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A Simple Method of Deriving Demand Curves

Author(s): E. J. BrosterSource: Journal of the Royal Statistical Society, Vol. 100, No. 4 (1937), pp. 625-641Published by: Wiley for the Royal Statistical Society

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1937] A

SimpleMi:ethod

f

Deriving

emand

Curves.

625

A SIMPLE METHOD

OF DERIVINGDEMAND CURVES.

By

E. J.

BROSTER.

To

manyeconomists,

nd

others

o whom

the

nature

of

the

demand

for

a

commodity

r

commodities s

of

interest

r

value,

the

assump-

tions and methods adopted by the mathematicalschool of statis-

ticians n the

derivation

of

demand

curves are

often

puzzling,

ome-

times

bewildering,

nd

occasionally

unconvincing.

The

principal

object

of

this

paper

is

to

suggest

a

simpler,

more

straightforward

method-less

refined

mathematically,

perhaps,

but

likely

to

give

more

satisfactory

esults-than those

usually applied

by

the

statis-

tician.

I

hope

to

show also

that when the

true demand

curve,

or

rather an arc

of the

true

demand

curve,

is

unobtainable

fromthe

observations, he constantelasticity unctions superior o thelinear

function

s a

guide

to

its

position.

It is

proposed

to illustratethe

arguments

by

actually

applying

the

simple

method to data which have

already

formed

he

basis

of

statistical

investigation.

And

there

seems

to

be

none

more

suited

to the purposethan those used by Professor chultz in deriving he

statistical

law

of

demand

for

sugar

in

the

United

States for

the

25

years

1890 to 1914.* For

he not

only

derived

a

demand curve

by

each

of

four different

methods-two

involving

the

use of

link

relatives

and

two the use

of trend

ratios-but he also

assumed

linearity of

the demand

functionwithin

the

limitsof the

observa-

tions,t

and criticized

Professor ehfeldt's

assumption that the

elas-

* Statistical aws ofDemand and Supply,1928.

t

The

problem

of

deriving

he

law

of

demand

for

sugar

reduces

to

the

problem

of

deriving

he

equation

of

the

'

best-fitting

straight

ine

(loc. cit.

p. 36).

It

is

evidentfrom he

scatter

diagram

Fig. 11)

and

from

he

usual

testsfor

inearity

f

regressionhat

the

regression

efore

s

[that

s,

the

demand

curvefor

ugar]

s

quite

inear

(loc.

cit.

pp.

57-58).

When

the

regressions

practically

inear,

s in

the

problem

t

hand,

the

use

of more

or

less

complex

equations

to

represent he aw

of

demand

may

give

mpossible

esults

(loc.

cit.

pp.

83-84). But

vide

also

p.

153,

n.:

.

.

.

the conclusions

eached

n

this

study

regarding

he

changes

n

the

elasticity

f demand

s

we

go from

ne

point

to another on the

demand

curve,

were

never

based

on

straight-line

emand

functionsonly. NeverthelessProfessorSchultz accepted the results of

straight-lineunctions

nly,

n

forming

ll

subsequentonclusions.







626 Miscellanea.

[Part

IV,

ticity of

demand

for

wheat was

constant

throughout

he observed

arc.*

The firstquestion to be answered is: What are we seeking?

Professor

chultz

deriveswhat he terms he dynamic

aw

of demand

from

which

we are supposed

to be

able

to determinenot

only

the

effect

n consumption

f a

given

change

n priceand vice

versa,

but

also the

normal trend

of

consumption

nd

price

fromyear to year.

However,

the trends

which he uses are-in his own words-

satis-

factory

only

within

the limits

of

observation. They

may

give

impossible

results

if extrapolated

beyond

these

limits (p.

93).

What purpose does a dynamic law of demand serve if we cannot

employ

t for

extrapolation-in

estimating he

demand next month

or next year, for

instance, at

any given price?

Purely

dynamic

equations

are of

no

practical

value unless

they

can

be

so

employed.

On the

other

hand,

thecoefficientf

the

elasticity

f

demand

for ny

price

within

a

given

range

has a number of

perfectly

egitimate

practical

uses.

Statistical

methods

of

deriving

demand

curves

usually

involve

theelimination f theseculartrend n bothpriceand quantityseries.

Where

the

response

to

price

change

is

quick,

or where

the secular

trend

n the

price

series s slight,

herecan

be no objection

to such

a

procedure,

except

that it

may

limit

the

range

of observations

to

a

very

narrow

one.

But where

the

response

s slow and the secular

trend

in the

series

steep,

only

a

part

of the effect

f short-period

price

movements

s reflected

n

the

short-period

hanges

of

con-

sumption,

with

the result

that

the

calculated

coefficient

f

demand

elasticity s less than the true coefficient.As we nearlyalways find

it impossible

to determine

whether he

response

to

price

change

is

quick

or

slow,

and

as a wide

range

of

price

observations

s

always

preferable

o

a

narrow

ange,

t

is

better

from hese

points

of

view to

use

a methodthat

does not involve

the

elimination

fseculartrend

in

the price

series.

II

Measurable

disturbing

lements

are

changes

in the real value

of

money ffectinghepriceseries, nd the trendofpopulationaffecting

the

quantity

series.

