Measuring poverty Measuring inequality
Advanced Economic Development – Recitation, Week #6
Poverty and Inequality 1. Measuring poverty and inequality
Claire Palandri
February 22, 2021
1 / 15 Measuring poverty Measuring inequality Measuring poverty
From the midterm review guide:
1 Measuring poverty ∗ Given data on income for a small population, you should be able to calculate the poverty measures P0, P1, and P2. 2 Measuring inequality ∗ Given data on share of total income for a number of income groups, you should be able to calculate Kuznets ratios. ∗ Given data on income for a number of income groups, you should be able to draw the Lorenz curve and calculate the Gini coefficient.
2 / 15 Measuring poverty Measuring inequality Outline
1 Measuring poverty
2 Measuring inequality
3 / 15 Measuring poverty Measuring inequality Measuring poverty
Drawing the poverty profile of a population: income vs. cumulative % of population, ranked by increasing income
p = poverty line (∼ arbitrary)
yi = household i’s income ⇒ (p − yi ) = “poverty gap” for yi < p n = population (nb of households) Pn 1 q = i=1 [yi < p] = nb of households below p
Poverty measures
– Does not capture the severity of poverty, only Head count ratio P = q 0 n counts the poor... – Does not capture the inequality among the poor 1 Pq p−yi Poverty gap index P1 = n i=1 p (considers only the total area between the income curve and p, not the shape of the curve)... Squared poverty severity index – Gives more weight to the wide gaps (considers (Foster-Greer- P = 1 Pq p−yi 2 2 n i=1 p inequality among the poor) Thorbecke measure of degree 2) 4 / 15 Measuring poverty Measuring inequality Measuring poverty
Drawing the poverty profile of a population: income vs. cumulative % of population, ranked by increasing income
p = poverty line (∼ arbitrary)
yi = household i’s income ⇒ (p − yi ) = “poverty gap” for yi < p n = population (nb of households) Pn 1 q = i=1 [yi < p] = nb of households below p
Poverty measures
– Does not capture the severity of poverty, only Head count ratio P = q 0 n counts the poor... – Does not capture the inequality among the poor 1 Pq p−yi Poverty gap index P1 = n i=1 p (considers only the total area between the income curve and p, not the shape of the curve)... Squared poverty severity index – Gives more weight to the wide gaps (considers (Foster-Greer- P = 1 Pq p−yi 2 2 n i=1 p inequality among the poor) Thorbecke measure of degree 2) 4 / 15 The headcount index does not take the intensity of poverty into account. Clearly there is greater poverty in country A, but the headcount index does not capture this. It does not indicate how poor the poor are, and hence does not change if people below the poverty line become poorer.
Measuring poverty Measuring inequality Measuring poverty
Compute P0, P1 and P2 for different income distributions (different countries):
1. Head count ratio P0
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Compute P0, P1 and P2 for different income distributions (different countries):
1. Head count ratio P0
The headcount index does not take the intensity of poverty into account. Clearly there is greater poverty in country A, but the headcount index does not capture this. It does not indicate how poor the poor are, and hence does not change if people below the poverty line become poorer.
5 / 15 Limitation: For countries A & B, the poverty gap rate is 0.10, but one could argue that country B has more serious poverty because it has an extremely poor member.
Measuring poverty Measuring inequality Measuring poverty
2. Poverty gap index P1 = mean proportionate poverty gap in the pop. Adds up the extent to which individuals on average fall below the poverty line, and expresses it as a % of the poverty line.
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2. Poverty gap index P1 = mean proportionate poverty gap in the pop. Adds up the extent to which individuals on average fall below the poverty line, and expresses it as a % of the poverty line.
Limitation: For countries A & B, the poverty gap rate is 0.10, but one could argue that country B has more serious poverty because it has an extremely poor member. 6 / 15 Measuring poverty Measuring inequality Measuring poverty
3. Foster-Greer-Thorbecke measure of degree 2 P2 or “Squared Poverty Gap Index” – It takes into account inequality among the poor, i.e. also the severity of poverty. – By squaring the poverty gap index, it implicitly puts more weight on observations that fall well below the poverty line.
7 / 15 Measuring poverty Measuring inequality Measuring poverty
3. Foster-Greer-Thorbecke measure of degree 2 P2 or “Squared Poverty Gap Index” – It takes into account inequality among the poor, i.e. also the severity of poverty. – By squaring the poverty gap index, it implicitly puts more weight on observations that fall well below the poverty line.
