Consumer returns reduction and information revelation mechanism for a supply chain

Abstract

In this paper, we develop two revelation mechanism models of a supply chain consisting of one manufacturer and one retailer under asymmetric information, where the retailer provides store assistance (SA) to reduce consumer returns rate and increase demand. Under full information, we find that a higher returns rate or returns handling cost increases the SA level if the market scale is sufficiently high. In the demand information asymmetry model, we find that: (i) the low-type retailer (facing a low demand) has no incentive to distort demand information while the high-type retailer may report wrong information; (ii) the manufacturer would like to design a menu of wholesale price-order quantity contract to induce truthful demand information and the manufacturer pays an information rent to the high-type retailer if the returns rate or returns handling cost for the retailer is sufficiently low; (iii) asymmetry of information does not change the monotonicity of the unit wholesale price in the retailer’s type, and information asymmetry decreases the retail price but increases the SA level. Unlike the demand information asymmetry model, a higher retailer’s returns handling cost expands the effects of information asymmetry on the retail price and the SA level, and using revelation mechanism decreases the channel profit if the retailer’s returns handling cost is sufficiently high under the returns rate information asymmetry model.

Appendix

Proof of Proposition 1

The Hessian matrix of the retailer’s profit over $(p,s)$ is

$$\begin{aligned} \left( {{\begin{array}{ll} {-2}&{} {(h_C -h_R )\lambda } \\ {(h_C -h_R )\lambda }&{} {-(\eta -2h_{C} h_R \lambda ^{2})} \\ \end{array} }} \right) \end{aligned}$$

which is negatively definite if $\eta >(h_C +h_R )^{2}\lambda ^{2}/2$. Solving the first-order conditions $\partial \pi _{Ri} /\partial p=0$ and $\partial \pi _{Ri} /\partial s=0$ for $(p,s)$, we have

$$\begin{aligned} p_i (w)&= \frac{(h_R -h_C )\eta \lambda +w(\eta -h_C h_T \lambda ^{2})+\overline{{v}}_i (\eta -h_R h_T \lambda ^{2})}{2\eta -h_T^2 \lambda ^{2}} \hbox { and } \nonumber \\ s_i (w)&= \frac{h_T \lambda (\overline{{v}}_i -w-h_T \lambda )}{2\eta -h_T^2 \lambda ^{2}}. \end{aligned}$$

Inserting $p_i (w)$ and $s_i (w)$ into (3), we have

$$\begin{aligned} \pi _{Mi} (w)=\frac{\eta (w-c)(\overline{{v}}_i -w-h_T \lambda )}{2\eta -h_T^2 \lambda ^{2}}, \end{aligned}$$

which is a concave function of $w$ following from $\eta >h_T^2 \lambda ^{2}/2$. Solving the first-order condition $\partial \pi _{Mi} (w)/\partial w=0$, we have $w_{i0}^{*} =(c+\overline{{v}}_i -h_T \lambda )/2$. Further, we can complete the proof. $\square$

Proof of Lemma 2

Differentiating $\pi _{Ri1} (s_{i1} ,w_j )$ twice with respect to $s_{i1}$, we have $\partial ^{2}\pi _{Ri1} (s_{i1} ,w_j )/\partial s_{i1}^2 =-\eta <0$, i.e., the second-order condition is satisfied. By solving the first-order condition $\partial \pi _{Ri1} (s_{i1} ,w_j )/\partial s_{i1} =h_T \lambda q_j (w_j )-\eta s_{i1} =0$, we have $s_{i1} (w_j )=h_T \lambda q_j (w_j )/\eta$. Further, we have

$$\begin{aligned} p_{i1} (w_j )=\overline{{v}}_i -(1-h_T \lambda q_j (w_j )/\eta )\lambda h_C -q_j (w_j ). \end{aligned}$$

Therefore, we can obtain the maximum profit of the type-$i$ retailer $\pi _{Ri1} (w_j )$.

