Dec13 JDN 2459197

This phenomenon has been particularly salient for me the last few months, but I think it’s a common experience for most people in my generation: Getting a job takes an awful lot of work.

Over the past six months, I’ve applied to over 70 different positions and so far gone through 4 interviews (2 by video, 2 by phone). I’ve done about 10 hours of test work. That so far has gotten me no offers, though I have yet to hear from 50 employers. Ahead of me I probably have about another 10 interviews, then perhaps 4 of what would have been flyouts and in-person presentations but instead will be “comprehensive interviews” and presentations conducted online, likely several more hours of test work, and then finally, maybe, if I’m lucky, I’ll get a good offer or two. If I’m unlucky, I won’t, and I’ll have to stick around for another year and do all this over again next year.

Aside from the limitations imposed by the pandemic, this is basically standard practice for PhD graduates. And this is only the most extreme end of a continuum of intensive job search efforts, for which even applying to be a cashier at Target requires a formal application, references, and a personality test.

This wasn’t how things used to be. Just a couple of generations ago, low-wage employers would more or less hire you on the spot, with perhaps a resume or a cursory interview. More prestigious employers would almost always require a CV with references and an interview, but it more or less stopped there. I discussed in an earlier post how much of the difference actually seems to come from our chronic labor surplus.

Is all of this extra effort worthwhile? Are we actually fitting people to better jobs this way? Even if the matches are better, are they enough better to justify all this effort?

It is a commonly-held notion among economists that competition in markets is good, that it increases efficiency and improves outcomes. I think that this is often, perhaps usually, the case. But the labor market has become so intensely competitive, particularly for high-paying positions, that the costs of this competitive effort likely outweigh the benefits.

How could this happen? Shouldn’t the free market correct for such an imbalance? Not necessarily. Here is a simple formal model of how this sort of intensive competition can result in significant waste.

Note that this post is about a formal mathematical model, so it’s going to use a lot of algebra. If you are uninterested in such things, you can read the next two paragraphs and then skip to the conclusions at the end.

The overall argument is straightforward: If candidates are similar in skill level, a complicated application process can make sense from a firm’s perspective, but be harmful from society’s perspective, due to the great cost to the applicants. This can happen because the difficult application process imposes an externality on the workers who don’t get the job.

All right, here is where the algebra begins.

I’ve included each equation as both formatted text and LaTeX.

Consider a competition between two applicants, X and Z.

They are each asked to complete a series of tasks in an application process. The amount of effort X puts into the application is x, and the amount of effort Z puts into the application is z. Let’s say each additional bit of effort has a fixed cost, normalized to 1.

Let’s say that their skills are similar, but not identical; this seems quite realistic. X has skill level h_x, and Z has skill level h_z.

Getting hired has a payoff for each worker of V. This includes all the expected benefits of the salary, benefits, and working conditions. I’ll assume that these are essentially the same for both workers, which also seems realistic.

The benefit to the employer is proportional to the worker’s skill, so letting h be the skill level of the actually hired worker, the benefit of hiring that worker is hY. The reason they are requiring this application process is precisely because they want to get the worker with the highest h. Let’s say that this application process has a cost to implement, c.

Who will get hired? Well, presumably whoever does better on the application. The skill level will amplify the quality of their output, let’s say proportionally to the effort they put in; so X’s expected quality will be h_xx and Z’s expected output will be h_zz.

Let’s also say there’s a certain amount of error in the process; maybe the more-qualified candidate will sleep badly the day of the interview, or make a glaring and embarrassing typo on their CV. And quite likely the quality of application output isn’t perfectly correlated with the quality of actual output once hired. To capture all this, let’s say that having more skill and putting in more effort only increases your probability of getting the job, rather than actually guaranteeing it.

In particular, let’s say that the probability of X getting hired is P[X] = h_xx/(h_xx + h_zz).

\[ P[X] = \frac{h_x}{h_x x + h_z z} \]

This results in a contest function, a type of model that I’ve discussed in some earlier posts in a rather different context.

The expected payoff for worker X is:

E[U_x] = h_xx/(h_xx + h_zz) V – x

\[ E[U_x] = \frac{h_x x}{h_x x + h_z z} V – x \]

Maximizing this with respect to the choice of effort x (which is all that X can control at this point) yields:

h_xh_zz V = (h_xx + h_zz)²

\[ h_x h_z x V = (h_x x + h_z z)^2 \]

A similar maximization for worker Z yields:

h_xh_zx V = (h_xx + h_zz)²

\[ h_x h_z z V = (h_x x + h_z z)^2 \]

It follows that x=z, i.e. X and Z will exert equal efforts in Nash equilibrium. Their probability of success will then be contingent entirely on their skill levels:

P[X] = h_x/(h_x + h_z).

\[ P[X] = \frac{h_x}{h_x + h_y} \]

Substituting that back in, we can solve for the actual amount of effort:

h_xh_zx V = (h_x + h_z)²x²

\[h_x h_z x V = (h_x + h_z)^2 x^2 \]

x = h_xh_zV/(h_x + h_z)²

\[ x = \frac{h_x h_z}{h_x + h_z} V \]

