UBC Theses and Dissertations

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UBC Theses and Dissertations

Price-sensitive inequality measurement Kwong, Sunny Kai-Sun 1985

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PRICE-SENSITIVE INEQUALITY MEASUREMENT BY SUNNY KAI-SUN KWONG B.Soc.Sc., The U n i v e r s i t y o f Hong Kong, 1979  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  IN THE FACULTY OF GRADUATE STUDIES (Department o f Economics)  We a c c e p t t h i s t h e s i s as conforming to the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA May  1985  (S) Sunny Kai-Sun Kwong, 1985  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may department or by h i s or her  be granted by  the head o f  representatives.  my  It i s  understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be  allowed without my  permission.  Department o f The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T  1Y3  Date  *  DE-6  (3/81)  written  Abstract  The  e x i s t i n g i n e q u a l i t y indexes  i n t h e economics  ( i n c l u d i n g the more s o p h i s t i c a t e d indexes o f Muellbauer Jorgenson-Slesnick  literature (1974) and  (1984)), are found t o be i n s e n s i t i v e t o r e l a t i v e  p r i c e changes o r are u n j u s t i f i a b l e i n terms o f s o c i a l e v a l u a t i o n e t h i c s o r both.'  The p r e s e n t r e s e a r c h f i l l s  t h i s gap i n t h e l i t e r a t u r e  by p r o p o s i n g a new index, named the I n d i v i d u a l E q u i v a l e n t Income (IEI) index.  A household  indirect u t i l i t y  f u n c t i o n i s h y p o t h e s i z e d which  i n c o r p o r a t e s c e r t a i n a t t r i b u t e parameters i n t h e form o f e q u i v a l e n c e scales.  These a t t r i b u t e s a r e demographic and e n v i r o n m e n t a l  t i c s s p e c i f i c t o a g i v e n household.  characteris-  This indirect u t i l i t y function  g i v e s a number which r e p r e s e n t s the u t i l i t y o f each member o f the household.  A p a r t i c u l a r l e v e l o f i n t e r p e r s o n a l comparison o f u t i l i t i e s  i s assumed which g i v e s r i s e t o an e x a c t i n d i v i d u a l u t i l i t y named e q u i v a l e n t income.  indicator  A d i s t r i b u t i o n o f these e q u i v a l e n t incomes  forms t h e b a s i s o f a p r i c e - s e n s i t i v e r e l a t i v e i n e q u a l i t y  index.  T h i s index can be implemented i n t h e Canadian c o n t e x t .  Pre-  f e r e n c e s a r e assumed t o be nonhomothetic t r a n s l o g and demand d a t a are d e r i v e d from c r o s s - s e c t i o n s u r v e y s and t i m e - s e r i e s a g g r e g a t e s .  Based on demand d a t a , the t r a n s l o g e q u i v a l e n t income f u n c t i o n can be e s t i m a t e d and e q u i v a l e n t incomes imputed t o a l l i n d i v i d u a l s i n society.  An A t k i n s o n i n d e x o f e q u i v a l e n t incomes i s then computed  t o i n d i c a t e t h e a c t u a l degree o f i n e q u a l i t y i n Canada.  The new common d a t a s e t .  I E I i n d e x i s compared w i t h o t h e r i n d e x e s based on a The main f i n d i n g s a r e : c o n v e n t i o n a l i n d e x e s g i v e  bad e s t i m a t e s o f t h e t r u e e x t e n t o f i n e q u a l i t y and t h e I E I i n d e x , w h i l e p r o v i d i n g a more a c c u r a t e e s t i m a t e , i n d i c a t e s d i s t r i b u t i v e p r i c e impact i n a p r e d i c t a b l e manner, i . e . , food p r i c e i n f l a t i o n w h i l e t r a n s p o r t a t i o n p r i c e i n f l a t i o n a m e l i o r a t e s the problem.  aggravates  inequality  Table  o f Contents  Chapter 1  Introduction  Chapter 2  Survey o f the L i t e r a t u r e Some simple  Chapter  3  indexes  13 17  Footnotes  28  A New Approach  Equivalent  30  scales  32  income  36  S o c i a l choice  42  I n e q u a l i t y measurement  47  Footnotes  54  Specification  58  Footnotes  Chapter  5  68  E s t i m a t i o n Method  69  Introduction  69  Cross-section estimation  71  Time-series  74  estimation  Footnotes  Chapter 6  12  Distributive price effects  Equivalence  Chapter 4  1  81  Implementation  82  Cross-section estimation  82  Time-series  86  Estimated  estimation  equivalence  Footnotes  scales  95 99  - iv -  Chapter 7  Applications  100  Introduction Estimated  Chapter 8  100  market e q u i v a l e n c e  scales  101  S e n s i t i v i t y o f the I E I index t o p°  107  Comparative study o f v a r i o u s measures  109  Distributive price effects  115  Inequality trend  116  Conclusion  121  Footnotes  123  Conclusion  125  Bibliography  130  Appendices  135  - v -  L i s t o f Tables  Table 1  Vector A  61  Table 2  T r a n s l o g Market E q u i v a l e n c e S c a l e s  102  Table 3  S t a t i s t i c s Canada Low-Income C u t - o f f R a t i o s  108  Table 4  S e n s i t i v i t y o f I E I Measure t o p °  110  Table 5  Comparison o f D i f f e r e n t Measures  111  Table 6  S e n s i t i v i t y o f I E I Measure t o p  117  Table 7  I n e q u a l i t y Trend  ( I E I Measure)  119  Table 8  I n e q u a l i t y Trend  (iIWR ( S t a t . Can) Measure)  120  - vi -  Acknowledgement  I wish t o express my deepest g r a t i t u d e t o members o f the a d v i s o r y committee: Donaldson  Charles Blackorby,  (thesis supervisor)  They have p l a y e d a c r i t i c a l  Erwin Diewert and David  f o r t h e i r continuous  guidance and s u p p o r t .  r o l e throughout the r e s e a r c h .  Thanks  are a l s o due t o Debra Glassman, Jon Kesselman and Hugh Neary f o r r e a d i n g an e a r l i e r d r a f t and f o r t h e i r e l a b o r a t e and e x p o s i t i o n .  comments on s t y l e  I should a l s o thank Jean Wu o f A r t s Computing, UBC  f o r her valuable consultancy  work i n computer  - vii -  operations.  - 1 -  CHAPTER 1  The the  INTRODUCTION  issue of i n e q u a l i t y —  the divergence  individuals i n a society —  c e r n t o economists.  has  i n w e l l - b e i n g among  t r a d i t i o n a l l y been o f g r e a t  T h i s i s h a r d l y s u r p r i s i n g because a b a s i c theme  i n economics i s the a l l o c a t i o n o f s o c i e t y ' s r e s o u r c e s bution of s o c i e t y ' s wealth. can be  con-  Indeed, s y s t e m a t i c  found as e a r l y as Cannan  (1914) and  and  the  distri-  study o f i n e q u a l i t y  Dalton  a s i d e the q u e s t i o n o f what causes i n e q u a l i t y and  (1920).  Putting  the more c o n t r o -  v e r s i a l i s s u e o f what i d e o l o g y j u s t i f i e s i n e q u a l i t y , on a p r a c t i c a l l e v e l , the measurement o f i n e q u a l i t y i s important reasons.  F i r s t , the government may  f o r at l e a s t  two  want t o know the i n e q u a l i t y  implications of alternative p o l i c i e s .  Second, i t i s i n t e r e s t i n g t o  compare the degree o f i n e q u a l i t y between d i f f e r e n t s o c i e t i e s contemporaneously  and  o f the p r e s e n t  i n time-series for a p a r t i c u l a r society.  The  objective  r e s e a r c h i s t o develop an i n e q u a l i t y index which i s  a v a s t improvement o v e r e x i s t i n g ones i n t h a t i t b r i n g s i n t o sharp f o c u s the n o t i o n o f i n d i v i d u a l w e l f a r e equality.  i n the measurement o f i n -  In p a r t i c u l a r , based on r e v e a l e d b e h a v i o u r a l  impact o f p r i c e changes on i n d i v i d u a l w e l f a r e  data,  the  i s incorporated into  i n e q u a l i t y measurement.  Echoing (Atkinson the  the i d e a o f D a l t o n  (1970) and B l a c k o r b y  f a c t t h a t u n d e r l y i n g every  (1920), r e c e n t r e s e a r c h  and  Donaldson  developments  (1978))have re-emphasized  i n e q u a l i t y index i s a s e t o f e t h i c s .  - 2 -  I t i s c l e a r t h a t i n e q u a l i t y measurement i s a normative endeavor r a t h e r than a p o s i t i v e one. welfare  The c l a i m t h a t one d i s t r i b u t i o n o f  (however d e f i n e d and measured) i s more unequal than another  d i s t r i b u t i o n i s contingent  on a s e t o f e t h i c s .  I t i s therefore  important t h a t the p a r t i c u l a r s e t o f e t h i c s i s made  The  main t a s k o f t h e p r e s e n t  p a r t i c u l a r e t h i c s t h a t one s h o u l d i s an i d e o l o g i c a l q u e s t i o n .  research  explicit.  i s not i n d i s p u t i n g the  choose t o measure i n e q u a l i t y .  The b a s i c l i n e o f a t t a c k  is:  This  what  s h o u l d be t h e b a s i c e n t i t i e s t h a t we use t o measure i n d i v i d u a l w e l l being?  L e t us look a t a common example. f o r N households  (for a recent  The A t k i n s o n  (1970) index i s ,  a p p l i c a t i o n and some ad hoc v a r i a n t s ,  see Beach, Card and F l a t t e r s (1981)),  (1.1)  v I : = 1 - (± A  N  (1.2)  where  : = 1  (y ,  -  TT  i=l  1/  N  Z (y /y) ) i=l r  ±  /  r  r < l ,  1/N ( y./u ) A /  r ^ O  r = 0.  1  y ) i s a d i s t r i b u t i o n o f household incomes and u i s the  mean o f t h e d i s t r i b u t i o n .  J u s t i f y i n g t h e i n e q u a l i t y i n d e x (1.1)  (1.2)  r s o c i a l welfare  i s t h e mean o f o r d e r  function  (provided  the household incomes are r e s t r i c t e d t o be p o s i t i v e ) which i s  that  - 3 -  c o m p l e t e l y e t h i c a l l y c h a r a c t e r i z e d i n B l a c k o r b y and Donaldson  There are two  o b j e c t i o n s t o the index  a r i s e from u s i n g household being.  First,  (1.1) , (1.2)  that  income as a measure o f i n d i v i d u a l  t h i s index i s i n s e n s i t i v e t o p r i c e  (1982).  well-  changes w h i l e , even  i n t u i t i v e l y , a change i n r e l a t i v e p r i c e s s h o u l d have d i s t r i b u t i o n a l impacts.  F o r example, an i n c r e a s e i n the p r i c e o f n e c e s s i t i e s  t o l u x u r i e s a f f e c t s t h e poor more than the r i c h . i n c r e a s e must aggravate increase to r e f l e c t  The was  this  Such a r e l a t i v e  change.  1  and  (1.2), t h a t the  o r i g i n a l l y t a k e n t o be a d i s t r i b u t i o n o f household  distribution  incomes.  i s c l e a r l y i n c o n s i s t e n t w i t h the s o c i a l w e l f a r e view o f  households.  household  them  The b a s i c q u e s t i o n "How  income o r e x p e n d i t u r e  not  V a r i o u s ad hoc m o d i f i c a t i o n s have  been made i n the l i t e r a t u r e though none o f 2 for details).  This  inequality  where i n d i v i d u a l s are viewed as the b a s i c e n t i t i e s i n s o c i e t y ,  Chapter  price  t h e i n e q u a l i t y s i t u a t i o n and the i n d e x s h o u l d  second o b j e c t i o n i s , i n (1.1)  the c o l l e c t i v e u n i t s —  relative  i s satisfactory s h o u l d we  (see  adjust  so t h a t an i n d i v i d u a l i n a f a m i l y o f  say, f o u r members, can be r e a s o n a b l y compared i n w e l f a r e terms w i t h an i n d i v i d u a l i n a f a m i l y o f one?" has not been a d e q u a t e l y d e a l t w i t h . T h i s i s r e f e r r e d t o below as t h e problem o f i n t e r p e r s o n a l comparison o f utility. The p r e s e n t r e s e a r c h attempts A new  to f i l l  t h i s gap  approach t o i n e q u a l i t y measurement i s developed  i n the  literature.  which d e a l s  - 4 -  w i t h these problems e x p l i c i t l y and t e s t o f t h i s new  s y s t e m a t i c a l l y . S i n c e the u l t i m a t e  index i s i n i t s p r a c t i c a l u s e f u l n e s s , the approach  has been implemented f o r Canada and the r e s u l t s are i n g e n e r a l v e r y appealing.  The  new  approach can be summarized as f o l l o w s .  Two  l i n e s of  r e s e a r c h are merged t o g e t h e r , namely, w e l f a r e measurement and t i o n , and demand system e s t i m a t i o n . a n a l y s i s i s extended household  utility  from the micro t o the macro.  This u t i l i t y  expenditure The  techniques,  As a f i r s t  expenditure  t o an o r d i n a l  f u n c t i o n ' s novel feature i s that  reason f o r t h i s s p e c i f i c a t i o n i s t h a t i n p r a c t i c e  e x p e n d i t u r e d a t a are not e a s i l y o b t a i n e d .  utility  household  i n s t e a d o f i n d i v i d u a l e x p e n d i t u r e e n t e r s the  function.  individual  However, as the o b j e c t i v e  of  f i n d i n g a n u m e r i c a l u t i l i t y r e p r e s e n t a t i o n i s t o measure  in  the aggregate,  inequality  the u t i l i t y number must be capable o f b e i n g  p r e t e d as t h e u t i l i t y o f each member i n the household.  parameter e s t i m a t e s o f the u t i l i t y  function.  Barten  inter-  Whether t h i s  i n t e r p r e t a t i o n i s a c c e p t a b l e o r not depends on the form and  s c a l e s p r o v i d e one  step,  i s measured by means o f an i n d i r e c t u t i l i t y f u n c t i o n  which maps p r i c e s and household number.  In u t i l i s i n g both  evalua-  the  equivalence  such form and t h e y are e s t i m a t e d t o g e t h e r w i t h  o t h e r parameters from demand d a t a .  Thus, by i n c o r p o r a t i n g f a m i l y  s i z e and o t h e r a t t r i b u t e s i n t o the u t i l i t y  f u n c t i o n , each  i s endowed w i t h a h o u s e h o l d - s p e c i f i c u t i l i t y  function.  household  - 5 -  The  o r d i n a l nature  ' problems.  o f the u t i l i t y  Subjecting a u t i l i t y  s p e c i f i c monotonic t r a n s f o r m  number g i v e s r i s e t o f u r t h e r  f u n c t i o n t o an a r b i t r a r y  y i e l d s t h e same s e t o f demand  Even i f a l l t h e parameters i n t h e u t i l i t y estimated  from demand d a t a , t h e u t i l i t y  number i s s t i l l  arbitrary..  ranking o f a single  But i n e q u a l i t y measurement i m p l i e s  measurement and comparison f o r a t l e a s t two i n d i v i d u a l s . a numerical  equations.  f u n c t i o n are a c c u r a t e l y  T h i s problem i s not s e r i o u s i f o n l y t h e u t i l i t y i n d i v i d u a l i s concerned.  individual-  representation of u t i l i t y  utility  Consequently,  i s o b t a i n e d by u s i n g a r e f e r e n c e  i n d i v i d u a l and assuming a p a r t i c u l a r l e v e l o f i n t e r p e r s o n a l u t i l i t y comparison.  T h i s r e p r e s e n t a t i o n , named t h e e q u i v a l e n t income o f each  member i n a s p e c i f i e d household i s t h e t o t a l e x p e n d i t u r e  that a  ' r e f e r e n c e household needs a t .reference p r i c e s i n o r d e r t h a t each member i n i t i s j u s t as w e l l o f f as each member i n t h e h o u s e h o l d s p e c i f i e d a t t r i b u t e s and p r i c e s . p r i c e s , expenditure,  E q u i v a l e n t income i s a f u n c t i o n o f  a t t r i b u t e s , r e f e r e n c e p r i c e s and r e f e r e n c e  a t t r i b u t e s , and i t i s e s t i m a b l e the reasonableness  with  e m p i r i c a l l y u s i n g demand d a t a . S u b j e c t t o  o f t h e parameter e s t i m a t e s ,  t h i s approach o f f e r s  a p a r t i a l s o l u t i o n t o t h e second problem mentioned above, and t o the e x t e n t t h a t e q u i v a l e n t income i s s e n s i t i v e t o p r i c e s , i t o f f e r s a s o l u t i o n t o t h e f i r s t problem. A d i s t r i b u t i o n o f i n d i v i d u a l e q u i v a l e n t incomes i s then aggregated by means o f a s o c i a l w e l f a r e  function.  A mean o f o r d e r r f u n c t i o n i s  used which has been c h a r a c t e r i z e d i n terms o f e t h i c a l axioms i n Blackorby  and Donaldson  (1982).  Adopting  t h e Atkinson-Kolm-Sen (AKS)  - 6 -  procedure,  an i n e q u a l i t y index i s c a l c u l a t e d which i s a c t u a l l y  an  A t k i n s o n index o f e q u i v a l e n t incomes.  The e s t i m a t i o n phase a l s o i n v o l v e s e x t e n s i o n from the micro t o the macro.  E x p e n d i t u r e share e q u a t i o n s are d e r i v e d from the  indirect u t i l i t y  household  f u n c t i o n s with the equivalence s c a l e s i n c o r p o r a t e d .  E s t i m a t i o n i s c a r r i e d out i n two o n l y micro household  s t a g e s : the f i r s t stage i n v o l v e s  commodity-expenditure  r e g r e s s e d u s i n g c r o s s - s e c t i o n household S i n c e some parameters  share e q u a t i o n s which are  expenditure survey data.  i n the e q u i v a l e n t income f u n c t i o n are not y e t  i d e n t i f i e d , the micro e q u a t i o n s are summed t o g e t h e r t o o b t a i n  aggregate  commodity-expenditure  share e q u a t i o n s which a l l o w the u t i l i z a t i o n  t i m e - s e r i e s aggregate  demand d a t a t o e s t i m a t e the r e m a i n i n g  i n the e q u i v a l e n t income  The  implementation  i n the Canadian  parameters  function.  and the r e s u l t s o f a p p l y i n g t h i s new  c o n t e x t are d e s c r i b e d l a t e r .  n e v e r t h e l e s s , t o mention  of  I t might be  index  helpful,  some o f t h e main c o n t r i b u t i o n s o f t h i s  research here.  1. U n l i k e a l o t o f o t h e r e m p i r i c a l demand s t u d i e s , the p r e s e n t does not assume the e x i s t e n c e o f an aggregate households  consumer.  are s p e c i f i c t o the e x t e n t t h a t t h e i r  The  sum  t o t a l o f a l l household  Instead,  characteristics  are c a p t u r e d by a t t r i b u t e v e c t o r s i n c o r p o r a t e d i n t o the function.  approach  demands y i e l d s  utility aggregate  - 7 -  demand which enables t h e u t i l i z a t i o n o f t i m e - s e r i e s aggregate T h i s i s not o n l y a t h e o r e t i c a l l y e x a c t approach,  data.  i t i s also  e m p i r i c a l l y s u p e r i o r , as t h e e s t i m a t i o n r e s u l t s show t h a t , based on b e h a v i o u r a l demand d a t a , meaningful inferred.  2. The new  w e l f a r e i n f o r m a t i o n can be  The e q u i v a l e n c e s c a l e e s t i m a t e s appear v e r y  reasonable.  index t r u l y c a p t u r e s d i s t r i b u t i v e p r i c e e f f e c t s , d e s p i t e  the margin o f e r r o r t h a t we might s u s p e c t i n t h i s type o f demand system s t u d i e s .  Indeed, r e s u l t s show t h a t commodities commonly  regarded as l u x u r i e s have an i n e q u a l i t y - r e d u c i n g p r i c e e f f e c t  while  the o p p o s i t e i s t r u e f o r n e c e s s i t i e s .  3. When we compare indexes t h a t v a r y from 0 t o 1, the d i s c r e p a n c y between two i n d e x e s i s e x p e c t e d  t o be s m a l l .  N e v e r t h e l e s s , the  new index t u r n s o u t t o be s u b s t a n t i a l l y d i f f e r e n t from a l l commonly used i n d e x e s .  We may conclude  picture o f the true inequality  4. Although it  t h i s methodology  t h a t these indexes g i v e a d i s t o r t e d situation.  i s developed  f o r i n e q u a l i t y measurement,  can be a p p l i e d w i t h some m o d i f i c a t i o n s t o o t h e r k i n d s o f w e l f a r e  analyses.  The framework i s q u i t e g e n e r a l .  F o r example, i n c o s t -  b e n e f i t a n a l y s e s , one f r e q u e n t l y l o o k s f o r a s o c i a l w e l f a r e measure as a judgment c r i t e r i o n when a l t e r n a t i v e  s t a t e s a r e b e i n g compared.  T h i s i s e a s i l y h a n d l e d w i t h i n the p r e s e n t framework, g i v e n p r i c e and e x p e n d i t u r e  i n f o r m a t i o n i n each  state.  - 8 -  T h i s d i s s e r t a t i o n i s o r g a n i z e d as f o l l o w s .  Chapter  2 surveys  c r i t i c a l l y the common i n e q u a l i t y indexes w i t h s p e c i a l emphasis on v a r i o u s measures o f u t i l i t y b, c) and Jorgenson l i m i n a r y attempts  used.  The works o f Muellbauer  and S l e s n i c k (1982  a,  a, b ) ( 1 9 8 4 ) , which are p r e -  t o capture p r i c e e f f e c t s , are e x p l a i n e d and  c r i t i c i z e d i n r e l a t i o n t o the p r e s e n t  Chapter  (1974  the  study.  3 i s t h e core c h a p t e r .  I t d e s c r i b e s the  r a t i o n a l e o f the s o c i a l e v a l u a t i o n framework and how  an  theoretical inequality  index i s c o n s t r u c t e d i n t h i s framework.  The e m p i r i c a l s p e c i f i c a t i o n o f the model i s p r e s e n t e d i n Chapter  4.  P r e f e r e n c e s are assumed t o be non-homothetic t r a n s l o g  B a r t e n e q u i v a l e n c e s c a l e s i n c o r p o r a t e d ) from which equations  (household and aggregate)  Chapter aggregate  5 e x p l a i n s how  expenditure  time-series data  faith  expenditure  are d e r i v e d .  t h e e s t i m a t i o n model o f h o u s e h o l d  and  shares can be e s t i m a t e d u s i n g c r o s s - s e c t i o n and  sequentially.  Canadian d a t a are used  for estimation.  Chapter  6.explains  how  p u b l i c l y a v a i l a b l e d a t a can be u t i l i z e d t o e s t i m a t e the model s e t out i n Chapter  5 and the e s t i m a t e d B a r t e n E q u i v a l e n c e S c a l e s are  and i n t e r p r e t e d .  The  presented  r e s u l t s are i n g e n e r a l v e r y a p p e a l i n g , l e n d i n g  f u r t h e r support t o t h e c r e d i b i l i t y o f the new  index.  - 9 -  Applications p r e s e n t e d i n Chapter  o f the e s t i m a t e s t o i n e q u a l i t y measurement are 7.  are used t o c a l c u l a t e  F i r s t l y , v a r i o u s i n d i v i d u a l w e l f a r e measures  i n e q u a l i t y u s i n g the same d a t a s e t and the same  formula f o r the i n e q u a l i t y i n d e x .  I t t u r n s out t h a t  the new  index  g i v e s s i g n i f i c a n t l y d i f f e r e n t answers from o t h e r commonly used Secondly,  indexes.  t o demonstrate q u a n t i t a t i v e l y the d i s t r i b u t i v e impacts  r e l a t i v e p r i c e changes, i n e q u a l i t y i s c a l c u l a t e d under h y p o t h e t i c a l intuition.  u s i n g the new  index  p r i c e i n c r e a s e s and the r e s u l t s conform w e l l  F i n a l l y , i n e q u a l i t y i n Canada i n 1975,  1979  of  with  and 1981  is  e s t i m a t e d t o r e f l e c t on the i n e q u a l i t y t r e n d i n the l a s t decade. Chapter  8 c o n c l u d e s the d i s s e r t a t i o n .  Conclusion  There i s no s a t i s f a c t o r y p r i c e - s e n s i t i v e i n e q u a l i t y index i n the l i t e r a t u r e and t h e need f o r f i l l i n g t h i s gap S i n c e p r e f e r e n c e s must be i n v o l v e d l o g i c a l way  to proceed  ture,  i n the e v a l u a t i o n  urgent.  process, a  i s to estimate hypothesized preferences  b e h a v i o u r a l demand d a t a . b e i n g no o b j e c t i v e  i s evidently  V a r i o u s problems a r i s e , however.  measure o f w e l f a r e , no d a t a on i n d i v i d u a l  no a p r i o r i dominating  rule of interpersonal  There expendi-  comparison are  j u s t some o f the problems t o which the p r e s e n t r e s e a r c h has  offered  solutions.  A new constructed.  p r i c e - s e n s i t i v e i n e q u a l i t y index i s s u c c e s s f u l l y Implementation  r e s u l t s show t h a t  from  the approach i s  - 10  p r a c t i c a l and r e a s o n a b l e .  -  I t i s s i g n i f i c a n t l y s u p e r i o r t o the  o t h e r i n d e x e s i n b o t h i t s t h e o r e t i c a l f o u n d a t i o n and e m p i r i c a l usefulness.  - 11 -  Chapter 1  1.  Jorgenson and S l e s n i c k  a price-sensitive  Footnote  (1984) have attempted t o c o n s t r u c t  index, b u t f o r reasons t h a t w i l l be made c l e a r i n  Chapter 2, t h e i r approach i s not c o m p l e t e l y s a t i s f a c t o r y .  -  CHAPTER 2  12  -  SURVEY OF THE LITERATURE  From a g e n e r a l p e r s p e c t i v e , i n e q u a l i t y measurement i s a s t a t i s t i c a l e x e r c i s e t h a t i s not c o n f i n e d w i t h i n the realm o f w e l f a r e economics. wealths,  Given  a d i s t r i b u t i o n o f numbers  (could be incomes,  o r s i z e o f f i r m s ) a s t a t i s t i c i a n t y p i c a l l y a p p l i e s an i n -  e q u a l i t y formula t o map t h i s d i s t r i b u t i o n t o an index  number.  T y p i c a l examples o f such indexes a r e the G i n i c o e f f i c i e n t , the c o e f f i c i e n t o f v a r i a t i o n and t h e A t k i n s o n index.  Per s e , the index  number does not have any s i g n i f i c a n c e b e s i d e s r e f l e c t i n g  certain  mathematical c h a r a c t e r i s t i c s o f the d i s t r i b u t i o n .  The  p r e s e n t t h e s i s , on the o t h e r hand, are m a i n l y  concerned  w i t h i n e q u a l i t y i n the d i s t r i b u t i o n o f w e l f a r e among i n d i v i d u a l s i n society.  In g e n e r a l , t h r e e c o n s i d e r a t i o n s are c e n t r a l i n any approach  to economic i n e q u a l i t y measurement.  F i r s t l y , what should be the  b a s i c e n t i t y t h a t r e f l e c t s i n d i v i d u a l w e l l - b e i n g and how i s i t obtained e m p i r i c a l l y ?  Secondly,  s i n c e i n e q u a l i t y measurement r e s t s  on a f o u n d a t i o n o f s o c i a l w e l f a r e e v a l u a t i o n , what framework s h o u l d one  adopt t o summarize the d i s t r i b u t i o n t o o b t a i n a s o c i a l  measure?  T h i r d l y , what s o c i a l w e l f a r e f u n c t i o n ( c h a r a c t e r i z e d by a  s e t o f e t h i c a l axioms) s h o u l d be used t o aggregate and what i n e q u a l i t y index employed?  welfare  the d i s t r i b u t i o n  ( r e l a t i v e , a b s o l u t e o r o t h e r s ) s h o u l d be  - 13 -  Interestingly,  the methods i n the l i t e r a t u r e do not f o l l o w  t h i s l o g i c a l procedure. used.  S e c t i o n 1 below d e s c r i b e s the common e n t i t i e s  Not o n l y a r e they i n a p p r o p r i a t e as measures o f i n d i v i d u a l  w e l f a r e , they a r e a l s o p r i c e - i n s e n s i t i v e , traditional  which e x p l a i n s why t h e  indexes are a l l i n c a p a b l e o f i n d i c a t i n g  price effects.  distributive  S e c t i o n 2 c i t e s some e v i d e n c e o f d i s t r i b u t i v e  e f f e c t s and d e s c r i b e s t h e essence o f the M u e l l b a u e r  price  (1974) method  and the J o r g e n s o n - S l e s n i c k (1984) method which a r e u n s a t i s f a c t o r y attempts t o c a p t u r e t h e s e p r i c e  Section 1  Some simple  effects.  