Abstract
This paper studies a principal–agent relation in which the principal’s private information about the agent’s effort choice is more accurate than a noisy public performance measure. For some contingencies the optimal contract has to specify ex post inefficiencies in the form of inefficient termination (firing the agent) or wasteful activities that are formally equivalent to thirdparty payments (money burning). Under the optimal contract, the use of these instruments depends not only on the precision of public information but also on job characteristics. Money burning is used at most in addition to firing and only if the loss from termination is small. The agent’s wage may depend only on the principal’s report and not on the public signal. Nonetheless, public information is valuable as it facilitates truthful subjective evaluation by the principal.
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Notes
Besides the obvious examples of layoffs and dismissals, also in some option contracts one party keeps the authority to terminate the relationship. See e.g. Lerner and Malmendier (2010) on the use of option contracts in biotechnology research.
An example of literal thirdparty payments is given by Fuchs (2007): some baseball teams can fine their players, and the fines are not paid to the club, but rather to a charity.
Rajan and Reichelstein (2006) show that this inefficiency decreases with the number of agents when the principal employs multiple agents. Fuchs (2007) and Chan and Zheng (2011) show that the efficiency loss can be mitigated in a (finite) long term relationship: the perperiod efficiency loss decreases in the number of periods.
Bester and Krähmer (2012) consider a buyer–seller relation where the buyer observes the seller’s quality choice, but his observation is not verifiable. They show that ‘exit option’ contracts, corresponding to the option of ‘firing’ in the present context, can implement the firstbest. Here and in Schmitz (2002) this is not possible because the agent’s (the seller’s) effort is not directly observable.
Obviously, firing might inflict a cost on the agent, and the threat of firing may be used to motivate the agent. The use of nonmonetary fines to overcome limited liability has been studied in Chwe (1990) and Sherstyuk (2000). To focus on the implications of firing versus money burning for the principal’s incentives, we assume that firing imposes no costs at all on the agent.
Our assumption that a constant fraction of output is lost upon termination implies that, the higher the output, the more costly it is for the principal to terminate the project. This drives our result that termination is preferred over money burning. We conjecture that similar result will hold in a more general model where the principal’s cost of firing is a strictly monotone increasing function of output.
For subjective evaluations in a different context where monetary transfers are not allowed, see Taylor and Yildirim (2011).
Instead of assuming that the principal loses \(\alpha x\), we could more generally assume that he loses \(\alpha (x)\), where \(\alpha (\cdot )\) is a strictly increasing function. Our simplification allows us to study the comparative statics of the optimal contract for changes in \(\alpha \).
As we show in Sect. 3, however, also wages that do not depend on termination are optimal. The reason is that in our model termination serves to induce truthful reporting by the principal rather than to incentivize the agent.
Note that whether the principal or the agent pays b plays no role because the wage payment can be adjusted accordingly. We assume that \(b \ge 0\) because any third party is covered by limited liability, just as the agent.
More formally, if \(w_{ij}^C\) is the wage when the project is completed and \(w_{ij}^T\) the wage after premature termination, we can simply replace these wages by \(w_{ij} \equiv (1 \theta _{ij}) w_{ij}^C + \theta _{ij} \,w_{ij}^T\).
We assume that the agent faces no risk of termination because he cares only about expected wages. If termination would inflict some cost on the agent, such as e.g. a reputation loss, then the threat of being fired would provide additional effort incentives.
There are contracts that achieve the firstbest, where payments depend on the principal’s report, but reporting is not truthful. Formally all four wage parameters could be different, but only two different wages will be paid with positive probability.
The argument for why \(b_{LL} = 0\) is a bit more subtle. However, if \(b_{LL}\) were positive, one could decrease it while simultaneously increasing \(b_{HL}\) and thereby increase the principal’s payoff.
See also our discussion at the end of Sect. 7.
The constraint \(0 \le e \le 1\) can be ignored because it is not binding.
Indeed, if the principal reports after having observed the public signal, any contract that solves the unlimited liability problem must ignore the principal’s information. This immediately follows from (29) because \(b = \theta = 0\).
This can be shown by differentiating them twice and using \(x_{H}>x_{L}\), \( \sigma >1/2\), and \(c^{\prime \prime \prime }\left( e\right) \ge 0\).
This can be shown by differentiating them twice and using \(1/2<\sigma< \bar{\sigma } < x_H/(x_H + x_L)\), \(c^{\prime \prime }\left( e\right) >0\) and \(c^{\prime \prime \prime }\left( e\right) \ge 0\).
