Question 1. Two tech companies, ABC and XYZ, are experiencing a rise and fall in their market share each year. Market research has determined the following pattern of how their respective portions of the market are redistributed at the end of each year: ABC retains 70% of its market share, and passes on 30% to XYZ. • XYZ retains 85% lof its market share, and passes on 15% to ABC. (a) Suppose that in 2022, ABC controlled 40% of the market, while XYZ controls the remaining 60%. How much market share does each company have in 2023? (b) Suppose that the market shares of ABC and XYZ in a given year are represented by variables uk and Uk respectively, while in the following year they are uk+1 and Uk+1. Express uk+1 and Uk+1 in terms of u and Uk, then rewrite your equations to involve a product of a matrix P and the vector k = - [u] (c) Is there a market share distribution between the two companies that will stay constant year over year? (This means their corresponding market shares do not change from one year to the next.) Explain how your answer relates to eigenvalues and eigenvectors.
Question 1. Two tech companies, ABC and XYZ, are experiencing a rise and fall in their market share each year. Market research has determined the following pattern of how their respective portions of the market are redistributed at the end of each year: ABC retains 70% of its market share, and passes on 30% to XYZ. • XYZ retains 85% lof its market share, and passes on 15% to ABC. (a) Suppose that in 2022, ABC controlled 40% of the market, while XYZ controls the remaining 60%. How much market share does each company have in 2023? (b) Suppose that the market shares of ABC and XYZ in a given year are represented by variables uk and Uk respectively, while in the following year they are uk+1 and Uk+1. Express uk+1 and Uk+1 in terms of u and Uk, then rewrite your equations to involve a product of a matrix P and the vector k = - [u] (c) Is there a market share distribution between the two companies that will stay constant year over year? (This means their corresponding market shares do not change from one year to the next.) Explain how your answer relates to eigenvalues and eigenvectors.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 1. Two tech companies, ABC and XYZ, are experiencing a rise and fall in their market share each year.
Market research has determined the following pattern of how their respective portions of the market are redistributed
at the end of each year:
4
ABC retains 70% of its market share, and passes on 30% to XYZ.
.XYZ retains 85% lof its market share, and passes on 15% to ABC.
(a) Suppose that in 2022, ABC controlled 40% of the market, while XYZ controls the remaining 60%.
How much market share does each company have in 2023?
(b) Suppose that the market shares of ABC and XYZ in a given year are represented by variables uk and Uk,
respectively, while in the following year they are uk+1 and Uk+1.
Express uk+1 and Uk+1 in terms of uk and vk, then rewrite your equations to involve a product of a matrix P
and the vector k =
Uk
Uk
(c) Is there a market share distribution between the two companies that will stay constant year over year? (This
means their corresponding market shares do not change from one year to the next.)
Explain how your answer relates to eigenvalues and eigenvectors.
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