Climate Policy under Sustainable Discounted Utilitarianism

is the discounted utilitarian (DU) SWF. Multiplying by 1 - δ ensures that the util-
ity weights 1 - δ, (1 - δ)δ, (1 - δ)δ², . . . add up to one. Such a normalisation is
essential for the analysis of this paper, as it makes the utility of each generation
comparable to the welfare of the stream.

The DU SWF is well-defined for the set of consumption streams which eventually
become constant. Furthermore, on this set, it is characterized by

w(tc) = (1 - δ)U(ct) + δw(t+1c)                         (w.1)

w(tc) = U(ct)     if tc is egalitarian .                    (w.2)

Clearly, (1) implies (w.1) and (w.2). Conversely, for any 0c = (c0, . . . , cτ , . . . ) with
cτ = cT for all τ ≥ T , it follows from (w.2) that

w(Tc) =U(cT) = (1 - δ)^X∞ δ^τ-TU(cτ).

τ=T

Repeated use of (w.1) for t = T - 1, T - 2, . . . , 1, 0, now yields (1) for t = 0.

The sustainable discounted utilitarian (SDU) SWF modifies DU by requiring that
an SDU SWF not be sensitive to the interests of the present generation if the present
is better off than the future:

(1 - δ)U (ct) + δW (t+1c)     if U(ct)≤W(t+1c)

(W.1)

W(t+1c)                      if U(ct) > W(t+1c),

W (tc) = U(ct)     if tc is egalitarian .                                  (W.2)

Condition (W.1) means that future utilities are not discounted (the discount factor is
set to 1) if the present is better off than the future. In this case, present utility is given
zero weight. The utility weights are still of the form 1 - δ, (1 - δ)δ, (1 - δ)δ², . . .
if generations with zero utility weight are left out, implying that the utility weights
add up to one also for the SDU SWF. This means that the utility of each generation is
comparable to the welfare of the stream and makes the comparison between U (ct) and
W (t+1 c) meaningful and independent of the period length. In particular, the welfare
of an egalitarian stream is equal to the utility of the constant level of consumption,
as specified by condition (W.2).

For any 0c = (c0, . . . , cτ, . . . ) with cτ = cT for all τ ≥ T , it follows from (W.2)
that W(Tc) = U(cT). Repeated use of (W.1) for t = T - 1, T - 2, . . . , 1, 0, now
allows us to recursively calculate W (T -1 c), W (T-2c), and so on, ending up with
W (0c). Hence, on our domain of eventually constant consumption streams, the SDU
SWF is uniquely determined.

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