The ‘Critical Threshold’ for Fiscal Projections and Policy

A Note prepared in October 2012.


Keywords: Macroeconomics & Money; Regulation & Taxation;




I have argued there is a critical threshold where if the fiscal deficit exceeds it public debt grows unsustainability faster than the ability of the economy to service it (usually measured by GDP). This note quantifies the threshold into a simple formula.


The threshold is related to the size of the government deficit. It should be clearly understood that crossing the threshold, so that the deficit is too large, does not immediately precipitate the government accounts into crisis like the Greek one. The arrival of the crisis is incremental. Each period it gets harder to reverse the trend: harder to cut back on expenditure, harder to raise revenue, while debt servicing remorselessly rises (and interest rates, but initially slowly). Eventually a second threshold is met; the point where lenders are no longer willing to advance  loans, even at very high interest rates, because they do not expect them to be repaid. Providing the fiscal deficit is below the critical threshold this wont happen.


The threshold is a medium-to-long term notion so that it might be temporarily exceeded at some point during the business cycle; that is not unsustainable if on average the fiscal position remains below the threshold.


Note a slightly confusing element is that the threshold is measured by the primary surplus (see below) which typically needs to be positive. Add in the debt servicing and the current account is likely to be in deficit. Assessed by the primary surplus the accounts need to be above the threshold (higher). Assessed by the total fiscal deficit the accounts need to be below the threshold (lower).


In a way all this paper does is to argue that we need to pay more attention to the primary surplus. This is not to say the total fiscal deficit is irrelevant; just that the measure of the primary surplus is a powerful assistance to understanding the debt path.




First a few simple symbols to simplify exposition (such mathematics there is, is in the appendix).

D         = the public debt at the beginning of the period;

D*       = the public debt at the end of the period;

PS        = the primary surplus in the period (to be explained below);

r           = the nominal interest rate on public debt;

g          = the nominal growth rate of GDP.


The primary surplus is the current revenue less current expenditure (this could include investment which does not directly generate revenue). Debt servicing is excluded, so the primary surplus is not the same thing as the fiscal surplus.


There are two reasons why the primary surplus might be of interest. First, if a country decides to abandon servicing its debt (not something one would recommend) then it needs a primary surplus.  A primary deficit means that the government has to borrow to cover its expenditures even though it is not debt servicing its debt; such borrowing is unlikely to be forthcoming if it refuses to service old debt.


Second, the mathematics (such as it is) is simpler.


The Total Fiscal Deficit is the primary surplus less the debt servicing (mainly interest paid on debt).


The Critical Threshold Formula


The critical threshold is given by


PS        = (r-g)D.


If the primary surplus exceeds (r-g)D then the debt path is sustainable, if the primary surplus is less than (r-g)D then the debt path is potentially unsustainable with net debt growing faster than GDP.


Intuitively the formula makes sense.


In order to prevent an unsustainable debt path, the primary surplus needs to be higher when

– the interest rate is higher;


– the net debt is higher.


On the other hand, if the economy is growing faster the primary surplus can be a little lower lower without compromising the debt path.


A useful variation of the formula is given by


PS/Y = (r-g)d


(Note that the appendix describes PS at the critical threshold as PS#.)



Y         = GDP

d          = the Net Debt to GDP ratio (= D/Y)


An Example


The current Treasury long-term projections have

r           = 6.0% p.a.

g          = 4.5% p.a. (made up of 1.5% p.a. for productivity growth, 1.0% p.a. for population growth and 2.0% p.a. for price growth.)


There is no projected net debt to GDP ratio, but the current government target is 20%.


So the critical threshold is given by (6% – 4.5%) times a fifth or 0.3%.


Given a GDP of about $210b This amounts to about $630m.


The primary surplus forecast for the 2013 year is around minus $4billion (that is Crown expenditures before debt servicing exceed current revenues by around $4billion). However the Crown accounts are not cast into a form which makes an exact measure easy to elicit. In any case the government would argue that this is a cyclical effect and the government accounts are on a track to return to the primary balance to above the critical threshold. Additionally the government is borrowing at lower interest rates than the long term projection but the economy is growing slower. Even so it might seem that the Crown’s primary balance is too negative.




There is a similar critical threshold for the economy as a whole. The equivalent of the Crown’s fiscal surplus is the current account surplus excluding debt servicing (basically exports of goods and services less imports of goods and services). Assuming the same interest rate (although the private borrowing rate is likely to be higher) and observing a net public and private debt to GDP ratio of about 75% the critical threshold for the economy as a whole is an external primary surplus of $2.4b. In fact the external primary surplus is currently close to zero.




Critical thresholds are one useful way for thinking about a sustainable debt path, for the government or for the economy as a whole. They are the point at which the debt is a constant proportion of GDP.


If the deficit is larger than that indicated by the critical threshold, then debt is rising faster than GDP; if the deficit remains in this larger state then eventually the debt will rise to the point where there will be a debt crisis (similar to the current experience of Greece).


If the deficit is smaller than that indicated by the critical threshold, then debt is rising more slowly than GDP.


For short term macroeconomic purposes – whether for monitoring the fiscal position or the external position – it is useful to compare the actual position with the threshold – perhaps even to plot it over time.


For long term fiscal projections, which tend to track linearly or exponentially/log linearly, the point in time at which the deficit crosses through the critical threshold is a way of summarising the sustainability of the fiscal path. Ideally, given that the long term track does not include the business cycle, the ideal is that the point should never exist.

APPENDIX: The mathematical derivation of the formula.


Given the definitions it follows that


D* = (1+r)D – PS,


The critical threshold is at the point where


D* = (1+g)D


Which gives


PS# = (r-g)D.


Where PS# is the primary surplus at the critical threshold.