Unit 2 Unemployment, wages, and inequality: Supply-side policies and institutions
2.2 Measuring the economy: Inequality
- Gini coefficient
- A measure of inequality of a quantity such as income or wealth, varying from a value of zero (if there is no inequality) to one (if a single individual receives all of it). It is the average difference in, say, income between every pair of individuals in the population relative to the mean income, multiplied by one-half. Other than for small populations, a close approximation to the Gini coefficient can be calculated from a Lorenz curve diagram. See also: Lorenz curve.
In this section, we introduce tools for assessing the amount of inequality of income or wealth in an economy. One useful tool is the Gini coefficient, named after the Italian statistician Corrado Gini (1884–1965): a numerical indicator that lies between zero (no inequality) and one (extreme inequality). The Gini coefficient for income, for example, measures the average of the differences in income between every pair of people in the economy, relative to the average income. It is equal to zero if everyone’s income is the same, and equal to one if a single person receives all the income and everyone else has nothing.
Building block
The exact method for calculating the Gini coefficient is introduced and explained, with examples, in Section 5.12 of the microeconomics volume
- Lorenz curve
- A graphical representation of the inequality of some quantity such as income or wealth. Taking income as an example, individuals in the population are arranged in ascending order of income. First we calculate the total income of the population. Then for each level of income, we plot the percentage of total income held by people at this income level or lower, against the percentage of people at this income level or lower. The area between the Lorenz curve and the 45-degree line, expressed as a fraction of the total area below the 45-degree line, is a measure of inequality. Other than for small populations, it is a close approximation to the Gini coefficient. See also: Gini coefficient.
For a more complete picture of how income or wealth is distributed among the members of a society, we can use a visual tool called the Lorenz curve (invented in 1905 by the American economist Max Lorenz (1876–1959) while he was still a student). It is related to the Gini coefficient, but it allows us to distinguish, for example, the difference between an economy that is unequal because a few people are very rich, and another that is unequal mainly because a small section of the population is extremely poor.
The Lorenz curve shows the entire population lined up along the horizontal axis from the poorest to the richest. The height of the curve at each point indicates the fraction of total income (or wealth) received by the fraction of the population at the corresponding point on the horizontal axis.
We can use the Lorenz curve to illustrate the distribution of income or wealth in the whole population of a country, but also within small communities. Imagine a village in which there are 10 landowners, each owning 10 hectares, and 90 others who farm the land and pay a certain proportion of the grain they produce to the landlord, but who own no land. The Lorenz curve for land ownership is the blue line in Figure 2.2. Lining the population up in order of land ownership, the first 90% of the population owns nothing, so the curve is flat. The remaining 10% owns 10 hectares each, so the ‘curve’ rises in a straight line to reach the point where 100% of people own 100% of the land.
Figure 2.2 A Lorenz curve for wealth ownership.
If, instead, each member of the population owned one hectare of land—perfect equality in land ownership—then the Lorenz curve would be a line at a 45-degree angle, indicating that the ‘poorest’ 10% of the population have 10% of the land, and so on (although in this case, everyone is equally poor, and equally rich).
The Lorenz curve allows us to evaluate how far a distribution departs from this perfect equality line. A vivid example comes from applying it to two comparable organizations in the eighteenth century—pirate ships and naval vessels. The articles of a pirate ship called the Royal Rover (shown in Section 5.1 of the microeconomics volume) described exactly how the booty from a captured ship was to be distributed among the crew. Figure 2.3 shows the resulting distribution of income. The Lorenz curve is very close to the 45-degree equality line, showing how the institutions of piracy allowed ordinary members of the crew to claim a large share of the prize.
In contrast, when the British navy’s ships Favourite and Active captured the Spanish treasure ship La Hermione, the division of the spoils on the two British men-of-war ships was far less equal. The Lorenz curves show that ordinary crew members received about a quarter of the income, with the remainder going to a small number of officers and the captain. The Favourite was more unequal than the Active, with a lower share going to each crew member. By the standards of the day, pirates were unusually democratic and fair-minded in their dealings with each other.
Figure 2.3 The distribution of spoils: pirates and the British navy.
