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

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

 

Thomas Piketty

Année universitaire 2008-2009

Academic year 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%