## Abstract

Ageing increases the income of a carbon tax *ceteris paribus* since energy consumption rises with age, as macro and micro data show. Ageing also increases some public expenditures, notably those of pay-as-you-go (PAYG) pension systems. Accordingly, there may be a case for recycling a carbon tax in an ageing context so as to finance ageing-related public expenditures. This article studies the interacting effects on intergenerational equity and growth of such a recycling. It relies on a general equilibrium model with overlapping generations parameterised with empirical data. Several results emerge. Implementing a carbon tax fully recycled through higher lump-sum pensions weighs relatively more on the intertemporal welfare of young and future generations. A carbon tax fully recycled through lower social contributions financing the PAYG bolsters the wellbeing of young and future generations but weighs on the welfare of baby-boomers and older cohorts. The redistributive effects of recycling a carbon tax can depend significantly on the way used to balance the PAYG regime.

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## Notes

- 1.
The model used here does not account explicitly for effects stemming from the external side of the economy. First, the main question that is addressed here focuses on the interrelated effects of recycling the carbon tax and balancing the PAYG regime. Accounting for external linkages would not modify substantially the answer to this question. It may smooth the dynamics of the variables but only to a limited extent. Home bias (the “Feldstein-Horioka puzzle”), exchange rate risks, financial systemic risk and the fact that many countries in the world are also ageing and thus competing for the same limited pool of capital all suggest that the possible overestimation of the impact of ageing on capital markets due to the closed economy assumption is small.

One implication of this closed-economy assumption is that the price of production of fossil fuels on world markets is considered as exogenous in the model.

- 2.
i.e. natural gas for households, natural gas for industry, automotive diesel fuel, light fuel oil, premium unleaded 95 RON, steam coal and coking coal.

- 3.
i.e. coal, natural gas, oil nuclear, hydroelectricity, onshore wind, offshore wind, solar photovolta ïc and biomass.

- 4.
The demand for these renewables substitutes is approximated, over the recent past, by demands for biomass, biofuels, biogas and waste.

- 5.
This simplification relies on the implicit assumption that the stock of biomass is sufficient to meet the demand at any time, without tensions that could end up in temporarily rising prices. It also avoids using unreliable (when not unavailable) time series for prices of renewables energies over past periods and in the future.

- 6.
The exact number of cohorts living at a given year depends on the year and each cohort’s life expectancy.

- 7.
This benchmark corresponds roughly to an actuarially fair penalty rate.

- 8.
This specification encapsulating the

*w*_{ t }’s ensures that the amount of non-ageing related public expenditures follows the same temporal trend as GDP, a trend which is related in the long run to annual TFP gains. Accordingly, non-ageing related public expenditures remain broadly constant as a fraction of GDP,*ceteris paribus*.The existence of such a public regime of redistribution with proportional taxes financing lump-sum expenditures involves some intergenerational redistribution among living cohorts. Indeed, the absolute amount of taxes paid is influenced by age (since

*τ*_{ t,N A }is a proportional rate that applies to a level of income which is linked to the number of units of efficient labour provided by households, which is related with age), while the absolute level of the lump-sum expenditure*d*_{ t,N A }, by definition, is not related with age*a*nor with the level of income of a household. - 9.
When the carbon tax is recycled through a rise in the level of pensions, this effect is assumed to be lump-sum. It could have been possible to assume that the recycling could trigger a rise in the replacement rate of the new retirees at any given year. However, this would have concentrated the favourable impact of the recycling of the carbon tax on a very small number of individuals, i.e. the annual cohort of new retirees. This would have entailed a rather volatile replacement rate over time.

- 10.
Before the announcement of a reform package in 2010, annual current welfare of one cohort is by assumption equal in all scenarios. Graphically, this involves a flat portion in the Lexis surface, at value 0. From 2010 onwards, the deformations of the Lexis surfaces mirror the influence of mechanisms of intergenerational redistribution as measured by their influence on current welfare.

- 11.
Since the net impact of recycling a carbon tax through higher future pensions is more negative for most cohorts when the PAYG is balanced through lower future pensions, it follows that the accumulation of capital in scenario F as compared to scenario D is lower than the accumulation of capital in scenario C as compared to scenario A. This is also observed in the model: Figure 2 shows that the interest rate in 2030 in scenario F is practically the same as the interest rate in scenario D at the same year; while the interest rate in 2030 in scenario C is significantly lower than the interest rate in scenario A in 2030, indicating a relatively higher effect of capital per unit of efficient labour.

