2. The model: a laisser-faire economy

We start by considering a world inhabited by selfish individuals interacting in competitive markets without any government intervention. We do not consider population growth and thus assume that population size, L, is constant. Without loss of generality we can normalise L = 1. A constant population implies that the birth rate equals the mortality rate and that fertility is at the replacement level. Let us, for simplicity, assume that the mortality rate, μ, is constant and independent of age. Although this is unrealistic,Footnote ¹ it is a standard assumption in continuous-time overlapping-generations models. More realistic models with age-dependent mortality do exist (e.g., Faruqee, Reference Faruqee2003; Heijdra and Romp, Reference Heijdra and Romp2008), but they are cumbersome to solve and have been used mainly for simple settings like the small open economy case, in which all prices are fixed exogenously. Moreover, the major effect driving the results of this paper, the so-called turnover effect, would be prevalent in more complex models as well. Regarding notation, we employ the convention that lowercase letters represent individual variables whereas the corresponding uppercase variables are the corresponding aggregates.

2.1 Households

An individual born at time v < t (v = vintage), expecting to die with probability μ, maximises the present value of his/her expected future welfare,

\[ \int_v^\infty \left( {\ln c\,(v,t) - \frac{{P\,{{(t)}^{1 + \varphi }}}}{{1 + \varphi }}} \right){e^{ - (\rho + \mu )\,(t - v)}} dt,\]

where ρ is the rate of time preference, $c\,(v,t)$ is the individual's consumption, $P\,(t)$ is the level of pollution and $\varphi \gt 0$ is a preference parameter determining the curvature of the environmental-damage function, which is increasing and strictly convex. Pollution is an externality and is taken as given by the individual. It will, however, play a role later in the paper, when optimal solutions are investigated. The individual owns capital – which includes insurance à la Yaari (Reference Yaari1965) and depreciates at rate δ – and is endowed with one unit of labour and self-employed in his/her own firm. This simplifying assumption, which does not affect the results, allows us to neglect the labour market. The individual owns capital $k(v,t)$, earns profits $\pi \,(t)$, a return on investment $r\,(t)\,k\,(v,t)$, a return on Yaari bonds $\mu \,(t)\,k\,(v,t)$, with the capital owned by dying individuals being redistributed to the surviving ones without transaction costs. Thus, the individual's wealth accumulates according to

\[ \dot{k}\,(v,t) = \pi \,(t) + r\,(t)\,k\,(v,t)-c\,(v,t)-\delta k\,(v,t) + \mu k\,(v,t).\]

Moreover, the initial level of wealth at birth is $k\,(v,v) = 0$. Inheritance motives are absent. Utility maximisation over the infinite horizon subject to the wealth constraint gives the usual Euler equation of individual consumption growth,

\[ \frac{{\dot{c}\,(v,t)}}{{c\,(v,t)}} = r\,(t) - \delta - \rho,\]

where the impact of mortality is cancelled out due to the existence of Yaari bonds (see appendix B for the derivation of the result). Thus, a change in longevity has no impact whatsoever on the individual's consumption and saving behaviour, although it affects the discounting of future utility.

2.2 Fossil fuels and global warming

Fossil fuels are extracted at a constant marginal extraction cost z measured in terms of GDP. Exhaustibility is neglected, which can be justified by the fact that, if coal deposits are taken into account, the supply of fossil fuels is abundant and the limits to using them are on the consumption side: climate change will be disastrous long before the resource base is exhausted. Units of measurement are chosen such that one unit of fossil fuels generates one unit of greenhouse gas emissions, E(t). Greenhouse gases are stock pollutants accumulating over time with a constant decay rate θ. Assuming an initial pollution level of $P\,(0) = P_0$, the change in the stock is

(1)\begin{equation} \dot{P}\,(t) = E\,(t)-\theta P\,(t). \end{equation}

2.3 Firms

Individuals own firms, in which they work as self-employed entrepreneurs. Firms are identical and each owner rents capital, $k\,(t)$, on the capital market and pays interest, $r\,(t)$. Note that $k\,(t)$ is the capital stock of a representative firm whereas $k\,(v,t)$ is the capital stock owned by an individual born on date versus From the fact that population size is unity, it follows that $k\,(t) = \int_{-\infty} ^t k\,(v,t)\,dv$. Firms maximise their profits,

\[ \pi \,(t) = f(K\,(t),k\,(t),e\,(t))-r\,(t)\,k\,(t)-ze\,(t),\]

