Redistributive effects of pension schemes if individuals differ by life expectancy

Transcription

1 Redistributive effects of pension schemes if individuals differ by life expectancy Miguel Sánchez-Romero 1,3, Ronald D. Lee 2 and Alexia Prskawetz 1,3 1 Wittgenstein Centre (IIASA, VID/ÖAW, WU) 2 University of California, Berkeley 3 Institute of Statistics and Mathematical Methods in Economics, Research Unit Economics, TU Wien, Austria The Economics of Ageing and Inequality, University of Hohenheim, Germany, May 4-5, 2018 SWM Economics ECON

2 Motivation Socio-economic differentials in life expectancy are widening in many countries. Pestieau and Ponthiere (2012) argue that Longevity inequalities across groups within nations may be as large if not larger than longevity inequalities between nations. 2 / 18

3 Motivation (cont d): Increasing longevity heterogeneity in the US (cohort and income) Chart 3. Cohort life expectancy at age 65 (and 95 percent confidence intervals) for male Social Security covered workers, by selected birth years and earnings group Years of life expectancy at age 65 Earnings in top half of distribution Earnings in bottom half of distribution Year SOURCE: Author s calculations using a matched 2001 Continuous Work History Sample. NOTE: Confidence intervals for 1912, 1917, and 1922 are so small that they are not visible on the chart. Source: Waldron, H. (2007). Trends in Mortality Differentials and Life Expectancy for Male Social Security-Covered Workers, by Socio-economic Status. Social Security Bulletin, 67(3), / 18

4 Motivation (cont d): Increasing longevity heterogeneity in the US (cohort and income) Quintile 1 Quintile 2 Quintile 3 Quintile 4 Quintile cohort 1960 cohort Figure 1: US Male Life Expectancy at Age 50 by Midcareer Average Labor Income Quintile, as Estimated by NRC (2015), for Birth Cohorts of 1930 and 1960 (Extrapolated) 4 / 18

5 Introduction Mortality differences interact with government programs for the elderly like the pension system and may reduce or even reverse the direction of redistribution 5 / 18

6 Introduction Mortality differences interact with government programs for the elderly like the pension system and may reduce or even reverse the direction of redistribution Expect behavioral responses (education, consumption, hours worked, retirement,... ) due to mortality differences 5 / 18

7 Introduction Mortality differences interact with government programs for the elderly like the pension system and may reduce or even reverse the direction of redistribution Expect behavioral responses (education, consumption, hours worked, retirement,... ) due to mortality differences Aim of the paper: - focusing on the redistributive effects of public pension systems (NDC, DB) and - studying the arising distortions of government programs for different longevity groups 5 / 18

8 Demographics Individuals are heterogeneous by their income (frailty) level I = {1, 2,..., I } denotes the set of I income levels 6 / 18

9 Demographics Individuals are heterogeneous by their income (frailty) level I = {1, 2,..., I } denotes the set of I income levels Probability of surviving to age x of an individual belonging to group i I: p i (x) = e x 0 µ i (t)dt, (1) with p i (0) = 1, p i (ω) = 0, ω (0, ) denotes the maximum age, and µ i (t) 0 is the mortality hazard rate at age t of an individual of group i 6 / 18

10 Demographics Individuals are heterogeneous by their income (frailty) level I = {1, 2,..., I } denotes the set of I income levels Probability of surviving to age x of an individual belonging to group i I: p i (x) = e x 0 µ i (t)dt, (1) with p i (0) = 1, p i (ω) = 0, ω (0, ) denotes the maximum age, and µ i (t) 0 is the mortality hazard rate at age t of an individual of group i Life expectancy at age x of an individual belonging to group i is defined as ω p i (t) e i (x) = dt. (2) x p i (x) 6 / 18

11 Demographics Individuals are heterogeneous by their income (frailty) level I = {1, 2,..., I } denotes the set of I income levels Probability of surviving to age x of an individual belonging to group i I: p i (x) = e x 0 µ i (t)dt, (1) with p i (0) = 1, p i (ω) = 0, ω (0, ) denotes the maximum age, and µ i (t) 0 is the mortality hazard rate at age t of an individual of group i Life expectancy at age x of an individual belonging to group i is defined as ω p i (t) e i (x) = dt. (2) x p i (x) Total population size at time t is ω P(t) = B(t) e nx p(x)dx, with p(x) = 1 0 I p i (x), (3) i I where B(t) is the total number of births at time t, p(x) is the average survival 6 / 18

