(a) In a Bertrand equilibrium, firms set prices equal to their marginal costs. Since both firms have a constant marginal cost of $4, the equilibrium prices can be found by setting the prices equal to $4.
Therefore, the Bertrand equilibrium prices are P(A) = $4 and P(B) = $4.
(b) To calculate the industry output, we need to find the quantities that correspond to the Bertrand equilibrium prices.
Using the inverse demand equation P = 80 - 4(QA + QB), we can substitute the equilibrium prices to find the corresponding quantities:
P(A) = 80 - 4(QA + QB)
$4 = 80 - 4(QA + QB)
Solving for QA + QB, we have:
QA + QB = (80 - $4) / 4
QA + QB = 76 / 4
QA + QB = 19
Thus, the industry output in the Bertrand equilibrium is 19.
(c) To solve for the Cournot equilibrium outputs, we need to maximize the total industry profits. Each firm chooses its quantity to maximize its profits, taking into account the quantity chosen by the other firm.
The profit function for firm A is (P - MC) x QA, where MC is the marginal cost. Substituting the inverse demand equation and the marginal cost, we have:
Profit(A) = (80 - 4(QA + QB) - 4) x QA
Profit(A) = (76 - 4QA - 4QB) x QA
Profit(A) = 76QA - 4QA^2 - 4QAA
Similarly, the profit function for firm B is:
Profit(B) = (80 - 4(QA + QB) - 4) x QB
Profit(B) = (76 - 4QA - 4QB) x QB
Profit(B) = 76QB - 4QBB - 4QB^2
To find the Cournot equilibrium outputs, we need to find the quantities QA* and QB* that maximize the total industry profit:
Total Profit = Profit(A) + Profit(B)
Total Profit = 76QA - 4QA^2 - 4QAA + 76QB - 4QBB - 4QB^2
To find the maximum, we need to differentiate the total profit with respect to QA and QB, and set the derivatives equal to zero:
dTotal Profit / dQA = 76 - 8QA - 4QB = 0
dTotal Profit / dQB = 76 - 8QB - 4QA = 0
Solving these equations simultaneously, we have:
8QA + 4QB = 76
8QB + 4QA = 76
Simplifying these equations, we get:
2QA + QB = 19
QA + 2QB = 19
Solving these equations, we find QA* = 5 and QB* = 7.
Therefore, the Cournot equilibrium outputs are QA* = 5 and QB* = 7.
To solve for the Stackelberg equilibrium, we need to consider a leader-follower model, where one firm acts as the leader and sets its quantity first, and the other firm acts as the follower and sets its quantity based on the leader's quantity. Let's assume firm A is the leader.
The follower maximizes its profit by taking the leader's quantity as given and setting its own quantity. The profit function for firm B is the same as before:
Profit(B) = (76 - 4QA - 4QB) x QB
To find the Stackelberg equilibrium outputs, we need to find the quantities QA* and QB* that maximize the total industry profit, considering the leader-follower model:
Total Profit = Profit(A) + Profit(B)
Total Profit = Profit(A) + (76 - 4QA - 4QB) x QB
Differentiating the total profit with respect to QB and setting the derivative equal to zero, we have:
dTotal Profit / dQB = -4QA - 8QB + 76 - 4QB + 76 - 2QA = 0
Simplifying this equation, we get:
-6QA - 12QB + 152 = 0
3QA + 6QB = 76
Substituting QA = 5 (the leader's quantity in the Cournot equilibrium), we can solve this equation to find QB*:
3(5) + 6QB = 76
15 + 6QB = 76
6QB = 76 - 15
6QB = 61
QB* = 10.17 (rounded to two decimal places)
Since the leader sets its quantity first, we have:
QA* = 5 (the same as in the Cournot equilibrium)
Therefore, the Stackelberg equilibrium outputs are QA* = 5 and QB* = 10.17.
Comparing the outputs, we have:
Bertrand equilibrium: QA = 9.50, QB = 9.50
Cournot equilibrium: QA* = 5, QB* = 7
Stackelberg equilibrium: QA* = 5, QB* = 10.17
In the Bertrand equilibrium, both firms produce the same quantity as they compete by setting prices. In the Cournot equilibrium, firm A produces a smaller quantity than in the Bertrand equilibrium, while firm B produces a larger quantity. In the Stackelberg equilibrium, firm A produces the same quantity as in the Cournot equilibrium, while firm B produces a larger quantity than in the Cournot equilibrium.