Short and long run minimum wage

On Wednesday, Erik Hurst presented a lovely paper, "The Distributional Impact of the Minimum Wage in the Short and Long Run," written with Elena Pastorino, Patrick Kehoe, and Thomas Winberry, at the Hoover Economic Policy Working Group seminar. Video (a great presentation) and slides here. 

This is a beautiful and detailed model, which won't try to summarize here. I write to pass on one central graph and insight. 

Suppose there is some "monopsony power," at the individual firm level. Don't argue about that yet. Erik and coauthors  put it in, so that there is a hope that minimum wages can do some good, and it is the central argument made by minimum wage proponents. In the paper it comes because people are uniquely suited to a particular job for personal reasons. Professors don't like to move, they've figured out the ropes at their current university, so the dean can get away with paying less than they could get elsewhere. Why this applies to MacDonalds relative to the Taco Bell next door is a good question, but again, the point is to analyze it not to argue about it. 

"Labor demand" here is the marginal product of labor. (\(f'(N)\) It's what labor demand would be in a competitive market. The monopsnists' demand is lower). Monopsony means that the "marginal cost of labor" rises with the number of employees. There is a core of people that really love the job that you can hire at low cost. As you expand, though, you have to hire people who aren't that attached to this particular job, so you have to pay more. And you have to pay everyone else more too, (by reasonable assumption -- no individually negotiated wages), so the average cost of labor rises. 

Thus, the monopsonies firm chooses to hire fewer people \(N_m\),  produce less, and pay them a wage \(W_n\) below their marginal product.  ("Average cost of labor" is really the labor supply curve, call it \(w=L(N)\). Then \(\max (f(N)-wN\) s.t. \(w=L(N)\) yields \(f'(N)=w+NL'(N)\). The "marginal cost of labor" in the graph is this latter quantity: the wage you pay the last worker, plus all workers times the extra wage you must pay them all. Disclaimer: the equations are me reverse-engineering the graph.) 

Now, add a minimum wage. As the minimum wage rises above \(W_m\), we initially see a rise in the number of workers, and their incomes. The firm moves along the arrow as shown. (\(\max f(N)-wN\) s.t. \( w \ge L(N)\), \( w \ge w^\ast\) gives \(w^\ast = L(N)\) .) 

Keep raising the minimum wage, though. Once we get past the point that labor supply ("average cost of labor") requires a wage greater than the marginal product of labor, the firm turns around and hires fewer people: 

(Really, the problem all along was \(\max_{w,N} f(N)-wN\) s.t. \( w \ge L(N)\), \( w \ge w^\ast\). Once the minimum wage rises enough, the solution \(w^\ast=L(N) \) has \(f'(N)<w^\ast\). The firm does better by hiring fewer people than are willing to work at that wage. With the second constraint slack, \(f'(N)=w^\ast\) is the optimum.) 

So, in this best case, minimum wages do first raise employment, and income. But if you keep going, they eventually turn around and lower employment and raise unemployment (people between the equilibrium and the "average cost of labor" curve want jobs but can't get them.)  We join the local "monopsony" view with the latter "neoclassical" view. 

The actual model is way more realistic, with multiple kinds of workers, firms that can substitute between workers, dynamics that include capital investment in worker-specific technologies, a search model for unemployment and more. Each seems to me just complicated enough to capture an important effect. Multiple kinds of workers is really important: a big part of the "labor demand" is not just a fixed marginal product of a given kind of worker, but the firm's ability to substitute other kinds of workers and machines for a given task. It's nicely calibrated to match the US economy. 

A bottom line: 

Start raising the minimum wage from $7.50.  At first, this raises employment of low-skilled workers, but the above mechanism. It does nothing to medium and high skill workers, since they are already being paid more than the minimum wage. (I'm not sure why we don't see substitution toward higher skills here.) As the minimum wage rises toward $10, however, we hit the neoclassical part of the low-skill curve, and it starts hurting low-skill employment. In their calibration, "monopsony" lowers wages by about 25%, so once the minimum wage has cured that, i.e. about $10 an hour, workers are being paid their marginal products, so requiring even more just quickly lowers their employment. 

Bit by bit the minimum wage starts to help each group as it hits the point between what they are actually paid and their marginal product. 

People whose marginal products are less than $7.50 an hour are missing from the picture. They were already driven out of the market by the current minimum wage.  (The conclusions about the optimum minimum wage are potentially flawed by this omission. It could be even less!) 

This is a lovely story. An obvious implication: Don't quickly generalize too far from local estimates or small interventions.  Big minimum wage changes can have the opposite effects as small ones! 

The big question of minimum wages is always which workers are helped vs hurt, not overall labor. Much of the other work on minimum wages (Jeff Clemens, for example) emphasizes that it helps a few, who can work the hours employers want, are already skilled, speak English, etc., at the cost of many others, who tend to be less well off to start. 

The dynamic part of the paper is great too. Minimum wages are like rent controls: the damage takes time to show up. In the model, dynamics show up as firms have structured their capital to the current employment mix. It takes time to put in, say, video screens to substitute away from order-takers. 

