Convergence of Discretized Value Function Iteration

We provide a proof that the computational solution from discretized value function iteration will converge uniformly to the true solution for both the value function and the optimal policy function. We allow for non-differentiable value functions, non-concave return functions, and non-convexities in the feasible choice set. This result fills an important gap in the literature for this commonly used numerical method as existing results assume differentiability of the value function, concavity of the return function, convexity of feasible choice sets, or simply do not consider the optimal policy function. Our results thus extend the existing literature to cover cases in which value function iteration becomes a common solution method and allow for economic applications such as modelling of technology adoption and payroll taxes not covered by previous results. Results on the convergence of the value function and optimal policy function allow for their use as the basis of studying convergence of computational solution, simulation, and estimation of more advanced Macroeconomic models.