Math 6.1-6.2

Gini Index

  • The Gini Index measures income concentration, expressed as a single number.
  • It quantifies inequality on a scale from $0$ to $1$, where:
    • A Gini index of $0$ indicates absolute equality (everyone has the same income).
    • A Gini index of $1$ indicates absolute inequality (one person has all the income).

Formula for Gini Index

  • The Gini index can be calculated using the formula:
    G=2∫01(x−f(x)) dx{G} = 2 \int_0^1 (x - f(x)) \, dx
    where $f(x)$ is the Lorenz curve.

Example: Gini Index Calculation for 2017

  • Given: Lorenz curve function for 2017 is $f(x) = x^{2.6}$.
  • To calculate the Gini index, we compute:
    G=2∫01(x−x2.6) dxG = 2 \int_0^1 (x - x^{2.6}) \, dx
  • This integral evaluates to:
    • G=2(12−13.6)G = 2 \left( \frac{1}{2} - \frac{1}{3.6} \right)
  • This results in an approximate value of G≈0.444G \approx 0.444.

Prediction for 2030

  • The predicted Lorenz curve for 2030 is $g(x) = x^{1.8}$.
  • Calculate the Gini index similarly:
    G=2∫01(x−x1.8) dxG = 2 \int_0^1 (x - x^{1.8}) \, dx
  • This results in G≈0.286G \approx 0.286.
  • Interpretation:
    • Decrease in Gini index from $0.444$ to $0.286$ indicates more equitable income distribution over time.

Income Flow from Vending Machine

  • Rate of flow of income is given by:
    f(t)=5000e0.04tf(t) = 5000 e^{0.04t}
    where $t$ is years since the installation.
  • Total income produced during the first five years is calculated as:
    ∫05f(t) dt\int_0^5 f(t) \, dt
  • Evaluate from $0$ to $5$:
    • Calculate:
      ∫055000e0.04t dt\int_0^5 5000 e^{0.04t} \, dt
    • Resulting total income: $27,675.

Future Value of Continuous Income

  • Future value formula for continuously compounded income: FV=e(rt)∫0tf(t)e−rt dtFV = e^{(rt)} \int_0^t f(t) e^{-rt} \, dt where:
    • $r = 0.12$ (12% interest rate)
    • $t = 5$ (years)
    • $f(t) = 5000 e^{0.04t}$
  • Compute:
    • Future value integral becomes:
      =e(0.12×5)∫055000e−0.04t dt= e^{(0.12 \times 5)} \int_0^5 5000 e^{-0.04t} \, dt
  • Resulting Future Value: approximately $37,545.
  • Interest earned:
    • $Interest = FV - PV = 37,545 - 27,675 = 9,870$.

Conclusion

  • The Gini index is a crucial measure of income inequality, showing trends over time.
  • Calculating total income from streams, like a vending machine, and determining future values helps understand investment growth and returns.