The following results come from Amershi and Hughes (1989) (AH) and we refer to that paper for proofs. AH focus on settings in which there is a vector of possible performance measures;; = (y^,...,;;^). They examine whether (and under what conditions) the principal is necessarily worse off if he receives less than a sufficient statistic for y. Proposition 18.2 establishes that the principal is never worse off with a sufficient statistic than with any other representation of the information. This follows from the fact that
cp (xi/1 a, ff)
L(ii/\a,rj'^ = = L(y\a,rj) if ^ = ^(y), g)(if/\a,rj'^)
which implies thatj; and ^result in the same compensation levels (i.e., allj; that result in the same statistic y/ also result in the same optimal compensation level).
Observe that the likelihood ratio L(-) (and the induced compensation func- tion c(-)) is a statistic, and we call it a sufficient "incentive" statistic since it pro- vides all the information necessary for specifying the optimal compensation level for a given action a. The question of whether a sufficient statistic is neces- sary for implementing the optimal compensation plan is equivalent to asking whether Z(-1 a, rf) is a minimal sufficient statistic (or whether Z(-1 a, rf) is invert- ible with respect to a minimal sufficient statistic).
Observe that the likelihood ratio L{-) is a one-dimensional statistic, i.e., L\
YxA ^ M. Consequently, it can only be a sufficient statistic for a family of distributions {(p.A) if the minimal sufficient statistic for that family is one- dimensional.
In examining the "necessity" of a sufficient statistic, AH pay particular attention to the exponential family of distributions (see Volume I, Chapter 2 for a characterization of the one-parameter family of exponential distributions).
Definition
A family of distributions {(p,A} is a member of the exponential family of rank r > I in Y if:
(a) There exist real-valued statistics i/zf. Y^ J? and parametric functions af.
A ^ M,i = I,..., r, such that (p(y\a) is of the form
(p{y\a) = 6(y)fi(a)Qxp
Yl «/(^) ¥i(y)
i=\
(b) The systems of functions {1, ^i(y),..., y/r(y)} ^^d {1, ai(a),..., aX^)}
are linearly independent over Y and ^ , respectively, where fi(a) is a scaling function that makes (p(y\a) = 1 under integration over 7. The functions a^ia) are called distribution parameters.
Observe that the rank of an exponential family is not the dimension of the signal y = fyi,..., y^), but rather the dimension of the minimal sufficient statistic i//(y)
= (^i(y),..., y^riy))' Effectively, this dimension depends on the number of "un- known parameters." For example, the normal distribution in which a influences only the mean is a member of exponential family of dimension 1, whereas the normal distribution in which a influences both the mean and the variance is a member of the exponential family of dimension 2. These characterizations hold for a signal that consists of sample size m, for any m > I. (See Volume I, Section 3.1.4.)
Proposition 18A.1 (AH, Prop. 1)
If {(p,A} belongs to the exponential family of rank one, then the principal strictly prefers every sufficient statistic to all nonsufficient statistics.
Proposition 18A.2 (AH, Theorem 1)
Assume {(p,A} is such that the density functions (p(y\a), a e A, are con- tinuous in y with fixed support 7. For all a e A, the likelihood ratios L{\i/{y) I a, fj''^ are strictly monotone in some one-dimensional minimal suffi- cient statistic y/iy) if, and only if, {(p,A} belongs to the exponential family of rank one.
The monotonicity of Z(^(y)|a,;/^) establishes its invertibility. Without mono- tonicity we have a setting in which two statistics y/^ and ^ can induce the same likelihood ratio (which implies the same compensation level) and, hence, the compensation function is based on "less" than a minimal sufficient statistic.
The above theorem establishes that of all the continuous distributions one can imagine, only those in the exponential family of rank one have one-dimensional sufficient statistics that result in monotone likelihood ratios.
Proposition 18A.3 (AH, Prop. 2)
If all actions in the set^ are "relevant," then in any agency characterized by {(p,A}, the principal strictly prefers a sufficient statistic to all nonsufficient statistics if, and only if, {(p,A} belongs to the exponential class of rank one.
Proposition 18A.4 (AH, Theorem 2)
Assume Y^Y^x...xY^ and the signals are independent, identically distribu- ted random variables with densities ^(y,!^), / = 1,..., m, that are continuous in y^ with fixed support for dXX a e A. If there exists a one-dimensional sufficient statistic, then {(p.A) belongs to the exponential family of rank one.
