Science:Math Exam Resources/Courses/MATH221/December 2009/Question 07 (b)

MATH221 December 2009

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Question 07 (b)
Two drink makers Hard and Soft compete for customers. Each year 20% of Hard’s customers shift to Soft and 30% of Soft’s customers shift to Hard. Assume that this year the drink makers have equal market share. What will be the market distribution in the distant future?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

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Solution

Whoops, no solution has been written yet. Be the first to submit a suggestion. Solving for eigenvalues $\displaystyle \lambda _{1}=1,\lambda _{2}=0.5$. We have n linearly independent eigenvectors so we can use Power Method. The eigenvector for $\displaystyle \lambda _{1}=1$ is ${\begin{bmatrix}1.5\\1\end{bmatrix}}$. We need to scale the vector into a market share vector $ANS={\begin{bmatrix}.6\\.4\end{bmatrix}}$