2.1 Households

We assume a stationary population.Footnote ⁷ Labor supply is inelastic and only depends upon productivity levels for young, $1-\phi _2$, and older people, $\phi _2$, respectively. Along the lines of Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021) and Hiraguchi and Shibata (Reference Hiraguchi and Shibata2015), we allow for uninsurable idiosyncratic productivity shocks, $\chi _{t+1}$, at old-age, which are independently and identically distributed across agents. Let $\Psi (\chi _{t+1})$ be the cumulative distribution function and $\psi (\chi _{t+1})$ the probability density function. Assuming that $\int \chi d\Psi = 1$, it turns out that the effective labor supply is equal to 1,

(1)\begin{equation} L_t = 1-\phi_2+\phi_2\int \chi\,{\rm d}\Psi = 1 \end{equation}

In order to obtain a closed-form solution for both the savings rate and the optimal carbon prices, we follow tradition in recent analytical macro-climate models, e.g, Golosov et al. (Reference Golosov, Hassler, Krusell and Tsyvinski2014), van den Bijgaart et al. (Reference van den Bijgaart, Gerlagh and Liski2016), Gerlagh and Liski (Reference Gerlagh and Liski2018a, Reference Gerlagh and Liski2018b), Iverson and Karp (Reference Iverson and Karp2021), and let preferences be represented by a logarithmic function,Footnote ⁸

(2)\begin{equation} W_t \equiv \log C_{1,t} + \beta \int \log C_{2,t+1}(\chi_{t+1})\,{\rm d}\Psi(\chi) \end{equation}

This utility function, (2), implies that the households receive utility only from consumption over their life-cycle, since we assume that labor supply is inelastic. Importantly, the future is discounted through the parameter, $\beta$, and consumption when they are old is uncertain due to the presence of idiosyncratic labor income risk in the second period of their lives. The sequence of budget constraints, when they are young and old, respectively, are as follows,

(3)\begin{align} C_{1,t}+K_{t+1}& = (1-\phi_2)(w_t+T_{1,t}^E) \end{align}

(4)\begin{align} C_{2,t+1}& =\phi_2\chi_{t+1}(w_{t+1}+T_{2,t+1}^E)+ (1-\tau_{t+1}^K)r_{t+1}K_{t+1}+ T_{2,t+1}^K \end{align}

where $\left \{C_{1,t},\,C_{2,t+1}\right \}$ are consumption levels for young and old people, $r_{t+1}K_{t+1}$ represents capital income, non-capital income is the sum of wages, $\left \{w_t,\,w_{t+1}\right \}$, and lump-sum transfers to households coming from climate policy revenues, $\left \{T_{1,t}^E,\,T_{2,t+1}^E\right \}$, respectively. Likewise, $T_{2,t+1}^K$ denotes lump-sum transfers to the old households coming from taxes on capital income, $\tau _{t+1}^K$, as in Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021). Since there is no labor choice, by solving this problem for capital holdings after replacing the budget constraints into the utility function, the first-order condition yields an usual Euler equation,

(5)\begin{equation} U_{C_{1,t}} = \beta (1-\tau_{t+1}^K)r_{t+1}\mathbb{E}[U_{C_{2,t+1}}] \end{equation}

This expression describes the trade-off between current and future consumption. Using the assumption of logarithmic preferences over consumption, the Euler equation can be rewritten as follows,

(6)\begin{equation} \frac{1}{C_{1,t}}=\beta(1-\tau_{t+1}^K)r_{t+1}\int \left(\frac{1}{C_{2,t+1}(\chi_{t+1})}\right)\,{\rm d}\Psi(\chi_{t+1}) \end{equation}

2.2 Firms

The representative firm uses a Cobb-Douglas technology, $F(\cdot )=\Omega (Z_t)K_t^\alpha [H_t (E_t,L_t)]^{1-\alpha }$, and profit maximization yields standard first-order conditions for capital $K_t$, aggregate labor $L_t$, and energy $E_t$, where the function $H_t(\cdot )$ is a composite labor-energy input in the spirit of Iverson and Karp (Reference Iverson and Karp2021) and Gerlagh and Liski (Reference Gerlagh and Liski2018b). There is full capital depreciation.Footnote ⁹ As in Gerlagh and Liski (Reference Gerlagh and Liski2018a, Reference Gerlagh and Liski2018b), production leads to $\text {CO}_2$ emissions (one-to-one with the use of energy) that increase the stock of pollution over time, $Z_t=\sum _{i=1}^\infty \theta _i E_{t-i}$, according to a reduced-form multi-box representation of the carbon cycle and temperature adjustments, $\theta _i$. High levels of atmospheric concentrations of $\text {CO}_2$ change the climate and reduce future output through a damage function, $\Omega _t=\text {exp}(-Z_t)$.Footnote ¹⁰