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We thank an associate editor and a referee for helpful comments. We also thank Jörg Budde, Daniel Krähmer, Matthias Lang, Bentley MacLeod, Patrick Schmitz, Roland Strausz, Peter Vida, and seminar participants at Freie Universität Berlin, Humboldt Universität Berlin, University of Cologne, University of Copenhagen, and WHU Vallendar for comments and interesting discussions. Support by the German Science Foundation (DFG) through SFB/TR 15 is gratefully acknowledged.
Appendix
Appendix
Proof of Proposition 1
Suppose that \(\theta =b=0\), \(w_{HH}=w_{HL}\), and \(w_{LH}=w_{LL}\). Then the principal’s incentive constraints (8) are obviously satisfied. Let the difference of the wages satisfy
Then by (13) the agent will choose the firstbest effort \(\tilde{e}\). In addition, by unlimited liability one can choose the wage \(w_{LL}\) such that the agent’s individual rationality constraint holds with equality:
This contract implements the firstbest effort \(\tilde{e}\). Moreover, the principal’s payoff is equal to the firstbest surplus \(S\left( \tilde{e}\right) \) because the agent receives his outside option payoff. Obviously, the payoff of the principal cannot be higher; thus the contract considered here is optimal. Moreover, any optimal contract must implement the firstbest effort \(\tilde{e}\), for otherwise the principal’s payoff must be lower than the firstbest surplus \(S\left( \tilde{e}\right) \).
It remains to show that \(\theta =b=0\) in any optimal contract. By assumption (1), \(\tilde{e}\in \left( 0,1\right) \). Since \(\sigma <1\), this implies that all four possible combinations of output and the public signal occur with positive possibility. Therefore, whenever \(\theta \not =0\) or \(b\not =0\), total surplus is below the firstbest surplus \(S\left( \tilde{e}\right) \), and hence the principal’s payoff is below \(S\left( \tilde{e} \right) \) as well. \(\square \)
Proof of Proposition 2
The agent’s utility is
Since \(w\ge 0\) and \(c\left( 0\right) =0\), \(U\left( \gamma ,0\right) \ge 0\). Thus, in what follows we can ignore constraint (14) because it is automatically satisfied.
If \(\left( \gamma ,e\right) \) solves problem (16), then obviously \(\gamma \) must maximize \(V(\gamma ,e)\) subject to the constraints in (16) when e is treated as a fixed parameter. The latter is a linear optimization problem since \(V\left( \gamma ,e\right) \) and all constraints are linear in \(\gamma \), and the Kuhn–Tucker conditions are both necessary and sufficient for a maximum. Following a standard method, we temporarily ignore that \(\gamma \) has to satisfy the inequality \(V_{L}\left( \gamma , \hat{x}_{L}\right) \ge V_{L}\left( \gamma ,\hat{x} _{H}\right) \) in (8), and show below that this constraint is automatically satisfied. Consider the Lagrangian
with \(\lambda \ge 0\). Note that \(\mu >0\) as the agent’s incentive constraint must be binding.
Straightforward differentiation shows that \(w_{HL}= w_{LL} = 0\) because
Moreover, \(\theta _{HH}=b_{HH}=\theta _{LH}=b_{LH}=0\) because
Finally, \(b_{LL}=0\) because \(\sigma > 1/2\) implies that
The above arguments prove part (b) of the proposition. To complete the proof of part (a), note that the agent’s incentive constraint (13) reduces to (17), because \(w_{HL}= w_{LL} = 0\). Changing \(w_{HH}\) and \(w_{LH}\) such that Eq. (17) continues to hold leaves the principal’s payoff constant and does not interfere with any of the constraints. Therefore, the optimal wages \(w_{HH}\) and \(w_{LH}\) are not unique.
To prove part (c), note that \(b_{HL}>0\) implies
and thus \(\lambda = {\left( 1e\right) \left( 1\sigma \right) }/{\sigma }\). This implies
Therefore \(b_{HL}>0\) implies \(\theta _{HL}=1\).
We now show that, as claimed above, only the second inequality in the ICP constraints (8) is binding. By parts (a) and (b), the principal’s incentive constraint \(V_{H}(\gamma ,\hat{x} _{H})\ge V_{H}(\gamma ,\hat{x}_{L})\) in (8) simplifies to
Suppose this constraint is not binding. Then \(\theta _{HL}\) must be strictly positive, since by limited liability the left hand side of (44) is nonnegative, and by part (c) the right hand side can be positive only if \(\theta _{HL}>0\). Thus one must have \({\partial L}/{\partial \theta _{HL}} \ge 0\), and by (42) this implies \(\lambda > 0\). This proves that the constraint (44) must be binding and \(V_{H}(\gamma ,\hat{x}_{H}) \ge V_{H}(\gamma ,\hat{x}_{L})\) must hold with equality.