Although

there

are rather serious

objections

to

adjusting

the

price

series

by

reference

o

an index

of

wholesale

prices,

t

is

proposed

to

use the

adjusted

data

compiled

by

Professor

*

Loc. cit.,

p.

211.

See

also R. A.

Lehfeldt,

Elasticityof

Demand

for

Wheat,

Economic

Journal,

Vol. 24, pp.

212 et eq.

It

is not intended hat

this

paper

be regarded

s an

appraisal

or criticism

f

Professor

chultz's

well-known

work.

Ifthere

s

any

criticism

t

all,

it s evelled

t the

mathematical

tatistical

methods

fderiving

emand

curves

n general

use, in

so far

as theysometimes

necessitate ssumptions hat are of questionablevalidity,and sometimesby

their

omplexity

onceal

the truth.







1937] A Simple

Method

of

Deriving Demand

Curves.

627

Schultz

(loc.

cit.

Appendix II,

Table

IA,

p.

214). These

are repro-

duced

in Table I

below.

The first tep is the eliminationfrom he consumption eriesof

that part of its

secular

trend

which is attributable

to

extraneous

factors.

As there

s no

evidence to

suggest

that

during the period

under review

the

rate of

changevaried

appreciably,

we may assume

for

the purposes

of a

first

approximation

that

if the

price

had

remained

constant

the

quantity series

would

have

been in geometric

progression

part

from accidental

variations

about the

norm.

In

order to determine he

common

ratio

of this trend,

t is

necessary

first o plot the two seriesshownin Table I, one against the other,

TABLE I.

The Per Capita

Consumption

nd theReal

Price of

Refined

ugar

in

the United

States

for

each Year

1890 to

1914.

Per

capita

Real

price

per

lb.*

Year.

Per

capita

Real

price

per

lb.*

Year. caniptita

ets

er

consumption.

cns

cs

.

lbs.

1890 52 8 6-643 1903 70 9 4 703

1891

66 3

5-061 1904

75-3

4

840

1892

63-8

5

053

1905 70

5 5

331

1893

64-4

5-484

1906

76-1

4-422

1894

66-7

5-209

1907 77-5

4

313

1895

63-4

5-171

1908 81-2

4-803

1896

62-5 5-901

1909

81 8

4

285

1897

64-8

5

863

1910

81-6

4-294

1898

61-5

6-183 1011

79-2

5

009

1899

62-6

5-720

1912

813

4

441

1900

65

2 5-727

1913

85-4

3

730

1901 68-7 5-574 1914 84 3 4-166

1902

72-8

4-626

*

Real price

money price

divided

by

Bureau

of

Labour Statistics

index number

of

wholesale

prices

'all

commodities,' verage

1900-1909

1-00.

prices

on the

ordinate

nd

quantities

on

the abscissa.

It is observed

from

his

that

the

points

for

the later

years

lie to the

right

of

those

for

earlier

years-from

which it follows

that the common

ratio

exceeds unity. Now, from he Table it will be seen that the prices

for

1891,

1892 and

1911

approximate

relatively

closely

one to

the

other,

but

that the

consumption

n 1892 was

probably

ubnormal.

If

the

consumptions

n

1891 and

1911 were

normal,

the rate of

in-

crease

between

the

two

years

would

give

the

general

trend

due

to

extraneous

factors.

The common

ratio, R,

is

1-009,

from

log

R

frlog

9t2i-

log

66c3

BEy

etermining

hetrend romhisratio

ommencing

ith

nity







628

Miscellanea.

