AGE-SEX EQUIVALENT SCALES
FOR UNITED STATES FOOD
EXPENDITURES—THEIR
COMPUTATION AND
APPLICATION

THESIS FOR THE DEGREE
OF PhD.
MICHIGAN STATE UNIVERSITY

By
DAVID WILLIAM PRICE
1965

 
    
    

  

TH ESlS

LIBRAR y

University

"I

l . .
I.__: I‘“

This is to certify that the

thesis entitled

Age-Sex Equivalent Scales for United States
Food Expenditures - Their Computation and
Application

presented by

David William Price

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Agricultural Economics

 

Date 196 5

0-169

(Jr‘
5'3".

2'
~..

figs-‘1

C ‘3

‘x

i

g “n
c

ROOM USE CHE)"

TH E!

 

 

ABSTRACT

AGE-SEX EQUIVALENT SCALES FOR UNITED STATES
FOOD EXPENDITURES--THEIR COMPUTATION
AND APPLICATION

by David William Price

Traditionally food expenditures have been expressed on a per capita
or a per household basis. Per capita expenditures do not reflect the dif-
ferences in the amount spent for food consumed by persons of differing ages
and sex. Consumption per household is likewise misleading, since much var-
iation in this variable is due to differences in the age, sex, and number
of members per household. This study will be concerned with estimating the
effect of age-sex differences upon food expenditures by constructing an age-
sex equivalent scale. This scale expresses the food expenditure of each.
individual age-sex type as a proportion of some "standard" type.

The problems involved in computing a scale based on actual expend-
itures and with the limitations that such a scale may have in expressing
the effects of age-sex composition have led to few attempts to estimate
them. It was the purpose of this study to explore the problems of come
puting an age-sex equivalent scale for United States food expenditures and
to evaluate the use of such a scale.

For the purpose of examining the nature of the variation in food
expenditures to be explained by the scale andto obtain coefficients to
be used in computing the scale, estimates of the income-food expenditure
relationship were made with age-sex composition held constant. Thirteen

household types, in which age-sex composition was approximately constant

David William Price

2
were selected from the urban sample of the 1955 United States Department
of Agriculture's Survey of Food Consumption of United States Households.
The income-food expenditure relationship was estimated for each household
type with the number of meals eaten at home used as an additional variable.
It was found that with the double-log form of this relationship, the in-
come elasticities and the residual variances differed significantly among
the household types. Therefore, for some purposes a scale may not be
adequate for specifying the effects of age-sex composition. However, for
many purposes its use may be an improvement over present methods.

To compute a scale, food expenditures were expressed as a function
of the number of persons in each of the eight age-sex types used in this
analysis. The ratios of the regression coefficients formed the scale. In
previous studies food expenditures had been adjusted for variations in in-
come by using a single income coefficient which was estimated by holding
age-sex composition constant. Since the income coefficients were found
to vary with household type it was necessary to generalize the procedure
so that an income coefficient for each type could be used to adjust food
expenditures. In addition, the number of meals was used as a second
"adjusting" variable. For the purpose of comparison, scales were esti-
mated with an adjustment using a single income coefficient and with a
coefficient for each household type. Scales were also estimated with and
without the adjustment for the number of meals.

In an application of the estimated scales to the income-food ex-
penditure relationship using the same cross-sectional data, the scales
gave better results in terms of the size of the coefficient of multiple
determination than did a per capita specification of food expenditures.

The scales estimated with the income coefficient for each household type

THE

 

 

David William Price

3

gave better results than those estimated with a single coefficient and the
scales estimated with the adjustment for the number of meals gave better
results than those without this adjustment.

The problem of applying a scale estimated from cross-sectional data
to time series data was examined. The algebraic formulation for the ap-
plication of a scale to time series data is not equivalent to the formula-
tion from which the scales were computed. The direct application of the
scales to United States time series data, therefore, biased the food expendi-
ture variable. However, in a direct application to the time series data
the scales gave better results in terms of the size of the coefficient of

correlation than a per capita specification of food expenditures.

TH!

 

 

AGE-SEX EQUIVALENT SCALES FOR UNITED STATES
FOOD EXPENDITURES--THEIR COMPUTATION

AND APPLICATION

BY

David William Price

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1965

ACKNOWLEDGMENTS

The author expresses his appreciation to Dr. James T. Bonnen who
suggested this topic and who provided a much needed source of inspiration
and dedication as well as advice for this study.

Dr. Lester V. Manderscheid gave advice freely and willingly on
the statistical portions of this study. His help was greatly appreciated.

The author wishes to thank the Washington State University Com-
puting Center for providing free computer time for this study. The help
of Mr. T. A. Van Wormer in writing the computer programs was also appre-
ciated.

The author acknowledges the Consumer and Food Economics Research
Division of the United States Department of Agriculture for providing the
data for this study.

Thanks are also due Dr. L. L. Boger and other members of the De-
partment of Agricultural Economics for their encouragement and inspiration

during the author's doctoral program.

ii

THE

 

 

TABLE OF CONTENTS

ACKNOWLEDGWNTS O O O O O O O O O O O O O O I O O O O O O O O O O 0

LI ST OF TABLES O O O O O O O O O O O O O O O O O O O O O O O O O 0

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter

I.

II.

III.

IV.

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . .

The Concept of the Age-Sex Equivalent Scale . . . . . . .

Uses and Importance of an Age-Sex Equivalent Scale . . . .
Objectives and Procedures . . . .,. . . . . . . . . . . . .
The Age-Sex Equivalent Scale as a Normative Concept . . . .

REV IEW OF LITERATURE C C O O O O O C O O O O O O C C O C O

The Development of the Procedure for Estimating Scales . .
The Equivalent Scale Hypothesis . . . . . . . . . . . . . .

ANALYSIS OF THE INCOME-FOOD EXPENDITURE RELATIONSHIP WITH
AGE-SEX COMPOSITION HELD CONSTANT o o o o o o a o o o o 0

Purpose of the Analysis . . . . . . . . . . . . . . . . . .
The Data . . . . . . . . . . . . . . . . . . . . . . . . .
The Formation of Household Types . . . . . . . . . . . . .
The Income-Food Expenditure Relationship within Household
Types . . . . . . . . . . . . . . . . . . . . . . . . . .
The Income-Food Expenditure Relationship between Household
Types . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

AN ALTERNATIVE METHOD OF SPECIFYING THE EFFECTS OF AGE-SEX
COMPOSITION US INC SCALES O O O C O O O O O O O O C O O O

The Conceptual Framework . . . . . . . . . . . . . . . . .
Estimates of the "Within-Type" Coefficients for Three

Alternative Models . . . . . . . . . . . . . . . . . . .
The Computation of an Age-Sex Equivalent Scale for

Food Expenditures . . . . . . . . . . . . . . . . . . . .
The Results . . . . . . . . . . . . . . . . . . . . . . . .
Reliability of the Estimated Scale Values . . . . . . . . .
Food Expenditure Scales with an Alternative Income Scale .
Summary and Conclusions . . . . . . . . . . . . . . . . . .

iii

Page

ii

vii

\IbNr-d

ll

11
23
3O
30
32
36
38
46
52
56
56
63
66
71
75

78
84

 

TH!

 

 

TABLE OF CONTENTS (Continued)

Page
V. EVALUATION OF THE AGE-SEX EQUIVALENT SCALES . . . . . . . . . . 85

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 85
An Application of the Food Expenditure Scales to Cross-

Sectional Data . . . . . . . . . . . . . . . . . . . . . . . 87
The Application of the Age-Sex Equivalent Scales to Time

Series Analysis . . . . . . . . . . . . . . . . . . . . . . . 95
Summary and Conclusions . . . . . . . . . . . . . . . . . . . . 107

VI. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . . . . . . . 109

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 109
The Nature of the Effects of Varying Age-Sex Composition

on United States Food Expenditures . . . . . . . . . . . v . 110
Computing an Age-Sex Equivalent Scale for Food Expenditures . . 114
Evaluation of the Age-Sex Equivalent Scales for United

States Food Expenditures . . . . . . . . . . . . . . . . . . 119
Limitations and Suggestions for Further Research . . . . . . . 124
Applications and Importance . . . . . . . . . . . . . . . . . . 127

BIBLIOGRAPHY O O O O I O O O O O O O O O O O O O O O O O O O O O O C O 130

iv

Table

10

11

12

13

14

15

LIST OF TABLES

Description of the Low-Income Households Eliminated from the
samp 1e 0 0 O O O O O O O O O O O O O O O O O O O O O O O O O O 0

Description of Household Types . . . . . . . . . . . . . . . . .
Scale Used to Adjust the Number of Meals Variable . . . . . . .

Description of the Food Expenditure-Income and Number of Meals
Relationship (Double-Log Form) . . . . . . . . . . . . . . . . .

Description of the Food Expenditure-Income and Number of Meals
RelationShip (semi-Log Form) 0 o o o o o o o o o o o o o o o 0

"Within Type" Estimates of the Income Elasticities and Constant
Terms for Three Alternative Models . . . . . . . . . . . . . . .

Description of the Individual Age-Sex Groups for Which Scales
were Computed O O O O O O O 0 O O O O O O O O O O O O O O O C 0

Correlation Matrix of the Independent Variables used in
computing scales 0 O I O O O O O O . C O O .7. ‘0 O C O O O O O O O 0
Description of the Four Models Used to Compute Age-Sex

Equivalent Scales . . . . . . . . . . . . . . . . . . . . . . .

Estimates of the Age-Sex Equivalent Scales for Food Expenditures
for Four Alternative Models using Per Capita Income . . . . . .

Estimates of the Regression Coefficients Expressing the Effects
of Age-Sex Composition for Four Models with Standard Errors . .

Estimates of the Standard Errors of the Age-Sex Equivalent
scales for Four MOdelS O O O O O O O O O O O O O O O O O O O O 0

"Estimates" of Specific Scales for Nine Expenditure Categories
and the Assumed Income Scale . . . . . . . . . . . . . . . . . .

Estimates of Age-Sex Equivalent Scales for Food Expenditures
for Three Models with Assumed Values for the Income Scale . . .

Results of Applying the Six Food Expenditure Scales and the
Per Capita Specification to the Income-Food Expenditure
Relationship using Cross-Sectional Data . . . . . . . . . . . .

Page

36
39

41

43

45

65

68

70

72

73

76

77

80

81

91

LIST OF TABLES (Continued)

Table Page
16 The Age-Sex Equivalent Scales used for the Analysis of
Time series Data 0 O O O O O O O O O O O O O O O O O O O O O O O 103
17 Results of Applying Three Scale Mbdels and the Per Capita
. . . . . 104

Deflation Procedure to Time Series Data .

vi

Figure

LIST OF FIGURES

Page

The Double-Log Form of the Food Expenditure-Income
RalationShip O O O O O O O I O O O O I I O I O O O O O O O O O 0 O 53

The Double-Log Form of the Food Expenditure-Income
Relationship with a Single Income Elasticity for all
Household Types . . . . . . . . . . . . . . . . . . . . . . . . . 54

vii

CHAPTER I
INTRODUCTION

The Concept of the Age-Sex Equivalent Scale

Tradiationally expenditures on specific commodities have been ex-
pressed on a per capita or per household basis. Per capita expenditures
do not reflect the differences in the amounts spent for commodities con-
sumed by persons of differing ages and sexes. For example, children under
12 years of age presumably do not consume tobacco products. For this particu-
lar product consumption per adult is a much more meaningful variable than
is per capita consumption. Consumption per household is likely misleading.
For example, consider a household composed of two adults and two children,
and one composed of four adults and no children. Households of the latter
type would be expected to have a higher average tobacco product consumption
than would households of similar size but which contain no more than two
adults.

In general, many differences exist in the amount Spent for commodi-
ties consumed by individuals of different age-sex composition. This study
will be concerned with estimating the effect of age-sex differences upon
food expenditures. The way in which the effects of these differences will
be measured Will be to construct an age-sex equivalent scale.1 This scale
expresses the food expenditure of each age-sex type as a proportion of
some "standard" type. If the adult male is taken as the standard, the

scale would be constructed by dividing the expenditure allocated to each

 

1The terms unit equivalent and adult equivalent have also been used
to designate this concept.
' l

2
of the various age-sex types by the expenditure allocated to the adult
male. Thus, it is the ratios of expenditures which will actually be es-
timated.

This concept can be applied to any commodity. If age-sex equiva-
lent scales are estimated for all commodites and these scales are com-
bined by weighting each one in proportion to the relative amount spent for
that commodity by the population under consideration, the result may be
termed an "income" or "level of living" scale. If, for example, a scale
was estimated for durable goods, non-durable goods, and services, these
three scales could be combined to estimate the proportion of household income

allocated to each age-sex type.

Uses and Importance'of an AgeJSex Equivalent Scale

The main objective of this study will be to estimate an age-sex
equivalent scale for food expenditures. No attempt will be made to esti-
mate a scale for other commodities or for all commodities (a level of liv-
ing scale). The scale for food expenditures will be based on actual food
expenditures allocated to individuals of varying age-sex composition.

With a scale of this type improvements in food expenditures data
are made possible. Food expenditure per equivalent adult is a more pre-
cise measurement than is food expenditure per capita or per household.
If, for example, a child under six years of age requires one half as much
food expenditure as an adult, the use of the age-sex equivalent scale
gives a much more accurate comparison of food expenditures between two
households, one consisting of two children and two adults, and the other
consisting of four adults. If the first household spent $30 per week for
food and the second household spent $40 per week for food, the expendi-

ture per equivalent adult would be $10 per week for each household. How-

TH

 

l

 

 

3

ever, per capita expenditures would be $7.50 for the household with chil-
dren and $10 for the other household. The use of the age-sex equivalent
scale, therefore, makes possible a more valid comparison of food expendi-
tures between households of varying age-sex composition. The same con-
clusion can be reached for groups of households. Therefore, with age-sex
compositional differences allowed for, more valid comparisons of food ex-
penditures can be made among, for example, households in different geo-
graphic regions, households of different ethnic background, households of
different social class, households with different income levels and house-
holds classified on any other basis.

The more explicit specification of food expenditures with scales
should prove to be valuable in any‘analysis where food expenditure data
is used. This includes both analyses with time series data and with cross-
sectional data. Changes in the proportion of persons of various ages have
been evident in the United States in recent years. Increases in life ex-
pectancy, the low birth rate during the depression of the 1930's and the
high birth rate following World War II have changed the age composition of
the populatibn of the United States. Any analysis of food expenditures
over time would be improved by specifying the effects of age composition.
An age-sex equivalent scale is one method of making such a specification.

The variation in the age composition of the population also af-
fects any analysis of demand. The quantity of food demanded per "equiva-
lent adult" is a more precise concept than is quantity per capita or per
household. For an economic analysis of the quantity of all food (though
not necessarily for individual food products) a price-weighted index is

2
more desirable than a quantity measure-constructed with nutritional weights.

 

2Marguerite C. Burk, Measures and Procedures for Analysis of U. S.
Food Consumption, U. S. Department of Agriculture, Agriculture Handbook No.
206 (Washington: U. 8. Government Printing Office, 1961) p. 12.

 

 

The pri
fore a

measure
a basis

sectiona

tion of
provides
with dif

stamp p1;

 

I
equivalen
penditure
authOf is

Uni ted St

4
The price-weighted index is essentially a food expenditure measure. There-
fore a scale based on food expenditures can be applied directly to this
measure of quantity. For all food, the age-sex equivalent scale thus gives
a basis for improved analyses of demand in both time series and cross-
sectional studies.

Under certain assumptions, which will be analyzed in the last sec-
tion of this chapter, an age-sex equivalent scale for food expenditures
provides a basis for allocating food welfare allotments among individuals
with different age-sex characteristics. Present programs such as the food

stamp plan are examples of programs where such a scale could be utilized.

Objectives and Procedures

 

The main objective of this study will be to estimate an age-sex
equivalent scale for food expenditures using United States data. Food ex-
penditure scales have been estimated with British data.3 However, this
author is not aware of any recent attempts to compute such a scale from
United States data.

The procedure used in the more recent British study was first to
estimate the relationship between income and food expenditures with house-
hold composition held constant.4 The estimate of this relationship was
then used to adjust food expenditures for variations in household income.
Least squares was used to estimate the scale. Food expenditures, with
the effect of income removed, formed the dependent variable and the number
of persons in the various age-sex categories formed the independent var-

iables. The scale was formed by the ratio of the regression coefficients,

 

3See, for example, S. J. Prais and H. S. Houthakker, The Analysis

of Family Budgets (Cambridge: Cambridge University Press, 1955), pp. 125-
145; and J. A. C. Brown, "The Consumption of Food in Relation to Household
Composition and Income," Econometrica, XII (October, 1954), pp. 444-460.

 

4Brown, op. cit., pp. 446-449.

 

 

with t

cedure
being
income
data.
valid
paris<
natin,

among

diffe
the s
Compo
the i
Cozpa
City

each

lent

pEDdi

the r

5
with the age-sex category used as the "standard" as the denominator.

The income-food expenditure relationship used in the above pro-
cedure was assumed to be of the double-log form with income elasticities
being constant among households of different age-sex composition. The equal
income elasticity assumption was found to have some support with British
data. However, Herrmann has tentatively found that this assumption is not
valid for United States data.5 But, his procedure of making pair-wise com-
parisons of household groups is subject to question. Thus, before esti-
mating a scale it is necessary to find out if income elasticities vary
among households of different age-sex composition.

This secondary objective of this study (testing for significant
differences in income elasticities) will be accomplished as follows. First,
the sample of households will be divided into types in which the age-sex
composition is held approximately constant. Then, a test of the equality of
the income elasticities of the different household types will be made by
comparing the error sum of squares resulting from a single income elasti-
city with that of the errdr sum of squares resulting from an elasticity for
each household type.

The main objective of this study is to estimate an age-sex equiva-
lent scale for food expenditures. The more recent method used by the
British authors has utilized the device of adjusting household food ex-
penditures for variations in income with a single income elasticity. Since
the results of this study have shown that income elasticities vary with

household composition, it would be more accurate to use a different income

 

5R. O. Herrmann, "An Investigation of Differences in Income
Elasticities of Demand for Food in Households of Differing Size and
Composition," (Unpublished Master's Thesis, Department of Agricultural
Economics, Michigan State University, 1961).

 

TH

 

 

 

elasticity for each household type in adjusting food expenditures for in-
come variations than it would be to use a single elasticity. The ramifica-
tions of this will be explored in this study.

Herrmann has found that the number of meals eaten at home is an im-
portant variable in explaining food expenditures. Thus, food expenditures
should be adjusted for variations in the number of meals as well as income
for the purpose of estimating scales. The ramifications of this adjustment
will also be explored in this study.

Another objective of this study is to evaluate the use of an age-
sex equivalent scale in analyses of food expenditure behavior. First,
the estimated scale will be used in a cross-section analysis of the income-
food expenditure relationship using the same data that was used to esti-
mate the scale. The results obtained with food expenditures per equiva-
lent adult will be compared with the results obtained with per capita food
expenditures. Since the scale will be estimated from cross-section data'
its application to time series data may involve some of the familiar prob-
lems encountered in the algebraic equivalence of time series and cross-
section relationships. This problem will be explored. Data published by
the United States Department of Commerce with respect to the number of
persons of various ages and sexes for each year, makes it possible to use
scales in an analysis of the United States income-food expenditure rela-
tionship over time.6 A final objeCtive will be to make such an applica-
tion and to compare the results with those obtained using per capita food

expenditures.

 

6U. S. Department of Commerce, Current Population Reports, Series

 

P-ZO.

TH

 

 

It—‘—

 

 

 

7
The Age-Sex Equivalent Scale as a Normative Concept

The age-sex equivalent scale has been conceived as describing the
proportion of expenditures (on a specific commodity or on all commodities)
actually allocated to the various individuals. The question now arises
regarding the conditions under which this allocation is optimal. This is
the normative problem of the optimum allocation of expenditures among the
various age-sex types. If the existing allocation approximates the "op-
timal" allocation the estimated age-sex equivalent scales form a basis for
distributing welfare payments among individuals of different ages and
sexes.

To explore this problem consider a single household composed of
three persons, each one being of a different age. Total expenditures (or
income) are a given quantity for the household. The first allocation prob-
lem is one of dividing the available income among the three individuals.
To maximize the total utility of the household, this division of income

would have to be such that the marginal utilities of income for each indi-

vidual be equal. That is,

MUI(1) = MUI(2) = MUI(3) (1)

To bring about this optimal allocation certain assumptions are
necessary.7 First, the principle of diminishing marginal utility of in-
come must hold for all members. This assumption follows from the assump-
tions inherent in indifference analysis.

With a given income, all the things bought give a greater

satisfaction for the money spent on them than for any of the other
things that could have been bought in their place but were not

 

7For a more detailed disCussion of these assumptions see A. P.
Lerner, The Economics of Control (New York: The MacMillan Co., 1944),
pp. 23-28.

TH

 

 

 

in

pt

80

Th

01'

an

pr

th

C0

to

of

’u
dh

8
bought for this very reason. From this it follows that if income
were greater the additional things that would be bought with the
increment of income would be things that are rejected when income
is smaller because they give less satisfaction; and if income
were greater still, even less satisfactory things would be bought.

The second assumption is that increments in utility arising from
increments in income can be measured for each individual and hence inter-
personal utility comparisons can be made. Lerner has called such a compari-
son impossible.9 However, he applied his analysis to the entire economy.
The household is a small group in which members express their satisfaction
or dissatisfaction to other members of the group to any particular distri-
bution of income. In addition, one member may receive some utility because
another member is enjoying the utility from some expenditure. That is,
preference functions of the household members may be interdependent. Given
these circumstances, there would be a tendency to distribute household in-
come among the individuals so that marginal utilities would be equated and
total utility of the household would be at a maximum.

Given an optimum allocation of income among the three individuals
of the household, there still exists the problem of allocating income among
alternative commodities. Consider the case in which two commodities exist--
food and non-food. For each individual, this is the problem which has been
handled by traditional indifference theory. The individual will have reached
an optimum allocation of his income if the ratio of his marginal utility for
food to his marginal utility for non-food (the marginal rate of substitution

of food for non-food) is equal to the ratio of the price of food to the price

of non-food.lo That is, for each individual,

 

81bid, pp. 26-27.

9Ibid, p. 28.

10For a more detailed discussion of indifference theory, see J. R.
Hicks, Value and Capital (2nd ed.; London: Oxford University Press, 1946),
pp. 11-250

 

__ = .2 (2)

If the conditions expressed by Equation (1) hold for each house-
hold and the condition expressed by Equation (2) holds for each member of
each household, the age-sex equivalent scales computed from actual data
will express the optimal allocation of expenditures among individuals of
varying age and sex. It is not likely that these conditions are ever met
for each household of any given population, but there probably is a tendency
for households to strive towards meeting these conditions. The present
state of knowledge also makes it possible to determine whether or not these
conditions are actually met.

One purpose of presenting these conditions was to show the rela-
tionship of age-sex equivalent scales to normative economic theory. To make
the age—sex equivalent scales a normative concept, marginal utilities of in-
come were equated within each household. The present state of knowledge
does not indicate if households act in such a manner. In fact, it is not
known if households act in approximately the manner in which indifference
theory dictates. However, there is no strong indication that they act in a
manner much differently than that of maximizing the total utility or satis-
faction of the household. There is at least an incentive to try to behave
in this manner, namely, the incentive of obtaining the greatest utility.
Therefore, there is no substantial obstacle to the belief that age-sex equi-
valent scales estimated from actual data are optimal with respect to the
allocation of goods among individuals.

