18.1.1 ^-informativeness
18.1.2 Second-order Stochastic Dominance with Respect to the Likelihood Ratio
In this section we summarize some results from Kim and Suh (1991) and Kim (1995). They focus on comparing the distribution function for the likelihood ratio L for alternative information structures. The agent is penalized (i.e., paid low values of c) if Z is small (i.e., very negative) and receives large bonuses (i.e., is paid high values of c) if Z is large. The point of their analysis is that greater variability in L permits more effective use of penalties and bonuses. In particular, if the distribution function for Z with rf second-order stochastically dominates that with rf, then if is preferred to rj^ by the principal for imple- menting a.
Kim (1995) provides an analysis in which he considers a set of information systems H ^ {^} in which each system produces a signal 3;.^^ His setting is one in which the outcome x is "owned" by the principal, but is not contractible information. The action set A is an interval of the real line and the incentive constraint is assumed to be characterized by the first-order condition of the agent's decision problem.
The optimal contract for inducing the agent to select action a using infor- mation system rj is characterized by (18.5) with explicit recognition of the fact that the optimal contract and the Lagrange multipliers depend on the informa- tion system in place, i.e., if c(y,;/) > c,
^ H79 uses a different approach to prove the necessity part of the proposition. He constructs a variation that induces an increase in the agent's effort without increasing the cost.
^^ Kim (1995) allows for multi-dimensional performance measures. However, this is of no real consequence in his analysis since the likelihood ratio is always single dimensional.
M(c(y,rj)) = l(a,rj) + ju(a,rj) L(y\a,rj), (18.11) where L(y \ a, rj)
(p{y\a,fj)
The likelihood ratio L plays a key role in the analysis. When viewed across the different performance measures, the likelihood ratio is a random variable. Let / = L{y I a, fj) denote this random variable, and let 0{l \ a, rj) denote the probability distribution function for /, i.e.,
0{l\a,fj) = Pr{Z < l\a,T]} = Y. ViyW^V)^
y 6 Y{l,a,rj)
where Y(l,a,rj) = {y\ L(y\a,rj) < I }.
Proposition 18.6 (Kim 1995, Prop. 1)
Assuming the first-order approach is valid, performance measurement system rj^ is strictly preferred to performance measurement system rj^ for implementing any a E (a, a], if 0(1 \a,rj^) strictly dominates 0(1 \ a, rf) in the sense of second-order stochastic dominance.
Proof: Consider a setting in which the principal will use performance measure- ment system rf with probability a e [0,1] and rj^ with probability \- a (see Section 18.4 for a similar setting in which this approach is further developed).
That is, we can formulate the principal's decision problem as in the basic model except that
U\c,a\a) = aUP(c\a\rj^) + (\ - a)U\c\a\rj\
U\c,a\a) = aU%c\a\rj^) + (\ - a)U\c\a\rj^),
where c^ and c^ are the incentive contracts for rj^ and rf, respectively. The Lagrangian in this setting is
a - a[UP(c\a\ff) +XU\c\a\ff) + juU^(c]a\rj^)]
+ (l-a)[UP(c\a\rj') + W(c\a\rj') + juU^(c\a\rj')] -XU.
The first-order conditions for the optimal incentive contracts with either rj^ or ff are similarly characterized by (18.5):^^
M(c^(y.)) = X + fiL{y^\a,fj\ i = 1,2, if the compensation is interior forj;. Otherwise, c'(y^) = c.
Note that both the principal's and the agent's expected utilities are linear in the probability a. This implies that the optimal probability a will always be a corner solution, i.e., a = 0 or a = 1. Differentiating the Lagrangian with respect to a yields the following expected marginal benefits of increasing a:
B ^ d^lda = [UP{c\a\rf) + XU\c\a\rf) + juU^(cla\rj^)]
- [UP(c\a\rj') +W%c\a\rj') + juU^(c\a\rj')]
= E [c'(y,)-u(c'(y,))M(c'(y,))](p(y,\a,rj')
- E [ ^'(yi) - <c'{y,)) M{c'{y,)) ] (p{y,\a,n\
where the equality is obtained by collecting terms and substituting in the M{-) based on the preceding first-order condition. Hence, it is optimal to choose a
= 1 (which is what we need to show) if, and only if, 5 > 0.
