Profitv:=α l+e (1-α)l-(1-e)θ-k*(e*e)-cv
Profiti:=r(α l+cs)-(1-e)β(α l+cs)
Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -r(α l+cs)+(1-e)β(α l+cs)

D[Profitv,e]

Solve[%==0,e]

e:=((l-l α+θ)/(2 k))

Profiti

Solve[%==0,r]

r:=((β (2 k-l+l α-θ))/(2 k))

Profits

Simplify[-cs-T-l α-(((cs+l α) β (2 k-l+l α-θ))/(2 k))+((((1+d-p) p-l (1-α)) (l-l α+θ))/(2 k))+(cs+l α) β (1-((l-l α+θ)/(2 k)))]

D[%,l]

Solve[%==0,l]

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

Solve[%==0,p]

p:=((1+d)/2)

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

l

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

e

Simplify[%26]

((-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α-4 θ+4 α θ)/(16 k (-1+α)))

r

Simplify[%28]

((β (8 k (-2+α)-2 d (-1+α)-(d^2) (-1+α)-(-1+α) (1+4 θ)))/(16 k (-1+α)))

Checking the Hessian matrix, we can find that Πs is concave in p and l if k > a2n/2, because the Hessian matrix is H = ⃒⃒⃒⃒ ,and the determinant is (-((-2 l (-1+α)+2 θ)/(2 k)))*(-((((-1+α))^2)/k))- (-((-(-1+d) (-1+α)-2 p (-1+α))/(2 k)))*(-((-(-1+d) (-1+α)-2 p (-1+α))/(2 k)))*() a2n).
 2nan an  k ⃒⃒⃒⃒ and the determinant is n(2k  a2n).

Profits

Simplify[-cs-T-l α-(((cs+l α) β (2 k-l+l α-θ))/(2 k))+((((1+d-p) p-l (1-α)) (l-l α+θ))/(2 k))+(cs+l α) β (1-((l-l α+θ)/(2 k)))]

D[%,p]

D[%,p]

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

D[%,l]

D[((l (-(1+d) (-1+α)+2 p (-1+α))+θ+d θ-2 p θ)/(2 k)),l]

ϕ:=((2 l (-1+α)-2 θ)/(2 k)) *(-((((-1+α))^2)/k))-((((((-1-d) (-1+α)+2 p (-1+α))/(2 k))))^2)

ϕ

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

y:=-(((((-1+α))^2) (1+(d^2)+d (2-4 p)-4 p+4 (p^2)+4 l (-1+α)-4 θ))/(4 (k^2)))

p:=((1+d)/2)

l:=((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2)))

y

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

Solve[-(-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α-4 θ+4 α θ)==0,α]

{{α->((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))}}
Solve[1-((-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α-4 θ+4 α θ)/(16 k (-1+α)))==0,α]

Solve[y==((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))-((1+2 d+(d^2)-16 k+4 θ)/(1+2 d+(d^2)-8 k+4 θ)),y]

We solve this game by backward induction. By checking the Hessian matrix, we find that πSA  is jointly concave in p and l, when α<(((((1+d))^2)+4 θ)/((((1+d))^2)+8 k+4 θ)).in this paper, we consider the case with  (((((1+d))^2)-16k+4 θ)/((((1+d))^2)-8 k+4 θ))<α<(((((1+d))^2)+4 θ)/((((1+d))^2)+8 k+4 θ)), which is the common situation because it means launch successfully propability is above zero and below one.
e = ((-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α-4 θ+4 α θ)/(16 k (-1+α))), that (ϕ/(16 k (1-α))), where  ϕ=1-2 d (-1+α)-(d^2) (-1+α)-α-8 k α+4 θ-4 α θ=(1-α)(((1+d))^2)-8αk-(1-α)4θ

r = β(1-(ϕ/(16 k (1-α)))),
p =(( 1+d)/( 2)),
l = ((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2))) = ((ϕ-8θ(1-α))/(8 (((1-α))^2)))

