Michigan’s Graduate Student Instructors (GSIs) receive student evaluations for every class they teach. The undergrads are invited to rate their GSIs using Likert Scales (aside: Likert had a degree in economics from the University of Michigan). Students see something like this:

The Registrar compiles this data, encoding “Strongly Disagree” as 1, “Strongly Agree” as 5, etc. They then provide GSIs with what is called their “Median score”, which is actually a slightly complicated interpolated form of the median.

The application of medians to discrete values was perplexing, so I asked the Office of the Registrar to provide me with the exact formula they use. They did so, and I reproduce it here in case any GSI in the future ever wonders as I once did. (I checked: the Registrar has no objection to this being made public knowledge.)

Their formula, which requires some lingo that is explained below, is:

\(\text{Median } = \text{Lower Real Limit } + \frac{0.5N – a}{b}\)

Calculate the range the median must lie in, e.g. somewhere between 2 and 3. Let’s call this range “the interval”. The Lower Real Limit is the midway point of the interval, i.e. 2.5.

The total number of observations (in the whole sample) is denoted \(N\). Denote as \(a\) the number of responses below the Lower Real Limit. In this example, \(a\) would be equal to the numbers of 1s and 2s.

Recall that our interval is the range between 2 and 3. The upper-limit of this is 3. Finally, denote as \(b\) the number of observations that choose the upper-limit of this interval, i.e. the number of people that chose 3.

Now for an example, suppose you have the following scores:

By inspection, the median cannot be greater than 4 and cannot be lower than 3. Therefore our interval is 3-4, with a Lower Real Limit of 3.5. There are 32 responses, so \(N=32\). Eleven people chose a score below the Lower Real Limit, so \(a=11\). Fourteen people chose the upper-limit of the interval, so \(b=14\). Plugging this into our equation above:

\(\text{Median } = 3.5 + \frac{0.5(32) – 11}{14} = 3.86\).

While watching football this morning, I also kept an eye on the betting odds. As soon one team scored, the odds changed.

Instead of introducing the efficient market hypothesis to students as a theory about stocks and bonds, we should perhaps start by drawing parallels to sports-betting. It is hard to consistently make money betting on Premier League results. It’s not impossible, but it takes a lot of skill and/or luck to pull it off.

The same logic applies to beating the general rate of return in a market. Financial markets tend to adjust prices so that there are no obvious bargains. This is not a claim that sporting events never throw up shock victories, nor does it imply that markets never crash. If anything, it suggests that predicting a stock market crash is about as difficult as predicting a shock sporting victory.

Like comparative advantage, it’s hard for undergrads to grasp this idea. I have found that the sporting analogy works well as a stepping stone.

(While on the topic of the EMH: there’s a fun contrast between the predicament of Cassandra, who could perfectly predict the future but nobody would believe her, and the EMH. The more people believe in EMH, the less likely they are to actively search for arbitrage opportunities. This will tend to leave market-beating returns uncollected, nullifying the EMH. The converse also holds. In this respect, EMH is a self-defeating prophecy.)

Undergrads are taught many formulae for the Price Elasticity of Demand. The most common one I have encountered is the particularly unwieldy “midpoint formula”:
\(\epsilon_D = \frac{ \frac{Q_2-Q_1}{(Q_1+Q_2)/2} }{\frac{P_2-P_1}{(P_1+P_2)/2}}\)

The elasticity at one point is a little more reasonable:
\(\epsilon_D = \frac{ \Delta Q }{ \Delta P }\times \frac{P}{Q}\)

There is an elegant simplification in the case of linear demand. With a demand curve like this:

We can use the formula above, and note that the (absolute value of) PED at point E is:
\(
\begin{align*}
\epsilon_D &= \frac{ \Delta Q }{ \Delta P }\times \frac{P}{Q} \\
&= \frac{b-d}{c} \times \frac{c}{d} \\
&= \frac{b-d}{d}\\
\end{align*}
\)
Or, equivalently:
\(
\begin{align*}
\epsilon_D &= \frac{ \Delta Q }{ \Delta P }\times \frac{P}{Q} \\
&= \frac{d}{a-c} \times \frac{c}{d} \\
&= \frac{c}{a-c}\\
\end{align*}
\)

I am pretty sure it was Steve Salant who taught me this.

Updated Feb 2014: Steve Salant adds two more points. Firstly, that this trick also applies to nonlinear demand curves. Draw a tangent to the curve at the point of interest. Its slope will equal the slope of the curve at the point of tangency, and you can then follow the same procedure as above. Secondly, that the elasticity can be expressed in terms of either \(a\) and \(c\), or \(b\) and \(d\). I have altered the post to reflect this.

Students ask me this all the time, so here’s my explanation of the three types of inefficiency.

1. Productive inefficiency. This is a supply-side idea. Mattie and Joe both produce bananas. The bananas are identical. It cost Mattie $12 to produce his last kilogram of bananas, and it cost Joe $10 to produce his last kilogram. They would both better off if, instead of producing that last unit for $12, Mattie paid Joe e.g. $11.50 to produce it. That way Mattie saves 50c and Joe makes a profit of $1.50.

The condition needed to satisfy productive efficiency is that the marginal cost for Company A equals the marginal cost for Company B. If the marginal costs differ, then there are gains from trade from the low marginal cost company (e.g. Joe) selling the item to the company with the high marginal cost (e.g. Mattie).

2. Distributive inefficiency. This is a consumer-side idea. Let’s say that for some reason bananas sell for $1 in the US and $3 in Canada. Then there are people in Canada who are willing to pay, say, $2.75 for bananas but do not receive them. Suppose some American is willing to pay up to (but no more than) $1.20 for bananas. If he sells the bananas to the Canadian for e.g. $2.50, then everyone is better off. Although the American was receiving 20c in consumer surplus by purchasing the bananas for $1, he is clearly better off by selling them for $2.50. (Obviously transport costs have to be small for this to hold.) The Canadian gets the bananas and 25c consumber surplus, so he is also better off. Therefore the $1 and $3 prices generate inefficiencies.

The condition needed to satisfy distributive efficiency is that the marginal benefit to Mr 1 equals the marginal benefit to Mr 2. If they differ, then there are gains from trade from the person with the low MB (e.g. $1) selling the item to the person with the high MB (e.g. $3).

3. Allocative inefficiency. This puts the consumer-side and the producer-side together. Suppose everything regarding productive efficiency (marginal costs for Company A and Company B are the same) and distributive efficiency (marginal benefits for Mr 1 and Mr 2 are the same) is just fine. You can still have inefficiency if the marginal benefits don’t equal the marginal costs. For example, if the costs of the last kg of bananas is $10 but consumers are only willing to pay $3 for it, then producing that last unit is inefficiently costly.

Thus the condition needed to satisfy allocative efficiency is that the marginal benefits (consumer side) equal the marginal costs (producer side).