12 General pension model Parametric components Assumption: Only old-age survivor s pensions no pensions for orphans or widows. The total number of pension points accumulated at the exact age x x 0 by an individual of type i I in any pension system can be formulated as follows x pp is (x; r) = φ e r(x t) p s (t) p s (x) τy i (t)dt with pp is (x 0; r) = 0, (4) x 0 7 / 18

13 General pension model Parametric components Assumption: Only old-age survivor s pensions no pensions for orphans or widows. The total number of pension points accumulated at the exact age x x 0 by an individual of type i I in any pension system can be formulated as follows x pp is (x; r) = φ e r(x t) p s (t) p s (x) τy i (t)dt with pp is (x 0; r) = 0, (4) x 0 where x 0 r p s (x) y i (t) τ φ minimum working age capitalization factor of pension points survival probability to age x used by the pension system labor income of a worker belonging to group i at age t contribution rate pension points earned per unit of social contribution paid 7 / 18

14 General pension model Parametric components Assumption: Only old-age survivor s pensions no pensions for orphans or widows. The total number of pension points accumulated at the exact age x x 0 by an individual of type i I in any pension system can be formulated as follows x pp is (x; r) = φ e r(x t) p s (t) p s (x) τy i (t)dt with pp is (x 0; r) = 0, (4) x0 where x 0 r p s (x) y i (t) τ φ minimum working age capitalization factor of pension points survival probability to age x used by the pension system labor income of a worker belonging to group i at age t contribution rate pension points earned per unit of social contribution paid, where { φ = 1 φ = ρ/τ in DC in DB 7 / 18

15 General pension model Parametric components Assumption: Only old-age survivor s pensions no pensions for orphans or widows. The total number of pension points accumulated at the exact age x x 0 by an individual of type i I in any pension system can be formulated as follows x pp is (x; r) = φ e r(x t) p s (t) p s (x) τy i (t)dt with pp is (x 0; r) = 0, (4) x0 where x 0 r p s (x) y i (t) τ φ minimum working age capitalization factor of pension points survival probability to age x used by the pension system labor income of a worker belonging to group i at age t contribution rate pension points earned per unit of social contribution paid, where { φ = 1 φ = ρ/τ in DC in DB Note: Second subscript s denotes the survival probability used by the social security system. 7 / 18

16 General pension model To calculate the pension benefit (b) of a retiree, the government applies a conversion factor, f is ( ), that transforms at age R i the pension points accumulated pp into pension benefits b is (R i, pp is (R i ; r)) = f is (R i, pp is (R i ; r))pp is (R i ; r). (5) 8 / 18

17 General pension model To calculate the pension benefit (b) of a retiree, the government applies a conversion factor, f is ( ), that transforms at age R i the pension points accumulated pp into pension benefits b is (R i, pp is (R i ; r)) = f is (R i, pp is (R i ; r))pp is (R i ; r). (5) DC system: where A s (R i, r) E i (R i ) f is (R i, pp is (R i ; r)) = (A s (R i, r)) 1 E i (R i ), (6) actuarial present value of a lifetime annuity paid from age R i correction factor for differences in life expectancy btw individuals 8 / 18

18 General pension model To calculate the pension benefit (b) of a retiree, the government applies a conversion factor, f is ( ), that transforms at age R i the pension points accumulated pp into pension benefits b is (R i, pp is (R i ; r)) = f is (R i, pp is (R i ; r))pp is (R i ; r). (5) DC system: f is (R i, pp is (R i ; r)) = (A s (R i, r)) 1 E i (R i ), (6) where A s (R i, r) { = 1 w/o E i (R i ) = A s (R i,r) A w/ i (R i,r) actuarial present value of a lifetime annuity paid from age R i correction factor for differences in life expectancy btw individuals 8 / 18

19 General pension model To calculate the pension benefit (b) of a retiree, the government applies a conversion factor, f is ( ), that transforms at age R i the pension points accumulated pp into pension benefits b is (R i, pp is (R i ; r)) = f is (R i, pp is (R i ; r))pp is (R i ; r). (5) DC system: where A s (R i, r) E i (R i ) f is (R i, pp is (R i ; r)) = (A s (R i, r)) 1 E i (R i ), (6) actuarial present value of a lifetime annuity paid from age R i correction factor for differences in life expectancy btw individuals DB system: f is (R i, pp is (R i ; r)) = ϕ(pp is (R i ; r))β(r i )E i (R i ), (7) 8 / 18

