As discussed in this chapter, real GDP per capita in the...

As discussed in this chapter, real GDP per capita in the United States grew from about $\$ 6,000$ in 1900 to about $\$ 50,010$ in $2014,$ which represents an average annual growth rate of 1.9 percent. If the U.S. economy continues to grow at this rate, how many years will it take for real GDP per capita to double? If government economic policies meant to stimulate economic growth result in the annual growth rate increasing to 2.2 percent, how many years will it take for real GDP per capita to double?

Key Concepts

Logarithmic Functions in Growth Calculations

Logarithmic functions are essential in calculating precise doubling times in exponential growth scenarios. The exact formula for doubling time, t = ln(2)/g (where g is the growth rate expressed as a decimal), uses the natural logarithm to determine the time required for a quantity to double, thereby accurately reflecting the compounding nature of growth over time.

Rule of 70

The Rule of 70 is a simple heuristic used to estimate the doubling time of a growing quantity. By dividing 70 by the annual percentage growth rate, one can quickly approximate how many years it will take for the initial value to double, offering an intuitive grasp on the speed of growth without requiring more complex calculations.

Exponential Growth

Exponential growth refers to the process where a quantity increases at a rate proportional to its current value. In the context of economic growth, this implies that real GDP per capita increases by a constant percentage over time, leading to compounding effects where the magnitude of growth becomes larger as the economy grows.

Doubling Time

Doubling time is the period it takes for a quantity that is growing exponentially to double in size. In economic analysis, it provides a tangible measure of how quickly an economy or any exponentially growing metric, such as real GDP per capita, increases relative to its current level.

Transcript

00:01 Okay, so for this problem, we're going to work through the first part, and you can use that as an example and use it as a method to solve the second part of the problem.

00:10 So the growth rate formula is fairly simple.

00:15 It says that the new value of gdp is equal to the old value of gdp multiplied by this special number e to the rt, so that's the real value of gdp, rate times the number of years.

00:35 I like to say that the new gdp is equal to the vert.

00:41 Value zero times e to the rt.

00:45 So in this problem, we have to identify what these values are.

00:50 Well, we start out with a gdp of 50 ,010.

00:56 We want a gdp that is double that.

01:00 So that's going to be 50 ,010.

01:02 So that's going to be 50 ,010.

01:03 Times 2, or 100 ,020.

01:09 Here we say that our old value is 50 ,010, and our new value is 100 ,020.

01:24 What else we need to know? well, we need to know our rate.

01:28 We know that our gdp is growing at a rate of 1 .9%.

01:33 Since this is a percent, we have to move the decimal place back until we get our r.