<p>The statement is not generally true.
For a counterexample check the function at <span class="math-container">$x_1=x_2=1$</span>. This is
<span class="math-container">$$
(\alpha \cdot 1 + (1- \alpha) \cdot 1)^{\frac{\sigma}{\sigma-1}} = 1
$$</span>
which is constant in <span class="math-container">$\sigma$</span>.</p>

CC BY-SA 4.0
2022-10-08T23:27:33.893Z
Thanks, but this is not the point. Let's say non-decreasing, then. I updated.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-09T02:09:07.480Z
Alright; then I recommend looking into weighted power means. https://planetmath.org/weightedpowermean Final property seems to be what you want.
Giskard
CC BY-SA 4.0
2022-10-09T08:56:09.700Z
Thanks a lot, Giskard! This is indeed what I wanted, but in my view, this is startling: it means that for the same allocation, a consumer with Leontief preferences has lower utility than a consumer with Cobb-Douglas preferences, hence that consumers dislike the fact that goods become more complementary.

This, in my view, doesn't make any sense (it makes sense for a CES production function, not for preferences). You're basically claiming that for the same allocation of, say, oranges and lemons and bike and wheels, consumer prefer the first, in gross surplus terms.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-09T14:00:46.560Z
It does not really make sense to compare utility from two different utility functions. Utility is not a real unit of measurement, it serves as a tool for ordinal comparisons *given* a utility function.
Giskard
CC BY-SA 4.0
2022-10-10T11:42:28.853Z
Well, it's the same utility function! CES preferences converge to a Leontief as sigma goes to 0, to a Cobb-Douglas as sigma goes to 1, and to linear preferences as sigma goes to infinity
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-10T12:24:21.893Z
I know this, but if you change the $\sigma$ parameter it is not the same utility function. It is the same utility function family.
Giskard
CC BY-SA 4.0
2022-10-10T12:25:58.640Z
If persons A and B had utilities of 10 and 15 respectively that does not imply A would be envious of B's basket. Comparing utilities from different utility functions does not make sense.
Giskard
CC BY-SA 4.0
2022-10-11T14:09:13.887Z
Yes, but it still means that if an exogenous shock would change sigma and make it larger (i.e., if he would perceive the two goods as being more "perfect substitutes" than before), he would be strictly better off all other things being equal. Again, it is equivalent to assuming that if goods become more substitutes, the same consumer is better off, namely that a consumer has a taste for substitutability. Has a sense for production, is strong (weird!) for utility.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-11T14:26:46.467Z
The way in which the parameter $\sigma$ enters CES preferences conveys meaning about the preferences of consumers for substitutability since this is what $\sigma$ captures. Claiming that, for ex, a consumer has CES preferences with parameter $\sigma_2$ for two goods produced by one firm and $\sigma_1 < \sigma_2$ for two goods produced by a competitor means that in gross surplus terms, ceteris paribus the same consumer prefers the good of the first firm because they are more substitutable.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-11T15:10:16.873Z
"*he would be strictly better off all other things being equal*" Again, no, this comparison is meaningless. Feel free to post this as a new question, you do not have to take my word for it.
Giskard
CC BY-SA 4.0
2022-10-11T15:12:13.953Z
"*the same consumer prefers the good of the first firm because they are more substitutable.*" I don't see how you arrive at this conclusion. The good**s** of the first firm are better substitutes, but that seems to be all we can say.
Giskard
CC BY-SA 4.0
2022-10-13T08:48:52.190Z
It's not "meaningless", you're basically claiming here that the parameter $\alpha$ of a linear utility function $U(x_1, x_2) = \alpha x_1 +(1-\alpha)x_2$ is "meaningless". Clearly, it is not: it conveys information on which of the two goods the consumer "consumes the most" (read, "prefers" all other things being equal).  Here, $\alpha$ is an intrinsic measure of quality; since every consumer is homogenous, only the "high quality good" will be consumed, i.e. for $\alpha > 1/2$ , the high quality good will be consumed, whereas for $\alpha < 1/2$ the other way around.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-13T09:00:06.510Z
Hence, once again, while in the example above $\alpha$ can be simply interpreted as preference for quality), by the same token $\sigma$ in the CES, which is an index of substitutability, conveys information on the preferences of consumers for substitutability. Since the utility this representative agent (in the case where the weights sum up to 1, otherwise the proof above is not valid) increases as $\sigma$ increases, this captures the fact consumers DO show a preference for substitutability, in the CES framework.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-13T14:41:48.847Z
This is not in fact what I have claimed; but I don't think this conversation is going anywhere. I am glad you found the answer to your question. Have a nice day!
Giskard
CC BY-SA 4.0
2022-10-08T08:19:05.170Z
2022-10-08T08:19:05.170Z
Giskard
<p>Ok, thanks to Giskard, I found the proof: I think it is so elegant it deserves to be shared.</p>
<p>Let's take <span class="math-container">$$ U(x_1, ..., x_L) = \left( \sum_{l=1}^{L}\alpha_l x_l^{\rho} \right)^{1/\rho} $$</span></p>
<p>With <span class="math-container">$\rho = \frac{\sigma-1}{\sigma}$</span>.</p>
<p>As <span class="math-container">$x_l \geq 0$</span>, our goal is simply to prove that, for any <span class="math-container">$\rho_2 &gt; \rho_1 &gt; 0$</span>,</p>
<p><span class="math-container">$$ \left( \sum_{l=1}^{L}\alpha_l x_l^{\rho_2} \right)^{1/\rho_2} \geq \left( \sum_{l=1}^{L}\alpha_l x_l^{\rho_1} \right)^{1/\rho_1} $$</span></p>
<p>Take a scalar <span class="math-container">$k&gt;1$</span>, then, for any <span class="math-container">$y\in \mathbb{R}_{+} \cup \{0\}$</span>, <span class="math-container">$f(y) = y^{k}$</span> is convex. Thus, by convexity and the Jensen's inequality, it holds that:</p>
<p><span class="math-container">$$ \sum_{l=1}^{L} \alpha_l y_l^k \geq \left(\sum_{l=1}^{L} \alpha_l y_l \right)^k $$</span></p>
<p>Substitute <span class="math-container">$y_l = x_l^{\rho_1}$</span> and <span class="math-container">$k = \rho_2/\rho_1$</span>, and take the <span class="math-container">$\rho_2$</span>th root, and the statement is proved!</p>
<p><span class="math-container">$$ \left( \sum_{l=1}^{L}\alpha_l x_l^{\rho_2} \right)^{1/\rho_2} \geq \left( \sum_{l=1}^{L}\alpha_l x_l^{\rho_1} \right)^{1/\rho_1} $$</span></p>
<p>As I commented below, I think that this feature of CES preferences makes them quite questionable at best, for most of the applications in Microeconomics. However, for trade theory and inter-temporal decisions, I think this can make sense.</p>

