Question

The market demand curve for a pair of duopolists is given as P=100- Q where Q= Q₁+ Q_2. The constant per unit marginal cost is 0 for firm 1 and c for firm 2 where c is some number. Find the equilibrium price, quantity and profit for each firm in the Bertrand model as a function of c

a. Equilibrium price equals P=0. Equilibrium quantity is Q₁=Q₂=10 with both earning Π₁=Π₂=0. Which one is correct?

---C= 0 OR C>0

b. Firm 2 produces zero Q₂=0 and earns Π₂= 0. Firm 1 charges P=c and produces Q₁=100-c earning Π₁=100 c – c²

^{Which one is correct?}

---C= 0 OR C>0

Is C=0 or C>0 for the two questions

Answer 1

Under Bertrand competition, firms choose the price at which they want to sell their goods, and the market determines the quantities which can be sold. Therefore, each firm will sell the goods at its marginal cost, in order to avoid a price competition. It is given that the firm l's marginal cost is O and firm 2’s marginal cost is c. Now, the demand curve is given as follows: P=100-Q, where Q is quantity and P is price. As the firms sell their goods at marginal cost, calculate the quantity sold for both the firms as follows: P=100-2 P=100-22-22 For firm 1, P=0 0=100-4-2 Q = 100-22

For firm 2, P=c c=100-9-22 Q2 =100-Q,-c Calculate the profits for the two firms as follows: Profits for Firm 1 = (P-MC)Q =(0-0)(100-22) = |SO Profits for Firm 2 = (P-MC)Q2 = (c-c)(100-22) = $0 Therefore, Firm 1's profit is $0 and Firm 2's profit is $0

If equilibrium price equals 0, and quantities sold equal 10 each, and the profits of the two firms are $0, then the price has to be equal to the marginal cost. This is because only when the price and the marginal cost are same can the firm earn 0 profits. Therefore, P=c=0 Thus, c= 0 is true. b. If Firm 2 sells 0 and makes O profits, then the price can be anything. It is given that firm 1 is charging P= c and the marginal cost is 0. Now, Profits for Firm 1 = (P-MC)Q. 100c-c? =(c-0)(100-22) 100c-c? =(0-0)(100-0) 100c-c? = 100c

Thus, c= 0 is true.