Dynamic Managerial Compensation: A Variational Approach Supplementary Material Daniel Garrett Toulouse School of Economics
Alessandro Pavan Northwestern University
March 10, 2015
This document contains proofs for Example 1, Propositions 6, and Proposition 7 of the manuscript "Dynamic Managerial Compensation: A Variational Approach" omitted in the main text. It also contains a brief description of the numerical analysis reported in the main text at the end of Section 4. All numbered items in this document contain the pre…x S. Any numbered reference without a pre…x refers to an item in the main text. Please refer to the main text for notation and de…nitions.
Proof of Example 1. Observe that the inverse hazard rate of the period-1 distribution is 8 32 > 1 5 ( 1 3 12 ) 1 0 1 1 F1 ( 1 ) < 32 8 (1 6 ) = 1 532 ( 3 +31 2 1 ) 1 > 1 1 1 f1 ( 1 ) 5 32 2 : 32 1 8 < 1 4 (6 ) 1
5
Next, observe that, for the e¤ort policy
R
2
to satisfy the integral monotonicity constraints
in (5), it must be that the function q( 1 ) de…ned by q( 1 )
1
(1 +
2 ) 1 F1 ( 1 ) f1 ( 1 )
is every-
where non-decreasing. It is easy to verify that the above condition fails under the proposed 2 [0; 1].1
distribution, for any
Finally, observe that the inverse hazard rate of the period-1 distribution reaches a maximum at
= 1=8 where it is equal to 5=16: Clearly, 5=16 = sup f[1
for any
F1 ( 1 )]=f1 ( 1 )]g < (1 + )=(1 +
2
)
1
1
= (1 + )=(1 +
2
)
1=4;
2 [0; 1]: Hence this distribution satis…es the condition in Part d(ii) of Proposition 5.
That " is drawn from a Uniform distribution in turn implies that the conditional distribution F2 ( j ) satis…es the monotone-likelihood-ratio property, which is the other condition in Part d(ii) of
Proposition 5. That expected e¤ort necessarily increases over time when Proposition 5. Q.E.D. 1
For example q(0) =
5 (1 32
+
2
) > q( 18 ) =
1 8
(1 +
2 10 ) 32 ,
1
for any :
< 1 then follows from
Let
Proof of Proposition 6.
;1 ;
;2
be the e¤ort policies sustained under any
optimal contract, when the manager’s preferences over consumption are represented by the function v , with the function family (v )
0
satisfying the properties described in the main text. Recall,
from Proposition 2, that such policies are essentially unique. Next, let K be the expected payo¤ ~ of the lowest period-1 type (i.e., E j 1 [V ( ~)]) under any optimal contract, when the manager’s riskaversion index is . Finally, let c
c
;1 ; c ;2
be a compensation policy sustained under an optimal
contract and recall that, again by virtue of Proposition 2, such a policy is also essentially unique when
> 0, i.e., when the manager is strictly risk averse. When
= 0, instead, the distribution
of consumption over the two periods is indeterminate, in which case let c0;1 ( 1 ) = 0 for all
1
and
then let c0;2 be given by (7). Proposition 5 implies that the expected power of incentives is strictly higher in period 2 than in period 1, when
= 0: Our goal is to show the existence of
> 0 such that, for any
2 [0; ],
the expected power of incentives under any optimal contract continues to be higher in period two than in period one. Suppose, for a contradiction, that no such
exists. Letting w denote the inverse of the function
v , we then have that the following is instead true. Claim S-A. For any n 2 N, there exists n 2 0; n1 such that i h h 0 ~1 )) ~1 ) w0 v (c ( ~) w0 n v n (c ( E E 0 ( n n ;2 n ;1 n ;1 n
( ~)) n ;2
i
.
(S1)
when = 0 (that is, when the manager is risk neutral), h On the other i hand, h we have that, i 0 0 ~ ~ E < E 0;1 ( 1 ) 0;2 ( ) , as established in Proposition 5. Given these observations,
below we establish a series of three properties that together imply that Claim S-A above is false. Property S-(i). For any , Z Z 1
2( 1)
Proof of Property S-(i). Let Z Z L 1
;1 ( 1 )
2(
;2 ( 1 ; 2 )
+
2 ;1 ( 1 )
1)
+
d 2d
1
2 ;2 ( 1 ; 2 )
2
b a
2
.
(S2)
d 2d 1.
(S3)
For arbitrary , consider the gain in expected pro…ts from using an optimal policy rather than simply paying the manager a constant wage equal to his outside option (assumed equal to zero), thus eliciting no e¤ort. Given that w lies nowhere below the identity function, and given the bounds on the densities over ; it is easy to see that this gain must be no greater than Z Z 1 2 2 b a L . ;1 ( 1 ) + ;2 ( 1 ; 2 ) d 2 d 1 2 1 2( 1) Then note that Z Z 1
2( 1)
;1 ( 1 ) +
;2 ( 1 ;
2) d 2d
1
Z
1
Z 2
2( 1)
;1 ( 1 )
+
;2 ( 1 ; 2 )
d 2d
1
L1=2 , (S4)
where the second inequality follows from Hölder’s inequality. Hence, the expected gain from using 1=2
1 2 2a L
the optimal policy is no greater than b2 L
, which is non-negative only if L
4
b 4 a .
