In these notes, we formally prove that the e¤ort and retention policies that maximize the …rm’s pro…ts are independent of realized cash ‡ows.
The proof is in two steps. The …rst step establishes that, in any mechanism that is incentive compatible and periodic individually rational for the managers, the …rm’s expected pro…ts 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 e¤ort and retention policies of Proposition 3 continue to solve the relaxed program also in this more general environment where e¤ort and retention decisions are allowed to depend also on observed cash ‡ows.
Step 1. Suppose that the …rm were to o¤er a mechanism h ; x; i where the e¤ort and retention policies t( t; t 1) and t( t; t) possibly depend on realized cash ‡ows — note that while the recommended e¤ort choice for each period t naturally depends only on past cash ‡ows, the retention decision may depend also on the cash ‡ow t observed at the end of the period.
Following arguments identical to those in the main text, one can then verify that, in any mech-anism h ; x; i that is incentive-compatible and periodic individually rational, each manager’s intertemporal expected payo¤ under a truthful and obedient strategy must satisfy
V ( 1) = V ( ) + Z 1
E~1>1;v1js
2 4
(s;~1>1;v1)
X
t=1
t 1J1t(s; ~t>1) 0( t(s; ~t>1; t 1(s; ~t 1>1 ; vt 1)) 3 5 dx
(19) where t 1( t 1; vt 1) denotes the history of cash ‡ows under a truthful and obedient strategy when the history of productivity realizations is t 1 and the history of transitory noise shocks is vt 1.
To see this, consider a …ctitious environment identical to the one de…ned in the main text.
In this environment, after any history ht = ( t; ^t 1; et 1; t 1); the manager can misrepresent his productivity to be ^t but then has to choose e¤ort equal to
e#t ( t; ^t; t 1) = ^t+ t(^t; t 1) t (20)
so that the distribution of period-t cash ‡ows t is the same as when the manager’s true period-t productivity is ^t and he follows the principal’s recommended e¤ort choice t(^t; t 1):
Now …x an arbitrary sequence of reports ^1 and an arbitrary sequence of true productivities
1 and let C(^1) denote the net present value of the stream of payments that the manager expects to receive from the principal when the sequence of reported productivities is ^1 and in each period he chooses e¤ort according to (20). Note that, by construction, C does not depend on the true productivities 1: Also note that the expectation here is over the transitory noise v1: For any
( 1; ^1), the manager’s expected payo¤ in this …ctitious environment is then given by and the noise terms vt 1, in each period s t 1; the manager follows the behavior described by (20).
As argued in the main text, it is easy to see that if the mechanism is incentive compatible and periodic individually rational in the original environment where the manager is free to choose his e¤ort after misreporting his type, it must also be in this …ctitious environment, where he is forced to choose e¤ort according to (20).
Next note that the assumption that is di¤erentiable and Lipschitz continuous implies that U is totally di¤erentiable in t, any t, and equi-Lipschitz continuous in 1 in the norm
jj 1jj P
t=1 tj tj:
Together with the fact that jj 1jj is …nite (which is implied by the assumption that is bounded) and that the impulse responses Jst( t) are uniformly bounded, this means that this …ctitious environment satis…es all the conditions in Proposition 3 in Pavan, Segal, and Toikka (2011). The result in that proposition in turn implies that, given any mechanism that is incentive compatible and periodic individually rational, the value function V ( 1) associated with the problem that consists of choosing the reports and then selecting e¤ort according to (20) is Lipschitz continuous and, at each point of di¤erentiability, satis…es
under truthtelling. Condition (19) then follows from the fact that
@U( 1; 1)
@ t = Ev1 t 1( t 1; t 1( t 1; ~vt 1)) 0( t( t; t 1( t 1; ~vt 1))) along with the independence between ~1 and ~v1:
Going back to the primitive environment, we then conclude that, given any mechanism that is incentive-compatible and periodic individually rational, the …rm’s expected pro…ts from each manager it hires are given by
or, equivalently, denoting by ( 1; v1) = infft : t( t; t( t; vt)) = 0g the stochastic length of the employment relationship,
E~1;~v1
2 64
(~1;~v1) X
t=1 t 1
( ~t+ t(~t; t 1(~t 1; ~vt 1)) + ~t ( t(~t; t 1(~t 1; ~vt 1))) (~1)J1t(~t) 0( t(~t; t 1(~t 1; ~vt 1))) (1 )Uo
)3 75(21)
+Uo V ( ) .
Step 2. Consider now the “relaxed program” that consists of choosing the policies ( t( ); t( ))1t=1 so as to maximize the sum of the pro…ts the …rm expects from each manager it hires, taking the contribution of each manager to be (21), and subject to the participation constraints of the lowest period-1 types V ( ) Uo.
To see that the policies and of Proposition 3 solve the relaxed program de…ned above, we then proceed as follows. First, …x an arbitrary e¤ort and retention policies and and note that for any ( 1; v1), any t < ( 1; ~v1) ; the ‡ow virtual surplus
t+ t( t; t 1( t 1; vt 1)) + vt ( t( t; t 1( t 1; vt 1))) ( 1)J1t( t) 0( t( t; t 1( t 1; vt 1))) (1 ) Uo
t+ t( t) + vt ( t( t)) ( 1)J1t( t) 0( t( t)) (1 ) Uo: This follows immediately from the fact that, for any t; any t2 t; the function
t+ e + vt (e) ( 1)J1t( t) 0(e) (1 ) Uo is maximized at e = t( t), where t( t) is as in Proposition 3.
Next let 0 be any retention policy that, given the e¤ort policy ; in each state ( 1; v1) implements the same retention decisions as the original rule : That is, denoting by t 1( t 1; vt 1) the equilibrium history of cash ‡ows under the e¤ort policy ; 0 is such that, for any t 1; any ( t 1; vt 1)
0t 1( t 1; t 1( t 1; vt 1)) = t 1( t 1; t 1( t 1; vt 1)):
It is then immediate that the …rm’s intertemporal expected pro…ts under the policies ( 0; ) are weakly higher than under the original policies ( ; ): In fact, while the separation decisions are the same under the two policies and 0, for each manager the …rm hires, and for each state ( 1; v1);
the virtual surplus in each period t prior to separation is higher under the e¤ort policy than under the policy :
Next, observe that, starting from the pair of policies ( ; 0); if the principal were to switch to the pair of policies ( ; ) of Proposition 3, then the …rm’s expected pro…ts would be higher. This follows from the fact that the ‡ow virtual surplus
t+ t( t) + vt ( t( t)) ( 1)J1t( t) 0( t( t)) (1 ) Uo (22)
that the …rm obtains from each manager it hires after any length t of employment and for each productivity history t is the same under ( ; 0) as it is under ( ; ); along with the fact that, by construction, the retention rule of Proposition 3 maximizes the …rm’s intertemporal payo¤ when the ‡ow virtual surpluses are given by (22).
We conclude that the policies ( ; ) of Proposition 3 solve the relaxed program de…ned above.
In turn, this implies that whenever there exists a compensation scheme x that implements these policies and gives the lowest period-1 type an expected payo¤ V ( ) equal to his outside option, then the …rm’s payo¤ cannot be improved by switching to any other mechanism, including those where the e¤ort and retention policies possibly depend on realized cash ‡ows.