Supply chain decisions with reference quality effect under the O2O environment

Abstract

The reference quality effect is an important factor that influences consumer purchasing behavior. To investigate how firms should incorporate the reference quality effect under different business models, we focus on a supply chain consisting of a supplier and a retailer where the retailer can be an offline store, a pure online store or a combination of offline and online stores within an offline to online model. We formulate the reference quality effect with a modified Nerlove-Arrow model and use two different sales functions to reflect the fact that the reference quality effect will affect consumers in various ways when the business model varies. Utilizing differential game theory, the equilibrium decisions of the channel members are derived, and the analysis illustrates how the firms should adjust their decisions when the retailers use different retail patterns. The basic model does not allow product returns, but then this assumption is relaxed. Comparison between the two cases shows under what conditions consumers will benefit from the retailer’s allowing product returns.

Appendix: Proof of Propositions 1, 2, 3, and 4

The analysis proceeds by backward induction, solving the retailer’s problem first.

The retailer’s objective is

$$\begin{aligned} V_R (G)=\mathop {\hbox {max}}\limits _{p\ge 0} \int _0^\infty {e^{-\rho t}} (p-w)[a-bp+\lambda kq+(\theta -\lambda k\xi )G]dt \end{aligned}$$

(36)

s.t. $\dot{G}(t)=\alpha q+\beta u-\delta G,G(0)=G_0 \ge 0.$

First, observe that p occurs only in the integrand and not in the dynamics of the goodwill, which allows us first to derive the best response retail $\hat{{p}}(G\left| {w,u,q)} \right.$ by maximizing the integrand in (36) with respect to p. It is evident that the result will be independent of u(G). We can thus denote by $\hat{{p}}(G\left| {w,q)} \right.$ the retailer’s best response retail price. It satisfies the first-order condition (FOC) of maximizing

$$\begin{aligned} (p-w)[a-bp+\lambda kq+(\theta -\lambda k\xi )G]. \end{aligned}$$

The FOC for a maximum is

$$\begin{aligned} -2bp+a+\lambda kq+(\theta -\lambda k\xi )G+bw=0, \end{aligned}$$

(37)

which gives

$$\begin{aligned} \hat{{p}}(G\left| {w,q)} \right. =\frac{a+\lambda kq+(\theta -\lambda k\xi )G+bw}{2b}. \end{aligned}$$

(38)

Here, we get Proposition 1.

We substitute $\hat{{p}}(G\left| {w,q)} \right.$ into the supplier’s problem (13) and obtain the following problem:

$$\begin{aligned} V_S (G)=\mathop {\hbox {max}}\limits _{w,u,q\ge 0} \int _0^\infty {e^{-\rho t}} \left\{ \frac{w[a+\lambda kq+(\theta -\lambda k\xi )G-bw]}{2}-\frac{1}{2}u^{2}-q^{2}\right\} dt, \end{aligned}$$

(39)

s.t. $\dot{G}(t)=\alpha q+\beta u-\delta G,G(0)=G_0 \ge 0.$

Note that the wholesale price does not enter the dynamics of goodwill, and can therefore be solved for separately. The optimal wholesale price $\hat{{w}}(G|q)$ satisfies the FOC

$$\begin{aligned} -2bw+a+\lambda kq+(\theta -\lambda k\xi )G, \end{aligned}$$

(40)

which gives

$$\begin{aligned} \hat{{w}}(G|q)=\frac{a+\lambda kq+(\theta -\lambda k\xi )G}{2b}. \end{aligned}$$

(41)

Inserting $\hat{{w}}(G|q)$ in (41) into (38) gives

$$\begin{aligned} \hat{{p}}(G\left| {q)} \right. =\frac{3[a+\lambda kq+(\theta -\lambda k\xi )G]}{4b}. \end{aligned}$$

(42)

We substitute $\hat{{w}}(G|q)$ into the supplier’s problem in (39) and then we can rewrite the supplier’s objective function as

$$\begin{aligned} V_S (G)=\mathop {\hbox {max}}\limits _{u,q\ge 0} \int _0^\infty {e^{-\rho t}} \left\{ \frac{[a+\lambda kq+(\theta -\lambda k\xi )G]^{2}}{8b}-\frac{1}{2}u^{2}-q^{2}\right\} dt, \end{aligned}$$

(43)

The Hamilton–Jacobi–Bellman (HJB) equation for the supplier can be written as

$$\begin{aligned} \rho V_S =\mathop {\hbox {max}}\limits _{u,q\ge 0} \left\{ \frac{[a+\lambda kq+(\theta -\lambda k\xi )G]^{2}}{8b}-\frac{1}{2}u^{2}-q^{2}+\frac{\partial V_S }{\partial G}(\alpha q+\beta u-\delta G)\right\} . \end{aligned}$$

