# Market equilibrium when marginal cost is decreasing

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!

• 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* ? Nov 2, 2021 at 14:25