Using part (a) and (b) the inequality \(V_{L}(\gamma ,\hat{x}_{L})\ge V_{L}(\gamma ,\hat{x}_{H})\) in (8) reduces to
Since (44) is binding,
where the inequality follows from \(w_{LH}\ge 0\). Subtracting the left hand side of (45) shows that
If \(\theta _{LL}=0\), this implies that (45) is satisfied. Similarly, if \(\left( 1\sigma \right) ^{2}x_{H}/\sigma \ge \sigma x_{L}\), (45) is satisfied. To complete the argument, we show that one cannot have that \(\left( 1\sigma \right) ^{2}x_{H}/\sigma <\sigma x_{L}\) and \(\theta _{LL}>0\). Indeed, \(\theta _{LL}>0\) implies
and so
By the first equality in (41) this implies
where the second inequality holds if \(\left( 1\sigma \right) ^{2}x_{H}/\sigma <\sigma x_{L}\). Since this would imply \(b_{HL} = \infty \), we have shown that \(\theta _{LL} = 0\) if \(\left( 1\sigma \right) ^{2}x_{H}/\sigma <\sigma x_{L}\).
Finally, statement (d) directly follows by combining (17) with (44), which holds with equality as shown above. \(\square \)
Proof of Lemma 1
(a) As shown in the last part of the proof Proposition 2, \(\theta _{LL} = 0\) if \(\left( 1\sigma \right) ^{2}x_{H}/\sigma <\sigma x_{L}\). As this inequality is equivalent to \(\sigma > \bar{\sigma }\), this proves part (a).
(b) Let \(b_{HL}>0\) and \(\sigma <\bar{\sigma }\). Then \({\partial L}/{\partial b_{HL}} = 0\) and so by the second equality in (36)
By the equality in (48) this implies
where the last inequality holds because \(\sigma <\bar{\sigma }\). Therefore, \(\theta _{LL}=1\).
As shown in the proof of Proposition 2, \(\theta _{LL}>0\) implies (49). Therefore, by (42)
Therefore \(\theta _{LL}>0\) implies \(\theta _{HL}=1\). \(\square \)
Proof of Proposition 3
We substitute out all choice variable except e from the principal’s problem, and then optimize with respect to e. By parts (a) and (b) of Proposition 2, the principal’s profit \(V\left( e,\gamma \right) \) equals
By Lemma 1, \(\theta _{LL}=0\) and hence by (19),
There are two possible cases. First, suppose that \(\sigma \alpha x_{H}\ge c^{\prime }\left( e\right) \). This is equivalent to \(e \le \hat{e}\), where \(\hat{e} \equiv c^{\prime 1}\left( \sigma \alpha x_{H}\right) \). In this case, (57) and Proposition 2 (c) imply that \(b_{HL}=0\) and \(\theta _{HL}=c^{\prime }\left( e\right) /\left( \sigma \alpha x_{H}\right) \). Profit equals
Second, suppose that \(\sigma \alpha x_{H} < c^{\prime }\left( e\right) ,\ \)or, equivalently, \(e>\hat{e}\). Then \(\theta _{HL}=1\) and so by (57) \(b_{HL} = {c^{\prime }(e)}/{\sigma }\alpha x_{H}>0\). In this case, the principal’s payoff is
Note that \(\phi _{1}\left( e\right) \le \phi _{2}\left( e\right) \) if and only if \(c^{\prime }\left( e\right) \le \sigma \alpha x_{H}\). Therefore, the principal’s payoff as a function of e can be written as
The functions \(\phi _{1}\) and \(\phi _{2}\) are strictly concave in e.^{Footnote 26} The minimum of two strictly concave functions is strictly concave, hence \(\tilde{V}\) is strictly concave.
Differentiating \(\phi _{2}\) yields
Define \(e_{2}^{*}\) implicitly by \(\phi _{2}^{\prime }\left( e_{2}^{*}\right) =0\). Since \(\phi _{2}\) is strictly concave, \(e_{2}^{*}\) is unique. Since \(c^{\prime \prime \prime }(e) \ge 0\) and \(c^{\prime }(0) = 0\), \(e c^{\prime \prime }(e) \ge c^{\prime }(e)\) for all e. By (1) therefore \(c^{\prime \prime }(1) \ge c^{\prime }(1) > x_H  x_L\). This implies \(\phi _{2}^{\prime }\left( 1\right) <0\) and so \(e_{2}^{*}<1\).
