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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 University of 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 of Economics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1985 (S) Sunny Kai-Sun Kwong, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date * DE-6 (3/81) Abstract The e x i s t i n g i n e q u a l i t y indexes i n the economics l i t e r a t u r e (including the more so p h i s t i c a t e d indexes of Muellbauer (1974) and Jorgenson-Slesnick (1984)), are found to be i n s e n s i t i v e to r e l a t i v e p r i c e changes or are u n j u s t i f i a b l e i n terms of s o c i a l evaluation e t h i c s or both.' The present research f i l l s t h i s gap i n the l i t e r a t u r e by proposing a new index, named the Individual Equivalent Income (IEI) index. A household i n d i r e c t u t i l i t y function i s hypothesized which incorporates c e r t a i n a t t r i b u t e parameters i n the form of equivalence scales. These a t t r i b u t e s are demographic and environmental c h a r a c t e r i s -t i c s s p e c i f i c to a given household. This i n d i r e c t u t i l i t y function gives a number which represents the u t i l i t y of each member of the household. A p a r t i c u l a r l e v e l of interpersonal comparison of u t i l i t i e s i s assumed which gives r i s e to an exact i n d i v i d u a l u t i l i t y i n d i c a t o r named equivalent income. A d i s t r i b u t i o n of these equivalent incomes forms the basis of 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. This index can be implemented i n the Canadian context. Pre-ferences are assumed to be nonhomothetic translog and demand data are derived from cross-section surveys and time-series aggregates. Based on demand data, 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 estimated and e q u i v a l e n t incomes imputed to a l l i n d i v i d u a l s i n s o c i e t y . An Atkinson index of e q u i v a l e n t incomes i s then computed to i n d i c a t e the a c t u a l degree of i n e q u a l i t y i n Canada. The new IEI index i s compared w i t h other indexes based on a common data s e t . The main f i n d i n g s are: conventional indexes give bad estimates of the true extent of i n e q u a l i t y and the IEI index, while p r o v i d i n g a more accurate estimate, 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 aggravates while 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 ameliorates the i n e q u a l i t y problem. Table of Contents Chapter 1 Introduction 1 Chapter 2 Survey of the L i t e r a t u r e 12 Some simple indexes 13 D i s t r i b u t i v e p r i c e e f f e c t s 17 Footnotes 28 Chapter 3 A New Approach 30 Equivalence scales 32 Equivalent income 36 S o c i a l choice 42 Inequality measurement 47 Footnotes 54 Chapter 4 S p e c i f i c a t i o n 58 Footnotes 68 Chapter 5 Estimation Method 69 Introduction 69 Cross-section estimation 71 Time-series estimation 74 Footnotes 81 Chapter 6 Implementation 82 Cross-section estimation 82 Time-series estimation 86 Estimated equivalence scales 95 Footnotes 99 - i v -Chapter 7 Applications 100 Introduction 100 Estimated market equivalence scales 101 S e n s i t i v i t y of the IEI index to p° 107 Comparative study of various measures 109 D i s t r i b u t i v e p r i c e e f f e c t s 115 Inequality trend 116 Conclusion 121 Footnotes 123 Chapter 8 Conclusion 125 Bibliography 130 Appendices 135 - v -L i s t of Tables Table 1 Vector A 61 Table 2 Translog Market Equivalence Scales 102 Table 3 S t a t i s t i c s Canada Low-Income Cut-off Ratios 108 Table 4 S e n s i t i v i t y of IEI Measure to p° 110 Table 5 Comparison of D i f f e r e n t Measures 111 Table 6 S e n s i t i v i t y of IEI Measure to p 117 Table 7 Inequality Trend (IEI Measure) 119 Table 8 Inequality Trend (iIWR (Stat. Can) Measure) 120 - v i -Acknowledgement I wish to express my deepest gratitude to members of the advisory committee: Charles Blackorby, Erwin Diewert and David Donaldson (thesis supervisor) for t h e i r continuous guidance and support. They have played a c r i t i c a l r o l e throughout the research. Thanks are also due to Debra Glassman, Jon Kesselman and Hugh Neary f or reading an e a r l i e r d r a f t and for t h e i r elaborate comments on s t y l e and exposition. I should also thank Jean Wu of Arts Computing, UBC for her valuable consultancy work i n computer operations. - v i i -- 1 -CHAPTER 1 INTRODUCTION The issue of i n e q u a l i t y — the divergence i n well-being among the i n d i v i d u a l s i n a society — has t r a d i t i o n a l l y been of great con-cern to economists. This i s hardly s u r p r i s i n g because a basic theme i n economics i s the a l l o c a t i o n of society's resources and the d i s t r i -bution of society's wealth. Indeed, systematic study of i n e q u a l i t y can be found as e a r l y as Cannan (1914) and Dalton (1920). Putting aside the question of what causes i n e q u a l i t y and the more contro-v e r s i a l issue of what ideology 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 of i n e q u a l i t y i s important for at l e a s t two reasons. F i r s t , the government may want to know the i n e q u a l i t y implications of a l t e r n a t i v e p o l i c i e s . Second, i t i s i n t e r e s t i n g to compare the degree of 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 contem-poraneously and i n time-series f o r a p a r t i c u l a r society. The objective of the present research i s to develop an i n e q u a l i t y index which i s a vast improvement over e x i s t i n g ones i n that i t brings i n t o sharp focus the notion of i n d i v i d u a l welfare i n the measurement of i n -e q u a l i t y . In p a r t i c u l a r , based on revealed behavioural data, the impact of p r i c e changes on i n d i v i d u a l welfare i s incorporated i n t o i n e q u a l i t y measurement. Echoing the idea of Dalton (1920), recent research developments (Atkinson (1970) and Blackorby and Donaldson (1978))have re-emphasized the f a c t that underlying every i n e q u a l i t y index i s a set of e t h i c s . - 2 -I t i s c l e a r that i n e q u a l i t y measurement i s a normative endeavor rather than a p o s i t i v e one. The claim that one d i s t r i b u t i o n of welfare (however defined 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 set of e t h i c s . I t i s therefore important that the p a r t i c u l a r set of et h i c s i s made e x p l i c i t . The main task of the present research i s not i n disputing the p a r t i c u l a r ethics that one should choose to measure i n e q u a l i t y . This i s an i d e o l o g i c a l question. The basic l i n e of attack i s : what should be the basic e n t i t i e s that we use to measure i n d i v i d u a l w e l l -being? Let us look at a common example. The Atkinson (1970) index i s , for N households (for a recent a p p l i c a t i o n and some ad hoc var i a n t s , see Beach, Card and F l a t t e r s (1981)), v N 1/ (1.1) I A : = 1 - (± Z ( y ± / y ) r ) / r r < l , r ^ O i = l N 1/N (1.2) : = 1 - TT ( y./u ) A / r = 0 . i = l 1 where (y , y ) i s a d i s t r i b u t i o n of household incomes and u i s the mean of the d i s t r i b u t i o n . J u s t i f y i n g the i n e q u a l i t y index (1.1) (1.2) i s the mean of order r s o c i a l welfare function (provided that the household incomes are r e s t r i c t e d to be po s i t i v e ) which i s - 3 -completely e t h i c a l l y characterized i n Blackorby and Donaldson (1982). There are two objections to the index (1.1) , (1.2) that ar i s e from using household income as a measure of i n d i v i d u a l w e l l -being. F i r s t , t h i s index i s i n s e n s i t i v e to p r i c e changes while, 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 should have d i s t r i b u t i o n a l impacts. For example, an increase i n the p r i c e of n e c e s s i t i e s r e l a t i v e to luxuries a f f e c t s the poor more than the r i c h . Such a r e l a t i v e p r i c e increase must aggravate the i n e q u a l i t y s i t u a t i o n and the index should increase to r e f l e c t t h i s change. 1 The second o b j e c t i o n i s , i n (1.1) and (1.2), that the d i s t r i b u t i o n was o r i g i n a l l y taken to be a d i s t r i b u t i o n of household incomes. This i s c l e a r l y i n c o n s i s t e n t with the s o c i a l welfare view of i n e q u a l i t y where i n d i v i d u a l s are viewed as the basic e n t i t i e s i n society, not the c o l l e c t i v e units — households. Various ad hoc modifications have been made i n the l i t e r a t u r e though none of them i s s a t i s f a c t o r y (see Chapter 2 f o r d e t a i l s ) . The basic question "How should we adjust household income or expenditure so that an i n d i v i d u a l i n a family of say, four members, can be reasonably compared i n welfare terms with an i n d i v i d u a l i n a family of one?" has not been adequately d e a l t with. This i s r e f e r r e d to below as the problem of interpersonal comparison of u t i l i t y . The present research attempts to f i l l t h i s gap i n the l i t e r a t u r e . A new approach to i n e q u a l i t y measurement i s developed which deals - 4 -with these problems e x p l i c i t l y and systematically. Since the ultimate t e s t of t h i s new index i s i n i t s p r a c t i c a l usefulness, the approach has been implemented for Canada and the r e s u l t s are i n general very appealing. The new approach can be summarized as follows. Two l i n e s of research are merged together, namely, welfare measurement and evalua-t i o n , and demand system estimation. In u t i l i s i n g both techniques, analysis i s extended from the micro to the macro. As a f i r s t step, household u t i l i t y i s measured by means of an i n d i r e c t u t i l i t y function which maps p r i c e s and household expenditure to an o r d i n a l u t i l i t y number. This u t i l i t y function's novel feature i s that household expenditure instead of i n d i v i d u a l expenditure enters the function. The reason f o r t h i s s p e c i f i c a t i o n i s that i n p r a c t i c e i n d i v i d u a l expenditure data are not e a s i l y obtained. However, as the objective of f i n d i n g a numerical u t i l i t y representation i s to measure i n e q u a l i t y i n the aggregate, the u t i l i t y number must be capable of being i n t e r -preted as the u t i l i t y of each member i n the household. Whether t h i s i n t e r p r e t a t i o n i s acceptable or not depends on the form and the parameter estimates of the u t i l i t y function. Barten equivalence scales provide one such form and they are estimated together with other parameters from demand data. Thus, by incorporating family siz e and other a t t r i b u t e s i n t o the u t i l i t y function, each household i s endowed with a household-specific u t i l i t y function. - 5 -The o r d i n a l nature of the u t i l i t y number gives r i s e to further ' problems. Subjecting a u t i l i t y function to an a r b i t r a r y i n d i v i d u a l -s p e c i f i c monotonic transform y i e l d s the same set of demand equations. Even i f a l l the parameters i n the u t i l i t y function are accurately estimated from demand data, the u t i l i t y number i s s t i l l arbitrary.. This problem i s not serious i f only the u t i l i t y ranking of a single i n d i v i d u a l i s concerned. But i n e q u a l i t y measurement implies u t i l i t y measurement and comparison f o r at l e a s t two i n d i v i d u a l s . Consequently, a numerical representation of u t i l i t y i s obtained by using a reference 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 of interpersonal u t i l i t y comparison. This representation, named the equivalent income of each member i n a s p e c i f i e d household i s the t o t a l expenditure that a 'reference household needs at .reference p r i c e s i n order that each member i n i t i s ju s t as well o f f as each member i n the household with 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 . Equivalent income i s a function of p r i c e s , expenditure, a t t r i b u t e s , reference p r i c e s and reference a t t r i b u t e s , and i t i s estimable e m p i r i c a l l y using demand data. Subject to the reasonableness o f the parameter estimates, 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 to the second problem mentioned above, and to the extent that equivalent income i s se n s i t i v e to p r i c e s , i t o f f e r s a solu t i o n to the f i r s t problem. A d i s t r i b u t i o n of i n d i v i d u a l equivalent incomes i s then aggregated by means of a s o c i a l welfare function. A mean of order r function i s used which has been characterized i n terms of e t h i c a l axioms i n Blackorby and Donaldson (1982). Adopting the 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 Atkinson index of equivalent incomes. The estimation phase also involves extension from the micro to the macro. Expenditure share equations are derived from the household i n d i r e c t u t i l i t y functions with the equivalence scales incorporated. Estimation i s c a r r i e d out i n two stages: the f i r s t stage involves only micro household commodity-expenditure share equations which are regressed using cross-section household expenditure survey data. Since some parameters i n the equivalent income function are not yet i d e n t i f i e d , the micro equations are summed together to obtain aggregate commodity-expenditure share equations which allow the u t i l i z a t i o n of time-series aggregate demand data to estimate the remaining parameters i n the equivalent income function. The implementation and the r e s u l t s of applying t h i s new index i n the Canadian context are described l a t e r . I t might be h e l p f u l , nevertheless, t o mention some of the main contributions of t h i s research here. 1. Unlike a l o t of other empirical demand studies, the present approach does not assume the existence of an aggregate consumer. Instead, households are s p e c i f i c to the extent that t h e i r c h a r a c t e r i s t i c s are captured by a t t r i b u t e vectors incorporated i n t o the u t i l i t y f unction. The sum t o t a l of a l l household demands y i e l d s aggregate - 7 -demand which enables the u t i l i z a t i o n of time-series aggregate data. This i s not only a t h e o r e t i c a l l y exact approach, i t i s also e m p i r i c a l l y superior, as the estimation r e s u l t s show that, based on behavioural demand data, meaningful welfare information can be i n f e r r e d . The equivalence scale estimates appear very reasonable. 2. The new index t r u l y captures d i s t r i b u t i v e p r i c e e f f e c t s , despite the margin of e r r o r that we might suspect i n t h i s type of demand system studies. Indeed, r e s u l t s show that commodities commonly regarded as luxuries have an inequality-reducing p r i c e e f f e c t while the opposite i s true f o r n e c e s s i t i e s . 3. When we compare indexes that vary from 0 to 1, the discrepancy between two indexes i s expected to be small. Nevertheless, the new index turns out to 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 indexes. We may conclude that these indexes give a d i s t o r t e d p i c ture of the true i n e q u a l i t y s i t u a t i o n . 4. Although t h i s methodology i s developed f o r i n e q u a l i t y measurement, i t can be applied with some modifications to other kinds of welfare analyses. The framework i s quite general. For example, i n cost-benefit analyses, one frequently looks f o r a s o c i a l welfare 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 states are being compared. This i s e a s i l y handled within the present framework, given p r i c e and expenditure information i n each state. - 8 -This d i s s e r t a t i o n i s organized as follows. 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 with s p e c i a l emphasis on the various measures of u t i l i t y used. The works of Muellbauer (1974 a, b, c) and Jorgenson and Slesnick (1982 a, b)(1984), which are pre-liminary attempts to capture p r i c e e f f e c t s , are explained and c r i t i c i z e d i n r e l a t i o n to the present study. Chapter 3 i s the core chapter. I t describes the t h e o r e t i c a l r a t i o n a l e of the s o c i a l evaluation framework and how an i n e q u a l i t y index i s constructed i n t h i s framework. The empirical s p e c i f i c a t i o n of the model i s presented i n Chapter 4. Preferences are assumed to be non-homothetic tr a n s l o g faith Barten equivalence scales incorporated) from which expenditure equations (household and aggregate) are derived. Chapter 5 explains how the estimation model of household and aggregate expenditure shares can be estimated using cross-section and time-series data sequentially. Canadian data 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 data can be u t i l i z e d to estimate the model set out i n Chapter 5 and the estimated Barten Equivalence Scales are presented and in t e r p r e t e d . The r e s u l t s are i n general very appealing, lending further support to the c r e d i b i l i t y of the new index. - 9 -Applications of the estimates to i n e q u a l i t y measurement are presented i n Chapter 7. F i r s t l y , various i n d i v i d u a l welfare measures are used to c a l c u l a t e i n e q u a l i t y using the same data set and the same formula for the i n e q u a l i t y index. I t turns out that the new index gives s i g n i f i c a n t l y d i f f e r e n t answers from other commonly used indexes. Secondly, to 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 of 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 using the new index under hypothetical p r i c e increases and the r e s u l t s conform well with i n t u i t i o n . F i n a l l y , i n e q u a l i t y i n Canada i n 1975, 1979 and 1981 i s estimated to r e f l e c t on the i n e q u a l i t y trend i n the l a s t decade. Chapter 8 concludes 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 the need f o r f i l l i n g t h i s gap i s evidently urgent. Since preferences must be involved i n the evaluation process, a l o g i c a l way to proceed i s to estimate hypothesized preferences from behavioural demand data. Various problems a r i s e , however. There being no objective measure of welfare, no data on i n d i v i d u a l expendi-ture, no a p r i o r i dominating rule of interpersonal comparison are j u s t some of the problems to which the present research has o f f e r e d s o l u t i o n s . A new 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 constructed. Implementation r e s u l t s show that the approach i s - 10 -p r a c t i c a l and reasonable. 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 to the other indexes i n both i t s t h e o r e t i c a l foundation and e m p i r i c a l usefulness. - 11 -Chapter 1 Footnote 1. Jorgenson and Slesnick (1984) have attempted to construct a p r i c e - s e n s i t i v e index, but f o r reasons that w i l l be made c l e a r i n Chapter 2, t h e i r approach i s not completely s a t i s f a c t o r y . - 12 -CHAPTER 2 SURVEY OF THE LITERATURE From a general perspective, 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 exercise that i s not confined within the realm of welfare economics. Given a d i s t r i b u t i o n of numbers (could be incomes, wealths, or s i z e of firms) a s t a t i s t i c i a n t y p i c a l l y applies an i n -equality formula to map t h i s d i s t r i b u t i o n to an index number. Ty p i c a l examples of such indexes are the Gi 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 of v a r i a t i o n and the Atkinson index. Per se, the index number does not have any s i g n i f i c a n c e besides r e f l e c t i n g c e r t a i n mathematical c h a r a c t e r i s t i c s of the d i s t r i b u t i o n . The present t h e s i s , on the other hand, are mainly concerned with 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 of welfare among i n d i v i d u a l s i n society. In general, three considerations are cen 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 basic e n t i t y that r e f l e c t s i n d i v i d u a l well-being and how i s i t obtained empirically? Secondly, since i n e q u a l i t y measurement r e s t s on a foundation of s o c i a l welfare evaluation, what framework should one adopt to summarize the d i s t r i b u t i o n to obtain a s o c i a l welfare measure? T h i r d l y , what s o c i a l welfare function (characterized by a set of e t h i c a l axioms) should be used to aggregate the d i s t r i b u t i o n and what i n e q u a l i t y index ( r e l a t i v e , absolute or others) should be employed? - 13 -I n t e r e s t i n g l y , the methods i n the l i t e r a t u r e do not follow t h i s l o g i c a l procedure. Section 1 below describes the common e n t i t i e s used. Not only are they inappropriate as measures of i n d i v i d u a l welfare, they are also p r i c e - i n s e n s i t i v e , which explains why the t r a d i t i o n a l indexes are a l l incapable of i n d i c a t i n g d i s t r i b u t i v e p r i c e e f f e c t s . Section 2 c i t e s some evidence of d i s t r i b u t i v e p r i c e e f f e c t s and describes the essence of the Muellbauer (1974) method and the Jorgenson-Slesnick (1984) method which are u n s a t i s f a c t o r y attempts to capture these p r i c e e f f e c t s . Section 1 Some simple indexes The most commonly used e n t i t y i n the measurement of economic in e q u a l i t y i s household income. The reason f or i t s widespread u t i l i z a t i o n i s probably that income .data are e a s i l y a v a i l a b l e . Household incomes are e a s i l y extracted from tax returns. Besides, r e l a t i v e l y speaking, they are quite r e l i a b l e i n accuracy terms. However, household income as a measure of i n d i v i d u a l u t i l i t y i s subject to a number of serious objections. (1) Individual u t i l i t y , i n r e l a t i o n to household income, depends very much on household s i z e and to a l e s s e r extent on household com-po s i t i o n , i . e . , the number of male and female adults, male and female ch i l d r e n i n the household. Using household income as a measure of i n d i v i d u a l u t i l i t y p r a c t i c a l l y means regardless of - 14 -household s i z e , household income indicates the ranking between , any two households i n welfare terms. (2) Consumers derive u t i l i t y from consumption rather than income r e c e i p t s . While income stream may be uneven over time, consumers tend to smooth out consumption by saving and dissaving. There-fore, u t i l i t y v a r i a t i o n s come c l o s e r to consumption v a r i a t i o n s than income v a r i a t i o n s . Furthermore, as obtained from a cross-section sample, income values are often negative ( p a r t i c u l a r l y for o l d consumers) a r i s i n g from c a p i t a l losses. These negative numbers create d i f f i c u l t i e s when aggregate s o c i a l welfare i s computed from i n d i v i d u a l incomes. In view of the second objection, the f i r s t i n e q u a l i t y measure to be computed i n Chapter 7 f o r comparative purposes i s the house-hold expenditure index (HEI). To each household i s imputed i t s t o t a l expenditure and i n e q u a l i t y i s ca l c u l a t e d based on the d i s t r i b u t i o n of household expenditures. Because of the f i r s t objection, HEI i s not j u s t i f i a b l e i n terms of normal s o c i a l e t h i c s , but i t i s worthwhile to 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 other measures. One simple and natural way to improve on the HEI i s by denominating household expenditure by household s i z e to a r r i v e at per ca p i t a expenditure. The l o g i c a l way to proceed then i s to impute per - 15 -c a p i t a expenditure to each i n d i v i d u a l of the household as a measure of i n d i v i d u a l u t i l i t y . Curiously, t h i s i s not what i s usually 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 instead of expenditure, what they would have done with expenditure would be to impute per c a p i t a expenditure to each house-hold rather than each i n d i v i d u a l , which i s again unjustiable i n terms of s o c i a l e t h i c s . To the extent that i n d i v i d u a l s constitute society, a l l i n d i v i d u a l welfares should have i d e n t i c a l weights i n s o c i a l wel-fare aggregation and not weights that vary with household s i z e . For example, i f the s o c i a l welfare function i s a d d i t i v e , such as mean of order r, each person i n an n-person household bears a weight of 1/n as opposed to 1. Therefore the acceptable way of imputing per capita expendi-ture as a measure of u t i l i t y i s to impute i t to each i n d i v i d u a l i n so c i e t y . This gives r i s e to the per capita expenditure (PCE) index which i s the second i n e q u a l i t y index computed for comparative purposes i n Chapter 7. The per-capita approach, as a method of approximating i n d i v i d u a l u t i l i t y using household expenditure, has been subject to c r i t i c i s m s . Wolfson (1979) points out that t h i s method ignores economies of scale i n the consumption of c a p i t a l s e r v i c e s . A better way i s , he suggests, to use "adult equivalents" i n place of family si z e to denominate household expenditure. - 16 -His method can be i l l u s t r a t e d as f o l l o w s i Let y, be the h t o t a l expenditure of household h. Let be the "low income c u t - o f f " l i n e for household h. 1 The welfare r a t i o , w , i s the r a t i o , y./L-n n n which i s imputed to household h as a measure of u t i l i t y . I f the in e q u a l i t y index i s r e l a t i v e ( i . e . , i t i s homogeneous of degree 0 i n the arguments), then t h i s welfare r a t i o approach i s i d e n t i c a l to using L h y, /-— , where L i s the "low income c u t - o f f " l i n e f o r a reference, h L q o say, one-adult-male, household. L, /L can be regarded as the number n o L h of equivalent-adults i n household h and y, /-— i s named " i n f l a t e d n L o welfare r a t i o ' 1 . 