Shifts and drifts of the Beveridge curve

As a follow up to my post on long term unemployment, I wanted to discuss the Beveridge curve. Gabriel Mathy discusses changes to it in his paper (ungated version here). He shows how there are differences in shifts in the Beveridge curve for short and long term unemployment (click to enlarge):

In an earlier version of the paper, he includes a graphical explanation:

The dynamic information equilibrium approach also describes the Beveridge curve with a formally similar "matching" framework described in my paper. However, one of the primary mechanisms for shifts of the Beveridge curve is actually just a mis-match in the (absolute value of the) dynamic equilibria, i.e.

\begin{eqnarray}
\frac{d}{dt} \log \frac{U}{L} = - \alpha + \sum_{i} \frac{d}{dt} \sigma_{i}(a_{i}, b_{i}; t-t_{i})\\
\frac{d}{dt} \log \frac{V}{L} = \beta + \sum_{j} \frac{d}{dt} \sigma_{j}(a_{j}, b_{j}; t-t_{j})
\end{eqnarray}

with $\alpha, \beta > 0$ — the difference in sign means you get a hyperbola. I can illustrate this using an idealized model with several shocks $\sigma_{i}(t)$. Let's keep $U$ constant, but change the relative parameters of $V$ (altering the dynamic equilibrium $\Delta \alpha$, the timing of the shocks $\Delta t$ and the amplitude of the shocks $\Delta t$). Here are $U(t)$ and $V(t)$ (click to enlarge):

If everything is the same ($\alpha = \beta$, $\Delta t = \Delta a = \Delta b = 0$), then you get the traditional Beveridge curve that doesn't shift:

Changing the dynamic equilibrium ($\alpha \neq \beta$) gives you the drift we see in the data:

This means the drift is due to the fact that (in the regular model) $\alpha$ = 0.084 and $\beta$ = 0.098 (vacancy rate increases at a faster rate than the unemployment rate falls). If we look at changes to the timing $\Delta t$ and amplitude $\Delta a$ of the shocks, we get some deviation but it is not as large as the change in dynamic equilibrium rate:

Combining the changes to the amplitude and timing also isn't as strong as changing the dynamic equilibrium ($\Delta \alpha$):

But if we do all of the changes, we get the mess of spaghetti we're used to:

...

PS I didn't change the widths of the shocks ($\Delta b$) because ... I forgot. I will update this later showing the effects of changing the widths. Or maybe I will remember to do it before this scheduled post auto-publishes (unlikely).

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Update

The $\Delta b$'s add adorable little curlicues (click to enlarge):