<h1>Distinguish between complements/substitutes in utility function or production function</h1>
<p>Hello everyone,</p>
<p>I would like to know if there exists some utility function <span class="math-container">$U(x)$</span> for <span class="math-container">$n$</span> goods that is able to distinguish between complements/substitutes at the same time. The reason is that in reality a consumer can consider good <span class="math-container">$A$</span> to be a complement (to some extent) to a good <span class="math-container">$B$</span> but the good <span class="math-container">$B$</span> can be a substitute (to some extent) to a good <span class="math-container">$C$</span>.</p>
<p>I am interested in a case of <strong>imperfect</strong> complements/substitutes rather than in their <em>perfect</em> variant since the <em>perfect</em> one is boring. That would be just something like: <span class="math-container">$U(x) = min\{x_1; x_2\} + x_3$</span>.</p>
<p>What I strive to find out is some <strong>generalization of CES function</strong> for <em>pairs of goods</em> that would provide the different elasticity of substitution for two chosen goods based on 1 parameter. Something which would work in a following way:</p>
<p><span class="math-container">$$ \{x_1; x_2 \}^{\rho_{1,2}}; \{x_1; x_3 \}^{\rho_{1,3}}; \{x_3; x_5 \}^{\rho_{3,5}} $$</span></p>
<ol>
<li>If <span class="math-container">$\rho_{1,2} &gt; 0$</span> goods <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are substitutes (meaning their elasticity of substitution <span class="math-container">$\sigma_{1,2} &gt; 1$</span>).</li>
<li>If <span class="math-container">$\rho_{3,5} &lt; 0$</span> goods <span class="math-container">$x_3$</span> and <span class="math-container">$x_5$</span> are complements (<span class="math-container">$\sigma_{3,5} &lt; 1$</span>)</li>
<li>etc...</li>
</ol>
<p>In a functional form it could work for example as this:</p>
<p><span class="math-container">$$
 U(x) = g \left[f_1(x_1;x_2)^{\rho_{1,2}}; f_2(x_1;x_3)^{\rho_{1,3}}; \dots; f_m(x_3;x_5)^{\rho_{3,5}} \right]
$$</span></p>
<p>Do you know some kind of elegant utility function (or production function) <span class="math-container">$U(x, \rho)$</span> with such property? Or whether some research of this topic is in process?</p>
<p><strong>Illustration:</strong> I am searching for a simple and flexible function <span class="math-container">$y = f(x_1; x_2; x_3)$</span> which would satisfy (at least to some extent, meaning <strong>it does not have to be exact</strong>) the following property per pairs of variables:</p>
<p><a href="https://i.sstatic.net/1CUzd.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1CUzd.png" alt="3d function with different contour shapes per pairs of variables" /></a></p>
<p>Thank you very much!</p>

CC BY-SA 4.0
2022-10-20T08:26:45.440Z
How do you define the UMO (Unidentified Math Objects) used in your inline equation? Most existing utility function are already compatible with commodities been substitutes or complements to some degree: translog, Box-Cox, or nonparametric utility functions.
Bertrand
CC BY-SA 4.0
2022-10-20T11:37:50.390Z
I am not sure if I understand correctly what you mean. The inline equation for distinguishing between perfect substitutes/complements is not that important. It is just to show that I can think of some function that would satisfy the requirement of having two goods complements and two other goods substitutes. I am searching for a mathematical function that would generalize the CES function for every pair of goods, meaning: I have 10 goods, then I would have $10!/(2!*8!)$ parameters $\rho_i$ which would give me elasticity of substitution between goods in each pair.
Athaeneus
CC BY-SA 4.0
2022-10-20T13:55:58.437Z
If you have $J$ commodities, then there are $J \times (J-1)$ elasticities of substitution. Each commodity can be substituted with the $J-1$ other commodity. These parameters are usually estimated from a demand system (AIDS, PIGLOG or nonparametrically),
Bertrand
CC BY-SA 4.0
2022-10-21T07:21:07.067Z
Yeah, I see... But with this question what I am concerned is rather theory than empirics. I am just looking for math expression (function) that I cannot find anywhere and even do not know how to construct. With this question I do not want to estimate, but rather to model preferences. Next what concerns the number of parameters, since $\sigma_{ij} = \sigma_{ji}$ I thought we could have only $n!/(2!*(n-2)!)$ of them. So in real, I search for some $U(x, \rho)$ such that the values of $\rho$ give elasticities of substitution between pairs of goods.
Athaeneus
CC BY-SA 4.0
2022-10-25T13:42:35.633Z
So, this infinite question could just be summed up in a more concise:

