# The Diamond-Mortensen-Pissarides Model
This note is a brief exposition of the Diamond-Mortensen-Pissarides model of job search as found in the Mortensen and Pissarides chapter in the first volume of the Handbook of Macroeconomics. The discussion broadly follows the article in a concise form, but with added notes and derivations. It is originally intended as a note to myself.

## The matching function

Let $\upsilon$ be the amount of job vacancies and $u$ be the amount of unemployed workers. Let the *match function*, $m(\upsilon, u)$ satisfy the usual assumptions of a production technology: increasing in both arguments, diminishing marginal products, and constant returns. The latter implies,
\begin{equation}
m(\upsilon, u) = m\left(1, \frac{u}{\upsilon}\right)\upsilon \equiv q(\theta)\upsilon,
\end{equation}
where $\theta \equiv \frac{\upsilon}{u}$ is the *market tightness*. The job filling rate $q(\theta)$ is decreasing in $\theta$. This is because an increase in $\theta$ means a decrease in $\frac{1}{\theta} = \frac{u}{\upsilon}$ and since $m(\cdot)$ is increasing in both its arguments, $m\left(1, \frac{u}{\upsilon}\right) = q(\theta)$ decreases. The job finding rate $\theta q(\theta)$ is increasing in $\theta$, since
\begin{align}
\theta q(\theta) = m\left(1, \frac{u}{\upsilon}\right)\frac{\upsilon}{u} = m\left(\frac{\upsilon}{u}, 1\right).
\end{align}
Since $q(\theta)$ and $\theta q(\theta)$ can be interpreted as duration hazards, their inverses are the mean durations of vacancies and unemployment respectively.
The productivity of a job is $px$, where $p$ is aggregate productivity common to all jobs and $x$ is an idiosyncratic productivity component with cdf $F(x)$. New shocks arrive at the Poisson rate $\lambda$. It is assumed that each job begins with $x = 1$, since employers are assumed to be free to choose their initial location, technology and product. There is a reservation threshold $R$ such that when $x < R$, the match is destroyed. Therefore, the average rate of transition from employment to unemployment is $\lambda F(R)$, i.e. the probability that a shock arrives $\lambda$ times the probability that this shock has a value below $R$.
The rate of change of unemployment is therefore
\begin{equation}
\dot{u} = \lambda F(R)(1 - u) - \theta q(\theta)u.
\end{equation}
In the steady state, $\dot{u} = 0$, so the expression for steady state unemployment is
\begin{equation}
u = \frac{\lambda F(R)}{\lambda F(R) + \theta q(\theta)}.
\label{eq:unemployment}
\tag{1}
\end{equation}
This figure is decreasing in $\theta$ and increasing in $R$ and defines the vacancy-unemployment schedule called the *Beveridge curve*.

## Job creation and job destruction
We now introduce wage determination into the model. We define a *wage contract* as a pair $(w_0, w(x))$, where $w_0 \in \mathbb{R}$ is a starting wage and $w: X \to \mathbb{R}$ is a function that maps any productivity contingency $x \in X$ to a wage.
A continuing match has capital value to an employer $J(x)$ which satisfies the Bellman equation
\begin{equation}
rJ(x) = px - w(x) + \lambda \int_R^1 \left[J(z) - J(x) \right]\mathrm{d}F(z) + \lambda F(R) \left[V - pT - J(x) \right],
\label{eq:bellmanemp}
\tag{2}
\end{equation}
where $r$ is the (risk free) real interest rate, $V$ is the value of a vacancy and $T \geq 0$ is the termination cost. The left hand side is the return of an existing match. The right hand side is the sum of current profits $px - w(x)$ plus the expected capital gain of the match given a new productivity shock $z$. The first part of the latter is the expected value if this shock is above $R$ and the second if it is below.
An analogous condition holds for the worker:
\begin{equation}
rW(x) = w(x) + \lambda \int_R^1 \left[W(z) - W(x) \right] \mathrm{d}F(z) + \lambda F(R)\left[U - W(x) \right],
\label{eq:bellmanwork}
\tag{3}
\end{equation}
where $W(x)$ is the value to the worker, and $U$ is the value of unemployment. The match is terminated if the new productivity shock $z$ is below $R = \max(R_e, R_w)$, where $J(R_e) = V - pT$ (the value of separation) and $W(R_w) = U$. If $R_e \neq R_w$, and without loss of generality $R_e > R_w$, then a productivity shock such that $R_e > z > R_w$ would imply that the match would be terminated, although the worker would gain surplus from continuation. If employers and workers are jointly rational, then the worker would be willing to forgo some of his wealth to maintain the employment relationship. Thus, joint rationality implies $R = R_e = R_w$ and $J(R) + W(R) = V - pT + U$. We will see in the next section what kind of wage contract satisfies these conditions.
Let $J_0$ be the expected profit of a new match. Then $V$ satisfies
\begin{equation}
rV = -pc + q(\theta)\left[J_0 - V - pC \right],
\end{equation}
where $pc$ is the recruiting cost per vacancy and $pC$ is the fixed cost of matching (e.g. hiring and training). The analogous equation holds for the worker:
\begin{equation}
rU = b + \theta q(\theta)\left[W_0 - U\right],
\end{equation}
where $b$ is the some form of unemployment benefit. Free entry requires that $V = 0$, otherwise more vacancies will be created until eventually the value of holding one is zero. The Bellman equation for V therefore implies
\begin{equation}
\frac{c}{q(\theta)} + C = \frac{J_0}{p}.
\label{eq:beveridge}
\tag{4}
\end{equation}
The left hand side is the expected cost of hiring a worker and the right hand side is the discounted benefit.

