Nelissen, J.H.M.

English

Research Papers in Economics

Economics  Letters  29  (1989)  189-193 
North-Holland 
189 
THE  DISTRIBUTION  OF  LIFETIME  EARNINGS  * 
J.H.M.  NELISSEN 
Tilburg  Unruersity,  5000  LE  Tilburg  The  Netherlands 
Received  29  August  1988 
Accepted  10  November  1988 
Assuming  that  earnings  at  a particular  age  are  distributed  log-normally,  it  is  shown  that  the  distribution  of  lifetime  earnings 
can  be  simulated.  For  that  purpose  estimations  for  the  course  of  p,  and  q2  are  made. 
1.  Introduction 
Little  is  known  about  the  real  distribution  of  lifetime  earnings.  Generally,  the  distribution  of 
lifetime  earnings  is  derived  from  panel  data  or  a  pooling  of  cohort  and  cross-section  data,  covering 
only  a  relatively  small  part  of  the  period  during  which  a  person  earns  money.  In  the  most  favourable 
cases  the  period  under  investigation  spans  about  ten  years  [e.g.,  Creedy  and  Hart  (1979)  and  Fase 
(1969)].  Starting  from  the  widely  accepted  assumption  that  at  a  particular  age,  the  earnings  are 
distributed  lognormally,  the  development  of  the  mean  and  variance  of  the  logarithms  of  earnings  to 
age  have  been  supposed  to  follow  a  quadratic  and  a  linear  function  in  the  variable  age,  respectively. 
The  functions  are  estimated,  and  this  way  the  distribution  of  lifetime  earnings  has  been  determined. 
In  a  similar  way  it  is  possible  to  describe  the  distribution  of  earnings  in  a  particular  year.  Besides 
that  these  functions  for  different  cohorts  are  based  on  (too)  few  observations,  there  is  another 
important  disadvantage.  It  is  hardly  possible  to  derive  the  distribution  of  earnings  for  the  future  or 
for  the  past.  For,  the  parameters  of  the  aforementioned  quadratic  and  linear  functions  for  the  mean 
pLt and  variance  a,* differ  between  generations  and  years.  This  is,  of  course,  because  these  functions 
are  not  based  on  theoretical  arguments,  but  only  on  empirical  insights,  although  human  capital 
theory,  for  example,  is  able  to  explain  the  quadratic  form  of  pl.  Our  purpose  now  is  to  estimate  pLt 
and  a,’  using  a  database  covering  an  unusually  long  period  and  including  as  many  theoretical 
considerations  as  possible. 
2.  The  data 
Since  1962,  the  Netherlands  Central  Bureau  of  Statistics  (henceforth,  NCBS)  provides  us  with 
(aggregated)  earnings  data  for  individuals.  ’  We  use 190  J.H.M.  Nelissen  /  The  distributmn  of lifetime  earnings 
as  the  income  concept.  This  means  that  we  take  the  labour  income  of  the  self-employed,  employees, 
and  managers  of  private  and  limited  liability  companies  and  corporations.  ’  From  the  NCBS  (1967a, 
b,  1972a,  b,  1977,  1979,  1980,  1982,  1984  and  1985)  we  have  data  for  different  age  groups  for  the 
years  1962,  1967,  1970,  1975,  1979  and  1981.  These  data  are  transformed  into  constant  prices  of 
1980. 
From  these  pooled  cross-section  data  we  can  derive  53  observations  for  the  mean  income,  and  31 
for  its  variance. 
3.  The  model 
Our  point  of  departure  is  the  statistical  model,  adopted  from  Aitchison  and  Brown  (1957)  and 
Fase  (1969),  which  describes  the  distribution  of  the  individual  income  by  age.  In  this,  the  individual 
income  has  been  considered  as  a  random  variable  related  to  age.  The  individual  is  assumed  to  start 
his  career  at  age  s,  at  which  he  or  she  receives  an  annual  income  equal  to  ys_ This  variable  ys  is 
assumed  to  follow  a  two-parameter  lognormal  distribution  and  age  is  considered  to  be  a  discrete 
variable.  If  we  also  assume  that  the  distribution  of  the  proportional  changes  in  income  with 
increasing  age  is  a  normal  one,  then  income  follows  a  lognormal  distribution  for  all  ages  [see  Hart 
(1973)].  Thus, 
with 
The  usual  procedure  is  to  determine  the  distribution  of  the  complete  age-income  profile  by 
introducing  a  quadratic  function  in  the  variable  age  for  CL,  and  a  linear  one  for  u,~, whereby  in  some 
cases  dummy  variables  for  different  cohorts  are  also  used  [e.g.,  Creedy  and  Hart  (1979)].  We  follow 
another  method  by  introducing  theoretical  interpretable  variables. 
