No Judge, No Job! Court errors and the contingent labor contract

Abstract

Judges are prone to error and misapprehension when they are verifying the facts of a legal case. We analyze the significance and scope of accurate court decisions and judicial error for labor contracting and identify the implications of these concepts on behavioral incentives and market outcomes. We find that imperfect judicial state verification and the diverging beliefs on a court ruling reduce the efficiency of contingent labor contracts and make them less effective in stipulating sufficient incentives for compliance. If increasing court accuracy in general is not feasible, the courts (and the legislator) should primarily mitigate type I errors. The common reversal of the burden of proof to the employer in labor laws reflects these insights. The model also indicates that the ability of judges to verify facts is a prerequisite for efficient law-making and contributes significantly to the economic role of courts.

Appendix

1.1 Risk averse agent and limited liability

Assume a monotonous Von-Neumann Morgenstern-utility function U. Beginning at Stage 4, the expectations of the agent in courtroom yield ${\text{E}}[{{\uppi}}_{\text{A}} ] = (1 - {\text{e}}_{\text{I}} ) \cdot {\text{U}}[{\text{W}}({\text{q}})] + {\text{e}}_{\text{I}} \cdot {\text{U}}[{\text{W}}({\bar{\text{q}}})] - {\text{U}}[{\text{L}}^{\text{A}} ]$. Given the monotony of the utility function, the incentive compatibility constraint remains unchanged. Then the principal maximizes his payoff as follows: ${{\uppi}}_{\text{P}} ({\text{q}}) = {\text{V}}({\text{q}}) - {\text{W}}({\text{q}}) - {\text{t}}\mathop \Rightarrow \limits_{\text{q}} \hbox{max} !$, given ${\text{W}}({\text{q}}) - {\text{W}}({\bar{\text{q}}}) \ge \frac{{{\text{C}}({\text{q}}) - {\text{C}}({\bar{\text{q}}})}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }}$  and$(1 - {\text{e}}{}_{\text{I}}) \cdot {\text{U}}[{\text{W}}({\text{q}})] + {\text{e}}_{\text{I}} \cdot {\text{U}}[{\text{W}}({\bar{\text{q}}})] - {\text{U}}[{\text{L}}^{\text{A}} ] \equiv {\text{U}}[{\text{C}}({\text{q}})] + {\text{U}}[{\text{t}}]$.

Assuming infinite risk aversion of the agent, the latter can be simplified to ${\text{W}}({\bar{\text{q}}}) - {\text{L}}^{\text{A}} = {\text{C}}({\text{q}}) + {\text{t}}$. This also holds in case of limited liability of the agent, which requires W(q) ≥ 0. The First-Order Condition follows as $\frac{{{\text{V}}^{\prime } ({\text{q}})}}{{{\text{C}}^{\prime } ({\text{q}})}} = \frac{{2 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }}$ and the wage function W*(q) is ${\text{W}}^{*} ({\text{q}}) = \left\{ {\begin{array}{*{20}c} {\frac{{{\text{C}}({\text{q}})}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} + {\text{C}}\left( {{\text{q}}^{*} } \right) + {\text{L}}^{\text{A}} + {\text{t}},\quad {\text{if}}\;{\text{q}} \le {\text{q}}^{*} } \\ {\frac{{(2 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} ) \cdot {\text{C}}\left( {{\text{q}}^{*} } \right)}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} + {\text{L}}^{\text{A}} + {\text{t}},\quad {\text{if}}\;{\text{q}} > {\text{q}}^{*} } \\ \end{array} \,\,\,\,\,\,\,\,\,} \right.$.

1.2 Succumbing party bears litigation cost

Consider the costs of enforcement L(q) contingent on quality to satisfy L′(q) < 0 and L(q*) = 0. Consequently, the successful party does not bear any trial costs. Beginning at stage 4, the expectations of the agent in courtroom yield ${\text{E}}[\pi_{\text{A}} ] = (1 - {\text{e}}_{\text{I}} ){\text{W}}({\text{q}}) + {\text{e}}_{\text{I}} {\text{W}}({\bar{\text{q}}}) - {\text{e}}_{\text{I}} {\text{L}}^{\text{A}} ({\bar{\text{q}}})$. The principal maximizes his payoff $\pi_{\text{P}} ({\text{q}}) = {\text{V}}({\text{q}}) - {\text{W}}({\text{q}}) - {\text{t}}\mathop \Rightarrow \limits_{\text{q}} \hbox{max} !$, given ${\text{W}}({\text{q}}) - {\text{W}}({\bar{\text{q}}}) \ge \frac{{{\text{C}}({\text{q}}) - {\text{C}}({\bar{\text{q}}})}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} - {\text{L}}^{\text{A}} ({\bar{\text{q}}})$ and $(1 - {\text{e}}_{\text{I}} ){\text{W}}({\text{q}}) + {\text{e}}_{\text{I}} {\text{W}}({\bar{\text{q}}}) - {\text{e}}_{\text{I}} {\text{L}}^{\text{A}} ({\bar{\text{q}}}) \ge {\text{C}}({\text{q}}) + {\text{t}}$. This determines the First-Order Condition as $\frac{{{\text{V}}^{\prime } ({\text{q}})}}{{{\text{C}}^{\prime } ({\text{q}})}} = \frac{{1 - {\text{e}}_{\text{II}} }}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }}$ and the profit maximizing wage function then is ${\text{W}}^{*} ({\text{q}}) = \left\{ {\begin{array}{*{20}c} {\frac{{{\text{C}}({\text{q}})}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} - \frac{{{\text{e}}_{\text{II}} {\text{C}}\left( {{\text{q}}^{*} } \right)}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} + {\text{L}}^{\text{A}} ({\text{q}}) + {\text{t}},\quad {\text{if}}\;0 \le {\text{q}} \le {\text{q}}^{*} } \\ {\frac{{(1 - {\text{e}}_{\text{II}} ) \cdot {\text{C}}\left( {{\text{q}}^{*} } \right)}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} + {\text{t}},\quad {\text{if}}\;{\text{q}} > {\text{q}}^{*} } \\ \end{array} \,\,\,\,\,\,\,\,\,} \right.$

