# B4 The Tax Timing Effect

## B4.1 The Welfare Implication of the Tax Timing Effect

Let us investigate the welfare aspect of the tax timing effect. In order to analyze the welfare of each generation explicitly, it is useful to employ the expenditure function approach as in Appendix A. The system is summarized by

where E[.] denotes the expenditure function and E_{2}[.] denotes the compensated demand function for second-period consumption. Differentiating Eqs. (9.B7) and (9.B8) comprehensively, we have

where *E _{u} = dE/du_{t}, E_{2u} = dE_{2}/du_{t}* and

*E*

_{22}= dE_{2}/d(^_{1}+

^{1}r ^. Hence,

where Д is the determinant of the matrix of the left-hand side of Eq. (9.B9). In addition, we have

Under the global stability condition, 0 < dr_{t+1}/dr_{t} < 1 at the steady state solution. Hence, Д> 0. The sign of [.] in Eq. (9.B10) is positive if the elasticity of substitution between labor and capital is large, which is consistent with the stability condition, Eq. (9.B3). In such an instance, a higher capital endowment given to an individual’s generation makes her or his lifetime utility higher. An increase in k_{t }raises w_{t} and lowers r_{t+1}. The former effect increases u_{t}, while the latter effect decreases u_{t}. If the elasticity of substitution is large, a decrease in r_{t} raises w_{t }significantly. The net effect is likely to increase u_{t} under the stability condition.

Thus, on the transitional growth process where capital accumulation is monotonously increased, each generation’s lifetime utility is monotonously increased. Note that this favorable tax timing effect works, irrespective of the sign of r — n. Consequently, generation j’s extra saving is favorable for those future generations that are close to generation j. For distant future generations, generation j’s extra saving is unimportant. In this sense, the temporary tax timing effect is relevant only to future generations that are close to the present. The utility of distant generations depends upon whether long-run equilibrium is closer to the golden rule because of the tax reform than before. Hence, if r > n, the tax reform (T^{1} ! *T ^{2})* is favorable for distant future generations from the viewpoint of the tax postponement effect and the permanent tax timing effect.