Question

The governmet injects 90 billion dollars into the economy. Each recipient spends 90% of the dollars received. In turn, the secondary recipients spend 90% of the dollars they recieve. etc. Use the expansion of 1/(1-x) in powers of x to come up with a clean measurement of the total spending that results from the 90 billion dollars injection.

Answer 1

So if we assume (not true, but not a gross assumption) that the money keeps turning over, the effect is
\[9(10^{10})\sum_{k=1}^{∞}(9/10)k=9(10^{10})(1/(1−0.9)−1)=8.1(10^{11})\]

Answer 2

The 90 billion subtracted as the last term is due to the fact that our formula sums from zero, and in this case, we only summed from one.

Answer 3

Detected another typo..... terms summed should be (9/10)^k.