<h1>Elasticity of substitution for 3 and more goods (interpretation)</h1>
<p>Hello everyone,</p>
<p>I have a problem regarding the understanding of how would the elasticity of substitution work in the case of function depending on multiple goods. Elasticity of substitution is defined for pair of goods the following way:</p>
<p><span class="math-container">$$
\sigma_{ij} = \frac{\frac{\partial (x_j/x_i)}{x_j/x_i}}{\frac{\partial MRS_{ij}}{MRS_{ij}}}
$$</span></p>
<p>Where <span class="math-container">$MRS_{ij} = MU_i/MU_j$</span>, which was deduced by applying total differencial on utility function <span class="math-container">$U(x_1;x_2)$</span> the following way:</p>
<p><span class="math-container">$$
dU = \frac{\partial U(x_1;x_2)}{\partial x_1}dx_1 + \frac{\partial U(x_1;x_2)}{\partial x_2}dx_2 
$$</span></p>
<p>Evaluated at <span class="math-container">$dU = 0$</span> and reshuffled for <span class="math-container">$-dx_2/dx_1$</span>.</p>
<p>The problem is that in reality, <strong>a decision maker usually consumes more than two goods</strong>, meaning his utility function would be something like <span class="math-container">$U(x_1; x_2;\dots ; x_n)$</span>. The thing is: <em>what should be done</em> with <span class="math-container">$\forall_{i \geq 3}: x_i $</span> <strong>to get the same phenomenon/interpretation as described by elasticity of substitution in case of two goods</strong>? The intuition behind <span class="math-container">$\sigma_{ij}$</span> is such that it measures the curvature of contour of utility function (indifference curve) between two goods, but such curvature would usually depend on the consumption of other goods (imagine 3D indifference curves), so it should be the function: <span class="math-container">$\sigma_{ij} = \sigma_{ij}(x_1; x_2; \dots ; x_n)$</span>.</p>
<p><strong>OPTION A:</strong></p>
<p>Now, should I consider them constant (same as some <em>tastes</em> parameters) in deducing <span class="math-container">$MRS_{ij}$</span>, meaning <span class="math-container">$U(x_1; x_2; \bar{x_3}; \bar{x_4}; \dots ; \bar{x_n})$</span>? That would change the total differential into the following form:</p>
<p><span class="math-container">$$
dU = \frac{\partial U(x_1; x_2; \bar{x_3}; \bar{x_4}; \dots ; \bar{x_n})}{\partial x_1}dx_1 + \frac{\partial U(x_1; x_2; \bar{x_3}; \bar{x_4}; \dots ; \bar{x_n})}{\partial x_2}dx_2 
$$</span></p>
<p>Should I <strong>leave it in the equation as a parameter</strong> then and compute with it as such, meaning I would get <span class="math-container">$\sigma_{ij} = \sigma_{ij}(x_1; x_2; \bar{x_3}; \bar{x_4}; \dots ; \bar{x_n})$</span>? By which I would know how the <span class="math-container">$\sigma_{ij}$</span> changes with the level of other goods?</p>
<p><strong>OPTION B:</strong></p>
<p>Or should I <strong>perform the total differential according to all goods</strong>? By which I would have:</p>
<p><span class="math-container">$$
dU = \frac{\partial U(x_1; x_2;\dots ; x_n)}{\partial x_1}dx_1 + \frac{\partial U(x_1; x_2;\dots ; x_n)}{\partial x_2}dx_2 + \dots + \frac{\partial U(x_1; x_2;\dots ; x_n)}{\partial x_n}dx_n
$$</span></p>
<p>Meaning I would miss the nice form for <span class="math-container">$MRS_{ij}$</span>?</p>
<p>Thank you very much!</p>
<hr />
<p><em>P.S. Although I work primarily with utility function, the same logic is applicable to production function as well.</em></p>

