Self-building or cooperating with a service platform: how should a dual-channel firm implement a trade-in program?

Abstract

Business practices demonstrate that firms can offer dual-channel trade-in programs by cooperating with a third-party trade-in service platform (i.e., cooperation mode) or on their own (i.e., self-building mode). This paper develops an analytical model to investigate the effect of a specific trade-in operation mode on the profitability of a dual-channel firm that implements a trade-in program. We characterize the equilibrium between the firm and the service platform. We identify the threshold of the self-building operational cost, above (below) which the firm’s profit under the self-building mode is less (greater) than that under the cooperation mode. We find that the cooperation mode is more beneficial to both consumers and a firm compared with the self-building mode when the consumer waiting cost is minor-to-moderate and the market potential of the cooperation mode is relatively high, which will improve social welfare due to the decrease in operational cost and product price. Finally, we consider a cooperative bargaining situation under the cooperation mode. We find that both the firm and the service platform will obtain more profits from the bargaining solution in a bilateral negotiation compared with the case of a dynamic game.

Appendix

Proof of Proposition 1

When ${c}_{t}<{c}_{w}<{c}_{t}+1$, we have ${d}_{s}^{1}={\int }_{0}^{{c}_{w}-{c}_{t}}{\int }_{\frac{p+{c}_{t}-t}{(1-\delta ){v}_{1}}}^{1}1d{\alpha }_{1}d{\alpha }_{2}=\left({c}_{w}-{c}_{t}\right)\left(1-\frac{p+{c}_{t}-t}{\left(1-\delta \right){v}_{1}}\right)$ and ${d}_{s}^{2}={\int }_{{c}_{w}-{c}_{t}}^{1}{\int }_{\frac{p+{c}_{t}-t}{(1-\delta ){v}_{1}}}^{1}1d{\alpha }_{1}d{\alpha }_{2}=\left(1-{c}_{w}+{c}_{t}\right)\left(1-\frac{p+{c}_{t}-t}{\left(1-\delta \right){v}_{1}}\right)$. When ${c}_{w}\le {c}_{t}$, we have ${d}_{s}^{1}=0$ and ${d}_{s}^{2}={\int }_{0}^{1}{\int }_{\frac{p+{c}_{w}-{\alpha }_{2}-t}{(1-\delta ){v}_{1}}}^{1}1d{\alpha }_{1}d{\alpha }_{2}=1-\frac{2\left(p-t+{c}_{w}\right)-1}{2\left(1-\delta \right){v}_{1}}$. Define $T=p-t$ and substituting ${d}_{s}^{1}$ and ${d}_{s}^{2}$ into the profit function ${\pi }_{s}^{a}=\left(p-t+s\right){d}_{s}^{1}+(p-t+s-c){d}_{s}^{2}$. Using the first-order conditions, $\frac{d{\pi }_{s}^{a}}{dT}=0$, we have ${T}^{*}=\frac{\left({c}_{t}-{c}_{w}+1\right)c}{2}-\frac{{c}_{t}}{2}+\frac{\left(-\delta +1\right){v}_{1}}{2}-\frac{s}{2}$ if ${c}_{t}<{c}_{w}<{c}_{t}+1$; ${T}^{*}=\frac{1}{2}(1-\delta ){v}_{1}+\frac{1}{2}c-\frac{1}{2}{c}_{w}-\frac{1}{2}s+\frac{1}{4}$ if ${c}_{w}\le {c}_{t}$. The second-order conditions are negative. And then we can obtain the equilibrium ${\pi }_{s}^{a*}$.

