outcome risk premium paid in the good event to induce the agent to jump.^^
Hence, the agent's trading opportunities make it more costly to induce him to jump because of the higher risk premium he must be paid in the good event (where his utility level is relatively higher than without trading opportunities).
c\a){c\a)) 0.159(0.218)
\u:{c,a\O)^<p{9)l(p{0)
c(x^,dg) 1.139
4xt,0g) -0.113 0.720x.5/.4 = 0.900
cipal is assumed to be risk neutral so the key aspect of the primary signal is its informativeness about the agent's action.
Basic Model Elements
We return to a setting in which a risk neutral principal "owns" the outcome x e X, which is the result of the action ae A implemented by a risk and effort averse agent with an additively separable utility function u^'ic.a) = u(c) - v(a), with u'
> 0 , ^/''<0, a n d v ' > 0 .
There are two possible signals, y^ e Y^ ^ndy2 e Y2. The primary signal j^^
is always reported, whereas the secondary signal 3;2 is only reported if the prin- cipal pays a cost K. The decision to incur the cost K may be contingent on the observed primary signal y^.
(p{yi,y2W) is the joint probability of the two signals given action a, and (piy^ I a) is the marginal probability of signal;;^, / = 1,2. The principal's posterior belief about 3;2 given j ; ! and a is given by (p{y2W->y\) " ^(yi?>^2k)/^(yi k)-^^
The principal's investigation strategy is denoted a\ Y^ ^ [0,1], where a{y^
is the probability that the principal "investigates," i.e., he pays to have3;2 report- ed, given the primary signal j ^ ^ It is important to observe that the analysis as- sumes that the principal commits to a particular investigation strategy a at the time he contracts with the agent. That is, the investigation is based on a fixed investigation rule, and is not based on an ex post decision by the principal.
Otherwise, it would be rational for the principal, ex post, not to investigate after the agent has taken his action (under the assumption that the investigation stra- tegy a will be implemented).
We represent the compensation contract as consisting of two components:
^(yi.3^2) = ^(yi) + ^(yi.3^2). where
c\ 7i - [0,oo) specifies the basic amount that is paid if only the primary signal is reported, i.e., there is no investigation.
d\ Y1XY2 ^ [-^(yi),~) specifies the "bonus" (possibly negative) that is paid if both j ; ! and3;2 ^^e reported, i.e., ifj^^ is reported and an investiga- tion to determine y2 is made.
Principars Decision Problem
We again focus on the first stage of the GH approach and determine the least cost strategy and compensation contract for inducing the agent to take an arbi- trary action a (which is not his least cost action). The principal's first-stage
^^ Baiman and Demski (1980a,b) assume independence, i.e., (p{yi,y2\ci) = (p(y\\^)(p(y2\^)^ but Lambert (1985) permits the two signals to be correlated.
decision problem is as follows, assuming that^ is convex and the incentive con- straint can be replaced by the agent's first-order condition:^^
c "^{a) = minimize c (a, a, c, d)
a,c,3
E
(l-«(y,))c(y,)+ «(yi) J2 [^(yO + ^(y^yi) + '^]^(yil^^yi) <p(yiWX subject to U''(a,a,c,S) > U,
U"(a,a,c,d) = 0,
c(y,)^0, Vj, 6 7„if«(y,)<l,
c(y,) + Siy^y^) > 0, V (y„y,) e Y, x Y„ if a(y,) > 0, a(y,)e [0,1], \/y, e Y„
where U%a, a, c, d)
[ 1 - a(yi)] u(c(y^))
E
3^2e^2
The Lagrangian (omitting constants and boundary conditions) for this decision problem is:
Sf = c(a,a,c,d) - lU^(a,a,c,d) - juU^(a,a,c,d).
The first-order conditions that characterize the two components of the optimal compensation contract (assuming an interior solution) are
no investigation: M(c(yi)) = 1 + juLiy^ \ a),
^^ See Jewitt (1988) for a discussion of conditions under which this can be done in the conditional investigation case considered by Baiman and Demski (1980).
investigation: M{c{yi,y^) = A + idL{yi,y2\a).
Observe that when there is investigation, the likelihood ratio can be written as L{y^.y2W) = ^ ^ " , I , + ^ -LiyiW) + L(y2\y^,a).