[Part IV,

in

the base year,

which

for our

purpose we

take as

1890, we

have

1890

R1 1000

1891 R1' 1-009

1892

R2 1-018

1914

R24- 1-230

Y

I I

I

I

I

65 *90

B

6-5A

Curve A-y

=

1311

-

01192x

Curve B-xy0 647 189-7

*98

60

6

.9~1

4

~~~~~~0

99

,5.5

-0L

0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~0

4-5

a

12

o

.

0~~~~~~~~~~~~~~~~~t

?10w09

40

B

*,A

35

I

I

1

E

50 55

60

65 70

75

x

Per

capita

consumption-lbs.

FIG.

1.-Scatter

diagram

of

the

real

price

n

each

year

plotted

against

the

per

capita

consumption

orrected or

the trend

due

to

extraneous

factors,

showing

he

best-fitting

ine,

Curve

B.

and

by

dividing

these

into

the

per

capita

consumptions

hown in

Table I, we have the serieswiththeassumedtrenddue to extraneous

factors

removed.

We

now test the

accuracy

of

the assumed

trend







1937] A

Simple Method, f

Deriving Demand Curves.

629

by

replotting,

ut the scatter

diagram

shows that

whereas in

the

first

iagramthe points for he

later years

ie

to

the

right f

those

for

earlier years, they now lie almost as far to the left. (The con-

sumption

n

1891 was

subnormal,

or

that for

1911

was

abnormal.)

A second

trial based

on a

common

ratio

of

i

oo5 appears

to

give as

perfect n

alignment s it is

possible to obtain.

The

resulting

eries

is

given

n

Table II and the

scatterdiagram

n

Fig. 1.

To

show

the

degree

of alignment

n

the

diagram,

he

year

is

given by

each

pair of

co-ordinates,*

TABLE

II.

The Per

Capita Consumption f

Refined ugar

in

theUnitedStates-

The

Secular Trend due to Extraneous Factors and

the

Adjusted

Per

Capita Consumptions.

(Common

ratio, R,

=

1.005.)

Per capita Per

capita

Secular rend

ue

consumption

Secular rend ue

Consumption,

Year.

to

extraneous

trend

emoved.

Year. to extraneous

trend

emoved.

factors.ROt-1)

t.

factors.

(t-1).

Rtl

lbs. lbs.

1890

1 000 52-8

1903

1-065 66-6

1891 1-005 66-0

1904 1-070

70 3

1892

1 010 63-2

1905

1-076

65-5

1893 1-015 63-5

1906 1-081

70 4

1894

1-020 65-4

1907 1-086

71-3

1895 1-025 61-9

1908 1-091

74 4

1896

1 030 60 7

1909 1-097 74 6

1897 1-035 62 6

1910

1-102

74 0

1898

1-040 59 1

1911

1-108

71-5

1899 1-045 59-9 1912 1-114 73 0

1900

1-050 62-1 1913 1-119

76-3

1901

1-055 65-1

1914 1-124

75 0

1902

1-060 68-7

It

is

clear that the general

course is curvilinear. It is equally

clear that a

curvecannotbe fitted

ntil greater egree

of smoothness

is

obtained. A process of averaging the

observations suggests

itself.

Apart possibly froma

tendency to push the curve slightly

*

The

correlation

etween the two series

s

high,

r

being

-

093

?

0-02.

According

o

Professor

chultz, he

correlationoefficient

or he real

price eries

and the

uncorrected er

capita

consumption

eries s

-

090

j

0

03,

which

s

considerably

higher

than

the

corresponding oefficient

or his

series of

link

relativesor trend

ratios,

he

highest

f

which s

-

0-80

?

0-05

see

pp.

78 and

80). He

explains p.

78) that

the

highcoefficientf

0f90

?

0-03

s

the

result

of spurious

correlation

attributable o

such factors s the

growing

aste

for

ugar.

Such

extraneous

actors,

owever,would

cause a

great

horizontal

spread

of

the

co-ordinates,nd

thereby onduce o

lower,

not

higher,

orrelation.

The

higherfigure btained after

the correction or

extraneousfactors

of

the

consumptioneries sthereforeonsistent ith xpectationsnd nconsistent ith

Schultz's

arguments.







630 Miscellanea. [Part IV,

to the right, uch

a course

seems

perfectly egitimate. There are,

however, wo difficultieso be considered.

First, the nature of the price data may not permitof averaging.

The underlyingmoney rice

in each

year

does

not appear to be the

average in the sense that price multiplied by consumptionequals

the total annual outlay of

consumers. The

figures re the average

annual wholesale prices at New

York

of refined ugar

(loc.

cit.

p. 35). But it is not

clear

whether hey are weighted. Probably

they

are

not, although

Professor chultz's methods

uggest hey

are.

To what extent the

basic

money price

in each

year differs rom he

average calculated from the total outlay divided by the total

consumption annot be determined, ut the error s likely o be small

unless the

price

fluctuations rom

week

to week or monthto

month

were arge. We may

in

any

case assume

that the weighted verage

real price in any two or more years

is

more accurate than the

unweighted verage.

Secondly,

f

we use the

averages

of

groups

of

years

n

chronological

order, the arc obtained

will

be too short to be

of value,

and

the

tendencyof the averagingprocessto push the curve to the rightof

its true position will

be

strengthened. Again, if

we

group the

observations in ascending or descending order according to the

magnitudeof the real prices, he implied assumption

will be

that all

deviations are horizontal-that is, that they lie in the consumption

series-while if we group

them

n

consumption rder, he assumption

will

be that all deviations are to

be

found n the price series. The

effect fusing one or the otherofthe two

series s a basis

forgrouping

is similar to the effect f obtaining the regressionof X on Y or

Y on X.

Neither assumption is

valid.*

Normally, the price of

a

com-

modity

ike

sugar is, cet. par.,

a

function f the

supply,

while con-

sumption, or rather the actual demand,

is

a

functionof

the

price.

Where the relation

between

actual

demand and

price

is

being

considered,

he latter

s the

independentvariable,

and

therefore

ny

*

That

is, neither

s valid so

far as we know. The

importance

f

having