indexes  The most commonly used e n t i t y inequality  i s household  income.  i n the measurement o f economic  The r e a s o n f o r i t s widespread  u t i l i z a t i o n i s p r o b a b l y t h a t income .data are e a s i l y Household  available.  incomes a r e e a s i l y e x t r a c t e d from t a x r e t u r n s .  r e l a t i v e l y speaking, they a r e q u i t e r e l i a b l e i n a c c u r a c y However, household  income as a measure o f i n d i v i d u a l  Besides, terms.  utility is  s u b j e c t t o a number o f s e r i o u s o b j e c t i o n s .  (1) I n d i v i d u a l u t i l i t y , much on household position,  i n r e l a t i o n t o household  income, depends v e r y  s i z e and t o a l e s s e r e x t e n t on household com-  i . e . , the number o f male and female a d u l t s , male and  female c h i l d r e n  i n the h o u s e h o l d .  measure o f i n d i v i d u a l  U s i n g household  income as a  u t i l i t y p r a c t i c a l l y means r e g a r d l e s s o f  - 14 -  household s i z e , household income i n d i c a t e s t h e r a n k i n g between , any two households i n w e l f a r e terms.  (2) Consumers d e r i v e u t i l i t y receipts.  from consumption  fore, u t i l i t y  by s a v i n g and d i s s a v i n g .  v a r i a t i o n s come c l o s e r t o consumption  than income v a r i a t i o n s .  for  than income  While income stream may be uneven over time, consumers  tend t o smooth o u t consumption  section  rather  sample,  There-  variations  Furthermore, as o b t a i n e d from a c r o s s -  income v a l u e s a r e o f t e n  negative ( p a r t i c u l a r l y  o l d consumers) a r i s i n g from c a p i t a l l o s s e s .  These n e g a t i v e  numbers c r e a t e d i f f i c u l t i e s when aggregate s o c i a l w e l f a r e i s computed from i n d i v i d u a l  incomes.  In view o f t h e second o b j e c t i o n ,  the f i r s t  i n e q u a l i t y measure  t o be computed i n Chapter 7 f o r comparative purposes i s t h e househ o l d e x p e n d i t u r e index  (HEI).  To each h o u s e h o l d i s imputed  e x p e n d i t u r e and i n e q u a l i t y i s c a l c u l a t e d household expenditures. justifiable  based on t h e d i s t r i b u t i o n o f  Because o f t h e f i r s t  i n terms o f normal  i t s total  objection,  HEI i s not  s o c i a l e t h i c s , b u t i t i s worthwhile t o  check i f i n p r a c t i c e HEI d i f f e r s s i g n i f i c a n t l y from o t h e r measures.  One simple and n a t u r a l  way t o improve on the HEI i s by  denominating household e x p e n d i t u r e by h o u s e h o l d s i z e t o a r r i v e a t p e r c a p i t a expenditure.  The l o g i c a l way t o proceed then i s t o impute p e r  - 15  -  c a p i t a e x p e n d i t u r e t o each i n d i v i d u a l o f the household of  individual u t i l i t y .  as a measure  C u r i o u s l y , t h i s i s not what i s u s u a l l y done.  A t y p i c a l example i s Beach, Card and F l a t t e r s  (1981).  Although  they  use income i n s t e a d o f e x p e n d i t u r e , what they would have done w i t h e x p e n d i t u r e would be t o impute per c a p i t a e x p e n d i t u r e t o each househ o l d r a t h e r than each i n d i v i d u a l , which i s a g a i n u n j u s t i a b l e i n terms of all  social ethics.  To the e x t e n t t h a t i n d i v i d u a l s c o n s t i t u t e  i n d i v i d u a l w e l f a r e s s h o u l d have i d e n t i c a l weights  f a r e a g g r e g a t i o n and not weights  society,  in social  t h a t v a r y w i t h household  size.  welFor  example, i f the s o c i a l w e l f a r e f u n c t i o n i s a d d i t i v e , such as mean o f o r d e r r , each p e r s o n i n an n-person household  bears a weight o f  1/n  as opposed t o 1.  T h e r e f o r e the a c c e p t a b l e way t u r e as a measure o f u t i l i t y society.  o f imputing per c a p i t a  i s t o impute i t t o each i n d i v i d u a l i n  T h i s g i v e s r i s e t o t h e per c a p i t a e x p e n d i t u r e  which i s the second i n Chapter  (PCE)  i n e q u a l i t y index computed f o r comparative  approach,  i n d i v i d u a l u t i l i t y u s i n g household Wolfson  as a method o f  to  purposes  approximating  e x p e n d i t u r e , has been s u b j e c t t o  (1979) p o i n t s out t h a t t h i s method i g n o r e s  economies o f s c a l e i n the consumption o f c a p i t a l s e r v i c e s . is,  index  7.  The p e r - c a p i t a  criticisms.  expendi-  he s u g g e s t s , t o use denominate household  A better  "adult equivalents" i n place of family size expenditure.  way  - 16 -  as f o l l o w s i  H i s method can be i l l u s t r a t e d t o t a l e x p e n d i t u r e o f household h. l i n e f o r household h . which i s imputed inequality  1  Let  L e t y, be the h  be the "low income c u t - o f f "  The w e l f a r e r a t i o , w , i s the r a t i o , y./Ln n n  t o household h as a measure o f u t i l i t y .  index i s r e l a t i v e  (i.e.,  I f the  i t i s homogeneous o f degree 0 i n  the arguments), then t h i s w e l f a r e r a t i o approach i s i d e n t i c a l t o u s i n g h y, / - — , where L i s the "low income c u t - o f f " l i n e h L o L  for a reference,  q  say, one-adult-male, household.  L, /L can be r e g a r d e d as the number n o  h o f e q u i v a l e n t - a d u l t s i n household h and y, / - — n L o L  welfare ratio' . 1  i s named  "inflated  Since the c u t - o f f values t y p i c a l l y e x h i b i t s  o f s c a l e , t h i s method i s an easy way  to capture these scale  Wolfson's method i s a l s o s u b j e c t t o the c r i t i c i s m w a i f a r e r a t i o s a r e imputed t o households r a t h e r than In Chapter 7,  effects.  that  individuals.  i t w i l l be demonstrated t h a t t h i s m i s - s p e c i f i c a t i o n does  make a s i g n i f i c a n t d i f f e r e n c e i n i n e q u a l i t y measurement. Chapter t h e t h i r d t o households  economies  In t h a t  i n d e x , which uses i n f l a t e d w e l f a r e - r a t i o s  (HIWR), i s s i g n i f i c a n t l y d i f f e r e n t from the  imputed  fourth  2 index which imputes  to i n d i v i d u a l s  (IIWR).  Furthermore, u s i n g w e l f a r e - r a t i o t o r e p r e s e n t i n d i v i d u a l utility  i s u n s a t i s f a c t o r y f o r t h r e e r e a s o n s even though  an improvement o v e r the  ,:  per c a p i t a " method.  i t i s already  F i r s t l y , w h i l e "economies  - 17  of  scale"  are  -  i n c o r p o r a t e d , i t has  economies o f s c a l e  i s the  been assumed t h a t the  same f o r a l l goods and  services.  u n r e a l i s t i c because, i n t u i t i v e l y , c a p i t a l s e r v i c e s and  transportation  degree o f  such as  This  housing  s h o u l d e x h i b i t h i g h e r degree o f economies o f  than consumption goods l i k e  food and  clothing.  d e f i n i t i o n of poverty i s c o n t r o v e r s i a l .  Secondly,  is  scale  the  Based on d i f f e r e n t d e f i n i t i o n s  o f p o v e r t y , t h e r e e x i s t t h r e e s e t s o f low-income c u t - o f f l i n e s i n Canada and  t h e r e i s no  s e t over the o t h e r two. r a t i o method  dominating e t h i c a l reason f o r p r e f e r r i n g (see Osberg  (1981)).  (as n o r m a l l y used) does not  by  Thirdly, t h i s welfare-  give r i s e to a p r i c e - s e n s i t i v e  i n e q u a l i t y measure which aims a t c a p t u r i n g In Canada, the  distributive price effects.  commonly used low-income c u t - o f f s are  S t a t i s t i c s Canada.  which does not  This  a f f e c t the  any  set i s revised  those p u b l i s h e d  annually only for  f i n a l i n e q u a l i t y measure i f the  inflation  index i s  3 relatxve,  i . e . , mean-independent.  non-relative, that  i t i s not  c l e a r why  i s c a l c u l a t e d based on  Section  2  On  the  o t h e r hand, i f the  i n f l a t i o n should a f f e c t  index i s  inequality  expenditure.  Distributive price  effects  I t i s somewhat o b v i o u s t h a t  r e l a t i v e p r i c e changes have  d i s t r i b u t i v e p r i c e e f f e c t s . ' In the Canadian c o n t e x t , a r e c e n t  attempt  t o study the w e l f a r e e f f e c t s o f p r i c e changes i s c o n t a i n e d i n Roberts the  (1982).  The  main o b j e c t i v e  e f f e c t o f food p r i c e changes on  o f h i s study i s t o  investigate  c o s t - o f - l i v i n g indexes o f  - 18  households  -  i n f i v e income q u i n t i l e s .  He uses f a m i l y e x p e n d i t u r e  survey d a t a f o r f i v e s e p a r a t e y e a r s t o e s t i m a t e a l i n e a r system o f e i g h t goods and  expenditure  s e r v i c e s f o r each income q u i n t i l e .  r e g r e s s i o n , p e r - c a p i t a e x p e n d i t u r e i n s t e a d o f household i s used t o a d j u s t f o r f a m i l y s i z e .  In the  expenditure  Exact c o s t - o f - l i v i n g indexes are  then computed  (being r a t i o s o f the minimum e x p e n d i t u r e t o a t t a i n  given u t i l i t y  l e v e l under two p r i c e s i t u a t i o n s )  income q u i n t i l e s .  The b a s i c f i n d i n g  tends t o i n c r e a s e the c o s t - o f - l i v i n g  f o r each o f the  a five  i s t h a t , food p r i c e i n f l a t i o n index f o r the lowest income  q u i n t i l e more than the h i g h e s t income q u i n t i l e .  S i n c e food  accounts  f o r a h i g h e r percentage o f t o t a l budget i n the poor, r e l a t i v e t o the rich, this finding interest  i s not s u r p r i s i n g a t a l l .  i s : the c o r o l l a r y  changes have a d e f i n i t e  of t h i s r e s u l t  What one  notes w i t h  i s that r e l a t i v e  impact on i n e q u a l i t y .  price  U n f o r t u n a t e l y , none  o f the indexes d e s c r i b e d so f a r i s c a p a b l e o f measuring  this  impact  because they are a l l c a l c u l a t e d based on income and e x p e n d i t u r e o r simple adjustments  There  o f income and  are two  capture p r i c e e f f e c t s ,  s t u d i e s i n the l i t e r a t u r e which attempt namely, M u e l l b a u e r  and S l e s n i c k  (1984).  attempts  unsatisfactory.  are  expenditure.  (1974, a,b,c) and  to  Jorgenson  However, as e x p l a i n e d i n the f o l l o w i n g ,  both  - 19 -  M u e l l b a u e r ' s method  Muellbauer e f f e c t s and a d j u s t  (1974 a,b,c)  attempted t o c a p t u r e r e l a t i v e p r i c e  income f o r f a m i l y s i z e and economies o f s c a l e i n  one c o h e r e n t model.  His method i s summarized  a household u t i l i t y  as f o l l o w s .  He  specifies  f u n c t i o n which has the image I  (2.1)  where  u = U(x/m, 1  (x,, I  n  of equivalent and S t a r k  , x  n  /m)  , x ) are h o u s e h o l d consumption and m i s t h e number  a d u l t s , taken a r b i t r a r i l y  (1972).  from P r e s t and S t a r k  The numbers are  Family size m  1 1  2 1.6  3 2.1  4 2.5  5  6  2.8  3.2  7 3.6  which as a sequence, shows economies o f s c a l e i n consumption. f o l l o w s from (2.1) t h a t the image o f t h e i n d i r e c t u t i l i t y  (2.2)  u = V(p , 1  , p , n  y/m)  and t h a t o f the c o s t f u n c t i o n i s  r  (2.3)  (1967)  y = C(u, mp^,  m  P ) n  8 4.0  It  function i s  - 20 -  Muellbauer  (1974 a,b,c) employs an a d a p t a t i o n o f the money-metric  u t i l i t y o f Samuelson  (1974) t o r e p r e s e n t household u t i l i t y .  concept i s f u r t h e r d i s c u s s e d i n Chapter utility  3, but b r i e f l y ,  i s the income t h a t enables an i n d i v i d u a l  This  money-metric  (a household  i n the  present context) to a r r i v e at a given l e v e l o f u t i l i t y  at reference  prices.  however, i s  M u e l l b a u e r ' s v e r s i o n o f money-metric u t i l i t y ,  r e p r e s e n t e d , f o r household h, by  ~h _,h oo y =C(u,mp ,  (2.4)  ,m  1  f,,, = C[V(p ,  p ,  1  where y  n  oo. p ) n  i s income o f household h, m  a d u l t s and m ° i s t a k e n as u n i t y ,  o\ p J  h . h. o y /m ), p±,  n  i s the number o f e q u i v a l e n t  (the number o f e q u i v a l e n t a d u l t s o f  4 a one-person  household).  Given a s o c i e t y o f H households,  suggests c a l c u l a t i n g money-metric u t i l i t y  f o r each household  computing  i n e q u a l i t y based on the d i s t r i b u t i o n  (2.5)  (y , 1  Muellbauer and  y ). H  T h i s method i s not e n t i r e l y s a t i s f a c t o r y , a l t h o u g h the index based on  (2.5) i s p r i c e - s e n s i t i v e .  Firstly,  a d u l t s i s taken from a s e p a r a t e s t u d y . representation  the s c a l e o f e q u i v a l e n t  Since u t i l i t y  i s given a  (2.1), the s c a l e numbers s h o u l d be e s t i m a t e d i n one  pass t o g e t h e r w i t h o t h e r parameters  i n the u t i l i t y  function.  -  21  -  Furthermore, as argued above, assuming the  same degree o f economies  of scale  very  f o r a l l goods and  Secondly, i t i s not  services  c l e a r what  M u e l l b a u e r seems t o suggest the o f the  utility  (2.5)  t o the  means.  number i n  In p a r t i c u l a r ,  (2.2)  household.  t o t a l number o f  (2.5), and  i s a measure  But  i f this is  An o b j e c t i v e  dimension A  related  comparison o f u t i l i t y  measure o f u t i l i t y  M u e l l b a u e r ' s money-metric u t i l i t y numerical representation  expand the  individuals i n society.  i s the problem o f i n t e r p e r s o n a l  p l e t e l y ignored.  realistic.  s h o u l d impute money-metric u t i l i t y o f a household  to each member i n the household i n  issue  (2.4)  u t i l i t y o f each i n d i v i d u a l i n the  h i s i n t e n t i o n , he  of  i s not  (2.4)  does not  r e p r e s e n t s one  i s comexist.  particular  o f each i n d i v i d u a l ' s u t i l i t y which must imply  a c e r t a i n underlying rule of interpersonal  comparison.  This  assump5  t i o n must be made c l e a r l y known i n any  i n e q u a l i t y measurement model.  In f a c t , Samuelson's money-metric u t i l i t y the  case o f a s i n g l e i n d i v i d u a l o n l y .  m u l t i - p e r s o n s i t u a t i o n needs more  i s applicable  d i r e c t l y to  E x t e n d i n g t h i s concept to  a  justification.  J o r g e n s o n - S l e s n i c k method  A r e c e n t attempt t o c o n s t r u c t index can (1982  be  a,b,c),  translog  found i n Jorgenson and (1983  a,b)).  household u t i l i t y  a price-sensitive  Slesnick  Their basic  inequality  (1984).. (see  strategy  also  i s : they s p e c i f y  f u n c t i o n which i n c o r p o r a t e s  commodity-  a  - 22 -  s p e c i f i c equivalence among households  s c a l e s t o account  for attribute  differences  and e s t i m a t e the parameters u s i n g demand d a t a .  Based on these e s t i m a t e s , they attempt  t o use u t i l i t y  numbers i n a  n o n - w e l f a r i s t s o c i a l w e l f a r e framework t o a r r i v e a t a measure o f inequality.  The d i r e c t household  (2.6)  utility  u = u f x /m. (A) , 1 1  , x /m (A)) n n '  v  where  (A) ,  m^(A)  f u n c t i o n has the image  are the c o m m o d i t y - s p e c i f i c  equivalence  which are f u n c t i o n s o f a t t r i b u t e s A. I t f o l l o w s from indirect u t i l i t y  (2.7)  scales,  (2.6) t h a t the  f u n c t i o n has t h e image  u = vfm. (A)p., 1 1  ..., m  v  n  (A)p , y) n ^  Assuming t r a n s l o g p r e f e r e n c e s t h e y c l a i m t h a t g i v e n the parameters i n v o l v e d i n (2.7), a u t i l i t y  number i s o b t a i n a b l e f o r each  g i v e n i t s a t t r i b u t e s , p r i c e s and t o t a l e x p e n d i t u r e . number i s t a k e n as a measure o f household household  (2.8)  utility,  h, u t i l i t y i s  u  h  = vfm.(A )p., ^ 1 1 h  m  (A )p , y ) . n n h  h  This  household utility  such t h a t f o r  - 23 -  In the aggregate,  the s o c i a l w e l f a r e framework i s unorthodox.  S o c i a l w e l f a r e w i s g i v e n by a n o n - w e l f a r i s t s o c i a l e v a l u a t i o n functional i . e . ,  w = Ea, ( x ) U ( x ) - y(x) (la,, (x) |u (x) - -u , n \ h '  (2.9)  h  |Pjl/P  h  where x i s a s t a t e v a r i a b l e and  (2.10)  u  x  :=  Ea^(x)U (x) , h n h  I t should be emphasized t h a t  i s n o n - w e l f a r i s t because a, , n 6 h H and y a r e f u n c t i o n s o f x. U (x) i n (2.9) and (2.10)  h=l, i s taken as  (2.8)  (2.9)  even though the number u  has no c a r d i n a l  signifi-  7 cance.  (2.11)  The  form f o r a, (x) i s assumed t o be h  a. (x) = m (p, A )/Em (p, A* ) , where h o o n  1  h  c(u ,m h  (2.12)  m  (p, A ) =  (A )p  ,  h  , m  X  C (u ,p , 1  One  may  r e c a l l that  f a c t o r o f household  (A* ) , i = 1, 1  h f o r good i .  a r e f e r e n c e household  (2.13)  m (A°) = 1 ±  (A )p ) h  n  , p ) n  . .., n i s the e q u i v a l e n c e s c a l e I f one  normalizes  the f a c t o r s f o r  (having a t t r i b u t e A°) t o be u n i t y , i . e . ,  i = 1,  ... , n  - 24 -  then ^ ( p , A ) can be i n t e r p r e t e d as a g e n e r a l market e q u i v a l e n c e s c a l e which measures t h e f r a c t i o n o f t o t a l e x p e n d i t u r e keep household level.  h and the r e f e r e n c e household  These s c a l e s d i f f e r  f rom  required to  a t the same u t i l i t y  those i n (2.6) i n t h a t they a r e  f u n c t i o n s o f p r i c e s as w e l l as a t t r i b u t e s .  y(x)  (2.14)  i n (2.9) i s assumed t o have t h e form,  y(x) =  1 + ({Ea, (x) }/a. ( x ) )  -1/P p  1  3  h  -1/P =  ( l+  a.(x)  ^  where  1 _ P  3  ) J  a. (x) = min a, (x) . 3  k v  Substituting  (2.11) (2.14) i n t o  (2.9), the f i n a l  social  welfare function i s  (2.15)  w = u  1-P 1 + a. 3  Em (p, A o  Given  -1/P (inMp,  A  h  ) | u  h (  x  )  - u  x  |  P  )  /  )  f i x e d s o c i e t y e x p e n d i t u r e , one may f i n d t h e d i s t r i b u t i o n o f house-  h o l d e x p e n d i t u r e which maximizes w i n (2.15). p r e f e r e n c e s and t h e assumption imply each household  Because o f t r a n s l o g  (2.11), t h e f i r s t o r d e r c o n d i t i o n s  i s endowed w i t h t h e same " e q u i v a l e n t income",  - 25 -  i.e.,  f o r household  h and k  h  k  (2.16)  — m (p, A ) o  (2.17)  y /Y h  k  =  ^ — m (p, A ) o  h,k = 1,  H  so t h a t  = m (p, A )/m (p, A ) . o o h  k  At the maximum, two households w i t h d i f f e r e n t a t t r i b u t e s w i l l be g i v e n d i f f e r e n t incomes a c c o r d i n g t o t h e g e n e r a l market e q u i v a l e n c e s c a l e s and g i v e n same income i f t h e y have t h e same a t t r i b u t e s . t h i s p o i n t , t h e second term i n (2.15) v a n i s h e s  At  so t h a t maximum s o c i a l  w e l f a r e e q u a l s u^, and a l l households a r e regarded  as e q u a l l y w e l l - o f f .  L e t t i n g w be t h e a c t u a l s o c i a l w e l f a r e g i v e n i n (2.15), an i n e q u a l i t y index i s proposed, i . e . ,  (2.18)  i  JS  =  i  - /u w  x  and  0 < I  JS  < 1  T h i s approach t o i n e q u a l i t y measurement i s not a c c e p t a b l e . The s o c i a l w e l f a r e framework d e s c r i b e d above i s open t o c r i t i c i s m s . The e t h i c a l reason c l e a r , although procedure.  f o r a d o p t i n g a n o n - w e l f a r i s t framework i s not  t h e y c i t e Sen's argument  (1979) t o support  Sen's argument a g a i n s t w e l f a r i s m i s based on t h e l a c k o f  consideration o f absolute r i g h t s i n a w e l f a r i s t s o c i a l function.  their  These a b s o l u t e r i g h t s r e f e r t o  welfare  equal-work-for-equal-pay,  - 26 -  freedom from e x p l o i t a t i o n and nor  social liberty.  i s i t r e a s o n a b l e t o assume t h a t any  plicitly  s o c i a l welfare  function  (2.9)  a ^ ( x ) , i s ad hoc  and  f u n c t i o n t h a t depends  misses.  (2.11), the  adults" according (2.12) .  But  social  welfare  Each household's  a weight p r o p o r t i o n a l t o the e s t i m a t e d number o f t o the e s t i m a t e d g e n e r a l  market  a d u l t s " i s much s m a l l e r  e t h i c a l b a s i s i s not c l e a r .  The  because, as emphasized i n Chapter 1,  (2.15).  room f o r improvement.  set  This i s serious  i n e q u a l i t y measurement i s a  normative judgemental e x e r c i s e t h a t i s c o n t i n g e n t o f e t h i c a l axioms.  households.  authors f a i l to give a f u l l  o f axioms t h a t c o m p l e t e l y c h a r a c t e r i z e  set  than  i t i s u n j u s t i f i a b l e t o a s s i g n s m a l l e r weights t o  members i n l a r g e households r e l a t i v e t o members i n s m a l l The  equi-  In case o f s i g n i f i c a n t economies o f s c a l e i n  consumption the number o f " e q u i v a l e n t family s i z e .  the  s a t i s f y anonymity, i . e . , each i n d i v i d u a l i s not  i s assigned  valence scales  i s an  Furthermore, the way  e q u a l l y important i n the s o c i a l r a n k i n g .  "equivalent  ex-  f a r away from Sen's o r i g i n a l i n t e n t i o n .  By the d e f i n i t i o n o f a, (x), h f u n c t i o n does not  imply,  c a p t u r e s t h e s e c h a r a c t e r i s t i c s , through  I t does not c a p t u r e what w e l f a r i s m  utility  does not  on s t a t e c h a r a c t e r i s t i c s (hence n o n - w e l f a r i s t )  improvement over a w e l f a r i s t f u n c t i o n .  Y (x) and  Sen  on an  Consequently, t h i s approach has  underlying l e f t some  - 27  To summarize  -  t h i s c h a p t e r , we have seen t h a t a l l t h e i n d e x e s  d e v e l o p e d i n the l i t e r a t u r e a r e u n s a t i s f a c t o r y .  They are n o t con-  s i s t e n t w i t h t h e s o c i a l w e l f a r e view o f economic i n e q u a l i t y .  In  a d d i t i o n , t h e r e i s no i n d e x t h a t can demonstrate t h e d i s t r i b u t i v e impact o f p r i c e changes. required to f i l l  this  gap.  T h e r e f o r e , a new approach i s u r g e n t l y  - 28 -  Chapter  1.  2.  Footnotes  S t a t i s t i c s Canada p u b l i s h e s low-income c u t - o f f l i n e s t h a t  are s p e c i f i c t o household C a t . No.  2  s i z e and  s i z e o f area o f r e s i d e n c e .  13-207.  I n f l a t e d w e l f a r e - r a t i o i s used  i n s t e a d o f Wolfson's w e l f a r e -  r a t i o t o make comparison w i t h o t h e r measures more immediate. index i s r e l a t i v e ,  3.  The  i n e q u a l i t y i s not  t a k i n g the average  once every f i v e y e a r s . household  20% o f the budget more than the average  expenditure  They are e s t i m a t e d  income o f those households  and a r e a on the n e c e s s i t i e s —  the  affected.  S t a t i s t i c s Canada c u t - o f f s are d e r i v e d from  surveys conducted  As  household  by  t h a t spend  o f the same s i z e  food, c l o t h i n g and s h e l t e r .  The  "mark-up" o f 20% i s a r b i t r a r y .  4.  As e x p l a i n e d above i t would be more a p p r o p r i a t e t o  use  e x p e n d i t u r e i n s t e a d o f income f o r y .  5.  By c o n t r a s t , the i n t e r p e r s o n a l comparison assumption  i n the  p e r - c a p i t a e x p e n d i t u r e method and w e l f a r e - r a t i o method i s e a s i e r t o see.  - 29 -  6.  A s o c i a l evaluation  functional  i s welfarist  i f the states  a f f e c t s o c i a l r a n k i n g o n l y through t h e i r e f f e c t s on i n d i v i d u a l utilities.  7. the  Any monotonic t r a n s f o r m a t i o n o f a u t i l i t y same demand e q u a t i o n s .  Empirical  J-S  does not e s t a b l i s h an o b j e c t i v e  fail  to point  representation  yields  e s t i m a t i o n o n l y y i e l d s enough  information to allow a ranking o f a l t e r n a t i v e (2.7)  function  price-income  situations.  s c a l e o f u t i l i t y measurement.  out that the choice o f the p a r t i c u l a r  (2.7) i s somewhat a r b i t r a r y .  numerical  - 30 -  CHAPTER 3  A NEW APPROACH  T h i s chapter  describes  the t h e o r e t i c a l background o f t h i s  new approach t o i n e q u a l i t y measurement. i l l u s t r a t e d by t h e c h a r t below.  represented  assume preferences  by  The b a s i c s t r a t e g y can be  Based on h y p o t h e s i z e d  derive  direct/indirect utility  preferences,  demand/expenditure share e q u a t i o n s  function  and cost  function  identification parameters  U-  econometric procedure  empirical data  one  can o b t a i n a f u n c t i o n a l r e p r e s e n t a t i o n by the d i r e c t and i n d i r e c t  utility  functions o r the cost f u n c t i o n .  f u n c t i o n s , demand and e x p e n d i t u r e  From any one o f these  share e q u a t i o n s can be d e r i v e d .  s u i t a b l e econometric procedure can t h e n be d e v i s e d estimates equations.  f o r the parameters i n the demand and e x p e n d i t u r e These e s t i m a t e s  functions that represent  preferences.  measuring i n e q u a l i t y u s i n g ,  empirical  share  can a l s o be used t o i d e n t i f y the o r i g i n a l I f these i n f o r m a t i o n a r e  a v a i l a b l e f o r a l l consumers, a c a s u a l o b s e r v e r  utility  to obtain  A  might contemplate  say, the image o f each consumer's i n d i r e c t  f u n c t i o n , g i v e n p r i c e s and nominal t o t a l e x p e n d i t u r e  distribution.  - 31 -  However, t h e r e are s e v e r a l problems i n v o l v e d i n t h i s  general  The  s p e c i f i c a t i o n o f i n d i v i d u a l p r e f e r e n c e s need  allow  for  t a s t e d i f f e r e n c e s a r i s i n g from v a r i o u s demographic  scheme. 1.  characteristics.  Furthermore,  o n l y household  to  expenditure  d a t a are a v a i l a b l e , as opposed t o i n d i v i d u a l e x p e n d i t u r e T h e r e f o r e , assumed p r e f e r e n c e s have t o c h a r a c t e r i s t i c s and  (1) i n c o r p o r a t e these  (2) employ household  expenditures  r e a s o n a b l e manner i n o r d e r t h a t meaningful about the i n d i v i d u a l s i n the household  data.  in a  welfare information  can be r e v e a l e d .  S e c t i o n 1 o f t h i s c h a p t e r suggests an e q u i v a l e n c e s c a l e s method t h a t d e a l s w i t h these problems d i r e c t l y .  2.  I t i s well-known t h a t any a r b i t r a r y monotonic t r a n s f o r m o f the d i r e c t or i n d i r e c t u t i l i t y equations.  