Product liability, multidimensional R&D and innovation

Abstract

This paper studies the effect of product liability cost on firms’ incentives to conduct R&D when innovation inherently consists a safety dimension and a novelty dimension. We consider a situation where a monopoly firm chooses both product novelty and product safety in an R&D stage followed by a production stage. While firms will invest more in product safety as product liability cost increases, their incentive for product novelty may increase or decrease, depending on the relative strengths of a demand-shifting effect and a cross-R&D effect identified in the model. Consequently, a higher product liability cost may decrease consumer welfare and total welfare. We extend the results to an oligopoly model with differentiated products and study the effects of competition. We find that equilibrium product novelty and safety levels decrease with the number of firms but exhibit non-monotonic relationships with the degree of product substitutability.

Appendices

Appendix 1

Proof of Proposition 1

Total differentiations of (5) and (6) with respect to L give

$$\begin{aligned}&\left( \frac{\partial A}{\partial \gamma }\frac{\partial \gamma }{\partial L} -\frac{\partial \varphi }{\partial s}\frac{\partial s}{\partial L}\right) \frac{\partial A}{\partial \gamma }+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}A}{\partial \gamma ^{2}} \frac{\partial \gamma }{\partial L}\\&\quad -2\left( \frac{\partial ^{2}x}{\partial \gamma ^{2}}\frac{\partial \gamma }{\partial L}+\frac{\partial ^{2}x}{ \partial \gamma \partial s}\frac{\partial s}{\partial L}\right) =0 \end{aligned}$$

and

$$\begin{aligned}&-\left( \frac{\partial A}{\partial \gamma }\frac{\partial \gamma }{\partial L }-\frac{\partial \varphi }{\partial s}\frac{\partial s}{\partial L}\right) \frac{\partial \varphi }{\partial s}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}\varphi }{\partial s^{2} }\frac{\partial s}{\partial L}\\&\quad +2-2\left( \frac{\partial ^{2}x}{\partial s\partial \gamma }\frac{\partial \gamma }{\partial L}+\frac{\partial ^{2}x}{ \partial s^{2}}\frac{\partial s}{\partial L}\right) =0. \end{aligned}$$

With rearrangement, we can further show that

$$\begin{aligned}&\left\{ \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{ \partial ^{2}A}{\partial \gamma ^{2}}-2\frac{\partial ^{2}x}{\partial \gamma ^{2}}\right\} \frac{\partial \gamma }{\partial L}\\&\quad +\left( -\frac{\partial \varphi }{\partial s}\frac{\partial A}{\partial \gamma }-2\frac{\partial ^{2}x}{\partial \gamma \partial s}\right) \frac{\partial s}{\partial L}=0 \end{aligned}$$

and

$$\begin{aligned}&\left( -\frac{\partial A}{\partial \gamma }\frac{\partial \varphi }{\partial s}-2\frac{\partial ^{2}x}{\partial s\partial \gamma }\right) \frac{\partial \gamma }{\partial L}+\left\{ \left( \frac{\partial \varphi }{\partial s} \right) ^{2}+\left[ A\left( \gamma \right) \right. \right. \\&\left. \left. \quad -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}\varphi }{\partial s^{2}}-2\frac{\partial ^{2}x }{\partial s^{2}}\right\} \frac{\partial s}{\partial L}=-2. \end{aligned}$$

Thus, applying Cramer’s rule, we can solve $\frac{\partial \gamma }{\partial L}$ and $\frac{\partial s}{\partial L},$ and obtain expressions as in (7) and (8).

The second-order conditions, which we assume to hold, are

$$\begin{aligned}&\left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}A}{ \partial \gamma ^{2}}-2\frac{\partial ^{2}x}{\partial \gamma ^{2}}<0 ,\left( \frac{\partial \varphi }{\partial s}\right) ^{2}\nonumber \\&\quad -\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}\varphi }{\partial s^{2}}-2\frac{\partial ^{2}x}{\partial s^{2}}<0 \end{aligned}$$