accurate basic data scarcelyneedsemphasizing. It is thereforeurprisinghat

Schultz

devotes so little

pace

to

a

description

f

their ources

nd

nature,

nd

especially

to

the method

of

obtaining

the

average price

in

each

year.

The

exact

significance

f either

eries

s not at all clear.

Apart

from he

possibility

of small

changes

n

the stocks

held

by consumers,

he

term

consumption

s a

synonym

f

demand.

In

so

far

as Schultz

derives he

law

of

demand-which

is a

function

f

price-from

these

data,

he

implies

that

the

figures

f

con-

sumption

re the

quantitiesdemanded.

In so

far as he

regardsconsumption

rather han

price

as the

independent ariable,

he

implies

hat

it

represents

he

supply,

of

which

price

is

a

function.

However,

as it is

necessary

o

confine

differences

n

the results btained o

the

differences

n

method,

he

accuracy

of

thedatamayremain nchallenged. At the sametime,wemust cceptSchultz's

dicturn

o the

effect hat

the

series re

mutually nterdependent.







1937]

A

Simple

Method of

Deriving Demand

Curves.

631

deviation or

error

s horizontal. We know,

however,

hat the

prices

under review

are subject

to error,as they

are

probably

not the

weighted averages; while in any case we cannot accept the con-

sumption

series as either

a dependent

or

an independent

variable

(loc.

cit. p. 35).

We

may overcome

these difficulties

y proceeding

as follows:

1. Arrange

the two

series in ascending

order

according

to real

price,

multiply

one by

the other, and thus

obtain

the real

outlay,

xy,

n

each year

(trend due to extranieous

actorsremoved).

2.

Calculate

the five-yearmoving

averages

of the

x series nd the

xy series,and divide the latter by the former o obtainthe five-year

moving

averages

of the

real prices.

3.

Rearrange

the

x series in descending

order

of magnitude,

calculate

the

five-year

movingaverages,

and place

this new

series

in

juxtaposition

to the

moving

averages of the

price,

y, series,

without

hanging he order,

egardless

f

whether he years correspond

or

not.

This

method

preserves

the

curvilinearity-if

any-of the

func-

tional relationship between the two series without placing the

burden

of the

deviations

entirely

n one or the other,

and

tends to

remove the residue

of

the trend

of consumption

attributable

to

extraneousfactors

t only

a slightrisk of

pushingthe

curve

from ts

true

position.*

By applying

t

to the

data

in

Table

II,

we

obtain

the

two

series

shown n

Table III.

III

A

scatter

diagram

of the

two series n

Table

III

is

given

n

Fig.

2.

For

the purposes

of

identification

nd comparison

the co-ordinates

are numbered to

2I,

as

in the table.

Each

pair

of

co-ordinates

represents

mixed

bag for

both series,

nd

a glance

at the

years

columns n the table suffices

o show

that the

obvious

curvature

s

not due to an

error

n the common

ratio of

I-005

used

for

removing

the trend

n

the

consumption

eries. We

must

necessarily

nfer

hat

the demand curvefor sugar in the United States duringthe period

1890-1914

was not

linear-in

spite

of

Professor

chultz's

protesta-

tions to the contrary.

But if

we cannot

accept

the

assumption

of

linearity,

hen

what

form does the equation

to the

curve

take?

Experience

in other

*

Where

he

secular

rend n

the

price

eries

s

negligible,

his

process

s

sufficient

n itself o remove

he

whole

f the

extraneous

rend n the con-

sumption

eries.

But

theresultingemand

urve

will

relate

o the

mid-year

observationsnlyfthey elate o consecutiveears; andweare eftwithoutthemeans fdeterminingtspositionnanyother ear.

z2







632

Miscellanea.

[Part IV,

fields of enquiry indicates

that the coefficient

f the elasticity

of

demand forany commodity

or service

normallyvaries

directlyas

the price-which is a characteristic f the linear function. On the

other hand,

general reasoning

upportsthe notion of curvature-of

upward concavity,

which s a characteristic

f the graphof

a constant

elasticity unction.

As

the

arc

we require

s clearly oncave upward,

TABLE III.

Five-Year Moving Averages

of the Per

Capita Consumption

Trend

Removed)

and the Real

Prices of RefinedSugar

in

the

United

States,from

890

to

1914, determinedfrom

he

Two

Series arranged

in

order

f

Magnitude.

eyto Real price per pouad.

Per

capita

consumption

trenid

Key

to

Raprcpepon.removed).

graph.

(Fig.

2).

Year. Cents. Year.

lbs.

_

I

_

1913

-

1913

1914 1914

1

1909

4-16

1909

74-9

2

1910

4-29

1908 74-2

3 1907 4-35 1910

73-5

4

1906

4-42

1912 72-8

5 1912 4.50

1911

72-0

6

1902

4-60

1907

71-3

7 1903 4-68

1906

70

4

8

1908

4-80

1904 69-5

9

1904

4-88

1902

68-4

10

1911 4-95 1903

67-4

11

1892

5-02 1891

66-4

12 1891 5.10 1905 65-7

13 1895 5-17 1894

65-1

14 1894

5-25

1901

64-5

15 1905 5-35

1893

64-0

16 1893

5-46

1892 63-3

17 1901

5-57 1897

62-7

18

1899

5-68

1900

62-1

19 1900

5-75

1895 61-5

20 1897

5-88

1896 60-7

21 1896

6-04

1899 58-9

1898

_

1898

_ 1890 1890

we conclude that it lies

between ts chord

and the arc

described

by

the constant elasticity function

based

on the same two

points of

intersection. The problem

now resolves

tself nto one of deciding

whether

the

chord or the arc of constant

elasticity

s the

closer

approximation

o the true

demand curve.

Line

A (Figs. 1 and 2)

is the chord of

an arc of the

true

demand

curve drawn so that it intersectshe curveat or near thelimitsofthe







1937] A

SimpleMethod

f

Deriving

emand

Curves.

633

Y

I

I

I

C

6-5

B\

Curve

A-y

=

13-11

01192x

Aw \

Curve

-xy0

647

189-7

Curve

C-xy05

= 148 8

650

21

20

19

17

o

5

o0

-

o

