income distribution: calculation of Gini coefficient and Lorenz curve analysis

Income Distribution, Development & the Full Cambridge 9708 Syllabus

Learning Objectives

By the end of this lesson students will be able to:

• Explain and draw the Lorenz curve and calculate the Gini coefficient from raw data.
• Interpret Gini values for low‑, middle‑ and high‑income economies.
• Analyse how income inequality interacts with population dynamics, sectoral structure, trade patterns and global links.
• Link inequality analysis to every required unit of the Cambridge AS & A Level Economics (9708) syllabus.
• Identify which Assessment Objectives (AO1‑AO3) are addressed by each activity.

1. Syllabus Coverage at a Glance

Unit	Title (Cambridge 9708)	Key Content (bullet points)	Key Cambridge Concepts Applied	Assessment Objectives Targeted
1	Basic Economic Ideas	Scarcity, choice, opportunity cost Production possibility frontier Economic systems & role of markets	Scarcity, choice, progress	AO1, AO2
2	Micro‑economics – Price System	Demand & supply, elasticity Market equilibrium, surplus/shortage Consumer & producer surplus	Equilibrium, marginal decision‑making	AO1, AO2, AO3
3	Government Intervention – Micro	Price controls, taxes & subsidies Market failure: externalities, public goods, information asymmetry Cost‑benefit analysis	Efficiency, role of government, marginal decision‑making	AO1, AO2, AO3
4	Macroeconomics – Aggregate Demand & Supply	AD/AS model, short‑run vs. long‑run Inflation, unemployment, output gap Multiplier effects	Equilibrium, time, progress	AO1, AO2, AO3
5	Macroeconomic Policy	Fiscal policy – government spending & taxation Monetary policy – interest rates, money supply Supply‑side policies	Role of government, marginal decision‑making, efficiency	AO1, AO2, AO3
6	International Economic Issues (AS)	Trade theory – comparative advantage Balance of payments, exchange rates Protectionism vs. free trade	Progress, time, role of government	AO1, AO2, AO3
7	Advanced Micro – Theory of Consumer Behaviour	Utility, marginal utility, indifference curves Budget constraint, consumer equilibrium	Marginal decision‑making, scarcity	AO1, AO2
8	Advanced Micro – Market Failure & Labour Market	Externalities, public goods, information problems Labour demand & supply, wage determination	Efficiency, role of government, equity	AO1, AO2, AO3
9	Advanced Micro – Market Structures	Perfect competition, monopoly, monopolistic competition, oligopoly Pricing & output decisions, welfare analysis	Equilibrium, efficiency, marginal decision‑making	AO1, AO2, AO3
10	Advanced Macro – Policy Inter‑relationships	Interaction of fiscal & monetary policy Supply‑side vs. demand‑side measures Stagflation, policy lags	Time, marginal decision‑making, role of government	AO1, AO2, AO3
11	International Issues – Development & Global Links (A‑level)	Development indicators, structural transformation Income distribution – Lorenz curve & Gini coefficient Foreign aid, FDI, debt, migration, globalisation	Progress, time, role of government, equity	AO1, AO2, AO3

2. Income Distribution – Lorenz Curve & Gini Coefficient

2.1 The Lorenz Curve

The Lorenz curve shows the cumulative share of national income (L(p)) earned by the cumulative share of the population (p), ordered from the poorest to the richest.

• The 45° line (line of perfect equality) represents a situation where each percentile of the population receives the same share of total income.
• Because the population is ordered from low to high income, the curve always lies on or below the line of perfect equality.
• The larger the vertical distance between the two lines, the greater the inequality.

2.2 Why Order Matters

Ordering from poorest to richest ensures that the cumulative income share at each point truly reflects the proportion of total income held by the bottom p % of the population. If the order were random, the curve would no longer represent a distribution function and the area‑based Gini measure would be meaningless.

2.3 The Gini Coefficient

The Gini coefficient (G) measures the area between the line of perfect equality and the Lorenz curve.

\( G = \dfrac{A}{A+B} \)

where:

• A = area between the 45° line and the Lorenz curve.
• B = area under the Lorenz curve.

