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

Part One
Single-Equation Regression Models
Thus, for small changes,
(ln
X
t
−
ln
X
t
−
1
)
≈
(
X
t
−
X
t
−
1
)
X
t
−
1
=
relative change in 
X
6A.4
Growth Rate Formulas
Let the variable 
Y
be a function of time, 
Y
f (
t
), where 
t
denotes time. 
The instantaneous (i.e., a
point in time) rate of growth of 
Y
, 
g
Y
is defined as
g
Y
=
dY
/
dt
Y
=
1
Y
dY
dt
(29)
Note that if we multiply 
g
Y
by 100, we get the percent rate of growth, where 
dY
dt
is the rate of change
of 
Y
with respect to time.
Now if we let ln 
Y
=
lnf(
t
), where ln stands for the natural logarithm, then
d
ln
Y
dt
=
1
Y
dY
dt
(30)
This is the same as Eq. (29).
Therefore, logarithmic transformations are very useful in computing growth rates, especially if 
Y
is a function of some other time-dependent variables, as the following example will show. Let
Y
=
X
·
Z
(31)
where 
Y
is nominal GDP, 
X
is real GDP, and 
Z
is the (GDP) price deflator. In words, the nominal GDP
is real GDP multiplied by the (GDP) price deflator. All these variables are functions of time, as they
vary over time.
Now taking logs on both sides of Eq. (31), we obtain:
ln 
Y
ln 
X
ln 
Z
(32)
Differentiating Eq. (32) with respect to time, we get
1
Y
dY
dt
=
1
X
d X
dt
+
1
Z
d Z
dt
(33)
that is, 
g
Y
=
g
X
+
g
Z
, where 
g
denotes growth rate.
In words, the instantaneous rate of growth of 
Y
is equal to the sum of the instantaneous rate of
growth of 
X
plus the instantaneous rate of growth of 
Z
. In the present example, the instantaneous rate
of growth of nominal GDP is equal to the sum of the instantaneous rate of growth of real GDP and
the instantaneous rate of growth of the GDP price deflator.
More generally, the instantaneous rate of growth of a product is the sum of the instantaneous rates
of growth of its components. This can be generalized to the product of more than two variables.
In similar fashion, if we have
Y
=
X
Z
(34)
1
Y
dY
dt
=
1
X
d X
dt
−
1
Z
d Z
dt
(35)
that is,
g
Y
=
g
X
−
g
Z
. In other words, the instantaneous rate of growth of
Y
is the difference between
the instantaneous rate of growth of
X
minus the instantaneous rate of growth of
Z
. Thus if
Y
=
per capita
income,
X
=
GDP and
Z
=
population, then the instantaneous rate of growth of per capita income is
equal to the instantaneous rate of growth of GDP minus the instantaneous rate of growth of population.
Now let 
Y
=
X
+
Z
. What is the rate of growth of 
Y
? Let 
Y
=
total employment, 
X
=
blue collar
employment, and 
Z
=
white collar employment. Since 
ln(
X
+
Z
) 
=
ln 
X
+
ln
Y
,
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Chapter 6
Extensions of the Two-Variable Linear Regression Model
187
it is not easy to compute the rate of growth of 
Y
, but with some algebra, it can be shown that
g
Y
=
X
X
+
Z
g
X
+
Z
X
+
Z
g
Z
(36)
That is, the rate of growth of a sum is a weighted average of the rates of growth of its components.
For our example, the rate of growth of total employment is a weighted average of the rates of growth
of white collar employment and blue collar employment, the weights being the share of each compo-
nent in total employment.