  This paper studies decision theoretic properties of benchmarked estimators
which are of some importance in small area estimation problems. Benchmarking is
intended to improve certain aggregate properties (such as study-wide averages)
when model based estimates have been applied to individual small areas. We
study decision-theoretic properties of such estimators by reducing the problem
to one of studying these problems in a related derived problem. For certain
such problems, we show that unconstrained solutions in the original
(unbenchmarked) problem give unconstrained Bayes and improved estimators which
automatically satisfy the benchmark constraint. Also, dominance properties of
constrained empirical Bayes estimators are shown in the Fay-Herriot model, a
frequently used model in small area estimation.