Note that the incentive contracts only depend on the performance measure- ment system rj' and the signals y through the likelihood ratio, i.e., we can write c\m{l)) = c(m(l)) where the likelihood measure m is defined by m(l) = 1 + jul (and m(l) = M(c) on the lower bound for c). Hence, if we define the function /(•)by
/ ( / ) = c(m(l)) - u(c(m(l)))m(l), we can write B as follows
B = '£f(l)<p(l\a,fj') - J2f(l)<p(l\a,fl').
I I
The function/(•) is a strictly concave function since
^^ We assume that the agent knows a when he selects a, but he does not yet know f]\ Hence, there is a single incentive constraint, implying that ju is not dependent on f]\ However, //' is contractible information, so that c depends on both y and rj\
f(l) = c'{m{l))fi - [u'{c{m{l)))c'{m{l))fim{l) + ^/(c(m(/)))//]
= - u(c(m(l)))ju,
f"(l) = - u'(c(m(l)))c'(m(l))ju^ <0,
where/' = - uju follows from the fact that m = l/u' and/" < 0 follows from the fact that u" < 0 implies c'(-) > 0 (for any interior compensation). Since/(•) is strictly concave and 0{l\a,ff) strictly dominates 0{l\a,ff) in the sense of second-order stochastic dominance, we get that^^
B = Y.N)(p{i\a,n') - J2f(0<p(iW,n')>o,
and, thus, a = 1 is optimal. Q.E.D.
Note that the random variable / always has a mean of zero. Hence, 0{l\a,ff) SS-dominates 0{l \ a, rf) if, and only if, the probability function (p{l \ a, rf) differs from (p{l\a,ff) by adding mean-preserving spreads.^^ This illustrates the point that more variation in the likelihood ratio is desirable - it implies that there is a wider range of information upon which to efficiently place penalties and re- wards (see also the hurdle model example in Chapter 17). Interestingly, the risk imposed on the agent (as indicated by the risk premium he is paid) decreases as the variation in the likelihood ratio increases.
In relating his analysis to that of GH, Gjesdal (1982) and Holmstrom (1979, 1982), Kim (1995) obtains the following results (see Kim, 1995, for proofs).
Proposition 18.7 (Kim 1995, Prop. 2, 4, and 5)
(a) If ff is more ^-informative than rf, then 0{l\a,ff) SS-dominates 0{l\a,ff'). However, the converse is not necessarily true.
^^ Strictly speaking, we here ignore the fact that the SS-dominance relation may be due to likelihood ratios where the compensation is on the boundary.
^^ Kim (1995) states his proposition in terms of mean-preserving spreads, and then interprets that to mean that 0(l\a,rj^) second-order stochastically dominates 0(l\a,rj^). The equivalence between the two concepts is given by the following result.
Lemma (Rothschild and Stiglitz 1970, Theorem 2)
The distribution for a random variable X second-order stochastically dominates the distri- bution for another random variable Y with the same mean, if, and only if, the probability function for Y differs from the probability function for X by adding mean-preserving spreads.
(b) Letj; = (yi^yi) ^nd assume rj^ only generates j^^, while rj^ generates both yi and3;2- Then 0(1 \a,rj^) SS-dominates 0(1 \ a, rf) for all ae (a, a], and 0(l\a,rj^) strictly SS-dominates 0(l\a,ff') for some a e (a,a] if and only if yi is not a sufficient statistic for j ; = (yi^yi)-
These results demonstrate that the SSD relation between the distribution func- tions of the likelihood ratios is a weaker condition than^-informativeness but, nevertheless, is sufficient for ranking the information systems by Proposition 18.6. In the case of the conditional value of an additional signal, the SSD rela- tion is equivalent to the Holmstrom (1979) result that the additional signal is incrementally informative about a.
