## An Economic Argument for a Mathematical Conclusion

27 04 2008

How valuable is an income stream that pays $1000 a year in perpetuity? Naively, one might suspect that since this stream will eventually pay out arbitrarily large amounts of money, it should be worth infinitely much. But of course, this is clearly not true – for a variety of reasons, future money is not as valuable as present money. (One reason economists focus on is the fact that present money can be invested and thus become a larger amount of future money. Another reason is that one may die at any point, and thus one may not live to be able to use the future money. Yet another reason is that one’s interests and desires gradually change, so one naturally cares less about one’s future self’s purchasing power as one’s current purchasing power.) Thus, there must be some sort of discount rate. For now, let’s make the simplifying assumption that the discount rate is constant over future years, so that money in any year from now into the future is worth 1.01 times the same amount of money a year later. Then we can calculate mathematically that the present value of an income stream of$1000 a year in perpetuity is given by the sum $\frac{1000}{1.01}+\frac{1000}{1.01^2}+\frac{1000}{1.01^3}\dots$. Going through the work of summing this geometric series, we find that the present value is $\frac{1000/1.01}{1-1/1.01}=\frac{1000}{1.01-1}=100,000$. However, there is an easier way to calculate this present value that is purely economic. The argument is not mathematically rigorous, but there are probably economic assumptions that could be used to make it so. We know that physical intuition can often suggest mathematical calculations that can later be worked out in full rigor (consider things like the Kepler conjecture on sphere packing, or the work that led to Witten’s Fields Medal) but I’m suggesting here that the same can be true for economic intuition (though of course the mathematical calculation I’m after is much simpler).

The economic argument goes as follows. If money in any year is worth 1.01 times money in the next year, then in an efficient market, there would be investments one could make that pay an interest of 1% in each year. Investing $100,000 permanently in this and taking out the interest each year gives rise to this income stream, and thus one can fairly trade$100,000 to receive this perpetual income stream, so they must be equal in value. We don’t need to sum the series at all.

Now perhaps there is a sense in which the mathematical argument given above and the economic argument given below can be translated into one another, but it’s far from clear to me. Thus, it looks like at least sometimes, economic intuition can solve mathematical problems. People often talk about the “unreasonable effectiveness of mathematics in the sciences”, but here I think I have another example of the unreasonable effectiveness of the sciences in mathematics.

### 3 responses

7 05 2008

When you assume the existence of a present value, you are allready assuming that your series converges. Given that, there is a very easy way to find a expression for the infinite sum. The details are here.

13 05 2008

Very interesting. Both this and the physical argument you suggest in yourreply to Michael worry me a bit.

Both arguments end in correct conclusions, and both arguments hide a lot of facts (empirical, mathematical) that might render one of their premises false. It’s this possibility that we want to avoid. It’s easy to generate wrong arguments for correct conclusions — and such arguments are completely uninformative.

What I would urge is plenty of caution with non-mathematical arguments for mathematical conclusions. I think that in many cases they can lead to new insight, and in some cases they do. But my suspicion is this: these non-mathematical tricks are instructive only insofar as they lead to good arguments.

For a similar discussion: a “biological proof” of the isoperimetric theorem in 3D.

16 07 2008

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