Longevity, social security and endogenous retirement theory and policy implications

LONGEVITY, SOCIAL SECURITY AND ENDOGENOUS RETIREMENT: THEORY AND POLICY IMPLICATIONS ZENG TING A THESIS SUBMITTED FOR FOR THE DEGREE OF MASTER OF SCOCIAL SCIENCES DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE 2011 ACKNOWLEDGEMENTS First of all, I owe my deepest gratitude to Professor Zhang Jie who guided me as my supervisor, and offered inspiration and encouragement throughout the process. I appreciate Dr. Zhu Shenghao for his suggestion on the improvement of my thesis, and my future research direction. I would also like to thank my fellow classmate, Mr. GaoXinwei, for his valuable comments, and administration staff, Ms Nicky Kheh and Ms Sagi Kaur, in the Economics Department for their kind help. And lastly, I want to extend my regards and blessings to my family and all those friends who supported me in any respect during the completion of the thesis. i TABLE OF CONTENTS 1. Introduction ........................................................................................................... 1 2. The theoretical model ........................................................................................... 8 2.1 Individual’s problem .................................................................................13 2.2 Comparative statics ...................................................................................14 2.3 Impacts on the economy ...........................................................................15 3. Calibration........................................................................................................... 20 4. Policy implications .............................................................................................. 26 5. 4.1 Universal vs. individual specific benefit plan ......................................26 4.2 Age-specific vs. uniform contribution schemes ..................................29 Concluding remarks ........................................................................................... 32 Bibliography ............................................................................................................... 34 Appendices ................................................................................................................ 36 ii SUMMARY Many existing studies predict that raising life longevity tends to increase retirement age. However, empirical observations of cross-country effective retirement age show the reverse trend. This inconsistency is believed to be caused by the existence of social security pension system. This paper employs a two-period overlapping generations (OLG) model to study the impact of an unfunded social security on retirement decision by incorporating uncertainty in life longevity. Analytical results confirm that retirement is negatively related to life longevity but positively to social security generosity. Numerical results from calibration illustrate the effect on retirement age, welfare and steady-state capital levels for various life longevity and payroll tax (pension benefit) levels. In addition to the baseline model, the paper also compares the retirement incentives induced by different social security contribution and benefit schemes, and thus draws implications for policy making on social security reform. iii LISTS OF TABLES Table 1 The historical U.S. social security contribution rates ............................ 5 Table 2 Quantitative impacts when   0.5 ................................................... 21 Table 3 Quantitative impacts when   0.75 ................................................. 21 Table 4 Quantitative impacts when   1 ....................................................... 22 iv LISTS OF FIGURES Figure 1 Trends in life expectancy at age 65 and at age 80, males and females, OECD average, 1970-2007 ................................................................. 2 Figure 2 OECD average effective age of retirement, 1970-2009 ....................... 4 Figure 3 Pension generosity and retirement age in OECD countries, 2009 ....... 6 Figure 4 Payroll tax and retirement decision .................................................... 22 Figure 5 Payroll tax and welfare ....................................................................... 23 Figure 6 Payroll tax and saving rate ................................................................. 23 Figure 7 Payroll tax and steady-state capital per worker .................................. 24 Figure 8 Payroll tax and steady-state output per capita .................................... 24 Figure 9 Individual vs. universal benefit plans ................................................. 29 Figure 10 The stability of the unique steady-state capital-labor ratio ................ 38 Figure 11 The impact of on k ...................................................................... 40 Figure 12 The impact of on y ...................................................................... 42 Figure 13 The impact of on U ......................................................................... 44 Figure 14 The impact of on U .......................................................................... 45 v LISTS OF SYMBOLS survival rate from young to old age zt 1 time spent on leisure during old age (retirement) for agents born at period t ct young age consumption dt 1 young age consumption st saving bt social security pension benefit in the baseline model payroll tax rate, or social security contribution rate in the baseline model wt wage rate at period t rt interest rate at period t Ut lifetime utility for agent born at period t coefficient of time preference, or discounting factor, exogenous  relative taste of leisure to consumption in old age, exogenous Yt aggregate output level at period t A coefficient of total factor productivity, exogenous vi Kt aggregate physical capital stock level at period t kt effective capital-labor ratio at period t Lt aggregate labor force at period t capital’s share of output  sw steady-state saving rate, defined as the ratio of saving to wage  sy saving to output ratio bt social security pension benefit in universal benefit plan z total retirement time under universal benefit plan bt social security pension benefit under the age-differentiated contribution scheme z total retirement time under the age-differentiated contribution scheme vii 1. Introduction The motivation of my paper is obtained from the 2010 French strikes and protests which were against the rise of the legal minimum retirement age from 60 to 62. People defended their rights for not working, but showed little concern about who pays for the early retirement. Pension promises are easy to make, but hard to keep due to the increasingly high cost of provision. The escalating costs of maintaining pension system have crowded out other public priorities with great importance. In 2010, the pension benefit paid in the state of California, the U.S., was over $6 billion, which exceeded what the state spent on higher education (Schwarzenegger, 2010). The huge pension burden is a common problem shared by most of the governments in developed countries. The severe budgetary problem cannot be resolved unless reforms take place. Among the potential reform approaches, increasing the official