Note that in our analysis, we only discuss the case when the vehicle’s profit is positive and the effort level is less than the upper bound.F is bounded above by ̄ F(or the vehicle manufacture will quit the launch project)

Profitv:=α l+e (1-α)l-(1-e)θ-k*(e*e)-cv
Profiti:=r(α l+cs)-(1-e)β(α l+cs)
Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -r(α l+cs)+(1-e)β(α l+cs)-T

e:=((l-l α+θ)/(2 k))

Profitv

Solve[%==0,l]

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

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

z:=((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2)))-((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))

z

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

Solve[%==0,α]

{{α->(-1-4 d-6 (d^2)-4 (d^3)-(d^4)+8 k+256 cv k+16 d k+8 (d^2) k-8 θ-16 d θ-8 (d^2) θ+160 k θ-16 (θ^2)-16 (Sqrt[(k^2)+4 d (k^2)+6 (d^2) (k^2)+4 (d^3) (k^2)+(d^4) (k^2)-192 cv (k^3)-192 (k^3) θ+48 (k^2) (θ^2)]))/(-1-4 d-6 (d^2)-4 (d^3)-(d^4)+16 k+256 cv k+32 d k+16 (d^2) k+192 (k^2)-8 θ-16 d θ-8 (d^2) θ+64 k θ-16 (θ^2))},{α->(-1-4 d-6 (d^2)-4 (d^3)-(d^4)+8 k+256 cv k+16 d k+8 (d^2) k-8 θ-16 d θ-8 (d^2) θ+160 k θ-16 (θ^2)+16 (Sqrt[(k^2)+4 d (k^2)+6 (d^2) (k^2)+4 (d^3) (k^2)+(d^4) (k^2)-192 cv (k^3)-192 (k^3) θ+48 (k^2) (θ^2)]))/(-1-4 d-6 (d^2)-4 (d^3)-(d^4)+16 k+256 cv k+32 d k+16 (d^2) k+192 (k^2)-8 θ-16 d θ-8 (d^2) θ+64 k θ-16 (θ^2))}}

t:=((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))-(-1-4 d-6 (d^2)-4 (d^3)-(d^4)+8 k+256 cv k+16 d k+8 (d^2) k-8 θ-16 d θ-8 (d^2) θ+160 k θ-16 (θ^2)+16 (Sqrt[(k^2)+4 d (k^2)+6 (d^2) (k^2)+4 (d^3) (k^2)+(d^4) (k^2)-192 cv (k^3)-192 (k^3) θ+48 (k^2) (θ^2)]))/(-1-4 d-6 (d^2)-4 (d^3)-(d^4)+16 k+256 cv k+32 d k+16 (d^2) k+192 (k^2)-8 θ-16 d θ-8 (d^2) θ+64 k θ-16 (θ^2))

t

Simplify[%31]

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

D[lVA,α]

Simplify[((-2 k+θ+(((Sqrt[k]) (2 cv (-1+α)+2 k α-θ))/(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))-((2 (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^3))]

k:=0.1
θ:=0.5
d:=0.9
Plot[((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2)),{α,0,0.1}{cv,0.1,1}]

Plot3D[((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2)),{α,0,1}, {cv,0,5}]

cv:=1
k:=0.1
θ:=0.5
d:=0.9
y2:=Plot[((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2))),{α,0,0.1}]
y2

Show[y1,y2]

Solve[((-2 k+θ+(((Sqrt[k]) (2 cv (-1+α)+2 k α-θ))/(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))+((2 (2 k α+θ-α θ-2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^3))==0,α]

α:=((1+2 d+(d^2)+4 θ-16k)/(1+2 d+(d^2)-8 k+4 θ))
lVA:=((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))

lVA

Simplify[%5]

l:=((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2)))
lVA:=((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))

Solve[lVA-l==0,cv]

cv:=(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2))

