WPS5470 Policy Research Working Paper 5470 Reform and Backlash to Reform Economic Effects of Ageing and Retirement Policy Svend E. Hougaard Jensen Ole Hagen Jorgensen The World Bank Latin American and Caribbean Region Economic Policy Sector November 2010 Policy Research Working Paper 5470 Abstract Using a stochastic general equilibrium model with margin, is found to simultaneously reduce labor supply overlapping generations, this paper studies (i) the effects at the intensive margin. This backlash to retirement on both extensive and intensive labor supply responses reform requires the statutory retirement age to increase to changes in fertility rates, and (ii) the potential of more than proportionally to fertility changes in order to a retirement reform to mitigate the effects of fertility compensate for endogenous responses of the intensity changes on labor supply. In order to neutralize the effects of labor supply. The robustness of this result is checked on effective labor supply of a fertility decline, a retirement against alternative model specifications and calibrations reform, designed to increase labor supply at the extensive relevant to an economic region such as Europe. This paper--a product of the Economic Policy Sector, Poverty Reduction and Economic Management in the Latin American and Caribbean Region--is part of a larger effort in the department to understand the macroeconomic implications of population dynamics. Policy Research Working Papers are also posted on the Web at http://econ.worldbank.org. The author may be contacted at ojorgensen@worldbank.org. The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The papers carry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors. They do not necessarily represent the views of the International Bank for Reconstruction and Development/World Bank and its affiliated organizations, or those of the Executive Directors of the World Bank or the governments they represent. Produced by the Research Support Team Reform and Backlash to Reform: Economic Effects of Ageing and Retirement Policy Svend E. Hougaard Jensen and Ole Hagen Jorgenseny October 13, 2010 Abstract Using a stochastic general equilibrium model with overlapping generations, this paper studies (i) the e€ects on both extensive and intensive labour supply responses to changes in fertility rates, and (ii) the potential of a retirement reform to mit- igate the e€ects of fertility changes on labour supply. In order to neutralize the e€ects on e€ective labour supply of a fertility decline, a retirement reform, designed to increase labour supply at the extensive margin, is found to simultaneously re- duce labour supply at the intensive margin. This backlash to retirement reform requires the statutory retirement age to increase more than proportionally to fer- tility changes in order to compensate for endogenous responses of the intensity of labour supply. The robustness of this result is checked against alternative model speci...cations and calibrations relevant to an economic region such as Europe. JEL Classi...cation: D91; E20; H55; J10; J26 Keywords: Population ageing; labour supply; welfare reform; fertility; retire- ment age; overlapping generations model. An earlier version of this paper was prepared for the WDA-HSG conference on "Economic E€ects of Low Fertility", University of St. Gallen, Switzerland, 11-12 April 2008. We thank David Bloom, Guen- ther Fink, Martin Flodén, Bo Sandemann Rasmussen, Lars Lønstrup, Casper Hansen, Tito Cordella, Michele Gragnolati, and Jan Walliser for helpful comments. y Jensen: Department of Economics, Copenhagen Business School, Porcelaenshaven 16A, 2000 Fred- eriksberg, Denmark. E-mail: shj.eco@cbs.dk. Jorgensen: The World Bank, Poverty Reduction and Eco- nomic Management, 1818 H Street, Washington DC, 20433, USA. E-Mail: ojorgensen@worldbank.org. 1 1 Introduction In modern economies, with a high level of welfare services and extensive income re- placement schemes, the tax burden is typically high. And with strong pay-as-you-go (PAYG) elements, in the sense that individuals on average are net recipients as old and net contributors when active in the labour market, changes in the age composition of the population can have dramatic consequences for public ...nances. Demographic shocks may thus translate into substantial changes in either taxes or welfare services. Since such changes often would be politically unacceptable, the main challenge is how the same welfare opportunities can be maintained for di€erent generations without causing substantial intergenerational redistribution (Andersen et al., 2008). To avoid escalating generational con icts and threatening ...scal sustainability in the wake of demographic shocks, pension reform - in a broad sense - seems almost inevitable. For example, when labour supply shrinks as a result of a lower fertility rate, an obvious response would be to raise the statutory retirement age. However, since decisions about labour supply ultimately rest with private households, the change in the e€ ective retirement age might well be much smaller than the change in the statutory retirement age. In fact, retirement is spread over a whole range of ages, re ecting that the e€ective retirement age is not under direct government control. There may thus be substantial backlashes to reform, and these should be accounted for in order to get a realistic assessment of what might be achieved through a reform process (Boersch- Supan and Ludvig, 2010). In this paper we study the scope for controlling the supply of labour through re- tirement reform. Our framework for addressing this important question is a dynamic stochastic general equilibrium model with overlapping generations. We model the link between the extensive and the intensive margins of labour supply, with the statutory retirement age serving as a proxy for the extensive margin. The novelty of this ap- proach is that it allows for deriving the implications on the intensive margin of labour supply to a change in the statutory retirement age. By assuming that the statutory retirement age is under government control, it is therefore possible to derive the change in the statutory retirement age which, under alternative demographic and economic contingencies, is needed to neutralize changes in the e€ective labour supply. Our main result is that an increase in the statutory retirement age has the potential to neutralize the fertility-induced decline in the labour force, provided that the statutory retirement age increases more than proportionally to the fall in fertility. The reason is that workers increase their demand for leisure, both when fertility falls and when the statutory retirement age increases. As a result, labour supply will fall not only due to low fertility but also as a side e€ect of the increase in the statutory retirement age. Consequently, policy makers should account for such reform backlashes when formulating the optimal policy to alleviate the impact of low fertility. The paper proceeds as follows. In the next section we present the model and the analytical solution method. Section 3 characterizes the market equilibrium and shows how key macroeconomic variables respond to a change in fertility. Section 4 considers the policy option of changing the statutory retirement age in order to neutralize the decline in the labour force. Finally, section 5 concludes and outlines some ideas for future research. 