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 economies of scale, t h i s method i s an easy way to capture these scale e f f e c t s . Wolfson's method i s also subject to the c r i t i c i s m that waifare r a t i o s are imputed to households rather than i n d i v i d u a l s . In Chapter 7, i t w i l l be demonstrated that 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. In that Chapter the t h i r d index, which uses i n f l a t e d welfare-ratios imputed to households (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 fourth 2 index which imputes to i n d i v i d u a l s (IIWR). Furthermore, using w e l f a r e - r a t i o to represent i n d i v i d u a l u t i l i t y i s unsatisfactory f o r three reasons even though i t i s already an improvement over the , :per c a p i t a " method. F i r s t l y , while "economies - 17 -of scale" are incorporated, i t has been assumed that the degree of economies of scale i s the same for a l l goods and services. This i s 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 services such as housing and transportation should e x h i b i t higher degree of economies of scale than consumption goods l i k e food and c l o t h i n g . Secondly, the 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 . Based on d i f f e r e n t d e f i n i t i o n s of poverty, there e x i s t three sets of low-income cut-off l i n e s i n Canada and there i s no dominating e t h i c a l reason for p r e f e r r i n g any set over the other two. (see Osberg (1981)). T h i r d l y , t h i s welfare-r a t i o method (as normally used) does not 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 at capturing 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 Canada, the commonly used low-income c u t - o f f s are those published by S t a t i s t i c s Canada. This set i s r e v i s e d annually only f o r i n f l a t i o n which does not a f f e c t the f i n a l i n e q u a l i t y measure i f the index i s 3 relatxve, i . e . , mean-independent. On the other hand, i f the index i s non-relative, i t i s not c l e a r why i n f l a t i o n should a f f e c t i n e q u a l i t y that i s c a l c u l a t e d based on expenditure. Section 2 D i s t r i b u t i v e p r i c e e f f e c t s I t i s somewhat obvious that 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 context, a recent attempt to study the welfare e f f e c t s of p r i c e changes i s contained i n Roberts (1982). The main objective of h i s study i s to investigate the e f f e c t of food p r i c e changes on c o s t - o f - l i v i n g indexes of - 18 -households i n f i v e income q u i n t i l e s . He uses family expenditure survey data for f i v e separate years to estimate a l i n e a r expenditure system of eight goods and services f or each income q u i n t i l e . In the regression, per-capita expenditure instead of household expenditure i s used to adjust f o r family s i z e . Exact c o s t - o f - l i v i n g indexes are then computed (being r a t i o s of the minimum expenditure to a t t a i n a 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 ) f or each of the f i v e income q u i n t i l e s . The basic f i n d i n g i s that, food p r i c e i n f l a t i o n tends to increase the c o s t - o f - l i v i n g index for the lowest income q u i n t i l e more than the highest income q u i n t i l e . Since food accounts for a higher percentage of t o t a l budget i n the poor, r e l a t i v e to the r i c h , t h i s f i n d i n g i s not s u r p r i s i n g at a l l . What one notes with i n t e r e s t i s : the c o r o l l a r y of t h i s r e s u l t i s that r e l a t i v e p r i c e changes have a d e f i n i t e impact on i n e q u a l i t y . Unfortunately, none of the indexes described so f a r i s capable of measuring t h i s impact because they are a l l c a l c u l a t e d based on income and expenditure or simple adjustments of income and expenditure. There are two studies i n the l i t e r a t u r e which attempt to capture p r i c e e f f e c t s , namely, Muellbauer (1974, a,b,c) and Jorgenson and Slesnick (1984). However, as explained i n the following, both attempts are uns a t i s f a c t o r y . - 19 -Muellbauer's method Muellbauer (1974 a,b,c) attempted to capture r e l a t i v e p r i c e e f f e c t s and adjust income for family s i z e and economies of scale i n one coherent model. His method i s summarized as follows. He s p e c i f i e s a household u t i l i t y function which has the image I (2.1) u = U(x/m, , x /m) 1 n where (x,, , x ) are household consumption and m i s the number I n of equivalent adults, taken a r b i t r a r i l y from Prest and Stark (1967) and Stark (1972). The numbers are Family si z e 1 2 3 4 5 6 7 8 m 1 1.6 2.1 2.5 2.8 3.2 3.6 4.0 which as a sequence, shows economies of scale i n consumption. I t follows from (2.1) that the image of the i n d i r e c t u t i l i t y function i s (2.2) u = V(p , , p , y/m) 1 n and that of the cost function i s r (2.3) y = C(u, mp^ , m P n ) - 20 -Muellbauer (1974 a,b,c) employs an adaptation of the money-metric u t i l i t y of Samuelson (1974) to represent household u t i l i t y . This concept i s further discussed i n Chapter 3, but b r i e f l y , money-metric u t i l i t y i s the income that enables an i n d i v i d u a l (a household i n the present context) to a r r i v e at a given l e v e l of u t i l i t y at reference p r i c e s . Muellbauer's version of money-metric u t i l i t y , however, i s represented, for household h, by ~h _ , h o o o o . (2.4) y = C ( u , m p 1 , ,m p n) f,,, h . h. o o\ = C[V(p 1, p n , y /m ), p±, p n J where y i s income of household h, m i s the number of equivalent adults and m° i s taken as unity, (the number of equivalent adults of 4 a one-person household). Given a society of H households, Muellbauer 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 and 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 , y H ) . This method i s not e n t i r e l y s a t i s f a c t o r y , although the index based on (2.5) i s p r i c e - s e n s i t i v e . F i r s t l y , the scale of equivalent adults i s taken from a separate study. Since u t i l i t y i s given a representation (2.1), the scale numbers should be estimated i n one pass together with other parameters i n the u t i l i t y function. - 2 1 -Furthermore, as argued above, assuming the same degree of economies of scale for a l l goods and services i s not very r e a l i s t i c . Secondly, i t i s not c l e a r what (2.4) means. In p a r t i c u l a r , Muellbauer seems to suggest the u t i l i t y number i n (2.2) i s a measure of the u t i l i t y of each i n d i v i d u a l i n the household. But i f t h i s i s h i s i n t e n t i o n , he should impute money-metric u t i l i t y of a household to each member i n the household i n (2.5), and expand the dimension of (2.5) to the t o t a l number of i n d i v i d u a l s i n society. A r e l a t e d issue i s the problem of interpersonal comparison of u t i l i t y i s com-p l e t e l y ignored. An objective measure of u t i l i t y does not e x i s t . Muellbauer's money-metric u t i l i t y (2.4) represents one p a r t i c u l a r numerical representation of 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 r u l e of interpersonal comparison. This assump-5 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 i s applicable d i r e c t l y to the case of a single i n d i v i d u a l only. Extending t h i s concept to a multi-person s i t u a t i o n needs more j u s t i f i c a t i o n . Jorgenson-Slesnick method A recent attempt to construct 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 index can be found i n Jorgenson and Slesnick (1984).. (see also (1982 a,b,c), (1983 a,b)). Their basic strategy i s : they specify a t r a n s l o g household u t i l i t y function which incorporates commodity-- 22 -s p e c i f i c equivalence scales to account f o r a t t r i b u t e d i f f e r e n c e s among households and estimate the parameters using demand data. Based on these estimates, they attempt to use u t i l i t y numbers i n a non-welfarist s o c i a l welfare framework to a r r i v e at a measure of i n e q u a l i t y . The d i r e c t household u t i l i t y function has the image (2.6) u = ufx /m. (A) , , x /m (A)) v 1 1 n n ' where (A) , m^(A) are the commodity-specific equivalence scales, which are functions of a t t r i b u t e s A. I t follows from (2.6) that the i n d i r e c t u t i l i t y function has the image (2.7) u = vfm. (A)p., ..., m (A)p , y) v 1 1 n n ^ Assuming translog preferences they claim that given the parameters involved i n (2.7), a u t i l i t y number i s obtainable f o r each household given 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 expenditure. This u t i l i t y number i s taken as a measure of household u t i l i t y , such that f o r household h, u t i l i t y i s (2.8) u h = v f m . ( A h ) p . , m (A h)p , y h ) . ^ 1 1 n n - 23 -In the aggregate, the s o c i a l welfare framework i s unorthodox. So c i a l welfare w i s given by a non-welfarist s o c i a l evaluation functional i . e . , (2.9) w = Ea, (x)U h(x) - y(x) (la,, (x) |uh(x) - u , n \ h ' - |Pjl/P where x i s a state v a r i a b l e and (2.10) u := Ea^(x)U h(x) x , h n I t should be emphasized that (2.9) i s non-welfarist because a, , n 6 h h = l , H and y are functions of x. U (x) i n (2.9) and (2.10) i s taken as (2.8) even though the number u has no c a r d i n a l s i g n i f i -7 cance. The form for a, (x) i s assumed to be h (2.11) a. (x) = m (p, A n)/Em (p, A*1) , where h o h o c(u h,m (A h)p , , m (A h)p ) (2.12) m (p, A ) = X n C (u ,p , , p ) 1 n One may r e c a l l that (A*1) , i = 1, . .., n i s the equivalence scale factor of household h for good i . If one normalizes the factors for a reference household (having a t t r i b u t e A°) to be unity, i . e . , (2.13) m ±(A°) = 1 i = 1, ... , n - 24 -then ^ ( p , A ) can be interpreted as a general market equivalence scale which measures the f r a c t i o n of t o t a l expenditure required to keep household h and the reference household at the same u t i l i t y l e v e l . These scales d i f f e r f rom those i n (2.6) i n that they are functions of p r i c e s as well as a t t r i b u t e s . y(x) i n (2.9) i s assumed to have the form, (2.14) y(x) = 1 + ({Ea, (x) }/a. ( x ) ) p 1  h 3 -1/P = ( l + a . ( x ) 1 _ P ) ^ 3 J -1/P where a. (x) = min a, (x) . 3 v k S u b s t i t u t i n g (2.11) (2.14) in t o (2.9), the f i n a l s o c i a l welfare function i s (2.15) w = u 1 + a. 3 1-P -1/P (inMp, A h ) | u h ( x ) - u x | P ) / Em (p, A ) o Given f i x e d society expenditure, one may f i n d the d i s t r i b u t i o n of house-hold expenditure which maximizes w i n (2.15). Because of translog preferences and the assumption (2.11), the f i r s t order conditions imply each household i s endowed with the same "equivalent income", - 25 -i . e . , for household h and k h k (2.16) — = ^ — h,k = 1, H so that m (p, A ) m (p, A ) o o (2.17) y h/Y k = m (p, A h)/m (p, A k ) . o o At the maximum, two households with d i f f e r e n t a t t r i b u t e s w i l l be given d i f f e r e n t incomes according to the general market equivalence scales and given same income i f they have the same a t t r i b u t e s . At t h i s point, the second term i n (2.15) vanishes so that maximum s o c i a l welfare equals u^, and a l l households are regarded as equally w e l l - o f f . L e t t i n g w be the actual s o c i a l welfare given i n (2.15), an in e q u a l i t y index i s proposed, i . e . , (2.18) i = i - w / u and 0 < I < 1 JS x JS This approach to i n e q u a l i t y measurement i s not acceptable. The s o c i a l welfare framework described above i s open to c r i t i c i s m s . The e t h i c a l reason f o r adopting a non-welfarist framework i s not c l e a r , although they c i t e Sen's argument (1979) to support t h e i r procedure. Sen's argument against welfarism i s based on the lack of consideration of 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 welfare function. These absolute r i g h t s r e f e r to equal-work-for-equal-pay, - 26 -freedom from e x p l o i t a t i o n and s o c i a l l i b e r t y . Sen does not imply, nor i s i t reasonable to assume that any function that depends ex-p l i c i t l y on state c h a r a c t e r i s t i c s (hence non-welfarist) i s an improvement over a w e l f a r i s t function. Furthermore, the way the s o c i a l welfare function (2.9) captures these c h a r a c t e r i s t i c s , through Y (x) and a^(x), i s ad hoc and far away from Sen's o r i g i n a l i n t e n t i o n . It does not capture what welfarism misses. By the d e f i n i t i o n of a, (x), (2.11), the s o c i a l welfare h function does not s a t i s f y anonymity, i . e . , each i n d i v i d u a l i s not equally important i n the s o c i a l ranking. Each household's u t i l i t y i s assigned a weight proportional to the estimated number of "equivalent adults" according to the estimated general market equi-valence scales (2.12) . In case of s i g n i f i c a n t economies of scale i n consumption the number of "equivalent adults" i s much smaller than family s i z e . But i t i s u n j u s t i f i a b l e to assign smaller weights to members i n large households r e l a t i v e to members i n small households. The e t h i c a l basis i s not c l e a r . The authors f a i l to give a f u l l set of axioms that completely characterize (2.15). This i s serious because, as emphasized i n Chapter 1, i n e q u a l i t y measurement i s a normative judgemental exercise that i s contingent on an underlying set of e t h i c a l axioms. Consequently, t h i s approach has l e f t some room for improvement. - 27 -To summarize t h i s chapter, we have seen t h a t a l l the indexes developed i n the l i t e r a t u r e are u n s a t i s f a c t o r y . They are not con-s i s t e n t w i t h the s o c i a l welfare view of economic i n e q u a l i t y . In a d d i t i o n , there i s no index t h a t can demonstrate the d i s t r i b u t i v e impact of p r i c e changes. Therefore, a new approach i s u r g e n t l y r e q u i r e d t o f i l l t h i s gap. - 28 -Chapter 2 Footnotes 1. S t a t i s t i c s Canada publishes low-income cut-off l i n e s that are s p e c i f i c to household si z e and s i z e of area of residence. Cat. No. 13-207. 2. I n f l a t e d w elfare-ratio i s used instead of Wolfson's welfare-r a t i o to make comparison with other measures more immediate. As the index i s r e l a t i v e , i n e q u a l i t y i s not a f f e c t e d . 3. The S t a t i s t i c s Canada c u t - o f f s are derived from expenditure surveys conducted once every f i v e years. They are estimated by taking the average household income of those households that spend 20% of the budget more than the average household of the same s i z e and area on the n e c e s s i t i e s — food, c l o t h i n g and s h e l t e r . The "mark-up" of 20% i s a r b i t r a r y . 4. As explained above i t would be more appropriate to use expenditure instead of income for y . 5. By contrast, the interpersonal comparison assumption i n the per-capita expenditure method and w e l f a r e - r a t i o method i s easier to see. - 29 -6. A s o c i a l evaluation functional i s w e l f a r i s t i f the states a f f e c t s o c i a l ranking only through t h e i r e f f e c t s on i n d i v i d u a l u t i l i t i e s . 7. Any monotonic transformation of a u t i l i t y function y i e l d s the same demand equations. Empirical estimation only y i e l d s enough information to allow a ranking of a l t e r n a t i v e price-income s i t u a t i o n s . (2.7) does not e s t a b l i s h an objective scale of u t i l i t y measurement. J-S f a i l to point out that the choice of the p a r t i c u l a r numerical representation (2.7) i s somewhat a r b i t r a r y . - 30 -CHAPTER 3 A NEW APPROACH This chapter describes the t h e o r e t i c a l background of t h i s new approach to i n e q u a l i t y measurement. The basic strategy can be i l l u s t r a t e d by the chart below. Based on hypothesized preferences, assume preferences represented by d i r e c t / i n d i r e c t u t i l i t y function and cost function derive demand/expenditure share equations i d e n t i f i c a t i o n parameters econometric U- procedure empirical data one can obtain a functional representation by the d i r e c t and i n d i r e c t u t i l i t y functions or the cost function. From any one of these functions, demand and expenditure share equations can be derived. A suit a b l e econometric procedure can then be devised to obtain empirical estimates f o r the parameters i n the demand and expenditure share equations. These estimates can also be used to i d e n t i f y the o r i g i n a l functions that represent preferences. I f these information are ava i l a b l e f o r a l l consumers, a casual observer might contemplate measuring i n e q u a l i t y using, say, the image of each consumer's i n d i r e c t u t i l i t y function, given p r i c e s and nominal t o t a l expenditure d i s t r i b u t i o n . - 31 -However, there are several problems involved i n t h i s general scheme. 1. The s p e c i f i c a t i o n of i n d i v i d u a l preferences need to allow for taste d i f f e r e n c e s a r i s i n g from various demographic c h a r a c t e r i s t i c s . Furthermore, only household expenditure data are a v a i l a b l e , as opposed to i n d i v i d u a l expenditure data. Therefore, assumed preferences have to (1) incorporate these c h a r a c t e r i s t i c s and (2) employ household expenditures i n a reasonable manner i n order that meaningful welfare information about the i n d i v i d u a l s i n the household can be revealed. Section 1 of t h i s chapter suggests an equivalence scales method that deals with these problems d i r e c t l y . 2. I t i s well-known that any a r b i t r a r y monotonic transform of the d i r e c t or i n d i r e c t u t i l i t y function y i e l d s the same demand equations. Consequently, even i f perfect parameter estimates are obtainable following the procedure described above, the images of the d i r e c t or i n d i r e c t u t i l i t y function are s t i l l a r b i t r a r y as u t i l i t y numbers. In the context of i n e q u a l i t y measurement, t h i s a r b i t r a r i n e s s cannot be allowed and the problem of interpersonal comparison of u t i l i t i e s has to be dealt with e x p l i c i t l y . Section 2 suggests that "equivalent income" as defined l a t e r , i s an acceptable measure of u t i l i t y for t h i s purpose. - 32 -Given a d i s t r i b u t i o n of acceptable measures of u t i l i t y , one can attempt to measure aggregate s o c i a l welfare. A s o c i a l welfare evaluation framework, can be constructed based on a set of e t h i c a l and informational assumptions. Described l a t e r i n Section 3 i s a wel-f a r i s t framework that i s argued to be appropriate i n the present context. In t h i s framework, a s o c i a l welfare function can be u t i l i z e d to aggregate i n d i v i d u a l u t i l i t i e s to a measure of s o c i a l welfare which then leads to the construction of a r e l a t i v e i n e q u a l i t y index i n Section 4. Section 1 Equivalence Scales Formally, the preferences of a society of H households can be 1 H represented by the u t i l i t y functions U , , U , with the i n t e r -p r e t a t i o n that (3.1) ^ = U h( x h ) i s the u t i l i t y of each member of household h and x*1 i s the consumption vector of household h."