# Fiscal arithmetic

I just came across this wonderfully concise explanation of the debt growth formula on Quora:

Notation:

D = debt, in % of GDP

r = real interest rate, in %

g = real growth rate of the economy

pd = primary deficit of the government budget, in % of GDP. Primary deficit does not include interest expenditures.

Say you start at your optimal debt level D. Your debt next year will be (1+r)D + pd. Your debt grows by the interest rate plus any additional borrowing pd. If you want to keep D constant relative to GDP, you want it to grow at rate g.

(1+r)D + pd = (1+g)D

solving for pd:

pd = (g-r)D

Now plug in values that were typical in developed countries in 80s and 90s. D=60, g=3, r=2. You get pd=0.6.

Add interest expenditure to the primary deficit: 0.02x60+0.6. You get headline deficit 1.8% GDP. Add some margin for errors and bad times, for example equal to normal interest expenditure. You get 3% deficit.

That explanation is from Martin Suster, and I like it a lot.

The point is that responsible leaders maintain debt (D) relatively stable (or at least sustainable). While it is the stock of debt that matters the most, flow quantities like the magnitude of the deficit tell us "where a country is going" unless an adjustment to revenues and spending is made soon.

I would just add that economic growth has salutary effects beyond the mechanical effect shown above: it not only raises the denominator (GDP), it also increases tax revenues, so decreasing the "pd" term is easier in years when GDP growth is high.

As the paper I wrote with Paolo Mauro shows, we can think of deviations from stable primary balances as "new measures" and assess their impact directly:

A more complete accounting of the impact of growth on the debt ratio recognizes that the primary surplus (as a share of GDP) itself depends on economic growth. Absent policy action, it is reasonable to assume that revenues rise in line with nominal GDP whereas primary expenditures rise in line with the GDP deflator. For example, a neutral policy approach would be for the government to raise civil servantsâ€™ wages and pensions in line with inflation ...

... If the primary surplus rises by more than implied by the expenditure ratio erosion term, policy measures (tax hikes
or real expenditure cuts) must account for the difference. Of course, higher revenues resulting from economic growth give governments the option to spend more: Increases in revenues are often accompanied by additional expenditures. But raising expenditures (as opposed to reducing the debt) is a policy choice enabled by economic growth. It is therefore appropriate to consider the erosion of the primary expenditure to GDP ratio attributable to economic growth as part of the overall contribution of economic growth to reducing the debt ratio.

### Reference

Paolo Mauro & Jan Zilinsky, 2016. "Reducing Government Debt Ratios in an Era of Low Growth," Policy Briefs PB16-10, Peterson Institute for International Economics.