α:=0.2
cv

k:=0.1
θ:=0.3
d:=1
Plot[(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)),{α,0,0.2}]

cv:=(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2))
α:=((1+2 d+(d^2)+4 θ-16k)/(1+2 d+(d^2)-8 k+4 θ))

cv

Simplify[%4]

cv:=(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2))
α:=((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))

cv

Simplify[%3]

cv:=(1/(256 k (((-1+α))^2)))(1+4 d+6 (d^2)+4 (d^3)+(d^4)-2 α-8 d α-12 (d^2) α-8 (d^3) α-2 (d^4) α+16 k α+32 d k α+16 (d^2) k α+(α^2)+4 d (α^2)+6 (d^2) (α^2)+4 (d^3) (α^2)+(d^4) (α^2)-16 k (α^2)-32 d k (α^2)-16 (d^2) k (α^2)-192 (k^2) (α^2)+8 θ+16 d θ+8 (d^2) θ-256 k θ-16 α θ-32 d α θ-16 (d^2) α θ+320 k α θ+8 (α^2) θ+16 d (α^2) θ+8 (d^2) (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2))

D[cv,α]

Simplify[%2]

Solve[%==0,α]

α3:=((1+2 d+(d^2)-12 θ)/(1+2 d+(d^2)+24 k-12 θ))

α1:=((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))

α2:=((1+2 d+(d^2)+4 θ-16k)/(1+2 d+(d^2)-8 k+4 θ))

Δα1:=α1-α3
Δα1

Simplify[-((1+2 d+(d^2)-12 θ)/(1+2 d+(d^2)+24 k-12 θ))+((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))]

Δα2:=α3-α2

Δα2

Simplify[((1+2 d+(d^2)-12 θ)/(1+2 d+(d^2)+24 k-12 θ))-((1+2 d+(d^2)-16 k+4 θ)/(1+2 d+(d^2)-8 k+4 θ))]

Simplify[-((16 k (1+2 d+(d^2)-24 k+12 θ))/((1+2 d+(d^2)+24 k-12 θ) (1+2 d+(d^2)-8 k+4 θ)))]

Δα3:=α2-α3

Δα3

Simplify[-((1+2 d+(d^2)-12 θ)/(1+2 d+(d^2)+24 k-12 θ))+((1+2 d+(d^2)-16 k+4 θ)/(1+2 d+(d^2)-8 k+4 θ))]

α:=((1+2 d+(d^2)-12 θ)/(1+2 d+(d^2)+24 k-12 θ))

cv

Simplify[%11]

α:=((1+2 d+(d^2)+4 θ)/(1+2 d+(d^2)+8 k+4 θ))

cv

Simplify[%3]

α:=((1+2 d+(d^2)+4 θ-16k)/(1+2 d+(d^2)-8 k+4 θ))

cv

Simplify[%6]

Factor[-3 k-((θ (1+2 d+(d^2)+4 θ))/(8 k))+(1/4) (1+2 d+(d^2)+8 θ)]

y:=(1+d)^4

y

Expand[(((1+d))^4)]

Δy:=((1+4 d+6 (d^2)+4 (d^3)+(d^4)-192 k θ+48 (θ^2))/(192 k))-(-3 k-((θ (1+2 d+(d^2)+4 θ))/(8 k))+(1/4) (1+2 d+(d^2)+8 θ))

Δy

Simplify[3 k+(1/4) (-1-2 d-(d^2)-8 θ)+((θ (1+2 d+(d^2)+4 θ))/(8 k))+((1+4 d+6 (d^2)+4 (d^3)+(d^4)-192 k θ+48 (θ^2))/(192 k))]

Solve[cv==0,α]

1. The optimal launch price is the l*

e:=((-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α-4 θ+4 α θ)/(16 k (-1+α)))
r:=((β (8 k (-2+α)-2 d (-1+α)-(d^2) (-1+α)-(-1+α) (1+4 θ)))/(16 k (-1+α)))
p:=((1+d)/2)
l:=((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2)))

Profitv:=α l+e (1-α)l-(1-e)θ-k*(e*e)-cv
Profiti:=r(α l+cs)-(1-e)β(α l+cs)
Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -r(α l+cs)+(1-e)β(α l+cs)
CS:=(1/2)(((1+d-p))^2)
SW:=Profits+Profitv+CS

