Consider a homogeneous product Cournot oligopoly consisting of n ex-ante symmetric firms who face the linear...

 

Transcribed Image Text:

Consider a homogeneous product Cournot oligopoly consisting of n ex-ante symmetric firms who face the linear inverse demand: p=a- 9i IEN where p is the price, q; is firm i's output and a > 0. The firms initially have zero fixed costs and identical constant returns-to-scale cost functions. Bilateral collaborations lower marginal costs, ci(g) = 70-di(g), i EN = {1,...., n}, where is a positive parameter representing a firm's marginal cost if it has no links and d; (g) is the number of links firm i has in g. Given any network g, the Cournot equilibrium output can be written as (a-70) +nydi (9)-7jid(9) (n+1) qi(g) = ,ie N. Assume that (a -70)-(n-1)(n-2) >0 to ensure that each firm produces a strictly positive quantity in equilibrium. Then, the Cournot profits for firm i E N are given by (ax - Yo) + mydi (9)-1jid(9)` (n+1) sti (9)) . Yi(g) = Leta - % = 1 and 0 < < 1/6. 1. Take n = 3. Which are the strongly efficient networks? Which are the Pareto-efficient networks? 2. Take n = 3. Which are the pairwise stable networks? Which are the strongly stable networks? Is there any conflict between efficiency and stability? Give some intuition. 3. Take n = 3. Which are the pairwise Nash stable networks of the Myerson simultaneous move linking game? Which are the strong Nash stable networks? Give a complete proof of your characterization. 4. Take n = 3. Which are the VNM farsighted stable sets? Give a complete proof of your characterization. 5. Take n = 3. Suppose now players 1 and 2 are myopic while player 3 is farsighted. Which are the myopic-farsighted stable sets? Why? Compare with the case where all players are either myopic or farsighted. What would happen if player 2 becomes farsighted? Consider a homogeneous product Cournot oligopoly consisting of n ex-ante symmetric firms who face the linear inverse demand: p=a- 9i IEN where p is the price, q; is firm i's output and a > 0. The firms initially have zero fixed costs and identical constant returns-to-scale cost functions. Bilateral collaborations lower marginal costs, ci(g) = 70-di(g), i EN = {1,...., n}, where is a positive parameter representing a firm's marginal cost if it has no links and d; (g) is the number of links firm i has in g. Given any network g, the Cournot equilibrium output can be written as (a-70) +nydi (9)-7jid(9) (n+1) qi(g) = ,ie N. Assume that (a -70)-(n-1)(n-2) >0 to ensure that each firm produces a strictly positive quantity in equilibrium. Then, the Cournot profits for firm i E N are given by (ax - Yo) + mydi (9)-1jid(9)` (n+1) sti (9)) . Yi(g) = Leta - % = 1 and 0 < < 1/6. 1. Take n = 3. Which are the strongly efficient networks? Which are the Pareto-efficient networks? 2. Take n = 3. Which are the pairwise stable networks? Which are the strongly stable networks? Is there any conflict between efficiency and stability? Give some intuition. 3. Take n = 3. Which are the pairwise Nash stable networks of the Myerson simultaneous move linking game? Which are the strong Nash stable networks? Give a complete proof of your characterization. 4. Take n = 3. Which are the VNM farsighted stable sets? Give a complete proof of your characterization. 5. Take n = 3. Suppose now players 1 and 2 are myopic while player 3 is farsighted. Which are the myopic-farsighted stable sets? Why? Compare with the case where all players are either myopic or farsighted. What would happen if player 2 becomes farsighted?