PreprintPDF Available

Non-parametric identification of the extent of downward wage rigidities

Authors:
Paris Nearchou
Paris Nearchou
Preprints and early-stage research may not have been peer reviewed yet.
Download file PDF
Read file
Download citation

Abstract

We study the problem of identification of measures of the extent of individual types of downward wage rigidity from micro-level data on nominal wage growth rates, in the context of a wage adjustment process that may feature any number such rigidity types. For that purpose we develop a comprehensive framework for the modelling and measurement of wage rigidities. We show that the presence of measurement error does not alter fundamentally the nature of this identification problem, and develop an identification strategy that is applicable with measurement-error-free and measurement-error-contaminated data. This relies on weaker restrictions than those usually employed in the literature, including being non-parametric.
Content uploaded by Paris Nearchou
Author content
All content in this area was uploaded by Paris Nearchou on Mar 05, 2019
Content may be subject to copyright.
Working Paper 18-2018
Non-parametric identification of the extent of
downward wage rigidities
Paris Nearchou
Department of Economics, University of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus
Tel.: +357-22893700, Fax: +357-22895028, Web site: http://www.ucy.ac.cy/econ/en
˙w∗∈R
˙w∈R
˙w
˙w∗
˙w
˙w∗
˙w∗=δ·˙wc∗+ (1 −δ) ˙wN∗,˙wc∗6= ˙wN∗
δ
δ= 1
˙wN∗∈R˙w∗
˙wc∗∈R
˙wN∗˙wc∗δ
f˙wN∗,˙wc∗,δ|R,x x∈ X
R∈ R
x
R
X
R x
f˙wN∗,˙wc∗,δ|R,x
f˙wN∗,˙wc∗,δ|R,x ˙wN∗,˙wc∗, δ|R, x=f˙wN∗|R,x ˙wN∗|R, x×
×f˙wc∗|˙wN∗,R,x ˙wc∗|˙wN∗, R, x
×Pr δ|˙wN∗,˙wc∗, R, x
˙w∗
x R
R x
˙wN∗f˙wN∗|R,x
