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73-20,399
SETTHAWONG, Phisit , 1941)AN ECONOMETRIC ANALYSIS Or ENGEL CURVES:
ON M.S.U. CONSUMER PANEL DATA.

BASED

Michigan State University, Ph.D., 1973
Economics, general

University Microfilms, A XEROX Company , A nn Arbor, M ichigan

AN

ECONOMETRIC

ANALYSIS

B A S E D ON M.S.U.

OF

CONSUMER

ENGEL CURVES:
PANEL

DATA

By

Phisit

Sotthawong

A THESIS

S u b m i t t e d to
M i c hi gan State U n i v e r s i t y
in p a r t i a l f u l f i l l m e n t o f t h e r e q u i r e m e n t s
f or the d e g r e e of

DOCTOR

OF

Department

PHILOSOPHY

of

1973

Economics

ABSTRACT

AN

ECONOMETRIC

ANALYSIS

BASED ON M.S.U.

OF

ENGEL CURVES:

CONSUMER

PANEL

DATA

By

Phisit

In

the p r e s e n t

M.S .U . C o n s u m e r Pan el
curves,

(2)

utility

functions

modities
fruit s ;

modify

under

curves,

and

the

Enge l

dairy

se t

out

forms

the E n g e l
double-log

studies

showed

"goodness
food,

highest

the

income

for
we r e

(3)

the

selected

forms.

The

that n o

functional

fit"

to

semi-log
elasticity

the

Engel

com­

a n d o i ls ;

on

th e

f o od s.

tr u e

func­

linear,

thirteen

cross-

form un iformly

the o b s e r v a t i o n s .
form d i d

the

and eggs.

studies

They w e r e

the

The

fats

f i sh ,

for

of

approximate

five c o m p o s i t e

curves.

of

use

curves.

products;

sectional

sectional

the

analyze

cro ss

and

give

(1)

thirteen

semi-log,

composite

to

poultry,

mathematical

each

made

meat,

Three

a better

author

and

were

gave

the
1958

Engel

studies were

curves

form of

d a t a of

by m e a n s of

Engel

tional

stud y ,

the

vegetables;
First,

Setthawong

For

consistently

estimates which were

Phisit

widely
log

different

forms.

for e a c h
log

Surprisingly,

composite

forms were

cross

from those

very

sectional

estimates

As

double-log

forms

were

the

income

based

the

on

.7052,

.2989,

.1885,

and

estimates

were

forms,

of

the

log,

linear,

and

forms w e r e

c o n s u m p t i o n was

the

The

of

.0938,

the i n c o m e

semi-log,
and

respec­

linear,

.2209,

estimates

The

and

respectively.

for f r u i t s
forms

figures

were
.4768,

based

on

the

semi-log,

As

forms were

Generally,

predominantly

the c r o s s
modified

for

estimates

the

on

linear,

etc.,

b a s e d on

the

.22 59 ,

and

the
semi­
and

income elasticity
b a s e d on e i t h e r

inelastic:

the

and

Engel's

negative

nonlinear

forms

func­

law of

income
were

infrequent.

sectional
Engel

meat,

.5428,

foods

Elastic

relatively

pooled.

and

linear,

income elasticity

found,

were

semi-log,

the

confirmed.

Secondly,

the

of

five c o m p o s i t e

were

on

mean

based

they

income e l a s t i c i t y

and d o u b l e - l o g

elasticity estimates
but

the

respectively.

double-log

for t h e

of

.2091,

elasticity

respectively.

estimates

the

respectively.

income

the t h i r t e e n

the m e a n

linear,

.3388,

all

.1009,

elasticity

for v e g e t a b l e s

mean

tional

and

.1988

double-log

.2353,

.5217,

semi-log,

over

based

on

double­

l i n e a r and d o u b l e ­

the

.2604,

based

linear a n d

on

a nd o i l s ,

estimates

the

elasticity estimates

The m ean

forms were
fat s

on

income

equal

products

elasticity

of

based

st u d i e s .

for

The m e a n

the

nearly

for d a i r y

and d o u b l e - l o g
tively.

food

based

Setthawong

and

curves

time s e r i e s d a t a

were

formulated

for

Phisit

estimating
composite
order

foods.

Engel

panel

actual
the

income

reported

food p r i c e

first order

were

for m e a t ,
the

income

and

oils;

.0928,
price

mates,

etc.

After

the

sistency

of

by

the

highly

signs

of
of

the p a n e l

the

dairy
meat,

data was

the

to but

for

were
As

-1.4543,

fats

and

products;

et c .

the

food s

close

.6966

on

Based on

autoregressive

etc.,

these combined

based

composite

products;

and

the

to r e p r e s e n t

respectively.

th e

first

for v e g e t a b l e s ;

income

they c o n f i r m e d

the

form o f

positive,

they were

and

th e

fiv e

a n d meat,

for

as

disturbances,

for d a i r y

.1472,

-.3580

of

products;

.6629

estimates,
and

the

five

households.

scheme

for

the

indices,

used

panel

estimates

and

price

were

f or

selected

functional

eliminating

vegetables;

.5031
effects,

fats

.0170,
f o r the

-1.5712,
fats

and

respectively.

price

elasticity

studies were

demand

theorem.

probably

the ma in

esti­

highly
The con­
reason

f or

success.
Lastly,

evidence
tions

prices,

vegetables;

results
as

M.S.U.

fruits;

.0533,

-.7424,

the

The

for d a i r y

elasticity

successful

this

.8284

fruits;

Regarding

true

They were

for

frui t s;

the

coefficients

elasticity

oils;

food

.7100

.1808,

-.8158,

fo r

elasticities

form was

autoregressive

t h a n one:
oils;

linear

faced

estimated.

less

and price

curves.

autoregressive

and

The

approximation

modified
the

both

Setthawong

on

based

on

Wald's

theorem,

the a p p r o x i m a t e d e t e r m i n a t i o n

by means

of

Engel

curves

was

some p r el im i n a r y
of

presented.

utility
Using

func­
the

Phisit

results
Engel

of

the

curves

illustrations
indicated
utility

thirteen

and

the M.S.U.

were

that,

cross

given.

under

functions

price
These

certain

could

be

sectional

studies

indices,

two

on

approximated.

th e

numerical

illustrations,

circumstances,

Setthawong

the

at

least,

empiriral

ACKNOWLEDGMENTS

The author
to Dr.

Anthony

support

Y.

C.

throughout
Special

wishes

Koo who

this

thanks

who approved

the

Agricultural

Economics,

this

Special

study.

Stephenson,

she id ,

and

made m a n y

are

Robert
valuable

for

for

thi s

making

to Dr.

grant

Michigan

due

appreciation

encouragement

a nd

from

is

supervisor,
to Drs.

Gustafson

Lester M a n d e r s h e i d

State

appreciation

Jan

the D e p a r t m e n t of
University,

given
and her

to J u d i t h
staff.

K m e n t a , Lester

who read

for

Mander­

the e a r l y d r a f t s

and

comments.

The Agricu l t u r a l
credit

has g i v e n

due

programming

are

his d e e p

s t u dy .

programming

Thanks

to e x p r e s s

Economics

the M . S . a .

Department deserved

Consumer

Panel

data

available

study.
Thanks

patience.

are

also due

to my w i f e ,

P e n s e e , for

her

TABLE OF

CONTENTS

Page
ACKNOWLEDGMENTS

..............................................

ii

LIST O F T A B L E S ..................................................

v

LIST O F F I G U R E S .................................................. vii
LIST O F A P P E N D I C E S ............................................. v i i i
INTRODUCTION

..................................................

1

Chapter
I.

REVIEW OF LITERATURE
1.

2.

3.
II.

III.

E n g e l C u r v e s .......................................
1.1.
H i s t o r i c a l R e v i e w ........................
1.2.
Cross Section
...........................
1 .3.
T i m e S e r i e s ...............................
1.4.
C o n s u m e r P a n e l ...........................
1.5.
C o n c e p t s of V a r i a b l e s ....................
P r i c e s and N o n - E c o n o m i c F a c t o r s
. . . .
2.1.
P r i c e s .......................................
2.2.
N o n - E c o n o m i c F a c t o r s ....................
F un ct i o n a l Forms of Engel Curves . . . .

REVIEW OF M.S.U.
1.
2.
3.

CONSUMERPANEL

SURVEY

.

.

.

G e n e r a l R e m a r k s ..................................
M e t h o d s of S e l e c t i o n ...........................
C o l l e c t i n g th e I n f o r m a t i o n ...................

CROSS
1.
2.

...............................

SECTIONAL

S T U D I E S ..............................

O b j e c t i v e s a nd S o m e G e n e r a l R e m a r k s . . .
Statistical Cross Sectional Models
. . .
2.1.
Estimation Procedure . . . . . .
2.2.
R e s u l t s of C r o s s S e c t i o n a l S t u d i e s
.
2.3.
Income Elasticity Estim at es .
. . .
2.3.1.
Dairy Products
................
2.3.2.
F a t s and O i l s ....................
2.3.3.
F r u i t s ...........................
2.3.4.
V e g e t a b l e s ........................
2.3.5.
M e a t , e t c .........................

4
4
4
7
11'
12
14
16
16
17
21
28
28
29
29
31
31
32
35
35
42
43
46
49
52
55

Page

Chapter
2.4.
2.5.

D i s t r i b u t i o n s of I n c o m e
E l a s t i c i t i e s ...........................
A C o m p a r i s o n o f " G o o d n e s s o f Fi t " .
.
2.5.1.
D i s t r i b u t i o n s of C o r r e c t
C o e f f i c i e n t of D e t e r m i n a ­
t i o n .......................
63

A p p e n d i x A:
A p p e n d i x B:
A p p e n d i x C:
A p p e n d i x D:
A p p e n d i x E:
A p p e n d i x F:
IV.

C O M B I N E D S T U D I E S .......................................
1.
2.

G:

Values

of

q k , p k , m, m / q k ,

•

SOME P R EL IM IN A R Y E V I D E N C E ON A P P R O X I M A T I N G
E M P I R I C A L U T I L I T Y F U N C T I O N S .........................
1.
2.
3.

O b j e c t i v e s ................................
94
Utility Functions and Eng el
Curves
. . .
A r e a s for F u t u r e R e s e a r c h .................109

BIBLIOGRAPHY

....................................................

iv

64
65
66
68
70
73
76

Objec t i v e s a n d Some Ge ne ralRemarks . . .
79
S t a t i s t i c a l C o m b i n e d M o d e l s ............
2.1.
E s t i m a t i o n P r o c e d u r e ...................
2.2.
R e s u l t s of C o m b i n e d S t u d i e s .
. . .
2.2.1.
I n c o m e and P r i c e
E l a s t i c i t i e s ...................

Appendix
V.

C o m p o s i t i o n of D a i r y P r o d u c t s .
.
C o m p o s i t i o n of F a t s and O i l s
.
.
C o m p o s i t i o n of F r u i t s .................
C o m p o s i t i o n of V e g e t a b l e s
. . .
C o m p o s i t i o n of_Me at, ^2.. . .
V a l u e s of Y k , M, a n d M / Y k
. . .

59
60

76
82
86
89
93

94

94

112

LIST OF

TABLES

E s t i m a t e s o f R e g r e s s i o n C o e f f i c i e n t s for
D a i r y P r o d u c t s Based o n A l t e r n a t i v e F u n c ­
t i o n a l F o r m s ..........................................

37

E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s for F a t s
and Oils Based on A l t e r n a t i v e Functio nal
Forms
..................................................

38

E s t i m a t e s of
Fruits Based

39

R e g r e s s i o n C o e f f i c i e n t s for
on Alt e r n a t i v e Functional Forms

E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s for
V egetables Based on Alternative Functional
Forms
..................................................

40

E s t i m a t e s o f R e g r e s s i o n C o e f f i c i e n t s f or
M e a t , etc. B a s e d o n A l t e r n a t i v e F u n c t i o n a l
Forms
..................................................

41

I n c o m e E l a s t i c i t y E s t i m a t e s for D a i r y
Products Based on A l t e r n a t i v e Functional
Forms
..................................................

44

I n c o m e E l a s t i c i t y E s t i m a t e s for F a t s
O ils Based on Al ternative Functional

47

and
Forms

.

I n c o m e E l a s t i c i t y E s t i m a t e s for F r u i t s B a s e d
o n A l t e r n a t i v e F u n c t i o n a l F o r m s ...................

50

I n c o m e E l a s t i c i t y E s t i m a t e s for V e g e t a b l e s
Bas ed on A l t e r n a t i v e Fu nc t i o n a l Forms.

53

I n c o m e E l a s t i c i t y E s t i m a t e s for M e a t , et c.
B ase d on A l t e r n a t i v e F u n c t i o n a l Forms.

56

T h e A c t u a l D a y s a n d H o l i d a y s I n c l u d e d in the
F o u r W e e k s o f E a c h P e r i o d f o r the Y e a r o f
1 9 5 8 ......................................................

58

D i s t r i b u t i o n s of I n c o m e E l a s t i c i t y E s t i m a t e s
for F o o ds B a se d on A l t e r n a t i v e F u n c t i o n a l
Forms
..................................................

59

v

Ta ble

Page

13.

Values

14.

D i s t r i b u t i o n s of C o r r e c t e d C o e f f i c i e n t
of
D e t e r m i n a t i o n Based on Al te rn a t i v e Fu nc ti on al
F o r m s ..................................................

63

P r i c e I n d i c e s of F i v e C o m p o s i t e F o o d s B a s e d
o n M . S . U . C o n s u m e r P a n e l D a t a o f 195 8
( 1 9 5 5 - 5 7 = 1 0 0 ) ......................................

80

E s t i m a t e s of A u t o c o r r e l a t e d C o e f f i c i e n t s for
F i v e C o m p o s i t e F o o d s ...............................

87

E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s for
F i v e C o m p o s i t e F o o d s a f t e r E l i m i n a t i n g the
A u t o r e g r e s s i v e E f f e c t s ...............................

88

I n c o m e an d P r i c e E l a s t i c i t y E s t i m a t e s for
F i v e C o m p o s i t e F o o d s a f t e r E l i m i n a t i n g the
A u t o r e g r e s s i v e E f f e c t s ...............................

90

15.

16.

17.

18.

of

an d

n

(Numb e r

vi

ofOb se rv at i o n s ) .

.

61

LIST

OF F I G U R E S

Figure
1.

2.

3.

4.

5.

6.

Page
A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y
E s t i m a t e s for D a i r y P r o d u c t s B a s e d on
A l t e r n a t i v e F u n c t i o n a l F o r m s .......................

45

A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y
E s t i m a t e s for F a t s a n d O i l s B a s e d on A l t e r n a ­
t i v e F u n c t i o n a l F o r m s ...............................

48

A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y
E s t i m a t e s for F r u i t s B a s e d o n A l t e r n a t i v e
F u n c t i o n a l F o r m s ......................................

51

A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y
E s t i m a t e s for V e g e t a b l e s B a s e d o n A l t e r n a ­
t i v e F u n c t i o n a l F o r m s ...............................

54

A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y
E s t i m a t e s for M e a t , etc. B a s e d o n A l t e r n a ­
t i v e F u n c t i o n a l F o r m s ...............................

57

A Diagram Showing Wald' s Method, with ThreeD i m e n s i o n a l C o m m o d i t y Space and Two
C o n s u m p t i o n E x p a n s i o n P a t h s .......................

99

LIST OF

APPENDICES

Page

Appendix
A.

Composition

of

B.

C o m p o s i t i o n of

Fats

C

Composition

Fruits

D

C o m p o s i t i o n of

E

C o m p o s i t i o n of M e a t ,

F

Values

of

Y

G

Values

o£

qk

of

k

Dairy

Products
and

Oils.

.

Vegetables.

M,

M/Yfc

p k , m,

etc..
.
m/c7k , p k / q k

viii

64
65

66

68
70
73
93

INTRODUCTION

The

functional

to e x p e n d i t u r e
good
the

on

(or q u a n t i t y

is g e n e r a l l y
functional

from

t he

budget
vided

loci

of

functional
is

the

with

fix ed

normally

circumstances,

the

Engel

between

and

the

Fitting
utility

Theoretically,
curve

the

is d e r i v a b l e

income,

But,

utility

the

search

curve

Engel

and

variable

a direct

Engel

income

a particular

indifference

is k n o w n .

of

of)

curve.

the

unknown,

relationship

inevitable.

Engel

prices

function

from disposable

purchased

ol

tangcncies

utility

function being

data

termed

relationship

surfaces
the

relationship

from

curve,

f u n c t i o n m i g h t be

on

pro­

the

the o b s e r v e d

in c e r t a i n
approximately

determined.
Most
ically
time
lar

used

previous
cross

to e s t i m a t e
functional

the M . S . U .
E nge l
mate

sectional
Engel

the

utility

on

Panel

(2) m o d i f y

Engel

curves

of a p a r t i c u l a r

income

elasticities.

selected

Engel

functions

the

data

for

the

class­
period

of

A particu­

functional

curves.

p re s e n t study,

Consumer

curves,

or

form w a s

r e l a t i o n s h i p of
In the

studies

the a u t h o r

data
the

of

Engel

by means

1

of

1958

will make

to

(1)

curves,
Engel

and

use

analyze
(3)

curves.

of
the

approxi­

2

In a n a l y z i n g

the

Engel

curves,

the

Engel

cu rv es are

sectional

studies

on

composite

foods .

Three

the

functional

way,

one

based

can

on

relationship

investigate

three

fer en t .

of

if

alternative

In a d d i t i o n ,

functional

functional

one

are

fo r

In

income e l a s t i c i t y
forms

a better

five

selected

curves.

investigate

gives

cross

set o u t

Engel

functional
can

form uni fo rmly

forms

the

the

thirteen

are

for
this

estimates

widely

dif­

if a p a r t i c u l a r
"goodness

of

fit"

to the o b s e r v a t i o n s .
In m o d i f y i n g
time

series dat a

ticities

for

are

five

Statistically,
reliable

tha n

studies,

since

tion of

limited on

is p o s s i b l e

lowing

is

are

sectional

and p r i c e

elas­

estimated.

e l a s t i c i t y e s t i m a t e s ar e
from

individual
are

used

in

cross

of

the

more

sectional

th e e s t i m a ­

the e s t i m a t i o n o f

because

and

price

continuous

collec­

data.

the p r e s e n t

sectional

on

of

the

utility

the o u t l i n e s

the o r d e r

study

studies

the e s t i m a t i o n

for

cross

income

foods

In a d d i t i o n ,

to a p p r o x i m a t e

As

Dot h

income

certain circumstances,
used

curves,

more observations

the p a n e l

cross

pooled.

those ob ta in ed

Lastly,
the

Engel

composite

these

tion p r o c ed ur e.
elasticities

the

attempts

the

Engel

income

fitted

to s h o w

curves

are

elasticities.

Engel

curves

that
not
In

might

be

functions.

of

the p r e s e n t

of discussion.

study,

the

fol­

3

In C h a p t e r
studies
sumer

are

III,

curves

set out

combined
out

the

Panel

Chapter
Engel

on

for

Engel

Survey

some

curves

is

is b r i e f l y

survey

of

s e t out.
reviewed

sectional

based

three

alternative

for

on

five
of

composite

composite

areas

f oo d s .

utility

for

future

Engel

Some

functions
research,

The

empirical
M.S.U.

studies

foods .

th e m o d i f i e d

th e

in C h a p t e r

cross

on a p p r o x i m a t i n g
and

a general

thirteen

analyses
five

I,

on

by means

of

given

In

the

In C h a p t e r
are

preliminary

are

II.

functional

curves

Con­

forms
IV,
set

findings
Engel

curves,

in C h a p t e r

V.

CHAPTER

REVIEW OF

1.
1.1.

Historical

household

has

a

lon g

and

Engel.^

Engel Curves

history,

relationships

study

was

based

which

had

been

and

the
was

The early
at l e n g t h in C. C.
L i v i n g (New Y or k :

dating

made

data

into three

public

relationships

expenditure

back

by E r n s t

Ducpetiax's

upon

functional

first and mo s t

classified

dependent

the

household

certainly

Perhaps

on

s t u d y of

income

these

families

LITERATURE

Review

The empirical
between

I

as

famous

Engel

for

far

153

in

as

LePlay

s t u d y of
1857.

Belgian

socio-economic

assistance;

on goods

families

2

The

families,
groups:
just

s t u d i e s of f a m i l y b u d g e t s a r e c o n s i d e r e d
Zimmerman, C o n s u m p t i o n and S t a n d a r d of
D. V a n N e s t r a n d , 1 93 6 ) .

^ E r n s t E n g e l ( 1 8 2 1 - 9 6 ) , D i r e c t o r of t h e P r u s s i a n
B u r e a u of S t a t i s t i c s , w a s an a d m i n i s t r a t o r w h o p u b l i s h e d
many prominent articles.
F o r a l i s t o f h is w o r k s , se e
J. A. S c h u m p e t e r , H i s t o r y of E c o n o m i c A n a l y s i s (New Y o r k :
O x f o r d U n i v e r s i t y P r e s s , 1954 ) , f o o t n o t e 14] p. 961.
It is i n t e r e s t i n g to not e t h a t E n g e l ’s w o r k w a s
m a i n l y i n f l u e n c e d b y two o f h i s c o n t e m p o r a r i e s . O n e w a s
the F r e n c h e n g i n e e r F. L e P l a y , w h o h a d c o l l e c t e d b u d g e t s
fro m h o u s e h o l d s a l l o v e r E u r o p e m o s t l y , it s e e m s , f r o m
h u m a n i t a r i a n interest.
The o ther was the B e l g i a n S t a t i s t i ­
c i a n y u e ' t e l e t , w h o w a s a f i r m p r o p o n e n t of the i d e a t h a t
h u m a n c h a r a c t e r i s t i c s , at l e a s t o n th e a v e r a g e , w e r e g o v ­
e r n e d by laws as d e f i n i t e as t h o s e w h i c h g o v e r n p h y s i c a l
p h e n o m e n a ; see H. S. H o u t h a k k e r , "An I n t e r n a t i o n a l C o m p a r i ­
son o f E x p e n d i t u r e P a t t e r n , C o m m e m o r a t i n g the C e n t e n a r y o f
E n g e l ' s L a w , " E c o n o m e t r i c a (October, 1957), p. 532.

4

5

able

to

live

comfortable
Eng el

the
to

a

the

states

to

increases.

originally

in

from

as

himself,

of

this

of

in

study,
a

family,

expenditure

food."

3

living

that

In o t h e r

expenditure

though

from

the

viewpoint

Eng el's works,

by

total

standard

anyone

economists.

results

its

nor

sional

economists

of

of

"The p o o r e r

p r o p o r t i o n of

the

Engel

budgets

revived

consumption:

families

of

words,

devoted

the h o u s e -

4

family

was

the

and

the b a s i s

1857,

importance

Aside

On

the p r o v i s i o n

that

Neither

its

l a w of

fo o d d e c r e a s e s

hold

assistance;

proportion

be d e v o t e d
la w

such

circumstances.

proposed

the g r e a t e r
must

without

did

obtained

from

of A l l e n

and

to u n d e r s t a n d
the

1935

analysis

that

Bowley.^
the

of

among

law

realized

theory.

analysis

attention

not until

his

to h a v e

economic

the q u a n t i t a t i v e

It w a s

begun

published

seemed

of

not a t t r a c t m u c h

the w o r k

have

else,

he

5

of
profes­

interest
Since

significance

then,
of

the

b u d g e t data.

3

G.
J. S t i g l e r , " T h e H i s t o r y o f E m p i r i c a l S t u d i e s
of C o n s u m e r B e h a v i o r , " J o u r n a l of P o l i t i c a l E c o n o m y , V o l . 42
{April, 1954 ), p. 98.
E n g e l a l s o a s s e r t e d tha t the w e a l t h i e r a n a t i o n ,
the s m a l l e r the p r o p o r t i o n of fo o d to t o t a l e x p e n d i t u r e .
I b i d ., f o o t n o t e 9.

Family

^ S . J.
Budgets

P r a i s a n d H.
(Cambridge:

^Schumpeter,

S. H o u t h a k k e r , T h e A n a l y s i s
U n i v e r s i t y P r e s s , 1971), p.

o p . c i t . , p.

961.

^ T h e s t u d y w a s d o n e by R. G.
B o w l e y , F a m i l y E x p e n d i t u r e (London:

D. A l l e n an d A.
S t a p l e , 1935),

L.

of
79.

6

As

time

it

is o f t e n

on

the

passed,

stated

average

expenditure

rate,

i.e.,

that

"percentage

on

income";

with

income,

is

inelastic

demand

slightly.

expenditure

f u n c t i o n of

increases
food

law changed

but

at

with

Now

food

7

is

or:

a

lesser

respect

to

„8
In the

devoted

as:

a decreasing

"food

income.

Engel's

to

last

35 y e a r s ,

the m e a s u r e m e n t
Q

Engel's

by

numbers

law

of

The

Wharton

cated

the

U. S . ,

7

the

elasticity

income

in

Schumpeter,

op.

h as

been

studies.^ ^

few exam pl es .
that,

effort

has

been
of

fo o d

1 A

consumption.
confirmed

of

tremendous

school

The

food

and

repeatedly

following

survey,

household

c i t . , p.

tested

in

1950,

arc

a

indi­

expenditures

to o k

961.

0
North

J. S.
Holland,

Cramer, Empirical
1 9 6 9 ) , p~ 135.

Econometrics

(Amsterdam:

9

Factors
p . 7 0.

M. C. B u r k , I n f l u e n c e s of E c o n o m i c a n d S o c i a l
o n U.S.
Food~Consumption (Minn. :
B u r g e s s , F $ 6 1) ,

K e c c n t s t u d i e s a l s o s h o w i n t e r e s t in o t h e r h o u s e ­
h o l d ite ms , s u c h as h o u s i n g , c l o t h i n g , h o u s e f u r n i s h i n g s ,
and services.
For a p a r t i a l l i s t of s t u d i e s of t h e s e s u b ­
ject s , see R. F e r b e r , " R e s e a r c h o n H o u s e h o l d B e h a v i o r , " in
S u r v e y s of E c o n o m i c T h e o r y , Vol . Ill, S u r v e y X I I (New Yo r k :
St. M a r t i n 's P r e s s , 196 6 ), f o o t n o t e 3, p. 138.
These
s t u d i o s y i e l d l o w i n c o m e e l a s t i c i t i e s for h o u s i n g , e l a s t i c i ­
tie s c l o s e to u n i t y for c l o t h i n g , a n d h i g h e r e l a s t i c i t i e s
for v a r i o u s t y p e s of r e c r e a t i o n , p e r s o n a l c a r e , h o m e o p e r a ­
tion, a n d o t h e r s e r v i c e s ; se e F e r b e r , I b i d . , p. 139.
^ S i m i l a r l a w s h a v e a l s o b e e n f o r m u l a t e d for o t h e r
i t e m s of e x p e n d i t u r e .
For e x a m pl e :
S c h w a b e ' s law s t a t e s
t h a t the p e r c e n t o f i n c o m e s p e n t for h o u s i n g d e c l i n e s as
i nc om e rises; see F e r be r, I b i d .
^ F o r a l i s t o f s t u d i e s , s e e the b i b l i o g r a p h y by
J. N. M o r g a n , "A R e v i e w o f R e c e n t R e s e a r c h o n C o n s u m e r
B e h a v i o r , " in C o n s u m e r B e h a v i o r :
Research on Consumer
R e a c t i o n s , e d . b y L. H. C l a r k (New Yor k:
Harper & Bros.,
1 9 5 8 ), pp. 9 3 - 2 1 9 .

