Optimal ordering and production decisions for remanufacturing firms with carbon options under demand uncertainty

Abstract

We study the role of carbon options in mitigating the risk of demand uncertainty for an emissions-dependent firm that conducts remanufacturing and then selling to consumers. Specifically, we first investigate the carbon option-void scenario as a benchmark where no carbon options are available under demand uncertainty in the emission trading market. Subsequently, the unidirectional carbon option scenario and the bidirectional carbon option scenario are introduced as alternatives to purchase carbon emission quotas. Through comparing the optimal ordering and production decisions under different scenarios, we demonstrate the positive role of carbon option contracts in improving the firm’s profits and, more importantly, co** with demand uncertainty. Among other results, we observe that the bidirectional carbon option contracts perform better than the unidirectional carbon option contracts. Under the two option-based scenarios, the firm is more sensitive to the carbon option price than the exercised price. In addition, the firm has incentives to remanufacture with a relatively high remanufacturing rate and a loose carbon emission policy.

Appendices

Appendix

Proof of lemma 1

From Eq. (2), we derive that $\frac{{\partial }^{2}{\pi }_{m}\left({q}_{b}\right)}{\partial {{q}_{b}}^{2}}=-\frac{\left(p-{c}_{R}+g\right){\varepsilon }^{2}}{{\left(e-k\right)}^{2}}f\left(\frac{\varepsilon {q}_{b}}{e-k}\right)<0$. Thus, ${\pi }_{m}\left({q}_{b}\right)$ is concave in ${q}_{b}$, and then, there exists a unique ${{q}_{b}^{\text{None}}}^{*}$ that maximizes ${\pi }_{m}\left({q}_{b}\right)$.

Proof of proposition 1

From Eq. (2), let $\frac{\partial {\pi }_{m}\left({q}_{b}\right)}{\partial {q}_{b}}=\frac{\left(p-{c}_{R}+g\right)\varepsilon }{e-k}-\frac{\left(p-{c}_{R}+g\right)\varepsilon }{e-k}F\left(\frac{\varepsilon {q}_{b}}{e-k}\right)-{\omega }_{b}=0$; then, we can obtain that ${{q}_{b}^{\text{None}}}^{*}=\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{\text{None}}\right)$, where ${a}^{\text{None}}=\frac{p-{c}_{R}+g-{\omega }_{b}\left(e-k\right)/\varepsilon }{p-{c}_{R}+g}$.

Proof of corollary 1

According to proposition 1, we can obtain that $\frac{\partial {{q}_{b}^{\text{None}}}^{*}}{\partial {\omega }_{b}}=-\frac{{\left(e-k\right)}^{2}}{{\varepsilon }^{2}\left(p-{c}_{R}+g\right)f\left({a}^{\text{None}}\right)}<0$ holds.

Proof of lemma 2

From Eq. (4), we derive that $\frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial {Q}^{2}}=-\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right)}{{\left(e-k\right)}^{2}}f\left(\frac{\varepsilon Q}{e-k}\right)<0$, $\frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial {{q}_{b}}^{2}}=-\frac{{\omega }_{e}\varepsilon }{e-k}f\left(\frac{\varepsilon {q}_{b}}{e-k}\right)<0$, and $\frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial Q\partial {q}_{b}}=\frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial {q}_{b}\partial Q}=0$. Furthermore, ${H}^{UO}=\left|\begin{array}{cc}\frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial {Q}^{2}}& \frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial Q\partial {q}_{b}}\\ \frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial {q}_{b}\partial Q}& \frac{{\partial }^{2}{\pi }_{m}\left(Q,{q}_{b}\right)}{\partial {{q}_{b}}^{2}}\end{array}\right|>0$ can be derived. Obviously, the Hessian matrix of ${\pi }_{m}\left(Q,{q}_{b}\right)$ is negative definite. Hence, ${\pi }_{m}\left(Q,{q}_{b}\right)$ is jointly concave in $Q$ and ${q}_{b}$, and the there exists a unique ${Q}^{UO*}$ and ${{q}_{b}^{UO}}^{*}$ that maximize ${\pi }_{m}\left(Q,{q}_{b}\right)$.