˙wc∗f˙wc∗|˙wN∗,R,x ˙wN∗
δ
ρR∗
x˙wN∗,˙wc∗≡Pr δ= 1|˙wN∗,˙wc∗, R, x
˙wN∗,˙wc∗
R˙wN∗,˙wc∗
˙w∗˙wN∗,˙wc∗, δ
R X
˙wN∗,˙wc∗
f˙wN∗,˙wc∗|R,x
˙wN∗˙wc∗
f˙wN∗,˙wc∗|R,x R x
f˙wN∗|R,x
˙wN∗R x f ˙wN∗|R,x
f˙wN∗|xWN∗
Rx =WN∗
xWN∗
x≡˙wN∗
x0,˙wN∗
x1f˙wN∗|x
f˙wN∗|R,x f˙wN∗|x
f˙wN∗|R,x f˙wN∗|x
R x ˙wc∗
f˙wc∗|˙wN∗,R,x
f˙wc∗|R,x ˙wN∗
Wc∗
Rx ≡[ ˙wc∗
Rx0,˙wc∗
Rx1]
˙wc∗
Rx1˙wc∗= ˙wc∗
Rx1
˙wN∗Wc∗
Rx ≡ { ˙wc∗
Rx1}
˙wc∗
Rx0= ˙wc∗
Rx1
WN∗
Rx
Wc∗
Rx Wc∗
Rx WN∗
Rx ˙wc∗
Rx1<˙wN∗
x0Wc∗
Rx
WN∗
Rx Wc∗
Rx ⊂ WN∗
Rx
WNc∗
Rx =˙wN∗
x0,˙wN∗
x1×[ ˙wc∗
Rx0,˙wc∗
Rx1]f˙wN∗,˙wc∗|R,x
R x ˙wN∗,˙wc∗∈ WNc∗
Rx
ρR∗
x˙wN∗,˙wc∗˙wN∗<˙wc∗
ρR∗
x˙wN∗,˙wc∗%R
x
ρR∗
x˙wN∗,˙wc∗=
%R
x≥0,˙wN∗<˙wc∗
0, o/w
f˙wc∗|R,x %R
x
˙w∗=
˙wN∗,˙wN∗≥˙wc∗
δ·˙wc∗+ (1 −δ) ˙wN∗,˙wN∗<˙wc∗
f˙wN∗|R,x
f˙wc∗|˙wN∗,R,x
R
˙wc∗
f˙wN∗|R,x
f˙wN∗,˙wc∗|R,x WNc∗
Rx
˙wc∗
WNc∗
Rx =˙wN∗
x0,˙wN∗
x1×[ ˙wc∗
Rx0,˙wc∗
Rx1]
ABB0A0WN c∗
Rx =˙wN∗
x0,˙wN∗
x1× { ˙wc∗
Rx1}
AB
ρR
x(·,·)
˙wN∗,˙wc∗, R, xR ρR
x˙wN∗,˙wc∗>0
f˙wc∗|˙wN∗,R,x
f˙wc∗|R,x f˙wc∗|R,x
f˙wc∗|˙wN∗,R,x ˙wN∗
'B
'A
C
'C
0Rx
¤c
w_1Rx
¤c
w_1x
¤N
w_
0x
¤N
w_0
o
45
AB
¤N
w_
¤c
w_
˙wc∗
B
A
o
45
0
0x
¤N
w_1x
¤N
w_
¤c
w_
¤N
w_
1Rx
¤c
w_
˙wc∗
f˙wN∗,˙wc∗|R,x Wc∗
Rx ⊂ WN∗
Rx
C∗
Rx ≡˙wN∗,˙wc∗∈ WNc∗
Rx :ρR
x˙wN∗,˙wc∗>0
f˙wN∗,˙wc∗|R,x
x
R
C∗
Rx ≡˙wN∗,˙wc∗∈ WNc∗
Rx : ˙wN∗<˙wc∗, %R
x>0
%R
x>0WNc∗
Rx
45 ˙wc∗
ACC0A0C C0
˙wc∗AC C ρR
x˙wN∗,˙wc∗
%R
x
˙w= ˙w∗+ ˙ε
˙ε
f˙ε
[−˙ε1,˙ε1]
f˙ε
˙ε
˙ε= 0 f˙ε
R
x
R %R
x
Rx
x
Rx≡R∈ R : Pr ˙wN∗,˙wc∗∈ C∗
Rx|R, x>0
Wc∗
Rx ⊂ WN∗
Rx R∈ Rx
Wc∗
Rx WN∗
Rx ˙wc∗
Rx1<˙wN∗
x0R /∈ RxR∈ Rx
%R
x≡Pr δ= 1|˙wN∗,˙wc∗∈ C∗
Rx, R, x
=Pr (δ= 1|R, x)
Pr (( ˙wN∗,˙wc∗)∈ C∗
Rx|R, x)
%R
x
R x
R x
Pr ˙wN∗,˙wc∗∈ C∗
Rx|R, xR
x
Pr δ= 1|˙wN∗,˙wc∗∈ C∗
Rx, R, x=Pr(δ=1,(˙wN∗,˙wc∗)∈C∗
Rx|R,x)
Pr(( ˙wN∗,˙wc∗)∈C ∗
Rx|R,x)
Pr δ= 1,˙wN∗,˙wc∗∈ C∗
Rx|R, x= Pr (δ= 1|R, x)R x
