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. Determinants of Household Health Expenditure: Case of Urban Orissa2. The name is absent
3. LOCAL PROGRAMS AND ACTIVITIES TO HELP FARM PEOPLE ADJUST
4. Staying on the Dole
5. The name is absent
6. Accurate, fast and stable denoising source separation algorithms
7. Empirically Analyzing the Impacts of U.S. Export Credit Programs on U.S. Agricultural Export Competitiveness
8. Education Research Gender, Education and Development - A Partially Annotated and Selective Bibliography
9. BILL 187 - THE AGRICULTURAL EMPLOYEES PROTECTION ACT: A SPECIAL REPORT
10. The name is absent