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 cost of $4 (fixed costs are zero).
(a)Write down the profit equations for firms A and B.
(b)Write down the marginal revenue equations for firms A and B.
(c)Write down the reaction functions for firms A and B.
(d)Graph the reaction functions for firms A and B.
(e)Calculate the Cournot equilibrium outputs for this market. Plot them on the graph.
(f)What is the price charged in this market?
(g)What is the industry output?
(h)Does the result of identical prices charged by the two firms depend on them having identical marginal costs? Explain.
(a) The profit equation for firm A can be calculated as follows:
Profit_A = Total Revenue_A - Total Cost_A
Total Revenue_A = Price_A * QA
Total Cost_A = (Marginal Cost_A + Total Fixed Cost_A) * QA
Since the fixed costs are zero for both firms, the profit equation for firm A simplifies to:
Profit_A = (P - Marginal Cost_A) * QA
Similarly, the profit equation for firm B can be calculated as:
Profit_B = (P - Marginal Cost_B) * QB
(b) The marginal revenue equation for firm A can be derived from the inverse demand equation:
P = 80 - 4(QA + QB)
To find the marginal revenue, we take the derivative of the total revenue equation with respect to QA:
MR_A = (∂TR_A / ∂QA) = (∂(P * QA) / ∂QA)
MR_A = P + QA * (∂P / ∂QA)
Since the inverse demand equation only depends on QA + QB, we can simplify:
MR_A = P + QA * (∂P / ∂(QA + QB)) * (∂(QA + QB) / ∂QA)
MR_A = P + QA * (∂P / ∂(QA + QB))
Using the inverse demand equation, the marginal revenue equation for firm A can be defined as:
MR_A = P + 4QA
The marginal revenue equation for firm B can be derived in a similar manner:
MR_B = P + 4QB
(c) The reaction function for firm A represents the output level (QA) that maximizes its profit, taking into account the output level chosen by firm B. To determine this, we set the derivative of firm A's profit equation with respect to QA equal to zero:
∂Profit_A / ∂QA = P - Marginal Cost_A + QA * ∂P / ∂(QA + QB) = 0
Simplifying, we find:
P - Marginal Cost_A + QA * (∂P / ∂Q) = 0
Using the inverse demand equation, we can rewrite this as:
P - Marginal Cost_A + QA * 4 = 0
Solving for QA, we find the reaction function for firm A:
QA = (Marginal Cost_A - P) / 4
Similarly, the reaction function for firm B can be determined by setting the derivative of firm B's profit equation with respect to QB equal to zero:
QB = (Marginal Cost_B - P) / 4
(d) To graph the reaction functions, we plot QA on the x-axis and QB on the y-axis. The reaction function for firm A is QA = (Marginal Cost_A - P) / 4, and the reaction function for firm B is QB = (Marginal Cost_B - P) / 4. Intersecting these two reaction functions will give us the Cournot equilibrium outputs.
(e) To calculate the Cournot equilibrium outputs, we equate the reaction functions for firms A and B:
(Marginal Cost_A - P) / 4 = (Marginal Cost_B - P) / 4
Solving for P, we find:
P = (Marginal Cost_A + Marginal Cost_B) / 2
Substituting this back into the reaction functions, we can find the equilibrium outputs:
QA = (Marginal Cost_A - (Marginal Cost_A + Marginal Cost_B) / 2) / 4
QB = (Marginal Cost_B - (Marginal Cost_A + Marginal Cost_B) / 2) / 4
(f) The price charged in this market is given by the inverse demand equation:
P = 80 - 4(QA + QB)
Substituting the Cournot equilibrium outputs into this equation, we can find the price.
(g) The industry output is the sum of the individual outputs of firm A and firm B at the Cournot equilibrium:
Industry Output = QA + QB
(h) No, the result of identical prices charged by the two firms does not depend on them having identical marginal costs. The result is determined by the equilibrium condition where the derivatives of the profit equations for both firms are set equal to zero. The price charged in the market and the equilibrium outputs are determined by the relationship between the marginal costs, demand, and the reaction functions of the firms. The identical prices charged by the two firms are the result of this equilibrium condition and do not depend on the firms having identical marginal costs.
(a) The profit equation for firm A can be calculated as follows:
Profit A = Total revenue A - Total cost A
Total revenue A = Price per unit * Quantity A = P * QA
Total cost A = Marginal cost * Quantity A = $4 * QA
Profit A = P * QA - $4 * QA = (P - $4) * QA
Similarly, the profit equation for firm B can be derived as:
Profit B = (P - $4) * QB
(b) Marginal revenue represents the change in total revenue resulting from a one-unit increase in output. It can be calculated by taking the derivative of the total revenue equation.
Marginal revenue for firm A = d(Total revenue A) / d(QA)
To find this, we differentiate the revenue equation:
Total revenue A = P * QA, so
Marginal revenue A = d(P * QA) / d(QA) = P
Similarly, marginal revenue for firm B is given by:
Marginal revenue B = P
(c) Reaction functions represent the optimal output decision of each firm given the output decision of the other firm. Firm A's reaction function can be derived by maximizing its profit with respect to QA while treating QB as a fixed variable.
To maximize Profit A = (P - $4) * QA, we differentiate it with respect to QA and set it equal to zero to find the maximum:
d(Profit A) / d(QA) = (P - $4) - 0 = 0
P - $4 = 0
P = $4
So, the reaction function for firm A is QA = 0.25 * (P - $4)
Similarly, the reaction function for firm B is QB = 0.25 * (P - $4)
(d) To graph the reaction functions, we plot QA on the x-axis and QB on the y-axis. The equations for the reaction functions are QA = 0.25 * (P - $4) and QB = 0.25 * (P - $4). We plug in different values of P and find the corresponding QA and QB values to obtain the reaction functions.
(e) To calculate the Cournot equilibrium outputs, we need to find the intersection point of the two reaction functions. Let's solve QA = QB using the reaction function equations:
0.25 * (P - $4) = 0.25 * (P - $4)
P - $4 = P - $4
This equation holds for any value of P.
So, QA = QB for all values of P. The Cournot equilibrium outputs in this market are the same for both firms.
(f) To find the price charged in this market, we substitute the Cournot equilibrium outputs (QA = QB) into the inverse market demand equation:
P = 80 - 4(QA + QB)
P = 80 - 4(2 * QA)
P = 80 - 8QA
(g) The industry output is the sum of the outputs of firm A and firm B:
Industry output = QA + QB = 2 * QA
(h) No, the result of identical prices charged by the two firms does not depend on them having identical marginal costs. The identical prices arise due to the assumption that both firms have the same reaction function, which is derived from the market demand curve and not the marginal cost. Even if the firms had different marginal costs, they would reach the same price and outputs as long as their reaction functions are the same.