In a market, there are two firms, each has a marginal cost c=30. Demand is given by

P=150-5Q where Q=q₁+q₂ and q_i is the quantity of firm i=1,2.

1. (Cournot) Assume firms simultaneously choose quantities, find the Cournot-Nash equilibrium quantities, and the equilibrium market price.

q₁ =q₂ =

P =

1. (Bertrand) Assume the firms simultaneously choose prices. Find the Bertrand Nash equilibrium.

p₁ =p₂ =

c.(Bertrand with capacity constraints) Suppose the firms have capacity constraints K₁=7 for firm 1 and K₂=7 for firm 2. The firm can produce any quantity smaller or equal its capacity at a cost of 30 per unit, but cannot produce more than its capacity. The firms simultaneously choose their prices.

(i)Find the equilibrium prices. [You may assume a pure strategy equilibrium exists].

p₁ =p₂ =

(ii)When firm 2 produces K₂=7 units, find firm 1's residual demand.

(iii) Verify that the prices in part (i) are equilibrium prices.