THE OUTCOME
18.3.1 Economy-wide and Firm-specific Risks^^
We now consider a setting in which the production technology is "owned" by a "principal" who is a partnership of well-diversified shareholders in an econ- omy where there are both economy-wide risk and firm-specific risk (see Section 5.4.2). It follows from the analysis in Section 5.4.2 that the principal's (i.e., the shareholders') preferences can be represented as if he is risk neutral with respect to the diversifiable firm-specific risk, whereas he is risk averse with respect to economy-wide risk.
We assume the economy-wide event 9^ e 0^ is contractible information and is not influenced by the agent's action, i.e., (p(0^ \ a) = (p{9^. The outcome rele- vant firm-specific events are not contractible, but the contractible performance measure y is influenced by both the firm-specific and economy-wide events as well as the agent's action, as represented by the joint conditional probability function (p{y,x \ a, 9^. If we ignore the possibility of a lower bound on compen- sation, the compensation contract is a function c\ 7x 0^ - M, where both 7 and 0g are assumed to be finite sets.
The objective of the principal is to maximize the market value of the firm net of compensation to the manager (agent). If the capital market is "effectively complete" with respect to the economy-wide events, there exists a unique risk- adjusted probability function for the economy-wide events (p{9) such that the market value of the firm is given by (see Section 5.4.2):
U\c,a,fj) - E E T.[^-^(y^Se)^V(yMa.9)c^{9),
eee yeY XEX
^^ The proof is basically the same as for Proposition 18.15.
^^ Ideally, the reader will have studied Volume I, Chapter 5 (or will have studied finance theory that deals with efficient risk sharing when there is diversifiable and non-diversifiable risk) before studying this section. That background would help you understand the assumptions made in this section. However, the material in this section can be read without that background.
That is, the market value is the risk-adjusted expected value of the conditional expected residual payoff to the shareholders given the economy-wide event.
The key here is that the risk adjustment of the probability function only pertains to the economy-wide events - the firm-specific risk can be diversified and, therefore, well-diversified shareholders do not require a risk premium for taking on that type of risk. We assume the capital market is large and competitive such that the agent's action has no impact on the risk-adjusted probabilities.
In a market setting the agent may also be able to trade. We assume that he is not able to trade in claims for his own firm. This would enable the manager to (partly) undo the firm-specific risk in his compensation and, thus, be detri- mental to incentives provided through his compensation. Of course, if the firm- specific events are publicly observable and the agent can trade in a complete set of firm-specific and economy-wide event claims, the first-best solution can be obtained by selling the firm to the manager and let him insure his risk through trading in the capital market. However, the capital market is typically incom- plete with respect to firm-specific claims.
On the other hand, it is unreasonable to assume that the agent cannot trade in diversified portfolios. Consequently, we assume that the agent can trade in a complete set of event claims for the economy-wide events. Hence, when designing the optimal compensation contract, the principal must consider both the agent's action choices and his trading in economy-wide event claims. The payoff from the portfolio acquired by the agent is denoted w = w(0^). We assume that the agent has no initial wealth so that the agent's portfolio problem given the compensation scheme c and action a can be formulated as
maximize UXc+w,a,rj) = Y. Y. u\c{y,9^)+w{9^),a)(p{y\a,9^)(p{9^),
subject to Y. J2 [c(yA)M^e)]'P(y\^AMOe)
9^E0 yeY
The first-order condition for the agent's position in the event claim for econ- omy-wide event 9^ is given by
U:(c+w,a\de^r])<p(9J - y<p\9J = 0, V^^, where y is the Lagrange multiplier for the agent's budget constraint and
U:{c^w,a\0,,n) - E ul{ciy,e^yw{9^),a)fiy\a,e^), Vd^.
;EY
If there is no firm-specific risk, so that the outcome x can be written as a func- tion of the agent's action and the economy-wide event 0^, i.e., x = x(a,0^), the first-best solution can be obtained by selling the firm to the agent.^^ In that case, the agent obtains x(a,0^) - F*, where F* is the first-best market value of the firm. It then follows from the agent's first-order condition for his portfolio choice that the agent's optimal portfolio of economy-wide event claims will be such that
ul(x(a,8J - r + w(8J,a)cp(8J = ycfiOJ, \/8^.