The shaded part is the duration of typical studies. Studies that examine the short run reactions to small minimum wage changes completely miss the long-run effect of large changes. 

Finally, once again, the minimum wage like so many other policies, is an answer in search of a question. If the issue is "how does policy address labor market monopsony," the minimum wage is a very ineffective answer to that question.  Once you spell out the nature of the actual problem, all sorts of other policies are more effective. If you fix the monopsony, wage subsidies are better. But starting with figuring out why there is monopsony in the first place and what policies are inadvertently supporting it is better still.  

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Update: "Minimum Wages, Efficiency and Welfare" by David Berger, Kyle Herkenhoff and Simon Mongey is a similar paper along these lines -- careful modeling of minimum wages with heterogeneity of workers and firms.  

This paper adds different kinds of firms: From Simon:

"when you start accounting for firms also being heterogeneous... a similar logic carries over. A small minimum wage lifts employment at the small firm with a slither of monopsony power before tanking them, while it's tanking them it starts raising employment at the slightly bigger firm, then tanks that. By the time you get up to the wages paid by any firm that might have considerable market power you've blown up employment at a whole load of firms. A perturbation argument essentially leads you to never increase the minimum wage."

Put another way, a minimum wage increase from $7.50 to $9.00 might actually increase employment at McDonalds... because it puts all the taco stands out of business. Then at $12.00, McDonalds goes out of business but Applebees expands, and so forth. (Or, "corner store" and "supermarket" in Simon's beautiful slides with lots of great supply and demand graphs.) 

They find that the efficiency maximizing minimum wage is close to where we are now. "Efficiency" means "offsetting monopsony." As in Hurst et al, only a small sliver of people are actually hurt by monopsony and helped by the minimum wage. Everyone whose productivity is below $7.50 an hour is already out of the labor force, and everyone whose productivity is higher than the proposed minimum wage is largely unaffected: 

 Again, "raise the minimum wage to offset labor monopsony" is an answer in search of a question. (They go on to evaluate redistribution, which I didn't look at. But I will ask the same question. "raise the minimum wage to redistribute income" sounds to me like an answer in search of a question; if the question is "redistribute income with minimum economic disincentive" I bet there are better answers.) 

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Update 2. Now, let's think a bit about this "monopsony" business. Both papers include monopsony really for good rhetorical reasons: Let's give the model some reason for minimum wages. Both cite long literatures. Hurst et al summarize that wages are about 25% less than marginal products. Really? At McDonalds? 

Without getting in to the weeds, think for a minute just how hard this is. What is the marginal product of workers at your job? The marginal product of an extra professor in your department? That's awfully hard to measure! Kudos to those who try. It's easier to measure average products: how much the company makes, divided by number of employees. But wages should be below average products. Someone has to pay for the other inputs and a competitive return to capital. Did we really tease out average vs. marginal products? Well, build a model, add lots of assumptions, and here we go. That's the best we can do, but recognize how hard it is. 

Pervasive monopsony means two things, both suspicious. First, it means that each company would have to pay more to hire more people, to do the exact same job as current people, and then it has to pay everyone more. The labor supply curve to the company is upward sloping. That's key in the graph above. Really? Do a restaurant really have to pay everyone more in order to get one more employee? Second, it means there are substantial "rents." Where does the extra 25% go? Not just to an ordinary return to capital, but to extraordinary profits. Together with the view that price markups over marginal costs are large, it's just hard to see large monopoly and oligopoly rents spewing out of businesses. 

I think this illustrates two problems in our general economic discourse. First, econ 101 tends to be a week of how a hypothetical free market works, and then a 9 week litany of market failures, each remediable by an omniscient "planner" -- monopoly, monopsony, externality, asymmetric information, and so on. Our students, like two year olds with hammers, go out and see those nails. But are they really there, and are the available instruments actually able to fix them? 

Second, there is a pervasive tendency for answers to search for questions. Clearly the minimum wage came first, a centuries old idea, long before monopsony. Monopsony is only the latest item in the shopping cart of reasons for a pre-exiting policy idea. As above, if the question is monopsony, however, the answer is not a minimum wage. This problem abounds. (If the question is how to raise GDP 5% in 100 years, there are 99 answers better than force everyone to buy electric cars today, for example.) 

Let's be honest. The idea behind minimum wages is to try to transfer income from businesses -- and thus from their customers, investors, and high-wage workers -- to low-wage employees. Mongey et al. explicitly consider "redistribution" as an objective, and this is the objective. The many unintended consequences -- more unemployment, lower employment, favoring the better off at the expense  of the most precarious of low-wage employees, etc. -- bear on that issue. Here to, though,  if the question is "how should we redistribute income to low-wage workers" or even "how should we improve the lot of low-skill workers," there are 100 better answers. The EITC looks better in Hurst et al, though it too has many problems including a very high marginal tax rate as it phases out. 

Economic discourse would be a lot more productive if instead of focusing on answers that have been around a long time -- let's find a new reason for minimum wages -- we focused on the question and the mechanisms.