Population Aging, Social Security and Fiscal Limits

We study the sustainability of pension systems using a life-cycle model with distortionary taxation that sets an upper limit to the real value of tax revenues. This limit implies an endogenous threshold dependency ratio, i.e. a point in the cross-section distribution of the population beyond which tax revenues can no longer sustain the planned level of transfers to retirees. We quantify the threshold using a computable life-cycle model calibrated on the United States and fourteen European countries which have dependency ratios among the highest in the world. We examine the effects on the threshold and welfare of a number of policies often advocated to improve the sustainability of pension systems. New tax data on dynamic Laffer effects are provided.


Introduction
Background Population aging is a major challenge for the public …nances of both advanced and developing economies. Longer life expectancy and declining birth rates are causing dependency ratios (the number of retirees as a proportion of the labor force) to rise world-wide. This is generating an increasing burden of taxation on the working population. It raises the issue of whether existing social security nets for older people are sustainable in the longer term and, if not, whether there are policy changes that would make them sustainable and what the welfare cost of this would be. The main reason why existing social security nets for older people may not be sustainable is that they cannot be funded under existing taxation policy. The question that then arises is whether it would be possible to increase tax revenues su¢ ciently to achieve sustainability. This paper investigates these problems using a life-cycle dynamic general equilibrium model with distortionary taxation that takes into account the possibility of an upper limit on the real value of tax revenues raised through direct taxation. The limit exists because tax revenues are subject to dynamic La¤er e¤ects (DLEs) due to the distortionary taxation of the factors of production.
The economic literature on DLEs typically employs models with in…nitelylived agents. As we are concerned with a generational issue, the sustainability of publicly funded support for older people, we use instead a life-cycle, multiperiod, overlapping generations model in the tradition of Auerbach, Kotliko¤, and Skinner (1983) and Auerbach and Kotliko¤ (1987).
We …nd that, in a life-cycle model, DLEs imply the existence of an upper bound, or threshold, on the dependency ratio. This threshold identi…es a critical point in the cross-section of the age-distribution of the population beyond which tax revenue from direct taxation can no longer sustain the planned level of transfers to retirees. We refer to this as the threshold dependency ratio. This is determined by the structure of the economy, the design of …scal policy and evolves over time due to demographic changes.
We show that the threshold dependency ratio is derived from a subset of the competitive equilibria achievable in a life-cycle economy. This subset includes all competitive equilibria in which the government chooses tax policy to maximize tax revenue. The threshold is then endogenously derived from the government budget constraint. We are interested in characterizing the level of the threshold in a given period and its projection over the medium and long term, with a view of comparing this against existing demographic projections.
Demographic projections possess a signi…cant degree of uncertainty. We exploit this to derive a statistical measure of the distance between the projected dependency ratio and the threshold. We use this distance in conjunction with the distribution of stochastic demographic forecasts to measure the probability of an economy reaching the threshold at some point in the future. The distance from the threshold indicates to what extent the government can exploit its ability to raise revenue through direct taxation in order to maintain current levels of publicly funded support for older people. The probability of reaching the threshold indicates how likely a government is to be able to sustain the pension system in the medium and long run. The probability of reaching the threshold also provides a direct comparison of the e¤ects of policy on the sustainability of a pension system, including changes in the retirement age.
The existence of the threshold a¤ects both the bene…ciaries of and the contributors to the social security net. Once the dependency ratio reaches the threshold -the distance is then zero -the government can no longer sustain the social security net for older people through an increase in direct taxation. It then faces a choice of either partially reneging on its social security commitments, for example, by reforming the pension system and making people retire later, or of increasing indirect taxation, or possibly reducing other types of public spending.
Quantitative studies on dynamic …scal policy based on large-scale (life-cycle) simulation models typically focus on single-country analysis and the majority of the literature considers the United States. We contribute to this literature by providing a multi-country analysis covering the United States and fourteen European (EU14) countries. In these EU14 countries, by 2010 dependency ratios reached some of the highest values in the world, and are projected to increase very rapidly by 2100. For each country, we start by quantifying the current size of the …scal space as measured by the potential increase in tax revenue that could be achieved if tax rates on income from capital and labor were set to maximize tax revenues. This gives an indication of a country's ability to sustain the pension system through increase in direct taxation alone. We then measure the threshold over the period 2010-2100 and use stochastic population forecasts to quantify the distance from the threshold and the probability of reaching the threshold in the medium and long run.
We consider four alternative policy scenarios. The …rst covers the case of nopolicy change (S1-NPC). The remaining three policy scenarios re ‡ect reforms typically advocated for improving the sustainability of existing pension systems (National Research Council, 2012): increasing in the consumption tax rate by 5 percentage points (S2-ICT), reducing the replacement ratio of pensions by 10 percentage points (S3-RRR) and increasing the retirement age from 65 to 70 (S4-IRA). We examine the contribution that these policy changes may make in increasing the distance from the threshold and/or reducing the probability of reaching the threshold in the medium and long term. We also rank these reforms based on their welfare e¤ects on the cohorts of individuals alive during 2010-2100.

