Fix critical style guide violations in wealth_dynamics.md

Co-authored-by: mmcky <8263752+mmcky@users.noreply.github.com>

Author: Copilot

lectures/wealth_dynamics.md

@@ -38,15 +38,13 @@ tags: [hide-output]
 
 ## Overview
 
-This notebook gives an introduction to wealth distribution dynamics, with a
-focus on
+This notebook gives an introduction to wealth distribution dynamics, with a focus on
 
 * modeling and computing the wealth distribution via simulation,
 * measures of inequality such as the Lorenz curve and Gini coefficient, and
 * how inequality is affected by the properties of wage income and returns on assets.
 
-One interesting property of the wealth distribution we discuss is Pareto
-tails.
+One interesting property of the wealth distribution we discuss is **Pareto tails**.
 
 The wealth distribution in many countries exhibits a Pareto tail
 
@@ -59,14 +57,13 @@ It also gives us a way to quantify such concentration, in terms of the tail inde
 
 One question of interest is whether or not we can replicate Pareto tails from a relatively simple model.
 
-### A Note on Assumptions
+### A note on assumptions
 
-The evolution of wealth for any given household depends on their
-savings behavior.
+The evolution of wealth for any given household depends on their savings behavior.
 
 Modeling such behavior will form an important part of this lecture series.
 
-However, in this particular lecture, we will be content with rather ad hoc (but plausible) savings rules.
+However, in this particular lecture, we will be content with rather *ad hoc* (but plausible) savings rules.
 
 We do this to more easily explore the implications of different specifications of income dynamics and investment returns.
 
@@ -82,14 +79,13 @@ from numba import jit, float64, prange
 from numba.experimental import jitclass
 ```
 
-## Lorenz Curves and the Gini Coefficient
+## Lorenz curves and the Gini coefficient
 
-Before we investigate wealth dynamics, we briefly review some measures of
-inequality.
+Before we investigate wealth dynamics, we briefly review some measures of inequality.
 
-### Lorenz Curves
+### Lorenz curves
 
-One popular graphical measure of inequality is the [Lorenz curve](https://en.wikipedia.org/wiki/Lorenz_curve).
+One popular graphical measure of inequality is the **Lorenz curve**.
 
 The package [QuantEcon.py](https://github.com/QuantEcon/QuantEcon.py), already imported above, contains a function to compute Lorenz curves.
 
@@ -116,21 +112,17 @@ plt.show()
 
 This curve can be understood as follows: if point $(x,y)$ lies on the curve, it means that, collectively, the bottom $(100x)\%$ of the population holds $(100y)\%$ of the wealth.
 
-The "equality" line is the 45 degree line (which might not be exactly 45
-degrees in the figure, depending on the aspect ratio).
+The "equality" line is the 45 degree line (which might not be exactly 45 degrees in the figure, depending on the aspect ratio).
 
-A sample that produces this line exhibits perfect equality.
+A sample that produces this line exhibits **perfect equality**.
 
 The other line in the figure is the Lorenz curve for the lognormal sample, which deviates significantly from perfect equality.
 
 For example, the bottom 80% of the population holds around 40% of total wealth.
 
-Here is another example, which shows how the Lorenz curve shifts as the
-underlying distribution changes.
+Here is another example, which shows how the Lorenz curve shifts as the underlying distribution changes.
 
-We generate 10,000 observations using the Pareto distribution with a range of
-parameters, and then compute the Lorenz curve corresponding to each set of
-observations.
+We generate 10,000 observations using the Pareto distribution with a range of parameters, and then compute the Lorenz curve corresponding to each set of observations.
 
 ```{code-cell} ipython3
 a_vals = (1, 2, 5)              # Pareto tail index
@@ -150,13 +142,11 @@ You can see that, as the tail parameter of the Pareto distribution increases, in
 
 This is to be expected, because a higher tail index implies less weight in the tail of the Pareto distribution.
 
