Dynamic Competition with Random Demand and Costless

Claim 1. In any equilibrium, we have  1  0. Proof. First, we show that in any equilibrium we must have ∗  0. Suppose to the contrary that we had ∗ = 0. Then by posting a price 0 ∈ ( ̄) in period 1, an inactive firm would be sure to sell, making the deviation profitable, a contradiction to equilibrium. Hence we must have ∗  0 . Next, suppose to the contrary to the statement of Claim 1 we had  1 = 0. We will argue that 2(2; 0) then cannot be constant on [0 ∗]. If 2(2; 0) were constant on [0 ∗], then in period 2 unit ∗ would sell in state , or else firms could increase revenues by setting 2(2; 0)−  for sufficiently small . Since ∗ sells in all states, if 2(2; 0)  ̄ held, a firm could increase revenues by raising its price and continuing to sell in all states   . If 2(2; 0) = ̄ held, a firm not producing could profitably enter, establishing a contradiction. Thus a range of prices is offered in period 2, i.e. 2(∗; 0)  2(0; 0) must hold. But then a firm posting 2(0; 0) in period 2 would have a profitable deviation to post 2(0; 0) + (̄)2 in period 1, because a range of consumers including type ̄ would be willing to purchase at that price. This contradiction establishes that  1  0.¥ Claim 2. In any equilibrium, there exists e  ≤ ̄ and a 1 function ̄1() defined on [ e ] satisfying ̄1   0 and ̄1(e ) =  1 .