2 2 The model In this section we present our analytical framework. This is a stochastic overlapping generations (OLG) model in line with Bohn (2001), here augmented by endogenous labour supply. The model consists of four di€erent building blocks: demographics, households, production, and social security. We ...rst describe each block, and then outline the solution method. 2.1 Demographics Individuals are assumed to be identical across cohorts, and to live for three periods: as children, adults and elderly, respectively. We denote the children born in period t as Ntc , where Ntc = bt Ntw and bt > 0 is the birth rate. Adults are denoted by Ntw and they are assumed to work for the full length of period t. During period t + 1 they are retired. The growth rate of the labour force, nw ( Ntw =Ntw 1 1), is given by the product of t the fertility rate in the previous period, bt 1 , and a factor denoting the length of the working period, t (i.e., nw bt 1 t ). A fall in the fertility rate in the previous period t thus implies a shrinking labour force in the present period. Figure 1 illustrates how adult lifetime is divided between work and retirement periods, respectively. Figure 1. Adult lifetime: work and retirement In a standard OLG framework, the working period, t, and the retirement period, t + 1, are assumed to be equally long. In reality, however, the length of the two periods might di€er considerably. For example, if the statutory retirement age increases, the share of life spent working goes up, provided that the total lifetime is unchanged. In order to allow for such a di€erence in the length of the two generational periods, we follow the standard practice of considering the overall time periods, t and t + 1, as . aggregate "supra-periods" divided into fractional "sub-periods" A similar technique has been applied by Auerbach and Hassett (2007), Chakraborty (2004) and Bohn (2001, 2002, 2006). The supra-periods are assumed to adjust to have the same lengths as the generational periods of work and retirement; the sub-periods of the total length of life merely may not be distributed equally across supra-periods, but consists of an unchanged total number of sub-periods.1 1 A supra-period could be de...ned to include any given number of smaller sub-periods without loss of consistency with the overall OLG model structure (Figure 1); all sub-periods remain of exactly the same length. Alternatively, the whole life could be considered as one single supra-period consisting of a number of sub-periods. The categorization into supra-periods is merely necessary in order to match the di€erential behavior of generations in their working and retirement periods with equivalent lengths of period t and t + 1-- irrespective of how many sub-periods each supra-period consists of. A change in the length of a generational period will, therefore, be accompanied by an equivalent change in the supra-period by simply changing the number of sub-periods a given supra-period consists of. This 3 The separation point between supra-period t and supra-period t + 1 is interpreted as an exogenous (statutory) retirement age where an agent changes status from having completed working sub-periods in supra-period t to entering a retirement supra- period t + 1 consisting of sub-periods. Therefore, an increase in is equivalent to an increase in the statutory retirement age, where workers would have to remain in the labour force for a longer time-period. Hence, the e€ective growth rate of the labour force, and the extensive margin of labour supply, increases. For a given length of total life, ; the total sum of sub-periods remains unchanged, and there will merely be fewer sub-periods in the retirement period t + 1, as can easily be seen t = t t 1 (1) The total length of life is assumed to comprise an expected and an unexpected component: t = e 1 u , where f g 2 (0; 2 ), while 2 (0; 1). Thus, the total length t t of life cannot be longer than the sum of sub-periods in two supra-periods. We assume that bt , t , and the components of t , are stochastic and identically and independently distributed. Workers are assumed to elastically supply labour, Lt , up to one unit, u 2 (0; 1), where Lt = ut Ntw , and ut = 1 lt is the intensity of labour supply in the working period. First period leisure therefore equals lt = 1 ut . Note that changes in e€ective labour supply can therefore be decomposed into three e€ects: ...rst, the e€ect from the exogenous extensive margin, ; second, the e€ect from the endogenous intensive margin, u; and, third, the e€ect from the exogenous growth in the number of workers, b. It is common in the literature to endogenise the intensity of labour supply, but to combine this with changes in labour supply at the extensive margin has, to our knowledge, not previously been attempted. 2.2 Households We adopt a log-utility function, displaying homothetic preferences over consumption and leisure, bearing in mind the well-known limitations of the log-speci...cation. lt ut = t 1(bt ) ln c1t + t ln + 2 Et [ t+1 ln c2t+1 ] (2) t We denote c1t and c2t+1 as ...rst and second period consumption, respectively. The discount rate on c2t+1 is 2 > 1, and > 0 is the relative weight on leisure in utility. Decisions about consumption for children are assumed to be made by parents, so children make no economic decisions and the intertemporal optimization by parents collapses to a two-period setting. An explicit formulation of the optimisation of parents' utility over their own consumption and that of their children is not important, since the optimisation problem would merely relate ...rst period consumption of the household to the weight that parents assign to consumption of their children. However, the childhood period is conceptually important in this model, since it is a change in fertility in period t 1 that a€ects the size of the labour force in period t.2 implies that two generations could be on the labor market simultaneously, which extends the extensive margin of labor supply as we structure the model. 