Economics of Inequality

Thomas Piketty

Course Notes: Factor Shares and Production Functions

Question: Are labor & capital shares (factor shares) stable in the long run, & why?

1. Standard theory for factor share stability: Cobb-Douglas production function

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 technology: behavior – either labor supply or saving elasticities – does not matter (note however that the assumption of competitive markets – firms maximize profits by taking prices as given – does matter)

2. Beyond Cobb-Douglas : CES 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

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)

3. Beyond Cobb-Douglas: Multi-Sector Production Functions

Another way to go beyond Cobb-Douglas production functions is to relax the homogenous good/single sector assumption. I.e. an important implicit assumption in the formulation Y=F(K,L) is that there exists a homogenous consumption and capital good, i.e. no long run divergence in relative prices; e.g. no divergence in the relative price of land, real estate, oil, services, etc.

For instance, it would probably make more sense to divide national income between a housing sector component (=rental value of housing) and a non-housing, “productive” sector component (=net output of other sectors; “productive” is the wrong term because housing also produces positive value):

Y = YH + YP

with

YH = F(KH) = value of housing services

YP = F(KP,L) = value of output from other productive sectors

On multi-sector growth models, see e.g. Baumol “The macroeconomics of unbalanced growth” AER 1967