Consequently,  f u n c t i o n y i e l d s the same demand even i f p e r f e c t parameter e s t i m a t e s  are o b t a i n a b l e f o l l o w i n g the procedure  d e s c r i b e d above, the  images o f the d i r e c t o r i n d i r e c t u t i l i t y a r b i t r a r y as u t i l i t y  numbers.  f u n c t i o n are  In the c o n t e x t o f  inequality  measurement, t h i s a r b i t r a r i n e s s cannot be a l l o w e d and problem o f i n t e r p e r s o n a l comparison o f u t i l i t i e s dealt with e x p l i c i t l y .  S e c t i o n 2 suggests  this  purpose.  the  has t o be  that "equivalent  income" as d e f i n e d l a t e r , i s an a c c e p t a b l e measure o f for  still  utility  - 32 -  Given a d i s t r i b u t i o n  o f a c c e p t a b l e measures o f u t i l i t y , one  can attempt t o measure aggregate  s o c i a l welfare.  A social  welfare  e v a l u a t i o n framework, can be c o n s t r u c t e d based on a s e t o f e t h i c a l and i n f o r m a t i o n a l assumptions.  Described l a t e r i n Section 3 i s a wel-  f a r i s t framework t h a t i s argued t o be a p p r o p r i a t e i n the p r e s e n t context.  In t h i s framework, a s o c i a l w e l f a r e f u n c t i o n can be u t i l i z e d  to aggregate  individual u t i l i t i e s  t o a measure o f s o c i a l w e l f a r e which  then l e a d s t o t h e c o n s t r u c t i o n o f a r e l a t i v e  i n e q u a l i t y index i n  S e c t i o n 4.  Section 1  Equivalence  Scales  F o r m a l l y , t h e p r e f e r e n c e s o f a s o c i e t y o f H households can be 1 r e p r e s e n t e d by the u t i l i t y  H  functions U ,  , U , with the i n t e r -  pretation that (3.1)  ^  = U ( x h  h  )  i s the u t i l i t y o f each member o f h o u s e h o l d v e c t o r o f household  h."*"  h and x* i s t h e consumption 1  I t i s assumed t h a t s t r i c t  e x i s t s i n a l l the h o u s e h o l d s .  equality of u t i l i t y  I t i s a l s o assumed i n the f o l l o w i n g  t h a t d i f f e r e n c e s i n p r e f e r e n c e s among households can be c a p t u r e d by a v e c t o r A which d e s c r i b e s household  attributes.  F o r m a l l y , t h i s means  t h a t the p r e f e r e n c e s o f s o c i e t y can be r e p r e s e n t e d by  -  U( x  (3.2)  The  utility  (3.3)  i.e.,  1  ,A  ),  1  33  -  ., U (x  ,A  H  H  ) .  o f an i n d i v i d u a l i , who b e l o n g s  u  = U( x , A h  ±  the u t i l i t y  i s household  t o household  h i s therefore  )  h  common t o a l l members i n h o u s e h o l d  h.  Note t h a t x  consumption and t h e r e f o r e the s e a r c h f o r an a p p r o p r i a t e  form f o r U( x^,A^ ) , h = 1, the i n t e r p r e t a t i o n  , H i s c r u c i a l i n order to j u s t i f y S u b j e c t t o t h i s r e s e r v a t i o n , one  (3.3).  utility  f u n c t i o n can now be a p p l i e d t o a l l i n d i v i d u a l s o f a l l households i n society.^ o r d e r t o i n t e r p r e t u^ i n ( 3 . 3 ) as i n d i v i d u a l u t i l i t y , the  In  p r e s e n t approach adopts scales.  (1964)  With n goods, t h e u t i l i t y  g i v e n by, from  (3.4)  Barten  ),  m  ;  L  ( A  h  ) ,  , m (A n  s c a l e s f o r household  o f each member o f h o u s e h o l d  h is  ,  xJj/m (A )) h  n  ) are the commodity-specific  equivalence  h, so t h a t x^ / m ( A* ) i s the e q u i v a l e n t consump1  t i o n o f good i f o r h o u s e h o l d whose s c a l e f a c t o r s  ( A ° ), 1  h, r e l a t i v e t o a r e f e r e n c e  Suppose  household  , m ( A ° ) a r e n o r m a l i z e d t o be 1 . n  For example, l e t f a m i l y s i z e be the o n l y demographic d e s c r i b e d by A.  equivalence  (3.3),  u^ = u ( x J /  where m, ( A 1  commodity-specific  characteristic  - 34 -  (3.5)  m_.(A°)=m (A°)=l, k  j ,k = 1, . . ., n  where A ° d e s c r i b e s a one-person-household.  Then these e q u i v a l e n c e  s c a l e s can be regarded as f a c t o r s t h a t d e f l a t e h o u s e h o l d to  a r r i v e a t e f f e c t i v e i n d i v i d u a l consumption.  As f a m i l y  consumption size  i n c r e a s e s , the s c a l e f a c t o r s should i n c r e a s e t o r e f l e c t the i n c r e a s i n g need f o r each good.  The r a t e o f i n c r e a s e , b e i n g s p e c i f i c t o t h e good,  depends on the c a p a b i l i t y o f s e c u r i n g economies o f s c a l e i n consumption.  By way o f example, c l o t h i n g s h o u l d have s m a l l e r economies o f  s c a l e than h o u s i n g . Thus, consumption  (3.4) i s a g e n e r a l s p e c i f i c a t i o n t h a t maps h o u s e h o l d  and a t t r i b u t e s  (through m^,  .. . ., m ) n  to a u t i l i t y  number  which can r e a s o n a b l y be r e g a r d e d as t h e u t i l i t y o f each member o f t h e household.  S p e c i a l cases o f (3.4) i n c l u d e t h e "head-counting" method  and E n g e l ' s method adopted by M u e l l b a u e r  (1974 a, b, c ) .  The head-  c o u n t i n g case i s ,  (3.6)  m  j  =  •^-'--'-Y s i z e , m a  j = 1,  n  which does not a l l o w f o r economies o f s c a l e i n consumption,  w h i l e E n g e l ' s case i s , l e t t i n g A r e p r e s e n t f a m i l y  (3.7)  (A) = ^ ( A )  j ,k = 1,  , n,  size;  - 35  i.e.,  common e q u i v a l e n c e  -  scale across  goods, which does not  d i f f e r i n g degrees o f economies o f s c a l e among goods. s p e c i a l cases are good approximations of checked by l o o k i n g a t a c t u a l e m p i r i c a l  Obviously,  the  for  Whether these  general  case can  be  estimates.  these B a r t e n e q u i v a l e n c e  household c h a r a c t e r i s t i c s o t h e r  allow  s c a l e s can  than f a m i l y s i z e .  accommodate  Consequently, i n  the  implementation o f t h i s model, f o u r c h a r a c t e r i s t i c s are i s o l a t e d :  the  s i z e o f the area o f r e s i d e n c e ,  f a m i l y s i z e and  the age  sex o f the household head,  o f the household head.  However, what the s i z e i s the o n l y  the  s c a l e s mean now  r e l e v a n t a t t r i b u t e , the  i s not as c l e a r . structure of these  I f family scales  r e f l e c t s the d i f f e r e n t degrees o f economies o f s c a l e o f d i f f e r e n t goods.. But  what does i t mean i f the  s c a l e f a c t o r f o r , say,  for  r u r a l households i s h i g h e r  than t h a t f o r urban households?  means t h a t k e e p i n g e f f e c t i v e consumption household) o f o t h e r  goods the  transportation  ( r e l a t i v e t o some  w e l l o f f as  s p e c i f i c a t i o n o f the d i r e c t u t i l i t y  f o l l o w s t h a t the  indirect u t i l i t y  u, = V f m h 1 v  (A  function  f u n c t i o n must i n c o r p o r a t e  s c a l e s by mark-ups i n p r i c e s , whose image i s " ^  (3.8)  t o be  urban j u s t as  before.  Given the it  reference  same, a household moving from an  area t o a r u r a l a r e a needs more t r a n s p o r t a t i o n i n o r d e r  It  )p . . . . , m ( A 1 n  )p ,y ) n '  (3.4), the  - 36 -  where y* i s household h's t o t a l e x p e n d i t u r e .  The c o s t f u n c t i o n , C i s  1  obtained  by i n v e r t i n g the i n d i r e c t u t i l i t y  functions  (3.8) and s o l v i n g  h  for  y .  [3.9)  y  Section  h  = c ( u . , n. ( A n 1  2  Equivalent  h  )p , 1  ,m(A n  h  )p) n  Income  In the l i t e r a t u r e , money-metric u t i l i t y was i n t r o d u c e d i n Samuelson  (1974) and V a r i a n  i n an i n d i v i d u a l ' s u t i l i t y ,  (1980) t o i n d i c a t e the d i r e c t i o n o f change and i s v e r y c l o s e t o the concepts o f  compensating v a r i a t i o n and e q u i v a l e n t surplus  literature.  Let  U  be  a  direct utility  regularity conditions eliminate  —  f u n c t i o n s a t i s f y i n g the u s u a l  continuity, positive s t r i c t  s a t i a t i o n ) and  (3.10)  m o n o t o n i c i t y (to  quasi-concavity.  u = U(x)  The c o r r e s p o n d i n g i n d i r e c t u t i l i t y have  v a r i a t i o n i n the consumer  images  (3.11)  u = V(p,y),  (3.12)  y = C(u,p)  and  f u n c t i o n V and c o s t f u n c t i o n C w i l l  - 37 -  Money-metric u t i l i t y  (1974) as  M( x,p° ) : = C ( u ( x ),p°)  (3.13)  where p  i s d e f i n e d i n Samuelson  o  i s a reference p r i c e vector.  " Since the c o s t f u n c t i o n C i s  i n c r e a s i n g i n u, money-metric u t i l i t y e s t a b l i s h e s a s c a l e t h a t measures u t i l i t y the  as an income concept.  T h i s i s j u s t i f i a b l e because  same r e f e r e n c e p r i c e p ° i s used f o r a l l s t a t e s , so t h a t M i s  o r d i n a l l y e q u i v a l e n t t o U, r e g a r d l e s s o f t h e c h o i c e o f p ° ,  i . e . ,  (3.14)  U( x  1  ) > U( x  Empirically, as K i n g into  2  ) <-  ->-M(x p ) > M ( x , p ° 1  (3.13) i s d i f f i c u l t  (1983) suggests,  0  2  /  to handle.  Alternatively,  (3.11) i n s t e a d o f (3.10) can be s u b s t i t u t e d  (3.12) which g i v e s r i s e t o a s o - c a l l e d r e a l income f u n c t i o n ,  (3.15)  C(p,y,p°)  := £(V( p,y ) ,p°)  C maps p r i c e s and e x p e n d i t u r e ,  given a reference p r i c e vector, to a  r e a l number, t h e r e a l income.  I t s nature  it  as a s o l u t i o n t o  (3.16)  V( p ° , y  e  ) = V(p,y )  i s made c l e a r by r e g a r d i n g  - 38 -  so  that  y  (3.17)  = c ( v ( p , y ),p°)  Q  ~e o y i s the amount o f e x p e n d i t u r e a t p t h a t w i l l keep the consumer j u s t as w e l l o f f as i n the s t a t e  (p,y).  Since p, y are r e a d i l y o b s e r v a b l e  *e variables, y  i s a p r a c t i c a l measure o f u t i l i t y .  However, a l t h o u g h  ~e y i s exact  i n showing the d i r e c t i o n o f u t i l i t y change, i t i s a r b i o 4 t r a r y i n a b s o l u t e q u a n t i t y as a r e s u l t o f p i n (3.15).  So f a r , o n l y one p e r s o n i s i n v o l v e d . the i d e a o f money-metric u t i l i t y  In t h e p r e s e n t  i s adapted t o take i n t o  households t h a t are i d e n t i f i e d by an a t t r i b u t e v e c t o r A.  model,  consideration Equivalent  e income, y  w i l l be used t o measure u t i l i t y which can be viewed as  a solution to  (3.18)  so  V(p°,y ,A° ) = V(p,y,A) e  that,  (3.19)  y  6  =  c(v(p,y,A ) ,p°A°)  = : E ( p , y , A , p ,A  where p ° , A° are r e f e r e n c e household.  Notice  that  )  p r i c e s and a t t r i b u t e v e c t o r o f a  (3.19) i s not a s t r a i g h t f o r w a r d  reference  extension  - 39 -  of  (3.17).  While  (3.16) j u s t compares u t i l i t i e s o f one p e r s o n ,  (3.18) r e p r e s e n t s e x p l i c i t i n t e r p e r s o n a l comparison o f u t i l i t i e s e between two households,  y  i s t h e household e x p e n d i t u r e  make each i n d i v i d u a l i n t h e household d e s c r i b e d by  that  will  A ° a t p ° j u s t as  w e l l o f f as each i n d i v i d u a l i n the household d e s c r i b e d by A a t p. Notice a l s o that  (3.18) r e p r e s e n t s one p a r t i c u l a r l e v e l o f  i n t e r p e r s o n a l comparison. of u t i l i t y .  E m p i r i c a l l y , t h e r e i s no o b j e c t i v e measure  I f V i s found t o be c o n s i s t e n t w i t h demand  so i s any h o u s e h o l d - s p e c i f i c monotonic t r a n s f o r m o f V.  behaviour, Indeed, con-  sumers c o u l d "announce" t h e i r own l e v e l s o f u t i l i t y a c c o r d i n g t o t h e i r own s c a l e s o f measurement so t h a t "announced" u t i l i t i e s be compared i n t e r p e r s o n a l l y . i n u t i l i t y measurement. transforms  y  cannot  i s n o t immune t o t h i s a r b i t r a r i n e s s  By a l l o w i n g h o u s e h o l d - s p e c i f i c monotonic  on V, a d i f f e r e n t e q u i v a l e n t income measure c o u l d be  o b t a i n e d by s o l v i n g f o r y i n e  (3.20)  $ ( v ( p , y , A ) , A ° ) = $ (v ( p , y 0  e  0  h  ,A ) , A ) h  h  so t h a t  (3.21)  "*e y i s equal  y  e  = C $  (.HV(p,y  ,A ) ,A j ,A J ,p ,A  e t o y i f and o n l y i f A  the i n t e r p e r s o n a l comparison  v  and A u  are i d e n t i c a l .  Therefore,  (3.18) i s a key assumption i n t h i s  - 40 -  approach t o i n e q u a l i t y measurement. and r e a l i s t i c ?  Is t h i s assumption  justifiable  I t depends on the v a l u e s o f the c o m m o d i t y - s p e c i f i c  e q u i v a l e n c e s c a l e s , which are e s t i m a t e d  from demand d a t a .  example, c o n s i d e r f a m i l y s i z e as t h e o n l y c h a r a c t e r i s t i c . v a l e n c e s c a l e f o r s i z e 1 i s n o r m a l i z e d t o be 1. s c a l e s f o r s i z e 2 s h o u l d f a l l between 1 and 2. indirect u t i l i t y  For The e q u i -  Then t h e e q u i v a l e n c e In o t h e r words, the  f u n c t i o n (3.8) has t o p l a y f u l l y the r o l e o f making  i n t e r p e r s o n a l comparison  (3.18) p o s s i b l e .  In the subsequent model o f i n e q u a l i t y measurement, e q u i v a l e n t income, b e i n g a r e s u l t o f t h e i n t e r p e r s o n a l comparison be used as a measure o f i n d i v i d u a l u t i l i t y .  (3.18),  will  T h i s i s p o s s i b l e because  by the d e f i n i t i o n o f t h e e q u i v a l e n t income f u n c t i o n (3.19) and the fact that C i s increasing i n u t i l i t y , ~ *  (3.22) /  -<  T  -  I  -  -  >  E ( p ,y ,A ,p ,A  \  ,-,,  1  1 1 1 V ( p ,y ,A  •  where households  A  1  )  1,1  o  )  and A  2  face  >  E ( p ,y ,A ,p A  >  2 2 2 V ( p ,y ,A )  ,o  ,  1 1 ( p ,y  „,  •  2  2,2  )  o,o,  2 2 ) and ( p ,y ) r e s p e c t i v e l y  and V ( p , y , A ) i s the u t i l i t y o f each member i n household i . 1  1  1  A corollary i s i fA  1  2 i s s e t e q u a l t o A , then  E p r e s e r v e s each i n d i v i d u a l ' s u t i l i t y  ranking.  (3.22) i m p l i e s  One can r e a d i l y  t h a t E i s i n f a c t a monotonic t r a n s f o r m o f V and a p p l y i n g Roy's Identity w i l l  y i e l d the same s e t o f demand f u n c t i o n s .  verify  - 41 -  e More i n s i g h t i n y  can be g a i n e d by r e f e r r i n g t o Deaton's  (1980) i n t e r p r e t a t i o n ,  e y can be expressed as  (3.23)  C<u,p,A)  y  e  =  C( u,p,A° )  C( u,p,A )  C( u,p°,A° )  C( u,p,A° )  II( u,p,p°,A° )S( u,A,p,A  0  where  (3.24)  n( u,p,p°,A° ) : = C( u,p,A° ) / C ( u,p°,A° )  i s a p r i c e i n d e x e v a l u a t e d a t u and A ° , and  (3.25)  S( u,A,p,A° ) : = C( u,p,A )/C ( u,p,A° )  i s a market e q u i v a l e n c e s c a l e c l e a r that y p.  i s sensitive  are e q u a l l y  of a t t r i b u t e s . a function  (and t o each i n d i v i d u a l  i t i s a function  of u t i l i t y .  w e l l - o f f w i l l have t h e same p r i c e  and a t t r i b u t e s . ^  c a p t u r e some d i s t r i b u t i v e p r i c e  effects.  therein)  Two households  index regardless  However, t h e market e q u i v a l e n c e s c a l e  o f both u t i l i t y  I t i s now  t o p because b o t h II and S a r e f u n c t i o n s o f  II i s s p e c i f i c t o a household  o n l y t o the e x t e n t t h a t that  f a c t o r e v a l u a t e d a t u and p.  S i s i n general  T h e r e f o r e , both II and S  - 42 -  To c o n c l u d e : e q u i v a l e n t i n s o c i a l welfare  evaluation.  income y  i s used as a u t i l i t y  The u t i l i t i e s  measure  of a l l individuals i n  a l l households are measured by a common y a r d s t i c k , namely, the t o t a l expenditure  that w i l l  keep each member o f a r e f e r e n c e household  as w e l l o f f , a t r e f e r e n c e  prices.  Therefore,  an e q u i v a l e n t  should be imputed t o each i n d i v i d u a l i n s o c i e t y .  just  income  In a s o c i e t y o f H  households and N i n d i v i d u a l s , H < N, the d i s t r i b u t i o n o f u t i l i t i e s for  welfare  (3.26)  e v a l u a t i o n purposes w i l l be  Cy®,  , y®)  e Since y i s p r i c e - s e n s i t i v e , t h e s o c i a l w e l f a r e  Section 3  indicator will  be a l s o .  S o c i a l Choice  This section introduces  the s o c i a l welfare  evaluation  frame-  work t h a t maps the d i s t r i b u t i o n o f i n d i v i d u a l e q u i v a l e n t incomes t o a s o c i a l welfare  number.  This  framework forms t h e b a s i s o f i n e q u a l i t y  measurement.  S o c i a l welfare problem.  A profile  e v a l u a t i o n can be l o o k e d  a t as an  i s a vector o f i n d i v i d u a l u t i l i t y  d e f i n e d over a s e t o f s o c i a l s t a t e s .  aggregation  functions  A social evaluation functional  i s then a mapping from such a p r o f i l e t o a s o c i a l o r d e r i n g o v e r the same s e t o f s t a t e s .  I n d e r i v i n g such a s o c i a l o r d e r i n g ,  two s e t s o f  - 43 -  assumptions are u s u a l l y i n v o l v e d .  The  f i r s t s e t i n v o l v e s the  axioms t h a t are argued as j u s t i f i a b l e and the weak Pareto  measurability  The and  the s o c i a l o r d e r i n g must rank s t a t e A  second s e t o f assumptions are the assumptions  interpersonal comparability  of u t i l i t i e s .  searches f o r s o c i a l w e l f a r e  r u l e s , forms t h a t can be  c h a r a c t e r i z e d by axioms are  favoured.  In w e l f a r e  e t h i c a l b a s i s s h o u l d be c l e a r , o t h e r w i s e , information  i s about i n d i v i d u a l w e l f a r e ,  the aggregate i s m e c h a n i c a l and  L e t T be f u n c t i o n s and (D b e i n g set  i.e.,  R  where U utility  k  u  one  completely  evaluation,  no matter how  valid  the the  the e v a l u a t i o n procedure i n  unjustifiable.  D the domain o f f - the s o c i a l e v a l u a t i o n f u n c t i o n a l o f T ) , RR  o f a l t e r n a t i v e s , X.  f :D  When  on  the s e t o f a l l p o s s i b l e p r o f i l e s o f i n d i v i d u a l u t i l i t y  a subset  (3.27)  For example,  r u l e i s commonly assumed, i . e . , i f e v e r y i n d i v i d u a l  p r e f e r s s t a t e A t o s t a t e B, above s t a t e B.  acceptable.  ethical  the  set o f a l l p o s s i b l e orderings  over  Then  > RR  = f(U  1  ,  i s i n d i v i d u a l k's  ,U  utility  N  ),  f u n c t i o n and  i n a p a r t i c u l a r s t a t e x i n X.  a s s o c i a t e d with  the p r o f i l e  (U  1  R^  , ...... U  i s the N  U  k  (x) i s h i s o r  social  ) , through f .  ordering  her  the  - 44 -  The f o l l o w i n g t h r e e axioms on f are commonly c a l l e d the "welfarism"  (1)  axioms,  U n r e s t r i c t e d Domain  T = D  (2)  Pareto I n d i f f e r e n c e  Let I  be the symmetric f a c t o r o f R . u  u  If  U(x) = U(y) ,  where  then  (3)  U(x): =  X  ^  V L  '  f  o  r  (u (x) ,  ., U ( x ) ) ,  1  a l  -  L  x  #  N  Y  i  n  x  a n  d a l l U i n D.  B i n a r y Independence o f I r r e l e v a n t A l t e r n a t i v e s  F o r a l l x, y i n X ;  U', U" i n D, i f  U'(x) = U"(x) and U'(y) = U " ( y ) , then R . and R „ must c o i n c i d e on u u  These w e l f a r i s m axioms important  v  ( x, y ) . '  (Blackorby, Donaldson and VJeymark (1983)) are  because they imply  strong n e u t r a l i t y  (SN), d e f i n e d as f o l l o w s :  - 45 -  Strong  Neutrality  For a l l w, x, y, z i n X and U', U" i n D, i f U'(x) = U"(w), u'(y) = U " ( z ) , xR ,y •* u  This property utilities  then  >- wR „z and yR ,x •* u u  i s very  strong.  »• zR „w u  In i t s e l f ,  i t means i n d i v i d u a l s '  are the o n l y d e t e r m i n a n t s o f s o c i a l w e l f a r e .  Anything that  can a f f e c t s o c i a l o r d e r i n g has t o "pass through" u t i l i t i e s . t h i s i n t e r p r e t a t i o n t h a t g i v e s t h e name " w e l f a r i s m " axioms.  (see Sen (1977)).  non-welfarist (1983  It i s  t o the t h r e e  T h i s framework c o n t r a s t s s h a r p l y w i t h the  framework o f Jorgenson and S l e s n i c k  a, b) where t h e s o c i a l w e l f a r e  (1982 a, b, c)  evaluation functional involves  parameters a and y t h a t are b o t h f u n c t i o n s o f s t a t e x.  I t can be shown t h a t a w e l f a r i s t f i m p l i e s and i s i m p l i e d by N the e x i s t e n c e o f an o r d e r i n g R on the r e a l E u c l i d e a n  space R  such  that (3.28)  It  xR y « u  • U(x)R U(y)  i s now p o s s i b l e t o p a r t i t i o n the s e t o f a l t e r n a t i v e s , X i n t o  s o c i a l l y i n d i f f e r e n t s e t s by r e f e r r i n g o n l y t o u t i l i t y  numbers.  An a d d i t i o n a l c o n t i n u i t y assumption on s o c i a l p r e f e r e n c e s ,  (namely  - 46 -  t h a t the " s o c i a l l y a t l e a s t as good a s " and " s o c i a l l y a t most as good as" s e t s are c l o s e d i n R^) w i l l  provide  f o r the e x i s t e n c e o f a 7  r e p r e s e n t i n g f u n c t i o n W, g e n e r a t i n g t h e same o r d e r i n g as R.  Wis  commonly r e f e r r e d t o as t h e Bergson-Samuelson s o c i a l w e l f a r e f u n c t i o n .  Sen has r a i s e d o b j e c t i o n s a g a i n s t w e l f a r i s m as an e v a l u a t i o n framework.  I f w e l f a r i s m i s assumed, i t i s n a t u r a l t o assume weak  Pareto as w e l l s i n c e o n l y u t i l i t i e s  determine s o c i a l o r d e r i n g .  In  some c a s e s , t h i s d e n i e s i n d i v i d u a l s a b s o l u t e r i g h t s a s s o c i a t e d w i t h such t h i n g s as freedom from e x p l o i t a t i o n and "equal work f o r e q u a l pay" which r e f e r t o s t a t e c h a r a c t e r i s t i c s n o t c a p t u r e d by u t i l i t i e s . c a l l e d these non-welfare  characteristics.  and weak P a r e t o , non-welfare  However, under w e l f a r i s m  c h a r a c t e r i s t i c s have no r o l e t o p l a y i n  d e t e r m i n i n g the s o c i a l o r d e r i n g .  It  Sen  Q  i s c l e a r t h a t t h e c h o i c e between w e l f a r i s m and non-  w e l f a r i s m depends on t h e type o f a n a l y s i s .  I n p o l i c y q u e s t i o n s where  p r i c e s and income a r e p a r t l y p o l i c y v a r i a b l e s , w e l f a r i s m i s adequate. Questions  l i k e the impact  o f t a x and t a r i f f  changes on economic  i n e q u a l i t y can be s e n s i b l y asked w i t h i n t h i s framework.  Welfarism i n  p r a c t i c e a l l o w s easy e s t i m a t i o n o f w e l f a r e i n d i c a t o r s s i n c e a l l t h a t is  r e q u i r e d i s computing i n d i v i d u a l measures o f u t i l i t y  p r i c e and income q u a n t i t i e s . of  non-welfare  from measurable  In t h i s type o f a n a l y s e s , i n c o r p o r a t i o n  c h a r a c t e r i s t i c s i s not r e l e v a n t .  9  - 47 -  Let W be a s o c i a l w e l f a r e f u n c t i o n equivalent  (3.29)  incomes; we measure  defined  on i n d i v i d u a l  s o c i a l w e l f a r e as  w  where N i s the number o f i n d i v i d u a l s i n s o c i e t y .  I t s h o u l d be noted  t h a t W i s not a Bergson-Samuelson s o c i a l w e l f a r e f u n c t i o n .  The o r d e r -  i n g generated by W depends on p ° and i n g e n e r a l on A ° as w e l l . Bergson-Samuelson f u n c t i o n also that  the e q u i v a l e n t  As they are f u n c t i o n s and  4  Inequality  This  section  incomes i n (3.29) are p o s i t i v e r e a l  o f p r i c e s , the s o c i a l o r d e r i n g  t h i s forms t h e b a s i s  Section  does n o t a l l o w t h i s a r b i t r a r i n e s s .  The Note  numbers.  depends on p r i c e s ,  of a price-sensitive inequality  index.  Measurement  d e s c r i b e s how a summary i n e q u a l i t y measure i s  computed u s i n g a d i s t r i b u t i o n o f i n d i v i d u a l e q u i v a l e n t The s o c i a l w e l f a r e f u n c t i o n  defined  anonymous mean o f o r d e r r . . ^  on  following  i n d e x i s then c o n s t r u c t e d the Atkinson-Kolm-Sen  (AKS) p r o c e d u r e .  The AKS i n d e x i s a c t u a l l y an A t k i n s o n i n d e x o f  equivalent  inequality.  income  (3.26).  (3.26) i s assumed t o be  An i n e q u a l i t y  from t h i s s o c i a l w e l f a r e f u n c t i o n ,  incomes,  - 48 Given a s o c i a l w e l f a r e  N f u n c t i o n W d e f i n e d on R , a typical e  element b e i n g a v e c t o r o f i n d i v i d u a l  e q u i v a l e n t incomes  ethically-indifferent-evenly-distributed  e  ( y^, . , . , y ^ ) ,  e q u i v a l e n t income £ can be  i m p l i c i t l y d e f i n e d as i n (3.30)  where  W( ? i ) = W( y®,  i : = (1,  )  , 1 ) , an N - v e c t o r .  Explicitly,  £ i s hence  d e f i n e d as  (3.31)  £ := E(  ,  )  £ i s t h a t l e v e l o f e q u i v a l e n t income which i f commanded by every individual  w i l l be e t h i c a l l y i n d i f f e r e n t  to the a c t u a l  F o l l o w i n g t h e AKS p r o c e d u r e , an i n e q u a l i t y index  (3.32)  i s then d e f i n e d as  I := 1 - £/y  N where  distribution.  Q  u = ( 1/N ) £ y , k=l k  i s mean e q u i v a l e n t income.  An i n e q u a l i t y index  i s a relative  i f i t i s mean-independent, i . e . , homogeneous o f degree 0. t o v e r i f y t h a t the AKS index 11 thetic.  index  I t i s easy  i s r e l a t i v e i f and o n l y i f W i s homo-  - 49 -  For N = 2, t h e p r e s e n t procedure e q u i v a l e n t income space. tribution income.  In t h e f o l l o w i n g diagram,  i s A where i n d i v i d u a l E i s the e g a l i t a r i a n  t o A.  the a c t u a l  situation  where each i n d i v i d u a l  dis-  enjoys  I f W(.) i s assumed t o be symmetric  (as drawn), then E i s unambiguously e t h i c a l l y  In f a c t ,  preferred  the same l e v e l o f s o c i a l w e l f a r e W can be a t t a i n e d by  a lower combined e q u i v a l e n t income a t p o i n t E . indifferent  d e p i c t e d i n the  1 enjoys a h i g h e r l e v e l o f e q u i v a l e n t  the mean o f t h e d i s t r i b u t i o n . quasi-concave  i s easily  E, b e i n g  t o A, i s c h a r a c t e r i z e d by E, as i n (3.30).  ethically  I t i s easy t o  see t h a t E, can be r e g a r d e d as a measure o f s o c i a l w e l f a r e and H(-) i n (3.31) i s o r d i n a l l y  e q u i v a l e n t t o W(-) i n (3.30).  measurement procedure  adopted  here, s i m i l a r  The i n e q u a l i t y  t o t h e AKS p r o c e d u r e ,  makes use o f t h e d i s c r e p a n c y between E and E .  The i n e q u a l i t y  index  i s d e f i n e d as t h e s h o r t f a l l o f E, r e l a t i v e t o u e x p r e s s e d as a p e r centage  of u  as i n (3.32).  