Actuarial Adjustment Aspects of Public Pension Schemes Kamil Galuščák
Copyright © 2000 Kamil Galuščák
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Contents Abstract ......................................................................................................................................................................................1 1. Introduction...........................................................................................................................................................................2 2. Survey of Literature .............................................................................................................................................................4 3. Analysis of the Actuarial Adjustment...............................................................................................................................6 3.1. General Framework ......................................................................................................................................................6 3.2. Uncertain Lifetime ........................................................................................................................................................8 4. Results ..................................................................................................................................................................................10 4.1. Calibration....................................................................................................................................................................10 4.1.1. Lifetime uncertainty...........................................................................................................................................10 4.1.2. Risk aversion .......................................................................................................................................................12 4.2. Optimal delayed retirement credit............................................................................................................................12 5. Policy Implications and Concluding Remarks ..............................................................................................................15 References................................................................................................................................................................................17 A. Appendix.............................................................................................................................................................................18 Labor Supply Incentives of the Pension Scheme in the Czech Republic ......................................................................32 Abstract................................................................................................................................................................................32 1. Introduction.....................................................................................................................................................................32 2. Description of the Pension System..............................................................................................................................32 2.1. The Czech Pension System After the 1996 Changes.......................................................................................33 2.2. The Pension System in the Czech Republic Prior to the 1996 Changes ......................................................34 3. Labor Supply Incentives of the Pension Scheme......................................................................................................35 3.1. Characteristics of the Pension Scheme ...............................................................................................................35 3.2. Analysis of Optimal Parameters in the Cases of Late Retirement and the Earnings Test.........................37 4. Concluding Remarks .....................................................................................................................................................39 References ...........................................................................................................................................................................40
1
Abstract
Abstract It has been shown in economic research that public pay-as-you-go defined-benefit pension plans penalize those who continue to work beyond a certain age by reducing the present discounted value of future retirement benefits. In discussions on the effectiveness of policies aimed at eliminating this age-dependency in worker retirement decisions, it is often assumed either that the benefits in all future periods have the same weight in the present discounted value or that the discount rate is close to unity due to low real interest rates used in this case. In this paper we show, using the example of the U.S. pension scheme, that discounting plays a crucial role, since the formula for the present discounted value of future retirement benefits is sensitive to the discount rate used. Using discount rates derived from real interest rates in 1997, we find that the annual delayed retirement credit by which retirement benefits should be increased to compensate for lost benefits is at least 25% greater than it is in the case when discounting is neglected. Moreover when accouting for risk aversion towards lifetime uncertainty, the optimal delayed retirement credit is increased by 15-25%. Our results indicate that the U.S. pension scheme is not age neutral between ages 62 and 65. This may explain the peak in labor force withdrawal observed at age 62.
2
1. Introduction
1. Introduction The labor market behavior of older workers is attracting the attention of many economists now that policies aimed at affecting such behavior have become an important part of policy-related debates. Declining labor force participation in the postwar U.S. economy (Peracchi and Welch, 1994) and in other developed ecomomies, along with increasing life expectancy, have caused fiscal tensions in financing public pensions.1 In order to combat these difficulties, policies to increase labor force participation of older workers and to encourage late retirement have been introduced in the last two decades.2 Recent evidence on the labor market behavior of older workers in the U.S. and some other OECD countries (Smeeding and Quinn, 1997) has shown that the downward trend in their labor force participation stopped in the mid-1980s and that even a slight increase was observed thereafter. This has revived discussions about the effectiveness of pro-work policies. Economic research has shown that public pay-as-you-go defined-benefit (DB) pension plans in which the amount of the benefit is determined by wages in the few years prior to retirement and the number of years of contributing to the pension plan have important retirement incentives or disincentives beyond a certain age (Stock and Wise, 1990). Many DB plans penalize workers who continue working beyond a certain age by reducing the present discounted value of their future retirement benefits. Pro-work policies, however, incorporate changes which eliminate these incentives to retire and may even introduce an additional incentive to stay in the labor market. The parameter of interest here is the actuarial adjustment rate at which the unpaid benefits of an individual who is eligible for and does not receive a pension are compensated in the future. The idea is to keep the present discounted value of future retirement benefits the same, eliminating its dependency on the age of retirement. The relation between discount rates and real interest rates can be explained both by investing and borrowing in capital markets.3 Assume the length of life is known. When real interest rates r are zero or negative, the discount rate β equals or is greater than unity, such that β = 1 / ( 1 + r ) . If there are positive real interest rates, individuals may prefer to have access to their financial sources in the current time period rather than in the future, and earn the real interest rate. The discount rate at which future retirement benefits are evaluated falls below unity in this case. We may conclude that retirement is less likely to be delayed when high positive real interest rates are observed. An alternative explanation of the relation between discount rates and real interest rates comes from the side of borrowing. An individual may choose to postpone retirement and borrow in capital markets to achieve his desired financial path. The rate at which future retirement benefits are discounted is determined by real interest rates on borrowing in this case. Higher real interest rates on borrowing decrease the willingness to delay retirement. These two explanations rely on three assumptions. Firstly, there are no fixed costs of work so that a worker can freely choose the labor supply. Secondly, there are no constraints on borrowing and lending such that a worker is able to achieve the desired financial path in capital markets, and finally, retirement benefits are indexed to inflation. Both the explanations provide the same predictions on the size of the discount rate if real interest rates on borrowing and lending are the same. Based on these assumptions, we may evaluate the costs and benefits of introducing a policy that removes the age dependency of the actuarial adjustment. Suppose that there is a disincentive to retire beyond a certain age such that all workers retire at that age. Suppose further that a delayed retirement credit such that the present discounted value of future retirement benefits does not depend on the age of retirement is introduced. In this case, there are no additional costs to introducing this policy. In fact, 1
World Bank Report (1994) describes potential problems of the ageing population in the world-wide context and presents proposals to reform pension systems. 2 3
For an overview of these policies introduced in the U.S., see Burkhauser and Quinn (1997).
Investing in capital markets means lending available financial sources by individuals in the form of deposits, for example. Borrowing, on the other hand, takes the form of loans and credits.
3
1. Introduction
the increased labor supply raises tax revenues from additonal work as well as state revenues from the social security insurance payments. This implies that such a pro-work policy should have favorable advantages for the state financing of pensions. The example given above does not account for post-retirement work. If a pensioner works, constraints on eligibility for retirement benefits are often imposed, for example limits on earnings (the earnings test). In this case, benefits are not paid if the income from post-retirement work exceeds a certain limit. Future retirement benefits are then increased to compensate for the unpaid benefits. The earnings test is in this way similar to late retirement. If the compensation credit is high enough such that the present discounted value of future retirement benefits is the same, no change in the labor supply is expected when the earnings test is introduced. In this paper we introduce risk aversion in the utility of lifetime and find that the effective time horizon is shorter for risk averse than for risk neutral individuals. Using the example of the U.S. pension scheme we show that discounting plays a crucial role since the formula for the delayed retirement credit by which future benefits are increased in the case of late retirement is sensitive to the discount rate used. Finally, we compute the optimal values of the delayed retirement credit for particular parameters in the case of risk neutrality and in the case of risk aversion. The paper is organized as follows. Section 2 provides a brief overview of the literature on this topic. The next section analyzes the actuarial adjustment mechanism and introduces risk aversion. The optimal parameters in the cases of risk neutrality and risk aversion are found using the example of the U.S. pension system in Section 4. The last section assesses policy-related findings and concludes the study.
4
2. Survey of Literature
2. Survey of Literature Although the age-neutral pension rule should not in theory affect the labor supply of older workers, numerous studies provide evidence to the contrary. The U.S. pension scheme is age neutral between ages 62 and 65,4 but a peak in the labor force withdrawal at the age 62 is still observed (Blau, 1994). Explanations of this phenomenon called "the retirement puzzle" either rely on liquidity constraints (Kahn, 1988, Rust and Phelan, 1997) or provide evidence that a lack of part-time jobs and the fixed costs of work are responsible for the withdrawal from the labor market (Hurd, 1996). Other studies provide evidence that the worker's perception of the adjustment mechanism is responsible for the labor force withdrawal. Thus, Burtless and Moffitt (1985) conclude that the adjustment rule for the foregone benefits is widely unrecognized. Reimers and Honig (1993) investigate the role of time preferences in the labor supply when the earnings test is in effect. They show that the effective time horizon is crucial in the labor supply decisions of older workers. They also find significant differences between male and female labor supply responses to the earnings test rule. Whitehouse (1990) analyzes the removal of the earnings test in the United Kingdom in 1989. He finds that the compensation for unpaid benefits in the cases of late retirement and the earnings test is fair for women, but not for men because the official retirement age is lower for females than for males, while women live longer than men in the U.K. His calculations are based on assumptions that there is no discounting of future retirement benefits, and that lifetime equals life expectancy. Attention should be devoted to questions related to discount rates and time preferences.5 In maximization problems of time separable utility functions of consumption, subjective time discount rates have traditionally been less than unity in life cycle models with certain or uncertain lifetimes. However, in an overlapping setting there are no restrictions on the size of the rate of time preferences. For example, Kocherlakota (1990) shows that equilibria with positive interest rates exist in infinite horizon growth economies in which per capita consumption grows over time. In these economies, individuals prefer consuming today rather than in the future, even though the subjective time discount rate is larger than one. The empirical evidence suggests that a subjective time discount rate greater than unity is plausible. Imrohoroglu, et al. (1998) provide an instructive overview of this topic. Incorporating mortality risk into a life cycle model and assuming an interest rate of 0.03, Hurd's (1989) nonlinear 2SLS estimates using panel data from the Retirement History Survey deliver a subjective time discount rate of 1.011. Hurd (1989) also discusses the relation between subjective time discount rates ρ and discount rates β derived from real interest rates. Although it is possible to claim that ρ < β, some studies of growth models imply that ρ > β (see for example Kocherlakota, 1990, or Hurd, 1989). Lifetime uncertainty has been introduced in life cycle models to reconcile the fact observed in cross sections that the elderly seem not to dissave at advanced ages. In his pioneering contribution Yaari (1965) finds that in an economy without social security annuities, the future is discounted more heavily than it would be with a certain lifetime if the marginal utility of consumption exceeds the marginal utility of bequests, while bequests are accidental due to the uncertainty of survival. The introduction of annuities is equivalent to the removal of lifetime uncertainty. However, lifetime uncertainty is important when evaluating the efficiency of the actuarial adjustment in the pension
4
The actuarial increase in retirement benefits is 8% per year between ages 62 and 65, and was 3% per year between ages 65 and 70 prior to 1990. According to the 1983 amendments to the Social Security Act, beginning in 1990 the actuarial increase has been gradually raised and will reach 8% per year by 2008. See, for example, Reimers and Honig (1993) for details. 5
The difference between time preferences and discount rates is described in Hurd (1989). Time preferences enter the utility maximization problem, while discount rates β derived from real interest rates as β= 1 / ( 1 + r ) are implemented in the budget constraint.
5
2. Survey of Literature
scheme, since the length of the period over which adjusted retirement benefits are received is crucial in assessing the efficiency of the adjustment mechanism.
3. Analysis of the Actuarial Adjustment
6
3. Analysis of the Actuarial Adjustment As we mentioned in Section 2, defined-benefit pension plans may have incentive or disincentive effects on workers' decisions to retire. The particular situation depends on the parameters of the pension scheme. In this section we analyze the impact of these parameters on the fairness of actuarial adjustment in the cases of late retirement and the earnings test. We introduce a general formula for adjusting the future benefits and analyze the implicit function for the optimal delayed retirement credit in the case of certain lifetime. We then model uncertain lifetime by introducing the risk premium into the utility associated with the length of life. We assess how the risk aversion to uncertain lifetime affects the individual effective time horizon.
