B Tax Smoothing under Certainty
Consider the simple model of Section 2 where, instead of uncertainty, it is known for certain that the additional cost of C will arise in the second period. In this case the tax rate in the first period is:
and the rate in the second period is:
The evaluation function is thus:
writing 
and Wi = W {Yi}, this is maximised when:
so that:
Furthermore:
The left hand side of (B.5) involves a ratio of marginal valuations for the two periods and the right hand side involves a ratio of ‘marginal costs’ of changing γ. In a two-dimensional diagram the optimum is represented by a tangency between an iso-welfare line and a nonlinear ‘relative price’ line. The value of γ that achieves this tangency solution gives the optimal degree of tax smoothing. This nonlinear equation can be solved numerically.