Interpreting contracts: the purposive approach and non-comprehensive incentive contracts

Abstract

Real world contracts often contain incentive clauses that fail to fully specify conditions triggering payments, giving rise to legal disputes. When complete contract generate Pareto efficient allocations the L&E literature advocates that courts should fill in the missing clauses. This logic does not directly extend to environments with moral hazard, where complete contracts result in constrained efficient allocations. Despite this inefficiency we find that when agency and marginal agency costs are congruent, the legal system can do no better than guide its courts to complete contracts according to the parties’ intentions.

Appendix

Proof of Proposition 1

Observe that due to the curvature assumption on $c(\cdot )$ and the CDFC requirement, the agent’s objective function is concave. Hence, the (IC) condition can be substituted for the first-order condition of the agent’s optimization problem. Initially ignoring the (PC) requirement and converting (I) into a maximization yields the Lagrangian:

$$\begin{aligned} {\mathcal {L}}= & {} -w_{\emptyset }(1-m)-m\int \limits _{0}^{{\overline{x}} }w(x)g(x;a)dx+\lambda \left( m\int \limits _{0}^{{\overline{x}} }w(x)g_{a}(x;a)dx-c^{\prime }(a)\right) \nonumber \\&+\int \limits _{0}^{{\overline{x}}}\xi (x)w(x)dx+\int \limits _{0}^{{\overline{x}}}\zeta (x)\left( w_{\emptyset }-w(x)\right) dx \end{aligned}$$

(15)

Taking the derivative with respect to $w_{\emptyset }$ and w(x), we obtain:

$$\begin{aligned} \left\{ \begin{array}{l} -(1-m)+\int \limits _{0}^{{\overline{x}}}\zeta (x)w_{\emptyset }dx = 0 \\ -mg(x;a)+\lambda mg_{a}(x;a)+\xi (x)-\zeta (x) = 0 \end{array} \right. \end{aligned}$$

(16)

which together with the complementary slackness conditions $\xi (x)w(x)=0,\ \zeta (x)\left( w_{\emptyset }-w(x)\right) =0$, the non-negativity requirements $\xi (x),\ \zeta (x)\ge 0$ as well as the constraints (AFC) and (PFC) for all $x\in [ 0;1]$ implicitly define the solution.

Claim 1 For any $a>0$ we have $\lambda \ne 0$.

Proof

Suppose to the contrary. Then the second equation in (16) implies that for all $x\in [ 0;1]$

$$\begin{aligned} mg(x;a)+\zeta (x)=\xi (x)>0 \end{aligned}$$

(17)

since $m>0$ and$\ g(x;a)>0$ over the support. Accordingly, by complementary slackness $w(x)=0$ over the support which violates the first-order condition of the agent’s (IC) for any $a>0$. $\square$

Claim 2 There can exist at most one point in [0, 1] with $\zeta (x)=\xi (x)=0$.

Proof

Suppose $\zeta (x)=\xi (x)=0$, then the second equation in (16) implies

$$\begin{aligned} -m+\lambda m\frac{g_{a}(x;a)}{g(x;a)}=0 \end{aligned}$$

(18)

verifying the claim by strict MLRP and $\lambda \ne 0$, $m>0$. $\square$

Since this potential one point is of measure zero, it is irrelevant to the optimization and will henceforth be ignored (together with the possibility $\zeta (x)=\xi (x)=0$).

Claim 3 There is no x in the support for which $\zeta (x),\ \xi (x)>0$.

Proof

Suppose to the contrary that at $x_{1}\in (0,{\overline{x}})$ we have $\zeta (x_{1}),\ \xi (x_{1})>0$. Then, by complementary slackness $w(x_{1})=w_{\emptyset }-w(x_{1})=0$. As a result, (PFC) implies that for all x, we have $0\le w(x)\le w_{\emptyset }=0$. Hence $w(x)=0$ for all $x\in (0,{\overline{x}})$ which violates the first-order condition of the agent’s (IC) for any $a>0$. $\square$

Taking $x\in (0,{\overline{x}})$, there are two possible cases remaining; either $\zeta (x)=0,\ \xi (x)>0$ or $\zeta (x)>0,\ \xi (x)=0$. Clearly each of these cases must occur over a subset of the support with positive measure. Suppose to the contrary that $\xi (x)>0$ almost everywhere (a.e. hereafter). This would imply $w(x)=0$ a.e. Similarly $\zeta (x)>0$ a.e. would yield $w(x)=w_{\emptyset }$ a.e. In either situation, setting positive incentives is not possible, leading to a contradiction.

