Self-Enforcing Intergenerational Social Contracts for Pareto Improving Pollution Mitigation

Abstract

We consider, in an overlapping generations model with an environmental externality, a scheme of contracts between any two successive generations. Under each contract, agents of the young generation invest a share of their labor income in pollution mitigation in exchange for a transfer in the second period of their lives. The transfer is financed in a pay-as-you-go manner by the next young generation. Different from previous work we assume that the transfer is granted as a subsidy to capital income rather than lump sum. We show that the existence of a contract which is Pareto improving over the situation without contract for any two generations requires a sufficiently high level of income. In a steady state with social contracts in each period, the pollution stock is lower compared to a steady state without contracts. Analytical and numerical analysis of the dynamics under Nash bargaining suggests that under reasonable conditions, also steady state income and welfare are higher. Delaying the implementation of a social contract for too long or imposing a contract with too low mitigation can be costly: Net income may inevitably fall below the threshold in finite time so that Pareto improving mitigation is no longer possible and the economy converges to a steady state with high pollution stock and low income and welfare. In the second part of the paper, we study a game theoretic setup, taking into account that credibly committing to a contract might not be possible. We show that with transfers granted as a subsidy to capital income, there exist mitigation transfer schemes which are both Pareto improving and give no generation an incentive to deviate from any of its contracts even in a dynamically efficient economy. Social contracts coexist with private savings.

Appendix

1.1 Utility gains from a social contract $(m_{t},\tau _{t+1}^{o})$

Generation t: By substituting the optimal choices $c_{t}^{y}$ and $c_{t+1}^{o}$ as characterized in (6) and (8) respectively into the utility function, we derive the perfectly foreseen indirect utility of agent t under the contract $(m_{t},\tau _{t+1}^{o})$,

$$\begin{aligned} V_{t}^{t+1}\left( I_{t},m_{t},\tau _{t+1}^{o}\right) =\Phi +\ln \left[ z\left( E_{t}\right) \alpha k_{t+1}^{\alpha -1}\left( 1+\tau _{t+1}^{o}\right) \right] ^{\beta }+\ln \left[ I_{t}\left( 1-m_{t}\right) \right] ^{1+\beta } \end{aligned}$$

(25)

where $\Phi =\ln \frac{1}{1+\beta }+\beta \ln \frac{\beta }{1+\beta }$. We denote by ${\tilde{k}}_{t+1}$ capital per capita if there is no social contract. The foreseen surplus that agent t gains from the social contract $(m_{t},\tau _{t+1}^{o})$ is

$$\begin{aligned} \triangle {\mathcal {V}}_{t}^{t+1}=\left( \alpha -1\right) \beta \ln \frac{k_{t+1}}{{\tilde{k}}_{t+1}}+\left( 1+\beta \right) \ln (1-m_{t})+\beta \ln \left( 1+\tau _{t+1}^{o}\right) \end{aligned}$$

From Eq. (7), we find that the relation between capital per capita under the social contract and without social contract is $\frac{{\tilde{k}}_{t+1}}{k_{t+1}}=\frac{1}{1-m_{t}}$. The surplus $\triangle {\mathcal {V}}_{t+1}^{t+1}$ can then straightforwardly be found to equal the expression in (12).

Generation $t+1$: The perfectly foreseen indirect utility of generation $t+1$ under the contract $(m_{t},\tau _{t+1}^{o})$ is

$$\begin{aligned} {\mathcal {V}}_{t+1}^{t+1}\left( I_{t+1},m_{t+1}^{e},\tau _{t+2}^{o,e}\right)= & {} \Phi +\ln \left[ z\left( E_{t+1}\right) \alpha \left( \frac{\beta I_{t+1}\left( 1-m_{t+1}^{e}\right) }{1+\beta }\right) ^{\alpha -1}\left( 1+\tau _{t+2}^{o,e}\right) \right] ^{\beta }\nonumber \\&+\ln \left[ I_{t+1}\left( 1-m_{t+1}^{e}\right) \right] ^{1+\beta } \end{aligned}$$

(26)

where $(m_{t+1}^{e},\tau _{t+2}^{o,e})$ is the foreseen social contract that generation $t+1$ will sign with the succeeding generation $t+2$. $I_{t+1}=z(E_{t})(1-\alpha )k_{t+1}^{\alpha }(1-\tau _{t+1}^{y})$ is net income of agent $t+1$ in period $t+1$, i.e. the income after paying the intergenerational transfer $\tau _{t+1}^{o}R_{t+1}k_{t+1}=z(E_{t})(1-\alpha )k_{t+1}^{\alpha }\tau _{t+1}^{y}$ under the social contract $(m_{t},\tau _{t+1}^{o})$.

The foreseen surplus that agent $t+1$ gains from the social contract $(m_{t},\tau _{t+1}^{o})$ with the preceding generation is

$$\begin{aligned} \triangle {\mathcal {V}}_{t+1}^{t+1}=(1+\alpha \beta )\ln \frac{I_{t+1}}{{\tilde{I}}_{t+1}}+\beta \ln \frac{z(E_{t+1})}{z({\tilde{E}}_{t+1})} \end{aligned}$$

(27)

where ${\tilde{I}}_{t+1}=z(E_{t})(1-\alpha ){\tilde{k}}_{t+1}^{\alpha }$ and ${\tilde{E}}_{t+1}$ are net income of agent $t+1$ and the stock of pollution, respectively, in period $t+1$ in the case of no contract $(m_{t},\tau _{t+1}^{o})$.

We know that

$$\begin{aligned} \frac{I_{t+1}}{{\tilde{I}}_{t+1}}=\frac{k_{t+1}^{\alpha }\left( 1-\tau _{t+1}^{y}\right) }{{\tilde{k}}_{t+1}^{\alpha }} \end{aligned}$$

where $k_{t+1}={\tilde{k}}_{t+1}(1-m_{t})=\frac{\beta I_{t}}{1+\beta }(1-m_{t}), {\tilde{P}}_{t+1}={\tilde{k}}_{t+1}, P_{t+1}=k_{t+1}-\gamma m_{t}I_{t}$, and $\tau _{t+1}^{y}=\frac{\alpha }{1-\alpha }\tau _{t+1}^{o}$. Upon substitution, we obtain:

$$\begin{aligned} \frac{I_{t+1}}{{\tilde{I}}_{t+1}}=\left( 1-\frac{\alpha \tau _{t+1}^{o}}{1-\alpha }\right) (1-m_{t})^{\alpha } \end{aligned}$$

We also know that

$$\begin{aligned} \frac{z(E_{t+1})}{z({\tilde{E}}_{t+1})}=\mathrm {e}^{{\tilde{E}}_{t+1}-E_{t+1}}=\mathrm {e}^{{\tilde{P}}_{t+1}-P_{t+1}}=\mathrm {e}^{\frac{\beta +\gamma +\gamma \beta }{1+\beta }m_{t}I_{t}} \end{aligned}$$

Upon substitution in Eq. (27), we obtain the expression in Eq. (13).

1.2 Definition of the Pareto Set and Proof of Compactness

Consider the set of contracts $(m_{t},\tau _{t+1}^{o})$ which do not make generation t worse off. We define this set ${\mathcal {S}}_{t}^{t+1}$ as

$$\begin{aligned} {\mathcal {S}}_{t}^{t+1}=\left\{ \left( m_{t},\tau _{t+1}^{o}\right) \in (0,{\bar{m}}(I_{t},E_{t})]\times {\mathbb {R}},\,m_{t}<1:\;\triangle V_{t}^{t+1}\ge 0\right\} \end{aligned}$$

(28)

And we define the indifference curve of agent t, i.e. the set of contracts $(m_{t},\tau _{t+1}^{o})$ for which agent t is indifferent between signing and not signing, as

$$\begin{aligned} \underline{{\mathcal {S}}}_{t}^{t+1}=\left\{ \left( m_{t},\tau _{t+1}^{o}\right) \in [0,\min \{1,{\bar{m}}_{t}\}]\times {\mathbb {R}},\,m_{t}<1:\;\triangle V_{t}^{t+1}=0\right\} \end{aligned}$$

which can be expressed as

$$\begin{aligned} \tau _{t+1}^{o}=\left( \frac{1}{1-m_{t}}\right) ^{\frac{1}{\beta }+\alpha }-1\equiv \Omega (m_{t})\quad \mathrm {with}\quad m_{t}\in [0,\min \{1,{\bar{m}}_{t}\}],\,m_{t}<1 \end{aligned}$$

(29)

It is straightforward from the last equation that $\Omega (m_{t})$ is strictly convex in $m_{t}\in [0,\min \{1,{\bar{m}}_{t}\}]$ and $m_{t}<1$.

Now consider the set of all pairs $(m_{t},\tau _{t+1}^{o})$ which do not make generation $t+1$ worse off. We define the set ${\mathcal {S}}_{t+1}^{t+1}$ as

$$\begin{aligned} {\mathcal {S}}_{t+1}^{t+1}=\left\{ \left( m_{t},\tau _{t+1}^{o}\right) \in (0,{\bar{m}}(I_{t},E_{t})]\times {\mathbb {R}},\,m_{t}<1:\;\triangle {\mathcal {V}}_{t+1}^{t+1}\ge 0\right\} \end{aligned}$$

(30)

And we define the indifference curve of agent t+1 as

$$\begin{aligned} \underline{{\mathcal {S}}}_{t+1}^{t+1}=\left\{ \left( m_{t},\tau _{t+1}^{o}\right) \in [0,\min \{1,{\bar{m}}_{t}\}]\times {\mathbb {R}},\,m_{t}<1:\;\triangle {\mathcal {V}}_{t+1}^{t+1}=0\right\} \end{aligned}$$

which can be expressed as

$$\begin{aligned} \tau _{t+1}^{o}= & {} \frac{1-\alpha }{\alpha }\left[ 1-\mathrm {e}^{-\frac{(\beta +\gamma +\gamma \beta )\beta m_{t}I_{t}}{(1+\beta )(1+\alpha \beta )}}(1-m_{t})^{-\alpha }\right] \nonumber \\&\equiv \psi (m_{t},I_{t})\quad \mathrm {with}\quad m_{t}\in [0,\min \{1,{\bar{m}}_{t}\}],\,m_{t}<1 \end{aligned}$$

(31)

We prove that $\psi (m_{t},I_{t})$ is concave in $m_{t}\in [0,\min \{1,{\bar{m}}_{t}\}]$. Indeed, we have

$$\begin{aligned} \psi _{mm}(m_{t},I_{t})=(\alpha -1)\frac{\mathrm {e}^{-am_{t}I_{t}}}{(1-m)^{\alpha }}\left[ \frac{1}{(1-m_{t})^{2}}+\left( \frac{aI_{t}}{\sqrt{\alpha }}-\frac{\sqrt{\alpha }}{1-m_{t}}\right) ^{2}\right] <0\quad \forall m_{t}\in [0,1) \end{aligned}$$

where $a=\frac{\beta (\beta +\gamma +\gamma \beta )}{\alpha (1+\beta )(1+\alpha \beta )}$.

The set of all contracts $(m_{t},\tau _{t+1}^{o})$ between generations t and $t+1$ which lead to a Pareto improvement over (0, 0) is defined as

$$\begin{aligned} {\mathcal {P}}^{t+1}={\mathcal {S}}_{t}^{t+1}\cap {\mathcal {S}}_{t+1}^{t+1}\setminus \left( \underline{{\mathcal {S}}}_{t}^{t+1}\cap \underline{{\mathcal {S}}}_{t+1}^{t+1}\right) \end{aligned}$$

Further, define the union of ${\mathcal {P}}^{t+1}$ with $\underline{{\mathcal {S}}}_{t}^{t+1}\cap \underline{{\mathcal {S}}}_{t+1}^{t+1}$, or the set of all contracts which do not make generations t and $t+1$ worse off compared to $\left( 0,0\right)$, as $\mathcal {{\overline{P}}}^{t+1}$:

$$\begin{aligned} \mathcal {{\overline{P}}}^{t+1}={\mathcal {P}}^{t+1}\cup \left( \underline{{\mathcal {S}}}_{t}^{t+1}\cap \underline{{\mathcal {S}}}_{t+1}^{t+1}\right) \end{aligned}$$

Lemma 2

$\mathcal {{\overline{P}}}^{t+1}$ is a compact set.

