6 The cake-eating problem Consider the optimal growth problem (discrete time) where: f(k) = k This problem is commonly called a "cake-eating" problem. The consumer starts with a certain amount of capital, and "eats" it over time. We will use this problem to try some dynamic programming. The planner's problem is to maximize: =0 log t t t c subject to the constraints: t t t k k c +1 kt 0 k0 > 0 given where log is the natural (base e) logarithm function. a) Write down Bellman's equation for this problem. b) First we will perform policy function iteration. First we guess at an optimal policy. I guess that the optimal policy is: c (k) (1 )k 0 = Write down the value of kt (for any t > 0) as a function of , 0 k , and , if this policy is followed. t c) Write down the value of ct as a function of , 0 k , and t if this policy is followed. d) Once you have a guess at the policy function, you calculate the value function under the assumption that this is the optimal policy. What is the value function if the equation above describes the optimal policy? (Just to keep things simple, feel free to drop any constant term). e) The next step is to calculate the optimal policy under the new value function. This will give you a new policy function, which we will call c1(k). Find c1(k). f) To find the true optimal policy, you keep applying these two steps until your policy function stops changing i.e., ci(k) = ci+1(k). What is the true optimal policy function? g) Next we will try value function iteration. First we guess at the form of the value function. Suppose that your initial guess for the value function is: V0(k) = log k Next, we calculate a new value function according to the formula: Vi+1(k) = max{log c + Vi(k c)} Calculate V1, V2, and V3. Feel free to throw out any constant terms. h) If you've done it right, you should see a pattern. Use this pattern to discern Vi for an arbitrary integer i. i) What is the optimal policy function when this is the value function?