3.1. General Framework Suppose a worker is eligible for retirement benefits B(R0 ) at the lowest retirement age R0 .6 If the worker retires at age R, such that R > R0 , he receives benefits B(R), such that B(R) > B(R0 ). The mechanism of this actuarial adjustment for the unpaid retirement benefits in the period from R0 to R is fair (or age-neutral) if the present discounted value of future retirement benefits at age R0 is the same as at age R.7 We now introduce a time variable t, such that t = 0 at age R0 . Denoting the number of years of late retirement r = R − R0 , we define B(0) ≡ B(R0 ) and B(r) = B(r + 0) ≡ B(r + R0 ) = B(R). Suppose the lifetime T is certain, and 0 < r < T,8 and denote β a discount rate, 0 < β ≤ 1. The actuarial adjustment is fair if and only if T −1
∑
B ( 0) βt =
t =0
T −1
∑ B (r )β , t
(3.1)
t=r
where B(r) are retirement benefits not dependent on period t.9 Let us introduce to the equation (3.1) the formula for retirement benefits B(r) in a general form B ( r ) = ( 1 + dr ) b.
(3.2)
The benefit is B(0) = b if retirement occurs at the age of retirement R0 . The benefit is increased in the case of retirement at an age beyond the lowest retirement age R0 by dr percent, where d is the delayed retirement credit. Substituting (3.2) in (3.1) and rearranging the terms yields r −1
T −1
t =0
t =r
∑ βt = ∑ dr βt ,
(3.3)
βr + dr βr − dr βT − 1 = 0,
(3.4)
1 + dr − dT = 0 .
(3.5)
from which it follows that
or for β = 1
6
We denote the lowest age at which an individual is eligible for old age benefits as the lowest retirement age. The official retirement age is the lowest retirement age specified in the law. The lowest retirement age is greater or equal to the official retirement age. 7
Individual subscripts are omitted.
8
Individuals may consider all known factors that enter their individual expected lifetime. In this work, we take T as the expected lifetime for the total population or for a particular population group. 9
B(r) is constant in real terms since it is indexed to inflation by assumption. In equation (3.1), the last period in which a person lives and receives retirement benefits is T — 1. All the following results can be reformulated by substituting T instead of T — 1.
3. Analysis of the Actuarial Adjustment
7
The equations (3.4) and (3.5) represent implicit functions for the optimal delayed retirement credit d and the optimal age of retirement R = R0 + r given the number of years r by which retirement is delayed, the discount rate β, and the lifetime T. Before analyzing the equations (3.4) and (3.5), it should be noted that the equation (3.4) is a transcendental function in the number of years of late retirement r and that the analytical solution for r is hard to find. Therefore, we focus on finding the value of the optimal delayed retirement credit d. Then we analyze this function with respect to parameters T, r, and β. The formula for the optimal delayed retirement credit can be easily written from equations (3.4) and (3.5) as d =
1 βr − 1 r βT − βr
,
(3.6)
and for β = 1 d =
1 T−r
.
(3.7)
Differentiating these equations with respect to lifetime T yields
∂d 1 βr − 1 =− βT ln β < 0 , ∂T r ( βT − βr ) 2
(3.8)
∂d 1 =− < 0. ∂T (T − r ) 2
(3.9)
and for β = 1
Both the equations (3.6) and (3.7) are decreasing functions in the life expectancy T. A longer period of collecting the adjusted benefits decreases the delayed retirement credit required. Similarly, differentiating the equations (3.6) and (3.7) with respect to r yields r T r ∂d 1 β −1 1 β ln β( β − 1) =− 2 T + > 0, ∂r r β − βr r ( βT − βr ) 2
(3.10)
∂d 1 = > 0. ∂r T − r
(3.11)
and for β = 1
The proof that the function (3.6) increases in r is given in the Appendix. For a particular discount rate and life expectancy, the delayed retirement credit that should be applied to compensate the pensioner for the unpaid benefits increases when a year of retirement is greater. Differentiating the equation (3.6) with respect to the discount rate yields r −1 T T −1 r ∂d r β ( β − 1) − T β ( β − 1) = < 0. 2 ∂β r ( βT − βr )
(3.12)
The proof that the equation (3.6) decreases in the discount rate β in the interval from zero to one is provided in the Appendix. This supports the explanation that an increasing discount rate lowers the optimal delayed retirement credit.
3. Analysis of the Actuarial Adjustment
8
3.2. Uncertain Lifetime We are now able to introduce lifetime uncertainty. In this work, we use a stochastic lifetime that enters a simple utility function. The expected utility from the stochastic lifetime is compared to the utility of a particular effective time horizon. In other words, we are looking for the certainty equivalent of the stochastic lifetime. Suppose the stochastic length of life in the time period t be T = T + ε , where T = E ( T | T > t ) is the conditional expected lifetime at time period t, and ε is a random term with zero mean and finite ~ ~ variance σε2 . Assume a twice differentiable utility u T is assigned to T . The key parameter is
( )
a risk premium ρ( T , ε) introduced as
(
)
( )
Eu (T ) ≡ Eu (T + ε) = u T − ρ( T , ε) ≡ E Teff .
(3.13)
The effective lifetime Teff = T − ρ(T , ε). is a certain lifetime whose utility is the same as the ~
expected utility of stochastic lifetime T . ~
The utility of lifetime T can be expanded using the Taylor series as u ( T + ε) ≈ u ( T ) + ε
∂u (T ) ε2 ∂ 2 u ( T ) + . ∂T 2 ∂T 2
(3.14)
Applying the expected values, the equation (3.14) becomes
σε2 ∂2 u (T ) Eu (T + ε) ≈ u (T ) + . 2 ∂T 2
(3.15)
Assuming ρ(T , ε) t ) − E ( T | T > t ) = E (T |T > t ) − . 2 E (T |T > t )
(3.21)
~
Assuming that the stochastic lifetime T can be described by a distribution function f ( T ) defined from zero to the terminal age Tm a x , the expected lifetime at time period t is Tmax
T =
∫ Tf ( T )dT
t Tmax
,
(3.22)
∫ f ( T )dT t
and the variance at period t equals 2
T ∫ Tf ( T )dT T f ( T ) dT ∫ σε2 = tT − Tt . ∫t f (T ) dT ∫t f ( T )dT T max
max
2
max
max
(3.23)
We have seen that the individual risk aversion towards lifetime uncertainty can be modelled using the risk premium in the utility of lifetime. It follows from the equation (3.20) that in the case of
(
)
no risk aversion γ = 0 , the effective time horizon faced by an individual equals the expected
(
)
lifetime, while risk averse individuals γ > 0 face an effective time horizon that is shorter than the expected lifetime.
4. Results
10
4. Results In this section we find the functional form and parameters of the lifetime distribution function f ( T ) used in the equations (3.22) and (3.23), and calculate from (3.20) the effective lifetime horizon for particular values of the coefficient of risk aversion. Then using equations (3.6) and (3.7), we calculate values of the optimal delayed retirement credit in the U.S. pension scheme for a particular range of discount rates and the number of years of late retirement. Finally, we derive values of discount rates from real interest rates.
4.1. Calibration 4.1.1. Lifetime uncertainty Most of the life tables used in demography utilize mortality rates observed in a short period of time to construct a hypothetical cohort that faces age-specific mortality rates in the period in question. The average remaining lifetime, also called life expectancy and reported at any given age, is the average number of years remaining to be lived by those from a hypothetical cohort surviving to that age. These current life tables are presented for example in the U.S. decennial life tables for 1989 1991 (National Center for Health Statistics, 1997) with mortality rates calculated up to the age of 109 using data from the 1990 Census. They are published every 10 years. In other years, National Center for Health Statistics releases life tables calculated up to 99 years using the full mortality data in a given year. Recent life tables for the U.S. population in year 1997 are provided in Anderson (1999). In order to calculate integrals in equations (3.22) and (3.23), we have to find the functional form of the probability density function f ( t ) at a particular age t and fit the parameters of this function using the data from life tables. Many studies use the Gompertz distribution, which assumes that the mortality hazard rate is an exponential function of age t. For example, Leung (1994) employs the parameters of the Gompertz distribution estimated in Wetterstrand (1981) from the U.S. life tables for 1969 - 1971. He uses these estimates in numerical examples of new features of the model proposed in Yaari (1965). Thatcher, Kannisto, and Vaupel (1998) use data from 13 countries over three decades from 1960 to 1990, and apply the maximum likelihood method to estimate the parameters of six functional forms of f ( t ) for ages from 80 to 98. Using these estimates, they fit mortality hazard rates in the age range from 99 to 109, and compare the goodness of fit for these models. The choice of the break point at age 98 was chosen to avoid any errors which may result from age heaping observed in the data around age 100, or from any inaccuracies in the data above age 98 due to a low number of observations. They find that the fitted values of the Gompertz model rise more rapidly than observed death rates, and that this departure becomes noticeble after age 95. Since our aim is to calculate the moments of the distribution of lifetime and not to fit mortality hazard rates at advanced ages, we use the Gompertz functional form in this work. Denoting parameters a and b and lifetime T, the mortality hazard rate µ( T ) becomes
µ( T ) = ae bT .
(4.1)
The simplicity of the Gompertz model appears to be an advantage in computing integrals in equations (3.22) and (3.23). Given the probability density function f ( t ) , the survivor function S ( T ) is S (T ) = 1 −
T
∫ f (t ) dt , 0
(4.2)
4. Results
11
from which it follows that dS ( T ) dT
= − f (T ).
(4.3)
The mortality hazard rate µ( T ) then equals
µ( T ) =
f (T )
S(T )
=−
d ln S ( T ) dT
.
(4.4)
By integrating the equation (4.4), we arrive at T
S(T ) = e
∫
− µ (t ) dt
.
0
(4.5)
Applying the Gompertz functional form from (4.1), the probability density function follows from (4.4) and (4.5) as T
f ( T ) = ae
∫
bT − a ebt d t
(4.6)
0
which can be solved as a
f ( T ) = ae b
a
+bT − e bT b
.
(4.7)
In order to calculate the expected lifetime and its variance from equations (3.22) and (3.23) using equation (4.7), we should estimate parameters a and b. For this purpose we rewrite equation (4.1) as ln µ( T ) = ln a + bT + ε,
(4.8)
where ε is a disturbance term distributed with zero mean and constant variance. We estimate the parameters of equation (4.8) using the method of least squares on the data from life tables.11 Table A.1. provides estimates of a and b with standard errors for 1990 as well as the expected lifetime T and variance σε2 at the age 65 calculated from equations (3.22) and (3.23). The parameters of a and b were estimated for mortality hazard rates from ages 65 to 109 using data from National Center for Health Statistics (1997). Since the aim is to have a good fit after age 65, we substracted the value of 65 from the actual age, so that the period in question is from 0 to 44. This transformation is necessary in order to avoid integrating in the equation (4.6) over the period for which the paremeters a and b were not estimated. In other words, we assume that the mortality hazard rates are zero for ages under 65.12 The expected lifetime and the variance in (3.22) and (3.23) were calculated at age 65 by integrating over the period from 0 to 44, which corresponds to ages from 65 to 109. In the last row of the table we show the value of the expected lifetime E(T) from the U.S. decennial life tables published in National Center for Health Statistics (1997). As we can see, the expected lifetimes calculated from the equation (3.22) are in accordance with the tabulated values. Using data from Anderson (1999), similar results are reported for year 1997 in Table A.2. Integrals from (3.22) and (3.23) were calculated over the age interval from 65 to 109 transformed to 11
The disturbance term e is autocorrelated in the equation (4.8) since non-linearities are neglected in this functional form. Since the Gompertz model departs significantly from the observed death rates only at advanced ages, the use of the least squares is justified. 12
Integrals
∫ f (T )dT
given in denominators of the equations (3.22) and (3.23) equal 0.9994 for the total
population, 0.9998 for males, and 0.9992 for females. These values are very close to unity, and the assumption that the mortality hazard rates are zero for ages before age 65 is therefore justified.
