Tacit collusion and liability rules

Abstract

This paper demonstrates that the likelihood of tacit collusion in a given oligopolistic industry may depend on the kind of liability rule applied to the industry. We study typical settings for the analysis of product liability and environmental liability. For the latter, it is established that tacit collusion is more likely under strict liability than under negligence. However, the two liability rules are equivalent with regard to their effects on tacit collusion in the model pertaining to product liability. This context-dependent impact on tacit collusion can be traced back to a difference in the shape of firms’ cost functions.

Appendix

In the main text, we analyze the case in which firms compete in quantities. As a check for robustness, we now briefly consider the scenario in which firms compete in prices (i.e., Bertrand competition). The individual demand for firm i is given by

$$ D_{i}=\left\{ \begin{array}{ll} D(p_{i}) &\hbox{if}\; p_{i} <\; \hbox{inf}(P_{-i}) \\ D(p_{i})/m &\hbox{if}\; p_{i}=\hbox{inf}(P_{-i}) \\ 0 &\hbox{if}\; p_{i}>\hbox{inf}(P_{-i}). \\ \end{array} \right.$$

(43)

where P _−i is the set of prices set by the other firms, m ≤ n is the number of firms setting the lowest price in the second line, and D(p) = a/b − p/b follows from the inverted demand function used in the main text.

Product liability:

Firms have the same constant marginal costs per unit of output, the level of which depends on the liability rule applied to the industry. Firms do not incur fixed costs and do not face capacity constraints. Denoting with π ^M _BP the monopoly profits in the Bertrand product liability game, the central condition defining the critical discount factor can be stated as

$$ \frac{\pi_{BP}^{M}}{n(1-\bar{\delta}_{BP})}=\pi_{BP}^{M}. $$

(44)

Loyalty to the cartel agreement implies that monopoly profits are shared equally between the firms, whereas the one-time deviation profit would imply that the deviating firm obtains all the monopoly profits for itself. The punishment phase in this context implies profits of zero since equilibrium prices are equal to constant marginal costs. The statement of Eq. (44) clearly shows that the discount factor is not a function of the level of marginal costs (i.e., it is not dependent on whether the industry is subject to strict liability or negligence).

Environmental liability:

In our environmental liability framework, symmetric firms have strictly convex costs, where levels of both costs and marginal costs are higher when firms are subject to strict liability for a given level of firm output. Starting from the general definition of individual profits,

$$ \bar{\pi}_{i}=D_{i} p_{i}-\kappa D_{i}^{2} $$

(45)

we derive that the price level that maximizes joint profits is subject to the condition that all firms actually produce in equilibrium as

$$ p_{BE}^{K}=\frac{a(bn+2 \kappa)}{2(bn+\kappa)}, $$

(46)

which in turn implies

$$ \pi^{K}_{BE}=\frac{a^{2}}{4\kappa+4bn}. $$

(47)

When firm i deviates and marginally undercuts p ^K _BE ,¹¹ we obtain

$$ \pi^{D}_{BE}=\frac{a^{2}n((b-\kappa)n+2\kappa)}{4(\kappa+bn)^2}. $$

(48)

The period profit from deviating always exceeds the firm profit from being in the cartel when b > κ. In this case, the critical discount factor follows from

$$ \frac{\pi_{BE}^{K}}{1-\bar{\delta}_{BE}}=\pi_{BE}^{D}, $$

(49)

when we assume that the equilibrium in the case of firm competition in prices implies zero firm profits. Weibull (2006) analyzes the case in which firms with convex costs compete in prices, establishing that there generally are different possible equilibria, including the equilibrium implying zero profits. Most important for the present study is that the critical discount factor $\bar{\delta}_{BE}=(\pi_{BE}^{D}-\pi_{BE}^{K})/\pi_{BE}^{D}$ decreases as the level of κ increases, since

$$ \frac{d \bar{\delta}_{BE}}{d \kappa}=\frac{b-bn}{((b-\kappa)n+2\kappa)^{2}}<0 $$

(50)

Consequently, the likelihood of tacit collusion is higher when firms are subject to strict liability (as was true in our model of Cournot competition).