2. Consider the following cake eating problem with uncertainty. The agent begins with K units of cake and it only lasts T periods. However, in each period, the agent has the apatite for cake only with probability p and enjoys c units with utility $u(c)$. Formulate the value functions. Suppose that $u(c) = \ln c$. Then solve for optimal $c$ as a function of number of periods left and cake left. Show that optimal consumption increases with the amount of cake left.
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The value function at time t, V(t, K), represents the maximum expected utility the agent can obtain from time t to T, given that there are K units of cake left at time t. The Bellman equation for the value function can be written as: V(t, K) = max{u(c) + (p * Show more…
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