12
4. Results
the range from 0 to 44. However, the fitted parameters a and b were estimated for ages from 65 to 99, because the life tables in Anderson (1999) do not provide mortality hazard rates after age 99. As we again observe, the fit of parameters a and b is good, and calculated values of expected lifetime T are in accordance with the tabulated values of life expectancy E ( T ) at age 65. As we have seen, the range of ages over which fitted parameters were estimated is not the same in 1990 and 1997. In order to examine how that change affects the size of expected lifetime T and variance σε2 , we reestimated the equation (4.8) for the range of ages from 65 to 99 using the data from 1990. The results are provided in Table A.3. Integrals given in (3.22) and (3.23) were calculated over the age interval from 65 to 109 transformed to 0 to 44. We can see that expected lifetimes do not change significantly when the range of fit is shorter, but the variances are smaller than in Table A.1.13
4.1.2. Risk aversion In order to proceed with numerical solutions, we must choose values for the coefficient of risk aversion. An important contribution to this topic was made by Hurd (1989). He estimates an extended life cycle model of consumption in which utility depends on the path of consumption and bequests. Using the same utility function as in equation (3.18), he obtains the value of 0.729 with standard error 0.091 using the nonlinear least squares on the panel data from the Retirement History Survey. Then he reestimates his model by nonlinear 2SLS with the result of 1.12 with standard error 0.074. As Hurd (1989) points out, his estimates are much smaller than those assumed in the previous literature. His estimates are used in Leung (1994) to complete the set of values chosen at 0.1, 0.5, 1 and 3 in a numerical analysis. Imrohoroglu (1998) employs values of 1.5 and 3 in his simulations. Based on this experience, we use Hurd's (1989) estimate of 1.12 as a benchmark of our model. In order to examine how the variation in the coefficient of risk aversion affects the effective time horizon, we also provide results for the other Hurd's (1989) estimate of 0.729, and for the value of 2.
4.2. Optimal delayed retirement credit In this subsection we provide the numerical results of the theoretical section with parameters found in the previous subsection. First, we compute the effective time horizon from equation (3.20). Regarding the expected lifetime, we do not use the calculated values T from equation (3.22), but employ the values from life tables given in the last row of Tables A.1. and A.2. The variance σε2 used is from Tables A.1. and A.2. Table A.4. summarizes the results for the total population, males, and females for the years 1990 and 1997. The calculations were performed for values of the coefficient of risk aversion of 0.729, 1.12, and 2. The case of γ = 0 represents no risk aversion, and values reported in this row help to demonstrate how the effective time horizon shortens with increasing risk aversion. In 1997, the effective time horizon is 17.7 years for the total population, but only 15.1 years when the most often cited estimate of the coefficient of risk aversion is taken into account. In 1997, the time horizon falls from 15.9 to 13.2 years for males and from 19.2 to 16.8 years for females. Table A.4. also demonstrates how the expected lifetime has changed between 1990 and 1997.14
13
As we mentioned, the Gompertz method provides fitted values of the logarithm of the mortality hazard rate higher than those observed at advanced ages. This implies that a shorter range of fit delivers a higher estimate of b in the equation (4.8). The probability density function f(T) given in (4.7) is narrower than for a longer range of fit, which implies that the variance is naturally lower. Later we will return to this point and investigate how the change in the variance affects our results. 14
Now we may go back to the point from the previous footnote and analyze how the change in the variance
affects results. We calculated values of the effective time horizon using the variance
σε2 and expected lifetime
E(T) from Table A.3. for 1990. For the coefficients of risk aversion 0.729, 1.12, and 2, the effective time
13
4. Results
With the results from Table A.4., we are able to calculate from equations (3.6) and (3.7) the optimal delayed retirement credit for particular discount rates and years of late retirement. Table A. 5. provides results for the case of no risk aversion ( γ = 0 ) , and for risk aversion ( γ = 1 . 1 2 ) for the total population in 1997. In the case of no risk aversion, the optimal delayed retirement credit at age 65 is 6.0% for unity discount rate if retirement is delayed by 1 year, while it is 6.4% and 6.8% when retirement is delayed by 2 or 3 years. When the discount rate is lower than unity, the optimal delayed retirement credit increases. When an individual delays retirement by 1 year, the optimal value changes from 6.0% to 6.5% while the discount rate decreases from 1 to 0.99, and to 7.1% for β = 0.98 . This significant impact is also noticeable for other values of years of late retirement, indicating that the optimal delayed retirement credit given in the equation (3.6) is sensitive to the variation in the discount rate. In order to illustrate this finding, Figure A.1. shows that the first derivative of the delayed retirement credit with respect to the discount rate declines as the discount rate decreases. Moreover, the first derivative is lower for more years of late retirement. When one accounts for risk aversion, the optimal delayed retirement credit increases from 6.0% to 7.1% when the discount rate is unity and retirement is delayed by 1 year. The ratio of the optimal credit in the cases with and without risk aversion is provided in Table A.6. The ratio for unity discount rate and 1 year late retirement is 1.184, but increases to 1.198 for 2 years, and even to 1.257 for 5 years late retirement. This implies that risk aversion has a more significant impact on the size of the optimal delayed credit the later an individual retires, although in the case of other values of discount rates, the effect of risk aversion diminishes. Thus, the ratio drops from 1.184 to 1.118, and 1.070 for discount rates 1, 0.95, and 0.90. This effect is also observed for other values of years of late retirement. Similar results for the optimal delayed retirement credit and for the effect of risk aversion are provided for males in Tables A. 7. and A. 8., and for females in Tables A. 9. and A. 10.. Since the effective time horizon is longer for females than for males as observed in Table A.4., the optimal delayed retirement credit is, not surprisingly, lower for females than for males for all values of discount rate and years of late retirement. As the results from Tables A.8. and A. 10. indicate, risk aversion has a less significant impact on delayed retirement credit for females than for males. Regarding the discount rates used to calculate the present discounted value of future retirement benefits, we are able to shed some light on their magnitude using the U.S. real interest rates. Figure A.2.15 shows real interest rates on 1-year Treasury Bills and Long-Term U.S. Government Securities in the period from 1986 to 1997. These interest rates range from 2% to 5% with the exception of the period from 1991 to 1993 when real interest rates from T-bills dropped to zero. In 1997, real interest rates on deposits were between 3% and 4.5%. This implies that relevant discount rates were between 0.96 and 0.97 in 1997. As we observe in Figure A.3.,16 real interest rates on borrowing were observed mostly between 4% and 6.5% during the period 1986 - 1997, and between 5% and 6.5% in 1997. These correspond to the values of the discount rate from 0.94 to 0.95 for 1997. As Tables A.5., A.7., and A.9. show, the optimal delayed retirement credit at age 65 is 6.0% with no discounting and 1 year late retirement for the total population in 1997. However when discounting is accounted for, the optimal delayed credit is between 7.8% and 9.9%. In the case of risk aversion, the delayed credit is 7.1% for no discounting, and between 8.9% and 11.0% for discount horizon is 15.5, 14.6 and 12.5 years in the case of the total population, 13.3, 12.4 and 10.2 years for males, and 17.4, 16.6 and 14.7 years for females. By comparing these values with the results for the year 1990 reported in Table A.4., we may conclude that the difference is negligible except the case of the coefficient of risk aversion γ= 2 when the difference is 0.3 years. This indicates that fitting the functional form of f(T) over the shorter age interval in 1997 is sufficient for the purpose of this work. 15
Source: Own calculations; data from Federal Reserve Board of Governors and U.S. Department of Labor, Bureau of Labor Statistics. 16
Source: Own calculations; data from Federal Reserve Board of Governors and U.S. Department of Labor, Bureau of Labor Statistics.
14
4. Results
rates derived from real interest rates in 1997. For males, the optimal delayed retirement credit is 6.7% for no discounting, and between 8.5% and 10.6% for discount rates from 0.94 to 0.97. In the case of risk aversion, delayed credits are 8.2%, and between 10.0% and 12.0%. The optimal values for females are 5.5%, 7.3% and 9.4% for no risk aversion, and 6.3%, 8.1% and 10.2% for risk averse females. All these results derived for 1 year late retirement and 1997 are repeated in Table A. 11. We may conclude that discounting has an important impact on the size of the optimal delayed retirement credit, since, when discounting is properly measured, the optimal delayed retirement credit may be at least 25% greater than in the case of no discounting. Moreover, risk aversion towards uncertain lifetime increases values of the optimal delayed retirement credit by 15% to 25%. The results reported in Table A. 11. were derived for age 65 and 1 year late retirement. Since the optimal value of the delayed retirement credit increases with the number of years of late retirement, see equations (3.10) and (3.11), the results in Table A. 11. are minimum values required in the actuarial adjustment scheme. In order to explain the U.S. "retirement puzzle" at age 62 (see Section 2), we must calculate the optimal delayed retirement credit at age 62 that provides an incentive not to retire before age 65. Table A. 12. summarizes the optimal delayed credit in the case of 3 years late retirement in year 1997.17 Since the actuarial increase is 8% between ages 62 and 65 in the U.S. pension scheme, we may conclude that when there is no discounting, the adjustment is fair between ages 62 and 65. The only exceptions are risk averse males for which the actuarial increase is not sufficient. However when discounting is considered in the calculations, the actuarial increase is only fair for some risk neutral females. Finally, we should assess the sensitivity of the results to alternative values of the coefficient of risk aversion. For this purpose we tabulate values of the optimal delayed credit at age 65 for the coefficient of risk aversion 0 (risk neutrality), 0.729, 1.12, and 2. The results for 1 year late retirement are presented in Table A. 13. for the total population, in Table A. 14. for males, and in Table A. 15. for females.18 In the case of the total population, for example, the delayed credit changes from 6.7% to 7.1% when the coefficient of risk aversion increases from 0.729 to 1.12 and the discount rate is unity. A similar slight increase in the optimal delayed retirement credit is observed for males and females. However when the coefficient of risk aversion increases from 1.12 to 2, the delayed credit rises from 7.1% to 8.3% for the total population, as seen in Table A. 13., and a similar significant increase is observed for males and females in Tables A. 14. and A. 15. This suggests that risk aversion should be properly measured in order to obtain reliable results for the optimal delayed retirement credit.
17
These results are almost identical to the results in Table A. 11. for age 65 and 1 year late retirement. We should note that the optimal delayed credit decreases with the life expectancy, but increases with the number of years of late retirement. 18
Results for γ= 1.12 are repeated from Tables A.5., A.7., and A.9.
