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The rich are different from us; so are the very poor. But in most countries most of us are in the middle. That leads to problems when you try to measure inequality. **Alex Cobham** and **Andy Sumner** have come up with a better way to show that some countries are more equal than others.

There are normative and there are instrumental reasons why inequality may be said to matter. Normative reasons are those that involve moral or value judgements: they include fairness and meritocracy. Most of us have a general feeling that fairness is a good thing, and that talented and hard-working people should not be kept down through having happened to be born in poverty. However, much global literature has taken an instrumentalist approach – that is, a practical as opposed to a moral one – as to why high or rising inequality can hinder development. As Nancy Birdsall, founder of the Centre for Global Development, has pointed out, income inequality in developing countries matters for at least three instrumental reasons: where markets are underdeveloped, inequality inhibits growth through economic mechanisms; where institutions of government are weak, inequality exacerbates the problem of creating and maintaining accountable government, increasing the probability of economic and social policies that inhibit growth and poverty reduction; and where social institutions are fragile, inequality further discourages the civic and social life that underpins the effective collective decision-making that is necessary to the functioning of healthy societies.

**How do you measure inequality? Not all ways are equal**

In fact, there is empirical research that high or rising national income inequality can have a negative effect on the rate of economic growth or the length of growth spells. Similarly, research has found that high or rising national income inequality is likely to be a drag on poverty reduction. So while it may be the case that growth (still) is good for the poor (or at least the poorest) in a general sense, growth is likely to be better for the poor in countries with lower initial income inequality or where income inequality is declining than in countries where the opposite is true.

There are various ways of measuring income inequality. The dominant one at present is the Gini. It was developed in the early 1900s – in fact about 100 years ago – by Italian statistician Corrado Gini. Another is the Theil, which we shall come to later. A century later we think that it may be time for a rethink on measuring inequality. In a new paperi we explore a different measure of income inequality or concentration. It is called the Palma.

The Palma is a particular specification within a family of inequality measures known as “inter-decile ratios”, of which the most commonly used is possibly the ratio of the bottom 20% to the top 20%, or its inverse. The Palma is the ratio of national income shares of the top 10% of households to the bottom 40%. If the richest 10% in a country earn between them half of the national income, and the poorest 40% earn one-tenth of the national income, the Palma ratio is 0.5 divided by 0.1, which is 5. It reflects an observation of Chilean economist José Gabriel Palma^{2} about the stability of the “middle” 50% share of income across countries. As Palma puts it: “A schoolteacher, a junior or mid-level civil servant, a young professional …, a skilled worker, middle-manager or a taxi driver who owns his or her own car, all tend to earn the same income across the world.” Such people between them, and in almost all countries, earn around 50% of their national income. And such people also almost always lie between the 40th and 90th deciles in their earning power – they are more than 40% of the way from the bottom and more than 10% away from the top. This has an important consequence. It means that differences in distribution of income are largely a question of the tails – of how the rest of the wealth is divided between the richest and the poorest.

We argue that as a measure of inequality the Palma has a number of advantages over the measure most commonly used, the Gini. Why?

The Gini (see box) reflects the difference between the actual cumulative distribution of income, or anything else in a population, and perfect equality (the yellow area in Figure 1). A Gini value of 0 would mean that the distribution is completely equal and a Gini value of 1 would mean that one person had all the income and everyone else nothing (i.e. all of the green area in Figure 1 would be yellow).

Simple, eh? So, what is the difference between country A with a Gini of 0.4 and country B with a Gini of 0.45? We can say country B is a bit less equal than country A. What we cannot say is where that inequality exists. Is it a squeezed middle – some rich, many poor, not many in between? Or is it at the poor man's end of the distribution – some rich, most doing OK, but an underclass of the very poor?

If you are 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 will not be a great deal of help.

It has also long been known, thanks to inequality guru Tony Atkinson (an economist who has an inequality measure of his own named after him), 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 is a problem because we care most about what happens at the top and bottom in developing countries. If teachers become slightly richer at the expense of taxi-drivers, or vice versa, it does not greatly matter; if a rich landowner becomes richer at the expense of a subsistence farmer whose family are already in want, it does.

So our working paper^{1}, for the Centre for Global Development in Washington DC, explored an alternative measure for policy, which is sensitive to exactly that.