2 This relation can be shown to enter into lifetime utility as a weight on ...rst period consumption, 1(bt ) > 0, that depends positively on the number of children, see Jensen and Jorgensen (2008). We 4 Second period consumption is scaled by the length of the retirement period.3 The higher is ; the longer period of time retirees can enjoy consumption. While the same argument also applies to the length of the ...rst period, , for consumption and leisure, we stress that if increases then some of the "sub-periods" in retirement, which are all composed by full leisure, will be substituted by sub-periods that consist of both labour and leisure in the working period. This has a negative impact on lifetime leisure. A novelty of our approach is to scale leisure by t to account for this e€ect. As a result, individuals can now account for the disutility of a fall in lifetime leisure, in case the retirement age should increase, by increasing leisure in their working period. In this case, the e€ective labour supply would initially rise by the full amount of the increase in the retirement age. But this e€ect will be counteracted if the disutility of less lifetime leisure induces workers to supply labour less intensively.4 The restrictions on c1t and c2t+1 are presented in (3) and (4), t c1t = (1 t ) (1 lt ) t wt St (3) Rt+1 c2t+1 = St + t+1 (1 lt+1 ) t+1 wt+1 (4) t+1 where t is the pension contribution rate, St is the level of savings, and t is the pension replacement rate. In terms of income in the working period, wt t , the wage rate in each sub-period (say, in each year) is denoted by wt , while t denotes how many sub-periods people have to work (say, the length of the working period in terms of years)5 . The gross return to the savings of retirees, Rt = (1 + rt ), is scaled by to account for the fact that savings must be spread across a given length of the retirement period. Combining c1t and c2t+1 over St yields the intertemporal budget constraint: t+1 t+1 t c1t + c2t+1 + (1 t ) wt t lt = (1 t ) wt t + t+1 ut+1 wt+1 t+1 (5) Rt+1 Rt+1 Note the roles of and as implicit prices on consumption and leisure: consump- tion and leisure must be spread across the lengths of working and retirement periods, respectively. Utility is therefore increasing in and , but so are the implicit prices on consumption and leisure. assume, however, that a 1% increase in fertility would increase 1(bt ) by 1%, because parents need to provide more consumption to more children in the household; i.e. 1(bt ) is normalised at 1. 3 Both Auerbach and Hassett (2007), Bohn (2001), and Chakraborty (2004) have incorporated the length of the retirement period (sometimes alternatively referred to as survival probabilities) into the utility function, but neither have incorporated the length of the working period. This is a novelty of our approach. Furthermore, in Bohn (2001), does not depend negatively to the retirement age, and in Chakraborty (2004), is endogenous to health expenditure and is incorporated so it encompasses both the discount rate and at the same time the length of total life. In our approach, however, is endogenous, and it depends on changes in the statutory retirement age or changes in the total length of adult life, i.e. = , and that could not be analysed by neither Bohn (2001) nor Chakraborty (2004). 4 By modelling the utility of leisure in this way we implicitly add the value of second period leisure into the utility function without having to maximise explicitly with respect to lt+1 . 5 If the retirement age increases, and the capital-labour ratio and the wage rate fall, then the income of workers may either increase or decrease depending on whether the drop in the wage rate across all sub-periods accounts for smaller fall in income than the increase in income induced by the additional sub-periods of work. 5 By maximising lifetime utility (2) subject to the intertemporal budget constraint (5), two ...rst order conditions are derived: ...rst, the Euler equation 1(bt ) c2t+1 c1t = Et (6) 2 Rt+1 and, second, the optimality condition for ...rst period consumption and leisure ! c1t lt = (7) 1(bt ) (1 t ) wt Note that households prioritise less consumption in the ...rst period if fertility de- creases. This will lower 1(bt ) in (6) and (7), such that c1t falls relative to c2t+1 and lt . If or changes, the optimality conditions will remain una€ected.6 2.3 Social security The economy is assumed to operate with a PAYG pension system, given by the following identity, w w t t ut wt Nt 1 = t ut wt Nt (8) where the left (right) hand side illustrates the pension bene...ts (contributions). Neither nor need to be ...xed, so the PAYG system can in principle display either de...ned ect bene...ts (DB) or de...ned contributions (DC) schemes. To re the empirical fact that the DB system is the most widespread PAYG arrangement (Gruber and Wise, 1999), we assume that bene...ts are held constant whereas the contribution rate may vary:7 t t 1 t = (9) 1 + nw t 2.4 Technology and resources Output, Yt , is assumed to be produced by ...rms with a Cobb-Douglas technology in terms of capital, Kt , and labour: Yt = Kt (At Lt )1 Productivity is denoted by At and is assumed to be stochastic and growing at a rate, at ; such that At = (1 + at ) At 1 , where at is assumed identically and independently distributed. The return to capital and the wage rate are standard and de...ned by rt (kt ) = f 0 (kt ) and Wt (kt ) = f (kt ) kt f 0 (kt ), and kt 1 Kt = (At 1 Lt 1 ) de...nes the capital-labour ratio over growth rates.8 By assuming that ...rms are identical, capital 6 The increase in utility of a longer working or retirement period is o€set by a corresponding increase in the implicit prices of consumption and leisure in the intertemporal budget constraint. 7 If the longevity of current retirees increases, the retirement period would residually increase, given that the retirement age remains unchanged, and this would call for a higher contribution rate. Similarly, an increase in the retirement age, given an unchanged length of life, would yield a lower contribution rate. Last, but not least, if fertility falls so will the growth in the number of workers and contributions need to rise to balance the PAYG budget. 8 Since a smaller labour force leads to an increase in the capital-labour ratio, changes in factor returns are likely to occur, see Kotliko€ et al. (2001), Murphy and Welch (1992) and Welch (1979). 6 will be accumulated through the savings of workers, i.e. Kt+1 = Ntw St . Furthermore, we assume that over one generational period (app. 30 years) capital fully depreciates. s The constraint on the economy' aggregate resources is, w w Yt Kt+1 = t Nt c1t + t Nt 1 c2t (10) which features the lengths of the working and retirement periods, respective, in con- nection with the sub-period rates of consumption. This completes the outline of the model. Next, we present our solution method. 