Idealized Unemployment

Unemployment sucks, let’s face it. We need the cash, so we need the work.

So let’s say you are going out and looking for jobs. What I am going to do is derive a limiting probability on how likely you are getting a job in this economic environment. This will be a heavily simplified idealization, but hopefully it will shed some insights. (This can also be generalized to applications of all nature.)

We assume that you apply to p positions/firms. For each firm, there are n applicants, each of equal merit and status, including yours. Each firm is going to only hire one person. For this setup the probability of not getting any positions is

(The probability of not getting any one position is n-1/n, and this is multiplied by itself p times since you apply for p positions – multiplication rule for probabilities. We assume that the number of applicants is the same for all firms, and that since everyone is equally qualified, the firm draws straws to choose who gets the job, so it’s completely random.)

Now let’s say you are desperate, and so you apply to many many many jobs. So is everyone else, so each firm gets many many many applications from everyone. Again we assume each application is equivalent. At this point, when both n and p become very large, their difference becomes negligible, so pn. Substituting n for p and taking the limit of the first equation as n becomes very very large yields the following expression:

This should be a familiar limit to anyone with calculus experience, and its value is 1/e, or around 0.368. In other words, you have about a 63% chance of landing a job. Not bad for if the chance of landing a single job was nearly zero.

The upshot is pretty interesting. Let’s say we have a country where nobody is employed. At once many entrepreneurs decide to build up businesses, each of which will hire k people (the generalized case, not just one), but since the population is huge in this country and nobody is educated or has work experience, the aforementioned idealizations still hold. If k = 1 then it is understood that each person has about 37% chance of not getting a job if everyone applies for every job. Put it in other terms, at the end of the day when decisions are made, 63% of the people have a job and the unemployment rate, people who don’t have a job, is 37%*. But now we look at the general case, and the probability becomes as follows:

This is striking in many ways. The unemployment rate goes markedly down if businesses hire more than one person. If k = 3 the rate drops down to 5%. When it is 6, the unemployment rate goes down to 0.24%.

According to the U.S. Bureau of Labor, the umemployment rate in November was nearly 10%. Broken down demographically, the statistics are as follows:

Among the major worker groups, the unemployment rates for adult men (10.0 percent), adult women (8.4 percent), whites (8.9 percent), and Hispanics (13.2 percent) edged up in November. The jobless rate for blacks (16.0 percent) showed little change over the month, while the rate for teenagers declined to 24.6 percent. The jobless rate for Asians was 7.6 percent, not seasonally adjusted. (See tables A-1, A-2, and A-3.)

So the question is why aren’t we reaching our idealized unemployment rate?

Well, first off, this is idealized, and the real data shows it. Not all applicants are equal. Some applicants are clearly more qualified, so they take the job; the less qualified candidate who had an equal chance in a random scenario sees his probability decreased in the real scenario. Some jobs are specialized so the unqualified candidate doesn’t have a rat’s ass chance of getting the job.

Second, some people are lazy and give up looking for a job after about 100 tries. The idealization requires everyone to try a very very very large amount of times.

There are other reasons I could expound upon but will not for the sake of brevity.

Still a nice theoretical result though, especially since we see the infamous e here again (which was the driving force for this post). I would argue that the idealized result is the best possible unemployment rate that can be achieved, which entails that unemployment, and all its evilness, can never be completely eradicated.

***NOTE: We ignore the possibility of one person getting two jobs. Let’s just say if one person gets a job from Company A and Company B both, he will reject one of the two offers wherehence the rejected company picks another candidate. In our scenario, the number of applicants is so large that the subtraction of one candidate has negligible effect. Later postscript: also note that this would be surely true if each person applies for a different set of jobs, which may be a more accurate model of what’s going on.
***EDIT #2: It might be hard to imagine unemployment issues when there are more companies/firms than people. It is also hard to imagine a “limiting unemployment rate” when an x number of firms can hire y amount of people and there are z people in the population. So with these stipulations let’s assume that
1) These ratios are ill-defined when the number of firms and number of people are truly infinite.
2) All probabilities are independent; each person applies to different firms where a large amount of people, let’s say of a different country, are also applying.
This should reduce some of the confusion. Many of my original points still stand, although I see some fault in my original country analogy now.

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