Quantitative Results
We …nd that the size of the …scal space in the United States ranges between 32 and 47 percent in 2010 (depending on whether the public sector is committed to maintain either the level or the replacement ratio of pensions, respectively) and is expected to grow over the period 2010-2100, though not fast. The threshold dependency ratio in the United States in 2010 is about three times larger than the actual dependency ratio (61% vs 22%). If no policy change is implemented, the probability of reaching the threshold is zero in 2050 but about 4% in 2100. Under the policy scenario S2-ICT the probability of reaching the threshold by 2100 declines to about 2%. Under the policy scenarios S3-RRR and S4-IRA the probability of reaching the threshold falls to zero by 2100.
The outlook is very di¤erent for the EU14 countries. Compared to the United States, they have, on average, narrower …scal spaces, more generous pension systems, are older (higher dependency ratios) and are expected to age much faster. On average across the EU14 countries the threshold dependency ratio is only 0.2 times larger than the actual dependency ratio in 2010. If no policy change is implemented, dependency ratios in all EU14 countries are expected to overtake the threshold well before 2100. The number of countries that are expected to overtake the threshold dependency ratio before 2100 reduces to thirteen, eleven and nine under the policy reform scenarios S2-ICT, S3-RRR and S4-IRA, respectively. The outlook is worst for Austria, Belgium, France, Greece, Italy, Spain and the three Scandinavian countries. If no change in policy is undertaken, on average, these countries are expected to overtake the threshold dependency ratio by 2030. This date is postponed by 5, 15 and 40 years under the policy reform scenarios S2-ICT, S3-RRR and S4-IRA, respectively. These results highlight how imminent is the need of signi…cant pension system reforms for the public …nances of EU14 countries.
The welfare analysis considers the e¤ects of changes in the consumption tax rate, pension contributions and the retirement age that yield the same distance from the threshold by 2050. For the United States we …nd that of the three policy reforms, the greatest welfare gains are obtained through an increase of the taxation of consumption, as this leads to the largest reduction of the distortionary taxation on income from capital and labor. A similar result for the United States is found by De Nardi et al. (1999), Kotlikof et al. (2007), and Conesa and Garriga (2015). In contrast, we …nd that this is not necessarily the best policy option for most of the EU14 countries, as increasing the retirement age and/or reducing pension contributions achieve greater welfare gains than increasing the taxation of consumption for the majority of these countries. These contrasting welfare results re ‡ect di¤erences in tax burdens, demographic structures and discount factors among the EU14 countries.
A by-product of our numerical analysis is the quanti…cation of revenue maximizing tax rates in a life-cycle model. This contributes to the existing quantitative literature on DLEs which is based on in…nitely-lived agent models. In particular, our life-cycle model calibrated on the United States highlights four new aspects of DLEs: (i) how the cross-section distribution of the tax burden changes once the economy moves to the peak of the La¤er hill, (ii) how the measurement of the …scal space depends on how tax revenue is shared among retirees, (iii) the extent to which changes in the demographic structure due to population aging impacts on the position and the shape of the La¤er curves and (iv) how uncertainty about demographic projections impacts on DLEs.
We also provide a new data set of revenue-maximizing tax rates on capital and labor for the United States and the EU14 countries based on a life-cycle model. When keeping constant the replacement ratio of pensions, these tax rates are generally in line with those obtained by Trabandt and Uhlig (2011) using in…nitely-lived agent models. The life-cycle model, however, gives signi…-cantly lower revenue-maximizing tax rates on capital and labor when the level of pension per-capita is kept constant.
Related Literature Our paper is related to the literature on the implications of aging for the sustainability of the social security systems based on multi-periods overlapping generation models as, for example, Auerbach, Kotliko¤ and Skinner (1983) and Auerbach and Kotliko¤ (1987), De Nardi et at (1999), Fuster et al. (2007, Kotliko¤ et al (2007), Heer and Irmen (2014), Conesa and Garriga (2016). 1 These studies evaluate how aging is likely to increase the tax burden required to fund the social security system over a given period of time and how the resulting welfare cost could be mitigated through various reforms of the social security system, including partial …nancing with a consumption tax, reduction of social security transfers or increase in the eligibility age. Quantitative analyses are typically concerned with the United States. Our paper contributes to this literature by providing a measure of the limits faced by tax policy in maintaining the sustainability of pension systems through the threshold dependency ratio and by assessing the probability that an economy will reach a point at which reforms will be inevitable. Our focus on a multi-country analysis, rather than the United States alone, also extends the scope of existing quantitative analyses in this literature.
Our paper is also related to the literature on the implications for public …nances and macroeconomic policy of DLEs, typi…ed by the works Davig, Leeper and Walker (2010), Trabandt and Uhlig (2011), Polito andWickens (2014, 2015), D'Erasmo, Mendoza and Zhang (2016). The common denominator among these studies is their use of in…nitely-lived agent models. We contribute to this literature by studying DLEs in a life-cycle model and by considering their implications for the sustainability of pension systems.
Two recent works, Holter, Krueger, Stepanchuk (2015) and Guner et al. (2016), also consider DLEs in a large-scale model of overlapping generations. Their aim is to quantify how much extra tax revenue can be generated in the United States by increasing the progressivity of the tax system. In these two studies, DLEs impose an upper bound on a government's ability to redistribute resources in the economy. The scope of our paper is di¤erent, as we are interested in how DLEs impose an upper bound on the government ability to sustain the pension system.
A number of issues concerning particular features of existing pension systems are beyond the scope of this paper. These include normative questions such as why we have the pension systems that we do and whether there is a socially optimal level of redistribution from workers to older people. 2 Our analysis is 2 Diamond (2004) and Diamond and Orszag (2005) present various economic arguments underpinning the existence of social security contributions. Shiller (2005) and Beetsma et al. (2011) survey advantages and disadvantages of individual savings accounts for social insurance. Volume 19, issue 2, of the Journal of Economic Perspectives has a series of di¤erent views on positive, being con…ned to the …nancial sustainability of a pension system in the presence of …scal limits, the policy changes that can be implemented to maintain the social security net for older people and the welfare costs that societies may incur in implementing these changes.
Paper Structure The paper is organized as follows. Section 2 provides a summary of global demographic trends and motivates our focus on advanced economies. Section 3 describes a stylized life-cycle model suitable for the analysis of dynamic …scal policy and derives the threshold dependency ratio, the distance from the threshold and the probability of reaching the threshold. We also employ a restricted version of the model to derive a closed-form solution for the threshold and examine its determinants more closely. Section 4 describes the assumptions made for the quantitative model, its calibration and the scope of the numerical analysis. Sections 5 and 6 present the results for the United States and the EU14 countries, respectively. Section 7 concludes. Appendix A describes the numerical algorithm, while Appendix B reports the parameter values calibrated in each country.
2 Demographic Trends Figure 1 shows the historic and projected evolution of dependency ratios over the period 1950-2100 in four regions: the world, the United States, Europe and fourteen European (EU14) countries. 3 We consider two measures of the dependency ratio endorsed by the United Nations (2015). The …rst is the Old-Age Dependency Ratio 2 (OADR2), which measures the number of people in the population aged 65 and above as a percentage of those aged between 20 and 64. The second is Old-Age Dependency Ratio 3 (OADR3) which measures the number of people aged 70 and above as a percentage of those aged between 20 and 70. 4 For the period 2015-2100, data for the world and Europe refer to the United Nations (2015) projections under the assumption of a medium fertility scenario, while data for the United States and the EU14 countries are based on the mean forecasts from the Bayesian hierarchical model underpinning the United Nations (2015)'s projections, see Alkema et al. (2011), Raftery et al. (2012), Raftery et al. (2013), Gerland et al. (2014) andUnited Nations (2015). 5 The Figure clearly shows that aging is a worldwide phenomenon. It is social security and reforms of social security systems. 3 The United States and Europe cover about 4.4 and 10 percent of world population in 2015. The EU14 countries cover about half of the European population. 4 Strictly speaking, the relevant indicator for our study is the retirees-to-workers ratio, de…ned as the number of retirees as a proportion of the labor force. We use the OADR2 because this is the closest proxy available the retiree-to-worker ratio, see National Research Council (2012) and forecasts of retirees-to-workers ratios are not available for the countries covered in our quantitative analysis. The OADR3 is the relevant dependency ratio for our analysis of reforms of the pension system based on increase of the retirement age to 70. 5 We thank Hana Sevcikova for providing the data on dependency ratio forecasts in the United States and the EU14 countries. however more relevant for advanced than developing economies as OADR2 and OADR3 for the United States and Europe are well above the world trend. Table 1 reports average OADR2 and OADR3 dependency ratios for the four regions and for each EU14 country in 1950, 2010, 2050 and 2100. The EU14 countries are ranked in descending order according to their OADR2 dependency ratios in 2050 (in bold). The upper part of the table shows that the dependency ratios for the four regions are generally expected to double over the period 2010-2100. There are, however, signi…cant di¤erences in the levels and the (expected) rates of change of dependency ratios across the EU14 countries. Italy, Germany, Portugal, Greece and Sweden have the highest dependency ratios in 2010. Those of Italy, Greece, Germany and Portugal are projected to remain above the average of the EU14 for 2100. Spain, Austria, Ireland and Portugal are projected to have very large increases in their dependency ratios over the 2010-2100 period. Greece and Italy are forecasted to have the highest dependency ratios by 2100. Compared to the United States, the EU14 countries are older and expected to age more rapidly over the period 2010-2100. Figure 2 shows the OARD2 in each EU14 country and in the United States over the period 2010-2100. In the …gure, data from 2015 onward refer to the mean and the two-standard-error bands from the empirical distribution of the forecasts for the OADR2. 6   over future demographic trends. The extent of this uncertainty tends to increase with the level of the OADR2, has shown by the amplitude of the error bands being larger over time and for countries higher forecasted value of the dependency ratio. We will exploit the uncertainty surrounding the dependency ratios forecasts to construct our statistical measure of the distance from the threshold and to quantify the probability of reaching the threshold.

The Model
The economy is described by a stylized life-cycle model comprising a large number of overlapping generations of households with a …nite life, a representative …rm and government. Each household includes one individual who makes consumption, saving and labor supply decisions to maximize lifetime utility. The …rm uses aggregate capital and labor to maximize pro…ts, while operating a neoclassical production technology. Consumption, income from labor and income from capital are subject to proportional taxes. The government uses tax revenue and issues debt to …nance the provision of public consumption goods and the social security system, which includes transfers to all individuals and

Demographic
In each period t 0 a new cohort of individuals is born and denoted by its date of birth. Individuals in each cohort live for J + 1 periods, with J 1. In t = 0, J cohorts of individuals are already alive, each indexed by their date of birth ( 1; 2; :::; J). We denote by j 0 t the age of an individual in t = 0, so that for any cohort born in t J, j 0 t = max f t; 0g. The probability of an age-j individual born in period t of surviving until j + 1 is t;j . 7 Without loss of generality, at this stage we assume t;j = 1 for j 2 j 0 t ; J 1 and t;J = 0. The population grows at the rate n > 1. The share of individuals of age j in the population, j , is given by j = 0 = (1 + n) j for j 2 (1; J), with P J j=0 j = 1. Individuals work in the …rst j R 1 periods of their life and retire from age j R onwards, with j R 2 (2; J). The dependency ratio d is thus de…ned as: (1) j denote the shares in the population of retirees and workers, respectively. The dependency ratio is determined by four factors: the maximum life duration J, the distribution of age-j individuals in the population, the growth rate of the population and the retirement age. The …rst three are a¤ected by population aging, through reductions in birth rates and increases in life expectancy. Given life expectancy, a decline in the birth rate results in a reduction of n that leads to an increase in the number of retirees relative to workers in the population. Given the birth rate, an increase in life expectancy, for example through a reduction in the mortality rate, leads to increase in the dependency ratio, as would a change in j for any given n. Without loss of generality, we abstract from exogenous changes in the crosssection distribution of the population due, for example, to migration. 8 We treat J, j 's and n as exogenous although, in practice, they could be related to the economic environment and policy, and hence be endogenous. Making these three variables endogenous would not a¤ect our qualitative results. The retirement age j R indicates the age from which individuals start receiving oldage social security contributions. This could be either an endogenous variable chosen by the individual or a policy parameter, depending on how social security eligibility is regulated in the economy. 9 The stylized life-cycle model of the economy described here is compatible with both these interpretations, since the existence of the threshold and its related statistics (distance and probability) do not depend on the mechanism underlying the choice of j R . We appraise the e¤ect of variation of j R in the quantitative analysis.