"How can I characterize preferences with non-constant elasticity of substitution?"
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-25T13:44:51.480Z
Well... No, it is a different question. The question might be summarized: **Is there a mathematical function that satisfies the coveted property?** When the property is that it gives you the different elasticity of substitution based on certain parameter inside the utility function.
Athaeneus
<microeconomics><mathematical-economics><utility><elasticity-of-substitution><ces-function>
Utility function distinguishing between complements/substitutes
Athaeneus
# Utility function distinguishing between complements/substitutes

(Asked by: username_0 on 2022-10-19)

# Distinguish between complements/substitutes in utility function or production function

Hello everyone,

I would like to know if there exists some utility function $U(x)$ for $n$ goods that is able to distinguish between complements/substitutes at the same time. The reason is that in reality a consumer can consider good $A$ to be a complement (to some extent) to a good $B$ but the good $B$ can be a substitute (to some extent) to a good $C$.

I am interested in a case of **imperfect** complements/substitutes rather than in their *perfect* variant since the *perfect* one is boring. That would be just something like: $U(x) = min\{x\_1; x\_2\} + x\_3$.

What I strive to find out is some **generalization of CES function** for *pairs of goods* that would provide the different elasticity of substitution for two chosen goods based on 1 parameter. Something which would work in a following way:

$$ \{x\_1; x\_2 \}^{\rho\_{1,2}}; \{x\_1; x\_3 \}^{\rho\_{1,3}}; \{x\_3; x\_5 \}^{\rho\_{3,5}} $$

1. If $\rho\_{1,2} > 0$ goods $x\_1$ and $x\_2$ are substitutes (meaning their elasticity of substitution $\sigma\_{1,2} > 1$).
2. If $\rho\_{3,5} < 0$ goods $x\_3$ and $x\_5$ are complements ($\sigma\_{3,5} < 1$)
3. etc...

In a functional form it could work for example as this:

$$
U(x) = g \left[f\_1(x\_1;x\_2)^{\rho\_{1,2}}; f\_2(x\_1;x\_3)^{\rho\_{1,3}}; \dots; f\_m(x\_3;x\_5)^{\rho\_{3,5}} \right]
$$

Do you know some kind of elegant utility function (or production function) $U(x, \rho)$ with such property? Or whether some research of this topic is in process?

**Illustration:** I am searching for a simple and flexible function $y = f(x\_1; x\_2; x\_3)$ which would satisfy (at least to some extent, meaning **it does not have to be exact**) the following property per pairs of variables:

[![3d function with different contour shapes per pairs of variables](https://i.sstatic.net/1CUzd.png)](https://i.sstatic.net/1CUzd.png)

Thank you very much!

**Tags:** microeconomics, mathematical-economics, utility, elasticity-of-substitution, ces-function

> **Comments:**
> **username_1**: How do you define the UMO (Unidentified Math Objects) used in your inline equation? Most existing utility function are already compatible with commodities been substitutes or complements to some degree: translog, Box-Cox, or nonparametric utility functions.  
> **username_0**: I am not sure if I understand correctly what you mean. The inline equation for distinguishing between perfect substitutes/complements is not that important. It is just to show that I can think of some function that would satisfy the requirement of having two goods complements and two other goods substitutes. I am searching for a mathematical function that would generalize the CES function for every pair of goods, meaning: I have 10 goods, then I would have $10!/(2!*8!)$ parameters $\rho_i$ which would give me elasticity of substitution between goods in each pair.  
> **username_1**: If you have $J$ commodities, then there are $J \times (J-1)$ elasticities of substitution. Each commodity can be substituted with the $J-1$ other commodity. These parameters are usually estimated from a demand system (AIDS, PIGLOG or nonparametrically),  
> **username_0**: Yeah, I see... But with this question what I am concerned is rather theory than empirics. I am just looking for math expression (function) that I cannot find anywhere and even do not know how to construct. With this question I do not want to estimate, but rather to model preferences. Next what concerns the number of parameters, since $\sigma_{ij} = \sigma_{ji}$ I thought we could have only $n!/(2!*(n-2)!)$ of them. So in real, I search for some $U(x, \rho)$ such that the values of $\rho$ give elasticities of substitution between pairs of goods.  
> **username_2**: So, this infinite question could just be summed up in a more concise:  "How can I characterize preferences with non-constant elasticity of substitution?"  
> **username_0**: Well... No, it is a different question. The question might be summarized: **Is there a mathematical function that satisfies the coveted property?** When the property is that it gives you the different elasticity of substitution based on certain parameter inside the utility function.  

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## 0 answers

