Fiscal policy/Tutorials

Fiscal sustainability

The debt trap identity

According to the debt trap identity, 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.

(for proof of the identity, see paragraph 2 of the addendum subpage[1])

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 diferent 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.