Public debt/Addendum

Let D and Y be the levels of public debt and GDP at the begin of a year; and,
let F be the primary, or discretionary budget deficit (the total deficit excluding interest payments) and,
let r be the annual rate of interest payable on the public debt.

ΔD/Y = (D + F + Dr)/{Y(1 + g)} - D/Y

let 1/{Y(1 + g)} = A

then:

= A(D + F + Dr) - D/Y

= A( D + F + Dr  - D/AY)

= A( D + F + Dr - D - Dg)

substituting for A:

= {F + D(r - g)}/{Y(1 + g)}

or, approximately:-

= {F + D(r - g)

= F/Y +(r - g)D/Y

putting  F/Y = f   and  D/Y = d:

= f + d(r - g)