15
5. Policy Implications and Concluding Remarks
5. Policy Implications and Concluding Remarks We have learned that the formula for the optimal delayed retirement credit is sensitive to the number of years of late retirement and to the life expectancy at the particular age of retirement. The effect of the number of years of late retirement is neglected when it is short enough with respect to the life expectancy. Regarding the effect of the life expectancy on the optimal delayed retirement credit, the key problem is that the life expectancy is different for males and females. This problem is even more significant in economies where the official retirement age is not the same for males and females. This is for example the case in the United Kingdom and in the transition economies in Central and Eastern Europe. In order to eliminate the age-dependency of retirement decisions of individuals, the optimal delayed retirement credit should be different for males and females. The formula for the delayed retirement credit is also sensitive to the discount rate used to calculate the present discounted value of future retirement benefits. Our results indicate that in economies with positive real interest rates, the optimal delayed retirement credit differs significantly from that which has been implemented under the assumption of no discounting. We may conclude that since the delayed retirement credit is institutionally determined, it can fairly reward unpaid benefits only when late retirement is short enough with respect to the life expectancy and when the discount rate is close to one. Only in this case is the optimal value of the delayed credit approximately stable. This implies that policies related to late retirement and the earnings test can be effective only in a short period beyond the official retirement age. We modelled lifetime uncertainty as the certainty equivalent to the stochastic lifetime. We demonstrated that risk aversion towards uncertain lifetime reduces the effective time horizon. Risk aversion has in this sense the same effect as a reduced lifetime. In order to account for the lifetime uncertainty, the optimal delayed retirement credit should be higher for risk averse individuals than for those who are risk neutral. In the empirical part of the paper, we found parameters of the lifetime probability distribution function, and using appropriate coefficients of risk aversion, we calculated the effective time horizon for the U.S. total population, males, and females at age 65 in 1997. Then we calculated the optimal delayed retirement credit for the same groups of population at age 65 in year 1997. From U.S. real interest rates, we found that discount rates were likely observed between 0.94 and 0.97 in year 1997. In the case of risk aversion and 1 year late retirement at age 65 in year 1997, we came to conclusion that the optimal values for the delayed credit are 7.1% for the total population, 8.2% for males, and 6.3% for females when discounting is neglected. However, when discount rates from real interest rates are implemented, the optimal values for the delayed retirement credit are between 8.9% and 11.0% for the total population, between 10.0% and 12.0% for males, and between 8.1% and 10.2% for females. We should keep in mind that these are the optimal values that should be implemented in the actuarial adjus-ment scheme in order to provide an incentive at age 65 to delay retirement by 1 year in year 1997. The most important finding is that optimal values for the delayed retirement credit are at least 25% greater than in the case of no discounting. Comparing these results with those for risk neutrality, we may also conclude that risk aversion increases the optimal delayed retirement credit by 15 - 25%. We found that discounting and risk aversion may explain "the retirement puzzle" which appears between ages 62 and 65 in the U.S. pension scheme. In order to encourage postponing retirement by 3 years at age 62, the optimal delayed retirement credit should be set between 8.7% and 11.2% for males, and between 7.4 - 10.0% for females in the case of risk neutrality based on 1997 data. When risk aversion is taken into account, the optimal values are between 10.3 - 12.8% for males, and between 8.2 - 10.8% for females. The 8% delayed retirement credit that is currently applied in the U.S. pension scheme between ages 62 and 65 does not compensate individuals deciding to delay retirement from age 62 to age 65 for their lost retirement benefits.
16
5. Policy Implications and Concluding Remarks
We obtained our results for discount rates equal to or lower than unity. The equation (3.6) indicates however that the optimal delayed retirement credit is defined even for values of discount rates greater than unity. This implies that our results could easily be extended to discount rates greater than unity. The results of this work are sensitive to how discount rates and risk aversion are measured. Our experience indicates that it may be difficult to find the optimal parameters of the pension scheme that would provide an incentive to retire later. Moreover, the decisions of older workers to retire and to undertake post-retirement work are affected by other factors that are beyond the scope of this study. An important incentive to retire later may be induced when the replacement rate between average net earnings in the years preceding retirement and retirement benefits is lower than unity. The indexation of old-age pensions to inflation as well as the volatility in real interest rates may also have significant effects on workers' decisions to retire.
17
References
References [1] Anderson, Robert N. (1999), "United States Life Tables, 1997," National Statistics Reports, vol. 47 no. 28, Hyattsville, Maryland: National Center for Health Statistics. [2] Blau, David M. (1994), "Labor Force Dynamics of Older Men," Econometrica 62(1): 117-156. [3] Burkhauser, Richard V. and Joseph F. Quinn (1997), "Implementing Pro-Work Policies for Older Americans in the Twenty-First Century," Syracuse University, Maxwell School Policy Brief. [4] Burtless, Gary and Robert A. Moffitt (1985), "The Joint Choice of Retirement Age and Postretiremen! Hours of Work," Journal of Labor Economics 3(2): 209-236. [5] Hurd, Michael (1989), "Mortality Risk and Bequests," Econometrica 57(4): 779-813. [6] Hurd, Michael (1996), "The Effect of Labor Market Rigidities on the Labor Force Behavior of Older Workers," in David A. Wise ed., Advances in the Economics of Aging, NBER, Chicago: The University of Chicago Press. [7] Imrohoroglu, Ayse, Selahattin Imrohoroglu, and Douglas H. Joines (1998), "The Effect of Tax-Favored Retirement Accounts on Capital Accumulation," The American Economic Review 88(4): 749-768. [8] Kahn, James A. (1988), "Social Security, Liquidity, and Early Retirement," Journal of Public Economics 35: 97-117. [9] Kocherlakota, Narayana R. (1990), "On the 'Discount' Factor in Growth Economies," Journal of Monetary Economics 25: 43-47. [10] Leung, Siu Fai (1994), "Notes and comments: Uncertain Lifetime, the Theory of the Consumer, and the Life Cycle Hypothesis," Econometrica 62(5): 1233-1239. [11] National Center for Health Statistics (1997), U.S.Decennial Life Tables forl989-1991. vol. 1 no. 2, Hyattsville, Maryland. [12] Peracchi, Franco and Finis Welch (1994), "Trends in Labor Force Transitions of Older Men and Women," Journal of Labor Economics 12(2): 210-242. [13] Reimers, Cordelia and Marjorie Honig (1993), "The Perceived Budget Constraint under Social Security: Evidence from Reentry Behavior," Journal of Labor Economics 11(1): 184-207. [14] Rust, John and Christopher Phelan (1997), "How Social Security and Medicare Affect Retirement Behavior in a World of Incomplete Markets," Econometrica 65(4): 781-831. [15] Schrooten Mechthild, Timothy M. Smeeding and Gert G. Wagner (1998), "Old-Age Security Reforms in Central-Eastern Europe: The Cases of Czech Republic, Slovakia, Hungary and Poland," Luxembourg Income Study Working Paper No. 189. [16] Smeeding, Timothy M. and Joseph F. Quinn (1997), "Cross-National Patterns of Labor Force Withdrawal," Luxembourg Income Study Working Paper No. 170. [17] Stock, James H. and David A. Wise (1990), "Pensions, the Option Value of Work, and Retirement," Econometrica 58(5): 1151-1180. [18] Thatcher, A. P., V. Kannisto and J. W. Vaupel (1998), The Force of Mortality at Ages 80 to 120, Odense: Odense University Press. [19] Wetterstrand, W. (1981), "Parametric Models for Life Insurance Mortality Data: Gompertz's Law Over Time," Transactions of the Society of Actuaries 33: 159-179. [20] Whitehouse, Edward (1990), "The Abolition of the Pensions 'Earnings Rule'," Fiscal Studies 11(3), 55-70. [21] World Bank (1994), Averting the Old Age Crisis: Policies to Protect the Old and Promote Growth, Policy Research Report Series. Oxford and New York: Oxford University Press for the World Bank. [22] Yaari, Menahem E. (1965), "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer," Review of Economic Studies 32(2): 137-150.
A. Appendix
18
A. Appendix Theorem A.I. It follows from (3.6) that r r T ∂d 1 β −1 1 β ( β − 1) ln β =− 2 T + > 0, ∂r r β − βr r ( βT − βr ) 2
where d is the delayed retirement credit, β is the discount rate, such that 0 < β < 1 , r is the number of years of late retirement, and T is the life expectancy, such that 0 < r < T . Proof. Let us start by proving the Theorem for r → 0 . Using the Taylor series, we can write β for small r as βr ≈ 1 + r ln β . Substituting βr into the equation yields r
T ∂d 1 r ln β 1 (1 + r ln β)( β − 1) ln β ≈− 2 T + = ∂r r β − 1 − r ln β r ( βT − 1 − r ln β)2
− ln β( βT − 1) + r (ln β) + ln β( βT − 1) + r ( ln β) ( βT − 1) 2
= =
(
2
)
r β − 1 − r ln β
(β
βT (ln β) T
2
=
2
− 1 − r ln β)
Going back to the Theorem,
T
> 0.
2
∂d > 0 if and only if ∂r
(
)
r T 1 β β − 1 ln β
r
(β
T
− βr )
2
βr − 1 > 2 T . r β − βr 1
Rearranging the terms in this inequality gives
(
)
r βr ln β βT − 1
>
(β
r
− 1)( βT − βr ),
r βr βT ln β − r βr ln β > r βT ln β − r ln β + βT − r + βr
βr βT − βr βr − βT + βr , > βT + 1.
Denote as L( r ) the left hand side and R the right hand side of the last inequality. Differentiating L( r ) with respect to r supplies
∂L (r ) = βT ln β − ln β − βT − r ln β + βr ln β = ∂r
(
)
= ln β ( βT − βT − r ) + ( βr − 1) > 0.
∂L ( r ) > 0 since ln β < 0 and both terms in the parentheses are negative. ∂r Keeping in mind that R is not a function in r, we conclude that if L ( r ) > R for some r, then L ( r ) > R for all r', such that r ′ > r . But we know that L ( r ) > R for r → 0 which completes the We have shown that
proof of the Theorem.
A. Appendix
19
Theorem A.2. It follows from (3.6) that r −1 T T −1 r ∂d r β ( β − 1) − T β (β − 1) = < 0, 2 ∂β r ( βT − βr )
where d is the delayed retirement credit, β is the discount rate, such that 0 < β < 1 , r is the number of years of late retirement, and T is the life expectancy, such that 0 < r < T . Proof. The first derivative of the delayed retirement credit with respect to the discount rate is negative if and only if
(
)
(
)
r βr −1 βT − 1 < T βT −1 βr − 1
or rearranging the terms r βr − 1
r , it is necessary to show that a function f ( x) =
x βx − 1
βx − 1
is increasing in the parameter x. Differentiating f ( x ) with respect to x yields x −1 x− 1 x x −1 x ∂f ( x ) (β + x β ln β)(β − 1) − x β β ln β = = ∂x ( βx − 1) 2
=
βx −1 (β x − 1) − x βx −1 ln β
(β
x
− 1)
2
.
The nominator in the last equation is positive if and only if
β x −1 ( βx − 1) > x βx −1 ln β, or
β x − 1 − x ln β = βx − 1 − ln β x > 0. Since the inequality becomes equality for x = 0 and since x > 0 , we should prove that the last inequality is increasing in x. Differentiating with respect to x supplies
β x ln β − ln β = ( βx − 1) ln β > 0 . since β x < 1 and ln β < 0 .
