A simple model of aggregate pension expenditure

A simple model of aggregate pension expenditure Angel de la Fuente* Instituto de Análisis Económico (CSIC) and Barcelona GSE April 2011 Abstract This paper develops a simple model that can be used to analyze the long-term sustainability of the contributive pension system and the steady-state response of pension expenditure to changes in some key demographic and economic variables, in the characteristics of the average pensioner and in the parameters that describe how pensions are calculated in Spain as a function of workers' Social Security contribution histories. Key words: pensions, sustainability, Spain JEL Classification: H55 * Financial support from BBVA Research and from the Spanish Ministry of Science and Innovation (under project ECO2008-04837/ECON) is gratefully acknowledged. I would like to thank Maite Naranjo for her help with some of the calculations. 1 1. Introduction This paper develops a simple model that can be used to analyze the long-term sustainability of the contributive pension system and the steady-state response of pension expenditure to changes in some key demographic and economic variables, in the characteristics of the average pensioner and in the parameters that describe how pensions are calculated in Spain as a function of workers' Social Security contribution histories. The model achieves tractability at the price of some very strong assumptions, including deterministic life spans and constant rates of growth of total employment and wages, ignores the heterogeneity of agents and the endogeneity of decisions to enter and exit the labor market, and does not take into account some important characteristics of the Spanish pension system, including the existence of caps and floors on contribution bases and pension levels and the possibility of early retirement. Under these assumptions, the model can be used to calculate the average pension and the ratio of this variable to the average salary, the ratio of pensioners to employed workers, the pension system's total current revenues and expenditures and its internal rate of return. It also provides two simple characterizations of the system's long-term financial sustainability: the contributive pension system will be sustainable in the long run if and only if its internal rate of return does not exceed the growth rate of aggregate wage income or, equivalently, if its initial replacement rate (the ratio between the initial pension and the wage at the time of retirement) does not exceed a critical value. In spite of its simplistic assumptions, the model highlights the main determinants of spending in contributory pensions and the necessary conditions for the system's sustainability. It can also be a useful complement of the standard short-cut procedure for projecting pension expenditure, which is based on a decomposition of this variable, measured as a fraction of GDP, into three factors that capture, respectively, the effects on pension outlays of demographics, labor market performance and the generosity of the pension system. 1 In particular, the model introduces a certain amount of discipline when projecting into the future the system's generosity factor (generally defined as the ratio between the average pension and average output per employed worker), which is the component of pension expenditure that is hardest to forecast directly. The remainder of the paper is divided into six sections and an appendix. Section 2 sets out the model's assumptions regarding demographics and the evolution of wages. Section 3 contains a simplified description of how retirement and widowers' pensions are set in Spain. In sections 4 and 5, wages and pensions are aggregated across individuals to calculate the key magnitudes of the pension system and two alternative characterizations of its long-term sustainability are derived. Section 6 contains a numerical analysis of the comparative statics of the model. Finally, section 7 concludes and the Appendix contains the details of the calculations. 