Optimal Incentives in a Principal-Agent Model with Endogenous Technology

One of the standard predictions of the agency theory is that more incentives can be given to agents with lower risk aversion. In this paper we show that this relationship may be absent or reversed when the technology is endogenous and projects with a higher e¢ ciency are also riskier. Using a modi(cid:133)ed version of the Holmstrom and Milgrom(cid:146)s (1987) framework, we obtain that lower agent(cid:146)s risk aversion unambiguously leads to higher incentives when the technology function linking e¢ ciency and riskiness is elastic, while the risk aversion-incentive relationship can be positive when this function is rigid.


Introduction
One of the main results of the agency theory is the trade-o¤ between incentives and insurance. Lower agent's risk aversion allows the principal to provide more incentives by making the payment of the agent more related to output while higher uncertainty increases the gains from insuring the agent and reduces the pay-for-performance sensitivity. The empirical works testing the link between uncertainty and incentives have found mixing results however (e.g., Rao and Hanumantha, 1971; Allen and Lueck, 1995; Aggarwal and Samwick, 1999;Core and Guay, 2002;Wulf, 2007). In many cases, the empirical …ndings are even in contradiction with the standard predictions of the theory as they document a positive (rather than negative) correlation between observed measures of uncertainty and the provision of incentives (see Prendergast, 2002, for an extensive discussion on this point).
Recently the matching literature (e.g., Wright, 2004;Legros and Newman, 2007;Serfes, 2005Serfes, , 2008; Li and Ueda, 2009) has attempted to provide a justi…cation of the above results based on the endogenous matching between principals and agents. For instance, Serfes (2005) shows that, whereas under e¢ cient positive assortative matching (in which higher risk-averse agents are optimally matched with riskier principals) the traditional trade-o¤ between risk and incentives holds, under e¢ cient negative assortative matching (lower risk-averse agents are matched with riskier principals) this trade-o¤ can fail to hold, in particular when matching curves are very steep. Li and Ueda's (2009) show, instead, that if the agents di¤er only in their productivity, safer …rms will o¤er highpowered incentives schemes, in this way capturing the higher productive workers at the endogenous matching.
While useful in disentangling the direct and indirect e¤ects of risk on equilibrium contracts, the endogenous matching models adopt a number of highly simplifying assumptions, as the monotonicity of equilibrium matching patterns. For this reason, in this paper we propose an alternative and simpler explanation of the relationship between risk and incentives, one based on the endogeneity of the technology. In particular, we show that the traditional relationship between agent's risk aversion and optimal incentive may be absent or reversed when the technology is endogenous and projects with a higher e¢ ciency are also riskier. More speci…cally, using a modi…ed version of the Holmstrom and Milgrom's (1987) framework, we obtain that lower agent's risk aversion unambiguously leads to higher incentives only when the technology function linking risk and e¢ ciency is elastic, while the risk aversion-incentive relationship can be positive when this function is rigid. This is because a lower risk aversion of the agent makes it optimal for the principal the adoption of a riskier and a more e¢ cient technology. While the higher e¢ ciency of the new technology (as well as the lower agent's risk aversion) allows the principal to give more incentives to the agent, its higher riskiness makes the provision of incentives more costly which works in the direction of reducing the optimal degree of the pay-for-performance sensitivity. We also show that di¤erently from the endogenous matching models (e.g. Serfes, 2005) the positive risk-incentive relationship is compatible with both positive and negative assortative matchings between principals and agents.
The paper is organized as follows. In Section 2 we describe the framework and Section 3 provides the solution of the model. Section 4 presents the comparative statics analysis of the e¤ect of a reduction of the agent's risk aversion on incentives. Section 5 concludes.

The Framework
We consider a moral hazard model as in Holmstrom and Milgrom (1987). The principal owns the technology and is risk neutral. The agent is risk averse and has a constant absolute risk aversion (CARA) utility function with a coe¢ cient of absolute risk aversion equal to r. Total output is equal to where e is the agent's action (e.g., e¤ort) and " is an (unobservable) random variable normally distributed with zero mean and variance 2 . The technology is characterized by quadratic costs, so that the agent's cost of action is where k is a constant representing the e¢ ciency of the technology employed. Better technologies are characterized by a lower k and vice-versa. The agent's reservation utility is equal to . We here modify the Holmstrom and Milgrom's framework by assuming the existence of a given set of technologies (or projects) with di¤erent levels of e¢ ciency and riskiness among which the principal can choose. In particular, we assume a trade-o¤ between e¢ ciency and riskiness so that technologies with a higher volatility 2 also have a lower marginal cost of e¤ort, i.e., where k > 0 for all 2 2 (0; 1). For simplicity, k( ) is assumed to be a function continuous and di¤erentiable in 2 .
In this framework, the principal decides the optimal technology and the agent's payment scheme; then, the agent optimally chooses the action. In the next sections, we determine these choices and analyze the e¤ects of a variation of the agents'risk aversion on the optimal payment scheme of the agent.

