M.P. Gos´albez, J.S. D´ıez J. of Economic Behavior Org. 42 2000 385–404 391
and e
∗SN 1
= V
1 − s
12
1 − ˜ q +
˜ q
1 − s
12 13
, e
∗NN 1
=V 1 − ˜
q + ˜
q 1 − s
12 13
, e
∗N 2
= V 1 − ˜
q + ˜
q 1 − s
12 23
, for the cases of disclosure and non-disclosure, respectively. Moreover, they satisfy
e
∗R 2
e
∗N 2
, e
∗R 1
e
∗SN 1
and e
∗SN 1
e
∗NN 1
. Proposition 3.1 shows
4
, first, that the equilibrium efforts are higher when firm 1 discloses the subsidy than when it does not. The reason is that under disclosure firm 2 infers a higher
effort from its partner and therefore, its best response is also to exert a higher effort. As a result, both firms exert higher efforts in equilibrium. Second, that firm 1’s equilibrium effort
is higher whenever it receives a subsidy, regardless of its disclosure decision. Finally, it is easy to check that firm 1’s expected profits increase with the size of the subsidy, because
both its direct effect that reduces firm 1’s RD costs as well as its strategic effect that leads firm 2 to exert a higher effort, positively affect those profits.
The Nash equilibrium efforts calculated in this section allow us to calculate the total expected profits, each firm’s expected profits and the transfer payment. The next section
goes into this question.
4. The transfer
Before choosing efforts, a transfer payment takes place in order the partners share the total expected profits of the project as agreed before, that is, firm 1 gets a fraction λ of those
profits and 1 − λ firm 2. Note that if firm 1 does not disclose it has a subsidy, the transfer is calculated using firm 2’s conditional beliefs ˜
q. Let T
A
represent the transfer firm 2 pays to firm 1 in case A, where A=N, R. It is calculated as the difference between λ times the
total expected profits and firm 1’s expected profits T
A
= λE5
A
− E5
A 1
, A = N, R.
4.1 Observe that the transfer may be either positive or negative, the latter case implying that it
is firm 1 that pays the transfer to firm 2. In the non-disclosure case, firm 2 infers lower total expected profits and firm 1’s expected
profits than the actual ones because they are calculated as an average between the expected profits if firm 1 has a subsidy and if it has no subsidy. This fact allows firm 1 to strategically
use its private information in order to extract, through the transfer payment, a greater share of the actual expected profits than the agreed proportion λ. In particular, for low values of λ,
4
A constraint on the values of V and s is required to guarantee that 0 P e
1
, e
2
1. A sufficient condition for that requirement to be satisfied is given by 0 s 1 − V
32
, which implies that we are constrained to values of V in the interval [0, 1].
392 M.P. Gos´albez, J.S. D´ıez J. of Economic Behavior Org. 42 2000 385–404
it is firm 1 that pays the transfer and this fact provides firm 1 with incentives not to disclose the subsidy in order to pay a lower transfer. On the other hand, for high values of λ, it is firm
2 that pays the transfer and, therefore, it is in the interest of firm 1 to disclose the subsidy in order to receive a higher payment from firm 2. If the efforts are strategic complements and
firm 1 has disclosed the subsidy, by direct substitution of the equilibrium efforts in 15 we obtain
T
R
= 2V
3
31 − s [2λ − 1].
4.2 On the other hand, if firm 1 has not disclosed the subsidy, the size of the transfer will
depend, among other things, on the conditional probability ˜ q
. In this case, direct substitution of the equilibrium values in 15 results in
T
N
= 2V
3
3 1 − ˜
q + ˜
q 1 − s
12 2
[2λ − 1]. 4.3
Notice that the transfer payments increase with both the subsidy and with the patent value. Moreover, for values of λ lower than 12,T
R
and T
N
become negative, implying that it is firm 1 that pays the subsidy in both cases. Finally, comparing both transfers we obtain
that T
R
− T
N
T 0 if λ T 1
2 ,
that is, from the point of view of the transfer stage and with strategic complements firm 1 has an incentive to disclose the subsidy only for high enough values of λ.
In the following section, we will analyze firm 1’s optimal decision on whether or not to disclose the subsidy to its partner. In order to take this decision, the firm will take into
account its effect on the transfer payment as well as on the equilibrium efforts.
5. The disclosure decision
