Your question about which country would “have the fastest growing standard of living” points to the importance of “normalizing” statistics so that you may make valid comparisons. Economists typically would normalize the comparison between country A and country B in your example by calculating the growth rate of output (GDP) on a per capita, or per person, basis. Per capita output is a more meaningful measure for comparing the standard of living of different countries. Moreover, by examining per capita output over time, economists also are better able to evaluate changes in living standards without the influence of changes in population growth.
In your example, basic mathematics indicate that country B would have the fastest growing standard of living, as measured by per capita output. This is because country B has a much slower population growth rate than country A. To understand how this works, let’s look at two extreme examples: a country with output growth and no population growth, and a country where output and population grow at the same rate.
In the first example, a country called “All Standard of Living Increase” is reporting a 6 percent annual rate of growth in output, but no growth in population. In this case, per capita output must increase by the full 6 percent, because the entire increase in output is divided over the existing population base. Over time, per capita output would climb rapidly in country “All Standard of Living Increase.”
In contrast, a country called “No Change in Standard of Living” is reporting 6 percent annual rates of growth in both output and population. In “No Change in Standard of Living” real per capita output would remain unchanged, since all the increase in output arose from population growth. Country “No Change in Standard of Living” would have little change in its standard of living over time, as long as the rate of population growth equals the rate of output growth.
Clearly, just looking at the growth rates for output for these two different countries would not be sufficient to answer your question. However, once the effects of population growth are included by using the normalized per capita output statistic the differences in standard of living growth are clear. This also can be shown in the Chart, which simulates per capita output over a ten year period where both countries start from the same output base but experience different population growth rates.
References
Blanchard, Oliver. 1997. Macroeconomics, Chapter 22. Upper Saddle River, New Jersey: Prentice Hall.
Greenwald, Douglas, Editor in Chief. 1994. The McGraw-Hill Encyclopedia of Economics. New York: McGraw-Hill, Inc.