2.5 Solving the model We solve the model analytically for the responses of economic variables to changes in fertility and the statutory retirement age. The solution method is designed to provide analytical elasticities of economic variables with respect to stochastic shocks, and it involves transforming the stochastic OLG model into a version that is log-linearised around the steady state of the model. Our analytical approach facilitates the isolation of the necessary response of the statutory retirement age that will o€set any negative responses of labour supply9 . A version of the method of undetermined coe¢ cients, which relies on Uhlig (1999) and extended by Jorgensen (2008) to accommodate an OLG model structure rather than the original real-business-cycle structure, is adopted to obtain the analytical solu- tion for the recursive equilibrium law of motion. The variables of the linearised model are stated in e¢ ciency units and in terms of percentage deviations from the steady state (marked with "hats")10 . A linear law of motion for the recursive equilibrium of the economy is conjectured, b x xt = Pbt 1 + Qbt z b x vt = Rbt 1 z + Sbt which is characterised by linear relationships between endogenous state variables in the b b vector xt and exogenous state variables (the shocks) in the vector zt . The non-state b endogenous (jump) variables are denoted by vt . The coe¢ cients in the matrices P, Q, R, and S are interpreted as elasticities. As an example of how a given endogenous variable is determined by changes in e.g. lagged fertility, bt 1 , or the statutory retirement age, b t , we illustrate the law of motion b for leisure, b= lt b lk kt 1 + lc2 b2t 1 c + b lb1 bt 1 + l bt (11) where, e.g., l denotes the elasticity ( ) of leisure (l) with respect to the retirement age ( )11 . 9 The advantage of an analytical, closed form, solution is that changes in any economic variable can be traced back to the underlying parameters and fundamental properties of the model. Thereby, valuable intuition on the impact of falling fertility on economic variables can be gained. 10 See Appendix A for more details on the solution technique. 11 All endogenous variables fbt ; b1t ; b2t ; b ; yt ; Rt ; wt ; bt g can be expressed in this fashion. The k c c lt b b b e e u complete vector of exogenous state variables is zt 2 fb t 1 ; b t ; bt ; bt 1 ; bt ; bt 1 ; bt ; bt g, but (11) only b a b b illustrates the shocks to lagged fertility and the statutory retirement age. The vector of endogenous state variables is fbt ; b2t g so these remain in equation (11) no matter which shocks are examined. k c 7 A key advantage of this analytical approach is that the impact on leisure of a change in the retirement age is stated in terms of an elasticity, l , the size of which, by construction, assumes a 1% shock to the statutory retirement age, b t . Therefore, we simply ask: "how will leisure change if there was suddenly an increase in the statutory retirement age of 1%?". Using this terminology, we basically make comparative statics with a model that is otherwise designed to be stochastic12 . This procedure is, by now, standard in the real-business-cycle literature (see, e.g., Uhlig, 1999). Our contribution, in this context, is to tailor the method in Uhlig (1999) to ...t a stochastic OLG model, which is complicated by changes in the retirement age that implies future changes in length of the retirement period. The elasticities can be interpreted (both analytically and numerically) and employed in connection with the design of policy rules for the retirement age when fertility has fallen and brought down the size of the labour force. We calibrate the analytical expressions of the model with values, in Table 1, that we trust are realistic to suit a large economic region such as Europe or the USA, and subsequently derive the numerical elasticities of the model. Importantly, we make robustness analyses with the weight on leisure in the utility function in section 4, since the model predictions depend crucially on the calibration of this parameter. Table 1. Parameter calibration Parameter Value Interpretation of steady state parameters 1=3 The capital share in output 0:35 The pension replacement rate a 0:40 The steady state growth rate of productivity 1 The rate of capital depreciation 1 The length of the working period 0:8 The length of total life b 0:1 The rate of growth in the number of children 1 The weight on leisure in the utility function 1(b) 1 The elasticity on the weight of ...rst period consumption in utility with respect to the birth rate 2 0:292 The consumption discount rate13 Note: The payroll tax rate will then be = ( ) = (1 + nw ) = 0:30. The calibration of the discount rate equals 0.960 per year or 0.292 over a 30 year period, and generates a savings rate of 20%. 3 Economic e€ects of low fertility In this section we analyse some economic e€ects of a shock to the birth rate in the previous period, bt 1 . The assessment focuses on the impact on leisure (b ), workers' b lt c c consumption (b1t ) and retirees'consumption (b2t ). Speci...cally, the economy is repre- sented by a linear law of motion in terms of elasticities for endogenous variables with 12 Note that the size of a stochastic shock to, e.g., fertility could be any value from a given pre-speci...ed distribution of innovations. 13 The calibration of the discount rate equals 0.960 per year or 0.292 over a 30 year period, and generates a savings rate of 20%. 8 respect to a fertility shock. These elasticities are reported in Table 2. The relevance of decomposing the net e€ect on each variable into various sub-e€ects is to obtain a better understanding of the magnitudes involved in the numerical simulations. Table 2. A shock to the fertility rate Variable Value Elasticity c1b1 = 0:02 = [ c2k Rk ] kb1 lb1 = 0:11 = c1b1 + 23 b1 22 wb1 c2b1 = 0:54 = [ 15 lb1 3 c1b1 5 kb1 2] = 4 12 wb1 21 lb1 8 kb1 = 0:02 = 9 wk 7 c2k + 12 Rk b1 20 lk The key issue is how a fertility decline a€ect the work-leisure choices. A number of counteracting forces are operating, and the net e€ect remains theoretically ambiguous. However, our numerical simulations imply that leisure will increase by 0.11% after a 1% fertility fall.14 The increase in leisure corresponds to a reduction in the intensity of labour supply, which will magnify the initial fertility-induced e€ect on the shrinking e€ective labour supply and the increasing capital-labour ratio. Changes in wages and pension contributions basically determine the e€ects on work- ers'consumption after the shock to fertility (see Jensen and Jorgensen, 2008). On the other hand, since labour supply is a choice-variable, consumption and leisure are in- terrelated and indirectly a€ect the capital-labour ratio: More leisure leads to an even higher capital-labour ratio, higher wages, and lower capital returns (see ...gures 2a and 2b). Therefore, by examining the intertemporal budget constraint in (5) we can analyse the substitution, income, and wealth e€ects on leisure.15 The substitution e€ect on leisure comes from a shrinking labour force that alters factor payments: Wages increase and the return to capital falls. The price (opportunity cost) of leisure thus increases so the substitution e€ect on the demand for leisure is negative. A given level of income can now buy less, resulting in negative income e€ects on all goods, including leisure. The wealth e€ect is positive for all goods, because the increased wage rate appears in lifetime income.16 The dynamics of leisure and retirees'consumption are illustrated by the simulated trajectories in ...gures 2c and 2d, respectively.17 In an in uencial paper, Weil (2006) ...nds that a key mechanism through which aggregate income and welfare are a€ected by population ageing is the distortion from taxes to fund PAYG pension systems. This mechanism is also present here: the price on at) leisure depends on ; i.e. the ( PAYG contribution rate. With labour supply being 14 Elasticities are, by construction, derived for a positive 1% shock to fertility. Therefore, the elas- ticities of economic variables with respect to a negative fertility shock must be interpreted with the opposite sign of those displayed in Table 2. 