Economics Stack Exchange is a question and answer site for those who study, teach, research and apply economics and econometrics. It only takes a minute to sign up.
Sign up to join this community
Given this question:
National income is increasing by 1.5% a year and population by 2.5% a year. What is the rate of growth of per capita income?
Attempt:
Since per capita income is GDP/ population. I divided 1.5 by 2.5 and got 0.6. Is this right? Thanks.
Consider: If national income is increasing at a slower rate than population growth, then intuitively per capita income will be falling. Here is a set-up for the rate of decline in per capital income.
$$\text{per capita income}_t = \frac{\text{GDP}(1.015)^t}{\text{population}(1.025)^t} \text{per capita income}_{t-1}$$
This is not correct. It will be a bit complicated to derive this formally, so I will provide a rule of thumb. For a time dependent variable that is a function of a ratio of two other time dependent variables: $$ y(t)=\frac{A(t)}{B(t)} $$ the growth rate of $y(t),$ call it $\tilde{y}=\tilde{A}-\tilde{B}$ where the RHS is the difference between the growth rates of $A(t)$ and $B(t)$. You can apply this to your question.
Here is the calculus that you can make. I note $\tau_{GDPc} $ the rate of change of the per capita GDP, $\tau_{pop}$ the one for the population and $\tau_{GDP}$ for the GDP rate of change.
A rate of change for any variable A is
$$ \tau _A = \frac{A(t)-A(t-1)}{A(t-1)}$$
t being the moment considered, $(t-1)$ the previous moment for the calculus of the rate of change. So knowing $GDPc = \frac{GDP}{pop}$ you can write: $$ \tau_{GDPc} = \frac{\frac{GDP(t)}{pop(t)} - \frac{GDP(t-1)}{pop(t-1)}}{ \frac{GDP(t-1)}{pop(t-1)}}$$ With the definition of the rate of change, you write $$pop(t)= pop(t-1)*(\tau_{pop}+1) $$ You can then simplify the $pop(t-1)$ term in the previous equation and you get
$$\tau_{GDPc} = \frac{1}{\tau_{pop}+1}\left[\frac{GDP(t)-GDP(t-1)}{GDP(t-1)}-\tau_{pop}\right] $$ $$\tau_{GDPc} = \frac{1}{\tau_{pop}+1}\left[\tau_{GDP}-\tau_{pop}\right]$$
It is indeed different than Brandon marcus answer.
We can make a try with simple numbers:
The rate of chage of the per capita income will be $\frac{0,99024-1}{1} \simeq -0,975 \%$.
With the formula I gave you:
$$\tau_{GDPc} = \frac{1}{1,025}(0,015 - 0,025) = -0,975 \% $$ Seems to work.
Rate of growth of per capita GDP is defined as the difference between the rate of growth of GDP and the rate of growth of population as Per Capita GDP = GDP/Population. So, the growth rate of per capita GDP = 1.5% - 2.5% = -1.0%
