8. Initially, two firms A and B (numbered 1 and 2) share the market for a...

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8. Initially, two firms A and B (numbered 1 and 2) share the market for a certain commodity. Firm A has 20% of the marked and B has 80%. In course of the next year, the following changes occur: A keeps 85% of its customers, while losing 15% to B B keeps 55% of its customers, while losing 45% to A We can represent market shares of the two firms by means of a market share vector, defined as a column vector s whose components are all nonnegative and sum to 1. Define the matrix T and the initial share vector s by T = (0.85 0.45) 0.15 0.55 S= 3 (0.2) 0.8 The matrix T is called the transition matrix. a) Compute the vector Ts, and show that it is also a market share vector. b) Find the eigenvalues and eigenvectors of T. c) Find a matrix P such that D = PTP is diagonal, and show that T" = PD"P-¹. d) Compute the limit of D" as n→∞ and use this to find the limit of T"s as n →∞. Explain that the we will approach an equilibrium, a situation where the market shares of A and B are constant. What are the equilibrium marked shares? 8. Initially, two firms A and B (numbered 1 and 2) share the market for a certain commodity. Firm A has 20% of the marked and B has 80%. In course of the next year, the following changes occur: A keeps 85% of its customers, while losing 15% to B B keeps 55% of its customers, while losing 45% to A We can represent market shares of the two firms by means of a market share vector, defined as a column vector s whose components are all nonnegative and sum to 1. Define the matrix T and the initial share vector s by T = (0.85 0.45) 0.15 0.55 S= 3 (0.2) 0.8 The matrix T is called the transition matrix. a) Compute the vector Ts, and show that it is also a market share vector. b) Find the eigenvalues and eigenvectors of T. c) Find a matrix P such that D = PTP is diagonal, and show that T" = PD"P-¹. d) Compute the limit of D" as n→∞ and use this to find the limit of T"s as n →∞. Explain that the we will approach an equilibrium, a situation where the market shares of A and B are constant. What are the equilibrium marked shares?