Consider the situation of a mass layoff (i.e., a factory shuts down) where 1,200 people become unemployed and now begin a job search. In this case there are two states: employed (E) and unemployed (U) with an initial vector x= [EU] = [0 1,200] Suppose that in any given period an unemployed person will find a job with probabil- ity .7 and will therefore remain unemployed with a probability of .3. Additionally, persons who find themselves employed in any given period may lose their job with a probability of .1 (and will have a .9 probability of remaining employed). (a) Set up the Markov transition matrix for this problem. (b) What will be the number of unemployed people after (i) 2 periods; (ii) 3 periods; (iii) 5 periods; (iv) 1.0 periods? (c) What is the steady-state level of unemployment?
Consider the situation of a mass layoff (i.e., a factory shuts down) where 1,200 people become unemployed and now begin a job search. In this case there are two states: employed (E) and unemployed (U) with an initial vector x= [EU] = [0 1,200] Suppose that in any given period an unemployed person will find a job with probabil- ity .7 and will therefore remain unemployed with a probability of .3. Additionally, persons who find themselves employed in any given period may lose their job with a probability of .1 (and will have a .9 probability of remaining employed). (a) Set up the Markov transition matrix for this problem. (b) What will be the number of unemployed people after (i) 2 periods; (ii) 3 periods; (iii) 5 periods; (iv) 1.0 periods? (c) What is the steady-state level of unemployment?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Please help to solve and explain into details. I want to understand the process. Thank you very much.
![### Markov Process for Job Market Dynamics
#### Problem Statement
Consider the situation of a mass layoff (i.e., a factory shuts down) where 1,200 people become unemployed and now begin a job search. In this case, there are two states: employed (E) and unemployed (U) with an initial vector
\[ x_0 = [ E \quad U ] = [0 \quad 1,200] \]
#### Assumptions
- In any given period, an unemployed person will find a job with a probability of 0.7 and will therefore remain unemployed with a probability of 0.3.
- Additionally, persons who are employed in any given period may lose their job with a probability of 0.1 and will have a probability of 0.9 of remaining employed.
#### Questions
1. **Markov Transition Matrix**
- Set up the Markov transition matrix for this problem.
2. **Number of Unemployed People Over Time**
- What will be the number of unemployed people after:
(i) 2 periods;
(ii) 3 periods;
(iii) 5 periods;
(iv) 10 periods?
3. **Steady-state Analysis**
- What is the steady-state level of unemployment?
#### Solutions
##### (a) Markov Transition Matrix
The transition probabilities can be arranged in a matrix form as follows:
\[
P = \begin{pmatrix}
0.9 & 0.7 \\
0.1 & 0.3 \\
\end{pmatrix}
\]
Where:
- Each column sums to 1, representing the total probability (i.e., an individual either remains in their current state or transitions to the other state).
##### (b) Number of Unemployed People
Using this transition matrix, the state vector after \( n \) periods can be calculated by multiplying the initial state vector \( x_0 \) with \( P^n \).
\( x_n = x_0 \cdot P^n \)
Let's calculate the number of unemployed people for the respective periods:
(i) **2 Periods**:
\[ x_2 = x_0 \cdot P^2 \]
(ii) **3 Periods**:
\[ x_3 = x_0 \cdot P^3 \]
(iii) **5 Periods**:
\[ x_5 = x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F362a762f-914f-42d1-b0fa-542c2b1764ea%2F3acca4af-b239-437b-ab75-869a18a572a4%2Flmgz1jd_processed.png&w=3840&q=75)
Transcribed Image Text:### Markov Process for Job Market Dynamics
#### Problem Statement
Consider the situation of a mass layoff (i.e., a factory shuts down) where 1,200 people become unemployed and now begin a job search. In this case, there are two states: employed (E) and unemployed (U) with an initial vector
\[ x_0 = [ E \quad U ] = [0 \quad 1,200] \]
#### Assumptions
- In any given period, an unemployed person will find a job with a probability of 0.7 and will therefore remain unemployed with a probability of 0.3.
- Additionally, persons who are employed in any given period may lose their job with a probability of 0.1 and will have a probability of 0.9 of remaining employed.
#### Questions
1. **Markov Transition Matrix**
- Set up the Markov transition matrix for this problem.
2. **Number of Unemployed People Over Time**
- What will be the number of unemployed people after:
(i) 2 periods;
(ii) 3 periods;
(iii) 5 periods;
(iv) 10 periods?
3. **Steady-state Analysis**
- What is the steady-state level of unemployment?
#### Solutions
##### (a) Markov Transition Matrix
The transition probabilities can be arranged in a matrix form as follows:
\[
P = \begin{pmatrix}
0.9 & 0.7 \\
0.1 & 0.3 \\
\end{pmatrix}
\]
Where:
- Each column sums to 1, representing the total probability (i.e., an individual either remains in their current state or transitions to the other state).
##### (b) Number of Unemployed People
Using this transition matrix, the state vector after \( n \) periods can be calculated by multiplying the initial state vector \( x_0 \) with \( P^n \).
\( x_n = x_0 \cdot P^n \)
Let's calculate the number of unemployed people for the respective periods:
(i) **2 Periods**:
\[ x_2 = x_0 \cdot P^2 \]
(ii) **3 Periods**:
\[ x_3 = x_0 \cdot P^3 \]
(iii) **5 Periods**:
\[ x_5 = x
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 8 images
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman