On January 30th of this year, one of the arms of the BBC reported a row at Sheffield University about an economics exam question. The offending exam question is reproduced below. Is the question, as one student suggested, indistinguishable from Chinese?

Consider a country with many cities and assume there are $N > 0$ people in each city. Output per person is $\sigma N^{0.5}$ and there is a coordination cost per person of $\gamma N^2$. Assume that $\sigma > 0$ and $\gamma > 0$.

a) What sort of things does the coordination cost term $\gamma N^2$ represent? Why does it make sense that the exponent on $N$ is greater than 1?

b) Draw a graph of per-capita consumption as a function of $N$ and derive the optimal city size $N$. How does it depend on the parameters $\sigma$ and $\gamma$? Provide intuition for your answers.

c) Describe which combination of $\sigma$ and $\gamma$ generate a peasant economy, meaning an economy with no cities (or 1-person cities). Why might the values of the parameters $\sigma$ and $\gamma$ have changed over time? What do these changes imply in terms of optimal city size.

Without knowing what was covered in classes and homework one cannot tell what kind of tacit knowledge/conventions the examiner was justified in assuming in posing the question. Its easy, with experience at these things, to guess what the examiner had in mind. Nevertheless, the question is badly worded and allows a `sea lawyer‘ of a student to get full marks.

First, the sentence does not assert a connection between output and coordination. Thus, the answer to (a) should be:

Without knowing the purpose of the coordination, it is impossible to answer this question.

A better first sentence would have been:

Consider a country with many cities and assume there are $N > 0$ people in each city. Output per person is $\sigma N^{0.5}$ and to achieve it requires a coordination cost per person of $\gamma N^2$.

Second, readers are not told the units in which output is denominated. Thus, part (b) cannot be answered unless one assumes that output has a constant dollar value. One might reasonably suppose this is not the case. The sea lawyer would answer:

As output can be generated at no cost, and is monotone in city size, the optimal size of the city is infinity. Note this does not depend on the values of $\sigma$ or $\gamma$.

The answer to part (c), consistent with the earlier answers:

From the answer to part (b) we see that no combination of parameters would generate a peasant economy.