7 / 15 Measuring poverty Measuring inequality Outline
1 Measuring poverty
2 Measuring inequality
8 / 15 Measuring poverty Measuring inequality Measuring inequality
Given a poverty profile, i.e., data on income for several of income groups, you should be able to quantify the inequality in the population, by:
computing Kuznets ratios income share of top x % Kuznets ratio x/y = income share of bottom y %
drawing the Lorenz curve cumulative share of total income vs. cumulative share of population
computing the Gini coefficient A how unequal it is Gini = = A + B how unequal it could have gotten ∈ [0, 1] 0 = complete equality (everyone has the same income) 1 = complete inequality (a single person has all the income)
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Example of poverty profile (Family Income Distribution, US 1983: from Kerbo 2000)
Population Percentile Share of Total Income (%) 0-20 4.7 20-40 11.1 40-60 17.1 60-80 24.4 80-100 42.7 Total 100
1. Calculate the Kuznets 20/40 ratio in the economy above (vs what would it be for an economy with absolute income equality?) 2. Draw its Lorenz curve 3. Calculate the Gini coefficient
10 / 15 income share of top 20 % 42.7 Kuznets = = ' 2.7 20/40 income share of bottom 40 % (4.7 + 11.1)
20 In an economy with absolute income equality: Kuznets = = 0.5 20/40 40 That economy with absolute income equality is the reference to which the GINI coefficient is calculated. It has all Kuznets ratio m/m = 1.
Measuring poverty Measuring inequality Measuring inequality
Population Percentile Share of Total Income (%) 0-20 4.7 20-40 11.1 40-60 17.1 60-80 24.4 80-100 42.7 Total 100
1. Calculate the Kuznets 20/40 ratio in the economy above (vs what would it be for an economy with absolute income equality?)
11 / 15 Measuring poverty Measuring inequality Measuring inequality
Population Percentile Share of Total Income (%) 0-20 4.7 20-40 11.1 40-60 17.1 60-80 24.4 80-100 42.7 Total 100
1. Calculate the Kuznets 20/40 ratio in the economy above (vs what would it be for an economy with absolute income equality?)
income share of top 20 % 42.7 Kuznets = = ' 2.7 20/40 income share of bottom 40 % (4.7 + 11.1)
20 In an economy with absolute income equality: Kuznets = = 0.5 20/40 40 That economy with absolute income equality is the reference to which the GINI coefficient is calculated. It has all Kuznets ratio m/m = 1.
11 / 15 Measuring poverty Measuring inequality Measuring inequality
Pop Percentile Share of Total Income (%) 0-20 4.7 20-40 11.1 40-60 17.1 60-80 24.4 80-100 42.7
2. Draw its Lorenz curve 3. Calculate the Gini coefficient
12 / 15 Measuring poverty Measuring inequality Measuring inequality
Pop Percentile Share of Total Income (%) Cumul. Pop (%) Cumul. Income (%) 0-20 4.7 20 4.7 20-40 11.1 40 15.8 40-60 17.1 60 32.9 60-80 24.4 80 57.3 80-100 42.7 100 100
2. Draw its Lorenz curve
3. Calculate the Gini coefficient 13 / 15 Measuring poverty Measuring inequality Measuring inequality – Lorenz curve & Gini coefficient
Method to compute the areas A & B 1×1 A = 2 − B Estimate B using a series of trapezoids:
area of a trapezoid = area of lower rectangle + area of top triangle