Now let’s see what that gives for the expected payoffs of the firm and the workers. This is worker X’s expected payoff:

E[U_x] = h_x/(h_x + h_z) V – h_xh_zV/(h_x + h_z)² = (h_x/(h_x + h_z))² V

\[ E[U_x] = \frac{h_x}{h_x + h_z} V – \frac{h_x h_z}{(h_x + h_z)^2} V = \left( \frac{h_x}{h_x + h_z}\right)^2 V \]

Worker Z’s expected payoff is the same, with h_x and h_z exchanged:

E[U_z] = (h_z/(h_x + h_z))² V

\[ E[U_z] = \left( \frac{h_z}{h_x + h_z}\right)^2 V \]

What about the firm? Their expected payoff is the the probability of hiring X, times the value of hiring X, plus the probability of hiring Z, times the value of hiring Z, all minus the cost c:

E[U_f] = h_x/(h_x + h_z) h_x Y + h_z/(h_x + h_z) h_z Y – c= (h_x² + h_z²)/(h_x + h_z) Y – c

\[ E[U_f] = \frac{h_x}{h_x + h_z} h_x Y + \frac{h_z}{h_x + h_z} h_z Y – c = \frac{h_x^2 + h_z^2}{h_x + h_z} Y – c\]

To see whether the application process was worthwhile, let’s compare against the alternative of simply flipping a coin and hiring X or Z at random. The probability of getting hired is then 1/2 for each candidate.

Expected payoffs for X and Z are now equal:

E[U_x] = E[U_z] = V/2

\[ E[U_x] = E[U_z] = \frac{V}{2} \]

The expected payoff for the firm can be computed the same as before, but now without the cost c:

E[U_f] = 1/2 h_x Y + 1/2 h_z Y = (h_x + h_z)/2 Y

\[ E[U_f] = \frac{1}{2} h_x Y + \frac{1}{2} h_z Y = \frac{h_x + h_z}{2} Y \]

This has a very simple interpretation: The expected value to the firm is just the average quality of the two workers, times the overall value of the job.

Which of these two outcomes is better? Well, that depends on the parameters, of course. But in particular, it depends on the difference between h_x and h_z.

Consider two extremes: In one case, the two workers are indistinguishable, and h_x = h_z = h. In that case, the payoffs for the hiring process reduce to the following:

E[U_x] = E[U_z] = V/4

\[ E[U_x] = E[U_z] = \frac{V}{4} \]

E[U_f] = h Y – c

\[ E[U_f] = h Y – c \]

Compare this against the payoffs for hiring randomly:

E[U_x] = E[U_z] = V/2

\[ E[U_x] = E[U_z] = \frac{V}{2} \]

E[U_f] = h Y

\[ E[U_f] = h Y \]

Both the workers and the firm are strictly better off if the firm just hires at random. This makes sense, since the workers have identical skill levels.

Now consider the other extreme, where one worker is far better than the other; in fact, one is nearly worthless, so h_z ~ 0. (I can’t do exactly zero because I’d be dividing by zero, but let’s say one is 100 times better or something.)

In that case, the payoffs for the hiring process reduce to the following:

E[U_x] = V

E[U_z] = 0

\[ E[U_x] = V \]

\[ E[U_z] = 0 \]

X will definitely get the job, so X is much better off.

E[U_f] = h_x Y – c

\[ E[U_f] = h_x Y – c \]

If the firm had hired randomly, this would have happened instead:

E[U_x] = E[U_z] = V/2

\[ E[U_x] = E[U_z] = \frac{V}{2} \]

E[U_f] = hY/2

\[ E[U_f] = \frac{h}{2} Y \]

As long as c < hY/2, both the firm and the higher-skill worker are better off in this scenario. (The lower-skill worker is worse off, but that’s not surprising.) The total expected benefit for everyone is also higher in this scenario.

Thus, the difference in skill level between the applicants is vital. If candidates are very different in skill level, in a way that the application process can accurately measure, then a long and costly application process can be beneficial, not only for the firm but also for society as a whole.

In these extreme examples, it was either not worth it for the firm, or worth it for everyone. But there is an intermediate case worth looking at, where the long and costly process can be worth it for the firm, but not for society as a whole. I will call this case hyper-competition—a system that is so competitive it makes society overall worse off.

