Appendix B: DETAILS ABOUT THE SIMULATION MODEL The simulaion model is carried ou on one spreadshee and has five modules, four of which are conained in lookup ables ha are all calculaed on an auxiliary spreadshee. 1. Populaion and Labor Force Module The lookup able conains he hree L-T populaion projecions (1994) for seleced years beween 2000 and 2050 for four age groups: 1-19, 20-64, over 64 and over 85. For he years beween hese poins, I assumed a consan growh rae of each populaion group - simplificaions o make inerpreaion of he resuls less problemaic. From hese daa I roughly esimaed he populaion wihin wo special age groups (20-24 and 65-69) by muliplying he relevan L-P age groupings by he fixed raios represening wha hese were in 1998. So ha he working age can sar a 25 and he reiremen age can be changed from 65 o 70. Such demographic approximaions have lile impac on he final resuls. The labor supply was calculaed separaely for he 25-64 and he 65-69 age groups. For he former, I merely muliplied he populaion in ha age group by a paricipaion raio which could be fixed or changed a a consan percenage incremens over ime. Workers in he 65-69 age group were esimaed in wo seps. If he reiremen age remained a 65, he number of workers in his cohor was zero; if he reiremen age was chosen o be 70 in 2050, I muliplied he populaion in he 65-69 age group by a fracion rising from 0 o 1 in regular incremens and hen by he same labor force paricipaion raio used for he 25-64 age group. 2. Saving Rae Module This calculaion cenered around a lookup able of he consumpion level of workers ha is mainained hroughou heir working lifeime so ha hey accumulae enough savings o finance consumpion during reiremen a some fracion of heir former consumpion level (consumpion-replacemen raio). The
consumpion level depends on he growh of he workers annual income, he ineres rae, heir year of reiremen, and heir year of deah. This consumpion level was deermined ieraively on he auxiliary spreadshee and he soluion required he worker o have exhaused all saving a he ime of deah. From he daa on he paern of consumpion and income of a single worker over a working lifeime, I calculaed he raio of he chosen consumpion level o he average aggregae oal income over he working life. Assuming he same number of workers in each age group wihin he cohor of workers, his was used as he saving rae for all workers in each year. A problem arose because I assumed ha he life expecancy beween 2000 and 2050 was increasing from 80 o 85 and ha, a leas in some simulaions, he age of reiremen was rising from 65 o 70 in he same period. To ake hese changes ino accoun, I calculaed wo ses of opimal savings raes, one assuming a life expecancy of 80 years and a reiremen age of 65, which is used o define he iniial saving rae; and anoher assuming a life expecancy of 85 years, and a reiremen age of eiher 65 or 70. These se he endpoin saving raes, wih he acual rae in he oher years rising a even incremens beween hese wo values. 3. Income Module The ne income of workers on which hey base heir saving decisions was se equal o heir work income (which increases a a consan annual rae) plus heir ineres income (or minus heir ineres paymens if hey are in deb). As noed in he ex, he ineres income on saving or ineres paymens on loans came from a source ouside he model. When calculaing he saving rae, I also calculaed he raio of oal aggregae income over he working lifeime o oal work income over he same period. This varied according o he ineres rae, growh rae, income replacemen rae, age of reiremen, and age of deah; much of he relevan informaion
was conained in a lookup able. 4. Module of Consumpion (Dissaving) Raes by Reired Workers This calculaion cenered around a lookup able providing raes of consumpion of reired workers wih a given life expecancy. This level was merely he level of consumpion mainained hrough he working lifeime imes he consumpion-replacemen raio and was par of he calculaions used o deermine he opimum consumpion level during a worker s lifeime. This dissaving needed, however, o be relaed o he curren level of income of acive workers, which was easily deermined by calculaing he raio of consumpion of a reired worker o he average income of curren workers. For simpliciy, I made hree calculaions: he raio of he consumpion of a worker who jus reired o he average income of workers 1 o 5 years in he pas; o 6 o 20 years in he pas, and 21 o 30 years in he pas. Using hese raios I could hen calculae from he curren income of a worker he consumpion level of reired workers in hese hree differen age brackes. Toal consumpion of for hese hree groups of reired workers could be calculaed by simply muliplying he average income of workers in he curren year imes hese raios imes he number of reired workers in he age group corresponding o he raio. Because life expecancy and, for some simulaions, he age of reiremen were increasing, I followed he same procedure as for saving, deermining he iniial consumpion raios using one se of assumpions abou age and reiremen and he end consumpion raios using anoher se of assumpions and hen creaing a weighed average ha increased in regular incremens of he 50-year period. 5. The Main Simulaion Since mos of he calculaions are carried ou in he saving, income, and dissaving modules, he calculaions conaining he aggregae resuls were simple and consised of he calculaion for income,
average consumpion of workers, aggregae saving of acive workers, aggregae dissaving of reired workers, and, finally, ne saving. Average income increased a a consan annual rae. Average consumpion was deermined by muliplying he income by he saving rae deermined in he saving module. Toal saving was calculaed by muliplying he average saving imes he number of acive workers. Toal dissaving was deermined by muliplying he number of reired workers imes he average curren consumpion of workers imes he consumpion-replacemen rae imes he raio of consumpion of reired o acive workers ha was calculaed in he dissaving module. Ne saving is simply he sum of aggregae saving of acive workers and aggregae dissaving of reired workers.
