<p>The exponents do not cancel each other out. E.g.,
<span class="math-container">$$
(a^2+b^2)^{1/2} \neq a + b 
$$</span>
because
<span class="math-container">$$
(a^2+b^2)^{1/2} \neq (a^2 + 2ab + b^2)^{1/2}.
$$</span></p>

CC BY-SA 4.0
2020-04-26T10:41:50.863Z
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 $\pi$?
David
CC BY-SA 4.0
2020-04-26T13:57:40.390Z
@David Your previous question was harder.
Giskard
CC BY-SA 4.0
2020-04-25T06:19:33.263Z
2020-04-25T06:19:33.263Z
Giskard
<p>I am trying to better understand the CES function:</p>

<p><span class="math-container">$$Y_{t}=C\left[\pi\left(A_{t}^{K} K_{t}\right)^{\frac{\sigma-1}{\sigma}}+(1-\pi)\left(A_{t}^{L} L_{t}\right)^{\frac{\sigma-1}{\sigma}}\right]^{\frac{\sigma}{\sigma-1}}$$</span></p>

<p>I understand the special cases where <span class="math-container">$\sigma=1$</span>, where <span class="math-container">$\sigma=0$</span> and where <span class="math-container">$\sigma\rightarrow0$</span>.</p>

<p>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 <span class="math-container">$\sigma$</span> and the expression reduces to:</p>

<p><span class="math-container">$$Y_{t}=C(\pi(A_{t}^{K} K_{t}) +(1-\pi)(A_{t}^{L} L_{t}))$$</span></p>

<p>This, of course, can't be right as <span class="math-container">$Y_t$</span> should vary for different values of <span class="math-container">$\sigma$</span>, no?</p>

<macroeconomics><microeconomics><labor-economics><elasticity>
Constant Elasticity of Substitution (non-special cases)
David
# Constant Elasticity of Substitution (non-special cases)

(Asked by: username_0 on 2020-04-25)

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?

**Tags:** macroeconomics, microeconomics, labor-economics, elasticity

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## 1 answers

### Answer by username_1 on 2020-04-25. Score: 3 | Accepted answer

The exponents do not cancel each other out. E.g.,
$$
(a^2+b^2)^{1/2} \neq a + b
$$
because
$$
(a^2+b^2)^{1/2} \neq (a^2 + 2ab + b^2)^{1/2}.
$$

> **Comments:**
> **username_0**: 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 $\pi$?  
> **username_1**: @David Your previous question was harder.  

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