The representative firm does not internalize these external marginal costs. Hence, the government would like to implement some sort of climate policy: for instance, a carbon price would aim to reduce the impacts of current emissions on carbon concentrations, changes in temperature, and future production losses due to global warming. The first-order conditions are thus given by,

(7)\begin{align} r_t& =F_{K_t} \end{align}

(8)\begin{align} w_t& =F_{L_t} \end{align}

(9)\begin{align} \tau_{t}^E& =F_{E_t} \end{align}

where $\tau _{t}^E$ denotes the carbon price and $F_{i,t}$ for $i \in \{K,\,L,\,E\}$ represents the marginal productivity of capital, labor, and energy, respectively. At the optimal allocation, all the inputs will be paid the value of their marginal productivity.

2.3 Government

The government has two main goals in this economy. First, due to uninsurable idiosyncratic labor income risk in the second period, households have the incentive to increase their precautionary savings but fail to internalize the general equilibrium effects on wages and capital returns of such decisions. The government can then use capital taxes to move the economy closer to the first best by reducing the level of aggregate capital. As in Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021), we assume that the government uses the revenues from capital taxation to finance transfers to the same generation to provide partial insurance against labor income risk.Footnote ¹¹

(10)\begin{equation} T_{2,t}^K=\tau_t^Kr_tK_t \end{equation}

Second, $\text {CO}_2$ emissions generate a climate externality in the economy by reducing future output. As an available instrument, the government can levy an excise tax on energy use to internalize the associated future marginal economic costs into firm's decisions. Since the government is constrained in the way it distributes resources across generations, we also assume that all tax revenues coming from carbon taxation are rebated to households as lump-sum transfers period-by-period as in Gerlagh et al. (Reference Gerlagh, Jaimes and Motavasseli2017).Footnote ¹² Here, since older people also supply labor, the revenues are allocated bearing in mind age-specific productivity levels, that is, transfers for the young and old people are constant fractions of total environmental revenues, formally,

(11)\begin{equation} \tau_t^E E_t = (1-\phi_2)T_{1,t}^E+\phi_2 T_{2,t}^E \end{equation}

2.4 Equilibrium

A competitive equilibrium for this economy can be written as follows,

2.5 The saving rate

We follow Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021) closely to derive an implementability condition for relaxing the government problem and solving for allocations instead of tax rates using the well-known primal approach in the optimal taxation literature (Lucas and Stokey, Reference Lucas and Stokey1983; Erosa and Gervais, Reference Erosa and Gervais2002). Since labor supply is exogenous and preferences are logarithmic, the government can determine allocations by choosing only saving rates. Below, we show that the optimal saving rate can be decentralized as a competitive equilibrium using capital income taxes. Let $s_t$ be the saving rate, which is a fraction of total income for the young, i.e., labor income and transfers from carbon taxation revenues,

(14)\begin{equation} s_t=\frac{K_{t+1}}{(1-\phi_2)(w_t+T_{1,t}^E)}\end{equation}

Using the sequence of budget constraints described above and the definition for the saving rate, we can rewrite the Euler equation as,

(15)\begin{equation} 1=\beta(1-\tau_{t+1}^K)\int \frac{1-s_t}{\frac{\phi_2\chi_{t+1}\hat{w}_{2,t+1}}{r_{t+1}(1-\phi_2)\hat{w}_{1,t}}+ (1-\tau_{t+1}^K)s_t+ \frac{T_{2,t+1}^K}{r_{t+1}(1-\phi_2)\hat{w}_{1,t}}}\,{\rm d}\Psi(\chi_{t+1})\end{equation}

where $\hat {w}_{i,t}=w_t+T_{i,t}^E$ for $i=\{1,\,2\}$ is the non-capital income for households as in Iverson and Karp (Reference Iverson and Karp2021). Given the assumption about technology and the energy–labor composite input, it can be shown that factor compensations are then a constant fraction of output, that is,