CC BY-SA 4.0
2022-10-09T10:50:56.747Z
2022-10-09T10:56:56.877Z
2022-10-09T10:56:56.877Z
Matteo Bulgarelli
<p>Is this possible? I've been trying so many times, without success, yet if I plot a two good CES on Wolfram, this seems true... I can't find a single numerical example where this does not hold, but I can't prove it either differentiating...</p>
<p>EDIT: The question is, can someone prove that the CES is increasing in the elasticity of substitution?</p>
<p>EDIT: I simply want to show whether the function <span class="math-container">$U(x_1,x_2;\sigma)$</span> increases as the parameter <span class="math-container">$\sigma$</span> does, which seems to be the case.</p>
<p><span class="math-container">$U(\bullet)$</span> is a standard two good CES.</p>
<p><span class="math-container">$$U(x_1,x_2;\sigma) = \left[\alpha x_1^{\frac{\sigma -1}{\sigma}} +(1-\alpha)x_2^{\frac{\sigma -1}{\sigma}} \right]^{\frac{\sigma}{\sigma -1}} $$</span></p>
<p><a href="https://i.sstatic.net/qQJBb.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qQJBb.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-10-07T09:26:58.607Z
Can you please edit your post so that it shows some of these plots & examples? I am not quite sure what you are trying to do as the elasticity of substitution in a CES function should be, by default, constant. You are probably changing some parameter, but I am not sure which one.
Giskard
CC BY-SA 4.0
2022-10-07T13:17:48.777Z
He seems to be showing how output/utility is increasing in the EOS..
Kwame Brown
<ces>
Showing that the CES is non-decreasing in the elasticity of substitution
Matteo Bulgarelli
# Showing that the CES is non-decreasing in the elasticity of substitution