The total area under the line of perfect equality is 0.5 (a right‑angled triangle with base = height = 1). Substituting A = 0.5 − B gives the widely‑used shortcut:

\( G = 1 - 2B \;=\; 1 - 2\int_{0}^{1}L(p)\,dp \)

Values range from 0 (perfect equality) to 1 (perfect inequality).

2.4 Step‑by‑Step Calculation from Tabular Data

1. Rank households (or income groups) from lowest to highest income.
2. Calculate cumulative population share P_i and cumulative income share Y_i for each group (include a 0,0 start point).
3. Approximate the area under the Lorenz curve using the trapezoidal rule:

   \( B \approx \displaystyle\sum_{i=1}^{n}\frac{Y_{i}+Y_{i-1}}{2}\,(P_{i}-P_{i-1}) \)

4. Compute the Gini coefficient: G = 1 − 2B.

2.5 Numerical Example

Five income groups each represent 20 % of the population. Their income shares are shown below.

Group	Population Share (%)	Income Share (%)	Cumulative Population Pi	Cumulative Income Yi
1 (Poorest)	20	5	0.20	0.05
2	20	10	0.40	0.15
3	20	20	0.60	0.35
4	20	30	0.80	0.65
5 (Richest)	20	35	1.00	1.00

Applying the trapezoidal rule (with \(Y_0=P_0=0\)):

\( B = \frac{0+0.05}{2}(0.20-0) + \frac{0.05+0.15}{2}(0.40-0.20) + \frac{0.15+0.35}{2}(0.60-0.40) + \frac{0.35+0.65}{2}(0.80-0.60) + \frac{0.65+1.00}{2}(1.00-0.80) = 0.340 \)

Hence,

\( G = 1 - 2B = 1 - 2(0.340) = 0.320 \)

A Gini coefficient of 0.32 indicates a moderate level of income inequality.

3. Development Context – How Inequality Relates to Other Syllabus Themes

3.1 Population & Demographic Profile (Unit 11.4)

• Population growth rate – High rates (common in low‑income countries) can stretch public services, increasing the gap between rich and poor.
• Age structure – A large youth cohort may raise unemployment, expanding the lower tail of the Lorenz curve.
• Urbanisation – Rural‑to‑urban migration creates informal‑sector jobs with low pay, temporarily widening inequality.
• International migration – Remittances lift incomes of poorer households, potentially lowering the Gini.

3.2 Economic Structure & Sectoral Employment (Unit 11.4)

Development Level	Primary (%)	Secondary (%)	Tertiary (%)	Typical Gini Range
Low‑income	≈55	≈30	≈15	0.45 – 0.55
Middle‑income	≈30	≈35	≈35	0.30 – 0.55
High‑income	≈5	≈25	≈70	0.25 – 0.35

During structural transformation, wages in emerging sectors (secondary, then tertiary) often rise faster than in the lagging primary sector, creating a temporary surge in inequality before the benefits diffuse.

3.3 Trade Patterns (Unit 11.4)

• Export composition – Primary‑commodity exporters face volatile prices, which can cause income swings for producers at the bottom of the distribution.
• Import of capital goods – Enables industrialisation, raising secondary‑sector wages and widening the upper tail of the Lorenz curve.
• Trade openness – Can reduce inequality by creating jobs in export‑oriented industries, but may increase it if gains accrue mainly to capital owners.

3.4 Global Links (Unit 11.5)

• International aid – Funds education, health and cash‑transfer programmes that lift the poorest, lowering the Gini.
• Foreign Direct Investment (FDI) – Raises demand for skilled labour, widening the top end of the Lorenz curve unless accompanied by training and inclusive policies.
• External debt – Heavy debt service may force cuts to social spending, pushing the Gini upward.
• Globalisation
  • “Race to the bottom” – competition can depress low‑skill wages.
  • Skill‑biased technological change – raises returns to education, widening the income gap.
• Migration – Remittances boost poorer households (lowering Gini); brain‑drain removes high‑skill workers (potentially raising Gini).

