Insurance Model
The first-best benchmark

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:=v-(α p+e(1-α)p+cs)

Profitc:=Profits+Profitv

D[Profitc,e]

-2 e k+θ

Solve[%==0,e]

e:=(θ/(2 k))

Profitc

(Subscript[Π, c])= -cs-cv-(e^2) k+v-(1-e) θ

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

-cs-cv+v-θ+((θ^2)/(4 k))

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

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

Assumption 1
The vehicle manufacture’s cost cv and his failure penalty level a satisfy 0 ≤ θ ≤ cv ≤ v+((θ^2)/(4 k))-cs-θ.

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)-(1-e)(α p+cs)

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))
r:=1-e

r

Profits

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

D[%,p]

Solve[%==0,p]

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

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)))]

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)))]

By substituting e into the vehicle manufacture’s payoff Profitv  it is easy to check that

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

Profitv:=α p+e (1-α)p-(1-e)θ-k*(e*e)-cv
Solve[Profitv==0,p]

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)))

Solve[((-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)))-((2 k e-θ)/(1-α))==0,e]

v:= cs+cv+e*e* k+(1-e) θ

e:=((p-p α+θ)/(2 k))

v

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

Solve[%-p==0,p]

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), equals the expected coverage 
(1-e) (α p+cs).
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.

Profiti

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

Solve[Profiti==0,p]

{{p->-(cs/α)},{p->((-2 k+2 k r+θ)/(-1+α))}}

When the constraint is satisfied, the equilibrium premium rate for any given p becomes r=1-((θ  + (1-α) p)/(2k))

Solve[Profiti==0,r]

{{r->((2 k-p+p α-θ)/(2 k))}}

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)))]

((-2 cs k+f (p-p α+θ)-p (p (((-1+α))^2)+2 k α+θ-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]

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

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

Profits

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

D[%,p]

Solve[%==0,p]

Profits

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

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

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

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

Solve[e-((p-p α+θ)/(2 k))==0,p]

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

Solve[cs+cv+k*e*e+(1-e)θ-((-2 e k+θ)/(-1+α))==0,e]

{{e->((((2 k)/(1-α))+θ-(Sqrt[-4 cs k-4 cv k+((4 (k^2))/(((1-α))^2))-4 k θ+(θ^2)]))/(2 k))},{e->((((2 k)/(1-α))+θ+(Sqrt[-4 cs k-4 cv k+((4 (k^2))/(((1-α))^2))-4 k θ+(θ^2)]))/(2 k))}}

Proof of Proposition 1. We prove the results by using e as the decision variable (instead of p). In preparation, let us transform the constraints in terms of e (instead of p). 

First, by considering insurer acceptance constraint  p>((-2 k+2 k r+θ)/(-1+α))holds if and only if ((2 k e - θ)/(1-α)) ≥ ((-2 k+2 k r+θ)/(-1+α)),
or, equivalently, e ≥ 1-r.

Second, we can apply e=((p-p α+θ)/(2 k))  to express p in terms of e so that p =(( 2k e -θ)/(1-α)). By using this expression of p, it is easy to check that the boundary constraint p ≤ v holds if and only if  e ∈ [(UnderBar[e]),  (Overscript[e, -])̄], where  (UnderBar[e]) =((((2 k)/(1-α))+θ-(Sqrt[-4 cs k-4 cv k+((4 (k^2))/(((1-α))^2))-4 k θ+(θ^2)]))/(2 k)), 
(Overscript[e, -]) = ((((2 k)/(1-α))+θ+(Sqrt[-4 cs k-4 cv k+((4 (k^2))/(((1-α))^2))-4 k θ+(θ^2)]))/(2 k))
By substituting p =(( 2ke -θ)/(1-α)) into the satellite owner’s  payoff (9) along with the transformed constraints in terms of e, the satellite owner’s profit can be written as:
max (Subscript[Π, S])= v-(α p+e(1-α)p+cs) -r(α p+cs)+(1-e)(α p+cs) = v-(e+r)cs +(((θ - 2k e )(α r + e ))/(1-α))

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

p :=(( 2k e -θ)/(1-α))

Profits

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

D[%,e]

Simplify[-((-4 e k+cs (-1+α)-2 k r α+θ)/(-1+α))]

Solve[((cs+4 e k-cs α+2 k r α-θ)/(-1+α))==0,e]

{{e->((-cs+cs α-2 k r α+θ)/(4 k))}}
Solve[((-cs+cs α-2 k r α+θ)/(4 k))-((((2 k)/(1-α))+θ+(Sqrt[-4 cs k-4 cv k+((4 (k^2))/(((1-α))^2))-4 k θ+(θ^2)]))/(2 k))==0,r]

{{e->((-cs+cs α-2 k r α+θ)/(4 k))}}
e:=((-cs+cs α-2 k r α+θ)/(4 k))
p :=(( 2k e -θ)/(1-α))

p

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

e:=((((2 k)/(1-α))+θ-(Sqrt[-4 cs k-4 cv k+((4 k*k)/((1-α)*(1-α)))-4 k θ+θ*θ]))/(2 k))

cs:=v+((θ*θ)/(4 k))-cv-θ

e

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

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

e:=((p-p α+θ)/(2 k))

Profits

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

D[%,p]

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

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

p

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

p :=(( 2k e -θ)/(1-α))

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

Profits

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

D[%,e]

Solve[%==0,e]

若考虑卫星提前签订了数据售卖合同，收益为F

Profits:=-α p +e [F-(1-α)p]-cs-r(α p +cs+F )+(1-e)(α p +cs+F )

e:=(((1-α)p+θ)/(2k))

r:=1-(((1-α) p+θ)/(2 k))

Profitc

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