An academic population model to distill the ‘PhD Problem’

by Andrew Tredennick

UPDATE: Wanted to add this link for an ESA lunch session on Aug. 6 called “Beyond Academia“.

In the past year or two there has been a deluge of articles (e.g., this, this, and this) and blog posts (e.g., this and this) written about the so-called “PhD Problem.” The essence of the problem is that we (academia) don’t have enough tenure-track jobs to possibly absorb all the PhD students. What this means is that many newly-minted PhD’s that want to go into academia (and even those that don’t want to go into academia, since our training is generally academic-oriented) have poor job prospects. Competition for a faculty position is very high, and good luck trying to choose where you end up (unless you’re a rock star). Academia is a pyramid scheme.

But, Jacquelyn Gill over at the Contemplative Mammoth wrote a great piece on how we don’t really have a PhD problem, but instead an attitude problem. The problem is that we are 1) not training graduate students well for positions outside of academia and 2) we aren’t doing a good job at selling our desirable skills outside of academia. Further, she suggests doing these things are critical because there are many PhD students that don’t even want to be a professor. In sum, Jacquelyn proposes that we do not need fewer PhD students, we just need them to leave academia afterward (which, I want to be clear, is not a judgmental statement; leaving academia for a different career path is not a “failure”, and it doesn’t mean leaving science).

I am all for this. It would help those who want a PhD and want to pursue work outside of  academia, and it might lessen the degree of our current PhD problem. But still, it seems to me we would need current graduate students to want to leave academia in a huge proportion. Some of my results below indicate the large gap in proportions that would need to be lessened. This means we need to focus on various fronts to tackle the PhD problem, both in terms of training and in terms of the rates at which people enter and leave academia. Sounds like a great problem for a model.

I’m going to build a very simple population dynamic model of academia. Let’s assume for simplicity that the limiting factor in all of this is money (after all, if there was just more money, universities could hire more faculty — not necessarily the case, but bear with me on this). The model is monetarily-implicit, just like how some population models of, say, plants are spatially-implicit — each person needs a “money patch” to survive, but we’re not actually going to model the transfer of money in any real way. All patches of money are always occupied by PhD students (S), postdoctoral researchers/fellows (D), or professors (P). The default state is the PhD student (S), so whenever a post-doc leaves his/her position (through attrition, not by moving up to faculty) or a professor retires, that money patch reverts to a PhD student. Basically, we can think of PhD students as some plant with a good seed source and good dispersal — if there is an open patch, a PhD student will be able to fill it!

To establish a monetary hierarchy, postdocs can only establish on PhD patches, whereas professors can establish on postdoc or PhD money patches. Basically, this assumes that money preferentially goes to professors, then post docs, the grad students. But, since it costs more to fund a postdoc or professor relative to a PhD student, a postdoc requires 2 student money patches and a professor requires 4 student money patches (and then a professor requires 2 postdoc money patches). Postdocs establish at rate a, which is dependent upon the relative abundance of PhD students, since they can only establish on those patches. Professors establish at rate b, which is dependent upon the relative abundance of PhD students + postdocs. Postdoc attrition rate is o, and professor retirement rate is v.

Let S = PhD students, D = postdoctoral researchers/fellows, and P = professor (Assistant through Full). The simple population dynamic model of academia is then three differential equations that represent the instantaneous rate of change for each of the three states:

\frac{dS}{dt} = vP + oD - a\frac{S}{2} - b\frac{S}{4}

\frac{dD}{dt} = a\frac{S}{2} - b\frac{D}{2} - oD

\frac{dP}{dt} = b(\frac{S}{4}+\frac{D}{2}) - vP

and the parameters, with ecological analogues in parentheses, are:

v = professor retirement rate (mortality of professors)
o = postdoctoral researcher attrition rate (mortality of postdocs)
a = postdoc establishment rate (birth rate of postdocs)
b = professor establishment (the birth rate of professors).

These are all annual rates, so v = 0.05 means a 5% per year retirement rate. The model is run on yearly time-steps.

Here’s a schematic of what the model looks like.