20 General pension model To calculate the pension benefit (b) of a retiree, the government applies a conversion factor, f is ( ), that transforms at age R i the pension points accumulated pp into pension benefits b is (R i, pp is (R i ; r)) = f is (R i, pp is (R i ; r))pp is (R i ; r). (5) DC system: where A s (R i, r) E i (R i ) f is (R i, pp is (R i ; r)) = (A s (R i, r)) 1 E i (R i ), (6) actuarial present value of a lifetime annuity paid from age R i correction factor for differences in life expectancy btw individuals DB system: where ϕ(pp) β(r i ) E i (R i ) f is (R i, pp is (R i ; r)) = ϕ(pp is (R i ; r))β(r i )E i (R i ), (7) denotes the replacement rate denotes an adjustment factor for early or late retirement correction factor for differences in life expectancy btw individuals 8 / 18

21 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as SSW is (x, R i ) = e r(r i x) p i (R i ) Ri p i (x) b is (R i, pp is (R i ; r))a i (R i, r) e r(t x) p i (t) x p i (x) τy i (t, R i )dt. (8) 9 / 18

22 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as Ri SSW is (x, R i ) = P is (x, R i )pp is (x; r) e r(t x) p i (t) x p i (x) t is (t, R i )y i (t, R i )dt, (8) 9 / 18

23 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as Ri SSW is (x, R i ) = P is (x, R i )pp is (x; r) e r(t x) p i (t) x p i (x) t is (t, R i )y i (t, R i )dt, (8) where P is (x, R i ) the price at age x of one pension point at age R figure t is (t, R i ) = τ (1 φp is (t, R i )) implicit tax/subsidy rate on work at age t x 9 / 18

24 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as Ri SSW is (x, R i ) = P is (x, R i )pp is (x; r) e r(t x) p i (t) x p i (x) t is (t, R i )y i (t, R i )dt, (8) where P is (x, R i ) the price at age x of one pension point at age R figure t is (t, R i ) = τ (1 φp is (t, R i )) implicit tax/subsidy rate on work at age t x The dynamics of P is is given by Ṗ is (x, R i ) P is (x, R i ) = (r r) + (µ i (x) µ s (x)). (9) 9 / 18

25 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as Ri SSW is (x, R i ) = P is (x, R i )pp is (x; r) e r(t x) p i (t) x p i (x) t is (t, R i )y i (t, R i )dt, (8) where P is (x, R i ) the price at age x of one pension point at age R figure t is (t, R i ) = τ (1 φp is (t, R i )) implicit tax/subsidy rate on work at age t x The dynamics of P is is given by Ṗ is (x, R i ) = (r r) + P is (x, R i ) (µ i (x) µ s (x)) }{{}. (9) Mortality differential effect 9 / 18

26 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as Ri SSW is (x, R i ) = P is (x, R i )pp is (x; r) e r(t x) p i (t) x p i (x) t is (t, R i )y i (t, R i )dt, (8) where P is (x, R i ) the price at age x of one pension point at age R figure t is (t, R i ) = τ (1 φp is (t, R i )) implicit tax/subsidy rate on work at age t x The dynamics of P is is given by (P evolves differently by life expectancy) figure Ṗ is (x, R i ) = (r r) + P is (x, R i ) (µ i (x) µ s (x)) }{{}. (9) Mortality differential effect 9 / 18

27 The pension wealth Assuming that an individual will retire at age R i, we define the social security wealth (SSW ) at age x R i of an individual of type i as Ri SSW is (x, R i ) = P is (x, R i )pp is (x; r) e r(t x) p i (t) x p i (x) t is (t, R i )y i (t, R i )dt, (8) where P is (x, R i ) the price at age x of one pension point at age R figure t is (t, R i ) = τ (1 φp is (t, R i )) implicit tax/subsidy rate on work at age t x The dynamics of P is is given by (P evolves differently by life expectancy) figure Ṗ is (x, R i ) = (r r) + P is (x, R i ) (µ i (x) µ s (x)) }{{}. (9) Mortality differential effect Note that an increase in P is has a negative impact on t is (t, R i ) and hence a positive impact on SSW is (x 0, R i ). 9 / 18

28 The pension model The average implicit tax/subsidy rate on work might be either positive (tax) or negative (subsidy) according to { > 0 if φp is (x, R i ) < 1, t is (x, R i ) (10) < 0 if φp is (x, R i ) > / 18