A scale that has been used to specify the differences that "should"
exist among different age-sex types with respect to food expenditures is

the "nutritional" scale. It has been based on the nutritional require-

10
ments of individuals of varying age and sex. This scale is normative
relative to nutritional requirements. However, the allocation of food
expenditures among individuals according to their nutritional requirements
does not necessarily mean that the normative criterion of maximum utility
is met. Certain age-sex types may have a higher preference for more ex-
pensive food than others. For example, processed baby food is probably
more expensive "per nutrient" than the food normally consumed by adults.11
Thus, the household probably would not allocate its food expenditures among
its individual members strictly according to nutritional requirements. In
addition, it may be argued that the allbcation shown by nutritional scales
is not optimal since preference differences among the age-sex types are

not taken into consideration.

 

11This is one of the most obvious examples. Another possible

example is that teenagers may have a higher preference for less expensive
meat products than adults.

CHAPTER II
REVIEW OF LITERATURE
The Development of the Procedure for Estimating Scales

The Problem

The first age-sex equivalent scales for food were estimated on
the basis of nutritional requirements. It was assumed that these scales
closely approximated the actual proportions of food expenditures allo-
cated to individuals of varying age and sex. It is not known how accur-
ately such a scale represents actual expenditure patterns. There are
reasons, given previously, for believing that there would be some dis-
crepancy. In addition, the assumption was made that the food nutrition
scale approximated the general level of living scale, or the scale for
all expenditures. This assumption is certainly subject to criticism since
it is reasonable to expect that the proportion of non-food expenditures
allocated to individuals of varying age and sex would be different than
that of food.12 However, a food expenditure scale may be a fairly good
estimate of a level of living scale for low income groups where food ex-
penditures account for a large portion of total expenditures.

The major reason for basing age-sex equivalent scales on nutri-
tional requirements is the difficulty of computing scales from actual
expenditure data. It is difficult and extremely costly to obtain the actual
quantity and cost of the food consumed by each individual member of a

12This assumption is, however, widely used in constructing total
consumer budgets.

ll

12

household with samples of sufficient size to estimate reliable scales.

To obtain data of this type the expenses of each household must be al-
located to the individual members. The amount of food an individual con-
sumes, for example, would have to be measured "at the table."

The type of data that is more commonly collected gives food ex-
penditures per household. This can be obtained from the amount spent for
food at the grocery store plus some additional allowances for food grown
at home. Data on the age and sex of each household member can also be
readily obtained. However, with food expenditures per household the fol-
lowing problem is encountered. A Scale could be computed by choosing
pairs of households who have equal levels of living and who differ in age-
sex composition only by one member. The amount of food expenditure, then,
to be allocated to that differing member, who must be of a constant age-
sex type, would be the difference in the food expenditure of the two house-
holds. The use of a large number of these pairs should yield a reliable
food expenditure scale. The problem is that a level of living scale would
be necessary before you could know when two households of differing com-
position had the same level of living. A scale representing total ex-
penditures could be computed by weighting scales for several individual
commodities. Thus, the route is circular. A level of living scale is
needed to compute scales for individual commodities and scales of indi-

vidual commodities are needed to compute a level of living scale.

The First Attempts by Sydenstricker and Kipg
The first persons to break this circle were Sydenstricker and

King.13 The source of their data was a survey of South Carolina Cotton

 

13Edward Sydenstricker and Willford I. King, "The Measurement of
the Relative Economic Status of Families," Oparterly Publication of the
American Statistical Association, XVII (September, 1921), pp. 842-857.

13

Mill Villages taken in 1916 and 1917. In essence, their method was first to
use nutritional scales to break the circle; then, correct the nutritional
scales for observed patterns of behavior. Actually, the nutritional scale was
used to obtain households with the same level of living. The next step was
to adjust the nutritional scale for observed expenditure patterns with re-
spect to sex.14

The nutritional scale was also used to deflate food expenditures
which held the size of the family approximately constant. Age was not held
constant but "it is almost certain that it was prOportionately divided ac-
cording to sexes and income groups."15 The proportion of females in each
family was computed and plotted against the deflated food expenditures. If
the nutritional scales were accurate in representing females, food expendi-
tures (which had been deflated with the nutritional scale) would have re-
mained constant as the proportion of females changed. Sydenstricker and
King actually found that the deflated food expenditure variable increased as
the proportion of females increased. The nutritional scale was then ad-
justed to eliminate this relationship. This also demonstrated that the
nutritional scale did not accurately estimate the expenditure scale.

After adjusting the nutritional scales for sex, the same pro-
cedure was used to adjust for age. The study did not stop with the es-
timation of food expenditure scales. Scales that represented age-sex
expenditure differences were computed for some other commodity groups.
A simpler procedure was utilized in obtaining these. The expenditures made
by each individual for such items as personal objects, clothing, etc.,
were recorded in the survey. Thus, scales were computed directly from

14For a brief review and evaluation of this study see William F.

0gburn, "A Device for Measuring the Size of Families, Invented by Edward
Sydenstricker and W. I. King," Methods in Social Science: A Case Book,
ed. Stuart A. Rice (Chicago: University of Chicago Press, 1931), pp.
211-219.

lslbid, p. 214.

 

 

14
this data.

With the food expenditure scale and the scales for some other
commodity groups in hand, a level of living scale was estimated through
a weighted aggregation of the various commodity scales. Expenditures
such as fuel, rent, etc., that could not be assigned to a particular
individual were disregarded. They consisted of only eleven per cent of
the total budget.

This "unsophisticated" method of Sydenstricker and King gives no
indication of the standard error of any of the estimates. However, their
basic idea of breaking the circularity inherent in the problems of estif
mating age-sex equivalent scales from household expenditures remains the
best approach. According to Prais and Houthakker, "In this paper (by
Sydenstricker and King) many of the problems of principle and practice
encountered in the work were expounded and given a satisfactory treat-
ment."16 However, methods used by recent authors are more refined sta-

tistically.

Introduction of Regression Techniques

 

The next step in the historical evolution of the method of com-
puting scales was the introduction of the multiple regression technique.
This is essentially a refinement of the basic idea of Sydenstricker and
King.

To evaluate the various techniques that have been proposed, it
is useful to introduce the general mathematical statement showing the
relationship between income, expenditure on a specific commodity, and

the age-sex composition of the household. For any commodity this rela-

16Prais and Houthakker, _p. cit., p. 126.

15

tionship can be expressed as:

___L_ = f(.__Y__) (3)
c N d N
,5 11 3": 11
X is the expenditure per household for a particular commodity (such

as food).
Y is income per household.
cj is the commodity expenditure scale value for each age-sex type.
dJ is the income or level of living scale value for each age-sex type.

N1 is the number of persons of each age-sex type.

If a linear regression form of the relationship is assumed, it
becomes:
X _ Y
'jsr-Egfi; - a + b ‘—§F-33fi3' + U (4)
J J
where U has zero mean and variance 0'2 ( a-z designates a variance un-

correlated with the dependent variable). By multiplying each term of

Equation (2) by. 2 CJNJ’ the relationship becomes

.1
2°95
X = a(EcN) + bY(-J—--—) + v (5)
JJJ sz
111

where v =- ( Z cJNJ)U. The variables X, Y, and Nj are to be observed and
a, b, cj, and dJ are parameters to be estimated (a is the constant term
and b is the income coefficient). With Equation (5) it is extremely
difficult to estimate all of these parameters with the least squares

technique because they are not linearly expressed.

16

The Assumption of No Income Effect--
Kemsley and Quenoville

 

If the income coefficient (b) is zero, the middle term of Equa-

tion (5) vanishes and the relationship becomes:

X

a(2c.N.) + v (6)
j J J

or X (ac.)N. + v
311

J

With this expression the parameters (acj)'s can be estimated by least
squares. The ratios of the (acj)'s will form the age-sex equivalent

scale for the specific commodity. That is, if the adult male is used
as the standard and is designated by the subscript number one, all

(ac )'s would be divided by (acl). Thus, if the income coefficient is

J
zero, to make a comparison of food expenditures between households dif-
fering by one individual of a given age-sex type, it does not matter if
two households have an unequal level of living.

17,18 The

This method has been used by Kemsley and by Quenoville.
use of least squares made it possible to estimate the standard errors
of the coefficients (i.e., of the acj's). This gave some indication of
the standard errors of the scale values.19 Although this method is ac-
ceptable for commodities with zero income coefficients, it can be shown

that it underestimates the scale values of commodities with income co-

efficients greater than zero.

 

17W. F. Kemsley, "Estimates of Cost of Individuals from Family
Data," Applied Statistics, I (November, 1952), pp. 192-201.

18M. H. Quenoville, "An Application of Least Squares to Family
Diet Surveys," Econometrica, XVIII (January, 1950), pp. 27-44.

 

 

19Since the scale values were formed from the ratio of the esti-
mates of two random variables, the standard errors of the scale values
cannot be easily obtained. In fact, the author knows of no way of com-
puting these values.

l7
Prais and Houthakker have demonstrated this bias with some man-
ipulation of the basic formulation as given by Equation (2).20 The in-

come elasticity %% is equal to

 

 

Y2 c.N
j J J
f' (7)
x2 NJ
J
where f' is the differential of f. The proportionate change in consumption
with a change in the age-sex composition of the household (31? ° :1: ) is
1

then obtained by differentiating X in Equation (1) with respect to Nj and

substituting for f from Equation (1) the f' from Equation (7). That is,

 

2%"1
1 +x j 1
———-—=<c.-Ede > . (8)
x J-Nj J j j §dJNj 2]: Cij
Thus c N
’ = I d __.LJ_
cj [(X d'NjV §chj] + dj e 2(1ij (9)
J

If the income elasticity is now assumed to be zero when it is

actually positive, the specific scale will be underestimated by the amount
c N
d e_j_.j_.
J

The Assumption that the Specific Scale is
Equal to the Income Scale--Friedman

Friedman has proposed a way of manipulating Equation (3) so that
the coefficients of the number of persons in each age-sex category (cj's)

can be estimated by least squares.21 The assumption is made that the

20

 

Prais and Houthakker, _p. cit., pp. 128-129.

21Milton Friedman, ”A Method of Comparing Incomes of Families of
Differing Compositions," Studies in Income and Wealth, XV (New York:
National Bureau of Economic Research, 1952), pp. 9-20.

18

scale for the particular commodity, for example, food, represents the
level of living scale. That is, cj = dj for all j. Equation (4) can then
be written as

_x -bY = a(E chJ.) + v. (10)
It is possible to estimate the parameter b independently of the other par-
j's) by holding chj constant. Thus, Equation (9) becomes X =

A + bY + v, where A = a(Z Cij)'

with the same age-sex composition. For example, households with one male

ameters (c

This can be done by choosing households

and one female age 20-40 years could be used to obtain an estimate of b and
households with one male age 20-40, one female age 20-40 and a child under
six years of age could be used to obtain another estimate of b.22 The
independent estimate of b can be used to adjust the specific commodity ex-
penditure (X) for differences in income, or level of living. The adjusted
expenditure (X - bY) can then be used to obtain the least squares estimates
of the coefficients of the number of persons of each age-sex type, which
are used to form the specific commodity scale.

Equation (9) can also be used to show the weakness in the Friedman
method. There is, of course, no inherent weakness in this method if the
level of living scale is equal to the scale for the specific commodity.

If this specific scale value is greater than the level of living scale

value (cj )'dj for any particular j), the specific scale value will tend

to be overestimated. Consider the last expression of Equation (9)

c N.
d e

1.1 .
j 2 d1”:
1

cj > dj’ the first term (dj) in the above expression will be larger than

The Friedman method assumes dj = cj, therefore, if

it should be and this will give the estimate of c a larger value. The

J

 

221t is assumed that the various estimates of b will not differ
significantly.

ten
con:

and

esti
beer
the

inde

squa

part

each
into
lEVe
the

Vari

CQul

19

term ( Z:dij) will have the opposite affect but with the values usually
consideied for the cj's and dj's the first effect will be the strongest
and the value of cj will be overestimated.

It should be noted that the Friedman method introduces the idea of
estimating certain parameters independently of others. This technique has
been used in a subsequent model which does not require the assumption that
the two types of scales are equal.23 If these parameters are not estimated
independently, then values must be assumed and a method of trial and error,
such as that used by Prais and Houthakker, utilized to obtain the least
squares estimates of the coefficients representing the scale values for a
particular commodity.24
Correctinggfor Income Variation by Assuming
an Income Scale--Prais and Houthakker

The method developed by Prais and Houthakker did not require
either the assumption of a zero income coefficient or the assumption that
each value of the specific commodity scale is equal to each value of the
income scale. Instead, to compute scales for specific commodities, a
level of living scale was assumed.25 Equation (3) was manipulated so that
the number of persons of each age-sex type (Nj) were the only dependent

variables, and, hence, least squares estimates of the scale values (Cj)

could be computed. That is

 

23Brown, pp, cit., pp. 444-460.

24Prais and Houthakker, _p. cit., p. 135.

25The method developed by Prais and Houthakker and their work on
the general theoretical development and application of age-sex equivalent
scales is discussed in (l) H. S. Houthakker, "The Econometrics of Family
Budgets," Journal of the Royal Statistical Society, Series A, CXX (Part I,
1952), pp. 1-21, (2) S. J. Prais, "The Estimation of Equivalent Adult
Scales from Family Budgets," Economic Journal, LXIII (December, 1953), pp.
791-810, and (3) Prais and Houthakker, The Analysis of Family Budgets, pp.
C_i_t_. , pp. 125-145.

20

__X___)
g; dij

X/f ( = ELN. (12)
j J J

Two forms of f were chosen, the semi-log and the double-log.. Thus,

_ _X____ _ ___L__
fl-a+blog(£dN)+u or fl—b(g+log2dN.)+u (13)
j JJ j JJ
where g = g and u has zero mean and variance 0-2.
_ Y e u
j .J J

where:.a is the logarithmic constant and u is the same as in Equation (13).

One of the parameters of f and f was merged with the c '5 since it is only

1 2 J
the ratio of the coefficients which form the scales. (For the semi-log
equation b was the merged parameter and for double-log equation a was merged

with the cj's.) The semi-log form thus becomes,

 

X . = £bc.N. + v. (15)
,[g + log (-———-———
§ 31

) are the parameters to be estimated; X, Y and N are ob-

J

have been assumed from prior knowledge; and

where g and (bc

J

served variables; values for d

J

v = u [g + log EEK—EV” ( 2 chj). The double-log form becomes

J
2 (1ij
x<—-l—Y—>e = < 2: ac.N.>._“ (16)
j J J
where e and (acj) are the parameters to be estimated; X, Y, Nj’ and dj are
the same as in Equation (15) and u and JL are the same as in Equation (13).26

 

26Note that the error term is not additive, as is usually assumed

for least squares, but is multiplicative. This is not discussed by Prais
and Houthakker. The same problem was encountered by Brown, _p. cit., p. 449,
and it will be discussed subsequently.

 

pro

W61

val.

gue
a g1

mod

21
The following procedure was used by Prais and Houthakker to obtain
least squares estimates of the coefficients of the number of persons of the

various age-sex types (ac 's or bcj's): (I) assume a value of g or e,

J
(2) calculate the values on the left hand side of Equation (11) or (12),
(3) compute the correlation between this variable and the Nj's, (4) repeat the
procedure for different values of g or e. This gives the relationship be-
tween the correlation coefficient and the values of g or e. (5) Choose the
values of the acj's or bcj's which maximize the correlation coefficient.

The greatest limitation of this procedure is the assumption or
guess that had to be made as to the level of living scale. However, if
a good guess could be made of this scale, scales for several specific com-
modities could be estimated. If these commodities accounted for a large
per cent of total expenditures, their scales could be weighted to form a
"reliable" estimate of a level of living scale. This scale could then be
used to compute revised estimates of the commodity scales and these again
would be used to estimate a revised level of living scale. With this itera-

tive procedure it should be possible to estimate both a level of living

scale and scales for various commodities.

Further Refinements--Brown

 

The method of Prais and Houthakker is somewhat tedious in that
several estimates of the specific commodity scale are required to obtain
an acceptable result. Brown found that the double log form could be man-
ipulated in such a way that the e coefficient (the income elasticity) could
be estimated independently of the cj's.27 By choosing households of the

same composition, that is, the same number of equivalent adult persons,

the variables Echj and E dij became constants. Thus, 2 chJ. = c
J J J

 

27Brown, _p. cit., pp. 447-449.

the}
The

051

the

Per

23.‘
int

:
“4;

t “I“. L

Ejef.

22

The expression,

X Y
W = “Tn->2”, (17>
j JJ - JJ
J
therefore can be written as X = (cad-e) Yéay. (18)

The logs of this expression can be fitted by least squares and estimates

of the income elasticity (e) and the constant term (cad-e) can be obtained.
Brown selected sixteen household types in which the age-sex compo-

sition was held approximately constant.28 Estimates of the income elastici-

ties were made for each type and within sampling error were found to be

equal. A "common" estimate of the income elasticity for all sixteen house-

hold types was then made. ' 2 d N
J J
Upon manipulation Equation (17) becomes X(--1---§---)e = 2:(achj)1P.
J

(This is Equation 16.) With the above estimate of the income elasticity

(e) and assumed values for the income scale (dj's) least squares estimates
of the acj's can be made with one modification.29 The error term is not
additive as is the usual case for least squares estimates but is multiplica-

tive. Therefore, the residual variance would be expected to be proportional

to the square of the expected size of the dependent variable.30 This vio-

 

28A household type will be defined as a group of households in which
the age and sex of each individual in those households is of a given type.
For example, one household type might be a male (age 20-40), a female (age
20-40) and a child (under six years of age). Each household of this particu-
lar type would have one and not more than one member of the three above
individual types.

29X, Y, and N 's are observed variables. The ratios of the (ac.)
coefficients will fo the specific commodity scale. J

30Brown actually made an approximate test of this hypothesis and
found it to be true, Ibid., p. 445.

23
lates the assumption of homoscedasticity inherent in the least squares
procedure and yields inefficient estimates of the coefficients (acj's).
To obtain more efficient estimates Brown weighted each observation in in-
verse proportion to the size of the dependent variable.

The methods of computing scales based on actual expenditures has
thus proceeded from the early unsophisticated method of Sydenstricker and
King to the use of least squares methods that utilized some dubious as-
sumptions and then finally to the relaxation of these assumptions. The
original idea of Sydenstricker and King that overcame the circularity
problem by assuming a set of scales as a starting point is still utilized
by the most recent methods. The assumption of a set of values for the level
of liVing scale is necessary to compute estimates of scales for Specific
commodities with the method developed by Brown.

There is a relationship between the level of living scale and any
given specific commodity scale as shown by Equation (8). Brown tested a
simplified version of this relationship, that is, a change in the level of
living scale multiplied by the income elasticity is approximately equal to

the change in the specific commodity scale.31 Algebraically,
lac. 5 e 23d.
J J

This approximation was found to be valid for food expenditures. Thus, any
scale for food expenditures based on a set of assumed values for the level

of living scale can be adjusted to conform to a new level of living scale.

The Equivalent Scale Hypothesis
The use of age-sex equivalent scales to specify the effects of

household composition has been subjected to much criticism. The position

 

,.

311bid., p. 455.

of t}
fies
ties.
pcthc

spec

24

of the proponents of scales has been that Equation (3) "adequately" speci-
fies the effect of household composition on expenditures for most commodi-
ties. This argument has been couched in terms of the equivalent scale hy-
pothesis. If for a given set of data age-sex equivalent scales "adequately"
specify the effects of household composition on the consumption of some
commodity, this gives support to the hypothesis. If, however, it is found
that age-sex equivalent scales do not "adequately" specify the effects of
household composition, the hypothesis is not confirmed.

The critics have had more than one point to attack. Allen, using'
American data, argued that economies of scale exist which cause the second
child of a family to have a different scale value than the first.32 He
demonstrated this by fitting Engel curves of different family types for
several commodities and comparing actual expenditures with those that would
be expected under the equivalent scale hypothesis. He concluded that:

The present evidence tends to suggest, therefore, that
"equivalent adult" scales are not capable of eliminating com-
pletely the effect on modes of expenditure of varying family
composition. The failure appears particularly between families
,comprising husband, wife, and children on the one hand, and
families with adult members (apart from man and wife) on the
other. It seems just possible that scales could be derived to
allow roughly for changes in family size due solely to varying
numbers of young children in the family. To do even this, the
scales may have to take account of differences between various
kinds of personal and household expenses, between occupational
groups of families and between the addition of the first child
and the addition of further children.
Therefore, Allen not only suggests that economies of scale exist but also
that age-sex equivalent scales may differ among various occupational groups.

It seems apparent that economies of scale should exist for many

commodities. All items used by a first child would not be "worn out" and

 

32R. D. G. Allen, "Expenditure Patterns of Families of Different
Types," Studies in Mathematical Economics and Econometrics, ed., Lange,
McIntyre, and Yntema (Chicago: University of Chicago Press, 1942), pp.
190-207.

 

1:1
to
f0
a:
ti
8‘;

I.»UIII

25
hence could be used by additional children. Likewise, the addition of an
adult member to a family may give rise to economies of scale, especially
among some durable goods and housing expenditures. It is not so apparent
that such economies of scale exist for food purchased in the modern super-
market.
The position of Houthakker was that the use of age-sex equivalent
scales offered some advantages over other alternatives.
The principle attraction of unit consumer scales, when

improved by distinguishing specific and income scales, is that

they permit all households to be dealt with on the same footing.

This is not the case if one divides families into types and

analyzes these separately, as was done by Allen, and Henderson,

and also by Nicholson for a selection from the U.K. working class

data.
In an actual analysis Prais and Houthakker used age-sex equivalent scales
to express the effects of household composition on expenditures for all
food and for expenditures on specific food products.34 They took the posi-
tion that economies of scale for food expenditures were not substantial
and, therefore, ignored them. It was their position that little modifica-
tion of their procedure would be required to account for economies of scale
and that equivalent scales would be a satisfactory method of specifying
the effects of age-sex composition on expenditures for other commodities.

One of the criteria that has been used to determine whether or not

the equivalent scale hypothesis is confirmed is the equality of income

 

33Houthakker, pp. 213., p. 11. The studies referred to were
(1) Allen, 122, £15., (2) A. M. HenderSon, "The Cost of Children,"
Population Studies, III (September, 1949), pp. 130-150, (3) A. M. Henderson,
"The Cost of Children," Pppulation Studies, IV (December, 1950), pp. 267-
298, (4) A. M. Henderson, "The Cost of a Family," Review of Economic Studies,
XVII (Part 2, 1949-1950), pp. 127-148, (5) J. L. Nicholson, "Variations in
WOrking Class Family Expenditure," Journal of the Royal Statistical Society,
Series A, CXII (Part IV, 1949), pp. 359-411).

34Prais and Houthakker, pp. cit., pp. 125-145.

'1")

26
elasticities for all household types. Scales specify only the differences
in the level of consumption among various types of individuals. They cannot
specify differences in reSponses of expenditure to income change among dif-
ferent household types. Therefore, if income elasticities do differ, equiva-
lent scales cannot "completely" specify the effects of age-sex composition
on expenditures. It is still open to argument whether or not the advantages
of using scales outweigh the disadvantages of not specifying completely the
effects of age-sex composition. The desicion as to the use of scales would
depend on the alternatives available and their advantages and shortcomings.
If differences in income elasticity do exist and scales are still determined
to be the best way of specifying the effects of age-sex composition, the
arguments over whether or not the equivalent scale hypothesis is confirmed
or disconfirmed seems to be irrelevant.

Brown tested the equality of income elasticities for food expendi-
tures with 1951 British working class data.35 As was stated previously, he
found significant differences at the one per cent level among the sixteen
household types.

