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I need to derive the profit function for the following CES function: $$ f(z) = (\sqrt{z_{1}^{\rho} + z_{2}^{\rho}})^{1/ \rho}$$ where $\rho \leq 1$. This is the answer that I am supposed to be getting:

If $ \rho <1$ then $$ \pi(w) = \begin{cases} \infty \;\;\;\;\; \text{if} & w_{1}^{\rho/(\rho -1)} + w_{2}^{\rho/(\rho -1)} <1 \\ 0 \;\;\;\;\; \text{if} & w_{1}^{\rho/(\rho -1)} + w_{2}^{\rho/(\rho -1)} \geq 1\ \end{cases} $$

If $\rho = 1$, then:

$$ \pi(w) = \begin{cases} 0 & \text{if} \;\ Min\{w_{1},w_{2}\} \geq 1 \\ \infty & \text{if} \;\; Min\{w_{1},w_{2}\} < 1 \\ \end{cases} $$

Normally I simply use profit maximization or cost minimization FOCs to obtain the profit function, as well as the supply one. But I have been trying to get around this problem and I really cannot get around these results or (of particular importance) how to derive them. That being said any tips, intuition, or just help with the derivation if you're feeling particularly selfless would be greatly appreciated.

Thanks in advance!

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  • $\begingroup$ Just in case someone has a problem similar to this one I followed the hint provided in the answer and simply used the condition that the ratio of the marginal product of the inputs be equal to the ratio of their marginal costs, and then used this to solve for $z_{1}$, and plugged it into the production function. Then, solving for $z_{2}$, which can be substituted into the cost function. [...] $\endgroup$ – user20105 May 9 at 11:34
  • $\begingroup$ [...] The marginal cost function can be obtained by differentiating the CES cost function with respect to output. The supply function (not included here), for competitive product market conditions, can be obtained by setting the marginal cost to P, the price of the output. $\endgroup$ – user20105 May 9 at 11:34
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Hint: Solving for the FOC's assumes that the solution is interior, in this case, that profits are positive and smaller than $\infty$. I would recommend you to derive the cost function $c(y)$ and then study its derivative. If the marginal cost is always smaller than the price of the good (probably it is assumed to be 1) then producing more is always better and profits are unbounded. However, if the marginal cost is always larger than the price, then the optimal is to produce nothing and get zero profits.

Hope this helps.

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  • $\begingroup$ Thank you! This was really useful $\endgroup$ – user20105 May 9 at 11:36

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