The result then follows from (S4). Property S-(ii). Let (
1 n )n=1
S-A (that is, for all n 2 N; N 2 N such that, for all n Z Z 2( 1)
1
n
be any sequence of real numbers satisfying the property in Claim
2 0; n1 is such that Condition (S1) holds). There exists
> 0 and
N, n ;1
( 1)
0;1 ( 1 )
+
n ;2
( 1;
2)
0;2 ( 1 ; 2 )
d 2d
> ,
1
(S5)
and Z
1
Z
2( 1)
n
n ;1
( 1)
2
0;1 ( 1 )
+
n ;2
( 1;
2)
0;2 ( 1 ; 2 )
2
o
d 2d
1
!1=2
> .
(S6)
Proof of Property S-(ii). For each , let inf l 2 R+ [ f+1g : w0 (u)
l
l; all u 2 R
1
and note that the assumptions on the function family (v ) start by proving the existence of
and N such that, for all n
contradiction that there are no such Z
1
Z
2( 1)
nk ;1
( 1)
0
imply that l ! 0 as N , (S5) holds.
Suppose for a
and N . Then there exists a subsequence ( 0;1 ( 1 )
+
nk ;2
( 1;
2)
0;2 ( 1 ; 2 )
d 2d
1
! 0. We
nk )
such that
!0
as nk ! 1. However, then note that 2
E4
0
nk
0 ~ ;1 ( 1 ) w n
k
v
nk (c
nk
~ ;1 ( 1 ))
0
nk
k
~ 0;1 0;2 ( ) 2 3 ~1 ) ~1 ) ~) ~) ( ( ( ( 0;1 0;2 nk ;1 nk ;2 6 7 6 0 ~ ~1 ) 7 = E 6 + w n v nk (c n ;1 ( 1 )) 1 ( 7 nk ;1 k k 4 5 ~) w0 n v nk (c n ;2 ( ~)) 1 ( nk ;2 k k 2 3 ~1 ) ~1 ) + ~) ~) ( ( ( ( ;1 0;1 ;2 0;2 nk nk 6 7 6 7 0 ~ ~ E 6 + w n v nk (c n ;1 ( 1 )) 1 ( 1) 7 nk ;1 k k 4 5 ~) + w0 n v nk (c n ;2 ( ~)) 1 ( nk ;2 k k Z Z n b2 ( 1) ( 1; 2) 0;1 ( 1 ) + 0;2 ( 1 ; nk ;1 nk ;2 1 2( 1) Z Z n o 2 + b l nk ( ) + ( ; ) d 2d 1. 1 1 2 ;1 ;2 n n 0
1
2( 1)
k
( ~1 )
0 ~ ;2 ( ) w n
k
3
v
nk (c
nk
~ ;2 ( ))
0
2)
o
d 2d
1
(S7)
3 5
The …nal expression converges to 0 as nk ! +1, the …rst integral by (S7), and the second by Property S-1 above along with the fact that l ! 0 as
! 0. It follows that, for any
exists N such that, for any nk > N; E
h
0
E
h
nk
0
0 ~ ;1 ( 1 ) w n
v
k
~
0
0;1 ( 1 )
( ~1 )) nk ;1 i ~) + . ( 0;2
The right-hand side is negative whenever E
0
nk (c
h
v
k
nk (c
nk
~ ;2 ( ))
i
is taken su¢ ciently small, since, as noted above, i
~ 0;1 ( 1 )
0
nk
0 ~ ;2 ( ) w n
> 0, there
h
0
~ 0;2 ( )
i
:
However, this contradicts the assumption that the original sequence (
n)
satis…es Condition (S1).
Finally, that (S6) is also true follows simply from (S5) using Hölder’s inequality. Now, for any h ( ; c1 ; K)
0, e¤ort policy ; …rst-period consumption policy c1 , and constant K h E ~1 +
~ + ~2 +
1( 1)
2(
~)
c1 ( ~1 )
w
W ( ~; ) + K
v (c1 ( ~1 ))
0, let
i
,
which is simply the principal’s payo¤s, as in (8). The …nal property we need to prove is the following.
Property S-(iii). Take any sequence ( h0
n
;c
n ;1
;K
n
! h0
0 ; c0;1 ; K0
1 n )n=1
as n ! +1.
such that, for all n 2 N;
n
2 0; n1 . Then
Proof of Property S-(iii). Suppose not. Then there must exist a subsequence (
nk )
and
> 0 such that h0
nk
;c
nk ;1
;K
nk
< h0
for all nk . Then note that, for any ( ; c1 ; K) ; any
0 ; c0;1 ; K0
0; h ( ; c1 ; K)
h0 ( ; c1 ; K), since w lies
everywhere above w0 . Hence, for all nk , h
nk
However, below we show that h the optimality of
nk
;c
nk ;1
;K
nk
nk nk
;c
nk ;1
;K
< h0
nk
0 ; c0;1 ; K0
! h0
0 ; c0;1 ; K0
0 ; c0;1 ; K0
for some nk su¢ ciently large.
