Measuring poverty: head-count ratio, poverty gap and the FGT family

Detail

The basic set-up: the poverty line (z)

• A poverty line (z) is a cut-off level of income or consumption. A person whose income or consumption (y) is below z is counted as poor.
• Once z is fixed, there are two separate questions:
• Identification: who is poor? (y < z)

• Aggregation: how do we add up all the poor into one number? The measures below answer this.

• How India set z (Tendulkar and Rangarajan):

• The Planning Commission set up an Expert Group under Prof. Suresh D. Tendulkar in December 2005. It reported in December 2009 [2].
• Tendulkar used a Mixed Reference Period (MRP) Poverty Line Basket. MRP means different recall periods for different items: 365 days for rarely bought items and 30 days for the rest. The basket was set to match an urban poverty ratio of 25.7% [2].
• Head-count results for 2004-05 under Tendulkar: rural 41.8%, urban 25.7%, all-India 37.2%, which is 407.6 million (40.76 crore) people [2].
• In June 2012 the Planning Commission set up an Expert Group under Dr. C. Rangarajan. It reported on 30 June 2014 [4].
• The Rangarajan poverty line was a monthly per capita expenditure of ₹1,407 (urban) and ₹972 (rural) [4].

Head-count ratio (HCR): how many are poor

• Head-count ratio (the share of people who are poor): H = q / n
• q = number of people below the poverty line

• n = total population

• It measures incidence (how widespread poverty is).

• Strengths:
• It is simple and easy to explain ("1 in 5 Indians is poor").

• It is the official Indian headline measure and the World Bank's headline measure.

• India's trend (Tendulkar method): the poverty ratio fell from 45.3% (1993-94) to 37.2% (2004-05) and then to 21.9% (2011-12) [3].

Why HCR alone misleads

• It ignores depth. A person ₹1 below the line counts the same as a person ₹500 below it.
• It ignores distribution among the poor. It cannot tell whether the poor are all close together or spread far apart.
• It breaks Sen's monotonicity axiom.
• Axiom (a basic rule a good measure should follow): if a poor person becomes poorer, measured poverty should go up.

• HCR does not change, because the person was already counted as poor.

• It breaks Sen's transfer axiom.

• Axiom: if income moves from a poorer person to a less-poor person, measured poverty should go up.

• Under HCR, poverty may stay the same. It may even fall, if the person who receives the money crosses the line.

• Perverse incentive (a reward for the wrong action):

• A government is judged on HCR → it helps people just below the line, because it is cheapest to push them over → HCR falls fast → the poorest are left behind.

• Ratio vs number. The ratio can fall while the absolute number of poor goes up, if population grows fast enough.

• Example: population 100 crore, HCR 30% → 30 crore poor.
• Later: population 125 crore, HCR 25% → 31.25 crore poor.
• The ratio fell, but 1.25 crore more people are poor. Always check both figures.

Poverty gap index (PGI): how deep is poverty

• Poverty gap index = the average shortfall of the poor from the line, as a share of the line. The non-poor count as a zero shortfall.

• PGI = (1/n) Σ (z − yᵢ)/z, summed over the poor only.

• It measures depth (how far below the line the poor are).

• Useful link: PGI = H × I. Here I is the income-gap ratio, which is the average gap of the poor alone, as a share of z.
• Cost of ending poverty: PGI × z × n = the minimum money needed to lift every poor person exactly to the line. This assumes perfect targeting, meaning every rupee reaches the right person and none leaks.

• Real schemes have leakages and admin costs, so the true cost is higher.

• PGI respects monotonicity: if a poor person gets poorer, PGI goes up.

• Limit: PGI does not care who among the poor gets the income. Moving ₹10 from the poorest person to a less-poor person who stays poor leaves PGI unchanged. So PGI does not fully respect the transfer axiom.

Squared poverty gap (SPG): how severe is poverty

• Squared poverty gap index: each person's gap ratio is squared before averaging.

• SPG = (1/n) Σ [(z − yᵢ)/z]²

• Squaring gives more weight to big gaps. A gap of 0.4 becomes 0.16, while a gap of 0.1 becomes only 0.01. So the poorest count far more.

• The World Bank calls it the poverty severity index (P₂). It is a weighted sum of poverty gaps, where the weights are the gaps themselves. It accounts for inequality among the poor [5].
• It measures severity. It respects both monotonicity and the transfer axiom.

The Foster-Greer-Thorbecke (FGT, 1984) family

• One formula produces all three measures:

Pα = (1/n) Σ (i = 1 to q) [(z − yᵢ) / z]^α - z = poverty line; yᵢ = income of poor person i; q = number of poor; n = total population - α (alpha) = a "poverty-aversion" setting. The higher the α, the more weight goes to the poorest.

α	Measure	Captures	Question it answers
0	Head-count ratio	Incidence	How many are poor?
1	Poverty gap index	Depth	How far below the line are they?
2	Squared poverty gap	Severity	How unequal is it among the poor?

• When α = 0, every gap raised to power 0 equals 1. The formula then just counts the poor, giving q/n = HCR.
• The World Bank confirms that the headcount, poverty gap and squared poverty gap all belong to the FGT class [5].
• Decomposability: FGT measures can be split by group (state, caste, rural/urban). National poverty = the population-weighted sum of group poverty. This makes it easy to see which group contributes most.

