<p>The OP has correctly defined the &quot;isoquant elasticity&quot;, for which one could argue that it should be the basic definition of &quot;elasticity of substitution&quot;.</p>
<p>The reason why Hicks originally defined the &quot;elasticity of substitution&quot; by the much more complicated formula that we know, is that this more complicated formula involves essentially <em>prices</em> (through the MRS and optimizing behavior) and hence it has to do with the function of markets and economic optimization and choices proper, while the much more intuitive and simpler &quot;isoquant elasticity&quot; is in principle a purely &quot;technical&quot; metric, related to technology, even if we consider a broader concept of &quot;technology&quot; than machines and recipes. And economics, even in production theory, is not about &quot;production engineering&quot;... It so happens that I have been sketching a paper on exactly this metric for some time now, and whether and how it could be useful... maybe in business administration, but no so much in economics.</p>

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2022-11-17T23:05:08.540Z
2022-11-18T19:49:55.640Z
2022-11-18T19:49:55.640Z
Alecos Papadopoulos
<p>Isoquants - the level sets of a production function <span class="math-container">$f$</span> - are very useful in microeconomics. For example, if we hold all but two inputs fixed, then the isoquant is a plane curve that quantifies the substitutability of those two inputs. Is there a standard term for the elasticity <span class="math-container">$E$</span> of an isoquant? This seems like a natural quantity to consider, since it tells you that a reduction of input 1 by a small percentage can be made up for by an increase of input 2 by <span class="math-container">$E$</span> times that percentage in order to hold output constant.</p>
<p>I know about two closely related concepts, but neither of them seem to be quite this:</p>
<ol>
<li>The <a href="https://en.wikipedia.org/wiki/Marginal_rate_of_technical_substitution" rel="nofollow noreferrer">marginal rate of technical substitution</a> (MRTS) gives the (negative) <em>slope</em> (i.e. derivative) of the isoquant, but not its elasticity.</li>
<li>The <a href="https://en.wikipedia.org/wiki/Elasticity_of_substitution" rel="nofollow noreferrer">elasticity of substitution</a> is similar, but (I believe) more complicated. As I understand it, it's the elasticity of the ratio of the ratio of inputs to the MRTS. This notion seems way more complicated than just the elasticity of the isoquant, which seems much more intuitive to me.</li>
</ol>
<p>Note that the elasticity of an isoquant of two inputs <span class="math-container">$x$</span> and <span class="math-container">$y$</span> equals
<span class="math-container">$$\frac{\partial y}{\partial x} \frac{x}{y} = -\mathrm{MRTS} \frac{x}{y} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} \frac{x}{y} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} \frac{\frac{x}{f}}{\frac{y}{f}},$$</span>
which is the (negative) ratio of the <a href="https://en.wikipedia.org/wiki/Output_elasticity" rel="nofollow noreferrer">partial output elasticities</a> with respect to the two inputs. So maybe there's no special term for this isoquant elasticity, since it's simply a ratio of two other standard quantities.</p>

<production-function><elasticity><marginal><elasticity-of-substitution>
Is there a standard term for the elasticity of an isoquant?
tparker
# Is there a standard term for the elasticity of an isoquant?

(Asked by: username_0 on 2022-11-13)

Isoquants - the level sets of a production function $f$ - are very useful in microeconomics. For example, if we hold all but two inputs fixed, then the isoquant is a plane curve that quantifies the substitutability of those two inputs. Is there a standard term for the elasticity $E$ of an isoquant? This seems like a natural quantity to consider, since it tells you that a reduction of input 1 by a small percentage can be made up for by an increase of input 2 by $E$ times that percentage in order to hold output constant.

I know about two closely related concepts, but neither of them seem to be quite this:

1. The [marginal rate of technical substitution](https://en.wikipedia.org/wiki/Marginal_rate_of_technical_substitution) (MRTS) gives the (negative) *slope* (i.e. derivative) of the isoquant, but not its elasticity.
2. The [elasticity of substitution](https://en.wikipedia.org/wiki/Elasticity_of_substitution) is similar, but (I believe) more complicated. As I understand it, it's the elasticity of the ratio of the ratio of inputs to the MRTS. This notion seems way more complicated than just the elasticity of the isoquant, which seems much more intuitive to me.

Note that the elasticity of an isoquant of two inputs $x$ and $y$ equals
$$\frac{\partial y}{\partial x} \frac{x}{y} = -\mathrm{MRTS} \frac{x}{y} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} \frac{x}{y} = -\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} \frac{\frac{x}{f}}{\frac{y}{f}},$$
which is the (negative) ratio of the [partial output elasticities](https://en.wikipedia.org/wiki/Output_elasticity) with respect to the two inputs. So maybe there's no special term for this isoquant elasticity, since it's simply a ratio of two other standard quantities.

**Tags:** production-function, elasticity, marginal, elasticity-of-substitution

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## 1 answers

### Answer by username_1 on 2022-11-17. Score: 2

The OP has correctly defined the "isoquant elasticity", for which one could argue that it should be the basic definition of "elasticity of substitution".

The reason why Hicks originally defined the "elasticity of substitution" by the much more complicated formula that we know, is that this more complicated formula involves essentially *prices* (through the MRS and optimizing behavior) and hence it has to do with the function of markets and economic optimization and choices proper, while the much more intuitive and simpler "isoquant elasticity" is in principle a purely "technical" metric, related to technology, even if we consider a broader concept of "technology" than machines and recipes. And economics, even in production theory, is not about "production engineering"... It so happens that I have been sketching a paper on exactly this metric for some time now, and whether and how it could be useful... maybe in business administration, but no so much in economics.


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