# Question

Consider the following data on real GDP per capita in the United States:

Year U.S. Real GDP Per Capita (2005 dollars)

1950 ............. $13,244

1960 ............. $15,773

1970 ............. $20,994

1980 ............. $25,752

1990 ............. $32,275

2000 ............. $39,744

2002 ............. $40,063

2003 ............. $40,703

2004 ............. $41,707

2005 ............. $42,572

2006 ............. $43,282

2007 ............. $43,785

2008 ............. $43,287

2009 ............. $41,377

2010 ............. $42,311

2011 ............. $42,733

(a) Calculate the percentage growth rates in real GDP per capita in each of the years 2003 through 2011, from the previous year.

(b) Now, instead of calculating the annual percentage growth rates in the years 2003 through 2011 directly, use as an approximation 100 Ã— (ln yt â€“ ln ytâ€“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)?

(c) Repeat parts (a) and (b), but now calculate the percentage rates of growth in real per capita GDP from 1950 to 1960, from 1960 to 1970, from 1970 to 1980, from 1980 to 1990, and from 1990 to 2000. In this case, how large an error do you make by approximating the growth rate by the change in the natural log? Why is there a difference here relative to parts (a) and (b)?

Year U.S. Real GDP Per Capita (2005 dollars)

1950 ............. $13,244

1960 ............. $15,773

1970 ............. $20,994

1980 ............. $25,752

1990 ............. $32,275

2000 ............. $39,744

2002 ............. $40,063

2003 ............. $40,703

2004 ............. $41,707

2005 ............. $42,572

2006 ............. $43,282

2007 ............. $43,785

2008 ............. $43,287

2009 ............. $41,377

2010 ............. $42,311

2011 ............. $42,733

(a) Calculate the percentage growth rates in real GDP per capita in each of the years 2003 through 2011, from the previous year.

(b) Now, instead of calculating the annual percentage growth rates in the years 2003 through 2011 directly, use as an approximation 100 Ã— (ln yt â€“ ln ytâ€“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)?

(c) Repeat parts (a) and (b), but now calculate the percentage rates of growth in real per capita GDP from 1950 to 1960, from 1960 to 1970, from 1970 to 1980, from 1980 to 1990, and from 1990 to 2000. In this case, how large an error do you make by approximating the growth rate by the change in the natural log? Why is there a difference here relative to parts (a) and (b)?

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