Description
This is my translation of a Commentary on Current Affairs written by François Ecalle on the French blog Fipeco, devoted to analyses of public finance and the economy in France.
In part I, Ecalle discussed how one might understand public debt. In part II, he explained why public debt cannot increase ceaselessly and discussed how the stabilizing of the debttoGDP ratio is a necessary (though not sufficient) condition for debt sustainability.
In this third part, Ecalle discusses the formula for a debtstabilizing primary balance and the limits to a stable public debt.
My thanks to François Ecalle for permission to publish this translation on my blog.
Publication details
French article by François Ecalle
English translation by Urmila Nair.
Website: https://www.fipeco.fr/
Date of publication: November 11, 2020.
To view the original French article, click HERE
The Translation, part III
(3) The primary surplus needed to stabilize public debt increases as the debt increases.
To stabilize the public debttoGDP ratio, the primary balance (that is, the fiscal balance excluding interest payments on debt) ^{1} should equal what is known as a “debtstabilizing primary balance” ^{2}. This may be expressed in terms of the following equation:
(S_{ps} / Y) = (i – g) (D / Y)
where
S_{ps} = the debtstabilizing primary balance;
Y = nominal GDP;
(S_{ps} / Y) = ratio of debtstabilizing primary balance to GDP;
D = public debt;
(D/Y) = public debttoGDP ratio;
i = the apparent interest rate on the public debt, also known as the implicit interest rate, and explained as the average interest rate that governments pay on their debt^{3};
g = the nominal GDP growth rate;
(i – g) = the interest rategrowth differential^{4}.
(See my note on the debtstabilizing primary balance for an explanation of this equation.)^{5}
The traditional hypothesis is that the apparent or implicit interest rate is greater than the GDP growth rate (i > g), even though the opposite has occurred frequently in the past. If this hypothesis holds, a primary surplus is required for debt stabilization. A larger debt would require a larger primary surplus in order to be stabilized. If the effective primary balance is higher than this stabilizing surplus required, the debt reduces. If not, the debt increases indefinitely under what is termed a “snowball effect”: the debt keeps increasing by virtue of the accumulation of interest.
Delays in implementing measures to stop debt increase would entail a need for stronger measures. Deficit and public debt remain necessary, nevertheless, when the economy is not doing well: they are needed to revive GDP growth. However, when the economy does well, they must be offset by a surplus that is greater than the debtstabilizing balance, and a debt reduction.
There is a limit to the primary surplus, because taxes cannot increase or because expenditure is incompressible. There exists, equally, a limit to public debt beyond which it cannot be stabilized. Drawing on the above equation, this limit value beyond which the debt cannot be stabilized (D_{max}) may be represented as:
(D_{max} / Y) = [(S_{pmax}) / Y] / (i – g)
where S_{pmax} is the corresponding maximum primary balance.
For example, for a maximum primary surplus of 2% of GDP, and an interest rategrowth differential (i – g) of 0.01, the debt can no longer be stabilized if it exceeds 200% of GDP.
Nevertheless, it is as difficult to determine the maximum primary balance as it is to estimate this maximum debt.
The debtstabilizing primary balance formula also indicates that inflation can facilitate stabilization or debt reduction by reducing the stabilizing primary balance since inflation translates into an increase in the nominal GDP growth rate (g) while i remains constant, at least initially. However, in such a situation, the central bank would have to increase the apparent or implicit interest rate (i) so that, in the long term, the interest rategrowth differential (i – g) would probably remain unchanged.
To be continued in On the limits of public debt, Part IV

Translator’s note on references:
The French term “solde primaire (hors intérêt)” is translated as “primary balance”. The equivalence is established, for instance, in an OECD document on “General government fiscal balance” available in English, and its translation, “Solde budgétaire des administrations publiques” in French.The vocabulary used in the above documents and elsewhere is, however, quite confusing, even misleading, for example: “primary balance – that is, the overall fiscal balance excluding net interest payments on public debt” or, in French, “Le solde primaire – c’estàdire le solde budgétaire global déduction faite de la charge nette des intérêts de la dette publique” (emphases added).
The following is, therefore, a useful clarificatory explanation of the term in English presented in an IMF document:
“Under the gross debt interpretation, interest payments are gross interest payments and the primary balance is defined as the overall balance plus gross interest payments (i.e., total revenue less expenditure excluding gross interest payments). Nevertheless, the debt could also be interpreted as net debt. In that case, interest payments would represent net interest payments (interest paid less interest received on assets) and the primary balance should be interpreted as the overall balance plus net interest payments.” (emphases added)
Thus, roughly speaking, if S is the fiscal balance, S_{p} the primary balance, i the interest rate on public debt and D that debt, then the relationship is as follows:
S_{p} = S + iD
(Cf. also Ecalle’s explanatory note where he uses this relation.)

Translator’s note on references:
The French term “solde primaire stabilisant” is translated as “debtstabilizing primary balance.” The following is a reference for the term in English from a World Bank document:“DebtStabilizing Primary Balance (for Public Debt) : The primary surplus needed to keep debttoGDP constant … proportional to the gap between the real interest rate and real growth rate (closed economy)."
(Cf. also slide 15 of this IMF presentation.) 
Translator’s note on references:
The French term “taux d’intérêt apparent” is translated as “apparent interest rate” or “implicit interest rate” which is explained as the “average cost of government debt” or “the average interest rate that governments pay on their debt”.(Cf. also the references mentioned in Translation Note 3 in Part I of my translation of this article.) ↩︎

Translator’s note on references:
The following are two references that offer a lucid discussion of the term “interest rategrowth differential (i – g)”:(i) https://www.suerf.org/policynotes/19679/interestrategrowthdifferentialsongovernmentdebtanempiricalinvestigationfortheeuroarea ;
(ii) https://www.ecb.europa.eu/pub/economicbulletin/focus/2019/html/ecb.ebbox201902_06~0c96ee6f7c.en.html. 
Translator’s synthetic note on the derivation of this equation:
Ecalle presents a highly simplified derivation of this equation in another note on the Fipeco website. His derivation is reproduced summarily below:
The debttoGDP ratio, Ecalle writes, is stable if the debt’s growth rate equals the nominal GDP growth rate, that is:(dD/D) = (dY/Y) —–(1)
where d represents the variation of the relevant variable (D or Y here) over a period.
Here,
dD = the fiscal balance at the end of the period, which may be represented as S in case of a surplus or (–S) in case of a deficit.
Substituting for dD and assuming a fiscal deficit, equation (1) becomes:
(–S/D) = (dY/Y)
Multiply and divide the lefthand side of the above equation by Y, and rearrange the terms. This gives us:
(–S/Y) (Y/D) = (dY/Y)
=>
(–S/Y) = (dY/Y) (D/Y)—–(2)
Now, the relationship between the fiscal balance (S) and the primary balance (S_{p}) is as follows:
S_{p} = (S + iD) (cf. footnote 1 above),
=>
S = (S_{p} – iD).
If the balance is a deficit, then as per our convention mentioned above, we use (S). We therefore have:
(–S) = (S_{p} – iD)—–(3)
If the primary balance or deficit here is the debtstabilizing value, then we may replace S_{p} by S_{ps} (viz. the debtstabilizing primary balance) and substitute equation (3) in (2). This gives us:
– ( S_{ps} – iD)/Y = (dY/Y) (D/Y)
which yields:
(S_{ps} / Y) = (iD/Y) – (dY/Y)(D/Y)
Now, (dY/Y) is g, the nominal GDP growth rate. Factoring out the common factor (D/Y) on the righthand side, we thus have the final equation:
(S_{ps} / Y) = (i – g ) (D/Y).