Product Liability and Strategic Delegation: Endogenous Manager Incentives Promote Strict Liability

Abstract

We derive the socially optimal allocation of liability for product-related accidents when firms delegate their output and safety choices to managers under a contract that depends on profits and revenues. With exogenous product risk, the optimal contract emphasizes revenue over profits as a way of inducing managers to increase output independently of the liability allocation. When product safety is endogenous, however, this strategy distorts managers’ product safety choice because the managers underweight the cost of safety relative to expected harm whenever consumers bear some share of liability. It is then socially optimal to hold firms fully liable for victim losses.

Appendices

Appendix 1: Derivations for the Stackelberg Protocol

Suppose that firm i is the Stackelberg leader and implements its safety and output level first in the market stage. The Stackelberg follower’s reaction function is stated in (13). Firm i’s manager anticipates this reaction function when choosing safety and output to maximize $O_i(x_i,q_i,q^{BR}_j(q_i))$. The equilibrium safety level $x_i^*$ and output level $q_i^*$ must satisfy:

$$\begin{aligned} \frac{\partial O_i}{\partial x_i}&=0 \end{aligned}$$

(34)

$$\begin{aligned} \frac{d O_i}{d q_i}&=\frac{\partial O_i}{\partial q_i}+\frac{\partial O_i}{\partial q_j}\frac{\partial q_j^*}{\partial q_i}=0. \end{aligned}$$

(35)

In the contract stage, both owners design managerial incentives simultaneously. The owner of firm i chooses $\alpha _i$ to maximize $\Pi _i(x_i^*,q_i^*,q_j^*(q_i^*))$, and thus solves:

$$\begin{aligned} \frac{\partial \Pi _i}{\partial x_i}\frac{\partial x_i^*}{\partial \alpha _i} + \left( \frac{\partial \Pi _i}{\partial q_i}+\frac{\partial \Pi _i}{\partial q_j}\frac{\partial q_j^*}{\partial q_i}\right) \frac{\partial q_i^*}{\partial \alpha _i}=0. \end{aligned}$$

(36)

At $\alpha _i=1$, the manager maximizes profits and the first-order condition (36) is satisfied thanks to (34)–(35). Thus, there is no reason for firm i’s owner to reduce $\alpha _i$ from one (i.e., to include revenue in managerial incentives). When $\alpha _i^*=1$, the manager chooses output to fulfill:

$$\begin{aligned} a-2bq_i-bq_j^*-h(x_s)-c(x_s)+\frac{1}{2}bq_i=0. \end{aligned}$$

(37)

The equilibrium output of firm i is then:

$$\begin{aligned} q_i^*=\frac{a+gh(x_j^*)+\alpha _j^*{\mathcal {C}}(x_j^*,g)-2[h(x_s)+c(x_s)]}{2b}. \end{aligned}$$

(38)

In equilibrium, reducing $\alpha _j$ will decrease firm i’s output, which enhances firm j’s profit. Thus, similar to Cournot competition, the owner of firm j will design manager incentives with $\alpha _j^*<1$. Plugging this value of $q_i^*$ into firm j’s reaction function, we obtain:

$$\begin{aligned} q_j^*=\frac{a-3gh(x_j^*)-3\alpha _j^*{\mathcal {C}}(x_j^*,g)+2[h(x_s)+c(x_s)]}{4b}. \end{aligned}$$

(39)

Now, consider again the benchmark case with safety that is fixed at the socially optimal level $x_s$. The maximization problem of firm j’s owner becomes:

$$\begin{aligned}&\frac{\partial \Pi _j}{\partial q_j}\frac{\partial q_j^*}{\partial \alpha _j} +\frac{\partial \Pi _j}{\partial q_i}\frac{\partial q_i^*}{\partial \alpha _j} =-[a-bQ-h(x_s)-c(x_s)-bq_j^*]\frac{3{\mathcal {C}}(x_s,g)}{4b}-bq_j^* \frac{2{\mathcal {C}}(x_s,g)}{4b}=0 \\&\alpha _j^*= \frac{4[h(x_s)+c(x_s)]-a-3gh(x_s)}{3{\mathcal {C}}(x_s,g)} \end{aligned}$$

At this level of $\alpha _j^*$, the equilibrium output levels of firm i and firm j no longer depend on g. The fact that increasing g and decreasing $\alpha _j^*$ have opposite impacts on output means that $\alpha _j^*$ will be matched with g to maintain the equilibrium pair $\{q_i^*,q_j^*\}$.