Profitv

Simplify[%21]

Profits

Simplify[%10]

Δ:=1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-256 cv k (((-1+α))^2)-2 α+16 k α+(α^2)-16 k (α^2)-192 (k^2) (α^2)+8 θ-256 k θ-16 α θ+320 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 k-4 θ)+4 θ)-(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-128 cs k (((-1+α))^2)-2 α-16 k α+(α^2)+16 k (α^2)+64 (k^2) (α^2)+8 θ-16 α θ+64 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (-1+α) (-1+α+8 k α-4 θ+4 α θ)+2 (d^2) (-1+α) (-3-4 θ+α (3+8 k+4 θ)))

Δ

Simplify[128 cs k (((-1+α))^2)-256 cv k (((-1+α))^2)+32 k α-32 k (α^2)-256 (k^2) (α^2)-256 k θ+256 k α θ-2 (d^2) (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 k-4 θ)+4 θ)-4 d (-1+α) (-1+α+8 k α-4 θ+4 α θ)-2 (d^2) (-1+α) (-3-4 θ+α (3+8 k+4 θ))]

CS

Simplify[(1/2) (((1+(1/2) (-1-d)+d))^2)]

SW

Simplify[%10]

(1/(256 k (((-1+α))^2)))(12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)-64 (k^2) (α^2)+3 (((-1+α))^2) (((1+4 θ))^2)-16 k (-1+α) (2+16 cs (-1+α)+16 cv (-1+α)-3 α-16 θ+12 α θ)+4 d (-1+α) (8 k (-2+3 α)+3 (-1+α) (1+4 θ))+2 (d^2) (-1+α) (8 k (-2+3 α)+3 (-1+α) (3+4 θ)))

Derivation of the sensitivity analysis for Model NG. How d,k,θ  and  will affect these optimal decisions:

e:=((-1+2 d (-1+α)+(d^2) (-1+α)+α+8 k α-4 θ+4 α θ)/(16 k (-1+α)))
r:=((β (8 k (-2+α)-2 d (-1+α)-(d^2) (-1+α)-(-1+α) (1+4 θ)))/(16 k (-1+α)))
p:=((1+d)/2)
l:=((1-2 d (-1+α)-(d^2) (-1+α)-4 θ+α (-1-8 k+4 θ))/(8 (((-1+α))^2)))
Profitv:=(1/(256 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-256 cv k (((-1+α))^2)-2 α+16 k α+(α^2)-16 k (α^2)-192 (k^2) (α^2)+8 θ-256 k θ-16 α θ+320 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)-2 (d^2) (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 d (-1+α) (1+α (-1+8 k-4 θ)+4 θ))
Profits:=(1/(128 k (((-1+α))^2)))(1+4 (d^3) (((-1+α))^2)+(d^4) (((-1+α))^2)-128 cs k (((-1+α))^2)-2 α-16 k α+(α^2)+16 k (α^2)+64 (k^2) (α^2)+8 θ-16 α θ+64 k α θ+8 (α^2) θ-64 k (α^2) θ+16 (θ^2)-32 α (θ^2)+16 (α^2) (θ^2)+4 d (-1+α) (-1+α+8 k α-4 θ+4 α θ)+2 (d^2) (-1+α) (-3-4 θ+α (3+8 k+4 θ)))
CS:=(1/8) (((1+d))^2)
SW:=(1/(256 k (((-1+α))^2)))(12 (d^3) (((-1+α))^2)+3 (d^4) (((-1+α))^2)-64 (k^2) (α^2)+3 (((-1+α))^2) (((1+4 θ))^2)-16 k (-1+α) (2+16 cs (-1+α)+16 cv (-1+α)-3 α-16 θ+12 α θ)+4 d (-1+α) (8 k (-2+3 α)+3 (-1+α) (1+4 θ))+2 (d^2) (-1+α) (8 k (-2+3 α)+3 (-1+α) (3+4 θ)))

d

D[e,d]

