# Convex CES Aggregator

I just find it seems that a CES aggregator e.g. $$\left[\sum_{j=1}^{J} N_{j}^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}$$ with $$\sigma<0$$ is called a general convex aggregator and its limit as $$\sigma \uparrow 0$$ is $$\max \left\{N_{1}, \ldots, N_{J}\right\}$$, which is exactly the inverse of the result that Leontief function is a special case of the CES aggregator when $$\sigma \downarrow 0$$.

Is there any easy way to prove this and more importantly what is the intuition behind these two distinguished results?

Let $$N=\max\{N_1,\ldots, N_J\}$$ and $$\sigma<0$$.
$$N=\left[N^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}\leq\left[\sum_{j=1}^{J} N_{j}^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}\leq \left[J N^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}=J^{\sigma /(\sigma-1)} N.$$ Since $$\lim_{\sigma\uparrow 0} J^{\sigma /(\sigma-1)}=1,$$ the result follows. The argument is adopted from here.
Similarly, let $$N'=\min\{N_1,\ldots, N_J\}$$, $$K>0$$ be such that $$KN'>N_j$$ for all $$j$$, and $$0<\sigma<1$$.
$$N'=\left[N'^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}\geq\left[\sum_{j=1}^{J} N_{j}^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}\geq \left[KJ N'^{(\sigma-1) / \sigma}\right]^{\sigma /(\sigma-1)}=[KJ]^{\sigma /(\sigma-1)} N'.$$
Since $$\lim_{\sigma\downarrow 0} [KJ]^{\sigma /(\sigma-1)}=1,$$ the result follows.