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?


1 Answer 1


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.

The difference between the two results should not be too surprising, since you work with positive and negative exponents, respectively, and they are clearly diametrically opposed.


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