(piyvyiW) v(yM) (p(y2\yv^)
Note also that E[Z(y2 bi?^) Ij^il " 0? which implies E[L(y^,y2 \ a) ly^] = L(y^ \ a) so that the likelihood ratio with investigation is a mean-preserving spread of the likelihood ratio without investigation. Lambert interprets this as implying that the additional incentive information provided by an investigation is not systema- tically favorable or unfavorable with respect to the agent's action.
Optimal Investigation Policy
In the principal's decision problem, the objective function and the participation and incentive constraints are all linear in a(yi), for QSichy^. This implies that the probability of investigation will always be a corner solution, i.e., for QSichy^ we have either afy^) = 0 or afy^) = 1 .^^ Differentiating the Lagrangian for the prin- cipal's decision problem with respect to aiy^) yields:
- [B(y^) -K]g)(y^\a%
where B(y^) = c(y^) - Y. ^(^1.3^2)^0^2^1.^) - K^CVi)) [^ + ML(yi\a)]
+ Y. ^(^(yi'>y2))[^ + ML(yi,y2\a)]g)(y2\yi,a).
The optimal investigation policy is determined by a trade-off between the cost and benefits of an investigation, so that a is either zero or one.
Proposition 18.18
The gross benefit of an investigation is positive for eachj;!, i.e., B{y^ > 0, and strictly positive if j^^ is not a sufficient statistic for (y 1,3^2) with respect to the agent's action. The optimal investigation policy is to investigate if, and only if, Bfy^) > K.
^^ This assumes that the optimal compensation contract is interior - otherwise, it may be optimal to use a randomized investigation strategy. In the subsequent analysis we assume the optimal compensation contract is interior.
Proof: It follows immediately from minimizing the Lagrangian that it is opti- mal to investigate with probability one if, and only if, B{y^ > K. Otherwise, no investigation is optimal.^"^
Next, show that 5(yi) > 0. Let m{l) = A + jul (= \lu'{c{m{l)))) denote the likelihood measure, and let (as in the proof of Proposition 18.6) the function/(-) be defined by
/ ( / ) = c{m{l)) - u{c{m{mm{l).
The gross benefits from an investigation can then be written as Biy^) - f{L{y^\a)) - Y. f{L{y^,y^\a)) (p{y^\a,y^).
As is shown in the proof of Proposition 18.6,/(-) is a strictly concave function of/. Hence, Jensen's inequality and E[L{y^,y2 \ a) \y^ = L{y^ \ a) imply that B{y^
> 0, with a strict inequality ifL(y^,y2\a) varies with3;2- Q.E.D.
Of course, if an investigation is costless (i.e., K = 0), it is optimal to investigate for all signals y^, since, at worst, the additional information in the secondary signal can be ignored. If j^^ is not a sufficient statistic for (y 1,3^2) with respect to the agent's action, there is a strict gain to an investigation. Hence, there is a non-trivial tradeoff between the gross benefits and the cost of an investigation.
This tradeoff depends on the factors affecting the gross benefits and, of course, on the acquisition cost. These factors are the agent's utility function, the likeli- hood ratio for the primary signal, L(yi\a), and the informativeness of the secondary signal about the agent's action giveny^^
Informativeness of Secondary Signal Independent of Primary Signal Initially, we consider the case in which the informativeness of the secondary signal about the agent's action is independent ofy^. Let 0(l2\a,yi) denote the conditional distribution function for the likelihood ratio for the secondary sig- nal, I2 = L(y2\a,yi), given the primary signal.
Proposition 18.19
Assume the informativeness of the secondary signal about a is independent of the primary signal, i.e., 0(l2 \ a,y^ is independent ofy^. The gross benefit of an investigation depends only on y^ through /^ = L{y^ \ a), and it is de-
^^ Note, however, that the benefit function itself depends on the optimal investigation strategy through the impact of this strategy on the multipliers X and ji. Hence, if there is some subset of primary signals for which Biy^ = /c, the optimal investigation strategy may be a non-trivial ran- domized strategy with a{y^) e (0,1).
creasing (increasing) in /^ if the optimal incentive contract with investiga- tion is such that the agent' s utility, i/ o c o m(-), is a concave (convex) function of the likelihood ratio, and independent ofl^ifu^c^ m{-) is a linear function of the likelihood ratio.
