Economics of Inequality 

Thomas Piketty

Academic year 2012-2013

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 (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

 

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  ?

life-cyle 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

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€

 

 

 

Research issues:

- dynamics of capital accumulation: k_Li ^t+1   =  k_Li ^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 = 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?)

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/(1-α)

 

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. Cobb-Douglas production functions and beyond: explaining α

 

Cobb-Douglas production function: Y = F(K,L) = K^αL^1-α

(typically, α = 0.25 and 1-α = 0.75)

 

>>> Then for any interest rate r and wage rate v, 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

 

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

See Course Notes on Factors Shares and Production Functions

7. Capital accumulation models: explaining β

 

Basic formula: β = s/g

If s=10% and g=1.5%, then β ≈ 600%

Other way to see the pb: β = α/r

If α=30% and r=5%, then β=600%

Long run stability or divergence?

See Course Notes on Models of Growth, Capital accumulation and Distribution