How can we measure growth over the very long run?

The poorest countries in the world have a per capita income of about $600 today. We can reasonably assume that it is nearly impossible to live on an income below half this level (below $300). Per capita income in the United States in 2009 was about $42,000. With this information in mind, consider the following questions.

(a) For how long is it possible that per capita income in the United States has been growing at an average annual rate of 2% per year?

(b) Some economists have argued that growth rates are mismeasured. For example, it may be difficult to compare per capita income today with per capita income a century ago when so many of the goods we can buy today were not available at any price then. Suppose the tru growth rate in the last century was 3% per year rather than 2%. What would the level of per capita income in 1800 have been in this case? Is this answer plausible?