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Math preparation
In this chapter we deal with the movement of the macroeconomic variables, such as GDP, over
time. There are two ways of dealing with time in macroeconomics:
discrete time
and
continuous
time
. Discrete-time models treat the time as integers:
t
= 0
,
1
,
2
, ...
. Continuous-time models
treat the time as real numbers. Here we will consider a continuous-time model.
In a continuous-time model, we view a particular macroeconomic variable as a
function of
time
. For example, when we consider GDP, instead of calling it as one number
Y
, we consider it
as a function of time
t
,
Y
(
t
). In this way, we can capture the movement of GDP over time. For
example, sometimes we want to see how much GDP changes per unit of time. Mathematically,
the change of GDP can be viewed as the
slope
of
Y
(
t
) with respect to
t
. And we know that
the slope can be expressed mathematically as a
derivative
. Therefore,
the change of GDP per unit of time, measured at time
t
=
dY
(
t
)
dt
.
Since we use this expression a lot, we use the short-cut expression by defining
˙
Y
(
t
)
≡
dY
(
t
)
/dt
.
In usual mathematics, the use of
is more common as a shortcut for a derivative (i.e.
Y
(
t
)
≡
dY
(
t
)
/dt
), but a convention in growth economics is to use the ˙ instead. For example, if the
GDP is a linear function of
t
, that is,
Y
(
t
) =
at
(where
a
is a constant number), then the
change of GDP per unit of time is equal to
˙
Y
(
t
) =
dY
(
t
)
/dt
=
a
. If GDP is an exponential
function of
t
, that is,
Y
(
t
) = exp(
bt
) (where
b
is a constant), then the change of GDP per unit
of time is equal to
˙
Y
(
t
) =
dY
(
t
)
/dt
=
b
exp(
bt
).
Note that there is a economic meaning of
˙
Y
(
t
). When
˙
Y
(
t
) = 0, it means that the GDP is
not changing over time. When
˙
Y
(
t
)
>
0, it means that the GDP is increasing over time. When
˙
Y
(
t
)
<
0, it means that the GDP is decreasing over time.
Since we analyze economic growth in this chapter, we will have many occasions where we
look at the growth rate. The
growth rate
of a variable is defined as the change of the variable
divided by the level of the variable. For example, the growth rate of
Y
(
t
),
g
Y
, is defined by
g
Y
≡
change of
Y
(
t
)
level of
Y
(
t
)
=
˙
Y
(
t
)
Y
(
t
)
.
For example, when
Y
(
t
) =
at
,
g
Y
=
˙
Y
(
t
)
/Y
(
t
) =
a/
(
at
) = 1
/t
.
When
Y
(
t
) = exp(
bt
),
g
Y
=
˙
Y
(
t
)
/Y
(
t
) =
b
exp(
bt
)
/
exp(
bt
) =
b
. Therefore, when
Y
(
t
) is an exponential function of
t
, the growth rate of
Y
(
t
) is constant.
There is another, very convenient, way of looking at the growth rate. Before explaining it,
let’s refresh our memory of calculus a little bit.
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