1.3 Diverging beliefs and litigation

As the principal believes the probabilities e ^P_I and e ^P_II to apply, the First-Order Condition (4) changes to $\frac{{{\text{V}}^{\prime } ({\text{q}})}}{{{\text{C}}^{\prime } ({\text{q}})}} = \frac{{1 - {\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}$ and he will offer the wage function ${\text{W}}^{*} ({\text{q}}) = \left\{ {\begin{array}{*{20}c} {\frac{{{\text{C}}({\text{q}})}}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }} - \frac{{{\text{e}}_{\text{II}}^{\text{P}} {\text{C}}\left( {{\text{q}}^{*} } \right)}}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }} + {\text{L}}^{\text{A}} + {\text{t}},\quad {\text{if}}\;0 \le {\text{q}} \le {\text{q}}^{*} } \\ {\frac{{\left( {1 - {\text{e}}_{\text{II}}^{\text{P}} } \right) \cdot {\text{C}}\left( {{\text{q}}^{*} } \right)}}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }} + {\text{L}}^{\text{A}} + {\text{t}},\quad {\text{if}}\;{\text{q}} > {\text{q}}^{*} } \\ \end{array} \,\,\,\,\,\,\,\,\,} \right.$.

Now, the participation constraint for the agent (3), given ${\bar{\text{q}}} = 0$, yields $\left( {1 - {\text{e}}_{\text{I}}^{\text{A}} } \right)\frac{{1 - {\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}{\text{C}}\left( {{\text{q}}^{*} } \right) - {\text{e}}_{\text{I}}^{\text{A}} \frac{{{\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}{\text{C}}\left( {{\text{q}}^{*} } \right) \ge {\text{C}}\left( {{\text{q}}^{*} } \right)$. This constraint is fulfilled, when the expectations about type I errors satisfy e ^P_I  ≥ e ^A_I . Also, the agent considers the incentive compatibility constraint (2), which changes to $\left( {1 - {\text{e}}_{\text{I}}^{\text{A}} } \right)\frac{{1 - {\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}{\text{C}}\left( {{\text{q}}^{*} } \right) - {\text{e}}_{\text{I}}^{\text{A}} \frac{{{\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}{\text{C}}\left( {{\text{q}}^{*} } \right) - {\text{C}}\left( {{\text{q}}^{*} } \right) \ge {\text{e}}_{\text{II}}^{\text{A}} \frac{{1 - {\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}{\text{C}}\left( {{\text{q}}^{*} } \right) - \left( {1 - {\text{e}}_{\text{II}}^{\text{A}} } \right)\frac{{{\text{e}}_{\text{II}}^{\text{P}} }}{{1 - {\text{e}}_{\text{I}}^{\text{P}} - {\text{e}}_{\text{II}}^{\text{P}} }}{\text{C}}\left( {{\text{q}}^{*} } \right)$This condition is fulfilled, when the expectations satisfy ${\text{e}}_{\text{II}}^{\text{P}} \ge {\text{e}}_{\text{II}}^{\text{A}} + {\text{e}}_{\text{I}}^{\text{A}} - {\text{e}}_{\text{I}}^{\text{P}}$.

1.4 Side conditions are binding in the optimum

Transform the participation constraint $(1 - {\text{e}}_{\text{I}} ){\text{W}}({\text{q}}) + {\text{e}}_{\text{I}} {\text{W}}({\bar{\text{q}}}) - {\text{L}}^{\text{A}} \ge {\text{C}}({\text{q}}) + {\text{t}}$ to the following binding equation $(1 - {\text{e}}_{\text{I}} ){\text{W}}({\text{q}}) + {\text{e}}_{\text{I}} {\text{W}}({\bar{\text{q}}}) - {\text{L}}^{\text{A}} = {\text{C}}({\text{q}}) + {\text{t}} + {\text{x}}$ with x ≥ 0 and solved to ${\text{W}}({\bar{\text{q}}})$: ${\text{W}}({\bar{\text{q}}}) = \frac{{{\text{C}}({\text{q}}) + {\text{t}} - (1 - {\text{e}}_{\text{I}} ){\text{W}}({\text{q}}) + {\text{L}}^{\text{A}} + {\text{x}}}}{{{\text{e}}_{\text{I}} }}$. Inserted into the incentive compatibility constraint (2), the side conditions require the wage payment to satisfy ${\text{W}}({\text{q}}) \ge \frac{{(1 - {\text{e}}_{\text{II}} ){\text{C}}({\text{q}}) - {\text{e}}_{\text{I}} {\text{C}}({\bar{\text{q}}})}}{{1 - {\text{e}}_{\text{I}} - {\text{e}}_{\text{II}} }} + {\text{L}}^{\text{A}} + {\text{t}} + {\text{x}}$. As the principal seeks to maximize his profits, he will stipulate the lowest wage payment that supports the optimal quality q*. Thus, x = 0 and the side conditions have to be binding in the optimum.