Suppose the inverse market demand equation is P = 80 ¡V 4(QA+QB), where QA is the output of firm A and QB is the output of firm B, and both firms have a constant marginal constant of $4.
(a)Write down the Bertrand equilibrium prices for this market.
(b)Calculate the industry output.
(c)Solve for the Cournot and Stackelberg equilibrium outputs, and compare to the Bertrand equilibrium.
(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.
To find the Bertrand equilibrium prices for this market, we need to find the values of QA and QB that maximize each firm's profit given the inverse market demand equation.
(a) In the Bertrand model, firms compete by simultaneously setting their prices. The firm with the lowest price wins all the market share. To find the Bertrand equilibrium prices, we need to determine the prices that make both firms have the same profit.
Let's denote the price for firm A as PA and the price for firm B as PB. The profit for firm A can be calculated as follows:
Profit(A) = (PA - 4QA) * QA
Similarly, the profit for firm B is:
Profit(B) = (PB - 4QB) * QB
To find the equilibrium prices, we need to set the profits of both firms equal to each other and solve for PA and PB:
(PA - 4QA) * QA = (PB - 4QB) * QB
From the inverse market demand equation, we know that PA = 80 - 4(QA + QB), substitute this into the above equation:
(80 - 4(QA + QB) - 4QA) * QA = (PB - 4QB) * QB
Now, we can solve for PA and PB by substituting the above equation into the inverse demand equation and solving for the equilibrium prices. However, the given question does not have the specific values of QA and QB, so we cannot calculate the exact equilibrium prices.
(b) To calculate the industry output, we need to find the total output of both firms at the equilibrium prices.
Industry output = QA + QB
Again, without the specific values of QA and QB, we cannot calculate the exact industry output.
(c) The Cournot equilibrium and Stackelberg equilibrium outputs can be found by assuming different strategies for the firms and solving for the equilibrium quantities.
In the Cournot model, firms choose their outputs simultaneously, taking into account the other firm's output as given. Each firm maximizes its profit by choosing its output.
To find the Cournot equilibrium output, we can differentiate the profit function of each firm with respect to its output, set the derivatives equal to zero, and solve for QA and QB. However, we do not know the specific values of QA and QB, so we cannot find the Cournot equilibrium output.
In the Stackelberg model, one firm, called the leader, sets its output first, and the other firm, called the follower, observes the leader's output and then chooses its output to maximize profit.
To find the Stackelberg equilibrium output, we need to determine the leader and follower, and solve for their outputs accordingly. Given that the leader sets its output first, the leader's profit-maximizing output can be found by differentiating its profit function with respect to its output and setting the derivative equal to zero. The follower then determines its profit-maximizing output by taking into account the leader's output as given.
Again, without the specific values of QA and QB, we cannot calculate the exact Cournot and Stackelberg equilibrium outputs, and compare them to the Bertrand equilibrium.