7

up a b o u t
income

was

$10,000.
about

30% of

income

$5,000,

12

40

household

surveys

le s s

equal;

the h i g h e s t

lowest

0.344

for

Food

ing e s t i m a t e s
in v a r i o u s
in

years

1960;

for

based

on

30 c o u n t r i e s ,

food

in

0.731

report

for

in

1962,

0.30
0.23

of

countries

were

all

but

not

similar,
Poland
17

and

the

expenditure
1955;

1965;

t he

Recently,

published

in

in

analyses

the

for

e l a s t i c i t y of

income was

that

were

1965

household

found

( m id d le c l a s s ) .

in B r i t a i n :

0.27

these

figure was

income

household

regression

They

Britain

income when

when

t h a n one .

Survey
of

21%

from a bout

elasticities

National

only

Houthakker,

significantly

0.25

but

disposable

0.28

and

the

follow­
on

fo o d

in 1958;

0.23

in

1 9 6 6 . 14
In o r d e r
Engel

curves,

and d at a

1.2.

that

Cross
The

sectional

it

to u n d e r s t a n d
is n e c e s s a r y

have

been widely

the

empirical

to r e v i e w

some

results

of

techniques

used.

Section
s t u d y of

data

Engel

obtained

curves

frequently

from a sample

of

uses

cross

households

for a

12

T h e U.S. B u r e a u of L a b o r S t a t i s t i c s f o r the
W h a r t o n S c h o o l of F i n a n c e , U n i v e r s i t y of P e n n s y l v a n i a :
S t u d y of C o n s u m e r E x p e n d i t u r e s , I n c o m e s a n d S a v i n g s ,
Vol. 1-2 (Uni versTi ty of P e n n s y l v a n i a P r e s s , 19 5 67 ■
13
'H. S. H o u t h a k k e r , "An I n t e r n a t i o n a l C o m p a r i s o n of
E x p e n d i t u r e P a t t e r s , C o m m e m o r a t i n g the C e n t e n a r y of E n g e l ' s
L a w , " E c o n o m e t r i c a , Vol . 25 (October, 1 9 5 7 ) , pp. 5 3 0 - 5 1 .
^ 4T h e N a t i o n a l F o o d S u r v e y C o m m i t t e e , D o m e s tic
F o o d C o n s u m p t i o n a n d E x p e n d i t u r e (London:
H.M.S.O.,
1965), p. 136.

8

particular
w ee k ,
The

period

a month,

techniques

tion,

and

th i s

arc

food

less

$6,000.

any other

may

be

convenient

single

a day,

interval.

regression

system of equ a t i o n s

a

have

16

equa­

been used

is

the o l d e s t

obtained

Lansing

$4,000;
-0.20

the

0.10

for

technique.

following

area:
for

0.26

those

households

17

West,

income
for

with

elas­

households
incomes

receiving

of

more

1R

Single
technique

The period

elasticity,

in th e

than
and

or

15

elasticities.

technique,

$4,000-56,000;

cross

of

elasticity

for

receiving

than

a ye a r ,

income

Arc

ticities

time.

simultaneous

to e s t i m a t e

using

of

regression

to i n v e s t i g a t e

sectional

stu dy ,

equation

is

income-food

the

Engel

the

most

popular

relationships.

curve

is g e n e r a l l y

In a
stated

15

T h e r e ar e a t l e a s t t h r e e p o s s i b l e m e t h o d s o f c o l ­
l e c t i n g the d a t a :
pe rso nal interview, mail, and telephone.
F o r a d i s c u s s i o n of r e l a t i v e m e r i t s of t h e s e m e t h o d s , s e e
R. F e r b e r a n d P. J. V e r d o o r n , R e s e a r c h M e t h o d s in E c o n o m i c s
a n d B u s i n e s s (New York:
M a c m i 1 lan , 1962) , p"! 2 0 9 - 1 3.
16

York:

.
K. A. Fox, I n t e r m e d i a t e E c o n o m i c
J o h n W i l e y & Sons , 1 9 6 8 ) , p. 72.

Statistics

^ F o r a d i s c u s s i o n of t h i s t e c h n i q u e , s e e W.
W a i t e a n d H. C. T r e l o g a n , A g r i c u l t u r a l M a r k e t P r i c e s
e d .; N e w York:
J o h n W i l e y - & S o n s , 19 51) , p^ 41.

(N^w

C.
(2nd

18

J. G. W es t , " E s t i m a t e s of I n c o m e E l a s t i c i t y of
C o n s u m e r P a n e l D a t a " ( u n p u b l i s h e d P h .D . d i s s e r t a t i o n , D e p a r t ­
m e n t of A g r i c u l t u r a l E c o n o m i c s , M i c h i g a n S t a t e U n i v e r s i t y ,
1958 ) .
In a c r o s s s e c t i o n a l s t u d y , p r i c e s a r e h e l d
reasonably constant.
E x p e n d i t u r e o n a p a r t i c u l a r g o o d is
p r o p o r t i o n a l to th e p h y s i c a l q u a n t i t y .
E i t h e r q u a n t i t y or

9

Y Xk =

+ U ik

whore

Y.,
lk

j

M^

is

U .

is

J- /C

s the

ihousehold

oxDenditure

the

i^?- h o u s e h o l d

income,

the

stochastic

the

effects

introduced

int o

the e q u a t i o n

in

the

function

the a s s u m p t i o n

that

except

differences

the

for

t he

stochastic

implies

that

much

the

the

on
same

It

B's.
is

estimates

of

mainly

the

in

obtained

good

0.56

it

are
in

household

of
and

Engel

curve.

the

homogeneous
and

assumption

average,

B does

on

the e q u a t i o n

homogeneity
on

explicitly

errors

the

both

is a p p r o p r i a t e

households

The

not

variable,

f o r m of

f^,

A would,

as

an d

in v a r i a b l e s

term.

household
k—

as

different

error

are

the d e p e n d e n t

functional

the k^il g o o d ,

representing

vnriable s that

is a s p e c i f i c

Regarding

term

of

measurements
f^

error

on

if A ' s

spend

as

income were

20

interesting
income

elasticity

neighborhood
for

to p o i n t o u t

the

of

United

for
0.50.

States

food

that

cross

in m o s t

sectional

studies

For example:
in

1941;

Stone,

fall

Tobin
et.

a l .,

e x p e n d i t u r e m a y bo use d a s the d e p e n d e n t v a r i a b l e .
This
is the t e c h n i q u e of s c a l i n g of v a r i a b l e s .
The r e g r e s s i o n
c o e f f i c i e n t s and t h e i r s t a n d a r d e r r o r s a r e d i f f e r e n t by
m e r e c h a n g e s of u n i t s .
F o r a full d i s c u s s i o n o f t h i s t e c h ­
n i q u e , see A. S. G o 1d b e r g e r , E c o n o m e t r i c T h e o r y (New Y o r k :
J o h n W i l e y & Sons, 1964), pp. 1 8 5- 8 6.

20

wood

L.
Cliffs:

R. K l e i n ,
Prentice

I n t r o d u c t i o n to E c o n o m e t r i c s
Hal 1, 196 2) , pT 54.

(Engle­

10

estimated
Jureen

0.53

obtained

estimated
0.58

for

average
over

for

0.40
the

ior

States

at d i f f e r e n t
The
upon

purely

exogenous.

more
tice,
thus

the

the

however,
far,

21

only

and

Clark,

United

States

in

1948;

Burk

cross

in

1950;

for

Snyder

fo o d of

section

0.54

family

1880-1950

(of

e t al . ,

obtained

obtained

an

calculated

budgets

from

studies

individual

in

cities,

21

structural

regression equation

If h o u s e h o l d

simultaneous

been

used

household

income

income

simultaneous

rarely

that

system

to e s t i m a t e

the

Wold

1933;

presupposition

appropriate

1938;

in

during

single

in

Sweden

States

dates).

priate

exogenous,

the

Kingdom

for

elasticity

hundred

the U n i t e d

United

0.51-0.53

United

income

three

the

is

not

is a p p r o ­

income
purely

of e q u a t i o n s

would

elasticities.

system

of

in

analysis

the

is

equations
of

be

In p r a c ­
has,
Ungel

J. T o b i n , "A S t a t i s t i c a l D e m a n d F u n c t i o n for
F o o d in t h e U . S . A . , " J o u r n a l of the R o y a 1 S t a t i s t i c a l
S o c i e t y , S e r i e s A (1
p . 119; iT! S t o n e , c t at. , T h e
M e a s u r e m e n t of C o n s u m e r E x p e n d i t u r e a n d B e h a v i o r in the
United Kingdom, 1920-38 ( C a m b r i d g e : U n i v e r s i t y P r e s s ,
1954) , p~! 3 27 ; FH W o l d a n d L. J u r e e n , D e m a n d A n a l y s i s
(New Y o r k :
J o h n W i l e y & S o ns , 1953 ), p^ 30 3 ; F . C l a r k ,
et al., " F o o d C o n s u m p t i o n o f U r b a n F a m i l i e s in t he U n i t e d
S t a t e s , " A g r i c u l t u r a l I n f o r m a t i o n B u l l e t i n No. 132
( W a s h i n g t o n ^ D . C . : U.S. D e p a r t m e n t of A g r i c u l t u r e , 1 9 54 ) ,
p. 39; M. C. Burk, " I n c o m e - F o o d R e l a t i o n s h i p s f r o m C r o s s
S e c t i o n a n d T i m e S e r i e s S u r v e y s , " P r o c e e d i n g s of the
A m e r i c a n S t a t i s t i c a l A s s o c i a t i o n ( B . E . S . S . , 1 9 57 ) , p. 103;
a n d LA W. S n y d e r , ''Long-term c h a n g e s a n d F a m i l y E x p e n d i ­
t u r e , " in C o n s u m e r B e h a v i o r :
R e sea rc h on C o n s u m e r
R e a c t i o n s , e d . by ET FT! C l a r k (New Y o r k : H a r p e r T B r os . ,
1958), pp. 3 6 1 - 6 2 .

1]

curves

mainly

because

specification

1.3 .

Time

of

data

curves.
gating

of

household
of

of

behavior

sectional

data,

also

been

used

in

this

type

are

expenditure

time.

observations

vkt

complexity

structural

cross

have

Da t a

periods

the

estimation

equations.

and

£

Series

besides
series

the

of

Th e

Engel

is g e n e r a l l y

the

the

analysis

normally

and

aggregate

obtained

household

c m ve

stated

of

based

on

time

Engel

by

income

ti m e

aggre­

over

scries

as:

= £k ' " t > + u k t

where

is

the

aggregate

expenditure

on

the

k ^ - good

at

- u

the
Mt

is

t—
the

period

of

aggregate

time,
income

at

the

t^- period

of

t imc ,
Uj,

is

the

f^

is

a specific

The
that

in

function

different

differences
the

in

the

stochastic

should

22

be

error

functional

f^

periods

Strictly
ture

stochastic

term,
form.

is a p p r o p r i a t e
of

time

explicit
error

on

the

homogeneous

variables

and

for

assumption
except

for

differences

term.

speaking,

converted

are

and

in a

time

to q u a n t i t y

series
since

study,
price

expendi­

varies

H.
S. H o u t h a k k e r a n d L. D. T a y l o r , C o n s u m e r
in th e U n i t e d S t a t e s :
A n a l y s e s a n d P r o j e c t i o n s (2nd e d .;
Harvard:
U n i v e r s i t y P r e s s , 1 9 7 0 ) , p. ?.

Deman

12

over

periods

s u c h as

food,

deflator.

time.

the

That

following
price

of

food

is,

i n d e x of

are d e f l a t e d

l ow e r

values

sectional

for

for

23

section
time

the

rather

series,

food of

0.27

of

and

0.23

Sweden;
period

1.4.

than

Tobin
for
0.28

Burk

Consumer

panel
23

footnote

For
17,

food

is

computed

where

as

a
in

P y t is

relationships,

for

seem

got

the
the

0.21

the

the

and

cross
estimates

of

using

elasticity
Jureen,

covered)

(depending on

States.

law,

cross

example:

and

period

vari­

Engel's

terms

income

Wold

0.2 3

United

series

For

of

States;
on

foot! t h a n

in

data.

where

substantially

to c o n f i r m

an e s t i m a t e

United

give

time

stated

series

(depending

for

values

fo r
the

24

Panel

Recently,
consumer

index

elasticity

time

for

used

( Y ^ ^ / P ^ t ) *100;

initially

obtained

covered)

be

price movements,

for

the

commodity

may

Nevertheless,

law wa s

a composite

food.

income

studies.

of

index

income-food

income elasticity

although

p.

k-—

series

ables

case

the q u a n t i t y
-

the

the

price

manner:

Time

of

In

a number

data.

25

of

This

an e x p l a n a t i o n

studies
type

of

of

this

have
data

made

use

of

is o b t a i n e d

phenomenon,

see

from

p.

90,

Tobin, o p . c i t ., p. 134; W o l d a n d J u r e e n , o p . c i t .,
303 ; Burk, " I n c o m e - F o o d R e l a t i o n s h i p s , " o p . c i t . , p . 10 3.

A c o n s u m e r p a n e l is e s s e n t i a l l y a s a m p l e of p e o p l e
w h o a r e i n t e r v i e w e d r e p e a t e d l y o v e r a p e r i o d of time.
S t r i c t l y s p e a k i n g , a s a m p l e b e c o m e s a p a n e l o p e r a t i o n if its
m e m b e r s a r e i n t e r v i e w e d in at l e a s t t w o d i f f e r e n t p o i n t s in
t i m e o n the s a m e g e n e r a l s u b j e c t .
If o n l y t w o o r t h r e e

13

the

same

provide

group
both

of

households

cross

Ferber

over

sectional

appraises

and

periods
time

the p a n e l

of

series

data

by

time.

The data

information.

stating

that

Time series aggre g at e s have serious disadvan t a ge s
b e c a u s e of the f r e q u e n t l y u n s t a b l e e s t i m a t e s of
i n c o m e e l a s t i c i t i e s . . . . On th e o t h e r h a n d , c r o s s
s e c ti o na l d a t a are e s s e n t i a l l y s t a t i c . . . Hence, a
c o m b i n a t i o n o f the t w o t y p e s o f d a t a w o u l d s e e m to
o f f e r a m u c h m o r e p o w e r f u l t e c h n i q u e for u n d e r s t a n d ­
ing c o n s u m e r b e h a v i o r .
By k e e p i n g
ture

c a n be

household's
which

obtained,

not

purchasing

decisions

data
Tg
parameters.*'

notes

records,

only

of

behavior,

arc made.

Marschuk
sectional

continuous

However,

but

factors
a lso of

pic­

influencing
the m a n n e r

a

in

27

that

increases

the

a comprehensive

the
the

combining
accuracy
combined

time
of

series

and

cross

the e s t i m a t e d

studies

face

some

" w a v e s " of i n t e r v i e w s a r e i n v o l v e d , t h e r e is a t e n d e n c y to
u s e the t e r m " r e i n t e r v i e w s a m p l e " r a t h e r t h a n " p a n e l . "
Then
a g a i n , if the s a m e p e o p l e a r e i n t e r v i e w e d s e v e r a l t i m e s o n
d i f f e r e n t a s p e c t s of the s a m e s u b j e c t , n e i t h e r " p a n e l " nor
" r e i n t e r v i e w s a m p l e " m a y be u s e d to d e s c r i b e the o p e r a t i o n .
For m o r e d i s c u s s i o n of the c o n s u m e r p a n e l , s e e F e r b e r a n d
V e r d o o r n , op. c i t . , pp. 2 6 7 - 7 6 ; G. G. Q u a c k e n b u s h a n d J. D.
Shaffer, " C o l l e c t i n g F oo d P u r c h a s e Data by C o n s u m e r Panel,
1951-58," T e c hnical Bulle ti n 2 7 9 , M.S.U. A g r i c u l t u r a l
E x p o r i m o n t S ta t"ion ( A u g u s t , 1 9 6 0 ) .
*■ F e r b e r ,

o p . ci t . , p.

2 7 l b i d . , p.
28

141.

145.

J. M a r s c h a k , " R e v i e w of S c h u l t z , T h e o r y a n d
M e a s u r e m e n t of D e m a n d ," E c o n o m i c J o u r n a l , Voir. 3*9 (19 39) ,
p . 4 87.

14

difficulty
function

in

f,

and

k

With
Consumer

interpretation
the

Data

income e l a s t i c i t y
Crockett,
Research

using

Concepts

regression
dependent

of

usinq
of

variable,

f ou n d

panel

America
for

expenditure.
independent

different

variables
Some
others

variable,

Nevertheless,

Sparks,

was about

consumer

analysis.

independent

the

the

food

the

that

of

M.S.U.

the e s t i m a t e

0.25.^®

data

for

using

Similarly,

the M a r k e t

1 9 5 1 - 5 3, o b t a i n e d

food

of

of

an

0.23.-*^

of V a r i a b l e s

ferent c o n ce pts

as

for

1955-58,

of

29

analysis,

income elasticity

Besides

total

of

Corporation

estimated

1.5.

disturbance.

the c o m b i n e d

Panel

and s p e c i f i c a t i o n

some

The

use

variable

several

other

are

techniques
also

studies

used

use

use

of o b s e r v e d

32

of

the

As

income,
or

has b e e n w i d e l y
concepts

in

data,

dif­

single

expenditure

use q ua ntity.
studies

and

for

as
the

others

current

the

use

income

accepted.

income

have

been

29

F o r a full d i s c u s s i o n of t h i s p r o b l e m , s e e E. K u h ,
Capital Stock Growth:
A Micro-Econometric Approach
( Am s te r d a m :
N o r t h H o l l a n d , 1 9 6 3 ) , p p . 1 1 6 - 4 0 a n d p p . 15 86 3; Also, J. K m e n t a , E l e m e n t s of E c o n o m e t r i c s (New Y o rk :
M a c m i l l a n , 1971), p p . 5 0 8 - 1 7 .
30

W. 11. S p a r k s , " E s t i m a t e s of th e D e m a n d f o r F o o d
f r o m C o n s u m e r P a n e l D a t a " ( u n p u b l i s h e d Ph.D. d i s s e r t a t i o n ,
Depart men t of A g r i cu lt u r a l Economics, M i c h i g a n St ate U n i ­
v e r s i t y , 1961).
31

J. C r o c k e t t , "A N e w T y p e of E s t i m a t e of t h e I n c o m e
E l a s t i c i t y of the D e m a n d f o r F o o d , " P r o c e e d i n g s o f the
A m e r i c a n S t a t i s t i c a l A s s o c a t i o n (B .E. S .S . , 5T9 5 7 ) , p p . 1 1 7 - 2 2 .
32
P . 11.

For

a discussion

of

this

problem,

s ee

p.

8 and

15

proposed.
relative

Among
income

them,
have

Briefly,
the

i—

t—

period

b u t on
of
the
on

household
of

the

time

ratio

the

would

The
variable
income

use

can

are

be

for

income

income

income.
of

hypothesis

on

k—

the
not

on

income

on

the

not o n

equals

at

its o b s e r v e d
and

and

that
the

income,

the m e a n

income

income

hypothesis,

k^- commodity

its

th e

states

commodity

income

permanent

and

permanent

income

observed
the

of

known.

depend

expenditure

observed

transitory

well

relative

between

ii-ii h o u s e h o l d
its p e r m a n e n t

been

concepts

expenditure

the g r o u p . A s

where
the

the

the

transitory

permanent

depends
income,

income

plus

34

tota l

expenditure

justified

unavailable,

or

on

as

the g r o u n d s

that

th e

the

independent

that

available

the d a t a
data

on

of

33
' T h e r e l a t i v e i n c o m e h y p o t h e s i s s e e m s to h a v e b e e n
f i r s t p r o p o u n d e d by P. B r a d y and R . F r i e d m a n .
Much a d d i ­
tional t h e o r e t i c a l a n d e m p i r i c a l s u p p o r t of t h i s h y p o t h e s i s
w a s p r o v i d e d b y the w o r k o f M o d i g l i a n i and o f D u e s e n b u r r y .
See D, S. B r a d y a n d K. F r i e d m a n , " S a v i n g s a n d the I n c o m e
D i s t r i b u t i o n , " N . B . E . R . , S t u d i e s in I n c o m e a n d W e a l t h ,
Vol. 10 (New York , 1947), pp. 2 4 7 - 6 5 ; F. M o d i g l i a n i , " F l u c ­
t u a t i o n s in th e S a v i n g - I n c o m e R a t i o :
A P r o b l e m in E c o n o m i c
Forecasting,"
N . B . E . R . , S t u d i e s in I n c o m e a n d W e a l t h , Vol.
11 (New Y o r k , 1949) , pp. 3 7 1 - 4 4 3 ; a n d J . D u e s e n b u r r y , I n c o m e ,
S a v i n g a n d the T h e o r y of C o n s u m e r B e h a v i o r ( C a m b r i d g e :
H a r v a r d U n i v e r s i t y P r e s s , 1952).
a

M. F r i e d m a n h a s g r e a t l y e l a b o r a t e d a n d t e s t e d th e
pe r m a n e n t income hypothes is .
M. D u n s i n g a n d M. G. R e i d ,
a n d M. N e r l o v e h a v e e x p l o r e d the a p p l i c a t i o n s of t h i s
h y p o t h e s i s to f o o d s .
See M. F r i e d m a n , A T h e o r y of t h e C o n ­
sumption F u n c t i o n (Princeton:
N a t i o n a l B u r e a u of E c o n o m i c
R e s e a r c h , 195 7); M. D u n s i n g and M. G. R ei d , " E f f e c t of V a r y ­
ing D e g r e e of T r a n s i t o r y I n c o m e o n I n c o m e E l a s t i c i t y o f
E x p e n d i t u r e s , " J o u r n a l of A m e r i c a n S t a t i s t i c A s s o c i a t i o n ,
Vol. 53 (June, 1958) , p p . 3 5 7-59; m T N e r l o v e , rnfEe I m p l i c a t i o n s o f F r i e d m a n ' s P e r m a n e n t I n c o m e H y p o t h e s i s for D e m a n d
A n a l y s i s , " A g r i c u l t u r a l E c o n o m i c R e s e a r c h ( J a n u a r y , 1 9 58 ).

16

income
it

are

total

highly

unreliable

expenditure

the e l a s t i c i t y

is u s e d

refers,

elasticity,

not

income

between

two

is

Stone,

the
et

ticities
income

al.,

of

be

slight.

For

that

reduced

and

influences

the

by

total

The

total

variable,
expenditure

difference

empirical

example.

expenditure

10?, in o r d e r

Prices
household

income-,

and

elas­

to a p p r o x i m a t e

factors

on

expenditure

household

Non-Kconomic

other

non-economic

might

factors

have

such

Factors
as

significant

behavior.

Prices
Prices

Enge l

the

Nevertheless,

independent

to

an

35

e l a s t i c i t i e s .^

Besides

2.1.

the

course,

2.

prices

as

distorted.

elasticity."*6

realized

should

and

curves,

variable

in

vary
prices

the

over

tim e .

should

regression

be

In a t i m e
introduced

analysis.

series
as

an

In a c r o s s

study

on

explicit
sectional

3^

T h e use of i n c o m e o b s e r v e d w i t h e r r o r a s the
i n d e p e n d e n t v a r i a b l e w i l l lead to s y s t e m a t i c u n d e r e s t i m a ­
ti o n of i n c o m e c o e f f i c i e n t by l e a s t s q u a r e s e s t i m a t e s .
For
a full d i s c u s s i o n o f th i s p r o b l e m , see C r a m e r , o p . c i t . ,
p . 139.
36

T h e use of t o t a l e x p e n d i t u r e as the i n d e p e n d e n t
v a r i a b l e m a y l e a d to i n c o n s i s t e n t e s t i m a t e s .
F o r a full
d i s c u s s i o n of t h i s p r o b l e m , see N. L i v i a t a n , " E r r o r s in
V a r i a b l e s an d E n g e l C u r v e A n a l y s i s , " E c o n o m e t r i c a , Vol. 29
(1961), pp. 336- 62 ; C r a m e r , o p . ci t . , p . TTiTJ a n d
R. S u m m e r s , "A N o t e o n L e a s t S q u a r e B i a s in H o u s e h o l d
E x p e n d i t u r e A n a l y s i s , " E c o n o m e t r i c a , 27 ( J a n u a r y , 1959),
p. 121.
37

Stone,

et

a 1 . , o p . c i t . , p.

312.

study,

prices

households

are

are

reasonably

likely

to

presumed

face

the

to b e

same

constant

since

s e t of market: p r i c e s

except,

o e r h a p s , for r e g i o n a l d i f f e r e n t i a l s o r p r i c e
*3Q
discrimination.
Principally, cross sectional studies
concentrate
Amonq
not

the

price

may

first,

from

and

much

of

the
the

to q u a l i t y

that

size

composition"
household.

bers

the

the m o s t

and

there

analysis
apparent

is

of price
price

differences.

39

is m e a n t

religion

the m e m b e r s

of

t he re a r e m a n y

household

consumption.

obvious

age

there

conditions

also

prices,

household

household.

occupation may
as

affect

Second,

external

of

inconif? a n d

probably

reflect

ol

since

elasticities.

in a s u r v e y ,

to p e r m i t

attributed

.actors

is h o u s e h o l d

we l l

income

n --E c o n o m ic F a c t n r s

economic

the

of

participating

variation

be

Apart

hold

estimation

particularly

variation

2 ..1.

the

households

enouah

effects,

on

influence

and va ri o u s

of

composition.
and

are

and

And,

cause

s e x of
regional

social

third,

household

the h o u s e h o l d .

I3y " h o u s e ­
of

factors, w h i c h
of

the

class

and

expenditure,

psychological

The

variations

the m e m b e r s

habits

social

non­

mem­

as

characteristics

40

38

L. R. Kle i n, A T e x t b o o k of E c o n o m e t r i c s ( E v a n s t o n :
Row, P e t e r s o n L Co., 195 3) , p . 213; S t o n e , e t al . , o p . ci t . ,
p. 312; P r a i s a n d H o u t h a k k e r , o p . c i t . , p. 110.
39
40

Houthakker

and

Taylor,

o p . c i t . , p.

254 .

