w2 q2 (w1 , w2 )) ,
where
(2 + θ)(1 − θ) − (2 − θ2 )w1 + θw2
,
(1 − θ2 )(4 − θ2 )
(2 + θ)(1 − θ) − (2 − θ2 )w2 + θw1
q2 (w1 , w2 ) =
.
(1 − θ2 )(4 − θ2 )
q1 (w1 , w2 ) =
After we account for symmetry, the first-order conditions corresponding to p0 and
9
wi become
2(1 − w)
2w
−
(2 − θ)(θ + 1) (2 − θ)(θ + 1)
−
(1 − w)2
− p0
(2 − θ)2 (θ + 1)
2(1−w)w
2(1 − w) p0 + (2−θ)(θ+1)
= 0,
(2 − θ)2 (θ + 1)
(1 − w
b∗ )2
2w
b∗ (1 − w
b∗ )
∗
= 0.
− 2b
p0 −
(1 + θ) (2 − θ)2
(1 + θ)(2 − θ)
Solving this system of equations, we obtain
w
b∗ =
and
1−θ
3 − 2θ
2θ − 1
.
2 (3 − 2θ)2 (1 + θ)
pb∗0 =
From the last equation, we can see that w
b∗ is always positive, but that pb∗0
is positive if θ > 1/2. Thus, the region of parameters for which the platform
provider subsidizes consumers is smaller in this case in comparison with the case
studied in the previous section.
Firm i ∈ {1, 2} then charges price
pb∗i =
and sells an amount of
qbi∗ =
2 (1 − θ)
3 − 2θ
1
(1 + θ) (3 − 2θ)
to each consumer. The total number of consumers is:
x
b∗0 =
1
.
2 (1 + θ) (3 − 2θ)
π
b∗i =
1−θ
.
(1 + θ)2 (3 − 2θ)3
Thus, the profit of firm i is
Both the profit generated by each consumer and the overall profit of firm i ∈ {1, 2}
10
decrease with θ. Finally, the profit of firm 0 is
π
b∗0 =
1
.
4 (3 + θ − 2θ2 )2
Comparing the equilibrium with that of the previous section, we can see that
the profit of firm 0 is larger when sellers can observe all royalties. Sellers’ profits,
on the other hand, may be larger or lower, depending on the value of θ. In
particular, there exists θ0 > 0, such that sellers’ profits are larger when they can
observe all royalties if and only if θ > θ0 .
To understand what is going on relative to the case with unobservability of
the other firm’s royalty fee, note that, if firm 0 kept w2 fixed and sought for the
optimal royalty fee to be charged to firm 1, it would choose to charge
w
b1∗ (w2 ) =
(2 + θ)(1 − θ) + 2θw2
.
2(2 − θ2 )
Because price competition between two firms selling substitutes displays strategic
complementarity, dw
b1∗ (w2 )/dw2 > 0 if θ > 0; if instead θ < 0, price competition between two firms selling complements displays strategic substitutability, so
dw
b1∗ (w2 )/dw2 < 0. Since |dw
b1∗ (w2 )/dw2 | < |dB(w2 )/dw2 |, it holds when firm 2
cannot observe the royalty offered to firm 1 that it forms beliefs that vary more
with the royalty fee it observes than the ones it should form if it could observe
firm 0’s offer to firm 1.
Under full observability, the fact that w
b1∗ (1/2) = 1/2 yields that, regardless
of the value taken by θ, firm 0 does not interfere with the way that firms 1 and
2 interact in the marketplace, and it acts as if they were selling independent
products, charging them w
b∗ = 1/2.3 Under unobservability, firms believe that
firm 0 does internalize the nature of competitive interaction when setting the
optimal royalty fees (unless of course θ = 0). Thus, when θ > 0, a firm that
observes a greater royalty fee than the one it was supposed to observe will believe
that firm 0 must be softening competition by raising both fees; as a result, it will
overreact relative to how it would if it could observe the royalty fee offered to the
other firm. When θ < 0, a firm that observes a greater royalty fee than the one it
3
If firm 0 fully internalized the profits of firms 1 and 2 when choosing the royalty fees, it
would always have an incentive to raise the royalty fees if θ > 0 so as to soften competition,
whereas it would always have an incentive to lower the royalty fees if θ < 0 so as to toughen
competition (it would set a royalty fee equal to 0 if θ = 0).
11
was supposed to observe will believe that firm 0 must be lowering the other firm’s
royalty fee. The assumption that firms act warily in this way when unobserving
2 − θ2
unexpected offers explains why w∗ =
Rw
b∗ = 1/2 if θ R 0.
4 − θ(1 + θ)
3.3
Observable royalties for firms and consumers
We conclude the case of royalty fees and n = 2 by comparing the equilibrium
outcome under unobservability with the equilibrium that holds when royalty fees
are public information. In this case, given that x(p0 ) consumers have traded with
firm 0, firm 1 charges price
p1 (w1 , w2 ) =
(2 + θ)(1 − θ) + 2w1 + θw2
,
(2 − θ)(2 + θ)
whereas firm 2 charges
p2 (w1 , w2 ) =
(2 + θ)(1 − θ) + 2w2 + θw1
(2 − θ)(2 + θ)
if their respective royalty fees equal w1 and w2 .
The overall utility expected by a consumer located at distance x from firm 0
equals
Ux∗ (p0 , w1 , w2 ) = u∗ (p1 (w1 , w2 ), p2 (w1 , w2 )) − x − p0 ,
where
u∗ (p1 , p2 ) =
2(1 − θ)(1 − p1 − p2 + p1 p2 ) + (p1 − p2 )2
.
2 (1 − θ2 )
As a result, firm 0 chooses p0 , w1 and w2 to maximize
2(1 − θ)[1 − p1 (w1 , w2 ) − p2 (w1 , w2 ) + p1 (w1 , w2 )p2 (w1 , w2 )] + [p1 (w1 , w2 ) − p2 (w1 , w2 )]2
− p0 ] ×
2 (1 − θ2 )
[p0 + w1 q1 (w1 , w2 ) + w2 q2 (w1 , w2 )],
[
where
q1 (w1 , w2 ) =
and
(2 + θ)(1 − θ) − (2 − θ2 )w1 + θw2
(1 − θ2 )(4 − θ2 )
(2 + θ)(1 − θ) − (2 − θ2 )w2 + θw1
q2 (w1 , w2 ) =
.
(1 − θ2 )(4 − θ2 )
12
The equilibrium price and royalty fees are
2θ − 1
2(1 + θ)(3 − 2θ)2
p∗∗
0 =
and
w∗∗ =
1−θ
,
3 − 2θ
so firm 0 earns
π0∗∗ =
1
4(1 +
θ)2 (3
− 2θ)2
,
and firm i ∈ {1, 2} charges
p∗∗
i = 1−
so as to earn
πi∗∗ =
1
3 − 2θ
1−θ
.
2(1 + θ)2 (3 − 2θ)3
Also,
qi∗∗ =
and
x∗∗ =
1
(1 + θ)(3 − 2θ)
1
.
2(1 + θ)(3 − 2θ)
Let us now compare the games under private information (secrecy/unobserbability)
and under public information. It holds that π0∗ = π0∗∗ if θ = b
θ ≈ −0.61803399
∗
∗∗
and π0 < π0 otherwise, so firm 0 is worse off when information is private. Also,
∗
∗∗
∗∗
∗
∗∗
b
b
b ∗
if
p∗0 > p∗∗
0 if θ < θ, p0 = p0 if θ = θ and p0 < p0 if θ > θ, whereas w < w
∗
∗∗
∗
∗∗
b
b
b
θ < θ, w = w if θ = θ and w > w if θ > θ. Under secrecy, firm 0 prefers
to tilt profit-making towards the opposite of what would happen without secrecy.
