The Simple Mathematics of Unemployment
Suppose there’s an illness for which the only known treatment is about 80% effective. Start with 100 patients - how many of those can we expect to cure with one course of treatment? The answer, obviously, is 80, leaving 20 patients who will require a second course of treatment. Of those, we can expect to cure 16 with another course, leaving 4 who will need a further course. And so on.
The mathematics of this example are very simple - it’s a straightforward geometric series. That is, each term in the series (tn) is given by the equation:
tn = tn-1*0.2
or more generally:
tn = t0*0.2n
and still more generally:
tn = t0*(1 - p)n
where p is the average probability that your treatment will cure each patient.
That mathematical simplicity comes at a cost - we have to ignore a lot of information about the nature of the disease and patient variation, lumping all that into p. But it’s still useful - if for example, you have a thousand patients with the disease and you want to know how many courses of treatment will be required to cure all of them, or an acceptably large percentage of them.
The same simple mathematics can be applied to unemployment - if 100 factory workers become unemployed on July 1st 2006, how many will still be unemployed on August 1st? On January 1st 2007? On July 1st 2007? Obviously that depends - but the best way to arrive at an answer is to ignore individual differences between workers and focus on the average likelihood that any of these workers will find work.
Note: three charts under the fold.
Before we go on to develop the model, we need to get some terminology clear. If you see the phrase “unemployment rate” you probably think “percentage of the working population which is unemployed” or just “unemployment” not “the rate at which people are losing their jobs”. It would muddy the waters a little if I started talking about the “employment rate” - that’s just an invitation to smart-arses to point out that as the unemployment rate for April was 5.1 the employment rate was obviously 94.9 (conveniently ignoring the participation rate).
So throughout this post I’ll be talking about the “re-employment rate”, that is, the rate at which the unemployed are getting back into work. I’ll take my chances with the smart-arses who want to argue that the model obviously doesn’t cover school-leavers and other new entrants to the workforce. And they can take their chances with me. I can’t say fairer than that now, can I?
Now it’s time to play the technocrat’s favourite game - let’s assume! So let’s assume that unemployment is very much like an illness, for which the “cure” is getting a job - re-employment. Let’s also assume that the “course of treatment” for this illness is a month of job hunting and the “cure rate” is 20%. So what happens in our example? Answer - by the end of July 2007, 80 workers will still be unemployed (100*(1-0.2)). By the end of August, 64 (80*(1-0.2)). On January 1st, 2007 26. And on July 1st 2007, 7 (6.8%). Fun wasn’t it?
It doesn’t take much more than half an hour to put together a spreadsheet to calculate the effects of changing the re-employment rate on the percentage of unemployed people who will stay unemployed for 6 months or longer. And it’s a perfectly respectable way to model economic situations - that’s how John Hewson got his Ph D. Or so I’ve heard.
With the help of this sophisticated technological approach, I came up with the following figures:
| Re-employment Rate: | 20% | 21% | 25% | 50% |
| 6 Months Unemployed: | 26% | 24% | 18% | 1.6% |
| 12 Months Unemployed: | 6.8% | 5.9% | 3.2% | 0.02% |
And this impressive graph:
I’ve anticipated a few objections to this model of long-term unemployment so I might as well deal with them now:
The whole approach is based on a fallacious analogy between illness and unemployment.
Bollocks. It’s based on a set of simplifying mathematical assumptions that work in both cases.
Those mathematical assumptions can’t be applied here.
Why not? I say they can, you say they can’t. That’s a take it or leave it proposition. So either come up with an argument to show that the mathematics doesn’t apply or bugger off.
You’re deliberately ignoring the effect of personal motivation on re-employment chances.
Damn straight I am - they don’t affect the overall outcome - they’re only of interest if you’re looking for an explanation of how particular individuals stay unemployed for 12 months and more.
You don’t have any supporting studies or data for the model.
Yes, that did give me a little trouble at first. My first thought was to check out the ABS’ Labour Force stats. But, although the stats are disaggregated in a number of interesting ways, duration of unemployment isn’t one of them. Centrelink doesn’t seem publish statistics on the breakdown of NewStart recipients either.
Of course, you could test the model experimentally - just find a factory with a hundred workers, close it down and then see how they make out in the labour market. Unfortunately you’d never get that idea past an ethics committee. Fortunately, I don’t have to - in September 2001, the free market did the experiment for me when Ansett Airlines went belly up tossing 16,000 workers onto the labourmarket.
