How do I find the number of employed workers and unemployed workers?

Question

Ellen is downloading labor market data for the most recent month, but her connection is slow. So far this is all she has been able to collect:

 

Unemployment Rate	5.8%
Participation Rate	62.5%
Not in the force	62.5 million

Find the working-age population, the labor force, the number of employed workers, and the number of unemployed workers.

Answer 1

The key to finding the missing data points is to understand the formulas for labor force, labor force participation rate, and the unemployment rate; then it is just algebra to solve for the unknowns.  There are three basic categories of the adult working age population:  employed, unemployed, and not in the labor force.  The four formulas you need to solve this problem are:

(1) labor force = employed + unemployed

(2) adult working age population = labor force + not in the labor force = employed + unemployed + not in the labor force

(3) labor force participation rate = labor force / adult working age population

(4) unemployment rate = unemployed / labor force

Now it is just algebra to solve.  From the data points, first use the labor force participation rate formula and plug in the known values:

62.5% = labor force / adult working age population

0.625 = labor force / (labor force + 62.5 MM)

0.625 * (labor force + 62.5 MM) = labor force

0.625 * 62.5 MM = labor force - 0.625 * labor force

39.0625 MM = (1 - 0.625) * labor force = 0.375 * labor force

labor force = 39.0625 MM / 0.375

labor force = 104.1667 MM

Now, the adult working age population = labor force + not in the labor force 

= 104.1667 MM + 62.5 MM 

= 166.6667 MM

Next, use the unemployment rate formula and plug in the known values:

unemployment rate = 5.8% = unemployed / labor force

0.058 = unemployed / 104.1667 MM

unemployed = 0.058 * 104.1667 MM = 6.0417 MM

Finally, employed + unemployed = labor force

employed + 6.0417 MM = 104.1667 MM

employed = 104.1667 MM - 6.0417 MM = 98.125 MM

So:  employed = 98.125 MM; unemployed = 6.0417 MM; labor force = 104.1667 MM; adult working age population = 166.6667 MM