In an analysis of 1955, urban United States data, Herrmann found
that income elasticities of food varied significantly with household compo-
sition.36. However, the procedure was to test differences between pairs
of household types, which may give misleading results. For example, if there
are ten separate t-tests of parameters coming from the same population
with a five per cent significance level, the probability that at least one

pair will be significantly different is .40.37 Therefore, the probability

 

35Brown,_p. cit., pp. 447-449.

6Herrmann, _p. cit.

37Wilfrid J. Dixon and Frank J. Massey, Introduction to Statistical
Analysis, 2nd ed. (New York: McGraw Hill, 1957), p- 139.

27
of making a Type I error is substantially increased if the hypothesis of
equality is rejected when one pair is significantly different. In addition,
there is a loss of precision in estimating the variance. The results of
Herrmann's study, therefore, still leave the equation of equal income
elasticities for food expenditures in the United States in doubt.

A recent attempt has been made to estimate expenditure scales for
several commodities from British data.38 Essentially the method developed
by Prais, Houthakker and Brown was utilized. The age-sex categories for
individuals were formed in such a way that economies of scale, believed
to be inherent in most commodities, were accounted for. To do this the
second child of a household was considered to be a different age-sex type
than the first. Thus, the second child would have a lower scale value
than the first if economies of scale existed. In the same way the third
child was considered to be a different individual than the second.

The equivalent scale hypothesis was then evaluated by comparing
the predicted expenditures with the actual ones. The predicted results
were those obtained by using the double-log form of Equation (16). Forsythe
concluded that this hypothesis

. . . . is much too simple to provide anything more than a very rough
approximation to the observed patterns of family expenditure. [The
graphs] . . . show that for housing, clothing, transport, entertain-
ment, and essential services, the slopes do indeed differ systemati-
cally and cannot be reconciled with a hypothesis of a common basic
expenditure-income relationship for all family types. When its

size changes, the family's tastes appear to change 33 more compli-
cated ways than those visualized by our hypothesis.

 

38F. G. Forsythe, "The Relationship Between Family Size and Family
Expenditure," Journal of the Royal Statistical Society, Series A, CXXVIII
(Part IV, 1960), pp. 367-393.

39Ibid., p. 386.

 

(.1.

Ci

CL

'56

ll)
H'

(1‘
m

“I
(ll!
1...

28

Even though the economies of scale problem has been overcome, the
use of scales left much variation in expenditures apparently caused by age-
sex composition to be explained. First, there is the problem of how well
the single parameter Engel curves fit the data when household composition
is held constant. Then there is the ability of the scales to explain the
differences between the income-expenditure relationships of households of
different age-sex composition. If, for example, the double-log form in-
come elasticities were equal for all household types and each type fitted
the double-log form in a satisfactory manner, the scales would be capable
of explaining much of the effects of age-sex composition.40 If, on the
other hand, the equations for each household type are of a different shape,
information in addition to age-sex equivalent scales would be needed to
explain a large proportion of the effects of household composition.

Forsythe found that for most commodities, the income-expenditure
curves were of such a nature that age-sex equivalent scales could explain
only a small proportion of their differences. However, food expenditures
were an exception and for this one commodity the equivalent scale hypothe-
sis was substantiated. That is, a large proportion of the effects of age-
sex composition was explained by scales.

Even though equivalent scales offer a "satisfactory" explanation
of the effects of age-sex composition on food consumption for Britain it
is not known whether they are satisfactory for United States data. The
finding of Herrmann gives some indication that the use of a food expendi-
ture scale may not give a "satisfactory" eXplanation of the effects of

age-sex composition. It is the purpose of this study to estimate an age-

‘

40Age-sex composition includes household size.

SEX

how

type

tize

29

sex equivalent scale for United States food consumption and to determine
how satisfactory this approach is for United States data. This will in-
clude an examination of the differences in income elasticities of household
types in which age-sex composition is held constant. An attempt will be
made to compare the use of the estimated age-sex equivalent scale with the
use of the per capita deflating procedure for food expenditures in both

time series and cross-section analysis.

CHAPTER III

ANALYSIS OF THE INCOME-FOOD EXPENDITURE RELATIONSHIPS
WITH AGE-SEX COMPOSITION HELD CONSTANT

Purpose of the Analysis

 

The results of previous studies have been inconclusive with respect
to the feasibility of using a scale to explain the variation in United States
food consumption caused by age-sex composition. If income elasticities
among households with different age-sex composition are not equal a scale
would not be capable of explaining all of this variation. In addition, if
the elasticities are not equal, the method of estimating scales used by
Prais and Houthakker and by Brown gives biased results.41

The latter statement perhaps needs some explanation. This method
of computing scales uses a common estimate of the income coefficient to
adjust food expenditures for differences in levels-of-living. (See Equations
number 14 and 15.) If some household type has a different income coeffi-
cient, the adjustment will be biased. That is, food expenditures for that
particular household type will not be correctly adjusted for differences in
the level-of-living if an income coefficient that is computed, say, from
the other household types, is applied to this one.

An analysis of the income-food expenditure relationship with age-

sex composition held constant is, therefore, important for at least two

reasons. When using a device like a scale it is essential to know if a

 

41Prais and Houthakker, _p. cit., pp. 125-145; Brown, _p. cit.,

pp. 444-460.

30

31
portion of the variation in expenditures caused by age-sex composition is
not explained and, of course, it is essential to know the characteristics
of this relationship in order to be able to compute scales. This type of
analysis is not new. Allen, Brown, Herrmann, and Forsythe have used it.42
More specifically, the first step of the analysis consiSts of

selecting household types that have a constant age-sex composition. For
example, one household type may consist of a male (age 20-49) and a female
(age 20-49). Another type might consist of a male (age 20-49), a female
(age 20-49), and a child (under six years of age).

Under the equivalent scale hypothesis, the income-expenditure re-

lationships for each of these household types is expressed as

X Y . .
2 c N — f( E d.N.) (this 13 Equation 2)
j J J j J J
With the age-sex composition held constant 2’6ij and Zdij are constants.
Thus, 23chj = c, and 22dij = d. These two constants can be combined‘

with the "usual constant" parameter of least squares. Thus, for each house-

hold type
X = f(Y) (19)

The double-log form of this relationship would be

x = a' Yes.“ (20)
where a' absorbs the effects of c and de and is therefore equal to 2&3
The other terms were defined in the previous chapter. With the usedof
least squares, estimates can be made of a' and e for each household type.

This method makes possible a "within household type" analysis. The

income-consumption relationship can be studied without the effects of age-

 

42Allen, loc. cit., Brown, loc. cit., Herrmann, loc. cit., and

Forsythe, loc. cit.

U)

’t)

In

(I)

(_I‘

(_1

32
sex composition. The question of how much of the variation in food ex-
penditures can be explained by income and whether each household type can
be fitted with the same equation form may be at least partially answered.

A "between household type" analysis also is essential. This con-
sists of comparing the way in which the different household types respond
to changes in income. Specifically it consists of a statistical compari-
son of the income elasticities. Such an analysis indicates the extent to
which an age-sex equivalent scale is capable of explaining the differences
that exist among household types.

The "within type" and the "between type" analyses indicate the ex-
tent that an age-sex equivalent scale can explain variations in United
States food consumption due to age-sex composition. If each household
type can be fitted successfully by the same equation form and if there is no
significant difference in the slopes of the income-food expenditure rela-
tionship between household types, an age-sex equivalent scale would be an
adequate measure of the effects of age-sex composition. The scale would
be capable of explaining the differences between income-food consumption
curves if these curves differed only in "height" or size of the constant
term (a'). If, however, the income-food expenditure relationships of each
of the household types differ substantially with respect to slope (or "best
fitting" form) it can be argued that the scale would not be an adequate.
measure of the effects of age-sex composition. The question of whether to
use such a scale would depend on the available alternatives and the purpose

of the proposed analysis.

The Data
The data requirements for this type of an analysis are met only by

large surveys. A considerable number of household types is needed for a

33
representative sample of the population, because the age-sex composition
of each type should be held approximately constant. Brown used sixteen
household types and Herrmann used twenty-five.43 With the variability of
United States cross-section data Herrmann's results indicate that at least
seventy observations of each household type are required to give estimates
of the parameters that would be reliable enought for this kind of analysis.

The survey of "Food Consumption of Households in the United States"
taken in 1955 by the United States Department of Agriculture includes
enough observations to meet these requirements.44 Data on the age and sex
of each member of each household, household expenditures on food consumed
at home, and household income were taken in this survey. These meet the
basic requirements for the proposed analysis. Other variables such as the
number of meals eaten at home by each household member and the degree of
urbanization will also be utilized in this study.

This survey was the source of the data used by Herrmann.45 Some
of the weaknesses of the data were made apparent through discussions with
him and his thesis advisor, James T. Bonnen, as well as by inspection of
the results he obtained.

Both rural and urban households were included in the survey. To
make food expenditures of rural households comparable to that of urban

households, the monetary value of food raised in gardens or on farms was

 

43Brown, loc. cit., and Herrmann, loc. cit.

44For a summary of the data see U. S. Department of Agriculture
publication, Food Consumption of Households in the United States, House-
hold Food Consumption Survey 1955, Report No. 1 (Washington: U. S. Govern-
ment Printing Office, 1956). The questionnaire was published as U. S.
Department of Agriculture, Schedule Used in the Survey of Food Consumption
of Households in the United States, Spring, 1955, Agricultural Marketing
Service 200 (Washington, July 1957).

 

5 .
Herrmann, loc. C1t.

I.“

34
included. Two variables, household expense of food purchased, and money
value of food used were tabulated from the survey. If the latter variable
had been recorded accurately, it would be possible to obtain satisfactory
results using both rural and the urban data. However, the difficulties of
obtaining an accurate measure of the value of home produced foods are well
known and discussions with others who have used this data indicate that
this problem was not really solved in this survey.46 The use of the urban
data only would avoid this problem.

The number of obsérVations of urban households was over 2,000 which
should be sufficient for the type of analysis proposed. These urban house-
holds should be more homogeneous with respect to food consumption habits
than a group of households that include,both urban and rural elements. The
urban data are probably more heterogeneOus than the data used by English
authors for similar analyses. English researchers have frequently used only
data from working class households. Herrmann, for similar reasons of
working with somewhat homogeneous data, restricted his analysis to the urban
data.

The income variable recorded in the survey was the 1954 yearly
household income. Yearly incomes are subject to wide fluctuations fbr
some groups and may not be the basis for expenditures on food. Some of
the households in the survey reported incomes that were less than their
yearly food expenditures. It is, therefore, necessary to work with a
somewhat unsatisfactory income variable.

One seriously erroneous observation may lead to misleading con-

clusions in an analysis of the kind proposed. The data will be separated

 

46
Such discussions were with Bonnen and Herrmann.

35
into groups which are perhaps less than one-tenth the size of the whole
sample. One erroneous observation thus has less chance of being Caveraged
out" by others. Herrmann found that the addition of one observation to a
household type was in some cases capable of changing a positive income
elasticity to a negative one.47 With this weakness of the income variable
it was decided that some of the low income observations should be eliminated.
This will not correct the basic weakness in the income variable, but will
only eliminate a few of the more .extreme observations that could make the
estimated parameters unrepresentative of the population.

For the purpose of eliminating some of the more extreme observa-
tions, urban data were first softed to select all one-person households
with incomes of less than $600 per year and all two-or-more-person house-
holds with incomes of less than $1,000 per year. One household had a nega-
tive income and was eliminAted from the sample. Sixty-eight households
"with positive insemes were examined. These households were segregated
into groups according to number of persons. Each group was examined for
gaps in yearly incomes. Yearly food expenditure was also computed for the
purpose of determining whether it exceeded income. All households with
incomes under a selected level were eliminated. The number eliminated from
the sample and the maximum income of the eliminated households are given
in Table l. A

A total of twenty-seven households were eliminated from the urban
sample of approximately 2,275 observations. Of these, 21 had yearly food
expenditures that exceeded yearly income. The six households with food

expenditures less than incomes spent so little for food that they were

47The negative elasticity was reported by Herrmann in this thesis.
He has indicated to me that the removal of the one observation changed the‘
elasticity from negative to positive.

vll)

In

In

36
either consuming an unusually low cost diet, an inadequate diet, eating

few meals at home or consuming food that was not purchased.

Table l.--Description of the Low-Income Households
Eliminated from the Sample '

 

“41.: #—

 

Number of Number of Number of Households Maximmm Income
Persons in Households Eliminated from of Eliminated
Household Analyzeda Sample Households

1 l9 6 $262

2 28 7 580

3 13 10 - 660

4 3 780

5 3 l 177

TOTAL 68 27

 

 

 

 

‘All one person households with incomes less than $600 were
analysed. All two or more person households with incomes less than
$1,000 were analysed.

The elimination of these households can thus be justified in more
than one way. It was felt that with these reasons and with the relatively
small number removed from the sample, there would be only a slight, but
unknown, bias introduced. on the other hand, the elimination will reduce
the variability of the observations and thus increase the efficiency of

the estimates.

The Formation of Household Types
The household types for this analysis should be formed so that
age-sex composition is held approximately constant. There is, however,
a way of handling small variations in size that will be explained later.
Due to the large variations in the data, the types must be such that their

size is large enough to provide reliable estimates of the parameters.

37
From previous experience with the data it was felt that each type should
include at least seventy observations.48

Household types for this type of analysis have been established by
Herrmann and by Brown.49 Herrmann's types were such that the size of
households were strictly constant. For example, a household with an adult
male, an adult female and two children under six years of age was a dif-
ferent type than one with two adults and one child under six. Brown al-
lowed some variation in size with his types. Both households in the above
examples would be included as one type in Brown's study. The addition of
a second independent variable--number of persons--to his analysis removed
the effect of this "within type" size variation.

Many of Herrmann's household types had fewer than seventy observa-
tions. Holding the "within type" size variation constant makes it diffi-
cult to select types with a sufficient number of observations. By allowing
some "within type" variation this problem is diminished.

Herrmann found that the number of meals eaten at home per house-
hold was highly significant in explaining food expenditures. This variable
directly accounts for the number of persons per household and therefore
will diminish the effects of within household size variation. In addition
it corrects for the variation in the proportion of meals eaten at home that
may exist between households.

One hypothesis that was kept in mind in forming the household
types was that the income-food expenditure relationship may vary with the

life cycle of the family. For example, the types were formed so that a

 

48Herrmann, loc. ci .

49Ibid.; and Brown, loc. cit.

38
female under 40 years of age was in a different type than a female over
40 years of agen-with the exception of Type 13. Types with children were
formed so that the life cycle changes that accompany the adolescense of
children could be observed.

With the above considerations in mind the household types were
formed. A brief description of each type is given in Table 2 as well as
the number of observations for each type. These types do not include all
the households of the urban sample. Approximately 300 observations did
not fit into these particular categories. It was found that these house-
holds were too heterogeneous to fit into types that would be as homogeneous
as those already formed.

The number of persons in the first two types is one. Types 3, 4,
and 5 contain two persons. Types 6, 7, 11, and 12 must have at least three
persons and Types 8, 9, 10, and 13 must have at least four persons. This
gives a variation in household size that should be sufficient to observe
any between-household differences that are due to size.

The Income-Food Expenditure Relationships
Within Household Types

 

The within household type analysis was made by fitting regression
equations to each household type. Household expense for food purchased in
one week was the dependent variable and number of meals eaten at home per
week and yearly household income were the independent variables. The forms
that have best fitted this relationship with the 1955 United States data
are the double-log, the semi-log, and analog inverse.50 The double-log
form is of particular interest. With it the equivalent scale hypothesis

can be readily evaluated. It also directly yields the income coefficients

 

50Herrmann, loc. cit.

39

Table 2.--Description of Household Types

 

 

a Mean No. Number of
Type No. Description of Persons Households
1 Female (20-64) 1 110
2 Female (65 and over) 1 73
3 Male (20-64), Female (20-39) 2 108
4 Male (20-64), Female (40-64) 2 280
5 Male (65 and over), Female (50 and over) 2 165
6 Male (20-64), Female (20-39), one or more
children (under 6) 3.69 297
7 Male (20—64), Female (20-39), one or more
children (6 to 11) 3.48 119
8 Male (20-64), Female (20-39), one or more
children (under 6) and one or more
children (6-11) 5.03 187
9 male (20-64), Female (20-39), one or more
children (12-19) or one or more children
(12-19) and one or more children (6-11) 4.23 134
10 Male (20-64), Female (40-64), one or more
children (6-11), or one or more children
(6-11) and one or more children (12-49) 4.32 106
11 Male (20-64), Female (40-64), one or more
children (12-19) 3.47 148
12 Male (20-64), Female (40-64), one or more
adults (20-64) 3.24 90
13 Male (20-64), Female (20-64), one or more
adults (20-64) and one or more children
(under 20) 5.55 117

 

 

 

 

8The age of the various individuals is given in parentheses.

4O
needed to compute scales with the method proposed by Brown.51 The semi-log
and the double-log equation forms have also been used by Brown and by Prais
and Houthakker.52 In order to keep the analysis from becoming too complex,
it was decided to use only the double-log and semi-log equation forms. The
information obtainable by using additional forms was believed negligible.

The number of meals variable was adjusted to account for age-sex
variation. If age-sex equivalent scales have any validity, a meal eaten
by a young child, for example, should be given less weight than a meal
eaten by an adult. The adjustment process was made in the following manner.
The number of meals eaten at home by a particular person was multiplied
by a scale value for that person. The value of this "new" variable on a
household basis was obtained by summing its values for the individuals in
the household. This variable will be termed "standard number of meal units"
or "standard number of meals" to distinguish it from the unadjusted number
of meals variable.

The scale used to adjust the number of meals variable was based on
the moderate food cost plan compiled by the U. S. Department of Agriculture.53
This was the only readily available source of an approximate scale based
on the 1955 household survey of food consumption. This scale was not com-
puted by the methods discussed in Chapter I, but was based partially on
nutritional requirements. The cost of a weeks food (see Table 3) was ob-
tained by utilizing the cost levels per family and the quantities of various

items consumed per family from the 1955 household survey. Quantities of

 

51Brown, loc. cit.

52Prais and Houthakker, loc. cit.

53Eloise Cofer, "Family Food Budgets, Revised 1957," Famin
Economics Review (October, 1957). p. 11.

41

food were assigned to various age-sex types by their nutritional require-
ments. Costs for each agedsex type were then made on the basis of the current

(1957) price of these items.

Table 3.-~Scale Used to Adjust the Number of Meals Variable‘

 

 

Age-Sex Type Foogosgbzf Weeks Scale

, crate Plan Value

Child under 1 year 3.50 .40
Child 1-5 years 4.65 .53
Child 6-11 years 6.54 .74
Male 12-14 years 8.75 .99
Female 12-14 years 8.00 .91
- Male 15-19 years 10.25 1.16
Female 15-19 years 8.05 .91
Male 20-49 8.80 1.00
Female 20-49 7.03 .80
Male 50-64 8.17 .93
Female 50-64 6.58 .75
Male 65 years and over 8.00 .91
Female 65 years and over 6.50 .74

 

 

 

aSource: Ibid.

The number of meals variable could be further refined if the dif-
ferences in the size of meal could be specified. That is, for most people
the morning and noon meals are less expensive than the evening meal. How-
ever, there is no readily available information pertaining to the size of
these cost differences. If there were, it is doubtful that the results
of this analysis would be affected significantly. Therefore, it was con-
cluded that this adjustment did not warrant the expense involved in

attempting it.

42
The double-log equation of the food expenditure-income and standard

number cf meals relationship.is

X = a'Y‘szLu (21)
where a', e and b2 are the parameters to be estimated; X, Y, and M are
observed variables, and u is assumed to have zero mean and variance 03.54

In hypothesis testing u is also assumed to be approximately normally disu

tributed. The semi-log equation of this relationship is

X = a" + b log Y and b log M + v (22)

1 2

where a", and b1 and b are the parameters to be estimated and X, Y, and

2
M are observed variables. The error term v is equal to cu, where u is
assumed to have zero mean and variance erg. c is constant for a particular
household type, but varies between types.

The results of fitting these equations for each household type
are given in Tables 4 and 5. The two independent variables have explained
approximately 30 per cent of the variation in food expenditures. This
varied between household types with the coefficient of multiple determin-
ation ranging from .171 to .409 with the double-log form and from .199
to .452 with the semi-log form. This is what might be expected with this
kind of cross-sectional data. The effects of such things as occupation,
social class, region, taste, etc., are just a few of the possible vari-
ables that have not been taken into consideration.

There was very little correlation between income and the standard

number of meals units. The absolute value of the estimate of rym was less

 

54X is weekly food expenditure per household, Y is disposable in-

come per household for the year 1954, and M is the standard number of
meals eaten at home per week per household.

.ahom moaueansoo on» now seamen a“ zw.wm «some comb» oopoaen
O

.oeuonuceasm a“ need» eke eeueawueo on» mo eaoauo caucqepwu

 

 

000.- com. .mmn. Aamo.v 00m. Ammo.0 mwm. ma
and. mmm. .msm. Amma.v «ma. Ammo.v mam. ma
Hm0.- 00m. mew. Ammo.v Ham. Aw:0.0 ~0H. Ha
0:0. mam. .mam. AmHH.0 mam. Am00.0 00m. 0a
000.- 000. .mmm. A00H.V 00m. “H00.0 man. 0
0:0.u mam. mam.‘ Amaa.v 00m. Awao.v mam. w
000. 50m. .00:. Amaa.v :00. A000.0 mam. a
400.- 00m. .cmm. Asmo.v 00m. A~:0.0 00m. 0
000.- Hmm. 00m. AHHH.V mam. A0m0.v mam. m
$50.- 00m. .Ham. AH00.V mmu. AH40.V mam. :
3N? Rm. RH. 3010 m3. $8.0 awn. m
Hmo. mom. awn. ham0.0 ~ma.a Ammo.v mmn. m
H00. Hos. new. Amoa.0 sen. Asm0.0 was. H
its
uoapuaus> Azw.xmv Azw.mmv passages “may snoaoacmooo has
unooeoeoueH opuaaumm mo unease assesses oases no nonasz unoaoaccooo .0: «may
doc-son noose undunupm co paoaoammuoo uunsueuom «deconu 0aoaonsam
no.“ ECHO-HHS

 

 

 

 

 

 

 

flange monuoapnonv mannuoaauaom uaaox no hopes: 00a
oaoonHueuspavconxm cook on» no nowunflaooenuu.: «Hana

an

.9703 mo gag macaque» 23 ea In

 

 

 

 

 

 

03m eon. om.~m 1. na
00% 804 00.3 A we
08m 30; Exam +m 3
09m .3m. 3.0m 1. 0a
008 03. 8.0m is m
000.. ~84 3.0m 1. 0
0R: Rmé o~.m~ +m a
000: 0:0. Sim +m m
000w $5. 8.3 m m
on? 3:. 05.9” m s
8.3 m3. 3.5 m m
003 2...... and a m
0me 30. 9.0 H a
Sousa 50335 08m 88: oauacaooa 3 30m 338.80 com .0: as
93: 3.3330 50: 020880 mucus? 383m no .02 38.038
38

3

0083801.: .35

#5

.0000 000-0000 000 000 000000 00 00000

00003 monk» ooponea
O

.eoeon0nohun q0 co>0m cud oepea0puo on» 00 000900 unuundume

 

 

00.:0 000. 00.00 000. 000.:0 00.00 000.00 00.00 00
00.00 000. :0.00 000. 000.:0 00.00 0:0.00 00.0 00
00.00 000. 00.0 .000. 000.00 00.00 00:.00 00.0 00
00.00 00:. 0:.00 000. 0::.:0 0:.:0 00:.00 00.00 00
00.00 000. 00.0 000. 000.00 :0.00 000.00 00.00 0
00.00 000. 00.00 .000. 0:0.00 ::.00 000.00 00.0 0
00.00 000. 00.0 0:0. 000.00 00.00 000.00 00.0 0
00.00 000. 0:.0 00:0. 000.00 00.00 0:0.00 00.0 0
00.00 000. ::.0 .00:. 000.00 00.00 000. 0 00.: 0
00.00 000. 00.0 :00. 000.00 00.00 000.00 00.0 :
00.00 000. :o.0 0000. 000.00 00.0 00:.00 00.0 0
00.0 0:0. 00.: .00:. 000.00 00.00 0:0. 0 00.0 0
00.0 000. 00.0 0000. 000.00 00.0 0:0. 0 00.0 0
000: 00 000.000 0:0.m00 0000 00000000000 0
00000000000 0000000000 00000000 0000000000000 0000: 0 00 .0: 0000
0000 000: 000000 00 00000 0000000: 00 00000000000 000000000
00000000 00 00000000000 000002 000000000 0000000

 

 

 

 

 

 

E
00000 000-00000 000000000000 0000: 00 000002

0nd eloocHIeaau00qomxn coon on» mo 000um0uouonuu.m 00009

than .
since

on the

demon:
(see ‘
tiple
than '

semi-

46
than .10 for eleven of the thirteen household types (see Table 4). Thus,

since the household size was nearly constant, income had very little effect
on the proportion of meals eaten at home.