29 The pension model The average implicit tax/subsidy rate on work might be either positive (tax) or negative (subsidy) according to { > 0 if φp is (x, R i ) < 1, t is (x, R i ) (10) < 0 if φp is (x, R i ) > 1. An actuarially fair pension system satisfies that φp is (x, R i ) = 1 10 / 18

30 Economic problem The individual optimally chooses the length of schooling, E i, the retirement age, R i, the consumption path, c i (t) c i (t, E i, R i ), and hours worked path, l i (t) l i (t, E i, R i ), maximizing the following lifetime expected utility ω V i (E, R, c, l) = x0 e ρ(t x0) p i (t) Ei p i (x U(c i (t))dt 0) Ri e ρ(t x0) p i (t) E i p i (x α i v(l i (t))dt 0) Ri x0 x 0 e ρ(t x0) p i (t) p i (x 0) ηdt e ρ(t x0) p i (t) ϕ(t)dt. (11) p i (x 0) 11 / 18

31 Economic problem The individual optimally chooses the length of schooling, E i, the retirement age, R i, the consumption path, c i (t) c i (t, E i, R i ), and hours worked path, l i (t) l i (t, E i, R i ), maximizing the following lifetime expected utility ω V i (E, R, c, l) = x0 e ρ(t x0) p i (t) Ei p i (x U(c i (t))dt 0) Ri e ρ(t x0) p i (t) E i p i (x α i v(l i (t))dt 0) Ri x0 x 0 e ρ(t x0) p i (t) p i (x 0) ηdt e ρ(t x0) p i (t) ϕ(t)dt. (11) p i (x 0) subject to ω e r(t x p 0) i (t) Ri p i (x c i (t)dt = e r(t x 0) p i (t) 0) E i p i (x (1 t is (t, E i, R i ))y i (t)dt, (12) 0) x0 11 / 18

32 Economic problem The individual optimally chooses the length of schooling, E i, the retirement age, R i, the consumption path, c i (t) c i (t, E i, R i ), and hours worked path, l i (t) l i (t, E i, R i ), maximizing the following lifetime expected utility ω V i (E, R, c, l) = x0 e ρ(t x0) p i (t) Ei p i (x U(c i (t))dt 0) Ri e ρ(t x0) p i (t) E i p i (x α i v(l i (t))dt 0) Ri x0 x 0 e ρ(t x0) p i (t) p i (x 0) ηdt e ρ(t x0) p i (t) ϕ(t)dt. (11) p i (x 0) subject to ω e r(t x p 0) i (t) Ri p i (x c i (t)dt = e r(t x 0) p i (t) 0) E i p i (x (1 t is (t, E i, R i ))y i (t)dt, (12) 0) x0 where ω maximum longevity ρ subjective discount factor p i (t) probability of surviving to age t by an individual of type i U(c) instantaneous utility function (with U (c) > 0 and U (c) < 0) η disutility of schooling (Sánchez-Romero et al., 2016) ϕ(t) disutility of labor is proportional to the mortality (Bloom et al., 2014) α i weight of the disutility of work on the lifetime utility v(l) disutility of work (with v (l) > 0 and v (l) > 0) y i (t, R i ) gross labor income of an individual of type i (=w(t)h i (t, E i )l i (t)) 11 / 18

33 Optimal decisions (consumption and labor supply) The optimal consumption path and labor supply, conditional on a length of schooling E i and a retirement age R i, are characterized by α i v (l i (t)) U (c i (t)) ċ i (t) = σ(r ρ), (13) c i (t) = (1 τ is (t, E i, R i ))w(t)h(t, E i ). (14) where σ is the inverse of the risk-aversion coefficient σ = U (c) cu (c) > 0, and where τ is (t, R i ) is the marginal implicit tax/subsidy rate on work for an individual of type i 12 / 18

34 Optimal decisions (consumption and labor supply) The optimal consumption path and labor supply, conditional on a length of schooling E i and a retirement age R i, are characterized by α i v (l i (t)) U (c i (t)) ċ i (t) = σ(r ρ), (13) c i (t) = (1 τ is (t, E i, R i ))w(t)h(t, E i ). (14) where σ is the inverse of the risk-aversion coefficient σ = U (c) cu (c) > 0, and where τ is (t, R i ) is the marginal implicit tax/subsidy rate on work for an individual of type i τ is (t, R i ) = { t is (t, R i ) if ε is (R i ) = 0, t is (t, R i ) + τφp is (t, R i )ε is (R i ) if ε is (R i ) > 0, (15) and ε is (R i ) is the elasticity between the replacement rate and pension points at age R i Example. 12 / 18