G e o m e t r i c a l l y , I can be e x p r e s s e d i n  terms o f d i s t a n c e s ,  I =  d( 0,E ) - d ( 0 , E ) d(0,E )  To implement t h i s p r o c e d u r e , necessary.  a specific  I t i s assumed t h a t W i s a symmetric mean o f o r d e r r t  function, i . e . ,  (3.33)  form f o r W(-) i s  V  N  }  =  $  ( * W  (y  1'  - 50 -  - 51  -  where  *  (3.34)  W( y  r  N  and  e . 1/N  r = 0  1  ...,y^ are d e f i n e d as t o t a l e x p e n d i t u r e , they are p o s i t i v e  as o b t a i n e d from a survey sample.  By  (3.23), i t f o l l o w s t h a t  e ....,y  are a l s o p o s i t i v e .  The mean o f o r d e r r f u n c t i o n N  (3.34) has d e s i r a b l e p r o p e r t i e s on R , ++  a d d i t i v e l y s e p a r a b l e , homothetic i s an o b v i o u s requirement present context. it  0  $ ( • ) > 0.  Since y^,  e y^,  f  f o r any  namely, i t i s a c o n t i n u o u s ,  and symmetric f u n c t i o n .  s o c i a l w e l f a r e f u n c t i o n i n the  (the i n f l u e n c e o f ) i n d i f f e r e n t  i . e . , the r a n k i n g o f any two  s t a t e s s h o u l d be independent  utility  l e v e l s enjoyed by the i n d i v i d u a l s who  the two  states  (see d'Aspremont and Gevers  (1982)).  Continuity  A d d i t i v e s e p a r a b i l i t y i s e t h i c a l l y d e s i r a b l e because  implies "elimination of  Donaldson  (3.33),  Homotheticity  r e l a t i v e , the importance  o f the  are i n d i f f e r e n t between  (1977), B l a c k o r b y  ensures t h a t the i n d e x  o f which w i l l be e x p l a i n e d l a t e r .  symmetry i m p l i e s "anonymity" which i s an e s s e n t i a l e t h i c a l i n i n e q u a l i t y measurement.  individuals",  and  (3.32) i s Finally, requirement  - 52 -  H i s t o r i c a l l y , the Lorenz on i n e q u a l i t y measurement.  c r i t e r i o n has a profound i n f l u e n c e  (see Sen  (1973)).  One would l i k e the  p r e s e n t i n e q u a l i t y index t o be c o n s i s t e n t w i t h i t , tribution y  eA  i . e . , i f the d i s -  i s L o r e n z - s u p e r i o r t o the d i s t r i b u t i o n y  s h o u l d be no g r e a t e r than I ( y  e B  ).  1  eB eA , then I ( y )  A s u f f i c i e n t condition i s that  2  the i n e q u a l i t y index i s S-convex, which r e q u i r e s r < 1 i n (3.34). Based on the mean o f o r d e r r f u n c t i o n (3.33) and i n d i f f e r e n t e q u i v a l e n t income i s e a s i l y computed,  (3.35)  £ = (( 1/N ) E ( y f )  r  )  1  A  ,  (3.34), 13  ethically-  i.e.,  r < 1 ,  r  f  0  N  =  , e l/N ( y ) k=l * (  T  Substituting  ,  (3.35) i n t o  r = 0  (3.32) y i e l d s a r e l a t i v e  inequality  index,  (3.36)  I  := 1 - (( 1/N ) E ( y ^ / y ) ) r  1  /  r  N :=l-TT(y/y) k=l  r < 1 ,  r ^ 0  r = 0  k  N where u = ( 1/N ) E y , and N i s the number o f i n d i v i d u a l s . k=l k  a c t u a l l y an A t k i n s o n i n d e x v a l e n t incomes.  In Chapter  (see A t k i n s o n  I  is  r  (1976)) on i n d i v i d u a l e q u i -  6 and 7, t h i s i n e q u a l i t y index, which i s  s e n s i t i v e t o p r i c e s , i s e s t i m a t e d f o r Canada.  - 53 -  Before  c l o s i n g t h i s c h a p t e r , i t i s important  t o emphasize t h e  d i f f e r e n c e between r e l a t i v e and a b s o l u t e i n d i c e s , and t h e j u s t i f i c a t i o n f o r a d o p t i n g t h e former r a t h e r than t h e l a t t e r i n t h e p r e s e n t context.  In c o n t r a s t w i t h a r e l a t i v e  index which i s i n v a r i a n t t o a  common r a t i o - s c a l e t r a n s f o r m on a d i s t r i b u t i o n , an a b s o l u t e index i s i n v a r i a n t t o a common t r a n s l a t i o n - s c a l e t r a n s f o r m , meaning t h a t  adding  the same q u a n t i t y t o each i n d i v i d u a l ' s e q u i v a l e n t income does n o t a f f e c t an a b s o l u t e index.  (3.37)  A( y  S  ) := y( y  F o r example, t h e p e r - c a p i t a index  6  ) - H( y ) e  i s an a b s o l u t e index i f W i s t r a n s l a t a b l e . e procedure drawback.  i n t r o d u c e d t o compute A ( y ) .  14  One c o u l d adopt the  But t h e r e i s one s e r i o u s  One can r e c a l l t h a t e q u i v a l e n t income as d e f i n e d i n (3.19)  i s sensitive to p°.  Since t h e f u n c t i o n E i s HD 1 i n p , y , p ° , i t  f o l l o w s t h a t measuring p,y,p° i n a d i f f e r e n t c u r r e n c y c o n s t i t u t e s a r e s c a l i n g o f e q u i v a l e n t income by an exchange r a t e f a c t o r .  However,  s i n c e t h e m u l t i p l e i s common among i n d i v i d u a l s a r e l a t i v e index, as.I  (such  i n (3.36)), i s immune t o t h i s type o f r e s c a l i n g , which by con-  t r a s t , a f f e c t s an a b s o l u t e  index.  - 54  Chapter  1.  The  3  -  Footnotes  s o c i a l c h o i c e problem o f Samuelson  i n d i v i d u a l p r e f e r e n c e s to household one  may  preferences i s ignored.  always assume t h a t w i t h i n each household  " p l a n n e r " who  2.  (1956) i n a g g r e g a t i n g  a l l o c a t e s consumption to e q u a l i z e  However,  there e x i s t s  a  utilities.  P o l l a k and Wales (1979) are s k e p t i c a l on t h i s  specification.  They argued t h a t t h i s i g n o r e s d i r e c t c o n t r i b u t i o n o f a t t r i b u t e s t o utility. and  This aspect o f u t i l i t y  i s ignored  3.  is difficult  to r e v e a l e m p i r i c a l l y  here.  T h i s i s r e a d i l y seen by w r i t i n g the budget c o n s t r a i n t o f each  household  m  as:  (A)p  1  Maximization  fx /m (A)) + L^ ± 1  of  (3.4) over  + m  n  (A)p  (x /m (A)) = y n ^ n n y  fx,/m,(A), . ...,x /m (A)] s u b j e c t t o the 1 1 n n v  1  c o n s t r a i n t would y i e l d the f o l l o w i n g s e t o f f i r s t o r d e r c o n d i t i o n s ,  U. fx,/m,(A), ....,x /m (A)| +Am.p. = 0 l 1 1 n n i x v  1  1=1,  ....,n  Em.(A)p. fx./m.(A)) = y  S u b s t i t u t i n g the s o l u t i o n s fx*/m.(A), 1 1 v  gives  (3.8) .  ....,x*/m (A)J i n t o n n '  (3.4)  - 55 -  4.  The  common consumer s u r p l u s e s , CV and EV are changes i n y  e v a l u a t e d a t f i n a l and i n i t i a l p r i c e s .  In g e n e r a l , CV and EV  are  not e q u a l , a l t h o u g h always take the same s i g n .  5.  The  s p e c i a l case o f homothetic p r e f e r e n c e s may  illustrate  this proposition:  V(p ,y ,A )  = $(y /n(p ,A ))(•)  >0  V(p ,y ,A )  = <&(y /n(p ,A ))  >0  1  1  2  then  1  2  2  E(p  2  2  Ete^y^A ,? ^ ) 1  1  1  0  0  1  (•)  2  =  y !! (p°,A°) /II ( p , A ) 1  1  1  2  2 ,2 o o, ,y ,A ,p ,A )  =  2„, o o. . 2 ,2, y n(p ,A )/n(p ,A )  1  1 ,1 o o. ,y ,A ,p ,A )  >  „, 2 2 ,2 o o. E(p ,y ,A ,p ,A )  >  y  >  V(p  therefore  E(p  1 1 1 Y /K(P ,A )  "«  •  <  *• V(p  6.  L  1 1 1 ,y ,A )  2  /n(p  2 2 ,A )  2 2 2 ,y ,A )  I f p r e f e r e n c e s are t r a n s l o g (see Chapter  a function of  utility.  4 ) , then S i s not  -  7.  -  T h i s i s an a p p l i c a t i o n o f Debreu  function representation.  8.  56  Sen ( 1 9 7 0 )  See Debreu  (1959)  (1959).  theorem o f u t i l i t y  Sec. 4 . 6 .  proves a L i b e r t a r i a n theorem s a y i n g t h a t un-  c o n d i t i o n a l L i b e r t a r i a n r u l e s are i n c o n s i s t e n t w i t h weak Pareto and u n l i m i t e d domain i n g e n e r a t i n g as axioms, a p r e f e r e n c e  9.  cycle r e s u l t s .  framework w i t h o u t  10. any  I f a l l are adopted  See a l s o Roberts  (1980).  The r e a d e r may r e c a l l t h a t i n Chapter 2 , i t has been argued  t h a t Jorgenson and S l e s n i c k  and  a social ordering.  (1984)  have made use o f a n o n - w e l f a r i s t  making i t c l e a r why such a framework i s n e c e s s a r y  justifiable.  A s o c i a l welfare  f u n c t i o n W i s anonymous i f and o n l y i f , f o r  two d i s t r i b u t i o n s o f u t i l i t i e s  u = (u. , I N  ,u )  and u' = (u ' , I  /U  ')  N  where one i s a p e r m u t a t i o n o f t h e o t h e r ,  W(u) = W(u').  11.  —  —  r*  B  \  I f W i s homothetic, then W(y) = II(W(y ) J , where II i s a mono* * t o n i c t r a n s f o r m , and W i s HD 1 . I t then f o l l o w s t h a t W(Ei) =  - 57 -  W(y^, HD  ,y ) d e f i n e s £, so t h a t E(y ) i s a l s o HD 1.  1, the i n d e x i s r e l a t i v e .  See a l s o Sen  12.  One  (1962), Dasgupta,  (1973) and B l a c k o r b y , Donaldson  suppose  mean u, then X curve f o r X  easily  verified.  (1973).  s h o u l d c o n s u l t Berge  (1973), Sen Briefly,  The converse i s now  Since u i s  and Auersperg  t h e r e are two d i s t r i b u t i o n s X , X E R a D  N  Starett (1981).  w i t h the same  i s s a i d t o be Lorenz s u p e r i o r t o X, i f the Lorenz b  a  l i e s c o m p l e t e l y i n s i d e t h a t o f X, . b  a  Sen and  In t h i s case X  a  =  BX^ where B i s a b i s t o c h a s t i c m a t r i x and i s not a p e r m u t a t i o n m a t r i x , and X  a  can be o b t a i n e d from X, by a f i n i t e b  N S :R  number o f t r a n s f e r s .  Then  *• R i s an S-concave f u n c t i o n i f S(X ) >_ SfX^) so t h a t i f fl  S i s a s o c i a l w e l f a r e f u n c t i o n , t h e n the AKS  index I  s  w i l l be S-convex,  where  I  s  (. X  a  ). < I (. X. ) — s b  since  l  - s( x ) / u < a  13.  One  l  - s( x ) / y D  a  can o b t a i n  (3.33) i s homothetic  (3.35) by an a l t e r n a t i v e r o u t e .  * (as W i s HD  1 and $'  E i s HD 1 and has t o be i d e n t i c a l t o  14.  For a f u l l  (•)  Since W i n  > 0), i t follows that  * W.  d i s c u s s i o n , see B l a c k o r b y and Donaldson  (1980).  - 58 -  CHAPTER 4  SPECIFICATION  In o r d e r t o a p p l y t h i s new approach, necessary.  specification i s  T h i s c h a p t e r d e s c r i b e s t r a n s l o g household  preferences  and t h e parameter r e s t r i c t i o n s n e c e s s a r y t o make e s t i m a t i o n f e a s i b l e . Because a l a r g e number o f parameters a r e i n v o l v e d , i n a d d i t i o n t o c r o s s - s e c t i o n d a t a , t i m e - s e r i e s aggregate  d a t a have t o be used.  a g g r e g a t i o n s t r u c t u r e w i l l be d e s c r i b e d which a l l o w s aggregate  An data  to be u t i l i z e d .  P r e f e r e n c e s a r e assumed t o be non-homothetic t r a n s l o g , f o l l o w i n g Jprgenson,  Lau and S t o k e r  f u n c t i o n i s a second-order  (1982).  A translog indirect  approximation  function at a single point.  o f any i n d i r e c t  or  Jin V ( p, y, A )  i n matrix  (4.2)  a  o  +  n £a.&n(m.p./y) + . . 1 1 1 i=l  form,  In V ( p , y, A )  utility  Incorporating commodity-specific  v a l e n c e s c a l e s , i t has t h e form, f o r n goods,  (4.1)  utility  a  + ( in mp/y )  T h ( An mp/y )  B  T  a  +  ( £n mp/y )  equi-  - 59  where a  i s a scalar, a p  o  matrix.  -  an n - v e c t o r , and B  By Roy's I d e n t i t y , e x p e n d i t u r e  (4.3)  e =  PP  an n x n symmetric  shares are  3 I n V /3 £n ( p/y ) i ( 3 £n V/3 In ( p/y )) T  .T where i  =  (1,  , 1 ) , i s an n - y e c t o r .  l o g , the e x p e n d i t u r e  (4 4) ' 1  shares take the form,  a+B  =  E  E  i  T  a  p  A p p l y i n g t h i s t o the t r a n s -  EE T +iB  ( £n m) + B ( £n p ) - B i ( £n y ) EE_ EE T T ( £n m ) + i B ( Jin p ) - i B i ( £ny) pp PP PP  S i n c e e i s homogeneous o f degree 0 i n the parameters a , B , i t is P PP u s u a l , as i n Jorgenson,  (4.5)  i a = P T  Lau and Stoker  (1982), t o n o r m a l i z e as f o l l o w s ,  -1  To o b t a i n the t r a n s l o g c o s t f u n c t i o n , the i n d i r e c t f u n c t i o n may  (4.6)  be i n v e r t e d t o s o l v e f o r y.  In V ( p, y, A ) = a o -  (4.2)  T + ( Jin mp ) a p  (£nmp) B T  utility  can be r e w r i t t e n as  T + Jin y + h. ( Jin mp ) B pp  (iilnY)+^(i£ny) B T  PP  which i s a q u a d r a t i c e q u a t i o n i n £n y .  PP  In o r d e r t o o b t a i n an  form f o r the c o s t f u n c t i o n , i t i s assumed t h a t .  ( In mp )  (i£ny)  explicit  - 60 -  (4.7)  i  T  B  i = 0  PP  so t h a t the second degree term i n (4.6) vanishes.''"  The  cost  function  V T T In u - [a + ( In mp ) a + h ( £n mp ) B = • 22 1 - ( in mp ) ( B i ) PP  ~\ ( £n mp )J  w i l l then have the form  (4.8)  taC(u,p,A)  where u i s a u t i l i t y  :  number.  The e q u i v a l e n c e s c a l e s have so f a r been l e f t  unspecified.  F o l l o w i n g Jorgenson, Lau and S t o k e r (1982) and Jorgenson and  Slesnick  (1982 a, b, c)(1983 a, b ) , the a t t r i b u t e v e c t o r A i s assumed t o be a v e c t o r o f dummy v a r i a b l e s , i . e . , the elements i n A are e i t h e r 0 o r 1. Four household a t t r i b u t e s are used t o d e s c r i b e each household, namely, the  a r e a o f r e s i d e n c e , the sex o f household head, the f a m i l y s i z e  the  age o f t h e h o u s e h o l d head,  and  so t h a t A i s an e l e v e n - v e c t o r , t o be  a s s i g n e d t o each household a c c o r d i n g t o T a b l e 1.  These  f o u r a t t r i b u t e s are thought t o be s i g n i f i c a n t  minants o f household consumption  pattern.  Indeed, t h e r e are r e l e v a n t  f a c t o r s which have been i g n o r e d h e r e , f o r example, household t i o n , education l e v e l , data a v a i l a b i l i t y .  race and c l i m a t e .  deter-  composi-  The b i n d i n g c o n s t r a i n t i s  As w i l l be seen i n Chapter 6, s t a t i s t i c s o f  e x p e n d i t u r e d i s t r i b u t i o n o v e r a t t r i b u t e groups are e s s e n t i a l f o r the  - 61 -  Table 1 : Vector A  S i z e o f Area o f Residence *  Sex o f Household  non-metropolitan  metropolitan  female  male  two persons  otherwise  three persons  otherwise  f o u r persons  otherwise  f i v e o r more p e r s o n s  otherwise  above 24 b u t  otherwise  Head  Family Size  Age o f Household  Head  34 o r below A,  above 34 b u t  otherwise  44 o r below A,  above 44 but  otherwise  54 o r below 10  above 54 but  otherwise  64 o r below 11  * Cities  above 64  w i t h p o p u l a t i o n above 30,000 are c l a s s i f i e d  otherwise  as m e t r o p o l i t a n .  - 62  -  s u c c e s s f u l e s t i m a t i o n o f t h i s model, and p u b l i c l y a v a i l a b l e  statistics  Canada d a t a do not a l l o w i n c o r p o r a t i o n o f more a t t r i b u t e s .  In p a r -  t i c u l a r , household to  composition  ( i . e . , the number o f a d u l t s , as opposed  c h i l d r e n i n the household) s h o u l d be an important  impact  o f household  sex o f household  composition  head.  i s r e f l e c t e d i n p a r t by the e f f e c t o f  By c o n v e n t i o n , a household  means e i t h e r an unattached  female  stituting  w i t h a female  o r a s i n g l e female-parent  Given any two multi-member households the same, the household  a t t r i b u t e . Here, the  household.  with a l l other a t t r i b u t e s being  w i t h a female head i m p l i e s i n most cases  a c h i l d f o r an a d u l t .  t h i s d i f f e r e n c e s h o u l d be r e f l e c t e d i n e s t i m a t e d e q u i v a l e n c e indeed been c o n f i r m e d by the e s t i m a t i o n r e s u l t s ,  i n Chapter  6 below) which p a r t i a l l y  A r e f e r e n c e household  j u s t i f i e s the whole  way  a, b, c)(1983  F o l l o w i n g Jorgenson  which has  (4.9)  P  A  i s an n x 11 m a t r i x , which  (4.10) P  A  = 0  24 o r and S l e s n i c k  a, b ) , the e q u i v a l e n c e s c a l e s are s p e c i f i e d i n a  t h a t e n a b l e s simple l i n e a r e s t i m a t i o n , t h a t i s ,  where B  scales.  approach.  i s d e f i n e d as the household  below, l i v i n g i n a m e t r o p o l i t a n a r e a .  and  (as d i s c u s s e d  a l l elements i n A e q u a l 0, i . e . , an u n a t t a c h e d male, o f age  (1982  sub-  Hence, i n t u i t i v e l y s p e a k i n g , house-  h o l d s w i t h female heads s h o u l d need more c l o t h i n g and l e s s f o o d ,  T h i s has  head  satisfies  - 63 -  T h i s l a s t assumption  i s necessary  i n making a g g r e g a t i o n a c r o s s i n -  d i v i d u a l expenditure  share e q u a t i o n s simple, because e i n (4.4) w i l l  now be l i n e a r i n a l l h o u s e h o l d - s p e c i f i c v a r i a b l e s . s h o u l d be p o i n t e d out t h a t , by o f the r e f e r e n c e household  (4.9), the e q u i v a l e n c e s c a l e  (4.11,  e  (4.5) (4.7) (4.9) and  s h a r e s can be e x p r e s s e d as  a + B A = J2—EA.  + B ( t o p ) -B i (toy ) EEL EE - 1+i  B (top) PP  which a r e s u i t a b l e f o r e s t i m a t i o n u s i n g c r o s s - s e c t i o n d a t a . B  r e p r e s e n t s t h e i n c r e m e n t a l e f f e c t s on e as a t t r i b u t e s  Given y, the r e f e r e n c e h o u s e h o l d central level  f o r e.  The  Note t h a t  change.  (whose A e q u a l s 0) e s t a b l i s h e s a  T h i s e f f e c t on e x p e n d i t u r e  i s then t r a n s l a t e d t o a p r i c e e f f e c t through B depends on demand  factors  are a l l equal t o u n i t y .  I n c o r p o r a t i n g t h e f o u r assumptions (4.10), t h e e x p e n d i t u r e  Furthermore, i t  pp  shares through B i n (4.9) which  elasticities.  aggregate  expenditure  share e q u a t i o n i s o b t a i n e d by  summing i n d i v i d u a l share e q u a t i o n s a c r o s s t h e e n t i r e p o p u l a t i o n . H be t h e t o t a l number o f households and  spending o f household  (4.12)  E =  pA  E ((y e h=l h  h  h.  and A , y  The aggregate  )/Ey ) h  Let  be the a t t r i b u t e v e c t o r  expenditure  s h a r e s , E, a r e  - 64 -  a P  + B E( y A PA  )/Y+ B (Inp)-(B pp PP - I + i B ( inp )  i )Z f y ( in y w  )) / Y .  T  p  where Y = are  ^ h Z y , the t o t a l h=l  spending.  2  p  These aggregate share e q u a t i o n s  s u i t a b l e f o r e s t i m a t i o n u s i n g t i m e - s e r i e s aggregate  data.  Notice  a l s o t h a t these e q u a t i o n s have the same form and i n v o l v e t h e same coefficients,  a , B , B , as i n the i n d i v i d u a l p pA pp  share e q u a t i o n s , so  t h a t d a t a from both sources can be combined, and i d e n t i f i c a t i o n o f a l l 3  parameters  i s obtained.  Using t h e d e f i n i t i o n  (3.19), t r a n s l o g e q u i v a l e n t income can  be e x p r e s s e d a s , r (4.13)  y  = exp  6  O T (co-w ) a P  T  oT o T \ B tt-o) B w ) + (1 - cu B i)(£ny) PP PP PP 1-0)  B l PP  where,  (4.14)  a) = k m  (A ) p  (4.15)  C J = Jlnm ( A ° ) p ° 0  To e s t i m a t e the e q u i v a l e n t income o f each i n d i v i d u a l , a  P  , B and B are r e q u i r e d . PP pA  estimates f o r  I t w i l l be shown i n Chapter 6 t h a t ,  - 65 -  by u s i n g both c r o s s - s e c t i o n and t i m e - s e r i e s aggregate d a t a , estimates  are o b t a i n a b l e .  I t i s e v i d e n t t h a t t r a n s l o g e q u i v a l e n t income o sensitive to p attached  (4.13) i s  o and the c h o i c e o f A .  F o r example, d e f i n i n g an un-  female, o f age 24 o r below, l i v i n g i n a m e t r o p o l i t a n  as the r e f e r e n c e household w i l l income v a l u e s .  give r i s e to d i f f e r e n t  However, i f the i n e q u a l i t y index  i n e q u a l i t y i s not s e n s i t i v e t o A ° .  T LD  y  = exp  a  + 3to  T B  to +  EE  i  1-0)  1 - to  but by  (4.15),  0)  B  l  B  PP  B  £P  i ) (&n  y)  l  (4.9) and  =  (_£nm (A )p J  =  (tam(A°)) B  B  T  1  (Inp  so t h a t  OT  - u)  B ^ i PP  and s u b s e q u e n t l y ,  PP  (1  oT , oT o a) a - *sa> B u'P PP  exp  O  T ) B  (4.16) can be w r i t t e n as  PP  equivalent  In o r d e r t o v e r i f y t h i s  T  3  P  PP  (4.10),  l  i+(£np°) B T  pp  i  area  i s relative,  (4.13) i s r e w r i t t e n as  (4.16)  these  pp  i  claim,  - 66 -  to  (4.17)  T  +  Sju)  T  Bp  p  oi  T + (l-o) B  p p  i) (Iny)  exp 1 - Unp°)  exp v  T  B i PP  oT . oT o - o) a - ^oi B o) P PP O T 1- ( t a p ) B i PP  Note t h a t o n l y t h e f i r s t  e x p o n e n t i a l term i s i n d i v i d u a l - s p e c i f i c but  i t does not i n v o l v e A ° .  The second e x p o n e n t i a l term, which i n v o l v e s  o A , i s a common s c a l a r m u l t i p l e on a l l i n d i v i d u a l e q u i v a l e n t incomes. T h e r e f o r e , the c h o i c e o f A ° cannot a f f e c t the r e l a t i v e  inequality  index. Adopting to  t h e t r a n s l o g s p e c i f i c a t i o n , i t would be  compute e m p i r i c a l l y the market e q u i v a l e n c e  s c a l e s (3.25).  s c a l e s can be compared w i t h t h e r a t i o s o f p o v e r t y S t a t i s t i c s Canada.  interesting  By d e f i n i t i o n , the p o v e r t y  These  l i n e s p u b l i s h e d by  income f o r h o u s e h o l d  his,  (4.18)  P,( p , A  h  ) = C( u,p,A ) h  where u i s the l e v e l o f s u b s i s t e n c e u t i l i t y . household, the p o v e r t y - l i n e r a t i o  (4.19)  U s i n g A° as a r e f e r e n c e  f o r household h i s  P(p,A )/P(p,A°) = C(u,p,A )/C(u,p,A°) h  h  - 67  -  which i s d i f f e r e n t from the market e q u i v a l e n c e (4.19) i s e v a l u a t e d market e q u i v a l e n c e  at u.  However, i t can be  s c a l e s are  scales only  shown t h a t the  translog  a c t u a l l y independent o f u so t h a t they  are i d e n t i c a l w i t h t r a n s l o g p o v e r t y - l i n e r a t i o s . t h a t the denominator o f  i n that  (4.8)  can be w r i t t e n  To  see  this,  notice  as  (4.20)  = 1 -  T ( Jin p ) B  because o f assumptions A,  and  ( Jin y  ( Jin u ) -  PP  (4.9)  i  and  (4.10), so t h a t i t i s independent o f  i n the numerator w i l l v a n i s h when the  Jin y° )  i s taken.  Consequently, the t r a n s l o g market e q u i v a l e n t p o v e r t y - l i n e r a t i o s can be  f  (4.21)  exp  -  difference  T (Jin m ) a  expressed  s c a l e s and  as  T ,T - ^(Jlnm) B (£nm)-(Jlnp) B (Jin m) PP PP T 1 - (Jlnp ) (B i) PP  These s c a l e s are e s t i m a t e d from demand d a t a and numbers i n Chapter 7.  translog  They p r o v i d e  valuable  study o f d i f f e r e n t i n e q u a l i t y i n d e x e s .  compared w i t h  published  i n s i g h t s i n a comparative  - 68 -  Chapter  1.  Incidentally,  expenditure  2.  4  Footnotes  t h i s assumption  share e q u a t i o n .  See  i s necessary  (4.4).  See a l s o  for a linear Diewert  N o t i c e the advantage o f l i n e a r i t y i n d e r i v i n g  (4.11), as a r e s u l t o f assumptions  (4.7) and  (4.12)  form can be aggregated.  "complete"  aggregation s t r u c t u r e regardless of d i s t r i b u t i o n s variables.  no l o n g e r be e s s e n t i a l .  3.  a  of  I f t h e s e were known, l i n e a r i t y would  See S t o k e r  (1983).  do not a l l o w the e x i s t e n c e o f a r a t i o n a l  aggregate  preferences  consumer.  E  shares o f a "consumer" whose p r e f e r e n c e s change  w i t h income d i s t r i b u t i o n .  However,  f o r e m p i r i c a l purposes,  p r e f e r e n c e s are too r e s t r i c t i v e i n f o r c i n g curves.  l i n e a r i t y ensures  I n Gorman's (1953) a g g r e g a t i o n framework, t r a n s l o g  r e p r e s e n t s aggregate  from  I n t h e o r y , any  functional  individual-specific  However,  (4.10).  (1974)  Gorman  p a r a l l e l and l i n e a r  Engel  - 69 -  CHAPTER 5  ESTIMATION METHOD  Section 1  Introduction  This s e c t i o n d e s c r i b e s the s t o c h a s t i c s t r u c t u r e o f the e s t i m a t i o n model and e x p l a i n s how t h e parameters e q u i v a l e n t income f u n c t i o n goods w i l l be d e f i n e d .  i n v o l v e d i n the  (4.13) can be e s t i m a t e d .  S i x composite  S i n c e 11 dummy v a r i a b l e s a r e u t i l i z e d t o  d e s c r i b e demographic a t t r i b u t e s , t h e r e a r e t o t a l l y 108 parameters t o be e s t i m a t e d : 6 parameters meters i n B . pA  i n a , 36 parameters i n B and 66 p a r a P PP U s i n g these e s t i m a t e s , t h e e a u i v a l e n c e s c a l e s c a n be  e s t i m a t e d which i n d i c a t e t h e a c t u a l r e l a t i o n s h i p between  utility,  consumption and t h e f o u r a t t r i b u t e s .  There a r e two ways t o e s t i m a t e t h i s model.  The f i r s t  approach,  named t h e p o o l e d e s t i m a t o r , has been used i n Jorgenson,  L a u and Stoker  (1982).  Basically,  The t e c h n i c a l d e t a i l s a r e n o t d e s c r i b e d h e r e .  they f o r m u l a t e a c o n s t r a i n e d m i n i m i z a t i o n problem w i t h an o b j e c t i v e f u n c t i o n b e i n g made up o f t h e sum o f squared s e c t i o n model  r e s i d u a l s i n the cross-  ( i . e . , composed o f t h e i n d i v i d u a l e x p e n d i t u r e  e q u a t i o n s ) and i n t h e t i m e - s e r i e s model gate share e q u a t i o n s ) .  share  ( i . e . , composed o f t h e aggre-  The s o l u t i o n 6* t o t h i s m i n i m i z a t i o n  problem  i s t h e s e t o f parameter e s t i m a t e s p r o v i d e d t h a t they a l s o s a t i s f y t h e symmetry and m o n o t o n i c i t y constraints."'" follows.  6* i s found by i t e r a t i o n as  The combined c r o s s - s e c t i o n and t i m e - s e r i e s model and t h e  - 70  c o n s t r a i n t s are point  & .  -  f i r s t - o r d e r approximated i n i t i a l l y  Then Liew's  (1976) i n e q u a l i t y c o n s t r a i n e d  squares method i s a p p l i e d t o generate 6^. o b j e c t i v e f u n c t i o n o f r e s i d u a l sum value. verges.  6^  three  stage l e a s t  i s substituted into  o f squares t o o b t a i n an  A l t h o u g h the e s t i m a t o r  i s b e l i e v e d t o be  because the  the  objective  T h i s p r o c e s s i s then r e p e a t e d u n t i l the o b j e c t i v e v a l u e  sample p r o p e r t i e s are unknown. and  around an a r b i t r a r y  consistent,  con-  small  Furthermore, the procedure i s c o s t l y  start-up value  6  q  i s a r b i t r a r y , an a c c u r a c y problem  might a r i s e .  A d i f f e r e n t approach i s adopted here which e s t i m a t e s c r o s s - s e c t i o n model and obtained  the t i m e - s e r i e s model s e q u e n t i a l l y .  i n the c r o s s - s e c t i o n are  s u b s t i t u t e d i n t o the  e q u a t i o n s as i f t h e y were t r u e v a l u e s .  On  the o t h e r  the  no i t e r a t i o n s  hand, however, r e l a t i v e t o the  approach, i t i s l e s s e f f i c i e n t . information  and  Estimates  time-series  T h i s procedure has  advantage t h a t i t i n v o l v e s o n l y l i n e a r e s t i m a t i o n are r e q u i r e d .  