Climate Policy under Sustainable Discounted Utilitarianism

The explicit algorithm defined by WT = U(cT) and repeated use of (5), with δt
determined by (3), is used for calculating SDU welfare in the empirical analysis. It
follows from (3) that the population-adjusted utility discount factor δt

• equals the unadjusted utility discount factor β if population is constant,

• exceeds β if there is positive population growth, and

• varies with time if population growth is not exponential.

Let ρ > 0 denote the unadjusted utility discount rate, where the relation between β
and ρ is given by

^β = 1+ρ.                            (6)

The theoretical presentation of SDU in this section is facilitated by using the utility
discount factor β , while the numerical results in Section 4 are easier to interpret in
terms of the utility discount rate ρ. Keeping in mind eq. (6), this should not create
confusion.

Now we turn to the analysis of uncertainty. To handle uncertainty, one can in
principle think of two polar approaches in a situation where there is a probability
distribution over consumption streams.³ One possibility is first to value each realiza-
tion and then assign probability weights to the different realizations. An alternative
approach is first to determine a certainty equivalent for each generation and then
value the stream of certainty equivalents. When applying SDU to uncertainty, the
choice between these approaches matters for policy evaluation: in the context of
climate change, the possibility of catastrophic consequences is assigned more weight
if the valuation is done first within each realization.

This point can be shown formally under the simplifying assumption that the
utility function U not only expresses aversion to inequality over time, but also
aversion to risk. By abstracting from population growth and writing V (u, w) :=
min{(1 - δ)u +δw, w} for the function that aggregates present utility and future
welfare, it follows from (W.1) that W (0c) = V (U (c0), W(1c)). Since V is a concave
function of u and w, it follows from Jensen’s inequality that

E(V(U(co),W(ic))) ≤ V(E(U(co)),E(W(ɪe))),

³In the empirical part this corresponds to the empirical distribution of 1000 random draws of a
Latin Hypercube sample.

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