A. Appendix
20
Table A.1. - Fitted parameters, age 65 - 109, year 1990 Total population
Males
Females
-3.943 (0.023)
-3.605 (0.029)
-4.261 (0.029)
b
0.0782 (0.0009)
0.0717 (0.0011)
0.0863 (0.0011)
R2
0.9944
0.9896
0.9926
a
0.0194
0.0272
0.0141
T
17.2
15.0
19.0
σε2
87.0
78.7
89.4
E (T )
17.3
15.1
19.0
ln a
Table A.2. - Fitted parameters, age 65 - 99, year 1997 Total population
Males
Females
-4.104 (0.010)
-3.826 (0.015)
-4.402 (0.013)
b
0.0868 (0.0005)
0.0820 (0.0007)
0.0956 (0.0006)
R2
0.9988
0.9972
0.9985
a
0.0165
0.0218
0.0123
T
17.6
15.9
19.0
σε2
82.2
76.2
81.9
E (T )
17.7
15.9
19.2
ln a
Table A.3. - Fitted parameters, age 65 - 99, year 1990 Total population
Males
Females
-4.017 (0.008)
-3.694 (0.014)
-4.353 (0.012)
b
0.0839 (0.0004)
0.0787 (0.0007)
0.0934 (0.0006)
R2
0.9992
0.9975
0.9986
a
0.0180
0.0249
0.0129
T
17.2
15.1
18.9
σε2
82.3
74.1
83.1
E (T )
17.3
15.1
19.0
ln a
A. Appendix
21
Table A.4. - The effective time horizon
γ
Total population
Males
Females
1990
1997
1990
1997
1990
1997
0
17.3
17.7
15.1
15.9
19.0
19.2
0.729
15.4
16.0
13.2
14.1
17.3
17.6
1.12
14.5
15.1
12.2
13.2
16.4
16.8
2
12.2
13.0
9.9
11.1
14.3
14.9
A. Appendix
22
Table A.5. - Optimal delayed retirement credit at age 65, risk aversion 0 and 1.12, U.S. total population, 1997 Number of years of late retirement (r)
β
1
2
3
4
5
noRA
RA
noRA
RA
noRA
RA
noRA
RA
noRA
RA
1.00
0.060
0.071
0.064
0.076
0.068
0.083
0.073
0.090
0.079
0.099
0.99
0.065
0.076
0.070
0.082
0.074
0.089
0.080
0.097
0.086
0.107
0.98
0.071
0.082
0.076
0.089
0.081
0.096
0.087
0.105
0.094
0.115
0.97
0.078
0.089
0.083
0.095
0.088
0.103
0.095
0.113
0.103
0.124
0.96
0.084
0.095
'0.090
0.103
0.096
0.111
0.104
0.122
0.112
0.134
0.95
0.091
0.102
0.098
0.110
0.105
0.120
0.113
0.131
0.122
0.145
0.94
0.099
0.110
0.106
0.119
0.114
0.129
0.123
0.141
0.133
0.156
0.93
0.107
0.118
0.115
0.127
0.124
0.139
0.134
0.152
0.145
0.168
0.92
0.116
0.126
0.124
0.137
0.134
0.149
0.145
0.164
0.158
0.182
0.91
0.125
0.134
0.134
0.146
0.145
0.160
0.158
0.177
0.173
0.196
0.90
0.134
0.144
0.145
0.157
0.157
0.172
0.172
0.190
0.188
0.212
0.89
0.144
0.153
0.156
0.168
0.170
0.185
0.186
0.205
0.205
0.229
0.88
0.155
0.163
0.168
0.179
0.184
0.198
0.202
0.220
0.223
0.247
0.87
0.166
0.174
0.181
0.191
0.198
0.212
0.219
0.237
0.243
0.267
0.86
0.177
0.185
0.194
0.204
0.214
0.227
0.237
0.255
0.264
0.288
0.85
0.189
0.196
0.208
0.218
0.231
0.244
0.257
0.274
0.287
0.311
0.84
0.201
0.208
0.223
0.232
0.248
0.261
0.278
0.295
0.312
0.336
0.83
0.214
0.221
0.239
0.247
0.267
0.279
0.300
0.317
0.340
0.363
0.82
0.228
0.234
0.255
0.263
0.287
0.298
0.324
0.341
0.369
0.392
0.81
0.242
0.247
0.272
0.280
0.308
0.319
0.350
0.366
0.401
0.424
0.80
0.256
0.261
0.290
0.297
0.330
0.341
0.378
0.393
0.436
0.458
A. Appendix
23
Table A. 6. - Ratio of risk averse to non-risk averse optimal delayed retirement credit at age 65, U.S. total population, 1997 r
β 1
2
3
4
5
1.00
1.184
1.198
1.215
1.234
1.257
0.99
1.169
1.183
1.199
1.219
1.241
0.98
1.155
1.169
1.185
1.204
1.226
0.97
1.142
1.155
1.171
1.189
1.211
0.96
1.129
1.142
1.158
1.176
1.197
0.95
1.118
1.130
1.145
1.163
1.184
0.94
1.107
1.119
1.133
1.151
1.171
0.93
1.096
1.108
1.122
1.139
1.159
0.92
1.087
1.098
1.112
1.128
1.148
0.91
1.078
1.089
1.102
1.118
1.137
0.90
1.070
1.081
1.093
1.108
1.126
0.89
1.063
1.073
1.084
1.099
1.116
0.88
1.056
1.065
1.077
1.090
1.107
0.87
1.050
1.058
1.069
1.082
1.099
0.86
1.044
1.052
1.062
1.075
1.090
0.85
1.039
1.047
1.056
1.068
1.083
0.84
1.034
1.041
1.050
1.062
1.076
0.83
1.030
1.037
1.045
1.056
1.069
0.82
1.026
1.032
1.040
1.050
1.063
0.81
1.023
1.028
1.036
1.045
1.057
0.80
1.020
1.025
1.032
1.040
1.052
A. Appendix
24
Table A.7. – Optimal delayed retirement credit at age 65, risk aversion 0 and 1.12, U.S. males, 1997 Number of years of late retirement (r)
β
1
2
3
4
5
noRA
RA
noRA
RA
noRA
RA
noRA
RA
noRA
RA
1.00
0.067
0.082
0.072
0.089
0.078
0.098
0.084
0.109
0.092
0.122
0.99
0.073
0.088
0.078
0.095
0.084
0.105
0.091
0.116
0.099
0.130
0.98
0.079
0.093
0.084
0.102
0.091
0.112
0.098
0.124
0.108
0.139
0.97
0.085
0.100
0.091
0.109
0.098
0.119
0.107
0.133
0.116
0.149
0.96
0.091
0.106
0.098
0.116
0.106
0.128
0.115
0.142
0.126
0.159
0.95
0.099
0.113
0.106
0.124
0.115
0.136
0.125
0.151
0.137
0.170
0.94
0.106
0.120
0.114
0.132
0.124
0.145
0.135
0.162
0.148
0.182
0.93
0.114
0.128
0.123
0.140
0.133
0.155
0.146
0.173
0.160
0.195
0.92
0.122
0.136
0.132
0.149
0.144
0.165
0.157
0.185
0.173
0.209
0.91
0.131
0.145
0.142
0.159
0.155
0.176
0.170
0.198
0.188
0.224
0.90
0.140
0.154
0.153
0.169
0.167
0.188
0.183
0.211
0.203
0.240
0.89
0.150
0.163
0.164
0.180
0.179
0.201
0.198
0.226
0.220
0.257
0.88
0.160
0.173
0.175
0.191
0.193
0.214
0.214
0.241
0.238
0.276
0.87
0.171
0.183
0.188
0.203
0.207
0.228
0.230
0.258
0.258
0.296
0.86
0.182
0.194
0.201
0.216
0.223
0.243
0.248
0.276
0.279
0.317
0.85
0.194
0.205
0.214
0.229
0.239
0.259
0.268
0.295
0.302
0.341
0.84
0.206
0.216
0.229
0.243
0.256
0.276
0.288
0.316
0.327
0.366
0.83
0.218
0.228
0.244
0.258
0.274
0.294
0.311
0.338
0.354
0.393
0.82
0.232
0.241
0.260
0.273
0.294
0.313
0.334
0.361
0.384
0.422
0.81
0.245
0.254
0.277
0.289
0.315
0.333
0.360
0.386
0.415
0.454
0.80
0.259
0.268
0.294
0.306
0.337
0.354
0.388
0.413
0.450
0.489
A. Appendix
25
Table A.8. - Ratio of risk averse to non-risk averse optimal delayed retirement credit at age 65, U.S. males, 1997 r
β 1
2
3
4
5
1.00
1.221
1.241
1.265
1.293
1.329
0.99
1.205
1.225
1.248
1.276
1.312
0.98
1.190
1.209
1.232
1.260
1.295
0.97
1.175
1.194
1.217
1.244
1.278
0.96
1.162
1.180
1.202
1.229
1.262
0.95
1.149
1.167
1.188
1.214
1.247
0.94
1.136
1.154
1.175
1.201
1.233
0.93
1.125
1.142
1.162
1.187
1.219
0.92
1.114
1.131
1.150
1.175
1.205
0.91
1.104
1.120
1.139
1.163
1.193
0.90
1.095
1.110
1.128
1.151
1.180
0.89
1.086
1.100
1.118
1.140
1.169
0.88
1.078
1.092
1.109
1.130
1.158
0.87
1.070
1.083
1.100
1.120
1.147
0.86
1.063
1.076
1.091
1.111
1.137
0.85
1.057
1.069
1.084
1.103
1.127
0.84
1.051
1.062
1.076
1.095
1.118
0.83
1.045
1.056
1.069
1.087
1.110
0.82
1.040
1.050
1.063
1.080
1.101
0.81
1.036
1.045
1.057
1.073
1.094
0.80
1.032
1.041
1.052
1.067
1.086
A. Appendix
26
Table A.9. - Optimal delayed retirement credit at age 65, risk aversion 0 and 1.12, U.S. females, 1997 Number of years of late retirement (r)
β
1
2
3
4
5
noRA
RA
noRA
RA
noRA
RA
noRA
RA
noRA
RA
1.00
0.055
0.063
0.058
0.068
0.062
0.072
0.066
0.078
0.070
0.085
0.99
0.060
0.069
0.064
0.073
0.068
0.079
0.072
0.085
0.077
0.092
0.98
0.066
0.075
0.070
0.080
0.075
0.086
0.080
0.092
0.082
0.100
0.97
0.073
0.081
0.077
0.087
0.082
0.093
0.087
0.100
0.094
0.109
0.96
0.079
0.088
0.084
0.094
0.090
0.101
0.096
0.109
0.103
0.118
0.95
0.087
0.095
0.092
0.102
0.098
0.109
0.105
0.118
0.113
0.129
0.94
0.094
0.102
0.101
0.110
0.107
0.118
0.115
0.128
0.124
0.140
0.93
0.103
0.110
0.110
0.119
0.117
0.128
0.126
0.139
0.136
0.152
0.92
0.111
0.119
0.119
0.128
0.128
0.139
0.138
0.151
0.149
0.165
0.91
0.121
0.128
0.129
0.138
0.139
0.150
0.150
0.163
0.163
0.179
0.90
0.130
0.137
0.140
0.149
0.151
0.162
0.164
0.177
0.179
0.195
0.89
0.140
0.147
0.152
0.160
0.164
0.174
0.179
0.192
0.196
0.212
0.88
0.151
0.157
0.164
0.172
0.178
0.188
0.195
0.207
0.214
0.230
0.87
0.162
0.168
0.177
0.184
0.193
0.202
0.212
0.224
0.234
0.250
0.86
0.174
0.179
0.190
0.197
0.209
0.218
0.230
0.242
0.255
0.271
0.85
0.186
0.191
0.205
0.211
0.226
0.234
0.250
0.262
0.278
0.294
0.84
0.199
0.203
0.220
0.226
0.244
0.252
0.271
0.282
0.304
0.319
0.83
0.212
0.216
0.235
0.241
0.262
0.270
0.294
0.305
0.331
0.346
0.82
0.226
0.229
0.252
0.257
0.283
0.290
0.319
0.329
0.361
0.376
0.81
0.240
0.243
0.269
0.274
0.304
0.311
0.345
0.355
0.393
0.407
0.80
0.254
0.258
0.287
0.292
0.326
0.333
0.373
0.382
0.428
0.442
A. Appendix
27
Table A. 10. - Ratio of risk averse to non-risk averse optimal delayed retirement credit at age 65, U.S. females, 1997 r
β 1
2
3
4
5
1.00
1.152
1.162
1.174
1.188
1.203
0.99
1.138
1.149
1.160
1.174
1.189
0.98
1.126
1.136
1.147
1.160
1.176
0.97
1.114
1.124
1.135
1.148
1.163
0.96
1.103
1.112
1.123
1.136
1.151
0.95
1.093
1.102
1.112
1.125
1.139
0.94
1.083
1.092
1.102
1.114
1.128
0.93
1.074
1.083
1.093
1.104
1.118
0.92
1.066
1.074
1.084
1.095
1.108
0.91
1.059
1.067
1.076
1.086
1.099
0.90
1.052
1.059
1.068
1.078
1.091
0.89
1.046
1.053
1.061
1.071
1.083
0.88
1.040
1.047
1.055
1.064
1.075
0.87
1.035
1.041
1.049
1.057
1.068
0.86
1.031
1.037
1.043
1.052
1.062
0.85
1.027
1.032
1.038
1.046
1.056
0.84
1.023
1.028
1.034
1.041
1.050
0.83
1.020
1.024
1.030
1.037
1.045
0.82
1.017
1.021
1.026
1.032
1.040
0.81
1.015
1.018
1.023
1.029
1.036
0.80
1.013
1.016
1.020
1.025
1.032
Table A. 11. - Optimal delayed retirement credit at age 65, risk aversion 0 and 1.12, 1 year late retirement, 1997 Total population
No discounting Discounting
Males
Females
noRA
RA
noRA
RA
noRA
RA
6.0%
7.1%
6.7%
8.2%
5.5%
6.3%
7.8-9.9%
8.9-11.0%
7.3-9.4%
8.1-10.2%
8.5-10.6% 10.0-12.0%
A. Appendix
28
Table A. 12. - Optimal delayed retirement credit at age 62, risk aversion 0 and 1.12, 3 years late retirement, 1997 Total population
No discounting Discounting
Males
Females
noRA
RA
noRA
RA
noRA
RA
5.9%
7.0%
6.7%
8.2%
5.4%
6.2%
7.9-10.5%
