We may explain the above arguments using government budget constraint. Thus,

where G is government spending, B is public debt, T is tax revenue, and r is the rate of interest. AB denotes the new issuance of public debt. In terms of per capita GDP (= Y), we have

where g = G/Y, b = B/Y, and t = T/Y. By definition, we have the following dynamics:

Thus, multiplying by b on both sides of Eq. (6.17), we have

Substituting Eq. (6.16) into Eq. (6.18), we finally derive the fundamental equation of dynamic government budget constraint:

where n = AY/Y is the growth rate. The first term of the right-hand side of Eq. (6.19) denotes the primary deficit and the second term denotes the difference between the interest rate and the growth rate multiplied by the debt/GDP ratio.

Some Special Cases

Based on Eq. (6.19), we consider some special cases.

g = t

In this case, Eq. (6.19) reduces to

If r > n, the government budget is unsustainable and vice versa. In order to maintain sustainability, we need the condition r < n. This is called the Domar condition. It clarifies the importance of the sign of r — n (see Domar 1944).

Figure 6.4a explains the dynamics of r > n. b increases as long as r > n. Further, Fig. 6.4b explains the opposite case of stable dynamics. Note that even if g > t and we have the primary deficit, the dynamics are qualitatively the same as long as g — t is fixed and independent of b. If g — t is fixed at any value, the same argument holds. However, if g — t is affected by b, the Domar condition is irrelevant. For this case, we need the Bohn condition.