- 12.
NEA [20] computes the supplementary network cost (in €/MWh) of a given rise in the penetration rate of intermittent sources of electricity.

- 13.
A proxy for the share of the inactive population that never receives a contributory pension is found in the ratio of inactive people aged 40–44 to inactive people aged 65–69 (in 2000). Distinguishing between pensioners and inactive people who never receive any pension is not only realistic but also important to get reasonable levels for the contribution rate balancing the PAYG regime.

- 14.
Introducing this parameter stabilises the ratio of the contributions of consumption and leisure to utility when technical progress is strictly positive. The Euler equation (infra) suggests that the annual growth rate of consumption is equal, at the steady-state, to the difference between the interest rate and the discount rate, which in turn is equal to annual TFP growth. See Broer et al. 1994.

- 15.
Remember that each cohort is a group of individuals born the same year, and is represented in the model by a representative individual whose economic life begins at 20 (

*a*=0) and ends up with certainty at Ψ_{ t,0}years (thus*a*=Ψ_{ t,0}−20), where Ψ_{ t,0}is the average life expectancy at birth for cohort born in*t*. - 16.
This value is obtained through the input-output matrix in national accounts. In the CES nest,

*C*_{ t }refers to GDP (i.e. added value) in volume, whereas*Y*_{ t }refers to aggregate production in volume, and thus takes account of inter mediate consumption (here,*B*_{ t }). Accordingly, the weighting parameter (*a*) should not be computed as the share of the value added of the energy sector in GDP but, preferably, as the share of intermediate consumption in energy items as a fraction of GDP.

## References

- 1.
Auerbach, A., & Kotlikoff, L. (1987).

*Dynamic fiscal policy*: Cambridge University Press. - 2.
Bovenberg, A.L. (1999). Green tax reforms and the double dividend: an updates reader’s guide.

*International Tax and Public Financ*,*6*, 421–443. - 3.
Bovenberg, A.L., & Goulder, L. (1996). Optimal environmental taxation in the presence of other taxes: general-equilibrium analyses.

*American Economic Review*,*86*(4), 985–1000. - 4.
Bovenberg, A.L., & Goulder, L.H. (2002). Environmental taxation In A. Auerbach, & M. Feldstein (Eds.),

*Handbook of Public Economics, North Holland, pp. 1471–1547*. - 5.
Böhringer, C., & Rutherford, T. (1997). Carbon taxes with exemptions in an open economy: a general equilibrium analysis of the German tax initiative.

*Journal of Environmental Economics and Management*,*32*, 189–203. - 6.
Broer, D.P., Westerhout, E., & Bovenberg, L. (1994). Taxation, pensions and saving in a small open economy.

*Scandinavian journal of economics*,*96*(3). - 7.
Brounen, D., Kok, N., & Quigley, J.M. (2012). Residential energy use and conservation: economics and demographics.

*European Economic Review*,*56*, 931–945. - 8.
Carbone, J.C., Morgenstern, R.D., Williams, R., & Burtraw, D. (2013). Deficit reduction and carbon taxes: budgetary, economic, and distributional impacts.

*Resources for the Future*. - 9.
Chiroleu-Assouline, M., & Fodha, M. (2006). Double-dividend hypothesis, golden rule and welfare distribution.

*Journal of Environmental Economics and Management*,*51*, 323–335. - 10.
Epstein, L.G., & Zin, S.E. (1991). Substitution, risk aversion and the temporal behavior of consumption and asset returns: an empirical analysis.

*Journal of Political Economy*,*99*. - 11.
Goulder, L. (1994). Environmental taxation and the double dividend: a reader’s guide.

*International Tax and Public Finance, August 1995*,*2*(2), 157–83. - 12.
Goulder, L. (1995). Effects of carbon taxes in an economy with prior tax distortions: an intertemporal general equilibrium analysis.

*Journal of Environmental Economics and Management, Part 1 November 1995*,*29*(3), 271–97. - 13.
Gourinchard, P.-O., & Parker, J. (2002). Consumption over the life-cycle.