To this end, for each , we …nd a lower bound on h
0 ; c0;1 ; K0
. as nk ! 1, which contradicts h0
0 ; c0;1 ; K0
. Recall that,
as a consequence of the result in Lemma 2 in the proof of Proposition 1, the function 0 0;1 ( 1 ) + h i ~ E 2 j 1 0 ( 0;2 ( 1 ; ~2 )) is bounded uniformly over 1 , and hence integrable; similarly, for each 1 , 4
0;2 ( 1 ;
) is uniformly bounded over h
0 ; c0;1 ; K0
h0
2
2 ( 1 );
and hence integrable. We then have that, for any ,
0 ; c0;1 ; K0
(
~ 0;1 ( 1 )) +
~ 0;2 ( ) h ~2 js 0( 0;1 (s) + E "Z ~
6 Z ~1 n 6 6 0 6 +K0 + l E6 1 6 Z ~2 6 4 + 0( ~ 0;2 ( 1 ; s))ds 2
2
E
2
~2 j ~1
2
io ~ ds 0;2 (s; 2 )) # 0( ( ~1 ; s))ds 0;2
~ 0;1 ( 1 )) +
~ 0;2 ( ) h ~2 js 0( (s) + E 0;1 "Z ~
(
3 7 7 7 7 7 7 7 5
6 Z ~1 n 6 io 6 0 ~2 )) ds 6 +K0 + (s; 0;2 l E6 1 # 6 Z ~ 6 2 2 ~ ~ 4 + 0( 0( ~ ~ 2j 1 0;2 ( 1 ; s)) ds + E 0;2 ( 1 ; s)) ds ~1 ) ~1 ) ( ( 2 2 3 2 ~ ( 0;1 ( ~1 )) + 0;2 ( ) 6 h i 7 6 7 1 F1 ( ~1 ) 0 0 ~ ~ + ( 0;2 ( )) 7 . +K0 + f ~ 1 l E6 0;1 7 6 1( 1) 4 5 1 F2 ( ~2 j ~1 ) 0( ( ~)) +2 f ~ j ~ 0;2 2( 2 1)
=
The …rst inequality follows because, for any y 2 R, any , jw (y)
yj
from integration by parts after noting that Z
1
1
n
0
0;1 (s)
Z
+ E 2
~2 js
0(
h
0
(
~
io
0;2 (s; 2 ))
3 7 7 7 7 7 7 7 5
l jyj. The equality follows
ds
ds is absolutely continuous in 2 for i h ~ each 1 (these observations follow in turn from the integrability of 0 0;1 ( 1 ) + E 2 j 1 0 ( 0;2 ( 1 ; ~2 )) , is absolutely continuous in
1 , and that
and from the integrability of
0;2 ( 1 ;
2(
1)
) for each
1 ).
0;2 ( 1 ; s))
Using the bounds on f1 and f2 , we then have
that 2
3 ~) ( 0;1 ( ~1 )) + ( 0;2 6 h i 7 6 7 1 F1 ( ~1 ) 0 0( ~ ~ 7 6 +K + + ( )) 1 E6 0 0;1 0;2 ~ f1 ( 1 ) 7 4 5 1 F2 ( ~2 j ~1 ) 0 ~ +2 f ~ j ~ ( 0;2 ( )) 2( 2 1) " Z Z 1 2 2 0;1 ( 1 ) + 0;2 ( 1 ; 2 ) + K0 2 b2 1 2 1 2( 1) 0;1 ( 1 ) + 0;2 ( 1 ; 2 ) + a 0;2 ( 1 ; a which, using the boundedness of (S3) together with (S2), is …nite. as nk ! +1 implies h
nk
0 ; c0;1 ; K0
h0
0 ; c0;1 ; K0
5
2)
#
d 2d
1
Thus the fact that l
nk
!0
! 0, which is what we wanted to show.
Now let N and
be the values de…ned in Property S-(ii). Note that, for any n > N , there
exists an incentive-compatible mechanism implementing the e¤ort policy
1 2 #
the manager’s period-1 compensation is given by an arbitrary function c for all
1,
n ;1
and under which the lowest period-1 type’s expected payo¤ is 12 K
of such a mechanism follows from Proposition 1; in particular, because
n
n
+
1 2 0,
under which
( ), say c#n ;1 ( 1 ) = 0 + 21 K0 (the existence
is quadratic,
1 2
+
n
1 2 0
satis…es Condition B(i) of this result). For such a mechanism, we have that h0
1 2
n
+
2
1 2 1
1 4 2 E 2 +
=
1 = E 8
n ;1
1 K 2
# 0 ; c n ;1 ;
n ;1
1 2
( ~1 ) +
n
1 ~ 2 0;1 ( 1 )
( ~) + n ;2
( ~1 )
1 + K0 2
~
2
1 ~ 2 0;2 ( )
0;1 ( 1 )
2
1 h0 2
+
1 2 2
n ;1
1 2
n ;2
n
;c
( ~1 )2
( ~)2 n ;2
( ~)
0;2 (
~)
n ;1
;K
n
1 ~ 2 2 0;1 ( 1 ) 1 ~2 2 0;2 ( )
1 + h0 2 3
a2 2 : 8
2
(S8)
0 ; c0;1 ; K0
5
where the inequality follows from (S6) along with the fact that f1 ( 1 ) f2 ( 2 j 1 ) > a2 for all 2
2
2 ( 1 ).
1
2
1,
That the property in Claim S-A is false then follows from the combination of the result in (S8)
along with Property S-(iii) above, which jointly imply that as n ! +1 h0
1 2
> h0
n
+
1 2
# 0 ; c n ;1 ;
1 K 2
n
1 + K0 2
1 h0 2
n
;c
n ;1
;K
1 + h0 2
n
0 ; c0;1 ; K0
+
a2 8
2
0 ; c0;1 ( 1 ) ; K0
thus contradicting the optimality of
0 ; c0;1 ; K0
.
Q.E.D. Proof of Proposition 7.