~~~~~~~~~~(0

4.5

4.0

CA

55

60

65

70 75

x

Per

capita

onsumption-lbs.

FiG.

2.-Scatter

diagram f the

five-year

verages

given

n

Table

III

showing

the

best-fitting

ine,

Curve

B,

and a

representative

ine

having

an

elasticity

coefficient

f 0-5,

Curve

C.

observations.

The

equation

to

the curve of constant

elasticity

akes

the

form

xyn

k

where n

is the

coefficient

f the

elasticity of

demand of

which

the

mathematical

oncept

s

*

y

x

x

dy

The

equation

may

be stated n

inear

orm

logx

=

log

k

-

n log

y

*

See Marshall, rinciples,

Note IV of the

MathematicalAppendix.







634

Miscellanea.

[Part

IV,

It follows hat the best means of

determining he values of n and k

is

to use the

logarithms f the series n

Table III.

The

equation to

the best-fittingtraight ine of these logarithmic eries s

log

x

=

2-278

06471 log y

The coefficient f the elasticityof

demand is therefore -647i, and

the value of

the logarithm f the constant,k, is

2'278,

the

equation

we

require

being

xyO

47

189.7

Curve B, Figs.

1

and 2 is the graph of this

function. It

is clear

from

Fig.

2

that

this curve provides as perfect a fit as any

other

con-

ceivably

continuous curve whosef (x) is

positive. It follows

that

although he

coefficientf the elasticity f demand may vary directly

as the price,

ts sensitiveness o price change s so slight

hat

it

may

be assumed to

remainconstantfor ll

practical purposesthroughout

the range of

observations.*

The

equation to the chord,Curve A,

is

y= 1311

-

01192x

fromwhich we have

ydx

y

xdy

0-1192x

From this we

obtain the following

alues of n-one near each

end

of the

observedrange of prices and one

midway:-

Real

price,

y.

Elasticity

Cents

per

lb.

coefficient,

.

4 350 0 4964

5 125 0

6416

5900 08179

Judgedby

a

linear

function, herefore,he coefficientf the elasticity

of demand is

highly sensitive to price

change. The ratio of the

highest o the

lowest of these prices s

I-36;

the ratio of the corre-

sponding lasticity oefficientss

i-65.t

*

Incidentally,the constant

elasticity

functionprovides a

form of

are

elasticitywhich has no superior. It conforms o the acceptedmathematical

concept

nd itsvalue provides

clue to the

pointelasticity

f demandespecially

for

shortarcs. It

conformslso to the

three essential

conditions

equiredof

any

form f arcelasticity. (See

R. G. D.

Allen,

The

Concept fArc

Elasticity

of

Demand.

Review fEconomic

tudies,

Vol. I, pp.

226-7.)

t

Where t is

safe to

suppose

that the constant

lasticity

unction

pplies,

the

averagingprocessmay be

omitted, nd

the demand

curveobtained direct

from

he

best-fittingtraight ine

of the

logarithms f the corrected er

capita

consumption nd

the real price

series. For the rest, s

the constant

lasticity

function

s a closer

pproximation

o the trueequation

than the

inear

function

in most

fnot all

cases, therewill

be lesstendency or he

arc to be

pushed from

its truestaticpositionfor hebase year ifthe averagingprocess s carried ut

onl

he

basis of the

geometric

nstead of thearithmetic

mean.







1937]

A

Simple Method of Deriving Demand Curves.

635

IV

As I have already pointedout,if we cannotuse a dynamic aw of

demand for he purpose of

estimating uture emand at a givenprice,

or

for estimating

he

price

that

will

dispose of

a

given supply,

t is

of

no

practical value.

However, extrapolation

n

some cases

may

be

permissible,

o that a

dynamic

aw would be useful. It

may easily

be derived.

The base year adopted

in

eliminating the

trend

of

demand

attributable to extraneous factors

is 1890.

The equation

to

the

demand curveconstitutes he staticlaw ofdemandin thatyear. In

eliminating

hat trend we

used

a

common

ratio, R,

of

o005,

so

that

Xt

x= x(1.005)(t-1).

.

(1)

where

xl

is the

per capita consumptionreading

on

Curve B

for

a

given price, and xt the per

capita consumption

n

the tth

year

for

the same

price.

For

changes

n

price

n the base

year

we have

xlY0=647

189.7

or xi

=

189.7y-0?647

. . .

(2)

By substituting

his

for

x1

n

Equation (1)

we

have a

dynamic

aw

of

demand,

viz.:

xt 189.7y-0O647(1005)(t-l) (3)

forestimating er capita

demand

in

the tth

year

at

a

given price; or

I

{189.7(1.005)(t-1)}

647

for

estimating

he

price

in

the tth

year

that

will

dispose of

a

given

per eapita supply,xt.

When

the

eliminated trend

of

'consumption

is

a

geometric

progression,

nd

the

equation

to

the

arc

of

the

demand

curve takes

the constant

elasticityform,

he

dynamic

aw

of

demand as

a

literal

equation

is

Xt kynR(t ).(3a)

SkR(t-1))

n

or

y

=

xt

I*}*.(4a)

Normallyby

far

the

most

mportant

xtraneous

actor

s

the trend

of net income.

It

follows hat

when

statistical estimates

of the net

national income are

available-as

they

are now

for

the

United

Kingdom

*-a

comparison

between the common

ratio

of

the

trend

*

See CohnC1ark, he National

ncome nd

Outlay.