*Econometrica*,*70*(1). - 14.
Hogan, W., & Manne, A. (1977). Energy economy interactions: the Fable of the Elephant and the Rabbit? In Hitch, C.J. (Ed.),

*Modeling energy-economy interactions: five approaches, Resources for the Future, Washington*. - 15.
Knopf, B., Edenhofer, O., Flachsland, C., Kok, M.T.J., Lotze-Campen, H., Luderer, G., Popp, A., & van Vuuren, D.P. (2010). Managing the low-carbon transition—from model results to policies.

*The Energy Journal*,*31*(1). - 16.
Kotlikoff, L., & Spivak, A. (1981). The family as an incomplete annuities market.

*Journal of political economy*,*89*(2). - 17.
Leimbach, M., Bauer, N., Baumstark, L., Lüken, M., & Edenhofer, O. (2010). Technological change and international trade–insights from REMIND-R.

*The Energy Journal*,*31*(1). - 18.
John, A., Pecchenino, R., Schimmelpfennig, D., & Schreft, S. (1995). Short-lived agents and the long-lived environment.

*Journal of Public Economics*,*58*, 127–141. - 19.
Miles, D. (1999). Modelling the impact of demographic change upon the economy.

*The Economic Journal*,*109*(452). - 20.
Nuclear Energy Agency (2012). Nuclear energy and renewables—system effects in low-carbon electricity systems, NEA/OECD.

- 21.
Normandin, M., & St-Amour, P. (1998). Substitution, risk aversion, taste shocks and equity premia.

*Journal of Applied Econometrics*,*13*. - 22.
OECD (2005). The impact of ageing on demand, factor markets and growth.

*OECD Economics Department Working Paper*,*420*. - 23.
Rasmussen, T. (2003). Modeling the economics of greenhouse gas abatement: an overlapping generations perspective.

*Review of Economic Dynamics*,*6*, 99–119. - 24.
Rausch, S. (2013). Fiscal consolidation and climate policy: an overlapping generations perspective.

*Energy Economics*,*40*, S134–S148. - 25.
Solow, R. (1974). Intergenerational equity and exhaustible resources.

*Review of Economic Studies, Special Issue*,*41*(128). - 26.
Tonn, B., & Eisenberg, J. (2007). The ageing US population and residential energy demand.

*Energy Policy*,*35*, 743–745. - 27.
Uzawa, H. (1961). Neutral inventions and the stability of growth equilibrium.

*Review of economic studies*,*28*(2).

## Author information

### Affiliations

### Corresponding author

## Electronic supplementary material

Below is the link to the electronic supplementary material.

## Appendix A: Description of the GE-OLG Model

### Appendix A: Description of the GE-OLG Model

The development of applied GE-OLG models, using empirical data, owes much to [1]. A more detailed, analytical presentation of our model is available upon request.

### A.1 The Energy Sector

The *end-use prices of natural gas, oil products and coal* (*q*
_{
i,t
} , *i*∈{1;2;3}) are computed as weighted averages of prices of different sub-categories of energy products: \(\forall i\in \left \{ 1;2;3\right \} , q_{i,t}=\sum \limits _{j=1}^{n}a_{i,j,t}q_{i,j,t}\). *q*
_{
i,j,t
} stands for the real price of the product *j* of energy *i* at year *t*. For natural gas (*i*=1), two sub-categories *j* are modeled: the end-use price of natural gas for households (*j*=1) and the end-use price of natural gas for industry (*j*=2). For oil products (*i*=2), three sub-categories *j* are modeled: the end-use price of automotive diesel fuel (*j*=1), the end-use price of light fuel oil (*j*=2) and the end-use price of premium unleaded 95 RON (*j*=3). For coal (*i*=3), two sub-categories *j* are modeled: the end-use price of steam coal (*j*=1) and the end-use price of coking coal (*j*=2). The *a*
_{
i,j,t
}’s weighting coefficients are computed using observable data of demand for past periods. For future periods, they are frozen to their level in the latest published data available: whereas the model takes account of interfuel substitution effects (cf. *infra*), it does not model possible substitution effects between sub-categories of energy products (for which data about elasticities are not easily available). The end-use prices of sub-categories of natural gas, oil or coal products (*q*
_{
i,j,t
}) are in turn computed by summing a real supply price with transport/distribution/refining costs and taxes, including a carbon tax computed by applying a tax rate to the carbon contained in one unit of volume of product *j* of energy *i*.

