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 (e_{K}) richer than the
top 1% labor earners (e_{L}) ?
e_{K} = top
1% capital share of the distribution of capital stock H(k)
e_{L} = top
1% labor income share of the distribution of labor income G(y)
γ = k/y_{L}
= aggregate capital/labor income ratio
Response : top
capital earners dominate if and only if r
γ e_{K} > _{ }e_{L}
In 2006 :
equilibrium between working rich and rentiers
r=4%, γ = 180 000€ / 24
000€ = 7,5, e_{K}=20%, e_{L}=7%
i.e.: rγ =30%,
r γ e_{K}=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, e_{K}=50%, e_{L}=7%
i.e.: rγ =30%, γe_{K}=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, e_{K}=20%,
e_{L}=7%
i.e.: rγ =14%,
rγe_{K}=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 (e_{K}=20%, rather than e_{K}=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 y_{L} :
1-G(y) = µ (y_{µ}/y)^{aL}
Distribution of capital stock k : 1-H(k) = µ (k_{µ}/k)^{aK}
b_{L} = a_{L }/ (a_{L}-1)
b_{K} = a_{K }/ (a_{K}-1)
γ_{µ}
= k_{µ }/ y_{µ}
In
practice, b_{K} > b_{L} (and a_{K} < a_{L} ) = wealth is more concentrated than labor
income
E.g. , b_{K}
=2.2, b_{L }=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) = y_{µ} (µ/ε)^{1-1/bL}
k(ε) = k_{µ} (µ/ε)^{1-1/bK}
Theorem. (a) If
b_{K} > b_{L} then
whatever µ , β_{µ} ,
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 γ_{µ})^{bK
bL / (bK – bL)}
Note: If γ_{µ}
= γ, i.e. same wealth/labor income ratio at the level of fractile µ as for
the average population, then r γ_{µ
}= r γ = α / (1-α)
E.g. r γ_{µ }= 43% if α = 30%, r γ_{µ }= 25% if α = 20%, r γ_{µ }= 11% if α = 10%, etc.
In any case, r γ_{µ }< 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 b_{K} and a decreasing
function of b_{L}
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
s = savings rate out of labor income
t = estate tax rate
d = number of
working years
γ e_{K} = (1-t) [ γ e_{K} + sde_{L}
]
i.e.: e_{K}
= sde_{L} / tγ
Assume that e_{K}=20%
is stable for e_{L}=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 e_{K}=20% to e_{K}=40%