A Bertrand competition is the type of firm interaction which competes over prices.

Cost of firm 1=c1>0

Cost of firm 1=c2>0

Also, c1<c2.

If both the firms set price equal to the marginal cost, no profits are earned. If one firm sets a price equal to the marginal cost and the other firm sets a price above the marginal/unit cost, then the second firm will earn nothing. So, the marginal cost is the equilibrium price in Bertrand duopoly and no other prices is equilibrium.If both the firms set prices above the marginal/unit cost, then there is an incentive for each firm to undercut by a very small amount and then occupy the entire market demand,

1)We know that c1 is less than c2, it implies marginal/unit cost of firm 1 is lower than the cost of firm 2. Firm 1 will capture the entire market. So, it is optimal for firm 2 to set price equal to c1 in order to equally distribute the share of the market. Otherwise it will lose all the market demand.Hence, (P1,P2)=(c1,c1) is a Nash Equilibrium.

2)As stated above c2 is greater than c1.So, if prices are set higher than the unit/marginal cost, each firm has an incentive to under cut the price by an arbitrarily small amount and capture the entire market.Hence,(P1,P2)=(c2,c2) cannot be Nash Equilibrium.

3)c(λ)= λc1+(1- λ)c2

To prove: (P1,P2)=(c(λ),c2) a Nash Equilibrium

In Nash equilibrium, both firms set price equal to the marginal unit cost.

c(λ)=c2

λc1+(1- λ)c2=c2

λc1+c2- λc2=c2

λc1=λc2

λ=1 for c1=c2.

4)No, the only Nash equilibrium in Bertrand is to set prices equal to the marginal/unit cost.