Environment
Households. Individuals within each cohort are the same. They are endowed with an initial allocation of assets in the …rst period of their life, a t;0 , and do not leave bequests. They are also endowed with one unit of time at each age of their life. This is shared between labor and leisure during the working age. No labor is supplied during retirement. Each unit of time devoted to labor provides z j 0 units of productivity.
Individual preferences depend on consumption and leisure. For any t J, these are ordered by the utility function: (2) where = (1 + ) 1 is the common discount factor, with denoting the discount rate; c t;j and l t;j are the consumption and the labor supply of an individual of age j born in period t, respectively. The utility function u is strictly 8 In the quantitative analysis we account for the impact on any factor in ‡uencing the demographic structure of the population, including migration. This is because our measure of the distance from the threshold and the probability of reaching the threshold depend on forecasts of dependency ratios in the medium and long term that account for these factors. 9 All the studies based on life-cycle models cited in the Introduction assume that the retirement age is exogenous. Fehr, Kallweit and Kindermann (2013) and Kitao (2014), among others, study a large-scale lfe-cycle model with endogenous retirement. increasing in consumption and leisure, twice continuously di¤erentiable, strictly concave and satis…es the Inada conditions. Individuals have perfect foresight. The budget constraints faced by individuals for j 2 j 0 t ; J are: in which Further, q t;j = 1 + c t;j , w t;j = 1 l t;j b w t+j and r t;j = 1 k t;j b r t+j are the after-tax prices of consumption, income from labor and income from capital, respectively; c t;j , l t;j and k t;j are the corresponding age-dependent tax rates; b w t+j and b r t+j denote the pre-tax prices of labor and capital; tr t;j are agerelated transfers; p t;j is the pension received by retired individuals. Without loss of generality, we do not include explicitly a payroll tax.
For an individual born in t of age j, the solution to the lifetime maximization problem is the sequence of allocations (c t;j ; l t;j ; a t;j+1 ) J j=j 0 t that for any t J satis…es the necessary and su¢ cient conditions: and the constraints in (3) -(6), where t;j is the Lagrange multiplier associated with an individual's budget constraint.
Firm. In each period t 0 there is a single produced good that can be used as private consumption, public consumption or capital. Goods are produced by a neoclassical production function with constant returns to scale, y t = f (k t ; l t ) k t , where y t and k t denote per-capita net output and capital, respectively; is the rate of physical depreciation and f is monotonically increasing, strictly concave and satis…es the Inada conditions. Factors of production are paid their marginal products. The before-tax prices of capital and labor are: respectively. Government. The government …nances an exogenous sequence of public consumption, transfers and pension payments, (g t ; tr t ; p t ) 1 t=0 , through revenue from taxation, (tax t ) 1 t=0 , and by issuing public debt, (b t ) 1 t=1 (all variables are in per-capita terms). The sequence of government budget constraints for t 0 is given by: where tax revenue in any t 0 is given by Without loss of generality, we abstract from a separate social security budget at this stage. Note how the dependency ratio is implicitly accounted for in the constraints faced by …scal policy through equations (12) and (13), since these depend on the same J, j J j=0 , n and j R that determine d in equation (1). This observation motivates the derivation of the threshold dependency ratio in the next section. Fiscal policy is subject to the solvency condition: Market-clearing and Feasibility. The equilibrium conditions for percapita labor, asset holdings and consumption are: respectively. The per-capita resource constraint requires Transfers and pension payments per-capita are tr t = P J j=0 j tr t j;j and p t = P J j=j R j p t j;j , respectively.

Threshold Dependency Ratio
First we de…ne the set of competitive equilibria. We then show that the dependency ratio can be derived endogenously as the unique number supporting a speci…c competitive equilibrium. The threshold dependency ratio is then simply a special case.
A competitive equilibrium can be computed in two sequential stages. The …rst consists in determining the sequence of individual allocations and the sequence of prices describing the private sector's optimal choices. At this stage some or all of the demographic variables determining d can be either taken as given or included among the set of choice variables. In the second stage, government policy is determined subject to the constraints set by the private sector choices, the dependency ratio and the government budget constraint. Crucially, one degree of freedom is missing at this stage, as the dependency ratio and the government policy have to satisfy the sequence of government budget constraints in (12) and (13). As a result, there are many competitive equilibria, each indexed with a di¤erent dependency ratio and government policy. This multiplicity implies that for any given …scal policy, the dependency ratio can be derived as a residual from the solution of the government budget constraint. This, however, would not uniquely identify d, since this is a highly nonlinear combination of the demographic parameters J; j 's, n and j R .
To highlight the relation between changes in tax revenue and the dependency ratio, consider a government implementing a new tax policy that delivers a higher level of tax revenue. In order for this new policy to be supported as a competitive equilibrium the government budget constraint has to be satis…ed. To this end, the additional tax revenue could be used to pay for a higher level of transfers to the existing cohort of retirees. It could also be used to maintain the current level of pensions per-capita while sustaining a higher number of bene…ciaries of the pension system. In this second case, increases in tax revenue can be associated with higher dependency ratios, while still be compatible with a competitive equilibrium. The threshold dependency ratio is a special case, being the dependency ratio d obtained when tax policy is set to maximize tax revenue given government spending and borrowing policy. In other words, it measures the maximum number of retirees per worker that the government could sustain through tax policy alone.
A maximum dependency ratio sustainable through changes in tax policy emerges naturally in a life-cycle model as long as there is an upper bound on tax revenue. This is provided by the dynamic La¤er e¤ect. The upper bound can be exploited in conjunction with the government budget constraint to give the threshold dependency ratio, d.
De…nition 1 implies that there is a competitive equilibrium where d = d. Still d is not uniquely determined being a nonlinear combination of the four parameters in equation (1).

An Illustrative Analytic Example
We illustrate the determinants of the threshold dependency ratio, and how this depends on direct and indirect taxation, using the following restrictions on the general model in Section 3.2: (i) individuals live for two periods (J = 1, thus j = 0; 1), they work in the …rst and retire in the second, j R = 1; (ii) labor productivity is normalized to one, z 0 = 1; (iii) there is no aggregate saving, a t;j = 0 for j = 0; 1 and t 0; (iv) there is no government consumption, g t = 0; (v) technology and utility are y t = !l t and U (c t;0 ; c t+1;1 ; l t ) = ln c t;0 + ln (1 l t ) + ln c t+1;1 , respectively, with ! 1 and 0. 10 With these assumptions, the dependency ratio can be solved directly from the government budget constraint as This illustrates a number of results about the threshold dependency ratio highlighted in the previous section using the general model. First, the dependency ratio can be derived as a residual from the solution of the government budget constraint. Second, the dependency ratio that can be supported as a competitive equilibrium increases the higher is total tax revenue. Third, a threshold dependency ratio exists as long as there is an upper bound on total tax revenue. Fourth, the dependency ratio is greater the lower are the levels of the pension p t and transfers tr t , and the higher is tax revenue. Fifth, although not explicitly modelled in this simpli…ed version of the general model which has a …xed time period for working, it is possible to infer the e¤ect of a later retirement age. This would, in e¤ect, raise both the labor input and the level of consumption of the working age group, and would reduce the consumption of the retirees. As a result, tax revenues from those who are working would be greater, which would raise the dependency ratio.
The optimality conditions for the consumption of workers and retirees and of labor are , respectively. These capture the trade-o¤s in equilibrium that give rise to DLEs. Consumption during the working age is a normal good. It reduces if tax rates on either consumption or labor increase. It is also positively related to government transfers. The labor supply is negatively related to the tax rate on labor income and transfers. It does not depend on the consumption tax rate. The demand for consumption goods during retirement is entirely exogenous, being (negatively) related to pensions and the rate of the consumption tax. Using these conditions, the equilibrium tax revenue is written as: The …rst term on the right side is the revenue from the labor income tax. This re ‡ects the typical dynamic La¤er trade-o¤. An increase in the rate of the labor income tax decreases the equilibrium supply of labor (income e¤ect) and thus the labor tax base. The second term on the right is the revenue generated by the consumption tax. If consumption is a normal good, as a higher labor income tax rate reduces disposable income, it also reduces the revenue from a consumption tax.
Di¤erentiating total tax revenue with respect to the labor income tax l t , gives the peak of the La¤er curve as l t = 1 Hence, the tax rate on income from labor that maximizes tax revenues depends negatively on the preference parameter, the level of transfers and the consumption tax rate, but positively on productivity. The negative dependence of l t on the consumption tax rate is due to the fall in consumption brought about by the increase in the labor tax rate. This further compounds the reduction in the total tax base due to the income e¤ect. Thus, the higher is the consumption tax rate, the lower is the labor tax rate at the peak of the La¤er curve.
Di¤erentiation of total tax revenue in equation (19) with respect to the consumption tax rate yields This result can be viewed as the analog, in a life-cycle model, of proposition 3.1 in Trabandt and Uhlig (2011)'s which states that, as long as c t is …nite and does not a¤ect the supply of labor, the government can generate an ever increasing revenue from the taxation of consumption. 11 After replacing the equilibrium conditions and the solution for l t in the government budget constraint, the threshold dependency ratio sustainable in equilibrium for any given c t is given by: The threshold is therefore entirely dependent on the parameters of the economy and the design of …scal policy. In particular, the threshold is unambiguously lower the higher is the level of government expenditure, whether through transfers or pension payments. The derivative of the threshold with respect to the consumption tax rate is generated by labor taxation due to the dynamic La¤er e¤ect, tax revenues from consumption are only constrained by technology and preferences.

