Chapter 6: Econoic Inequality We are interested in inequality ainly for two reasons: First, there are philosophical and ethical grounds for aversion to inequality per se. Second, even if we are not interested in inequality at an intrinsic level, we ay care about inequality at a functional level (i.e. inequality ay be iportant not for its own sake, but because it has an ipact other econoic features that you care about, such as growth). Question: How should we think about inequality at a conceptual level? We will focus on inequality of incoe or wealth, although, as we entioned before, inequality is intiately linked to concepts such as lifeties, personal capabilities and political freedos. We will focus on inequality of incoe or wealth, not because they represent all differences, but because they represent an iportant coponent of those differences. Short-ter vs. long-ter considerations: Inequality in current incoe ay be harless, both fro an ethical point of view and fro the point of view of their effects on the econoy. For exaple, suppose in society A that soe people earn $2000/onth while others earn $3000/onth and there is perfect iobility. In society B, soe earn $000/onth while others earn $4000/onth but people exchange jobs every onth. At any one point in tie, society A shows up as ore equal, but in society B average annual earnings are the sae for everyone. (Data probles.) Personal vs. functional distribution of incoe: How uch people earn vs. how they earn it (rent, interest, wage). Related to the ownership pattern of productive factors (land, capital, labor). Looking at incoe inequality ay ask serious differences, such as the well-being of an uneployed person who receives his incoe fro charity and a person who earns hiself the sae aount of incoe. Also, judge whether two iddle-class failies with the sae aount of incoe should be deeed equal (one receives rent incoe whereas the other receives wage incoe). What to use as an indicator of resources? Incoe vs. wealth vs. expenditures: If incoe is chosen as the indicator, it is usually the disposable incoe that is used. Disposable incoe is the incoe that households have available for spending and saving after incoe taxes have been accounted for. Wealth can be defined as net worth, which is the su of assets (including financial and nonfinancial assets inus the liabilities of the econoic unit. It is a suary easure of all accuulated resources at a given tie. Soe researchers think that incoe is an indirect easure of the standard of living and thus prefer to use expenditures as the indicator. Expenditures include current expenditures (purchases of goods and services, such as food, rent, utilities, entertainent and personal care) and capital expenditures (such as the repayent of the principal on own hoe, the purchase of investent properties, hoe iproveents and life insurance). Both incoe and expenditures are flow variables, whereas wealth is a stock variable. Question: How should we easure inequality?
Measuring Econoic Inequality Suppose that there are only two people. If they share a cake 50-50 then the distribution is equal, otherwise it is not. Furtherore, we can easily see that a 40-60 split is ore equal than a 30-70 split. However if there are ore than two people, then aking a judgent becoes hard. How do you copare a 20-30-50 division with a 22-22-56 division? We need a easure that will help us rank different distributions. Let (, I( y, yn easure, defined as a function of the incoe distribution. y be an incoe distribution and I ) be an inequality Question: What properties should a desirable inequality easure satisfy? Should we have weak or strict criteria? Four criteria for inequality easureent:. Anonyity principle (soeties called the syetry principle), which says that it does not atter who receives the incoe. Perutations of incoes aong people should not atter for inequality judgents. This eans that we can always arrange the incoe distribution of n people so that y y2... y n, where y is incoe. If the prie inister trades his incoe with one of yours, incoe inequality in Turkey should not change. 2. Population principle, which says that cloning the entire population and their incoes should not change inequality. According to this principle, population size does not atter. All that atters are the proportions of the population that earn different levels of incoe. For exaple, suppose that we arrange and categorize the incoes so that we can deterine the percentage of the population earning incoe in each category. Percentage of the Incoe range ($) population 0-200 8 20-400 30 40-600 33 60-800 25 80 + 4 Here, for inequality easureent purposes, we do not know (and do not care) how any people live in the country or who they are (the anonyity principle). All that we care about is the percentage of the population earning different levels of incoe. Forally, we can state this principle as I( y, I( y, yn; y, yn ) 2
