Double Free-Riding in Innovation and Abatement: A Rules Treaty Solution

Abstract

In addressing climate change, both abatement itself and the innovation of superior abatement technologies are exposed to free-riding. To examine this double free-riding problem, we develop a multi-country model with an international market for emission permits and licenses for abatement technologies. We show that the two problems are mutually reinforcing. To address the double free-riding problem we propose a rules treaty for innovation and abatement that consists of two rules, an allocation rule and a refunding rule. The allocation rule determines the share of issued emission permits that each country can directly allocate to its domestic firms, while the remainder is handed over to an international agency. The refunding rule determines how the agency’s revenues from selling these permits to firms are redistributed. A fraction is given to those countries that successfully develop superior abatement technologies provided they license the technology free of charge to all countries. The remaining revenues are redistributed to all countries. These rules can approximate globally optimal abatement and innovation levels.

Appendices

A Appendix: Table of Variables

Symbol	Meaning
n	Number of countries
i, j	Country indices
$\overline{e}$	Business-as-usual emissions
$e_i$	GHG emissions of Country i
$\overline{E}$	Global business-as-usual emissions
E	Global GHG emissions
$\epsilon _i$	Emission permits issued by local planner to domestic firm
$\mathcal {E}$	Global emission permits issued
$\alpha$	Innovation cost parameter
$\beta$	(Environmental) damage parameter
$\overline{\chi }$	Business-as-usual abatement technology
$\chi _i^\prime$	Pre-trade abatement technology of Country i
$\chi _i$	Post-trade abatement technology of Country i
p	Price of emission permit
$\pi$	Price of abatement technology
$K_i$	Local costs of Country i
$\mu$	Allocation parameter
$\rho$	General refunding parameter
$\rho ^{CT}$	Clean-tech refunding parameter
GSO	Global social optimum
NTO	No-treaty outcome
RTIA	Rules treaty for innovation and abatement
RTA	Refunding treaty for abatement only
INN	NTO with cooperation in innovation only
ABA	NTO with cooperation in abatement only
AUT	Autarky

B Appendix: Proofs

1.1 B.1 Proof of Proposition 1

Let $\lambda$ denote the Lagrange multiplier associated with the constraint. The Karush–Kuhn–Tucker conditions read

$$\begin{aligned} \frac{\partial GSC}{\partial e}&= -\chi (\overline{e}-e) + n \beta = 0,\\ \frac{\partial GSC}{\partial \chi }&= \frac{n}{2} (\overline{e}-e)^2 - 2 \alpha \frac{\overline{\chi }^2}{\chi ^3} + \lambda = 0, \\ \chi - \overline{\chi }&\le 0, \quad \lambda \ge 0, \quad \lambda ~ (\chi - \overline{\chi }) = 0. \end{aligned}$$

Summing up over all countries and rearranging terms yields

$$\begin{aligned} E&= \overline{E} - \frac{n^2 \beta }{\chi },\\ \chi ^3&= \frac{4 \alpha \overline{\chi }^2}{n (\overline{e}-e)^2 + 2 \lambda }. \end{aligned}$$

Suppose the constraint is non-binding. In this case, the complementarity slackness condition yields $\lambda = 0$, and since $\overline{e}-e = \frac{n \beta }{\chi }$, we get $\chi = \frac{4 \alpha \overline{\chi }^2}{n^3 \beta ^2}$. This a solution to the optimization problem if and only if it satisfies the constraint $\chi - \overline{\chi } \le 0$, which is true as long as $4 \alpha \overline{\chi } \le n^3 \beta ^2$. Otherwise, the globally optimal technology level is $\chi = \overline{\chi }$. $\square$

1.2 B.2 Proof of Equations (12a) and (12b)

Let $\lambda _1$ denote the Lagrange multiplier associated with the constraint. The Karush–Kuhn–Tucker conditions are given by

$$\begin{aligned}&\frac{n \beta ^2}{\chi _1^2} - \frac{\beta ^2}{2 \chi _1^2} - 2 \alpha \frac{\overline{\chi }^2}{\chi _1^3} + (n-1) \frac{\beta ^2}{2 \chi _1^2} + \lambda _1 = 0,\\&\chi _1 - \overline{\chi } \le 0, \quad \lambda _1 \ge 0, \quad \lambda _1 ~ (\chi _1 - \overline{\chi })=0. \end{aligned}$$

