However, what is more interesting is that the expected change in the stationary
equivalent under SDU holds up to a greater extent than under DU, so that the dif-
ference between the two evaluation principles grows. When ρ = 0, the two principles
yield an identical evaluation. The reason for this can readily be seen by comparing
(w.1) and (W.1) in Section 2: when the discount factor approaches unity, the SDU
algorithm approaches the DU algorithm.14 However, as ρ increases, the two algo-
rithms can yield different results depending on the probability of falling consumption
per capita, and Figure 2 bears this out. For both policies, the expected change in the
stationary equivalent falls and eventually becomes negative under DU, but remains
positive under SDU. As ρ rises, the far-off future matters less and less under DU,
and it is in the far-off future that the benefits of abatement accrue. However, under
SDU the far-off future can continue to receive significant weight, if at some point in
time future discounted utility is below present utility. We know from Figure 1 that
this is the case.
4.2 Optimal policies
Finally, rather than evaluating exogenous policy settings, it is informative to com-
pare the optimal schedule of emissions abatement under SDU and DU. To do this,
we set ρ = 0.02, and follow Nordhaus (2008) in simultaneously solving a schedule
of emissions control rates {μt} (where for each t, μt ∈ [0,1]) that maximises the ex-
pectation of SDU and DU respectively. Thus the emissions control rate is a number
between 0 and 1, which controls the emissions intensity of output, and a control rate
of μt results in a fraction 1 — μt of output contributing to emissions.
In an integrated assessment model such as DICE, and especially in running risk
analysis, solving this optimisation problem is a non-trivial computational challenge.
However, we are able to find a solution using a genetic algorithm (Riskoptimizer)
and with two modifications to the basic optimisation problem.15 Figure 3 presents
14In the limit, as ρ → 0, or equivalently, δ → 1, it follows from (w.1), (w.2), (W.1) and (W.2)
that DU and SDU welfare are determined only by the eventual constant part of 0c beyond T , where
ct = cT for all t ≥ T . Then both DU and SDU welfare become insensitive to present wellbeing, as
w(0c) = W (0c) = U(cT), illustrating a problematic aspect of undiscounted utilitarianism.
15First, we only solve μt from 2005 to 2245 inclusive, rather than all the way out to 2395. This
considerably reduces the scope of the optimisation problem, in return for making little difference
to the results, since, in the standard version of DICE, μt = 1, t > 2245 (i.e. abatement yields
high benefits relative to costs in the far-off future). Our own results also show that μt → 1 as
t → 2245. Second, we guide the optimisation by imposing the soft constraint that μt is non-
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