Climate Policy under Sustainable Discounted Utilitarianism

(i) the stream of per-capita consumption tc = (ct, . . . , cτ, . . . ),

(ii) the development of population tN = (Nt, . . . , Nτ , . . . ), and

(iii) the utility discount factor β ∈ (0, 1) used to discount the product of population
size and the utility derived from per-capita consumption.

It can be expressed by

P∞=t β ^τ Nτ U (cτ )
w^t     P∞=tβ^τ Nτ

where the normalisation ensures that the weights assigned to utility derived from
per-capita consumption add up to one when summed over all present and future
individuals. In particular, the DU welfare of an egalitarian stream, with per-capita
consumption being c at all times, equals U (c). Hence, DU welfare is made comparable
to the utility derived from per-capita consumption when being expressed by (2).

By defining a time-dependent population-adjusted utility discount factor δt by

∑∞=t+ιβ^τ Nτ        β^tNt

P∞=tβ^τNτ       P∞=tβ^τNτ ,

we get the following recursive formula for classical DU welfare:

^{β t}NtU (ct)    P∞=t+1^{β τ} NT U (cτ )

P∞=tβ^τ Nτ ⁺   P∞=tβ^τ NT

If 0c = (c0 , . . . , cT, . . . ) is an infinite stream of per-capita consumption, where there
exists T ≥ 0 such that cT = cT for all τ ≥ T , then wT = U(cT ) combined with
repeated use of (4) allow us to calculate classical DU welfare at time 0.

Likewise, if 0c = (c0, . . . , cT, . . . ) is an infinite stream of per-capita consumption,
with cT = c_T for all τ ≥ T , then W_T = U(c_T) combined with repeated use of
allow us to calculate SDU welfare at time 0. It is also the case that with the recursive
formula (5) for SDU welfare, the weights assigned to utility derived from per-capita
consumption add up to one when summed over all present and future individuals.
Thus, the normalisation ensures that SDU welfare is comparable to the utility de-
rived from per-capita consumption, making the comparison between U(ct) and Wt+1
meaningful.

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