Debt and Time: Another Puzzle

Again I don’t have time to work this out properly; again I’m throwing it up for helpful comments/corrections.

Here is a very simple textbook model for asset pricing. I won’t go through the whole thing, but here is the crucial equation giving us the price of an asset — say a loan:

Pt is the current price, and (24) tells us that this will equal the expected time-discounted value of the whole return (the first term on the right-hand side: D is the return and E is the expectation operator) plus the expected time-discounted price of the asset at the final period (the second term on the right-hand side). Now comes a crucial assumption. Here is what the textbook says:

The assumption is that the second term — the expected ‘final’ price of the asset must converge towards zero over time. And the ‘logic behind this assumption’ is that if the asset generates no return at all, it must have no current value. People can’t just keep selling on the asset for some positive price even though it generates no return. That is, people can’t just keep selling on a debt that never gets repaid to cover the value of the original loan.

Now: why not? The ‘logic’ in question involves, I think, a fallacy: a fallacy of trying to pivot out from a finite time case to one of infinite time.

Start with the finite-time case. The very last person to buy the asset, provided she is rational and she knows she is the last buyer, will buy it only for the return it generates. There is no price for which she can sell it on, because she can’t sell it on. So she will discount the price of the asset to the value of its expected return. But then if the person she buys it from is also rational, and knows all this, he too will buy it only for the value of its expected return. He won’t buy it for more than its expected return, since he knows the next buyer won’t buy it for more. Running then by backwards induction to the current price of the asset, we can eliminate the second term altogether. We end up with the equation in (26) (I don’t like the notation representing an ‘infinite sum’ there, for reasons I’ve discussed elsewhere, but it’s not a big deal — you could rewrite (26) as a proposition of analysis, as I’d prefer; my problem is with that proposition).

So much for the finite-time case. Can we simply port this logic over to the infinite-time case? I say no, because that logic depended on backwards induction. With the infinite-time case there is no final point from which to induce backwards. You might be reminded of Wittgenstein’s story about a old man saying “. . . 9, 5, 1, 4, 1, 3, done!” and claiming to have just finished reciting the decimal expansion of π backwards. Impossible, of course, because where did he start? Likewise with the backwards induction that got us, in the finite-time case, to the result that the second term in (24) must disappear altogether.

Why do economic models use these infinite-time horizons? If I understand rightly, they use infinite time to proxy for indefinite time. Capitalism will one day come to an end — when human life ends if not mercifully before. But we don’t know when that will be, and so no expected end can feature in our rational expectations models.

We could, if we wanted, try to represent this uncertainty in the logic. Suppose, looking back at (24), we know that somewhere between t and t+N (t+N might be the heat death of the universe), capitalism will be overthrown, or all debts will be cancelled, or something in any case will make it impossible to collect any more return from the asset. But nobody knows when this will be. Well then, it can’t happen at t+N, because then the people at t+N would know it, contrary to the hypothesis. But then it can’t happen at t+N-1 either (I’m using discrete time here — Brian Romanchuk has a post defending the use of discrete-time models in economics). Otherwise the people at t+N-1 would know it — we’ve already eliminated t+N as the terminal point, so if we get to t+N-1 it remains as the only possibility, and then the terminal point would be known, again contrary to the hypothesis. By the same logic, it can’t be t+N-2, and so on. By backwards induction, we arrive at the result that the terminal point can’t be any of the periods from t to t+N. But now that we’ve arrived at that result, the terminal point can be any of the periods from t to t+N, since nobody rational thinks — indeed nobody rational can think—that the termination will come at any of those periods.

In other words, we’ve found our way to the Surprise Exam Paradox. It might be inconvenient to have found a paradox right at the heart of what is meant to be a practically convenient economic model, but, well, life is paradoxical, isn’t it? Or, again, am I missing something?

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