(16)\begin{align} & w_t L_t + \tau_t^E E_t = (1-\alpha)Y_t \end{align}

(17)\begin{align} & r_t K_t = \alpha Y_t \end{align}

Hence, by substituting prices and transfers in terms of aggregate variables into the Euler equation, assuming that the government distributes carbon tax revenues bearing in mind productivity levels, and given an initial capital stock, $K_0$, one can obtain an expression for the dynamics of capital as follows, which also hinges on the level of idiosyncratic labor income risk in old age, $\Phi$,

(18)\begin{equation} 1=\alpha \beta (1-\tau_{t+1}^K) \left[\frac{(1-\phi_2)(1-\alpha)Y_t-K_{t+1}}{K_{t+1}}\right]\underbrace{\int \frac{1}{\phi_2 \chi_{t+1}(1-\alpha)+\alpha}\,{\rm d}\Psi(\chi_{t+1})}_{\Phi = \Phi(\phi_2,\alpha;\Psi)} \end{equation}

Notice that since the term in the integral is convex in $\chi _{t+1}$, $\Phi$ is increasing in the level of idiosyncratic risk. For vanishingly small spread, then it converges to $\frac {1}{\phi _2 \chi _{t+1}(1-\alpha )+\alpha }$. Finally, along the lines of Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021), for a given tax policy $\tau _{t+1}^K \in (-\infty,\,1]$, we can solve for the saving rate in competitive equilibrium to obtain,

Proposition 1 The saving rate in competitive equilibrium is given by,

(19)\begin{equation} s_t = \frac{1}{1+\frac{1}{\alpha \beta (1-\tau_{t+1}^K)\Phi(\phi_2,\alpha;\Psi)}} \end{equation}

This constant saving rate is unique and independent of climate policies and capital stocks, but dependent of the capital income tax policy and the level of idiosyncratic risk. Specifically, a positive tax on capital reduces the incentives for saving but labor income risk increases it. As one may expect, a higher subjective discount factor, $\beta$, and a larger capital share, $\alpha$, also increase the saving rate. This will be crucial for the setting of fiscal and climate policies as discussed below. Indeed, as shown in Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021), and following the primal approach, the government can then solve directly for the saving rate and then decentralize it by choosing capital income taxes.

2.6 First-best allocations

To begin with, we solve for the first-best allocations under the case of idiosyncratic labor income risk and global warming. We then perform a comparative analysis about the implications of constraints on policy instruments. In the first best, the government would like to: (i) internalize future climate damages by means of a carbon tax, and (ii) provide full insurance to households facing productivity shocks, and in order to do so, it should be able to transfer resources across generations. By solving the social planner's problem, it follows that,

Proposition 2 The unconstrained government finds it optimal to implement a constant saving rate,

(20)\begin{equation} s^{*}=\frac{\alpha \gamma}{(1-\phi_2)(1-\alpha)} \end{equation}

and a carbon price given by,

(21)\begin{equation} \tau_t^E= Y_t \sum_{i=1}^\infty \gamma^i \theta_i \end{equation}

where $\gamma$ is the intergenerational discount factor used for the social planner in its welfare maximization problem.

As expected, given the assumptions about technology and preferences, the saving rate, (20), is constant over time and increasing in the intergenerational discount factor, $\gamma$, the capital share, $\alpha$, and the productivity for the old people, $\phi _2$. As in recent literature related to simple formulae for the social costs of carbon, i.e., as in Golosov et al. (Reference Golosov, Hassler, Krusell and Tsyvinski2014), the optimal carbon price, (21), is proportional to output (and grows at the same rate) and rises with higher future weights and damages.

In the first best, the carbon price thus equals both the market costs of carbon, i.e., the present value of future marginal damages discounted using the market interest rate, and the Pigouvian tax. Since the government would like to avoid intertemporal distortions, by providing full insurance to households through transfers, optimal policies yield a zero capital income tax rate and a carbon price that fully internalizes the future marginal damages of one additional unit of $\text {CO}_2$ emissions today. The key difference here is the use of a distinct discount factor which depends upon government preferences and not on the households’ discount factor. If we assume that those discount factors coincide, we are back into the seminal simple formula for the carbon price derived by Golosov et al. (Reference Golosov, Hassler, Krusell and Tsyvinski2014) using an infinitely-lived representative agent model and a particular climate module, that is,

Corollary 1 If the government uses the same discount factor as the households, $\gamma =\beta,$ then the optimal carbon price resembles the one in Golosov et al. (Reference Golosov, Hassler, Krusell and Tsyvinski2014),

(22)\begin{equation} \tau_t^E= Y_t \sum_{i=1}^\infty \beta^i \theta_i \end{equation}

2.7 Second-best allocations

When the government cannot either implement a carbon tax or transfer resources across generations for risk-sharing the first-best allocation could not be attained. In a second-best scenario, the problem then is to choose optimal policies given the set of policy instruments available to the government. In particular, we assume that the government can implement carbon taxes to internalize climate externalities and rebate such fiscal revenues to household via lump-sum transfers, and tax capital income in order to make lump-sum transfers to members of the same generation as a means of partially insuring people against idiosyncratic labor income risk.