$\square$

Proof of Proposition 2

Part (i) Differentiating $\pi _{Ri1} (w_j )$ twice with respect to $w_j$, we have

$$\begin{aligned} \hbox {d}^{2}\pi _{Ri1} (w_j )/\hbox {d}w_j^2 =\eta /(2\eta -h_T^2 \lambda ^{2})>0, \end{aligned}$$

i.e., $\pi _{Ri1} (w_j )$ is a convex function of $w_j$. Note that $\pi _{Ri1} (\overline{{v}}_j -\lambda h_T )=0$,

$$\begin{aligned} \hbox {d}\pi _{RL1} (\overline{{v}}_H -{h}_{T}\lambda )/\hbox {d}w_H = \eta (\overline{{v}}_H \!-\!\overline{{v}}_L )/(2\eta \!-\! h_T^2 \lambda ^{2})\!>\!0 \hbox { and } \hbox {d}\pi _{RH1} (\overline{{v}}_L \!-\!\lambda h_T )/\hbox {d}w_L \!<\!0. \end{aligned}$$

Further, we have $\pi _{RL1} (w_H )>0$ for $w_H <\hat{{w}}_{H-}$ and $\pi _{RH1} (w_L )>0$ for $w_L <\overline{{v}}_L -{h}_{T}\lambda$. Note that the manufacturer always offers a unit wholesale price $w_L <\overline{{v}}_L -{h}_{T}\lambda$ to induce a positive order of the retailer. Thus, we have $\pi _{RH1} (w_L )>0$.

Part (ii) From Proposition 1, we know that the unit wholesale price for the type-$H$retailer should be higher than that for the type-$L$ retailer under the asymmetric market scale, i.e., $w_H >w_L$. If $w_H \ge \hat{{w}}_{H-}$, we have $\pi _{RL1} (w_H )\le 0$, where $\pi _{RL} (w_L )>\pi _{RL1} (w_H )$. From Proof of Proposition 2(i), we see that if $w_H <\hat{{w}}_{H-}$, $\pi _{RL1} (w_H )$ is a decreasing function of $w_H$. Thus, we have $\pi _{RL1} (w_L )>\pi _{RL1} (w_H )$. However, from Lemmas 1 and 2, it follows that $\pi _{RL} (w_L )-\pi _{RL1} (w_L )=\frac{(\overline{{v}}_H\,-\,\overline{{v}}_L )^{2}\eta }{2(2\eta \, -\,h_T^2 \lambda ^{2})}>0$. Thus, we have $\pi _{RL} (w_L )>\pi _{RL1} (w_L )>\pi _{RL1} (w_H )$, i.e., the type-$L$ retailer has no incentive to report wrong demand information.

Part (iii) We see that the type-$H$retailer has an incentive to report wrong market scale information if $\pi _{RH} (w_H )<\pi _{RH1} (w_L )$, i.e., $\frac{\eta (\overline{{v}}_H\, -\,w_H\, -\,h_T \lambda )^{2}}{2(2\eta \, -\,h_T^2 \lambda ^{2})}<\pi _{RH1} (w_L )$. According to $\eta >h_T^2 \lambda ^{2}/2$, $\overline{{v}}_H -w_H -h_T \lambda >0$, and $\pi _{RH1} (w_L )>0$, we can rewrite it as

$\overline{{v}}_H -w_H -h_T \lambda <\sqrt{2(2-h_T^2 \lambda ^{2}/\eta )\cdot \pi _{RH1} (w_L )}.$ $\square$

