Inequality and the Bill Gates effect
(Note: I haven't gotten all the kinks worked out in what follows below. So maybe I've missed something or said something wrong)
The recurrent topic of which economy's dad can beat up which economy's dad - the US' or the EU's - recently popped up again here and there. Roughly speaking, the US has lots of inequality but higher per capita income, whereas the EU (here, as often, basically meaning France, Italy and Germany) has lower per capita income but a lot less inequality. In fact if you take the OECD countries, minus the late joiners, and slap'em up on a scatter plot what you'll see is a (pretty rough - there's all kinds of measurment issues here) positive relationship between per capita income and the level of inequality as measured by the Gini coefficient.
These differences start people talking about the Anglo-Saxon vs. European models of the economy, how the US and Europe have different economic structures which produce these results and how either Europe is quickly falling into the dustbin of history, or the US is becoming a place where folks starve in the gutter because they have no access to health insurance and evil capitalist stole their puppies. A possible question one might ask however is whether this kind of relationship - high income and inequality in US, low income and inequality in Europe - could arise simply by chance. To do that let's delve a bit deeper into the measurement of inequality and income distributions. What I'm gonna argue is that despite the fact that it may not seem like it, this kind of relationship could very well arise by chance and that relatedly, there probably isn't that much of a difference between the economic structure of the US and that of the EU. In fact, a positive relationship between per capita income and inequality is PRECISELY what you would expect if THERE ARE NO DIFFERENCES in the underlying structure of these economies (given certain assumptions of course).
Ok, so the Gini coefficient is a measure of inequality and it is given by:
 = \frac {1} {K} (\frac {2*\sum_{i=1}^K i*y_i} {\sum_{i=1}^K *y_i}-(K+1)) $)
where y_i is the income of person i and the y's are arranged in an ascending order, that is:
The Gini has the familiar interpretation as twice the area between the diagonal and the Lorenz curve, and it is equal to 0 if everyone has the same income, and 1 if one person has all the income. Higher levels of G indicate higher inequality.
One of the key properties of the Gini (or any half way decent measure of inequality for that matter) is that it is scale independent. What this means is that if we multiply everyone's income by some positive number (say double everyone's income) then the amount of inequality, as measured by the Gini, will not change. In other words, richer economies are not automatically considered to be more unequal than poor ones. (It also means that the Gini is a unit-less measure, independent of whether we measure incomes in US dollars or rupees or World of Warcraft gold or whatever)
Mathematically it means that G is a function homogenous of degree 0 in all the y's
 = G(y_1,y_2,...,y_K) \forall A>0 $)
So, you would expect that if you have a whole bunch of economies, all of which have the same "underlying economic structure" then, on average, there should be no relationship between per capita income and inequality. Some economies might randomly end up with high inequality and high per capita income, some randomly end up with high inequality and low per capita income, some with low income/low inequality and some with low income/high inequality. But on average there should be no pattern for "similar" economies.
In fact there's a whole literature on the relationship between inequality and economic development (as measured by level of per capita income) going back to Nobel prize winner Simon Kuznets. Kuznets, famously claimed to have discovered the so called Kuznets curve which says that the relationship between inequality and per capita income is reversed n-shape. That is, in poor countries everyone's dirt poor but they're all equally dirt poor. As the economy begins to develop and per capita income rises however, some portion of the population begins pulling away from everyone else and inequality rises. In developed economies the rest of the population has increasing incomes along with the very top and inequality actually falls. There's been a lot of work looking into whether the Kuznets curve actually exists in the data (it sort of looks like it, except that most middle income countries tend to be in Latin America which has high inequality for historical reasons, hence the relationship might not be driven by changes in income), controlling for reverse causality and other factors. A typical approach is to run a regression like this
^2 + \gamma*x+e$)
where G is the gini, y_A is per capita income, x is a set of control variables and the betas and gammas are parameters to be estimated. Then if the estimation produces beta_1>0 and beta_2<0 then this is taken as evidence in support of the Kuznets hypothesis.
Relatedly, we might expect that if we were to do a regression such as the one above and get beta_1=beta_2=0 then all the economies in the sample have roughly the same "underlying economic structure" (except for variation in the x's) and the differences among them in per capita income and inequality have arisen purely by chance.