For future periods, it is assumed that the economy modeled here is too small to influence the prices of fossil fuels on world energy markets. Thus, the production price of fossil fuels on world markets is exogenous in the model for oil, natural gas and coal.

The *real end-use price of electricity* is computed as a weighted average of prices of electricity for households and industry \( (i=4); q_{4,t}=\sum \limits _{j=1}^{2}a_{4,j,t}q_{4,j,t}\). *q*
_{4,j,t
} stands for the end-use real price, at year *t* , of the product *j* of electricity. Two sub-categories *j* are modeled: the end-use price of electricity for households (*j*=1) and the end-use price of electricity for industry (*j*=2). The *a*
_{4,j,t
}’s weighting coefficients are computed using observable data of demand for past periods, and frozen to their level in the latest published data available for future periods. Real end-use prices of electricity are equal to ∀*j*,*q*
_{4,j,t
}=*q*
_{4,t,s
}+*q*
_{4,j,t,c
}+*q*
_{4,j,t,τ
} , i.e. they are the sum of an endogenously generated (structural) wholesale market price of production of electricity (*q*
_{4,t,s
}), of network costs of transport and distribution (*q*
_{4,j,t,c
}) and different taxes (VAT, excise, tax financing feed-in tariffs for renewables, price of carbon emission quotas...)(*q*
_{4,j,t,τ
}). The wholesale market price of production of electricity (*q*
_{4,t,s
}) is computed from an endogenous average peak price of electricity and a peak/offpeak spread. The peak market price of production of electricity derives from costs of production of electricity among different technologies, weighted by the rates of marginality in the electric system of each production technology. The costs of producing electricity (*ϱ*
_{
e
l,x,t,p
r
o
d
}) are computed for 9 different technologies *x*: coal (*x*=1), natural gas (*x*=2), oil (*x*=3), nuclear (*x*=4), hydroelectricity (*x*=5), onshore wind (*x*=6), offshore wind (*x*=7), solar photovoltaïc (*x*=8) and biomass (*x*=9). Each is computed as the sum of variable costs (i.e. fuel costs and operational costs) and fixed (i.e. investment) costs of production.

The model takes account of the downward effect on the market price associated with the development of intermittent producers of electricity (onshore wind, offshore wind and solar PV). The cost of transport and/or distribution of electricity (*q*
_{4,j,t,c
}) takes account of the effects of the development of renewables on network costs using the orders of magnitude computed by [20].^{Footnote 12} The amount, in real terms, of taxes paid by an end-user of electricity (*q*
_{4,j,t,τ
}) at year *t* takes account of the effect of the development of renewables on the rate of the tax financing feed-in tariffs for producers of electricity out of renewables energies.

“Renewables substitutes” in the model are defined as a set of sources of renewable energy whose price of production (*q*
_{5,t
}) is not influenced in the long-run by an upward Hotelling-type trend; nor by a strongly downward learning-by-doing related trend; and which, eventually, does not contain (much) carbon and/or is not affected by any carbon tax. The demand for these renewables substitutes is approximated, over the recent past, by demands for biomass, biofuels, biogas and waste. Given this definition, the real price of renewables substitutes is set at 1 and remains constant through time. In other words, it is assumed that the price of renewable substitutes (excluding wind and PV in the electric sector) rises in the long run as inflation. Since inflation is zero in this model where all prices are expressed in real terms, then ∀*t*,*q*
_{5,t
}=1.

*Energy demand in volume* over the past is broken up into demand for coal (*D*
_{
c
o
a
l,t
}), demand for oil (*D*
_{
o
i
l,t
}), demand for natural gas (*D*
_{
n
a
t
g
a
s,t
}), demand for electricity (*D*
_{
e
l,t
}) and demand for renewable substitutes (*D*
_{
r
e
n
e
w,t
} , which covers, over the recent past, demand and supply for biomass, biofuels, biogas and waste). Data stem from OECD/IEA databases. In this model, they are used mainly to compute the average weighted real energy price for end-users (*q*
_{
e
n
e
r
g
y,t
}) in the past, following the above mentioned formula \(q_{energy,t}=\sum \limits _{i=1}^{5}D_{i,t-1}q_{i,t}\). As concerns the structure of the energy demand in the future, the modeling framework used here follows the literature (see for instance [17]) which usually computes future energy mix using a nest of interrelated CES functions. This nest allows for the relative importance in the future of each component of the energy mix—i.e. *D*
_{
c
o
a
l,t
}, *D*
_{
o
i
l,t
}, *D*
_{
n
a
t
g
a
s,t
}, *D*
_{
e
l
e
c,t
} and *D*
_{
r
e
n
e
w,t
}—to vary over time according to changes in their relative prices (i.e. *q*
_{1,t
}, *q*
_{2,t
}, *q*
_{3,t
}, *q*
_{4,t
} and *q*
_{5,t
}) and according to exogenous decisions of public policy.