(25)

and

$$\begin{aligned} \Phi= & {} \left\{ \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{ \partial ^{2}A}{\partial \gamma ^{2}}-2\frac{\partial ^{2}x}{\partial \gamma ^{2}}\right\} \nonumber \\&\left\{ \left( \frac{\partial \varphi }{\partial s}\right) ^{2}-\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}\varphi }{\partial s^{2}}-2\frac{\partial ^{2}x}{\partial s^{2}}\right\} \nonumber \\&-\left( \frac{\partial A}{\partial \gamma }\frac{\partial \varphi }{ \partial s}+2\frac{\partial ^{2}x}{\partial \gamma \partial s}\right) ^{2}>0. \end{aligned}$$

(26)

From (25) and (26), we have $\frac{\partial s}{ \partial L}>0.$ Moreover, $\frac{\partial \gamma }{\partial L}\gtreqless 0$ if $\Delta \gtreqless 0.$ Finally, if $\frac{\partial ^{2}x}{\partial \gamma \partial s}=0,$ $\Delta >0$ and thus $\frac{\partial \gamma }{\partial L} >0.$ $\square$

Proof of Proposition 2

Note that

$$\begin{aligned} \frac{dV}{dL}=\frac{1}{2}q\left( \frac{\partial A}{\partial \gamma }\frac{ \partial \gamma }{\partial L}-\frac{\partial \varphi }{\partial s}\frac{ \partial s}{\partial L}\right) . \end{aligned}$$

Under the linear consumer demand and quadratic R&D cost, we have

$$\begin{aligned} \frac{dV}{dL}= & {} \left( \frac{1}{2}q\right) \left( -\frac{2}{\Phi }\right) \left[ \left( -\delta +2\sigma \right) -\left( -\delta \right) \left( 1-2\alpha \right) \right] \\= & {} \left( \frac{1}{2}q\right) \left( -\frac{2}{\Phi }\right) \left( 2\sigma -2\delta \alpha \right) . \end{aligned}$$

Thus, $\frac{dV}{dL}>(<)0$ if $\sigma <(>)\alpha \delta .$ Moreover, by envelop theorem, we have

$$\begin{aligned} \frac{\partial \pi }{\partial L}=-\left( 1-s\right) . \end{aligned}$$

Therefore, total welfare, $W=V+\pi +\left( 1-s\right) L,$ increases (decreases) with $\sigma$ if $\sigma <(>)\alpha \delta .$ $\square$

Proof of Proposition 3

We first consider the choice of output quantity, $q_{i},$ given product novelty, $\gamma _{i}$ and $\gamma _{j},$ and safety level, $s_{i}$ and $s_{j}$. Firm i maximizes profit,

$$\begin{aligned} \pi _{i}=\left( p_{i}-c\right) q_{i}-\left( 1-s_{i}\right) L-x\left( \gamma _{i},s_{i}\right) . \end{aligned}$$

First order condition, $\frac{\partial \pi _{i}}{\partial q_{i}}=0,$ implies

$$\begin{aligned} q_{i}=\frac{A\left( \gamma _{i}\right) -\varphi \left( s_{i},\delta \right) -c-\frac{\theta }{2+\theta \left( n-1\right) }\Sigma _{j=1}^{n}\left[ A\left( \gamma _{j}\right) -\varphi \left( s_{j},\delta \right) -c\right] }{ \left( 2-\theta \right) }. \end{aligned}$$

Next, we consider the choice of product novelty and safety. Note that the profit can be written as

$$\begin{aligned} \pi _{i}=\left( q_{i}\right) ^{2}-\left( 1-s_{i}\right) L-x\left( \gamma _{i},s_{i}\right) . \end{aligned}$$

First order conditions with respect to $s_{i}$ and $\gamma _{i},$ respectively, yield

$$\begin{aligned} \frac{\partial \pi _{i}}{\partial s_{i}}=2q_{i}\frac{\partial q_{i}}{ \partial s_{i}}+L-\frac{\partial x}{\partial s_{i}}=2q_{i}\frac{2+\left( n-2\right) \theta }{\left( 2-\theta \right) \left[ 2+\left( n-1\right) \theta \right] }\left( -\frac{\partial \varphi }{\partial s_{i}}\right) +L- \frac{\partial x}{\partial s_{i}}=0 \end{aligned}$$

and

$$\begin{aligned} \frac{\partial \pi _{i}}{\partial \gamma _{i}}=2q_{i}\frac{\partial q_{i}}{ \partial \gamma _{i}}-\frac{\partial x_{i}}{\partial \gamma _{i}}=2q_{i} \frac{2+\left( n-2\right) \theta }{\left( 2-\theta \right) \left[ \left( 2+\left( n-1\right) \theta \right) \right] }\frac{\partial A}{\partial \gamma _{i}}-\frac{\partial x}{\partial \gamma _{i}}=0. \end{aligned}$$