CC BY-SA 4.0
2022-10-25T10:26:51.317Z
What *should be done* always depends on *what your goal is*. So what exactly are you trying to accomplish here?
Giskard
CC BY-SA 4.0
2022-10-25T10:32:08.750Z
To get the right/proper elasticity of substitution for a utility/production function depending on 3 or more goods... Because elasticity of substitution is always defined for two goods functions in textbooks, so I wonder how it changes with multiple goods.
Athaeneus
CC BY-SA 4.0
2022-10-25T10:33:55.443Z
Okay, well elasticity of substitution (in fact, substitution) is defined for two goods, so the problem is solved.
Giskard
CC BY-SA 4.0
2022-10-25T10:35:08.423Z
You can always come up with a new definition, but whether it makes any sense will most likely depend on what you want to use it for.
Giskard
CC BY-SA 4.0
2022-10-25T10:35:11.670Z
Well, it is not... The thing is how to get it from utility function with multiple goods.
Athaeneus
CC BY-SA 4.0
2022-10-25T10:36:14.893Z
You are asking for the definition of an uncharacterized concept.
Giskard
CC BY-SA 4.0
2022-10-25T10:39:06.307Z
I know I can come up with new definition, but the thing is that elasticity of substitution has some interpretation so I wonder, how to get the same interpretation for a slightly different function. You say it is uncharacterized, but if I connected goods into two composite goods, it would work (but that is easy and might not give the full picture). There might be a way how to work with it further... To get the same interpretation.
Athaeneus
CC BY-SA 4.0
2022-10-25T11:15:46.753Z
Okay, then perhaps write down this "same interpretation" in a very clear and precise way?
Giskard
CC BY-SA 4.0
2022-10-25T13:27:28.997Z
Are you asking "how can I characterize an elasticity of substitution that is not constant"? Because the elasticity of substitution is by definition between two goods.

With a utility function with multiple goods, you get it EXACTLY as everyone told you here, and you reported! You simply take the ratio between good i and good j given the UMP (or cost minimization, for inputs), and then compute $d \ln(x_j/x_i)/d \ln MRS_{i,j}$.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-25T13:37:30.473Z
Yes and no, to be clear. First off... There is a problem with computing $MRS_{ij}$, since it was deduced from total differential applied on utility function with two arguments only. So I am asking how to properly apply it on utility function with multiple arguments such that I can get reasonably interpretable $MRS_{ij}$. I would like to see the logic behind it: how should we approach it if utility function had more arguments. Then I suppose that in such a case, elasticity of substitution would not be constant, so I also implicitly ask, what to do in that situation.
Athaeneus
CC BY-SA 4.0
2022-10-25T13:50:17.137Z
$MRS_{ij}$ is simply the ratio of the marginal utilities, has nothing to do with total differentiation. It is simply the amount of good j you are willing to give up for an additional unit of good i (that keeps you on the same indifference curve). The MRS of good i and j can depend of course also on all the other goods that are neither i nor j your utility function is some kind of polynomial (even a 2nd order polynomial satisfies this condition). Something like $U(x_1,...,x_l) = \alpha(\sum_{l}^L \gamma_l x_l) + \beta(\sum_{l}^L \gamma_l x_l)^2$ could work, I think.
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-25T13:57:09.203Z
What you are saying about $MRS_{ij}$ is not true. The reasoning behind it comes from total differential of utility function. See here: [link](https://www.economicsdiscussion.net/cardinal-utility-analysis/marginal-rate-of-substitution-principle-reasons-and-relationship-between-mrs-and-marginal-utilities/954). It was even in some textbook, but I cannot find it right now (probably Varian, which I do not have here).

Yes, polynomials are functions I work with on this topic. The only thing in the way is how to get the proper $MRS_{ij}$.
Athaeneus
CC BY-SA 4.0
2022-10-25T14:57:47.473Z
MRS$_{ij}$ is quite well defined for spaces with more than two goods. You calculate the total differential w.r.t. the two goods, assuming that the quantities of all other goods are held equal.
Giskard
CC BY-SA 4.0
2022-10-25T15:08:10.500Z
No, it doesn't come from the total differentiation of the utility function (it does, in two dimensions only, though!). The guy is using total differentiation to show you that in two dimensions, the MRS is exactly the slope of the indifference curves, and this should be the mathematical intuition even when you think about it in L dimensions. So, once again: It is simply the ratio of the marginal utilities.

