Platform or direct channel: government-subsidized recycling strategies for WEEE

Abstract

We investigate how government subsidies affect pricing and service-quality strategies under different online-recycling channel structures. We consider two cases: the monopoly case, where the manufacturer recycles by itself, and the coopetition case, where a platform and the manufacturer compete for used products while the platform provides services to the manufacturer. We find the optimal price and service-quality strategies in these two cases with or without government subsidy. We also examine the recycling outcomes in terms of total recycling quantities and the parties’ profits. We find that with government subsidy, in the monopoly case, the manufacturer is motivated to increase both acquisition price and service quality and thus achieves higher recycling quantities and profit. In the case of coopetition, the manufacturer enjoys similar benefits from subsidy, but the platform suffers. The manufacturer gains competitive advantage from subsidy by offering a higher acquisition price, which forces the platform to increase acquisition price as well. To compensate for this increased cost, the platform must lower service quality, which leads to lower recycling quantities and profits. Only when the subsidy is high enough can the platform benefit. We also show that a subsidy can increase the total recycling quantity of the system only when consumers are fairly insensitive to service quality. Our study contributes to the understanding of how subsidy policy interacts with different online channel structures.

Appendix: Proofs

Proof of Lemma 1

The Hessian matrix of $\prod_{m}^{M}$ with respect to $p$, $P_{m}$, and $s$ is

$$H_{1} = \left[ {\begin{array}{*{20}c} { - 2} & 0 & 0 \\ 0 & { - 2} & { - \beta } \\ 0 & { - \beta } & { - 1} \\ \end{array} } \right].$$

$H_{1}$ is a negative-definite matrix. Therefore, a unique optimal solution exists. By the first-order conditions of $\prod_{m}^{M}$ with respect to $p$, $P_{m}$, and $s$, we have

$$\begin{aligned} \frac{{\partial \prod_{m}^{M} }}{\partial p} & = c + \varphi - 2p = 0 \\ \frac{{\partial \prod_{m}^{M} }}{{\partial P_{m} }} & = \delta + g - 2P_{m} - \beta s = 0 \\ \frac{{\partial \prod_{m}^{M} }}{\partial s} & = (\delta + g - P_{m} )\beta - s = 0. \\ \end{aligned}$$

Based on the first equation, we can derive the optimal price $p_{{}}^{M*} = (c + \varphi )/2$. Based on the second and third equations, we can derive optimal acquisition price $P_{m}^{M*}$ and the manufacturer's optimal service level $s^{M*}$ as in Lemma 1. Substituting $p_{{}}^{M*}$, $P_{m}^{M*}$, $s^{M*}$ into $Q_{m}$ and $\prod_{m}$ in Eqs. (1) and (4), we can derive the manufacturer’s (which is also the system’s) optimal recycling quantity $Q^{M*}$ and the manufacturer’s optimal profit $\prod_{m}^{M*}$ as in Lemma 1.

Proof of Proposition 1

We use superscript N to denote the equilibrium outcome for the case with no government subsidy. By Lemma 1, we can obtain $P_{m}^{M*} - P_{m}^{NM*} = \frac{{g(1 - \beta^{2} )}}{{2 - \beta^{2} }} > 0$, $s^{M*} - s^{NM*} = \frac{g\beta }{{2 - \beta^{2} }} > 0$, and $Q_{{}}^{M*} - Q_{{}}^{NM*} = \frac{g}{{2 - \beta^{2} }} > 0$.

Proof of Lemma 2

We first derive the Hessian matrix of $\prod_{t}^{MP}$ with respect to $P_{t}$ and $s$:

$$H_{2} = \left[ {\begin{array}{*{20}c} { - 2} & { - \beta } \\ { - \beta } & { - 1} \\ \end{array} } \right].$$

$H_{2}$ is a negative-definite matrix. Therefore, a unique optimal solution exists. By the first-order condition of maximizing the platform’s profit, we can derive her optimal price and service level, given the manufacturer’s price:

$$\begin{aligned} P_{t}^{*} (P_{m} ) & = \frac{{\alpha P_{m} - \beta^{2} v - \beta^{2} \lambda - \alpha \lambda + v}}{{2 - \beta^{2} }} \\ s^{*} (P_{m} ) & = \frac{{ - \alpha \beta P_{m} + \beta v + 2\beta \lambda + \alpha \beta \lambda }}{{2 - \beta^{2} }}. \\ \end{aligned}$$

