A Model of Optimal Corporate Bailouts - Faculty of Business and ...

A.4 Proof of Lemma 2If the replacement manager has the same effort cost as the initial manager, then the social continuationvalue at the reinvestment stage is identical with either manager and is sv ∗ 2 (g∗ , T ∗ c R 3·R + 4·S2) = e∗2·(R + S) −22·e∗2 − I 2 = ·− I 2 ,2·c 4because T ∗ 2= 0 by Theorem 1. The government’s objective is to maximizeSV = e 1·(R + S) − c 2·e2 1 + (1 − e 1)·sv ∗ 2 (g∗ , T ∗ 2 ) − I 1 ,and∂ SV= (R + S)·(1 − e ∗ 2∂ e ) − c·e∗ 1 + c 2·e∗2 2 + I 21 R − T1> (R + S)·(1/2) − c· + c 2·c 2·e∗2 2 + I 2> 0 ,where the inequality follows from (i) e ∗ 2 = R/(2·c) < 1/2 because c > R, and (ii) e∗ 1 = R−T 12·cif the initialmanager is replaced and e ∗ 1 = R−T 1−R2if the initial manager is not replaced. The result follows2·c 16·c 2because social value is increasing in initial managerial effort and e 1 is higher when the initial manager isreplaced.A.5 Proof of Lemma 3If π ∗ 2 ( 0, 0 ) = I 2 − R2< 0 and there is no government intervention, we showed that the firm does not4·creinvest and its maximized profits areΠ| ρ2 =OUT= R24·c − I 1 .In this case, the incumbent manager receives the expected payoff M = R28·c .If π ∗ 2 ( 0, 0 ) = I 2 − R2< 0 and there is government intervention, the incumbent manager is fired, and4·cthe firm’s maximized profits are 2 R − T1Π| F G=FIRE( e ∗ 1 ) = − I 1 .4·cIn this case, the incumbent manager receives the expected payoff M = (R−T 1) 2. The results follow8·cimmediately.33