Corollary (AH, Corollary 1)
Assume {(p,A} satisfies the assumptions of Proposition 18A.2. Every non- sufficient statistic is also globally ''incentive'' insufficient (see the earlier Holmstrom definition) if, and only if, {(p.A) belongs to the exponential family of rank one.
Corollary (AH, Prop. 3)
For every agency with statistical structure {(p,A} that belongs to the expo- nential family of rank r > 1, an optimal incentive contract is always a non- sufficient statistic.
The key factor that leads to the last result is that if more than one parameter is influenced by the agent's action a, the sufficient statistic has more than one dimension. However, while the likelihood ratio is monotonic in each compo- nent of that sufficient statistic, there is more than one sufficient statistic that results in the same likelihood ratio. Hence, neither the compensation level nor the likelihood ratio that generated it is sufficient to infer even a minimal suffi- cient statistic. That is, in this setting, the principal does not use all the ''infor- mation '' provided by a sufficient statistic in constructing an optimal compen- sation contract. Of course, he can always use a sufficient statistic in construct- ing the optimal compensation contract, since he can "ignore" any information he does not require.
REFERENCES
Amershi, A. H., R. D. Banker, and S. M. Datar. (1990) "Economic Sufficiency and Statistical Sufficiency in the Aggregation of Accounting Signals," The Accounting Review 65,113-130.
Amershi, A. H., and J. S. Hughes. (1989) "Multiple Signals, Statistical Sufficiency, andPareto Orderings of Best Agency Contracts," Rand Journal of Economics 20, 102-112.
Baiman, S., and J. S. Demski. (1980a) "Variance Analysis as Motivational Devices," Manage- ment Science 26, 840-848.
Baiman, S., and J. S. Demski. (1980b) "Economically Optimal Performance Evaluation and Control Sy stems J' Journal of Accounting Research 18 (Supplement), 184-220.
Banker, R. D., and S. M. Datar. (1989) "Sensitivity, Precision, and Linear Aggregation of Signals for Performance Evaluation," Journal Accounting Research 27, 21-39.
Christensen, P. O., and H. Frimor. (1998) "Multi-period Agencies with and without Banking,"
Working Paper, Odense University.
Feltham, G. A., and P. O. Christensen. (1988) "Firm-Specific Information and Efficient Resource Allocation," Contemporary Accounting Research 5, 133-169.
Gjesdal, F. (1981) "Accounting for Stewardship," Journal of Accounting Research 19,208-231.
Gjesdal, F. (1982) "Information and Incentives: The Agency Information Problem," i?ev/ew o/
Economic Studies 49, 373- 390.
Grossman, S. J., and O. D. Hart. (1983) "An Analysis of the Principal-Agent Problem,"
Econometrica 51, 7-45.
Holmstrom, B. (1979) "Moral Hazard and Observability," ^e/ZJowrwa/o/^'cowow/c^ 10,74-91.
Holmstrom, B. (1982) "Moral Hazard in Teams," Bell Journal of Economics 13, 324-340.
Jewitt, I. (1988) "Justifying the First-Order Approach to Principal-Agent Problems," Econo- metrica 56, 1177-1190.
Kim, S. K. (1995) "Efficiency of an Information System in an Agency Model," Econometrica 63,89-102.
Kim, S. K., and Y. S. Suh. (1991) "Ranking Accounting Information Systems for Management Control," Journal of Accounting Research 29, 386-396.
Kim, S. K., and Y. S. Suh. (1992) "Conditional Monitoring Under Moral Hazard," Management 5'c/ewce 38, 1106-1120.
Lambert, R. (1985) "Variance Investigation in Agency Settings," Journal of Accounting Research 23, 633-647.
Rothschild, M., and J. E. Stiglitz. (1970) "Increasing Risk I: A Definition," Journal of Economic Theory 2, 225-243.
Sinclair-Desgagne, B. (1994) "The First-Order Approach to Multi-Signal Principal-Agent Problems," Econometrica 62, 459-466.
Young, R. A. (1986) "A Note on 'Economically Optimal Performance Evaluation and Control Systems': The Optimality of T^o-Tdi\\QdlnYQ^t\gdX\on^,'' Journal of Accounting Research 24,231-240.