A Comment On Ramanan's Critique Of Fullwiler's Debt Sustainability Analysis

I was contacted by Ramanan, who alerted me to his critique of Scott Fullwiler's work on debt sustainability. It is the article "Wynne Godley And The Dynamics Of Deficits And Debts." There were two legs to the argument: a mathematical one, and one about Wynne Godley's analysis of the interplay between the fiscal deficit and the external deficit. For simplicity, I will only comment on the mathematical complaint; I will need time to digest the external constraint arguments. (I will be writing about this at length when I get back to my Modern Monetary Theory (MMT) primer.) In summary, I agree with Fullwiler's stance, but if we want to express it mathematically, we might need to use tighter language.

(Yes, this article is attempting to get away from the ongoing worries about the pandemic.)

The Mathematical Critique

I will just quote Ramanan's article to explain the issue.

The initial background is as follows.

Fullwiler’s error is a simple mathematical one. He sums the series for debt-sustainability equation and shows the the public debt/gdp ratio converges to

– pb/(g – r)

where pb is the primary balance/gdp ratio, g is the growth rate of output and r the interest rate. [notations are changed somewhat without any effect on conclusions]

He then observes:

Now this doesn’t make sense. The claim that “any” level of primary deficit can converge if the interest rate is below the growth rate is incorrect. For example, if we have primary balances pb0, pb1, pb2 and so on and each of them is growing sufficiently faster, the debt/gdp ratio is explosive even if interest rate is less than the growth rate of output. His result is valid if each of the balances pb0, pb1, pb2 … are equal to each other and not in general.

This is argument is technically correct. However, we could tighten up Fullwiler's statements, and get a mathematically clean version. If I were reviewing Fullwiler's article at one of control systems journals where I used to be a reviewer, yes, I would have thrown one of my little pedantic fits. However, given my distaste for the direction that neoclassical academia went, I am happy that heterodox authors went to for ease of understanding over mathematical argle-bargle.

The Mathematics of Convergence

What does it mean for a series to converge? I will give the formal definition in a mathematical appendix, but I will outline why we would never expect a debt/GDP ratio to converge under realistic circumstances.

Let us imagine that we have a country that has a steady level of GDP, but swings back and forth between deficit and surplus. In particular, the debt/GDP ratio is the following sequence:

101%, 99%, 101%, 99%, ... (repeat forever).

The debt-to-GDP ratio does not converge to anything; it is oscillating back and forth. For example, you cannot say that it converges to 99%. Yes, one year it will jump to 99%. Thing is, the next year it jumps away to 101%.

This lack of convergence will always happen in any mathematical model where there is cyclic behaviour, or random disturbances to model variables.

This means that we need to throw out using "converges" if we want to be mathematically proper.

Rephrasing Proper Like

The concept we are interested in is whether or not the debt-to-GDP ratio remains bounded. This is equivalent to the phrasing "does not go to infinity," although "going to infinity" is somewhat vaguely defined in mathematics.

Take my example debt-to-GDP ratio. Although it is not going towards a particular value, it remains bounded, and so it is not a debt trajectory that is worrisome.

So even if Fullwiler used the term "converges" or similar, we just need to translate to "remains bounded."

Dealing with Ramanan's Critique

If we wanted to deal with the mathematical niceties, we stop worrying about the exact level of the debt-GDP ratio. Instead, we calculate an upper bound (and possibly a lower bound, if we had some fear that government debt can be arbitrarily negative).

To save the -pb/(g – r) equation, we replace the actual primary balance with a lower bound with the balance, call it pb*. So long as a finite pb* exists such that pb* < pb(t), then we have an upper bound for the debt-to-GDP ratio.

Given that it seems hard to see how the primary balance can be sustainably larger than GDP, we should expect that such a bound exists.

However, trying to make a general statement is very awkward. In addition to the primary deficit moving, GDP growth and the real rate are also not constant. We would need to sandwich all these variables within bounds, and then pin down the debt-to-GDP ratio within an inequality, and then take the limit. This would take a fair amount of algebra (and I grew algebra-intolerant after I finished grad school).

Concluding Remarks

Writing down the conditions for the existence of an upper bound of the debt ratio has a lot of hidden complexities. There is a lot to be said about assuming a steady state, and working with constant growth rates -- which is what Fullwiler did.

Ramanan's discussion of Godley's work on the external constraint is interesting, but it will have to wait for a follow up article.

Technical Appendix: Convergence

The usual formalism for defining convergence of a sequence is as follows. (Trust me, I pulled out Rudin to make sure!)

A sequence is a function that is defined on the set of all positive integers. (E.g., if it is a time series, there is a value for all integer times.) We will assume that the function maps to the real line.

A sequence {p_n} converges to a point p on the real line if for any e>0 there exists an integer N such that  |p_n - p| < e for all n >= N.

It is entirely possible that the above statement made no sense. In plain English, for an margin of error that is greater than zero, there is a time point N for which all points of the sequence are within the margin of error of the limit. The example sequence failed, because the series is always jumping by 2%, and so it is not arbitrarily close to a single point.

(c) Brian Romanchuk 2020