*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) = Y_{K} + Y_{L}
= output = income = capital income + labor income

K = capital stock

L = labor input

Y_{K} = rK = capital income

Y_{L} = 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

y_{K} =Y_{K }/N = rk
= average capital income

y_{L} = Y_{L }/N = wl
= average labor income

__2. The labor side: distribution of individual labor income____ y _{Li} __

Individual labor supply = l_{i}

Individual labor income y_{Li }= wl_{i}

One should view l_{i} as the
number of efficiency labor units

E.g. l_{i} = e_{i} x h_{i}

With e_{i} = labor hours
(part-time, full-time, etc.)

h_{i} = 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 l_{i}
, and therefore differ in their labor income y_{L}

Implicit assumption = all types of
labor are perfect substitutes, all what matters is the total number of
efficiency labor units

Distribution of labor income y_{L}
:

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 y_{L} =
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 y_{L}^{a} / y_{L}
= Pareto coefficient β = about 1.7-

(y_{L}^{a} = average
income above income threshold y_{L})

>>> 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,L_{U},L_{S}) etc.)

- human capital vs labor market
institutions: minimum wage, unions, governance rules for executive compensation

-dynamics of labor income: y_{Li }^{t+1}_{ } = y_{Li
}^{t} ?

life-cyle dimensions (shocks,
training, unemployment, retirement), intergenerational dimensions
(intergeneration transmission of human capital)

__4. The capital side__

Individual capital stock = k_{i}

Individual capital income y_{Ki }= rk_{i}

One should view individual capital
stock k_{i} 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 y_{K }= rk_{}

Exemple:

If k = 1 000 000€ and r =
5%, y_{K }= rk = 50 000€_{}

If k = 240 000€ and r = 5%, y_{K }= rk = 12 000€_{}

__5. Opening the capital side black box__

- dynamics of
capital accumulation: k_{Li }^{t+1}_{ } = k_{Li
}^{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 = y_{L} + y_{K}
= y_{L} + 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(y_{L})

(ii)
inequality
of capital stock h(k)

(iii)
relative
importance of capital vs labor income: α = rk/y

(iv)
correlation
µ between between g(y_{L}) 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/y _{L} = 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) = K^{α}L^{1-}^{α }

(typically, α = 0.3 and
1-α = 0.7)

**>>> Then for any interest rate r and
wage rate w, Y _{K} = αY & Y_{L} = (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 F_{K} = r &
F_{L} = w

F_{K}
= r means α K^{α-1 }L^{1-}^{α }= r

I.e. αY/K = r

I.e. Y_{K}
= rK = αY

[Alternatively, F_{L}
= w means (1-α) K^{α }L^{-}^{α }= w , i.e.
(1-α)Y/L = w, i.e. Y_{L} = 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 = Y_{H}
+ Y_{P} , with Y_{H} = F(K_{H}) vs Y_{P} = F(K_{P},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 = F_{K} = 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 β^{1-1/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)