Suppose the constraint is non-binding. Then, $\lambda _1 = 0$. Solving for $\chi _1$ yields

$$\begin{aligned} \chi _1 = \frac{4 \alpha \overline{\chi }}{(3n-2) \beta ^2} ~ \overline{\chi }. \end{aligned}$$

This is a solution as long as $\chi _1 \le \overline{\chi }$, which is true if $4 \alpha \overline{\chi } \le (3n-2) \beta ^2$. Otherwise, $\chi _1 = \overline{\chi }$. Hence, $\chi _1 = \min \left\{ \frac{4 \alpha \overline{\chi }}{(3n-2) \beta ^2}, 1 \right\} ~ \overline{\chi }$.

Let $\lambda _2$ denote the Lagrange multiplier associated with the constraint. The Karush–Kuhn–Tucker conditions read

$$\begin{aligned}&- 2 \alpha \frac{\overline{\chi }^2}{\chi _2^{\prime 3}} + \frac{\beta ^2}{2 \chi _2^{\prime 2}} + \lambda _2 = 0\\&\chi _2^\prime - \overline{\chi }\le 0, \quad \lambda _2 \ge 0, \quad \lambda _2 ~ (\chi _2^\prime - \overline{\chi }) = 0 \end{aligned}$$

Suppose the constraint is non-binding. Then, $\lambda _2 =0$. Solving for $\chi _2^\prime$ yields

$$\begin{aligned} \chi _2^\prime = \frac{4 \alpha \overline{\chi }^2}{\beta ^2}, \end{aligned}$$

which is a solution as long as $\chi _2^\prime \le \overline{\chi }$ or, equivalently, $4 \alpha \overline{\chi } \le \beta ^2$. Otherwise, $\chi _2^\prime = \overline{\chi }$. Hence, $\chi _2^\prime = \left\{ \frac{4 \alpha \overline{\chi }}{\beta ^2}, 1 \right\} ~ \overline{\chi }$. $\square$

1.3 B.3 Proof of Corollary 1

The cost functions are

$$ \begin{aligned} K_1^{NTO}&= \beta \left( \overline{E}- \frac{n \beta }{\chi _1}\right) + \frac{\beta ^2}{2 \chi _1} + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{1}}\bigg )^2-1\bigg ) - (n-1) ~ \pi (\chi _1, \chi _2^\prime ),\\ K_2^{NTO}&= \beta \left( \overline{E} - \frac{n \beta }{\chi _1}\right) + \frac{\beta ^2}{2\chi _1} + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{2}^\prime }\bigg )^2-1\bigg ) + \pi (\chi _1, \chi _2^\prime ),\\ K_j^{NTO}&= \beta \left( \overline{E} - \frac{n \beta }{\chi _1}\right) + \frac{\beta ^2}{2\chi _1} + \pi (\chi _1, \chi _2^\prime ),\\ K^{NoR \& D}&= \beta \left( \overline{E} - \frac{n \beta }{\overline{\chi }}\right) + \frac{\beta ^2}{2 \overline{\chi }}. \end{aligned}$$

For convenience, define $\gamma = \frac{4 \alpha \overline{\chi }}{(3 n - 2) \beta ^2}$. There are three possible cases to consider:

Case 1 :

Suppose $4 \alpha \overline{\chi } \ge (3n-2) \beta ^2$. This implies $\chi _1 = \chi _2^{\prime } = \chi _j^{\prime } = \overline{\chi }, ~ \forall j = 3,\ldots ,n$. Because there are no innovation effort, this directly leads to $K_j^{NTO} = K_2^{NTO} = K_1^{NTO} = K^{NoR \& D}, ~ \forall j = 3,\ldots ,n$.