*" I t i s assumed that s t r i c t e q u a l i t y of u t i l i t y e x i s t s i n a l l the households. I t i s also assumed i n the following that differences i n preferences among households can be captured by a vector A which describes household a t t r i b u t e s . Formally, t h i s means that the preferences of society can be represented by - 3 3 -( 3 . 2 ) U( x 1 , A 1 ) , . , U ( x H , A H ) . The u t i l i t y of an i n d i v i d u a l i , who belongs to household h i s therefore ( 3 . 3 ) u ± = U( x h,A h ) i . e . , the u t i l i t y common to a l l members i n household h. Note that x i s household consumption and therefore the search f o r an appropriate form f o r U( x^,A^ ) , h = 1, , H i s c r u c i a l i n order to j u s t i f y the i n t e r p r e t a t i o n ( 3 . 3 ) . Subject to t h i s reservation, one u t i l i t y function can now be applied to a l l i n d i v i d u a l s of a l l households i n socie t y . ^ In order to 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 present approach adopts Barten ( 1 9 6 4 ) commodity-specific equivalence sc a l e s . With n goods, the u t i l i t y of each member of household h i s given by, from ( 3 . 3 ) , ( 3 . 4 ) u^ = u ( x J / m ; L ( A h ) , , x J j / m n ( A h ) ) where m, (A ), , m (A ) are the commodity-specific equivalence 1 n scales f o r household h, so that x^ / m ( A*1 ) i s the equivalent consump-t i o n of good i f o r household h, r e l a t i v e to a reference household whose scale factors ( A° ), , m ( A° ) are normalized to be 1 . 1 n For example, l e t family si z e be the only demographic c h a r a c t e r i s t i c described by A. Suppose - 34 -(3.5) m _ . ( A ° ) = m k ( A ° ) = l , j ,k = 1, . . ., n where A° describes a one-person-household. Then these equivalence scales can be regarded as factors that d e f l a t e household consumption to a r r i v e at e f f e c t i v e i n d i v i d u a l consumption. As family s i z e increases, the scale factors should increase to r e f l e c t the increasing need f o r each good. The rate of increase, being s p e c i f i c to the good, depends on the c a p a b i l i t y o f securing economies of scale i n consump-t i o n . By way of example, c l o t h i n g should have smaller economies of scale than housing. Thus, (3.4) i s a general s p e c i f i c a t i o n that maps household consumption and a t t r i b u t e s (through m^ , .. . ., mn) to a u t i l i t y number which can reasonably be regarded as the u t i l i t y of each member of the household. Special cases o f (3.4) include the "head-counting" method and Engel's method adopted by Muellbauer (1974 a, b, c ) . The head-counting case i s , (3.6) m j = •^ am-'--'-Y s i z e , j = 1, n which does not allow for economies of scale i n consumption, while Engel's case i s , l e t t i n g A represent family s i z e ; (3.7) (A) = ^ ( A ) j ,k = 1, , n, - 35 -i . e . , common equivalence scale across goods, which does not allow for d i f f e r i n g degrees of economies of scale among goods. Whether these s p e c i a l cases are good approximations of the general case can be checked by looking at actual empirical estimates. Obviously, these Barten equivalence scales can accommodate household c h a r a c t e r i s t i c s other than family s i z e . Consequently, i n the implementation of t h i s model, four 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 of the area of residence, the sex of the household head, family s i z e and the age of the household head. However, what the scales mean now i s not as c l e a r . I f family si z e i s the only relevant a t t r i b u t e , the structure of these scales r e f l e c t s the d i f f e r e n t degrees of economies of scale of d i f f e r e n t goods.. But what does i t mean i f the scale f a c t o r f o r , say, t r a n s p o r t a t i o n fo r r u r a l households i s higher than that for urban households? I t means that keeping e f f e c t i v e consumption ( r e l a t i v e to some reference household) of other goods the same, a household moving from an urban area to a r u r a l area needs more transportation i n order to be j u s t as well o f f as before. Given the s p e c i f i c a t i o n of the d i r e c t u t i l i t y function (3.4), i t follows that the i n d i r e c t u t i l i t y function must incorporate the scales by mark-ups i n p r i c e s , whose image is"^ (3.8) u, = V f m (A )p . . . . , m (A )p ,y ) h v 1 1 n n ' - 36 -where y*1 i s household h's t o t a l expenditure. The cost function, C i s 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 sol v i n g h for y . [3.9) y h = c(u. , n. ( A h )p , , m ( A h ) p ) n 1 1 n n Section 2 Equivalent Income In the l i t e r a t u r e , money-metric u t i l i t y was introduced i n Samuelson (1974) and Varian (1980) to i n d i c a t e the d i r e c t i o n of change i n an i n d i v i d u a l ' s u t i l i t y , and i s very close to the concepts of compensating v a r i a t i o n and equivalent v a r i a t i o n i n the consumer surplus l i t e r a t u r e . Let U be a d i r e c t u t i l i t y function s a t i s f y i n g the usual r e g u l a r i t y conditions — c o n t i n u i t y , p o s i t i v e s t r i c t monotonicity (to eliminate s a t i a t i o n ) and quasi-concavity. (3.10) u = U(x) The corresponding i n d i r e c t u t i l i t y function V and cost function C w i l l have images (3.11) u = V(p,y), and (3.12) y = C(u,p) - 37 -Money-metric u t i l i t y i s defined i n Samuelson (1974) as (3.13) M( x,p° ) : = C ( u ( x ),p°) o " where p i s a reference p r i c e vector. Since the cost function C i s increasing i n u, money-metric u t i l i t y e stablishes a scale that measures u t i l i t y as an income concept. This i s j u s t i f i a b l e because the same reference p r i c e p° i s used f o r a l l states, so that M i s o r d i n a l l y equivalent to U, regardless of the choice of p°, i . e . , (3.14) U( x 1 ) > U( x 2 ) <- ->-M(x 1 /p 0) > M(x 2,p° Em p i r i c a l l y , (3.13) i s d i f f i c u l t to handle. A l t e r n a t i v e l y , as King (1983) suggests, (3.11) instead of (3.10) can be substituted i n t o (3.12) which gives r i s e to a s o - c a l l e d r e a l income function, (3.15) C(p,y,p°) := £(V( p,y ) ,p°) C maps p r i c e s and expenditure, given a reference p r i c e vector, to a r e a l number, the r e a l income. I t s nature i s made c l e a r by regarding i t as a sol u t i o n to (3.16) V( p°,y e ) = V(p,y ) - 38 -so that (3.17) yQ = c(v(p,y ),p°) ~ e o y i s the amount of expenditure at p that w i l l keep the consumer j u s t as well o f f as i n the state (p,y). Since p, y are r e a d i l y observable *e va r i a b l e s , y i s a p r a c t i c a l measure of u t i l i t y . However, although ~e y i s exact i n showing the d i r e c t i o n of 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 absolute quantity as a r e s u l t of p i n (3.15). So f a r , only one person i s involved. In the present model, the idea of money-metric u t i l i t y i s adapted to take into consideration households that are i d e n t i f i e d by an a t t r i b u t e vector A. Equivalent e income, y w i l l be used to measure u t i l i t y which can be viewed as a so l u t i o n to (3.18) V(p°,y e,A° ) = V(p,y,A) so that, (3.19) y 6 = c(v(p,y,A ) ,p°A°) = : E(p,y,A,p ,A ) where p°, A° are reference p r i c e s and a t t r i b u t e vector of a reference household. Notice that (3.19) i s not a straightforward extension - 39 -of (3.17). While (3.16) j u s t compares u t i l i t i e s of one person, (3.18) represents e x p l i c i t interpersonal comparison of u t i l i t i e s e between two households, y i s the household expenditure that w i l l make each i n d i v i d u a l i n the household described by A° at p° just as well o f f as each i n d i v i d u a l i n the household described by A at p. Notice also that (3.18) represents one p a r t i c u l a r l e v e l of interpersonal comparison. E m p i r i c a l l y , there i s no objective measure of u t i l i t y . I f V i s found to be consistent with demand behaviour, so i s any household-specific monotonic transform of V. Indeed, con-sumers could "announce" t h e i r own l e v e l s of u t i l i t y according to t h e i r own scales of measurement so that "announced" u t i l i t i e s cannot be compared i n t e r p e r s o n a l l y . y i s not immune to t h i s a r b i t r a r i n e s s i n u t i l i t y measurement. By allowing household-specific monotonic transforms on V, a d i f f e r e n t equivalent income measure could be obtained by s o l v i n g f o r y e i n (3.20) $(v(p 0,y e,A 0),A°) = $ (v ( p,y h ,Ah ) , A h) so that (3.21) y e = C "*e e v u y i s equal to y i f and only i f A and A are i d e n t i c a l . Therefore, the interpersonal comparison (3.18) i s a key assumption i n t h i s $ ( . H V(p , y ,A ) ,A j ,A J ,p ,A - 40 -approach to i n e q u a l i t y measurement. Is t h i s assumption j u s t i f i a b l e and r e a l i s t i c ? It depends on the values of the commodity-specific equivalence scales, which are estimated from demand data. For example, consider family si z e as the only c h a r a c t e r i s t i c . The equi-valence scale for size 1 i s normalized to be 1. Then the equivalence scales f o r size 2 should f a l l between 1 and 2. In other words, the i n d i r e c t u t i l i t y function (3.8) has to play f u l l y the role of making interpersonal comparison (3.18) p o s s i b l e . In the subsequent model of i n e q u a l i t y measurement, equivalent income, being a r e s u l t of the interpersonal comparison (3.18), w i l l be used as a measure of i n d i v i d u a l u t i l i t y . This i s possible because by the d e f i n i t i o n of the equivalent income function (3.19) and the fa c t that C i s inc r e a s i n g i n u t i l i t y , ~ * / T - I - - > \ , - , , 1 1,1 o ,o , „, • 2 2,2 o,o, (3.22) E ( p ,y ,A ,p ,A ) > E ( p ,y ,A ,p A ) 1 1 1 2 2 2 -< • V ( p ,y ,A ) > V ( p ,y ,A ) 1 2 1 1 2 2 where households A and A face (p ,y ) and (p ,y ) r e s p e c t i v e l y and V ( p 1 , y 1 , A 1 ) i s the u t i l i t y of each member i n household i . 1 2 A c o r o l l a r y i s i f A i s set equal to A , then (3.22) implies E preserves each i n d i v i d u a l ' s u t i l i t y ranking. One can r e a d i l y v e r i f y that E i s i n f a c t a monotonic transform of V and applying Roy's Identity w i l l y i e l d the same set of demand functions. - 41 -e More i n s i g h t i n y can be gained by r e f e r r i n g to Deaton's e (1980) i n t e r p r e t a t i o n , y can be expressed as (3.23) y e = C<u,p,A) 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,A0 where (3.24) n( u,p,p°,A° ) : = C( u,p,A° )/C ( u,p°,A° ) i s a p r i c e index evaluated at u and A°, and (3.25) S( u,A,p,A° ) : = C( u,p,A )/C ( u,p,A° ) i s a market equivalence scale f a c t o r evaluated at u and p. I t i s now cl e a r that y i s s e n s i t i v e to p because both II and S are functions of p. II i s s p e c i f i c to a household (and to each i n d i v i d u a l therein) only to the extent that i t i s a function of u t i l i t y . Two households that are equally w e l l - o f f w i l l have the same p r i c e index regardless of a t t r i b u t e s . However, the market equivalence scale S i s i n general a function of both u t i l i t y and a t t r i b u t e s . ^ Therefore, both II and S capture some d i s t r i b u t i v e p r i c e e f f e c t s . - 42 -To conclude: equivalent income y i s used as a u t i l i t y measure i n s o c i a l welfare evaluation. The u t i l i t i e s of a l l i n d i v i d u a l s i n a l l households are measured by a common yardstick, namely, the t o t a l expenditure that w i l l keep each member of a reference household just as well o f f , at reference p r i c e s . Therefore, an equivalent income should be imputed to each i n d i v i d u a l i n soc i e t y . In a society of 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 of u t i l i t i e s for welfare evaluation purposes w i l l be (3.26) Cy®, , y®) e Since y i s p r i c e - s e n s i t i v e , the s o c i a l welfare i n d i c a t o r w i l l be al s o . Section 3 S o c i a l Choice This section introduces the s o c i a l welfare evaluation frame-work that maps the d i s t r i b u t i o n of i n d i v i d u a l equivalent incomes to a s o c i a l welfare number. This framework forms the basis of i n e q u a l i t y measurement. Soci a l welfare evaluation can be looked at as an aggregation problem. A p r o f i l e i s a vector of i n d i v i d u a l u t i l i t y functions defined over a set of s o c i a l s t a t e s . A s o c i a l evaluation functional i s then a mapping from such a p r o f i l e to a s o c i a l ordering over the same set of stat e s . In d e r i v i n g such a s o c i a l ordering, two sets of - 43 -assumptions are usually involved. The f i r s t set involves the e t h i c a l axioms that are argued as j u s t i f i a b l e and acceptable. For example, the weak Pareto rule i s commonly assumed, i . e . , i f every i n d i v i d u a l prefers state A to state B, the s o c i a l ordering must rank state A above state B. The second set of assumptions are the assumptions on measurability and interpersonal comparability of u t i l i t i e s . When one searches for s o c i a l welfare r u l e s , forms that can be completely characterized by axioms are favoured. In welfare evaluation, the e t h i c a l basis should be c l e a r , otherwise, no matter how v a l i d the information i s about i n d i v i d u a l welfare, the evaluation procedure i n the aggregate i s mechanical and u n j u s t i f i a b l e . Let T be the set o f a l l possible p r o f i l e s of i n d i v i d u a l u t i l i t y functions and D the domain o f f - the s o c i a l evaluation functional (D being a subset of T), RR the set of a l l possible orderings over the set of a l t e r n a t i v e s , X. Then (3.27) f :D > RR 1 N i . e . , R = f ( U , ,U ), u k k where U i s i n d i v i d u a l k's u t i l i t y function and U (x) i s h i s or her u t i l i t y i n a p a r t i c u l a r state x i n X. R^ i s the s o c i a l ordering 1 N associated with the p r o f i l e (U , ...... U ) , through f. - 44 -The following three axioms on f are commonly c a l l e d the "welfarism" axioms, (1) Unrestricted Domain T = D (2) Pareto Indifference Let I be the symmetric fa c t o r of R . If u u U(x) = U(y) , where U(x): = (u1 (x) , ., U N ( x ) ) , then X ^ L V ' f o r a l- L x# Y i n x a n d a l l U i n D. (3) Binary Independence of Irrelevant A l t e r n a t i v e s For 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 coincide on ( x, y ) . u u v ' These welfarism axioms (Blackorby, Donaldson and VJeymark (1983)) are important because they imply strong n e u t r a l i t y (SN), defined as follows: - 45 -Strong N e u t r a l i t y 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), then xR ,y •* >- wR „z and yR ,x •* »• zR „w u u u u This property i s very strong. In i t s e l f , i t means i n d i v i d u a l s ' u t i l i t i e s are the only determinants of s o c i a l welfare. Anything that can a f f e c t s o c i a l ordering has to "pass through" u t i l i t i e s . I t i s t h i s i n t e r p r e t a t i o n that gives the name "welfarism" to the three axioms. (see Sen (1977)). This framework contrasts sharply with the non-welfarist framework of Jorgenson and Slesnick (1982 a, b, c) (1983 a, b) where the s o c i a l welfare evaluation f u n c t i o n a l involves parameters a and y that are both functions of state x. I t can be shown that a w e l f a r i s t f implies and i s implied by N the existence of an ordering R on the r e a l Euclidean space R such that (3.28) xR y « • U(x)R U(y) u I t i s now possible to p a r t i t i o n the set of 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 sets by r e f e r r i n g only to 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 preferences, (namely - 46 -that the " s o c i a l l y at l e a s t as good as" and " s o c i a l l y at most as good as" sets are closed i n R^) w i l l provide for the existence of a 7 representing function W, generating the same ordering as R. W i s commonly referred to as the Bergson-Samuelson s o c i a l welfare function. Sen has r a i s e d objections against welfarism as an evaluation framework. I f welfarism i s assumed, i t i s natural to assume weak Pareto as well since only u t i l i t i e s determine s o c i a l ordering. In some cases, t h i s denies i n d i v i d u a l s absolute r i g h t s associated with such things as freedom from e x p l o i t a t i o n and "equal work f o r equal pay" which r e f e r to state c h a r a c t e r i s t i c s not captured by u t i l i t i e s . Sen c a l l e d these non-welfare c h a r a c t e r i s t i c s . However, under welfarism and weak Pareto, non-welfare c h a r a c t e r i s t i c s have no r o l e to play i n Q determining the s o c i a l ordering. I t i s c l e a r that the choice between welfarism and non-welfarism depends on the type of a n a l y s i s . In p o l i c y questions where pr i c e s and income are p a r t l y p o l i c y v a r i a b l e s , welfarism i s adequate. Questions l i k e the impact of tax and t a r i f f changes on economic i n e q u a l i t y can be sens i b l y asked within t h i s framework. Welfarism i n pr a c t i c e allows easy estimation of welfare i n d i c a t o r s since a l l that i s required i s computing i n d i v i d u a l measures of u t i l i t y from measurable p r i c e and income q u a n t i t i e s . In t h i s type of analyses, incorporation 9 of non-welfare c h a r a c t e r i s t i c s i s not relevant. - 47 -Let W be a s o c i a l welfare function defined on i n d i v i d u a l equivalent incomes; we measure s o c i a l welfare as (3.29) w where N i s the number of i n d i v i d u a l s i n soc i e t y . I t should be noted that W i s not a Bergson-Samuelson s o c i a l welfare function. The order-ing generated by W depends on p° and i n general on A° as w e l l . The Bergson-Samuelson function does not allow t h i s a r b i t r a r i n e s s . Note also that the equivalent incomes i n (3.29) are p o s i t i v e r e a l numbers. As they are functions of p r i c e s , the s o c i a l ordering depends on p r i c e s , and t h i s forms the basis of 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 index. Section 4 Inequality Measurement This section describes how a summary i n e q u a l i t y measure i s computed using a d i s t r i b u t i o n of i n d i v i d u a l equivalent incomes, (3.26). The s o c i a l welfare function defined on (3.26) i s assumed to be anonymous mean of order r . . ^ An i n e q u a l i t y index i s then constructed from t h i s s o c i a l welfare function, following the Atkinson-Kolm-Sen (AKS) procedure. The AKS index i s a c t u a l l y an Atkinson index of equivalent income i n e q u a l i t y . - 48 -N Given a s o c i a l welfare function W defined on R , a t y p i c a l e e element being a vector of i n d i v i d u a l equivalent incomes ( y^, . , . , y ^ ) , e t h i c a l l y - i n d i f f e r e n t - e v e n l y - d i s t r i b u t e d equivalent income £ can be i m p l i c i t l y defined as i n (3.30) W( ? i ) = W( y®, ) where i : = (1, , 1 ), an N-vector. E x p l i c i t l y , £ i s hence defined as (3.31) £ := E( , ) £ i s that l e v e l of equivalent income which i f commanded by every i n d i v i d u a l 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 actual d i s t r i b u t i o n . Following the AKS procedure, an i n e q u a l i t y index i s then defined as (3.32) I := 1 - £/y N Q where u = ( 1/N ) £ y , k=l k i s mean equivalent income. An i n e q u a l i t y index i s a r e l a t i v e index i f i t i s mean-independent, i . e . , homogeneous of degree 0. I t i s easy to v e r i f y that the AKS index i s r e l a t i v e i f and only i f W i s homo-11 t h e t i c . - 49 -For N = 2, the present procedure i s e a s i l y depicted i n the equivalent income space. In the following diagram, the actual d i s -t r i b u t i o n i s A where i n d i v i d u a l 1 enjoys a higher l e v e l of equivalent income. E i s the e g a l i t a r i a n s i t u a t i o n where each i n d i v i d u a l enjoys the mean of the d i s t r i b u t i o n . I f W(.) i s assumed to be symmetric quasi-concave (as drawn), then E i s unambiguously e t h i c a l l y preferred to A. In f a c t , the same l e v e l of s o c i a l welfare W can be attained by a lower combined equivalent income at point E. E, being e t h i c a l l y i n d i f f e r e n t to A, i s characterized by E, as i n (3.30). I t i s easy to see that E, can be regarded as a measure of s o c i a l welfare and H(-) i n (3.31) i s o r d i n a l l y equivalent to W(-) i n (3.30). The i n e q u a l i t y measurement procedure adopted here, s i m i l a r to the AKS procedure, makes use of the discrepancy between E and E. The i n e q u a l i t y index i s defined as the s h o r t f a l l of E, r e l a t i v e to u expressed as a per-centage of u as i n (3.32). Geometrically, I can be expressed i n terms of distances, To implement t h i s procedure, a s p e c i f i c form f o r W(-) i s necessary. I t i s assumed that W i s a symmetric mean of order r t function, i . e . , I = d( 0,E ) - d(0,E ) d(0,E ) (3.33) VN } = $ ( W* ( y 1' - 50 -- 51 -where (3.34) W( y * r f 0 N e . 1/N r = 0 and $ 1(•) > 0. Since y^, ...,y^ are defined as t o t a l expenditure, they are p o s i t i v e as obtained from a survey sample. By (3.23), i t follows that e e y^, ....,y are also p o s i t i v e . The mean of order r function (3.33), N (3.34) has desirable properties on R + +, namely, i t i s a continuous, a d d i t i v e l y separable, homothetic and symmetric function. Continuity i s an obvious requirement for any s o c i a l welfare function i n the present context. Additive s e p a r a b i l i t y i s e t h i c a l l y desirable because i t implies "elimination of (the influence of) i n d i f f e r e n t i n d i v i d u a l s " , i . e . , the ranking of any two states should be independent of the u t i l i t y l e v e l s enjoyed by the i n d i v i d u a l s who are i n d i f f e r e n t between the two states (see d'Aspremont and Gevers (1977), Blackorby and Donaldson (1982)). Homotheticity ensures that the index (3.32) i s r e l a t i v e , the importance of which w i l l be explained l a t e r . F i n a l l y , symmetry implies "anonymity" which i s an e s s e n t i a l e t h i c a l requirement i n i n e q u a l i t y measurement. - 52 -H i s t o r i c a l l y , the Lorenz c r i t e r i o n has a profound influence on i n e q u a l i t y measurement. (see Sen (1973)). One would l i k e the present i n e q u a l i t y index to be consistent with i t , i . e . , i f the d i s -eA eB - eA t r i b u t i o n y i s Lorenz-superior to the d i s t r i b u t i o n y , then I ( y ) should be no greater than I( y e B ) . 