An academic population dynamic model. Amazingly over-simplified, I know. But I think serves our purposes here. One thing to make slightly more realistic is to have actual PhD establishment that is linked to the relative abundance of professors.

Remember, the flows aren’t necessarily stage-structured. For example, aS does not really show the recruitment of a PhD student to the post-doc stage, but more explicitly a post-doc establishing on a PhD “money patch.” This can be interpreted as a PhD leaving the PhD cohort and moving to the post-doc stage, but we aren’t following individuals here — the flow of a PhD to post-doc (aS) DOES NOT leave an empty patch for a new PhD. New PhDs only establish when a professor retires or a post-doc leaves the system — this is of course not how it really works, but I think the model can still pick up the general population dynamics.


Figure 1. Results of running the model through time at “ideal” rates that are meant to mirror the real situation in academia. Under these assumptions the equilibrium of the model is a high proportion of professors with lower proportions of PhD students and postdocs.

Since this model is so simple*, it reaches equilibrium really fast on yearly timesteps. So, I’ll just show some numerical results of running the model through time. Let’s think about an ideal world in terms of the rates abo, and v. Ideally, at least in my thinking, we all would want relatively high establishment rates, and low attrition and retirement rates. From that statement alone I bet you can guess the problem! Here is the equilibrium result when a = 0.5, b = 0.3, and o = v = 0.05 (i.e., 5% retirement and attrition rate; Figure 1).

Uh-oh! If this is the ideal, and probably pretty close to actual rates, then that means our actual situation (high proportion of PhD students) is not at equilibrium and wildly unstable (which we knew, right)! What kind of values would it take to get a proportion more like what we actual have in academia?

You guessed it, higher attrition and retirement rates. Here is the result (Figure 2) with higher attrition and mortality rates (a and b are the same as above). Keeping a and b constant, I had to bump the retirement and attrition rates of professors and postdocs, respectively (and respectfully of course) over 50% in order to get the propotion of PhD students to be greater than the proportion of professors (our current situation). That means 50% of professors would have to retire every year! Not likely, and as someone that wants to be a professor, that doesn’t sound good.


Figure 2. Trying to get the proportions a little bit more like real life.

So, what can we do? Well, the great thing about using simple models like this is that it is easy to see where we need to make changes. Obviously, we need to increase the rates o and v. We can also just reduce the number of PhD students to reduce their contribution to the population percentages. Which of these are actionable? In my mind, reducing the number of PhDs (either through stricter standards or having them leave academia for other fulfilling employment) and increasing the retirement rate of professors are the two really actionable items. By the time someone is a postdoc they are pretty committed to academia, so it is doubtful we can raise the attrition rate there.

Are these things we really want to do? I’d argue that there is no downside to making it harder to get a PhD. Eventually there is a bottle neck where competition for employment within the academy is insanely high. Right now it is at the postdoc-to-faculty stage. Why not make it sooner so people can get on with their lives before becoming overly committed to academia?

Now, what about a mandatory retirement age for faculty? This will be met with uproar! As someone who wants to be a professor and work well into my old age, I don’t even like this one (I like it right now, of course, because it would benefit me getting a job. But after that I would want to repeal the retirement clause!). But maybe these are conversations we need to be having. Another option, even more incendiary, that some people I’ve chatted with have mentioned is getting rid of tenure and treating the academic world more like the results-based private sector.

What do you think? How do we go about solving this problem?

Do you think this model gets at the important rates? Or, should it be more mechanistic, with PhDs and postdocs being tied to individual professors through the money they bring in?

*Yes, I completely recognize just how simple and unrealistic the model is, but let’s run with it for now. In the comments, let me know what you think should be added (Lotka-Volterra type dynamics with density-dependence? Stage-structure?). You can access the basic model used in this post here.


  1. Ed

    I wouldn’t be so sure about the difficulty of raising the attrition rate of post-docs. My ‘desire for attrition” (oi maybe) waxes and wanes on annual, seasonal and even diel cycles. If the right graduate student bought me the right pint of beer on the right day I might just walk out and head straight for the goat farm.

    • Andrew Tredennick

      I can see the subsequent post-doc ad to fill your position now: “…Desired Qualifications: NO interest in goat farming.”