In October 2002, Michael Webber And Sally Weller of the University of Melbourne prepared an extensive report on The post-retrenchment labour market experiences of Ansett workers (PDF) for the Department of Treasury and Finance Victoria. The report includes a section dealing with the Re-employment of the retrenched workers, including the delay between retrenchment and their first new jobs. The report includes a graph showing the time from retrenchment to the first post-Ansett job:
The report doesn’t include the raw data from which the graph was derived but, with some straight edge assisted eyeballing I was able to extract approximate figures for the numbers of retrenchees in the survey who found jobs within a month of retrenchment (Month 1), within 2 months (Month 2) and so on (those who found new jobs before retrenchment were ignored). From there, it was a simple enough matter to calculate percentages remaining unemployed after those periods. If we plot these “massaged data” points and the series we would expect with a re-employment rate (R) of 30% on the same graph, the result is very impressive indeed:
Of course, that’s only one example, based on very soft data. But it is a useful reminder that while you have unemployment, you’re going to have long-term unemployment and the longer-term unemployed. Sheeting the blame for long-term unemployment home to Government or to the unemployed themselves is purely a matter of personal ideological preference.
Update: thanks to commenter JC for pointing out the possibility that the phrasing of the first paragraph might cause, or allow, some readers to a misread this post.
Excellent post. GT. But (and forgive me for making this totally pedestrian, as I know nothing about mathematics or economic models) I wonder if a variability factor can be included. In the medical example for those who don’t respond to medication, this might include diet, environment, biochemistry etc. But in the unemployment calculations, there are so many more external variables, of course. Policy, investment, industry trends, stock prices, that sort of thing. Can you calculate formula for those, too, based on historical data? Or is your idea here to make a “base” formula?
I liked your premise, that there will always be long-term unemployed. I’m sure that you’ve proved it. I also liked your graphs of the Ansett workers. It shows that a highly motivated and educated workforce can get work at a re-employment rate of 30%. People reading this should not take it that anybody who is unemployed can be re-employed at anywhere near that rate. For example, in our area there are a relatively large number of people who cannot read or write—about 35 (possibly as many as 50) in a population of 4000—and you would have to say that their re-employment rate is close to zero percent while the agricultural sector is shrinking. That’s another reason why there are long-term unemployed. That no one is trying to teach such people, even if they were willing, is another reason for the long-term unemployed. But I do like your explanation and intend to use it wherever I can!
Weathergirl,
I did come up with a more elaborate model that posited a steady decline in re-employment chances over a period of unmployment - the result is a higher tail at the right side of the chart.
But in this case it’s not needed - and I question whether the received wisdom that the longer you’re unemployed, the greater your chances of staying that way is correct. The example of the Ansett workers, suggests otherwise.
Calling into question the standard advice given to retrenchees (and others) - get out htere and get and get a job quick! That week off to clean up the house and sample the delights of day-time television could cost you big-time in the long run! Maybe it doesn’t.
But let’s remember - this is only one study. The results need to be replicated a few times before we start drawing too many inferences.
linm -
Ta. You might want to adopt a new moniker for posting, to protect yourself from spamming. So I’ve changed your nick for the time being.
Trotsky
Your maths in wrong. You can’t conclude 80 will get better and 20 won’t. Those are probability estimates. You could very well have all 100 people fail with the medication while it is still a success in a larger population context.
You forgot to take into account sympathy for the Ansett workers so they could regain a job
Anyhow, all a little over my head. What are you saying? WorkChoices wont statistically hurt?
You need to look at in a probability context if you’re gonna go down this road, Trotsky. In fact it’s the only way to arrive at reasonable estimates.
Go and ask the resident math wiz, Tim Lambert; he’ll tell you I’m right. I’m sure he’ll be only to happy to help you out.
JC:
“Your maths in (sic) wrong. You can’t conclude 80 will get better and 20 won’t. Those are probability estimates. You could very well have all 100 people fail with the medication while it is still a success in a larger population context.”
Thank you for pointing out that statistical probability doesn’t equal fact. I’m sure that will be news to GT, who will forthwith only rely on hard data from his own experience. That might make for some pretty tedious posts, but at least we won’t have people trying to make predictions based on observed phenomena. When did that ever advance understanding?
Joe,
Congratulations - you came up with a criticism I hadn’t anticipated, by the simple expedient of ignoring the fact that this is all based on probabilities - as I’ve made pretty explicit throughout the post.
All the same, I think I’d better rephrase this question for your benefit:
“how many of those will be cured by one course of treatment?”
I think changing it to
“how many of those can we expect to be cured by one course of treatment?”
ought to satisfy you.
Now, perhaps you can come up with an explanation for the fairly close eyeball fit between the mathematical prediction (at a re-employment rate of 30%) and the available data on the Ansett workers?
Trotsky
I don’t want to get into a math quizz debate with you as that isn’t the intention of your post.
The problem you seem to have, like a lot of people, when looking at data is the tyranny of aggregates. This of course is the biggest problem that slavish followers Keynes like youself seem to suffer from.
