metadata: authors: 0: author: Jensen,Svend Erik Hougaard
1: author: Jorgensen,Ole Hagen
count: World
docna: 0: docna: Reform and backlash to reform : economic
            effects of ageing and retirement policy
docty: Policy Research Working Paper
subtopic: Pensions & Retirement Systems,Labor Markets,Population & Development,Demographics
teratopic: Macroeconomics and Economic Growth,Social Protections and Labor,Health, Nutrition and Population
abstracts: cdata!: Using a stochastic general equilibrium
            model with overlapping generations, this paper studies (i)
            the effects on both extensive and intensive labor supply
            responses to changes in fertility rates, and (ii) the
            potential of a retirement reform to mitigate the effects of
            fertility changes on labor supply. In order to neutralize
            the effects on effective labor supply of a fertility
            decline, a retirement reform, designed to increase labor
            supply at the extensive margin, is found to simultaneously
            reduce labor supply at the intensive margin. This backlash
            to retirement reform requires the statutory retirement age
            to increase more than proportionally to fertility changes in
            order to compensate for endogenous responses of the
            intensity of labor supply. The robustness of this result is
            checked against alternative model specifications and
            calibrations relevant to an economic region such as Europe.
repnme: repnme: Reform and backlash to reform : economic
            effects of ageing and retirement policy
display_title: Reform and backlash to reform :
            economic effects of ageing and retirement policy
listing_relative_url: /research/2010/11/13044137/reform-backlash-reform-economic-effects-ageing-retirement-policy
topicv3: Demographic Change m1327040 1448,Aging Population m1327275 643,Retirement m1504007 2896,Labor Supply m1504055 2751,Fertility Rate m1504357 2662
publishtoextweb_dt: 2010-07-01T00:00:00Z
historic_topic: Macroeconomics and Economic Growth,Social Protections and Labor,Health, Nutrition and Population
no_of_pages: 21
author_keys: 0: author_key: 1131250
1: author_key: 805008
model_extractions: 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 overlapping generations, this paper studies (i) the effects on both extensive and intensive labor supply responses to changes in fertility rates, and (ii) the potential of a retirement reform to mitigate the effects of fertility changes on labor supply. In order to neutralize the effects on effective labor supply of a fertility decline, a retirement reform, designed to increase labor supply at the extensive margin, is found to simultaneously reduce labor supply at the intensive margin. This backlash to retirement reform requires the statutory retirement age to increase more than proportionally to fertility changes in order to compensate for endogenous responses of the intensity of labor supply. The robustness of this result is checked against alternative model specifications and calibrations 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
# 1 Introduction

In modern economies, with a high level of welfare services and extensive income replacement 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 finances. 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 different generations without causing substantial intergenerational redistribution (Andersen et al., 2008).

To avoid escalating generational conflicts and threatening fiscal 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 effective 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, reflecting that the effective 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 (BoerschSupan and Ludvig, 2010).

In this paper we study the scope for controlling the supply of labour through retirement 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 approach 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 effective 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 effect 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
Adult lifetime (¢) LL he Work (y) Retirement (A)
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 χ working sub-periods in supra-period t to entering a retirement supraperiod t + 1 consisting of λ sub-periods. Therefore, an increase in χ 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 effective growth rate of the labour force, and the extensive margin of labour supply, increases. For a given length of total life, φ, 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 The total length of life is assumed to comprise an expected and an unexpected component: φt = φ<sup>e</sup> t−1<sup>φu</sup> t<sup>, where {φ} ∈(0, 2−χ), while χ ∈(0, 1).Thus, the total length</sup> of life cannot be longer than the sum of sub-periods in two supra-periods. We assume that bt, χt, and the components of φt, are stochastic and identically and independently distributed.

Workers are assumed to elastically supply labour, Lt, up to one unit, u ∈ (0, 1), where Lt = utNt<sup>w,andut=1 −ltistheintensityoflaboursupplyintheworking</sup> period. First period leisure therefore equals lt = 1 − ut. Note that changes in effective labour supply can therefore be decomposed into three effects: first, the effect from the exogenous extensive margin, χ; second, the effect from the endogenous intensive margin, u; and, third, the effect 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-specification.