Proof

The statement is trivially true for the case ${\mathcal {P}}^{t+1}=\oslash$. If ${\mathcal {P}}^{t+1}\ne \oslash$,

$$\begin{aligned} \mathcal {{\overline{P}}}^{t+1}={\mathcal {P}}^{t+1}\cup \left( \underline{{\mathcal {S}}}_{t}^{t+1}\cap \underline{{\mathcal {S}}}_{t+1}^{t+1}\right) ={\mathcal {S}}_{t}^{t+1}\mathcal {\cap S}_{t+1}^{t+1}\cup \left\{ (0,0)\right\} \end{aligned}$$

It is obvious that ${\mathcal {S}}_{t}^{t+1}\mathcal {\cap S}_{t+1}^{t+1}\cup \left\{ (0,0)\right\}$ contains its boundary. Therefore $\mathcal {{\overline{P}}}^{t+1}$ is closed. Moreover, since $\psi (m_{t},I_{t})$ is continuous over $m_{t}\in [0,1)$ and $\psi (0,I_{t})=0, \underset{m_{t}\rightarrow 1^{-}}{\lim }\psi (m_{t},I_{t})=-\infty$ for all $I_{t}<+\infty$, it is true that $\exists N<+\infty$ such that $\psi (m_{t},I_{t})<N$ for all $m_{t}\in [0,1)$ and $I_{t}<+\infty$. Thus $\mathcal {{\overline{P}}}^{t+1}$ is always bounded by the ball $B_{R}(0,0)$ of center (0, 0) and some finite radius $R\ge \sqrt{1+N^{2}}$. Therefore, $\mathcal {{\overline{P}}}^{t+1}$ is compact. $\square$

1.3 Proof of Proposition 1

We see from (29) and (31) that

$$\begin{aligned} \Omega (0)=\psi (0,I_{t})=0 \end{aligned}$$

(32)

Now we take into account the slopes of $\Omega (m_{t})$ and $\psi (m_{t},I_{t})$ as $m_{t}$ approaches 0 from the right. We have

$$\begin{aligned} \Omega ^{\prime }(0^{+})=\underset{m_{t}\rightarrow 0^{+}}{\lim }\frac{1+\alpha \beta }{\beta }\left( \frac{1}{1-m_{t}}\right) ^{1+\alpha +\frac{1}{\beta }}=\frac{1+\alpha \beta }{\beta } \end{aligned}$$

and

$$\begin{aligned} \psi _{m}(0^{+},I_{t})= & {} \frac{1-\alpha }{\alpha }\underset{m_{t}\rightarrow 0^{+}}{\lim }\left[ \frac{(\beta +\gamma +\gamma \beta )\beta I_{t}}{(1+\beta )(1+\alpha \beta )}-\frac{\alpha }{1-m_{t}}\right] \mathrm {e}^{-\frac{(\beta +\gamma +\gamma \beta )\beta m_{t}I_{t}}{(1+\beta )(1+\alpha \beta )}}(1-m_{t})^{-\alpha }\\= & {} \frac{1}{\frac{\alpha }{1-\alpha }(1+\alpha \beta )}\left[ \frac{(\beta +\gamma +\gamma \beta )\beta I_{t}}{1+\beta }-\alpha (1+\alpha \beta )\right] \end{aligned}$$

Due to the convexity of $\Omega (m_{t})$, the concavity of $\psi (m_{t},I_{t})$, and (32), ${\mathcal {P}}^{t+1}\ne \oslash$ if, and only if,

$$\begin{aligned} \Omega ^{\prime }(0^{+})= & {} \frac{1+\alpha \beta }{\beta }<\frac{1}{\frac{\alpha }{1-\alpha }(1+\alpha \beta )}\left[ \frac{(\beta +\gamma +\gamma \beta )\beta I_{t}}{1+\beta }-\alpha (1+\alpha \beta )\right] =\psi _{m}(0^{+},I_{t})\nonumber \\\Longleftrightarrow & {} \quad I_{t}>\frac{\alpha (1+\alpha \beta )(1+\beta )^{2}}{(1-\alpha )(\beta +\gamma +\gamma \beta )\beta ^{2}}={\hat{I}} \end{aligned}$$

(33)

1.4 Proof of Proposition 2

Since both $\triangle V_{t}^{t+1}$ and $\triangle {\mathcal {V}}_{t+1}^{t+1}$ are well-defined and continuous over the set $\mathcal {{\overline{P}}}^{t+1}$, the product $\triangle V_{t}^{t+1}\triangle {\mathcal {V}}_{t+1}^{t+1}$ is continuous over $\mathcal {{\overline{P}}}^{t+1}$. So the existence of $(m_{t}^{*},\tau _{t+1}^{o*})$ is guaranteed by the compactness of $\mathcal {{\overline{P}}}^{t+1}$ which has been proven in Lemma 2 in appendix section “Definition of the Pareto Set and Proof of Compactness”. It is trivial to rule out the case where at least one of the constraints $\triangle V_{t}^{t+1}\ge 0$ and $\triangle {\mathcal {V}}_{t+1}^{t+1}\ge 0$ is binding: In this case the objective function $\triangle V_{t}^{t+1}\triangle {\mathcal {V}}_{t+1}^{t+1}=0$ while any other point $(m_{t},\tau _{t+1}^{o})\in {\mathcal {P}}^{t+1}\setminus {(}\underline{{\mathcal {S}}}_{t}^{t+1}\cup \underline{{\mathcal {S}}}_{t+1}^{t+1}{)}$ in the interior of the Pareto improvement set yields $\triangle V_{t}^{t+1}\triangle {\mathcal {V}}_{t+1}^{t+1}>0$. So at the maximum none of these two constraints is binding. Hence, the Lagrangian for the optimization is

$$\begin{aligned} {\mathcal {L}}\left( m_{t},\tau _{t+1}^{o},\mu \right) =\triangle V_{t}^{t+1}\triangle {\mathcal {V}}_{t+1}^{t+1}-\mu (m_{t}-{\bar{m}}_{t}) \end{aligned}$$

where $\mu \ge 0$ is the Lagrangian multiplier for the constraint $m_{t}-{\bar{m}}_{t}\le 0$. The first-order Kuhn-Tucker conditions at the optimal point $(m_{t}^{*},\tau _{t+1}^{o*})\in {\mathcal {P}}^{t+1}$ are

$$\begin{aligned} {\mathcal {L}}_{m}\left( m_{t}^{*},\tau _{t+1}^{o*},\mu \right)= & {} \mu \!-\! \frac{1\!+\!\alpha \beta }{1\!-\!m_{t}^{*}}\triangle {\mathcal {V}}_{t+1}^{t+1,*}\!+\!\triangle V_{t}^{t+1,*}\left[ \frac{\beta +\gamma +\gamma \beta }{1+\beta }\beta I_{t}\!-\!\frac{\alpha \left( 1\!+\!\alpha \beta \right) }{1-m_{t}^{*}}\right] =0 \nonumber \\ \end{aligned}$$

(34)

$$\begin{aligned} {\mathcal {L}}_{\tau }\left( m_{t}^{*},\tau _{t+1}^{o*},\mu \right)= & {} \frac{\beta }{1+\tau _{t+1}^{o*}}\triangle {\mathcal {V}}_{t+1}^{t+1,*}-\triangle V_{t}^{t+1,*}\frac{\alpha \left( 1+\alpha \beta \right) }{1-\alpha \left( 1+\tau _{t+1}^{o*}\right) }=0 \nonumber \\ \mu \left( m_{t}^{*}-{\bar{m}}_{t}\right)= & {} 0 \end{aligned}$$

(35)

(i) If $m_{t}^{*}={\bar{m}}_{t}=\frac{(1-\delta )(1+\beta )E_{t}/I_{t}+\beta }{\beta +\gamma +\gamma \beta }$, then from (35)

$$\begin{aligned} Q\left( \tau _{t+1}^{o*}\right)\equiv & {} \frac{\beta }{1+\tau _{t+1}^{o*}}\left\{ \left( 1+\alpha \beta \right) \ln \left[ \left( \frac{1-\alpha \left( 1+\tau _{t+1}^{o*}\right) }{1-\alpha }\right) \left( 1-{\bar{m}}_{t}\right) ^{\alpha }\right] +\frac{\beta +\gamma +\gamma \beta }{1+\beta }\beta {\bar{m}}_{t}I_{t}\right\} \nonumber \\&-\left\{ \left( 1+\alpha \beta \right) \ln \left( 1-{\bar{m}}_{t}\right) +\beta \ln \left( 1+\tau _{t+1}^{o*}\right) \right\} \frac{\alpha \left( 1+\alpha \beta \right) }{1-\alpha \left( 1+\tau _{t+1}^{o*}\right) }=0 \end{aligned}$$

(36)

where $Q(\tau _{t+1}^{o*})$ is decreasing in $\tau _{t+1}^{o*}$, and

$$\begin{aligned} Q\left( 0\right)= & {} \beta ^{2}\frac{\beta +\gamma +\gamma \beta }{1+\beta }{\bar{m}}_{t}I_{t}-\frac{\alpha \left( 1+\alpha \beta \right) ^{2}}{1-\alpha }\ln \left( 1-{\bar{m}}_{t}\right) >0\qquad \mathrm {and}\\&\quad \underset{\tau _{t+1}^{o*}\rightarrow \left( \frac{1-\alpha }{\alpha }\right) ^{-}}{\lim }Q\left( \tau _{t+1}^{o*}\right) =-\infty \end{aligned}$$

Hence, there always exists a unique $\tau _{t+1}^{o*}\in (0,\frac{1-\alpha }{\alpha })$ solving (36), implying that there always exits a unique $(m_{t},\tau _{t+1}^{o})=({\bar{m}}_{t},\tau _{t+1}^{o*})$ solving the Nash bargaining problem in case the constraint $m_{t}\le {\bar{m}}_{t}$ is binding.

(ii) If $m_{t}^{*}<{\bar{m}}_{t}$, then $\mu =0$ and from (34) and (35), we find that

$$\begin{aligned} \frac{1+\tau _{t+1}^{o*}}{1-m_{t}^{*}}\frac{1+\alpha \beta }{\beta }= & {} \left[ \beta \frac{\beta +\gamma +\gamma \beta }{1+\beta }I_{t}-\frac{\alpha \left( 1+\alpha \beta \right) }{1-m_{t}^{*}}\right] \frac{1-\alpha \left( 1+\tau _{t+1}^{o*}\right) }{\alpha \left( 1+\alpha \beta \right) }\nonumber \\&\Longrightarrow 1+\tau _{t+1}^{o*}=\frac{\beta ^{2}\frac{\beta +\gamma +\gamma \beta }{1+\beta }I_{t}\left( 1-m_{t}^{*}\right) -\alpha \beta \left( 1+\alpha \beta \right) }{\alpha \left( \beta ^{2}\frac{\beta +\gamma +\gamma \beta }{1+\beta }I_{t}\left( 1-m_{t}^{*}\right) +1+\alpha \beta \right) } \end{aligned}$$

(37)

It can be shown that Eq. (37) denotes the Pareto frontier. Since $\tau _{t+1}^{o*}>0$, it follows from (37) that

$$\begin{aligned} 1-m_{t}^{*}>\frac{\alpha (1+\alpha \beta )(1+\beta )^{2}}{(1-\alpha )(\beta +\gamma +\gamma \beta )\beta ^{2}I_{t}}\quad \mathrm {i.e.}\quad m_{t}^{*}<1-\frac{\alpha (1+\alpha \beta )(1+\beta )^{2}}{(1-\alpha )(\beta +\gamma +\gamma \beta )\beta ^{2}I_{t}}=\hat{m_{t}}\nonumber \\ \end{aligned}$$

(38)

Substituting (37) into (34), we find $m_{t}^{*}$ as the solution to

$$\begin{aligned} {\hat{Q}}(m_{t};I_{t})= & {} \ln \left[ (1-m_{t})^{1+\alpha \beta }\left( \frac{\beta bI_{t}(1-m_{t})-\alpha \beta (1+\alpha \beta )}{\alpha \left( \beta bI_{t}(1-m_{t})+1+\alpha \beta \right) }\right) ^{\beta }\right] ^{bI_{t}-\frac{\alpha (1+\alpha \beta )}{1-m}}\nonumber \\&-\frac{1+\alpha \beta }{1-m_{t}}\left\{ \ln \left[ \frac{(1-\alpha ^{2}\beta ^{2})(1-m_{t})^{\alpha }}{(1-\alpha )\left( \beta bI_{t}(1-m_{t})+1+\alpha \beta \right) }\right] ^{1+\alpha \beta }+bm_{t}I_{t}\right\} =0\nonumber \\ \end{aligned}$$

(39)

where $b=\frac{\beta (\beta +\gamma +\gamma \beta )}{1+\beta }$.

