Macroeconomics — a view from the peanut gallery

Update: I run a more detailed version of the puzzle I find here in this post — some thoughts Brian Romanchuk helped to provoke. See also his trilogy of posts on this, starting here. See also C Trombley’s contribution here.

I’ve always liked microeconomics, aware as I am of all the shortcomings of Rational Choice Theory. I see it as largely a branch of logic. Adam Smith was a professor of logic, as was J.S. Mill, as was Stanley Jevons — Maynard Keynes was the son of a logician and his own treatise on probability is really an attempt to formalise a class of inductive inferences. Joan Robinson was basically a philosopher.

I don’t know about the predictive power of microeconomics. I don’t know about its status as a scientific theory. But I find microeconomic theories helpful for clarifying some questions in intriguing and surprising ways — just like the best sorts of logic puzzles. The theory of opportunity cost allows you to see how even if littering is bad, reducing littering to zero might be worse. Harberger triangles help you to see how the cost of taxation isn’t what comes out of your pay, and how paying tax can make you better off in some circumstances.

Microeconomic models give the answer “it depends” to just about every question of economic policy and justice. Again, the parallel with logic is clear. Still, I think it’s very helpful to be reminded that it depends, in a world full of pundits confidently proclaiming that this tax will hurt the middle classes, these immigrants will drive down wages, etc. etc.

I Don’t Get Macro

Macroeconomics, by contrast, I just don’t get. I called this post “a view from the peanut gallery”, because I want to be clear that when it comes to macroeconomics I have only a distant, side-balcony, heavily-obscured view.

Some macroeconomists have been very generous in trying to explain their theories to me. But it’s never really ‘clicked’. I can pick up an intermediate textbook in micro and happily work through the exercises. Every beginner-level macro textbook I’ve looked at has left me more confused than before (I mean ‘orthodox’ macro — I’ll say something about ‘heterodox’ macro in a bit).

Macro just operates in such a weird world. Infinitely-lived households with rational expectations… Dynamic Stochastic General Equilibrium… even the name I struggle to understand.

Example: ‘Transversality’

Here’s an example: the ‘transversality condition’ on government debt. Everyone talks about government debt — it’s always a big decider in elections. Your natural impulse is to go and see what macroeconomists have to say about it. But what you find when you do so is so bizarre it’s hard to know what to do with it.

Here’s an equation from a macro textbook:

You’ll have to click the link to have all the variables defined. But what this effectively tells you is that the value of the government’s debt at time t is the value of all the future surpluses that will be run to pay it off, plus any new debt the government takes on to refinance its old debt.

The model then imposes the condition that, roughly, the government does not keep borrowing to pay off its debt forever. I say “roughly” because it gets a bit more complicated when we think about time-discounting. The condition is known as the ‘transversality condition’. In the model, this condition is imposed in infinite time (the chapter explains why it’s pretty near impossible to make sense of this model in a finite time framework). In infinite time, the transversality condition requires that the final term in the equation converges to zero as the total future time periods (N) run to infinity. So the condition is just:

The second term from the original equation has gone to zero and disappeared.

To satisfy the transversality condition, it is necessary that government debt grows at a rate below the real rate of interest — that is to say, the rate at which people ‘discount the future’.

Why is there a transversality condition at all? Well, the model makes the present market value of the government’s debt equal to the discounted stream of primary surpluses that will be run to pay it off. If the government borrows at a rate that exceeds the discount rate, the value of its debt falls below zero — the chapter maps out various scenarios to show this. And then, since purchasers of government debt are assumed to have rational expectations, nobody would buy the debt.

I like to try to explain economic models (to myself) in plain language, but here I find myself struggling. This is the best I can do right now. I don’t know if this is right!

Assume that there is just one infinitely-lived household, lending to the government (where the household gets the money to lend is none of your concern). Now the household is impatient, so that it has a ‘time discount rate’: $1 today is worth >$1 tomorrow. Let’s say that $1 today is worth $1.10 next year, and that the discount rate is constant like this. This yields a series of ‘exchange rates’ by which we can convert future real dollars into subjective dollars — what the future dollars are subjectively worth to the impatient household. For this year the XRT is 1/1, but for next year it is 1/1.1; for two years’ time it is 1/1.21, etc.