Proof of proposition 2

From Eq. (4), let $\frac{\partial {\pi }_{m}}{\partial Q}=\frac{\left(p-{c}_{R}+g\right)\varepsilon }{e-k}-\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right)\varepsilon }{e}F\left(\frac{\varepsilon Q}{e-k}\right)-\left({\omega }_{e}+{\omega }_{o}\right)=0$ and $\frac{\partial {\pi }_{m}}{\partial {q}_{b}}=-{\omega }_{e}F\left(\frac{\varepsilon {q}_{b}}{e-k}\right)-{\omega }_{b}+\left({\omega }_{e}+{\omega }_{o}\right)=0$, we can obtain that ${{q}_{b}^{UO}}^{*}=\frac{e-k}{\varepsilon }{F}^{-1}\left({b}^{UO}\right)$, ${Q}^{UO*}=\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{UO}\right)$, and ${{q}_{o}^{UO}}^{*}={Q}^{UO*}-{{q}_{b}^{UO}}^{*}=\frac{e-k}{\varepsilon }\left[{F}^{-1}\left({a}^{UO}\right)-{F}^{-1}\left({b}^{UO}\right)\right]$, where ${a}^{UO}=\frac{p-{c}_{R}+g-\left({\omega }_{e}+{\omega }_{o}\right)\left(e-k\right)/\varepsilon }{p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon }$, and ${b}^{UO}=\frac{{\omega }_{e}+{\omega }_{o}-{\omega }_{b}}{{\omega }_{e}}$.

Proof of corollary 2

According to proposition 2, we can obtain that (i) $\frac{\partial {{q}_{b}^{UO}}^{*}}{\partial {\omega }_{b}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({b}^{UO}\right)}{\partial {\omega }_{b}}=-\frac{e-k}{{\varepsilon \omega }_{e}f\left({b}^{UO}\right)}<0$, $\frac{\partial {{q}_{b}^{UO}}^{*}}{\partial {\omega }_{o}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({b}^{UO}\right)}{\partial {\omega }_{o}}=\frac{e-k}{{\varepsilon \omega }_{e}f\left({b}^{UO}\right)}>0$, $\frac{\partial {{q}_{b}^{UO}}^{*}}{\partial {\omega }_{e}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({b}^{UO}\right)}{\partial {\omega }_{e}}=\frac{\left(e-k\right)\left({\omega }_{b}-{\omega }_{o}\right)}{{{\varepsilon \omega }_{e}}^{2}f\left({b}^{UO}\right)}>0$.

(ii) $\frac{\partial {Q}^{UO*}}{\partial {\omega }_{b}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{UO}\right)}{\partial {\omega }_{b}}=0$, $\frac{\partial Q^{UO\ast}}{\partial\omega_o}=\frac{\frac{e-k}\varepsilon F^{-1}\left(a^{UO}\right)}{\partial\omega_o}=-\frac{\left(e-k\right)^2}{\left[\varepsilon^2\left(p-c_R+g\right)-{\varepsilon\omega}_e\left(e-k\right)\right]f\left(a^{UO}\right)}<0$, $\frac{\partial Q^{UO\ast}}{\partial\omega_e}=\frac{\frac{e-k}\varepsilon F^{-1}\left(a^{UO}\right)}{\partial\omega_e}=-\frac{\omega_o\left(e-k\right)^3}{\varepsilon\left[\varepsilon\left(p-c_R+g\right)-\omega_e\left(e-k\right)\right]^2f\left(a^{UO}\right)}<0$.