The existence of a solution to (39) is proven by the existence of a solution $(m_{t}^{*},\tau _{t+1}^{o*})$ to the Nash bargaining problem. The uniqueness of $m_{t}^{*}$ is guaranteed by the monotonicity of the function ${\hat{Q}}(m_{t}^{*};I_{t})$. We now prove that for all $I_{t}>{\hat{I}}, {\hat{Q}}(m_{t}^{*};I_{t})$ is decreasing in $m_{t}^{*}$. From (39), it is sufficient to prove that $\frac{(1-m_{t})^{\alpha }}{\beta bI_{t}(1-m_{t})+1+\alpha \beta }$ is increasing in $m_{t}$ for all $I_{t}>{\hat{I}}$ and $m_{t}<\hat{m_{t}}$. In effect, its derivative with respect to $m_{t}$ is

$$\begin{aligned} \frac{(1-\alpha )(1-m_{t})\beta bI_{t}-\alpha (1+\alpha \beta )}{(1-m_{t})^{1-\alpha }\left[ \beta bI_{t}(1-m_{t})+1+\alpha \beta \right] ^{2}}>0\qquad \forall m_{t}<\hat{m_{t}},\,\forall I_{t}>{\hat{I}} \end{aligned}$$

Hence, there exists a unique $m_{t}^{*}=m(I_{t})\in (0,{\hat{m}}_{t})$ that solves (39), i.e. there exists a unique interior pair $(m_{t}^{*},\tau _{t+1}^{o*})\equiv (m(I_{t}),\tau (I_{t}))\in {\mathcal {P}}^{t+1}$ that solves the Nash bargaining problem. Since $(m_{t}^{*},\tau _{t+1}^{o*})$ is the unique pair solving the first order conditions (34) and (35), it is the unique stationary point of the function $\triangle V_{t}^{t+1}\triangle {\mathcal {V}}_{t+1}^{t+1}$ defined over the set ${\mathcal {P}}^{t+1}$.

1.5 Steady State with Zero Pollution

In this section, we study the dynamics of the model assuming that parameters satisfy the following sufficient condition for the mitigation share to be increasing in $I_{t}$ for all $I_{t}\in ({\hat{I}},{\bar{I}})$:

Assumption 2

$\frac{\beta +\gamma +\gamma \beta }{\alpha (1+\beta )}\left[ \frac{\beta (1-\alpha )}{1+\beta }\right] ^{\frac{2-\alpha }{1-\alpha }}<1$.

Under Assumption 2, we can prove the existence of an area in the North-East quadrant of the $I-E$ space such that under period-by-period social contracts an economy starting from any point in this area may converge to a steady state with zero pollution stock $E_{*}=0$, and high net income $I_{*}$. We call this area the “ideal area”. Under Assumption 2, for sufficiently high income, we may find

$$\begin{aligned} m_{t}^{*}=\frac{(1-\delta )(1+\beta )E_{t}+\beta I_{t}}{(\beta +\gamma +\gamma \beta )I_{t}}\equiv {\bar{m}}_{t} \end{aligned}$$

(40)

where

$$\begin{aligned} m_{t}^{*}\in \underset{m_{t}}{\arg \max }\left\{ \triangle V_{t}^{t+1}\triangle {\mathcal {V}}_{t+1}^{t+1}\right\} \end{aligned}$$

(41)

subject to $\triangle V_{t}^{t+1}\ge 0$ and $\triangle {\mathcal {V}}_{t+1}^{t+1}\ge 0$. Note that in this optimization, we ignore the constraint $m_{t}\le {\bar{m}}_{t}$ in order to find the condition under which this constraint is just binding. And note also that we substitute $\tau _{t+1}^{o}$ as a function of $m_{t}$ from (37) in the appendix section ”Proof of Proposition 2”.

Similar to the proof of Proposition 2, the optimization problem (41) has a unique interior solution and it holds that²²

$$\begin{aligned} {\hat{Q}}(m_{t}^{*};I_{t})= & {} \ln \left[ (1-m_{t}^{*})^{1+\alpha \beta }\left( \frac{\beta bI_{t}(1-m_{t}^{*})-\alpha \beta (1+\alpha \beta )}{\alpha \left( \beta bI_{t}(1-m_{t}^{*})+1+\alpha \beta \right) }\right) ^{\beta }\right] ^{bI_{t}-\frac{\alpha (1+\alpha \beta )}{1-m_{t}^{*}}}\nonumber \\&-\frac{1+\alpha \beta }{1-m_{t}^{*}}\left\{ \ln \left[ \frac{(1-\alpha ^{2}\beta ^{2})(1-m_{t}^{*})^{\alpha }}{(1-\alpha )\left( \beta bI_{t}(1-m_{t}^{*})+1+\alpha \beta \right) }\right] ^{1+\alpha \beta }+bm_{t}^{*}I_{t}\right\} =0\nonumber \\ \end{aligned}$$

(42)

with $b=\frac{\beta (\beta +\gamma +\gamma \beta )}{1+\beta }$. We find that ${\hat{Q}}_{m}(m_{t}^{*};I_{t})<0$ and, under Assumption 2, ${\hat{Q}}_{I}(m_{t}^{*};I_{t})>0$ for all $I_{t}\in ({\hat{I}},{\bar{I}})$, where ${\bar{I}}=\left[ A(1-\alpha )\right] ^{\frac{1}{1-\alpha }}\left( \frac{\beta }{1+\beta }\right) ^{\frac{\alpha }{1-\alpha }}$ is conditional steady state income in the case of no social contract with the pollution stock set at $E=0$. By applying the implicit function theorem we have

$$\begin{aligned} \forall I_{t}\in ({\hat{I}},{\bar{I}}),\quad m_{t}^{*}=m(I_{t})\quad \mathrm {and}\quad m^{\prime }(I_{t})>0 \end{aligned}$$

We can now prove the following lemma:

Lemma 3

Under Assumption 2, if $m({\bar{I}})>\frac{\beta }{\beta +\gamma +\gamma \beta }$, there exist $\underline{I}\in ({\hat{I}},{\bar{I}})$ and ${\hat{E}}_{t}={\hat{E}}(I_{t})$ for $I_{t}\in (\underline{I},{\bar{I}})$ such that $m_{t}^{*}={\bar{m}}_{t}$ (or equivalently $E_{t+1}=0$) if, and only if, $E_{t}\le {\hat{E}}(I_{t})$. Moreover, ${\hat{E}}(\underline{I})=0$, and ${\hat{E}}^{\prime }(I_{t})>0$ for $I_{t}\in [\underline{I},{\bar{I}}]$.

Proof

Since $m({\hat{I}})=0, m({\bar{I}})>\frac{\beta }{\beta +\gamma +\gamma \beta }$ and $m^{\prime }(I_{t})>0$ for all $I_{t}\in ({\hat{I}},{\bar{I}})$ , there exists a unique $\underline{I}\in ({\hat{I}},{\bar{I}})$ such that $m(\underline{I})=\frac{\beta }{\beta +\gamma +\gamma \beta }$. From (40), it follows that

$$\begin{aligned} m(I_{t})-\frac{(1-\delta )(1+\beta )E_{t}+\beta I_{t}}{(\beta +\gamma +\gamma \beta )I_{t}}=0 \end{aligned}$$

(43)

For all $I_{t}\in [\underline{I},{\bar{I}})$, there exists a unique ${\hat{E}}_{t}$ solving (43), and

$$\begin{aligned} {\hat{E}}_{t}=\frac{(\beta +\gamma +\gamma \beta )m(I_{t})-\beta }{(1-\delta )(1+\beta )}I_{t}\equiv {\hat{E}}(I_{t}) \end{aligned}$$

where ${\hat{E}}(\underline{I})=0$ and ${\hat{E}}^{\prime }(I_{t})>0$ for all $I_{t}\in [\underline{I},{\bar{I}}]$. $\square$

The proof of existence of this steady state is fairly straightforward because starting from any point $(I_{t},E_{t})$ in the ideal area leads to a social contract with $m_{t}=\frac{(1-\delta )(1+\beta )E_{t}+\beta I_{t}}{(\beta +\gamma +\gamma \beta )I_{t}}$. Hence, from $t+1$ onward $E=0$ and $m=\frac{\beta }{\beta +\gamma +\gamma \beta }$. The existence and uniqueness of the transfer $\tau _{t+1}^{o*}$ are proved in Proposition 2. Since $I_{t+1}=z(E_{t})\left( \frac{\beta (1-m_{t})I_{t}}{1+\beta }\right) ^{\alpha }\left[ 1-\alpha (1+\tau _{t+1}^{o})\right]$, the steady state is characterized by

$$\begin{aligned} I_{*}^{1-\alpha }-A\left( \frac{\gamma \beta }{\beta +\gamma +\gamma \beta }\right) ^{\alpha }\left[ 1-\alpha (1+\tau (I_{*}))\right] =0 \end{aligned}$$

which always guarantees the existence and uniqueness of the steady state.

1.6 Proof of Proposition 3

From Eqs. (18) and (19), we can derive the steady state of the dynamic system in the case of no social contracts by setting $E_{t+1}=E_{t}=E$ and $I_{t+1}=I_{t}=I$ and substituting $(m_{t},\tau _{t+1}^{o})=(0,0)$ for all t. The steady state is characterized by the following function

$$\begin{aligned} \varphi (E)\equiv E-\frac{1}{\delta }\left[ \frac{\beta (1-\alpha )}{1+\beta }A\mathrm {e}^{-E}\right] ^{\frac{1}{1-\alpha }}=0 \end{aligned}$$

(44)

with

$$\begin{aligned} \varphi ^{\prime }(E)= & {} 1+\frac{1}{(1-\alpha )\delta }\left[ \frac{\beta (1-\alpha )}{1+\beta }A\mathrm {e}^{-E}\right] ^{\frac{1}{1-\alpha }}>0\\ \varphi (0)= & {} -\frac{1}{\delta }\left[ \frac{\beta (1-\alpha )}{1+\beta }\right] ^{\frac{1}{1-\alpha }}<0\qquad \mathrm {and}\qquad \underset{E\rightarrow +\infty }{\lim }\varphi (E)=+\infty \end{aligned}$$

Hence, there exists a unique steady state which is characterized by (44). The steady state income of the agent in this case is

$$\begin{aligned} {\tilde{I}}=A^{\frac{1}{1-\alpha }}\mathrm {e}^{-\frac{{\tilde{E}}}{1-\alpha }}\left( \frac{\beta }{1+\beta }\right) ^{\frac{\alpha }{1-\alpha }}(1-\alpha )^{\frac{1}{1-\alpha }} \end{aligned}$$

The Jacobian matrix associated with the dynamic system (18) and (19) with $(m_{t},\tau _{t+1}^{o})=(0,0)$ for all t and evaluated at the steady state $({\tilde{I}},{\tilde{E}})$ is

$$\begin{aligned} {\tilde{J}}=\left( \begin{array}{ccc} \alpha &{} &{} -\frac{1+\beta }{\beta }\delta {\tilde{E}}\\ \\ \frac{\beta }{1+\beta } &{} &{} 1-\delta \end{array}\right) \end{aligned}$$

Its determinant and trace are

$$\begin{aligned} \det ({\tilde{J}})=\alpha (1-\delta )+\delta {\tilde{E}}>0;\qquad \mathrm {Tr}({\tilde{J}})=\alpha +1-\delta >0 \end{aligned}$$

and the characteristic function is

$$\begin{aligned} C(\lambda )=\lambda ^{2}-\mathrm {Tr}({\tilde{J}})\lambda +\det ({\tilde{J}}) \end{aligned}$$

(i) If $\mathrm {Tr}({\tilde{J}})^{2}>4\det ({\tilde{J}})$, we have

$$\begin{aligned} C(-1)=1+\mathrm {Tr}({\tilde{J}})+\det ({\tilde{J}})>C(1)=\delta (1-\alpha )+\delta {\tilde{E}}>0 \end{aligned}$$

So, we have two distinct eigenvalues $\lambda _{1},\lambda _{2}\in (0,1)$. The steady state $({\tilde{I}},{\tilde{E}})$ is a stable node.