1 See for instance Jimeno, Rojas and Puente (2008) and Doménech and Melguizo (2008). For an application that combines the decomposition sketched in the text with the model developed in this paper in order to quantify the effects of Spain's most recent pension reform, see de la Fuente and Doménech (2011). 2 2. Demographic assumptions and the evolution of wages The model economy is populated by overlapping generations of a continuum of finitely-lived homogeneous agents. The number of births grows over time at a constant exponential rate, n, so that the number of agents born at time s is equal to (1) L(s)=ens An individual born at time s enters the labor market at s+E and starts to work immediately, retires at s+J and dies at s+Z. With probability π , he leaves behind a spouse who survives until s+Z2. Figure 1: Breakdown of the population at time t by its economic status I will treat wages as exogenous. It will be assumed that average real wages increase over time at a constant rate due to technical progress and capital accumulation and that individual wages rise with experience as well. The real wage at time t![s+E,s+J] of a worker born at time s will be given by (2) W(s,t)=Ate ! (t"(s+E)) =Aoegt e ! (t"s"E)=Aoe(g+ ! )te" ! (s+E) where At=Aoegt captures the effects of technical progress and capital accumulation on average wages and the term e ! (t"s"E) is the experience premium. To simplify the calculations, I have assumed that the experience premium grows at a constant rate, ν , and does not therefore display the hump-shaped pattern that is usually found in the data. 3. Pension determination I will assume that pensions are set using the rules that are currently applied in Spain. The starting pension of an individual born at time s who retires at s+J is given by (3) P(s,s+J,C,N)= ! (C)B(s,s+J,N) 3 where φ () is a percentage that depends on the number of years the agent has paid Social Security contributions,2 (4) C = JE, and B(), the so-called regulatory base of the pension, is an average of the agent’s past wages calculated over the last N years prior to retirement. I will refer to N as the pension’s calculation period. Assuming that wages are valued in real terms in the calculation (which is approximately true in Spain), the regulatory base is given by (5) B(s,s+J,N)=1 NW(s,t)dt s+J!N s+J "= 1!e!(g+ # )N ( ) (g+ # )NW(s,s+J)$b(N)W(s,s+J) Hence, the regulatory base can be written as a fraction b() of retirement wages, W(s, s+J). It is easy to check (see the Appendix) that this fraction is a decreasing function of (g+ ν )N, where N is the length of the calculation period and g+ ν the growth rate of individual real wages. Notice that we can write the initial pension in the form (3') P(s,s+J,C,N)= ! (C)b(N)W(s,s+J)" # (C,N)W(s,s+J) Hence, ρ () = φ ()b() is the ratio between the wage at retirement and the starting pension. I will refer to this quantity as the initial replacement rate. Once its initial value is set, it will be assumed that an individual’s pension (P) grows over time at a constant rate ω in real terms. If pensions are indexed to consumer prices, as is the case in Spain, we will have ω = 0 and the real value of individual pensions will remain constant over time. In the general case, the real pension at time t of a worker who has retired at s+J will be given by (6) P(s,t,C,N)=P(s,s+J,C,N)e ! (t"(s+J)) = # (C,N)W(s,s+J)e ! te" ! (s+J)= # (C,N)Aoe $ Ce ! te(g" ! )(s+J) for t!s+J,s+Z [ ] . If the pensioner leaves a widower when he or she dies (which happens with probability π ), the surviving spouse will enjoy a widower’s pension (PV) for the rest of his or her life. Assuming widowers’ pensions are set at a constant fraction φ v (= 0.52 in Spain) of the deceased spouse’s pension at the time of death and grow at the same rate as retirement pensions, the real value of the widower’s pension at time t will be given by (7) PV (s,t,C,N)= ! vP(s,t,C,N)= ! v " (C,N)W(s,s+J)e # te$ # (s+J) for t!s+Z,s+Z2 [ ] Figure 2 shows how wages and pensions change across cohorts, indexed by their time of birth (s), at a given point in time (t). If there is a positive experience premium ( ν > 0) wages rise with age and are therefore a decreasing function of the time of birth. If productivity growth is faster 2 Under the current system, φ () is a piecewise linear function of the number of years of contribution, C. A minimum of 15 years is required for access to a contributory pension and entitles the worker to a pension equal to 50% of the regulatory base. This percentage rises by 3 points per year of contribution up to 25 years and by 2 points for each additional year thereafter, reaching 100% after 35 years. 