Consider a Bertrand duopoly where the 2 firms produce a homogenous product. Assume the market demand curve is y = y1+ y2 = 1 - p, where p is the relevant market price, y is the total amount demanded at that price, y1 and y2 are the output levels of firm (1) and (2) respectively.
Assume that the firm's cost functions are C(yi;) = \(\frac{1}{2}\)yi; for i = 1, 2. The rules of the pricing game are as follows: Each firm must simultaneously quote a price in the interval [0, 1]. If the prices are different, the firm with the lower price sells all the units demanded at that price. If they quote the same price, the amount demanded at that price is splited equally between the two firms. What would be the Nash equilibrium price and quantities?
1
P1 = P2 = 1; Y1 = Y2 = \(\frac{1}{2}\) Y; is one of the Multiple Nash Equilibria
2
P1 = P2 = \(\frac{1}{2}\) ; Y1 = Y2 = \(\frac{1}{2}\) Y; is the Unique Nash Equilibrium
3
P1 > P2 = \(\frac{1}{2}\); Y1 = 0, Y2 = Y; is one of the Multiple Nash Equilibria
4
P1 < P2 = \(\frac{1}{2}\); Y1 = Y, Y2 = 0; is one of the Multiple Nash Equilibria