Carbon price interaction between allocated permits and generated offsets

Abstract

The Kyoto Protocol established targets for curbing greenhouse gas emissions in order to mitigate climate change, and it introduced two kinds of market-based mechanisms: the emission allowance market and the carbon offset market. We identify stylized features of the two mechanisms with a partial equilibrium model. Our work is the first to derive a closed form solution incorporating most policy instruments, such as abatement and offset usage, and delivery risks in offsets. We show that policy changes will impact one market directly and the other indirectly, generating unequal price responses that affects the spread between the two compliance instruments. We show how the price spread between allowances and offsets is affected by market conditions such as the offset import limit, abatement and offset cost, penalty rate, emission cap, and baseline emissions.

Appendices

Appendix 1: Comparative statics

Here, we apply the implicit function theorem, and apply the envelope conditions to derive solutions for the comparative statics analysis. From the first order conditions in Eqs. (8) and (9), the interior solution $\left( {u^{*} , q^{*} } \right)$ is defined as below:

$$\frac{\partial TAC\left( u \right)}{\partial u} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{\partial u} = 0$$

(16)

$$\frac{\partial C\left( q \right)}{\partial q} + \frac{\partial M\left( q \right)}{\partial q} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{\partial q} = 0$$

(17)

The Eq. (16) implicitly defines $u^{*} \left( {P_{penalty} |\overline{\Omega } } \right), u^{*} \left( {y|\overline{\Omega } } \right), u^{*} \left( {L|\overline{\Omega } } \right), u^{*} \left( {\theta |\overline{\Omega } } \right)$, and the Eq. (17) implicitly defines $q^{*} \left( {P_{penalty} |\overline{\Omega } } \right), q^{*} \left( {y|\overline{\Omega } } \right), q^{*} \left( {\theta |\overline{\Omega } } \right), q^{*} \left( {L|\overline{\Omega } } \right)$, where we assume the other parameters, $\Omega$, are held constant the assumption of Ceteris Paribus.

Regarding the effect of $P_{penalty}$, redefine (16) and (17) as

$$\frac{{\partial TAC\left( {u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {P_{penalty} |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}=0$$

(18)

$$\frac{{\partial C\left( {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}} + \frac{{\partial M\left( {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}} = 0$$

(19)

To see how $P_{penalty}$ affects the equilibria, we totally differentiate Eqs. (18) and (19),

$$\begin{aligned} &\left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {P_{penalty} |\Omega } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {P_{penalty} |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }}} \right\}du^{*} \left( {P_{penalty} |\overline{\Omega } } \right) \hfill \\ &+ \left\{ {\frac{{\partial E max\left[ {y - L - u^{*} \left( {P_{penalty} |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}} \right\}dP_{penalty} = 0 \hfill \\ \end{aligned}$$

$$\to \frac{{du^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}{{dP_{penalty} }} = - \frac{{\left\{ {\frac{{\partial E \hbox{max} \left[ {y - L - u^{*} \left( {P_{penalty} |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}} \right\}}}{{\left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E \hbox{max} \left[ {y - L - u^{*} \left( {P_{penalty} |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }}} \right\}}}$$

And,

$$\left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }}} \right\}dq^{*} \left( {P_{penalty} |\overline{\Omega } } \right) + \left\{ {\frac{{\partial E max\left[ {y - L - u - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}} \right\}dP_{penalty} = 0$$

$$\to \frac{{dq^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}{{dP_{penalty} }} = - \frac{{\left\{ {\frac{{\partial E \hbox{max} \left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}} \right\}}}{{\left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E \hbox{max} \left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)\varepsilon ,L\theta } \right],0} \right]}}{{\partial q^{*} \left( {P_{penalty} |\overline{\Omega } } \right)^{2} }}} \right\}}}$$

Since the objective function is a convex function with respect to $\left( {u, q} \right)$ as a cost-minimizing objective function, the denominators of $\frac{{du^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}{{dP_{penalty} }}$ and $\frac{{dq^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}{{dP_{penalty} }}$ are positive; whereas, the nominators are negative.

