# The Government's Budget Constraint

Following this definition, permanent levels of government spending G and taxes T are given respectively as G_{p} and T_{p}, which satisfy the following equations.

where G_{1} and T_{1} are government spending and taxes for period 1 respectively, and G2 and T2 are government spending and taxes for period 2 respectively.

Now, the government’s budget constraints in both periods are given as

where B is one-period maturity public debt.

When a budget deficit occurs in period 1 as an amount of G_{1} — T_{1}, one-period public debt B is issued in accordance with the amount. Since one-period maturity public debt has to be redeemed in period 2, the amount of (1 + r)B is needed for debt redemption. In addition, government spending in period 2, G_{2}, needs money. The total spending, (1 + r)B + G_{2} in period 2 must be financed by raising taxes. Since period 2 is the end period in the two-period model, the government cannot issue new bonds in period 2.

Eliminating B from Eqs. (3.2) and (3.3), government budget constraint in the present value is given as

Using the notation of permanent variables, this equation is equivalent to

The permanent level of government spending must be equal to the permanent level of tax revenue, irrespective of the size of bond issuance.

Figure 3.2 shows this relationship. In this figure, point G on line AB associated with Eq. (3.4) denotes the actual combination of government spending (G_{1}, G_{2}), while point T shows the actual combination of taxes (T_{1}, T_{2}). Point E denotes the permanent level. Since the present value is the same between spending and taxes, the permanent level is also the same between spending and taxes. The government may choose any combination of government spending and taxes on line AB. Point G could be different from point T; however, Eqs. (3.4) or (3.4') requires that G and T are on the same budget line AB. The government may choose B under this constraint.

Fig. 3.2 **The government’s budget constraint**

Debt issuance, B, corresponds to the gap between point G and point T. By changing B, the government may choose any point T on AB at the given level of G. However, since E does not change, we always have G_{p} = T_{p}. This means that the present value amount of the tax burden is determined by the present value amount of government spending, independent of the amount of public debt.

As explained later, rational households are indifferent about the financing method as long as government spending is fixed. As explained in Chap. 4, this corresponds to the Ricardian neutrality theorem of public debt.