retirement age, the age when workers have the first access to their pension benefit, seems to be the most feasible and least costly method in the short term. Raising the legal retirement age would delay pension pay-outs, which buys governments extra time to recover the pension fund. Individual retirement decision, together with the decisions on life-time consumption and labor supply, is largely affected by the expected life longevity. We observed a significant improvement of life expectancy since the mid of the twentieth century. The upward lines in figure 1 show the life expectancy has increased steadily in OECD countries, truncated at age 65 and at age 80 for both males and female in recent four decades. The first question that I intend to answer is whether pushing backward 1 retirement is a natural response by individuals given the rising life longevity. If it is true, then raising the official retirement age should be considered as a fair treatment. Figure 1 Trends in life expectancy at age 65 and at age 80, males and females, OECD average, 1970-2007 Females aged 65 Females aged 80 Years Males aged 65 Males aged 80 21 18 15 12 9 6 3 1970 1975 1980 1985 1990 1995 2000 2005 Source: OECD Health Data (2009). Many existing literatures have already studied the relationship between life longevity and retirement in various settings. Ferreira and Pessôa (2007) studied a finite life economy in which higher life expectancy explains the increases in schooling and retirement age. By using a continuous time framework and assuming certain life with exogenous life longevity, their simulation shows that although the total time spent on retirement would increase, the increment is less than that of the life longevity. So their model predicts that retirement age will be pushed backward as a result of the rising life longevity. Zhang and Zhang (2009) adopt a simple two periods overlapping generations (OLG) model to explain the impact of life longevity on retirement and capital 2 accumulation. They interpret life longevity as the contingent survival rate from young to old age, so the life expectancy is determined by the survival rate despite a fixed maximum age. It has been shown that the retirement age is also increasing in life longevity. Similar to interpretation of life expectancy by Zhang and Zhang (2009), d’Albis, Lau and Sánchez-Romero (2010) characterize the recent rise of life expectancy as a successively reduction in mortality rates at older ages, but in an age-dependent fashion. They studied how a mortality change at an arbitrary age affects the optimal retirement age, which also predict that a mortality decline at an older age unambiguously leads to a later retirement age. However, the predictions mentioned do not appear consistent with the empirical observations. Over the last four decades, despite a robust gain in life expectancy, workers today retire earlier than they would few generations ago, evident by a decline of OECD average effective retirement age1, with only a small rebound since the early 2000s. The transition of OECD average effective retirement age from 1970 to 2009 is shown in Figure 2. 1 The average effective age of retirement is calculated as a weighted average of withdrawals from the labor market at different ages over a 5-year period for workers initially aged 40 and over (OECD, 2010). 3 Figure 2 OECD average effective age of retirement, 1970-2009 70 68 Male Female 66 64 62 60 1970 1975 1980 1985 1990 1995 2000 2005 Source: author’s calculation based on data from OECD (2010). I do not intend to override the studies above; rather, I am going to show that the impact of life longevity on retirement in my lifecycle model agrees with the results in those literatures suggest. Rather, to explain the puzzle of historical change of retirement, we must take into account some other institutional factors in one’s retirement decision: such as the pension adequacy. Table 1 shows the historical change of pension contribution in the United State. As the taxable earnings pool enlarges, the total contribution rate has increased remarkably since the establishment of the social security. Meanwhile, the dependency ratio2 has increased steadily from around 0.15 in 1950s, to 0.21 by 2010, and projected to reach 0.3 by early 2020s (OECD data, 2009). It is very suspicious that the expansion of social security system may be an important causal factor for the declining retirement age. 2 The old age dependency ratio is the number of dependents above age 65 per 100 persons of working age. 4 Table 1 The historical U.S. social security contribution rates Year Maximum Taxable Earnings (Dollars) Combined Employer and Employee Tax (%)3 1937 3,000 2.00 1950 3,000 3.00 1960 4,800 6.00 1970 7,800 8.40 1980 29,700 10.16 1990 51,300 12.40 2000 76,200 12.40 2006 94,200 12.40 2010 106,800 12.40 Source: Office of the Chief Actuary, Social Security Administration. In a horizontal comparison, the difference between individual pension systems can explain the across-country differences in retirement behavior. A rough test of the relationship between pension generosity and effective retirement age using 2009 OECD data is presented in Figure 3. The strong negative correlation gives us a possible candidate that should be responsible for the early retirement. 3 Note: These rates do NOT include the payroll tax used to finance Medicare, which is 1.45% each on employers and employees. There is no ceiling for that tax. 5 Figure 3 Pension generosity and retirement age in OECD countries, 2009 Average effective age of retirement 75 C 70 65 60 55 30 40 50 60 70 80 90 100 Gross pension replacement rates for median earner Source: base on data from OECD (2010). Social security is a major source of income in one’s old age, and it often constitutes a large share of family wealth. Using the wealth of recent data through the Health and Retirement Survey (HRS), Coile and Gruber (2000) confirm the deterministic role of social security in one’s retirement behavior. In a forward-looking model, they find that the individual’s retirement decision appears to be made based on all the future streams of social security income, not just the wealth level or income in the next few years. What explains the historical change of actual retirement age? How does the individual make retirement decision, given the prevalent social security system? What is the fair retirement age if we incorporate the rising life longevity? Do the existing policies today distort the retirement decisions by provoking unnecessarily early retirement? What are the long term impacts to the economy? These are indeed the central questions my paper is attempting to answer. 6 The remaining part of the paper proceeds as follows. In section two, I build a two-period OLG model based on Zhang and Zhang (2009) with the Pay-as-you-go (PAYG) unfunded social security as the new element. Subsequently, the impacts of rising life longevity and greater pension generosity on the economy will be examined carefully. To quantify these impacts, a calibration is followed by using realistic parameters. The estimation results are presented in section three. I believe a minor difference of policy instrument even within the same PAYG system may bring vast different retirement incentives. Based on this idea, section four compares the retirement incentives brought by various benefit and contribution schemes, which aims to draw implications for policy making on social security reform. 