Herwaarden, F.G. van

Kapteyn, A.J.

Praag, B.M.S. van

English

Research Papers in Economics

Economics  Letters  1 (1978)  173-177 
0  North-Holland  Publishing  Company 
173 
INDIVIDUAL  WELFARE  FUNCTIONS  AND  SOCIAL  REFERENCE  SPACES  * 
Arie  KAPTEYN,  Bernard  M.S. VAN  PRAAG  and  Floor  G. VAN  HERWAARDEN 
University  of  Leyden,  The  Netherlands 
Center  for  Research  in Public  Economics,  Leyden,  The Netherlands 
Received  June  1978 
A model  of interdependent  welfare  functions  is developed.  The relationship  between  the  param- 
eters  of an individual’s  welfare  function  and  the  income  distribution  in his Social  Reference 
Space  is established.  Results  based  on  Dutch  data  are presented. 
In  this  paper  we present  a study  of utility  interdependence,  both  theoretically  and 
empirically. 
1. Social  reference  spaces 
Let  s2 be  a population  of individuals  o.  The  behaviour  of an  individual  w E Sz in 
a given  period  may  be  described  by  a vector  x(w).  For  instance,  if the  elements  of x 
denote  the  levels  of consumption  of  different  goods,  then  x(w)  describes  w’s con- 
sumer  behaviour. 
Generally,  the  behaviour  of individuals  is perceived  by  other  individuals.  Thus,  an 
individual  o0  will  perceive  a whole  distribution  of behaviours  x(w)  (w  E a&,). 
This  perception  depends  upon  two  phenomena:  (1)  on  the  value  of the  vector  x(o) 
for  each  w E fi\o,;  (2)  on  the  weight  which  individual  w.  assigns  to  each  individual 
o,  to  be  denoted  by  d#(wloo),  If d#(wlwo)  = 0 for  some  o,  w.  does  not  attach 
any  weight  to  w.  We require  J{,  E a\wo}  d@(olwo)  = 1 [so d@(wloe)  is a density- 
element].  We call  d$(wlwo)  the  reference  weight (RW)  which  w.  attaches  to  w. 
The  function  d@(olwo)  on  fi  defines  a normed  measure  vu0  on  Q.  We call  (a, 
“!&  vuo)  the  Social Reference  Space  (SRS)  of  w.  , where  “a  is a u-algebra  on  s2 
with  individuals  as atoms. 
After  the  introduction  of x(o)  and  d#(olwo)  we can  define  the  density-element 
* This  study  has been  supported  by  a grant  from  the  Netherlands  Organization  for  the  Advance- 
ment  of Pure  Research  (ZWO) and by  the  kind  cooperation  of  the  Dutch  Consumer  Union. 
It is a part  of  the  Leyden  Income  Evaluation  Project  of the  Center  for  Research  in Public 
Economics  of  Leyden  University.  A more  extensive  version  of this  paper  is available  under 
the  same  title  as Report  76.01  of CERPEC. 174  A.  Kapteyn  et al. /Individual  welfare  functions 
d$(xlwo)  of the  behaviour  distribution  perceived  by  w.  as 
W&loo)  =  s  d@(wlwo). 
{w E ci\wolx(w)=x} 
(1) 
Probably  drl/( * Iwo) influences  individual  we’s welfare  function.  This  will  be  investi- 
gated  in  the  sequel. 
2. The  individual  welfare  function  of income 
We assume  that  an  individual  is able  to  evaluate  income  levels  z on  a  [O,l]-scale. 
These  evaluations  are described  by  a so-called  individual  welfare  function  of  income 
WFI).  An  individual’s  WFI  is measured  by  asking  him  the  following  question: 
In answering the following question it is advisable to start with the underlined  words. Try 
at any  rate  to fill in  all  amounts  asked  for  to  the  best  of  your  judgement. 
Taking  into  account  my  (our)  present  living  circumstances,  I would  regard  a net  weekly/ 
monthly/yearly  (encircle  the  period)  family  income  as: 
excellent 
good 
amply  sufficient 
sufficient 
barely  suffcien  t 
insufficient 
very  insufficient 
bad 
very  bad 
if  it  were  above 
if  it  were  between 
if  it  were  between 
if  it  were  between 
if  it  were  between 
if  it  were  between 
if  it  were  between 
if  it  were  between 
if  it  were  below 
and 
.  and 
.  and 
and 
and 
.  and 
.  and 
We call  this  the  income-evaluation  question. 
The  verbal  evaluations  (excellent,  good,  amply  sufficient,  etc.)  are transformed 
into  numbers  on  a zero-one  scale by  identifying  these  evaluations  with  equal  quan- 
