The impossible economics exam

Students at Sheffield University are complaining that the questions in their finals were simply impossible to answer. They’d not been taught the material, they were flummoxed, all most unfair:

Final year economics students at Sheffield University are furious after an exam this week contained questions they found “impossible”.

The paper, on the economics of cities, contained compulsory questions on topics they had never been taught, say the students.

More than 90% of those who took the exam have now signed an online petition demanding the university investigate.

The university said all questions were based on topics taught in the course.

But, in a tweet, one candidate complained: “Question three may as well have been in Chinese.”

Another asked: “How can they write a paper and include questions on something we haven’t been taught, or told to research?”

Just over 100 students took the exam on Wednesday.

Well, here is question 3. And it’s 30 years since I did any formal economics (or, indeed, any algebra and I never really did cotton on to that anyway). But I reckon that anyone reasonably attentive should be able to get 50% on this question in about 5 minutes.

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Coordination costs are just what they say on the tin. There’s value to the division and specialisation of labour (derived from Adam Smith). There’s value to comparative advantage and trade in the resultant production (Smith and Ricardo). However, there’s obviously costs associated with finding the people one is going to divide and specialise with, who one is going to trade with, discovering what is comparative and so on. These are coordination costs. A close analogue is that we know that there are economies of scale at times: but we also should be aware that there are diseconomies of scale.

That the exponent on N is greater than 1 is simply a reflection of the thought that coordination costs rise with the number of people being coordinated with. To assume otherwise would be to assume that we had declining coordination costs with scale: that 2 billion people could work out how to divide, specialise and trade more easily than 2 people could. This is neither a reflection of the world we see out the window nor a reasonable starting assumption. Therefore we don’t make it.

The graph, well, I can’t work out how to draw an electronic graph. But axes, two lines. Optimal city size is where the lines cross. Must be: when the rise in coordination costs is lower than the extra production then the people in the city will be richer in a larger one. Where the marginal coordination costs are higher than the extra production then richer in a smaller one. What that actual size is depends upon what the actual values of the parameters are.

c) would take a bit more thinking about (hey, you have a go!). Just the above would provide a pass I’m pretty sure. Probably a Desmond these days actually.

And seriously folks, you really don’t need to be in your final year of an economics degree in order to get the above right.

 

42 thoughts on “The impossible economics exam”

  1. Indeed, I never studied economics at all but A-level maths gives you a fairly good start on that question

  2. I have never studied economics formally or informally, but wouldn’t per capita consumption be output per person less cost (i.e. “useful production”, in layman speak?) i.e. sigma N^0.5 – gamma N^2.

    Most efficient city size is found by identifying the maximum of that graph (i.e. differentiate and solve). So N is (sigma/2gamma)^(2/3).

    “Where the lines cross” would be where co-ordination costs use up all the output of the city.

    That might be bollocks, though.

  3. Well, it depends whether they were taught it or not. Exams are supposed to examine the person on the subjects the course covered. If you don’t know what a coordination cost is, you might guess, but that isn’t very fair since often jargon means something different to the vernacular usage of a word. If the students are correct that this was not part of the course, it is unfair.

  4. Provide intuition for your answers.

    What’s the answer to this? “I’m a woman and therefore just knew it”?

  5. “Exams are supposed to examine the person on the subjects the course covered. ”

    Aah, such an Anglo approach. According to my French colleague, in France, they are to show how clever the exam setter is and how stupid the students are. Hence, asking stuff off-curriculum that only the very brightest stand a chance of working out from first principles is entirely standard.

    When my colleague studied at one of the London universities, he found the British approach refreshing, together with the approachability of lecturers. In France, “la bibliothèque se trouve là-bas” is the standard answer to questions.

  6. If I remember my exams correctly they tended to be on what the course covered rather than what was taught in lectures.
    The reading was a whole lot more than the lectures.

    Would this also be something that would have come up in reading a few times?

  7. There are costs other than co-ordination costs though. Is the idea that they are fixed per person, so not dependent on city size? Still you need to know that other factor to calculate your optimum city size.

    The exponent on N gives a co-ordination cost per capita proportional to the number of individuals. I know this is just playing around but that seems very high. 1.1 would be better for the sake of the example (even for innumerate economics students).

    For your peasant economy, sigma<4gamma/sqrt(2). How am I doing?

  8. Your explanation only explains why the exponent is above 0. Why should the per capita coordination cost scale superlinearly?

  9. I briefly taught Economics to first years at Oxford, as well as doing some admissions interviews. No exaggeration, that question would be fair game for an admissions interview to PPE if you gave it to them 30 mins in advance. I have posed harder, with reading time.

    The difference is probably not about ability, but is that they were expecting to see something new. My guess is that seeing something even mildly unfamiliar threw these finalists into a panic. I don’t many attempted to reason it through. It’s a shame that they are leaving university with so little tolerance for ambiguity.

    Regardless of if they were taught the material or not, the professors have done then a disservice by allowing them to expect such spoonfeeding and simple assessment.

  10. Isn’t the Murphmeister an expert on economics?

    Perhaps he could have a go?

