VM provide Insurance Model

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

The VM’s Effort Under Insurance Model

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+cs β+θ)/(2 k))

Profitv

Simplify[-cv+cs rv+p α+((p (1-α) (p-p α+cs β+θ))/(2 k))-((((p-p α+cs β+θ))^2)/(4 k))-cs β (1-((p-p α+cs β+θ)/(2 k)))-θ (1-((p-p α+cs β+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)-(((1-α))^2)k rv Cs+(1-α)kCs]) -2k α -(1-α)(θ+β Cs))/(((1-α))^2))

The SO’s Price

Profits:=-α p+e(f-(1-α)p)-cs -rv(α p+cs+f)+(1-e)β cs
e:=((p-p α+cs β+θ)/(2 k))

Profits

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

D[%,p]

Solve[%==0,p]

{{p->((f-f α-2 k α-2 k rv α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))}}

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

Δp:=((f-f α-2 k α-2 k rv α-2 cs β+2 cs α β-θ+α θ)/(2 (((-1+α))^2)))-(1/(1-2 α+(α^2)))(-2 k α-cs β+cs α β-θ+α θ+2 √(cv k-cs k rv-2 cv k α+2 cs k rv α+cv k (α^2)+(k^2) (α^2)-cs k rv (α^2)+cs k β-cs k α β+k θ-k α θ))

Δp

Solve[%==0,f]

1. if  (p^s)>(p^VA)

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

e

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

(1/4) ((f/k)+((2 (1+rv) α)/(-1+α))+(θ/k))

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

Profitv

Simplify[%31]

Profits

Simplify[%33]

The profit

1.  If the (p^S)>(p^VA),  which satisfied with F≥((-2 k α(1-rv)-(1-α) θ+4 (Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α )-(((1-α))^2)rv Cs+(1-α)Cs])))/(1-α)) (the dark region) then the effort is e=(1/4) (((f+θ)/k)-((2 α(1+rv))/(1-α))), the contract [p,α] satellite owner offered is p=(((f-θ)(1-α)-2 k α (1+rv)-(1-α)2β Cs)/(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 [3(α^2)k +  θ (1- α)(4-α)  - F (1-α)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 ((15(((F+θ))^2)(((1-α))^2) + 20(α^2)(k^2)+ 4k (1- α)(4θ + αθ - 3αF)  )/(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)),
  the effort is e=((-k α+(Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α) )]))/(k (1-α)))
  The premium rate is r=((k-(Sqrt[k (cv (((1-α))^2)+k (α^2)+θ(1-α ))]))/(k (1-α)))
  the price is ((2(Sqrt[k cv·(((1-α))^2) + (α^2)(k^2) + θ k(1-α)]) -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

Δ:=16 cs k rv+(f^2) (((-1+α))^2)-16 cv k (((-1+α))^2)-32 cs k rv α-12 (k^2) (α^2)+16 cs k rv (α^2)-8 (k^2) rv (α^2)+4 (k^2) (rv^2) (α^2)-16 cs k β+16 cs k α β-16 k θ+20 k α θ-4 k rv α θ-4 k (α^2) θ+4 k rv (α^2) θ+(θ^2)-2 α (θ^2)+(α^2) (θ^2)+2 f (-1+α) (2 k (-1+rv) α+(-1+α) θ)-((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)

Δ

Simplify[16 cs k rv-32 cs k rv α+16 cs k rv (α^2)-8 (k^2) rv (α^2)+4 (k^2) (rv^2) (α^2)-16 cs k β+16 cs k α β-4 k rv α θ+4 k rv (α^2) θ+2 f (-1+α) (2 k (-1+rv) α+(-1+α) θ)+2 f (-1+α) (2 k α+θ-α θ)]

Simplify[%2]