A note on optimal contracting with public ex post information under limited liability

Abstract

Riordan and Sappington (J Econ Theory 45:189–199, 1988) show that in an agency relationship in which the agent’s type is correlated with a public ex post signal, the principal may attain first best (full surplus extraction and efficient output levels) if the agent is faced with a lottery such that each type is rewarded for one signal realization and punished equally for all the others. Gary-Bobo and Spiegel (RAND J Econ 37:431–448, 2006) show that there exist locally incentive compatible lotteries such that limited liability constraints are satisfied, only if these constraints are satisfied when that kind of lottery is used. We explore how lotteries should be designed to attain not only local but also global incentive compatibility when the agent is protected by limited liability. The main issue with global incentive compatibility rests with intermediate types being potentially attractive reports to both lower and higher types. A lottery including three levels of profit (rather than only two) is found to be optimal under limited liability, in that it is most likely to be globally incentive compatible, if local incentive constraints are strictly satisfied. Conditions for first-best implementation are identified.

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Notes

  1. 1.

    Reynès (2017) shows that the Cobb–Douglas function approximates a large class of functions in a simpler and, possibly, more general framework.

  2. 2.

    As is now well known, in single-product industries economies of scale over the relevant range of production is a sufficient condition for natural monopoly, which is the traditional normative argument for the regulation of public utilities (see Joskov 2007 and Decker 2015, among others). There are a number of empirical studies that detect economies of scale in regulated industries, using increasing returns to scale as a measure for them. To illustrate, we mention a few studies which rely on a Cobb–Douglas functional form for cost estimation. Fraas and Munley (1984) detect strong economies of scale using data provided by U.S. EPA on 178 municipal wastewater treatment plants. Fraquelli and Giandrone (2003) reach a similar conclusion using data on 103 urban wastewater plants in Italy. Farsi et al. (2010) analyse the cost structure of the French electricity distribution sector prior to the reforms initiated in 2005, being based on 279 observations from 93 units operating within Electricité Réseau Distribution France in the period 2003–2005. They also find that most units exploit economies of scale.

  3. 3.

    One may also consider the situation, already explored by the literature on incentive problems, in which the agent’s cost is composed of a fixed cost and a privately known variable cost that are inversely related. This is the case, for instance, when the fixed cost is, in fact, an opportunity cost of renouncing to alternative businesses, which naturally increases with the efficiency of the agent in the trade with the principal. Obviously, depending on the shape of the two components, the total cost may take any shape with respect to the type (Maggi and Rodriguez-Clare 1995). In particular, the total cost is concave in the type when, as is highly plausible, the opportunity cost increases with the efficiency of the agent at a decreasing rate (Lewis and Sappington 1989).

  4. 4.

    Considering the same kind of signals, Kessler et al. (2005) point out: “Such additional ex post information seems to be the rule instead of the exception in many empirically relevant circumstances” (p.1208).

  5. 5.

    Laffont and Tirole (1983) cite early applications of this ”yardstick competition” approach in the Spanish power sector as well as in Illinois. In the Netherlands it has been applied to regulated electricity distribution companies for all over a decade. Further experiences are found in gas and water industries in Britain and Austria (see Decker 2015).

  6. 6.

    See Demougin and Garvie (1991) and Kessler et al. (2005) for agency problems in which limited liability takes the form of a bound on the transfer to the agent rather than on his profit.

  7. 7.

    Remark that this expression would still result if \(z_{s}\left( \theta \right) \) were to include a fixed component related to the type and a stochastic component conditional on the signal realization, as considered by Bose and Zhao (2007).

  8. 8.

    A more general formulation of the objective function in the programme of P would be \(\int _{\theta }\left( S(q\left( \theta \right) )-C\left( q\left( \theta \right) ,\theta \right) -\left( 1-\omega \right) \sum _{s=1}^{n}\pi _{s}\left( \theta \right) p_{s}\left( \theta \right) \right) f\left( \theta \right) d\theta ,\) for some \(\omega \in \left[ 0,1\right) \). With this formulation, results would be qualitatively the same in that P dislikes conceding surplus to the agent and would attempt to set profits such that \(\sum _{s=1}^{n}\pi _{s}\left( \theta \right) p_{s}\left( \theta \right) \) is as low as possible. Accounting for this, we simply take \(\omega =0\).

  9. 9.

    For instance, in regulated industries, in which this is common practice, L could be interpreted as an indicator of financial viability, beyond which the regulated firm would go bankrupt. Remark that the limited liability constraints are required to hold as long as the agent does not conceal information. This involves that, even if the profits are set such that the agent does not lose more than L in equilibrium, he might still incur a greater loss, should he decide to deliver an out-of-equilibrium report.

  10. 10.

    The “only if” proof of CM shows that if the vector \(\mathbf {h(}\theta _{T}\mathbf {)}\) does not exist, then it is impossible to ensure that \(\theta _{T}\) is not an attractive report to any type \(\theta _{t}<\theta _{T}.\) Notice however that the existence of \(\mathbf {h(}\theta _{t}\mathbf {)}\) is not necessary for all types. In particular, it does not need to hold for type \(\theta _{1}.\) This paves the way for the results drawn in the study of RS, in which first-best implementation does not necessarily depend on the existence of \(\mathbf {h(}\theta _{t}\mathbf {)}\), \(\forall t\). Bose and Zhao (2007) show that Proposition 1 in RS implies that first best might be effected when the full-rank condition is violated.