9.1-11.6%
8.7-11.2%
10.3-12.8% 7.4-10.0%
8.2-10.8°
Table A. 13. - The effect of risk aversion on the optimal delayed retirement credit at age 65, 1 year late retirement, total population 1997
β
γ=0
γ=0.729
γ=1.12
γ=2
1
0.060
0.067
0.071
0.083
0.99
0.065
0.072
0.076
0.089
0.98
0.071
0.078
0.082
0.095
0.97
0.078
0.084
0.089
0.101
0.96
0.084
0.091
0.095
0.108
0.95
0.091
0.098
0.102
0.115
0.94
0.099
0.106
0.110
0.122
0.93
0.107
0.113
0.118
0.129
0.92
0.116
0.122
0.126
0.138
0.91
0.125
0.131
0.134
0.146
0.90
0.134
0.140
0.144
0.155
A. Appendix
29
Table A. 14. - The effect of risk aversion on the optimal delayed retirement credit at age 65, 1 year late retirement, males 1997
β
γ=0
γ=0.729
γ=1.12
γ=2
1
0.067
0.076
0.082
0.099
0.99
0.073
0.082
0.088
0.105
0.98
0.079
0.088
0.093
0.111
0.97
0.085
0.094
0.100
0.117
0.96
0.091
0.101
0.106
0.123
0.95
0.099
0.108
0.113
0.130
0.94
0.106
0.115
0.120
0.137
0.93
0.114
0.123
0.128
0.145
0.92
0.122
0.131
0.136
0.153
0.91
0.131
0.139
0.145
0.161
0.90
0.140
0.148
0.154
0.170
Table A.15. - The effect of risk aversion on the optimal delayed retirement credit at age 65, 1 year late retirement, females 1997
β
γ=0
γ=0.729
γ=1.12
γ=2
1
0.055
0.060
0.063
0.072
0.99
0.060
0.066
0.069
0.077
0.98
0.066
0.072
0.075
0.083
0.97
0.073
0.078
0.081
0.090
0.96
0.079
0.085
0.088
0.096
0.95
0.087
0.092
0.095
0.103
0.94
0.094
0.099
0.102
0.111
0.93
0.103
0.107
0.110
0.118
0.92
0.111
0.116
0.119
0.127
0.91
0.121
0.125
0.128
0.135
0.90
0.130
0.135
0.137
0.145
30
Figure A.1. - First derivatives of the delayed retirement credit
Figure A.2. - Real interest rates on deposits in the United States
A. Appendix
31
Figure A.3. - Real interest rates on loans in the United States
A. Appendix
32
Labor Supply Incentives of the Pension Scheme in the Czech Republic
Labor Supply Incentives of the Pension Scheme in the Czech Republic Abstract We describe the pension scheme in the Czech Republic before and after January 1996 when changes to the scheme were introduced. We analyze formulas determining retirement benefits and examine the relation between earnings before retirement and retirement benefits. We investigate how the size of the delayed retirement credit by which future retirement benefits are increased in the case of late retirement and the earnings test affects the labor supply of older workers. Our results indicate that replacement rates between pre-retirement earnings and retirement benefits are lower for persons with higher earnings. Although replacement rates are greater in 1996 than in 1995, they are still far below unity. In 1996, men are likely to retire at the lowest retirement age, while some women face an incentive to delay retirement by up to 6.9 years beyond the lowest retirement age. In order to provide an incentive to delay retirement by 2 years, the optimal delayed retirement credit is between 5.1% and 7.6% for men and between 3.2% and 5.7% for women in 1996.
1. Introduction In the Czech Republic, changes to the pay-as-you-go pension scheme were introduced in January 1996. After that date, rules determining retirement benefits were changed and gradual postponing of the lowest retirement age was implemented. In addition, a limit on earnings from postretirement work above which retirement benefits are not payable was also introduced.19 In this work we describe the pension scheme before and after January 1996 and analyze formulas for calculating retirement benefits. Using a framework developed in Galuscak (2000), we derive the optimal delayed retirement credit for men and women. Using discount rates from real interest rates and the life expectancy in 1996, we investigate the labor supply incentives induced by the pension scheme by calculating the optimal age of retirement. We also provide results on the optimal delayed retirement credit by which retirement benefits are increased in the case of late retirement and the earnings test. The paper is organized as follows. First, we provide an overview of the Czech pension system. In the following section we evaluate the labor supply incentives of the scheme by calculating the discount rates from real interest rates and deriving the optimal age of retirement for men and women as well as the optimal delayed retirement credit. The last section concludes the study.
2. Description of the Pension System Old-age insurance is mandatory and is financed by employees, employers and the state. According to the legislative framework, a worker is eligible for retirement benefits if he or she has been insured for a particular period and reaches the lowest retirement age specified in law.20 If the worker turns the lowest retirement age21 and is eligible for old-age pensions, he or she may either retire and collect retirement benefits or continue to work. Retirement benefits accepted after the labor market withdrawal depend on the number of years of insurance prior to the age at which the worker is eligible for retirement benefits, and on earnings over a particular period before retirement. If the worker retires later than at the age at which he is eligible for retirement benefits, the period beyond that age is accounted in the benefits at more favorable conditions. 19
Prior to 1996, restrictions on early retirement were introduced to the Czech pension system in March 1994. The delayed retirement credit applied in the case of late retirement was virtually unchanged before and after January 1996. The pension system is described in Section 2. 20
The worker may also claim early pensions prior to reaching the lowest retirement age if he meets legal requirements. 21
Since January 1996, the lowest retirement age increases gradually in the Czech Republic each year by 2 months for men and by 4 months for women until 2006.
Labor Supply Incentives of the Pension Scheme in the Czech Republic
33
This simple explanation neglects, however, any labor market activity of persons receiving retirement benefits. If a retired individual works, restrictions on eligibility for the benefits may be imposed. In the Czech Republic prior to 1996, no limits on earnings from post-retirement work were imposed. The only requirement was that the validity of employment contracts was limited to a period of one year. Restrictions in the case of post-retirement work were introduced in the Czech Republic in January 1996. After that date, benefits are paid only if earnings do not exceed a certain limit. This earnings test is imposed for the period of two years beyond the lowest age at which an individual is eligible for retirement benefits. If benefits are reduced due to excess in earnings, subsequent benefits are increased to compensate for the unpaid benefits. This compensation is effective when postretirement earnings are higher than a limit after January 1996, and when the employment was longer than 1 year in the period before 1996. Benefits are increased in these cases by the same credit as in the case of late retirement.
2.1. The Czech Pension System After the 1996 Changes Suppose an individual i reaches the age of eligibility for retirement benefits Ri 0 . The amount of the retirement benefit Bi at the age of retirement Ri , such that Ri ≥ Ri 0 and ri = Ri − Ri 0 , can be described as
(
)
B i = b0 + 0 .015 * y i + dri I i .
(2.1)
The benefit consists of two parts. The fixed part b0 is the same for every pensioner and is determined by law. The income related part22 is a function of the income-related income base I i calculated from earnings in the period before the retirement, as defined later, and a number of years of insurance y i prior to Ri 0 . As defined in law, each year of insurance y i contributes to the benefit by 1.5% of the income base I i . The term dri I i , is related to the case of late retirement. If the individual retires at Ri > Ri 0 , the income related part is increased by dri I i , where d is the annual delayed retirement credit that equals to 4% according to law, and ri is a number of years by which the individual retires later, such that ri = Ri − Ri 0 . The income base I i can be expressed as a function of the personal income base Pi , that is defined below, using a formula
(
)
{
(
}
)
(
)
I i = min Pi , a 1 + 0.3 min max 0 , Pi − a 1 , a 2 − a 1 + 0.1 max 0, Pi − a 2 ,
(2.2)
where both a 1 and a 2 are parameters specified by law and a 2 > a 1 > 0 . The equation (2.2) reveals a redistributive aspect of the pension system as the value of Pi between a 1 and a 2 is mapped into I i with the weight 0.3, while Pi exceeding a 2 is multiplied by only 0.1. The personal income base Pi is finally calculated from Pi =
t R i −1
12 Yt −1 W ∑ ∑ . M i t =1 9 8 6 m =1 imt Yt 1
Ri
(2.3)
The annual sum of gross earnings Wimt in month m and year t is adjusted to the present value using the economy-level average earnings Yt and Yt R −1 , respectively.23 Given that M i is the number i
of months in the period from 1986 until t Ri − 1 , the personal income base Pi is then a monthly
22
The minimum level is guaranteed.
23
Index t Ri , denotes a year t in which an individual turns age R i .
