Skip to main content
Financial Innovation

Table 1 The optimal profits under the two mechanisms and types of markets

From: Competition vs cooperation: renewable energy investment under cap-and-trade mechanisms

 

\({\pi }^{*}\)

MG

\(\frac{\left\{\begin{array}{c}4{B}^{2}{\widetilde{E}}_{1}w-{A}_{1}^{2}{c}_{1}-{A}_{2}^{2}{c}_{2}+2{A}_{1}\left(C-Bw{e}_{1}+{c}_{1}{F}_{1}\right)\\ +{F}_{1}[2Bw{e}_{1}-({c}_{1}+{d}_{1}){F}_{1}]+2Bw{e}_{2}{F}_{2}-{c}_{2}{F}_{2}^{2}+2{A}_{2}(C-Bw{e}_{2}+{c}_{2}{F}_{2})\end{array}\right\}}{2{B}^{2}}\)

MB

\(\frac{\left\{\begin{array}{c}{d}_{2}{\left({F}_{2}+{c}_{1}{d}_{1}{e}_{0}w{c}_{2}\right)}^{2}+{d}_{1}{\left({F}_{1}+{c}_{1}{d}_{2}{e}_{0}w{c}_{2}\right)}^{2}+{c}_{1}{\left({A}_{1}-{F}_{1}+{d}_{1}{d}_{2}{e}_{0}w{c}_{2}\right)}^{2}-2B\left({F}_{1}+{c}_{1}{d}_{2}{e}_{0}w{c}_{2}\right)w{e}_{0}-\\ 2\left\{C-bw{e}_{0}\left\{{c}_{2}{d}_{1}{d}_{2}+{c}_{1}\left[{c}_{2}{d}_{1}+\left({c}_{2}+{d}_{1}\right){d}_{2}\right]\right\}\right\}\left\{{A}_{1}+{A}_{2}+w{e}_{0}\left\{{c}_{1}{c}_{2}{d}_{1}+{d}_{2}\left[{c}_{1}{c}_{2}+\left({c}_{1}+{c}_{2}\right){d}_{1}\right]\right\}\right\}+\\ 2Bw\left(-{e}_{0}+{e}_{1}\right)\left\{{A}_{1}-{F}_{1}+{d}_{2}\left[-{c}_{1}{e}_{0}w{c}_{2}+{c}_{2}\left({c}_{1}+{d}_{1}\right)w{e}_{0}\right]\right\}\\ -2Bw\left({e}_{0}-{e}_{2}\right)\left\{-{A}_{2}+{F}_{2}+{c}_{1}{d}_{1}\left[{e}_{0}w{c}_{2}-\left({c}_{2}+{d}_{2}\right)w{e}_{0}\right]\right\}+{c}_{2}{\left\{{A}_{2}-{F}_{2}+{c}_{1}{d}_{1}\left[-{e}_{0}w{c}_{2}+\left({c}_{2}+{d}_{2}\right)w{e}_{0}\right]\right\}}^{2}\end{array}\right\}}{2{B}^{2}}\)

CG

\({\pi }_{1}^{CG*}=\frac{2{D}^{2}Rw-{d}_{1}{G}_{1}^{2}+2Dw{e}_{1}({G}_{1}-{T}_{1})-{c}_{1}{({G}_{1}-{T}_{1})}^{2}+2J{T}_{1}}{2{D}^{2}}\);

\({\pi }_{2}^{CG*}=\frac{2{D}^{2}Rw-{d}_{2}{G}_{2}^{2}+2Dw{e}_{2}({G}_{2}-{T}_{2})-{c}_{2}{({G}_{2}-{T}_{2})}^{2}+2J{T}_{2}}{2{D}^{2}}\)

CB

\({\pi }_{1}^{CB}=\frac{\left\{\begin{array}{c}2Dw{e}_{0}\left\{{G}_{1}+{c}_{1}w{e}_{0}\left[b{d}_{2}+{c}_{2}\left(b+{d}_{2}\right)\right]\right\}-2Dw\left({e}_{0}-{e}_{1}\right)\left\{{G}_{1}-{T}_{1}-{d}_{1}w{e}_{0}\left[b{d}_{2}+{c}_{2}\left(b+{d}_{2}\right)\right]\right\}\\ +{c}_{1}{\left\{-{G}_{1}+{T}_{1}+{d}_{1}w{e}_{0}\left[b{d}_{2}+{c}_{2}\left(b+{d}_{2}\right)\right]\right\}}^{2}\\ +2\left(J-M{e}_{0}\right)\left\{{T}_{1}+w{e}_{0}\left({c}_{1}+{d}_{1}\right)\left[b{d}_{2}+{c}_{2}\left(b+{d}_{2}\right)\right]\right\}+{d}_{1}{\left\{{G}_{1}+{c}_{1}w{e}_{0}\left[b{d}_{2}+{c}_{2}\left(b+{d}_{2}\right)\right]\right\}}^{2}\end{array}\right\}}{2{D}^{2}}\);

\({\pi }_{2}^{CB}=\frac{\left\{\begin{array}{c}2Dw{e}_{0}\left\{{G}_{2}+{c}_{2}w{e}_{0}\left[b{d}_{1}+{c}_{1}\left(b+{d}_{1}\right)\right]\right\}\\ -{d}_{2}{\left\{{G}_{2}+{c}_{2}w{e}_{0}\left[b{d}_{1}+{c}_{1}\left(b+{d}_{1}\right)\right]\right\}}^{2}+2Dw\left({e}_{0}-{e}_{1}\right)\left\{{G}_{2}-{T}_{2}-{d}_{2}w{e}_{0}\left[b{d}_{1}+{c}_{1}\left(b+{d}_{1}\right)\right]\right\}\\ -{c}_{2}{\left\{-{G}_{2}+{T}_{2}+{d}_{2}w{e}_{0}\left[b{d}_{1}+{c}_{1}\left(b+{d}_{1}\right)\right]\right\}}^{2}-2\left(J-M{e}_{0}\right)\left\{{T}_{2}+w{e}_{0}\left[b{d}_{1}+{c}_{1}\left(b+{d}_{1}\right)\right]\left({c}_{2}+{d}_{2}\right)\right\}\end{array}\right\}}{2{D}^{2}}\)

Back to article page