The Equilibrium
We solve the problem by determining the optimal payment scheme and the agent's action for a given technology. 1 Then, we determine the optimal technology choice of the principal. Holmstrom and Milgrom (1987) show that a linear payment is optimal in the above framework, so that the agent's payo¤ can be written as s (y) = y + , where and are constants optimally chosen by the principal that have to be determined. Taking into account (1), (2) and the distribution of the shock, the agent's expected utility is and therefore his maximization problem can be written as The …rst order condition of this problem is = ek. Substituting this condition into (4) and then setting the expression (the agent's certainty equivalent) equal to gives = (1=2)ke 2 + (1=2)r 2 2 + . Hence, the principal's maximization problem becomes which gives the following well-known second best solution for the agent's action 2 Using the fact that = ek, it follows that the optimal share of output paid to the agent is and the optimal …x payment is Let now 2 denote the variance of the optimal project. This is the solution of the following maximization problem of the principal subject to the technological constraint (3). 3 The …rst order condition of this problem is and therefore the variance 2 of the optimal project is implicitly de…ned by the following equation where k k ( 2 ) and k 0 k 0 ( 2 ). The e¤ort cost parameter at the optimal technology follows from (3) and it is k( 2 ). 4 In order to have unique maximum, which will be useful for the comparative static analysis, we restrict the attention to functions of the technology k ( 2 ) such that F in (11) is strictly concave. This requires that the following condition is always satis…ed The …rst component of (12) is positive (as k 0 < 0), the second is negative while the third one has the opposite sign of k 00 . Therefore, while k( 2 ) can generally be concave or convex, a su¢ cient condition for (12) to hold is that k is su¢ ciently convex, i.e., that k 00 is positive and large enough. The following proposition summarizes these results.

Proposition 1
The principal chooses the technology with the variance 2 implicitly de-…ned by equation (11) and e¢ ciency k( 2 ) as in (3). The agent optimally chooses the action e reported in (6) and the coe¢ cients of the linear payment scheme and are de…ned respectively by (7) and (8) with k k( 2 ) and 2 2 .