Case 2 :

Suppose $\beta ^2 \le 4 \alpha \overline{\chi } < (3n-2) \beta ^2$. This implies $\chi _1 < \chi _2^{\prime } = \chi _j^{\prime } = \overline{\chi }, ~ \forall j = 3,\ldots ,n$. Because Country 2 exerts no innovation effort, $K_j^{NTO} = K_2^{NTO}, ~ \forall j = 3,\ldots ,n$. First, the cost difference between Country 2 and Country 1 is given by

$$\begin{aligned} K_2^{NTO} - K_1^{NTO}&= \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{2}^\prime }\bigg )^2 - \bigg (\frac{\overline{\chi }}{\chi _{1}}\bigg )^2 \bigg ) + n ~ \pi (\chi _1, \chi _2^\prime )\\&= \frac{\beta ^2}{4 \gamma \overline{\chi }} (1-\gamma ) (2 n - (1+\gamma ) (3n-2)). \end{aligned}$$

Since $\gamma \in (0,1]$, the first two factors are unambiguously positive. The third factor attains its maximum as $\gamma \rightarrow 0$ and amounts to $2-n$, which is non-positive as long as $n \ge 2$. This establishes $K_2^{NTO} < K_1^{NTO}$. Second, the cost difference between Country 1 and the benchmark without innovation reads

$$ \begin{aligned} K_1^{NTO} - K^{NoR \& D}&= \left( \frac{\beta ^2}{2} - n \beta ^2 \right) \bigg ( \frac{1}{\chi _1} - \frac{1}{\overline{\chi }} \bigg ) + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{1}}\bigg )^2-1\bigg ) \\&\quad - (n-1) ~ \pi (\chi _1, \chi _2^\prime ) \\&= -\frac{(\gamma -1)^2 (3n-2) \beta ^2}{4 \gamma \overline{\chi }} < 0, \end{aligned}$$

which establishes $K_1^{NTO} < K^{NoR \& D}$.

Case 3 :

Suppose $\beta ^2 > 4 \alpha \overline{\chi }$. This implies $\chi _1< \chi _2^{\prime } < \chi _j^{\prime } = \overline{\chi }, ~ \forall j = 3,\ldots ,n$. First, for an arbitrary Country $j = 3,\ldots ,n$, the cost difference to Country 2 is given by

$$\begin{aligned} K_j^{NTO} - K_2^{NTO} = - \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{2}^\prime }\bigg )^2-1\bigg ) < 0, \end{aligned}$$

which establishes $K_j^{NTO} < K_2^{NTO}, ~ \forall j = 3,\ldots ,n$. Second, the cost difference between Country 2 and Country 1 is given by

$$\begin{aligned} K_2^{NTO} - K_1^{NTO}&= \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{2}^\prime }\bigg )^2 - \bigg (\frac{\overline{\chi }}{\chi _{1}}\bigg )^2 \bigg ) + n ~ \pi (\chi _1, \chi _2^\prime )\\&= - \frac{3 (n-1)^2 \beta ^2}{4 \gamma (3n-2) \overline{\chi }} < 0, \end{aligned}$$

which establishes $K_2^{NTO} < K_1^{NTO}$. Third, the cost difference between Country 1 and the benchmark without innovation reads

$$ \begin{aligned} K_1^{NTO} - K^{NoR \& D}&= \left( \frac{\beta ^2}{2} - n \beta ^2 \right) \bigg ( \frac{1}{\chi _1} - \frac{1}{\overline{\chi }} \bigg ) + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{1}}\bigg )^2-1\bigg ) \\&\quad - (n-1) ~ \pi (\chi _1, \chi _2^\prime ) \\&= \frac{\beta ^2}{4 \gamma \overline{\chi }} \left( -(3n-2) \gamma ^2 - 2 (1-2 n) \gamma - n -6 \frac{(n-1)^2}{3n-2}\right) . \end{aligned}$$

The term in brackets is a quadratic equation in $\gamma$ that attains its maximum at $\gamma = \frac{2n-1}{3n-2} > 0$. At its maximum the value of the large parenthesis is negative. Hence, $K_1^{NTO} < K^{NoR \& D}$.

Summing up, $K_j^{NTO} \le K_2^{NTO} \le K_1^{NTO} \le K^{NoR \& D}, ~ \forall j = 3,\ldots ,n$. $\square$

1.4 B.4 Proof of Table 1

In general the local costs function for Country 1 is:

$$\begin{aligned} K_1 = \beta E + \frac{p^2}{2 \chi _1} + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _{1}}\bigg )^2-1\bigg ) - (n-1) ~ \pi . \end{aligned}$$

Case AUT :

For autarky, $p = \beta$, $E = \overline{E}- \frac{p}{\chi _1}$ and $\pi = 0$. The cost minimization problem with respect to the abatement technology reads

$$\begin{aligned}&\min _{\chi _1} \bigg \{ \beta \left( \overline{E}- \frac{\beta }{\chi _1}\right) + \frac{\beta ^2}{2 \chi _1} + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi _1}\bigg )^2-1\bigg ) \bigg \}. \end{aligned}$$