Thus, we can conclude that $\frac{{du^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}{{dP_{penalty} }} > 0$ and $\frac{{dq^{*} \left( {P_{penalty} |\overline{\Omega } } \right)}}{{dP_{penalty} }} > 0$

Regarding the effect of $y$, redefine (16) and (17) as

$$\frac{{\partial TAC\left( {u^{*} \left( {y |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {y |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)}} = 0$$

(20)

$$\frac{{\partial C\left( {q^{*} \left( {y |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y |\overline{\Omega } } \right)}} + \frac{{\partial M\left( {q^{*} \left( {y |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {y |\overline{\Omega } } \right)}} = 0$$

(21)

To see how $y$ affects the equilibria, we totally differentiate Eqs. (20) and (21).

$$\begin{aligned} & \left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {y |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {y |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)^{2} }}} \right\}du^{*} \left( {y |\overline{\Omega } } \right) \hfill \\ & + \left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {y |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)\partial y}}} \right\}dy = 0 \hfill \\ \end{aligned}$$

$$\to \frac{{du^{*} \left( {y |\overline{\Omega } } \right)}}{dy} = - \frac{{\left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {y |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta \overline{\Omega } } \right], 0} \right]}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)\partial y}}} \right\}}}{{\left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {y |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {y |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {y |\overline{\Omega } } \right)^{2} }}} \right\}}}$$

And,

$$\begin{aligned} & \left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {y|\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {y|\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y|\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }}} \right\}dq^{*} \left( {y|\overline{\Omega } } \right) \hfill \\ &+ \left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y|\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)\partial y}}} \right\}dy = 0 \hfill \\ \end{aligned}$$

$$\to \frac{{dq^{*} \left( {y|\overline{\Omega } } \right)}}{dy} = - \frac{{\left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y|\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)\partial y}}} \right\}}}{{\left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {y|\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {y|\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y|\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{2} }}} \right\}}}$$

Since the objective function is a convex function with respect to $\left( {u, q} \right)$ as a cost-minimizing objective function, the denominators of $\frac{{du^{*} \left( {y |\overline{\Omega } } \right)}}{dy}$ and $\frac{{dq^{*} \left( {y |\overline{\Omega } } \right)}}{dy}$ are positive; whereas, the nominators are negative.

Thus, we can conclude that $\frac{{du^{*} \left( {y |\overline{\Omega } } \right)}}{dy} > 0$ and $\frac{{dq^{*} \left( {y |\overline{\Omega } } \right)}}{dy} > 0$

Regarding the effect of $L$, redefine (16) and (17) as

$$\frac{{\partial TAC\left( {u^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {L |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)}} = 0$$

(22)

$$\frac{{\partial C\left( {q^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)}} + \frac{{\partial M\left( {q^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {L |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)}} = 0$$

(23)

To see how $L$ affects the equilibria, we totally differentiate Eqs. (22) and (23).

$$\begin{aligned} & \left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {L |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta \overline{\Omega } } \right], 0} \right]}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)^{2} }}} \right\}du^{*} \left( {L |\overline{\Omega } } \right) \hfill \\ & + \left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {L |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)\partial L}}} \right\}dL = 0 \hfill \\ \end{aligned}$$

$$\to \frac{{du^{*} \left( {L |\overline{\Omega } } \right)}}{dL} = - \frac{{\left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {L |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{\partial u\partial L}} \right\}}}{{\left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {L |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {L |\overline{\Omega } } \right)^{2} }}} \right\}}}$$

And,

$$\left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {L |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {L |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)^{2} }}} \right\}dq^{*} \left( {L |\overline{\Omega } } \right) + \left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {L |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {L |\overline{\Omega } } \right)\partial L}}} \right\}dL = 0$$

$$\to \frac{{dq^{*} \left( {y|\overline{\Omega } } \right)}}{dy} = - \frac{{\left\{ {\frac{{\partial^{2} P_{penalty} \cdot E\hbox{max} \left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y|\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)\partial y}}} \right\}}}{{\left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {y|\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {y|\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {y|\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E \hbox{max} \left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {y|\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{2} }}} \right\}}}$$