If \(e_{2}^{*}>\hat{e}\), then \(e_{2}^{*}\) maximizes the principal’s payoff \(\tilde{V}\left( e\right) \). Moreover, if the optimal contract involves \(b_{HL}>0\), then \(e_{2}^{*}> \hat{e}\). We use the intermediate value theorem to show that \(e_{2}^{*}> \hat{e}\) if and only if \(\alpha \) is strictly smaller than a critical value \(\bar{\alpha }\in \left( 0,1\right) \). The argument proceeds in three steps:

1.
At \(\alpha =0\), \(\phi _{2}^{\prime }\left( 0\right) >0\) by assumptions (1) and (2). Moreover, if \(\alpha =0\), the critical value \(\hat{e}\) equals zero. Therefore, if \(\alpha =0\), then \(e_{2}^{*} > \hat{e}\).

2.
The critical value \(\hat{e}\) is continuous and strictly increasing in \(\alpha \). Moreover, \(e_{2}^{*}\) is continuous and strictly decreasing in \(\alpha \):
$$\begin{aligned} \frac{de_{2}^{*}}{\mathrm{d}\alpha }=\frac{{\partial \phi _{2}^{\prime } \left( e_{2}^{*}\right) }/{\partial \alpha }}{\phi _{2}^{\prime \prime } \left( e_{2}^{*}\right) }=\frac{\left( 1\sigma \right) \left( x_{H}x_{L}\right) }{\phi _{2}^{\prime \prime }\left( e_{2}^{*}\right) } <0. \end{aligned}$$(62) 
3.
If \(\alpha =1\), \(e_{2}^{*}\) solves
$$\begin{aligned} \frac{2\sigma 1}{\sigma }c^{\prime }\left( e_{2}^{*}\right) +\left[ e_{2}^{*}+\left( 1e_{2}^{*}\right) \frac{\left( 1\sigma \right) }{ \sigma }\right] c^{\prime \prime }\left( e_{2}^{*}\right) =\sigma \left( x_{H}x_{L}\right) . \end{aligned}$$(63)Since \(c^{\prime \prime \prime }\left( e\right) \ge 0 = c^{\prime }\left( 0\right) \), we have \(ec^{\prime \prime }\left( e\right) \ge c^{\prime }\left( e\right) \) and thus
$$\begin{aligned} \left[ \frac{2\sigma 1}{\sigma }+1\right] c^{\prime }\left( e_{2}^{*}\right) <\sigma \left( x_{H}x_{L}\right) . \end{aligned}$$(64)By \(\sigma >1/2\), it follows that \(c^{\prime }\left( e_{2}^{*}\right) <\sigma x_{H}\). We conclude that, if \(\alpha =1\), \(c^{\prime }\left( e_{2}^{*}\right) <\sigma \alpha x_{H}\) and thus \(e_{2}^{*}<\hat{e}\).