the  first  For the p o o l e d e s t i m a t i o n ,  from both s o u r c e s i s p o o l e d t o g e t h e r  and  since  estimates  g e n e r a t e d i n a s i n g l e p a s s , even the parameters t h a t are  estimable  u s i n g o n l y c r o s s - s e c t i o n d a t a are e s t i m a t e d u s i n g a d d i t i o n a l t i m e series information. because the  T h i s c o n s t i t u t e s some e f f i c i e n c y g a i n .  s i z e o f the  sample i s l a r g e i n the c r o s s - s e c t i o n  s m a l l i n the t i m e - s e r i e s , The  e s t i m a t e s w i l l be  However,  the e f f i c i e n c y g a i n i s l i k e l y to be  dominated by the c r o s s - s e c t i o n d a t a .  r e s u l t s i n Jorgenson, Lau  and  and  Stoker  (1982) s u b s t a n t i a t e  small.  The  this  claim,  - 71 -  namely, except  f o r the p r i c e c o e f f i c i e n t s which do not e n t e r the  c r o s s - s e c t i o n model, the e s t i m a t e s f o r B  and a  pA  p  o b t a i n e d from c r o s s -  s e c t i o n alone are v e r y c l o s e t o the p o o l e d e s t i m a t e s .  The procedure  adopted here i s a s e q u e n t i a l one.  The  first  step i n v o l v e s e s t i m a t i n g those parameters t h a t are i d e n t i f i e d i n d i v i d u a l expenditure section data only.  share e q u a t i o n s , i . e . , a  , using cross-  p pA Because o f the l a c k o f p r i c e v a r i a t i o n i n the  c r o s s - s e c t i o n d a t a , the p r i c e c o e f f i c i e n t s , B The  and B  i n the  second s t e p i n v o l v e s e s t i m a t i n g B  PP  are not  identified.  u s i n g the aggregate  expendi-  t u r e share e q u a t i o n s and t i m e - s e r i e s d a t a o n l y , p r o c e e d i n g as i f the e s t i m a t e s o b t a i n e d i n the f i r s t In  o t h e r words, the aggregate  and B  step f o r a  and B  pA  were t r u e v a l u e s .  e q u a t i o n s are e s t i m a t e d s u b j e c t t o a  P  b e i n g e q u a l t o t h e i r c r o s s - s e c t i o n e s t i m a t e d v a l u e s , as w e l l  as the u s u a l symmetry c o n d i t i o n s on B PP  Section 2  The In  p  2  Cross-section Estimation  i n d i v i d u a l share e q u a t i o n  a family expenditure  each household  faces.  a c r o s s the h o u s e h o l d s . year f o r the p r i c e  (4.11) i s n o n - l i n e a r i n  (£np).  survey, t h e r e i s no i n f o r m a t i o n on the p r i c e I t w i l l be assumed t h a t p r i c e s are The  uniform  survey year i s taken as the r e f e r e n c e  s e r i e s so t h a t the p r i c e v e c t o r i s (1,  , 1),  3 an n - v e c t o r .  The purpose i s t o a v o i d  s e c t i o n share e q u a t i o n i s t h e r e f o r e ,  (inp)  i n (4.11).  The c r o s s -  - 72 -  (5.1)  e  where y , n  survey.  (5.2)  Vi  = - a  P  - B  PA  A  1*1  + B  pp  h i(£ny)  h = 1,  ..., H  i s taken as the t o t a l e x p e n d i t u r e o f household  h i n the  I t then f o l l o w s t h a t the i t h r e g r e s s i o n e q u a t i o n i s ,  e  h  I  =-a.  - Z B., A, + 6. ( «.ny ) + e , , ikx I I k=l h  I  h  h  i = 1,  ..., n  h = 1,  . . ., H  Y^ where e ^ i s the e x p e n d i t u r e share o f the i t h good f o r household i s the i t h component i n a , 6 ^  h, Y^ is  i s the i k t h element i n p » B  A  the k t h element i n A , 9. i s the i t h component i n B i , o r the sum o f l pp h the i t h row (or i t h column) o f B , e. i s a d i s t u r b a n c e term o f the pp i i t h e x p e n d i t u r e share e q u a t i o n f o r household  h.  As i s t r u e f o r any consumption a l l o c a t i o n i s redundant  i n (5.1).  model, one  equation  Summing up the e x p e n d i t u r e shares i n the  left-  hand s i d e g i v e s 1 i d e n t i c a l l y ,  and so must the r i g h t - h a n d s i d e , by  restrictions  This implies that, i n  on a , B , B . P PA PP  (5.2)  the d i s -  turbances are l i n e a r l y dependent because  (5.3)  n . Z E = 0 i=l  h = 1,  ...,  H  1  so t h a t the c o v a r i a n c e m a t r i x  (5.4)  var  U  ) = v a r (e, , . . . . , e ) 1 n  =1  h = l ,  the  ...,H  - 73  must be  singular.  -  I t i s assumed t h a t the d i s t u r b a n c e term  the f o l l o w i n g assumptions, f o r a l l h = 1,  (5.5)  E(e )  = 0  (5.6)  var(e ) = $ .  (5.7)  var(e ,  h  satisfies  H,  n  h  e )  1  H  where $ i s o f rank n-1  and  T  = $&I„ H  d i s t u r b a n c e s are independent a c r o s s  households.  A c t u a l e s t i m a t i o n i n v o l v e s n-1 rank n-1,  equations.  Since | i s of  and each r e g r e s s i o n e q u a t i o n i n v o l v e s the same e x p l a n a t o r y  v a r i a b l e s , the J o i n t G e n e r a l i z e d L e a s t Squares e s t i m a t o r T h e i l ) i s i d e n t i c a l to equation-by-equation  OLS  (Zellner,  e s t i m a t o r which i s  4 best l i n e a r unbiased. estimates  By r e s t r i c t i o n s ,  (4.5),  (4.10) and  f o r the o m i t t e d e q u a t i o n , say the nth one,  (4.7), the  are o b t a i n e d  follows, (5.8)  a  = 1 -  n-1 E a. i=l 1  (5.9)  6  (5.10)  9  nk  • = -  = -  n-1 E § . , lk i=l  n-1^ E 9. i=l 1  k = 1,  K  as  - 74 -  where a s u p e r s c r i p t  " i n d i c a t e an e s t i m a t e .  5  Because the c r o s s -  s e c t i o n sample i s u s u a l l y v e r y l a r g e , the e s t i m a t e s o b t a i n e d f o r a , P B and B i s h o u l d be v e r y a c c u r a t e . However, B i s not e s t i m a b l e pA pp pp 1  because o f the l a c k o f p r i c e v a r i a t i o n i n the sample.  Section 3  Time-series Estimation  The  s e q u e n t i a l approach adopted  o n l y i n the second  step.  The aggregate  here r e q u i r e s t i m e - s e r i e s d a t a share e q u a t i o n s are e s t i m a t e d  s u b j e c t t o the e s t i m a t e d v a l u e s f o r a , B , and B i o b t a i n e d i n the P PA pp c r o s s - s e c t i o n and the symmetry c o n d i t i o n s on B . The main concern PP here i s t o e s t i m a t e the i n d i v i d u a l elements i n B PP  The t i m e - s e r i e s model i s d e r i v e d from summing up all  the households  so t h a t y  i n society.  (4.11) a c r o s s  L e t s u b s c r i p t t denote time p e r i o d ,  i s the t o t a l e x p e n d i t u r e o f household  a d d i t i o n , the f o l l o w i n g short-hand  h i n period t.  n o t a t i o n s are adopted,  In  for t = l ,  • . . , T,  T  (5.11)  D (p  ) = -1 + i  B  (!np  ), a s c a l a r ,  (5.12)  S  yAt  = Zy^A /Y , a K - v e c t o r , where K = 11 from T a b l e ^ t t  (5.13)  S  yyt  = Eyk( to v? ) /Y t t t  h  h  . a scalar  1,  -  (5.14)  =  where  Y  ^h t t^ t' v  H X y h=l  =  £  Y  a  75 -  n  n  -  v  e  c  t  o  r  h  The t i m e - s e r i e s r e g r e s s i o n e q u a t i o n s can then be e x p r e s s e d a s , i n m a t r i x form.  (5.15)  E  fc  t  = D(p.) t  ( a +B S +B ( i n p j - (B i ) S J + E p pA yAt pp t pp yyt^ t  1  K  t = 1,  where e  i s t h e d i s t u r b a n c e term.  (5.16)  E. it  = D(p. ) [a. + ^t i 1  3  where S (Inp  t  i yyJ S  +  k  =  £  The i t h e q u a t i o n w i l l  K EB.,S,.,+ l k yAkt  1 =  be  l b . . (£np.J xj  1  i t '  T  1  0  t  =  1  T  , i s t h e k t h element i n S , (£np . ) i s t h e j t h element i n yAkt yAt jt  ) and  e. i s the i t h element i n £ . i t t  covariance structure o f e  i s s t a t i o n a r y t h r o u g h time, i . e . ,  (5.17)  E(E^)= E ( E ) =0 t s  (5.18)  var(E  H  t  ) = var(e  I t i s assumed t h a t the  s  ) = t  s , t = 1,  T  s , t = 1,  T  - 76  where \ i s o f rank n-1.  (5.19)  E  (e ) t  -  I t then f o l l o w s  = E(E y^e^/Y )  = 0  T  h  that  t = 1,  T  and,  var'(e )  (5.20)  =  (z (y£)  2  var ( e " ) )  /  Y '  t = 1,  s i n c e household d i s t u r b a n c e s are assumed t h a t  (5.21)  e  i s not  , e )  S i n c e $ i s o f rank n-1,  constraints, explanatory  correlated.  T  It i s  further  s e r i a l l y correlated, i . e . ,  var ( e ,  i s redundant and  not  ...,  =  ft  i s a l s o o f rank n-1.  One  should be o m i t t e d because o f c r o s s - e q u a t i o n  despite  the  variables.  f a c t t h a t a l l e q u a t i o n s have the  equation symmetry  same  In the a c t u a l e s t i m a t i o n , are  imposed,  (5.22)  a  (5.23)  B  (5.24)  (B i ) = PP  and  the f o l l o w i n g parameter r e s t r i c t i o n s  P  pA  = a  P  = B  K  pA  by symmetry,  T i B  (5.25)  PP  ~T = 6  where a s u p e r s c r i p t  " denotes an e s t i m a t e d v a l u e from the c r o s s - s e c t i o n .  These r e s t r i c t i o n s  (5.26)  where  E  E  t  t  = B  p p  (kp ) t  can be s u b s t i t u t e d i n t o  + D  (P )e t  t  = D(p )E - (o +B S ^ - 8S t t p pA yAt Y A  v  (5.15) t o o b t a i n  y t  )  D(p ) = -1 + G ( J l n p ) T  t  As d e f i n e d  e a r l i e r , b.. i s t h e i j t h element i n B 1 3  e s t i m a t e d sum o f the i t h column  PP  (or i t h row) o f B  PP  and 8. i s the i .  The i t h e q u a t i o n  - 78 -  in  (5.26) i s  (5.27)  E. it  =  E b. . ( i n p . J . in it 3=1  By r e s t r i c t i o n  (5.24), however,  (5.28)  n-1 = 0. -.• E b. . i ID = 1  b. m  + D(p.) e. t i t u  j  T h e r e f o r e , b. , f o r a l l i , s h o u l d be s u b s t i t u t e d out o f the system, in In d o i n g so, the i t h e q u a t i o n becomes,  Q. it  (5.29)  =  n-1 E b. . ((Anp.. ) - U n p . )) + D(p. ) E . . . ]t nt ' t i t 1=1 I T  ,  C" = E _ - 6. ( k p ) it i t i nt  where  I t s h o u l d be noted t h a t the d i s t u r b a n c e term i n (5.29), D(p )e. i s not c l a s s i c a l t i t (5.20), i t can e a s i l y corrected  (5.30)  so  i n structure.  be v e r i f i e d  I t v a r i e s with time.  t h a t h e t e r o s c e d a s t i c i t y can be  f o r by the f a c t o r  p  = D(p )" (Y 1  t  t  2 t  /  E h=l  {y\) ) 2  h  that  (5.31)  E(p (D(p )e )) t  t  t  = 0  Given  t = 1,  T  - 79 -  (5.32)  -var(p  (5.33)  v a r (  P  (D(p )e )) = $  L  ( D (  P  L  )  E  I  ) ,  t = 1,  , p^, (D (p^,) e ^)) t  T  = | Q I ^  A f t e r a l l these m a n i p u l a t i o n s , t h e t i m e - s e r i e s e s t i m a t i o n problem i s t o e s t i m a t e n-1 l i n e a r e q u a t i o n s , each c o n t a i n i n g t h e same explanatory v a r i a b l e s .  The i t h e q u a t i o n o f t h e system i s , f o r  period t ,  (5.34)  p Q t  ± t  = p  t  n-1 Z b j=l  (Unp  ) - Unp^)) + P D(P )e. t  K Q • ^ = D (p ) E - (a. + E S . . S it t i t *• I l k yAkt  where  A l  x  To e s t i m a t e  t  t  -9.S t l - e . ( 2 , n p ) l yytI nt 1  (5.34), J o i n t G e n e r a l i z e d L e a s t Squares can be a p p l i e d  s u b j e c t t o t h e symmetry c o n s t r a i n t s on p # i . e . . B  D  (5.35)  b.. = b.. 13  The  Di  i , j = 1, ..., n-1  number o f parameter e s t i m a t e s o b t a i n e d d i r e c t l y i s  (n-1) X (n-1).  From these e s t i m a t e s , t h e n t h c o e f f i c i e n t i n each o f  the n-1 e q u a t i o n s and a l l t h e c o e f f i c i e n t s i n t h e n t h e q u a t i o n can be d e r i v e d as f o l l o w s .  By  (5.28),  - 80 -  (5.36)  b. i  = 6. n  1  n-1 E h. . j=l 1 3  By the symmetry c o n s t r a i n t s , one  obtains  (5.37)  n-1  and  b  . = b. i = 1, ni in  ...,  finally.  (5.38)  b  nn  = 6  n  -  n-1 E b . . , ni 3=1  To summarize t h i s c h a p t e r , by u s i n g a two-step approach which i n v o l v e s e s t i m a t i n g the c r o s s - s e c t i o n and tially,  t i m e - s e r i e s models sequen-  a l l the parameters i n the e q u i v a l e n t income f u n c t i o n can  estimated.  T h i s means t h a t , g i v e n A* , 1  y* and 1  (p, p°) , e q u i v a l e n t  income can be imputed t o each i n d i v i d u a l i n household dent i n t e r e s t i s the c o m m o d i t y - s p e c i f i c s c a l e s can be e x p r e s s e d a s , from  (5.39)  m = expfB  1  PP  B  pA  be  h.  Of  indepen-  e q u i v a l e n c e s c a l e s m.  These  (4.9),  A) ;  A complete s e t o f e q u i v a l e n c e s c a l e s f o r d i f f e r e n t c o n f i g u r a t i o n s o f A can then be e s t i m a t e d u s i n g e s t i m a t e d v a l u e s f o r B  pp  and  B  pA  - 81 -  Chapter  1.  Since a  p  5  Footnotes  and B , e n t e r both c r o s s - s e c t i o n and t i m e - s e r i e s pA  models, the two terms i n t h e o b j e c t i v e f u n c t i o n s h o u l d be minimized together.  2.  The m o n o t o n i c i t y c o n d i t i o n s i n Jorgenson,  Lau and S t o k e r  (1982)  b e i n g i m p l i e d by i n t e g r a b i l i t y o f demand f u n c t i o n s a r e not imposed. These c o n d i t i o n s ensure  negative semi-definiteness o f the Jacobian  matrix o f the cost f u n c t i o n only i f expenditure to be non-negative.  shares a r e r e s t r i c t e d  The f o c u s a t p r e s e n t i s on e s t i m a t i n g t h e p a r a -  meters i n t h e e q u i v a l e n t income f u n c t i o n r a t h e r than r e c o v e r i n g unknown p r e f e r e n c e s from h y p o t h e s i z e d demand f u n c t i o n s . r e s t o f t h e i n t e g r a b i l i t y c o n d i t i o n s : summability,  However, the  homogeneity and  symmetry a r e imposed.  3.  Rigorously speaking, t h i s i n v o l v e s r e d e f i n i n g the p h y s i c a l  u n i t s i n measuring q u a n t i t i e s o f commodities.  4.  See T h e i l  5.  S i n c e t h e r e a r e no c r o s s - e q u a t i o n c o n s t r a i n t s , an e q u i v a l e n t  procedure  (1970) Chapter  7.  w i l l be t o e s t i m a t e a l l n e q u a t i o n s  independently.  - 82 -  CHAPTER 6  IMPLEMENTATION  This chapter  d e s c r i b e s how t h e e s t i m a t i o n model i n Chapter 5  can be implemented i n t h e Canadian c o n t e x t , data.  using p u b l i c l y a v a i l a b l e  The c r o s s - s e c t i o n model r e q u i r e s e x p e n d i t u r e  survey micro  whereas t h e t i m e - s e r i e s model r e q u i r e s aggregate t i m e - s e r i e s  data,  statistics  t h a t a r e not r e a d i l y a v a i l a b l e from t h e p u b l i c a t i o n s o f S t a t i s t i c s Canada.  These s t a t i s t i c s have t o be s p e c i a l l y computed.  The sequen-  t i a l e s t i m a t i o n approach has been c a r r i e d o u t and the e s t i m a t i o n r e s u l t s can be found i n Appendix A and Appendix D. are t h e e s t i m a t e d  equivalence  Of s p e c i a l  s c a l e s , while b e i n g a g o v e r n i n g  interest factor  i n making i n t e r p e r s o n a l comparison p o s s i b l e , p l a y a c r u c i a l r o l e i n the d e t e r m i n a t i o n  o f e q u i v a l e n t incomes.  i n Appendix E and a r e i n t u i t i v e l y v e r y  Section 1  These s c a l e s can be found  appealing.  Cross-section Estimation  The  s e q u e n t i a l approach adopted here c a l l s f o r , i n t h e f i r s t  s t e p , e s t i m a t i o n o f t h e m i c r o i n d i v i d u a l household e x p e n d i t u r e  share  equation,  (6.1)  e  h  x  =  - a. - E ^ J . , A + 8. ( U n y ) + £ x k = l xk K x x h  h  h  i=l,  n  - 83 -  where e^ i s the e x p e n d i t u r e share o f the i t h good f o r household A, \  a. i s the i t h element i n a , 3 . , i s the i k t h element i n B . i p ik pA i s the k t h component i n A i t h component i n B  h,  (as a l r e a d y d e f i n e d i n T a b l e 1 ) , 0^ i s the 1*1  i , and e. i s the d i s t u r b a n c e term. PP i  r e g r e s s i o n o f the e x p e n d i t u r e  (6.1) i s a  share o f good i on an i n t e r c e p t  a s e t o f 11 dummy v a r i a b l e s and the l o g a r i t h m o f t o t a l  expenditure.  In so f a r as e x p e n d i t u r e shares d i f f e r a c r o s s households a t t r i b u t e s , these d i f f e r e n c e s wi'll be accounted  term,  of different  f o r by the  coeffi-  c i e n t s g. , ' s . ik  The d a t a s e t f o r t h i s r e g r e s s i o n i s d e r i v e d from the F a m i l y E x p e n d i t u r e Survey  1978.  T h i s s u r v e y p r o v i d e s micro d a t a f o r  on the e x p e n d i t u r e p a t t e r n s and household  characteristics of a  r e p r e s e n t a t i v e sample o f a p p r o x i m a t e l y 10,000 households.  The  i n f o r m a t i o n a v a i l a b l e a l l o w s a c l a s s i f i c a t i o n o f 6 composite d e f i n e d as  (6.2)  1978  goods,  follows,  "Food"  =  food p r e p a r e d a t home and o u t s i d e + tobacco and a l c o h o l i c  beverages  (6.3)  "Clothing"  =  a l l c l o t h i n g and  footwear  (6.4)  "Recreation"  =  r e c r e a t i o n and e n t e r t a i n m e n t + r e a d i n g materials + education + g i f t s contributions  and  - 84 (6.5)  " P e r s o n a l and  =  medical care"  (6.6)  a l l p e r s o n a l maintenance needs + medical  "Shelter"  =  treatment  r e n t + payment f o r h o u s i n g mortgages + water + f u e l and e l e c t r i c i t y + household  operations +  f u r n i s h i n g s and  (6.7)  "Transportation"  =  automobile purchased  The  household  equipment  and t r u c k s e r v i c e s  +  transportation  survey p r o v i d e s i n f o r m a t i o n on the amount o f money each  spends i n each o f these 6 consumption c a t e g o r i e s . e x p e n d i t u r e s f o r household h i s taken as y  The  sum  household o f these  i n the r e g r e s s i o n ( 6 . 1 ) .  Not a l l the r e c o r d s c o n t a i n e d i n the survey e n t e r the d a t a set  for regression.  Seventy households  have been e x c l u d e d because  t h e y are c l a s s i f i e d i n the survey as roomers and they d i d not pay r e n t i n 1978.  These households  might e x h i b i t spending b e h a v i o u r  d e v i a t e s from t h e norm and s h o u l d be d i s c a r d e d . this  Consequently,  any that  after  s c r e e n i n g , the sample s i z e f o r the c r o s s - s e c t i o n r e g r e s s i o n i s  9285.  Since e x p e n d i t u r e shares always sum  t o 1, the  e q u a t i o n , o r indeed any one o f the 6 e q u a t i o n s , can be Parameters  transportation omitted.  i n the t r a n s p o r t a t i o n e q u a t i o n can be d e r i v e d from the  - 85  -  e s t i m a t e s o f the o t h e r 5 e q u a t i o n s and  (4.10).  t o the e q u a t i o n - b y - e q u a t i o n  ordinary least  can be a p p l i e d on each o f the remaining  independently can be  (4.5),  (4.7)  Since the Z e l l n e r and T h e i l G e n e r a l i z e d L e a s t Squares  method reduces method, OLS  u s i n g assumptions  5  u s i n g the data s e t o f 9285 households.  found i n Appendix A.  t i v e l y appealing.  Out o f the t o t a l  s i g n i f i c a n t a t the 95%  equations The  These c o e f f i c i e n t s s h o u l d be  a c c u r a t e because t h e sample s i z e i s so l a r g e .  level.  The  squares  results very  They are a l s o i n t u i -  78 c o e f f i c i e n t s , 66 o f them are c o e f f i c i e n t of  ( l o g y) i s v e r y  s i g n i f i c a n t i n a l l e q u a t i o n s , i m p l y i n g t h a t h o m o t h e t i c i t y i s an unreasonable  restriction.  to be most important plays reasonable  Of a l l the a t t r i b u t e s , f a m i l y s i z e seems  i n a f f e c t i n g expenditure  trends.  shares.  It also dis-  For example, i n c r e a s i n g f a m i l y s i z e  leads  to i n c r e a s i n g food share and d e c r e a s i n g t r a n s p o r t a t i o n s h a r e .  This  i s c o n s i s t e n t w i t h the n o t i o n t h a t food i s a n e c e s s i t y and t r a n s portation i s a luxury.  i d e n t i f i c a t i o n of a , p T row i s (B i ) , the  Thus, u s i n g c r o s s - s e c t i o n data e n a b l e s B pp i and B pA', i . e .>, i n Appendix A,, the f i r s t T second t o the second l a s t rows form ( - B ) pA - T -a P  p  and  p  the l a s t row  is  - 86 -  Section 2  Time-series Estimation  In o r d e r t o e s t i m a t e e q u i v a l e n t income i n (4.13), for B  PP  estimates  are r e q u i r e d i n a d d i t i o n t o the e s t i m a t e s i n Appendix  However, these e s t i m a t e s can be  A.  s u b s t i t u t e d i n t o the aggregate  e q u a t i o n s as i f t h e y were t r u e v a l u e s t o generate  estimates  share  for B PP  In t h i s second aggregate  s t e p o f the s e q u e n t i a l approach,  d a t a are used.  The  only time-series  i t h r e g r e s s i o n e q u a t i o n i n the  time-  s e r i e s model i s , a f t e r s u b s t i t u t i o n o f a , B and 6 (being a v e c t o r p pA of estimates f o r B i ) , PP n-1 (6.8) p Q. = P . ^ ( ( ^ P ) - ( £ n p ) ) p D ( p ) . t  t  t  i  j  j  t  n  1,  t  +  t  t  e  t  ...... T  where,  (6.9)  (6.10)  The  Q. . xt  =  x  D(p ) t  =  -1  +  )E.  xt  -  v  (a.  x  +  E K.—-L  6., xk  S  yAkt  estimation of  J  -  3. ( £ n p . ) x nt  t  ,(  tap  ,  , Jtnp  )  u s i n g p u b l i s h e d d a t a from S t a t i s t i c s Canada.  t h a t generate (6.8)  9. S • J x yyt  T  usable t i m e - s e r i e s runs  The procedures  -  6 (£np )  b a s i c problem then i s t o compute p , Q.  and D ( p ^ ) i n (6.8) The  D(p. t  from 1971  t o 1981,  the aggregate  are d e s c r i b e d below.  i . e . , 11 o b s e r v a t i o n s .  statistics  for successful  - 87 -  1.  Aggregate e x p e n d i t u r e  The  aggregate  shares, E  share d a t a are d e r i v e d from P e r s o n a l E x p e n d i t u r e  on Consumer Goods and S e r v i c e s i n C u r r e n t D o l l a r s , i n N a t i o n a l Income and E x p e n d i t u r e Accounts  Catalogue No. 13-201.  Some minor a d j u s t -  ments are n e c e s s a r y t o regroup those e x p e n d i t u r e items so t h a t the c l a s s i f i c a t i o n i n the t i m e - s e r i e s i s c o n s i s t e n t w i t h t h a t i n the cross-section.  The groupings are shown as f o l l o w s , the numbers i n  p a r e n t h e s e s b e i n g t h e account  numbers.  (6.11)  (1) Food, beverages and tobacco +  Food  h x (45) E x p e n d i t u r e s on r e s t a u r a n t s and h o t e l s  (6.12)  Clothing  (6.13)  Recreation  =  (5) C l o t h i n g and  footwear  (36) R e c r e a t i o n , e n t e r t a i n m e n t , t i o n and c u l t u r a l  educa-  services +  h x (45) E x p e n d i t u r e s on r e s t a u r a n t s and h o t e l s + (48) Net e x p e n d i t u r e abroad  (6.14)  P e r s o n a l and  (24) M e d i c a l care and h e a l t h s e r v i c e s +  M e d i c a l Care  (43) T o i l e t a r t i c l e s , c o s m e t i c s + (44) P e r s o n a l c a r e  - 88 -  (6.15)  Shelter =  ((9) Gross r e n t , f u e l and power - (10) Gross imputed  rent  No. o f households w i t h o u t mortgage i n 1981 ^ No. o f households w i t h owned accommodation i n 1981 ' (16) F u r n i t u r e , f u r n i s h i n g s , e t c . + watches and r e p a i r s +  (42) J e w e l l e r y ,  (46) F i n a n c i a l , l e g a l  and  o t h e r services."*"  (6.16)  T r a n s p o r t a t i o n = (29) T r a n s p o r t a t i o n and  Communication  As i n the c r o s s - s e c t i o n e s t i m a t i o n , t o t a l aggregate t u r e , Y ,• i s taken as the sum  expendi-  o f t h e e x p e n d i t u r e s on the 6 goods i n  year t .  2.  Expenditure/attribute d i s t r i b u t i o n s t a t i s t i c ,  This s t a t i s t i c b u t i o n o f aggregate  S  yAt  i s a summary s t a t i s t i c r e f l e c t i n g the  distri-  e x p e n d i t u r e o v e r the s p e c i f i e d a t t r i b u t e  groups.  Formally,  (6.17)  A  S  y A t  =  ^ A  h  / Y  t  i s an e l e v e n - v e c t o r , and so i s ^  see t h a t the f i r s t  s  A t  -  .For example, i t i s easy t o  component i s s i m p l y the t o t a l e x p e n d i t u r e o f a l l  n o n - m e t r o p o l i t a n households population expenditure.  (which have A  = 1) d i v i d e d by  In o t h e r words, each  +  total  component i n S ^  the p r o p o r t i o n o f e x p e n d i t u r e i n y e a r t t h a t i s accounted  A t  is  f o r by a  - 89 -  group o f households h a v i n g a common a t t r i b u t e . expenditure information income c o u n t e r p a r t ,  Unfortunately,  o f t h i s s o r t , as opposed t o i t s a f t e r - t a x  i s not a v a i l a b l e on a t i m e - s e r i e s  basis.  An  a c c e p t a b l e a p p r o x i m a t i o n , however, i s t o use a v a i l a b l e a f t e r - t a x income d i s t r i b u t i o n d a t a and d e r i v e e x p e n d i t u r e d i s t r i b u t i o n s by observing  the r e l a t i o n s h i p between a f t e r - t a x income and e x p e n d i t u r e  i n the c r o s s - s e c t i o n specific.  More s p e c i f i c a l l y ,  sumption f u n c t i o n s using  sample, a l l o w i n g  the r e l a t i o n s h i p t o be a t t r i b u t e -  the f o l l o w i n g f o u r c r o s s - s e c t i o n  con-  corresponding to the four a t t r i b u t e s are estimated  the c r o s s - s e c t i o n d a t a  set.  Area o f Residence  (6.18)  Sex  y  h  o f Household Head  (6.19)  y  h  Household  (6.20)  Age  =  y  =  a  + b  2  2  z  + y A 2  + v  2  2  Size  h  =  a  3  +  b  o f Household Head  3  z  + Y  3 1  A  3  + Y  3 2  A  4  + Y  3 3  A  5  +  y  3  4  A  6  +  v  3  - 90 -  ( 6  -  Y  2 1 )  =  h  a  +  +  y  and z  b  Y  4  h  +  45 l l  +  4  z  A  41 7  Y  A  V  +  Y  42 8  h  A^ t o A ^  +  Y  43 9 A  +  Y  44 10 A  4  are the t o t a l e x p e n d i t u r e  h respectively.  A  and a f t e r - t a x income o f household  are the dummy v a r i a b l e s d e f i n e d i n T a b l e  1.  h  v^,  , v^ are t h e d i s t u r b a n c e terms.  I t i s assumed t h a t i n each  o f the 4 e q u a t i o n s t h e r e i s no contemporaneous c o v a r l a n c e s i n the d i s t u r b a n c e terms so t h a t OLS i s b e s t l i n e a r unbiased.  The e s t i m a t e d  c o e f f i c i e n t s f o r these 4 r e g r e s s i o n s can be found i n Appendix B. The e s t i m a t e d c o e f f i c i e n t s f o r (6.18) t o (6.21) are u t i l i z e d t o map  the a f t e r - t a x income d i s t r i b u t i o n s e r i e s , 1971-1981 t o a  corresponding procedure  s  ^  A t  expenditure  i s as f o l l o w s .  distribution series.  Take t h e f i r s t  component  The mapping of S  as an yAt  example.  I t i s r e q u i r e d to estimate the p r o p o r t i o n o f t o t a l p o p u l a t i o n  e x p e n d i t u r e t h a t i s accounted  f o r by n o n - m e t r o p o l i t a n  households.  Suppose the e s t i m a t e d consumption f u n c t i o n f o r n o n - m e t r o p o l i t a n households  (6.22)  y  k  a c c o r d i n g t o (6.18) i s  =  (a  + y ) + b  where k i s a n o n - m e t r o p o l i t a n an e s t i m a t e d v a l u e .  z  k  household  and a s u p e r s c r i p t " i n d i c a t e s  (6.22) r e p r e s e n t s the r e l a t i o n s h i p  between  - 91  -  a f t e r - t a x income and e x p e n d i t u r e f o r a n o n - m e t r o p o l i t a n Summing  (6.22) a c r o s s a l l n o n - m e t r o p o l i t a n households  estimated t o t a l expenditure o f non-metropolitan  (6.23)  K , E y = K(a k=l  + y )+ b  1  1  household.  