7 2. The theoretical model First of all, I would like to give a brief overview of the theoretical model. The model has infinitely many periods and overlapping generations with identical agents who may live for a maximum of two periods. Young workers supply labor inelastically, while the old agent may choose time spent between working and leisure (retirement). This assumption is based on the observation of high and stable labor force participation rates for both men and women between ages 25-50, but high labor force exit rates for ages thereafter in the United States (the U.S. Census Bureau, 2000). We can complete a general equilibrium analysis without labor and capital income uncertainty. As the agents value leisure only in their old age, retirement in my model is thus a work-life balance choice4. According to characteristics of recent demographic transition, the concept of life expectancy in this two-period OLG model is equivalent to the chance of survival from young to old age. A simple two-period lifecycle model with analytical solutions can be sufficient for us to understand the relations between behavior and policy motivations. Retirement and saving decisions without the existence of social security can be found in Zhang and Zhang (2009). It is served as a benchmark model to compare the impacts brought by social security. Please note that only the unfunded PAYG social security system where the benefit is financed by a payroll tax is in my research interest. It would be less meaningful 4 In a model with labor income uncertainties, retirement (exiting labor market with social security benefits) can be considered as an optimal choice for risk averse agents. 8 to study an individual’s behavior in a funded system with nonbinding contribution obligation, because he or she will act exactly in the same way as if there were no social security system at all. Now let’s set up the model formally. Consider a simple two-period model with a constant size of the young population, and each agent is endowed with 1 unit of time each period. A representative agent is a working adult in period 1, and become old in the second period. Assuming the agent survives for sure upon birth, but the survival from adulthood to old-age is uncertain with an exogenous probability  (0,1) . An increase of means a rise of the survival rate or a rise of life longevity, so life expectancy can be represented as 1 . This definition of life longevity in terms of the survival rate is particularly suitable in multiple-period model, where the increase of survival chance in each period is essentially an extension of the life-span. For simplicity, we normalize the size of the working adult to be unit 1 in each period, the size of the old-age population is then , and the total population size is therefore 1 . In period t , a young worker allocates his labor income for young age consumption ct and saving st , which is a source of his old age consumption dt 1 . When he gets old in period t  1 , he allocates his time endowment for leisure zt 1 and labor 1  zt 1 , where zt 1 [0,1] . The concept of retirement can be directly interpreted as time spent on leisure in one’s old age. Since there is life uncertainty and no bequest motives by assumption, we need a redistributive mechanism which transfers the assets (i.e. savings) of the deceased to those 9 who are still alive. A complete and competitive life annuity market is therefore assumed to be in its place functioning both as transfer mechanism and a channel between savings and capital investment. If survival is uncertain, it can be shown that a non-altruistic individual’s optimal choice is always to purchase life annuity with all his saving st 5. The annuity intermediary invests st in final goods production, and receives a total return of st 1  rt 1  in the following period. Subsequently the size of old agents receives annuity payment I t 1 conditional on survival is equal to 1  r  , which is derived from zero profit condition st 1  rt 1   It 1 . Notice that a rise of life longevity will lower the annuity return, because there would be more beneficiaries alive in the second period. We assume a PAYG social security system exists in the economy. Upon survival from young to old age, the agent can draw total benefit bt throughout his old age. Due to the limitation of the two-period model, we can only assume there is no liquidity constraint of assessing the social security benefit, so the agent makes fully informed choices by expecting his lifetime resources. The social security benefit paid to the old is financed by a payroll tax at a rate of on wage income for all workers regardless of age. The budget must be always balanced in all periods t :  wt 1  1  zt   bt . 5 (1) Because it offers higher returns than the non-annuity saving when   (0,1) . 10 Notice that the above equation can be also interpreted as the benefit formula6 of a representative agent, since we have assumed the size of young working adult to be unity. Let us further assume that the benefit received by the agent at period t is legislated to depend on his own old age labor supply zt . As we will see soon in section IV, a minor difference of the benefit formula provides different retirement incentives. The budget constraints face by the representative agents in both periods are as follows: ct  (1   )wt  st , dt 1  (1  zt 1 )(1   )wt 1  (1  rt 1 )st   bt 1 , (2) (3) where w is the wage rate per unit of labor, and is the payroll tax rate. The agent values only consumption in young age, but both consumption and leisure in old age. This setting is reasonable because weaker health and lower productivity are usually experienced among the elderly, leisure is essential for an aged person. Besides, we can focus on old-age labor-retirement decision by simplifying the problem in the younger age. A logarithm utility function is a good candidate to describe an agent’s lifetime preference as it can give concavity on all arguments and tractable closed form solutions: Ut  ln ct   (ln dt 1   ln zt 1 ) , 0    1, and   0 , (4) So replacement rate   bt / wt   1  (1  zt )  . 6 11 where  is the relative taste of leisure to consumption in old age, and is the discounting factor or a relative weight of old age welfare to young age welfare. The aggregate production function exhibits a Cobb-Douglas form as follows: Yt  AKt L1t  , A 0 , 0    1 , (5) where K is the aggregate physical capital stock, L is the aggregate labor supply which consists of 1 unit of young adult labor and 1  z the coefficient of total factor productivity, and unit supplied by old workers, A is is the parameter indicates capital’s share of output. The Cobb-Douglas form of production function also has advantages of giving concavity on all factor inputs and satisfying the Inada condition. For simplicity, we assume full depreciation of physical capital for all periods. Factor prices are determined in a competitive way. Assuming full depreciation of capital, and competitive market implies wt  (1   ) Akt , and (6) 1  rt   Akt 1 , (7) where kt  Kt / Lt is the effective capital-labor ratio, and recall that effective labor force Lt  1  (1  zt ) . The capital market clears if the aggregate savings and investments are equal for one or some factor prices: st  Kt 1  kt 1  Lt 1  kt 1 1  (1  zt 1 ) . (8) 12 2.1 Individual’s problem The typical young agent takes (wt , , bt 1 ) as given, choosing ( st , zt 1 ) to obtain their optimal lifetime consumption and retirement (ct , dt 1 , zt 1 ) in terms of . The first order conditions of individual problem are derived as follows: 1 1  rt 1 ,  ct dt 1 (9) wt 1  .  