T h e r e h a v e b e e n n u m b e r s of s t u d i e s o f n o n - e c o n o m i c
factors.
In o r d e r to l i m i t the s u r v e y , t h i s s t u d y w i l l
c o n c e n t r a t e on h ou s e h o l d size and h o u s e h o l d co m p o s i ti on .
For

18

Of
household

all

the

composition

Household
usually measured
or

unit

each

say,

by r e q a r d i n q
s ome

by

where

used

the

consumed

by

an

adult

by a c h i l d

less

and

an

includes ho u s e h o l d

are

scales,

values,

and

man

some

"standard"

the

scale

female

as

the v a l u e

3.5.

is

year

constructed

equivalent

In fac t,

labelled

than one

sectional
size

composition

41

a p r o p o r t i o n of

who

adult male

intensively.

expendi­

male.^

as

and

food

adult

Engel

size

the

T h u s,

an

detined

In a c r o s s

stated

as

household
most

scale expresses

que t was

3.1 q u e t s ,

studied

household

male.

Ernst

factors,

equivalent

ty p e

a child or

t r a c t i o n of

already

This

adult

been

and

adult

aqe-sex

the

have

size

consumers.

ture of
type,

non-economic

this

scale was

the u n i t
of

food

ol d ,

to

a

" q u e t" ;

that was

an a d u l t

female

43

stu d y,

the

household

Engel

curve

that

compos!tion may

be

as:

more1 s u r v e y s a n d d i s c u s s i o n s of o t h e r n o n - e c o n o m i c f a c t o r s ,
see R. O. H e r m a n n , " H o u s e h o l d S o c i o - E c o n o m i c a n d D e m o g r a p h i c
C h a r a c t e r i s t i c s as D e t e r m i n a n t s of F o o d E x p e n d i t u r e
B e h a v i o r " ( u n p u b l i s h e d Ph.D. d i s s e r t a t i o n , D e p a r t m e n t of
A g r i c u l t u r a l E c o n o m i c s , M i c h i g a n S t a t e U n i v e r s i t y , 1 9 6 4 );
Burk, U.S. F o o d C o n s u m p t i o n , o p . ci t . , pp. 5 3-64 ; a n d
F e r b e r , o p . c i t . , pp. 1 2 6 - 3 4 .
^ I t s h o u l d be p o i n t e d ou t t h a t h o u s e h o l d s i z e and
c o m p o s i t i o n v a r y f r o m h o u s e h o l d to h o u s e h o l d , y e t t h e y v a r y
v e r y l i t t l e f r o m y e a r to yea r.
Thus, t h es e f a c to rs m i g h t
be o m i t t e d in a ti me s e r i e s s t u d y .
See H o u t h a k k e r a n d
T a y l o r , o p . ci t . , p. 2 7 5 .
42
43
Economy

Prais

and

Houthakker,

op.

c i t . , p.

126.

C . S . B e 1 1 , Consumer Choice m
the A m e r i c a n
(New Y o r k :
R a n d o m H o u s e , 1 9 6 7 ) , p~! 1 0 4 .

19

Y,, =
] 1,

f.
v

(M. , lc
N. .) + U iv
i
, ik : n
1K

where

N. . is
11

the

number

in

the

i—

c.-j, is

is

persons

of

the

of

the

j—

age-sex

type

household ,

the e x p e n d i t u r e

type
Y.

of

person

scale

for

summation

of

the

th
j—
age-sex

t hi
k—
good , and

the

over

value

-j .

J

J n t his
lent

adult

scale

lor

seale .

is d i s t i n g u i s h e d

of

a

Ji i.f.oror.i

for

each

th
tlic k~
c e m m o d i ty may

be

scale

of

commodity,

and

th
k—

t c-rim:-i the

equiva­
the

specific

44
The

(the

formuiatlon,

scale

varying

s e t of
that
aye

values

is b a s e d
and

sex),

may

be

the

nutritional

on

the

nutritional

or

the

actual

scale

requirements

expenditure

scale
A [*

that

is c o n s t r u c t e d

practice,
person

and

a s e t of
for

each

from

the

actual

nutritional
commodity

observed

scales
is

for

data.

each

type

In
of

chosen.

44
S o m e r e s e a r c h e r s e x p e c t tha t t h e s c a l e s o f
e q u i v a l e n t a d u l t s w i l l be s i m i l a r for all c o m m o d i t i e s , so
t h a t it w i l l n o t be n e c e s s a r y in p r a c t i c e to d i s t i n g u i s h
a s c a l e for e a c h c o m m o d i t y .
45

F o r a full d i s c u s s i o n o f c o m p u t a t i o n a n d a p p l i c a ­
t i o n of a g e - s e x e q u i v a l e n t s c a l e s , see D. W. P r i c e ,
" A g e - S e x E q u i v a l e n t S c a l e s tor U n i t e d S t a t e s F o o d E x p e n d i t u r e s — Their C o m p u t a t i o n and Application" (unpublished
Ph.D. d i s s e r t a t i o n , D e p a r t m e n t o f A g r i c u l t u r a l E c o n o m i c s ,
M i c h i g a n S t a t e U n i v e r s i t y , 1965).

20

Traditionally,
i, k .

*1

income

In m o s t

are high l y

hypothesis
household
depends
Eng el

survey

only on

curve

may

is

to

the
be

for

studies

data,

positive

allowing
size

most

effects

suppose

stated

household

correlated.

the

level

assume

of

The

per

and

fo r al l

household

simplest

variations

that consumption
income

= 1

ik

siz e

47

of

c

in

per p e r s o n

person.

The

as:

46
T h u a s s u m p t i o n i m p l i e s th a t t h e r e is no d i f f e r ­
e n c e in t h e a m o u n t s p e n t for a n y c o m m o d i t i e s c o n s u m e d by
p e r s o n s o f d i f f e r i n g a g e a n d sex.
In g e n e r a l , m a n y
d i f f e r e n c e s e x i s t in the a m o u n t s p e n t for c o m m o d i t i e s
consuiru J by i n d i v i d u a l s of d i l t e r e n t a g e - s e x c o m p o s i t i o n .
H o w e v e r , t .is a s s u m p t i o n is a c c e p t a b l e w h e n i n f o r m a t i o n
r e g a r d i n g a g o - s o x c o m p o s i ti tin of i n d i v i d u a l h o u s e h o l d s
is u i u v d i lab l o .
47
It s h o u l d be n o t e d t h a t the p o s i t i v e c o r r e l a t i o n
is not d u e to a d i r e c t c a u s a l lin k b e t w e e n the two v a r i ­
a b l e s b u t to f o r t u i t o u s c h a r a c t e r i s t i c s of the e x i s t i n g
s o c i a l s t r u c t u r e , as m a y be i l l u s t r a t e d by the two e x t r e m e
i n s t a n c e s w h i c h l a r g e l y d e t e r m i n e the o b s e r v e d c o r r e l a ­
tion.
A t o n e e n d of the s c a l e we h a v e h o u s e h o l d s of
one or t w o p e r s o n s , w h i c h u s u a l l y r e p r e s e n t the v e r y
y o u n g - - b a c h e l o r s a n d y o u n g c o u p l e s - - o r the old; b o t h
c a t e g o r i e s t e n d to h a v e s u b s t a n t i a l l y l o w e r i n c o m e s t h a n
the a c t i v e a d u l t p o p u l a t i o n .
A t the o t h e r e n d of the
scale, v e r y l a r g e f a m i l i e s of e i g h t o r m o r e p e r s o n s
often include m o r e than on e wag e earner, either b e c a u s e
the y a re in f a c t c o m p o s i t e h o u s e h o l d s , or b e c a u s e o f
the n a t u r a l a g e s t r u c t u r e of f a m i l i e s w i t h six o r m o r e
children.
S e e C r a m e r , o p . c i t . , p. 162.

2]

This
t ion

hypothesis

of c o n s t a n t

returns

obviously
to

scale

corresponds
often

made

to
in

the a s s u m p ­
the

theory

of p r o d u c t i o n . ^

r u n ci;iona I F o r m s
Perhaps
ii’.u'i

the

most

of

difficult

'-nr f>s ■;r. to (-■'[ cess

the

En gc i C u r v e s
part

function

of

the

analysis

in an

of

appropriate

48 ,
.
ilio a s s u m p t i o n <«! cni.stunt r e t u r n s to s c a l e is
l i k e l y i m p r o p e r s i n c e a l a r g e h o u s e h o l d m a y be a b l e to
a t t a i n a h i g h e r l e v e l of p e r c a p i t a c o n s u m p t i o n t h a n a
smaller household.
P a r t i c u l a r l y w i t h fo o d , e c o n o m i e s m a y
ii i.u ■ in p u r c h a s i n g , s t o r a g e , a n t i p r e p a r a t i o n of food.
S e v e r a l e m p i r i c a l s t u d i o s h a v e a t t e m p t e d to i n v e s t i g a t e
the d e g r e e of e c o n o m i c s ot sca le .
For a c l a s s i c e x a m p l e ,
P r u i s set up the f o l l o w i n g m o d e l :

where

f^ and

g^

are

undefined

functional

forms.

In P r u i s ' s m o d e l , if the r e w e r e n o e c o n o m i e s of
scale, g^ is zero.
N e v e r t h e l e s s , the m o d e l f a c e s t wo d i f ­
ficulties.
T h e y are :
if f^ is n o t s p e c i f i e d c o r r e c t l y ,
then p a r t of the v a r i a n c e p r o p e r l y a s c r i b a b l e to M y / N y is
i sc r i b e d to 1^ , a n d v a l u e s of My a n d Ny t e n d to b e
c o r r e l a t e d so t h a t the c o e f f i c i e n t s of th e r e g r e s s i o n are
i m p r e c i s e b e c a u s e the s t a n d a r d e r r o r s of the r e g r e s s i o n
c o e f f i c i e n t s b e c o m e large.
P r a i s u s e d t h e s e m i - l o g form;
w i t h the B r i t i s h d a t a of 19 38, he f o u n d t h a t the e c o n o m i e s
of s c a l e a p p e a r e d , b u t w e r e v e r y s m a l l .
With a belief
that the d i s a d v a n t a g e s of s m a l l h o u s e h o l d s m a y t o d a y bo
not so g r e a t , the a s s u m p t i o n of c o n s t a n t r e t u r n s to s c a l e
may b e t a k e n as s u b s t a n t i a l l y c o r r e c t in the f o r m u l a t i o n
of E n g e l c u r v e s .
F o r a f u l l e r d i s c u s s i o n of t h i s p r o b l e m ,
see S. J. P r a i s , " W o n - L i n e a r E s t i m a t e s of the E n g e l
C u r v e , " R e v i e w of E c o n o m i c S t u d i e s , Vol. 20 ( 1 9 5 2 - 5 3 ) .

22

algebraic

functional

households
all

were

households

tion a n d

were

to d e r i v e

regarded

a9

as

Theoretically
the

approximately
faced

by

functional

particular

form.

f o r m of

the

forms

the

fundamental

had

the

speaking,
consumer

same p r e f e r e n c e

same p ric es ,

it w o u l d

of

curves

all

preference

Engel

function were

be

if
units,
func­

possible

provided
given.

a

50

4n

'1ho !>i>.blem of 1 : n 1 inu t h e m o s t ■:o p ropr i a te fo rm
of E ng e l c u r v e s is an o l d o n e in e c o n o m e t r i c s .
As yet, n o
s o l u t i o n appeal s to h-ive f o u n d g e n e r a l a c c e p t a n c e .
Gener­
al ly s p e a k i n g , it is p r o b a b l y true t h a t the i n v e s t i g a t i o n
of the f o r m of E n g e l c u r v e s has a t t r a c t e d l e s s a t t e n t i o n
tha n h a v e m e t h o d s of e s t i m a t i n g p a r a m e t e r s f o r s p e c i f i e d
cqii'il ion.cj.
Sec C. E. V, L o n e r , " f o r m s of E n g e l F u n c t i o n s , "
E c o n o m e t r i c a , 11
( October, 1963 ) , p. 694.
50

1

u

D e n o t e q , ..., q* be a s e t of n g o o d s p u r c h a s e d
by the r o p r c s c n t « t i v e consuim i d
a p e r i o d o f time; p ,
p n be the c o r r e s p o n d i n g s e t of p r i c e s ; m is t h e d i s p o s a b l e
income; u ( q , ..., q n ) the u t i l i t y i n d i c a t o r .
G i v e n p's
and m at a p e r i o d
of time, t h e fir s t c o n d i t i o n for m a x i m i z ­
ing the u t i l i t y s u b j e c t e d to the b u d g e t c o n s t r a i n t is
f u l f i l l e d if the c o n s u m e r p u r c h a s e d the q u a n t i t i e s s u c h as
rtU
iiq

, l
T p
=

n
y. p k q k
k=l

jU

, 2 ...
rd1, n
" .................. '
aq

aq

- m.

S o l v i n g the a b o v e e q u a t i o n s , o n e g e t s th e q u a n t i ­
ties p u r c h a s e d a s f u n c t i o n s of p r i c e s a n d i n c o m e s .
In a
cross s e c t i o n w i t h c o n s t a n t prices, q u a n t i t i e s p u r c h a s e d
w i l l d e p e n d o n l y o n the i n c o m e ; th a t is,
q 1 - f 1 (m) ,
or

qk

-

.

fk (m)

for

These

Engel
yk

By
curves

- fk (m)

,

q n = fn (m) ,
k = 1,

are,

the
can

in

2,

...,

fact ,

n.

the E n g e l

technique of scaling
b e r e w r i t t e n as:

of

curves.
variables,

th e

23

Unfortunately,
function

ence

ilonc

could

m a t i ca l

is

not

lation.

As

and

Enqnl

Needless

curves.

functional
among

Some

say,

Both economic

"The

the

theory,

prefer­

economic
of
and

of

theory

the m a t h e ­
statistical

alqebraic

choice

in p r a c t i c e ,

. . . economic

the

knowledge

t h e c h o i c e of

form,

f o r m of

to

appropriate

influence

s i m p l i c i t y ."

element

functional

Goldberqor mentions,

appropriate
comoromiso

unknown.
provide

f o r m of

considerations

the

formu­

an

involves

goodness

a

of

fits,

b1

researchers

from e c o n o m i c

believe

theory

that

that

can

there
be

is

taken

at

least one

over

for
c2

lovmulating

where

the

yk = pk

*

alnebraie

for k =

functional

1,

2,

...,

f o r m of

Engel

curves.

n.

F o r a lul l diGcustiinn of thi s p r o b l e m , s e e A. W a l d ,
"The A p p r o x im.it e D e t e r m i n a t i o n o f I n d i f f e r e n c e S u r f a c e s b y
M e a n s of E n g e l C u r v e s , " K c o n o m o t r i c a , Vol. 8 (1940),
p p . 14 4-46.
It is i n t e r e s t i n g to p o i n t o u t that, k n o w i n g th e
s h a p e s of E n g e l c u r v e s , the f u n c t i o n a l form o f i n d i f f e r e n c e
s u r f a c e s m i g h t be a p p r o x i m.j te J y d e t e r m i n e d .
See W ol d and
J u r e e n , o p . ci t . , pp. 1 3 0 - 3 1 ; W a l d , o p . c i t . , pp. 14 4-7 5 ;
and M. T. D a v i s , T h e T h e o r y of E c o n o m e t r i c s ( B l o o m i n g t o n ,
Ind.:
P r i n c i p i a P r e s s T 1941), p p ”
5 -68™.
51
52

Goldberger,

o p . c l t . , p.

217.

Tf p o s s i b l e , t h e r e is a n o t h e r e l e m e n t :
the
s a t u r a t i o n p o i n t o r the p o i n t of z e r o m a r g i n a l u t i l i t y .
T h e o r e t i c a l l y , th e s a t u r a t i o n p o i n t is v e r y u n l i k e l y to b e
reached.
In p r a c t i c e , h o w e v e r , m o s t r e s e a r c h e r s b e l i e v e
that if a c o m m o d i t y is a s p e c i f i c item, for an i n d i v i d u a l
h o u s e h o l d , the s a t u r a t i o n p o i n t is l i k e l y to o c c u r at a
h i g h l e v e l of i n c o m e .
O n t h e o t h e r h a n d , if a n u m b e r of
c o m m o d i t i e s are a g g r e g a t e d as a c o m p o s i t e c o m m o d i t y , o r
if a g r o u p of h o u s e h o l d s r a t h e r t h a n an i n d i v i d u a l h o u s e ­
h o l d is i n v e s t i g a t e d , the .satiety lev el is n o t l i k e l y to
be r e a d i e d .
See P r a i s .and H o u t h a k k e r , op, cit. , pp. 1 6 - 1 7 ;
and C r a m e r , o p . c i t . , p. 149.

24

Assuminq

or

no

saving,

t he . o . i i i i j

f't

.1

that

ti.q i

■:

element

is

. Y, >
,

’ *■

the

budget

whi ch

restriction

a Y ik

implies

/.

whoro

IK

.jvcrayt'

4. i( n i

/r»M

-j^ ttin

1

the

t fa
k—
commodity.

11

1 he

number of

restricted

to

to

is b o u g h t ,

tha t cumiiioui ty
additional
the

diminish
the

the

steep

as

slope

fit

and

for

to m a k e

Engel
income
As

unity.

commodities

coirunod 1 1 l e s

that

is

for

tiro

room,
are

the
If

are

is

very

slope
income

bought,

mrur*

the

of

As

as

for

increases.
statistical
some

on

the

l u d g e f rcaf.ri.c-

it w e r e ,

income

with

the

low,

only

the

Engel

rises,
then,

rises,

moste l e ­
one
curve

it

is

apparent

gradually

for n e w e n t r a n t s .

r -j
J it

is

all c o m m o d i t i e s

for

successively

first commodity will

substitutes,

curves

simplicity),

starting

income

then

1 ;

incur, e cc n sutipfiun is

commodities.

Thu s ,

situation when

commodity

that

number

enter.

Fneel

c inim- 1i L j e s in

vdi y , at. low

small

new c o m m o d i t i e s
mentary

<^f the

for

-if l o w e d

■- =

k dMi

k ±k

valid

to

become

If

suppose
less

54

considerations
researchers

(goodness

believe

that

of
the

53

B u t t h e r e is no n e c e s s a r y r e a s o n f o r all c o m m o d i ­
t i e s to be s u b s ti L u t e s .
Tire i n t r o d u c t i o n of a n e w
c o m m o d i t y i n t o t h e b u d g e t r e s t r i c t i o n m i g h t c a u s e the
s l o p e s of its c o m p l e m e n t s to ris e.
If s e v e r e , t h e s u c c e s ­
s i v e n e w e n t r a n t s m i g h t r e p l a c e the a l r e a d y p r e s e n t
c o m m o d i t i e s e n t i r e l y , so t h a t t h e s h a p e s of E n g e l c u r v e s
m i g h t be k i n k e d o r d i s c o n t i n u o u s o v e r c e r t a i n r a n g e s of
income.
54

Cramer,

P r a i s a n d H o u t h a k k e r , o p . c i t . , pp.
o p . c i t . , pp. 1 4 7 - 4 9 .

15-17;

25

functional
data
may

and

f o r m of

reproduce

possess.

The

transformation
be

applied

tions.

th at
are

of

F.nqol

curves

any marked
form can

should

curvature

be m a d e

the d a t a ,

broadly

so

linear

that

to

the

correspondingly

There

are

dozens

the

the

observations

by

linear

fi t

a simple

regression

transformed

can

obscrva-

i>5

have
some

been

proposed

mathematical

empirical

studios

semi-log,

doublc-loq,

The

of

o.
lor

forms
this
and

iustificaticn

a first

order

fined.

Tho

35
J Cramer,

of

Engel
that

lunotional

cuives.
have

suhicet.

5 r,

been

They

forms

The

used

ar c :

lollowing
in m o s t

linear,

log-normal.
of

approximation

linearity

algebraic

the

linearity

to a n y

Engel

o p . c i t . , p.

function

curves

147;

is

tha t

which

it

is u n d e ­

is a c c e p t a b l e

Loser,

is

o p . c i t .,

when

p.

694 .

Jblt m i g h t be t h o u g h t that the p r o b l e m of s e a r c h i n g
for the a p p r o p r i a t e for m of E n g e l c u r v e s is t r i v i a l , s i n c e
a p o l y n o m i a l o f s u f f i c i e n t l y h i g h d e g r e e c a n a s s u m e any
required shape.
H o w e v e r , the f l e x i b i l i t y of a p o l y n o m i a l
is o n l y a n a d v a n t a g e if the d e g r e e of s c a t t e r of t he
o b s e r v a t i o n s is s m a l l e n o u g h to a l l o w a p r e c i s e d e t e r m i n a ­
t i o n of the p a r a m e t e r s o f t h e p o l y n o m i a l .
The data
p r o v i d e d b y f a m i l y b u d g e t s do n o t .seem to h a v e s u f f i c i e n t
r e g u l a r i t y to m a k e t n i s a d v a n t a g e p o s s i b l e .
It is t h e r e ­
fore n e c e s s a r y to c h o o s e a f o r m w h i c h s u b s t a n t i a 1 ly
r e p r e s e n t s the r e q u i r e d form.
See P r a i s a n d H o u t h a k k e r ,
o p . c i t . , p . 86.
F o r s o m e r e s e a r c h e r s , the c h o i c e of th e a p p r o ­
p r i a t e f o r m is i g n o r e d .
G e n e r a l ly, t h e y b e l i e v e tha t it
will be m o r e i m p o r t a n t to h a v e r e l a t i o n s h i p s w h i c h are
c o n v e n i e n t for o n e o r the o t h e r p u r p o s e .
S e e C. E . V.
L o s e r , " F a m i l y B u d g e t d a t a a n d P r i c e E l a s t i c i t i e s of
D e m a n d , " R e v i e w o f E c o n o m i c S t u d i e s , Vo l . 9 (1941), p. 47.

26

the o b s e r v a t i o n s
narrow

interval

The

of

income

where

confined

curvature

curvature

.since s u r v e y d a t a

are

of

Engel

generally

to a r e l a t i v e l y

matters

curves

cover

little

cannot

or

be

not

at

neglected

a considerable

income

I'Q
ranqe.
the

Amonq

British

semi-loq

fit"

data,

forms.^

form of

to h i s

used
57

found

in m o s t

Cramer,

forms,
out

was

Prais

that

curves

household
that

in

the

studies

o p . c i t .,

and
the

prelorable

Similarly,

Engel

Israeli

Ferber
curves

found

relationship

functional
semi-log

several

case

of

to o t h e r

Liviatan

gave

Houthakker,

the

foods

a

alternative

found

best

using

that

the

"goodness

of

budgets.
functional

was

pp.

form of

essentially

146-47;

the

Allen

Engel
same

and

as

Bowley,

o p . cit.

S t u v o l and d a m e s , in t h e i r s t u d y of h o u s e h o l d
e x p e n d i t u r e o n food in H o l l a n d , s h o w e d tha t n e i t h e r the
lino ir n o r the e x p o n e n t i a l f o r m s g e n e r a l l y u s e d in e s t i m a ­
ting i n c o m e e l a s t i c i t i e s w e r e a p p r o p r i a t e f o r the w h o l e
r a n g e of b u d g e t s in t h a t c o l l e c t i o n .
See G. S t u v e l a n d
S, F. J a m e s , " H o u s e h o l d E x p e n d i t u r e on F o o d in H o l l a n d , "
J
j—o u r n a l of R o y a l S t a t i s t i c S o c i e t y , S e r i e s A, 113 (1950),
69

Prais

and

Houthakker,

o p . c i t . , p.

166.

L i v i a t a n , C o n s u m p t i o n P a t t e r n s in I s r a e l
(Jerusalem:
F a l k , 1964) , p p . 29- 3 0.
B e s i d e s its s i m p l i c i t y , the s e m i - l o g f u n c t i o n h a s
the f o l l o w i n g p r o p e r t i e s :
t h e s l o p e of the c u r v e a n d t h e
i n c o m e e l a s t i c i t y d e c l i n e w i t h the r i s e of t h e i n c o m e l e v e l ;
the c u r v e h a s no s a t i e t y l e v e l ; the c u r v e d o e s n o t p a s s
t h r o u g h its o r i g i n b u t i n t e r c e p t s a p o s i t i v e l e v e l of
income.
Se e J. J o h n s t o n , E c o n o m e t r i c M e t h o d s (New Y o r k :
M c G r a w - H i l l , 1 9 6 3) , p. 47; G o l d b e r g e r , o p . c i t . , p. 214.

27

used

by

Ernst

main

reason

explained

E n q e l , namely,

tor

by

the d o u b l e - l o g

t h e p o p u l a r i t y of

Ezekiel

and

Fo x,

is

form.6 ^

the d o u b l e - l o g

The

form,

as

that

T h o u g h the i n c o m e e l a s t i c i t y f o r a c o m m o d i t y m a y
c h a n q o f r o m o n e i n c o m e level to a n o t h e r , it is o f t e n
m o r e d e s i r a b l e to o b t a i n an a v e r a g e e l a s t i c i t y o v e r
some s p eci fi ed range of incomes.
In fact, t h i s is
e q u i v a l e n t to a s s u m e t h a t the i n c o m e e l a s t i c i t y is
m n s t i n t ov e r the r a n o o in q u e s t i o n .
Another
fhe

i nt e g r a l

proposed

by

asymptote

log-normal
Aitchison

and

Nevertheless,
requires
ently
Danish

classic

at

the

this

iterative

possible,

functional

f o r m of

Engel

curve

sigmoid

response

and
same

curve

nr a

Bro wn .
time
is

methods,

as J o r g e n s e n

C3

This

passes

not e a s y
and

curve

through
to

fit,

in h i s

has

is

curve
an

upper

the o r i g i n .
in

nonconvergence

found

curves

that

it

is a p p a r ­

analysis

of

budgets.^

F e r b e r , o p . c i t . , p. 138.
The d o u b l e - l o g form
p r o v i d e s for a c o n s t a n t
income elasticity.
It p a s s e s
t h r o u g h the o r i g i n , a n d
it h a s an u p w a r d r a t h e r t h a n a
downward curvature when
the e l a s t i c i t y is g r e a t e r t h a n one.
See J o h n s t o n , op. c i t . , p. 48; G o l d b e r g e r , o p . c i t . , p. 215.
rn
M. E z e k i e l and K. A. Fox, M e t h o d s o f C o r r e l a t i o n
and R e g r e s s i o n A n a l y s i s {3rd e d .; N e w Y o r k : J o h n W i l e y &
Sons, 1 9 5 9) , p. 110.
t ■>
F o r a full d i s c u s s i o n of thi s c u r v e , see
J. A i t c h i s o n a n d J. A. C. B r o w n , T h e L o g - N o r m a l D i s t r i b u t i o n
(Cambridge:
U n i v e r s i t y P r e s s , 1957) .
64

of

Danish

1 9 6 5 ),

p.

E. J o r g e n s e n , I n c o m e - E x p e n d i t u r e R e l a t i o n s h i p s
Wage and Salary Earner (Copenhagen:
Kobenhaven,
55'.

CHAPTER

REVIEW

OF M . S . U .

1_.
Since
Consumer
bri ef

data

r ev ie w of

this

The p a n e l
holds

selected

households

The p a n e l
through

household

in t h e

for home

expenditure
or
the

14

composite

fourteen

fru it s ;
etc.;
sugar,

operation

panel

sweets,

of

ending

in

weekly

item.