We denote c1t and c2t+1 as first and second period consumption, respectively. The discount rate on c2t+1 is ρ2 > −1, and β > 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 first 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 affects the size of the labour force in period t.<sup>2</sup>

> 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 first period consumption, ρ1(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.<sup>3</sup> The higher is λ, the longer period of time retirees can enjoy consumption. While the same argument also applies to the length of the first period, χ, for consumption and leisure, we stress that if χ 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 χt to account for this effect. 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 effective labour supply would initially rise by the full amount of the increase in the retirement age. But this effect will be counteracted if the disutility of less lifetime leisure induces workers to supply labour less intensively.<sup>4</sup> The restrictions on c1t and c2t+1 are presented in (3) and (4), where κt is the pension contribution rate, St is the level of savings, and γt is the pension replacement rate. In terms of income in the working period, wtχt, the wage rate in each sub-period (say, in each year) is denoted by wt, while χt denotes how many sub-periods people have to work (say, the length of the working period in terms of years)<sup>5</sup> . The gross return to the savings of retirees, Rt = (1 + rt), is scaled by λ 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: Note the roles of χ and λ as implicit prices on consumption and leisure: consumption and leisure must be spread across the lengths of working and retirement periods, respectively. Utility is therefore increasing in χ and λ, but so are the implicit prices on consumption and leisure.

assume, however, that a 1% increase in fertility would increase ρ1(bt) by 1%, because parents need to provide more consumption to more children in the household; i.e. ρ1(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), λ does not depend negatively to the retirement age, and in Chakraborty (2004), λ 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, λ is endogenous, and it depends on changes in the statutory retirement age or changes in the total length of adult life, i.e. λ = φ − χ, 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 first order conditions are derived: first, the Euler equation and, second, the optimality condition for first period consumption and leisure Note that households prioritise less consumption in the first period if fertility decreases. This will lower ρ1(bt) in (6) and (7), such that c1t falls relative to c2t+1 and lt. If χ or λ changes, the optimality conditions will remain unaffected.<sup>6</sup>

# 2.3 Social security

The economy is assumed to operate with a PAYG pension system, given by the following identity, where the left (right) hand side illustrates the pension benefits (contributions). Neither κ nor γ need to be fixed, so the PAYG system can in principle display either defined benefits (DB) or defined contributions (DC) schemes. To reflect the empirical fact that the DB system is the most widespread PAYG arrangement (Gruber and Wise, 1999), we assume that benefits are held constant whereas the contribution rate may vary:<sup>7</sup>

# 2.4 Technology and resources

Output, Yt, is assumed to be produced by firms with a Cobb-Douglas technology in terms of capital, Kt, and labour: 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 defined by rt (kt) = f<sup>′</sup> (kt) and Wt (kt) = f (kt) − ktf<sup>′</sup> (kt), and kt−1 ≡ Kt/ (At−1Lt−1) defines the capital-labour ratio over growth rates.<sup>8</sup> By assuming that firms are identical, capital

> 6 The increase in utility of a longer working or retirement period is offset 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 Kotlikoff et al. (2001), Murphy and Welch (1992) and Welch (1979).

6
will be accumulated through the savings of workers, i.e. Kt+1 = Nt<sup>wSt.Furthermore,</sup> we assume that over one generational period (app. 30 years) capital fully depreciates. The constraint on the economy’s aggregate resources is, which features the lengths of the working and retirement periods, respective, in connection 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 offset any negative responses of labour supply<sup>9</sup> .

A version of the method of undetermined coeffi 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 solution for the recursive equilibrium law of motion. The variables of the linearised model are stated in effi ciency units and in terms of percentage deviations from the steady state (marked with "hats")<sup>10</sup> . A linear law of motion for the recursive equilibrium of the economy is conjectured, which is characterised by linear relationships between endogenous state variables in the vector x�t and exogenous state variables (the shocks) in the vector z�t. The non-state � endogenous (jump) variables are denoted by vt. The coeffi 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,<sup>�</sup> bt−1, or the statutory retirement age, χt, we illustrate the law of motion for leisure, where, e.g., πlχ denotes the elasticity (π) of leisure (l) with respect to the retirement age (χ)<sup>11</sup> .