dt 1 zt 1 (10) Notice that the intertemporal allocation of consumption in equation (9) is not affected by the probability of survival , because the chance to enjoy second period consumption (in the utility function) is offset by the same chance of receiving annuity payment (in the second budget constraint). Therefore, individual agent would make intertemporal consumption decision as if the life longevity were certain. Equation (10) shows that the marginal cost of the leisure (in terms of the wage-income-equivalent consumption measured in utility) must be equal to its marginal benefit (measured in utility directly). As people make retirement decision after the survival has been realized, the decision is no longer affected by the survival rate. Solving (10) with equilibrium conditions (6) and (7), we get the equilibrium solution for total retirement time: zt 1  z       (1   ) ,  1      (11) which is strictly positive given all the parameters are between zero and one. 13 Corner solution z  1 exists if    1      , i.e., the old agent has no  1    incentive to work at all if the pension benefit is over-generous or the payroll tax rate is too high. Also notice that when  0 , the total retirement time becomes: z  (  ) .  (1     ) The scenario is then identical to the case without social security, or a funded social security system with non-binding private saving. The result is identical to that in Zhang and Zhang (2009). 2.2 Comparative statics How does individual’s retirement decision change in response to a change in life longevity and social security? When interior solution of z exists as in equation (11), time spent on retirement, z, is decreasing in the rate of survival,  , and increasing in the payroll tax rate, . The result is summarized in result 1, and proved in appendix A. RESULT 1. The individual retirement age is positively affected by life longevity , and negatively affected by social security contribution rate . The intuition follows naturally. Rising life longevity lowers the return of life annuity (saving), people have to work more and save more to meet the increased needs for old age consumption. As we shall see later, higher saving level tends to raise the wage rate, which also induces a later retirement. However, a more generous social security plan (characterized by a higher replacement rate or a higher payroll tax) encourages early 14 retirement. On the one hand, higher payroll tax rate lowers the real return of labor and cost of old-age leisure; on the other hand, higher social security income reduces the necessity of work in the old age. 2.3 Impacts on the economy We can also investigate the impacts on the economy brought by rising longevity and social security system. If we define saving as a fraction of wage income, i.e., st   sw  wt , then the saving rate can be obtained from equation (9):  sw   1    1  1  z   st .  wt  1  z 1   1     1  1  z         1     Saving to output ratio  sy   sy  (12) st st wt 1     sw  is then Yt wt Yt 1  1  z   1   1    .  1  z 1   1     1  1  z         1     (13) Notice that the two indicators of savings differ in the term of 1    in the numerator, and  sw   sy , since wt  1    Yt / Lt and wt / Yt  1 . Result 2 can be shown easily (see appendix B). RESULT 2: Both saving rate  sw and saving-to-output ratio  sy are increasing in life longevity , but decreasing in social security contribution rate or generosity . 15 Despite a lower return of saving, individuals save more when they expect to live longer. As Bloom, Canning, and Jamison (2004) note: the idea of planning for retirement occurs only when mortality rates become low enough for retirement to be a realistic prospect. Rising longevity increases the incentive to save, and provides an incentive that can have dramatic effects on national saving rates. The success stories of East Asia Miracle can be a good footnote of this point. The region’s capital accumulation rate is driven by high household saving levels which often exceed 30 percent of income. The rise of life expectancy from 39 in 1960 to 69 in 1990 has largely contributed to the region’s rapid economic growth. However, this incentive of saving for retirement can be weakened by a PAYG social security by the provision of retirement income. As a consequence, capital accumulation is also slowed down by this policy. Since st  Kt 1  kt 1  Lt 1  kt 1 1  (1  zt 1 ) , capital-labor ratio evolves according to kt 1  st Lt 1  swwt Lt  sw (1   ) Akt 1  (1  z) , (14) and per worker capital stock converges to the steady state level 1  A sw 1    1 k    .  1 (1 ) z     (15) The existence and uniqueness of the steady state is evident through the explicit form of k∞ in Equation (15). The stability and characteristics of the steady state can be shown in Appendix C. 16 RESULT 3: There exists a unique steady state capital-labor ratio k∞, which is stable, decreasing in life longevity  and payroll tax rate  . From equation (15), we can tell two opposite impacts of life longevity on physical capital accumulation. Firstly, rising life longevity increases the needs of saving through the term  sw , which equivalently boosts the per capita investment. Secondly, higher old-age labor supply 1  z and a larger old-age potential labor force  collectively lower the capital-labor ratio. The dominant effect determines the net effect from life longevity. The steady-state per-capita output level can be found as: y  Y Ak [1   (1  z)]   , 1  1  (16) The discussion on y shown in Appendix D is summarized in Result 4 below. RESULT 4: The steady-state per capita output level y is ambiguously affected by life expectancy , but is always decreasing in payroll tax rate . The indeterminacy of dy d is caused by different impacts of rising life longevity on per capita income. First of all, rising life expectancy increases the capital level through saving and investment, which directly contributed to the final goods production. In addition, higher survival chance from young to old age generates larger old-age labor force, which is also an important factor of production. Both effects increase aggregate output level without any ambiguity. However, larger population reduces the 17 output in per capita terms. The overall effect depends on amplitude of the three effects. The hump shape of y on shows the diminishing return of capital and labor. When life expectancy is too low, a small increase in capital and labor force due to a small increase in life expectancy gives high return on capital and labor inputs. As more and more people are able to live longer, abundant saving lowers the return of physical capital and labor in this neoclassical model where final good production is the only channel of investment. The existence of PAYG social security affects the steady-state per-capita stock in an adverse way. It is reduced by higher payroll tax rate by all means. Although income per capita is a good