There were

and

rep o rt .

dairy

food;

bakery

and

nuts

of

28

February,

o n all

a

300

house­

to

30,000

100,000

The

food

50 0

fa t s

and
Each

pur­

and
food

following

items
were

a n d o i ls :

and e g g s ;

cereal

and n u t

1951,

1958.

price,

about

fish,

and

cooking

^For m o r e d i s c u s s i o n
Shaffer, o p . cit

about

products,-

poultry,

and m i n e r a l s ;

of

quantity,

meat;

cand y ;

25,000

in D e c e m b e r ,

each

baby

curves,

excluded.

reported

g r o up s :

the M . S . U .

be g i v e n . ^

a city

started

in e a c h

Engel

about

th e

foods

use o f

of a p p r o x i m a t e l y

giving

food

prepared

first

use,

vegetables;

vitamins

and

for

will

was

a period

make

in a n a l y z i n g

Michigan,

Lansing

SURVEY

Re m a r k s

representative

East

continued

1958

was c o m p o s e d

as

PANEL

s t u d y wi l l

survey

in L a n s i n g ,

inhabitants.

chased

of

CONSUMER

Ge n o r a j

t.hc p r e s e n t

Pane l

]I

jam,

products;

products;

beverages;

aids.

this

survey,

see

Quackenbush

29

In a d d i t i o n ,
after

federal

income

number of

meals

away

home

from

censuses
don e

of

m

sample

on

street and

was

sample

census.

Size

homemaker,

car d

sorting

was d r a w n

the
the

to

selected

those

from

refusing

The
the

were

1,885

needed

fo r

families

panel

family

dropping

and

survey

to

the p a n e l

asked

was

the panel

panel
in

were
use of

then

each

oriqinal

panel

sam­

the

1950

educa­
used

by

as

punch
N—

family

member.

an d

refusals

characteristics

similar

out.

Collecting

questionnaires
were

the

the

Directory.

homemaker,

replacements

with

3.
The

of

households,
in

M.S.U.

households

age

an

City

house­

Each

of

address

serialized

as

were

sampled

sample

original

of

of

Lansing

controls,

listing

or

the

income

four

Substitutes
were

of

sample

respectively.

a random

family,

families

on

from

of

for m e a l s

censuses

number

residential

from

an d

periodic

Sample

2,103,

excluded.

drawn

composition,

Selection

The

of

section

323 w a s

All

and

every

p l e of

income

expenditure

involved

L956.

composed

using

and

Lansing.

1,775,

address

controls.

of

of
and

its

basis.

Methods

1954,

reported

in h o u s e h o l d

quests,

a weekly

City

East L a n s i n g w as

tion o f

to

the

1,885,

by

change

procedure

census

obtained

tax,

samplina

1950,

holds wa s

household

served

2.
The

each

carried

out

households.

to r e p o r t

each week

the

Information

through
All

mailing

households

by m a i l i n g

in

in

30

a

purchased

food

previous

wouldn't
a diary.

panel
and

was

for g et ,

no

neqlect,

An e v e n g r e a t e r

households
weekly

visits were

indicated
in c o m e .

had

assurance

current dispos abl e

check

These

which

been

mailed

to

them

the

week.
Ther e

weekly

diary

were

or

the

income

high

the

data.

annually

aqainst
well

panel

to e n t e r

p r o b l e m was

generally

a reasonably

refuse

visited

reports

that

annual

m

some

items

collection

About

half

in o r d e r
reports

received.

accuracy

members

the

of

in
the

the

to v e r i f y
of

income.

This

rapport

data

on

observed

CHAPTER

CROSS

SECTIONAL

1.
Most
sectional
Enqel

or

Enqel

previous

data

of

studies

on

a particular

for

the

true

and

Enqel

period

e l a s t i c i t i e s .^

selected

Snme

G e n e r a 1 Remarks

curves
of

used

time

A particular
functional

cross

to e s t i m a t e
functional

f o r m of

the

curves.
In

tional

STUDIES

O b j e c t ivos

income

for m w a s

III

this

studies

composite
selected

chapter,
on

foods.

the

Engel

Three

for

the

true

In t h i s w a y ,

the

income

composite

f oo d s ,

forms,

ar e

income

elasticity

functional

based

obtained.

forms

thirteen

successive

curves

are

alternative

functional

on

three

Thus,

estimates
are w i d e l y

one

set o ut

functional

f o r m of

elasticity

the

estimates

alternative
can

based

cross

on

different.

five

forms

Engel
for

are

curves.

the

five

functional

investigate
three

for

sec­

if

th e

alternative

In a d d i t i o n ,

one

F o r p u r e e c o n o m i c t h e o r y , an i n c o m e e l a s t i c i t y h a s
l ong b e e n u s e d to i n d i c a t e if a c e r t a i n c o m m o d i t y is a
luxury, a n e c e s s i t y , or an i n f e r i o r g o o d .
For business
firms, an i n c o m e e l a s t i c i t y for a p a r t i c u l a r g o o d m a y be
used as an i n d e x o f d e m a n d , or m a r k e t p o t e n t i a l sa l e .
For
e c o n o m i c p o l i c y , i n c o m e e l a s t i c i t i e s m i g h t b e u s e d for
adopting various possible policies.
See J. M. S l a t e r ,
"Regional Cons u m e r Exp e n d i t u r e Studies Using Nat io na l Food
S u r v e y D a t a , " J o u r n a l of A g r i c u l t u r a l E c o n o m i c s (May, 1 9 6 9 ) ,
p. 197; G. T i n t n e r , E c o n o m e t r i c s fNew Y o rk :
John Wiley &
Sons, 1952 ), pp. 5 7 - 6 2.

31

32

can

investigate

gives

a better

if a p a r t i c u l a r
"goodness

2.
The
P ane l d a t a

data
of

tnirteen

periods

fats
and

and

the

of

as

to

oils;

taken

the

weekly

in 1 9 5 8 ,

curves

These

fruits;

are

the

Consumer

grouped

Since

there

thirteen

cross

sectional

to

be

foods

vegetables;

are

Models

t i me .

able

composite

Sectional
M.S.U.

reports
of

form uniformly

given o b s e r v a t i o n s .

Cross

from

a period

time

Engel

f o o d s.

are
Four

treated

set o u t

are:

and meat,

were

for

dairy

five

products;

poultry,

fi s h ,

e g g s .^
In e a c h

ticular period
assumed
per

used

1958.

and

composite

fit "

S t a t i s t ical

together

studies on

of

functional

to be

capita

cross
of

sectional

time,

all

homogeneous

disposable

study

households

except

income,

and

belonging
in

for p e r

to

each

the p a n e l
capita

par­

are

expenditure,

stochastic error.^

2

T h e o r e t i c a l l y , d i f f e r e n t v a r i e t i e s of g o o d s c a n be
g r o u p e d as a s i n g l e c o m p o s i t e g o o d if t h e r e l a t i v e p r i c e s
r e m a i n fixed, or t h e y a r c c o n s u m e d in f i x e d p r o p o r t i o n s .
For a full d i s c u s s i o n of t h i s p r o b l e m , s e e D. P a t i n k i n ,
M o n e y , I n t e r e s t , a n d P r i c e s (2nd e d .; N e w York:
Harper &
iiow, 1965) , p p . 4 1 1 - 1 6 .
T h e e x a c t c o m p o s i t i o n of t h e s e g r o u p e d g o o d s is
g i v e n in A p p e n d i c e s A, D, C, D, a n d E.
3

In a c r o s s s e c t i o n a l s t u d y , p r i c e s ar e h e l d c o n ­
stant.
Th e y are o m i t t e d f r o m the f o r m u l a t i o n of E ng el
curves.
T h e p e r c a p i t a h y p o t h e s i s is a d o p t e d to c o p e w i t h
the i n f l u e n c e of h o u s e h o l d s i z e o n h o u s e h o l d e x p e n d i t u r e
behavior.
T h e o m i s s i o n of h o u s e h o l d s i z e in t h e f o r m u l a t i o n
of E n g e l c u r v e s w i l l r e s u l t in b i a s e d a n d i n c o n s i s t e n t
estimates of income e l a s t i c i t y since hou s e h o l d incom e and
h o u s e h o l d s i z e , in m o s t s u r v e y d a t a , a r e h i g h l y p o s i t i v e
correlated.
T h e p e r c a p i t a h y p o t h e s i s m a y be t a k e n as s u b ­
s t a n t i a l l y c o r r e c t in t h e f o r m u l a t i o n of E n g e l c u r v e s .
See
pp. 1 7 - 2 1 a n d f o o t n o t e 48, C h a p t e r I.

33

In o t h e r w o r d s ,
posite

food,

yi k /Ni

■ V

where

Y-,
1K

in e a c h

'

is

Mj

cross

curve

sectional

for th e

com­

s t u d y , c a n be

s t a t e d as

th
i—
household

aggregate

expenditure

on

the

food ,

is

th
i—
h o u s o h o ] d uyyi e y a te

the

federal
lb

Enqel

* u ik

W

the

k—

the

is

the

income

taxes,

average

number

of

income

persons

in

after

the

i^il

household,
lhk

is

the

the

stochastic

effects

besides

is

In

the

of c i t h e r

household

urement of
f,
k

error

the

representing

non-economic

size,

and

reyressand,

undefined

this chapter,

term

functional

the

both

factors

e r r o r of m e a s ­

and
form,

the

functional

forms

selected

an d

double-log

f o tm s .

for

4

f

arc

Engel

linear,
curve

for

semi-log,
the

th
k—
foo d

is e x p r e s s e d

^

Linear:

Semi-log:

Double-log:

Y i k / N i = a k2

log

=

a k 3

as

Th u s ,

the

follows:

4 Pk 2 (Mi/ N i ) + U ik

+

3

k2

+

^ 3

lo9

<M i / N i>

+ U ik

lo9

(M ^ / N ^ ) + U ik

4

in

T h e s e t h r e e f u n c t i o n a l for ms h a v e b e e n w i d e l y u s e d
t h e a n a l y s i s of E n g e l c u r v e s ; see pp. 26-27 , C h a p t e r I.

34

where

ot^ 1 s an d

terms,

and

6 ^ ' s ar e

the

income

on a l t e r n a t i v e

the

coefficients

functional

Regarding

t he

and

the

assumptions

(i)

Normality:

(ii)

Zero mean:

(iii)

the v a l u e s

is
^^ik^

Homoskedasticity:

(iv)

No

(v)

exogenous

of

are

the

the

constant

k-^- food,

based

distribution

of

the e x p l a n a t o r y

the d i s ­

variable,

assumed:

normally distributed;
“

2
2
E(U.. ) = o ;
I K

interdependence:

The

fo r

i.e.,

forms.

probability

turbance
following

parameters,

K.

E ^u i k U jk^

variable,

K^/N^,

=

® ^or

^

is m e a s u r e d

without

error.

Considering
(i),

(ii),

(iii)
d oe s

is

since

not

not

mates.

and

(iii)

the a b o v e

are a s s u m e d .

fulfilled,

affect

th e

panel

the a s s u m p t i o n

(iv)

households

property
is

likely

were

( v ) , the p r e s e n t

simplyass u m e s

theme as ur ed

BLUE.
5

study,

the
the

assumptions
assumption

of

heteroskedasticity

of

least

to be

randomly

incomes

the a b o v e a s s u m p t i o n s , t h e o r d i n a r y
tors a r e

Though

the e x i s t e n c e

unbiased

The assumption
the

assumptions,

esti­

satisfied,

selected.

like

many

areaccurate.
least

squares

As

for

others,
Based

on

squares estima-

5

T h e a s s u m p t i o n s (ii) t h r o u g h (v) s u f f i c e to e s t a b ­
lish t h a t the l e a s t s q u a r e s e s t i m a t e s a r e B L U E .
The
a s s u m p t i o n s (ii) a n d (v) a s s u r e the u n b i a s e d n e s s o f l e a s t
squares estimators.
T h e a s s u m p t i o n (i) s e r v e s to e s t a b l i s h

35

2

.

i

P'. *

i ' i * i (Mi

The
mate

the

icoression

Before
computation

zero v a l u e s of
S inc e

log

of

least

analyzing

over

is

all

used

the o n e

to e s t i ­
hundred

the m a i n

results,

one

difficulty

It

household

on a c o m p o s i t e

expenditure

this c r e a t e s
those pairs

outcome

of

part o f

the s a m p l e .

observations

They qive
and

on
no

that

should

not

composite

be

food.

problem.

for w h i c h

information

of

In

the

fo o d

regarding

are
the

counted

as

and

inco m e

7

Cross

Th e e s t i m a t e s
for

of

is the p r o b l e m

a computational

expenditure

the e x p e r i m e n t

R e s u l t s of

alternative

method

s h o u l d be m e n t i o n e d .

household

coefficients

squares

coefficients

zero a r c e x c l u d e d . ^

2.2.

r d i] i t

regressions.

0 =

this c h a p t e r ,
values

<

ordinary

Linu n i n e t y - f i v e

of

1 if

the

Sectional
of

five

functional

t he

Studies

constant

composite

f o r m s,

over

ter m s

foods,
thirteen

th e

based
cross

on

t hr e e

sectional

an i d e n t i t y b e t w e e n l e a s t s q u a r e s a n d m a x i m u m l i k e l i h o o d
e s t i m a t e s and to j u s t i f y s t r i c t l y the use of t, F, a n d z
test p r o c e d u r e s .
F o r a fu ll d i s c u s s i o n of t h i s p r o b l e m ,
see E. M a l i n v a u d , S t a t i s t i c a l M e t h o d s of E c o n o m e t r i c s (2nd
e d .; N e w York:
A m e r i c a n E l s e v i c e , 197 0) , pp. 8 4 -8 6 ; a l s o
E. J. Kane, E c o n o m i c S t a t i s t i c s & E c o n o m e t r i c s (New Yor k:
H a r p e r & Row, 1 9 6 8 ) , pp. 3 5 5- 5 7.
^Some studies assigned
zero o b s e r v a t i o n s of h o u s e h o l d
H o u t h a k k e r , op. c i t . , p. 50.
7

Kmenta,

op.

c i t . , p.

an a r b i t r a r i l y low v a l u e for
e x p e n d i t u r e ; s e e P r a i s and

337.

36

studies

f igu r e

ii*1 p r e s e n t e d

in

Table

in p a r e n t h e s e s

is

t he

income

mated

coefficient.

corresponding
ferent

from

economic

zero

sense,

coefficient
pensity
based

this

1 through
{1)

Mpe

income
is

based

level
been

the

dairy
the

@ indicates

is

well

not

known
form

whereas

form

some
Q
5.

is the

that

is

the

products,
form

estimates

is

bas ed

esti­

that

the

In an

the

income

income

mean

pro­

coefficient

elasticity.

might

of

.0024;
on

th e

the m a r g i n a l

income

the

The

significantly dif­

conclusions

linear

of

of s i q n i f i c a n c e .

linear

(MPC);

Table

elasticity

MPC

For

based on

fats
th e

elasticity

be drawn

from

the estimates

the m e a n

of

the double-log

the

form

and

oils,

linear

form

estimates

the m e a n
is

of

.0014;

based on

the

estimates

the m e a n

of

the doubl e - l o g

the

form

.2209.
(3)

basedon

For

the

elasticity

based

elasticity

f r u it s ,

linear
est imates

(4)
MPC

on

The mark

error

5.

.0038.

income
is

on

point,

For

(2)
oj

5%

it h a s

based

standard

coefficient

the d o u b l e - l o g
At

ol

at

to c o n s u m e

on

Table

income

Table

1 through

For

on

the

form

the m e a n of
is

based

.004 3;
on

vegetables,
linear

estimates

is

based

the

mean of

the d o u b l e - l o g
the m e a n

.0026;
on

the e s t i m a t e s

the

of

the

form

of M P C
i n co m e
is

.3388.

the e s t i m a t e s

mean

of

the d o u b l e - l o g

the
form

of

income
is

.1988.

Q
It s h o u l d be n o t e d that t h e s e r e s u l t s , f r o m T a b l e 1
t h r o u g h T a b l e 5, p r o v i d e t h e i n f o r m a t i o n n e e d e d to a p p r o x i ­
mate u t i l i t y funct io ns b y Wald's me tho d.
Some preliminary
e v i d e n c e on t h i s s u b j e c t is g i v e n in C h a p t e r V.

37

TABLE

1 . — E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s f o r D a i r y
P r o d u c t s Based on A l t e r n a t i v e F u n c t i o n a l F o r ms

Period
of

1

"'^ii.nihcr. of Constant Terms
T3nr<“,d on
T.i ne,ir
Ocmi-Log Eoublo-Log
3.878

1 .976

.4 07

1.710

2 .203

.373

3. 7r.Q

.5 37

'i
■1

3. 28 0

.7^2

.194

5

3.443

1 .08 2

.257

6

3.114

- .541

.105

7

3.430

1 .575

. 35 6

8

3.125

.4 57

.199

n

?. 055

.684

.204

10

3.157

.729

.18 2

11

3. 365

1 .RB9

.34 9

12

3 . 350

3 .763

.522

11

3. 355

3.611

.506

Mean

Remarks:

Estimates of
I.inea r

Income Coefficients
Pared nn
Eerni- I.oq
Dnublo-Loq

.00110
{.0008)
.00050
(.0005)
.0025
{.000°)
.003 3
(.0011)
.0025
( .0010)
,0048
( .0012)
.0015 O
(.0011)
.0034
(.0012)
.0029
(.0011)
.0030
£.0010)
.0020
f.0010)
.0026
(.0011)
.0015 0
(.0009)

.8887
(.4017)
.7584
(.3769)
.003°^
(.0059)
1.4550
(.4471)
1.3089
£.4167)
2.OR 5 0
(.4618)
.9947
( .4234)
1 .5176
(.4528)
) .34 56
( . 3945)
1.3760
(.3R90)
.8589
(.3448)
-.00020
(.0044)
.0024 9
(.0060)

.06570
(.0423)
.0 7940
( .0420)
.00070
(.0006)
.1576
(.0542)
.1 325
( .0495)
.2041
(.05 3 3)
.06919
( .0517)
. 1451
{.0561)
.1323
(.0569)
.1538
(.0494)
,080 3«
(.0422)
-.00010
(.0005)
.00010
(.0007)

.0024

.9GR8

.0938

The mark 0 indicates that the income coefficient is not s i g ­
nificantly different from zero at 5% level of significance.
T h e till p e r i o d of time i m p l i e s
s tudy, s i n c e a c r o ss s e c t i o n a l
p e r i o d of time.

the t
cross
study belongs

s e c t i on a l
to a particular-

38

TABLE

2.

Period
of
Time

E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s
O r i s B a s e d on A l t e r n a t i v e F u n c t i o n a l

Intimates of Constant Terms
Based on
Linear
Pemi-Loq Doublc-Log

1

.901

-.210

-.545

*-

].020

-.09]

-.097

T

.°04

1.077

-.07?

4

.627

-1.034

-.79'.

5

.855

-.704

-.682

r,

.712

-.706

-.057

7

.830

.004

-.49 7

8

.773

-.509

-.756

9

.719

-.305

-.051

10

.835

-.025

-.484

11

.750

-.028

-.690

12

.760

1.052

-.070

13

.852

1.037

-.00?

Mean

Remarks:

for F a t s
Forms

and

Estimates of Income Coefficients
Bared on
Li near
.Semi-Log
Double-Log
.0009
f.0004)
.0007
(.0002)
.00079
f.0004)
.0028
(.0005)
.001 7
(.0004)
.0020
(.0005)
.0007 9
(.0004)
.0017
(.0005)
.0016
(.0004)
.0011
(.0004)
.00] 5
( .0004)
.0018
(.0004)
.0010
{.0004)

.5953
(.1997)
.nr r,7
(. 1607)
- .00209
(. T077)
.ri970
(.1925)
.868 3
(.1963)
.81 74
(.2032)
.4490
(.1854)
.7334
(.1885)
.6037
( .1709)
.4900
(.1 706)
.57 28
(.1642)
.0021?
(.0020)
-. 0 0 4 1 o
(.0028)

.2279
(.0592)
. 216°
(. 0 C8 V)
-,000 9 0
f.0006)
. 3486
(.0627)
. 3039
(.0652)
. 36 28
(.0734)
.1739
(.0750)
.3202
( .0733)
. 3441
(.0730)
.1899
( .0617)
.28 12
(.0591)
.0011?
(.0007)
-.0014 0
( .0009)

.0014

.5382

.2 209

The mark @ indicates that the income coefficient is not significantly different from zero at 5*. lu:vel of significance.
Thu t—
p e r i o d of tim<> i m p l i e s
s tudy, s i n c e a c r o s s s e c t i o n a l
p e r i o d o f time.

the
cross
study belongs

sectional
to a p a r t i c u l a r

39

TABLE

3 . — E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s for F r u i t s
Bused on A l t e r n a t i v e F u n c t i o n a l Forms

Period

of
Time

Estimates of Constant Terms
Based on
Li near
Semi-Ixjg Double-Log

)

1.459

-1.649

-, 616

2

1 .867

-2.117

-.533

3

1 .794

2.119

.229

4

1 .362

-1.800

-. 609

5

1 .517

-2.533

-.917

6

1 .593

-2.251

- .703

7

1 .958

-2.313

- .680

8

1.402

-3.776

- .896

9

1 .531

-2.420

-.817

CO
V
in

-1.221

- .506

1 .389

-1.127

-.552

12

1 .752

2 .156

.230

13

1 .694

2. 279

.244

10
11

Mean

Remarks;

Estimates of Income Coefficients
Based on
Double-Log
Semi-Log
Linear
.0029
(.0007)
.0021
(.0005)
.0020
(.0007)
.0046
(.0008)
.0051
(.0009)
.0050
(.0010)
.0054
{.0012)
.0075
(.0011)
.0058
(.0010)
.0048
(.0009)
.0049
(.0009)
.0025
(.0009)
.0034
(.0009)

.0043

1.6923
{.3326)
2.0468
(.3724)
.00570
(.0048)
1.8 399
(.3542)
2.2980
(.3611)
2.1390
(.396 2)
2.4292
{.4546)
2.9988
(.4403)
2.304 3
(.3863)
1.6702
(.35 20)
1 .5710
(.3218)
.002 30
(.0037)
-.00189
( .0061)

1.6153

.3718
(.0737)
.3605
(.0756)
.00180
(.0010)
.3887
( .0753)
.5518
(.0774)
.4594
(.0779)
.4775
(.0814)
.5586
(.0807)

.5058
(.0858)
.3594
(.0737)
.3675
(.0737)
.001 7
(.0007)
-.0006
{.0010)

.3 388

The mark 0 indicates that the income coefficient is not sig­
nificantly different from zero at 5% level of significance.
T h e t-^- p e r i o d of time impl ies the t ^ - c r o s s
study, since a c r o s s s ec t io na l s t u d y b e l o n g s
p e r i o d o f time.

s ec tio na l
to a p a r t i c u l a r

40

TABLE
wmm

I

ml

4.—

•m m m

Period
of
Time

E s t i m a t e s o f R e g r e s s i o n C o e f f i c i e n t s for V e g e t a b l e s
Based o n A l t e r n a t i v e F u n c t i o n a l F o r m s
- v

Estimates of Constant Terms
Based on
Linear
Semi-Log Double-Log

1

1. 779

-.392

- .211

2

1.980

-.506

-.227

3

1.864

2.164

.260

4

1. 599

-1. 243

-.313

5

2.069

-1.236

-.289

6

1.959

-.176

-.162

7

1.864

-.549

- .277

8

1. 579

-1.328

-.416

9

1.432

-.629

-.354

10

1 .548

-.049

-.232

11

1 .476

- .521

-.351

12

1 .628

2.023

.224

13

1.640

1.850

.180

Mean

Remarks:

Estimates of Income Coefficients
Based on
Linear
Semi-Log
Double-Log
.0024
( .0006)
.0013
{.0004)
.0019
(.0006)
.004 2
(.0008)
.0036
(.0009)
.0028
(.0008)
.0029
(.0010)
.0039
{.0008)
.0027
(.0007)
.0021
(.0007)
.0028
(.0007)
.0024
(.0007)
.0012
{.0006)

.0026

1.2070
(.3058)
1.2793
(.3061)
.00240
(.0045)
1.6580
(.3454)
1.8400
(.3594)
1.2149
(.3252)
1.3650
(.3545)
1.6651
(.3067)
1.1850
(.2615)
.9197
( .2667)
1.1742
( .2289)
.00420
(.0030)
.00190
( .0042)

1.0397

.2257
(.0562)
.2371
(.0551)
.00030
(.0009)
.2805
(.0604)
.3034
(.0578)
.2277
(.0560)
.2699
(.0603)
.3213
(.063B)
.2552
(.0651)
.1982
(.0664)
.2639
(.0569)
.00120
(.0006)
.00020
(.0009}

.1988

The mark @ indicates that the income coefficient is not significantly different from zero at 5% level o f significance.
th
tli
T h e t- "- period of time implies the t—
cross sectional
study, since a cross sectional study belongs to a p articular
period of time.

41

TABLE

5 . - - E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s for
Based o n A l t e r n a t i v e F u n c t i o n a l F o r m s

Period
of
Time

Estimates of Constant Terms
Based on
Linear
S e m i - L o g Double-Log
5.993

-5.586

.134

5 .984

-15.359

.133

3

A. 167

7 . 382

.806

4

5 .792

-4.034

.14R

5

5.621

-6.806

.050

6

6 .74B

- .295

.308

7

5 .011

-7.032

.142

8

5. 371

-3.654

.070

9

5.447

-2.720

.149

10

6. 179

- .440

.330

11

5 .499

-2.293

.235

12

6. 79 3

8 .128

.845

13

6. 7B1

7 .799

.827

1

Mean

Remarks:

Meat,

Etc.

Estimates of Income Coefficients
Based on
Linear
Semi-Log
Double-Log
.0112
(.0022)
.0125
{.0036)
.0077
{.0018)
.0125
( .0023)
.0157
(.0026)
.0075
(.0022)
.01 73
{.0035)
.0129
(.0023)
.0109
(.0021)
.0095
(.0023)
.0138
(.0022)
.0083
(.0024)
.0060
{.0022)

6.3161
(1.0157)
11.0688
(2.5591)
.00320
(.0124)
5.5704
(.9136)
7.0562
(1.0665)
3.6459
(.8532)
1.9938
(1.3370)
5.2173
(.8841)
4.6853
(.7671)
3.8448
(.8519)
4.7586
(.7477)
.0213
{.0099)
-.00090
(.0145)

.3285
(.0503)
. 3232
(.0566)
- .00000
( .0008)
. 3202
(.0585)
. 3723
{.0568)
.2403
(.0563)
.3127
(.0614)
.3446
(.0628)
.3021
{.0612)
.2345
{.0531)
.2783
( .0490)
.0019
(.0006)
-.00020
(.0008)

.0112

4.1678

.2353

The mark @ indicates that the income c oefficient is not significantly different f rom zero at 5% level of significance.
T h e t— p e r i o d of time i m p l i e s
study, s i n c e a c r o s s s e c t i o n a l
p e r i o d o f t im e .

the t—
cross
study belongs

sectional
to a p a r t i c u l a r

42

(5)
MPC

based

on

elasticity

2.3.