Distance and Probability
We are interested in measuring the distance between any forecast of the dependency ratio at some point in the future and the threshold dependency ratio at that point of time and the probability of reaching the threshold, or equivalently exhausting the distance, at some point in the future. Consider writing the dependency ratio in t + 1 as where E t d t+1 is the expected value of the dependency ratio by the end of period t + 1 conditional on information available in t, and t+1 = t+1 is the corresponding innovation in period t + 1, with t+1 being an independent and identically distributed disturbance, t+1 i:i:d: (0; 1). The h-period ahead dependency ratio is therefore where t+h is the h-period ahead innovation. It follows that t+h i:i:d: 0; h 2 , where t+h = P h j=1 t+j and the h-period ahead conditional variance of the dependency ratio is V ar t+h = P h j=1 V ar t+j = h 2 . The forecast error associated with the h-period ahead dependency ratio can be written as t+h = P h j=1 t+j = P h j=1 t+j = u t+h : The probability that the h-period ahead dependency ratio exceeds the threshold dependency ratio d, Pr d; t + h , is therefore written as In the special case of t N (0; 1), then u t+h is also normally distributed. More generally Pr d; t + h can be computed for any stochastic distribution of the expected dependency ratio. We de…ne the distance from the threshold, D d; t + h as the number of standard deviations that the h-period ahead dependency ratio is from the dependency ratio threshold d. 12 This is given by: The probability of exceeding the threshold dependency ratio Pr d; t + h is therefore a function of the distance from the threshold D d; t + h . It increases as the gap between the threshold and the forecasted dependency ratios widens, and the uncertainty surrounding the dependency ratio forecast increase. This probability changes over time due to changes in the base year changes and to new information which a¤ect the forecast of the dependency ratio, its uncertainty and the threshold.
4 Quantitative Analysis

Assumptions
Demographic. Each period, t, corresponds to …ve years. Newborns have a real-life age of 20-24 (j = 0), retire at age 64 (j R = 8) and live age 94 (J = 14). The survival probability in each age j is non-zero, other than in the last period. Households. Preferences are described by the expected lifetime utility where t;s denotes the conditional probability of surviving up to age t + s. Following Trabandt and Uhlig (2011), the instantaneous utility is speci…ed as: wherec = c t =A t is stationary consumption, with A t denoting the technology level. We make four modi…cations to household budget constraint in equation (3). First, taxes are age-independent. Second, households contribute to the pension system through a proportional social security tax levied on wage income at the rate p t . Thus the after-tax labor income is (1 for t 0. Third, transfers are also age-independent, tr t;j = tr t for any t 0 and j 2 j 0 t ; J . Fourth, pension payments are also made ageindependent, being set as a constant proportion (replacement ratio) of the average labor income in the economy b w t l t = W , p t = b w t l t = W for any t 0 and j 2 (j R ; J).
In equilibrium the household is indi¤erent between holding assets in the form of physical capital or government debt, since both yield the same (certain) after-tax return. With a single household living for two periods the proportion of asset holdings would be the same at the household and the aggregate level, but with many periods, the portfolio allocation is indeterminate. Consequently, without loss of generality, we assume that each household holds the two assets in the same …xed proportions.
Firm. Production is described by a Cobb-Douglas function with laboraugmenting technological progress, y t = k t (A t l t ) 1 k t . Technology A t grows over time at the exogenous rate g A 0, which is also equal to the balanced-growth rate of the economy.
Government. Government expenditures (consumption and transfers) grow at the exogenous balanced-growth rate. Government revenue is augmented to include all accidental bequests from households that do not survive. There is a separate balanced-budget for pensions, so that aggregate expenditure on pensions is equal to the aggregate revenue raised through the social security tax: p w t l t = p t R . Stationary equilibrium. All variables, other than labor, are made stationary by expressing them as a proportion of technological progress.
Computation. The main focus of our quantitative analysis is on the sustainability of public pension systems in aging economies. Quantitative studies employing large-scale models with overlapping generations are often concerned with the distributional e¤ects of various forms of macroeconomic policy interventions. These studies account for di¤erent forms of heterogeneity among agents (for example, with regard to income shocks, …nancial wealth distribution, education attainments, health, disability status, sex, marital status and household composition), other than age and productivity.
In our judgment, we can, without any loss of generality, dispense with most of these features when de…ning the threshold dependency ratio in Section 3. In principle, all these forms of heterogeneity could be included in our quantitative analysis, depending on the availability and comparability of these data for each country. Doing so would, however, add signi…cantly to the computation time required in a multi-country analysis. We estimate that with existing computer technologies and a version of the model including all the forms of heterogeneity described above, the solution of a single equilibrium takes not less than one hour, and iteration of the algorithm over the tax rates grid would take about four weeks. Added to this, the time required to iterate over the 2010-2100 period and for each country. Under our speci…cation, the solution of a single equilibrium takes about one second and the search over the three-dimensional grid for n , k and d takes about two hours for each country, depending on the grid size and the number of years considered. Our speci…cation is an attempt to balance the accuracy of the results with the feasibility of their computation. 13

