Public debt/Addendum: Difference between revisions
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==Proof of the debt trap identity== | |||
<small> | <small> | ||
Let D and Y be the levels of public debt and GDP at the beginning of a year; and,<br> let F be the primary, or discretionary budget deficit (the total deficit excluding interest payments) and,<br> let r be the annual rate of interest payable on the public debt; | Let D and Y be the levels of public debt and GDP at the beginning of a year; and,<br> let F be the primary, or discretionary budget deficit (the total deficit excluding interest payments) and,<br> let r be the annual rate of interest payable on the public debt; | ||
then the public debt at the end of the year is D<sub>1</sub> = D + F +Dr; the GDP at the end of the year is Y<sub>1</sub> = Y(1 + g); <br>and the ratio of public debt to GDP has risen from D/Y to (D + F + Dr)/{Y(1 + g) | - then the public debt at the end of the year is D<sub>1</sub> = D + F +Dr; the GDP at the end of the year is Y<sub>1</sub> = Y(1 + g); <br>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 | :::Δ(D/Y) = (D + F + Dr)/{Y(1 + g)} - D/Y | ||
Let 1/{Y(1;+ g)} = A | |||
 <br> | |||
- then:<br> | |||
  | |||
then: | |||
:::Δ(D/Y) = A(D + F + Dr) - D/Y | :::Δ(D/Y) = A(D + F + Dr) - D/Y | ||
::::= A( D + F + Dr - D/AY) | ::::= A( D + F + Dr - D/AY) | ||
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:::Δ(D/Y) = {F + D(r - g)}/{Y(1 + g)} | :::Δ(D/Y) = {F + D(r - g)}/{Y(1 + g)} | ||
or, approximately:- | or, approximately:- | ||
:::Δ(D/Y) = {F + D(r - g) | :::Δ(D/Y) = {F + D(r - g)}/Y | ||
Let f = F/Y ,and d = D/Y<br> | |||
- 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 | |||
</small> | </small> | ||
Revision as of 06:00, 21 March 2009
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
 
- then:
- Δ(D/Y) = A(D + F + Dr) - D/Y
- = A( D + F + Dr - D/AY)
- = A( D + F + Dr - D - Dg)
- Δ(D/Y) = A(D + F + Dr) - D/Y
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