Proof: When the conditional distribution of/2 = L{y2 \ a,y^ is independent ofj^^, the gross benefit of an investigation can be written as
B{yi) = fill) - Y.f(li+L(y^\a,y^))(p(y^\a,y^)
where/(•) is defined as in the proof of Proposition 18.18. Hence, the gross benefit of an investigation depends only ony^ through Z^, and
where / ' ( / ) = - u(c(m(l))) ju.
Since any ju satisfying the incentive compatibility constraint on the agent's ac- tion choice and the first-order condition for an optimal incentive contract is positive (see Proposition 17.8), the claim follows from using Jensen's inequality
and the fact that E[l^ + l2\a] = l^. Q.E.D.
In this case the additional information provided by an investigation about the agent's action is independent ofy^. Hence, the benefit of an investigation is highest for those y^ where the risk premium for imposing incentive risk on the agent is lowest. This risk premium depends on the utility function and the level of expected utility given jv^p If the agent's utility, u°c°m{-), is a concave (con- vex) function of the likelihood ratio, this risk premium is increasing (decreas- ing) in the level of expected utility (see Proposition 17C.2).^^ Moreover, the
^^ Note that if the agent's utiHty is a concave function of/, an investigation is "bad news" for the agent, since the additional risk in the Hkehhood ratio, L{y2 \ a), caused by the investigation is a fair gamble. On the other hand, if the agent's utility is a convex function of/, an investigation is
"good news" for the agent.
level of expected utility given j ; ! is increasing in the likelihood ratio for the first signal, L{y^ \ a). Hence, the risk premium is lowest (highest) for small values of L{y^\a) if the agent's utility is a concave (convex) function of the likelihood ratio. However, if the agent's utility is a linear function of the likelihood ratio, the benefit of an investigation is independent of L{y^\a) (and, thus, also ofj^^).
If MLRP holds for L{y^ \ a), then the above result can be applied directly to y^. In this setting, we have "lower-tailed" investigation when i/ocom(-) is con-
cave in the likelihood ratio, and "upper-tailed" investigation when i/ocom(-) is convex in the likelihood ratio.^^ Ifu^c^m{-) is linear in the likelihood ratio, the optimal investigation policy is independent of the primary signal (i.e., only the total probability of investigation matters).^^
Note that the investigation region has nothing to do with whether the values ofy^ are unusual or not. MLRP is merely a condition on the likelihood ratios.
While the upper and lower tails represent unusual events for a normal distribu- tion, one can construct distributions in which much of the mass is in one of the tails and yet the MLRP condition holds.
If the agent's utility function is a member of the HARA class, we can use Proposition 17C.1 to relate the benefits of an investigation to the agent's risk cautiousness.^^
Corollary
If the agent's utility function for consumption is a member of the HARA class, then the gross benefit of an investigation is decreasing (increasing) in L{y^\a) if the agent's risk cautiousness is less (more) than 2, and independ- ent of L{y^\a) if the agent's risk cautiousness is equal to 2 (i.e., square-root utility).
Informativeness of Secondary Signal Depends on Primary Signal
When the informativeness of3;2 depends ony^, an optimal investigation is not only determined by the likelihood ratio for the primary signal as in the previous analysis, but also by how the informativeness of the investigation varies with the primary signal. Note that by Proposition 18.6, we can rank the informative- ness of an investigation for different primary signals y^ by a SSD relation be- tween the conditional distributions for the likelihood ratio for the secondary sig-
^^ Young (1986) considers two utility functions for which the agent's utiHty is a concave function of / for small / and a convex function of / for large / resulting in a "two-tailed" investigation policy.
^^ This can include null and full investigation, but also a randomized investigation strategy independent of the primary signal (see the hurdle model example below).
^^ Proposition 17C. 1 is stated in terms of the likelihood measure m{l) = X + jil instead of directly in terms of the likelihood ratio /. However, note that m{l) is linear such that u°c° m(l) is concave (convex) in / if, and only if, u°c{m) is concave (convex) in m.
nal given the primary signal. Of course, if the agent has square-root utility, the benefits of an investigation do not depend on the likelihood ratio L{y^ \ a) per se, but only on how the informativeness of3;2 about a given j ; ! varies withy^.^^ In general, the two effects interact and the optimal investigation policy is deter- mined by the relative sizes of those effects. However, if the two effects go in the "same direction," lower- or upper-tailed investigation can be sustained as optimal investigation policies.
