# Talk:Elasticity of substitution

This post does not make it clear that the MRS is the ratio of the marginal utilities WRT the two goods, so it would benefit from clarification of the notation.

## Missing a Minus?

I believe the definition of elasticity of substitution typically includes a minus sign. That is, I think for two goods ${\displaystyle x_{i}}$ and ${\displaystyle x_{j}}$ and utility function ${\displaystyle u}$ that the definition ought to be

${\displaystyle \xi _{ij}=-{\frac {\partial \ln(x_{i}/x_{j})}{\partial \ln(u_{i}/u_{j})}}}$

where ${\displaystyle u_{i}={\frac {\partial u}{\partial x_{i}}}}$ as per MWG page 97. However, I may be wrong, or there may be notational differences, so I'm writing here instead of just making the change on the page.Coleca 06:13, 13 May 2007 (UTC)

## Limit definition?

This notation has always confused me. At least with CRRA (constant relative risk aversion) utility, I prefer (Uc1/Uc2)/(c2/c1) / (d(Uc1/Uc2)/d(c2/c1)) At least, I could write this in terms of the limit definition of a derivative. Could the current notation be written in such a way to make clear How the derivative is taken? —Preceding unsigned comment added by 24.184.26.164 (talk) 21:52, 8 December 2008 (UTC)

## The full name & geometric meaning

I will refine this later. It should be relative price elasticity of relative demand. the marginal rate of substitution (as the denominator) is equal to the relative price. Why relative because we are considering 2 demands rather that 1 demand, and we also have 2 prices rather than 1 price.

I often forgot how the shit is defined because of the name is not good. Analog to price elasticity of demand which is defined as ${\displaystyle {\frac {\partial \ln Q}{\partial \ln P}}}$, then how should ${\displaystyle {\frac {\partial \ln {\frac {c_{2}}{c_{1}}}}{\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}}}$ be named? (The marginal rate of substitution is always ${\displaystyle =-{\frac {dc_{2}}{dc_{1}}}}$ or its inverse.) If we draw the curve between ${\displaystyle {\frac {c_{2}}{c_{1}}}}$ and ${\displaystyle -{\frac {dc_{2}}{dc_{1}}}}$. Let's also write down its inverse ${\displaystyle {\frac {\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}{\partial \ln {\frac {c_{2}}{c_{1}}}}}}$, then the latter one should be named as the relative allocation elasticity of the marginal rate of substitution, rather than "Elasticity of complementarity". If there is a term as the quantity elasticity of price, then the former should be named as the marginal rate of substitution elasticity of the relative allocation, rather than "elasticity of substitution" which is very confusing and can not be easily understood.

Another way to remember the definition is to associate the two with their geometric meaning. Let's restate its geometric meaning: given a indifference curve with 2 variables, for any point on the curve, we draw 2 lines. 1 line is from the point to the origin, another line is the tangent line through the point.

• ${\displaystyle {\frac {\partial \ln {\frac {c_{2}}{c_{1}}}}{\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}}}$ says along the indifference curve, the percentage change of the 1st slope (to the origin) when the 2nd slope (the tangent slope) change 1%.
• ${\displaystyle {\frac {\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}{\partial \ln {\frac {c_{2}}{c_{1}}}}}}$ says along the indifference curve, the percentage change of the 2nd slope when the 1st slop change 1%.

We define these two animals because 2 variables are of the same nature, while for the demand curve, 2 variables are not of the same nature, so the meaning of the above 2 definitions is hard to interpret.

As for the price elasticity of a demand, its geometric meaning answers the following question: Along the demand curve, 1% change on the price, how much percentage change in quantity will occur?

The question is that, then, why for the indifference don't we define something similar to the price elasticity of a demand? If we do, the economic meaning is still easy to interpret. There is no reason not to use it, if you prefer, you can use it. But a constant price elasticity demand is simple, i.e. ${\displaystyle Q=aP^{c}}$ where a & c are parameter, and the constant price elasticity is c and ${\displaystyle c\leq 0}$. In terms of indifference curve, the curve should be ${\displaystyle c_{2}=ac_{1}^{c}}$.

For CRRA utility, ${\displaystyle u\left(c\right)={\frac {c^{1-\rho }}{1-\rho }}}$ with ${\displaystyle u^{'}\left(c\right)=c^{-\rho }}$. At first, we should construct an indifference curve, ${\displaystyle u\left(c_{1},c_{2}\right)=u\left(c_{1}\right)+\beta u\left(c_{2}\right)}$.

Then with ${\displaystyle 0=du\left(c_{1},c_{2}\right)=u^{'}\left(c_{1}\right)dc_{1}+\beta u^{'}\left(c_{2}\right)dc_{2}}$, we have ${\displaystyle {\text{MRS}}={\frac {u^{'}\left(c_{1}\right)}{\beta u^{'}\left(c_{2}\right)}}=-{\frac {dc_{2}}{dc_{1}}}={\frac {c_{1}^{-\rho }}{\beta c_{2}^{-\rho }}}={\frac {1}{\beta }}\left({\frac {c_{2}}{c_{1}}}\right)^{\rho }}$, so ${\displaystyle \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)=-\ln \beta +\rho \ln {\frac {c_{2}}{c_{1}}}}$, and ${\displaystyle {\frac {\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}{\partial \ln {\frac {c_{2}}{c_{1}}}}}=\rho }$.

And with ${\displaystyle 0=du\left(c_{1},c_{2}\right)=u^{'}\left(c_{1}\right)d\ln \left(c_{1}\right)+\beta u^{'}\left(c_{2}\right)d\ln \left(c_{2}\right)}$, i.e. ${\displaystyle 0=c_{1}^{1-\rho }d\ln \left(c_{1}\right)+\beta c_{2}^{1-\rho }d\ln \left(c_{2}\right)}$, so ${\displaystyle {\frac {d\ln \left(c_{2}\right)}{d\ln \left(c_{1}\right)}}=-{\frac {1}{\beta }}\left({\frac {c_{1}}{c_{2}}}\right)^{1-\rho }}$.

Jackzhp (talk) 00:18, 18 April 2010 (UTC)