∗
∗∗
∗∗
∗
∗∗
b ∗
b
Also, note that p∗i < p∗∗
i and qi > qi if θ < θ, pi = pi and qi = qi if θ = θ and
∗
∗∗
∗
∗∗
∗∗
b
b ∗
b
p∗i > p∗∗
i and qi < qi if θ > θ, whereas πi > πi if θ < θ, πi = πi if θ = θ and
πi∗ < πi∗∗ if θ > b
θ.
It also holds that
x∗∗ − x∗ =
1
5 − 2θ2
,
−
2(1 + θ)(3 − 2θ) 2(1 + θ)[4 − θ(1 + θ)]2
so x∗∗ = x∗ if θ = b
θ ≈ −0.61803399 and x∗∗ > x∗ otherwise, so fewer platforms
are sold under secrecy. It remains to examine what happens with overall welfare,
so as to drawn policy implications regarding whether or not contracts should be
13
public.
Total consumer welfare under secrecy can be shown to equal
CS ∗ =
(5 − 2θ2 )2
,
8(1 + θ)2 [4 − θ(1 + θ)]4
whereas consumer welfare when information is public equals
CS ∗∗ =
(3 − 2θ)2
.
8(1 + θ)2 (3 − 2θ)4
It holds that CS ∗ = CS ∗∗ if θ = b
θ ≈ −0.61803399 and CS ∗ < CS ∗∗ otherwise,
so consumers are worse off when information is private. Also, the overall profit
attained by firms 1, . . . , n is greater under secrecy if and only if θ < b
θ. Finally,
social welfare under secrecy equals
W∗ =
8(1−θ)(5−2θ2 )
8(1+θ)2 (4−θ(1+θ))4
+
(5−2θ2 )2
8(1+θ)2 (4−θ(1+θ))4
+
2(5−2θ2 )2
8(1+θ)2 (4−θ(1+θ))4
=
(23−8θ−6θ2 )(5−2θ2 )
,
8(1+θ)2 (4−θ(1+θ))4
whereas social welfare when information is observed by all parties equals
W ∗∗ =
8(1−θ)(3−2θ)
8(1+θ)2 (3−2θ)4
+
(3−2θ)2
8(1+θ)2 (3−2θ)4
+
2(3−2θ)2
8(1+θ)2 (3−2θ)4
=
(17−14θ)(3−2θ)
,
8(1+θ)2 (3−2θ)4
so W ∗ > W ∗∗ if and only if θ < b
θ. Note that θ ≥ 0 would imply that secrecy
makes all the parties worse off when royalty fees are used (this result persists if
firm 0 could offer fixed fees as well, see last section).
3.4
General case under unobservability
Let n ≥ 2. Taking into account that
n
P
1 − θ − (1 + (n − 2)θ)pi + θ
qi (p1 , . . . , pi , . . . , pn ) =
j=1;j6=i
(1 + (n − 1)θ)(1 − θ)
pj
,
firm i ∈ {1, . . . , n} chooses pi to maximize (pi − w)Q
b i (p1 , . . . , pi , . . . , pn ) given that
pj = p(B(w))
b for all j ∈ {1, .., n}, j 6= i. Therefore,
1 − θ + (1 + (n − 2)θ)w
b − 2(1 + (n − 2)θ)pi (w)
b + θ(n − 1)p(B(w))
b = 0.
Let us now turn to analyzing the first period of play. Regardless of the price p0
14
that consumers observe, they believe that firm i ∈ {1, . . . , n} is charged a royalty
fee of w∗ , so they believe that they will face a price
pC
i =
1 − θ + (1 + (n − 2)θ)w∗
2 + (n − 3)θ
for each unit they purchase from firm i ∈ {1, . . . , n} in the second period. The
overall utility expected by a consumer located at distance x from firm 0 equals
Ux∗ (p0 ) =
n[1 + (n − 2)θ]2 (1 − w∗ )2
− x − p0 .
2[1 + (n − 1)θ][2 + (n − 3)θ]2
As a result, firm 0 attracts
x(p0 ) =
n[1 + (n − 2)θ]2 (1 − w∗ )2
− p0
2[1 + (n − 1)θ][2 + (n − 3)θ]2
consumers when charging p0 .
Firm 0’s total profit if it charges p0 and makes a private offer of wi to firm
i ∈ {1, . . . , n} is therefore as follows:
π0 (w1 , . . . , wn , p0 ) = x(p0 )[p0 +
n
X
wi qi (pi (wi ))].
i=1
The inference made by firm n about the other firms’ royalty fees upon observing a
price of p0 and an offer of w
bn must be such that B(w
bn ) = arg max π0 (w1 , . . . , wn−1 , w
bn , p0 )
w1 ,...,wn−1
in order for it to form wary beliefs. Since
∂π0 (w1 ,...,wn−1 ,w
bn ,p0 )
∂w1
1−θ−(1+(n−2)θ)p1 (w1 )+θ
=
n−1
P
j=2
pj (wj )+θpn (w
bn )−[(1+(n−2)θ)w1 −θ
Pn−1
i=2
wi −θw
bn ](dp1 (w1 )/dw1 )
(1+(n−1)θ)(1−θ)
the following condition must hold in a symmetric equilibrium:
1 − θ − p(B(w
bn )) + θp(w
bn ) + [θw
bn − B(w
bn )]
dp(B(w
bn ))
= 0.
dw
Using symmetry, the following condition must also hold
1 − θ + (1 + (n − 2)θ)w
bn − 2(1 + (n − 2)θ)p(w
bn ) + θ(n − 1)p(B(w
bn )) = 0.
Let us seek for linear symmetric PBE sustained by wary beliefs, so let us
15
,
conjecture that p(w) = δ + λw and B(w) = µ + τ w for some parameters δ, λ, µ
and τ to be determined. The following should hold:
(1 − θ)(1 − δ) − 2λµ + 2λ(θ − τ )w
bn = 0
and
1 − θ + θ(n − 1)λµ − (2 + (n − 3)θ)δ + {[1 + (n − 2)θ](1 − 2λ) + λθ(n − 1)τ }w
bn = 0.
Since these two conditions should be satisfied for all w
bn , we must have
(1 − θ)(1 − δ) − 2λµ = 0,
2λ(θ − τ ) = 0,
1 − θ + θ(n − 1)λµ − (2 + (n − 3)θ)δ = 0
and
[1 + (n − 2)θ](1 − 2λ) + λθ(n − 1)τ = 0,
so
δ=
(1 − θ)[2 + θ(n − 1)]
,
4(1 + (n − 2)θ) − (n − 1)(1 + θ)θ
λ=
µ=
1 + (n − 2)θ
,
2[1 + (n − 2)θ] − (n − 1)θ2
(1 − θ)[2(1 + (n − 2)θ) − (n − 1)θ2 ]
4(1 + (n − 2)θ) − (n − 1)(1 + θ)θ
and
τ = θ.
In equilibrium, it must hold that beliefs are fulfilled, so w∗ = B(w∗ ) implies
that
2(1 + (n − 2)θ) − (n − 1)θ2
w∗ =
,
4(1 + (n − 2)θ) − (n − 1)(1 + θ)θ
2−θ
. Also, firm 1 should find it optimal to choose p0 = p∗0 and
3−θ
wi = w∗ for all i ∈ {1, . . . , n}, so (w∗ , . . . , w∗ , p∗0 ) ∈ arg maxπ0 (w1 , . . . , wn , p0 ).
so limn→∞ w∗ =
{wi }n
i=1 ,p0
Hence,
p∗0 =
n[1 + (n − 2)θ]2 (1 − w∗ )2
nw∗ (1 − δ − λw∗ )
−
.
