LINEAR CONTRACTS
19.1 LINEAR SIMPLIFICATIONS
In this section we consider a setting sometimes referred to as the LEN frame- work which stands for "Linear contracts", "Exponential utility", and "Normally distributed performance measure."^ That is, compensation contracts are exoge- nously restricted to the class of linear contracts, the agent's preferences are represented as a multiplicatively separable exponential utility function in c and
The following analysis is similar to that found in Hughes (1988).
a, and the performance measure is normally distributed. The search is for an optimal contract within the class of linear contracts. No mention is made of overall optimality at this point. An attractive feature of this approach is that it gives a very simple characterization of the "optimal" contract. To summarize, our basic assumptions in this section are:
(a) The principal must choose the compensation contract within the class of linear contracts, i.e., within the class of functions
iff + vj; e C and c{y) = c otherwise }.
(b) The principal is risk neutral and the agent has a multiplicatively sepa- rable, negative exponential utility function with c = -oo^ i.e.,
u''{c,a) = -exp[-r(c - K(a))] = u(c)k(a),
withu(c) = - Qxp[-rc],k(a) = exp[r7c(a)], K'(a)>0,K"(a)>0,K"'(a)
> 0, which implies
M(c) = — exp[rc].
(c) The contractible performance measure is normally distributed with mean a and variance a^ (which is in the one-parameter exponential family of distributions), i.e.,3; ~ N(a,cr^), which implies that^
L(y\a) = —-(y-a).
o
In general, an optimal compensation contract is characterized by M = Xk{a) + ju[k(a)L + k^a)].
^ Note that if the agent's action only affects the mean of a normally distributed performance measure, we can always express the action as equal to the mean of that measure and adjust the agent's cost function accordingly. That is, there is an indeterminancy in how we express the agent's cost function and how the action affects the mean of the performance measure.
Assumptions (b) and (c) imply that^
c{y) = K(a) + — In
r r\ 1 + ju[rK'(a) + — ( y - ^ ) ] a'
Hence, an optimal contract with assumptions (b) and (c) is a strictly concave function. On the other hand, if we also impose the linear contract assumption (a), we obtain a significant simplification in the analysis."^ The key here is that if c is a linear function ofj; andj; is normally distributed, then c is normally distributed. Hence, we can then use a slightly generalized version of the rela- tion in Proposition 2.7 to obtain
U\c,a)= -Qxp[-rCE(v,fa)l where the agent's certainty equivalent is given by
CE{v,fa) = va + f - Virv^o^ - K(a).
The sum of the first two terms in the certainty equivalent is the expected com- pensation, the third term is the risk premium, and the fourth term is the mone- tary cost of effort.
With a linear contract, the agent's incentive constraint can be expressed as the first-order condition based on the certainty equivalent CE(), i.e.,
CEJ^v,fa) = V - K'{a) = 0 - v = K'{a). (19.2) Assume the agent's reservation utility has a certainty equivalent of zero, i.e., U
= - 1. Hence, given a and v, the contract acceptance constraint can be expressed as the requirement that
f-K{a) ^Virv^G^ -va. (19.3) The principal's decision problem is
^ If c = -oo, the Mirrlees condition discussed in Proposition 17.10 applies even with a multi- plicatively separable utility function. That is, an optimal contract does not exist and a penalty contract can be used to obtain a result that is arbitrarily close to the first-best result. We exo- genously exclude this possibility here, for example by assuming that the lower bound on con- sumption is finite.
^ Note that when we restrict the contracts to be linear, we cannot use penalty contracts as in the Mirrlees argument to get arbitrarily close to first-best even though c = -0°. That is, there generally exists an optimal contract within the class of linear contracts which is not first-best.
maximize U^(c,a) = b(a) - (va + / ) subject to (19.2) and (19.3),
v,f,a
where b(a) is the expected outcome to the principal, i.e., b(a) = E[x \ a], and we assume b"(a) < 0. Of course, if the performance measure is the outcome, then b(a) = a, but that does not generally have to be the case.
Incentive constraint (19.2) specifies the incentive wage parameter v required to induce a given action a, and participation constraint (19.3) specifies the fixed component/required to satisfy the agent's reservation utility.^ Hence, we can substitute them into the objective function to simplify the principal's decision problem to an unconstrained optimization problem,
maximize b{a) - V2r[K'(a)f'a^ - K(a).
Differentiating with respect to a provides
bXa) - K'{a) - rK'{a)K"{a)G^ = 0, and the optimal action satisfies
K'(a)[l + rK"(a)a^] = b\a). (19.4) A common cost function used in the literature is K{a) = Via^. In this case K '(a)
= a and K"{a) ^\, which implies that the optimal action (given linear contracts) is given by
a-v-b\a)l{\+rG^).
Proposition 19.2 (Hughes 1988, Prop. 8.1 and 8.2)
An agent who either faces more risk or is more risk averse will be induced to work less hard and will get a lower incentive wage.
Proof: L e t ^ = rcr^, i.e., the agent's risk times his risk aversion. Totally differ- entiating first-order condition (19.4) with respect to a and q yields
[K"{a)[\ + qK"{a)\ + qK'{a)K"'{a) - b"{a))da + K'{a)K"{a)dq = 0.
^ Note that the assumption of a multipHcatively separable exponential utility function is of utmost importance for this simple separation of incentive and contract acceptance concerns, since it is the only concave utility function that has no wealth effects.
With the assumptions that TC^^^) > 0, K\a) > 0, K'\a) > 0, and b"(a) < 0, this implies that
da dq
<o,
which by (19.2) implies that
— < 0. Q.E.D.
dq
Of course, the key here is that as ^ = ra^ increases so does the risk premium for a fixed incentive wage. Hence, the principal's trade-off between providing incentives for action choices and the risk premium paid to the agent changes such that it is optimal to lower the induced action in order to reduce the risk pre- mium. However, note that if the risk in the performance measure, cr^, increases with the risk aversion r fixed, the agent's compensation risk, v^cr^, and, thus, the risk premium may decrease or increase. This depends on the specifics of the problem.^