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I'm trying to solve this problem:


Technology for producing $q$ gives rise to the cost function $c(q) = aq + bq^2$. The market demand for $q$ is $p = \alpha - \beta q$

(a) If $a > 0$, if $b < 0$, and if there is $J$ firms in the industry, what is the short-run equilibrium market price and number of firms?

(b) If $a > 0$, if $b < 0$, what is the long-run equilibrium market price and number of firms?

(c) If $a > 0$, if $b > 0$,what is the long-run equilibrium market price and number of firms?


Two questions:

Regarding (a): First order condition for max profit is $ p = \frac{dc}{dq}$, so $q^* = \frac{p-a}{2b}$. But if $b<0$, then $c''(q) = 2b < 0$, and the firm would be minimizing profits at $q^*$. I'm tempted to simply answer that the problem is undefined. What do you think?

Regarding (c): The book gives two conditions for finding long-run equilibrium: market clearing and zero profit. To find the profit function for a given firm, I integrated the supply function, using Hotelling's lemma $\frac{\partial \pi (p,w)}{\partial p} = y(p,w)$. Am I on the right track?

Thanks!

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  • $\begingroup$ With b<0 , then as q increases the cost will go to zero at q*=-a/b , and then becomes negative for larger q. Which doesn't make sense. Unless perhaps the firms are capacity-constrained so total production is less than q* ? $\endgroup$
    – Daniel
    Nov 2, 2021 at 14:25

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