M.P. Gos´albez, J.S. D´ıez J. of Economic Behavior Org. 42 2000 385–404 387
The remainder of the paper is organized as follows: Section 2 presents the model. Sections 3–5 solve the stage of efforts, analyze the participation decision and enter into firm 1’s
disclosure decision respectively, for the case of efforts being strategic complements. Section 6 provides a numerical simulation for the case of strategic substitutes. Section 7 analyzes
some welfare implications of the asymmetry of information. Finally, Section 8 concludes. All formal proofs are relegated to the Appendices A and B.
2. The model
We consider two symmetric, risk neutral firms denoted by i = 1, 2. Firm i is located in country i. They have the possibility to undertake a cooperative RD project trying to
obtain an innovation. Let V be the known present value of the innovation for each firm. For example, we can consider that the innovation will be sold in geographically separated
markets, each of them earning those profits.
Firm 1 may receive with probability q a cost subsidy from its government that covers a fraction s of the total RD costs of that firm.
1
On the other hand, country 2 is assumed not to subsidize firm 2.
If firm 1 receives a subsidy s, it may decide whether or not to disclose this private information to its partner. This information is assumed to be verifiable: firm 1 can always
let firm 2 see the official documents granting the subsidy. However, we do not allow firm 1 to reveal in a credible way that it has no subsidy. In other words, firm 1 can always hide
that it has a subsidy but it cannot credibly announce that it has one when this is not true. The announcement of a subsidy without showing the official documents has no signaling
power since it has no cost for firm 1.
Let us denote by superscript R the case in which firm 1 has a subsidy and reveals it, by SN that it has a subsidy but it has not disclosed it, by NN that firm 1 has no subsidy and it
can not credibly disclose that information. On the other hand, only two possibilities exist for firm 2: either firm 1 reveals it has a subsidy, which we denote by R, or not, which we
denote by N. In the last case, firm 2 cannot distinguish between the NN and SN cases, that is, it does not know whether firm 1 has no subsidy or it is hiding that information.
The probability of achieving the innovation depends on the efforts the two firms devote to the project. Let P e
1
, e
2
be the probability function, where e
1
and e
2
are the efforts exerted by firms 1 and 2, respectively, and e
1
, e
2
∈ [0, 1]. That function is assumed to be increasing in both efforts. We will use the above notation to denote the equilibrium
efforts in each of the possible situations regarding the disclosure of the subsidy. Effort is a non-verifiable variable, which implies the existence of a double moral hazard problem
between the partners. Therefore, each firm will decide how much effort to put into the project in order to maximize own profits. We will distinguish two possibilities regarding the
1
We consider cost subsidies, as they are the only type of subsidies used in Europe. On the other hand, in our setting, a subsidy is always profitable for firm 1 and, therefore, this firm has always an incentive to apply for it.
However, in a different framework from ours, Kauko 1996 shows that, depending on the size of the strategic effect, a subsidy could hurt the subsidized firm. In that case, firm 1’s decision on whether or not to apply for the
subsidy should also be taken into account.
388 M.P. Gos´albez, J.S. D´ıez J. of Economic Behavior Org. 42 2000 385–404
strategic relationship between efforts. They can be either strategic complements or strategic substitutes. This characteristic will affect the total expected profits from the venture, the
optimal behavior of the firms and, in particular, firm 1’s decision about whether or not to disclose the subsidy to its partner. We will analyze what the optimal decision is in both cases.
RD costs are captured by the following increasing and convex cost function:
2
ce
i
= e
i 3
3 ,
On the other hand, if firm 1 receives a cost subsidy s, its actual RD costs are given by ce
1
= 1 − s
e
1 3
3 ,
2.1 where s ∈ [0, 1].
Now we can define the firm’s expected profits from the venture contingent upon the existence of the subsidy and firm 1’s disclosure decision. They are given by
E5
NN 1
= P e
NN 1
, e
N 2
V − e
NN 1
3
3 ,
2.2 E5
SN 1
= P e
SN 1
, e
N 2
V − 1 − s
e
SN 1
3
3 ,
2.3 E5
R 1
= P e
R 1
, e
R 2
V − 1 − s
e
R 1
3
3 ,
2.4 E5
N 2
= 1 − ˜ q
P e
NN 1
, e
N 2
V − e
N 2
3
3 + ˜
q P
e
SN 1
, e
N 2
V − e
N 2
3
3 ,
2.5 E5
R 2
= P
e
R 1
, e
R 2
V − e
R 2
3
3 ,
2.6 where the meaning of the superscripts is already known. Parameter ˜
q represents the condi-
tional probability firm 2 assigns to firm 1 being subsidized when the latter has not disclosed that information. In order to define more accurately that conditional probability, let us de-
note by t ∈ 0, 1 the probability that firm 1 discloses the subsidy. In particular, t will be 0 when firm 1 prefers not to disclose, 1 when it prefers to disclose and somewhere between 0
and 1 when it is indifferent between the two options. On the other hand, we have to consider firm 2’s beliefs only if firm 1 does not disclose the subsidy because, otherwise, the asym-
metric information problem would disappear. Effectively, if firm 1 decides not to disclose the subsidy, firm 2 cannot distinguish whether firm 1 does not have a subsidy or simply is
not interested in revealing its existence. In this case, firm 2 updates its beliefs by using the Bayes rule, resulting in the conditional probability ˜
q = 1 − t q1 − tq + 1 − q.