> 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 {�kt, �c1t, �c2t, �lt, y�t, R�t, w�t, κ�t} can be expressed in this fashion. The � � � e e u complete vector of exogenous state variables is zt ∈{χt−1, χt, �at,<sup>�</sup> bt−1,<sup>�</sup> bt, φ<sup>�</sup> t−1<sup>,φ�</sup> t<sup>,φ�</sup> t<sup>},but(11)only</sup> illustrates the shocks to lagged fertility and the statutory retirement age. The vector of endogenous state variables is {<sup>�</sup> kt, �c2t} so these remain in equation (11) no matter which shocks are examined.

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, πlχ, the size of which, � by construction, assumes a 1% shock to the statutory retirement age, χt. 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 stochastic<sup>12</sup> . 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 fit 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|
|---|---|---|
|α|1/3|The capital share in output|
|γ|0.35|The pension replacement rate|
|a|0.40|The steady state growth rate of productivity|
|δ|1|The rate of capital depreciation|
|χ|1|The length of the working period|
|φ|0.8|The length of total life|
|b|0.1|The rate of growth in the number of children|
|β|1|The weight on leisure in the utility function|
|πρ1(b)|1|The elasticity on the weight of frst period|
|||consumption in utility with respect to the birth rate<br>|
|ρ2|0.292|The consumption discount rate<sup>13</sup>|

Note: The payroll tax rate will then be κ = γ (φ − χ) / (1 + n<sup><u>w</u></sup> ) = 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 effects of low fertility

In this section we analyse some economic effects of a shock to the birth rate in the previous period,<sup>�</sup> bt−1. The assessment focuses on the impact on leisure (<sup>�</sup> lt), workers’ � � consumption (c1t) and retirees’consumption (c2t). Specifically, the economy is represented 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-specified 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 effect on each variable into various sub-effects 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|
|---|---|---|---|
|πc1b1 =|−0.02|=|[πc2k−πRk]πkb1|
|πlb1 =|−0.11|=|πc1b1+ Λ23πκb1−Λ22πwb1|
|πc2b1 =|0.54|=|[Λ15πlb1−Λ3πc1b1−Λ5πkb1−Λ2]/Λ4|
|πkb1 =|−0.02|=|Λ12πwb1−Λ21πlb1−Λ8πκb1<br>Λ9πwk−Λ7πc2k+Λ12πRk−Λ20πlk|

The key issue is how a fertility decline affect the work-leisure choices. A number of counteracting forces are operating, and the net effect remains theoretically ambiguous. However, our numerical simulations imply that leisure will increase by 0.11% after a 1% fertility fall.<sup>14</sup> The increase in leisure corresponds to a reduction in the intensity of labour supply, which will magnify the initial fertility-induced effect on the shrinking effective labour supply and the increasing capital-labour ratio.

Changes in wages and pension contributions basically determine the effects on workers’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 interrelated and indirectly affect the capital-labour ratio: More leisure leads to an even higher capital-labour ratio, higher wages, and lower capital returns (see figures 2a and 2b). Therefore, by examining the intertemporal budget constraint in (5) we can analyse the substitution, income, and wealth effects on leisure.<sup>15</sup> The substitution effect 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 effect on the demand for leisure is negative. A given level of income can now buy less, resulting in negative income effects on all goods, including leisure. The wealth effect is positive for all goods, because the increased wage rate appears in lifetime income.<sup>16</sup> The dynamics of leisure and retirees’consumption are illustrated by the simulated trajectories in figures 2c and 2d, respectively.<sup>17</sup> In an influencial paper, Weil (2006) finds that a key mechanism through which aggregate income and welfare are affected by population ageing is the distortion from taxes to fund PAYG pension systems. This mechanism is also present here: the price on leisure depends on κ, i.e. the (flat) PAYG contribution rate. With labour supply being

> 14 Elasticities are, by construction, derived for a positive 1% shock to fertility. Therefore, the elasticities 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 effect 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 first-period consumption is identical to the simulated trajectory for leisure, though larger numerically.