3. Relative incoe principle, which says that only relative incoes, and not the absolute levels of these incoes atter. For exaple, an incoe distribution over two people of (500, 000) has the sae inequality as (2000, 4000), and regardless of the currency the incoes are denoinated in. This principle is actually not very easy to buy. Absolute incoes are iportant deterinants of overall econoic developent. Later in the course, we will ake up by focusing explicitly on absolute incoes!! With the relative incoe principle added, we do not need absolute incoe inforation. We can now define incoe categories in ters of quintiles (or deciles or percentiles, depending on data availability). Percentage of total Incoe Quintile incoe earned Poorest quintile 9 Second quintile 5 Third quintile 20 Fourth quintile 25 Richest quintile 3 Forally, we can state this principle as I( y, I( y,, for every positive nuber. 4. The Dalton principle (also known as The Pigou-Dalton transfer principle ): Let ( y, be an incoe distribution and consider two incoes yi and y i y j. A transfer of incoe fro the not richer individual to the not poorer individual will be called a regressive transfer. (The opposite of a regressive transfer is a progressive transfer.) The Dalton principle states that if one incoe distribution can be generated fro another via a sequence of regressive transfers, then the original distribution is ore equal than the other. y j, Forally, we say that I satisfies the Dalton principle if for every incoe distribution, I( y,..., yi,..., y j,..., I( y,..., yi,..., y j,...,, for every positive nuber. The Lorenz Curve: First we rank the econoic units according to incoe (or another easure of resources) fro the sallest to the largest. Then we plot the cuulative shares of units versus the cuulative share of these units in total incoe. In particular, the horizontal axis shows the cuulative percentages of the population arranged in increasing order of incoe. On the vertical axis, we show the percentage of total incoe received by a fraction of the population. The convex shaped curve thus drawn is called the Lorenz curve. For exaple if the (20%, 6%) point is on the Lorenz curve, then this eans the poorest 20% of the population receives 6% of total incoe. This also eans that the richest 80% of the population receives 94% of total incoe. 3
The greater the level of inequality, the farther the Lorenz curve will be fro the 45 degree line. In the figure, L2 lies to the right of L everywhere, so we would expect an inequality index to indicate greater inequality in the L2 case. Another way to see this is that, the poorest x% of the population will always have an equal or greater share of incoe under L than under L2, regardless of what x is. This is called the Lorenz criterion (or Lorenz doinance criterion). Forally, an inequality easure I is consistent with the Lorenz criterion, if for every pair of incoe distributions ( y, y 2,..., y n) and ( z, z 2,..., z ), I ( y, y2,..., y n ) I ( z, z 2,..., z ), whenever the Lorenz curve of ( y, y 2,..., y n) lies to the right of ( z, z 2,..., z ) everywhere. It turns out that an inequality easure is consistent with the Lorenz criterion if and only if it satisfies all of the four principles that we stated above: the anonyity, population, relative incoe and Dalton principles. Therefore, the Lorenz criterion captures our four criteria in one stateent (and in graphical for). If the Lorenz curves of two distributions do not cross, then all inequality easures that are consistent with the Lorenz criterion yield the sae ranking of these distributions. If two Lorenz curves cross, then inequality easures that are consistent with the Lorenz criterion ay yield different rankings. Exercise: Show on a graph that if an incoe distribution satisfies the Dalton principle (along with the other three principles), it is also consistent with the Lorenz criterion. (Hint: Iagine a regressive incoe transfer and show how it affects the Lorenz curve.) Question: What happens if Lorenz curves cross? Answer: In this case, the Lorenz criterion does not apply. When there is a Lorenz curve crossing, we cannot go fro one distribution to another via a sequence of regressive transfers; we ust have at least one progressive transfer. Exaple: Suppose that a society consists of four individuals with incoes (75, 25, 200, 600). Consider a second incoe distribution (25, 75, 400, 400). How can we travel fro the first to the second? 4
