3.1 Optimal Policy
Suppose that, as before, y1 = 5000, b1 = 2000, β = 600 and gy = gb = r. Hence τ1 = τ2 = 0.4. There are known probabilities that an event will take place costing either CL = 600 and CH = 1000. These are 11 per cent and nineteen per cent respectively of income, y2 = 5250, in the second period. For γ > 0.6, the fund available in the second period exceeds the low-cost contingency.
Figure 4 shows the optimal values of γ for variations in the two probabilities, p1 and p2, for a risk aversion parameter of ε = 0.1. Again each three-dimensional diagram shows two perspectives of the same results. The diagram show that, even with very high p2, the fund never exceeds the smallest of the two possible costs, CL. When p2 is low there is a range of values of p1 for which γ = 0 is optimal. As with the simpler model of the previous section, comparisons (not shown here) show that the degree of risk aversion makes very little difference. The main influence on the optimal value of γ is the probability, p2. Figure 5 shows, for the corresponding case, the variation in the overall tax rate in the first period. As mentioned above, a value of γ = 0 corresponds to τ1 = 0.4. As in the simple model considered earlier, the optimal value of γ shows very little sensitivity with respect to risk aversion.
- Figure 4: Optimal Gamma for Variations in Two Probabilities: Risk Aversion of 0.1
- Figure 5: Optimal Tax Rate in Period 1: Risk Aversion of 0.1
- Figure 6: Optimal Gamma with p1+p2=1
The analysis has not restricted the two probabilities to sum to 1. A modification is to suppose that an event will certainly happen in period that gives rise to an uncertain cost. Figure 6 shows the results from setting p2 = 1 - p1, so it is possible to show the effect on optimal γ of the joint variation in p1 and ε. The results are, perhaps not surprisingly, similar to the case of a single cost above, except that γ > 0 is optimal for all combinations. Again risk aversion has little effect and p2 dominates.