From steps 1–3, it follows that there exists a critical value \(\bar{\alpha } \in \left( 0,1\right) \) such that \(e^{*}>\hat{e}\) holds if and only if \(\alpha <\bar{\alpha }\). As argued above, this implies that \(b_{HL} > 0\) if and only if \(\alpha <\bar{\alpha }\). \(\square \)
Proof of Proposition 4
There are three cases corresponding to statements (a), (b) and (c). First, suppose that \(c^{\prime }\left( e\right) \le \sigma \alpha x_{H}\), or equivalently, \(e\le \hat{e}=c^{\prime 1}\left( \alpha \sigma x_{H}\right) \). Then, by Proposition 2 (c) and (d), \(\theta _{HL}=c^{\prime }\left( e\right) /\left( \sigma \alpha x_{H}\right) \) and \(\theta _{LL}=b_{HL}=0\). Profit equals \(\phi _{1}\left( e\right) \), as defined in the proof of Proposition 3. Second, suppose that \(\sigma \alpha x_{H}<c^{\prime }\left( e\right) \le \alpha x_{H}\). This is equivalent to \( e\in \left( \hat{e},\bar{e}\right] ,\ \)where \(\bar{e}:=c^{\prime 1}\left( \alpha x_{H}\right) \). In this case \(\theta _{HL}=1\), and \(b_{HL}=0\), and \( \theta _{LL}=\left[ c^{\prime }(e)\sigma \alpha x_{H}\right] /\left[ \left( 1\sigma \right) \alpha x_{H}\right] \). Using (56), the principal’s profit is
Third, suppose that \(c^{\prime }\left( e\right) >\alpha x_{H}\), or, equivalently, \(e>\bar{e}\). Then \(\theta _{LL}=\theta _{HL}=1\) and \( b_{HL}=\left( c^{\prime }(e)\alpha x_{H}\right) /\sigma \), and profit equals
Note that \(\phi _{1}\left( e\right) \le \phi _{3}\left( e\right) \) if and only if \(c^{\prime }(e) \le \sigma \alpha x_H\). Moreover \(\phi _{3}\left( e\right) \le \phi _{4}\left( e\right) \) if and only if \(c^{\prime }(e) \le \alpha x_H\). Thus \(c^{\prime }\left( e\right) \le \alpha \sigma x_{H}\) if and only if \(\phi _{1}\left( e\right) =\min \left\{ \phi _{1}\left( e\right) ,\phi _{3}\left( e\right) ,\phi _{4}\left( e\right) \right\} \). Moreover, \(\sigma \alpha x_{H}<c^{\prime }\left( e\right) \le \sigma x_{H}\) if and only if \(\phi _{3}\left( e\right) <\phi _{1}\left( e\right) \) and \(\phi _{3}\left( e\right) \le \phi _{4}\left( e\right) \). Therefore, the principal’s payoff can be written
The functions \(\phi _{1}\), \(\phi _{3}\), and \(\phi _{4}\) are strictly concave in e.^{Footnote 27} Therefore, \(\tilde{V}\) is strictly concave in e.
Define \(e_{4}^{*}\) implicitly by \(\phi _{4}^{\prime }\left( e_{4}^{*}\right) =0\). Since \(\phi _{4}\) is strictly concave, \(e_{4}^{*}\) is unique. By assumption (1), \(e_{4}^{*}<1\). If \(e_{4}^{*} > \bar{e}\), then \(e_{4}^{*}\) maximizes \(\tilde{V}\left( e\right) \). Moreover, if the optimal contract involves \(b_{HL}>0\), then \(e_{4}^{*}>\bar{e}\).
We use the intermediate value theorem to show that \(e_{4}^{*}>\bar{e}\) if and only if \(\alpha \) is strictly smaller than a critical value \(\bar{\alpha } _{1}\in \left( 0, 1\right) \). The argument proceeds in three steps:

1.
If \(\alpha =0\), \(\phi _{4}^{\prime }\left( 0\right) >0 = \bar{e}\). Thus if \(\alpha =0\), \(e_{4}^{*}>\bar{e}\).

2.
\(e_{4}^{*}\) is continuous and strictly decreasing in \(\alpha \). Moreover, \(\bar{e}=c^{\prime 1}\left( \alpha x_{H}\right) \) is continuous and strictly increasing in \(\alpha \).

3.
At \(\alpha =1\), \(e_{4}^{*}\) solves
$$\begin{aligned} x_{H}=c^{\prime }\left( e_{4}^{*}\right) + \left[ e_{4}^{*}\sigma +\left( 1e_{4}^{*}\right) \left( 1\sigma \right) \right] c^{\prime \prime }\left( e_{4}^{*}\right) \frac{\sigma }{2 \sigma 1}. \end{aligned}$$(68)Therefore, at \(\alpha =1\), \(\alpha x_{H}>c^{\prime }\left( e_{4}^{*}\right) \) and so \(e_{4}^{*} < \bar{e}\).
Thus there exists an \(\bar{\alpha }_{1}\in \left( 0,1\right) \) such that if \(\alpha =\bar{\alpha }_{1}\), \(e_{4}^{*}\) \(=\bar{e}\). For all \(\alpha <\bar{\alpha }_{1}\), the optimal effort is \(e_{4}^{*}>\bar{e};\) moreover \( b_{HL}>0\) and \(\theta _{HL}=\theta _{LL}=1\). On the other hand, for all \(\alpha \ge \bar{\alpha }_{1}\), \(b_{HL}=0\).