1 2 A s u f f i c i e n t condition i s that the i n e q u a l i t y index i s S-convex, which requires r < 1 i n (3.34). Based on the mean of order r function (3.33) and (3.34), e t h i c a l l y -13 i n d i f f e r e n t equivalent income i s e a s i l y computed, i . e . , (3.35) £ = (( 1/N ) E ( yf ) r ) 1 A , r < 1 , r f 0 N , e ( l / N = T ( y ) , r = 0 k=l * Substituting (3.35) in t o (3.32) y i e l d s a r e l a t i v e i n e q u a l i t y index, (3.36) I := 1 - (( 1/N ) E ( y ^ / y ) r ) 1 / r r < 1 , r ^ 0 N : = l - T T ( y / y ) r = 0 k=l k N where u = ( 1/N ) E y , and N i s the number of i n d i v i d u a l s . I i s k=l k r a c t u a l l y an Atkinson index (see Atkinson (1976)) on i n d i v i d u a l equi-valent incomes. In Chapter 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 to p r i c e s , i s estimated f o r Canada. - 53 -Before c l o s i n g t h i s chapter, i t i s important to emphasize the difference between r e l a t i v e and absolute i n d i c e s , and the j u s t i f i c a -t i o n for adopting the former rather than the l a t t e r i n the present context. In contrast with a r e l a t i v e index which i s i n v a r i a n t to a common r a t i o - s c a l e transform on a d i s t r i b u t i o n , an absolute index i s inva r i a n t to a common t r a n s l a t i o n - s c a l e transform, meaning that adding the same quantity to each i n d i v i d u a l ' s equivalent income does not a f f e c t an absolute index. For example, the per-capita index (3.37) A( y S ) := y( y 6 ) - H( y e ) 14 i s an absolute index i f W i s t r a n s l a t a b l e . One could adopt the e procedure introduced to compute A ( y ). But there i s one serious drawback. One can r e c a l l that equivalent income as defined i n (3.19) i s s e n s i t i v e to p°. Since the function E i s HD 1 i n p,y,p°, i t follows that measuring p,y,p° i n a d i f f e r e n t currency constitutes a r e s c a l i n g of equivalent income by an exchange rate f a c t o r . However, since the multiple 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, (such as.I i n (3.36)), i s immune to t h i s type of 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 absolute index. - 54 -Chapter 3 Footnotes 1. The s o c i a l choice problem of Samuelson (1956) i n aggregating i n d i v i d u a l preferences to household preferences i s ignored. However, one may always assume that within each household there e x i s t s a "planner" who a l l o c a t e s consumption to equalize u t i l i t i e s . 2. Pollak and Wales (1979) are s k e p t i c a l on t h i s s p e c i f i c a t i o n . They argued that t h i s ignores d i r e c t c ontribution of a t t r i b u t e s to u t i l i t y . This aspect of u t i l i t y i s d i f f i c u l t to reveal e m p i r i c a l l y and i s ignored here. 3. This i s r e a d i l y seen by w r i t i n g the budget constraint of each household as: m (A)p fx /m (A)) + + m (A)p (x /m (A)) = y 1 L ^  ± 1 n n ^ n n y Maximization of (3.4) over fx,/m,(A), . ...,x /m (A)] subject to the v 1 1 n n 1 constraint would y i e l d the following set of f i r s t order conditions, U. fx,/m,(A), ....,x /m (A)| +Am.p. =0 1 = 1 , ....,n l v 1 1 n n 1 i x Em.(A)p. fx./m.(A)) = y Su b s t i t u t i n g the solutions fx*/m.(A), ....,x*/m (A)J into (3.4) v 1 1 n n ' gives (3.8) . - 55 -4. The common consumer surpluses, CV and EV are changes i n y evaluated at f i n a l and i n i t i a l p r i c e s . In general, CV and EV are not equal, although always take the same sign. 5. The s p e c i a l case of homothetic preferences may i l l u s t r a t e t h i s p roposition: V(p 1,y 1,A 1) = $ ( y 1 / n ( p 1 , A 1 ) ) ( • ) >0 V(p 2,y 2,A 2) = <&(y 2 /n(p 2,A 2)) (•) >0 then E t e ^ y ^ A 1 , ? 0 ^ 0 ) = y1!! (p°,A°) /II (p 1,A 1) 2 2 ,2 o o, 2„, o o. . 2 ,2, E(p ,y ,A ,p ,A ) = y n(p ,A )/n(p ,A ) therefore 1 1 ,1 o o. „, 2 2 ,2 o o. E(p ,y ,A ,p ,A ) > E(p ,y ,A ,p ,A ) 1 1 1 2 2 2 "« • Y /K(P ,AL) > y /n(p ,A ) 1 1 1 2 2 2 < *• V(p ,y ,A ) > V(p ,y ,A ) 6. I f preferences are translog (see Chapter 4), then S i s not a function of u t i l i t y . - 56 -7 . This i s an a p p l i c a t i o n of Debreu ( 1 9 5 9 ) theorem of u t i l i t y function representation. See Debreu ( 1 9 5 9 ) . Sec. 4 . 6 . 8 . Sen ( 1 9 7 0 ) proves a L i b e r t a r i a n theorem saying that un-cond i t i o n a l L i b e r t a r i a n rules are in c o n s i s t e n t with weak Pareto and unlimited domain i n generating a s o c i a l ordering. I f a l l are adopted as axioms, a preference cycle r e s u l t s . See also Roberts ( 1 9 8 0 ) . 9 . The reader may r e c a l l that i n Chapter 2 , i t has been argued that Jorgenson and Slesnick ( 1 9 8 4 ) have made use of a non-welfarist framework without making i t c l e a r why such a framework i s necessary and j u s t i f i a b l e . 1 0 . A s o c i a l welfare function W i s anonymous i f and only i f , f o r any two d i s t r i b u t i o n s of u t i l i t i e s u = (u. , ,u ) and u' = (u ' , /U ') I N I N where one i s a permutation of the other, W(u) = W(u'). — — r* B \ 1 1 . I f W i s homothetic, then W(y) = II(W(y ) J , where II i s a mono-* * to n i c transform, and W i s HD 1 . I t then follows that W(Ei) = - 57 -W(y^, ,y ) defines £, so that E(y ) i s also HD 1. Since u i s HD 1, the index i s r e l a t i v e . The converse i s now e a s i l y v e r i f i e d . See also Sen (1973). 12. One should consult Berge (1962), Dasgupta, Sen and S t a r e t t (1973), Sen (1973) and Blackorby, Donaldson and Auersperg (1981). N B r i e f l y , suppose there are two d i s t r i b u t i o n s X , X E R with the same a D mean u, then X i s s a i d to be Lorenz superior to X, i f the Lorenz a b curve f o r X l i e s completely ins i d e that of X, . In t h i s case X = a b a BX^ where B i s a b i s t o c h a s t i c matrix and i s not a permutation matrix, and X can be obtained from X, by a f i n i t e number of t r a n s f e r s . Then a b N S : R *• R i s an S-concave function i f S(Xf l) >_ SfX^) so that i f S i s a s o c i a l welfare function, then the AKS index I w i l l be S-convex, s where I (. X ). < I (. X. ) s a — s b since l - s( x ) / u < l - s( x D ) / y a a 13. One can obtain (3.35) by an a l t e r n a t i v e route. Since W i n * (3.33) i s homothetic (as W i s HD 1 and $' (•) > 0), i t follows that * E i s HD 1 and has to be i d e n t i c a l to W. 14. For a f u l l d iscussion, see Blackorby and Donaldson (1980). - 58 -CHAPTER 4 SPECIFICATION In order to apply t h i s new approach, s p e c i f i c a t i o n i s necessary. This chapter describes translog household preferences and the parameter r e s t r i c t i o n s necessary to make estimation f e a s i b l e . Because a large number of parameters are involved, i n addi t i o n to cross-section data, time-series aggregate data have to be used. An aggregation structure w i l l be described which allows aggregate data to be u t i l i z e d . ing Jprgenson, Lau and Stoker (1982). A translog i n d i r e c t u t i l i t y function i s a second-order approximation of any i n d i r e c t u t i l i t y function at a single point. Incorporating commodity-specific equi-valence scales, i t has the form, f o r n goods, Preferences are assumed to be non-homothetic translog, follow-n (4.1) Jin V ( p, y, A ) a + £a.&n(m.p./y) + o . . 1 1 1 i = l or i n matrix form, (4.2) In V (p, y, A ) a + ( in mp/y ) a + T h ( An mp/y ) B ( £n mp/y ) T - 59 -where a i s a s c a l a r , a an n-vector, and B an n x n symmetric o p PP matrix. By Roy's Identity, expenditure shares are 3InV /3 £n ( p/y ) (4.3) e = i T ( 3 £n V/3 In ( p/y )) . T where i = (1, , 1), i s an n-yector. Applying t h i s to the trans-log, the expenditure shares take the form, a + B ( £n m) + B ( £n p) - B i ( £n y ) (4 4) E = E EE EE_ EE 1 ' T T T T i a + i B ( £n m ) + i B ( Jin p ) - i B i ( £ n y ) p pp PP PP Since e i s homogeneous of degree 0 i n the parameters a , B , i t i s P PP usual, as i n Jorgenson, Lau and Stoker (1982), to normalize as follows, (4.5) i T a = -1 P To obtain the t r a n s l o g cost function, the i n d i r e c t u t i l i t y function may be inverted to solve for y. (4.2) can be rewritten as T T (4.6) In V ( p, y, A ) = a + ( Jin mp ) a + Jin y + h. ( Jin mp ) B ( In mp ) o p pp - ( £ n m p ) T B ( i i l n Y ) + ^ ( i £ n y ) T B ( i £ n y ) PP PP which i s a quadratic equation i n £n y. In order to obtain an e x p l i c i t form for the cost function, i t i s assumed that. - 60 -(4.7) i T B i = 0 PP so that the second degree term i n (4.6) vanishes.''" The cost function w i l l then have the form V T T ~\ In u - [a + ( In mp ) a + h ( £n mp ) B ( £n mp )J (4.8) taC(u,p,A) = - - : • 22 1 - ( in mp ) ( B i ) PP where u i s a u t i l i t y number. The equivalence scales have so f a r been l e f t unspecified. Following Jorgenson, Lau and Stoker (1982) and Jorgenson and Slesnick (1982 a, b, c)(1983 a, b), the a t t r i b u t e vector A i s assumed to be a vector of dummy va r i a b l e s , i . e . , the elements i n A are e i t h e r 0 or 1. Four household a t t r i b u t e s are used to describe each household, namely, the area of residence, the sex of household head, the family s i z e and the age of the household head, so that A i s an eleven-vector, to be assigned to each household according to Table 1. These four a t t r i b u t e s are thought to be s i g n i f i c a n t deter-minants of household consumption pattern. Indeed, there are relevant factors which have been ignored here, for example, household composi-t i o n , education l e v e l , race and climate. The binding co n s t r a i n t i s data a v a i l a b i l i t y . As w i l l be seen i n Chapter 6, s t a t i s t i c s of expenditure d i s t r i b u t i o n over 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 Size of Area of Residence * non-metropolitan metropolitan Sex of Household Head female male Family Size two persons otherwise three persons otherwise four persons otherwise f i v e or more persons otherwise Age of Household Head A, A, 10 above 34 or above 44 or above 54 or above 64 or 24 but below 34 but below 44 but below 54 but below otherwise otherwise otherwise otherwise 11 above 64 otherwise * C i t i e s with population above 30,000 are c l a s s i f i e d as metropolitan. - 62 -successful estimation of t h i s model, and p u b l i c l y a v a i l a b l e s t a t i s t i c s Canada data do not allow incorporation of more a t t r i b u t e s . In par-t i c u l a r , household composition ( i . e . , the number of adults, as opposed to children i n the household) should be an important a t t r i b u t e . Here, the impact of household composition i s r e f l e c t e d i n part by the e f f e c t of sex of household head. By convention, a household with a female head means e i t h e r an unattached female or a single female-parent household. Given any two multi-member households with a l l other a t t r i b u t e s being the same, the household with a female head implies i n most cases sub-s t i t u t i n g a c h i l d f o r an adult. Hence, i n t u i t i v e l y speaking, house-holds with female heads should need more cl o t h i n g and l e s s food, and t h i s difference should be r e f l e c t e d i n estimated equivalence scales. This has indeed been confirmed by the estimation r e s u l t s , (as discussed i n Chapter 6 below) which p a r t i a l l y j u s t i f i e s the whole approach. a l l elements i n A equal 0, i . e . , an unattached male, of age 24 or below, l i v i n g i n a metropolitan area. Following Jorgenson and Slesnick (1982 a, b, c)(1983 a, b), the equivalence scales are s p e c i f i e d i n a way that enables simple l i n e a r estimation, that i s , A reference household i s defined as the household which has (4.9) where B P A i s an n x 11 matrix, which s a t i s f i e s (4.10) PA = 0 - 63 -This l a s t assumption i s necessary i n making aggregation across i n -di v i d u a l expenditure share equations simple, because e i n (4.4) w i l l now be l i n e a r i n a l l household-specific v a r i a b l e s . Furthermore, i t should be pointed out that, by (4.9), the equivalence scale factors of the reference household are a l l equal to unity. Incorporating the four assumptions (4.5) (4.7) (4.9) and (4.10), the expenditure shares can be expressed as a + B A + B ( t o p ) -B i ( to y ) (4.11, e = J2—EA. EEL EE - 1 + i B ( t o p ) PP which are sui t a b l e f o r estimation using cross-section data. Note that B represents the incremental e f f e c t s on e as a t t r i b u t e s change. Given y, the reference household (whose A equals 0) establishes a ce n t r a l l e v e l for e. This e f f e c t on expenditure shares through B pA i s then t r a n s l a t e d to a p r i c e e f f e c t through B i n (4.9) which pp depends on demand e l a s t i c i t i e s . The aggregate expenditure share equation i s obtained by summing i n d i v i d u a l share equations across the e n t i r e population. Let H be the t o t a l number of households and A , y be the a t t r i b u t e vector and spending of household h. The aggregate expenditure shares, E, are (4.12) E = E ( ( y h e h )/Ey h) h=l - 64 -a + B E ( y A ) / Y + B ( I n p ) - ( B i ) Z f y ( i n y )) / Y P PA pp PP w . - I + i T B p p ( inp ) ^ h 2 where Y = Z y , the t o t a l spending. These aggregate share equations h=l are suitable for estimation using time-series aggregate data. Notice also that these equations have the same form and involve the same c o e f f i c i e n t s , a , B , B , as i n the i n d i v i d u a l share equations, so p pA pp that data from both sources can be combined, and i d e n t i f i c a t i o n of a l l 3 parameters i s obtained. Using the d e f i n i t i o n (3.19), translog equivalent income can be expressed as, (4.13) y 6 = exp r O T T oT o T \ (co-w ) a B tt-o) B w ) + (1 - cu B i)(£ny) P PP PP PP 1-0) B l PP where, (4.14) a) = k m ( A ) p (4.15) CJ 0 = Jlnm ( A° ) p° To estimate the equivalent income of each i n d i v i d u a l , estimates f o r a , B and B are required. I t w i l l be shown i n Chapter 6 that, P PP pA - 65 -by using both cross-section and time-series aggregate data, these estimates are obtainable. I t i s evident that translog equivalent income (4.13) i s o o se n s i t i v e to p and the choice of A . For example, d e f i n i n g an un-attached female, of age 24 or below, l i v i n g i n a metropolitan area as the reference household w i l l give r i s e to d i f f e r e n t equivalent income values. However, i f the i n e q u a l i t y index i s r e l a t i v e , i n e q u a l i t y i s not s e n s i t i v e to A°. In order to v e r i f y t h i s claim, (4.13) i s rewritten as (4.16) y = exp T T T LD a + 3 3 t o B to + (1 - u) B i ) (&n y) P EE £P i OT 1 - 0 ) B l PP exp oT , oT o a) a - *sa> B u'-P PP 1 - to B ^ i PP but by (4.15), and subsequently, (4.9) and (4.10), 0) B l = (_£nm (A )p J B l PP PP = (tam(A°)) TB i + ( £ n p ° ) T B i 1 pp pp O T (Inp ) B i PP so that (4.16) can be written as - 66 -(4.17) exp T T T to + Sju) Bp p oi + (l - o ) B p p i ) (Iny) 1 - U n p ° ) T B i PP exp oT . oT o - o) a - ^oi B o) P PP O T 1- ( t a p ) B i v PP Note that only the f i r s t exponential 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 involve A°. The second exponential term, which involves o A , i s a common scalar multiple on a l l i n d i v i d u a l equivalent incomes. Therefore, the choice of A° cannot a f f e c t the r e l a t i v e i n e q u a l i t y index. Adopting the translog s p e c i f i c a t i o n , i t would be i n t e r e s t i n g to compute e m p i r i c a l l y the market equivalence scales (3.25). These scales can be compared with the r a t i o s of poverty l i n e s published by S t a t i s t i c s Canada. By d e f i n i t i o n , the poverty income f o r household h i s , (4.18) P,( p,A h ) = C( u,p,Ah ) where u i s the l e v e l of subsistence u t i l i t y . Using A° as a reference household, the poverty-line r a t i o f o r household h i s (4.19) P(p,A h)/P(p,A°) = C(u,p,A h)/C(u,p,A°) - 67 -which i s d i f f e r e n t from the market equivalence scales only i n that (4.19) i s evaluated at u. However, i t can be shown that the translog market equivalence scales are a c t u a l l y independent of u so that they are i d e n t i c a l with translog poverty-line r a t i o s . To see t h i s , notice that the denominator of (4.8) can be written as because of assumptions (4.9) and (4.10), so that i t i s independent of A, and ( Jin u ) i n the numerator w i l l vanish when the d i f f e r e n c e Consequently, the translog market equivalent scales and t r a n s l o g poverty-line r a t i o s can be expressed as (4.20) = 1 - ( Jin p ) B i PP T ( Jin y - Jin y° ) i s taken. f T - (Jin m ) a - ^(Jlnm) B ,T (£nm)-(Jlnp) B (Jin m) PP T (4.21) exp PP 1 - (Jlnp ) (B i ) PP T These scales are estimated from demand data and compared with published numbers i n Chapter 7. They provide valuable i n s i g h t s i n a comparative study of d i f f e r e n t i n e q u a l i t y indexes. - 68 -Chapter 4 Footnotes 1. I n c i d e n t a l l y , t h i s assumption i s necessary for a l i n e a r expenditure share equation. See (4.4). See also Diewert (1974) 2. Notice the advantage of l i n e a r i t y i n d e r i v i n g (4.12) from (4.11), as a r e s u l t of assumptions (4.7) and (4.10). In theory, any func t i o n a l form can be aggregated. However, l i n e a r i t y ensures a "complete" aggregation structure regardless of d i s t r i b u t i o n s of i n d i v i d u a l - s p e c i f i c v a r i a b l e s . I f these were known, l i n e a r i t y would no longer be e s s e n t i a l . See Stoker (1983). 3. In Gorman's (1953) aggregation framework, translog preferences do not allow the existence of a r a t i o n a l aggregate consumer. E represents aggregate shares of a "consumer" whose preferences change with income d i s t r i b u t i o n . However, f o r empirical purposes, Gorman preferences are too r e s t r i c t i v e i n f o r c i n g p a r a l l e l and l i n e a r Engel curves. - 69 -CHAPTER 5 ESTIMATION METHOD Section 1 Introduction This section describes the stochastic structure of the estimation model and explains how the parameters involved i n the equivalent income function (4.13) can be estimated. Six composite goods w i l l be defined. Since 11 dummy var i a b l e s are u t i l i z e d to describe demographic a t t r i b u t e s , there are t o t a l l y 108 parameters to be estimated: 6 parameters i n a , 36 parameters i n B and 66 para-P PP meters i n B . Using these estimates, the eauivalence scales can be pA estimated which indi c a t e the actual r e l a t i o n s h i p between u t i l i t y , consumption and the four a t t r i b u t e s . There are two ways to estimate t h i s model. The f i r s t approach, named the pooled estimator, has been used i n Jorgenson, Lau and Stoker (1982). The t e c h n i c a l d e t a i l s are not described here. B a s i c a l l y , they formulate a constrained minimization problem with an objective function being made up of the sum of squared r e s i d u a l s i n the cross-section model ( i . e . , composed of the i n d i v i d u a l expenditure share equations) and i n the time-series model ( i . e . , composed of the aggre-gate share equations). The so l u t i o n 6* to t h i s minimization problem i s the set of parameter estimates provided that they also s a t i s f y the symmetry and monotonicity constraints."'" 6* i s found by i t e r a t i o n as follows. The combined cross-section and time-series model and the - 70 -constraints are f i r s t - o r d e r approximated i n i t i a l l y around an a r b i t r a r y point & . Then Liew's (1976) i n e q u a l i t y constrained three stage l e a s t squares method i s applied to generate 6^. 6^  i s substituted into the objective function of r e s i d u a l sum of squares to obtain an objective value. This process i s then repeated u n t i l the objective value con-verges. Although the estimator i s believed to be consistent, small sample properties are unknown. Furthermore, the procedure i s c o s t l y and because the start-up value 6 q i s a r b i t r a r y , an accuracy problem might a r i s e . A d i f f e r e n t approach i s adopted here which estimates the cross-section model and the time-series model sequ e n t i a l l y . Estimates obtained i n the c r o s s - s e c t i o n are substituted into the time-series equations as i f they were true values. This procedure has the advantage that i t involves only l i n e a r estimation and no i t e r a t i o n s are required. On the other hand, however, r e l a t i v e to the f i r s t approach, i t i s l e s s e f f i c i e n t . For the pooled estimation, since information from both sources i s pooled together and estimates generated i n a s i n g l e pass, even the parameters that are estimable using only cross-section data are estimated using a d d i t i o n a l time-serie s information. This c o n s t i t u t e s some e f f i c i e n c y gain. However, because the s i z e of the sample i s large i n the cross-section and small i n the time-series, the e f f i c i e n c y gain i s l i k e l y to be small. The estimates w i l l be dominated by the cross-section data. The r e s u l t s i n Jorgenson, Lau and Stoker (1982) substantiate t h i s claim, - 71 -namely, except for the p r i c e c o e f f i c i e n t s which do not enter the cross-section model, the estimates for B and a obtained from cross-pA p section alone are very close to the pooled estimates. The procedure adopted here i s a sequential one. The f i r s t step involves estimating those parameters that are i d e n t i f i e d i n the i n d i v i d u a l expenditure share equations, i . e . , a and B , using cross-p pA section data only. Because of the lack of p r i c e v a r i a t i o n i n the cross-section data, the p r i c e c o e f f i c i e n t s , B are not i d e n t i f i e d . PP The second step involves estimating B using the aggregate expendi-ture share equations and time-series data only, proceeding as i f the estimates obtained i n the f i r s t step for a and B were true values. p pA In other words, the aggregate equations are estimated subject to a P and B being equal to t h e i r cross-section estimated values, as well 2 as the usual symmetry conditions on B PP Section 2 Cross-section Estimation The i n d i v i d u a l share equation (4.11) i s non-linear i n (£np). In a family expenditure survey, there i s no information on the p r i c e each household faces. I t w i l l be assumed that p r i c e s are uniform across the households. The survey year i s taken as the reference year for the p r i c e s e r i e s so that the p r i c e vector i s (1, , 1), 3 an n-vector. The purpose i s to avoid (inp) i n (4.11). The cross-section share equation i s therefore, - 72 -Vi 1*1 h (5.1) e = - a - B A + B i ( £ n y ) h = 1, . . . , H P PA pp where y n , i s taken as the t o t a l expenditure of household h i n the survey. I t then follows that the i t h regression equation i s , (5.2) e h =-a. - Z B., A,h + 6. ( «.nyh ) + e h i = 1, . . . , n I I , , i k x I I k=l h = 1, . . ., H Y^ where e^ i s the expenditure share of the i t h good for household h, Y^ i s the i t h component i n a , 6 ^ i s the i k t h element i n B p A » i s the kth element i n A , 9. i s the i t h component i n B i , or the sum of l pp h the i t h row (or i t h column) of B , e. i s a disturbance term of the pp i i t h expenditure share equation f o r household h. As i s true for any consumption