Calibration
The benchmark calibration of the model parameters is reported in Table ?? in Appendix B. As in Holter et al (2015), the parameters describing preferences, production and most of the …scal policy variables are the same as those used by Trabandt and Uhlig (2011) for studying DLEs within in…nitely-lived-agent models. By using the same calibration, we can better appraise how DLEs in in…nitely-lived-agent models change due to the life-cycle structure of the economy and aging. As Trabandt and Uhlig's calibration is based on data averages for 1995-2007, the measurement of stationary equilibria is not a¤ected by the large ‡uctuations in …scal and real variables brought about by the Great Reces-sion of 2007-2010. 14 The remaining parameters are calibrated as follows. The discount rate is calibrated so that the annual return on bonds is equal to 4% in all countries, as in Trabandt and Uhlig (2011). The steady-state growth rate of the population and the (5-year) survival probabilities for each age group of the population are calibrated using 20-years rolling windows averages from the data of the United Nations (2015). 15 Consequently, the steady-state dependency ratio is calculated as an implied residual. Government transfers (tr=y) and the social security tax rate ( p ) are determined endogenously to satisfy the the government budget constraint and the social security budget constraint. Pension payments are computed using data on the gross replacement ratios for pensions from the OECD (2015). 16 Trabandt and Uhlig (2011) employ e¤ective tax rates on labor income which already include social security contributions. We therefore restrict our tax rate on labor income l = w + p to have the same value of the labor tax rate l used in Trabandt and Uhlig (2011). We use the humpshaped age-productivity pro…les estimated by Hansen (1993) to measure the labor productivity z j for the United States. Labor productivity is set equal to 1 for all j 2 (0; J) in all other countries. 17 5 United States 5.1 Fiscal Space Table 2 shows the distribution of the tax burden between workers and retirees when the tax rates on labor income and capital are set as in the benchmark calibration or at the values that maximize tax revenue that correspond with the peak of the La¤er hill on tax revenue raise from capital and labor income. 18 The …rst two columns report the results for 2010. In the last two columns the model is re-calibrated for the demographic structure in 2050. 19 1 4 Despite being based on assumptions that are fairly common in quantitative macro models on the e¤ects of taxation, the calibration could be is not however free from controversy. For example, D'Erasmo et al. (2016) use a two-country model with limited depreciation tax allowance and endogenous capacity utilization to match empirical estimates of the capital income tax base short-run elasticity 1 5 We do not report the survival probabilities for reason of space. These are available upon request from the authors.
1 6 These are based on the percentage of pre-retirement income for men. 1 7 To the best of our knowledge, no data is readily available on age-productivity pro…les for the majority of European countries. 1 8 For tax revenue maximization, we choose the combination of these two tax rates that maximizes tax revenue as a proportion of GDP, with transfers adjusting to balance the government budget. Alternatively, the budget can be balanced by increasing either government consumption or social security spending. The results are not signi…cantly di¤erent from those presented in the main text.
1 9 For the 2050 simulation, we keep the parameters for preferences, production and government spending at the same values of 2010. The survival probabilities for each age group of the population are calculated using averages over 2030-2050 from the United Nation (2015), while the steady-state growth rate of the population is calculated as an implied residual given the mean forecast of the 2050 dependency ratio.  A number of features are revealed about DLEs in a life-cycle model. Under the benchmark calibration, the largest contribution to tax revenue is made by workers through the taxation of their income from labor (column 1). The government could increase in tax revenue by further shifting the tax burden on to workers. This could be accomplished by increasing the taxation of labor income relative to that of capital income and consumption (column 2). As a result of population aging the proportion of workers in the economy falls relative to that of retirees as does the tax burden both from the benchmark and the La¤er calibrations. Figure 3 shows the cross-section distribution of the tax burden among the cohorts of individuals in the population when the tax rates on income from labor and capital are set as in the benchmark calibration or at the values that maximize tax revenue in 2010 (left panels) and in 2050 (right panels).
The left panels show that the burden of the labor income tax is slightly higher for the cohorts of those aged between 30 and 50, since labor productivity is hump-shaped in the data for the United States. The burden of the capital income tax falls mainly on the middle-age cohorts (those aged between 40 and 70), as these are the individuals with the largest amount of savings. 20 The burden of the consumption tax is more equally distributed among the cohorts than the capital income tax burden as consumption is more uniformly distributed in the cross-section of individuals than is saving. When taxation is at the peak of the La¤er hill, the skewness in the cross-section distribution increases, due to the higher tax burden on workers aged between 30 and 50. The majority of the increase is generated by the higher revenue from the labor income tax. The taxation of capital is largely una¤ected, while the taxation of consumption falls for all generations. However, it falls more for workers due to the negative income e¤ect caused by the higher labor income tax, which reduces consumption. These patterns for 2010 are not signi…cantly altered under the demographic structure of 2050. Both under the benchmark calibration and at the peak of the La¤er hill, total tax revenue declines over time. Table 3 reports the tax rates on income from labor and capital at the peak of the La¤er hill in 2010, together with the implied measure of the size of the …scal space (F S). We consider two cases: a constant replacement ratio ( ) and a constant pension level (p). For convenience we also report the tax rates under the benchmark calibration. We highlight three main results. 21 The main result in the table is that DLEs in a life-cycle model depend on how tax revenue is shared among retirees, as shown by the peaks of the La¤er curves being higher and the size of the …scal space being larger for the case of constant replacement ratio. This is because pensions fall in absolute value as the tax rates increase and the government is committed to maintain a constant replacement ratio. According to life-cycle theory, workers will therefore increase savings in order to smooth their consumption when retired. This leads to additional capital accumulation that partly o¤sets the negative e¤ect from a higher tax burden. In contrast, when the pension level is …xed workers no longer need to increase  saving. There is then no additional capital accumulation to partly o¤set the negative e¤ect from a higher tax burden. Additional capital accumulation is a feature speci…c to a life-cycle model; it is absent in an in…nitely-lived agent model, where agents can change their labor supply in every period of their life.
It is useful to relate the results in Table 3 to other works in the literature. First, the tax rate on income from labor could be increased by as much as 115-130%. This increase is larger than the increases in the labor tax rate measured by De Nardi et al (1999) or by Kotliko¤ et al. (2007). This is not surprising as these studies consider the additional tax burden required to sustain a given demographic structure, whereas the peak of the La¤er hill corresponds to the maximum tax burden sustainable for that demographic structure. Second, the rates of the labor income tax at the peak of the La¤er hill reach values similar (60%) to those calculated by Trabandt and Uhlig (2011) using a neoclassical growth model with in…nitely-lived agents and by Holter, Krueger and Stepanchuk (2015) using an multi-period overlapping generations model. Third, when the government maintains a constant pension level the capital income tax rate is on the "slippery" side of the La¤er hill beyond the peak, as also found by Trabandt and Uhlig (2011). Unless it is explicitly stated, all subsequent results are based on the assumption of a constant replacement ratio. Table 4 shows how population aging impacts on DLEs, by reporting the tax rates on labor and capital income and the …scal space (F S) at the peak of the La¤er hill in 2050, using the mean forecast, the upper and lower two-standarderror bands for the 2050 dependency ratio. 22 For reference we also include the pension contribution rate that, due to population aging, has to increase to balance the social security budget. We highlight two main results. First, the size of the …scal space declines the more pessimistic is the demographic projection. Second, for any given demographic forecast, the size of the …scal space increases over time, from about 47% in 2010 (see Table 3) to 55-66% in 2050. This is because due to population aging tax revenue under the benchmark calibration declines more rapidly than the maximum tax revenue . Our analysis of the threshold dependency ratio in the next section illustrates to what extent   a larger …scal space is likely to sustain the increasing cost of the pension system in the United States. Figure 4 displays the La¤er curves for labor and capital income taxes in 2010 (top panel) and 2050 (bottom panels). 23 La¤er curves for the capital income tax are ‡atter than those for the labor income tax, as also found by Trabandt and Uhlig (2011) using a neoclassical growth model with in…nitely-lived agents. This suggests that the slope of the La¤er curve is not qualitatively altered when accounting for the life-cycle structure of individuals in the economy. Comparison of the three panels shows that demographic uncertainty a¤ects the position of the La¤er curve but does not signi…cantly alter their shapes.
In summary, these results highlight four aspects of DLEs in the a life-cycle model: (i) as an economy moves towards the peak of the La¤er hill the tax burden shifts further towards workers, with the largest increase in tax revenue generated through the labor income tax; (ii) revenue-maximizing tax rates are higher when the government keeps the replacement ratio constant rather than the level of pensions, as this induces further private sector saving and capital accumulation which partly compensates for the negative e¤ects of an increase in taxation; (iii) population aging a¤ects more the position of the La¤er curves than their shape; and (iv) uncertainty about demographic projections has a signi…cant impact on the predicted position of La¤er curves.

