Price And Quantity Determination Under Duopoly - SS2 Economics Lesson Note

In a duopoly, there are only two firms that dominate the market and compete against each other. Each firm must determine the price and quantity of goods to produce based on what their competitor is doing.

To determine the quantity of goods to produce, each firm will consider the other's response. If one firm decides to produce more goods, the other firm may follow suit to avoid losing market share and value. This leads to a situation where both firms end up producing less than what they would produce if they were a monopoly.

The firms will need to consider the price set by their competitor before setting their own price. If one firm sets a higher price, it may lose customers to the other firm that offers a lower price. On the other hand, if both firms set a low price, they may end up in a price war where neither firm earns a profit. This leads to a Nash equilibrium, where both firms set a price that is lower than what they would have set in a monopoly but higher than what they would have set in a price war.

Both firms need to consider the potential reactions of their competitor before setting their own strategy. This leads to a situation where both firms may end up producing less and charging a higher price than what they would have in a monopoly.

 

Word problems on the determination of equilibrium price and quantity under Duopoly

Example 1:

Suppose there are two firms, A and B, in a duopoly market. The market demand and cost functions are given as follows:

Demand function: P = 80 - Q

Cost function: C = 20Q

where P is the market price, Q is the total quantity of goods produced by both firms, and C is the total cost of production. Assuming that firm A and firm B have equal production costs, what is the Nash equilibrium price and quantity in this duopoly market?

Solution:

Determine the total quantity produced by both firms. Since each firm has equal production costs, they will produce the same quantity. Therefore, the total quantity produced, Q, can be written as:

Q = QA + QB

Determine each firm's best response function. To do this, we need to find each firm's profit-maximizing quantity given the other firm's quantity. Since the firms have equal costs, we can assume that they will split the market equally. Therefore, each firm's best response function is:

QA = (80 - QB - 20)/2

QB = (80 - QA - 20)/2

Find the Nash equilibrium. The Nash equilibrium is the point where both firms produce the same quantity and charge the same price. Therefore, we need to solve for QA and QB where QA = QB. Substituting QB = QA into the best response function for firm A, we get:

QA = (80 - QA - 20)/2

2QA = 60

QA = 30

Similarly, we can find QB = 30.

Calculate the market price. The market price can be found by substituting Q = 60 into the demand function:

P = 80 - Q

P = 80 - 60

P = 20

Therefore, the Nash equilibrium price and quantity in this duopoly market are P = 20 and Q = 60, respectively. 

 

Example 2:

Suppose there are two firms, A and B, in a duopoly market. The market demand and cost functions are given as follows:

Demand function: P = 100 - Q

Cost function: C = 10Q

where P is the market price, Q is the total quantity of goods produced by both firms, and C is the total cost of production. Assuming that firm A and firm B have equal production costs, what is the Nash equilibrium price and quantity in this duopoly market?

Solution:

We need to determine the total quantity produced by both firms. Since each firm has equal production costs, they will produce the same quantity. Therefore, the total quantity produced, Q, can be written as:

Q = QA + QB

We need to find each firm's profit-maximizing quantity given the other firm's quantity. Since the firms have equal costs, we can assume that they will split the market equally:

QA = (100 - QB - 10)/2

QB = (100 - QA - 10)/2

To find the Nash equilibrium, we need to solve for QA and QB where QA = QB. Substituting QB = QA into the best response function for firm A, we get:

QA = (100 - QA - 10)/2

2QA = 90

QA = 45

Similarly, we can find QB = 45.

We calculate the market price, which can be found by substituting Q = 90 into the demand function:

P = 100 - Q

P = 100 - 90

P = 10

Thus, the Nash equilibrium price and quantity in this duopoly market are P = 10 and Q = 90, respectively.