(ii) If $\mathrm {Tr}({\tilde{J}})^{2}=4\det ({\tilde{J}})$, we have a pair of repeated real eigenvalues $\lambda =\frac{\alpha +1-\delta }{2}\in (0,1)$. The steady state $({\tilde{I}},{\tilde{E}})$ is stable.

(iii) If $\mathrm {Tr}({\tilde{J}})^{2}<4\det ({\tilde{J}})$, we have two complex eigenvalues. It is obvious that

$$\begin{aligned} \varphi \left( \frac{1-\alpha (1-\delta )}{\delta }\right) >0\quad \Longrightarrow \quad {\tilde{E}}<\frac{1-\alpha (1-\delta )}{\delta } \end{aligned}$$

Hence

$$\begin{aligned} \det ({\tilde{J}})=\alpha (1-\delta )+\delta {\tilde{E}}<1 \end{aligned}$$

Therefore in this case, the steady state $({\tilde{I}},{\tilde{E}})$ is a spiral sink.

1.7 Isoclines and Directions of Motion

To study the dynamics with social contracts, we substitute the Nash bargaining solution $(m_{t},\tau _{t+1}^{o})=(m_{t}^{*},\tau _{t+1}^{o*})=(m(I_{t}),\tau (I_{t}))$ into the dynamic system (18)–(19) in every period t. Note that whenever net income falls short of the threshold (16), i.e. $I_{t}\le {\hat{I}}$, we set $(m_{t}^{*},\tau _{t+1}^{o*})=(0,0)$. From the dynamic system, we then define the sets II and EE where income and the pollution stock, respectively, do not change over time, as follows:

$$\begin{aligned} II\equiv & {} \left\{ (I_{t},E_{t})\in \mathfrak {R}_{+}^{2}:\,I_{t+1}=I_{t}\right\} \nonumber \\ \mathrm {i.e.}\quad E_{t}= & {} \ln A+\alpha \ln \left[ \frac{\beta (1-m(I_{t}))}{1+\beta }\right] +\ln \left[ 1-\alpha (1+\tau (I_{t}))\right] -(1-\alpha )\ln I_{t}\equiv \Gamma (I_{t})\nonumber \\ \end{aligned}$$

(45)

and

$$\begin{aligned}&EE\equiv \left\{ (I_{t},E_{t})\in \mathfrak {R}_{+}^{2}:\,E_{t+1}=E_{t}\right\} \nonumber \\&\mathrm {i.e.}\qquad E_{t}=\frac{\beta -(\beta +\gamma +\gamma \beta )m(I_{t})}{\delta (1+\beta )}I_{t}\equiv \Lambda (I_{t})\qquad \end{aligned}$$

(46)

Without social contracts, $\Gamma (I_{t})$ is monotonously decreasing and $\Lambda (I_{t})$ monotonously increasing in income $I_{t}$. We can easily see this by substituting (0, 0) for $(m(I_{t}),\tau (I_{t}))$ into (45) and (46). In the case of social contracts, with $m(I_{t}),\tau (I_{t})>0$, the functional relation between the pollution stock and income along the curves depends also on the derivatives $m^{\prime }(I_{t}),\tau ^{\prime }(I_{t})$. The curves in Figs. 3 and 4 are drawn under Assumption 2 that the mitigation share $m(I_{t})$ increases in income for all $I_{t}\in ({\hat{I}},{\bar{I}})$. For any $m(I_{t}),\tau (I_{t})>0$, both the II-curve and the EE-curve lie below their respective counterparts without contracts for all $I_{t}>{\hat{I}}$. The dynamics on and offside the II and EE loci are described by the following lemma:

Lemma 4

For the dynamic system $(I_{t},E_{t})_{t}$ characterized by Eqs. (18)–(19), it holds that:

$$\begin{aligned} (\mathrm {i})\quad I_{t+1}\!-\!I_{t} {\left\{ \begin{array}{ll}\! \begin{array}{lll}>0 &{} \mathrm {if} &{} 0<E_{t}<\Gamma (I_{t})\\ =0 &{} \mathrm {if} &{} E_{t}=\Gamma (I_{t})\\<0 &{} \mathrm {if} &{} E_{t}>\Gamma (I_{t}) \end{array}&\quad and \end{array}\right. }\quad (\mathrm {ii})\quad E_{t+1}-E_{t} {\left\{ \begin{array}{ll} \begin{array}{lll}>0 &{} \mathrm {if} &{} 0<E_{t}<\Lambda (I_{t})\\ =0 &{} \mathrm {if} &{} E_{t}=\Lambda (I_{t})\\ <0 &{} \mathrm {if} &{} E_{t}>\Lambda (I_{t}) \end{array} \end{array}\right. } \end{aligned}$$

Proof

The proof for this lemma is fairly straightforward. $\square$

1.8 Proof of Proposition 4

(i) Existence We prove existence for the case shown in Fig. 3, where $m({\overline{I}})<\frac{\beta }{\beta +\gamma +\gamma \beta }$. For the case $m({\overline{I}})>\frac{\beta }{\beta +\gamma +\gamma \beta }$, existence of a steady state ($I_{*},0$) is proved in the appendix section “Steady State with Zero Pollution”. If $m({\overline{I}})<\frac{\beta }{\beta +\gamma +\gamma \beta }$, it is straightforward that $\Lambda ({\overline{I}})>0>\Gamma ({\overline{I}})$: The EE-curve lies above the II-curve for large incomes. Now define the EE-curve and II-curve without social contracts as $EE(0,0)=\Lambda ^{NC}(I_{t})$ and $II(0,0)=\Gamma ^{NC}(I_{t})$. At the income threshold ${\hat{I}}, m({\hat{I}})=0$, so that $\Lambda ({\hat{I}})=\frac{\beta }{\delta (1+\beta )}{\hat{I}}=\Lambda ^{NC}({\hat{I}})<\frac{\beta }{\delta (1+\beta )}{\tilde{I}}={{\Lambda ^{NC}({\tilde{I}})=}}\Gamma ^{NC}({\tilde{I}})<\Gamma ^{NC}({\hat{I}})=\Gamma ({\hat{I}})$. The equality ${{\Lambda ^{NC}({\tilde{I}})=}}\Gamma ^{NC}({\tilde{I}})$ uses that the EE- and II-loci intersect in the steady state without contracts. The inequality $\Gamma ^{NC}({\tilde{I}})<\Gamma ^{NC}({\hat{I}})$ holds because $\Gamma ^{\prime }(I_{t})<0$. Finally, $\Gamma ^{NC}({\hat{I}})=\Gamma ({\hat{I}})$ because $m({\hat{I}})=0$. From the continuity of the EE-curve and II-curve, it follows that they intersect at least once for $I_{t}\in ({\hat{I}},{\overline{I}})$.

Steady state pollution stock, income and welfare From the dynamic system (18)–(19), a steady state with social contracts is characterized by

$$\begin{aligned} I^{1-\alpha }= & {} A\mathrm {e}^{-E}\left[ \frac{\beta (1-m)}{1+\beta }\right] ^{\alpha }\left[ 1-\alpha (1+\tau ^{o})\right] >{\hat{I}}^{1-\alpha }\qquad \mathrm {and}\\ \qquad E= & {} \frac{\beta -(\beta +\gamma +\gamma \beta )m}{\delta (1+\beta )}I\ge 0 \end{aligned}$$

where $(m,\tau ^{o})\in {\mathcal {P}}\ne \oslash$ is the social contract at steady state. Since ${\hat{I}}>0$, the two inequality signs imply that $\tau ^{o}<\frac{1-\alpha }{\alpha }$ and $m\le \frac{\beta }{\beta +\gamma +\gamma \beta }$. The stock of pollution at the steady state with social contracts is characterized by

$$\begin{aligned} \mathrm {e}^{\frac{E}{1-\alpha }}E=A^{\frac{1}{1-\alpha }}\frac{\beta -(\beta +\gamma +\gamma \beta )m}{\delta (1+\beta )}\left[ \frac{\beta (1-m)}{1+\beta }\right] ^{\frac{\alpha }{1-\alpha }}\left[ 1-\alpha (1+\tau ^{o})\right] ^{\frac{1}{1-\alpha }} \end{aligned}$$

(47)

It is follows from (47) that E is decreasing in both m and $\tau ^{o}$ so that

$$\begin{aligned} E=E(m,\tau ^{o})<E(0,0)={\tilde{E}} \end{aligned}$$

As a steady state (I, E) with social contracts may fall in the area $I_{t}\in ({\hat{I}},{\tilde{I}})$, steady state income I and therefore welfare may be lower with than without social contracts.

Non-existence of the no-contract steady state $({\tilde{I}},{\tilde{E}})$ The result follows straightforwardly because $\Gamma$ decreases in both m and $\tau ^{o}$, so that $\Gamma (I_{t})<\Gamma ^{NC}(I_{t})$ for all $I_{t}\in ({\hat{I}},{\overline{I}})$.

(ii) Steady state pollution stock, income and welfare The proof that $E<{\tilde{E}}$ from (i) still applies. Now as ${\hat{I}}>{\tilde{I}}$ and $I>{\hat{I}}$, it follows straightforwardly that $I>{\tilde{I}}$. From (25), we can derive lifetime indirect utility of a generation at a steady state (I, E) under social contracts. It is given by

$$\begin{aligned} V(I,E,m,\tau ^{o})=\Phi ^{\prime }+\ln z(E)+\left( 1+\alpha \beta \right) \ln I+\left( 1+\alpha \beta \right) \ln (1-m)+\beta \ln (1+\tau ^{o}) \end{aligned}$$

where $\Phi ^{\prime }=\Phi +\beta \ln \alpha +\beta \left( \alpha -1\right) \ln \frac{\beta }{1+\beta }$. Similarly, lifetime utility at the steady state $({\tilde{I}},{\tilde{E}})$ without social contracts is given by

$$\begin{aligned} V({\tilde{I}},{\tilde{E}},0,0)=\Phi ^{\prime }+\ln z({\tilde{E}})+\left( 1+\alpha \beta \right) \ln {\tilde{I}} \end{aligned}$$

Now as $E<{\tilde{E}}$ and $I>{\tilde{I}}, \ln z(E)>\ln z({\tilde{E}})$ and $\left( 1+\alpha \beta \right) \ln I>\left( 1+\alpha \beta \right) \ln {\tilde{I}}$. It follows that $V(I,E,m,\tau ^{o})>V({\tilde{I}},{\tilde{E}},0,0)$ if $\left( 1+\alpha \beta \right) \ln (1-m)+\beta \ln (1+\tau ^{o})\ge 0$, which is guaranteed as it is the condition $\triangle V_{t}^{t+1}\ge 0$ at steady state.