  11. 11.

    In particular, considering a discrete set of types - as in RS and CM - it can be shown that as long as property (1) holds, \({\mathbf {p}}\left( \theta _{1}\right) \) and \({\mathbf {p}}\left( \theta _{T}\right) \) do not lie in the convex hull generated by the probability vectors of the other types. Moreover, there exist vectors \({\mathbf {p}}\left( \theta _{t}\right) ,\)\(t\ne 1,T,\) which lie in the convex hull generated by the probability vectors of the other types and do not violate (1). The formal proof, here omitted, is available with the authors.

  12. 12.

    We let \(\theta ^{-}\) and \(\theta ^{+}\) denote types respectively below and above \(\theta \), but not necessarily limit values around \(\theta \).

  13. 13.

    The standard procedure would be to derive the local incentive constraint directly from (12) by taking the limit for \(\theta ^{\prime }\rightarrow \theta .\) The expression of \(\pi _{n}\left( \theta \right) \) in (16) would result after manipulating the local incentive constraint. We follow a somewhat different procedure that is functional to our analysis.

  14. 14.

    Specifically, (21) reduces to (6) if \(N=\left\{ 1,n\right\} \) so that \(p_{n}\left( \cdot \right) =1-p_{1}\left( \cdot \right) \).

  15. 15.

    Whereas, as mentioned in Introduction, many empirical studies on regulated industries identify increasing returns to scale, in theoretical studies these are often captured by assuming concavity of the cost function with respect to output. One example is found in Schmalensee (1978). Referring to the distribution of a public utility service in a given geographical area, the author allows for the transmission cost function to be strictly concave in the service flow. As a result, the average cost is a decreasing function of the flow, and there are increasing returns to scale in transmission. Greve and Keiding (2016) study regulated competition with an increasing returns to scale technology, which they represent by taking the cost function to be concave in output. Further cases are reported by Armstrong and Sappington (2007), who review the literature on optimal monopoly regulation, referring to a concave cost function in several stages of the analytical development (see point \(\left( ii\right) \) of Proposition 4, for instance).

  16. 16.

    In a second-best environment with three types, it can be shown that, just as in the first-best environment, the optimal vector of profits either includes only two possible levels, or it includes three levels. The optimal choice will depend on what incentive constraint is tighter in the principal’s problem. Mathematical details are available with the authors.

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Correspondence to Daniel Danau.

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An earlier version was presented at the 74th IIPF Congress (Tampere), the 18th PET Meeting (Paris), and the 5th World Congress of the Game Theory Society (Maastricht). We thank participants for their comments. The usual disclaimer applies.

Appendices

Derivation of (14) and (15)

Using \({\widetilde{\pi }}_{s}\left( \theta \left| \theta ^{\prime }\right. \right) =z_{s}\left( \theta \right) -C\left( q\left( \theta \right) ,\theta ^{\prime }\right) \) and \(\pi _{s}\left( \theta \right) ={\widetilde{\pi }}_{s}\left( \theta \left| \theta \right. \right) ,\) we have

$$\begin{aligned} {\mathbb {E}}_{s}\left[ {\widetilde{\pi }}_{s}\left( \theta \left| \theta ^{\prime }\right. \right) \right] =\sum _{s=1}^{n}\pi _{s}\left( \theta \right) p_{s}\left( \theta ^{\prime }\right) +C\left( q\left( \theta \right) ,\theta \right) -C\left( q\left( \theta \right) ,\theta ^{\prime }\right) . \end{aligned}$$

Using (3), \({\mathbb {E}}_{s}\left[ {\widetilde{\pi }}_{s}\left( \theta \left| \theta ^{\prime }\right. \right) \right] \) is rewritten as (12). Rewriting (3) as \(\pi _{1}\left( \theta \right) =-\sum _{s=2}^{n}\pi _{s}\left( \theta \right) \frac{p_{s}(\theta )}{p_{1}(\theta )}\), (12) is rewritten as

$$\begin{aligned} \Delta C\left( \theta ,\theta ^{\prime }\right)&\le \sum _{s\ne 1,n}\pi _{s}\left( \theta \right) p_{s}\left( \theta \right) \left( \frac{p_{1}\left( \theta ^{\prime }\right) }{p_{1}(\theta )}-\frac{p_{s}\left( \theta ^{\prime }\right) }{p_{s}(\theta )}\right) \\&\quad +\pi _{n}\left( \theta \right) p_{n}(\theta )\left( \frac{p_{1}\left( \theta ^{\prime }\right) }{p_{1}(\theta )}-\frac{p_{n}\left( \theta ^{\prime }\right) }{p_{n}(\theta )}\right) , \end{aligned}$$

hence as

$$\begin{aligned}&\pi _{n}\left( \theta \right) p_{n}(\theta )\left( \frac{p_{1}\left( \theta ^{\prime }\right) }{p_{1}(\theta )}-\frac{p_{n}\left( \theta ^{\prime }\right) }{p_{n}(\theta )}\right) \ge \Delta C\left( \theta ,\theta ^{\prime }\right) \nonumber \\&\quad -\sum _{s\ne 1,n}\pi _{s}\left( \theta \right) p_{s}\left( \theta \right) \left( \frac{p_{1}\left( \theta ^{\prime }\right) }{p_{1}(\theta )}\quad -\frac{p_{s}\left( \theta ^{\prime }\right) }{p_{s}(\theta )}\right) . \end{aligned}$$
(29)

