<p>Found my mistake. I guess I'm not allowed to move the thetas to the other side.</p>

<p>Define $q^{-1}$ be the inverse of $q$. Then,</p>

<p>$$ \theta = q^{-1} \left( \frac{c}{J} \right)$$</p>

<p>$$ \frac{W_1}{W_0} = \frac{f(\theta_0)}{f(\theta_1)} = \frac{f \left( q^{-1} \left(\frac{c}{J_1}\right)\right)}{f \left( q^{-1} \left(\frac{c}{J_0}\right)\right)}$$ </p>

<p>$J_1 &gt; J_0$, hence $\theta_1 &gt; \theta_0$ and $W_1 &lt; W_0$.</p>

CC BY-SA 3.0
2015-04-21T16:40:50.260Z
2015-04-21T16:40:50.260Z
FooBar
<p>Consider the following extension of the standard DMP model, in which we have two labor markets with different returns, indexed $i$. The markets are completely separated and firms and workers can only go to one market. It might help to think of them as different countries, for example. </p>

<p><strong>I am ultimately interested in the different waiting time of unemployed to find a job</strong>.</p>

<p>As usual, we match using a CRS function $M(u,v)$, such that the likelihoods of finding a job and finding a worker are given by $f(\theta) = M(u,v)/u = M(1, \theta) $, and $q(\theta) = M(u, v)/v = M(\theta^{-1}, 1)$.</p>

<p>The vacancy-value equation is </p>

<p>$$ \rho V_i = -c q(\theta_i) (J_i - V_i)$$</p>

<p>Free entry implies $V_i = 0$. Manipulating the equations we get</p>

<p>$$ \frac{q(\theta_1)}{q(\theta_0)} = \frac{J_0}{J_1}$$</p>

<p>which makes sense - the job-filling-rates are inverse to the value of the jobs. Say $J_1 &gt; J_0$. Then, market $1$ is tighter, leading to a lower job-filling-rate for an individual firm.</p>

<p>Using the earlier definitions, I can use that $f(\theta) = q(\theta)\theta$:</p>

<p>$$ \frac{f(\theta_1)}{f(\theta_0)} = \frac{J_0}{J_1}\frac{\theta_1}{\theta_0}$$</p>

<p>Let $W_i$ denote the average waiting time of an unemployed in market $i$. As we're using Poisson rates, $W_i = 1/f(\theta_i)$:</p>

<p>$$ \frac{W_0}{W_1} = \frac{J_0}{J_1}\frac{\theta_1}{\theta_0}$$</p>

<p>Which doesn't make sense anymore. Consider again $J_1 &gt; J_0$: A worker in market $1$ is much more valuable to firms than a worker in market $0$. However, the relative waiting time for workers in market $1$ <em>increases</em> in the relative value of workers in market $1$.</p>

<p>I don't understand this. Was my derivation wrong? Or am I not allowed to to the ceteris paribus argument while ignoring $\theta_1/\theta_0$ on the right-hand-side? </p>

CC BY-SA 3.0
2015-04-21T16:00:46.577Z
On the fly, maybe this explains why middle-level executives find it hard to find a job after leaving one (i.e. they have to wait longer compared to non-managerial employees)? (I am only half-joking, I will return here)
Alecos Papadopoulos
<macroeconomics><labor-economics><unemployment>
DMP: Two Labor Markets
FooBar
# DMP: Two Labor Markets

(Asked by: username_0 on 2015-04-21)

Consider the following extension of the standard DMP model, in which we have two labor markets with different returns, indexed $i$. The markets are completely separated and firms and workers can only go to one market. It might help to think of them as different countries, for example.

**I am ultimately interested in the different waiting time of unemployed to find a job**.

As usual, we match using a CRS function $M(u,v)$, such that the likelihoods of finding a job and finding a worker are given by $f(\theta) = M(u,v)/u = M(1, \theta) $, and $q(\theta) = M(u, v)/v = M(\theta^{-1}, 1)$.

The vacancy-value equation is

$$ \rho V\_i = -c q(\theta\_i) (J\_i - V\_i)$$

Free entry implies $V\_i = 0$. Manipulating the equations we get

$$ \frac{q(\theta\_1)}{q(\theta\_0)} = \frac{J\_0}{J\_1}$$

which makes sense - the job-filling-rates are inverse to the value of the jobs. Say $J\_1 > J\_0$. Then, market $1$ is tighter, leading to a lower job-filling-rate for an individual firm.

Using the earlier definitions, I can use that $f(\theta) = q(\theta)\theta$:

$$ \frac{f(\theta\_1)}{f(\theta\_0)} = \frac{J\_0}{J\_1}\frac{\theta\_1}{\theta\_0}$$

Let $W\_i$ denote the average waiting time of an unemployed in market $i$. As we're using Poisson rates, $W\_i = 1/f(\theta\_i)$:

$$ \frac{W\_0}{W\_1} = \frac{J\_0}{J\_1}\frac{\theta\_1}{\theta\_0}$$

Which doesn't make sense anymore. Consider again $J\_1 > J\_0$: A worker in market $1$ is much more valuable to firms than a worker in market $0$. However, the relative waiting time for workers in market $1$ *increases* in the relative value of workers in market $1$.

I don't understand this. Was my derivation wrong? Or am I not allowed to to the ceteris paribus argument while ignoring $\theta\_1/\theta\_0$ on the right-hand-side?

**Tags:** macroeconomics, labor-economics, unemployment

> **Comments:**
> **username_1**: On the fly, maybe this explains why middle-level executives find it hard to find a job after leaving one (i.e. they have to wait longer compared to non-managerial employees)? (I am only half-joking, I will return here)  

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## 1 answers

### Answer by username_0 on 2015-04-21. Score: 3

Found my mistake. I guess I'm not allowed to move the thetas to the other side.

Define $q^{-1}$ be the inverse of $q$. Then,

$$ \theta = q^{-1} \left( \frac{c}{J} \right)$$

$$ \frac{W\_1}{W\_0} = \frac{f(\theta\_0)}{f(\theta\_1)} = \frac{f \left( q^{-1} \left(\frac{c}{J\_1}\right)\right)}{f \left( q^{-1} \left(\frac{c}{J\_0}\right)\right)}$$

$J\_1 > J\_0$, hence $\theta\_1 > \theta\_0$ and $W\_1 < W\_0$.


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