UNIVERSITÀ DEGLI STUDI DI ROMA “LA SAPIENZA”
DIPARTIMENTO DI ECONOMIA PUBBLICA
Working Paper No. 46
Susanna Mancinelli e Maria Augusta Miceli
IRRELEVANCE OF PROFIT SHARING IN THE
PRINCIPAL-AGENT MODEL
Roma, novembre 2001
Abstract
The aim is to investigate the difference in the functional dependence between
incentives based on output and incentives based on an actual share of profit.
Although the incentive role of profit sharing in contracts is an established result,
we were looking for a functional solution form, able to implement this contract
under moral hazard. In order to do this, we added the possibility of profit sharing in
the classical principal-agent incentive model. Surprisingly the moral hazard setting
lead to negative results both under a ''principal-agent'' and under a ''collusive''
relationship between the two contract parts. The main results may be summarized
as follows. (i) Profit and utility function maximization lead to a unique optimal
level of both revenues independent of sources: profit sharing has no role
independently of contract relationships and, more importantly, independently of
relative degrees of risk aversion. (ii) Being revenues the same, the optimal choice
for the equilibrium effort level is constant as well. (iii) Collusion under moral
hazard has no role, since there is redundancy of either the binding incentive
constraint or the collusive role of the agent.
J.E.L. Classification: D82, J30, J33.
Keywords: Moral Hazard, Principal-Agent Models, Compensation, Profit sharing.
IRRELEVANCE OF PROFIT SHARING IN THE
PRINCIPAL-AGENT MODEL∗
Susanna Mancinelli†
Maria Augusta Miceli‡
Introduction
In this paper we are looking for a worker optimal incentive contract, contingent on profit sharing, in a moral hazard context. We started the analysis
in the principal-agent model, since it represents the established benchmark in
the incentive literature. We wanted to investigate the difference in the functional dependence between incentives based on output and incentives based
on an actual share of profit. We were trying to provide theoretical evidence
to the common intuition in favour of profit sharing1 : while contingency on
output pushes the agent compensation toward a wage as high as possible, the
profit share incentives should moderate this tendency.
The result is unexpectedly ”irrelevance” of the contingency on profit sharing with respect to contingency on output in the PA model. In other words the
model is unable to distinguish the source of incentive income, being only able
to determine a unique optimal amount for agent and principal gross revenue
in every state of nature, given the constraints. The result is also independent
on assumptions about principal and agent relative degree of risk aversion and
it does not even vary by changing the position of the two subjects: we will see
how even allowing collusion the result does not change.
Strikingly not only the optimal revenues do not change by changing the
incentive scheme or by changing the contract position, but the same is true
∗
We deeply thank Domenico Tosato, the University of Ferrara ”Compensation System”
team, the two anonimous referees and participants to the Delft (NL) and Milano Catholique
University for helpful comments to a previous version of the paper. We can’t avoid to express
our gratitude to Dr. Gabriele Susinno for his crucial contribution in keeping us ”inside the
margins ...”.
†
Facoltà di Economia, Università di Ferrara, [email protected].
‡
Dipartimento di Economia Pubblica, Università di Roma ”La Sapienza”,
[email protected]
1
For example Meade (1986) and Weitzman (1985).
1
for the optimal effort offered by the agent. Along the paper we will show the
sources of these results and the possibles ways out to construct a model that
better represents (i.e. the role of wrong and right underlying assumptions)
what we have in mind as a ”profit-sharing” model.
This negative result is more important in weakening the importance of the
PA model as a valid benchmark than in considering the model answers as true.
The paper is structured as follows. In Section 1 the benchmark model
is introduced, and then studied when the worker revenue includes a share of
the firm profit. Section 2 studies equilibrium conditions, when the bargaining
structure changes, by allowing the agent to share the maximization problem
with the principal. Conclusions follow, underlying weaknesses of the model
and suggesting improvements, that will be correctly analyzed only in a further
work.
1. The Principal-Agent Model under Moral Hazard
We start our analysis from the principal-agent (P-A from now on) model
under moral hazard, as it represents the optimal incentive contract for a wage
contingent on output. Our purpose is to analyze whether in this approach
contingency on profits may have some relative importance with respect to
contingency on output on level of effort and welfare.
To achieve our purpose, we first solve the standard problem under no profit
sharing, we then introduce a share of the principal profit into the agent revenue
in every state of nature.
1.1. Assumptions and definitions in the P-A Model.
This section provides a tedious but necessary list of the necessary definitions and assumptions for the PA model that will underlie all the results of
this work.
Definition 1. Output is determined by the agent’s effort level e ∈ [e0 , e1 ] and
by a random variable s ∈ S ≡ [0, 1] which defines the states of nature
y = y(e, s) ≡ ys (e) ∈ [y0 , y1 ]
Definition 2. Each effort level implies a distribution function of outcomes
contingent on s
F [ys , e]
2
Assumption 1.
Fe [ys , e] < 0
The increase in effort reduces the probability of getting an output smaller
than the specified level.
Assumption 2. Convex distribution function condition (CDFC)
Fee ≥ 0
(1)
Again, an increase in effort reduces the probability of getting an output
smaller than any specified level, but does so at a decreasing rate.
Definition 3. The distribution F [ys , e00 ] stochastically dominates the distribution F [ys , e0 ], if for e00 > e0
£
£
¤
¤
F ys , e00 ≤ F ys , e0 , ∀ y ∈ [y0 , y1 ]
Definition 4. The probability density function contingent on e is:
∂F [ys , e] def
= f (ys , e)
∂y
Assumption 3. Monotone likelihood ratio condition (MLRC)
fe (ys , e)
is non-decreasing in y
f (ys , e)
(2)
fey f − fe fy
∂ (fe /f)
≥0
=
∂ys
f2
(3)
or:
Corollary 1. MLRC implies w0 (ys ) ≥ 0.
Proof.
See the derivation of the first order conditions characterizing
the optimal contract under incomplete information.
Definition 5. The agent’s utility function contingent on the state of nature
is:
def
Us = u [w(ys )] − c (e)
3
Therefore we assume that the agent’s utility function is additively separable in the components w (wage or pay-off) and e.
