if  the premium rate is exoginex

Profitv:=α p+e (1-α)p-(1-e)θ- k*(e*e)-cv
Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)

D[Profitv,e]

Solve[%==0,e]

the government compensate the launch insurance:

Profitv:=α p+e (1-α)p-(1-e)θ- k*(e*e)-cv
Profits:=-α p+e(f-(1-α)p)-cs -(r-g)(α p+cs+f)+(1-e)(α p+cs+f)

D[Profitv,e]

Solve[%==0,e]

e:=((p-p α+θ)/(2 k))
r:=1-((p-p α+θ)/(2 k))

Profits

Simplify[-cs-p α+(((f-p (1-α)) (p-p α+θ))/(2 k))+(cs+f+p α) (1-((p-p α+θ)/(2 k)))-(cs+f+p α) (1-g-((p-p α+θ)/(2 k)))]

D[%,p]

Simplify[((f (1-α)-2 p (((-1+α))^2)+2 (-1+g) k α-θ+α θ)/(2 k))]

Solve[%==0,p]

{{p->((f-f α-2 k α+2 g k α-θ+α θ)/(2 (((-1+α))^2)))}}
p:=((f-f α-2 k α+2 g k α-θ+α θ)/(2 (((-1+α))^2)))

e

Simplify[((θ+((f-f α-2 k α+2 g k α-θ+α θ)/(2 (((-1+α))^2)))-((α (f-f α-2 k α+2 g k α-θ+α θ))/(2 (((-1+α))^2))))/(2 k))]

without the government compensating

Profitv:=α p+e (1-α)p-(1-e)θ- k*(e*e)-cv
Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)

D[Profitv,e]

Solve[%==0,e]

e:=((p-p α+θ)/(2 k))

Profits

D[%,p]

Solve[%==0,p]

p:=((cs-cs α+2 k r α+θ)/(2 (-1+α)))

e

Simplify[((θ+((cs-cs α+2 k r α+θ)/(2 (-1+α)))-((α (cs-cs α+2 k r α+θ))/(2 (-1+α))))/(2 k))]