Job Assignments under Moral Hazard - School of Economics and ...

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2 The static job assignment model 2.1 The model A risk neutral principal (‘she’) needs to assign an agent (‘he’) to one of two jobs, j ∈ {l, h}. He is protected by limited liability and has a reservation utility of zero. In each job he produces a verifiable and observable output. Output can either be high (revenue for the principal normalized to 1) – or low (revenue normalized to 0). The probability of high output, fj(e, θ), depends on the job j ∈ {l, h}, the agent’s ability θ ∈ [θ, ¯ θ], and his effort e ∈ {e, ē}. We assume that f(·, θ) is twice continuously differentiable in θ, and that higher effort leads to a higher success probability, i.e., fj(ē, θ) > fj(e, θ) ∀ θ. The utility cost of effort is c if the agent provides high effort, and zero otherwise. Job l entails ‘safe’ tasks, where a low ability agent can do no harm because the success probability is insensitive to ability. Hence, we refer to it as the low ability job hereafter and simplify notation to fl(e) ≡ fl(e, θ). In the high ability job h, expected output increases with higher ability, and the impact of ability on output also (weakly) increases with effort (i.e., ability and effort are weak complements). The following sensitivity to ability (SA) assumption captures this: 4 0 = fl θ(e, θ) = fl θ(ē, θ) < fh θ(e, θ) ≤ fh θ(ē, θ). (SA) Additionally, we assume that for a given effort, an agent with the highest ability level ( ¯ θ) is most productive in job h, and an agent with the lowest ability level (θ) is most productive in job l. The following crossing property (SP) expresses this formally: fh(e, θ) < fl(e, θ) and fl(e, ¯ θ) < fh(e, ¯ θ) for e ∈ {e, ē}. (SP) Together with Condition (SA) this implies that for each pair of equal effort levels in the two jobs (e, e) ∈ {(e, e), (ē, ē)} there exists a unique interior threshold ˆ θ(e, e) such that all types with θ ≥ ˆ θ(e, e) have a higher success probability in job h than in job l, and all types with θ < ˆ θ(e, e) have a lower one. 5 Figure 1 illustrates this general property with a linear example. It also shows that the success probability functions for the two jobs need not cross for unequal effort levels: in the example with θ = 0, there is a crossing point ˆ θ(ē, e) but no crossing point ˆ θ(e, ē) in the range θ ∈ [θ, ¯ θ]. The timing is as follows. First, the principal offers a contract, specifying the job j ∈ {l, h} and an output-contingent wage scheme. Both the principal and the agent know the 4 The subscript θ denotes the partial derivative with respect to θ. 5 On its own, Condition (SP) guarantees that at least one such threshold exists (by the intermediate value theorem). Condition (SA) gives a unique threshold: it implies that fh(e,θ) fl(e,θ) higher ability the agent becomes relatively more productive in job h. 6 is increasing in θ – with
Figure 1: Illustration of the single crossing property (SP) with a linear example [fh(e, θ) = qh(e) + θ, qh(ē) = 0.4, qh(e) = 0; fl(e) = ql(e), ql(ē) = 0.7, ql(e) = 0.35] agent’s ability θ ∈ [θ, ¯ θ]. The contract fixes a wage ¯wj(θ) after a high output and w j(θ) after a low one. If the agent accepts the contract, he then provides unobservable effort e ∈ {e, ē}. At the last stage, output and payoffs are realized: the principal receives the revenue net of the wage to the agent; the agent receives utility equal to the wage less his effort cost. If the agent rejects the contract, the game ends with zero payoffs for all players. 2.2 Observable effort (first best) In the benchmark case where the principal can observe the agent’s effort, she simply compensates him for his effort cost. That is, given job j and type θ her expected profit F B F B is either Πj (ē, θ) = fj(ē, θ) − c for high effort, or Πj (e, θ) = fj(e, θ) for low effort. To make the problem interesting, we assume that first-best profits for high effort are larger than the ones for low effort: Assumption 1 High effort is efficient in jobs j = l, h: fj(ē, θ) − fj(e, θ) ≥ c ∀θ ∈ [θ, ¯ θ]. The optimal job assignment rule is that the principal places an agent with type θ in job h if and only if the maximized profit in job h is higher than in job l, which is equivalent to fh(ē, θ) ≥ fl(ē). So, unsurprisingly, the first-best job assignment rule places the agent in the job where he is most productive: job h if θ ≥ θ F B = ˆ θ(ē, ē) and job l otherwise. 7
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Page 11 and 12: 2.4 Discussion: Biases in hierarchi
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Page 25 and 26: fh(ē, ˜ θ)−Ch(ē, ˜ θ) = fl(
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Job Assignments under Moral Hazard - School of Economics and ...

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