Maintaining monopoly power: entry deterrence $\text{'}vs\text{'}$ buyout

Consider the market for a homogeneous good with demand given $byP(Q)=34-4Q.$

Let there $be$ a single incumbent firm $(I)in$ the market and a single potential entrant $(E).$

Each firm $i=$I $orE$ has access $to$ the same technology characterised $bycost$ function

$C_i(q_i)=2q_i+F,$

with $2<F<16(the$ constraint $2<F$ ensures that entry deterrence $is$ preferred $to$

accommodation and $F<16$ ensures that both firms can reap positive profits under

Stackelberg competition$).$ Consider two scenarios:

Scenario $1(limit$ pricing$)$: The incumbent can commit $to$ a post$-$entry quantity and

the entrant believes that the incumbent will carry out its threats. However, the

incumbent $is$ not allowed $to$ buy out the entrant.

Scenario $2(buyout)$: The incumbent cannot commit $to$ a post$-$entry quantity; the

entrant rationally believes that the incumbent will accommodate entry once the

entrant $isin.$ However, the incumbent has the option $to$ buy out the entrant before

production begins. $If$ the incumbent buys out the entrant, $it$ becomes a monopoly.

$We$ want $to$ know which $of$ the two scenarios the incumbent prefers depending $onF.$

$(a)10$ marks Solve for the Stackelberg equilibrium $in$ this market. What are the two

firms' equilibrium profits $as$ a function $ofF?$

$(b)$ Suppose $we$ are $in$ Scenario $1.$ The timing $of$ the events $isas$ follows. First, the

incumbent announces the quantity $it$ will produce. Second, given this, the entrant

decides whether $to$ enter and how much $to$ produce $ifit$ does enter.

$i.$ What $is$ the minimum quantity that the incumbent needs $to$ produce

$to$ keep the entrant out $of$ the market $(thelimit$ quantity$)?$

$ii.5$ marks What $is$ the profit $of$ the incumbent $ifit$ chooses $to$ deter entry $by$

producing the $limit$ quantity?

$(c)$ Suppose $we$ are $in$ Scenario $2.$ The timing $of$ the events $isas$ follows. First, the

incumbent decides whether $or$ not $to$ make a buyout offer $to$ the entrant and how

much $to$ offer $ifit$ does. Second, the entrant decides whether $to$ accept the offer.

$If$ the entrant accepts, the incumbent pays $it$ its offer and becomes a monopolist.

$If$ the entrant rejects $(orno$ offer $is$ made $in$ the first place$),$ the firms compete $by$

playing the Stackelberg game $(with$ the incumbent $as$ the leader$).$

$i.$ How much must the incumbent offer the entrant $to$ make $it$ stay out?

$ii.If$ the incumbent makes this minimum offer, what will $be$ its final

profit $(netof$ the price $it$ paid $to$ buy out the entrant$)?$

$(d)$ For what values $ofF$ does the incumbent prefer Scenario $1to$ Scenario $2?$