<p>Elasticity of substitution is a property of a function, like slope or curvature. Constant elasticity of substitution means that this property has the same value at all arguments of the function.</p>

<p>There are many explanations of the elasticity of substitution online, e.g. <a href="http://en.wikipedia.org/wiki/Elasticity_of_substitution" rel="nofollow">http://en.wikipedia.org/wiki/Elasticity_of_substitution</a>
If after reading these, you are still confused, you can ask a more specific question.</p>

CC BY-SA 3.0
2015-04-20T07:06:33.670Z
Why did this get down-voted?? It seems precisely correct to me.
jmbejara
CC BY-SA 3.0
2015-04-20T08:19:24.730Z
An alternative reference, which I found to be quite clear is here: http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/elasticity%20of%20substitutionrevised.tex.pdf
Ubiquitous
CC BY-SA 3.0
2015-04-20T03:47:53.347Z
2015-04-20T03:47:53.347Z
Sander Heinsalu
<p>Well, you didn't get the math <em>all</em> right, otherwise you might have seen it yourself! </p>

<p>By Elasticity of substitution, you <em>should</em> refer to</p>

<p>$$ \sigma(x,y) = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U_y}{U_x}\right)} $$</p>

<p>Then, constant elasticity of substitution is a property of a function $U$ such that for some constant $\bar \sigma$,</p>

<p>$$ \sigma(x,y) = \bar\sigma \, \forall y, x$$</p>

<p>That is, the elasticity doesn't depend on $x,y$. What you showed as $U$ is a function that has exactly that feature. If you compute $\sigma(x,y)$ (do it!), you will see that</p>

<p>$$ \sigma(x,y) = \bar\sigma = \frac{1}{1+\rho}$$</p>

<p>That is, the elasticity of your $U$ function is constant for any allocation. Hence, we refer to that class of function (for different $\rho$) to Constant-Elasticity-of-Substitution-Functions (with share parameters $\alpha$). Cobb-Douglas preferences would be a special (limiting) case of that, with $\sigma(x,y) = 1$</p>

<h3>Example</h3>

<p>Say, we produce happiness ($U$) with apples ($x$) and oranges ($y$). Lets look at the allocation $\{5,1\}$. We only have one orange, but we would <em>really</em> want more. Hence, we are willing to give up 2 apples to get one orange, and we would be indifferent:</p>

<p>$$ U(5,1) = U(3,2)$$</p>

<p>It is <em>very easy</em> to substitute apples with oranges, we have a high elasticity of substitution. </p>

<p>Now look at the allocation $\{1,6\}$. Do you think that you would again be willing to lose $40\%$ of $x$ to get an increase of $50\%$ in $y$? Is it true that</p>

<p>$$ U(1,6) = U(0.6, 12)$$</p>

<p>If yes, that means that the elasticity of substitution of $U$ (or my rather handwaving approximation to it) is the same at the two allocations $\{1,6\}$ and $\{5,1\}$. If that was true for <em>all</em> possible allocations, $U$ was CES.</p>

<p>There are some reasons why you would think some preferences are perhaps not CES: For example, if $x$ and $y$ are good complements. Then, when you have little of $x$, you're willing to give up a lot for the other, but that elasticity might decrease the closer you are to equality between the two inputs.</p>

CC BY-SA 3.0
2016-10-02T01:20:43.660Z
@FooBar Upvote, your example explains very clearly what is the interpretation of constant elasticity of substitution utilities/preferences.
deps_stats
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2015-04-20T13:56:24.593Z
But what does that mean? I got that answer. Like I don't understand why the $U$ function has to be so ugly looking.
Stan Shunpike
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2015-04-20T14:08:13.557Z
@StanShunpike I've added an example, hope that helps.
FooBar
CC BY-SA 3.0
2015-04-20T15:18:50.353Z
@StanShunpike it doesn't *have* to be so ugly looking, that's just a very general form. You can set $A = 1$, remove the $\alpha$, $1-\alpha$ terms and still have CES.
FooBar
CC BY-SA 3.0
2015-04-20T16:19:47.790Z
Thanks for your replies. I still am confused, so let me try to be more specific: What does $A$ do? What is $\alpha$ do? Is $\alpha$ related to strict convexity? Why do we the the $-\rho$ power of the goods?
Stan Shunpike
CC BY-SA 3.0
2015-04-20T16:59:35.653Z
@StanShunpike I suggest you open up a new question about this, as this one was about the difference between ES and CES and I feel like we're diverging.
FooBar
CC BY-SA 3.0
2015-04-20T17:03:25.670Z
Okay, I will do that.
Stan Shunpike
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2015-04-20T13:55:00.257Z
2015-04-20T14:07:59.507Z
2015-04-20T14:07:59.507Z
FooBar
<p>What is the difference between constant elasticity of substitution and elasticity of substitution? Are these formulas related? How are they different concepts? <strong>I just did a whole problem set involving these and got the right answers. But my thought process was all math. I don't feel like I understand them economically. And that's what I want explained.</strong></p>

<p>By CES, I mean
$$U(x,y) = A \left(\alpha x^{-\rho} + (1-\alpha) y^{-\rho}\right)^{-\frac{1}{\rho}}$$</p>

<p>By ES, I mean </p>

<p>$$\sigma = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U_y}{U_x}\right)}$$</p>

<elasticity>
What is the difference between constant elasticity of substitution and elasticity of substitution?
Stan Shunpike
# What is the difference between constant elasticity of substitution and elasticity of substitution?