tiles.  ’ That  is the  qualification  “excellent”  is identified  with  0.888,  the  qualifica- 
tion  good  is identified  with  0.777,  etc.  Denoting  the  amount  in  the  left-hand  column 
in  the  ith  row  of the  income  evaluation  question  by  zi and  the  corresponding  numer- 
ical  evaluation  by  U(zi),  we obtain  a sequence  {(zi,  V(Zi))} f= 1, where  U(zi)  = 
(9 -  i)/9,  i = 1, . .  . . 8. (Note  that  the  amount  in  the  ninth  row  may  be  discarded 
because  it will  be  equal  to  the  amount  in  the  eighth  row.) 
According  to  the  theory  outlined  in Van  Praag  (1968),  the  answers  to  the  income- 
evaluation  question  will  follow  a definite  pattern.  More  precisely,  the  evaluation 
1 This  transformation  rests  upon  an  information  maximization  argument  developed  by  Van 
Praag  (1971)  and  generalized  by  Kapteyn  (1977). A.  Kapteyn  et al. /Individual  welfare  functions  175 
U(z)  of an  income  z is fairly  well  approximated  by 
u(z)  = L  _f  1  exp  1-i  [(In(t)  -  ~)/a]  *}  dt 
042ir  o  t 
= A@;  c130)  , 
the  lognormal  distribution  function  with  parameters  p  and  u. * 
The  parameters  p and  u can  be  estimated  per  individual  from  the  eight  points 
{(zi,  U(zi))}  by  means  of simple  regression.  If individual  o.  has  a higher  /J [and 
consequently  a hi&er  expb)]  than  individual  w 1, then  w.  needs  more  income  to 
reach  a certain  evaluation  level  than  does  w1  . The  quantity  exp@)  has been  called 
the  natural  unit  of  income  [for  a motivation  of the  term,  see Van  Praag  (1968, 
p. 37)].  The  parameter  u determines  the  slope  of the  WFI  about  the  median  value 
exp(p).  The  smaller  an individual’s  u, the  steeper  his WFI  will  be.  The  parameter  u 
has been  called  the  welfare  sensitivity  [Van  Praag  (1968,  p. 38)]. 
Over  a five-year  period,  WFIs  of  about  12,000  individuals  were  measured,  and  a 
number  of  attempts  were  made  to  explain  individual  welfare  parameters  p and  u 
from  individuals’  personal  and  social  circumstances.  [Van  Herwaarden,  Kapteyn 
and  Van  Praag  (1977)  give a short  review  of results.]  It  appears  that  an individual’s 
own  actual  income  y  and  his  family  size fs  are the  most  important  factors.  In  the 
present  study  we extend  the  explanation  of the  welfare  parameters  by  taking  into 
account  reference  group  effects. 
We assume  that  an individual  we’s  WFI  depends  on  his  income  y(oo),  the  num- 
ber  of equivalent  adults  in his  family  fs(wo)  [for  details,  see Kapteyn  and  Van  Praag 
(1976)]  and  on  the  income  distribution  in  LA  as it is perceived  by  wo,  J/(  . IWO), 3 
~(Zl~o)  =  U?Ifi(~O)~Y(~O),  J/( . Iwo))  1  (3) 
where  U(zlw,)  is individual  we’s evaluation  of an  income  level  z. 
3. The  relation  between  an  individual’s  WFI  and  his SRS 
Characterizing  $(  . Iwo) by  its  two  first  log-moments, 
m&0)=  J  In  Y(W)  dO(4wo)  , 
n\wo 
?(wo)  =  s  Uny(w)  -  m@)12  dN44, 
n\wo 
(4) 
(5) 
* It  should  be  stressed  that  in  the  present  context  the  log-normal  distribution  function  has  no 
probability  theoretical  meaning. 
3 The  quantity  $ (  IWO)  has  been  defined  by  (l),  be  it  that  the  vector  x(w)  is  now  replaced  by 
the  scalar  y(w),  the  net  after  tax  income  of  individual  w. 176  A.  Kapteyn  et al. /Individual  welfare  functions 
we specify  the  following  relationships: 
&we)=Po+  Pl~fi(~o)+Pz  ~JJ(oo)+Psm(wo)  +u(mo)  2  (6) 
a2(oo)=  {a0  +  als2(wo)+~z~(wo)-  t~~(w~)]~}  eUcWo)  ,  (7) 
where  u(wo)  and  u(wo)  are i.i.d.  error  terms  and  PO, fir, f12,  B3, ao, al,  a2 are param- 
eters.  The  relations  (6)  and  (7)  are theoretically  motivated  in  Kapteyn  (1977). 
The  main  problem  in  the  empirical  research  is to  specify  d$(wloo),  which  defines 
m(wo)  and  s2(oo),  as a function  of  a modest  number  of parameters  which  can  be 
estimated  along  with  the  other  parameters  in  model  (6)  and  (7).  To  simplify  mat- 
