Profits:=-αp+e(F-(1-α)p)-cs

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv

Profitv

D[Profitv,e]

Simplify[-2 e k+p (1-α)+θ]

Solve[-2 e k+p (1-α)+θ==0,e]

e:=((p-p α+θ)/(2 k))

e

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv

Profitv

Simplify[-cv+p α+((p (1-α) (p-p α+θ))/(2 k))-((((p-p α+θ))^2)/(4 k))-θ (1-((p-p α+θ)/(2 k)))]

Factor[((-4 cv k+(p^2) (((-1+α))^2)+θ (-4 k+θ)+2 p (2 k α+θ-α θ))/(4 k))]

Factor[-cv+p α+((p (1-α) (p-p α+θ))/(2 k))-((((p-p α+θ))^2)/(4 k))-θ (1-((p-p α+θ)/(2 k)))]

Simplify[-((4 cv k-(p^2)-4 k p α+2 (p^2) α-(p^2) (α^2)+4 k θ-2 p θ+2 p α θ-(θ^2))/(4 k))]

Profits:=-αp+e(F-(1-α)p)-cs

Profits

Simplify[-cs-αp+(((F-p (1-α)) (p-p α+θ))/(2 k))]

D[Profits,α]

Solve[-((p (F-p (1-α)))/(2 k))+((p (p-p α+θ))/(2 k))==0,α]

Profits:=-αp+e(F-(1-α)p)-cs
e:=((p-p α+θ)/(2 k))

D[Profits,p]

Solve[(((F-p (1-α)) (1-α))/(2 k))+(((-1+α) (p-p α+θ))/(2 k))==0,p]

Solve[{((p (F-p (1-α)))/(2 k))+((p (p-p α+θ))/(2 k))==0,(((F-p (1-α)) (1-α))/(2 k))+(((-1+α) (p-p α+θ))/(2 k))==0},{α,p}]