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                 Δd  =  f + d(r - g)

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

The effect of financial support

(Guidance based on Government Finance Statistics Manual (GFSM 2001)) The following is the recommended treatment of the impact on the government balance of the main financial support operations: Capital grants: Increase the deficit by the amount of the grant. Equity purchases: Have no impact on the fiscal balance, if purchase is at market value, but increase government gross debt. Raise the deficit by any marked/undisputable excess of what the government pays over the value of the equity. Asset purchases/swaps: Same as equity purchases. Loans: Have no immediate impact on the fiscal balance if there is no inherent subsidy, but increase government debt. Reduce the balance by any amount that the government cannot expect to be repaid. Guarantees: Have no immediate impact on the fiscal balance or debt unless there is a significant probability the guarantee will be called (in practice when a reserve has been created). In other cases, the fiscal balance would weaken and debt increase if and when the guarantee is called. Associated fees, interest, and dividends: Affect the deficit in the same way as other government income or expense. Central bank operations: Are reflected in its own balance sheet and income statement, rather than those of the government. However, losses on these operations will affect the budget over time, as they affect profit transfers or necessitate recapitalization. For transparency and to facilitate policy decision making, these operations should be disclosed, possibly as complementary information in the budget.