Blockchain Model--SA pay for BEL
The VM’s Effort Under Insurance Model

Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv
Profitib:=rb(α pb+cs+f)-(1-eb)(α pb+cs+f)
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rb(α pb+cs+f)+(1-eb)(α pb+cs+f)-csb -cvb

D[Profitvb,eb]

Solve[%==0,eb]

We now solve the Stackelberg game  by using backward induction. First, given any contingent price (p^B) , by considering the first-order condition of Profitv, the vehicle manufacture’s best response is given as
e^B=(((1-α)p+θ)/(2(k^B)))

eb:=((pb-pb α+θ)/(2 kb))

Profitvb

Simplify[-cv+pb α+((pb (1-α) (pb-pb α+θ))/(2 kb))-((((pb-pb α+θ))^2)/(4 kb))-θ (1-((pb-pb α+θ)/(2 kb)))]

Solve[%==0,pb]

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

The IC’s Premium Rate Under Blockchain 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(1- α)+θ)/(2 (k^B))) , the insurance company breakeven condition can be satisfied if and only if p ≥ (p^IA) ≡((2 k^B (1-r)- θ)/((1-α) )) ,which we refer as the insurance company’s underwriting constraint.  And the premium rate satisfied with r=((2 (k^B)-p(1- α)-θ)/(2 (k^B)))

Profitib:=rb(α pb+cs+f)-(1-eb)(α pb+cs+f)

Solve[Profitib==0,rb]

The SO’s Optimal Contract when she pay for the BEL

eb:=((pb-pb α+θ)/(2 kb))
rb:=((2 kb-pb+pb α-θ)/(2 kb))
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rb(α pb+cs+f)+(1-eb)(α pb+cs+f)-csb-cvb

Profitsb

Simplify[-cs-csb-cvb-pb α-(((cs+f+pb α) (2 kb-pb+pb α-θ))/(2 kb))+(((f-pb (1-α)) (pb-pb α+θ))/(2 kb))+(cs+f+pb α) (1-((pb-pb α+θ)/(2 kb)))]

D[%,pb]

Solve[%==0,pb]

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

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

Δp

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

Solve[%==0,f]

kb:=60
k:=100
cv:=30
cvb:=10
θ:=80
f1:=((-((kb α)/(((-1+α))^2))-(θ/(2 (((-1+α))^2)))+((α θ)/(2 (((-1+α))^2)))+((2 (Sqrt[kb (cv (((-1+α))^2)+cvb (((-1+α))^2)+kb (α^2)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))
f2:=((-((kb α)/(((-1+α))^2))-(θ/(2 (((-1+α))^2)))+((α θ)/(2 (((-1+α))^2)))+((2 (Sqrt[kb (cv (((-1+α))^2)+kb (α^2)+θ-α θ)]))/(((-1+α))^2)))/((1/(2 (((-1+α))^2)))-(α/(2 (((-1+α))^2)))))

pf3=Plot[{f1,f2},{α,0,0.8},Filling->{1->
{2}},GridLines->{{0.8},{}}]

The profit:

eb:=((pb-pb α+θ)/(2 kb))
pb:=((f-f α-2 kb α-θ+α θ)/(2 (((-1+α))^2)))
rb:= 1-eb
Profitvb:=α pb+eb (1-α)pb-(1-eb)θ-kb*(eb*eb)-cv
Profitib:=rb(α pb+cs+f)-(1-eb)(α pb+cs+f)
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rb(α pb+cs+f)+(1-eb)(α pb+cs+f)-csb-cvb

eb

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

rb

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

rb:=1+(1/4) (((2 α)/(1-α))-((f+θ)/kb))
eb:=(1/4) (((f+θ)/kb)-((2 α)/(1-α)))
pb:=((f-f α-2 kb α-θ+α θ)/(2 (((-1+α))^2)))
Profitsb:=-α pb+eb(f-(1-α)pb)-cs -rb(α pb+cs+f)+(1-eb)(α pb+cs+f)-csb-cvb

Profitsb

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