Friday, February 24, 2023

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.  


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.) 


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. 


  1. Regarding minimum wage laws, is it better to have no (zero ) minimum wage laws or state mandated law or federal law ?

  2. Fastest way to kill two economists is to throw a minimum wage paper between them and watch them fight to the death.

  3. Christina D. Romer, “The Business of the Minimum Wage,” New York Times, March 2, 2013.

    She did not seem to think much of the monopsony argument and thought that EITC was better


    "If these new workers are typically more affluent — perhaps middle-income spouses or retirees — and end up taking some jobs held by poorer workers, a higher minimum could harm the truly disadvantaged.

    Another reason that employment may not fall is that businesses pass along some of the cost of a higher minimum wage to consumers through higher prices. Often, the customers paying those prices — including some of the diners at McDonald’s and the shoppers at Walmart — have very low family incomes. Thus this price effect may harm the very people whom a minimum wage is supposed to help."

  4. "Let's be honest. The idea behind minimum wages is to try to transfer income from businesses..."

    Actually, the disemployment effects of the minimum wage were recognized at the Progressive era inception. It was intended: Blacks and Yellows who worked hard were to be excluded from competing with Whites.

  5. If developed nations carefully calibrate immigration, there will be little need for a minimum wage.

  6. I think the justification is mostly felt for 2 reasons:

    1) Virtue signaling. Taking from evil corporations and giving it back to the working class
    2) This policy does not go on budget. It becomes a shadow price with unclear costs that are largely not considered until you suffer an India like rupee crisis and it falls on the laundry list of macroeconomic reforms mandated by the world bank.

  7. Monopsony and oligopsony are models ('ideals') of firm behavior that are, for the purpose of mathematical economics, ideally suited to the purpose of demonstrating the trade analyses that should be brought to address the question of whether a $15 per hour federal minimum wage is social welfare improving or not.

    Both papers are far above the undergraduate 2nd year economic's syllibus. The authors endeavor to tie their models to earlier papers and empirical data, as they should.

    On a generalist's level, i.e., the level of an experienced business executive or government policy maker who doesn't necessarily have the academic credentials of an academic specialized in mathematical economics, it doesn't take all that much effort to put the results and the effects cited in the papers into a practical framework based on his/her business or government experience.

    A brief review of an undergraduate microeconomics textbook on labor economics and firm type quickly brings the experienced executive or government decision-maker up to speed with the concepts and the terminology of the problem that the two papers seek to address. The brief review also serves to make intelligible the several charts that appear in the blog post above. Also, Joan Robinson's 1933 paper is available as are Frank P. Ramsey's papers published in the late 1920s. Contrary to John's assertion, a 2nd-year or higher undergraduate course in microeconomics does indeed help, and at least it can't hurt.

    Kudos to John for bringing these papers to our attention. The federal minimum wage proposal is topical, and the papers do point out certain fallacies that the proponents of the federal minimum wage level increase gloss over or ignore.

  8. Can someone in the halls of academy explain why professionally trained economists like the minimum wage? Even if you think the empirics are murkey; it has the dubious distinction of

    a) flying in the face of basic economic theory. Literally day 1 stuff in economics

    b) There are 9 million better/more effective anti poverty policies out there. Picking this one out of a hat makes 0 sense to me.

    1. If you consider a union that has negotiated a labor contract that specifies wage rates for each non-exempt position with the firm that employs union members, then you have the answer you are seeking.

      In the case of the minimum wage, replace the union with the local or state government, and replace the firm with all firms doing business in the local or state government's jurisdiction, and you have the answer.

      Minimum wage increases the minimum-wage employees' surplus from the labor they supply firms that employ them. The surplus comes out of the firms' operating margin.

      Up to a certain wage rate threshold, the minimum wage is social welfare improving. Beyond that threshold wage rate, the minimum wage is social welfare diminishing. John's second chart shows the effect, although not as clearly as might.

      Caveat: The firms must be operating in an economic sector dominated by imperfect competition.

  9. (posting this comment in 2 parts since it was too long...)

    IMO the minimum wage debate is less settled than you'd think based on the above discussion. and not in a "politically controversial" sense, but genuinely scientifically unsettled

    first of all, there is some very strong recent econometric evidence against large employment effects of the minimum wage, even in the medium-run (2-7 years). the strongest evidence is the recent trio of papers by Attila Lindner and co-authors. each papers contains strong evidence of plausible mechanisms by which labour markets adjust to minimum wage hikes other than through employment. the lack of employment response documented in these papers is not a "miracle", but rather a plausible story about how markets can adjust on more than one margin

    1. in "Who Pays for the Minimum Wage?" (AER, 2019) with P├ęter Harasztosi, they find a negative but small employment elasticity (a 10% increase in labour costs decreases employment by less than 2%) up to 4 years after Hungary raised its federal minimum wage from about the level of the US to the level of France. firms raised prices in response to the shock and increased capital stock. the large shock and large sample size allow for very precise estimates. in my opinion, this result should "move your priors" more than the noisy CPS state panels people argue over in the US literature

