Insurance Model
The first-best benchmark -- centre decision

By analyzing a centralized controlled supply chain. Without the need to consider payment to the vehicle manufacture and deal with insurancing within a centralized, controlled system, the expected savings associated with launching vie an internal spacecraft are equal to

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

D[%,e]

Solve[%==0,e]

e:=((f+θ)/(2 k))
Profits

D[%,p]

-α+(((-1+α) (f+θ))/(2 k))
Profitv

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

Solve[%==0,p]

{{p->((-(f^2)-4 cv k-4 k θ+(θ^2))/(2 (-f+f α-2 k α-θ+α θ)))}}
p:=((-(f^2)-4 cv k-4 k θ+(θ^2))/(2 (-f+f α-2 k α-θ+α θ)))

p:=(((f+θ)(f-θ)+4k(cv+θ))/(2 (f+θ)(1-α)+2 k α))

Profitv

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

Profits

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

Profits

Simplify[-cs-((α (4 k (cv+θ)+(f-θ) (f+θ)))/(2 k α+2 (1-α) (f+θ)))+(((f+θ) (f-(((1-α) (4 k (cv+θ)+(f-θ) (f+θ)))/(2 k α+2 (1-α) (f+θ)))))/(2 k))]

Profitc

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

((((f+θ))^2)/(4 k))-(cs+cv+θ)

Profitv

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

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

Cancel[-((α ((f^2)+4 cv k+(4 k-θ) θ))/(4 (f (-1+α)-k α+(-1+α) θ)))]

((α [(f-θ)(f+θ)+4 cv k+4 k θ])/(4 (f(1- α)+k α+θ(1-α ))))

Profitc

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

((((f+θ))^2)/(4 k))-(cs+cv+θ)

Lemma 1
In a centralized chain, the satellite owner contract with the vehicle manufacture if and only if ((((f+θ))^2)/(4 k))≥ cs + cv + θ   . The resulting launch probability is ((f+θ)/(2k)), and the corresponding chain payoff is ((((f+θ))^2)/(4 k)) - cs - cv - θ.

It follows from the lemma1  that, in order to avoid trivial cases, we assume that k >(f+θ)/2 and ((((f+θ))^2)/(4 k)) ≥ cs + cv + θ   throughout this paper.

Assumption 1
The vehicle manufacture’s cost cv and his failure penalty level a satisfy 0 ≤ θ ≤ cv

Insurance Model
The first-best benchmark - 2

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)
Profiti:=r(α p+cs+f)-(1-e)(α p+cs+f)

Profitc:=Profits+Profitv

D[Profitc,e]

Solve[%==0,e]

{{e->((-cs-p α+θ)/(2 k))}}

The VM’s Effort Under Insurance Model

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)

Profiti:=r(α p+cs+f)-(1-e)(α p+cs+f)

D[Profitv,e]

Solve[%==0,e]

We now solve the Stackelberg game  by using backward induction. First, given any contingent price p , by considering the first-order condition of Profitv, the vehicle manufacture’s best response is given as

e:=((p-p α+θ)/(2 k))

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)))]

((([(1-α)p + θ])^2)/(4k))-cv-θ+2p

Plot[Profitv,{p,0,200}]
α:=0.2
k:=100
θ:=50
cv:=50

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)))]

Solve[%==0,p]

Hence the vehicle manufacture’s participation condition, i.e., Profitv≥0, can be written as 
p ≥ ((2(Sqrt[k cv·(((1-α))^2) + (α^2)(k^2) + θ k(1-α)]) -2k α -(1-α)θ)/(((1-α))^2))

e: transform the constraints in terms of e (instead of p)

Solve[e-((p-p α+θ)/(2 k))==0,p]

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv
p:=((-2 e k+θ)/(-1+α))
Profitv

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

Solve[%==0,e]

it is easy to check that the boundary constraint Profitv  ≥ 0,holds if and only if  e ≥   (Overscript[e, -]), where

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

Plot[Profitv,{e,0,1}]
α:=0.2
k:=100
θ:=50
cv:=50

Show[%27,AxesLabel->{HoldForm[e],HoldForm[Profit]},PlotLabel->HoldForm[the Porfit of Vehicle Manufacture],LabelStyle->{FontFamily->"Al Bayan",10,GrayLevel[0]}]

