Advanced course « Economics of
Inequality » (Master APE, M2 year)
Thomas Piketty
Année universitaire/Academic year
2009-2010
Course Notes A :
Basic Models of Inequality – Capital
vs Labor
1. Notations
Y = F(K,L) = YK + YL
= output = income = capital income + labor income
K = capital stock
L = labor input
YK = rK = capital income
YL = wL = labor income
r = interest rate = average return
to capital
w = wage rate = average labor
compensation
Population i = 1, …, N
y = Y/N = average income
k = K/N = average capital stock
l = L/N = average labor input
yK =YK /N = rk
= average capital income
yL = YL /N = wl
= average labor income
2. The labor side: distribution of individual labor income yLi
Individual labor supply = li
Individual labor income yLi = wli
One should view li as the
number of efficiency labor units
E.g. li = ei x hi
With ei = labor hours
(part-time, full-time, etc.)
hi = human capital (measured
in labor productivity units)
I.e. everybody gets the same wage
rate w, but individuals differ by their number of efficiency labor units li
, and therefore differ in their labor income yL
Implicit assumption = all types of
labor are perfect substitutes, all what matters is the total number of
efficiency labor units
Distribution of labor income yL
:
g(y) = density function
G(y) = distribution function = % of
population with labor income < y
1-G(y) = % of population with labor income
> y
Exemple France 2008:
To fix ideas, let’s say l = 1
corresponds to full-time, minimum-wage labor
France 2008 : w = 12 000€
(annual wage, net of social contributions)
About 10% of workers are at the
minimum wage: G(12 000€) = 10%
Median labor income = about 20 000€ :
G(20 000€) = 50%
Average labor income yL =
about twice the minimum wage = 24 000€
Top 10% = workers above about
48 000€ : G(48 000€) = 90%
Top 1% = workers above about
96 000€ : G(96 000€) = 99%
Ratio yLa / yL
= Pareto coefficient β = about 1.7-
(yLa = average
income above income threshold yL)
>>> Top 1% share = about
7%-8% of total income in France 2008
>>> Top 10% share = about
32%-34% of total income in France 2008
(
(all these numbers are approximative
and illustrative)
>>> See Weeks 3 & 4 on top income shares
3. Opening the labor side black box
- labor supply behaviour, labor
supply elasticity, work incentives for low income vs high income, optimal
redistributive taxation of labor income
>>> see Week 5
- men vs women labor supply and
labor income inequality: female participation, discrimination, assortative
mating according to human K
>>> see Week 8
- investment in human capital:
returns to education, school inputs, governance and financing of higher
education
>>> see Week 8
- more complex production functions,
relaxation of the perfect substitutability assumption between unskilled and
skilled labor (Y = F(K,LU,LS) etc.)
- human capital vs labor market
institutions: minimum wage, unions, governance rules for executive compensation
-dynamics of labor income: yLi t+1 = yLi
t ?
life-cyle dimensions (shocks,
training, unemployment, retirement), intergenerational dimensions
(intergeneration transmission of human capital)
4. The capital side
Individual capital stock = ki
Individual capital income yKi = rki
One should view individual capital
stock ki as the sum of all types of wealth owned by the individual:
stock, bonds, savings accounts, housing, etc.
Implicit assumption = all types of
capital are perfect substitutes and get the same return r, all what matters is
total capital stock
Distribution of capital stock k :
h(k) = density function
H(k) = distribution function = % of
population with capital stock < k
1-H(k) = % of population with
capital stock > k
The distribution of capital stock
h(k) translates mechanically into a distribution of capital income yK = rk
Exemple:
If k = 1 000 000€ and r =
5%, yK = rk = 50 000€
If k = 240 000€ and r = 5%, yK = rk = 12 000€
5. Opening the capital side black box
- dynamics of capital accumulation: kLi t+1 = kLi t ?
life-cycle capital vs inherited
capital, age structure of wealth
>> see Week 2
- optimal taxation of capital and
capital income
>> see Week 6
- financial intermediation, long
chain between household capital and firm ownership, financial regulation,
wealth inequality and efficiency, family firms
>> see Week 7
6. Putting the labor side and the capital side together
Total income y = yL + yK
= yL + rk
Distribution of total income y :
s(y) = density function
S(y) = distribution function = % of
population with total income < y
1-S(y) = % of population with total
income > y
Total inequality S(y) depends on several
factors:
(i)
inequality
of labor income g(yL)
(ii)
inequality
of capital stock h(k)
(iii)
relative
importance of capital vs labor income: α = rk/y
(iv)
correlation
µ between between g(yL) and h(k)
(i.e. to what extent top labor income earners and top capital holders are
the same people?)
α =
rk/y = capital income share in total income (α = capital share, 1-α =
labor share)
β =
k/y = capital/output ratio (i.e. capital stock = how many years of income
flows?)
γ = k/yL = capital/labor income
ratio (i.e. capital stock = how many years of labor income flows)
By definition: α =
r β
γ = β/(1-α)
Exemple:
If
capital/output ratio β = 6 and interest rate r=5%, then capital share
α = 30%, labour share 1-α = 70%, γ = 6/0.7 = 8.6
7. Cobb-Douglas production functions
Cobb-Douglas production function: Y
= F(K,L) = KαL1-α
(typically, α = 0.3 and
1-α = 0.7)
>>> Then for any interest rate r and
wage rate w, YK = αY & YL = (1-α)Y
Intuition:
with an elasticity of substitution between K and L equal to 1, the substitution
effect exactly compensates the price effect
Demonstration:
Take r and w as given. Then profit maximization leads to FK = r &
FL = w
FK
= r means α Kα-1 L1-α = r
I.e. αY/K = r
I.e. YK
= rK = αY
[Alternatively, FL
= w means (1-α) Kα L-α = w , i.e.
(1-α)Y/L = w, i.e. YL = wL = (1-α)Y]
[Putting
the capital demand and labor demand equations together : K/L = [α/(1-α)] w/r, i.e. if the relative price w/r rises by
1%, the capital-labor ratio increases by 1%, i.e. annihilates the price effect]
>>> with a Cobb-Douglas production
function, the capital and labor shares are entirely determined by the production
function
8. Beyond Cobb-Douglas production functions
In
practice, F(K,L) does not seem to be exactly Cobb-Douglas: historically, capital
share was lower when capital/output was lower >>> this suggests that
the elasticity of substitution is above 1
(or that multi-sector
model: Y = YH
+ YP , with YH = F(KH) vs YP = F(KP,L),
etc.)
Y =
F(K,L) = [(1-a) L(γ-1)/ γ + a K(γ-1)/γ]γ/(γ-1)
= CES
production function with elasticity of substitution between K and L = γ
Then if
competitive markets r = FK = a K-1/γ Y-1/γ
I.e.
α = capital share = rK/Y = a (K/Y)1-1/γ
i.e. if
we note β=K/Y, we have:
r = a β-1/γ
α =
a β1-1/γ
I.e. r is
always a declining function of β, but α is an increasing function of
β if and only if γ>1, i.e. elasticity of substitution higher than
1
If
γ=1, then Cobb-Douglas production function, α = a does not depend on
β: price and quantity effects exactly offset each other
If γ
is infinite, then linear production function, i.e. fixed capital return r, so
that capital share increases proportionally with β
If
γ=0, the putty-clay production function, i.e. extra capital is useless
(and conversely capital destructions are devastating: when K/Y is divided by 2,
output should be divided by 2)