Public debt/Addendum

Proof of the debt trap identity

Let D and Y be the levels of public debt and GDP at the beginning 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;

- then the public debt at the end of the year is  D₁ = D + F +Dr; the GDP at the end of the year is   Y₁ = Y(1 + g);
and the ratio of public debt to GDP has risen from  D/Y to  (D + F + Dr)/{Y(1 + g);

- thus the increase in the ratio of public debt to GDP in the course of a year is:

Δ(D/Y) = (D + F + Dr)/{Y(1 + g)} - D/Y

Let 1/{Y(1;+ g)} = A &nbsp
- then:

Δ(D/Y) = A(D + F + Dr) - D/Y

=  A( D + F + Dr  - D/AY)

=  A( D + F + Dr - D - Dg)

substituting for A:

Δ(D/Y) = {F + D(r - g)}/{Y(1 + g)}

or, approximately:-

Δ(D/Y) = {F + D(r - g)}/Y

Let  f = F/Y ,and d = D/Y

- then                 Δ(D/Y)%  =  f + d(r - g)

where f is the primary budget deficit as a percentage of GDP, and d is the initial public debt as a percentage of GDP