Since the objective function is a convex function with respect to $\left( {u, q} \right)$ as a cost-minimizing objective function, the denominators of $\frac{{du^{ *} \left( {L | {\bar{{\Omega }}}} \right)}}{dL}$ and $\frac{{{\text{dq}}^{ *} \left( {{\text{L}}|\overline{\Omega }} \right)}}{\text{dL}}$ are positive; also as the previous cases, the nominator of $\frac{{{\text{du}}^{ *} \left( {{\text{L}}|\overline{\Omega }} \right)}}{\text{dL}}$ is positive. In case of the nominator of $\frac{{{\text{dq}}^{ *} \left( {{\text{L}}|\overline{\Omega }} \right)}}{\text{dL}}$, Note the sign of total differentiation depends on the parameter, $\uptheta$. When the import limit is low, the value of available offsets, the value of $\hbox{min} \left[ {{\text{q }}\upvarepsilon,{\text{L}}\uptheta} \right]$ is defined just as the quantity limit, ${\text{L}}\uptheta$. Then we see a sequential partial differentiation with respect to L and q leads the nominator negative. Whereas, when the import limit is not binding and high enough, the value of $\hbox{min} \left[ {{\text{q}}\upvarepsilon,{\text{L}}\uptheta} \right]$ is characterized as a successfully delivered offset quantity. Note that the partial derivative,$\frac{{\partial {\text{q}}}}{{\partial {\text{L}}}}$ is negative. Then we see a sequential partial differentiation with respect to L and q leads the nominator positive; then the total differentiation, $\frac{{{\text{dq}}^{ *} \left( {{\text{L}}|\overline{\Omega }} \right)}}{\text{dL}}$, describes the case where an increase in free permits would decrease the amount of offsets consumed. Thus, we can derive different directions that $\frac{{{\text{du}}^{ *} \left( {{\text{L}}|\overline{\Omega }} \right)}}{\text{dL}} < 0$ and $\frac{{{\text{dq}}^{ *} \left( {{\text{L}}|\overline{\Omega }} \right)}}{\text{dL}} > or \le 0$.

Regarding the effect of $\theta$, redefine (16) and (17) as

$$\frac{{\partial TAC\left( {u^{*} \left( {\theta | {\overline{{\Omega }}}} \right)} \right)}}{{\partial u^{*} \left( {\theta | {\overline{{\Omega }}}} \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {\theta | {\overline{{\Omega }}}} \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {\theta | {\overline{{\Omega }}}} \right)}} = 0$$

(24)

$$\frac{{\partial C\left( {q^{*} \left( {\theta |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)}} + \frac{{\partial M\left( {q^{*} \left( {\theta |\overline{\Omega } } \right)\overline{\Omega } } \right)}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)}} + \frac{{\partial P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {\theta |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)}} = 0$$

(25)

To see how $\theta$ affects the equilibria, we totally differentiate Eqs. (24) and (25),

$$\left\{ {\frac{{\partial^{2} TAC\left( u \right)}}{{\partial u^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {\theta |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }}} \right\}du^{*} \left( {\theta |\overline{\Omega } } \right) + \left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {\theta |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {\theta |\overline{\Omega } } \right)\partial \theta }}} \right\}d\theta = 0$$

$$\to \frac{{du^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta } = - \frac{{\left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {\theta |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {\theta |\overline{\Omega } } \right)\partial \theta }}} \right\}}}{{\left\{ {\frac{{\partial^{2} TAC\left( {u^{*} \left( {\theta |\overline{\Omega } } \right)} \right)}}{{\partial u^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u^{*} \left( {\theta |\overline{\Omega } } \right) - \hbox{min} \left[ {q\varepsilon ,L\theta } \right], 0} \right]}}{{\partial u^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }}} \right\}}}$$

And,

$$\begin{aligned} & \left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {\theta |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {\theta | {\bar{{\Omega }}}} \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {\theta |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {\theta | {\bar{{\Omega }}}} \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {\theta |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }}} \right\}dq^{*} \left( {\theta |\overline{\Omega } } \right) \hfill \\ & + \left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {\theta |\overline{\varOmega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)\partial \theta }}} \right\}d\theta = 0 \hfill \\ \end{aligned}$$

$$\to \frac{{dq^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta } = - \frac{{\left\{ {\frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {\theta |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)\partial \theta }}} \right\}}}{{\left\{ {\frac{{\partial^{2} C\left( {q^{*} \left( {\theta |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} M\left( {q^{*} \left( {\theta |\overline{\Omega } } \right)} \right)}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }} + \frac{{\partial^{2} P_{penalty} \cdot E max\left[ {y - L - u - \hbox{min} \left[ {q^{*} \left( {\theta |\overline{\Omega } } \right)\varepsilon ,L\theta } \right], 0} \right]}}{{\partial q^{*} \left( {\theta |\overline{\Omega } } \right)^{2} }}} \right\}}}$$

Since the objective function is a convex function with respect to $\left( {u, q} \right)$ as a cost-minimizing objective function, the denominators of $\frac{{du^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta }$ and $\frac{{dq^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta }$ are positive. Contrary to previous cases, the nominator of $\frac{{du^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta }$ is positive, whereas, the nominator of $\frac{{dq^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta }$ is negative.

