An Ultraminimalist Model of the Beveridge Curve, or, How I Learned to Start Worrying and Love Structural Unemployment

Where do businesses find people to hire?  A few new employees – graduating students, for example – are recruited from outside the labor force, but I’m going to ignore them (as later I will also ignore retirees, figuring that they roughly offset each other).  Most new employees come either from among the unemployed or from other firms.  Hiring the unemployed is easy inasmuch as they’re usually knocking at your door asking for jobs.  On the other hand, the selection process might be difficult, since they aren’t doing a job now, so you have to make an educated guess as to whether they’ll be good at the job for which you’re hiring.  Hiring people from other firms is difficult in that you have to go out and actively recruit them, as well making an offer that justifies leaving their old job, but the selection process is easier, because all you have to do is find someone who is already doing a job similar to the one for which you’re hiring. 

So there is a tradeoff.   Presumably the terms of this tradeoff depend on how many people are unemployed:  if only a few people are unemployed, then the number of qualified unemployed applicants will be low, and they’ll be in demand from other firms, so you’ll have to pay them well, so you might as well just try to poach someone directly from another firm; if a lot of people are unemployed, the number of qualified unemployed applicants will be high, and they’ll be willing to accept less attractive offers, so poaching might not be worth it.  So here’s the crux of my model:  the fraction of new hires that comes from the unemployed depends on how many unemployed there are.  Using “H” for total hires, “He” for “hires out of employment,” and “Hu” for “hires out of unemployment,” we have Hu/H=f(U), where f() is some increasing function, and for adding-up, we have H=He+Hu.

Now, why do people quit their jobs?  Some retire, but I’ve already said I’m going to ignore them.  Some quit for other personal reasons, and I’m going to ignore them too.  A few people quit, especially when the labor market is strong, because they don’t like their job and figure it will be easy enough to find a new one.  I’m also going to ignore them.  Most people who quit, I believe, quit because they already have another job lined up.  In other words, if we ignore all the categories I’m ignoring, then the number of quits equals the total number of new hires minus the number of new hires that are hired out of unemployment, or Q=H-Hu.  Putting this proposition together with the one at the end of the last paragraph and solving simultaneously, we get Q=H*(1-f(U)).

OK, what about layoffs?  It may sound crazy at first, but I’m essentially going to ignore layoffs.  I’m going to assume that they happen at a constant rate.  We do know that layoffs tend to spike during the early part of a recession (or in the case of the recent recession, in the middle, when the “Great Recession” took over from the “little recession” that was already in progress).  But the typical spike is fairly small compared to the total number of layoffs.  (We notice those layoffs more because they result in significant spells of unemployment, whereas non-recession layoffs often result in just changing jobs, or in brief spells of unemployment that often aren’t long enough to justify filing an unemployment claim.)  So the “constant layoffs” assumption isn’t too far from the truth.  Also, layoff spikes are clearly “disequilibrium” phenomena that induce changes in the unemployment rate rather than explaining how a given unemployment rate is maintained.  In thinking about the Beveridge curve, I’m interested in the equilibrium relationship between unemployment and job openings.

And here’s the equilibrium condition.  I’ll ignore longer run changes in the labor force and the capital stock and define equilibrium as constant total employment (which implies constant total unemployment, since I’m ignoring labor force changes).  Constant employment implies that hires equal separations.  I’ll ignore “other separations” and assume all the separations are either quits or layoffs.  Then we have H=Q+L (where L stands for “layoffs,” not “labor”).

Since I’ve assumed that layoffs are constant, we have three variables here, U, H, and Q.  We’re more interested in hires than quits, so we can solve to eliminate Q, and we get H=L /f(U).  Since f() is an increasing function, this gives as an inverse equilibrium relationship between hires and unemployment.

Ultimately we want to relate unemployment to job openings, since that’s what the Beveridge curve is about.  How do hires relate to job openings?  One traditional approach is to fit a “matching function” in which hires are an increasing function of both job openings and unemployment.  The theory is that it should be easier to fill job openings when there are a lot of unemployed people looking for jobs.  I tried fitting such a function using JOLTS data, and the coefficient on unemployment consistently came out with the wrong sign, no matter how many polynomial time trends or dummy variables I put in, and even when I included an interaction term between unemployment and the availability of extended unemployment benefits.  Actually, that result is what motivated this model.  While obviously a high unemployment rate will reduce the number of people who quit their jobs in order to fill job openings, it does not apparently result in those openings being filled any more quickly.  So my matching function is a one-variable function. H=m(V), where V (“vacancies”) is the number of job openings.

