Income Distribution & Percentiles: Lorenz Curves, Gini Coefficient & Tourism Applications

TAKEAWAYS:

• Percentiles divide data into 100 equal parts (e.g., the 75th percentile is the income below which 75% of people earn).
• Lorenz curves visually compare income inequality by plotting cumulative % of income vs. % of households.
• The Gini coefficient (0 = perfect equality, 1 = perfect inequality) quantifies inequality using the Lorenz curve’s area.
• Income distribution tables (like those for hotel staff or tour guides) reveal disparities between high/low earners.
• Percentiles in tourism help set fair wages (e.g., "60% of guides earn between Rs. 40k–80k/month").
• Case studies (e.g., NEPSE stockholders vs. daily-wage laborers) show how these tools inform policy.

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1. Income Distribution: Definitions and Types

Income distribution describes how total income is spread across individuals or households in an economy. In tourism, this matters for:

• Hotel staff wages (front desk vs. housekeeping).
• Tour guide earnings (seasonal vs. year-round).
• Small business profits (homestays vs. luxury resorts).

Key Terms

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2. Percentiles: Splitting the Data

Percentiles divide data into 100 equal parts. For example:

• 25th percentile (Q1): 25% of people earn ≤ this value.
• 50th percentile (Median): Half earn ≤ this value.
• 75th percentile (Q3): 75% earn ≤ this value.

Worked Example: Tour Guide Earnings

Given income data for 100 guides (in Rs. thousands/month):

Income Range:  30–40 | 40–50 | 50–60 | 60–70 | 70–80 | 80–90 | 90–100
No. of Guides: 10   | 20   | 30   | 25   | 10   | 4    | 1

Find: a) Income of the richest 10% (90th percentile). b) Income range for the middle 60% (20th–80th percentiles).

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Solution:

1. Cumulative Frequency Table:

   Income Range	No. of Guides	Cumulative Frequency
   30–40	10	10
   40–50	20	30
   50–60	30	60
   60–70	25	85
   70–80	10	95
   80–90	4	99
   90–100	1	100

2. 90th Percentile (Richest 10%):

   • Locate the 90th position in cumulative frequency (90th percentile = 90th guide).
   • The 90th guide falls in the 80–90 range (since cumulative frequency reaches 99 at this range).
   • Answer: The richest 10% earn ≥ Rs. 80,000/month.

3. Middle 60% (20th–80th Percentiles):

   • 20th percentile: 20th guide → 40–50 range.
   • 80th percentile: 80th guide → 60–70 range.
   • Answer: Middle 60% earn Rs. 40,000–70,000/month.

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3. Lorenz Curve and Gini Coefficient

Lorenz Curve

Plots cumulative % of income vs. cumulative % of households. A 45° line (perfect equality) is the benchmark.

Key Observations:

• The curve bows away from the equality line → inequality exists.
• Area between the curve and equality line = Gini coefficient (scaled to 0–1).

Gini Coefficient Calculation

$$\text{Gini}=\frac{\text{Area between equality line and Lorenz curve}}{\text{Total area under equality line}}$$ Example: If the area between the curve and line is 0.25 (out of 0.5 total), then: $$\text{Gini}=\frac{0.25}{0.5}=0.5$$ Interpretation:

• 0 = Perfect equality (everyone earns the same).
• 1 = Perfect inequality (one person earns everything).
• 0.5 = High inequality (like Nepal’s tourism sector).

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4. Applications in Tourism and Business

Case Study 1: Wage Disparity in Hotels

Scenario: A 5-star hotel in Kathmandu pays:

• Housekeeping: Rs. 25,000/month.
• Managers: Rs. 150,000/month. Analysis:

1. Calculate percentiles for 100 staff:
   • Median (50th percentile): Rs. 40,000.
   • Top 10% (90th percentile): Rs. 120,000.
2. Gini Coefficient: If the Lorenz curve shows a steep bow, Gini > 0.4 → high inequality. Solution: Adjust wages to reduce disparity (e.g., raise housekeeping to Rs. 35,000).

