Product liability under ambiguity

Abstract

The introduction of new varieties of goods increases welfare under certainty and perfect competition. However, when the quality of new goods is uncertain, the need for a regulatory regime on liabilities and hazards arises. We examine the optimality of the regulatory mechanisms of quality under ambiguity (non-uniqueness of the probability distribution). We develop a model showing that product liability does not lead to optimality under ambiguity and so it constitutes an inadequate instrument for controlling the potential damages caused by innovative products. The level of precaution will be larger, equal or less than the optimal level and will decrease with the degree of optimism and will increase with the degree of pessimism. Consequently the price will not reflect the actual product risk and consumers will buy either an insufficient or an excessive amount, according to the case. We present some considerations on the adequate institutional design, capturing the insights obtained in the comparison between regulatory regimes.

Appendix

Proposition 4

$V_{\pi } \left( x \right)$ is continuous and defined over a compact $\left[ {0, \bar{x}} \right]$ and reaches a maximum in this interval.

The vertices of the graph of $V_{\pi } \left( x \right)$ form a set of isolated points because the functions in $\left( {\bar{\pi } \left( x \right), \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi} \left( x \right)} \right)$ are continuous and decreasing and, because of the single-crossing property, intersect only at a finite number of points. Thus, in a neighborhood of a vertex there will not exist another vertex (by the continuity of the functions). The measure of a set of isolated points in the compact $\left[ {0, \bar{x}} \right]$ is 0, which in our case implies that the probability of the optimum being at a vertex is null. In contrast, the generic case corresponds to the claim that, with probability 1, no vertex is an optimum. Thus, $V\left( x \right)$ is regular and derivable except in a set of measure zero.

If $x^{*} = argmax_{{\left[ {0,\bar{x}} \right]}} V\left( x \right)$ then

$$ \left[ {p - c - x^{*} } \right] - \left[ {\alpha \bar{\pi }\left( {x^{*} } \right) + \left( {1 - \alpha } \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi} \left( {x^{*} } \right)} \right]l \ge \left[ {p - c - x} \right] - \left[ {\alpha \bar{\pi }\left( x \right) + \left( {1 - \alpha } \right)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi} \left( x \right)} \right]l $$

(13)

and therefore,

$$ \left( {x - x^{ *} } \right) + l\left[ {\alpha \left( {\bar{\pi }\left( x \right) - \bar{\pi }\left( {x^{ *} } \right)} \right) + \left( {1 - \alpha } \right)(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi} \left( x \right) - \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi}\left( {x^{ *} } \right)} \right] \ge 0 $$

(14)

If $\Delta x = x - x^{ *}$, $\Delta \bar{\pi } = \bar{\pi } \left( x \right) - \bar{\pi } \left( {x^{ *} } \right)$ and $\Delta \pi = \pi \left( x \right) - \pi \left( {x^{ *} } \right)$ expression (14) can be written as

$$ x + l\left[ {\alpha\Delta \bar{\pi } + \left( {1 - \alpha } \right)\Delta\pi } \right] \ge 0 $$

(15)

Rearranging terms again,

$$ \left[ {\alpha \Delta\bar{\pi } + \left( {1 - \alpha } \right)\Delta\pi } \right] \ge {\raise0.7ex\hbox{${ - \pi x}$} \!\mathord{\left/ {\vphantom {{ -\Delta \pi x} l}}\right.\kern-0pt} \!\lower0.7ex\hbox{$l$}} $$

(16)

$$ \left[ {\alpha {\Delta \bar{\pi }}/{\Delta x}} + \left( {1 - \alpha } \right){\Delta \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi } }/{x\Delta } \right] \ge -1/l $$

(17)

And thus, if we denote $\left[ \alpha {\Delta \bar{\pi }}/\Delta x+ \left( {1 - \alpha } \right){\Delta \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\pi } }/{\Delta x} \right] = {\Delta \tilde{\pi }}/\Delta x$, we have that $\Delta\tilde{\pi } \ge {\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} l}}\right.\kern-0pt} \!\lower0.7ex\hbox{$l$}}\Delta x$ where ${\raise0.7ex\hbox{${ - 1}$} \!\mathord{\left/ {\vphantom {{ - 1} l}}\right.\kern-0pt} \!\lower0.7ex\hbox{$l$}}$ is the first-order approximation to the lower linear support of $\tilde{\pi }$ at $x^{ *}$.