Threshold Dependency Ratio
Due to population aging, the cost of current pension systems is expected to rise over time. We therefore consider the consequences of four di¤erent policy scenarios that are commonly suggested to make pension systems …scally sustainable, for example De Nardi et. al. (1999), Kotlikof et al. (2007), Nishiyama andSmetters (2007), National Research Council (2012) and Conesa and Garigga (2016). Under the …rst scenario the government maintains the replacement ratio of pension as in 2010 and …nances increases in the cost of pensions over time by raising the social security tax rate p . This quanti…es the threshold dependency threshold ratio under a scenario of no policy change, and is referred to as S1-NPC. The second scenario, S2-ICT, considers the e¤ect of increasing the consumption tax rate by 5 percentage points, from 5% to 10% in the case of the United States. The third scenario, S3-RRR, considers the e¤ect of reducing the replacement ratio of pensions by 10 percentage points, from 35.2% to 25.2% in the case of the United States. The fourth scenario, S4-IRA, considers the e¤ect of increasing the retirement age to 70. Figure 5 shows the evolution of the threshold dependency ratio under the four policy scenarios over the period 2010-2100. The dotted lines in the panels denote the dependency ratio forecasts (OADR2 for S1-NPC, S2-ICT and S3-RRR; OADR3 for S4-IRA). In 2010, the threshold dependency ratio under S1-NPC is about three times the actual dependency ratio. Doubling the consumption tax rate (S2-ICT) would change this only marginally. The threshold would be about 4 times higher than the actual dependency ratio following either a reduction in the replacement ratio by 10 percentage points (S3-RRR) or an increase in the retirement age to 70 (S4-IRA).
Over the period 2010-2100, threshold dependency ratios increase, but very little. In contrast, the OADR2 and the OADR3 are forecasted to increase very rapidly over the same period of time. Consequently, the gap between the forecasted dependency ratio and the threshold narrows under each policy scenario. The upper two-standard-error band of the dependency ratio forecast rises above the threshold under S1-NPC from 2085, while it never reaches the threshold under the other three policy scenarios.    Table 5 reports the threshold dependency ratios and the OARD2 and OADR3 forecasts for 2010, 2015, 2050 and 2100. On average, across the four policy scenarios, the thresholds increase by about one percentage point between 2010 and 2015, by about three further percentage points until 2050, and by about two more percentage points by 2100. These increases are consistent with the size of the …scal space increasing over this period (compare Tables 3 and 4). Under S4-IRA both the threshold dependency ratio and the projected OADR3 are much lower. Therefore, the e¤ects of the four policy scenarios are not comparable when considering the level of the thresholds in isolations, which motivates the use of our measures of distance and probability. 2050-2100. With regard to the distance, we observe two main results. First, the distance declines under each policy scenario. This is because the forecasted dependency ratios increase more rapidly than the thresholds over the period 2050-2100 which reduces the numerator in equation (21). At the same time, the standard deviation of the dependency ratio forecasts also increases over time, thereby increasing the denominator. Second, the distance increases when moving from S1-NPC to S2-ICT to S3-RRR. This is consistent with the increases in the threshold levels under these three policies reported in Table 5. The distance increases even more under S4-IRA. This implies that increasing the retirement age in the United States to 70 would improve the sustainability of the pension system more than would doubling the taxation of consumption or reducing the replacement ratio by 10 percentage points. The probability of reaching the threshold is strictly positive from 2085 onwards and reaches about 4.5% by 2100. This suggests that without any change in policy, there is a probability of about 5% that the cost of the pension system will become unsustainable in the United States over the medium and long run. Under S2-ICT, the probability of reaching the threshold is positive from 2090 onward, and reaches around 2.5% by 2100. For the other two policy scenarios, the threshold is reached with probability zero.

Welfare Analysis
The choice of policy to increase the distance of the economy from the threshold could have signi…cant welfare implications. We therefore compare the implications of changing in 2010 the consumption tax rate, the replacement ratio of pension and the retirement age in order to deliver the same distance from the threshold dependency ratio in stationary equilibrium in 2050. As shown in Fig-S2A  ure 6, if the retirement age were to increase to 70 the distance from the threshold dependency ratio would be of about 17.66 standard deviations by 2050. The same distance would also result from an increase in the consumption tax rate to 25.4% or a reduction in the replacement ratio to 24.1% We refer to these two adjusted policy scenarios as S2A-ICT and S3A-RRR, respectively.

Steady state
We begin by calculating the welfare of the newborn cohorts under S2A-ICT, S3A-RRR and S4-IRA assuming that the economy is in steady state in 2050 with survival probabilities as predicted for that year by the United Nations (2015). In each policy scenario, the labor income tax rate and the pension contribution rate are adjusted to balance the general and the social security government budgets, respectively. We then compute the steady-state life-cycle pro…le of consumption and leisure over the 15 and 9 (10 under S4-IRA) lifetime periods. Instantaneous utility in each lifetime period and lifetime utility are then calculated using equations (23) and (22), respectively. Table 6 presents the steady-state values of aggregate consumption, aggregate labor, the tax rate on income from labor and steady-state welfare (U ) under the three policies for the newborn generation in 2050. The last row reports the percentage consumption equivalent change ( ) required for welfare under S4-IRA to be the same as under S2A-ICT and S3A-RRR. The results show that the policy reform based on increasing the consumption tax gives higher steady-state welfare than the policy based on reducing the replacement ratio or increasing the retirement age. As also found by De Nardi et al. (1999), Kotlikof et al (2007) and Conesa and Garriga (2016), this policy ranking is related to that of the tax rate on labor, which is lower under S2A-ICT compared to S3A-RRR and S4-IRA. Thus the policy that achieves the lowest distortion on the production factors brings also higher welfare gains. A drawback of this steady-state welfare analysis is that it neglects the (potentially important) welfare e¤ects while the economy is transiting between steady-states. We address this issue in the next sub-section.

Transition experiment
The transition experiment follows a protocol similar to that of Conesa and Garriga (2016). This is based on the following assumptions: the economy is in steady state in 2010; between 2015 and 2050 the demographic variables are the same as those forecasted by the United Nations (2015); starting from 2050, the demographic variables remain constant; the government announces the policy change (either an increase in the consumption tax rate as in S2A-ICT, or a reduction of the replacement ratio as in S3A-RRR, or an increase in the retirement age as is S4-IRA) in the second transition period, corresponding to the year 2015, and the policy becomes e¤ective in that year; the policy change is unanticipated and agents adjust their behavior accordingly; government consumption and transfers are …xed throughout the whole transition period, while the labor income tax and the pension contribution rates adjust to balance the general and the social security government budgets, respectively. For comparison, we also consider the transition of the economy when the government …nances the higher pension burden by an increase in the taxation of labor without implementing any policy change, as in S1-NPC. 24 Figure 7 plots the evolution of aggregate capital, aggregate labor, aggregate consumption, and the tax rate on income from labor l during the transition. The results are broadly in line with those of similar transition experiments in the literature. The economy transits to a steady state with the lowest labor supply under S1-NPC, as this yields the highest labor income tax and, consequently, the lowest after-tax income and consumption. In contrast, S2A-ICT generates the highest labor supply due to lowest labor income tax required to …nance public spending. The higher level of taxation on consumption under S1-NPC induces an increase in savings and a higher level of capital accumulation than the other three policy scenarios. Under S3A-RRR, because pensions are lower, individuals work more as they need extra savings in old age. This increase in the labor supply is also supported by the partial reduction l , due to the reduction in p . As a result, capital is almost as high as in S2A-ICT. Labor supply and consumption are similar to those achieved under S2A-ICT. 25 Under S4-IRA the labor supply is high because individuals work for …ve extra years. Lifetime income and hence consumption are the highest. Individuals do not save as much under this policy scenario as under S2A-ICT and S3A-RRR. The main observations from these transition dynamics is that all three policy change scenarios result in a signi…cant fall in the rate of tax on labor income throughout the whole transition period.
2 4 The transition experiment could be based on a di¤erent timing protocol. For example, policy changes could be pre-announced and/or implemented gradually, as in De Nardi et al. (1999) and Kotlikof et al (2007). The protocol of policy does not alter the long-run results, but it does impact on the transition dynamic adjustments and the measurement of welfare in the short run.
2 5 The relative di¤erences between the transtion dynamics under S2A-ICT and S3A-RRR depend on the amount of income and substitution e¤ects. The orders may change for different calibrations of the Frisch labor supply elasticity and the intertemporal elasticity of substitution.  Figure 8 shows the cross-section distribution for the consumption-equivalent changes under S2A-ICT, S3A-RRR and S4-IRA relative to S1-NPC for the cohorts alive in 2010 and 2100. Under our policy protocol, none of the cohorts alive in 2010 anticipate the policy change. In contrast, the 2010 policy changes are fully incorporated in the dynamic responses of all those alive in 2100. In addition, the capital stock has not fully adjusted to the new steady state by 2100. Thus, the results in the …gure illustrate how policy changes impact on di¤erent cohorts depending on whether or not they are fully anticipated and by how much the transition is completed.
The top panel shows that in 2010 the cohorts of younger individuals (those aged less than 40) gain from the unanticipated reforms under policies S2A-ICT, S3A-RRR and S4-IRS due to a lower tax burden on income from labor. As they are already working when the policy change takes place, these cohorts can adjust their labor supply and savings over a relatively long life-time horizon. The middle-aged cohorts (those aged between 40 and 65) face a shorter adjustment horizon. This is less of a problem under scenario S4-IRA, since households can adjust their labor supply at age 65-69. In contrast, middle-aged households under policies S2A-ICT and S3A-RRR are a¤ected much more severely: had they known that they would be able to a¤ord lower consumption in old age (due to higher consumption taxes under policy S2A-ICT or lower pensions under policy S3A-RRR) these cohorts would have adjusted their labor supply when younger. For retirees in 2010, the policy change has less e¤ect due to their lower life expectancy. As shown in Figure 8, the welfare losses for older cohorts under all policy-change scenarios are smaller and they decrease with age. 26 The bottom panel shows that in the long run all age groups make considerable welfare gains from the policy changes made in 2010, although these gains decline gradually with age. This re ‡ects the ability to optimally adjust behavior by 2100 when fully informed beforehand about policy changes. The welfare gains are inversely related to the burden of distortionary taxation on labor income. Raising the tax on consumption as in S2A-ICT generates an increase in welfare of about 33-36% higher than reducing the retirement age as in S4-IRA and, depending on age, about 5-8% higher than lowering the replacement ratio as in S3A-RRR. As in Conesa and Garriga (2016), those who face a lower tax burden gain most from the policy reforms.
In summary, these results show that reforms designed to improve the sustainability of the pension system can bring welfare gains for all age groups in the long run, but only the young bene…t in the short run. The larger bene…ts in the long run re ‡ect the advantages of having full information about policy changes and therefore being able to respond to them. The greatest welfare gains (for all cohorts in the long run and for the young cohorts in the short run) are obtained through raising consumption taxes.