Empirically, I fit H=m(V) as H=a*V^b, where a and b are fitted constants.  (Why do I use that form? Tradition, I suppose:  it just seemed reasonable.  It allows for the intuitive special case where b=1, so that job openings fill at a constant rate, but one casual look at the data will tell you that b<1 in reality: openings fill more quickly when there are fewer of them.)  The fit is pretty good (“log V” explains 78% of the variance in “log H” with a slope coefficient of about 0.5, implying that H is proportional to the square root of V), but there is an obvious pattern in the residuals.  (The Durbin-Watson statistic is a mere 0.7 – in case this post isn’t wonkish enough already.)  The cumulative sum of the residuals peaks in July 2006, suggesting that there may be a structural break in August.  A casual look at the residuals strongly suggests another structural break in July 2010.  Both purported structural breaks go in the same direction:  a decline in the number of hires associated with any given number of job openings.  So, contrary to what I said in 2010, it does look like we are seeing more structural unemployment now than in the past.  (In my defense, the first break occurs long before the recession, so I was right to assert that recession had not produced an increase in structural unemployment; and the second break occurs just when I was making that assertion, so I had no data from after the break.)

After I had done all this pseudo-theorizing, I decided to do a little pseudo-test of my pseudo-model, and it actually holds up surprisingly well (allowing the function f(U) to have the same form as m(V), because don’t all functions have that form, damn it!).   There is a nice, linear-looking, downward-sloping relationship between the log of hires and the log of unemployment.  Log unemployment explains almost 90% of the variance in log hires, with a slope coefficient of -0.37, and the coefficients are robust to the inclusion of an ARMA(1,1) residual process that results in a Durbin-Watson statistic of precisely 2.0.  (Ooooh, talk nerdy to me, Baby!)  There is no obvious pattern in the residuals.  Surprisingly, there are only 3 significant outliers (March 2003, November 2008, and May 2010; call them “Iraq War,” “Post-Lehman,” and “Census”).  At least I find that surprising, because this is an equilibrium model of a system that obviously, in real life, is subject to shocks that move it out of equilibrium – as we know from the fact that the unemployment rate changes a lot.  If you take this test at face value, it suggests that the equilibrating forces (which I haven’t tried to model) are very strong.

So what does all this imply about the natural rate of unemployment?  To answer that question we need a model of aggregate supply, and I happen to have one up my sleeve.  Here’s my model:  there’s a constant natural rate of job openings.  That’s it.  If firms have an unusually large number of positions to fill, they bid up wages, and you get accelerating inflation.  If firms have an unusually small number of positions to fill (like right now, but even more like three years ago), they start to let wages erode, and you get decelerating inflation (although research now suggests that it’s very difficult to erode wages that already aren’t rising, so this won’t work very well unless there is some substantial inflation or productivity growth to begin with – but all that belongs in another post).  Somewhere there’s a happy medium rate of job openings, such that wages tend to continue rising at a rate consistent with the expected rate of inflation.  That’s the natural rate of jobs openings.  Or the Non-Accelerating Inflation Rate of Job Openings (NAIRJO).   Or the Non-Accelerating Inflation Rate of Vacancies (NAIRV).

If the relationship between hiring and unemployment is stable, as it appears to be, then my model implies that shifts in the matching function will determine a shifting relationship between the (assumed constant) NAIRV and the NAIRU (Non-Accelerating Inflation Rate of Unemployment, a.k.a. the natural rate of unemployment).  For what it’s worth, my estimates suggest that the hypothesized August 2006 and July 2010 shifts in the matching function would, collectively, increase the NAIRU by a factor of about one and a third.  So if the NAIRU was 4.5% (my best guess, which happens to be conveniently divisible by 3) in July 2006, it is 6% now.  Of course, by the time the unemployment rate gets down to 6%, there’s a good chance that the matching function will have shifted again, but as for which direction and how far, your guess is as good as mine.

UPDATE: Posted scatter of log hires vs log unemployment on Twitter.

UPDATE2: Posted graph of series hires/sqrt(openings) on Twitter.

DISCLOSURE: Through my investment and management role in a Treasury directional pooled investment vehicle and through my role as Chief Economist at Atlantic Asset Management, which generally manages fixed income portfolios for its clients, I have direct or indirect interests in various fixed income instruments, which may be impacted by the issues discussed herein. The views expressed herein are entirely my own opinions and may not represent the views of Atlantic Asset Management. This article should not be construed as investment advice, and is not an offer to participate in any investment strategy or product.