measure of the social well beings, a lower income per capita does not necessarily imply a lower social welfare. In reality, individual may values longer lifespan despite fewer resources they might hold. To investigate the impacts of life longevity and social security on welfare, we shall complete the following procedures. Substituting consolidated lifetime budget constraint into the Euler equation (9), we can solve for the optimal consumption allocation for a representative agent: ct  1  mt 1   mt   , 1  rt 1    dt 1      mt (1  rt 1 )   mt 1  , (17) (18) where mt and mt 1 are the total (non-saving) income of young and old generations respectively, i.e.: mt  (1   )wt , and 18 mt 1  (1  z)(1   ) wt 1  bt 1  (1  z)(1   ) wt 1   wt 1 1  (1  z)   . To find the relationship between welfare and social security, we can substitute (ct , dt 1 , zt 1 ) in (17), (18) and (11) by replacing saving rate,  sw , in (12), steady state capital per worker, k in (14), together with the competitive prices wt , and rt in (6) and (7), into utility function (4). We are able to write down the welfare expression explicitly, but the tediousness prevents us to see its relationships to life longevity and social security contribution rate. Therefore, a numerical discussion in Appendix E is adopted to show Result 5. RESULT 5: Social welfare improves as life longevity increases, and decline with social security contribution rate . Longer life longevity always gives an agent higher utility level, although the per capita output is lower when life longevity reaches certain level. However, a higher social security contribution rate used to finance higher pension benefits always leads to a lower welfare. Since agents are homogeneous in this model, the Result 5 is applicable to the entire population. For a social planner, the Result 5 suggests that the expansion of social security is harmful since it weakens the working incentive and hence lowers the national output. In other words, if the society values old-age leisure, it will be welfare improving if the society can weaken the role of PAYG social security system, or replace the PAYG by funded social security system which provides greater incentive for savings and self-reliance working. 19 3. Calibration In this section, we substitute parameters with their plausible numerical values to see the quantitative effects we discussed in section 2. Case 1:   0.35 ,   0.95 , A 1.74726990 ,   0.5 ,   0.5 . is set as 0.35 which is closed to the long-term capital’s share of output in the United States. Discounting factor is chosen as 0.95 to show an agent’s impatience. Total factor of productivity (TFP) parameter A is calibrated in the way such that the aggregate output in steady state without social security is normalized to 1, given the size of labor force as well as the capital and labor’s share of output. The relative taste of leisure to consumption in old age is arbitrarily chosen as 0.5, meaning the old agent values his leisure only half as much of the consumption. In the first case, we assume only half of the young agents are able to survive to their old age, i.e. . In the subsequent cases, we will see the same impacts of social security by varying life longevity levels one at a time. 20 Table 2 Quantitative impacts when   0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z U 0.7391 0.7957 0.8522 0.9087 0.9652 1 1 1 1 1 1 -0.4085 -0.5690 -0.7504 -0.9575 -1.1974 -1.4888 -1.8547 -2.3196 -2.9654 -4.0534 -∞ sw 0.2812 0.2355 0.1950 0.1592 0.1274 0.0988 0.0734 0.0514 0.0321 0.0151 0 k∞ y 0.6667 0.5989 0.5346 0.4734 0.4146 0.3587 0.3056 0.2522 0.1958 0.1304 0 0.1430 0.1132 0.0881 0.0672 0.0498 0.0346 0.0219 0.0126 0.0061 0.0019 0 sy 0.1617 0.1389 0.1180 0.0989 0.0814 0.0642 0.0477 0.0334 0.0209 0.0098 0 The numerical simulation confirms our Results One to Five in the previous section. For a given level of life longevity , total time spent on retirement is increasing in tax rate , and gives a corner solution since onwards. Welfare, two types of saving rates, steady-state output per capita and capital-labor ratio are all decreasing in . Case 2:   0.35 ,   0.95 , A 1.74726990 ,   0.5 ,   0.75 . Table 3 Quantitative impacts when   0.75 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 z U 0.6377 0.6754 0.7130 0.7507 0.7884 0.8261 0.8638 0.9014 0.9391 0.9768 1 -0.2177 -0.3703 -0.5437 -0.7427 -0.9747 -1.2511 -1.5912 -2.0315 -2.6536 -3.7190 -∞ sw 0.3378 0.2881 0.2429 0.2016 0.1639 0.1296 0.0984 0.0701 0.0443 0.0210 0 sy 0.1726 0.1506 0.1299 0.1104 0.0920 0.0745 0.0580 0.0424 0.0276 0.0134 0 k∞ y 0.6660 0.6050 0.5460 0.4885 0.4322 0.3766 0.3209 0.2641 0.2039 0.1347 0 0.1582 0.1283 0.1021 0.0795 0.0600 0.0435 0.0296 0.0182 0.0094 0.0031 0 21 Case 3:   0.35 ,   0.95 , A 1.74726990 ,   0.5 ,   1 . Table 4 Quantitative impacts when   1 z 0.5870 0.6152 0.6435 0.6717 0.7000 0.7283 0.7565 0.7848 0.8130 0.8413 0.8696 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 U -0.0396 -0.1884 -0.3578 -0.5525 -0.7801 -1.0521 -1.3877 -1.8235 -2.4411 -3.5020 -∞ sw k∞ 0.6528 0.5974 0.5429 0.4892 0.4358 0.3823 0.3279 0.2716 0.2111 0.1405 0 sy 0.3811 0.3288 0.2802 0.2351 0.1932 0.1544 0.1184 0.0851 0.0544 0.0260 0 0.1753 0.1543 0.1343 0.1151 0.0966 0.0789 0.0619 0.0455 0.0298 0.0146 0 y 0.1620 0.1332 0.1075 0.0847 0.0648 0.0474 0.0326 0.0204 0.0106 0.0035 0 We see stylized results from the comparison of all the three cases in Table 2, 3 and 4. To illustrate them more clearly, I reconstruct the results into the following diagrams. Figure 4 Payroll tax and retirement decision 1.1 1.00 1.0 0.97 0.94 0.91 0.9 0.90 0.86 0.85 z 0.8 0.83 0.80 0.7 0.79 0.75 0.74 0.71 0.68 0.64 0.59 0.6 0.62 0.64 0.98 0.67 0.70 0.73 0.76 0.78 0.81 0.84 0.87 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 For each life longevity level  , total time spent on retirement is increasing in payroll tax rate  , and may even reach the maximum retirement time z  1 when  is 22 too high. And for each payroll tax rate  , people recess less or retire later when they expect to live longer. This numerical result is also consistent with the theoretical results in many existing literatures. Figure 5 Payroll tax and welfare 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -2 U -3 -4 -5 -6 Welfare shows a clear downward trend in payroll tax rate, but it is always higher when people live longer despite a possible lower income per capita as we will see shortly. Figure 6 Payroll tax and saving rate 0.1 0.3 45% 40% 35% 30% 25% sw 20% 15% 10% 5% 0% 0 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1 23 Figure 6 confirms the Result 2 which predicts that people save more when they expect to live longer; but larger scale of the PAYG social security definitely lowers the saving rate. Figure 7 Payroll tax and steady-state capital per worker 18% 16% 14% 12% 10% k 8% 6% 4% 2% 0% 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Besides a clear downward line in tax rate, Figure 7 shows a positive relationship between steady-state capital-labor ratio and life expectancy. However, as life expectancy becomes higher and higher, the marginal increment in k becomes smaller and smaller due to the diminishing product of capital. Figure 8 Payroll tax and steady-state output per capita 0.8 0.7 0.6 0.5 0.4 y 0.3 0.2 0.1 0.