3. Greedy monopolist The market for a certain product is governed by the equation d\"n+1 = dn + 05(er _ 3n) + (3n _ 3n+1): where (in and SR represent the market demand and supply levels at year in, E N, respectively. The parameter 0: corresponds to the rate of change of the demand with respect to the excess demand for the previous year and the parameter 6 measures the market reaction to the supply uctuations. A corporation took over control of all supply resources and decided to implement a greedy approach: to set the level of supply every year such that the revenue is maximised. Assume that the price is proportional to the demand and the production costs remain unchanged so the revenue at year it is given by the equation Tn = snhdn 5). In this question you nd out whether the greedy approach performs well in a long term for this model. In the following set so do 10,cr i, 6 %, y 2, 6 1. (a) Given xed (in, an, show that Hm) = 33 (7(0'.\" + Dawn 7 an) + 6(sn 7 m 7 6) has unique global maximum. This point is the level of supply s+1 at year it + 1. (b) Find a matrix A and a vector b such that (:\""1) = A (8") + b' 1171 (c) Find a vector u that u = An + b. Show that (SW1) 7 u = A\"+1 ((30) 7 u) . [1] Write A2 and A3 as scalars times A . Guess a general formula for A\" for all n 2 1 and prove that your formula is correct. Find expressions for an and d\". Using the formula 2:720 M = 11?? write an exact explicit expression for the total revenue of the corporation over the rst ten years Rgreed = 2:; shdn 7 6). Compute the revenue Requil that the corporation can get (over the same period) by the equilibrium approach: setting 3n = .30 for all n, = 1, . . . ,10. Approximate Rgreed /Requi1 to two decimal places. How can the corporation outperform the equilibrium approach over the same period? Design a strategy that would bring more than Requ over the rst ten years. [3] [3] [2] [1] [2]