There are many ways we can think about the job market. I believe that I said it before that we should build a model of whatever we are studying only after we specify what exactly we are interested in. And when it comes to the job market there are many things we might be interested in. For instance, we could be especially interested in unemployment: What determines whether someone is unemployed? What consequences does unemployment have for other family members? How do unemployed people find a job again? We could also be especially interested in how many members of a household work and how much and how this is decided at the household level. Why do some people work full time and some part time? We could also be especially interested in how people prepare themselves for the job market. How do people decide which career path to choose? How do they decide what to learn? There are so many things we could be interested in and I believe that each different question will need its own model, where one focuses on the salient features of the job market for that particular question.

In this post I am perhaps being a bit eclectic but I want to think about which person gets which job. That is, I want to think about how the job market allocates people to jobs.

And I will do this in a highly idealized setting. So the key feature of the world that I will focus on are these: In this model different people have very different skills and one person’s job cannot easily be done by a random other person. I keep the rest of the model as simple and idealized as possible. Jobs are just there, for instance. Nobody creates these. There is no innovation either. Everyone is super active on the job market, constantly checking to see if there are better jobs out there, and everyone cares only about money. These are huge simplifications and in their extreme form all these assumptions are empirically wrong. But this is how modelling works, and if you are unhappy with something, especially with the final insight this model provides, then have a go at creating your own model. This is in fact how scientific progress is made and how research in economics works.

With these excuses let me turn to the story I want to tell here. You can see for yourself in the end if you feel that you learnt something from it. Let me consider a slice of the world (somewhat independent of the rest) of three jobs and three people. The people are called Yvonne, Jacqueline, and Babette, and the jobs that they could have (we need one person doing each job) are manager, electro technician, and recycling expert.

As I said, I want these three people to be differently good at the various possible jobs, so this table above captures this. In this table, each cell describes the value added (recall the post on GDP) in thousands of euros a year that would result if a person does a particular job. So for instance, Yvonne would generate a value added of 120 thousand euros a year if she works as a manager and 160 thousand euros a year if she works as a technician, whereas Babette would generate 100 thousand euros a year if she worked as a manager and 180 thousand euros a year if she worked as a recycling expert.

So who will get which job in this job market? Let us begin by considering the following possible allocation of people to jobs: Yvonne is manager, Jacqueline technician, and Babette recycling expert. Suppose furthermore that employer and employee divide the generated value added equally. In other words Yvonne and her employer get 60 thousand a year each, Jacqueline and her employer 110 thousand a year each, and Babette and her employer 90 thousand a year each. Do we think that this is a “stable” job market outcome? Or is there a pair of person and employer that could negotiate a better deal between them and, thus, “block” the current situation. You may want to think about it for yourself for some time.

Well? Consider Jacqueline and the management position. In this position Jacqueline could generate a value added of 200 thousand a year much more than the 120 thousand a year that Yvonne can generate in this position. To attract Jacqueline to the management position she would need to be paid her current salary of at least 110 thousand a year. Suppose the management position offers Jacqueline 125. This still leaves 75 for the employer who at the moment only receives 60 by employing Yvonne. So this new situation is an improvement for both Jacqueline and her new employer. In this fictitious world in which the two only care about money (and the job market is as flexible as we assume here) they would then probably go through with this change.

Let us suppose Jacqueline does move to the management position. What do we have now? Yvonne is unemployed (earning zero), Jacqueline works in the management position and earns 125 thousand a year, Babette works as a recycling expert and earns 90 thousand a year. On the employer’s side we have that the employer seeking a manager employs Jacqueline and earns 75 thousand a year, the employer of the technician has nobody at the moment (earning zero), and the employer of the recycling expert, currently Babette, earns 90 thousand a year. So what happens to Yvonne and the employer seeking a technician? Well, they could team up. How much would Yvonne have to get paid so that she does not go back to knock on the door of the management position to accept the job at lower salary? How much does Yvonne’s new employer need to earn so that they don’t call up Jacqueline again and offer her her old job back at a higher salary? Well, looking at the numbers Yvonne needs to earn at least 45 thousand a year, otherwise she could offer a deal to the management employer that this employer would find attractive. Yvonne’s new employer would request at least 95 thousand a year, otherwise they would offer a new deal to Jacqueline that Jacqueline would find attractive. But this could work out. Together they create an added value of 160 thousand a year, so they could for instance split it 55 thousand for Yvonne and 95 for the employer. This way neither of them can find a better opportunity elsewhere.

Is this new allocation of people to jobs with the given salaries stable? What do we need for a job market outcome to be stable in our fictitious job market?

