Kooreman, P.

English

Research Papers in Economics

European Economic Review 27 (1985) 391-394.  North-Holland 
SUBSTITUTION  IN  CONSUMPTION;  AN  APPLICATION  TO  THE 
ALLOCATION  OF  TIME 
A Comment 
Peter KOOREMAN* 
Tilburg University, 5000  LE Tilburg,  The Netherlands 
Received January 1984, final version received October 1984 
1.  Introduction 
The  paper  by  S.  Krsenne  (1983)  focuses  on  the  relation  between  the 
commodity-price  effect  and  the  good-price  effect  within  the  Becker  frame- 
work  of  the  allocation  of  time.  Assuming  input  separability  in  linear 
homogeneous  production  functions,  the  author  derives  the  matrix  of price 
elasticities of the demand for market goods for a  2-commodity-4-goods case. 
It turns out that  these elasticities can be expressed as simple functions of the 
elasticity  of factor substitution  (which  only depends  on  technology) and  the 
shadow price elasticity of the demand for commodities (which only depends 
on  preferences).  In  section  3  of  the  paper  these  elasticities  appear  as 
parameters  in  a  double  logarithmic  equation,  which  is  estimated  using 
Belgian  time  series  data  on  the  allocation  of time.  Through  this  approach, 
the  author  claims  to  be  able  to  disentangle  the  effects  of  tastes  and 
technology on consumer allocation of time. 
In  this  note,  I  will  argue  that  on  the  basis  of the  data  available  and  the 
assumptions made, only the parameters  of the production function of leisure 
activity  can  be  estimated.  Moreover,  this  is  possible  only  because  the 
assumptions  are  so special  that  there  is  no  room left for  tastes  to  influence 
behavior. 
2.  The model 
The empirical  model starts  from  the (unspecified) utility  function u(al,a2), 
where al  is a  leisure activity and a2 is a  so-called semi-leisure activity. Both 
al  and  az  are  unobservable.  The  linear  homogeneous  production  functions 
*I wish to thank Professors A. Kapteyn and J. Waelbroeck for helpful comments. 
0014-2921/85/$3.30  ￿9  1985, Elsevier Science Publishers B.V. (North-Holland) 392  P. Kooreman, Substitution in consumption, A comment 
al(ql,ti)  and a2(q2, t2) are  assumed to be of the CES type, i.e., 
a,(qi, ti) = E~q; P' + (1 -  cq)tT'-I -  l/p,,  i = 1,2,  (1) 
where  qi  is  good  input  and  ti  is  time  input,  for  the  ith  activity  and  p~= 
(1-a~)/a~,  with a~ the elasticity of factor substitution between  q~ and  t~. The 
parameters ~  measure the degree to which technology is good intensive. 
The time-income constraint is 
Piql + P2q2 d- w(t I q- t2) =  m,  (2) 
where Pl  and P2  are prices  of q~  and q2  respectively, w is the wage rate  and 
m is full income. 
Krsenne  assumes  that  semi-leisure  good  q2  and  semi-leisure  time  t2  are 
constant over the sample period, since they only represent the most essential 
consumption  goods  and  the  time  needed  to  consume  them.  As  a  conse- 
quence, a2 is also constant. Hence, maximizing u(al,a2)  subject to (1) and (2) 
is equivalent to maximizing ax(ql, tl) subject to, 
Plql  +  wtl =  m* -  m--  P2q2 -- wt2.  (3) 
Note  that  by these  assumptions  tastes  have  become  irrelevant  for  observed 
behavior. 
Solving the maximization problem yields: 
t,  =  m*{(1  -- ~,)lw}:'l[{(1  -  ~31w} <"" w +  {~l/pl }:" pl]-  (4) 
The parameters  ex  and el  describe the production of a~  but do not provide 
any information on  the  structure of tastes. There  is no  relation between the 
shadow price elasticity of the demand for commodities (which only depends 
on tastes) and 0t I and al. 
Even if qz  and ta  are not constant and observations  on these goods were 
available,  there  are  some  difficulties  with  Krsenne's  approach,  as  will  be 
explained  below.  Krsenne  estimates  the  following equation  by  OLS,  after 
taking first differences: 
In tl =~+flln m + ~hi[k  i lnpl +(1 -kl) In w] 
q- 1712 [k2 In P2 + (1 -  k 2) In w] + ~ 1 Ek l(ln pl -- In w)],  (5) 
where  r#l i is  the  elasticity of the  demand for al  with respect  to  the  shadow 