(Asked by: username_0 on 2022-10-07)

Is this possible? I've been trying so many times, without success, yet if I plot a two good CES on Wolfram, this seems true... I can't find a single numerical example where this does not hold, but I can't prove it either differentiating...

EDIT: The question is, can someone prove that the CES is increasing in the elasticity of substitution?

EDIT: I simply want to show whether the function $U(x\_1,x\_2;\sigma)$ increases as the parameter $\sigma$ does, which seems to be the case.

$U(\bullet)$ is a standard two good CES.

$$U(x\_1,x\_2;\sigma) = \left[\alpha x\_1^{\frac{\sigma -1}{\sigma}} +(1-\alpha)x\_2^{\frac{\sigma -1}{\sigma}} \right]^{\frac{\sigma}{\sigma -1}} $$

[![enter image description here](https://i.sstatic.net/qQJBb.png)](https://i.sstatic.net/qQJBb.png)

**Tags:** ces

> **Comments:**
> **username_1**: Can you please edit your post so that it shows some of these plots & examples? I am not quite sure what you are trying to do as the elasticity of substitution in a CES function should be, by default, constant. You are probably changing some parameter, but I am not sure which one.  
> **username_2**: He seems to be showing how output/utility is increasing in the EOS..  

---

## 2 answers

### Answer by username_0 on 2022-10-09. Score: 2 | Accepted answer

Ok, thanks to username_1, I found the proof: I think it is so elegant it deserves to be shared.

Let's take $$ U(x\_1, ..., x\_L) = \left( \sum\_{l=1}^{L}\alpha\_l x\_l^{\rho} \right)^{1/\rho} $$

With $\rho = \frac{\sigma-1}{\sigma}$.

As $x\_l \geq 0$, our goal is simply to prove that, for any $\rho\_2 > \rho\_1 > 0$,

$$ \left( \sum\_{l=1}^{L}\alpha\_l x\_l^{\rho\_2} \right)^{1/\rho\_2} \geq \left( \sum\_{l=1}^{L}\alpha\_l x\_l^{\rho\_1} \right)^{1/\rho\_1} $$

Take a scalar $k>1$, then, for any $y\in \mathbb{R}\_{+} \cup \{0\}$, $f(y) = y^{k}$ is convex. Thus, by convexity and the Jensen's inequality, it holds that:

$$ \sum\_{l=1}^{L} \alpha\_l y\_l^k \geq \left(\sum\_{l=1}^{L} \alpha\_l y\_l \right)^k $$

Substitute $y\_l = x\_l^{\rho\_1}$ and $k = \rho\_2/\rho\_1$, and take the $\rho\_2$th root, and the statement is proved!

$$ \left( \sum\_{l=1}^{L}\alpha\_l x\_l^{\rho\_2} \right)^{1/\rho\_2} \geq \left( \sum\_{l=1}^{L}\alpha\_l x\_l^{\rho\_1} \right)^{1/\rho\_1} $$

As I commented below, I think that this feature of CES preferences makes them quite questionable at best, for most of the applications in Microeconomics. However, for trade theory and inter-temporal decisions, I think this can make sense.


