Expansion and Recession

If an economy is currently in an expansion, there's a 90% chance the next period will also be an expansion and a 10% chance it will be a recession. If the economy is currently in a recession, there's a 60% chance the next period will be a recession and a 40% chance it will shift to expansion. If the economy is currently in an expansion, on average, how many periods will we see until the next expansion? Please round to the nearest hundredth.

Answer

Using the law of total expectation, we have

E = 0.9\cdot 1 + 0.1\cdot(1+E_r),

where $$E_r$$ is the expected number of periods from a recession until an expansion. Since, in a recession, the probability of shifting to expansion is 0.4, we model the waiting time as a geometric random variable with mean

E_r = \frac{1}{0.4} = 2.50.

Thus,

E = 0.9\cdot 1 + 0.1\cdot(1+2.50) = 0.9 + 0.1\cdot3.50 = 0.9 + 0.35 = 1.25.

Notes and comments

Comment 1: Note that we can use the "geometrically distributed RV" argument as long as the probability of success stays constant UP UNTIL the success itself. But we already knew that from the Collector's Coupon Problem and other problems.

Note that as long as the success probability remains constant up until the success itself, the waiting time follows a geometric distribution. We used this fact to obtain $E_r = \frac{1}{0.4} = 2.50$ for the recession phase.