    I’m sure his answer would involve raising taxes on rent-seekers

  11. An economics degree would probably have few actual lectures and seminars. Most of the ‘week’ would be considered reading time where you’d be expected to read around the subject.

    That’s how things were in the days of yore, anyway.

  12. Aah, such an Anglo approach. According to my French colleague, in France, they are to show how clever the exam setter is and how stupid the students are. Hence, asking stuff off-curriculum that only the very brightest stand a chance of working out from first principles is entirely standard.

    That explains their conduct in meetings and presentations, where the aim of most attendees is to ask utterly irrelevant questions hoping to catch each other off guard, and hence look awfully clever in front of their boss. The worst thing is that this is how careers appear to be made.

  13. Good attempts so far.

    @Manhattan Brit you lost a factor of 2 somewhere. But correct that “Where the lines cross” would be where co-ordination costs use up all the output of the city.

    @Bloke in Germany, you’ve lost a 2 too. We just need sigma1 exponent on costs (perhaps the infrastructure is fixed). A simple interpretation would be congestion in the city. Each additional person trying to make a meeting makes me more likely to miss it, and I lose business.

    b) The graph of total consumption per capita. Just take one from the other:

    per capita consumption = sigma*N^0.5 – gamma * N^2

    The graph starts at zero, goes up, has a single peak, and then falls back to zero again.

    For “derive the optimal city size”, this is simple calculus, we just take the first derivative of per capita consumption with respect to N and set it equal to zero:

    0.5 sigma / N^0.5 – 2gamma *N = 0

    sigma / N^(0.5) = 4*gamma*N

    N^(3/2) = sigma / (4*gamma)

    Nmax = (sigma/4*gamma)^2/3

    (For bonus marks, also show this stationary point is a maximum and is unique in the given parameter range)

    c) Just means that the Nmax is greater than 1. We can see that Nmax is determined by the ratio of sigma to gamma. Eyeball the expression for Nmax long enough, and it should be clear that if (sigma/4*gamma) is less than 1, then Nmax is less than 1. That is, sigma needs to be at least four times greater than gamma to prevent peasant economies . Otherwise, the coordination costs added by just the second person coming along is more than they produce, and it isn’t worth having a second person in the city.

    The parameters sigma and gamma are functions of the productive technology of the society, so obviously they change over time. As alluded to above, a reasonable interpretation is that sigma is the capital stock that labour is applied, and gamma is the inverse of infrastructure (so a bigger gamma means a worse infrastructure), where infrastructure is how we coordinate – roads, postal service, telecoms etc.

  14. Sorry, trouble with greater than, less than symbols in html: try again:

    Good attempts so far.

    @Manhattan Brit you lost a factor of 2 somewhere. But correct that “Where the lines cross” would be where co-ordination costs use up all the output of the city.

    @Bloke in Germany, you’ve lost a 2 too. We just need sigma less than 4gamma for a peasant economy.

    Spoilers and full answers:

    a) Basically, marginal output is falling (perhaps the stock of capital is fixed), hence the 1/2 exponent on output per person. But marginal costs are rising, hence the >1 exponent on costs (perhaps the infrastructure is fixed). A simple interpretation would be congestion in the city. Each additional person trying to make a meeting makes me more likely to miss it, and I lose business.

    b) The graph of total consumption per capita. Just take one from the other:

    per capita consumption = sigma*N^0.5 – gamma * N^2

    The graph starts at zero, goes up, has a single peak, and then falls back to zero again.

    For “derive the optimal city size”, this is simple calculus, we just take the first derivative of per capita consumption with respect to N and set it equal to zero:

    0.5 sigma / N^0.5 – 2gamma *N = 0

    sigma / N^(0.5) = 4*gamma*N

    N^(3/2) = sigma / (4*gamma)

    Nmax = (sigma/4*gamma)^2/3

    (For bonus marks, also show this stationary point is a maximum and is unique in the given parameter range)

    c) Just means that the Nmax is greater than 1. We can see that Nmax is determined by the ratio of sigma to gamma. Eyeball the expression for Nmax long enough, and it should be clear that if (sigma/4*gamma) is less than 1, then Nmax is less than 1. That is, sigma needs to be at least four times greater than gamma to prevent peasant economies . Otherwise, the coordination costs added by just the second person coming along is more than they produce, and it isn’t worth having a second person in the city.

    The parameters sigma and gamma are functions of the productive technology of the society, so obviously they change over time. As alluded to above, a reasonable interpretation is that sigma is the capital stock that labour is applied, and gamma is the inverse of infrastructure (so a bigger gamma means a worse infrastructure), where infrastructure is how we coordinate – roads, postal service, telecoms etc.

  15. It may be the case that the structure of University teaching leads to a set of familiar analysis of familiar subjects however that question can be tackled purely mathematically because the question deliberately poses a task of equating abstract numbers out of any context, and If you cant do maths then you are not going to be able to do economic scrutiny.

  16. To add to Matt’s quite excellent answer.

    (c) Why have gamma and sigma changed over time? Technology would be a standard answer, but we can expand that to include organisational technologies – the creation of a church, the creation of the nation state, the rule of law, through printing press. These decrease coordination costs and increase production. Technologies such as the plough make it possible for families to produce surpluses that can then be taxed or traded. etc.