15 In the case where labour supply is exogenous (see, e.g., Jensen and Jorgensen, 2008), the only e€ect on the capital-labour ratio originates from the lower fertility rate. 16 See the right-hand side of the intertemporal budget constraint in equation (5). 17 The dynamics of ...rst-period consumption is identical to the simulated trajectory for leisure, though larger numerically. 9 endogenous, this distorting tax rate implies that the positive wealth e€ect will more than o€set the (negative) sum of substitution and income e€ects (i.e. lb1 = 0:11)18 . Figure 2. Economic e€ects of a fall in fertility a. Wage rate b. Return to capital Response to a one percent decline in fertility, lagged Response to a one percent decline in fertility, lagged 0.35 0 Wage rate 0.3 -0.1 Percent deviation from steady state Percent deviation from steady state 0.25 -0.2 0.2 -0.3 0.15 -0.4 0.1 -0.5 0.05 -0.6 Returns 0 -0.7 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 Generational periods after shock Generational periods after shock c. Workers'leisure d. Retirees'consumption -4 Response to a one percent decline in fertility, lagged Response to a one percent decline in fertility, lagged x 10 6 0.05 0 5 Percent deviation from steady state Percent deviation from steady state Leisure -0.05 4 -0.1 3 -0.15 -0.2 2 -0.25 1 -0.3 Second period consumption 0 -0.35 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 Generational periods after shock Generational periods after shock There are additional e€ects to consider in order to obtain a complete analysis of the impacts of low fertility. Due to, ...rst, a changing capital-labour ratio and, second, the presence of distortionary taxation we have to consider "factor price e€ects" and "...scal e€ects", respectively: a negative fertility shock implies that each worker (in the smaller labour force) must pay more taxes (because the bene...ts to retirees are assumed ...xed in a DB system). Thus the ...scal e€ect is negative. In addition, workers will receive higher wages due to the higher capital-labour ratio so the factor price e€ect is positive. This net e€ect is caused by a direct e€ect and an indirect e€ect: The population growth rate falls which directly reduces the size of the labour force. The indirect e€ect is due to the endogenous response of leisure ( lb1 > 0) which has a reinforcing negative e€ect on labour supply. The implication for e€ective labour supply is, therefore, that the initial negative e€ect from lower fertility is ampli...ed by lower intensity of labour supply due to the demand for more leisure as a consumption-equivalent good. 18 The distorting e€ects increase with the size of the pension system, so the larger is the larger is, the larger is lb1 . If taxation was lump sum and not distortionary these three e€ects will o€set each other so the net e€ect on leisure is zero, given that intertemporal elasticity of substitution equal to one, as in our case. 10 The net e€ect on consumption is consequently ambiguous, but our simulations show that consumption increases for a negative fertility shock: c1b1 = 0:02 and c2b1 = 0:54, such that workers gain in terms of consumption and leisure and retirees lose in terms of consumption. Thus, there will be an uneven intergenerational distribution of the economic e€ects. While such welfare implications will not be pursued further in this paper, it is an interesting topic for future research. 4 Policy reform The statutory retirement age can be used as a policy instrument to increase e€ective labour supply by retaining workers in the labour force for a longer period of time ­ and denying them PAYG pension bene...ts until this later date. Such changes will have economic implications that should be well understood by policy makers before designing a policy rule for the retirement age. The purpose of this section is to present a positive analysis on how changes in the retirement age a€ect key economic variables. While the statutory retirement age is exogenous to the consumer, this is not the case for the e€ ective retirement age since the intensity of labour supply is assumed endogenous to the household. If the statutory retirement age increases, no matter why, households may thus decide to supply less labour. If this reduction in labour supply takes place towards the end of households'working life (rather than being spread across all sub-periods of the working period), the reduction re ects the fact that people may retire earlier based on their own savings and thus represent a fall in the e€ective retirement age. An exogenous increase in the statutory retirement age will tend to directly increase labour supply and lower the length of the retirement period, which is in line with our speci...cation of the length of the retirement period is residually determined by the length of the working period ( = ). As a result, workers need to save less for a shorter retirement period. Table 3. A shock to the statutory retirement age Variable Value Elasticity c1 = 1:06 = [ c2k Rk ] k + ( c2 1 R 1) [ c2k Rk ] k +( c2 1 R 1 )+( 23 + 22 11 ) l = 0:04 = 1+ 22 11 c2 = 0:42 = [ 9 wk 7 c2k + 12 Rk 20 lk ] k 21 l + 12 w 8 15 l 3 c1 4 c2 2 k = 0:31 = 5 The change in leisure is determined through the same channels as a fertility shock: the substitution e€ect, the income e€ect, the wealth e€ect, and the ...scal and factor price e€ects, respectively. These dynamics are all intertwined through both exogenous and endogenous changes in the capital-labour ratio and changes in pension contributions and bene...ts. The net e€ect on the capital-labour ratio is negative if the intensity of labour supply does not endogenously fall more than the retirement age has increased. 