where $f\,(.,.,.)$ is a constant returns to scale (CRS) production function and its first argument, $K\,(t) = L\,(t)\,k\,(t) = k\,(t)$, is the economy-wide capital stock, generating a Romer-type externality (Romer, Reference Romer1986). The capital market is in equilibrium, i.e., $K\,(t) = \int_{-\infty} ^t k\,(v,t)\,dv. e\,(t)$ denotes a representative firm's use of fossil fuels, with one unit of fossil fuels generating one unit of emissions, and $z$ is the market price of fossil fuels, equaling the constant marginal extraction cost of the resource. Firms pay $z$, but they do not internalise the social cost of pollution. Profit maximisation yields $f_k = r\,(t)$ and $f_e = z$. With CRS, marginal productivities are homogenous of degree zero and the factor-price frontier is constant, i.e., $r\,(t)$ is a function of z. Moreover, the ratio of $k\,(t)$ and $e\,(t)$ is fixed: $e\,(t) = bk\,(t)$.

2.4 Aggregation

Let uppercase letters denote the aggregate levels of the variables denoted by corresponding lowercase variables on the individual level. Since the mass of firms is 1, total emissions equal those of the individual firm, $E\,(t) = e\,(t)$. Regarding consumption, we have to take into account that dying individuals have other consumption levels than the newborn individuals replacing them. As population is constant, each dying individual is replaced by exactly one newborn and the rate of replacement is μ. Newborns have no wealth and finance their consumption, $c\,(t,t)$, from current income only, whereas the average individual in the economy has accumulated some wealth and therefore has a higher consumption level, $C\,(t)$. This implies that

\[ \frac{{\dot{C}\,(t)}}{{C\,(t)}} = r - \delta - \rho - \mu \frac{{C\,(t) - c\,(t,t)}}{{C\,(t)}}.\]

Since in models with logarithmic utility functions, the share of consumption derived from wealth is proportional to the wealth, the proportionality factor being the individual discount rate, $\rho + \mu$, we have

(2)\begin{equation} \frac{{\dot{C}\,(t)}}{{C\,(t)}} = r - \delta - \rho - \mu \,(\rho + \mu )\frac{{K\,(t)}}{{C\,(t)}}. \end{equation}

The last term on the right-hand side is called the turnover effect in the literature (Heijdra, Reference Heijdra2017, chapter 16). This suggests that the higher the mortality rate, the larger the turnover effect and the smaller the consumption growth rate. However, this is not totally clear since a change in μ does not leave $c\,(t)$ and $K\,(t)$ unaffected. Aggregation of the capital stocks of individual firms yields

(3)\begin{equation} \dot{K}\,(t) = f\,(K(t),K\,(t),\; E\,(t))-C\,(t)-\delta K\,(t)-zE\,(t). \end{equation}

Note that there is no term $\mu K\,(t)$ in this equation since Yaari bonds are pure transfers from the dying to the survivors. Using the proportionality of $E\,(t)$ and $K\,(t)$, we can rewrite (3) as:

(3')\begin{equation} \dot{K}\,(t) = AK\,(t)-C\,(t)-\delta K\,(t) + zbK\,(t). \end{equation}

Finally, the stock of greenhouse gases evolves according to

(1')\begin{equation} \dot{P}\,(t) = bK\,(t)-\theta P\,(t). \end{equation}

In a long-run steady state, consumption and capital grow at the same rate and are proportional to each other. Let this rate be g and let $q = C\,(t)/K\,(t)$ be the consumption-capital ratio. Then,

(4)\begin{equation} g = r - \delta - \rho - \frac{{\mu \,(\rho + \mu )}}{q}, \end{equation}

and

(5)\begin{equation} g = A-\delta -zb-q.{\rm \;} \end{equation}

Total differentiation of (4) and (5) yields

\[ \frac{{dg}}{{d\mu }} = \frac{{ - q\left( {\rho + 2\mu } \right)}}{{{q^2} + \mu \left( {\rho + \mu } \right)}} = - \frac{{dq}}{{d\mu }},\]

i.e., the economic growth rate is negatively affected by an increase in general mortality and the consumption-to-capital ratio is positively affected. See Prettner (Reference Prettner2013) for the same result in a different modelling context.

Since long-run growth of the stock of pollutants is affected by the long-run growth of emissions, which grow at the same rate as the capital stock, we have the following proposition.

Proposition 1. An increase in longevity leads to a faster growth of greenhouse gas emissions and to a faster growth of the stock of greenhouse gases in the atmosphere.

