Product recall with symmetric uncertainty and multiunit purchases

Abstract

This paper considers product recall under a perfect regime of strict liability. We show that safety is under supplied given output due to an under internalization of infra-marginal units. If we add a mandatory refund with a possible penalty fee in the event that the product turns out to be unsafe, then while the price increases, there is no change in the allocation, utility, profit or total welfare. The recall procedure is then neutral. We then extend the model to examine optimal fines and minimum output taxes, endogenous proclivity to return a product, endogenous decision to sue in the event of damage and the effects of having the consumer under estimate expected damages.

Appendix

Numerical example (NE). Let $u(q)=q^{\alpha }$, $\alpha \in (0,1)$, and $c(h)=\{Ln[1/(1-h)]-h\}.$

1.1 The (SOC) for profit maximization

The second order condition (SOC) (iii) for profit maximization simplifies to

$$\begin{aligned} \frac{1}{(1-h)^{2}}>\frac{H^{2}(1-\alpha )}{\hat{h}}q^{\alpha -1}. \end{aligned}$$

From (FOC) (3), $q^{\alpha -1}=\frac{1}{\alpha H}(\frac{h}{1-h}-(r+HD)).$ Substituting this expression for $q^{\alpha -1}$ into the specialized (SOC), the (SOC) becomes

$$\begin{aligned} \frac{\hat{h}\alpha -(1-\alpha )hH(1-h)+H(r+HD)(1-\alpha )(1-h)^{2}}{ (1-h)^{2}\hat{h}\alpha }>0. \end{aligned}$$

This condition is met if, for example, $\alpha >0.5,$ by $\hat{h} >h$ and $H(1-h)\in (0,1).$

1.2 The (SOC) for welfare maximization

The (SOC) (iii) for welfare maximization is

$$\begin{aligned} \frac{1}{(1-h)^{2}}>\frac{H^{2}(1-\alpha )q^{\alpha -1}}{\hat{h}\alpha }. \end{aligned}$$

Use (FOC) (5) to substitute for $q^{\alpha -1}=\frac{1}{H}(\frac{h}{1-h} -(r+HD)),$ and the (SOC) is

$$\begin{aligned} \frac{\hat{h}\alpha -hH(1-h)(1-\alpha )+H(r+HD)(1-\alpha )(1-h)^{2}}{\hat{h} \alpha (1-h)^{2}}>0. \end{aligned}$$

The latter condition is met if $\alpha >0.5,$ by $\hat{h}>h$ and $H(1-h)\in (0,1).$

Proof of Proposition 2:

From Proposition 1, it is clear that the recall procedure is neutral. The firm’s profit is $\pi (h,q)=\hat{h} u^{\prime }(q)q-c(h)q-(1-h)(HD+r)q,$ with (FOC)

$$\begin{aligned} \pi _{h}= & {} Hu^{\prime }(q)q-c^{\prime }(h)q+(HD+r)q=0, \\ \pi _{q}= & {} \hat{h}(u^{\prime \prime }(q)q+u^{\prime }(q))-c(h)-(1-h)(HD+r)=0. \end{aligned}$$

We have that $\pi _{qq}=\hat{h}(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))<0,\pi _{hh}=-c^{\prime \prime }q<0,$ and $\pi _{hq}=H(u^{\prime \prime }(q)q+u^{\prime }(q))-c^{\prime }(h)+(HD+r).$ Using the local condition $\pi _{h}=0,$ $\pi _{hq}=Hu^{\prime \prime }(q)q<0.$ We assume that the Hessian $|\Delta |$ $=-\hat{h}(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))c^{\prime \prime }(h)q-(Hp^{\prime }q)^{2}>0,$ so that the local SOC are met. Employing the usual comparative static techniques,

$$\begin{aligned} \frac{\partial q}{\partial r}= & {} (|\Delta |)^{-1}[-(1-h)c^{\prime \prime }(h)q+Hu^{\prime \prime }(q)q]<0, \\ \text { }\frac{\partial h}{\partial r}= & {} (|\Delta |)^{-1}[-\hat{h} q(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))-(1-h)Hu^{\prime \prime }(q)q]>0, \\ \frac{\partial q}{\partial D}= & {} (|\Delta |)^{-1}[-(1-h)Hc^{\prime \prime }(h)q+H^{2}u^{\prime \prime }(q)q^{2}]<0,\text { and } \\ \frac{\partial h}{\partial D}= & {} (|\Delta |)^{-1}[-\hat{h}Hq(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))-(1-h)H^{2}u^{\prime \prime }(q)q]>0. \end{aligned}$$