a l l o c a t i o n model, one equation i s redundant i n (5.1). Summing up the expenditure shares i n the l e f t -hand side gives 1 i d e n t i c a l l y , and so must the right-hand side, by the r e s t r i c t i o n s on a , B , B . This implies that, i n (5.2) the d i s -P PA PP turbances are l i n e a r l y dependent because n . (5.3) Z E = 0 h = 1, ..., H i = l 1 so that the covariance matrix (5.4) var U ) = var (e, , . . . . , e ) = 1 h = l , ...,H 1 n - 73 -must be singular. I t i s assumed that the disturbance term s a t i s f i e s the following assumptions, for a l l h = 1, H, (5.5) E(e h) = 0 n (5.6) var(e h) = $ . (5.7) v a r ( e 1 , e H ) T = $&I„ H where $ i s of rank n-1 and disturbances are independent across households. Actual estimation involves n-1 equations. Since | i s of rank n-1, and each regression equation involves the same explanatory v a r i a b l e s , the J o i n t Generalized Least Squares estimator (Zellner, Theil) i s i d e n t i c a l to equation-by-equation OLS estimator which i s 4 best l i n e a r unbiased. By r e s t r i c t i o n s , (4.5), (4.10) and (4.7), the estimates for the omitted equation, say the nth one, are obtained as follows, n-1 (5.8) a = 1 - E a. i = l 1 • n-1 (5.9) 6 = - E § k = 1, K nk . , l k i = l n-1^ (5.10) 9 = - E 9. i = l 1 - 74 -5 where a superscript " indicate an estimate. Because the cross-section sample i s usually very large, the estimates obtained for a , P B and B i should be very accurate. However, B i s not estimable pA pp 1 pp because of the lack of p r i c e v a r i a t i o n i n the sample. Section 3 Time-series Estimation The sequential approach adopted here requires time-series data only i n the second step. The aggregate share equations are estimated subject to the estimated values f o r a , B , and B i obtained i n the P PA pp cross-section and the symmetry conditions on B . The main concern PP here i s to estimate the i n d i v i d u a l elements i n B PP The time-series model i s derived from summing up (4.11) across a l l the households i n s o c i e t y . Let subscript t denote time period, so that y i s the t o t a l expenditure of household h i n period t . In addition, the following short-hand notations are adopted, for t = l , • . . , T, T (5.11) D (p ) = -1 + i B (!np ), a s c a l a r , (5.12) S = Zy^Ah/Y , a K-vector, where K = 11 from Table 1, yAt ^ t t (5.13) S = Eyk( to v? ) /Y . a s c a l a r yyt h t t t - 75 -(5.14) = ^ h v t £ t ^ Y t ' a n n - v e c t o r H h where Y = X y h=l The time-series regression equations can then be expressed as, i n matrix form. (5.15) E f c = D(p.) 1 ( a +B S + B ( i n p j - (B i ) S J + E t t K p pA yAt pp t pp yyt^ t t = 1, T where e i s the disturbance term. The i t h equation w i l l be (5.16) E. = D(p. )  1[a. + E B . , S , . , + l b . . (£np.J i t ^ t i k = 1 l k yAkt xj K 3 i S y y J + £ i t ' 1 = 1 0 t = 1 T where S , i s the kth element i n S , (£np . ) i s the j t h element i n yAkt yAt j t (Inp ) and e. i s the i t h element i n £ . I t i s assumed that the t i t t covariance structure of e i s stationary through time, i . e . , (5.17) E ( E ^ ) = E ( E H ) = 0 s,t = 1, T t s (5.18) v a r ( E ) = var(e ) = t s,t = 1, T t s - 76 -where \ i s of rank n-1. I t then follows that (5.19) E ( e t ) = E ( E h y ^ e ^ / Y T ) = 0 t = 1, T and, (5.20) var'(e ) = (z (y£) 2 var (e") ) / Y ' t = 1, ..., T since household disturbances are not c o r r e l a t e d . I t i s further assumed that e i s not s e r i a l l y c o r r e l a t e d , i . e . , (5.21) var ( e , , e ) = Since $ i s of rank n-1, ft i s also of rank n-1. One equation i s redundant and should be omitted because of cross-equation symmetry constraints, despite the f a c t that a l l equations have the same explanatory v a r i a b l e s . In the actual estimation, the following parameter r e s t r i c t i o n s are imposed, (5.22) a = a P P (5.23) B = B K pA pA (5.24) (B i ) = PP and by symmetry, T ~T (5.25) i B = 6 PP where a superscript " denotes an estimated value from the cross-section. These r e s t r i c t i o n s can be substituted i n t o (5.15) to obtain (5.26) E t = B p p ( k p t ) + D ( P t ) e t where E = D(p )E - (o +B S A^ - 8 S ) t t t v p pA yAt Y y t D(p ) = -1 + G T(Jlnp t) As defined e a r l i e r , b.. i s the i j t h element i n B and 8. i s the 1 3 PP i estimated sum of the i t h column (or i t h row) of B . The i t h equation PP - 78 -i n (5.26) i s (5.27) E. = E b. . ( i n p . J + D(p.) e.u i t . in i t t i t 3=1 By r e s t r i c t i o n (5.24), however, n-1 (5.28) b. = 0. -.• E b. . m i j = 1 ID Therefore, b. , for a l l i , should be substituted out of the system, i n In doing so, the i t h equation becomes, n-1 (5.29) Q . = E b. . ((Anp.. ) - U n p . )) + D(p. ) E . . i t . , I T ] t nt ' t i t 1=1 where C" = E_ - 6. ( k p ) i t i t i nt I t should be noted that the disturbance term i n (5.29), D(p )e. i s not c l a s s i c a l i n structure. I t varies with time. Given t i t (5.20), i t can e a s i l y be v e r i f i e d that h e t e r o s c e d a s t i c i t y can be corrected f o r by the factor (5.30) p t = D ( p t ) " 1 ( Y t 2 / E {y\)2)h h=l so that (5.31) E ( p t ( D ( p t ) e t ) ) = 0 t = 1, T - 79 -(5.32) -var(p (D(p )e )) = $ t = 1, T (5.33) v a r ( P L ( D ( P L ) E I ) , , p^ , (D (p^ ,) e t^)) = | Q I ^ A f t e r a l l these manipulations, the time-series estimation problem i s to estimate n-1 l i n e a r equations, each containing the same explanatory v a r i a b l e s . The i t h equation of the system i s , f o r period t , n-1 (5.34) p tQ ± t = p t Z b (Unp ) - U n p ^ ) ) + P t D ( P t ) e . t j=l K where Q • ^  = D (p ) E - (a. + ES..S A l -9.S t l - e.(2,np ) x i t t i t *• I lk yAkt l yyt-1 I nt To estimate (5.34), J o i n t Generalized Least Squares can be applied subject to the symmetry constraints on B Dp# i . e . . (5.35) b.. = b.. i , j = 1, ..., n-1 1 3 D i The number of parameter estimates obtained d i r e c t l y i s (n-1) X (n-1). From these estimates, the nth c o e f f i c i e n t i n each of the n-1 equations and a l l the c o e f f i c i e n t s i n the nth equation can be derived as follows. By (5.28), - 80 -n-1 (5.36) b. = 6. - E h. . i n 1 j=l 1 3 By the symmetry constraints, one obtains (5.37) b . = b. i = 1, ..., n-1 n i i n and f i n a l l y . n-1 (5.38) b = 6 - E b . nn n . , ni 3=1 To summarize t h i s chapter, by using a two-step approach which involves estimating the cross-section and time-series models sequen-t i a l l y , a l l the parameters i n the equivalent income function can be estimated. This means that, given A*1, y*1 and (p, p°) , equivalent income can be imputed to each i n d i v i d u a l i n household h. Of indepen-dent i n t e r e s t i s the commodity-specific equivalence scales m. These scales can be expressed as, from (4.9), (5.39) m = expfB 1 B A) PP pA ; A complete set of equivalence scales for d i f f e r e n t configurations of A can then be estimated using estimated values for B and B pp pA - 81 -Chapter 5 Footnotes 1. Since a and B , enter both cross-section and time-series p pA models, the two terms i n the objective function should be minimized together. 2. The monotonicity conditions i n Jorgenson, Lau and Stoker (1982) being implied by i n t e g r a b i l i t y of demand functions are not imposed. These conditions ensure negative semi-definiteness of the Jacobian matrix of the cost function only i f expenditure shares are r e s t r i c t e d to be non-negative. The focus at present i s on estimating the para-meters i n the equivalent income function rather than recovering unknown preferences from hypothesized demand functions. However, the re s t of the i n t e g r a b i l i t y conditions: summability, homogeneity and symmetry are imposed. 3. Rigorously speaking, t h i s involves r e d e f i n i n g the p h y s i c a l units i n measuring q u a n t i t i e s of commodities. 4. See T h e i l (1970) Chapter 7. 5. Since there are no cross-equation constraints, an equivalent procedure w i l l be to estimate a l l n equations independently. - 82 -CHAPTER 6 IMPLEMENTATION This chapter describes how the estimation model i n Chapter 5 can be implemented i n the Canadian context, using p u b l i c l y a v a i l a b l e data. The cross-section model requires expenditure survey micro data, whereas the time-series model requires aggregate time-series s t a t i s t i c s that are not r e a d i l y a v a i l a b l e from the publications of S t a t i s t i c s Canada. These s t a t i s t i c s have to be s p e c i a l l y computed. The sequen-t i a l estimation approach has been c a r r i e d out and the estimation r e s u l t s can be found i n Appendix A and Appendix D. Of sp e c i a l i n t e r e s t are the estimated equivalence scales, while being a governing factor i n making interpersonal comparison p o s s i b l e , play a c r u c i a l role i n the determination of equivalent incomes. These scales can be found i n Appendix E and are i n t u i t i v e l y very appealing. Section 1 Cross-section Estimation The sequential approach adopted here c a l l s f o r , i n the f i r s t step, estimation of the micro i n d i v i d u a l household expenditure share equation, (6.1) e h = - a. - E ^ J . , A h + 8. ( U n y h ) + £ h i = l , n x x k=l xk K x x - 83 -where e^ i s the expenditure share of the i t h good for household h, a. i s the i t h element i n a , 3 . , i s the i k t h element i n B . A, i p i k pA \ i s the kth component i n A (as already defined i n Table 1), 0^ i s the 1*1 i t h component i n B i , and e. i s the disturbance term. (6.1) i s a PP i regression of the expenditure share of good i on an int e r c e p t term, a set of 11 dummy variables and the logarithm of t o t a l expenditure. In so far as expenditure shares d i f f e r across households of d i f f e r e n t a t t r i b u t e s , these di f f e r e n c e s wi'll be accounted f o r by the c o e f f i -cients g. , ' s . i k The data set for t h i s regression i s derived from the Family Expenditure Survey 1978. This survey provides micro data f o r 1978 on the expenditure patterns and household c h a r a c t e r i s t i c s of a representative sample of approximately 10,000 households. The information a v a i l a b l e allows a c l a s s i f i c a t i o n of 6 composite goods, defined as follows, (6.2) "Food" = food prepared at home and outside + 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" = recreation and entertainment + reading materials + education + g i f t s and contributions - 84 -(6.5) "Personal and = a l l personal maintenance needs + medical care" medical treatment (6.6) "Shelter" = rent + payment for housing mortgages + water + f u e l and e l e c t r i c i t y + household operations + household furnishings and equipment (6.7) "Transportation" = automobile and truck services + purchased transportation The survey provides information on the amount of money each household spends i n each of these 6 consumption categories. The sum of these expenditures for household h i s taken as y i n the regression (6.1). Not a l l the records contained i n the survey enter the data set f o r regression. Seventy households have been excluded because they are c l a s s i f i e d i n the survey as roomers and they did not pay any rent i n 1978. These households might e x h i b i t spending behaviour that deviates from the norm and should be discarded. Consequently, a f t e r t h i s screening, the sample siz e f o r the cross-section regression i s 9285. Since expenditure shares always sum to 1, the transportation equation, or indeed any one of the 6 equations, can be omitted. Parameters i n the transportation equation can be derived from the - 85 -estimates of the other 5 equations using assumptions (4.5), (4.7) and (4.10). Since the Z e l l n e r and T h e i l Generalized Least Squares method reduces to the equation-by-equation ordinary l e a s t squares method, OLS can be applied on each of the remaining 5 equations independently using the data set of 9285 households. The r e s u l t s can be found i n Appendix A. These c o e f f i c i e n t s should be very accurate because the sample s i z e i s so large. They are also i n t u i -t i v e l y appealing. Out of the t o t a l 78 c o e f f i c i e n t s , 66 of them are s i g n i f i c a n t at the 95% l e v e l . The c o e f f i c i e n t of (log y) i s very s i g n i f i c a n t i n a l l equations, implying that homotheticity i s an unreasonable r e s t r i c t i o n . Of a l l the a t t r i b u t e s , family s i z e seems to be most important i n a f f e c t i n g expenditure shares. I t also d i s -plays reasonable trends. For example, increasing family s i z e leads to increasing food share and decreasing transportation share. This i s consistent with the notion that food i s a necessity and trans-port a t i o n i s a luxury. Thus, using cross-section data enables i d e n t i f i c a t i o n of a , p T B i and B , i . e . , i n Appendix A, the f i r s t row i s (B i ) , the pp pA' > , p p T second to the second l a s t rows form ( - B ) and the l a s t row i s pA - T -a P - 86 -Section 2 Time-series Estimation In order to estimate equivalent income i n (4.13), estimates for B are required i n ad d i t i o n to the estimates i n Appendix A. PP However, these estimates can be substituted into the aggregate share equations as i f they were true values to generate estimates for B PP In t h i s second step of the sequential approach, only time-series aggregate data are used. The i t h regression equation i n the time-seri e s model i s , a f t e r s u b s t i t u t i o n of a , B and 6 (being a vector p pA of estimates for B i ) , PP n-1 (6.8) p tQ. t = P t . ^ i j ( ( ^ P j t ) - ( £ n p n t ) ) + p t D ( p t ) e . t 1, ...... T where, (6.9) Q. . = D(p. ) E . - ( a . + E 6., S - 9. S • J - 3. ( £np . ) x x t t xt v x xk yAkt x yytJ x nt K.—-L (6.10) D ( p t ) = -1 + 6 T ( £ n p t ) The basic problem then i s to compute p , Q. , ( tap , , Jtnp ) and D(p^ ) i n (6.8) using published data from S t a t i s t i c s Canada. The usable time-series runs from 1971 to 1981, i . e . , 11 observations. The procedures that generate the aggregate s t a t i s t i c s for successful estimation of (6.8) are described below. - 87 -1. Aggregate expenditure shares, E The aggregate share data are derived from Personal Expenditure on Consumer Goods and Services i n Current D o l l a r s , i n National Income and Expenditure Accounts Catalogue No. 13-201. Some minor adjust-ments are necessary to regroup those expenditure items so that the c l a s s i f i c a t i o n i n the time-series i s consistent with that i n the cross-section. The groupings are shown as follows, the numbers i n parentheses being the account numbers. (6.11) Food (1) Food, beverages and tobacco + h x (45) Expenditures on restaurants and hotels (6.12) Clothing = (5) Clothing and footwear (6.13) Recreation (36) Recreation, entertainment, educa-t i o n and c u l t u r a l services + h x (45) Expenditures on restaurants and hotels + (48) Net expenditure abroad (6.14) Personal and Medical Care (24) Medical care and health services + (43) T o i l e t a r t i c l e s , cosmetics + (44) Personal care - 88 -(6.15) Shelter = ((9) Gross rent, f u e l and power - (10) Gross imputed rent No. of households without mortgage i n 1981 ^ + No. of households with owned accommodation i n 1981 ' (16) Furniture, furnishings, etc. + (42) Jewellery, watches and rep a i r s + (46) F i n a n c i a l , l e g a l and other services."*" (6.16) Transportation = (29) Transportation and Communication As i n the cross-section estimation, t o t a l aggregate expendi-ture, Y ,• i s taken as the sum of the expenditures 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 , S yAt This s t a t i s t i c i s a summary s t a t i s t i c r e f l e c t i n g the d i s t r i -bution of aggregate expenditure over the s p e c i f i e d a t t r i b u t e groups. Formally, (6.17) S y A t = ^ A h / Y t A i s an eleven-vector, and so i s s ^ A t - .For example, i t i s easy to see that the f i r s t component i s simply the t o t a l expenditure of a l l non-metropolitan households (which have A = 1) divided by t o t a l population expenditure. In other words, each component i n S^ A t i s the proportion of expenditure i n year t that i s accounted for by a - 89 -group of households having a common a t t r i b u t e . Unfortunately, expenditure information of t h i s s o r t , as opposed to i t s a f t e r - t a x income counterpart, i s not ava i l a b l e on a time-series b a s i s . An acceptable approximation, however, i s to use ava 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 data and derive expenditure 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 expenditure i n the cross-section sample, allowing the r e l a t i o n s h i p to be a t t r i b u t e -s p e c i f i c . More s p e c i f i c a l l y , the following four cross-section con-sumption functions corresponding to the four a t t r i b u t e s are estimated using the cross-section data set. Area of Residence (6.18) y h = Sex of Household Head (6.19) y h = a 2 + b 2 z + y 2 A 2 + v 2 Household Size (6.20) y h = a 3 + b 3 z + Y 3 1 A 3 + Y 3 2 A 4 + Y 3 3 A 5 + y 3 4 A 6 + v 3 Age of Household Head - 90 -( 6 - 2 1 ) Y h = a 4 + b 4 z h + Y 4 1 A 7 + Y 4 2 A 8 + Y 4 3 A 9 + Y 4 4 A 1 0 + Y45 A l l + V4 y and z are the t o t a l expenditure and a f t e r - t a x income of household h r e s p e c t i v e l y . A^ to A ^ are the dummy variables defined i n Table 1. h h v^, , v^ are the disturbance terms. I t i s assumed that i n each of the 4 equations there i s no contemporaneous covarlances i n the disturbance terms so that OLS i s best l i n e a r unbiased. The estimated c o e f f i c i e n t s f o r these 4 regressions can be found i n Appendix B. The estimated c o e f f i c i e n t s f o r (6.18) to (6.21) are u t i l i z e d to 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 to a corresponding s ^ A t expenditure d i s t r i b u t i o n s e r i e s . The mapping procedure i s as follows. Take the f i r s t component of S as an yAt example. I t i s required to estimate the proportion of t o t a l population expenditure that i s accounted f or by non-metropolitan households. Suppose the estimated consumption function f or non-metropolitan households according to (6.18) i s (6.22) y k = (a + y ) + b z k where k i s a non-metropolitan household and a superscript " in d i c a t e s an estimated value. (6.22) represents the r e l a t i o n s h i p between - 91 -after-tax income and expenditure for a non-metropolitan household. Summing (6.22) across a l l non-metropolitan households gives the estimated t o t a l expenditure of non-metropolitan households K , K (6.23) E y = K(a + y ) + b E z k=l 1 1 ^=1 where K i s the number of non-metropolitan households. This quantity can be found i f K and Ez are a v a i l a b l e . These can be found i n "Income After-tax, D i s t r i b u t i o n s by Size i n Canada" Cat. No. 13-210, 1971-81. The t o t a l expenditure of metropolitan households can be computed using a s i m i l a r procedure. For a metropolitan household j , the estimated consumption function i s , corresponding to (6.22), (6.24) y 3 = a + b^ z? because A^ = 0. Accordingly, t o t a l expenditure i s , for J metropo-l i t a n households. J E j=l - ~ j = l (6.25) E y3 = Ja + b z? which i s obtainable given J and 1.2? from the same data source. Therefore, the f i r s t component of S i s just the share of yAt non-metropolitan expenditure (6.23) i n the sum of non-metropolitan expenditure (6.23) and metropolitan expenditure (6.25), i n year t . Other components of S can be c a l c u l a t e d by a s i m i l a r procedure. yAt - 92 -3. Expenditure d i s t r i b u t i o n s t a t i s t i c s S yyt This i s a s t a t i s t i c that depends on both the d i s t r i b u t i o n and the magnitude of expenditure. Formally, (6.26) S = EyJ( iny!| )/Y. yyt J t Jt t Since expenditure i s used for y , as opposed to income, a s i m i l a r procedure of mapping after-tax income d i s t r i b u t i o n to expenditure d i s t r i b u t i o n i s required. However, the cross-section consumption function formulated f o r t h i s purpose does not include any dummy variables since S y y t ignores household a t t r i b u t e s . Consequently, the regression equation i s simply h h h (6.27) y = a 5 + b 5 z + v 5 where y , z and v,_ are the expenditure, a f t e r - t a x income and d i s -turbance r e s p e c t i v e l y of household h, h = 1, H. I t i s assumed that there i s no contemporaneous covariances i n the disturbance term so that OLS i s best l i n e a r unbiased and t h i s regression equation i s estimated using the cross-section data set. The r e s u l t s are found i n Appendix B, column 5. The appropriate income data can be found i n "Income After-tax, D i s t r i b u t i o n s by Size i n Canada" No. 13-210, 1971-81. In each year, - 9 3 -the income spectrum i s divided into income brackets and the number of households i n each bracket i s provided. The mapping procedure i s therefore: assume that every household i n a p a r t i c u l a r bracket receives the mid-point income, and estimate i t s expenditure by using the estimated equation, from (6.27) (6.28) y h = a c + b c z h D 0 Assuming that a l l households i n that bracket have the same expenditure, 1*1 h a number f o r the sum of y log y over a l l households i n that bracket i s c a l c u l a t e d . The procedure i s then repeated f o r other income h li brackets, and Zy l o g y i s obtained by summing over the p a r t i a l sums i n a l l brackets. F i n a l l y , s y y t I s obtained by d i v i d i n g the o v e r a l l sum by the t o t a l expenditure i n a l l brackets. This process i s admittedly a rough approximation but since the range of expenditure i s rather small and S , not very s e n s i t i v e to expenditure d i s t r i b u -yyt t i o n , the margin of e r r o r involved i s not l i k e l y to be s i g n i f i c a n t . 4. Price indexes, p. i t The 6 composite goods c l a s s i f i e d i n the present model are s i m i l a r to those c l a s s i f i e d i n the p r i c e s e r i e s published i n "The Consumer Price Index" No. 62-001. Almost no recompilation i s required for the p r i c e s e r i e s , although one exception i s that i n that p r i c e - 94 -s e r i e s , "food" and "tobacco and a l c o h o l " are c l a s s i f i e d as separate goods. To combine them in t o one composite good, the p r i c e index for food i s obtained as a weighted-average of the p r i c e indexes for "food" and "tobacco and a l c o h o l " , the weights being the expenditures on the two goods divided by the sum, for a p a r t i c u l a r 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 normalized so that the p r i c e indexes for the survey year, 1978, are a l l unity, as mentioned i n Chapter 4. 