Appendix C: AN ALGEBRAIC DEMONSTRATION OF SOME SIMULATION RESULTS The discussion in he ex is based on inuiive argumens and numbers derived from he simulaions. Neverheless, if we assume a world wihou an ineres rae, he various resuls can be derived from a simple algebraic model ha provide more rigor o he argumen ha ne saving falls beween 2000 and 2050. Le S = ne saving, he sum of he saving of acive workers and he dissaving of he reired workers. The saving of acive workers in ime period is SA. Equaion C1: SA =. (ó Y ) (a A ), where ó = he saving rae, Y = average income, a = percenage of adul populaion who are acive workers, and A = adul populaion. The expression in he firs parenheses is he saving of one worker and he expression in he second parenheses is he number of acive workers. Dissaving by reired workers = consumpion by reired workers is SA. Equaion C2: SA = (ñ (1 - ó ) Y z ) ((1-a ) A ), where ñ = consumpion-replacemen raio, and z = raio of income on which saving decisions of reired workers were based o curren income of acive workers. The expression in he firs brackes is he dissaving of a single worker and he expression in he second brackes is he number of reired workers. Thus, in any given year, Equaion C3-a: S = (ó Y ) (a A ) - (ñ (1 - ó ) Y z ) ((1-a ) A ) Since we are ineresed in he ne saving raio, ha is, he raio of ne saving over oal income, his expression can be arranged for easier analysis: Equaion C3-b: (S / a A Y ) = ó - (ñ (1 - ó ) z )((1-a )/a ).
Given he assumed growh of 1.8% a year of income and assuming ha he populaion in each year cohor beween reiremen and deah is he same, z =.60. Since ñ = 1, he expression reduces o: Equaion C3-c: (S / a A Y ) = ó - (.6 (1 - ó )) ((1-a )/a ). Since ó depends on he age of reiremen and he life expecancy, wo variables which may change over ime, i is necessary o know how ó will change when hese wo variables change. Since lifeime saving (oal average annual savings imes number of work years) = oal dissaving in reiremen (annual consumpion imes number of reiremen years). Leing k = percenage of adul years spen working and K = oal adul years, hen: Equaion C4-a: ó Y k K = (ñ (1 - ó ) Y)((1-k) K). Rearranging and simplifying: Equaion C4-b: ó = ñ (1-k)/ (k - ñk +ñ). Given he assumpions of he model, his can be simplified o: Equaion C4-c: ó = (1-k). Since life expecancy rises from 80 o 85 and he reiremen age from 65 o 70, k changes from 0.727 o 0.750, and ó falls from 0.273 o 0.250 (in he simulaions). This should be obvious since he number of reiremen years is he same, bu workers have more years o accumulae he necessary savings. Given he assumpions of he model, he quesion is how he aggregae saving rae changes is simple o derive. From he L-T daa we deermine ha a falls from 0.820 o 0.766 and, as a resul, (1-a)/a rises from 0.219 o 0.305 (in he simulaions i falls somewha less because I am defining he adul labor force as 25 o reiremen, raher han 20 o reiremen). Because he expressions in boh of he brackes in equaion 1a are increasing and because ó is falling, he saving rae falls. Alhough he numerical resuls are sensiive o he populaion esimaes, hey do no affec he
qualiaive resuls. Wih he SSA esimaes, he same fall in (1-a)/a occurs, bu wih he Census esimaes here is a sligh increase. Neverheless, in he laer case he resul of muliplying he wo brackeed expressions in equaion 1a sill shows an increase.