4 than the real rate of appreciation of pensions (g > ω ) then pensions rise as we move to the right to younger cohorts (or decrease with age, as we move to the left). Figure 2: Pensions and wages at time t as a function of the date of birth of each cohort It will be useful to compute the following three magnitudes as of time t. By (2), the starting salary at time t, i.e. the wage earned by a worker with no experience who has just entered the labor market, will be given by (8) Ws(t)=Aoegt To calculate the retirement wage at t, notice that a worker who retires at that time must have been born at s = t – J. Using (2) again, this implies that (9) Wr(t)!W(t"J,t)=Ate # (t"(t"J+E)) =Aoegt e # (J"E)=Aoegt e # C=Ws(t)e # C Finally, the starting pension of this worker will be equal to (10) Ps(t)!P(t"J,t)= # (C)b(N)W(t"J,t)= $ (C,N)Aoegt e % C= $ (C,N)Wr(t) The internal rate of return of the pension system From the point of view of a worker, the public system of contributory pensions can be seen as an investment vehicle that allows him to obtain a retirement annuity in return for a flow of contributions during his working life. The internal rate of return (IRR) of this investment can be calculated in the standard way. The expected net present value of the investment, calculated as of time E, is given by (11) V(r) = ! " W(t)e!r(t!E)dt s+E s+J #+P(t)e!r(t!E)dt s+J s+Z #+ $ PV (t)e!r(t!E)dt s+Z s+Z2 # where τ is the Social Security contribution rate (defined as the sum of the rates paid by the worker and by his employer), π the probability that the worker is survived by his spouse and r 5 the discount rate. For convenience, I have suppressed all arguments of the functions W(), P() and PV() except for t. Substituting (2), (6) and (7) into (11), we have V(r)=! " Aoe! # (s+E)erE e!(r!g! # )tdt s+E s+J $+ % (C,N)W(s,s+J)e! & (s+J)erE e!(r! & )tdt s+J s+Z $+ '( ve!(r! & )tdt s+Z s+Z2 $ ) * +, - . The IRR of the pension system from the point of view of the representative worker/pensioner is the value of r that makes V(r) equal to zero. Setting the previous expression equal to zero and using (2) to write (12) W(s,s+J)=Aoe(g+ ! )(s+J)e" ! (s+E) we have ! e"(r"g" # )tdt s+E s+J $= % (C,N)e(g+ # " & )(s+J)e"(r" & )tdt s+J s+Z $+ '( ve"(r" & )tdt s+Z s+Z2 $ ) * +, - . Solving the integrals that appear in this expression and simplifying the result, we arrive at the following equation, which can be solved numerically for r: (13) ! e(r"g" # )C"1 ( ) r"g" # = $ (C,N)1"(1 " %& v)e"(r" ' )X" %& ve"(r" ' )(X+X2) r" ' where (14) X = Z – J, X2 = Z2 – Z and C = J – E are, respectively, the number of years that a retirement and a widower’s pension will be collected and the length of the agent’s working career (or the number of years he will have contributed to the Social Security system at the time of retirement). 4. Aggregate magnitudes To calculate total pension expenditure and other economy-wide aggregates, we need to add things up across all agents who are either employed workers or pensioners at a given point in time. To make the exercise tractable, I will assume that nothing changes over time or across individuals. In addition to constant values of g, n and ν , this means that the parameters of the system (including the contribution rate, τ , the retirement age, J, the pension calculation period, N, and the rules for computing the percentage φ of the regulatory base that is paid out as pension) remain constant over time and that all pensioners have the same characteristics both within and across generations (and, in particular, the same number of contribution years, C, and the same life expectancy, Z and Z2). Hence, I am essentially solving directly for a steady state of the model with constant life expectancy. As a result, the solution I will obtain will describe the equilibrium point to which the system will converge if we let it run undisturbed during a sufficiently long period under stationary circumstances, but it will tell us nothing about the transition path it will follow to reach this target from given initial conditions. 