Thus, we can derive different directions that $\frac{{du^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta } < 0$ and $\frac{{dq^{*} \left( {\theta |\overline{\Omega } } \right)}}{d\theta } > 0$.

Appendix 2: Beta probability density function

2.1 Probability density function of baseline emissions (y)

2.2 Probability density function of the CER default factor ($\varepsilon$)

Appendix 3

The analytic solutions can be constructed by solving simultaneous Eqs. (10) and (11), and the following shows the closed form solutions written in Mathematica code with conditions on the import limit domains. Only assumptions given to this closed form solutions are the parameters that characterizes the specific Beta distributions in “Appendix 2”. The rest of policy or cost parameters are given in Table 1.

$${\text{u}}^{*} = - \frac{{P\left( {L - \bar{y}} \right)\left( {5P + 252\left( {k + m} \right)\bar{y}} \right)}}{{5P^{2} + 36\left( {5c + 7\left( {k + m} \right)} \right)P\bar{y} + 252c\left( {k + m} \right)\bar{y}^{2} }}$$

where $1 \ge \theta \ge \frac{{210cP\bar{y}\left( { - L + \bar{y}} \right)}}{{L\left( {5P^{2} + 36\left( {5c + 7\left( {k + m} \right)} \right)P\bar{y} + 252c\left( {k + m} \right)\bar{y}^{2} } \right)}}$

$${\text{u}}^{ *} = \bar{y} - \frac{1}{7}L\left( {7 + 6\theta } \right) - \frac{{6\left( {k + m} \right)\bar{y}{\text{Root}}\left[ {35L^{12} P^{2} \theta^{12} + \left( {210cL^{7} P\bar{y}\theta^{6} - 210cL^{6} P\bar{y}^{2} \theta^{6} - 30L^{7} P^{2} \theta^{7} + 180cL^{7} P\bar{y}\theta^{7} } \right)\# 1^{5} + \left( {252kP\bar{y} + 252mP\bar{y} + 252ck\bar{y}^{2} + 252cm\bar{y}^{2} } \right)\# 1^{12} \& ,2} \right]^{7} }}{{5L^{6} P\theta^{6} }}$$

where $0 \le \theta < \frac{{210cP\bar{y}\left( { - L + {\text{ybar}}} \right)}}{{L\left( {5P^{2} + 36\left( {5c + 7\left( {k + m} \right)} \right)P\bar{y} + 252c\left( {k + m} \right)\bar{y}^{2} } \right)}}$

$$q^{*} = \frac{{210cP\bar{y}\left( { - L + \bar{y}} \right)}}{{5P^{2} + 36\left( {5c + 7\left( {k + m} \right)} \right)P\bar{y} + 252c\left( {k + m} \right)\bar{y}^{2} }}$$

where $1 \ge \theta \ge \frac{{210cP\bar{y}\left( { - L + \bar{y}} \right)}}{{L\left( {5P^{2} + 36\left( {5c + 7\left( {k + m} \right)} \right)P\bar{y} + 252c\left( {k + m} \right)\bar{y}^{2} } \right)}}$

$$q^{*} = {\text{Root}}\left[ {35L^{12} P^{2} \theta^{12} + \left( {210cL^{7} P{\text{ybar}}\theta^{6} - 210cL^{6} P{\text{ybar}}^{2} \theta^{6} - 30L^{7} P^{2} \theta^{7} + 180cL^{7} P{\text{ybar}}\theta^{7} } \right)\# 1^{5} + \left( {252kP{\text{ybar}} + 252mP{\text{ybar}} + 252ck{\text{ybar}}^{2} + 252cm{\text{ybar}}^{2} } \right)\# 1^{12} \& ,2} \right]$$

where $0 \le \theta < \frac{{210cP\bar{y}\left( { - L + \bar{y}} \right)}}{{L\left( {5P^{2} + 36\left( {5c + 7\left( {k + m} \right)} \right)P\bar{y} + 252c\left( {k + m} \right)\bar{y}^{2} } \right)}}.$