(27)

We focus on the symmetric equilibrium in which each firm chooses product novelty $\gamma ,$ safety s and output q. It follows that

$$\begin{aligned} q=\frac{A\left( \gamma \right) -\varphi \left( s,\delta \right) -c}{2+\theta \left( n-1\right) }. \end{aligned}$$

(28)

Thus,

$$\begin{aligned} \frac{\partial \pi }{\partial \gamma }=\xi \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial A}{\partial \gamma }-\frac{\partial x}{\partial \gamma }=0 \end{aligned}$$

(29)

and

$$\begin{aligned} \frac{\partial \pi }{\partial s}=\xi \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( -\frac{\partial \varphi }{\partial s }\right) +L-\frac{\partial x}{\partial s}=0 \end{aligned}$$

(30)

where

$$\begin{aligned} \xi =\frac{2\left[ 2+\left( n-2\right) \theta \right] }{\left( 2-\theta \right) \left[ 2+\left( n-1\right) \theta \right] ^{2}}. \end{aligned}$$

Note that second order conditions, which we assume to hold, are

$$\begin{aligned}&\xi \left[ \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{ \partial ^{2}A}{\partial \gamma ^{2}}\right] -\frac{\partial ^{2}x}{\partial \gamma ^{2}}<0,\nonumber \\&\quad \xi \left[ \left( \frac{\partial \varphi }{\partial s}\right) ^{2}-\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}\varphi }{\partial s^{2}} \right] -\frac{\partial ^{2}x}{\partial s^{2}}<0 \end{aligned}$$

(31)

and

$$\begin{aligned} \Omega= & {} \left[ \xi \left( \left( \frac{\partial A}{\partial \gamma } \right) ^{2}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}A}{\partial \gamma ^{2}}\right) -\frac{\partial ^{2}x}{\partial \gamma ^{2}}\right] \nonumber \\&\left[ \xi \left( \left( \frac{\partial \varphi }{\partial s}\right) ^{2}-\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}\varphi }{\partial s^{2} }\right) -\frac{\partial ^{2}x}{\partial s^{2}}\right] \nonumber \\&-\left[ \xi \left( \frac{\partial A}{\partial \gamma }\frac{\partial \varphi }{\partial s}\right) +\frac{\partial ^{2}x}{\partial s\partial \gamma }\right] ^{2}>0. \end{aligned}$$

(32)

Total differentiation of (29) and (30) with respect to L,  with rearrangement, yield,

$$\begin{aligned}&\left[ \xi \left( \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+ \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}A}{\partial \gamma ^{2}}\right) -\frac{\partial ^{2}x}{ \partial \gamma ^{2}}\right] \frac{\partial \gamma }{\partial L}\\&\quad +\left[ \xi \left( -\frac{\partial \varphi }{\partial s}\frac{\partial A}{\partial \gamma }\right) -\frac{\partial ^{2}x}{\partial \gamma \partial s}\right] \frac{\partial s}{\partial L}=0 \end{aligned}$$

and

$$\begin{aligned}&\left[ \xi \left( \frac{\partial A}{\partial \gamma }\frac{\partial \varphi }{\partial s}\right) -\frac{\partial ^{2}x}{\partial s\partial \gamma } \right] \frac{\partial \gamma }{\partial L}+\left[ \xi \left( \left( \frac{ \partial \varphi }{\partial s}\right) ^{2}+\left[ A\left( \gamma \right) \right. \right. \right. \\&\left. \left. \left. \quad -\varphi \left( s,\delta \right) -c\right] \left( -\frac{\partial ^{2}\varphi }{\partial s^{2}}\right) \right) -\frac{\partial ^{2}x}{\partial s^{2}}\right] \frac{\partial s}{\partial L}=-1. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \frac{\partial s}{\partial L}=-\frac{1}{\Omega }\left[ \xi \left( \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}A}{ \partial \gamma ^{2}}\right) -\frac{\partial ^{2}x}{\partial \gamma ^{2}} \right] \end{aligned}$$

