# A3 Government Debt and Intergenerational Transfer

## A3.1 The Transfer Program

We shall assume that the government issues debt b_{t} to the younger generation in period t. This debt has one-period maturity and will be repaid in the next period with interest at the same rate of return as on capital. b can be negative, in which case b means “negative debt”; namely, the government lends b to each individual of the younger generation and will recover this credit with interest.

Let us denote the (per-capita) lump sum tax levied on the younger generation and the older generation in period t by T^{1} and Tj^{2} respectively. Suppose for simplicity that the government does not make any public expenditure. Then, the government budget constraint in period t is

where N_{t} is the number of people in generation t.

The following cases are of considerable interest.

- (a) T
^{2}= 0. The tax collected to finance interest costs minus new debt issuance is a lump sum tax on the younger generation. This debt issue corresponds to Diamond’s internal debt. - (b) T
^{1}= 0. The tax collected to finance interest costs minus new debt issuance is a lump sum tax on the older generation. - (c) b = 0. The government does not issue debt. The government levies the lump sum tax T
^{1}on the younger generation and transfers it to the older generation in the same period. This corresponds to the unfunded pay-as-you-go system.

The private budget constraints of generation t, (4.A1) and (4.A2), are rewritten as follows:

Each individual’s lifetime disposable income (W_{t}) is given by (w_{t} — T^ and her or his disposable income in the younger period t minus (T^{2}t+1/(1 + r_{t+1})) the present value of the tax in the older period t+ 1. Thus, the lifetime budget constraint (4.A3) is rewritten as

^{where} w t = *W _{t}*

^{—}Г,

^{1 —}^

^{t2}+1.

Considering (4.A3^{0}), capital accumulation equation (4.A11) may be rewritten as

Let us define effective taxes by

t^{1} and t^{2} are net receipts from the young and old. These two equations (4.A14.1 and 4.A14.2) are government budget constraints in period t and period t+1. Thus, dynamic equilibrium can be summarized by the following two equations:

Equation (4.A15) is the government budget constraint. Equation (4.A16) comes from Eq. (4.A13) and is the capital accumulation equation. b, T^{1}, and T^{2} do not appear in these two equations.

In other words, fiscal action is comprehensively summarized by a sequence of effective taxes {t/} and {t2}. One of b, T^{1}, and T^{2} is redundant in order to attain any fiscal policy. The three cases (a), (b), and (c) are equivalent so long as two of b, T^{1}, and T^{2} are adjusted to attain the same {t/ } and {tj^{2}}. In cases (a) and (b), the government budget is not balanced. But in case (c), the government budget is balanced since b = 0. This means that the government deficit is not a useful policy indicator to summarize fiscal action. This is supported by Kotlikoff (1992), who said that if lump sum taxes are appropriately adjusted, debt policy is not effective and the government deficit is a meaningless policy indicator.