  This paper extends the classical contraction mapping theorem to the Cartesian
product of complete metric spaces. The modulus $\beta<1$ of a contraction
mapping is replaced by the condition $\rho(B)<1$, 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
dynamic programming and asset pricing are discussed.
