Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
∂ lg k j (x)
2.6. Appendix
Hence, we get
⎛
∂xi
= −δij ⎝
f
l=j
′ j1(xj, xl) lg l ⎞
j(x) ⎠ lg k j (x) − (1 − δij)f ′ j2(xj, xi) lg i j(x) lg k j (x)
+ (1 − δjk) δijf ′ j1(xj, xk) lg k j (x) + δikf ′ j2(xj, xk) lg k
j (x) .
Similarly, the second order derivative is given by ∗
∂2 lg k ⎛
j (x)
= −δij
⎝δjm f
∂xm∂xi
l=j
′′
j11(xj, xl) lg l j +(1 − δjm)f ′′
j12(xj, xm) lg m j +
f
l=j
′ j1(xj, xl) ∂ lgl ⎞
j ⎠ lg
∂xm
k j
⎛
− δij ⎝
f
l=j
′ j1(xj, xl) lg l ⎞
⎠
j
∂ lgkj ∂xm
δjmf −(1−δij)
′′
j21(xj, xi) + δimf ′′
j22(xj, xi) lg i j lg k j +f ′ j2(xj, xi) ∂ lgi j
lg
∂xm
k j +f ′ j2(xj, xi) lg i ∂ lg
j
k
j
∂xm
f ′′
+ (1 − δjk)δij j11(xj, xk)δjm + f ′′
k
j12(xj, xk)δkm lgj +f ′ j1(xj, xk) ∂ lgk
j
∂xm
f ′′
+ (1 − δjk)δik j21(xj, xk)δjm + f ′′
k
j22(xj, xk)δim lgj +f ′ j2(xj, xk) ∂ lgk
j
.
∂xm
Portfolio size function
We recall that the expected portfolio size of insurer j is defined as
(x) +
ˆNj(x) = nj × lg j
j
l=j
nl × lg j
l (x),
where nj’s denotes last year portfolio size of insurer j and lg k j is defined in equation (2.1).
The function φj : xj ↦→ lg j
j (x) has the following derivative
φ ′ j(xj) =
∂ lgj
j (x)
∂xj
For the two considered price function, we have
f ′ 1
j1(xj, xl) = αj
⎛
= − ⎝
f ′ j1(xj, xl) lg l ⎞
j(x) ⎠ lg j
j (x).
xl
l=j
and ˜ f ′ j1(xj, xl) = ˜αj,
which are positive. So, the function φj will be a decreasing function.
(x) has the following derivative
For l = j, the function φl : xj ↦→ lg j
l
φ ′ l (xj) =
∂ lgj
l (x)
∂xj
= −f ′ j2(xl, xj) lg j
l (x) lgj l (x)+f ′ j2(xl, xj) lg j
l (x) = f ′ j2(xl, xj) lg j
l (x)(1−lgj l (x)).
∗. We remove the variable x when possible.
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