## Recovering from the recession

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I have a Modest Proposal for how to get the US (and World) economy out of the current recession.

It came to me last Friday, on our last day of classes, as I was walking across campus to Calculus II.  I wanted to talk about cool applications of the course content, as a way of summing up (so to speak) the semester: a course on techniques of integration, applications of integration, and infinite series.  I decided to go with something simple:  applications of geometric series.  And one of my favorites is computing the total amount of economic activity that ensues from a single injection of economic stimulus from the government.

Here’s the (fairly standard) story:  Suppose that each individual saves 10% of each dollar they receive, and they spend (recirculate) the other 90%.  Then for each $1000 (say) of government stimulus,$100 gets saved, and $900 is spent, becoming additional income for other individuals in the economy. But now consider that$900 on the rebound.  10% of it ($90) goes into savings, and the other 90% ($810) gets spent again, this time by its second owners.  And now of the $810, 10% is saved, 90% spent. And so it goes… This story leads to the following calculation. A stimulus of$1000 will lead to a total amount of economic activity equal to

$1000 + 1000(.9) + 1000(.9)^2 + 1000(.9)^3 + \cdots$

an infinite geometric sum, where each summand is 90% of the previous term.  Now a geometric series $a + ar + ar^2 + ar^3 + \cdots$ with common ratio $r$ converges to a finite sum provide $|r|<1$, and in that case the limiting value is $\frac{a}{1-r}$.   In the case of the $1000 above, the total amount of economic activity is $\frac{1000}{1 - .9}$, or$10,000.  [So the $700 billion stimulus package, under these assumptions, could lead to$7 trillion in economic activity, or roughly half of the US Gross National Product.]

The brilliant thought I had whilst crossing campus:  what if we drop the savings rate from 10% to a smaller number?  If instead of recirculating 90% of our income, each of us went out and spent 95%?  or 99%, or even… more???  This model predicts that the total amount of economic activity from a given stimulus is the amount of that stimulus divided by 1 – (the proportion recirculated).

So, what would happen if no one put any money into savings, and ALL of our income went directly into consumer spending?  In that case, we have a common ratio of $r=1$ in the geometric series, which now diverges, and the model predicts that any amount of government stimulus leads to an infinite amount of economic activity!

Woo hoo!  Let’s send out 5 cents in economic stimulus, and watch the American electorate spend us out of recession!!!

[inhale, exhale]

Ok, so that’s obviously wrong, which means the original discussion (standard fodder for all the calculus and precalculus texts I’ve seen in recent years) is also flawed.  And seeing what’s wrong is easier now in this extreme case.  Suppose the government gives Joe the Drummer a $10 stimulus check. Joe goes out and buys new drumsticks; the music store spends all of Joe’s$10 on rent; the landlord spends all of the $10 on utilities; the utility company etc…. What is missing in the series is the issue of time. Joe might take a day or two between getting his government rebate check and actually spending it. The music store owner won’t spend the$10 until the utilities are due on the 15th of the month; and so it goes.   $10 in stimulus in theory leads to an unending succession of financial activity, but it cannot do so in a finite period of time. What can happen in a finite period of time is a finite number of transactions. If we assume that over the course of the month (say), each dollar spent in stimulus changes hands a total of ten times, we get a truncated geometric series (again, with a$1000 initial stimulus, and 90% recirculation rate):

$1000 + 1000(.9) + 1000(.9)^2 + \cdots +1000(.9)^9 = \frac{1000( 1 - (.9)^{10})}{1-.9} \simeq \ 6513$

This is quite a bit less than the infinite sum ($10,000). However, to some extent the predicted multiplier effect is real, if not quite as dramatic as one gets with infinite series. Now if the savings rate drops toward 0, and the recirculation rate increases toward 100%, the simplification on the right-hand side of the equation no longer works, and we instead decide that after 10 transactions, a stimulus of$1000 with all of it being recirculated 10 times leads to a total of \$10,000 in total economic activity.

So much for my 5 cent solution.

[Although it *does* optimize the amount of economic activity in a fixed number of transactions.  Just sayin’.  That Vox AC-30 amp would look mighty good under the tree this year….  Spending more this season just might be patriotic!]