Proposition 18.20
Suppose MLRP holds for L(y^ \ a), and let 0(l2 \ a, l^ denote the conditional distribution function for the likelihood ratio for the secondary signal, I2 = Lfyil^^yi)^ given the likelihood ratio for the first signal /^ = Lfy^la).
(a) Ifu^co m(-) is a concave function of the likelihood ratio, and 0(l2 \ a, //') SS-dominates 0(l2\aJ{) for all // < //', lower-tailed investigation is optimal.
(b) If i/ocom(-) is a convex function of the likelihood ratio, and 0(l2\aJI) SS-dominates 0{l2\af(') for all // < //', upper-tailed investigation is optimal.
(c) lfu°c°m{-) is a linear function of the likelihood ratio, and 0{l2\aJ(') SS-dominates 0{l2\aJ(), the benefit of investigation is higher for //
than for //'.
Proof: We only show (a) since the proofs of (b) and (c) are similar. Since MLRP holds for L{y^ \ a) there is a one-to-one correspondence between y^ and /i = L{y^ I a) so we can write the benefits of an investigation as
h
where/(•) is defined as in the proof of Proposition 18.18. For // < I" we get 5(//) - 5(/i")
/(/,')-/(/,")
Y.f{i{^i^q>{i^\a,ii) - Y.m' ^h)<pihwj;')
^^ Note that in this setting, Proposition 18.8 imphes that the benefits of an investigation is increasing in the conditional variance of the likelihood ratio for the second signal given j^^ (see also Lambert 1985, Prop. 3).
= \Ai;)-Y.N;^h)<pihw;)\ - /(//') - Y.N;'^h)v{hw;)\
L ;, J L ;^ J
where the first inequality follows from Proposition 18.19, and the second in- equality follows from 0Q2 \ a, //') SS-dominating 0{l2 \ a, //) and the fact that/(-)
is concave as shown in the proof of Proposition 18.6. Q.E.D.
A Hurdle Model Example
The hurdle model in Section 18.1.3 provides a setting in which we can illustrate the preceding results. The primary signal is the outcome x e {x^,x^} with (p{Xg\a) = a, and the secondary signal is one of two equally likely signals;; e {yL^yu}^ with posterior distributions for the hurdle given by
[(1 + ^) - 2kh ify = y ^ , cp{h\y) =
[(I- k) ^ 2kh ify =y^,
with the informativeness parameter k e [0,1]. The likelihood ratio for the secondary signal given the primary signal is L(y\x,a) = L(x,y\a) - L(x\a), and are shown in Figure 18.3 for a = /4.
NotethatZ(y^|x^,a=/4) = -Z(y^|x^,a = /4),Z(y^|x^,a=/4) = -Z(y^|x^,a=/4), and (p(yjj\Xg, a=V2) = cpiy^ | x^, ^=V2). Hence, the distribution of the likelihood ratio for the secondary signal given x^, is the same as the distribution of minus the likelihood ratio for the secondary signal given x^. This implies that the informa- tiveness of the secondary signal is independent of the outcome x such that Proposition 18.19 applies.
In order to illustrate the results in Proposition 18.19 and its corollary we use a power utility_function, u{c) = c^, with risk cautiousness equal to 1/(1 -p).
Furthermore, U = 2, v(a) = a/(l- a), and the informativeness parameter is ^ = V2. Table 18.5 summarizes the optimal investigation policies for varying values of the utility power/? and the investigation cost K.
Note that for a = Vi, both outcomes are equally likely. Hence, consistent with the corollary to Proposition 18.19, the gross benefit from investigation is decreasing (increasing) in the outcome ioxp = .45 (/? = .55), whereas it is the same for both outcomes in the square-root utility case. If the investigation cost is low {K = .15), full investigation is optimal with/? = .45, whereas upper-tail investigation is optimal with/? = .55.
2 1.5 1 0.5 0 -0.5 -1 -1.5
L{yjj\x^,a=V2\
^L(yL\Xg,a=y2)
0 0.2 0.4 0.6 0.8
Informativeness parameter k
Figure 18.3: Likelihood ratios for secondary signal for a ^Vi with varying informativeness parameter k.