For
the

linear

estimates

Income

estimates

for

alternative
ring

to

income

based

the

five

the estimates
elasticity

etc.,

mean of

the m e a n

on

double-log

th e

of

the

the

estimates

income

form

is

.2353.

Estimates
Table

composite
forms,
of

5,

the

foods,

income

based

can easily

income

estimates

functional

the

.0112;

1 through

functional

alternative

is

Elasticity

F r o m Table

meat,

be

coefficients

are

computed

three

derived.

at m e a n v a l u e s ,

form s ,

on

elasticity

Refer­

(S^'s) , the
based

as

on

th e s e

follows:

9

L i n ea r:
Semi-log:

^k2^Yk

Double-log:

whore

M

is

the

sample

disposable
Y^

is

the

mean of

income

sample

expenditure

at

mean

on

the

a period

of

the

th
k—

the

household
of

time,

household

food

at

per c a p i t a

pe r

a period

capita
of

time,

and
1s are
j .

the

For

Appendix

The
F.

based

income

on a l t e r n a t i v e

each composite

at m e a n
9

of

coefficients

for

the

functional

f o r ms .

L.

food,

mates

estimates

values,

values

of

based

foo d,
on

_
„
Y. , M,

the

three

income

elasticity

alternative

_ _
a n d M/Y.

are

esti­

functional

set out

in

of

43

forms,

are

widely d i f f e r e n t . ^

tently g i v e s

the h i g h e s t

The

income

elasticity

double-log

The

income

elasticity estimates

form a r e

fairly

stable

s e c t io n al
signs of

studies.
the

alternative
To
estimates

income

clarify
at

mean

Figure

also

the

values

are

graphically

which

an d
based

different

cross

studies,

based

on

the

three

different.
the

income

elasticity

composite

for m s,

10.

over

linear

sectional

for e a c h

functional

the

estimates

analysis,

are

These

based

separately

income

presented

food,

on

set o u t

elasticity esti­

in F i g u r e

1 through

5.

2.3.1.
presentation
p roducts,
given

cross

forms

through Table

6

mates a r e

some

elasticity

functional

three a l t e r n a t i v e
in Table

In

on

estimates

from

on the l i n e a r

based

form consis­

are v e r y d i f f e r e n t
forms.

those

semi-log

Dairy

of

based

in T a b l e
The

on
6

three

an d

gives

for e a c h p e r i o d
The

of

the
time

values

elasticity

alternative

Figure

first point

consistently

periods.

the i n c o m e

P r o d u c t s .- - T h e

to

estimates
functional

and

a graphical

fo r d a i r y
forms,

are

1.
notice

highest
except

based

values

on

is

that

income
for
the

the

the

semi-log

elasticity
3—

semi-log

, 12^-,

form

estimates
and

13^-

form widely

^ T h e d i f f e r e n c e s between the income e l a s t i c i t y
e s t i m a t e s a r e g r e a t e r w h e n e s t i m a t e d at a n y p o i n t a w a y f r o m
the mean, s i n c e e a c h f u n c t i o n a l f o r m m a k e s d i f f e r e n t
a s s u m p t i o n s as to the w a y in w h i c h t h e e l a s t i c i t y v a r i e s .
For a n u m e r i c a l i l l u s t r a t i o n of t h i s p r o b l e m , s e e P r a i s
and H o u t h a k k e r , op. c i t ., p. 94.

44

TABLE

6 . - - I n c o m e E l a s t i c i t y E s t i m a t e s for D a i r y
P r o d u c t s Based on A l t e r n a t i v e Functional
F o r m s (at M e a n V a l u e s )

Period
of
Ti m e
1
2

3
4
5
6

7
8

9
10

Semi-Log

. 0 4 59 0
. 0 2 15 0
. 1048
. 1 335
.0993

.2302
.1995
.0 0 1 0 ^
.38 28
.3417

.06570
.07940
.000 3 0
.1576
.1325

.1881
. 0 61 0 0
.1430
.1251
.1305

.54 15
.27 2 5
.4146
.38 44
.3822

.2041
.06910
.1451
.1323
. 15 3 8

.0793
.1097
. 07 0 2 0

11
12

13

. 1009
.0015

Mean
S. D.

Remarks :

Linear

.2353
- . 0 0 0 0 0

Double-Log

.08030
- . 0 0 0 1 0

.0006

. 0 0 0 1 0

.26 0 4
.0265

.0938
.0035

The m a r k 0 i n d i c a t e s t ha t the in co me e l a s ­
t i c i t y (or the c o r r e s p o n d i n g i n c o m e
c o e f f i c i e n t ) is n ot s i g n i f i c a n t l y d i f f e r e n t
f r o m z e r o at 5% l eve l of s i g n i f i c a n c e .
S.D.

=

Standard

Deviation

_.
^ th
. .
th
cross
The t—
p e r i o d o f time i m p l i e s t h e t —
s e c t i o n a l s tu d y , s i n c e a c r o s s s e c t i o n a l
s t u d y b e l o n g s to a p a r t i c u l a r p e r i o d of t i m e .

45

In c o m o
K *it i in.1 1 c s

\

Semi-log

^

/
/

V
■Doublo-log

V
T

T

0

2

1

4

5
]>tir

Figure

i oil

T

T

(>

7

o1

T im e

'T
9

9

10

11

12

1

3

1.--A G r a p h i c P r e s e n t a t i o n of Income E l a s t i c i t y E s t i ­
m a t e s for D a i r y P r o d u c t s B a s e d o n A l t e r n a t i v e
F u n c t i o n a l F o r m s (at M e a n V a l u e s )

46

fluctuate

and

other

tw o

forms.

mates

based

deviation

are

on

is

very different
The mean of

the

semi-log

from

the

is

those

income

.2604

and

based

on

the

elasticity e sti­
the

standard

.0265.

Secondly,

the

e s t i m a t e s b a s e d on the s e m i - l o g a n d
L
the 1 2 —
pe riod are negative, w h e r e a s
i-

double-log

forms

the e s t i m a t e

at

based

Thirdly,
the d o u b l e - l o g
v er y

nearly

on

the

except

and

equal

linear

periods

over

l i n e ar

f o r m arc

of

time.

form

is

oils,

based

valu e s,

are

of
on

and

based

Fats
income

shown

the

different
values

based

3E l a n d
based on

1 2

—

the

the

same

estimates
over

the e s t i m a t e s
standard

estimates

functional

forms

noticed

that ,

e x c e p t for

elasticity

those
the

periods
linear

semi-log

f o r m are

two

double-log

are n e ga ti ve ,

whereas

positive.

on

is

for

the

.0015.

values

and

a graphical

fats a n d

at m e a n

the

The

fluctuate

the other

on

2.

estimates.

and

are

based

based

3El,

form c o n s istently

form w i d e l y
based on

that

different

deviation

and O i l s . - - T h e

semi-log

periods,

time.

Figure

the

13—

estimates

7 and

semi-log

on

of

the

, and

in T a b l e

income

from

, 12 —

give

elasticity

alternative

t ll

on

the

—

is p o s i t i v e .

elasticity

reasonably

.1009

13c±L p e r i o d s ,

the h i g h e s t

periods

form
3

forms

The mean of

It w i l l be
and

the

income

2.3.2.
presentation

for

linear

F o u r t h l y , the
the

linear

The

gives

values

and

are

very

forms.
forms
the

12 —

The
at

the

estimates

linear

and

,

TABLE

7 . - - I n c o m e E l a s t i c i t y E s t i m a t e s for F a t s a n d O i l s
B a s e d o n A l t e r n a t i v e F u n c t i o n a l F o r m s {at
Mean Values)

Period
of
Time

Linear

Semi-Log

. 1365
.0 9 9 7
.1079G
.403 3
.2311

. 5724
.7 6 6 9
-.0019(3
.944 7
.7 8 2 2

.2279
. 3169
-.00090
. 3485
. 3039

. 2936
.111313
. 2528
.2 5 0 2
.1725

.8 0 9 3
.4776
. 712 0
.6354
.4900

.3628
.1739
. 3202
.3441
. 1B 9 9

. 5844

.2812

. 0 0 2 0 0

. 0 0 1 1 0

13

.2223
. 27 3 3
. 1638

- .00390

-.00140

Mean
S .D .

. 2091
. 00 6 5

.521 7
.0997

. 2209
.0169

1
2

3
4
5
6

7
8

9
10

11
12

Remark s :

Double-Log

T h e m a r k 0 i n d i c a t e s t h a t the i n c o m e
e l a s t i c i t y (or t h e c o r r e s p o n d i n g i n c o m e
c o e f f i c i e n t ) is n o t s i g n i f i c a n t l y d i f f e r e n t
f r o m z e r o a t 5% l e v e l of s i g n i f i c a n c e .
S .D.

= Standard

De vi a t i o n

The t—
p e r i o d o f ti m e i m p l i e s the t —
cross
s e c t i o n a l study, s ince a c r os s s e c t i o n a l
s t u d y b e l o n g s to a p a r t i c u l a r p e r i o d of time.

48

Tn come
K 1 ,v: I 1 •' i t y
r. s t i m a t o s

r\

9

I \
^ s
R
'

/

si'Dii-loq

\

i

A .

/

\

\

/

\ /
V

x

A

V

\
\

. D o u b l i* - 1 o y

V-"

Li near

1

Period

Figure

- •

f.

■1

of

~T~

7

1■- 1

b

9

‘t
JO

1

11

‘n

12

1—
11

Tim e

2.--A Graphic P r e s e n t a t i o n of Income E l a s t i c i t y E s t i ­
m a t e s for F a t s and O i l s B a s e d o n A l t e r n a t i v e
F u n c t i o n a l F o r m s (at M e a n V a l u e s )

49

double-loa
are v e r y

nearly

periods.
linear,

for m s

and

equal

except

for

forms

.0997,

arc

would c l a i m

forms

the

and

income

form are

income

elasticity

and

3.

in T a b l e
One

8

can

time

quite

forms.

Both

th e

double-log

on

semi-log

the

3^-,

the s e m i - l o g
fr o m

non-linear

estimated
linear

for

give
the

income
forms

form gives
than

at

income

unity.

the

based

of

a graphical

fruit s ,

values,

based

are

set

form consistently

12^-,

based

and

at e a c h
13—

periods.

fluctuate

on

the o t h e r

two

negative

estimates

at

3—

, 12^^.,

very

close

and

13^-

base d

on

to e a c h

the
other.

i. y.

5-^- a n d

elasticity

The mean

for

elasticities

are

and

estimates

4-

higher

One

different

form w i d e l y

those

forms

Except

In a d d i t i o n ,
log

and d o u b l e - l o g

values

at m ean

elasticity

for

different

and

the

income

period.

periods,

that

except

based

and are

Figure

see

highest

The v a l u e s

13—

.2209,

income e l a s t i c i t y

over

estimates

out

the

1 3 -

semi-log,

and

estimates

same

F r u i t s . --The

forms

of

on the

respectively.

elasticity

functional

period

linear,

the

, and

.2091,

of

that

time.

of

the

based

.5217,

.016 9 ,

on a l t e r n a t i v e

gives

estimates

3— ., 12 —

deviation

almost

2.3 .3 .
entation

are

semi-log,

.0065,
the

the

the e s t i m a t e s

standard

on

tha t

linear
of

of

The

based

periods

elasticity

double-log

estimates

the

inco m e

The m e a n

respectively.

on

give

8

—

periods,

estimates
the

income

the

that a r e
elasticity

semi­

pres­

50

TABLE

0 . - - I n c o m e E l a s t i c i t y E s t i m a t e s for F r u i t s B a s e d
on A l t e r n a t i v e F u n c t i o n a l F o r m s (at M e a n
Values)

Period
of
Time

Linear

1
2

4
5
6

7
0

9
10

Semi-Log

Double-Log

. 2396
.1554
. 1499
. 3309
.3304

.8768
.9261
.0026(3
.8845
1.1003

.3718
.3605
.0018
.3887
.5518

. 3200
. 2899
.4526
. 3665
. 3287

.9063
.8801
1.1668
.9561
.7230

.4594
.4775
.5586
.5058
.3594

13

. 3301
. 1856
. 2525

-.00070

.3675
.0017
-.00060

Mean
S.D.

. 2889
.0041

.7052
.1753

.3388
.0376

11
12

Remarks:

.7445

0

. 0 0 1 0 0

The m a r k @ i n d i c a t e s t h a t the i n c o m e
e l a s t i c i t y (or the c o r r e s p o n d i n g i n c o m e
c o e f f i c i e n t ) is n o t s i g n i f i c a n t l y d i f f e r e n t
f r o m z e r o at 5% lev el of s i g n i f i c a n c e .
S.D.

= Standard

Deviation

th
_
.
th
The t
p e r i o d o f t i m e i m p l i e s the t—
cross
s e c t i o n a l study, s i n c e a c r o s s s e c t i o n a l
s t u d y b e l o n g s to a p a r t i c u l a r p e r i o d of time.

51

1n c otno
!" 1 n r 1' i

r -1

ty

1 < * in.it r';

i

0

R

6
Doulili'-lcg -

4.

1

0

]

3
I’ u r i o d

Figure

7

6
of

3.— A Graphic Presentation
mates for Fruits Based
F o r m s (at M e a n V a l u e s )

8

0

1 0

1 1

1?

13

T i me

of Income Elasticity E s t i ­
on Alternative Functional

52

estimates

based

forms

0.7052,

are

standard
on

the

the

and

semi-log,

0.2889,

deviation

semi-log,

.0041,
that

on

of

income

linear,

.0376,

and

income

elasticity

form are

fairly

stable

bles,

based

values,

elasticity

The

periods

on

income

elasticity

alternative

given

in T a b l e

9 and

The

first

point

to

notice

form c o n s i s t e n t l y

gives

the

of

ti m e

The e s t i m a t e s
and a r e
other

except
based

considerably

two

for

on

except

the d o u b l e - l o g

t hat

very

nearly

Thirdly,
on

0.4768,

the

for
and

—

,

the

values

forms

—

linear

and

for

a graphical

vegeta­

at m e a n

4.

that

12

from

the

semi-log

, and

at

each

13—

periods.

form widely

fluctuate

those

based

on

, 12 —

, and

1 3 -

the

semi-log,

forms

Lastly,

the

standard
on

.0003,

income

linear,

0.1988,

.1064,

3—

give

estimates

equal.

the m e a n o f

based

the

linear

and

are

3

on

estimates

semi-log

0.1885,

ity e s t i m a t e s
forms

the

the

is

indicate

time.

Figure

highest

different

periods,

based

.1753,

are

forms.

Secondly,

are

forms

estimates

functional

are

period

based

based
of

The

estimates

V e g e t a b l e s .- - T h e

of

double-log

results

estimates

over

an d

respectively.

and d o u b l e - l o g

the

presentation

0.3388,

respectively.

2 .3. 4 .

linear,

the
and

elasticity

and d o u b l e - l o g

estimates
forms

are

respectively.
deviation

semi-log,
.0118,

of

income

linear,

and

respectively.

elastic­
double-log
One

53

TABLE

9 . - - I n c o m e E l a s t i c i t y E s t i m a t e s for
Based on A l t e r n a t i v e F u n c t i o n a l
{at M e a n V a l u e s )

Period
of
Time

Linear

Semi-Log

Vegetables
Forms

Double-Log

. 1765
. 09 6 8
.1391
. 2870
. 206 3

.5587
. 584)
.0 0 1 1 ^
. 7368
.6996

.2257
. 23 7 1
.00030
.2 8 0 5
. 3034

. 1774
. 18 4 7
. 2753
. 2198
.1740

.5104
. 5934
. 7638
.6405
. 4866

.2277
.2699
.3213
.2 5 5 2
.1982

.6212

.2 6 3 9

. 0 0 2 0 0

. 0 0 1 2 0

13

. 214 5
.1885
. 10 8 7

. 0 0 1 0 0

. 0 0 0 2 0

Mean
S.D.

.1885
.0 0 0 3

. 476 8
.1064

.1988
.0118

1
2

3
4
5
6

7
8

9
10

11
12

Remarks:

T h e m a r k 0 i n d i c a t e s t h a t the i n c o m e
e l a s t i c i t y (or t h e c o r r e s p o n d i n g i n c o m e
c o e f f i c i e n t ) is n o t s i g n i f i c a n t l y d i f f e r e n t
f r o m z e r o at 5% lev el o f s i g n i f i c a n c e .
S.D.

=

Standard

Deviation

T h e t— — p e r i o d of tim e i m p l i e s t h e t —
cross
s e c t i o n a l study, since a c ros s s e c t i o n a l
s t u d y b e l o n g s to a p a r t i c u l a r p e r i o d of time.

54

I 11c;Oi»i;
1:1 a n L i c i
I t .

t:

y

i in a L e s

\Somi-log

3

0

Double-log

1

3

A
P e r i o d

Figure

G

7

o f

T i m e

H

9

1 0

1 1

4 . - - A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y
ma te s for V e g e t a b l e s Based on A l t e r n a t i v e
F u n c t i o n a l F o r m s (at M e a n V a l u e s )

1 2

13

Esti­

55

would
on

notice

the

that

linear

periods

of

forms

and

estimates
forms

highly

F is h ,

presentation

e t c . , based

values,

are

would

gives

the

highest

3^-^,

12 —

,

based

are v e r y

forms.

estimates

over

based

different

and
of

E g g s . --The

income elasticity

on alt e r n a t i v e

functional

set

out

in T a b l e

10

At

notice

that

the

semi

form c o nsis­

and 13—

on

2—

estimates

periods.

and

from those

are
3—

and

log

form g i v e s

estimates

that

the

3—

, 12 —

income

t he

greater

At

, and

13—

on

the

double-log

estimates

based

double-log

forms

.5428,

The

standard

estimates

based

forms

.1385,

are

results

are

indicate

on

and

the

th e

the

the o t h e r

unity

of

an d

linear,

.0178,

double­
for

form gives

th e

to

th o s e

income

linear,

.2353,

income

form

Except

semi-log,

.2259,

income

the

close

of

two

and negative,

linear

mean

elasticity

semi-log

are ver y

the

semi-log,

.0011,

on

except

fluctuate

negative.

The

deviation

the

that

ar e

that

on

income

1 3 ^ - periods,

form.

elasticity

tively.

based

than

periods,

elasticity estimates

The

estimates

form w i d e l y

and 13— periods,

that

log

elasticity

th e s e m i - l o g

different

the

income

respectively.

based

stable

Poultry,

One

the

gives

Meat,

for meat,

estimates
and

elasticity

5.

tently
for

are

a graphical

at m e a n

Fxgure

income

time.

2.3.5.
values

the

and

respec­

elasticity
an d

double-log

respectively.

elasticity

The

estimates

56

TABLE

10.—

I n c o m e E l a s t i c i t y E s t i m a t e s for M e a t ,
B a s e d on A l t e r n a t i v e F u n c t i o n a l F o r m s
(at M e a n V a l u e s )

Period
of
Time

Linear

Semi-Log

Etc.

Double-Log

1
2
3
4
5

.2299
.2526
.1647
.2478
.2 9 7 8

.8 0 9 7
1.3801
.0 004 0
.7224
.8 7 9 8

. 32B5
. 3232
-.00000
. 3203
. 3723

6
7
8
9
10

. 1532
. 3382
.2694
.2321
.1938

.4 9 4 0
.2 6 3 0
. 70 7 9
.6608
.5012

.2403
. 3127
. 3446
. 3021
.2345

11
12
13

.2663
.162 0
.1293

.6353
.0026
-.00010

. 27 8 3
.0019
- .0 0 0 2 0

Mean
S.D.

.2259
.0011

.5428
.1385

. 23 5 3
.0178

Remark s :

The m a r k @ i n d i c a t e s that the income
elu:' v i c i t y (or the c o r r e s p o n d i n g i n c o m e
c. it r - ■oient) is n o t s i g n i f i c a n t l y d i f f e r e n t
f r o m z e r o at 5% level o f s i g n i f i c a n c e .
S.D.

= Standard

Deviation

T h e t—
p e r i o d of t i m e i m p l i e s the t—
cross
s e c t i o n a l study, since a cross s e cti on al
s t u d y b e l o n g s to a p a r t i c u l a r p e r i o d o f time.

57

T n c ( i n't e■

Klasticity
Estimates
■i

3

1

0

9
H
7
r>

4
D o u b le -lo g

1

0

1

3

1

T o r i oil

Figure

9

G
<3 f

10

11

12

1

3

T i me

5 . - - A G r a p h i c P r e s e n t a t i o n of I n c o m e E l a s t i c i t y E s t i ­
m a t e s for Meat, P o u l t r y , F i s h an d Eggs Based on
A l t e r n a t i v e F u n c t i o n a l F o r m s (at M e a n V a l u e s )

58

based on
periods

the
of

linear

elasticity

Perhaps
on

Table

estimates
lower

are

11,

periods

proceeding,

at

equal

the

one

one m i g h t

the

3—

can

five

, 12—

answer

influenced

below,

are

f or

the

one p o s s i b l e

foods

nearly

over

different

time.

Before

siderably

form are

by

is

composite

, and
that

that

the
foods

households'

the

income
are

con­

132-21 p e r i o d s .

the h o l i d a y s

se e

pre-Easter,

ask w h y

of

3^-,

Thanksgiving,

the

expenditures
year.

12^-21, a n d
and

From
13^21

Christmas

holidays.

TABLE

Year

1958

1 1 . --The Ac t u a l Days and H o l id ay s
W e e k s o f e a c h P e r i o d f o r the

Period

1
2
3
4
5
6
7
8
9
10
11
12
13

Month

and

Date

1-1 to 1 - 2 9
1-3 0 to 2 - 2 6
2 - 2 7 to 3-2 6
3-27 to 4 - 2 3
4-24 to 5- 2 1
5 - 2 2 to
6 - 1 9 to
7 - 1 7 to
8-14 to
9 - 1 1 to

6-18
7-16
8-13
9-10
10-8

1 0 - 9 t o 11 - 5
1 1 - 6 to 1 2 - 3
12-4 to 1 2 - 3 1

I n c l u d e d in t h e
Ye ar of 1958

Four

Holidays

New

Years

Easter

M e m o r i a l Day
F o u r t h of July
Labor

Day

Thanksgiving
Christmas

59

2.4.

D i s t r i b u t i o n a of I n c o m e
Elasticity" Estimate's
From

quency
foo ds

Table

6

distributions
as

a whole,

through
of

income

based

Such d i s t r i b u t i o n s

ar c

Table

10, o n e

elasticity

on a l t e r n a t i v e
set out

can

derive

estimates

functional

in T a b l e

fre­
fo r

f o r ms .

12.

TABLE

12 .- - D i s t r i b u t i o n s of I n c o m e E l a s t i c i t y E s t i m a t e s
Foods Based on A l t e r n a t i v e F u nctional Forms

Range

of

......
,.,
Elasticities

-1
0
. 10
. 20
. 30

to
to
to
to
to

0
.10
. 20
. 30
. 40

.40
.50
.60
. 70
. B0

to
to
to
to
to

. 50
.60
. 70
.80
.90

. 90
1 .00

to
to

1.00
2.00

Income
Linear

Elasticity
Semi-Log

8
27
21
7
2

income
are
Of

based
lie

elasticity

greater
the

would

on

notice

and

income

the d o u b l e - l o g

in the

range

5
10
1
4
4

6
13
8
13
20

5
8
6
10
5

2
3

0 to

the

that

the

semi-log
are

rest

form,

six
All

form gives

negative,

in

elasticity

0.60.

65

65

that

estimates

t h a n one,

sixty-five

Estimates
Double-Log

4
3

65

One

for

the

range

estimates

are

negative

income

three
0 to

for
and

elasticity

five
that
1.00,

foods
the

rest

estimates

60

based on
the

the

range

0

to

In
foo ds

linear

form arc

on

the

either

inelastic:

Engel's

Elastic and

negative

the

a nd

concentrate

in

0.50.

summary,

based

positive

nonlinear

income

functional

l a w of

are

form

but

for

predominantly

is c o n f i r m e d .

elasticity

f o un d ,

estimates

are

consumption

income

forms

elasticity

estimates

they

are

based

on

comparatively

infrequent.

2.5.

A Comparison
of F i t "
Up

tigate
gives

if

to
the

of

R

2

an

semi-log

the d o u b l e - l o g

3—

11

atte mpt will

fit"

13

the

coefficient

functional

Table

to

that all

and c l o s e

to

zero.

values

, and

a better

forms
13—

differ

periods,

"goodness

of

only

used

the
K

2

slightly

where
fi t "

of

the

to t h e

inves­

uniformly
The

of d e t e r m i n a t i o n ,

values

noticeable

to

observations.

the

be

functional

form

are

shows

The

be m a d e

forms

It w i l l

, 12 —

gives

of

, the c o r r e c t e d

determination.

tive

or

"goodness

based on a l t e r n a t i v e

small

"Goodness

this point,

a better

values

of

of

for

such

—o
R .

values

of

— 2

R

are

based on a l t e r n a ­
except

linear

for

the

form o b v i o u s l y

observations.

T h e c o r r e c t e d c o e f f i c i e n t of d e t e r m i n a t i o n R
is
used to d e s c r i b e h o w w e l l the s a m p l e r e g r e s s i o n li ne
fits
the o b s e r v e d d a t a .
T hi s m e a s u r e takes into a c c o u n t the n u m ­
ber o f e x p l a n a t o r y v a r i a b l e s in r e l a t i o n to t he n u m b e r o f
observations.
N e e d l e s s to say, the p u r p o s e o f R
is to
f a c i l i t a t e c o m p a r i s o n s o f the " g o o d n e s s o f f i t " o f s e v e r a l
r e g r e s s i o n e q u a t i o n s t h a t v a r y w i t h r e s p e c t to the n u m b e r
of e x p l a n a t o r y v a r i a b l e s a n d t h e n u m b e r o f o b s e r v a t i o n s .
For a full d i s c u s s i o n of thi s p r o b l e m , s e e K m e n t a , op. c i t . ,
pp. 2 2 9 - 3 5 a n d p. 365.

61

TABl.t’
. 13. ■ -V.iluer, of
mination)
Period
of
Time

R
(the C o r r e c t e d
a n d n (the N u m b e r

Composite
Food

Linear

1

D a i r y P ro d .
Fats & Oils
Frui ts
Veqetab]es
Meat, etc.

.002
.011
.051
.03 9
.078

2

D a i r y Prod .
Fats & Oils
Fruits
Vege tables
M e a t , etc .

3

C o e f f i c i e n t of D e t e r ­
of O b s e r v a t i o n s )

Based
Semi-Log

on
Double-Log

n

.014
.028
.084
.050
.120

.005
.048
.083
.052
.131

275
269
270
27 3
275

.000
.0 2 4
.052
.030
.04 0

.01 3
.076
.097
.057
.061

.009
.096
.074
.060
. 105

272
267
270
2 71
270

D a i r y Prod.
Fats & Oils
Fruits
Vegetables
Meat, etc.

