4 Extension
While the loss function considered so far assumed a finite dimensional deci-
sion vector, macroeconomists tend to use infinite horizon models with infinite
dimensional decision vectors. In this section we show that the results of the
previous section extend in a natural way to the infinite horizon problems with
discounting.
Consider the following loss function
∞
L(x,s) = ^ βtl(xt,s)
¢=0
where xt ∈ Rn denotes the period t decision, the vector x = (x0, x'r,... )' the
stacked period decisions, and β < 1 a discount factor. The period loss function
l(∙,s) is assumed to be strictly convex and twice continuously differentiable for
all s. The period decision xt must be chosen from a compact and convex set of
feasible decisions Ωt that might depend on past decisions. Furthermore, there
is a compact set Ωx C Rn such that Ωt C Ωx for all t.
The robust decision maker minimizes
∞
(7)
min max) βt l(xt,s)
{xt∖xt∈Ωt} s∈Ω, ¢=0
To construct the transformed Bayesian problem it might seem natural at first
to transform the period loss function l(∙, ∙) to preserve the time separability of
the objective function, e.g. to let the Bayesian minimize
min
{ιt∣xt∈Ωt}
i=l t=0
(8)
However, the solution to this problem will not necessarily converge to the so-
lution of the robust decision problem as к increases without bound. This is
the case because a marginal change of some decision might have its strongest
impact for a state s⅛ that differs from the worst-case state s* associated with
the robust decision. When, in addition, the sign of the utility change for s⅛ is
opposite to the sign of the utility change for s*, then the Bayesian decisions for
(8) fails to converge as к → ∞. This is illustrated in the following example.
Convergence will be slower, the less weight is attached to the worst state associated with the
robust decision.