-### The Gini Coefficient
+### The Gini coefficient
 
-The definition and interpretation of the Gini coefficient can be found on the corresponding [Wikipedia page](https://en.wikipedia.org/wiki/Gini_coefficient).
+The definition and interpretation of the **Gini coefficient** can be found on the corresponding [Wikipedia page](https://en.wikipedia.org/wiki/Gini_coefficient).
 
-A value of 0 indicates perfect equality (corresponding the case where the
-Lorenz curve matches the 45 degree line) and a value of 1 indicates complete
-inequality (all wealth held by the richest household).
+A value of 0 indicates perfect equality (corresponding the case where the Lorenz curve matches the 45 degree line) and a value of 1 indicates complete inequality (all wealth held by the richest household).
 
 The [QuantEcon.py](https://github.com/QuantEcon/QuantEcon.py) library contains a function to calculate the Gini coefficient.
 
@@ -166,8 +156,7 @@ $$
 G = 1 - 2^{-1/a}
 $$
 
-Let's see if the Gini coefficient computed from a simulated sample matches
-this at each fixed value of $a$.
+Let's see if the Gini coefficient computed from a simulated sample matches this at each fixed value of $a$.
 
 ```{code-cell} ipython3
 a_vals = range(1, 20)
@@ -190,7 +179,7 @@ plt.show()
 
 The simulation shows that the fit is good.
 
-## A Model of Wealth Dynamics
+## A model of wealth dynamics
 
 Having discussed inequality measures, let us now turn to wealth dynamics.
 
@@ -212,28 +201,26 @@ where
 Letting $\{z_t\}$ be a correlated state process of the form
 
 $$
-z_{t+1} = a z_t + b + \sigma_z \epsilon_{t+1}
+z_{t+1} = a z_t + b + σ_z ε_{t+1}
 $$
 
 we’ll assume that
 
 $$
-R_t := 1 + r_t = c_r \exp(z_t) + \exp(\mu_r + \sigma_r \xi_t)
+R_t := 1 + r_t = c_r \exp(z_t) + \exp(μ_r + σ_r ξ_t)
 $$
 
 and
 
 $$
-y_t = c_y \exp(z_t) + \exp(\mu_y + \sigma_y \zeta_t)
+y_t = c_y \exp(z_t) + \exp(μ_y + σ_y ζ_t)
 $$
 
-Here $\{ (\epsilon_t, \xi_t, \zeta_t) \}$ is IID and standard
-normal in $\mathbb R^3$.
+Here $\{ (ε_t, ξ_t, ζ_t) \}$ is IID and standard normal in $\mathbb R^3$.
 
-The value of $c_r$ should be close to zero, since rates of return
-on assets do not exhibit large trends.
+The value of $c_r$ should be close to zero, since rates of return on assets do not exhibit large trends.
 
-When we simulate a population of households, we will assume all shocks are idiosyncratic (i.e.,  specific to individual households and independent across them).
+When we simulate a population of households, we will assume all shocks are *idiosyncratic* (i.e.,  specific to individual households and independent across them).
 
 Regarding the savings function $s$, our default model will be
 
@@ -245,12 +232,11 @@ s(w) = s_0 w \cdot \mathbb 1\{w \geq \hat w\}
 
 where $s_0$ is a positive constant.
 
-Thus, for $w < \hat w$, the household saves nothing. For
-$w \geq \bar w$, the household saves a fraction $s_0$ of
-their wealth.
+Thus, for $w < \hat w$, the household saves nothing.
 
-We are using something akin to a fixed savings rate model, while
-acknowledging that low wealth households tend to save very little.
+For $w \geq \bar w$, the household saves a fraction $s_0$ of their wealth.
+
+We are using something akin to a *fixed savings rate model*, while acknowledging that low wealth households tend to save very little.
 
 ## Implementation
 
@@ -346,7 +332,7 @@ class WealthDynamics:
         return wp, zp
 ```
 
-Here's function to simulate the time series of wealth for in individual households.
+Here's a function to simulate the time series of wealth for individual households.
 
 ```{code-cell} ipython3
 
@@ -373,7 +359,7 @@ def wealth_time_series(wdy, w_0, n):
     return w
 ```
 
-Now here's function to simulate a cross section of households forward in time.
+Now here's a function to simulate a cross section of households forward in time.
 