You will notice that the answer will include at least one regressive and one progressive transfer. In other words, we will not be able to copare the two distributions by using the four principles. What we need here is another principle that allows us to weigh the costs of a regressive transfer with the benefits of a progressive transfer. This requires a value judgent which akes it ipossible to quantify the trade-offs in a way that is approved by everyone! Coplete easures of inequality: Although Lorenz curve is a nice tool to visually exaine the degree of inequality, often there is a need to suarize the inforation in a nuber. A nuber is ore concrete and quantifiable than a picture. Moreover, this nuber can be used to ake a judgent when Lorenz curves cross. Let us suppose that there are n individuals and distinct incoes. In each incoe class j, the nuber of individuals earning that incoe is n j. Therefore, n n j. j Define the overall ean incoe as n j y j. n j Soe coplete easures of inequality that are often used are: Range: R ( y y) The Kuznets ratios: The ratio of the shares of incoe of the richest x% to the poorest y%. For exaple, the ratio of the incoe of the richest 0% to the incoe of the poorest 40% can tell us soething about distribution when ore detailed inforation is not available. The range and the Kuznets ratios are crude but nevertheless useful indicators of inequality when detailed data are not available. The ean absolute deviation: Express average deviation fro ean incoe as a fraction of total incoe. M n j y j n j Although it takes into account the entire incoe distribution (unlike the previous two), the ean absolute deviation is actually not a very good easure of inequality. It has one serious drawback; oreover it does not have a copensatory feature. The drawback is that it often does not satisfy the Dalton principle. To see this, take two incoes, both either above the ean or below the ean. A regressive transfer between the two (so that both incoes reain either above or below the ean, as before) will leave the ean absolute deviation unchanged. The coefficient of variation: It is the ratio of the standard deviation to the ean. C n j ( y j ) 2 n j 5
This easure satisfies all four properties, so it is Lorenz-consistent. The Gini coefficient: Very widely used in epirical work. It takes the absolute differences between all incoe pairs and sus the up. The su of differences is divided by n 2 (since there are n 2 such pairs), by the ean (so that the coefficient is sensitive to only relative changes in incoes and not to absolute changes), and by two (since each incoe pair appears twice). G n j n k y j y k 2n 2 j k This easure satisfies all four properties, so it is Lorenz-consistent. Check and ake sure! We also know that the Gini coefficient is the ratio of the area between the Lorenz curve and the 45-degree line to the area of the triangle below the 45-degree line. The last two easures of inequality, naely the coefficient of variation and the Gini coefficient are Lorenz-consistent. Therefore, both are satisfactory according to our criteria. Which one should we use? The crucial point is that both easures will give the sae ranking as long as Lorenz curves can be ranked, i.e. they do not cross. When the Lorenz curves cross, it is possible for the two easures to give different rankings. In such cases, we should rely on not one but a set of easures, and probably study the Lorenz curves as well. (Note: If there is no Lorenz crossing, then Gini and CV (and other easures that are Lorenzconsistent) agree. If Gini and CV do not agree, then there is Lorenz curve crossing. If Gini and CV agree, there ay or ay not be Lorenz curve crossing.) Country /date Gini Coefficient of variation Incoe share of the richest 5% (%) Incoe share of the poorest 40% (%) Puerto Rico 953 0.45.52 23.4 5.5 963 0.449.035 22.0 3.7 Argentina 953 0.42.62 27.2 8. 959 0.463.887 3.8 6.4 96 0.434.605 29.4 7.4 Mexico 950 0.526 2.500 40.0 4.3 957 0.55.652 37.0.3 963 0.543.380 28.8 0. Source: Fields (980). In the table above, we can see that in Puerto Rico, both the poorest and the richest lost incoe share. This is a clear sign of Lorenz crossing. Gini and CV disagree in this case, although a Lorenz curve crossing does not necessarily ean that Gini and CV disagree. In Argentina, in 953-6, Gini and CV do not agree, therefore there ust have been a Lorenz crossing, although the population shares do not show this to us. In Mexico, in 957-63, Gini and CV agree, but the changes in population shares indicate that there has been a Lorenz curve crossing. 6