Assumption 4. The utility function of the agent is concave in the pay-off:
u0 (.) > 0, u00 (.) ≤ 0
the agent may be either risk-neutral or risk-averse.
Assumption 5. The disutility. function of the agent is convex in the effort:
c0 (.) > 0, c00 (.) ≥ 0
Definition 6. The principal’s profit function contingent on the state of nature
is:
def
πs = π [ys − w(ys )]
Assumption 6. The profit function of the principal is concave:
π0 (.) > 0, π00 (.) ≤ 0
the principal may be either risk-neutral or risk-averse.
Definition 7. The expected utility function of the agent is:
def R y
EU = y01 u [w (y)] f (y, e) dy − c(e) =
Ry
= u [w (y)] F [y, e]yy10 − y01 u0 [w (y)] F [y, e] dy − c (e) =
Ry
= u [w(y1 )] − y01 u0 [w (y)] w0 (y) F [y, e] dy − c (e)
Definition 8. The expected profit function of the principal is:
def
R y1
π [y − w (y)] f [y, e] dy = π [y − w (y)] F [y, e]yy10 +
R y1 0
− y0 π [y − w (y)] [1 − w0 (y)] F [y, e] dy =
Ry
= π [y1 − w (y1 )] − y01 π0 [y − w (y)] [1 − w0 (y)] F [y, e] dy
Eπ =
y0
Proposition 1. Under the assumptions (1) and (2) the expected profit function and the expected utility function are concave in the effort.
4
Proof.
∂Eπ
=−
∂e
Z
y1
y0
£
¤
π0 [y − w (y)] 1 − w0 (y) Fe [y, e] dy ≥ 0
because Fe < 0 and MLRC implies w0 (y) ≥ 0.
Z y1
£
¤
∂ 2 Eπ
=
−
π0 [y − w (y)] 1 − w0 (y) Fee [y, e] dy ≤ 0
2
∂e
y0
because Fee ≥ 0.
∂Eu
=−
∂e
Z
y1
y0
u0 [w (y)] w0 (y) Fe [y, e] dy ≥ 0
because Fe < 0 and MLRC implies w0 (y) ≥ 0.
Z y1
∂ 2 Eu
=−
u0 [w (y)] w0 (y) Fee [y, e] dy ≤ 0
∂e2
y0
(4)
(5)
(6)
(7)
because Fee ≥ 0, and c00 (e) ≥ 0.
Definition 9. Define the ”constant contract under complete information” the
pair:
λCI , eCI
Definition 10. Define the ”not revealing output level” (yni ), that level of
output for which the likelihood ratio is zero:
yni :
fe (yni , e)
=0
f
and such that the agent’s pay-off is constant and equal to the contract under
complete information
¡
¢
Corollary 2. The function fe y, eCI is such that:
¡
¢
ª
©
Y − := ©y ∈ [y0 , yni ) : fe ¡ys , eCI ¢ < 0ª
Y + := y ∈ (y0 , yni ] : fe ys , eCI > 0
that is, given a signal of output less than yni every increase in the effort implies
a decrease in the probability and vice versa.
5
1.2. The P-A. Model under ”No profit-sharing”.
We set the standard principal-agent model under moral hazard with one
principal and one agent as benchmark of our analysis. The principal maximizes
his expected profit subject to the participation constraint and to the incentive
compatibility constraint:
Z y1
π [y − w (y)] f (y, e) dy
max Eπ =
w,e
y0
( Ry
1
u [w (y)] f (y, e) dy − c(e) ≥ UR
Ryy01
(8)
s.t.
0
y0 u [w (y)] fe (y, e) dy − c (e) = 0
Notice that the incentive constraint imposes that the agent maximizes his
objective function with respect to e. The Lagrangian is:
¶
µZ y1
Z y1
π [y − w (y)] f (y, e) dy + λ
u [w (y)] f (y, e) dy − c(e) − UR +
L=
y0
y0
¶
µZ y1
u [w (y)] fe (y, e) dy − c0 (e)
+µ
y0
The problem becomes a system of four first order conditions with respect to
w, e, λ and µ, in four unknowns (w, e, λ, µ). Since the FOCs with respect to
λ and µ are just the participation and the incentive constraints, we will focus
the analysis only on the FOCs with respect to w and e.
1.2.1. FOC w.r.t. w.
∂u (w)
∂L ∂π [y − w] ∂ [y − w]
:
f (y, e) + λ
f (y, e) +
∂w ∂ [y − w]
∂w
∂w
∂u [w (y)]
fe (y, e) = 0, w > 0
+µ
∂w (y)
(9)
where ∂[y−w]
= −1.
∂w
Dividing through by ∂u(w)
∂w and f (y, e), and under the condition of binding
constraint, we get the standard P-A incentive contract:
∂π
∂[y−w]
∂u
∂w
=λ+µ
6
fe (y, e)
f (y, e)
(10)
Since the likelihood ratio, ffe , is non decreasing in output (see equation (2)),
the contract is always increasing in the obtained output. In the special case
where the principal is assumed to be risk neutral, the numerator of the LHS
is constant,
1
u0 (w)
=λ+µ
fe (y, e)
f (y, e)
hence the common intuition is that the LHS can be approximated as the
compensation itself
w 'λ+µ
fe (y, e)
.
f (y, e)
where the first term on the RHS is the wage component independent on output,
while the second term is the component dependent on output.
1.2.2. Risk Sharing.
We are interested in studying how the compensation moves w.r.t. the
realization of output. To study this we differentiate the compensation scheme
w.r.t. the output.
Result 1. The revenue of the agent, ceteris paribus, is increasing in the realization of output, the less he is risk-averse. Formally:
0
e /f )
ρP + µ πu0 ∂(f
dw
∂y(s)
=
dy
ρP + ρA
(11)
Proof. By differentiating (9) with respect to y, and denoting by ρP =
the principal’s measure of absolute risk-aversion, and by ρA = −u00 /u0
the agent’s measure of absolute risk-aversion, we get the above result.
The above proposition states that the more the agent is risk-averse, the
less he likes the dependence of his wage on output.