Let
;1 ;
;2
be the (essentially unique) e¤ort policy
sustained under any optimal contract, when the persistence of the productivity process ish . Let i ~ K be the optimal choice of the expected payo¤ for the lowest period-1 type (i.e., E j 1 V ( ~) )
when the persistence of the process is .
Finally, let c
c
;1 ; c ;2
be a compensation policy
sustained under an optimal contract and recall that, by virtue of Proposition 2, such a policy is also essentially unique. Part (a). Consider the case of = 1. From (13), note that, for almost all 1 2 1 ; h i ~ E j 1 0 1;2 ( ~) w0 v(c1;2 ( ~)) = 0 1;1 ( 1 ) w0 v c1;1 ( 1 ) 2 3 00 ~) Z 2 n o ( 1;2 ~ w0 v(c1;2 ( ~1 ; r)) E j 14 w0 v(c1;1 ( ~1 )) f2 (rj ~1 )dr5 . ~ ~2 f2 ( 2 j ~1 )
(S9)
We now establish that, whenever v is strictly concave, then with probability one (that is, for all but a zero-measure set of ), Z 2 m( 2 ; 1 ) w0 v c1;2 ( 1 ; r)
w0 v c1;1 ( 1 )
2
6
f2 (rj 1 ) dr
0.
(S10)
To see this, note that, for all @m( 2 ; @ 2
1)
1,
almost all
2,
w0 v c1;2 ( 1 ;
=
w0 v c1;1 ( 1 )
2)
Next, recall from (12) that, with probability one, w0 v c1;1 ( 1 )
f2 ( 2 j 1 ) .
= E
~2 j
1
(S11)
h i w0 v(c1;2 ( 1 ; ~2 )) .
Moreover, c1;2 ( 1 ; ) must be non-decreasing (this follows from the fact that incentive compatibility requires that 2 ( 1 ; ) be non-decreasing, as established in Proposition 1). Therefore, there ^2 ( 1 ) and c ( 1 ; 2 ) > exists ^2 ( 1 ) 2 2 ( 1 ) ; 2 ( 1 ) such that c1;2 ( 1 ; 2 ) c1;1 ( 1 ) for 2 1;2 c ( 1 ) for 2 > ^2 ( 1 ). Using the property that w0 (v ( )) is increasing, together with (S11), 1;1
we then have that the function m( on 2 ( 1 ) ; 2 ( 1 ) . Finally, h ; 1 ) must be quasi-concave i ~2 j 1 0 0 ~ note that w v c1;1 ( 1 ) = E w v(c1;2 ( 1 ; 2 )) ; implies that m( 2 ( 1 ) ; 1 ) = 0. That 2 ( 1) ; 1)
m(
= m(
2 ( 1) ; 1)
lish the claim in (S10). c1;2 ( 1 ; ) is constant over
= 0, along with the property that m( ;
1)
is quasi-concave, estab-
Similarly, it is easy to see that the inequality in (S10) is strict, unless 2 ( 1) ; 2 ( 1)
.
Combining (S9) with (S10) permits us to conclude that, when
= 1; the expected power of
incentives is weakly lower in period 2 than in period 1 (strictly lower, unless, with probability one, c1;2 ( 1 ; ) is constant over
2 ( 1) ; 2 ( 1)
).
Next consider Part (b). Suppose the result is not true. Then the following must be true. 1 n )n=1 ,
Claim S-B. There exists a sequence (
with
below and such that, for all n; E
h
( ~1 ) w0 v(c n ;1
0
( ~1 )) n ;1
i
E
h
n
0
all n 2 N, converging to 1 from
( ~1 ; ~1 + "~) w0 v(c n ;2
0
i
( ~1 ; ~1 + "~)) n ;2
Below we show that Claim S-B is inconsistent with the fact that, by assumption, when E
h
0
~
1;1 ( 1 )
w0 v(c1;1 ( ~1 ))
i
>E
h
~
0
1;2 ( 1 ;
~1 + "~) w0 v(c ( ~1 ; ~1 + "~)) 1;2
: (S12)
= 1, i
.
(S13)
We establish the inconsistency by means of three properties that jointly lead to a contradiction of the claim. First note that, by assumption, each
n
and
1
are uniformly bounded, with the bound M
uniform over n. This last property, along with (3) and (12), in turn imply that there exists C > 0 such that c
n ;1
; c1;1 ; c
n ;2
; c1;2
C almost everywhere, and uniformly over n. Furthermore,
from the optimality of the policies, one can easily see that there must exist K > 0 such that K
n
K for all n. The following must then be true. Property S-(iv). Assume Claim S-B is true. Then there exist 7
> 0 and N 2 N such that,
for all n
N , at least one of the following holds: Pr Pr
~1 ;
n ;2
~1
n ;1
~ + "~
n 1
Pr
1;2
c
~1
>
,
~1 ; ~1 + "~
>
,
c1;1 ~1
>
, or
1;1
~1
n ;1
K
K1
n
.
