A4.2 The Elasticity Term
Assume for simplicity of interpretation that the cross-substitution effects are zero ( Ey = 0 for i = j). Then, from Eq. (9.A21), we have in the elasticity form
where ei is the effective tax rate (ti/qi) and оу is compensated elasticity (qjEij/Ei).
If labor supply is completely inelastic (along the compensated supply curve), the optimal tax on second-period consumption is zero, while the tax on labor income is equivalent to a lump sum tax and could be set arbitrarily high. If, however, the demand for future consumption is inelastic, the argument is reversed, and future income is the ideal tax base from an efficient view.
In general, the optimal rate of effective tax, ei, depends upon the relative magnitudes of the elasticities. There is no particular reason to believe that the optimal rate should be the same for the three sources of the tax base. This interpretation carries over, with appropriate modifications, to the situation of non-zero cross-elasticities.
A4.3 The Implicit Separability Condition
Considering the homogeneity condition in elasticity terms,
2, 3), Eq. (9.A21) may be reduced to
If ff13 = ff23, Eq. (9.A23) is reduced to
which implies ^e2 = e1. Considering Eqs. (9.A9.1), (9.A9.2), and (9.A16), we obtain q1 = (1 +r)q2. Substituting Eqs. (9.A5.1) and (9.A5.2) into the above equation, we finally have в = 0. Thus, the optimal tax on interest income is zero. ff13 = ff23 is called the implicit separability condition. If this condition is satisfied, the government should not impose interest income tax.