2.5 The Role of Risk Aversion
This subsection investigates the role of risk aversion in influencing the optimal degree of tax smoothing. The analysis proceeds by considering two basic relationships. The first is between tax rates in the two periods which arises from the budget constraint. The second relationship is between the two rates, for which the expected social welfare function is constant: it gives what may be referred to as iso-welfare or social indifference curves. The optimal γ is characterised by a point of tangency between these two lines.
For the budget constraint, first rearrange (8):
(10)
Substitute this into (6) so that:
(11)
Consider the case where gy = r, so that, since y2 = y1 (1 + gy):
(12)
The relationship between the two rates is thus a downward sloping 45 degree line.
- Figure 2: Slope of Indifference Curves for Alternative Risk Aversion Parameters
To obtain the slope of social indifference curves, first use (10) to substitute in (7) so that:
(13)
(14)
Define the terms:
(15)
(16)
(17)
Expected social welfare is:
(18)
Consider variations in τ2H and τ1H which leave E unchanged:
(19)
Rearranging gives:
(20)
The optimal solution is the value of γ for which:
(21)
The sensitivity of the optimal value of γ to ε is indicated by the extent to which the right hand side of (20) changes as ε changes, holding the values of τ constant at their optimal values.
Figure 2 shows, for the first case considered above (where C = 1500), with p = 0.6, the variation in the slope of the indifference curves as risk aversion increases. Each line shows the slope, plotted against gamma, for a given value of ε. The flatter line is for lower risk aversion. The optimal value of γ corresponding to the point where the line intercepts the horizontal line at -1. It is clear from these schedules that the insensitivity of the degree of tax smoothing with respect to ε arises largely from the nature of the budget constraint: that is, irrespective of the value of C and of the other tax-financed expenditure, the constraint is a downward sloping 45 degree line. The degree of convexity of indifference curves increases as ε increases, corresponding to the greater concavity of the welfare function, but there is little variation where the slope is -1. As can be seen from (11), the slope of the budget line can differ from -1 where r≠gy.