# Why is the competitive equilibrium price not well defined in this question?

Consider a market demand function p = 100 − q, where p is market price and q is aggregate demand. There is a large number of firms with identical cost functions

ci(qi) = 10 + 2qi, if qi > 0
0, otherwise


Solving the cost function, I am getting that Marginal Cost is equal to 2. For a competitive equilibrium price, we set Price equal to Marginal Cost. So, price = 2. But the answer says that competitive equilibrium price is not well defined.

Could anyone please explain this to me? Why is it not well defined?

• Competitive Equilibrium does not exist in this case. This is because at $p=2$, firm will choose to produce $0$ and that is not equal to the demand which is $98$.
– Amit
Commented Apr 16 at 18:17
• Can you please explain why would the firm choose to produce 0 units at price = 2? Commented Apr 16 at 18:24
• @Shreya because by producing $98$ (or any non zero quantity), it would have a profit of $-10$, which is worse than having a profit of $0$ which is achieved by producing $0$. Commented Apr 16 at 20:01

Solution: Your approach is right but it wont work in this situation as there are certain avoidable fixed costs associated with your cost function.

we are given the demand function as, $$p^d(q)=100-q$$ (i.e., $$q^d(p)=100-p$$)and the cost function as $$c_i(q_i)= (10+2q_i)\mathbb{I}_{(q_i \in (0,\infty))}$$ which tells us that there are avoidable fixed costs of 10.

Now we can setup the firm's optimization problem, $$\max_{\{q_i\}} \hspace{4mm} pq_i-(10+2q_i) \mathbb{I}_{(q_i \in (0,\infty))}$$

if we solve the above problem, we will get the supply as a function of price(p) for firm i,

$$q_i^s(p)= \cases{\{0\}, & \text{p \le 2}\\ \infty, & \text{otherwise} }$$

Notice that the markets wont clear for any price $$p \le 2$$ as the firms wont be willing to produce anything and for price $$p >2$$,the firms are ready to produce $$+\infty$$ amounts of quantity, Thus even in this case, the market wont clear.

Hence, There doesn't exist a competitive equilibrium.

• The cost functions precise definition makes it clear that these are quasi-fixed costs, not fixed costs, which would occur even when $q=0$. Commented Apr 17 at 8:28
• Yeah you are right. I should have used the term "avoidable fixed costs", so do you also mean the same thing by quasi-fixed costs?
– SGP
Commented Apr 17 at 8:35
• Yes. There is one more occurence "which tells us that there are fixed costs of 10" Commented Apr 17 at 8:40
• Yeah thanks for your inputs @Giskard
– SGP
Commented Apr 17 at 8:46