Estimates

re

givenfor

each year 1924 to 1935 inclusive. Some information or1936 is givenby the

same

author n

the

Economic

Journal, 937, pp.

308

et

seq.







636

Miscettanea.

[Part IV,

of consumption nd

the change

fromyear

to year in the total

net

income

may constitute

soundbasis for he

derivation

f an equation

expressingthe total demand for the commodityor service as a

function

of

the total net income.* By

embodying

this equation

instead

of a commonratio of trend

n the dynamic

aw,

we are likely

to obtainmore accurate

results.

In

order

to demonstrate

he

use,

and

presumably to show the

accuracy,of

his

equations,

Professor

chultz

estimatesfrom hemthe

price

in 1914

that

would

in

effectdispose

of

a

supply

equal

to

the consumption

n that

year.

Against

the

actual

money price

of

4-683centsper lb., he obtains by the two firstmethods the prices

of4-539

cents

and

4-533

cents-a difference

n

each case of 3 per cent.

By the

two other

methods

he obtains the

real

prices

of 3'947 cents

and

4'I23

cents

compared

with the actual

real

price of

4'I66 cents-

a

difference

n

the

former

f

5 per

cent.

and

in the, atter

of per

cent.t

These

differences

re

remarkably

mall. But if

it was indeed his

purpose

partly

to

show the

degree

of

accuracy

achieved,

then it

should

have

been

pointed

out that

the data

for

1914

conform

more

closelyto thenorm ndicatedby the best-fitine than those ofother

years.

If

we take

the

other

extreme,

we

find that the data

most

distant

from

the

norm

are those for

1908.

For

the

per capita

consumption

f

8I

2

lbs.

in that

year,

the estimated

price

based

on

his

best

method

1

s

4-232

cents,compared

with

the

actual

real

price

of 4-803

cents-a

difference

f

3 per

cent.

Let

us

compare

these resultswiththose obtainable

from

quation

(4)

above.

For

xt

substitutethe

actual

per capita

consumption

n

1914 (that is, 84.3) and fort substitute

25-1914

beingthe twenty-

*

It

is

worth

noting

here

that family

xpenditure

n

any

commodity

s

a

Ainear

unction

f the total

family

ncome.

See

Allen

and Bowley,

F,amily

Expenditure.

In

determining

nd

using

such

an

equation,

the

investigator

ill

be

confronted

y

a

number

of

difficulties,

ut

they

are not

insurmountable.

See,

for

instance,

my

article

on

Railway

Passenger

Receipts

and

Fares

Policy,

Economic

Journal,

ept.

1937.

In

this

nvestigation,

or given

real

-price,

l,

the

relation

between

he

real

expenditure

f consumers nd the

real

national

income

takes

the

form

xoy1

aIo

+ .

When the

real

national

income

s

I,

we

have

Xtyi

alg+

P.

Then

Xty1 xoy1

a(It

=

I),

and

a(-It

-.(i))

Xt

xO

+ al-1) . . . . . . (i)

Yi

Since

x0yn

k,

xo0

ky.

Substitute

his

for

xo

n

Equation

(i).

Then

xt

=

ky-

+ a(lt

-o)

.

. .

. .

(ii)

which

s

the

law

of

demand

expressing

he

quantity

demanded

as

a

function

both

of

price

and

income.

t

Loc.

cit.,

pp.

42,

60,

74

and

86.

1

Vide

Equation

(15),

loc.

cit.,

p.

85,

the

best

method

eing

hat

based on

the

trend atiosofadjusteddata,whichgivethehighest orrelationr

=-080

i

0-05).







1937]

A Simple

Method f

Deriving emandCurves.

637

fifth

year

counting

the base year, 1890,

as

the first-so

that

t

-

1

=

24.

Theni

log

Y

0 647

(log

189-7

+

24

log

1P005

log

84

3)

0-6249

y

=

4216 (cents per lb.).

The difference

etween

this and the actual

real price

of 4 i66 cents

in 1914

is

i

per cent., the estimate

being

as good as

the best of

Schultz's. For 1908

the

estimate based

on my Equation (4) is

slightlymore accuratethan that obtainablefrom chultz's equation.

His estimate

s 4-232

cents and

mine 4-264,comparedwith

the actual

real price of 4-803 cents.

In this

case also

the data for 1908 are the

most distant from

he norm. (See

Fig. 1.)

In

order o determine

he demand

n

1914

at the givenreal price.

substitute

-i66 for

y in Equation

(3) above.

Then log xt log

189-7

0647

log 4-166

+

24

log

1-005

and

.

. x

85-0 (lbs. per

head of

population).

Between this

and the actual consumptionof 84-3 lbs. there is a

difference

f per cent.

These

figures

how that no matter

which

method we use, the

margin

f possible error nvolved

does

not fall

far hort

of 5 per cent.

They

also

indicate that even

without the elaboration

of Equations

(3a)

and

(4a)

now

made possible

by

the

publication

of

comparable

statistics

of

the national

ncome, quations

of

that

type

will

generally

give

at least

as

accurate

results

as

those derived

by

much more

complicated methods. It should be emphasized that the small

proportional

differences etween

the

estimates

and the actual

figures

n 1914

are in

all

cases

accidental.

The

data for

that

year

conform

ery

closely

to the norm.

(See

Fig. 1.)

