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 $$ (a^2+b^2)^{1/2} \neq (a^2 + 2ab + b^2)^{1/2}. $$

  • $\begingroup$ 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 $\pi$? $\endgroup$
    – David
    Apr 26 '20 at 10:41
  • 1
    $\begingroup$ @David Your previous question was harder. $\endgroup$
    – Giskard
    Apr 26 '20 at 13:57

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