But this expectation is actually wrong. The reason for this is that both statistics - the Gini and the per capita income - are themselves constructed from the same underlying data, the individual level or household level incomes, which themselves are produced through some partly random process.
To explain and illustrate it's actually easier to do a simulation (and like I said, I haven't worked out all the kinks).
I'm assume here that within each economy (simulation) individual incomes are draws from a log-normal distribution with parameters mu and sigma where
*exp{2\mu+\sigma^2} $)
Why a log normal distribution? Well, for most real life economies the income distribution is skewed to the left (meaning median income is less than average income) and generally a log normal is a pretty good fit. Another potential candidate could be the Pareto distribution but that actually turns to be "too skewed" - overall it's something like mostly log-normal with Pareto at the very top. At any rate, what actually matters is that the incomes are generated by a data generating process which is left skewed.
(See here. Actually, I had a better reference somewhere but managed to loose it among my bookmarks)
I'm gonna keep stressing this throughout but the key is that all the simulated economies have incomes generated by the SAME process, that is the same mu and sigma.
So we generate, say a 1000 draws from a log normal distribution and then repeat this, say, 250 times (should do more than 1K but that was enough to considerably slow down my computer and anyway 1K obs might very well be more than what you have to work with in the real world when calculating Gini's). Then, for each economy (each set of 1K draws) we calculate the Gini, per capita income and some other statistics. Then we look at the relationship between the Gini and per capita income. And we see something like this...
(mu=10.3, sigma=.7, implying on average median income of about 29K, per capita income of 38K and an average Gini of .378)
or something like this...
(mu=10.15, sigma=.8307; I was trying to calibrate these roughly to US and European data but admittedly there's so many measurement issues here that's pretty hard to actually know what the median income of ... individuals?, households?, workers? ... is).
So. What we end up getting is a POSITIVE relationship between inequality and per capita income EVEN THOUGH we assumed that all the economies had the same "underlying economic structure". What "randomness" produces here is not a lack of a relationship but a fairly strong positive one. By this interpretation it could very well be pure luck that US ended up with higher per capita income and higher inequality than Europe, rather there being any actual "structural" differences between them.
But what is the intuition for these results? Well, it's Bill Gates. With a log normal distribution (or any left skewed one) once in a while, PURELY BY CHANCE, you will get a Bill Gates in your economy. A really really rich person out in the right tail end of the distribution. His appearance is purely random, but given enough economies to observe we should observe Bill Gates' in at least some of them.
What happens when a Bill Gates (in real life, a few thousand of them) appears? Well, a very very rich person has the effect of pulling up the average income in an economy but leaving the median income unchanged. And with a log normal distribution the resulting Gini is in large part a measure of the difference between the median and the average. In fact if we regress the ginis from out simulations on the means and the medians (or their ratio) we can explain 2/3 of the variation in the gini:
If instead the data generating process which throws up the initial incomes was symmetric (or close to it) then for every Bill Gates, on average, there'd be an anti-Bill Gates, a really really poor person in the left tail of the distribution to offset the effect of the other on the mean (more precisely there'd be on average same amounts of really really rich and really really poor). In that case you WOULD actually expect no relationship between income and gini for similar economies. But if the individual incomes comes from a skewed distribution then you will get a positive relationship.
Ok, so what does it mean and is it plausible? Well, unless I'm missing something, first thing it means is that there's plenty of reason to be suspicious of all them studies which regress ginis on incomes to uncover relationships between inequality and development. For the topic at hand it could mean that there isn't that much *real* difference between US and EU. By pure chance, in US a few hundred more Bill Gates appeared than in EU and as a result both per capita income and level of inequality got pulled up.
We have this tendency to look for "structural" explanations when confronted with patterns in the data. And so we start talking about the "Anglo-Saxon model" or "European social democracy" but we forget that these patterns could be purely random. Add to that the fact that usually these kind of debates serve ideological and political purposes and it's easy to see why folks are so quick to jump to conclusions. But the above would actually be bad news for both Euro-bashers - lower incomes in Europe is just a matter of luck - and US-haters - higher inequality in US would also not be a result of some rat race soulless system (at least not any less soulless than the European one) but again, an artifact of random chance.
