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



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