35 Optimal decisions (length of schooling and retirement) An optimal length of schooling satisfies r h (E i ) = (1 T E is (E i, R i )) r(e i, R i ) + η α i v(l i (E i )) W (E i, R i )U (c i (E i )). (16) 13 / 18

36 Optimal decisions (length of schooling and retirement) An optimal length of schooling satisfies r h (E i ) = (1 T E is (E i, R i )) r(e i, R i ) + η α i v(l i (E i )) W (E i, R i )U (c i (E i )). (16) An interior optimal retirement age satisfies ( ) U (c i (R i ))y i (R i, E i ) 1 T R is (E i, R i ) = α i v(l i (R i )) + ϕ(r i ), (17) 13 / 18

37 Optimal decisions (length of schooling and retirement) An optimal length of schooling satisfies r h (E i ) = (1 T E is (E i, R i )) r(e i, R i ) + η α i v(l i (E i )) W (E i, R i )U (c i (E i )). (16) An interior optimal retirement age satisfies ( ) U (c i (R i )) y i (R i, E i ) 1 T R is }{{} (E i, R i ) = α i v(l i (R i )) + ϕ(r i ), (17) }{{}}{{}}{{} Income effect disut. of work disut. of not Pension effect where being retired r h (E i ) r(e i, R i ) Tis E (E i, R i ) Tis R (E i, R i ) returns to education at E i years of schooling marginal cost of education (financial) is the implicit tax/subsidy on education is the implicit tax/subsidy on retirement 13 / 18

38 PAYG pension systems Table 1: Modeled PAYG pension systems Pension system Acronym Replacement Rate Life Expectancy Correction Defined Contribution NDC-I No Defined Contribution NDC-II Yes (at retirement) Defined Contribution NDC-III Yes (all ages) Defined Benefits DB-I Constant No Defined Benefits DB-II Progressive No Defined Benefits DB-III Progressive Yes (at retirement) Notes: (i) DB-II case matches the US pension system, (ii) NDC-II and DB-III cases implements the proposal of Ayuso et al. (2017). 14 / 18

39 Redistributive effects of each pension system: Internal rate of return by income quintile and pension system Internal rate of return (in %) Actuarially fair pension system (NDC III) Constant Replacement (DB I) USA OAS (DB II) USA OAS corrected by LE (DB III) q1 q2 q3 q4 q5 Figure 2: US males, 1930 cohort irr NDCs 15 / 18

40 Redistributive effects of each pension system: Internal rate of return by income quintile and pension system Internal rate of return (in %) Actuarially fair pension system (NDC III) Constant Replacement (DB I) USA OAS (DB II) USA OAS corrected by LE (DB III) q1 q2 q3 q4 q5 Figure 2: US males, 1960 cohort irr NDCs 15 / 18

41 Implicit tax on work by income quintile and pension system Pension system, DB I Pension system, DB II Pension system, DB III q1 q2 q3 q4 q5 q1 q2 q3 q4 q5 q1 q2 q3 q4 q Implicit tax/saving rate per $1.00 of contribution E R E R E R E R E R Age E R E R E R E R E R Age E R E R E R E R E R Age Figure 3: US males, 1930 cohort Work NDCs, Recall: τ is (t, R i ) = τ(1 φp is (t, R i )(1 ε is (R i ))) 16 / 18

42 Implicit tax on work by income quintile and pension system Pension system, DB I Pension system, DB II Pension system, DB III q1 q2 q3 q4 q5 q1 q2 q3 q4 q5 q1 q2 q3 q4 q Implicit tax/saving rate per $1.00 of contribution E R E R E R E R E R Age E R E R E R E R E R Age E R E R E R E R E R Age Figure 3: US males, 1960 cohort Work NDCs, Recall: τ is (t, R i ) = τ(1 φp is (t, R i )(1 ε is (R i ))) 16 / 18

43 Implicit tax on retirement by income quintile and pension system Tax/subsidy rate (in %) Actuarially fair pension system (NDC III) Constant Replacement (DB I) USA OAS (DB II) USA OAS corrected by LE (DB III) q1 q2 q3 q4 q5 Figure 4: US males, 1930 cohort Retirement NDCs 17 / 18