The social welfare function is $W(h,q)\ =\hat{h}\int \limits _{0}^{q}u^{ \prime }(z)dz-c(h)q-(1-h)(HD+r)q,$ and the (FOC) for social optimality are

$$\begin{aligned} W_{h}= & {} H\int \limits _{0}^{q}u^{\prime }(z)dz-c^{\prime }(h)q+(HD+r)q=0, \\ W_{q}= & {} \hat{h}u^{\prime }(q)-c(h)-(1-h)(HD+r)=0. \end{aligned}$$

The second order derivatives of social welfare are $W_{qq}=\hat{h}u^{\prime \prime }(q)<0,W_{hh}=-c^{\prime \prime }(h)q<0,$ and $W_{hq}=Hu^{\prime }(q)-c^{\prime }(h)+(HD+r).$ Using $W_{h}=0,$ $W_{hq}=-HS(q)/q<0.$ Assuming that the relevant Hessian determinate is positive, $|\Delta ^{o}|$ $=$ $- \hat{h}u^{\prime \prime }(q)c^{\prime \prime }(h)q-(\frac{HS(q)}{q})^{2}>0,$ $\partial q/\partial i=(1/|H^{o}|)[-(1-h)c^{\prime \prime }(h)q-S(q)]<0$

$$\begin{aligned} \partial q/\partial r= & {} (1/|\Delta ^{o}|)[-(1-h)c^{\prime \prime }(h)q-HS(q)]<0, \\ \partial h/\partial r= & {} (1/|\Delta ^{o}|)[-\hat{h}u^{\prime \prime }(q)q+(1-h)HS(q)/q]>0, \\ \partial q/\partial D= & {} (1/|\Delta ^{o}|)[-(1-h)Hc^{\prime \prime }(h)q-H^{2}S(q)]<0,\text { and} \\ \partial h/\partial D= & {} (1/|\Delta ^{o}|)[-\hat{h}Hu^{\prime \prime }q+H^{2}(1-h)S(q)/q]>0. \end{aligned}$$

The remainder of the proof is contained in the text. $\square$

Proof of Proposition 3:

From Proposition 1, it is clear that the recall procedure is neutral. Consider the contingent lump sum fine first. The firm’s expected profit is $\pi (h,q)=\hat{h}u^{\prime }(q)q-c(h)q-(1-h)HDq-(1-h)F,$ and the (FOC) for profit maximization are

$$\begin{aligned} Hu^{\prime }(q)q-c^{\prime }(h)q+HDq+F= & {} 0\text {, and } \\ \hat{h}(u^{\prime \prime }(q)q+u^{\prime }(q))-c(h)-(1-h)HD= & {} 0. \end{aligned}$$

The under supply of q given h and h given q are apparent from these conditions. The SOC are similar to the original problem with $\pi _{qq}=\hat{ h}(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))<0,\pi _{hh}=-c^{\prime \prime }(h)q<0,$ and, using $\pi _{h}=0,$ $\pi _{hq}=(Hu^{\prime \prime }(q)q-\frac{F}{q})<0.$ Note that that q and h are again substitutes. As before we assume that the relevant Hessian, denoted $\hat{\Delta },$ has a positive determinate: $|\hat{\Delta }|$ $=(\hat{ h}(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q)))(-c^{\prime \prime }(h)q)-(Hu^{\prime \prime }(q)q-\frac{F}{q})^{2}>0.$ Using the usual comparative static techniques on the (FOC) we obtain $\partial q/\partial F= \frac{1}{|\hat{\Delta }|}(Hu^{\prime \prime }(q)q-\frac{F}{q})<0$ and $\partial h/\partial F=\frac{-\hat{h}(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))}{|\hat{\Delta }|}>0.$ Finally, it is clear that the social optimum is unaffected by the lump sum fine.

