Economics of Inequality
Thomas
Piketty
Academic
year 2010-2011
Course Notes : 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
= 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
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
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,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)
3. 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
Research
issues:
-
dynamics of capital accumulation: kLi t+1 = kLi
t ?
life-cycle
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 = 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 ΅ 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?)
θ = k/yL
= capital/labor income ratio (i.e. capital stock = how many years of labor
income flows)
By definition: α =
r β
θ =
β/(1-α)
Exemple: France 2010
y = 33 000
yL = 25 000
yk = 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. Cobb-Douglas production functions:
explaining α
Cobb-Douglas
production function: Y = F(K,L) = KαL1-α
(typically,
α = 0.25 and 1-α = 0.75)
>>>
Then for any interest rate r and wage rate v, 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 = v
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 = v
, i.e. (1-α)Y/L = v,
i.e. YL = vL = (1-α)Y]
[Putting the capital demand and labor demand equations
together : K/L = [α/(1-α)]
v/r, i.e. if the relative price v/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
6. 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
F(K,L) = KαL1-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 fixed-coefficient (putty-clay)
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