1
c ^ {a,a) p = 0.45, K = 0.15 p = 0.45; K = 0.25 p =0.50; K = 0.15 p = 0.50; K = 0.25 p =0.55; K = 0.15 p =0.55; K = 0.25
a(Xg) = 0 a(Xb) = 0
18.375 18.375 13.000 13.000 9.829 9.829
a(Xg) = 0
a{xt) = 1 18.171 18.346 12.946 12.996
9.837 9.887
«(Xy) = 1
«fe) = 0
18.255 18.430 12.946 12.996
9.815 9.865
«(Xy) = 1
a{xt) = 1 18.069 18.419 12.900 13.000
9.827 9.927 Table 18.5: Minimum expected compensation cost, c\a,a), for inducing a =
Vi for varying investigation policies, utility functions, and infor- mation costs.
On the other hand, if the investigation cost is high (K = .25), lower-tail investi- gation is optimal with/? = .45, and null investigation is optimal with/? = .55.
In the square-root utility case, full investigation is optimal for a low investi-
gation cost, whereas for a high investigation cost lower- and upper-tail investi- gation dominates both null and full investigation. In the latter case, only the total probability of an investigation matters, i.e., a{x^ + «(x^), and not how this total probability is split between the two outcomes.
In all the reported cases in Table 18.5, the optimal compensation is strictly positive for all signals and, thus, the contract is interior. Hence, the optimality of lower-tail versus upper-tail investigation depends exclusively on whether the agent's utility is a concave or convex function of the likelihood ratio for the pri- mary signal. Table 18.6 summarizes the optimal investigation policy in a set- ting in which the non-negativity constraint onthe agent's compensation is bind- ing for the bad outcome {p = .45; K = 1.25; U = 0; k = Vi).
c^{a,a) 10.884
aiXg)
0.441
aix^) 0.000
c(x^) 20.765
cixt) 0.000
<Xg,yH)
35.038
ciXg,yL)
13.840 Table 18.6: Optimal contract with binding boundary conditions.
First, note that there is upper-tail investigation even though the risk cautiousness is less than 2. Of course, the reason is that investigation for the bad outcome cannot impose any additional penalties on the agent (but only reward the agent).
Secondly, a non-trivial randomized investigation policy is used for the good outcome.
When the induced jump is a = /4, the informativeness of the secondary signal is independent of the primary signal, but that only holds for this particular induced action. Figure 18.4 shows the likelihood ratios for the secondary signal given the primary signal for a = % and varying informativeness parameter k.
The distributions of the likelihood ratios for the two primary signals cannot be ranked on the basis of SSD. However, it appears that the likelihood ratio for j ; ^ given the good outcome is significantly higher than minus the likelihood ratio for y^ given the bad outcome, so that there is more variation in the likelihood ratio for the good outcome than the bad outcome. Table 18.7 summarizes the optimal investigation policies for ^ = 1, TC = 0.15, and varying values of the utility power/?.
Note that upper-tail investigation is optimal with square-root utility. In this setting, the cost of imposing additional risk on the agent is independent of the primary signal and, hence, the additional variation in the likelihood ratio for the good outcome compared to the bad outcome makes it optimal to investigate for the good outcome and not for the bad outcome. For a risk cautiousness moder- ately less than 2 (p = .45), upper-tail investigation still dominates lower-tail investigation even though the cost of imposing additional risk on the agent is higher for the good outcome than the bad outcome. If the risk cautiousness is decreased further (p = .40), the impact of the varying cost of imposing addi-
tional risk on the agent dominates the impact of the variation in the likelihood ratios such that lower-tail investigation dominates upper-tail investigation.
When the risk cautiousness is higher than 2, the two effects both go in the direc- tion of upper-tail investigation (even though no investigation is optimal when the agent has a sufficiently high risk cautiousness).
L{yL\xt,,a=V4)
0.2 0.4 0.6 0.8 Informativeness parameter k
Figure 18.4: Likelihood ratios for secondary signal for a = % with varying informativeness parameter k.
c^{a,a) p = 0.40 p = 0.45 p = 0.50 p = 0.55 p = 0.60
a(Xg) = 0
«(XA) = 0
10.114 7.568 6.037 5.038 4.344
a(Xg) = 0
9.947 7.540 6.072 5.105 4.429
a(Xg) = 1
«fe) = 0
9.999 7.515 6.017 5.037 4.356
a(Xg) = 1
9.874 7.510 6.066 5.113 4.446 Table 18.7: Minimum expected compensation cost, c'^(a,a), for inducing a
% for varying investigation policies and utility functions.