To conclude the proof, observe that $\lambda$ must be positive. Suppose it is negative, then the contract would pay $w_{\emptyset }>0$ for small realization of x and 0 for large realization of x whereby the critical $x^{c}$ would solve $-1+\lambda \frac{g_{a}(x;a)}{g(x;a)}=0$. But then setting incentives is not feasible.

Altogether, we now know that the contract which solves the simplified principal’s problem where (PC) has been ignored is a bonus scheme paying B if $x\ge z$ for some critical value which partitions the support and pays 0 otherwise. Accordingly, we can rewrite the simplified problem as:

$$\begin{aligned} C(a)=\min _{B,z}\ [1-mG(z;a)]B\quad \text {s.t. }-mG_{a}(z;a)B-c^{ \prime }(a)=0 \end{aligned}$$

(IV)

Substituting B, we define $C^{P}(a;z)=\frac{1-mG(z;a)}{-mG_{a}(z;a)} c^{\prime }(a)$.⁴⁹ Observe that $C^{P}(0,z)=0$ by $c^{\prime }(0)=0$. Moreover, we have

$$\begin{aligned} \frac{\partial C^{P}}{\partial a}(a,z)=c^{\prime }(a)+\left[ \frac{ 1-mG\left( z;a\right) }{-mG_{a}\left( z;a\right) }\right] \left( c^{\prime \prime }(a)-\frac{G_{aa}\left( z;a\right) }{G_{a}\left( z;a\right) } c^{\prime }(a)\right) \ . \end{aligned}$$

(19)

Strict MLRP, CDFC and the convexity of $c\left( \cdot \right)$ imply $\frac{ \partial C^{P}}{\partial a}(a,z)>c^{\prime }(a)$ for $a>0$. Finally, note that by the envelope theorem, $C^{\prime }(a)=\frac{\partial C^{P}}{\partial a}(a,x^{c})$. Hence, the simplified problem also satisfies the (PC) requirement implying that its solution is identical to that of problem (I) thereby verifying the claim of Proposition 1.$\square$

Proof of Proposition 2

The agent’s rent is given by the difference

$$\begin{aligned} R(a,z)=C^{P}(a,z)-c\left( a\right) . \end{aligned}$$

(20)

From the proof of the foregoing proposition, we have $R(0,z)=0$, $\frac{ \partial R}{\partial a}(a,z)>0$ and $R(a,z)>0$ for all $a>0$, thus, verifying the claim of Proposition 2.$\square$

Lemma 7

The set ${\mathcal {S}}(a)$ defined by (8) is an interval endogenously defined by a critical value $z^{BoP}\left( a\right)$, i.e. ${\mathcal {S}}(a)=[z^{BoP}\left( a\right) ,{\overline{x}}]$.

Proof of Lemma 7

Suppose $x\in {\mathcal {S}} (a)$ i.e.

$$\begin{aligned} \forall a^{\prime }\le a,\quad \frac{g\left( x;a\right) }{g\left( x;a^{\prime }\right) }\ge 1 . \end{aligned}$$

(21)

Moreover, MLRP states

$$\begin{aligned} \frac{d}{dx}\left[ \frac{g\left( x;a\right) }{g\left( x;a^{\prime }\right) } \right] >0\ . \end{aligned}$$

(22)

Hence for any $y>x$ we have:

$$\begin{aligned} \frac{g\left( y;a\right) }{g\left( y;a^{\prime }\right) }>\frac{g\left( x;a\right) }{g\left( x;a^{\prime }\right) } \end{aligned}$$