2 5. Heteroscedasticity c o r r e c t i o n f a c t o r p As explained i n Chapter 5, the following f a c t o r i s necessary to correct for h e t e r o s c e d a s t i c i t y i n the time-series model, (6.29) p t = D ( p t ) " 1 ( ( Y t ) 2 / Z ( y ^ ) 2 ) i 5 -1 2 The computation of D(p^ ) and ( ) i s s t r a i g h t forward. The h 2 computation of E ( y ) i s performed by using the expenditure informa-t i o n generated i n the course of computing S . Again, each household yyt within an income bracket i s assumed to receive the mid-point income. Using the estimated consumption function (6.28), the corresponding expenditure i s obtained which i s then squared and m u l t i p l i e d by the h 2 estimated number of households i n that bracket. E ( y ) i s then obtained by summing "the sum of squares" over a l l brackets. This - 95 -completes the discussion of time-series data generation. A complete set of time-series data can be found i n Appendix C. Since the covariance matrix i n the time-series model i s singular, the transportation equation i s dropped. Time-series estimation i s performed by applying J o i n t Generalized Least Squares on a system of 5 equations, (5.34) or (6.8), each having the same explanatory v a r i a b l e s , subject to the following symmetry constraints, (6.30) b 1 2 = b 2 1 , b 1 3 = b 3 1 , = b 4 1 , b 1 5 = b 5 1 (6.31) b 2 3 = b 3 2 , b 2 4 = b 4 2 , b 2 5 = b 5 2 (6.32) b 3 4 = b 4 3 , b 3 5 = b 5 3 (6.33) b 4 5 = b 5 4 where 11 observations 1971 - 1981, are used. The estimation r e s u l t s , namely B , can be found i n Appendix D. PP Section 3 Estimated Equivalence Scales Usinq (5.39) and estimated values for B and B . the pp pA commodity-specific equivalence scales can be estimated. Since there are 11 dummy variables d e s c r i b i n g 4 a t t r i b u t e s , there are 120 possible - 96 -configurations for A. The corresponding 120 sets of equivalence scales can be found i n Appendix E. They are i n t u i t i v e l y very appealing. One can r e a d i l y notice that, as a function of A, the estimated scales display c e r t a i n general trends. Comparing the scales f o r metropolitan area of residence with that for non-metropolitan area of residence, the metropolitan scales are higher for every good except transportation. The apparent reason i s that, i n urban centres, the c o s t - o f - l i v i n g i s higher and, i n addition, c e r t a i n goods and services i n the r u r a l areas are home-produced. However, f a c i l i t i e s i n r u r a l areas are l e s s concentrated so that r u r a l households need more transportation s e r v i c e s . Comparing the scales f o r male head with that f o r female head shows that households with male heads have higher needs i n food, she l t e r and transportation but lower needs i n c l o t h i n g , r e c r e a t i o n and personal/medical care. These dif f e r e n c e s are probably due to the way "sex of household head" i s defined. Households with female heads usually imply single-parent f a m i l i e s , so that other members i n these f a m i l i e s are probably c h i l d r e n . Therefore, other a t t r i b u t e s being equal, compared with a male-headed household, a female-headed household involves s u b s t i t u t i n g a c h i l d for an adult. - 97 -Comparing the scales f o r d i f f e r e n t family sizes i s most i n t e r e s t i n g . As an informal t e s t of the structure of the present model, the scales not only have to be increasing with family s i z e but must f a l l between c e r t a i n ranges i n absolute magnitude. The scales estimated here are, i n general increasing with family s i z e , showing strong economies of scale and are a l l i n t u i t i v e l y appealing 3 i n magnitude. In the order of decreasing degree of economies of scale, the 6 goods can be ordered as follows: transportation (strongest), food, s h e l t e r , recreation, personal/medical care and c l o t h i n g (weakest). Age of the household does not a f f e c t the scales very much, although there i s a noticeable increase from the f i r s t age group (below 24) to the fourth age group (between 44 and 54) and a decline t h e r e a f t e r . Although not very s i g n i f i c a n t , t h i s trend could be explained by the d i f f e r i n g l e v e l s of a c t i v i t y and needs associated with d i f f e r e n t age groups. To summarize t h i s chapter: i t has been shown that the trans-log equivalent income, being the measure of u t i l i t y used i n the proposed new i n e q u a l i t y index, i s estimable using a micro-macro sequential approach. Only survey and aggregate time-series demand data are required for successful estimation. These data are e i t h e r d i r e c t l y obtainable i n p u b l i c f i l e s or i n d i r e c t l y a f t e r some simple - 9 8 -computations. The estimation r e s u l t s are however very encouraging judging from the estimated equivalence scales i n Appendix E. This f i n d i n g d e f i n i t e l y supports the e n t i r e approach to i n e q u a l i t y measurement. - 99 -Chapter 6 Footnotes 1. Information on the proportion of households without mortgage i s taken from Household F a c i l i t i e s and Equipment 1977-81, Cat. No. 64-202 (occasional). This s t a t i s t i c i s only a v a i l a b l e for 1981. The adjustment i s necessary because (9) includes imputed rent as owned housing. 2. Estimated t o t a l expenditures c a l c u l a t e d according to d i f f e r e n t a t t r i b u t e s may deviate within 5% which would not a f f e c t time-series estimation r e s u l t s s i g n i f i c a n t l y . 3. The only exception i s i n the transportation scales and where family size changes from 3 to 4 — causing a s l i g h t decrease. But since transportation shows the strongest economies of s c a l e , t h i s i s not e n t i r e l y s u r p r i s i n g and contradictory to common sense. - 100 -CHAPTER 7 APPLICATIONS Section 1 Introduction I t might be useful to r e c a l l the development through the previous chapters here. In Chapter 2, i t has been es t a b l i s h e d that a new i n e q u a l i t y index which i s able to measure d i s t r i b u t i v e p r i c e e f f e c t s within a rigorous s o c i a l welfare evaluation framework i s urgently needed. The f a i l u r e of conventional indexes has also been pointed out. A new index i s introduced i n Chapter 3 which i n theory f i l l s the gap i n the l i t e r a t u r e and should be a s i g n i f i c a n t improve-ment over the e x i s t i n g indexes. Chapter 4 and 5 provide s p e c i f i c a t i o n for preferences and estimation algorithms to make'implementation of the index p o s s i b l e . Chapter 6 describes the actual estimation process, the handling of data and the intermediate r e s u l t s of estimated equivalence s c a l e s . Although the estimation 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 reasonable, whether the new index performs well i n pr a c t i c e has yet to be seen. In p a r t i c u l a r , we want to know i f i n p r a c t i c e , i t r e a l l y d i f f e r s s i g n i f i c a n t l y from the other indexes and captures p r i c e e f f e c t s i n a predictable way. This chapter provides the necessary t e s t s . I t i s organized as follows. Section 2 discusses the estimated market equivalence scales (3.25) using the estimates obtained i n Chapter 6. Due to the translog preferences s p e c i f i e d i n Chapter 4, these scales are independent of u t i l i t y and hence equal to the poverty-line r a t i o s (4.21). These scales are compared with the - 101 -low-income cu t - o f f r a t i o s of S t a t i s t i c s Canada to provide i n s i g h t s to the subsequent comparative studies. Section 3 attempts to show that the new index i s not very s e n s i t i v e to 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 serious. Section 4 i s a comparative study of the new index with other major indexes, using the 1978 expenditure survey sample as the common data set. Section 5 studies 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 . A hypothetical increase of 10% i n the p r i c e of each good i s considered i n turn. F i n a l l y , Section 6 applies t h i s new index to measure i n e q u a l i t y i n Canada f or 1975, 1979 and 1981, using income survey data. For com-parison, the same data sets are employed using the we l f a r e - r a t i o approach with S t a t i s t i c s Canada low-income cu t - o f f values. Section 2 Estimated Market Equivalence Scales As explained i n Chapter 4, the translog s p e c i f i c a t i o n of t h i s model implies that market equivalence scales (3.25) and poverty l i n e r a t i o s (4.19) are equal and can be expressed as (4.21). Using the parameter estimates obtained, the estimated scales are presented i n Table 2 for a l l a t t r i b u t e configurations from 1971 to 1981. In Table 2, the reference household i s an unattached male, of age below 24, l i v i n g i n a metropolitan area, and h i s income i s used as the denominator i n (3.25) to generate these r a t i o s . According to (3.25) the r a t i o f o r t h i s reference household i s 1 for a l l years. URBAN MALE HOUSEHOLD HEAD FAM. SIZE URBAN 1 1 1 1 2 MALE HOUSEHOLD HEAD FAM. SIZE URBAN 1 1 1 2 2 MALE HOUSEHOLD HEAD HEAD AGED 24 1971 .0000 .4043 .7458 .9525 .6712 HEAD AGED 24-1971 .0469 . 4709 .8292 .0437 . 7931 HEAD AGED 34-TABLE 2: TRANSLOG MARKET EQUIVALENCE SCALES OR BELOW . 1972 1.0000 1 .4039 1 .7447 1.9508 2.6671 34 1972 1 .0477 1 .4716 1 .8294 2.0434 2.7909 1973 1.0000 1.4030 1.7429 1.9484 2.6617 1973 1 .0486 1 .4719 1.8291 2.0427 2.7877 1974 1.0000 1.4032 1.7434 1.9485 2.6625 1974 1 .0480 1 .4713 1 .8286 2.0417 2.7871 1975 1.0000 1.4028 1.7424 1.9467 2.6594 1975 1.0478 1.4705 1.8272 2.0393 2.7832 1976 1.0000 1.4029 1.7423 1.9456 2.6554 1976 1.0492 1.4726 1.8295 2.0409 2.7827 1977 1.0000 1.4028 1.7421 1.9446 2.6519 1977 1.0504 1.4743 1.8314 2.0422 2 . 7823 1978 1.0000 1.4024 1.7409 1 .9421 2.6463 1978 1.0511 1 .4747 1 .8314 2.0409 2 . 7782 1979 1 .OOOO 1 .4033 1 .7429 1.9439 2.6503 1979 1.05O0 1 .4742 1 .8315 2.0407 2.7796 1980 1.0000 1.4045 1.7456 1.9469 2.657 1 1980 1.0484 1 .4733 1.8317 2.0408 2.7826 1981 1.0000 1.4051 1.7468 1.9465 2.6567 1981 1.0479 1.4731 1 .8319 2.0393 2.7806 44 FAM. SIZE 1 2 3 4 . 5 URBAN 1 1 2 2 3 MALE HOUSEHOLD HEAD FAM. SIZE URBAN 1 1 2 2 3 MALE HOUSEHOLD HEAD FAM. SIZE URBAN 1 1 1 2 3 MALE HOUSEHOLD HEAD FAM. SIZE 1 0 2 1 3 1 4 1 5 2 1971 . 1821 .6617 .0666 . 3204 .1811 HEAD AGED 1971 .3162 .8506 . 3012 . 5835 . 5438 HEAD AGED 1971 . 1320 .5917 .9796 .2107 .0244 HEAD AGED 1971 .9635 .3546 .6852 .8793 .5692 1972 1 . 1825 1 .6618 2.0660 2.3192 3.1773 44-54 1972 1 .3158 1.8495 2.2991 2.5805 3.5373 54-64 1972 1.1319 1 .5910 1 .9781 2.2086 3.0194 OVER 64 1972 0.9638 1.3545 1.6846 1.8783 2.5660 1973 1.1829 1.6614 2.0647 2.3170 3.1718 1973 1 .3152 1 .8475 2.2957 2.5760 3.5282 1973 1.1316 1 .5896 1 .9755 2.2052 3.0124 1973 0.9641 1 .354 1 1.6834 1.8765 2.5616 1974 1.1824 1.6608 2.0643 2.3161 3.1713 1974 1 .3154 1 .8479 2.2966 2.5764 3.5296 1974 1.1319 1.5902 1.9766 2.2060 3.0142 1974 0.9642 1 .3543 1.6840 1 .8768 2.5626 1975 1. 1820 1.6599 2.0625 2.3132 3.1666 1975 1.3156 1.8477 2.2958 2.5744 3.5261 1975 1.1327 1.5909 1.9769 2.2054 3.0128 1975 0.9649 1.3549 1.6842 1.8763 2.5615 1976 1 .1826 1.6608 2.0634 2.3131 3. 1635 1976 1 . 3145 1 .8463 2.2937 2 . 57 10 3.5180 1976 1.1319 1.5899 1.9754 2.2027 3.0062 1976 0.9647 1.3547 1.6838 1.8750 2.5572 1977 1 . 1832 1 .6615 2.0641 2 .3130 3.1607 1977 1 .3136 1 .8450 2 . 2917 2.5678 3.5107 1977 1.1312 1.5888 1.9738 2.2001 3.0002 1977 0.9645 1 .3545 1.6833 1 .8737 2.5533 1978 1.1832 1 .6610 2.0628 2.3100 3.1539 1978 1.3131 1 .8436 2.2893 2.5633 3.5016 1978 1.1313 1.5884 1.9727 2. 1975 2.9942 1978 0.9650 1 . 3547 1 .6830 1.8722 2.5493 1979 1.1824 1.6609 2.0636 2.3104 3.1564 1979 1 .3135 1.8455 2.2927 2.5665 3.5082 1979 1.1318 1.5902 1.9759 2.2006 3.0002 1979 0.9649 1.3554 1 .6847 1.8737 2.5529 1980 1.1813 1.6609 2.0650 2.3119 3. 1617 1980 1 .3142 1.8481 2.2974 2.5718 3.5190 1980 1.1324 1.5924 1.9800 2.2051 3.0095 1980 0.9646 1.3562 1 .6868 1.8760 2 . 5586 1981 1 . 1803 1.6601 2.0646 2.3095 3.1585 198 1 1 . 3140 1 .8484 2.2986 2.5709 3.5178 1981 1.1329 1.5937 1.9821 2.2056 3.0102 1981 0.9648 1.3569 1 .6882 1 .8759 2.5585 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW FAM. SIZE 1971 TABLE 2 (CONTINUED) URBAN 0.8885 1.2500 1.5542 1.7681 2.4394 FEMALE HEAD 1972 0.8887 1.2499 1.5535 1.7669 2.4359 HOUSEHOLD HEAD AGED 24-34 FAM. SIZE 1 2 3 4 5 URBAN 1971 0.9301 1.3092 1.6283 1.8505 2.5505 FEMALE HEAD 1972 0.9310 1.3101 1.6288 1.8506 2.5488 HOUSEHOLD HEAD AGED 34-44 FAM. SIZE 1 2 3 4 5 URBAN 197 1 1.0616 1.4951 1 .8596 2 . 1240 2.9365 1972 1.0622 1.4954 1.8594 2.1231 2.9331 FEMALE HEAD HOUSEHOLD HEAD AGED 44-54 FAM. SIZE 1 2 3 4 5 URBAN 1971 1 . 1791 1 .6609 2.0656 2.3589 3.2630 FEMALE HEAD 1972 1.1790 1.6602 2.0641 2.3564 3.2573 HOUSEHOLD HEAD AGED 54-64 FAM. SIZE 1 2 3 4 5 URBAN 1971 1.0014 1.4105 7545 1.9931 2.7497 FEMALE HEAD 1 . 1972 1.0015 1 .4103 1 .7536 1 .9915 2.7455 HOUSEHOLD HEAD AGED OVER 64 FAM. SIZE 1 2 3 4 5 1971 0.8500 1.1970 1.4894 1.6896 2.3293 1972 0.8504 1 .1973 1.4893 1.6889 2.3268 1973 0.8888 1.2492 1 .5520 1.7644 2.4305 1973 0.9320 1.3105 1.6287 1.8497 2.5454 1973 1 .0626 1.4950 1.8582 2.1207 2.9271 1973 1.1785 1.6584 2.0610 2.3518 3.2480 1973 1.0014 1.4092 1.7515 1 .9883 2.7388 1973 0.8508 1.1971 1.4884 1.6873 2.3225 1974 0.8877 1.2477 1.5504 1.7621 2.4278 1974 0.9302 1.3082 1.6262 1.8463 2.5412 1974 1.0607 1.4925 1.8553 2. 1 168 2.9223 1974 1 . 1771 1 .6565 2.0590 2.3489 3.2445 1974 1.0004 1.4079 1.7503 1.9863 2.7366 1974 0.8498 1.1958 1.4870 1.6852 2.3202 1975 0.8862 1.2453 1.5470 1.7574 2.4208 1975 0.9284 1.3053 1.6221 1.8409 2.5332 1975 1.0586 1.4891 1.8506 2.1104 2.9129 1975 1.1753 1 .6536 2.0548 2.3430 3.2356 1975 0.9994 1.4061 1.7475 1.9824 2.7305 1975 0.8490 1.1943 1.4848 1.6820 2.3151 1976 0.8870 1.2465 1.5483 1.7583 2.4197 1976 0.9305 1.3084 1.6257 1.8443 2.5355 1976 1 .0601 1 .4913 1.8532 2.1126 2.9133 1976 1.1754 1.6539 2.0550 2.3423 3.2318 1976 0.9996 1.4065 1 .7478 1 .9819 2.7274 1976 0.8496 1.1952 1.4858 1.6824 2.3136 1977 0.8877 1.2475 1.5494 1 .7588 2.4183 1977 0.9324 1.3110 1.6288 1.8470 2.5371 1977 1.06 15 1.4932 1 .8553 2.1141 2 .9129 1977 1.1756 1 .6540 2.0549 2.3412 3.2276 1977 0.9998 1.4068 1 .7479 1.9812 2.7241 1977 0.8502 1.1960 1.4866 1.6826 2.3120 1978 0.8868 1.2458 1 .5467 1 .7543 2.4100 1978 0.9320 1.3100 1 .6270 1 .8435 2.5300 1978 1.0603 1 .4910 1 .8520 2.1085 2.9024 1978 1.1738 1.6509 2.0503 2.3341 3 . 2146 1978 0.9989 1.4050 1.7451 1.9765 2.7151 1978 0. 8498 1 . 1950 1 .4848 1 .6794 2.3054 1979 0.8854 1.2447 1.5461 1 . 7532 2.4098 1979 0.9297 1.3075 1 .6246 1.8403 2.5272 1979 1.0579 1.4886 1 .8498 2.1055 2.9000 1979 1.1724 1.6500 2.0501 2.3332 3.2153 1979 0.9979 1.4045 1.7453 1.9762 2.7163 1979 0.8484 1.1939 1.4841 1.6781 2.3050 1980 0.8841 1.2439 1.5462 1 . 7532 2.4125 1980 0.9269 1.3048 1.6223 1.8377 2.5263 1980 1.0554 1.4864 1.8483 2.1037 2.9006 1980 1.1712 1 .6499 2.0513 2.3346 3.2206 1980 0.9969 1.4043 1.7463 1.9772 2.7206 1980 0.8468 1 . 1927 1 .4836 1 .6776 2.3067 1981 .8818 . 24 12 . 5433 .7436 .4061 1981 0.9240 1.3013 1.6184 1 .8318 2.5183 1981 1.0518 1 .4820 1.8433 2.0965 2.8908 198 1 . 1680 .6460 .0472 . 3280 .2117 198 1 0.9947 1.4018 1.7437 1 .9727 2.7 146 1981 0.8448 1.1903 1 .481 1 1 . 6733 2.3010 RURAL MALE HOUSEHOLD HEAD FAM. SIZE TABLE 2 (CONTINUED) RURAL O 1 1 1 2 MALE HOUSEHOLD HEAD FAM. SIZE RURAL 0 1 1 1 2 MALE HOUSEHOLD HEAD FAM. SIZE RURAL 1 1 1 1 2 MALE HOUSEHOLD HEAD FAM. SIZE RURAL 1 1 1 2 3 MALE HOUSEHOLD HEAD FAM. SIZE RURAL 0 1 1 1 2 MALE HOUSEHOLD HEAD FAM. SIZE 1 0 2 1 3 1 4 1 5 2 HEAD AGED 1971 . 8648 .2133 .5077 .6743 .2815 HEAD AGED 1971 . 9058 .2715 .5805 .7534 . 3869 HEAD AGED 1971 .0181 .4299 . 7775 .9818 .7062 HEAD AGED 1971 . 1345 .5938 .9810 . 2083 .0171 HEAD AGED 1971 .9810 . 3782 .7133 .8998 . 5887 HEAD AGED 1971 . 8361 . 1744 . 4604 .6172 . 2021 24 OR BELOW 1972 0.8646 1.2126 1.5063 1 .6725 2.2776 24-34 1972 0.9062 1.2717 1.5803 1.7528 2.3846 34-44 1972 1.0182 1 .4297 1.7766 1.9804 2.7025 44-54 1972 1 . 1340 1 .5924 1 .9787 2.2053 3.0111 54-64 1972 0.9807 1.3772 1.7115 1 .8975 2.5840 OVER 64 1972 0.8362 1 .1740 1.4595 1.6159 2.1989 1973 0.8643 1 .2115 1.5043 1.6700 2.2725 1973 0.9067 1.2716 1.5795 1.7517 2.3813 1973 1.0183 1 .4289 1.7750 1.9781 2.6973 1973 1 . 1331 1.5902 1.9751 2.2009 3.0027 1973 0.9801 1.3755 1.7087 1 .8941 2.5774 1973 0.8361 1.1732 1.4579 1.6139 2.1945 1974 0.8647 1 .2122 1.5055 1 .6710 2 . 2744 1974 0.9067 1.2717 1.5799 1 .7518 2.3820 1974 1.0184 1 .4292 1.7756 1.9784 2.6983 1974 1.1338 1 .5914 1 .9769 2.2024 3.0056 1974 0.9808 1 .3767 1.7105 1.8957 2.5803 1974 0.8366 1.1740 1.4591 1.6149 2.1966 1975 0.8651 1 .2124 1.5053 1.6702 2.2729 1975 0.9068 1.2716 1.5793 1.7505 2.3798 1975 1.0185 1 .4290 1 . 7749 1.9768 2.6957 1975 1 . 1345 1.5920 1.9771 2.2018 3.0041 1975 0.9819 1.3779 1.7115 1.8962 2.5803 1975 0.8375 1 . 1750 1.4600 1 .6153 2.1966 1976 0.8650 1 .2124 1.5051 1.6690 2.2690 1976 0.9080 1.2733 1.5812 1.7516 2.3790 1976 1.0189 1.4295 1.7754 1.9764 2.6925 1976 1.1334 1.5905 1.9751 2.1984 2.9966 1976 0.9811 1.3768 1.7100 1.8935 2.5742 1976 0.8373 1.1748 1.4595 1.6139 2. 1925 1977 0.8647 1 .2120 1.5044 1 .6678 2.2655 1977 O.9088 1.2744 1.5824 1.7524 2.3781 1977 1.0191 1.4298 1.7755 1.9759 2.6895 1977 1.1323 1.5890 1.9729 2.1952 2.9897 1977 0.9802 1.3756 1.7082 1.8909 2.5685 1977 0. 8369 1. 1742 1.4586 1 .6124 2.1888 1978 0.8649 1.2119 1.5038 1.6661 2.2615 1978 0.9095 1 .2750 1 .5827 1 .7517 2.3754 1978 1.0194 1 . 4298 1.7749 1 . 9740 2 .6848 1978 1.1322 1 .5882 1 .9713 2.1921 2.9831 1978 0.9805 1.3755 1.7075 1 .8891 2.5641 1978 0.8375 1.1746 1.4587 1.6116 2.1859 1979 0.8656 1 .2136 1.5066 1.6690 2.2667 1979 0.9093 1.2755 1 .5840 1.7529 2.3785 1979 1 .0195 1 .4308 1 . 7770 1 .9759 2.6891 1979 1.1334 1 .5910 1.9757 2.1966 2.991 1 1979 0.98 17 1.3781 1.7116 1.8933 2.57 13 1979 0.8380 1 . 1762 1 .4612 1.6141 2.1908 1980 0.8664 1.2158 1.5103 1.6730 2.2746 1980 0.9088 1 . 2760 1.5856 1.7546 2.3832 1980 1 .0195 1 . 4321 1.7797 1 .9789 2.6959 1980 1.1350 1.5947 1 .9816 2.2031 3.0030 1980 0.9831 1 .3813 1.7167 1.8988 2.5815 1980 0.8385 1.1779 1.4644 1.6175 2.1976 1981 0.8674 1.2 178 1.5132 1.6747 2.2769 1981 O.9094 1 . 2773 1.5878 1.7554 2.3843 1981 1.0198 1 . 4332 1 .7816 1.9792 2.6964 1981 1 . 1362 1.5970 1.9850 2.2049 3.0055 1981 0.9848 1.384 1 1.7207 1.9015 2.5852 1981 0.8397 1.1800 1 .4674 1 .6194 2.2002 RURAL HOUSEHOLD FAM. S I Z E 1 2 3 4 5 RURAL HOUSEHOLD FAM. S I Z E 1 2 3 4 5 RURAL HOUSEHOLD FAM. S I Z E 1 2 3 4 5 RURAL HOUSEHOLD FAM. S I Z E 1 2 3 4 5 RURAL HOUSEHOLD FAM. S I Z E 1 2 3 4 5 RURAL HOUSEHOLD FAM. S I Z E 1 2 3 4 5 FEMALE HEAD HEAD AGED 24 OR BELOW 1971 1972 0 . 7 5 4 9 0 . 7 5 4 9 1.0610 1.0608 1.3186 1.3179 1.4896 1.4883 2 . 0 4 7 0 2.0439 FEMALE HEAD HEAD AGED 24-34 1971 1972 0 . 7 9 0 6 0 . 7 9 1 3 1.1119 1 . 1124 1.3823 1.3825 1.5598 1.5597 2 . 1 4 1 4 2.1397 FEMALE HEAD HEAD AGED 34-44 1971 1972 0 . 8 9 8 3 0.8987 1.2640 1.2641 1.5715 1.5711 1.7822 1.7813 2 . 4 5 4 2 2.4512 FEMALE HEAD HEAD AGED 4 4 - 5 4 1971 1972 0 . 9 9 8 6 0.9983 1.4053 1.4045 1.7469 1.7454 1.9809 1.9786 2 . 7 2 9 3 2.7243 FEMALE HEAD HEAD AGED 54-64 1971 1972 0 . 8 5 2 6 0.8526 1.1999 1.1995 1.4918 1.4908 1.6827 1.6811 2 . 3 1 2 3 2.3086 FEMALE HEAD HEAD AGED OVER 64 TABLE 2 (CONTINUED) 1973 0 . 7 5 4 9 1.0600 1.3164 1.4862 2.0392 1973 O. 7919 1.1126 1.3822 1.5588 2.1368 1973 0 . 8 9 8 9 1.2636 1.5698 1.7792 2.4462 1973 .9978 . 4028 . 7426 .9747 .7165 1973 0 . 8 5 2 3 1.1983 1.4888 1.6783 2.3028 1971 0 . 7 2 4 6 1.0196 1 .2681 1.4284 1.9614 1972 0.7249 1.0196 1 .2677 1 .4276 1.9591 1973 0 . 7251 1.0193 1.2668 1.4261 1.9553 1974 0 . 7 5 4 3 1.0594 1.3158 1.4852 2.0383 1974 0 . 7 9 0 9 1.1113 1.3808 1 .5569 2. 1346 1974 0 . 8 9 7 8 1.2622 1.5684 1 .7770 2.4438 1974 0.9971 1.4020 1 .7419 1.9734 2.7153 1974 0 . 8 5 1 9 1.1979 1.4885 1.6776 2.3023 1974 0 . 7 2 4 6 1.0187 1.2663 1.4252 1.9546 1975 0 . 7 534 1.0578 1.3135 1.4820 2.0334 1975 O. 7898 1.1094 1.3780 1.5531 2.1289 1975 0 . 8 9 6 5 1.2599 1.5651 1.7726 2.4371 1975 0.9961 1.4002 1.7392 1.9695 2.7093 1975 0 . 8 5 1 5 1. 1970 1.4870 1.6751 2.2984 1975 0 . 7 2 4 3 1.0180 1.2650 1.4231 1.9513 1976 0 . 7 5 4 0 1.0586 1.3144 1.4823 2.0319 1976 0 . 7 914 1.1118 1.3808 1.5555 2.1303 1976 0.8976 1.2616 1.5670 1.7739 2.4368 1976 0 . 9 9 6 0 1.4002 1.7389 1.9684 2.7053 1976 0 . 8 5 1 5 1.1970 1.4869 1.6743 2.2952 1976 0.7247 1 .0185 1.2656 1 .4231 1.9495 1977 0 . 7 5 4 5 1.0593 1 .3150 1 .4824 2.0304 1977 0 . 7 9 2 8 1.1137 1.3831 1 .5575 2.1312 1977 0 . 8 9 8 6 1.2629 1.5684 1.7749 2 . 4 3 6 0 1977 0 . 9 9 5 9 1.4000 1.7385 1.9671 2.7013 1977 0 . 8 5 1 5 1.1970 1.4867 1.6734 2 . 2 9 2 0 1977 0 . 7 2 5 0 1.0190 1.2660 1.4231 1.9478 1978 0 . 7 5 3 9 1.0582 1.3133 1 .4794 2 . 0 2 4 5 1978 O.7928 1.1133 1 .3821 1 .5553 2. 1263 1978 0 . 8 9 7 9 1 . 2 6 16 1 .5663 1 . 771 1 2 . 4 2 8 6 1978 0 . 9 9 4 8 1.3980 1.7354 1.9621 2 . 6 9 1 9 1978 0 . 8 5 1 0 1 . 1959 1.4849 1.6702 2 . 2 8 5 5 1978 0 . 7 2 4 9 1 .0185 1 . 2 6 5 0 1.4209 1 .9431 1979 0 . 7 534 1.0582 1.3138 1.4796 2.0261 1979 0 . 7 9 1 4 1.1121 1 .3813 1 .5540 2.1258 1979 0 . 8 9 6 7 1.2606 1.5658 1.7701 2.4287 1979 0 . 9 9 4 5 1.3984 1.7367 1.9631 2 .6949 1979 0 . 8 5 0 9 1 . 1965 1 .4862 1 .6713 2 . 2 8 8 5 1979 0 . 7 2 4 3 1.0184 1 .2654 1 .4210 1 .9445 1980 O.7529 1.0584 1.3151 1 .4810 2 . 0 3 0 0 1980 0 . 7 8 9 8 1.1108 1.3805 1.5531 2.1268 1980 0 . 8 9 5 3 1.2598 1.5658 1.7701 2 . 4 3 1 3 1980 0 . 9 9 4 3 1.3995 1.7392 1.9658 2.7016 1980 0 . 8 5 0 7 1.1974 1.4883 1.6736 2.294 1 1980 O.7236 1.0183 1.2661 1 .4218 1.9476 1981 0 . 7 5 1 9 1.0574 1.314 1 1 .4787 2 . 0 2 6 9 1981 O.7882 1.1091 1.3788 1.5499 2 . 1 2 2 5 1981 0 . 8 9 3 3 1.2575 1.5634 1.7660 2 . 4 2 5 7 1981 0 . 9 9 2 8 1.3978 1.7377 1.9625 2.6971 1981 0 . 8 4 9 9 1.1967 1.4879 1.6718 2 . 