4[1 + (n − 1)θ][2 + (n − 3)θ]2
2[1 + (n − 1)θ]
16
It readily follows that
p∗0 = −
n{3[1 + (n − 1)θ]2 − 8θ[1 + (n − 1)θ] + θ(2 + 3θ) − 2(n − 1)(n − 2)θ3 }
,
4[1 + (n − 1)θ][4(1 + (n − 2)θ) − (n − 1)(1 + θ)θ]2
so limn→∞ p∗0 = −
3 − 2θ
, which means that limn→∞ p∗0 < 0 if and only if
4θ(3 − θ)2
θ > 0. Also,
x∗ =
n{5[1 + (n − 1)θ]2 − 12θ[1 + (n − 1)θ] + θ(2 + 5θ) − 2(n − 1)(n − 2)θ3 }
,
4[1 + (n − 1)θ][4(1 + (n − 2)θ) − (n − 1)(1 + θ)θ]2
so x∗ > 0 if either i t holds that θ > 0 or it holds that θ < 0 together with
5 − 2θ
n < (θ − 1)/θ.4 As a result, limn→∞ x∗ =
, which is positive if and only
4θ(3 − θ)2
2−θ
if θ > 0. Also, limn→∞ p∗i =
, so limn→∞ (p∗i −w∗ ) = limn→∞ qi∗ = 0 regardless
3−θ
of θ > 0: firm 0 induces monopolistic competition if there are many complementors
5 − 2θ 2
that sell imperfect substitutes of each other. Finally, limn→∞ π0∗ = [
]
4θ(3 − θ)2
and limn→∞ πi∗ = 0 for firm i ∈ {1, . . . , n}, with limn→∞ nπi∗ = 0 as well.
4
Resolution of the model: Fixed fees when entry is endogenous
Once we have solved the model in which firm 0 charges private royalty fees to firm
i ∈ {1, . . . , n}, we will examine the cases in which firm 0 charges fixed fees to firm
i ∈ {1, . . . , n} with the aim of deciding how many firms should be active.
Letting n = 2,5 suppose that firm 0 has to determine whether both firms 1 and
2 should be active, or just one of them, by simply charging a fixed fee denoted by
fi . Once they have accepted/rejected the contracts offered by firm 0, we assume
that firms 1 and 2 always observe against how many firms they are competing.
Such firms observe p0 also at that point in time. Let m ∈ {1, 2} denote the number
4
Since
f (n) ≡ 5[1 + (n − 1)θ]2 − 12θ[1 + (n − 1)θ] + θ(2 + 5θ) − 2(n − 1)(n − 2)θ3
is strictly convex with f (1) = 5(1 − θ)2 > 0 and df (1)/dn = 2θ(1 − θ)(5 − θ) > 0, it holds that
θ < 0 requires that n < (θ − 1)/θ in order for x∗ > 0 to be positive, whereas θ > 0 requires that
n > (θ − 1)/θ in order for x∗ > 0 to be positive. Note that (θ − 1)/θ < 0 whenever θ > 0.
5
Probably solving n = 3 will capture all the essential ingredients of the general case, but this
is left aside for now.
17
of firms that are induced to enter by firm 0 in equilibrium. In particular, suppose
that firm 0 induces entry by firms 1, . . . , m (this is without loss of generality given
the symmetry of firms), where m ∈ {1, 2}. We first consider the case in which
m = 1 and we then turn to the case in which m = 2.
Let us try to sustain an equilibrium in which m = 1 (i.e., f2 = ∞, say). If
firm 1 is the only active firm, then its demand function is Q11 (p1 ) = x(p0 )(1 − p1 )
and therefore finds it optimal to charge p11 = 1/2 so as to make profit equal to
π11 = x1 (p0 )/4. If consumers expect a monopoly, then they expect a price equal
to p11 , so that the utility of a consumer at distance x ≥ 0 from firm 0 equals
Ux1 (p0 ) = 1/8 − x − p0 , whence it follows that consumer demand is
x1 (p0 ) =
1
− p0 .
8
Firm 1 worries that it pays a high fixed fee in the expectation that it will be in
monopoly when firm 0 has instead secretly allowed firm 2 to enter. A wary firm
such as 1 will only be convinced to enter as a monopolist by a fixed fee f ≥ 0
that makes deviations by firm 0 unprofitable. Optimality then requires that firm
0 sets a price of p10 = 1/16 to consumers so as to earn π01 = f + 1/256. It remains
to pin down the values of f , if they exist.
Therefore, if firm 1 believes that it is the sole firm being invited to enter, then
it should observe f ; any other offer will make it believe either that it is not invited
to enter or that firm 2 is being invited to enter as well. There are two possible
types of deviations: (a) those in which firm 1 observes an offer such as the one it
expects in equilibrium, but firm 2 does not; and (b) those in which neither firm
observes an offer such as the one it expects in equilibrium. Recalling that beliefs
are wary, it is clear in case (a) that firm 1 accepts, whereas firm 2 is induced to
accept in the (correct) belief that it will be active together with firm 1; in case
(b), it is clear too that both firms must be induced to accept in the (correct) belief
that both will be active.
Consider case (a). Firm 0 must be choosing p0 , f1 and f2 to maximize
π
b20 (p0 , f1 , f2 ) = x1 (p0 )p0 + f1 + f2
subject to the constraints that f1 = f and f2 ≤
18
(1 − θ)x1 (p0 )
. Therefore, firm
(1 + θ)(2 − θ)2
0 must choose
pb20 =
and
(1 + θ)(2 − θ)2 − 8(1 − θ)
16(1 + θ)(2 − θ)2
(1 − θ)[(1 + θ)(2 − θ)2 + 8(1 − θ)]
fb22 =
16(1 + θ)2 (2 − θ)4
so as to earn
π
b20 = (
(1 + θ)(2 − θ)2 + 8(1 − θ) 2
3
+ f.
) +
2
16(1 + θ)(2 − θ)
64
Since π
b20 > π01 for all θ ∈ (−1, 1), it follows that there can exist no equilibrium
in which only one firm enters. (Even though this is firm 0’s optimal deviation,
it suffices to consider deviations that involve keeping p0 fixed and some positive
but small f2 , since firm 2 will accept them even if it expects that it will be active
together with firm 1, and this clearly makes firm 0 better off.)
So let us now try to sustain an equilibrium in which m = 2. In this case, firm
1−θ
i ∈ {1, 2} anticipates charging a price equal to p2i =
, so it foresees making
2−θ
the following profit after paying the fixed fee:
πi2 (p0 ) =
where
x2 (p0 ) =
(1 − θ)x2 (p0 )
,
(1 + θ)(2 − θ)2
1
− p0
(1 + θ) (2 − θ)2
because consumers believe that firm 0 is allowing both firms 1 and 2 to enter.
Therefore, firm 0 must be choosing p0 , f1 and f2 to maximize
π02 (p0 , f1 , f2 ) = x2 (p0 )p0 + f1 + f2
subject to the constraints that fi ≤ πi2 (p0 ) for all i ∈ {1, 2}. It follows that firm
0 must choose a price
2θ − 1
p20 =
2 (1 + θ) (2 − θ)2
and charge a fixed fee of
f2 =
(1 − θ)(3 − 2θ)
2 (1 + θ)2 (2 − θ)4
19
so as to earn
π02 =
(3 − 2θ)2
.