34
Labor Supply Incentives of the Pension Scheme in the Czech Republic
average of the sum of the adjusted annual earnings from 1986 until the year preceding the year of retirement, t Ri − 1 .24 If the individual retires at the age Ri , the amount of retirement benefits received is computed from the equations (2.1), (2.2) and (2.3). As we have mentioned, the pensioner may be engaged in a work activity in the period after retirement. Suppose that in month m and year t the pensioner earns Wimt ′ from post-retirement work. According to law, retirement benefits are not provided between the age Ri 0 and the age Ri 0 + 2 in periods where Wimt ′ > We′ , where We′ is the earnings limit. In this case, however, future retirement benefits are adjusted for each year or quarter of the year in which the pensioner works and in which Wimt ′ > We′ . The benefits are calculated from
{
(
B i = b0 + 0.015 * y i + d ri + ri e
e
)}I ,
(2.4)
i
where ri e is a number of years in which the pensioner does not receive retirement benefits due to the earnings test. The value of I i in the equation (2.4) is given in (2.2) and (2.3), respectively. It should be noted that the value of I i is the same in (2.1) and (2.4) since it is calculated from earnings prior to Ri , while the earnings test is applied in the period beyond the age Ri if Ri ≤ Ri 0 + 2 .
2.2. The Pension System in the Czech Republic Prior to the 1996 Changes In this part we describe the pension scheme in the Czech Republic before January 1996. Denoting the variables related to the period prior to 1996 with the superscript b , the amount of the retirement benefit can be written as
R −1 B = 0.5 * I + 0.01 * ∑ 1 + dri I ib . y =2 6 i0
b i
b i
(2.5)
b i
The benefit equals 50% of the individual income base I ib plus 1% of I ib for every year of employment y ib exceeding 25 years25 and before the lowest age of retirement Ri 0 . In the case of late retirement, years of work beyond the age of eligibility for retirement benefits Ri 0 are rewarded in the same way as in the period after January 1996. The delayed retirement credit d is 4% as in the equation (2.1). For every year by which an individual retires later, the retirement benefits are increased by d percent of the income base I ib computed from the earnings in the period prior to Ri . The income base I ib from the equation (2.5) is calculated from
) ( ) { ( + 0.1 min{ max ( 0 , P − a ), a − a } , (
)
}
I ib = min Pi b , a 1b + 1 3 min max 0 , Pi b − a 1b , a 2b − a 1b + b
i
b
b
b
2
3
2
(2.6)
where a b3 > a 2b > a 1b > 0 as defined in law. The equation (2.6) differs from (2.2) in that it imposes the maximum level of the retirement benefits.
24
The period over which the earnings are taken into account is 30 years by law, but years prior to 1986 are excluded. 25
The 1996 changes define retirement in terms of insurance. Before 1996, years of employment instead of years of insurance have been in use since there has been the full employment in the economy prior to the end of the communist era in 1989.
Labor Supply Incentives of the Pension Scheme in the Czech Republic
35
The personal income base Pi b from the equation (2.6) equals the average gross monthly earnings from 5 years with the highest earnings in the period of 10 years before the age of retirement R i . In other words,
1 12 Pi = ∑ ∑ Wimt 5 j =1 12 m =1 1
b
5
(2.7)
j
{
}
for t ∈ t R i −1 0 , K , t Ri −1 , where
1
12
∑ 12 m =1
Wimt j >
1
∑ Wimt ′ 12 m =1 12
j
if j < j ′ for j , j ′ ∈{1, 2 , K ,10} . A retired person may be engaged in a work activity. A pensioner who works receives retirement benefits only if the employment is limited to the period of one year. When the employment is longer than one year, retirement benefits are not paid, but future benefits are adjusted as
R −1 = 0.5 * I + 0 .01 * ∑ 1 + d (ri + ri e ) I ib , y =2 6 i0
B
b ,e i
b i
(2.8)
b i
where ri e is a number of years in which the pensioner does not receive retirement benefits. It should be pointed out that the income base I ib is the same as in the equation (2.5) since it is calculated from earnings before retirement at the age Ri .
3. Labor Supply Incentives of the Pension Scheme We are now able to analyze incentives of the pension scheme outlined in the previous part on older workers' labor supply in the Czech Republic. We describe formulas for retirement benefits effective before and after January 1996, and derive the amount of retirement benefit for a typical person in both periods. Using parameters from year 1996, we calculate the optimal delayed retirement credit for particular discount rates and the number of years of late retirement. Using real interest rates, we derive the optimal number of years of late retirement and the optimal size of the delayed retirement credit. Finally, we assess how the introduction of the earnings test affects retired persons' labor supply.
3.1. Characteristics of the Pension Scheme The personal income base Pi given in the equation (2.3) adjusts gross earnings from the period before retirement to the present equivalent using the economy-level average earnings. Before 1996, the personal income base Pi b is given by the average monthly gross earnings in the period preceding retirement at the age Ri , see the equation (2.7). The earnings used in this case are not adjusted to the present equivalent. Although the period from which the personal income base is calculated is shorter in the scheme before 1996 than after, the inflation associated with the economic transition starting from the early 1990s reduced the value of the personal income base before 1996. Moreover, the 1996 legislative change postulates that after that date, a pensioner receives the maximum calculated from equations (2.1) and (2.5). This implies that nobody is worse-off under the 1996 pension scheme than before 1996. Both the pension schemes before and after 1996 exhibit a lower ratio of retirement benefits to pre-retirement earnings for high income than for low income workers. It follows from the equations (2.6) and (2.2) that a ratio of income base I i or I ib to personal income base Pi or Pi b is lower for an individual with higher personal income base than for an individual with lower personal income base.
Labor Supply Incentives of the Pension Scheme in the Czech Republic
36
This implies that persons with a higher personal income base receive less from the system than they contribute relatively to persons with the lower personal income base. Finally, we are able to calculate the amount of retirement benefit for a typical person retiring at the lowest retirement age in the Czech Republic before and after the 1996 changes. Suppose that an individual starts working at the age 20 and that there are no spells out of work until the lowest retirement age Ri 0 = 60 .26 It then follows from the equation (2.5) that before-the-change retirement benefits are B ib = 0 .5 * I ib + 0.15 * I ib = 0 .65 * I ib ,
(3.1)
where I ib is the income base from the equation (2.6). A similar result can be obtained from the equation (2.1) for the period after January 1996.
(
)
Denoting b1 = 0. 015 * y i * I i , we know that b0 + b1 b 0 = 7 .12 in 1996.27 This implies for after-thechange retirement benefits that B i = 0.1 * I i + 0.6 * I i = 0 .7 * I i
(3.2)
for a person who works from the age 20 and retires at the age 60, where I i is the income base from (2.2). Table 4.1. provides old-age pensions in years 1995 and 1996 calculated for the personal income base from 3000 to 15000 CZK. Pensions are derived from equations (3.1) and (3.2) using the income base from (2.6) and (2.2).28 As we observe in Table 4.1., pensions increase with the personal income base, while replacement rates decline. In year 1995, we observe that the ceiling imposed on old-age pensions affected those with personal income base greater or equal to 10000 CZK. Finally, replacement rates are greater in 1996 than in 1995. This indicates that the new pension rules implemented in 1996 exhibit an improved replacement between personal income base and oldage pension. In order to calculate replacement rates between earnings and retirement benefits for a person earning average wage prior to his retirement, we should analyze formulas for personal income base in (2.7) and (2.3). After January 1996, the personal income base is given by earnings from the particular period prior to retirement adjusted to one year before retirement. Therefore, we may assume that the personal income base in 1996 equals the average wage in 1995. On the other hand before 1996, the personal income base is given by the average gross monthly earnings from 5 years with the highest earnings in the period of 10 years before the age of retirement, see equation (2.7). This implies that we may use the average of the average wage in years 1990 to 1994 for the personal income base in 1995. Based on this finding, the personal income base is 4887 CZK in 1995 and 8172 CZK in 1996. Given these values, we are able to derive the income base from (2.6) for 1995 and from (2.2) for 1996. Table 4.2. then supplies average old-age pensions calculated from equations (3.1) and (3.2) for a person earning the average wage and retiring at the lowest retirement age. The pension is 2142 CZK in 1995 and 4166 CZK in 1996 with replacement rates 26.2% and 43.1%, respectively. The difference 26
This assumption holds as most of the work history of people retiring in the 1990s has been experienced during the communist period of full employment before 1989. The fixed part of the benefit b0 is 680 CZK in 1996 by law. We do not know the average retirement benefit of persons retiring at the lowest retirement age, but the average benefit of those retiring at or after the lowest retirement age was 4842 CZK in 1996 (source: The Czech Statistical Office). This implies that 27
(b
0
28
+ b1 ) b 0 = 7 .12 .
b
Parameters used in equations (2.6) and (2.2) are a 1
a 2 = 10000 CZK by law.
= 2500 , a 2b = 6000 , a 3b = 10000 , a 1 = 5000 and
Labor Supply Incentives of the Pension Scheme in the Czech Republic
37
between pensions in these two years is explained by the increase in the personal income income base that rose by 67%, while the average wage increased by 18% between years 1995 and 1996. This implies that the before 1996 pension rule failed to account for pre-retirement wages in old-age pensions. Table 4.2. also shows average old-age pensions of persons retiring at or after the lowest retirement age during the particular year. The average new pension was 3396 CZK in 1995 and 4842 CZK in 1996. These values are greater than what we derived since we employed economy-level average wages in calculating the personal income base that are lower than wages before retirement.
3.2. Analysis of Optimal Parameters in the Cases of Late Retirement and the Earnings Test In the case of late retirement, the delayed retirement credit d is the same according to law before and after January 1996. In both periods, retirement benefits are increased by 4% of the income base I ib or I i for every year, or by 1% of I ib or I i for every 90 days by which a person retires later, see the equations (2.1) and (2.5). In order to analyze incentives to retire later or to work when the earnings test rule is in effect, we should examine whether this delayed retirement credit compensates fully for unpaid benefits in the case of late retirement and the earnings test. According to Galuscak (2000), the actuarial adjustment is fair if and only if T −1
∑
B ( 0 ) βt =
t =0
T −1
∑ B ( r )β , t
(3.3)
t =r
where r is the number of years by which an individual retires later than at the lowest retirement age, T is the life expectancy, such that 0 < r < T , β is the discount rate, such that 0 < β ≤ 1 , and B ( r ) is the retirement benefit. We are now able to introduce into the equation (3.3) a formula for retirement benefits B( r ) and analyze the optimal delayed retirement credit. Introducing a parameter k , such that 0 < k ≤ 1 , retirement benefits can be expressed as B ( r ) = ( k + dr ) I ,
(3.4)
where I is the income base, d is the delayed retirement credit, and r is the number of years beyond the age Ri 0 by which a worker retires later.29 Substituting (3.4) into (3.3) and rearranging the terms yields r −1
∑
kI βt =
t =0
T −1
∑ drI β , t
(3.5)
t= r
from which follows that d =
k βr − 1 r β −β T
r
,
(3.6)
or for β = 1 d =
k T−r
.
(3.7)
Using βr ≈ 1 + r ln β as an approximation for values of r near zero in the equation (3.6), we arrive at the formula for the optimal year of retirement 29
Alternatively, r is the number of years in which a pensioner does not receive benefits due to the earnings test.