For  the  variable  p1 we  use  the  following  variables: 
(i)  The  development  in  the  logarithm  of  national  income,  corrected  for  the  working  population  in 
proportion  to  the  total  population,  NI. 
(ii)  Because  female  labour  force  participation  rates  differ  from  male  ones,  and  because  of  their 
development  in  the  course  of  time,  these  labour  force  participation  rates  (corrected  for  part-time 
participation),  LFF,  are  inserted. 
(iii)  A  higher  level  of  education  of  the  population  under  consideration  can  influence  the  proportion 
of  national  income  which  the  working  population  receives.  Therefore,  the  proportion  of  people 
with  secondary  education,  EDS,  has  been  used. 
(iv)  Unemployment  can  have  the  opposite  effect,  which  argues  for  the  insertion  of  the  unemploy- 
ment  rate,  UN.  This  variable  could  also  correct  for  the  employment  benefits  (see  footnote  2). 
*  Some  benefits  for  the  unemployed  are  included  and  cannot  be  isolated.  However,  their  impact  during  the  period  under 
investiagation  was  limited. J.H.M.  Nelissen  /  The  distribution  of lifetime  earnings  191 
(v)  Human  capital  theory  shows  a  relationship  between  income  and  age.  We  insert  this  by  using 
dummy  variables  for  different  age  groups,  AGExx.  3 
The  difference  with  the  afore-mentioned  method  is  that  the coefficients  of  AGExx  are  constant  in 
time  and  that  changes  in  the  income-age  profile  are  the  consequence  of  changes  in  other  variables. 
The  GLS  estimation  result  is (standard  errors  are  shown  in  parentheses)  4 
log  (Y  =  11.281  -  0.145  CT2  + 0.653  NI  -  0.967  LFF -  0.017  UN 
(0.286)  (0.043)  (0.071)  (0.172)  (0.006) 
+ 1.612  EDS  +  1.265  AGE20  +  1.624  AGE30  +  1.777  AGE40 
(0.472)  (0.218)  (0.167)  (0.163) 
+  1.785  AGE50  +  1.713  AGE60  +  1.109  AGE70 
(0.166)  (0.165)  (0.259)  (2) 
R*adj  +  +  0.986,  a, = 0.064. 
Here,  an  extra  dummy  variable  CT2,  which  equals  one  for  observation  before  1971,  has  been 
inserted  to correct  for  a shift  in  mean  income  after  1970.  The  goodness  of  fit,  as measured  by  R*adj, 
is very  high.  All  variables  are  significant  at  the  0.05  level  and  have  the  expected  sign. 
This  equation  has  been  used  to  simulate  the  mean  of  the  age-earnings  profile  of  individuals  for 
the  year  1950.  The  result  is  given  in  the  second  column  of  table  1.  The  third  column  shows  the 
average  household  income  for  that  year.  Data  on  the  mean  individual  income  are  not  available. 
Using  female  labour  force  participation  rates  we have  made  a correction  for  this,  as  given  in  the  last 
column.  Opposite  the  data  used  for  the  estimation  (and  consequently  for  the  simulation),  the  1950 
data  are  not  adjusted  for  persons  who  have  a  job  during  only  part  of  the  year  (school-leavers, 
Table  1 
Simulation  of  the  mean  age-earnings  profile  for  the  year  1950  (in  constant  prices  of  1980). 
Age  Model 
<  20  5000 
20-24  9100 
25-29  14100 
30-34  16 200 
35-39  17500 
40-44  18600 
45-49  18800 
50-54  18600 
55-59  18300 
60-64  16 200 
Income  hh  a  Idem,  corrected 
4441  4441 
8489  7900 
13600  12600 
18853  16 500 
23 089  19 600 
23 084  20 000 
20837  18900 
a Source;  NCBS  (1954). 
3  For  age 30,  for  example,  AGE30  takes  the value  of  1, whereas  for  age 27,  AGE20  takes  the  value  of 0.3  and  AGE30  a value 
of  0.7. 
4  The  dependant  variable  is not  the mean  of the  logarithm  of income,  p,,  but  the logarithm  of  the  mean  of  the  income,  log  (Y!, 
which  can  be  transformed  to  pI  via  pLr  =  log  a,  -  0.5  0,‘.  The  reason  for  this  is  that  we have  more  observations  for  a,. 192  J.H.M.  Nelissen  /  The  distribution  of  lifetime  earnings 
vacation  workers,  and  so  on)  and  they  do  include  capital  income  of  workers.  This  means  that 
observed  income  has  been  underestimated  in  the  younger  age  groups  and  overestimated  in  the  older 
age  groups.  Taking  this  into  account,  we are  allowed  to  conclude  that  the  simulation  results  fit  very 
well.  We  also  made  a number  of predictions  for  the  year  2000  (for  different  growth  scenarios)  which 
led  to very  plausible  results  [see  Nelissen  (1988)]. 
In  a  similar  way  a regression  equation  for  the  coefficient  for  variation  a,/~,  has  been  estimated. 
