TIONS
HARA utility functions were introduced in Volume I, Chapter 2. If the agent's utility for consumption is HARA, then
-lie-^'P u{c) ~ ^ ^ ^^
1
a -1 [ac^P} \-\la
if a = 0, yg>0, if a = 1, c + ^>0, if a ^ 0,1, ac + yff > 0,
where a is the agent's risk cautiousness. The analysis in this chapter establishes that, if the principal is risk neutral, optimal contracts take the general form:
c(m(x))
M-\m{x)) if mix) > M{c\
c otherwise,
where m{x) is a linear function of the likelihood ratios for x given the induced action a relative to the alternative actions for which the incentive constraints are binding (see, for example, (17.6), (17.6'), and (17.6")).
Observe that with HARA utility functions:
M{c)
u'{c)
c + j3
if a = 0, yg>0, c>c, if a = 1, c > c> -yff, if a ^ 0, 1, c > c> -fila.
Hence, for m = m(x) > M(c),
fiXnm M~\m) = \m -/]
a-'(m''-j3)
if a=0, yg>0, m>e-^>0, if a = 1, m > c + fi>0, if a ^ 0 , 1 , m> [ac + yg]^^^>0.
Furthermore, the relation between the agent's utility for consumption and the likelihood measure m is
u(M-\m))
-pm-^
\nm
[ a-l
if a = 0, yff >0, m > e cip if a = 1, m> c + fi>0, if a ^ 0 , 1 , m> [ac + yg]i/«>0.
From the above we can readily characterize how the agent's compensation and utility vary with the likelihood measure m for m > M(c). Of course, for m <
M(c), the compensation is equal to c.
Proposition 17C.1
If the agent has separable utility with HARA utility u(c) for consumption, then form > M(c):
(a) the agent's compensation is a strictly concave (convex) function of the likelihood measure m if the agent's risk cautiousness a is less (more) than 1, and is linear if a = 1;
(b) the agent's utility is a strictly concave (convex) function of the likeli- hood measure m if the agent's risk cautiousness a is less (more) than 2, and is linear if a = 2.
Proof: In the proof we assume that the set of possible values of m is a convex set on the real line, so that c(m) and u''M~^(m) are continuously differentiable functions. The results also hold if the set of possible values ofm is finite.
(a): Recall that c(m; =M~\m).
c'{m)
Pm-^ if a = 0 , yg>0, m>e-^, 1 i f a = l, m > c + yff>0, m^"^ if a ^ 0 , 1 , m> [ac+ yg]^^^ > 0.
(b):
c"{m)
-jim if a = 0, yg>0, m>e-^, 0 if a = 1, m>c + /]>0, (a-l)m«-^ if a ^ 0 , 1 , m> [ac+ yg]i/« > 0.
du(M-\m)) dm
fim-^ if a=0, yg>0, m > e"^^, m
m a-2
d^u(M-\m)) dm^
-ipm -3
-m~^
(a-2)m oc-3
if a = 1, m> c + ^>0, if a ^ 0 , 1 , m> [ac + yg]i/«>0.
if a = 0 , yg>0, m>e-^^, if a = 1, m > c + fi>0, if a ^ 0 , 1 , m> [ac + yg]i/«>0.
Q.E.D.
Observe that if there exist likelihood measures m < M(c), then the compensation and utility levels are flat, with c = c and u(c) = u(c) for those values of m. This does not disturb the convexity of either c(m) or uoM~\m). However, the linear cases become piecewise linear, and the concave functions are not concave over the entire range.
Most analytical research is based on a general concave utility function or assumes the utility function is either exponential or square-root. The exponen- tial utility function has a = 0, which implies that the optimal compensation and utility functions are strictly concave functions of the likelihood measure for m
> M(c). The square-root utility function, on the other hand, has a = 2 (since 1 -V2 = /4), which implies the optimal compensation is a strictly convex function of the likelihood measure, while the utility function is linear (or piecewise linear if there exists m < M(c)).
In the first-stage of the GH approach we minimize the expected compen- sation cost to induce a given action. This is equivalent to minimizing the risk premium paid to the agent, since the risk premium is given by
7r{c,a) = E[c|a] - CE{c,a),
where the certainty equivalent is given by the participation constraint as the first-best cost of implementing a (provided the participation constraint is bind- ing), i.e.,
CE(c,a) = w ; ^ ,
^ k{a) '
where w(-) = u~^{') denotes the inverse of the agent's utility for consumption.
In subsequent analyses, with additive separable utility functions of the HARA class, we use properties of the change in risk premium that occurs when the level of utility is increased by the same amount for all outcomes. That is, for a given compensation contract c that implements a we consider another com- pensation contract c^ defined by
u{c^{x)) = u(c(x)) + A, \/ X e X.
Clearly, if c implements a, c^ also implements a since^^
argmax f u(c(x))d0(x\a) - v(d) = argmax f [u(c(x)) + l]d0(x\a) - v(d).
aeA '^ aeA '^
X X
The risk premium paid to the agent for contract c^ is given by
7r(c^,a) = (w{u{c{x)) +X)d0{x\a) - wU [u{c{x)) + X'\d0{x\a)y
Increasing the level of utility, increases the variance of the compensation and, therefore, one might think that the risk premium paid to the agent also increases.
