Final answer:
In the Cournot duopoly model, each firm sets its quantity to maximize profit, assuming the other firm's quantity remains constant. The Cournot outcome in this case is a price of 0, firm 1 produces 13.33 units, firm 2 produces 16.67 units, and both firms earn zero profit.
Step-by-step explanation:
In a duopoly, both firms set their quantities and compete for market share. The Cournot model assumes that each firm sets its quantity taking the other firm's quantity as given. In this case, both firms have zero marginal costs, and the market demand is given by P=30-Q, where Q is the total quantity produced by both firms. The Cournot outcome occurs when each firm sets its quantity to maximize its profit based on the assumption that the other firm's quantity remains constant.
To find the Cournot outcome, we can start by finding the reaction functions of each firm. The reaction function shows the quantity that a firm would choose given the quantity of the other firm. In this case, the reaction function can be derived by solving each firm's profit maximization problem.
The profit function of firm 1 is given by
π1 = (P - MC1) * Q1,
where MC1 is the marginal cost of firm 1. By substituting the market demand function into the profit function and simplifying, we get
$(1) Q1 = rac{1}{2} (30 - Q2)$.
Similarly, the profit function of firm 2 is given by π2 = (P - MC2) * Q2, and substituting the market demand function, we get $(2) Q2 = rac{1}{2} (30 - Q1)$.
Now we can solve these two equations simultaneously to find the Cournot equilibrium.
By substituting equation $(2)$ into equation $(1)$, we get
$(3) Q1 = rac{1}{2} (30 - rac{1}{2} (30 - Q1))$.
Simplifying, we find $(4) Q1 = rac{4}{3} * 10 = 13.33$.
Substituting this value back into equation $(2)$, we find $(5) Q2 = 16.67$.
Finally, we can find the price by substituting the quantity values into the market demand function, which gives
$(6) P = 30 - Q = 30 - (13.33 + 16.67) = 30 - 30 = 0$.
Therefore, the Cournot outcome in this case is a price of 0, firm 1 produces 13.33 units of the good, firm 2 produces 16.67 units of the good, and both firms earn zero profit.