Question

(Markov Matrices): The manufacturer of a product (P) currently controls 20% of the market in a particular country, while the rival brand (Q) controls the rest of the market. Data from the previous year show that 75% of P's customers remained loyal, while 25% switched to the rival brand. In addition, 50% of the competition's customers did not switch to P during the year while the other 50% did. (a) Representing the market share of each brand as a row vector (whose components are fractions), determine the switching or transition matrix so that the problem of determining the market share next year, given the market share this year, can be reduced to determining the product of the row vector and the transition matrix (b) Ditto, but now represent the market share of each brand as a column vector and market share next year is the product of the transition matrix times the current market share vector. (c) use either of the above approaches to determine the market share of each brand after two years (d) Determine the market share of the two products in the long run (e.g. after 20 years) by formulating it as a solution to simultaneous linear equations.

          (Markov Matrices): The manufacturer of a product (P) currently
controls 20% of the market in a particular country, while the rival
brand (Q) controls the rest of the market. Data from the previous
year show that 75% of P's customers remained loyal, while 25%
switched to the rival brand. In addition, 50% of the competition's
customers did not switch to P during the year while the other 50%
did. (a) Representing the market share of each brand as a row
vector (whose components are fractions), determine the switching or
transition matrix so that the problem of determining the market
share next year, given the market share this year, can be reduced
to determining the product of the row vector and the transition
matrix (b) Ditto, but now represent the market share of each brand
as a column vector and market share next year is the product of the
transition matrix times the current market share vector. (c) use
either of the above approaches to determine the market share of
each brand after two years (d) Determine the market share of the
two products in the long run (e.g. after 20 years) by formulating
it as a solution to simultaneous linear equations.

Added by Kathleen B.

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