Moral hazard with non-additive uncertainty: When are actions implementable?
Summary (2 min read)
- This paper aims to characterize conditions for a two-part, piecewise-constant contract (‘flat payment plus bonus’, FPB henceforth) to be used to implement actions in moral hazard problems with non-additive uncertainty.
- Lopomo et al. (2011) provide sufficient conditions for FPB contracts to be uniquely optimal under Bewley-type preferences on a finite outcome space, when Principal has more precise information than Agent.
- The authors consider three different formulations of ambiguity-sensitive preferences on a continuum of outcomes, and identify sufficient conditions for FPB contracts to solve the implementation problem for any action other than the least costly one, regardless of whether Agent knows more or less than Principal.
- Section 2 lays out the preliminaries, Section 3 presents the implementation results and the authors conclude with a discussion of their sufficient conditions in relation to the literature.
- The first author gratefully acknowledges the financial support from the Ministerio Economia y Competitividad through grants ECO2014-55953-P and MDM 2014-0431.
- Let B(Y) denote the Borel σ-algebra on Y , ∆(Y) denote the set of Borel distributions on Y with the weak topology and K∆(Y) denote the class of non-empty, compact, convex subsets of ∆(Y).
- The authors formal set up can accommodate asymmetry of uncertainty, in particular both of the following cases: (I) Agent has more precise knowledge about technology than Principal: QA(a) ⊂ QP (a) for each a ∈ A, and (II) Principal has more precise knowledge about technology: QP (a) ⊂ QA(a) for each a ∈ A.
- The authors results are interesting when Agent’s information sets are sufficiently rich, in that they are not singletons (i.e. when Agent is not a standard expected utility maximizer).
- Cores of convex capacities are well-defined (Schmeidler (1986)).
- Ch is the Choquet integral (Choquet (1954)).
- If a is the least cost action, then a flat payment of g(a) would implement it for any of the three kinds of objective functions under consideration.
- These implementation conditions, (2) - (3), (7) - (8), (12) - (13), depend only on the payment scheme w(y) and Agent’s perception QA but neither on Principal’s perceived ambiguity nor her ambiguity attitude.
- Here the last inequality shows that aj is rational for Agent, while the strict inequality, which follows from gk > gj, together with the first inequality, which follows from the fact QA(aj) ⊂ QA(ak) and that minimum of a non-negative-valued function does not get smaller over a larger set, establishes that Agent would prefer aj rather than ak.
- The next Proposition shows that strengthening the necessary condition to stochastic dominance, and imposing bounds on the rate at which costlier actions improve outcomes becomes sufficient for implementation.
- 3.2. Implementation under α−MEU Preferences.
- The authors provide conditions for two-part, piecewise-constant contracts to implement actions in a moral hazard setting with non-additive uncertainty, under three different formulations of ambiguity attitude for Agent.
- The authors necessary conditions for implementation are comparable to the one in Hermalin and Katz (1991) for the standard Bayesian model, and those in Ghirardato (1994) and Lopomo et al. (2011) for non-additive models.
- All these conditions stipulate a necessary amount of ‘disjointedness’ between the sets of probabilities generated by different actions.
- The authors stochastic dominance assumption is comparable to the MLRP condition on extreme points in Lopomo et al. (2011), and helps make downward deviations unattractive enough.
- Given that these contracts provide just the minimum level of variability needed for implementation, their results suggest they could be robustly optimal across different formulations of ambiguity attitude.
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"Moral hazard with non-additive unce..." refers background in this paper
...For a convex capacity C, the Choquet integral is given by ∫ Ch fdC = minq∈core (C) ∫ fdq....
...C1 dominates C2 with respect to a family of Borel measurable functions F , denoted C1 %F C2, if∫ Ch fdC1 ≥ ∫ Ch fdC2 ∀f ∈ F (1) where ∫ Ch is the Choquet integral (Choquet (1954))....
"Moral hazard with non-additive unce..." refers background or methods in this paper
...Cores of convex capacities are well-defined (Schmeidler (1986))....
...We also use the following result (Proposition 3 in Schmeidler (1986), proof omitted)....
...‘α-max-min’ or Hurwicz criterion (Hurwicz (1951)); 3. probability-set-dominance criteria (Bewley (2002))....
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