Recall that, by assumption, \(\frac{p_{1}\left( \theta ^{\prime }\right) }{p_{1}(\theta )}>\frac{p_{n}\left( \theta ^{\prime }\right) }{p_{n}(\theta )}\) if and only if \(\theta ^{\prime }>\theta .\) Using this property for \(\theta ^{-}<\theta \) and \(\theta ^{+}>\theta ,\) (29) is respectively rewritten as (14) and (15).

Proof of Lemma 1

Recall \({\widetilde{\pi }}_{s}\left( \theta \left| \theta ^{\prime }\right. \right) =z_{s}\left( \theta \right) -C\left( q^{fb}\left( \theta \right) ,\theta ^{\prime }\right) \) and

$$\begin{aligned} {\mathbb {E}}_{s}\left[ {\widetilde{\pi }}_{s}\left( \theta \left| \theta ^{\prime }\right. \right) \right] \equiv \sum _{s=1}^{n}\left( z_{s}\left( \theta \right) -C\left( q^{fb}\left( \theta \right) ,\theta ^{\prime }\right) \right) p_{s}\left( \theta ^{\prime }\right) . \end{aligned}$$
(30)

The first-order condition of the agent’s problem, evaluated at \(\theta ^{\prime }=\theta ,\) is given by:

$$\begin{aligned} \sum _{s=1}^{n}\left( z_{s}^{\prime }\left( \theta \right) -C_{q}\left( q^{fb}\left( \theta \right) ,\theta \right) \left( q^{fb}\left( \theta \right) \right) ^{\prime }\right) p_{s}\left( \theta \right) =0. \end{aligned}$$
(31)

From \(z_{s}\left( \theta \right) =\pi _{s}\left( \theta \right) +C\left( q^{fb}\left( \theta \right) ,\theta \right) ,\) we compute \(z_{s}^{\prime }\left( \theta \right) =\pi _{s}^{\prime }\left( \theta \right) +C_{q}\left( q^{fb}\left( \theta \right) ,\theta \right) \left( q^{fb}\left( \theta \right) \right) ^{\prime }+C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right) ,\) which we then replace into (31) to find

$$\begin{aligned} C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right) =-\sum _{s=1}^{n}\pi _{s}^{\prime }\left( \theta \right) p_{s}\left( \theta \right) . \end{aligned}$$
(32)

Using (3), we obtain \(-\sum _{s=1}^{n}\pi _{s}^{\prime }\left( \theta \right) p_{s}\left( \theta \right) =\sum _{s=1}^{n}\pi _{s}\left( \theta \right) p_{s}^{\prime }\left( \theta \right) \). Using this equality, (32) is rewritten as

$$\begin{aligned} C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right) =\sum _{s=1}^{n}\pi _{s}\left( \theta \right) p_{s}^{\prime }\left( \theta \right) \end{aligned}$$
(33)

Suppose that \(\pi _{1}\left( \theta \right) \) is changed by \(\varepsilon _{1}.\) Accordingly, \(\pi _{s}\left( \theta \right) \) is changed by \(\varepsilon _{s}\) and \(\pi _{n}\left( \theta \right) \) by \(\varepsilon _{n}\) such that (PC) is still saturated and the right-hand side of (33) does not vary. Dropping the argument \(\theta \) everywhere for the sake of shortness, this requires

$$\begin{aligned} \varepsilon _{s}p_{s}= & {} -\varepsilon _{1}p_{1}-\varepsilon _{n}p_{n}\Leftrightarrow \varepsilon _{s}=-\varepsilon _{1}\frac{p_{1}}{p_{s}}-\varepsilon _{n}\frac{p_{n}}{p_{s}} \end{aligned}$$
(34)
$$\begin{aligned} \varepsilon _{n}p_{n}^{\prime }= & {} -\varepsilon _{s}p_{s}^{\prime }-\varepsilon _{1}p_{1}^{\prime }\Leftrightarrow \varepsilon _{n}=-\varepsilon _{s}\frac{p_{s}^{\prime }}{p_{n}^{\prime }}-\varepsilon _{1}\frac{p_{1}^{\prime }}{p_{n}^{\prime }}. \end{aligned}$$
(35)

