Incremental Risk Vulnerability
16
= Ey
Ee u '( w + y + e )
Ey+e u ' ( w + y + e )ee w y ,
where re (w + y) is as defined on page 9. Hence
^^ / ∖ ʌ / \
r( w ) — r( w )
Ey
Ee u '( w + y + e )
Ey+e u '( w + y + e )
Ey
u '( w + y ) , , .
---(—---(r-( r e( w + y )
Ey u' ( w + y )V
— r(w + y))
Condition (4) implies that the second term is positive or zero. The first term is similar
to a covariance term since the term in ( ) has zero expectation. Hence the first term is
nonnegative if the term in ( ) is single crossing downwards and re (w + y) is declining in y .
The latter is implied by condition (5). Therefore, to complete the proof we have to establish
the single crossing downward property. For notational simplicity, let Z(w + y) denote the
term in ( ),
Z(w + y)=
Ee u '( w + y + e )
a
u '( w + y )
b
with a and b being appropriately defined constants.
Differentiating with respect to y yields
Z '( w + y )
Ee u "( w + y + e )(1 + dy ) u "( w + y )
--------------------------- — -----:-----
ab
Ee u'( w + y + e ) u'( w + y ) Eeu ( w + y + e ) ∂y
---------------r e( w + y )+--;----r ( w + y )+---
ab a
For Z = 0 it follows that sgnZ`(w + y) = sgn[r(w + y) — re(w + y) + [Eeu'(w + y +
e)]-1 Eeu"(w + y + e)(∂e∕∂y)]. Hence condition (4) implies Z'(w + y) ≤ 0 at a crossing point
if e is distributed independently of y, i.e. ∂e∕∂y ≡ 0. Then only one crossing point exists,
therefore Z(w + y) is downward sloping. If the distribution of e improves with increasing y
according to second-order stochastic dominance, then Eeu"(w + y + e)(de/dy) < 0 if u111 > 0.
u`" > 0 follows from condition (5) because dre(w + y)/dy ≤ 0 holds for a small risk only if
r' < 0. Hence, at a crossing point, Z'(w + y) ≤ 0. □
More intriguing information
1. Mortality study of 18 000 patients treated with omeprazole2. The Context of Sense and Sensibility
3. The Response of Ethiopian Grain Markets to Liberalization
4. The name is absent
5. Placenta ingestion by rats enhances y- and n-opioid antinociception, but suppresses A-opioid antinociception
6. Investment in Next Generation Networks and the Role of Regulation: A Real Option Approach
7. Economies of Size for Conventional Tillage and No-till Wheat Production
8. Segmentación en la era de la globalización: ¿Cómo encontrar un segmento nuevo de mercado?
9. Using Surveys Effectively: What are Impact Surveys?
10. Altruism with Social Roots: An Emerging Literature