# Cours avancé « Economie des inégalités » (Master APE, année M2)

Advanced course « Economics of Inequality » (Master APE, M2 year)

Thomas Piketty

Année universitaire 2008-2009

Course Notes C :

Top labor incomes vs Top capital incomes: Who are the richest?

Some simple arithmetic of working rich vs rentiers

1. Basics: Top 1% labor earners vs top 1% capital earners

Question : under what conditions are the top 1% capital earners (eK) richer than the top 1% labor earners (eL) ?

eK = top 1% capital share of the distribution of capital stock H(k)

eL = top 1% labor income share of the distribution of labor income G(y)

γ = k/yL = aggregate capital/labor income ratio

Response : top capital earners dominate if and only if  r γ eK >  eL

In 2006 : equilibrium between working rich and rentiers

r=4%, γ = 180 000€ / 24 000€ = 7,5, eK=20%, eL=7%

i.e.: rγ =30%, r γ  eK=6%

Top 1% labor income earners make 168 000€ (7 x 24 000€)

Top 1% capital holders own  3 600 000€  and make 144 000€ (6 x 24 000€)

In 1914 : a society of rentiers

r=4%,  γ =7,5, eK=50%, eL=7%

i.e.: rγ =30%,  γeK=15%

Top 1% capital earners make twice as much money as top 1% labor income earners

Post WW2 period : a society of working rich

r=4%, γ =3,5, eK=20%, eL=7%

i.e.: rγ =14%, rγeK=2,8%,

Top 1% labor earners make twice as much money as top 1% capital earners

Conclusion :

The working rich society corresponds to very specific parameter combinations:

Moderate wealth concentrationl (eK=20%, rather than eK=50%)

AND

Low capital/labor ratio (γ =3-4, rather than γ =6-8)

2. Generalization : for what fraction of the population is capital income > labor income ?

Distribution of labor income yL : 1-G(y) = µ (yµ/y)aL

Distribution of capital stock k : 1-H(k) = µ (kµ/k)aK

bL  = aL / (aL-1)

bK  = aK / (aK-1)

γµ = kµ / yµ

In practice, bK  > bL  (and aK  < aL )  = wealth is more concentrated than labor income

E.g. , bK =2.2,  bL =1.8, µ=10% (i.e. Pareto approximation OK for top 10%)

For any 0<ε<µ, define y(ε) and k(ε) s.t. 1-G(y(ε)) = ε and 1-H(k(ε)) = ε

We have:

y(ε) = yµ  (µ/ε)1-1/bL

k(ε) = kµ  (µ/ε)1-1/bK

Theorem. (a) If bK > bL  then whatever   µ , βµ , r,  there exists 0<ε*<µ such that: (i) If ε<ε*, then r k(ε) > y(ε) ; (ii) If ε>ε*, then r k(ε) < y(ε)

(b) ε* is given by: ε* = µ  / (r γµ)bK bL / (bK – bL)

Note: If γµ = γ, i.e. same wealth/labor income ratio at the level of fractile µ as for the average population, then r γµ  = r γ = α / (1-α)

E.g. r γµ  = 43% if α = 30%, r γµ  = 25% if α = 20%, r γµ  = 11% if α = 10%, etc.

In any case,  r γµ  < 1 (otherwise this would imply that capital income is already larger than labor income at the level of fractile µ)

>>> therefore ε* is an increasing function of bK and a decreasing function of bL

 µ= 10% 10% 10% 10% 10% bl= 1.60 1.60 1.60 1.60 1.60 bk= 2.00 2.25 2.50 2.75 3.00 bk bl /(bk-bl) = 8.00 5.54 4.44 3.83 3.43 r βµ = ε* = 10% 0.00% 0.00% 0.00% 0.00% 0.00% 20% 0.00% 0.00% 0.01% 0.02% 0.04% 30% 0.00% 0.01% 0.05% 0.10% 0.16% 40% 0.01% 0.06% 0.17% 0.30% 0.43% 50% 0.04% 0.22% 0.46% 0.71% 0.93% 60% 0.17% 0.59% 1.03% 1.42% 1.74% 70% 0.58% 1.39% 2.05% 2.55% 2.94% 80% 1.68% 2.91% 3.71% 4.26% 4.65% 90% 4.30% 5.58% 6.26% 6.68% 6.97%

>>> the key point is that the fraction ε* varies in a highly non-linear way: for instance, it is multiplied by 100 (from 0.01% to over 1%) when rβµ is multiplied by 3 (from 20% to 60%); note that it is even more non-linear when the labor and capital Pareto coefficient are close to one another

3. A simple model illustrating the importance of multiplicative dynamic effects

s = savings rate out of labor income

t = estate tax rate

d = number of working years

γ eK = (1-t) [ γ eK + sdeL ]

i.e.: eK = sdeL / tγ

Assume that eK=20% is stable for eL=7%, r=4%, γ =7,5, t=20%, i.e. sd = 3,5 (10% savings rate during 35 years)

Then if the estate tax rate goes from t=20% to t=10%, long term wealth concentration goes from eK=20% to eK=40%