# Piketty's explanation of elasticity of substitution (from his book Capital in the 21st century)

Economics Asked by Nasan on July 31, 2020

I have some trouble following the explanation of the elasticity of substitution between capital and labor and its implications on p189.

Take this part:

The relevant question is whether the elasticity of substitution
between labor and capital is greater or less than one. If the
elasticity lies between zero and one, then an increase in the
capital/income ratio β leads to a decrease in the marginal
productivity of capital large enough that the capital share α = r × β
decreases (assuming that the return on capital is determined by its
marginal productivity).

Specifically, when he talks about that elasticity, does he mean something like (dL/L)/(dK/K) with L for labor and K for capital? Assuming that, how exactly does a higher elasticity translate in a lower fall in marginal product of capital?

Thanks

The following is from Thomas Piketty and Gabriel Zucman (2015, From Handbook of Income Distribution, Volume 2, Chapter 15, Part 15.5.3 which is hard to link to directly but get it here):

Take a CES production function $$Y=F(K,L)=(a⋅K^{frac{sigma-1}{sigma}}+(1−a)⋅L^{frac{sigma-1}{sigma}})^{frac{sigma}{1-sigma}}$$

Me: $$sigma$$ is the elasticity of substitution. They have a small typo here with the outer exponent of $$frac{sigma-1}{sigma}$$ instead of $$frac{sigma}{1-sigma}$$, but I'm pretty sure that is wrong so I corrected it here. It doesn't change the rest of what I'm doing.

Back to Piketty and Zucman:

The rate of return is given by $$r = F_{K} = a cdot beta^{frac{-1}{sigma}}$$ with $$beta = K/Y$$.

The capital share is given by $$alpha = rcdotbeta = acdotbeta^{frac{sigma-1}{sigma}}$$

Me: Now take the partial derivative of $$alpha$$ with respect to to $$beta$$ $$frac{partial alpha}{partial beta} = a cdot (1-1/sigma) cdot beta ^{-1/sigma}$$

By assumption $$a$$ is always positive. Because $$K$$ and $$Y$$ are always positive so is $$beta ^{-1/sigma}$$. That means the sign of $$alpha$$ is the sign of $$1-1/sigma$$. For $$0 you can see that $$1-1/sigma < 0$$ and therefore $$frac{partial alpha}{partial beta} < 0$$.

Answered by BKay on July 31, 2020

## Related Questions

### Theoretical models of copyright infringement and efficiency implications

0  Asked on May 22, 2021

### Is there any pressure to keep currency exchange rates around 1?

2  Asked on May 21, 2021 by andreitoroplean

### How to define the natural rate output

1  Asked on May 21, 2021

### Can anyone help me understand the Motrtensen-Pissarides model?

1  Asked on May 20, 2021

### Models with Learning by Doing and Knowledge Spillovers – Barro, Sala-i-Martin (2003)

1  Asked on May 20, 2021

### Where can I retrieve a complete history of macro economic data annoucements?

1  Asked on May 20, 2021 by hariboy

### Why high levels of inflation doesn’t leads to high levels of employment?

1  Asked on May 19, 2021 by vernica-rmz

### An Extension to CES Demand

0  Asked on May 19, 2021 by alalalalaki

### Why can the job offer have a negative relationship between salary and quantity of work?

1  Asked on May 19, 2021

### Measurement for the downstream use of a commodity

1  Asked on May 18, 2021 by mgint

### What is used for calculating equity multiplier between total liabilities and total debts from Datastream?

0  Asked on May 17, 2021 by beautifulmindset

### Choice between dummy variables and Likert scale in Linear Regression

1  Asked on May 17, 2021

### Applying modern economics to Edgeworth’s Hedonical Calculus

1  Asked on May 17, 2021

### Regressing (Very) Smooth Time Series

1  Asked on May 17, 2021 by chopschoc

### What does “steep incentive contract” mean in the context of adverse selection?

1  Asked on May 16, 2021

### Dynamic model calibration of rate parameters

0  Asked on May 16, 2021

### prodest package in R, TFP growth estimation

0  Asked on May 14, 2021

### K-shaped recovery – is there a benign explanation for it?

0  Asked on May 14, 2021

### Is it true that if marginal cost is constant, then average variable cost is also constant and equals marginal cost?

1  Asked on May 14, 2021

### Would someone be able to help me solve capital per capita in the steady state (check my work)

1  Asked on May 14, 2021