3 Uncertain Costs
The above simple model can be extended to allow for the case where there is some uncertainty about the cost of the event in addition to whether or not it occurs. Suppose that there is a probability p1 of the event taking place with a relatively low cost of CL, and a probability of p2 of the event taking place with a higher cost of CH. To allow for the chance that the event may not take place, p1 + p2 < 1. It was seen in the previous section that, since γ can be varied continuously from γ = 0, it is only necessary to consider the result of acting immediately. Here, γ is expressed as a proportion of the highest cost. Producing a fund of γCH in the second period requires saving γCH/(1 + r) in the first period. It is first necessary to consider the different tax rates. The tax rate in the first period is:
(24)
If the event does happen, the tax rate applicable to the low-cost outcome is:
(25)
and in the case where γCH > CL, then τL,2 < τ2. For the high-cost outcome:
(26)
If the event does not occur at all, the tax rate in the second period is:
(27)
The expected value of the welfare function (using the basic iso-elastic function of subsection 2.1) is:
(28)