Economics of Inequality
Thomas
Piketty
Academic
year 20102011
Course Notes : 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}
= vL = labor income
r
= interest rate = average return to capital
v
= 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 (parttime, fulltime, 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
1G(y)
= % of population with labor income > y
Research
issues:

labor supply behaviour, labor supply elasticity, work incentives for low income
vs high income, optimal redistributive taxation of labor income

men vs women labor supply and labor income inequality: female participation,
discrimination, assortative mating according to human K

investment in human capital: returns to education, school inputs, governance
and financing of higher education

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} ?
lifecyle
dimensions (shocks, training, unemployment, retirement), intergenerational
dimensions (intergeneration transmission of human capital)
3. 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
1H(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_{}
Research
issues:

dynamics of capital accumulation: k_{Li }^{t+1}_{ } = k_{Li
}^{t} ?
lifecycle
capital vs inherited capital, age structure of wealth

optimal taxation of capital and capital income

financial intermediation, long chain between household capital and firm
ownership, financial regulation, wealth inequality and efficiency, family firms
4. 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
1S(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 ΅ 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, 1α = labor share)
β = k/y = capital/output ratio (i.e. capital
stock = how many years of income flows?)
θ = k/y_{L}
= capital/labor income ratio (i.e. capital stock = how many years of labor
income flows)
By definition: α =
r β
θ =
β/(1α)
Exemple: France 2010
y = 33 000
y_{L} = 25 000
y_{k} = 8 000
α =
24%
k =
182 000
β =
5.6
θ =
7.3
r = 4.4%
See Distribution of Income & Wealth in France
2010
5. CobbDouglas production functions:
explaining α
CobbDouglas
production function: Y = F(K,L) = K^{α}L^{1}^{α }
(typically,
α = 0.25 and 1α = 0.75)
>>>
Then for any interest rate r and wage rate v, Y_{K} = αY & Y_{L}
= (1α)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} = v
F_{K} = r means α K^{α1 }L^{1}^{α }=
r
I.e. αY/K = r
I.e. Y_{K} = rK = αY
[Alternatively,
F_{L} = w means (1α) K^{α }L^{}^{α }= v
, i.e. (1α)Y/L = v,
i.e. Y_{L} = vL = (1α)Y]
[Putting the capital demand and labor demand equations
together : K/L = [α/(1α)]
v/r, i.e. if the relative price v/r rises by 1%, the capitallabor ratio
increases by 1%, i.e. annihilates the price effect]
>>> with a CobbDouglas
production function, the capital and labor shares are entirely determined by
the production function
6. Beyond CobbDouglas production
functions
In practice, F(K,L) does not seem to be exactly
CobbDouglas: historically, capital share was lower when capital/output was
lower >>> this suggests that the elasticity of substitution is above 1
(or that multisector model: Y = Y_{H}
+ Y_{P} , with Y_{H} = F(K_{H}) vs Y_{P} = F(K_{P},L),
etc.)
Y = F(K,L) = [(1a) L^{(γ1)/ γ }+ a
K^{(γ1)/γ}]^{γ/(γ1)}
= 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)^{11/γ}
i.e. if we note β=K/Y, we have:
r = a β^{1/γ}
α = a β^{11/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 CobbDouglas production function
F(K,L) = K^{α}L^{1}^{α },
α = a does not depend on β: price and quantity effects exactly offset
each other
If γ is infinite, then linear production function
F(K,L) = rK+vL, i.e. fixed capital return r and labor productivity v (labor can
produce output without capital, and conversely), so that capital share
increases proportionally with β
If γ=0, then fixedcoefficient (puttyclay)
production function F(K,L) = min(rK,vL), where r and v are entirely given by
technology: one hour of work produces v units of output iff only we have
exactly v/r units of capital per hour of work, i.e. extra capital is useless; and
conversely capital destructions are devastating: when K is divided by 2, then Y
should be divided by 2 (half of labor becomes useless)
7. Next
step: explaining β
Basic formula: β = s/g
Long run stability or divergence?
See Course
Notes on Models of Growth, Capital accumulation and Distribution