Consequently, for an optimal portfolio the agent's marginal utility of consump- tion is proportional to the risk-adjusted probabilities for the economy-wide events. That is, the sharing of the economy-wide risk is efficient and the agent's action choice is first-best since the agent bears all the costs and benefits of his action. Therefore, there must be firm-specific risk for an incentive problem to exist]
Suppose (c,w,a) is an optimal contract. Now consider the compensation contract c^ defined by
c^iyA) = c(yA) ^ HOel
This contract gives the agent the same consumption possibilities in all contin- gencies (y,^e) and, therefore, leaves the agent's action incentives and expected utility unchanged compared to (c,w). Moreover, it follows from the agent's budget constraint that c and c^ are equally costly to the principal. However, since w solves the agent's portfolio problem given the compensation scheme c, the agent's optimal portfolio choice with c^ is not to trade in any of the event claims. Hence, we can assume without loss of generality that the principal chooses among compensation contracts for which the agent has no incentive to trade. Note that this does not imply that the agent's portfolio choice is a non- binding incentive constraint.
We can now formulate the principal's decision problem for inducing a parti- cular action a as follows, assuming the first-order conditions are sufficient con- ditions for the agent's incentive constraints.
^^ Alternatively, in this case, the first-best solution might also be achievable with a penalty contract that severely punishes the agent if the outcome reveals that he has not taken the first-best action.
(18.r') c \a, rj) = minimize J ] Yl ^CV' 6>^) ^ (v I ^^ ^e) ^ (^e)'
O,E0^ yEY
subject to U^{c,a,fj) > U, (18.2^0
Ul{c,aM9^cp{9^-yc^{9^ = 0, V ^, e 0 „ (18.3p'0
U^(c,a,T]) = 0. (18.3c'')
where c\a,fj) is the market value of the market value minimizing compensa- tion contract that implements a given performance measurement system rj.
There are two main differences between this decision problem and those considered earlier with a risk neutral principal. Firstly, the principal and the agent use different probabilities for the economy-wide events. The principal uses the risk-adjusted probabilities for the economy-wide events reflecting the risk premiums attached to those events. The agent uses the unadjusted probabil- ities, since his marginal utility of consumption is affected by the firm-specific risk and is, therefore, not proportional to the risk-adjusted probabilities. Sec- ondly, there is an additional incentive constraint for the agent's portfolio choice.
This may be a binding constraint, since the agent has the possibility of mitigat- ing the impact of the economy-wide events on his compensation through his portfolio choice of economy-wide event claims.
Assuming the agent has a separable utility function, the first-order condition for an optimal compensation contract is given by^^
M(c(y,^J) = k{a) X + d{e^
fi k(a) (p(y\a,6) (18.19) where A, S(0^), and ju are the Lagrange multipliers for the corresponding con- straints in the principal's decision problem. The impact of the agent's no- trading constraint appears as a term related to his risk aversion, whereas the impact of the differences in the beliefs for the economy-wide events enters as a simple multiple of the ratio between the agent's probability and the investors' risk-adjusted probability (i.e., the inverse of the valuation index for the econ-
Note that (p(6^) and (p(6) have the same support (see Chapter 5).
omy-wide events). When the risk-adjusted probability is relatively low, i.e., aggregate consumption is relatively high, the agent receives a relatively high compensation, and vice versa. That is, the principal sets the compensation such that it is positively "correlated" with aggregate consumption. This occurs for two related reasons. Firstly, the market value of a compensation contract is lower, the more positively correlated it is with aggregate consumption, ceteris paribus. Secondly, since the agent can trade in economy-wide event claims and
the compensation contract must be such that he has no incentive to trade, he must have a relatively low conditional expected marginal utility for the econ- omy-wide events for which the event prices are relatively low.
In order to disentangle these two effects and to abstract from the effects of variations in the agent's risk aversion, we assume that the agent has an exponen- tial utility function which is either additively or multiplicatively separable, i.e.,
u''{c,a) = - exp [- r(c-K(a)) ] - v(a), so that k(a) = exp [ rK(a) ] with multiplicatively separable: K\ K" > 0 and v{a) = 0,
additively separable: K(a) = 0 and v' > 0, v" > 0.
