<p>They consider a model with two islands: a Production island and a Leisure island.</p>
<p>Every transition from a period <span class="math-container">$t$</span> to a period <span class="math-container">$t+1$</span> is split into two parts.</p>
<ol>
<li>People who are in the Production island at the end of period <span class="math-container">$t$</span>, start on the Leisure island in the beginning of period <span class="math-container">$t+1$</span> with probability <span class="math-container">$\sigma$</span>, and stay on the Production island with probability <span class="math-container">$(1-\sigma)$</span>.</li>
<li>At the start of period <span class="math-container">$t+1$</span>, everyone on the leisure island (which are the people who were on leisure island in period <span class="math-container">$t$</span> plus the ones who moved to the Leisure island in step 1) move to the Production island with probability <span class="math-container">$\lambda_w$</span> and stay on the Leisure island with probability <span class="math-container">$1 -\lambda_w$</span>.</li>
</ol>
<p>I tried to schematize the transition in the figure below.
<a href="https://i.sstatic.net/cnh7j.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cnh7j.png" alt="enter image description here" /></a></p>
<p>Given this, we have that the transition from the Leisure island to the Production island happens at the rate <span class="math-container">$\lambda_w$</span> which the authors put at 0.2.</p>
<p>The transition from the Production island to the Leisure island, which the authors set at <span class="math-container">$0.016$</span> (<span class="math-container">$1.6\%$</span>) is equal to:
<span class="math-container">$$
\sigma(1 - \lambda_w) = 0.016,\\
\to \sigma \times 0.8 = 0.016,\\
\to \sigma = 0.02.
$$</span></p>
<p>So these are the people who moved from the Production island to the Leisure island (at rate <span class="math-container">$\sigma$</span>) and subsequently did not move back to the Production island, i.e. only a fraction <span class="math-container">$(1-\lambda_w)$</span> stays at the Leisure island.</p>

CC BY-SA 4.0
2021-05-03T16:52:53.247Z
Very clear explanation. Thank you very much. And taking advantage of this question, on that sampe page 9, when the authors say that they set $d(1)=-2.3\times log(1-1/3)$, do you know the meaning/origin of that expression?
Alex Ruiz
CC BY-SA 4.0
2021-05-03T06:00:49.153Z
2021-05-03T06:00:49.153Z
tdm
<p>I want to refer to the paper <a href="https://www.sciencedirect.com/science/article/pii/S0304393208000469" rel="nofollow noreferrer">Aggregate implications of indivisible labor, incomplete markets, and labor market frictions</a>. In Footnote 9, there is a brief explanation of how the separation rate is calibrated. I want to quote what it says:</p>
<p>&quot;See Hobijn and Sahin (2007, Table 3). They report that the transition rate from unemployment to employment is on average 20% for 1976-2005. Consistent with this, we set <span class="math-container">$\lambda_w=0.2$</span> for our benchmark calibration. Hobijn and Sahin also report that employment to unemployment transition rate is on average 1.6% for the same sample period. Since <span class="math-container">$\lambda_w=0.2$</span> fraction of the unemployed workers find jobs in the same period, we set <span class="math-container">$\sigma=0.02$</span> which is consistent with a transition rate of 1.6%.&quot;</p>
<p><span class="math-container">$\lambda_w$</span> is an exogenous job arrival rate and <span class="math-container">$\sigma$</span> is the exogenous separation rate. My question is , how can one get the value of <span class="math-container">$\sigma$</span>? I mean, how can we use the 1.6% to get the <span class="math-container">$\sigma=0.02$</span>? It makes sense that the trasition from unemployment to employment is 20%, so that <span class="math-container">$\lambda_w=0.2$</span>, but I just cannot see this logic for <span class="math-container">$\sigma$</span>.</p>
<p>Thank you for your help/comments in advance.</p>

<macroeconomics><dsge>
Separation rate
Alex Ruiz
# Separation rate

(Asked by: username_0 on 2021-04-28)

I want to refer to the paper [Aggregate implications of indivisible labor, incomplete markets, and labor market frictions](https://www.sciencedirect.com/science/article/pii/S0304393208000469). In Footnote 9, there is a brief explanation of how the separation rate is calibrated. I want to quote what it says:

"See Hobijn and Sahin (2007, Table 3). They report that the transition rate from unemployment to employment is on average 20% for 1976-2005. Consistent with this, we set $\lambda\_w=0.2$ for our benchmark calibration. Hobijn and Sahin also report that employment to unemployment transition rate is on average 1.6% for the same sample period. Since $\lambda\_w=0.2$ fraction of the unemployed workers find jobs in the same period, we set $\sigma=0.02$ which is consistent with a transition rate of 1.6%."

$\lambda\_w$ is an exogenous job arrival rate and $\sigma$ is the exogenous separation rate. My question is , how can one get the value of $\sigma$? I mean, how can we use the 1.6% to get the $\sigma=0.02$? It makes sense that the trasition from unemployment to employment is 20%, so that $\lambda\_w=0.2$, but I just cannot see this logic for $\sigma$.

Thank you for your help/comments in advance.

**Tags:** macroeconomics, dsge

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## 1 answers

### Answer by username_1 on 2021-05-03. Score: 3 | Accepted answer

They consider a model with two islands: a Production island and a Leisure island.

Every transition from a period $t$ to a period $t+1$ is split into two parts.

1. People who are in the Production island at the end of period $t$, start on the Leisure island in the beginning of period $t+1$ with probability $\sigma$, and stay on the Production island with probability $(1-\sigma)$.
2. At the start of period $t+1$, everyone on the leisure island (which are the people who were on leisure island in period $t$ plus the ones who moved to the Leisure island in step 1) move to the Production island with probability $\lambda\_w$ and stay on the Leisure island with probability $1 -\lambda\_w$.

I tried to schematize the transition in the figure below.
[![enter image description here](https://i.sstatic.net/cnh7j.png)](https://i.sstatic.net/cnh7j.png)

Given this, we have that the transition from the Leisure island to the Production island happens at the rate $\lambda\_w$ which the authors put at 0.2.

The transition from the Production island to the Leisure island, which the authors set at $0.016$ ($1.6\%$) is equal to:
$$
\sigma(1 - \lambda\_w) = 0.016,\\
\to \sigma \times 0.8 = 0.016,\\
\to \sigma = 0.02.
$$

So these are the people who moved from the Production island to the Leisure island (at rate $\sigma$) and subsequently did not move back to the Production island, i.e. only a fraction $(1-\lambda\_w)$ stays at the Leisure island.

> **Comments:**
> **username_0**: Very clear explanation. Thank you very much. And taking advantage of this question, on that sampe page 9, when the authors say that they set $d(1)=-2.3\times log(1-1/3)$, do you know the meaning/origin of that expression?  

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