2.6 An Alternative Welfare Function
A feature of the welfare function used above is that the measure of relative risk aversion, ε, is directly linked to the intertemporal elasticity of substitution, which is equal to 1/ε. An increase in risk aversion (which might, ceteris paribus, be thought to increase tax smoothing) is therefore accompanied by a reduction in the intertemporal elasticity of substitution, which in turn makes consumption less sensitive to the interest rate and weakens the motive for consumption (and hence tax smoothing). The fact that these effects oppose each other largely explains the insensitivity observed in the previous subsection. An alternative specification of the welfare function follows what are generally referred to as Epstein and Zin (1989) preferences. In this case preferences, for example over consumption in two periods, are expressed as a constant elasticity of substitution (CES) aggregate of consumption in the first period and the certainty-equivalent value of consumption in the second period. The certainty equivalent, the value that if know for certain would give the same welfare as the uncertain prospect, is the power mean given in equation (3). [15]
- Figure 3: Choice of Gamma with Epstein-Zin Preferences: c = 600 and α = 0.5
Using the notation in the previous subsection, the relevant certainty equivalent, denoted Y ε, is:
(22)
Letting α denote the intertemporal elasticity of substitution, the welfare function can be written as:
(23)
Consider again the illustrative case examined above where y1 = 5000, b1 = 2000, β = 600 and gy = gb = r = 0.05. Letting C = 600, Figure 3 shows the resulting choice of γ for variations in ε and p and a value of α = 0.5. In this case the optimal choice, and hence degree of tax smoothing, does depend on the degree of risk aversion, particularly for values of p around 0.5. [16]
Notes
- [15]An early proposal to have one concave function to generate the certainty equivalent and another concave function to describe intertemporal substitution between current consumption and the certainty equivalent was made by Selden (1978). For a review of a range of ‘exotic' preference functions, see Backus et al. (2004).
- [16]This value of p effectively represents the point of maximum uncertainty.