On the Relation between Robust and Bayesian Decision Making

Maximizing (20) delivers the same solution as maximizing (6). The limit of
—к ,  . ч _   .           .        ...     .

L (x*) for к → ∞ exists and is given by:

Iim L^k(x*_r) =        pi

k→∞          .

ⁱ∈ ^max

Next, consider L(x* + d) with d ∈ Rⁿ and ∣∣d∣∣ = δ, δ sufficiently small. From
lemma 4 and (20) it follows that

__{k r           e}^k(b(≈i.sι)+ε)

^{L (x}_r ^{+ d)} >   gkL(^*,s₁) ^Pmin

where p_m∙_in = mini Pi. Therefore, there exists a к < ∞ such that for all к > к

L^k(x^t_r + d) >L^k(x*)

__          .         .                 .        _ — к                         .        .        . .             . „ — к , ч

From the strict convexity of L (∙) it follows that the minimum x_k of L (∙) must
be within distance δ from x^t_r for all к > к, which establishes the claim.

References

Blinder, Alan, Central Banking in Theory and Practice, MIT Press, 1998.

Chamberlain, Gary, “Econometric Applications of Maxmin Expected Util-
ity,” Journal of Applied Econometrics, 2000, 15, 625-644.

Gilboa, Izhak and David Schmeidler, “Maxmin Expected Utility Theory
with Non-Unique Prior,” Journal of Mathematical Economics, 1989, 18,
141-153.

Hansen, Lars P. and Thomas J. Sargent, “Wanting Robustness in Macroe-
conomics,” 2000. Standord University, mimeo.

Hansen, L.P., T.J. Sargent, and T. Tallarini, “Robust Permanent Income
and Pricing,” Review of Economic Studies, 1999, 66, 873-907.

___, ___, G.A. Tirmuhambetova, and N. Williams, “Robustness and
Uncertainty Aversion,” 2002. Stanford University Mimeo.

Onatski, Alexei, “Minimax Analysis of Monetary Policy under Model Uncer-
tainty,” 2000. Harvard University, Dept. of Economics, mimeo.

___ and James Stock, “Robust Monetary Policy under Model Uncertainty
in a Small Model of the U.S. Economy,” NBER Working Paper 7f90, 2000.

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