We study a class of corner growth models in which the weights are either all exponentially or all geometrically distributed. The parameter of the distribution at site $(i, j)$ is $a_i+b_j$ in the exponential case and $a_ib_j$ in the geometric case, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 1}$ are themselves drawn randomly at the outset from ergodic distributions. These models are inhomogeneous generalizations of the much studied exactly solvable models in which the parameters are the same for all sites. Our motivation is to understand how inhomogeneity influences the limit shape and the corresponding limit fluctuations. We obtain a simple variational formula for the shape function and prove that it is strictly concave inside a cone (possibly the entire quadrant) but is linear outside. This is in contrast with the situation in the models with i.i.d. weights in which the shape function is expected to be strictly concave under mild assumptions. For the directions inside the cone, we show that the limit fluctuations are governed by the Tracy-Widom GUE distribution and derive bounds for the deviations of the last-passage times above the shape function. To obtain the shape result, we couple the model with an explicit family of stationary versions of it. For the fluctuation and large deviation results, we perform steepest-descent analysis on an available Fredholm determinant formula for the one-point distribution of the last-passage time. We also develop a detailed appendix on the steepest-descent curves of harmonic functions of two real variables and approximate the contour integral of an arbitrary meromorphic function along such curves.