User:Nick Gardner/Public debt/Tutorials

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Implications of the debt trap identity

The 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,

(for proof of the identity, see 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 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, by definition, be self-financing and would therefore not require a surplus for sustainability. Thus the debt trap applies only to that part of the debt that is undertaken for other reasons.

International comparisons

( % of GDP )

	Japan	Italy	France	Germany	United States	United Kingdom	China	Australia
2007	195	107	65	65	62	44	20	9
2014 est	222	118	79	77	100	76	19	4

(Source: IMF [2])