# Constant Elasticity of Substitution (non-special cases)

I am trying to better understand the CES function:

$$Y_{t}=C\left[\pi\left(A_{t}^{K} K_{t}\right)^{\frac{\sigma-1}{\sigma}}+(1-\pi)\left(A_{t}^{L} L_{t}\right)^{\frac{\sigma-1}{\sigma}}\right]^{\frac{\sigma}{\sigma-1}}$$

I understand the special cases where $$\sigma=1$$, where $$\sigma=0$$ and where $$\sigma\rightarrow0$$.

I am, for some reason, having a really hard time developing intuition for other cases. Don't the exponents cancel for all other values of $$\sigma$$ and the expression reduces to:

$$Y_{t}=C(\pi(A_{t}^{K} K_{t}) +(1-\pi)(A_{t}^{L} L_{t}))$$

This, of course, can't be right as $$Y_t$$ should vary for different values of $$\sigma$$, no?

The exponents do not cancel each other out. E.g., $$(a^2+b^2)^{1/2} \neq a + b$$ because $$(a^2+b^2)^{1/2} \neq (a^2 + 2ab + b^2)^{1/2}.$$
• Ah, yes, I feel embarrassed about the silliness of my question. May I ask a simple follow-up? In the case where $$A_{t}^K K_{t} = A_{t}^L L_{t}$$ should we then be indifferent to the selection of values of $\pi$? Apr 26 '20 at 10:41