Existence of the no-contract steady state $({\tilde{I}},{\tilde{E}})$ Proof follows directly from $\Gamma (I_{t})=\Gamma ^{NC}(I_{t})$ and $\Lambda (I_{t})=\Lambda ^{NC}(I_{t})$ for $I{{\in }}(0,{\hat{I}})$.

1.9 Incentive Constraints

Denote expected indirect utility from compliance with the strategy as $V^{C}({\hat{m}}_{t},\tau _{t}^{y}\,|\,p(h_{t}),\tau _{t+1}^{o,e}={\hat{\tau }}_{t+1}^{o})$, where $\tau _{t}^{y}{{\in }}\left[ 0,{\hat{\tau }}_{t}^{y}\right]$ depending on the state $p(h_{t})$ of the game. Expected indirect utility from deviation is $V^{D}(0,0\,|\,p(h_{t}),\tau _{t+1}^{o,e}=0)$.

Incentive constraint when the game is in compliance phase

When the game is in a compliance phase ($p(h_{t})=C$) and generation t follows strategy s and complies with the contracts with generations $t-1$ and $t+1$ it is involved in, indirect utility is:

$$\begin{aligned} V^{C}\left( {\hat{m}}_{t},{\hat{\tau }}_{t}^{y}\,|\,p(h_{t}\right)= & {} C,\tau _{t+1}^{o,e}={\hat{\tau }}_{t+1}^{o})=\Phi +\beta \ln \left[ z(E_{t})\alpha \left( \frac{\beta I_{t}^{C}(1-{\hat{m}}_{t})}{1+\beta }\right) ^{\alpha -1}\right] \\&+\,(1+\beta )\left[ \ln I_{t}^{C}+\ln (1-\hat{m_{t}})\right] +\beta \ln \left( 1+{\hat{\tau }}_{t+1}^{o}\right) \end{aligned}$$

where $\Phi =\ln \frac{1}{1+\beta }+\beta \ln \frac{\beta }{1+\beta }$ is a constant and $I_{t}^{C}=z(E_{t-1})(1-\alpha )k_{t}^{\alpha }(1-{\hat{\tau }}_{t}^{y})$ is net income of agent t in period t after paying $z(E_{t-1})(1-\alpha )k_{t}^{\alpha }{\hat{\tau }}_{t}^{y}={\hat{\tau }}_{t}^{o}R_{t}k_{t}$ to the old generation $t-1$ in t. When complying with the contracts, generation t expects to receive the perfectly foreseen transfer ${\hat{\tau }}_{t+1}^{o}$ from generation $t+1$. Recall that the pollution stocks $E_{t-1}$ and $E_{t}$ as well as the capital stock $k_{t}$ are given in period t.²³

If generation t deviates, indirect utility is

$$\begin{aligned} V^{D}\left( 0,0\,|\,p(h_{t})=C,\tau _{t+1}^{o,e}=0\right) =\Phi +\beta \ln \left[ z(E_{t})\alpha \left( \frac{\beta I_{t}^{D}}{1+\beta }\right) ^{\alpha -1}\right] +(1+\beta )\ln I_{t}^{D} \end{aligned}$$

where $I_{t}^{D}=z(E_{t-1})(1-\alpha )k_{t}^{\alpha }$ is net income if generation t does not pay the transfer (${\hat{\tau }}_{t}^{y}=0$).

There is no incentive to deviate if and only if the difference $\triangle V(\hat{m_{t}},{\hat{\tau }}_{t}^{y}\,|\,p(h_{t})=C,{\hat{\tau }}_{t+1}^{o})$ in indirect utilities is non-negative. Using the relation $z(E_{t-1})(1-\alpha )k_{t}^{\alpha }{\hat{\tau }}_{t}^{y}={\hat{\tau }}_{t}^{o}R_{t}k_{t}$ between the transfer received by the old generation in t and the transfer paid by the young generation in t, the condition $\triangle V(\hat{m_{t}},{\hat{\tau }}_{t}^{y}\,|\,p(h_{t})=C,{\hat{\tau }}_{t+1}^{o})\ge 0$ can be shown to be (20).

Incentive constraint when the game is in punishment phase

Indirect utility from following strategy s when the game is in punishment phase $(p(h_{t})=P$) is

$$\begin{aligned} V^{C}({\hat{m}}_{t},0\,|\,p(h_{t})= & {} P,\tau _{t+1}^{o,e}={\hat{\tau }}_{t+1}^{o})=\Phi +\beta \ln \left[ z(E_{t})\alpha \left( \frac{\beta I_{t}^{C}(1-{\hat{m}}_{t})}{1+\beta }\right) ^{\alpha -1}\right] \\&+\,(1+\beta )\left[ \ln I_{t}^{C}+\ln (1-\hat{m_{t}})\right] +\beta \ln (1+{\hat{\tau }}_{t+1}^{o}) \end{aligned}$$

where $I_{t}^{C}=z(E_{t-1})(1-\alpha )k_{t}^{\alpha }$ is net income if the agent born in t chooses compliance and punishes generation $t-1$.

Indirect utility from deviation is

$$\begin{aligned} V^{D}(0,0\,|\,p(h_{t})=P,\tau _{t+1}^{o,e}=0)=\Phi +\beta \ln \left[ z(E_{t})\alpha \left( \frac{\beta I_{t}^{D}}{1+\beta }\right) ^{\alpha -1}\right] +(1+\beta )\ln I_{t}^{D} \end{aligned}$$

where $I_{t}^{D}=z(E_{t-1})(1-\alpha )k_{t}^{\alpha }=I_{t}^{C}$. Note that net income is the same whether generation t follows its strategy or deviates because in a punishment phase, generation t will not pay the transfer in any case.

There is no incentive for deviation if and only if $\triangle V({\hat{m}}_{t},0\,|\,p(h_{t})=P_{t},{\hat{\tau }}_{t+1}^{o})\ge 0$ which yields (21).

1.10 Proof of Proposition 5

The set of stationary self-enforcing contracts is given by:

$$\begin{aligned} S^{IC}=\left\{ (m,\tau {}^{o})\in [0,1)\times [0,\tau ^{o,\max }):\;m\le 1-\frac{1}{(1-\frac{\alpha }{1-\alpha }\tau {}^{o})\left( 1+\tau {}^{o}\right) ^{\frac{\beta }{1+\alpha \beta }}}\right\} \end{aligned}$$

(48)

We define as $\tau ^{o,\max }\equiv \frac{1-\alpha }{\alpha }$ the maximum transfer reconcilable with non-negative net labor income (i.e. $\tau _{t}^{y}\le 1)$ of agent t. In the text, we motivated the following lemma:

Lemma 5

A self-enforcing contract scheme $(m_{t},\tau _{t+1}^{o})_{t=T}^{\infty }$ must converge to or fluctuate without trend around a constant $(m,\tau ^{o})$. The set of stationary self-enforcing pairs $(m,\tau ^{o})$ is characterized by (48).

Proof

The proof is contained in the text. $\square$

It can be concluded from Lemma 5 that a self-enforcing contract scheme with $m_{t}>0,\tau _{t+1}^{o}>0\text { for every }t=T,\ldots ,\infty$ exists if and only if the stationary set $S^{IC}$ is non-empty, $S^{IC}\ne \oslash$, with $S^{IC}$ given by (48).

As can be seen from (22), the function $m^{IC}(\tau {}^{o})$ delineating the boundary of $S^{IC}$ is concave, with $m^{IC}(0)=0$. $S^{IC}\ne \oslash$ if and only if the slope of the IC-curve at the origin is positive. Formally, this is expressed by the condition $\frac{\partial m^{IC}}{\partial \tau {}^{o}}(0)=\frac{\beta -\frac{\alpha }{1-\alpha }\left( 1+\alpha \beta \right) }{1+\alpha \beta }>0$ which yields condition (23). If condition (23) does not hold, only negative values of $\tau {}^{o}$ satisfy the incentive constraint. To see that condition (23) is also sufficient, note that the IC-curve is continuous in $\tau {}^{o}$ for $\tau {}^{o}{{\in }} [0,\tau {}^{o,\max })$. A sustainable equilibrium path with positive transfer payments and positive mitigation investment in each period will therefore exist under condition (23) for sufficiently small mitigation levels $m_{t}$.

Under condition (23), $m^{IC}=0$ not only at the origin but also for some positive ${\overline{\tau }}{}^{o}<\tau {}^{o,\max }$ and $\underset{\tau {}^{o}\rightarrow \tau {}^{o,\max }}{\lim }m^{IC}(\tau {}^{o})=-\infty$ . Hence, a positive mitigation share $m>0$ can only be sustained for $0<\tau {}^{o}<{\overline{\tau }}{}^{o}$. Further, there exists a maximum sustainable mitigation share m, as the function $m^{IC}(\tau {}^{o})$ is increasing for small but decreasing for large values of $\tau {}^{o}$. The maximum sustainable m is derived by setting $\frac{\partial m^{IC}}{\partial \tau {}^{o}}=0$. Solving for $\tau {}^{o}$ and substituting the solution back into (22), we obtain:

$$\begin{aligned} m^{\max }=1-\left( \frac{\alpha }{1-\alpha }\frac{1+\alpha \beta }{\beta }\right) ^{\frac{1+\alpha \beta }{\beta }}\left( \left( 1-\alpha \right) \frac{\beta +1+\alpha \beta }{1+\alpha \beta }\right) ^{1+\frac{1+\alpha \beta }{\beta }}<1 \end{aligned}$$

(49)

$m^{\max }$ is strictly smaller than one.

1.11 Heuristic Derivation of Condition (23)

Condition (23) can be derived by explicitly comparing returns from pure capital investment and capital investment combined with the transfer scheme: Assume that there is no social contract, i.e. $m_{t}=\tau _{t}^{o}=\tau _{t+1}^{o}=0$. Marginally increasing the transfer $\tau _{t}^{o}$ to the current old generation yields a perfectly foreseen return of $R_{t+1}^{e}k_{t+1}d\tau _{t+1}^{o,e}=R_{t+1}k_{t+1}d\tau _{t+1}^{o}$ in period $t+1$. Further, an agent of generation t takes into account the equilibrium effect of slower capital accumulation due to lower income on the perfectly foreseen return to capital, $\frac{\partial R_{t+1}}{\partial k_{t+1}}\frac{\partial k_{t+1}}{\partial \tau _{t}^{o}}d\tau _{t}^{o}>0$, which is associated with a total change in return of $\frac{\partial R_{t+1}}{\partial k_{t+1}}\frac{\partial k_{t+1}}{\partial \tau _{t}^{o}}d\tau _{t}^{o}k_{t+1}=(1-\alpha )\frac{\beta }{1+\beta }R_{t+1}R_{t}k_{t}d\tau _{t}^{o}$. On the other hand, increasing the transfer $\tau _{t}^{o}$ reduces income in period t by $R_{t}k_{t}d\tau _{t}^{o}$. Investing this income in physical capital instead would yield an expected return of $R_{t+1}R_{t}k_{t}d\tau _{t}^{o}$ (as $\tau _{t+1}^{o,e}=0$ without social contract). The agent will prefer to combine capital investment with investing in the transfer system and the social contract over investing only in capital if and only if

$$\begin{aligned} R_{t+1}k_{t+1}d\tau _{t+1}^{o}+(1-\alpha )\frac{\beta }{1+\beta }R_{t+1}R_{t}k_{t}d\tau _{t}^{o}-R_{t+1}R_{t}k_{t}d\tau _{t}^{o}>0 \end{aligned}$$

Assuming a stationary transfer system, so that $d\tau _{t}^{o}=d\tau _{t+1}^{o}$, this condition becomes

$$\begin{aligned} R_{t+1}k_{t+1}+(1-\alpha )\frac{\beta }{1+\beta }R_{t+1}R_{t}k_{t}-R_{t+1}R_{t}k_{t}>0. \end{aligned}$$

After some simplification, taking into account $\tau _{t}^{y}=0$, so that $R_{t}=\alpha z(E_{t-1})k_{t}^{\alpha -1}$ and $k_{t+1}=\frac{\beta }{1+\beta }z(E_{t-1})(1-\alpha )k_{t}^{\alpha },$ the condition is equivalent to

$$\begin{aligned} 1-\frac{\alpha }{(1-\alpha )\frac{\beta }{1+\beta }}\left[ 1-(1-\alpha )\frac{\beta }{1+\beta }\right]>0\quad \Longleftrightarrow \quad \beta >\frac{\alpha }{1-\alpha }\left( 1+\alpha \beta \right) \end{aligned}$$

which is condition (23).

