Limit of CES function as $p$ goes to $- \infty$

Question

I am trying to evaluate the limit of the CES Production function:

$$Y=(aK^p + bL^p)^{1/p} $$

when $p$ goes to -infinity.

It first yields the indeterminate form $0^0$. I tried solving the problem by taking the log of the function, but then the limit gave resulted in $0\cdot(-\infty)$.

Thanks in advance.

Answer 1

I think this is an easier way to solve it.

When $L<K$ we can factorize $L$ as follows:

$Y=\Big[bL^p \Big(\frac{aK^p}{bL^p}+1\Big)\Big]^{1/p}=b^{1/p} L \Big[\frac{a}{b}\big(\frac{K}{L}\big)^p +1\Big]^{1/p}=b^{1/p} L \Big[\frac{a}{b}\big(\frac{L}{K}\big)^{-p} +1\Big]^{1/p}$

$lim_{p \rightarrow - \infty} Y =L$ ,

since $b^{1/-\infty}=1$, and $lim_{p \rightarrow - \infty} \big(\frac{L}{K}\big)^{-p}=\big(\frac{L}{K}\big)^{\infty}=0$.

By symmetry, when $K<L$ (factoring $K$ instead of $L$), we obtain:

$lim_{p \rightarrow - \infty} Y =K$ .

Answer 2

Let be $Y=(aK^p+bL^p)^{1/p}$ the CES function with $a+b=1$. Take the logarithm $$ \log Y=\frac{\log(aK^p+bL^p)}{p} $$ and use the L'Hopital's rule with $p$ as variable: $$ \lim_{p\to -\infty}\frac{\log(aK^p+bL^p)}{p}=\lim_{p\to -\infty}\frac{aK^p\log K+bL^{p}\log L}{aK^p+bL^p} $$ Dividing both the numerator and denominator by $M^p$ where $M := \min(K,L)$ $$\lim_{p\to -\infty}\frac{aK^p\log K+bL^{p}\log L}{aK^p+bL^p}= \lim_{p\to -\infty}\frac{a(K/M)^p\log K+b(L/M)^{p}\log L}{a(K/M)^p+b(L/M)^p}=\log M $$ We get $(K/M)^p\to 0$ as $p\to -\infty$ for $K>M$ and $(L/M)^p\to 0$ as $p\to -\infty$ for $L>M$; and $K/M=1$ if $K=M$ or $L/M=1$ if $L=M$.

Subsequently, we obtain the corresponding Leontief function for $Y$ $$ \lim_{p\to -\infty}Y=M=\min(K,L). $$