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I have a maximization problem as shown in the picture below. The output has a common shock z that follows a two-state Markov chain with a transition matrix Π. Does anyone know how I would go about solving this by discrete-state-space dynamic programming?

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Best I know as I used in my own paper is MDP toolbox which proposes functions related to the resolution of discrete-time Markov Decision Processes: backwards induction, value iteration, policy iteration, linear programming algorithms with some variants.

Can be downloaded from Toolbox page: http://www.inra.fr/mia/T/MDPtoolbox or the following address:(Marie-Josee Cros (2023). Markov Decision Processes (MDP) Toolbox) https://www.mathworks.com/matlabcentral/fileexchange/25786-markov-decision-processes-mdp-toolbox)

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