1.12 Proof of Proposition 6

Fulfillment of condition (24) in period T is necessary and sufficient for the set $S^{IC}\cap {\mathcal {P}}^{T+1}$ to be non-empty in the starting period of the contract scheme. If the condition holds also for every $t>T$, then there exists some pair $(m,\tau ^{o}), m>0,\tau ^{o}>0$ in the set $S^{IC}\cap {\mathcal {P}}^{T+1}$ which will also lie in every set $S^{IC}\cap {\mathcal {P}}^{t+1}$ for $t>T$. Such a pair can be maintained as incentive compatible, Pareto improving contract for $t\rightarrow \infty$.

Condition (24) is derived as follows: The boundary of the set $S^{IC}$ is the stationary incentive constraint $m^{IC}(\tau ^{o})$ in (22), while the boundaries of the Pareto improvement set ${\mathcal {P}}^{t+1}$ are given by $\Omega (m_{t})$ and $\psi (m_{t};I_{t})$ in (31) and (29). Because the stationary incentive constraint is defined as $m^{IC}(\tau ^{o})$ while the boundaries of the Pareto improvement set are defined as functions of m, we first invert the function $m^{IC}(\tau ^{o})$ over the interval $\tau ^{o}\epsilon \left[ 0,\tau ^{o}(m^{\max })\right]$, where $m^{\max }$ is defined in (49). Denote the inverse by $\tau ^{o,IC}(m)$.

In $(m_{t},\tau _{t+1}^{o})$-space, $\tau ^{o,IC}(m)$ lies above the lower boundary $\Omega (m_{t})$ of the Pareto improvement area for $m_{t},\tau _{t+1}^{o}>0, \forall t$. $\tau ^{o,IC}(m)$ and the upper boundary $\psi (m_{t};I_{t})$ of the Pareto improvement area intersect at the origin $(m_{t},\tau _{t+1}^{o})=(0,0)$ for all t. Given the functional forms of the two curves, they enclose a non-empty set of pairs $(m_{t},\tau _{t+1}^{o}) ,m_{t},\tau _{t+1}^{o}>0$ (so that ) in a period t if and only if at the origin, the slope of $\tau ^{o,IC}(m)$ is flatter than the slope of $\psi (m_{t};I_{t})$, i.e. $\tau _{m}^{o,IC}(0)<\psi _{m}(0^{+},I_{t}^{g})$.

From (22), it follows that the derivative $\tau _{m}^{o,IC}(0)$ is:

$$\begin{aligned} \tau _{m}^{o,IC}(0)=\frac{1+\alpha \beta }{\beta -\frac{\alpha }{1-\alpha }\left( 1+\alpha \beta \right) } \end{aligned}$$

The numerator gives the absolute value of the marginal utility loss from an increase in the mitigation share m from zero. The denominator contains the marginal utility gain from an increase in $\tau ^{o}$ net of the marginal utility loss incurred by an increase in the transfer paid, $\tau ^{y}$. Under condition (23), the denominator and therefore $\tau _{m}^{o,IC}(0)$ is strictly positive.

The slope of the upper boundary $\psi (m_{t};I_{t})$ at the origin was shown to be

$$\begin{aligned} \psi _{m}(0^{+},I_{t})=\frac{1}{\frac{\alpha }{1-\alpha }(1+\alpha \beta )}\left[ \frac{(\beta +\gamma +\gamma \beta )\beta I_{t}}{1+\beta }-\alpha (1+\alpha \beta )\right] \end{aligned}$$

which, setting $\tau _{t}^{o}=0$, becomes $\psi _{m}(0^{+},I_{t}^{g})=\frac{1}{\frac{\alpha }{1-\alpha }(1+\alpha \beta )}\left[ \frac{(\beta +\gamma +\gamma \beta )\beta I_{t}^{g}}{1+\beta }-\alpha (1+\alpha \beta )\right] .$

The relation

$$\begin{aligned} \tau _{m}^{o,IC}(0)<\psi _{m}\left( 0^{+},I_{t}^{g}\right) \end{aligned}$$

is satisfied if and only if

$$\begin{aligned} \frac{1+\alpha \beta }{\beta -\frac{\alpha }{1-\alpha }\left( 1+\alpha \beta \right) }< & {} \frac{1}{\frac{\alpha }{1-\alpha }(1+\alpha \beta )}\left[ \frac{(\beta +\gamma +\gamma \beta )\beta I_{t}^{g}}{1+\beta }-\alpha (1+\alpha \beta )\right] \nonumber \\\iff & {} I_{t}^{g}>\frac{(1-\alpha )\frac{\beta }{1+\beta }\left( 1+\alpha \beta +\beta \right) }{\beta -\frac{\alpha }{1-\alpha }\left( 1+\alpha \beta \right) }\underset{{\widehat{I}}}{\underbrace{\frac{\alpha \left( 1+\alpha \beta \right) \left( 1+\beta \right) ^{2}}{(1-\alpha )\left( \beta +\gamma +\gamma \beta \right) \beta ^{2}}}}>{\widehat{I}}\nonumber \\ \end{aligned}$$

(50)

which is condition (24). The new threshold is larger than ${\widehat{I}}$.

1.13 Simulation

We illustrate numerically that through Pareto improving intergenerational social contracts, an economy may have a chance to converge to a better steady state with lower stationary pollution stock and higher stationary income compared to the case without social contracts. We set the share of capital, $\alpha$, to 0.3. For the rate of time preference, $\beta$, we choose the value 0.7 which yields a plausible savings rate of households around 40%. Without loss of generality we set the effectiveness of mitigation to $\gamma =1$. Note that these parameter values guarantee that the condition in Assumption 2 holds. With the above parameter values we compute ${\hat{I}}=\frac{\alpha (1+\alpha \beta )(1+\beta )^{2}}{(1-\alpha )(\beta +\gamma +\gamma \beta )\beta ^{2}}\simeq 1.2744$. We run the simulation with different levels of technology A corresponding to the two distinct cases ${\tilde{I}}<{\hat{I}}$ and ${\tilde{I}}>{\hat{I}}$ in the following subsections (Figs. 6, 7, 8).

Fig. 6

Dynamics in the case ${\hat{I}}<{\tilde{I}}$. a Income dynamics with and without social contracts, b Pollution stock dynamics with and without social contracts, c Mitigation and transfer

Fig. 7

Dynamics in the case ${\hat{I}}>{\tilde{I}}$ with early signing of contracts. a Income dynamics with and without social contracts, b Pollution stock dynamics with and without social contracts, c Mitigation and transfer

Fig. 8

Dynamics in the case ${\hat{I}}>{\tilde{I}}$ with delayed signing of contracts. a Income dynamics with and without social contracts, b Pollution stock dynamics with and without social contracts, c Mitigation and transfer

1.14 Robustness

Although functional forms used in this paper are standard in the literature, they are specific—in particular when they are combined. We adopt these standard functional forms in order to keep our model tractable. They allow us to characterize conditions for the existence of Pareto improving, self-enforcing social contracts. In this section, we discuss the implications of changes in functional forms for the existence of such social contracts. We first consider the following changes: (i) a rate of capital depreciation $\lambda \in (0,1)$; (ii) a production function of the constant elasticity of substitution (CES) form with capital intensive externality on labor productivity; and (iii) a utility function of the Constant Relative Risk Aversion (CRRA) form combined with a CES production function as in (ii).

We can derive sufficient conditions on income for the existence of Pareto improving and self-enforcing social contracts which are in general similar to the benchmark model. Differences arising from changes in functional forms will be pointed out in this section.

1.14.1 Capital Depreciation Rate $\lambda \in (0,1)$

We define the aggregate per capita production function as follows

$$\begin{aligned} {\hat{F}}(k_{t},E_{t-1})=z(E_{t-1})\left[ k_{t}^{\alpha }+(1-\lambda )k_{t}\right] ;\quad k_{t}=K_{t}/L_{t} \end{aligned}$$

The disposable income and gross return to capital that an agent t receives in period t and $t+1$, respectively, under the social contracts $(m_{t-1},\tau _{t}^{o})$ and $(m_{t},\tau _{t+1}^{o})$ are

$$\begin{aligned} I_{t}\left( 1-m_{t}\right)= & {} \left( 1-\alpha \right) z\left( E_{t-1}\right) k_{t}^{\alpha }\left( 1-\frac{\alpha \tau _{t}^{o}}{1-\alpha }\right) \left( 1-m_{t}\right) \\ {[}R_{t+1}\left( 1+\tau _{t+1}^{o}\right) +z\left( E_{t}\right) \left( 1-\lambda \right) ]k_{t+1}= & {} z\left( E_{t}\right) \left[ \alpha k_{t+1}^{\alpha }\left( 1+\tau _{t+1}^{o}\right) +\left( 1-\lambda \right) k_{t+1}\right] \end{aligned}$$

The welfare surplus under the social contract $(m_{t},\tau _{t+1}^{o})$ of agent t is

$$\begin{aligned} \triangle V_{t}^{t+1}=(1+\alpha \beta )\ln (1-m_{t})+\beta \ln \frac{\alpha (1+\tau _{t+1}^{o})+(1-\lambda )\left[ \frac{\beta }{1+\beta }I_{t}(1-m_{t})\right] ^{1-\alpha }}{\alpha +(1-\lambda )\left[ \frac{\beta }{1+\beta }I_{t}\right] ^{1-\alpha }} \end{aligned}$$

Setting $\triangle V_{t}^{t+1}=0$ allows us to determine the indifference curve of agent t

$$\begin{aligned} \tau _{t+1}^{o}=\frac{1+\frac{1-\lambda }{\alpha }\left( \frac{\beta I_{t}}{1+\beta }\right) ^{1-\alpha }}{(1-m_{t})^{\alpha +\frac{1}{\beta }}}-\frac{1-\lambda }{\alpha }\left( \frac{\beta I_{t}}{1+\beta }(1-m_{t})\right) ^{1-\alpha }-1\equiv {\hat{\Omega }}(m_{t},I_{t}) \end{aligned}$$

where ${\hat{\Omega }}_{m}(m_{t},I_{t})>0, {\hat{\Omega }}_{mm}(m_{t},I_{t})<0, {\hat{\Omega }}(0,I_{t})=0$ and

$$\begin{aligned} {\hat{\Omega }}_{m}(0^{+},I_{t})=\frac{1+\alpha \beta }{\beta }+\frac{1-\lambda }{\alpha }\left( \frac{1+\beta }{\beta }\right) ^{\alpha }I_{t}^{1-\alpha } \end{aligned}$$

Different from the indifference curve $\Omega (m_{t})$ of agent t in the benchmark model, the corresponding curve ${\hat{\Omega }}(m_{t},I_{t})$ now, due to the appearance of vintage capital, depends on net income $I_{t}$, and so does its slope at the origin ${\hat{\Omega }}_{m}(0^{+},I_{t})$.

