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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.

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  • $\begingroup$ What should be done always depends on what your goal is. So what exactly are you trying to accomplish here? $\endgroup$
    – Giskard
    Commented Oct 25, 2022 at 10:26
  • $\begingroup$ 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. $\endgroup$
    – Athaeneus
    Commented Oct 25, 2022 at 10:32
  • $\begingroup$ Okay, well elasticity of substitution (in fact, substitution) is defined for two goods, so the problem is solved. $\endgroup$
    – Giskard
    Commented Oct 25, 2022 at 10:33
  • $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Oct 25, 2022 at 10:35
  • 2
    $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Oct 25, 2022 at 14:57

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