Case Study 2: Seasonal Earnings of Trekking Guides

Data: Earnings in peak (Nov–May) vs. off-season (Jun–Oct).

• Off-season has higher inequality (Gini = 0.45 vs. 0.35).
• Policy: Offer subsidies or part-time work to stabilize incomes.

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5. Income Distribution vs. Percentiles vs. Gini

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6. Practical Calculation: Mean, Median, Mode, and Gini

Given:

• Mean income = Rs. 50,000.
• Variance = 250,000 (SD = Rs. 500).
• Coefficient of Variation (CV) = (SD/Mean) × 100 = (500/50,000) × 100 = 1%. Interpretation:
• Low CV (1%) → incomes are very similar (e.g., a small homestay business with uniform wages).

Worked Example: NEPSE Stockholders Data: Income of 50 NEPSE shareholders (in Rs. lakhs):

Income Range:  0–5 | 5–10 | 10–15 | 15–20 | 20–50
No. of People: 10  | 15   | 15    | 8     | 2

Find: a) Median income. b) Gini coefficient (assume Lorenz curve area = 0.3).

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Solution:

1. Median (25th Person):

   • Cumulative frequency reaches 25 at the 10–15 range.
   • Median income = Rs. 12.5 lakhs.

2. Gini Coefficient:

   • Area between Lorenz curve and equality line = 0.3.
   • Total area under equality line = 0.5.
   • Gini = 0.3 / 0.5 = 0.6 → High inequality (typical of stock markets).

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In the Real World

1. eSewa/Khalti (Digital Payments)

   • Idea Used: Income distribution analysis for merchant commissions.
   • How: eSewa tracks transaction volumes across merchants (e.g., small shops vs. large hotels). If 80% of transactions are <Rs. 5,000 but 20% are >Rs. 50,000, the Gini coefficient reveals high inequality in merchant earnings. This helps design tiered commission structures to support small businesses.

2. Pathao (Ride-Hailing)

   • Idea Used: Percentiles for driver pay.
   • How: Pathao’s algorithm sets fares based on demand. The 75th percentile fare (Rs. 250/trip) ensures most drivers earn enough, while the 90th percentile (Rs. 400/trip) caps high-earning outliers. Drivers in the bottom 20% (earning <Rs. 150/trip) get bonuses to reduce inequality.

3. NTC (Telecom Subsidies)

   • Idea Used: Lorenz curves for subsidy targeting.
   • How: NTC’s Rs. 200/month internet subsidy targets low-income users. By plotting cumulative % of users vs. cumulative % of data usage, NTC identifies that 60% of users consume <1GB/month (bottom 40% of income earners). This justifies targeted subsidies to bridge the digital divide.

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Exam Tip

1. Percentiles:

   • Always sort data and use cumulative frequency.
   • For grouped data, use the formula: $$P_k=L+\left(\frac{\frac{kN}{100}-C}{f}\right)\times h$$ Where:
     • $L$ = lower boundary of the percentile class.
     • $k$ = percentile (e.g., 75 for Q3).
     • $N$ = total frequency.
     • $C$ = cumulative frequency before the class.
     • $f$ = frequency of the class.
     • $h$ = class width.

2. Gini Coefficient:

   • If the Lorenz curve is given, estimate the area between it and the equality line.
   • Shortcut: If the curve is a straight line from (0,0) to (100,100), Gini = 0. For a steep bow, Gini > 0.4.

3. Real-World Links:

   • Tourism: Always relate to wage gaps (guides vs. managers) or seasonal income (trekking vs. off-season).
   • Business: Use NEPSE, Daraz seller data, or bank loan defaults for case studies.
   • Policy: Connect to subsidies (NTC), minimum wages (hotels), or tax brackets.

4. Common Mistakes:

   • Ignoring cumulative frequency in percentile calculations.
   • Misinterpreting Gini: Higher Gini = more inequality, not "better" distribution.
   • Forgetting units: Always label income in Rs./month or lakhs/year.

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