The coefficients of multiple determination (R2 '3) did not

X'YM
demonstrate that one equation form was generally preferable to the other
(see Tables 4 and 5). Using the double-log form the coefficients of mul-
tiple determination were higher for seven of the thirteen household types
than with the semi-log form. There was some indication, however, that the
semi-log form gave better results for the one and two person households
(Types 1 through 5), while the double-log form generally gave better re-
sults for the three or more person households (Types 6 through 13). The
semi-log form yielded a higher coefficient of multiple determination for
four out of the five households with one or two persons. Therefore, these
findings indicate that the "best fitting" form may not be the same for all
household types.
The Income-Food Expenditure Relationships
Between Household Types
The question central to this analysis is, can a single equation
"adequately” express the income-food expenditure relationship of all house~
hold types by deflating the variables with age-sex equivalent scales. Two
criteria for the double log form to be "adequate" expressing are (1) that
the income coefficients of the various household types be equal and (2) that
the residual variances (SZX'YM) be equal. Thus, if the only difference in
the equations of the various household types is the constant term, these
equations can be combined, with the use of a scale, into a single equation.
This single equation would then be just as adequate as a number of equations

for expressing the income—food expenditure relationship.

47

It is obvious from the results shown in Table 5 that the semi-log
form does not meet these criteria. Both the income coefficient and the
residual variances appear to be correlated with size of the household or
amount of food expenditures. The correlation of the residual variances
with the size of household would be expected since the term 2: CJNJ (or:

c for each household type) was merged with the error term (see Equation 22).
That is, the error term v was cu where u was assumed to have a variance
uncorrelated with the size of household. The value of c would by its
definition change with household size. In addition, in the derivation of
Equation (22) from an equation of the form of number (3), c was merged with
the income coefficient of the latter to form the income coefficient for
Equation (22). This would then explain the increase in the size of the
income coefficient with the increase in household size. It is thus more
difficult to determine whether or not the equations of the semi-log form
can be combined into a single equation which will "adequately" express

the effects of age-sex composition. The criteria would be different than
those for the double-log form.

To test the ability of the doubleulog form to meet the above
criteria, of course, requires that both the residual variances and the
income coefficients be tested for equality. If the residual variances are
unequal, a more satisfactory test of the equality of income coefficients
can be made by weighting each observation in inverse proportion to the
size of the standard error of the estimate of food expenditures (SX-YM)'
Therefore, a test of the equality of the residual variances was made first.

The residual variances (Table 4) do show a pattern of being nega-
tively correlated with the size of household. The variances of the one

person household were larger than those of the two person households and

48
the variances of the two person households were larger than those of the
' households with three or more persons.

The most common test for equality of variances is Bartlett's. This
test, however, has been subjected to the criticism that it is highly sen-
sitive to kurtosis.55 Scheffe has proposed an alternative procedure that
is alike in principle of simple analysis of variance.56 To test for dif-
ferences in variances of, for example, thirteen groups, each group is first
divided randomly into two or more sub-groups. Variances of each sub-
group are then estimated. These estimates are used to estimate a standard
error of the variances that is due only to chance. This "within type"
variation is then compared with the "between type" variation by forming
the usual ratio which has the F distribution. Thus, if the "between type"
variation is also due to chance the hypothesis of equality will not be
rejected.

The procedure actually used in this study to form sub-groups was
as follows: the data cards for each household type were first sorted by
income. Each group (or household type) was divided into two sub-groups
by putting every other card into one sub-group and the remainder into the
second sub-group. This was not a strictly random process. However, upon
examination there was no evidence that the residuals were correlated with
income. It is possible that this procedure might produce less "within
type" variation than a strictly random selection. If this occurred, the
probability of making a Type I error would be greater and the probability
of a Type II error less than that which would be obtained with the strictly

random process. The reduction of the probability of the Type II error is

_A_4_

55Henry Scheffe, Analysis of Variance (New York: Wiley and Sons,
1959), p. 83.

56Ib1de , pp. 84-870

 

the

was

49
perhaps more important with this type of analysis, since it is the hypothe-
sis of equality that is of major concern.57
Scheffe's test showed that there is a significant difference among
the residual variances at the one per cent level. vThe computed F ratio

was ¢.25 which is greater than F of 3.96. Bartlett’s test gave the

01
same result; the computed X2 value was 64.307 which is greater than the

2

X 1 of 26.205.

.0
These findings are contrary to those of Brown.58 He found no
significant difference with Bartlett's test at the twenty per cent level.
More than one reason for this difference is apparent. First, it is ex-
pected that urban United States households have many characteristics that
differ from British working class households. Some of these characteris-
tics may affect the income-food expenditure relationship. Another reason
stems from the difference in the household types analysed. The one and
two person households had higher residual variances than the others. Brown's
household types contained a smaller proportion of these types than did this
analysis. Hence, the way in which Brown formed household types may have
obscured the differences in variance.

It should be noted that the high residual variances of the one and

two person households (Types 1 through 5) may be related to the form of the

 

57Briefly, the reason for greater importance attached to the re-
duction in the probability of making a Type II error is this: in this
study the primary concern is with whether the variances or coefficients
are equal. The methods of hypothesis testing have been constructed pri-
marily for tests of inequality. The probability of making a Type I error
is the probability that the hypothesis of equality is rejected when it
should be accepted, and therefore it is of primary importance that it
should be minimized for tests of inequality. The probability of making a
Type II error is the probability that the hypothesis of equality is ac-
cepted when it should have been rejected. It is of primary importance
that this error be minimized when testing for equality.

58Brown, 22, gig., p. 452.

50
equation. The double-log form yielded lower coefficients of multiple de-
termination than did the semi-log form for these households. However, the
variances of the log of the food expenditure variable (3:), before the
effects of income and the number of meals were removed were also highest
for the one and two person households. Thus, a larger amount of variation
would have to be explained by the two independent variables (for these
household types) for the residual variances of the doublealog form to be
equal.

The question that will now be considered is, are the differences
between the double-leg income coefficients of the thirteen household types
significant? The test that was used consists of comparing the error sum
of squares of an equation with one income coefficient with the error sum
of squares of an equation with an income coefficient for each household
type.59 The difference in these error sums of squares (adjusted for de-
grees of freedom) divided by the error sums of squares of the latter equa-

tion (adjusted for degrees of freedom) forms a ratio having an F distribu-

tion. That is,

 

F = 33531 :ESSZ /....._.2_
«1.13.1 d.f.2

where ESS1 is the error sum of squares of the equation with one income
coefficient and E582 is the error sum of squares of the equation with
thirteen income coefficients.

The test of the equality of the income coefficients was actually
made by first fitting a single equation of the double-log form with

thirteen constants, thirteen income coefficients and one standard number

 

59A general description of this procedure is given in Richard J.
Foote, Analytical Tools for Studying_Demand and Price Structures, U. S.
Department of Agriculture, Agricultural Handbook No. 1461 (washington:
U. 8. Government Printing Office, 1958), pp. 180-182.

 

51

of meals coefficient. The error sum of squares from this equation was
compared to that resulting from.an equation fitted with the same number
of constants and number of meals coefficients but with one income coeffi-

cient. Since it was shown that the residual variances (S2 's) were un-

X'YM
equal, each observation was weighted in inverse proportion to the size of

the standard error of the estimate (S These weights were obtained

x-YM)'
from the estimates of the double-log equations representing each house-
hold type.

The equation with thirteen income coefficients yielded an error
sum of squares that was significantly lower at the five per cent level
than that yielded by the equation with a single income coefficient. The
computed F value was 1.79 compared with the F.05 value of 1.75. However,
there was not a significant difference at the one per cent level (F.01 =
2.18). Since the test of the income coefficients was one of equality and
not inequality, it is important to minimise the probability of making a
Type II error. The probability of making a Type II error would be less at
the five per cent level than at the one per cent level.60 It is, thus,
preferable to use the five per cent level of significance and reject the
hypothesis of equality.

It should be noted that the test for equality of income coefficients
was made by comparing error sums of squares from equations with a constant
term for each household type. It was expected that the constant terms

would be correlated with household size since the household size term (c)

was merged with the constant term of the doublewlog form of Equation (2).

6OLester V. Manderscheid, An Introduction to Statistical Hypothesis
Testin , Michigan State University, Department of Agricultural Economics
867u-Revised (East Lansing: 1964), pp. 2- 3.

52
(See Equation 20, a' = git) Since income elasticities are less than one,
the term c would be greiter than de. By holding the number of meals con-
stant at the mean for each household type a constant term that would re-
flect this relationship was computed (see Table 4). Although much varia-
tion exists, the smaller households generally have smaller constant terms
than the larger ones. A graphic demonstration of the difference in the
size of these constant terms and the differences in income elasticities
is given by Figure 1.61 If the income (and the number of meals) coefficients
are equal, the only difference in the curves for the household types would
be the size of the constant terms. These curves are presented in Figure 2.

They also demonstrate that the constant term varied directly with household

size.

Summary

The results of this analysis of the thirteen household types
demonstrate that an age-sex equivalent scale is not capable of explaining
all of the variation in United States food expenditures that stem from
ageesex composition. There was some indication that the "best fitting"
form varies with household size since the semi-log form yielded higher
coefficients of multiple determination (R2) than the double-log form for
the one and two member households. The reverse was true for the households
having three or more members. The residual variances of the double-log
equations varied significantly with size of household. In addition, the
income coefficients of the various household types were significantly

different.

 

W

61The data for this figure were obtained by holding the number of
meals constant at the mean.

53

.mwnooogoaoh
oaooowlousuwooomxo woo.“ on» M0 3.3.“ moanofloboo coal...” .wfim

Anson non 5.305 8803” oaosoosom
ooom 000w coon ooom 000m 000: ooom ooom 08H 0
pl!» p P p p b P L b

 

 

 

 

 

q‘ 4 a 1 u n J] u .q o
.oz 05
7v hurl! .. 0H
well
3003
.rom non 93.305
oguwooofinm

econ 30538

 

 

5h

.oomho caofloooo: add how huwofipooao oaoomfi camofio u mean mammooau
leach oaoonfiuoumoaonomxo 000% on» mo Show woanoaoaoo oga||.N .mah

A900» hoe uneaaonv oaoonH oaonomoom
000m 000w 0005 0000 000m 000: 000m OOON OOOH o

F
1 q q q u

.02 cake
oaonoosom
N‘Dlll AUOH

.filvqu
m Axooi
all 1:

Mlv ON mom nHoHHOQV

: nousowunomxm
coon cHonousom

NH.“

0" 1'8

malt

 

 

 

55

An age-sex equivalent scale would be capable of explaining the
differences in the heights but not the differences in the slopes of the
curves expressing the income-food expenditure relationship. Both the
variation in "best fitting" form and the differences in income elastici-
ties demonstrate varying slopes. If the slopes of the household curves
were equal (such as those in Figure 2), the individual curves could be
combined into a single equation with only one income coefficient. The agen
sex equivalent scale would account for differences in height. However,
with unequal residual variances the error term of this equation would vary
systematically with household size; and, thus, it would not be strictly
random.

The findings of this analysis make the method of computing age-
sex equivalent scales developed by Prais, Houthakker, and Brown subject
to bias.62 Their method utilized a single income coefficient to adjust
specific expenditures for variations in income. Unequal income coeffi-
cients would make this adjustment subject to bias. The findings of this
analysis, however, do not mean that it is impossible to compute age-sex
equivalent scales. Such scales must still exist. In addition, these
scales are capable of explaining some and perhaps most of the variation

in United States food expenditures caused by varying age-sex composition.

 

62Brown, _p, cit., pp. 444-460; and Prais and Houthakker, gp, cit.

pp. 125-145.

CHAPTER IV

AN ALTERNATIVE METHOD OF SPECIFYING THE EFFECTS OF
AGE-SEX COMPOSITION USING SCALES

The Conceptual Framework

 

The question now arises: should the method of specifying the
Aeffects of age~sex composition on United States food consumption with an
age-sex equivalent scale be dropped from consideration or should this method
be retained since a scale will explain some of the effects of household
composition? If a scale is computed, some modification of the method of
computing it should be made since the methods developed by Prais and
Houthakker and by Brown assumed equal income elasticities among the house-
hold types.63 The author attempted to modify the method of computing
scales; then, evaluate the usefulness of the scale for United States food
expenditures.

Modification of the Basic Income-Food
Expenditure Relationship

 

Under the assumptions of equal income elasticities and equal residual
variances, the doubleulog form of the basic income-food expenditure rela-
tionship is

X e u

‘57s— =a<"§‘¥‘aT>fl--

X is expenditure per household fora particular commodity

(23)

 

Y is income per household

63Ibid.

56

57

c is the commodity expenditure scale value for each age-sex type

1
dj is the level of living scale value for each age-sex type
NJ is the number of persons of each age-sex type

.1 is the natural log constant
a is the income elasticity
a is the usual regression constant
and u is the error term assumed to have mean of zero and variance gr}.
If, however, income elasticities and residual variances are unequal
the basic relationship for each of the k household types would be
ek uk

X Y
3‘1“: = ““5“” L ' (24)

The income-food expenditure relationship as expressed by Equation
(23) implies that, with the use of scales, this relationship for all
household types can be expressed with an equation having a single income
elasticity and a single constant term. When graphed, this relationship
would be a single line. The relationship expressed by Equation (24)
would have k income elasticities and a single constant term. A graph of
this relationship would show k lines stemming from a single point. This
point is the constant term and would be the implied amount of food ex-
penditure at the zero income level. With different income elasticities,
but with a single constant term, the lines of the food expenditure-income
relationship for each household type would be at different levels for in—
come values greater than zero. For example, consider household Types
1 and 2. These income elasticities were .158 and .328 respectively. If
a single constant term (say .236) is used for these two types, the pre-
dicted level of food expenditure at the $1,000 income level would be $3.78
for Type 1 and $12.20 for Type 2. These two types are similar in that

each one consists of a single adult female. Therefore, 23chJ and z:dij

58

would be approximately one for each type (using the adult male as the
standard). I

The income-food expenditure relationship should be formulated in
such a way that the differences in predicted food expenditures between
households of equal size and composition, say Type 1 and Type 2, should
be at a minimum. If Equation (24) is modified so that there is a constant
term for each type, the distances between the curves at points where in-
come is greater than zero would be the minimum possible with different
income elasticities. For example, with different constant terms the pre—
dicted food expenditure of household Type 1 would be $7.63 with an income

64

level of $1,000 and that of Type 2 would be $6.03. Thus, Equation (24)

would become

(25)

Estimation of the Parameters with Age-Sex
Composition Held Constant

 

The work of Brown has shown that it is possible to manipulate
Equation (22) in such a way that the least squares estimate of the income

elasticities can be obtained independently of the "scale" parameters

(c 's and d 's).65

J J

penditure variable for income, and least squares estimates of a food

This estimate can then be used to adjust the food ex-

expenditure scale that is free from the effects of income can be made.
The question that must be examined is, does Equation (25) lend itself
to the same kind of manipulation?

64Some of this difference in the amount of food expenditures for
these two types might be explained by an ageusex equivalent scale. The

scale value for a female, age 65 years and over (household Type 2) may
be smaller than that for a female age 20-64 years (household Type 1).

 

65Br0wn, _p. 335., p. 448.

59

The income elasticity of the relationship with a single elasticity
for all household types (equation 23) can be estimated by holding the
age-sex composition of the sample constant. This has been done by divid-
ing the sample into various household types in which the age-sex composi-
tion was approximately constant. If no significant differences exist,
some weighted average of the income elasticities of the various types
would yield the "over-all" estimate (e in Equation 23). Note that this
is a "within-type" estimate and is conceptually different than a "between!
type" estimate. This latter estimate would result if all household types
were included in an equation where food expenditure was the dependent
variable and income was the independent variable. The difference in the
two estimates would stem from.the effects of age-sex composition. The
"between-type" estimate would be obtained by simply fitting a regression
equation in which food expenditures and income would be on a per household
basis. Graphically, such a relationship would differ from the relation-
ships shewn in Figures 1 and 2 in that a line would begin at the lower
left portion of the graph and extend upwards at a steeper angle than
the individual lines shown in these graphs. The income elasticity of the
"between-type" estimate would, therefore, be greater than that of the
"over~all within-type" estimate.

Another way of obtaining the "over-all within-type" estimate of
the income elasticity is to fit an equation to all household types with
s constant term for each type. Equation (23) can be manipulated to obtain

3’3 “J“:
‘< )5 dJ‘Nj)‘
J

 

X Y. on. (26)

By fitting a single equation with a constant term for each household type,

the terms ZcJN‘1 and ZdJNj become constants. That is, 23c1N_1 a ck and

60

deN1 = dk. Thus,
X a a‘k Ye.au (27)
ac
where a'k - —¥% .66 The constant terms of Equation (27) will thus absorb
d

the effects of age-sex composition. Graphically, this means that the
lines of a graph of this relationship will have equal slopes but different
heights. Figure 2 shows this type of relationship for the thirteen house-
hold types of this study. The size of the constant terms will increase
as the number of equivalent adults in each household type increases.
Conceptually, the process of manipulating Equation (25) to obtain

estimates of the income effects that are independent of the effects of
age-sex composition is identical to that used for the above (Equation 17).
Thus, upon manipulation Equation (25) becomes

2 “J":

6k “k
at ( 2 dij)ek Y

JL . (28)

 

By fitting an equation with a constant term for each household type 2:ch]

and 23dij become constants. Thus,

e u
x = wkyk 1“ (29)
‘1: Ck 67
where a"k = -—:;T-. The income elasticities and the constant terms can
d

k .
thus be estimated independently of the food expenditure and level-of-

living scales.

 

66The logarithm of Equation (27) will actually be used to obtain
the least squares estimates. Thus, logeX = log a a‘ + e legaY + u. The
dummy variable Zk will be a vector for each observation. For household
Type 1, for example, = (1:1 1 . . . l), for household Type 2, -
(l.a l l . . . 1). Thus, log.a Zk = (l O O . . . O), (01 OO . . O . . .
(0 O O . . . 01). .

67In actually fitting Equation (29), Y is a vector for each ob-
servation. For Type 1, for example, Y # (Yi 1 l . . . 1). Thus, log.a
Y (log.a Y1 0 O O . . . 0), (O log.a Yi 0 0‘0 . . . 0), etc.

Prawn 3'

61

Estimating Scales with the
Modified Relationship

 

 

The method of computing scales with the relationship expressed by
Equation (23) (the single income elasticity relationship) was as follows:
first, obtain an estimate of the income elasticity (e). Equation (23)
can be manipulated so that the specific scale forms the independent var-

iables. Thus,

 

2I‘dJNJ .
X( )e= ( 2 a c N )1". (30) ,
Y J J
J i
If values of the income scales are now assumed, food expenditures can be 1-

adjusted for differences in household income levels. Food expenditures
(X‘s), income (Y’s) and the number of persons of different ages and sexes
(Nj's) are observed variables. Thus, least squares estimates of the
(a cj's) can be computed. The ratios of these coefficients would then
form a food expenditure scale.
.A similar manipulation is possible with the modified income-food

expenditure relationship. Equation (25) would become
2:djN

Y

Jek

X ( = ak ( ZcJNJ)2uk. (31)

The constant term (ck), however, is not a constant when Equation (31) is
applied to all household types. It will vary with the size of the house-
hold. Therefore, it cannot be combined with the cj's in the same manner
as in Equation (30). In addition, it was previously argued that with a
different income elasticity for each household type a different constant
term was also needed in order to specify the effects of income. Similarly,
both an income elasticity and a constant term are needed to adjust food

expenditure for differences in household income levels. Thus, the equa-

tion for computing scales should be

will! |(|I| lll‘l'lll'

62

Ed»:
x—l- (——-—“—J-—)ek = (2cN)Luk. <32)
ak Y j j

By making independent estimates of the ak's and the ek's with Equation
(29), food expenditures can be adjusted for differences in levels of
household income.68 This makes it possible to compute a scale for food
expenditures from the modified expenditure-income relationship.

One additional modification was made to this basic relationship.

The standard number of meal units consumed by the individual households

was included as a second independent variable. The relationship would

then be
x Y 9k m' buk
———-=a( ) (————-)1.. (33)
c N k d N c N
211 211 JJ
where M' is the number of meals consumed per household and b is an unknown
parameter. The term "standard number of meal units" is applied to the
M.
expression.-Sr;;fi; since the number of meals is adjusted for the age-sex

composition of the person actually consuming the meal.

By holding the age-sex composition constant in the same manner as

 

Equation (29) the equation for estimating the parameters ek's, b, and ak's
is '
e u
x = akak M'b 111‘. (34)
“k °k
where a"& = e b' The equation for estimating scales is thus,
d k c
k k .
d N c N
1 3 ek 2 b “k
x;- <———Y-Li> <—-3,—.-L> = <ch15): . <35)
R

It should be noted that values for the c 's must be assumed in order to

J

adjust food expenditures for differences in the standard number of meals.

 

 

 

68The constant term estimated with Equation (29) is actually
a c d 6k
n k k n k
a k = e . Thus, a k c = ak.
d k k

k

63
This, however, is no more of a limitation than is the assumption of the

values for the level of living scale.

Estimates of the "Within-Type" Coefficients
for Three Alternative Modgls

 

The first step in computing scales is to estimate the income
elasticity (or elasticities) and the constant terms used in adjusting
specific expenditures. If income elasticities among household types are
equal, scales can be estimated with an equation having a single income
elasticity, a single standard number of meals coefficient and a constant

term for each household type.69 That is, for the double-log form

‘5‘“.

x = a(1)k Y“ Mb 1.“. (36)

This will be designated as Model 1. It expresses the income-food expendi-
ture relationship by letting the vector of the constant term absorb the
effects of household composition.

A more general model for expressing this relationship is one in
which the income elasticities can differ with household type. (See
Equation 34.) This will be designated as Model 2. The "constant term
vector" will also absorb much of the effect of household composition.
However, since the income elasticities vary with household type they also
would absorb some of the effects of household composition.

Model 2 has but one standard number of meals coefficient. A more
general model would be one in which the number of meals coefficient would
vary among household types as well as the constant term and the income
coefficient. This will be designated as Model 3. In essence, it consists

of fitting a separate equation to each household type. Keeping the nota-

I
69The standard number of meals variable C-J!‘--
3 “a”:

eliminate the variations in age-sex composition that still exist among the
various household types.

