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!

  • 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
    Commented 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
    Commented 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
    Commented 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
    Commented 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?" $\endgroup$ Commented Oct 25, 2022 at 13:42


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