6 The average wage Under these assumptions, it is easy to calculate aggregate magnitudes by integrating over the time of birth, s. Let us start with the working population. At time t, the labor force is composed of all the agents who entered the labor market between t-C and t, and were therefore born between t-C-E and t-E. Hence, the labor force at time t is given by (15) LF(t)=L(s)ds t!C!E t!E "=ens ds t!C!E t!E "= enC !1 ( ) n ent e!n(C+E) The aggregate wage bill (WB) is the sum of the earnings of all employed workers, that is, (16) WB(t)=L(s)W(s,t)ds t!C!E t!E "=ens Aoe! # (s+E)e(g+ # )tds t!C!E t!E "=Aoe(g+n)te!nE 1!e!(n! # )C ( ) n! # Hence, the average salary is (17) W(t)=WB(t) LF(t)=Aoegt ( ) n n! " 1!e!(n! " )C 1!e!nC # $ %& ' ()Ws(t)Dw(n, " ,C) where Ws(t) is the starting wage at time t and Dw(n, ! ,C) a correction factor that captures the effect of the age distribution of the population on average wages. In the absence of an experience premium ( ν = 0), all workers who are active at time t earn the same wage regardless of their date of birth and the correction factor collapses to 1 independently of the age distribution of the working population. When ν > 0, however, wages rise with age, making the average wage higher than the current starting wage (Dw>1) , and the demographic structure of the population matters. As the experience premium ( ν ) rises, the upward sloping wage-age profile becomes steeper and the average wage rises relative to the starting wage. Similarly, when ν > 0, an increase in the length of the working career, C, raises the average wage (relative to the starting wage). Finally, as the rate of population growth (n) increases, the relative weight of younger workers in the labor force increases. If these workers have lower wages than older ones (i.e. if ν > 0), then the average wage falls. Hence, we have3 (18) !Dw ! " >0 , !Dw !n <0 if ν > 0 and !Dw !C >0 if ν > 0 The average pension Next, we consider the population of pensioners (LP). Referring to Figure 1, we see that at time t the population is comprised of those agents born between t-Z and t. Of these, those born between t-Z and t-J are retired. In addition to them, a fraction π of those born between t-Z2 = tZ-X2 and t-Z have spouses that are still alive and are drawing a widower’s pension. Hence, the 3 See the Appendix for a proof. 7 pensioner population at time t, including widows (or rather, the number of pensions, since widowers may be counted twice) is given by (19) LP(t)=L(s)ds + ! L(s)ds t"Z2 t"Z # t"Z t"J #=ens ds + ! ens ds t"Z2 t"Z # t"Z t"J #=en(t"J)1"(1 " ! )e"nX " ! e"n(X+X2) n Adding up over living pensioners, including widowers, total pension expenditure (PE) at time t is given by (20) PE(t)=L(s)P(s)ds + ! L(s)PV (s)ds t"Z2 t"Z # t"Z t"J # where I have suppressed all arguments of P() and PV() except for s, the time of birth of the (original) beneficiary. Using (6), (7) and (12), this expression becomes PE(t)=L(s)P(s)ds + ! L(s)PV(s)ds t"Z2 t"Z # t"Z t"J # =Ao $ (C,N)e % te(g" % )Je & Ce(n+g" % )sds + !' ve(n+g" % )sds t"Z2 t"Z # t"Z t"J # ( ) *+ , - and, solving the integrals inside the parentheses and simplifying (see the Appendix for details), (21) PE(t)=Ao ! (C,N)e " Ce(g+n)te#nJ 1#(1 # $% v)e#(n+g# & )X# $% ve#(n+g# & )(X+X2) n+g# & Hence, the average pension is given by (22) P(t)=PE(t) LP(t) =Ao ! (C,N)e " Cegt 1#(1 # $% v)e#(n+g# & )X# $% ve#(n+g# & )(X+X2) 1#(1 # $ )e#nX # $ e#n(X+X2) n n+g# & ' ( )* + , -Ps(t)Dp(n,g# & ) where Ps(t) is the starting pension at time t and Dp a correction factor that depends on the age distribution of pensioners and on how pensions vary with age at a given point in time. If productivity does not grow over time, pensions are indexed to consumer prices and widowers inherit their spouse’s full pension (i.e. if g = ω = 0 and ! v =1) then