Agent' s risk aversion and the provision of incentives
We now analyze how a variation in the agent's risk aversion a¤ects the provision of incentives when, as in our framework, such a variation also induces a change in the technology adopted. By applying the implicit function theorem to equation (11), we obtain that as the denominator is negative from the second order condition of maximization problem (9) and the numerator is also negative since the …rst order condition (11) implies that 2rkk 0 2 k 2 = k 0 =r < 0. This means that a reduction in the agent's risk aversion 3 The maximized expected pro…t (for a given technology) is obtained from the substitution of (6) into (5). 4 Note that the …rst two components of (11) are positive while the third one is negative.
increases the riskiness 2 as well as the e¢ ciency (k( 2 ) goes down) of the technology chosen by the principal.
We will now show that while the reduction of the agent's risk aversion induces the principal to provide more incentives by increasing the agent's payment related to the output for any given technology (it is immediate from (7) that is decreasing in r), this may no longer hold if the lower risk aversion of the agent leads the principal to change the technology employed (i.e., its e¢ ciency and riskiness). In this case the characteristics of the new technology may a¤ects the optimal provision of incentives in ways that counterbalance the former e¤ect.
The total e¤ect of a reduction of the agent's risk aversion on the optimal share of output paid to the agent is obtained by total di¤erentiation of (7) which gives The …rst component in (14) represents the direct e¤ect of a reduction of r on , namely the e¤ect on if the same technology is employed. This component is equal to and it is always negative as a lower risk aversion makes it optimal for the principal to give more incentives and less insurance to the agent, which requires increasing the payment related to output. The other two components in (14) represent the indirect e¤ect of the reduction of r on , i.e. the e¤ect caused by a change in the technology employed by the principal. The new technology is characterized by a higher e¢ ciency and a higher riskiness which generate two opposing e¤ects on . The higher riskiness 2 of the project makes it optimal the provision of more insurance and less incentives to the agent, and this implies that the payment related to output decreases (we can call this the riskiness e¤ect). Indeed, we obtain that On the other hand, the new technology is also characterized by a higher e¢ ciency (i.e., a lower cost of e¤ort k), which makes it optimal an increase of incentives as 5 @ @k = r 2 (1 + rk 2 ) 2 < 0: This means that increases as r goes down. We call this the e¢ ciency e¤ect and it goes in the same sign of the direct e¤ect. 6 Therefore, the net indirect e¤ect due to the change of technology may in general lead to an increase or a decrease in . We now try to understand under what conditions there is a de…nite sign in the relationship between r and .
Let us …rst analyze the case where the net indirect e¤ect has the same sign of the direct e¤ect, so that d =dr is always negative and, therefore, a lower agent's risk aversion leads to more incentives. From (14) it is immediate that this is the case when (@ =@ 2 ) + (@ =@k)k 0 0 since @ 2 =@r is always negative. Using (16) and (17), we obtain that this condition is satis…ed when the elasticity E k of the technology with respect to the volatility is weakly greater than 1, i.e., The intuition for this result is the following. If the function k( 2 ) is elastic, then the increased e¢ ciency of the technology (i.e., the reduction of k) associated to a given increase in its riskiness 2 is relatively large. This implies that the e¢ ciency e¤ect dominates the riskiness e¤ect. Therefore, under Condition 1, the indirect e¤ect has a negative sign and the reduction of the agent's risk aversion r always leads to an increase of , which means that the principal will provide more incentives to the agent.
When k( 2 ) is rigid and therefore Condition 1 does not hold, the e¢ ciency e¤ect is small relative to the riskiness e¤ect and the indirect e¤ect will be positive. As the direct e¤ect has a negative sign, the total e¤ect of a reduction in r on will generally be ambiguous. However, if the increased riskiness of the new technology is su¢ ciently strong, then the lower agent's lower may induce a reduction of incentives. The following proposition summarizes these results.
Proposition 2 A reduction in the agent's risk aversion r generates two e¤ects on the optimal share of output paid to the agent. The direct e¤ect always increases while the indirect e¤ect due to the change of technology can lead to an increase or a decrease of . When Condition 1 is satis…ed, both the direct and indirect e¤ects have the same sign and a lower risk aversion r unambiguously increase (i.e., @ =@r < 0). When Condition 1 does not hold, the total e¤ect of r on is generally ambiguous.
It can be interesting to compare the above results with those usually obtained in the endogenous matching models. However, since the two models strongly di¤er in their assumptions, they are hardly comparable. What can be done here is to consider the basic version of the matching model introduced by Serfes (2005) and add to it, as in our model, an inverse link between technology and risk. As in Serfes (2005), we can assume that the principals are uniformly distributed according to their riskiness in the interval [ 2 L ; 2 H ] and the agents in [r L ; r H ] according to their risk aversion. 7 Now, by taking into account the e¤ect of function k(:), the condition ensuring a positive (negative) assortative matching becomes @ @ 2 @r ( )0 , (18) instead of (as in Serfes 2005, p. 346): @ @ 2 @r ( ) 0 , kr 2 1 2 (kr 2 + 1) 3 ( ) 0 holding for r 2 ( ) Comparing (18) and (19) helps to make clear that, in a matching model modi…ed to include an inverse relationship between technology and risk, the condition required for a positive (negative) assortative matching becomes more (less) demanding than when technology is given. As a result, it becomes now harder, in this model, to obtain a well-behaved (negative) relationship between risk and incentives. 8 However, comparing our model without endogenous matching with the modi…ed version of Serfes's (2005) model, it can be seen that Condition 1 is in general compatible with both positive and negative assortative matchings. This is not surprising since, di¤erently from the usual matching models in which only the direct and riskiness e¤ect are at work, our model introduces an additional e¤ect (denoted e¢ ciency e¤ect) that works in the same direction as the direct e¤ect (see expression 4.2 and footnote 5 on this point).
To see this point, let us consider a speci…c functional form for the relationship between the cost parameter k of the agent and the risk of the project expressed by 2 . In particular, let us assume that this technology function has a constant elasticity and it is given by k = ( 2 ) , with 2 (0; 1=2) and 2 2 (0; 1) so that k is …nite and positive for all 2 . Then, k 0 = k( 2 ) 1 < 0 and k 00 = ( + 1)k( 2 ) 2 > 0. The …rst order condition (11) of the principal's maximization problem can be rewritten as which implies that the variance of the optimal technology is equal to 9 2 = r (1 2 ) 1 1 : (21) From < 1=2 follows that Condition 1 is not satis…ed (as E k = < 1) and the indirect e¤ect is positive, i.e., the change of technology induced by the lower agent's risk aversion r leads to a reduction of (the riskiness e¤ect dominates the e¢ ciency e¤ect). This indirect e¤ect opposes to the direct e¤ect which instead pushes for an increase in . The total e¤ect of a reduction of r on can be computed by substituting (15), (16), (17) and @ 2 =@r (which is obtained from (21)) into (14). This leads to @ =@r = 0 which means that, in this special case, the direct and indirect e¤ect of a change in r on exactly o¤set each other and therefore that a reduction in the agent's risk aversion leaves the fraction of output paid to the agent unchanged. Moreover, from (20) it is obtained that @ @ 2 @r = 2 1 + (2 1) which can be either positive or negative for < 1=2 (e.g., if 2 = 1, the expression is positive for > 1=3 and negative for < 1=3) and, therefore, is in general compatible with both positive and negative assortative matchings, as discussed above.