# 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 e_{i} is the effective tax rate (t_{i}/q_{i}) and оу is compensated elasticity (qjE_{i}j/E_{i}).

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, e_{i}, 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 ff_{13} = ff_{23}, Eq. (9.A23) is reduced to

which implies ^e_{2} = e_{1}. Considering Eqs. (9.A9.1), (9.A9.2), and (9.A16), we obtain q_{1} = (1 +r)q_{2}. 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. ff_{13 }= ff_{23} is called the implicit separability condition. If this condition is satisfied, the government should not impose interest income tax.