Appendix 1: Debt dynamics
Sovereign debt dynamics relate to a range of interrelated variables.
To remain solvent a government must ensure that the net present value (NPV) of government revenue net of any initial indebtedness exceeds the NPV of its primary expenditure. Primary expenditure is expenditure less debt servicing costs.
(1)
Where E is government expenditure, r is the real interest rate, Ris revenue, and D is debt.
The solvency constraint binds the government over long time periods. In the interim the change in government debt as a proportion of GDP (Y) can be modelled as a function of interest rates (r), growth rates (g), initial debt levels as a percent of GDP, and the primary balance (pb). The primary balance is revenue less spending net of interest payments.
(2)
To stabilise debt as a proportion of GDP a government requires a primary balance that covers the growth adjusted interest payments on debt. If growth is positive debt can still grow in nominal terms, while remaining stable as a proportion of GDP.
(3)
The calculation can be rearranged to calculate the primary balance needed to stabilise debt over a specific time frame. Let:
(4)
To reach a specific primary balance (
) in a specific (n) number of periods the government would need to run a primary balance sufficient to satisfy equation (5)
(5)
The impact of growth can be seen in equation (2). Growth increases the denominator (Y) which lowers debt as a proportion of GDP. Conversely policies that decrease growth can create potentially explosive debt dynamics.
Y = C + I + G + Ex - Im (6)
GDP in equation (6) is defined as the sum of consumption, investment, net government spending, and exports less imports. An economic shock as occurred in 2007/08 can lower consumption, investment, and net trade receipts. As a result government debt as a percent of GDP increases as output declines.
A government response, all else equal, that reduces net government spending (G) further decreases economic output. The final impact of fiscal consolidation on growth depends on the extent to which fiscal consolidation drives changes in other variables in equation (6). For example, lower government spending may reduce crowding out creating an offsetting increase in net investment.
