On inequality, let’s do the Palma (because the Gini is so last century)

What better place than South Africa to run an inequality week on the blog? Today’s guest post from Alex Cobham (left) and Andy Sumner (right) summarizes their Andy Sumner mugnew paper on inequality – got a feeling this one might be quite important. Tomorrow, Brazil v South Africa. There’s one measure of inequality that gets all the attention – the Gini index. The Gini was developed in the early 1900s – in fact about 100 years ago – by Italian Statistician, Corrado Gini (see pic, looks like a real party animal). A century later our paper argues that it may be time for a rethink on measuring inequality. Why? [caption id="attachment_13985" align="alignleft" width="160" caption="it's no fun being a guru"]it's no fun being a guru[/caption] The Gini reflects the difference between the actual cumulative distribution of income, or anything else in a population, and perfect equality (the yellow area in the graphic). A Gini value of zero would mean that the distribution is completely equal and a Gini value of one would mean that one person had all the income and everyone else nothing (i.e. all of the green area would be yellow). Simple, eh? So, what’s the difference between a country A with a Gini of 0.4 and country B with a Gini of 0.45? We can say country B (0.45) is a bit less equal than country A (0.4). What we can’t is where that inequality exists. Is it a squeezed middle? Or is it at the poor’s end of the distribution? So if you’re a policy maker working for an incoming president elected on a mandate to address inequality and increase the share of income to the poor, the Gini won’t be a great deal of help. It’s also long been known thanks to inequality guru Tony Atkinson that the Gini is over-sensitive to changes in the middle of the distribution – and, as a consequence, insensitive to changes at the top and bottom. That’s a problem because we care most about what happens at the top and bottom in developing countries. So, we’ve just put out a new paper, predictably titled ‘Putting the Gini back in the Bottle?’, exploring an alternative measure for policy, which is sensitive to exactlygini area that. We’ve called it the ‘Palma’ as it is based on the research of Chilean economist, Gabriel Palma (below). When Palma started looking at the finer grain of inequality, rather than just the Gini, he made a startling observation (see Duncan’s take on it here). He found that the ‘middle classes’ – more accurately the middle income groups between the ‘rich’ and the ‘poor’ (defined as the five ‘middle’ deciles, 5 to 9) – tend to capture around half of GNI – Gross National Income wherever you live and whenever you look. The other half of national income is shared between the richest 10% and the poorest 40% but the share of those two groups varies considerably across countries. Palma suggested distributional politics is largely about the battle between the rich and poor for the other half of national income, and who the middle classes side with. So, we’ve given this idea a name – ‘the Palma’ (brilliant eh?) or the Palma Ratio. It’s defined as the ratio of the richest 10% of the population’s share of gross national income (GNI), divided by the poorest 40% of the population’s share. We think this might be a more policy-relevant indicator than the Gini, especially when it comes to poverty reduction. inequality cartoon 2 In the paper, we do a few things. First, we confirm the robustness of Palma’s main results over time: the remarkable stability of the middle class capture across countries, coupled with much greater variation in the 10/40 ratio. Second, we suggest that the Palma might be a better measure for policy makers to track as it is intuitively easier to understand for policy makers and citizens alike. For a given, high Palma value, it is clear what needs to change: to narrow the gap, by raising the share of national income of the poorest 40% and/or reducing the share of the top 10%. Third, we also present some tentative but striking evidence of a link between countries’ Palma and their rates of progress on the major Millennium Development Goal (MDG) poverty targets. More work is needed, and there are all sorts of caveats, but the results indicate that countries that reduced their Palma exhibit mean rates of progress which, compared to countries with rising Palmas, are three times higher in reducing extreme poverty and hunger, twice as high in reducing the proportion of people lacking access to improved water sources, and a third higher in reducing under-five mortality. If that isn’t worth a closer look we don’t know whatGabriel Palma is. Of course not everyone is going to like our paper – we sent it around the great and the good of the inequality world and really got a ‘marmite effect’ –people love or hate it (Andy’s New Bottom Billion paper on poverty in middle income countries got much the same initial response). Who likes it? Without naming names, here are the main love/hate responses – stylized – for a few salient groupings (those who commented on the paper shouldn’t get too hung up on this table).

