metadata: authors: 0: author: Jensen,Svend Erik Hougaard
1: author: Jorgensen,Ole Hagen
count: World,Brazil
docna: 0: docna: Labor supply and retirement policy in an
            overlapping generations model with stochastic fertility
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 a
            policy rule for the retirement age aiming at offsetting the
            effects on the supply of labor following fertility changes.
            The authors find that the retirement age should increase
            more than proportionally to the direct fall in labor supply
            caused by a fall in fertility. The robustness of this result
            is checked against alternative model specifications and
            parameter values. The efficacy of the policy rule depends
            crucially on the link between the preference for leisure and
            the response of the intensive margin of labor supply to
            changes in the statutory retirement age. The model has
            subsequently been calibrated for Brazil by Jorgensen (2010),
            in the context of the Brazil Aging Study.
geo_reg_key: 119228,517191
repnme: repnme: Labor supply and retirement policy in an
            overlapping generations model with stochastic fertility
display_title: Labor supply and retirement policy
            in an overlapping generations model with stochastic fertility
listing_relative_url: /research/2010/07/12588833/labor-supply-retirement-policy-overlapping-generations-model-stochastic-fertility
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: WPS5382 Policy Research Working Paper 5382 Labor Supply and Retirement Policy in an Overlapping Generations Model with Stochastic Fertility _Ole Hagen Jorgensen Svend E. Hougaard Jensen_ The World Bank Latin American and Caribbean Region Economic Policy Sector July 2010
Policy Research Working Paper 5382

# **Abstract**

Using a stochastic general equilibrium model with overlapping generations, this paper studies a policy rule for the retirement age aiming at offsetting the effects on the supply of labor following fertility changes. The authors find that the retirement age should increase more than proportionally to the direct fall in labor supply caused by a fall in fertility. The robustness of this result is checked against alternative model specifications and parameter values. The efficacy of the policy rule depends crucially on the link between the preference for leisure and the response of the intensive margin of labor supply to changes in the statutory retirement age. The model has subsequently been calibrated for Brazil by Jorgensen (2010), in the context of the Brazil Aging Study.

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 economic implications of population dynamics in less developed economies, as well as being a contribution to the Brazil Aging Study (http://go.worldbank.org/ NMFNG40F30). 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
Labor Supply and Retirement Policy in an Overlapping Generations Model with Stochastic Fertility<sup>�</sup> Ole Hagen Jorgensen and Svend E. Hougaard Jensen<sup>y</sup> July 27, 2010 Abstract JEL Classi...cation: D91; E20; H55; J10; J26 Keywords: Labor supply; fertility; retirement age; overlapping generations; method of undertermined coe¢ cients.

> �For valuable comments to this paper, we would like to thank David Bloom, Guenther Fink, Martin Flodén, Bo Sandeman Rasmussen, Michele Gragnolati, Tito Cordella, Jan Walliser, Rodrigo Chaves, Makhtar Diop, Lars Lønstrup, and Casper Hansen.

> yJorgensen: The World Bank, Poverty Reduction and Economic Management, LCSPE, 1818 H Street, Washington D.C., 20433, E-Mail: ojorgensen@worldbank.org. Jensen: CEBR, Copenhagen Business School, Porcelaenshaven 16A, DK-2000 Frederiksberg, E-mail: shj.cebr@cbs.dk.

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# 1 Introduction

As a result of low fertility rates since the 1960s, the labor supply in most developed countries is likely to shrink over the coming decades. Since a smaller labor force increases the capital-labor ratio during the transition to a new steady state, an upward pressure on wages will arise, so the work-leisure choice becomes important.<sup>1</sup> The fertility rates did bounce back, however, so the nature of the shock is temporary. As a result, labor forces will eventually come close to their previous sizes, but during the transition – which will be about two to three decades – there is likely to be important transitional dynamics. In fact, workers might react quite strongly to a higher opportunity cost of leisure by demanding less of it, thus countering the e¤ects of the fertility fall. However, with a strong income e¤ect, the net e¤ect of higher wages might be an increased demand for leisure, so the e¤ective supply of labor would be reduced even further.