 

) is used here to

64

tion of Model 1 and 2

e b u
(2)k YkMknk.

Each of these models can be used to study the income elasticity

X = a (37)
for food expenditures without the effects of age-sex composition. The
differences among the three models provide an insight into the differences
in the United States food expenditure-income relationship under the vary-
ing assumptions underlying each model.

'Very little difference exists between the income elasticities of .

Model 2 and those of Model 3 (see Table 6). The same is true with respect

 

 

to the constant terms which were computed by adding the "mean" effect of
the standard number of meals variable. That is, the mean standard number
of meals for each household type was multiplied by its coefficient and
added to the "usual" constant tenm. In contrast, the constant terms of
Model 1 differ markedly from those of Models 2 and 3. The constant terms
of Models 2 and 3 changed with changes in income elasticity. Therefore,
to specify the food expenditure-income relationship with differing in-
come elasticities, the constant terms must change by an amount different
from that due to changes in household size. Thus, this shows the superior-
ity of Equation (25) over that of (24) in specifying the income-food ex-
penditure relationship.

The correlation coefficients show that the most variation in food
expenditures was explained by Model 3 and the least by Model 1, which
would be expected. The size of the correlation coefficients were .289,
.297, and .307 for Models 1, 2, and 3. In the last chapter a test of
the error sum of squares showed that Model 2 explained a significantly
larger amount of variation in food expenditures than did Model 1. A

similar test showed that Model 3 explained a larger amount of this

65

.3393 one 2&3 m3» mo «coupon 0.38 .Ho 30 non» 033.35 H .3 0

Id

 

 

 

 

 

 

son. in. on.H mmm. mmm. mum. Homvo .p .3o..o~v a .368 a .3ouomo 2 MH
Hmo.H MHo.H new. msm. mm~. new. Asonomv.n_.x3onosv a .Asosomo = NH
mmo.H moo.H mom. mmH. mmH. mum. AmHumHV.m..A3onoso a .Asonomv x HH
33m. mom. mmo.H mom. non. mum. AmHano.w
one HHHuov b .3onoi .H .3onowv : 2
m3. 8:. Ros 3,... Han. mam. 378 o. as 873 .o.
no meumHo m .Ammnomv a .Asouomv : m
mmo.H oHo.H Hmo.H mmm. 3mm. new. 3T8 m can GVV w .Ammnomv .H .3963 2 w
He~.H HoH.H hem. mom. mew. mum. HHHuov.w..Amn.o~o a .3onomv : 5
So. new. mom. mom. mom. new. 8! w .Hmmnomo .H .3ouomv : o
moo. mHm. Hmm. mom. mom. mum. 5+ one a .A+ moo z m
mos. mos. nHo. . omm. 3mm. mum. Asonosv m .Hsonomo : 3
mmH. moo. 3mm. mum. com. mom. Ammuomv a .Asouomo z m
03... m3... coo... wwm. mmn. new. T. m8 m m
Hem. Hoe. moo. mmH. me. mom. Asonomv a H
n Hoooz m Hoooz H Hose: m Hooox m Hoooz H Hoooz

 

see: as made: no

.0: 5H: canoe ufionooo

 

hvfioa puma 08005“

_ s25 3388a

 

 

“I!

 

"

uHooo: 838.334 é no.“ manna pfionooo
one 533qu scan 25 co 8»qu ..oEunE»§l.o «Home

Ill,'lll'lelll‘ ‘I‘lllll.'lllel

66

variation than did Model 2. Thus, the standard number of meals coeffi-
cients were also significantly different among the household types.

The results of Table 6 show that few differences exist between
the income coefficients of Model 2 and Model 3. Thus, it makes little
difference in specifying the food expenditure-income relationship if the
less general Model 2 is used. The main purpose of this analysis was to
specify this relationship with the standard number of meals being con—
sidered as an aid in improving it. In addition, the use of Model 3 would
have further complicated a relationship that was not yet tested. That is,
the implications of the use of different income coefficients for each
household type have not been explored. Different standard number of meals
coefficients may complicate the relationship to such a degree that the
basic idea would be "lost in the fog." Therefore, Model 2 was used as
the basis for computing a food expenditure scale.

The Computation of an Age-Sex Equivalent Scale
for Food Expenditures

Formation of the Independent
Variables

The independent variables used in computing an age-sex equivalent
scale are the number of persons in the various age-sex groups (Nj's).7o

Before computing a scale it is necessary to discuss the problems encountered

in the formation of these groups. These types (or groups) must be formed

 

7OThe agebsex groupsare "individual" age-sex groups. These groups
consist of individual persons who have approximately the same food con-
sumption levels. For example, a child, age 6-11 years, might be considered
as one group or type. Note the relationship to a household type. A house-
hold type would consist of one or more individual types. Individual types
would be constant for a single household type. That is, a household type
might include individual Types 1, 2, and 3, but no others. Generally, the
number of persons of these individual types would not change within a
household type.

lull-III Illul [I ll“

67
in such a way that the assumptions of regression analysis are not markedly
violated. In addition, other criteria must be satisfied.

One obvious criterion is that there must be a sufficient number of
observations of each individual type to make reliable estimates of the
scale values. Another criterion, which tends to lead to a small number
of observations of each type, is that each individual type should be as
homogeneous as possible. For example, a type that included children
under six year of age would be more homogeneous than a type that in-
cluded children under twelve years of age. The more homogeneous the type,
the greater will be the accuracy of specifying the effects of age-sex
composition.

Another consideration in the formation of these types is the
amount of correlation among the independent variables. The amount of
intercorrelation among some of the adult types is very high. For example,
if a male, age 20-49 years, is present in a given household there is a
high probability that a female, age 20—49 years, will be present. There
is also a high probability that a male or a female age 50 and over would
not be present. To eliminate thishigh intercorrelation previous authors
have utilized a rather heterogeneous type consisting of two persons-~the
first male and female, age 20 or over, of the household. That is, this
individual type would include an adult male and an adult female.71 The
presence of adults other than the first two occurs more or less randomly.
Therefore, types composed of these adults will not be highly correlated
with other individual types.

The individual age-sex types formed for this analysis are described

in Table 7. The one-person households cannot be readily used with this

 

71Brown,_p_. cit., p. 450.

We infirm—Elf

68
classification of types. That is, Type 1 includes the first male and
female of the household and the person of a one-person household does not
fit this classification. They were, therefore, eliminated from the sample.
Since there were only 183 persons in these households, it was thought that

their exclusion would not alter the results significantly.

Table 7.--Description of the Individual Age-Sex Groups

 

 

for which Scales were Computed L
Individual Descri tion Number of Individuals E
Type No. 9 in the Sample g
l
1 First male and female of the house-
hold (age 20 and over) 3,502
2 Child (under 6 years) 872
3 Child (6-11 years) 754
4 Child (12-14 years) 271
5 Male (15-19 years) 188
6 Female (15-19 years) 189
7 Male (20 and over) other than
the one included in Type 1 124
8 Female (20 and over) other than
the one included in Type 1 136

 

 

 

It should be noted that the Individual Type 1 has no variation.
Since the one-person households were eliminated from the sample and since
the other households included in the sample all have an adult male and an
adult female in the household, the number of persons in Type 1 is two for
each household. If the regression is fitted with the "usual" constant
term this type will be perfectly correlated with that term. Therefore,
the scales were computed by forcing the regression "line" through the

origin.

69

As a check on the amount of intercorrelation that exists among
the independent variables (Nj's), the correlation matrix for the individual
age-sex Types 2 through 8 was computed (see Table 8). Type 1 was not
included since it is the constant term. The results showed that the
amount of intercorrelation among the independent variables is low and
will not seriously affect the statistical efficiency of the estimated
scale values.

Adjustinggthe Oflservations to Account
for Unequal Residual Variances

 

The equation to compute a scale derived from the income-food
expenditure relationship with a single income coefficient and equal re-
sidual variances (Equation 30) has an error term that is not additive
but multiplicative. If the variance of the error term (u) is equal for
each household type, the variance of u' (u' = 1?) would be proportional
to the expected size of the dependent variable. Since the number of
persons is expressed linearly in Equation (30), it would not be fitted
by taking logarithms and hence the error term would be u'. Brown ac—
counted for this by weighting each observation in inverse proportion to
the size of the mean of the dependent variable for each household type.72

The results of this study have shown that the residual variances
of the u's were inversely proportional to the size of the dependent var-
iable (see Table 4). Thus, this would tend to counteract the effects of

the multiplicative error term. In order to provide an approximate check

on the above effects on the error term, the standard deviation of the

d N e c N
dependent variable of Equation (35) [fig-(—22§J-J) k’(-g:ifl—Jfib] was computed
k

 

721bid., p. 455.

w

7O

 

.oanownob unaccomoocw no nos .osnu .ow ono announce c ow H .oz omhao

 

 

 

 

 

 

H as
com. H a
mmo. moH. H m
soc. sec. omH. H m
moo. 3Ho. 55H. HnH. H 3
_mHo.u Hmo.u Hoo.- Hmo. msH. H m
moo.u ooo.n mMH.- mmH.u me.- moo. H m

m A o m 3 m m .mmmwmmw

 

 

 

 

 

«meadow mnwunmaoo ca coon Am.

n

zv moHosscs> poooooooonH

on» mo seeps: nooosHonsoouu.m oHnse

71
for each household type. It was found to be roughly proportional to the
size of the dependent variable. It was, therefore, concluded that the
variance of the error term would be roughly proportional to the size of
the dependent variable.

If no adjustment is made to account for unequal residual variances,
the estimates of the parameters (cJ's) would be consistent and unbiased,
but inefficient.73 To yield more efficient estimates, each observation
was weighted in inverse proportion to the size of the standard deviation

of the dependent variable for a particular household type.

The Results

 

For computing age-sex equivalent scales, household food expendi-
tures have been adjusted for variations in income and number of meals. If
significant differences in income elasticities among household types do
not exist, the income adjustment can be made by the use of a single income
elasticity. If these differences do exist, a scale can be computed by
using an income elasticity for each household type and a constant term for
each type (Equation 35). It would be possible to compute scales with
several models, although they would not all be appropriate. The use of
different models makes it possible to observe the effects that the various
adjustments on the dependent variable have on the scales. Four different
models were used in this analysis. A description of these is given in
Table 9.

In computing the scale values for Models 2A and ZB the following
problem was encountered. The constant terms estimated with Equations (29)

and (34) were a" and a"' respectively. The equations for estimating

k k'

 

'73
p. 49.

Stefan Valvanis, Econometrics (New York: McGraw Hill, 1959),

 

72

scales (numbers 32 and 35) utilize the constant terms a For Model 2A,

 

k
d ‘k e an: d Bk ckb
a, =-3L-jL- and for Model 23, a = k k . The values for a
k ak ck k ak ck k

were estimated by assuming a value for ck. To obtain the values of the
two scales presented in Table 10 an iterative procedure was used. Scales
were computed using the assumed values for ck. The means of the computed
scale values and the assumed scale values were obtained. These means were

then used to estimate a new value of c and another set of scales were

k
H“
computed. This process was repeated until the difference between the a
computed scale value and the one used to compute ak was less than .01. is

Table 9.--Description of the Four Models Used to Compute
Age-Sex Equivalent Scales

 

 

Single Income Elas- Number
Model Income ticity for of Meals Dependent
Elasticity Each House- Adjustments Variables
~ hold Type
2‘“ e
1A x X ("-ij-JO
c N
1B x x(_._E__.j_—j_)e(l)b a
X Y M
ZCN e
2A x xi— _....._.LJ.) k
ak Y
2° CN 8
2B x x X-—- (----—-‘1--'1)(k 11b a

 

 

 

 

 

3The actual number of meals eaten at home was used to compute
scales instead of the standard number of meals. It was thought that the
difference between these two variables was not significant.

The age-sex equivalent scales computed from each of the four models
are presented in Table 10. Solutions obtained by weighting each observa-

tion in inverse proportion to the size of the standard deviation are given

73

 

.usdoflMfinmamnw on on onwnonu one moanoanmb as» ones» noosuon oceanommwo one .oHoo

«0 homes: uncommon on» no cocoon“ coo: no: names mo Manama Hooves on» moacoo on» mmupsmaoo nH

p

.n enema eon monk» omen» no mowumwhomoo ooodnaoo once a homo

 

 

mm. Hm. mm. mm. no. 3m. on. we. ApocHo can»
. noose + owe a

wm. mm. mO.H Hm. Nu. mm. mm. mm. Apmhfim nun»
noose + omo z
om.H om.H mm.H 3m.H mm. on. mm. on. AmHumHv a
mm.H mH.H 3m.H oH.H om. mm. mm. so. AmHumHV :
Ho.H oo.H oo.H me. me. .50. mm. om. asHumHo o
:w. :B. :m. MB. :5. no. H5. om. adanmv U
Nhe Nme p0 Nme 8e NW0 wme HMe AWV v U

oo.~ oo.m oo.~ oo.~ oo.m oo.~ oo.m oo.~ A+ omv
mazfifia

mm mm «N «m mH mH «H «H some
pHooo: oHoooz Hoooz Hooo: Hoooz Hooo: Hoooz Hoooz exomuowe

ooosmau: nauseous amasmaus oooomHoa

 

 

 

 

 

 

 

 

g
oaoonH avenue new woman mHoooz ooapcmhova< hook

you eohnpaonoekfl oooh you meadow psoao>wswm xomlom< on» mo oopuaaummnl.oa edema

74
as well as the unweighted solutions. It should be noted that a per capita
income variable was used for the results given in Table 10. That is, all
dJ's were assumed to have a value of one.

The addition of the adjustment for the number of meals had the
effect of increasing the scale values for the individual age-sex types of
persons other than Type 1 (the first male and female of the household).
This would be expected since all but one of these types would be either
school children or persons in the work force who would have a higher
probability of eating the noon meal away from home than the first female
of the household. The one exception was the child under six years of age.
However, the change in the scale values of this latter type when the number
of meals adjustment is added was very small.

The change from the model with one income coefficient and one
constant term to the model with an income coefficient and a constant term
for each household type also had the effect of increasing most of the
scale values other than the first male and female of the household. This
difference arises from the effects of the different constant terms and
income elasticities on the various household types. That is, more weight
is given to the household types with three or more persons, than is given
them with the single elasticity adjustment.

The procedure of weighting each observation in inverse proportion
to the size of the standard deviation of the independent variable also
increased the size of the scale values for the age-sex types other than
the first male and female of the household. The reason for this increase
was not obvious and it is doubtful that it is important enought to warrant

serious consideration.

T““'"‘““F

75
Reliability of the Estimated Scale Values
The age-sex equivalent scale values are, of course, the ratios of
the regression coefficients of the independent variables--the number of
persons of the various age-sex types (Nj's). To obtain a "rough" estimate
of the reliability of the scale values the standard errors of the regression
coefficients from which they were formed were examined (see Table 11). The
standard error of a scale value involves the standard errors of the ratio

of two approximately independent variables. The author knows of no way to F“

“a

compute such a standard error.

The standard errors of the regression coefficients vary with the
number of observations of each age-sex type. The greater the number of
observations, the smaller the standard error. The coefficient used as the
denominator in the formation of a scale is the first male and female of
the household. There were a large number of observations of this type and
hence this estimate is quite reliable. If this estimate is treated as a
constant, estimates of the standard errors, which are in the same "units"
as the scales, can be computed (see Table 12). This will probably under-
estimate the actual standard errors but it can be defended pragmatically
since it is better than no estimate.

A comparison of the errors of the scale values obtained with the
different models is possible with these estimates. The addition of the
number of meals variable increased the accuracy of the scale values in
each case. The errors of Model 1B were lower than those of 1A and the
errors of Model 2B were lower than those of 2A. The models with an in-
come coefficient for each household type, however, yielded larger standard
errors than the models with a single income elasticity. That is, the

errors from Models 2A and 2B were larger than those from Models 1A and 1B.

76

 

 

 

Hm3oo. V Ammoo. V HeHmo. V Ao3mo. V Ameo. V AooHo. V HomH. V HoHH. pmcHe use»

33mo. mono. on. sow. oHH. ooH. 3mm. 3Ho. assoc A+ omV a

Am3oo. V Hemoo. V AeHmo. V “ammo. V HmmHo. V AmeHo. V “NMH. V HHNH. omcHo ass»

33mo. ammo. enm. 3HN. 3mH. mmH. 0mm. one. noose H+ omV z

338. V Homoo. V 330. V HHHmo. V GoHo. V 338. V GHH. V 82.

mo3o. om3o. mom. wow. moH. mmH. 3Ho.H o3m. HmHanV a

:38. V 83o. V Ammmo. V HHHmo. V EoHo. V 338. V HRH. V 32.

3e3o. ~n3o. new. new. aoH. 3mH. HHo.H o3m. HmHanV z

3m8. V Smoo. V Homo. V RmHo. V 838. V 3mHo. V SHH. V Koo.

oomo. mono. mwm. mwm. mmH. mnH. oom. Noe. A3HumHV o

3H8. V 3H8. V GHHo. V “88. V 3.8. V :68. V ammo. V 33o.

ommo. Homo. mmH. HeH. o3H. HMH. mow. Hum. AHHro o

AmHoo. V AmHoo. V Amoco. V Homoo. V Ammoo. V Amoco. V Ammo. V HH3o.

mmwo. ammo. moH. 33H. NHH. moH. moo. Hmo. Ho V.V o

380. V 2.08. V Knee. V “Hmoo. V 2.80. V Ammoo. V 38. V 38. 3. 03 oHeaoe

omno. mono. NNN. mum. omH. meH. eH~.H mem.H so oHe: onsHa

mm mm «N «m. mH mH 3H 4H oaks

Hoooz H28: Hone: Hoes. H80: H80: Hoooz Hooox «can»?

oouames ooosoHoa coonoHos oosaoHaa

 

 

 

 

 

 

 

 

E
Aeoeoounohom mu Pug unconfined naeoo: mach no.“ oowfimonaoo womlem<

no 9.83m as» masons eunoHoHoeooo oonnonmom as» no moouaHEHIHH Home

 

.nowawnwmoo he mobwm one cameo name mo
mosao> on» ocean oaoeoosoe one no oHoaom one ease vases eunam one mom code no: opcawpoo can
.HP. Hm
m...~ u w you an n m
m .3 page .AHH Home 83 3233s
on» mo odesom no case woman on» you menofiowmmooo nowoooawoh on» he add oases oooV monowowumooo nowmoonwoa
on» no enonuo uncommon on» mdwow>wo he oouonaoo shoe monaob mason can no oucnuo venomous omam

 

 

 

mmH. omo. n3H. moH. woo. 3mo. 8H. 39 "She Sufi
nose... 3 o~V m

mmH. HoH. m3H. moH. moo. omo. moH. moo. 35 use»
noose A+ omV z

mHH. wwo. oNH. ooo. omo. moo. moo. omo. AoHumHV .m
mHH. moo. mmH. omo. omo. moo. omo. omo. HmHumHV z
moo. Hmo. 3oH. m8. omo. moo. omo. moo. 3HumHV o
39 who. woo. 30. 3o. omo. 33o. Ro. 378 o
33. mno. omo. Ro. omo. omo. omo. mmo. 3 VV o
no ow «N «m an mH «H «H 25
H36: H26: H80: Hoooz Hoooz Hoooz H80: Hooo: oxomnoma

823.: 25:33 3332. cannons

 

 

 

 

 

 

 

 

eeHooox noon com 378 332,35 «owns: 23 .Ho 38am Ego on» no noogommI.~H «Home

78

Thus, the modified income-food expenditure relationship requires a larger
number of observations to obtain the same degree of accuracy as that ex-
pressed by a single income coefficient. It should also be noted that the
addition of an extra variable, such as the number of meals, to the basic
relationship makes it possible to obtain the same degree of accuracy with
fewer observations.

There is no objective way to determine when the standard error of a
coefficient is so large that it makes the coefficient useless. The stand»

ard errors of scale values for the two types of adolescents (male, age

 

15-19), and female (age 15-19), and the two types of adults other than the
first two (Types 7 and 8) may be judged to be too large for some pur-
poses. However, if the alternative to using these scale values is a per
capita specification, it may be concluded that the scale values would give

better results.

Food Expenditure Scales with an Alternative Income Scale

 

It was indicated in the above discussion that the food expenditure
scales were estimated with a value of one given to all income scale values

(d 's). This is unrealistic, since the proportion of household income

J
spent on a child is less than that spent on an adult. Thus, the income
or level-of-living scale value for a child would be less than that of an
adult. The author is not aware of any estimates of an income scale that
have been made in recent years.

Brown found that any food expenditure scale based on one income

scale can be adjusted to conform to another income scale by the simple

formula74

ch ° e ( Adj) (38)

 

74Brown, _p_. 5.1.5., p. 455.

79

This is actually a simplification of a more complex relationship (see Equa-
tion 8, Chapter I). To test this relationship and also make an estimate

of a food expenditure scale that is based on a more realistic income scale,
a set of food consumption scales was computed from an alternative income
scale.

To obtain an alternative income scale total expenditures were divided
into nine categories according to the Department of Commerce classification.
A scale value for each expenditure category was then "estimated" on the
basis of the judgement of the author, his wife and a professional colleague.
These scale values are given in Table 13. For some classifications, such
as alcoholic beverages, the addition of a child would be expected to have
no effect on consumption, and hence the child was given a scale value of
zero. The income scale was then obtained from a weighted mean of the speci-
fic scales. The weights given each scale value were the per cent of total
expenditures.75

Estimates of the age-sex equivalent scales for food consumption were
computed with the income scale given in Table 13 for Models 1A, 2A, and ZB
(see Table 14).76 The scale values were smaller than those computed with
the per capita income variable (see Table 10) for the individual age-sex
types other than the first male and female of the household. The difference
between the estimates for Model lA (model with the single income elasticity)
obtained with the "assumed" level—of—living scale and the level-of—living
scale which assumed a value of one for each age-sex type, did not substan-
tiate the use of the adjustment proposed by Brown. [His adjustment was

that the change in the income scale value is approximately equal to the

 

75The per cents were those given by U. S. Department of Commerce,
Business Statistics, XI (1957), pp. 44-45, for the year 1955.

76Estimates for Model 1B were not given because of a programming error.

 

80

 

 

 

 

 

 

 

 

 

 

 

 

mm. 3m. on. 03. mm. mm.
mosHe> oHoom oaoonH
o.ooH Hosea
o.H o.H m. n. m. m. o.mH nooHssom guano
m. m. m. mH. mo. 0. m.m uoHospnoenqssa
m. m. m. m. m. mH. m.HH ooHooom
m. m. m. m. m. m. ~.m ooHoouoco oHooousom
w. m. 3. m. mH. H. o.m uoHpensouooz nospo
m. m. m. H. o. o. 3.m HHo one «so
o.H o.H o. o. o. o. o.3 «escapee oHHosooH<
o.H o.H o.H m. m. w. H.m muasooHo
mm. oo.H ~.H m. m. m. 3.mm oooe
oo=Ho> oaoom onouuomomwm camaoomw
oueHo one» omsHm one» AmHumHV o H3HumHV o AHHnoV o Hoqu o mossoaonomxm
gonna A+ omV henna A+ omV h no : Hoops hhomopou
h 2 mo pnoonom ohnpwonomnm

 

 

 

momma nomuome HseoH>Hqu

 

 

 

oaoom oaoonH ooanom< on» one

ooHeooooso ousoHoaomxm oaHz you uoHaom oHeHooam mo zuoosaHoumzua.mH «Home

81

.mum momma you 4mm. a

n

n

a one .m o5 m 3.5. too m3H. u .x fit. .85 8338 one 8H oHpsa

oomV capewho> oaooca madame you s new: covenaoo oosaob emcee scam cosmopno one: mendeb enemas

 

 

 

 

 

 

 

 

mm. No. Hm. mm. mm. om. 3. 3.5 55
henna A+ ONV h

mm. mm. 3w. mm. mm. 30. mm. out—Ho use»
.HOfiO A... ONV 8

mm. sH.H HH.H oo.H oH.H 3m. mo. AmHumHV e
mm. mH.H oo.H oo.H ~o.H 3m. 8. HmHumHV 2
on. No. mm. mm. mm. mm. 3. 3H-~HV o
mw. 8. 3m. 8. 3m. on. m3. AHHnoV o
3m. mm. 33. 3. m3. 3. on. G v V o
oo.~ oo.~ 8.~ oo.~ oo.~ oo.m .H one 2 subs

he . mm mm am <~ H «H as
so» one Hooo: Hooo: Hoooz Hoooz Hoooz H802 samuome

0.85333 3332. onions . monsoon...»