The IC’s Premium Rate Under Insurance Model

Observing the contract price p selected by the satellite owner, the insurance company can anticipate the vehicle manufacture’s effort e as given above. Operating in a competitive insurance market, the insurance company sets its premium rate r to breakeven in expectation. In other words, under the premium rate r that it offers, the insurance company’s expected profit,r(α p+cs+f), equals the expected coverage 
(1-e) (α p+cs+f).
Substituting e given ine:=((p-p α+θ)/(2 k)) , the insurance company breakeven condition can be satisfied if and only if p ≥ (p^IA) ≡((2 k (1-r)- θ)/((1-α) )) ,which we refer as the insurance company’s underwriting constraint.  And the premium rate satisfied with r=((2 k-p+p α-θ)/(2 k))

Profiti:=r(α p+cs+f)-(1-e)(α p+cs+f)

Profiti

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

Solve[%==0,r]

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

The SO’s Optimal Contract Under Insurance Model

e:=((p-p α+θ)/(2 k))

r:=((2 k-p+p α-θ)/(2 k))

Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)

Profits

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

D[%,p]

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

Solve[%==0,p]

D[((-2 cs k+f (p-p α+θ)-p (p (((-1+α))^2)+2 k α+θ-α θ))/(2 k)),α]

Solve[%==0,α]

α:=((-f-2 k+2 p+θ)/(2 p))

p:=((f-f α-2 k α-θ+α θ)/(2 (((-1+α))^2)))

p

for the satellite owner, her unconstrained optimal contract (p,α) can be written as p=(((f-θ)(1-α)-2 k α )/(2 (((1-α))^2)))
s.t.  p ≥ ((2(Sqrt[k cv·(((1-α))^2) + (α^2)(k^2) + θ k(1-α)]) -2k α -(1-α)θ)/(((1-α))^2)) ( the level of price that vehicle manufacture will accept)

Δp:=(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))

Δp

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

((f-f α+2 k α+θ-α θ-4 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(2 (((-1+α))^2)))
Solve[%==0,f]

((f-f α+2 k α+θ-α θ-4 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(2 (((-1+α))^2)))

{{f->((-((k α)/(((-1+α))^2))-(θ/(2 (((-1+α))^2)))+((α θ)/(2 (((-1+α))^2)))+((2 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))}}
Simplify[((-((k α)/(((-1+α))^2))-(θ/(2 (((-1+α))^2)))+((α θ)/(2 (((-1+α))^2)))+((2 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))]

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

k:=100
cv:=30
θ:=80
f1:=((-((k α)/(((-1+α))^2))-(θ/(2 (((-1+α))^2)))+((α θ)/(2 (((-1+α))^2)))+((2 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))
pf1=Plot[f1,{α,0,1},GridLines->{{1},{}}]

f2:=1800
pf2=Plot[f2,{α,0,1}]
Show[pf1,pf2,Filling->Axis]

Plot[{f1,f2},{α,0,1},Filling->{1->{2}},Filling->Axis,GridLines->{{1},{}}]

Show[%49,ImageSize->Full]

Show[%49,ImageSize->Medium]

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

e

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

(1/4) (((f+θ)/k)-((2 α)/(1-α)))

p:=((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))
e:=((p-p α+θ)/(2 k))
e

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

((k α-(Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(k (-1+α)))

k:=100
cv:=30
cs:=20
θ:=80
p:=(((f-θ)(1-α)-2 k α)/(2 (((1-α))^2)))
e:=((p-p α+θ)/(2 k))
r:=1-e
Profits:=-α p+e(f-(1-α)p)-cs -r(α p+cs+f)+(1-e)(α p+cs+f)
Profits

Simplify[-20-((((-80+f) (1-α)-200 α) α)/(2 (((1-α))^2)))+(1/200) (f-(((-80+f) (1-α)-200 α)/(2 (1-α)))) (80+(((-80+f) (1-α)-200 α)/(2 (((1-α))^2)))-((((-80+f) (1-α)-200 α) α)/(2 (((1-α))^2))))]

