3.Entry deterrence in Stackbelberg model

Suppose the inverse demand is given byP(Q) = 56 2Q, whereQ=q1+q2.

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The cost function is

C(qi)=20qi+f,

wherefis the fixed cost of entry. Firm 1, the leader/incumbent firm, first choosesq1. Potential entrant Firm 2 then choosesq2after observingq1. Like the lecture notes, we assume only Firm 2 needs to pay this entry cost in the event that it enters the industry.

1. (a)Find the backward-induction equilibrium assuming thatf= 0. (Be sure to state the full strategy profile, i.e., the quantity each firm plans to choose at every decision node, no matter whether it is reached or unreached. The same remark applies to parts (c) and (d) below).
2. (b)Find the respective limit output for fixed costfequal to50,18, and2.
3. (c)What is the backward-induction equilibrium for each case off= 50,f= 18, andf= 2?
4. (d)Fixf= 2. Suppose in addition to Firm 2, there is another potential entrant Firm 3 that shares the same cost function as Firm 2. The modified game has Firm 2 and Firm 3 simultaneously decide their respective quantities after observingq1. What is the subgame-perfect Nash equilibrium? (Note: you may focus on pure-strategy equilibria.)