I am interested in a slightly different issue. In problem 11 of this file, ignoring the population size and growth, and focusing on how one could price the asset instead, I would like to find an expression for the price at equilibrium. In particular, after having solved the maximization problem thanks to the Bellman equation and having derived a correct Euler Equation, I have also found an expression for price with an infinite sum. The problem is how to obtain a closed form of it. In particular, is the price a Markov process? At time t, the investor may expect two possibilities with probabilities pi and 1-pi, but what are the expectations for two periods ahead? The dividend ultimately depend also on previous periods, since in case of a bad event the economy will never recover. So, how is this summation developed and later solved in order to find a price set depending on exogenous parameters?