Farmer Giles makes hay each year and he makes far more than he could possibly store and use himself, but he does not always sell it all.. He has decided to offer incentives for people to buy large quantities so it does not sit in his field deteriorating. He has devised the following "discount" scheme. He has a Base price B of 8 per bale. Then he has three levels of discount: Good price G is a 10% discount, Loyalty price L is a 20% discount and Super price S is a 25% discount on the Base price. Customers who increase their order compared with last year move to one higher discount level, or remain at level S. Customers who maintain their order from last year stay at the same discount level. Customers who reduce their order from last year drop one level of discount or remain at level B provided that they maintained or increased their order the previous year. Customers who reduce their order from last year drop two levels of discount if they also reduced their order last year, subject to remaining at the lowest level B. (i) Explain why a process with the state space (B.G.L.S) does not display [2] the Markov property. (ii) (a) Define any additional state(s) required to model the system with the Markov property. (b) Construct a transition graph of this Markov process, clearly labelling all the states. [3] Farmer Giles thinks that each year customers have a 60% likelihood of increasing their order and a 30% likelihood of reducing it, irrespective of the discount level they are currently on. (iii) (a) Write down the transition matrix for the Markov process. (b) Calculate the stationary distribution. (c) Hence calculate the long-run average price he will get for each bale of hay. [8] (iv) Calculate the probability that a customer who is currently paying the Loyalty price L will be paying L in two years' time. [3] (v) Suggest reasons why the assumptions Farmer Giles has made about his customers' behaviour may not be valid. [3] [Total 19]