In: Economics

# Consider the following data on real GDP per capita in: Year Per Capita Real GDP 1950...

Consider the following data on real GDP per capita in:

 Year Per Capita Real GDP 1950 14 339 1960 17 351 1970 23 790 1980 30 732 1990 35 868 2000 43 288 2010 46 406 2011 47 554 2012 47 741 2013 48 066 2014 48 780

a) Calculate the percentage growth rates in real GDP per capita in each of the years 2011 through 2014, from the previous year.

b) Now, instead of calculating the annual percentage growth rates in the years 2011 through 2014 directly, use as an approximation

100×logyt-logyt-1

where yt is real per capita GDP in year t. How close does this approximation come to the actual growth rates you calculated in part (a)?

## Solutions

##### Expert Solution

a)

The annual percentage growth rates in real GDP per capita is given by [(Yt - Yt-1)/Yt-1]*100%

Thus, the annual percentage growth rates in real GDP per capita for each of the years 2011 through 2014, from the previous year is:

 Year Per Capita Real GDP The percentage growth rates in real GDP per capita 1950 14339 1960 17351 1970 23790 1980 30732 1990 35868 2000 43288 2010 46406 2011 47554 2.47% 2012 47741 0.39% 2013 48066 0.68% 2014 48780 1.49%

b)

Using as an approximation 100×logyt-logyt-1, where yt is real per capita GDP in year t, the result is:

 Year Per Capita Real GDP (Yt) The percentage growth rates in real GDP per capita Per Capita Real GDP (Yt-1) 100×logyt-logyt-1 1950 14339 1960 17351 14339 1970 23790 17351 1980 30732 23790 1990 35868 30732 2000 43288 35868 2010 46406 43288 2011 47554 2.47 46406 1.06 2012 47741 0.39 47554 0.17 2013 48066 0.68 47741 0.29 2014 48780 1.49 48066 0.64

It shall be noted that when 100×logyt-logyt-1 is used as an approximation, the result is not close to the annual percentage growth rates in real GDP per capita. This is because, the approximation uses log, which is log to the base of 10.

It shall be noted that instead natural log, that is log to the base of e if used, the approximation would take the form

100×lnyt-lnyt-1

The result is:

 Year Per Capita Real GDP (Yt) The percentage growth rates in real GDP per capita Per Capita Real GDP (Yt-1) 100×logyt-logyt-1 100×lnyt-lnyt-1 1950 14339 1960 17351 14339 1970 23790 17351 1980 30732 23790 1990 35868 30732 2000 43288 35868 2010 46406 43288 2011 47554 2.47 46406 1.06 2.44 2012 47741 0.39 47554 0.17 0.39 2013 48066 0.68 47741 0.29 0.68 2014 48780 1.49 48066 0.64 1.47

It shall be noted that excel function log(yt) would provide the value of log to the base of 10, whereas Ln(yt) would provide the value of log to the base of e

Thus, the approximation 100×lnyt-lnyt-1 provides an estimate that is close to the annual percentage growth rates in real GDP per capita

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