Individuals face the problem of allocating personal wealth between investment and consumption to maximize their lifetime utility. Suppose that we represent wealth by a value in 10, m), and when the individual allocates x units of wealth to consumption in a particular month, he receives log(x + 1) units of utility. Wealth not allocated to consumption is invested for one period and appreciates or depreciates at interest rate p E [ – 1, MI which varies according to a probability distribution F(p) where M
a. Formulate this as a continuous-state infinite-horizon discounted Markov decision problem in which the objective is to maximize expected total discounted utility. To derive transition probabilities, first give a dynamic equation which models .the change in system state.
b. Verify that an appropriate generalization of Assumptions 6.10.1 and 6.10.2 hold for this model.
c. Speculate on the form of an optimal policy for this model.
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