Proof of lemma 3

From Eq. (6), we derive that $\frac{{\partial }^{2}{\pi }_{m}\left({q}_{b},{q}_{o}\right)}{\partial {{q}_{b}}^{2}}=\frac{{\partial }^{2}{\pi }_{m}\left({q}_{b},{q}_{o}\right)}{\partial {{q}_{o}}^{2}}=-\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right){\varepsilon }^{2}}{{\left(e-k\right)}^{2}}f\left(\frac{\varepsilon \left({q}_{b}+{q}_{o}\right)}{e-k}\right)-\frac{{\omega }_{e}\varepsilon }{e-k}f\left(\frac{\varepsilon \left({q}_{b}-{q}_{o}\right)}{e-k}\right)<0$, $\frac{{\partial }^{2}{\pi }_{m}\left({q}_{b},{q}_{o}\right)}{\partial {q}_{b}\partial {q}_{o}}=\frac{{\partial }^{2}{\pi }_{m}\left({q}_{b},{q}_{o}\right)}{\partial {q}_{o}\partial {q}_{b}}=-\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right){\varepsilon }^{2}}{{\left(e-k\right)}^{2}}f\left(\frac{\varepsilon \left({q}_{b}+{q}_{o}\right)}{e-k}\right)+\frac{{\omega }_{e}\varepsilon }{e-k}f\left(\frac{\varepsilon \left({q}_{b}-{q}_{o}\right)}{e-k}\right)$. Let $A=\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right){\varepsilon }^{2}}{{\left(e-k\right)}^{2}}f\left(\frac{\varepsilon \left({q}_{b}+{q}_{o}\right)}{e-k}\right)$, $B=\frac{{\omega }_{e}\varepsilon }{e-k}f\left(\frac{\varepsilon \left({q}_{b}-{q}_{o}\right)}{e-k}\right)$. Furthermore, we obtain ${H}^{BO}=\left|\begin{array}{cc}-A-B& -A+B\\ -A+B& -A-B\end{array}\right|$, with ${H}_{1}<0$, and ${H}_{2}=4AB>0$. Obviously, the Hessian matrix of ${\pi }_{m}\left({q}_{b},{q}_{o}\right)$ is negative definite. Hence, ${\pi }_{m}\left({q}_{b},{q}_{o}\right)$ is jointly concave in ${q}_{b}$ and ${q}_{o}$, and the there exists a unique ${{q}_{b}^{BO}}^{*}$ and ${{q}_{o}^{BO}}^{*}$ that maximize ${\pi }_{m}\left({q}_{b},{q}_{o}\right)$.

Proof of proposition 3

From Eq. (6), let $\frac{\partial {\pi }_{m}}{\partial {q}_{b}}=\frac{\left(p-{c}_{R}+g\right)\varepsilon }{e-k}-\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right)\varepsilon }{e-k}F\left(\frac{\varepsilon \left({q}_{b}+{q}_{o}\right)}{e-k}\right)-{\omega }_{e}F\left(\frac{\varepsilon \left({q}_{b}-{q}_{o}\right)}{e-k}\right)-{\omega }_{b}=0$ and $\frac{\partial {\pi }_{m}}{\partial {q}_{o}}=\frac{\left(p-{c}_{R}+g\right)\varepsilon }{e-k}-\frac{\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right)\varepsilon }{e-k}F\left(\frac{\varepsilon \left({q}_{b}+{q}_{o}\right)}{e-k}\right)+{\omega }_{e}F\left(\frac{\varepsilon \left({q}_{b}-{q}_{o}\right)}{e-k}\right)-\left({\omega }_{o}+{\omega }_{e}\right)=0$, we can obtain that ${{q}_{b}^{BO}}^{*}=\frac{e-k}{2\varepsilon }\left[{F}^{-1}\left({a}^{BO}\right)+{F}^{-1}\left({b}^{BO}\right)\right]$, ${{q}_{o}^{BO}}^{*}=\frac{e-k}{2\varepsilon }\left[{F}^{-1}\left({a}^{BO}\right)-{F}^{-1}\left({b}^{BO}\right)\right]$, and ${Q}^{BO*}={{q}_{b}^{BO}}^{*}+{{q}_{o}^{BO}}^{*}=\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{BO}\right)$, where ${a}^{BO}=\frac{2\left(p-{c}_{R}+g\right)-\left({\omega }_{b}+{\omega }_{e}+{\omega }_{o}\right)\left(e-k\right)/\varepsilon }{2\left(p-{c}_{R}+g-{\omega }_{e}\left(e-k\right)/\varepsilon \right)}$, and ${b}^{BO}=\frac{{\omega }_{e}+{\omega }_{o}-{\omega }_{b}}{2{\omega }_{e}}$.