Proof of Lemma 1

When ${c}_{t}<{c}_{w}<{c}_{t}+1$, we have ${d}_{c}^{1}=\gamma {\int }_{0}^{{c}_{w}-{c}_{t}}{\int }_{\frac{p+{c}_{t}-t}{(1-\delta ){v}_{1}}}^{1}1d{\alpha }_{1}d{\alpha }_{2}=\gamma \left({c}_{w}-{c}_{t}\right)\left(1-\frac{p+{c}_{t}-t}{\left(1-\delta \right){v}_{1}}\right)$ and ${d}_{c}^{2}=\gamma {\int }_{{c}_{w}-{c}_{t}}^{1}{\int }_{\frac{p+{c}_{t}-t}{(1-\delta ){v}_{1}}}^{1}1d{\alpha }_{1}d{\alpha }_{2}=\gamma \left(1-{c}_{w}+{c}_{t}\right)\left(1-\frac{p+{c}_{t}-t}{\left(1-\delta \right){v}_{1}}\right)$. When ${c}_{w}\le {c}_{t}$, we have ${d}_{c}^{1}=0$ and ${d}_{c}^{2}=\gamma {\int }_{0}^{1}{\int }_{\frac{p+{c}_{w}-{\alpha }_{2}-t}{(1-\delta ){v}_{1}}}^{1}1d{\alpha }_{1}d{\alpha }_{2}=\upgamma \left(1-\frac{2\left(p-t+{c}_{w}\right)-1}{2\left(1-\delta \right){v}_{1}}\right)$. For a given $\tau$, substituting ${d}_{c}^{1}$ and ${d}_{c}^{2}$ into the profit function ${\pi }_{c}^{a}=\left(p-t+s\right){d}_{c}^{1}+\left(p-t+s-\tau \right){d}_{c}^{2}$ and using the first-order conditions $\frac{d{\pi }_{c}^{a}}{dp}=0$, we have ${p}^{*}=\frac{\left(1-{c}_{w}+{c}_{t}\right)\tau }{2}+\frac{\left(1-\delta \right){v}_{1}}{2}-\frac{s+{c}_{t}-2t}{2}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$; ${p}^{*}=\frac{\tau }{2}+\frac{\left(1-\delta \right){v}_{1}}{2}-\frac{s+{c}_{w}-2t}{2}+\frac{1}{4}$ if ${c}_{w}\le {c}_{t}$. The second-order conditions are negative. And then we can get $\frac{{dp}^{*}}{d\tau }=\frac{1-({c}_{w}-{c}_{t})}{2}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$; $\frac{{dp}^{*}}{d\tau }=\frac{1}{2}$ if ${c}_{w}\le {c}_{t}$.

Proof of Proposition 2

Following Lemma 1, plugging ${p}^{*}$ into ${\pi }_{c}^{b}=\tau {d}_{c}^{2}$ and using the first-order conditions $\frac{d{\pi }_{c}^{b}}{d\tau }=0$, we have ${\tau }^{*}=\frac{\left(1-\delta \right){v}_{1}-{c}_{t}+s}{2\left(1-{c}_{w}+{c}_{t}\right)}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$; ${\tau }^{*}=\frac{\left(1-\delta \right){v}_{1}+s-{c}_{w}}{2}+\frac{1}{4}$ if ${c}_{w}\le {c}_{t}$. The second-order conditions are negative.

Proof of Proposition 3

By comparing the firm’s profits under the self-building and cooperation modes, we have $\frac{{\pi }_{s}^{a*}}{{\pi }_{c}^{a*}}=\frac{4{\left({v}_{1}\left(1-\delta \right)-c(1-{c}_{w}+{c}_{t})+s-{c}_{t}\right)}^{2}}{{\gamma \left(\left(1-\delta \right){v}_{1}-{c}_{t}+s\right)}^{2}}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$; $\frac{{\pi }_{s}^{a*}}{{\pi }_{c}^{a*}}=\frac{4{[{2v}_{1}\left(1-\delta \right)-2c-2{c}_{w}+2s+1]}^{2}}{{\gamma [{2v}_{1}\left(1-\delta \right)-2{c}_{w}+2s+1]}^{2}}$ if ${c}_{w}\le {c}_{t}$. When ${c}_{t}<{c}_{w}<1+{c}_{t}$, since ${u}_{1}\ge 0$ and $c<s$, it is easy to verify that, if the inequality $c\ge \frac{(2-\sqrt{\gamma })(\left(1-\delta \right){v}_{1}-{c}_{t}+s)}{2(1-{c}_{w}+{c}_{t})}$ holds, then $\frac{{\pi }_{s}^{a*}}{{\pi }_{c}^{a*}}\le 1$; otherwise, $\frac{{\pi }_{s}^{a*}}{{\pi }_{c}^{a*}}>1$. When ${c}_{w}<{c}_{t}$, since ${u}_{3}\ge 0$ and $c<s$, it is easy to verify that, if the inequality $c\ge \frac{(2-\sqrt{\gamma })(2\left(1-\delta \right){v}_{1}-2{c}_{w}+2s+1)}{4}$ holds, then $\frac{{\pi }_{s}^{a*}}{{\pi }_{c}^{a*}}\le 1$; otherwise, $\frac{{\pi }_{s}^{a*}}{{\pi }_{c}^{a*}}>1$.