ters,  individuals  are characterized  by  a number  of social characteristics (education; 
job;  degree  of urbanization;  age; geographical  location;  working  environment,  i.e., 
whether  working  in  private  firms,  self-employed,  or not  employed  at  all). 
Next,  the  RWs d@(wloo)  are specified  as a function  of the  social  characteristics 
of individuals  o  and  oo.  The  whole  model  specifies  the  RWs d$(wloe)  for  all w 
and  w.  as a function  of 20 unknown  parameters. 
4. Results 
The  parameters  in  (6)  and  (7)  [including  the  20  parameters  inherent  in  m(wo) 
and  s2(wo)]  have  been  estimated  by  means  of Gallant’s  non-linear  least  squares 
method  from  a sample  of 2,774  members  of the  Dutch  Consumer  Union,  drawn  in 
1971.  The  estimated  counterparts  of (6)  and  (7)  read  (standard  errors  in  paren- 
theses) 
M(w~) = 1.94  + 0.12  In fs(wo)  + 0.49  In y(wo)  + 0.29  m(wo)  ,  R * = 0.647, 
(0.32)  (0.02)  (0.01)  (0.03)  N  = 2,774, 
(8) 
a*(wo)  = 0.12  + 0.53  ?(wo)  + 0.21  [p(we)  -  m(wo)]2  )  R2  = 0.064, 
(0.01)  (0.10)  (0.03)  N  = 2,774. 
(9) 
The  meaning  of (8)  is illustrated  by  the  example  in  fig.  1 [the  numbers  do  not  fol- 
low  exactly  from  (8)  but  are merely  illustrative;  the  argument  w.  is omitted]. 
Let  individual  w.  with  income  y(wo)  and  WFI A  at a certain  moment  expect  an 
income  increase  by  a factor  (1 + o).  He evaluates  his present  income  by  0.70.  The 
expected  future  income  y(wo)(l  + ol) is evaluated  by  0.95.  Once  he  receives  the 
income  y(wo)(  1 + a),  eq.  (8)  implies  that  his WFI  shifts  to  position  B so ex post 
he  evaluates  the  new  income  by  0.85.  The  phenomenon  that  the  WFI  shifts  with 
income  has  been  called  the preference  drift effect  [Van  Praag  (1971)].  If,  moreover, 
all other  individuals  receive  the  same  income  increase,  m(oo)  rises to  m(oo)  + 
ln( 1 + cr), and  the  WFI  shifts  to  position  C implying  a welfare  evaluation  of the  new 
income  by  only  0.75.  The  phenomenon  that  an  individual’s  WFI  shifts  with  incomes  in A. Kapteyn et al. /Individual welfare functions  177 
/  0.49  ; 
P 
(I  +cr)  pi.,8 
(I  +Q)  a 
Fig.  1.  Ilhrstration  of  eq.  (8). 
his SRS  has been  called  the  reference  drift  effect  [Kapteyn  (1977)].  The  positive 
coefficient  offi  finally  implies  that  with  a larger  family  the  WFI  lies more  to 
the  right.  Hence  a larger  income  is required  to  attain  a certain  evaluation  level.  This 
observation  allows  for  the  construction  of constant  welfare  family  income  equiv- 
alence  scales  [Kapteyn  and  Van  Praag  (1976)].  Regarding  (9)  similar  interpretations 
may  be  provided. 
References 
Kapteyn,  A.,  1977,  A  theory  of  preference  formation,  Ph.D.  thesis  (Leyden). 
Kapteyn,  A.  and  B.M.S.  Van  Praag,  1976,  A  new  approach  to  the  construction  of  family  equiv- 
alence  scales,  European  Economic  Review  7,  313-335. 
Van  Herwaarden,  F.G.,  A.  Kapteyn  and  B.M.S.  Van  Praag,  1977,  Twelve  thousand  individual 
welfare  functions,  European  Economic  Review  9,  283-300. 
Van  Praag,  B.M.S.,  1968,  Individual  welfare  functions  and  consumer  behavior  (North-Holland, 
Amsterdam). 
Van  Praag,  B.M.S.,  1971,  The  welfare  function  of  income  in  Belgium:  An  empirical  investiga- 
tion,  European  Economic  Review  2,  337-369. 
Van  Praag,  B.M.S.  and  A.  Kapteyn,  1973,  Further  evidence  on  the  individual  welfare  function 
of  income:  An  empirical  investigation  in  The  Netherlands,  European  Economic  Review  4, 
33-62. 

A theory of preference formation,

Further evidence on the individual welfare function of income: An empirical investigation in The Netherlands,

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The welfare function of income in Belgium: An empirical investigation,

Twelve thousand individual welfare functions,

Individual welfare functions and social reference spaces.

Individual welfare functions and social reference spaces.

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