4(1 + θ)2 (2 − θ)4
In order for this to constitute an equilibrium, firm 0 should have no incentive
to restrict entry or try to fool firm 1 into believing that it is the sole entrant when
it is not. Upon observing an unexpected entry offer, firms 1 and 2 must believe
either that firm 0 intends to exclude one of them or that it is trying to trick them
into accepting an offer in the belief that they will be alone. We proceed to show
that wary firms will not be fall into any of these beliefs.
The first aspect to notice regarding the formation of wary beliefs is that beliefs
are not a function of the exact fee that is observed off the equilibrium path. To see
this, suppose that firm 1 receives an unexpected offer of f10 6= f1∗ . If firm 1 believes
that firm 0 is excluding firm 2, then firm 0 would be earning f10 +maxp0 {p0 x2 (p0 )},
where
1
− p0 .
x2 (p0 ) =
(1 + θ)(2 − θ)2
If firm 1 conjectured that firm 0 is not excluding firm 2, then it would believe that
firm 0 must be earning f10 + maxf2 ,p0 {f2 + p0 x2 (p0 )} subject to the constraint that
f2 does not exceed what firm 1 conjectures that firm 2 could earn if firm 2 believed
it is alone or if instead firm 2 believed it is active with 1. When comparing this
profit with f10 + maxp0 {p0 x2 (p0 )}, it is immediate that the comparison does not
depend on f10 . As a result, wary beliefs do not depend on f10 .
In what follows, let Π0 denote the profit attained by firm 0 when it controls
f2 and p0 to maximize f2 + p0 x2 (p0 ) given that f2 cannot exceed what firm 1
conjectures that firm 2 could earn if firm 2 believed it is alone. Similarly, let
Π00 denote the profit attained by firm 0 when it controls f2 and p0 to maximize
f2 +p0 x2 (p0 ) given that f2 cannot exceed what firm 1 conjectures that firm 2 could
earn if firm 2 believed it is not alone. Finally, let Π000 denote the profit attained
by firm 0 when it controls p0 to maximize p0 x2 (p0 ).
Recalling that firms act symmetrically upon observing an unexpected fixed
fee in a symmetric equilibrium, note now that no firm can believe that Π000 >
max(Π0 , Π00 ) upon observing an unexpected fee. Otherwise, if firm 1 observed
such a fee, it would conjecture that f20 6= f2∗ and would conclude that firm 2 must
be entering in the belief that it is alone, thus contradicting its belief. As a result,
wary beliefs require that either Π0 > max(Π00 , Π000 ) or Π00 ≥ max(Π0 , Π000 ). Suppose
now that a firm believed that Π0 > max(Π00 , Π000 ) when observing an unexpected
20
fixed fee. If firm 1 observed such a fee, it would believe that firm 2 has been made
an offer f20 6= f2∗ (otherwise, firm 1 would not form the conjecture that firm 2 holds
the belief that it is alone). As a result, both firms would conclude that the belief
that the other firm is excluded from the market has no basis, which would again
result in an inconsistency. It follows that a firm that observes an unexpected fixed
fee believes that Π00 ≥ max(Π0 , Π000 ), that is, forming wary beliefs requires that no
firm believes that the other firm is being excluded from the market.
Given these beliefs formed by firms 1 and 2 off the equilibrium path, it is
clear that firm 0 cannot do better by deviating from what equilibrium behavior
prescribes. Note that f 2 > 0, whereas p20 < 0 if and only if θ < 1/2. Also, π02
decreases with θ.
Under endogenous entry with firm 0 charging solely fixed fees, the unique
equilibrium is the same that would obtain if firm 0 were committed to letting
both firms enter. Such result arises here because firm 0 cannot credibly convince
firm i ∈ {1, 2} that entry will be restricted: such a firm would be (correctly) wary
that firm 0 would hold it up and secretly allow for more firms to enter, so it is not
willing to pay the fixed fee that firm 0 demands (this holds even if firm 0 could
commit to choosing a certain p0 in the contract it signs with firm 1). This result
driven by the lack of firm 0’s commitment not to hold up firm i ∈ {1, 2} should be
more or less easy to extend to any n ≥ 2 (one can then take the asymptotic limit
and examine monopolistic competition). This hold-up problem is also likely to
arise when royalty fees are used, as well as both fixed fees and royalty fees. Wary
beliefs held by consumers may act as a commitment device for firm 0: if upon
observing a different price from the one they expect, they (correctly) anticipate
firm 0’s intentions to cheat everybody, they will probably decrease the profitability
of deviations, thus allowing firm 0 to commit more easily.
5
Robustness check
We show in this section that the comparison between public information and
secrecy does not depend on whether a fixed fee is charged by firm 0 given that it
charges a royalty fee to each firm. So suppose that firm 0 offers a secret contract
(wi , fi ) to firm i ∈ {1, 2}. Firm 0’s total profit if it charges p0 and makes a private
21
offer of (w1 , f1 ) and (w2 , f2 ) to firms 1 and 2, respectively, is therefore as follows:
π0 (w1 , f1 , w2 , f2 , p0 ) = x(p0 )[p0 + w1 q1 (p1 (w1 ), p2 (w2 )) + w2 q2 (p1 (w1 ), p2 (w2 ))] + f1 + f2 .
x(p0 )[p0 + wq1 (p1 (w), p2 (w2 )) + w2 q2 (p1 (w), p2 (w2 )) + (p1 (w) − w)q1 (p1 (w),
In order for firm 2 to form wary beliefs, the inference made by firm 2 about firm
1’s contract upon observing a price of p0 and an offer of (w2 , f2 ) must be such
that B(w2 ) maximizes π0 (w, f, w2 , f2 , p0 ) with respect to w and f subject to the
constraint that f ≤ (p1 (w) − w)x(p0 )q1 (p1 (w), p2 (B(w))).
Taking into account that the constraint must bind at the optimum, the firstorder condition yields that
0=
∂π0 (w, f, w2 , f2 , p0 )
=
∂w
∂q1 (p1 (w), p2 (w2 )) dp1 (w
∂q2 (p1 (w), p2 (w2 ))
+w
]
∂p1
∂p1
dw
∂q1 (p1 (w), p2 (w2 )) dp1 (w) ∂q1 (p1 (w), p2 (B(w))) dp2 (B
+(p1 (w) − w)[
+
∂p1
dw
∂p2
dB
dp1 (w)
+q1 (p1 (w), p2 (B(w)))(
− 1).
dw
q1 (p1 (w), p2 (w2 )) + [w2
Using the fact that q1 (p1 (w), p2 (B(w))) = (p1 (w) − w)/(1 − θ2 ) (see (1)) when
evaluating it at w = B(w2 ), we have that the following condition must hold:
dp1 (B(w2 ))
dp2 (B(w2 )) dB(B(w
)−[p1 (B(w2 ))−B(w2 )][1−θ(
dw
dw
dw
(4)
Let us seek for linear symmetric PBE sustained by wary beliefs, so let us
conjecture that pi (w) = Θ + Σw (i ∈ {1, 2}) and B(w) = Γ + Φw for some
parameters Θ, Σ, Γ and Φ to be determined. Using conditions (1) and (3) yield
that the following should hold:
1−θ−p1 (B(w2 ))+θp2 (w2 )+(θw2 −B(w2 ))(
(1−θ)(1−Θ)−2ΣΓ+(Θ+ΣΓ−Γ)(θΦΣ−1)+[2Σ(θ−Φ)+Φ(Σ−1)(θΦΣ−1)]w2 = 0
and
1 − θ + θΣΓ − (2 − θ)Θ + (1 − 2Σ + θΣΦ)w2 = 0.
Since these two conditions should be satisfied for all w2 , we must have
(1 − θ)(1 − Θ) − 2ΣΓ + (Θ + ΣΓ − Γ)(θΦΣ − 1) = 0,
22
2Σ(θ − Φ) + Φ(Σ − 1)(θΦΣ − 1) = 0,
1 − θ + θΣΓ − (2 − θ)Θ = 0
and
1 − 2Σ + θΣΦ = 0.