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 24 Although the aggregate output level is always higher when life expectancy increases, the difference in per capita terms can be very minimum for various life longevity levels. As to be shown in Figure 12, the steady-state per capita output may exhibit a hump shape in life longevity for some low tax rate levels, but increasing in life longevity when it becomes higher. The ambiguousness explains the intersections of the three lines which represents three different life longevity levels when at some points when tax rate is low. The numerical results also confirms the second part of the Result 4 which says income steady-state per capita output is declining in payroll tax rate , for every level of life longevity. 25 4. Policy implications The rising life longevity and declining fertility today give increasing pressure on the social security system. Among the fierce policy debates on various approaches of social security reform, the most feasible solution seems to be pushing backward the official retirement age when first allow access to social security benefit. A recent issue of The Economist confirms this idea in its special report on pensions (2011): “The first step is to increase the present retirement age. The second step is to halt the widespread practice of retiring long before the official pension age.” As shown in the Result 5, the welfare loss from working longer can be compensated by the gain from a raised longevity. Despite of the readily available solution, there are some challenging questions need to be addressed: how to reshape people’s behavior to induce more labor supply while also to ensure them a sufficient financial security in their old age? Furthermore, how can we achieve these targets without introducing dramatic institutional change which may hurt the public interests? This section discusses a contribution and a benefit schemes as an illustration that a small mechanism may affect retirement behavior for any given demographic and taxation environment. 4.1 Universal vs. individual specific benefit plan Universal pension benefit plan means every eligible elderly person receive same amount of pension benefit. In the universal benefit plan, pension benefit is calculated on the basis of average social retirement age and fiscal affordability, rather than on the 26 individual’s past earnings or duration of lifetime working. The universal pension benefit is particularly popular in the developing countries with the belief that everyone receives equal benefit can reduce population poverty. Its simplicity is also attractive to many developed countries, such as the United Kingdom (called universal state pension) and Canada, which introduced it as a key component of their comprehensive pension systems. Now let us consider a universal benefit plan, where the benefit formula is based on the social average of retirement age. The government budget is balanced if:  wt 1   (1  zt )  bt , (19) where bt is the universal benefit level, and zt is the average social retirement age. The rest part of the problem set up is the same as the baseline case in section 2. The first order condition w.r.t. zt 1 then becomes: 1    wt 1  dt 1  zt 1 . (20) Comparing to the baseline model FOC equation (10), the difference between the two FOCs is only the term in the parentheses. The universal benefit plan lowers the marginal cost of retirement (LHS) by a fraction of  , so it tends to induce early retirement. Solving above equation equilibrium for all t , we have: z       1      1   1       . (21) 27 Recall the retirement decision under individual benefit scheme in the equation (11),       1     . zt 1  z    1      With the term 1    presents in the denominator of equation (21), we can easily deduce that z  z for all the parameters holding constant. It implies that the universal benefit formula introduces a bias towards early retirement. This is because the universal pension benefit cuts the linkage between individual income level and their own working effort in the old age. Individuals would then be encouraged to supply less labor. We can then obtain Result 6. RESULT 6: Universal pension benefit plan induces early retirement compared to an individual specific pension benefit plan. To see the effects quantitatively in a graph, let us suppose there are total 30 years in one’s old age; and half of the young workers survive into the old age, i.e. =0.5. The effects of individual and universal pension benefit plans on retirement are shown in Figure 9. 28 Figure 9 Individual vs. universal benefit plans 0.5 0.4 Individual Universal 0.3 0.2 0.1 0 30 29 28 27 26 Total retirement years 25 24 23 Under universal benefit plan, total retirement time is more responsive (elastic) to the payroll tax rate. For each payroll tax rate level, an individual benefit scheme induces less retirement (more old age labor supply) than a universal benefit scheme. In another word, to delay the retirement by one year, the cut in payroll tax rate must be greater in the individual benefit plan than that in the universal benefit plan. Even under the same PAYG social security system, how the benefits are formulated matters for individual retirement decision. From above analysis, universal benefit plan which weaken the work incentive by breaking down the linkage between personal income and labor supply should be avoided during policy making. 4.2 Age-specific vs. uniform contribution schemes In reality, social security contribution rates may differ among different age groups. Many countries adopts age regressive contribution scheme for the concern of intergenerational equity. One common myth is that the implicit tax on continued work 29 should be responsible for the widely observed trend towards early retirement. Does it imply that the elderly workers should be taxed less heavily than the young in order to delay the retirement? This subsection takes a look at the age-differentiated contribution system, and attempts to show its effects on retirement behavior. Now consider an age-differentiated contribution rate,  1 on the young, and  2 on the old, with  1  0 ,  2  0 and 1   2 . Other than this differentiated payroll tax, the rest settings are the same as the baseline model. So the budget balance changes to: wt 1   2 (1  zt )  bt , (22) where bt is the new benefit level under differentiated tax system. It gives the same F.O.Cs compares to the base line model, which means the marginal trade-off of old age leisure are the same under the two different contribution schemes: wt 1  .  dt 1 zt 1 (23) Solving above equation by letting zt 1  z in equilibrium for all t, we have: z      1 (1   ) ,  1      (24) which is independent of  2 , because the pension benefit paid by the elderly workers are fully offset by the tax on continued work, leaving the real old-age income unchanged. The net effect on retirement behavior is thus only influence by the benefit from the other source, namely, the current young generation. If the tax rate in the uniform contribution 30 scheme is equal to the tax rate imposed on the young workers under the age-differentiated contribution scheme, then the retirement behavior induced by the two schemes are actually exactly the same. RESULT 7: For a given life longevity level, age-differentiated contribution scheme does not change individual retirement behavior, if the young workers are taxed at the same rate. Contrasting to the pervasive view which attributes the prevalent early retirement to the high tax rate on continued work, the effect of old age income tax is neutral as the taxed away income can be fully compensated by the pension benefit. To change the retirement behavior, policy makers should adjust the pension benefit by altering the payroll tax rate imposed on the young workers. The mechanism design of age differentiated contribution fails to meet its purpose of inducing or discouraging continued work due to such neutrality of old age contribution. 