(Asked by: username_0 on 2015-04-20)

What is the difference between constant elasticity of substitution and elasticity of substitution? Are these formulas related? How are they different concepts? **I just did a whole problem set involving these and got the right answers. But my thought process was all math. I don't feel like I understand them economically. And that's what I want explained.**

By CES, I mean
$$U(x,y) = A \left(\alpha x^{-\rho} + (1-\alpha) y^{-\rho}\right)^{-\frac{1}{\rho}}$$

By ES, I mean

$$\sigma = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U\_y}{U\_x}\right)}$$

**Tags:** elasticity

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## 2 answers

### Answer by username_4 on 2015-04-20. Score: 4 | Accepted answer

Well, you didn't get the math *all* right, otherwise you might have seen it yourself!

By Elasticity of substitution, you *should* refer to

$$ \sigma(x,y) = \frac{\mathrm{d}\log \left(\frac{x}{y}\right)}{\mathrm{d}\log \left(\frac{U\_y}{U\_x}\right)} $$

Then, constant elasticity of substitution is a property of a function $U$ such that for some constant $\bar \sigma$,

$$ \sigma(x,y) = \bar\sigma \, \forall y, x$$

That is, the elasticity doesn't depend on $x,y$. What you showed as $U$ is a function that has exactly that feature. If you compute $\sigma(x,y)$ (do it!), you will see that

$$ \sigma(x,y) = \bar\sigma = \frac{1}{1+\rho}$$

That is, the elasticity of your $U$ function is constant for any allocation. Hence, we refer to that class of function (for different $\rho$) to Constant-Elasticity-of-Substitution-Functions (with share parameters $\alpha$). Cobb-Douglas preferences would be a special (limiting) case of that, with $\sigma(x,y) = 1$

### Example

Say, we produce happiness ($U$) with apples ($x$) and oranges ($y$). Lets look at the allocation $\{5,1\}$. We only have one orange, but we would *really* want more. Hence, we are willing to give up 2 apples to get one orange, and we would be indifferent:

$$ U(5,1) = U(3,2)$$

It is *very easy* to substitute apples with oranges, we have a high elasticity of substitution.

Now look at the allocation $\{1,6\}$. Do you think that you would again be willing to lose $40\%$ of $x$ to get an increase of $50\%$ in $y$? Is it true that

$$ U(1,6) = U(0.6, 12)$$

If yes, that means that the elasticity of substitution of $U$ (or my rather handwaving approximation to it) is the same at the two allocations $\{1,6\}$ and $\{5,1\}$. If that was true for *all* possible allocations, $U$ was CES.

There are some reasons why you would think some preferences are perhaps not CES: For example, if $x$ and $y$ are good complements. Then, when you have little of $x$, you're willing to give up a lot for the other, but that elasticity might decrease the closer you are to equality between the two inputs.

> **Comments:**
> **username_0**: But what does that mean? I got that answer. Like I don't understand why the $U$ function has to be so ugly looking.  
> **username_4**: @StanShunpike I've added an example, hope that helps.  
> **username_4**: @StanShunpike it doesn't *have* to be so ugly looking, that's just a very general form. You can set $A = 1$, remove the $\alpha$, $1-\alpha$ terms and still have CES.  
> **username_0**: Thanks for your replies. I still am confused, so let me try to be more specific: What does $A$ do? What is $\alpha$ do? Is $\alpha$ related to strict convexity? Why do we the the $-\rho$ power of the goods?  
> **username_4**: @StanShunpike I suggest you open up a new question about this, as this one was about the difference between ES and CES and I feel like we're diverging.  
> **username_0**: Okay, I will do that.  
> **username_5**: @FooBar Upvote, your example explains very clearly what is the interpretation of constant elasticity of substitution utilities/preferences.  

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### Answer by username_1 on 2015-04-20. Score: 2

Elasticity of substitution is a property of a function, like slope or curvature. Constant elasticity of substitution means that this property has the same value at all arguments of the function.

There are many explanations of the elasticity of substitution online, e.g. <http://en.wikipedia.org/wiki/Elasticity_of_substitution>
If after reading these, you are still confused, you can ask a more specific question.

> **Comments:**
> **username_2**: Why did this get down-voted?? It seems precisely correct to me.  
> **username_3**: An alternative reference, which I found to be quite clear is here: http://www.econ.ucsb.edu/~tedb/Courses/GraduateTheoryUCSB/elasticity%20of%20substitutionrevised.tex.pdf  

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