.026
.008
.025
.024
.056

.000
.0 00
.001
.000
.000

.000
.000
.008
.000
.000

2 68
2 63
265
268
265

4

D a i r y Prod.
Fats & Oils
Fruits
Vegetables
M e a t , et c.

.028
. 103
.093
.079
.095

.034
.08 7
.088
.075
.119

.026
.100
.087
.070
.097

272
267
268
272
268

5

D a i r y Prod.
Fats & Oils
Fruits
Vegetables
Meat, etc.

.018
.041
. 103
. 054
. 112

.032
.066
. 129
.085
.139

.022
.073
. 157
.09 0
. 136

26 9
26 3
267
269
266

6

D a i r y P ro d .
Fats & Oils
Fruits
Vegetables
M e a t , et c .

.052
.044
.076
.034
.036

.067
.054
.095
.045
. 060

.048
.081
.112
.054
.060

271
264
268
271
268

7

D a i r y Prod.
Fats & Oils
F r u i ts
Vegetables
Meat, etc.

.003
.005
.066
.029
.080

.016
.019
.096
.050
.092

.003
.017
. 113
.068
.087

264
248
261
26 0
259

62

'''o]i t. i nued )

Composite
Food

Linear

^
Based on
Semi-Log
Double-Log

D a i r y Prod.
F a t s & Oils
Fruits
V e g o tables
Men t , e t c .

.027
.041
. 1 35
.077
. 1 00

.037
.052
. 149
.097
. 114

.021
.066
.15 3
.084
.1 0 0

265
255
259
2C5
262

D a i r y Prod.
F a t s & Oils
Fruits
Vegetables
Moat , e t c .

.023
.036
.098
.048
.0 B 4

.039
. 043
. 119
.069
.123

.016
.076
.116
.05 2
.0 8 3

262
256
257
261
259

D a i r y Prod.
F a t s & OiIs
Fruits
Vegetables
M e a t , etc.

.026
.017
.090
.030
.058

.042
.027
.078
. 040
. 070

.032
.032
.082
.029
.0 6 7

259
256
25 3
259
256

D a i r y Prod.
F a t s S. O i l s
Fruits
V e g e t a b l es
Meat, etc.

.010
.042
.092
.059
. 125

.019
.050
.083
.090
.135

.010
.080
. 114
.074
. 109

257
247
25 1
25 5
254

D a i r y Prod.
Fa t s & Oils
Fruits
Vegetables
M e a t , etc.

.017
.046
.024
.037
.038

. 000
.000
. 000
.003
.014

.0 0 0
.0 0 6
.017
.008
.035

26 1
258
25 5
261
25 7

D a i r y Prod.
F a t s & Oi l s
F r u i ts
V e g e t a b l es
M e a t , etc.

.006
.019
. 047
.010
. 024

.000
.004
.000
.000
.000

.000
.0 0 4
.0 0 0
.0 0 0
.0 0 0

26 0
25 0
25 5
25 8
256

The t—
p e r i o d o f time i m p l i e s the t —
cross
s ec tional study, since a cross sec tio na l study
b e l o n g s to a p a r t i c u l a r p e r i o d of time .

n

63

2 ._5. 1 .

Distributions

D e t e r m i n a t i o n ■— B e f o r e
" g o o d n e s s of
alternative
Such

fit,"

From T able
R

based
.20.

form

uniformly

These

of

fit"

the

that the

tc
to
to
to
to
to

of

„2
R

.02
.04
.06
.08
. 10
.20

would

be

about
R

of

a

, based

from

on

Table

13.

14.

notice

that

forms

indicate

nonlinear

of

derived

highest values

form ,

to the

conclusions

in T a b l e

functional

14.- - D i s t ributions
m i n a t i o n {R^)
Forms

Range

0
.02
. 04
.06
. OR
. 10

one

will

distributions

the d o u b l e - l o g

"goodness

TABLE

forms,

14,

any

Coefficient

distributions

are g i v e n

gives

cannot c l a i m

log o r

frequency

on al t e r n a t i v e

0 to

one

drawing

functional

distributions

o f Corrected

the

values

of

lie

in

the

that

no

functional

_2
R .

of

form,

either

uniformly gives

range

Obviously,
the

semi­

a better

observations.

of Co r r e c t e d C o e f f i c i e n t of D e t e r ­
B as ed on A l t e r n a t i v e F u n c t i o n a l

Linear

V a l u e s of R^_______
______
Semi-Log
Double-Log

13
19
15
5
7
6

20
7
10
8
11
9

20
7
7
8
12
11

65

65

65

APPENDIX

DAIRY
Fresh

A

PRODUCTS

Milk
H o m o q e n i z e d - - V i t . D.
Multiple Vitamin Milk
H o m o g e n i z e d - - P i n in
Regular Pasteurized
J er se y or G u e r n s e y
Buttermilk
Chocolate
Skim Milk
Sour M il k
Egg Nog, etc .
Other Milk

Cream

Canned

Cofee C r e a m
Whipping Cream
Sour C r e a m
(Liquid)
Evaporated--Unsweetened
C ondensed— Sweetoned
Canned--Baby Formulas

Dried
Powdered--Skim Milk
Powdered--Whole Milk
P o w d e r e d — Baby Formulas
Ice C r e a m M i x
Sher be t Mix
Malted Milk Powder
Ice C r e a m
H a n d P a c k e d Ice C r e a m
P r e - P a c k a g e d Ice C r e a m
O t h e r Ice C r e a m
S h e r b e t s a n d Ices
D a i r y Q u e e n , F r o s t i e , etc.
Cheese
Natural A m e r i c a n (Cheddar, etc.)
P r o c e s s e d A m e r i c a n ( V e l v e e t a , etc.)
Swiss C he e s e
Cheese Spread
C r e a m C h e e s e ( P h i l a d e l p h i a , etc.)
Cottage Cheese
Other C h e es e

64

APPENDIX

FATS

B

AND OILS

Butter
Oleomargarine
Lard
Swiltninq
Vegetable Shortening
O the r Fats

(Cnsco,

Oi 1 s
C o o k i n g Oils
Mayonnaise
Salad Dressing
Roquefort Dressing
S a l a d Oils, et c.
F r e n c h D r e s s i n g , et c .
Sandwich Spread, Tartar
Whips
O ther Oils

65

Sauce

Spry,

e tc . )

APPENDIX

C

FRUITS

Merr i os
B l u o b e r r i os
C v a n b n r r i os
Cu r rants
D e w b e r r i e s anc3 B 1 a * k bcr r io ^
Raspberries
Strawberries
Berry Juice
Other Berrios
Ci t r u s

Other

Grapefruit
Lemons
Lemonade
Lemon Juice
Gr apef ruit Juice
Limes
Lime Juice
Limeade
Oranges
Orange Juice
O r a n g e D r i n k (Hi-C, etc.)
Tangerines
Ta n g e r i n e Juice
Mixed Citrus Fruits
Mix ed Citrus Juice
Other Citrus
O th er Ci t r u s Juice
Fruits
Apples
A p p l e s a u c e and Apple Butter
Apple Cider
Apple Juice
Apricots
Apricot Nectar
Avocados
Bananas
Cherries--Maraschino
Cherries--Sour
Cherries--Sweet
Da tes
Figs

66

67

other

Fruits

( C ontinued)

Grape Juice
C a n t a l o u p e a nd M u s k m e l o n
Watermelon
Nectarines
01 iv e s
Persimmons
Peaches
Pears
Pineapple
Iineapple Juice
Plums
Prunes
Prune Juice
Raisins
Rhubarb
H a w a i i a n P u n c h Base
Mixed Fruits
Fruit Cocktail
F r u i t Pie M i x
Mixed Fruit Juice
Fruit Gelatin Salad--Prepared
Powdered Juice
Candied Fruit
Fruit Pickles
Other Fruits
Other Fruit Juice

APPENDIX

D

VEGETABLES

Green

I,eafy V e g e t a b l e s
Brussel Sprouts
Cabbage
Cabbage Salad
Sauerkraut
Celery Cabbage
Endive, Chicory, Escaro le
G r e e n s - - B e e t , M u s t a r d , etc.
Lettuce--Head
L e t t u c e — Leaf
Lettuce--Bib
Parsley, Swiss Chard, Water Cress
Spi n a c h
Mixed Leafy Vegetables
Other Leafy Vegetables
Green and Y e l l o w V e g e t a b l e s
Artichokes
Asparagus
Beans--Lima
Beans--Snap
Bean Sprouts
Broccoli
Carrots
Corn--Sweet
Peas
Peppers
Pumpkin
S g u a sh
Soy Steak and C h o pl ets
M i x e d G r e e n an d Y e l l o w V e g e t a b l e s
Others
All O t h e r V e g e t a b l e s
Beans --N av y, Baked, White
Pork and B eans
Beans--Kidney
Beets
Cauliflower
Cucumbers
Cucumber Pickles
Relish
Egg Plant
G a r 1 ic
68

69

Horseradish
Mushrooms
Onions--Mature
On i o n s - - G r e e n
Parsnips
Pimentoes
Michigan Potatoes
Maine Potatoes
Idaho Potatoes
California Potatoes
Other Potatoes
P o t a t o e s — F r e n c h Fries
Potato Chips
Potato Sticks
Potato Salad
M a s h e d P o t a t oe s or Patties
S w e e t P o t a t o e s a n d Yams
Rad i s h e s
Tomatoes
Tomato Catsup
To mato Juice
T u r n i p s an d R u t a b a g a s
Prepared Vegetable Gelatin Salad
Mixed Vegetables
C h o p Suey, C h o w M e i n , w i t h o u t M e a t
Mixed V e g e t a b l e Juice
Other Vegetables

APPENDIX

MEAT,

POULTRY,

E

F I S H , AND

EGGS

Beef
C a n n e d Beef
C o r n e d Beef
C h i p p e d Beef
G r o u n d Beef, H a m b u r q e r
G r o u n d R o u n d Steak, L e a n G r o u n d Beef
Beef Liver and Baby Beef Liver
Heart, Tongue, other O rgan Parts
C h u c k R o a s t (Pot R o as t )
Rib Roast
Other Roast
Round and S w i s s Steak
Sirloin Steak
P or t e r h o u s e and T- Bone Steak
O t h e r Steak
S t e w i n g B e e f ( B on e l e s s )
Boiling Beef or Short Ribs
All Other Beef
Pork
Bacon
Canadian Bacon
C a n n e d Pork
Chops
Steaks
Ham--Center Slice
H a m - - W h o l e o r Half
Ham--Canned
Ham--Other
P i c n i c Ham, C u r e d B u t t s
Pork Liver
Heart, Tongue, other O r g a n
Roast--Fresh
Sausage--Link
Sausage
Spax~eribs
S i d e or S a l t P o r k
O t h e r Pork
Lamb-Mutton
Chops, Steaks
R o a s t (Leg, e t c . )
Other Lamb— Mutton

70

Parts

71

Veal
C u t J e t s , Chops, steaks
Ground Veal
Calf Liver
City Chicken
Roast
Stewing, Soup Veal
Other Veal
O t h e r Meat and Meat M i x t u r e s
W i e n e r s a n d F r a n k s , etc.
B o l o g n a — R i n g or L a r g e R o u n d
Other Cold Cuts
Prem, S p a m , T r e e t , etc.
Rabbit, Domestic
V e n i s o n and O t h e r Game A n i m a l s
Chop Suey Meat and Kabobs
Bouillon Cubes
Beef Stew
Chile Con Carne
Hash
Mincemeat
M e a t B a l l s and S p a g h e t t i
Ravioli and Tamales
C h o p Sue y , C h o w M e i n w i t h M e a t
Potted Meat
Meat Spreads
Por k a n d B e a n s
Others
Chicken
B r o i l e r s or F r y e r s
Roas ters
Stewing
Barbecued Chicken
Turkey
Duck

Other

Poultry
Game Birds
Mixtures--Chiefly Chicken
Chicken Noodle Dinner
C h i c k e n a la K i n g
C h i c k e n C h o p S u e y , etc.
Others
F is h and Sea F o od
Tuna
Salmon
Fish Sticks
Other Fish

72

Fish

Eggs

and S e a F o o d ( C o n t i n u e d )
Lobster, Lobstertail
Oysters
Oys t e r Stew
Seallops
Shrimps
T u n a Pi e or C a s s e r o l e
S a r d i n e s in O i l
S a r d i n e s in S a u c e

APPENDIX

VALUES

Period
oi Tinie
Parry

Y k , M,

y

AND

M/Yk

m

k

M/Y,
K

P i cjcU j c ts
1
2
3
4
5

3.86
3.80
3.77
3.8 0
3.83

1 b i .09
163.74
158.17
153.79
152.24

41.73
43.08
41.95
40.47
39.74

6

9
10

3.8 5
3.65
3.6 6
3.5 0
3.60

150.89
148.63
153.95
151.10
156.63

39.19
40.72
42.06
43.17
43.50

11
12
13

3.66
3. 7 6
3.61

144.8 0
158.70
168.99

39.67
42.20
46.81

1
2
3
4
5

1.04
1.13
1. 0 2
1.05
1.11

157 . 81
161.06
157.31
151.25
150.95

151.74
142.53
154.22
144.04
135.99

6

9
10

1.01
.94
1.0 3
.95
1. 0 0

148.29
149.50
153.23
148.59
156.82

146.82
159.04
148.76
156.41
156.82

11
12
13

.98
1.05
1.03

145.30
159.47
168.81

148.26
151.87
163.89

7
8

Fats

OF

F

and

O iIs

7
8

73

74

P e r iocf
Y,
M
M/Y.
________ of T i m e _______________ h_____________________________________ lL
Frui ts
1
2
3
4

1.93
2.21
2.12
2.08
2.29

159.49
16 3.63
158.9 4
153.29
151.52

82.63
74.04
74.97
73.69
66.16

2.36
2.76
2.57
2. 4 1
2.31

151.44
148.19
155.11
152.33
158.20

64.16
53.69
60.35
63.20
68.48

12
13

2.11
2.15
2.27

145.65
159.64
168.64

69.02
74.25
74.29

1
2
3
4
5

2.16
2.19
2.16
2.25
2.63

158.94
163.24
158.17
153.79
152.24

73.58
74.52
73.22
68.35
57.88

6

9
10

2.38
2.30
2.18
1.85
1.89

150.89
146.53
153.95
15 0.61
156.63

63.39
63.70
70.61
8 1. 4 1
82.87

11
12
13

1.89
2.02
1.85

144.83
158.70
167.62

76.62
78.56
90.60

5

6

7
8

9
10
11

Vege tables

7
8

75

Period
of T i m e
Meat,

y

,
k

M

M/Y.
1

etc.
1
2

3
4
5
6

7
8

9
10

H oma rk :

7 . 80
.0 2
7 . 38
7.71
8 . 0 2

16 1.09
162.12
158.00
152.96
152.16

7 . 38
7 . 58
7 . 37
7. 09
7 . 67

150.80
148.23
154.00
151.03
156.59

20.43
19 . 5 5
20 .89
21. 30
20 .41

144.56
158.62
167.90

19 . 30
19.53
21.55

8

11

7.49

12

8 .12

13

7.79

The t —
p e r i o d of t i m e i m p l i e s
s e c t i o n a l study, since a c r o s s
b e l o n g s to a p a r t i c u l a r p e r i o d

20 .53
20 .21

21.40
19 . 8 3
18.97

the t —
cross
sectional study
of time.

CHAPTER

COMBINED

I.

combining

earlier

cross
used

are
cross

income

sectional
in

model
ferent

many

the

with

the

regression,

several
of

elast.ici t i c s .

As

and

Prais

time

since

and

Engel

series

used

than

more

cross

be

General

any

R e m a rks

curves

dat a.

regression will

procedure.*

may

Some

studying

estimates

successive

prices,

and

ior

combined

elasticity

estimation

sets

reasons

sectional

mentioned,

reliable

STUDIES

Objectives

There
by

IV

As

give

more

individual

observations

are

Moreover,

the

sections,

or w i t h

to e s t i m a t e

price

Houthakker

combined
dif­

mention:

T h e d e r i v a t i o n of p r i c e e l a s t i c i t i e s . . . h a s
b e c o m e p o s s i b l e f o l l o w i n g the c o l l e c t i o n of f a m i l y
b u d g e t s o n a c o n t i n u o u s b a s i s for a l e n g t h p e r i o d .
The analysis . . .
is n o t d i f f e r e n t f r o m t h a t
c l a s s i c a l l y a p p l i e d to t i m e s e r i e s , b u t the r e s u l t s
r e c ent ly a c h i e v e d using family bu d g e t records
appear more successful.
T h e c o n s i s t e n c y of the
d a t a is p r o b a b l y the m a i n r e a s o n for g r e a t e r s u c ­
c e s s , in t h a t b o t h p r i c e s a n d q u a n t i t i e s are
c o l l e c t e d s i m u l t a n e o u s l y , u s i n g p r e c i s e l y the s a m e
e o m m o d i t y - d e f i n i t i o n s and m eth od s of o b s e r v a t i o n ,
and over a lengthy period.

*See

p.

13,

Chapter

I.

■^Prais a n d H o u t h a k k e r , o p . ci t . , p. x x v i .
The deri­
v a t i o n of p r i c e e l a s t i c i t y e s t i m a t e s F r o m t h e f a m i l y b u d g e t
r e c o r d s is a l s o m e n t i o n e d in K l e i n , I n t r o d u c t i o n to E c o n o ­
m e t r i c s , op. c i t . , p. 62, f o o t n o t e 2^~.

76

77

T h o o r e t i c a 1 ly , the
is e s s e n t i a l
behavior

lar

for e x p l a i n i n g

changes

commodity.

When

commodity

is n e i t h e r

price

unity,

the

esting
for

regardless

estimate

price

elasticity

to e c o n o m i s t s

the

good.

who

engage

the

U.S.

price*?.

This

may

be d o n e

of p r i c e

inelastic,

estimates

that

estimates

on

expenditures.

instance,

lood

lower

the d e m a n d

the

same

t he

than

price,

When

it

the

the

is

lower

the

3

also

be

inter­

in e c o n o m i c p o l i c y .

are

amount
When

greater

government decides

parameters

a

value,

and

may

by p ri ce

of

a particu­

changes.

the

is

to t a l

price

and

expenditures

the

is,

price
for

is n u m e r i c a l l y

is e l a s t i c ,

the d e m a n d

in the

estimate

That
of

elasticities

expenditure

in a b s o l u t e

spent

smaller

The

unity

inelastic.

tot al

unity,

to

nor

elasticity

the

than

price,

elasticity

the d e m a n d

greater
less

the

price

price

how household
to v a r i a t i o n s

is e q u a l

be

of

according

elastic

of m o n e y w i l l
the

estimation

fixing.

reliable,

the

to

Assume,

raise
If

the

quantity

T h e a b o v e s t a t e m e n t c a n be p r o v e d as f o l l o w s :
the t o t a l a m o u n t s p e n t for a c o m m o d i t y is g i v e n b y PQ;
w h e r e P is the p r i c e , a n d 0 is the q u a n t i t y p u r c h a s e d of
the g o o d . T h u s ,

where

dP
n is the

= Q(1

+ n) ;

price

elasticity

=

^

.

T h i s e x p r e s s i o n is n e g a t i v e f o r v a l u e s of n b e t w e e n
- i" a n d - 1 , z e r o f o r n e q u a l to - 1 , a n d p o s i t i v e for v a l u e s
of b e t w e e n -1 a n d zero.
T h a t is, t h e t o t a l a m o u n t s p e n t
increases, re m a in s constant, or d e c r e a s e s wh en pr ic e
d e c r e a s e s ; a c c o r d i n g l y , t h e p r i c e e l a s t i c i t y o f d e m a n d is
n u m e r i c a l l y g r e a t e r than, e q u a l to, o r less t h a n , u n i t y .
See J. M. H e n d e r s o n a n d R. E. Q u a n d t , M i c r o e c o n o m i c T h e o r y
(2nd od. ; N e w Y o r k :
M c G r a w - H i l l , 1 9 7 l T ^ p~. 2*7.

78

of

the p r o d u c t s

be c o n s i d e r e d
which

are

stances,

desirable

pursued

in

the d e c l i n e

negligible,
from

demanded

the

compared

increase

in e c o n o m i c

in

can b e
or not

expected.
depending

The
on

economic policy.Under
in c o n s u m p t i o n o f
with

the b e n e f i t

prices.

policy may

be

Various

results may

the

social

ends

certain c i r c u m ­

f o o d s t u f f s is
accruing
social

in c o n f l i c t .

to p r o d u c e r s

ends pursued

Yet,

Tintner men-

tions:
E c o n o m e t r i c s c a n c o n t r i b u t e n o t h i n g as f a r as the
c h o i c e of a c o n c r e t e p o l i c y b a s e d u p o n t h e s o c i a l
e n d s is c o n c e r n e d .
But e c o n o m e t r i c s c a n p e r h a p s
c o n t r i b u t e s o m e t h i n g in g i v i n g e c o n o m i s t s n u m e r i c a l
e s t i m a t e s of the r e s u l t s of t h e a d o p t i o n of v a r i ­
ous p o s s i b l e p o li c i e s .
For
sectional
curves
price
us e d
As

in

and

will

reasons

time

are

taken

the

and

stayed

from

and

T h er e are 212*

the
th e

as

returned
are

13 =

Tintner,

be

five c o m p o s i t e
M.S.U.

Consumer

fo u r w e e k l y

a period

of

the p a n e l

selected
2756

will

for e s t i m a t i n g

chapter,

treated

p e r i o d s of t i m e

A

for

previous

m e n t i o n e d , ' ’ in

se ri es data

be m o d i f i e d

parameters

together
that

the

this

chapter,

pooled.
both

The

income

foods.
Panel

reports

for t he s a m p l e

an d

d a t a of
are

Those
all
of

Engel

The d a t a

reports

time.

cross

195 8 .

grouped

households

thirteen

observations.

o b s e r v a t i o n s for e a c h c o m p o s i t e food.*’

o p . c i t . , p.

12.

T o e s t i m a t e p r i c e e l a s t i c i t i e s a n d to o b t a i n m o r e
r e l i a b l e e s t i m a t e s o f i n c o m e e l a s t i c i t i e s f o r the five
c o m p o s i t e foods.
that
teen

^ T w o h u n d r e d t w e l v e is t h e n u m b e r o f h o u s e h o l d s
s t a y e d a n d r e p o r t e d their e x p e n d i t u r e s o v e r all t h i r ­
p e r i o d s of t i m e in ]958.

79

2.
by
the p r i c e
able

pooling
of

the

cross

foo d

in d e t e r m i n i n g

Different
prices.
by

Statistical

cross
Th e

Wang

sectional

concerned

household

sections

M.S.U.

will

be

used

loccc

by

the p a n e l

an

expenditure

series

to

data,
vari­

behavior.
sets

indices

chapter

Models

important

to d i f f e r e n t

food p r i c e

in t h i s

time

becomes

belong

retail

and

Combined

of

constructed

represent

the

food

7

prices
of

tJie fi v e

composite

In a c r o s s
reasonably

analysis.

quantity
ture

food

°itv. =

the

price

out

in

where

the

proper

The

i^- household

quantity

tn e

t— — periou

time

indices

lb.

prices

are

is

usually

used

regression

analyses,

to q u a n t i t y
computed

price

Table

sectional

combined

simply

These

expenditure

in a c r o s s

for

is

set

study,

be c o n v e r t e d

purchased

by

at

must

are

household

variable

However,

expenditure

foods

sectional

constant,

as a d e p e n d e n t

households.

prices

vary;

purchased.

by d e f l a t i n g

The

expendi­

index.

cf

purchased

is c o m p u t e d

as

on

the

k—

follows:

(yit.k * 1 0 0 ) / P tk
th
t ^ie i—
household

where
the
P ^

is

k^ii f o o d
the p r i c e

period
7

of

at

the

index

aggregate
t^l period

of

the

expenditure
of

k^Jl f o o d

time,
at

on

and

the

t—

time .

H.
F . Wang, "Retail F oo d P r i c e I ndex Based
M . S . U . C o n s u m e r P a n e l " ( u n p u b l i s h e d Ph.D. d i s s e r t a t i o n ,
M i c h i g a n S t a t e U n i v e r s i t y , 1960).

on

80

'I’M-; f

■Price I n d i c e s of Five
M.S.U. C o n s u m e r Panel
(1955-57 = 100)

Period
of
Time
1

->
■1

Da iry
Prod .

Fats &
Oi Is

100 .9
98 . 8
1 0 0 .2
98 .8
98 . 1

101.3
98 . 3
95 .9

97.4
97.9
97 .6
99 .2
98 . 2

6

7
8

o
1 0

98 .2
97 . 7
96.1

1 1
12

13

Source:

on

Fruits

Vege­
tables

Hea t,
etc.

93.4
105 . 8
1 00. 5
115.4
112.5

115.9
117.5
1 29 . 1
1 36 . 6
1 30. 4

105 .0
107 .4
106 .4
1 1 0 .2
1 1 1 .4

119.6

.8
.4
.3
.4

119.8
11 4 . 5
91 .1
82.5
76.1

113. 7
114 .7
113. 2
Ill .7
I l l .e

97 .6
95.0
96 . 4

85. 2
83.1
82 . 3

1 0 0 .0

100.7
1 0 0 . 1

98
97
97
96

1 1 2 . 2

92.2
3 .5
71.2
7

79 .5
93.8
1 0 1 . 0

109 .6
Ill .6
1 1 0 .0

H . F. W a n g , "R e t a i 1 F o c d Pr i c e Index B a s e d on
M . S . U. C o n s u m e r P a n e l " ( u n p u b l i s h e d Ph.D. dissert.i tion, M i c h i g a n S t a t e U n i v c r s i ty , 1960), T a b l e 1 0 ,
p p . 146-47.

Tii this m a n n e r , o n e
price multiplied
Based
Fngel

C o m p o s i t e For Is B a s e d
D a t a of 1958

curve

on

for

by q u a n t i t y
the c e t e r i s
,i
the k _ fo od

c an

notice

equaling
paribus

that

the

identity

expenditure
assumption,

of

is p r e s e r v e d
the m o d i f i e d

8

ca n

be

stated

as:

O
T h i s m o d i f i e d E n g e l c u r v e is s i m i l a r to the
th
M a r s h a l l i a n d e m a n d f u n c t i o n w h e r e o n l y the p r i c e of the k.bii
f o o d a n d per c a p i t a d i s p o s a b l e i n c o m e are a l l o w e d to v a r y a n d
all o t h e r p r i c e s are h e l d fixed.
Under certain assumptions,
M a r s h a l l d e d u c e d the s o - c a l l e d " l a w of d e m a n d , " in w h i c h he
s t a t e d t h a t the s l o p e of h i s d e m a n d c u r v e w i t h r e s p e c t to
p r i c e is a l w a y s n e g a t i v e .
F o r a full d i s c u s s i o n o f M a r s h a l l ' s
law of d e m a n d , s e e D. W. K a t z n e r , S t a t i c D e m a n d T h e o r y (New
York:
M a c m i l l a n , 1970), pp. 58-59,

81

°itv

M it

“ it

w it

whore

4 V itV

''tv.'