(44)

We can derive the best advertising level $u^{*}(G)$ and quality level $q^{*}(G)$ by maximizing the right-hand side in (A.9) with respect to u and q, which gives

$$\begin{aligned} u^{*}(G)= & {} \beta (\partial V_S /\partial G), \end{aligned}$$

(45)

$$\begin{aligned} q^{*}(G)= & {} \frac{\lambda k[a+(\theta -\lambda k\xi )G]+4b\alpha (\partial V_S /\partial G)}{8b-(\lambda k)^{2}} \end{aligned}$$

(46)

Substituting $q^{*}(G)$ in (46) into (41) and (42), we get

$$\begin{aligned} w^{*}(G)= & {} 2\frac{2[a+(\theta -\lambda k\xi )G]\hbox {+}\alpha \lambda k(\partial V_S /\partial G)}{8b-(\lambda k)^{2}}. \end{aligned}$$

(47)

$$\begin{aligned} p^{*}(G)= & {} 3\frac{2[a+(\theta -\lambda k\xi )G]\hbox {+}\alpha \lambda k(\partial V_S /\partial G)}{8b-(\lambda k)^{2}}. \end{aligned}$$

(48)

Inserting $u^{*}(G)$, $q^{*}(G)$ in (45) and (46) into the HJB equation in (44), we get

$$\begin{aligned} \rho V_S= & {} \frac{[a+(\theta -\lambda k\xi )G]^{2}}{8b-(\lambda k)^{2}}+\frac{\lambda k[a+(\theta -\lambda k\xi )G-bc](\partial V_S /\partial G)}{8b-(\lambda k)^{2}} \nonumber \\&+\frac{2b(\partial V_S /\partial G)^{2}}{8b-(\lambda k)^{2}}+\frac{1}{2}\beta ^{2}(\partial V_S /\partial G)^{2}-\delta G(\partial V_S /\partial G) \end{aligned}$$

(49)

Let $V_S =\kappa _1 G^{2}+\omega _1 G+\varepsilon _1$. Then, $\partial V_S /\partial G=2\kappa _1 G+\omega _1$. Substituting these into expressions (45)–(49), the equilibrium feedback advertising, quality, and price policies become

$$\begin{aligned} u^{*}(G)= & {} 2\beta \kappa _1 G+\beta \omega _1 , \end{aligned}$$

(50)

$$\begin{aligned} q^{*}(G)= & {} \frac{\lambda ka+4b\alpha \omega _1 }{8b-(\lambda k)^{2}}+\frac{\lambda k(\theta -\lambda k\xi )+8b\alpha \kappa _1 }{8b-(\lambda k)^{2}}G, \end{aligned}$$

(51)

$$\begin{aligned} w^{*}(G)= & {} 2\frac{2a+\alpha \lambda k\omega _1 }{8b-(\lambda k)^{2}}+4\frac{\theta -\lambda k\xi +\alpha \lambda k\kappa _1 }{8b-(\lambda k)^{2}}G, \end{aligned}$$

(52)

$$\begin{aligned} p^{*}(G)= & {} 3\frac{2a\hbox {+}\alpha \lambda k\omega _1 }{8b-(\lambda k)^{2}}+6\frac{\theta -\lambda k\xi \hbox {+}\alpha \lambda k\kappa _1 }{8b-(\lambda k)^{2}}G. \end{aligned}$$

(53)

Equating like powers of G in the HJB equation, we can express all the unknowns in terms of $\kappa _1$, which itself can be explicitly solved. That is,

$$\begin{aligned} \kappa _1= & {} \frac{M-\sqrt{M^{2}-8(\theta -\lambda k\xi )^{2}N}}{4N},\nonumber \\ \omega _1= & {} \frac{2a(\theta -\lambda k\xi +\alpha \lambda k\kappa _1 )}{(\rho +\delta )[8b-(\lambda k)^{2}]-\alpha \lambda k(\theta -\lambda k\xi )-2N\kappa _1 },\\ \sigma _1= & {} \frac{2a^{2}+2\alpha \lambda ka\omega _1 +N\omega _1 ^{2}}{2\rho [8b-(\lambda k)^{2}]},\nonumber \end{aligned}$$

(54)

where $M=(\rho +2\delta )[8b-(\lambda k)^{2}]-2(\theta -\lambda k\xi )\alpha \lambda k$ and $N=4b(\alpha ^{2}+2\beta ^{2})-(\beta \lambda k)^{2}$.

The sufficient conditions for $u^{*}(G)\ge 0$, $q^{*}(G)\ge 0$, $w^{*}(G)\ge 0$, $p^{*}(G)\ge 0$ are given by $a>0$, $8b-(\lambda k)^{2}>0$, $\theta -\lambda k\xi \ge 0$, $\kappa _1 \ge 0$, and $\omega _1 \ge 0$.