Peter T. Leeson and R. Beatson. 1804. ‘Invisible Hook: The Law and Economics of Pirate Tolerance’. In Naval and Military Memoirs of Great Britain, from 1727 to 1783 (vol. 3). Longman, Hurst, Rees and Orme.
Using the Lorenz curve to measure the Gini coefficient
The Lorenz curves in Figure 2.3 show that more unequal distributions have a greater area between the Lorenz curve and the 45-degree equality line. It turns out that this area gives us a close approximation to the Gini coefficient, calculated as the ratio of the area to the area of the whole of the triangle under the 45-degree line.
For example, we can calculate the approximate Gini for land ownership in Figure 2.4a as area A, between the Lorenz curve and the perfect equality line, as a proportion of area (A + B), the triangle under the 45-degree line:
\[\text{Gini} = \frac{\text{A}}{\text{A + B}}\]You can easily calculate the areas yourself to check that this gives a Gini coefficient of exactly 0.9. To calculate the Gini accurately, you need to find the average difference in wealth across all pairs of individuals in the population; this is a bit more fiddly to do, but it turns out to be 10/11 (0.9091 to four decimal places). In this example, the total population is 100; in general, the area method gives more accurate approximations for larger populations.
Figure 2.4a The Lorenz curve and Gini coefficient for wealth ownership.
Figure 2.4b shows the Gini coefficients calculated by the area method for each of the Lorenz curves we have drawn so far.
| Distribution | Gini |
|---|---|
| Pirate ship Royal Rover | 0.06 |
| British Navy ship Active | 0.59 |
| British Navy ship Favourite | 0.6 |
| The village with landowners and landless farmers | 0.9 |
Figure 2.4b Comparing Gini coefficients.
Market and disposable income distributions
- market income
- Market income is income before the payment of taxes or the receipt of transfers from the government; it includes earnings (wages and salaries from employment) as well as income from self-employment and from the ownership of assets (interest, rents, or dividends). See also: disposable income.
- disposable income
- A household’s disposable income is the maximum the household can spend (‘dispose of’) without borrowing or using savings, after paying tax and receiving transfers (such as unemployment insurance and pensions) from the government.
To assess income inequality within a country, we can either look at total market income (all earnings from employment, self-employment, savings, and investments), or disposable income, which better captures living standards. Disposable income is what a household can spend after paying tax and receiving transfers (such as unemployment benefit and pensions) from the government:
Figure 2.5 Market income and disposable income.
Figure 2.6 shows the Lorenz curve for the distribution of market income and disposable income across the population of the Netherlands in 2020.
With a large population, the Lorenz curve would be smooth if we had data for each individual. Here, we have plotted it using the level of income at each decile, so there are kinks at these points.
- decile
- A subset of observations, formed by ordering the full set of observations according to the values of a particular variable, and then splitting the set into ten equally-sized groups. For example, the 1st decile refers to the smallest 10% of values in a set of observations.
The Gini coefficient for market income is 0.40, so by this measure it has greater inequality than the Royal Rover, but less than the British navy ships. The analysis in Figure 2.6 shows how redistributive government policies result in a more equal distribution of disposable income than of market income.
Figure 2.6 Distribution of market and disposable income in the Netherlands (2020).
World Inequality Database. 2023.
The Lorenz curve for market income
The Gini for market income
Disposable income
The Gini for disposable income
In the Netherlands, almost 10% of the households have a near-zero market income. The extent of redistribution that allows households with very low market income to survive and possibly live comfortably is illustrated by the fact that the poorest one-fifth of the population with only 3.3% of market income receives about 7.1% of all disposable income.
Exercise 2.1 Inequalities among your classmates
- Record the heights of yourself and your classmates in an Excel spreadsheet. Now, make a table with percentiles (0–100, in increments of 1) in one column and the corresponding cumulative share of the population in another column. Use this table to draw a Lorenz curve for height. (For help on drawing a Lorenz curve in Excel, follow this tutorial).
- Using a Gini coefficient calculator, calculate the degree of inequality of height (in centimetres) among your classmates. Check that the Lorenz curve is similar to the one you drew in Question 1.