(23)

so that ${\mathcal {S}}(a)$ is an interval verifying the Lemma 7. $\square$

Proof of Proposition 6

First, observe that the equation system (4) which defines $(a^{C},x^{C})$ under comprehensive contracting can be rewritten as:

$$\begin{aligned} \left\{ \begin{array}{l} v^{\prime }\left( a\right) -c^{\prime }(a)-\left[ \dfrac{1-mG\left( z;a\right) }{-mG_{a}\left( z;a\right) }\right] \left( c^{\prime \prime }(a)- \dfrac{G_{aa}\left( z;a\right) }{G_{a}\left( z;a\right) }c^{\prime }(a)\right) = 0 \\ -\dfrac{\partial }{\partial z}\left[ \dfrac{1-mG\left( z;a\right) }{ -mG_{a}\left( z;a\right) }\right] c^{\prime }(a)= 0 \end{array} \right. \end{aligned}$$

(24)

In the case where the regulator sets z, the pair $\left( a^{R},z^{R}\right)$ which maximizes the regulator’s objective is implicitly defined as the solution to the $2\times 2$ system

$$\begin{aligned} \left\{ \begin{array}{l} v^{\prime }\left( a\right) -c^{\prime }(a)-\left[ \dfrac{1-mG\left( z;a\right) }{-mG_{a}\left( z;a\right) }\right] \left( c^{\prime \prime }(a)- \dfrac{G_{aa}\left( z;a\right) }{G_{a}\left( z;a\right) }c^{\prime }(a)\right) = 0 \\ \dfrac{\partial }{\partial z}\left( \left[ \dfrac{1-mG\left( z;a\right) }{ -mG_{a}\left( z;a\right) }\right] \left( c^{\prime \prime }(a)-\dfrac{ G_{aa}\left( z;a\right) }{G_{a}\left( z;a\right) }c^{\prime }(a)\right) \right) = 0 \end{array} \right. \end{aligned}$$

(25)

where the second equation follows by applying the implicit function theorem with respect to z on the first-order condition of (III). Intuitively, the regulator selects $z^{R}$ to minimize the principal’s marginal costs. The requirement (10) ensures that the systems (24) and (25) yield the same solution verifying the claim of Proposition 6. $\square$

The LiCalzi and Spaeter distributions. For the sake of completeness, we briefly reproduce from LiCalzi and Spaeter (2003) the conditions characterizing the two distribution families satisfying MLRP and CDFC used in the Sect. 5.2. For the first family described by the generic form (11), the functions $\beta \left( \cdot \right)$ and $\gamma \left( \cdot \right)$ must satisfy:

• 1. $\beta (x)$ is a positive and concave function on the support $x\in \left[ 0,1\right]$ such that $\lim _{x\downarrow 0}\ \beta (x)=\lim _{x\uparrow 1}\ \beta (x)=0$ and $\left| \beta ^{\prime }(x)\right| \le 1$ for all $x\in \left( 0,1\right)$;

• 2. $\gamma (a)$ is a decreasing and convex function for all $a\ge 0$ such that $\left| \gamma (a)\right| <1$.

Moreover, for the second family given by (13), the functions $\delta \left( \cdot \right) ,\ \beta (\cdot )$ and $\gamma (\cdot )$ are required to satisfy:

• 1. $\beta (x)$ is a non-constant, negative, increasing, and convex function on the support $x\in \left[ 0,1\right]$ such that $\lim _{x\uparrow 1}\ \beta (x)=0$;

• 2. $\gamma (a)$ is a strictly positive, increasing, and concave function for all $a\ge 0$;

• 3. $\delta (x)$ is a positive, strictly increasing, and concave function on the support $x\in \left[ 0,1\right]$ such that $\lim _{x\downarrow 0}\ \delta (x)=0$ and $\lim _{x\uparrow 1}\ \delta (x)=1$.

Note that these are not the only distribution families for which MLRP and CDFC hold. For instance, the family of distributions $G\left( x;a\right) = \left[ F(x)\right] ^{\gamma \left( a\right) }$ where $F\left( \cdot \right)$ is a CDF defined over $x\in \left[ 0,1\right]$ and $\gamma (\cdot )$ a strictly increasing and concave function also has the desired properties. Moreover, in line with the observation of footnote 46 that class does not satisfy (10).