Love it Hate it
Inequality gurus and wonks Those who appreciate the point about communicability and policymaker accountability Those who feel the mathematical properties of an inequality measure are more important
Other wonks Those who feel tackling inequality (or at least, vertical inequality) is central to development Those who prioritise other aspects, e.g. the 0.7 target for aid
Economists More ‘political’ economists, philosophers More technical economists
We think there’s an important debate to be had on measuring inequality, especially if the post-2015 discussions take it into central account. So let’s start with a vote (see right), preferably after you’ve taken a look at our paper. Andy Sumner is co-director of the newly established, International Development Institute at King’s College London, that is seeking to study development from a different angle. Alex Cobham is a research fellow at the Center for Global Development in Europe and a member of the advisory group of the UN consultation on inequalities in post-2015.]]>

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Comments

17 Responses to “On inequality, let’s do the Palma (because the Gini is so last century)”
  1. Chris

    wow, those are some really striking statistics regarding “the Palma” and the MDGs, it would be definitely be interesting to see some more indepth research on this. If they follow this trend and the Palma gains traction it could provide a great motivator to put inequality high-up on the development agenda, both at the international and national level.

  2. This is great! In the water sector, in the post 2015 discussions on how to measure the reduction in inequalities in access to water and sanitation we did look at the Gini. A couple of experts got to the conclusion that it was not very helpful. The Palma does look promising – will look further into applying it.

  3. max

    really great blog. palma it is then.one question-how would palma also be used to measure wealth? as I understand it the focus on income has hugely under-represented inequality over the years. can we have a wealth palma?
    and also when can we start fighting inequality and not just measuring and talking about it?

  4. Alex Cobham

    Thanks for comments and votes – more please!
    Chris, we are certainly pursuing the result. Watch this space…
    Catarina, as we discuss a little in the paper, part of our thinking comes from exploring the set of group inequality measures that seem likely to be part of the post-2015 framework (e.g. on gender and ethnolinguistic group), and seeing some value in having a comparable measure for economic inequality. There’s a little discussion of that in the synthesis report of the post-2015 inequalities consultation report too – but I know your sector has done a good deal of thinking about measurement issues already, in fact I’ve drawn on a bit of it for a DHS-based paper that’s in the works.
    Max, no reason not to use this for wealth – am also trying that in the aforementioned DHS bit of work. Though of course we need much better data on wealth, it makes the income distribution data look quite good which is no mean feat.
    As to when we can start fighting inequality, not just measuring and talking about it: our hope is that the intuitive simplicity of the Palma can make this whole area resonate much more readily than the Gini with citizens and policymakers alike. Bring it on!

  5. On a more philosophical note, I’m a little bit concerned that inequality is becoming so in vogue.
    What about opportunity? It’s a bit more sketchy, but doesn’t it trump inequality, justice-wise?
    Take a busker and a banker – no reason they shouldn’t have lots of inequalities, right? If they both went to Eton, and had a similar enough menu of life choices – good; if the busker wants to be a banker (or vice-versa), not so good.
    The problem comes when you start to wish your past menu choices were different, but can’t retread (path dependence). For me, for example, dropping maths means I’m unlikely to get an Economics PhD. Some people are never given the opportunity to be musical (a much harsher deprivation, in my view).

