18.1.1 ^-informativeness
18.2 RISK AVERSE AGENT "OWNS" THE OUTCOME
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.
bound on the agent's consumption, which is c = x - niy)?^ Hence, the princi- pal's decision problem is:
maximize U^{n,a,f]) = ^ 7t(y) (p(y\a,rj), (18.T)
7i(y),a yeY
subject to U%7t,a,rj) = X^ X^ u(x-7t(y)) (p(x,y\a,rj) - v(a) > U, (18.2')
xeX yeY
a E argmax U\n,a',fi), (18.3')
a' EA
We assume that^ is convex and constraint (18.3') can be represented by
U:{n,a,fj)-0, (18.3c0 Forming the appropriate Lagrangian and differentiating with respect to n{y)
provides:
Y, u'i^ - ^(y))
XEX
A + Jd
(p{y\a,fj) (p(y\a,fj)
1. (18.50 Observe that if x andj; are independent, i.e., (p{x,y \a,rj) = (p{x \ a, rj) (p{y \ a, rj), then y reveals nothing about x and cannot be used for risk sharing. In that setting, 7t(y) = TT^, a constant, and the induced action is
a"" e argmax U^{7f,a\f]^) = ^ u(x-7i^) (p(x\a') - v(a'),
a'eA XEX
i.e., the result is the same as if there is no contractible information.
Pure Insurance Informativeness
We first consider information that reveals nothing about the agent's action, but is informative about the uncontrollable events that influence the outcome x. We assume that events 9 e 0 define an outcome adequate partition on the state space S, so that we can express the outcome as a function x = x{9,a).
^^ If there is a lower bound c_ and j^ does not reveal x, we must either restrict niy) to be such that X - niy) > c for all x andy such that (p(x,y\a)> 0 orwQ must introduce the possibility that the agent can declare bankruptcy if x - 7t(y) < c, possibly with a deadweight bankruptcy cost being borne by the principal.
Definition Exclusively 0-informative
Performance measurement system rj is exclusively ©-informative if (p{y\9,a) -(p{y\9\ V a E A,
i.e., conditional on 0 the action does not influence the signal;;, and (p(y\9) ^ (p(y)^ for some (y,^),
i.e., the signal j ; is not independent of ^.
Recall that in Chapter 3 we introduced the concepts of an outcome relevant par- tition of the state space S (the coarsest outcome adequate partition) and the informativeness relation between two information systems. We now introduce the concepts of payoff relevance and 9 informativeness for a given action.
Definition Payoff Relevance for Action a
©{a) is a payoff relevant partition of S for action a if it is the coarsest partition such that for each 9 e ©{a)
x{s\a) = x(s]a), V s\s^ E 9.
Definition At least as ©(a)-informative
Performance measurement system rf is at least as ©(a)-informative as rf if, and only if, there exists a Markov matrix B such that
where x\ ^ [(p(y\9)\0^^^\^\Y\ ^^d B ^ [ % ^ | / ) ]|^2|,|^i|.
Proposition 18.14
If the agent "owns" x and is strictly risk averse, a system that is exclusively
©-informative has positive value (relative to no information). Furthermore, if a^ would be implemented with rj^ and system rf is at least as ©(a^)- informative as //^ then rf is at least as preferred as ;/\ with strict preference if//^ is not at least as 0(a^)-informative as rf.
Proof: Set n{y^) such that
E u{x{0,a)-n{y^))cp{0\y^)
9E0{a)
- E E u{x{9,a)-n'{y'))b{y'\/)(p{9\/).
This new contract has the same incentive properties as TI" and provides U to the agent. By Jensen's inequality, it provides the principal with at least the same level of utility. Strict preference follows \fb(y \y^) e (0,1) for some3;/,3;2 such that n^iyl) ^ n^iyl)- Alternatively, ifj;^ is a function of^ (i.e., rf is a collapsing of ;/^) 3ndyi,y2 are two signals such that3;^(yi) ^y^(yl) and (p{x\yl,a) ^ (p{x\yl,a) for some x e X, then n^iy^) cannot satisfy the first-order conditions for bothj;!^
3ndy2 (except in anomalous cases). Q.E.D.
The key here is that tj provides a basis for insurance without raising any moral hazard problems (e.g., hail insurance)?^ There is no need here for the agent to be effort averse to obtain the above result.
Insurance/Incentive Informativeness
Now consider the case in which rj is not a pure insurance reporting system (i.e., it is not exclusively 0-informative). If x (i.e., 9) is revealed by j ; , then (p{x\y,a)
= 1 if X = x{y,a), and the first-order condition becomes
^n(y I ^) M{x{y)-n{y)) -X + / / ^ ^ V T T -
(p(y\a)
Hence, if two systems both reveal x, then we can compare those systems on the basis of their relative ^-informativeness, and we will get the same results as if a risk neutral principal "owns" the outcome. Therefore, we focus here on cases in which j ; does not fully reveal x. Of course, the system must reveal something about X, otherwise it has no value.
Definition Insurance/Incentive Informativeness
Performance measurement system rj isXa-informative (insurance/incentive informative) if ^(xlj;,^) ^ (p(x\a), for at least somej; e 7, and isXA-infor- mative if it is Xa-informative for dXXa e A.
ff is at least as XA-informative as rf if, and only if, there exists a Markov matrix B such that
^^ Hail storms are a major risk for the crops on the prairies, but farmers can insure themselves against that risk by buying hail insurance. The contract is such that the farmer buys insurance for a nominal amount per acre, for example, $1,000 per acre. In case of a hail storm, the contract is settled by paying the farmer the nominal amount per acre times the number of acres insured times the average percentage of the crop destroyed in those acres. A key feature of this contract is that the insurance payment is independent of the value of the crop and, hence, the payment is inde- pendent of the farmer's skills and effort.
where ^ ^ [(p(y\x,a,rj)\^^^,^^^^,^Y^ and B ^ [ % ^ | / ) ]|^2|,|^i|.
Proposition 18.15 (Gjesdal 1982, Prop. 2)
If the agent "owns" x and is strictly risk and effort averse, then rj has posi- tive value if it is X4-informative (and has zero value if it is notXa-inform- ative for any a). Furthermore, if//^ is at least as X4-informative as rj\ then fj^ is at least as preferred asrj\ with strict preference if//Ms not at least as X4-informative as tj^.
Proof: Set 7t(y^) such that
X) u(x-7t(y^))(p(x\y^a) = Y. Yl u(x -7t^(y^)) b(y^\y^)(p(x\y^,a).
XEX y^eY^ ^^^
This new contract has the same incentive properties as n^ and provides U to the agent. By Jensen's inequality, it provides the principal with at least the same level of utility.
Strict preference follows if b{y^ \y^) e (0,1) for some3;/,3;2 such that 7t^(yl)
^ ^^(yi)' Alternatively, ifj;^ is a function of^ diwdy^^yl are two signals such that3;^(yi^) ^ y^iyi) and (p{x\yl,a) ^ (p{x\y2,a) for some x eX, then n^iy^) cannot satisfy the first-order conditions for bothj;!^ ^^^yl (except in anomalous cases).
Q.E.D.
Observe that informativeness about the outcome is crucial, because the primary purpose of the contract is to reduce the risk that must be borne by the agent.
However, if a signal used for risk sharing is influenced by the agent's action, then a comparison of one signal to another must simultaneously include bothX- and ^-informativeness.