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  • How to measure income inequality in macroeconomics

Gini Coefficient

• The Gini Coefficient is essentially how much disparity there is between the equal and inequal curves

\begin{document}
	\begin{tikzpicture}[scale=6]
	  % Draw axes
	  \draw[->] (0,0) -- (1.1,0) node[right] {Cumulative \% of Population};
	  \draw[->] (0,0) -- (0,1.1) node[above] {Cumulative \% of Wealth};
	  
	  % Draw grid
	  \foreach \x in {0,0.2,0.4,0.6,0.8,1.0}
	    \draw[gray!30, very thin] (\x,0) -- (\x,1);
	  \foreach \y in {0,0.2,0.4,0.6,0.8,1.0}
	    \draw[gray!30, very thin] (0,\y) -- (1,\y);
	  
	  % Axis tick labels
	  \foreach \x in {0,20,40,60,80,100}
	    \node[below] at (\x/100,-0.02) {\tiny \x};
	  \foreach \y in {0,20,40,60,80,100}
	    \node[left] at (-0.02,\y/100) {\tiny \y};
	  
	  % Line of Perfect Equality (diagonal)
	  \draw[blue, thick, dashed] (0,0) -- (1,1) node[above right, pos=0.8] {\small Perfect Equality};
	  
	  % Lorenz Curve showing inequality
	  % Using a smooth curve that bows below the diagonal
	  \draw[red, thick] (0,0) 
	    .. controls (0.2,0.05) and (0.4,0.15) .. (0.5,0.25)
	    .. controls (0.6,0.35) and (0.7,0.50) .. (0.8,0.65)
	    .. controls (0.9,0.82) .. (1,1);
	  
	  % Label for Lorenz Curve
	  \node[red] at (0.65,0.35) {\small Inequality};
	  \node[red] at (0.65,0.31) {\small (Lorenz Curve)};
	  
	  % Shaded area between curves (representing inequality)
	  \fill[red!10, opacity=0.5] (0,0) 
	    .. controls (0.2,0.05) and (0.4,0.15) .. (0.5,0.25)
	    .. controls (0.6,0.35) and (0.7,0.50) .. (0.8,0.65)
	    .. controls (0.9,0.82) .. (1,1)
	    -- (0,0);
	  \fill[blue!10, opacity=0.3] (0,0) -- (1,1) -- (1,0) -- cycle;
	  
	  % Origin label
	  \node[below left] at (0,0) {0};
	  
	  % Data points for perfect equality
	  \foreach \p in {0,10,20,30,40,50,60,70,80,90,100}
	    \fill[blue] (\p/100,\p/100) circle (0.01);
	    
	\end{tikzpicture}
\end{document}

• Here's our Lorenz Curve.
• $\text{Gini}=\frac{A}{A+B}\in [0,1]$
  • Let $0\mathrm{\%}$ be $0$ and $100\mathrm{\%}$ be $1$
  • The area under the line of perfect equality is a triangle:
    • $A+B=\frac{1}{2}$
  • $\text{Gini}=\frac{A}{\frac{1}{2}}=2A$
  • $2A=1-2B$
• $L(x)$ is our Lorenz Curve
• $A={\int }_0^1[x-L(x)]\text{differential}x$
• $B={\int }_0^{1}L(x)\text{differential}x$

Discrete

• $\text{Gini}=1-\sum _{i=1}^n(x_i-x_{i-1})(y_i-y_{i-1})$
• Example:
  • $$$$

\text{Population %} & & \text{Income %} & \
\text{Cumulative} & \text{Margin} & \text{Cumulative} & \text{Margin} \
0 & - & 0 & - \
20 & 20 & 10 & 10 \
50 & 30 & 31 & 21 \
60 & 10 & 40 & 9 \
100 & 400 & 100 & 6
\end