Similarly, we obtain the welfare surplus of agent $t+1$ under the social contract $(m_{t},\tau _{t+1}^{o})$

$$\begin{aligned} \triangle {\mathcal {V}}_{t+1}^{t+1}= & {} \left( 1+\alpha \beta \right) \ln \frac{I_{t+1}}{{\tilde{I}}_{t+1}}+\beta \ln \frac{z\left( E_{t+1}\right) }{z\left( {\tilde{E}}_{t+1}\right) }\\&+\,\beta \ln \frac{\alpha \left( 1+\tau _{t+2}^{o,e}\right) +\left( 1-\lambda \right) \left[ \frac{\beta I_{t+1}\left( 1-m_{t+1}^{e}\right) }{1+\beta }\right] ^{1-\alpha }}{\alpha \left( 1+\tau _{t+2}^{o,e}\right) +\left( 1-\lambda \right) \left[ \frac{\beta {\tilde{I}}_{t+1}\left( 1-m_{t+1}^{e}\right) }{1+\beta }\right] ^{1-\alpha }} \end{aligned}$$

where ${\tilde{I}}_{t+1}$ and ${\tilde{E}}_{t+1}$, as in the benchmark model, are respectively net income and the stock of pollution in the case of no social contract, i.e. $(m_{t},\tau _{t+1}^{o})=(0,0)$. We can prove that:

$$\begin{aligned} \triangle {\mathcal {V}}_{t+1}^{t+1}\in \left( (1+\beta )\ln \frac{I_{t+1}}{{\tilde{I}}_{t+1}}+\beta \ln \frac{z(E_{t+1})}{z({\tilde{E}}_{t+1})},\;(1+\alpha \beta )\ln \frac{I_{t+1}}{{\tilde{I}}_{t+1}}+\beta \ln \frac{z(E_{t+1})}{z({\tilde{E}}_{t+1})}\right) \end{aligned}$$

This allows us to determine the lower bound and upper bound for the indifference curve of agent $t+1$ respectively:

$$\begin{aligned} \tau _{t+1}^{o}= & {} \frac{1-\alpha }{\alpha }\left[ 1-\mathrm {e}^{-\frac{\beta (\beta +\gamma +\gamma \beta )}{(1+\beta )^{2}}m_{t}I_{t}}(1-m_{t})^{-\alpha }\right] \equiv {\underline{\psi }}(m_{t},I_{t})\\ \tau _{t+1}^{o}= & {} \frac{1-\alpha }{\alpha }\left[ 1-\mathrm {e}^{-\frac{(\beta +\gamma +\gamma \beta )\beta m_{t}I_{t}}{(1+\beta )(1+\alpha \beta )}}(1-m_{t})^{-\alpha }\right] \equiv {\bar{\psi }}(m_{t},I_{t}) \end{aligned}$$

The slopes at the origin are respectively:

$$\begin{aligned} {\underline{\psi }}_{m}(0^{+},I_{t})= & {} \frac{1-\alpha }{\alpha }\left[ \frac{\beta (\beta +\gamma +\gamma \beta )}{(1+\beta )^{2}}I_{t}-\alpha \right] \end{aligned}$$

(51)

$$\begin{aligned} {\bar{\psi }}_{m}(0^{+},I_{t})= & {} \frac{1-\alpha }{\alpha }\left[ \frac{\beta (\beta +\gamma +\gamma \beta )}{(1+\beta )(1+\alpha \beta )}I_{t}-\alpha \right] \end{aligned}$$

(52)

It is fairly straightforward to show that ${\underline{\psi }}(m_{t},I_{t})$ and ${\bar{\psi }}(m_{t},I_{t})$ are strictly concave and it is trivial that ${\underline{\psi }}(0,I_{t})={\bar{\psi }}(0,I_{t})=0$. By comparing the slopes ${\underline{\psi }}_{m}(0^{+},I_{t})$ and ${\hat{\Omega }}_{m}(0^{+},I_{t})$ at the origin, we can derive a sufficient condition on net income, ${\hat{I}}^{*}>0$, above which ${\mathcal {P}}^{t+1}\ne \oslash$. Similarly, by comparing ${\bar{\psi }}_{m}(0^{+},I_{t})$ and ${\hat{\Omega }}_{m}(0^{+},I_{t})$ we also can derive a sufficient condition on net income, ${\hat{I}}^{**}\in (0,{\hat{I}}^{*})$, below which ${\mathcal {P}}^{t+1}=\oslash$.

Contrary to the benchmark model which allows us to derive necessary and sufficient conditions on net income for the existence of the set of Pareto improving social contracts, we can now only derive sufficient conditions on net income which guarantee the existence of this set. Nevertheless, the qualitative results for the existence of this set in this extension do not change crucially compared to those from the benchmark model.

Pareto improving and self-enforcing social contracts

Crucial for the existence of a scheme of self-enforcing contracts is the incentive constraint for generation t in compliance phase, given by the following condition

$$\begin{aligned} \triangle V^{C}\left( m_{t},\tau _{t+1}^{o};I_{t}^{D}\right)= & {} (1+\alpha \beta )\ln \frac{I_{t}^{C}(1-m_{t})}{I_{t}^{D}}\nonumber \\&+\,\beta \ln \frac{\alpha \left( 1+\tau _{t+1}^{o}\right) +(1-\lambda )\left[ \frac{\beta I_{t}^{C}(1-m_{t})}{1+\beta }\right] ^{1-\alpha }}{\alpha +(1-\lambda )\left[ \frac{\beta I_{t}^{D}}{1+\beta }\right] ^{1-\alpha }}\ge 0 \end{aligned}$$

(53)

with $I_{t}^{C}=(1-\frac{\alpha }{1-\alpha }\tau _{t}^{o})I_{t}^{D}$, where $I_{t}^{C}$ and $I_{t}^{D}=I_{t}^{g}$, respectively, denote the income of agent t in the case of complying with and deviating from strategy s when the game is in compliance phase.²⁴

Unlike in the benchmark model, the incentive constraint now depends on gross income $I_{t}^{D}=I_{t}^{g}$ of agent t. The gain from a given self-enforcing social contract $(m_{t},\tau _{t+1}^{o})$, when it exists, decreases in gross income $I_{t}^{g}$. From equation (53) we can find that, for any given $(m_{t},\tau _{t+1}^{o})\in (0,1)\times \left( 0,\frac{1-\alpha }{\alpha }\right)$ and $\lambda \in (0,1), \triangle V^{C}(m_{t},\tau _{t+1}^{o};I_{t}^{g})<0$ whenever gross income $I_{t}^{g}$ is sufficiently high. In this case self-enforcing social contracts cannot exist. That is because the appearance of vintage capital increases the agent’s old-age consumption more in the case of defaulting on the contract than in the case of complying. The remaining amount of capital, indeed, depends linearly and positively on gross income. So higher gross income $I_{t}^{g}$ makes the existence of self-enforcing social contracts less likely.

We can derive a condition on gross income $I^{g}$ for the existence of a non-empty set $S^{IC}$ of stationary self-enforcing contracts by requiring the slope of the boundary of this set at the origin to be strictly positive. Applying the implicit function theorem, we find that:

$$\begin{aligned} s_{|(0,0)}^{IC}=-\frac{\triangle V_{m}^{C}(0,0;I^{g})}{\triangle V_{\tau }^{C}(0,0;I^{g})}=\frac{(1-\alpha )\left[ 1+\alpha \beta +\frac{1+\beta }{\alpha }(1-\lambda )\left( \frac{\beta }{1+\beta }I^{g}\right) ^{1-\alpha }\right] }{(1-\alpha )\beta -\alpha (1+\alpha \beta )-(1+\beta )(1-\lambda )\left( \frac{\beta }{1+\beta }I^{g}\right) ^{1-\alpha }}>0 \end{aligned}$$

For the condition to be satisfied, the denominator must be strictly positive, hence it must hold that

$$\begin{aligned} \beta >\frac{\alpha }{1-\alpha }(1+\alpha \beta )+\frac{(1+\beta )(1-\lambda )}{1-\alpha }\left( \frac{\beta }{1+\beta }I^{g}\right) ^{1-\alpha } \end{aligned}$$

(54)

Comparing condition (54) with condition (23) from the benchmark model, we can identify the effect of slower capital depreciation. Given that $\beta >\frac{\alpha }{1-\alpha }(1+\alpha \beta )$, we can derive a condition on gross income guaranteeing the existence of the set of stationary self-enforcing contracts $S^{IC}$:

$$\begin{aligned} I^{g}<\left[ \frac{\beta (1-\alpha )-\alpha (1+\alpha \beta )}{(1+\beta +\alpha \beta )(1-\lambda )}\right] ^{\frac{1}{1-\alpha }}\frac{1+\beta }{\beta } \end{aligned}$$

(55)

The right hand side increases unboundedly when $\lambda$ increases and approaches 1. Therefore the higher the depreciation rate of capital $\lambda$, the more likely it is that $S^{IC}\cap {\mathcal {P}}\ne \oslash$. That is because the higher $\lambda$ decreases the slope of the stationary incentive constraint at the origin and relaxes the condition on income (55) for the existence of the set of stationary self-enforcing contracts $S^{IC}$. So it can be predicted that when $\lambda$ is sufficiently high and gross income $I_{t}^{g}$ is sufficiently high as well, so that ${\underline{\psi }}_{m}(0^{+},I^{g})>s_{|(0,0)}^{IC}$ then $S^{IC}\cap {\mathcal {P}}\ne \oslash$.

However, one could argue that in a persistently growing economy, when gross income $I^{g}$ exceeds some high threshold, condition (55) cannot be satisfied any longer and thus self-enforcing social contract schemes may not exist. Notwithstanding, persistent growth is driven by technological progress which also enhances the longevity of agents through its positive effects on medical technology and nutrition, for example. It thus lengthens the working-age and old-age periods. The longer working-age period means capital is used longer, making the depreciation rate of capital over the whole lengthened period higher as well. So as is obvious from (55), the condition on gross income for the existence of self-enforcing contract schemes is relaxed unboundedly when technological progress enhances longevity. Intergenerational social contracts which are simultaneously Pareto improving and self-enforcing can thus exist even in a growing economy.

1.14.2 Production with Capital Intensive Externality

We argue that the qualitative results of the benchmark model do not change crucially when we adopt logarithmic preferences and the following CES production function featuring an externality of capital on labor productivity:

$$\begin{aligned} Y_{t}=z(E_{t-1})\left[ \alpha K_{t}^{\rho }+(1-\alpha )(k_{t}L_{t})^{\rho }\right] ^{\frac{1}{\rho }};\quad \rho \in (-\infty ,1],\,k_{t}=K_{t}/L_{t} \end{aligned}$$

(56)

For this production function, the marginal returns to labor and capital become

$$\begin{aligned} w_{t}=(1-\alpha )z(E_{t-1})k_{t}\quad \text {and}\quad R_{t}=\alpha z(E_{t-1}) \end{aligned}$$

In this case the computations to obtain sufficient conditions on net income that guarantee the existence of a non-empty intersection between the Pareto improvement set and the set of stationary self-enforcing contracts are similar to what we have done in the benchmark model. Along with the CES production function, we can also take into account less than full capital depreciation at rate $\lambda \in (0,1)$ in characterizing the sufficient conditions for the existence of a non-empty intersection between the Pareto improvement set and the set of stationary self-enforcing contracts. The procedures and mathematical structures are similar to those in the previous subsection.