Replacing first \(\varepsilon _{n}\) from (35) in (34), then \(\varepsilon _{s}\) from (34) in (35), and rearranging we obtain

$$\begin{aligned} \varepsilon _{s}=-\varepsilon _{1}\frac{p_{1}}{p_{s}}\frac{\frac{p_{1}^{\prime }}{p_{1}}-\frac{p_{n}^{\prime }}{p_{n}}}{\frac{p_{s}^{\prime }}{p_{s}}-\frac{p_{n}^{\prime }}{p_{n}}},\quad \varepsilon _{n}=\varepsilon _{1}\frac{p_{1}}{p_{n}}\frac{\frac{p_{1}^{\prime }}{p_{1}}-\frac{p_{s}^{\prime }}{p_{s}}}{\frac{p_{s}^{\prime }}{p_{s}}-\frac{p_{n}^{\prime }}{p_{n}}}, \end{aligned}$$
(36)

from which we deduce that \(\varepsilon _{s}\) and \(\varepsilon _{n}\) have different signs.

Hence, any change in \(\pi _{1}\left( \theta \right) \) must be matched with an opposite change in \(\pi _{s}\left( \theta \right) \) and in \(\pi _{n}\left( \theta \right) \). It follows that, starting from \({\varvec{\pi }}^{1}\left( \theta \right) \), where \(\pi _{s}^{1}\left( \theta \right) =\pi _{n}^{1}\left( \theta \right) ,\)\(\forall s\ne 1,n,\) a new vector of profits, such that (3) and (32) are both satisfied, can be reached only if (LL) is tightened.

Proof of Lemma 2

For the purpose of this proof, we denote

$$\begin{aligned} \Omega \left( \theta ^{\prime },\theta \right) \equiv \psi _{s,n}\left( \theta ^{\prime },\theta \right) -\frac{\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{s}^{\prime }\left( \theta \right) }{p_{s}(\theta )}}{\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}},\quad \forall \theta ,\theta ^{\prime },\quad \forall s\ne 1,n. \end{aligned}$$

Taking the expression of \(\pi _{n}\left( \theta \right) p_{n}(\theta )\) from (16), plugging into (14), and making use of the inequalities \(\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}>\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}\) and \(\frac{p_{n}\left( \theta ^{-}\right) }{p_{n}(\theta )}>\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}\) to rearrange, (14) is rewritten as

$$\begin{aligned} C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right)\ge & {} \left( \frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}\right) \left( \frac{\Delta C\left( \theta ,\theta ^{-}\right) }{\frac{p_{n}(\theta ^{-})}{p_{n}(\theta )}-\frac{p_{1}(\theta ^{-})}{p_{1}(\theta )}}\nonumber \right. \\&\left. +\,\sum _{s\ne 1,n}\pi _{s}\left( \theta \right) p_{s}(\theta )\Omega \left( \theta ^{-},\theta \right) \right) ,\quad \forall \theta ^{-}<\theta . \end{aligned}$$
(37)

Similarly, (15) is rewritten as

$$\begin{aligned} C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right)\le & {} \left( \frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}\right) \left( \frac{-\Delta C\left( \theta ,\theta ^{+}\right) }{\frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}}\right. \nonumber \\&\left. +\,\sum _{s\ne 1,n}\pi _{s}\left( \theta \right) p_{s}(\theta )\Omega \left( \theta ^{+},\theta \right) \right) ,\quad \forall \theta ^{+}>\theta . \end{aligned}$$
(38)

From (37) and (38), the necessary condition (20) is derived. To show sufficiency of (20), we compute

$$\begin{aligned} \underset{\theta ^{\prime }\rightarrow \theta }{\lim }\psi _{s,n}\left( \theta ^{\prime },\theta \right) =\underset{\theta ^{\prime }\rightarrow \theta }{\lim }\frac{\frac{\frac{p_{1}\left( \theta ^{\prime }\right) -p_{1}(\theta )}{\theta ^{\prime }-\theta }}{p_{1}(\theta )}-\frac{\frac{p_{s}\left( \theta ^{\prime }\right) -p_{s}(\theta )}{\theta ^{\prime }-\theta }}{p_{s}(\theta )}}{\frac{\frac{p_{1}\left( \theta ^{\prime }\right) -p_{1}(\theta )}{\theta ^{\prime }-\theta }}{p_{1}(\theta )}-\frac{\frac{p_{n}\left( \theta ^{\prime }\right) -p_{n}(\theta )}{\theta ^{\prime }-\theta }}{p_{n}(\theta )}}=\frac{\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{s}^{\prime }\left( \theta \right) }{p_{s}(\theta )}}{\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}}. \end{aligned}$$
(39)

Taking \(\theta ^{+}\rightarrow \theta \) and using (39), (20) is rewritten as (37). Similarly, taking \(\theta ^{-}\rightarrow \theta \) and using (39), (20) is rewritten as (38). Hence, (20) boils down to either (37) or (38), hence to either (14) or (15), entailing that it is sufficient for either such condition to hold.