Proof of Proposition 3

From (5), it follows that the Hessian matrix of the manufacturer’s profit over $(w_L ,w_H )$ is

$$\begin{aligned} \left( {{\begin{array}{l@{\quad }l} {-2\alpha \eta /(2\eta -h_T^2 \lambda ^2 )}&{} 0 \\ 0&{} {-2\eta (1-\alpha )/(2\eta -h_T^2 \lambda ^2 )} \\ \end{array} }} \right) , \end{aligned}$$

which is negatively definite following from $\eta >h_T^2 \lambda ^2 /2$ and $0<\alpha <1$. That is, the second-order condition is satisfied. Similar to Proposition 1, we obtain the unit wholesale price for the type-$L$ retailer $w_{L0}^{*}$. When constraint (6) is binding, we have $\hat{{w}}_H =\overline{{v}}_H -h_T \lambda -\sqrt{2(2-h_T^2 \lambda ^{2}/\eta )\cdot \pi _{RH1} (w_{L0}^{*} )}$.

From Lemma 1, $\eta >h_T^2 \lambda ^{2}/2$ and $\overline{{v}}_H -w_H -h_T \lambda >0$, it follows that $\pi _{RH} (w_H )<\pi _{RH} (w_{L0}^{*} )$ for $w_H >w_{L0}^{*}$. Moreover, we have

$$\begin{aligned} \pi _{RH} (w_{L0}^{*} )-\pi _{RH1} (w_{L0}^{*} )=\frac{(\overline{{v}}_H -\overline{{v}}_L )^{2}\eta }{2(2\eta -h_T^2 \lambda ^{2})}>0. \end{aligned}$$

Further, it follows that $\hat{{w}}_H >w_{L0}^{*}$ because $\pi _{RH} (w_H )$ decreases with $w_H$.

Now, we show that the manufacturer has an incentive to design a revelation mechanism. If the manufacturer does not design the revelation mechanism, the type-$H$ retailer may have an incentive to choose the contract $(w_{L0}^{*} ,q_L (w_{L0}^{*} ))$. Note that $\min \{w_{H0}^{*} ,\hat{{w}}_H \}>w_{L0}^{*}$ and the type-$L$ retailer chooses the right contract. We only need to show that

$$\begin{aligned} q_H (\min \{w_{H0}^{*} ,\hat{{w}}_H \})\!=\!\frac{\eta (\overline{{v}}_H -\min \{w_{H0}^{*} ,\hat{{w}}_H \}-h_T \lambda )}{2\eta -h_T^2 \lambda ^{2}}>q_L (w_{L0}^{*} )\!=\!\frac{\eta (\overline{{v}}_L \!-\!w_{L0}^{*} -h_T \lambda )}{2\eta -h_T^2 \lambda ^{2}}. \end{aligned}$$

We show this inequality as follows: From Proposition 1, we see that $q_H (w_{H0}^{*} )=q_{H0}^{*} >q_{L0}^{*} =q_L (w_{L0}^{*} )$ because of $\overline{{v}}_H >\overline{{v}}_L$ and $\eta >h_T^2 \lambda ^{2}/2$. Further, we have $q_H (\min \{w_{H0}^{*} ,\hat{{w}}_H \})>q_H (w_{H0}^{*} )>q_L (w_{L0}^{*} )$. $\square$

Proof of Proposition 4

Similar to Lemma 2, when the type-$i$ retailer reports wrong type information, its optimal profit is

$$\begin{aligned} \overline{{\pi }}_{Ri2} (w_j )=\overline{{q}}_j (w_j )[2\eta (\overline{{v}}-w_j -h_T \lambda _i )-\overline{{q}}_j (w_j )(2\eta -h_T^2 \lambda _i^2 )]/(2\eta ), \end{aligned}$$

which is a convex function of $w_j$ following from $\eta >h_T^2 (2\lambda _H^2 -\lambda _L^2 )/2$ and $\lambda _H >\lambda _L >0$.

Similar to Proposition 2(i), we can show that $\overline{{\pi }}_{RH2} (w_L )$ is positive and decreases with $w_L$ for $w_L <\hat{{\overline{{w}}}}_{L-}$, and $\overline{{\pi }}_{RL2} (w_H )>0$, where $\hat{{\overline{{w}}}}_{L-}$ is the minimum positive root of $\overline{{\pi }}_{RH2} (w_L )=0$.

