# Cours avancé « Economie des inégalités » (Master APE, année M2)

Advanced course « Economics of Inequality » (Master APE, M2 year)

Thomas Piketty

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

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

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

>>> France 2007 : capital share α = 30%, capital/output ratio β = 7.6, average capital return r = α/β = 3.9%

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%)

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