Math  /  Data & Statistics

QuestionUsing data from a nation's census, an economist produced the following Lorenz curves for the distribution of that nation's income in 1962 and 1972. f(x)=12x+12x2 Lorenz curve for 1962g(x)=25x+35x2 Lorenz curve for 1972\begin{array}{ll} f(x)=\frac{1}{2} x+\frac{1}{2} x^{2} & \text { Lorenz curve for } 1962 \\ g(x)=\frac{2}{5} x+\frac{3}{5} x^{2} & \text { Lorenz curve for } 1972 \end{array}
Find the Gini index of income concentration for each Lorenz curve and interpret the results.
Identify the integrand for the computation of the Gini index for 1962 and 1972.
The Gini index for 1962 is given by 201(d2x2 \int_{0}^{1}\left(\square d^{2} x\right. and the Gini index for 1972 is given by 201dx2 \int_{0}^{1} \square d x.

Studdy Solution

STEP 1

What is this asking? We need to calculate the Gini index for income distribution in 1962 and 1972 using the given Lorenz curves and then explain what those numbers mean. Watch out! Don't mix up the Lorenz curve formulas!
Also, remember the Gini index formula uses the *difference* between the line of equality (xx) and the Lorenz curve.

STEP 2

1. Identify the integrands.
2. Calculate the Gini index for 1962.
3. Calculate the Gini index for 1972.

STEP 3

The Gini index measures income inequality.
A Gini index of **zero** represents perfect equality (everyone earns the same), while a Gini index of **one** represents perfect inequality (one person earns everything).

STEP 4

The Gini index is calculated as twice the area between the line of equality (xx) and the Lorenz curve.
The line of equality represents a perfectly equal income distribution.

STEP 5

For **1962**, the integrand is the line of equality (xx) *minus* the Lorenz curve f(x)f(x): xf(x)=x(12x+12x2)=12x12x2 x - f(x) = x - \left(\frac{1}{2}x + \frac{1}{2}x^2\right) = \frac{1}{2}x - \frac{1}{2}x^2

STEP 6

For **1972**, the integrand is the line of equality (xx) *minus* the Lorenz curve g(x)g(x): xg(x)=x(25x+35x2)=35x35x2 x - g(x) = x - \left(\frac{2}{5}x + \frac{3}{5}x^2\right) = \frac{3}{5}x - \frac{3}{5}x^2

STEP 7

**Set up the integral:** The Gini index is twice the integral of the integrand from 0 to 1: 201(12x12x2)dx 2 \int_{0}^{1} \left(\frac{1}{2}x - \frac{1}{2}x^2\right) dx

STEP 8

**Compute the integral:** 2[12x2212x33]01 2 \cdot \left[ \frac{1}{2} \cdot \frac{x^2}{2} - \frac{1}{2} \cdot \frac{x^3}{3} \right]_0^1 2[x24x36]01 2 \cdot \left[ \frac{x^2}{4} - \frac{x^3}{6} \right]_0^1

STEP 9

**Evaluate the integral:** Substitute the **upper limit** (x=1x = 1) and the **lower limit** (x=0x = 0): 2[(124136)(024036)] 2 \cdot \left[ \left(\frac{1^2}{4} - \frac{1^3}{6}\right) - \left(\frac{0^2}{4} - \frac{0^3}{6}\right) \right] 2[(1416)0] 2 \cdot \left[ \left(\frac{1}{4} - \frac{1}{6}\right) - 0 \right] 2(312212)=2112=16 2 \cdot \left(\frac{3}{12} - \frac{2}{12}\right) = 2 \cdot \frac{1}{12} = \frac{1}{6}

STEP 10

**Set up the integral:** The Gini index is twice the integral of the integrand from 0 to 1: 201(35x35x2)dx 2 \int_{0}^{1} \left(\frac{3}{5}x - \frac{3}{5}x^2\right) dx

STEP 11

**Compute the integral:** 2[35x2235x33]01 2 \cdot \left[ \frac{3}{5} \cdot \frac{x^2}{2} - \frac{3}{5} \cdot \frac{x^3}{3} \right]_0^1 2[3x210x35]01 2 \cdot \left[ \frac{3x^2}{10} - \frac{x^3}{5} \right]_0^1

STEP 12

**Evaluate the integral:** Substitute the **upper limit** (x=1x = 1) and the **lower limit** (x=0x = 0): 2[(31210135)(30210035)] 2 \cdot \left[ \left(\frac{3 \cdot 1^2}{10} - \frac{1^3}{5}\right) - \left(\frac{3 \cdot 0^2}{10} - \frac{0^3}{5}\right) \right] 2[(31015)0] 2 \cdot \left[ \left(\frac{3}{10} - \frac{1}{5}\right) - 0 \right] 2(310210)=2110=15 2 \cdot \left(\frac{3}{10} - \frac{2}{10}\right) = 2 \cdot \frac{1}{10} = \frac{1}{5}

STEP 13

The Gini index for 1962 is 160.167 \frac{1}{6} \approx 0.167 .
The Gini index for 1972 is 15=0.2 \frac{1}{5} = 0.2 .
This means that income inequality *increased* from 1962 to 1972.

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