$(25$ points$)$ Consider a two$-$period model with two firms having identical

initial cost functions $c(q)=15+20q.$ In the first period, the incumbent

firm has the option of investing in $R$&$D$ in order to reduce its second$-$

period marginal costs to $$2$ per unit. Let the cost of investing in $R$&$D$

be $$63\mathrm{.}5.$ The other firm $($the entrant$)$ may enter the market in the

second period. If two firms coexist in the market, they use Cournot

strategies. The linear market demand curve is $p=74-9Q.$

a$.$ Find the profits of the incumbent and the entrant under the fol$-$

lowing four scenarios:

i The incumbent does not invest in $R$&$D,$ and the entrant en$-$

ters.

ii The incumbent invests in $R$&$D,$ and the entrant enters.

iii The incumbent does not invest in $R$&$D,$ and the entrant does

not enter.

iv The incumbent invests in $R$&$D,$ and the entrant does not

enter.

b$.$ What's the outcome of this model based on your answers in part

a$?$