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: ππ = π ′ (π‘). ππ‘ Based on this notion of “change”, we can also define growth: πΊπππ€π‘β ππ π = ππ = 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 ππ = πΆβππππ ππ ππ’πππ‘ππ‘π¦ = 11 − 10 ππ‘ = πΆβππππ ππ π‘πππ = 1 π = ππ’πππ‘ππ‘π¦ = 10. Therefore, if the definition is correct, then we should see: ππ 1 π ′ (π‘) ππ‘ = = 1 = 10%. π(π‘) π 10 Funny fact: If you take the derivative of ln π(π‘), then the answer is ππππ(π‘) π ′ (π‘) = . ππ‘ 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‘) × π΅(π‘). Then the growth rate of π can be calculated as ln π = ln π΄ + πππ΅. Take the time derivative of both sides to find the growth rate of π: ππ = ππ΄ + ππ΅ . Solow’s Model Consider an economy where the production technology is Cobb-Douglas: π = (πΎ)π (π΄πΏ)1−π where π is output, πΎ is capital, πΏ is labor, and π΄ is the average (per worker) information or knowledge. π΄ can also be interpreted as “technology”. In fact, π΄πΏ is called “effective labor”. We assume that the growth rates of π΄ and πΏ are constant, denoted by ππ΄ > 0 and ππΏ > 0. This means technological growth and population growth are exogenous. Now let us see the growth rate of πΎ. The rate of change in πΎ is ππΎ = π ×π−πΏ×πΎ ππ‘ where π ∈ [0,1] is the saving rate, and πΏ ∈ [0,1] is the depreciation rate. To find the growth rate of πΎ, simply divide the rate of change by πΎ to obtain ππΎ ππ‘ = π = π π − πΏ. πΎ πΎ πΎ In real life π πΎ ≈ 1/3. Definition: In economics, long-run is when endogenous variables are constant. Example: In Solow’s model, the growth rate of πΎ is endogenous so it would be constant in the longrun. Theorem: ππΎ = ππ in the long-run. Proof: First note that ππΎ is a constant in the long-run. This means π π −πΏ πΎ is a constant. However, by assumption, π and πΏ are constants. Deduce that π πΎ can be neither increasing nor decreasing over time. This means ππππ π‘πππ‘ = π πΎ ln(ππππ π‘πππ‘) = ln π πΎ π dln(ππππ π‘πππ‘) π ln πΎ = ππ‘ ππ‘ 0= π π ln πΎ ππ‘ = π(πππ − ln πΎ) ππππ ππππΎ = − = ππ − ππΎ . ππ‘ ππ‘ ππ‘ Conclude ππ = ππΎ . End of proof. As a consequence, the following result ensues: Theorem: ππ = ππ΄ + ππΏ in the long-run. Proof: Recall that π = (πΎ)π (π΄πΏ)1−π . Therefore, take the logarithm of this expression to see: ln π = π ln πΎ + (1 − π)(ln π΄ + ln πΏ). Now take the time derivative to find the growth rates: ππ = πππΎ + (1 − π)(ππ΄ + ππΏ ). Since ππΎ = ππ in the long-run, this expression simplifies to ππ = πππ + (1 − π)(ππ΄ + ππΏ ) ππ − πππ = (1 − π)(ππ΄ + ππΏ ) (1 − π)ππ = (1 − π)(ππ΄ + ππΏ ) ππ = ππ΄ + ππΏ which is the desired result in the long-run. This completes the second proof. Our final result is this. Theorem: ππ = ππ΄ in the long run. πΏ Proof: Note that the growth of GDP per capita is dln π/πΏ dln π dln πΏ = − = ππ − ππΏ = ππ΄ + ππΏ − ππΏ = ππ΄ . ππ‘ ππ‘ ππ‘ Conclude ππ = ππ΄ . πΏ This completes our final result.