Proof of Property S-(iv). Suppose Property S-(iv) is false. Then there exists a subsequence (
nk )
such that
and c
nk ;2
( 1;
nk ;1
n 1
( 1)
1;1 ( 1 )
+ ")
c1;2 ( 1 ;
, 1
nk ;2
( 1;
+ ")
n 1
1;2 ( 1 ; 1
+ ") ; c
nk ;1
( 1)
c1;1 ( 1 ) ,
+ ") all converge in probability to zero, which, given the
boundedness of the policies, implies that (S12) and (S13) are mutually inconsistent. Now, abusing the notation introduced in the proof of Proposition 6, we let h ( ; c1 ; K)
h E ~1 +
~ + ~2 +
1( 1)
2(
~)
c1 ( ~1 )
w W ( ~; ) + K
i
v(c1 ( ~1 ))
,
denote the …rm’s expected pro…ts under the policies ( ; c1 ); when the lowest period-1 type’s expected payo¤ is K: Note that the dependence on is both directly through the fact that ~2 = ~1 + "~ as well as through the function W ( ; ) that, along with ( ; c1 ; K) ; determines the period-2 compensation policy c2 ( ) according to (7). Note that h is strictly concave in , v (c1 ) and K (this follows straightforwardly from the convexity of w and
). Strict concavity of h1 ( ; c1 ; K), in particular,
implies the following property (the result is obvious and hence the proof omitted). Property S-(v). There exists a function Take any
> 0 and any pair (
0 ; c0 ; K 0 ) 1
and (
Pr 0 2
Pr
0 1
~1 ; ~1 + "~ Pr
: R++ ! R++ satisfying 00 ; c00 ; K 00 ) satisfying at least 1 ~1
00 1
~1
>
,
00 2
~1 ; ~1 + "~
>
,
c001 ~1
>
, or
c01 ~1
K0 Let ( Then
000 ; c000 ; K 000 ) be de…ned 1 000 h1 ( 000 ; c000 ( 1 ;K )
by
000
=
1 0 1 00 2 +2 ,
c000 1 =w
1 0 2 v (c1 )
K 00
the following property. one of the following
.
+ 12 v (c001 ) , and K 000 = 12 K 0 + 12 K 00 .
).
Next, we use the boundedness of the optimal policies to establish the following property. Property S-(vi). Assume Claim S-B is true. For all n ;1
h1
( 1 ), 0
n
0
n ;2
( ) =
; c0 n ;1 ; K 0 n ! h1
n ;2
( 1;
n 1
1 ; c1;1 ; K1
+
2
1 ),
c0
n ;1
as n ! +1. 8
( 1) = c
2 n ;1
; all n 2 N, let
( 1 ), and
K0
n
0
n ;1
( 1) =
= K n.
Then
Proof of Property S-(vi). Our approach to the proof is as follows. We construct, for each n, #
a policy
n
; c#n ;1 ; K #n which (together with c#n ;2 de…ned by (7)) is implementable when #
We choose
n
; c#n ;1 ; K #n
# n
n
; c#n ;1 ; K #n ! h1
1 ; c1;1 ; K1 ##
as n ! 1. Similarly, for each n, we construct a policy c## de…ned by (7)) is implementable for n ;2 ##
h1
n
n
; c## ; K ## !h n n ;1
n
n
n
; c## ; K ## n n ;1
h1
1 ; c1;1 ; K1
h
n
n
;c
n ;1
n
;c
(S14)
; c## ; K ## n n ;1
##
= 1. Moreover,
as n ! 1. These observations, together with the fact that h and h1
n.
in particular so that h
##
=
n ;1
; c## ; K ## is chosen so that n n ;1
;K
# n
n
which (together with
(S-15)
n
; c#n ;1 ; K #n
h
n
n
;c
n ;1
;K
n
for each n, then imply that ;K
! h1
n
1 ; c1;1 ; K1
(S16)
as n ! +1. The result in Property S-(vi) then follows from (S16) by considering the functional ^ ^ ( ; c1 ; K; ) h 2 0
6 6 6 6 = E6 6 6 6 4
B B B wB B @
3 ~ + ^ ~1 + "~ + 2 ( ~1 ; ~1 + "~) c1 ( ~1 ) 1 7 Z 1 7 0 " ~ 0 ~ ~ ( 1 (s)) + ^ E [ ( 2 (s; s + "~))] ds C 7 ~) + 2( 1; 1+" C 7 1 Z "~ . C 7 C 7 7 0 ( ( ~ ; ~ + s))ds " ~ 0 ~ ~ E ( ( ; + s))ds C 7 2 1 1 2 1 1 A 5 " +K v(c1 ( ~1 ))
~1 + ( 1 ( ~1 )) + Z +
"~
"
1( 1)
^ ^ ( ; c1 ; K; ) is continuous in ^ uniformly over In particular, it follows from observing that h [ ^ h
0 ; 1] n
and over policies ( ; c1 ; K) satisfying the aforementioned bounds, and that h ^1 ;c ; K ; n while h1 0 ; c0 ; K 0 = h ;c ;K ; n : n ;1
n
n
Our construction of #
Then, let
n ;1
( 1) =
c1;1 . Note that, since # n
#
;c
n ;1
;K
# n
n ;1
n
# n
#
;c
n ;1
n 1;1 ( 1 )
;K
and
1 ; c1;1 ; K1
n
n
n ;1
n
n
n
# n ;2
( 1;
2)
=
1;2 ( 1 ; 1
n ;1
;K
+(
1 #
n 1 )).
2
n) M
Finally, let c
n ;1
, together with (7), de…nes an implementable policy when
also de…nes an implementable policy when
n
=
n
for each n is as follows. Let K #n = K1 +2 (1
#
;c
2
=
n.