Part (i) The type-$L$retailer has a incentive to report wrong type information if $\overline{{\pi }}_{RL} (w_L )<\overline{{\pi }}_{RL2} (w_H )$, i.e.,

$$\begin{aligned} \frac{\eta (\overline{{v}}-w_L -h_T \lambda _L )^{2}}{2(2\eta -h_T^2 \lambda _L^2 )}<\overline{{\pi }}_{RL2} (w_H ). \end{aligned}$$

(11)

Since $\eta >h_T^2 \lambda _L^2 /2$ and $\overline{{v}}-w_L -h_T \lambda _L >0$, we can rewrite (11) as

$$\begin{aligned} \overline{{v}}-w_L -h_T \lambda _L <\sqrt{2(2-h_T^2 \lambda _L^2 /\eta )\cdot \overline{{\pi }}_{RL2} (w_H )}. \end{aligned}$$

Part (ii) Note that under full basic returns rate information, the unit wholesale price for the type-$L$ retailer is higher than that for the type-$H$ retailer. Thus, under the revelation mechanism, the unit wholesale prices should satisfy $w_L >w_H$. Note that $\overline{{\pi }}_{RH2} (w_L )\le 0$ for $w_L \ge \hat{{\overline{{w}}}}_{L-}$. We only need to consider the case with $w_L <\hat{{\overline{{w}}}}_{L-}$. Note that $\overline{{\pi }}_{RH2} (w_L )$ decreases with $w_L$ for $w_L <\hat{{\overline{{w}}}}_{L-}$. Thus, we have $\overline{{\pi }}_{RH2} (w_H )>\overline{{\pi }}_{RH2} (w_L )$. From Lemma 3 and $\eta >h_T^2 \lambda _H^2 /2$, we have

$$\begin{aligned} \overline{{\pi }}_{RH} (w_H )\!-\!\overline{{\pi }}_{RH2} (w_H )\!=\!\frac{\eta h_T^2 (\lambda _H -\lambda _L )^{2}[2\eta +h_T^2 \lambda _L \lambda _H -h_T (\overline{{v}}-w_H )(\lambda _H +\lambda _L )]^{2}}{2(2\eta \!-\!h_T^2 \lambda _H^2 )(2\eta -h_T^2 \lambda _L^2 )^{2}}\!>\!0 \end{aligned}$$

Thus, we have $\overline{{\pi }}_{RH} (w_H )>\overline{{\pi }}_{RH2} (w_H )>\overline{{\pi }}_{RH2} (w_L )$. $\square$

Proof of Proposition 5

From (9), we see that the Hessian matrix of the manufacturer profit over $(w_L ,w_H )$ is

$$\begin{aligned} \left( {{\begin{array}{l@{\quad }l} {-2k\eta /(2\eta -h_T^2 \lambda _L^2 )}&{} 0 \\ 0&{} {-2\eta (1-k)/(2\eta -h_T^2 \lambda _H^2 )} \\ \end{array} }} \right) , \end{aligned}$$

which is negatively definite following from $\eta >h_T^2 (2\lambda _H^2 -\lambda _L^2 )/2>h_T^2 \lambda _H^2 /2$ and $0<k<1$. That is, the second-order condition is satisfied.

Similar to Proposition 1, we obtain the unit wholesale price for the type-$H$ retailer $\overline{{w}}_{H0}^{*}$. Inserting $\overline{{w}}_{H0}^{*}$ into (10), and then solving the binding constraint (10), we obtain $\hat{{\overline{{w}}}}_L$. Similar to Proposition 3, we can show that $\hat{{\overline{{w}}}}_L >\overline{{w}}_{H0}^{*}$ and the manufacturer has an incentive to design a revelation mechanism. $\square$