Suppose that real GDP per capita in the United States is $49,000. If the long-term growth rate of real GDP per capita is 1.6% per year, how many years will it take for real GDP per capita to reach $98,000?
62.5
To calculate the number of years it will take for real GDP per capita to reach $98,000, we can use the formula for compound interest. In this case, the real GDP per capita is growing at a constant annual rate of 1.6%.
The formula for compound interest is:
A = P(1+r)^n
Where:
- A is the final amount
- P is the initial amount
- r is the growth rate (as a decimal)
- n is the number of years it takes to reach the final amount
Let's solve the equation for this specific problem:
$98,000 = $49,000(1+0.016)^n
First, let's simplify the expression inside the parentheses:
1+0.016 = 1.016
Now, divide both sides of the equation by $49,000:
($98,000 / $49,000) = (1.016)^n
2 = (1.016)^n
To solve for n, we need to take the logarithm of both sides. Since the base is 1.016, we will use the natural logarithm (ln):
ln(2) = ln(1.016)^n
Using the property of logarithms that states ln(a^b) = b * ln(a), we can rewrite the equation as:
ln(2) = n * ln(1.016)
Now, divide both sides by ln(1.016):
n = ln(2) / ln(1.016)
Using a calculator, we can then perform the division and find:
n ≈ 43.23
Since n represents the number of years, we round up to the nearest whole number. Therefore, it will take approximately 44 years for real GDP per capita to reach $98,000 with a growth rate of 1.6% per year.