In this paper a problem of consumption and investment is presented as a model of a discounted Markov decision process with discrete-time. In this problem, it is assumed that the wealth is affected by a production function. This assumption gives the investor a chance to increase his wealth before the investment. For the solution of the problem there is established a suitable version of the Euler Equation (EE) which characterizes its optimal policy completely, that is, there are provided conditions which guarantee that a policy is optimal for the problem if and only if it satisfies the EE. The problem is exemplified in two particular cases: for a logarithmic utility and for a Cobb-Douglas utility. In both cases explicit formulas for the optimal policy and the optimal value function are supplied.
discounted Markov decision processes, differentiable value function, differentiable optimal policy, stochastic Euler equation, consumption and investment problems