Substituting $P_{t}^{*} (P_{m} )$ and $s^{*} (P_{m} )$ into $\prod_{m}^{MP}$, we get the Hessian matrix of $\prod_{m}^{MR}$:

$$H_{3} = \left[ {\begin{array}{*{20}c} { - 2} & 0 \\ 0 & {\frac{{2\alpha^{2} + 2\alpha \beta^{2} + 2\beta^{2} - 4}}{{2 - \beta^{2} }}} \\ \end{array} } \right].$$

We notice that $\prod_{m}^{MP}$ is jointly concave on $p$ and $P_{m}^{MP}$ when the condition $2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} > 0$ is satisfied.

Under this condition, based on the first-order conditions of $\prod_{m}^{MP}$ with respect to $p$ and $P_{m}^{MP}$, we can derive the equilibrium selling price $p^{MP*}$ and the equilibrium acquisition price $P_{m}^{MP*}$ as in Lemma 2. By substituting $p^{MP*}$ and $P_{m}^{MP*}$ into $P_{t}^{*} (P_{m} )$ and $s^{*} (P_{m} )$, we can derive the platform’s equilibrium acquisition price, $P_{t}^{MP*}$, and service level, $s^{MP*}$, as in Lemma 2. By substituting $P_{m}^{MP*}$, $P_{t}^{MP*}$, and $s^{MP*}$ into Eqs. (2) and (3), we can derive $Q_{m}^{MP*}$ and $Q_{t}^{MP*}$ as in Lemma 2. Similarly, by substituting the equilibrium prices and service levels into Eqs. (5) and (6), we can derive $\Pi_{m}^{MP*}$ and $\Pi_{t}^{MP*}$ as in Lemma 2.

Proof of Proposition 2

We use superscript N to denote the equilibrium outcome for the case with no government subsidy. By Lemma 2, we have $P_{m}^{MP*} - P_{m}^{NMP*} = g/2 > 0$, and

$$Q_{m}^{MP*} - Q_{m}^{NMP*} = \frac{{g(2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} )}}{{2(2 - \beta^{2} )}} > 0.$$

In addition, we have

$$\begin{aligned} P_{t}^{MP*} - P_{t}^{NMP*} & = \frac{\alpha g}{{2(2 - \beta^{2} )}} > 0 \\ s^{MP*} - s^{NMP*} & = - \frac{g\alpha \beta }{{2(2 - \beta^{2} )}} < 0 \\ \end{aligned}$$

and

$$Q_{t}^{MP*} - Q_{t}^{NMP*} = - \frac{\alpha g}{{2(2 - \beta^{2} )}} < 0.$$

Proof of Proposition 3

We use superscript N to denote the equilibrium outcome for the case without government subsidy. By Proposition 1, a government subsidy increases recycling quantity in Model M.

In Model MP, we have

$$\begin{aligned} Q^{MP*} - Q^{NMP*} & = (Q_{m}^{MP*} + Q_{t}^{MP*} ) - (Q_{m}^{NMP*} + Q_{t}^{NMP*} ) \\ & = (Q_{m}^{MP*} - Q_{m}^{NMP*} ) + (Q_{t}^{MP*} - Q_{t}^{NMP*} ) \\ & = \frac{{g(2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} - \alpha )}}{{2(2 - \beta^{2} )}}. \\ \end{aligned}$$

(10)

Therefore, $Q^{MP*} - Q^{NMP*} > 0$ if and only if $(2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} - \alpha ) > 0$, which is equivalent to the condition prescribed in the proposition.

Proof of Proposition 4

(a) In Model M, by Lemma 1, we have

$$\prod_{m}^{M*} - \prod_{m}^{NM*} = \frac{g(2\delta + g)}{{2(2 - \beta^{2} )}} > 0.$$

In Model MP, by Lemma 2, we have.

$\prod_{m}^{MP*} - \prod_{m}^{NMP*} = \frac{{g[K_{1} (2\delta + g) + 2\lambda K_{4} (1 + \alpha ) - 2vK_{3} ]}}{{4(2 - \beta^{2} )}} > 0$.

