Question

Two manufacturers, denoted 1 and 2, are competing for 100 identical customers. Each manufacturer chooses both the price and quality of its product, where each variable can take any nonnegtive real number. Let pi and xi denote, respectively, the price and quality of manufacturer i's product. The cost to manufacturer i of producing for one customer is 10+5xi . Note in this expression that the cost is higher when the quality is higher. If manufacturer i sells to qi customers, then its total cost is qi(10+5xi). Each cutomer buys from the manufacturer who offers the greatest value, where the value of buying from manufacturer i is 1000+ xi - pi ; higher quality and lower price means more value. A manufacturer's is qi( pi- 10 - 5xi ). If both manufacturer offers the same value, then 50 customers buy from each manufacturer. If one manufacturer offers higher value, then 100 customers buy from it.

Find all symmetric Nash equilibria.

Answer 1

Nash equilibrium will occur at the following conditions P1 = P2 = 10 and x1 = x2 = 0.

Step-by-step explanation:

The term or concept known as the Nash equilibria is very important and it is often used in the determination of the kind of price strategies companies that are competing against one another will use in order to acquire more customers than the others.

So, in this question/problem we are given that there are two manufacturer that is manufacturer 1 and manufacturer 2. Also, the total number of customers both manufacturers are competing for is equal to 100.

Kindly note that we are given from the question that ''Each manufacturer chooses both the price and quality of its product, where each variable can take any non-negative real number''

If each of the manufacturer has 50 customers each that is symmetric condition.

Assuming we have a condition or situation where p1 is less than p2 for manufacturer 1, it means that manufacture 1 lessens its price, therefore manufacturer 1 will have all all the profit = 100(p1 - 10 - 5x1).

Assuming manufacturer 1 reduces both the quality and the price this time around to the point that it is justifiable to lower the price because of the quality , it means that we will have 1000 + (x1 = 0) + (p1 - compensation m).

For any of the manufacturer, If m> x' and we have that x1 = x'>0[ which is for the quality], then, the profit will be 100(10 + 5x'- m -10).

Also, For any of the manufacturer, if we have x'<m<5x' and x1 for the representation of quality, then, Customers will buy from both manufacturer making m<5x'.

Therefore, Nash equilibrium will occur at the following conditions: P1 = P2 = 10 and x1 = x2 = 0.