It remains to show that there exists \(\bar{\alpha }_{2}\in \left( \bar{\alpha } _{1},1\right) \) such that \(\theta _{LL}>0\) if and only if \(\alpha <\bar{\alpha }_{2}\). Note that \(\theta _{LL}>0\) if and only \(\phi _{3}^{\prime }\left( \hat{e}\right) >0\). To see this, first suppose that \(\phi _{3}^{\prime }\left( \hat{e}\right) \le 0\). Then \(\tilde{V}\left( e\right) < \tilde{V}\left( \hat{e}\right) \) for all \(e>\hat{e}\) since \(\tilde{V}(\cdot )\) is strictly concave. Hence the optimal effort is no larger than \(\hat{e}\) and \(\theta _{LL}=0\). Second, if \(\phi _{3}^{\prime }\left( \hat{e}\right) >0\), then the optimal effort is strictly bigger than \(\hat{e}\) and \(\theta _{LL}>0\).
We use the intermediate value theorem to show that there exists \(\bar{\alpha }_{2} \in \left( \bar{\alpha }_{1}, 1\right) \) such that \(\phi _{3}^{\prime }\left( \hat{e}\right) >0\) if and only if \(\alpha <\bar{\alpha }_{2}\). The argument proceeds in three steps:

1.
By definition of \(\bar{\alpha }_{1}\), if \(\alpha =\bar{\alpha }_{1}\), then \(\phi _{4}^{\prime }\left( e_{4}^{*}\right) =\phi _{4}^{\prime }\left( \bar{e}\right) =0\). Since \(\phi _{3}\) and \(\tilde{V}\) are strictly concave, and \(\bar{e}>\hat{e}\), it follows that \(\phi _{4}^{\prime }\left( \bar{e} \right) \le \phi _{3}^{\prime }\left( \bar{e}\right) <\phi _{3}^{\prime }\left( \hat{e}\right) \). Therefore, if \(\alpha =\bar{\alpha }_{1}\), then \(\phi _{3}^{\prime }\left( \hat{e}\right) >0\).

2.
\(\phi _{3}^{\prime }\left( \hat{e}\right) \) is continuous and strictly decreasing in \(\alpha :\)
$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}\alpha }\phi _{3}^{\prime }\left( \hat{e}\right) =\left[ \frac{ \partial }{\partial \alpha }\phi _{3}^{\prime }\left( e\right) \right] _{e= \hat{e}}+\phi _{3}^{\prime \prime }\left( \hat{e}\right) \frac{\mathrm{d}\hat{e}}{ \mathrm{d}\alpha }<0. \end{aligned}$$(69) 
3.
Suppose \(\alpha =1\). Since \(e c^{\prime \prime }\left( e\right) \ge c^{\prime }\left( e\right) \),
$$\begin{aligned} \phi _{3}^{\prime }\left( \hat{e}\right)= & {} x_{H}x_{L}x_{L}\frac{2\sigma 1}{1\sigma }\left[ 1\frac{\sigma x_{L}}{\left( 1\sigma \right) x_{H}} \right] c^{\prime }\left( \hat{e}\right) \end{aligned}$$(70)$$\begin{aligned}&\left[ \hat{e}+\left( 1\hat{e}\right) \frac{\sigma x_{L}}{\left( 1\sigma \right) x_{H}}\right] c^{\prime \prime }\left( \hat{e}\right) \end{aligned}$$(71)$$\begin{aligned}< & {} x_{H}x_{L}x_{L}\frac{2\sigma 1}{1\sigma }\left[ 2\frac{\sigma x_{L} }{\left( 1\sigma \right) x_{H}}\right] \sigma x_{H} \end{aligned}$$(72)$$\begin{aligned}= & {} x_{H}2\sigma x_{H}\sigma x_{L}<0. \end{aligned}$$(73)Hence if \(\alpha =1\), then \(\phi _{3}^{\prime }\left( \hat{e}\right) <0\).