Let’s now see what the optimal condition, with regards to effort, shows.
1.2.3. FOC w.r.t. e.
−π00 /π0
7
Since the dependence on effort is only in the probability, the derivative
affects only the density function. We get:
µZ y1
¶
Z y1
∂L
0
:
π [y − w (y)] fe (y, e) dy + λ
u [w (y)] fe (y, e) dy − c (e) +
∂e
y0
y0
(12)
µZ y1
¶
+µ
u [w (y)] fee (y, e) dy − c00 (e) = 0, e > 0
y0
We immediately see that the second term is zero because of the incentive
constraint, while the third term is negative for the incentive constraint to hold.
Rearranging terms and knowing that the second derivatives in the RHS is
negative by second order conditions, we realize that the derivative of expected
profits w.r.t. the level of effort, calculated at the equilibrium point is positive:
¯
· 2
¸
∂ Eu
∂Eπ ¯¯
00
− c (e) ≥ 0
= −µ
(13)
∂e ¯ ∗
∂e2
e=e
i.e. the principal is never satisfied with the agent’s equilibrium effort, he
always wants more. This is simply another way to state that partecipation
constraint (PC from now on) is always binding, as in the literature2 .
Let’s now see what kind of considerations can be made about welfare.
1.2.4. Welfare.
Since welfare is given by the sum of principal and agent expected values
of the objective functions at the equilibrium point, we have to calculate these
values at w, e, λ and µ equilibrium values. Agent’s expected utility, by participation constraint, is always equal to the reservation utility. For principal’s
expected profit we integrate (9), along the {w∗ (y)} equilibrium path
¸
·
∗
∗
∗ fe
u [w∗ (y)]
π [ys − w (y)] = λ + µ
f
and then over the realizations of output, getting
Z y1
Z y1
π [ys − w∗ (y)] f (ys , e∗ ) dy = λ∗
u [w∗ (y)] f (ys , e∗ ) dy +
y0
y
Z y01
+ µ∗
u [w∗ (y)] fe (ys , e∗ ) dy
y0
2
See Grossman and Hart (1983), Holmström (1979) and Kreps (1990).
8
and knowing that the expected value of the utility function, by the participation constraint, is equal to UR + c(e∗ ). Hence:
Eπ [ys − w∗ (y)] = λ∗ [UR + c(e∗ )] + µ∗ M (e∗ ).
(14)
The integral M(e∗ ) adds on the terms fe (ys , e) . These terms are typically negatives for ”low” levels of output (an increase of effort decreases the probability
of having a low output) and positive for ”high” levels of output (an increase of
effort increases the probability of having a high output). The quality ”high”
ˆ
or ”low” depends for y being larger or smaller
³ˆ
´than the level y such that the
∗
∗
derivative w.r.t. effort nullifies, i.e. fe ys , e = 03 . This position depends
on the density function selected by the equilibrium level of effort. Hence the
integral M(e∗ ) may have any sign depending on effort. We just know that
M(e∗ ) is non decreasing in effort by MLRP and (3). However as we will see
this term does non change under the different model conditions, so we will not
have problems in comparisons.
Welfare, finally, results:
W (w∗ , e∗ ) = EU P S + EπP S
= UR + λ∗ [UR + c(e∗ )] + µ∗ M(e∗ )
1.3. The P-A. Model under ”Profit-sharing”.
We now want to analyze what happens to the agent and principal revenue,
to equilibrium level of effort and to the principal and agent welfare, when
we introduce a share of the principal profit in the agent revenue. The agent
remuneration includes now the standard incentive
and
of ¤ª
the
£ a share
© contract
P
S
P
S
profit. The worker revenue changes from w into w ©(y) + θ y£ − w (y) ¤ª,
while the principal revenue changes from (y − w) into (1 − θ) y − wP S (y)
3
Referring to the above following assumption:
©
¡
¢
ª
Y − := © y ∈ [y0 , yni ) : fe ¡ys , eCI ¢ < 0ª
Y + := y ∈ (y0 , yni ] : fe ys , eCI > 0 )
9
in every state of nature. The problem becomes:
Z y1
©
£
¤ª
max Eπ (w, e | y) =
π (1 − θ) y − wP S (y) f [y, e] dy
w,e
y0
( Ry ©
£
¤ª
1
P
S
P S (y)
u
w
(y)
+
θ
y
−
w
f [y, e] dy − c(e) ≥ UR
y
£
¤ª
R y01 © P S
s.t.
P
S
(y) + θ y − w (y) fe [y, e] dy − c0 (e) = 0
y0 u w
The Lagrangian is:
¶
µZ y1
Z y1
£ PS¤
£ P S¤
L=
f (y, e) dy + λ
f (y, e) dy − c(e) − UR +
π x
u z
y0
y0
¶
µZ y1
£ PS¤
0
u z
+µ
fe (y, e) dy − c (e) ,
y0
where:
Definition 11. the ”principal revenue under profit sharing” is
£
¤
def
xP S = (1 − θ) y − wP S (y)
Definition 12. the ”agent revenue under profit sharing” is
£
¤
def
z P S = wP S (y) + θ y − wP S (y) = (1 − θ) wP S (y) + θy.
Following the structure of the previous section, we concentrate our analysis
only on the FOCs with respect to wP S and e.