The sufficient conditions for $\kappa _1 \ge 0$ are given by $8b-(\lambda k)^{2}>0$ and $(\rho +2\delta )[8b-(\lambda k)^{2}]-2\alpha \lambda k(\theta -\lambda k\xi )>0.$

The sufficient conditions for $\omega _1 \ge 0$ are given by $a\ge 0$, $8b-(\lambda k)^{2}>0$, $\theta -\lambda k\xi \ge 0$, and $\kappa _1 \ge 0$.

We can conclude that the sufficient conditions for $u^{*}(G)\ge 0$, $q^{*}(G)\ge 0$, $w^{*}(G)\ge 0$, $p^{*}(G)\ge 0$ are given by

• (1) $a>0$;

• (2) $\theta -\lambda k\xi \ge 0$;

• (3) $0\le k<2[\sqrt{\alpha ^{2}\theta ^{2}+8b(\rho +2\delta )(\rho +2\varphi )}-\alpha \theta ]/(\rho +2\varphi )$;

• (4) $k<\sqrt{8b}/\lambda$.

Thus, we get Proposition 2.

Under the above four conditions, substituting $u^{*}(G)$ and $q^{*}(G)$ into Equation (3), we have

$$\begin{aligned} \dot{G}(t)= & {} \frac{\alpha \lambda ka}{8b-(\lambda k)^{2}}+\left[ \frac{4b\alpha ^{2}}{8b-(\lambda k)^{2}}+\beta ^{2} \right] \omega _1 \nonumber \\&+\left\{ \frac{\alpha \lambda k(\theta -\lambda k\xi )}{8b-(\lambda k)^{2}} +2\left[ \frac{4b\alpha ^{2}}{8b-(\lambda k)^{2}}+\beta ^{2}\right] \kappa _1 -\delta \right\} G \end{aligned}$$

(55)

The general solution of (55) is

$$\begin{aligned} G^{*}(t)=G_{\textit{ss}} +(G_0 -G_{\textit{ss}} )e^{-mt} \end{aligned}$$

(56)

where

$$\begin{aligned} m=-\frac{1}{2}\rho +\frac{\sqrt{M^{2}-8(\theta -\lambda k\xi )^{2}N}}{2[8b-(\lambda k)^{2}]} \end{aligned}$$

and

$$\begin{aligned} G_{\textit{ss}} =\frac{a[\alpha \lambda k(\rho +\delta )+2(\alpha ^{2}+2\beta ^{2})(\theta -\lambda k\xi )]}{[8b-(\lambda k)^{2}]m(m+\rho )}. \end{aligned}$$

Here, we get Proposition 3.

Substituting $G^{*}(t)$ in Eq. (56) into Eqs. (50)–(53), we get

$$\begin{aligned} u^{*}(t)= & {} u_{\textit{ss}} +2\beta \kappa _1 (G_0 -G_{\textit{ss}} )e^{-mt}, \\ q^{*}(t)= & {} q_{\textit{ss}} +\frac{(\theta -\lambda k\xi )\lambda k+8b\alpha \kappa _1 }{8b-(\lambda k)^{2}}(G_0 -G_{\textit{ss}} )e^{-mt}, \\ w^{*}(t)= & {} w_{\textit{ss}} +4\frac{\theta -\lambda k\xi +\alpha \lambda k\kappa _1 }{8b-(\lambda k)^{2}}(G_0 -G_{\textit{ss}} )e^{-mt},\\ p^{*}(t)= & {} p_{\textit{ss}} +6\frac{\theta -\lambda k\xi +\alpha \lambda k\kappa _1 }{8b-(\lambda k)^{2}}(G_0 -G_{\textit{ss}} )e^{-mt}, \end{aligned}$$

where

$$\begin{aligned} u_{\textit{ss}}= & {} \frac{2\beta \delta (a-bc)(\theta -\lambda k\xi )}{[8b-(\lambda k)^{2}]m(m+\rho )},\\ q_{\textit{ss}}= & {} \frac{a\delta [\lambda k(\rho +\varphi )+\alpha \theta ]}{[8b-(\lambda k)^{2}]m(m+\rho )},\\ w_{\textit{ss}}= & {} \frac{4a\delta (\rho +\delta )}{[8b-(\lambda k)^{2}]m(m+\rho )},\\ p_{\textit{ss}}= & {} \frac{6a\delta (\rho +\delta )}{[8b-(\lambda k)^{2}]m(m+\rho )}. \end{aligned}$$

Thus, Proposition 4 is proved.

Proofs for Proposition 5, 6 and 7 are similar to those above and are omitted.