Notice that ˜ q =
1 only if q = 1, that is, if firm 1 receives the subsidy with probability
2
This specific cost function allows us to obtain interior solutions.
M.P. Gos´albez, J.S. D´ıez J. of Economic Behavior Org. 42 2000 385–404 389
one. On the other hand, ˜ q =
0 if t = 1, that is, if firm 1 prefers to disclose the subsidy. Finally, if t = 0 firm 1 prefers not to disclose the subsidy and therefore, not to receive an
announcement is not informative for firm 2, implying ˜ q = q.
Before the project starts, the partners must negotiate how to share the total expected profits from the project. Given that firm 2 only observes whether or not firm 1 discloses the
subsidy, the total expected profits to be shared can only depend on that fact. Therefore, we must calculate the total expected profits as perceived by firm 2, thus taking into account the
subsidy only if firm 1 reveals its existence. Notice that firm 1’s decision on whether or not to disclose the subsidy will depend, among other things, on how that decision affects firm
2’s expectation about the total expected profits and therefore, to the part of those profits each firm gets. Let us denote by
E5
R
= E5
R 1
+ E5
R 2
2.7 and
E5
N
= 1 − ˜ qE5
NN 1
+ ˜ qE5
SN 1
+ E5
N 2
2.8 the total expected profits from the project as perceived by firm 2 in case of disclosure and
non-disclosure, respectively. Therefore, before the project starts the partners negotiate how to share the total expected
profits and we assume that, as a result, firm 1 gets a fraction λ and firm 2 a fraction 1 − λ of those profits, with λ ∈ [0, 1]. We consider λ as an exogenous variable and do not make it
depend on whether or not firm 1 discloses the subsidy. Otherwise, an interesting bargaining problem would arise, but this is outside the scope of this work. We are interested in analyzing
firm 1’s incentives to disclose the subsidy for different values of the exogenous parameter λ
.
3
The timing of the game is as follows: in the first stage, firm 1 receives with probability q a cost subsidy s from its government. If the subsidy is granted, in the second stage it decides
whether to disclose that information to its partner. Next, a transfer payment takes place in order to share the total expected profits as agreed before. Let T
A
denotes the transfer firm 2 pays to firm 1, where A=R if firm 1 disclosed the subsidy and A=N if it did not. A negative
T
A
would imply that it is firm 1 that pays the transfer to firm 2. Now, we can define the actual expected profits for each firm, that is, taking into account
the transfer payment. They are E ¯
5
R 1
= E5
R 1
+ T
R
, 2.9
E ¯ 5
SN 1
= E5
SN 1
+ T
N
, 2.10
E ¯ 5
NN 1
= E5
NN 1
+ T
N
, 2.11
E ¯ 5
R 2
= E5
R 2
− T
R
, 2.12
E ¯ 5
N 2
= E5
N 2
− T
N
. 2.13
3
We interpret λ and 1−λ as the bargaining power of firms 1 and 2, respectively. In that case, the Nash bargaining solution would yield those shares of the total expected profits.
390 M.P. Gos´albez, J.S. D´ıez J. of Economic Behavior Org. 42 2000 385–404
Fig. 1. Timing of the game.
In the fourth stage of the game, each firm chooses simultaneously the effort that maxi- mizes own profits. As a result, and only if the project is successful, they obtain the innovation.
Fig. 1 displays the timing of the game. We will obtain the subgame perfect Bayesian Nash equilibrium of the game, solving by
backward induction in two different situations regarding the relationship between efforts, namely, either strategic complements or strategic substitutes. In Sections 3–5 we explicitly
solve the game for the case of strategic complements. For the case of strategic substitutes, however, and given the complexity of the explicit expressions we obtain, a numerical sim-
ulation illustrates this case in Section 6.
3. Stage of efforts
