Ayudantia 10 Solución

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PONTIFICIA UNIVERSIDAD CATÓLICA DE CHILE
Facultad de Ciencias Económicas y Administrativas
EAA200B-3 Fundamentos de Dirección de Empresas
1 Managerial Incentives
Next, we investigate the properties of a very popular compensation scheme for
executives. We use the setting with CARA utility for the Agent. As before, the
manager takes an action e which a¤ects the output of the �rm. Suppose that
the output signal takes the form:
y = e+ "y
At the same time, the action of the manager also a¤ects the price of the �rm,
p; through the formula:
p = e+ "p
"i � N
�
0; �2i
�
; where i = y; p: The two noise terms are independent. The
Agent is risk-averse with CARA preferences:
u (w; e) = � exp f�� [w � (e)]g
where � > 0 is the agent�s coe¢ cient of absolute risk-aversion. The Principal is
risk-neutral. Assume for simplicity that (e) = 12ce
2: The Principal considers
only linear contracts, i.e.:
w = �+ �y + 
p
Suppose that the Principal cares only about the dividend �ow from the �rm, so
that her objective function is the �ow of pro�ts (under some assumptions, this
is no very restrictive assumption: what assumptions?):
E (y � w)
Recall that for " � N
�
0; �2
�
;
E (exp f
"g) = exp
�
1
2

2�2
�
which implies that
E (exp f��"yg) = exp
�
1
2
�2�2�2y
�
1
and
E (exp f�
"pg) = exp
�
1
2
�2
2�2p
�
Then maximizing the Agent�s utility is equivalent to maximizing the certainty
equivalent de�ned by:
� exp
�
��
�
�+ (� + 
) e� 1
2
ce2 � 1
2
��2�2y �
1
2
�
2�2p
��
= � exp f��w (e)g
In other words, maximizing the objective function of the Agent is equivalent to
maximizing
�+ (� + 
) e� 1
2
ce2 � 1
2
��2�2y �
1
2
�
2�2p
Let w be the outside o¤er of the Agent.
1. Solve the problem of the Agent.
Solution: The modi�ed problem of the Agent is:
max
e
�
�+ (� + 
) e� 1
2
ce2 � 1
2
��2�2y �
1
2
�
2�2p
�
The �rst-order condition for this problem is:
e =
� + 
c
2. Use the solution to simplify the problem of the Principal and then solve
for the optimal linear contract.
Solution: Substituting this expression for e¤ort into the Principal�s prob-
lem, we obtain:
max
�;�
8<: E(y)| {z }
revenues
� [� (E (y)) + 
 (E (p)) + �]| {z }
costs
9=;
Taking expectations:
max
�;�
8>><>>:
�
� + 
c
�
| {z }
revenues
�
�
�
�
� + 
c
�
+ 
�
� + 
c
�
+ �
�
| {z }
costs
9>>=>>;
The problem of the shareholders is:
max
�;�
8<:(1� � � 
)
�
� + 
c
�
� �
9=;
2
subject to
�+ (� + 
)
�
� + 
c
�
� 1
2
c
�
� + 
c
�2
� 1
2
��2�2y �
1
2
�
2�2p = w
We can use the IR constraint to express � in terms of � :
� = w �
 
1
2
(� + 
)
2
c
� 1
2
��2�2y �
1
2
�
2�2p
!
The the problem of the Principal can be solved using unconstrained optimization
methods:
max

;�
(
(1� � � 
)
�
� + 
c
�
�
 
w �
 
1
2
(� + 
)
2
c
� 1
2
��2�2y �
1
2
�
2�2p
!!)
The �rst-order condition for this problem yields the following solution for �
and 
 :
� =
�2p
�2p + �
2
y + �c�
2
p�
2
y

 =
�2y
�2p + �
2
y + �c�
2
p�
2
y
3. Study how the optimal � and 
 depends on the variance of y and p, the
cost of e¤ort c; and the Agent�s risk-aversion.
Solution: The base pay � depends on the reservation wage w and �: The
characteristics of the solution for � are:
� The bonus rate is equal to a fraction of the total output (or revenue after
the normalization of the price of the product to 1).
� As �2p increases, the incentives for high output increase while the incentives
for high price decrease. Similarly for the e¤ect of more volatile y: the
incentives for high output decrease while the incentives for high stock
price increase. The incentives to reward high performance decrease as risk-
aversion increases, which re�ects the underlying tension between insurance
and incentives.
� The incentives to reward high performance decrease as the subjective cost
of e¤ort c increases.
� The incentives to reward high performance decrease as the noise in the
output signal increases.
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