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Math Help - How long does it take real GDP per person to double?

  1. #1
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    How long does it take real GDP per person to double?

    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.
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  2. #2
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    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.
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  3. #3
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    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!
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  4. #4
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    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)}

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  5. #5
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    Re: How long does it take real GDP per person to double?

    Quote Originally Posted by SpringFan25 View Post
    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.
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