 Note the use of parallelization to speed up computation.
 
@@ -406,16 +392,13 @@ def update_cross_section(wdy, w_distribution, shift_length=500):
     return new_distribution
 ```
 
-Parallelization is very effective in the function above because the time path
-of each household can be calculated independently once the path for the
-aggregate state is known.
+Parallelization is very effective in the function above because the time path of each household can be calculated independently once the path for the aggregate state is known.
 
 ## Applications
 
-Let's try simulating the model at different parameter values and investigate
-the implications for the wealth distribution.
+Let's try simulating the model at different parameter values and investigate the implications for the wealth distribution.
 
-### Time Series
+### Time series
 
 Let's look at the wealth dynamics of an individual household.
 
@@ -435,12 +418,11 @@ Notice the large spikes in wealth over time.
 
 Such spikes are similar to what we observed in time series when {doc}`we studied Kesten processes <kesten_processes>`.
 
-### Inequality Measures
+### Inequality measures
 
 Let's look at how inequality varies with returns on financial assets.
 
-The next function generates a cross section and then computes the Lorenz
-curve and Gini coefficient.
+The next function generates a cross section and then computes the Lorenz curve and Gini coefficient.
 
 ```{code-cell} ipython3
 def generate_lorenz_and_gini(wdy, num_households=100_000, T=500):
@@ -457,7 +439,7 @@ def generate_lorenz_and_gini(wdy, num_households=100_000, T=500):
 
 Now we investigate how the Lorenz curves associated with the wealth distribution change as return to savings varies.
 
-The code below plots Lorenz curves for three different values of $\mu_r$.
+The code below plots Lorenz curves for three different values of $μ_r$.
 
 If you are running this yourself, note that it will take one or two minutes to execute.
 
@@ -475,7 +457,7 @@ gini_vals = []
 for μ_r in μ_r_vals:
     wdy = WealthDynamics(μ_r=μ_r)
     gv, (f_vals, l_vals) = generate_lorenz_and_gini(wdy)
-    ax.plot(f_vals, l_vals, label=fr'$\psi^*$ at $\mu_r = {μ_r:0.2}$')
+    ax.plot(f_vals, l_vals, label=fr'$\psi^*$ at $μ_r = {μ_r:0.2}$')
     gini_vals.append(gv)
 
 ax.plot(f_vals, f_vals, label='equality')
@@ -485,29 +467,23 @@ plt.show()
 
 The Lorenz curve shifts downwards as returns on financial income rise, indicating a rise in inequality.
 
-We will look at this again via the Gini coefficient immediately below, but
-first consider the following image of our system resources when the code above
-is executing:
+We will look at this again via the Gini coefficient immediately below, but first consider the following image of our system resources when the code above is executing:
 
-Since the code is both efficiently JIT compiled and fully parallelized, it's
-close to impossible to make this sequence of tasks run faster without changing
-hardware.
+Since the code is both efficiently JIT compiled and fully parallelized, it's close to impossible to make this sequence of tasks run faster without changing hardware.
 
 Now let's check the Gini coefficient.
 
 ```{code-cell} ipython3
 fig, ax = plt.subplots()
 ax.plot(μ_r_vals, gini_vals, label='gini coefficient')
-ax.set_xlabel(r"$\mu_r$")
+ax.set_xlabel(r"$μ_r$")
 ax.legend()
 plt.show()
 ```
 
-Once again, we see that inequality increases as returns on financial income
-rise.
+Once again, we see that inequality increases as returns on financial income rise.
 
-Let's finish this section by investigating what happens when we change the
-volatility term $\sigma_r$ in financial returns.
+Let's finish this section by investigating what happens when we change the volatility term $σ_r$ in financial returns.
 
