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I am a little bit curious about the following problem:

If we have multiple goods (at least 3 or more)... And we know that $x_1$ and $x_2$ are substitutes and $x_1$ and $x_3$ are also substitutes, does it imply that $x_2$ and $x_3$ are substitutes as well? I would say so, just intuitively: If I do not mind consuming $x_1$ instead of $x_2$ and I do not mind consuming $x_3$ instead of $x_1$, I would also have to be indifferent (utility-wise) between $x_2$ and $x_3$ through the mentioned path...

BUT are there any robust conditions, for example in case of 4 goods and more?

  • $x_1$ and $x_2$ substitutes
  • $x_1$ and $x_3$ complements
  • $x_3$ and $x_4$ substitutes

Can I say in such a case that $x_2$ is also a complement with $x_3$ or $x_1$ is a complement to $x_4$ etc...?

What would it be in case of imperfect substitutability? Let's say $x_1$ and $x_2$ have elasticity of substitution equal to 3, what would it imply about elasticity of substitution for $x_3$ and $x_2$ etc...? Is there some math behind it what conditions must hold in these cases?

Thank you very much!

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