Rey and Verg´e (2004) show that there exists a unique tuple (Θ, Σ, Γ, Φ) that solves
these equations and the required second-order conditions for firm 0’s maximization program, with the equilibrium royalty fee w∗ such that w∗ = Γ/(1 − Φ).
Also, firm 1 should find it optimal to choose p0 = p∗0 and w1 = w2 = w∗ , so
(w∗ , w∗ , p∗0 ) ∈ arg max{π0 (w1 , w2 , p0 )}. Note that the optimal choices of w1 and
w1 ,w2 ,p0
w2 do not depend on the choice of p0 , so firm 0 can maximize with respect to
w1 and w2 ignoring the value of p0 ; the analysis above leadind to expression (3)
shows that private offers are chosen optimally, since second-order conditions are
satisfied as Rey and Verg´e (2004) show. As for the optimal choice of p0 given that
firm i ∈ {1, 2} receives an offer equal to w∗ , we need that
x(p0 ) + [p0 + 2p1 (w∗ )q1 (p1 (w∗ ), p2 (w∗ ))]
so
p∗0 =
dx(p0 )
= 0,
dp0
2(Θ + Σw∗ )(1 − Θ − Σw∗ )
(1 − w∗ )2
−
.
2(1 + θ)(2 − θ)2
2(1 + θ)
In what follows, we numerically solve the problem, which yields the following
solution:6
θ = −0.9999 → Θ = 0.50006247, Σ = 0.99980009, Γ = 0.49991253, Φ =
−0.99990004, w∗ = 0.24996876, p∗i = 0.74998126, p∗0 = −1562.5468, x∗ = 2187.6405, π ∗ =
4785771.2, CS ∗ = 2392885.6, W ∗ = 7178656.8
θ = −0.99 → Θ = 0.50593158, Σ = 0.98084115, Γ = 0.49150203, Φ = −0.99037063, w∗ =
0.24693995, p∗i = 0.74814044, p∗0 = −15.670971, x∗ = 22.014294, π ∗ = 484.62913, CS ∗ =
242.31457, W ∗ = 726.9437
θ = −0.90 → Θ = 0.54160281, Σ = 0.85500667, Γ = 0.42800426, Φ = −0.92268721, w∗ =
0.22260733, p∗i = 0.73193356, p∗0 = −1.6027702, x∗ = 2.3213664, π ∗ = 5.3887417, CS ∗ =
2.6943709, W ∗ = 8.0831126
6
Note that x∗ =
∗
∗
(1 − w∗ )2
∗
∗
∗ pi (1 − pi )2
−
p
,
π
=
x
(
+ p∗0 ) and CS ∗ =
0
(1 + θ) (2 − θ)2
1+θ
1
(1 − w∗ )2
(
− p∗0 )2 .
2 (1 + θ) (2 − θ)2
23
θ = −0.80 → Θ = 0.56243901, Σ = 0.76365517, Γ = 0.36857404, Φ = −0.86313554, w∗ =
0.19782460, p∗i = 0.71350879, p∗0 = −0.81687694, x∗ = 1.227263, π ∗ = 1.5061745, CS ∗ =
0.75308726, W ∗ = 2.2592618
θ = −0.70 → Θ = 0.57320217, Σ = 0.69665596, Γ = 0.3124193, Φ = −0.80653034, w∗ =
0.17293886, p∗i = 0.69368106, p∗0 = −0.55190667, x∗ = 0.86467765, π ∗ = 0.74766743, CS ∗ =
0.37383372, W ∗ = 1.1215012
θ = −0.60 → Θ = 0.57721690, Σ = 0.64431268, Γ = 0.25669747, Φ = −0.74659753, w∗ =
0.14697002, p∗i = 0.67191155, p∗0 = −0.41656351, x∗ = 0.68566859, π ∗ = 0.47014141, CS ∗ =
0.23507071, W ∗ = 0.70521212
θ = −0.50 → Θ = 0.5758547, Σ = 0.60211272, Γ = 0.20050479, Φ = −0.67836283, w∗ =
0.11946451, p∗i = 0.6477858, p∗0 = −0.33226387, x∗ = 0.58037355, π ∗ = 0.33683346, CS ∗ =
0.16841673, W ∗ = 0.50525019
θ = −0.40 → Θ = 0.56967993, Σ = 0.56770330, Γ = 0.14430145, Φ = −0.5962912, w∗ =
0.090397949, p∗i = 0.62099914, p∗0 = −0.27256397, x∗ = 0.51196672, π ∗ = 0.26210992, CS ∗ =
0.13105496, W ∗ = 0.39316488
θ = −0.30 → Θ = 0.55885804, Σ = 0.53996073, Γ = 0.090293664, Φ =
−0.49337825, w∗ = 0.060462689, p∗i = 0.59150552, p∗0 = −0.22598981, x∗ = 0.46437230, π ∗ =
0.21564163, CS ∗ = 0.10782082, W ∗ = 0.32346245
θ = −0.20 → Θ = 0.54341286, Σ = 0.51874340, Γ = 0.043294031, Φ =
−0.36132309, w∗ = 0.031802907, p∗i = 0.55991041, p∗0 = −0.18696415, x∗ = 0.42906271, π ∗ =
0.18409481, CS ∗ = 0.092047403, W ∗ = 0.27614221
θ = −0.10 → Θ = 0.52354620, Σ = 0.50490649, Γ = 0.010952235, Φ =
−0.19435259, w∗ = 0.0091700182, p∗i = 0.5281762, p∗0 = −0.15321917, x∗ = 0.40057217, π ∗ =
0.16045806, CS ∗ = 0.080229031, W ∗ = 0.24068709
θ = −0.01 → Θ = 0.50248731, Σ = 0.50004999, Γ = 0.00010145142, Φ =
−0.019994004, w∗ = 0.000099462761, p∗i = 0.50253705, p∗0 = −0.12753421, x∗ =
0.37750329, π ∗ = 0.14250873, CS ∗ = 0.071254367, W ∗ = 0.21376310
θ = 0.01 → Θ = 0.49748768, Σ = 0.50004999, Γ = 0.000098453644, Φ =
0.019994004, w∗ = 0.00010046229, p∗i = 0.49753792, p∗0 = −0.12253452, x∗ = 0.37250298, π ∗ =
0.13875847, CS ∗ = 0.069379234, W ∗ = 0.2081377
θ = 0.1 → Θ = 0.47390099, Σ = 0.50490649, Γ = 0.0081575902, Φ = 0.19435259, w∗ =
0.010125509, p∗i = 0.47901343, p∗0 = −0.10349642, x∗ = 0.35024825, π ∗ = 0.12267383, CS ∗ =
0.061336918, W ∗ = 0.18401075
θ = 0.2 → Θ = 0.44586530, Σ = 0.51874340, Γ = 0.024651293, Φ = 0.36132309, w∗ =
0.038597439, p∗i = 0.46588747, p∗0 = −0.088498530, x∗ = 0.32622870, π ∗ = 0.10642516, CS ∗ =
24
0.053212581, W ∗ = 0.15963774
θ = 0.3 → Θ = 0.41563117, Σ = 0.53996073, Γ = 0.040577002, Φ = 0.49337825, w∗ =
0.080093289, p∗i = 0.4588784, p∗0 = −0.078386711, x∗ = 0.30362716, π ∗ = 0.092189451, CS ∗ =
0.046094725, W ∗ = 0.13828418
θ = 0.4 → Θ = 0.38243328, Σ = 0.56770330, Γ = 0.052374402, Φ = 0.5962912, w∗ =
0.12973312, p∗i = 0.4560832, p∗0 = −0.071534694, x∗ = 0.28285290, π ∗ = 0.080005763, CS ∗ =
0.040002881, W ∗ = 0.12000864
θ = 0.5 → Θ = 0.34523836, Σ = 0.60211272, Γ = 0.059316254, Φ = 0.67836283, w∗ =
0.18441977, p∗i = 0.45627985, p∗0 = −0.066848497, x∗ = 0.26393623, π ∗ = 0.069662335, CS ∗ =
0.034831168, W ∗ = 0.1044935
θ = 0.6 → Θ = 0.30267626, Σ = 0.64431268, Γ = 0.061426593, Φ = 0.74659753, w∗ =
