Abstract
Government incentives play an important role in the development of the remanufacturing industry. It remains a challenge to determine an optimal policy (subsidy and/or tax refund) and how firms (manufacturers and retailers) can integrate it into their pricing decisions. We analyze the impacts of government financial incentives on manufacturers’ and retailers’ pricing decisions in terms of corporate profits and social welfare under different scenarios. We find that government incentives increase the recycling price and availability of used products, while the wholesale and retail prices of new products remain unchanged. Government incentives also significantly increase the manufacturer’s profits and enhance social welfare.
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Notes
Cai Shui [2015] No.78, http://www.chinatax.gov.cn/chinatax/n810341/n810825/c101434/c1519869/content.html.
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Acknowledgements
We are grateful for the editor’s and anonymous reviewers’ constructive comments and suggestions which have improved the quality of this paper. This paper was partially supported by the following fund projects: National Natural Science Foundation of China (No.71761004, 71864003, 71763003, 72243002); China Humanities and Social Sciences Youth Fund Project of the Ministry of Education (No.17XJC630006, 18YJA630063); China Postdoctoral Science Foundation (No. 2017M612868); Key Research Base of Humanities and Social Sciences in Guangxi Universities (No. 2020GDSIYB01); the Interdisciplinary Scientific Research Foundation of Applied Economics of GuangXi University (Grant No. 2023JJJXA01).
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Appendices
Appendix
Appendix A. Proof of Proposition 1
The social welfare function under Model S is given by \(\mathop {\max }\limits_{s} TS^{S} = \pi_{m}^{*S} + \pi_{r}^{*S} + CS^{S} - s\left( {k + hb_{r} } \right)\), where \(\pi_{m}^{*S} + \pi_{r}^{*S} = \frac{{3\left[ {\alpha - \beta c_{m} \left( {1 + t} \right)} \right]^{2} }}{{16\beta \left( {1 + t} \right)}} + \frac{{3\left[ {h\left( {\Delta + s} \right) + k} \right]^{2} }}{16h},\)\(CS^{S} = \frac{1}{2\beta }\left[ {\frac{{\alpha - \beta c_{m} \left( {1 + t} \right)}}{4}} \right]^{2}\) and \(s\left( {k + hb_{r}^{*S} } \right) = \frac{{\left[ {k + h\left( {\Delta + s} \right)} \right]s}}{4}\). Based on the maximization of the social welfare function, we take first derivatives of \({\text{TS}}^{S}\) with respect to \(s\), and let \(\frac{{{\text{dTS}}^{S} }}{{{\text{d}}s}} = 0\). Then, we obtain the equilibrium solution: \(s^{*} = \frac{\Delta h + k}{h}\).
Appendix B. Proof of proposition 2
\(r^{*R} - t = \frac{{\left( {k + \Delta h} \right)\left( {1 + t} \right)}}{k + 2\Delta h} - \frac{{t\left( {k + 2\Delta h} \right)}}{k + 2\Delta h} = \frac{{k + \Delta h\left( {1 - t} \right)}}{k + 2\Delta h},\) because \(t < 1\), then \(r^{*R} - t > 0,\) i.e., \(r^{*R} > t\).
Appendix C. Proof of Proposition 3
The proof of proposition 3 can be easily shown in a similar manner as the proof of proposition1. Note that the social welfare \({\text{TS}}^{{{\text{SR}}}}\) is monotonically increasing on \(\left( {0,\;r^{{*{\text{SR}}}} } \right)\), monotonically decreasing on \(\left( {r^{{*{\text{SR}}}} ,\;t} \right)\) if and only if \(0 \le r^{{*{\text{SR}}}} \le t\), that is, \(x \ge \frac{{\Delta h\left( {1 - t} \right) + k}}{{ht\left( {1 + t} \right)}}\).
