I've written multiple blogs about my attempts to run interactive classroom exercises, mainly for my first year students, where I get them to guess the level of income inequality. It's part of a longer lecture on inequality and income (re)distribution in general and the idea is that if you want to have a serious discussion about income (re)distribution, you need to have a fairly correct estimate of the existing income inequality.

As blogged previously, I started out using Norton and Ariely's (2011) approach of simply asking what the students thought what percentage of the total income was being earned by various quintiles of the population (ie. what percentage of total income is being earned by the bottom/top 20% of the income distribution?). This consistently leads to students underestimating the income inequality but as Eriksson and Simpson (2011) suggest, this could be caused by an anchoring effect and when I tried their method of instead asking for the average income of the various quintiles, the effect disappeared in my classroom as well.

Although this exercise started out as an attempt to show that students underestimate income inequality, my experience with trying out these activities in the classroom and doing a bit of reading around the topic, is that students don't necessarily underestimate inequality. In fact, they are often surprised by how relatively little you have to earn to be in the top 10% or top 1% earners of the country. Or think that the average UK income is much higher than it really is.

Also, most of the designs of the exercises that are described in the literature and that I have tried out, ask the students to put amounts (or percentages) to different points on the distribution. My thinking is that in the real world we often approach the question from the other direction. We observe an amount – a friend tells us their salary or we see an amount offered in a job advert - and we compare this with other salaries to evaluate if this is relatively high or low; basically, we try and guess where in the income distribution someone earning that particular income might be placed.

Inspired by these two observations I devised a completely new version of the exercise in my Principles of Economics class this year. At the beginning of the segment on income inequality I briefly introduce the exercise as an activity to see what they knew about the income distribution for both the UK (where we are) and the world (where, ehm, we also are). I presented them with a link to the online questionnaire (QR code, I am so modern!) and gave them some background about what I was exactly asking.

I used the data from the World Inequality Database because that was the only source I could find that also included a worldwide measure of income inequality. This database measures income on an individual level but does so by dividing household income by the number of members of the household. Even (young) children have a positive income this way. Also, as far as I can tell they use pre-tax income but do count any pension and unemployment benefits as income.

As you can see above, for the UK I asked the students to guess where in the income distribution we would place someone earning £8.000, £16.000, £24.000, £45.000 and £80.000. For the worldwide comparison I asked slightly different amounts: £1.000, £2.000, £9.000, £24.000 and £45.000. For both the UK and the world I also asked how much an individual would need to earn to be in the top 1% of income distribution (per year, in pounds). Below are the average guesses and the actual numbers.

The results seem to confirm my hypothesis that students actually overestimate income inequality. The estimates for the amounts needed to reach the top 1% are a bit silly because skewed by a couple of extreme outliers but in the other guesses they also almost consistently underestimate the place within the income distribution of various different hypothetical income levels and as such overestimate how much you need to earn to be in the upper reaches of the distribution. This makes for a slightly different discussion compared to the situation where we started with the results of Norton and Ariely's (2011) approach, where students seemed to underestimate inequality.

Ok, so now for the technical bit that is a bit dodgy but bear with me. One way of visualizing the income distribution for a particular country or society is by using a Lorenz curve. Such a graph shows the cumulative population on the horizontal axis and the cumulative share of income on the vertical axis. The graph below shows three examples. You can read off how much the bottom 20% of Norwegians earn and, say, the bottom 60% (about 9% and 38% of the total income). The top 20% earned about 40% of the total. The shape says something about the level of inequality. In particular, the closer the curve is to the 45 degree line, the more equal a society (on the 45 degree line the share of both the bottom and the top 20% will both be 20%). The US and Brazil are more unequal than Norway because the share the bottom 20% and 40% earn is smaller and their Lorenz curve is further from the 45 degree line.

Drawing and interpreting the Lorenz curve is an important part of the lecture on inequality. So it would be nice if I could turn the students’ guesses into a Lorenz curve as an easy (visual) way of comparing their estimations with the actual data. However, because I ask for percentages I don't end up with the neat 20/20/20/20/20 buckets that the standard Lorenz curve assumes.

I figured that I could use the percentages that the students guessed to create my own buckets. It does require a couple of leaps of our imagination. What I did was the following: since my students thought someone earning £8000 was earning more than 17% of the UK population and someone earning £16.000 more than 29%, I took this £8000 as an approximation of what they think the average wage is for someone in the bottom 23% of the population (the halfway point between 17% and 29%). And £16.000 is their approximation of the average earnings of someone between 23% and 35.5% (halfway between 29% and 42%%). At the top end I took their guess of how much you need to earn to be in the top 1% as their approximation of the average earning in the top 10.5% (89.5 being the halfway point between 79% and the upper end of the distribution). Based on these wonky, asymmetric buckets I then calculated the average income share for each of the buckets and used them to draw a Lorenz curve.

Now, this is obviously not the proper way to do economics. For a start, using that estimate for the top 1% (especially if it is as silly as it was in this instance) as the amount for the top 10.5% will sharply overestimate the inequality. On the other hand, if the guesses for the other salaries are all below their actual value, they will actually bring down the total income I am basing my shares on, so that will actually decrease the inequality I calculate using my method. But yes, it’s not ideal but it’s the only way I could do it (I think) and I really *really* wanted to be able to draw some Lorenz curves.

Below is the one for the UK distribution with the students’ guesses in blue and the actual numbers in orange. Because of my wonky calculation it doesn’t make sense to compare their shape with the ones for Norway (and the US and Brazil) above but I like to think that comparing the students' guesses with the actual numbers side by side like this is still interesting.

The ones for the worldwide income distribution look like this. Again, not really *proper* Lorenz curves but as way of showing that the students overestimated the income inequality I think they kind of work.

But yeah, despite the problems with drawing the Lorenz curves I am pretty happy with this approach. Will definitely use it again next year. Comments and suggestions for improvement are more than welcome.

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