Proof of Proposition 1

Derivation of (25)

Because \(\frac{d\psi _{s,n}\left( \theta ^{+},\theta \right) }{d\theta ^{+}}<0\) and \(\frac{d\psi _{s,n}\left( \theta ^{-},\theta \right) }{d\theta ^{-}}<0,\) where \(s\ne n,\) the difference \(\psi _{s,n}\left( \theta ^{-},\theta \right) -\psi _{s,n}\left( \theta ^{+},\theta \right) \) has a lower bound for \(\theta ^{-}\rightarrow \theta \) and \(\theta ^{+}\rightarrow \theta \). Using (39), the lower bound is found to be equal to zero. Hence, \(\psi _{s,n}\left( \theta ^{-},\theta \right) -\psi _{s,n}\left( \theta ^{+},\theta \right) \ge 0\), \(\forall \left\{ \theta ^{-},\theta ,\theta ^{+}\right\} \), and (20) is weakest if \(\pi _{s}\left( \theta \right) =-L,\)\(\forall s\ne 1,n.\) Replacing in (20) and rearranging yields (25).

Derivation of (23) and (24)

Setting \(\pi _{s}\left( \theta \right) =\pi _{n}\left( \theta \right) \) in (14), we see that \(\pi _{n}\left( \theta \right) \ge -L\) if and only if (24) is satisfied. It follows immediately from Lemma 1 that if \({\varvec{\pi }}^{1}\left( \theta \right) \) violates (LL), then no other vector of incentive compatible profits can be found such that (LL) is satisfied.

Setting \(\pi _{s}\left( \theta \right) =\pi _{n}\left( \theta \right) \) in (16), then plugging the resulting expression of \(\pi _{n}\left( \theta \right) ,\) we see that (14) and (15) are jointly satisfied if and only if (23) is satisfied.

Proof of Corollary 1

Using \(\pi _{s}\left( \theta \right) =-L\) in (16), \(\pi _{n}\left( \theta \right) \) is rewritten as

$$\begin{aligned} \pi _{n}\left( \theta \right) =\frac{L\sum _{s\ne 1,n}p_{s}(\theta )\left( \frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{s}^{\prime }\left( \theta \right) }{p_{s}(\theta )}\right) -C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right) }{p_{n}(\theta )\left( \frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}\right) }. \end{aligned}$$

Replacing \(\sum _{s\ne 1,n}p_{s}(\theta )=1-\left( p_{1}(\theta )+p_{n}(\theta )\right) \) and \(\sum _{s\ne 1,n}p_{s}^{\prime }(\theta )=-\left( p_{1}^{\prime }(\theta )+p_{n}^{\prime }(\theta )\right) ,\)\(\pi _{n}\left( \theta \right) \) is further rewritten as \(\pi _{n}^{*}\left( \theta \right) \). Recalling from (3) that \(\pi _{1}\left( \theta \right) =-\sum _{s=2}^{n}\pi _{s}\left( \theta \right) \frac{p_{s}(\theta )}{p_{1}(\theta )}\) and using \(\pi _{s}\left( \theta \right) =-L\) together with \(\pi _{n}^{*}\left( \theta \right) \) in the expression of \(\pi _{1}\left( \theta \right) \), we obtain

$$\begin{aligned} \pi _{1}\left( \theta \right) =L\sum _{s\ne 1,n}\frac{p_{s}(\theta )}{p_{1}(\theta )}-\left( \frac{L\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right) }{p_{n}(\theta )\left( \frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}\right) }-L\right) \frac{p_{n}(\theta )}{p_{1}(\theta )}. \end{aligned}$$

Replacing again \(\sum _{s\ne 1,n}p_{s}(\theta )=1-\left( p_{1}(\theta )+p_{n}(\theta )\right) ,\)\(\pi _{1}\left( \theta \right) \) is further rewritten as \(\pi _{1}^{*}\left( \theta \right) \).

We are left with verifying that \(\pi _{1}^{*}\left( \theta \right) \ge -L\) and \(\pi _{n}^{*}\left( \theta \right) \ge -L.\) The former is true because \(p_{n}^{\prime }\left( \theta \right) <0.\) The latter is implied by the property \(\frac{p_{1}^{\prime }\left( \theta \right) }{p_{1}(\theta )}>\frac{p_{n}^{\prime }\left( \theta \right) }{p_{n}(\theta )}\) together with \(C_{\theta }\left( q^{fb}\left( \theta \right) ,\theta \right) \frac{p_{1}(\theta )}{p_{1}^{\prime }\left( \theta \right) }\le L,\) which is implied by (24) in turn.

Proof of Corollary 2

Using (39) and \(\frac{d\psi _{s,n}\left( \theta ^{+},\theta \right) }{d\theta ^{+}}<0\) (from the proof of Proposition 1), we see that \(\Omega \left( \theta ^{+},\theta \right) <0\), \(\forall \theta ^{+}>\theta \), so that the term \(\sum _{s\ne 1,n}\pi _{s}\left( \theta \right) p_{s}(\theta )\Omega \left( \theta ^{+},\theta \right) \) in the right-hand side of (38) increases as \(\pi _{s}\left( \theta \right) \) is decreased. Hence, (38) is relaxed. Further using (39) and \(\frac{d\psi _{s,n}\left( \theta ^{-},\theta \right) }{d\theta ^{-}}<0\) (from the proof of Proposition 1), we see that \(\Omega \left( \theta ^{-},\theta \right) >0\), \(\forall \theta ^{-}<\theta \), so that the term \(\sum _{s\ne 1,n}\pi _{s}\left( \theta \right) p_{s}(\theta )\Omega \left( \theta ^{-},\theta \right) \) in the right-hand side of (37) increases as \(\pi _{s}\left( \theta \right) \) is decreased. Hence, (37) is tightened.