  17. And to add to Ken, what do the changes imply in terms of the optimal city size? Well, you would expect sigma to increase over time (representing greater worker efficiency) and gamma to decrease (representing greater co-ordination efficiency). As such, the optimal city size should increase over time.

    (That said, the implicit assumption here is zero communication between cities)

  18. “Aah, such an Anglo approach. According to my French colleague, in France, they are to show how clever the exam setter is and how stupid the students are. Hence, asking stuff off-curriculum that only the very brightest stand a chance of working out from first principles is entirely standard.”

    Not completely foreign to Britain. The more fiendish Cambridge maths exams of the 19th century were often used to introduce new and interesting ideas to the world, as a bizarre alternative to publishing them. That’s the ultimate “unseen exam” – stuff that wasn’t in the lectures or in the reading, because it’s only just been discovered. If I recall correctly this was where Stokes’ theorem in vector calculus got its first airing, for instance.

  19. @MyBurningEars

    In a reversal of that, Oxford’s standard for a high First class mark on any one economics paper was, until very recently, that it be of a “publishable standard”

    That is, your hand-written 4 essays in 3 hours had to be of a journal quality.

  20. That question should be fair game for 3rd year Econ finals, it requires pure understanding. If students have been taught it then its a fairy trivial test of diligence, but if the concepts are “new” then its a good test of fundamental understanding of the mathematical concepts of wealth creation.

    I can only suppose that when I was young and nimble minded I would have murdered that question at the end of my physics degree and that it would not have been suitable for my economics finals as its far too easy.

    A good friend of mine did economics at Sheffield in the mid eighties, I can only suppose he also would have murdered that question. Am I correct in thinking that economics has become a girls subject? Do we have any sort of breakdown of Sheff Econ by sex, and likewise of the protesters?

  21. “the more fiendish Cambridge maths exams of the 19th century were often used to introduce new and interesting ideas to the world”: when I used to set uni exams I would forever pop new results in. Not earth-shaking, but new; it would require the buggers to think rather than memorise. Can’t think in an exam? Bugger off to Sheffield.

  22. So Much for Subtlety

    MyBurningEars – “Not completely foreign to Britain. The more fiendish Cambridge maths exams of the 19th century were often used to introduce new and interesting ideas to the world, as a bizarre alternative to publishing them.”

    Not completely foreign to America either. Kip Thorne has a story somewhere about how he was working on a time-traveling wormhole in space. So he put it in the undergraduate exam. Oddly enough none of them recognised it for what it was.

  23. KevinB, Andy C and others:

    Candidly, you are all wrong. The question appears to be another attempt to indoctrinate students in neoliberal thinking without offering them an alternative. Real economics can only be absorbed through ‘The Courageous State’, and indeed its successor, the forthcoming ‘The Joy of Tax’, which are both online from all good bookstores and as ebooks.

  24. >> Provide intuition for your answers.

    >What’s the answer to this? “I’m a woman and therefore just knew it”?

    It’s poorly expressed, but basically it means explain the reasoning in a non-technical way so that it seems plausible, or (as modern Humanities academics say) ‘intuitive’.

  25. >the professors have done then a disservice by allowing them to expect such spoonfeeding and simple assessment.

    That’s the way modern Universities are. Students expect to be spoonfed. The academics no longer have a say, because if you don’t spoonfeed the students then your student satisfaction scores will go down and then the University higher-ups will crush you.

  26. it is perhaps poorly phrased (and a little confusing, in so far as the fact there is more than one city appears to play not role)

    output per person isn’t sigma*N^0.5

    it’s sigma*N^0.5 – gamma * N^2

    but otherwise this is classic shit student behaviour – the better students won’t be complaining about questions like this. They will have seen production functions with exponents, they will have seen increasing returns to scale before etc. they are only being asked to extend familiar ideas in some unfamiliar directions.

  27. Ummm, I’ve never studied the subject but it does help explain a few things if economists think 0.5 is greater than 1…

  28. As cities grow with economic development, Europe in the 19th century; Africa now, gamma is falling.
    My question (asked in genuine ignorance) is ‘why?’ What does and does not get included in coordination costs?

  29. Sorry, hadn’t read Matt Moor or Ken.

    But to add, it could of course be that sigma is rising. Again though, the same question: why? Setting aside the maths, we know that social organisations such as churches, mutual socities etc help cities grow. We know that public health and hygiene, eg. the Victorian London sewerage systems, help cities grow. Where though so these things fit into sigma or gamma?

  30. @Ironman: Churches and mutual societies probably reduce gamma, by providing trust frameworks which allow quicker co-ordination. Public health probably increase sigma, by reducing the proportion of N who are undproductively ill.

    Things that will reduce both sigma and increase gamma include civil unrest, deadly or vastly debilitating pandemics, communist governments, natural disasters, over-regulation, general war…

  31. NB: By public health I mean stuff like sanitation and vaccination, not worrying about whether Coca-Cola and Irn-Bru can vie for my pound via advertising…

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