44 Implicit tax on retirement by income quintile and pension system Tax/subsidy rate (in %) Actuarially fair pension system (NDC III) Constant Replacement (DB I) USA OAS (DB II) USA OAS corrected by LE (DB III) q1 q2 q3 q4 q5 Figure 4: US males, 1960 cohort Retirement NDCs 17 / 18

45 Conclusions We have developed a general framework for analyzing any pension system Within this framework we study the redistributive effect of the pension system when there exists differential mortality across income groups We are taking into account the behavioral reactions of each pension system (on savings, education, and labor supply) The widening of the longevity gap between the birth cohorts of 1930 and 1960 leads to much greater inequalities in the pension 18 / 18

46 Conclusions We have developed a general framework for analyzing any pension system Within this framework we study the redistributive effect of the pension system when there exists differential mortality across income groups We are taking into account the behavioral reactions of each pension system (on savings, education, and labor supply) The widening of the longevity gap between the birth cohorts of 1930 and 1960 leads to much greater inequalities in the pension 18 / 18

47 Conclusions We have developed a general framework for analyzing any pension system Within this framework we study the redistributive effect of the pension system when there exists differential mortality across income groups We are taking into account the behavioral reactions of each pension system (on savings, education, and labor supply) The widening of the longevity gap between the birth cohorts of 1930 and 1960 leads to much greater inequalities in the pension 18 / 18

48 Conclusions We have developed a general framework for analyzing any pension system Within this framework we study the redistributive effect of the pension system when there exists differential mortality across income groups We are taking into account the behavioral reactions of each pension system (on savings, education, and labor supply) The widening of the longevity gap between the birth cohorts of 1930 and 1960 leads to much greater inequalities in the pension 18 / 18

49 Thank you! We thank Bernhard Hammer, Robert Holzmann, Michael Kuhn, Edward Palmer, and Alexia Prskawetz. We also thank Arda Aktas, Gretchen Donehower, and Miguel Plobete-Cazenave for providing valuable data. This project has received funding from the European Union s Seventh Framework Program for research, technological development and demonstration under grant agreement no : Ageing Europe: An application of National Transfer Accounts (NTA) for explaining and projecting trends in public finances. This project has also being partly financed by the Austrian National Bank (OeNB) under Grant no Ronald Lee s research was supported by the grant NIA 5R24 AG / 18

50 US OAI pension system (DB-II) Replacement rate, ψ(p) p:= y:= Pension earnings or Average Indexed Monthly Earnings (AIME) Average Labor Income y/6 y 2y p (or AIME) Figure 5: Old-Age Insurance replacement rate in the US Note: AIME is calculated as 1/12 of the mean of the 35 highest labor incomes over the working life, measured in real terms. optimal conditions 18 / 18

51 Table 2: Alternative PAYG pension systems and their impact on the social security wealth at age x 0 by life expectancy e i Avg. implicit tax Soc. sec. wealth Defined Contribution (DC) Avg. Life Table (LT) Corrected Avg. LT i-th LT Symbol NDC-I NDC-II NDC-III tis SSWis (x0) { > 0 for ei < es, < 0 for ei > es. { < 0 for ei < es, > 0 for ei > es. { 0 for ei < es, 0 for ei > es. { < 0 for ei < es, > 0 for ei > es. 0 0 Avg. implicit tax Soc. sec. wealth Defined Benefit (DB) Non-Progressive Progressive Corrected-Progressive Symbol { DB-I DB-II DB-III > 0 tis for ei < es, 0 0 < 0 for ei > es. { SSWis (x0) < 0 for ei < es, > 0 for ei > es. 0 0 Notes: All the calculations are done under the following assumptions: (a) the life expectancy is positively correlated with the income level, (b) the market interest rate r is equal to r = n + g, (c) the pension replacement rate ϕ is equal to (τ/ϱ)/as (R, r) so as to coincide with the defined contribution system, and (d) the retirement age is fixed at the normal retirement age for all population groups, which implies that β(r) = 1. Ai (R, r) denotes the actuarial present value of an individual of type i at the exact age R when the effective interest rate is r. 18 / 18

52 Redistributive effects of each pension system: Internal rate of return by income quintile and pension system Internal rate of return (in %) Actuarially fair pension system (NDC III) NDC I NDC II NDC III q1 q2 q3 q4 q5 Figure 6: US males, 1930 cohort irr DBs 18 / 18