Proof of Corollary 3

Condition (26)

Using the definition of \(\psi _{s,n}\left( \theta ^{+},\theta \right) \), \(\frac{d\psi _{s,n}\left( \theta ^{+},\theta \right) }{d\theta ^{+}}<0\) is equivalent to

$$\begin{aligned} \frac{\frac{p_{1}^{\prime }\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{s}^{\prime }\left( \theta ^{+}\right) }{p_{s}(\theta )}}{\frac{p_{1}\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{s}\left( \theta ^{+}\right) }{p_{s}(\theta )}}<\frac{\frac{p_{1}^{\prime }\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta ^{+}\right) }{p_{n}(\theta )}}{\frac{p_{1}\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{n}\left( \theta ^{+}\right) }{p_{n}(\theta )}}. \end{aligned}$$
(40)

Multiplying the numerator by \(\left( \theta ^{+}-\theta \right) \) in both sides, subtracting 1 from each side and manipulating further, (40) becomes

$$\begin{aligned}&\frac{\frac{p_{s}\left( \theta ^{+}\right) -p_{s}^{\prime }\left( \theta ^{+}\right) \left( \theta ^{+}-\theta \right) }{p_{s}(\theta )}-\frac{p_{1}\left( \theta ^{+}\right) -p_{1}^{\prime }\left( \theta ^{+}\right) \left( \theta ^{+}-\theta \right) }{p_{1}(\theta )}}{\frac{p_{1}\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{s}\left( \theta ^{+}\right) }{p_{s}(\theta )}}\\&\quad <\frac{\frac{p_{n}\left( \theta ^{+}\right) -p_{n}^{\prime }\left( \theta ^{+}\right) \left( \theta ^{+}-\theta \right) }{p_{n}(\theta )}-\frac{p_{1}\left( \theta ^{+}\right) -p_{1}^{\prime }\left( \theta ^{+}\right) \left( \theta ^{+}-\theta \right) }{p_{1}(\theta )}}{\frac{p_{1}\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{n}\left( \theta ^{+}\right) }{p_{n}(\theta )}}. \end{aligned}$$

Using the definition of \(\rho _{s}\left( \theta ^{\prime },\theta \right) ,\) this is rewritten as

$$\begin{aligned} \frac{\rho _{s}\left( \theta ^{+},\theta \right) -\rho _{1}\left( \theta ^{+},\theta \right) }{\frac{p_{1}\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{s}\left( \theta ^{+}\right) }{p_{s}(\theta )}}<\frac{\rho _{n}\left( \theta ^{+},\theta \right) -\rho _{1}\left( \theta ^{+},\theta \right) }{\frac{p_{1}\left( \theta ^{+}\right) }{p_{1}(\theta )}-\frac{p_{n}\left( \theta ^{+}\right) }{p_{n}(\theta )}}, \end{aligned}$$
(41)

which is satisfied by assumption. Similarly, using the definition of \(\psi _{s,n}\left( \theta ^{-},\theta \right) ,\) we see that \(\frac{d\psi _{s,n}\left( \theta ^{-},\theta \right) }{d\theta ^{-}}<0\) if and only if

$$\begin{aligned} \frac{\frac{p_{1}^{\prime }\left( \theta ^{-}\right) }{p_{1}(\theta )}-\frac{p_{s}^{\prime }\left( \theta ^{-}\right) }{p_{s}(\theta )}}{\frac{p_{s}\left( \theta ^{-}\right) }{p_{s}(\theta )}-\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}}>\frac{\frac{p_{1}^{\prime }\left( \theta ^{-}\right) }{p_{1}(\theta )}-\frac{p_{n}^{\prime }\left( \theta ^{-}\right) }{p_{n}(\theta )}}{\frac{p_{n}\left( \theta ^{-}\right) }{p_{n}(\theta )}-\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}}. \end{aligned}$$
(42)

Multiplying the numerator by \(\left( \theta -\theta ^{-}\right) \) in both sides, subtracting 1 from either side and rearranging, (42) becomes

$$\begin{aligned}&\frac{\frac{p_{1}\left( \theta ^{-}\right) +p_{1}^{\prime }\left( \theta ^{-}\right) \left( \theta -\theta ^{-}\right) }{p_{1}(\theta )}-\frac{p_{s}\left( \theta ^{-}\right) +p_{s}^{\prime }\left( \theta ^{-}\right) \left( \theta -\theta ^{-}\right) }{p_{s}(\theta )}}{\frac{p_{s}\left( \theta ^{-}\right) }{p_{s}(\theta )}-\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}}\\&\quad >\frac{\frac{p_{1}\left( \theta ^{-}\right) +p_{1}^{\prime }\left( \theta ^{-}\right) \left( \theta -\theta ^{-}\right) }{p_{1}(\theta )}-\frac{p_{n}\left( \theta ^{-}\right) +p_{n}^{\prime }\left( \theta ^{-}\right) \left( \theta -\theta ^{-}\right) }{p_{n}(\theta )}}{\frac{p_{n}\left( \theta ^{-}\right) }{p_{n}(\theta )}-\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}}. \end{aligned}$$