g i v e s the  households  K E z ^=1  where K i s the number o f n o n - m e t r o p o l i t a n households.  This quantity  can be found i f K and  found i n "Income  Ez  are a v a i l a b l e .  These can be  A f t e r - t a x , D i s t r i b u t i o n s by S i z e i n Canada" C a t . No. The  t o t a l e x p e n d i t u r e o f m e t r o p o l i t a n households  using a s i m i l a r procedure.  can be  F o r a m e t r o p o l i t a n household  e s t i m a t e d consumption f u n c t i o n i s , c o r r e s p o n d i n g t o  (6.24)  y  3  = a  (6.25)  computed j , the  (6.22),  + b^ z?  because A^ = 0. litan  13-210, 1971-81.  A c c o r d i n g l y , t o t a l e x p e n d i t u r e i s , f o r J metropo-  households.  E y j=l  3  = Ja  + b -  J E z? ~j=l  which i s o b t a i n a b l e g i v e n J and 1.2? from the same d a t a s o u r c e . T h e r e f o r e , the f i r s t non-metropolitan expenditure expenditure  component o f S  yAt  (6.23) i n the sum  (6.23) and m e t r o p o l i t a n e x p e n d i t u r e  Other components o f S  yAt  can be  i s j u s t the share o f of  non-metropolitan  (6.25), i n y e a r t .  c a l c u l a t e d by a s i m i l a r  procedure.  - 92  3.  Expenditure  -  distribution statistics  S  yyt  T h i s i s a s t a t i s t i c t h a t depends on both the and the magnitude o f e x p e n d i t u r e .  (6.26)  S  =  EyJ( t  yyt  Since expenditure procedure  iny!| t  J  J  distribution  Formally,  )/Y. t  i s used f o r y , as opposed t o income, a s i m i l a r  o f mapping a f t e r - t a x income d i s t r i b u t i o n t o e x p e n d i t u r e  distribution i s required.  However, the c r o s s - s e c t i o n consumption  f u n c t i o n f o r m u l a t e d f o r t h i s purpose does not i n c l u d e any dummy variables since y S  y t  i g n o r e s household  attributes.  Consequently,  the r e g r e s s i o n e q u a t i o n i s s i m p l y  (6.27)  y  h  where y , z turbance  =  a  5  + b  5  z  h  +  h v 5  and v,_ are the e x p e n d i t u r e , a f t e r - t a x income and  r e s p e c t i v e l y o f household  h, h = 1,  H.  dis-  I t i s assumed  t h a t t h e r e i s no contemporaneous c o v a r i a n c e s i n the d i s t u r b a n c e so t h a t OLS  i s b e s t l i n e a r unbiased and t h i s r e g r e s s i o n e q u a t i o n i s  e s t i m a t e d u s i n g the c r o s s - s e c t i o n d a t a s e t . i n Appendix B, column  The  term  The  r e s u l t s are  found  5.  a p p r o p r i a t e income d a t a can be  D i s t r i b u t i o n s by S i z e i n Canada" No.  found i n "Income A f t e r - t a x ,  13-210, 1971-81.  In each y e a r ,  -  the income spectrum o f households  9 3  -  i s d i v i d e d i n t o income b r a c k e t s and the number  i n each b r a c k e t i s p r o v i d e d .  t h e r e f o r e : assume t h a t every household r e c e i v e s the m i d - p o i n t  y  h  =  a  c  D  + b  c  z  h  0  i n t h a t b r a c k e t have the same e x p e n d i t u r e ,  1*1  a number f o r the sum  of y  Zy  i n a l l brackets. sum  log y  The procedure h  b r a c k e t s , and  using  (6.27)  Assuming t h a t a l l households  i s calculated.  is  i n a p a r t i c u l a r bracket  income, and e s t i m a t e i t s e x p e n d i t u r e by  the e s t i m a t e d e q u a t i o n , from  (6.28)  The mapping procedure  h  o v e r a l l households  i n that bracket  i s then r e p e a t e d f o r o t h e r income  li logy  i s o b t a i n e d by summing o v e r the p a r t i a l sums  Finally,  s y  by the t o t a l e x p e n d i t u r e  y  t  I  s  i n a l l brackets.  a d m i t t e d l y a rough a p p r o x i m a t i o n i s r a t h e r s m a l l and  o b t a i n e d by d i v i d i n g the  overall  This process i s  but s i n c e the range o f  expenditure  , not v e r y s e n s i t i v e t o e x p e n d i t u r e d i s t r i b u yyt t i o n , the margin o f e r r o r i n v o l v e d i s not l i k e l y t o be s i g n i f i c a n t .  4.  S  P r i c e i n d e x e s , p.  i t  The 6 composite  goods c l a s s i f i e d i n the p r e s e n t model are  s i m i l a r t o those c l a s s i f i e d Consumer P r i c e Index" No.  i n the p r i c e s e r i e s p u b l i s h e d i n  62-001.  f o r the p r i c e s e r i e s , a l t h o u g h one  "The  Almost no r e c o m p i l a t i o n i s r e q u i r e d exception i s that i n that price  - 94  series,  "food" and  goods.  To combine them i n t o one  -  "tobacco and a l c o h o l " are c l a s s i f i e d as composite  separate  good, the p r i c e index f o r  food i s o b t a i n e d as a weighted-average o f the p r i c e indexes f o r "food" and  "tobacco and a l c o h o l " , the weights  on the two  goods d i v i d e d by the sum,  for a particular  year.  A f i n a l note about the p r i c e  s e r i e s used here  i s that a l l  p r i c e indexes have been n o r m a l i z e d survey year, 1978,  5.  b e i n g the  expenditures  so t h a t the p r i c e indexes  are a l l u n i t y , as mentioned i n Chapter  Heteroscedasticity correction factor p  As e x p l a i n e d i n Chapter  f o r the  4.  2  5, the f o l l o w i n g f a c t o r i s n e c e s s a r y  to c o r r e c t f o r h e t e r o s c e d a s t i c i t y i n the t i m e - s e r i e s model,  (6.29)  The  p  t  =  computation  D(p )" ((Y ) /Z(y^) ) 1  of D(p^ )  of E ( y  t i o n generated  2  i 5  t  h computation  2  t  -1  and  (  )  2  i s straight  forward.  The  2 )  i s performed  by u s i n g the e x p e n d i t u r e  i n the course o f computing S  .  A g a i n , each  informahousehold  yyt w i t h i n an income b r a c k e t i s assumed t o r e c e i v e the m i d - p o i n t U s i n g the e s t i m a t e d consumption f u n c t i o n (6.28), the expenditure  i s o b t a i n e d which i s then squared  e s t i m a t e d number o f households o b t a i n e d by summing "the sum  income.  corresponding  and m u l t i p l i e d by h 2  i n that bracket.  E(y  )  the  i s then  o f s q u a r e s " over a l l b r a c k e t s .  This  - 95 -  completes t h e d i s c u s s i o n o f t i m e - s e r i e s data g e n e r a t i o n . set o f t i m e - s e r i e s data can be found  A complete  i n Appendix C.  Since t h e c o v a r i a n c e m a t r i x i n t h e t i m e - s e r i e s model i s s i n g u l a r , the t r a n s p o r t a t i o n e q u a t i o n i s dropped.  Time-series  e s t i m a t i o n i s performed by a p p l y i n g J o i n t G e n e r a l i z e d L e a s t Squares on a system o f 5 e q u a t i o n s ,  (5.34) o r (6.8), each h a v i n g t h e same  e x p l a n a t o r y v a r i a b l e s , s u b j e c t t o t h e f o l l o w i n g symmetry c o n s t r a i n t s ,  (6.30)  b  1  2  = b  2 1  , b  1  3  = b  3 1  ,  (6.31)  b  2  3  = b  3 2  , b  2  4  = b  4 2  , b  (6.32)  b  3 4  =b  , b  3  5  =b  (6.33)  b  4  = b  5  4 3  5  2  = b  5  , b  1 5  = b  5  1  2  4  namely B , can be found PP  Usinq  4 1  5 3  where 11 o b s e r v a t i o n s 1971 - 1981,  Section 3  5  = b  a r e used.  The e s t i m a t i o n r e s u l t s ,  i n Appendix D.  Estimated Equivalence  Scales  (5.39) and e s t i m a t e d v a l u e s f o r B and B . the pp pA  commodity-specific  equivalence  s c a l e s can be e s t i m a t e d .  Since  there  are 11 dummy v a r i a b l e s d e s c r i b i n g 4 a t t r i b u t e s , there a r e 120 p o s s i b l e  -  c o n f i g u r a t i o n s f o r A.  The  96  -  c o r r e s p o n d i n g 120  s c a l e s can be found i n Appendix E.  sets of equivalence  They are i n t u i t i v e l y v e r y  appealing.  One  can r e a d i l y n o t i c e t h a t , as a f u n c t i o n o f A, the  estimated  scales display c e r t a i n general trends.  Comparing the s c a l e s f o r m e t r o p o l i t a n a r e a o f r e s i d e n c e w i t h that f o r non-metropolitan  a r e a o f r e s i d e n c e , the m e t r o p o l i t a n s c a l e s  are h i g h e r f o r every good e x c e p t t r a n s p o r t a t i o n . The  apparent  i s t h a t , i n urban c e n t r e s , the c o s t - o f - l i v i n g i s h i g h e r and, a d d i t i o n , c e r t a i n goods and produced.  in  s e r v i c e s i n the r u r a l areas are home-  However, f a c i l i t i e s  so t h a t r u r a l households  reason  i n r u r a l areas are l e s s  concentrated  need more t r a n s p o r t a t i o n s e r v i c e s .  Comparing the s c a l e s f o r male head w i t h t h a t f o r female shows t h a t households  w i t h male heads have h i g h e r needs i n food,  s h e l t e r and t r a n s p o r t a t i o n but lower needs i n c l o t h i n g , and p e r s o n a l / m e d i c a l c a r e . the way  "sex o f household  heads u s u a l l y imply  recreation  These d i f f e r e n c e s are p r o b a b l y due head" i s d e f i n e d .  Households w i t h  to  female  s i n g l e - p a r e n t f a m i l i e s , so t h a t o t h e r members i n  these f a m i l i e s are p r o b a b l y c h i l d r e n .  Therefore, other  b e i n g e q u a l , compared w i t h a male-headed household, household  head  involves substituting a c h i l d  attributes  a female-headed  f o r an a d u l t .  - 97  Comparing the interesting. model, the  -  s c a l e s f o r d i f f e r e n t f a m i l y s i z e s i s most  As an i n f o r m a l t e s t o f the s t r u c t u r e o f the  s c a l e s not o n l y have t o be  i n c r e a s i n g with family s i z e  but must f a l l between c e r t a i n ranges i n a b s o l u t e s c a l e s e s t i m a t e d here a r e ,  i n general  showing s t r o n g economies o f s c a l e and  present  magnitude.  The  i n c r e a s i n g with family are a l l i n t u i t i v e l y  size,  appealing  3  i n magnitude.  In the o r d e r o f d e c r e a s i n g  degree o f economies o f  s c a l e , the 6 goods can be o r d e r e d as f o l l o w s : ( s t r o n g e s t ) , food, clothing  s h e l t e r , r e c r e a t i o n , personal/medical  o f the household does not  although there (below 24)  explained  i s a noticeable  t o the  thereafter.  f o u r t h age  a f f e c t the  increase  group  Although not v e r y  from the  and  To  proposed new sequential  54)  needs  and  group a decline  could  be  associated  groups.  income, b e i n g  i t has  been shown t h a t the  the measure o f u t i l i t y  i n e q u a l i t y index, i s e s t i m a b l e  approach.  required  f i r s t age  s i g n i f i c a n t , t h i s trend  summarize t h i s c h a p t e r :  equivalent  s c a l e s v e r y much,  (between 44 and  by the d i f f e r i n g l e v e l s o f a c t i v i t y and  w i t h d i f f e r e n t age  data are  care  (weakest).  Age  log  transportation  Only survey and  d i r e c t l y obtainable  used i n the  u s i n g a micro-macro  aggregate t i m e - s e r i e s  for successful estimation.  trans-  These data are  demand either  i n p u b l i c f i l e s o r i n d i r e c t l y a f t e r some simple  -  computations.  98  -  The e s t i m a t i o n r e s u l t s are however v e r y e n c o u r a g i n g  j u d g i n g from t h e e s t i m a t e d e q u i v a l e n c e s c a l e s i n Appendix E. f i n d i n g d e f i n i t e l y s u p p o r t s the e n t i r e approach t o i n e q u a l i t y measurement.  This  - 99  Chapter 6  1.  -  Footnotes  I n f o r m a t i o n on the p r o p o r t i o n  o f households w i t h o u t mortgage  i s taken from Household F a c i l i t i e s and 64-202 ( o c c a s i o n a l ) . The  Equipment 1977-81, Cat.  This s t a t i s t i c i s only available for  adjustment i s n e c e s s a r y because  (9) i n c l u d e s  No.  1981.  imputed r e n t  as  owned h o u s i n g .  2.  Estimated t o t a l expenditures c a l c u l a t e d according  a t t r i b u t e s may estimation  3.  The  to d i f f e r e n t  d e v i a t e w i t h i n 5% which would not a f f e c t t i m e - s e r i e s  results significantly.  only exception  i s i n the  f a m i l y s i z e changes from 3 t o 4 —  t r a n s p o r t a t i o n s c a l e s and  causing  s i n c e t r a n s p o r t a t i o n shows t h e s t r o n g e s t i s not e n t i r e l y s u r p r i s i n g and  a s l i g h t decrease.  where But  economies o f s c a l e , t h i s  c o n t r a d i c t o r y t o common sense.  - 100 -  CHAPTER 7  APPLICATIONS  Section 1  Introduction  I t might be u s e f u l t o r e c a l l t h e development through t h e previous chapters here.  I n Chapter  2, i t has been e s t a b l i s h e d t h a t  a new i n e q u a l i t y i n d e x which i s a b l e t o measure d i s t r i b u t i v e  price  e f f e c t s w i t h i n a r i g o r o u s s o c i a l w e l f a r e e v a l u a t i o n framework i s u r g e n t l y needed. pointed out. fills  The f a i l u r e o f c o n v e n t i o n a l indexes has a l s o been  A new index i s i n t r o d u c e d i n Chapter  3 which i n t h e o r y  the gap i n t h e l i t e r a t u r e and s h o u l d be a s i g n i f i c a n t  ment over t h e e x i s t i n g i n d e x e s .  Chapter  4 and 5 p r o v i d e  improve-  specification  f o r p r e f e r e n c e s and e s t i m a t i o n a l g o r i t h m s t o make'implementation o f the index p o s s i b l e .  Chapter  6 describes the actual estimation process,  the h a n d l i n g o f d a t a and the i n t e r m e d i a t e r e s u l t s o f e s t i m a t e d equivalence s c a l e s .  Although  the e s t i m a t i o n r e s u l t s are s a t i s f a c t o r y  and i n t u i t i v e l y r e a s o n a b l e , whether t h e new index performs p r a c t i c e has y e t t o be seen.  In p a r t i c u l a r , we want t o know i f i n  practice, i treally differs significantly  from t h e o t h e r indexes and  c a p t u r e s p r i c e e f f e c t s i n a p r e d i c t a b l e way. the n e c e s s a r y t e s t s .  well i n  T h i s chapter provides  I t i s o r g a n i z e d as f o l l o w s .  Section 2 discusses  the e s t i m a t e d market e q u i v a l e n c e s c a l e s (3.25) u s i n g t h e e s t i m a t e s o b t a i n e d i n Chapter i n Chapter  6.  Due t o the t r a n s l o g p r e f e r e n c e s  4, these s c a l e s a r e independent  t o the p o v e r t y - l i n e r a t i o s  (4.21).  specified  o f u t i l i t y and hence e q u a l  These s c a l e s a r e compared w i t h the  - 101 -  low-income c u t - o f f r a t i o s o f S t a t i s t i c s Canada t o p r o v i d e to t h e subsequent comparative  studies.  insights  S e c t i o n 3 attempts  t o show  t h a t the new index i s not v e r y s e n s i t i v e t o changes i n p ° e m p i r i c a l l y , thus the a r b i t r a r i n e s s problem i s not s e r i o u s . comparative  Section 4 i s a  study o f t h e new index w i t h o t h e r major i n d e x e s , u s i n g  the 1978 e x p e n d i t u r e  survey sample as t h e common d a t a s e t .  s t u d i e s d i s t r i b u t i v e p r i c e e f f e c t s on i n e q u a l i t y .  Section 5  A hypothetical  i n c r e a s e o f 10% i n t h e p r i c e o f each good i s c o n s i d e r e d i n t u r n . F i n a l l y , S e c t i o n 6 a p p l i e s t h i s new index t o measure i n e q u a l i t y i n Canada f o r 1975, 1979 and 1981, u s i n g income survey d a t a .  F o r com-  p a r i s o n , t h e same d a t a s e t s a r e employed u s i n g t h e w e l f a r e - r a t i o approach w i t h S t a t i s t i c s Canada low-income c u t - o f f v a l u e s .  Section 2  E s t i m a t e d Market E q u i v a l e n c e  As e x p l a i n e d i n Chapter  Scales  4, t h e t r a n s l o g s p e c i f i c a t i o n o f  t h i s model i m p l i e s t h a t market e q u i v a l e n c e s c a l e s (3.25) and p o v e r t y line ratios  (4.19) a r e e q u a l and can be e x p r e s s e d  as (4.21).  Using  the parameter e s t i m a t e s o b t a i n e d , t h e e s t i m a t e d s c a l e s a r e p r e s e n t e d i n Table 2 f o r a l l a t t r i b u t e c o n f i g u r a t i o n s from 1971 t o 1981.  In T a b l e 2, the r e f e r e n c e household o f age below 24, l i v i n g  i s an unattached  i n a m e t r o p o l i t a n a r e a , and h i s income i s  used as the denominator i n (3.25) t o generate to  male,  these r a t i o s .  (3.25) the r a t i o f o r t h i s r e f e r e n c e household  According  i s 1 f o r a l l years.  URBAN MALE HEAD HOUSEHOLD HEAD AGED 24 FAM. SIZE 1971 1.0000 1.4043 1.7458 1.9525 2 .6712 URBAN MALE HEAD HOUSEHOLD HEAD AGED 241971 FAM. SIZE 1.0469 1. 4709 1.8292 2 .0437 2 . 7931 URBAN MALE HEAD HOUSEHOLD HEAD AGED 34-  TABLE 2: TRANSLOG MARKET EQUIVALENCE SCALES OR BELOW . 1972 1973 1.0000 1.0000 1 .4039 1.4030 1 .7447 1.7429 1.9508 1.9484 2.6671 2.6617  1974 1.0000 1.4032 1.7434 1.9485 2.6625  1975 1.0000 1.4028 1.7424 1.9467 2.6594  1976 1.0000 1.4029 1.7423 1.9456 2.6554  1977 1.0000 1.4028 1.7421 1.9446 2.6519  1978 1.0000 1.4024 1.7409 1 .9421 2.6463  1979 1 .OOOO 1 .4033 1 .7429 1.9439 2.6503  1980 1.0000 1.4045 1.7456 1.9469 2.657 1  1981 1.0000 1.4051 1.7468 1.9465 2.6567  34 1972 1 .0477 1 .4716 1 .8294 2.0434 2.7909  1973 1 .0486 1 .4719 1.8291 2.0427 2.7877  1974 1 .0480 1 .4713 1 .8286 2.0417 2.7871  1975 1.0478 1.4705 1.8272 2.0393 2.7832  1976 1.0492 1.4726 1.8295 2.0409 2.7827  1977 1.0504 1.4743 1.8314 2.0422 2 . 7823  1978 1.0511 1 .4747 1 .8314 2.0409 2 . 7782  1979 1.05O0 1 .4742 1 .8315 2.0407 2.7796  1980 1.0484 1 .4733 1.8317 2.0408 2.7826  1981 1.0479 1.4731 1 .8319 2.0393 2.7806  1973 1.1829 1.6614 2.0647 2.3170 3.1718  1974 1.1824 1.6608 2.0643 2.3161 3.1713  1975 1. 1820 1.6599 2.0625 2.3132 3.1666  1976 1 .1826 1.6608 2.0634 2.3131 3. 1635  1977 1 . 1832 1 .6615 2.0641 2 .3130 3.1607  1978 1.1832 1 .6610 2.0628 2.3100 3.1539  1979 1.1824 1.6609 2.0636 2.3104 3.1564  1980 1.1813 1.6609 2.0650 2.3119 3. 1617  1981 1 . 1803 1.6601 2.0646 2.3095 3.1585  1973 1 .3152 1 .8475 2.2957 2.5760 3.5282  1974 1 .3154 1 .8479 2.2966 2.5764 3.5296  1975 1.3156 1.8477 2.2958 2.5744 3.5261  1976 1 . 3145 1 .8463 2.2937 2 . 57 10 3.5180  1977 1 .3136 1 .8450 2 . 2917 2.5678 3.5107  1978 1.3131 1 .8436 2.2893 2.5633 3.5016  1979 1 .3135 1.8455 2.2927 2.5665 3.5082  1980 1 .3142 1.8481 2.2974 2.5718 3.5190  198 1 1 . 3140 1 .8484 2.2986 2.5709 3.5178  1973 1.1316 1 .5896 1 .9755 2.2052 3.0124  1974 1.1319 1.5902 1.9766 2.2060 3.0142  1975 1.1327 1.5909 1.9769 2.2054 3.0128  1976 1.1319 1.5899 1.9754 2.2027 3.0062  1977 1.1312 1.5888 1.9738 2.2001 3.0002  1978 1.1313 1.5884 1.9727 2. 1975 2.9942  1979 1.1318 1.5902 1.9759 2.2006 3.0002  1980 1.1324 1.5924 1.9800 2.2051 3.0095  1981 1.1329 1.5937 1.9821 2.2056 3.0102  1973 0.9641 1 .354 1 1.6834 1.8765 2.5616  1974 0.9642 1 .3543 1.6840 1 .8768 2.5626  1975 0.9649 1.3549 1.6842 1.8763 2.5615  1976 0.9647 1.3547 1.6838 1.8750 2.5572  1977 0.9645 1 .3545 1.6833 1 .8737 2.5533  1978 0.9650 1 . 3547 1 .6830 1.8722 2.5493  1979 0.9649 1.3554 1 .6847 1.8737 2.5529  1980 0.9646 1.3562 1 .6868 1.8760 2 . 5586  1981 0.9648 1.3569 1 .6882 1 .8759 2.5585  44  FAM. SIZE 1971 1972 1 1. 1821 1 . 1825 2 1.6617 1 .6618 3 2 .0666 2.0660 4 2 . 3204 2.3192 . 5 3 .1811 3.1773 URBAN MALE HEAD HOUSEHOLD HEAD AGED 44-54 1971 1972 FAM. SIZE 1 .3158 1.3162 1.8495 1.8506 2.2991 2 . 3012 2.5805 2 . 5835 3.5373 3 . 5438 URBAN MALE HEAD HOUSEHOLD HEAD AGED 54-64 1971 FAM. SIZE 1972 1.1319 1. 1320 1 .5910 1.5917 1 .9781 1.9796 2.2086 2 .2107 3.0194 3 .0244 URBAN MALE HEAD HOUSEHOLD HEAD AGED OVER 64 1971 1972 FAM. SIZE 0.9638 1 0 .9635 1.3545 2 1 .3546 1.6846 3 1 .6852 1.8783 4 1 .8793 .5692 2.5660 5 2  URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW FAM. SIZE 1971 1972 0.8887 0.8885 1.2499 1.2500 1.5535 1.5542 1.7669 1.7681 2.4359 2.4394 FEMALE HEAD URBAN HOUSEHOLD HEAD AGED 24-34 FAM. SIZE 1971 1972 1 0.9301 0.9310 2 1.3092 1.3101 3 1.6283 1.6288 4 1.8505 1.8506 5 2.5505 2.5488 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 34-44 FAM. SIZE 197 1 1972 1 1.0616 1.0622 2 1.4951 1.4954 3 1 .8596 1.8594 4 2 . 1240 2.1231 5 2.9365 2.9331 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 44-54 FAM. SIZE 1971 1972 1 1 . 1791 1.1790 2 1 .6609 1.6602 3 2.0656 2.0641 4 2.3589 2.3564 5 3.2630 3.2573 FEMALE HEAD URBAN HOUSEHOLD HEAD AGED 54-64 FAM. SIZE 1971 1972 1 1.0014 1.0015 2 1.4105 1 .4103 3 1 .7536 1 7545 . 4 1.9931 1 .9915 5 2.7497 2.7455 FEMALE HEAD URBAN HOUSEHOLD HEAD AGED OVER 64 FAM. SIZE 1971 1972 1 0.8500 0.8504 2 1.1970 1 .1973 3 1.4894 1.4893 4 1.6896 1.6889 5 2.3293 2.3268  TABLE 2 (CONTINUED) 1973 0.8888 1.2492 1 .5520 1.7644 2.4305  1974 0.8877 1.2477 1.5504 1.7621 2.4278  1975 0.8862 1.2453 1.5470 1.7574 2.4208  1976 0.8870 1.2465 1.5483 1.7583 2.4197  1977 0.8877 1.2475 1.5494 1 .7588 2.4183  1978 0.8868 1.2458 1 .5467 1 .7543 2.4100  1979 0.8854 1.2447 1.5461 1 . 7532 2.4098  1980 0.8841 1.2439 1.5462 1 . 7532 2.4125  1973 0.9320 1.3105 1.6287 1.8497 2.5454  1974 0.9302 1.3082 1.6262 1.8463 2.5412  1975 0.9284 1.3053 1.6221 1.8409 2.5332  1976 0.9305 1.3084 1.6257 1.8443 2.5355  1977 0.9324 1.3110 1.6288 1.8470 2.5371  1978 0.9320 1.3100 1 .6270 1 .8435 2.5300  1979 0.9297 1.3075 1 .6246 1.8403 2.5272  1980 0.9269 1.3048 1.6223 1.8377 2.5263  1981 0.9240 1.3013 1.6184 1 .8318 2.5183  1973 1 .0626 1.4950 1.8582 2.1207 2.9271  1974 1.0607 1.4925 1.8553 2. 1 168 2.9223  1975 1.0586 1.4891 1.8506 2.1104 2.9129  1976 1 .0601 1 .4913 1.8532 2.1126 2.9133  1977 1.06 15 1.4932 1 .8553 2.1141 2 .9129  1978 1.0603 1 .4910 1 .8520 2.1085 2.9024  1979 1.0579 1.4886 1 .8498 2.1055 2.9000  1980 1.0554 1.4864 1.8483 2.1037 2.9006  1981 1.0518 1 .4820 1.8433 2.0965 2.8908  1973 1.1785 1.6584 2.0610 2.3518 3.2480  1974 1 . 1771 1 .6565 2.0590 2.3489 3.2445  1975 1.1753 1 .6536 2.0548 2.3430 3.2356  1976 1.1754 1.6539 2.0550 2.3423 3.2318  1977 1.1756 1 .6540 2.0549 2.3412 3.2276  1978 1.1738 1.6509 2.0503 2.3341 3 . 2146  1979 1.1724 1.6500 2.0501 2.3332 3.2153  1980 1.1712 1 .6499 2.0513 2.3346 3.2206  1973 1.0014 1.4092 1.7515 1 .9883 2.7388  1974 1.0004 1.4079 1.7503 1.9863 2.7366  1975 0.9994 1.4061 1.7475 1.9824 2.7305  1976 0.9996 1.4065 1 .7478 1 .9819 2.7274  1977 0.9998 1.4068 1 .7479 1.9812 2.7241  1978 0.9989 1.4050 1.7451 1.9765 2.7151  1979 0.9979 1.4045 1.7453 1.9762 2.7163  1980 0.9969 1.4043 1.7463 1.9772 2.7206  198 1 0.9947 1.4018 1.7437 1 .9727 2.7 146  1973 0.8508 1.1971 1.4884 1.6873 2.3225  1974 0.8498 1.1958 1.4870 1.6852 2.3202  1975 0.8490 1.1943 1.4848 1.6820 2.3151  1976 0.8496 1.1952 1.4858 1.6824 2.3136  1977 0.8502 1.1960 1.4866 1.6826 2.3120  1978 0. 8498 1 . 1950 1 .4848 1 .6794 2.3054  1979 0.8484 1.1939 1.4841 1.6781 2.3050  1980 0.8468 1 . 1927 1 .4836 1 .6776 2.3067  1981 0.8448 1.1903 1 .481 1 1 . 6733 2.3010  1981 .8818 . 24 12 . 5433 .7436 .4061  198 1 . 1680 .6460 .0472 . 3280 .2117  TABLE 2 (CONTINUED) RURAL MALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW FAM. SIZE 1971 1972 O . 8648 0.8646 1 .2133 1.2126 1 .5077 1.5063 1 .6743 1 .6725 2 .2815 2.2776 RURAL MALE HEAD HOUSEHOLD HEAD AGED 24-34 1971 1972 FAM. SIZE 0.9062 0 . 9058 1.2717 1 .2715 1.5803 1 .5805 1.7528 1 .7534 2.3846 2 . 3869 RURAL MALE HEAD HOUSEHOLD HEAD AGED 34-44 1971 FAM. SIZE 1972 1.0182 1 .0181 1 .4297 1 .4299 1.7766 1 . 7775 1.9804 1 .9818 2.7025 2 .7062 RURAL MALE HEAD HOUSEHOLD HEAD AGED 44-54 1971 1972 FAM. SIZE 1 . 1340 1 . 1345 1 .5924 1 .5938 1 .9787 1 .9810 2.2053 2 . 2083 3.0111 3 .0171 RURAL MALE HEAD HOUSEHOLD HEAD AGED 54-64 1971 1972 FAM. SIZE 0.9807 0 .9810 . 3782 1.3772 1 1.7115 1 .7133 1 .8975 1 .8998 2.5840 2 . 5887 MALE HEAD RURAL HOUSEHOLD HEAD AGED OVER 64 1971 1972 FAM. SIZE 0.8362 1 0 . 8361 1 .1740 2 1 . 1744 . 4604 1.4595 3 1 1.6159 4 1 .6172 2.1989 5 2 . 2021  1973 0.8643 1 .2115 1.5043 1.6700 2.2725  1974 0.8647 1 .2122 1.5055 1 .6710 2 . 2744  1975 0.8651 1 .2124 1.5053 1.6702 2.2729  1976 0.8650 1 .2124 1.5051 1.6690 2.2690  1977 0.8647 1 .2120 1.5044 1 .6678 2.2655  1978 0.8649 1.2119 1.5038 1.6661 2.2615  1979 0.8656 1 .2136 1.5066 1.6690 2.2667  1980 0.8664 1.2158 1.5103 1.6730 2.2746  1981 0.8674 1.2 178 1.5132 1.6747 2.2769  1973 0.9067 1.2716 1.5795 1.7517 2.3813  1974 0.9067 1.2717 1.5799 1 .7518 2.3820  1975 0.9068 1.2716 1.5793 1.7505 2.3798  1976 0.9080 1.2733 1.5812 1.7516 2.3790  1977 O.9088 1.2744 1.5824 1.7524 2.3781  1978 0.9095 1 .2750 1 .5827 1 .7517 2.3754  1979 0.9093 1.2755 1 .5840 1.7529 2.3785  1980 0.9088 1 . 2760 1.5856 1.7546 2.3832  1981 O.9094 1 . 2773 1.5878 1.7554 2.3843  1973 1.0183 1 .4289 1.7750 1.9781 2.6973  1974 1.0184 1 .4292 1.7756 1.9784 2.6983  1975 1.0185 1 .4290 1 . 7749 1.9768 2.6957  1976 1.0189 1.4295 1.7754 1.9764 2.6925  1977 1.0191 1.4298 1.7755 1.9759 2.6895  1978 1.0194 1 . 4298 1.7749 1 . 9740 2 .6848  1979 1 .0195 1 .4308 1 . 7770 1 .9759 2.6891  1980 1 .0195 1 . 4321 1.7797 1 .9789 2.6959  1981 1.0198 1 . 4332 1 .7816 1.9792 2.6964  1973 1 . 1331 1.5902 1.9751 2.2009 3.0027  1974 1.1338 1 .5914 1 .9769 2.2024 3.0056  1975 1 . 1345 1.5920 1.9771 2.2018 3.0041  1976 1.1334 1.5905 1.9751 2.1984 2.9966  1977 1.1323 1.5890 1.9729 2.1952 2.9897  1978 1.1322 1 .5882 1 .9713 2.1921 2.9831  1979 1.1334 1 .5910 1.9757 2.1966 2.991 1  1980 1.1350 1.5947 1 .9816 2.2031 3.0030  1981 1 . 1362 1.5970 1.9850 2.2049 3.0055  1973 0.9801 1.3755 1.7087 1 .8941 2.5774  1974 0.9808 1 .3767 1.7105 1.8957 2.5803  1975 0.9819 1.3779 1.7115 1.8962 2.5803  1976 0.9811 1.3768 1.7100 1.8935 2.5742  1977 0.9802 1.3756 1.7082 1.8909 2.5685  1978 0.9805 1.3755 1.7075 1 .8891 2.5641  1979 0.98 17 1.3781 1.7116 1.8933 2.57 13  1980 0.9831 1 .3813 1.7167 1.8988 2.5815  1981 0.9848 1.384 1 1.7207 1.9015 2.5852  1973 0.8361 1.1732 1.4579 1.6139 2.1945  1974 0.8366 1.1740 1.4591 1.6149 2.1966  1975 0.8375 1 . 1750 1.4600 1 .6153 2.1966  1976 0.8373 1.1748 1.4595 1.6139 2. 1925  1977 0. 8369 1. 1742 1.4586 1 .6124 2.1888  1978 0.8375 1.1746 1.4587 1.6116 2.1859  1979 0.8380 1 . 1762 1 .4612 1.6141 2.1908  1980 0.8385 1.1779 1.4644 1.6175 2.1976  1981 0.8397 1.1800 1 .4674 1 .6194 2.2002  TABLE RURAL F E M A L E HEAD H O U S E H O L D H E A D A G E D 2 4 OR BELOW FAM. SIZE 1971 1972 1 0.7549 0.7549 2 1.0610 1.0608 3 1.3186 1.3179 4 1.4896 1.4883 5 2.0470 2.0439 RURAL F E M A L E HEAD HOUSEHOLD HEAD AGED 2 4 - 3 4 FAM. S I Z E 1971 1972 1 0.7906 0.7913 2 1.1119 1 . 1124 3 1.3823 1.3825 4 1.5598 1.5597 5 2.1414 2.1397 RURAL F E M A L E HEAD HOUSEHOLD HEAD AGED 3 4 - 4 4 FAM. S I Z E 1971 1972 1 0.8983 0.8987 2 1.2640 1.2641 3 1.5715 1.5711 4 1.7822 1.7813 5 2.4542 2.4512 RURAL F E M A L E HEAD HOUSEHOLD HEAD AGED 4 4 - 5 4 FAM. SIZE 1971 1972 1 0.9986 0.9983 2 1.4053 1.4045 3 1.7469 1.7454 4 1.9809 1.9786 5 2.7293 2.7243 RURAL F E M A L E HEAD HOUSEHOLD HEAD AGED 5 4 - 6 4 FAM. S I Z E 1971 1972 1 0.8526 0.8526 2 1.1999 1.1995 3 1.4918 1.4908 4 1.6827 1.6811 5 2.3123 2.3086 RURAL F E M A L E HEAD H O U S E H O L D H E A D A G E D OVER 6 4 FAM. S I Z E 1971 1972 0.7246 1 0.7249 1.0196 1.0196 2 1 .2681 1 .2677 3 1.4284 1 .4276 4 1.9614 1.9591 5  2  (CONTINUED)  1973 0.7549 1.0600 1.3164 1.4862 2.0392  1974 0.7543 1.0594 1.3158 1.4852 2.0383  1975 0.7534 1.0578 1.3135 1.4820 2.0334  1976 0.7540 1.0586 1.3144 1.4823 2.0319  1977 0.7545 1.0593 1 .3150 1 .4824 2.0304  1978 0.7539 1.0582 1.3133 1 .4794 2.0245  1979 0. 