all pensions paid out at a given point in time (including widowers’ pensions) are equal and the correction factor collapses to 1 for any value of n. Otherwise, real pensions vary with the date of birth of the original beneficiary and the age distribution of the pensioner population matters. Under normal circumstances, older pensioners will have lower pensions than younger ones due to productivity growth and to the fact that survivors’ pensions are only a fraction of the original retirement pension. As a result, the average pension will be below the current starting pension (i.e. Dp<1 if g – ω > 0 and/or ! v < 1). Under these conditions, moreover, an increase in the rate of population growth (n) will increase the average pension by raising the relative weight of younger individuals, who have higher than average pensions, in the stock of live pensioners. As productivity growth (g) rises (or ω declines), the downward sloping pension-age profile 8 becomes steeper and the average pension falls relative to the starting pension. Finally, the average pension rises with the generosity of widowers' pensions, measured by ! v . Hence, we have4 (23) !Dp !(g" # ) <0 and !Dp ! " v >0 and, (24) !Dp !n >0 if g – ω > 0 and/or ! v < 1 The components of pension expenditure as a fraction of the wage bill Finally, pension expenditure as a fraction of the wage bill can be written (25) EXPW =PE(t) WB(t) =LP LF *P W !DEMLAB *GENW The first term on the right-hand side of this expression, DEMLAB, gives the number of pensions per employed worker, a useful summary of the joint impact of demographics and labor market conditions on the pension system. (Notice that the labor market component of this factor is trivial in the current model since it assumes that all agents are continuously employed during their entire active life). Using (15) and (19), this ratio can be written (26) DEMLAB !LP LF = en(t"J)1"(1 " # )e"nX " # e"n(X+X2) n enC "1 ( ) n ent e"n(C+E) =1"(1 " # )e"nX " # e"n(X+X2) enC "1 Recalling that C = J – E and X = Z - J, it is easy to see that DEMLAB increases with life expectancy (Z) and decreases with the retirement age (J) and with the growth rate of population (n). The second factor in (25), GENW, is the ratio between the average pension and the average wage. I will refer to this term as the generosity factor of the pension system (defined in terms of the average wage). Using (17) and (22), GENW will be given by (27) GENW (t)=P(t) W(t) = ! (C,N)e " CDp(n,g# $ ) Dw(n, " ,C) Our earlier results about the comparative statics of the numerator and denominator of this ratio imply that GENW will be a decreasing function of the rate of productivity growth and an increasing function of the rate of population growth. 4 See the Appendix for a proof. 15 APPENDIX 1. Some useful results This section collects the results of some simple calculations that will be useful later on. • In this paper we often have to calculate integrals of a certain type. Introducing a change of variable, it is easy to show that (A.1) e ! t a b "dt = e ! (b#a)#1 ( ) e ! a ! = 1#e# ! (b#a) ( ) e ! b ! for ! $0 b#a for ! =0 % & ' ( ' Proceeding in the same manner with ! ="r#0 , we have (A.2) e!rt a b "dt = 1!e!r(b!a) ( ) e!ra r • For any x, it is easy to show that (A.3) ! (x)"e#x(1 +x)$1 (A.4) µ (x)!ex(x"1) # "1 and that both expressions hold as strict inequalities for x!0 . • Next, consider the function (A.5) h ! ,D ( ) = 1"e" ! D ! for ! #0 D for ! = 0 $ % & ' & with D > 0 It is easy to show that h() is always positive, tends to ! as ! " #$ and to 0 as ! " # , is continuous at 0 and increases with D. Differentiating with respect to each argument and using (A.3) we have (A.6) hD ! ,D ( ) ="e" ! D(" ! ) ! =e" ! D>0 h !! ,D ( ) =#h ! ,D ( ) # ! =e" ! D(1 + ! D)"1 ! 2= $ ( ! D)"1 ! 2<0 for ! "0 . Notice that this expression can also be written (A.7) h !! ,D ( ) = ! De" ! D"(1 "e" ! D) ! 2=De" ! D ! "(1 "e" ! D) ! 