1
=
= 1,
This is veri…ed with respect
to the conditions in Proposition 1. The only condition that is not immediate to check is B(i), or (using that is quadratic) that, for all 1 ; ^1 , Z 1n Z 1n h io h io # # "~ # "~ # ^ ^ ^ ( ) + + E ( ; s + " ~ ) ds (s) + s + E (s; s + " ~ ) ds, 1 1 n 1 n n n n;2 n ;1 n ;2 n ;1 ^1
^1
which, substituting for #n , we can rewrite as Z 1n h "~ ^ ^ ^ ~ + (1 n 1;1 ( 1 ) + 1 + E 1;2 ( 1 ; n s + " ^1 n
Z
1
^1
1;1 (s)
+ s + E"~
1;2 (s; s
+ "~)
ds:
9
^ n) 1)
io
ds
(1
n)
Z
1
^1
s
^1 ds (S17)
.
To see that (S17) must hold, note that, because decreasing in Z
n
=
1
^1 n
Z
n ^1
2,
n
1;2 ( 1 ; 2 )
h
1;2
"~ ^ ^ 1;1 ( 1 ) + 1 + E
h
i ( ^1 ; s + "~) + (1 1;2
n)
io ( ^1 ; s + "~) ds
o
^1
s
(1
n)
Z
2
ds
(1
1
since
1
1
^1
n
"~ ^ ^ 1;1 ( 1 ) + 1 + E
# n
h
; c#n ;1 ; K #n 2
6 6 6 6 6 6 ~1 ;~ ") 6 ( h ( ; c1 ; K) = E 6 6 6 6 6 4 d ( 1;
1
is non-
1;2
Z
io ( ^1 ; s + "~)
1
1;1 (s)
^1
1
^1 ds
s
^1
+ s + E"~
1,
1;2 (s; s
i.e.
+ "~)
ds,
is an implementable policy.
To see that
Let
2
^1 ds
s
^1
n)
Z
That (S17) holds then follows because condition B(i) of Proposition 1 holds for Z
+
and so the left-hand side is no greater than
"~ ^ ^ 1;1 ( 1 ) + 1 + E 1
is implementable,
1
( 1 ; ") =
1;2 ( 1 ; 1
satis…es (S14), let ~1 +
( 1 ; ") =
~ ~1 + "~ + 1( 1) +
2 ( 1;
1
+ ") and note that
~1 ; "~
c1 ( ~1 )
3
1 7 7 7 C 7 Z ~1 C 7 C 7 0 ( (s)) + E"~ [ 0 ( (s; " + ~))] ds C 7 1 C 7 C 7 Z "~ 1 Z "~ C 7 0 " ~ 0 7 + ( ( ~1 ; s))ds E ( ( ~1 ; s))ds C C 7 " " A 5 +K v(c1 ( ~1 ))
0
( 1 ( ~1 )) +
B B B B B wB B B B @
( ~1 ; "~)
(S18)
; c1 ; K) .
#
+ ") and, for each n, let
be the space of …rst-period e¤ort policies
1
n
#
( 1 ; ") =
n ;2
( 1;
n 1
+ "). Now let E1 (M )
bounded by M , and endow this space with the sup norm.
Let Z (M ) denote the space of functions ( 1 ; ") (essentially) bounded by M , and let C1 C denote
the space of functions c1 ( 1 ) (essentially) bounded by C. Then note that d ( 1 ; ; c1 ; K) is continuous in ( ;
1 ; K)
uniformly over [ 0 ; 1]
E1 (M )
C1 C
Z (M )
Moreover, by construction, for all n, and for all ( 1 ; ") 2 c
n; 1
#(
1)
["; "],
= c1;1 ( 1 ). These observations, together with the fact that
uniformly to 1;
1;1 ; K1
Next, we construct and
1
## ( 1; 2) n ;2
=
n ;2
# n
( 1 ; ") = n;
# n ;1
;K
1 # n
1
1
.
( 1 ; "), and converges
, then imply (S14). ## n
( 1;
; c## ; K ## . Let n n ;1 n 1
+(
2
1 )).
K ## = K1 + n Finally, let c## = c1;1 . Note that, since n ;1 mentable policy when
0) M
0; K + 2 (1
=
n,
## n
## ( 1) n ;1
=
1
n ;1
n
( 1 )+ 2M b (1
n)
+
1
n n
1
Let 1 1
n
; c## ; K ## n n ;1
n
1
;c
n ;1
M.
n
;K
n
, together with (??), de…nes an imple-
also de…nes an implementable policy when 10
= 1.
Again, this is veri…ed by considering Proposition 1. The only condition which is not immediate to check is B(i), or (using that is quadratic) that, for all 1 ; ^1 , Z
1
^1
n
## ^ ( 1) n ;1
+ ^1 + E"~ ##
which, substituting for Z
1
1
^1
Z
1
^1
n ;1
n
Z
io + "~) ds
## ^ ( 1; s n ;2
n
1
^1
## (s) n ;1
+ s + E"~
h
## (s; s n ;2
+ "~)
io
ds, (S19)
, we can rewrite as
( ^1 ) + ^1 + 2 ^1 M b (1 n ;1
n 1
n
h
(s) + s + 2sM b (1
"~ n) + E n)
h
^1 ;
n ;2
+ E"~
n ;2
(s;
~ + (1 ns + "
ns
+ "~)
n) s
^1
i
ds
ds:
Then note that, for any ^1 , s, E"~ =
h
n ;2
Z
^1 ;
ns
maxf"+(1
minf"+(1
2M b (1
+ "~ + (1 ^1 );"g
(s
n)
^1 );"g
(s
n)
s
( ^1 ; n ;2
^1 ,
s
n)
n)
i
^1
E"~
h
( ^1 ;
n ;2
h s + ")d G " n
(1
ns
+ "~)
i
^1
n) s
i G (") (S20)
where the inequality follows because the density of "~ is bounded by b (which is equivalent to our requirement that the density f2 ( 2 j 1 ) is bounded). The inequality (S20), together with 1 n
Z
1
^1
1 n
(which holds, since
1
Z
n
^ + ^1 +
"~ nE
( 1) n ;1 1
^1
n ;1
(s) + s +
h
^
( 1; ns n ;2
+ "~)
(s;
+ "~)
"~ nE
n ;2
ns
io
ds
ds
is an implementable e¤ort policy), implies (S19).