V

The

implied

assumption

of

the

foregoing quations

is

that

the

elasticity

of

demand

for

ny given

price

remained

constant

through-

out the periodofobservation. Therearegoodreasonsfor upposing,

however,

that

for

any commodity

he

elasticity

of

demand varies

with

the

passage

of time.

So

far

as I

am

aware,

a

satisfactory

method of determining

he trend has not

yet

been devised. Even

its

direction-upward

or downward-is uncertain. As

distinct

rom

such

a

special

factor

as

the invention

of

a

substitute,

which

would

tend

to

cause the

elasticity

coefficiento

rise,

the

principal

normal

continuous

actor

s the trend

f

per capita

ncome.

This

stimulates

change in each directionthroughtwo different hannels. First,







638

Miscellanea.

[Part IV,

in the words of Mr.

R. G.

D. Allen

and

Professor

Bowley,

. . . the

price

elasticity

of

demand

is

only

modified

by changes

in the

substitution actoras income changes. It is to be expected,more-

over,

that substitution

ecomes

more

easy formost goods

as

income

rises.

The

larger expenditure

s

spread

over

a

wider range

of items

and the possibilities

f

substituting

ther tems for

a

given

tem are

thereby

ncreased.

It follows

hat the elasticity

of demand

forany

item with respect

to changes

in

its price

is

likely

to

increase with

income.

Demands tend

to become

more elastic

as

the

income

level

rises.

*

Secondly,

with

rising

income the marginal

utility

of money tendsto fall,so that for given real price,the commodity

becomes

cheaper

in

the eyes

of the

consumer. A

fall

in

the

marginal utility

of money

stiinulates

the production

of

new kinds

of commodities

nd

services, he

elasticity f

demand

for

which,

s

a

whole,

is above unity.

As that

for

all commodities

and services

remains onstant

t

unity, t follows

hat

as incomerises he

elasticity

of demand

for

old

commodities

nd

services

tends

to

fall.

What

is the net effect? As

I have

already

stated,

no satisfactory

answer has yet been given; but as the variation, in whichever

direction

t

may be,

is

necessarily slight,

it

is

a

safe assumption

that

over

a

period

of

even

such

a

length

as

25

years,

the elasticity

of demand

remains

constant.

The coefficient

we have

derived-

that is, o647-may

be

applied

equally

to 1890

and 1914

as

to the

average

of the

period.

Special allowance,

however,

should

always

be made when

a

substitute

has

been

rapidly

developed

during

the

period

of

observation.

Professor Schultz's interpretationof his demand curves is

interesting.

It suggests

that

the

coefficient f

the elasticity

of

demand

is a functionof deviations-

rom

the

long-period

norm

of

consumption-that

in

effect

he

coefficient

emainsconstant

so

long

as

consumptionconforms

to the

norm

indicated by

the trend,

regardless

f

the xtent o

whtch

his

trend s influenced y

secular

price

chavnge.t

As this

is

necessarily

an

underlying

ssumption

of

the

methods

employed,

ts

subsequent

proof begs the

question.

It is,

in any case, invalid. While it is true that the periodicvariation

in

the elasticity

of

demand

for

a

commodity

s

partly

a

function

f

changes

in

consumption

attributable

to non-price

factors

only,

there

s,

as I have

shown,

no reason

for upposing

that it

is

entirely

*

Family Expenditure,.

125.

t

Loc.

cit.

pp.

66 and 91. For

example,

when

the

consumption

rendratio

is 1-0, he coefficient

s

0-51,when

t is 0-9,the coefficients 0-67. When

the

consumption or ny

year is

'

normal,'

that is to

say, when t is equal to

that

indicated

by the trend

of consumption

or he same year . .

. then n

=

0-5

(p. 66). I have alteredthe symbolto conform o my own, and shownthe

coefficient

f elasticity

with positive

nstead ofa negative ign.







1937] A Simple Method of Derivinig

emand

Curves. 639

so. As the trend

of sugar prices

duringthe period was downward,

Schultz's

assumption

involves

a

rising

demand

elasticity for any

given price with the passage of time. But the Allen-Bowley

substitution

factor

was

scarcely applicable

to refined

ugar,

and

the other factor,

the'

diminishingutility of

money,

would

have

caused

a

falling

demand

elasticity.

The four

methods

used

by

ProfessorSchultz

yield remarkably

similar results,

the

average

coefficient

f

the

elasticity

of

demand

ranging

between o46

and

o052,

with

an

all-round

average,

as he

himself uts it,

of

o

5.

In orderto illustrate he difference

etween

this last figure nd that ofo647, CurveC, witha constantelasticity

of

o

5,

is shown

n

Fig.

2. The

equation

to this curve

s

xy05

=

148-8

The difference,

hich is

appreciable,

requires explanation.

In

the

first lace,

at

least

a

part

of

the

difference

s

attributable o

Professor

Schultz's

elimination

f

the ultimate

effects n

consumption f price

change.