Here,  the  dummy  variables  for  age,  AGExx,  are  inserted  to  cover  variance  increasing  elements  with 
growing  age.  Economic  growth  is  assumed  to diminish  income  inequality.  Therefore,  the variable  NZ 
is included  in  the  equation.  The  same  holds  true  for  a better  educated  population.  This  is  embodied 
in  the  variable  EDS.  Unemployment  is  expected  to  raise  income  inequality  and  the  variable  LFF is 
used  for  the  effect  of  female  labour  force  participation.  The  sign  of  LFF is  not  known  beforehand. 
Theory  is  not  unequivocal.  Also,  the  labour  force  as  a  proportion  of  the  total  population,  LF,  is 
inserted  in  the  regression  equation.  A  relatively  large  working  population  could  diminish  income 
inequality,  because  the  possibility  for  shortages  in  labour  force  are  minor.  The  regression  result  is 
a//.~= 3.271-13.079  LF-0.477NZ+0.149LFF+0.048  UN 
(1.172)  (3.573)  (0.064)  (0.093)  (0.012) 
-  5.737EDS-O.O79AGE30-0.093  AGE40-  0.062  AGESO, 
(3.464)  (0.033)  (0.027)  (0.031) 
(3) 
R2=0.894,  a,  =  0.041. 
The  adjusted  R2  is rather  high,  all  coefficients  are  significant  and  they  have  the  expected  sign.  In 
the same way as we did for nt  it is now possible  to derive  the coefficient  of variation  for,  for  example, 
1950  and  2000. 
The  complete  distribution  of  the age-earnings  profile  up to age x  for  a specific  cohort  or year  can 
now be  derived,  after  solving  pL,  and  a,  for  each  t  from  eqs.  (2)  and  (3)  by  means  of  convolution  of 
the  distributions  for  each  age  (taking  account  of  the  probability  that  an  individual  will  survive  and 
participate  in earning  money  at that  age)  or by  simulation.  Given  the  analytical  calculation  problems 
inherent  to  the  first  method,  the  simulation  approach  seems  to be  preferable.  In  Nelissen  (1988),  the 
income  distribution  is implicitly  derived  within  a micro-simulation  context. 
4.  Conclusion 
Starting  from  the  statistical  model  used by Aitchison  and  Brown  (1957)  and  Fase  (1969)  and  using 
an  explanatory  model  for pL,  and  ut/pr it is possible  to derive  the  distribution  of  lifetime  income  of  a 
cohort  or  the  distribution  of  income  in  a particular  year  in  the  future. 
References 
Aitchison,  J.  and  J.A.C.  Brown,  1957,  The  lognormal  distribution  (Cambridge  University  Press,  Cambridge). 
Creedy,  J.  and  P.E.  Hart,  1979,  Age  and  the  distribution  of  earnings,  Economic  Journal  89,  280-293. 
Fase,  M.M.G.,  1969,  An  econometric  model  of  age-income  profiles,  Ph.D.  dissertation  (Universitaire  Pers  Rotterdam, 
Rotterdam). J.H.M.  Nelissen  /  The  distribution of lifetime earnings  193 
Hart,  P.E.,  1973,  The  comparative  statics  and  dynamics  of  income  distribution,  Discussion  paper,  series  A,  no.  49  (University 
of  Reading,  Reading). 
NCBS,  1954,  Inkomensverdeling  1950;  Aanvullende  gegevens  (de  Haan,  Utrecht). 
NCBS,  1967a,  Inkomensverdeling  1962  en  vermogensverdeling  1963  (Staatsuitgeverij,  The  Hague). 
NCBS,  1967b,  Inkomensverdeling  van  werkende  gehuwde  vrouwen  1962  (Staatsuitgeverij,  The  Hague). 
NCBS,  1972a,  inkomensverdeling  1967  en  vermogensverdeling  1968  (Staatsuitgeverij,  The  Hague). 
NCBS,  1972b,  Inkomensverdeling  van  werkende  gehuwde  vrouwen  1967  (Staatsuitgeverij,  The  Hague). 
NCBS,  1977,  Inkomensverdeling  1970  en  vermogensverdeling  1971  (Staatsuitgeverij,  The  Hague). 
NCBS,  1979,  Inkomensverdeling  van  werkende  gehuwde  vrouwen  1970  en  1973  (Staatsuitgeverij,  The  Hague). 
NCBS,  1980,  De  personele  inkomensverdeling  1975,  deel  I  (Staatsuitgeverij,  The  Hague). 
NCBS,  1982,  De  personele  inkomensverdeling,  Werkende  gehuwde  vrouwen  1975  en  1976  (Staatsuitgeverij,  The  Hague). 
NCBS,  1984,  De  personele  inkomensverdeling  1979  (Staatsuitgeverij,  The  Hague). 
NCBS,  1985,  The  personele  inkomensverdeling  1981,  Individuen  (Staatsuitgeverij,  The  Hague). 
Nelissen,  J.H.M.,  1988,  A  model  for  the  age-earnings  profile,  Working  paper  (Tilburg  University,  Tilburg)  forthcoming. 
Nelissen,  J.H.M.,  1989,  The  redistributive  impact  of  social  security  settlements  on  lifetime  income,  Ph.D.  dissertation, 
forthcoming. 

The distribution of lifetime earnings.

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