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In a competitive market, is it possible to know the change (increase/decrease) in number of firms in the Long run with a positive shift in demand for increasing costs case?

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Yes, it is possible. In the long run, firms enter until they break even. Suppose firms are symmetric. Then for each firm the break even condition is that the average costs equal the price. This is because the price is equal to the average revenue. The average revenue is given by $px/x=p$ where $p$ is price and $x$ is quantity. If revenue equals costs on average, then there are no profits left to be made and this is the equilibrium number of firms.

Further, in a competitive market, price equals marginal costs. Hence, the break even condition there is also price equals marginal costs. Alternatively, in this case the condition is Average Costs equal Marginal Costs. From this condition you can find the break even quantity. From there find the break even price, by setting price=marginal costs. Then plug in the break even price and quantity into the demand function to determine how many firms must enter until the market equilibrium price and quantity are equal to the break even price and quantity. The result is the long run equilibrium number of firms.

Note that the question is only reasonable in the presence of fixed costs. If there are no fixed costs, the long run number of firms is infinite and we have a trivial answer. However, the treatment of this topic in textbooks as well as academic papers alwa assumes the presence of fixed costs.

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    $\begingroup$ Is this independent of the returns to scale of firms' technology? I had the idea that CRS does not allow you to pin down number of firms, as scale is irrelevant. $\endgroup$ – luchonacho Mar 30 '17 at 18:56
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    $\begingroup$ It is possible to pin down if there are fixed costs, which are usually assumed in such models. Otherwise, with CRS there is an infinite (or undefined) number of firms. $\endgroup$ – BB King Mar 30 '17 at 21:02
  • $\begingroup$ "Yes, it is possible" - Shouldn't this be "Yes, it is possible in some circumstances"? The circumstances being symmetric firms with some fixed costs. $\endgroup$ – Adam Bailey Apr 1 '17 at 9:08
  • $\begingroup$ The firms need not necessarily be symmetric. This is just the easiest case to illustrate and the only case used in undergraduate studies, which is the impression I got from OP. Questions about the number of firms always include fixed costs. Papers on the subject always assume it. Technically, it is even easier to find the long run number of firms without fixed costs. It is infinite. I can add my comment on this issue to the answer. $\endgroup$ – BB King Apr 1 '17 at 9:19
  • $\begingroup$ Actually, the solution of this question lies in this other question posed by me: economics.stackexchange.com/questions/16028/… $\endgroup$ – Akash Malhotra Apr 2 '17 at 18:43

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