1.3.1. FOC w.r.t. wP S .
∂L
∂π ∂xP S
∂u ∂z P S
:
f
(y,
e)
+
λ
f (y, e) +
∂wP S ∂xP S ∂wP S
∂z P S ∂wP S
∂u ∂z P S
+µ P S
fe (ys , e) ≤ 0
wP S ≥ 0
∂z ∂wP S
and since
∂xP S
= − (1 − θ) ;
∂wP S
∂z P S
= (1 − θ) ,
∂wP S
10
(15)
we get:
∂L
∂π
∂u
: − P S (1 − θ) f (y, e) + λ P S (1 − θ) f (y, e) +
P
S
∂w
∂x
∂z
∂u
+µ P S (1 − θ) fe (ys , e) ≤ 0
wP S ≥ 0
∂z
The term (1 − θ) cancels out leading to:
·
¸
∂L
fe (y, e)
∂π
∂u
≤0
: − PS + PS λ + µ
∂wP S
∂x
∂z
f (y, e)
wP S ≥ 0
(16)
(17)
From equation (17), and under the condition of binding constraint, the PA
incentive contract under profit sharing is, again:
∂π
∂xP S
∂u
∂z P S
=λ+µ
fe
f
(18)
The contract defining the solution for the new definitions of revenue z P S and
xP S is the same as the contract defining w in the previous model under no
profit sharing. Hence, surprisingly
Lemma 1. In the PA model, profit sharing in every state of nature does not
change the agent equilibrium revenue
¡
¢
(19)
w = z P S ≡ wP S + θ y − wP S
Therefore
¡
¢
(i) the contingent wage wP S varies according to keep z P S ≡ wP S +θ y − wP S =
w;
(ii) The optimal level of share, θ∗ , is uniquely determined by equation (19).
Proof.
The FOC w.r.t. w in the profit sharing case (17) or (18) is
identical to the FOC w.r.t. w in the no profit sharing case (9) or (10) once
agent and principal revenues are redefined.
11
Remark 2. The
components, the contingent wage wP S and the
¢
¡ twoP Srevenue
are perfect substitutes
profit share θ y − w
1
dwP S
=−
d [θ (y − wP S )]
1−θ
(20)
¢
¡
Proof.
Let’s rewrite the agent revenue z P S = wP S + θ y − wP S =
(1 − θ) wP S + θy. By totally differentiating the (16) w.r.t. the two arguments
(1 − θ) wP S and θy,
#
"
∂ 2 us
∂ 2 us
∂2πs
f (ys , e) + λ
f (ys , e) + µ
fe (ys , e) dwsP S +
(1 − θ) −
∂ (xP S )2
∂ (z P S )2
∂ (z P S )2
#
"
¢¤
£ ¡
∂ 2 us
∂ 2 us
∂ 2 πs
+λ
f (ys , e) + µ
fe (ys , e) d θ ys − wsP S = 0
+ −
2
2
2
∂ (xP S )
∂ (z P S )
∂ (z P S )
we obtain the result.
The result that profit sharing implies no change in agent revenue, implies
also that:
Corollary 3. In the PA model, profit sharing doesn’t change
(i) neither the principal equilibrium revenue in every state of nature, which
remains constant. Formally
xP S = (y − w)
¤
PS
£
where xP S ≡ (1 − θ) yP S − w
(ii) nor the expected value of principal profits
Z
¤
£
(21)
(1 − θ) π y P S − wP S f (y, e∗ ) dy ≡ Eπ P S =
y
Z
= Eπ ≡ π (y − w) f (y, e∗ ) dy
y
Proof.
By lemma 1 we established that w = (1 − θ) wP S + θy P S , hence:
wP S =
θ PS
w
−
y
(1 − θ) 1 − θ
12
Substituting in the principal revenue
·
½
PS
PS
x = (1 − θ) y −
θ
w
−
yP S
(1 − θ) 1 − θ
n
h
w
−
xP S = (1 − θ) y P S − (1−θ)
ª
© PS
= y −w
θ
PS
1−θ y
¸¾
io
but since the equilibrium
© P S effortªhas not changed, the density function of output
is the same, hence y − w = {y − w} . Being the equilibrium effort the
same, the equilibrium density function is the same, then the expected value
of profits does not change, leading to (21) above.
1.3.2. Risk sharing
To investigate the behavior of the compensation as the result in output
changes, we again differentiate the FOC w.r.t. y.
Result 2. Risk sharing when a part of worker wage is due to profit sharing
is
ρP +
dw
=
dy
µ∗ u0 ∂(fe /f )
(1−θ) π 0 ∂y(s)
ρP + ρA
θ
− ρA 1−θ
increasing convexly in θ4 . The wage request in the likelihood ratio increases
with the share of profit θ.
Proof. Usual. See appendix.
This result, when compared with the no profit sharing case (11), has a new
third negative term, while the second is divided by (1 − θ) . The second term
shows an increased ”insurance effect”. The higher θ, the more dependence
on the likelihood ratio the agent gets and hence the higher wage component
4
0
e /f )
ρA (θ − 1) + ρA θ − µ∗ uπ0 ∂(f
d2 w
∂y(s)
=
>0
dydθ
(ρP + ρA ) (1 − θ)2
with positive higher derivatives.
13
he pretends. On the contrary, the more risk averse the agent is, the larger
θ
with negative sign, requiring less dependence from output,
is the term ρA 1−θ
proportionally to θ.
Remarks
1. The proposition is equivalent to the no-sharing case for θ = 0.
2. The dependence on the LR is maximized when ρA = 0 (i.e. the agent is
RN)
¯
u0 ∂(fe /f )
µ∗
dw ¯¯
=1+
¯
dy ρA =0
(1 − θ) ρP π0 ∂y(s)
3. On the contrary, when the principal is RN the first term disappears,
decreasing the multiplier
¯
dw ¯¯
u0 ∂(fe /f)
µ∗
θ
+
.
=−
¯
dy ρP =0
(1 − θ) (1 − θ) ρA π0 ∂y(s)
1.3.3. FOC w.r.t. e.
µZ y1
¶
Z y1
£ PS¤
£ PS¤
∂L
0
=
fe (y, e) dy + λ
fe (y, e) dy − c (e) +
π x
u z
∂e
y0
y0
¶
µZ y1
£
¤
u z P S fee (y, e) dy − c00 (e) ≤ 0,
e≥0
+µ
y0
Also in this case the second term is zero because of the incentive constraint,
and the third term is negative for the incentive constraint to hold. Hence:
¯
· 2
¸
∂Eπ ¯¯
∂ Eu
00
− c (e) ≥ 0
= −µ
∂e ¯e=e∗
∂e2
which is the ”same condition as in the P-A” model under no profit sharing as
in (13).
Moreover, we can prove that the level of effort chosen by the agent under
profit sharing is the same as the one chosen under no profit sharing.
14
¡
¢
Lemma 3. e z P S∗ ≡ e (w∗ ) .
Proof.
