Negligence and two-sided causation

Abstract

We extend the economic analysis of negligence and intervening causation to “two-sided causation” scenarios. In the two-sided causation scenario the effectiveness of the injurer’s care depends on some intervention, and the risk of harm generated by the injurer’s failure to take care depends on some other intervention. We find that the distortion from socially optimal care is more severe in the two-sided causation scenario than in the one-sided causation scenario, and generally in the direction of excessive care. The practical lesson is that the likelihood that injurers will have optimal care incentives under the negligence test in the presence of intervening causal factors is low.

Appendix

Definition 1

The random variable X is said to have a Beta type I distribution with parameters $\left( {a,b} \right), \, a > 0,b > 0$ denoted as $X\sim B^{I} \left( {a,b} \right),$ if its p.d.f. is given by

$$\left\{ {B\left( {a,b} \right)} \right\}^{ - 1} x^{a - 1} \left( {1 - x} \right)^{b - 1} , \, 0 < x < 1,$$

where, $B\left( {a,b} \right)$ is the Beta function given by

$$B\left( {a,b} \right) =\Gamma \left( a \right)\Gamma \left( b \right)\left\{ {\Gamma \left( {a + b} \right)} \right\}^{ - 1}$$

and the gamma function $\Gamma \left( n \right) = \left( {n - 1} \right)!.$

Definition 2

The random variable X is said to have a hypergeometric function type I distribution, denoted by $X\sim H^{I} \left( {\nu ,\alpha ,\beta ,\gamma } \right),$ if its p.d.f. is given by

$$\frac{{\Gamma \left( {\gamma + \nu - \alpha } \right)\Gamma \left( {\gamma + \nu - \beta } \right)}}{{\Gamma \left( \gamma \right)\Gamma \left( \nu \right)\Gamma \left( {\gamma + \nu - \alpha - \beta } \right)}}x^{\nu - 1} \left( {1 - x} \right)^{\gamma - 1} {}_{2}F_{1} \left( {\alpha ,\beta ;\gamma ;1 - x} \right), \, 0 < x < 1,$$

where, ${}_{2}F_{1} \left( {a,b;c;z} \right) = 1 + \frac{ab}{1!c}z + \frac{{a\left( {a + 1} \right)b\left( {b + 1} \right)}}{{2!c\left( {c + 1} \right)}}z^{2} + \cdots = \sum\limits_{n = 0}^{\infty } {\frac{{\left( a \right)_{n} \left( b \right)_{n} }}{{\left( c \right)_{n} }}\frac{{z^{n} }}{n!},}$

$$\gamma + \nu - \alpha - \beta > 0{\text{ and }}\nu > 0.$$

Definition 3

Let $X_{1}$ and $X_{2}$ be independent, $X_{i} \sim B^{I} \left( {a_{i} ,b_{i} } \right), \, i = 1,2.$ Then, $X_{1} X_{2} \sim H^{I} \left( {a_{1} ,b_{2} ,a_{1} + b_{1} - a_{2} ,b_{1} + b_{2} } \right).$

See Zarrazola and Nagar (2009).

• 1. Single Signal Case (Beta Distribution)

$$\begin{aligned} E\left( s \right) & = \frac{a}{a + b} \\ G\left( {E\left( s \right)} \right) & = \int_{0}^{E\left( s \right)} {\frac{1}{{B\left( {a,b} \right)}}x^{a - 1} \left( {1 - x} \right)^{b - 1} dx} \\ \end{aligned}$$

To compute the value, we used “betacdf” in Matlab.

$$D = \left( {\frac{r - w}{r}} \right) - \frac{{\left( {1 - G\left( {E\left( s \right)} \right)} \right)}}{E\left( s \right)}$$

where $r,w$ are given.

• 2. Two Signal Case (Beta Distribution)

  $$E\left( s \right) = \frac{a}{a + b}, \, E\left( q \right) = \frac{c}{c + d}$$

Based on the Definition 3 above, the product of independent Beta variables follows $X_{1} X_{2} \sim H^{I} \left( {a,d,a + b - c,b + d} \right).$

$$H\left( {E\left( s \right)E\left( q \right)} \right) = \int_{0}^{E\left( s \right)E\left( q \right)} {\frac{{\Gamma \left( {b + a} \right)\Gamma \left( {d + c} \right)}}{{\Gamma \left( {b + d} \right)\Gamma \left( a \right)\Gamma \left( c \right)}}x^{\nu - 1} \left( {1 - x} \right)^{\gamma - 1} {}_{2}F_{1} \left( {d,a + b - c;a;1 - x} \right)dx}$$

To compute the values for the simulation, we used “int” in Matlab and assuming n = 3 in ${}_{2}F_{1} \left( {a,b;c;1 - x} \right).$