EU14 Countries
In this section we extend our quantitative analysis of the threshold dependency ratio to the EU14 countries. First, we provide a cross-country comparison of the size of their …scal space and highlight how this is related to di¤erences in the labor tax rate, the replacement ratio, the dependency ratio and aging. We then present our measurement of the level of the threshold, the distance form the threshold and the probability of reaching the threshold in each country under the same four policy scenarios considered for the United States. We conclude our assessment with a welfare analysis. Table 7 reports the tax rates on labor income and capital at the peak of the La¤er hill and the size of the …scal space (F S) for each EU14 country assuming either a constant replacement ratio ( ) or a constant pension level (p) in 2010. 27 For reference we also report tax rates under the benchmark calibration. With a constant pension level, the payroll tax rate p adjusts to balance the social security budget. Numbers in bold indicate revenue-maximizing rates that are lower than the corresponding rates under the benchmark calibration. These highlight instances where current tax rates are on the slippery side of the La¤er hill.

Fiscal Space
The main result emerging from the table is that the revenue-maximizing tax rates and the size of the …scal space are signi…cantly higher in the case of constant replacement ratio. As found for the United States, this is because pensions fall in absolute value as tax rates increase and the government maintains a constant replacement ratio. This induces workers to increase savings in order to smooth their consumption when retired, thereby leading to additional capital accumulation that is absent when the government maintains instead a …xed pension level.
In particular, the results with a constant replacement ratio are generally in line with those of Trabandt and Uhlig (2011). The labor income tax rates for all countries are lower than at the peak of the La¤er hill. For the capital income tax rate we …nd that only Denmark and Great Britain are on the slippery side of the La¤er hill. The average size of the …scal space for the EU14 countries in 2010 is 22%. There are, however, signi…cant cross-country di¤erences. Portugal has the largest …scal space of about 70%; Belgium and the three Scandinavian countries have the lowest …scal spaces due to their high tax rates.
The results for a constant pension level show that for the labor income tax, all countries but Sweden are on the "right" side of the La¤er hill. For the capital income tax rate, we …nd that 8 of the 14 countries have a tax rate higher than at the peak of the La¤er hill. The average size of the …scal space with constant pensions is about a quarter of that with constant replacement ratio. Although there are still cross-country di¤erences in the size of the …scal space, these are less pronounced. Figure 9 relates EU14 countries'the …scal space (under constant replacement ratio) to the tax rate on labor income, the replacement ratio of pensions, the dependency ratio (OADR2 in 2010) and aging (change in the OADR2 between 2010 and 2050). The United States is included for comparison. In each panel,  the vertical line indicates the average …scal space and the horizontal line is the average value of the variable on the vertical axis. The negative relation between the …scal space and the labor tax rate in the top-left panel helps to explain the cross-country di¤erences in the size of the …scal space observed in Table 7. 28 The top-right panel shows that the four EU14 countries with the largest …scal space (Portugal, Spain, Greece and the Netherlands) also have among the highest replacement ratios. Six countries (Austria, Denmark, Italy, Sweden, Finland and France) have high replacement ratios but relatively small …scal spaces. The bottom-left panel shows a negative relation between the …scal space and the age structure of the population in 2010. Eight countries are concentrated in the top-left corner of the panel. They have relatively high dependency ratios and narrow …scal spaces. The four countries with the largest …scal space also have relatively high dependency ratios. The bottom-right panel shows a positive relation between the size of the …scal space and the forecasted increase in the OADR2 between 2010 and 2050. The four countries with the largest …scal space are located towards the top-right corner. This suggests that countries that in 2010 have a relatively large …scal space are likely to exhaust it relatively quickly. From this perspective, the …scal outlooks for Italy, Austria and Germany are the most precarious as they have a relatively narrow …scal space and are projected to age very rapidly over the period 2010-2050.
In summary, when compared to the United States, the EU14 countries have, on average, narrower …scal spaces, higher replacement ratios, higher dependency ratios and are expected to age much faster. Figure 10 shows the evolution of the threshold dependency ratio in the EU14 countries over 2010-2100. In each panel, the dotted lines denote the OADR2 forecasts (mean and two-standard-deviation bands); the solid line denotes the threshold dependency ratio under the no-policy-change scenario (S1-NPC); the dashed-dotted and dashed lines denote the threshold obtained when the consumption tax rate is increased by 5 percentage points (S2-ICT) and the replacement ratio is reduced by 10 percentage points (S3-RRR), respectively. Figure  11 shows the evolution of the threshold dependency ratio when the age of retirement is increased from 65 to 70 (S4-IRA) in each EU14 country together with the OADR3 forecasts over the period 2010-2100.

Threshold Dependency Ratio
We …nd that under S1-NPC the majority of the EU14 countries have threshold dependency ratios below the mean forecast of the OARD2 for the largest part of the 2010-2100 period. Under S2-ICT and S3-RRR there is a very modest increase in the threshold in all EU14 countries, other than Great Britain. The outlook appears to improve under S4, though in no country is the threshold above the higher error band of the OADR3.  Table 8 reports our estimate of the year when the OADR2 and OADR3 mean forecast are expected to be higher than the threshold dependency ratio for each EU14 country under the four policy scenarios. Under S1-NPC, the OADR2 is expected to overtake the threshold for all EU14 countries before 2100. Under S2-ICT, S3-RRR and S4-IRA the number of countries overtaking the threshold before 2100 reduces to thirteen, eleven and nine, respectively. The outlook is therefore particularly concerning for this last group of nine, that comprises Austria, Belgium, France, Italy, Greece, Spain and the three Scandinavian countries. On average, under S1-NPC these nine countries are expected to overtake the threshold by 2030. This date is postponed by 5, 15 and 40 years under S2-ICT, S3-RRR and S4-IRA, respectively. Table 9 reports the distance from the threshold and probability of reaching the threshold in each EU14 country in 2050 under the four policy scenarios. The last two rows report the EU14 average and the United States for reference. Under S1-NPC, for all countries other than Great Britain the distance from the threshold in 2050 is negative implying the dependency ratio exceeds the threshold. The distance is still negative under S2-ICT, though on average smaller than S1-NPC. It becomes positive on average under S3-RRR and S4-IRA. The probability of reaching the threshold by 2050 declines on average from about 90 to about 20 percent when moving from S1-NPC to S4-IRA. In contrast, there is a zero probability of the United States reaching the threshold by 2050 even under the scenario of no change in policy. These results highlight how imminent reforms of the pension system are in the EU14 countries. They also highlight that the extent of these reforms in EU14 countries should be more radical than for the United States. Table 10 reports our results on the welfare e¤ects of alternative policy changes designed to satisfy the threshold dependency ratio for the EU14 countries based on the stationary lifetime utility of the 2050 newborn generation. The …rst two columns report for each country the targeted distance from the threshold in 2050 and the corresponding probability (these are the same as in the last two columns of Table 9). The next two columns report the tax rate on consumption and the pension replacement ratio required to achieve the targeted distance in each country under S2A-ICT and S3A-RRR, respectively. The last two columns report the percentage consumption change ( ) required for welfare under S4-IRA to be equal to welfare under S2A-ICT and S3A-RRR, respectively. The results for the United States are also reported in the last row for comparison. Welfare calculations are not provided for Italy and Spain as the distance remains negative for these two countries. We highlight two main results. First, the increases in the consumption tax rate and/or reduction in the replacement ratio required to achieve the targeted distances are signi…cantly higher than those calculated for the United States. The average consumption tax rate under the benchmark calibration of the twelve countries that have a positive distance is about 21%. This needs to increase on average across these countries to 43%. Similarly the replacement ratio needs to be reduced from 57 to 43%, on average. ---USA ----Notes: '-'indicates threshold not reached before 2100. Table 8: Years when forecasted dependency ratios are estimated to overtake the thresholds under S1-NPC, S2-ICT, S3-RRR and S4-IRA, EU14 countries.