Solving the first order condition

$$\begin{aligned}&\frac{\beta ^2}{\chi _1^2} - \frac{\beta ^2}{2 \chi _1^2} - 2 \alpha \frac{\overline{\chi }^2}{\chi _1^3}=0 \end{aligned}$$

for $\chi _1$ yields the autarky technology level given in the table.³⁹

Case INN :

For cooperation with respect to innovation only, $p = \beta$, $E = \overline{E}- n \frac{p}{\chi _1}$ and $\pi = 0$. Furthermore we take into account that Country 1 considers global damages and global abatement costs. The cost minimization problem with respect to the abatement technology reads

$$\begin{aligned} \min _{\chi } \bigg \{ n\Big (\beta \left( \overline{E}- n\frac{\beta }{\chi }\right) + \frac{\beta ^2}{2 \chi }\Big )+ \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi }\bigg )^2-1\bigg ) \bigg \}. \end{aligned}$$

Solving the first order condition

$$\begin{aligned}&n^2\frac{ \beta ^2}{\chi ^2} - n\frac{\beta ^2}{2 \chi ^2} - 2 \alpha \frac{\overline{\chi }^2}{\chi ^3}= 0 \end{aligned}$$

for $\chi _1$ yields the cooperation with respect to innovation-only technology level given in the table.

For both cases, the global emissions can be calculated using $E=\overline{E}-n\frac{p}{\chi }$. $\square$

1.5 B.5 Proof of Equation (22)

The respective cost minimization problem reads

$$\begin{aligned}&\min _{\chi } \bigg \{ n \bigg ( \frac{{p^{RTIA}}^2}{2 \chi } + \beta \bigg (\overline{E} - n\frac{p^{RTIA}}{\chi }\bigg ) \bigg ) + \alpha \bigg (\bigg (\frac{\overline{\chi }}{\chi }\bigg )^2-1\bigg ) \bigg \} \end{aligned}$$

subject to

$$\begin{aligned}&\chi - \overline{\chi } \le 0. \end{aligned}$$

Let $\lambda$ denote the Lagrange multiplier associated with the constraint. The Karush–Kuhn–Tucker conditions read

$$\begin{aligned}&-\frac{n {p^{RTIA}}^2}{2 \chi ^2} + \beta \frac{n^2 p^{RTIA}}{\chi ^2} - 2 \alpha \frac{\overline{\chi }^2}{\chi ^3} + \lambda = 0, \\&\chi - \overline{\chi } \le 0, \quad \lambda \ge 0, \quad \lambda ~ (\chi - \overline{\chi }) = 0. \end{aligned}$$

Suppose the constraint is non-binding. In this case, the complementarity slackness condition yields $\lambda = 0$ and we get

$$\begin{aligned} \chi ^*&= \frac{4 \alpha \overline{\chi }^2}{n p^{RTIA} (2 n \beta - p^{RTIA})}. \end{aligned}$$

This is a solution to the optimization problem if and only if it satisfies the constraint $\chi - \overline{\chi } \le 0$, which is true as long as $4 \alpha \overline{\chi } \le n^3 \beta ^2 (1+\frac{(n-1)\mu }{(n-1)\mu +1})$. Otherwise, the globally optimal technology level is $\chi ^*= \overline{\chi }$. $\square$

1.6 B.6 Proof of Proposition 3

First, we determine the first-order condition of local costs of Country 1 (13a) with respect to the technology level under a RTIA $\{\mu ,\rho ,\rho ^{CT},F\}$. Using Eqs. (1), $E = \overline{E}- \frac{n p}{\chi _1}$ and (21a) together with $\frac{\partial E}{\partial \chi _1}= \frac{n p}{\chi _1^2}$, $\frac{\partial e}{\partial \chi _1}= \frac{ p}{\chi _1^2}$, and $\frac{\partial \epsilon _1}{\partial \chi _1} = \frac{p}{\chi _1^2} - 2(n-1) \frac{1-\mu }{\mu } \rho ^{CT} \frac{p}{\chi _1^2}$, the first-order condition reads

$$\begin{aligned}&\frac{n \beta p}{\chi _1^2} - \frac{p^2}{2 \chi _1^2} - 2 \alpha \frac{\overline{\chi }^2}{\chi _1^3} + p \bigg ( \frac{p}{\chi _1^2} - \mu \big ( \frac{p}{\chi _1^2} - 2(n-1) \frac{1-\mu }{\mu } \rho ^{CT} \frac{p}{\chi _1^2} \bigg ) \bigg ) ~ - \\&\quad (\rho +\rho ^{CT}) (1-\mu ) p \frac{n p}{\chi _1^2} = 0. \end{aligned}$$

