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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.
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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.
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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
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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
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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.
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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.
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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
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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
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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.
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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.
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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.
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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).
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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,
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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.
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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.
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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
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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.
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