2=De" ! D ! " h ! ,D ( ) ! <0 This implies that (A.8) h ! ,D ( ) >De" ! D for γ > 0 and h ! ,D ( ) <De" ! D for γ < 0 16 When ! =0 , moreover, we have ! "0 limh ! =0#1+1 0 = ! "0 lim # ! D2e# ! D 2 ! =#D2 2 <0 by L’Hopital’s rule. • Similarly, the function (A.9) m ! ,D ( ) = e ! D"1 ! for ! #0 D for ! = 0 $ % & ' & with D > 0 takes on only positive values regardless of the sign of γ , is continuous at zero, tends to 0 as ! " #$ and to infinity as ! " # and increases with D. Differentiating with respect to γ and D and using (A.4) we have (A.10) !m " ,D ( ) ! " =e " D( " D#1) +1 " 2= µ ( " D)+1 " 2>0 !m " ,D ( ) !D=e " D " " =e " D>0 for ! "0 . For ! =0 we have ! "0 limm ! =0+1#1 0 = ! "0 lim D2e# ! D 2 =D2 2 >0 • Next, consider the function (A.11) f( ! ,p,X,Y)=1"(1"p)e" ! X"pe" ! Y ! with 0<X!Y and p!0,1 [ ] and observe that it can be written (A.12) f( ! ,p,X,Y)=(1"p)h( ! ,X)+ph( ! ,Y) Differentiating this expression and using previous results we have (A.13) f ! ( ! ,p,X,Y)=(1"p)h ! ( ! ,X)+ph ! ( ! ,Y)<0 fX( ! ,p,X,Y)=(1"p)hX( ! ,X)=(1 "p)e" ! X>0 fY( ! ,p,X,Y)=phY( ! ,Y)=pe" ! Y>0 fp( ! ,p,X,Y)="h( ! ,X)+h( ! ,Y)=h( ! ,Y)"h( ! ,X)#0 where the last inequality holds because h() is increasing in its second argument and Y!X. • Some useful bounds on the exponential function Let f:R!R be n+1 times differentiable on some open interval, I. It is well known that for any a,x!I , f can be written in the form of a Taylor polynomial with remainder, 17 f(x)=f(a)+f(k)(a) k! k=1 n !x"a ( ) k+f(n+1) (b) (n+1)! x"a ( ) n+1 where b is some point between a and x and f(k)(a) is the k-th derivative of f() evaluated at a. Letting f(x)=ex!1 , we have f(k)(x)=ex for all k and, setting a to zero and n to 2, we can write ex!1=e0!1 ( ) +e0x+e0x2 2+ebx3 3! =x+x2 2+ebx3 6 for some b between 0 and x. If we constrain x to be positive, the remainder (the last term on the right-hand side of this expression) will also be positive and this implies that (A.14) ex!1>x+x2 2 for all x > 0 If x < 0, on the other hand, the remainder will be negative and we will have (A.14') ex!1<x+x2 2 for all x < 0 Proceeding in a similar way with f(x)=1!e!x we have f(0) =0, f'(x)=e!x, f"(x)=!e!x and f'''(x)=e!x and therefore 1!e!x=1!e!0 ( ) +e!0x+!e!0x2 2+ebx3 3! =x!x2 2+ebx3 6 from where (A.15) 1!e!x>x!x2 2 for all x > 0 and 1!e!x<x!x2 2 for x < 0 Using these inequalities, it is easy to establish the following result, which will be useful below. • Claim 1: ex!1 ( ) 2 "x2ex with strict inequality whenever x!0 Proof: We want to show that (A.16) v(x)!ex"1 ( ) 2#x2ex!u(x) with strict inequality whenever x!0 . Notice that (A.16) holds as an equality for x = 0 since v(0) =e0!1 ( ) 2=0=0e0=u(0) 18 Next, we need to compare the functions v(x) and u(x) for x > 0. Since both functions vanish when x = 0, their relative size will depend on that of their derivatives. Formally, since we can write v(x)=v(0) +v'(s) 0 x !ds =v'(s) 0 x !ds and u(x)=u'(s) 0 x !ds a sufficient condition for (A.16) to hold for all x > 0 is that (A.17) v'(x)>u'(x) for all x > 0 Calculating the relevant derivatives, (A.18) v'(x)=2ex!1 ( ) ex u'(n)=x2ex+2xex=x2+2x ( ) ex this condition can be written ex!1 ( ) >x+x2 2 which is true for all x > 0 by (A.14). Finally, assume that x < 0. As before, we need to compare the functions v(x) and u(x) using their derivatives. Notice, however, that the direction of the inequality between the relevant derivatives will be reversed as we cross the origin. Since v(x) = u(x) = 0, v(x) will lie above u(x) as we move from 0 to the right if v() is steeper than u(). As we move from 0 to the left, however, we need v() to be flatter than u() in order to obtain the same result. Hence, in the first case we need v'(x) > u'(x) for x > 0 and in the second v'(x) < u'(x) for x < 0. Formally, when x < 0 we can write v(x)=v'(s) 0 x !ds ="v'(s) x 0 !ds and u(x)=u'(s) 0 x !ds ="u'(s) x 0 !ds so a sufficient condition for (A.16) to hold for all x < 0 is (A.17') !v'(x)>!u'(x)"v'(x)<u'(x) for all x < 0 or, using (A.18), ex!1 ( ) <x+x2 2 which we know to hold for all x < 0 by (A.14'). ! Letting z = -x, claim 1 can be rewritten (A.19) e!z!1 ( ) 2"(!z)2e!z#1!e!z ( ) 2"z2e!z with strict inequality whenever z!0 . If we define the function A(z) by 19 (A.20) A(z)=1!e!z ( ) 2!e!zz2 we have by (A.19) that (A.21) A(0) =0 and A(z)>0 for z!0 Differentiating A(), we have A'(z)=2e!z1!e!z ( ) !z!z2 2 " # $% & ' ( ) * * + , - - Using (A.15), this expression implies that (A.22) A'(z)= >0 for z>0 <0 for z<0 ! " # or A’(z)z > 0 for all z!0 Finally, notice that A'(z)+2A(z)=2e!zez!1 ( ) !z+z2 2 " # $% & ' ( ) * * + , - - Using (A.14) and (A.14’) we see that (A.23) A'(z)+2A(z)=>0 for z>0 <0 for z<0 ! " # $ A'(z)+2A(z) [ ] z>0 for all z%0 2. Calculation and comparative statics of the regulatory base of the pension The regulatory base of the pension is defined as the average wage of the worker calculated over the N years prior to retirement: B(s,s+J,N)=1 NW(s,t)dt s+J!N s+J "=1 NAoe(g+ # )te! # (s+E)dt s+J!N s+J "=1 NAoe! # (s+E)e(g+ # )tdt s+J!N s+J "= =1 NAoe! # (s+E)1!e!(g+ # )N ( ) e(g+ # )(s+J) g+ # = 1!e!(g+ # )N ( ) (g+ # )NAoe! # (s+E)e(g+ # )(s+J) = 1!e!(g+ # )N ( ) (g+ # )NW(s,s+J)$b(g+ # )N ( ) W(s,s+J) or B(s,s+J,N)=b(g+ ! )N ( ) W(s,s+J) with b(g+ ! )N ( ) =1"e"(g+ ! )N (g+ ! )N To abbreviate, define x=(g+ ! )N and write b() in the form b x ( ) =1!e!x x 20 Differentiating this function and using (A.3), we have b'x ( ) =xe!x!(1!e!x) x2=e!x(1 +x)!1 x2= " (x)!1 x2<0 for x!0 . Hence, the ratio b() is a decreasing function of N and g+ ν . 3. Calculation of the IRR of the pension system The IRR of the pension system is the value of r that solves the following equation ! e"(r"g" # )tdt s+E s+J $= % (C)b(N)e(g+ # " & )(s+J)e"(r" & )tdt s+J s+Z $+ '% ve"(r" & )tdt s+Z s+Z2 $ ( ) *+ , - Solving the integrals that appear in this expression and operating, we have e!(r! " )tdt s+J s+Z #+ $% ve!(r! " )tdt s+Z s+Z2 #= 1!e!(r! " )(J!Z) ( ) e!(r! " )(s+J) r! " + $% v 1!e!(r! " )(Z2!Z) ( ) e!(r! " )(s+Z) r! " =e!(r! " )(s+J)1!e!(r! " )X ( ) + $% v1!e!(r! " )X2 ( ) e!(r! " )[(s+Z)!(s+J)] r! " =e!(r! " )(s+J)1!(1! $% v)e!(r! " )X! $% ve!(r! " )(X+X2) r! " and e!(r!g! " )tdt s+E s+J #=e!(r!g! " )(s+E)1!e!(r!g! " )C ( ) r!g! " where X = Z – J X2 = Z2 – Z and C = J – E. Collecting results, the IRR of the system is the value of r that solves the following equation: ! e"(r"g" # )(s+E)1"e"(r"g" # )C ( ) r"g" # = $ (C)b(N)e(g+ # " % )(s+J)e"(r" % )(s+J)1"(1 " &$ v)e"(r" % )X" &$ ve"(r" % )(X+X2) r" % which can be somewhat simplified to ! e(r"g" # )C"1 ( ) r"g" # = $ (C,N)1"(1" %& v)e"(r" ' )X" %& ve"(r" ' )(X+X2) r" ' 4. Aggregate magnitudes The average wage The labor force at time t is given by LF(t)=L(s)ds t!C!E t!E "=ens ds t!C!E t!E "= enC !1 ( ) en(t!C!E) n= enC !1 ( ) n ent e!n(C+E) 21 The aggregate wage bill (WB) is the sum of the earnings of all employed workers, that is, WB(t)=L(s)W(s,t)ds t!C!E t!E "=ens Aoe! # (s+E)e(g+ # )tds t!C!E t!E "=Aoe(g+ # )te! # Ee(n! # )sds t!C!E t!E "= =Aoe(g+ # )te! # E1!e!(n! # )C ( ) e(n! # )(t!E) n! # =Aoe(g+n)te!nE 1!e!(n! # )C ( ) n! # Hence, the average salary is W(t)=WB(t) LF(t)= Aoe(g+n)te!nE 1!e!(n! " )C ( ) n! " enC !1 ( ) nent e!n(C+E) =Aoegt ( ) n n! " 1!e!(n! " )C 1!e!nC # $ %& ' ()Ws(t)Dw(n, " ,C) Notice that Dw(n, ! ,C)=1"e"(n" ! )C n" ! n 1"e"nC =h(n" ! ,C) h(n,C) where h() has been defined above in (A.5). Since h() is decreasing in its first argument, it follows that Dw > 1 for ν > 0 and !Dw ! " =1 h(n,C) !h(n# " ,C) !(n# " )(#1) >0 so the average wage increases with the experience premium. As we increase n the weight of the younger workers in the labor force increases. If these workers have lower wages than older ones, i.e. if ν > 0, then the average wage falls with n. Hence, we have • Claim 2: Given ν > 0, we have !Dw(n, " ,C) !n <0 for all n!0 Proof: Fix ν > 0 and define the functions F() and g() by F(n)!ln Dw(n, " ,C)=ln h(n# " ,C)#ln h(n,C)!g(n# " )#g(n) Since ln() is an increasing function, the derivative of f() will have the same sign as that of Dw() . Hence, the desired result will follow if we can show that F'(n)=g'(n! " )!g'(n)<0#g'(n! " )<g'(n) for ν > 0 i.e. that g’() is an increasing function. Hence, it will be sufficient to show that g"(n)>0 To continue, we need to distinguish two cases depending on the sign of n. • Case i) Assume n > 0. Then we can write 22 (A.24) g(n)=ln h(n,C)=ln1!e!nC n=ln 1!e!nC ( ) !ln n and differentiating this expression g'(n)=e!nCC 1!e!nC !1 n=C enC !1 !1 n g"(n)=!C2enC enC !1 ( ) 2+1 n2 To establish the desired result we need