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Multi-task agency with unawareness

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Abstract

The paper introduces the problem of unawareness into multi-dimensional Principal–Agent theory. We introduce two key parameters to describe the problem, the extent and the effect of unawareness, show under what conditions it is optimal for the Principal to propose an incomplete or a complete contract, and characterize the incentive power of optimal linear contracts. If Agents differ in their unawareness, optimal incentive schemes can be distorted for both aware and unaware Agents, because, different from standard contract theory, the single-crossing property fails to hold. In this case, even aware Agents can be subject to inefficiently high or low incentives.

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Notes

  1. 1.

    In von Thadden and Zhao (2012), we study the properties of optimal contracts in the classic one-dimensional setting. However, in the general model it is difficult to characterize the incentive power of contracts and their comparative statics, which is the main focus of this paper.

  2. 2.

    We assume this form of contract because it is simple and captures two important elements of incentive contracting. It is worth noting that Holmström and Milgrom (1987) provide a foundation for this assumption in a dynamic setting.

  3. 3.

    Because \(\pi ^{\mathrm{A}}>0\), whenever \(\pi ^{\mathrm{U}}>\pi ^{\mathrm{A}}\), we get \(\pi ^{\mathrm{U}}>0\). Hence, the Principal always gains from proposing a contract.

  4. 4.

    This tradeoff, which also appears in von Thadden and Zhao (2012), provides a new perspective on the foundations of contract incompleteness, different from classical approaches such as verifiability (Grossman and Hart 1986; Hart and Moore 1990), signaling (Aghion and Bolton 1987; Spier 1992), explicit writing costs (Dye 1985; Anderlini and Felli 1999; Battigalli and Maggi 2002), strategic incompleteness (Bernheim and Whinston 1998; Dessi 2009, or limited cognition (Bolton and Faure-Grimaud 2010; Tirole 2009).

  5. 5.

    Note that \(2/(2+\sigma ^{2})\in \left( \tau _{\min },\tau _{\max }\right) \) defined in (18).

  6. 6.

    Without loss of generality, our figures focus on the case \(1<\sigma ^{2}<2\).

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Acknowledgments

The authors thank Bruno Biais, Patrick Bolton, Roberta Dessí, Mathias Dewatripont, Kfir Eliaz, Klaus Schmidt, Dagmar Stahlberg, and Jidong Zhou for useful discussions and comments. They also thank the German Science Foundation (DFG) and National Natural Science Foundation of China (Grant No.71303245) for financial support.

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Correspondence to Xiaojian Zhao.

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This paper builds on an earlier paper entitled “Incentives for Unaware Agents”.

Appendix

Appendix

Proof of Proposition 2

Proposition 2 (with explicit expressions for the optimal contracts)

Let

$$\begin{aligned} \underline{\tau }&= \underline{\tau }(\lambda )=\frac{3(1-\lambda )(\sigma ^{2}+1)+1+\lambda }{(\sigma ^{2}+2)(\lambda +2(1-\lambda )(\sigma ^{2}+1))}\\ \overline{\tau }&= \overline{\tau }(\lambda )=\frac{(1-\lambda )(\sigma ^{2}+1)+3\lambda -1}{\lambda (\sigma ^{2}+2)} \end{aligned}$$

We have \(\frac{\mathrm{d}}{\mathrm{d}\lambda }\underline{\tau }>0, \frac{\mathrm{d}}{\mathrm{d}\lambda } \overline{\tau }<0\) for all \(\lambda \in [0,1], \underline{\tau } (0)<\tau _{\min }<\tau _{\max }<\overline{\tau }(0)\) and \(\underline{\tau }(1)=\overline{\tau }(1)=t_{2\mathrm{F}}^{\mathrm{A}}.\) The solution of the problem (22)-(ICU) is unique and given as follows:

  1. 1.

    If \(\tau <\underline{\tau }\) or \(\tau >\overline{\tau }\) the incentive constraint (ICU) is slack and the solution is separating, with

    $$\begin{aligned} \alpha ^{\mathrm{A}}=\frac{2}{2+\sigma ^{2}},\qquad \text { } \beta ^{\mathrm{A}}&= \frac{4\left( \sigma ^{2}-2\right) }{2(2+\sigma ^{2})^{2}}+\frac{1}{2}\left( 1-\lambda \right) ^{2}\frac{\left( 1-(\sigma ^{2}+1)\tau -1\right) ^{2}}{\left( 1+\sigma ^{2}(1-\lambda )\right) ^{2}},\\ \alpha ^{\mathrm{U}}\!=\!\frac{1+\lambda (\tau -1)}{1+\sigma ^{2}(1-\lambda )},\text { } \beta ^{\mathrm{U}}&= \frac{1}{2}\tau ^{2}\!-\!\tau \frac{1+\lambda (\tau -1)}{1+\sigma ^{2}(1-\lambda )}-\frac{(1-\sigma ^{2})\left( 1\!+\!\lambda (\tau -1)\right) ^{2} }{2\left( 1+\sigma ^{2}(1-\lambda )\right) ^{2}}. \end{aligned}$$
  2. 2.

