The PhilJobs appointment dataset has both (i) the year of hiring and (ii) the year the appointee earned their PhD. We can use that to start getting some clarity on when staleness sets in.
Using the data I downloaded from PhilJobs at the end of April, here's a set of counts:
1/3 of junior placements posted to PhilJobs get a job the same year they finish their PhD. And, as you can see, things drop off quickly after that. Less than 1% of posters list a job after 8 years post-PhD. There are two hypotheses possible here, which are not mutually exclusive:
(1) very few people get a job after 8 years post-PhD
(2) some people change jobs and don't post their new jobs
(3) as people get further out from their PhD, they are less likely to post a job
EDIT 9:43 am PDT:
(4) X people drop out of the market after N years (thanks to Marcus Arvan for this suggestion)
If I had to put money on it, I'd bet (1) does more explaining of the phenomenon than (2) or (3). In the case of (2), it's difficult to reliably track individuals across positions in PhilJobs, so it's hard to be certain. I'm not sure how we'd rule out (3). I'd love to hear readers' thoughts, though.
Here's a thought about (4). We have two explanations for the drop off: PhD's get stale or people drop out of the market. Clearly, these aren't independent of one another: stale PhD's are more likely to drop out of the market and folks who drop out of the market thereby have stale PhD's. My hunch is that these hypotheses would be really hard to tease apart. What would we need? One step towards answering it would be the number of people applying for jobs 1 year out, 2 years out, etc. We want to know (e.g.) the number of 8-years-out job posters out of the number of 8-years-out applicants.
Here's another informative plot: the differences between successive years in job postings. That is, the drop in ratio of postings from 0 years from PhD to 1 year from PhD, from 1 year from PhD to 2 years from PhD, etc. (If you were to measure the gaps between the tops of the bars in the previous plot, you'd get this one.)
So there you have it. I think this gives us some idea of what it means to say a PhD is stale. The takeaways are:
1. there are levels of staleness: 8+ years is the most stale, 4-7 years is 2nd most stale, 1-3 years is the least stale, 0 years is baby fresh.
2. decay in job placements is exponential. The costs of not placing are greatest earliest on. After a while, the costs level out, but that's largely because jobs aren't being won.
Thanks for Greta Turnbull, Tim Weidel, and Maria Howard for making sure I didn't ramble incoherently.