9
endogenous, this distorting tax rate implies that the positive wealth effect will more than offset the (negative) sum of substitution and income effects (i.e. πlb1 = 0.11)<sup>18</sup> .

Figure 2. Economic effects of a fall in fertility <!-- Start of picture text --> a. Wage rate b. Return to capital<br>Response to a one percent decline in fertility, lagged Response to a one percent decline in fertility, lagged<br>0.35 0<br>Wage rate<br>0.3 -0.1<br>0.25 -0.2<br>0.2 -0.3<br>0.15 -0.4<br>0.1 -0.5<br>0.05 -0.6<br>Returns<br>0 -0.7<br>-1 0 1 2 3 4 5 -1 0 1 2 3 4 5<br>Generational periods after shock Generational periods after shock<br>c. Workers’leisure d. Retirees’consumption<br>6 x 10-4 Response to a one percent decline in fertility, lagged 0.05 Response to a one percent decline in fertility, lagged<br>0<br>5<br>Leisure -0.05<br>4<br>-0.1<br>3 -0.15<br>-0.2<br>2<br>-0.25<br>1<br>-0.3<br>Second period consumption<br>0 -0.35<br>-1 0 1 2 3 4 5 -1 0 1 2 3 4 5<br>Generational periods after shock Generational periods after shock<br>Percent deviation from steady state Percent deviation from steady state<br>Percent deviation from steady state Percent deviation from steady state<br><!-- End of picture text --> There are additional effects to consider in order to obtain a complete analysis of the impacts of low fertility. Due to, first, a changing capital-labour ratio and, second, the presence of distortionary taxation we have to consider "factor price effects" and "fiscal effects", respectively: a negative fertility shock implies that each worker (in the smaller labour force) must pay more taxes (because the benefits to retirees are assumed fixed in a DB system). Thus the fiscal effect is negative. In addition, workers will receive higher wages due to the higher capital-labour ratio so the factor price effect is positive. This net effect is caused by a direct effect and an indirect effect: The population growth rate falls which directly reduces the size of the labour force. The indirect effect is due to the endogenous response of leisure (πlb1 > 0) which has a reinforcing negative effect on labour supply. The implication for effective labour supply is, therefore, that the initial negative effect from lower fertility is amplified by lower intensity of labour supply due to the demand for more leisure as a consumption-equivalent good.

> 18 The distorting effects increase with the size of the pension system, so the larger κ is the larger is, the larger is πlb1. If taxation was lump sum and not distortionary these three effects will offset each other so the net effect on leisure is zero, given that intertemporal elasticity of substitution equal to one, as in our case.

10
The net effect on consumption is consequently ambiguous, but our simulations show that consumption increases for a negative fertility shock: πc1b1 = 0.02 and πc2b1 = −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 effects. 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 effective labour supply by retaining workers in the labour force for a longer period of time — and denying them PAYG pension benefits 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 affect key economic variables.

While the statutory retirement age is exogenous to the consumer, this is not the case for the effective 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 reflects the fact that people may retire earlier based on their own savings and thus represent a fall in the effective 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 specification of the length of the retirement period is residually determined by the length of the working period (λ = φ − χ). As a result, workers need to save less for a shorter retirement period.

Table 3. A shock to the statutory retirement age The change in leisure is determined through the same channels as a fertility shock: the substitution effect, the income effect, the wealth effect, and the fiscal and factor price effects, respectively. These dynamics are all intertwined through both exogenous and endogenous changes in the capital-labour ratio and changes in pension contributions and benefits. The net effect 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 effect on capital returns remains positive and the wage rate will fall. This will indeed be the case since effective labour supply increases by 0.96% because leisure increases by 0.04% for each 1%−increase in the retirement age increases (see table 3).<sup>19</sup> Regarding the fiscal effect: workers now face more subperiods during which they work and has to contribute to the fixed PAYG benefits of retirees. This implies less need for savings to finance a shorter retirement period, so workers save less and free resources for leisure and first-period consumption. So, the fiscal effect is positive.

