Question

Consider the situation of a mass layoff (i.e., a factory shuts down) where 1,200 people become unemptoyed and now begin a job search. In this case there are two states: employed (E) and unemployed (U) with an initial vector $$x_{0}^{\prime}=\left[\begin{array}{ll} E & U \end{array}\right]=\left[\begin{array}{ll} 0 & 1,200 \end{array}\right]$$ Suppose that in any given period an unemployed person will find a job with probability .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) 10 periods? (c) What is the steady-state level of unemployment?

Short answer

Use transition matrix to find future unemployment levels.

Step-by-step solution

Define the Transition Matrix

We define the transition matrix based on the given probabilities. In each period, the probability of an unemployed person finding a job is 0.7, and the probability of remaining unemployed is 0.3. Conversely, the probability of an employed person losing their job is 0.1, and the probability of remaining employed is 0.9. The transition matrix, denoted as \( P \), is:\[ P = \begin{pmatrix} 0.9 & 0.1 \ 0.7 & 0.3 \end{pmatrix} \]

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with the analysis of random phenomena. Fundamentally, it allows us to model the likelihood of various outcomes in uncertain situations, such as the probability of finding a job if you're unemployed. In this exercise, probabilities are used to simulate the job search process for unemployed individuals who have been laid off due to a factory shutdown.

Probability values range from 0 to 1, where 0 indicates an impossible event, and 1 indicates certainty. For instance, the probability that an unemployed person will find a job is given as 0.7, which is relatively high, suggesting that a majority of these individuals might secure employment.

• Probability of employment for unemployed: 0.7
• Probability of staying unemployed: 0.3
• Probability of losing job if employed: 0.1
• Probability of staying employed: 0.9

Understanding these probabilities is crucial when constructing transition matrices, which help predict future states of employment, as seen in Markov models.

Steady-State Level

In probability theory and Markov processes, the steady-state level refers to a point where the probabilities of being in different states stabilize and remain constant over time. This means the system has reached equilibrium, and the proportion of employed and unemployed workers will remain the same in subsequent periods.

To find the steady-state level of unemployment, we use the transition matrix to simulate the employment system iteratively until changes between periods become negligible. Mathematically, this involves solving for the steady-state vector in the equation\[\mathbf{x} = \mathbf{x}P,\]where \( \mathbf{x} \) is the steady-state vector and \( P \) is the transition matrix. The vector \( \mathbf{x} \) provides the long-term probabilities of being employed or unemployed.

Understanding steady-state levels is essential for predicting long-term trends in unemployment, especially after significant economic events such as mass layoffs.

Unemployment Modeling

Unemployment modeling uses mathematics to understand and predict unemployment trends based on various parameters. In this exercise, the Markov model is employed to simulate how a group of unemployed individuals transition between employment and unemployment over time.

By setting up a transition matrix, we can apply this model to calculate how many people will remain unemployed after several periods. For instance, with the initial unemployment of 1,200 people and known transition probabilities, we iteratively apply the matrix over multiple time periods to project future unemployment figures.

This type of modeling helps in:

• Assessing the impact of economic policies.
• Planning social welfare measures.
• Forecasting labor market trends.

Markov models are particularly useful in fields like economics and sociology, where they assist in understanding dynamic systems.

Mathematical Economics

Mathematical economics utilizes mathematical methods to represent economic theories and analyze problems. It provides economists with the tools necessary to comprehend complex systems, such as the employment dynamics seen in this exercise.

In the case of unemployment due to a factory closure, mathematical economics helps evaluate how initial conditions (like the number of people laid off) and transition probabilities (chances of getting or losing jobs) influence employment trends over time. The use of mathematical models enables more precise predictions and policy planning.

Some benefits include:

• Quantifying economic relationships.
• Formulating predictions based on mathematical equations.
• Analyzing the stability and efficiency of markets.

By using mathematics, economists can better understand and respond to challenges within the economy, making mathematical economics a critical component of effective economic analysis and planning.