Abstract
The model allows the manager to recognize a fraction of next period's income in this period (i.e., ask the customers to pay in advance for deliveries in January and recognize revenue on a cash basis) and to postpone the recognition of some income items (for example, bill the customers in January for shipments made in December and recognize revenue on a cash basis). For companies that produce to order it is not unreasonable to assume the company knows next period's revenue when periods are relatively short. Defense contractors, for example, know relatively well what next quarter's revenue will be. Hence the model describes income smoothing rather than earnings management.
In order to solve the model numerically, the validity of the Bellman's equation is shown. (The difficulty consists of keeping the state space small, so that the Bellman's equation can be computed numerically. Otherwise, one could simply set the state space equal to the set all possible histories, and then the Bellman's equation would immidiately follow. An added advantage of looking at stationary strategies for a ``small'' state space is that it restricts the principal and agent to simpler strategies.) The fact that the principal does not observe income smoothing, but forms beliefs over the variables not observed, complicates the proof.
Trueman and Titman [Trueman and Titman1988] model income smoothing in a
similar fashion in a model of adverse selection (without any moral
hazard). In their model there are only two periods, so smoothing only
occurs in period 1 while period 2 is used to ``add up" accounts. Also,
in their model, next period's output, 
, is not revealed before
next period, but its (unconditional) expectation is and that is the
figure that Trueman and Titman [Trueman and Titman1988] use.
Further, they require that the
manager (weakly) overreports if realized income is below its expected
value and underreports if the realized income is above its expected
value. Our model does not put such restrictions on the income report.
In proving the results, we used techniques that we saw in other papers. The idea of using the agent's continuation value as a state variable came from [Spear and Srivastava1987]. The idea of using self-generating sets (as defined in [Abreu, Pearce, and Stachetti1990]) in this context came from [Wang1993]. The idea of using the principal's beliefs in the state variable came from [Rhenius1974].
Section 2 lays out the principal-agent model. Section 3 describe a recursive principal-agent model. The theorem at the end of the Section 3 states the equivalence of these models. The appendix containts the proof of the theorem.