Problem 2. We have m advertisement spots, which we want to sell over the next / time periods. At each period, we may sell at most one spot, and the probability of sale is equal to e-", where u is the price set for this time. Our objective is to find the optimal policy of setting prices at each time period, so as to maximize the expected revenue over / periods. We assume that N 2 m. (a) Formulate the corresponding Markov decision problem. Clearly define the state space, con- trol space, transition probabilities, and the reward function. (b) Show that the optimal value function v.(x) is nondecreasing with respect to the number of items on hand x, for all r = 1,2, ..., N. (c) Fort = 1, 2. ..., N and all x > 0, show the equations: 1, (x) = 1+vi+1(x) - viti(x-1), vi (x) = vitI(x)te() where v, (x) is the optimal value function for state x at time / and u. (x) is the optimal control for state x at time r