# Appendix A: Optimal Taxation in an Overlapping-Generations Economy

# A1 The Optimal Tax Rule

## A1.1 Overlapping-Generations Growth Model

The theory of optimal taxation is one of the oldest topics of public finance. Originally, the studies centered on the theory of optimal consumption taxation. Ramsey (1927) published the first theoretical result, which proved to be the starting point of a great many studies on the subject and is well known today as the Ramsey tax rule, as explained in the main text of this chapter. This appendix examines the optimal combination of consumption taxes, labor income taxes, and capital income taxes in an overlapping-generations growth model, based on Diamond (1965). In the context of economic growth, we must also consider dynamic efficiency; namely, the golden rule. After investigating the first best solution, this appendix intends to clarify the relationship between the Ramsey rule and the golden rule when lump sum taxes are not available.

We apply the overlapping-generations model of the advanced study from Chap. 5, in which every individual lives for two periods. We extend this basic model by incorporating several distortionary taxes and by allowing for endogenous labor supply; otherwise, a labor income tax becomes a lump sum tax.

An individual living in generation t has the following utility function:

where c^{1} is the individual’s first-period consumption, c^{2} is second period consumption, and x = (Z — l) — Z *=—l* is first-period net leisure. Z is the initial endowment of labor supply.

The individual’s consumption, saving, and labor supply programs are restricted by the following first- and second-period budget constraints:

where т is the consumption tax rate, *у* is the tax rate on labor income, w is the real wage rate, s is the individual’s real saving, r is the real rate of interest, and в is the tax rate on capital income. T^{1} is the lump sum tax levied on the young in period t and TjHs the lump sum tax levied on the old in period t.

From Eqs. (9.A2) and (9.A3), the individual’s lifetime budget constraint reduces to

where q_{t} = (q_{lt}, q_{2t}+i, q_{3t}) is the consumer price vector for generation t. Thus, we have the following relationships between consumer prices and tax rates:

The present value of a lifetime lump sum tax payment on the individual of generation t (T_{t}) is given as

Equilibrium in the capital market is simply

where n is the rate of population growth and k is the capital/labor ratio.

The feasibility condition is in period t:

where g is the government’s expenditure per individual of the younger generation. The government budget constraint in period t is given as

Equation (9.A8) may be rewritten in terms of tax wedge:

where tj is a tax wedge and is given as

The tax wedge t_{i} is the difference between the consumer price q_{i} and the producer price. Note that labor income taxation means t_{3} < 0 since x < 0. Capital income taxation (q_{2} > 1/(1 + r)) means that the consumer price of c^{2} is greater than the producer price (t_{2} > 0). 1/q_{2} — 1 is the after-tax net rate of the return on saving. Thus, for T^{2} = 0, t_{2} is given as

On the left-hand side, we multiply (1 + n) because t_{2} is an effective tax rate on second-period consumption; hence, it is relevant for the older generation.

Observe that the government budget constraint, Eq. (9.A8), is consistent with the production feasibility condition, Eq. (9.A7). Namely, one of the three equations, Eqs. (9.A4), (9.A7), and (9.A8), is not an independent equation that can be derived by the other two equations.

## A1.2 Dual Approach

It is useful to formulate the optimal tax problem by using the dual approach. Let us write E(q_{t}, u_{t}) for the minimum expenditure necessary to attain utility level u_{t} when prices are q_{t} = (q_{1t}, q_{2t}+_{1}, q_{3t}). Then, we have

which implicitly defines the utility level of generation t as a function of consumer prices: q_{t} and lump sum taxes T_{t}. Eq. (9.A10) incorporates the lifetime budget constraint, Eq. (9.A4).

However, as in the advanced study of Chap. 5, Appendix, the factor price frontier is written as

where

From Eqs. (9.A3), (9.A5.4), (9.A6), (9.A10) and (9.A11), we can express the second-period budget constraint in terms of compensated demands as

I

E_{i} denotes the partial derivatives for the expenditure function with respect to price q_{i} (i = 1, 2, 3). Note that E_{3} = x = —l *<* 0. We call Eq. (9.A12) the compensated capital accumulation equation.

The production feasibility condition, Eq. (9.A7), is also rewritten in terms of compensated demands as