# Growth

```Growth
Consider any variable π which depends on time, π‘. Then we can write π(π‘). Then the rate of change of
π(π‘) is simply is the time derivative:
ππ
= π ′ (π‘).
ππ‘
Based on this notion of “change”, we can also define growth:
πΊπππ€π‘β ππ π = ππ =
π ′ (π‘)
.
π(π‘)
Why is this the definition of the growth rate?
Example: Suppose that GDP per capita in Turkey is \$10K this year. If the GDP per capita will rise to
\$11K next year, then
ππ = πΆβππππ ππ ππ’πππ‘ππ‘π¦ = 11 − 10
ππ‘ = πΆβππππ ππ π‘πππ = 1
π = ππ’πππ‘ππ‘π¦ = 10.
Therefore, if the definition is correct, then we should see:
ππ
1
π ′ (π‘)
ππ‘
=
= 1 = 10%.
π(π‘)
π
10
Funny fact: If you take the derivative of ln π(π‘), then the answer is
ππππ(π‘) π ′ (π‘)
=
.
ππ‘
π(π‘)
That is because, if you take the derivative of an expression in logarithms, then you should first take the
derivative of the original expression and divide the derivative by the original expression.
Remark: If you want to calculate the growth rate of any variable, then you can calculate the derivative
of its logarithm.
Application: Assume that π(π‘) = π΄(π‘) &times; π΅(π‘). Then the growth rate of π can be calculated as
ln π = ln π΄ + πππ΅.
Take the time derivative of both sides to find the growth rate of π:
ππ = ππ΄ + ππ΅ .
Solow’s Model
Consider an economy where the production technology is Cobb-Douglas:
π = (πΎ)π (π΄πΏ)1−π
where π is output, πΎ is capital, πΏ is labor, and π΄ is the average (per worker) information or knowledge.
π΄ can also be interpreted as “technology”. In fact, π΄πΏ is called “effective labor”.
We assume that the growth rates of π΄ and πΏ are constant, denoted by ππ΄ &gt; 0 and ππΏ &gt; 0. This means
technological growth and population growth are exogenous. Now let us see the growth rate of πΎ. The
rate of change in πΎ is
ππΎ
= π &times;π−πΏ&times;πΎ
ππ‘
where π  ∈ [0,1] is the saving rate, and πΏ ∈ [0,1] is the depreciation rate. To find the growth rate of πΎ,
simply divide the rate of change by πΎ to obtain
ππΎ
ππ‘ = π = π  π − πΏ.
πΎ
πΎ
πΎ
In real life
π
πΎ
≈ 1/3.
Definition: In economics, long-run is when endogenous variables are constant.
Example: In Solow’s model, the growth rate of πΎ is endogenous so it would be constant in the longrun.
Theorem: ππΎ = ππ in the long-run.
Proof: First note that ππΎ is a constant in the long-run. This means
π
π
−πΏ
πΎ
is a constant. However, by assumption, π  and πΏ are constants. Deduce that
π
πΎ
can be neither increasing nor decreasing over time. This means
ππππ π‘πππ‘ =
π
πΎ
ln(ππππ π‘πππ‘) = ln
π
πΎ
π
dln(ππππ π‘πππ‘) π ln πΎ
=
ππ‘
ππ‘
0=
π
π ln πΎ
ππ‘
=
π(πππ − ln πΎ) ππππ ππππΎ
=
−
= ππ − ππΎ .
ππ‘
ππ‘
ππ‘
Conclude ππ = ππΎ . End of proof.
As a consequence, the following result ensues:
Theorem: ππ = ππ΄ + ππΏ in the long-run.
Proof: Recall that
π = (πΎ)π (π΄πΏ)1−π .
Therefore, take the logarithm of this expression to see:
ln π = π ln πΎ + (1 − π)(ln π΄ + ln πΏ).
Now take the time derivative to find the growth rates:
ππ = πππΎ + (1 − π)(ππ΄ + ππΏ ).
Since ππΎ = ππ in the long-run, this expression simplifies to
ππ = πππ + (1 − π)(ππ΄ + ππΏ )
ππ − πππ = (1 − π)(ππ΄ + ππΏ )
(1 − π)ππ = (1 − π)(ππ΄ + ππΏ )
ππ = ππ΄ + ππΏ
which is the desired result in the long-run. This completes the second proof.
Our final result is this.
Theorem: ππ = ππ΄ in the long run.
πΏ
Proof: Note that the growth of GDP per capita is
dln π/πΏ dln π dln πΏ
=
−
= ππ − ππΏ = ππ΄ + ππΏ − ππΏ = ππ΄ .
ππ‘
ππ‘
ππ‘
Conclude
ππ = ππ΄ .
πΏ
This completes our final result.
```