This is the first of the challenge questions for the course. They are meant to be challenging and stretch your skills, but remember you can always ask for help or collaborate with your classmates. There are two articles at the bottom which will help provide broader context for the questions we’re studying here.
Consider the wealth and labor income distributions for a single generation of a simple society, described in the dataset simple_society.csv. People in this society work, save, and accumulate wealth over generations. For simplicity, we’ll assume that the labor income for a given household is constant over time and that every household saves a fixed portion of their total income. A household’s wealth accumulation in this society is characterized by two equations:
\[\begin{align} \text{(Total income)}~~~~~~~~ Y_t &= rW_t + i \\ \text{(Wealth accumulation)}~~~ W_{t+1} &= sY_t + (1+x_t)W_t, \end{align}\]
where \(Y_t\) is total household income in generation \(t\), \(W_t\) is total household wealth in generation \(t\), \(W_{t+1}\) is household wealth in generation \(t+1\), \(i\) is household labor income, \(r\) is the rate of interest on assets, \(s\) is the savings rate, and \(x_t\) is a random variable representing other factors (we’ll call \(x_t\) “luck”).1 (The second component is expressed as \(1+x_t\) so that we can interpret \(x_t\) as the non-savings-related growth/decline in wealth, i.e. \(x_t=1\) means wealth grew by 100%, \(x_t=-1\) means 100% of wealth was lost, and \(x_t=0\) means no change.)
The first equation describes a household’s total income for one generation. In words, it says, “total income for the household in generation \(t\) is the sum of capital rents earned on their wealth (\(rW_t\)) and labor income (\(i\))”.
The second equation describes how wealth accumulates. In words, it says, “wealth for the household in generation \(t+1\) is the sum of income that was saved by generation \(t\) (\(sY_t\)) and any other losses/gains (\((1+x_t)W_t\)).”
We’re going to track this society over two generations and see how the distributions of wealth and total income evolve. For simplicity, the amounts have been normalized so that wealth in the first generation ranges from 50 to 150. The savings rate is 20% (i.e., \(s=0.2\)) and the interest rate is 8% (i.e., \(r=0.08\)). “Luck” (\(x_t\)) is a random variable distributed uniformly over \([-1,1]\), i.e., \(x_t \sim U[-1,1]\).
To start, we’ll load tidyverse and cowplot, set the savings rate and interest rate parameters, and set the “seed” for random number generation later:
library(tidyverse)
library(cowplot)
interest_rate <- 0.08
savings_rate <- 0.2
set.seed(210)The seed is a parameter that controls random number generation for simulations. By setting the seed to a specific number (the number itself doesn’t matter), we ensure that our results can be replicated later and by others.
read_csv. If each row is a household, how many households are in this society? Assign the number of households to a variable n_households for later use.Hint: You can use nrow() to calculate the number of rows.
summary function to display summary statistics about the first generation.Hint: Use mutate.
Hint: Use the plot_grid function from the cowplot package.
Hint: Use the quantile() function.
Hint: Use runif() to generate luck as draws from a uniform distribution with min -1 and max 1.
Hint: To iterate it forward one generation, you’ll need to calculate \(W_{t+1}\) and then \(Y_{t+1}\). Supposing you read the data in as the object data, then you would calculate \(W_{t+1}\) as
data <- data %>% mutate(second_generation_wealth = savings_rate*total_income + (1 + luck)*wealth)To make the overlays easier to read, set the alpha to 0.25 on the second generations’ plots.
Read the following articles by John Cassidy (pdf version) and Benjamin Wallace-Wells (pdf version). Finally, purely for your edification, check out this article about the magnitude of income inequality in the US over the last few decades.
A random variable is just a variable whose outcome is uncertain. For example, the outcome of a coin toss before you flip it is a random variable: it could come up heads, it could come up tails. Or the amount of money you get from a lottery ticket: before you scratch the ticket, you don’t know how much you’ll get.↩︎