4 Suppose that firms in a particular industry fall into one of three size categories: large, medium and small. If a firm is large one year, the probabilities that it will remain large, fall into the medium size category, or become small in the next year are, respectively, 0.7, 0.2 and 0.1. For a firm of medium size, the corresponding probabilities are, respectively, 0.1, 0.8 and 0.1. For a small firm, the probabilities are, respectively, 0, 0.1 and 0.9. (a) Let Ln, Mn and Sn represent the number of firms in each category after n years have elapsed. Express the above information in the matrix equation form Lin Zn+1 = TZn [] (You may wish to draw a network diagram to represent the above situation.) where Zn =
4 Suppose that firms in a particular industry fall into one of three size categories: large, medium and small. If a firm is large one year, the probabilities that it will remain large, fall into the medium size category, or become small in the next year are, respectively, 0.7, 0.2 and 0.1. For a firm of medium size, the corresponding probabilities are, respectively, 0.1, 0.8 and 0.1. For a small firm, the probabilities are, respectively, 0, 0.1 and 0.9. (a) Let Ln, Mn and Sn represent the number of firms in each category after n years have elapsed. Express the above information in the matrix equation form Lin Zn+1 = TZn [] (You may wish to draw a network diagram to represent the above situation.) where Zn =
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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4
Transcribed Image Text:(b) Find the transition matrix T and then express it in the form QDQ-¹, where D is a diagonal matrix.
(c) Suppose that the total number of firms in the industry remains fixed at 4000. By considering what
happens to D" as n→ ∞o, determine how many firms fall into each category in the long term.
Transcribed Image Text:4 Suppose that firms in a particular industry fall into one of three size categories: large, medium and small.
If a firm is large one year, the probabilities that it will remain large, fall into the medium size category, or
become small in the next year are, respectively, 0.7, 0.2 and 0.1.
For a firm of medium size, the corresponding probabilities are, respectively, 0.1, 0.8 and 0.1.
For a small firm, the probabilities are, respectively, 0, 0.1 and 0.9.
(a) Let Ln, Mn and Sn represent the number of firms in each category after n years have elapsed. Express
the above information in the matrix equation form
Zn+1 = TZn, where Zn =
Lin
Mn
Sn
(You may wish to draw a network diagram to represent the above situation.)
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