### A.2 Demographics

The model embodies around 60 cohorts each year (depending on the average life expectancy), thus capturing in a detailed way changes in the population structure. Each cohort is characterised by its age at year *t*, has *N*
_{
t,a
} members and is represented by one average individual. The average individual’s economic life begins at 20 (*a*=0) and ends with certain death at Ψ_{
t,0} (*a*=Ψ_{
t,0}−20), where Ψ_{
t,0} stands for the average life expectancy at birth of a cohort born in year *t*. In each cohort, a proportion *ν*
_{
t,a
} of individuals are working while *μ*
_{
t,a
} are unemployed and receive no income. The inactive population is divided into two components. A first component corresponds to individuals who never receive any contributory pension during their lifetime.^{Footnote 13} The proportion *π*
_{
t,a
} of pensioners in a cohort is then computed as a residual. Future paths for the labour force and the working population over the simulation period are in line with a rise in the average *effective* age of retirement of 1.25 year per decade from 2010 on, following a reform of the PAYG pension regime implemented by the goverment from 2010 on. Accordingly, future age-specific participation and employment rates of workers above 50 years of age increase in line with the changes in the age of retirement.

### A.3 The Private Agents’ Maximizing Behaviour

The household sector is modelled by a standard, separable, time-additive, constant relative-risk aversion (CRRA) utility function and an inter-temporal budget constraint. This utility function has two arguments, consumption and leisure. The labour supply of the representative individual of a whole cohort (*ℓ*
_{
t,a
}∈[0;1]) is \(1-\ell _{t,a}=\nu _{t,a}(1-\ell _{t,a}^{\ast } )+(1-\nu _{t,a})=1-\nu _{t,a}\ell _{t,a}^{\ast } \leq 1\) where *ν*
_{
t,a
} is the fraction of working individuals in a cohort aged *a* in year *t* and \(\ell _{t,a}^{\ast } \) is the optimal fraction of time devoted to work by the working sub-cohort. The objective function over the lifetime of the average *working* individual of a cohort of age *a* born in year *t* is:

where \(c_{t+j,j}^{\ast } \) is the consumption level of the average individual of the working sub-cohort of age *j* in year *t*, *ρ* is the subjective rate of time preference, *σ* is the relative-risk aversion coefficient, \(V_{t,j}=\left ((c_{t+j,j}^{\ast } )^{1-1/\xi } +\eta \left (H_{j}\left (1-\ell _{t+j,j}^{\ast } \right ) \right )^{1-1/\xi } \right )^{\frac {1}{1-1/\xi }}\) is the CES instantaneous utility function at year *t*, ϰ is the preference for leisure relative to consumption, 1/*ξ* the elasticity of substitution between consumption and leisure in the instantaneous utility function, and *H*
_{
j
} a parameter whose value depends on the age of an individual and whose annual growth rate is equal to the annual TFP growth rate (with *H*
_{0}=1).^{Footnote 14} The intertemporal budget constraint for the working sub-cohort of age 20 (i.e. a=0) in year *t* is:

Parameter *ω*
_{
t+j,j
} is the after-tax income of a working individual per hour worked such that *ω*
_{
t+j,j
}=*w*
_{
t
}
*ε*
_{
a
}(1−*τ*
_{
t,P
}−*τ*
_{
t,N
A
})+*d*
_{
t,N
A
}−*d*
_{
t,e
n
e
r
g
y
}. *w*
_{
t
} stands for the gross wage per efficient unit of labour. The parameter *ε*
_{
a
} links the age of a cohort to its productivity. Following [19], a quadratic function is used: \(\varepsilon _{a}(a)=e^{0.05(a+20)-0.0006(a+20)^{2}}\). Parameter *τ*
_{
t,P
} stands for the proportional tax rate financing the PAYG pension regime paid by households on their labour income. *τ*
_{
t,N
A
} stands for the rate of a proportional tax levied on labour income and pensions to finance public non ageing-related public expenditure *d*
_{
t,N
A
}. *d*
_{
t,N
A
} stands for the non-ageing related public spending that one individual consumes irrespective of age and income. This variable is used as a monetary proxy for goods and services in kind bought by the public sector and consumed by households.

