Theory of Job Search
Given a certain number of job vacancies $A$ and job seekers $Q$, we define the vacancy ratio as $j = A/Q$. The matching function tells us how many jobs are formed given this
E = e \cdot m(Q,A)
From this, we can determine the respective probabilities that a worker finds a job
p_c = \frac{E}{Q} = \frac{e \cdot m(Q,A)}{Q} = e \cdot m(1,A/Q) = e \cdot m(1,j)
and that a firm finds a worker
p_f = \frac{E}{A} = \frac{e \cdot m(Q,A)}{A} = e \cdot m(Q/A,1) = e \cdot m(1/j,1)
Notice that we used constant returns to scale to divide through. Given unemployment benefits $b$ and productivity $z$, we set the wage according to
w = a z + (1-a) b
where $a$ is the bargaining power of the workers. Now we can find the expected return to the work from searching
EV_c \align = p_c w + (1-p_c) b \\
\align = b + p_c (w-b) \\
\align = b + p_c (az+(1-a)b-b) \\
\align = b + p_c a (z-b) \\
\align = b + e \cdot m(1,j) a (z-b)
and to the firm from posting a job vacancy
EV_f \align = p_f (z-w) \\
\align = p_f (z-az-(1-a)b) \\
\align = p_f (1-a) (z-b) \\
\align = e \cdot m(1/j,1) (1-a) (z-b)
We assume that firms pay a cost $k$ to post a job vacancy. Thus in equilibrium this marginal cost should equal the marginal benefit $EV_f$. Therefore the equilibrium value for $j$ should satisfy
k = EV_f = e \cdot m(1/j,1) (1-a) (z-b)
Using this value for $j$ we can then find the equilibrium number job searchers $Q$
Q = N \cdot F(EV_c) = N \cdot F(b + e \cdot m(1,j) a (z-b))
Since any worker with value of staying at home $s_i \lt EV_c$ will choose to search for a job. Here $F$ is the cumulative distribution function over $s_i$ and $N$ is the number of potential workers.
A Specific Example
A common functional form to use for the matching function is
E = e \cdot m(Q,A) = e \cdot Q^{\beta} A^{1-\beta}
This leads to simple expressions for the matching probabilities
p_c \align = \frac{E}{Q} = \frac{e \cdot Q^{\beta} A^{1-\beta}}{Q} \\
\align = e \cdot Q^{\beta-1} A^{1-\beta} = e \cdot \left(\frac{A}{Q}\right)^{1-\beta} \\
\align = e \cdot j^{1-\beta}
and
p_f \align = \frac{E}{A} = \frac{e \cdot Q^{\beta} A^{1-\beta}}{A} \\
\align = e \cdot Q^{\beta} A^{-\beta} = e \left(\frac{A}{Q}\right)^{-\beta} \\
\align = e \cdot j^{-\beta}
Turning to the firms optimality equation, $k=EV_f$, we find
\align k = p_f (1-a) (z-b) \\
\Rightarrow \align k = e \cdot j^{-\beta} (1-a) (z-b) \\
\Rightarrow \align j^{\beta} = \frac{e(1-a)(z-b)}{k} \\
\Rightarrow \align j = \left[\frac{e(1-a)(z-b)}{k}\right]^{1/\beta}
Mapping to the Data
We are interested in observable quantities like the unemployment rate and the vacancy rate. In this model, the unemployment rate is the number of people who couldn't find a job divided by the total number of job seekers
u = \frac{Q-E}{Q} = 1 - \frac{E}{Q} = 1 - p_c = 1 - e \cdot m(1,j)
So for our specific example we get
u = 1 - e \cdot j^{1-\beta}
Similarly, the vacancy rate is the number of job openings divided by the sum of created jobs and job openings
v = \frac{A}{A+E} = \frac{1}{1+E/A} = \frac{1}{1+p_f} = \frac{1}{1+e \cdot m(1/j,1)}
In the specific example
v = \frac{1}{1+e \cdot j^{-\beta}}
Using these two equations, we can derive a relationship between $u$ and $v$
\align u = 1 - e \cdot j^{1-\beta} \\
\Rightarrow \align j^{1-\beta} = \frac{1-u}{e} \\
\Rightarrow \align j = \left(\frac{1-u}{e}\right)^{\frac{1}{1-\beta}} \\
\Rightarrow \align v = \frac{1}{1+e \cdot j^{-\beta}} \\
\Rightarrow \align v = \frac{1}{1+e \cdot \left(\frac{e}{1-u}\right)^{\frac{\beta}{1-\beta}}} \\
\Rightarrow \align v = \frac{1}{1+\left[\frac{e}{(1-u)^{\beta}}\right]^{\frac{1}{1-\beta}}}
This is known as the Beveridge Curve. We can see that there is a negative relationship between $v$ and $u$. So when $u$ is at its lowest value $0$, $v$ is at its highest value $1/(1+e^{1/(1-\beta)})$. When $u$ is at its highest value $1$, then $v$ is zero.