In a second-best world, the government now chooses carbon taxes and capital income taxes such that the resulting allocation maximizes the welfare of current and future generations. From the previous section, we can solve a relaxed problem by choosing the saving rate directly instead of tax rates. By solving the government maximization problem the saving rate, the optimal capital tax, and the optimal carbon tax are given by,

Proposition 3 The constrained government finds it optimal to implement a constant saving rate,

(23)\begin{equation} s^{*}=\frac{\alpha(\beta+\gamma)}{1+\alpha \beta} \end{equation}

a constant capital income tax,

(24)\begin{equation} \tau^K=1-\frac{\beta+\gamma}{(1-\alpha \gamma)\beta \Phi(\phi_2,\alpha; \Psi)} \end{equation}

and a carbon price given by,

(25)\begin{equation} \tau_t^E= Y_t \sum_{i=1}^\infty \hat{\gamma}^i \theta_i, \text{ with } \hat{\gamma} \equiv \frac{(\beta+\gamma)(1-\phi_2)(1-\alpha)}{1+\alpha \beta} \end{equation}

From proposition 3, it turns out that the optimal saving rate, (23), is constant over time, it depends on the capital share as before, and positively on both the subjective and intergenerational discount factors, but it is independent of idiosyncratic risk. The government can decentralize this saving rate using a constant capital income tax rate (24), which is increasing in the level of idiosyncratic risk, $\Phi$ and decreasing concerning the discount factors. The intuition for this expression is as follows. If households face a higher level of idiosyncratic labor income risk in old age, they would like to save more to insure against that risk. Hence, the aggregate capital would be higher than the efficient level. The government can correct this inefficiency by taxing capital income.

Furthermore, the carbon price in the second best, (25), takes a similar form as before, but now it uses a modified discount factor, $\hat {\gamma }$, for the Ramsey planner. It happens because capital income taxes distort an intertemporal margin, i.e., how much to consume in each period. Recall also that climate externalities also distort an intertemporal margin. Hence, the government finds it optimal to implement a lower carbon price to move the economy closer to its efficient level. Importantly, in the second best, the adjusted-discount factor, $\hat {\gamma }$, depends on the capital share, the productivity of the old, and both the subjective and intergenerational discount factors. As in the first best, note that we can get a simple formula for the carbon price because the saving rate is constant and we obtain a closed-form solution for the market interest rate.

3.1 Climate module

We follow Gerlagh and Liski (Reference Gerlagh and Liski2018b) and van den Bijgaart et al. (Reference van den Bijgaart, Gerlagh and Liski2016) closely in modeling climate change and use a multi-box representation for the carbon cycle and temperature adjustments. As shown in Gerlagh et al. (Reference Gerlagh, Jaimes and Motavasseli2017), the response function to current emissions, $\theta _i$, can be written as,

(26)\begin{equation} \theta_{i} = \sum \nolimits_{j} \sum \nolimits_{k} a_j b_k \pi \varepsilon_k \frac{(1-\eta_j)^i-(1-\varepsilon_k)^i}{\varepsilon_k-\eta_j}, \end{equation}

where $\eta _j$ presents the rates for atmospheric depreciation factors, $\varepsilon _k$ denotes the temperature adjustment speeds, $a_j$ sets the shares of carbon emissions entering the reservoirs considered i.e., atmosphere and upper ocean, biomass and deep oceans, $b_k$ designates how temperature responds to changes in carbon stocks, and $\pi =0.0167$ is the climate sensitivity which comes from a linear approximation to the relationship between carbon concentrations, temperature changes, and damages. Below, we determine the paths of this response function for the remainder of this century, by using the procedure explained in Gerlagh et al. (Reference Gerlagh, Jaimes and Motavasseli2017), when the saving rate depends only on a few parameters, i.e., it is constant over time, and the parameter values for the climate system as reported in table 1. Calibrated parameters come from van den Bijgaart et al. (Reference van den Bijgaart, Gerlagh and Liski2016).Footnote ¹³