7534 1.0582 1.3138 1.4796 2.0261  1980 O.7529 1.0584 1.3151 1 .4810 2.0300  1981 0.7519 1.0574 1.314 1 1 .4787 2.0269  1973 O. 7 9 1 9 1.1126 1.3822 1.5588 2.1368  1974 0.7909 1.1113 1.3808 1 .5569 2. 1346  1975 O. 7 8 9 8 1.1094 1.3780 1.5531 2.1289  1976 0.7914 1.1118 1.3808 1.5555 2.1303  1977 0.7928 1.1137 1.3831 1 .5575 2.1312  1978 O.7928 1.1133 1 .3821 1 .5553 2. 1263  1979 0.7914 1.1121 1 .3813 1 .5540 2.1258  1980 0.7898 1.1108 1.3805 1.5531 2.1268  1981 O.7882 1.1091 1.3788 1.5499 2.1225  1973 0.8989 1.2636 1.5698 1.7792 2.4462  1974 0.8978 1.2622 1.5684 1 .7770 2.4438  1975 0.8965 1.2599 1.5651 1.7726 2.4371  1976 0.8976 1.2616 1.5670 1.7739 2.4368  1977 0.8986 1.2629 1.5684 1.7749 2.4360  1978 0.8979 1 . 2616 1 .5663 1 . 771 1 2.4286  1979 0.8967 1.2606 1.5658 1.7701 2.4287  1980 0.8953 1.2598 1.5658 1.7701 2.4313  1981 0.8933 1.2575 1.5634 1.7660 2.4257  1973 .9978 .4028 .7426 .9747 .7165  1974 0.9971 1.4020 1 .7419 1.9734 2.7153  1975 0.9961 1.4002 1.7392 1.9695 2.7093  1976 0.9960 1.4002 1.7389 1.9684 2.7053  1977 0.9959 1.4000 1.7385 1.9671 2.7013  1978 0.9948 1.3980 1.7354 1.9621 2.6919  1979 0.9945 1.3984 1.7367 1.9631 2 .6949  1980 0.9943 1.3995 1.7392 1.9658 2.7016  1981 0.9928 1.3978 1.7377 1.9625 2.6971  1973 0.8523 1.1983 1.4888 1.6783 2.3028  1974 0.8519 1.1979 1.4885 1.6776 2.3023  1975 0. 8515 1. 1 9 7 0 1.4870 1.6751 2.2984  1976 0.8515 1.1970 1.4869 1.6743 2.2952  1977 0.8515 1.1970 1.4867 1.6734 2.2920  1978 0. 8 5 1 0 1 . 1959 1.4849 1.6702 2.2855  1979 0.8509 1 . 1965 1 .4862 1 .6713 2.2885  1980 0.8507 1.1974 1.4883 1.6736 2.294 1  1981 0.8499 1.1967 1.4879 1.6718 2 . 2916  1973 0. 7251 1.0193 1.2668 1.4261 1.9553  1974 0.7246 1.0187 1.2663 1.4252 1.9546  1975 0.7243 1.0180 1.2650 1.4231 1.9513  1976 0.7247 1 .0185 1.2656 1 .4231 1.9495  1977 0.7250 1.0190 1.2660 1.4231 1.9478  1978 0. 7249 1 .0185 1 . 2650 1.4209 1 .9431  1979 0.7243 1.0184 1 .2654 1 .4210 1 .9445  1980 O.7236 1.0183 1.2661 1 .4218 1.9476  1981 O.7227 1.0174 1.2654 1.4199 1.9449  O  - 106  -  Table 2 shows t h a t , i n g e n e r a l , f o r each a t t r i b u t e  configura-  t i o n , market e q u i v a l e n c e s c a l e s are not v e r y s e n s i t i v e t o the p r i c e changes e x p e r i e n c e d i n 1971 - 1981. attributes.  The  However, they do change w i t h  s c a l e s are i n c r e a s i n g w i t h f a m i l y s i z e a l t h o u g h  d e c r e a s i n g increments  by  (except f o r the change from s i z e 4 t o s i z e 5  because the l a t t e r c a t e g o r y i n c l u d e s a l l l a r g e r f a m i l y s i z e s ) . Furthermore,  the s c a l e s are i n g e n e r a l s l i g h t l y h i g h e r f o r  w i t h male heads than those w i t h female  heads.  goods c o n s i d e r e d , f a m i l i e s w i t h female  heads have s l i g h t l y  consumption r e q u i r e m e n t s .  T h i s i s a reasonable  w i t h female heads are u s u a l l y s i n g l e - a d u l t n o r m a l l y consume l e s s .  One  T h i s means t h a t , a l l  r e s u l t as  f a m i l i e s and  families  children  than n o n - m e t r o p o l i t a n  ( r e f l e c t i n g the higher " c o s t - o f - l i v i n g " —  i n the urban c i t i e s ) .  lower  a l s o notes w i t h i n t e r e s t t h a t the s c a l e s  are h i g h e r f o r m e t r o p o l i t a n households holds  households  house-  broadly defined  —  F i n a l l y , the s c a l e s v a r y w i t h age o f head i n  the same manner as the c o m m o d i t y - s p e c i f i c i . e . , i n c r e a s e s l i g h t l y w i t h age  e q u i v a l e n c e s c a l e s do,  up t o the 44-54 b r a c k e t and  decrease  thereafter.  Because o f t r a n s l o g p r e f e r e n c e s , market e q u i v a l e n c e s c a l e s and p o v e r t y - l i n e r a t i o s are i d e n t i c a l as g i v e n by  (4.21).  Thus  Table 2 can be r e g a r d e d as the e s t i m a t e d t r a n s l o g p o v e r t y - l i n e r a t i o s . I n t u i t i v e l y , these r a t i o s i n d i c a t e the number o f e q u i v a l e n t maleadults  (of age below 24 i n a m e t r o p o l i t a n area)  f o r each  household.  - 107  -  There are o t h e r d a t a sources from which one lent-adults ratios.  As suggested  i n Wolfson  can d e r i v e these (1979), one  can use  low-income c u t - o f f v a l u e s p u b l i s h e d by S t a t i s t i c s Canada. o f f v a l u e s are c l a s s i f i e d by f a m i l y s i z e and Using a m e t r o p o l i t a n unattached one  u s i n g c u t - o f f v a l u e s p u b l i s h e d f o r 1975  the  These c u t -  s i z e of area of residence.  i n d i v i d u a l as the r e f e r e n c e  can compute s i m i l a r p o v e r t y - l i n e r a t i o s .  equiva-  Table  and 1981."'"  household,  3 g i v e s the  results  These r a t i o s show  s i g n i f i c a n t economies o f s c a l e and t h a t n o n - m e t r o p o l i t a n  households  have o v e r a l l lower-consumption needs.  The  r a t i o s i n T a b l e 2 and Table  3 can be compared,  although  the d i f f e r e n c e i n the c l a s s i f i c a t i o n o f a t t r i b u t e s p r e c l u d e s on a one-to-one b a s i s . d e r i v e d from t h e two  comparison  However, the g e n e r a l i m p r e s s i o n i s the  ratios  d i s t i n c t l y d i f f e r e n t approaches are r a t h e r c l o s e .  T h i s f i n d i n g has s i g n i f i c a n t b e a r i n g on the f o l l o w i n g comparative study o f v a r i o u s i n e q u a l i t y  Section 3  indexes.  S e n s i t i v i t y o f the IEI index t o  p°  Before comparing t h e v a r i o u s i n e q u a l i t y i n d e x e s , i t i s u s e f u l t o f i r s t examine t h e s e n s i t i v i t y o f the new Equivalent-Income  index, named the  (IEI) i n e q u a l i t y index, t o changes i n the  Individual arbitrary  o price vector p  empirically.  The  by u s i n g the F a m i l y E x p e n d i t u r e 9285 households  and  sensitivity analysis i s  Survey  27651 i n d i v i d u a l s .  1978  undertaken  sample which c o n t a i n s  F o l l o w i n g the  procedure  - 108 -  Table 3 Statistics  Family Size  Canada Low-Income C u t - o f f R a t i o s  1975  1981  Metropolitan  NonMetropolitan  Metropolitan  NonMetropolitan  1  1.0000  0.8241  1.0000  0.8255  2  1.4496  1.1951  1.3165  1.0828  3  1.8496  1.5250  1.7604  1.4514  4  2.1997  1.8131  2.0326  1.6776  5  2.4588  2.0275  2.3610  1.9483  6  2.6996  2.2252  2.5770  2.1258  7  2.9697  2.4395  2.8403  2.3433  Source:  Income D i s t r i b u t i o n by S i z e i n Canada 1975, 1981. Catalogue  Notes:  No. 13-207  (annual)  (1) Given f a m i l y s i z e and a r e a o f r e s i d e n c e , a low-income  cut-off  v a l u e i s d e r i v e d a r b i t r a r i l y by s e t t i n g i t e q u a l t o t h e average  observed  income o f those households  more than t h e average  Canadian household  who spend 20%  does on food,  c l o t h i n g and s h e l t e r . (2) The c u t - o f f v a l u e s f o r m e t r o p o l i t a n households  a r e taken as  the means o f t h e c u t - o f f v a l u e s f o r the 3 p o p u l a t i o n b r a c k e t s : over 500,000, 100,000-499,999 and 30,000-99,9999. values f o r non-metropolitan of  households  t h e c u t - o f f v a l u e s f o r the r e m a i n i n g  l e s s than 30,000 and r u r a l .  The c u t - o f f  a r e taken as t h e means 2 population brackets:  - 109 -  e x p l a i n e d i n Chapter  3, e q u i v a l e n t income i s imputed  d u a l t a k i n g 1978 p r i c e s as p. s e t equal t o 0.5, -1 and  t o each  indivi-  A t k i n s o n index i s then computed f o r r Computation i s r e p e a t e d f o r p ° s e t e q u a l  2 t o the a c t u a l p r i c e s e x p e r i e n c e d i n v a r i o u s y e a r s from 1971 t o 1981.  The  r e s u l t s a r e p r e s e n t e d i n T a b l e 4.  A l t h o u g h t h e r e i s no  benchmark by which assessment o f s e n s i t i v i t y can be made, t h e g e n e r a l i m p r e s s i o n i s t h a t the new I E I i n d e x i s n o t v e r y s e n s i t i v e t o p ° . I t ranges  from,  f o r r = 0.5, 0.046909 f o r 1981 p r i c e s t o a maximum  o f 0.049461 f o r 1971 p r i c e s , i . e . , a d e v i a t i o n range  Section 4  Comparative  o f ±2.6%.  Study o f V a r i o u s Measures  As d e s c r i b e d i n Chapter  2, v a r i o u s measures o f u t i l i t y  have  been used f o r i n e q u a l i t y measurement and because they a r e a l l uns a t i s f a c t o r y , a new i n d e x , t h e I E I i n d e x , i s developed i n Chapter 3. I t i s i n t e r e s t i n g , however, t o compare e m p i r i c a l l y these indexes and see i f they a r e r e a l l y s i g n i f i c a n t l y  different.  To c a r r y o u t t h i s comparative the F a m i l y E x p e n d i t u r e Survey 9285 households  study, a common d a t a s e t , namely  1978 sample, i s used which c o n t a i n s  and 27651 i n d i v i d u a l s ,  p i s taken as 1978 p r i c e s and  p ° , i f a p p l i c a b l e i s taken as 1971 p r i c e s . i n f o r m a t i o n on household  The sample c o n t a i n s  e x p e n d i t u r e s and a t t r i b u t e s f o r each house-  h o l d , so t h a t e q u i v a l e n t income each o f the 27651 i n d i v i d u a l s .  (3.19) o r (4.13) can be imputed t o The r e s u l t s a r e p r e s e n t e d i n T a b l e 5.  Table  4:  Sensitivity  of  IEI measure  r=.5  p  Data  r=  P°  u  €  I  €  I  1971  4915.5G  4672.43  .049461  4009.63  . 184299  565.24  .885011  1973  5469.58  5207.04  .048000  4489.12  . 179256  650.93  .880991  1975  6688.20  6371.17  .047401  5503.17  . 177182  807.35  .879288  1977  7775.47  7405.16  .047626  6391.74  . 177962  933.59  .879932  1978  8411.36  8019.14  .046629  6943.50  .174507  1034.23  .877044  1979  9155.78  8731.40  .046351  7566.92  .173536  1133.31  .876220  1981  11315.12  10784.35  .046909  9329.60  .175475  1382.OO  .877863  : 1978  No.  to  prices  of entries Set:Family  : 27651 Expenditure  Survey  1978  I  Table  1.  "Income"  No.  definition  entries  5:  Comparison  of  of D i f f e r e n t  r=.5  Measures  r=0  ^  r=-1  5  r=-2 I  <=  r = -oo I  €  I  Household Expenditure (HE)  9285  15159.73  14055.78  .072821  12920.24  .147726  10602.41  .300620  8451.11  .442529  2155.00  .857847  2. P e r C a p i t a Expenditure (PCE)  27651  5100.91  4765.96  .065700  4461.80  .125295  3923.42  .230839  3454.99  .322673  808.20  .841558  9285  9119.93  8622.06  .054591  8146.76  .106707  9119.93  .204757  6433.84  .294529  1170.00  .871710  Individual Inflated Welfare27651 Ratio(IIWR) (Stat.Can. )  9006.28  8586.14  .046650  8183.37  .091371  7417.71  .176385  6698.10  .256285  1170.00  .870091  Individual Inflated WelfareRatio(IIWR) (estimated)  27651  8411.33  8019.11  .046630  7645.73  .091020  6943.50  .174507  6289.10  .252306  1034.23  .877044  9285  5081.52  4797.56  .055880  4529.06  .108719  4029.71  .206986  3573.10  .296844  565.24  .888767  27651  4915.56  4672.43  .049461  4441.74  .096392  4009.63  .184299  3608.94  .265813  565.24  .885011  3.  4.  5.  6.  7.  Household Inflated WelfareRatlo(HIWR) (Stat.Can.)  Household Equivalent Income(HEI ) Individual Equivalent Income(IEI)  p ° : 1971 Data  prices  Set:Family  Expenditure  Survey  1978  - 112 -  In  each case, i n e q u a l i t y i s c a l c u l a t e d u s i n g the A t k i n s o n index w i t h  r s e t equal t o 0.5, 0, -1, -2 and -<».  u i s the mean o f the d i s t r i b u -  t i o n and £ i s the e v e n l y - d i s t r i b u t e d e q u i v a l e n t o f t h e d i s t r i b u t i o n w h i l e I i s the A t k i n s o n index d e f i n e d i n (3.32) o r (3.36) (3.37) w i t h the c o r r e s p o n d i n g u t i l i t y measure i n each c a s e .  In  Table 5, the f i r s t i n e q u a l i t y measure computed i s t h e  Household E x p e n d i t u r e  (HE) index which n e g l e c t s f a m i l y s i z e , even  d i s t r i b u t i o n o f s o c i a l weights  and o t h e r household  attributes.  The  mean o f t h e d i s t r i b u t i o n s i s 15159.73. F o r r = 0.5,5 i s 14055.78 g i v i n g an i n e q u a l i t y measure o f 0.072821. second  An improved index i s t h e  index computed, namely t h e P e r C a p i t a E x p e n d i t u r e  T h i s method imputes p e r - c a p i t a e x p e n d i t u r e r i s e t o a s i g n i f i c a n t decrease  index has i g n o r e d t h i s  The  t o each i n d i v i d u a l  o f t h e mean.  s u g g e s t i n g t h a t l a r g e r households  (PCE) i n d e x .  Inequality also  giving drops  have h i g h e r incomes w h i l e . t h e HE  correlation.  PCE index i g n o r e s economies o f s c a l e and o t h e r r e l e v a n t  a t t r i b u t e s t h a t might a f f e c t p r e f e r e n c e s . Card and F l a t t e r s  Wolfson  (1979) and Beach,  (1981) employ w e l f a r e r a t i o s as u t i l i t y measures t o  capture the scale e f f e c t s .  Although t h e y use income d a t a , what they  would have done w i t h e x p e n d i t u r e d a t a would be t o d i v i d e  household  e x p e n d i t u r e by t h e p o v e r t y income f o r the a p p r o p r i a t e a t t r i b u t e s and i n e q u a l i t y i s computed f o r a d i s t r i b u t i o n o f t h e s e  welfare-ratios.  - 113  -  To make comparisons with o t h e r indexes more immediate, w e l f a r e - r a t i o s " are used here  "inflated  i n place of welfare-ratios.  "Inflated  w e l f a r e - r a t i o s " are w e l f a r e r a t i o s m u l t i p l i e d by the p o v e r t y income o f a r e f e r e n c e household, metropolitan area.  namely, an unattached  individual in a  S i n c e the A t k i n s o n index i s r e l a t i v e , the  i s not a f f e c t e d by t h i s m o d i f i c a t i o n .  I f one  uses the c u t - o f f  p u b l i s h e d by S t a t i s t i c s Canada, then the " i n f l a t e d w i l l be j u s t household i n Table  index values  welfare-ratios"  e x p e n d i t u r e d i v i d e d by the a p p r o p r i a t e r a t i o s  3.  In T a b l e 5, the t h i r d index, namely Household I n f l a t e d Ratio  (HIWR) ( S t a t . Can.)  index, i s computed by imputing  w e l f a r e r a t i o s " t o each o f t h e 9285 households ratios  based  ( i d e n t i c a l with t h o s e p u b l i s h e d f o r 1978)  the method a c c o r d i n g t o Wolfson shows a f u r t h e r decrease  impute t o a l l 27651 i n d i v i d u a l s .  1975  3.  This i s  (1979) and Beach e t . a l . (1981).  i n inequality.  However, as e x p l a i n e d i n Chapter  "inflated  on the  i n Table  Welfare-  I i s 0.054591 a t r =  It  0.5.  3, a more a c c e p t a b l e procedure  i s to  Making t h i s a l t e r a t i o n , the f o u r t h  index named the I n d i v i d u a l I n f l a t e d Welfare  Ratio  (IIWR) ( S t a t .  Can.)  index i s computed which i n t e r e s t i n g l y , i s a p p r e c i a b l y s m a l l e r than the HIWR ( S t a t . Can.)  index.  I i s 0.046650 a t r = 0.5.  This  t h a t the c o n v e n t i o n a l w e l f a r e r a t i o index o f f e r s a d i s t o r t e d o f the a c t u a l i n e q u a l i t y s i t u a t i o n , hence s h o u l d be  avoided.  suggests picture  - 114  The  -  " i n f l a t e d w e l f a r e r a t i o s " are a c t u a l l y  denominated by the p o v e r t y - l i n e r a t i o s . income c u t - o f f s are a r b i t r a r i l y d e r i v e d . to  The  expenditures  S t a t i s t i c s Canada  It i s therefore interesting  employ the e s t i m a t e d t r a n s l o g p o v e r t y - l i n e r a t i o s f o r 1978,  as  g i v e n i n T a b l e 2 and r e p e a t the above i n e q u a l i t y computation. g i v e s r i s e t o the I n d i v i d u a l I n f l a t e d W e l f a r e - R a t i o index i n T a b l e 5.  As shown, the two  close empirically. T a b l e 2 and T a b l e different.  (IIWR)  IIWR indexes are a c t u a l l y  3, the p o v e r t y - l i n e r a t i o s are not  r e g u l a r l y updated f o r r e l a t i v e p r i c e  significantly  i n d e x e s , the IIWR (est.) i s  l a s t two  indexes computed make use o f the e q u i v a l e n t 3.  The  sixth  E q u i v a l e n t incomes are imputed t o households,  opposed t o i n d i v i d u a l s .  I t i s p r e s e n t e d here  e m p i r i c a l l y i t does make a d i f f e r e n c e i f one  index as  j u s t t o show t h a t imputes t o  r a t h e r than i n d i v i d u a l s , r e g a r d l e s s o f the u t i l i t y —  not  changes.  income measure o f u t i l i t y e x p l a i n e d i n Chapter  concept  households  measurement  e q u i v a l e n t income o r i n f l a t e d w e l f a r e r a t i o .  The  last  index, the I n d i v i d u a l E q u i v a l e n t Income (IEI) index i s the new proposed  very  T h i s comes as no s u r p r i s e because comparing  In c h o o s i n g between the two  i s erroneous.  This  (estimated)  p r e f e r r e d because S t a t i s t i c s Canada low-income c u t - o f f s are  The  low-  i n the p r e s e n t t h e s i s .  e x p l a i n e d i n Chapter  3.  index  I t s s o c i a l welfare foundation i s  F o r r = 0.5,  i s a p p r e c i a b l y d i f f e r e n t from the HEI  the I E I index i s 0.049461 which index.  However, one  cannot  - 115 -  conclude t h a t I E I and IIWR (estimated) are e m p i r i c a l l y The  different.  r e a s o n i s t h e I E I index depends on p ° which i s somewhat a r b i t r a r y .  I f p ° i s taken as 1978 p r i c e s , then i t i s easy t o see t h a t , i n (3.23), IT (•)  i n the denominator w i l l v a n i s h so t h a t e q u i v a l e n t income i s  i d e n t i c a l t o an i n f l a t e d w e l f a r e r a t i o .  I n t h a t c a s e , the I E I  index and t h e IIWR (estimated) indexes w i l l c o i n c i d e .  Therefore,  based on T a b l e 5, one can a s s e r t t h a t , the IIWR ( S t a t . Can.) i n d e x , the IIWR ( e s t . ) index and the proposed  IEI index are s i m i l a r e m p i r i -  c a l l y b u t as a group a r e d i f f e r e n t from t h e o t h e r f o u r i n d e x e s .  However, t h e proposed two  IIWR i n d e x e s .  Firstly,  I E I index has two advantages o v e r t h e  i t i s j u s t i f i a b l e i n terms o f s o c i a l  3 welfare e v a l u a t i o n .  Secondly, i t c a p t u r e s q u a n t i t a t i v e l y  t i v e p r i c e e f f e c t s on i n e q u a l i t y . both S(') and TT (•)  distribu-  R e f e r r i n g a g a i n t o (3.23), s i n c e  a r e s e n s i t i v e t o p r i c e s , t h e I E I i n d e x i s more  p r i c e - s e n s i t i v e than the o t h e r two i n d e x e s .  Section 5  Distributive Price  The poor households,  effects  r e l a t i v e t o the r i c h households,  spend  a l a r g e r p r o p o r t i o n o f t h e i r household budgets on t h e n e c e s s i t i e s . Roberts  (1982) i s an attempt  t o show t h a t i n c r e a s e s i n food p r i c e  a f f e c t the c o s t - o f - l i v i n g indexes o f t h e poor more than t h a t o f the rich.  I t seems r e a s o n a b l e t o c o n j e c t u r e t h a t food p r i c e  inflation  might have a n e g a t i v e impact on t h e i n e q u a l i t y s i t u a t i o n . IEI index i s a v a l u a b l e t o o l t o demonstrate  t h i s impact  The new  empirically.  - 116  -  To examine the p r i c e - s e n s i t i v i t y o f i n e q u a l i t y , each good i s r a i s e d i n t u r n by s e t i s a g a i n the taken as 1971  The  i n t e r e s t i n g r e s u l t i s that increases  2.3%  0.9%  broadly consistent  increase  i n recreation In the  luxuries.  On  the  m e d i c a l care and i s not  6  IIWR (est.) index, the  the that  l a t t e r two  c a s e s , the  F o r example, a 10%  prices  i f one  and  IEI  f i x e s p° and  are  increase  i n food,  decrease i n i n e q u a l i t y .  This  0.5, is  and  personal/  inequality.  c l a s s i f i e d as n e c e s s i t i e s o r  This  aggreluxuries.  Trend  empirically  the  very close  IIWR ( S t a t . Can.) t o the proposed  reason b e i n g t h a t p° i s a r b i t r a r y and  IIWR (est.)  rich  for r =  been shown i n T a b l e 5 t h a t  p r i c e s , TT (•) i n the  price  causes r e s p e c t i v e l y ,  s h e l t e r have n e g l i g i b l e e f f e c t s on  indexes are  increases  transportation  o t h e r hand, p r i c e changes i n c l o t h i n g ,  Inequality  most  usual c l a s s i f i c a t i o n of n e c e s s i t i e s  cannot r e a s o n a b l y be  I t has  1978  p r i c e and  is  The  s u r p r i s i n g because these consumption items are h i g h l y  g a t e d and  Section  p°  t h a n the poor, a l t h o u g h everyone i n  d e c r e a s e , 1.6% w i t h the  sample and  of  data  i n food p r i c e a c t u a l l y  transportation  increase,  The  an  the p r i c e i n c r e a s e s  and  level.  shown i n T a b l e 6.  s o c i e t y i s i n e v i t a b l y worse o f f . recreation  1978  r e s u l t s are  a c t u a l l y decrease i n e q u a l i t y . h u r t more by  from the  F a m i l y E x p e n d i t u r e Survey 1978  prices.  i n e q u a l i t y but  10%  the p r i c e  denominator i n index i d e n t i c a l .  when s e t e q u a l  (3.23) v a n i s h e s  and IEI to  rendering  However, i t i s p l a u s i b l e  examines i n e q u a l i t y t r e n d  on  a  time-series  Table  G:  Sensitivity  of  IEI measure  r=.5  r = -1  r=0  -2  -  CO  p  «•  I  «  I  4915.56  4672.43  .049461  4441.74  .096392  4009.63  .184299  3608.94  .265813  565.24  .885011  Food  4795.16  4552.59  .050587  4322.73  .098523  3892.88  188165  3495.09  .271121  535.52  .888112  Clothing  4877.85  4636.89  .049399  4408.28  .096266  3980.17  184031  3583.32  .265390  564.27  .884320  Recreation  4863.29  4624.65  .049069  4398.10  .095654  3973.49  182962  3579.50  .263976  563.56  .884120  P/M  4893.39  4651.13  .049507  4421.26  .096482  3990.73  184464  3591.54  .266042  562.26  .885098  Shelter  4725.69  4491.60  .049537  4269.41  .096553  3853.02  184664  3466.64  .266427  539.96  .885740  Trans.  4838.22  4602.69  .048682  4379.06  .094905  3959.84  181551  3570.75  .261971  567.98  .882605  Food  4683.66  4441.87  .051625  4213.02  .100486  3785.72  .191718  3391.02  .275990  511.12  .890870  Commodity 0%  10%  to p  Care  I 20%  I  P" No.  1971 of  prices  entries  : 27651  - 118 -  b a s i s , the two methods might suggest m u t u a l l y c o n f l i c t i n g t r e n d s . T h i s would most l i k e l y happen i f the e x p e n d i t u r e d i s t r i b u t i o n i s s t a b l e o v e r time w h i l e r e l a t i v e p r i c e s a r e w i d e l y f l u c t u a t i n g , because TT (•) makes the I E I i n d e x more p r i c e - s e n s i t i v e .  An i n f o r m a l t e s t can be c a r r i e d o u t t o a s c e r t a i n i f t h i s c o n j e c t u r e h o l d s i n Canada.  S i n c e no a d d i t i o n a l e x p e n d i t u r e  survey  d a t a are a v a i l a b l e i n the t i m e - s e r i e s p e r i o d , income survey d a t a have t o be used f o r t r e n d a n a l y s e s .  Used here are t h e income s u r v e y 4  samples o f 1975, 1979 and 1981. they were e x p e n d i t u r e s . are brought  A f t e r - t a x income a r e t r e a t e d as i f  Furthermore,  i n t o computation.  o n l y p o s i t i v e a f t e r - t a x incomes  S i n c e a f t e r - t a x income has a much  h i g h e r v a r i a n c e than e x p e n d i t u r e , i n e q u a l i t y measures computed here are much h i g h e r than those computed u s i n g e x p e n d i t u r e s .  The I E I  indexes can be found i n T a b l e 7 and t h e IIWR ( S t a t . Can.) measures i n T a b l e 8. As e v i d e n t i n Table 7 and T a b l e 8, both t h e I E I i n d e x and the IIWR ( S t a t . Can.) index suggest t h a t i n e q u a l i t y i s h i g h e s t i n 1979, f o l l o w e d by 1981 and lowest i n 1975.  The t r e n d s s u g g e s t e d by  the two indexes a r e c o n s i s t e n t w i t h one a n o t h e r . is:  The p r o b a b l e  reason  a l t h o u g h the I E I index i s more s e n s i t i v e t o p, the changes i n  r e l a t i v e p r i c e s e x p e r i e n c e d i n Canada i n the l a s t decade are n o t d r a s t i c enough t o a l l o w the p r i c e  index TT(*) i n (3.23) t o a f f e c t the  i n e q u a l i t y measure so much t h a t i t c o n t r a d i c t s t h e s i m p l i s t i c ( S t a t . Can.) i n d e x i n a t r e n d  comparison.  IIWR  Table  Year  size  No. o f  of  sample  Entrles  7:Inequality  trend  r = .5  l>  ( I E I measure)  r=0 I  I  1975  2G5G9  78640  5 0 0 6 .92  4 6 5 3 . 22  .070642  4 2 7 4 ..72  146238  3146.69  .371532  920.48  .816159  1979  39906  105785  5 3 8 5 ..98  4 9 7 8 .82  .075600  4 5 3 2 . 98  158374  3074.36  .429192  639.43  .881279  1981  40308  103961  5 5 1 0 . 25  51 15 .82  .071581  4 6 9 5 ..11  147932  3416.67  .379943  801.54  .854537  p°  :1971  prices  Table  Year  size  of  8:Inequality  No.of  trend  (IIWR(Stat.  r = .5  Can.)  measure)  r = -1  r=0  r = -2  p  «  I  5  I  «  I  78640  7386.01  6869.75  .069897  6315.03  .145001  4707.72  .362617  39906  105785  10747.35  9949.89  .074201  9072.94  .155798  6264.40  40308  103961  14452.25  13406.39  .072366  12287.10  .149814  8927.17  sample  Entries  1975  265G9  1979 1981  *  1  1677.28  .772911  .417122  1566.59  .854235  .382299  2248.46  .844422  to o  - 121  -  Conclusion  In Chapter 3, i t has  been e s t a b l i s h e d  that  the proposed  IEI  index i s s u p e r i o r  t o a l l o t h e r e x i s t i n g indexes i n i t s s o c i a l e v a l u a -  t i o n foundation.  In t h i s c h a p t e r , i t i s shown t h a t  extends t o the e m p i r i c a l empirically,  the  the  The  IEI index e x h i b i t s  from those by the convincingly  scene.  empirical  this superiority  evidence i n d i c a t e s  a different inequality  o t h e r major i n d e x e s .  d i s t r i b u t i v e p r i c e e f f e c t s on  inequality.  IIWR (est.)  indexes  not  sense where the  as c l e a r - c u t .  inflation  i n d e x are  future.  j u s t i f i a b i l i t y o f the  index i s s t i l l  the  preferred  Even i n the  the  IEI  dynamic  index and  a l t h o u g h such a One  Can.)  Wolfson  compared, the p r i c e changes are  i s l i k e l y i f p r i c e changes are l a r g e enough. changes i n the  IIWR ( S t a t .  IEI index c l o s e l y .  