Since the agent revenue does not change as proved by Lemma
1, the agent maximizing choice of effort cannot change, leading to the same
equilibrium probability distribution f (y, e∗ ) .
Hence, we get the following result.
Lemma 4. λP S∗ ≡ λ∗ , µP S∗ ≡ µ∗ .
Proof. Given the same probability distribution, f (y, e∗ ), and the same
agent revenue, z P S , the system of the four FOC, stated in gross revenues, is
the same as the system of FOCs in the ”no-sharing” case, leading therefore to
the same solution values λ∗ and µ∗ .
We are now able to state the first ”Irrelevance” proposition.
Proposition 2. Under the assumptions of section 1.1. and Lemmas 1, 2 and
3, there is irrelevance of profit sharing in the principal agent model: profit
sharing is unable to change agent and principal equilibrium revenues, the
equilibrium level of effort and the lagrange multipliers
z P S = w,
xP S = {y − w}
eP S∗ ≡ e∗
λP S∗ ≡ λ∗ ,
µP S∗ ≡ µ∗
Proof. Consider the equilibrium four FOCs defining the no profit sharing P-A equilibrium: {w∗ , e∗ , λ∗ , µ∗ } . Consider ©now the equilibrium four
ª
FOCs defining the profit sharing P-A equilibrium z P S∗ , eP S∗ , λP S∗ , µP S∗ .
We first notice that z P S∗ solves (17) as w∗ solved (9). Hence the agent total revenue does not change and also the principal revenue is the same too
(corollary 3). By lemma 3 we showed that being the agent revenue the same,
same maximizing choice for effort will be done. Having changed the revenues
definitions, the two constraints are the same, hence same are the multipliers,
as in lemma 4.
1.3.4. Welfare.
The invariance of the equilibrium solutions imply invariance of welfare.
15
Corollary 4. Proposition 2 implies irrelevance of profit sharing in increasing
wealth in the PA model
£ P S ¡ P S P S ¢ P S∗ ¤
PS
z
w ,θ
,e
Wpa (w∗ , e∗ ) = Wpa
Proof.
Following the same reasoning as in the no-sharing case, by
integrating the FOC w.r.t. w (17) we get:
£
¤
EπP S = λ∗ UR + c(eP S∗ ) + µ∗ M(eP S∗ )
which is exactly the same condition as the ”no-sharing” case (14). Moreover,
since equilibrium effort is the same as in the no profit sharing case by lemma
3, the term M(eP S∗ ) is the same as before. Hence:
£ P S ¡ P S P S ¢ P S∗ ¤
PS
Wpa
z
w ,θ
,e
= EU P S + EπP S
¤
£
= UR + λ∗ UR + c(eP S∗ ) + µ∗ M(eP S∗ )
= UR + λ∗ [UR + c(e∗ )] + µ∗ M(e∗ )
= Wpa (w∗ , e∗ )
where the subscript ”pa” denotes the principal agent setting.
We are then able to assert that in the Principal Agent model the introduction of a share of the principal profits into the agent revenue has no effects
neither on the level of effort delivered by the agent, nor on the welfare.
Conclusion 1. In the P-A model agent, revenue contingent on output already
represents the optimal incentive contract and the introduction of a share of
profits is not able to add anything more. The reason for this irrelevance
result is (i) homogeneity of the two sources of revenue and (ii) the fact that
they enter the utility and profit function linearly , hence they are perfect
substitutes. These facts lead to the striking result that the model optimizes
the argument of the utility and the profit functions, although we derive w.r.t.
the wage. This was not expected and depends on the additivity of the two
objects: the proper wage and the share of profit.
2. Collusion under Moral Hazard
We now wonder whether there is a role for profit sharing in a contract
structure, where the agent is allowed to share decisions with the principal.
16
Having both subjects with decision power is here represented through a collusive setting, in which both the principal and the agent maximize a joint
objective function, given by the sum of the expected profit function and the
expected utility function.
We will notice that in this framework the incentive constraint loses meaning. There is an inner contradiction in having the agent optimizing twice: both
at the participation constraint and at the objective function levels. Interpreting the PA model as a leader-follower framework to be solved backwards, the
contradiction is evident. The collusive setting with incentive constraint may
be interpreted like having the agent as follower and both of them as leaders:
this is actually meaningless and we will discuss this issue in the following section. In this section, we are interested to keep the same framework, to be
able to directly compare the role of an increased agent decision power in the
contract. The problem is first solved under no-sharing, then under sharing.
2.1 Collusion under No Profit-Sharing.
Allowing the agent to collude with the principal, both the participants
maximize a joint objective function given by the sum of the expected profit
function and the expected utility function. The new problem is defined as:
Z y1
Z y1
π [y − w (y)] f (y, e) dy +
u [w (y)] f (y, e) dy − c(e)
(22)
max
w,e y
y0
0
( Ry
1
u [w (y)] f (y, e) dy − c(e) ≥ UR
Ryy01
s.t.
(23)
0
y0 u [w (y)] fe (y, e) dy − c (e) = 0
where the Lagrangian is:
¶
µZ y1
Z y1
π [y − w (y)] f (y, e) dy + (1 + λ)
u [w (y)] f (y, e) dy − c(e) − UR +
L=
y0
y0
¶
µZ y1
u [w (y)] fe (y, e) dy − c0 (e)
+µ
y0
2.1.1. FOC w.r.t. w.
∂u [w (y)]
∂L ∂π [y − w (y)] ∂ [y − w (y)]
:
f (y, e) + (1 + λ)
f (y, e) +
∂w ∂ [y − w (y)]
∂w (y)
∂w (y)
∂u [w (y)]
fe (y, e) = 0, w > 0
+µ
∂w (y)
17
(24)
where
∂[y−w(y)]
∂w(y)
= −1.
Dividing through by ∂u[w(y)]
∂w(y) and f (y, e), and under the condition of binding constraint, we get the collusive incentive contract:
∂π
∂[y−w(y)]
∂u
∂w(y)
= 1+λ+µ
fe
f
(25)
Remark 5. We have a term (1 + λ) substituting the term λ of the previous
setting. This may lead to believe that π 0 is a higher multiple of u0 , and hence
the worker revenue is a higher multiple of the principal revenue (because of decreasing marginal utilities). Unfortunately, as it will be shown, in equilibrium
(1 + λ)∗CL = λ∗P A .