It is simply the amount of good j you are willing to give up for an additional unit of good i (that keeps you on the same indifference curve
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-25T17:45:45.193Z
Seems like @Giskard provided the answer I was searching for. According to his answer the presented **OPTION A** should be the correct approach to this problem. I will try to think about it a little bit more, but it seems to make sense. So thank you a lot everyone.
Athaeneus
<microeconomics><mathematical-economics><utility><production-function><elasticity-of-substitution>
Elasticity of substitution for 3 and more goods (interpretation)
Athaeneus
# Elasticity of substitution for 3 and more goods (interpretation)

(Asked by: username_0 on 2022-10-25)

# Elasticity of substitution for 3 and more goods (interpretation)

Hello everyone,

I have a problem regarding the understanding of how would the elasticity of substitution work in the case of function depending on multiple goods. Elasticity of substitution is defined for pair of goods the following way:

$$
\sigma\_{ij} = \frac{\frac{\partial (x\_j/x\_i)}{x\_j/x\_i}}{\frac{\partial MRS\_{ij}}{MRS\_{ij}}}
$$

Where $MRS\_{ij} = MU\_i/MU\_j$, which was deduced by applying total differencial on utility function $U(x\_1;x\_2)$ the following way:

$$
dU = \frac{\partial U(x\_1;x\_2)}{\partial x\_1}dx\_1 + \frac{\partial U(x\_1;x\_2)}{\partial x\_2}dx\_2
$$

Evaluated at $dU = 0$ and reshuffled for $-dx\_2/dx\_1$.

The problem is that in reality, **a decision maker usually consumes more than two goods**, meaning his utility function would be something like $U(x\_1; x\_2;\dots ; x\_n)$. The thing is: *what should be done* with $\forall\_{i \geq 3}: x\_i $ **to get the same phenomenon/interpretation as described by elasticity of substitution in case of two goods**? The intuition behind $\sigma\_{ij}$ is such that it measures the curvature of contour of utility function (indifference curve) between two goods, but such curvature would usually depend on the consumption of other goods (imagine 3D indifference curves), so it should be the function: $\sigma\_{ij} = \sigma\_{ij}(x\_1; x\_2; \dots ; x\_n)$.

**OPTION A:**

Now, should I consider them constant (same as some *tastes* parameters) in deducing $MRS\_{ij}$, meaning $U(x\_1; x\_2; \bar{x\_3}; \bar{x\_4}; \dots ; \bar{x\_n})$? That would change the total differential into the following form:

$$
dU = \frac{\partial U(x\_1; x\_2; \bar{x\_3}; \bar{x\_4}; \dots ; \bar{x\_n})}{\partial x\_1}dx\_1 + \frac{\partial U(x\_1; x\_2; \bar{x\_3}; \bar{x\_4}; \dots ; \bar{x\_n})}{\partial x\_2}dx\_2
$$

Should I **leave it in the equation as a parameter** then and compute with it as such, meaning I would get $\sigma\_{ij} = \sigma\_{ij}(x\_1; x\_2; \bar{x\_3}; \bar{x\_4}; \dots ; \bar{x\_n})$? By which I would know how the $\sigma\_{ij}$ changes with the level of other goods?

**OPTION B:**

Or should I **perform the total differential according to all goods**? By which I would have:

$$
dU = \frac{\partial U(x\_1; x\_2;\dots ; x\_n)}{\partial x\_1}dx\_1 + \frac{\partial U(x\_1; x\_2;\dots ; x\_n)}{\partial x\_2}dx\_2 + \dots + \frac{\partial U(x\_1; x\_2;\dots ; x\_n)}{\partial x\_n}dx\_n
$$

Meaning I would miss the nice form for $MRS\_{ij}$?

Thank you very much!