In terms of substitution, income and wealth effects on leisure, we find that the substitution effect is negative due to the net increase in the price on leisure. The dynamics of factor payments therefore generate a positive wealth effect (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 effect (an unchanged level of income can buy less consumption and leisure since leisure has become more expensive). The positive wealth effect offsets the negative sum of substitution and income effects, partly due to distortionary taxation, so the effect on leisure is positive.<sup>20</sup> A particularly important mechanism in this model is that we account for the disutility 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 effective 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 effect; the fiscal effect; and the endogenous intensity of labour supply (determined, in turn, by substitution, income and wealth effects). 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.<sup>21</sup> 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 confirmed by the elasticities of the wage rate and capital returns with respect to the retirement age (πwχ = α (1 − πlχ) = −0.32; πRχ = (1 − α) (1 − πlχ) = 0.64) which represents a negative (positive) factor price effects for workers (retirees). The direct effect on, e.g., capital returns is (1 − α) due to the fall in the capital-labour ratio, while the indirect effect originating from endogenous labour supply is (1 − πlχ).

> 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 first-period consumption theoretically ambiguous. We find that πc1χ = 1.06 is positive, however, and that it depends, especially, on the need for less savings to finance 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 effect is ambiguous, but our simulations show an increase in πc2χ.

> 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 effects<sup>22</sup> .

The effective labour supply comprises three elements: first, the fertility rate,<sup>�</sup> bt−1; second, the extensive margin limited by the retirement age, χ�t; and third, the intensity � with which workers work (the intensive margin, ut = −<sup>�</sup> lt). The effective labour supply is dt = (1 + n<sup>w</sup> t<sup>) (1 −lt),orinlog-deviationsfromsteadystate:</sup> Assume first 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 � statutory retirement age, which would offset the fertility decline, i.e. d<sup>�</sup> t = χt +<sup>�</sup> bt−1 ≡ 0, � would just be a proportional increase of χt = 1%. However, if the intensity of labour supply is in fact endogenous, so<sup>�</sup> lt = 0, then clearly the response of χ�t would have to be different from 1%. In our case, the initial effect from the fertility decline on the effective labour supply will be reinforced because leisure increases, so the statutory retirement � age would have to increase even more than 1%. To derive the offsetting response of χt we insert the linear law of motion for<sup>�</sup> lt to obtain: From (13) isolate �χt, and insert the numerical elasticities, πlb1 and πlχ, and the negative fertility shock,<sup>�</sup> bt−1 = −1: Observe that if πlb1 < πlχ the optimal response is χ�t > 1. So, we conclude that the statutory retirement age has to increase more than fertility fell in order to offset the negative impact on the effective labour force. The offsetting response of the statutory retirement age, when<sup>�</sup> bt−1 = −1% and the weight on leisure in utility is β = 3, amounts � to χt = 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 amplified. Since the weight that households 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 suggests various values for β generally within the range β ∈{1; 9} (see, e.g., Blackburn and Cipirani, 2002; Cardia, 1997; Chari et al., 2000; Jonsson, 2007). We have calibrated our model with β = 3, as an example, and found the offsetting response of the � statutory retirement age to be larger than the fertility rate (χt = 1.15). In terms of robustness analysis, however, we simulate the value for χ�t given alternative values for β and illustrate the results in figure 3.

> 22 Jensen and Jorgensen (2008) evaluates the attractiveness of an uneven distribution of the economic effects associated with low fertility in a model with exogenous labour supply, while Jorgensen (2008) does so in a model with endogenous labour supply.

13
<!-- Start of picture text --> Offsetting response<br>of the retirement age re<br>: OFFSET<br>0 B<br>1.6 Preference for leisure<br>approx.<br><!-- End of picture text -->
more time has been allocated to human capital investment, by spending more years within the educational system, the empirical evidence clearly suggests that the preference 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 offsetting 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 significant 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 finding 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 offsetting 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 finding, that leisure may increase when the statutory retirement age increases, could be interpreted as an endogenous drop in the voluntary early retirement age, financed 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 offsetting increase in the statutory retirement age.

The analytical framework is subject to a number of limitations. The utility function 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 specifications 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-specific, foundation for designing a policy rule for the retirement age. Last but not least, human 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 offset the smaller labour force. These issues may modify our results, and are interesting subjects for future research.

15