And once you pause and think about it this way you realize that the European economies and the "Anglo-Saxon" ones have way way more in common with each other than they do with all the other economies in the world. They're all rich, essentially capitalist (don't kid yourself, European economies are capitalist, Scandinavia included) economies and hence any real differences between them are bound to be minimal.
(Note also that the above explanation is pretty inline with the paper and comments cited at Marginal Revolution linked to above)
The recurrent topic of which economy's dad can beat up which economy's dad - the US' or the EU's - recently popped up again here and there. Roughly speaking, the US has lots of inequality but higher per capita income, whereas the EU (here, as often, basically meaning France, Italy and Germany) has lower per capita income but a lot less inequality. In fact if you take the OECD countries, minus the late joiners, and slap'em up on a scatter plot what you'll see is a (pretty rough - there's all kinds of measurment issues here) positive relationship between per capita income and the level of inequality as measured by the Gini coefficient.
These differences start people talking about the Anglo-Saxon vs. European models of the economy, how the US and Europe have different economic structures which produce these results and how either Europe is quickly falling into the dustbin of history, or the US is becoming a place where folks starve in the gutter because they have no access to health insurance and evil capitalist stole their puppies. A possible question one might ask however is whether this kind of relationship - high income and inequality in US, low income and inequality in Europe - could arise simply by chance. To do that let's delve a bit deeper into the measurement of inequality and income distributions. What I'm gonna argue is that despite the fact that it may not seem like it, this kind of relationship could very well arise by chance and that relatedly, there probably isn't that much of a difference between the economic structure of the US and that of the EU. In fact, a positive relationship between per capita income and inequality is PRECISELY what you would expect if THERE ARE NO DIFFERENCES in the underlying structure of these economies (given certain assumptions of course).
Ok, so the Gini coefficient is a measure of inequality and it is given by:
where y_i is the income of person i and the y's are arranged in an ascending order, that is:
The Gini has the familiar interpretation as twice the area between the diagonal and the Lorenz curve, and it is equal to 0 if everyone has the same income, and 1 if one person has all the income. Higher levels of G indicate higher inequality.
One of the key properties of the Gini (or any half way decent measure of inequality for that matter) is that it is scale independent. What this means is that if we multiply everyone's income by some positive number (say double everyone's income) then the amount of inequality, as measured by the Gini, will not change. In other words, richer economies are not automatically considered to be more unequal than poor ones. (It also means that the Gini is a unit-less measure, independent of whether we measure incomes in US dollars or rupees or World of Warcraft gold or whatever)
Mathematically it means that G is a function homogenous of degree 0 in all the y's
So, you would expect that if you have a whole bunch of economies, all of which have the same "underlying economic structure" then, on average, there should be no relationship between per capita income and inequality. Some economies might randomly end up with high inequality and high per capita income, some randomly end up with high inequality and low per capita income, some with low income/low inequality and some with low income/high inequality. But on average there should be no pattern for "similar" economies.
In fact there's a whole literature on the relationship between inequality and economic development (as measured by level of per capita income) going back to Nobel prize winner Simon Kuznets. Kuznets, famously claimed to have discovered the so called Kuznets curve which says that the relationship between inequality and per capita income is reversed n-shape. That is, in poor countries everyone's dirt poor but they're all equally dirt poor. As the economy begins to develop and per capita income rises however, some portion of the population begins pulling away from everyone else and inequality rises. In developed economies the rest of the population has increasing incomes along with the very top and inequality actually falls. There's been a lot of work looking into whether the Kuznets curve actually exists in the data (it sort of looks like it, except that most middle income countries tend to be in Latin America which has high inequality for historical reasons, hence the relationship might not be driven by changes in income), controlling for reverse causality and other factors. A typical approach is to run a regression like this
where G is the gini, y_A is per capita income, x is a set of control variables and the betas and gammas are parameters to be estimated. Then if the estimation produces beta_1>0 and beta_2<0 then this is taken as evidence in support of the Kuznets hypothesis.
Relatedly, we might expect that if we were to do a regression such as the one above and get beta_1=beta_2=0 then all the economies in the sample have roughly the same "underlying economic structure" (except for variation in the x's) and the differences among them in per capita income and inequality have arisen purely by chance.
