Liability, morality, and image concerns in product accidents with third parties

Abstract

This paper explores how consumers’ moral and image concerns influence the equilibrium in a product-accident model in which third parties incur harm. We differentiate results according to whether the product is supplied by a monopolistic firm or competitive ones. Assuming incomplete compensation of third parties, we find that both moral and image concerns of consumers increase product safety in the context of a competitive market, while the monopolist’s product safety level varies exclusively with consumers’ morality. Comparing market outcomes, we find that the monopolist’s product safety levels may induce greater welfare than a competitive industry.

Appendix

1.1 1.1 No voluntary compensation under Assumption 1

We will rule out the possibility of voluntary compensation through parameter restrictions on $\theta _i$ and $\beta$. The proof follows Deffains and Fluet (2013). Consider that harm has already occurred and that the consumer may choose between voluntary compensation and no compensation of the injured party, where the choice is perfectly observed by the public. Assume that a liability system as described in Sect. 2 applies, with $\gamma =0$, i.e. strict consumer liability. The utility in case of no compensation payment is $U_{NC}=v-(1-\tau )h\theta _i+\tau h+\beta \theta _{NC}-P_i$, with $\theta _{NC}$ denoting the belief about consumers’ type when no compensation is payed. Consumer’s utility in case of compensation amounts to $U_{C}=v-h+\beta \theta _{C}-P_i$, with subscript $\theta _C$ denoting public’s belief about consumers’ type when compensation is payed. To rule out voluntary compensation, it has to hold that $U_{NC}\ge U_{C}$. Simplifying yields $(1-\tau )h(1-\theta _i)\ge \beta [\theta _{C}-\theta _{NC}]$. The highest (lowest) possible belief the population may hold about the moral type is $\theta _H$ ($\theta _L$). Thus, $[\theta _{C}-\theta _{NC}]\le \theta _H-\theta _L$. It follows that consumers do not voluntarily compensate when $\theta _H\le \bar{\theta}_H$.

1.2 1.2 Proof of Proposition 2

We prove the second claim, namely that the competitive firm will underprovide safety compared to the socially efficient level. Comparing conditions (4) and (5), we find that the competitive firm overprovides safety if $\beta \Delta >(1-\tau )h$, which cannot hold under Assumption 1. From Appendix 1.1, we know that $(1-\tau )h(1-\theta _H)\ge \beta [\theta _H-\theta _L]$. Because $\Delta \le \theta _H-\theta _L$, it follows that $\beta \Delta <(1-\tau )h$.

1.3 1.3 Comparative statics

We first show the effects of a larger $\beta$ and $\tau$ for the competitive market. Define the condition for the optimal safety levels in the competitive market, condition (5), by:

$$\begin{aligned} X_i=-\pi '(x_i) \left( (\tau +(1-\tau )\theta _i)h+\beta \Delta \right) =k, \end{aligned}$$

with $i=L,H$. Using the implicit function theorem gives the effect of an increase in $\beta$ and $\tau$ on the respective safety level in the competitive market, with $i,j=L,H$ and $i\ne j$:

$$\begin{aligned} \frac{d x_i^C}{d\beta }&=-\frac{\frac{\partial X_i}{\partial \beta }\frac{\partial X_j}{\partial x_j}-\frac{\partial X_j}{\partial \beta }\frac{\partial X_i}{\partial x_j}}{\frac{\partial X_L}{\partial x_L}\frac{\partial X_H}{\partial x_H}-\frac{\partial X_H}{\partial x_L}\frac{\partial X_L}{\partial x_H}}, \end{aligned}$$

(10)

$$\begin{aligned} \frac{d x_i^C}{d\tau }&=-\frac{\frac{\partial X_i}{\partial \tau }\frac{\partial X_j}{\partial x_j}-\frac{\partial X_j}{\partial \tau }\frac{\partial X_i}{\partial x_j}}{\frac{\partial X_L}{\partial x_L}\frac{\partial X_H}{\partial x_H}-\frac{\partial X_H}{\partial x_L}\frac{\partial X_L}{\partial x_H}}. \end{aligned}$$