Sen index (1976)

• S = H [I + (1 − I) Gₚ]
• H = head-count ratio (incidence)
• I = average income-gap ratio of the poor (depth)

• Gₚ = Gini coefficient among the poor (inequality among the poor; 0 = all poor have equal income, 1 = extreme inequality)

• It puts incidence, depth and inequality into one number.

• Limit: unlike FGT, it cannot be neatly split into groups.

Worked example (all measures)

Poverty line z = 100. Four people have incomes 60, 90, 120 and 150.

Person	Income	Poor?	Gap ratio (z − y)/z	Squared gap
1	60	Yes	0.40	0.16
2	90	Yes	0.10	0.01
3	120	No	0	0
4	150	No	0	0

• HCR = 2/4 = 50%
• PGI = (0.4 + 0.1)/4 = 0.125
• Check: I = (0.4 + 0.1)/2 = 0.25, so H × I = 0.5 × 0.25 = 0.125 ✓

• Cost of ending poverty = 0.125 × 100 × 4 = 50 (= 40 for person 1 + 10 for person 2)

• SPG = (0.16 + 0.01)/4 = 0.0425

• Sen index: Gₚ for incomes 60 and 90 = (2 × 30)/(2 × 2² × 75) = 0.1 (simple Gini formula)
• S = 0.5 × [0.25 + 0.75 × 0.1] = 0.1625

Test 1: monotonicity. Person 1 falls from 60 to 40.

• HCR stays at 50%, so it does not notice.
• PGI rises to (0.6 + 0.1)/4 = 0.175.
• SPG rises to (0.36 + 0.01)/4 = 0.0925.
• Lesson: only the gap measures notice that the poorest person got poorer.

Test 2: transfer axiom. Take 11 from person 1 (60 → 49) and give it to person 2 (90 → 101).

• HCR falls to 1/4 = 25%. "Poverty fell", yet the poorest person is now worse off.
• PGI = 0.51/4 = 0.1275, a small rise.
• SPG = 0.2601/4 = 0.065, a sharp rise, so it catches the harm clearly.

From one dimension to many: the MPI link

• Income measures miss deprivation in health, education and living standards.
• India's National Multidimensional Poverty Index (MPI) calls a person "MPI poor" if they are deprived in one-third or more of 12 weighted indicators [6].
• Like the Sen index, the MPI combines a head-count (incidence) with the intensity of deprivation among the poor. It is the same idea of "how many" plus "how badly".
• Multidimensional poverty fell from 29.17% (2013-14) to 11.28% (2022-23). 24.82 crore people escaped it in these 9 years [6].
• The biggest declines were in Uttar Pradesh (5.94 crore), Bihar (3.77 crore), Madhya Pradesh (2.30 crore) and Rajasthan (1.87 crore) [6].

Prelims Hooks

• FGT (1984): α = 0 → head-count ratio (incidence); α = 1 → poverty gap index (depth); α = 2 → squared poverty gap (severity).
• Squared poverty gap is the only FGT measure (of the three) that captures inequality among the poor. The World Bank calls it the "poverty severity index" [5].
• Head-count ratio violates both Sen's monotonicity and transfer axioms. The poverty gap index satisfies monotonicity but not the transfer axiom.
• PGI × poverty line × population = the minimum cost of ending poverty with perfect targeting.
• Sen index (1976): S = H[I + (1 − I)Gₚ]. It combines incidence, depth and the Gini among the poor.
• Trap: a falling poverty ratio does not mean the number of poor is falling. Population growth can push the absolute number up.
• Tendulkar Expert Group: set up December 2005, reported December 2009, used the Mixed Reference Period. All-India HCR in 2004-05 was 37.2% [2].
• Rangarajan Expert Group: set up June 2012, reported 30 June 2014. Poverty line was ₹1,407/month urban and ₹972/month rural [4].
• National MPI: a person is poor if deprived in ≥ 1/3 of 12 weighted indicators. MPI poverty was 11.28% in 2022-23 [6].

Mains Points

• Headline-measure bias in policy: India targets and reports on the HCR. This rewards schemes that lift the "near-poor" over the line.
• A shift to reporting PGI and SPG alongside HCR would direct welfare (PDS, MGNREGA, DBT) towards the poorest.

• This fits the Antyodaya idea of reaching the last person first.

• Budget planning: the poverty gap gives the minimum fiscal cost of ending poverty (PGI × z × n).

• The gap between this minimum and actual spending measures targeting inefficiency: leakages, exclusion errors and inclusion errors.

• This supports the case for Aadhaar-linked DBT and better beneficiary lists.

• Line-setting debates: HCR depends heavily on where z is drawn.

• The Tendulkar line (2009) and the Rangarajan line (2014) produced different poverty counts [2][4].

• This affects how many people are entitled to schemes and how central funds are shared among states. A mix of depth and multidimensional measures (MPI) reduces this dependence on one line [6].

• Growth vs distribution: HCR can fall through growth alone even while inequality among the poor rises. Severity measures (SPG, Sen index) show whether growth is truly inclusive.