# How long does it take real GDP per person to double?

• Dec 6th 2011, 06:29 PM
jwxie
How long does it take real GDP per person to double?
Quote:

Japan's real GDP was 525 trillion yen in 2009, and 535 trillion yen in 2010. Japan's population was 127.6 million in 2009 and 127.5 million in 2010. Calculate

a. The economic growth rate
b. The growth rate of real GDP per person.

c. The approximate number of years it takes for real GDP per person in Japan to double if the real GDP economic growth rate returns to 3 percent a year and the population growth rate is maintained.
Okay.

Answer to A is [(535-525)/525] * 100 = 1.9%

Answer to B is "Growth rate of real GDP - Growth rate of population".

The growth rate of population is [(127.5-127.6)/127.6] * 100 = -0.07837%.
Hence, the answer is 1.9% - (-0.07837%) = 1.97837% per person.

I need help with solving part c.

My attempt:

Use Rule of 70. First we have to recalculate the growth rate of real GDP per person since the economic growth rate is different.

Growth rate of real GDP per person = 3% - (-0.07837%) = 3.07837%

So the years it takes is 70 / 3.07837 = 22.74 years

Is this correct?

Thanks.
• Dec 7th 2011, 05:45 AM
SpringFan25
Re: How long does it take real GDP per person to double?
your method for b and c is based on approximations so its hard to comment. If those methods are allowed by your teacher then your answers are fine.
• Dec 7th 2011, 08:08 AM
jwxie
Re: How long does it take real GDP per person to double?
Hi SprinFn25,
Thanks.
Do you have another way to solve b and c?
Thanks!
• Dec 7th 2011, 08:36 AM
SpringFan25
Re: How long does it take real GDP per person to double?
without approximations:

part b
The growth in GDP per capita is:
$\frac{\frac{535}{127.5}}{\frac{525}{127.6}} -1 \approx 0.019846872$

part c
The growth in GDP per capita is:
$g = \frac{1.03}{\frac{127.5}{127.6}} -1 \approx 0.03080$

an exact formula for the doubling time is:
$\frac{\ln 2}{\ln(1 + g)} \approx 22.844$

derivation of the above formula is:
Spoiler:

suppose annual growth rate g, and current value is X. We want to find the doubling time (t)
this will satisfy the following equation:
$X(1+g)^t = 2X$

cancel the X's
$(1+g)^t = 2$

take logs
$t \ln(1+g) = \ln 2$

$t = \frac{\ln 2}{\ln(1+g)}$

• Dec 9th 2011, 03:39 AM
dofdiamond
Re: How long does it take real GDP per person to double?
Quote:

Originally Posted by SpringFan25
without approximations:

part b
The growth in GDP per capita is:
$\frac{\frac{535}{127.5}}{\frac{525}{127.6}} -1 \approx 0.019846872$

part c
The growth in GDP per capita is:
$g = \frac{1.03}{\frac{127.5}{127.6}} -1 \approx 0.03080$

an exact formula for the doubling time is:
$\frac{\ln 2}{\ln(1 + g)} \approx 22.844$

derivation of the above formula is:

Spoiler:

suppose annual growth rate g, and current value is X. We want to find the doubling time (t)
this will satisfy the following equation:
$X(1+g)^t = 2X$

cancel the X's
$(1+g)^t = 2$

take logs
$t \ln(1+g) = \ln 2$

$t = \frac{\ln 2}{\ln(1+g)}$

Thanks for sharing, this is perfect method.