Resting on the definition of \(\rho ,\) this is rewritten as

$$\begin{aligned} \frac{\rho _{s}\left( \theta ^{-},\theta \right) -\rho _{1}\left( \theta ^{-},\theta \right) }{\frac{p_{s}\left( \theta ^{-}\right) }{p_{s}(\theta )}-\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}}<\frac{\rho _{n}\left( \theta ^{-},\theta \right) -\rho _{1}\left( \theta ^{-},\theta \right) }{\frac{p_{n}\left( \theta ^{-}\right) }{p_{n}(\theta )}-\frac{p_{1}\left( \theta ^{-}\right) }{p_{1}(\theta )}}, \end{aligned}$$
(43)

which is satisfied by assumption.

Conditions (i) and (ii)

\(\left( a\right) \)\(\rho _{n}\left( \theta ^{\prime },\theta \right) >\rho _{1}\left( \theta ^{\prime },\theta \right) \) and \(\rho _{1}\left( \theta ^{\prime },\theta \right) >\rho _{s}\left( \theta ^{\prime },\theta \right) \). It is immediate to see that (26) is satisfied. We next check situations where at least one of these inequalities is not satisfied.

\(\left( b\right) \)\(\rho _{n}\left( \theta ^{\prime },\theta \right) >\rho _{1}\left( \theta ^{\prime },\theta \right) \) and \(\rho _{1}\left( \theta ^{\prime },\theta \right) <\rho _{s}\left( \theta ^{\prime },\theta \right) \). Using these inequalities in (26) and rearranging, we obtain (27). Because property (1) implies that the right-hand side of (27) is above one, (27) is satisfied only if \(\rho _{n}\left( \theta ^{\prime },\theta \right) -\rho _{1}\left( \theta ^{\prime },\theta \right)>\rho _{s}\left( \theta ^{\prime },\theta \right) -\rho _{1}\left( \theta ^{\prime },\theta \right) \Leftrightarrow \rho _{n}\left( \theta ^{\prime },\theta \right) >\rho _{s}\left( \theta ^{\prime },\theta \right) \).

\(\left( c\right) \)\(\rho _{n}\left( \theta ^{\prime },\theta \right) <\rho _{1}\left( \theta ^{\prime },\theta \right) \) and \(\rho _{1}\left( \theta ^{\prime },\theta \right) >\rho _{s}\left( \theta ^{\prime },\theta \right) \). Using these inequalities in (26) and rearranging, we obtain the converse of (27). Because property (1) implies that the right-hand side of (27) is above one, the converse of (27) is satisfied if \(\rho _{1}\left( \theta ^{\prime },\theta \right) -\rho _{n}\left( \theta ^{\prime },\theta \right) <\rho _{1}\left( \theta ^{\prime },\theta \right) -\rho _{s}\left( \theta ^{\prime },\theta \right) \Leftrightarrow \rho _{n}\left( \theta ^{\prime },\theta \right) >\rho _{s}\left( \theta ^{\prime },\theta \right) \).

\(\left( d\right) \)\(\rho _{n}\left( \theta ^{\prime },\theta \right) <\rho _{1}\left( \theta ^{\prime },\theta \right) \) and \(\rho _{1}\left( \theta ^{\prime },\theta \right) <\rho _{s}\left( \theta ^{\prime },\theta \right) \). It is immediate to see that (26) is violated.

Taking \((a)-(d)\) altogether, (26) holds only if \(\rho _{s}\left( \theta ^{\prime },\theta \right) <\max \left\{ \rho _{1}\left( \theta ^{\prime },\theta \right) ,\right. \)\(\left. \rho _{n}\left( \theta ^{\prime },\theta \right) \right\} .\) From (a) and (c),  it is sufficient to have either \(\rho _{n}\left( \theta ^{\prime },\theta \right)>\rho _{1}\left( \theta ^{\prime },\theta \right) >\rho _{s}\left( \theta ^{\prime },\theta \right) \) or \(\rho _{1}\left( \theta ^{\prime },\theta \right)>\rho _{n}\left( \theta ^{\prime },\theta \right) >\rho _{s}\left( \theta ^{\prime },\theta \right) \), hence \(\min \left\{ \rho _{1}\left( \theta ^{\prime },\theta \right) ,\rho _{n}\left( \theta ^{\prime },\theta \right) \right\} >\rho _{s}\left( \theta ^{\prime },\theta \right) \). From (b),  it is necessary and sufficient that (27) holds if \(\rho _{n}\left( \theta ^{\prime },\theta \right)>\rho _{s}\left( \theta ^{\prime },\theta \right) >\rho _{1}\left( \theta ^{\prime },\theta \right) .\) From \(\left( c\right) ,\) it is necessary and sufficient that the converse of (27) holds if \(\rho _{1}\left( \theta ^{\prime },\theta \right)>\rho _{s}\left( \theta ^{\prime },\theta \right) >\rho _{n}\left( \theta ^{\prime },\theta \right) \). From \(\left( d\right) ,\) (26) is violated if \(\rho _{n}\left( \theta ^{\prime },\theta \right)<\rho _{1}\left( \theta ^{\prime },\theta \right) <\rho _{s}\left( \theta ^{\prime },\theta \right) \).

