# Short, Medium and Long-Run Profit Maximization

Suppose that, in a perfectly competitive industry, the firms' technology have the following cost function: $$C(x) = 100 + 3x + 0.04x^2$$. Assume the fixed costs are sunken.

a) If the demand for the product is given by $$D(p) = 200(10-p)$$, what is the long-run equilibrium for this industry? (price, quantity demanded and consumed, number of firms).

In the long-run equilibrium, the price should equal the minimum average cost. That happens when $$MC = AC$$, which means: $$3 + 0.08x = \frac{100}{x} + 3 + 0.04x \implies x = 50, x > 0$$

For this value of $$x$$, we have, equating $$p = CMg$$: $$p = 3 + 0.08 \cdot 50 = 7$$ and, therefore, $$D(7) = Q^D = 600$$ which means that the number of firms is $$n = 600/50 = 12$$.

b) Under a negative and permanent demand shock, it dislocates to $$D(p) = 200(8-p)$$. In the very-short run, the firms cannot change their production plans. What is the new equilibrium price? In the medium-run, however, the firms can adjust their production plans. Find the medium-run equilibrium (price, quantity produced and consumed, number of firms).

If in the very-short run the productions plans cannot be changed, let's assume the total market output is still 600. Using the new demand function: $$600 = 200(8-p) \implies p = 5$$

In the medium-run, the firms can adjust their production plans and therefore we should find the value of $$q$$ that satisfies $$MC = AVC$$: $$3 + 0.08x = 3 + 0.04x \implies x = 0$$ And so: $$p = 3 + 0.08 \cdot 0 \implies p = 3$$ and therefore $$D(3) = Q^D = 1000$$ However, since the best option for the firms is to not produce, we can (can we?) suppose that all of them leave the market, resulting in $$n = 0$$.

c) Find the new long-run equilibrium (price, quantity produced and consumed, number of firms). Compare your results with the medium-run equilibrium and interpret.

In the new long-run, we will again have $$MC = AC$$, resulting in a price of $$p = 7$$ and $$x = 50$$ again. However, given the change in the demand function, the quantity demanded now will be $$D(7) = 200(8-7) = 200$$ which means that the new number of firms is $$n = 200/50 = 4$$, an increase for the medium-run, due to the absence of fixed-costs in the long-run.

Are my answers correct? I had never encountered a question asking about medium-run before. Also, is the profit maximization level of production different from the equilibrium level of production?

Thanks!

• b) looks wrong to me. The way I read the assignment firms are in the medium term able to adjust quantity but firms are not able to enter or exit market. So p=MC and Supply(p) = Demand(p), with supply being 12*(p-3)/0.08 = 12x ... 12 firms all same production technology/cost function. – Jesper Hybel Mar 14 at 21:29
• That was very helpful. Following that path, I found x = 70, p' = 3.8 and q' = 840. How to interpret the change in the number of firms? Some firms decided to leave the market in the long-run after the reduction in demand? Thanks! – Pedro Cunha Mar 14 at 23:49