However, due to wealth effects on the agent's risk aversion, the relation be- tween the utility level and the risk premium is more complicated than suggested by this intuition. The following proposition demonstrates that the risk premium increases with X if the agent's utility is a concave function of the likelihood measure (or, equivalently, the derivative of the inverse utility function, i.e.,
^^ In this analysis we do not consider the impact on the participation constraint. In subsequent applications we consider cases in which the level of utility is increased for outcomes that are affected by the agent's action and decreased correspondingly for outcomes that are not affected by the agent's action. The variation is such that it leaves both incentives and the agent's expected utility unchanged.
w'{'),^^ is convex). On the other hand, if the agent's utiHty is a convex function of the HkeHhood measure, the risk premium decreases as the utility increases.
Proposition 17C.2
If the agent has an additively separable utility function, the risk premium n{c^,a) is increasing (decreasing) in X if the agent's utility is a concave (convex) function of the likelihood measure m.
Proof: From the definition of the risk premium and Jensen's inequality we get that
M^iA = fwXu(c(x)) +X)d0(x\a) - w'[f[u(c(x)) +l]d0(x\a)) >(<) 0,
if, and only if, w'(-) is convex (concave). Now recall that w'(u(c(m))) = M(c(m))
= m. Hence, w'(-) is the inverse function of i/oc(-) so that w'(') is convex (con-
cave) if, and only if, i/oc(-) is concave (convex). Q.E.D.
Of course, if the agent's utility for consumption is HARA we can use Propo- sition 17C. 1 to obtain the result that the risk premium is increasing (decreasing) in 1 if the risk cautiousness is less (more) than 2, and independent of A if a =2.
REFERENCES
Arya, A., R. A. Young, and J. C. Fellingham. (1993) "Preference Representation and Randomi- zation in Principal-Agent Contracts," Economic Letters 42, 25-30.
Fellingham, J. C , Y. Kwon, and D. P. Newman. (1984) "Ex ante Randomization in Agency ModQh,'' RandJournal of Economics 15, 290-301.
Gjesdal, F. (1982) "Information and Incentives: The Agency Information Problem," Review of Economic Studies 49, 373-390.
Grossman, S. J., and O. D. Hart. (1983) "An Analysis of the Principal-Agent Problem," Econo- metrica 51, 7-45.
Holmstrom, B. (1979) "Moral Hazard and Observability," BellJournal of Economics 10,74-91.
Innes, R. D. (1990) "Limited Liability and Incentive Contracting with Ex-ante Action Choices,"
Journal of Economic Theory 52, 45-67.
^^ Note that the derivative of the inverse utility function is equal to the marginal cost of providing utility to the agent.
Jewitt, I. (1988) "Justifying the First-Order Approach to Principal-Agent Problems," Econome- trica 56, 1X11-1190.
Mirrlees, J. A. (1999) "The Theory of Moral Hazard and Unobservable Behaviour: Part I,"
Review of Economic Studies 66, 3-21. (Working paper originally completed in 1975.) Rogerson, W. (1985) "The First-Order Approach to Principal-Agent Problems," Econometrica
53, 1357-1368.
Ross, S. (1973) "The Economic Theory of Agency: The Principal's Problem," American Economic Review 63, 134-139.
£ A : P 0 5 r PERFORMANCE MEASURES
In Chapter 17 we assume that the action a and the event 6 are not observable, but there is a verified report of the final outcome x. Hence, incentive contracts can be based on the reported outcome. In this chapter we allow for the possibil- ity that the outcome may not be contractible. If it is not, then inducing more than the agent's least cost action will require the use of incentives based on some alternative performance measure that is contractible and is influenced by the agent's action. Furthermore, it is potentially valuable to use more than one performance measure. This chapter explores the relation of the characteristics of alternative performance measures to the principal's expected payoff, and the form of the optimal incentive contract.
We continue to focus on a single-task setting, so that the key benefit from a superior set of performance measures takes the form of a reduction in the risk premium the agent must be paid for taking a given level of induced effort. Of course, a reduction in the risk premium may lead the principal to offer a contract that induces more effort.
Since the outcome x is not necessarily contractible information, it is impor- tant to designate whether the principal or the agent is the residual "owner" of that payoff That "ownership" may derive from legal or physical consider- ations. For example, the principal may own the production technology and will ultimately receive the final payoff, even though that payoff may not be realized until sometime subsequent to the termination of the compensation contract with the agent. On the other hand, the agent may physically control the payoff such that he can consume the difference between the outcome received and the amount he is contracted to pay to the principal.
We first (Section 18.1) consider the setting in which a risk neutral principal
"owns" the outcome. In that setting all risk is ideally borne by the principal and a performance measure is beneficial if it permits the principal to impose less risk on the agent while still inducing a given action (or permits inducement of a more preferred action). In Section 18.2, a risk averse agent "owns" the outcome.
