Question

Two firms simultaneonsly and independently select the number of a good to produce. Let q₁ denote firm 1 's quantity and q₂ denote firm 2's quantity. Assume that q₁ ​,q₂ ≥0. The total output in the industry is then q₁+q₂. Suppose the price is given by the simple function p=10−q₁−q₂​. But there's a twist, suppose firm 1 must pay a production cost of $1 per product. And firm 2 must pay a production cost of $4 per product. Firms wish to maximize their profits. (Hint The profit of firm 2 can be written as π₂​=pq₂−4q₂ , where 4q₂ is firm 2 's total cost) 1 (a) Find the best response for two firms

Answer 1

The best response for firm 1 is to produce 1 unit of the good, and the best response for firm 2 is also to produce 1 unit of the good.

Step-by-step explanation:

A perfectly competitive firm determines its best response by maximizing its profits, which occurs when its marginal cost (MC) is equal to the market price. In this case, firm 1's profit can be calculated using the equation π₁ = p*q₁ - c₁*q₁, where c₁ is the production cost per unit for firm 1. Similarly, firm 2's profit can be calculated using the equation π₂ = p*q₂ - c₂*q₂, where c₂ is the production cost per unit for firm 2.

To find the best response, firms need to choose the quantity (q₁, q₂) that maximizes their respective profits. This occurs when the marginal cost (MC) is equal to the market price (p) for each firm.

Let's solve an example using the given information. Consider a scenario where the market price (p) is 10. As per the hint, firm 1 has a production cost (c₁) of $1 per product, while firm 2 has a production cost (c₂) of $4 per product.

For firm 1, the profit equation becomes π₁ = 10*q₁ - 1*q₁ = 9*q₁. To maximize profits, we set MC = p, and we have MC₁ = 9. Therefore, firm 1's best response is q₁ = 1.

For firm 2, the profit equation becomes π₂ = 10*q₂ - 4*q₂ = 6*q₂. Again, setting MC = p, we have MC₂ = 6. Therefore, firm 2's best response is q₂ = 1.

So, the best response for firm 1 is to produce 1 unit of the good, and the best response for firm 2 is also to produce 1 unit of the good.