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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. 3