Advanced course « Economics of
Inequality » (Master APE, M2 year)
Thomas Piketty
Année universitaire/Academic year
2008-2009
Course Notes A :
Basic Models of Inequality – Capital
vs Labor – Road Map of the Course
1. Basics
Y = F(K,L) = YK + YL
= output = 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
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
Simplifying 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:
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%
(all these numbers are approximative
and illustrative)
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.
Simplifying 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
- capital structure: housing capital
vs productive capital, capital intensity if capital goods production vs
consumption goods production, etc.
>>> virtually no research;
things to do on housing vs productive capital? (YH = F(KH)
vs YP = F(KP,L), etc.)
- endogenous returns to capital: yki
= ri ki
>>> virtually no research,
things to do?
- dynamics of capital accumulation: kLi t+1 = kLi t ?
life-cycle capital vs inherited
capital, age structure of wealth
>> see Week 6
- men vs women capital stock,
assortative mating according to k
>>> virtually no research,
things to do on assortative mating (wealth and wedding, e.g. with EP 2004)
- financial intermediation, long
chain between household capital and firm ownership, financial regulation,
wealth inequality and efficiency, family firms
>> see Week 7
- global perspective on world wealth
distribution
>> see Week 4
- capital supply elasticity, optimal
taxation of capital and capital income
>> see Week 6
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
= exactly the capital and labor
shares that what one would get with a Cobb-Douglas production function Y =
F(K,L) = KαL1-α , with α = 0.3 and
1-α = 0.7
I.e. with a Cobb-Douglas production
function, the capital and labor shares are entirely determined by the
production function
Note that the capital/output β depends upon other parameters, e.g.
dynamic savings behaviour. For instance, with a dynastic utility
function Ut = ∑t≥0 U(ct)/(1+θ)t,
where θ is the rate of time preference, the steady-state interest rate r*
is entirely determined by the Golden rule of capital accumulation: r* = θ.
The steady-state capital/output ratio is then entirely determined by β =
α / r .
>>> see Week 2-4
7. Illustration of the basic concepts using
National Accounts
Simple computations using National Income and Capital Accounts
[Tableau économique d'ensemble, TEE, France 2007 (in excel format)] [(in pdf format)]
French
GDP, 2007 = 1 892.2 billions €
By
definition, GDP = Gross Domestic Product = gross value-added (PIB = Produit intérieur brut = valeur ajoutée brute)
[more
precisely, GDP = sum of gross value-added from the various sectors (corporate
sector, public sector, household sector)
+ product taxes (VAT) ]
Capital
depreciation (Consommation de capital
fixe) = 252.2 billions € ( =
typically 10-15% of GDP)
NDP = Net
National Product = GDP – Capital depreciation (PIN = Produit intérieur net = PIB – CCF) = 1 892.2 – 252.2 =
1 640.1 billions €
= net value-added (valeur ajoutée nette)
Capital
and labor shares α and 1-α in value-added (national economy):
Total
gross value-added = Wage bill (Rémunération des salariés) + Gross
profits (Excédent brut d’exploitation +
Revenu mixte brut) + Product taxes
I.e.
1 892.2 = 976.3 + 661.5 (537.7 +
123.9) + 254.4 (289.7 - 35.3)
>>>
Labor share = 976.3/(976.3+661.5) = 60%
Capital share = 661.5/(976.3+661.5) =
40%
Total net
value-added = Wage bill (Rémunération des salariés) + Net
profits (Excédent brut d’exploitation +
Revenu mixte brut - CCF) + Product taxes
I.e.
1 640.1 = 976.3 + 409.4 (302.0 +
107.4) + 254.4 (289.7 - 35.3)
>>> Labor share = 976.3/(976.3+409.4) =
70% = 1-α
Capital share = 409.4/(976.3+409.4) = 30% = α
Capital
and labor shares α and 1-α in value-added (corporate sector,
non-financial + financial):
Corporate
gross value-added = Wage bill (Rémunération des salariés) + Gross
profits (Excédent brut d’exploitation)
+ Product taxes
I.e.
1 035.1 (957.1 + 78.1) = 672.1
(622.8 + 49.3) + 322.5 (299.0 + 23.5) + 40.6 (51.6 + 5.7 -16.4 -0.4)
>>>
Labor share = 672.1/(672.1+322.5) = 68%
Capital share = 322.5/(672.1+322.5) =
32%
Corporate
net value-added = Wage bill (Rémunération des salariés) + Net
profits (Excédent brut d’exploitation -
CCF) + Product taxes
I.e.
878.0 (810.9 + 67.1) = 672.1 (622.8 +
49.3) + 165.3 (152.8 + 12.5) + 40.6 (51.6 + 5.7 -16.4 -0.4)
>>>
Labor share = 672.1/(672.1+165.3) = 80%
Capital share = 322.5/(672.1+165.3) =
20%
Capital/output ratio β
(national economy)
Net wealth (Valeur nette du patrimoine) = 12 512.3 billions €
Net
wealth / GDP = 12 512.3/1 892.2 = 6.6
Net
wealth / NDP = 12 512.3/1 640.1 = 7.6 = β
>>>
Capital/output ratio β
(household sector)
Net household wealth (Valeur nette du patrimoine) = 9 389.9 billions €
Gross
household income (Solde des revenus
primaires bruts) = 1 399.6 billions €
(1 399.6
= 984.5 + 290.9 (167.0+123.9) + 124.3 (161.6-27.3) >>> labor share =
984.5/1399.6 = 70%, capital share = 30%)
Net
household income (Solde des revenus
primaires nets) = 1 352.0 billions €
(1 352.0
= 984.5 + 243.2 (135.9+107.4) + 124.3 (161.6-27.3) >>> labor share =
984.5/1352.0 = 73%, capital share = 27%)
Total
adult population (20-year-old and over) = about 45 millions
Average wealth per adult = 9 389.9 billions
/ 45 millions = 209 000 €
Average income per adult = 1 352.2 billions
/ 45 millions = 30 000 €
Wealth/income ratio = 9 389.9/1 352.0 =
209 000/30 000 = 6.9