# What's special about Cobb–Douglas utility relative to the rest of the CES family?

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:

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

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?