In this paper we present a model that captures these economic relationships. The analytical framework is a dynamic stochastic general equilibrium (DSGE) model with overlapping generations. In a three-period setup, we formulate the explicit relationship between the extensive and the intensive margin of labor supply, where the statutory retirement age functions as a proxy for the extensive margin. In line with Chakraborty (2004), we make the length of the retirement period residually determined by, ...rst, the length of the working period and, second, the total length of life. The novelty of this approach is the feasibility of deriving the implications on the intensive margin of labor supply to a change in the statutory retirement age.

The analytical framework is augmented by a policy rule for the statutory retirement age that e¤ectively counteracts the decline in the size of the labor force caused by a lower fertility rate. Indeed, a straightforward policy response would be to increase the statutory retirement age in order to retain workers in the labor force for a longer period of time. The paper studies how such a link between changes in fertility and the statutory retirement age can be established. Our approach facilitates an analytical presentation, where the role of each model parameter can be identi...ed. For example, analytical expressions can be derived that links the e¤ect on leisure to changes in, respectively, the statutory retirement age, the fertility rate and the preference for leisure.

By assuming that the statutory retirement age is under government control, it is possible to analyze the o¤setting response of the statutory retirement age under alternative demographic and economic contingencies. Our main result is that an increase in the retirement age has the potential to o¤set the (fertility induced) decline in the labor force, provided that the retirement age increases more than proportionally to the fall in fertility. The reason is that workers substitute for leisure both when fertility falls and the retirement age increases. Furthermore, an implication of the increase in the retirement age is that lifetime leisure will fall, and this further increases the demand for leisure during the working period. As a result, labor supply will fall not only due to low fertility but also as a side e¤ect of the increase in the retirement age. Consequently, policy makers should account for this endogenous response of labor supply when formulating the optimal policy to alleviate

> 1 Weil (2006) ...nds that the distortion created by taxes needed to fund PAYG pension systems is a key channel through which a higher dependency ratio a¤ects aggregate output and welfare.

2
=X x Adult lifetime (¢) aWork (7) rs Retirementees(1) x
one unit, u 2 (0; 1), where Lt = utNt<sup>w,andutistheintensityoflaborsupplyinthe</sup> working period. First period leisure therefore equals lt = 1 � ut.

The aggregate measure of adult lifetime, �, is the sum of the lengths of the working period and the retirement period, respectively. Thus, retirement, � 2 (0; 1), is residually determined as: More speci...cally, the total length of life, �; comprises an expected component and an unexpected component, i.e. �t = �<sup>e</sup> t�1<sup>�u</sup> t<sup>,wheref�g2(0; 2).Weassumethat</sup> these components �t are stochastic and identically and independently distributed. The same is the case for bt and �t. Changes in � will exclusively a¤ect � if � remains constant. Based on this formulation, we assume that an increase in � is equivalent to an increase in the retirement age. Note that changes in e¤ective labor 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, ut; and, third, the e¤ect from the exogenous growth in the number of workers, bt�1. It is common in the literature to endogenize the intensity of labor supply, but to combine this with changes in labor supply at the extensive margin has, to our knowledge, not previously been attempted<sup>3</sup> .

# 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<sup>4</sup> .

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<sup>5</sup> .

> 3 As a result, if fertility falls by, for instance, 1% and the respons by the government is to increase the statutory retirement age by 1% both these events may lead to an increased demand for leisure by households and thus a fall in labor supply. The 1% increase in the retirement age may therefore not be enough to counteract the general equilibrium e¤ects that induce workers to reduce their labor supply. This result is clearly ambiguous and depends on parameter calibration as well as substitution, income and wealth e¤ects, respectively.

> 4 A CES utility function could be speci...ed and experiments be made with alternative values for the elasticity of substitution.

> 5 An explicit formulation of the optimisation of parents’ utility over their own consumption and that of their children is not necessarily important. This is because the optimisation problem would merely relate ...rst period consumption of the household to the weight that parents assign to consumption of their children in utility. The childhood period is conceptually necessary in this model, though, in order to study a change in fertility in period t � 1 that a¤ects the size of the labor force in period t. 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 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.

