4 An Option to Reduce other Expenditure

Consider the simple model of section (2) but suppose that the extra cost C, if the need arises, can be met either from additional taxation in the second period or from a combination of reducing b₂ and raising additional tax revenue in period 2. If the choice is made to take precautionary action in the first period by contributing to a fund obtained from additional tax in the first period, the fund may be designed to meet a only proportion, γ, of the possible cost, C, the remaining cost being met from additional taxation in the second period if the event actually occurs, or a combination of reduced b₂ and extra tax. [17] The reduction in b₂, by a proportion, η, clearly involves a sacrifice, so it is necessary to attach a value to this loss. Suppose the welfare function for each period, i, is similar to that used above, but involves a weighted geometric mean of net income, y_i (1 - τ_i), and tax-financed expenditure, b_i, with weights ξ and 1 - ξ respectively Hence:

      (31)

and, as before, suppose the judge displays constant relative risk aversion of ε. Hence, welfare in each period is now written as:

      (32)

As γ can be varied continuously from zero, it is only necessary to consider the case where action is taken immediately, by saving enough in the first period to finance a proportion, γ, of the anticipated cost. Thus, in the first period a tax rate of τ_γ,1 is imposed, given by:

      (33)

If the event does happen, and it is possible to reduce expenditure on other items by a proportion, η, then the rate in the second period becomes, τ_γ,2, given by:

      (34)

Clearly, b-type expenditure cannot be cut beyond the amount needed to finance the additional C-type cost. Hence τ_γ,2 ≥ τ₂ and η must be restricted to the range:

      (35)

In the second period, if the event does not happen, the revenue can be used to lower the tax rate below the planned rate needed to finance b₂. Allowing for the additional possibility that b₂ is reduced below the value it would have without the potential event, then:

      (36)

It is assumed that b₁ is not adjusted.

Expected social welfare from acting immediately is:

      (37)

The optimal policy therefore involves finding combinations of the two policy variables, η and γ, which maximise (37).

Figure 8: Optimal Values of Gamma and Eta

Figure 8 plots combinations of optimal η and γ for alternative values of p and ε, for the following values of other variables: y₁ = 6000; C = 2000; b₁ = 2100; β = 900; α = 0.5; r = g_y = g_b = 0.05. As before, the optimal choice is not sensitive to ε but is sensitive to p, with η and γ moving in opposite directions, as expected. In varying these other variables, the results are relatively sensitive to variations in α: raising α leads to higher optimal γ and lower optimal η.

As before, this can be extended to allow for 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 C_L, and a probability of p₂ of the event taking place with a higher cost of C_H. Suppose it is planned to accumulate a proportion, γ, of the anticipated higher cost in the first period. Thus, the tax rate in the first period is:

      (38)

If funds are needed, the alternative tax rates for low and high cost outcomes respectively are:

      (39)

If more than C_L is accumulated, then of course τ_L,2 is accordingly lower than otherwise.

      (40)

These expressions reflect the reduction in planned expenditure in period 2 to help finance the contingency (again assuming that b₁ is not also adjusted). As in the simple case of one known cost level, the value of η must be restricted to ensure that τ_L,2 > τ₂ and τ_H,2 > τ₂. If the funds are not needed, then:

      (41)

Expected utility is:

      (42)

Figure 9: Optimal Eta for alternative Risk Aversion

Figure 10: Optimal Gamma for Alternative Risk Aversion

Figures 9 and 10 show optimal values of η and γ for variations in the probabilities p₁ and p₂, in each case for two different values of risk aversion, ε, for the following values: y₁ = 6000; C_H = 2500; C_L = 1500; b₁ = 3000; β = 900; α = 0.4; r = g_y = g_b = 0.05. In this case higher risk aversion does lead to higher amounts of prefunding with consequently lower reductions in the second period other expenditure. Again higher values of γ are found to be associated with higher ε and lower η.