Substituting for the permit price (27) and using the fact that $\rho ^{CT} = 1-n \rho$ yields

$$\begin{aligned} \rho ^{CT} = \frac{4\alpha \frac{ \overline{ \chi }^2}{\chi _1}-p(2n \beta - p)}{2p^2(1- \mu ) (n-1)}. \end{aligned}$$

Suppose there is an interior solution of (22). Setting $\chi _1= \chi ^*(\mu )= \frac{4 \alpha \overline{\chi }^2}{n p (2 n \beta - p)}$ yields the clean-tech parameter that ensures that $\chi ^*(\mu )$ is the minimum of Country 1’s local costs

$$\begin{aligned} {\rho ^{CT}}^*= \frac{(n-1) \mu + 0.5}{1-\mu }. \end{aligned}$$

Plugging ${\rho ^{CT}}^*$ and $\chi _1= \chi ^*(\mu )= \frac{4 \alpha \overline{\chi }^2}{n p (2 n \beta - p)}$ into the local costs functions (13a) and (13b) and using the constrained optimal amount of emission permits (21a) and (21b) yields

$$\begin{aligned} K_1^{RTIA}(\chi _1= \chi ^*(\mu )) =&-\frac{n^2 p^2 (2 n \beta - p)^2}{16 \alpha \overline{\chi }^2} - \alpha + \beta \overline{E}+F,\\ K_2^{RTIA}(\chi _1= \chi ^*(\mu )) =&\beta \overline{E} -\frac{F}{n-1}. \end{aligned}$$

Next, given $\mu$, we define $\overline{F}(\mu )$ as the lump-sum parameter that equalizes local costs in countries 1 and 2: $K_1^{RTIA}(\chi _1 = \chi ^*(\mu )) = K_2^{RTIA}(\chi _1 = \chi ^*(\mu ))$.

$$\begin{aligned} \overline{F}(\mu )= \frac{n(n-1){p(\mu )}^2 (2 n \beta - p(\mu ))^2}{16 \alpha \overline{\chi }2} +\frac{\alpha (n-1)}{n}. \end{aligned}$$

Consider a RTIA $\{\mu , \rho , \rho ^{CT} ,F\}$. If $F < \overline{F}(\mu )$ an equilibrium that implements $\chi ^*(\mu )$ does not exist. The proof is as follows: Suppose there are two countries A and B, and Country A is the technology leader with $\chi _A=\chi ^*(\mu )$. Since $F < \overline{F}(\mu )$, local costs in Country B are higher than local costs in Country A. By exerting slightly more innovation effort than Country A, Country B would steal the complete clean-tech refund and lower its local costs. Clearly, country A responds to increase its innovation effort as well. This process continues until costs are equalized between Country A and B. As a consequence, one country, say A, would exert innovation effort while the other, say B, would not innovate. However, this cannot be an equilibrium because given that Country B does not exert innovation effort, Country A chooses $\chi _A = \chi ^*(\mu )$. Thus, for an equilibrium to exist, $F \ge \overline{F}(\mu )$.

Furthermore, if F is sufficiently large, the costs for Country 1 can exceed the costs for either leaving the treaty or the costs for selling the technology in Stage 2. Thus, in our treaty proposal, we restrict to $F = \overline{F}(\mu )$ since this always guarantees the existence of a constrained optimal subgame perfect equilibrium by construction. $\square$

1.7 B.7 Proof of Proposition 5

Because $K_i^{NTO} \le K_2^{NTO} \le K_1^{NTO} \le K^{NoR \& D}, ~ \forall i = 3,\ldots ,n$ (see Corollary 1) and $K_j^{RTIA} = K^{RTIA}, ~ \forall j = 1,\ldots ,n$ (see Proposition 3), it suffices to show that $K_i^{NTO} \ge K^{RTIA}$.