- Explain any differences between the Gini coefficient and Lorenz curve you found for height, those for the distribution of spoils in Figure 2.3, and those for market and disposable income in Figure 2.6.
Question 2.1 Choose the correct answer(s)
In a typical Lorenz curve diagram, A represents the area enclosed by the perfect equality line and the Lorenz curve, and B represents the area underneath the Lorenz curve (as in Figure 2.6). Read the following statements about the Lorenz curve diagram for income and choose the correct option(s).
- If area A increases, then inequality (as measured by the Gini coefficient) rises.
- The proportion in the numerator should be the area A, not the area B.
- Countries with lower Gini coefficients have lower inequality (by this measure), hence a more equal income distribution.
- The coefficient takes the value zero when all have the same income (the Lorenz curve is on the perfect equality line).
Extension 2.2 Owners, workers, and the profit share
In subsequent sections, we use the Lorenz curve and Gini coefficient alongside the WS–PS model so that we can analyse employment, unemployment, and income inequality together. In this extension, we use a simple example to illustrate how the level of the profit share in the economy affects inequality.
Using simple algebra, we obtain a formula in which the Gini coefficient for income depends on the proportions of owners and workers, and the profit share, \(\sigma\).
Think about a population of 100 in which a fraction n are workers, and the others (1 – n) are owners of firms employing the workers. Output per worker is \(λ\); of this, the employer gets \(σλ\), and the worker’s income is \((1 \ – \ σ)λ\).
There are \(100n\) workers, so the total income of the \(100(1 \ – \ n)\) owners is \(100n × σλ\), and each owner gets \(σnλ/(1 \ – \ n)\).
Figure E2.1 presents the Lorenz curve and the perfect equality line. To calculate the Gini coefficient, we have divided the areas below the 45-degree line into triangles and a rectangle.
Figure E2.1 Calculating the Gini coefficient using the Lorenz curve diagram.
The slope of the line separating area A from B1 is \((1 \ – \ σ)/n\) (the fraction of total output that each worker receives in wages), and the slope of the line separating area A from B3 is \(σ/(1 \ – \ n)\), the fraction of total output that each owner receives. We can approximate the Gini coefficient by the expression A/(A + B), where in the figure, B = B1 + B2 + B3.
Note that the area of the entire square is 1, while the area (A + B) under the perfect equality line is \(½\). The area A is ½ – B. Then we can write the Gini coefficient as
\[g = \frac{0.5 - (\text{B}_1 + \text{B}_2 + \text{B}_3)}{0.5} = 1 - 2(\text{B}_1 + \text{B}_2 + \text{B}_3)\]The figure shows that
\[\begin{align*} \text{B}_1 &= \frac{n(1-\sigma)}{2} \\ \text{B}_2 &= (1-n)(1-\sigma) \\ \text{B}_3 &= \frac{(1-n)\sigma}{2} \end{align*}\]So,
\[\begin{align*} g &= \frac{0.5-( \text{B}_1+ \text{B}_2+ \text{B}_3)}{0.5}=1-2( \text{B}_1+ \text{B}_2+ \text{B}_3)\\ & = 1 - 2\left(\frac{n(1-\sigma)}{2} + (1-n)(1-\sigma) + \frac{(1-n)\sigma}{2}\right) \\ &= \sigma - (1-n) \end{align*}\]This means that the Gini coefficient in this simple example is just the profit share minus the fraction of the total population who are owners.
Inequality will increase in this model economy if:
- the fraction of owners in the economy shrinks but the share \(σ\) of output they receive remains unchanged; this would be the case if ownership in the economy became concentrated among fewer owners or
- the wage share falls, that is, the profit share, \(\sigma\), increases.
Exercise E2.1 Building a Gini coefficient calculator
Using the Gini coefficient formula given in this extension, build a simulator (in a software of your choice) that can do the following:
- Calculate the Gini coefficient if you are given the fraction of workers in the economy (\(n\)) and the profit share of employers (\(\sigma\)).
- Draw the corresponding Lorenz curve as in Figure E2.1 (without the shaded areas).