 

g

ooasom oaoonH on» new nosHo> oomswn< noes oaooo: comma
you moaneasunoo oooh you condom anodebdovm Romnom< no noeeadpomul.:a edema

82
change in the specific scale multiplied by the income elasticity (Equa-
tion 38).] The change in the food consumption scale per one per cent
change in the income scale-was 0.202. This is less than the income elas-
ticity of 0.275 which is applicable to this model. In addition, there was
a difference between the individual age-sex types with respect to this
relationship. The change in the food consumption scale per unit change
in the income scale averaged 0.145 for the two adult types other than the
first male and female of the household (Types 7 and 8) and 0.224 for the
other types (2 through 6).

Since the scale values of Models 2A and 2B were computed with an
income elasticity for each household type, the relationship between the
income scale and the food expenditure scale would be more complex than was
indicated by Equation 8 (Chapter II). This equation was derived under the
assumption of a single income elasticity for each household type. Thus,
the relationship proposed by Brown would not be expected to hold. With
Models 2A and 28 the income elasticity would be different for each indi-
vidual age-sex type since different income elasticities for each house-
hold type were used. A food expenditure scale for a child under six years
of age, for example, would be estimated with a different income elasticity
than that for a child from 12 to 15 years of age.

Even though the above problems exist, there was some indication
that a relationship existed between the food expenditure scale and the in-
come scale that is simple enough and yet reliable enough to make adjustments
in the food consumption scale for alternative values of the income scale.
The differences that were found to exist between the scales of Models 2A
and 2B computed with the per capita income variable and the "scaled" income

variable indicate that the following relationship can be utilized.77

 

 

77The "scaled" income variable refers to household income deflated
by the "assumed" income scale given in Table 13.

83

c N”
)__2__.L.L. (39)

Ac 5 k (Ad EdJNJ

J J J

A change in a specific scale value is thus approximately equal to a change

in the income scale value multiplied by a factor k which has a different

J
c N
value for each individual age-sex type, and the factor —§fdlfil—' This latter
JJ

factor would, of course, be equal to one if the income scale values are

equal to the specific scale values (cJ = d1, for all j). The values for '

k are given in Table 14.78 These values take into consideration the dif— if

J

ferences in income elasticities used to estimate the various scale values. r

n."
O 9 § .

The k 's also include effects other than those resulting from the income

J

elasticities, since they were estimated from empirical differences. Of
course, these estimates (kj's) were made from very few observations and

may be subject to the errors usually associated with such estimates. The
c N ‘

(__.L.1
551%

was found to be important in explaining the differences between the scale

last term of Equation (39), ), was included in Equation (8). It
values computed with the per capita income variable and the "scaled" in-
come variable.

The relationship expressed by Equation (39) could not be substantiated
with the weighted Model 2A and the weighted Model 2B scales. The values of
the kj's computed from the estimated scales of these models differed sub-
stantially from those computed from the scales of the two unweighted models.
In addition, substantial differences were noted between the values of the
kj's of these two weighted models. It is, therefore, not possible to ac-
curately adjust the weighted Model 2A and ZB scales for a change in the

value of the level-of-living scale.

 

78The k 's were computed from Equation (39). Given values for cj,

d Nj’ ch, add 'Adj’ the kJ's are the only unknowns.

j!

84

Summarygand Conclusions

 

The age-sex equivalent scales for United States food expenditures
computed with the four alternative models were acceptable from the stand-
point of a few simple criteria. That is, none of the scale values for the
individual age-sex types were negative and generally less money was Spent
for the food consumed by younger children than for the food consumed by
adults. The scale values for the models in which food expenditures were
adjusted with an income elasticity for each household type were smaller Ed
relative to the "standard" (the first male and female of the household) 3
than those computed with the models in which food expenditure was adjusted
with a single income elasticity. The question of whether the scales com-
puted from the models with income elasticities for each household type are
capable of explaining more of the variation in food expenditures than the
scale computed from the models with a single income elasticity, at this
point, is open to question. Likewise, it is not known if these scales
computed by adjusting food expenditures for the number of meals and those
computed by weighting each observation in inverse proportion to the standard
deviation to account for unequal residual variances are "better" than those
computed without these adjustments. One way of finding out which scales
are "best" is to apply these scales in analyses of food expenditures.
However, it can be argued that on the basis of the conceptual framework
of this chapter the scale computed with the income elasticity for each
household type, with the adjustment for number of meals, and with the

"weighting procedure is "best."

CHAPTER V

EVALUATION OF THE AGE-SEX EQUIVALENT SCALES

Introduction

 

The analysis of Chapter IIIhas indicated that the income elastici-
ties of United States food expenditures vary with household composition.
Subsequent analyses (Chapter IV) have shown that scales for specific ex-
penditures can be computed even though income elasticities differ among
household types. The method developed by Brown was modified so that in-
stead of adjusting food expenditures for differences in income with a single
income elasticity, the adjustment was made with a different elasticity for
each household type.

With different income elasticities for the various household types,
the use of an age-sex equivalent scale to Specify the effects of the age-
sex composition leaves some of the variation in expenditures caused by
age-sex composition unexplained. A scale would, however, express the rela-
tive amounts of expenditures used by individuals of varying age and sex
with a given population. A change in the age-sex composition of the popu-
lation that would change the proportion of persons in the various household
types would change the relationship between income and food consumption.

A scale would not be capable of explaining this change.

A change in the proportion of persons in the various household
types may have an effect on the scale values themselves. Such a change
would change the income elasticities used to compute the scale values, and

depending on other accompanying changes, the scale values may be altered.

85

 

86
Thus, the scales for United States food expenditures may not be stable.
A change in the age-sex composition of the population may have some effect
on the scale values. The important question is whether a scale is an im-
provement over other methods of Specifying the effects of age-sex composi-
tion. The answer will, of course, depend on the alternative method and
the purpose for which the specification is to be used.

One alternative is to use a per capita specification. With this
alternative a value of one is given to the scale values for all individual
age-sex types. The use of an age-sex equivalent scale, even though its
values may be unstable, should be superior to the use of per capita food
expenditures. That is, only a small change in the scale values would be
expected with relatively large changes in the age-sex composition of the
population. The age-sex equivalent scales should, thus, be capable of
explaining more of the variation in age-sex composition than a per capita
specification.

The analysis of the last chapter included several alternative
models for computing an age-sex equivalent scale for food expenditures.
The analysis did not show which one of these scales was "optimal" or what
differences exist among the scales with respect to their "performance” in
any given analysis of food expenditures.

The purpose of the analysis of this chapter will be to demonstrate
the differences that exist in two analyses of food expenditures using the
various scales and a per capita Specification. One application of the
scales will be made to cross sectional and the other to time series data.
In addition, an attempt will be made to explain the algebraic difference
that exists between the application of scales to time series and cross-

sectional data.

 

I l I ll ll '1“
l I '11 Ill'l II 'II. III I

87

An Application of the Food Expenditure Scales
to Cross- Sectional Data

 

 

To demonstrate the differences that exist between deflating food
expenditures with an age-sex equivalent scale and deflating food expendi-
tures with the number of persons (yielding per capita food expenditures)
least squares estimates were made of the food expenditure, income, number
of meals relationship. The simple relationship which is expressed with a
single constant term, a single income coefficient, and a single number of

meals coefficient was utilized. This relationship is not as "complete" as

l—
. _rrl
I.

the more complex one in which income is expressed with a coefficient for
each household type. It was used because of its simplicity and also to
observe the results of applying the scales to a relationship that is more
commonly used. For many applications this simple relationship gives a
sufficient amount of information.

The independent variable, household income, was deflated by the
income scale "assumed" in the previous chapter (see Table 13). It was
thought that this would be more realistic than using per capita income.

The dependent variable, household food expenditure, was deflated by seven
different‘"scales":

(l) A "per capita" scale in which all scale values had a value of one.

(2) The scale which was obtained by adjusting food expenditures with
a single income coefficient and a single number of meals coefficient
(Model 1B).

(3) The scale obtained by weighting the observations of the previous
model (Model 1B) in inverse proportion to the standard deviation
of the dependent variable.

(4) The scale obtained by adjusting food expenditures with an income

coefficient and a constant term for each household type (Model 2A).

88

(5) The scale obtained by weighting the observations of Model 2A in
inverse proportion to the size of the standard deviation of the
dependent variable.

(6) The scale obtained by adjusting food expenditures with an income
coefficient and a constant term for each household type and a single
number of meals coefficient (Model 2B).

(7) The scale obtained by weighting the observations of Model 2B in
inverse proportion to the size of the standard deviation of the
dependent variable.

The food expenditure scales with the income adjustment with a single coef-
ficient and with no adjustment for the number of meals were not considered
in this analysis. It was thought that the inclusion of them would add very
little to the analysis.

The second independent variable, number of meals eaten at home per
household, was deflated by the "weighted Model 2B" food expenditure scale.
The weighted Model 23 scale was used because the logic of the analysis of
Chapter IV indicated that it would be the "best" scale. Deflating the num-
ber of meals per household with a scale puts this variable on a per adult
equivalent basis as was done with the other two variables of this relation-
ship.

The algebraic formulation of the relationship is:

) (40)

where X is food expenditure per household
Y is disposable income per household

M is number of meals eaten at home per household

 

89

c is the value of the ith food expenditure scale for the jth

1]
individual age-sex type (i = 1, 2, 3, 4, 5, 6, or 7)

d is the value of the income scale for the jth individual age-sex

J
type
N1 is the number of persons per household of the jth individual
age-sex type.
Two forms of the above relationship, the linear and the double-

log were fitted. Although other forms might yield as good or better re- I"
sults than these (for example, the semi-log form), it was thought that
these two were sufficient to demonstrate the differences among the four

scales and the per capita deflating procedure. The equations for these

two forms are:

X Y M

2
where u is assumed to have zero mean and variance or , and

, Y b' M b' u'
5 ‘1ij = a (E dJNJ) 1 (Z °7JNJ) 21 ’ (A2)
J J

 

where u' is assumed to have zero mean and variance (7&2. It should be
noted that variance of u’ in Equation (42) would not be expected to be con-
stant among household types since the analysis of individual household
types (Chapter EU) showed it to be unequal. However, no provision was
made for weighting the observations. It was thought that the purposes of
this analysis could be accomplished without it.

The data that were used were the 1751 observations of United
States urban households used to compute the age-sex equivalent scales.

This application then provides an opportunity to observe the differences

90

in the scales without any change in the sample. Thus, other variables
affecting food expenditures and perhaps the scales themselves would not
change.

The use of the same sample together with the use of the same
scales to deflate the dependent variables should at least partially isolate
the effects of the four different scales and the per capita specification
on the food expenditure variable. This isolation would not be complete
since there is a relationship between the income scale and the food ex-

penditure scale. That is, the income scale by definition is composed of

I...—
I. N t

the scales for specific commodities. A change in the values of the food
expenditure scale would, thus, change the values of the income scale.

The results of fitting the above equations show that the use of
the Model 2B scale to deflate food expenditures yielded the highest co-
efficient of multiple determination (R2) (see Table 15). The R2 values
are, of course, an approximate measure of the proportion of the variance
in the dependent variable explained by the independent variables.79 Thus,
more of the variation in food expenditures was explained with the use of
the scale of the unweighted Model 2B than with either of the other three
scales or the per capita specification. The Model 2B scales were com-
puted by adjusting food expenditures with an income elasticity and a con-
stant term for each type as well as for variations in the number of meals.

It is possible that the model with the highest R2 value is not the
one in which the sum of the squared deviations from the regression line

are the smallest. The most obvious case of a low R2 value also associated

with a low value of these squared deviations is the one in which the re-

 

The true proportion would be given by?2 = 1 - (1 - R2) figfiéf'

where N = number of observations and P = number of independent variables.

79

91

one: nuance one one no monanb any

.AmmV noHnnnnm nan

monam> canon oaoonw on» no omcoeo one mom ooumnwoo one: oaoonfi oewmoo hoe now: ooumnnoo wagon oehnpdonomxo

oooo nan mH Hone: non oonnanno noHnon on»

”humans mewsoflaom on» no ooqaoono one: home

.uopenmo puma

on» ow oopnoaoo you once oaooo second :oopoawpoo: om» no comma oaooos omen» how eozdnb cameo ogao

 

 

 

 

 

 

 

 

 

 

moo3. mm3m. o3om. moo. omo. omo. 3o~.~u mm Hone: a
mmo3. ommm. ommm. omo. mmm. mmm. «Hm.mu mm Hone:

mmo3. mo3m. momm. mmo. Hmm. Hmm. mom.mu am Hoooz n
mmo3. 3wnm. m3mm. m3o. own. mmm. mm~.mr 3m Hooo:

ommm. HH3m. mom. mom. o~m . own. $9.? an Hoooz a
moon. 3o3m. new. moo. mHm. «Hm. Hoo.mr an Hoooz
mmm3. moon. mmmm. oom. mmm. mmm. now.H- nanno use

nnmmwmnmmummmmmW

Nmo.3 3mm.m mn3m. Hom. m3m. moHoo. mom. mm Hone: a
oMH.3 mmm.m ommm. mom. mom. 3oHoo. oom. mm Hooo:

omo.3 mmm.m mm3m. Hem. mom. moHoo. 3cm. em Hooo: n
m3H.3 mom.m mmmm. mom. 3mm. 3oHoo. now. am Hone:

moH.3 mmo.m mom. mom. Hmm. 3oHoo. Hem.H an Hoooz a
oom.3 mmo.n mom. Hmm. 3mm. 3oHoo. nom.H an Hana:
mom.m Hom.n m3mH. omH. oom. 3oHoo. mmm.H nanno non

anon neonHH

nonnannonum onnannm AmmV “.mn onn mnV anHoHnnnHm a.Ho one HnV H.n one nV monnnHononxm

once on» noHnnnannonon nnoHoHooooo onoonH nnoHoHnnooo anon none onnnHmon

no noHnanan mo nonnm oHanHn: on mHnnz on .02 oonnanmm snoonH nnnnnnoo on noon oHnom

ovheoaopm pohoonopm anowowmmooo ooenswumm oopnawomm oopnaauom

 

 

open Hdnoapoomuomoao women monocouusaom oesofioeomxm ooohuoaoonH emu o»

noHnnoHcHoonm nanno use one one noHnom onnnHononxm once me on» mnHmHnme no nannom--.mH oHnna

 

n n n n
.Buom won 0.38m. on» now A 2 “WV moa u .h was Show .80an on» no.“ balm“. u .5 once: .hm

 

0

, n n w
.58 mo.” 3350.0 on» how AIZMIWMV MOH u Non odd .Alzlnwlwv mad u Hun . WWW-IV

 

 

 

 

 

 

 

 

 

 

n n n n n n
woa u .5 one anon noon: on» no.“ 2:0 W a Non one . z W N u ax . 2 MW a In cheek muddlhm n
mom. mom. awn. How. mom. Now. 0.3. oo.~ ma Homo: a
oo.H. Nam. mom. o8. ohm. «mm. :wm. oo.m m." Homo:
no.3.“ ween
menu 9050 Home henna Aoaumd .m 373V 2 AJHINC o 3.73 o GVV o A... 03 .m one
3 omV m 3 o~V : x 33m

Auofivaoov II . ma 0.369

93

gression line is perfectly horizontal. In addition, the values of the de-
pendent variable must be subject to some variation so that all observed
values do not fall on the regression line. A steeper regression line could
have a higher R2 value but the same sum of the squared deviations from the
line. One purpose of deflating household food eXpenditures by a scale is
to improve the estimate of food expenditures. The standard error of the

'x x ) provides such a measure.80 The results shown in Table
1 2

15 indicate that with the linear form weighted Model 2B is optimal with

estimate (Sy'

respect to this measure, and with the double-log form Model 2A is optimal.

The standard error of the estimate (Sy'-x x ) may be correlated
l 2

with the size of the dependent variable (y'). In this case it would not be
the one which gives the "best estimate" of food expenditures. The standard
error would change with the size of the sum of the scale values. The stand-
ard deviations of the dependent variables (Sy.'s) indicate that with the
linear form there would be a positive correlation between the size of the
dependent variable and its standard error. With the double-log form this
correlation would be negative. Therefore, with the linear form, it can be
concluded that the weighted Model 2B scale yields better results than the
per capita specifications. The above correlation prohibits any further
conclusions. With the doubleulog form it can be concluded that the Model
2A and the Model 28 scales give better results than the Model 1B scale.

The conclusions reached by examining the standard errors of the estimates,

 

80 X Y M
Y. = “'::jf“ , X1 = -—_?f7§__ , and x2 = -f—-7;73- for the linear
2 J J 2 J J 2 J J
form and y' = log(-J&-—-—), x1 = log(--X--), and x = log(-¥fl--) for
zchj 2(1ij 2 2 Cij

the double-log form.

 

94
thus, do not conflict with those reached by examining the values of R2.

A few additional aspects of fitting these models to the two forms
should be noted. The double-log form gave better results in terms of R2
values than did the linear form. The coefficients of multiple determina-
tion ranged from .198 to .266 with the linear form and from .232 to .304
with the double-log form. These results agree with those obtained in a
previous study in which household composition was held constant by es-
timating separate equations for each household type.81

Another result that should be noted is the lower values for the
income elasticities obtained with "scaled" food expenditures than with
per capita food expenditures. The income elasticities of the "scaled"
food expenditure models varied in proportion to the sum of the scale
values, and, thus, inversely proportional to deflated food expenditures.
That is, Model 1B had the lowest sum and the lowest elasticity while
weighted Model 28 had the largest sum and the highest elasticity. In
contrast to the income elasticities, the "scaled" food expenditure models
yielded higher coefficients for the number of meals. Therefore, more of
the variation in food expenditures was explained by the number of meals
and less with income with the "scaled" models than with the "per capita"
model. Among the equations in which food expenditures were deflated by
the four scales, the size of the number of meals coefficients was not
obviously correlated with the size of the sum of the scale values. The
largest coefficient was obtained with the unweighted Model 1B, and not
the weighted Model 1B which had the largest sum of the scale values.

The application of the four scales and the per capita specifica-

tion to the income, food expenditure, number of meals relationship demon-

 

 

81Herrmann,_p_. cit.

95

strated that for this particular situation the Model 2B was optimal

with respect to the largest coefficient of multiple determination. The
standard errors of the estimates did not invalidate this conclusion. The
weighted Model 2B scale was considered to be the optimal scale from a
priori considerations. For example, the weighting procedure applied to
the various scales was intended to increase the efficiency of the esti-
mates. Thus, the weighted Model 2B scale should from a priori considera-
tions give better results than the unweighted Model ZB scale.

The sample that was used for this analysis was the same one that
was used to compute the scales, and hence, the results do not aid in
generalizing the use of these scales to another‘population. The results
have demonstrated that "scaled" food expenditure is a more relevant
variable on which households base their purchasing decisions than is per
capita expenditure.

The Application of Age-Sex Equivalent
Scales to Time Series Analysis

 

 

The age-sex equivalent scales computed for United States food
expenditures were obtained from cross-sectional data. Can these scales
be used to improve analyses of time series data? With time series data
income and food expenditures are given for aggregates of households. This
is in contrast to cross-sectional data where the amount of food expendi-
tures and the amount of income are given for each household. These are,
of course, the usual differences between cross-sectional and the time
series data. The following is a discussion of some of the problems en-
countered in applying scales computed from cross—sectional data to analy-
ses of time series data.

‘Algebraically, the time series food expenditure variable for any

given year can be expressed as 2) Xi’ where 1 goes from 1 to H and H is
i

 

96
the total number of households in the population.' Similarly, income can
be expressed as 23Y1. The time series data on the number of persons of
the various individual age-sex types are also in the form of population
totals and not household totals. Algebraically the number of persons of
a given age-sex type can be expressed as 2>Nij' The income-food expendi-

i
ture relationship for time series data would be

2x1 ZEY

A ll , _—_ lay—L

The "cross-sectional" relationship from which the food expenditure scales

 

 

a.
c...

04

z
I‘-
u.

were computed was

X Y1
fi— = £( ) (44)
3’3 J 1:

 

EdN
3311

An aggregate of this relationship could be obtained by summing over all

households in the population and dividing by the total number of house-

holds That is,

X Y

§(-§—-%3§E)/ H = f(§(-§rfim)/H) (45)
.1' J

The food expenditure and income scales computed from cross-sectional
data cannot be directly applied to Equation (45). If Equation (43) were
equivalent to Equation (45) food expenditure and income scales could be
applied to the "time series" relationship.

It can be shoWn that under certain conditions Equation (43) is
equivalent to Equation (45). One condition is that the terms ( 2:°ij)

J

) be constants. To prove this it is sufficient to show that
X §)JX
i
2( )/H=
c N c
133111 331

and ( ,Zfd N
111 1

N
§1J

 

97
since the same type of expressions are on the right hand side of Equations

(43) and (45). That is, if the equality of the food expenditure variables

are proven, -the proof of the equality of the income variables is essentially

the same.
If 2 chj = K, than
.1
z: hxi 2 x1 §X1
(—-----)/H= (--->/H -—-—..
i 2 chij 1 K HR
.1
Also, the denominator of the "time series" expression is Be] 2 Nij
. J 1

c1(N11+:N21+ . . . 4-NH1)+. . . +ck (N1K+N2K+. . . +NHK). By
multiplying and sorting this eXpression becomes, (clNll + clez + . . .

+ CKNIK) + . . . + (clNHl + cZNH2 1? .7 . . + ckNHK), which is equal to
(3}cJNlj + chjuzj + . . . + chjNHj' Since §chlj = 32?ch“ = . . .

ECJNZH = 2 chj’ from the assumed condition, the above entpression is
equgl to H 2.313% = HK. Thus, under the assumption that the age-sex
composition of all households is equal the time series expression is
equivalent to the cross-sectional expression. This is by itself a very
unrealistic assumption. The purpose for computing scales to begin with
was to specify variations in age-sex composition. However, this assump-
tion can be used along with another condition to form a more realistic
condition which will show some of the difference between the results ob-

tained with the time series and the cross-sectional formulations.