Take the tech crash as an example.
The rate of unemployment in the tech sector was much higher than other areas of the economy- both here and the US. However the aggregate disguised the rate of unemploment in that sector.
Same with what’s happening in the economy now. The Australian purchasing managers indeax is now below 50. This is used as a good barometer of that if it holds below that recession may not be too far away. Yet the GDP number is still showing a growth rate of 3.5% for the next 12 months mostly on the back of commodity prices (which are now falling in a big way).
That’s what I mean, Trostsky. That’s what I mean about the tyranny of aggregation.
So your assumptions about unemployment and the likelyhood of getting a job by examining it from a math perspective isn”t that good.
Shorter JC
I bummed out on the mathematics, so now I’m going after the methodology based on the assumption that Gummo is a slave to Keynesianism.
Try this as a simpler explanation of what I’m doing - Gummo got curious about whether a simple mathematical model might account for durations of unemployment. And your remarks on the tech sector are similar to objections I anticipated here:
You’re deliberately ignoring the effect of personal motivation on re-employment chances.
Damn straight I am - they don’t affect the overall outcome - they’re only of interest if you’re looking for an explanation of how particular individuals stay unemployed for 12 months and more.
Yes, I’m looking at aggregates and reaching conclusions about how the total pool of unemployed will be distributed by length of unemployment. Debate on unemployment in all the op-eds and in policy formulation is dominated by the same slavery of the aggregate- we look at total employment rates, total unemployment rates and total participation rates.
When we get away from the aggregates we go to the other extreme - we get into exceptionalism - what is it about this person, or these people that makes them unemployed? Industry sector? Age, or the lack of it? Motivation, or the lack of it?
As for the various indices that are used as barometers of the economy - what are these if not attempts to define aggregates which can be used as reliable predictors of where the economy will go?
Oh, and you still haven’t addressed the problematic little matter of the fit between the model and the availabe data.
Scrap all employment agency funding immmediately!!! Put them all on the dole!!! I’ve been twenty years on the dole and they have never ever help me get a job as kind as each of them may have been.
The long term unemployed should randomly shoehorned into jobs. And neither the employers nor the long term unemplyed should have any bloody choice. This way no one could complain about the effect on competitiveness by having to deal with the village idiot becuae all employers would have to deal with it sometime. They should just make it work.
Thats what mutual responsibility and community means.
Trosky
I did economic maths as part of a commerce degree a long time ago. I didn’t bum out of math, Trotsky as you suggest. I’m not a mathmetician and it seems neither are you.
It wasn’t clear in you piece that you anticiptated probabilities coming into play. It was only after I informed you about the weakness of your model that it was suddenly part of the picture. And then you get all antsy because I brought it up.
Trotsky Says:
“When we get away from the aggregates we go to the other extreme”
No you don’t, Trotsky. Aggregation can throw you off centre with data analysis. Your defense of this simplistic way of looking at things is by saying newspapers etc. do htis sort of thing all the time. Of course they do but that doesn’t make it right does it?
Right now maunfacturing labor is shrinking, while retail labor is expanding. This doesn’t pertend to good times ahead on a probility basis. Yet you think that we should look into such numers to give clues.
Joe,
What precisely has you so upset about this post that you persistently misconstrue my responses to your comments?
First, on the subject of going to the other extreme, that was a statement on the state of debate - swinging from what you call the slavish Keynesian’s “tyranny of the aggregate” to an equally fallacious “tyranny of the exceptional”.
If my explicit use of an agregating method in this post is so upsetting to you, why do you reply with arguments based on the use of aggregated indexes, such as the purchasing managers’ index?
My statement that you bummed out on the mathematics referred to your specific comment that I got the maths wrong and the arguments you mounted on that basis - not the state of your maths education.
And on the manufacturing sector and retail sectors - what you might expect to find in such a situation, as people move from one employment sector to another, is a lot of “frictional” unemployment as former manufacturing workers come to terms with their new jobs. And possibly a short term downward shift in the aggregate re-employment rate of all unemployed people. But that’s purely speculative.
Whoops - forget the scare quotes around frictional.
Trotsky
I specifically made mention of the PMI manufacturing part, which is slowing.
Manufacturing jobs slowing and retail jobs increasing is very ominous because it means that jobs are moving to the final stages of prodiction.
Oh , and nothing particulalry upset me about your post. I just considered your assumption to be incorrect which you then corrected in a back handed way after i made mention of the problem.
Joe,
OK, I think I owe you an apology for that snarky update. Enough of the hairy masculist tub-thumping circle stuff.
I think a follow up post on this topic might be worthwhile once we’ve had a chance for some more discussion and head-banging. But for now, I think it’s time for me to stop running up my phone-bill by jumping into the webs everytime I think of something new that has to be said now rather than later and take the rest of the week off for off-line writing.
