Brian Romanchuk has written a very interesting reply to my last post. He takes issue, not with my argument, but with the textbook model I refer to. I think in a way we have the same issues with the model, but he expresses himself differently. There is more to come from him, but I’d like to see how far our criticisms line up and if they might be synthesised.
The first thing that Brian notes is very important (he calls me ‘Professor Douglas’ in his post, but I’ve asked if we can switch to first names). In real analysis, infinity doesn’t exist. I’m going to say some obvious and pedantic (or possibly wrong) things about analysis now, but bear with me — you’ll see where it’s going.
The function I drew from the textbook is this:
Now the ‘transversality condition’ the model wants to impose on this function is that as N approaches ∞, the second term converges to zero. Brian points out that there is an implicit domain restriction in this model: i and N are defined over the domain of positive integers, which of course does not include ∞. So we’re looking for the condition that:
And this can be rewritten without reference to ∞; it tells us that, to adapt Brian’s words, for any non-zero error tolerance (less than b_t plus the highest absolute value of any finite summation — call that A), we can guarantee that all sufficiently long finite summations lie within that error bound of b_t. That is:
Brian then says: “We can always write down an infinite summation, but we need to validate that it converges.” Well, here comes the pedantry. Where you have convergence, you can write down what notationally looks like an infinite summation, but is actually a proposition of analysis of the sort seen above.
When we somewhat hastily say that the sum f(1) + f(2) + . . . f(N) goes to some value x as N goes to ∞, we are not giving the value of an infinite sum: there is no definable infinite sum since N ranges over a domain of finite numbers. What we are giving is a property of the (infinite) set of finite summations up to various values for N: namely that it is guaranteed to include a proper subset that gets within any error bound of x you care to give, no matter how small (as long as it isn’t too big).
Why, I hear you scream, am I droning on about elementary principles of analysis? Well, it’s important for the interpretation the textbooks want to give of the model. Here is where I think the real trouble lies.
We’re meant to think of an household facing the possibility of lending to the government over a certain time, and deciding whether it’s worth it, given a real discount rate and a rate at which government debt will grow.
In finite-time cases, the transversality condition sounds plausible enough: nobody wants to hold useless bonds on the last day, so all the bonds will get retired before the last trumpet sounds (of course the household must know in advance when that day will be).
(Some people have commented that this model leaves out the fact that the government has the power to create “high-powered money” with which to retire its bonds. It’s true that the textbook model doesn’t include the creation of HPM — of course the sophisticated models in the literature do. But given the optimisation assumptions, I don’t think that changes much. HPM is also just a liability of the state, and the household doesn’t want idle IOUs in its pocket at the end of time. So it will spend up all the HPM before the final moment the way you spend up all your foreign currency on the last day of the holiday — the HPM will return to the government in tax payments and get destroyed.)
In this sense the transversality condition is fulfilled automatically: the initial bond issuance is automatically valued to equal the time-discounted stream of all future surpluses, and the time-discounted future bond holdings must be, by accounting identity, such as to preserve this equality. This is how we ‘prove’ convergence of the final term in the first equation. Importantly, this result is trivial — it follows directly from the first-period accounting identity and has nothing to do with household optimisation; Brian has a nice inductive proof here.
What economists call the “infinite-time” case is not, for the reasons I’ve given, really a case of infinite time. Instead, the limit-function is a proposition about all finite series: we can always find a finite series where the second term equals ϵ for any ϵ>0.
What then explains the alleged difference between the finite and ‘infinite’ cases? In the ‘infinite’ case, we are told, the transversality condition does not imply that the second term sums to zero; it requires only that debt grows slower than the household’s discount rate. Why didn’t the discount rate appear in the description of the finite case? Well, even if debt grows slower than the discount rate, the discounted value of the debt need not reach zero in any finite period. And zero is the value the household ‘wants’, to the extent that its future bond purchases must be such as to ensure that that value is reached.
The discounted value of the initial bond outlay can, however, converge towards zero as the series is extended indefinitely. But why is that enough for the household in the ‘infinite’ case? The trick lies in the misleading notation of:
That looks like it says that the infinite sum equals b_t. But that isn’t true, since there is no infinite sum — again, i ranges only over a domain of finite numbers. Again, a less misleading notation would have:
Now what is the household doing when it ‘optimises’? Well, in the model it is maximising some utility function including its discount rate. But you can see from the formula above that in the ‘infinite’ case there might be no maximum. The household wants to get the smallest possible ϵ in:
But, assuming convergence and with indefinitely extendible finite time, there might well be no smallest possible ϵ. For any (finite) N, there might be some (finite) Q>N such that:
The household can’t solve its optimisation problem since the relevant maximum (or minimum, depending on how you look at it) doesn’t exist.
Roy Sorensen puts it nicely in the article “Necessary Waste” in his Cabinet of Philosophical Curiosities:
My contentment with the economist’s optimal trade-off did not last. Mathematicians asked: What guarantees the existence of an optimum?
They have learned a lesson in the nineteenth century. Karl Weierstrass demanded existence proofs of minima and maxima. This struck his colleagues as excessively sceptical.
. . . Weierstrass gave memorable illustrations of optimising problems with false existential presuppositions. Suppose a pilot needs to fly from a point directly above A to point B. Which is the shortest path? As the candidate points of descent lose altitude their paths to B shorten. But there is no shortest path.
Well there you go. There might be no optimum in the ‘infinite time’ case, and so the model with an optimising household might have no solution. I have no idea how economists would prove the existence of an optimum here, but certainly there being no optimum is consistent with the convergence they assume.
Should we then move back to the finite-time case? Well, its assumptions are quite wrong. There is no fixed end-date known by those who lend to the government today. The future of the government’s fiscal operations are indefinite, and indefiniteness is what is actually captured by the mislabeled ‘infinite time’ case.
In that case there is no provable optimum and the transversality condition (whose justification is — I think — meant to arise from some household optimisation problem) need not hold. It might well be simply impossible for the household lending to the government to achieve its objective of full repayment, even with the assumed convergence in place. In the model, the household won’t lend to the government: there will be no way for the present value of its bonds to be positive (they’ll always be a bit negative — if only ‘infinitesimally’ so).
I think it’s possible to prove that in fact the transversality condition can only be really satisfied in the ‘finite-time’ case — where the government is bound to retire all its bonds before the end of time or else nobody now would be lending to it. And what’s the problem with that? Well, it’s certainly predictively powerful! But the prediction doesn’t seem awfully plausible. . .