0.24240724, p∗i = 0.45886232, p∗0 = −0.063682934, x∗ = 0.24670168, π ∗ = 0.060861719, CS ∗ =
0.030430860, W ∗ = 0.091292579
θ = 0.7 → Θ = 0.25274407, Σ = 0.69665596, Γ = 0.058580442, Φ = 0.80653034, w∗ =
0.30278878, p∗i = 0.46368368, p∗0 = −0.061684427, x∗ = 0.2308816, π ∗ = 0.053306314, CS ∗ =
0.026653157, W ∗ = 0.079959471
θ = 0.8 → Θ = 0.19211663, Σ = 0.76365517, Γ = 0.049989759, Φ = 0.86313554, w∗ =
0.36525011, p∗i = 0.47104176, p∗0 = −0.060701672, x∗ = 0.21614435, π ∗ = 0.04671838, CS ∗ =
0.02335919, W ∗ = 0.07007757
θ = 0.9 → Θ = 0.11417480, Σ = 0.85500667, Γ = 0.033258060, Φ = 0.92268721, w∗ =
0.43017540, p∗i = 0.48197764, p∗0 = −0.060790321, x∗ = 0.20344345, π ∗ = 0.040812362, CS ∗ =
0.020407186, W ∗ = 0.061219548
θ = 0.99 → Θ = 0.014461238, Σ = 0.98084115, Γ = 0.0047432494, Φ =
0.99037063, w∗ = 0.49258149, p∗i = 0.49760543, p∗0 = −0.062208099, x∗ = 0.18904242, π ∗ =
0.035737036, CS ∗ = 0.017868518, W ∗ = 0.053605554
θ = 0.9999 → Θ = 0.00014994254, Σ = 0.99980009, Γ = 0.000049972520, Φ =
0.99990004, w∗ = 0.49992517, p∗i = 0.49997517, p∗0 = −0.062496918, x∗ = 0.18751558, π ∗ =
0.035162093, CS ∗ = 0.017581047, W ∗ = 0.05274314
This outcome is to be compared with the one that arises when contracts are
observable to everybody. In the absence of any private information, firm 0 chooses
p0 , w1 and w2 to maximize
2(1 − θ)[1 − p1 (w1 , w2 ) − p2 (w1 , w2 ) + p1 (w1 , w2 )p2 (w1 , w2 )] + [p1 (w1 , w2 ) − p2 (w1 , w2 )]2
[
− p0 ] ×
2 (1 − θ2 )
[p0 + p1 (w1 , w2 )q1 (w1 , w2 ) + p2 (w1 , w2 )q2 (w1 , w2 )],
25
where
q1 (w1 , w2 ) =
(2 + θ)(1 − θ) − (2 − θ2 )w1 + θw2
,
(1 − θ2 )(4 − θ2 )
q2 (w1 , w2 ) =
(2 + θ)(1 − θ) − (2 − θ2 )w2 + θw1
,
(1 − θ2 )(4 − θ2 )
p1 (w1 , w2 ) =
(2 + θ)(1 − θ) + 2w1 + θw2
(2 − θ)(2 + θ)
p2 (w1 , w2 ) =
(2 + θ)(1 − θ) + 2w2 + θw1
.
(2 − θ)(2 + θ)
and
The first-order conditions wrt w1 and p0 after making use of the fact that
w1 = w2 are:
3θ − 4p0 − 6wθ + 12wp0 − 4θp0 + 3w2 θ + 3θ2 p0 + 2θ3 p0 − θ4 p0 + 6w2 − 4w3 − 9wθ2 p0 + 3wθ3 p0 − 2
(θ + 1)2 (θ − 2)4
3w2 − 2wθ − 2w + 2θ − 1 − 2(1 + θ)(2 − θ)2 p0
(1 + θ) (θ − 2)2
Solutions to the first-order conditions are w1 = w2 = 1, w1 = w2 = −(1−θ) < 0
and w1 = w2 = −(3 − 2θ) < 0, so we must have that w∗ = −(1 − θ) < 0 and
1
1
, so that firms price at 0. Firm 0 makes sales of
and earns
p∗0 =
2(1 + θ)
2(1 + θ)
1
1
3
. Consumer surplus equals
, so total welfare equals
.
2
2
4(1 + θ)
2(1 + θ)
4(1 + θ)2
Irrespective of the value taken by θ, secrecy can be easily shown to result in lower
profits for firm 0 and in a lower consumer surplus than when private information
is inexistent. Therefore, public information leads to greater welfare for each party
than private information when firm 0 sets both royalty fees and fixed fees.
FIXED FEES AND ROYALTIES UNDER ENDOGENOUS ENTRY
We assume that firms always observe against how many firms they are competing, but they can never observe each other’s royalty fee. Assume that n = 2,
and firm 0 has to determine whether both firms 1 and 2 should be active, or just
one of them.
If firm i ∈ {1, 2} is the only active firm and is charged a royalty fee of wi , then
its demand function is Qm (pi ) = x(p0 )(1 − pi ) and therefore finds it optimal to
charge pi = (1 + wi )/2 so as to make profit equal to πim = x(p0 )(1 − wi )2 /4. We
now turn to the first period. If firm 1 believes that it is the sole firm being invited
26
to enter, then it should observe wm (p0 ) = 0 and f m (p0 ) = x(p0 )/4, since this is the
only way to maximize firm 0’s profit, π0m (p0 , f, w) = x(p0 )(p0 + w(1 − w)/2) + f ,
with respect to w and f subject to the constraint that x(p0 )(1−w)2 /4−f ≥ 0. To
pin down the optimal value of p0 , we need to examine the consumers’ problem. If
they expect a monopoly, then they expect a price equal to 1/2, so that the utility
of a consumer at distance x ≥ 0 from firm 0 equals Uxm (p0 ) = 1/8 − x − p0 . As a
result, maximizing
1
1
π0m (p0 , f m (p0 ), wm (p0 )) = ( − p0 )(p0 + )
8
4
with respect to p0 yields that pm
0 = −1/16. Therefore, if firm 1 believes that it is
the sole firm being invited to enter, then it should observe wm = 0 and f m = 3/64;
any other offer will make it believe either that it is not invited to enter or that
firm 2 is being invited to enter as well.
Let us try to sustain an equilibrium in which only firm 1 is induced to enter.
We just showed that, if such an equilibrium exists, it must involve pm
0 = −1/16,
wm = 0 and f m = 3/64, so that firm 0 earns π0m = (3/16)2 . There are two possible
types of deviations: (a) those in which firm 1 observes an offer such as the one it
expects in equilibrium, but firm 2 does not and; (b) those in which neither firm
observes an offer such as the one it expects in equilibrium. In case (a), it is clear
that firm 1 accepts, whereas firm 2 is induced to accept in the (correct) belief that
it will be active together with firm 1. In case (b), it is clear too that both firms
must be induced to accept in the (correct) belief that both will be active.