Similar to the Cournot case, the endogeneity of safety imposes a cost for distorting managerial incentives: $\alpha _j^*$ cannot be decreased to the same extent as in the benchmark. Firm j’s reaction curve is thus shifted inward when g increases, which increases firm i’s output and decreases firm j’s output. The changes in output levels are shown in the term $gh(x_j^*)+\alpha _j^*{\mathcal {C}}(x_j^*,g)$, which clearly is more influential in $q_j^*$ than in $q_i^*$. As a result, the total output decreases when g increases. At $g=0$, the total output is the highest, but still lower than the socially optimal level:

$$\begin{aligned} Q^*=\frac{5[a-h(x_s)-c(x_s)]}{6b}<Q_s \end{aligned}$$

(40)

Because the socially optimal safety is chosen by firm j when $g=0$, we again conclude that strict liability with full compensation is the socially optimal loss allocation.

Appendix 2: Derivations for Bertrand Competition

Consider Bertrand competition between two firms with differentiated products facing the demand functions:

$$\begin{aligned} q_i=A-B{\mathcal {P}}_i+Y{\mathcal {P}}_j \end{aligned}$$

(41)

where $A=a/(b+y)$, $B=b/(b^2-y^2)$ and $Y=y/(b^2-y^2)$. With $\alpha _i \in [0,\infty )$, we state managerial incentives as:

$$\begin{aligned} O_i=\left[ p_i-\alpha _i(1-g)h(x_i)-\alpha _i c(x_i)\right] q_i. \end{aligned}$$

(42)

The first-order conditions result as:

$$\begin{aligned} \frac{\partial O_i}{\partial x_i}&=-\alpha _i\left[ (1-g)h'(x_i)+c' (x_i)\right] q_i - B\left[ p_i-\alpha _i(1-g)h(x_i)-\alpha _i c(x_i)\right] gh'(x_i)=0 \end{aligned}$$

(43)

$$\begin{aligned} \frac{\partial O_i}{\partial p_i}&=q_i-B\left[ p_i-\alpha _i(1-g)h(x_i)-\alpha _i c(x_i)\right] =0. \end{aligned}$$

(44)

Thus, the equilibrium care level $x_i^*$ chosen by the manager of firm i must satisfy:

$$\begin{aligned} -\alpha _i\left[ (1-g)h'(x_i^*)+c'(x_i^*)\right] - gh'(x_i^*)=0. \end{aligned}$$

(45)

When $\alpha _i>1$ (see below), the equilibrium care level will decrease in response to an increase in either $\alpha _i$ or g. The socially optimal care level can be achieved only when $\alpha _i=1$ or $g=0$. Firm i’s best-response function in terms of price is:

$$\begin{aligned} p_i^{BR}&= \frac{q_i}{B} +\alpha _i(1-g)h(x_i)+\alpha _i c(x_i)=\frac{A + Y{\mathcal {P}}_j - Bgh(x_i^*)+B\alpha _i\left[ (1-g)h(x_i^*)+c(x_i^*)\right] }{2B} \end{aligned}$$

(46)

Using the best-response function, we have the equilibrium price and full price levels as:

$$\begin{aligned} p_i^*=&\frac{(2B+Y)A-(2B^2-Y^2)gh(x_i^*)+2B^2\alpha _i {\mathcal {C}}(x_i^*,g) +YBgh(x_j^*)+YB\alpha _j {\mathcal {C}}(x_j^*,g)}{4B^2-Y^2} \end{aligned}$$

