StudySpace: Economics in Development, 6th Edition

Worked Example: The Lorenz Curve and the Gini Ratio

Table 6–1 presents data on the size distribution of income in Brazil and Hungary (both for 1989). The contrasts are unusually stark because Brazil is a world leader in inequality, whereas Hungary has one of the world’s most equal income distributions.

Click here for Table 6-1

The Lorenz curve and Gini concentration ratio for Brazil are derived here. The corresponding derivations for Hungary are deferred to Exercise 1. The first step in constructing a Lorenz curve is to calculate the income share accruing to any given cumulative percentage of households. Taking data on income shares, starting with Brazil’s poorest households, one finds

The poorest 20 percent receive 2.1 percent of total income. The poorest 40 percent receive 2.1 + 4.9 = 7.0 percent of total income. The poorest 60 percent receive 7.0 + 8.9 = 15.9 percent of total income. The poorest 80 percent receive 15.9 + 16.8 = 32.7 percent of total income. All 100 percent of the households receive 100.0 percent of total income.

The Lorenz curve is simply a graph showing these data points. The horizontal axis measures the cumulative percentage of recipient units (households here) covered, and the vertical axis shows the corresponding share of total income earned by those households. Figure 6 –2 shows the Lorenz curve drawn from the Brazilian data.

How can one calculate the Gini concentration ratio? Just apply the formula Gini = A/(A + B), where B is the area under the Lorenz curve, and A is the area between the Lorenz curve and the diagonal line. As a first step, convert all the percentages into decimal units. In other words, consider 40 percent as 0.40. In this way the values on both axes range from 0.00 (that is, 0 percent) to 1.00 (that is, 100 percent). The box in which the Lorenz curve is drawn is thus a unit square, which always has an area of 1.00 (length times width = 1 × 1 = 1). The area under the diagonal must be half this.

Thus, A + B = 0.50.

Area A can be computed geometrically using familiar formulas for the areas of rectangles and triangles, but the calculation is a bit messy. In this case, area AA works out as 0.285. (It is easiest to calculate area B and then take advantage of the equation A = 0.5 – B to compute area A.). Then one can calculate the Gini ratio for Brazil from the formula
Gini = A/(A + B) = 0.285/0.50 = 0.57.

Exercises

1. It is your turn to derive a Lorenz curve and calculate a Gini concentration ratio.

a. Use the data on income shares in Hungary from Table 6–1 to calculate the cumulative income shares. Start the cumulative calculation with the poorest households.

20 percent of the households receive percent of total income.
40 percent of the households receive percent of total income.
60 percent of the households receive percent of total income.
80 percent of the households receive percent of total income.
100 percent of the households receive percent of total income.

b.  Plot the points corresponding to these five data observations on Figure 6–2, and draw the Lorenz curve for Hungary.

c. The area under the Lorenz curve for Hungary can be calculated by marking off rectangles and triangles and then applying formulas from geometry. It is B = 0.39. Knowing the area under the Lorenz curve, find the value of area A and the value of the Gini ratio.

Area A = .
Gini concentration ratio = .

d. Study the Lorenz curve for Hungary in comparison with the one for Brazil. What do the curves reveal about the size distribution of income in these two countries.

e.Compare Gini ratios for Hungary and Brazil. Do the Gini ratios reflect the same differences that you observed in the Lorenz curves?

2. This exercise illustrates some of the problems encountered in measuring and interpreting income distribution statistics. The subject is Pauvritania, a very small country that uses the rupee as the national currency (as in India, Pakistan, and Nepal).

a. Pauvritania contains five households. A recent field census produced the following data on household incomes:

Household A income = 500 rupees.
Household B income = 700 rupees.
Household C income = 900 rupees.
Household D income = 1,100 rupees.
Household E income = 3,000 rupees.
Total income, all households = 6,200 rupees.

(i) What share of total income goes to the following households in Pauvritania?

Poorest 20 percent: percent of total income
Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income

(ii) The government's national income accounts are derived from a wide variety of regular data reports on production, sales, earnings, and trade. According to these accounts, income in Pauvritania totals Rs8,000, which is nearly 30 percent higher than the total reported by the field census. What do you suppose is the reason for this difference?

b. The textbook says that the best ranking criterion is household income per capita. Let's see what difference this makes. The census data indicate that there are two people in household A, three people in B, four people in C, five people in D, and six people in E.

(i) Knowing income and the number of people in each household, you can calculate income per capita for each household:

Family A: per capita income = rupees.
Family B: per capita income = rupees.
Family C: per capita income = rupees.
Family D: per capita income = rupees.
Family E: per capita income = rupees.

(ii) When the households are ranked by per capita income,

Household is poorest.
Household is second poorest.
Household is richest.

(iii) Has the ranking changed now that we are evaluating household income on a per capita basis?

c. The poverty line in Pauvritania is 250 rupees per person.

(i) Which households have incomes at or below the poverty line?

Households .

(ii) How many individuals are living in poverty? (Assume that each household member lives in poverty if the household's per capita income is at or below the poverty line.)

individuals.

(iii) What percentage of the population lives in poverty?

% of the population.

(iv) A World Bank economist has just completed a study of poverty in Pauvritania. She concludes that the government has mismeasured the poverty line by using out-of-date statistics on the cost of living. The revised poverty line is 225 rupees. With this revised figure, what percentage of the population lives in poverty?