Finally, (S-15) follows by arguments analogous to those for (S14).
1=2 1=2 ( ) = 12 0 n ;2 ( ) + 12 1;2 ( ), c1; n ( 1 ) = w 21 v c0 n ;1 ( n ;2 1 0 ; c0 ; K 0 n is de…ned in Property S-(vi). n 2 K1 , where n ;1
1=2 ( 1) n ;1
1 0 1 2 n ;1 ( 1 ) + 2 1;1 ( 1 ), 1=2 1 and K n = 12 K 0 n + 1 ) + 2 v c1;1 ( 1 ) ## ## Recall the construction of ## n ; c n ;1 ; K n
We are now ready to establish that Claim S-B is false. Let
=
in the proof of Property S-(iii), and recall that this policy is implementable when = 1. Then 1=2 1=2 ^ 1=2 + 12 1 , c^1; n = w 21 v c## + 21 v c1;1 and K = 12 K ## + 12 K1 . Using let ^ n = 12 ## n n n n ;1 that
1=2 ^ 1=2 ^1=2 ^1; n ; K and n ;c n
1=2 n
1=2
; c1; n ; K
1=2 n
are uniformly (essentially) bounded (and hence the
continuity of h1 ( ; ; ) over the bounded policies), we have 1=2 ^ 1=2 h1 ^1=2 ; c^1; n ; K n n
h1
as n ! 1. 11
1=2 n
1=2
; c1; n ; K 1=2 !0 n
(S21)
Now note that, if Claim S-B were true, by virtue of Properties S-(iv) and S-(v), we would have that h1 for all n
N.
1=2 n
1 h1 2
1=2
; c1; n ; K 1=2 n
0
n
1 ; c0 n ;1 ; K 0 n + h1 2
By the inequality (S22), the fact that
1 ; c1;1 ; K1
+ ( )
(S22)
( ) > 0, Property S-6, and (S21), we
conclude that, for all large enough n, 1=2 ^ 1=2 h1 ^1=2 ; c^1; n ; K > h1 n n
1 ; c1;1 ; K1
.
1=2 1=2 ^ 1=2 de…nes (together with (7)) an implementable policy for However, note that ^ n ; c^1; n ; K n
(this follows because both
## n
; c## ; K ## n n ;1
and
1 ; c1;1 ; K1
by the conditions in Proposition 1; in particular, because
are implementable for
=1
= 1, and
is quadratic, the convex combination of
any two e¤ort policies satisfying condition B(i) in Proposition 1 continues to satisfy this condition). This contradicts the optimality of
1 ; c1;1 ; K1
. That Claim S-B is false then implies the result
in Part (b) in the proposition is true, which concludes the proof.
Q.E.D.
Numerical Analysis of Section 4. As explained in the main text, in the numerical exercises at the end of Section 4, we assume that, for all c
0; v (c) = c1
Hence, for any v
) + 1)1=(1
1=(1
); w (v) = (v (1
)
1 = (1
), with
. We also assume that
2 [0; 1=2].2 1
is drawn
from a uniform distribution with support [0; 1=2], while " is drawn from a uniform distribution with support [ :5; :5]. We then characterize the optimal contract in terms of three policies
1
: [0; 1=2] ! R
^2 : [0; 1=2]
[ :5; :5] ! R
c1 : [0; 1=2] ! R for di¤erent values of the persistence parameter, , and coe¢ cient of relative risk aversion, : Note that, contrary to the main text, here we …nd it convenient to express period-2 e¤ort as a function ^2 ( 1 ; ") of period-1 productivity, 1 ; and the period-2 shock, ": Obviously, this alternative representation is inconsequential for the results (for each ( 1 ; ^ 2( 1; 2) = 2( 1; 2 1 ):). 2
2)
with
2
2
1 ( 1 );
simply let
Note that, contrary to what assumed in the model setup in the main text, this felicity function is not surjective
and Lipschitz continuous over the entire real line. However, the numerical results do not hinge on the lack of these properties. In fact, under the optimal policies identi…ed in the numerical analysis, consumption is bounded away from zero from below. One can then construct extensions v^ of the assumed felicity function v such that (a) v^(c) = v(c) for all c > c0 > 0, (b) the numerical solutions under v^ coincide with those under v; and (c) v^ satis…es all the conditions in the model setup.