This implies that
M(c) = r~^ exp [re], — — = - ^^ and — — = rK'(a).
u'{c) k(a) Hence, by taking logs of both sides of (18.19) and rearranging terms, the first- order condition becomes
r ln| + In l(a,e^) + ju^-—-— | + K(a) (18.20)
where X{a,9^) = X - rd{9^) + rjUK'(a); K(a) = ln(r) + rK(a).
Proposition 18.17
Suppose the agent has either an additively or multiplicatively separable exponential utility function. Ifj; and 0^ are independent, i.e., (p(y\a,0^) = (p{y\a), then
(a) the agent's no-trading constraint (18.3p") is not binding, and (b) the agent's compensation is additively separable inj; and 9^.
Proof: To show (a) suppose (c, a) is an optimal contract for the principal' s deci- sion problem in which the agent cannot trade in economy-wide event claims.
The optimal compensation contract is determined by an equation similar to (18.20) except that the ^(^g)-term is fixed at zero so that )i{a,9^) does not depend on 9^. We now show that this contract leaves the agent with no incen- tive to trade, i.e., there exists a Lagrange-multiplier y independent of the econ- omy-wide event such that the agent's no-trade constraint (18.3p") is satisfied.
Inserting the structure of the optimal contract given by (18.20) using the assumption thatj; and 9^ are independent, we get that
U^(c,a,T]\9J = 2^ ^—77- ^(^) +/^ , , , Qxp(-K(a) +rK(a)) (p(y\a) yeY (p(9J V (P(y\ci) )
Hence, defining y by
1^ U(a) + // , , , ^(yl4
shows that the agent has no incentive to trade. Since the principal can do at least as well with the imposition of a no-trading constraint as with permitting agent trading, and {c,a) is feasible with agent trading, {c,a) is also optimal with trading permitted.
(b) follows immediately from (18.20), given independence and (a).
Q.E.D.
The proposition demonstrates that if the economy-wide event is not informative about either the agent's action or the agent's conditional marginal utility of consumption given 9^, the variation in the agent's compensation due to the economy-wide events is solely derived from an efficient risk sharing of the economy-wide risk between the principal and the agent. That is, the sharing of the economy-wide risk and the provision of incentives through the firm-specific risk are separable. In order to minimize the market value of the compensation contract the principal chooses the compensation so that it is highly correlated with aggregate consumption. If the agent cannot trade, he requires a risk premium for taking on that type of risk. That tradeoff is precisely such that the marginal rates of substitutions for the economy-wide events are equated for the
well-diversified shareholders and the agent so that the agent has no incentive to do any additional trading in economy-wide event claims.
At first glance it may seem surprising that the no-trading constraint is not binding when the agent has additively separable exponential utility since this utility function exhibits wealth effects. However, recall that the agent has only one consumption date. Hence, his trading only affects the variation in his con- sumption across the economy-wide events and not the level of consumption. If the agent has an initial consumption date (at the contracting date) so that he can shift the level of consumption between consumption dates, the no-trading con- straint will be binding for the additively separable exponential utility function.
In that case there will be a tension between the intertemporal allocation of con- sumption and optimal incentives (which we explore in Chapter 24). However, the no-trading constraint will still be non-binding for the multiplicatively sepa- rable exponential utility function with multiple consumption dates since the level of consumption has no impact on action choices for this utility function.
In general, we expect the performance measure y to be correlated with the economy-wide event, and to be influenced by the agent's action. Consequently, the economy-wide event is expected to be insurance informative. For example, knowledge of the economy-wide event can be helpful in making inferences about whether a good outcome is due to the agent working hard or to favorable market conditions. In such cases, there will be tension between the sharing of economy-wide risk, the agent's trading, and optimal incentives. Intuitively, if the agent can trade in economy-wide event claims, the principal cannot as efficiently allocate incentive bonuses and penalties across the economy-wide events as would be possible if the agent could not trade in these claims. When the agent can trade in these claims, he will have an incentive to "insure" (i.e.,
"smooth") these bonuses and penalties through his trading. We illustrate this in the following section using the hurdle model.