  6. gawain kripke

    In general, i’m sold on la palma. wondering a couple things:
    1. did the World Bank in setting it’s strategic goal on the growth of the bottom 40% go for la palma and then lose their nerve before they got to the top 10%. what gives? how do you interpret that?
    2. I wonder if the top 10% is actually a good enough selection. After all, in the US, it’s hardly the growth of the top 10% that’s driving inequality. Even the top 1% is probably too expansive. the big driver of inequality is the extraordinary, outrageous growth in the top 0.01%. NOt sure how applicable that is in other countries/contexts. But politically, it’s important. If new taxes and other confiscations are necessary, we want to do good and effective targeting. And, not trivially, we don’t want to make unnecessary enemies. So, how about a “super Palma” = income share of 1%/40%? How different would that be? Would it have any analytical or other value?

  7. Alex Cobham

    Thanks Gawain.
    1. Ha! Not sure; Adam Wagstaff points out that the leaked documents are only drafts and can change, so I’m hoping that means the Palma might yet feature.
    blogs.worldbank.org/developmenttalk/should-inequality-be-reflected-in-the-new-international-development-goals
    2. I like it – although the data challenges are significant. It’s probably worth mentioning that some defenders of the Gini have criticised the Palma precisely because the data on the top end of the distribution is bad. (As if it somehow speaks in favour of the Gini that it is insensitive to a badly measured bit of the distribution.)
    There isn’t the same stability of the omitted bit of the distribution as the Palma seems to have, which makes it less attractive. But on the other had, we could call it ‘extreme Palma’, or Palmax…

  8. One thing that is really cool about ratios is that you can combine them and compare them. Perhaps we need not consider this a dichotomous choice.
    In particular, because they are both generated from the same data (income distributions), it would not be difficult to track them side by side. Then one could determine which one is more interesting to a given topic at hand using statistical analysis to form a theory about which type of inequality is most pertinent.
    Although the normative assumptions about the Gini coefficient are important to consider, we might also consider whether it is of practical importance if it gives us information about where the Palma is going. If the Gini coefficient is more sensitive to changes in the middle, then we might ask what these changes indicate, and what this can tell us about other issues, rather than dismissing this as less useful than a different type of predictive power.
    Although your results for the Palma and MDGs are really interesting, I think that your note that “Given the findings of section 3, it follows that similar results are likely for the Gini
    also.” is more critical than the parentheses imply. The question seems less either/or and more, which does a better job predicting what?
    It might be that different inequality distributions have different importance in different situations, just like we measure both blood pressure and heart rate, and these are different in men and women.
    Congratulations on this excellent paper and thanks for your work. I hope that the development community can take a balanced approach to putting another tool in the toolkit of poverty analysis and reduction.

  9. Mark

    Hi Duncan, some thoughts
    1. Income is presumably easier to collect data on than wealth. But in places like SA and Kenya, where the wealth (real estate etc) has historically been in the hands of a tiny minority (an elite, in fact – rather different characteristics in the two countries), and where there has been massive asset inflation, the share of a nation’s wealth that stays in the hands of this tiny minority merits attention (not least because it is not earned)
    2. I agree that, from a communications point of view, the Palma is much more compelling than the Gini. But I still think the focus on relative income shares (ie proportions), as opposed to share of income in absolute terms, misses an important part of the story.
    From the tables it looks as though the top 10%’s share of income in Kenya declined from about 50% to about 40% over the 20 years, where the bottom 40%’s share increased from about 10% to about 13% (sorry, can’t see the exact figures).
    Let’s assume Kenya’s national income was 100 in 1990 and grew at 3% compound for the next 20 years (recent growth has been faster, of course), it would now be 180. The top guys’ share would have gone up from 50 to 72 (+22), whereas the bottom guys’ share would have gone up from 10 to 23 (+13). Big deal, one might say, in the context. I guess this speaks to the issue that the paper raised about the difficulty of determining what an acceptable level of inequality is, or, for that matter, what we should consider to be an acceptable rate at which inequality declines.