to show that (A.25) g"(n)=!C2enC enC !1 ( ) 2+1 n2>0"1 n2>C2enC enC !1 ( ) 2 "v(n)#enC !1 ( ) 2>n2C2enC #u(n) for all n > 0, which holds by claim 1 with x = nC. • Case ii) We now consider the case when n < 0. Then, we need to write (A.24) in a slightly different way (A.24') g(n)=ln h(n,C)=ln e!nC !1 !n=ln e!nC !1 ( ) !ln(!n) to ensure that we are taking logs of positive numbers. This does not change the derivative of g(), however, so we still need to show that (A.25) holds for all n < 0, which is true by claim 1. ! • Claim 3: Given ν > 0, we have !Dw(n, " ,C) !C >0 . Proof: Fix ν > 0 and define the functions F() and g() by F(n,C)!ln Dw(n, " ,C)=ln h(n# " ,C)#ln h(n,C)!g(n# " ,C)#g(n,C) We want to show that FC(n,C)!gC(n" # ,C)"gC(n,C)>0$gC(n" # ,C)>gC(n,C) for ν > 0 i.e. that gC() is a decreasing function of its first argument. Hence, it will be sufficient to show that gCn (n,C)<0 Differentiating g(n,C)=ln h(n,C)=ln1!e!nC n 23 we have gC(n,C)=hC(n,C) h(n,C) =e!nC 1!e!nC n =n enC !1 and gCn (n,C)= enC !1 ( ) !nenC C enC !1 ( ) 2=!enC (nC !1) !1 enC !1 ( ) 2=! µ (nC)!1 enC !1 ( ) 2<0 where the inequality follows by multiplying both sides of (A.4) by -1. ! The average pension The pensioner population at time t, including widows, is given by LP(t)=L(s)ds + ! L(s)ds t"Z2 t"Z # t"Z t"J #=ens ds + ! ens ds t"Z2 t"Z # t"Z t"J #= 1"e"n(Z"J) ( ) en(t"J) n+ ! 1"e"n(Z2"Z) ( ) en(t"Z) n= =en(t"J)1"e"nX ( ) + ! 1"e"nX 2 ( ) e"nX n=en(t"J)1"(1 " ! )e"nX " ! e"n(X+X2) n Adding up over live pensioners, total pension expenditure (PE) at time t is given by PE(t)=L(s)P(s)ds + ! L(s)PV (s)ds t"Z2 t"Z # t"Z t"J # Using equations (6), (7) and (12) in the text, this expression can be written PE(t)= ! (C,N)e " tensW(s,s+J)e# " (s+J)ds + $% v ! (C,N)e " tensW(s,s+J)e# " (s+J)ds t#Z2 t#Z & t#Z t#J & =Ao ! (C,N)e " te(g# " )Je ' Ce(n+g# " )sds + $% ve(g+n# " )sds t#Z2 t#Z & t#Z t#J & ( ) *+ , - Now, the term in parentheses becomes e(n+g! " )sds + #$ ve(g+n! " )sds t!Z2 t!Z % t!Z t!J %= 1!e!(g+n! " )(Z!J) ( ) e(g+n! " )(t!J) g+n! " + #$ v 1!e!(g+n! " )(Z2!Z) ( ) e(g+n! " )(t!Z) g+n! " =e(g+n! " )te!(g+n! " )J1!e!(g+n! " )X ( ) + #$ v1!e!(g+n! " )X2 ( ) e!(g+n! " )X g+n! " =e(g+n! " )te!(g+n! " )J1!(1 ! #$ v)e!(g+n! " )X! #$ ve!(g+n! " )(X+X2) g+n! " Substituting this into the previous expression, PE(t)=Ao ! (C,N)e " te(g# " )Je $ Ce(n+g# " )sds + %& ve(g+n# " )sds t#Z2 t#Z ' t#Z t#J ' ( ) *+ , - =Ao ! (C,N)e $ Ce(g+n)te#nJ 1#(1 # %& v)e#(g+n# " )X# %& ve#(g+n# " )(X+X2) g+n# " 24 Hence, the average pension is given by P(t)=PE(t) LP(t) = Ao ! (C,N)e " Ce(g+n)te#nJ 1#(1 # $% v)e#(g+n# & )X# $% ve#(g+n# & )(X+X2) g+n# & en(t#J)1#(1 # $ )e#nX # $ e#n(X+X2) n =Ao ! (C,N)e " Cegt 1#(1 # $% v)e#(g+n# & )X# $% ve#(g+n# & )(X+X2) 1#(1 # $ )e#nX # $ e#n(X+X2) n g+n# & ' ( )* + ,-Ps(t)Dp(n,g# & ) where Ps(t) is the starting pension at time t and Dp a correction factor that depends on the age distribution of pensioners and on how pensions vary with age at a given point in time. Notice that Dp can be written Dp(n,g! " , # , $ v,X,X+X2) =f(g+n! " , #$ v,X,X+X2) f(n, # ,X,X+X2) where f( ! ,p,X,Y)=1"(1"p)e" ! X"pe" ! Y ! =(1 "p)h( ! ,X)+ph( ! ,Y) has been defined above in (A.11). Since f() is decreasing in its first argument and increasing in the second (see A.13), it follows that !Dp !(g" # )=1 f(n, $ ) !f(n+g" # , $% v) !(n+g" # )*1 <0 !Dp ! % v =1 f(n, $ ) !f(n+g" # , $% v) !( $% v) $ >0 i.e. the average pension, written as a fraction of the starting pension, decreases with g, increases with ω and increases with the generosity of widower pensions, φ v. Notice that Dp = 1 when g – ω = 0 and φ v = 1. Combining this with the signs of the two partial derivatives we have just calculated, we see that Dp < 1 for g – ω > 0 and/or φ v < 1. Finally, it can be shown that Dp is an increasing function of n. • Claim 4: Given g - ω > 0, we have !Dp(n,g" # ) !n >0 for all n!0 Proof: To simplify a bit the notation, we can assume ω = 0 without loss of generality and work with g rather than g - ω . We are interested in the function Dp(n,g, ! , " v,X,Y)=1#(1 # !" v)e#(g+n)X# !" ve#(g+n)Y 1#(1 # ! )e#nX # ! e#nY n g+n=f(g+n, !" v,X,Y) f(n, ! ,X,Y) with