    If \(\underline{\tau }\le \tau \le \frac{2}{2+\sigma ^{2}}\), the incentive constraint (ICU) is binding, with \(\alpha ^{\mathrm{A}}-\tau =\tau -\alpha ^{\mathrm{U}}\) and

    $$\begin{aligned} \alpha ^{\mathrm{A}}&= \frac{1}{1+2\lambda +\sigma ^{2}}\left( 2\tau (1+\sigma ^{2})(1-\lambda )-1+3\lambda +\tau \lambda \right) ,\\ \beta ^{\mathrm{A}}&= \frac{1}{2}\left[ \frac{1}{1+2\lambda +\sigma ^{2}}\left( 1-3\lambda +(5+2\sigma ^{2})\tau \lambda \right) \right] ^{2} \\&-\frac{\tau }{1+2\lambda +\sigma ^{2}}\left( 1-3\lambda +(5+2\sigma ^{2})\tau \lambda \right) \\&+\frac{\tau ^{2}}{2}-\frac{1}{2}(2-\sigma ^{2})\left[ \frac{1}{1+2\lambda +\sigma ^{2}}\left( 2\tau (1\!+\!\sigma ^{2})(1-\lambda )-1\!+\!3\lambda +\tau \lambda \right) \right] ^{2},\\ \alpha ^{\mathrm{U}}&= 2\tau -\alpha ^{\mathrm{A}}=\frac{1}{1+2\lambda +\sigma ^{2}}\left( 1-3\lambda +(5+2\sigma ^{2})\tau \lambda \right) ,\\ \beta ^{\mathrm{U}}&= \frac{1}{2}\tau ^{2}-\frac{1}{2}(1-\sigma ^{2})\left[ \frac{1}{1+2\lambda +\sigma ^{2}}\left( 1-3\lambda +(5+2\sigma ^{2})\tau \lambda \right) \right] ^{2}\\&-\frac{\tau }{1+2\lambda +\sigma ^{2}}\left( 1-3\lambda +(5+2\sigma ^{2} )\tau \lambda \right) . \end{aligned}$$
  3. 3.

    If \(\frac{2}{2+\sigma ^{2}}\le \tau \le \overline{\tau }\) the solution is pooling, with

    $$\begin{aligned} \alpha ^{\mathrm{A}}&= \alpha ^{\mathrm{U}}=\frac{1+\lambda +\tau \lambda }{1+2\lambda +\sigma ^{2} },\\ \beta ^{\mathrm{A}}&= \beta ^{\mathrm{U}}=\frac{1}{2}\tau ^{2}-\tau \frac{1+\lambda +\tau \lambda }{1+2\lambda +\sigma ^{2}}-\frac{1}{2}\frac{\left( 1+\lambda +\tau \lambda \right) ^{2}}{\left( 1+2\lambda +\sigma ^{2}\right) ^{2}}\left( 1-\sigma ^{2}\right) . \end{aligned}$$

Proof

Using Lemmas 2 and 3, one can eliminate the fixed payment \(\beta \) from the problem and express the contracting problem solely in terms of the incentive component \(\alpha \):

$$\begin{aligned}&\max _{\alpha ^{\mathrm{A}},\alpha ^{\mathrm{U}}}\lambda \left[ 4\alpha ^{\mathrm{A}}\!-\!(\sigma ^{2} +2)(\alpha ^{\mathrm{A}})^{2}\!+\!2\tau \alpha ^{\mathrm{U}}-(\alpha ^{\mathrm{U}})^{2}\right] +(1-\lambda )\left[ 2\alpha ^{\mathrm{U}}\!-\!(\sigma ^{2}\!+\!1)(\alpha ^{\mathrm{U}})^{2}\right] \nonumber \\\end{aligned}$$
(23)
$$\begin{aligned}&\quad \text {s.t. }\,\,(\alpha ^{\mathrm{A}}-\tau )^{2}\ge (\alpha ^{\mathrm{U}}-\tau )^{2} \end{aligned}$$
(24)