% of the population.

d. The textbook says that "lifetime income" would be an even better measure of the income distribution. Let's consider how life-cycle income patterns affect the measurement of inequality. In each household, just one person is out earning income. The five household heads are absolutely identical in terms of lifetime earnings patterns and demographic behavior. Specifically, each household head marries at age 20 and faces the following life cycle: How does this information alter the meaning of the census data on income distribution and poverty in Pauvritania?

3. This exercise examines how economic development can produce an inverted-U pattern of changes in income inequality.

a. Indozania is a country with a fixed population of five people. In the days before development began, all five people were dreadfully poor subsistence farmers, each with a subsistence income of $100 per year. The five incomes were 100, 100, 100, 100, and 100. Each number represents the income of one person. Under these initial conditions, what share of total income accrued to the poorest 40 percent of the population? The richest 20 percent?

Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income

b. Indozania then started on the path of growth and structural change. One foreign-owned factory opened up in the city, employing one worker. To lure the worker away from her farm, the factory paid twice the subsistence income, or $200. The income of the farmers stayed the same. At this point in the development process the income levels of the five people were 100, 100, 100, 100, and 200. What share of total income accrued to the poorest 40 percent of the population? The richest 20 percent?

Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income

c. Later a second factory opened up creating a second job at a wage of $200. This brought income levels to 100, 100, 100, 200, and 200.

(i) At this stage of development, what share of total income accrued to the poorest 40 percent of the population? The richest 20 percent?

(ii) To make a long story short, however, eventually there were five factories in operation providing five jobs, at $200 per worker. This brought the income distribution to 200, 200, 200, 200, and 200. What share of total income accrued to the poorest 40 percent of the population? The richest 20 percent?

Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income
Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income

(iii) Review how income distribution changed during the course of economic development in Indozania. What happened to the income share of the poorest 40 percent? What happen to income inequality? What happened to the extent of poverty?

d. Let's add two details to make the model more realistic, at the cost of some extra complexity.

(i) First, let the factories be owned locally. One person is the capitalist. Suppose that each factory creates $600 of value added, of which the capitalist retains the residual after paying wages. With one factory operating, the incomes of the four workers plus the one capitalist are 100, 100, 100, 200, and 400. Note that the capitalist earns $600 – $200 from one factory. At this stage of development, what share of total income accrues to the poorest 40 percent of the population? The richest 20 percent?

Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income

(ii) Next, recall (from the two-sector growth model in Chapter 3) that rural incomes ultimately rise as workers move off the farms. When the economy reaches this turning point, the wage in industry also rises. Figure 4–3 shows this as a rise in the supply curve of labor to industry once the third worker is drawn out of agriculture.

FIGURE 6–3

With three factories operating, the wage in industry is $400 per worker; the hardy soul who remains in agriculture now earns $200. The five income levels are 200, 400, 400, 400, and 600. What share of total income accrues to the poorest 40 percent of the population? The richest 20 percent?

Poorest 40 percent: percent of total income
Richest 20 percent: percent of total income

(iii) Taking into account the income accruing to the capitalist as well as the rising wages once labor markets reach the turning point, explain how modern-sector growth generates an inverted-U pattern of income inequality in Indozania.

4. Advanced. This exercise tests Kuznets’s hypothesis that income inequality first rises and then falls during the process of economic development. The textbook presents an equation to show the empirical relationship between the Gini ratio and per capita GDP. The equation is repeated here.

Gini = –1.072 + 0.395 ln Y – 0.026 (ln Y )2. [6–1]

Logarithms (base e) are used to convert per capita GDP into proportional differences. This is a statistical regression equation based on cross-section data for 65 countries. The regression equation traces the curve that most closely fits the data points. In this case, a quadratic term is introduced to generate a curvilinear relationship, since a simple linear equation cannot capture the pattern being tested. Numerous computer packages, including all the popular spreadsheets, can compute such regression equations.

a. Table 6–3 shows per capita income and the Gini ratio.

(i)  Fill in column 3 by computing for each country the natural logarithm of GDP per capita. Most hand calculators perform this computation easily.

(ii)  Fill in column 4 with the square of the ln Y for each country.

(iii) Plug into Equation 4–1 these values for ln Y and (ln Y )2 for each country to obtain the predicted Gini ratio. Taking Ethiopia as an example,

Predicted Gini = –1.072 + 0.395(ln 340) – 0.026(ln 340)2 = 0.347. Record the results of your calculations in column 5.

b. Now examine the regression relationship graphically.

(i)  Plot the predicted Gini ratio for each country in Figure 6–6 and connect the points to form a graph of the regression equation.

(ii)  Describe the basic shape of this curve. Is it consistent with Kuznets's inverted-U hypothesis? Explain.

FIGURE 6–6

c. The text reports that R2 = 0.15 for the regression equation. This means that differences in ln Y explain just 15 percent of the observed variations in the Gini ratio across the sample. There is lots of unexplained variance in the data, so other country-specific factors strongly influence the degree of income inequality.

(i)  Plot the actual data point for the Gini ratio and per capita GDP for each country in Table 6–3.

(ii)  How well do the actual data points conform to the predicted values?

(iii) Which countries have Gini ratios that are close to the predicted value, say, within plus or minus 0.05 of the predicted Gini? (Refer to your answers to Table 6–3 for exact figures.)

For which three countries are the actual Gini ratios farthest from the predicted value on the high side? On the low side? (Refer to your answers to Table 6–3 for exact figures.)

 

---

 

Fill out the information and press the button to Print.

Your Name:
Professor's Name:
Class Name: