I want to refer to the paper Aggregate implications of indivisible labor, incomplete markets, and labor market frictions. 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.


1 Answer 1


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

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.

  • $\begingroup$ 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? $\endgroup$
    – Alex Ruiz
    May 3, 2021 at 16:52

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