price  Pai of commodity i  and  k~ is  the  share  of the cost  of one  good in  the 
cost of producing commodity i, which K~senne calculates as P. Kooreman, Substitution in consumption, A comment  393 
kl -  Plql  k2 =  P2q2  (6) 
Pxql + Wtl '  P2q2 -1- wt2" 
Eq. (5) is obtained as follows. First, the author shows that 
Or1  .Pl =kl(?/ll  q-trl) , 
]~=-Op I  tl 
_  Otl  . P2 =  k2ql2,  (7) 
OP2  tl 
2-- 
Ot I  w 
￿9  -- = (1 -  kl)r/11 -  kiai + (1 -  k2)rh 2. 
aw  tl 
Next,  he  assumes  ~,  6  and  2  to  be constant,  so  that  he  specifies  a  double 
logarithmic  relation 
lntl=ct+fllnm+ylnpl+61np2+21nw.  (8) 
Substitution of (7) into (8) and rearranging  terms yields (5). 
There  are two problems with eq. (5). The first one concerns the estimation 
method.  Since kl  and  k2 depend on the endogenous variables ql,  q2,  tl  and 
t2,  OLS will  generally yield inconsistent  estimates, whereas the small  sample 
properties  of  the  OLS  estimator  in  this  case  are  unknown.  The  second 
problem concerns the double logarithmic  specification. One might  argue that 
this  specification  serves  as  a  reasonable  approximation  to  a  generally 
unknown relation.  However, in the household production model the demand 
for  goods  equations  represent  the  influence  of both  preferences  and  tech- 
nology. Consequently,  by explicitly assuming  CES production  functions and 
double  logarithmic  demand  for  goods  equations,  the  author  implicitly 
assumes  a  certain  functional  form  for  the  utility  function.  It  is  at  least 
questionable  whether  this  utility  function  has  reasonable  properties,  especi- 
ally in view of the fact that  double logarithmic  demand equations satisfy the 
theoretical  restrictions  of demand theory if and  only if all income elasticities 
are  unity,  all  own  prices  elasticities  are  minus  one  and  all  cross  price 
elasticities are zero. 
A better procedure would be to specify a  utility function and to derive the 
demand for goods equations by maximizing  the utility function subject to (1) 
and  (2). However, as  has  been argued  convincingly  by Pollak  and  Wachter 
(1975)  such  an  approach  has  no  empirically  testable  implications,  unless 
outputs  of  household  production  are  measurable.  In  Krsenne's  example 394  P. Kooreman, Substitution in consumption, A  comment 
outputs cannot  be measured, as is usually the case in empirical work. I In the 
absence of direct measures of the commodities produced by the household, it 
is  basically  impossible  to  distinguish  between  a  production  function  and  a 
utility function interpretation  of the household's behavior. Consequently, any 
conclusions on the separation  of tastes and technology are entirely dependent 
on non-testable assumptions such as input separability and  linear  homogene- 
ity  of  production  functions.  Indeed,  as  argued  above  with  respect  to  (4), 
K6senne's assumptions effectively eliminate any possible role for tastes. 
In  view  of these  facts,  we must  conclude  that  K6senne's  approach  is  not 
(and  cannot  be) capable of disentangling  the effects of tastes and  technology 
on consumer allocation of time. 
References 
K6senne,  S.,  1983,  Substitution  in  consumption;  An  application  to  the  allocation  of  time, 
European Economic Review 23, 231-239. 
Pollak,  R.A.  and M.L.  Wachter,  1975, The relevance of the household production function and 
its implications for the allocation of time, Journal of Political Economy 83, 255-277. 
Rosenzweig,  M.R.  and  T.P.  Schultz,  1983,  Estimating  a  household  production  function: 
Heterogeneity,  the demand  for health inputs,  and  their effects  on  birth weights, Journal  of 
Political Economy 91,723-746. 
~An exception  is  the  paper  by  Rosenzweig  and  Schultz  (1983)  who  use  birth  weight  as  an 
indicator of the output of the household health production function. 

Substitution in consumption: An application to the allocation of time: A comment.

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