Bugger of a thing that - I wanted to jump into Casuistry Challenge big-time this week.
Oh and on the snarky update - I won’t delete it because my personal policy is once it’s posted it’s posted, give or take the occasional problem with the blogger archive.
Trotsky
What arrrrre you talking about and then referring me to some post. Sorry but you lost here. Seriously.
Gummo, everything in your post is perfectly valid — but in the end it doesn’t really say much. Even from 1st-year macroeconomics, I remember the lecturer explaining the different varieties of unemployment: the short-term (frictional) unemployment of a few months or less, and the ‘tail’ of long-term (structural) unemployment that can last for years.
In a relatively healthy economy, the free market can do a good job of dealing with the short-term variety; the big question is what you do with the long-term sort.
My solution is to remove as many impediments as possible to part-time work, and perhaps find ways for government to actually encourage it. In my experience, many long-term unemployed don’t really want to work a 40 or 50 hour week, particularly not in some brain-deadening entry-level retail or manufacturing job. But they might be quite happy to work 10 or 20 hours, and the $ would go a long way to lifting them over the poverty line. Plus, if they later wished to become a full-time wage slave again, they’d have the benefit of the part-time experience rather than having a blank CV.
But of course they face high EMTRs on their $, plus the general attitude of society against part-time work. We are now, grudgingly, coming to terms with the concept for women with children, but what about the single person who would like to work part-time hours for lifestyle reasons?
So, we need to spread work around and become more flexible in how work is organised. The ‘9-to-5′ dinosaur needs to die a quick and painful death.
JC - would you by any chance consider yourself a follower of the ‘Austrian’ school of economics? Just curious.
Paulus
“JC - would you by any chance consider yourself a follower of the ‘Austrian’ school of economics? Just curious.”
Yes I do. Found it by myself. Amazed we were never taught it at Uni and barely touched it in economic history.
I find Mises “Human Action’ to be the
giant of All economic books. Nothing comes close to it.
Since Joe asked I’ll comment on Gummo’s maths. Under the stated assumption that the probability is the same for everybody his calculation is correct. If p=0.8 then the number cured will not necessarily by 80 out of 100, but it will be approximately that and the expected number is 80.
However under the reasonable assumption that the probabilty is different for different people and is just 0.8 on average, the tail of the distribution will be fatter.
I agree Tim and on reflection Trotsky did actually present the point pretty well seeing he used the word expect and expected etc., which does accurately describe liklihhod.
It’s just that Trotsky is so wrong on so many things that I went into autodrive without thinking he could get one right.
It’s painful to say this but you were right this time Trotsky and I jumped the gun. Don’t get too overconfident Trotsky as there’s always next time.
part-time employment does not lead to ful-time work for long term unemployed
in non word of mouth job applications applicants are first culled, in this your application is always half empty, a half-empty cv will be culled earlier and never be regarded as half full
becuase employers are lazy scaredy cats
Paulus,
I remember the first year eco lecture too. A couple of points:
Most of the policy - particularly CentreLink policy - is geared to the assumption that there is no structural unemployment. And the assumption that people are looking for permanent ful-time work.
I think it’s maybe a mistake to consider the tail of the distribution the structural part, while the head represents frictional unemployment - people between jobs for whatever reason.
Moving on:
The picture is one of duration of employment distributed according to a pattern which is consistent with random chance. So if random chance is enough to explain the distribution of unemployment durations in the Ansett case, do we need to start looking for exceptionalist explanations for why Tom is unemployed after 12 months, while Dick and Harry - with the same skills etc - got jobs within a month? Or can we say, in all fairness, they’re just unlucky bastards?
On EMTRs - yes, they’re a big disincentive to part-time work seekers. As soon as you get over $31 of part-time job income you get hit. My housemate was doing his sums on his current part-time job and worked out that a second wouldn’t be worth his while because, with EMTRs, it would only pay him 5c in the dollar and actually cost him more money in transport and leave him in debt to the ATO. So he’d pay society (not the mill owner) for the privilege of going to work.
Finally (way over my three paras) - the results are trivial, but then I remember lots of time in that government department where we spent a lot of time on worrying about Tom (and Dick and Harry) when a simple look at the numbers, in the right light, would have shown us that something very simple and trivial was going on.
As for you Joe - damn you for denying me my first chance to use one of those spiffy new cards!
why is Gummo Trotsky russian around?
must be the ice (pick) in his veins
“As for you Joe - damn you for denying me my first chance to use one of those spiffy new cards!”
Trotsky, what is it you’re trying to say here? What’s with the spiffy cards business.
I will assume I am correct unless corrected everywhere I contribute.