38
Labor Supply Incentives of the Pension Scheme in the Czech Republic
r≈
βT − 1 k − . ln β d
(3.8)
Similarly it follows from (3.7) that r=T−
k d
(3.9)
if β = 1 . The equations (3.6) and (3.7) represent expressions for the optimal delayed retirement credit that depend on the number of years of late retirement, while (3.8) and (3.9) show the number of years of late retirement for a particular delayed credit. Tables 4.3. and 4.4. show the results for the Czech Republic in year 1996 for
k = 0. 7 according to the equation (3.2), and for the life expectancies T = 16.14 for men and T = 24 .40 for women.30 Results provide conclusions similar to those drawn in the case of the United
States, see Galuscak (2000). The optimal delayed credit rises moderately with increasing number of years of late retirement for values of the discount rate near unity for both men and women. However, the optimal credit increases strongly as the discount rate departs from unity. There are also differences in the optimal delayed retirement credit between men and women since the life expectancy is longer for women than for men, while the lowest retirement age is lower for women than for men. The pension scheme provides a sufficient compensation for the unpaid benefits only if the optimal delayed credit is lower or equal to the official one. Since the official delayed retirement credit is 4% in the Czech Republic, we can see in Table 4.3. that the scheme does not fully compensate men even if β = 1 and r = 1 . This suggests that there is not an incentive induced by the pension scheme to retire later in the case of men. However, as seen in Table 4.4., the scheme compensates women for their unpaid benefits if β ≥ 0.99 or β = 0.98 for r = 1 . Assuming that women apply discount rates from these intervals,31 the size of the delayed retirement credit in the pension scheme provides an incentive for women to retire later than at the lowest retirement age. The last parameter of interest is the discount rate used by those willing to postpone retirement or to be engaged in post-retirement work. Table 4.5. and Figure 4.1. show nominal and real interest rates on credits and deposits of commercial banks in the Czech Republic in the period between 1995 and 1997. Since a 15% tax is imposed on yields from deposits, after tax real interest rates on deposits are also shown. As we observe in Table 4.5., there is a significant margin between interest rates on deposits and credits amounting to around 4%. Real interest rates on credits reach values between 4% and 5%. Real interest rates on deposits fluctuate between 0% and 1%, while the after tax real interest rates on deposits fall into negative values, see Figure 4.1. A positive peak observed in May 1997 was caused by the turbulences in the Czech financial market. Discount rates derived from real interest rates between zero and 5% are between 0.95 and unity. As results in Table 4.4. indicate, some Czech women face an incentive to retire later than at the lowest retirement age. Based on these results, we are able to calculate the optimal year of retirement r from the equations (3.8) and (3.9). Having in mind that d = 0.04 by law, and using k = 0.7 and the life 30
The lowest age of retirement was 60 years and 2 months for men and 55 years and 4 months for women in 1996. Life expectancies were T6 0 = 16.25 and T6 1 = 15.59 years for men and T5 5 = 24. 69 and
T5 6 = 23.82 years for women in that year (source: "Podrobne umrtnostni tabulky za CR v roce 1996," Czech Statistical Office, 1997). We used linear approximation to compute life expectancies at the lowest age of retirement. Moreover, since the women's lowest retirement age depends on the number of children in the Czech Republic, we employed the case of a woman with two children. Real interest rates that correspond to the discount rates β = 0.99 and β = 0.98 equal 1.0% and 2.0%, respectively. 31
39
Labor Supply Incentives of the Pension Scheme in the Czech Republic
expectancy for year 1996, Czech women with 2 children retire at the lowest retirement age if β = 0.95 , but by 6.9 years later than at the lowest retirement age if β = 1 .32 Now we are also able to find from Tables 4.3. and 4.4. the optimal delayed retirement credit for men and women. In order to provide an incentive to delay retirement by 2 years beyond the lowest retirement age, the optimal delayed retirement credit should be between 5.1% and 7.6% for men and between 3.2% and 5.7% for women.33 Our results should be interpreted in the way so that the delayed retirement credit is the only parameter determining workers' decisions to delay retirement for particular discount rates and the life expectancy. Decisions of older workers between work and retirement are, however, also affected by the replacement rate between earnings and retirement benefits. As we have seen in Table 4.2., replacement rates between earnings of employees and retirement benefits were below unity in year 1996. This implies that our findings on the optimal delayed retirement credit hold for those persons who have already entered or decided to enter retirement. This is for example the case of retired individuals who are potentially affected by the earnings test implemented after January 1996. Since the earnings test is applied in the period of 2 years beyond the lowest age of retirement R i 0 , the delayed retirement credit should compensate for the unpaid benefits if r ≤ 2 . As it follows from Tables 4.3. and 4.4., this is the case only for women if β ∈ 0.98 ,1 . Since we know that Czech women apply discount rates from 0.95 to 1, this implies that those retired women who apply discount rates between 0.95 and 0.97 and whose earnings exceed the limit imposed by the earnings test are likely to change their labor supply after January 1996. On contrary, the compensation for the unpaid benefits is not fair for retired men regardless the discount rate they apply. This suggests that retired men earning more than the earnings limit are likely to change their labor supply after the introduction of the earnings test.
4. Concluding Remarks The 1996 changes to the pension scheme improve the relation between wages before retirement and the personal income base from which retirement benefits are calculated. Our results indicate that the personal income base of a person earning average wage before retirement is 4887 CZK in 1995, while it is 8172 CZK in 1996. Replacement rates between personal income base and pension are lower for greater personal income base in both schemes. Nevertheless, these replacement rates are greater in 1996 than in 1995. We found that for discount rates derived from real interest rates in 1996, the pension scheme does not provide men an incentive to retire later than at the lowest retirement age. However, the scheme motivates women to delay retirement by up to 6.9 years. The delayed retirement credit that provides an incentive to delay retirement by 2 years beyond the lowest retirement age is between 5.1 and 7.6% for men and between 3.2 and 5.7% for women in 1996. Based on these results, we were able to draw predictions on pensioners' labor supply after the earnings test was introduced in January 1996. Men earning more than the earnings limit are likely to change their labor supply. The labor supply of some retired women earning more than the limit is likely not affected since the pension scheme provides them with a full compensation for their lost benefits.
32
In order to show the size of the error in the approximation used in (3.8), we can write an expression for the approximated optimal delayed retirement credit from the equation (3.8) and compare the result with the outcome of (3.6) for the same parameters. The error of the result r for women in the Czech Republic is around 2% in this case. 33
We use this interval of r since the earnings test is imposed there.
Labor Supply Incentives of the Pension Scheme in the Czech Republic
40
References [1] Galuscak, Kamil (2000), "Actuarial Adjustment Aspects of Public Pension Schemes," CERGE-EI Working Paper No. 161.
Table 4.1. - Old-age pensions and replacement rates between personal income base (PIB) and old-age pensions in 1995 and 1996. Pensions (CZK)
Repl. rate (%)
PIB (CZK) 1995
1996
1995
1996
3000
1733
2100
0.578
0.700
4000
1950
2800
0.487
0.700
5000
2167
3500
0.433
0.700
6000
2383
3710
0.397
0.618
7000
2448
3920
0.350
0.560
8000
2513
4130
0.314
0.516
9000
2578
4340
0.286
0.482
10000
2643
4550
0.264
0.455
11000
2643
4620
0.240
0.420
12000
2643
4690
0.220
0.391
13000
2643
4760
0.203
0.366
14000
2643
4830
0.189
0.345
15000
2643
4900
0.176
0.327
Labor Supply Incentives of the Pension Scheme in the Czech Republic
41
Table 4.2. - Replacement rates between average monthly gross wages of employees and average monthly new old-age pensions. Pensions (CZK) Year
Replacement rate (%)
Wages (CZK) Observed
Predicted
Observed
Predicted
1990
3286
1999
60.8
1991
3792
2301
60.7
1992
4644
2271
48.9
1993
5817
2799
48.1
1994
6894
2722
39.5
1995
8172
3396
2142
41.6
26.2
1996
9676
4842
4166
50.0
43.1
1997
10691
5764
53.9
1998
11693
6237
53.3
Source: Statistical Yearbook of the Czech Republic, and Czech Statistical Office
Labor Supply Incentives of the Pension Scheme in the Czech Republic
42
Table 4.3. - Optimal delayed retirement credit for men, Czech Republic 1996
β
Number of years of late retirement 1
2
3
4
5
1.00
0.048
0.051
0.055
0.059
0.065
0.99
0.052
0.055
0.059
0.064
0.070
0.98
0.056
0.060
0.064
0.070
0.076
0.97
0.060
0.065
0.070
0.075
0.082
0.96
0.065
0.070
0.075
0.082
0.089
0.95
0.070
0.076
0.081
0.088
0.097
0.94
0.076
0.081
0.088
0.096
0.105
0.93
0.081,
0.088
0.095
0.104
0.114
0.92
0,087
0.094
0.102
0.112
0.123
0.91
0.094
0.101
0.110
0.121
0.133
0.90
0.100
0.109
0.119
0.131
0.145
0.89
0.107
0.117
0.128
0.141
0.157
0.88
0.115
0.125
0.138
0.152
0.170
0.87
0.122
0.134
0.148
0.165
0.184
0.86
0.131
0.144
0.159
0.178
0.199
0.85
0.139
0.154
0.171
0.191
0.216
0.84
0.148
0.164
0.183
0.206
0.234
0.83
0.157
0.175
0.197
0.222
0.253
0.82
0.166
0.187
0.211
0.240
0.275
0.81
0.176
0.199
0.226
0.258
0.297
0.80
0.186
0.212
0.242
0.278
0.322
Labor Supply Incentives of the Pension Scheme in the Czech Republic
43
Table 4.4. - Optimal delayed retirement credit for women, Czech Republic 1996
β
Number of years of late retirement 1
2
3
4
5
1.00
0.031
0.032
0.034
0.035
0.037
0.99
0.035
0.036
0.038
0.040
0.042
0.98
0.039
0.041
0.043
0.045
0.047
0.97
0.044
0.046
0.048
0.050
0.053
0.96
0.049
0.051
0.054
0.056
0.060
0.95
0.054
0.057
0.060
0.063
0.067
0.94
0.061
0.063
0.067
0.071
0.075
0.93
0.066
0.070
0.074
0.078
0.083
0.92
0.073
0.077
0.082
0.087
0.093
0.91
0.080
0.085
0.091
0.097
0.103
0.90
0.087
0.093
0.100
0.107
0.115
0.89
0.095
0.102
0.109
0.118
0.127
0.88
0.103
0.111
0.120
0.130
0.141
0.87
0.112
0.121
0.131
0.143
0.155
0.86
0.121
0.131
0.143
0.156
0.171
0.85
0.130
0.142
0.156
0.171
0.189
0.84
0.140
0.153
0.169
0.187
0.207
0.83
0.149
0.165
0.183
0.204
0.228
0.82
0.160
0.177
0.198
0.222
0.250
0.81
0.170
0.190
0.214
0.241
0.274
0.80
0.181
0.204
0.231
0.262
0.299
Labor Supply Incentives of the Pension Scheme in the Czech Republic
44
Table 4.5. - Average nominal and real interest rates of commercial banks in the Czech Republic Nom. int. rates (%) Date
Real int. rates (%) CPI (%)+
Credits*
Deposits**
Cred.
Dep.
Dep. after tax
III.95
13.28
9.54
9.6
3.36
-0.05
-1.36
VI.95
12.89
9.62
10.0
2.62
-0.35
-1.66
IX.95
13.09
9.79
8.6
4.14
1.10
-0.25
XII.95
12.90
9.69
7.9
4.63
1.66
0.31
III.96
12.68
9.62
8.9
3.47
0.66
-0.67
VI.96
13.13
9.24
8.4
4.36
0.77
-0.50
IX.96
13.83
9.40
8.9
4.52
0.46
-0.84
XII.96
13.65
9.33
8.6
4.65
0.68
-0.61
III.97
13.53
9.26
6.8
6.30
2.30
1.00
VI.97
21.09
13.00
6.8
13.38
5.81
3.98
IX.97
15.72
10.98
10.3
4.91
0.62
-0.87
XII.97
16.53
11.58
10.0
5.94
1.44
-0.14
Source: Czech National Bank, own calculations + Note: * new short term credits, ** short term deposits, with respect to the same period in previous year.
Figure 4.1: Real interest rates in the Czech Republic