4
Second period consumption is scaled by the length of the retirement period<sup>6</sup> . 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 labor 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 labor 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 labor less intensively<sup>7</sup> . The restrictions on c1t and c2t+1 are presented in (3) and (4), 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)<sup>8</sup> .

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. In that way, if the retirement age increases so the retirement period is residually reduced, there will be more second period income in each sub-period in retirement. 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 paper, however, � is also endogenous, but it depends on changes in the retirement age or changes in the total length of adult life, i.e. � = � � �. Changes in � is therefore seen to a¤ect �, and that could not be analyzed by neither Bohn (2001) nor Chakraborty (2004).

Combining c1t and c2t+1 over St yields the intertemporal budget constraint: Note the roles of � and � as implicit prices on consumption and leisure: consumption and leisure must be spread across the lengths of working and retirement

> 6 Both Chakraborty (2004) and Bohn (2001) have incorporated the length of the retirement period into the utility function, but neither have incorporated the length of the working period.

> 7 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.

> 8 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
periods, respectively. Utility is therefore increasing in � and �, but so are the implicit prices on consumption and leisure.

By maximizing lifetime utility (2) subject to the intertemporal budget constraint (5), two ...rst order conditions are derived: ...rst, the Euler equation in (6), and, second, the optimality condition for ...rst period consumption and leisure in (7): Note, that households prioritise less consumption in the ...rst period if fertility decreases. 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<sup>9</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 bene...ts (contributions). Neither � nor � need to be ...xed, so the PAYG system can in principle display either de...ned bene...ts (DB) or de...ned contributions (DC) schemes. To re‡ect 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<sup>10</sup> :

# 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 labor: 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<sup>0</sup> (kt) and Wt (kt) = f (kt) � ktf<sup>0</sup> (kt), and kt�1 � Kt= (At�1Lt�1) de...nes

> 9 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.

> 10 Evidently, 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.

6
the capital-labor ratio over growth rates.<sup>11</sup> By assuming that ...rms are identical, capital will be accumulated through the savings of workers, i.e. Kt+1 = Nt<sup>wSt.</sup> Furthermore, 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 loglinearized 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 labor supply<sup>12</sup> .

A version of the method of undetermined coe¢ cients, which relies on Uhlig (1999) and extended by Jorgensen (2008), is adopted to obtain the analytical solution for the recursive equilibrium law of motion. The variables of the linearized model are stated in e¢ ciency units and in terms of percentage deviations from the steady state (marked with "hats")<sup>13</sup> . A linear law of motion for the recursive equilibrium of the economy is conjectured, which is characterized by linear relationships between endogenous state variables in the vector xbt and exogenous state variables (the shocks) in the vector zbt. The b non-state 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 b e.g. lagged fertility,<sup>b</sup> bt�1, or the statutory retirement age, �t, we illustrate the law of motion for leisure, where, e.g., �l� denotes the elasticity (�) of leisure (l) with respect to the retirement age (�)<sup>14</sup> .

> 11 Since a smaller labor force leads to an increase in the capital-labor ratio, changes in factor returns are likely to occur, see Kotliko¤ et al. (2001), Murphy and Welch (1992) and Welch (1979).

> 12 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.

> 13 See Appendix A for more details on the solution technique.

> 14 All endogenous variables fbkt; bc1t; bc2t; blt; ybt; Rbt; wbt; �btg can be expressed in this fashion. The b b b b e e u complete vector of exogenous state variables is zt 2 f�t�1; �t; at;<sup>b</sup> bt�1;<sup>b</sup> bt; �<sup>b</sup> t�1<sup>;�b</sup> t<sup>;�b</sup> t<sup>g,but</sup>

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 b which, by construction, assumes a 1% shock to the statutory retirement age, �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 stochastic<sup>15</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 ...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 labor force. We calibrate the analytical expressions of the model with values, in table 1, that we trust are realistic, and subsequently derive the numerical elasticities of the model. Importantly, we make robustness analyzes with the weight on leisure in the utility function in section 4, since the model predictions depend crucially on the calibration of this parameter.