2.1.2. Risk Sharing
The result in risk sharing is exactly equal to the principal agent setting.
Result 3. Given the assumptions of section 1.1., the revenue of the agent,
ceteris paribus, is increasing in the realization of output, the less she is riskaverse. Formally:
0
e /f )
ρP + µ πu0 ∂(f
dw
∂y(s)
=
dy
ρP + ρA
Proof.
By differentiating (24) w.r.t. y.
2.1.3. FOC w.r.t. e.
µZ y1
¶
Z y1
∂L
0
π [y − w (y)] fe (y, e) dy + (1 + λ)
u [w (y)] fe (y, e) dy − c (e) +
:
∂e
y0
y0
(26)
µZ y1
¶
+µ
u [w (y)] fee (y, e) dy − c00 (e) = 0, e > 0
y0
Once again the second term is zero because of the incentive constraint, and
importantly, it nullifies the role of the term ”1” which is the component due
to decision sharing.
18
The third term is negative since it is the second order condition at a maximum, needed for the incentive constraint to hold. Rearranging terms, as
before, we get:
¸
· 2
∂ Eu
∂Eπ
00
− c (e) ≥ 0
= −µ
∂e
∂e2
Remark 6. The only innovation coming from collusion, (1 + λ) instead of λ,
is nullified by the incentive constraint, leaving the same first order condition
with respect to effort.
The equality between the FOCs w.r.t. e in collusion and in the PA-A
model leads to the following irrelevance result of decision sharing.
Proposition 3. Under the assumptions of section 1.1. there is ”irrelevance
of collusion under moral hazard”
∗
∗
= wcl
;
wpa
e∗pa = e∗cl ;
λ∗pa = (1 + λ∗cl ) ;
µ∗pa = µ∗cl .
Proof. The system of four FOCs under collusion [(24) , (26) , (23)] and
the system of four FOCs under the leader-follower framework5 [(9) , (12) , (8)]
is the same and hence leads to the same solution, provided (1 + λ∗CL ) = λ∗P A .
We have only to prove that (1 + λ∗CL ) = λ∗P A . But the scalar 1 is a constant
added to the constant λCL , wherever the term λCL appears, hence the role
of λ∗pa is substituted by (1 + λ∗cl ) which has necessarily, ceteris paribus, to be
solved for the same values. This implies that agent and principal revenues are
the same. Equilibrium effort is the same. since agent revenue is the same as
in the P-A model and the new term (1 + λ) is nullified, the solution of the
equation (26) is the same of the equation (12) in the P-A model.
2.1.4. Welfare.
By the same argument as before (see proof of corollary 4), being the solution revenues the same, same will be welfare.
Corollary 5. Invariance of welfare under change of decision sharing under
moral hazard
Wcl = Wpa
5
Or principal-agent framework.
19
Same a before. See appendix.
Proof.
Remark 7. The reason for irrelevance of decision sharing is due to the fact
that the incentive constraint implies the agent to be already effort maximizer.
Hence, he cannot do better. The problem for the collusive contract is represented by this kind of structure. It is meaningless that, although the agent
is maximizing at the incentive constraint level, he maximizes at the objective
function level, too. This second step is shown to be irrelevant under asymmetric information, when the incentive constraint has to be binding.
2.2. Collusion under Profit-Sharing.
We wonder now, whether profit sharing may have some relevance in the
new contract form.
Introducing a share of the principal profit into the agent revenue under
collusion leads to the following problem:
Z y1
©
£
¤ª
π (1 − θ) y − wP S (y) f [y, e] dy +
max
w,e y
0
Z y1
©
£
¤ª
u wP S (y) + θ y − wP S (y) f [y, e] dy − c(e)
+
y
( 0R y ©
£
¤ª
1
u wP S (y) + θ y − wP S (y) f [y, e] dy − c(e) ≥ UR
y
0
R y1 © P S
£
¤ª
s.t.
(y) + θ y − wP S (y) fe [y, e] dy − c0 (e) = 0
y0 u w
and
L=
Z
y1
y0
+µ
µZ
£ S¤
π xPCL
f (y, e) dy + (1 + λ)
y1
y0
µZ
y
y1
¶ 0
£ PS¤
u zCL fe (y, e) dy − c0 (e) ,
¶
£ PS¤
u zCL f (y, e) dy − c(e) − UR +
where again:
£
¤
xPcl S = (1 − θ) y − wP S (y)
is the principal revenue in collusion under profit sharing, and
£
¤
PS
zcl
= wP S (y) + θ y − wP S (y) = (1 − θ) wP S (y) + θy
20
is the agent revenue in collusion under profit sharing.
2.2.1. FOC w.r.t. wP S .
Since again
∂xPcl S
= − (1 − θ) ;
∂wP S
PS
∂zcl
= (1 − θ) ,
∂wP S
we get:
∂L
∂π
∂u
: − P S (1 − θ) f (y, e) + (1 + λ) P S (1 − θ) f (y, e) +
P
S
∂w
∂xcl
∂zcl
∂u
wP S ≥ 0
+µ P S (1 − θ) fe (ys , e) ≤ 0
∂zcl
and still under collusion the term (1 − θ) cancels out leading to:
·
¸
fe (y, e)
∂π
∂u
∂L
: − PS + PS 1 + λ + µ
≤0
wP S ≥ 0
∂wP S
f (y, e)
∂xCL ∂zCL
(27)
(28)
¿From equation (28), and under the condition of binding constraints, the
collusive incentive contract under profit sharing is:
∂π
S
∂xP
cl
∂u
PS
∂zcl
=1+λ+µ
fe
f
(29)
P S is the same as
Also under collusion the contract defining the solution for zCL
the contract defining w in the collusive setting under no profit sharing. Hence
we are able to establish the following irrelevance result.
Lemma 8. Also under collusion, profit sharing in every state of nature doesn’t
change the agent equilibrium revenue. Formally:
¡
¢
PS
≡ wP S + θ y − wP S
w = zcl
Proof.