(33)

and

$$\begin{aligned} \frac{\partial \gamma }{\partial L}=\frac{1}{\Omega }\left[ \xi \left( - \frac{\partial \varphi }{\partial s}\frac{\partial A}{\partial \gamma } \right) -\frac{\partial ^{2}x}{\partial \gamma \partial s}\right] . \end{aligned}$$

(34)

From (31) and (32), we have $\frac{\partial s}{ \partial L}>0.$ Moreover, $\frac{\partial \gamma }{\partial L}\gtreqless 0$ if $\Sigma \gtreqless 0$ where

$$\begin{aligned} \Sigma =\xi \left( -\frac{\partial \varphi }{\partial s}\frac{\partial A}{ \partial \gamma }\right) -\frac{\partial ^{2}x}{\partial \gamma \partial s}. \end{aligned}$$

$\square$

Proof of Lemma 1

• (i) From (16),

  $$\begin{aligned} \frac{\partial \xi }{\partial n}=-\frac{2\theta \left[ \left( n-3\right) \theta +2\right] }{\left( 2-\theta \right) \left[ \left( n-1\right) \theta +2 \right] ^{3}}<0 \end{aligned}$$

  (35)

  for $\theta \in (0,1).$

• (ii) Note that

  $$\begin{aligned} \frac{\partial \xi }{\partial \theta }=\frac{4\left( n-1\right) }{\left( \theta -2\right) ^{2}\left[ \left( n-1\right) \theta +2\right] ^{3}}\lambda \left( \theta ,n\right) \end{aligned}$$

  (36)

  where $\lambda \left( \theta ,n\right) =\left( n-2\right) \theta ^{2}+\left( 5-n\right) \theta -2.$ Hence,

  $$\begin{aligned} sign\left( \frac{\partial \xi }{\partial \theta }\right) =sign\left( \lambda \left( \theta ,n\right) \right) . \end{aligned}$$

  Moreover, $\lambda \left( \theta ,n\right) <0$ if $\theta =0$ and $\lambda \left( \theta ,n\right) >0$ if $\theta =1.$ It follows that there exists a unique ${\hat{\theta }}>0$ such that $\frac{\partial \xi }{\partial \theta }<0$ for $\theta \in (0,{\hat{\theta }}]$ and $\frac{\partial \xi }{\partial \theta } >0$ for $\theta \in ({\hat{\theta }},1).$ $\square$

Proof of Proposition 4

In the symmetric equilibrium, product novelty and safety are given by (29) and (30). Total differentiations of (29) and (30) with respect to $\xi ,$ with rearrangement, give

$$\begin{aligned} \frac{\partial \gamma }{\partial \xi }=\frac{1}{Z}\left\{ \begin{array}{c} \left[ -\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c \right] \frac{\partial A}{\partial \gamma }\right] \left[ \xi \left( \frac{ \partial \varphi }{\partial s}\right) ^{2}+\xi \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( -\frac{\partial ^{2}\varphi }{\partial s^{2}}\right) -\frac{\partial ^{2}x}{\partial s^{2}} \right] \\ -\left[ \xi \left( -\frac{\partial \varphi }{\partial s}\right) \frac{ \partial A}{\partial \gamma }-\frac{\partial ^{2}x}{\partial \gamma \partial s}\right] \left[ -\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( -\frac{\partial \varphi }{\partial s}\right) \right] \end{array} \right\} \end{aligned}$$

(37)

and

$$\begin{aligned} \frac{\partial s}{\partial \xi }=\frac{1}{Z}\left\{ \begin{array}{c} \left[ \xi \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\xi \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{ \partial ^{2}A}{\partial \gamma ^{2}}-\frac{\partial ^{2}x}{\partial \gamma ^{2}}\right] \left[ -\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( -\frac{\partial \varphi }{\partial s}\right) \right] \\ -\left[ -\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c \right] \frac{\partial A}{\partial \gamma }\right] \left[ \xi \frac{\partial A}{\partial \gamma }\left( -\frac{\partial \varphi }{\partial s}\right) - \frac{\partial ^{2}x}{\partial s\partial \gamma }\right] \end{array} \right\} \end{aligned}$$