Plot3D[((f^2)/800)+((f (-2+7 α))/(10 (-1+α)))-((2 (6-32 α+(α^2)))/(((-1+α))^2)),{f,0,2231.6228},{α,0,1},AxesLabel->{F,α,(Subscript[Π, s])}]

1

Plot3D[((f^2)/800)+((f (-2+7 α))/(10 (-1+α)))-((2 (6-32 α+(α^2)))/(((-1+α))^2)),{f,0,2231.6228},{α,0,0.8},AxesLabel->{F,α,(Subscript[Π, s])}]

Plot3D[((f^2)/800)+((f (-2+7 α))/(10 (-1+α)))-((2 (6-32 α+(α^2)))/(((-1+α))^2)),{f,0,2231.6228},{α,0,0.5},AxesLabel->{F,α,(Subscript[Π, s])}]

Plot3D[((f^2)/800)+((f (-2+7 α))/(10 (-1+α)))-((2 (6-32 α+(α^2)))/(((-1+α))^2)),{f,0,2231.6228},{α,0,0.2},AxesLabel->{F,α,(Subscript[Π, s])}]

Show[%25,ImageSize->Medium]

Show[%19,ViewPoint->{1.3`,-2.4`,2.`},AxesLabel->{F,α,(Subscript[Π, s])}]

Show[%19,Boxed->False]

Plot3D[((f^2)/800)+((f (-2+7 α))/(10 (-1+α)))-((2 (6-32 α+(α^2)))/(((-1+α))^2)),{f,0,2231.6228`},{α,0,0.5},PlotTheme->Automatic]

Plot3D[((f^2)/800)+((f (-2+7 α))/(10 (-1+α)))-((2 (6-32 α+(α^2)))/(((-1+α))^2)),{f,0,2231.6228`},{α,0,1},PlotTheme->Automatic]

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

Profitv

Simplify[%112]

Plot3D[((f^2)/1600)-((f (2+3 α))/(20 (-1+α)))+((-106+152 α-121 (α^2))/(((-1+α))^2)),{f,0,2231.6228},{α,0,0.8},AxesLabel->{F,α,(Subscript[Π, v])}]

Plot3D[((f^2)/1600)-((f (2+3 α))/(20 (-1+α)))+((-106+152 α-121 (α^2))/(((-1+α))^2)),{f,0,2231.6228},{α,0,0.5},AxesLabel->{F,α,(Subscript[Π, v])}]

Show[%26,ImageSize->Large]

Show[%28,ImageSize->Tiny]

Show[%28,ImageSize->Medium]

