Separation rate

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 $$\sigma$$.

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. 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.

• 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? May 3 '21 at 16:52