Contracts in informed-principal problems with moral hazard

Abstract

In many cases, an employer has private information about the potential productivity of a worker, who in turn has private information about the effort she exerts on the job. Much of the analysis of this environment in the literature restricts the employer to offer contracts that depend only on observable outcomes (e.g., profit). This paper studies the advantages to the employer of offering the worker a set of potential contracts from which the employer will choose after the worker has accepted the offer, so called menu-contracts. Specifically, in a two-state principal-agent problem with moral hazard, I show when the principal can obtain strictly higher expected payoffs than the restricted contracts of the literature by offering a menu-contract.

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Fig. 1

Notes

  1. 1.

    Chade and Silvers (2002) do consider more general contracts as a robustness check, but mainly focus on point-contracts. See below for more details.

  2. 2.

    When no separating equilibrium exists, the set of point-contract equilibrium payoffs may coincide with the set of menu-contract equilibrium payoffs.

  3. 3.

    Their analysis allows the principal to offer menu-contracts, but the same payoffs can be achieved using point-contracts.

  4. 4.

    Where \(\cdot \) denotes the inner product operator.

  5. 5.

    The formal timing of the game is outlined in Sect. 2.4.

  6. 6.

    Thus, the agent cannot extract any information from the principal’s offer; the principal is inscrutable at the time of the agent’s acceptance decision.

  7. 7.

    If the choice of effort is the same for both principal types, each will offer \((\mathbf {w}^{*L},a^{*L})\). In some cases, the high type principal may find it profitable to induce effort \(a_2\) while the low type principal prefers \(a_1\); then the type-L principal offers \((\mathbf {w}^{*L},a^{*L})\) and the type-H principal offers \((\mathbf {w}^{*L}(a_2),a_2)\), that is, the type-L optimal wage when effort is fixed at \(a_2\). Notice that if a wage profile satisfies AIC(0, 1) then it also satisfies AIC(1, 0), since \(\pi ^H_s(a_2)-\pi ^H_s(a_1)>\pi ^L_s(a_2)-\pi ^L_s(a_1)\). Moreover, for any wage profile \(\mathbf {w}\) that satisfies \(AIC(\varvec{\rho })\) for any \(\varvec{\rho }\), it must be that \(w_s>w_f\), since \(a_2>a_1\). Therefore, since \(\pi ^H(a_2)>\pi ^L(a_2)\), if a wage profile satisfies IR(0, 1) then it also satisfies IR(1, 0).

  8. 8.

    Rothchild-Stiglitz-Wilson is a reference to the similar least-cost-separating contracts developed in the insurance models of Rothschild and Stiglitz (1976) and Wilson (1977).

  9. 9.

    The RSW menu-contract always exists and is unique.

  10. 10.

    In terms of Myerson (1983), any feasible solution to the RSW problem is safe. The RSW menu for the type-i principal is her best safe menu.

  11. 11.

    Due to the linearity of the principal’s indifference curves (in particular, the fact that they posses the single crossing property) computing the RSW contract can be simplified, as I show in Lemma 1 in the Appendix. This lemma extends Proposition 2 in Maskin and Tirole (1992).

  12. 12.

    Otherwise, one can show that menu- and point- contract equilibrium payoffs coincide.

  13. 13.

    In this case, the pooling contract is not an equilibrium in menu-contracts because the type-H principal gets a payoff that is lower than \(\hat{v}^H\). Ex ante, the pooling contract may give a higher payoff than the least-cost-separating contract. This demonstrates that a menu-contract is not a generalization of a point-contract.

  14. 14.

    The quantity \(\Delta \) is chosen so that the type-L principal extracts all of the agent’s ex ante surplus.

  15. 15.

    Note that \(w_f^{L'}(w_s;\text{ AIC })=h'\left( U(w_s)-\frac{a_2-a_1}{(\pi ^L_s(a_2)-\pi ^L_s(a_1))}\right) U'(w_s)> 0.\)

  16. 16.

    One could also make both types of principal strictly better off.

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Acknowledgments

I would like to thank Charles Zheng for his advice and an anonymous referee for helpful comments.

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Correspondence to Nicholas Charles Bedard.

Additional information

The author gratefully acknowledges financial support from the Social Sciences and Humanities Research Council of Canada, the Ontario Graduate Scholarship, and the European Research Council (ERC Advanced Investigator Grant, ESEI-249433).