#### The Gini index

Suppose we add up the income and the individual in sequence from the poorest to the richest in a country: so we add person 1, then person 2, and so on, and at the same time we add the income of person 1 to the income of person 2, and so on, until we have added up the whole group. As we go along we plot the results on a graph. The horizontal axis is the proportion of the population; the vertical axis is the proportion of national income (see Figure 1). The result is called the *Lorenz curve*. It illustrates for any point what percentage of income the poorest percentage of the population between them receive. In the example in Figure 1 the poorest 50% of the population own between them about 20% of the nation's wealth. The curve shows in a single line the extent to which income or wealth is distributed unequally in society or a group. If everyone in a nation had exactly the same income, the curve would be the diagonal straight line which forms the 45° line of equality on the graph.

The yellow area between the equality line and the Lorenz curve is the “area of inequality”. The greater the yellow area, the more inequality there is. This can give us a single number that measures inequality. It is called the Gini index, and it is the yellow area divided by the total area under the line of equality – that is, yellow/(yellow + green). Since it is a proportion, it ranges between 0 (perfect equality: the yellow area disappears) and 1 (perfect inequality: the rightmost person has all the income, and the whole triangle is yellow).

In practice, Gini numbers tend to range between 0.2 and 0.6. Sweden has a Gini index of 0.23, the UK is at 0.34, the USA is at 0.45, and South Africa, at 0.65, is the most unequal of all. Gini coefficients in most advanced and emerging economies have been increasing in the last thirty years, with some exceptions. The Gini index of the UK increased most in the 1980s; you could say that the Thatcher era was when the gap between rich and poor widened most quickly.

The Palma is based, as we said, on the research of Chilean economist, José Gabriel Palma. When Palma started looking at the finer grain of inequality, rather than just the Gini, he made a startling observation. 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 gross national income – and that holds good 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 in that battle.

So we have given this idea a name – the “Palma ratio” or just Palma. Its definition is nice and simple and easy to understand: as we said above, it is the income share of the top 10% divided by the income share of the poorest 40%. We think this might be a more policy-relevant indicator than the Gini, especially when it comes to poverty reduction.

**The people in the middle own half of a nation's wealth. Who owns the other half – the rich or the poor?**

In the paper^{1}, 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%.

Of course not everyone likes our paper. We sent it around the great and the good of the inequality world and really got a “Marmite effect” – people love it or hate it. Who likes it? Without naming names, Table 1 shows the main love/hate responses – stylised – for a few salient groupings. (Those who commented on the paper should not get too hung up on this table.)

Table 1. The Palma's strengths and weakesses, as shown by those who love it and those who hate it*Group* | *Love it* | *Hate it* |
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Inequality gurus and wonks | Those who appreciate the point about communicability and policy-maker 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.

The paper makes three main points. First, inequality measures for policy frameworks such as the UN's post-2015 Development Agenda must be considered on the basis of policy criteria, not only the common technical criteria – in other words, we need to think about whether measures are useful for policy processes, not only whether they exhibit certain mathematical properties.

A technically perfect measure which is unintelligible to most people (or which requires significant explanation) is highly unlikely to form the basis for policy-maker accountability. So we propose five policy axioms:

- That the value judgements of using this indicator are sufficiently explicit.
- That it is clear what signal is being given to policy-makers on the preferred direction of change of inequality (improving or worsening).
- That it is clear to a public (i.e. non-technical) audience what has changed and what it means.
- That the policy response is sufficiently clear to policy-makers (meaning how policies do or do not influence the indicator).
- That it is possible to capture horizontal (e.g. gender and ethno-linguistic group) as well as vertical inequality in the indicator.

So what? We think the Palma is much more meaningful to policy-makers and otherwise normal people than the Gini and Theil measures, which are both relatively obscure statistical constructs. The Theil, for example, measures inequality as the maximum possible entropy of the data minus the observed entropy – which is clear to those who love it, but not to the man or woman on the Clapham omnibus.

Second, income inequality is in the “tails” – the rich and the poor. That is, most of the difference when you compare countries, or the same country over time, is in what happens at the top and bottom of society rather than in the middle. For example, the middle 50% of the population have about half of national income in both Honduras and Morocco; but the poorest 40% in Morocco have more than twice the share of national income as in Honduras, and the top 10% have correspondingly less (see Table 2). Most people would consider Morocco the more equal country (see Figure 2).

