3. Let s, and do denote the supply and demand at year n of the market, respectively, for certain product. We assume that dat1 = dn + a(dn - Sn) + B(Sn - Sat1), where o = 4, B = ;. We also assume the revenue of the company at year n is given by "n = Sn(ydn - 6), where y = 2, 6 = 1. In the following set so = do = 10. (a) Given fixed d, and s, show that [1] f(x) = x (y(dn + a(dn - Sn) + 8(sn - x)) - 5) has a unique global maximum. This point is the level of supply sat at year n + 1. We call this the greedy approach. (b) Find a matrix A and a vector b such that (27+1 = A ( ) +6. [2] (c) Find a vector u that u = Au + b. Show that ( Satl ) - u = 4+ ((50) -2) [4] (d) Write A' and As as scalars times A . Guess a general formula for A" for all n 2 1 and [3] prove that your formula is correct. Find expressions for s, and da. (e) Using the formula Ex = 1-14+1 1-x to write an exact explicit expression for the total [3] revenue of the company over the first ten years Rgreed = = Ta. (f) Compute the revenue Requi that the company can get (over the same period) by the [2] equilibrium approach: setting s, = so for all n = 1, ..., 10. Approximate Agreed/ Requil to two decimal places. (g) We assume S, = so for all n = 1, ....9. How can the company outperform the [1] equilibrium approach over the same period? (h) We assume s, = so for all n = 1, ....9, and we can choose any non-negative dio and $10. Design a strategy that would bring more than 6Requ (computed from (f)) over the [2] first ten years