 
 ```{code-cell} ipython3
@@ -520,7 +496,7 @@ gini_vals = []
 for σ_r in σ_r_vals:
     wdy = WealthDynamics(σ_r=σ_r)
     gv, (f_vals, l_vals) = generate_lorenz_and_gini(wdy)
-    ax.plot(f_vals, l_vals, label=fr'$\psi^*$ at $\sigma_r = {σ_r:0.2}$')
+    ax.plot(f_vals, l_vals, label=fr'$\psi^*$ at $σ_r = {σ_r:0.2}$')
     gini_vals.append(gv)
 
 ax.plot(f_vals, f_vals, label='equality')
@@ -535,30 +511,26 @@ We see that greater volatility has the effect of increasing inequality in this m
 ```{exercise}
 :label: wd_ex1
 
-For a wealth or income distribution with Pareto tail, a higher tail index suggests lower inequality.
+For a wealth or income distribution with **Pareto tail**, a higher tail index suggests lower inequality.
 
-Indeed, it is possible to prove that the Gini coefficient of the Pareto
-distribution with tail index $a$ is $1/(2a - 1)$.
+Indeed, it is possible to prove that the **Gini coefficient** of the Pareto distribution with tail index $a$ is $1/(2a - 1)$.
 
 To the extent that you can, confirm this by simulation.
 
-In particular, generate a plot of the Gini coefficient against the tail index
-using both the theoretical value just given and the value computed from a sample via `qe.gini_coefficient`.
+In particular, generate a plot of the Gini coefficient against the tail index using both the theoretical value just given and the value computed from a sample via `qe.gini_coefficient`.
 
 For the values of the tail index, use `a_vals = np.linspace(1, 10, 25)`.
 
 Use sample of size 1,000 for each $a$ and the sampling method for generating Pareto draws employed in the discussion of Lorenz curves for the Pareto distribution.
 
-To the extent that you can, interpret the monotone relationship between the
-Gini index and $a$.
+To the extent that you can, interpret the monotone relationship between the Gini index and $a$.
 ```
 
 ```{solution-start} wd_ex1
 :class: dropdown
 ```
 
-Here is one solution, which produces a good match between theory and
-simulation.
+Here is one solution, which produces a good match between theory and simulation.
 
 ```{code-cell} ipython3
 a_vals = np.linspace(1, 10, 25)  # Pareto tail index
@@ -575,8 +547,7 @@ ax.legend()
 plt.show()
 ```
 
-In general, for a Pareto distribution, a higher tail index implies less weight
-in the right hand tail.
+In general, for a Pareto distribution, a higher tail index implies less weight in the right hand tail.
 
 This means less extreme values for wealth and hence more equality.
 
@@ -589,26 +560,25 @@ More equality translates to a lower Gini index.
 :label: wd_ex2
 ```
 
-The wealth process {eq}`wealth_dynam_ah` is similar to a {doc}`Kesten process <kesten_processes>`.
+The wealth process {eq}`wealth_dynam_ah` is similar to a **Kesten process**.
 
 This is because, according to {eq}`sav_ah`, savings is constant for all wealth levels above $\hat w$.
 
-When savings is constant, the wealth process has the same quasi-linear
-structure as a Kesten process, with multiplicative and additive shocks.
+When savings is constant, the wealth process has the same quasi-linear structure as a Kesten process, with multiplicative and additive shocks.
 
-The Kesten--Goldie theorem tells us that Kesten processes have Pareto tails under a range of parameterizations.
+The **Kesten--Goldie theorem** tells us that Kesten processes have Pareto tails under a range of parameterizations.
 
 The theorem does not directly apply here, since savings is not always constant and since the multiplicative and additive terms in {eq}`wealth_dynam_ah` are not IID.
 
 At the same time, given the similarities, perhaps Pareto tails will arise.
 
-To test this, run a simulation that generates a cross-section of wealth and
-generate a rank-size plot.
+To test this, run a simulation that generates a cross-section of wealth and generate a rank-size plot.
 
 If you like, you can use the function `rank_size` from the `quantecon` library (documentation [here](https://quanteconpy.readthedocs.io/en/latest/tools/inequality.html#quantecon.inequality.rank_size)).
 
-In viewing the plot, remember that Pareto tails generate a straight line.  Is
-this what you see?
+In viewing the plot, remember that Pareto tails generate a straight line.
+
+Is this what you see?
 
 For sample size and initial conditions, use