(11)

Simplifying conditions (10) and (11) yields:

$$\begin{aligned} \frac{d x_i^C}{d\beta }=\frac{-\Delta \pi '(x_i) \pi ''(x_j) \left( \tau h+(1-\tau )\theta _j h+ \beta \Delta \right) }{\Psi } \end{aligned}$$

and

$$\begin{aligned} \frac{d x_i^C}{d\tau }=\frac{-h \pi '(x_i) \left( (1-\theta _i) \pi ''(x_j) \left( \tau h+(1-\tau )\theta _j h+ \beta \Delta \right) +(2\ell -1)\beta (\theta _H-\theta _L) \pi '(x_j) \frac{\partial \Delta }{\partial x_j}\right) }{\Psi }, \end{aligned}$$

with $\ell =0$ for $i=H$ and $\ell =1$ for $i=L$ and $\Psi =\frac{\partial X_L}{\partial x_L}\frac{\partial X_H}{\partial x_H}-\frac{\partial X_H}{\partial x_L}\frac{\partial X_L}{\partial x_H}$. It is easy to see that the numerator of $\frac{dx_i^C}{d\beta }$ is positive, whereas it is unclear for $\frac{dx_i^C}{d\tau }$ as $\frac{\partial \Delta }{\partial x_L}<0$ and $\frac{\partial \Delta }{\partial x_H}>0$. Next we show that $\Psi >0$ in the equilibrium.

In general the sign of $\Psi$ is unclear, because $\frac{\partial X_H}{\partial x_H}$ is unclear. However, we can show that for an equilibrium to exist, it has to hold that $\Psi >0$. Consider the ’best response safety level’ of the type-i product on the safety level of the type-j product. To derive the slope of the best response safety level, note that in the equilibrium we obtain:

$$\begin{aligned} \frac{dx_i}{dx_j}=-\frac{\frac{\partial X_i}{\partial x_j}}{\frac{\partial X_i}{\partial x_i}}. \end{aligned}$$

It holds that $\frac{dx_L}{dx_H}>0$ and $\frac{dx_H}{dx_L}>(<)0$ if $\frac{\partial X_H}{\partial x_H}>(<)0$. Figure 3 depicts the best response functions of both product types. For an equilibrium to exist, it has to hold that $|\frac{dx_L}{dx_H}|>|\frac{dx_H}{dx_L}|$ which yields: $\frac{\partial X_L}{\partial x_L}\frac{\partial X_H}{\partial x_H}-\frac{\partial X_H}{\partial x_L}\frac{\partial X_L}{\partial x_H}>0$. Thus, it follows that $\Psi >0$ in the equilibrium.

Fig. 3

Illustration of best response functions. Notes: The left- (right-)hand side depicts the best response function of the H type for $\frac{\partial X_H}{\partial x_H}>(<)0$

For the monopolistic market, the optimality condition for the respective safety levels are:

$$\begin{aligned} X_L^M&=- \pi '(x_L) \left( \tau +(1-\tau )\theta _i \right) h=k,\\ X_H^M&=- \pi '(x_H) \left( \tau +(1-\tau )\theta _H +(1-\tau )\frac{\alpha _L}{\alpha _H}(\theta _H-\theta _L)\right) h=k. \end{aligned}$$

Because both conditions are independent on $\beta$ it follows that $\frac{d x_i^M}{d \beta }=0$. Using the implicit function theorem, it follows that:

$$\begin{aligned} \frac{dx_L^M}{d \tau }&=-\frac{ \pi '(x_L) (1-\theta _L)}{\pi ''(x_L) \left( \tau +(1-\tau )\theta _L \right) }>0,\\ \frac{dx_H^M}{d \tau }&=-\frac{\pi '(x_H) (1-\theta _H-\frac{\alpha _L}{\alpha _H}(\theta _H-\theta _L))}{ \pi ''(x_H) \left( \tau +(1-\tau )\left[ \theta _H+\frac{\alpha _L}{\alpha _H}(\theta _H-\theta _L)\right] \right) }. \end{aligned}$$