Appendix D. Proof of Proposition 4
The proof of part (a) of proposition 4 can be given as follows. First, in Table 2, \(b_{r}^{*S} = b_{r}^{{*{\text{SR}}}}\) is obvious. To prove \(b_{r}^{*R} < b_{r}^{*S}\), we have to show that \(\frac{{\Delta h\left( {1 + t} \right) - 3k}}{4h} < \frac{\Delta h - k}{{2h}}.\) After simplification, this reduces to showing that \(\frac{{\Delta h\left( {1 + t} \right) - 3k}}{4h} - \frac{\Delta h - k}{{2h}} = \frac{{ - \Delta h\left( {1 - t} \right) - k}}{4h} < 0\), which is true. To prove \(b_{r}^{*B} < b_{r}^{*R}\), we have to show that \(\frac{\Delta h - 3k}{{4h}} < \frac{{\Delta h\left( {1 + t} \right) - 3k}}{4h},\) i.e., \(\frac{\Delta h - 3k}{{4h}} - \frac{{\Delta h\left( {1 + t} \right) - 3k}}{4h} = \frac{ - \Delta ht}{{4h}} < 0\), which is true. The proof of \(G\left( {b_{r}^{*B} } \right) < G\left( {b_{r}^{*R} } \right) < G\left( {b_{r}^{*S} } \right) = G\left( {b_{r}^{{*{\text{SR}}}} } \right)\) can be easily shown in a similar manner.
Second, we examine the effect of \(s\) and \(r\) on the recycling price. Hence, the proof follows from the fact that \(\frac{{\partial b_{r}^{S} }}{\partial s} > 0\),\(\frac{{\partial b_{r}^{R} }}{\partial r} > 0\) and \(\frac{{\partial b_{r}^{{{\text{SR}}}} }}{\partial r} > 0\). To show these statements, note that \(\frac{{\partial b_{r}^{S} }}{\partial s} = \frac{1}{4} > 0\), for a given \(s\), the sign of \(\frac{{\partial b_{r}^{S} }}{\partial s}\) is always positive. Similarly,
\(\frac{{\partial b_{r}^{{{\text{SR}}}} }}{\partial r} = \frac{{\left[ {xh\left( {1 + t} \right) + 3k} \right]\Delta 4h\left( {1 + t - r} \right) - \left[ {h\left( {1 + t} \right)\left( {\Delta + xr} \right) - 3k\left( {1 + t - r} \right)} \right]\Delta \left( { - 4h} \right)}}{{16h^{2} \left( {1 + t - r} \right)^{2} }} = \frac{{\left( {1 + t} \right)\left[ {\Delta + x\left( {1 + t} \right)} \right]}}{{4\left( {1 + t - r} \right)^{2} }} > 0\) The proof of part (b) of Proposition 4. Omit.
Appendix E. Proof of Proposition 5
We divide the proof into three parts:
(i) \(\pi_{m}^{*B} < \pi_{m}^{*R} < \pi_{m}^{*SR} < \pi_{m}^{*S}\), and \(\frac{{\partial \pi_{m}^{S} }}{\partial s} > 0\), \(\frac{{\partial \pi_{m}^{R} }}{\partial r} > 0\), \(\frac{{\partial \pi_{m}^{SR} }}{\partial r} > 0\). First, to prove \(\pi_{m}^{*SR} < \pi_{m}^{*S}\), we have to show that \(2X + \frac{{\left( {\Delta h + k} \right)^{2} \left[ {hx\left( {1 + t} \right) + \Delta h} \right]}}{{2h\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}} < 2X + \frac{{\left( {\Delta h + k} \right)^{2} }}{2h}.\) After simplification, this reduces to showing that \(\frac{{\left( {\Delta h + k} \right)^{2} \left[ {hx\left( {1 + t} \right) + \Delta h} \right]}}{{2h\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}} < \frac{{\left( {\Delta h + k} \right)^{2} \left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}}{{2h\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}}\), which is true. To prove \(\pi_{m}^{*R} < \pi_{m}^{{*{\text{SR}}}}\), we have to show that \(2X + \frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{{8h\left( {1 + t} \right)}} < 2X + \frac{{\left( {\Delta h + k} \right)^{2} \left[ {hx\left( {1 + t} \right) + \Delta h} \right]}}{{2h\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}},\) i.e., \(\frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{{8h\left( {1 + t} \right)}} < \frac{{\left( {\Delta h + k} \right)^{2} \left[ {hx\left( {1 + t} \right) + \Delta h} \right]}}{{2h\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}}.\) This reduces to showing that \(\frac{{\left[ {\Delta h\left( {1 + t} \right) + k\left( {1 + t - r} \right)} \right]^{2} }}{{8h\left( {1 + t} \right)\left( {1 + t - r} \right)}} < \frac{{\left[ {h\left( {1 + t} \right)\left( {\Delta + xr} \right) + k\left( {1 + t - r} \right)} \right]^{2} }}{{8h\left( {1 + t} \right)\left( {1 + t - r} \right)}}\), which is true. To prove \(\pi_{m}^{*B} < \pi_{m}^{*R}\), we have to show that \(2X + \frac{{\left( {\Delta h + k} \right)^{2} }}{8h} < 2X + \frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{{8h\left( {1 + t} \right)}}.\) After simplification, this reduces to showing that \(\frac{{\left( {\Delta h + k} \right)^{2} }}{8h} - \frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{{8h\left( {1 + t} \right)}} = \frac{{\left[ {k^{2} - \left( {\Delta h} \right)^{2} \left( {1 + t} \right)} \right]t}}{{8h\left( {1 + t} \right)}} < 0,\) which is true (because \(\Delta h > 3k\), then \(k^{2} < \left( {\Delta h} \right)^{2} \left( {1 + t} \right)\)). Second, we examine the effect of \(s\) and \(r\) on the profit of the manufacturer. Hence, the proof follows from the fact that \(\frac{{\partial \pi_{m}^{S} }}{\partial s}\), \(\frac{{\partial \pi_{m}^{R} }}{\partial r}\) and \(\frac{{\partial \pi_{m}^{SR} }}{\partial r} > 0\). To show these statements, note that \(\frac{{\partial \pi_{m}^{S} }}{\partial s} = \frac{{h\left( {\Delta + s} \right) + k}}{4}\), for a given \(s\), the sign of \(\frac{{\partial \pi_{m}^{S} }}{\partial s}\) is always positive. Similarly,
\(\frac{{\partial \pi_{m}^{R} }}{\partial r} = \frac{{\left[ {\Delta h\left( {1 + t} \right) + k\left( {1 + t - r} \right)} \right]\left[ {\Delta h\left( {1 + t} \right) - k\left( {1 + t - r} \right)} \right]}}{{8h\left( {1 + t} \right)\left( {1 + t - r} \right)^{2} }}\). Because \(\Delta h > 3k > k > 0\) and \(1 + t > 1 + t - r\), then \(\Delta h\left( {1 + t} \right) > k\left( {1 + t - r} \right),\) i.e., \(\frac{{\partial \pi_{m}^{R} }}{\partial r} > 0\).
\(\frac{{\partial \pi_{m}^{{{\text{SR}}}} }}{\partial r} = \frac{{\left[ {h\left( {1 + t} \right)\left( {\Delta + xr} \right) + k\left( {1 + t - r} \right)} \right]\left\{ {xh\left( {1 + t} \right)\left[ {2\left( {1 + t} \right) - r} \right] + \left[ {\Delta h\left( {1 + t} \right) - k\left( {1 + t - r} \right)} \right]} \right\}}}{{8h\left( {1 + t} \right)\left( {1 + t - r} \right)^{2} }}.\) Because \(2\left( {1 + t} \right) > r\) and \(\Delta h\left( {1 + t} \right) > k\left( {1 + t - r} \right)\), then \(\frac{{\partial \pi_{m}^{{{\text{SR}}}} }}{\partial r} > 0\).