---

*P.S. Although I work primarily with utility function, the same logic is applicable to production function as well.*

**Tags:** microeconomics, mathematical-economics, utility, production-function, elasticity-of-substitution
> **Comments:**
> **username_1**: What *should be done* always depends on *what your goal is*. So what exactly are you trying to accomplish here?  
> **username_0**: To get the right/proper elasticity of substitution for a utility/production function depending on 3 or more goods... Because elasticity of substitution is always defined for two goods functions in textbooks, so I wonder how it changes with multiple goods.  
> **username_1**: Okay, well elasticity of substitution (in fact, substitution) is defined for two goods, so the problem is solved.  
> **username_1**: You can always come up with a new definition, but whether it makes any sense will most likely depend on what you want to use it for.  
> **username_0**: Well, it is not... The thing is how to get it from utility function with multiple goods.  
> **username_1**: You are asking for the definition of an uncharacterized concept.  
> **username_0**: I know I can come up with new definition, but the thing is that elasticity of substitution has some interpretation so I wonder, how to get the same interpretation for a slightly different function. You say it is uncharacterized, but if I connected goods into two composite goods, it would work (but that is easy and might not give the full picture). There might be a way how to work with it further... To get the same interpretation.  
> **username_1**: Okay, then perhaps write down this "same interpretation" in a very clear and precise way?  
> **username_2**: Are you asking "how can I characterize an elasticity of substitution that is not constant"? Because the elasticity of substitution is by definition between two goods.  With a utility function with multiple goods, you get it EXACTLY as everyone told you here, and you reported! You simply take the ratio between good i and good j given the UMP (or cost minimization, for inputs), and then compute $d \ln(x_j/x_i)/d \ln MRS_{i,j}$.  
> **username_0**: Yes and no, to be clear. First off... There is a problem with computing $MRS_{ij}$, since it was deduced from total differential applied on utility function with two arguments only. So I am asking how to properly apply it on utility function with multiple arguments such that I can get reasonably interpretable $MRS_{ij}$. I would like to see the logic behind it: how should we approach it if utility function had more arguments. Then I suppose that in such a case, elasticity of substitution would not be constant, so I also implicitly ask, what to do in that situation.  
> **username_2**: $MRS_{ij}$ is simply the ratio of the marginal utilities, has nothing to do with total differentiation. It is simply the amount of good j you are willing to give up for an additional unit of good i (that keeps you on the same indifference curve). The MRS of good i and j can depend of course also on all the other goods that are neither i nor j your utility function is some kind of polynomial (even a 2nd order polynomial satisfies this condition). Something like $U(x_1,...,x_l) = \alpha(\sum_{l}^L \gamma_l x_l) + \beta(\sum_{l}^L \gamma_l x_l)^2$ could work, I think.  
> **username_0**: What you are saying about $MRS_{ij}$ is not true. The reasoning behind it comes from total differential of utility function. See here: [link](https://www.economicsdiscussion.net/cardinal-utility-analysis/marginal-rate-of-substitution-principle-reasons-and-relationship-between-mrs-and-marginal-utilities/954). It was even in some textbook, but I cannot find it right now (probably Varian, which I do not have here).  Yes, polynomials are functions I work with on this topic. The only thing in the way is how to get the proper $MRS_{ij}$.  
> **username_1**: MRS$_{ij}$ is quite well defined for spaces with more than two goods. You calculate the total differential w.r.t. the two goods, assuming that the quantities of all other goods are held equal.  
> **username_2**: No, it doesn't come from the total differentiation of the utility function (it does, in two dimensions only, though!). The guy is using total differentiation to show you that in two dimensions, the MRS is exactly the slope of the indifference curves, and this should be the mathematical intuition even when you think about it in L dimensions. So, once again: It is simply the ratio of the marginal utilities.  It is simply the amount of good j you are willing to give up for an additional unit of good i (that keeps you on the same indifference curve  
> **username_0**: Seems like @Giskard provided the answer I was searching for. According to his answer the presented **OPTION A** should be the correct approach to this problem. I will try to think about it a little bit more, but it seems to make sense. So thank you a lot everyone.  

---

## 0 answers

