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
In the last stage of the game, each firm decides how much effort to exert in the project in order to maximize own expected profits. Notice that the transfer payment of the previous
stage does not affect firms’ decisions on efforts at this stage. In this section, we will calculate the Nash equilibrium in efforts for the case of strategic complements.
If the efforts are strategic complements, an increase decrease in firm i’s effort is best responded with an increase decrease in firm j ’s effort in terms of reaction curves this
implies they are positively sloped. In order to obtain explicit solutions we will use the following probability function:
P e
1
, e
2
= e
1
e
2
. 3.1
This function could represent a situation in which the partners have complementary research abilities and they divide the project in two different parts, each firm undertaking the part
in which it has a comparative advantage. Therefore, only if the two parts are successfully finished is the patent obtained. The following proposition shows the Nash equilibrium
efforts.
Proposition 3.1. The Nash equilibrium in efforts for the case of strategic complements is given by
e
∗R 1
= V
1 − s
23
, e
∗R 2
= V
1 − s
13
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