First, suppose $\chi ^{NTO} = \overline{\chi }$ and $\chi ^{RTIA} = \overline{\chi }$. This directly yields $K_i^{NTO} \ge K^{RTIA}$. Next, suppose $\chi ^{NTO} = \overline{\chi }$ and $\chi ^{RTIA} < \overline{\chi }$. Due to optimization, we immediately know that $K^{RTIA}(\chi ^{RTIA}) < K^{RTIA}(\overline{\chi }) = K_i^{NTO}$. Hence, it remains to show that for interior solutions of the technology level in the NTO and the RTIA, $K_i^{NTO} \ge K^{RTIA}$.

The cost under RTIA is given by

$$\begin{aligned} K^{RTIA}&= \frac{\chi }{2} ~ (\overline{e}-e)^2 + \beta ~ E + p^{RTIA} ~(e-\mu ~\epsilon _j) - \rho ~(1-\mu ) ~ p^{RTIA} ~ \mathcal {E} \\&= \frac{p^{RTIA 2}}{2 \chi } + \beta ~ \bigg ( \overline{E} - \frac{n p^{RTIA}}{\chi } \bigg ) + (1-\mu ) ~ \rho ^{CT} ~\frac{p^{RTIA 2}}{\chi } \\&= \beta ~\overline{E} - \frac{\alpha }{n} - \frac{n p^{RTIA 2} (2 n \beta - p^{RTIA})^2}{16 \alpha \overline{\chi }^2}, \end{aligned}$$

where the second line follows from the equilibrium conditions on the emission market, zero-profits for the agency controlling the refunding scheme, and the optimal amount of emission permits issued by Countries $j = 2,\ldots ,n$. The last line additionally uses the optimal technology level and the optimal clean-tech refund for a given $\mu$. Since $p^{RTIA}$ is decreasing in $\mu$, the cost function $K^{RTIA}$ is increasing in $\mu$. Hence, it suffices to show that $K^{RTIA}|_{\mu = 1} \le K_i^{NTO}$. Evaluating the cost at $\mu = 1$ yields

$$\begin{aligned} K^{RTIA}|_{\mu = 1} = \beta ~\overline{E} - \frac{\alpha }{n} - \frac{n (2 n - 1)^2 ~\beta ^4}{16 ~ \alpha ~ \overline{\chi }^2}. \end{aligned}$$

The cost function under NTO is given by

$$\begin{aligned} K_3^{NTO}&= \frac{\chi }{2} ~ (\overline{e}-e)^2 + \beta ~ E + p^{NTO} ~ (e-\epsilon ) + \pi (\chi ^{NTO},\chi _2^{NTO}) \\&= \beta ~ \overline{E} - \frac{\beta ^2}{\chi ^{NTO}} ~ (n-1) - \frac{\beta ^2}{2} \frac{1}{\chi _2^{NTO}} \\&= \beta ~ \overline{E} - \frac{4 ~(n-1) ~(3n-2) ~\beta ^2}{16 ~ \alpha ~ \overline{\chi }^2} - \frac{\beta ^2}{2} \frac{1}{\chi _2^{NTO}}, \end{aligned}$$

where the second line follows from the equilibrium conditions on the emission market, the optimal amount of emission permits issued by countries $i = 3,\ldots ,n$ and the equilibrium price for the frontier technology. The last line additionally uses the optimal technology level.

Case 1 :

Suppose $4 \alpha \overline{\chi } < \beta ^2$. Then, $\chi _2^{NTO} = 4 \alpha \overline{\chi }^2/\beta ^2$ and

$$\begin{aligned} K_3^{NTO} - K^{RTIA}|_{\mu = 1} = \frac{\alpha }{n} - \frac{\beta ^4}{16 ~ \alpha ~ \overline{\chi }^2} ~ [10 - 21 ~ n + 16 ~ n^2 - 4 ~ n^3]. \end{aligned}$$

The term in square brackets is decreasing in n and equals to zero for $n = 2$. Hence, the cost difference is strictly positive.

