1. **Problem Statement:** We need to find the Bertrand equilibrium prices for two firms where firm 1 has a marginal cost of $30$ per unit and firm 2 has a marginal cost of $10$ per unit. 2. **Bertrand Model Basics:** In the Bertrand competition model, firms compete by setting prices simultaneously. The firm with the lower price captures the entire market, assuming identical products and no capacity constraints. 3. **Key Rule:** In equilibrium, no firm can profitably undercut the other's price. Prices tend to be driven down to the marginal cost of the more efficient firm. 4. **Step-by-step Solution:** - Firm 2 has a lower marginal cost ($10$) than firm 1 ($30$). - If firm 1 sets a price above $10$, firm 2 can undercut slightly and capture the whole market. - Firm 2 will set price equal to its marginal cost $p_2 = 10$ to prevent firm 1 from undercutting. - Firm 1 cannot profitably set a price below $10$ because its marginal cost is $30$. - Therefore, the Bertrand equilibrium prices are: $$p_1 = 30, \quad p_2 = 10$$ - Firm 2 captures the entire market at price $10$. 5. **Interpretation:** The firm with the lower marginal cost sets the price equal to its marginal cost and captures the market. The other firm cannot compete profitably. **Final answer:** $$p_1 = 30, \quad p_2 = 10$$