Where's the production function?

Should small schools be closed? Is the government right to build giant prisons? These two questions share a common theme - the theme being that the government's pretensions to technocratic expertise are a sham.
In principle, the question of what size schools or prisons should be is a narrow technical - mere mathematical -  one. The question is simply: what is the production function?
Take prisons. A desirable output might be a low recidivism rate. Our independent variables might include prison size, guard-prisoner ratio, money spent on therapies and education, diet, controls for the type of prisoner and so on. We then regress the recidivism rate upon these variables across all prisons, and estimate the coefficients. We can then see the impact prison size has upon recidivism.
Prison size, then, is not a political question, but a mathematical one.
So, why are questions of school or prison size political ones, rather than mere maths? Here are some possibilities:
1. There are multiple desirable products of schools or prisons. For example, in schooling the desire to maintain community cohesion might argue for small schools, whilst exam grades might argue for larger ones. The production function approach still has a role here, however, as it allows us to better estimate the trade-offs between these objectives. It allows us to say (for example): "keeping a village school open costs one GCSE per student."
2. The notion of a production function might be incoherent. There might be more than one optimum technique. For example, big prisons might work well with big spending on education, but worse than small prisons if there's little such spending. 
3. There might be insufficient variation in inputs to allow us to estimate coefficients well - for example, if spending per pupil varies little from school to school.
4. Politicians - and unions - might not like what production functions say. For example, there's evidence (pdf) that spending on schools on its own doesn't (pdf) do much to improve outcomes.
5. Drawing attention to production functions forces politicians to look at outputs, not inputs. Their boasts of spending billions more on education (or health) become vacuous. And they have to do the grunt work of actually making things work.
Whatever the reason, the fact's the same. In not thinking about the maths of production functions, politicians  - unless they have Sraffian doubts about their efficiacy, which I doubt - are like cargo cults. They practice a sham ritualistic imitiation of rational management, not the substance of it.

Comments

[Jonathan]

Be careful what you wish for here. Try this, a Dept of Education commissioned report which estimates production functions for higher education. The existence of economies of scale and scope would be important to determine the optimal means if HE is to be expanded signficantly:

http://www.dfes.gov.uk/research/data/uploadfiles/RR641.pdf

Plenty of discussion of the technicalities of estimating such functions. But then look at the fine print. The teaching output measure is some crude quality indicator with degree classifications relative to A'level scores. The research output measure is in terms of research funding raised - yes, an output measured by an input! 3rd stream/knowledge transfer estimated by other services income (possibly least problematic of the 3). If you were being harsh - as I'm inclined to be - no matter how sophisticated the econometrics it's garbage in garbage out.

The research indicator here is particularly worrisome. Back in the late 80s came the big push to concentrate research on big institutions, even though evidence at the time that this was effective was pretty scant (see Jennifer Platt, 'Research Policy in British Higher Education and its Sociological Assumptions', Sociology 1988 22: 513-529. Apologies, probably need academic library to access it; it isn't sociological theory incidentally). Having done this measuring research output in terms of funding input would appear to give economies of scale but probably just reflects funding bodies' policy.

BTW, you don't even need Sraffian stuff here (powerful though it is); aggregation issues with production functions are a nightmare that analysts usually ignore.

[Bob B]

As mentioned before, the London Borough of Sutton is (again) top of local education authorities league table for England based on average candidate attainment in last summer's GCSE exams:
http://news.bbc.co.uk/1/hi/education/7180228.stm

As for ranking LEAs in England by spending on education per pupil, try this:
http://www.telegraph.co.uk/education/main.jhtml?xml=/education/2007/12/24/nschools124.xml

There's evidently not much correlation between spending and outcomes but then you'd expect LEAs with a record of low achieving to spend more in the (perhaps misguided) belief of improving attainment. Sutton's secret recipe is that it has a cluster of outstanding, maintained (meaning, non fee-paying) selective schools.

[Mark Harrison]

I'm willing to believe that the number of schools in the UK is high enough to give statistical validity to a production function.

I'm afraid I have NO IDEA how many prisons we have, not even the order of magnitude, but given the number of people I know who have even been to prison (two), I have to assume that it's several orders of magnitude lower than the number of schools.

My concern about production functions on small sample sizes is, of course, confidence (in the statistical sense, not the vernacular sense.)

[Luis Enrique]

Can't a prison production function include an 'external' variable like spending on education?

[dearieme]

We really should worry about a school system that results in an intelligent chap believing that a heap of correlations will naturally explain causation.

[Mark Harrison]

dearieme,

Scientists worry about WHY things work.

Engineers worry about making things work reliably, even if they don't understand why.

This is why we trust engineers to design bridges, cars, office blocks, planes, and all the other things that could kill us if they failed.

I think we should worry about a school system that relates in a huge bunch of politicians being prepared to ignore heaps of correlations all pointing in the same direction, because either (A) their ideology of (B) their belief of their potential voters' ideologies point in a different direction.

[Not intended to "disrespect Scientists" in any way - their skills are equally needed.]