The market demand in a Bertrand duopoly is P = 15 - 4Q, and the marginal costs are $3. Fixed costs are zero for both firms. Which of the following statement(s) is/are true?
a.P = $3.
b.P = $10.
c.P = $15.
d.None of the statements associated with this question are correct.
hjvuk
To determine the correct statement(s), we need to find the equilibrium price (P) in the Bertrand duopoly. The equilibrium price occurs when both firms set their prices based on their marginal costs and the market demand.
First, let's find the equilibrium quantity (Q):
The total quantity demanded in the market is the sum of the quantities demanded by both firms. So we have:
Q = Q1 + Q2
Each firm's quantity demanded is determined when its price equals the market demand. Thus, we have:
Q1: P1 = 15 - 4Q
Q2: P2 = 15 - 4Q
Setting the prices equal to marginal costs (MC) gives us:
P1 = MC1 = $3
P2 = MC2 = $3
Substituting the prices into the quantity-demand equations:
Q1 = 15 - 4Q
Q2 = 15 - 4Q
Now we can solve for the equilibrium quantity. Adding the quantity demanded by both firms:
Q = Q1 + Q2
Q = (15 - 4Q) + (15 - 4Q)
Q = 30 - 8Q
Simplifying the equation:
9Q = 30
Q = 30/9
Q ≈ 3.33
Now that we have the equilibrium quantity (Q), we can find the equilibrium price (P) by substituting it into either Q1 or Q2's price-demand equation. Let's use P1 = 15 - 4Q:
P = 15 - 4(3.33)
P ≈ $1.67
As a result, none of the statements (a, b, c) are true. The correct statement is:
d. None of the statements associated with this question are correct.