Managerial Turnover in a Changing World: Supplementary Material

In these notes, we formally prove that the effort and retention policies that maximize the firm’s profits are independent of realized cash flows. The proof is in two steps. The first step establishes that, in any mechanism that is incentive compatible and sequentially individually rational for the managers, the firm’s expected profits from each manager it hires are given by a formula analogous to that in Proposition 2 in the main text. The second step then shows that the effort and retention policies of Proposition 3 continue to solve the relaxed program in this more general environment where effort and retention decisions are allowed to depend on observed cash flows. This last result in turn implies that, when the policies of Proposition 3 are implementable (which is guaranteed in the main text by the conditions in Propositions 4 and 5), then there is no way the principal can raise the firm’s payoff by conditioning effort and retention decisions on realized cash flows. Step 1. Suppose that the firm were to offer a mechanism Ω ≡ 〈ξ, x, κ〉 where the effort and retention policies ξt(θ , πt−1) and κt(θ, πt) possibly depend on realized cash flows – note that while the recommended effort choice for each period t naturally depends only on past cash flows, the retention decision may depend also on the cash flow πt observed at the end of the period. Following arguments identical to those in the main text, one can then verify that, in any mechanism Ω ≡ 〈ξ, x, κ〉 that is incentive-compatible and sequentially individually rational, each manager’s intertemporal expected payoff under a truthful and obedient strategy must satisfy