Explanation of the Lorenz Curve Simulator

1. Gibrat's Law (also known as the Law of Proportional Effect)
This law states that the growth rate of a firm or the population growth rate of a city is determined independently and randomly, regardless of its current size. While there is debate regarding whether this can be extended to apply to general household income, we define it here as follows:
X(i,t): Income of household *i* at term *t* (with *t* extending up to 100 years).
Here, we set:
X(i,0) = 1: The initial income for all households is set to 1.
i = 1 to N: N = 1000 (households).
Then:
xx(i,t + 1) = X(i,t) * (1 + Norm(1,a)) * (1 + g)
In each term, household incomes disperse according to a normal distribution with a standard deviation of *a*. Here, *g* represents the economic growth rate of the entire region, which we assume to be constant for the time being.
We use the notation xx(i,t) here because we intend to make adjustments to address the aforementioned "debatable points."

2. Adding Financial Assets
Based on the sound principle of allocating surplus funds to investment:
Max(0, xx(i,t + 1) - b * M(t)) * r
Here:
M(t) is the average income of the previous year: sum(X(i,t))/N (i = 1 to N). Note that M(0) = 1.
*b* is defined as the "surplus fund ratio"; income exceeding *b* times the average income is treated as surplus funds available for investment.
*r* is the financial interest rate; like *g*, it is assumed to remain constant over the 100-year period.
Using the Max() function prevents households without surplus funds from engaging in speculative, gambling-like investments. 3. Levying taxes on the wealthy
Max(0, xx(i, t + 1) - c*M(t)) * t
Here, 'c' is referred to as the minimum taxation threshold ratio; a wealth tax at rate 't' is levied on income exceeding this level. The Max() function is used here.

4. Establishing a safety net
The value d*M(t) is defined as the minimum amount required to lead a healthy and cultured life. X(i, t+1) = xx + [Return on surplus funds] - [Tax on the wealthy] + Max(0, d*M(t) - xx)
Under this formula, households with income below d*M(t) effectively receive d*M(t).

5. Summary of coefficients
Ultimately, the equations are summarized as follows:
xx(i,t + 1) = X(i,t)*(1 + Norm(1,a))*(1 + g)
X(i,t + 1) = xx(i,t + 1) + Max(0, xx(i,t+1) - b*M(t)) * r - Max(0, xx(i,t + 1) - c*M(t)) * t + Max(0, d*M(t) - xx(i,t + 1))
M(t) = sum(X(i,t))/N : Average income of the previous year, M(0)=1

These coefficients can be adjusted using sliders:
a = 0.0 to 0.3: Implements Gibrat's Law with a mean of 1 and standard deviation of a
g = -0.05 to 0.05: Regional economic growth rate
r = 0.01 to 0.09: Financial interest rate (investment return rate)
t = 0.1 to 0.5: Tax rate for the wealthy
b = 2 to 10: Surplus funds ratio (investment threshold)
c = 1 to 10: Minimum taxation threshold ratio (taxation start line)
d = 0 to 0.5: Social welfare (subsistence) line

6. Lorenz curve and Gini coefficient
Based on X(i, 100) at term 100, a graph (Lorenz curve) is plotted with the cumulative proportion of households on the horizontal axis and the cumulative proportion of income on the vertical axis, and the Gini coefficient is calculated.
Clicking the calculation start button updates the Lorenz curve and the Gini coefficient. Since the calculations are based on aggregated household income data, the figures fluctuate each time you press a button—offering a chance to reflect on the nature of life itself.

7. Scenario Presets
Try out the preset buttons for "Standard," "Widening Inequality," "Redistribution-Focused," and "Equality & Stability." You will see that any type of economic policy is possible.
It makes you wonder why the average income ranks 15th under the "Widening Inequality" scenario, yet 2nd under "Equality & Stability."

Needless to say, this does not reflect reality. Please enjoy it simply as a toy.