  10. Michael Ossipoff

    The Palma is undeniably better than the Gini. The Gini is riddled with unnecessary faults.
    If the Palma is the current competitor to the Gini, then immediately switch to the Palma. Plenty of time later to consider further refinements and improvements.
    The Gini is a good measure of inequality.
    …until you compare it to any one of the other measures of inequality.
    Regarding the Gini, the good news is that there’s a way in which it could make some (but not much) sense as a sum of weighted departures from the mean. The bad news is that it doesn’t, because its response to above-mean incomes makes no sense. (So disregard the above-mean income, or do include them via a separate Gini calculation that responds to them in keeping with the stated purpose of inequality-measures.
    What’s wrong with the simple, natural and obvious CV?
    …or just the bottom 10%’s percentage of its equal-share?

  11. Michael Ossipoff

    The Palma proposition’s astonishing accuracy makes the Palma Index an inequality-index that should be reported. …one that tells something meaningful. It seems to me that the Palma Proposition sheds some light on the power-dynamics of what’s going on in countries’ income-structures.
    .
    But not by itself. I’d add Theil-L &/or Theil-T (preferably the latter), with their definitions broadly-interpreted in the versions that I propose. I’ll get back to that a little bit farther below in this note.
    .
    In the bottom 40%. The people between the 39th percentile and the 40th percentile could take everything away from everyone below the 39th percentile, and the Palma wouldn’t notice.
    .
    The Theil-L inequality-index is the logarithm of the geometric-mean of the ratio between each income and the mean income. …i.e. the ratio of mean income divided by income i.
    .
    Why the geometric-mean instead of the arithmetical-mean?
    .
    Because we’re averaging ratios, (mean income)/(income # i). That ratio is the multiplicative-factor relating the two incomes. To fully & genuinely meaningfully average multiplicative-factors, they should be multiplicatively-averaged. …by multiplying all N of them together, and taking the Nth root of the resultl (…i.e. finding the number such at N factors of itself gives a number equal to the product of those N numbers).
    .
    If one of the ratios increases by a certain factor, then the product increases by that same factor. The geometric-mean averages in terms of, and measures by, what those ratios are…multiplicative-factors.
    .
    Why the logarithm?
    .
    Multiplying many numbers together can be most easily accomplished by adding together the logarithms of those numbers. …and then taking the antilogarithm. That’s what logarithms were invented and introduced for. That method was probably introduced in the 1600s.
    .
    The common, or Briggs, logarithm consists of the power of 10 that that number is. The Briggs logarithm of 10 is 1. The log of 100 is 2. The log of 1000 is 3. …etc. The log of 1 is 0, because there are no muliplications by 10.
    .
    With Theil-L, if money is treansferred between two incomes, the smaller income’s income-ratio changes more than that of the larger income. Thus the transfer-principle is satisfied by Theil.
    .
    The Theil-T inequality-index uses the same income-ratio, but with each term weighted by the value of that income-ratio, so that, e.g., if an income-ratio is equal to 2, then the overall value of the index is affected as if there were two of that person.
    .
    As I define them, either of those two Theil-Indices uses the ratio between the mean income, and a particular person’s income. …that ratio consisting of either one of those incomes divided by the other. i.e. an income-ratio written with either income on top, divided by the other.
    .
    And to get one Theil number, I’d only apply the method on one side of the mean income. Separate applications of Theil, reporting separate numbers, could be used for above-mean and below-mean regions.
    .
    Below the mean, I’d evaluate Theil from the bottom, up to the 10th percentile, the 40th percentile, the 50th percentile, and the mean income.
    .
    Above the mean, it could be evaluated for the top 10%, the top 1 %, and the top .01 %.
    .
    Another difference in how I’d suggesting using Theil is: For Theil-L, instead of the logarithm of the geometric mean of the income-ratio, I’d report the geometric mean itself.
    .
    Likewise, I’d convert the Theil-T result to a geometric mean of all the income-ratios over the persons in the income-range looked-at. …including the effectively-added-persons resulting from the weighting.
    .
    …a reasonable weighting if it is judged that the importance of each income-ratio is proportional to the value of that ratio…a ratio between a particular income and the mean-income.
    .
    Those Theil inequality indices look at all incomes in any income-range to which they’re applied, and meet the transfer-principle everywhere.
    .
    …and measure inequality based on what we all know is the important thing: The ratio between an income and the value it would have with complete equality (the mean income).
    .
    The Gini doesn’t look at that. It judges by absolute changes in incomes. Taking $1000 away from a working-poor person isn’t the same as taking $1000 away from a millionaire. …but the Gini doesn’t recognize that.
    .
    The Theil looks at ratios, and it doesn’t miss anything at any part of the income-range over which it is evaluated.
    .
    Details of Theil-T, when modified to be a geometric-mean:
    .
    Where Ii is the particular income # i, and Iav is the arithmetical-mean income:
    .
    …and where i = 1 and i = N are the endpoints of whatever income-range over which the Theil geometric-mean is being reported:
    .
    Sum, over i = 1 to N, of (Iav/Ii)log(Iav/Ii)
    .
    …that sum divided by:
    .
    Sum, over 1 = 1 to N, of (Iav/Ii).
    .
    The index is the antilog of the result of the above.
    ————–
    Writen all together, as a single formula:
    .
    The Theil version that I propose reports the following:
    .
    Antilog{[Sum, over i = 1 to N, of ((Iav/Ii)log(Iav/Ii))]/[Sum, over i = 1 to N, of (Iav/Ii)]}
    —————
    The Self-Weighted Geometric Mean of (Iav/Ii).