The profit

1.  If the (p^S)>(p^VA),  which satisfied with F≥((-2 k α-(1-α) θ+4 (Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α )])))/(1-α)) (the dark region) then the effort is e=(1/4) (((f+θ)/k)-((2 α)/(1-α))), the contract [p,α] satellite owner offered is p=(((f-θ)(1-α)-2 k α)/(2 (((1-α))^2))). The profit of vehicle manufacture is (1/(16 k (((-1+α))^2)))((f^2) (((1-α))^2)-16 cv k (((1-α))^2)-12 (k^2) (α^2)-16 k θ+20 k α θ-4 k (α^2) θ+(((θ-α θ))^2)-2 f (-1+α) (2 k α+θ-α θ))
which can be written as (((((F+θ))^2)(((1-α))^2) + 4k [ F (1-α)α- 3(α^2)k - θ (1- α)(4-α)  ])/(16k (((1-α))^2)))-Cv
The profit of satellite owner is (1/(8 k (((-1+α))^2)))((f^2) (((-1+α))^2)-8 cs k (((-1+α))^2)+2 f (-1+α) (2 k α+(-1+α) θ)+(((2 k α+θ-α θ))^2))
wich can be written as (((((F+θ))^2)(((1-α))^2)+4kα [αk +(θ -F)(1-α) ])/(8k (((1-α))^2)))-Cs
The premium rate is r = 1+(1/4) (((2 α)/(1-α))-((f+θ)/k))
the whole chain payoff is ((3(((F+θ))^2)(((1-α))^2) -4(α^2)(k^2)+ 4k (1- α)(3θ α-α F- 4θ )  )/(16k (((1-α))^2)))-Cv-Cs
2. If the (p^S)<(p^VA), which satisfied with F<((-2 k α-(1-α) θ+4 (Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α )])))/(1-α))=((-2 k α-(1-α) θ+4 A)/(1-α))
  the effort is e=((-k α+(Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α) )]))/(k (1-α)))=((-k α+A)/(k (1-α)))
  The premium rate is r=((k-A)/(k (1-α)))
  the price is ((2A-2k α -(1-α)θ)/(((1-α))^2))
  The profit of satellite owner is -(1/(k (((-1+α))^2)))(2 cv k+cs k (((-1+α))^2)-4 cv k α+f k α+2 cv k (α^2)-f k (α^2)+2 (k^2) (α^2)+2 k θ-2 k α θ-f (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])+f α (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])-2 k α (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])-θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])+α θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))
   which can be written as 
   (([(F+θ)(1-α)+2αk]*(Sqrt[k[((Cv(1-α))^2)+((kα)^2)+θ(1-α)]])- k (1-α) (α F+2θ ) - 2 (k^2)(α^2) )/((k(1-α))^2))
- (Cs+2Cv)
   令 A=(Sqrt[k[((Cv(1-α))^2)+((kα)^2)+θ(1-α)]])
     (([(F+θ)(1-α)+2αk]*A - k (1-α) (α F+2θ ) - 2 (k^2)(α^2) )/((k(1-α))^2)) - (Cs+2Cv)
   the profit of vehicle manufacture is 0
  the whole chain payoff is (([(F+θ)(1-α)+2αk]*A - k (1-α) (α F+2θ ) - 2 (k^2)(α^2) )/((k(1-α))^2))- Cs-2Cv

e:=(1/4) (((f+θ)/k)-((2 α)/(1-α)))
p:=((f-f α-2 k α-θ+α θ)/(2 (((-1+α))^2)))
r:=1-e
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)
Profiti:=r(α p+cs+f)-(1-e)(α p+cs+f)

Profitv

Simplify[-cv+((α (f-f α-2 k α-θ+α θ))/(2 (((-1+α))^2)))+(((1-α) (f-f α-2 k α-θ+α θ) (-((2 α)/(1-α))+((f+θ)/k)))/(8 (((-1+α))^2)))-(1/16) k (((-((2 α)/(1-α))+((f+θ)/k)))^2)-θ (1+(1/4) (((2 α)/(1-α))-((f+θ)/k)))]

(1/(16 (((α-1))^2) k))((α^2) (θ^2)-2 α (θ^2)-16 (((α-1))^2) cv k+(((α-1))^2) (f^2)-2 (α-1) f (-(α θ)+θ+2 α k)+(θ^2)-12 (α^2) (k^2)-4 (α^2) θ k+20 α θ k-16 θ k)

Profits

Simplify[-cs-((α (f-f α-2 k α-θ+α θ))/(2 (((-1+α))^2)))+(1/4) (-((2 α)/(1-α))+((f+θ)/k)) (f-(((1-α) (f-f α-2 k α-θ+α θ))/(2 (((-1+α))^2))))]

(1/(8 k (((-1+α))^2)))((f^2) (((-1+α))^2)-8 cs k (((-1+α))^2)+2 f (-1+α) (2 k α+(-1+α) θ)+(((2 k α+θ-α θ))^2))

Profiti

r

Profit:=(1/(16 (((α-1))^2) k))((α^2) (θ^2)-2 α (θ^2)-16 (((α-1))^2) cv k+(((α-1))^2) (f^2)-2 (α-1) f (-(α θ)+θ+2 α k)+(θ^2)-12 (α^2) (k^2)-4 (α^2) θ k+20 α θ k-16 θ k)+(1/(8 k (((-1+α))^2)))((f^2) (((-1+α))^2)-8 cs k (((-1+α))^2)+2 f (-1+α) (2 k α+(-1+α) θ)+(((2 k α+θ-α θ))^2))
Profit

Simplify[%3]

e:=((p-p α+θ)/(2 k))
p:=((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))
r:=1-e

e

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

((k α-(Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(k (-1+α)))
r

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

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)
Profiti:=r(α p+cs+f)-(1-e)(α p+cs+f)