## Generalized Nash bargaining

We now introduce a specific kind of bargaining arrangement that allows for wage determination. The initial wage determined by a generalized Nash bargain is
\begin{equation}
w_0 = \arg \max \left\{\left[W_0 - U \right]^{\beta} \left[S_0 - (W_0 - U) \right]^{1 - \beta} \right\},
\end{equation}
where $S_0 \equiv J_0 - pC - V + W_0 - U$ and $\beta$ is the bargaining power of the worker. The continuing wage determined by a generalized Nash bargain is
\begin{equation}
w(x) = \arg \max \left\{\left[W(x) - U \right]^{\beta} \left[S(x) - (W(x) - U) \right]^{1 - \beta} \right\},
\end{equation}
where $S(x) \equiv W(x) - U + J(x) - V + pT$.
The difference between the two bargaining problems lies in the surplus definitions. $S_0$ contains the fixed cost $pC$, whereas $S(x)$ does not because it is sunk. $S(x)$ contains the termination cost, whereas $S_0$ does not because there is no termination cost to not hiring.
We will solve the first problem, the second follows analogously. Plug in the definition of $S_0$ to the objective function to get
\begin{equation}
w_0 = \arg \max \left\{\left[W_0 - U \right]^{\beta} \left[J_0 - pC - V \right]^{1 - \beta} \right\}.
\end{equation}
Differentiate with respect to $W_0$ and $J_0$ and set to zero to get the first order conditions
\begin{align}
\beta \left[\frac{J_0 - pC - V}{W_0 - U}\right]^{1 - \beta} &= 0,\\
(1 - \beta) \left[\frac{J_0 - pC - V}{W_0 - U}\right]^{- \beta} &= 0.
\end{align}
Setting these two expressions equal yields the first order condition
\begin{equation}
\beta (J_0 - pC - V ) = (1 - \beta)(W_0 - U),
\end{equation}
and the analogous one for the continuing wage
\begin{equation}
\beta (J(x) + pT - V ) = (1 - \beta)(W(x) - U).
\end{equation}
It follows from the latter first order condition that $R$ is jointly rational. This is because if $R_e > R_w$, then $J(R) = J(R_e) = V - pT$ and therefore $W(R) = U$ (since $0 < \beta < 1$). The other case follows analogously.
Our goal is now to solve for the surplus function $S(x)$ and continuation wage $w(x)$. To that end, rewrite equations $\eqref{eq:bellmanemp}$ and $\eqref{eq:bellmanwork}$ as
\begin{equation}
(r + \lambda)(J(x) - V + pT) = px - w(x) - r(V - pT) + \lambda \int_R^1 \left[J(z) - V + pT\right] \mathrm{d}F(z),
\label{eq:J}
\tag{5}
\end{equation}
and
\begin{equation}
(r + \lambda)(W(x) - U) = w(x) - rU + \lambda \int_R^1 \left[W(z) - U\right] \mathrm{d}F(z).
\label{eq:W}
\tag{6}
\end{equation}
(To verify these, it is easier to go from the above to equations $\eqref{eq:bellmanemp}$ and $\eqref{eq:bellmanwork}$.) Sum these two together and use the definition of $S(x)$ to get
\begin{align}
S(x) &= \frac{px - r(U + V - pT) + \lambda \int_R^1 S(z)\mathrm{d}F(z)}{r + \lambda},\\
&= \frac{px - r(U + V - pT) + \lambda \int_0^1 \max (S(z), 0)\mathrm{d}F(z)}{r + \lambda}.
\label{eq:s}
\tag{7}
\end{align}
This is a functional equation in $S(x)$. It can be shown that it satisfies the Blackwell sufficient conditions for a contraction mapping, which means it has a solution (a fixed point). Evaluating equation $\eqref{eq:s}$ at $x = R$ and subtracting from equation $\eqref{eq:s}$ reveals that $S(x) = \frac{p(x - R)}{r + \lambda}$. Plugging this back into equation $\eqref{eq:s}$ and using $S(R) = 0$, we get
\begin{equation}
pR + \left(\frac{\lambda}{r + \lambda}\right)p \int_R^1 (z - R)\mathrm{d}F(z) = r(U + V - pT).
\label{eq:option}
\tag{8}
\end{equation}
This equation states that the return on continuing the match to both the employer and worker (the right hand side) must be equal to the reservation product, $pR$, plus the *option value* of a possible increase in productivity in the future. More on the intuition behind this later.