|Parameter|Value|Interpretation of steady state parameters|
|---|---|---|
|�|1=3|The capital share in output<br>|
|�|0:35|The pension replacement rate<sup>16</sup>|
|a|0:40|The steady state growth rate of productivity|
|�|1|The rate of capital depreciation|
|�|1|The length of the working period|
|�|1|The length of the retirement period|
|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 rate<sup>17</sup>|

Table 1. Parameter calibration

# 3 Equilibrium with Shocks

How will a fall in fertility a¤ect the e¤ective supply of labor? Obviously, the size of the labor force falls with the decline in past fertility. Labor may be supplied by households more or less intensively, though, when fertility has fallen. In addition, if the retirement age rises in response to a shrinking labor force this, in itself, may also lead to a more or less intensive supply of labor.

> (11) only illustrates the shocks to lagged fertility and the statutory retirement age. The vector of endogenous state variables is f<sup>b</sup> kt; bc2tg so these remain in equation (11) no matter which shocks are examined.

> 15 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.

> 16 The payroll tax rate will then be � = � (� � �) = (1 + nw) = 0:30.

> 17 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
b Two shocks are examined, namely an exogenous shockb to the lagged birth rate, bt�1, and an exogenous change in the retirement age, �t. We the derive the elasticities of economic variables with respect to these two shocks. In this section we assume that changes in the statutory retirement age are outside government control –as if the changes were stochastic. This provides insights as to how the economy and intergenerational distributions of welfare are a¤ected by demographic shocks under a passive policy framework. In section 4, however, the statutory retirement age is assumed to be under government control and used as a policy instrument<sup>18</sup> . We limit the focus on leisure (<sup>b</sup> lt) and consumption (bc1t) for workers, as well as retirees’ b consumption (c2t).

# 3.1 E¤ects of Low Fertility

In this section we analyze the macroeconomic impacts of the signi...cant historical declines in fertility rates. A lagged shock to fertility is therefore the focus. The economy is represented by a linear law of motion in terms of elasticities for endogenous variables with respect to a positive 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 get a better understanding of the magnitudes involved in the numerical simulations.

Table 2. A fall in fertility

|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|
|�kb1 =|�0:02|=|�12�wb1��21�lb1��8��b1<br>�9�wk��7�c2k+�12�Rk��20�lk|

The key issue is how work-leisure choices will be determined subsequent to the fertility decline. This result is subject to a number of counteracting e¤ects and remains theoretically ambiguous. Our simulations imply, however, that leisure will increase by 0.11% after a 1% fertility fall<sup>19</sup> . The increase in leisure corresponds to a reduction in the intensity of labor supply, which will magnify the initial fertilityinduced e¤ect on the shrinking e¤ective labor supply and the increasing capital-labor ratio.

Changes in wages and pension contributions basically determine the e¤ects on workers’consumption after the shock to fertility (see Jensen and Jorgensen, 2008). On the other hand, since labor supply is a choice-variable, consumption and leisure are interrelated and indirectly a¤ect the capital-labor ratio: more leisure leads to an

> 18 In the present section, we are interested only in the e¤ects of a change in the retirement age and not in what causes that change. It is often more natural to think of a shock to the statutory retirement age as exogenous to the agent and controlled by the government.

> 19 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.

9
even higher capital-labor ratio, higher wages, and lower capital returns (see ...gures 2a and 2b). Therefore, by examining the intertemporal budget constraint in (5) we can analyze the substitution, income, and wealth e¤ects on leisure<sup>20</sup> .

Figure 2a. Wage rate Figure 2b. Return to capital <!-- Start of picture text --> 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>Figure 2c. Workers’leisure Figure 2d. 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 --> The substitution e¤ect on leisure comes from a shrinking labor force that alters factor payments: wages increase; the return to capital falls. The price (opportunity cost) of leisure thus increases so the substitution e¤ect on 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<sup>21</sup> . The dynamics of leisure, as well as retirees’consumption, are illustrated by the simulated trajectories in ...gures 2c and 2d, respectively<sup>22</sup> .

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 leisure depends on �; i.e. the (‡at) PAYG contribution rate. With labor supply being endogenous, this distorting tax rate implies that the positive wealth

> 20 In the case where labor supply is exogenous (see, e.g., Jensen and Jorgensen, 2008), the only e¤ect on the capital-labor ratio originates from the lower fertility rate.

