Then =1-91=1(3+:2)(3+2n 可=-1-3)=1(4-+2)(1-2+a With 1 <a2 re have Ox,>0 O: Thus, in equilibrium, we must have ai=.2. In fact, the two firms must sit in the middle By Proposition 2.1, Pi=p?=c Discussion 1. Nonexistence of Nash equilibrium if n>3 2. Existence of a reactive equilibrium for any n>1 3. What will happen if n identical firms are located on a circle? 5.9. Entry Cost A monopolist industry can be a result of entry costs. Consider Stage 1. All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost K>0 Stage 2. All firms that have entered play a Cournot game p(y=a-y where a>0 and c>0 are two constants In stage 2, with n firms in the industry, each firm considers the following problem nax a-c L Us VhThen, π∗ 1 = (p∗ 1 − c)ˆz = t 2 3 + x1 + x2 3  2 3 + x2 + x1 6  , π∗ 2 = (p∗ 2 − c)(1 − zˆ) = t 4 3 − x1 + x2 3  1 3 − x2 + x1 6  . With x1 < x2, we have ∂π∗ 1 ∂x1 > 0, ∂π∗ 2 ∂x2 < 0. Thus, in equilibrium, we must have x1 = x2. In fact, the two firms must sit in the middle: x∗ 1 = x∗ 2 = 1 2 . By Proposition 2.1, p∗ 1 = p∗ 2 = c. Discussion: 1. Nonexistence of Nash equilibrium if n ≥ 3. 2. Existence of a reactive equilibrium for any n ≥ 1. 3. What will happen if n identical firms are located on a circle? 5.9. Entry Cost A monopolist industry can be a result of entry costs. Consider Stage 1. All potential firms simultaneously decide to be in or out. If a firm decides to be in, it pays a setup cost K > 0. Stage 2. All firms that have entered play a Cournot game. Let ci(y) = cy, pd (y) = a − y, where a > 0 and c ≥ 0 are two constants. In stage 2, with n firms in the industry, each firm considers the following problem πi = max yi # a − c −[n j=1 yj $ yi. 2 — 11