# Still on the arithmetic of public debt

Gigi left a new comment on your post "The Marattineide, or the glitch. Tragi-comic drama in an impure act.":

So to reduce the debt/GDP ratio must we continue to make negative primary balances?

Published by Gigi on Goofynomics on 16 May 2023, 1:20 pm

I realize that navigating through the vast amount of material presented by this blog is quite complicated. The *tagcloud* at the bottom of the page (on PC, not on mobile) can help up to a certain point, the introductory pages need to be rewritten because I haven't touched them in years, etc. However, we have illustrated the formula that governs the trend of the debt/GDP ratio here and it is this:

where *d* is the debt/GDP ratio, *r* the *ex post* real interest rate on public debt securities (constructed as the ratio between interest expenditure at time *t* divided by the debt stock at time *t* -1 minus the rate of change in the GDP deflator), and *a* is the primary surplus (the primary budget balance, which if positive is a surplus and if negative, a deficit).

First observation: given that *a* (as surplus) enters the formula with a minus sign, other things being equal, a primary surplus will decrease the change in the debt/GDP ratio, bringing it towards negative territory. A fact that conforms to common sense.

Second observation: a positive primary balance is neither a necessary nor a sufficient condition for the change in public debt to be negative. It is not necessary because the change in public debt can be negative even if the primary balance is negative. It is not sufficient because it is not enough for the primary balance to be positive to have a negative change in public debt.

To stick to the current orders of magnitude, those resulting from the DEF (which you find in its place, i.e. here ), and without particular pretension of precision, considering 2023 as time *t* (and therefore 2022 as time *t-* 1), the nominal interest rate in 2023 is equal to 2.71% (resulting from the flow of interest expenditure in 2023, equal to 74.67 billion, divided by the debt stock at the end of 2022, equal to 2756.9). Subtracting the rate of change of the GDP deflator, equal to 4.8, we get a real interest rate of -2.1%. With a real growth rate of 0.9%, the coefficient *rn* is therefore equal to -0.03 (3%). The product of the debt/GDP ratio in *t* -1 and this coefficient is equal to -4.32% (0.0432). On this purely algebraic basis it is therefore perfectly possible that the debt will decrease even with a primary deficit of around 4%.

Naturally, the limitation of this purely accounting approach is that it does not take into account the relationships between the variables involved. Another limitation of this abstract exercise is that the cost of debt is estimated in an approximate way: the MEF has micro-simulation models that track the performance of each security issued. However, if the meaning of the question was: "Is it necessary to make a deficit to bring down the debt/GDP ratio?", the answer is no. If, on the other hand, the meaning was: "do we have to make a surplus to bring down the debt/GDP ratio?", the answer is still no. Inflation helps us, and real growth would help us, as illustrated over and over again.

*(… please do the math again because this was a post interruptus: first I had to go to XIV to vote on an opinion, then to der Palla's birthday lunch, then to the MEF with my minister, then to VI to vote on the basic text of the tax delegation: it's easy that I got lost along the way. But in short: whatever you ask, the answer is no!… )*

*This is a machine translation of a post (in Italian) written by Alberto Bagnai and published on Goofynomics at the URL https://goofynomics.blogspot.com/2023/05/ancora-sullaritmetica-del-debito.html on Thu, 18 May 2023 14:40:00 +0000. Some rights reserved under CC BY-NC-ND 3.0 license.*