Matching theory and employment in information equilibrium

In reading up for this piece on Roger Farmer's "post-Keynesian DSGE" theory, I noted his microfoundations: search/matching theory. Here:

I provide a foundation—Keynesian search theory—to the Keynesian theory of aggregate supply. This new theory is rooted firmly in the microeconomic theory of behavior.

and here:

By modelling the process by which unemployed workers are matched with jobs, we can use search theory (for which Dale Mortensen, Chris Pissarides and Peter Diamond were awarded the 2010 Nobel prize) to understand how unemployment varies over time.

Now I have presented information equilibrium as a kind of search/matching process before (demand events matching with supply events forming transaction events), but I thought I'd look at the job openings and hires data at FRED to try to put together a general theory of employment dynamics.

Let's posit that hires (H) are in information equilibrium with unemployed people (U) and job vacancies (V, aka job openings): $H \rightleftarrows U$, and $H \rightleftarrows V$. From the differential equations we are able to derive the Cobb Douglas form:

$$
\log H = a \log U + b \log V + c
$$

From the individual information equilibrium relationships, we can surmise that the rate of change is constant (except for a finite number of shocks):

$$
\begin{align}
\frac{d}{dt} \log \frac{U}{H} & \approx \; \text{const}\\
\frac{d}{dt} \log \frac{V}{H} & \approx \; \text{const}
\end{align}
$$

The derivation is here (and we also look at U/V in the JOLTS data). We can fit to the data using the same procedure as those posts:

The resulting logarithmic derivative (slope) is -0.17/y for $\log U/H$ and 0.08/y for $\log V/H$. Now Petrongolo and Pissarides (2001) [pdf] survey of estimations of the cobb Douglas form generally support constant returns to scale such that $a + b = 1$; this is not required in the information equilibrium model, but gives us a neat trick to use to determine the Cobb Douglas exponents $a$ and $b$.

If we start with the equation above and take $1 = a + b$, we can say (taking the derivative)

$$
\begin{align}
(a + b) \log H & = a \log U + b \log V + c\\
0 & = a \log \frac{U}{H} + b \log \frac{V}{H} + c\\
0 & = a \frac{d}{dt}\log \frac{U}{H} + b \frac{d}{dt}\log \frac{V}{H} \\
a \frac{d}{dt}\log \frac{U}{H} & = - b \frac{d}{dt}\log \frac{V}{H} \\
a \frac{d}{dt}\log \frac{U}{H} & = - (1-a) \frac{d}{dt}\log \frac{V}{H}
\end{align}
$$

Solving for $a$ using the results above gives us $a \simeq 0.32$ and $b = 1 - a \simeq 0.68$. These values are consistent with some of the results from P & P (2001) linked above. If we look at the logarithm of the data we used:

And if we look at $a \log U + b \log V$ (gray) versus the hires data (yellow) we see:

Note that the difference isn't constant! The thing is that the dynamic equilibrium models with constant slopes have non-equilibrium shocks. During these shocks the derivation above does not hold and therefore the matching function shouldn't hold either. But the good news is that the aforementioned difference is pretty well approximated by a logistic curve (a smoothed step function) ‒ the difference is constant except during a shock:

This tells us that during "normal times" (non-recessions) we can use a Cobb-Douglas matching function to understand the constant rate of change of the unemployment rate. However, during recessions, the shocks cause this picture to fail.

Now this doesn't mean matching theory fails during recessions. It just means a simple matching model fails and that the matching function should include its own shocks:

$$
M_{t}(U, V) = c U_{t}^{a} V_{t}^{b} + \epsilon_{t}
$$

It would be interesting to see if the matching function shocks have any relationship to the shocks to $U/H$ and $V/H$ e.g. are they basically equal to the latter, in which case high unemployment can be seen as a shock to vacancies/openings? The data series aren't long enough to resolve it at this point (only one and a half recessions since the hires and openings data started).