The inequality is because $2[\delta K_{1} + \lambda K_{4} (1 + \alpha ) - vK_{3} ] > 0$ based on $Q_{m}^{NMP*} > 0$.

(b) By Lemma 2, we have.

$\prod_{t}^{MP*} - \prod_{t}^{NMP*} = \frac{{\alpha^{2} gK_{1} (2\delta + g) + 2g\lambda (2K_{1}^{2} - \alpha^{2} K_{2} ) - 2\alpha gv(K_{1} + K_{5} )}}{{8K_{1} (2 - \beta^{2} )}}$.

Because the denominator is positive, the above is positive if and only if the numerator is positive, which is equivalent to the condition prescribed in the proposition.

Proof of Proposition 5

(a) By Lemma 1, we have

$$\frac{{\partial P_{m}^{M*} }}{\partial g} = \frac{{1 - \beta^{2} }}{{2 - \beta^{2} }} > 0.$$

By Lemma 2, we have $\partial P_{m}^{MP*} /\partial g = 1/2 > 0$, and

$$\frac{{\partial P_{t}^{MP*} }}{\partial g} = \frac{\alpha }{{2(2 - \beta^{2} )}} > 0.$$

In addition, we have

$$\frac{{\partial P_{m}^{MP*} }}{\partial g} - \frac{{\partial P_{m}^{M*} }}{\partial g} = \frac{{\beta^{2} }}{{2(2 - \beta^{2} )}} > 0.$$

(b) By Lemma 1, we have

$$\frac{{\partial s_{{}}^{M*} }}{\partial g} = \frac{\beta }{{2 - \beta^{2} }} > 0.$$

By Lemma 2, we have

$$\frac{{\partial s_{{}}^{MP*} }}{\partial g} = - \frac{\alpha \beta }{{2(2 - \beta^{2} )}} < 0.$$

(c) By Lemma 1, we have

$$\frac{{\partial Q_{{}}^{M*} }}{\partial g} = \frac{1}{{2 - \beta^{2} }} > 0.$$

By Eq. (10), we have

$$\frac{{\partial Q_{{}}^{MP*} }}{\partial g} = \frac{{2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} - \alpha }}{{2(2 - \beta^{2} )}}.$$

Therefore, $\partial Q_{{}}^{MP*} /\partial g > 0$ if and only if $(2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} - \alpha ) > 0$, which is equivalent to the condition prescribed in the proposition. Moreover, we have

$$\frac{{\partial Q_{{}}^{M*} }}{\partial g} - \frac{{\partial Q_{{}}^{MP*} }}{\partial g} = \frac{1}{{2 - \beta^{2} }} - \frac{{2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} - \alpha }}{{2(2 - \beta^{2} )}} = \frac{{\alpha^{2} + \beta^{2} + \alpha \beta^{2} + \alpha }}{{2(2 - \beta^{2} )}} > 0.$$

Proof of Proposition 6

(a) By Lemma 1, we have

$$\begin{aligned} \frac{{\partial P_{m}^{M*} }}{\partial \delta } & = \frac{{1 - \beta^{2} }}{{2 - \beta^{2} }} > 0 \\ \frac{{\partial s^{M*} }}{\partial \delta } & = \frac{\beta }{{2 - \beta^{2} }} > 0 \\ \frac{{\partial Q_{m}^{M*} }}{\partial \delta } & { = }\frac{1}{{2 - \beta^{2} }} > 0. \\ \end{aligned}$$

(b) By Lemma 2, we have

$$\begin{aligned} \frac{{\partial P_{m}^{MP*} }}{\partial \delta } & = \frac{1}{2} > 0 \\ \frac{{\partial P_{t}^{MP*} }}{\partial \delta } & = \frac{\alpha }{{2(2 - \beta^{2} )}} > 0 \\ \frac{{\partial Q_{m}^{MP*} }}{\partial \delta } & = \frac{{2 - \alpha^{2} - \beta^{2} - \alpha \beta^{2} }}{{2(2 - \beta^{2} )}} > 0 \\ \frac{{\partial s^{MP*} }}{\partial \delta } & = - \frac{\alpha \beta }{{2(2 - \beta^{2} )}} < 0 \\ \frac{{\partial Q_{t}^{MP*} }}{\partial \delta } & = - \frac{\alpha }{{2(2 - \beta^{2} )}} < 0. \\ \end{aligned}$$