53 Redistributive effects of each pension system: Internal rate of return by income quintile and pension system Internal rate of return (in %) Actuarially fair pension system (NDC III) NDC I NDC II NDC III q1 q2 q3 q4 q5 Figure 6: US males, 1960 cohort irr DBs 18 / 18

54 Implicit tax on work by income quintile and pension system Pension system, NDC I Pension system, NDC II Pension system, NDC III q1 q2 q3 q4 q5 q1 q2 q3 q4 q5 q1 q2 q3 q4 q E R E R E R E R E R Age E R E R E R E R E R Age E R E R E R E R E R Age Figure 7: US males, 1930 cohort Work DBs 18 / 18

55 Implicit tax on work by income quintile and pension system Pension system, NDC I Pension system, NDC II Pension system, NDC III q1 q2 q3 q4 q5 q1 q2 q3 q4 q5 q1 q2 q3 q4 q E R E R E R E R E R Age E R E R E R E R E R Age E R E R E R E R E R Age Figure 7: US males, 1960 cohort Work DBs 18 / 18

56 Implicit tax on retirement by income quintile and pension system Tax/subsidy rate (in %) Actuarially fair pension system (NDC III) NDC I NDC II corrected by LE NDC III q1 q2 q3 q4 q5 Figure 8: US males, 1930 cohort Retirement DBs 18 / 18

57 Implicit tax on retirement by income quintile and pension system Tax/subsidy rate (in %) Actuarially fair pension system (NDC III) NDC I NDC II corrected by LE NDC III q1 q2 q3 q4 q5 Figure 8: US males, 1960 cohort Retirement DBs 18 / 18

58 Optimal retirement ages Table 3: Optimal length of schooling by income quintile (R i ), US male birth cohorts 1930 and 1960 Defined Contribution (NDC) Defined Benefit Avg. LT Corrected i th LT Non Progressive Progressive Avg. LT progressive Corrected NDC-I NDC-II NDC-III DB-I DB-II DB-III Cohort 1930 Quintile Quintile Quintile Quintile Quintile Cohort 1960 Quintile Quintile Quintile Quintile Quintile / 18

59 Optimal retirement ages Table 4: Optimal retirement age by income quintile (R i ), US male birth cohorts 1930 and 1960 Defined Contribution (NDC) Defined Benefit Avg. LT Corrected i th LT Non Progressive Progressive Avg. LT progressive Corrected NDC-I NDC-II NDC-III DB-I DB-II DB-III Cohort 1930 Quintile Quintile Quintile Quintile Quintile Cohort 1960 Quintile Quintile Quintile Quintile Quintile / 18

60 Price of a pension point, Pis (x, R) Ṗ is (x,r) P is (x,r) = (r r) + (µi (x) µs (x)) Downward slope Mag. benefit of education tis (t, R) < 0 higher subsidy µlow (x) < µavg (x) contrib. rate accrual rate = τ τ = 1 actuarially fair r = r tis (t, R) > 0 higher tax Upward slope Mag. benefit of education µhigh(x) > µavg (x) 0 x0 R Age x Figure 9: Case NDC-I, Corrected Average Life Table 18 / 18

61 Price of a pension point, Pis (x, R) Ṗ is (x,r) P is (x,r) = (r r) + (µi (x) µs (x)) Downward slope Mag. benefit of education tis (t, R) < 0 higher subsidy µlow (x) < µavg (x) contrib. rate accrual rate = τ τ = 1 actuarially fair r = r µhigh(x) > µavg (x) tis (t, R) > 0 higher tax Upward slope Mag. benefit of education 0 x0 R Age x Figure 10: Case NDC-II, Corrected Average Life Table Back 18 / 18

62 Price of a pension point, Pis (x, R) Ṗ is (x,r) P is (x,r) = (r r) + (µi (x) µs (x)) contrib. rate accrual rate = τ τ = 1 actuarially fair r = r tis (t, R) > 0 higher tax tis (t, R) < 0 higher subsidy 0 x0 R Age x Figure 11: Case NDC-III, ith Life Table 18 / 18

63 Price of a pension point, Pis (x, R) Ṗ is (x,r) P is (x,r) = (r r) + (µi (x) µs (x)) Downward slope Mag. benefit of education tis (t, R) < 0 higher subsidy µlow (x) < µavg (x) contrib. rate accrual rate = τ ρ actuarially fair r = r µhigh(x) > µavg (x) tis (t, R) > 0 higher tax Upward slope Mag. benefit of education 0 x0 R Age x Figure 12: Case DB-I, Non-Progressive (Note that this is similar to NDC-I) 18 / 18

