Fixed Deficit Ratio

Abstract

To illustrate the basic idea, consider a simple model of public debt dynamics. The government borrows a certain fraction of national income B = bY. Here B denotes the budget deficit, Y is national income, and b is the deficit ratio. Strictly speaking, the government fixes the deficit ratio. The budget deficit in turn augments public debt $\dot{D}=B$, where D is public debt and the dot symbolizes the time derivative. This implies $\dot{D}=bY$ . Now it is convenient to do the analysis in per capita terms. N stands for labour, d = D/N is public debt per head and y = Y/N is income per head. Next take the time derivative of public debt per head $\dot{d}=\dot{D}/N-D\dot{N}/{{N}^{2}}$, observing $\dot{D}=bY$ . Moreover let labour grow at a constant rate N = nN, with n being the natural rate. From this follows:

$$\dot{d}=by-nd$$

(1)

Here b, n and y are given exogenously. In the steady state, public debt per head does no longer move $\dot{d}=0,$ which yields:

$$d=by/n$$

(2)

Finally have a look at stability. Differentiate (1) for d to get $d\dot{d}/dd=-n<0$. Therefore the steady state will be restored automatically. In the real world, however, income per head appears to be endogenous. An increase in the deficit ratio, for example, will reduce investment, capital and thus output.