Profits

-cs-((α (-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ])))/(1-2 α+(α^2)))+(1/(2 k))(f-(1/(1-2 α+(α^2)))(1-α) (-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))) (θ+((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))-((α (-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ])))/(1-2 α+(α^2))))

Simplify[%12]

-(1/(k (((-1+α))^2)))(2 cv k+cs k (((-1+α))^2)-4 cv k α+f k α+2 cv k (α^2)-f k (α^2)+2 (k^2) (α^2)+2 k θ-2 k α θ-f (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])+f α (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])-2 k α (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])-θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)])+α θ (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))

Profitv

Simplify[%14]

Plot[-(x^2),{x,0,1},Filling->Axis]

Solve[(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))==0,cv]

α:=0.2
f:=50
cs:=10
k:=100

α:=0.2
k:=100
θ:=50
cv:=50

cv1:=(((f^2)-2 (f^2) α+4 f k α+(f^2) (α^2)-4 f k (α^2)-12 (k^2) (α^2)+2 f θ-16 k θ-4 f α θ+20 k α θ+2 f (α^2) θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2))/(16 k (((-1+α))^2)))
cv2:=((((f+θ))^2)/(4k))-cs-θ
θ:=2k-f

y1=Plot[cv1,{θ,-100,500}]
y2=Plot[cv2,{θ,-100,500}]


Show[y1,y2]

α:=0.2
k:=100
θ:=50
cs:=50
cv:=(((((1-α))^2) k+2α(1-α)k-3α k-4(1-α)θ)/(4(((1-α))^2)))
Plot[cv,{θ,0,10}]

cv1:=(((f^2)-2 (f^2) α+4 f k α+(f^2) (α^2)-4 f k (α^2)-12 (k^2) (α^2)+2 f θ-16 k θ-4 f α θ+20 k α θ+2 f (α^2) θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2))/(16 k (((-1+α))^2)))
cv2:=((((f+θ))^2)/(4k))-cs-θ
θ:=2k-f
α:=0.2
k:=100
θ:=50
cs:=50

Plot[cv1,{f,0,1000}]

Plot[cv2,{f,0,500}]

Show[cv1,cv2]

Solve[(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))==0,cv]

{{cv->(((f^2)-2 (f^2) α+4 f k α+(f^2) (α^2)-4 f k (α^2)-12 (k^2) (α^2)+2 f θ-16 k θ-4 f α θ+20 k α θ+2 f (α^2) θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2))/(16 k (((-1+α))^2)))}}
Simplify[(((f^2)-2 (f^2) α+4 f k α+(f^2) (α^2)-4 f k (α^2)-12 (k^2) (α^2)+2 f θ-16 k θ-4 f α θ+20 k α θ+2 f (α^2) θ-4 k (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2))/(16 k (((-1+α))^2)))]

(((f^2) (((-1+α))^2)-12 (k^2) (α^2)-4 k (4-5 α+(α^2)) θ+(((-1+α))^2) (θ^2)-2 f (-1+α) (2 k α+θ-α θ))/(16 k (((-1+α))^2)))

Solve[(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))==0,f]

{{f->2 (1-α) (((k α)/(((1-α))^2))+(θ/(2 (1-α)))+((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2))))}}
Simplify[2 (1-α) (((k α)/(((1-α))^2))+(θ/(2 (1-α)))+((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2))))]

f1:=((2 k α+θ-α θ-4 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(-1+α))

Solve[(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))==0,k]

Solve[(((1-α) (f-θ)-2α k)/(2(((1-α))^2)))-((-2 k α-θ+α θ+2 (Sqrt[cv k-2 cv k α+cv k (α^2)+(k^2) (α^2)+k θ-k α θ]))/(1-2 α+(α^2)))==0,θ]

k:=100
θ:=50
cv:=50

Plot[f1,{α,0,0.4}]

α:=0.2
k:=100
θ:=50
cv:=50
f1:=((2 k α+θ-α θ-4 (Sqrt[k (cv (((-1+α))^2)+k (α^2)+θ-α θ)]))/(-1+α))
f2:=k-cv
Plot[f1,{cv,0,50}]
Plot[f2,{cv,0,50}]