11 In that case the net e€ect on capital returns remains positive and the wage rate will fall. This will indeed be the case since e€ective labour supply increases by 0:96% because leisure increases by 0:04% for each 1% increase in the retirement age increases (see table 3).19 Regarding the ...scal e€ect: workers now face more subperiods during which they work and has to contribute to the ...xed PAYG bene...ts of retirees. This implies less need for savings to ...nance a shorter retirement period, so workers save less and free resources for leisure and ...rst-period consumption. So, the ...scal e€ect is positive. In terms of substitution, income and wealth e€ects on leisure, we ...nd that the substitution e€ect is negative due to the net increase in the price on leisure. The dynamics of factor payments therefore generate a positive wealth e€ect (lifetime income increases dispropotionally to the fall in the wage rate but proportionally to the increase in the statutory retirement age) and a negative income e€ect (an unchanged level of income can buy less consumption and leisure since leisure has become more expensive). The positive wealth e€ect o€sets the negative sum of substitution and income e€ects, partly due to distortionary taxation, so the e€ect on leisure is positive.20 A particularly important mechanism in this model is that we account for the dis- utility of work in terms of less lifetime leisure when the retirement age increases, i.e. workers will be induced to supply labour less intensively when the sub-periods of full leisure in retirement are reduced. An increase in the retirement age does not yield an equal increase in e€ective labour supply when fertility has declined. This complicates the analysis of an mitigating policy rule for the statutory retirement age. That is precisely why it is crucial to emphasize the dynamics of the intensive margin of labour supply relative to the extensive margin. This is what we study next. We have seen that three main forces are operating when fertility or the statutory retirement age change: the factor price e€ect; the ...scal e€ect; and the endogenous intensity of labour supply (determined, in turn, by substitution, income and wealth ef- fects). In this section, we make use of our general equilibrium framework to derive how much the statutory retirement age should increase in order to neutralize the decline in the labour force caused by low fertility in the past.21 It is important, though, which role one assigns to the statutory retirement age, and we operate under the explicit assumption that the retirement age is an exogenous variable that is under government control. Note, that our analyses are independent of the social desirability of any inter- 19 This is also con...rmed by the elasticities of the wage rate and capital returns with respect to the retirement age ( w = (1 l ) = 0:32; R = (1 ) (1 l ) = 0:64) which represents a negative (positive) factor price e€ects for workers (retirees). The direct e€ect on, e.g., capital returns is (1 ) due to the fall in the capital-labour ratio, while the indirect e€ect originating from endogenous labour supply is (1 l ). 20 As a result of the dynamics above, workers receive a lower wage rate over a longer working period, which renders the net impact on ...rst-period consumption theoretically ambiguous. We ...nd that c1 = 1:06 is positive, however, and that it depends, especially, on the need for less savings to ...nance a shorter retirement period and a higher lifetime income due to more sub-periods of work. Retirees tend to gain in terms of consumption. The net e€ect is ambiguous, but our simulations show an increase in c2 . 21 Proposals for using the retirement age as a policy instrument are found in, e.g., de la Croix et al. (2004) and Andersen, Jensen and Pedersen (2008). Also, Cutler (2001) recommends an extention of Bohn (2001) to incorporate "the length of the period where people work". 12 generational (welfare) distribution of the associated e€ects22 . The e€ective labour supply comprises three elements: ...rst, the fertility rate, bt 1 ; b second, the extensive margin limited by the retirement age, b t ; and third, the intensity with which workers work (the intensive margin, ut = b ). The e€ective labour supply b lt w ) (1 is dt = (1 + nt lt ), or in log-deviations from steady state: b b dt = bt 1 + bt b lt (12) Assume ...rst that the intensity of labour supply is exogenous and that we examine a 1% decline in fertility. It is then clear from (12) that the necessary response of the b statutory retirement age, which would o€set the fertility decline, i.e. dt = b t +bt 1 0, b would just be a proportional increase of b t = 1%. However, if the intensity of labour supply is in fact endogenous, so b 6= 0, then clearly the response of b t would have to be lt di€erent from 1%. In our case, the initial e€ect from the fertility decline on the e€ective labour supply will be reinforced because leisure increases, so the statutory retirement age would have to increase even more than 1%. To derive the o€setting response of b t we insert the linear law of motion for b to obtain: lt bt = [ b lb1 bt 1 + l bt ] bt b 1 (13) From (13) isolate b t , and insert the numerical elasticities, lb1 and l , and the negative fertility shock, bt 1 = 1: b 1 lb1 bt bt = b 1 = 1:15 (14) 1 l Observe that if lb1 < l the optimal response is b t > 1. So, we conclude that the statutory retirement age has to increase more than fertility fell in order to o€set the negative impact on the e€ective labour force. The o€setting response of the statutory retirement age, when bt 1 = 1% and the weight on leisure in utility is = 3, amounts b to b t = 1:15%. These dynamics are due to the choice of leisure by individuals, which will increase both when fertility falls and when the statutory retirement age increases. Thus, the negative fertility-impact on labour supply is ampli...ed. Since the weight that house- holds place on leisure is so crucial to the macroeconomic dynamics when the labour force shrinks, this weight should be tested for alternative values. The literature sug- gests various values for generally within the range 2 f1; 9g (see, e.g., Blackburn and Cipirani, 2002; Cardia, 1997; Chari et al., 2000; Jonsson, 2007). We have cali- brated our model with = 3, as an example, and found the o€setting response of the statutory retirement age to be larger than the fertility rate (b t = 1:15). In terms of robustness analysis, however, we simulate the value for b t given alternative values for and illustrate the results in ...gure 3. 22 Jensen and Jorgensen (2008) evaluates the attractiveness of an uneven distribution of the economic e€ects associated with low fertility in a model with exogenous labour supply, while Jorgensen (2008) does so in a model with endogenous labour supply. 