    2. "Reallocation Effects of the Minimum Wage" (QJE, 2022) with Christian Dustmann, Uta Schonberg, Matthias Umkehrer, and Philipp vom Berge takes another massive European Federal Minimum Wage (Germany's in 2014) and studies it using comprehensive employer-employee matched data. the mechanism here points to re-allocation across firms. some small unproductive firms that paid below the MW exit, and more productive higher-wage firms expand. results extend to several years after the minimum wage comes into effect.

    doesn't this contradict Berger, Mongey and Herkenhoff's model? there, firms' wage-setting power comes from their size (firms are basically Cournot oligopsonists for labour). however, in a different class of models where market power comes from workers' heterogeneous preferences, even small firms can exercise wage-setting power. i go back and forth on whether this is counterintuitive. try introspective thought experiment and ask yourself "if your boss cut your pay by 5%, would you leave your job?" i love my job; i'd practically do it for free! if McDonalds paid 9.50$ per hour instead of 10$, would all of its employees leave? some would, but how many? how much labour did McD poach from Burger King when it raised its minimum wage to 13$ a couple years ago? some economists have actually gone out and randomized wages, and they found considerable wage setting power. whether firms actually exercise this power by setting the wage at the optimal markdown below the wage is another question. i agree with John that profit margins put some bound on the extent to which employers can be marking down labour below MPL--i'm just not sure where the bound is

  10. ... (part 2)

    3. in "The Effect of Minimum Wages on Low Wage Jobs" (QJE, 2019) with Doruk Cengiz, Arindrajit Dube, and Ben Zipperer the authors present a nice "sanity" check for minimum wage effects. if not all jobs below the MW are lost, we should expect them to show up above the new MW. the authors calculate the "excess mass" above the MW and the "missing mass" below using hourly wage data in surveys and administrative records. the employment effect should be close to the difference between these two. the authors show that in many previous studies, negative employment effects are driven by employment in jobs with wages far above the MW. the results show no discernible employment effect 5 or even 7 years out using this method. this one's US data

    i think it is unwise to ignore this econometric evidence. these papers are not saying "set a really high minimum wage!" (although of course many people will interpret them that way). rather, they are teaching us something about the ways that labour markets operate

    as an intellectual history aside, it's amusing to see John call the assumption of no individually negotiated wages "reasonable". the OG "wage posting" model of Burdett and Mortensen was first released as a working paper in 1989 but was rejected from many journals (and all top 5 journals) for nearly a decade before being published in 1998 (and mostly languished in obscurity until it was revived by Alan Manning's 2003 book Monopsony in Motion and structural empirical work by Jean-Marc Robin and co-authors). it's now a centerpiece of most PhD labour courses. as i understand, it was precisely the "no bargaining" assumption that lead to its rejection. why would firms not bargain, if a deal can be made? times change! see "Firms' Choices of Wage-Setting Protocols" by Christopher Flinn and Joseph Mullins for a beautiful micro-foundation of wage posting

  11. What about (K) in the long run contributing to hiring less as MRTS gets some traction? Growth for a monopsony doesn't just worry about labor and it doesn't make sense to "grow" in perpetual SR thinking.

  12. The Wall Street Journal published an editorial outlining the Biden administration's approach to industrial planning: "The Chips Act Becomes Industrial Social Policy -- Gina Raimondo uses semiconductor subsidies to impose progressive priorities via corporations." Feb. 28, 2023.

    Part of the regulations being imposed in exchange for federal subsidies are minimum wage rates based on union labor wages. Additional regulations tied to the grant of federal subsidies require establishment of child-care services for construction workers and the eventual workforce employed by the semiconductor fab owners.

    Union wage rates have the same effect on employment as minimum wage rates established by the states and the federal government, under conditions of imperfect competition.

  13. Minimum wage in Switzerland is about $23 per hour. They have Obamacare-like health insurance, except that it is more generous and covers everyone. Poverty rate there is 0.2% (1.7% in the US). College education is almost free (730 CHF per semester at ETH). Current research indicates that when you take US minimum wage in the 60s, account for inflation and raise in productivity, current minimum wage should be about $23. This means that money has been transferred from employees upwards.

    1. Switzerland doesn't have a national minimum wage. The $23 per hour wage you cite is from Geneva, which is one of the most expensive parts of the country to live in.

  14. Correction to my post above. Poverty rate in the U.S. is about 11.6%, not 1.7%

  15. Yet another correction. Poverty rate in Switzerland is between 6.6 and 10% depending on the source. Too many mistakes.


Comments are welcome. Keep it short, polite, and on topic.

Thanks to a few abusers I am now moderating comments. I welcome thoughtful disagreement. I will block comments with insulting or abusive language. I'm also blocking totally inane comments. Try to make some sense. I am much more likely to allow critical comments if you have the honesty and courage to use your real name.