See proof of Lemma 1 in the PA model case.
Hence, also in the collusive model under
sharing,
the contingent wage
¢
¡ profit
P
S
P
S
P
S
= w.
varies according to keep zcl ≡ w + θ y − w
21
Proposition 4. Since the system of FOC is the same as the case under nosharing, once the problem is set in terms of gross revenues of the two subjects,
there is irrelevance of profit sharing in the collusive model:
1.
PS
= wcl ;
zcl
xPcl S = {y − w}cl
2.
ePcl S∗ ≡ e∗cl
3.
λPcl S∗ ≡ λ∗cl ,
Proof.
µPcl S∗ ≡ µ∗cl
As above. See Appendix.
Corollary 6.
PS
= Wpa
WclP S = Wcl = Wpa
Proof.
As above. See appendix.
Hence, also in the collusive model the introduction of a principal profit
share into the agent revenue has no effects neither on the level of effort, nor on
the welfare. Moreover, since in the previous section we have proved the irrelevance of decision sharing in the P-A model under incomplete information, we
can assert that, under incomplete information, the collusive equilibrium under
profit sharing coincides with the P-A equilibrium under no profit sharing.
CONCLUSIONS
Although the incentive role of profit sharing in contracts is an established
common sense, we were looking for a closed form solution able to show the
additional incentive role of profit sharing to a wage which is already contingent
on output. In order to do this, we added the possibility of profit sharing in
the classical principal-agent incentive model. Surprisingly the moral hazard
setting lead to negative results both under a leader-follower and under a collusive relationship between the two contract parts. The main results may be
summarized as follows:
22
1. Profit sharing under moral hazard is irrelevant both under a leaderfollower (principal-agent model) and under a collusive setting. Maximization of profit and utility function leads to a unique optimal level
of both revenues independent of incentive sources. The existence of the
profit share in worker revenue decreases his need of incentive wage.
2. Being revenues the same, the optimal choice for the equilibrium effort
level is constant as well.
3. Collusion under moral hazard has no role, since there is redundancy of
either the binding incentive constraint or the collusion role of the agent.
4. Hence the final result is that the Principal-Agent model under moral
hazard determines a unique optimal amount of revenue for a two side
contract. Of course the equilibrium revenues depend on the roles assigned to the two side of the contract: entrepreneur as principal and
worker as agent. In a further work we investigate what happens allowing the possibility of changing the roles.
The first result is robust under a multi-agents (multi-workers) contract to
the extent that they are symmetric. Relevance of profit sharing starts when
agents are different, as shown in a further work.
In order to establish a role for collusion the necessary condition is for the
incentive constraint to disappear. In other words, collusion is Pareto improving for the joining agents, if and only if it is possible to assume a complete
information setting or perfect monitoring. In this context we have shown
[Mancinelli and Miceli (2000)] that collusion under complete information leads
to a higher equilibrium effort level. Of course, canceling incentive constraints
while asymmetric information problems exist, implies free-riding problems,
to be eliminated through a right incentive structure, using for example the
argument of increasing returns to scale [Caravani and Miceli (1993)].
References
[1] Caravani, P. and M.A. Miceli (1993) ”Game Theoretic Regulation to
Oligopoly” in Dynamic Optimal Control and Economic Applications,
Feuchtinger H. Ed., North Holland.
23
[2] Fershtman, C. and K. Judd (1987) ”Equilibrium Managerial Incentives
in Oligopoly”, American Economic Review, vol. 77, pp. 927-40.
[3] Fudenberg D. and J. Tirole(1990) ”Moral Hazard and Renegotiation in
Agency Contracts”, Econometrica, vol.58, n.4, pp.1279-1319.
[4] Gravelle, H. and R. Rees, (1992), Microeconomics, Longman.
[5] Grossman S. and O. Hart (1983) ”An Analysis of the Principal-Agent
Problem”, Econometrica, vol.56, n.1, pp.7-45.
[6] Holmström B. (1979) ”Moral Hazard and Observability”, Bell Journal of
Economics, no. 10, pp.74-91.
[7] Jewitt I. (1988) ”Justifying the First Order Approach to Principal-Agent
Problems”, Econometrica, vol.51, n.5, pp.1177-1190.
[8] Kreps, D. M. (1990) A Course in Microeconomic Theory, Princeton University Press.
[9] Laffont J.J. and J. Tirole (1988) ”The Dynamics of Incentive Contracts”,
Econometrica, vol.56, n.5, pp.1153-1175. University Press.
[10] Mancinelli, S. and M. A. Miceli (2000) ”Profit and Decision Sharing in
the Principal Agent Model”, WP, Università di Ferrara.
[11] Meade J. (1986), Alternative Systems of Business Organization and of
Workers’ Remuneration, London, Allen & Unwin.
[12] Rogerson W. P. (1985) ”The First Order Approach to Principal-Agent
Problems”, Econometrica, vol.53, n.6, pp.1357-1367.
[13] Shavell S. (1979) ”Risk Sharing and Incentives in the Principal-Agent
Relationship”, Bell Journal of Economics, no. 10, pp. 55-73.
[14] Weitzman, M. L. (1985) ”Profit Sharing as Macroeconomic Policy”,
American Economic Review, 75, May, pp. 937-953.
[15] Weitzman M.L. and L. Kruse (1990) ”Profit Sharing and Productivity”,
in Blinder A. Ed. (1990), Paying for Productivity: a Look at the Evidence,
Washington D.C, The Brookings Institutions.
24
APPENDIX
Proof of Result 2.
By differentiating the FOC over the wage (16) w.r.t. y, and recalling that
ρP = −π00 /π0 and ρA = −u00 /u0 . To simplify notation we omit the
subscript P S, writing w instead of wP S in the following.