(38)

where

$$\begin{aligned} Z= & {} \left[ \xi \left( \frac{\partial A}{\partial \gamma }\right) ^{2}+\xi \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \frac{\partial ^{2}A}{\partial \gamma ^{2}}-\frac{\partial ^{2}x}{\partial \gamma ^{2}}\right] \\&\left[ \xi \left( \frac{\partial \varphi }{\partial s} \right) ^{2}+\xi \left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( -\frac{\partial ^{2}\varphi }{\partial s^{2}} \right) -\frac{\partial ^{2}x}{\partial s^{2}}\right] \\&-\left[ \xi \frac{\partial A}{\partial \gamma }\left( -\frac{\partial \varphi }{\partial s}\right) -\frac{\partial ^{2}x}{\partial s\partial \gamma }\right] \left[ \xi \left( -\frac{\partial \varphi }{\partial s} \right) \frac{\partial A}{\partial \gamma }-\frac{\partial ^{2}x}{\partial \gamma \partial s}\right] . \end{aligned}$$

Under the specifications, (37) and (38) become

$$\begin{aligned} \frac{\partial \gamma }{\partial \xi }=\frac{\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( \beta -\delta \sigma \right) }{\left( \xi -\alpha \right) \left( \xi \delta ^{2}-\beta \right) -\left( \xi \delta -\sigma \right) ^{2}} \end{aligned}$$

and

$$\begin{aligned} \frac{\partial s}{\partial \xi }=\frac{\left[ A\left( \gamma \right) -\varphi \left( s,\delta \right) -c\right] \left( \alpha \delta -\sigma \right) }{\left( \xi -\alpha \right) \left( \xi \delta ^{2}-\beta \right) -\left( \xi \delta -\sigma \right) ^{2}}. \end{aligned}$$

From (32), $\left( \xi -\alpha \right) \left( \xi \delta ^{2}-\beta \right) -\left( \xi \delta -\sigma \right) ^{2}>0.$ Moreover, from (31), $\xi <\min \left\{ \beta /\delta ^{2},\alpha \right\} .$ Thus, together with the assumption $\sigma \le \xi \delta ,$ we have $\sigma <\min \left\{ \beta /\delta ,\alpha \delta \right\} .$ Hence, by Lemma 1, results in Proposition 4 follow. $\square$

Appendix 2

In this appendix, we consider a variant of our main model in which a monopoly firm’s liability cost is linear in quantity sold. Compared to the main model, there will be an additional quantity effect in the variant considered here because product quantity will depend on liability cost. The analysis will become more complicated. Nevertheless, we will show that the main insight that a higher product liability cost may induce a lower product novelty and consequently, lead to a lower consumer welfare will continue to hold under some parameter regions.

The setup is the same as in the main model except that the product liability cost L is equal to lq where l measures the strength of liability. The following analysis is similar to that in the main model. In particular, the firm chooses q to maximize profit,

$$\begin{aligned} \pi =\left( p-c\right) q-\left( 1-s\right) lq-x\left( \gamma ,s\right) . \end{aligned}$$

The resulting profit-maximizing quantity is

$$\begin{aligned} q=\frac{A\left( \gamma \right) -\varphi \left( s,\delta \right) -c-\left( 1-s\right) l}{2}. \end{aligned}$$

and thus we can rewrite the profit function as

$$\begin{aligned} \pi =\left[ q\left( \gamma ,s,l\right) \right] ^{2}-x\left( \gamma ,s\right) . \end{aligned}$$

(39)

The first-order conditions of (39) with respect to $\gamma$ and s,  respectively, yield

$$\begin{aligned} q\frac{\partial A}{\partial \gamma }-\frac{\partial x}{\partial \gamma }=0 \end{aligned}$$

(40)

and

$$\begin{aligned} q\left( -\frac{\partial \varphi }{\partial s}+l\right) -\frac{\partial x}{ \partial s}=0. \end{aligned}$$

(41)

To conduct tractable analysis, we focus on the case with linear demand and quadratic cost function as considered in Corollary 1. In particular, from the total differentiations of (40) and (41 ) with respect to l,  we can obtain that

$$\begin{aligned} \frac{\partial s}{\partial l}=\frac{1}{\Theta }\left\{ \left( \alpha -\frac{1 }{2}\right) \left[ q-\frac{\left( 1-s\right) }{2}\left( \delta +l\right) \right] +\frac{\left( 1-s\right) }{2}\left[ \sigma -\frac{1}{2}\left( \delta +l\right) \right] \right\} \end{aligned}$$

