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In economic theory we know that with the use of some calculus, Hotellings Lemma and Sheppards lemma we can derive a given firms supply function and in term its Profit function.

With data of a given firms costs, can we in fact get an accurate estimate of its profit function and supply function?

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    $\begingroup$ Great question. I hope someone pulls data and answers this. $\endgroup$ – 123 Apr 2 '17 at 23:44
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    $\begingroup$ Maybe this and this could help. The latter says that the estimation of cost function from profit function is better than the reverse. $\endgroup$ – luchonacho Apr 4 '17 at 9:28
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    $\begingroup$ As you stated in the question, theoretically if cost is known, then supply and profit are known. So the estimate is perfectly accurate. To clarify the question, are you asking how empirically accurate is cost estimation usually? Or if costs are mis-estimated, then what is the error with which the supply and profit are estimated? $\endgroup$ – Sander Heinsalu Sep 28 '18 at 1:10
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Here's a paper link where they picked up a $R^2$ of over .99 I don't think this is particularly unusual or strange of a result, either. In estimating costs for firms in public utilities, I recall doing numerous exercises with real data from various utilities. I typically got $R^2$'s in the 0.95's or higher, so I feel they are fairly accurate within sample.

This means economists are fairly good at estimating what a firm's per-unit costs are if we have a good knowledge of their total capital stock and expenditures, and we stay within the bounds of our sample.

However, if you are asking about very far out-of-sample predictions, or systemic shifts in the market, then you may find less success.

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