问题：Consider a consumer with the preferences given by P∞t=0 βtlog (ct), where ct is
Consider a consumer with the preferences given by P∞ t=0 β t log (ct), where ct is consumption in period t and log ct represents the flow utility from consumption in period t. The discount factor β = 1 1+ρ ∈ (0, 1) where ρ > 0 is time preference rate. The consumer has a bucket of ice cream of size w0 at the beginning of period 0. The consumer can eat some of the ice cream and save the remainder at each point of time t = 0, 1, 2, · · ·. The ice cream melts at rate δ, thus its evolution is governed by wt+1 = (1 − δ) wt − ct. Write the consumer’s problem in a recursive form to set up the dynamic programming. What is her state variable? What is her control (choice) variable?