Parameter *d*
_{
t,e
n
e
r
g
y
} stands for the energy expenditures paid by households, such that \(d_{t,energy}=C_{age}C_{en}\frac {\sum \nolimits _{a} [w_{t}\varepsilon _{a}\upsilon _{t,a}N_{t,a}+{\Phi }_{t,a}\pi _{t,a}N_{t,a}]} {\sum \nolimits _{a}N_{t,a}}\frac {q_{energy,t}E_{t}}{A_{t}}\) where [*w*
_{
t
}
*ε*
_{
a
}
*υ*
_{
t,a
}
*N*
_{
t,a
}+Φ_{
t,a
}
*π*
_{
t,a
}
*N*
_{
t,a
}] is the aggregate tax base, *C*
_{
e
n
} is a constant of calibration and \(\frac {q_{energy,t}E_{t}}{A_{t}}\) measures the dynamics of energy expenditures as a share of income. *C*
_{
a
g
e
} is a constant depending on age that capture the rising share of energy in income when age increases. Its value, depending on age, is in line with [22] which suggests that the share of energy in income is close to 4 % for French households under 30; 5 % for households between 30 and 44; 6 % for households between 45 and 59; and close to 10 % for households over 60.

The first-order condition for the intratemporal optimization problem is \( 1-\ell _{t,a}^{\ast } =\left (\frac {\varkappa } {\omega _{t,a}}\right )^{\xi } \frac {c_{t,a}^{\ast } }{H_{a}}>0\). A higher after-taw work income per hour worked (*ω*
_{
t,a
}) prompts less leisure (\(1-\ell _{t,a}^{\ast } \)) and more work (\(\ell _{t,a}^{\ast } \)). Thus, it captures the distorsive effect of a tax on labour supply. The first-order condition for the inter-temporal optimization problem is (with *κ*=1/*σ*): \(\frac {c_{t,a}^{\ast }}{ c_{t-1,a-1}^{\ast } }=\left (\frac {1+r_{t}}{1+\rho } \right )^{\kappa } \left (\frac {1+\varkappa ^{\xi } \omega _{t,a}^{1-\xi } }{1+\varkappa ^{\xi } \omega _{t-1,a-1}^{1-\xi }}\right )^{\frac {\kappa -\xi } {\xi -1}}\). Plugging this expression back into the budget constraint yields the initial level of consumption for the working cohort aged *a* at year *t* (\(c_{t,0}^{\ast } \)). The optimal consumption path for each working sub-cohort is derived from the optimal value of \(c_{t,0}^{\ast } \) and the Euler equation. The paths of the labour supplies of the working cohorts (\(\ell _{t,a}^{\ast } \)) are then derived from the values (\(c_{t,a}^{\ast } \)) using the intra-temporal first-order condition. Eventually, one can derive the optimal labour supply of the average individual of a whole cohort (i.e. *ℓ*
_{
t,a
} such that \(1-\ell _{t,a}=1-\nu _{t,a}\ell _{t,a}^{\ast } \)). Knowing the optimal paths (*ℓ*
_{
t,a
}) simplifies the computation of the optimal level of consumption of the average individual representative of a *whole* cohort. The values (*c*
_{
t,a
}) are obtained by maximising the utility function of the average individual of a whole cohort, where the labour supply \(1>\ell _{t,a}=\nu _{t,a}\ell _{t,a}^{\ast } \geq 0\) is already known. The total income net of taxes of the average individual representative of a whole cohort includes Φ_{
t,a
}, the pension income received by the retirees of a cohort (see main text).

This two-steps resolution method avoids some drawbacks associated with the use of shadow wages during the retirement period (see for instance [6]). While in principle mathematically correct, this method may not be very intuitive from an economic point of view since it assumes that agents keep optimising between work and leisure even during the retirement period. Furthermore, this approach makes it practically impossible to derive an analytical solution to the model and complicates its numerical solution.