³
´
∂2L
00 [.] 1 − dw(y) (1 − θ)
:
−π
dy
∂w(y)∂yi
h
o
n
∗ u00 (1 − θ) dw + θ fe + µ∗ u0 fey f −fe fy =
+
θ
+
µ
+λ∗ u00 [.] (1 − θ) dw
dy
dy
f
f2
dw(y)
∗ 00
= −π00 (1 − θ) + π00 (1 − θ) dw(y)
dy + λ u [.] (1 − θ) dy +
∗ 00 fe
∗ 0 fey f −fe fy
+λ∗ u00 [.] θ + µ∗ u00 ffe (1 − θ) dw(y)
dy + µ u f θ + µ u
f2
dividing through by (1 − θ)
dw (y)
dw (y)
θ
+ λ∗ u00
+ λ∗ u00 [.]
+
dy
dy
1−θ
fe dw (y)
fe θ
u0 fey f − fe fy
+ µ∗ u00
+ µ∗
=0
+µ∗ u00
f dy
f 1−θ
1−θ
f2
³ 0
´
since λ∗ = πu0 − µ∗ ffe in equilibrium
µ 0
¶
µ 0
¶
π
π
θ
00
00 dw (y)
∗ fe
00 dw (y)
∗ fe
−µ
−µ
+
+
+
u
u00 [.]
−π + π
0
0
dy
u
f
dy
u
f
1−θ
fe dw (y)
fe θ
u0 fey f − fe fy
+ µ∗ u00
+ µ∗
=0
+µ∗ u00
f dy
f 1−θ
1−θ
f2
−π00 + π00
dw (y) π0 00 dw
fe dw (y)
− µ∗ u00
+ 0u
dy
u
dy (s)
f dy
0
θ
fe θ
π
− µ∗ u00 [.]
+
+ 0 u00 [.]
u
1−θ
f 1−θ
fe dw (y)
fe θ
u0 fey f − fe fy
+ µ∗ u00
+ µ∗
=0
+µ∗ u00
f dy
f 1−θ
1−θ
f2
−π00 + π00
25
and finally dividing by π0
−
π00 π00 dw (y) u00 dw (y) u00 θ
µ∗ u0 fey f − fe fy
+
+
+
+
=0
π0
π0 dy
u0 dy
u0 1 − θ 1 − θ π0
f2
¸
·
dw (y) π00 u00
π 00 u00 θ
µ∗ u0 fey f − fe fy
−
=
−
+
+
+
=0
dy
π0
u0
π0
u0 1 − θ 1 − θ π0
f2
we get the result.
0
e /f )
θ
ρP − ρA 1−θ
+ µ∗ πu0 ∂(f∂y
dw
=
dy
ρP + ρA
¥.
Proof to Corollary 5 Let’s consider (24). By integrating it with respect to
the {w∗ (y)} sequence and over the states of nature we get:
Z y1
Z y1
∗
π [ys − w (y)] f (ys , e) dy = (1 + λcl )
u [w∗ (y)] f (ys , e) dy +
y0
y0
Z y1
u [w∗ (y)] fe (ys , e) dy
+µ
y0
but, we know that the expected value of the utility function, by the
participation constraint is equal to UR + c(e∗ ), hence
Eπ [ys − w∗ (y)] = (1 + λ∗CL ) [UR + c(e∗ )] + µ∗ M(e∗ )
where M(e∗ ) has³positive
´ or negative sign depending on where the point
ˆ
ˆ
y such that fe∗ ys , e∗ = 0. However, since equilibrium effort is the
same as in the PA model, the term M(e∗ ) is the same as before, and
26
since we have proved that (1 + λCL ) = λP A , we can also assert that
welfare in collusion is the same as in the P-A setting:
Wcl = EU + Eπ
= UR + (1 + λ∗CL ) [UR + c(e∗ )] + µ∗ M(e∗ )
= UR + λ∗P A [UR + c(e∗ )] + µ∗ M (e∗ )
= Wpa . ¥
Proof of Prop. 4
1.
¡
¢
PS
w = zCL
≡ wP S + θ y − wP S
As in the PA model case, the FOC w.r.t. w in the profit sharing case
(28) or (29) is identical to the FOC w.r.t. w in the no profit sharing case
(24) or (25) once agent and principal revenues are redefined.
2.
xPclS = {y − w}cl
To be followed the same proof as in Corollary 2.As a consequence we
have that also in the collusive setting the expected value of profits under
profit sharing is the same as under no profit sharing.
¡ P S∗ ¢
≡ e (w∗ ) .
3. Also under collusion e zCL
By FOC w.r.t. effort
µZ y1
¶
Z y1
£ S¤
£ PS¤
∂L
=
fe (y, e) dy + (1 + λ)
fe (y, e) dy − c0 (e) +
π xPCL
u zCL
∂e
y0
y
¶0
µZ y1
£ P S¤
u zCL fee (y, e) dy − c00 (e) ≤ 0,
e≥0
+µ
y0
the second term is zero because of the incentive constraint, and the third
term is negative for the incentive constraint to hold. Thus
¸
· 2
P S∗ )
∂ Eu(zCL
∂Eπ
00
− c (e) ≥ 0
= −µ
∂e
∂e2
27
we get the same condition as before, since the agent revenue is the same
as before. Hence we can prove that also in the collusive setting the best
response schedule e(w), chosen by the agent under profit sharing, is the
same as the one chosen under no profit sharing.
4. λPcl S∗ ≡ λ∗cl ,
µPcl S∗ ≡ µ∗cl
See proof of Proposition 4 in the P-A setting.
¥
Proof to Corollary 6
PS
= Wpa
WclP S = Wcl = Wpa
We proved that also under collusion equilibrium revenues under profit
sharing are coincident with the no profit sharing case. Moreover in
Corollary 5 we proved, in the no-sharing case, that welfare under collusion replicates welfare under principal-agent setting, therefore the thesis.
¥
Corollaries to Prop. 4
P S and the profit
1. The two¡ revenue ¢components, the contingent wage wCL
P
S
share θ y − wCL are hence still perfect substitutes.
Proof.
Same as proof of corollary 2 in the P-A model case.
2. The risk sharing is allocated as follows.
dw
=
dy
Proof.
ρP −
θ
(1−θ) ρA
+
µ∗ u0 ∂
(1−θ) π 0 ∂y
ρP + ρA
³ ´
fe
f
As the case of PA model with profit sharing (Prop. 2).
28