Note that and you can , that’s, try decreasing for the x
There exists a unique x1 , where 0 ? x1 ? 1 , given that Vt(q1, q2) is an increasing function in q1 .
Proof. When the strategic RA (RA1) gets a bad project, it will get pay-off if it gives the project a GramsR, and if it refuses rating. 1 and is increasing in x1 . Given that Vt(q1, q2) is increasing in q1 , it is easy to see that ?(lie) is decreasing in x1 and that ?(honest) is increasing in x1 . Thus, if we define x1 such that
dos Proof Suggestion dos
Proof. Suppose that the strategic RA (RA1) gets a good project and that its strategy is x1 . Let us examine whether RA1 wants to deviate:
•if x1 = 1 , we have ?(lie) ? ?(honest) , or . If the RA1 gives NR to the good project, it will get and otherwise. Since RA1 does not want to deviate.
•if x1 = 0 , , hence reputation becomes irrelevant and the RA does not have an incentive to give NR to the good project.
•if 0 < x1 < 1 , we have ?(lie) = ?(honest) , so , and hence RA1 does not want to deviate.
step three Evidence of Corollary step one
Proof. Suppose that the equilibrium strategy is x1 = 0 . Then and we must have I + ?Vt(q1, q2) ? ?Vt(q1, q2) . This is impossible as long as I > 0 . Hence, x1 = 0 cannot be an equilibrium strategy.
4 Proof of Corollary dos
Suppose the model ends in period T. Then the equilibrium strategy of the strategic RA is xt = 1 at t = T ? 1, T tinychat .
Proof. At t = T , the strategic RA does not have any reputational concerns. This implies that the strategy of strategic RA will be to always give GR if the project is bad, that is, xT = 1 .
Furthermore, on t = T ? 1 , this new strategic RA are often rest. Suppose that a detrimental investment relates to proper RA, say RA1. The fresh new requested spend-away from RA1 is
Although in this case RA1 does have reputational concerns, these are not sufficient to prevent RA1 from being lax and not giving GR to bad projects.