\(\square \)
Proof of Proposition 5
The principal’s expected payoff \(\tilde{V}\left( e\right) \) defined in (60) and (67) is strictly increasing in \(\sigma \) because the functions \(\phi _1, \phi _2, \phi _3\) and \(\phi _4\) are strictly increasing in \(\sigma \). If \(\theta _{HL} < 1\), then \(\tilde{V}\left( e\right) = \phi _1(e)\), as defined in (58), and therefore \(\partial \tilde{V}\left( e\right) /\partial \alpha = 0\). If \(\theta _{HL} = 1\), then \(\tilde{V}\left( e\right) = \phi _2(e)\) in the case of Proposition 3 and \(\partial \tilde{V}\left( e\right) /\partial \alpha > 0\) because \(\phi _2\) is strictly increasing in \(\alpha \). In the case of Proposition 4, \(\tilde{V}\left( e\right) = \min \left\{ \phi _3(e), \phi _4(e)\right\} \). As \(\sigma < \bar{\sigma }\), \(\phi _3\) and \(\phi _4\) are strictly increasing in \(\alpha \) and so \(\partial \tilde{V}\left( e\right) /\partial \alpha > 0\). \(\square \)
Proof of Proposition 6
Suppose \(\sigma >\bar{\sigma }\). By Proposition 3, \(b_{HL}\) is strictly positive if and only if \(\alpha <\bar{\alpha }\). Moreover, the proof of Proposition 3 establishes that, when \(\alpha <\bar{\alpha }\), the profit maximizing effort is equal to \(e_{2}^{*}\) which is strictly decreasing in \(\alpha \) [see Eq. (62)], and \(\theta _{HL}=1\) and \(\theta _{LL}=0\). By Eq. (19) it follows that
which is strictly decreasing in \(\alpha \).
By the proof of Proposition 3, the principal’s profit from implementing effort e is \(\tilde{V}\left( e\right) =\phi _{1}\left( e\right) \) if \(e\le \hat{e}\equiv c^{\prime 1}\left( \sigma \alpha x_{H}\right) \), and \(\tilde{V}\left( e\right) =\phi _{2}\left( e\right) \) if \(e>\hat{e}\). We show that there exists an \(\varepsilon ^{\prime }>0\) such that, if \(\alpha \in \left( \bar{\alpha },\bar{\alpha }+\varepsilon ^{\prime }\right) \), the profit maximizing effort is equal to \(\hat{e}\) and thus strictly increasing in \(\alpha \).
Since \(\tilde{V}\left( e\right) \) is strictly concave in e, it is sufficient to prove that \(\phi _{1}^{\prime }\left( \hat{e}\right)>0>\phi _{2}^{\prime }\left( \hat{e}\right) \) if \(\alpha \in \left( \bar{\alpha } ,\bar{\alpha }+\varepsilon ^{\prime }\right) \) for some \(\varepsilon ^{\prime }>0\). Straightforward calculations show that
Since \(\phi _{1}\left( \hat{e}\right) =\phi _{2}\left( \hat{e}\right) \),
Moreover, if \(\alpha =\bar{\alpha }\), then \(\phi _{2}^{\prime }\left( \hat{e}\right) =0\) by construction of \(\bar{\alpha }\). Thus \(\alpha =\bar{\alpha }\) implies
By continuity, there exists \(\varepsilon ^{\prime }>0\) such that \(\phi _{1}^{\prime }\left( \hat{e}\right) >0\) when \(\alpha \in \left( \bar{\alpha },\bar{\alpha }+\varepsilon ^{\prime }\right) \). It remains to show that \(\phi _{2}^{\prime }\left( \hat{e}\right) <0\) when \(\alpha \in \left( \bar{\alpha },\bar{\alpha }+\varepsilon ^{\prime }\right) \). Equation (61) implies that \(\phi _{2}^{\prime }\left( e\right) \) is strictly decreasing in \(\alpha \) (holding e constant). Moreover, \(\hat{e}\) is strictly increasing in \(\alpha \), and \(\phi _{2}^{\prime \prime }\left( e\right) <0\). Since \(\alpha =\bar{\alpha }\) implies \(\phi _{2}^{\prime }\left( \hat{e}\right) =0\), it follows that \(\phi _{2}^{\prime }\left( \hat{e}\right) <0\) when \(\alpha \in \left( \bar{\alpha },\bar{\alpha }+\varepsilon ^{\prime }\right) \).
Suppose \(\sigma <\bar{\sigma }\). By Proposition 4, \(b_{HL}\) is strictly positive if and only if \(\alpha <\bar{\alpha }_{1}\). Moreover, the proof of Proposition 4 establishes that, when \(\alpha <\bar{\alpha }_{1}\), the profit maximizing effort is equal to \(e_{4}^{*}\) which is strictly decreasing in \(\alpha \), and \(\theta _{LL}=\theta _{HL}=1\). By Eq. (19), it follows that
which is strictly decreasing in \(\alpha \).