2 9 1 6 1981 O.7227 1.0174 1.2654 1.4199 1.9449 O - 106 -Table 2 shows that, i n general, for each a t t r i b u t e configura-t i o n , market equivalence scales are not very s e n s i t i v e to the p r i c e changes experienced i n 1971 - 1981. However, they do change with a t t r i b u t e s . The scales are increasing with family si z e although by decreasing increments (except f o r the change from si z e 4 to si z e 5 because the l a t t e r category includes a l l l a r g e r family sizes) . Furthermore, the scales are i n general s l i g h t l y higher f o r households with male heads than those with female heads. This means that, a l l goods considered, f a m i l i e s with female heads have s l i g h t l y lower consumption requirements. This i s a reasonable r e s u l t as f a m i l i e s with female heads are usually s i n g l e - a d u l t f a m i l i e s and c h i l d r e n normally consume l e s s . One also notes with i n t e r e s t that the scales are higher f o r metropolitan households than non-metropolitan house-holds ( r e f l e c t i n g the higher " c o s t - o f - l i v i n g " — broadly defined — i n the urban c i t i e s ) . F i n a l l y , the scales vary with age of head i n the same manner as the commodity-specific equivalence scales do, i . e . , increase s l i g h t l y with age up to the 44-54 bracket and decrease thereafter. Because of t r a n s l o g preferences, market equivalence scales and poverty-line r a t i o s are i d e n t i c a l as given by (4.21). Thus Table 2 can be regarded as the estimated translog poverty-line 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 of equivalent male-adults (of age below 24 i n a metropolitan area) for each household. - 107 -There are other data sources from which one can derive these equiva-lent-adults r a t i o s . As suggested i n Wolfson (1979), one can use the low-income cut-off values published by S t a t i s t i c s Canada. These cut-o f f values are c l a s s i f i e d by family size and si z e of area of residence. Using a metropolitan unattached i n d i v i d u a l as the reference household, one can compute s i m i l a r poverty-line r a t i o s . Table 3 gives the r e s u l t s using c u t - o f f values published for 1975 and 1981."'" These r a t i o s show s i g n i f i c a n t economies of scale and that non-metropolitan households have o v e r a l l lower-consumption needs. The r a t i o s i n Table 2 and Table 3 can be compared, although the difference i n the c l a s s i f i c a t i o n of a t t r i b u t e s precludes comparison on a one-to-one b a s i s . However, the general impression i s the r a t i o s derived from the two d i s t i n c t l y d i f f e r e n t approaches are rather close. This f i n d i n g has s i g n i f i c a n t bearing on the following comparative study of various i n e q u a l i t y indexes. Section 3 S e n s i t i v i t y of the IEI index to p° Before comparing the various i n e q u a l i t y indexes, i t i s useful to f i r s t examine the s e n s i t i v i t y of the new index, named the Individual Equivalent-Income (IEI) i n e q u a l i t y index, to changes i n the a r b i t r a r y o p r i c e vector p e m p i r i c a l l y . The s e n s i t i v i t y a n a l y s i s i s undertaken by using the Family Expenditure Survey 1978 sample which contains 9285 households and 27651 i n d i v i d u a l s . Following the procedure - 108 -Table 3 S t a t i s t i c s Canada Low-Income Cut-off Ratios Family Size 1 2 3 4 5 6 7 1975 1981 Non- Non-Metropolitan Metropolitan Metropolitan Metropolitan 1.0000 1.4496 1.8496 2.1997 2.4588 2.6996 2.9697 0.8241 1.1951 1.5250 1.8131 2.0275 2.2252 2.4395 1.0000 1.3165 1.7604 2.0326 2.3610 2.5770 2.8403 0.8255 1.0828 1.4514 1.6776 1.9483 2.1258 2.3433 Source: Income D i s t r i b u t i o n by Size i n Canada 1975, 1981. Catalogue No. 13-207 (annual) Notes: (1) Given family si z e and area of residence, a low-income cut-off value i s derived a r b i t r a r i l y by s e t t i n g i t equal to the average observed income of those households who spend 20% more than the average Canadian household does on food, c l o t h i n g and s h e l t e r . (2) The cu t - o f f values f o r metropolitan households are taken as the means of the cu t - o f f values f o r the 3 population brackets: over 500,000, 100,000-499,999 and 30,000-99,9999. The cu t - o f f values for non-metropolitan households are taken as the means of the c u t - o f f values f o r the remaining 2 population brackets: l e s s than 30,000 and r u r a l . - 109 -explained i n Chapter 3, equivalent income i s imputed to each i n d i v i -dual taking 1978 p r i c e s as p. Atkinson index i s then computed for r set equal to 0.5, -1 and Computation i s repeated for p° set equal 2 to the actual p r i c e s experienced i n various years from 1971 to 1981. The r e s u l t s are presented i n Table 4. Although there i s no benchmark by which assessment of s e n s i t i v i t y can be made, the general impression i s that the new IEI index i s not very s e n s i t i v e to p°. It ranges from, f o r r = 0.5, 0.046909 for 1981 p r i c e s to a maximum of 0.049461 f o r 1971 p r i c e s , i . e . , a deviation range of ±2.6%. Section 4 Comparative Study of Various Measures As described i n Chapter 2, various measures of u t i l i t y have been used for i n e q u a l i t y measurement and because they are a l l un-s a t i s f a c t o r y , a new index, the IEI index, i s developed i n Chapter 3. It i s i n t e r e s t i n g , however, to compare e m p i r i c a l l y these indexes and see i f they are r e a l l y s i g n i f i c a n t l y d i f f e r e n t . To carry out t h i s comparative study, a common data set, namely the Family Expenditure Survey 1978 sample, i s used which contains 9285 households 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 applicable i s taken as 1971 p r i c e s . The sample contains information on household expenditures and a t t r i b u t e s f or each house-hold, so that equivalent income (3.19) or (4.13) can be imputed to each of the 27651 i n d i v i d u a l s . The r e s u l t s are presented i n Table 5. T a b l e 4: S e n s i t i v i t y o f I E I m e a s u r e t o r = . 5 r = P° u € I € 1971 4 9 1 5 . 5 G 4 6 7 2 . 4 3 .049461 4 0 0 9 . 6 3 1973 5 4 6 9 . 5 8 5 2 0 7 . 0 4 .048000 4 4 8 9 . 1 2 1975 6 6 8 8 . 2 0 6 3 7 1 . 1 7 .047401 5 5 0 3 . 1 7 1977 7 7 7 5 . 4 7 7 4 0 5 . 1 6 .047626 6 3 9 1 . 7 4 1978 8 4 1 1 . 3 6 8 0 1 9 . 1 4 .046629 6 9 4 3 . 5 0 1979 9 1 5 5 . 7 8 8 7 3 1 . 4 0 .046351 7 5 6 6 . 9 2 1981 1 1 3 1 5 . 1 2 1 0 7 8 4 . 3 5 .046909 9 3 2 9 . 6 0 p : 1978 p r i c e s N o . o f e n t r i e s : 27651 D a t a S e t : F a m i l y E x p e n d i t u r e S u r v e y 1978 I . 184299 . 179256 . 177182 . 177962 .174507 .173536 .175475 5 6 5 . 2 4 6 5 0 . 9 3 8 0 7 . 3 5 9 3 3 . 5 9 1034.23 1133.31 1382.OO I .885011 .880991 .879288 .879932 .877044 . 8 7 6 2 2 0 . 8 7 7 8 6 3 T a b l e 5: C o m p a r i s o n o f D i f f e r e n t M e a s u r e s " I n c o m e " No. o f d e f i n i t i o n e n t r i e s ^ 1. H o u s e h o l d E x p e n d i t u r e 9 2 8 5 15159.73 ( H E ) 2 . P e r C a p i t a E x p e n d i t u r e 2 7 6 5 1 5 1 0 0 . 9 1 ( P C E ) 3 . H o u s e h o l d I n f l a t e d W e l f a r e - 9 2 8 5 9 1 1 9 . 9 3 R a t l o ( H I W R ) ( S t a t . C a n . ) 4 . 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 - 2 7 6 5 1 9 0 0 6 . 2 8 R a t i o ( I I W R ) ( S t a t . C a n . ) 5 . 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 - 2 7 6 5 1 8 4 1 1 . 3 3 R a t i o ( I I W R ) ( e s t i m a t e d ) 6 . H o u s e h o l d E q u i v a l e n t 9 2 8 5 5 0 8 1 . 5 2 I n c o m e ( H E I ) 7. I n d i v i d u a l E q u i v a l e n t 2 7 6 5 1 4 9 1 5 . 5 6 I n c o m e ( I E I ) r=.5 r = 0 r=-1 r=-2 5 I <= I r = -oo € I 1 4 0 5 5 . 7 8 .072821 12920.24 .147726 10602.41 .300620 8 4 5 1 . 1 1 .442529 2 1 5 5 . 0 0 .857847 4 7 6 5 . 9 6 .065700 4 4 6 1 . 8 0 .125295 3 9 2 3 . 4 2 .230839 3 4 5 4 . 9 9 .322673 8 0 8 . 2 0 .841558 8 6 2 2 . 0 6 .054591 8 1 4 6 . 7 6 .106707 9 1 1 9 . 9 3 .204757 6 4 3 3 . 8 4 .294529 1 1 7 0 . 0 0 . 8 7 1 7 1 0 8 5 8 6 . 1 4 .046650 8 1 8 3 . 3 7 .091371 7 4 1 7 . 7 1 .176385 6 6 9 8 . 1 0 .256285 1 1 7 0 . 0 0 .870091 8 0 1 9 . 1 1 .046630 7 6 4 5 . 7 3 .091020 6 9 4 3 . 5 0 .174507 6 2 8 9 . 1 0 .252306 1 0 3 4 . 2 3 .877044 4 7 9 7 . 5 6 .055880 4 5 2 9 . 0 6 .108719 4 0 2 9 . 7 1 .206986 3 5 7 3 . 1 0 .296844 5 6 5 . 2 4 .888767 4 6 7 2 . 4 3 .049461 4 4 4 1 . 7 4 .096392 4 0 0 9 . 6 3 .184299 3 6 0 8 . 9 4 .265813 5 6 5 . 2 4 .885011 p° : 1971 p r i c e s D a t a S e t : F a m i l y E x p e n d i t u r e S u r v e y 1978 - 112 -In each case, i n e q u a l i t y i s calculated using the Atkinson index with r set equal to 0.5, 0, -1, -2 and -<». u i s the mean of the d i s t r i b u -t i o n and £ i s the evenly-distributed equivalent of the d i s t r i b u t i o n while I i s the Atkinson index defined i n (3.32) or (3.36) (3.37) with the corresponding u t i l i t y measure i n each case. In Table 5, the f i r s t i n e q u a l i t y measure computed i s the Household Expenditure (HE) index which neglects family s i z e , even d i s t r i b u t i o n of s o c i a l weights and other household a t t r i b u t e s . The mean of the d i s t r i b u t i o n s i s 15159.73. For r = 0.5,5 i s 14055.78 giving an i n e q u a l i t y measure of 0.072821. An improved index i s the second index computed, namely the Per Capita Expenditure (PCE) index. This method imputes per-capita expenditure to each i n d i v i d u a l g i v i n g r i s e to a s i g n i f i c a n t decrease of the mean. Inequality also drops suggesting that larger households have higher incomes while.the HE index has ignored t h i s c o r r e l a t i o n . The PCE index ignores economies of scale and other relevant a t t r i b u t e s that might a f f e c t preferences. Wolfson (1979) and Beach, Card and F l a t t e r s (1981) employ welfare r a t i o s as u t i l i t y measures to capture the scale e f f e c t s . Although they use income data, what they would have done with expenditure data would be to divide household expenditure by the poverty income f o r the appropriate a t t r i b u t e s and in e q u a l i t y i s computed f or a d i s t r i b u t i o n of these w e l f a r e - r a t i o s . - 113 -To make comparisons with other indexes more immediate, " i n f l a t e d w e lfare-ratios" are used here i n place of w e l f a r e - r a t i o s . " I n f l a t e d w elfare-ratios" are welfare r a t i o s m u l t i p l i e d by the poverty income of a reference household, namely, an unattached i n d i v i d u a l i n a metropolitan area. Since the Atkinson index i s r e l a t i v e , the index i s not af f e c t e d by t h i s m o dification. I f one uses the c u t - o f f values published by S t a t i s t i c s Canada, then the " i n f l a t e d w e l f a r e - r a t i o s " w i l l be just household expenditure divided by the appropriate r a t i o s i n Table 3. In Table 5, the t h i r d index, namely Household I n f l a t e d Welfare-Ratio (HIWR) (Stat. Can.) index, i s computed by imputing " i n f l a t e d welfare r a t i o s " to each of the 9285 households based on the 1975 r a t i o s ( i d e n t i c a l with those published for 1978) i n Table 3. This i s the method according to Wolfson (1979) and Beach e t . a l . (1981). I t shows a further decrease i n i n e q u a l i t y . I i s 0.054591 at r = 0.5. However, as explained i n Chapter 3, a more acceptable procedure i s to impute to a l l 27651 i n d i v i d u a l s . Making t h i s a l t e r a t i o n , the fourth index named the Individual I n f l a t e d Welfare Ratio (IIWR) (Stat. Can.) index i s computed which i n t e r e s t i n g l y , i s appreciably smaller than the HIWR (Stat. Can.) index. I i s 0.046650 at r = 0.5. This suggests that the conventional welfare r a t i o index o f f e r s a d i s t o r t e d p i c t u r e of the actual i n e q u a l i t y s i t u a t i o n , hence should be avoided. - 114 -The " i n f l a t e d welfare r a t i o s " are a c t u a l l y expenditures denominated by the poverty-line r a t i o s . The S t a t i s t i c s Canada low-income cu t - o f f s are a r b i t r a r i l y derived. I t i s therefore i n t e r e s t i n g to employ the estimated translog poverty-line r a t i o s f or 1978, as given i n Table 2 and repeat the above i n e q u a l i t y computation. This gives r i s e to the Individual I n f l a t e d Welfare-Ratio (IIWR) (estimated) index i n Table 5. As shown, the two IIWR indexes are a c t u a l l y very close e m p i r i c a l l y . This comes as no surprise because comparing Table 2 and Table 3, the poverty-line r a t i o s are not s i g n i f i c a n t l y d i f f e r e n t . In choosing between the two indexes, the IIWR (est.) i s preferred because S t a t i s t i c s Canada low-income cu t - o f f s are not r e g u l a r l y updated for r e l a t i v e p r i c e changes. The l a s t two indexes computed make use of the equivalent income measure of u t i l i t y explained i n Chapter 3. The s i x t h index i s erroneous. Equivalent incomes are imputed to households, as opposed to i n d i v i d u a l s . I t i s presented here just to show that e m p i r i c a l l y i t does make a diffe r e n c e i f one imputes to households rather than i n d i v i d u a l s , regardless of the u t i l i t y measurement concept — equivalent income or i n f l a t e d welfare r a t i o . The l a s t index, the Individual Equivalent Income (IEI) index i s the new index proposed i n the present t h e s i s . I t s s o c i a l welfare foundation i s explained i n Chapter 3. For r = 0.5, the IEI index i s 0.049461 which i s appreciably d i f f e r e n t from the HEI index. However, one cannot - 115 -conclude that IEI and IIWR (estimated) are e m p i r i c a l l y d i f f e r e n t . The reason i s the IEI 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 to see that, i n (3.23), IT (•) i n the denominator w i l l vanish so that equivalent income i s i d e n t i c a l to an i n f l a t e d welfare r a t i o . In that case, the IEI index and the IIWR (estimated) indexes w i l l c oincide. Therefore, based on Table 5, one can assert that, the IIWR (Stat. Can.) index, the IIWR (est.) index and the proposed IEI index are s i m i l a r empiri-c a l l y but as a group are d i f f e r e n t from the other four indexes. However, the proposed IEI index has two advantages over the two IIWR indexes. F i r s t l y , i t i s j u s t i f i a b l e i n terms of s o c i a l 3 welfare evaluation. Secondly, i t captures q u a n t i t a t i v e l y 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 . Referring again to (3.23), since both S(') and TT (•) are s e n s i t i v e to p r i c e s , the IEI index i s more p r i c e - s e n s i t i v e than the other two indexes. Section 5 D i s t r i b u t i v e Price e f f e c t s The poor households, r e l a t i v e to the r i c h households, spend a larger proportion of t h e i r household budgets on the n e c e s s i t i e s . Roberts (1982) i s an attempt to show that increases 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 of the poor more than that of the r i c h . I t seems reasonable to conjecture that food p r i c e i n f l a t i o n might have a negative impact on the i n e q u a l i t y s i t u a t i o n . The new IEI index i s a valuable t o o l to demonstrate t h i s impact e m p i r i c a l l y . - 116 -To examine the p r i c e - s e n s i t i v i t y of i n e q u a l i t y , the p r i c e of each good i s r a i s e d i n turn by 10% from the 1978 l e v e l . The data set i s again the Family Expenditure Survey 1978 sample and p° i s taken as 1971 p r i c e s . The r e s u l t s are shown i n Table 6. The most i n t e r e s t i n g r e s u l t i s that an increase i n food p r i c e a c t u a l l y increases i n e q u a l i t y but increases i n r e c r e a t i o n p r i c e and transportation p r i c e a c t u a l l y decrease i n e q u a l i t y . In the l a t t e r two cases, the r i c h are hurt more by the p r i c e increases than the poor, although everyone i n society i s i n e v i t a b l y worse o f f . For example, a 10% increase i n food, recreation and transportation p r i c e s causes r e s p e c t i v e l y , for r = 0.5, 2.3% increase, 0.9% decrease, 1.6% decrease i n i n e q u a l i t y . This i s broadly consistent with the 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 and l u x u r i e s . On the other hand, p r i c e changes i n c l o t h i n g , personal/ medical care and 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 i n e q u a l i t y . This i s not s u r p r i s i n g because these consumption items are h i g h l y aggre-gated and cannot reasonably be c l a s s i f i e d as n e c e s s i t i e s or l u x u r i e s . Section 6 Inequality Trend I t has been shown i n Table 5 that the IIWR (Stat. Can.) and IIWR (est.) indexes are e m p i r i c a l l y very close to the proposed IEI index, the reason being that p° i s a r b i t r a r y and when set equal to 1978 p r i c e s , TT (•) i n the denominator i n (3.23) vanishes rendering the IIWR (est.) and IEI index i d e n t i c a l . However, i t i s p l a u s i b l e that i f one f i x e s p° and examines i n e q u a l i t y trend on a time-series T a b l e G: S e n s i t i v i t y o f I E I m e a s u r e t o p 0 % C o m m o d i t y p 4 9 1 5 . 5 6 r = . 5 «• I 4 6 7 2 . 4 3 .049461 r = 0 « I 4 4 4 1 . 7 4 .096392 F o o d 4 7 9 5 . 1 6 C l o t h i n g 4 8 7 7 . 8 5 10% R e c r e a t i o n 4 8 6 3 . 2 9 P/M C a r e 4 8 9 3 . 3 9 S h e l t e r 4 7 2 5 . 6 9 T r a n s . 4 8 3 8 . 2 2 4 5 5 2 . 5 9 .050587 4 6 3 6 . 8 9 .049399 4 6 2 4 . 6 5 .049069 4 6 5 1 . 1 3 .049507 4 4 9 1 . 6 0 .049537 4 6 0 2 . 6 9 .048682 4 3 2 2 . 7 3 .098523 4 4 0 8 . 2 8 .096266 4 3 9 8 . 1 0 .095654 4 4 2 1 . 2 6 .096482 4269.41 .096553 4 3 7 9 . 0 6 .094905 2 0 % F o o d 4 6 8 3 . 6 6 4 4 4 1 . 8 7 .051625 4 2 1 3 . 0 2 .100486 P" N o . 1971 p r i c e s o f e n t r i e s : 27651 r = -1 -2 - C O 4 0 0 9 . 6 3 .184299 3 6 0 8 . 9 4 .265813 5 6 5 . 2 4 .885011 3 8 9 2 . 8 8 3 9 8 0 . 1 7 3 9 7 3 . 4 9 3 9 9 0 . 7 3 3 8 5 3 . 0 2 3 9 5 9 . 8 4 188165 184031 182962 184464 184664 181551 3 7 8 5 . 7 2 .191718 3 4 9 5 . 0 9 .271121 3 5 8 3 . 3 2 .265390 3 5 7 9 . 5 0 .263976 3 5 9 1 . 5 4 .266042 3 4 6 6 . 6 4 .266427 3 5 7 0 . 7 5 .261971 3 3 9 1 . 0 2 .275990 5 3 5 . 5 2 .888112 5 6 4 . 2 7 . 8 8 4 3 2 0 5 6 3 . 5 6 5 6 2 . 2 6 5 3 9 . 9 6 5 6 7 . 9 8 5 1 1 . 1 2 . 8 8 4 1 2 0 .885098 . 8 8 5 7 4 0 . 8 8 2 6 0 5 . 8 9 0 8 7 0 I I - 118 -basis, the two methods might suggest mutually c o n f l i c t i n g trends. This would most l i k e l y happen i f the expenditure d i s t r i b u t i o n i s stable over time while r e l a t i v e p r i c e s are widely f l u c t u a t i n g , because TT (•) makes the IEI index more p r i c e - s e n s i t i v e . An informal t e s t can be c a r r i e d out to ascer t a i n i f t h i s conjecture holds i n Canada. Since no ad d i t i o n a l expenditure survey data are av a i l a b l e i n the time-series period, income survey data have to be used f o r trend analyses. Used here are the income survey 4 samples of 1975, 1979 and 1981. After-tax income are treated as i f they were expenditures. Furthermore, only p o s i t i v e a f t e r - t a x incomes are brought into computation. Since a f t e r - t a x income has a much higher variance than expenditure, i n e q u a l i t y measures computed here are much higher than those computed using expenditures. The IEI indexes can be found i n Table 7 and the IIWR (Stat. Can.) measures i n Table 8. As evident i n Table 7 and Table 8, both the IEI index and the IIWR (Stat. Can.) index suggest that i n e q u a l i t y i s highest i n 1979, followed by 1981 and lowest i n 1975. The trends suggested by the two indexes are consistent with one another. The probable reason i s : although the IEI index i s more s e n s i t i v e to p, the changes i n r e l a t i v e p r i c e s experienced i n Canada i n the l a s t decade are not d r a s t i c enough to allow the p r i c e index TT(*) i n (3.23) to a f f e c t the ine q u a l i t y measure so much that i t contradicts the s i m p l i s t i c IIWR (Stat. Can.) index i n a trend comparison. T a b l e 7 : I n e q u a l i t y t r e n d ( I E I m e a s u r e ) Y e a r s i z e o f No. o f r = .5 r=0 s a m p l e E n t r l e s l> I I 1975 2G5G9 7 8 6 4 0 5006 .92 4 6 5 3 . 22 .070642 4274 . .72 146238 1979 3 9 9 0 6 1 0 5 7 8 5 5 3 8 5 . .98 4 9 7 8 .82 .075600 4 5 3 2 . 98 158374 1981 4 0 3 0 8 103961 5 5 1 0 . 25 51 15 .82 .071581 4 6 9 5 . .11 147932 p° :1971 p r i c e s 3 1 4 6 . 6 9 .371532 3 0 7 4 . 3 6 .429192 3 4 1 6 . 6 7 .379943 9 2 0 . 4 8 .816159 6 3 9 . 4 3 .881279 8 0 1 . 5 4 .854537 T a b l e 8 : I n e q u a l i t y t r e n d ( I I W R ( S t a t . C a n . ) m e a s u r e ) Y e a r s i z e o f N o . o f s a m p l e E n t r i e s p 1975 265G9 7 8 6 4 0 7 3 8 6 . 0 1 1979 3 9 9 0 6 1 0 5 7 8 5 10747.35 1981 4 0 3 0 8 103961 14452.25 r= .5 « I 6 8 6 9 . 7 5 .069897 9 9 4 9 . 8 9 .074201 1 3 4 0 6 . 3 9 .072366 r = 0 5 I 6 3 1 5 . 0 3 .145001 9 0 7 2 . 9 4 .155798 1 2 2 8 7 . 1 0 .149814 r = -1 « I 4 7 0 7 . 7 2 .362617 6 2 6 4 . 4 0 .417122 8 9 2 7 . 1 7 .3 82299 r = -2 * 1 1677.28 .772911 1566.59 .854235 2 2 4 8 . 4 6 .844422 to o - 121 -Conclusion In Chapter 3, i t has been established that the proposed IEI index i s superior to a l l other e x i s t i n g indexes i n i t s s o c i a l evalua-t i o n foundation. In t h i s chapter, i t i s shown that t h i s s u p e r i o r i t y extends to the empirical scene. The empirical evidence in d i c a t e s that e m p i r i c a l l y , the IEI index e x h i b i t s a d i f f e r e n t i n e q u a l i t y scenario from those by the other major indexes. More importantly, i t shows convincingly the 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 . Food p r i c e i n f l a t i o n aggravates i n e q u a l i t y while transportation p r i c e i n f l a t i o n ameliorates i n e q u a l i t y ! Other r e s u l t s are not as c l e a r - c u t . The IIWR (Stat. Can.) and IIWR (est.) indexes (both being improvements over the Wolfson (1979) index) approximate the IEI index c l o s e l y . Even i n the dynamic sense where the i n e q u a l i t y trends i n d i c a t e d by the IEI index and the IIWR (Stat. Can.) index are compared, the p r i c e changes are not d r a s t i c enough to cause a c o n f l i c t i n trend although such a c o n f l i c t i s l i k e l y i f p r i c e changes are large enough. One cannot rule out such p r i c e changes i n the future. Therefore, considering p r i c e - s e n s i t i v i t y and the j u s t i f i a b i l i t y of the s o c i a l - e v a l u a t i o n procedure, the IEI index i s s t i l l the p r e f e r r e d index. - 122 -One can conclude that, judging from empirical evidence and the s o c i a l welfare foundation, the IEI index proposed i n Chapter 3 should be adopted i n place of a l l other e x i s t i n g indexes. - 123 -Chapter 7 Footnotes 1. S t a t i s t i c s Canada update the c u t - o f f values every year for i n f l a t i o n only so that the poverty-line r a t i o s are constant. However, major r e v i s i o n s are done a f t e r each family expenditure survey. The 1969 survey implies r a t i o s f o r 1971 - 1979, while the 1978 survey implies r a t i o s f o r 1980 - 1982. 2. Since the Atkinson index i s r e l a t i v e , only r e l a t i v e p r i c e s i n p° matter. Therefore choosing actual 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 that the i n f l a t e d welfare 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 u t i l i t y function. I t can be obtained by s o l v i n g f o r y* i n , V( y, p, A ) = V( y*, p, A° ) so that y* = C (V ( y, p, A ) , p, A° ) However, l e t t i n g , y* = C(V( y±, P l , A), p x , A°) y* = c(v( y 2 , p 2 , A ) , p 2 , A°) i t i s not true that v * > V * Y l = Y2 - 124 -i f and only i f V( P ± , A ) > V( Y 2 , p 2 , A ) unless p^ = p^. This f a i l s y* as an exact u t i l i t y i n d i c a t o r . 