(47)

$$\begin{aligned} {\mathcal {P}}_i^*&= \frac{(2B+Y)A + 2B^2gh(x_i^*)+2B^2\alpha _i {\mathcal {C}}(x_i^*,g)+YBgh(x_j^*)+YB\alpha _j {\mathcal {C}}(x_j^*,g)}{4B^2-Y^2} \end{aligned}$$

(48)

A change in firm j’s equilibrium care level $x_j^*$ does not affect firm i’s equilibrium price $p_i^*$ thanks to (45), which renders a positive impact of $\alpha _j$ on $p_i^*$:

$$\begin{aligned} \frac{\partial p_i^*}{\partial \alpha _j}=\frac{YB {\mathcal {C}}(x_j^*,g)}{4B^2-Y^2}>0. \end{aligned}$$

(49)

In stage 1, the owner chooses $\alpha _i$ to maximize the firm’s profit:

$$\begin{aligned} \frac{\partial \Pi _i}{\partial x_i}\frac{\partial x_i^*}{\partial \alpha _i} + \frac{\partial \Pi _i}{\partial p_i}\frac{\partial p_i^*}{\partial \alpha _i} + \frac{\partial \Pi _i}{\partial p_j}\frac{\partial p_j^*}{\partial \alpha _i} =0. \end{aligned}$$

(50)

The left-hand side is reduced to its third term and is positive when evaluated at $\alpha _i=1$. Thus, in equilibrium, the owner will choose $\alpha _i^*>1$.

Considering the symmetric equilibrium and focusing on the full price level, we have:

$$\begin{aligned} {\mathcal {P}}_i^*={\mathcal {P}}_j^*={\mathcal {P}}^*=\frac{A+Bgh(x^*)+B\alpha ^*\left[ (1-g)h(x^*)+ c(x^*)\right] }{2B-Y} \end{aligned}$$

(51)

Exogenous care: When safety is fixed at a specific level—for example $x_s$—the chosen managerial incentives must satisfy:

$$\begin{aligned} \frac{\partial \Pi _i}{\partial p_i}\frac{\partial p_i^*}{\partial \alpha _i} + \frac{\partial \Pi _i}{\partial p_j}\frac{\partial p_j^*}{\partial \alpha _i} =0 \end{aligned}$$

(52)

which results in symmetric equilibrium as:

$$\begin{aligned}{}[A-(2B-Y)p^*-(B-Y)gh(x_s)+B{\mathcal {C}}]2B+Y^2(p^*-{\mathcal {C}})&=0 \\ 2AB-(2B^2-2BY)gh(x_s)+(2B^2-Y^2){\mathcal {C}}&=(4B^2-2BY-Y^2)p^* \\ \frac{2AB-(2B^2-2BY)gh(x_s)+(2B^2-Y^2){\mathcal {C}}}{4B^2-2BY-Y^2}&=p^* \end{aligned}$$

The equilibrium full price becomes:

$$\begin{aligned} {\mathcal {P}}^*=\frac{2AB+(2B^2-Y^2)[h(x_s)+c(x_s)]}{4B^2-2BY-Y^2} \end{aligned}$$

(53)

which is independent of g and $\alpha ^*$: Similar to the Cournot case, the owner will adjust managerial incentives in response to g, so that the full price and the level of output stay the same.

Endogenous care: When care is endogenous, it imposes a cost on increasing $\alpha ^*$, as doing so would drive the care level further downward from the profit-maximizing level. Thus, the firm will not increase $\alpha ^*$ to the level that would stabilize the full price. As g increases, the full price decreases and the output level increases, which increases social welfare. When firms compete in prices, the socially optimal level of g cannot be equal to zero because the social planner prefers to increase g marginally as the marginal welfare effect from safety is zero (envelope theorem) whereas the marginal welfare effect from output is strictly positive.