Proposition 3 Let x*k denote the solution to the transformed Bayesian decision
problem (11) with prior probabilities pi > 0 (i = l,...n). Let xtr denote the
solution to the robust decision problem (7). Then
Iim ∣∣4 - xrllβ = 0
κ→∞
The proof of proposition 3 is identical to the one of proposition 1 with the
exception that one has to substitute expressions involving the standard vector
norm by the ’discounted’ norm (12).
As a final remark, I want to stress that the transformed Bayesian utility
function fails to be time separable.6 Marginal utility for the Bayesian problem
is given by
∂E ∖Tk(L(x,s))l . ,∣∞∞ fl⅛,z x
---______ = kβt y^l(xt,si)ek^h=oβ i(^ħ,si))pi (13)
∂Xt ‘ .
Since this expression does not converge for k → ∞, consider the ratio of marginal
utilities instead:
∂E[T fc(L(x,s))]
______∂xt_______
∂E[T k(L(x,s))]
∂xt+j
(14)
The limit of (14) for k → ∞ depends on the states s that maximize 4∞=oβhl(xh,s),
where the latter expression is the term showing up in the exponent of (13). Since
the states that maximize this expression depend on the whole decision vector x,
a decision change in some period other than t or t + j may well alter this ratio.
5 Appendix
This section proves proposition 1. Rename states s such that at x*
T(x*, si) ≥ T(x), s2) ≥ ... ≥ L(xtr,s1 )
and let
Ωmax = {i∖L(x*r,si) = L(x*,sι)}
I first prove the following auxiliary result:
6The subsequent arguments assume a non-atomistic decision maker who takes into account
that the maximizing states are a function of his/her own decision x. See section 6 in Hansen
et al. (2002) for further discussion.
11