12
Then let ^ =
D
E ^ , and, for all ( 1 ; ") 2 [0; 1=2]
1; 2
^ ( 1 ; ") ; ^ W
[ :5; :5] and
2 [0; 1], let
h io ^2 ( 1 ; ")2 Z 1 n "~ ^ + (s) + E (s; " ~ )) ds 1 2 2 2 0 "Z # Z " "~ ^2 ( 1 ; y)dy E"~ ^2 ( 1 ; y)dy + 2 1( 1)
+
1 2
1 2
( Z ^2 ( 1 ; ")2 Z 1 = + + 1 (s) + 2 2 0 Z 1 Z " Z " 2 ^ ^2 ( 1 ; y)dyd". + 2 ( 1 ; y)dy 2 1( 1)
1 2
1 2
1 2 1 2
^2 (s; y)dy
)
ds
1 2
^ is the analog of the function W in the main text, but with arguments Note that the function W ^ and W are related by the condition ( 1 ; ") as opposed to ( 1 ; 2 ): Again, the two functions W D E ^ ( 1 ; ") ; 1 ; ^2 W = W ( 1 ; 1 + "; h 1 ; 2 i) all ( 1 ; "): The objective of the numerical analysis is to maximize by means of the functions
^
1 ; 2 ; c1
the …rm’s expected pro…ts, which, by virtue of Lemma 1 in the main text, can be expressed as follows: h i ~1 ; "~ ; ^ ^ E ~1 + 1 ( ~1 ) + ~1 + "~ + ^2 ( ~1 ; "~) c1 ( ~1 ) w W v(c1 ( ~1 )) h i = E (1 + ) ~1 + "~ Z 1=2 Z 1 h i 2 ^ ( 1 ; ") ; ^ c1 ( 1 ) w W v(c1 ( 1 )) 2d"d 1 : + 1 ( 1 ) + 2 ( 1 ; ")
(0.1)
1 2
0
The Euler conditions for this problem are the analogs of those in Proposition 8 in the main text, expressed in terms of ( 1 ; ") as opposed to ( 1 ;
2 ):
0 1 ( 1 ) w (v (c1 ( 1 ))) = 1
^2 ( 1 ; ") w0 W ^ (( 1 ; ") ; ^) Z
"
^ (( 1 ; r) ; ^) w0 W
0
w (v (c1 ( 1 ))) = We arrive at the policies
R ^R R 1 ; 2 ; c1
Z
1=2
w0 (v (c1 (r))) dr,
1
Z
v(c1 ( 1 )) = 1
1=2 n
and
Z
1 2 1 2
v(c1 ( 1 ))
1=2
w10 (v (c1 (r))) dr
1
o w0 (v (c1 ( 1 ))) dr,
^ ( 1 ; ") ; ^ w0 W
v(c1 ( 1 )) d".
by maximizing (0.1) over the set of policies that solve the
above Euler conditions. We then verify that the policies 13
R ^R R 1 ; 2 ; c1
can indeed be sustained in a
mechanism that is IR and IC for the managers. For this purpose, we verify that the policies identi…ed in the numerical analysis, along with the period-2 consumption policy de…ned by cR 2 ( 1; ^R ^ ( 1; 2 w(W v(c1 ( i1 ))) satisfy conditions (a)-(e) below: 1 ); h) R R ^ (a) 1 ( 1 ) + E 2 ( 1 ; "~) nondecreasing in 1 ; (b) ^R ( 1 ; ") nonincreasing in "; all 1 2 [0; 1=2]; (c)
(d) (e)
2 R ( ; ") + " nondecreasing in "; all 1 1i 2 [0; 1=2]; 2 h R R ^ ~ + 2 ( 1 ; "~) nondecreasing in 1 ; 1 + 1 ( 1) + E 1+" R R ^R cR 0; all ( 1 ; ") 2 [0; 1=2] [ 1 ( 1 ); c2 ( 1 ; "); 1 ( 1 ); 2 ( 1 ; ")
2)
=
1=2; +1=2]:
R One can easily verify that the above conditions imply that the policies 1R ; 2R ; cR 1 , with 2 de…ned by 2R ( 1 ; 2 ) = ^2R ( 1 ; 2 1 ) all ( 1 ; 2 ) with 2 2 2 ( 1 ); along with the period-2 comR ^R ) v(c1 ( 1 ))) satisfy all the conditions in Proposition ^ ( 1; 2 pensation c ( 1 ; 2 ) = w(W 1 ); 2
1 in the main text (with K = 0), and hence are implementable. Finally, observe that Figures 1, 2, and 4 in the main text depict the distortions D1 ( 1 ) = 1 Z ~j 1 ~ E [D2 ( )] = for di¤erent values of
1;
R 0 1 ( 1) w 1 2 1 2
h 1
v cR 1 ( 1)
^R ( 1 ; ") w0 W ^ (( 1 ; ") ; ^R ) 2
v(cR 1 ( 1 ))
i
d"
whereas Figure 3 depicts the e¤ort functions R 1 ( 1)
and E[ ^2R ( 1 ; "~)] =
Z
1 2 1 2
^R ( 1 ; ") d": 2
Finally, Figures 5 and 6 in the main text depict the unconditional expected di¤erence between period-2 and pariod-1 distortions h i h R ~ ~1 ^ ( ~1 ; "~ ; ^R ) v(cR ( ~1 )) Dif f = E 1 ^2R ~1 ; "~ w0 W E 1 w0 v cR 1 1 1 1 h i ~1 ^R ~1 ; "~ w0 W ^ ( ~1 ; "~ ; ^R ) v(cR ( ~1 )) = E 1R ~1 w0 v cR 1 1 2 ) Z 1=2 ( Z 1 2 R 0 R ^R ( 1 ; ") w0 W ^ (( 1 ; ") ; ^R ) v(cR =2 2 1 ( 1 )) d" d 1 : 1 ( 1 ) w v c1 ( 1 ) 0
1 2
14
i