But this expectation is actually wrong. The reason for this is that both statistics - the Gini and the per capita income - are themselves constructed from the same underlying data, the individual level or household level incomes, which themselves are produced through some partly random process.
To explain and illustrate it's actually easier to do a simulation (and like I said, I haven't worked out all the kinks).
I'm assume here that within each economy (simulation) individual incomes are draws from a log-normal distribution with parameters mu and sigma where
Why a log normal distribution? Well, for most real life economies the income distribution is skewed to the left (meaning median income is less than average income) and generally a log normal is a pretty good fit. Another potential candidate could be the Pareto distribution but that actually turns to be "too skewed" - overall it's something like mostly log-normal with Pareto at the very top. At any rate, what actually matters is that the incomes are generated by a data generating process which is left skewed.
(See here. Actually, I had a better reference somewhere but managed to loose it among my bookmarks)
I'm gonna keep stressing this throughout but the key is that all the simulated economies have incomes generated by the SAME process, that is the same mu and sigma.
So we generate, say a 1000 draws from a log normal distribution and then repeat this, say, 250 times (should do more than 1K but that was enough to considerably slow down my computer and anyway 1K obs might very well be more than what you have to work with in the real world when calculating Gini's). Then, for each economy (each set of 1K draws) we calculate the Gini, per capita income and some other statistics. Then we look at the relationship between the Gini and per capita income. And we see something like this...
(mu=10.3, sigma=.7, implying on average median income of about 29K, per capita income of 38K and an average Gini of .378)
or something like this...
(mu=10.15, sigma=.8307; I was trying to calibrate these roughly to US and European data but admittedly there's so many measurement issues here that's pretty hard to actually know what the median income of ... individuals?, households?, workers? ... is).
So. What we end up getting is a POSITIVE relationship between inequality and per capita income EVEN THOUGH we assumed that all the economies had the same "underlying economic structure". What "randomness" produces here is not a lack of a relationship but a fairly strong positive one. By this interpretation it could very well be pure luck that US ended up with higher per capita income and higher inequality than Europe, rather there being any actual "structural" differences between them.
But what is the intuition for these results? Well, it's Bill Gates. With a log normal distribution (or any left skewed one) once in a while, PURELY BY CHANCE, you will get a Bill Gates in your economy. A really really rich person out in the right tail end of the distribution. His appearance is purely random, but given enough economies to observe we should observe Bill Gates' in at least some of them.
What happens when a Bill Gates (in real life, a few thousand of them) appears? Well, a very very rich person has the effect of pulling up the average income in an economy but leaving the median income unchanged. And with a log normal distribution the resulting Gini is in large part a measure of the difference between the median and the average. In fact if we regress the ginis from out simulations on the means and the medians (or their ratio) we can explain 2/3 of the variation in the gini:
If instead the data generating process which throws up the initial incomes was symmetric (or close to it) then for every Bill Gates, on average, there'd be an anti-Bill Gates, a really really poor person in the left tail of the distribution to offset the effect of the other on the mean (more precisely there'd be on average same amounts of really really rich and really really poor). In that case you WOULD actually expect no relationship between income and gini for similar economies. But if the individual incomes comes from a skewed distribution then you will get a positive relationship.
Ok, so what does it mean and is it plausible? Well, unless I'm missing something, first thing it means is that there's plenty of reason to be suspicious of all them studies which regress ginis on incomes to uncover relationships between inequality and development. For the topic at hand it could mean that there isn't that much *real* difference between US and EU. By pure chance, in US a few hundred more Bill Gates appeared than in EU and as a result both per capita income and level of inequality got pulled up.
We have this tendency to look for "structural" explanations when confronted with patterns in the data. And so we start talking about the "Anglo-Saxon model" or "European social democracy" but we forget that these patterns could be purely random. Add to that the fact that usually these kind of debates serve ideological and political purposes and it's easy to see why folks are so quick to jump to conclusions. But the above would actually be bad news for both Euro-bashers - lower incomes in Europe is just a matter of luck - and US-haters - higher inequality in US would also not be a result of some rat race soulless system (at least not any less soulless than the European one) but again, an artifact of random chance.