  12. Michael Ossipoff

    My vote in the Inequality-Measures poll:
    .
    Agreed, the Gini has got to go. There are much better inequality-indices.
    .
    The Palma-Index should be reported, because of the astonishing accuracy of the Palma-Proposition. The Palma tells the general account of how income is shared between the two variable classes, the top & bottom, classes.
    .
    But, because inequality does its worst damage and harm at the very bottom, then, at the least, the income of the bottom 10%, divided by what it would be at equality (i.e., divided by the arithmetical-mean income) should be reported too.
    .
    Those are the two essential numbers that should be reported: The Palma, and the bottom 10%’s share, as a fraction of what its equal share would be.
    .
    But, ideally, I’d like more:
    .
    Because the Palma-Propositions shows that the bottom 40% is the large underclass, then I’d like a direct report of their share, divided what it would be under equality (…in other words,divided by the arithmetical-mean income).
    .
    Likewise for the bottom 50%, because the poorest half are significant. Likewise for the population-segment below the arithmetical-mean income, because they’re the overall population-segment who are getting less than their equal-share.
    .
    But, especially for the large population-segments, such as the bottom 40%, the bottom 50%, and the people below the arithmetical-mean income, I’d prefer a measure that tells something about the distribution *within* those groups. …a self-weighted-average of each person’s income divided by the arithmetical-mean income.
    .
    I’d use self-weighting. Each person’s income-ratio (income/average-income) would be self-weighted, meaning that if a person’s income-ratio is N, then the index would be affected as if there were N of that person.
    .
    For the purpose of the averaging, all N of those copies of that person would be counted among the total number of people, so that the result would remain a genuine average.
    .
    Using that self-weighting, I’d report the average income in the population-segment of interest (such as bottosm 10%, bottom 40%, bottom 50%, and people below the arithmetical-mean).
    .
    For that average, because ratios are being averaged, it would be best to use the geometric-mean instead of the arithmetical-mean.
    .
    That self-weighted geometric-mean of income-ratio is a form of the Theil-T index.
    .
    I’d apply it to the bottom 10%, bottom 40%, bottom 50%, and to the people below the arithmetical-mean income.
    .
    Of course the arithmetical-mean could be used instead of the geometric-mean. It wouldn’t be quite as meaningful for averaging ratios, but it would okay. It could be argued that people would accept it better because of its greatrer simplicity. But, not only is the geometric-mean more meaningful, but it also has more precedent: What I’ve described above, the self-weighted geometric-mean of the income-ratio, is a form of the popular Theil-T index. Because Theil-T is already in wide use, and has many proponents, that geometric-mean would likely get better acceptance than an arithmetical-mean.
    .
    I’d like to answer the argument that the Theil is too complicated or wonky for acceptance. No, not in this form. When you tell someone that this number is the average income-ratio (people’s received-percent of their equal-share) over the bottom 10 or 40 or 50 percent of the population, or over the below-mean incomes…weighted to emphasize what happens at lower-incomes, they’ll know that means, and they’ll know why it matters.
    .
    If we agree that the importance, and deservance of notice, of an income-ratio is equal to the factor by which it differs from unity, then the self-weighted geometric mean of the income-ratio gives a number that tells the genuine merit of the distribution, with everyone’s income-ratio weighted by its importance. …and has precedent as a form of Theil-T.
    .
    So my vote:
    .
    The Palma, along with the bottom 10%’s collective income divided by what its collective equal share would be.
    Or, preferably, additionally include the bottom 40%, the bottom 50%, and the population-segment with income below the arithmetical-mean income.
    .
    And, preferably report the self-weighted geometric-mean of the individual income-ratios within each of those population-segments, instead of just the collective income-ratio of each of those population-segments, to give a measure of how people are treated *throughout* each of those population-segments, for the overall-merit of that part of the distribution.
    ——————
    Of course, if desired, something similar could be reported for the top 1%, the top .1%, and the top .01%.

