Notes: In some country the average annual growth rate of GDP was 5% in each of the first 42 years, and in each of the next 35 years was 6%. By how much...

Sunday, November 25, 2012

In some country the average annual growth rate of GDP was 5% in each of the first 42 years, and in each of the next 35 years was 6%. By how much...

Hello!

If a constant annual growth rate $\displaystyle{p}%$ of GDP is known, the rule of 70 gives an approximation of how many years n it will take to double GDP. Specifically,

$\displaystyle{n}\approx\frac{{70}}{{p}}.$

As you understand, actually $\displaystyle{{\left({1}+\frac{{p}}{{100}}\right)}}^{{n}}={2},$ take the natural logarithm of both sides and obtain $\displaystyle{n}{\ln{{\left({1}+\frac{{p}}{{100}}\right)}}}={\ln{{2}}},$ or $\displaystyle{n}=\frac{{{\ln{{2}}}}}{{\ln{{\left({1}+\frac{{p}}{{100}}\right)}}}}.$ For small $\displaystyle{p}$ 's it is approximately $\displaystyle\frac{{{\ln{{2}}}}}{{{\left(\frac{{p}}{{100}}\right)}}}=\frac{{{100}{\ln{{2}}}}}{{p}}\approx\frac{{70}}{{p}}.$

In our problem, $\displaystyle{p}_{{1}}={5}%,$ so GDP will double after about $\displaystyle\frac{{70}}{{5}}={14}$ years. It will double $\displaystyle\frac{{42}}{{14}}={3}$ times, i.e. will be multiplied by $\displaystyle{2}\cdot{2}\cdot{2}={8}.$

After that, $\displaystyle{p}_{{2}}={6}%$ and GDP will double after about $\displaystyle\frac{{70}}{{6}}=\frac{{35}}{{3}}$ years. Thus GDP will double $\displaystyle\frac{{35}}{{{\left(\frac{{35}}{{3}}\right)}}}={3}$ times and will be also multiplied by $\displaystyle{8}.$
The combined growth will be $\displaystyle{8}\cdot{8}={64}$ times (using the 70 rule). Actually it will be $\displaystyle{{\left({1}+\frac{{5}}{{100}}\right)}}^{{42}}\cdot{{\left({1}+\frac{{6}}{{100}}\right)}}^{{35}}\approx{60}$ times.