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

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an interesting direction for future research. Appendix A Job assignments given inefficient effort levels This section builds intuition for additional effects that inefficient second-best effort levels can have on the job assignment rule. To illustrate these effects, we assume that job h is locally more informative about effort than job l (Condition 2). We discuss here the case where the principal implements the inefficient low effort in job l (e SB l Condition 1 is violated for job l). The driving forces for the other case (e SB l = e, i.e., = ē) are similar to those in Section 2, and we leave the details for the proof of Proposition 5. Suppose first that in job h Condition 1 is also violated for all ability levels, i.e., e SB l e SB h (θ) = e. Then no incentives are needed because the principal implements low effort in either job. Hence, the job assignment problem reduces to maximizing the expected revenue fj(e, θ) over j, resulting in threshold θ SB = ˆ θ(e, e) > θ F B = ˆ θ(ē, ē). In other words, the downward distortion of effort in both jobs pushes the job assignment threshold above the first-best level. As we saw in Section 2, if second-best effort levels in both jobs are efficient (i.e., Condition 1 holds for all ability levels in both jobs), the principal distorts the job assignment threshold in the opposite direction: assignment to job h leads to an implementation cost saving effect that pulls the threshold below the first-best level: θ SB < ˆ θ(ē, ē). We expect both forces to be at work whenever Condition 1 holds for job h, but not for job l. To illustrate that this is indeed the case, and to see how these two forces interact, suppose that for job h Condition 1 holds for all θ. 16 Because no incentives are needed in job l, placing the agent into job h cannot lead to cost savings. But it may be cheaper to implement high effort in job h than it would be in job l. For that reason, assigning the agent to job h and then implementing high effort might be worthwhile for some θ < ˆ θ(e, e). Indeed, it is even possible that θ SB is lower than θ F B . When does this occur? Assigning the agent to the higher level job and implementing high effort increases output by fh(ē, θ) − fl(e). This has to outweigh the increase in implementation costs of Ch (ē, θ). Hence, if there is a strict gain in profit from assigning an agent with ability θ = θ F B to job h we know that θ SB < θ F B (by continuity). The following proposition, which we prove formally in Appendix B, summarizes our results. 16 Dropping the qualifier “for all θ”, implies that multiple thresholds are possible. Then θ SB in the explanations and results below refers to the lowest threshold for assigning the agent to job h (see the proof of Proposition 5). 22 =
Proposition 5 Suppose job h is locally more informative than job l (Condition 2). In case multiple cutoffs exist, θ SB refers to the lowest second-best threshold. 1. Suppose the second-best effort level in job l is inefficient (e SB l is violated for job l). Then, = e, i.e., Condition 1 (a) θSB ≤ ˆ θ(e, e), and θSB is higher than it would be if eSB h (θ) = eSB l = ē ∀ θ; (b) θ SB < θ F B if and only if fh(ē, ˆ θ(ē, ē)) − fl(e) > Ch(ē, ˆ θ(ē, ē)). (Condition 3) 2. Suppose the second-best effort level in job l is efficient (e SB l holds for job l), then θ SB < θ F B . B Proofs Proof of Lemma 1. = ē, i.e., Condition 1 d d θ ΠSB h θ (ē, θ) = fh θ(ē, θ) − Ch θ(ē, θ) = fh θ(ē, θ) − fh(ē, θ) fh θ(e, θ) − fh(e, θ) fh θ(ē, θ) [fh(ē, θ) − fh(e, θ)] 2 c > fh θ(ē, θ) − fh(ē, θ) fh θ(e, θ) c > 0, [fh(ē, θ) − fh(e, θ)] 2 where the last inequality follows from fh θ(ē, θ) ≥ fh θ(e, θ) (Assumption SA) and [fh(ē, θ) − fh(e, θ)] 2 > fh(ē, θ) c (Condition 1). This shows Part (a). Part (b) follows. Proof of Proposition 1. We prove one side of the inequality here, the reverse direction is analogous. Π SB h (ē, ˆ θ(ē, ē)) − Π SB l (ē) > 0 ⇔ fh(ē, ˆ θ(ē, ē)) − Ch(ē, ˆ θ(ē, ē)) > fl(ē) − Cl(ē) ⇔ Ch(ē, ˆ θ(ē, ē)) < Cl(ē) ⇔ fh(ē, ˆ θ(ē, ē)) − fh(e, ˆ θ(ē, ē)) > fl(ē) − fl(e) ⇔ fh(e, ˆ θ(ē, ē)) < fl(e), using repeatedly fh(ē, ˆ θ(ē, ē)) = fl(ē). θ SB < θ F B follows because θ F B = ˆ θ(ē, ē) and the profit difference is strictly increasing in θ (Lemma 1). Proofs of Propositions 2-4. See Appendix C. Proof of Proposition 5. Part (1a): Denote by ˜ θ the lowest ability level where expected profits in jobs h and l would be equalized if high effort was implemented in both jobs. That is, Π SB h (ē, ˜ θ) = 23
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Page 3 and 4: in the hierarchy often put low dema
Page 5 and 6: promotion. 3 Bernhardt (1995) forma
Page 7 and 8: Figure 1: Illustration of the singl
Page 9 and 10: If this fails to hold the principal
Page 11 and 12: 2.4 Discussion: Biases in hierarchi
Page 13 and 14: 3.1.2 The continuous effort model O
Page 15 and 16: eSB h (θ) = 0.5 F B k eh (θ) +
Page 17 and 18: 3. In the lower ability range of jo
Page 19 and 20: Figure 5: Expected earnings and cos
Page 21: Applying our simple model to a prom
Page 25 and 26: fh(ē, ˜ θ)−Ch(ē, ˜ θ) = fl(
Page 27 and 28: if he were instead assigned to job
Page 29 and 30: Clark, D. J., and C. Riis (2001):
Page 31: Rosen, S. (1982): “Authority, Con

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