# Cost function and short run supply

Suppose there is a number of identical firms producing diamonds. Let the total cost function for each firm be given by $C(q, w)$, where q is the firm’s output level and w is the wage rate of diamond cutters.

I'm supposed to solve for a firm's (short-run) supply curve, given a wage rate $w$, and the industry's (total) supply curve).

I'm very confused with the relationship between the cost function and the supply curve. For part a this question, I get the marginal cost from taking the derivative of the cost function -- is the marginal cost also the supply curve? We know the cost function of the firm, but how can I arrive at the industry's supply curve?

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Hi (again). In order to better understand where your problem lies (and to avoid us doing your homework) we ask that you show your own work. Otherwise this question will be closed. – denesp Oct 21 '15 at 6:27
You should reduce this question by a lot, only showing us the particular part you're having problems with. We don't need any numbers or formulas in your question. I'll have an attempt at simplifying this, revert if you dislike it. – FooBar Oct 21 '15 at 8:35
@denesp In my opinion the OP has included information on what she has done on the problem. As regards this aspect, I don't see why the question should be closed. – Alecos Papadopoulos Oct 21 '15 at 10:30
@AlecosPapadopoulos Then you should vote "Leave open". Perhaps meta would be a better place for this discussion. – denesp Oct 21 '15 at 11:07

## 1 Answer

Just like a Production function, in Economics, a Cost function is not an "engineering relation", but a "best response", reflecting minimum cost to produce a given quantity $q$.

Cost-minimizing firms are at first assumed not to determine their scale of operations -what they decide upon is the demand for production inputs,given the inputs' prices.

Still, by duality, the profit-maximizing condition should also hold -these are for-profit private companies, not, say, some public utility without a profit motive. If the firms are price-takers in the diamonds market, this relation is well-known, and provides the answer to a) (remember that there are functions, and then there are inverse functions).

As for b), I see in the question that the suppliers are assumed identical.

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