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...internal audits, hoc personalized evaluations, or fast tracks). Yet, there is currently no theoretical mean to rank the information systems induced by cost-equivalent contingent monitoring policies.
Early research on ranking information systems centered on developing orderings based on statistical decision theory--notably on Blackwell's theorem and its fineness criterion. In the principal-agent (hence, game-theoretic) framework, the informativeness criterion was introduced by Holmstrom (1979). This criterion conveys the intuitive requirement that, to be valuable, an amendment must bring information about the manager's actions beyond what can already be gathered with the current information system. One shortcoming, however, is that it does not allow comparison of the relative contribution of different amendments. Developed more recently by Kim (1995), the MPS criterion--which ranks information systems according to the mean-preserving spread relation between their respective likelihood ratio distributions--is so far the ordering that best deals with this issue. (1) This criterion renders the intuition that an information system is better when the firm can infer the manager's actions with greater accuracy. While it embodies the previous classifications, it allows the comparison of systems that are not nested.
Among the possible information system amendments, however, one type still eludes the MPS criterion: those that make the acquisition of further information conditional upon observing certain results. As Holmstrom (1979, p. 87) and others pointed out, such strategies of contingent monitoring should nonetheless remain significant in practice because obtaining more data often involves unsunk additional expenses. This paper's objective is now to improve the comparison of information systems in this setting.
The text unfolds as follows. The next section lays out a standard principal-agent model with contingent monitoring, linear monitoring cost, and conditionally independent signals. Unlike several works on this topic that build on Baiman and Demski's (1980) seminal article, we do not require that the agent's utility function belong to the hyperbolic absolute risk aversion (HARA) family. (2) The section ends with a reformulation of the principal-agent model as a one-person decision problem as in Grossman and Hart (1983). Using this, [section]3 then presents the intuition why the MPS criterion does not provide a useful ranking of information systems in this context. Basically, implementing a new contingent monitoring policy while keeping the ex ante probability of investigation constant amounts to making mean and variance-preserving transformations of specific lotteries. This suggests (see Menezes et al. 1980) that the information systems generated by contingent monitoring must now be sorted, not according to second-order stochastic dominance (which is the same here as the mean-preserving-spread ordering), but via third-order stochastic dominance. This is shown formally in Theorem 1 of [section]4. Balancing the pros and cons of various contingent monitoring policies thus involves choices with respect to added local risk, a behavior that is actually captured by the concept of prudence (Eeckhoudt and Schlesinger 2005). Theorem 2 of [section]5 provides an extension of Kim's (1995) criterion which conveys this notion and so deals with mean and variance-preserving amendments of information systems. This yields a generalization (Proposition 1) of Baiman and Demski's (1980) characterization of optimal audits. Further implications and generalizations when signals are correlated (Proposition 2), as in Lambert (1985), and where the precision of additional data is endogenous (Proposition 3), as in Kim and Suh (1992), are also derived in [section]6. Section 7 concludes the paper.
2. The Model
Consider a one-period relationship between a principal and an agent. An amount of effort a [member of] [0, [infinity]) is expected from the latter. This effort, however, is only imperfectly observable through some random variables, X and Y. We assume (until [section]6) that X and Y are conditionally independent, so for a given effort a the realizations x and y of the random variables obey the conditional distributions F(x, a) and G(y, a), respectively. These distributions have respective densities f(x, a) and g(y, a) that exhibit constant bounded supports [[GAMMA].sub.X] (with bounds [x.bar] and [bar.x]) and [[GAMMA].sub.Y], and are twice differentiable in a.
The likelihood ratios associated with X and Y will now be respectively denoted [L.sub.X](x, a) = [f.sub.a](x, a)/f(x, a) and [L.sub.Y](y, a) = [g.sub.a](y, a)/g(y, a). (3) A standard assumption is that these ratios satisfy the monotone likelihood ratio property (MLRP), that is: [L.sub.X](x, a) and [L.sub.Y](y, a) increase in x and y, respectively, for every a. Clearly, [L.sub.X] and [L.sub.Y] are themselves random variables, and their respective distribution--called a likelihood ratio distribution--constitutes a formal representation of an accounting information system. (4) It is well known that all likelihood ratio distributions have the same mean E[[L.sub.X]] = 0. The variance of, say, [L.sub.X] is then given by Var([L.sub.X]) = E[([L.sub.X])[.sup.2]]; it is often denoted [I.sub.X] and called the "Fisher information index" associated with X. (5)
The risk-neutral principal routinely observes the value of X. Based on this, she may either compensate the agent immediately according to a wage schedule w(X), or she may monitor the agent further at a constant cost K--thereby also gathering signal Y--and pay him according to a sharing rule s(X, Y). We suppose that the principal can commit to a probability m(x) of further monitoring after observing X = x. Her expected cost when the agent delivers effort a is therefore given by
EC = [[integral].sub.[GAMMA].sub.X][[integral].sub.[GAMMA].sub.Y]{(1 - m(x))w(x) + m(x)s(x, y)}dF(x, a)dG(y, a) + K [[integral].sub.[GAMMA].sub.X]m(x)dF(x, a). (1)
The latter integral M(a) = [[integral].sub.[GAMMA].sub.X] m(x)dF(x, a) gives the expected monitoring probability (which we also call the monitoring intensity) under a contingent monitoring policy m(X).
The agent's preferences are assumed to be additively separable in effort and wealth. The cost of effort is scaled so that its first-order derivative is equal to one. The agent's attitude with respect to uncertain variations of his wealth exhibits risk aversion and is represented by a positive, strictly concave and three-times continuously differentiable Von Neumann-Morgenstern (VNM) utility index...
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