In today's lab, you will be computing rates of population change, and using these rates to project the size of future populations and to compute a population doubling time.   In order to compute population change, the first step is to compute the rate of change. This requires having populations for two time periods. These can be any two points in time since the rate of change will be based on the number of years between the two time periods.

Rate of Change: D P = (P _{t 2}/P_t1 – 1)

Percentage Rate of Change: D P = (P _{t 2}/P_t1 – 1) x 100%

D (triangle) is the Greek symbol Delta which means change.

When computing rates of change and population projections, there are 2 key assumptions.   First, the rate of change over 10 years is assumed to be equally divided across each of the 10 years, and second, the population is assumed to continue to grow at the same rate as it has in the past.

For this lab, you will be using the 1990 and 2000 census data to compute a 10-year rate of change for the populations of the five boroughs, New York State, and the United States. Use these rates to project each region’s population for 2010 and 2020, using both the geometric and exponential projections. The difference between the projections is the steepness of the curves.  These equations are adaptations of the formulas used to compute compounded interest.  The slope of the curve on a geometric projection is less steep, so it would be a more conservative projection, while the exponential would be the higher projection.  In slower growing countries, there will be very little difference between your results.  The more rapid the rate, the greater the difference.

Geometric rate of change: P_t+n = P_t (1 + r)ⁿ

Exponential rate of change: P_t+n = P_t e^r*n

Where t = the first time period, n = the number of time periods relative to the number of years in computing the rate of change. For example, 10 years equals one time period. Thus, if you are using 2000 as your starting population for computing 2010, n=1. 2020 will be 2 10-year time periods from 2000.

As mentioned in class and in your textbook, a population's doubling time in the number of years it takes for a population to double, or the number of years that it would take for P_t+n = 2*P_t using the exponential rate of change.  Using algebra to solve for this results in a formula for doubling time that is ln2/r, but r needs to be computed at 1-year rates.  What that means is that if you have computed a rate of change using ten years, divide by 10 to get a 1-year rate, assuming that growth was equally distributed across the ten years.  The natural log of 2 equals .6931.  

An interesting note on doubling time is that a population has to be growing in order to double.  Those countries in Stage V of Demographic Transition that have negative rates of natural increase and are not growing, or whose rate of growth is miniscule, will have doubling times that either cannot be computed or that are in the thousands of years.

Geometric Growth:
The formula for geometric growth is:

P_t+n = P_t (1 + r)ⁿ

Where:

1. Pt+n is the year you are projecting to, in this case, 2010; and

2. Pt is the ending point for computing the rate of change, in this case, 2000; and

3. r is the rate of change computed above; and

4. n=the number of time periods you are projecting forward.

In this case, you have computed a ten-year rate of change and are computing a projection for 10 years later, so n=1. If you were using the same rate of change (1990-2000) to compute a projected population for 2030, n would equal 3 (3 ten-year time periods).

"n" is an exponent. That means whatever number is in the parentheses is multiplied by itself "n" times. If n=1, then the value of 1+r is raised to the first power. If n=2 (if computing a projection for 2020), then the value of 1+r would be raised to the second power or squared.

In Excel, exponents are computed using ^. Remember, all formulas are computed from the inside out and must be inputted into the computer from the inside out.

Exponential Growth:
Exponential growth is a bit more complex because it uses "e." "e" is a constant with a value of approximately 2.71828. The formula for computing exponential growth is:

P_t+n = P_t e^r*n

Again where

1. Pt+n is the year you are projecting to, in this case, 2010; and

2. Pt is the ending point for computing the rate of change, in this case, 2000; and

3. r is the rate of change computed above; and

4. n=the number of time periods you are projecting forward.

When computing regular (base 10) exponents in Excel, you use ^ as described above for geometric growth. When using "e", you use the command EXP.

For this assignment:

Create a table that lists the following for the regions listed above:

Rate of Change
Populations in 1990, 2000
Geometric Projections for 2020, 2050
Exponential Projections for 2020, 2050
Doubling time

Write a 1-2 page essay discussing your observations and the impact/importance of examining projected growth at different scales.  Remember the assumptions in projecting growth.