---

### Answer by username_1 on 2022-10-08. Score: 0

The statement is not generally true.
For a counterexample check the function at $x\_1=x\_2=1$. This is
$$
(\alpha \cdot 1 + (1- \alpha) \cdot 1)^{\frac{\sigma}{\sigma-1}} = 1
$$
which is constant in $\sigma$.

> **Comments:**
> **username_0**: Thanks, but this is not the point. Let's say non-decreasing, then. I updated.  
> **username_1**: Alright; then I recommend looking into weighted power means. https://planetmath.org/weightedpowermean Final property seems to be what you want.  
> **username_0**: Thanks a lot, username_1! This is indeed what I wanted, but in my view, this is startling: it means that for the same allocation, a consumer with Leontief preferences has lower utility than a consumer with Cobb-Douglas preferences, hence that consumers dislike the fact that goods become more complementary.  This, in my view, doesn't make any sense (it makes sense for a CES production function, not for preferences). You're basically claiming that for the same allocation of, say, oranges and lemons and bike and wheels, consumer prefer the first, in gross surplus terms.  
> **username_1**: It does not really make sense to compare utility from two different utility functions. Utility is not a real unit of measurement, it serves as a tool for ordinal comparisons *given* a utility function.  
> **username_0**: Well, it's the same utility function! CES preferences converge to a Leontief as sigma goes to 0, to a Cobb-Douglas as sigma goes to 1, and to linear preferences as sigma goes to infinity  
> **username_1**: I know this, but if you change the $\sigma$ parameter it is not the same utility function. It is the same utility function family.  
> **username_1**: If persons A and B had utilities of 10 and 15 respectively that does not imply A would be envious of B's basket. Comparing utilities from different utility functions does not make sense.  
> **username_0**: Yes, but it still means that if an exogenous shock would change sigma and make it larger (i.e., if he would perceive the two goods as being more "perfect substitutes" than before), he would be strictly better off all other things being equal. Again, it is equivalent to assuming that if goods become more substitutes, the same consumer is better off, namely that a consumer has a taste for substitutability. Has a sense for production, is strong (weird!) for utility.  
> **username_0**: The way in which the parameter $\sigma$ enters CES preferences conveys meaning about the preferences of consumers for substitutability since this is what $\sigma$ captures. Claiming that, for ex, a consumer has CES preferences with parameter $\sigma_2$ for two goods produced by one firm and $\sigma_1 < \sigma_2$ for two goods produced by a competitor means that in gross surplus terms, ceteris paribus the same consumer prefers the good of the first firm because they are more substitutable.  
> **username_1**: "*he would be strictly better off all other things being equal*" Again, no, this comparison is meaningless. Feel free to post this as a new question, you do not have to take my word for it.  
> **username_1**: "*the same consumer prefers the good of the first firm because they are more substitutable.*" I don't see how you arrive at this conclusion. The good**s** of the first firm are better substitutes, but that seems to be all we can say.  
> **username_0**: It's not "meaningless", you're basically claiming here that the parameter $\alpha$ of a linear utility function $U(x_1, x_2) = \alpha x_1 +(1-\alpha)x_2$ is "meaningless". Clearly, it is not: it conveys information on which of the two goods the consumer "consumes the most" (read, "prefers" all other things being equal).  Here, $\alpha$ is an intrinsic measure of quality; since every consumer is homogenous, only the "high quality good" will be consumed, i.e. for $\alpha > 1/2$ , the high quality good will be consumed, whereas for $\alpha < 1/2$ the other way around.  
> **username_0**: Hence, once again, while in the example above $\alpha$ can be simply interpreted as preference for quality), by the same token $\sigma$ in the CES, which is an index of substitutability, conveys information on the preferences of consumers for substitutability. Since the utility this representative agent (in the case where the weights sum up to 1, otherwise the proof above is not valid) increases as $\sigma$ increases, this captures the fact consumers DO show a preference for substitutability, in the CES framework.  
> **username_1**: This is not in fact what I have claimed; but I don't think this conversation is going anywhere. I am glad you found the answer to your question. Have a nice day!  

---