(xj −xj−1)(yj −yj−1) = (xj − xj−1)yj−1 + 2 yj yj−1 = (xj − xj−1)(yj−1 + 2 − 2 ) 1 = 2 (xj − xj−1)(yj−1 + yj ) 1 = 2 (bin width)(yj−1 + yj )
J trapezoids J trapezoids X 1 1 X B = 2 × bin width × (yj−1 + yj ) = 2 × bin width × (yj−1 + yj ) j=1 j=1
14 / 15 1 2 × 0.2 × ([0 + 0.047] + [0.047 + 0.158] + ... + [0.573 + 1]) 0.2 = 2 (2 × 0.047 + 2 × 0.158 + 2 × 0.329 + 2 × 0.573 + 1) 0.2 = 0.2 (0.047 + 0.158 + 0.329 + 0.573) + 2 = 0.3214 1×1 1 A = 2 − B = 2 − 0.3214 = 0.1786
A 0.1786 Gini = = = 0.3572 A + B 0.5
Measuring poverty Measuring inequality Measuring inequality – Lorenz curve & Gini coefficient
Pop Percentile Share of Total Income (%) Cumul. Pop (%) Cumul. Income (%) 0-20 4.7 20 4.7 20-40 11.1 40 15.8 40-60 17.1 60 32.9 60-80 24.4 80 57.3 80-100 42.7 100 100 1 B = × bin width × PJ trapezoids(y + y ) 2 j=1 j−1 j =
15 / 15 1×1 1 2 − B = 2 − 0.3214 = 0.1786
A 0.1786 Gini = = = 0.3572 A + B 0.5
Measuring poverty Measuring inequality Measuring inequality – Lorenz curve & Gini coefficient
Pop Percentile Share of Total Income (%) Cumul. Pop (%) Cumul. Income (%) 0-20 4.7 20 4.7 20-40 11.1 40 15.8 40-60 17.1 60 32.9 60-80 24.4 80 57.3 80-100 42.7 100 100 1 B = × bin width × PJ trapezoids(y + y ) 2 j=1 j−1 j 1 = 2 × 0.2 × ([0 + 0.047] + [0.047 + 0.158] + ... + [0.573 + 1]) 0.2 = 2 (2 × 0.047 + 2 × 0.158 + 2 × 0.329 + 2 × 0.573 + 1) 0.2 = 0.2 (0.047 + 0.158 + 0.329 + 0.573) + 2 = 0.3214 A =
15 / 15 A 0.1786 Gini = = = 0.3572 A + B 0.5
Measuring poverty Measuring inequality Measuring inequality – Lorenz curve & Gini coefficient
Pop Percentile Share of Total Income (%) Cumul. Pop (%) Cumul. Income (%) 0-20 4.7 20 4.7 20-40 11.1 40 15.8 40-60 17.1 60 32.9 60-80 24.4 80 57.3 80-100 42.7 100 100 1 B = × bin width × PJ trapezoids(y + y ) 2 j=1 j−1 j 1 = 2 × 0.2 × ([0 + 0.047] + [0.047 + 0.158] + ... + [0.573 + 1]) 0.2 = 2 (2 × 0.047 + 2 × 0.158 + 2 × 0.329 + 2 × 0.573 + 1) 0.2 = 0.2 (0.047 + 0.158 + 0.329 + 0.573) + 2 = 0.3214 1×1 1 A = 2 − B = 2 − 0.3214 = 0.1786
15 / 15 A 0.1786 = = 0.3572 A + B 0.5
Measuring poverty Measuring inequality Measuring inequality – Lorenz curve & Gini coefficient
Pop Percentile Share of Total Income (%) Cumul. Pop (%) Cumul. Income (%) 0-20 4.7 20 4.7 20-40 11.1 40 15.8 40-60 17.1 60 32.9 60-80 24.4 80 57.3 80-100 42.7 100 100 1 B = × bin width × PJ trapezoids(y + y ) 2 j=1 j−1 j 1 = 2 × 0.2 × ([0 + 0.047] + [0.047 + 0.158] + ... + [0.573 + 1]) 0.2 = 2 (2 × 0.047 + 2 × 0.158 + 2 × 0.329 + 2 × 0.573 + 1) 0.2 = 0.2 (0.047 + 0.158 + 0.329 + 0.573) + 2 = 0.3214 1×1 1 A = 2 − B = 2 − 0.3214 = 0.1786
Gini =
15 / 15 Measuring poverty Measuring inequality Measuring inequality – Lorenz curve & Gini coefficient
Pop Percentile Share of Total Income (%) Cumul. Pop (%) Cumul. Income (%) 0-20 4.7 20 4.7 20-40 11.1 40 15.8 40-60 17.1 60 32.9 60-80 24.4 80 57.3 80-100 42.7 100 100 1 B = × bin width × PJ trapezoids(y + y ) 2 j=1 j−1 j 1 = 2 × 0.2 × ([0 + 0.047] + [0.047 + 0.158] + ... + [0.573 + 1]) 0.2 = 2 (2 × 0.047 + 2 × 0.158 + 2 × 0.329 + 2 × 0.573 + 1) 0.2 = 0.2 (0.047 + 0.158 + 0.329 + 0.573) + 2 = 0.3214 1×1 1 A = 2 − B = 2 − 0.3214 = 0.1786
A 0.1786 Gini = = = 0.3572 A + B 0.5
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