Now the household is deciding on what terms to lend to the government, and it’s reckoning all this in terms of subjective dollars. Suppose it’s deciding whether to let the government borrow now and then just keep refinancing to infinity. The government offers a deal of endlessly refinanced debt growing at 5% per year in real dollars. Thus the household will lend the government $1 this year and roll this over to $1.05 next year, $1.10 the year after, $1.15 the year after, etc. But to convert these amounts to subjective dollars (s$) it must divide through by the XRT series generated by its discount rate. Thus the first loan is worth s$1 ($1/1), next year’s loan is worth s$0.95 ($1.05/1.1), the year after loan is worth s$0.91 ($1.10/1.21), etc.

So even though the household is lending the government more and more real dollars to refinance, it is lending it fewer and fewer subjective dollars. If I can put it like this: the government isn’t paying back its debt in real dollars, but it is paying back its debt in subjective dollars. As time approaches infinity, the outstanding debt (in subjective dollars) from the government to the household converges to zero. At the end (which never comes) the government has fully repaid the household in subjective dollars, even if the amount of real dollars borrowed spirals up to infinity.

This is all very clever and interesting. But does it tell us anything useful about the real world? It is hard to see what. It seems to tell us that if anyone is buying government debt today (rather than being paid to take it), they must believe that the government will run enough primary surpluses in the future (adjusting for time-discounting) to pay it off. But, well, really? Surely most bondholders believe that the government will collapse at some point, or that an asteroid will hit, or that something will happen sometime in the future before the government has run a stream of surpluses equal in time-discounted value to what they’ve paid for the bonds. But they lend anyway.

That, of course, is because they’re finitely-lived, and they don’t have rational expectations. They don’t (and couldn’t) internalise an economic model with an infinite time horizon. Now you might say that they approximate the infinitely-lived household with its rational expectations, since they are concerned with the future of their descendants (you could even set the discount rate to reflect the level of their concern). But it seems to me that the weakness of the analogy is more than amply demonstrated by the plain facts that:

(a) People lend to the government, and

(b) They don’t care one bit whether or not the ‘transversality condition’ holds over all time.

This is far from being the only thing that makes the model of very limited (infinitely limited?) applicability to the real world. I think Brian Romanchuk’s posts on this are very good — try here, here, and here. You can also try this paper from Scott Fullwiler or this one from Jamie Galbraith for (similar?) criticisms.

Update: I also have a concern that the optimisation assumed in the model isn’t possible when the set of solutions isn’t closed — see next post.

And I can’t resist adding that I’m not even sure the mathematics of the model are right. The problem is that thinking about the limit of a series as time approaches infinity is not the same as thinking about what happens in the case of infinite time. Calculus is good for limits — what Aristotle called ‘potential infinity’. But if you’re talking about actual infinity — that is, a set of time-periods with a cardinality of aleph-null — then things are different and calculus is no help. It seems to me that, whether reckoning in real or subjective dollars, a household facing actually infinite time could reason just like Bertrand Russell did in his Tristram Shandy paradox, to the effect that the government will repay all its debt, no matter how much faster the debt grows than the rate of repayment. The transversality condition becomes trivial, or, to express myself poetically:

Had we but world enough and time,
A Ponzi scheme would be just fine.

‘Heterodox’ Macro

What about ‘heterodox’ macro? The sort I’m most familiar with is ‘stock-flow consistent modelling’. I find that stuff quite useful for debunking certain misunderstandings people can have about debt (indeed I have had some): for instance the belief that if debt is always growing then a future default is inevitable. The mathematics is logical and elegant, with no wild assumptions about infinite lifespans and rational expectations, nor mysteries of subjective dollars, nor Tristram Shandy paradoxes. Instead you just trace flows around with the clarity found in electrical engineering or fluid dynamics models.

But there is a shortcoming: as far as I know, there is no welfare dimension to SFC modelling. We can look at the models to see how systems are likely to evolve over time. And we can use our intuition to think about how desirable those modelled outcomes are. For instance, if a certain solution has an increasing share of income going to holders of government bonds over time, that looks like a system assignment to avoid (but the models don’t tell us whether everyone in the population gets a turn being in that group). But what you don’t have (again so far as I know, which is not very far) is a way of defining some social welfare function in terms of people’s preferences and seeing how this might be maximised.

Is that a problem? Maybe not. Maybe social welfare functions haven’t proven all that helpful in economic policymaking (again I just don’t know). Many famous models don’t actually define any social welfare function — they just assume that there is one. Certainly the preference-satisfaction theory of welfare is crude and deeply limited, as Dan Hausman and Michael McPherson have compellingly argued.

All the same, a system of economics that doesn’t offer some principled way of deciding what sorts of arrangements might make people better or worse off, even if the answer to any practical question is always “it depends”, seems limited in some way.

But all this is subject to the caveat that I can only say what I see from the peanut gallery.