Table 1. Median parameter values for the climate system

3.2 Optimal carbon and capital taxes

This subsection describes the calibration procedure and presents the numerical exercise using the simple formulae derived above for the capital income tax and the carbon price in both a first-best and a second-best world. Nonetheless, notice that the purpose is not to provide a completed calibration and characterization of policies but to present a quantitative illustration of the possible impacts of the given externalities on climate and fiscal policy in our simplified model.Footnote ¹⁴

As a benchmark, we only look at the situation when the government uses the same discount factor as the households, $\gamma =\beta$, and we set $\beta =0.985$, as in Eggertsson et al. (Reference Eggertsson, Mehrotra and Robbins2019), which is a standard value for OLG models.Footnote ¹⁵ Following Krueger et al. (Reference Krueger, Ludwig and Villalvazo2021), we also assume a value for old age productivity of $\phi _2=0.3117$, a capital share, $\alpha =0.2071$, a total labor supply normalized to 1, full depreciation of capital, a constant elasticity of intertemporal substitution equal to one, and the other calibrated parameters for the mean and variance of idiosyncratic labor income risk as described in the table 2.

Table 2. Calibrated parameters for the benchmark model

Note: The value for $\mu _{\ln \chi }$ is set such that $\mathbb {E}\chi =1$.

Figure 1 shows the path for carbon prices in the first-best and second-best world for the remainder of the century. Two features stand out. On the one hand, carbon taxes grow at 2 per cent per year. That is, at the same growth rate we assume for the economy, because carbon prices are proportional to output. On the other hand, in contrast to Barrage (Reference Barrage2018), even if the government resembles the same level of impatience than the households, the optimal carbon taxes in a second-best world are approximately 7.5 per cent lower than those derived in the first-best scenario. The intuition for this is straightforward. Since the government cannot provide full insurance to households and the capital income tax is not able to replicate completely the first best, the government finds it optimal to adjust downwards the carbon price accordingly.

FIG. 1. Optimal carbon taxes, 2010–2100. Notes: Optimal carbon taxes in a first-best world vs. second-best policy. The value of standard (private) discounting is 0.985.

A closer look at capital and carbon taxes across different scenarios yields interesting insights. Under the baseline case in which $\beta =\gamma$, table 3 reports saving rates for the competitive equilibrium without fiscal and climate policy, the first-best allocation and the constrained efficient allocation or second-best with both capital and carbon taxes.Footnote ¹⁶

Table 3. Optimal climate and fiscal policy, 2020

Notes: Values for $\beta =0.985^{30}$, $\gamma =0.985^{30}$, $\alpha =0.2082$, $\phi _2=0.3117$, and $\Phi (\phi _2,\,\alpha ;\Psi (\chi ))=3.55$ where $\chi \sim LN(-0.125,\,0.25)$. Carbon taxes in €/$\text {tCO}_2$.

In the first case, as one may expect, the presence of idiosyncratic labor income risk creates a motive for precautionary savings and therefore a saving rate of 0.319, a rate that is too high from the perspective of a social planner. In contrast, the optimal saving rate in a world where the government can provide full insurance to households is just 0.243, while at the same time being able to implement an optimal carbon tax (at its Pigouvian level, since there are not intertemporal distortions coming from taxing capital) of 26.27€/$\text {tCO}_2$ at 2020. However, when the government can provide only partial insurance, due to the presence of incomplete markets, it finds it optimal to tax capital at a rate of 35 per cent, yielding a saving rate of $0.234$, and to set a lower carbon tax of 24.33€/$\text {tCO}_2$ relative to the first best.

Despite of being an analytical climate-economy model, this framework is able to provide reasonable values for macroeconomic policies. For example, according to empirical estimates for a set of countries, in her analysis, Barrage (Reference Barrage2020) uses as baseline for the capital tax a value of 33.40 per cent. Similar values for baseline capital taxes, 41.1 per cent for US and 36.8 per cent for Europe, can be found in Trabandt and Uhlig (Reference Trabandt and Uhlig2012) and Trabandt and Uhlig (Reference Trabandt and Uhlig2011). Likewise, the level of the carbon prices (26.27€/$\text {tCO}_2$ is approximately equal to 110 USD/tC, bearing in mind the fact that 1 $\text {tCO}_2$ = 3.67 tC and 1€ is about 1.142 USD) reported in table 3 are in line, for instance, with some of the estimations made by Barrage (Reference Barrage2020) and Nordhaus (Reference Nordhaus2017) using infinitely-lived representative agent models.