  13. Michael Ossipoff

    Can I say one more thing, one clarification? In calculating the average of the logarithms of the individual income-ratios, the usual Theil-T index divides the sum of those weighted logarithms by the number of people.
    .
    That means that, for each person, if that person’s income-ratio differs from unity by a factor F, and the log of F is weighted by multiplying it by F, and the sum of the logs is divided by the number of people, then the Theil is treating that person as one person, F times agrieved.
    .
    I averaged differently. I divided the sum of the weighted logarithms by the sum of all the weights, instead of by the number of people.
    .
    That means that, for each person, if that person’s income-ratio differs from unity by a factor of F, I treated that person as if there were F of that person. …counting that person as F people, each with the income ratio that that person has.
    .
    I felt that my way of doing it makes it more validly an average. But I realize that either way is a valid average. There’s nothing wrong with the Theil way of doing it, and, in fact, it lets the weighting more dramatically affecf the index’s value.
    .
    In fact, a Theilist could say that my way of averaging waters-down the strength of the weighting.
    .
    Partly for that reason, but largely because it’s better to not unnecessarily change an existing inequality-index, I hereby change my proposal to average in the usual Theil manner…dividing the sum of the weighted logarithms by the number of people, instead of by the sum of the weightings.
    .
    For one thing, it’s simpler.
    .
    So now I can say that the only difference between my version of Theil-T, and the usual one, is that I report the self-weighted geometric mean, itself, rather than its logarithm.
    .
    By the way, when showing or publishing the Theil-T numbers, in the version that I propose (whether currently or previously), one needn’t bring up the matter of logarithms, or even say “geometric-mean”. One need only refer to it as an average of each person’s received percentage of their equal share, that average being calculated over some specified population-range (such as the bottom 10%, 49%, 50%, etc.), and weighted to preferentially represent the income-ratios differing from untity by the largest factors.
    .
    I should add that, as I understand it, there might be one other difference between my proposal and the usual Theil-T: I would never report just one number to aggregate the inequality over the entire population. I’d only apply and report the index separately over the following population-ranges:
    .
    Bottom 10%
    .
    Bottosm 40%
    .
    Bottom 50%
    .
    The people below the arithmetic mean income
    .
    The top 1%
    .
    The top .1%
    .
    The top .01%
    .
    Michael Ossipoff

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