**Calculating Geometric Growth**

We are familiar with geometric growth in the context of compound interest. You put $1000 in a savings account. Suppose that the account pays 4 percent interest annually. How much will you have in the bank after 10 years?

The basic equation for growth is Y_{t}** = **Y_{0}(1+r)^{t}** **

where Y_{0} is the initial
amount ($1000 in this example), r is the growth rate expressed as a decimal
(.04 in this example), and t is the number of years of growth (10 in this
example). The amount you will end up
with after 10 years is $1000(1 + .04)^{10} = $1480.24.

You will want to be very familiar with this equation. You need to know how to solve for: the final valueY_{t}; the cumulative growth rate Y_{t}/Y_{0};
the periodic growth rate r; or the time t.

__Solving for the Final Value__

Suppose we had a population of 100 gerbils growing at a rate
of 24 percent per year. After four
years, how many gerbils would we have? Y_{0} = 100, r=0.24, and t =
4. Therefore,

Y_{t}
= 100(1 + .24)^{4} = 236 gerbils.

__Solving for Cumulative Growth__

Suppose that we want to measure cumulative growth. That is, we want to measure the total growth that has taken place from a base year (year 0) to a later year (year t). Then we take the percentage change in Y from year 0 to year t.

100 * (Y_{t}/Y_{0} - 1)

For example, if we start with 1000 gerbils and after five years we have 1800 gerbils, then the cumulative growth is 100 * (1800/1000 - 1) = 80 percent. We say that the population of gerbils has grown by 80 percent in five years.

__Solving for Average Growth__

Often, we want to convert a cumulative growth rate to an
average growth rate. In that case, in
the basic growth rate equation we know Y_{t}/Y_{0} and we know
t, and we need to know r. Solving for r,
we obtain

r = (Y_{t}/Y_{0})^{(}^{1/t)}
- 1

If we start with 1000 gerbils and after five years we have
1800 gerbils, then the average growth rate, r, is (1800/1000)^{1/5} - 1
= .125, or 12.5 percent per year.

__Solving for t__

Suppose that we start with a population of 1000 gerbils, with an annual growth rate of 24 percent. We want to know how many years it will take for the gerbil population to reach 4000. To do this, we use something I’ll bet you never thought you would use: logarithms! Taking the log of both sides of the basic growth equation gives

log(Y_{t})
= log(Y_{0}) + t(log[1+r])

Solving for t gives

t = [log(Y_{t})
- log(Y_{0})]/log(1+r) = log(Y_{t}/Y_{0})/log(1+r)

In our example, we want to know how many years it will take for the gerbil population to get from 1000 to 4000 at an annual growth rate of 24 percent. We have

t = log(4000/1000)/log(1 + 0.24) = 6.44 years.

**Practice Exercises**

1. Suppose that you start with $1000 in the bank. The bank pays 6 percent interest per year. How much money will you have after 6 years? After 8 years?

2. Suppose that real GDP is $2 trillion. It is growing at a rate of 1.5 percent per year. What will real GDP be after 10 years? After twenty years?

3. Suppose that real GDP is $2 trillion. It is growing at a rate of 1.5 percent per year. How long will it take for real GDP to reach $2.5 trillion per year?

4. Suppose that real GDP is $2 trillion. It is growing at a rate of 2.0 percent per year. How long will it take for real GDP to reach $2.5 trillion per year?

5. If real GDP is $100 billion in 1950 and $230 billion in 2000, what is the cumulative growth of real GDP? What is the average annual growth rate?

6. If nominal GDP is $220 billion in 1950 and $680 billion in 2000, what is the cumulative growth rate of nominal GDP? What is the average annual growth rate?

7. Suppose that in the year 2000, nominal GDP is $800 billion and real GDP is $757 billion. In 2003 nominal GDP is $920 billion and real GDP is $810 billion. What is the value of the GDP deflator in 2000? What is the value of the GDP deflator in 2003? Calculate the cumulative growth from 2000 to 2003 of nominal GDP, real GDP, and the deflator. The growth rate of the GDP deflator is called inflation. Which is a better approximation:

(a) growth in nominal GDP = (growth in real GDP) x (inflation); or

(b) growth in nominal GDP = (growth in real GDP) + (inflation)

8. Using the figures in problem (7) calculate the average annual growth rate of nominal GDP, real GDP, and inflation. Compare the approximations (a) and (b) above for the average annual growth rates.

9. Suppose that real GDP grows at an average rate of 2 percent per year, and population grows at an average rate of 1.5 percent per year. What would you guess is the average growth rate of GDP per capita? Check your answer by letting GDP start at $100 billion and population start at 10 million. After one year, suppose that GDP has increased by 2 percent and population has increased by 1.5 percent. Calculate the ratio of GDP to population in each year. What is the growth rate of this ratio?

10. Suppose that real GDP grows at an average rate of 3 percent per year, and the total number of hours worked grows at a rate of 1 percent per year. What would you guess is the average growth rate of productivity (output per hour)? Check your answer by letting GDP start at $100 billion and hours worked equal 7.2 billion. Calculate output per hour, which is the ratio of GDP to hours worked. Then let GDP grow by 3 percent and let hours worked grow by 1 percent. What is the growth rate of the ratio of GDP to hours worked?

11. Recall that GDP per capita is equal to output per hour times the number of hours worked per capita. If output per hour grows at 2.5 percent, but the number of hours worked per capita falls by 1 percent, what do you think will happen to real GDP per capita? Check your answer by letting real GDP start at $100 billion, hours worked equal 7.2 billion, and population equal 10 million. Compute the ratios of GDP per capita, GDP per hour, and hours worked per capita. Let GDP per hour grow by 2.5 percent and let hours worked per capita grow by (- 1) percent. Finally, compute the new ratio of GDP per capita.