### A.4 The Production Function

The K-L module of the nested production function is \(C_{t}=\left [\alpha K_{t}^{1-\frac {1}{\beta } }+(1-\alpha )\left [ A_{t}\bar {\varepsilon }_{t}{\Delta }_{t}L_{t}\right ]^{1-\frac {1}{\beta }}\right ]^{\frac {1}{1-\frac {1} {\beta }}}\) where the variables are defined in the main text. The parameter \(\bar {\varepsilon }_{t}=\sum \limits _{a}^{\max (a,t)}\varepsilon _{a}\frac {\nu _{t,a}N_{t,a}}{L_{t}}\) links the aggregate productivity of labour force at year *t* to the average age of active individuals at this year. max(*a*,*t*) stands for the age of the older cohort in total population at year *t*. Parameter *ε*
_{
a
} is the productivity of an individual as function of his/her age *a*. Following [19], it is defined using a quadratic form: \(\varepsilon _{a}=e^{0.05(a+20)-0.0006(a+20)^{2}}\) which yields its maximum at 42 years of age when individual productivity is 32 % higher than its level for age 20. *N*
_{
t,a
} is the total number of individuals aged *a* at year *t*.^{Footnote 15} The variable \({\Delta }_{t}=\sum \limits _{a}\ell _{t,a}^{\ast } \frac {\nu _{t,a}N_{t,a}}{L_{t}}\) is the aggregate parameter corresponding to the average working time across working sub-cohorts in *t* (where \(\ell _{t,a}^{\ast } \) is the optimal fraction of time devoted to work by the working sub-cohort). Thus, \(A_{t}\bar { \varepsilon }_{t}{\Delta }_{t}L_{t}\) is the optimal total labour supply. This labour supply is endogenous since the \(\ell _{t,a}^{\ast } \)’s (and thus Δ_{
t
}) are endogenous in the model. Profit maximization of the production function in its intensive form yields optimal factor prices, namely, the equilibrium cost of physical capital and the equilibrium gross wage per unit of efficient labour. The long-run equilibrium of the model is characterised by a constant capital per unit of efficient labour *k*
_{
t
} and a growth of real wage equalising annual labour productivity gains. The model is built on real data exclusively: the price of the good produced out of physical capital and labour \(p_{c_{t}}\) is constant and normalized to 1.

In the previous CES production function, *C*
_{
t
} stands for an aggregate of production in volume. However, since intermediate consumptions do not appear in its expression, they are implicitly neglected and *C*
_{
t
} equivalently stands for the GDP in volume. Introducing energy demand (*E*
_{
t
}) in a CES function, as [25], yields a more realistic production function *Y*
_{
t
} , again in volume, associated with the value-added which remunerates labour and capital: \(Y_{t}=\left [a\left (B_{t}E_{t}\right )^{\gamma _{_{en}}}+(1-\alpha )\left [ C_{t}\right ]^{\gamma _{_{en}}}\right ]^{\frac {1} {\gamma _{_{en}}}}\) where *a* is a weighting parameter; \(\gamma _{_{en}}\) is the elasticity of substitution between factors of production and energy (with \(\gamma _{_{en}}\)=1-1/elasticity); *E*
_{
t
} is the total demand of energy; and *B*
_{
t
} stands for an index of (increasing) energy efficiency. The cost function is the solution of \(\underset {E_{t},C_{t}}{\min } q_{t}B_{t}E_{t}+p_{C_{t}}C_{t}\) SC \(Y_{t}^{\gamma _{_{en}}}=a\left (B_{t}E_{t}\right )^{\gamma _{_{en}}}+(1-a)\left [ C_{t}\right ]^{\gamma _{_{en}}}\). It is worth noting that in the latter expression, *q*
_{
t
} refers to the price of energy services, these services being measured by (*B*
_{
t
}
*E*
_{
t
}). The price of energy services (*q*
_{
t
}) is related to the price of energy computed in the energy module (*q*
_{
e
n
e
r
g
y,t
}) by the relation: *q*
_{
t
}=*B*
_{
t
}
*q*
_{
e
n
e
r
g
y,t
}. Optimization yields the optimal total energy demand *E*
_{
t
} after some manipulations: \(E_{t}=\frac {q_{t}^{\frac {1}{\gamma _{_{en}}-1}}a^{\frac {-1}{\gamma _{_{en}}-1}}C_{t}}{p_{C_{t}}^{\frac {1}{\gamma _{_{en}}-1}} \left (1-a\right )^{\frac {-1}{\gamma _{_{en}}-1}}}\).

