# A3 Second Best Solution

We are now ready to investigate normative aspects of distortionary tax policy. From this point on, we do not impose lump sum taxes: T = *T ^{2} =* 0. First, let us investigate the situation where all the consumer prices, q

_{1}, q

_{2}, and q

_{3}, are optimally chosen. In other words, we assume that the government can choose consumption taxes, wage income taxes, and capital income taxes optimally although lump sum taxes are not available.

The maximization problem may be solved in two stages. In the first stage, one can choose {r_{t+1}} and {q_{t} = (q_{1t}, q_{2t+1}, q_{3t})} (t = 0, 1, ...) so as to maximize W. In the second stage, one can choose (т, *у,* в) to satisfy Eqs. (9.A5.1), (9.A5.2), and (9. A5.3). Thus, our main concern here is with the first stage problem. The optimization problem is solved in terms of the consumer price vector. The actual tax rates affect the problem only through the consumer price vector.

In other words, the problem is to maximize

Equations (9.A10) and (9.A13) both have zero degree with respect to the q vector, but Eq. (9.A12) does not. We consider the problem as follows. The maximum W is subject to Eqs. (9.A10) and (9.A13), and q_{t} is uniquely determined to a proportionality. Then, Eq. (9.A12) gives the level of q_{2t+1}, which is consistent with the solution of our main problem. Thus, we obtain

Differentiating the Lagrangian function, (9.A17), with respect to q_{1t}, q_{2t+1}, and q_{3t}, we have

Differentiating with respect to r_{t+1}, we obtain
In a steady state, Eq. (9.A20) means

Considering Eq. (9.A20) and the homogeneity condition (^^= qj-Ey *=* 0), Eq. (9. A19) in the steady state reduces to

or

Equation (9.A21) is the modified Ramsey rule, an extension of the standard Ramsey rule for the intertemporal setting with p. Note that p appears in the second-period excess burden in Eq. (9.A21). Hence, when all consumer prices are available, the optimality condition is given by the modified golden rule, Eq. (9.A16), and the modified Ramsey rule, Eq. (9.A21) (see Ihori 1981).