Add lecture on inequality measures

inequality.md

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+# Measuring Inequality
+
+
+## Overview
+
+Readers will have some intuitive understanding of the term ``inequality''.
+
+Many economic policies, from taxation to the welfare state, are clearly
+aimed at addressing inequality.
+
+However, debate on inequality is often tied to political beliefs.
+
+This is dangerous for us because allowing political beliefs to
+shape our findings reduces objectivity.
+
+To bring a scientific perspective to the topic of inequality we must start
+with careful definitions.
+
+In this lecture we discuss measures of inequality used in economic research.
+
+We will also look at inequality in 
+
+
+## Inequality in Distributions
+
+When we consider inequality, we are looking at distributions where some people
+have more than others.
+
+Below we look at some examples from both simulated and real data.
+
+### An Illustration
+
+For example, suppose we have a population
+
+TODO 
+
+* show two histograms from pre-generated data, interpret as income
+distributions of some population
+* which of these distributions is more ``unequal''?
+
+* discuss top shares
+
+
+### Wealth and Income Shares in the US
+
+TODO 
+
+* plot the top income and wealth shares in the US over time
+
+Use the file ./temp/gini_lorenz_us.ipynb but cut unnecessary.
+
+Replicate the relevant figures f:gini_lorenz_us_x in ch_income_wealth.tex of
+macro dynamics text --- ask JS.
+
+Explain carefully the pandas steps.
+
+
+## The Lorenz Curve
+
+Another popular measure of inequality is the Lorenz curve.
+
+The Lorenz curve is more complex than measuring specific shares but also more
+informative.
+
+The Lorenz Curve takes a sample $w_1, \ldots, w_n$ and produces a curve $L$.
+
+The curve $L$ is just a function $y = L(x)$ that we can plot and interpret.
+
+The steps are as follow.
+
+We suppose that the sample $w_1, \ldots, w_n$ has been sorted from smallest to largest
+
+Then we generate data points $(x_i, y_i)$ for the Lorenz curve according to
+
+\begin{equation*}
+    x_i = \frac{i}{n},
+    \quad
+    y_i = \frac{\sum_{j \leq i} w_j}{\sum_{j \leq n} w_j},
+    \quad i = 1, \ldots, n
+\end{equation*}
+
+Now the Lorenz curve $L$ is formed from these data points using linear interpolation.
+
+(If we plot the points with Matplotlib, the interpolation will be done for us.)
+
+The meaning of the curve is:  $y = L(x)$ indicates that the lowest $(100
+\times x)$\% of people have $(100 \times y)$\% of all wealth.
+
+Let's look at some examples and try to understand what this means.
+
+In the next figure, we generate 
+$n=2000$ draws from the standard lognormal distribution.  
+
+The straight line corresponds to perfect equality.  
+
+The lognormal draws produce a less equal distribution.  
+
+For example, if we imagine these draws as being observations of wealth across
+a sample of households, then the dashed lines show that the bottom 80\% of
+households own just over 40\% of total wealth.
+
+
+```{code-cell} ipython3
+n = 2000
+fig, ax = plt.subplots()
+sample = np.exp(np.random.randn(n))
+
+f_vals, l_vals = qe.lorenz_curve(sample)
+ax.plot(f_vals, l_vals, label=f'lognormal sample', lw=2)
+
+ax.plot(f_vals, f_vals, label='equality', lw=2)
+ax.legend(fontsize=12)
+
+ax.vlines([0.8], [0.0], [0.43], alpha=0.5, colors='k', ls='--')
+ax.hlines([0.43], [0], [0.8], alpha=0.5, colors='k', ls='--')
+
+ax.fill_between(f_vals, l_vals, f_vals, alpha=0.06)
+
+ax.set_ylim((0, 1))
+ax.set_xlim((0, 1))
+
+plt.show()
+```
+
+
+
+## The Gini Coefficient
+
+The Lorenz curve is a useful visual representation of inequality in a
+distribution.
+
+Another popular measure of income and wealth inequality is the Gini coefficient.
+
+The Gini coefficient is just a number, rather than a curve.
+
+As before, suppose that the sample $w_1, \ldots, w_n$ has been sorted from smallest to largest
+
+The Gini coefficient is defined for the sample above as 
+
+\begin{equation}
+    \label{eq:gini}
+    G :=
+    \frac
+        {\sum_{i=1}^n \sum_{j = 1}^n |w_j - w_i|}
+        {2n\sum_{i=1}^n w_i}.
+\end{equation}
+
+
+The Gini coefficient is closely related to the Lorenz curve.
+
+In fact, it can be shown that its value is twice the area between the line of
+equality and the Lorenz curve (e.g., the shaded area in Figure~????).
+
+The idea is that $G=0$ indicates complete equality, while $G=1$ indicates complete inequality.
+
+
+## Dynamics of the Gini Coefficient
+
+
+TODO 
+
+Plot the top income and wealth shares in the US over time
+
+Discuss with Shu
+
+Use the file ./temp/gini_lorenz_us.ipynb but cut unnecessary, use quantecon
+functions to compute gini coefficient
+
+Replicate the relevant figures f:gini_lorenz_us_x in ch_income_wealth.tex of
+macro dynamics text --- ask JS.
+
+Carefully discuss the pandas code.
+
+
+## Exercises
+
+NOTE fix up ex 1 and add more exercises -- perhaps empirical
+
+Ex 1
+
+Using simulation, compute the Lorenz curves and Gini coefficients for the
+collection of lognormal distributions associated with the random variables
+$w_\sigma = \exp(\mu + \sigma Z)$, where $Z \sim N(0, 1)$ and $\sigma$ varies
+over a finite grid between $0.2$ and $4$.  
+
+As $\sigma$ increases, so does the variance of $w_\sigma$.  
+
+To focus on volatility, adjust $\mu$ at each step
+    to maintain the equality $\mu=-\sigma^2/2$.
+
+(Confirm: this implies that the mean of $w_\sigma$ does not change with $\sigma$.) 
+
+For each $\sigma$, generate 2,000 independent draws of $w_\sigma$ and
+calculate the Lorenz curve and Gini coefficient.  
+
+Confirm that higher variance
+generates more dispersion in the sample, and hence greater inequality.
+
+
+Solution to Ex 1
+
+```{code-cell} ipython3
+k = 5
+σ_vals = np.linspace(0.5, 2, k)
+
+ginis = []
+
+fig, ax = plt.subplots()
+
+ax.plot(f_vals, f_vals, label='equality')
+
+for σ in σ_vals:
+    μ = -σ**2/2
+    y = np.exp(μ + σ * np.random.randn(n))
+    f_vals, l_vals = qe.lorenz_curve(y)
+    ginis.append(qe.gini_coefficient(y))
+    ax.plot(f_vals, l_vals, label=f'$\sigma = {σ:.1f}$')
+
+ax.legend(fontsize=12)
+
+plt.show()
+```