Appendix: Proofs

Appendix: Proofs

Lemma 1

Suppose Assumption 1 holds. The RSW equilibrium is the least-cost separating equilibrium that has type-L principal offering \((\mathbf {w}^{*L}, a^{*L})\) and the type-H principal offering the solution to

$$\begin{aligned} I^{\text {RSW}}: \begin{array}{llll} \max \limits _{\left( \mathbf {w}^H, a^H\right) }&\varvec{\pi }_H(a^H)\cdot \left( \mathbf {q}-\mathbf {w}^H\right)&\text {s.t.}&IR(1,0),\ \text{ AIC }(1,0)\text { and } \text{ PIC }(L,H). \end{array} \end{aligned}$$

The RSW contract always exists.

Proof

Let \(\left( \tilde{\mathbf {w}}^H,\tilde{a}^H\right) \) be a solution to \(I^\text{ RSW }\). First, I claim that the constraint \(\text{ AIC }\left( 1,0\right) \) in problem \(I^{\text{ RSW }}\) must bind. Suppose \(\text{ AIC }\left( 1,0\right) \) holds with strict inequality and let \(\pi ^H_s\ge \pi ^H_f\). Then, decrease \(w_s^H\) and increase \(w_f^H\) slightly to \((\tilde{w}_s^H-\epsilon _s,\tilde{w}_f^H+\epsilon _f)\) for small \(\epsilon _s,\epsilon _f>0\) so that IR\(\left( 1,0\right) \) and \(\text{ AIC }\left( 1,0\right) \) still hold. Since \(\pi ^H_s>\pi ^L_s\), \((\epsilon _s,\epsilon _f)\) can be chosen such that the right hand side of \(\text{ PIC }(L,H)\) (possibly weakly) decreases while the objective function strictly increases. If \(\pi ^H_s<\pi ^H_f\), increase \(w_s^H\) and decrease \(w_f^H\) slightly to \((\tilde{w}_s^H+\epsilon _s,\tilde{w}_f^H-\epsilon _f)\) for small \(\epsilon _s,\epsilon _f>0\) so that \(IR\left( 1,0\right) \) and \(\text{ AIC }\left( 1,0\right) \) still hold. Since \(\pi ^H_s>\pi ^L_s\), \((\epsilon _s,\epsilon _f)\) can again be chosen such that the right hand side of \(\text{ PIC }(L,H)\) (possibly weakly) decreases while the objective function strictly increases.

Second, I claim that \(\left( (\tilde{\mathbf {w}}^H,\tilde{a}^H),(\mathbf {w}^{*L},a^{*L})\right) \) is incentive compatible. This is vacuously true for the type-L principal since \(\text{ PIC }(L,H)\) is imposed in problem \(I^{\text {RSW}}\) and \(\mathbf {w}^{*L}\) is incentive compatible for the agent by construction. Further, \(\text {AIC}(1,0)\) is imposed in problem \(I^\text {RSW}\). It remains to show that

$$\begin{aligned} \varvec{\pi }^H(\tilde{a}^H)\cdot \left( \mathbf {q}- \tilde{\mathbf {w}}^H\right) \ge \varvec{\pi }^H(a^{*L})\cdot \left( \mathbf {q}-\mathbf {w}^{*L} \right) . \end{aligned}$$
(5)

I claim that (5) holds with strict inequality. Note that the curve in \((w_s,w_f)\) space implicitly defined by the agent’s RSW incentive compatibility constraint for the type-H principal,

$$\begin{aligned} \left( \pi ^H_s(\tilde{a}^H)-\pi ^H_s(a_1)\right) \left[ U(\tilde{w}_s^H)-U(\tilde{w}_f^H)\right] = \tilde{a}^H-a_1, \end{aligned}$$

is strictly above that of the type-L principal,

$$\begin{aligned} \left( \pi ^L_S( a^{*L})-\pi ^L_S(a_1)\right) \left[ U(w_s^{*L})-U(w_f^{*L})\right] = a^{*L}-a_1 \end{aligned}$$

due to Assumption 1. Further, the indifference curves of the type-H principal are steeper than the type-L principal’s. Therefore, the indifference curves possess the single crossing property. If \(\text{ PIC }(L,H)\) holds with equality, \(\tilde{\mathbf {w}}^H\) lies to the north-west of \(\mathbf {w}^{*L}\) in \((w_s,w_f)\)-space which implies that (5) strictly holds. Otherwise, \(\tilde{\mathbf {w}}^H=\mathbf {w}^{*H}\) and (5) strictly holds since \(\pi ^H_s(a^{*H})>\pi ^L_s(a^{*L})\).