Welfare Analysis
Second, the ranking implied of the three policy reform scenarios is very different across countries. In particular, the partial …nancing with the consumption tax, which is the preferred policy change for the United States, does not yield the higher long-run welfare gains for the majority of EU14 countries. The policy reform S3A-RRR yields higher welfare gains in the long run for Austria, Belgium, Finland, Germany and Great Britain, while S4-IRA yields higher welfare gains for Denmark, France and the Netherlands.
To shed light on some of the factors determining cross-country di¤erences in the welfare ranking, we calculate how the measured consumption compensations correlate with the deep parameters that determine the distance from the threshold in each country. 29 The following patterns emerge. Policy reforms based on increasing the taxation of consumption tend to yield higher welfare gains in countries with relatively high debt-to-GDP ratios. Under this policy the government can reduce the level of public spending since the equilibrium stock of capital is high, thus leading to a reduction in the cost of servicing public debt. Policy reforms based on reduction of pension contributions tend to be preferred to those based on higher taxation of consumption in countries where the taxation of income from labor is relatively high, since the consumption tax rate has the e¤ect of further increasing the tax wedge. For the same reason, in countries with relatively high S1-NPC  taxation of income from labor, policy reforms based on the reduction of pension contributions tend to be preferred also to those based on increasing the retirement age. Cross-country di¤erences are also a¤ected by di¤erences in discount factors and the taxation of saving. Under policy S4-IRA there is a reduction in public transfers at age 65-70 (the pension for the cohort that is required to work under S4-IRA). Under policy S3-RRR, there is on average a reduction in transfers at a later point in life for those age 65-95. Di¤erent discount factors across countries will weight these two policies di¤erently. For any given policy individuals in countries with high capital tax rates have lower consumption than those living in countries with low capital tax rates in order to build up their savings. Accordingly, for a lower discount rate and capital income tax rate, policy S3-RRR is preferred to policy S4-IRA.

Conclusion
Population aging is a major challenge for the sustainability of public pension systems around the world. The challenge is particularly signi…cant for advanced economies, where the ratios between the bene…ciaries of and the contributors to pension systems are among the highest in the world and are projected to double over the next 85 years.  The paper develops a stylized multi-period overlapping generations model that explicitly accounts for a limit on the ability of the government to increase tax revenue due to the distortionary taxation of income from capital and labor. This …scal limit implies the existence of an endogenous threshold dependency ratio, i.e. a point in the cross-section age distribution of the population beyond which the government can no longer extract further revenue from the taxation of income to …nance the cost of pensions. We characterize the level of the threshold as a competitive equilibrium solution to the model. We also derive statistical measures of the distance of the economy from the threshold and the probability of reaching the threshold at some point in the future. From a policy perspective, a zero-distance from the threshold dependency ratio identi…es a point in which reforms of the pension system cannot longer be postponed.
We quantify the threshold level, distance and probability for the United States and a sample of fourteen European countries over the period 2010-2100. We study how these three measures change under four alternative policy scenarios: no-policy change, partial …nancing with the taxation of consumption, reduction of pension contributions and increase in the retirement age. In all these countries we …nd that the threshold is increasing but not as rapidly as the forecasted dependency ratio over the 2010-2100 period. For this reason the distance from the threshold is found to decline very quickly and the probability of reaching the threshold is positive and increasing over the same period of time.
We …nd signi…cant cross-country di¤erences in thresholds levels, distances and probabilities. The outlook is particularly concerning for the European countries. Compared to the United States, they have, on average, narrower …scal spaces, more generous pension systems, are older and are expected to age much faster. For these reasons, the European countries are found to be much closer to the threshold than the United States in 2010. The majority of these countries is also predicted to reach the threshold well before 2050, while the United States is found to maintain a positive distance until 2100. Either increasing the taxation of consumption by 5 percentage, reducing the replacement ratio of pension by 10 percentage points, or increasing the retirement age to 70 raise the distance from the threshold and leads to zero probability of reaching the threshold in the United States. In contrast, the outlook for the EU14 countries improves only marginally under these three policy change scenarios, therefore highlighting how imminent and drastic pension system reforms are for EU14 countries. We also study the welfare implications of changes in the consumption tax rate, replacement ratio and retirement age designed to maintain a given distance from the threshold at some point in the future. In contrast with the existing evidence in the literature for the United States, we …nd that there is not unique policy that is preferred for the European countries, as the welfare ranking depend on country-speci…c characteristics, such as the design of the tax system, the current level of public spending, private sector preferences and productivity.
A further contribution of the paper is the quanti…cation of DLEs in multiperiod overlapping generations model. This exercise highlights how DLEs in a life-cycle model change the cross-section distribution of the tax burden, depend on how tax revenue is shared among retirees, and change over time due to population aging.
We believe that while the quantitative results presented in the paper are based on a very stylized model of the economy, they are however indicative current policy concerns with regard to urgency and extent of pension system reforms in advanced economies. Our results do not account for many further e¤ects that population aging may have on the macro-economy and public …nances, like for example (i) the political feasibility for the government to extract the maximum revenue from the taxation of income in the economy; (ii) the cost of non-pension-related components of public spending; and (iii) long-term rates of economic growth. We conjecture that most of these side-e¤ects are likely to strengthen our main …nding that a large number of European countries are likely to reach their threshold dependency ratio within the next 20-30 years.

A.1 Steady state
In order to compute the steady state, we solve a non-linear equations problem in 28 variables consisting of the 14 individual asset levels, a j = k j + b j , j = 0; : : : ; 14, (with a 0 = 0), the 9 individual labor supplies, l j , j = 0; : : : ; 8, and the aggregate variables K, N , A, p , and T r.
The non-linear equations system consists of the 23 …rst-order conditions (the 14 Euler conditions and the 9 …rst-order conditions of the household with respect to the labor supply): All other variables, e.g. individual consumption, factor prices, and aggregate bequests and taxes, can be computed with the help of the 28 endogenous variables. For example, for the computation of individual consumption levels c j , we can use the individual budget constraint. For the computation of the factor prices, we use the …rst-order conditions of the …rms. We solve this non-linear equations problem with a modi…ed Newton-Rhapson algorithm as described in Section 11.5.2 and applied to a large-scale OLG model in Section 9.1.2 in Heer/Maussner (2009). The main challenge for the solution is to come up with a good initial value for the individual and aggregate state variables.
Therefore, we started from a simple 15-period OLG model with exogenous labor where all cohorts are workers. The exogenous labor supply is set equal to 0.3 and the initial value for the aggregate capital stock is set equal to the corresponding value in the Ramsey model. Thereafter, we added one additional cohort of retirees in each step and used the solution of the model in the previous step as an input for the initial value of the next step. Finally, we introduced endogenous labor in the model. During these initial computations, we computed the solution for the individual optimization problem in an inner loop and updated the aggregate capital variables in an outer loop with a dampening iterative scheme as described in Section 3.9 of Judd (1998) that helped to ensure convergence. For the …nal calibration and the computation of the steady states for di¤erent tax rates, we applied the modi…ed Newton-Rhapson algorithm to the complete set of the 28 individual and aggregagte equilibrium conditions. The Gauss computer programs are available from the authors upon request.

A.2 La¤er curve and threshold dependency ratio
In order to …nd the revenue-maximizing tax rates ( l , k ), we employ a nested procedure. In the inner loop, we use a maximization routing (Golden Section search) in order to …nd the revenue-maximizing capital income tax rate k for given labor income tax rate l . In the outer loop, we use a grid search over l over the interval [20%, 70%] with an accuracy equal to 0.01%. In order to …nd the threshold dependency ratio in a particular year, we iterate over the dependency ratio starting from a low value. As initial value, we choose the dependency ratio that is associated with the survival probabilities of this year and a population growth rate of 2%. We slowly increase the dependency ratio and compute the revenue maximizing tax combination ( l , k ) in each step as described in the last paragraph. When we …nd a tax combination that …nances the government expenditures in the particular year, we continue to increase the dependency ratio. We stop the computation, when we cannot …nd a tax combination (the La¤er peak) that is able to …nance government expenditures.