'*'S t ^°

household

on

the

k 1^22. f o o d

at

is

the

i±£L h o u s e h o l d

per

the

capita

t-—

consumption

period

of

time,

I- L

at

Engel

u .j

is

the

disturbance,

t,

is

the

undefined

prices
Engel

curve
are

defined
held

gives

in

the

constant,

of

disposable

time,

and

f u n c t i o n a l form.

curve

is c o n s i s t e n t w i t h

previous

chapter.

the m o d i f i e d

As

Engel

lon g

curve

is

the
as
the

c u r v e.
Regarding

curve,

the

Engel

capita

th
t-— p e r i o d

income

This m o d i f i e d

per

since

neither

a better

the

thirteen

the-

Linear

the

the

"goodness

cross

form

functional

f o r m of

semi-log
of

fit"

sectional

is a d o p t e d

in

nor
to

this

the

the

studies

the

in

modified

double-log

En g e l
form

observations

based

the

chapter,

chapter

previous
as

the

first

on

order

•
« •
approxlina
t ion . 9

u

T h e a d o p t i o n of l i n e a r r e l a t i o n s h i p s is a p r o p e r
procedure.
As
a p p l i e d to th e m e a s u r e m e n t o f d e m a n d o f f o o d
in t er n s of
p r i c e p^ a n d i n c o m e
m, t h i s i n v o l v e s the
T a y l o r ' s series a p p r o x i m a t i o n ar ound any g i v e n point
0
o
0 .
- q ]:,
ni^) or

n
qk " q k +

3qk 0
o
(^ F P ) (m“m 1 +

3qk 0
o
(3p“ J (pk " p ' + r e m a i n d e r *

As l o n g as t h e p r i c e an d i n c o m e c h a n g e s w e r e s m a l l ,
the r e m a i n d e r e r r o r t e r m c a n be n e g l e c t e d .
S ee P. A. S a m u e l s o n , " S o m e I m p l i c a t i o n s of ' L i n e a r i t y , ' " T h e R e v i e w of
E c o n o m i c S t u d i e s , 1 9 4 7 - 4 8 , r e p r i n t e d in T h e C o l l e c t e d S c i ­
e n tific Papers
of P. A. S a m u e l s o n ,e d . by
J . EL S t i g l i t z
T m . 1 .T . P r e s s , 1^6 6 ) , p"! 6 1 .

82

T h u s r the m o d i f i e d
expressed

as

— rr-—

+

=

Ni

where

Cl■,

k

R

—

k U ifc

jm a t i o n

pooled,
of

when

of

periods

of

relics

as

s u m m a r y of

factors

which
effect
par t,
Kmenta

that

ar e
of

P, ,

ar e
and

+

tk

the

k - ^ food

is

1

price

i.e. ,

constant

coefficient,

fo r

probability distribution

various

time.

a

on

large

enter

successive

obviously
that

into

The
the

the

factors
to

the

cross

the

term,
k^-

the

disturbance

with other

common

of

the

sections

ca n a r i s e . ^

food.

Th e

dis­
are

existence

occurring

at

disturbances

at

belief

in

the a u t o c o r r e l a ­

interpretation

of

the d i s t u r b a n c e

number of

no t m e a s u r a b l e . ^

carry over

tk

is c o r r e l a t e d

largely

thes e

U. , ,

parameters,

implies

time

tion
a

k

autocorrelation

period

other

Y

the

autoregression

one

for

Procedure

Regarding
turbance-

+

y.

coefficient,

2.1.

curve

follows:

uv , R , and

income

Enael

random and

relationship
Then

independent

under

one w o u l d

operating

in one

following

periods.

study,

suspect

period
As

but

that

would,

in

Professor

mentions:

A u t o r e g r e s s i o n o f the d i s t u r b a n c e s c a n be c o m p a r e d
w i t h the s o u n d e f f e c t of t a p p i n g a m u s i c a l s t r i n g :
w h i l e the s o u n d is l o u d e s t a t the t i m e o f i m p a c t , it

*°Kuh,

o p . c i t . , p.

98.

^ F o r a full d i s c u s s i o n
J o h n s t o n , o p . c i t . , pp. 177-99.

of

autocorrelation,

see

the

83

d o e s not s t o p i m m e d i a t e l y but l i n q e r s o n for a t i m e
u*:tii :t f i n a l l y Hi e s off.
T h i s m a y a l s o he the
c h a r a c t e r i s t i c of the d i s t u r b a n c e , s i n c e its e f f e c t
m a y lin g er for some t i m e a f t e r its o c c u r r e n c e .
But
w h i l e the e f f e c t of o n e d i s t u r b a n c e l i n g e r s on,
o t h e r d i s t u r b a n c e s ta k e p l a c e, as if th e m u s i c a l
s t r i n g w e r e t a p p e d o v e r a n d over , s o m e t i m e s h a r d e r
t h a n at o t h e r times.
The .shorter the ti m e b e t w e e n
t h e t a p p i n g s , the g r e a t e r the l i k e l i h o o d t h a t the
p r e c e d i n g s o u n d ca n s t i l l be h e a r d .
S i m i l a r l y , the
s h o r t e r the p e r i o d s of i n d i v i d u a l o b s e r v a t i o n s , the
g r e a t e r the l i k e l i h o o d of e n c o u n t e r i n g a u t o r e g r e s ­
sive d i s t u r b a n c e s .
In r e c e n t
how

to cope w i t h

Most of
of

the

w ill

y ea r s ,

the a u t o c o r r e l a t i o n

the p r o p o s e d

corrections

variance-covariance

s e l d o m be
This

s t u dy ,

The d i s t u r b a n c e
wil l

be

of

order

and

first

the v a l u e s

characterized

(ii)

Zero m e a n :

E (u itk^

Homoskedasticity:

13

upon

as

assumes

exact

that

autoregressive
of

which

the
scheme.

the e x o g e n o u s

vari­

is n o r m a l l y d i s t r i b u t e d ;
= 0;
2
E ^U itk^

o p . c i t ■ , p.

on

knowledge

fo l l o w s :

2
= °k;

i n t e r d e p e n d e n c e :E ( U . ,U. , ) =
ltk jtk

Kmenta,

literature

the d i s t u r b a n c e

the

U •

No

depend

others,

Normality:

(iv)

of

has b e e n a c c u m u l a t e d .

like m a n y

(i)

(iii)

matrix

body

13

kno w n.

au to c o r re la ti o n has

ables

a substantial

0 for

i ^ j;

270.

K u h , o p . c i t . , p. 99.
For a rec en t survey of
l i t e r a t u r e o n tKe s p e c i f i c a t i o n of a u t o r e g r e s s i v e s c h e m e ,
see G. T i n t n e r a n d J. K. S e n g u p t a , S t o c h a s t i c E c o n o m i c s
(New York:
A c a d e m i c Pres s , 1972), pp. 12-21.

84

*v'

Firr^

order

autoregression:
^

E < u it-]kv jtk’ =

(vi)

The e x o g e n o u s

Vf'P

t 1 Ml I 1 least

sauarcs

<*o n s ist en t , yet

mal e s

they

nsymptotica1

two-stage

Orcntt

wiH

(1 ) A p p l y

of

From

1

os

a re

th e

0

error.

) , tlie c o n ­

u n b i a sod

o f f ici e n t
of

without

^

(f,k

and

a sympt o l ical

nor
least

1

1

y

s q u a res os t i -

an d

can

be

these
14
1

S

used

to

consists

th e

least

(i.e.,

asymptotically

suggested

the

of

the

squares

t )l

k-rii food.

coefficients
calculate

residuals,

For

method

at

least,
consis­
normal),

by C o c h r a n e

and

15

for

regression

tha t ,

y efficient,

used.

curve

estimates

properties

estimation

be

the

asymptotical

^ho o r d i n a r y

Engel

th e

and

j;

are m e a s u r e d

variances

The p r o c e d u r e

fied

not

to o b t a i n

the d e s i r a b l e

tent,
the

*•

.are b i a s e d . ^
In o r d e r

have

a11

cot re 1 at ion

estima

are

A n d , th e

efficient.

£or

variables

is s o r i a 1

T f th e r e

0

U . , . = nv U. ,
+V. . ;
lit'.
K lt'lK
] tk

are

the

following
method

The

*

regression
t he

the m o d i -

resulting

unbiased
3

to

two s t a g e s :

and

*

estimates

consistent,

residuals

one

can

obtain

estimate

proofs,

see

K m e n t a , o p . c i t . , p.

of

U ., , .
i tk
by

269-97.

S e e D. C o c h r a n e a n d G. H. O r c u t t , " A p p l i c a t i o n
of L e a s t S q u a r e s R e g r e s s i o n s to R e l a t i o n s h i p s C o n t a i n i n g
A u t o c o r r e l a t e d E r r o r T e r m s , " J o u r n a l o f the A m e r i c a n S t a ­
t i s t i c a l A s s o c i a t i o n , Vol . 44 ( M a r c h , 19 49) , p p . 32-61;
K m e n t a , o p . c i t . , pp. 2 8 7 - 8 8 a n d pp. 5 0 9 - 1 2 .

85

°k

212

13

l-l
2T 2

t= 2
13

.
1

where
of

=] r= 2

212

is

V .t , U .
it!; u

lk

.
U it-lk

the n u m b e r

of h o u s e h o l d s ,

and

13

is

the

number

periods.
IF*

Obviously,
(2 )

r-ing

the

p^

pv
to

is a c o n s i s t e n t
transform

estimator

of

.

the o b s e r v a t i o n s :

° t t k = "k + "kM i t + v k p * k + u *tk

where ° i tk ■ °itk/ N it - ek ° i t - i k / N it-i
M *t ■ M u /Ni t - K
- p tk ^itk

i =

1

t =

2,

The
regressive.
again,

~ ^itk

the

,

2

p t-ik

K

p k U it- lk
,

3,

. . . , 2 1 2

...,

disturbance
Applying
estimators

13.

Uf
*lK

is a s y m p t o t i c a l l y

the o r d i n a r y
of a*,
K
.

asymptotic

properties;

i.e.,

efficient,

and a s y m p t o t i c a l l y

p

K

least

, and

y.

K

consistent
normal.

nonauto-

squares
have

method

the d e s i r a b l e

asymptotically

86

2 .2.

R e sults
At

ting

the

of

th e

Combined

first

stage

autoregressive

estimates

of

regression

Studies
of

computation,

effects,

the

coefficients

without

least
are

elimina­

squares

obtained

as

foilow s :

bn

1

ry

Pt od u c t s :

q.,

= 4.0498

11 1

=

S =
Fats

and

Oils:

q

+

.00 1 7 m-.
,
it
(.0003)

+

.0 0 1 1 m . . - .0057 P
(. 0 0 0 1 )
(.0086) U

+

.0 0 3 9 m- .0215 P
(.0002)
C
(.0023)

xt

Frui t s :

.0 2 0 7
.8 6 0 9

q.
= 3.8473
lfc

R2 =

.0058 P.
t
( .0217)

.0112
1.8163

= 1.4148

R2 =
S =

-

.0854

S = 1.8020
Vegetables:

q.

=

3.1244

+

lt:
R2 =
S =
M e a t , e t c .:

q .

=

R2
S =
where

q..
at

=

m it
— 2

R

it

it

. 0 0 2 3 m.

-

(.0 0 0 2 )

.01 3 3

P.

(.0 0 1 1 )

.0 7 8 7
1 . 3354
12 . 2750

+

. 0 0 8 9 m. .
(.0006) lt:

.0621 P t
(. 0262)

. 0699
3.7902

;

M. . / N . .

it

is

it

the c o r r e c t e d

S is t h e

standard

coefficient
error

of

of

determination;

estimate.

and

B7

Using
cients,

the

the e s t i m a t e s

the

fiv e

are

presented

composite
in

to b u t

less

.6629,

and

cifnts

for d a i r y

and

meat,

.8284

resulting

than
.50 3 1

of

foods

regression

16.

one.

The

They

are

figures

products;

fats

products,

and

all

.8284,

the e s t i m a t e s

of

These

for

estimates

positive,
.6966,

close

.7100,

autocorrelated

and

oils;

The

highest

the

coeffi­

coefficients

are c a l c u l a t e d .

etc. , r e s p e c t i v e l y .

fo r d a i r y

of

autocorrelated

Table

are

estimates

lowest

fruits;

is

coeffi-

vegetables;

estimate
.5031

is

for

me a t , etc .

TABLE

15.

E s t i m a t e s of A u t o c o r r e l a t e d
Five Com p o s i t e Foods

Foods

Dairy
Fats

Estimates

of

Autocorrelated

Products
and

Coefficients

Oils

.6966
.7100

Vegetables

.66

Mea t,

.5031

etc.

Statistically,
coefficient
for

figure

is c l o s e

the

relationship

the

positive

f or a p a r t i c u l a r

disturbances
which

Coefficients

.8284

Fruits

ted

for

that

food

ar e

to o n e

between

29

estimate

food

indicates

positively

indicates

of

that

the

correlated.

that

the d i s t u r b a n c e s

autocorrela­

is

t^e

degree

fairly

The
of

high.

88

At
gressive

the

final

effects

staqe

of co m p u t a t i o n ,

are e l i m i n a t e d .

regression

The

least

squares

of

foods

are

obtained

TABLE

1 7 . - - E s t i m a t e s of R e g r e s s i o n C o e f f i c i e n t s for F i v e
C o m p o s i t e F o o d s a f t e r E l i m i n a t i n g the A u t o r e ­
gressive Effects
Constant
Te r m

Food s

Dairy

Fats

Products

and

shown

in

fo r

autore­

estimates

and

coefficients

the

Table

the

fiv e

composite

17.

C o e l.f .

Price
Coe ff .

1 . 5 01 1

.0 0 0 4
{ .0 0 0 2 )

-.0553
(.0 2 0 0 )

.0038

1.0029

.7698

.0006
(. 0 0 0 1 )

-.0165
( .0066)

.0104

.5589

Oils

1 ncom.e

S

R2

F r u i ts

1 .1656

.0 0 2 7
(.0003)

-.0200
( .0025)

.0494

1.2784

Vegetables

1. 16 0 0

.0 0 0 7
(. 0 0 0 2 )

-.0147
(.0016)

.0318

.9 8 7 2

Meat,

4 .0680

.0 0 6 3
(.0007)

- . 0 2 2 1 0

.0262

3 . 12 6 3

etc.

Rema r k a ■

T h e m a r k @ i n d i c a t e s t h a t the r e g r e s s i o n c o e f f i ­
c i e n t is not s i g n i f i c a n t l y d i f f e r e n t f r o m z e r o at
5% l e v e l of s i g n i f i c a n c e .
R

2

is

S is

t he
the

From Table
mates
are

of

income

positive

level
income

of

(.0395)

and

corrected
standard

17,

one

coefficient

error

would

coefficients

fo r

coefficients

a re

Needless
more

determination.

of e s t i m a t e .

notice
the

to

that

five

significantly different

significance.

of

say,

reliable

all

the

composite
from
these

than

esti­
foods

z e r o at

5%

estimates

those

of

estimated

89

from

individual

tions

are

used

As
five

for

composite

meat,

etc.,

cantly

cross
in

the

the

the

estimates

obtained
ticity

as

ze ro

Tncome

and

shown

income

At

and

of

the

the m e a n

price

price
all

at

5%

observa­

of

price

E s t i m a t c s .—

and

k—

are

are

price

foods

the

for
signifi­

coefficients

composite

estimates

are

the

significance.

income

for

for

Except

coefficients

the

values, ^

elasticity

negative.

E l a s t i c i ty

and

five

more

coefficients

level

Price

17,

since

procedure.

price

income

in T a b l e
for

o s t - i m o t e s

derived.

of

from

th<-' e s t i m a t e s

of

th ey a r e

estimates

different

studies,

estimation

foods,

/.2.1.
When

sectional

can

elas­

bo

food,

easily

the

calculated

as

follows:

where

q^

the

income

the

price

is

the

elasticity
elasticity

average

estimate
estimate

value

of

=

(m/q^)

■-

^Pk^k^

h o u s e h o l d s ’ per

,
;

capita

11
.

quantity
is

the

k—
m

purchased

average

on

the

*

k—

food,

value

of

price

indices

for

value

of

h o u s e h o l d s ’ pe r

th e

food,

is the

average

disposable

incom e,

and

are

the

estimated

cients

f or

the

k—

capita

and
income

and

price

coeffi-

^ U

Appendix

* ^Th e
G.

values

food,

of q. , p. , m,

respectively.

and

m / q k are

given

in

90

The
values

lor

income
the

five

and

price

elasticity

composite

foods

are

estimates
shown

at m e a n

in T a b l e

18,

• i
17
below.

TABLE

1H. - - i n c o m e and P r i c e E l a s t i c i t y E s t i m a t e s f o r F i v e
C o m p o s i t e F o o d s a f t e r E l i m i n a t i n g the A u t o c o r r e l a
te d E f f e c t s

Fooas

Income

Elasticity

Price

Elasticity

.0170

-1 . 4 5 4 3

.0928

-1.5712

Frui ts

.1808

-.8158

Vegetables

.0 5 3 3

-.7424

Meat, e tc.

.14 72

-.3580

Dairy
Fats

Pioduct.s
and

Oils

^ T h e s e co mb ined studie s give s u b s t a n t i a l l y lower
v a l u e s of i n c o m e e l a s t i c i t i e s t h a n c r o s s s e c t i o n a l s t u d i e s
in the p r e v i o u s c h a p t e r .
O n e p o s s i b i l i t y to e x p l a i n t h i s
p h e n o m e n o n is t h e i m p l i c a t i o n s of F r i e d m a n ' s p e r m a n e n t
income hypothe si s.
F r i e d m a n h a s d e m o n s t r a t e d t h a t the e l a s ­
t i c i t y of c o n s u m p t i o n w i t h r e s p e c t to m e a s u r e d i n c o m e
sep arates into two e l a s tici ti es :
the e l a s t i c i t y of c o n ­
s u m p t i o n w i t h r e s p e c t to p e r m a n e n t i n c o m e a n d t h e e l a s t i c i t y
of p e r m a n e n t i n c o m e w i t h r e s p e c t to m e a s u r e d i n c o m e .
Given
the a s s u m p t i o n s of the p e r m a n e n t i n c o m e h y p o t h e s i s , F r i e d m a n
d e m o n s t r a t e s t h e e q u i v a l e n c e of the e l a s t i c i t y of p e r m a n e n t
i n c o m e w i t h r e s p e c t to m e a s u r e d i n c o m e a n d the e l a s t i c i t y
of c o n s u m p t i o n o n m e a s u r e d i n c o m e .
Thus, it w o u l d s e e m
that the income e l a s t i c i t y e s t i m a t e d from cr os s s e c t i o n al
d a t a is a r e a s o n a b l e a p p r o x i m a t i o n of th e e l a s t i c i t y o f the
p e r m a n e n t income.
As m o r e t i m e s e r i e s d a t a a r e i n t r o d u c e d ,
the p e r m a n e n t i n c o m e c o m p o n e n t o f the m e a s u r e d i n c o m e is
reduced.
T h i s r e s u l t s in l o w e r v a l u e s of i n c o m e e l a s t i c i ­
ties d e r i v e d f r o m the c o m b i n e d studies.
For more
d i s c u s s i o n of this problem, see Friedman, o p . c i t ., S e c ­
t i o n 2, C h a p t e r V I I I , p. 206.

91

From
elasticity

for

The

for

are

highest

fruits,

the

Lansing

household
fats

per

per

by

cent,

composite

for

fats

products,

and

products

cent.

fruits,

.0928,

income
.0533,

elas­
and
.1472
products.

things

disposable
on the

are all

is

other

a n d meat,

income

average,

in
the

products;

etc.

wo u l d

.0533,

and

.1472

estimates

for

the f i v e

.1808,

widely

etc.,

-.7424,

The p r i c e

elasticity

an
or

oils

increase
fats

and

for

1 per cent

that

different.

fruit s ;

vegetables,

or

for

elasticity

other

in

food w o u l d

vegetables;

and

estimates

for

for d a i r y
meat,

etc.,

d a ir y

of

price of

1 per

cent

other

the p r i c e
meat,

being
be

in t h e

things

p r o d u c t would be

loss

b e i n g equal,
greater

elasticity

etc.

the p r i c e

things

-.3580

figures

In an

that

and

and

The

are h i g h l y e l a s t i c .

oils,

Alternatively,

of

demand

foods

p u r c h a s e d on d a i r y

and oils ;

in d e m a n d

increase
meat,

that,

1 per c e n t ,

-.8158,

fats and

sense,

a decrease

for

income

.0170 for d a i r y

per ca p i t a

they a r e

- 1. 5 7 1 2 ,

respectively.

1 per

is

vegetables;

the p r i c e

foods,

products;

dairy

.1808,

imply

capita q u a n t i t y

.0170,

The

the

e l as t ic i t y estimate

lowest

by

fruits;

composite

.0928,

figures

rises

about

-1.4543,

economic

five

that

respectively.

As

are

area

notice

E n g e l ’s law.

if h o u s e h o l d

and oils;

increase

the

income

etc.,and the

equal,

would

.0170,

Tht’o r c t.ioa 1 ly , these
being

one

They confirm

estimates

.1808.

18,

estimates

inelastic.
ticity

Table

of

are

equal,
th a n

estimates

inelastic.

fruits

or

An

vegetables

a decrease
1 per

than

cent.

in

92

Jn
of

income

foods,
fied

and

the

Engel

the d e m a n d
probably

summary,

regardinq

the

price elasticities

r e s u l t s of

these

curves

highly

arc

theorem.

the m a i n

The

reason

siqns
for

combined

this

the

five

studies

successful

consistency
for

the

of

of

as

estimates
composite

on

they

the m o d i ­
confirm

the p a n e l

success.

data

is

APPENDIX G

VALUES

OF

F oo d s

Dairy

qk ,

, m, m / q ^ , A N D

Pk

Products

Py/qy

m/qj.

P k /(\

3.74

98 . 39

159 . 40

42.62

26 . 30

1 .03

98.09

159.40

1 5 4 . 75

95.23

Fruits

2.38

97.09

159.40

66 .97

40. 79

Vegetables

2 .09

105.57

159 .40

76 . 26

50 .51

Meat,

6 .82

110.51

159 .40

23. 37

16 . 20

Fats

and

Oils

etc.

93

CHAPTER V

SOME

PRELIMINARY EVIDENCE
EMPIRICAL UTILITY

ON APPROXIMATING
FUNCTIONS

1.
In

th i s c h a p t e r ,

approximating

empirical

Engel

based

curves,

In a d d i t i o n ,
study

will

be

2,

some

utility

areas

Utility

tiome p r e l i m i n a r y

Wald's

proposed

In t h i s
approximating

on

O b i c c t ivcs

theorem,

that w e r e
for

future

Functions

section,

empirical

functions

some

on

by m e a n s

of

will

omitted

be

from

g i v e n . ’'
t he

present

research.

aad

Engel

preliminary

utility

evidence

functions

Curves
evidence

on

by

of

means

E ng e l

F o r a full d i s c u s s i o n o f W a l d ' s t h e o r e m , see
A. Wald, op. c i t . ,
pp. 1 4 4 - 5 5 .
T h i s t h e o r e m is a l s o m e n ­
t i o n e d in Z. H e l l w i g , L i n e a r R e g r e s s i o n a n d Its A p p l i c a t i o n
t o E c o n o m i c s (New Y o r k :
M a c m i l l a n , 1963), pp. 6 2 - 6 3 ;
1’i n t n e r , E c o n o m e t r i c s , op. cit. , pp. 6 0 - 6 1 ; G. T i n t n e r ,
M e t h o d o l o g y of M a t h e m a t i c a l E c o n o m i c s a n d E c o n o m e t r i c s
Tchicago:
U n i v e r s i t y Pres s , 1968 ), p p . 21- 23 ; a n d TH T.
D a v i s , o p. c i t . , p . 168.
M o r e r e s e a r c h in t h i s f i e l d of d e t e r m i n i n g e m p i r i ­
c a l u t i l i t y f u n c t i o n s and c o n d i t i o n s of i n t e g r a b i 1 ity a r e
being u n d e r t a k e n at M i c h i g a n St at e U n i v e r s i t y un de r
P r o f e s s o r A. Y. C. K o o 's l e a d e r s h i p .
F o r s o m e of P r o f e s s o r
K o o ' s w o r k s , see A. Y. C. Koo, "An E m p i r i c a l T e s t of
R e v e a l e d P r e f e r e n c e T h e o r y , " E c o n o m e t r i c a , 31 ( O c t o b e r ,
1 963), pp. 6 4 6 - 6 4 ; A. Y. C. Koo", h R e v e a l e d P r e f e r e n c e :
A
S t r u c t u r a l A n a l y s i s , " E c o n o m e t r i c a , 39 ( J an u a r y , 1 9 7 1 ) ,
pp. 8 9 -9 7; a n d A. Y. C~ K o o a n d G. H a s e n k a m p , " S t r u c t u r e
of R e v e a l e d P r e f e r e n c e :
Some Pr eliminary Evid ence,"
J o u r n a l of P o l i t i c a l E c o n o m y , V o l . 80 ( J u l y / A u g u s t , 1 9 7 2 ),
p p . 724-44.

94

9 l>

curves

will

be p r e s e n t e d .

Lhe-wi _ ic-1
utility
which
the

and

piaccicul

functions.

can

be

determination
Also,

e n .h 3 i 3

c ;,e to
It

cross

in Chnptoi

the

most

knows

the d e m a n d

calculate

should

ITT,

of

if o n e

of

the

will

the

note d

studies

and

IV,

be

to say,

impox Lance

on

i n d e x of

the

the
Engel

given M.S.U.
provide

important

the

all

empirical

function

for

cost

great

problems
is

consumers'

utility
of

function
l i v i ng .

results

of

the

curves

as

shown

price

the

is of

the

the u t i l i t y

of

that

it

Lo k n o w

functions

the d e t e r m i n a t i o n

sectional

Chapter

One

solved

goods.

Needless

indices

as

information

2

thirteen
in

shown

needed

to

3

approximate

the u t i l i t y

Before
will

bo g i v e n

presenting

P

(p

the
u(q

by

1

, ..., p

, ...,

utility

q -

(q^,

.

q

n

theorem,

q n ) be

a representative

disposable
1

Wald's

by W a l d ' s

method.

some

notations

first.