Kim (1995) also relates his results to the Blackwell-theorem (see Propo- sition 3.7) in the sense of ^-informativeness. Proposition 18.1 establishes the sufficiency of the Blackwell relation in an agency setting. The question raised by Kim (1995) is whether it is necessary for such a Markov kernel to exist to ensure that rf is preferred over rj^ for all agency preferences. In addressing this issue, recall that the structure of the agency problem is restricted to be one in which the principal is risk neutral and the agent is risk averse with additively separable preferences. Hence, Kim loses some of the generality that pertains to the original Blackwell result - which considered any payoff function. Given that limitation in the analysis, he then claims that the "necessary part of Blackwell's theorem does not hold in the agency model" (see Kim 1995, Proposition 3). He provides an example as his proof. Essentially the point is that, by Proposition 18.7(a), the SSD relation can hold even if the Blackwell relation does not and, by Proposition 18.6, the SSD relation is sufficient for rf to be less costly than rj^ for all agency problems (that satisfy Kim's basic as- sumptions).
The above results provide a partial ordering between information systems that are distinguished by ^-informativeness or mean-preserving spreads of the probability functions for the likelihood ratios. Kim and Suh (1991) seek a com- plete ordering in terms of a simple measure such as the variance of the likeli- hood ratio. They accomplish this by either restricting the agent's utility for compensation (to be a square-root function) or by restricting the underlying probability functions (to normal, log-normal, or Laplace families).
Proposition 18.8 (KS, Prop. 1)
(a) lfu(c) = 2c^' and the optimal incentive contracts are interior, then tj^ is more valuable than tj^ in inducing action a if, and only if, Var(/|a,;/^)
<Var(/|a,;/').
(b) Assume that cpiy \a,rj^) and (p(y \ a, rf) belong to the normal, log-normal, or Laplace families. Then rf is more valuable than rj^ in inducing ac- tion a if, and only if, Var(/|a,;/^) < YdiX(l\a,ff')}^
Proof:
(a): A key feature of the square-root utility function is that it results in a com- pensation function that is the square of a linear function of / (see Appendix
17C), i.e., M(c) = c^' =^ c(l,a,fj) = [l(a,fj) + ju(a,fj)l]^. Since /has zero mean, it follows that
E[c|a,;/] = [>1(^,//)]^ + [ju(a,fj)]^YsiY(l\a,fj).
Since the optimal incentive contracts are assumed to be interior, the agent's par- ticipation constraint is satisfied as an equality. Using again that / has mean zero, it follows from the participation constraint that >l(a,;/) = V2[U+ v(a)], and from the incentive compatibility constraint that//(a,;/) Var(/1 a,;/) = V2v'(a) for rj = fj\
if. Hence,
E[c|a,;/] = - k/j + v(a) + ^v'(a)ju(a,T]).
Finally, ju(a,rj^) > id(a,ff) ^ Var(/|a,;/^) < Var(/|a,;/^).
(b): KS demonstrate that for these distributions, Var(/|a,;/^) < Var(/|a,;/^) if, and only if, 0(1 \a,ff) strictly SS-dominates 0(1 \ a, rf), and then Proposition 18.6
gives the result. Q.E.D.
^^ The three classes of distributions have the following characteristics:
Normal Log-normal Laplace
(P
Var(/)
exp
a"
-iy-y{a)y
\y-y{a)]
- e x p
y -M\^y-y(a))^ exp --^|j^-K«)
^[\ny-y{a)] ±'-y'{a)
y'{a) o
12 y\d)
\ P \
2
y{a) = E[y|a] E[lnj^|a] E[y|a]