The long-run sustainability of the government’s fiscal position depends on the interest rate it has to pay on its debt, the size of the government’s deficit after excluding interest payments, and the growth of the country’s nominal GDP. To express this formally let the current value of a country’s nominal debt at the end of the previous period be *D*(t-1). Let the country’s nominal interest rate be *i* and the growth in its nominal GDP be *g*. And let the government’s primary budget balance in any period be, in nominal terms, a constant proportion, b, of nominal GDP in that period. Note that a positive primary balance means a surplus, a negative balance a deficit. The country’s debt in nominal terms will therefore evolve according to the equation:

*D*(t) = [1+*i*]**D*(t-1) – b**GDP*(t)

Dividing through by *GDP*(t) gives the process for the debt to GDP ratio, *d*(t), as,

*d*(t) = [1+*i*]**D*(t-1)/*GDP*(t) – b

If GDP grow at the constant rate *g* then we can write *GDP*(t)= [1+*g*]**GDP*(t-1). So we can re-write the process for *d*(t) as,

*d*(t) = [1+*i*]**d*(t-1)/[1+*g*] – b

The change in the debt to GDP ratio, *d*(t)-*d*(t-1) is therefore,

*d*(t)-*d*(t-1) = *d*(t-1)*[*i*–*g*]/[1+*g*] – b

The debt ratio will be constant if *d*(t)-*d*(t-1) = 0. So the requirement for a constant long-term debt ratio is that,

0 = *d*(t-1)*[*i*–*g*]/[1+*g*] – b

which we can re-write as,

b = *d*(t-1)*[*i*–*g*]/[1+*g*]

This implies the following:

- If the country’s interest rate exceeds the growth rate of its nominal GDP then for sustainability b must be positive, i.e. the government must run a primary surplus. The required surplus depends upon
*i*, g and the existing debt ratio. If the primary surplus is less than the required amount the debt ratio will continually rise and eventually become unsustainable. - If the country’s interest rate is less than the growth rate of its nominal GDP then the government can run a primary deficit such that its debt ratio neither rises nor falls. If its primary deficit is smaller than this then the debt ratio will gradually fall to zero.