# Public Debt

## Ricard's Neutrality Theorem

### A Two-Period Model

In order to investigate the burden of public debt, it is useful to explain Ricard’s neutrality theorem. We employ a simple two-period model as in Chap. 3. In this regard, a household optimizes consumption for two periods, namely period 1 (current period) and period 2 (future period).

The household’s utility function is given as

where c1 represents consumption in period 1 and c2 represents consumption in period 2. The agent earns income Y_{1} in period 1 and consumes or saves. In period 2, the agent consumes from savings and interest on savings.

The government issues public debt in period 1 and the private agents buys it. In period 2, the government redeems the public debt with interest. The government imposes taxes in period 1, T_{1}, and period 2, T_{2}. For simplicity, the government does not spend. Thus, G_{1} = G_{2} = 0. This is not a crucial assumption. As long as government spending is fixed, the analytical result should be the same.

Thus, the budget constraints for the agent and government are written as follows.

© Springer Science+Business Media Singapore 2017

T. Ihori, *Principles of Public Finance,* Springer Texts in Business and Economics, DOI 10.1007/978-981-10-2389-7_4

where s denotes savings for private assets and b denotes public debt issuance; Tj and T_{2} denote taxes in period 1 and period 2 respectively; Yj denotes income in period 1; and r is the rate of interest. Eqs. (4.2) and (4.3) summarize the private budget constraints and Eqs. (4.4) and (4.5) summarize the government budget constraints in the two periods.

Households buy public debt as a means of saving. Since we assume that public bonds and private saving are perfect substitutes without uncertainty for the agent, public debt should have the same rate of return as a private asset in the market. Thus, the rate of interest for public debt is equal to the rate of interest for a private asset. Consequently, households are indifferent about using either public debt or capital as a means of saving.

Ifb > 0, from the government budget constraint in period 1, Eq. (4.4), Tj < 0. In other words, the government issues debt so as to reduce taxes (or give subsidies) in period 1 and raises taxes in period 2 so as to redeem debt (T_{2} > 0). Note that the government does not spend at all: G_{1} = G_{2} = 0.

From Eqs. (4.2), (4.3), (4.4), and (4.5), household budget constraint and government budget constraint in terms of present value are given respectively as

Substituting Eq. (4.7) into Eq. (4.6), we have as the integrated budget constraint

A household determines optimal saving/consumption behavior so as to maximize the lifetime utility given by Eq. (4.1) subject to Eq. (4.8). Since public debt b does not appear in Eq. (4.8), any changes in b would not affect the consumption/ saving behavior of households at all. In other words, public debt is neutral with respect to economic variables. This is called the debt neutrality theorem or Ricardian equivalence.

Consider the following numerical example. Suppose T_{1} = —30, T_{2} = 33, G_{1} = G_{2} = 0, and r = 0.1. Thus, from the government budget constraint, b = 30. Further, the permanent tax revenue T_{p} is 0. The government then reduces T_{1} from —30 to —50 by raising debt issuance b from 30 to 50. T_{2} becomes 55 but does not affect T_{p}; hence, consumption does not change either.

Suppose that T_{1} = 10, T_{2} = 50, G_{1} = 30, G_{2} = 30, and r = 0. Then, from the government budget constraint, b = 20. Further, the permanent tax revenue T_{p} is 30. The government then reduces T_{1} from 10 to 5 by raising debt issuance b from 20 to 25. This does not affect T_{p}; hence, consumption does not change either. In this circumstance, only private demand for b is raised by the increase of 5 in the amount of new debt issuance.