Simplify[((2 (-1+α)+2 d (-1+α))/(16 k (-1+α)))]

((1+d)/(8 k))

D[r,d]

Simplify[(((-2 (-1+α)-2 d (-1+α)) β)/(16 k (-1+α)))]

-(((1+d) β)/(8 k))

D[l,d]

Simplify[((-2 (-1+α)-2 d (-1+α))/(8 (((-1+α))^2)))]

D[p,d]

D[Profits,d]

Simplify[(1/(128 k (((-1+α))^2)))(12 (d^2) (((-1+α))^2)+4 (d^3) (((-1+α))^2)+4 (-1+α) (-1+α+8 k α-4 θ+4 α θ)+4 d (-1+α) (-3-4 θ+α (3+8 k+4 θ)))]

D[Profitv,d]

Simplify[(1/(256 k (((-1+α))^2)))(12 (d^2) (((-1+α))^2)+4 (d^3) (((-1+α))^2)-4 d (-1+α) (3+α (-3+8 k-4 θ)+4 θ)-4 (-1+α) (1+α (-1+8 k-4 θ)+4 θ))]

D[CS,d]

D[SW,d]

Simplify[%12]

k

D[e,k]

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

-((1+2 d+(d^2)+4 θ)/(16 (k^2)))

D[r,k]

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

((β (1+2 d+(d^2)+4 θ))/(16 (k^2)))

D[l,k]

D[p,k]

0

D[Profits,k]

Simplify[%36]

D[Profitv,k]

Simplify[%9]

D[CS,k]

D[SW,k]

Simplify[%14]

θ

D[e,θ]

Simplify[((-4+4 α)/(16 k (-1+α)))]

D[r,θ]

D[l,θ]

Cancel[((-4+4 α)/(8 (((-1+α))^2)))]

D[p,θ]

D[Profits,θ]

Simplify[%27]

Simplify[((8-16 α+64 k α+8 (α^2)-64 k (α^2)+4 d (-1+α) (-4+4 α)+2 (d^2) (-1+α) (-4+4 α)+32 θ-64 α θ+32 (α^2) θ)/(128 k (((-1+α))^2)))]

D[Profitv,θ]

Simplify[%29]

Simplify[((8-256 k-4 d (4-4 α) (-1+α)-2 (d^2) (4-4 α) (-1+α)-16 α+320 k α+8 (α^2)-64 k (α^2)+32 θ-64 α θ+32 (α^2) θ)/(256 k (((-1+α))^2)))]

D[Profits,θ]

Simplify[((8-16 α+64 k α+8 (α^2)-64 k (α^2)+4 d (-1+α) (-4+4 α)+2 (d^2) (-1+α) (-4+4 α)+32 θ-64 α θ+32 (α^2) θ)/(128 k (((-1+α))^2)))]

D[CS,θ]

D[SW,θ]

Simplify[%16]

2. The optimal launch price is the (l^VA)

when cv > H(α)(Subscript[|, α < α < (Subscript[α, 0])  ]), where H(α)=  (((((1+d))^4)(((1-α))^2)+(((1+d))^2)*8(1-α)[2αk+θ(1-α)]+16(θ^2)(((1-α))^2)-192(α^2)(k^2)-64θk(1-α)(4-α))/(256 k (((-1+α))^2)))     ,
α = (((((1+d))^2)+4θ-16k)/((((1+d))^2)+4θ-8k)) , (Subscript[α, 0]) = (((((1+d))^4)+8(((1+d))^2)(θ-k)+16θ(θ-10k)+16k(Sqrt[(((1+d))^2)+48θ(θ-4k)]))/((((1+d))^4)+8(((1+d))^2)(θ-2k)+16θ(θ-4k)-192(k^2))), the optimal launch price is less than the VM acceptable price, so the price should be (l^VA)=((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))

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

e:=((l-l α+θ)/(2 k))
r:=((β (2 k-l+l α-θ))/(2 k))

e

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

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

r

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

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

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

Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -r(α l+cs)+(1-e)β(α l+cs)

Profits

Simplify[%12]

