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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

Steady state analysis

The debt trap identity establishes as 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).

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.

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

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 identity applies only to that part of the debt that yields social rather than financial returns This qualification is important because failure to take up financially successful investment opportunities in order to reduce the national debt future generations imposes opportunity costs, and may reduce fiscal stability. That may also be true in the longer term of successful social expenditures that yield gains in human capital and social capital.

However the debt trap identity, upon which those conclusions depend, applies only to a steady-state situation in which there are no significant changes in its variables of growth rate, debt level and discount rate. The following paragraph considers the problem of maintaining sustainability under other circumstances.

Dynamic effects

During periods of economic stability, and when liquidity is plentiful and domestic interest rates are low, investors tend to seek profit opportunities abroad, as a result of which debtor governments find it easy to borrow at modest rates of interest. However, an international economic downturn, or a credit crunch, or discount rate increase in their creditors' countries can threaten the fiscal sustainability, even of countries with relatively modest levels of national debt. The economic downturn may be transmitted to their economies and raise their budget deficits through the operation of their automatic stabilisers. A credit crunch or discount rate increase may make investors reluctant to roll-over their short-term debt, and the resulting fall in demand may raise the interest rate necessary to continue borrowing. In some cases the resulting reduction in their fiscal sustainability may prompt investors to demand the addition of a risk premium to the interest rate that they would otherwise accept - thus detracting further from their fiscal sustainability. And in extreme cases rumours of of impending sovereign default spread by a government's critics can generate a herding response that leads to a rapid escalation of their risk premium and a precipitous loss of sustainability.

Appendix: 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