The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

FIGURE 6.3
Constant elasticity
model.
10
The elasticity coefficient, in calculus notation, is defined as (
dY
/
Y
)
/
(
d X
/
X
)
=
[(
dY
/
d X
)(
X
/
Y
)]
.
Readers familiar with differential calculus will readily see that 
β
2
is in fact the elasticity coefficient.
A technical note:
The calculus-minded reader will note that 
d
(ln
X
)
/
d X
=
1
/
X
or 
d
(ln
X
)
=
d X
/
X
,
that is, for infinitesimally small changes (note the differential operator 
d
) the change in ln
X
is equal
to the relative or proportional change in 
X
. In practice, though, if the change in 
X
is small, this rela-
tionship can be written as: change in ln
X
=
.
relative change in 
X
, where 
=
.
means approximately.
Thus, for small changes,
(ln
X
t
−
ln
X
t
−
1
)
=
.
(
X
t
−
X
t
−
1
)
/
X
t
−
1
=
relative change in
X
Incidentally, the reader should note these terms, which will occur frequently: (1) 
absolute change,
(2) 
relative
or 
proportional change,
and (3) 
percentage change,
or 
percent growth rate.
Thus, (
X
t
−
X
t
−
1
) represents absolute change, (
X
t
−
X
t
−
1
)
/
X
t
−
1
=
(
X
t
/
X
t
−
1
−
1) is relative or
proportional change, and [(
X
t
−
X
t
−
1
)
/
X
t
−
1
]100 is the percentage change, or the growth rate.
X
t
and 
X
t
−
1
are, respectively, the current and previous values of the variable 
X
. 
11
A constant elasticity model will give a constant total revenue change for a given percentage change
in price regardless of the absolute level of price. Readers should contrast this result with the elasticity
conditions implied by a simple linear demand function, 
Y
i
=
β
1
+
β
2
X
i
+
u
i
.
However, a simple linear
function gives a constant quantity change per unit change in price. Contrast this with what the log-
linear model implies for a given dollar change in price.
12
Concerning the nature of the bias and what can be done about it, see Arthur S. Goldberger, 
Topics
in Regression Analysis
, Macmillan, New York, 1978, p. 120.
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Chapter 6
Extensions of the Two-Variable Linear Regression Model
161
In the two-variable model, the simplest way to decide whether the log-linear model fits
the data is to plot the scattergram of ln
Y
i
against ln
X
i
and see if the scatter points lie
approximately on a straight line, as in Figure 6.3
b
. 
A cautionary note: 
The reader should be aware of the distinction between a percent
change and a percentage point change. For example, the unemployment rate is often
expressed in percent form, say, the unemployment rate of 6%. If this rate goes to 8%, we say
that the percentage point change in the unemployment rate is 2, whereas the percent change
in the unemployment rate is (8
−
6)
/
6, or about 33%. So be careful when you deal with
percent and percentage point changes, for the two are very different concepts.
13
Durable goods include motor vehicles and parts, furniture, and household equipment; nondurable
goods include food, clothing, gasoline and oil, fuel oil and coal; and services include housing, elec-
tricity and gas, transportation, and medical care.
EXAMPLE 6.3
Expenditure
on Durable
Goods in
Relation to
Total Personal
Consumption
Expenditure
Table 6.3 presents data on total personal consumption expenditure (PCEXP), expenditure on
durable goods (EXPDUR), expenditure on nondurable goods (EXPNONDUR), and expendi-
ture on services (EXPSERVICES), all measured in 2000 billions of dollars.
13
Suppose we wish to find the elasticity of expenditure on durable goods with respect
to total personal consumption expenditure. Plotting the log of expenditure on durable
goods against the log of total personal consumption expenditure, you will see that 
the relationship between the two variables is linear. Hence, the double-log model may be
appropriate. The regression results are as follows:
ln EXDUR
t
= −
7.5417
+
1.6266 ln PCEX
t
se
=
(0.7161)
(0.0800)
(6.5.5)
t
=
(
−
10.5309)*
(20.3152)*
r
2
=
0.9695
where * indicates that the 
p 
value is extremely small.