The underlying reason is the turnover effect. If a smaller number of dying wealthy people are replaced by poorer newborns in every period, the growth of aggregate consumption is larger. And if the economy is growing faster, this has a negative effect on the global environment. Note that, in addition, the population will grow during the transition from high to low mortality if fertility is at the replacement level as it has been assumed here. This will of course exert additional pressure on the environment.

3. Environmental policy: an exogenously given target

The Paris Accord has the goal of keeping human-generated global warming well below 2°C compared to pre-industrial times and strives for a target of 1.5°C. Translated into the terminology of our model, this implies restriction of the pollution stock such that $P\,(t) \le \bar{P}$. It follows that, in the long run, $E\,(t) \le \theta \bar{P}$.Footnote ² Since emissions would grow to infinity under laisser faire, this is a binding constraint. Therefore $E\,(t)$ is replaced by $\theta \bar{P}$ in the production function. It follows that $\; f_e \gt z$. The difference between marginal productivity and the fuel price, $\; f_e-z$, is the permit price in a cap-and-trade system with cap $\theta \bar{P}$ or the carbon tax implemented by the government to keep emissions at $\theta \bar{P}$. With a fixed energy input, the production function exhibits decreasing returns to scale and capital accumulation and consumption growth will go to zero. From (2) and (3), we have the long-term steady-state conditions:

\[ \dot{C} = 0:\quad {f_k}\,(K,\,K,\,\theta \bar{P}) = \delta + \rho + \mu \,(\rho + \mu )\frac{K}{C}\]

and

\[ \dot{K} = 0:\quad C = f\,(K,\,K,\,\theta \bar{P})-\delta K-z\theta \bar{P}.\]

These two conditions are isoclines in a (K, C) phase diagram and standard procedures show that the optimum solution is an increasing saddle path approaching the unique intersection point of the two curves (Blanchard, Reference Blanchard1985: 232; Heijdra, Reference Heijdra2017: 569), the only difference being that in our case the production function contains an externality, which, however, does not affect the qualitative result. Figure 1 depicts the isoclines and the equilibrium.

Figure 1. Increased longevity and economic growth when emissions are constant.

The $(\dot{C} = 0)$ line approaches the vertical golden-rule line, $f_k = \delta + \rho$, asymptotically for $C\to \infty$. It can be seen that the golden rule does not hold in the equilibrium. This is a standard result of the Blanchard-Yaari model and it is due to the turnover effect. The turnover term affects aggregate consumption growth negatively, and therefore less capital is accumulated in the long run. The dashed line shows location of $\dot{C} = 0$ for a decrease in mortality and it can be seen that with lower mortality, long-run capital and long-run consumption are larger. Since an increase in capital implies that the marginal productivity of fossil fuels is increased (due to $f_{ek} \gt 0$ and $f_{eK} \gt 0\;$), it follows that the carbon tax or emission permit price, $\; f_e \gt z$, will also rise.

A tightening of the environmental standard, i.e., a reduction in $\bar{P}$, would shift the $(\dot{K} = 0)$ line downwards and would lead to less capital accumulation and lower consumption in the long run.

4. Optimal policies

Let us assume that there is a social planner discounting future welfare at the same rate as the representative individual.Footnote ³ The objective is to maximise

\[ \int_{ - \infty }^\infty \left( {\int_v^\infty \left( {\textrm{ln}\,c\,(v,t) - \frac{{P\,{{(t)}^{1 + \varphi }}}}{{1 + \varphi }}} \right){e^{ - (\rho + \mu )\,(t - v)}}dt} \right){e^{ - \rho v}}dv,\]

subject to

\[ \dot{K}\,(t) = f\,(K(t),\,K\,(t),\; \,E\,(t))-\int_{-\infty} ^t c\,(v,t)\,e^{-\mu \,(t-v)}dv-\delta K\,(t)-zE\,(t)\]

and

\[ \dot{P}\,(t) = E\,(t)-\theta P\,(t).\;\]