Proof of Proposition 4

When ${c}_{t}<{c}_{w}<1+{c}_{t}$, we have ${CS}_{s}^{1}={\int }_{0}^{{c}_{w}-{c}_{t}}{\int }_{\frac{p+{c}_{t}-t}{(1-\delta ){v}_{1}}}^{1}((1-{\delta )\alpha }_{1}{v}_{1}-p-{c}_{t}+t)d{\alpha }_{1}d{\alpha }_{2}+{\int }_{{c}_{w}-{c}_{t}}^{1}{\int }_{\frac{p+{c}_{t}-t}{(1-\delta ){v}_{1}}}^{1}((1-\delta ){\alpha }_{1}{v}_{1}-p-{c}_{w}+{\alpha }_{2}+t)d{\alpha }_{1}d{\alpha }_{2}=\frac{\left(\left(\delta -1\right){v}_{1}+p-t+{c}_{t}\right)\left(\left(\delta -1\right){v}_{1}-{{c}_{t}}^{2}+\left(2{c}_{w}-1\right){c}_{t}-{{c}_{w}}^{2}+p-t+2{c}_{w}-1\right)}{2\left(1-\delta \right){v}_{1}}$. When ${c}_{w}\le {c}_{t}$, we have ${CS}_{s}^{2}={\int }_{0}^{1}{\int }_{\frac{p+{c}_{w}-t-{\alpha }_{2}}{(1-\delta ){v}_{1}}}^{1}((1-\delta ){\alpha }_{1}{v}_{1}-p-{c}_{w}+{\alpha }_{2}+t)d{\alpha }_{1}d{\alpha }_{2}=\frac{-3{\left(\delta -1\right)}^{2}{v}_{1}^{2}-6\left(p-t+{c}_{w}-\frac{1}{2}\right)\left(\delta -1\right){v}_{1}-3{c}_{w}^{2}+\left(3-6(p-t)\right){c}_{w}-3{(p-t)}^{2}+3(p-t)-1}{6\left(\delta -1\right){v}_{1}}$. Following Proposition 1 and plugging ${T}^{*}$ into ${CS}_{s}^{1}$ and ${CS}_{s}^{2}$, we can get ${CS}_{s}^{1*}=\frac{\left(\left(c+1\right){c}_{t}+\left(1-{c}_{w}\right)c+\left(\delta -1\right){v}_{1}-s\right)\left(-2{c}_{t}^{2}+ \left(c+4{c}_{w}-3\right){c}_{t}+\left(1-{c}_{w}\right)c-2{c}_{w}^{2}+4{c}_{w}+\left(\delta -1\right){v}_{1}-s-2\right)}{8\left(1-\delta \right){v}_{1}}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$; ${CS}_{s}^{2*}=\frac{12{\left(\delta -1\right)}^{2}{v}_{1}^{2}+24\left(c-s+{c}_{w}-\frac{1}{2}\right)\left(\delta -1\right){v}_{1}+12{c}_{w}^{2}-\left(12-24c+24s\right){c}_{w}+12{\left(c-s\right)}^{2}-12c+12s+7}{96\left(1-\delta \right){v}_{1}}$ if ${c}_{w}\le {c}_{t}$. Following the same approach as in the case of self-building mode, we can derive ${CS}_{c}^{1*}$ and ${CS}_{c}^{2*}$ under the cooperation mode.