Consider case (b) and focus on symmetric continuation equilibria once firm
0 has deviated (noting that it is common knowledge that firm 0 has deviated).7
Symmetry simply implies that firms use the same belief function when forming
conjectures about the other firm’s royalty fee. Firm 1 forms a belief B(w
bi ) about
the other firm’s royalty fee, so its first-order condition becomes
1−θ+w
b1 − 2p1 (w
b1 ) + θp2 (B(w
b1 )) = 0.
Firm 1 believes it makes the following profit after paying the fixed fee:
π1d (p0 , w
b1 ) = x(p0 )(p1 (w
b1 ) − w
b1 )q1 (p1 (w
b1 ), p2 (B(w
b1 ))).
7
Such symmetric behavior is probably without loss of generality.
27
Using symmetry, a similar result holds for firm 2:
b2 ) = x(p0 )(p2 (w
b2 ) − w
b2 )q2 (p1 (B(w
b2 )), p2 (w
b2 )).
π2d (p0 , w
Let us now examine what happens with firm 0’s pricing behavior if it induces
entry by both firms, where the latter may have paid the fixed fee in the belief
that they would be alone.
Regarding the first period, let 1i be an indicator function that takes value of
1 if firm 0 induces entry by firm i ∈ {1, 2}, and takes value of 0 otherwise. Upon
observing p0 , w
bi and fbi , firm i forms a belief A(p0 , w
bi , fbi ) ∈ {0, 1} about the value
taken by 1j , as well
F (p0 , w
bi , fbi ) about the fixed fee charged to the other firm as well as a belief
B(w
bi ) about the royalty fee that the other firm will pay if it becomes active.
Conditional on firm 1 believing that firm 0 is inviting it to enter, it holds if firm 1
believes that firm 2 is invited to enter as well that A(p0 , w
b1 , fb1 ), F (p0 , w
b1 , fb1 ) and
B(w
b1 ) maximize
x(p0 )[p0 + w
b1 q1 (p1 (w
b1 ), p2 (w)) + wq2 (p1 (w
b1 ), p2 (w))] + fb1 + f
with respect to 12 , f and w subject to the constraints that
(F (p0 , w
bi , fbi ), B(w
bi )) ∈ arg max{x(p0 )[p0 +]}
f,w,1j
If firm 1 believes that it is the sole firm being invited to enter, then it should
observe w∗ (p0 ) = 0 and f (p0 ) = x(p0 )/4, since this is the only way to maximize
firm 0’s profit, π0m (p0 , f, w) = x(p0 )(p0 + w(1 − w)/2) + f , with respect to w
and f subject to the constraint that x(p0 )(1 − w)2 /4 − f ≥ 0. To pin down
the optimal value of p0 , we need to examine the consumers’ problem. If they
expect a monopoly, then they expect a price equal to 1/2, so that the utility of
a consumer at distance x ≥ 0 from firm 0 equals Uxm (p0 ) = 1/8 − x − p0 . As a
result, maximizing
1
1
π0m (p0 , f (p∗0 ), w∗ (p0 )) = ( − p0 )(p0 + )
8
4
with respect to p0 yields that p∗0 = −1/16. Therefore, if firm 1 believes that it is
the sole firm being invited to enter, then it should observe w∗ = 0 and f ∗ = 3/64;
28
any other offer will make it believe that it is either not invited to enter or firm 2
is being invited to enter as well.
Suppose that firm 1 ob
Suppose that a firm has paid a fixed fee equal to Fb. Because firm 0 must
behave optimally, its belief about what the other entrants are paying must be
such that π(n) = B(Fb), so any firm must believe that n
b ≡ π −1 (B(Fb)) entrants
are active upon observing Fb. Given this belief,
so that it believes that n
b firms have entered.
Best way is to assume that there exist N firms, and that those who do not
pay the fixed fee choose to sell nothing by charging an infinite price: π(n).
Firm 0’s total profit if it charges p0 and makes a private offer of Fbi to firm
i ∈ {1, . . . , N } so that n firms accept it is as follows:
π0 (F1 , . . . , FN , p0 ) = x(p0 )p0 +
N
X
Fi 1i (Fi ),
i=1
where 1i (Fi ) = 1 if firm i enters and pays Fi and 1i (Fi ) = 0 otherwise. The
wary beliefs formed by firm 1 about the other firms’ fixed fee upon observing a price of p0 and a fee of Fb1 must be such that (B2 (Fb1 ), . . . , BN (Fb1 )) =
P
arg maxπ0 (Fb1 , . . . , FN , p0 ) subject to the constraint that Fi ≤ π( N
j=1 1j (Bj (Fi ))).
F2 ,...,FN
.
q2
q3
1
q2 + q3 − (q22 + q32 + 2θq2 q3 ) − p2 q2 − p3 q3
2
1 − θ − p2 + θp3
=
1 − θ2
1 − θ − p3 + θp2
=
1 − θ2
x
b(p2 , p3 )q2 (p2 , p3 )(p2 −c2 ), where x
b(p2 , p3 ) =
u
b(p2 ,p3 )−p1
t
and q2 =
1 − θ − p2 + θp3
1 − θ2
(p2 ,p3 )
[b
x(p2 , p3 ) ∂q2∂p
− q22 (p2 , p3 )](p2 − c2 ) + x
b(p2 , p3 )q2 (p2 , p3 )
2
∂q2 (p2 ,p3 )
2
[b
x(p2 , p3 ) ∂p2 − q2 (p2 , p3 )](p2 − c2 ) + x
b(p2 , p3 )q2 (p2 , p3 )
WARY BELIEFS
Suppose consumers believe that firm i ∈ {2, 3} is charged a royalty fee of
C
Bi (p1 ) given that they observe a price of p1 for the platform. Consumers then
29
believe that firm i ∈ {2, 3} chooses pi to maximize (pi − BiC (p1 ))qi (p2 , p3 ). It
follows that consumers believe that firm 2 chooses an equilibrium price
pC
2 (p1 ) =
(2 + θ)(1 − θ) + 2B2C (p1 ) + θB3C (p1 )
,
(2 − θ)(2 + θ)
whereas firm 3 chooses
pC
3 (p1 ) =
(2 + θ)(1 − θ) + 2B3C (p1 ) + θB2C (p1 )
.
(2 − θ)(2 + θ)
The overall utility expected by a consumer located at distance x from firm 1 equals
Ux∗ (p1 , B2C (p1 ), B2C (p1 )) = u∗ (B2C (p1 ), B3C (p1 )) − tx − p1 ,
where
u∗ (B2C (p1 ), B3C (p1 )) =
2
C
C
C
C
C
2(1 − θ)[1 − pC
2 (p1 ) − p3 (p1 ) + p2 (p1 )p3 (p1 )] + (p2 (p1 ) − p3 (p1 ))
.
2 (1 − θ2 )
Consumers act in the belief that firm 1 has chosen offers w
b and w
b0 to maximize
its overall profit:
(B2C (p1 ), B3C (p1 )) = arg max{
(w,
bw
b0 )
+w
b0 [
(2 − θ)(2 + θ)(1 − θ) − (2 + θ)(1 − θ)2 − (2 −
p1 [u∗ (w,
b w
b 0 ) − p1 ]
+ w[
b
t
(1 − θ2 )(2 − θ)(2 + θ)
(2 − θ)(2 + θ)(1 − θ) − (2 + θ)(1 − θ)2 − (2 − θ2 )w
b0 + θ w
b
]}.