Table 2. Gini and Palma ratios for selected countries. On both Gini and Palma scales a low score equates to more equality of income within the country. Note that South Africa ranks as the world's most unequal country on both scales. Some middle-income countries, such as Brazil and Honduras, are particularly unequal in their wealth distribution. There is more Palma inequality in the US than in the UK. The US is a more unequal country than India or Tanzania in its distribution of wealth*Country* | *Gini* | *Palma* |
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Bangladesh | 0.33 | 1.27 |

Brazil | 0.54 | 2.23 |

Denmark | 0.24 | 0.92 |

Honduras | 0.57 | 5.21 |

India | 0.34 | 1.39 |

Morocco | 0.41 | 1.96 |

Nepal | 0.32 | 1.30 |

Nigeria | 0.44 | 1.84 |

Russia | 0.42 | 1.88 |

South Africa | 0.63 | 7.05 |

Tanzania | 0.38 | 1.65 |

UK | 0.34 | 1.62 |

US | 0.45 | 1.85 |

In the paper we provide substantial new evidence for Palma's finding that the “middle” 50% of the population has a strikingly stable share of national income (around 50% in fact) – not only in countries at different income levels, but also in any given country over time, and in addition through the fiscal stages of taxation and transfers. This demonstrates very clearly that inequality is about how much the rich (the top 10%) and poorest (the bottom 40%) get, or what are known as the “tails”.

But the Gini is overly sensitive to the middle of the distribution, so it is not well equipped to address this type of inequality – while the Palma is designed to do just that.

Third, the Palma has been criticised for relying on only two points of the distribution, and thus ignoring too much information about inequality. We show that in regression analysis the same two points of the distribution (i.e. the income shares of the top 10% and bottom 40%) can perfectly explain the Gini also – so that in practice it contains no more information than does the Palma.

The difference is that while the Palma is the simple ratio of the two, the Gini has the following form:

Hardly intuitive to the person in the street, is it?

In practice, the Ginis that are used in much analysis contain no more information than the Palma, but the Palma is transparent about this, and we think intuitively clear – take what the rich get and divide it by what the poor get.

We are not trying to completely replace the Gini; we are just saying that the Palma has something to add that is understandable to more people. Our view is that multiple measures should be used to monitor income inequality for policy purposes – including not only the Palma but also, for example, the median income; and, if you must, the Gini. But if one measure alone is to be used, please do not let it be the Gini.

As many have made clear, it is important to recognise that no measure of inequality, including the Gini, is “neutral”: the best we can do is to be explicit about the value-based decision being taken in the choice of any given measure. This is certainly true of the Palma.

We have corroborated the surprising stability of “middle-class capture” across countries, and across time, while confirming much greater variation in the Palma ratio of the top 10% and bottom 40% income shares, and we have also found the Palma and the Gini to have a near-perfect fit – suggesting that much of the same information is captured by the two measures. Indeed, the components of the Palma ratio alone are able to “explain” between 99% and 100% of Gini variation. In practice, we find that no more information is contained in the Gini – a measure of the entire income distribution – than in the Palma ratio, which excludes completely the fifth to ninth deciles.

**No one measure should be used to monitor income inequality. But if one measure alone is used, please let it not be the Gini**

Further research will be needed to evaluate the extent to which this finding is driven by Palma's stylised fact of the “homogeneous middle” of the distribution, and to what extent the finding results from over-simplistic calculation methods used to generate the most widely used Gini series. Even simple rules of thumb based on a single point of the distribution seem able to predict the Gini with an accuracy approaching 100%. The same holds for the Palma, if the income share of the bottom 40% is used; but in this case the finding is by construction. In the case of the Gini, the results reveal a hitherto hidden lack of depth.

We would conclude that the robustness of Palma's thesis and the intuitive nature of the Palma ratio provide a strong case for further exploration of the Palma. We would argue that the Palma may be a better measure for policy-makers and citizens to track as it is intuitively easier for policy-makers and citizens to understand; that it is a more policy-relevant measure of inequality because, given the observed stability of the middle income deciles, it is clear what change is implied by a desire to change the Palma; and that it is explicit about the assumed preferences in regard to inequality.

An obvious criticism of the Palma is that it only considers half of the income distribution; for which reason we consider a measure of concentration rather than the full distribution. However, since it turns out that the Gini in practice does not capture any additional information, and moreover that it does this in an opaque and hitherto undiscovered way (as far as we are aware), we consider this an argument in favour of the Palma. Following a similar line of thought, it is worth noting that the Palma does directly expose the top decile somewhat – which in many countries may not be appreciated, by those in that decile at least – but it is the Palma's simplicity which may be its greatest strength. A Gini coefficient of 0.5 implies serious inequality but yields no intuitive statement for a non-technical audience. In contrast, the equivalent Palma of 5.0 can be directly translated into the statement that the richest 10% of the nation earn five times the income of the poorest 40%.