Social welfare under the contingent lump sum fine is given by consumer surplus, $[\hat{h}S(q)+(1-h)F],$ plus profit, $[\hat{h}u^{\prime }(q)q-c(h)q-(1-h)HDq-(1-h)F].$ At the profit maximal solution, the direct impact on profit of a change F is $-(1-h^{*})<0,$ by the envelope theorem. The change in consumer surplus is given by $\frac{\partial (\hat{h }^{*}S(q^{*})+(1-h^{*})F)}{\partial h^{*}}\frac{\partial h^{*}}{\partial F}+\frac{\partial (\hat{h}^{*}S(q^{*}))}{ \partial q^{*}}\frac{\partial q^{*}}{\partial F}=(HS(q^{*})-F) \frac{\partial h^{*}}{\partial F}+[-\hat{h}^{*}u^{\prime \prime }(q^{*})q^{*}]\frac{\partial q^{*}}{\partial F}$, where $\frac{ \partial h^{*}}{\partial F}>0$ and $\frac{\partial q^{*}}{\partial F} <0.$ The first term is indeterminate and the second is negative. Thus, the effect is indeterminate. The overall effect on welfare of a change in F is

$$\begin{aligned} (HS(q^{*})-F)\frac{\partial h^{*}}{\partial F}+[-\hat{h}^{*}u^{\prime \prime }(q^{*})q^{*}]\frac{\partial q^{*}}{\partial F} -(1-h^{*}), \end{aligned}$$

where the second and third terms are negative and the first is indeterminate.

A per unit fine f generates expected profit $\pi (h,q)=\hat{h}u^{\prime }(q)q-c(h)q-(1-h)(f+HD)q.$ This is analogous to the introduction of recall costs from Proposition 2. The (FOC) to the firm’s profit maximization problem are

$$\begin{aligned} \pi _{h}= & {} Hu^{\prime }(q)q-c^{\prime }(h)q+(HD+f)q=0, \\ \pi _{q}= & {} \hat{h}(u^{\prime \prime }(q)q+u^{\prime }(q))-c(h)-(1-h)(HD+f)=0. \end{aligned}$$

The under supply of q given h and h given q are apparent from these conditions. The second order conditions and the Hessian are identical to the case of recall costs. We have

$$\begin{aligned} \frac{\partial q}{\partial f}= & {} (|\Delta |)^{-1}[-(1-h)c^{\prime \prime }(h)q+Hu^{\prime \prime }(q)q]<0,\text { }\frac{\partial h}{\partial f} =(|\Delta |)^{-1}[-\hat{h}q(u^{\prime \prime \prime }(q)q\\&+2u^{\prime \prime }(q))-(1-h)Hu^{\prime \prime }(q)q]>0, \end{aligned}$$

where $|\Delta |$ $=-\hat{h}(u^{\prime \prime \prime }(q)q+2u^{\prime \prime }(q))c^{\prime \prime }(h)q-(Hu^{\prime \prime }(q)q)^{2}>0.$ Because the revenue from the per unit fine is lump sum transferred to the consumer, the social optimal allocation is not affected.

The effects on profit and consumer surplus of a change in f are shown in the text. $\square$

The overly optimistic consumer: profit maximization and social welfare. Maximization of (17) over a choice of h and q,  respectively, yields the first order conditions

$$\begin{aligned} Hu^{\prime }(q)q-c^{\prime }(h)q+H[s\overline{D}+\hat{s}(L-\underline{D})+ \underline{D}]q+rq=0, \end{aligned}$$

(i)

$$\begin{aligned} \hat{h}(u^{\prime \prime }(q)q+u^{\prime }(q))-c(h)-(1-h)H[s\overline{D}+ \hat{s}(L-\underline{D})+\underline{D}]-(1-h)r=0. \end{aligned}$$

(ii)

Denote the profit maximal solution as $(h^{*},q^{*})$. Social welfare is $W(h,q)=\hat{h}\int \limits _{0}^{q}u^{\prime }(z)dz-c(h)q-(1-h)H[s \overline{D}+s(L-\underline{D})+\underline{D}]q-(1-h)rq,$ so that (i) implies

$$\begin{aligned} W_{h}(h^{*},q^{*})=H[S(q^{*})+(s-\hat{s})(L-\underline{D} )q^{*}]>0, \end{aligned}$$

while (ii) implies

$$\begin{aligned} W_{q}(h^{*},q^{*})= & {} \hat{h}^{*}u^{\prime }(q^{*})-c(h^{*})-(1-h)H[s\overline{D}+s(L-\underline{D})+\underline{D}]-(1-h)r \\= & {} u^{\prime }(q^{*})[\hat{h}^{*}\frac{-u^{\prime \prime }(q^{*})q^{*}}{u^{\prime }(q^{*})}-(1-\hat{h}^{*})\frac{(s-\hat{s})(L- \underline{D})}{u^{\prime }(q^{*})}]. \end{aligned}$$