> 21 See the right-hand side of the intertemporal budget constraint in equation (5).

> 22 The dynamics of ...rst-period consumption is identical to the simulated trajectory for leisure, though larger numerically.

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e¤ect will more than o¤set the (negative) sum of substitution and income e¤ects (i.e. �lb1 = 0:11)<sup>23</sup> .

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-labor 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 labor 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-labor 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 (1 + n<sup>w</sup> t<sup>)fallswhichdirectlyreducesthesize</sup> of the labor force. The indirect e¤ect is due to the endogenous response of leisure (�lb1 > 0) which has a reinforcing negative e¤ect on labor supply. The implication for e¤ective labor supply is, therefore, that the initial negative e¤ect from lower fertility is ampli...ed by lower intensity of labor supply due to the demand for more leisure as a consumption-equivalent good.

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.

# 3.2 E¤ects of a Higher Statutory Retirement Age

The statutory retirement age can be used as a policy instrument to increase e¤ective labor supply by retaining workers in the labor 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. We are only interested in the e¤ects of an exogenous change in the retirement age and not in what causes the change<sup>24</sup> . When we are well informed about the implications of a change in the retirement age we move on, in the next section, to present a normative analysis of the retirement age –based on the assumption that the statutory retirement age is under government control.

An increase in the retirement age will tend to directly increase labor supply and lower the length of the retirement period, which is in line with our speci...cation of

> 23 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.

> 24 While we assume, in this section, that the statutory retirement age is exogenous to the consumer, this will not be the case for the e¤ective retirement age, because the intensity of labor supply is endogenous to the household in this paper. If the statutory retirement age increases, no matter why, households may decide to supply less labor. If one assumes that this reduction in labor 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.

11
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.

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-labor ratio and changes in the pension contributions and bene...ts. The net e¤ect on the capital-labor ratio is negative if the intensity of labor supply does not endogenously fall more than the retirement age has increased. In that case the net e¤ect on capital returns remains positive and the wage rate will fall. This will endeed be the case since the e¤ective labor supply increases by 0:96% because leisure increases by �l� = 0:04% for each 1%�increase in the retirement age increases (see table 3)<sup>25</sup> .

Table 3. An increase in the retirement age

|Variable|Value||Elasticity|
|---|---|---|---|
|�c1�=|1:06|=|[�c2k��Rk]�k�+ (�c2�1��R�1)|
|�l�=|0:04|=|[�c2k��Rk]�k�+(�c2�1��R�1)+(�23���+�22�11)<br>1+�22�11|
|�c2�=|0:42|=|[�9�wk��7�c2k+ �12�Rk��20�lk]�k�<br>��21�l�+ �12�w���8���|
|�k�=|�0:31|=|�15�l���3�c1���4�c2���2<br>�5|

Regarding the ...scal e¤ect: workers now face more subperiods during which they work and have 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. Thus, a positive ...scal e¤ect.

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 generates 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<sup>26</sup> .

> 25 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-labor ratio, while the indirect e¤ect originating from endogenous labor supply is (1 � �l�).

> 26 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�.

12
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 labor 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 labor supply when fertility has declined. This complicates the analysis of an o¤setting policy rule for the statutory retirement age. That is precisely why it is crucial to emphasize the dynamics of the intensive margin of labor supply relative to the extensive margin. This is the topic of the next section.

# 4 Policy Reform

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 labor supply (determined, in turn, by substitution, income and wealth e¤ects). In this section, we make use of our general equilibrium framework to derive how much the statutory retirement age should increase in order to o¤set the decline in the labor force caused by low fertility in the past<sup>27</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 analyzes are independent of the social desirability of any intergenerational (welfare) distribution of the associated e¤ects<sup>28</sup> .

