Question

There are only two firms in an industry producing a homogeneous product and having identical production technology. The cost function of firm $i$ is \[ C_i(q_i) = q_i^2, \quad \text{for } i = 1,2; \] where $q_i$ is the quantity produced by firm $i$. The market demand for the product is $p = 100 - q$, where $p$ is the unit price and $q = q_1 + q_2$ is the aggregate quantity. Assuming the firms are price takers, the competitive equilibrium solution of $p$ and $q$ in this market is

• A. $p = 80, \ q = 20$
• B. $p = 20, \ q = 80$
• C. $p = \dfrac{200}{3}, \ q = \dfrac{100}{3}$
• D. $p = 50, \ q = 50$

Hint:

In perfect competition, equilibrium occurs when $P = MC$. If multiple firms have identical cost functions, the market supply is the sum of individual outputs where $P = MC_i$.

Answer 1

The Correct Option is C

Step 1: Determine market equilibrium condition.
Under perfect competition, firms are price takers, so price equals marginal cost (MC).
Step 2: Find marginal cost for each firm.
\[ C_i(q_i) = q_i^2 \Rightarrow MC_i = \frac{dC_i}{dq_i} = 2q_i. \] At equilibrium, $p = MC_i = 2q_i$. Hence, each firm produces $q_i = \frac{p}{2}$.
Step 3: Market supply and demand.
Aggregate supply: $q = q_1 + q_2 = p$. Market demand: $p = 100 - q$. Equating supply and demand: \[ p = 100 - p \Rightarrow 2p = 100 \Rightarrow p = 50. \] Then, $q = p = 50$. Wait — this gives option (D). But this represents the **competitive equilibrium** only if we assumed perfect competition. The question says “firms are price takers,” which fits this case, so option (D) seems correct.
Step 4: Verification.
Each firm produces $q_i = 25$ (since $q = 50$ total), and $p = 50$ equals $MC = 2(25) = 50$. Hence, equilibrium is consistent.
Step 5: Conclusion.
Competitive equilibrium solution: $p = 50, \ q = 50$.