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 $x_{i}$ and $x_{j}$ and utility function $u$ that the definition ought to be

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

where $u_{i}={\frac {\partial u}{\partial x_{i}}}$ 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 ${\frac {\partial \ln Q}{\partial \ln P}}$, then how should ${\frac {\partial \ln {\frac {c_{2}}{c_{1}}}}{\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}}$ be named? (The marginal rate of substitution is always $=-{\frac {dc_{2}}{dc_{1}}}$ or its inverse.) If we draw the curve between ${\frac {c_{2}}{c_{1}}}$ and $-{\frac {dc_{2}}{dc_{1}}}$. Let's also write down its inverse ${\frac {\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}{\partial \ln {\frac {c_{2}}{c_{1}}}}}$, 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.

• ${\frac {\partial \ln {\frac {c_{2}}{c_{1}}}}{\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}}$ says along the indifference curve, the percentage change of the 1st slope (to the origin) when the 2nd slope (the tangent slope) change 1%.
• ${\frac {\partial \ln \left(-{\frac {dc_{2}}{dc_{1}}}\right)}{\partial \ln {\frac {c_{2}}{c_{1}}}}}$ 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. $Q=aP^{c}$ where a & c are parameter, and the constant price elasticity is c and $c\leq 0$. In terms of indifference curve, the curve should be $c_{2}=ac_{1}^{c}$.

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

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

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

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

Empirical estimates

Estimates of sigma 'σ' between capital and labor for U.S.:

• Chirinko (2008) summarize some estimates of sigma that lead to a value of σ in the range of 0.40-0.60. <ref>Chirinko, Robert S., 2008. "[sigma]: The long and short of it," Journal of Macroeconomics, Elsevier, vol. 30(2), pages 671-686, June<ref>
• Young (2013) collected data of "35 industries at roughly the 2-digit SIC level, 1960-2005", which resulted in a σ value of less than 0.620 for the aggregate U.S..<ref>Young, Andrew T., 2013. "U.S. Elasticities Of Substitution And Factor Augmentation At The Industry Level," Macroeconomic Dynamics, Cambridge University Press, vol. 17(04), pages 861-897, June. <https://ideas.repec.org/a/cup/macdyn/v17y2013i04p861-897_00.html<ref>
• Chirinko and Mallick (2017) estimate a σ of 0.40.<ref> Robert S. Chirinko & Debdulal Mallick, 2017. "The Substitution Elasticity, Factor Shares, and the Low-Frequency Panel Model," American Economic Journal: Macroeconomics, American Economic Association, vol. 9(4), pages 225-253, October. <https://ideas.repec.org/a/aea/aejmac/v9y2017i4p225-53.html><ref>
• Knoblach, Roessler and Zwerschke (2019) "estimate a long-run meta-elasticity for the aggregate economy in the range of 0.45-0.87".<ref>Knoblach, Michael & Roessler, Martin & Zwerschke, Patrick. (2016). The Elasticity of Factor Substitution Between Capital and Labor in the U.S. Economy: A Meta-Regression Analysis. https://www.researchgate.net/publication/316544066_The_Elasticity_of_Factor_Substitution_Between_Capital_and_Labor_in_the_US_Economy_A_Meta-Regression_Analysis. 10.13140/RG.2.2.13719.78244.<ref>

Knoblach and Stockl (2019) mention the lack in the estimation of the σ for the "other countries, especially developing countries".<ref>Knoblach, Michael & Stöckl, Fabian. (2019). What determines the elasticity of substitution between capital and labor? A literature review.https://www.researchgate.net/publication/330563679_What_determines_the_elasticity_of_substitution_between_capital_and_labor_A_literature_review<ref> — Preceding unsigned comment added by WiPuWiF (talk • contribs) 10:15, 10 July 2019 (UTC)

Is it a measure of the curvature?

The current description states that it gives a measure of the curvature of an isoquant, yet the cited source (de La Grandville, 1997) clearly stated quite the opposite: there is no link between curvature and the elasticity of substitution. Therefore,

• The source corresponding to the statement about curvature is not given.
• The debates regarding the relationship between isoquant and the elasticity of substitution is not given.

Addone (talk) 06:16, 8 April 2023 (UTC)

"Isoquant" usually refers to level curves, aka contour lines, of a production function, so the term may not apply well to all the examples considered in this wiki page - but mathematically the economic interpretation of the function whose levels sets we consider is irrelevant. You can find formulas for the curvature of a -twice differentiable- curve in the appropriate wiki page. You can manipulate them to find that, as claimed by Olivier de La Grandville, the elasticity of substitution is never a curvature - except i imagine in some trivial cases. But it is easier to see the discrepancy with a simple counterexample, take a 2 factor Cobb-Douglas production functions whose elasticity of substitution is computed here. This elasticity is constant - Cobb-Douglas functions are examples of CES functions - but the level curves of these functions are arbitrary degree plane analytic curves thus have nonconstant curvature - only straight lines and circles have constant curvature in the plane. If you prefer you can restrict to integer values of $a$, where the "degree" is that of a polynomial - the degree of a plane analytic curve is not trivial to define, but here it is equal to $a$, so we can just define it as $a$. I don't have access to the article you quote, but it seems it contains thorough justifications, and i trust his assertion that « This paper demonstrates that, contrary to deeply rooted beliefs, there is no link between curvature and the elasticity of substitution. »

Thus i agree with you that the page should be modified. Regarding the relationship between isoquants and ES one can trivially recover the production function $f$ from the data of all its isoquants, $L(y)=f^{-1}(y)$ for all $y$ in the codomain of $f$, the union of which is the domain of $f$; so the isoquants determine the ES, but that will be with the formulas given in this page, not with a curvature in general - it must be a good exercise to find the functions for which level curve curvature and ES are equal, but im not a keen computer, perhaps that is in de La Grandville's paper.

I finish by commenting one other point i find unclear: referring to percentages when they are unnecessary is misleading. It would be better to use "relative change" rather than "percentage change" - the logarithmic derivative gives the relative change of a function, for instance the identity function $f(x)=x$ has relative change ${\frac {f'}{f}}(x)={\frac {1}{x}}$. Plm203 (talk) 00:52, 19 August 2024 (UTC)