The e¤ective labor supply comprises three elements: ...rst, the fertility rate,<sup>b</sup> bt�1; second, the extensive margin limited by the retirement age, �bt; and third, the intenb sity with which workers work (the intensive margin, ut = �<sup>b</sup> lt). The e¤ective labor supply is dt = (1 + n<sup>w</sup> t<sup>) (1 �lt),orinlog-deviationsfromsteadystate:</sup> Assume ...rst that the intensity of labor supply is exogenous and that we examine a 1% decline in fertility. It is then clear from (12) that the necessary responseb of the statutory retirement age, which would o¤set the fertility decline, i.e. dt = b b �t +<sup>b</sup> bt�1 � 0, would just be a proportional increase of �t = 1%. However, if the intensity of labor supply is in fact endogenous, so<sup>b</sup> lt = 0, then clearly the response of b �t would have to be di¤erent from 1%. In our case, the initial e¤ect from the fertility decline on the e¤ective labor 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 �bt we insert the linear law of motion for<sup>b</sup> lt to obtain: From (13) isolate �bt, and insert the numerical elasticities, �lb1 and �l�, and the negative fertility shock,<sup>b</sup> bt�1 = �1:

> 27 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".

> 28 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 labor supply, while Jorgensen (2008) does so in a model with endogenous labor supply.

13
_— TT B x ~~——~~ xX — ~~|~~

_ x xX <!-- 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 --> B x —K Wx
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 labor 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 ...gure 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, labor 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 labor 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 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 labor force, will generate more demand for leisure as real wages increase. This implies that the o¤setting response of the labor force will be in a more than 1:1 relationship to the contraction in the labor force. A model which does not incorporate labor supply as a choice variable may fail to capture some important macroeconomic dynamics. The ability to analyze the impacts of shrinking labor forces for various values for the preference for leisure thus marks a signi...cant extension of the framework used by, e.g., Bohn (2001). Such an analysis would not be feasible without the explicit relationship in the model between the extensive and intensive margins of labor supply.

# 5 Conclusion

This paper has developed an intertemporal setting in which retirement policy can be used to correct for fertility-induced changes in the supply of labor. 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 labor 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 labor 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 15
an analytical framework linking the endogenous intensive margin to the extensive margin of labor supply, this analysis would not be feasible.

An additional ...nding is that leisure may increase when the statutory retirement age increases. This 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 function has been modeled in accordance with our best beliefs of how to incorporate the value of leisure and the lengths of periods. However, the robustness of our result could be examined in greater detail for alternative speci...cations of the utility function: for example, by adopting a CES speci...cation that allows for robustness analyzes with respect to the elasticity of substitution. In addition, we assume that the economic impacts of changes in dependency ratios can be analyzed in a linearized 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, 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 labor may incorporate a higher productivity. Thus, there may be less need for the statutory retirement age to increase to completely o¤set the smaller labor force. These issues may modify our results, and are interesting subjects for future research.

# References

- [1] Andersen, T. M., S. E. H. Jensen and L. H. Pedersen (2008), The Welfare State and Strategies towards Fiscal Sustainability in Denmark, in Neck, R. and J.-E. Sturm (ed.), Sustainability of Public Debt, MIT Press.

- [2] 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.

- [3] Bohn, H. (2001), Social Security and Demographic Uncertainty: the RiskSharing Properties of Alternative Policies, in J. Campbell and M. Feldstein (ed.), Risk Aspects of Investment Based Social Security Reform, University of Chicago Press, 2001, pp. 203-241.

- [4] Cardia, E. (1997), Replicating Ricardian Equivalence Tests with Simulated Series, The American Economic Review, Vol. 87, Issue 1, pp. 65-79.

- [5] Chakraborty, S. (2004), Endogenous Lifetime and Economic Growth, Journal of Economic Theory, Vol. 116, pp. 119-137.

- [6] 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, No. 5, pp. 1151-1179.

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- [7] Cutler, D. (2001), Comment to Bohn, 2001, in J. Campbell and M. Feldstein (ed.), Risk aspects of investment based social security reform, University of Chicago Press, 2001, p. 245.

- [8] 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(4), pp. 607-636.

- [9] Diamond, P. A. (1965), National Debt in a Neoclassical Growth Model, The American Economic Review, Vol. 55 (December), pp. 1126-1150.

- [10] Gruber, J. and D. Wise (1999) Social Security Around the World, The University of Chicago Press.

