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Consider a firm with the following CES production function, which utilizes only two production factors (capital and labor) whose prices are, respectively, $r > 0$ and $w > 0$:

$$ y = \gamma \cdot \left\{ \delta \cdot K^{-\rho} + (1-\delta) \cdot L^{-\rho} \right\}^{-\frac{V}{\rho}} + 100\epsilon $$

where $\gamma > 0, V > 0, \rho \geq -1$ and $0 < \delta < 1$.

a) Determine the value of the parameter $\epsilon$ and justify your answer.

Here's my thought: based on the "no free-lunch" property, a firm cannot produce while using zero inputs, which would mean that:

$$ y(K=0,L=0) = 0 \implies \gamma \cdot \left\{ \delta \cdot 0^{-p} + (1-\delta) \cdot 0^{-p} \right\}^{-\frac{V}{p}} + 100\epsilon = 0 \Leftrightarrow \epsilon = 0 $$

However, for that to be valid, $-\rho > 0 \therefore \rho < 0 $, because otherwise we would have a division by 0. So should the correct definition of the interval where $\rho$ belongs be $-1 \leq \rho < 0 $ ?

On the same topic, can a production function ever have a constant term? I'd think not because that would violate the "no free-lunch" property.

I appreciate any inputs!

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