Online Appendix to “Honest equilibria in reputation games: The role of time preferences”

PROOF: Let {wt , bt}∞t=0 denote the set of contracts in the optimal hybrid equilibrium, and let et denote the effort implemented at t (note that {wt , bt} is offered and accepted provided that the principal has not defaulted at any τ < t). Obviously, u0 = ū 1−δ must hold, otherwise w0 can be reduced by a small amount and make both types better off. Next, suppose towards a contradiction that ut+1 > ū 1−δ at some t ≥ 0. Note that either (i) bt > δlπl,t+1 − δlπ d l,t+1; or (ii) bt ≤ δlπl,t+1 − δlπ l,t+1, where πl,t+1 represents the payoff of a low type principal who has never defaulted until t + 1 (by an abuse of notation), and πd θ,t+1 represents the punishment payoff of a type-θ principal who defaulted at t . If (i) holds, then ut+1 > ū 1−δ cannot be optimal for any t ≥ 0. The reason is as follows. In this case, the low type strictly prefers defaulting at t . Consider the modified hybrid contract: bt is increased by a small amount δε > 0, and wt+1 is reduced by ε; thus, ut+1 reduces by ε whereas πh,t+1 and πl,t+1 both increase by ε. The bonus reward bt + δε is still contingent on et as in the original contract, and everything else remains the same. This modified hybrid contract strictly increases the payoff of the high type, whereas the low type principal and the agent are unaffected. To see why, first note that the agent’s participation constraint is still satisfied at every t (since ε is small enough), and