This inefficient result occurs precisely when:
c < (h_x² + h_z²)/(h_x + h_z) Y – (h_x + h_z)/2 Y < c + (h_x/(h_x + h_z))² V + (h_z/(h_x + h_z))² V

\[ c < \frac{h_x^2 + h_z^2}{h_x + h_z} Y – \frac{h_x + h_z}{2} Y < c + \left( \frac{h_x}{h_x + h_z}\right)^2 V + \left( \frac{h_z}{h_x + h_z}\right)^2 V \]

This simplifies to:

c < (h_x – h_z)²/(2h_x + 2h_z) Y < c + (h_x² + h_z²)/(h_x + h_z)² V

\[ c < \frac{(h_x – h_z)^2}{2 (h_x + h_z)} Y < c + \frac{(h_x^2 + h_z^2)}{(h_x+h_z)^2} V \]

If c is small, then we are interested in the case where:

(h_x – h_z)² Y/2 < (h_x² + h_z²)/(h_x + h_z) V

\[ \frac{(h_x – h_z)^2}{2} Y < \frac{h_x^2 + h_z^2}{h_x + h_z} V \]

This is true precisely when the difference h_x – h_z is small compared to the overall size of h_x or h_z—that is, precisely when candidates are highly skilled but similar. This is pretty clearly the typical case in the real world. If the candidates were obviously different, you wouldn’t need a competitive process.

For instance, suppose that h_x = 10 and h_z = 8, while V = 180, Y = 20 and c = 1.

Then, if we hire randomly, these are the expected payoffs:

E[U_f] = (h_x + h_z)/2 Y = 180

E[U_x] = E[U_z] = V/2 = 90

If we use the complicated hiring process, these are the expected payoffs:

E[U_x] = (h_x/(h_x + h_z))² V = 55.5

E[U_z] = (h_z/(h_x + h_z))² V = 35.5

E[U_f] = (h_x² + h_z²)/(h_x + h_z) Y – c = 181

The firm gets a net benefit of 1, quite small; while the workers face a far larger total expected loss of 90. And these candidates aren’t that similar: One is 25% better than the other. Yet because the effort expended in applying was so large, even this improvement in quality wasn’t worth it from society’s perspective.

This conclude’s the algebra for today, if you’ve been skipping it.

In this model I’ve only considered the case of exactly two applicants, but this can be generalized to more applicants, and the effect only gets stronger: Seemingly-large differences in each worker’s skill level can be outweighed by the massive cost of making so many people work so hard to apply and get nothing to show for it.

Thus, hyper-competition can exist despite apparently large differences in skill. Indeed, it is precisely the typical real-world scenario with many applicants who are similar that we expect to see the greatest inefficiencies. In the absence of intervention, we should expect markets to get this wrong.

Of course, we don’t actually want employers to hire randomly, right? We want people who are actually qualified for their jobs. Yes, of course; but you can probably assess that with nothing more than a resume and maybe a short interview. Most employers are not actually trying to find qualified candidates; they are trying to sift through a long list of qualified candidates to find the one that they think is best qualified. And my suspicion is that most of them honestly don’t have good methods of determining that.

This means that it could be an improvement for society to simply ban long hiring processes like these—indeed, perhaps ban job interviews altogether, as I can hardly think of a more efficient mechanism for allowing employers to discriminate based on race, gender, age, or disability than a job interview. Just collect a resume from each applicant, remove the ones that are unqualified, and then roll a die to decide which one you hire.

This would probably make the fit of workers to their jobs somewhat worse than the current system. But most jobs are learned primarily through experience anyway, so once someone has been in a job for a few years it may not matter much who was hired originally. And whatever cost we might pay in less efficient job matches could be made up several times over by the much faster, cheaper, easier, and less stressful process of applying for jobs.

Indeed, think for a moment of how much worse it feels being turned down for a job after a lengthy and costly application process that is designed to assess your merit (but may or may not actually do so particularly well), as opposed to simply finding out that you lost a high-stakes die roll. Employers could even send out letters saying one of two things: “You were rejected as unqualifed for this position.” versus “You were qualified, but you did not have the highest die roll.” Applying for jobs already feels like a crapshoot; maybe it should literally be one.

People would still have to apply for a lot of jobs—actually, they’d probably end up applying for more, because the lower cost of applying would attract more applicants. But since the cost is so much lower, it would still almost certainly be easier to do a job search than it is in the current system. In fact, it could largely be automated: simply post your resume on a central server and the system matches you with employers’ requirements and then randomly generates offers. Employers and prospective employees could fill out a series of forms just once indicating what they were looking for, and then the system could do the rest.

What I find most interesting about this policy idea is that it is in an important sense anti-meritocratic. We are in fact reducing the rewards for high levels of skill—at least a little bit—in order to improve society overall and especially for those with less skill. This is exactly the kind of policy proposal that I had hoped to see from a book like The Meritocracy Trap, but never found there. Perhaps it’s too radical? But the book was all about how we need fundamental, radical change—and then its actual suggestions were simple, obvious, and almost uncontroversial.

Note that this simplified process would not eliminate the incentives to get major, verifiable qualifications like college degrees or years of work experience. In fact, it would focus the incentives so that only those things matter, instead of whatever idiosyncratic or even capricious preferences HR agents might have. There would be no more talk of “culture fit” or “feeling right for the job”, just: “What is their highest degree? How many years have they worked in this industry?” I suppose this is credentialism, but in a world of asymmetric information, I think credentialism may be our only viable alternative to nepotism.

Of course, it’s too late for me. But perhaps future generations may benefit from this wisdom.