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My question concerns CES utility functions, which have the form $$u(x) = \left(\sum_{j=1}^n a_j x_j^{\rho} \right)^{1/\rho}$$ for utility parameters $a_j$, an elasticity parameter $\rho$, and some allocation of goods given by $x$.

When $\rho=1$ we have linear utility, or perfect substitution, with $$u(x) = a^T x$$ The level sets of $u$ are straight lines, and it is always possible to achieve the same utility by exchanging one good for another in some fixed ratio. I can see this arising in real life: Two bags of flour are perfectly interchangeable, and the only thing that matters to the buyer is their size and cost.

When $\rho \to -\infty$ we have Leontief utility, or perfect complementarity, with $$u(x) = \min_j a_j x_j$$ The level sets of $u$ are L-shaped, and optimal utility is always achieved by purchasing goods in a target ratio given by $a$. This also makes sense: If I am making a recipe that calls for two units of egg for every unit of sugar, having twice as much sugar makes no difference unless I also have twice as much egg.

Most situations are more complicated than this, however, and the nice thing about the CES form and its $\rho$ parameter is that it lets us capture situations that don't sit at one of these extremes. A baker can replace sugar with brown sugar, but not arbitrarily. Setting $\rho$ to some value between $1$ and $-\infty$ captures this imperfect substitutability. But the precise value of $\rho$ depends on the situation at hand: if the ingredients are pretty good substitutes, maybe $\rho = 0.8$; if they are fairly poor substitutes, maybe $\rho = -10$.

If we take the limit as $\rho \to 0$, we get a utility function that has a nice name, the Cobb–Douglas utility function. It also has a nice form:

$$u(x) = \prod_j x_j^{a_j}$$

But is there anything that's truly special about this case?

From my standpoint, it looks like the Cobb–Douglas utility is just one arbitrary point along the continuum running from linear to Leontief. Perhaps it gets a name because it was discovered before the generalized CES family, but I can't see why there's anything especially meaningful about it beyond the simplicity of the form. Is this argument fair?

Is there a reason why we might assume $\rho$ is close to zero if we have ruled out the linear and Leontief cases?

Does Cobb–Douglas utility correspond to an "idealized" situation like the flour or egg/sugar examples above?

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This is not supposed to be an answer, just a comment: Cobb-Douglas function is named after the paper https://www.jstor.org/stable/1811556 (1928).

Most cases of production functions have returns to scale equal to one, which is the Cobb-Douglas form. It is not special, it is common.

CES function was generalized by Dixit and Stiglitz in the paper https://www.jstor.org/stable/1831401 (1977).

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