D[%,p]

Solve[%==0,p]

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

p:=((1+d)/2)

Profitv:=α l+e (1-α)l-(1-e)θ-k*(e*e)-cv
Profiti:=r(α l+cs)-(1-e)β(α l+cs)
Profits:=-α l+e(p(1+d-p)-(1-α)l)-cs -r(α l+cs)+(1-e)β(α l+cs)

Profitv

Simplify[%21]

Profits

Simplify[%23]

Simplify[(1/((Sqrt[k]) (((-1+α))^3)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs (((-1+α))^2)+α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-(1/((Sqrt[k]) (-1+α)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2)))]

Simplify[-cs+((α (2 k α+θ-α θ-2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))-(1/(4 (Sqrt[k]) (((-1+α))^2)))(-(Sqrt[k]) α+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (-1+2 d (-1+α)+(d^2) (-1+α)+α-8 k α-4 θ+4 α θ+8 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))+(1/((Sqrt[k]) (((-1+α))^3)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs (((-1+α))^2)+α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))-(1/((Sqrt[k]) (-1+α)))β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2)))]

Simplify[((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs (((-1+α))^2)+α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))))/((Sqrt[k]) (((-1+α))^3)))-((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (cs+((α (-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))))/((Sqrt[k]) (-1+α)))]

Simplify[-cs+((α (2 k α+θ-α θ-2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))-(1/(4 (Sqrt[k]) (((-1+α))^2)))(-(Sqrt[k]) α+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (-1+2 d (-1+α)+(d^2) (-1+α)+α-8 k α-4 θ+4 α θ+8 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))]

CS:=(1/8) (((1+d))^2)

SW:=CS+Profits

SW

Simplify[%31]

Derivation of the sensitivity analysis for Model NG. How d,k,θ  and  will affect these optimal decisions:

l:=((-2 k α+(-1+α) θ+2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(((-1+α))^2))
p:=((1+d)/2)
e:=(((Sqrt[k]) α-(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/((Sqrt[k]) (-1+α)))
r:=((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/((Sqrt[k]) (-1+α)))
Profits:=-cs+((α (2 k α+θ-α θ-2 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(((-1+α))^2))-(((-(Sqrt[k]) α+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])) (-1+2 d (-1+α)+(d^2) (-1+α)+α-8 k α-4 θ+4 α θ+8 (Sqrt[k]) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(4 (Sqrt[k]) (((-1+α))^2)))
CS:=(1/8) (((1+d))^2)
SW:=(1/(8 (Sqrt[k]) (((-1+α))^2)))(-16 (k^(3/2)) (α^2)-(Sqrt[k]) (-1+α) (1+2 d+(d^2)+8 cs (-1+α)+16 cv (-1+α)-3 α-6 d α-3 (d^2) α-16 θ)+16 k α (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])-2 (-1+α) (1+2 d+(d^2)+4 θ) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))

d

D[e,d]

D[r,d]

D[p,d]

D[l,d]

D[Profits,d]

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

D[SW,d]

Simplify[((-(Sqrt[k]) (-1+α) (2+2 d-6 α-6 d α)-2 (2+2 d) (-1+α) (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(8 (Sqrt[k]) (((-1+α))^2)))]

D[CS,d]

k

D[e,k]

Simplify[(((α/(2 (Sqrt[k])))-((α^2)/(2 (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))))/((Sqrt[k]) (-1+α)))-(((Sqrt[k]) α-(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ]))/(2 (k^(3/2)) (-1+α)))]

D[r,k]

Simplify[((β (-(1/(2 (Sqrt[k])))+((α^2)/(2 (Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))))/((Sqrt[k]) (-1+α)))-((β (-(Sqrt[k])+(Sqrt[cv (((-1+α))^2)+k (α^2)+θ-α θ])))/(2 (k^(3/2)) (-1+α)))]

D[l,k]

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

D[Profits,k]

Simplify[%60]

θ

D[e,θ]

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

D[r,θ]

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

D[l,θ]

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

D[Profits,θ]

Simplify[%17]