Now we can use the first order conditions in the bargaining problem to conclude that equation $\eqref{eq:J}$ multiplied by $\beta$ is equal to equation $\eqref{eq:W}$ multiplied by $(1 - \beta)$. So
\begin{align}
\beta px - &\beta w(x) - \beta r(V - pT) + \lambda \int_R^1 \beta \left[J(z) - V +pT \right]\mathrm{d}F(z)\\
&= (1 - \beta)w(x) - (1 - \beta)rU + \lambda \int_R^1 (1 - \beta) \left[W(z) - U\right] \mathrm{d}F(z).
\end{align}
The two integrals cancel out because of the first order conditions of the bargaining problem. Therefore, after solving for $w(x)$ we get
\begin{equation}
w(x) = rU + \beta[px - r(V - pT) - rU].
\label{eq:wage}
\tag{9}
\end{equation}
We can write the time $0$ analogues to equations $\eqref{eq:J}$ and $\eqref{eq:W}$ as
\begin{align}
(r + \lambda)(J_0 - V + pT) = p& - w_0 - rV - (r + \lambda)pC - \lambda pT\\ &+ \lambda \int_R^1 \left[J_0 - V + pT\right] \mathrm{d}F(z),
\end{align}
and
\begin{equation}
(r + \lambda)(W_0 - U) = w_0 - rU + \lambda \int_R^1 \left[W_0 - U\right] \mathrm{d}F(z).
\end{equation}
Summing these two equations together we get
\begin{align}
(r + \lambda)S_0 &= p - r(U + V) - (r + \lambda)pC - \lambda pT + \lambda \int_R^1 S(z)\mathrm{d}F(z) \\
&= p(1 - R) - (r + \lambda)p(C + T) \\
&= (r + \lambda)(S(1) - pC - pT),
\end{align}
where the first equation uses the definition of $S_0$, the second the reservation threshold equation $\eqref{eq:option}$ after plugging in $S(z) = \frac{p(z - R)}{r + \lambda}$ and the third uses $S(1) = \frac{p(1 - R)}{r + \lambda}$. Finally, we can use equation $\eqref{eq:beveridge}$ to get one of the key equations in this model, the *job creation curve*:
\begin{align}
\frac{pc}{q(\theta)} &= J_0 - pC\\
&= (1 - \beta)S_0 \\
&= (1 - \beta)p\left(\frac{1 - R}{r + \lambda} - C - T\right).
\label{eq:jobcreation}
\tag{10}
\end{align}
This equation says that the expected gain from the match for the firm equals the expected cost of searching and hiring another worker. It implicitly defines an equilibrium relationship between market tightness, $\theta$ and the reservation product $R$.
Analogously to the continuing case, we can derive the time $0$ analogue of equation $\eqref{eq:wage}$ to be
\begin{equation}
w_0 = rU + \beta \left[p - r(V + U) - (r + \lambda)pC - \lambda pT \right].
\end{equation}
Now, to get an expression for the equilibrium wage contract $(w_0, w(x))$, we need an expression for $rU$ in terms of our parameters. Recall that the value of unemployment, $U$, solves
\begin{equation}
rU = b + \theta q(\theta)[W_0 - U].
\end{equation}
Since $W_0 - U = \beta S_0$ by the first order conditions from the bargaining problem, we get
\begin{equation}
rU = b + \beta \theta q(\theta) S_0 = b + \left(\frac{pc \beta}{1 - \beta}\right)\theta,
\end{equation}
where we have used $q(\theta)S_0 = \frac{pc}{1 - \beta}$ by virtue of equation $\eqref{eq:jobcreation}$. Plugging this into the expressions for $w_0$ and $w(x)$ we get the equilibrium wage contract
\begin{equation}
w_0 = \beta p \left[1 + c\theta - (r + \lambda)C - \lambda T\right] + (1 - \beta)b,
\label{eq:w0eq}
\tag{11}
\end{equation}
and
\begin{equation}
w(x) = \beta p (x + c\theta + rT) + (1 - \beta)b.
\label{eq:wxeq}
\tag{12}
\end{equation}
Finally, we also get a nice expression for the reservation threshold equation:
\begin{align}
p\left(R + \frac{\lambda}{r + \lambda} \int_R^1 (x - R)\mathrm{d}F(x)\right) &= r(U - pT) \\
&= b - rpT + \left(\frac{\beta}{1 - \beta}\right)pc\theta.
\label{eq:jobdestruction}
\tag{13}
\end{align}
This equation defines the *job destruction curve*, which defines a positive relationship between the reservation product $R$ and market tightness $\theta$.