In the second

place,

it is not at all certain whether

the

average of

o

5

is

the coefficientf

elasticity

t

any point

on

the arc

of

the

true

demand

curve. The

representative oint

on

the

best-fit

straight

ine

of

either

he

link relatives

or the trendratios

s

obscured

by

the

wide

scatter

of

the

observations.*

From

Schultz's

results,

the

only

safe

conclusion

we

can

draw is

that

the

coefficient f

demand

elasticity ay

somewhere

etween

o03

and

o-7.t

VI

Comparedwith mathematicalmethods,the principaladvantage

of

the

simple

method

outlined above

is

that the

slope

and

position

of any point

on

the

resulting

urve

are

determined

y

the

observa-

tions,

and

not

by

an

arbitrarily

elected curve that

conforms

o,

or

is

a

part of,

mathematical

cheme. If

it

fails

to

reveal the

equation

to

the true

demand

curve,

at least

it

gives

us

the next

best

thing:

the coefficient f

the

elasticity

of

demand

for

any

price

within

the

limits

of

the

observations.

And

for

the observed

arc,

it

indicates

whetherthe linear function s preferableor not to the constant

elasticity

unction,

r

to

some

other unction

uch

as

Moore's

typical

equation

to the

law of

demand.

I

The

fundamental

objection

to the

simple

method is

the

means

adopted for

ascertaining

the trend

in

the

quantity

series

that

is

*

Loc.

cit.

Figs.

5, 11,

15

and

22.

t

Loc.

cit.

Cf.

pp.

66 and

91.

1

See Schultz,

oc. cit.,p.

56,

and

H. L.

Moore, Elasticity

of

Demand

and

Flexibilityof Prices, Joutrnalfthe- merican tatisticalAssociation,March

1922.







640 Miscellanea.

[Part IV,

attributable

to non-price

factors-the need for assuming

that it is

in geometric rogression.

It is merely means

of surmounting

he

principaldifficultyfderiving emandcurves from ime series. But

whether we

assume the trend to be

in

geometric

or arithmetic

progression,

he error nvolved, f any, s

to a great extent

eliminated

in the subsequent process

of

smoothing.

In

turn,

the

process

of smoothing

may

call

for

criticism n the

groundthat

it is crude. Its simplicity

s appreciated;

the only real

objection

to it

is the

danger

that, ies

in

the

averaging of averages.

Apart from

he

population

factor

n the

consumption,eries,which

s

negligible n this respect,the danger has been avoided. Another

possible

objection

s

the

use

of

moving

verages for

a

purpose

not

in

keeping

with he

normal

tatistical

bject

of

this

device.

My purpose

herehas been to preserve

he

maximumnumberof observations,

nd

thus

to

reduce the

risk

of

producing

inaccurate results.

Curve

fitting

s

largely

a imatter f

personal udgment-more

or less as the

degree of alignment

o which

we

reduce the

observations

s

low

or

high.

One important conclusion we have arrived at is that the

coefficient

f

the

elasticity

f

demand for

ugar

n

the

United

States

during

he

years

1890-1914

was

to

all

intents nd

purposesthe

same

for

all

prices

within

the

limits

of

the

observations. Can we

derive

from

his

a

law

of

general

application?

Prima

facie,

the law of the

elasticity

f demand

may

be

expressed

in

the following

erms: although

the coefficient

f

the elasticityof

demand

tends

to

vary directly

s

the

price of

the

commodity,

he

variation is slight,so that to all intents and purposes it remains

constant during

small but finite

price

movements. It

is

not

clear,

however,

what

constitutes the limit

to

a

small

change

in

price.

Refined ugar,

the

subject

of

the

illustration,

as two

characteristics

that

have

an

important

earing

on the

problem:

it is

a

necessary

of

life,

and it therefore

tands

high

in

the scale

of

urgency;

and it

does

not

possess

a

substitute

f

any

consequence.

These character-

istics

account

for

he

inelasticnature

of

the demand

for

t.

If it werea luxury inwhichcase itwould be in competitionwith

other uxuries),

or if

there

were

good

substitutes vailable at

similar

prices,

its coefficient f

demand

elasticity

would

be

high.

Other

thingsbeing equal,

with

every

decline in its

price

it

would

in

the

former

ase

fall

in the

scale

of

esteem and

ultimately

become first

comfort

nd

later

a

necessary;

and

in

the

latter,

ts

position

would

approach

more

and

more

to one

of perfectmonopoly.

In

either

case the

demand

for

it would become less

elastic,

the coefficient

fallingrapidlyat.firstbecause the marketwould stillbe influenced

by

competition;

but with further

rice

fall, competition

would

at







1937] A SimpleMethod f

Deriving emandCurves. 641

length disappear,

and

with

t

the agency throughwhich movements

in

its price affect he elasticity fthe demand for t.

For this purpose, therefore,he limit of a small change in price

must be judged by

the

elasticityof

demand. Where t is high, t is

more sensitive o price-change han

where t is low. It will be seen

in Table

III

that the highest

observed average price

was

6-o4 cents

per

lb.

and

the lowest

41i6

cents

per

lb.

The ratio

of

the

former

o

the latter is 1 45. For commodities he

demand for which has an

elasticity

coefficient

f

not

more

than, say,

o

65,

we may therefore

state the law of the elasticity of demand

in

more concrete

terms,

viz. for practical purposes the coefficientf the elasticity fdemand

remains

constant during any

change

in

price

where

such

change

expressed

as a

ratio

of

the

higherprice

to the lower does

not

exceed

I45. The higher the coefficient,he smaller is the proportional

change

in

price for

whichwe

may

safely

ssume constant

elasticity.

a simple method of deriving demand curve jstor.pdf

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