2.4 Optimal Policy
Suppose that y1 = 5000, b1 = 2000, β = 600 and gy = gb = r. These assumptions imply that the tax rate needed to finance the bs is constant at τ1 = τ2 = 0.4. Let C = 1500; this is chosen to be a large proportion - just over twenty eight per cent - of the tax base in the second period. Hence if no action is taken in the first period (γ = 0) a very large increase is required in the second period tax rate if C needs to be financed. The top segment of Figure 1 shows the way in which the optimal value of γ varies as the values of ε and p vary. This clearly shows that the decision depends strongly on the probability of the event taking place, and is not sensitive to the degree of risk aversion. Even for high values of ε, the optima value of γ remains low for lower values of p. Hence in this simple framework it appears that the attitude of the judge to risk is less important than the perceived degree of uncertainty attached to the outcome.
- Figure 1: Optimal Gamma for Variations in Probability and Risk Aversion: Cost of 1500 and 500
Reductions in the potential cost, C, have a large effect on the optimal policy. For example, the lower segment of Figure 1 shows the results if C is reduced to 500. It can be seen that the optimal γ remains at zero for p values up to 0.3, and even with p = 0.75 gamma is only around 0.23 with tax rates in first and second periods of 0.425 and 0.47. Again there is little variation with respect to risk aversion. Here the certainty case does not imply complete tax smoothing, as the optimal rates in the two periods are 0.435 and 0.46. A higher value of C = 800 implies that for p = 0.75 the optimal γ = 0.32, with tax rates of 0.45 and 0.50 in first and second periods respectively.
The effect of a change in the rate of interest has opposing tendencies. Despite the higher discounting of the second period, the higher rate means that, for the same value of γ, a larger fund is accumulated in the second period. This means that tax rates can be lower, hence excess burdens are lower, and the overall effect is to raise the optimal value of γ very slightly. [14]
Notes
- [14] A further characteristic of the models examined here is that, despite the fact that they display unique optimal policies (for given parameters), the concavity of the expected welfare function is quite low. This suggests that the cost of a sub-optimal policy may not be large. Armstrong et al. (2007, pg.31), when considering demographic uncertainty, found that, ‘the extent to which the government can mitigate its effects is relatively small'.