4. The data f i l e s are known as Economic Family Incomes, 1975; Census Family Incomes, 1979; Census Family Incomes, 1981. The difference i n family d e f i n i t i o n s i s not believed to a f f e c t measured in e q u a l i t y s i g n i f i c a n t l y . - 125 -CHAPTER 8 CONCLUSION Every economics student knows that, through the budget cons-t r a i n t , attainable u t i l i t y depends on p r i c e s . Since the r i c h consume more luxuries r e l a t i v e to n e c e s s i t i e s than the 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 l e v e l s d i f f e r e n t l y . I t then follows i n t u i t i v e l y that 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 e f f e c t s , hence a f f e c t i n e q u a l i t y . However, i t i s somewhat s u r p r i s i n g that despite the existence of empirical evidence substantiating 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 able to 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 of the e x i s t i n g indexes are calculated based on d i s t r i b u t i o n s of incomes or expenditures or some simple adjustments of the two. Two notable exceptions are the Muellbauer (1974) approach and the Jorgenson-Slesnick (1984) approach which, while being worthwhile attempts, are not completely 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 evaluation frameworks. By contrast, the present research r e s u l t s i n the establishment of a new index that i s not subject to these c r i t i c i s m s . The i n e q u a l i t y i m p l i c a t i o n of s o c i a l choice theories i s c l e a r . As a r e s u l t , the IEI 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 welfare evaluation foundation. What i s required to generate a new index i s the following: 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 numerical measure of u t i l i t y , an appro-p r i a t e s o c i a l welfare function and a formula for i n e q u a l i t y measure-ment. The second requirement above poses the greatest challenge because of the absence of an objective scale of u t i l i t y measurement and the absence of behavioral data such as i n d i v i d u a l demand data and the fac t that c e r t a i n human and environmental c h a r a c t e r i s t i c s 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 . To cope with these d i f f i c u l t i e s , the present model incorporates a t t r i b u t e parameters into the u t i l i t y function and assumes a p a r t i c u l a r l e v e l of interpersonal comparison of u t i l i t i e s , which r e s u l t i n a numerical representation of u t i l i t y . This measure i s named equivalent income. An equivalent income measure i s imputed to each i n d i v i d u a l i n society. A d i s t r i b u t i o n of equivalent incomes then form the basis of i n -equality measurement i n a w e l f a r i s t s o c i a l welfare framework. 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 of providing an i n -e q u a l i t y index that i s based on a rigorous s o c i a l welfare framework, the present research i s also marked by i t s impressive empirical r e s u l t s . Numerically, i t i n d i c a t e s a d i f f e r e n t i n e q u a l i t y scenario from those indicated by the major e x i s t i n g indexes. Apparently, the t h e o r e t i c a l m i s - s p e c i f i c a t i o n problem that plagues these indexes has turned them into unworthy empirical t o o l s . Furthermore, the proposed IEI index s u c c e s s f u l l y 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 p r i c e increases do have an aggravating impact on i n e q u a l i t y while - 127 -the opposite i s true for transportation. This f i n d i n g conforms reasonably with the common c l a s s i f i c a t i o n of luxuries and n e c e s s i t i e s . In addition, one should r e a l i z e that these are not a r t i f i c i a l mechanical r e s u l t s . Neither the translog equivalent scales s p e c i -f i c a t i o n nor any s t r u c t u r a l assumption i n the estimation model necessarily drives 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 . The most fundamental one i s that p° i n the equivalent income function i s a r b i t r a r y . In i n e q u a l i t y measurement, i t becomes an extra parameter i n a d d i t i o n to r, the degree of i n e q u a l i t y aversion. While measured in e q u a l i t y i s not very s e n s i t i v e to the choice of p° e m p i r i c a l l y , nevertheless, i t i s impossible to pinpoint p r e c i s e l y the degree of i n e q u a l i t y which creates some vagueness i n exercises such as i n t e r -country comparisons. Furthermore, the estimation model, while being quite apart from the i n e q u a l i t y measurement framework, could be improved i n several d i r e c t i o n s . F i r s t l y , to avoid simultaneous equation bias, the production side of the economy could be incorporated. The model adopted here i s a limited-information model which does not make use of supply data. Secondly, i n the s p e c i f i c a t i o n of pre-ferences, assumptions are imposed to a r r i v e at l i n e a r expenditure share equations. While they s i m p l i f y estimation, these assumptions might not be consistent with actual consumer behavior. T h i r d l y , data a v a i l a b i l i t y imposes severe constraints on the number of a t t r i b u t e s - 128 -that can be incorporated i n t o the s p e c i f i c a t i o n of preferences. The present research, for the sake of c r e d i b i l i t y , only makes use of p u b l i c l y a v a i l a b l e data. But the estimation r e s u l t s w i l l de-f i n i t e l y b enefit from a d d i t i o n a l micro data concerning other relevant a t t r i b u t e s . For example, household composition i s an a t t r i b u t e that 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 u t i l i t y . One should notice the implementation advantages of the IEI index. I t might seem that t h i s index i s very c o s t l y to implement, i n view of the complexities of the model. This i s not true. When implementing t h i s index, the a d d i t i o n a l work that i t requires i s i n improving the estimates as new demand data become a v a i l a b l e . This i s not c o s t l y because a l l the procedures involved can be executed by computer programs, the f e a s i b i l i t y of which have been demonstrated i n Chapter 6. However, the small increase i n cost gains i n return a much improved measure of i n e q u a l i t y — i n i t s s o c i a l welfare foundation, e t h i c a l s i g n i f i c a n c e and p r i c e - s e n s i t i v i t y . F i n a l l y , the scope of a p p l i c a t i o n of t h i s research i s extremely wide. On one hand, as already explained, i t gives r i s e to an i n e q u a l i t y index that can i n d i c a t e the e f f e c t s of tax and t a r i f f changes on i n e q u a l i t y . This i s a valuable p o l i c y t o o l . On the other hand, the approach to u t i l i t y measurement and s o c i a l welfare aggregation i s a p p l i c a b l e to other d i s c i p l i n e s such as cost-benefit - 129 -analyses and s o c i a l planning. This research represents an important step towards i n t e g r a t i n g s o c i a l choice theories with p r a c t i c a l p o l i c y evaluation. 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(1979) "Wealth and the d i s t r i b u t i o n of income, Canada 1968-70" International Assoc. for Research i n Income and Wealth Series 25, No. 2, June, p. 129-140. - 135 -Appendix A Cross-section Results FOOD CLOTHING RECREATION Variable coeff. t - r a t i o c o e f f . t - r a t i o coeff. t - r a t i o log y -0 .1209 -59 .02 0 .0140 13 .23 0 .0472 25 .28 A l -0 .0119 -6 .28 0 .0021 2 .12 0 .0009 0 .52 A2 -0 .0364 -14 .02 0 .0169 12 .54 0 .0006 0 .24 A 3 0 .0382 12 .66 0 .0086 5 .51 -o .0321 -11 .67 A4 0 .0636 18 .29 0 .0161 8 .95 -0 .0502 -15 .83 A5 0 .0731 19 .79 0 .0207 10 .84 -0 .0464 -13 .80 A 6 0 .1109 28 .30 0 .0293 14 .42 -0 .0560 -15 .69 A 7 0 .0002 0 .04 -0 .0095 -4 .90 -0 .0110 -3 .20 A 8 0 .0148 3 .64 -0 .0052 -2 .45 -0 .0078 -2 .10 A 9 0 .0343 8 .53 -0 .0023 -1 .09 -0 .0036 -0 .98 A i o 0 .0205 5 .18 -0 .0085 -4 .17 0 .0030 0 .84 A l l 0 .0006 0 .15 -0 .0145 -7 .05 0 .0124 3 .45 Intercept 1 .3593 70 .84 -0 .065 3 -6 .57 -o .3076 -17 .60 R 2 .3165 .1203 .0756 - 136 -Appendix A (continued) P/M CARE SHELTER TRANSPORTATION Variable coeff. t - r a t i o coeff. t - r a t i o coeff.* t - r a t i o log y -0 .0037 -5 .63 -0 .0290 -10 .53 0 .0924 35 .55 A l -0 .0030 -4 .88 -0 .0229 -8 .98 0 .0348 14 .47 A2 0 .0072 8 .60 0 .0386 11 .03 -0 .0268 -8 .12 A 3 0 .0078 8 .00 -0 .0059 -1 .45 -0 .0166 -4 .35 A4 0 .0065 5 .81 -0 .0110 -2 .36 -0 .0251 -5 .68 A 5 0 .0085 7 .16 -0 .0136 -2 .73 -0 .0424 -9 .10 A 6 0 .0088 6 .92 -0 .0357 -6 .76 -0 .0573 -11 .52 A 7 0 .0002 0 .18 0 .0473 9 .37 -0 .0272 -5 .71 A8 0 .0048 3 .63 0 .0274 5 .01 -0 .0341 -6 .60 A 9 0 .0097 7 .45 -0 .0168 -3 .10 -0 .0213 -4 .18 A i o 0 .0094 7 .37 -0 .0211 -3 .98 -0 .0032 -0 .64 A l l 0 .0046 3 .61 -0 .0022 -0 .41 -0 .0010 -0 .20 Intercept 0 .0674 10 .88 0 .6451 25 .00 -0 .6990 -28 .70 R .0369 .1123 .1727 * These are derived estimates. Appendix B  Cross-section Consumption Function coeff. t - r a t i o coeff. t - r a t i o coeff. t - r a t i o coeff. t - r a t i o coeff. t - r a t i o y A, A, 10 A l l Intercept 0.7461 131.81 -2.5252 -0.02 0.7294 123.34 0.6820 110.42 0.6986 116.11 0.7461 133.28 -1148.3 -8.50 819.90 2054.70 2983.20 3581.50 3126.3 26.95 3620.5 30.55 2410.2 5.36 11.71 16.78 19.23 350.55 801.64 24.42 -1605.1 -2550.7 18.82 4344.3 1.69 3.69 0.11 -7.26 -11.85 22.33 3125.1 30.17 A p p e n d i x C: T i m e - s e r i e s D a t a E x p e n d i t u r e S h a r e s Y e a r F o o d C l o t h i n g R e c r e a t i o n P/M C a r e S h e l t e r 1971 0 . 2 6 6 0 5 9 2 0 . 0 7 9 4 6 6 7 0 . 1 3 7 7 7 6 9 0 . 0 4 9 0 6 4 9 0 . 3 1 3 9 1 5 7 1972 0 . 2 6 3 6 3 5 4 0 . 0 7 7 8 9 2 2 0 . 1 4 3 4 5 8 7 0 . 0 4 8 6 1 8 4 0 . 3 1 1 8 0 8 8 1973 0 . 2 6 3 8 8 7 6 0 . 0 7 6 3 3 3 6 0 . 1 4 5 2 2 7 6 0 . 0 4 8 0 8 1 2 1974 0 . 2 6 0 3 9 9 2 0 . 0 8 1 6 9 0 8 0 . 1 4 6 1 4 4 1 1975 0 . 2 6 2 0 3 6 9 0 . 0 7 8 3 4 8 2 0 . 1 4 9 7 6 5 1 1976 0 . 2 5 0 1 9 3 5 0 . 0 7 7 7 2 3 0 0 . 1 5 6 8 7 9 9 0 . 0 5 0 3 3 6 9 0 . 3 0 7 8 6 2 1 1977 0 . 2 4 8 2 9 7 4 0 . 0 7 5 9 1 7 9 0 . 1 5 9 3 9 0 4 0 . 0 5 0 4 4 1 7 1978 0 . 2 5 0 0 8 6 2 0 . 0 7 4 5 5 3 0 0 . 1 5 7 9 2 8 1 0 . 0 5 1 1 0 5 7 0 . 3 1 1 0 6 4 9 1979 0 . 2 5 0 4 0 3 2 0 . 0 7 4 8 9 4 8 0 . 1 5 2 2 7 6 3 0 . 0 5 0 7 4 7 0 0 . 3 1 3 0 1 9 9 1980 0 . 2 5 0 2 9 1 7 0 . 0 7 3 2 4 6 4 0 . 1 5 2 3 5 2 2 0 . 0 5 1 4 9 8 9 0 . 3 1 5 8 8 7 4 1981 0 . 2 5 0 4 5 0 0 0 . 0 7 2 3 8 0 0 0 . 1 4 8 9 5 0 0 0 . 0 5 2 4 8 0 0 0 . 3 1 5 2 2 0 0 0 . 3 0 9 1 6 5 9 0 . 0 4 9 6 1 0 7 0 . 3 0 7 2 1 9 9 0 . 0 4 9 2 7 5 6 0 . 3 0 4 0 7 4 5 0 . 3 0 7 8 6 2 1 0 . 3 1 0 0 0 6 1 T r a n s . 0 . 1 5 3 7 1 6 3 0 . 1 5 4 5 8 6 2 0 . 1 5 7 3 0 3 8 0 . 1 5 4 9 3 4 9 0 . 1 5 6 4 9 9 4 0 . 1 5 7 0 0 4 2 0 . 1 5 5 9 4 6 3 0 . 1 5 5 2 6 1 9 i 0 . 1 5 8 6 5 8 4 to 00 0 . 1 5 6 7 2 3 2 | 0 . 1 6 0 5 2 0 0 A p p e n d i x C ( c o n t i n u e d ) P r i c e I n d e x e s Y e a r F o o d C l o t h . R e c r n . P/M C. S h e l t . 1971 0 . 5 1 4 9 0 . 6 8 3 1 0 . 6 7 4 8 0 . 6 0 1 7 0.5744 1972 0 . 5 4 6 9 0 . 7 0 0 8 0 . 6 9 3 7 0 . 6 3 0 6 0 . 6 0 1 4 1973 0 . 6 1 0 2 0 . 7 3 5 7 0 . 7 2 2 7 0 . 6 6 0 7 0 . 6 3 9 9 1974 0 . 6 9 4 1 0 . 8 0 6 0 0 . 7 8 6 1 0 . 7 1 8 4 0 . 6 9 6 2 1975 0 . 7 8 3 2 0 . 8 5 4 5 0.8671 0 . 8 0 0 2 0.7651 1976 0 . 8 1 2 0 0 . 9 0 1 6 0 . 9 1 9 0 0 . 8 6 8 2 0.8501 1977 0 . 8 7 6 9 0 . 9 6 3 1 0 . 9 6 2 9 0 . 9 3 2 6 1978 1.0000 1.0000 1.0000 1.0000 1979 1.1205 1.0922 1.0688 1.0903 1980 1.2400 1.2206 1.1707 1.1992 1981 1.3841 1.3074 1.2888 1.3297 0 . 9 2 9 9 1.0000 1.0695 1.1568 1.3004 T r a n s . 0 . 6 1 6 5 0 . 6 3 2 6 0 . 6 4 9 2 0 . 7 1 3 9 0.7978 0 . 8 8 3 5 0.9451 1.0000 1.0974 1.2374 1.4649 0J Y e a r 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 SAR 0 . 3 3 3 7 0 7 9 0 . 3 3 7 1 3 2 6 0 . 3 4 3 0 3 7 3 0 . 3 2 9 9 6 4 4 0 . 3 0 9 1 9 7 9 0 . 3 1 2 4 1 3 5 0 . 3 2 6 9 9 0 7 O . 3 2 0 3 5 5 0 0 . 3 1 2 5 6 5 4 0 . 3 0 5 6 5 7 3 0 . 3 0 2 9 5 0 3 A p p e n d i x C ( c o n t i n u e d ) E x p e n d i t u r e / A t t r i b u t e S t a t i s t i c s SOH F a m i l y S i z e O .1096137 0 . 2 0 1 3 6 7 7 0 . 1 7 1 6 4 0 6 0 . 1 9 9 2 4 9 8 0 . 1 1 6 5 1 7 7 0 . 2 0 5 3 7 7 1 0 . 1 7 0 0 9 4 3 0 . 2 0 1 8 3 3 3 0 . 1 2 1 4 1 1 4 0 . 2 1 1 3 0 5 4 0 . 1 6 9 5 5 8 5 0 . 2 0 9 4 4 2 2 0 . 1 2 0 6 9 0 1 0 . 2 1 9 4 0 1 5 0 . 1 6 6 9 1 8 2 0 . 2 1 2 7 4 9 4 0 . 1 2 1 5 9 6 4 0 . 2 2 9 2 6 2 0 0 . 1 6 7 2 5 4 8 0 . 2 1 9 1 9 7 7 0 . 1 2 4 9 3 2 6 0 . 2 2 3 8 1 1 9 0 . 1 7 5 5 7 5 1 0 . 2 2 3 7 5 9 8 0 . 1 3 0 4 9 3 0 0 . 2 2 5 7 1 2 5 0 . 1 7 6 2 7 9 2 0 . 2 2 7 1 4 8 7 0 . 1 2 8 1 5 7 4 0 . 2 3 0 6 6 2 0 0 . 1 8 1 8 3 0 1 0 . 2 2 4 9 7 6 7 0 . 1 3 6 0 8 9 4 0 . 2 3 6 4 7 8 5 0 . 1 8 3 2 7 8 0 0 . 2 2 5 1 6 6 9 0 . 1 3 3 8 7 3 5 0 . 2 4 7 3 1 6 3 0 . 1 8 7 1 6 2 4 0 . 2 2 0 9 0 4 1 0 . 1 3 8 4 5 8 8 0 . 2 4 6 6 7 2 1 0 . 1 8 9 9 5 7 5 0 . 2 3 2 3 4 9 4 0 . 2 9 2 6 3 7 2 0 . 2 8 3 1 2 9 3 O . 2 6 8 3 0 4 0 0 . 2 5 3 9 1 5 4 0 . 2 3 6 6 3 1 8 0 . 2 2 8 7 2 3 7 0 . 2 2 1 9 2 3 0 0 . 2 0 8 0 3 9 0 O.1951508 0 . 1 8 4 3 0 0 5 O.1750336 0 . 2 2 6 8 6 5 1 O.2304947 0 . 2 3 6 6 6 5 4 0 . 2 4 5 9 2 1 9 0 . 2 5 7 9 6 3 5 0 . 2 5 6 6 2 2 2 0 . 2 5 5 0 1 4 9 O .2540106 0 . 2 5 6 1 9 7 7 0 . 2 5 1 5 9 0 3 0 . 2 4 7 9 3 8 8 Age 0 . 2 4 8 6 5 0 9 0 . 0 . 2 4 4 9 6 4 1 0 . 0 . 2 3 8 2 7 4 2 O. 0 . 2 3 5 4 5 0 8 0 . 0 . 2 3 1 1 3 1 0 O. 0 . 2 2 6 6 4 0 3 0 . 0 . 2 2 1 1 7 5 9 0 . 0 . 2 2 5 8 7 6 0 0 . 0 . 2 2 7 5 6 1 2 O. 0 . 2 3 0 5 6 7 6 0 . 0 . 2 3 4 5 8 7 1 0 . o f H e a d 2 1 6 8 1 1 0 215631 1 2 1 5 0 1 3 6 2 0 8 2 9 6 2 2 0 5 1 6 0 6 2 0 9 3 4 0 9 2 1 4 4 7 3 6 2 1 0 6 1 1 2 2 0 3 2 4 2 3 1990146 2 0 0 8 2 0 2 0 . 1 3 O 3 1 9 0 0 . 1 3 0 4 7 1 6 0 . 1317976 0 . 1292976 0 . 1 2 6 6 7 8 0 0 . 1 3 0 8 2 7 4 0 . 1 3 3 8 6 2 4 0 . 1 3 5 5 0 6 0 O . 1 3 6 4 6 2 5 0 . 1 4 2 0 2 7 8 0 . 1 4 3 2 1 0 2 0 . 0 8 6 1 0 4 7 0 . 0 9 2 0 8 4 5 0 . 0 9 3 5 0 2 8 0 . 0 9 6 2 1 8 1 0 . 0 9 4 0 3 6 8 0 . 0 9 2 6 5 8 9 0 . 0 8 9 9 0 9 5 0 . 0 9 5 3 0 1 5 0 . 0 9 8 3 1 3 5 0 . 1 0 5 4 0 9 2 0 . 1 0 4 5 2 6 8 O A p p e n d i x C ( c o n t i n u e d ) Y e a r Sy y t 1971 9 . 1 6 7 8 3 8 1 0 5 6 4 6 2 5 8 1972 9 . 2 2 3 2 7 7 1 0 5 7 6 7 0 6 5 1973 9 . 3 1 2 9 3 5 8 0 5 8 1 3 4 8 9 1974 9 . 4 2 4 3 3 9 3 0 6 0 7 6 6 8 6 1975 9 . 5 1 6 7 9 8 0 0 6 2 2 7 1 5 4 1976 9 . 6 1 1 7 0 7 7 0 6 3 0 3 1 7 0 1977 9 . 6 8 2 6 0 6 7 0 6 4 4 2 3 4 1 1978 9 . 7 5 3 5 1 2 4 0 6 6 1 7 8 3 5 1979 9 . 8 0 9 0 0 6 7 0 6 8 1 3 2 5 7 1980 9 . 9 2 7 2 6 1 4 0 6 6 3 2 3 2 5 1981 1 0 . 0 2 9 2 6 4 5 0 6 9 3 2 6 0 5 h-1 I - 142 -Appendix D Time-series Results, B PP Food Clothing Recreation P/M Care Shelter Trans* Food -0.11148 -0.02224 -0.04682 0.00026 (-4.6197) (-0.9363) (-1.7898) (0.0069) 0.14293 -0.08358 (5.5466) Clothing -0.02224 0.16707 (-0.6298) (1.9605) -0.15309 -0.04365 -0.11494 (-1.8222) (-0.3949) (-1.7470) 0.18089 Recreation -0.04682 -0.15 309 (-0.3139) (-0.4752) -0.03234 0.06491 0.48122 -0.26672 (-0.0853) (0.1506) (1.9837) P/M Care 0.00026 -0.04365 (0.0066) (-0.5615) 0.06491 -0.06829 0.11340 (0.8211) (-0.3026) (1.0731) -0.07037 Shelter 0.14293 -0.11494 0.48122 0.11340 -0.98008 (0.9977) (-0.4672) (2.0343) (0.2018) (-2.9720) 0.32847 Trans. -0.08358 0.18089 -0.26672 -0.07037 0.32847 0.00374 * Estimates i n the l a s t row and l a s t column are derived estimates, (See Chapter 5) Numbers i n parentheses are the t - r a t i o s Time-series: 1971 - 1981 A p p e n d i x E URBAN MALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW B a r t e n E q u i v a l e n c e S c a l e s FAM. S I Z E 1 2 3 4 5 FOOO 1.OOOO 1.3836 1.7147 1.8045 2.3943 CLTH 1.0000 1.3831 1.6748 2.9199 4 .9348 URBAN MALE HEAD HOUSEHOLD HEAD AGED 24-34 FAM. S I Z E FOOO 1.0388 1 .4374 1 .7813 1 .8745 2 . 4 8 7 3 MALE HEAD CLTH 1.0250 1.4176 1.7166 2.9928 5 . 0 5 8 0 URBAN HOUSEHOLD HEAD AGED 34-44 FAM. S I Z E FOOD 1 1.1342 2 1.5693 2 3 1.9448 2 4 2.0466 4 5 2.7156 7 URBAN MALE HEAD HOUSEHOLD HEAD AGED 4 4 - 5 4 FAM. S I Z E FOOD CLTH 1.5474 .1403 .5916 . 5 1 8 3 .6364 1 1.2838 2 1.7763 2 3 2.2012 2 4 2.3165 4 5 3.0738 8 URBAN MALE HEAD HOUSEHOLD HEAD AGED 54-64 FAM. S I Z E FOOD CLTH 1.6544 .2882 .7708 .8306 B.1642 URBAN 1.1565 1.6003 1.9831 2.086 9 2.7692 MALE HEAD CLTH 1.0419 1 .4410 1 .7449 3.0421 5 . 1414 HOUSEHOLD HEAD AGED OVER 64 FAM. S I Z E 1 2 3 4 5 FOOD 0 . 9 9 4 9 1.3765 1.7059 1.7952 2 . 3 8 2 0 CLTH 0 . 8 3 8 2 1.1593 1.4037 2.4473 4.1362 RCRN 1.0000 1.4226 1.8084 2.4953 3 . 9 3 1 0 RCRN 0.9884 1.4061 1.7874 2.4664 3.8854 RCRN 1.2937 1.8403 2.3395 3.2281 5.0854 RCRN 4541 0 6 8 6 6296 6284 7 1 6 0 RCRN 1.0682 1 .5195 1 .9317 2.6654 4 . 1 9 8 9 RCRN 8682 2351 5701 1665 4 1 3 0 MEDC 1.0000 1.6080 2.0387 2.6513 3.9552 MEDC 1 .0196 1.6396 2.0787 2.7033 4.0329 MEDC 1 .3196 2 . 1220 2.6903 3.4987 5.2193 MEDC 1.5298 2.4599 3.1187 4.0558 6 . 0 5 0 6 MEDC 1 . 1733 1.8868 2.3921 3.1109 4.6408 MEDC 0. 9 1 0 3 1.4638 1.8559 2.4135 3.6005 SHTR 1.0000 1.4035 1.7459 1.8894 2.5028 SHTR 1.0756 1.5095 1.8778 2.0322 2.6919 SHTR 1.1777 1.6529 2.0561 2.2252 2.9475 SHTR 1.2771 1 .7925 2.2297 2.4131 3.1964 SHTR 1.1161 1.5664 1.9485 2.1088 2.7933 SHTR 0 . 9 6 3 9 1.3529 1.6829 1.8213 2.4125 TRAN 1.0000 1.3787 1.6960 1 .4648 1 . 7 2 3 5 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 . 0 5 3 5 1 .7735 2.0868 TRAN 1.0799 1 .4888 1 .8315 1 .5818 1 .8612 A p p e n d i x E URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW FAM. S I Z E FOOD CLTH RCRN 1 0 . 7 3 7 4 2 . 5 2 7 3 1.3832 2 1.0203 3.4955 1.9677 3 1.2644 4 . 2 3 2 7 2.5013 4 1.3306 7.3794 3.4515 5 1.7656 12.4718 5.4372 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 24-34 FAM. S I Z E FOOD CLTH RCRN 1 0 . 7 6 6 0 2.5904 1 .3672 2 1.0599 3.5828 1.9449 3 1.3135 4.3384 2.4724 4 1.3823 7.5636 3.4115 5 1.8341 12.7832 5.3742 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 34-44 FAM. S I Z E FOOD CLTH RCRN 1 0 . 8 3 6 3 3.9109 1 .7894 2 1 . 1572 5.4091 2.5455 3 1.4340 6 . 5 4 9 9 3.2359 4 1.5091 11.4192 4.4651 5 2.0025 19.2994 7.0340 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 4 4 - 5 4 FAM. S I Z E FOOD CLTH RCRN 1 0 . 9 4 6 6 4.1812 2.0113 2 1.3098 5 . 7 8 3 0 2.8612 3 1.6232 7.0026 3.6372 4 1.7082 1 2 . 2 0 8 5 5.0188 5 2.2666 2 0 . 6 3 3 4 7.9063 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED 5 4 - 6 4 FAM. S I Z E FOOD CLTH RCRN 1 0.8528 2.6331 1.4775 2 1.1800 3.6418 2.1018 3 1.4623 4 . 4 0 9 9 2.6718 4 1.5389 7.6883 3.6867 5 2.0419 12.9939 5.8078 URBAN FEMALE HEAD HOUSEHOLD HEAD AGED OVER 64 FAM. S I Z E FOOD CLTH RCRN 1 0 . 7 3 3 6 2.1183 1.2009 2 1.0150 2.9298 1.7084 3 1.2579 3.5477 2.1718 4 1.3237 6.1851 2.9967 5 1.7565 10.4534 4.7208 ( c o n t i n u e d ) MEDC 1.3608 2.1882 2.7742 3.6078 5.3822 SHTR 0.8447 1.1856 1.4748 1.5961 2 . 1 142 TRAN 0 . 4 7 4 6 .6543 .8049 .6951 .8179 0 . 0 . 0 . 0 . MEDC 1.3875 2.2312 2.8287 3.6787 5.4879 SHTR 0 . 9 0 8 6 1.2752 1.5863 1.7167 2 . 2 7 4 0 TRAN 51 19 7057 8681 7 4 9 8 8 8 2 2 MEDC 1.7957 2.8875 3.6609 4.7609 7.1024 SHTR 0.9948 1.3962 1.7369 1.8797 2.4899 TRAN 0 . 4 8 8 8 0 . 6 7 3 9 0 . 8 2 9 0 0 . 7 1 5 9 0.8424 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 . 7 6 2 5 .9381 .8101 .9533 0 . 0 . O. 0 . MEDC 1.5967 2.5675 3.2551 4.2332 6.3151 SHTR 0 . 9 4 2 8 1.3232 1 .6460 1.7814 2.3596 TRAN 0 . 5746 0 . 7 9 2 2 0 . 9 7 4 5 0 . 8 4 1 6 0 . 9 9 0 3 MEDC 1.2387 1.9919 2.5254 3.2843 4.8995 SHTR 0 . 8 1 4 3 1 . 1428 1 .4216 1.5385 2 . 0 3 8 0 TRAN 0 . 5 1 2 5 0 . 7 0 6 5 0 . 8 6 9 2 0 . 7 5 0 6 0 . 8 8 3 3 A p p e n d i x E RURAL MALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW FAM. S I Z E FOOD CLTH RCRN 1 0 . 9 1 6 5 0 . 5 4 3 7 0.7126 2 1.2681 0 . 7 5 2 0 1.0138 3 1.5715 0 . 9 1 0 5 1 .2887 4 1.6538 1.5874 1.7782 5 2.1944 2 . 6 8 2 9 2.8013 RURAL MALE HEAD HOUSEHOLD HEAD AGED 24-34 FAM. S I Z E FOOD CLTH RCRN 1 0.9521 0 . 5 5 7 2 0.7044 2 1.3174 0 . 7 7 0 7 1.0020 3 1.6325 0 . 9 3 3 3 1.2738 4 1.7180 1.6271 1.7576 5 2.2796 2 . 7 4 9 9 2.7689 RURAL MALE HEAD HOUSEHOLD HEAD AGED 34-44 FAM. S I Z E FOOD CLTH RCRN 1 1.0395 0 . 8 4 1 3 0.9219 2 1.4383 1.1636 1.3115 3 1.7824 1.4090 1.6672 4 1.8757 2 . 4 5 6 5 2.3005 5 2 . 4 8 8 9 4 . 1 5 1 7 3.6240 RURAL MALE HEAD HOUSEHOLD HEAD AGED 44-54 FAM. S I Z E FOOD CLTH RCRN 1 1 . 1766 0 . 8 9 9 5 1.0362 2 1.6280 1.2440 1.4741 3 2 . 0 1 7 5 1.5064 1.8739 4 2.1231 2 . 6 2 6 3 2.5857 5 2.8171 4 . 4 3 8 6 4.0734 RURAL MALE HEAD HOUSEHOLD HEAD AGED 54-64 FAM. S I Z E FOOD CLTH RCRN 1 1.0600 0 . 5 6 6 4 0 . 7 6 1 2 2 1.4666 0 . 7 8 3 4 1.0829 3 1.8175 0 . 9 4 8 6 1.3766 4 1.9127 1.6539 1.8994 5 2 . 5 3 8 0 2 . 7 9 5 2 2.9923 RURAL MALE HEAD HOUSEHOLD HEAD AGED OVER 64 FAM. S I Z E FOOD CLTH RCRN 1 0 . 9 1 1 8 0 . 4 5 5 7 0.6187 2 1.2616 0 . 6 3 0 3 0 . 8 8 0 2 3 1.5634 0 . 7 6 3 2 1.1189 4 1.6453 1.3305 1.5439 5 2 . 1832 2.2487 2.4322 ( c o n t f n u e d ) MEDC 0.7 4 2 5 1 . 1940 1.5138 1.9687 2.9369 SHTR 0 . 8 7 5 8 1.2291 1.5290 1.6547 2.1919 TRAN 1.1027 1.5203 1.8702 1 .6152 1.9005 MEDC 0.7571 1.2175 1.5435 2.0073 2.9946 SHTR 0 . 9 4 2 0 1.3220 1.6445 1.7798 2.3575 TRAN 1.1894 1.6398 2.0172 1.7422 2.0499 MEDC 0. 9 7 9 9 1.5756 1.9976 2.5979 3.8755 MEDC 1.1359 1.8266 2.3158 3.0116 4.4928 SHTR 1.0314 1 .4475 1.8007 1 .9487 2.5814 SHTR 1 . 1 185 1.5698 1.9528 2.1133 2.7994 TRAN 1.1357 1.5658 1.9262 1.6636 1.9574 TRAN 1 .2852 1 .7719 2.1797 1.8825 2 . 2 1 5 0 MEDC 0.87 1 2 1 .4010 1.7762 2.3099 3.4460 SHTR 0 . 9 7 7 4 1.3718 1.7065 1.8468 2.4463 TRAN 1.3351 1.8408 2.2644 1.9557 2.3012 MEDC 0. 6 7 5 9 1.0869 1.3780 1.7921 2.6735 SHTR 0 . 8 4 4 2 1.1848 1.4739 1.5951 2.1128 TRAN 1.1908 1.6418 2.0196 1.7442 2.0524 A p p e n d i x E RURAL FEMALE HEAD HOUSEHOLD HEAD AGED 24 OR BELOW FAM. S I Z E FOOD CLTH RCRN 1 0.6758 1.3740 0.9857 2 0.9351 1.9004 1.4022 3 1.1588 2.3012 1.7825 4 1.2195 4 . 0 1 2 0 2.4596 5 1.6181 6 . 7 8 0 6 3.8747 RURAL FEMALE HEAD HOUSEHOLD HEAD AGED 24-34 FAM. S I Z E FOOD CLTH RCRN 1 0.7021 1.4083 0 . 9 7 4 3 2 0.9714 1.9479 1.3860 3 1.2038 2.3586 1.7619 4 1.2668 4.1121 2.4311 5 1.6810 6 . 9 4 9 9 3.8299 RURAL FEMALE HEAD HOUSEHOLD HEAD AGED 34-44 FAM. S I Z E FOOD CLTH RCRN 1 0 . 7 6 6 5 2 . 1 2 6 2 1 .2752 2 1.0606 2.9408 1.8140 3 1.3143 3 . 5 6 1 0 2 . 3 0 6 0 4 1.3831 6 . 2 0 8 3 3.1819 5 1.8353 10.4926 5.0126 RURAL FEMALE HEAD HOUSEHOLD HEAD AGED 44-54 FAM. S I Z E FOOD CLTH RCRN 1 0 . 8 6 7 6 2.2732 1.4333 2 1.2004 3 . 1 4 4 0 2 . 0 3 9 0 3 1.4876 3.8071 2 . 5 9 2 0 4 1.5655 6 . 6 3 7 4 3.5766 5 2.0773 11.2178 5.6343 RURAL FEMALE HEAD HOUSEHOLD HEAD AGED 54-64 FAM. S I Z E FOOD CLTH RCRN 1 0.7816 1.4315 1.0529 2 1.0815 1.9800 1.4978 3 1.3402 2 . 3 9 7 5 1.9040 4 1.4104 4 . 1 7 9 9 2.6273 5 1.8715 7.0644 4 . 1 3 8 9 RURAL FEMALE HEAD HOUSEHOLD HEAD AGED OVER 64 FAM. S I Z E FOOD CLTH RCRN 1 0 . 6 7 2 3 1.1517 0 . 8 5 5 8 2 0 . 9 3 0 3 1.5928 1.2175 3 1.1529 1.9288 1.5477 4 1.2132 3.3627 2.1356 5 1.6098 5.6832 3.3642 ( c o n t 1 n u e d ) MEDC 1.0104 1.6248 2 . 0 6 0 0 2 . 6 7 9 0 3.9965 SHTR 0.7398 1.0383 1.2916 1.3978 1 .8516 TRAN O. 5 2 3 3 0 . 7 2 1 5 0 . 8 8 7 5 0 . 7 6 6 5 0 . 9 0 1 9 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 . 5 6 4 4 0 . 7 7 8 2 0 . 9 5 7 3 0 . 8 2 6 8 0 . 9 7 2 8 MEDC 1.3334 2 . 1441 2.7183 3.5352 5.2738 SHTR 0 . 8 7 1 3 1.2228 1 .521 1 1 .6462 2.1806 TRAN 0 . 5 3 9 0 0.7431 0.9141 0 . 7 8 9 5 0 . 9 2 8 9 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 . 6 0 9 9 0 . 8 4 0 9 1.0344 0.8934 1.0512 MEDC 1.1856 1.9065 2.4170 3. 1433 4.6892 SHTR 0.8257 1.1588 1.4415 1.5601 2.0665 TRAN 0 . 6 3 3 6 0 . 8 7 3 6 1.0746 0.9281 1.0920 MEDC 0.9198 1 .4791 1 .8752 2.4387 3.6381 SHTR 0.7131 1.0009 1 . 2 4 5 0 1.3474 1.7848 TRAN 0.5651 0.7791 0 . 9 5 8 4 0 . 8 2 7 7 0 . 9 7 4 0 

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