By straightforward differentiation, the unconstrained solution to the maximization problem (23) and (24) is

$$\begin{aligned} \alpha ^{\mathrm{A}}=\frac{2}{2+\sigma ^{2}},\quad \alpha ^{\mathrm{U}}=\frac{1+\lambda (\tau -1)}{1+\sigma ^{2}(1-\lambda )} \end{aligned}$$
(25)

This solution satisfies the constraint (24) strictly if and only if

$$\begin{aligned} (\tau (\sigma ^{2}+2)-2)^{2}(\lambda +(1-\lambda )(\sigma ^{2}+1))^{2} >(1-\lambda )^{2}(\tau (\sigma ^{2}+1)-1)^{2}(\sigma ^{2}+2)^{2} \end{aligned}$$

Viewed as a quadratic inequality in \(\tau \), this is equivalent to \(\tau <\underline{\tau }\) or \(\tau >\overline{\tau }\). Hence, (25) yields the separating solution of the proposition.

If (ICU) is binding, there are two possibilities: \(\alpha ^{\mathrm{U}}=\alpha ^{\mathrm{A}}\) (pooling) or \(\alpha ^{\mathrm{U}}+\alpha ^{\mathrm{A}}=2\tau \) (constrained separating). Direct comparison shows that when \(\frac{2}{2+\sigma ^{2}}\le \tau \le \overline{\tau }(\lambda )\) we have the pooling solution, and when \(\underline{\tau } (\lambda )\le \tau \le \frac{2}{2+\sigma ^{2}}\) we have the constrained separating solution.

The monotonicity of \(\underline{\tau }(\lambda )\) and of \(\overline{\tau }(\lambda )\) follows by differentiation, and the statements about \(\underline{\tau }(0),\overline{\tau }(0),\underline{\tau }(1)\), and \(\overline{\tau }(1)\) by direct computation. \(\square \)

Proof of Proposition 3

Proof

We must compare the value \(\pi _{\mathrm{S}}(\lambda ,\tau )\) of the screening problem solved in Proposition 2 to the profit from making all Agents aware, \(\pi ^{\mathrm{A}}=\frac{2}{2+\sigma ^{2}}\). We do this by discussing the three possible cases derived in Proposition 2 in turn.

  • (1) The separating case: Straightforward computation shows that \(\pi ^{\mathrm{A}} >\pi _{\mathrm{S}}(\lambda ,\tau )\) if and only if

    $$\begin{aligned} \tau >R_{1}=\frac{X_{1}+Y_{1}}{Z_{1}} \end{aligned}$$

    or

    $$\begin{aligned} \tau <L_{1}=\frac{X_{1}-Y_{1}}{Z_{1}} \end{aligned}$$

    where

    $$\begin{aligned}&\displaystyle X_{1}=\left( 2+\sigma ^{2}\right) \left( 1+\lambda +\sigma ^{2}\left( 1-\lambda \right) \right) ,\\&\displaystyle Y_{1}=\sigma ^{2}\sqrt{\left( 2+\sigma ^{2}\right) \left( 1-\lambda \right) \left( 1+\sigma ^{2}\left( 1-\lambda \right) \right) }, \end{aligned}$$

    and

    $$\begin{aligned} Z_{1}=\left( 2+\sigma ^{2}\right) \left( 1+\lambda +\sigma ^{2}\right) . \end{aligned}$$

    Comparing these boundaries to those of Proposition 2, it is straightforward to show that \(R_{1}>\overline{\tau }\) if and only if \(\lambda >\frac{1}{2}\), and that \(L_{1}<\underline{\tau }\) if and only if \(\lambda >\frac{1}{2}\). Thus, in the separating case, when \(\lambda >\frac{1}{2},\) the Principal makes all Agents aware if and only if \(\tau >R_{1}\) or \(\tau <L_{1}\). When \(\lambda <\frac{1}{2},\) the Principal makes all Agents aware.

