  We generalize the exactly solvable corner growth models by choosing the rate
of the exponential distribution $a_i+b_j$ and the parameter of the geometric
distribution $a_i b_j$ at site $(i, j)$, where $(a_i)_{i \ge 1}$ and $(b_j)_{j
\ge 1}$ are jointly ergodic random sequences. We identify the shape function in
terms of a simple variational problem, which can be solved explicitly in some
special cases.