### A.5 The Public Sector

See main text.

### A.6 Aggregation, Convergence and Parameterisation of the Model

Capital supplied by households is \(W_{t}=\sum \limits _{a}{\Omega }_{t,a}N_{t,a} \). Total efficient labour supply is \(A_{t}\bar {\varepsilon }_{t}{\Delta }_{t}L_{t}\). Numerical convergence applies to both \(\left ({\Xi }_{t}\right )_{d}=K_{t}/A_{t}\bar {\varepsilon }_{t}{\Delta }_{t}L_{t}\) and \( \left ({\Xi }_{t}\right )_{s}=W_{t}/A_{t}\bar {\varepsilon }_{t}{\Delta }_{t}L_{t}\) , i.e. the demand and supply of capital per unit of efficient labour respectively.

Demographic data for the period 2000 to 2050 stem from OECD data and projections, and take account of a rise in life expectancy. After 2050, population level and structure by age groups are assumed to be constant.

As in [19], there is no depreciation of capital, an assumption which has no consequence for the dynamics of the model and the interest rate in a model. The TFP gains (*A*
_{
t
}) are set to 1.5 % per year from 1975 to 2000, and from 2020 onwards. It is set to 1.0 % per year from 2000 to 2020 to take account of recent observed data and the probable effect of the financial crisis on TFP. The model is back to its economic long-run steady-state in 2080. The model is calibrated on a real average rate of cost of capital of 6.0 % in the base year.

The weighting parameter *α* in the production function is set at 0.3. The elasticity of substitution between capital and labour is set at 0.8. The households’ psychological discount rate is 2 % per annum [13]. Parameter ϰ—the preference for leisure relative to consumption—is set to 0.25. The elasticity of substitution between consumption and leisure in the instantaneous utility function (1/*ξ*) is equal to 1 (so as to avoid a temporal trend in the conditions for the optimal working time, cf. [1], p. 35). The average effective age of retirement *ζ*
_{
t
} increases in the model from 61 to 65 over the next decades (+ 1.25 year per decade). The level of the average replacement rate (*p*
_{
t
}) is set at 62 %. The risk-aversion parameter *σ* in the CRRA utility function is assumed to be equal to 1.33 (implying an intertemporal substitution elasticity of 0.75) (cf. [10, 16]). The elasticity of substitution between energy and capital (*γ*
_{
e
n
}) is set at 0.4. The weighting parameter (*a*) in the CES production function with energy is set at 0.1.^{Footnote 16} We calibrate the values of interfuel elasticities mainly so as to reproduce observed evolutions of the energy sector. The elasticity of substitution between oil and gas is set at 0.3. Coal is assumed not to be substituable to oil and gas. The elasticity of substitution between electricity and renewables is set at 0.15. The elasticity of substitution of renewables substitutes to fossil fuels is set at 0.1. In the model parameterized on French data, these values allows for reproducing in the simulations of the model well-known characteristics of the energy sector in this country. As concerns the gains or losses of productivity for different technology, we use *ϱ*
_{
e
l,4,t,p
r
o
d
l
o
s
s
}=5*%* per year from 2013 to 2025 for nuclear (with a negative sign); for onshore wind: *ϱ*
_{
e
l,6,t,l
e
a
r
n
i
n
g
}=2*%* per year up to 2025; for offshore wind: *ϱ*
_{
e
l,7,t,l
e
a
r
n
i
n
g
}=1*%* per year up to 2025; for solar photovoltaïc: *ϱ*
_{
e
l,8,t,l
e
a
r
n
i
n
g
}=10*%* per year up to 2025; for biomass: *ϱ*
_{
e
l,9,t,l
e
a
r
n
i
n
g
}=4*%* per year up to 2020.

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### Cite this article

Gonand, F. The Carbon Tax, Ageing and Pension Deficits.
*Environ Model Assess* **21, **307–322 (2016). https://doi.org/10.1007/s10666-015-9482-2

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### Keywords

- Carbon tax
- Intergenerational redistribution
- Overlapping generations
- PAYG pension reform
- General equilibrium