3.5 Linking Inequality to Other Development Indicators

• Human Development Index (HDI) – Generally, lower Gini values accompany higher HDI, though outliers (e.g., the United States) exist.
• Structural transformation – Early stages raise inequality; later stages, when wages rise across all sectors, tend to reduce it.
• Institutional quality – Secure property rights, transparent governance and effective redistribution lower the Gini.
• Social cohesion – High inequality can fuel unrest, undermining political stability and long‑term growth.

4. Comparative Gini Data (Illustrative)

Country	Income Level (World Bank)	Gini Coefficient (latest)	Key Development Features	External Links (Aid / FDI / Debt)
Niger	Low‑income	0.43	Predominantly agrarian, low HDI, limited education.	High aid receipt; negligible FDI; external debt ≈ 30 % GNI
India	Middle‑income	0.35	Rapid industrialisation, large informal sector, expanding social programmes.	Moderate aid; growing FDI (≈ 2 % GDP); external debt ≈ 20 % GNI
Brazil	Upper‑middle‑income	0.53	High GDP per capita but pronounced regional gaps; recent redistributive policies.	Low aid; substantial FDI; external debt ≈ 30 % GNI
Germany	High‑income	0.29	Strong welfare state, progressive taxation, high HDI.	Minimal aid; high‑tech FDI (net inflow); external debt ≈ 70 % GNI
Sweden	High‑income	0.27	Extensive universal services, low poverty, very low inequality.	Negligible aid; selective FDI; external debt ≈ 35 % GNI

5. Summary Checklist for Students (AO1‑AO3)

• Draw and label a Lorenz curve, identifying the line of perfect equality.
• Calculate the Gini coefficient using both the area method and the trapezoidal approximation (AO1, AO2).
• Explain why a higher Gini indicates greater inequality and discuss possible impacts on economic growth, social cohesion and development (AO3).
• Compare Gini values of low, middle and high‑income countries and relate them to population trends, sectoral employment, trade patterns and global links (AO3).
• For each syllabus unit, indicate which key Cambridge concepts are being applied (e.g., marginal decision‑making in fiscal policy, equilibrium in AD/AS) (AO1‑AO3).

6. Suggested Classroom Activities (with AO Mapping)

1. Data‑driven Lorenz construction (AO1, AO2, AO3) – Provide a fictitious country’s household‑income dataset. Students must:
   • Rank the data, compute cumulative shares, plot the Lorenz curve.
   • Calculate the Gini coefficient using the trapezoidal rule.
   • Interpret the result in relation to the country’s stage of development.
2. Policy‑impact debate (AO3) – Split the class into “government” and “market” teams. Each team evaluates a policy (e.g., progressive tax, universal basic income, minimum wage) on its likely effect on the Lorenz curve and Gini coefficient, linking the discussion to the relevant key concepts.
3. Country case‑study carousel (AO2, AO3) – Stations with data on Niger, India, Brazil, Germany and Sweden. Students rotate, fill a comparative matrix (population growth, sectoral shares, trade openness, Gini) and present one policy recommendation per country.
4. Cross‑unit mind‑map (AO1, AO2) – Students create a visual map that connects the Lorenz‑curve analysis (Unit 11) to earlier units (e.g., how fiscal policy (Unit 5) can shift the Lorenz curve, how labour‑market structures (Unit 8) affect income distribution).

7. Assessment‑Objective (AO) Quick‑Reference Table

Activity	AO1 – Knowledge & Understanding	AO2 – Application & Analysis	AO3 – Evaluation
Lorenz‑curve drawing	✓	✓
Gini calculation (trapezoidal)	✓	✓
Policy‑impact debate		✓	✓
Country case‑study carousel	✓	✓	✓
Cross‑unit mind‑map	✓	✓	✓

8. Linking Back to the Rest of the Syllabus (Brief Recap)

Use the table in Section 1 as a reference point. Whenever you discuss a new topic (e.g., elasticity, AD/AS shifts, externalities), ask yourself:

• Which key Cambridge concept does this illustrate?
• How might this affect the distribution of income (i.e., the Lorenz curve)?
• Which assessment objective does the related activity target?

This systematic approach ensures full coverage of the 9708 syllabus while keeping inequality analysis central to the development narrative.