Well, an allocation (often also called matching) of people to jobs is considered stable if there is no pair of employee and employer that could block the allocation by getting together and that way both improving their income.

If you think about it, this means that the matching has to be such that employees and employers get so much that for any pair of them the sum of what they get is at least as much as they could both get together if they work together. This is not an easy sentence to read. So let us go through some examples.

Let us first go back to the beginning where we had Yvonne as manager, Jacqueline as technician, and Babette as the recycling expert such that in each employee employer pair they split the value added by them equally. That meant that Yvonne and her employer have 60 thousand a year each, Jacqueline and her employer 110 thousand a year each, and Babette and her employer 90 thousand a year each.

So why was this not stable? This is so because Jacqueline (getting 110) and the employer of the manager (getting 60) together could create a value added of 200, which is more than the sum of what they both currently make. This is why they could come to a mutually beneficial agreement to block the current matching. This is why the original matching is not stable.

What about the final matching that we found? How was it? We had Jacqueline (125) as manager (75), Yvonne (55) as technician (95), and Babette (90) as recycling expert (90), with numbers in brackets the amount of money the employee or employer makes. Is this stable now? To see this we need to check how much any pair of employee and employer could generate together and compare this with what they currently make. Let us go through all of these cases.

Jacqueline (125) and the employer of the technician (95) have a joint income of 125+95=220, which is just as much as they could generate if they were to get together. So they cannot strike a mutually beneficial deal. In any deal in which Jacqueline gets more than 125 the employer gets less than 95 as the sum has to be 220.

Jacqueline (125) and the employer of the recycling expert (90) have a joint income of 125+90=215, which is more than the 140 that they could generate together. So they cannot strike a mutually beneficial deal. In any deal in which Jacqueline gets more than 125 the employer gets less than 15 as the sum has to be 140.

These are the two possibilities Jacqueline has. Let us turn to Yvonne.

Yvonne (55) and the employer of the manager (75) have a joint income of 55+75=130, which is more than the 120 that they could generate together. So they cannot strike a mutually beneficial deal. In any deal in which Yvonne gets more than 55 the employer gets less than 65 as the sum has to be 120.

Yvonne (55) and the employer of the recycling expert (90) have a joint income of 55+90=145, which is more than the 40 that they could generate together. So they cannot strike a mutually beneficial deal. There is not even a possible deal in which Yvonne could get more than 40.

We can finally turn to Babette.

Babette (90) and the employer of the manager (75) have a joint income of 90+75=165, which is more than the 100 that they could generate together. So they cannot strike a mutually beneficial deal. In any deal in which Babette gets more than 90 the employer gets less than 10 as the sum has to be 100.

Babette (90) and the employer of the technician (95) have a joint income of 90+95=185, which is more than the 100 that they could generate together. So they cannot strike a mutually beneficial deal. In any deal in which Babette gets more than 90 the employer gets less than 10 as the sum has to be 100.

We therefore have a stable matching of people to jobs. But is this the only stable matching? It turns out that, actually yes, this is the only one. What is not uniquely determined, however, is the set of salaries and employer earnings. But none of the other 5 possible allocation of people to jobs can be made stable with any feasible salary – profit arrangement. I would encourage you to try to do this for yourself (for instance in a spreadsheet document).

But is there an easier way to find the stable matching in this and other cases? Can we find another (easier to check and possible insightful) property that a matching needs to have to be stable?

Yes! In fact a matching has to be Pareto-efficient, which, given we have money in this problem, means that it must maximize the total sum of value added over all possible matchings. To see this for our specific problem let us have a look at the table of all possible added values generated by different pairs of employee and employer.

Using what I hope are self-explanatory abbreviations, we can compute the total added value generate by any matching in the following table.

We can see that the unique total added value maximizing matching is the matching of Yvonne to the technical job, Jacqueline to the management job, and Yvonne to the recycling job. And this is also the only stable matching in this job market, and is therefore supposedly the outcome of this (admittedly highly fictitious) job market.

I would like to leave it to you to think about why this always has to be so in such a highly fictitious job market: only Pareto-efficient (that is overall added value maximizing) matchings can be stable.

Please note that, as we had so often before in this class, that Pareto-efficiency has again nothing to do with fairness. While this example gives us some hope that the real job market might allocate people to jobs perhaps close to a Pareto-efficient manner, we should note that it does not necessarily do so in a fair manner. While I do not here want to take a stance on what might be considered fair, it is at least clear that in this fictitious job market not everyone can expect the same salary. You can play with this example as much as you want but you will always find that to get a stable job market outcome you need to give Jacqueline a higher salary than Yvonne.

There are many more things we can discuss using this example, but I will leave this to separate posts.

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