3
$\begingroup$

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

Thank you very much!

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ 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. $\endgroup$
    – Bertrand
    Oct 20, 2022 at 8:26
  • $\begingroup$ 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. $\endgroup$
    – Athaeneus
    Oct 20, 2022 at 11:37
  • $\begingroup$ 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), $\endgroup$
    – Bertrand
    Oct 20, 2022 at 13:55
  • $\begingroup$ 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. $\endgroup$
    – Athaeneus
    Oct 21, 2022 at 7:21
  • $\begingroup$ So, this infinite question could just be summed up in a more concise: "How can I characterize preferences with non-constant elasticity of substitution?" $\endgroup$
    – Matteo Bulgarelli
    Oct 25, 2022 at 13:42

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.