(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 toHe then observes:

–pb/(g – r)

wherepbis the primary balance/gdp ratio,gis the growth rate of output andrthe interest rate. [notations are changed somewhat without any effect on conclusions]

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 balancesThis 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.pb0, pb1, pb2and 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 balancespb0, pb1, pb2… are equal to each other and not in general.

## 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

Hi Brian

ReplyDeleteI find this whole thing a bit weird. All I did in that paper was use the neoclassical view of debt sustainability that you find in any grad textbook and show that (1) it is all about i vs g (not really pathbreaking, but also not what was being emphasized at the time by neoclassicals like Gokhale and Smetters that I was responding to there), (2) that they don't know how interest rates are set. Any problem with language or math isn't a problem with me or my paper on but with the language and math of neoclassicals I cite there. It was their math and their language. And, yes, their math obviously has assumed cycles or no deviations from long run GDP path ... again,just see the Blanchard et al 1990 paper (a classic in that literature) that I cite frequently in my paper. That was the entire point ... on their own terms they are wrong. It's not surprising that Ramadan would build such a straw man though as that and misrepresentation are really all he ever does.