Problem 1

Consider a market with only 2 firms that compete on the quantity sold. We use the Cournot model of duopoly to model this scenario. The inverse demand curve of the market is given by

p = 100 - q

Where p indicates the market price, q = q1 + q2 is the market quantity, q1 indicates the quantity sold by

firm 1 and q2 is the quantity sold by firm 2. Firms have the same total cost function

TC1 = 10q1

TC2 = 10q2

A. Compute the Cournot-Nash equilibrium of the game between the two firms. How much does each firm

produce in equilibrium? What is the equilibrium price? What is the profit of each firm?

B. The scientist Sheldon Cooper developed a new machine to produce the good, which would decrease

variable cost of production. Sheldon approaches the manager of Firm 1, to sell his innovation. If Firm 1

buys the innovation, its total variable cost of production will be

TV C1 = 5q1

Sheldon asks for a price of F = 200 to sell his product. So if Firm 1 wants to use the innovation, it must

spend 200.

Consider the following game. Before starting the Cournot game, Firm 1 can buy Sheldon's invention,

spending F = 200, and its total cost function will be TC1 = 200 + 5q1. If Firm 1 does not buy Sheldon's

machine, its cost is the same as in Part A, i.e. TC1 = 10q1. Firm 2 does not have this possibility. Solve this

game by backward induction. What are the equilibrium quantities, price and profits of the firms?