13 Figure 3. Robustness analysis For = 0, the analysis for the o€setting response of b t corresponds to the exogenous labour supply scenario. The 1% fall in fertility can therefore be exactly o€set by a 1% increase in the statutory retirement age. For small values of there is a tendency for the o€setting response of the statutory retirement age to be even less than the fertility-induced fall in labour supply. This means that a contraction in the labour force combined with an increase in the statutory retirement age increases the intensity of labour supply (reduces leisure). The large (net) increase in the price on leisure, (1 ) w , when fertility falls and the statutory retirement age increases, drives the substitution and income e€ects to outweigh the wealth e€ect so the intensity of labour supply increases. As the weight on leisure increases beyond app. 1.6 this trend is reversed. Households now value leisure to such a high extent that substitution and income e€ects no longer dominate the decision to "purchase" leisure. The higher the preference for leisure the greater the tendency to substitute for leisure, and this trend exerts downward pressure on the intensity of labour supply. As a result, the o€setting response of the statutory retirement age becomes increasingly larger than the fall in fertility (the grey area in ...gure 3). An important question now arises: what is the empirical trend in the preference for leisure? If households over the past decades have had a tendency to substitute for more leisure as real wages (and, thus, the price on leisure) have increased, then the o€setting response of the statutory retirement age is likely to equal a value on the curve in the grey area of Figure 3. In that case, policy makers should take the resulting dynamics into account when designing policy rules for the retirement age in order to overcome the problems for welfare arrangement when fertility, and thus, labour supply has fallen. According to Pencavel (1986), the share of life that men spend at work for pay has fallen signi...cantly. In fact, workers are retiring from the labour force at younger ages, the number of hours worked per day or per week has fallen, and the number of holidays has increased - and holidays have become longer. Schmidt-Sørensen (1983) ...nds for Denmark that the number of working hours per week fell by 25% over the period 1911-83, and by 15% over the period 1955-83. Similarly, the number of working hours per year fell by 34% over the period 1911-81. While the fraction of lifetime spent at market work may also have fallen because 14 more time has been allocated to human capital investment, by spending more years within the educational system, the empirical evidence clearly suggests that the prefer- ence for leisure has been increasing for decades. It is therefore likely that the dynamics of the economy, when facing a shrinking labour force, will generate more demand for leisure as real wages increase. This implies that the o€setting response of the labour force will be in a more than 1 : 1 relationship to the contraction in the labour force. A model which does not incorporate labour supply as a choice variable may fail to capture some important macroeconomic dynamics. The ability to analyse the impacts of shrinking labour forces for various values for the preference for leisure thus marks a signi...cant extension of the framework used by, e.g., Auerbach and Hassett (2007) and Bohn (2001). Such an analysis would not be feasible without the explicit relationship in the model between the extensive and intensive margins of labour supply. 5 Conclusion This paper has developed an intertemporal setting in which retirement policy can be used to mitigate the fertility-induced changes in the supply of labour. Our main ...nding is that the retirement age should increase more than proportionately to a fertility decline in order to account for negative responses of the intensity of labour supply. However, this result depends crucially on the preference for leisure by households. In line with empirical evidence there has been a tendency for leisure to rise when real wages increase. And real wages tend to increase when labour supply shrinks as a result of a fertility decline. Therefore, the necessesary o€setting response of the stautory retirement age is likely to be even higher than previously believed. Without an analytical framework linking the endogenous intensive margin to the extensive margin of labour supply, this analysis would not be feasible. The ...nding, that leisure may increase when the statutory retirement age increases, could be interpreted as an endogenous drop in the voluntary early retirement age, ...nanced by workers'own savings. This is exactly the opposite of what is intended by the policy rule of increasing the statutory retirement age. This counteracting mechanism is part of the underlying reason why we derive a more-than-proportionate o€setting increase in the statutory retirement age. The analytical framework is subject to a number of limitations. The utility func- tion has been modelled in accordance with our best beliefs of how to incorporate the value of leisure and the length of periods. However, the robustness of our result could be examined in greater detail for alternative speci...cations of the utility function. In addition, we assume that the economic impacts of changes in dependency ratios can be analysed in a linearised model. Simulation excercises with CGE models should, in the future, be performed to yield a more empirically accurate, and country-speci...c, foundation for designing a policy rule for the retirement age. Last but not least, hu- man capital accumulation may have the implication that workers choose to invest in education to a higher extent when fertility is low because they receive higher wages. As a result, the supply of labour may incorporate a higher productivity. Thus, there may be less need for the statutory retirement age to increase to completely o€set the smaller labour force. These issues may modify our results, and are interesting subjects for future research. 15 References [1] Andersen, T. M., S. H. Jensen and L. H. Pedersen (2008), The Welfare State and Strategies towards Fiscal Sustainability in Denmark, in Neck, R. and J.-E. Sturm (eds.), Sustainability of Public Debt, MIT Press. [2] Auerbach, A. J. and K. Hassett (2007), Optimal Long-Run Fiscal Policy: Con- straints, Preferences and the Resolution of Uncertainty, Journal of Economic Dy- namics and Control, vol. 31, pp. 1451-1472. [3] Blackburn, K. and G. P. Cipirani (2002), A Model of Longevity, Fertility and Growth, Journal of Economic Dynamics and Control, vol. 26, pp. 187-204. [4] Bloom, D., D. Canning and G. Fink (2010), The Graying of Global Population and Its Macroeconomic Consequences, WDA-HSG Discussion Paper Series No. 2010/4. [5] Boersch-Supan, A.H. and A. Ludwig (2010), Old Europe Ages: Reforms and Re- form Backlashes, NBER Working Paper Series No. 15744. [6] Bohn, H. (2001), Social Security and Demographic Uncertainty: The Risk-Sharing Properties of Alternative Policies, in J. Campbell and M. Feldstein (eds.), Risk Aspects of Investment Based Social Security Reform, University of Chicago Press. [7] Bohn, H. (2002), Retirement Savings in an Aging Society: A Case for Innovative Government Debt Management, in A. Auerbach and H. Herrman (eds.), Ageing, Financial Markets and Monetary Policy, Springer. [8] Bohn, H. (2007), Optimal Private Responses to Demographic Trends: Savings, Be- quests, and International Mobility, Working Paper, University of California Santa Barbara [9] Cardia, E. (1997), Replicating Ricardian Equivalence Tests with Simulated Series, The American Economic Review, vol. 87, pp. 65-79. [10] Chakraborty, S. (2004), Endogenous Lifetime and Economic Growth, Journal of Economic Theory, vol. 116, pp. 119-137. [11] Chari, V. V. , P. J. Kehoe and E. T. McGrattan (2000), Sticky Price Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem?, Econometrica, vol. 68, pp. 1151-1179. [12] Cutler, D. (2001), Comment to Bohn (2001), in Campbell, J. and M. Feldstein (ed.), Risk Aspects of Investment Based Social Security Reform, University of Chicago Press. [13] de la Croix, D., G. Mahieu and A. Rillares (2004), How Should the Allocation of Resources Adjust to the Baby-Bust?, Journal of Public Economic Theory, vol. 6, pp. 607-636. 