d r a s t i c enough t o cause a c o n f l i c t i n t r e n d  the  The  i n e q u a l i t y t r e n d s i n d i c a t e d by  IIWR ( S t a t . Can.)  and  price  (both b e i n g improvements o v e r the  (1979) index) approximate the  price  Food p r i c e  inequality!  Other r e s u l t s are and  scenario  More i m p o r t a n t l y , i t shows  i n f l a t i o n aggravates i n e q u a l i t y w h i l e t r a n s p o r t a t i o n ameliorates  that  the  not conflict  cannot r u l e out  such  Therefore, considering p r i c e - s e n s i t i v i t y social-evaluation index.  p r o c e d u r e , the  IEI  -  One  can conclude  122  -  t h a t , j u d g i n g from e m p i r i c a l e v i d e n c e  the s o c i a l w e l f a r e f o u n d a t i o n , the I E I index proposed i n Chapter s h o u l d be adopted i n p l a c e o f a l l o t h e r e x i s t i n g  indexes.  and 3  - 123 -  Chapter  1.  7  Footnotes  S t a t i s t i c s Canada update the c u t - o f f v a l u e s every  f o r i n f l a t i o n o n l y so t h a t the p o v e r t y - l i n e r a t i o s  year  are c o n s t a n t .  However, major r e v i s i o n s are done a f t e r each f a m i l y e x p e n d i t u r e survey.  The 1969 survey i m p l i e s r a t i o s  1978 survey i m p l i e s r a t i o s  2.  f o r 1971 - 1979, w h i l e the  f o r 1980 - 1982.  S i n c e the A t k i n s o n index i s r e l a t i v e , o n l y r e l a t i v e  i n p ° matter.  prices  T h e r e f o r e c h o o s i n g a c t u a l p r i c e s i s not too r e s t r i c t i v e  i n studying s e n s i t i v i t y .  3.  I t can e a s i l y be v e r i f i e d t h a t the i n f l a t e d w e l f a r e r a t i o i s  not o r d i n a l l y  equivalent to the i n d i r e c t  o b t a i n e d by s o l v i n g f o r y* i n , V( y, p, A ) = V( y*, p, A ° ) so t h a t y* = C (V ( y, p, A ) , p, A ° ) However, l e t t i n g , y* = C ( V ( y , ±  y* it  Y  *  l  , A), p , A°) x  = c(v( y , p , A ) , p , A°) 2  i s not t r u e t h a t v  P l  >  V  = 2 Y  *  2  2  utility  function.  I t can be  - 124 -  i f and o n l y i f V(  P , ±  u n l e s s p^ = p^.  4.  A ) > V(  Y ,  This f a i l s  The d a t a f i l e s  2  p , A ) 2  y* as an e x a c t u t i l i t y  indicator.  are known as Economic F a m i l y Incomes, 1975;  Census F a m i l y Incomes, 1979; Census Family Incomes, 1981.  The  d i f f e r e n c e i n f a m i l y d e f i n i t i o n s i s n o t b e l i e v e d t o a f f e c t measured inequality  significantly.  - 125 -  CHAPTER 8  CONCLUSION  Every economics s t u d e n t knows t h a t , through the budget traint, attainable u t i l i t y  depends on p r i c e s .  cons-  S i n c e t h e r i c h consume  more l u x u r i e s r e l a t i v e t o n e c e s s i t i e s than t h e poor, changes i n r e l a t i v e p r i c e s w i l l a f f e c t persons on d i f f e r e n t u t i l i t y differently.  levels  I t then f o l l o w s i n t u i t i v e l y t h a t r e l a t i v e p r i c e  have d i s t r i b u t i v e e f f e c t s , hence a f f e c t i n e q u a l i t y .  changes  However, i t i s  somewhat s u r p r i s i n g t h a t d e s p i t e t h e e x i s t e n c e o f e m p i r i c a l  evidence  s u b s t a n t i a t i n g t h i s claim, a s a t i s f a c t o r y i n e q u a l i t y index that i s a b l e t o capture these e f f e c t s i s absent i n the l i t e r a t u r e .  Most o f  the e x i s t i n g indexes a r e c a l c u l a t e d based on d i s t r i b u t i o n s o f incomes or  e x p e n d i t u r e s o r some simple adjustments  exceptions are the Muellbauer Slesnick  o f the two.  (1974) approach  (1984) approach which,  Two n o t a b l e  and t h e J o r g e n s o n -  w h i l e b e i n g worthwhile  attempts, a r e  n o t c o m p l e t e l y s a t i s f a c t o r y i n t h e i r somewhat ad hoc s o c i a l frameworks.  By c o n t r a s t , t h e p r e s e n t r e s e a r c h r e s u l t s i n t h e  e s t a b l i s h m e n t o f a new index t h a t i s n o t s u b j e c t t o these  The clear.  evaluation  criticisms.  i n e q u a l i t y i m p l i c a t i o n o f s o c i a l choice theories i s  As a r e s u l t , t h e I E I index proposed  i n t h i s t h e s i s i s based  bn an e x p l i c i t s o c i a l w e l f a r e e v a l u a t i o n f o u n d a t i o n .  What i s  r e q u i r e d t o generate a new index i s t h e f o l l o w i n g : a s o c i a l  evaluation  - 126 -  framework, a p r i c e - s e n s i t i v e  n u m e r i c a l measure o f u t i l i t y ,  priate  s o c i a l welfare function  ment.  The second r e q u i r e m e n t above poses t h e g r e a t e s t  an appro-  and a formula f o r i n e q u a l i t y  because o f the absence o f an o b j e c t i v e  scale  measure-  challenge  of u t i l i t y  measurement  and  t h e absence o f b e h a v i o r a l d a t a such as i n d i v i d u a l demand d a t a  and  the f a c t t h a t  c e r t a i n human and e n v i r o n m e n t a l  characteristics  a f f e c t the r e l a t i o n s h i p between consumption and u t i l i t y . w i t h these d i f f i c u l t i e s ,  t h e p r e s e n t model i n c o r p o r a t e s  parameters i n t o t h e u t i l i t y of interpersonal  attribute  and assumes a p a r t i c u l a r l e v e l  comparison o f u t i l i t i e s ,  representation of u t i l i t y . An  function  To cope  which r e s u l t i n a n u m e r i c a l  T h i s measure i s named e q u i v a l e n t income.  e q u i v a l e n t income measure i s imputed t o each i n d i v i d u a l i n s o c i e t y .  A d i s t r i b u t i o n o f e q u i v a l e n t incomes then form t h e b a s i s equality  measurement i n a w e l f a r i s t  s o c i a l welfare  Besides the t h e o r e t i c a l c o n t r i b u t i o n equality the  index t h a t  of providing  i s based on a r i g o r o u s s o c i a l w e l f a r e  Numerically, i t indicates  from those i n d i c a t e d  problem t h a t  t u r n e d them i n t o unworthy e m p i r i c a l index s u c c e s s f u l l y  price  an i n framework,  empirical  a different inequality  by t h e major e x i s t i n g i n d e x e s .  theoretical mis-specification  IEI  framework.  p r e s e n t r e s e a r c h i s a l s o marked by i t s i m p r e s s i v e  results.  of i n -  tools.  scenario  A p p a r e n t l y , the  p l a g u e s these i n d e x e s has Furthermore, the proposed  measures d i s t r i b u t i v e p r i c e e f f e c t s .  Food  i n c r e a s e s do have an a g g r a v a t i n g impact on i n e q u a l i t y  while  - 127  the o p p o s i t e  i s true  -  for transportation.  This  f i n d i n g conforms  r e a s o n a b l y w i t h the common c l a s s i f i c a t i o n o f l u x u r i e s and In a d d i t i o n , one  should  mechanical r e s u l t s . f i c a t i o n nor  any  r e a l i z e t h a t these are not  Neither  the  necessities.  artificial  translog equivalent  scales speci-  s t r u c t u r a l assumption i n the e s t i m a t i o n  model  n e c e s s a r i l y d r i v e s these r e s u l t s .  However, t h i s approach does have i t s l i m i t a t i o n s . fundamental one arbitrary.  i s t h a t p ° i n the e q u i v a l e n t  most  income f u n c t i o n i s  In i n e q u a l i t y measurement, i t becomes an e x t r a parameter  i n a d d i t i o n to r , the degree o f i n e q u a l i t y a v e r s i o n . i n e q u a l i t y i s not v e r y nevertheless,  s e n s i t i v e t o the c h o i c e  i t i s impossible  i n e q u a l i t y which c r e a t e s c o u n t r y comparisons. quite apart  from the  While measured  of p° e m p i r i c a l l y ,  t o p i n p o i n t p r e c i s e l y the  degree o f  some vagueness i n e x e r c i s e s such as  Furthermore, the e s t i m a t i o n  e q u a t i o n b i a s , the p r o d u c t i o n  Firstly,  to a v o i d  inter-  model, w h i l e  i n e q u a l i t y measurement framework, c o u l d  improved i n s e v e r a l d i r e c t i o n s .  The  The  being  be  simultaneous  s i d e o f the economy c o u l d be  incorporated.  model adopted here i s a l i m i t e d - i n f o r m a t i o n model which does  make use ferences,  o f supply  data.  assumptions are  share e q u a t i o n s . might not be  Secondly, i n the s p e c i f i c a t i o n o f  not  pre-  imposed t o a r r i v e a t l i n e a r e x p e n d i t u r e  While t h e y s i m p l i f y e s t i m a t i o n ,  these assumptions  c o n s i s t e n t w i t h a c t u a l consumer b e h a v i o r .  a v a i l a b i l i t y imposes severe c o n s t r a i n t s on  T h i r d l y , data  the number o f a t t r i b u t e s  - 128  -  t h a t can be i n c o r p o r a t e d i n t o the s p e c i f i c a t i o n o f p r e f e r e n c e s . The p r e s e n t r e s e a r c h , f o r the sake o f c r e d i b i l i t y , o n l y makes use of p u b l i c l y a v a i l a b l e data.  But the e s t i m a t i o n r e s u l t s w i l l  de-  f i n i t e l y b e n e f i t from a d d i t i o n a l micro d a t a c o n c e r n i n g o t h e r r e l e v a n t attributes.  For example, household  composition  i s an a t t r i b u t e t h a t  p l a u s i b l y a f f e c t s the r e l a t i o n s h i p between consumption and  One index.  s h o u l d n o t i c e the implementation  utility.  advantages o f the  IEI  I t might seem t h a t t h i s i n d e x i s v e r y c o s t l y t o implement,  i n view o f the c o m p l e x i t i e s o f the model. implementing improving  T h i s i s not t r u e .  When  t h i s index, the a d d i t i o n a l work t h a t i t r e q u i r e s i s i n  the e s t i m a t e s as new  demand d a t a become a v a i l a b l e .  i s not c o s t l y because a l l t h e p r o c e d u r e s  This  i n v o l v e d can be e x e c u t e d  by  computer programs, the f e a s i b i l i t y o f which have been demonstrated i n Chapter 6.  However, the s m a l l i n c r e a s e i n c o s t g a i n s i n r e t u r n  a much improved measure o f i n e q u a l i t y — f o u n d a t i o n , e t h i c a l s i g n i f i c a n c e and  Finally, extremely  wide.  i n i t s social  welfare  price-sensitivity.  the scope o f a p p l i c a t i o n o f t h i s r e s e a r c h i s On one  hand, as a l r e a d y e x p l a i n e d , i t g i v e s r i s e  t o an i n e q u a l i t y i n d e x t h a t can i n d i c a t e the e f f e c t s o f t a x tariff  changes on i n e q u a l i t y .  and  This i s a valuable p o l i c y t o o l .  the o t h e r hand, the approach t o u t i l i t y measurement and s o c i a l a g g r e g a t i o n i s a p p l i c a b l e t o o t h e r d i s c i p l i n e s such as  On welfare  cost-benefit  - 129 -  a n a l y s e s and s o c i a l p l a n n i n g .  T h i s r e s e a r c h r e p r e s e n t s an  important  s t e p towards i n t e g r a t i n g s o c i a l c h o i c e t h e o r i e s w i t h p r a c t i c a l evaluation. to be  policy  I t d e f i n i t e l y opens up new r e s e a r c h areas t h a t have y e t  explored.  - 130  -  BIBLIOGRAPHY  Atkinson,  A.  Atkinson,  A.B. (1982) "A b i b l i o g r a p h y o f p u b l i s h e d measurement o f i n e q u a l i t y and p o v e r t y  Barten,  A.P.  (1976) The  Economics o f I n e q u a l i t y C l a r e n d o n , O x f o r d . r e s e a r c h on the 1970-1982".  (1964) "Family c o m p o s i t i o n , p r i c e s , and expenditure p a t t e r n s " i n P. H a r t , G. M i l l s , and J.K. W h i t t a k e r , eds. Econometric A n a l y s i s f o r N a t i o n a l Economic P l a n n i n g : 16th Symposium o f the C a l s t o n S o c i e t y , London, B u t t e r w o r t h , p. 277-292. 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(1979) "Wealth and the d i s t r i b u t i o n o f income, Canada 1968-70" I n t e r n a t i o n a l A s s o c . f o r Research i n Income and Wealth S e r i e s 25, No. 2, June, p. 129-140.  - 135 -  Appendix A Cross-section  FOOD Variable log y A  l  A  2  A  3  A  4  A  5  A  6  A  7  A  8  A  9  A  io  A  l l  Intercept  R  2  Results  CLOTHING  RECREATION  coeff.  t-ratio  coeff.  t-ratio  coeff.  t-ratio  -0 .1209  -59 .02  0 .0140  13 .23  0 .0472  25 .28  -0 .0119  -6 .28  0 .0021  2 .12  0 .0009  0 .52  -0 .0364  -14 .02  0 .0169  12 .54  0 .0006  0 .24  0 .0382  12 .66  0 .0086  5 .51  -o .0321  -11 .67  0 .0636  18 .29  0 .0161  8 .95  -0 .0502  -15 .83  0 .0731  19 .79  0 .0207  10 .84  -0 .0464  -13 .80  0 .1109  28 .30  0 .0293  14 .42  -0 .0560  -15 .69  0 .0002  0 .04  -0 .0095  -4 .90  -0 .0110  -3 .20  0 .0148  3 .64  -0 .0052  -2 .45  -0 .0078  -2 .10  0 .0343  8 .53  -0 .0023  -1 .09  -0 .0036  -0 .98  0 .0205  5 .18  -0 .0085  -4 .17  0 .0030  0 .84  0 .0006  0 .15  -0 .0145  -7 .05  0 .0124  3 .45  1 .3593  70 .84  -0 .065 3  -6 .57  -o .3076  -17 .60  .3165  .1203  .0756  - 136 -  Appendix A  P/M CARE Variable log y A  l  A  2  A  3  A  4  A  5  A  6  A  7  A  8  A  9  A  io  A  l l  Intercept  (continued)  SHELTER  TRANSPORTATION  coeff.  t-ratio  coeff.  t-ratio  coeff.*  t-ratio  -0 .0037  -5 .63  -0 .0290  -10 .53  0 .0924  35 .55  -0 .0030  -4 .88  -0 .0229  -8 .98  0 .0348  14 .47  0 .0072  8 .60  0 .0386  11 .03  -0 .0268  -8 .12  0 .0078  8 .00  -0 .0059  -1 .45  -0 .0166  -4 .35  0 .0065  5 .81  -0 .0110  -2 .36  -0 .0251  -5 .68  0 .0085  7 .16  -0 .0136  -2 .73  -0 .0424  -9 .10  0 .0088  6 .92  -0 .0357  -6 .76  -0 .0573  -11 .52  0 .0002  0 .18  0 .0473  9 .37  -0 .0272  -5 .71  0 .0048  3 .63  0 .0274  5 .01  -0 .0341  -6 .60  0 .0097  7 .45  -0 .0168  -3 .10  -0 .0213  -4 .18  0 .0094  7 .37  -0 .0211  -3 .98  -0 .0032  -0 .64  0 .0046  3 .61  -0 .0022  -0 .41  -0 .0010  -0 .20  0 .0674  10 .88  0 .6451  25 .00  -0 .6990  -28 .70  R  * These a r e d e r i v e d  .0369  estimates.  .1123  .1727  Appendix B C r o s s - s e c t i o n Consumption  coeff. y A,  t-ratio  0.7461  131.81  -2.5252  -0.02  coeff. 0.7294  -1148.3  t-ratio 123.34  coeff. 0.6820  Function  t-ratio 110.42  819.90  5.36  2054.70  11.71  2983.20  16.78  3581.50  19.23  10 l l  Intercept  3126.3  26.95  3620.5  0.6986  t-ratio 116.11  coeff. 0.7461  t-ratio 133.28  -8.50  A,  A  coeff.  30.55  2410.2  18.82  350.55  1.69  801.64  3.69  24.42  0.11  -1605.1  -7.26  -2550.7  -11.85  4344.3  22.33  3125.1  30.17  Appendix  C:  Time-series  Expenditure Year  Food  Clothing  Recreation  P/M  Data  Shares  Care  Shelter  Trans.  1971  0.2660592  0.0794667  0.1377769  0.0490649  0.3139157  0.1537163  1972  0.2636354  0.0778922  0.1434587  0.0486184  0.3118088  0.1545862  1973  0.2638876  0.0763336  0.1452276  0.0480812  0.3091659  0.1573038  1974  0.2603992  0.0816908  0.1461441  0.0496107  0.3072199  0.1549349  1975  0.2620369  0.0783482  0.1497651  0.0492756  0.3040745  0.1564994  1976  0.2501935  0.0777230  0.1568799  0.0503369  00..33007788662211  0.1570042  1977  0.2482974  0.0759179  0.1593904  0.0504417  0.3100061  0.1559463  1978  0.2500862  0.0745530  0.1579281  0.0511057  0.3110649  0.1552619  i  1979  0.2504032  0.0748948  0.1522763  0.0507470  0.3130199  0.1586584  to  1980  0.2502917  0.0732464  0.1523522  0.0514989  0.3158874  1981  0.2504500  0.0723800  0.1489500  0.0524800  0.3152200  00  0.1567232 0.1605200  |  Appendix Price  C (continued) Indexes  Year  Food  Cloth.  Recrn.  P/M  C.  Shelt.  Trans.  1971  0.5149  0.6831  0.6748  0.6017  0.5744  0.6165  1972  0.5469  0.7008  0.6937  0.6306  0.6014  0.6326  1973  0.6102  0.7357  0.7227  0.6607  0.6399  0.6492  1974  0.6941  0.8060  0.7861  0.7184  0.6962  0.7139  1975  0.7832  0.8545  0.8671  0.8002  0.7651  0.7978  1976  0.8120  0.9016  0.9190  0.8682  0.8501  0.8835  1977  0.8769  0.9631  0.9629  0.9326  0.9299  0.9451  1978  1.0000  1.0000  1.0000  1.0000  1.0000  1.0000  1979  1.1205  1.0922  1.0688  1.0903  1.0695  1.0974  1980  1.2400  1.2206  1.1707  1.1992  1.1568  1.2374  1981  1.3841  1.3074  1.2888  1.3297  1.3004  1.4649  0J  Appendix  C  (continued)  Expenditure/Attribute  Statistics  Year  SAR  SOH  Family  Size  1971  0.3337079  O.1096137  0.2013677  0.1716406  0.1992498  0.2926372  0.2268651  0.2486509  0 .2 1 6 8 1 1 0  0.13O3190  0.0861047  1972  0.3371326  0.1165177  0.2053771  0.1700943  0.2018333  0.2831293  O.2304947  0.2449641  0 .2 1 5 6 3 1 1  0.1304716  0.0920845  1973  0.3430373  0.1214114  0.2113054  0.1695585  0.2094422  O.2683040  0.2366654  0.2382742  O.2 1 5 0 1 3 6  0. 1317976  0.0935028  1974  0.3299644  0.1206901  0.2194015  0.1669182  0.2127494  0.2539154  0.2459219  0.2354508  0 .2 0 8 2 9 6 2  0 . 1292976  0.0962181  1975  0.3091979  0.1215964  0.2292620  0.1672548  0.2191977  0.2366318  0.2579635  0.2311310  O. 2 0 5 1 6 0 6  0. 1266780  0.0940368  1976  0.3124135  0.1249326  0.2238119  0.1755751  0.2237598  0.2287237  0.2566222  0.2266403  0 .2 0 9 3 4 0 9  0.1308274  0.0926589  1977  0.3269907  0.1304930  0.2257125  0.1762792  0.2271487  0.2219230  0.2550149  0.2211759  0 .2 1 4 4 7 3 6  0.1338624  0.0899095  1978  O.3203550  0.1281574  0.2306620  0.1818301  0.2249767  0.2080390  O.2540106  0.2258760  0 .2 1 0 6 1 1 2  0.1355060  0.0953015  1979  0.3125654  0.1360894  0.2364785  0.1832780  0.2251669  O.1951508  0.2561977  0.2275612  O.2 0 3 2 4 2 3  O.1364625  0.0983135  1980  0.3056573  0.1338735  0.2473163  0.1871624  0.2209041  0.1843005  0.2515903  0.2305676  0.1990146  0.1420278  0.1054092  1981  0.3029503  0.1384588  0.2466721  0.1899575  0.2323494  O.1750336  0.2479388  0.2345871  0 .2 0 0 8 2 0 2  0.1432102  0.1045268  Age  of  Head  O  Appendix Y e a r  S  C  (continued)  yyt  1971  9.16783810  5646258  1972  9.22327710  5767065  1973  9.31293580  5813489  1974  9.42433930  6076686  1975  9.51679800  6227154  1976  9.61170770  6303170  1977  9.68260670  6442341  1978  9.75351240  6617835  1979  9.80900670  6813257  1980  9.92726140  6632325  1981  10.02926450  6932605  h-  1  I  - 142 -  Appendix D Time-series Results, B  Food  Clothing  Recreation  P/M Care  Shelter  Trans.  PP  Food  Clothing  Recreation  P/M Care  Shelter  Trans*  -0.11148  -0.02224  -0.04682  0.00026  0.14293  -0.08358  (-4.6197)  (-0.9363)  (-1.7898)  (0.0069)  (5.5466)  -0.02224  0.16707  -0.15309  -0.04365  -0.11494  (-0.6298)  (1.9605)  (-1.8222)  (-0.3949)  (-1.7470)  -0.04682  -0.15 309  -0.03234  0.06491  0.48122  (-0.3139)  (-0.4752)  (-0.0853)  (0.1506)  (1.9837)  0.00026  -0.04365  0.06491  -0.06829  0.11340  (0.0066)  (-0.5615)  (0.8211)  (-0.3026)  (1.0731)  0.14293  -0.11494  0.48122  0.11340  -0.98008  (0.9977)  (-0.4672)  (2.0343)  (0.2018)  (-2.9720)  -0.08358  0.18089  -0.26672  -0.07037  0.32847  * E s t i m a t e s i n t h e l a s t row and l a s t column a r e d e r i v e d e s t i m a t e s , (See Chapter 5) Numbers i n p a r e n t h e s e s a r e t h e t - r a t i o s T i m e - s e r i e s : 1971 - 1981  0.18089  -0.26672  -0.07037  0.32847  0.00374  Appendix E Barten Equivalence Scales URBAN MALE HEAD H O U S E H O L D H E A D AGED 2 4 OR BELOW FAM. S I Z E FOOO CLTH RCRN MEDC SHTR 1 1.OOOO 1.0000 1.0000 1.0000 1.0000 2 1.3836 1.3831 1.4226 1.6080 1.4035 3 1.7147 1.6748 1.8084 2.0387 1.7459 4 1.8045 2.9199 2.4953 2.6513 1.8894 5 2.3943 4 .9348 3.9310 3.9552 2.5028 URBAN M A L E HEAD H O U S E H O L D H E A D AGED 2 4 - 3 4 FAM. S I Z E FOOO CLTH RCRN MEDC SHTR 1.0388 1.0250 0.9884 1 .0196 1.0756 1 .4374 1.4176 1.4061 1.6396 1.5095 1 .7813 1.7166 1.7874 2.0787 1.8778 1 .8745 2.9928 2.4664 2.7033 2.0322 2.4873 5.0580 3.8854 4.0329 2.6919 MALE HEAD URBAN H O U S E H O L D H E A D AGED 3 4 - 4 4 FAM. S I Z E FOOD RCRN CLTH MEDC SHTR 1.2937 1 1.1342 1.5474 1 .3196 1.1777 1.8403 2 1.5693 2 2. 1 4 0 3 1.6529 2 . 1220 2.3395 2 2.0561 3 1.9448 2. 5 9 1 6 2.6903 3.2281 4.5183 2.2252 4 2.0466 3.4987 5.0854 77. 6 3 6 4 2.9475 5 2.7156 5.2193 URBAN MALE HEAD HOUSEHOLD HEAD AGED 4 4 - 5 4 FAM. S I Z E FOOD CLTH RCRN SHTR MEDC 1.6544 4541 1.2771 1 1.2838 1.5298 0686 1 .7925 2.4599 2 1.7763 2. 2 8 8 2 6296 2.2297 3.1187 3 2.2012 2. 7 7 0 8 4.8306 6284 2.4131 4.0558 4 2.3165 4 B.1642 7160 3.1964 6.0506 5 3.0738 8 URBAN M A L E HEAD H O U S E H O L D H E A D AGED 5 4 - 6 4 SHTR FAM. S I Z E FOOD CLTH RCRN MEDC 1.1161 1.1565 1.0419 1.0682 1 . 1733 1.5664 1.6003 1 .4410 1 .5195 1.8868 1.9485 1.9831 1 .7449 1 .9317 2.3921 2.1088 2.0869 3.0421 2.6654 3.1109 2.7933 2.7692 5 . 1414 4.1989 4.6408 MALE HEAD URBAN H O U S E H O L D HEAD AGED OVER 6 4 RCRN SHTR FAM. S I Z E FOOD CLTH MEDC 8682 0.9639 1 0.9949 0.8382 0.9103 2351 1.3529 2 1.3765 1.1593 1.4638 5701 1.6829 3 1.7059 1.4037 1.8559 1665 1.8213 4 1.7952 2.4473 2.4135 4130 2.4125 5 2.3820 4.1362 3.6005  TRAN 1.0000 1.3787 1.6960 1 .4648 1 .7235  TRAN 1.0786 1 .4871 1.8293 1 .5799 1 .8590  TRAN 1 .0299 1 .4200 1 .7468 1.5086 1.7751  TRAN 1.1655 1.6068 1.9767 1 .7071 2.0O87  TRAN 1 .2108 1 .6693 2.0535 1 .7735 2.0868  TRAN 1.0799 1 .4888 1 .8315 1 .5818 1 .8612  Appendix URBAN F E M A L E HEAD H O U S E H O L D H E A D A G E D 2 4 OR BELOW FAM. S I Z E FOOD CLTH 1 0.7374 2.5273 2 1.0203 3.4955 3 1.2644 4.2327 4 1.3306 7.3794 5 1.7656 12.4718 URBAN F E M A L E HEAD H O U S E H O L D H E A D AGED 2 4 - 3 4 FAM. S I Z E FOOD CLTH 1 0.7660 2.5904 2 1.0599 3.5828 3 1.3135 4.3384 4 1.3823 7.5636 5 1.8341 12.7832 URBAN F E M A L E HEAD H O U S E H O L D H E A D AGED 3 4 - 4 4 FAM. S I Z E FOOD CLTH 1 0.8363 3.9109 2 1 . 1572 5.4091 3 1.4340 6.5499 4 1.5091 11.4192 5 2.0025 19.2994 URBAN F E M A L E HEAD H O U S E H O L D H E A D AGED 4 4 - 5 4 FAM. S I Z E FOOD CLTH 1 0.9466 4.1812 2 1.3098 5.7830 3 1.6232 7.0026 4 1.7082 12.2085 5 2.2666 20.6334 URBAN F E M A L E HEAD H O U S E H O L D H E A D AGED 5 4 - 6 4 FAM. S I Z E FOOD CLTH 1 0.8528 2.6331 2 1.1800 3.6418 3 1.4623 4.4099 4 1.5389 7.6883 5 2.0419 12.9939 URBAN F E M A L E HEAD H O U S E H O L D H E A D AGED OVER 6 4 FAM. S I Z E FOOD CLTH 1 0.7336 2.1183 2 1.0150 2.9298 3 1.2579 3.5477 4 1.3237 6.1851 5 1.7565 10.4534  E  (continued)  RCRN 1.3832 1.9677 2.5013 3.4515 5.4372  MEDC 1.3608 2.1882 2.7742 3.6078 5.3822  SHTR 0.8447 1.1856 1.4748 1.5961 2 . 1 142  RCRN 1 .3672 1.9449 2.4724 3.4115 5.3742  MEDC 1.3875 2.2312 2.8287 3.6787 5.4879  SHTR 0.9086 1.2752 1.5863 1.7167 2.2740  TRAN 51 19 7057 8681 7498 8822  RCRN 1 .7894 2.5455 3.2359 4.4651 7.0340  MEDC 1.7957 2.8875 3.6609 4.7609 7.1024  SHTR 0.9948 1.3962 1.7369 1.8797 2.4899  TRAN 0.4888 0.6739 0.8290 0.7159 0.8424  RCRN 2.0113 2.8612 3.6372 5.0188 7.9063  MEDC 2.0817 3.3474 4.2439 5.5191 8.2335  SHTR 1.0789 1.5142 1.8836 2.0384 2.7002  TRAN 0.5531 0 .. 7 6 2 5 0 .. 9 3 8 1 O.. 8 1 0 1 0 .. 9 5 3 3  RCRN 1.4775 2.1018 2.6718 3.6867 5.8078  MEDC 1.5967 2.5675 3.2551 4.2332 6.3151  SHTR 0.9428 1.3232 1 .6460 1.7814 2.3596  TRAN 0. 5746 0.7922 0.9745 0.8416 0.9903  RCRN 1.2009 1.7084 2.1718 2.9967 4.7208  MEDC 1.2387 1.9919 2.5254 3.2843 4.8995  SHTR 0.8143 1 . 1428 1 .4216 1.5385 2.0380  TRAN 0. 5125 0.7065 0.8692 0.7506 0.8833  TRAN 0.4746 0 .. 6 5 4 3 0 .. 8 0 4 9 0 .. 6 9 5 1 0 .. 8 1 7 9  Appendix RURAL MALE HEAD H O U S E H O L D HEAD AGED 2 4 OR BELOW FAM. S I Z E FOOD CLTH 1 0.9165 0.5437 2 1.2681 0.7520 3 1.5715 0.9105 4 1.6538 1.5874 5 2.1944 2.6829 RURAL M A L E HEAD H O U S E H O L D HEAD A G E D 2 4 - 3 4 FAM. S I Z E FOOD CLTH 1 0.9521 0.5572 2 1.3174 0.7707 3 1.6325 0.9333 4 1.7180 1.6271 5 2.2796 2.7499 RURAL M A L E HEAD H O U S E H O L D HEAD A G E D 3 4 - 4 4 FAM. S I Z E FOOD CLTH 1 1.0395 0.8413 2 1.4383 1.1636 3 1.7824 1.4090 4 1.8757 2.4565 5 2.4889 4.1517 RURAL M A L E HEAD H O U S E H O L D H E A D AGED 4 4 - 5 4 FAM. S I Z E FOOD CLTH 1 1 . 1766 0.8995 2 1.6280 1.2440 3 2.0175 1.5064 4 2.1231 2.6263 5 2.8171 4.4386 RURAL M A L E HEAD H O U S E H O L D HEAD AGED 5 4 - 6 4 FAM. S I Z E FOOD CLTH 1 1.0600 0.5664 2 1.4666 0.7834 3 1.8175 0.9486 4 1.9127 1.6539 5 2.5380 2.7952 RURAL M A L E HEAD H O U S E H O L D HEAD AGED OVER 6 4 FAM. S I Z E FOOD CLTH 1 0.9118 0.4557 2 1.2616 0.6303 3 1.5634 0.7632 4 1.6453 1.3305 5 2 . 1832 2.2487  E  (cont f nued)  RCRN 0.7126 1.0138 1 .2887 1.7782 2.8013  MEDC 0.7425 1 . 1940 1.5138 1.9687 2.9369  SHTR 0.8758 1.2291 1.5290 1.6547 2.1919  TRAN 1.1027 1.5203 1.8702 1 .6152 1.9005  RCRN 0.7044 1.0020 1.2738 1.7576 2.7689  MEDC 0.7571 1.2175 1.5435 2.0073 2.9946  SHTR 0.9420 1.3220 1.6445 1.7798 2.3575  TRAN 1.1894 1.6398 2.0172 1.7422 2.0499  RCRN 0.9219 1.3115 1.6672 2.3005 3.6240  MEDC 0.9799 1.5756 1.9976 2.5979 3.8755  SHTR 1.0314 1 .4475 1.8007 1 .9487 2.5814  TRAN 1.1357 1.5658 1.9262 1.6636 1.9574  RCRN 1.0362 1.4741 1.8739 2.5857 4.0734  MEDC 1.1359 1.8266 2.3158 3.0116 4.4928  SHTR 1 . 1 185 1.5698 1.9528 2.1133 2.7994  TRAN 1 .2852 1 .7719 2.1797 1.8825 2.2150  RCRN 0.7612 1.0829 1.3766 1.8994 2.9923  MEDC 0.8712 1 .4010 1.7762 2.3099 3.4460  SHTR 0.9774 1.3718 1.7065 1.8468 2.4463  TRAN 1.3351 1.8408 2.2644 1.9557 2.3012  RCRN 0.6187 0.8802 1.1189 1.5439 2.4322  MEDC 0.6759 1.0869 1.3780 1.7921 2.6735  SHTR 0.8442 1.1848 1.4739 1.5951 2.1128  TRAN 1.1908 1.6418 2.0196 1.7442 2.0524  Appendix RURAL F E M A L E HEAD H O U S E H O L D H E A D AGED 2 4 OR BELOW FAM. S I Z E FOOD CLTH 1 0.6758 1.3740 2 0.9351 1.9004 3 1.1588 2.3012 4 1.2195 4.0120 5 1.6181 6.7806 RURAL F E M A L E HEAD H O U S E H O L D H E A D AGED 2 4 - 3 4 FAM. S I Z E FOOD CLTH 1 0.7021 1.4083 2 0.9714 1.9479 3 1.2038 2.3586 4 1.2668 4.1121 5 1.6810 6.9499 RURAL F E M A L E HEAD H O U S E H O L D H E A D AGED 3 4 - 4 4 FAM. S I Z E FOOD CLTH 1 0.7665 2.1262 2 1.0606 2.9408 3 1.3143 3.5610 4 1.3831 6.2083 5 1.8353 10.4926 RURAL F E M A L E HEAD H O U S E H O L D H E A D AGED 4 4 - 5 4 FAM. S I Z E FOOD CLTH 1 0.8676 2.2732 2 1.2004 3.1440 3 1.4876 3.8071 4 1.5655 6.6374 5 2.0773 11.2178 RURAL F E M A L E HEAD H O U S E H O L D H E A D AGED 5 4 - 6 4 FAM. S I Z E FOOD CLTH 1 0.7816 1.4315 2 1.0815 1.9800 3 1.3402 2.3975 4 1.4104 4.1799 5 1.8715 7.0644 RURAL F E M A L E HEAD H O U S E H O L D H E A D AGED OVER 6 4 FAM. S I Z E FOOD CLTH 1 0.6723 1.1517 2 0.9303 1.5928 3 1.1529 1.9288 4 1.2132 3.3627 5 1.6098 5.6832  E (cont1nued)  RCRN 0.9857 1.4022 1.7825 2.4596 3.8747  MEDC 1.0104 1.6248 2.0600 2.6790 3.9965  SHTR 0.7398 1.0383 1.2916 1.3978 1 .8516  TRAN O. 5 2 3 3 0.7215 0.8875 0.7665 0.9019  RCRN 0.9743 1.3860 1.7619 2.4311 3.8299  MEDC 1 .0303 1.6567 2.1004 2.7315 4.0749  SHTR 0.7957 1 . 1 168 1.3892 1.5035 1.9915  TRAN 0.5644 0.7782 0.9573 0.8268 0.9728  RCRN 1 .2752 1.8140 2.3060 3.1819 5.0126  MEDC 1.3334 2 . 1441 2.7183 3.5352 5.2738  SHTR 0.8713 1.2228 1 .521 1 1 .6462 2.1806  TRAN 0.5390 0.7431 0.9141 0.7895 0.9289  RCRN 1.4333 2.0390 2.5920 3.5766 5.6343  MEDC 1.5457 2.4856 3.1513 4.0982 6.1 137  SHTR 0.9448 1 .3261 1.6496 1.7852 2.3648  TRAN 0.6099 0.8409 1.0344 0.8934 1.0512  RCRN 1.0529 1.4978 1.9040 2.6273 4.1389  MEDC 1.1856 1.9065 2.4170 3. 1433 4.6892  SHTR 0.8257 1.1588 1.4415 1.5601 2.0665  TRAN 0.6336 0.8736 1.0746 0.9281 1.0920  RCRN 0.8558 1.2175 1.5477 2.1356 3.3642  MEDC 0.9198 1 .4791 1 .8752 2.4387 3.6381  SHTR 0.7131 1.0009 1 .2450 1.3474 1.7848  TRAN 0.5651 0.7791 0.9584 0.8277 0.9740  

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