## Some intuition

For many readers, intuition can get lost among the sea of algebra in the previous sections. Therefore, let's dedicate this section for providing some intuition. The solved DMP model consists of the steady state unemployment equation $\eqref{eq:unemployment}$, the equilibrium wage contract $\eqref{eq:w0eq}$ and $\eqref{eq:wxeq}$, and the job creation and job destruction curves $\eqref{eq:jobcreation}$ and $\eqref{eq:jobdestruction}$. For a nice overview, let's collect those equations here
\begin{align}
u &= \frac{\lambda F(R)}{\lambda F(R) + \theta q(\theta)} \tag{1}\\
w_0 &= \beta p \left[1 + c\theta - (r + \lambda)C - \lambda T\right] + (1 - \beta)b \tag{11}\\
w(x) &= \beta p (x + c\theta + rT) + (1 - \beta)b \tag{12}\\
\frac{pc}{q(\theta)} &= (1 - \beta)p\left(\frac{1 - R}{r + \lambda} - C - T\right) \tag{10} \\
p\left(R + \frac{\lambda}{r + \lambda} \int_R^1 (x - R)\mathrm{d}F(x)\right) &= b - rpT + \left(\frac{\beta}{1 - \beta}\right)pc\theta \tag{13}
\end{align}
Let's begin with the job creation and job destruction curves. Plotted in $R$-$\theta$ space, the two curves would look like this:
The job creation curve is decreasing because the higher is the reservation threshold, the shorter is the expected duration of the match (recall that the average rate of transition from employment to unemployment is given by $\lambda F(R)$), and thus firms will not be as incentivized to create new jobs, so they will decrease their vacancies and market tightness will fall as a result. The job destruction curve is increasing because the higher is market tightness, the better the outside opportunities for the worker in terms of wages and thus the higher the reservation threshold. It is called the *job destruction* curve because a higher reservation threshold implies more job destruction in response to a productivity shock. The integral expression on the left hand side is the option value to the employer that a currently occupied job sees an increase in productivity. It is increasing in $\lambda$, because that means productivity changes more frequently, thus implying option value of a possible increase in productivity is higher. It is decreasing in $r$ because of discounting. All of this means that the employer will choose to keep some unproductive jobs occupied, precisely because of the existence of this option value. These two curves together determine the equilibrium pair $(R^{*}, \theta ^{*})$, which together determine the unemployment rate through the *Beveridge curve* in equation $\eqref{eq:unemployment}$.
Finally, the equilibrium wage contract $(w_0, w(x))$ is straightforward to interpret. Clearly, the market power of the worker, $\beta$, determines the portion of the match that they earn. Furthermore, a higher market tightness means more jobs are arriving for workers than workers are filling in vacancies, which implies higher bargaining power for the worker and thus a higher wage.

## Conclusion
This note presented the DMP model with endogenous job destruction. As we will see in a later forthcoming note, this model turns out to be rather useful in explaining some of the key properties of labour markets and business cycles in developed countries.