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Fiscal sustainability

Overview

The ultimate limit upon the size of the national debt is reached when more money is required for its repayment than the government can raise from taxation - at which point, the only alternative to a default amounting to national insolvency is by creating money for the purpose of repayment. Money creation aside, national insolvency is, in fact, an inevitable long-term outcome if national debt persistently grows faster than gdp. That is known as the debt trap, and its avoidance is the economic policy objective known as "fiscal sustainability". Fiscal sustainability normally requires the maintenance of a surplus of tax revenue over public expenditure, when expressed as fraction of the national debt, has to average a percentage of gdp least equal to the difference between the interest rate payable and the gdp growth rate.

Although that identity-based criterion would ensure fiscal sustainability in a stable, risk-free environment, it is generally accepted that a more stringent criterion is needed in order to guard against operating risks. A government's ability to market bonds offering any particular rate of return depends upon the willingness of operators in the bond market to accept that rate of return, and that willingness is influenced by market expectations. A comparatively recent development is the availability of credit default swaps that enable market operators to insure against the risk of sovereign default. Operators in the bond market may be expected to require an increase in the rate of return on a government's bonds sufficient to offset the premium (or sovereign spread} on such insurance. Sovereign spreads tend to be greater, the greater the level of national debt as a percentage of gdp, but they are also influenced by expectations of increases in that level, and by other factors, including expectations of gdp growth or decline. Most of the sovereign defaults of recent years have been due to banking and currency crises rather than to a previously excessive level of public debt

The debt trap identity

According to the debt trap identity (proved below), the annual increase in public debt as a percentage of GDP is given by:

Δd = f + d(r - g)

where d is public debt as a percentage of GDP
and f is the primary budget deficit (shown with a negative negative sign if a surplus) as a percentage of GDP,
and r is the interest rate payable on government debt.

Sustainability

A necessary condition for long-term sustainability is that Δd does not consistently exceed zero - since otherwise the interest due would eventually amount to a greater percentage of GDP than could conceivably be financed from taxation. The dept trap, implies, therefore, that
-   if the interest rate is greater than the growth rate, sustainability requires an average budget surplus ratio equal to at least d(r-g) and
-   if the growth rate exceeds the interest rate, it requires that the budget deficit ratio does not on average exceed d(g-r).

However, the identity embodies the implicit assumptions that deficits earn no return, and that they do not affect growth rates or interest rates.

Since many different combinations of r, g are possible the debt trap identity does not define a unique relation between the the debt/gdp ratio, d and the minimum value of surplus/gdp (or maximum value of the deficit/gdp) ratio, f that is necessary for sustainability, even under the assumptions that have been implicitly adopted.

Some light can nevertheless be thrown on the issues by inserting some typical values for r and g and by qualifying the implicit assumptions.

Interest rates on government bonds are usually greater than gdp growth rates, so an average budget surplus will usually be required for sustainability.
If, for example, r were 5% and g were 2% then - on the original assumptions - a debt of 50% of gdp would require an average surplus of 1.5% of gdp a debt of 100% of gdp would require an average surplus of 3% of gdp, and so forth.

The first qualification to those conclusions is that a deficit devoted exclusively to investments having positive net present values in financial terms would eventually, by definition, be self-financing and would therefore not require a surplus for sustainability. That means that the debt trap applies only to that part of the debt that is undertaken for other reasons. This qualification is important because failure to take up successful investment opportunities in order to reduce the national debt imposes an opportunity cost on future generations.

The second qualification concerns the possibility the growth rate, g, could be influenced by a deficit. At times of impending recession any avoidance of a drop in growth that could be achieved by a deficit would at least partially offset the increase in surplus that would subsequently be necessary for sustainability. (It could conceivably even reduce the required surplus. If, for example, an increase in the debt ratio from 50% to 100% averted the replacement of a 2% growth rate by a 2% rate of decline and the interest rate remained at 5%, it would reduce the required surplus from 3.5% to 3%.)

A third qualification is that r may not be independent of d. The size of the public debt may influence the interest rate that would have to be paid on it. That can happen if operators in the market for government bonds believe that there is an increased probability of default, in which case they will require a risk premium in addition to the rate of return that they would otherwise require. The size of the premium is indicated by the country's sovereign spread.

Lastly, it may be necessary to take account of expectations, both rational and irrational. A rationally-formed expectation of a reduction in g may increase the market value of r because of its effect upon default risk. More importantly, a rumour of increased default risk may be self-fulfilling because of herding behaviour in the bond market.

Proof of the debt trap identity

Let D and Y be the levels of public debt and GDP at the beginning of a year; and,
let F be the primary, or discretionary budget deficit (the total deficit excluding interest payments) and,
let r be the annual rate of interest payable on the public debt;
and assume that  F, r, and g are all mutually independent.

- then the public debt at the end of the year is  D₁ = D + F +Dr; the GDP at the end of the year is   Y₁ = Y(1 + g);
and the ratio of public debt to GDP has risen from  D/Y to  (D + F + Dr)/{Y(1 + g);

- thus the increase in the ratio of public debt to GDP in the course of a year is:

Δ(D/Y) = (D + F + Dr)/{Y(1 + g)} - D/Y

Let 1/{Y(1;+ g)} = A  andso that AY = 1/(1 + g) ,and  1/AY = 1 + g  
- then:

Δ(D/Y) = A(D + F + Dr) - D/Y

=  A( D + F + Dr  - D/AY)

- and substituting 1 + g for 1/AY:

=  A( D + F + Dr - D - Dg)

substituting for A:

Δ(D/Y) = {F + D(r - g)}/{Y(1 + g)}

or, approximately:-

Δ(D/Y) = {F + D(r - g)}/Y

= F/Y + (r - g)D/Y

Let  f = F/Y ,and d = D/Y

- then                 Δd  =  f + d(r - g)

where f is the primary budget deficit as a percentage of GDP, and d is public debt as a percentage of GDP