Classify an individual’s wealth as low (L), medium (M), or high (H). In state L, the individual has insufficient wealth to invest, so each period he consumes all earned income and as a consequence receives one unit of utility and remains in state L. In state M, three consumption options are available. Consuming at a high rate yields a utility of three units; at a moderate level, a utility of two units, and at a low level a utility of 1.5 units. If an individual in state M consumes at a low rate, with probability 0.95 he remains in state M, and with probability 0.05 his wealth rises to level H at the next period; if he consumes at a moderate rate, with probability 0.7 his wealth remains at a medium level, with probability 0.01 it rises to a high level, and with probability 0.29 it falls to a low level; and, finally, if he consumes at a high level, with probability 0.9 his wealth falls to a low level in the subsequent period and otherwise remains at a medium level. If an individual’s wealth is high, he may consume at a moderate level or a high level. If he consumes at a moderate level, he receives two units of utility per period, and if he consumes at a high level, he receives three units of utility. By consuming at a moderate level, his wealth remains at a high level with probability 0.9, and falls to a moderate level with probability 0.1. Consuming at a high level, his wealth remains at a high level with probability 0.5, and falls to a moderate level with probability 0.5.
a. Formulate the individual’s consumption level choice problem as a Markov decision process.
b. Verify that the model is neither unichain nor communicating.
c. Find a policy that maximizes the individual’s expected average utility using multichain policy iteration and linear programming.
d. Suppose the individual discounts utility at rate A, and uses the expected total discounted utility optimality criterion. Investigate the effect of the discount rate on the optimal consumption policy.
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