      Yeah, I may be wrong about the postdoc attrition rate; I have less of a feel for it. There has to be data out there on this, but I’m not sure where to find it.

  2. Nice model, interesting. I’ve just read and posted on Jacquelyn Gill’s blog about this problem. From the UK, we have similar issues but I think we’re a little bit more relaxed about PhD students and even post-docs stepping out into the wider world. I still think some employers perceive a PhD as meaning the individual has become too academic rather than commercially minded, and this can be a problem. However, on the flip side there are many transferable skills in a PhD that are directly usable in a commercial or policy setting- problem solving, organisation and project management, literature and communication skills. Its interesting that the higher echelons of large companies are increasingly dominated by people with PhDs and even in small companies and start-ups, the executive board members often all have PhDs related to the field of the company. So it shouldn’t all be doom and gloom.

    I work in a chemistry department at a British university. Some of my friends and colleagues decided that they wanted to find out about jobs opportunities outside of academia, so they organised a Career’s event. We had people from industry, science policy, patent law, teaching and even a serial entrepreneur, all with chemistry PhDs behind them, come and talk. More info here

    • Andrew Tredennick

      I agree there are many transferable skills, regardless of discipline. But I wonder how opportunities outside of academia vary with discipline. For example, I would imagine there is a wider breadth of opportunities for a chemistry PhD than an ecology PhD. However, maybe the problem, as Jacquelyn points out, is that we are just not very good at selling ourselves.

      • You are right to point out that there is a difference in disciplines and I think between countries/cultures as to the value of a PhD, particularly perceived outside of academia. Interestingly, I was at a talk just now by the Corporate Responsibility Manager for a large energy company in the UK. Her PhD was in soil science/sedimentology, she then did a post-doc before going to work in a water company and then the energy company-I think she shows you can go a long way with a PhD, even if its not related to your eventual career. The energy company she works for are actually German- she described a big cultural difference between Germany and the UK around PhDs. In Germany, they are revered (many politicians in Germany have PhDs, for example) whereas in the UK people tend to keep higher qualifications quieter except in technical fields. This was illustrated when the company merged its email servers and all staff found that their email signature contained their qualifications- apparently many in the UK complained and asked for the “Dr.” to be taken off!

        As for PhD students “selling themselves”, you are right. I think many PhD students fail to frame their research experiences in a context that an employer will appreciate or understand. But in many cases while the specifics do not correlate, the generalities do- take the ability to record, manage and manipulate large amounts of data as an example! I also think there is a tendency for employers to write off PhD students as “academic”, and some education needs to go on- this is where professional bodies and universities should come in. I know the University I work in has a specific careers advisor for PhD students, and part of her job is to interact with business and industry to promote the skills of her constituency. We also have a Graduate School that represents PhD and post-doctoral researchers across the University- providing careers advice, training and other networks. It is, perhaps, harder for academics to do this type of careers training themselves. In my experience, those academics who have had industrial experience or collaborated with industry tend to offer some good advice, but those who’ve remained confined to the ivory tower struggle.

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  4. I don’t agree that “there is no downside to making it harder to get a PhD.” For one anecdote: I was a marginal admit to a physics program, and remained a marginal producer of new physics as a grad student, but I’m an excellent teacher and I’m very happy sharing the experience and perspective of someone who can bridge “real physics” to students at the high school and community college levels. Many of my friends won’t get TT jobs or “push the frontiers” but they’re doing excellent work in industry and elsewhere. And we did do good science while grad students — there would be a lot less science done with fewer grad students. I think the world is better off for us having had the opportunity to taste the edge in grad school, and so are we — but we wouldn’t have been able to if it was harder to get a PhD. Physics grad school was tough enough as it was.

    Of course, I do think it’s wrong to recruit so many graduate students into programs that have no hope of placing them into academic positions, when the programs are so heavily skewed in emphasis, skills training, experience, etc. toward producing clones of the PIs. It’s less actionable than making standards tougher, but better for everyone, I think, if the focus is on changing programs to make jumps to industry and to “lower status” academic jobs more acceptable and easier to do.