(ii) \(\pi_{r}^{*B} < \pi_{r}^{*R} < \pi_{r}^{{*{\text{SR}}}} = \pi_{r}^{*S}\), and \(\frac{{\partial \pi_{r}^{S} }}{\partial s} > 0\), \(\frac{{\partial \pi_{r}^{R} }}{\partial r} > 0\), \(\frac{{\partial \pi_{r}^{{{\text{SR}}}} }}{\partial r} > 0\). First, in Table 2, \(\pi_{r}^{{*{\text{SR}}}} = \pi_{r}^{*S}\) is obvious. To prove \(\pi_{r}^{*R} < \pi_{r}^{*SR}\), we have to show that \(\frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{16h} < \frac{{\left( {\Delta h + k} \right)^{2} }}{4h},\) i.e., \(\frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{16h} - \frac{{\left( {\Delta h + k} \right)^{2} }}{4h} = \frac{{\left( {\Delta h} \right)^{2} \left( {t + 3} \right)\left( {t - 1} \right) + 2\Delta hk\left( {t - 3} \right) - 3k^{2} }}{16h} < 0,\) which is true (because \(t < 1\)). To prove \(\pi_{r}^{*B} < \pi_{r}^{*R}\), we have to show that \(\frac{{\left( {\Delta h + k} \right)^{2} }}{16h} < \frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]^{2} }}{16h}\), which follows from simple algebra. Second, we examine the effect of \(s\) and \(r\) on the profit of the retailer. Hence, the proof follows from the fact that \(\frac{{\partial \pi_{r}^{S} }}{\partial s}\), \(\frac{{\partial \pi_{r}^{R} }}{\partial r}\) and \(\frac{{\partial \pi_{r}^{{{\text{SR}}}} }}{\partial r} > 0\). To show these statements, note that \(\frac{{\partial \pi_{r}^{S} }}{\partial s} = \frac{{h\left( {\Delta + s} \right) + k}}{8}\), for a given \(s\), the sign of \(\frac{{\partial \pi_{r}^{S} }}{\partial s}\) is always positive. Similarly,
\(\frac{{\partial \pi_{r}^{R} }}{\partial r} = \frac{{\Delta \left( {1 + t} \right)\left[ {\Delta h\left( {1 + t} \right) + k\left( {1 + t - r} \right)} \right]}}{{8\left( {1 + t - r} \right)^{3} }} > 0\) and \(\frac{{\partial \pi_{r}^{{{\text{SR}}}} }}{\partial r} = \frac{{\left( {1 + t} \right)\left[ {\Delta + x\left( {1 + t} \right)} \right]\left[ {h\left( {1 + t} \right)\left( {\Delta + xr} \right) + k\left( {1 + t - r} \right)} \right]}}{{8\left( {1 + t - r} \right)^{3} }} > 0\).
(iii) Because \(\pi_{m}^{*B} < \pi_{m}^{*R} < \pi_{m}^{{*{\text{SR}}}} < \pi_{m}^{*S}\) and \(\pi_{r}^{*B} < \pi_{r}^{*R} < \pi_{r}^{{*{\text{SR}}}} = \pi_{r}^{*S}\), it trivially follows that \(\pi_{T}^{*B} < \pi_{T}^{*R} < \pi_{T}^{{*{\text{SR}}}} < \pi_{T}^{*S}\).
Appendix F. Proof of Proposition 6
First, in terms of the social welfare, in Table 2, \({\text{TS}}_{ }^{{*{\text{SR}}}} = {\text{TS}}_{ }^{*S}\) is obvious. To show \({\text{TS}}_{ }^{*R} < {\text{TS}}_{ }^{{*{\text{SR}}}}\), we have to show \(Y + \frac{{\left[ {\left( {\Delta h + k} \right) + \Delta ht} \right]\left[ {3\left( {\Delta h + k} \right) - \Delta ht} \right]}}{16h} < Y + \frac{{\left( {\Delta h + k} \right)^{2} }}{4h},\) i.e., \(\frac{{\left[ {\left( {\Delta h + k} \right) + \Delta ht} \right]\left[ {3\left( {\Delta h + k} \right) - \Delta ht} \right]}}{16h} - \frac{{\left( {\Delta h + k} \right)^{2} }}{4h} = \frac{{ - \left( {\Delta h} \right)^{2} \left( {1 - t} \right)^{2} - 2\Delta hk\left( {1 - t} \right) - k^{2} }}{16h} < 0\), which is always true. To show \({\text{TS}}_{ }^{*B} < {\text{TS}}_{ }^{*R}\), we have to show \(\frac{{3\left( {\Delta h + k} \right)^{2} }}{16h} < \frac{{\left[ {\left( {\Delta h + k} \right) + \Delta ht} \right]\left[ {3\left( {\Delta h + k} \right) - \Delta ht} \right]}}{16h},\) i.e., \(\frac{{3\left( {\Delta h + k} \right)^{2} }}{16h} - \frac{{\left[ {\left( {\Delta h + k} \right) + \Delta ht} \right]\left[ {3\left( {\Delta h + k} \right) - \Delta ht} \right]}}{16h} = \frac{{\Delta ht\left[ {\Delta h\left( {t - 2} \right) - 2k} \right]}}{16h} < 0\), which always holds. For the government expenditure, to prove \(g\left( {r^{{*{\text{SR}}}} } \right) < s^{*} G\left( {b_{r}^{*S} } \right)\), we have to show.\(g\left( {r^{{*{\text{SR}}}} } \right) - s^{*} G\left( {b_{r}^{*S} } \right) = \frac{{\left[ {x\left( {1 + t} \right) + \Delta } \right]\left( {\Delta h + k} \right)^{2} }}{{2\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}} - \frac{{\left( {\Delta h + k} \right)^{2} }}{2h} = \frac{{ - \left( {\Delta h + k} \right)^{2} }}{{2h\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}} < 0\),which always holds. To show \(f\left( {r^{*R} } \right) < g\left( {r^{{*{\text{SR}}}} } \right)^{ }\), we have to show that\(f\left( {r^{*R} } \right) - g\left( {r^{{*{\text{SR}}}} } \right) = \frac{{\left[ {\Delta h\left( {1 + t} \right) + k} \right]\left[ {\Delta h\left( {1 + t} \right) - k} \right]t}}{{8h\left( {1 + t} \right)}} - \frac{{\left[ {x\left( {1 + t} \right) + \Delta } \right]\left( {\Delta h + k} \right)^{2} }}{{2\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}} = \frac{Z}{{8h\left( {1 + t} \right)\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right]}} < 0,\) where \(Z = t\left\{ {\left[ {\Delta h\left( {1 + t} \right)} \right]^{2} - k^{2} } \right\}\left[ {hx\left( {1 + t} \right) + 2\Delta h + k} \right] - 4\left( {1 + t} \right)\left[ {hx\left( {1 + t} \right) + \Delta h} \right]\left( {\Delta h + k} \right)^{2}\). Let \(Z = A + B\), where \(A = t\left[ {hx\left( {1 + t} \right) + \Delta h} \right]\left\{ {\left( {\left[ {\Delta h\left( {1 + t} \right)} \right]^{2} - k^{2} } \right) - 4\left( {\Delta h + k} \right)^{2} } \right\} = t\left[ {hx\left( {1 + t} \right) + \Delta h} \right]\left\{ {\left( {\Delta h} \right)^{2} \left[ {\left( {1 + t} \right)^{2} - 4} \right] - 5k^{2} - 8\Delta hk} \right\} < 0\) (because \(0 < t < 1\), then \(\left( {1 + t} \right)^{2} - 4 < 0,\) i.e., \(A < 0\)), and \(B = \left( {\Delta h + k} \right)\left\{ {t\left( {\Delta h} \right)^{2} \left( {1 + t} \right)^{2} - tk^{2} - 4\left( {\Delta h + k} \right)\left[ {hx\left( {1 + t} \right) + \Delta h} \right]} \right\} = \left( {\Delta h + k} \right){\text{\{ }}\left( {\Delta h} \right)^{2} [t\left( {1 + t} \right)^{2} - 4{]} - tk^{2} - 4\left[ {\Delta h\Delta hx\left( {1 + t} \right) + k} \right[hx\left( {1 + t} \right) + \Delta h]]\} < 0\) (note that \(0 < t < 1\), hence, \(t\left( {1 + t} \right)^{2} < \left( {1 + t} \right)^{2} - 4 < 0\), i.e., \(B < 0\)). Therefore, \(Z = A + B < 0\), that is, \(g\left( {r^{{*{\text{SR}}}} } \right) < s^{*} G\left( {b_{r}^{*S} } \right)\) always holds.
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Hao, H., Ran, G., Liu, Hm. et al. Pricing strategies for remanufacturing with government incentives. Neural Comput & Applic 36, 2187–2200 (2024). https://doi.org/10.1007/s00521-023-08804-6
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DOI: https://doi.org/10.1007/s00521-023-08804-6