The second condition is that

X X X

 

 

—-———1--—- = --—2—-——— =. . . = . H and
JEClej ? CJNZJ 12 chHj

Y Y Y

1 2 _ _ H
21} dlej SJ: djNZj zjjchH'j

98
That is, if food expenditures and income per adult equivalent were
equal for all households the time series and crosswsectional formulations
would be equivalent. To prove that under this assumption Equation (43) is

equivalent to Equation (45) 1': will be sufficient to show that

X X

 

 

 

 

 

 

 

 

 

1 _ 1
2 c N ’ 2 c SN
1 J 11 J J j 13
since under the assumed condition,
2 x1 x1 x2 xH
( ) / H = .. = ----- = . . . = .
1 SJ °jN13 '32 ch11 23; oJN21 §CJNHJ
Thus,
x1< 23“J"2J> x1<§°JN31> ~x1< §CJNH1>
x2= ch 'X3= ch ""XH‘ ‘
J J” 15” v 2 °JNIJ
’3 X1 3
The time series term [2 cj 2 N13 1 is equal to
j i
N
2 °j 23) 2c N
[X+x1(j , +...+§1(J)HJ)1/ 2c2N.
l E? chlj ?? chlj .j j 1 ij

This expression is equal to

Xl(c1N11 + ... + c N ) + X1(c

KlK +eee+cKN2K)+ees+X1(CN +eee+CN )

1N12 1 H1 k HK

 

 

(§:cJNIJ) (gyENU)

By multiplying and regrouping this expression is equal to

 

x1( 515 312%) ' x1
(ijchlJXZ-‘ljc‘1 §Nij) EchJNlj—o

The assumption of no variation in income and food expenditures

per equivalent adult is also unrealistic. If there were no variations in

99
these variables there would be no way of estimating the income elasticity
or the constant term of the income-food expenditure relationships. In—
come and food expenditures would be constants and not variables. Either
condition 1 or condition 2 is sufficient for making the time series formu-
lation equivalentto the cross-sectional formulation, but they are both
unrealistic. This study has used the device of formulating household types

in which age-sex composition was held constant, and, thus, condition 1 was

met for each type. If condition 2 is applied to these groups of house- Sf
holds and not to individual households the following condition can be de- y

rived: if the expected value of income and food expenditure per equiva-
lent adult are equal for groups of households in which the age-sex compo-
sition and the number of persons per household remain constant; the time
series formulation is equivalent to the cross-section formulation.

Thus, the two unrealistic assumptions may be utilized to form an
assumption that may be approximately met in some populations. The devia-
tions from this condition help to explain the possible discrepancies be-
tween the results of time series analyses and cross-sectional analyses.
For United States households it is not expected that this single condition
will be met. Households with children usually have lower incomes per
adult equivalent than do households without children. This would also
make a difference in the expected food expenditure levels (per equivalent
adult) of these two groups since food expenditures are correlated with in-
come. The difference between the income and food expenditure levels would
be less in countries that have allowances for children and, hence, the
difference between the time series and cross-sectional formulations would
be less.

Other differences,of course, exist in the results obtained from

using cross~sectiona1 and from using time series data. The number of

100

meals variable which has been important in explaining the variation in
food expenditures between households may not be significant when applied
to time series data. If there were no change in the average proportion
of meals eaten away from home by all members of the population over time
the coefficient would be zero. Other differences also exist between
1'time series” and "cross-sectional" analyses but they will not be dis-
cussed here.

It has been shown that the application of the age-sex equivalent
scales to United States time series data is strictly applicable only
under certain conditions which have not been met. But what can be done
with time series data to specify the effects of age-sex composition? Will
the application of an age-sex equivalent scale computed from cross-
sectional data be an improvement over other methods? If so, then even
though the conditions under which these scales are applicable are not
exactly met, it can be defended pragmatically as the "best" known method.

One alternative would be to use per capita food expenditures. To
test the proposition that the use of a "scaled" food expenditure variable
would give better results than this, the United States income-food expendi-
ture relationship over the fourteen year period, 1948-1961, was examined.
The years immediately preceding this period were affected by the ration-
ing and price policies of World War II. Data giving the number of persons
of each age-sex type were not available on a yearly basis prior to 1940
which prevents the use of a period in which low birth rates had a sub-
stantial effect on the age composition of the population.

The relationship that was used was, algebraically,

§x1 XY

"‘7‘?" = a+b
12151311

 

+ u. (46)

 

lOl
Disposable income (2 Y1) was deflated by the consumer price index'to
remove the effects of infaltion from this variable. Food expenditures
( 2X1) (excluding alcoholic beverages, but including food eaten away
frim home) were deflated by the retail price of food which yielded a
measure of the Quantity of food purchased. It was suspected that this
quantity was affected by the price of food but since the purpose of this

analysis was to compare age-sex equivalent scales and the per capita

specification, this variable was omitted. 'The justification for this is

03“." m1

that the price of food should have approximately the same effect on each

scale.
The number of persons of the various ages and sexes ( ZINij‘s)
i
were obtained from.the Department of Commerce estimates.82 The age-groups

used by the Department of Commerce were not the same as the ones of this
study but they were similar enough to be used. By applying the scale
values computed for a child less than six years of age to the number of
children under five years of age, and by applying the mean of the scale
values of the child age 6~ll years and that of a child 12-14 years to the
number of children age 5-14 years, the discrepancy in children's age
groups was eliminated. The number of females or the number of males other
than the first ones of a household is not given by the Department of
Commerce. The discrepancy between the first and the second male or be—
tween the first and the second female that was prevalent for some scales
was therefore, of necessity, ignored. The scale values used for the adult
male and the adult female were formed so that their mean was one. The
difference between the sexes was obtained from the differences in the

scale values between the adult male and the adult female other than the

 

fiv—v— v

82H. S. Department of Commerce, Current Population Reports, Series

P'ZO.

102
first ones of a household. The age-sex types used for the time series
analysis are given in Table 16.

Three food expenditure scales were used for the analysis (see
Table 16). These three were selected because of their range of values.
That is, the scale values of Model 1A were generally the smallest and
the values of weighted Model ZB were the largest with respect to the
types other than the adult male and adult female. These three scales
were those computed with the use of the assumed income scale and not the
per capita income variable. The income variable was also deflated by
the income scale assumed in the previous chapter. In addition, equations
were fitted in.which food expenditure and income were on a per capita
basis.

The linear form of the income-food expenditure relationship was
used throughout. Upon examination of a graph of this relationship it was
decided that this form.would fit as well as a curvilinear form. However,
the relationship might be of a curvilinear form if a longer time span
were used.

The results showed that the highest proportion of the variation
in food expenditure was explained when it is deflated with the scale
values of Model lA (see Table 17). This indicates that, over time, this
dependent variable is a "better" indicator of how consumers react to a
change in income than is food expenditure deflated by any of the other
"scale" models or per capita food expenditure. It should be noted that
this variable was optimal by the criterion of the highest coefficient of
multiple determination, (R2), if income was deflated by the assumed scale
or was on a per capita basis. Per capita food expenditures yielded the

lowest R2 values.

 

103

 

 

mm. mm. mm. 0a. A... 03 .m
NO.H mood mo.H OH.H A+ Omv :
mm. NH.H HH.H mm. AmHImHv h
mm. wflofi mO.H mm. anIMHV S
mm. in am. me. $75 0
mm. mm. +3. on. Amvv 0

came» mm MN 4H omha

oaoonH Home: Home: Home: Momlom<
depends: 3335

 

 

 

 

3.60m. 593mg cook
E

38 not.» 25. no mange 8: soc some 828 oqoagaswm xomuome canted canon.

 

104

 

 

 

 

 

 

 

 

 

 

 

3 an a n «a a
. IE- " HM 6nd E u .h 22.3 Hanoinm.

Hh N Hun W a
ooe.mm mom.em new. ammo. moon. omnm.mm now. Home. amen. ooaaoo com
mme.wm 0mm.mm mmn. ammo. amen. mom.mm Hem. ammo. omen. mm Hoses“:
omw.uaa :Hm.um mom. mmmo. comm. umm.am 5mm. mono. ammo. mm Hocoz
womoana 533mm m3. wad—u. mmmm. mom.om 3m. snub. mmmw. .3 Homes

oaposuo> consano> hoaoaomoam conseno> apaoaomoam m

oanmmmomon camsoamom oaoomH 9 mm mausmauom oaoouH n N uohdufiomomxm
mo - ooom oooecon op

ooqmwno> sowpmo hem oaoomH aquambwsnm pauc¢ Mom oaoonH coop Home: oamom

 

 

soon ooaaom oaaa op onsooooam aoaooacon
avenue com one oaooo: oaoom ooaga muahacm<_mo moaouomuu.ea oases

 

105
It should be noted that the residual variances (82y.,x ) were low-
est for the per capita food expenditure variable.83 However, with the
linear form there was reason to' believe that the residual variance. was
correlated with the size of the dependent variable. The variances of the
dependent variables give some support to this. By using the double-log
form the food expenditure deflated by the scales of Model 1A should yield

the lowest residual variances and, hence, be optimal with respect to this

criterion. E
The optimal "scale" model for the cross-section analysis was uny- 3
weighted M’d‘l 13‘ Th? "1““ ‘f 3113 “‘1. (other than the adult male k‘

and female) were higher than those of'Modsl Hi. What are the reasons for
the discrepancy between the "time series" results and the "cross sectional"
results? Several explanations of this phenomenon could be put forth. The
most obvious one from the viewpoint of this analysis is the failure of
United States food consumption data to meet the condition necessary to make
the time series formulation algebraically equivalent to the cross-sectional
formulation. To obtain some indication of. this difference, food expendi-i
tures for the 1751 households used for the'cross section analysis were
deflated by both formulations. With the crosspsoctienal formulation,

§( Kiln”) / H, the use of the Model 23 scale yielded a mean value for

food expenditure of 9.460. The use of?!“ same scale and the same data

xi

( , V
§ aJ § N1.1
8.571. Therefore, if the Model 23 scale was applied to time series data,

with the time series formulation, - ), yielded a value of

it would underestimate the value of the dependent variable. The scale

83y' is equal to the per capita or per adult equivalent food ex-

penditure variable and X1 is equal to the per capita or per adult equiva—
lent income variable.

106

values for Model 1A other than the adult male and female are smaller than
those of Model ZB and their use would tend to correct for this under-
estimation.

This, of course, is not the only source of the difference between
the results obtained with time series data and with cross-sectional data.
Consumers may react differently to changes over time than the differences
between households indicate. Another source of discrepancy in this com-
parison is that the scales were computed from urban data. The application
to time series data implies that these scales also adequately specify the
age-sex composition of rural households. Still another difference is that
the cross-sectional data covered a week, whereas the time series was an
annual average. To these sources of discrepancy must also be added the
problem of estimation error. The "time series" analysis was based on only
fourteen observations. An analysis that covered a different time span 9
might give different results arising from nothing but random errors. The
problem of estimation error also applies to the cross-sectional analysis
and to the scales even though 1751 observations were used.

One of the results of this analysis that is difficult to explain
is the difference between the coefficient of multiple determination (R2)
obtained with the use of per capita income and the "scaled" income variable.
The use of the per capita variable yielded slightly higher R2 values. There
are many possible explanations for this, such as, the income scales (which
were assumed and not estimated from actual data) were not accurate, and
the error that arises from not meeting the condition necessary for applying
the scales estimated from cross-sectional data to time series analyses.
This will not be explored further since the primary purpose of this analy-

sis was to study the food expenditure scales.

 

107

Another aSpect of this analysis that should be noted is the dif-
ference in the size of the income elasticity obtained with the optimal
scale (Model 1A) and with per capita food expenditures. The use of the
Model 1A scale yielded an income elasticity of either .347 or .413 depend—
ing on which income variable was used. However, the use of per capita
food expenditures yielded an income elasticity of either .205 or .245
depending on which income variable was used. This was in contrast to the
"cross-sectional" estimates where the use of the per capita food expendi- F”
ture variable yielded the highest income elasticities. The explanation

of this phenomenon was thought to be beyond the scope of this study.

Summary and Conclusions

In this chapter two applications of the scales computed from the
1955 survey of urban United States households were made. The first appli—
cation was made by applying the scales to the same cross-sectional data
that they were computed from. The second application was made by applying
the scales to a time series analysis of the income food expenditure re-
lationship of the United States for the fourteen year period 1948-1961.

In both applications the use of scales was compared to the use of the per
capita deflating procedure.

In both the time series and the cross-sectional applications the
use of scales for deflating food expenditures gave better results in terms
of the proportion of variation in food expenditures explained than did the
per capita deflating procedure. The "optimal" scale with respect to the
above criterion for the cross-sectional analyses was unweighted Model ZB.
The scales computed with an income elasticity for each household type gave
better results than did the ones computed with a single elasticity. It

was expected on the basis of the analysis of Chapters III and IV that the

108
weighted Model 2B scale would be'"optima'l."'1 However, this single appli-
cation only proved that the "unweighted" scale would sometimes give better
results than the “weighted" one.

The optimal scale with the time series analysis was unweighted
Model 1A. The values of this scale (other than the first adult male and
female of the household) were smaller than those of Modal 2B. One dis-
crepancy between the results of the "time series" and Vtross-sectional“
analyses was the algebraic difference in the incomedfOod expenditure re-
lationships. The two relationships would have been equivalent if the mean
income and food expenditures per equivalent adult for groups of households
with a constant age-sex composition had been equal. A rough test of this
was made for United States food expenditure data and it was found not to
hold. Thus, some of the discrepancy in the results stems from the algo-
braic unequivalence of the time series and cross-sectional formulations.
One of the needs for further study is to find a way to adjust the food
expenditure scales, computed from cross-sectional data, so that they can
be accurately applied to time series data.

The above two applications are just two examples of analyses to
which the food expenditure scales can be applied.‘ The improvement in the
results from using scales over using per capita variables indicates that
improvements in demand analysis can be made by using an age-sex equivalent
scale. In fact, anywhere that food expenditure should be expressed on a
per adult equivalent basis, the scales usually would be an improvement

over specifying food expenditures on a per capita basis.

 

CHAPTER VI

SUMMARY AND CONCLUSIONS

Introduction

 

The age-sex equivalent scale is a method of specifying the effects
of age-sex composition on the consumption of a particular commodity. A
scale would express the quantity of some commodity consumed by the persons
of various ages and sexes as a proportion of that consumed by some "standard"
person. In addition to specific commodities, this concept can be applied
to an aggregate of all commodities. This is called an income or level-of-
living scale.

Some of the age-sex equivalent scales for food have been based on
nutritional requirements. Others have been based on actual expenditures
for food. For most economic analyses the latter scale would be preferable
to one based on nutritional requirements. However, the problems involved
in computing a scale based on actual expenditures together with the limita-
tions that such a scale would have in expressing the effects of varying
age-sex composition have led to few attempts to compute them. Under such
conditions, their use has also been limited. It was the purpose of this
study to explore the problems of computing an age-sex equivalent scale for
United States food expenditures and if possible to compute such a scale.
Another purpose was to evaluate the computed scale by comparing it to other
methods of specifying the effects of age-sex composition.

With age-sex equivalent scales the relationship between income and

food expenditures (with cross-sectional data) can be stated as

109

 

110

Y
——- = f < > <47)
3‘3 “J”: 2, ‘5“:

where X is food expenditure per household
Y is disposable income per household

N is the number of persons of a particular age and sex per household

J
J

c is the scale value for food expenditures for a person of a particu-
lar age and sex
d is the scale value for income for a person of a particular age
and sex.
This relationship states that food expenditures per equivalent adult are
a function of income per equivalent adult. This is the basic relationship
for this study. From it formulas for analyzing the income-food expendi-

ture relationships were derived as well as formulas for computing scales

for specific commodities.

The Nature of the Effects of Varying Age-Sex Composition on

f United States Food Expenditures

It has been found that the double-log form of the above basic
relationship can be manipulated so that the income coefficient (or elas-
ticity for this equation form) can be estimated independently of the in-
come and food expenditure scales.84 This is done by selecting groups of
households in which the age-sex composition is approximately constant.
Each group of households in which the age-sex composition is constant is
referred to as a household type. For each of these types the terms,

§CJNJ and ifdij’ become constants and will be absorbed in the usual

constant of regression analysis. This device makes it possible to analyze

 

‘fii

8“Brown, 22. 513., p. 448.

 

111
the income-food expenditure relationship within household types, and the
differences in this relationship between the household types. This type of
analysis will indicate the extent to which an age-sex equivalent scale is
capable of explaining the effects of age-sex composition on food expendi-
tures.

For the scale computed from the double-log equation form to be
a "satisfactory" explanation of this variation, the income coefficients of
the different household types (these would have varying age-sex composi-
tion) must be equal. The differences that the scale would be capable of
explaining are the differences in "levels" of expenditure between such
household types. With the double-log form this would be differences in the
size of the constant terms. For example, consider a household type con-
sisting of an adult male and an adult female and another type consisting
of an adult male, an adult female and a child under six years of age. For
the scale to "satisfactorily" specify the effects of age-sex composition,
the curves expressing the income-food expenditure relationships for the
two household types would have the same slope and differ in height by the
amount of expenditure necessary to "compensate" for the child under six
years of age.

The procedure for this study was to select thirteen household types
from the urban sample of the 1955 United States Department of Agriculture
Survey of Food Consumption of United States Households. Within each of
the thirteen household types the age-sex composition was approximately
constant. The basic income-food expenditure relationship was modified
to include an additional independent variable, the standard number of

meal units.85 In addition to expressing the effects of variations in the

 

85This is the number of meals eaten at home per household adjusted

for age-sex composition with a scale based on food nutritional requirements.

 

112
raumber of meals eaten at home, this variable absorbed some of the varia-
‘tion in age-sex composition within the household types that remained after
:forming only thirteen types. Additional household types could have re—
rnoved much of this variation, but they were not made because of making
‘the sample size for each type too small.
Algebraically, the double—log equation fitted to the data for

each household type was

X = aYeMyzp (48)
vihere M is the standard number of meal units; a, e, and b are the popu—
lation parameters to be estimated and u is the error term assumed to have
zero mean and constant variance 0’2. For purposes of comparison, the
Asemi-log fonm

X = a + b

log Y +vb log M-+ V (49)

l 2

twas fitted to the data for each type. (a, b1, and b are the population

2
,parameters to be estimated and V = uk, where k is a constant for each
Thousehold type but varying with changes in household size.) The results
indicated that significant differences existed among the thirteen types
vsith respect to the residual variances for both forms. For the semi-log
form this would be expected from the definition of the error term. With
the double-log form the residual variances were negatively correlated
‘vith the size of the household and with the semi-log form they were posi-
tively correlated with household size. Both Bartlett's and Scheffe's test
indicated that for the double-log form the differences were significant

.at.the one per cent level.86 The differences with the semi-log form were

even larger.

868cheffe, 92. cit., p. 83.

 

113

The test made to determine whether the income coefficients of the
double-log equation were equal consisted of a comparison of the error sum
of squares. That is, the error sum of squares of an equation with a con-
stant term and an income coefficient for each household type, and a single
standard number of meals coefficient was compared to the error sum of
squares of an equation with a constant term for each household type and a
single income coefficient and a single standard number of meals coeffi-
cient. Both equations were of the double-log form. Since the residual
variances had been found to be unequal, each observation was weighted in
inverse proportion to the size of standard error of the estimate (square
root of the residual variance) for that particular household type.

The results of this test showed that the income coefficinets (or
elasticities) were significantly different at the five per cent level.
Thus, it was concluded that income elasticities do vary with changes in
household (or age-sex) composition. This means that an age-sex equivalent
scale applied to the basic relationship (Equation 47) for United States
food expenditure would not give as complete an explanation of the income-
food expenditure, age-sex composition relationship as an equation for each
household type. It can thus be argued that the use of an age~sex eqpiva-
lent scale would not be adequate for some purposes. However, if the use
of a scale gives better results then present methods of specifying the
effects of age-sex composition its use can be defended. The author then
decided to compute an age-sex equivalent scale for the United States food
expenditures and evaluate this scale by comparing it with the procedure

of using a per capita deflating procedure for food expenditures.

 

114

Computing an Age-Sex Equivalent Scale
for Food EXpenditures

 

 

The problems of computing an age-sex equivalent scale for a com-
modity such as food expenditures are the nature of the data and "circular-
ity." Data for food expenditures are obtained on a per household basis and
not on an individual basis. The latter method would be too difficult and
costly. With expenditures per household it would be possible to compute a

scale by averaging the differences in expenditure between households that

35

differ in age-sex composition by only one member. For example, a scale

mm! mm Iii.
I

value for a child under six years of age could be computed from the differ-
ences in expenditures between households with two adults and households
with two adults and a child under six years of age. However, since the
income elasticity for food is greater than zero, these households would
have to be chosen so that the households without children would be on the
same level of living as the households with the child under six. To obtain
such information an income scale would be needed. The only known method of
computing an income scale is to compute scales for several commodities and
to form the income scale from a weighted average of these commodities.
Thus, the income scale is needed to compute scales for specific commodities
and the scales for specific commodities are needed to compute the income
scale. This circle can be broken by assuming an income scale, computing
commodity scales, forming a new income scale, computing new commodity scales,
etc. This process should converge fairly rapidly to produce an income scale
and scales for several commodities.

This study was confined to computing a scale for food expenditures.
A recent study has shown that the problems of computing age-sex equivalent

scales for non-food expenditures are more complex than those for food.87

L

87Forsythe, 22, cit., pp. 367-393.

115

The method used in this study was to compute an age-sex equivalent scale for
food expenditures with one income scale, change the values of the income
scale and compute another food expenditure scale. It has been shown in a
previous studythat there is a relationship between the income scale and
any specific expenditure scale so that the specific scale can be adjusted
for any new values assumed for the income scale.88 By computing the food
expenditure scales with different values of the income scale, this relation-
ship can be tested. If a stable relationship exists, it can be used for
such an adjustment.

Regression analysis has been used in recent years to compute the
age-sex equivalent scales for specific commodities. The most convenient
and most accurate technique was developed by Brown.89 Consider the double—

log form of the basic income, food expenditure relationship.

X _ Y e u
ZcJNJ ’ ‘( EdJNJ ) 3‘ (50)

By algebraic manipulation an expression can be formed in which the Nj's
(the number of persons in the various individual age-sex types) are the
only independent variables and, therefore, the ratios of their coefficients

will form estimates of scale values for the specific commodity. That is,

d N '
x<—2—YJ-J—>‘ = aQ) cJNJ).¢u. (51)
The income elaSticity (e) was computed by holding the age-sex composition

(XCJNJ and EdJNJ) constant. It was assumed that the income elasticity

rused in this formulation did not differ significantly among households with

88Brown,_p_. cit., p. 455.

89Ibid., pp. 446-449.

116
different age-sex composition. This is contrary to the findings of this
study. If income elasticities differ among the various household types, use
of a common estimate of the income elasticity would give biased results.
Therefore, the formulation was modified to take these differences into con-v

sideration. The modified formulation is

x [:1: 1 <————-1—-L>ek ]= < zcjnjuuk . <52)

The R subscripts indicate that the values of a, e, and u are different for
different household types. It should be noted that with the modified formu-
lation the constant term was also used to adjust food expenditures for
variations in income levels. With different income elasticities for each
household type, a different constant term was found to be necessary to
specify the effects of income.

One additional modification was made. The number of meals variable
was found to be important in explaining the variation in food expenditures
among the households with the same age-sex composition. With the double-
log form it is possible to add this variable as an additional adjustment to

food eXpenditures. The formulation then becomes

d N
x [$12 <—2—Y—J—i>ek (fi17>b]= (zcjujnf‘k (53>

where M' is the number of meals not adjusted for differences in the age
and sex of the persons consuming them. (The adjustment factor EcJN.1 could
also be applied here to the number of meals but its use was not expected to
alter the results significantly.)

The error term of the equations used to compute scales is not
additive but is multiplicative. Hence, if the variance of the error term

(u) of the original formulation (Equation 49) is independent of the size

 

117

of the dependent variable, the variance of the error term that results from
applying the linear regression technique to Equations (50), (51), or (52) will
be correlated with the dependent variable. The error term of the original
formulation was actually found to be negatively correlated with the dependent
variable.‘ These two sources of unequal residual variances did not cancel
each other. The estimates would, thus, be unbiased and consistent but in-

efficient. Therefore, to increase the statistical efficiency of the estimates,

' T

scales were computed in which each observation was weighted in inverse pro-

portion to size of the standard deviation of the dependent variable for each

 

—. _- -- a 1:99:

household type.

