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 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 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. Thus the debt trap applies only to that part of the debt that is undertaken for other reasons, and that failure to take up successful investment opportunities imposes an opportunity cost on future generations.

The second qualification concerns the possibility the growth rate, g, could be influenced by deficits contributing to an increase in public debt. At times of impending recession

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])