2
(1 − θ )(2 − θ)(2 + θ)
(2 + θ)(1 − θ) − 2(2 − θ2 )w
b + 2θw
b0 p1 (4 − 3θ2 )(1 − w)
b − θ3 (1 − w
b0 )
−
=0
(1 − θ2 )(2 − θ)(2 + θ)
t
(1 − θ2 )(2 − θ)2 (2 + θ)2
t(2 − θ)
→ BiC (p1 ) = 1 +
for i ∈ {2, 3}
p1 − 2t(2 − θ)
So upon observing price p1 , firm i ∈ {2, 3} knows that consumers will believe
t(2 − θ)
that each firm has been offered a royalty fee equal to B C (p1 ) = 1 +
,
p1 − 2t(2 − θ)
and a consumer at distance x ∈ [0, 1) from firm 1 expects to attain a utility of
Ux∗ (p1 ) =
t2
− tx − p1 ,
(1 + θ)[p1 − 2t(2 − θ)]2
30
so firm 1 attracts
x(p1 ) =
t
p1
−
2
(1 + θ)[p1 − 2t(2 − θ)]
t
consumers when charging p1 .
In what follows, let p2 (w)
b denote firm 2’s strategy in the second-period subgame if it has observed price p1 and an offer of w,
b and let p3 (w
b0 ) denote firm 3’s
strategy if it has observed price p1 and an offer of w
b0 . Having observed w,
b firm 2
chooses p2 to maximize (p2 − w)x(p
b
b
so its first-order condition
1 )q2 (p2 , p3 (B(w))),
is
1−θ+w
b − 2p2 (w)
b + θp3 (B(w))
b = 0.
Similarly, if firm 3 observes w
b0 , then its first-order condition is
1−θ+w
b0 − 2p3 (w
b0 ) + θp2 (B(w
b0 )) = 0.
PASSIVE BELIEFS
Suppose consumers believe that firm i ∈ {2, 3} is charged a royalty fee of
C
Bi (p1 ) = w∗ given that they observe a price of p1 for the platform. Consumers
then believe that firm i ∈ {2, 3} chooses pi to maximize (pi − w∗ )qi (p2 , p3 ). It
follows that consumers believe that firm i ∈ {2, 3} chooses an equilibrium price
pC
i =
1 − θ + w∗
2−θ
regardless of the price p1 she observes. The overall utility expected by a consumer
located at distance x from firm 1 equals
Ux∗ (p1 ) =
(1 − w∗ )2
− tx − p1 .
(1 + θ) (2 − θ)2
As a result, firm 1 attracts
x(p1 ) =
(1 − w∗ )2
p1
−
2
t (1 + θ) (2 − θ)
t
consumers when charging p1 .
In what follows, let p2 (w)
b denote firm 2’s strategy in the second-period subgame if it has observed price p1 and an offer of w,
b and let p3 (w
b0 ) denote firm 3’s
strategy if it has observed price p1 and an offer of w
b0 . Having observed w,
b firm 2
chooses p2 to maximize (p2 − w)x(p
b
b
so its first-order condition
1 )q2 (p2 , p3 (B(w))),
31
is
1−θ+w
b − 2p2 (w)
b + θp3 (B(w))
b = 0.
Similarly, if firm 3 observes w
b0 , then its first-order condition is
1−θ+w
b0 − 2p3 (w
b0 ) + θp2 (B(w
b0 )) = 0.
In a symmetric equilibrium sustained by wary beliefs, the inference made by
firm 3 upon observing a price of p1 and an offer of w
b0 must be such that
B(w
b0 ) = arg max {x(p1 )[p1 + wq2 (p2 (w), p3 (w
b0 )) + w
b0 q3 (p2 (w), p3 (w
b0 ))]} .
w
Hence, the following must hold:
q2 (p2 (w), p3 (w
b0 )) + [w
b0
∂q2 (p2 (w), p3 (w
b0 )) dp2 (w)
∂q3 (p2 (w), p3 (w
b0 ))
+w
]
= 0.
∂p2
∂p2
dw
Equivalently,
[θw
b0 − B(w
b0 )]
dp2 (B(w
b0 ))
+ 1 − θ − p2 (B(w
b0 )) + θp3 (w
b0 ) = 0.
dw
Conjecture that p2 (w
b0 ) = Θ + Σw
b0 and B(w
b0 ) = Γ + Φw
b0 . Then
(θw − Γ − Φw)Σ + (1 − θ)(1 − Θ) − ΣΓ − ΣΦw + θΣw + 2X(Θ + ΣΓ + ΣΦw − Γ − Φw)(Σ
b
− 1) =
1 − θ + w − 2Θ + θΘ + θΣΓ + θΣΦw − 2Σw =
1 + θΣΦ − 2Σ =
1 − θ − 2Θ + θΘ + θΣΓ =
θ
2
(2 + θ)(1 − θ)
, then Σ =
, Θ =
and Γ =
2
(2 + θ)(2 − θ)
4 − θ − θ2
(1 − θ)(2 − θ)(2 + θ)
(1 − θ)(2 + θ)
. The condition that w∗ = B(w∗ ) implies that w∗ =
.
2
2(4 − θ − θ )
4 − θ − θ2
Still need that firm 1 finds it optimal to choose p1 = p∗1 and w
b=w
b0 = w∗ in a
(linear) symmetric PBE sustained by wary beliefs:
Solution: Φ =
(p∗1 , w∗ , w∗ ) ∈ arg max{x(p1 )[p1 + wq
b 2 (p2 (w),
b p3 (w
b0 )) + w
b0 q3 (p2 (w),
b p3 (w
b0 ))]}.
pb1 ,w,
bw
b0
32
Equivalently:
x0 (p1 )[p1 + 2w∗ q2 (p2 (w∗ ), p3 (w∗ ))] + x(p1 ) = 0,
so
(1 − w∗ )2
2w∗ (1 − Θ − Σw∗ )
−
2(1 + θ)(2 − θ)2
2(1 + θ)
3
2
2(−θ + θ + 4θ − 3)
=
(1 + θ)(2 − θ)2 (4 − θ − θ2 )2
p∗1 =
p∗1
equilibrium profits for firm 1 are
2
(−θ3 + θ2 + 4θ − 5)
4
(θ + 1)2 (−θ3 + θ2 + 6θ − 8)4
2(−θ3 +θ2 +4θ−3)
(1+θ)(2−θ)2 (4−θ−θ2 )2
∂Π1 (w,
b w
b0 )
∂q2 (p2 (w),
b p3 (w
b0 ))
∂q3 (p2 (w),
b p3 (w
b0 )) dp2 (w)
b
= q2 (p2 (w),
b p3 (w
b0 ))+[w
b
+w
b0
]
∂w
b
∂p2
∂p2
dw
b
Since
∂ 2 Π1 (w,
b w
b0 )
2Σ
=−
2
∂w
b
1 − θ2
33
and
∂ 2 Π1 (w,
b w
b0 )
2θΣ
=
,
0
∂ w∂
b w
b
1 − θ2
it follows from the fact that Σ ≥ 0 that
∂ 2 Π1 (w,
b w
b0 )
≤0
∂w
b2
and
(
b w
b0 ) ∂ 2 Π1 (w,
b w
b0 ) ∂ 2 Π1 (w,
b w
b0 )
4Σ2
∂ 2 Π1 (w,
b w
b0 ) ∂ 2 Π1 (w,
−
)(
+
)
=
≥ 0.
∂w
b2
∂ w∂
b w
b0
∂w
b2
∂ w∂
b w
b0
1 − θ2
References
McAfee, R. P. and M. Schwartz (1994): “Opportunism in multilateral vertical contracting: Nondiscrimination, exclusivity, and uniformity,” The American
Economic Review, 210–230.
´ (2004): “Bilateral control with vertical contracts,”
Rey, P. and T. Verge
RAND Journal of Economics, 728–746.
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