1.14.3 Constant Relative Risk Aversion (CRRA) Utility Function

With a CRRA utility function,²⁵

$$\begin{aligned} u_{t}=\frac{\left( c_{t}^{y}\right) ^{1-\theta }-1}{1-\theta }+\beta \frac{\left( c_{t+1}^{o}\right) ^{1-\theta }-1}{1-\theta };\quad \theta >0 \end{aligned}$$

there are several challenges in deriving the sufficient conditions on income for the existence of the Pareto improvement set as well as the set of stationary self-enforcing contracts. It is therefore not straightforward to reach a conclusion about the exact difference between this generalized model and the benchmark one. That is because the conditions now depend not only on income but also on the stock of pollution through complex exponential functions. In order to make useful analyses, we adopt the production function in (56) and assume that capital fully depreciates after a period of use. With this utility function, the optimal consumption plans of agent t under a contract $(m_{t},\tau _{t+1}^{o})$ are

$$\begin{aligned} c_{t}^{y}= & {} \frac{R_{t+1}\left( 1+\tau _{t+1}^{o}\right) }{R_{t+1}\left( 1+\tau _{t+1}^{o}\right) +\left[ \beta R_{t+1}\left( 1+\tau _{t+1}^{o}\right) \right] ^{1/\theta }}I_{t}\left( 1-m_{t}\right) \equiv c^{y}\left( m_{t},\tau _{t+1}^{o},I_{t},R_{t+1}\right) \nonumber \\ \end{aligned}$$

(57)

$$\begin{aligned} c_{t+1}^{o}= & {} \frac{\left[ \beta R_{t+1}\left( 1+\tau _{t+1}^{o}\right) \right] ^{1/\theta }R_{t+1}\left( 1+\tau _{t+1}^{o}\right) }{R_{t+1}\left( 1+\tau _{t+1}^{o}\right) +\left[ \beta R_{t+1}\left( 1+\tau _{t+1}^{o}\right) \right] ^{1/\theta }}I_{t}\left( 1-m_{t}\right) \equiv c^{o}\left( m_{t},\tau _{t+1}^{o},I_{t},R_{t+1}\right) \nonumber \\ \end{aligned}$$

(58)

The condition for agent t not to suffer a welfare loss from a social contract $(m_{t},\tau _{t+1}^{o})$ is

$$\begin{aligned} \triangle V_{t}^{t+1}= & {} \frac{c^{y}\left( m_{t},\tau _{t+1}^{o},I_{t},R_{t+1}\right) ^{1-\theta }-\left( {\tilde{c}}_{t}^{y}\right) ^{1-\theta }}{1-\theta }\nonumber \\&+\,\beta \frac{c^{o}\left( m_{t},\tau _{t+1}^{o},I_{t},R_{t+1}\right) ^{1-\theta }-\left( {\tilde{c}}_{t+1}^{o}\right) ^{1-\theta }}{1-\theta }\ge 0 \end{aligned}$$

(59)

where ${\tilde{c}}_{t}^{y}=c^{y}(0,0,I_{t},R_{t+1})$ and ${\tilde{c}}_{t+1}^{o}=c^{o}(0,0,I_{t},R_{t+1})$.

The existence of such a contract $(m_{t},\tau _{t+1}^{o})$ is guaranteed by the strictly positive slope at the origin (0, 0) of the indifference curve $\triangle V_{t}^{t+1}=0, s_{t|(0,0)}^{t+1}$. By applying the implicit function theorem, we find:

$$\begin{aligned} s_{t|(0,0)}^{t+1}= & {} -\frac{\partial \triangle V_{t}^{t+1}/\partial m_{t}}{\partial \triangle V_{t}^{t+1}/\partial \tau _{t+1}^{o}}_{|(0,0)}\\= & {} \frac{\left( 1+\beta (\beta R_{t+1})^{\frac{1-\theta }{\theta }}\right) \left( R_{t+1}+(\beta R_{t+1})^{\frac{1}{\theta }}\right) }{\frac{\theta -1}{\theta }(\beta R_{t+1})^{\frac{1}{\theta }}R_{t+1}+\beta (\beta R_{t+1})^{\frac{1-\theta }{\theta }}\left( \frac{1}{\theta }R_{t+1}+(\beta R_{t+1})^{\frac{1}{\theta }}\right) }\qquad \end{aligned}$$

It is trivial that $s_{t|(0,0)}^{t+1}>0$ for all $\theta >1$. For $\theta \in (0,1)$, it is easy to find a condition on $R_{t+1}$ making the denominator of the last equation strictly positive to guarantee $s_{t|(0,0)}^{t+1}>0$.

Similarly, we determine the condition for agent $t+1$ not to suffer a welfare loss from the social contract $(m_{t},\tau _{t+1}^{o})$, given the perfectly foreseen social contract $(m_{t+1}^{e},\tau _{t+2}^{o,e})$, as

$$\begin{aligned} \triangle {\mathcal {V}}_{t+1}^{t+1}= & {} \frac{c^{y}\left( m_{t+1}^{e},\tau _{t+2}^{o,e},I_{t+1},R_{t+2}\right) ^{1-\theta }-c^{y}(m_{t+1}^{e},\tau _{t+2}^{o,e},{\tilde{I}}_{t+1},{\tilde{R}}_{t+2})^{1-\theta }}{1-\theta }\\&+\,\beta \frac{c^{o}(m_{t+1}^{e},\tau _{t+2}^{o,e},I_{t+1},R_{t+2})^{1-\theta }-c^{o}( m_{t+1}^{e},\tau _{t+2}^{o,e},{\tilde{I}}_{t+1},{\tilde{R}}_{t+2})^{1-\theta }}{1-\theta }\ge 0 \end{aligned}$$

where ${\tilde{I}}_{t+1}$ and ${\tilde{R}}_{t+2}$ are respectively net income in period $t+1$ and the return to capital in period $t+2$ in the case of no social contract $(m_{t},\tau _{t+1}^{o})$ between generations t and $t+1$.

The slope of the indifference curve $\triangle {\mathcal {V}}_{t+1}^{t+1}=0$ at the origin (0, 0) is

$$\begin{aligned} s_{t+1|\left( 0,0\right) }^{t+1}=-\frac{\partial \triangle {\mathcal {V}}_{t+1}^{t+1}/\partial m_{t}}{\partial \triangle {\mathcal {V}}_{t+1}^{t+1}/\partial \tau _{t+1}^{o}}_{|\left( 0,0\right) }=\frac{1-\alpha }{\alpha }\left( \frac{\beta \left[ \frac{\left( \beta R_{t+1}\right) ^{\frac{1}{\theta }}}{R_{t+1}+\left( \beta R_{t+1}\right) ^{\frac{1}{\theta }}}+\gamma \right] I_{t}}{\left[ \beta {\tilde{R}}_{t+2}\left( 1+\tau _{t+2}^{o,e}\right) \right] ^{\frac{\theta -1}{\theta }}+\beta }-1\right) \end{aligned}$$

(60)

It is easy to find a condition on income under which $s_{t+1|(0,0)}^{t+1}>0$. Moreover, comparing the two slopes $s_{t+1|(0,0)}^{t+1}$ and $s_{t|(0,0)}^{t+1}$, we can see that $s_{t|(0,0)}^{t+1}$ is independent of net income $I_{t}$ while $s_{t+1|(0,0)}^{t+1}$ increases unboundedly in $I_{t}$. That is to say when $I_{t}$ is sufficiently high, then

$$\begin{aligned} s_{t+1|(0,0)}^{t+1}>s_{t|(0,0)}^{t+1} \end{aligned}$$

which guarantees the existence of Pareto improving social contracts, i.e. ${\mathcal {P}}^{t+1}\ne \oslash$.

Different from the benchmark model, the threshold of net income now not only depends on the parameters of the model but also on state variables, i.e. the stock of pollution (note that $R_{t+1}=\alpha z(E_{t}))$, and on the forseen next social contract $(m_{t+1}^{e},\tau _{t+2}^{o,e})$. That is because unlike with logarithmic preferences, the savings decision with CRRA preferences depends on the return to capital. Hence, the variables above appear in the welfare gain of agent $t+1$.

Pareto improving and self-enforcing social contracts

The incentive constraint for generation t in compliance phase is

$$\begin{aligned} \triangle V^{C}\left( m_{t},\tau _{t+1}^{o};I_{t}^{D}\right)= & {} \frac{c^{y}\left( m_{t},\tau _{t+1}^{o},I_{t}^{C},R_{t+1}\right) ^{1-\theta }-c^{y}\left( 0,0,I_{t}^{D},R_{t+1}\right) ^{1-\theta }}{1-\theta }\nonumber \\&+\,\beta \frac{c^{o}\left( m_{t},\tau _{t+1}^{o},I_{t}^{C},R_{t+1}\right) ^{1-\theta }-c^{o}\left( 0,0,I_{t}^{D},R_{t+1}\right) ^{1-\theta }}{1-\theta }\ge 0\nonumber \\ \end{aligned}$$

(61)

with $I_{t}^{C}=(1-\frac{\alpha }{1-\alpha }\tau _{t}^{o})I_{t}^{D}$, where $I_{t}^{C}$ and $I_{t}^{D}=I^{g}$ are respectively the income of agent t in the case of complying with and deviating from strategy s when the game is in compliance phase.

We focus on the set of stationary self-enforcing social contracts as characterized by

$$\begin{aligned} \triangle V^{C}(m,\tau ^{o},I^{g})= & {} \frac{c^{y}(m,\tau ^{o},I^{C},R)^{1-\theta }-c^{y}(0,0,I^{g},R)^{1-\theta }}{1-\theta }\\&+\,\beta \frac{c^{o}(m,\tau ^{o},I^{C},R)^{1-\theta }-c^{o}(0,0,I^{g},R)^{1-\theta }}{1-\theta }\ge 0 \end{aligned}$$

The existence of this set is guaranteed by the condition that the slope of the stationary incentive constraint $\triangle V^{C}(m,\tau ^{o},I^{g})=0$ at the origin be strictly positive, i.e.

$$\begin{aligned} s_{|(0,0)}^{IC}=-\frac{V_{m}^{C}(m,\tau ^{o},I^{g})}{V_{\tau }^{C}(m,\tau ^{o},I^{g})}>0 \end{aligned}$$

(62)

where

$$\begin{aligned} V_{m}^{C}(m,\tau ^{o},I^{g})= & {} -\left[ c^{y}(0,0,I^{g},R)^{1-\theta }+\beta c^{o}(0,0,I^{g},R)^{1-\theta }\right] <0\\ V_{\tau }^{C}(m,\tau ^{o},I^{g})= & {} c^{y}(0,0,I^{g},R)^{-\theta }\left[ c_{\tau }^{y}(0,0,I^{g},R)-\frac{\alpha I^{g}}{1-\alpha }c_{I}^{y}(0,0,I^{g},R)\right] \\&+\,\beta c^{o}(0,0,I^{g},R)^{-\theta }\left[ c_{\tau }^{o}(0,0,I^{g},R)-\frac{\alpha I^{g}}{1-\alpha }c_{I}^{o}(0,0,I^{g},R)\right] \end{aligned}$$

Condition (62) holds if and only if $V_{\tau }^{C}(m,\tau ^{o},I^{g})>0$, which boils down to

$$\begin{aligned} 1-(1-\theta )(\beta R)^{\frac{1}{\theta }}-\frac{\alpha \theta }{1-\alpha }\left[ R+(\beta R)^{\frac{1}{\theta }}\right] >0 \end{aligned}$$

We next find a condition under which there exist stationary social contracts that are both Pareto improving and self-enforcing, i.e. under which $S^{IC}\cap {\mathcal {P}}\ne \oslash$. We do so by comparing the slope of the “stationary” incentive constraint, $\triangle V^{C}(m,\tau ^{o},I^{g})=0$, at the origin with the slope defined in (60) at the origin. Using the property that consumption in both periods of life is linear in income, as is obvious from (57) and (58), we can easily prove that the slope $s_{|(0,0)}^{IC}$ is independent of gross income $I^{g}$. Further, the slope $s_{|(0,0)}^{IC}$ is bounded, while the slope $s_{t+1|(0,0)}^{t+1}$, as defined in (60), increases unboundedly in income I. This suggests that whenever gross income $I^{g}$ is sufficiently high, then $S^{IC}\cap {\mathcal {P}}\ne \oslash$.