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To clarify this is for an economics course.

$$ x \in \mathbb R^n $$ For a firm with two outputs with cost function where

$$ C(q)=q_1^2+q_2^2+10 $$ given output levels $q=(q_1,q_2) \ge 0$.

If output prices given and fixed, find profit maximizing solution.

I tried to find the global max, but ended up with global min at $(0,0)$, and am not sure how to proceed given just this one equation. Any help would be greatly appreciated.

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  • $\begingroup$ Sorry for the formatting, was trying to figure out how to do subscript and super script. $\endgroup$ – Shaner Oct 9 '16 at 21:47
  • $\begingroup$ What eactly is the profoit in this case? I only see a cost function. $\endgroup$ – 6005 Oct 9 '16 at 21:49
  • $\begingroup$ @Shaner I changed the formatting to Latex $\endgroup$ – Jannik Pitt Oct 9 '16 at 21:49
  • $\begingroup$ Thanks for the help with the formatting! $\endgroup$ – Shaner Oct 9 '16 at 21:50
  • $\begingroup$ I have written the problem exactly as I have it. I can add more assumptions , if more information is necessary please let me know and I'll search my book. $\endgroup$ – Shaner Oct 9 '16 at 21:53
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The profit is maximal, where the cost is minimal, which happens at the origin.

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You can't solve this problem without knowing the output prices. The higher the prices, the higher the manufactured quantities should be. You are minimizing the cost function, which you have done correctly. The minimum cost is $10$ at $q_1=q_2=0$ To maximize profit, you need to be able to compute it. Alternately, you could assume the price of item $1$ is $p_1$ and the price of item $2$ is $p_2$, find that revenue is $p_1q_1+p_2q_2$ and the profit is $p_1q_1+p_2q_2-q_1^2-q_2^2-10$, and find the maximum the same way you did viewing $p_1,p_2$ as parameters. You should find $q_1=p_1/2, q_2=p_2/2$

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  • $\begingroup$ But isn't the profit always as high as possible when the cost is the lowest? The question doesn't state that you actually need to calculate the final price. $\endgroup$ – Jannik Pitt Oct 12 '16 at 19:11
  • $\begingroup$ @JannikPitt: no, increasing the quantity increases the cost, but it also increases the revenue. The profit is as high as possible at a constant revenue when the cost is lowest. The profit is maximal when the marginal cost equals the marginal revenue. $\endgroup$ – Ross Millikan Oct 12 '16 at 21:54

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