Calvo and Obstfeld (Reference Calvo and Obstfeld1988) have shown that $\partial c\,(v,t)/dv = 0$ in the optimum, i.e., a transfer scheme has to be implemented such that all individuals living at time t enjoy the same level of consumption irrespective of their age. This is very intuitive since otherwise marginal utilities would differ across individuals and huge welfare gains could be made by transferring income from persons with low marginal utilities to persons with high marginal utilities. The potential of welfare gain is exhausted if all marginal utilities are equal, i.e., if all individuals consume at the same level. Moreover, Calvo and Obstfeld (Reference Calvo and Obstfeld1988: 416) have shown that the social planner's problem can be regarded as a two-stage optimisation problem, the first stage being the allocation of consumption across cohorts at each moment of time and the second one being the optimal intertemporal choice of the aggregate variables. Thus, the second-stage problem is to maximise

\[ \int_v^\infty \left( {\ln C\,(t) - \frac{{P\,{{(t)}^{1 + \varphi }}}}{{1 + \varphi }}} \right){e^{ - \rho t}}dt,\]

with respect to the stock pollution constraint, (1), and aggregate capital accumulation, (4). Since neither the objective function nor the constraints contain μ, it follows that the optimal policy is independent of mortality. The Euler equation now is

\[ \frac{{\dot{C}\,(t)}}{{C\,(t)}} = {f_k} + {f_K} - \delta - \rho,\]

and does not contain a turnover effect. The additional term $f_K$ indicates that the planner takes the knowledge spillover into account and that this raises the growth rate (Acemoglu, Reference Acemoglu2009, chapter 11.4).

Intergenerational transfers that shift huge incomes from older cohorts – which have been economically active as workers and entrepreneurs for many years – to young ones without similar merits, may be difficult to implement as they are likely to be regarded as unjust. So let us look at a planner without access to such intergenerational transfers. Appendix B derives the optimality conditions and the corresponding dynamics of the economy are characterised by

(1)\begin{align} \dot{P} &= E-\theta P. \end{align}

(6)\begin{align} \dot{K} &= F\,(K,\, E)-C-\delta K-zE, \end{align}

(7)\begin{align} \frac{{\dot{C}}}{C} &= {F_K}\,(K,E) - \delta - \rho - \mu \,(\rho + \mu )\frac{K}{C}, \end{align}

(8)\begin{align} \dot{F}_E &= (F_E-z)\,(F_K-\delta + \mu + \theta ) + CP^\varphi, \end{align}

where the time argument has been omitted for the sake of brevity and where $F\,(K,E)\equiv f\,(K,K,E)$,$\; F_E\,(K,E) = f_e\,(K,K,E)$ and $F_K\,(K,E) = f_k\,(K,K,E) + f_K\,(K,K,E)$. From (8) we have that the optimal environmental policy in the long run must satisfy

(9)\begin{equation} {F_E}\,(K,E) - z = \frac{{{P^\varphi }C}}{{{F_K}\,(K,E) - \delta + \mu + \theta }}. \end{equation}

The left-hand side of (9) is the marginal productivity of fossil fuels minus the marginal cost of providing them. This not only measures the marginal benefit to society from using additional fossil fuels, it also measures the cost arising from foregoing the use of a marginal unit, i.e., the marginal abatement cost. In the optimum, the marginal abatement cost must equal the present value of marginal environmental damage. The marginal environmental damage at time t is the marginal rate of substitution between pollution and consumption, $P^\varphi C$. Future damages are discounted at the gross rate of interest, $F_K\,(K,E)-\delta + \mu$, including the return on Yaari bonds, plus the natural rate of decay of pollution, $\theta$. Given that $\mu$ is part of the discount rate, it follows that increased longevity (a lower $\mu$) raises the willingness to pay for avoiding climate change. However, there is also an indirect effect as a change in longevity affects aggregate savings behaviour and this affects the long-run interest rate.

Before this indirect effect is determined, note that (9) is incompatible with balanced growth, i.e., with a scenario in which all variables grow at the same positive rate. In this case, the left-hand side would be constant (since $F_E\,(K,E)$ is homogenous of degree zero under constant returns to scale), whereas the right-hand side would go to infinity. Thus, condition (9) can only be satisfied as a long-run optimality condition if the growth rate goes to zero and the economy approaches a steady state in the far future. The underlying reason is simple: the marginal benefits from growth are decreasing since the marginal utility of consumption declines, whereas the marginal cost of growth is increasing due to increasing marginal environmental damage.

By setting $\dot{P} = \dot{K} = \dot{C} = 0$ in (1), (6) and (7) and using (9), the steady state is determined. Appendix C shows that the comparative statics with respect to μ are ambiguous.

One of the reasons for the ambiguities is that the effects of changes in emissions and capital on condition (9) are unclear. An increase in capital raises the marginal abatement cost due to $F_{KE} \gt 0$, but it also reduces the rate at which future environmental damages are discounted since $F_{KK} \lt 0$ and thus raises the present value of environmental damages as well. A similar argument holds for increases in emissions. In appendix C it is shown that the direct effect has the expected sign if the determinant of the Jacobian matrix is positive. The indirect effect is ambiguous even in this case.