    • Andrew Tredennick

      Interesting take on that — I hadn’t thought about it that way. My thoughts on this are obviously skewed to reflect my own field, ecology. I’m convinced there is little downside in ecology (where a Master’s degree can get you many good jobs outside of academia, in general), and maybe there should be two PhD tracks: research (aimed at people wanting to go TT at an R1 institution) and teaching (aimed at those that want to be experts and be able to teach at a high level. Indeed, this is an institutional problem as well — maybe we shouldn’t be asking people that want to do “big” research to teach (if they don’t want to or are bad at it) and there should be more professors who are really focused on education. Not very tenable from a monetary standpoint, but much better for education overall I would imagine.

      • You’re still only talking only about academia (TT at R1 vs Teaching). I’m a microbial ecologist doing work at a government (DOE lab) that employs 1000s of PhDs doing ‘real’, fundamental, leading edge research. Actually DOE labs kind of sit between academia/government/industry since none of us are gov employees. Biotechnology (ag and healthcare) employs a lot of people doing “real” research some of it often fairly fundamental in addition to applied.

        Maybe there are some science disciplines where there is not much outside of academia possible. But life and environmental sciences is generally not one of them.

        To me the Ph.D. is about learning how to identify and solve problems in using new approaches. If you stick to that as a first principle for education of science Ph.D.s they will be employable, and employable on the widest swath of the market, inside and outside of academia.

  5. I agree with Mark Betnel. Not only are PhDs potentially useful in society, many are useful as PhD students doing real research and real teaching / tutoring, running labs, etc. Also, many postdocs are happy to postdoc but don’t really want to have to spend the rest of their lives writing grants and teaching. I think we could increase postdoc attrition by working to increase the profile of PhD-qualified researchers in business and government, note this would also increase the attrition of professors. We might also do it if we were more tolerant and encouraging of people going back and forth between industry, government and academia. I do think we might want to increase the attrition rate of PhD students by not treating graduation rate as a metric of success (as the UK does), but rather encouraging and expecting some to realise they are on the wrong track. We should allow them out with MScs, MPhils and such-like qualifications, or even putting “research assistant” rather than “PhD student ABD” on their CVs, not coerce them into writing full dissertations.

  6. Andrew Tredennick

    “many are useful as PhD students doing real research and real teaching / tutoring, running labs, etc.”

    I agree, but we need to be careful we aren’t exploiting graduate students. If their contribution to science is so great, but the prospect for a job so low, perhaps money should re-apportioned to research technicians that get paid accordingly. I’m not saying I feel exploited — I LOVE being a graduate student and have had a great time (even with full recognition of job prospects) — but we need to be careful of arguments that argue for cheap, ephemeral scientific labor.

  7. Since no one else wants to talk about model, I’ll have a go: Most egregious, I think, is that you let dS/dt have a positive vP term. In words, students and professors are in competition for the same money, and that is just not how things work. Students are clearly not hired dependent on professors leaving or vice versa. Rather, the relationship is more accurately the exact opposite, because profs bring in money to hire students (and postdocs, who *are* in competition with students for money, so the oD term gets a pass). If you model this dependence of students on profs, then I suspect the results will be quite different. And you can call this pedantry, but since the discussion is at least partly based on your results, I think an update is warranted.

  8. Andrew Tredennick

    Thanks for your thoughtful comments on the model, Bjørn. I agree the vP term is inherently wrong in terms of how things actually work. Given the simple model structure I chose, it had (well, not had, but basically had) to be positive to balance things out. Now, if I had done a stage-structured model I think the problem could be side-stepped — but, admittedly, I didn’t want to put forth the effort at the time! My main goal was to get people thinking about the rates as opposed to just states of academia. So, I am glad you picked up on the model formulation itself!

    One thought though, including graduate student dependence on profs would create a positive feedback that would actually produce “worse” results (“worse” in terms of jobs and the rates needed to overcome our PhD “debt”) than I have here, right? That would be interesting to do.

    Do you think it would be fruitful to include this dependence within the model structure already outlined above? Is there a simple way to do so? In my mind it would require re-tooling the whole approach toward more mechanism — but I may be missing something (very likely).

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