In ianksalter/bistr:

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(ggplot2)
library(pracma)
library(EnvStats)
source("../R/distributions.R")

The OECD lists Gini and Mean Income data for before and after the application of tax and benefits. From this data, assuming a given probability distribution, it is possible to derive a simple model of a net income function that shows the net income for very possible gross income value. This approximation is named a Basic Income Simple Tax (BIST) net income function.

Here we consider two possible probability distributions as possible models of income distribution: The Log Normal Distribution and the Pareto Distribution.

The Log Normal Distribution is defined in terms of two parameters $\mu$, the mean, and $\sigma$, the standard deviation of the variables natural logarithm (sdlog). The following plot shows the Probability Density Function for the logormal distribution with $\mu = 0$ and various values of $\sigma$:

p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
p +
  stat_function(fun = dlnorm, args = list(sdlog = 0.25), colour = "red") +
  geom_text(aes(x= 2.2, y = 1.5), label = "sdlog = 0.25", size = 3, colour = "red") +
  stat_function(fun = dlnorm, args = list(sdlog = 0.5), colour = "green") +
  geom_text(aes(x= 2.5, y = 0.5), label = "sdlog = 0.5", size = 3, colour = "green") +
  stat_function(fun = dlnorm, args = list(sdlog = 1), colour = "blue") +
  geom_text(aes(x= 2.8, y = 0.25), label = " sdlog = 1", size = 3, colour = "blue") +
  xlim(0, 5) +
  ylim(0, 2) +
  ylab("PDF")

The Pareto Distribution is defined in terms of two parameters $x_{m}$, the scale (or location) and $\alpha$, the shape, The following plot shows the Probability Density Function for the pareto distribution for $x_{m} = 1$ and various values of $\alpha$:

p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
p +
  stat_function(fun = dpareto, args = list(location = 1, shape = 3), colour = "red") +
  geom_text(aes(x= 2.2, y = 1.5), label = "shape = 3", size = 3, colour = "red") +
  stat_function(fun = dpareto, args = list(location = 1, shape = 2), colour = "green") +
  geom_text(aes(x= 2.5, y = 0.5), label = "shape = 2", size = 3, colour = "green") +
  stat_function(fun = dpareto, args = list(location = 1, shape = 1), colour = "blue") +
  geom_text(aes(x= 3.5, y = 0.25), label = " shape = 1", size = 3, colour = "blue") +
  xlim(0, 5) +
  ylim(0, 3) +
  ylab("PDF")

Given values of $gini$ and mean $\mu$ we can calculate the parameters for both lognormal and pareto distributions For the lognormal distribution:

$\mu = \mu$

$\sigma = 2erf^{-1}(gini)$

For the pareto distribution (assuming gini is not 1):

$x_{m} = \frac{(\alpha - 1)\mu}{\alpha}$

$\alpha = \frac{1 + gini}{2gini}$

For 2017 we have the following OECD data for the UK:

total_earnings = 20051
capital_income = 1223
self_emp_income = 3156

gini_net = 0.357
mean_net = 22879
gini_gross = 0.506
mean_gross = total_earnings + capital_income + self_emp_income

• Gini (disposable income, post taxes and transfers): r format(gini_net)
• Mean disposable income (current prices): r format(mean_net)
• Gini (market income before taxes and transfers) : r format(gini_gross)
• Mean Income Before Tax: r format(mean_gross) calculated by summing
• Total Earnings (current prices): r format(total_earnings)
• Capital Income (current prices): r format(capital_income)
• Income from self-employment and goods produced for own consumption (current prices): r format(self_emp_income)

Figure 1 shows log normal probability distributions for the UK in 2017 where the OECD data:

lnorm_sdlog(gini_net)
probability_distribution <- function(x){
  dlnorm(x)
}
modified_distribution <- function(x){
  dlnorm(x, sdlog = lnorm_sdlog(gini_net))
}

p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
p +
  stat_function(fun = probability_distribution) +
  stat_function(fun = modified_distribution,
                mapping = aes(linetype = "dotted")) +
                xlim(0, 5) +
                ylim(0, 1)

net_income_distribution <- function(x){
  dlnorm(x, meanlog = mean_net, sdlog = lnorm_sdlog(gini_net))
}
gross_income_distribution <- function(x){
  dlnorm(x, meanlog = mean_gross, sdlog = lnorm_sdlog(gini_gross))
}
p <- ggplot(data = data.frame(x = 0), mapping = aes(x = x))
p +
  stat_function(fun = net_income_distribution) +
  stat_function(fun = gross_income_distribution) +
  xlim(0, 1000000) +
  ylim(0, 0.000000000000000000001)

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output: 
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plot(1:10)
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output:
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ianksalter/bistr documentation built on May 4, 2020, 12:16 a.m.