18.1.1 ^-informativeness
18.1.4 Linear Aggregation of Signals
sensitivity of the agent's compensation with respect to variations in the likeli- hood ratio (i.e., //(A:)) goes down as k increases.^^
k 0.00 0.25 0.50 0.75 1.00
c ^{a, k) 13.000 12.937 12.750 12.437 12.000
cixg,y,) 25.000 22.562 20.250 18.062 16.000
c{xt,yd 1.000 0.562 0.250 0.063 0.000
c(.Xg,yH) 25.000 27.562 30.250 33.063 36.002
cix^^yH) 1.000 1.563 2.250 3.062 3.999
X{k) 6.0 6.0 6.0 6.0 6.0
fiik) 2.000 1.969 1.875 1.719 1.500 Table 18.1: Optimal incentive contracts for inducing a ^Vi for varying
informativeness of j ; .
MLRP with respect to each of the elements of j ; . The optimal compensation contract is then assumed to be characterized by^^
M{c{y)) ^X + juL(y\a) (18.12) subject to boundary conditions in which c(y) = c if A + juL(y\a) < M(c).
Definition
The optimal compensation contract is based on a linear aggregate of (the elements of) y (where j ; eY^ W) if there exist weights 6^, ...,d^ and a con- tract c'^'.R^C such that
c(y) = c^iyfiy)) and ^(y) - ^ d.y.. (18.13)
i = \
BD are particularly interested in settings in which the signal weights are inde- pendent of the utility function u(c), although they can depend upon the action a that is to be implemented.^^
Proposition 18.9 (BD, Prop. 1)
When the principal is risk neutral, a sufficient condition for the optimal compensation contract for inducing a to be based on a linear aggregation of the signals y, represented by
i = \
with y/(-) independent of the agent's utility function, is that the joint density function is of the form:
(p(y\a) = exp f g{y/(y,aXa)da + t(y) (18.14)
where g(-), ^i(-)? •••? ^n(')? ^^^ K') ^^e arbitrary functions. Further, in this case,
^^ For ease of notation, we suppress the dependence on the performance measurement system rj, which is kept constant in this analysis.
^^ We refer to BD for proofs.
dc{y)ldy. _ d^
dciy)ldyj " dfa)'
The key characteristic of distributions satisfying (18.14) is that the likelihood ratio can be expressed as a function of a linear function of the signals, i.e.,
L(y\a) = g(y/(y,a),aX
so that the optimal compensation contract is based on a linear aggregate of j ; . Note that this does not imply that the compensation contract itself is a linear function ofj;. A broad subclass of joint density functions satisfying (18.14) is given by:
cp(y\a) = cxp[j2U ^i(^)yi - r(a) + t(y)]. (18.15) This subclass includes, for example, a multivariate normal distribution in which
a influences the means of the distributions of each variable.
Corollary
If (p(y I a) satisfies (18.15), then the optimal compensation contract for indu- cing a can be written as c^(^), where i//(-) is a linear function of j ; (and the action to be implemented).
Proof: (18.15) is a special case of (18.14) ifdi(a) = A-{a), giy/.a) = i// - r'{a),
and t{y) is the constant of integration. Q.E.D.
The following proposition shows that the joint density satisfying (18.14) is also a necessary condition for the optimal compensation contract to be based on a linear aggregate of j ; if the result must hold for all actions in A?^
Proposition 18.10 (BD, Prop. 2)
A necessary (as well as sufficient) condition for the optimal compensation contract to be written as c^(^(y,a)), y/(y,a) = Si(a)yi + ... + S„(a)y„ for in- ducing all a E A,is that the joint density function is of the form in (18.14).
In the above analysis the weights on the signals can depend on the action to be implemented. BD also consider the conditions under which S^(a) = d^,\/ a E A.
^^ BD give an example in which the optimal compensation contract for inducing the optimal action is based on a linear aggregate of j^ even though the joint density does not satisfy (18.14).
This holds if ij/iy) = S^ S^y^ is a sufficient statistic for j ; with respect to a, i.e., there exists a function g(y) such that
Relative Signal Weights
BD examine the relation between a signal's "precision" and "sensitivity" and the relative weight it is given in the linear aggregation of the signals.