Proof of corollary 4

According to proposition 3, for given ${\omega }_{b}>{{\omega }_{b}}^{*}$, we can obtain that (i) $\frac{\partial {{q}_{b}^{BO}}^{*}}{\partial {\omega }_{b}}=-\frac{\left(e-k\right)}{4\varepsilon }\left[\frac{\left(e-k\right)}{\left[\varepsilon \left(p-{c}_{R}+g\right)-{\omega }_{e}\left(e-k\right)\right]f\left({a}^{BO}\right)}+\frac{1}{ef\left({b}^{BO}\right)}\right]<0$, $\frac{\partial {{q}_{o}^{BO}}^{*}}{\partial {\omega }_{o}}=-\frac{\left(e-k\right)}{4\varepsilon }\left[\frac{\left(e-k\right)}{\left[\varepsilon \left(p-{c}_{R}+g\right)-{\omega }_{e}\left(e-k\right)\right]f\left({a}^{BO}\right)}+\frac{1}{ef\left({b}^{BO}\right)}\right]<0$, $\frac{\partial {{q}_{b}^{BO}}^{*}}{\partial {\omega }_{e}}=\frac{\left(e-k\right)}{4\varepsilon }\left[\frac{\left(e-k\right)\left[\varepsilon \left(p-{c}_{R}+g\right)-\left({\omega }_{b}+{\omega }_{o}\right)\left(e-k\right)\right]}{{\left[\varepsilon \left(p-{c}_{R}+g\right)-{\omega }_{e}\left(e-k\right)\right]}^{2}f\left({a}^{BO}\right)}+\frac{{\omega }_{b}-{\omega }_{o}}{{e}^{2}f\left({b}^{BO}\right)}\right]>0$.

(ii) $\frac{\partial {Q}^{BO*}}{\partial {\omega }_{b}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{BO}\right)}{\partial {\omega }_{b}}=-\frac{{\left(e-k\right)}^{2}}{2\varepsilon \left[\varepsilon \left(p-{c}_{R}+g\right)-{\omega }_{e}\left(e-k\right)\right]f\left({a}^{BO}\right)}<0$, $\frac{\partial {Q}^{BO*}}{\partial {\omega }_{o}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{BO}\right)}{\partial {\omega }_{o}}=-\frac{{\left(e-k\right)}^{2}}{2\varepsilon \left[\varepsilon \left(p-{c}_{R}+g\right)-{\omega }_{e}\left(e-k\right)\right]f\left({a}^{BO}\right)}<0$, $\frac{\partial {Q}^{BO*}}{\partial {\omega }_{e}}=\frac{\frac{e-k}{\varepsilon }{F}^{-1}\left({a}^{BO}\right)}{\partial {\omega }_{e}}=\frac{{\left(e-k\right)}^{2}\left[\varepsilon \left(p-{c}_{R}+g\right)-\left({\omega }_{b}+{\omega }_{o}\right)\left(e-k\right)\right]}{2\varepsilon {\left[\varepsilon \left(p-{c}_{R}+g\right)-{\omega }_{e}\left(e-k\right)\right]}^{2}f\left({a}^{BO}\right)}>0$.