Inequality and Distribution

Sources: Wikipedia (CC BY-SA 4.0) · David Ricardo, Principles of Political Economy (1817, public domain)

The Lorenz curve plots cumulative income share against cumulative population share. The Gini coefficient measures the gap between the Lorenz curve and perfect equality. It ranges from 0 (everyone has the same) to 1 (one person has everything).

The Lorenz curve

Sort everyone by income, poorest to richest. The x-axis is cumulative population share, the y-axis is cumulative income share. Perfect equality is the 45-degree line. The further the Lorenz curve bows below the diagonal, the greater the inequality. The Gini coefficient is the area between the diagonal and the Lorenz curve, divided by the total area under the diagonal.

Scheme

; Gini coefficient from income list
; Gini = (sum |xi - xj|) / (2 * n^2 * mean)

(define (gini incomes)
  (let* ((n (length incomes))
         (mean (/ (apply + incomes) n))
         (pairs
           (apply append
             (map (lambda (x)
               (map (lambda (y) (abs (- x y))) incomes))
             incomes)))
         (sum-diffs (apply + pairs)))
    (/ sum-diffs (* 2 n n mean))))

; Example: five-person economy
(define equal-economy (list 50 50 50 50 50))
(define moderate (list 20 30 50 70 80))
(define unequal (list 5 10 20 40 175))

(display "Equal economy:    Gini = ")
(display (gini equal-economy)) (newline)
(display "Moderate economy: Gini = ")
(display (/ (round (* (gini moderate) 1000)) 1000)) (newline)
(display "Unequal economy:  Gini = ")
(display (/ (round (* (gini unequal) 1000)) 1000))

The Pareto distribution

Vilfredo Pareto observed that income follows a power law: the fraction of people with income above x is proportional to x^-alpha. With alpha around 1.5-2, this generates the "80-20 rule": 20% of people hold 80% of the wealth. The Pareto distribution has a fat tail; extreme wealth is not an anomaly but a structural feature of the distribution.

Scheme

; Pareto distribution: P(X > x) = (x_min / x)^alpha
; For income: x_min = minimum income, alpha = shape parameter

(define x-min 10000)  ; minimum income

(define (pareto-survival x alpha)
  (expt (/ x-min x) alpha))

; What fraction earns more than $100,000?
(display "alpha=1.5: P(X > $100k) = ")
(display (/ (round (* (pareto-survival 100000 1.5) 10000)) 10000)) (newline)

(display "alpha=2.0: P(X > $100k) = ")
(display (/ (round (* (pareto-survival 100000 2.0) 10000)) 10000)) (newline)

; 80-20 check: top 20% share with alpha=log(5)/log(4)
(define alpha-8020 (/ (log 5) (log 4)))
(display "80-20 alpha = ") (display (/ (round (* alpha-8020 1000)) 1000)) (newline)

; Gini from Pareto: Gini = 1 / (2*alpha - 1)
(display "Gini at alpha=1.5: ")
(display (/ 1 (- (* 2 1.5) 1))) (newline)
(display "Gini at alpha=2.0: ")
(display (/ (round (* (/ 1 (- (* 2 2.0) 1)) 1000)) 1000))

Income vs wealth inequality

Income is a flow (what you earn per year). Wealth is a stock (what you own). Wealth inequality is always larger than income inequality because wealth compounds: r > g (Piketty's observation that returns on capital exceed economic growth) means the wealthy pull further ahead each generation. The US Gini is roughly 0.39 for income but 0.85 for wealth.

Scheme

; r > g: wealth divergence over generations
; Two families: one starts with $100k capital, one with $0
; Both earn $50k/year, save 20%
; Capital earns r=7%, wages grow at g=2%

(define (simulate-generations n capital-0 wage-0 r g savings-rate)
  (define (loop gen capital wage)
    (if (> gen n)
        (list capital wage)
        (let* ((income (+ wage (* capital r)))
               (savings (* income savings-rate))
               (new-capital (+ capital savings))
               (new-wage (* wage (+ 1 g))))
          (loop (+ gen 1) new-capital new-wage))))
  (loop 1 capital-0 wage-0))

; Family A: starts with $100k capital
(define family-a (simulate-generations 30 100000 50000 0.07 0.02 0.20))

; Family B: starts with $0 capital
(define family-b (simulate-generations 30 0 50000 0.07 0.02 0.20))

(display "After 30 years:") (newline)
(display "Family A wealth: $") (display (round (car family-a))) (newline)
(display "Family B wealth: $") (display (round (car family-b))) (newline)
(display "Ratio: ") (display (/ (round (* (/ (car family-a) (car family-b)) 100)) 100))

; r > g: wealth divergence over generations
; Two families: one starts with $100k capital, one with $0
; Both earn $50k/year, save 20%
; Capital earns r=7%, wages grow at g=2%

(define (simulate-generations n capital-0 wage-0 r g savings-rate)
  (define (loop gen capital wage)
    (if (> gen n)
        (list capital wage)
        (let* ((income (+ wage (* capital r)))
               (savings (* income savings-rate))
               (new-capital (+ capital savings))
               (new-wage (* wage (+ 1 g))))
          (loop (+ gen 1) new-capital new-wage))))
  (loop 1 capital-0 wage-0))

; Family A: starts with $100k capital
(define family-a (simulate-generations 30 100000 50000 0.07 0.02 0.20))

; Family B: starts with $0 capital
(define family-b (simulate-generations 30 0 50000 0.07 0.02 0.20))

(display "After 30 years:") (newline)
(display "Family A wealth: $") (display (round (car family-a))) (newline)
(display "Family B wealth: $") (display (round (car family-b))) (newline)
(display "Ratio: ") (display (/ (round (* (/ (car family-a) (car family-b)) 100)) 100))

Neighbors

Foundations (Wikipedia)

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