In order to compare the results of using the various formulations
for computing scales, four different "models" were used. First only a single
income elasticity was used to adjust food expenditures for differences in
income levels (Equation 51). This was referred to as Model 1A. Another
model (1B) was formed by adding to this adjustment, an adjustment for the
number of meals. A third model (2A) included an adjustment for differing
income levels by using an income coefficient and a constant term for each
household type (Equation 52). The final model (2B) included all modifica-
tions of the basic relationship (Equation 53). Age-sex equivalent scales
for food expenditures were computed for each of these models without the
weighting procedure previously described as well as with the weighting pro-
cedure.

These eight scales were first computed by using per capita income to
adjust food expenditures. A second set of scales for Models 1A, 2A, ZB, and
weighted Models 2A and 2B was computed with an assumed income scale in which
the scale value for a child under six years of age was approximately one

third that of an adult. A comparison of these two sets of scales was made

118
and a procedure was outlined to adjust the food expenditure scale for any
change in the.income scale. The formula for adjusting is

N.
Ac? 9. 2.1.1

3 - (Adj) KJ< 2“)“3) (54)

where Kj's are a set of constants that allow for variations in income
elasticities among household types.90
To apply regressing analysis to the above formulation without having

high intercorrelation among the dependent variables, individual age-sex types

were formed that are in some respects unsatisfactory. To prevent a high inter- ‘

 

correlation the somewhat heterogeneous type consisting of the first male and
the first female of the household was formed. With this type no distinctions
could be made between the food consumption of adults over 50 years of age and
adults under 50. Also no’"good" distinction could be made between males and
females. Some indication of the difference between the scale value of an adult
male and that of an adult female was observed in the individual types that in-
cluded either a male or female other than the first ones of the household.
However, the small number of observations of these types prevented a reliable
estimate of this difference.

The models with an incomplete coefficient for the number of meals gen-
erally yielded higher scale values for the age-sex types other than the "stand-
ard" type.91 The scale value for a child under six years of age was the ex-
ception. This would be expected since children 6-14 years of age and adults
other than the first male and female of the household eat a greater propor-

tion of their meals away from home. The models with income coefficients for

L #4

90This formula was "satisfactory" for adjusting the unweighted food
expenditure scales for different income scales but its accuracy in adjusting
the weighted scales was questionable.

91The "standard" type for this analysis was the first adult male
and the first female of the household.

119
each household type also yielded higher scale values for the types other
than the "standard," than did the models with a single income coefficient.
This then demonstrates that the unmodified relationship of Equation (50) under-
estimates the scale values for the individualage-sex types other than the
"standard" type. The weighted models also yielded higher scale values for
these types than the unweighted models.

The results demonstrated that it is possible to modify the simple
income-food expenditure relationship of the double-log form to allow for
differences in income elasticities. It was shown that a food expenditure
scale could be computed from such a formulation that was satisfactory with
respect to a few simple criteria such as not having any negative values and
having smaller values for younger children than for older children and
adults. The successful use of the standard number of meals adjustment
demonstrates that with the double-log form food expenditures can be adjusted
by more than a single variable for the computation of scales. There is no
reason why additional variables affecting food expenditures cannot be used.

Evaluation of the Age-Sex Equivalent Scales
for United States Food Expenditures

 

 

It was demonstrated with the analysis of the income-food expenditure
relationships of the individual household types that the application of an
age-sex equivalent scale to the basic relatiOnship, having a single income
coefficient, did not give as complete an explanation of the effects of age-
sex composition on United States food consumption as did an equation for
each household type. However, for many applications the "single income
coefficient" relationship may be adequate. The extra expense required to
make the more complete specification may not be justified.

One alternative to using the age—sex equivalent scale would be to

deflate food expenditures by the total number of persons giving per capita

120

food expenditures. But, does the use of an age-sex equivalent scale im-
prove the results of any analysis of food expenditures over those obtained
with a "per capita" specification to a sufficient degree to warrant their
use? Some indication of the improvement was obtained by fitting regression
"lines" to a single income-food expenditure, number of meals relationship
by deflating food expenditures with six alternative ”scales" and the number
of persons. This also would demonstrate which of the six "scale models"
give the best results.

The double-log and linear form were both used for the analysis.

The relationships were

 

X Y e M b u'
3 ‘11“: 3 “—53.5? (m) 1 ' '(55)
J J
and
x Y M '
-——-—- = a + b (--——-) + b (——"'—) + u (56)
§ (21ij 1 32(1ij 2 chUNJ

The independent variable, income, was deflated with the income scale values
assumed for this analysis and not the "per capita" procedure. The number
of meals was deflated with the weighted Model 2B scale. The six scales
used to deflate food expenditures were Model 1B, weighted Model 18, Model
2A, weighted Model 2A, Model 2B, and weighted Model 2B.

The results showed the equation in which food expenditures were
deflated by the "unweighted Model 2B" scale gave the best results in terms
of the largest coefficient of multiple determination (R2). With the double-
log form the R2 with this scale was .295 compared with a value of .228 for
the equation with per capita food expenditures. An R2 value of .254 was

obtained with the Model 2B scale using the linear form whereas this value

was .195 with per capita food expenditures. With results such as these the

VIN: ‘ V1 _ne .

 

Illlll'll'l
it!

{I ll‘ |.| .IIII l l L! r III II: III. 'III I

121
use of scales should give better results than with per capita variables
in terms of the amount of food expenditures being explained in other cross-
sectional analyses.

The results indicated that the scales computed with the income
coefficient for each household type give better results than do the scales
computed with a single income elasticity. However, the "weighted Model 2B
scale" did not give as good a result as the unweighted Model 2B scale. One
can conclude that the use of the income elasticity for each household type
in adjusting food expenditure for differences in income levels did improve

the results. From previous indications it can also be concluded that the

 

adjustment for number of meals in computing scales improved the resulting
scale values. Thus, it has been shown that the simple formulation for
computing scales by adjusting food expenditures with a single income co-
efficient can be generalized in two "dimensions." Variables other than
income can be successfully used to adjust food expenditures, and the co-
efficients of the "adjusting" variables can vary with household type (or
age-sex composition).

Analyses of food expenditures are not only made with cross-sectional
data but it is common to "run" time series analyses. The characteristics
of time series data are in many respects different from those of cross-
sectional data. The characteristic of time series data that is most im-
portant when specifying the effects of age-sex composition is the way in
which the number of persons of the various ages and sexes are given. With
cross sectional data they are given on a per household basis. With time
series data they are given on a per "population" basis. The time series
data would be applicable to the following formulation of the income-food

expenditure relationship.

l lllll. II I. n

 

 

2x1 2Y1

1 - £< 3 f 1) <57)
2.2»: ' zdzN
§ 1, 11 J J, 11

An aggregate of the cross-sectional formulation (Equation 47) which would
be comparable with the time series formulation is

Y

[2<—————-— >]/H 1 ——-————>]/H} <58)

$2”: J “11 l

where H is the total number of households in the population. To make these ,

{£25.

two formulations equivalent, that is, for the scales computed from the cross-

sectional data to be directly applicable to the "time series" formulation, y“

 

certain conditions must hold. It was shown that if the terms ( 2:61NJ
' J

) are con-

) and
(2d ) are constants or if the terms (_—2(—N—) and (--¥-—

1 3 N3 53 ‘1”: 3’3 d1”:
stants, Equation (56) will be algebraically equivalent to Equation (57).
Thus, the conditions are (1) if all households are of the same age-sex
composition, or (2) if food expenditure and income per equivalent adult are the
same for all households. Both these conditions are unrealistic and would
not be expected to hold. By applying condition 2 to groups of households
for which condition 1 holds the following more realistic condition is ob-
tained: if the average food expenditure and income per equivalent adult
are equal for groups of households in which the age-sex composition and
the number of persons per household remain constant, the two formulations
are algebraically equivalent.

A priori reasoning suggests that income per equivalent adult would

be negatively correlated with the number of children per household in the
United States. This would also cause food expenditure per equivalent adult
to have the same correlation. Thus, unless the population was from a country

in which families are monetarily compensated for raising children, the

III‘ I II! | .

123
application of "cross-sectional" scales to time series data would lead to
some discrepancies in the results.

To obtain some indication of the size of the discrepancy caused
by the above source, the food expenditure, income relationship for the
United States was examined for the fourteen year period, 1948-1961. Food
expenditures were deflated by the number of persons (to yield a per capita
variable) and by applying three "scale models" to the time series Equation
(56). These models were (1) Model 1A, (2) Model 2B, and (3) weighted
Model 2B. Income was deflated by the number of persons to form one income
variable and an alternative variable was formed with the assumed income
scale. The linear form of the relationship was used.

The results indicated that the equation with the food expenditure
variable deflated with the scale of Model 1A yielded the highest R2 value.
This was in contrast to the results of the cross-sectional analysis in which
the use of Model 2B scale gave the highest R; values. The difference be-
tween deflating food expenditures by the "time series" formulation and the
cross-sectional formulation was examined by using the data from the 1751
households used elsewhere in this analysis. It was found that the value
of food expenditure when deflated by the "time series" formulation was lower
than when deflated by the "cross-sectional" formulation. Thus, this dif-
ference would be one source of the discrepancy between the results obtained
with the time series data and those obtained with the cross-sectional data.
Actually the scale values (for each age-sex type except the "standard")
were smaller for Model 1A than for Model 2B. This is but one of several
factors that lead to discrepancies between cross-sectional and time series
analysis of the income-food expenditure relationship.

All three of the age-sex equivalent scales for food expenditures

 

124

gave better results in the time series analysis with respect to the size

of the coefficient of multiple determination (R2) than did the "per capita

deflation” method. However, the use of the income variable formed by

deflating with the assumed income scale gave slightly lower R2 values

than did the use of per capita income. In view of the limited information

available with respect to the income variable and the possible explanations

that could be put forth, an explanation of this phenomenon was thought to I

be outside the scope of this study. “-
The results of this single application of food expenditure scales

to time series data indicate that the scales do give better results than K‘J

 

per capita food expenditures. The problem with the difference in the

"time series" and "cross sectional" formulations plus other problems with

the difference between time series and cross sectional analyses prevent form-
ing a definite conclusion that the use of scales is superior to the per
capita deflation procedure. If a method is devised to adjust a scale that
has been computed from cross-sectional data for the discrepancy between the
two formulations, the possibility of making such a conclusion would be in-
creased. Two other limitations of the time series application should also

be noted. (1) The time period (14 years) was fairly short and (2) only one
independent variable was used. The use of another time period or the

addition of other independent variables may alter the results.

Limitations and Suggestions for Further Research

 

 

Many factors and variables affecting the relationship of the
demographic variables of age and sex to food expenditures remain to be ex-
plored. One of the most obvious limitations of this study has been the

lack of an income scale. In a recent study it has been concluded that the

125

scales for non-food expenditures cannot be successfully computed with
present knowledge.92 This then prevents forming an income (or level-of-
living) scale from a weighted average of the scales for specific commodi-
ties. However, no attempts have been made to apply the method developed
in this study to non-food commodities (the method of computing a scale
from a formulation in which income coefficients vary with changes in age-
sex composition and adjusting food expenditures for variables other than I
income). This technique, along with the method used by Forsythe to ac- “

count for economies of scale, may make it possible to compute such scales

.4;
s

and to form an income scale. '

 

An almost direct application of the technique of this study to
expenditures or quantities for specific food commodities such as dairy
products, fruits, vegetables, and meat should be possible. The charac—
teristics of many of these commodities with respect to economies of scale
and their relationship to income and other variables that affect food
expenditures should not be substantially different than total food expendi-
ture.

The food expenditure scales computed in this study do not account
for variations in food expenditures due to age differences among persons
20 years old and over. In addition, the difference between the adult male
and the adult female with respect to the amount of food expenditures has
not been measured accurately. It should be possible to devise techniques
for measuring these differences.

It is obvious that all the implications of applying age-sex equiva-
lent scales computed from cross~sectional data to analyses of time series

data have not been explored. More work is needed on the algebraic differences

 

92Forsythe, _p, cit., pp. 367-393.

126

that exist between the "time serieS" and "cross sectional" formulations.
It should be possible to devise a method to either adjust the number of
persons of each age-sex type given by the time series data or to adjust the
"cross sectional" scale so that the two formulations will be equivalent.
One of the characteristics of adult equivalent scales that was not
considered in this study is their stability. Many factors could be sources
of instability. One obvious element is the introduction of processed baby 1
"fl

food. Changes in the proportion of adult males engaged in physical labor m
I
is another. Both of these factors have been changing in recent years and '

e a-

probably have increased the scale values for individual types other than

 

the adult male, relative to the value for the adult male. This is one ex-
planation of why the scales obtained in this study were larger than the
scale values obtained in previous studies (referring to scale values of
persons other than the adult male). One of the previous studies used
British working class data.93 TherefOre, it would be expected that the
scales obtained from urban United States data would be larger.

The income variable used in this analysis is subject to some
limitations. If Friedman's permanent income hypothesis is valid, the use
of some average income over more than one year would give better results.94
Another characteristic of income to be considered is the amount actually
disposable for food purchases. The income variable used for this study
was the income available to consumers after taxes. In addition to the re-
moval of taxes from gross income, the amounts committed to the payment of

"time" purchases or other fixed committments should also be removed. A

third problem of the income variable that is closely related to the concerns

93

 

Brown, _p. cit., p. 454.

94Milton Friedman, A Theory of the Consumption Function (Princeton:
Princeton University Press, 1957).

 

127
of the permanent income hypothesis is the question of the lag between the
receipt of income and consumption. In this study 1954 annual disposable
income was associated with food consumption in the spring of 1955. These
factors probably have contributed to the relatively small amount of the
variation in food expenditures that was explained by the income variable in
the cross-sectional analysis.

These limitations have been viewed by the author as areas in which
the estimates of and use of age-sex equivalent scales for food expendi-
tures can be improved. They do not imply that the estimates of this study
are of little use. It has been shown that for one example using cross-
sectional data and another using time series data that the use of scales

has given better results than the per capita specification.

Applications and Importance

In this study a methOd has been devised to compute an age-sex
equivalent scale fer United States food expenditures. This method was a
generalization of one devised to compute scales for British data. The
more general method of this study consists of adjusting one variable (food
expenditures) for variations in more than one other variable (income and
number of meals). These other variables could have coefficients that
varied with some classification of another characteristic (household type).
The adjusted variable (food expenditure) could then be utilized as a
dependent variable in a regression with independent variables (the number
of persons of the various ages and sexes) other than those used to adjust
the dependent variable. This was, thus, a more generalized method of
handling variables that enter a relationship in a way that cannot be handled
by a usual regression analysis. This method may have applications other

than that of computing scales.

 

128

An age-sex equivalent scale for United States food expenditures
makes it possible to refine many analyses that have been based on per capita
or per household food expenditures. Food expenditure per equivalent adult
is a more precise measurement than is food expenditure per capita or per
household. This makes possible more valid comparisons of food expenditures
between households or between groups of households. For example, more valid
comparisons can be made between food expenditures of households in different
geographic regions, households of different ethnic background, households
of different social class, households with different income levels, etc.

If the assumption that food expenditures and income are allocated among the
members of households in such a way that total satisfaction of the house-
hold is maximized the food expenditure scale provides a normative criterion
for allocation of food among households of different age-sex composition.

The application of food expenditure scales computed from cross-
sectional data to time series data is not as direct since the problem of
the algebraic equivalence of the two formulations exists. However, the
results of the time series application indicate that some improvement may
be made in time series analyses, such as demand analyses, with the use of
a scale for food expenditures. It is possible to use the scale as an
independent variable as well as a "deflator" for food expenditures. With
the double-log form it can be shown that if the coefficient of this inde-
pendent variable is one, then the uses would be equivalent. That is, as
a deflator
2x1 BY

1
i .__
§ cJ¥Nij 12 dj§Nij

and as a second independent variable,

>51.“ (59)

 

 

= a(

 

129

 

i b u
)(Ec EN) (60)
J§Nis in 13 1

If the coefficient of the variable ( 2 CJ §Nij)’ b, is equal to one, the
two formulations would be equivalent.J The advantage of Equation (59)
would be that the b coefficient could have values other than one, which
might account for some factors omitted with Equation (58). Another ad-
vantage would be that the significance of this variable could be more
readily evaluated. A disadvantage would be its possible correlation with
other independent variables.95

The use of scales in forming the independent variable offers some
advantages over specifying age-sex composition with the proportion of a
certain age or sex group.96 The use of a scale in the terms (2cJ 2 Nij)
combines several such age and sex groups. The use of the propoitioniof a
certain age or sex group as an independent variable, however, relates vari-
ation in food expenditures to a particular group.

All the implications and uses of scales have not been explored in
this study. The results of the two applications and the possibilities for

other applications that have been mentioned make the age-sex equivalent

scale for United States food expenditures a potentially valuable tool.

 

95The same procedure could be used for "cross sectional" analyses.

6For example, Herrmann has used the proportion of the population
under 20 years of age as an additional independent variable in an analysis
of the demand for milk. See Robert O. Herrmann, Changes in Population
Composition as an Explanation of Changes in the Demand for Food: The Case
of Fluid Milk, Ag. Econ. 866 (Michigan State University, Department of
Agricultural Economics, April 1962).

 

BIBLIOGRAPHY

Books

David, Martin H., Family Composition and Consumppion, Amsterdam: North
Holland Publishing Co., 1962.

Dixon, Wilfrid J., and Massey, Frank J., Introduction to Statistical Analypis,
2nd ed., New York: McGraw Hill, 1957.

Fraser, D. A. 8., Statistics, an Introduction, New York: John Wiley and
Sons, 1958.

Friedman, Milton, A Theory of the Consumption Function, Princeton: Princeton
University Press, 1957. '

Hicks, J. R., Value and Capital, 2nd ed., London: Oxford University Press,
1946.

Lerner, Abba P., The Economics of Control, New York: The MacMillan Co.,
1944. '

Prais, S. J., and Houthakker, H. S., The Analysis of Family Budgets, Cambridge:
Cambridge University Press, 1955.

Scheffe', Henry, Analysis of Variance, New York: John Wiley and Sons, 1959.
Valvanis, Stefan, Econometrics, New York: McGraw Hill, 1959.
Walker, Helen M., and Lev, Joseph, Statistical Inference, New York: Henry
Holt and Co., 1953.
Articles and Periodicals
Allen, R. D. C., "Expenditure Patterns of Families of Different Types,"
Studies in Mathematical Economics and Econometrics, ed. Lange, McIntyre,

and Yntema, Chicago: University of Chicago Press, 1942.

Brown, J. A. C., "The Consumption of Food in Relation to Household Composi-
tion and Income," Econometrica, XII (October 1954), 444-460.

Cofer, Eloise, "Family Food Budgets, Revised 1957," Family Economics Review,
(October 1957), l-ll.

 

Forsythe, F. C., "The Relationship Between Family Size and Family Expenditure,"

Journal of the Royal Statistical Sociepy, Series A, CXXVIII (Part IV,
1960), 367-393.

130

 

131

Friedman, Milton, "A Method of Comparing Incomes of Families of Differing
Composition," Studies in Income and Wealth, XV, New York: National
Bureau of Economic Research, 1952.

Henderson, A. M., "The Cost of Children," Population Studies, 111 (September
1949),'130-1so.

Henderson, A. M., "The Cost of Children," Population Studies, IV (December
1950), 267-298.

Henderson, A. M., "The Cost of a Family," Review of Economic Studies, XVII,
(Part 2, 1949-50), 127-148.

 

Houthakker, H. S., "The Econometrics of Family Budgets," Journal of the
Royal Statistical Society, Series A, CXX (Part I, 1952), 1-21.

Kemsley, W. F., "Estimates of Cost of Individuals from Family Data," Applied
Statistics, I (November 1952), 192-201.

0gburn, William F., "A Device for Measuring the Size of Families, Invented
by Edward Sydenstricker and W. I. King," Methods in Social Science:
A Case Book, ed., Stuart A. Rice, Chicago: University of Chicago
Press, 1931.

 

Prais, S. J., "The Estimation of Equivalent Adult Scales from Family Budgets,"
Economic Journal, LXIII (December 1953), 791-810.

Nicholson, J. L., "Variations in Working Class Family Expenditure," Journal
' of the Reyal Statistical Society, Series A, CXII (Part IV, 1949), 359-411.

Quenoville, M. H., "An Application of Least Squares to Family Diet Surveys,"
' Econometrica, XVIII (January 1950), 27-44.

Sydenstricker, Edward, and King, W. I., "The Measurement of the Relative
Economic Status of Families,"_gparterly Pgtlication of the American
Statistical Association, XVIII (September 1921), 842-857.

Woodbury, Robert M., "Economic Consumption Scales and Their Uses," Journal
of the American Statistical Association, XXXIX (December 1944), 455-468.

Public Documents

Burk, Marguerite C., Measures and Procedures for Analysis of U. S. Food
Consumption, U. S. Department of Agriculture, Agricultural Handbook
No. 206 (washington: U. S. Government Printing Office, 1961).

Foote, Richard J., Analytical Tools foru§tudying Demand and Price Sttuctures,
U. S. Department of Agriculture, Agricultural Handbook No. l46l'(Wash-
ington: U. S. Government Printing Office, 1958),

Rockwell, G. R., Jr., Income and Househpld Size, Their Effect on Food Con-
sumption, U. S. Department of Agriculture, Agricultural Marketing

Research Report No. 340 (Washington: U. S. Government Printing Office,
1959).

132

‘U. S. Department of Agriculture, Food Consumption of Households in the
United States, Household Food Consumption Survey 1955, Report No. 1
(Washington: U. S. Government Printing Office, 1956).

'U. S. Department of Agriculture, Food Consumption and Dietary Levels of
Households of Different Sizes, United States--Bytgggions, Household
Food Consumption Survey 1955, Report No. 17 (Washington: U. S.
Government Printing Office, 1963).

U. S. Department of Agriculture, Schedule Used in the Survey of Food Con-
sumption of Households in the United States, Spring 1955, Agricultural
Marketing SerVice 200 (Washington, 1957).

U. S. Department of Commerce, Business Statistics, XI (1957). %

U. S. Department of Commerce, Current Popplation Reports, Series P-20.

 

 

Unpublished Ma terial i

Herrmann, Robert 0., "An Investigation of Differences in Income Elasticities
of Demand for Food in Households of Differing Size and Composition,"
(Unpublished Master's Thesis, Department of Agricultural Economics,
Michigan State University, 1951).

Herrmann, Robert 0., Changes in Population Composition as an Explanation
of Chapges in the Demand for Food: The Case of Fluid Milk, Michigan
State University, Department of Agricultural Economics 866 (East
Lansing, 1962).

Herrmann, Robert 0., "Household Socio-Economic and Demographic Characteris-
tics as Determinants of Food Expenditure Behavior," (Unpublished Ph.D.
Dissertation, Department of Agricultural Economics, Michigan State
University, 1963).

Manderscheid, Lester V., An Introduction to Statistical Hypothesis Testing,
Michigan State University, Department of Agricultural Economics 867--
Revised (East Lansing, 1964).

Sparks, Willard Robert, "Estimates of the Demand for Food from Consumer
Panel Data," (Unpublished Ph.D. Dissertation, Department of Agricultural
Economics, Michigan State University, 1961).

2
1
0
6
5
7
1
3
0
3
9
2
1
3