Proof of Proposition 5

As $\Omega ={({\pi }_{c}^{a}-{\pi }_{c}^{a0})}^{1-\beta }{({\pi }_{c}^{b}-{\pi }_{c}^{b0})}^{\beta }={({\pi }_{c}^{a}-{\pi }_{s}^{a*})}^{1-\beta }{({\pi }_{c}^{b}-0)}^{\beta }$, we can take the logarithm of $\Omega$ as $\mathrm{ln\Omega }=\left(1-\beta \right)\mathrm{ln}\left({\pi }_{c}^{a}-{\pi }_{s}^{a*}\right)+\beta \mathrm{ln}({\pi }_{c}^{b})$ and define $F\triangleq \mathrm{ln\Omega }$. For tractability, we define ${k}_{0}\triangleq s-{c}_{t}$, ${k}_{1}\triangleq \left(1-\delta \right){v}_{1}$, ${k}_{2}\triangleq 1-{c}_{w}+{c}_{t}$, ${k}_{3}\triangleq \left({c}_{w}-1\right)c+s$, and ${k}_{4}\triangleq s-{c}_{w}+\frac{1}{2}$. Since ${\pi }_{c}^{a}$ and ${\pi }_{c}^{b}$ are concave functions of $T$ and $\tau$ (proofs of Lemma 1 and Proposition 2), we can use the first-order conditions ($\frac{\partial F}{\partial \tau }=0$ and $\frac{\partial F}{\partial T}=0$) to obtain ${\widetilde{\tau }}^{*}=\frac{\beta \left(\left(\gamma -1\right){k}_{1}^{2}+2{k}_{1}\left(c{k}_{2}+\left(\gamma -1\right){k}_{0}\right)-{c}^{2}{k}_{2}^{2}+2c{k}_{2}{k}_{0}+(\gamma -1){k}_{0}^{2}\right)}{2\gamma {k}_{2}({k}_{1}+{k}_{0})}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$; ${\widetilde{\tau }}^{*}=\frac{\beta \left(\left(\gamma -1\right){k}_{1}^{2}+2{k}_{1}\left(c+(\gamma -1){k}_{4}\right)+\gamma {k}_{4}^{2}-{(c-{k}_{4})}^{2}\right)}{2\gamma ({k}_{1}+{k}_{4})}$ if ${c}_{w}\le {c}_{t}$. And then we can derive the equilibrium ${\widetilde{T}}^{*}$, ${\widetilde{\pi }}_{c}^{a*}$ and ${\widetilde{\pi }}_{c}^{b*}$ using the equilibrium commission fee ${\widetilde{\tau }}^{*}$.

Proof of Proposition 6

Following Table 2 and Proposition 5, after solving these two inequalities $\left\{\begin{array}{c}{\widetilde{\pi }}_{c}^{a*}\ge {\pi }_{c}^{a*}\\ {\widetilde{\pi }}_{c}^{b*}\ge {\pi }_{c}^{b*}\end{array}\right.$, we have $\frac{\eta }{2}\le \beta \le \frac{3\eta }{4}$, where $\eta =\frac{\gamma {({k}_{1}+{k}_{0})}^{2}}{\left(\gamma -1\right){k}_{1}^{2}+2{k}_{1}\left(c{k}_{2}+\left(\gamma -1\right){k}_{0}\right)-{c}^{2}{k}_{2}^{2}+2c{{k}_{0}k}_{2}+(\gamma -1){k}_{0}^{2}}$ if ${c}_{t}<{c}_{w}<1+{c}_{t}$ and $\eta =\frac{\gamma {({k}_{1}+{k}_{4})}^{2}}{\left(\gamma -1\right){k}_{1}^{2}+2{k}_{1}\left(c+\left(\gamma -1\right){k}_{4}\right)+\gamma {k}_{4}^{2}-{(c-{k}_{4})}^{2}}$ if ${c}_{w}\le {c}_{t}$.