Proof of Corollary 4

Replacing \(\pi _{s}\left( \theta \right) =-L\) in (20) and rearranging, (20) is rewritten as (28).

Proof of Corollary 5

(25) implies (24) if and only if

$$\begin{aligned} \frac{\frac{\Delta C\left( \theta ,\theta ^{-}\right) }{\frac{p_{n}(\theta ^{-})}{p_{n}(\theta )}-\frac{p_{1}(\theta ^{-})}{p_{1}(\theta )}}-\frac{-\Delta C\left( \theta ,\theta ^{+}\right) }{\frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}}}{\sum _{s\ne 1,n}p_{s}(\theta )\left( \psi _{s,n}\left( \theta ^{-},\theta \right) -\psi _{s,n}\left( \theta ^{+},\theta \right) \right) }>\frac{\Delta C\left( \theta ,\theta ^{-}\right) p_{1}(\theta )}{p_{1}(\theta )-p_{1}(\theta ^{-})}. \end{aligned}$$

Using the definition of \(\psi _{s,n}\left( \theta ^{\prime },\theta \right) \) and grouping the terms that include \(\Delta C\left( \theta ,\theta ^{-}\right) \), this becomes

$$\begin{aligned}&\Delta C\left( \theta ,\theta ^{-}\right) \left[ \frac{1}{\frac{p_{n}(\theta ^{-})}{p_{n}(\theta )}-\frac{p_{1}(\theta ^{-})}{p_{1}(\theta )}}\right. +\frac{p_{1}(\theta )}{p_{1}(\theta )-p_{1}(\theta ^{-})}\\&\qquad \left( \frac{\frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}\sum _{s\ne 1,n}p_{s}(\theta )-\sum _{s\ne 1,n}p_{s}(\theta ^{+})}{\frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}}\right. \\&\qquad \left. \left. -\frac{\frac{p_{1}(\theta ^{-})}{p_{1}(\theta )}\sum _{s\ne 1,n}p_{s}(\theta )-\sum _{s\ne 1,n}p_{s}(\theta ^{-})}{\frac{p_{1}(\theta ^{-})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{-})}{p_{n}(\theta )}}\right) \right] \\&\quad > \frac{-\Delta C\left( \theta ,\theta ^{+}\right) }{\frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}} \end{aligned}$$

Using \(\sum _{s\ne 1,n}p_{s}(\cdot )=1-p_{1}\left( \cdot \right) -p_{n}\left( \cdot \right) \) and rearranging further yields

$$\begin{aligned} \frac{\Delta C\left( \theta ,\theta ^{-}\right) p_{1}(\theta )}{p_{1}(\theta )-p_{1}(\theta ^{-})}\left( p_{n}(\theta )+\frac{1-p_{n}(\theta )-\frac{p_{1}(\theta )}{p_{1}(\theta ^{+})}\left( 1-p_{n}(\theta ^{+})\right) }{\frac{p_{1}(\theta )}{p_{1}(\theta ^{+})}\left( \frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}\right) }\right) >\frac{-\Delta C\left( \theta ,\theta ^{+}\right) }{\frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}}.\nonumber \\ \end{aligned}$$
(44)

Take the expression in brackets in the left-hand side of (44) and factorize \(p_{n}(\theta )\) to develop

$$\begin{aligned} p_{n}\left( \theta \right) \left( 1+\frac{1-p_{n}(\theta )-\frac{p_{1}(\theta )}{p_{1}(\theta ^{+})}\left( 1-p_{n}(\theta ^{+})\right) }{\frac{p_{1}(\theta )}{p_{1}(\theta ^{+})}\left( p_{n}\left( \theta \right) \frac{p_{1}(\theta ^{+})}{p_{1}(\theta )}-p_{n}(\theta ^{+})\right) }\right) =\frac{p_{1}(\theta ^{+})-p_{1}(\theta )}{p_{1}(\theta ^{+})-p_{1}(\theta )\frac{p_{n}(\theta ^{+})}{p_{n}(\theta )}}. \end{aligned}$$

The converse of (23) is obtained by using this in (44) and rearranging.

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Danau, D., Vinella, A. A note on optimal contracting with public ex post information under limited liability. Int J Game Theory 49, 47–74 (2020). https://doi.org/10.1007/s00182-019-00674-4

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Keywords

  • Informative signals
  • Limited liability
  • Incentive compatibility

JEL Classification

  • D82