(42)

and

$$\begin{aligned} \frac{\partial \gamma }{\partial l}=\frac{1}{\Theta }\left\{ \frac{\left( 1-s\right) }{2}\left[ \frac{1}{2}\left( \delta +l\right) ^{2}-\beta \right] - \left[ \frac{1}{2}\left( \delta +l\right) -\sigma \right] \left[ \frac{ \left( 1-s\right) }{2}\left( \delta +l\right) -q\right] \right\} \end{aligned}$$

(43)

with

$$\begin{aligned} \Theta =\left[ \frac{1}{2}-\alpha \right] \left[ \frac{1}{2}\left( \delta +l\right) ^{2}-\beta \right] -\left[ \sigma -\frac{1}{2}\left( \delta +l\right) \right] ^{2}. \end{aligned}$$

Moreover, assuming $\gamma$ and s are interior solution, we have, from the second-order condition, $\alpha >\frac{1}{2}$, $\beta >\frac{1}{2}\left( \delta +l\right) ^{2}$ and $\Theta >0.$

Next, we can show that consumer welfare is

$$\begin{aligned} V=\frac{1}{2}\left[ \frac{A\left( \gamma \right) -\varphi \left( s,\delta \right) -c-\left( 1-s\right) l}{2}\right] ^{2}+y. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{dV}{dl}=\frac{1}{2}q\left[ \frac{\partial \gamma }{\partial l}+\left( \delta +l\right) \frac{\partial s}{\partial l}-\left( 1-s\right) \right] . \end{aligned}$$

(44)

We have the following result.

Proposition 6

Suppose that $\left( \gamma ^{*},s^{*}\right) ,$ which is an interior solution to (40) and (41), exists. (i) $s^{*}$ increases in l and $\gamma ^{*}$ decreases in l if $\sigma >\alpha \left( \delta +l\right)$ and $\beta >\sigma \left( \delta +l\right) .$ Moreover, if $\beta >2\sigma \left( \delta +l\right) ,$ then consumer surplus decreases in l.

Proof

From (42),

$$\begin{aligned} \frac{\partial s}{\partial l}=\frac{1}{\Theta }\left\{ \frac{\left( 1-s\right) }{2}\left[ \sigma -\alpha \left( \delta +l\right) \right] +\left( \alpha -\frac{1}{2}\right) q\right\} . \end{aligned}$$

Hence, if $\sigma >\alpha \left( \delta +l\right) ,$ then $\frac{\partial s}{ \partial l}>0$ because $\alpha >\frac{1}{2}.$

Moreover, from (43),

$$\begin{aligned} \frac{\partial \gamma }{\partial l}=\frac{1}{\Theta }\left\{ \frac{\left( 1-s\right) }{2}\left[ \sigma \left( \delta +l\right) -\beta \right] +\left[ \frac{1}{2}\left( \delta +l\right) -\sigma \right] q\right\} . \end{aligned}$$

Thus, if $\sigma >\alpha \left( \delta +l\right)$ and $\beta >\sigma \left( \delta +l\right) ,$ then $\frac{\partial \gamma }{\partial l}<0.$

Finally,

$$\begin{aligned} \frac{dV}{dl}= & {} .\frac{1}{2}q\left[ \frac{1}{\Theta }\left\{ \frac{\left( 1-s\right) }{2}\left[ 2\sigma \left( \delta +l\right) -\beta -\alpha \left( \delta +l\right) ^{2}\right] \right. \right. \\&\left. \left. +\left[ -\sigma +\left( \delta +l\right) \alpha \right] q\right\} -\left( 1-s\right) \right] \end{aligned}$$

If $\sigma >\alpha \left( \delta +l\right)$ and $\beta >2\sigma \left( \delta +l\right) ,$ then we have $\frac{dV}{dl}<0.$ $\square$

The intuition behind Proposition 6 is similar to that in the main model. Specifically, an increase in l directly increases the marginal benefit of improving product safety. However, when the cross-R&D effect is large ($\sigma$ is relatively large), an increase product safety induces a decrease in product novelty. When the reduction in product novelty is large it will offset the benefit of an increase in product safety and consequently, consumer welfare will decrease.