In this setting a performance measure has two potential roles: as a mechanism to facilitate the sharing of the agent's outcome risk, and as a mechanism to provide incentives for the agent's action. Section 18.3 considers the setting in which a risk averse principal "owns" the outcome. This provides results similar to those in Section 18.2. However, in that section we focus on a setting in which there are both economy-wide and firm-specific risks and the principal is
a partnership of well-diversified shareholders. While well-diversified share- holders are risk neutral with respect to firm-specific risk, they are risk averse with respect to the economy-wide risk. We show how their risk preferences with respect to economy-wide risk can be represented in a simple way by using risk-adjusted probabilities for the economy-wide events, and illustrate how this translates into an optimal compensation scheme. Section 18.4 considers optimal costly acquisition of a secondary performance measure conditional on a primary performance measure. Section 18.5 concludes the chapter with some brief remarks.
18.1 RISK NEUTRAL PRINCIPAL "OWNS" THE OUTCOME
The simplest case to consider is one in which the principal is risk neutral and he ultimately receives the output x, so that there are no risk sharing concerns - only incentive issues. In Section 18.3 we consider the setting in which the principal is risk averse.
Basic Elements of the Model
The agent again chooses an action a e A, which generates an outcome x e X.
The contractible information is denoted j ; e 7, which is the outcome of an infor- mation or performance measurement system rj. It can be multi-dimensional and may include x, but we allow for the possibility that x may not be part of j ; . We assume Xand Fare finite sets to avoid potential technical problems associated with convex sets. However, given suitable regularity, the analysis can be ex- tended to settings in which X and Fare convex sets - and much of the literature assumes that to be the case.
The joint probability function over Xx Y given action a and performance measurement system rj is denoted (p{x,y\a,fi), and the marginal probability functions are (p{x\a) and (p{y\a,fi). We assume the cost of the information system is separable, so that rj does not affect the gross payoff x.
The principal is assumed to be risk neutral, while the agent is risk and effort averse with an additively separable utility function:^
i/{n) =7r =x - c, u\c,a) = u(c) - v(a), u' > 0, u" < 0, v' > 0, v" > 0.
Principars Decision Problem
For the main part of the analysis we focus on the first stage of the Grossman and Hart (1983) (GH) approach in which we identify the least expected cost contract
The analysis can be readily extended to consider a multiplicatively separable utility function.
for inducing an action a that is not the agent's least costly action, i.e., there is at least one other action ae A such that v{a) < v{a).
The principal's decision problem is essentially the same as in Chapter 17, except that in this setting the agent's compensation contract is defined over the anticipated contractible information;;, i.e., c\Y^ C= [c,oo), and the probability function over the contractible signals depends on the performance measurement system that is used.
c\a,f]) = minimize J^ c(y) (p(y\a,rj), (18.1)
c(y) yeY
subject to
U\c,a,n) - E u{c{y))(p{y\a,n) - v{a) > U, (18.2)
yeY
a E argmax U\c,a',fi), (18.3)
a' EA
c{y)>c, yyeY. (18.4) We assume that^ is an interval on the real line, i.e., A = [a,a], and that (18.3)
can be represented by the first-order condition for the agent's decision problem, i.e.,^
^ Jewitt (1988) identifies conditions under which the first-order incentive constraint is appropriate in a setting in which j^ = (3^1,3^2) ^^^ (p(y I ^) = (p(yi I ^) (piyi I <^)^ i-^-? the two signals are independent (see his Theorems 2 and 3). Theorem 3 invokes conditions (b) and (c) from his Theorem 1 (see Proposition 17.9) and requires that 0{y^\a) be quasi-convex in (y^,^), / = 1, 2. Quasi-convexity implies that if 0(y^i Ia^) < 0(y^21'^^)'then 0{Xy^^+{\ - X)y^2\^a^ +{\ -X)a^) < 0(y-2\a^) for all j^^^, y^2 ^ Y-, a\a^ e A, and A e [0,1].
Sinclair-Desgagne (1994) identifies conditions under which the first-order incentive con- straint is appropriate in a setting in which j^ = (y^,..., y^), x is a function of j^, and Y^ is finite and ordered. Sinclair-Desgagne demonstrates that the use of the first-order incentive constraint results in identification of the second-best contract and action if the following conditions on (p(y \ a) hold:
(a) MLRP: cpiy \ d)l(p{y \ a^) is nondecreasing in y whenever d > a^.
(b) SDC: For at least one dimension h (with j^ = iyn^y-i)) Qiyh^y-h\^) = J2 (pit,y_h\a)
is nondecreasing in a, i.e., g^ > 0 at every a, yj^, midy_,^, where Q is the upper cumulative probability of signal yj^ at y _,^.
(continued...)
U:(c,a,rj) = 0. (18.3c) The first-order condition characterizing the optimal incentive contract (for c(y)
> c) is^
M(c(y)) = X + fiL{y\a,rj\ (18.5)
\ (p (y I a, fj)
where M(c) = , and L(y\a,T]) = .
u'(c) (p(y\a,rj)