Denote
chased

functions

n

) be

income

} be

function.

an

consumer

a set

the

total

indicator

of

n goods

at a p e r i o d

the c o r r e s p o n d i n g
or

of

set

of

pur­

t i m e;

of p r i c e s ;

m

be

expenditure;
a well-defined

Given p and m at

a period

of

total
time,

the

^ F o r m o r e d i s c u s s i o n o f t h e a p p l i c a t i o n s of t h e
u t i l i t y f u n c t i o n s , se e Wald, o p . c i t . , p. 1 7 1- 7 5; a n d
G. J. S t i g l e r , T h e T h e o r y of P r i c e s T3rd. e d .; N e w Y o r k :
M a c m i l l a n , 196 67 ^ p p . T l - 8 3 .
3

F o r a c r i t i c a l e v a l u a t i o n of the p r o b l e m c o n n e c t e d
w i t h the e m p i r i c a l d e r i v a t i v e s of i n d i f f e r e n c e s u r f a c e s ,
se e W. A. W a l l i s a n d M. F r i e d m a n , "The E m p i r i c a l D e r i v a ­
t i o n of I n d i f f e r e n c e F u n c t i o n s , " in S t u d i e s in M a t h e m a t i c a l
E c o n o m i c s a n d E c o n o m e t r i c s , e d . b y 0~ Lanqe", e t al.
T c h i c a g o : U n i v e r s i t y P r e s s , 1 9 4 2 ) , pp. 1 7 5 - 8 9 .

96

or n e c c s ' v r v

firr+

subjected

to

the

budget

consumer purchases
3u , 1
— t /p
=
33

for m a x i m i z i n g

constraint

the q u a n t i t i e s

is

the

fulfilled

such

utility
if the

as:

3u . n
. . . = -^/p
3q

h K
[■ a

?.

condition

m

k= 1

^o.lvma
p u r c h a s e d as
period

oi

1

= f

functions

time w i t h

c h a s e d will

q

1 , i
(m),

depend

. . . q

These
loci
the

the e q u a t i o n s ,

only on

beionys

are

expansion
a certain

Assuming
function,
cally

thus,

shown

of

belonging

in C h a p t e r

For

a given

quantities

th a t

pur­

is.

curves

have

represent

which

To e a c h

En g e l

curves

tim e

space

( C).

of

an d

a

is c a l l e d

s y s t e m of

curves.

the

sam e

preference

c a n be d e t e r m i n e d e m p i r i ­

by o b ser vi ng

to d i f f e r e n t

income

the c o n s u m p t i o n
levels,

as

III.

Considering

the

, . . . , CT belonging
generally,

path
set

Engel

in e a c h p e r i o d

consumers

the

income;

En g e l

al l c o n s u m e r s
the

income.

pri c es ,

the

the q u a n t i t i e s

„n , .
= f (m)

functions

consumption

more

and

in the n - d i m e n s i o n a 1 q u a n t i t y

prices

of

of p r i c e s

constant

n

^no n e t s

to

consumption

ex pansion paths

to t h e p e r i o d s

the d i f f e r e n t

price

t j , . . . tT or,
situations.

If all

97

Fngel
C

curves

(t = 1,

points;

are

lin ea r,

. . ., T)

each

can

be d e t e r m i n e d

by

two of

path

its

say,

qt =

' * *' ^t ^ a

tor

the

tn d err'to q.
prices

consumption expansion

p,
b

- p

sake of

by
t

^t =

simplil ication,

n and
i.L <.

, the

^ t f * * *' ^ t ^ *

q

t,

symbols

by q

of

2t 1

it

.

is a d v a n t a g e o u s

Tor

p.,, 9 a n d
t■
’
“►

p

the

set
are

2L-1

of
also

used .
Considering
point

q^ and

terminal

Wald's T h e o r e m :
q n ) which
the

If

there

paths

an a r b i t r a r y
If
the p o i n t s

of

second

the

linea r

space

2,

C T , then

. .

formula

the

. .

initial
2T-1).

i n d i c a t o r u (q1 , . . .,
degree

determined

this

sa i d

the

in q
by

ove r

the

consumption

indicator

is

consumption expansion

proportionality

p aro

is

the

factor

and

constant.

considered,

F or

an

arbitrary

additive

vector coordinates

4

q fc (t = 1,

in S by the

from an

A^,

- qgq,. w i t h

exists

Cj , . .

uniquely determined
apart

v

vn
, asso ci ated with
*-1 1

1

following

point

dimensional

v,, . .

expansion

paths,

vector

is a p o l y n o m i a l

(2T-1)

vectors

the

proof,

c o o r d i n a t e s of

a n d A is an a r b i t r a r y

the

A^,

VGCtor

i n d i c a t o r as a

. .

constant,

function of

the

4
^2t - 1 :

see W a l d ,

o p . c i t . , pp.

146-53.

the

98

) = I

.T-l;

1

.
where

v

p

=

t-’

2T-1

t = 1

S=1

5.

2

!3
l
k=1

p

for

t = 0,

__

j.

k k
v
=
t s

t,s

> >

t s

2T-1

-t

;;

o-

fc=1

0

t

f

a

k
k k.
I
p.(q - q n >
k ^i
t
s 0
1,

. .

f’3,2(fjl,3/ p 3, 1*

wt =

2T-1;

s = 1,

. .

2T-1

= “

t ,s

w

(a

+ a

S ,t

the m a r g i n a l

s y s t e m of

for

Q,s

t,s

2

is

the

“ ,V2 , 3 (P1 ,2/ p 2 ,1 J

(wi p i,t " D o,t + p o , i )/pt,i

‘if
q) = W «- ,s
c
L '!
t Pit

prices

for 11 = 2 ........ 2T_1

t ' s = 1»

2,

2T-1

)

utility
p

Practically,

of m o n e y

(t - 0,

one

1,

wants

at

the

.

point q^

The

function

to h a v e

function

♦-rar.siormation of

u(q^,

1

as a

1

V l
n

q n ) can

4
+

1

1
2 V 2

A

•

to z e r o ,

be m a d e

• • +

n

the

indicator

.

. +

obtains

*2T-1^
as

= q

n

X^

= g

of
,

(q

the

2

, q

into

0

,

q

f ° rm

1

- <30
n

X 2t — 1 V 2 T - 1

q

of

follows:

1

T-1V 2T-1

the d e t e r m i n a n t

one

quantities

1
2

,

XfVi + X j ^2 + .

II

f ( X^ ,

X
1

of

under

2 T - 1).

I
utility

,

>

,t

, ° ° > 2 n0 . i + P 2 , 3 (o0 , l ~ P 0 f2 ?/° 2 , l “ p 3 t2 (p0 >l P Q , 3 ) / p 3,l
1

n

2T-1

“

q

these
1

2

, q

n
"

^0

equations
, ,..,

q

is

n, ^
) for

no t

equal

99

t ~ 1,

2,

2T-1.

i‘{-'■j f •••/
cator

of

X 2 ^_ ^ ) , one o b tains

the u t i l i t y
To

functions

by means

and

of

for

method

Engel
the

the se

X

u f q 1 , . ..,

f u n c t i o n as a

present Wald's

su c h m e t h o d o l o g y
space

Substituting

q r')f

of a p p r o x i m a t i n g
Figure

two p e r i o d s

of

time.

6

indi­
. ••, q n .
utility

roughly

three-dimensional

two c o n s u m p t i o n e x p a n s i o n p a t h s

bci.orujj.ng to

the

f u n c t i o n of q \

curves,

c a s e of

into

shows

commodity

and

5

c“

Figure

6

.- -A D i a g r a m S h o w i n g W a l d ' s M e t h o d , w i t h T h r e e D i me n si on al C o m m o d i t y Space and T w o C o n s u m p t i o n
Expansion Paths

For

a numerical

c o m m o d i t i e s , no.

1

illustration,

= dairy

products,

consider
no.

2

the

= fats

three
a n d oils,

C
F o r n u m e r i c a l i l l u s t r a t i o n s of W a l d ' s m e t h o d of
a p p r o x i m a t in g utility functions by means of Engel curves,
see Wald, o p . c i t . , pp. 1 5 3 -5 5; a n d T i n t n e r , E c o n o m e t r i c s ,
o p . c i t . , pp. 60-61.

100

no.
t2

3 = fruits,
- period

2

and

t wo p e r i o d s

pj

=

P J2

- 97<>,

1

where

. pj =

p^

period

P2

1

, p3 =

“

*970,

is the

the

time,

t^ = p e r i o d

1

,

as

these

thr ee

commodities

in

follows:^

;

p

price

1,

3

- 1.132;

i n d e x of

the

good

at

the

t—

.0009 in2

1. 4 5 9 +

.0029 m x

is g i v e n

by

at period
the

consumption expansion

set uf

En g e l

path

is

curves:^

=

3.798 +

.0005 m 2

q2

=■ 1.0 5 1 +

.0007 m 2

q

=

.0018 m 2

1.649 +

1

.

2,

following

q3

^For
that p e r i o d

the

.0011 m-^

+

And

3

1

following

3.67 a +

q 3 =_ .yoi
q3 =

of

oJ. time.

g i v e n by

-

i n d i c e s of

are g i v e n

At p er i o d

q^

two p e r i o d s

.

The p r i c e
the

the

the c o n s u m p t i o n
set of

Engel

expansion

path C 2

curves:

;

the p r i c e i n d i ce s , see T a b l e
is the b a s e d p e r i o d .

15,

p.

80.

Assume

2Fo r the E n g e l c u r v e s , see T a b l e 1 t h r o u g h T a b l e 5,
pp. 37-41.
In o r d e r to o b t a i n the q u a n t i t i e s p u r c h a s e d q
as a f u n c t i o n of m, the e s t i m a t e s of the r e g r e s s i o n c o e f f i ­
c i e n t s a re d e f l a t e d by the p r o p e r p r i c e i n d ic es .

101
V
q v is

where

the

good,
m^

is

per

capita

quantity

capita

disposable

On

the

per

two p o i n t s

C ] , the

income

points

£15 0

sponding

to

»5o E' P 1

='

p 2 -- p 3 li
o

^1 "
q2 -

the

on e a c h
qg

C

ft =

the

disposable

on

the

income

th
k—

the

at

period

function

by Wald's

,2 ) h a v e

to be

to

points

£160

and

the

chosen.
disposable

and
£1 7 0

t.

q-^ c o r r e ­

are

chosen.

p

's:

obtains

tli-

1,

1)

(.979,

.970,

1 . 1 32 )

(3.8430,

1 .0360,

1 .8940)

(3.8540,

1.0450,

1 .9230}

f7 .8 7 9 7 ,

1.1665,

1 .9448)

(3.8849,

1.1738,

1 . 96 33)

The

1

income

corresponding

S]6 Q , and

one

utility

and

and

Hence,

of

and

To approximate
method,

purchased

followinq

figures

are

the

values

of

t ,s

t
0
1
2

3

s

1

2

3

.0490
.0490
.0524
. 0524

.2181
.2 1 8 1
.2 2 0 2
.2 2 0 2

.24 9 0
.2490
.2 5 3 2
.2 5 3 2

T h e s e l e v e l s o f i n c o m e a r e a r o u n d the m e a n
observ ed per c a p i t a d i s p o s a b l e income; see A p p e n d i x
p p . 7 3-75.

of
F,

the

102

As
money

at

for

the

the p o i n t

values

of w^,

q t under

the

the m a r g i n a l
s y s t e m of

u t i l i t y of

prices

p t , one

obtains :
w^

- 1.0065

w

2

-

.9215

w

3

=

.9193
Tiic l o i i o w i n y

t

s

3

One would
Thus,

fun''Mr n as

of

a t , s ’s:

3

notice

-.0008
-.015?
-.0157

that

the v a l u e s

Hence,

of

a

. 5 (- . 000 2

t

A^

= a

f 5

<7
t,s

o ne can o b t a i n

a f u n c t i o n of

f ( X l ,?'2'? 3 ) =

= —

(n

) - n

indicator

1^ »

*3

as

A^

-

.0162

.0008

A ^

-

.0009

A 3 A3 -

.0157

A 3 X 2 )+

+

.2181

A2

+ .2490

A3

s,t

of

the

=
t,s

utility

foll ow s:

A1 A^ -

transformation

+ a

t,s

.0009

the

t,s

the

A ^ - .0152

to o b t a i n

. for all

S ^ i

2

-

A

3 -

.0157

.0490

.0008

AjA2

A2 A 3

Aj

+ A

indicator

as a f u n c t i o n of q u a n t i t i e s

following

-.0009
-.0157
- .0162

-

In o r d e r

the

the v a l u e s

-.0002
-.0008
-.0009

1

function

are

1

2

1,2,3.

figures

is made:

of

the u t i l i t y

purchased

q

1

7

, q*, q

3

,

103

.0110

+

.0367

A

+

.0419 >, = q 1

- 3 . 843 0

+

. 1 378 *

- 1 . 0360

.0 O 9 L

»
■x ^ .1305

A2

.0290

\1 +

X2

.0508

Since
equa l

+

. 06 9 3

the v a l u e s

- q2

A 3 - q 3 - 1.8940

th e d e t e r m i n a n t

to zero,

3

of

of

those

equations

X 2 , anc^

^3

are

is not
calculated

as :

A

=

287.2330

q1

-

58.9996 q 2 - 56.1575

X2

-

471.4688

q

-

63.0785 q 2 - 159.2569 q

A3

- -465.5462

1

q1

+ 70.8890 q 2

Substituting
gets
of

the

in

i n d i c a t o r of

f(A^,

the

q

3

3

- 661.292

- 212.207

+ 154.5865 q 3 + 454.696

A 2 ' * 3 ) f ° r *]»

utility

*2'

*3'

one

f u n c t i o n as a f u n c t i o n

q 1, q 2, q 3.
Wi t h an

can

be

an d

periods

As
no.

= fruits,
three
period

computer,

a p p r o x i m a t e d by e x t e n d i n g

T-periods

ties,

electronic

of

of

the

utility

number

to n - d i m e n s i o n a l

functions

of c o m m o d i t i e s

commodity

space

and

time.

for a n o t h e r
1

= dairy

numerical

illustration,

p r o d u c t s , no.

no. 4 = v e g e t a b l e s , no.

periods
3,

time

the

of

time,

tj_ = p e r i o d

are considered.

2

five

= fats a n d o i l s ,

5 = meat,
1

,t 2

=

commodi­
no.

3

etc., a n d
period

2

, t3

=

104

T he
three

Pi

price

periods

lf Pi

=

p4
2 = 1.013,
4

q1
+

=

by

+

q4

=

the

1,

.0018 m

3

q

+ . 0 0 0 5 m 2 ,q

+

1.954

.0025

, q4

=

=

.993,
k

commodities
1

follows:

*9 7 9 '

where

=
p2

is

=

p^

=

-9 7 0 ' P 2
.946 ,

1

in

the
2

, p^

=

1.07 6 ,

index

of

the

s e t of E n g e l

=

2

+

.901

by

m 3#
1.67 4

= 1.051

+

,q

by the

set

q2

+ .0007

+

+

is

+

5

1.649
.0 1 2 2 m 2 .

Engel

.0007 m 3 , q 3

.0017 m 3 , q

= 6.087

ni! .

curves:

m2, q3

s e t of

1.459

+ .0 1 1 2

Engel

= 5.855

5

following

= .955

of

=

3

= 5.993

5

.0012 m 2 , q

the

path

curves:

+ . 0 0 0 9 rr^ , q

. 0024

2

expansion

curves:

=
+

1.667
. 00 7 6 m
1

. i_

where

q

good,

and

period

is

the per

capita quantity

is

per capita d i s p o s a b l e

the

purchased

o n t he

income

k—

at

t.
On

disposable

, the p o i n t s
income

1

= 3 *1 3 2 '

p3 =

the price

consumption

2 is g i v e n

is q i v e n

= 3.389

five

given as

P2 =

the

1 . 779

=

these

t.

.0 0 1 1

.0018 m 2 , q 4

And

q1

3.798

I?

1.013;

st p e r i o d

-

are

following

+

.0029 m 1#

q1

=

of

1.022; p*

period

the

3.678

The

+

5

=

=

at p e r i o d
At

given

time

Pi
p|

- 1.1]3,

k — . q oo d

of

indices

$150 a n d

q^ and q ^
$160;

on

corresponding
C 2 * the p o i n t s

to
q2

the
and

;

,

105

q^

corresponding

and

on

the

disposable

points

income

Hence,

po = P1 =
P2 ~

J3
o

1!

P4 -

*1

=

to the d i s p o s a b l e

one

$160

a n d q,- c o r r e s p o n d i n g

$170

and

$180,

are

and
to

the

chosen.

1,

1,

1)

1,

;3 -

(. 979 ,

.970 ,

1 .J 3?,

3.013,

3 .022 )

Pj. -

( .99 3 ,

.946 ,

1 . 076 ,

1 .1 1 3 ,

1.013)

(3.8430,

1.0360,

1.8940,

2.1390,

7.6730)

(3.8540,

1.0450,

1 .923 0 ,

2. 1 6 3 0 ,

7 .7850)

q2 ~

(3.8797,

1.1665,

1 .9448 ,

2.1582,

7.8056)

q3 =

(3.8849,

1.1738,

1.9633,

2.1710,

7.9278)

(3.8175,

1 .0806,

1.9832,

1 . 9634 ,

7.3776)

(3.8427,

1 .0880 ,

2 . 0 01 8,

1 . 9804 ,

7.4536)

q4 "
q5 =

The

t

following

s

$170;

obtains

1,

C

income

figures

are

the

v a l u e s of

„ 1s :

1

2

3

2

.18 50
. 185 0
.1 9 1 2

. 3699
. 3699
. 3753

.5 358
. 5358
.5463

-.3627
-.3627
-.3608

-.2185
- . 2185
-.2129

3
4
5

. 19 1 2
.1909
. 190 9

. 3753
. 3705
. 3705

.5463
. 5405
.5405

-.3608
-.3820
-. 3820

-.2129
-.2340
-.2340

0
1

As for
money

at

o b ta i ns :

th e v a l u e s

the p o i n t

of w^,

under

the

4

p.

the m a r g i n a l

5

utility

system of prices

of

p t , one

106

1.0911

w] =

1 . 143b
II

S
CJ

W 2

1.2225
.7962

W4 =

.8650

w5 =

ihese
u t i l i t y of
income

s

values

2

3
4
5

The

s

1
2

3
4
5

is the

pe r

t ,s

ps a r e

as

lh.it

the m a r g i n a l

capita

disposable

.0168
.0 337
. 048 8
-.0330
-.0199

one

. 0337
. 05 9 3
.0889
-.0749
-.0495

4

are

the

v a l u e s of

A^,

the
A0 ,

u

t ,s

1s

;

5

- .0330
-.0624
-.0919
.0585
.0322

.0488
.0889
.1321
-.0919
-.0550

can obtain

-.0199
-.0250
-.0418
.0321
.0160

4

3

. 0337
. 05 93
. 0889
-.0624
-.0372

5

- .033 0
-.0499
- . 07 8 4
.0585
.0322

.0488
,0889
.1321
-.1055
-.0683

figures

a function of

follows:

3

2

1

as

a

2

following

T hu s ,
function

of

1

.0168
.0337
.0488
-.0330
- .0199

1

t

m o n e y dec:linos

t.e i n d i c a t e

increases.
The

t

l j<juic.‘b t e n d

indicator

of

-.0199
-.0372
-.0550
.0322
.0160

the

> 4 , A^ as

utility

follows:

107

f (> 1 , ' *o , > , , > , i r ) =

. 5 ( .01

6

R A^ +

.0 5 9 3 i 2

+ . 1 3 21 > 2

J

+ .0 5 8 5 A ^

In o r d f r
func’ t-i on

4
t] , and

.0110

A

as

+

.0160

+

.0337

+

.0488

A.X_

+

.0337

a

~

.0372

A2A5

+

? ■>A 4

“

.0550

Ay

0624

A^A-, -

.0199

A.A. 5 1

+

.0322

A 5 A 4 ) + .1850

+

.5358

A3

to o b t a i n

a function

-

>2

1 3

the

.0330

>

+

-

A, A
14

.0889

a

.04b8A3A1

A1 A2

-

.0199

A A

A -

.0624

a

A4

+

.0869

A 3A2

“

.0530

X4 A 3

.0919

A4 A 3 +

. 0 3 22

A4 A5

.0372

A-A, 5 2

.0550

Ac A
5 3

.3627

> 4

indicator

of q u a n t i t i e s

5

Ax +

-

of

.3699

.2185

A2

A

3 A

the u t i l i t y

purchased

q

1

2
, q

, q

3

,

r

q

, the

+

.0367

following

A2

+

transformation

.0419

A3

-

.0255

is made:

-

.0003

q1

.0090

1 5

X1 +

.1305

A2

+• .1375

A3

+

.0446

A4

+

.0520

A5

-

A5

=

3.8 4 30

-

q 2 - 1.0360

108

0290

X

+

.0508

X2

+

.06 93

> 3

+

.0892

+

.1078

q

0 24 0

X. +
1

.0192

^

A, +

.0 3 2 0
3

X, -

.1756
4

L

-

5

X5

=

- 1.89 4 0

3

.1586

X,

„ 4 _ 2.1390

112 0

+

.1326

X? +

.2548

X3 -

.2954

X

-

.2194

X5

q

Sinco
equal

to zero,

lated

as :

XL

-36.7916

A2

>

=

^

=

=

the d e t e r m i n a n t
the v a l u e s

q

q5

-.9020 q 1

+ 8 .2 2 6 8

- 1 7 . 72 8 4

q5

+

+ 17.5 04 5

q5

-57.0921 q

1

=

57.5593 q 1

+ 1 . 5990 q 5

these

equations

\ 2 , X 3 , X 4 , A5

q2

+ 52.9775

q

3

- 7.6730

5

is

not

are

calcu­

+ 49.8635

q4

- 27.9322

q2

+ 27.9770

q3

+ 46.2554

q

4

+ 20.9552

.7735 q 2

- 1 . 74 4 5 q 5

X5

- 6.0435

1

- 1 1 . 37 4 8

.0177 q 1

of

of

=

- 30.7245

q

3

- 44.8605

q

4

- 18.9673

+ 10.666

q2

+ 12.8541

q

3

+ 14.7763

q4

- 165.7879

- 11.5761

q2

+ 164.6116

- 9.0355

q

3

- 18.5843

q4

109

Substituting
' -j, ) q , > c , one

gets

as a f u n c t i o n of q
utility

function
The

least,

that

functions
theorem
cal

in f{

, q

3

, q

4

, q

5

the u t i l i t y

.

Of

complicated

course,

than

A^,

th is

the p r e v i o u s

approximate

determination

of

of

Engel
Mo r e

curves could

research on

is n e e d e d

i n t e g r a b i l i t y , and

of e q u i l i b r i u m ar e

needed

for

to bo
the

the

at

if W a l d ' s

f i e l d of e m p i r i ­

undertaken.

sufficient

further

one.

utility

be m a d e ,

this

A2 ,

function

indicate,

functions

of

, q

A j. ) for

illustrations

is a d o p t e d .

conditons

i n d i c a t o r of

A^,

numerical

by means

utility

2

is m o r e

above
the

1

the

A ^ , A 2 , A^,

The

condition

empirical

test before
9

any

economic

policy

3.
Besides
be p r o p o s e d
area,
to

recommendation

Areas

some

in t h i s

areas

Box-Cox model

Future

mentioned

section

in p a r t i c u l a r ,

supplement

for

c a n bo d r a w n .

for

c o u l d be

It is

to

linearity

of

an

area will

research.^

undertaken

study.

the

earlier,

future

the p r e s e n t
test

Research

an d
the

Engel

This

le a d d i r e c t l y
us e

of

the

curves.

Th e f u l f i l l m e n t o f the s u f f i c i e n t c o n d i t i o n of
e q u i l i b r i u m i m p l i e s that the e q u i l i b r i u m p o s i t i o n is the
m a x i m u m one.
T h e f u l f i l l m e n t of the i n t e g r a b i 1 ity c o n d i ­
t i o n s m e a n s t h a t t h e r e e x i s t s one a n d o n l y o n e i n d i c a t o r of
u t i l i t y s u c h that a l o n g the g i v e n c o n s u m p t i o n e x p a n s i o n
p a t h s the n e c e s s a r y c o n d i t i o n s for the e q u i l i b r i u m p o s i t i o n
ar e f u l f i l l e d .
F o r the e m p i r i c a l t e s t s of t h e s e areas, see
Wald, op. cit.
^ O t h e r p o s s i b l e a r e a s of r e s e a r c h a r e the s i m u l t a n e ­
o u s e q u a t i o n s , t h e d y n a m i c m o d e l s , the p r o j e c t i o n s of
h o u s e h o l d s ' e x p e n d i t u r e b e h a v i o r , the e s t i m a t i o n of c o e f f i ­
c i e n t s of e c o n o m i e s of s c a l e , etc.
For more specif ic areas
of r e s e a r c h , see Q u a c k e n b u s h a n d S h a f f e r , o p . cit. , pp. 46- 5 1.

110

Consider

y

i

the

following

function:

)

~

m . -

1

—

= r,k

where

is

m.

>

Bk

4

the

the
is

1

u ik

4

household

k^-

food

at

per c a p i t a

a period

th
i—
household

the

of

pe r

expenditure

on

time ,

capita

disposable

i

income

at

t>.y , tiy , a no
U.,
ik

is

tor

lyik

-

C

where

which

k

/ -

4

"k

-

■

it* --k
is a

y ik

which

6

*

>. ar e

1

, one

4

. k

P

- ..k

k

error

and

term,

4

1(

u

i

4

“ ik

k

1

regression

model.

one obtains

log

is a d o u b l e - l o g

functional

"

linear

In g e n e r a l ,
ferent

i

B, •+
k

A - 0,

+ ek

m

time,

obtains

+ V " !

;

of

parameters,

stochastic

simple

For

loq

the

a period

mi

+ u ik

regression model .

different

values

specification

of

of

the

A lead

to

regression

dif­
equation

Ill

This

allows

alternative

v

> - 1

l!a :

> *

and

its

along

one

to

test

the

h y p o t h e s i s .^^

linear

hypothesis

against

the

Formally,

1

To carry

out

standard

error.

with

the

the

other

test,

one

Obviously,

parameters

by

needs

an e s t i m a t e

> ran
the

be

of

1

estimated

maximum

likelihood

me thorl.
The

likelihood

function

for y ^ j , . .

yn^

is

y X _x

0.-1):: log y ik - | log 211 - £ l o g o 2 2o
n,X-l
-

-

The
found

with

standard

V - r r

appropriate

2
•

11

maximizing

values

an e l e c t r o n i c

errors

T. I (“ ^ -- )
i

can

of

A,

computer,

be e s t i m a t e d

by

^ , and
and

the

o** c a n

be

respective

reference

to t h e

information matrix.

F o r m o r e d i s c u s s i o n of thi s p r o b l e m , s e e G. E. P.
B o x a n d D. R. C o x , "An A n a l y s i s of T r a n s f o r m a t i o n s , "
J o u r n a l o f the R o y a l S t a t i s t i c a l S o c i e
, S e r i e s B, V o l . 26
(11)64) , p p . 2 1 1 - 4 3; a l s o , K m e n t a , o p . c i t . , pp. 46 7- 6 8 .
It s h o u l d b e n o t e d that the m a i n t a i n e d h y p o t h e s i s
c o u l d b e the d o u b l e - l o g f o r m , or Hq : A = 0 a g a i n s t H 2 : A ^ 0.

BI BL IO G R A P H Y

112

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