  • (2) The pooling case: We have \(\pi ^{A}>\pi _{S}(\lambda ,\tau )\) if and only if

    $$\begin{aligned} \tau >R_{2}=\frac{X_{2}+Y_{2}}{Z_{2}} \end{aligned}$$

    or

    $$\begin{aligned} \tau <L_{2}=\frac{X_{2}-Y_{2}}{Z_{2}} \end{aligned}$$

    where

    $$\begin{aligned}&\displaystyle X_{2}=\left( 2+\sigma ^{2}\right) \left( 1+\lambda \left( 2-\lambda \right) +\sigma ^{2}\left( 1-\lambda \right) \right) ,\\&\displaystyle Y_{2}=\left( 1-\lambda \right) \sigma ^{2}\sqrt{\left( 2+\sigma ^{2}\right) \left( 1+2\lambda +\sigma ^{2}\right) }, \end{aligned}$$

    and

    $$\begin{aligned} Z_{2}=\left( 2+\sigma ^{2}\right) \left( 1+\lambda \left( 2-\lambda \right) +\sigma ^{2}\right) . \end{aligned}$$

    Direct computation shows that \(\overline{t}_{2}^{\mathrm{A}}>L_{2}\). Because necessarily \(\tau \ge \overline{t}_{2}^{\mathrm{A}}\) under pooling, \(\tau <L_{2}\) is impossible. Furthermore, \(R_{2}<\overline{\tau }\) if and only if \(\lambda <\frac{1}{2}\). Hence, if pooling is optimal in the screening problem, when \(\lambda <\frac{1}{2}\) the Principal makes all Agents aware if and only if\(\ \tau >R_{2}\). When \(\lambda >\frac{1}{2},\) the Principal only uses the pooling solution.

  • (3) The constrained separating case: We have \(\pi ^{\mathrm{A}}>\pi _{\mathrm{S}}(\lambda ,\tau )\) if and only if

    $$\begin{aligned} \tau >R_{3}=\frac{X_{3}+Y_{3}}{Z_{3}}\end{aligned}$$

    or

    $$\begin{aligned} \tau <L_{3}=\frac{X_{3}-Y_{3}}{Z_{3}} \end{aligned}$$

where

$$\begin{aligned}&\displaystyle X_{3}=\left( 2+\sigma ^{2}\right) \left( \sigma ^{2}\left( 6\lambda +1\right) \left( \lambda -1\right) +9\lambda ^{2}-10\lambda -1\right) ,\\&\displaystyle Y_{3}=\left( 1-\lambda \right) \sigma ^{2}\sqrt{\left( 2+\sigma ^{2}\right) \left( 1+2\lambda +\sigma ^{2}\right) }, \end{aligned}$$

and

$$\begin{aligned} Z_{3}=\left( 2+\sigma ^{2}\right) \left( \sigma ^{2}\left( 4\lambda \sigma ^{2}\left( \lambda -1\right) +12\lambda ^{2}-12\lambda -1\right) +9\lambda ^{2}-10\lambda -1\right) . \end{aligned}$$

We have \(\overline{t}_{2}^{\mathrm{A}}<R_{3}\) Since necessarily \(\tau \le \overline{t}_{2}^{\mathrm{A}}, \tau >R_{3}\) is impossible. Furthermore, \(L_{3}>\underline{\tau }\) if and only if\(\ \lambda <\frac{1}{2}\). Hence, in the constrained-separating case, when \(\lambda <\frac{1}{2}\) the Principal makes all Agents aware if and only if\(\ \tau <L_{3}\). When \(\lambda >\frac{1}{2},\) the Principal only uses the constrained separating solution.

The above three case discussions establish the awareness-thresholds for each \(\lambda \). Formally, we have \(\tau _{R}(\lambda )=R_{1}\) and \(\tau _{L} (\lambda )=L_{1}\) for \(\lambda \ge \frac{1}{2}\), and \(\tau _{R}(\lambda )=R_{2}\) and \(\tau _{L}(\lambda )=L_{3}\) for \(\lambda \le \frac{1}{2}\). It is straightforward to show that these \(\tau _{R}\) and \(\tau _{L}\) are continuous at \(\frac{1}{2}\), that \(R_{1}\) and \(R_{2}\) are decreasing in \(\lambda \), and \(L_{1}\) and \(L_{3}\) are increasing in \(\lambda \).

To complete the picture given in Fig. 9, for \(\lambda =1\) one calculates \(R_{1}=L_{1}=\frac{2}{2+\sigma ^{2}}=\overline{t}_{2}^{A}\), hence \(\tau _{R}(1)=\tau _{L}(1)=\overline{t}_{2}^{A}\). Similarly, for \(\lambda =0, R_{2}=\tau _{\max }\) and \(L_{3}=\tau _{\min }\). \(\square \)

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von Thadden, EL., Zhao, X. Multi-task agency with unawareness. Theory Decis 77, 197–222 (2014). https://doi.org/10.1007/s11238-013-9397-9

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Keywords

  • Multi-task agency
  • Unawareness
  • Moral hazard
  • Screening
  • Incomplete contracts

JEL Classification

  • D01
  • D86
  • D82
  • D83