The RSW problem for the type-H principal is more constrained than \(I^\text {RSW}\), but \(\left( \tilde{\mathbf {w}}^H,\tilde{a}^H\right) \) solves the latter problem and satisfies all the constraints of the former (with the type-L contract specified as \((\mathbf {w}^{*L},a^{*L})\)). Therefore, it solves the RSW problem for the type-H principal: \(\left( \hat{\mathbf {w}}^H,\hat{a}^H\right) =\left( \tilde{\mathbf {w}}^H,\tilde{a}^H\right) \). Similarly, the RSW problem for the type-L principal is more constrained than the public information problem, but \(\left( \mathbf {w}^{*L},a^{*L}\right) \) solves the latter problem and satisfies all the constraints of the former (with the type-H contract specified as \((\hat{\mathbf {w}}^{*H},a^{*H})\)). Therefore, it solves the RSW problem for the type-L principal.

To see that this menu-contract exists I first claim that \(\mathbf {w}^{*L}(a)\) exists for any a. For \(a=a_1\), \(\mathbf {w}^{*L}(a)=(h(\bar{U}+a_1),h(\bar{U}+a_1))\). For \(a=a_2\), the constraints \(\text{ AIC }(0,1)\) and \(IR(a_2;\{0,1\})\) will be satisfied with equality. Since \(w_f^L(\cdot ;IR)\) is strictly decreasing and \(w_f^L(\cdot ;\text{ AIC })\) is strictly increasing, they must intersect exactly once in \(\mathbb {R}^2\). Denote this intersection point \((w_s',w_f')\). If \((w_s',w_f')\in [\underline{w},\infty )^2\) I am done: \(\mathbf {w}^{*L}=(w_s',w_f')\). Otherwise, the solution is \(\mathbf {w}^{*L}=(w_s'',\underline{w})\) where \(w_s''\) satisfies \(w^L_f(w_s'',\text{ AIC })=\underline{w}\). If the type-L principal is indifferent between \(a_1\) and \(a_2\), set \(a^{*L}=a_2\).

\(I^\text {RSW}\) can be broken down into separate problems of minimizing the cost of implementing each effort then choosing most profitable effort. Note that \(a^{*H}=a_1\) implies that \(a^{*L}=a_1\) since the expected payoff from the agent’s effort is strictly higher for they type-H principal. Thus, if \(a^{*H}=a_1\), \(\hat{\mathbf {w}}^H=(h(\bar{U}+a_1),h(\bar{U}+a_1))\) which satisfies all the constraints of \(I^\text {RSW}\) given our previous statement.

If \(a^{*H}=a_2\) and \(\text{ PIC }(L,H)\) does not bind, the solution to \(I^\text {RSW}\) is simply \(\mathbf {w}^{*H}\), which exists by an argument analogous to the previous one for the existence of \(\mathbf {w}^{*L}\). Otherwise, the solution to \(I^\text {RSW}\) is defined by

$$\begin{aligned}&\left( \pi ^H_s(a_2)-\pi ^H_s(a_1)\right) \left( U(\hat{w}_s^H)-U(\hat{w}_f^H)\right) = a_2-a_1 \end{aligned}$$
(6)
$$\begin{aligned}&\varvec{\pi }^L\left( a^{*L}\right) \cdot \left( \mathbf {q}-\mathbf {w}^{*L}\right) = \varvec{\pi }^L(a_2)\cdot \left( \mathbf {q}-\hat{\mathbf {w}}^H\right) . \end{aligned}$$
(7)

Equation (6) implicitly defines a strictly increase line in \((w_s,w_f)\)-space while equation (7) defines a strictly decreasing line in \((w_s,w_f)\)-space. These lines therefore intersect exactly once in \(\mathbb {R}^2\). Denote this intersection point \((w_s',w_f')\). If this \((w_s',w_f')\in [\underline{w},\infty )^2\) I am done: \(\tilde{\mathbf {w}}^H=(w_s',w_f')\) . Otherwise, the solution is \(\mathbf {w}^{*L}=(w_s'',\underline{w})\) where \(w_s''\) satisfies equation (6) with \(\hat{w}_f^H=\underline{w}\). \(\square \)

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Bedard, N.C. Contracts in informed-principal problems with moral hazard. Econ Theory Bull 5, 21–34 (2017). https://doi.org/10.1007/s40505-016-0093-0

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Keywords

  • Informed principal
  • Moral hazard
  • Menu-contracts
  • Principal agent

JEL Classification

  • D82
  • D86