31 5. Concluding remarks In this paper, we have studied the effect of the rising life longevity and unfunded social security on retirement behavior and capital accumulation. The analyses are built upon a two-period overlapping generations (OLG) model in a neoclassical framework. We have shown that the retirement age is increasing in life longevity and decreasing in social security pension contribution (and benefit) by fixing the maximum life span, defining the life expectancy in terms of surviving chance from young to old age, and interpreting retirement as the old age leisure. The life longevity also has positive effect on saving rate, steady-state capital-labor ratio and individual welfare, but may exhibit net negative effect on steady-state per capita output if the life longevity is high enough. However, unfunded social security has negative impacts on all these measures. Numerical calibrations are then followed. The quantitative effects on retirement age, welfare, saving rates, steady-state capital labor ratio and per capita output for various life longevity and payroll tax (pension benefit) levels confirm the results in the analytical part. Policy implications can be drawn upon the theoretical model. Even within the same PAYG unfunded social security system, a difference of formula design may or may not provides different retirement incentives. We found that a universal benefit plan breaks down the connection of individual income and labor, thus discourage continued work; 32 however, an age-differentiated contribution scheme is essentially the same as the uniform contribution scheme. Such designs of the social security pension system should therefore be avoided, if the policy makers wish to push backward the voluntary retirement age as a solution to the social security crisis. 33 Bibliography Bloom, David E., David Canning, and Dean T. Jamison. 2004. Health, wealth, and welfare. Finance and development, March. Coile, Courtney, and Jonathan Gruber. 2000. Social security and retirement. NBER Working Paper No. 7830. Available at: http://www.nber.org/papers/w7830 d'Albis, Hippolyte, S. Paul Lau, and Miguel Sanchez-Romero. 2010. Mortality Transition and Differential Incentives for Early Retirement. Working Paper 10.21.327, LERNA, University of Toulouse. Ferreira, Pedro Cavalcanti, and Samuel de Abreu Pessôa. 2007. The effects of longevity and distortions on education and retirement. Review of Economic Dynamics 10 (2007): 472–493. Available at: http://www-wds.worldbank.org/external/default/main?pagePK=64193027&piPK= 64187937&theSitePK=523679&menuPK=64187510&searchMenuPK=64187283 &siteName=WDS&entityID=000009265_3970311123336 Imrohoroglu, Ayse, Selahattin Imrohoroglu, and Douglas H. Joines. 1995. A Life Cycle Analysis of Social Security. Economic Theory, vol. 6(1), pages 83-114. OECD. 2011. http://www.oecd.org Office of the Chief Actuary, Social Security Administration. 2011. http://www.ssa.gov 34 Schwarzenegger, Arnold. 2010. Public Pensions and Our Fiscal Future. In The Wall Street Journal, August 27, http://online.wsj.com/article/SB1000142405274870344700457544981307170951 0.html World Bank, The. 1994. Averting the old age crisis: policies to protect the old and promote growth. Washington, D.C.: The World Bank: Oxford University Press. Zhang, Jie, and Junsen Zhang. Longevity, retirement, and capital accumulation in a recursive model with an application to mandatory retirement. Macroeconomic Dynamics, 13 (2009), 327–348. 35 Appendices Appendix A Proof of Result 1 RESULT 1. The individual retirement age is positively affected by life longevity , and negatively affected by social security contribution rate . Proof. Differentiating in equation (11) with respect to  , we get:     1     dz   2  0. d  1      Total time spent on retirement is decreasing in , equivalently, retirement age 2   is increasing in . Differentiating z in equation (11) with respect to , we get:  1    dz   0 , given the range of all the parameters and variables. d  1      Q.E.D. 36 Appendix B Proof of Result 2 RESULT 2: Both saving rate  sw and saving-to-output ratio  sy are increasing in life longevity , but decreasing in social security contribution rate or generosity . Proof. given all the parameters and variables are between zero and one, 2  1     2 1  z   1         2 1  z    sw      1  z 1   1     1   1  z         1       2  1    1  1  z   sw    1  z 1   1     1  1  z        1     0, 2   sy     1   1     1      1   2 1  z     2  1   1    1  1  z   sw    1  z 1   1     1  1  z        1      0,  0 , and   1  z 1   1     1  1  z        1    2  2  0. Q.E.D. 37 Appendix C Discussion of Result 3 RESULT 3: There exists a unique steady state capital-labor ratio k∞, which is stable, decreasing in life longevity  and payroll tax rate  . Proof. Rewriting (14) by substituting (12): kt 1   A sw 1     kt , 1  1  z  A 1   1     1  z 1   1     1  1  z         1    kt . Subsequently, we can find the first and second order derivatives: A 2 1   1    dkt 1  kt 1  0 , dkt  1  z 1   1     1  1  z         1     A 2 1   1     2 kt 1   kt 2  0 . kt2  1  z 1   1     1  1  z        1    2 Figure 10 The stability of the unique steady-state capital-labor ratio 45 Therefore, the unique steady-state equilibrium capital-labor ratio is globally stable. 38 To see the relationship between life longevity and steady-state equilibrium capital-labor ratio, we can do the following decomposition because k is defined by  sw , z and  , which are all affected by  : dk k  sw k z k ,    d  sw  z   1  A sw 1    1 k k 1 where   0,    (1   ) sw  sw (1   ) sw  1  (1  z)  1  A sw 1    1  k k    0,    z 1   1  (1  z)   1  (1  z)  1   1  (1  z)  1  A sw 1    1 k (1  z)k 1 z   0,     1   1  (1  z)   1  (1  z)  1   1  (1  z)   sw z  0 by result 2, and  0 by result 1.   Then, 2  1     2 1  z   1         2 1  z   dk k    d (1   ) sw  1  z 1   1     1   1  z         1           2     1     (1  z)k  k  2 1   1   (1  z)   1      1   1   (1  z)    2 1  z   1         2 1  z 2              k  sw    1    1   1  z   1        (1   ) 1   1  z      (1  z)  The overall sign of dk d can be determined by introducing the parameterization. Substituting the numerical values of all the parameters suggested in the next section, we can see that dk d  0 for the whole range of   [0,1] . 