By the proof of Proposition 4, the principal’s profit from implementing effort e is \(\tilde{V}\left( e\right) =\phi _{1}\left( e\right) \) if \(e\le \hat{e}\), \(\tilde{V}\left( e\right) =\phi _{3}\left( e\right) \) if \(\hat{e}<e\le \bar{e}\equiv c^{\prime 1}\left( \alpha x_{H}\right) \), and \(\tilde{V}\left( e\right) =\phi _{4}\left( e\right) \) if \(e>\bar{e}\). We show that there exists an \(\varepsilon ^{\prime \prime }>0\) such that, if \(\alpha \in \left( \bar{\alpha }_{1},\bar{\alpha }_{1} +\varepsilon ^{\prime \prime }\right) \), the profit maximizing effort is equal to \(\bar{e}\) and thus strictly increasing in \(\alpha \).
Since \(\tilde{V}\left( e\right) \) is strictly concave in e, it is sufficient to prove that \(\phi _{3}^{\prime }\left( \bar{e}\right)>0>\phi _{4}^{\prime }\left( \bar{e}\right) \) if \(\alpha \in \left( \bar{\alpha } _{1},\bar{\alpha }_{1}+\varepsilon ^{\prime \prime }\right) \) for some \(\varepsilon ^{\prime \prime }>0\). Straightforward calculations show that
Since \(\phi _{3}\left( \bar{e}\right) =\phi _{4}\left( \bar{e}\right) \),
Suppose that \(\alpha =\bar{\alpha }_{1}\). Then \(\phi _{4}^{\prime }\left( \bar{e}\right) =0\) by construction of \(\bar{\alpha }_{1}\). Thus
which is strictly positive since \(\sigma <\bar{\sigma }\). By continuity, there exists some \(\varepsilon ^{\prime \prime }>0\) such that \(\phi _{3}^{\prime }\left( \bar{e}\right) >0\) when \(\alpha \in \left( \bar{\alpha }_{1},\bar{\alpha }_{1}+\varepsilon ^{\prime \prime }\right) \). It remains to show that \(\phi _{4}^{\prime }\left( \bar{e}\right) <0\) when \(\alpha \in \left( \bar{\alpha }_{1},\bar{\alpha }_{1}+\varepsilon ^{\prime \prime }\right) \). Equation (66) implies that \(\frac{\partial }{\partial \alpha }\phi _{4}^{\prime }\left( e\right) =x_{L}\frac{1\sigma }{\sigma }x_{H}<0\), where the inequality follows from \(\sigma <\bar{\sigma }\). That is, \(\phi _{4}^{\prime }\left( e\right) \) is strictly decreasing in \(\alpha \) (holding e constant). Moreover, \(\bar{e}\) is strictly increasing in \(\alpha \), and \(\phi _{4}^{\prime \prime }\left( e\right) <0\). Since \(\alpha =\bar{\alpha }_{1}\) implies \(\phi _{4}^{\prime }\left( \bar{e}\right) =0\), it follows that \(\phi _{4}^{\prime }\left( \bar{e}\right) <0\) when \(\alpha \in \left( \bar{\alpha }_{1},\bar{\alpha }_{1}+\varepsilon ^{\prime \prime }\right) \).
Setting \(\varepsilon :=\min \left\{ \varepsilon ^{\prime },\varepsilon ^{\prime \prime }\right\} \) concludes the proof. \(\square \)
Proof of Proposition 7
Since \((\gamma , e)\) solves problem (16), \(\gamma \) satisfies (18) and \(w_{LL} = w_{HL} = 0\) by Proposition 2 (a). This reduces the constraints in (29) to
By (85) \(w_{LH} \le \alpha \theta _{LL} x_H\). Suppose that this inequality is strict, i.e. \(w_{LH} < \alpha \theta _{LL} x_H\). Then (86) implies that \(w_{HH} > \alpha \,\theta _{HL}\, x_H + b_{HL}\), a contradiction to (84). This proves that the constraints in (29) cannot be satisfied if \(w_{LH} \ne \alpha \theta _{LL} x_H\).
Now let \(w_{LH} = \alpha \theta _{LL} x_H\). Then (82) and (85) hold because \(x_L < x_H\). As \(w_{LH} = \alpha \theta _{LL} x_H\), (86) implies
Thus the equality holds in (84), and the strict inequality holds in (83) because \(x_L < x_H\). This proves that (82)–(85) are satisfied if \((\gamma , e)\) solves problem (16) with \(w_{LH} = \alpha \theta _{LL} x_H\) and \(w_{HH} = \alpha \,\theta _{HL}\, x_H + b_{HL}\). \(\square \)
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Bester, H., Münster, J. Subjective evaluation versus public information. Econ Theory 61, 723–753 (2016). https://doi.org/10.1007/s0019901609538
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DOI: https://doi.org/10.1007/s0019901609538