  This paper shows the usefulness of Perov's contraction principle, which
generalizes Banach's contraction principle to a vector-valued metric, for
studying dynamic programming problems in which the discount factor can be
stochastic. The discounting condition $\beta<1$ is replaced by $\rho(B)<1$,
where $B$ is an appropriate nonnegative matrix and $\rho$ denotes the spectral
radius. Blackwell's sufficient condition is also generalized in this setting.
Applications to asset pricing and optimal savings are discussed.