39 Figure 11 The impact of on k =0.3 for illustration, the rest parameterization follows the calibration in the next section. d d Similarly, we can decompose since is defined by functions of as well: and , which are dk k  sw k z ,   d  sw  z  2   1    1   1  z   k    (1   ) sw   1  z 1   1     1   1  z         1          1     k  1   1   (1  z)   1         2     1       k   sw   2 1     1   (1  z)     (1   ) 1     where  sw k k  0 by result 2, and  0 from the previous part; and  0 and  z  sw z  0 by result 1. After substituting the plausible parameters, the sign of dk d can  be determined as negative. Q.E.D. 40 Appendix D Discussion of Result 4 RESULT 4: The steady-state per capita output level y is ambiguously affected by life expectancy , but is always decreasing in payroll tax rate . Proof. Since y is defined by k , z and  , the effect of  on y can be found through the following decomposition: dy y k y z y ,    d k  z   where and y  A1   (1  z)   1  y y A  k    0, k  0 ,  k k 1  1  z z k y zA  0 by result 1, and  0 after parameterization  k  0 ; 2     1    in result 3. Due to the ambiguousness in k , the sign of dy is also depending on the level of life d expectancy  and the tax rate  . As we can see from figure 12, for some constant parameters and low survival rates  , steady-state per capita output level rises sharply initially as life expectancy  increases, the increment gradually slows down and eventually moves in the reverse direction when  becomes large enough. This changing impact is shown in panel (a) of Figure 12, with   0.1 as an illustration. However, as the tax rate level goes higher, say   0.5 as shown in panel (b) of Figure 12, y exhibits monotonic relationship with  . 41 Figure 12 ∞ The impact of d d on y ∞ (a) = 0.1 ∞ d d ∞ (b) = 0.5 The steady-state per capita output level y is indirectly influenced by tax rate  through k and z: dy y k y z .   d k  z  42 Recall that k z y y 0,  0 from  0 from the previous part, and 0,   z k result 3 and 1 respectively. The overall sign of dy is undetermined until submitting all d the plausible parameters, and we can find that dy  0 for    0,1 when all parameters d stay constant. Q.E.D. 43 Appendix E Discussion of Result 5 RESULT 5: Social welfare improves as life longevity increases, and decline with social security contribution rate . Proof. With the aid of computational software, we can find the trend of variables for the whole spectrum of values. Assigning all the parameters with the same values as before, the effect of life longevity on welfare yields the same shape for   0,1 . As shown in Figure 13, individual welfare U is always increasing in life longevity , holding other variables and parameters constant. Figure 13 The impact of on U =0.3 for illustration, the rest parameterization d d follows the calibration in the next section. Similarly, Figure 14 shows a negative relationship between payroll tax rate  and individual welfare. The result is valid for all    0,1 . 44 Figure 14 The impact of on U d d =0.5 for illustration, the rest parameterization follows the calibration in the next section. Q.E.D 45 [...]... every level of life longevity 25 4 Policy implications The rising life longevity and declining fertility today give increasing pressure on the social security system Among the fierce policy debates on various approaches of social security reform, the most feasible solution seems to be pushing backward the official retirement age when first allow access to social security benefit A recent issue of The... 0 , the total retirement time becomes: z  (  )  (1     ) The scenario is then identical to the case without social security, or a funded social security system with non-binding private saving The result is identical to that in Zhang and Zhang (2009) 2.2 Comparative statics How does individual’s retirement decision change in response to a change in life longevity and social security? When... prices wt , and rt in (6) and (7), into utility function (4) We are able to write down the welfare expression explicitly, but the tediousness prevents us to see its relationships to life longevity and social security contribution rate Therefore, a numerical discussion in Appendix E is adopted to show Result 5 RESULT 5: Social welfare improves as life longevity increases, and decline with social security. .. Retirement and saving decisions without the existence of social security can be found in Zhang and Zhang (2009) It is served as a benchmark model to compare the impacts brought by social security Please note that only the unfunded PAYG social security system where the benefit is financed by a payroll tax is in my research interest It would be less meaningful 4 In a model with labor income uncertainties, retirement. .. Retirement Survey (HRS), Coile and Gruber (2000) confirm the deterministic role of social security in one’s retirement behavior In a forward-looking model, they find that the individual’s retirement decision appears to be made based on all the future streams of social security income, not just the wealth level or income in the next few years What explains the historical change of actual retirement age? How does... bring vast different retirement incentives Based on this idea, section four compares the retirement incentives brought by various benefit and contribution schemes, which aims to draw implications for policy making on social security reform 7 2 The theoretical model First of all, I would like to give a brief overview of the theoretical model The model has infinitely many periods and overlapping generations... generosity and retirement age in OECD countries, 2009 Average effective age of retirement 75 C 70 65 60 55 30 40 50 60 70 80 90 100 Gross pension replacement rates for median earner Source: base on data from OECD (2010) Social security is a major source of income in one’s old age, and it often constitutes a large share of family wealth Using the wealth of recent data through the Health and Retirement. .. interior solution of z exists as in equation (11), time spent on retirement, z, is decreasing in the rate of survival,  , and increasing in the payroll tax rate, The result is summarized in result 1, and proved in appendix A RESULT 1 The individual retirement age is positively affected by life longevity , and negatively affected by social security contribution rate The intuition follows naturally Rising... more and save more to meet the increased needs for old age consumption As we shall see later, higher saving level tends to raise the wage rate, which also induces a later retirement However, a more generous social security plan (characterized by a higher replacement rate or a higher payroll tax) encourages early 14 retirement On the one hand, higher payroll tax rate lowers the real return of labor and. .. has increased remarkably since the establishment of the social security Meanwhile, the dependency ratio2 has increased steadily from around 0.15 in 1950s, to 0.21 by 2010, and projected to reach 0.3 by early 2020s (OECD data, 2009) It is very suspicious that the expansion of social security system may be an important causal factor for the declining retirement age 2 The old age dependency ratio is the ... paper also compares the retirement incentives induced by different social security contribution and benefit schemes, and thus draws implications for policy making on social security reform iii LISTS... sufficient for us to understand the relations between behavior and policy motivations Retirement and saving decisions without the existence of social security can be found in Zhang and Zhang (2009) It... 25 Policy implications The rising life longevity and declining fertility today give increasing pressure on the social security system Among the fierce policy debates on various approaches of social