Case 2 :

Suppose $4 \alpha \overline{\chi } \ge \beta ^2$. Then, $\chi _2^{NTO} = \overline{\chi }$ and

$$\begin{aligned} K_3^{NTO} - K^{RTIA}|_{\mu = 1} = \frac{\alpha }{n} - \frac{\beta ^4}{16 ~ \alpha ~ \overline{\chi }^2} ~ [8 - 21 ~ n + 16 ~ n^2 - 4 ~ n^3] - \frac{\beta ^2}{2 ~ \overline{\chi }}. \end{aligned}$$

The term in square brackets is unambiguously negative and since $4 \alpha \overline{\chi } \ge \beta ^2$, we get

$$\begin{aligned} K_3^{NTO} - K^{RTIA}|_{\mu = 1}&\ge \frac{\alpha }{n} - \alpha ~ [8 - 21 ~ n + 16 ~ n^2 - 4 ~ n^3] - \frac{\beta ^2}{2 ~ \overline{\chi }}\\&\ge \frac{\alpha }{n} - \alpha ~ [8 - 21 ~ n + 16 ~ n^2 - 4 ~ n^3] - 2 ~ \alpha \\&\ge \frac{\alpha }{n} - \alpha ~ [10 - 21 ~ n + 16 ~ n^2 - 4 ~ n^3]. \end{aligned}$$

The expression in square brackets is decreasing in n and equals to zero for $n = 2$. Hence, the cost difference is strictly positive.

Combining case 1 and case 2 completes our proof. $\square$

1.8 B.8 Proof of Equation (23)

To derive Eq. (23), we consider the case in which only one country innovates, i.e. $\chi _2'=\overline{\chi }$. Countries are symmetric and $\rho ^{CT} = 0$. Thus $\rho =\frac{1}{n}$ and $e^{RTA}= \epsilon ^{RTA}$. For this reason the local costs of trading, $p^{RTA} (e^{RTA} - \mu \epsilon ^{RTA})$ and the refund revenue, $\rho (1-\mu )p^{RTA} E^{RTA}$, add up to zero irrespective of whether a country buys the technology or not. As a consequence, we can directly apply Eq. (10) to calculate the technology price $p = p^{RTA}$. $\square$

C Appendix: Negative Willingness to Pay in the RTA

We discuss the rather surprising result that in the RTA as $\mu$ decreases, the buyer country’s willingness to pay for the superior technology decreases and finally even turns negative.

Consider Eq. (23). The first term in the first bracket is positive and linear in the permit price, as it stems from the local damages reduction. The second term in the brackets is negative and quadratic in the permit price, as it stems from the cost increase in the abatement costs. Surprisingly, although the superior technology decreases abatement cost for a given amount of abatement, the total abatement costs increase. The reason is that better technology induces higher abatement levels, which overcompensates the cost decrease for a certain level. As a consequence, for $p^{RTA}$ sufficiently high, the quadratic term dominates and buying the technology makes the country worse off. This result clearly depends on how the damage and abatement costs are modeled and, in our case, is driven by the assumption of linear damages on the one side and quadratic abatement costs on the other side.

To get further intuition, consider a setting where a firm faces the decision of either buying (or not selling) permits or reducing emissions. Suppose that permit prices are fixed and the value of the permits the firm is being grandfathered is sunk. Hence, a firm minimizes $pe +\frac{\chi }{2} (\overline{e}-e)^2$, and with $e=\overline{e}-\frac{p}{\chi }$ we get $p\overline{e}-\frac{p^2}{2\chi }$. The firms’ costs unambiguously decrease in the technology level. The firm level, however, is not the case we are considering. Instead, our focus is on the national level where the local damages, $\beta E$, are also accounted for. In addition, in our setting there are no costs associated with buying emission permits as the local planner creates exactly the amount needed by the firm. Hence, the term pe is dropped and other terms related to permit trading or refunding do not enter either.⁴⁰ Knowing that he can create the amount of permits needed, the local planner only cares about the balance between abatement costs on the one side and environmental damages on the other side.

The result hinges on the assumption that the firms always use the best available technology. To see this, consider the case where the willingness to pay is negative ($p^{RTA}>2 \beta$). If the local planner could influence the firms’ decision, he would promt the firm to use $\overline{\chi }$ instead of $\chi ^{RTA}$, even though the technology leader does not charge anything for it. This influence could be direct via a command and control approach. A more indirect way would be to promise the firms that, for whatever they emit, they receive both the fraction of permits the local planner is allowed to keep and, in addition, the revenues of the refunding. Thus, the only costs the firm would be left with are the abatement costs and would thus voluntarily use $\overline{\chi }$. The RTA has to be set up in a way that prevents such a counterproductive behavior of countries.