Question

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)?

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

Rahul Sunny answered 5 months ago

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