And once you pause and think about it this way you realize that the European economies and the "Anglo-Saxon" ones have way way more in common with each other than they do with all the other economies in the world. They're all rich, essentially capitalist (don't kid yourself, European economies are capitalist, Scandinavia included) economies and hence any real differences between them are bound to be minimal.
(Note also that the above explanation is pretty inline with the paper and comments cited at Marginal Revolution linked to above)
posted by YouNotSneaky! at 3:38 PM
32 Comments:
Awesome.
Piketty and/or Saez (there's been a series of papers) find the top 1% (or 0.01%) income earners have doubled their share of income in the capitalist puppy stealing countries but stayed near constant in "Europe" (France and the Netherlands, but not, interestingly, Sweden).
I think their results suggest a structural relationship between growth and inequality. You wouldn't expect decades long trends if these results were driven by randomness.
Actually I think US is a bit too much of an outlier for it to come from exactly the "same" distribution as say France. Although it could be - see that lonely little dot up there in the second graph that popped up just by chance? But yeah, the probability is probably low. Still, it does suggest that at least some of the diff between US and EU is just randomness. Probably more of the diff than most people suppose.
Also I did calculate share of top 1% for each sim and you also get a positive relationship - as you'd expect with Bill Gates - though it's weaker (as you'd also expect since it ignores what's going on in other parts of the distribution)
How, in general, do we address criticisms like your's though?
I was at a talk the other day about Farewell to Alms and the point was made that in the study of the industrial revolution we have exactly one data point. A theory with 17 explanatory variables will be just as good as a model one variable will be just as good as a random number generator at drawing a line through that dot.
We only observe one instance of reality (how cool would it be if that weren't true!) so at some level we can never know if the relationship between inequality and growth (or any other two things) is spurious.
Sorry to butt in – I did a year of economics at university and you guys are way out of my league, so can I drag this back to something that is missing here: politics.
The ‘anglo saxon’ model refers, basically to the US, with a bit of UK thrown in. The UK gets dragged into it because of the Thatcher reforms in the 1980s, and her defeat of organized labour, and her questioning the post WW II consensus in Europe – welfare capitalism.
Since Thatcher we have seen a new consensus, economically, in the UK of a flexible economy, lower tax, flexible labour force etc.....
We have also seen a constant angst over the welfare state (and much change) but also an increase in inequality in the UK (and a decrease in social mobility, interestingly).
I appreciate what you are saying about the Bill Gates Effect, but those gorgeous economic models and measurement obscure the point about inequality and reactions to it – it’s the politics, stupid!
Will,
Well, I don't think the situation here's as bad as with trying to explain the IR, where you do have basically just one data point. The Gini that you get from the simulations has a distribution - normal, by the CLT - so if you observe an economy with a gini in the upper tail you can actually state with what probability it came from log normal with a particular mu and sigma. Of course what you'd really wanna look at is the join distribution of gini, median, mean and whatever else but it might be possible to calculate the probability that any two economies came from the same process. That's sort of the comment about the unworked out kinks. Also it's all assuming that the log normal is the distribution for all economies and the difference is just do to diff mu's, sigma's and chance. But that's still something. For instances, it's very unlikely that Latin American countries come from the same mu and sigma as their ginis/avg incomes don't line up positively with the OECD countries.
Beatroot - If you look at it from point of view of politics then yeah, you might see a lot of difference. But the question always is to what extent various policies translate into observable effects on the ground that can be linked to those politics and to what extent could it be just chance. One of the things that makes economists so annoying to everyone else is that we tend to think that unless some particular set of policies is very extreme and enforced through draconian measures then people will try and find ways around these policies and you won't observe that much variation in outcomes. This is basically just another way of saying that US and Western Europe have a lot more in common with each other than they do with the rest of the world. Actually quite often even economists forget that insight.
As a political economist, I chafe at the suggestion that economic outcomes--any outcomes, really--are purely random. Institutional variations abound across the Atlantic, and many of them have been shown to contribute to patterns of economic growth, inequality, the endogeneity of trade policy, etc. Simple variations in the taxation structure of a country--again, endogenous to political contestation, which is shaped by the institutional environment--can have measurable and substantive effects on wealth accumulation and resultant inequalities and these, too, are shaped by self-interested politicians (just look at the difference between the Swedish and American system). Simple variations in the degree and composition of public spending, for instance, can have important effects on levels of inequality. European countries, which by and large have proportional representation systems, has been shown to be highly correlated with public goods expenditures--education, welfare, health, etc. I'm not sure inequality is completely independent of these social outcomes, which leads me to wonder what is lurking in your error term and whether they are fully independent.
