RENEGOTIATION IN SINGLE-PERIOD SETTINGS
16.4 MULTI-PERIOD/SINGLE-AGENT SETTINGS
16.4.3 Full Commitment with Interdependent Periods
Chapter 27 considers settings in which there is stochastic and technological interdependence across periods. This occurs if the uncontrollable events are correlated across periods and the actions in one period affect performance measures beyond the current period. Also, we consider settings in which the
performance measure in one period is informative about the marginal product- ivity of effort in a subsequent period.
Orthogonalized and Normalized Performance Statistics
In Chapter 27 we use orthogonalization and normalization to modify the repre- sentation of normally distributed performance measures. Orthogonalization transforms stochastically interdependent reports into stochastically independent performance statistics. The resulting statistic for each period only reveals the
"new" information provided in that period. Interestingly, the orthoganalized statistics are generally technologically interdependent even if the initial repre- sentations of the reports were technologically independent.
Normalization uses the principal's conjecture with respect to the agent's actions to construct performance statistics that have zero mean if the agent' s ac- tions are equal to the principal's conjecture. That is, a normalized statistic is ef- fectively equal to the difference between the realized value of a report and a standard or budget that is equal to its (conditional) expected value if the agent takes the conjectured action. In equilibrium, the agent's action choice equals the principal's conjecture, but in choosing his action the agent considers the possibility of deviating from the principal's conjecture.
The induced effort in any given period depends on direct and indirect incen- tives. The former refer to the incentive rates applied to the statistics directly af- fected by the action. The indirect incentives arise from the fact that the agent's action affects the reports that will be used in producing the orthogonalized sta- tistics and in determining the posterior means used in producing the normalized statistics.
Information Contingent Actions
In the basic multi-periodZ£7V model the only source of uncertainty is the addi- tive noise in the performance measures. This additive structure plus the lack of a wealth effect (due to exponential AC-EC preferences), and the restriction to linear contracts, results in second-period incentive rates and, thus, second-period actions that are independent of the first-period performance reports. In section 27.3 we first use a first-order approach to obtain insight into the characteristics of an optimal contract based on the stochastically independent performance sta- tistics (not constrained to be linear). Even though there are no wealth effects and the first-period performance report is uninformative about the second-period effort productivity, the characterization of the optimal contract shows that in contrast to the multi-period Z£7Vmodel the second-period action varies with the first-period performance report. This is due to the fact that the variation in the second-period contract induces positive indirect first-period incentives. The characterization of the optimal first-period action shows that it is influenced by direct first-period incentives and two types of indirect incentives. The first type is referred to as an indirect "posterior mean" incentive. It is due to the impact
of the first-period action on the principal's beliefs about the second-period per- formance measure when the two performance measures are correlated. The second type is referred to as an indirect "covariance" incentive. This incentive is due to the impact of the first-period action on the covariance between the first-period performance report and the agent's conditional expected utility of the second-period contract. The latter incentive is only present if the second- period contract varies with the first-period performance report.
Second, we consider modifications to the linear contracts that capture these key aspects of the optimal contract, yet are analytically tractable. The central element in these changes is to allow the second-period incentive rate to vary linearly with the first-period report. This causes the second-period effort cost and risk premium to vary, creating effort-cost risk and risk-premium risk. Two quadratic functions based on the agent's conjectured actions are introduced to insure the agent against these two risks. We refer to this as a QEN contract.
Varying the second-period incentives with the first-period report creates costs in the second period (i.e., the effort-cost and risk-premium risks intro- duced above), but those costs are offset by the benefits of the indirect first- period covariance incentives created by this variation. Interestingly, while increased positive covariance between the two performance measures has a negative effect with a Z^'A^ contract, it has a positive effect with a g£7V contract.
Learning about Effort Productivity
In Section 27.4 we consider two settings in which the first-period report is informative about the output productivity of the agent's second-period effort.
The first is an extension of the LEN model, which we call the QEN-P model.
The preferences and performance measures are the same as in the Z£7V model, but the second-period productivity is random and correlated with the first-period report. A ig£7V contract is used. In this case there are two reasons for letting the second-period incentive rate vary with the first-period report. First, it creates indirect first-period covariance incentives of the type described above. Second, it provides more efficient direct second-period incentives, i.e., the induced second-period effort is positively correlated with its second-period output pro- ductivity.
Our second setting uses a two-period model in which the first-period action influences the information revealed by the first-period report about the second- period productivity. Optimal contracts (that are not constrained to be linear) are identified. A key feature of this example is that the optimal first-period effort reflects both its output productivity and its impact on the informativeness of the first-period report about the second-period productivity.