Definition
ThQ precision of signal;;^ is h^ia) = 1/Var(y^|a) and its sensitivity is y^J^a)
= dE\yi\a]/da.
Proposition 18.11 (BD, Prop. 3)
If the joint density function ofj; = (y^, ...,y„) is of the form
(p(y\a) = exp 'E(A,(a)y, ^ t,(y))-r(a)
i = \
(18.16)
S.(a) h.(a)y.^(a) then = = .
dj(a) hj(a)yj^(a)
That is, if thej^/s are independent (which is implied by (18.16)), then the rela- tive weights assigned to a pair of signals is equal to the relative value of the pre- cision of the signal times its sensitivity to changes in a, where the precision and sensitivity are evaluated at the action to be implemented. The following propo- sition considers a case in which the signals are correlated.
Proposition 18.12 (BD, Prop. 4 and Corr. 2)
If the joint density function ofj; = iyi.yi) is of the form (p{y\a)
= exp[zfi(a)3;i ^ A^{a)y^^ t^{y;) ^ t^{y^-yy;) - r{a)\ y ^ 0, (18.17) dAa) hAa) \y^(a) - yo(^)yoJ^)l
then ^ — = iv )v^uy ) /2V )yia\ n^ ^^^^^^
d^{a) h^(a) {y^^(a) - y^(a)y^^(a)]
Coy(y^,y2\a) a^(a) where y.(a) = = Coniy^.y^la) ^ - - ,
Var(y. | a) o^a)
and ^/^(^) = Var(y^|a), which implies that y^a) = y.
BD refer to the expressions in the square brackets in (18.18) as the adjusted sensitivity of the signals. It reflects the fact that the information contained in one signal is partially reflected in the other signal if the signals are not inde- pendent.
Now consider the special case in which y^J^a) > 0 and 3^2/^) " ^•> i-^-? the action influences the first signal but not the second.
Proposition 18.13 (BD, Prop. 5)
Ifthejoint density function of3; = (yi,3;2) belongs to the class in (18.17), and y\a<~^) ^ ^' >^2a(^) " ^' hi{a) > 0, and h2{a) > 0, then
8^{a) CoY{y^,y^\a) G^{a)
= = -CovY{y^,y^\a) .
8^{a) Var(y21 d) o^d) Observe that d^id) is nonzero ifj^^ and3;2 ^^e correlated. Hence, even though 372 reveals nothing about a directly, it is used in deriving the optimal performance measure because it is informative about the uncontrollable factors influencing y^ (which is influenced by the action d). Further observe that ifj^^ ^^dy2 are positively (negatively) correlated, then y2 will be given negative (positive) weight. That is, if the two signals are positively correlated, the agent will receive higher compensation if he obtains a high value ofj^^ with a low value of 3^2 than with a high value of 3;2- This is consistent with the concept of basing compensation on how well the agent does relative to some other "standard" or other measure that reveals whether the uncontrollable factors were favorable or unfavorable. That is, the agent receives higher compensation if he obtains a
"high" outcome in "bad" times than in "good" times and, conversely, he is not penalized as severely for a "low" outcome in "bad" times as he is for a "low"
outcome in "good" times.
BD make the observation that two signals, y^ 3ndy2, should be aggregated into a single measure j^^ + y2 if, and only if, the intensity (sensitivity times preci- sion) of the individual components are equal.
Impact of Changes in the Scale of a Signal
Consider a pair of signals y = (yi,>^2)? ^^^ assume that the second signal is re- placed by a linear transformation of that signal, i.e., 3/ = (yi,y2) where y2 = ky2
+ b. Observe that changing the scale of a signal does not change its informa- tiveness. In particular, it is relatively straightforward to prove that using 3/ in-
stead ofj; will result mprecisely the same action choice and compensation cost - the optimal incentive contracts will have the following relation:
Observe that transforming y2 will change both the precision and the sensitivity of the second signal. In particular,
h^Xa) = h^ia)/]^
and 3^2a(^)= ^^2/^)-
Furthermore, the transformation will change the relative weight assigned to the two signals:
dl{a) ^2(^)
If ^ > 1, then the contract based on the transformed signal will place relatively more weight on the first signal - but that is merely an offsetting adjustment.
The transformed second signal is more sensitive than the untransformed signal, but that is offset by the decreased precision.