- [11] Jensen, S. E. H. and O. H. Jorgensen (2008), Uncertain demographics, longevity adjustment of the retirement age and intergenerational risk-sharing, in Alho, J., S. E. H. Jensen and J. Lassila (eds.), Uncertain Demographics and Fiscal Sustainability, Cambridge University Press.

- [12] Jonsson, M. (2007), The Welfare Cost of Imperfect Competition and Distortionary Taxation, Review of Economic Dynamics, Vol. 10, pp. 576-594.

- [13] Jorgensen, O. H. (2008), Retirement Indexation in a Stochastic Model with Overlapping Generations and Endogenous labor Supply, CEBR Discussion Paper, No. 2008-03. Centre for Economic and Business Research, Copenhagen, Denmark.

- [14] Jorgensen, O. H. (2010), Aging and Inequality in Brazil: The Implications for Saving and Growth, The World Bank, PREM (Paper for the Brazil Aging Study)<sup>29</sup> .

- [15] Keilman, N., H. Cruijsen and J. M. Alho (2008), Changing views on future demographic trends, in Alho, J., S. E. H. Jensen and J. Lassila (eds.), Uncertain Demographics and Fiscal Sustainability, Cambridge University Press.

- [16] Kotliko¤, L. J., K. Smetters and J. Walliser (2001), Finding a Way Out of America’s Demographic Dilemma, NBER Working Paper, No. 8258, April 2001.

- [17] Murphy, K. and F. Welch (1992), The Structure of Wages, Quarterly Journal of Economics, Vol. 107, pp. 407-437.

- [18] Pencavel, J. (1986), labor Supply of Men: A Survey, in Ashenfelter, O. and R. Layard, Handbook of Labor Economics, Vol. 1, Elsevier Science

- [19] Schmidt-Sørensen, J. B. (1985), Arbejdstidsforkortelse i historisk perspektiv, Økonomi & Politik, Vol. 59(1), 2-11.

- [20] 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.

- 29 Available upon request from the author: Ole Hagen Jorgensen, ojorgensen@worldbank.org, or

- at: http://go.worldbank.org/NMFNG40F30

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- [21] Weil, D. (2006), Population Aging, NBER Working Paper, No. 12147, March 2006.

- [22] 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.

# 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-linearized model, ebt 2 f<sup>b</sup> kt; bc2t; bc1t;<sup>b</sup> lt; ybt; R<sup>b</sup> t; wbt; �btg, are written as linear functions of a vector of endogenous and exogenous state variables, respectively. The vector of endogenous state variables is bxt 2 f<sup>b</sup> kt; bc2tg of size m�1<sup>30</sup> , the vector of endogenous non-state variables is vbt 2 fbc1t;<sup>b</sup> lt; ybt; R<sup>b</sup> t; wbt; �btg of size b e j �1, while the vector of exogenous state variables is bzt 2 f�t�1; b�t; bat;<sup>b</sup> bt�1;<sup>b</sup> bt; �<sup>b</sup> t�1<sup>;</sup> �bet<sup>;�b</sup> ut<sup>gofsizeg �1.Thelog-linearizedequationsareinwrittenmatrixnotationin</sup> the following equilibrium relationships, where C is of size h�j, where h denotes the number of non-expectational equations. b b In this particular OLG model h = j, due to the de...nition of xt = f<sup>b</sup> kt; c2tg, because with merely the capital stock as a state variable h < j, and the system cannot not be solved<sup>31</sup> . 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 between endogenous variables in the vector ebt, and state variables (endogenous and exogenous, respectively) in the vectors vbt and zbt. The conjectured linear law of motion is written as, 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 alternatively be written out for each variable in ebt., as e.g for leisure,<sup>b</sup> lt,

> 30 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).

> 31 Note that if h > j the equations in this section become slightly more complicated, see Uhlig (1999), but a solution is still feasible.

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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.

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: 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 expression �<sup>�1</sup> � for eigenvalues-eigenvectors, as in (20). Then, �<sup>�1</sup> � is diagonalized in (21) since each eigenvalue, �i, can be associated with a given eigenvector, qm.

The matrix �<sup>�1</sup> � =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