Marshall,
Of course an important consideration is whether one is talking about pre-tax/redistribution inequality or post. With respect to pre, this would suggest simply that all the differences in labor and other factor market structures between US and EU (strength of labor unions, min wages, etc.) don't really matter that much for observed inequality. But implied positive relationship between mean income and inequality would also matter when looking at post redistribution incomes - basically you'd have to see a relationship more positively sloped than the one implied by the simulations in order to be able to say that the re distributive policies make a difference. In fact, like I said before, you do sort of see a more positive relationship, but this still implies that policy differences account for a lot less variation than is commonly supposed.
And of course one can chafe at randomness, but the possibility that a particular result is random is generally the benchmark that we judge our theories (empirical or otherwise) by. Basically you can say "it's the policies that cause the difference" but then I'll reply with "no, this is exactly what we'd expect to see if this was just chance".
And I know that there's been a lot of work which claims to show that institutional variations contribute to variations in growth or inequality between US and EU. But just because you see a pattern in the data doesn't mean the variables are causally related. In fact that's sort of the point of these simulations - those studies estimate biased effects which are merely artifacts of chance.
And since these are simulations, my error terms are fine by construction (except for the usual fact that those random number generators aren't really random).
Notsneaky,
I meant to ask but I always forget (busy times!), could you upload the XLS for this? (It is Excel, isn't it?)
Your formula for mean is wrong. 0.5 sigma^2 should be part of the exponent.
There's a huge lag between the time you post and the time I get to seriously read what you're posting :-(
There is no denying that if the distribution is skewed the variance (measure of inequality) will always be correlated with the mean. So in its essence this post is correct.
But here are some alternative titles for this blog post that certainly fit the model (data points "drawn" from far out in the right tail) but somehow just don't seem to convey the same meaning:
"Inequality and the _______ Effect"
where ______ might be:
Mobuto Sese Seko
General Marcos
General Suharto
Carlos Slim*
Russian Oligarchs
Nigerian Generals
Saudi Royals
* == 2007 richest man in the world, Mexican owner of Telmex and other companies, and very effective government lobbyist.
You're right up to a point - this is why I was comparing US and EU, not US and Zaire.
And there is other questions. If Bill Gates "produced" at least a good portion of his income then both Gini and mean will go up in US. But if Mobutu merely stole his income from his people you'd see Gini go up and mean income stay the same or probably go down. This is of course true (maybe it could be even used as an indirect measure of "appropriation" and "rent seeking" in some economies) but it doesn't affect the point made in the post.
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The regression of Gini on income or on any factor derived from income is logically circular and thus a fatal fallacy.
The Gini for a uniform (flat or horizontal) distribution is zero percent. The Gini for a distribution of a sole individual with a positive value among many individuals (a point or vertical distribution) is 100 percent.
QUESTION: What is the Gini for a Normal (mu, sigma) or Logarithm Normal (mu, sigma) distribution? Is it invariant? Or, does it vary with the mean and standard deviation?
Thanks for any clarification you may provide.
Well, they're just two different ways of summarizing a distribution of income so it's not circular. But if both the Gini and the per capita income are calculated using the same sample then the sampling is not random (one of the variables you calculate will not be based on a random sample, since the probability that the sample distribution for one variable is the same as the sample distribution for the other is 1). Basically you need to separate datasets to calculate the Gini and the per capita income.
For the log normal distribution the Gini is given by
G=2*Pi(s/(2^.5))-1 where s is the standard deviation and Pi(.) is the normal distribution.
The gini for a uniform distribution is not zero, since you still have income inequality (there's just the same amount of people at each level of income). So if y(i)=a+bi for i from iLO to iHI you can calculate the Gini from that and maybe simplify it to put it in terms of mean and variance.
For the normal distribution I don't know off the top of my head though it might be straightforward.
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what I always admired about Bill Gates (besides the millions of dollars on his account) is that he is always worried about social causes. Great character!
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