# DMP Model Simulation Tool

This page contains for simulating the model described in the article Revisiting Unemployment in Intermediate Macroeconomics: A New Approach for Teaching Diamond-Mortensen-Pissarides by Arghya Bhattacharya, Paul Jackson, and Brian C Jenkins. The homepage for the project is at the following URL:

To help keep this page at least somewhat self-contained, we provide a brief overview of the DMP model. Click here to jump down to the simulation tool.

#### Overview of the DMP Model

Here we provide a quick over view of the DMP model. For more details, please check out our article. We begin by assuming that the number of new hires over a given time period $$H_t$$ is linked to the number of unemployed persons $$U_t$$ and the number of job vacancies $$V_t$$ by the matching function: \begin{align} H_t = A\sqrt{U_t\cdot V_t}, \end{align} where $$A$$ reflects the efficiency of the matching process. The job finding rate $$f_t$$ is the fraction of unemployed workers that find jobs in the period and is given \begin{align} f_t & = \frac{H_t}{U_t} = \frac{A\sqrt{U_t\cdot V_t}}{U_t} = A\sqrt{\theta_t}, \end{align} where $$\theta \equiv V_t/U_t$$ reflects market tightness from the perspective of firms making hiring decisions. The number of unemployed workers in period $$t+1$$ is equal to the uemployed workers form period $$t$$ who did not find a job plus the employed workers from period $$t$$ who became separated from their jobs: \begin{align} U_{t+1} & = (1-f_t)U_t + \lambda E_t, \end{align} where $$\lambda$$ denotes reflects the exogenous separation rate. Using the matching fuction to elminimate the job finding rate and assuming that the size of the labor force is constant, rewrite the law of motion for unemployment in terms of the uemployment rate: \begin{align} u_{t+1} & = \left(1 - A\sqrt{\theta_t} - \lambda\right) u_t + \lambda, \end{align} Then, in the steady state we have the Beveridge curve relationship expressing the steady state unemployment rate as a decreasing function of market tightness: \begin{align} \boxed{u = \frac{\lambda}{A\sqrt{\theta} + \lambda} \vphantom{\bigg(}}.\label{bc}\tag{BC} \end{align} Equation $$(\ref{bc})$$ forms the first block of the DMP model. In the tool below, this curve will be plotted as the downward-sloping curve in the left panel.

Next, the probability $$q_t$$ that a firm fills a vacancy in a given period is found by using the mathing function: \begin{align} q_t & = \frac{H_t}{V_t} = \frac{A\sqrt{U_t\cdot V_t}}{V_t} = \frac{A}{\sqrt{\theta_t}}. \end{align} A firm with a filled vacancy receives a flow profits until the job is destroyed. An employed worker produces $$y$$ units of output each period and is paid a wage $$w$$ and so the period profit to a firm from a filled job is $$y - w$$. Firms incur a cost $$\kappa$$ each period that they advertise a job vacancy. In equilibirum, firms will create new job vacancies until the expected profit from doing so falls to equal the cost of posting a vacancy: \begin{align} \kappa & = q_t \left( y - w \right) \frac{1}{\lambda}, \end{align} where $$1/\lambda$$ is the expected life of filled vacancy. Substituting for $$q_t$$, find the following steady state relationship between the downward-sloping relationship between market tightness and the real wage: \begin{align} \boxed{\theta = \left[\frac{A}{\kappa}\left( \frac{y-w}{\lambda}\right) \right]^2 \vphantom{\bigg(} }, \label{vs}\tag{VS} \end{align} Equation $$(\ref{vs})$$ forms the second block of the DMP model. In the tool below, this curve will be plotted as the downward-sloping curve in the right panel.

Finally, the third block of the model is a steady state wage setting equation: \begin{align} \boxed{w = \beta (y + \theta \kappa) + (1-\beta) b \vphantom{Bigg(}}. \label{ws}\tag{WS} \end{align} Equation $$(\ref{ws})$$ captures the outcome of wage bargaining between workers and firms. Firms receive value from having a job filled equal to $$y + \theta \kappa$$ beacuse the firm gets the output of the worker $$y$$ and the firm doesn't have to incur a cost $$\theta \kappa$$ to advertise a vacancy. The exogenous variable $$b$$ is the value to the worker of not being employed and reflects, among other things, the amount of unemployment benefits provided by government policy. The exogenous variable $$\beta\in[0,1]$$ reflects workers' degree of bargaining power in wage negotiations. In the tool below, Equation $$(\ref{ws})$$ will be plotted as the upward-sloping curve in the right panel.

Steady state equilibrium in the DMP model is characterized by equations $$(\ref{bc})$$, $$(\ref{vs})$$, and $$(\ref{ws})$$. Together, equations $$(\ref{vs})$$ and $$(\ref{ws})$$ determine the steady state values of the wage and market tightness. Then, equation $$(\ref{bc})$$ determines the steady state unemployment rate given steady state market tightness.

#### Simulation Tool

The tool below can be used to simulate the effect on steady state labor market equilibrium following a one-time change in one of the model's exogenous variables. Select which exogenous variable you want to change and whether you want to the chosen exogenous variable to increase or decrease in value. To see how the unemployment rate transitions to the new steady state, select "yes" under "Show unemployment transition". Click "Submit" to see the change in the equilibrium and click "Reset" to start over. The figures can be downloaded by using the menus in the top right corners of each.

Simulation settings