16 [14] Diamond, P. A. (1965), National Debt in a Neoclassical Growth Model, The Amer- ican Economic Review, vol. 55, pp. 1126-1150. [15] Gruber, J. and D. Wise (1999), Social Security Around the World, The University of Chicago Press. [16] Jensen, S. H. and O. H. Jorgensen (2008), Uncertain Demographics, Longevity Ad- justment of the Retirement Age and Intergenerational Risk-Sharing, in Alho, J., S. H. Jensen and J. Lassila (eds.), Uncertain Demographics and Fiscal Sustainability, Cambridge University Press. [17] Jonsson, M. (2007), The Welfare Cost of Imperfect Competition and Distortionary Taxation, Review of Economic Dynamics, vol. 10, pp. 576-594. [18] Jorgensen, O. H. (2008), Retirement Indexation in a Stochastic Model with Over- lapping Generations and Endogenous Labour Supply, Discussion Papers on Busi- ness and Economics, No. 1/2008, Department of Business and Economics, Univer- sity of Southern Denmark. [19] Keilman, N., H. Cruijsen and J. M. Alho (2008), Changing views on future demo- graphic trends, in Alho, J., S. H. Jensen and J. Lassila (eds.), Uncertain Demo- graphics and Fiscal Sustainability, Cambridge University Press. [20] Kotliko€, L. J., K. Smetters and J. Walliser (2001), Finding a Way Out of Amer- s ica' Demographic Dilemma, NBER Working Paper No. 8258. [21] Murphy, K. and F. Welch (1992), The Structure of Wages, Quarterly Journal of Economics, vol. 107, pp. 407-437. [22] Pencavel, J. (1986), Labour Supply of Men: A Survey, in Ashenfelter, O. and R. Layard (eds.), Handbook of Labor Economics, vol. 1, Elsevier Science [23] Schmidt-Sørensen, J. B. (1985), Arbejdstidsforkortelse i historisk perspektiv, Økonomi & Politik, vol. 59, 2-11. [24] Uhlig, H. (1999), A Toolkit for Analysing Nonlinear Dynamic Stochastic Models Easily, in Marimon, R. and A. Scott (ed.), Computational Methods for the Study of Dynamic Economies, Oxford University Press. [25] Weil, D. (2006), Population Aging, NBER Working Paper No. 12147. [26] Welch, F. (1979), E€ects on Cohort Size on Earnings: The Baby Boom Babies' Financial Bust, Journal of Political Economy, vol. 85, pp. S65-S97. 17 A The solution method The way we apply the method of undetermined coe¢ cients relies on Uhlig (1999). The method is adapted, though, to the stochastic OLG structure of our model in line with Jorgensen (2008). This appendix provides a brief overview of the solution method, but we refer to the aforementioned authors for more details. All endogenous variables from the log-linearised model, et 2 fbt ; b2t ; b1t ; b ; yt ; Rt ; wt ; bt g, are written as linear b k c c lt b b b functions of a vector of endogenous and exogenous state variables, respectively. The vector of endogenous state variables is xt 2 fbt ; b2t g of size m 123 , the vector of b k c endogenous non-state variables is vt 2 fb1t ; b ; yt ; Rt ; wt ; bt g of size j 1, while the b c lt b b b e e u vector of exogenous state variables is zt 2 fb t 1 ; b t ; bt ; bt 1 ; bt ; bt 1 ; bt ; bt g of size b a b b g 1. The log-linearised equations are in written matrix notation in the following equilibrium relationships, x x 0 = Abt + Bbt 1 v z + Cbt + Dbt (15) x x x 0 = Et [Fbt+1 + Gbt + Hbt 1 v v z z + Jbt+1 + Kbt + Lbt+1 + Mbt ] (16) b z zt+1 = Nbt + "t+1 ; Et ["t+1 ] = 0 (17) where C is of size h j, where h denotes the number of non-expectational equations. In this particular OLG model h = j, due to the de...nition of xt = fbt ; b2t g, because b k c with merely the capital stock as a state variable h < j, and the system cannot not be solved24 . The matrix F is of size (m + j h) j, and it is assumed that N has only stable eigenvalues. The recursive equilibrium is characterized by a conjectured linear law of motion b between endogenous variables in the vector et , and state variables (endogenous and b b exogenous, respectively) in the vectors vt and zt . The conjectured linear law of motion is written as, b x xt = Pbt 1 + Qbt z (18) b x vt = Rbt 1 z + Sbt (19) where the coe¢ cients in the matrices P, Q, R, and S are interpreted as elasticities. These linear relationships between endogenous variables and state variables could al- ternatively be written out for each variable in et ., as e.g for leisure, b , b lt b = lt b lk kt 1 + lc2 b2t 1 c + l 1 bt 1 + l bt + la bt a + b lb1 bt 1 + b + lb bt be be l e1 t 1 + l e t + bu l u t where e.g. la denotes the elasticity ( ) of leisure (l) with respect to productivity (a). The stability of the system is determined by the stability of the matrix P, given the assumptions on the matrix N. 23 In order to solve the model it is necessary to have at least as many state variables as there are expectational equations in the model (h j). 24 Note that if h > j the equations in this section become slightly more complicated, see Uhlig (1999), but a solution is still feasible. 18 The stable solution for this system boils down to solving a matrix-quadratic equation in line with Uhlig (1999). The matrix-quadratic equation can be solved as a generalized eigenvalue-eigenvector problem, where the generalized eigenvalue, , and eigenvector, q, of matrix with respect to are de...ned to satisfy: q= q 0=( )q For this particular stochastic OLG model is invertible so the generalized eigenvalue problem can be reduced to a standard eigenvalue problem of solving instead the ex- pression 1 for eigenvalues-eigenvectors, as in (20). Then, 1 is diagonalized in (21) since each eigenvalue, i , can be associated with a given eigenvector, qm . 1 I q=0 (20) 1 1 P= (21) The matrix 1 =diag ( ; :::; m ) then contains the set of eigenvalues from which a saddle path stable eigenvalue can be identi...ed, and the matrix = [q1 ; :::; qm ] contains the characteristic vectors. Ultimately, the matrix P, governing the dynamics of the OLG model, is derived, and the system can be "unfolded" to provide the elasticities in the matrices Q; R; and S. For more detail on the solution technique for RBC models we refer to Uhlig (1999). 19