For given utility discount factor δ, the concavity of the U -function and W (tc)
being the minimum of (1 - δ)U (ct) +δW(t+1c) and W(t+1c) (cf. (W.1)) both express
a preference for consumption smoothing over time. The Ramsey model of optimal
growth can be used to show that these instruments are not redundant (see Asheim
and Mitra, 2010, Section 4). If initial capital productivity is higher than the one
corresponding to the modified Golden Rule, then the SDU-optimal stream coincides
with the increasing DU-optimal stream and the concavity of the U -function smooths
consumption as for the DU-optimal stream. However, if initial capital productivity
is lower than the one corresponding to the modified Golden Rule, then (W.1) implies
that the SDU-optimal stream coincides with the efficient egalitarian stream. For
such initial conditions, DU leads to a decreasing optimal stream for any choice of
strictly increasing and strictly concave U -function, thereby representing preferences
in conflict with the condition of “Hammond Equity for the Future”.
The following proposition establishes as a general result that SDU welfare coin-
cides with DU welfare on the set of non-decreasing streams. Also, it shows that SDU
welfare is a non-decreasing function of time and bounded above by DU welfare.
Proposition 1 Assume that 0c is eventually constant.
(i) For all t ≥ 0, W (0c) ≤ W (tc) ≤ w(tc)
(ii) If 0c is non-decreasing, then W (0c) = w(0c).
Proof. This is a special case of Asheim and Mitra (2010, Proposition 2). ■
Part (ii) means that SDU welfare differs from DU welfare only if the consumption
stream is not non-decreasing. Hence, existence of some t ≥ 0 such that ct > ct+1
is a necessary, but insufficient, condition for SDU welfare being strictly below DU
welfare, and emphasis will be placed on this possibility in the empirical analysis.
On the set of non-in creasing streams, SDU coincides with both maximin and nu-
merically representable criteria that depend only on the streams’ limiting properties.
While such criteria satisfy the condition of “Hammond Equity for the Future” by giv-
ing priority to the future in conflicts where the future is worse off than the present,
they are extremely insensitive of the interests of generations and yield very different
conclusions from SDU for streams that are not non-increasing. In particular, max-
imin assigns no weight to any generation but the worst off, while criteria that depend
on the limiting properties assign no weight to any finite subset of generations.