64 Price of a pension point, Pis (x, R) Ṗ is (x,r) P is (x,r) = (r r) + (µi (x) µs (x)) tis (t, R) < 0 higher subsidy Downward slope Mag. benefit of education µhigh(x) > µavg (x) contrib. rate accrual rate = τ ρ actuarially fair r = r tis (t, R) > 0 higher tax Upward slope Mag. benefit of education µlow (x) < µavg (x) 0 x0 R Age x Figure 13: Case DB-III, Corrected-Progressive. Both curves can be shifted up or down depending on the degree of progressivity. DB-II coincides with DB-III, although it is not guaranteed that at retirement the value of P is above (resp. below) the actuarial fairness line for the high (resp. low) mortality group. 18 / 18

65 Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 x0 x R R + 1 R R + n Age x pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

66 Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 b = f b = f... b = f x0 x R R + 1 R R + n Age x pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

67 Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 b = f b = f... b = f x0 x R R + 1 R R + n Age x pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

68 fi (R, )Ai (R, r) Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 b = f b = f... b = f x0 x R R + 1 R R + n Age x pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

69 fi (R, )Ai (R, r) Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 e r(r x) p i (R) p i (x) b = f b = f... b = f x0 x R R + 1 R R + n Age x pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

70 e r(r x) p i (R) p fi (R, )Ai (R, r) i (x) fi (R, )Ai (R, r) Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 e r(r x) p i (R) p i (x) b = f b = f... b = f x0 x R R + 1 R R + n Age x pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

71 e r(r x) p i (R) p fi (R, )Ai (R, r) i (x) fi (R, )Ai (R, r) Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 e r(r x) p i (R) p i (x) b = f b = f... b = f x0 x R R + 1 R R + n Age x e r(r x) p s (R) p s (x) pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

72 e r(r x) p i (R) p fi (R, )Ai (R, r) i (x) fi (R, )Ai (R, r) Pis (x, R) = e r(r x) pi (R) p i (x) f i (R, )A i (R,r) e r(r x) p s (R) p s (x) 1 e r(r x) p i (R) p i (x) b = f b = f... b = f x0 x R R + 1 R R + n Age x e r(r x) p s (R) p s (x) e r(r x) p s (R) p s (x) 1 pp = 1 Figure 14: The price at age x of one pension point at age R, P is(x, R) Back 18 / 18

73 Table 5: Model parameters Parameter Symbol Value Parameter Symbol Value Demographics Preferences First age at entrance x0 14 Subjective discount factor ρ 0,005 Maximum age ω 115 Utility cost of not being retired ϕ(x) 92e x Annual population growth n 0,005 Labor elasticity of substitution σ 0,33 Minimum length of schooling E 10 Utility weight of labor cost α(q1) 140 Maximum length of schooling Ē 20 α(q2) 120 α(q3) 80 Technology α(q4) 50 Market interest rate r 0,030 α(q5) 85 Labor-aug. tech. progress growth rate g 0,015 Education Social security system Returns of scale in education γ 0,65 Minimum retirement age R 55 Disutility of schooling η 3 Maximum retirement age R 71 Mincerian eq. β0 0,07 Capitalization factor r 0,02 β Accrual rate in DB systems φ 1/45 Learning ability θ(q1) 0,113 Avg. replacement rate in DB systems f (pp) 0,4167 θ(q2) 0,113 Social contribution rate (US pension) θ(q3) 0,113 Cohort 1930 τ1930 0,1043 θ(q4) 0,114 Cohort 1960 τ1960 0,1087 θ(q5) 0, / 18

74 Table 6: Sensitivity analysis: Behavioral differences (Case, DB-II) PVB 50 (in 1, 000) IRR (in %) Length of education Retirement Benchmark Same Benchmark Same Benchmark Same Benchmark Same preferences preferences preferences preferences Cohort 1930 Quintile ,8 1, Quintile ,0 1, Quintile ,0 2, Quintile ,0 2, Quintile ,1 2, q5-q ,3 0, Cohort 1960 Quintile ,0 0, Quintile ,3 1, Quintile ,1 2, Quintile ,2 2, Quintile ,3 2, q5-q ,3 1, Higher inequality (IRR) Lower difference in PVB at age 50 due to the lower education of q5 individuals. 18 / 18