δ= 1
˙wN∗,˙wc∗∈ C∗
Rx R∈ R \ Rx
Rx=R∈ R : Pr ˙wN∗<˙wc∗|R, x>0
%R
xR∈ Rx
%R
x= Pr δ= 1|˙wN∗<˙wc∗, R, x
=Pr (δ= 1|R, x)
Pr ( ˙wN∗<˙wc∗|R, x)
˙wc∗
R x
%R
x%R
x
( ˙w, x)x∈ X ˙w
x %R
xR∈ Rx
f˙w|xx∈ X f˙w|x
x
˙wN≡˙wN∗+ ˙ε˙wc≡˙wc∗+ ˙ε
˙w=δ·˙wc+ (1 −δ) ˙wN,˙wc6= ˙wN
˙wN˙wcδ
R x
f˙wN,˙wc,δ|R,x ˙wN,˙wc, δ|R, x=f˙wN|R,x ˙wN|R, x×
×f˙wc|˙wN,R,x ˙wc|˙wN, R, x×
×Pr δ|˙wN,˙wc, R, x
ρR
x˙wN,˙wc≡Pr δ= 1|˙wN,˙wc, R, x
R˙wN,˙wcf˙wN,˙wc|R,x
WNc
Rx
WNc
Rx =˙wN∗+ ˙ε, ˙wc∗+ ˙ε:˙wN∗,˙wc∗∈ WN c∗
Rx ,˙ε∈[−˙ε1,˙ε1]
˙wNR x f ˙wN|R,x f˙wN|x
f˙wN∗|x
f˙wN|x( ˙w|x) = ˆ˙ε1
−˙ε1
f˙wN∗|x( ˙w−|x)f˙ε()d
f˙wN|xWN
x≡˙wN
x0,˙wN
x1=
˙wN∗
x0−˙ε1,˙wN∗
x1+ ˙ε1
E˙wN|x=E˙wN∗|x
V ar ˙wN|x=V ar ˙wN∗|x+V ar ( ˙ε)≥V ar ˙wN∗|x
R x ˙wc∗= ˙wc∗
Rx1˙wN∗˙wc= ˙wc∗
Rx1+ ˙ε
f˙wc|˙wN,R,x f˙wc|R,x
f˙wc|R,x ( ˙w|R, x) = f˙ε( ˙w−˙wc∗
Rx1)
f˙wc|R,x Wc
Rx ≡[ ˙wc
Rx0,˙wc
Rx1] = [ ˙wc∗
Rx1−˙ε1,˙wc∗
Rx1+ ˙ε]
E( ˙wc|R, x) = ˙wc∗
Rx1
V ar ( ˙wc|R, x) = V ar ( ˙ε)≥0
R x ˙wN˙wc∗˙wc= ˙wc∗+ ˙ε
f˙wc|˙wN,R,x f˙wc∗|˙wN∗,R,x
f˙wc|˙wN,R,x ˙w|˙wN, R, x=ˆ˙ε1
−˙ε1
f˙wc∗|˙wN∗,R,x ˙w−|˙wN∗−, R, xf˙ε()d
f˙wc|˙wN,R,x Wc
Rx ≡[ ˙wc
Rx0,˙wc
Rx1] =
[ ˙wc∗
Rx0−˙ε1,˙wc∗
Rx1+ ˙ε1]
E˙wc|˙wN, R, x=E˙wc∗|˙wN∗, R, x
V ar ˙wc|˙wN, R, x=V ar ˙wc∗|˙wN∗, R, x+V ar ( ˙ε)
≥V ar ˙wc∗|˙wN∗, R, x
ρR
x˙wN,˙wc˙wN<˙wc%R
x
ρR
x˙wN,˙wc=
%R
x≥0,˙wN<˙wc
0, o/w
˙w=
˙wN,˙wN≥˙wc
δ·˙wc+ (1 −δ) ˙wN,˙wN<˙wc
x f ˙wN|xf˙wc|˙wN,R,x f˙wN∗|x
f˙wc∗|˙wN∗,R,x
f˙wN,˙wc|R,x WNc
Rx f˙wN∗,˙wc∗|R,x WN c∗
Rx
˙wN˙wc
WNc
Rx WNc∗
Rx
AB
abb0a0WNc∗
Rx
f˙ε
˙ε= 0
f˙wN∗,˙wc∗|R,x
ρR
x˙wN,˙wc
ρR∗
x˙wN∗,˙wc∗˙wN,˙wc
˙wN∗,˙wc∗ ˙wN∗+ ˙ε, ˙wc∗+ ˙ε: ˙ε∈[−˙ε1,˙ε1]
'c
c
0Rx
c
w_
1x
N
w_
1x
¤N
w_
{
0x
N
w_0x
¤N
w_
{
1Rx
¤c
w_
1Rx
c
w_N
w_;
¤N
w_
0
o
45
'b
'a
ab
1
"_1
"_
AB
c
w_;
¤c
w_
1
"_
{
{
1
"_
f˙wN,˙wc|R,x
c
w_;
¤c
w_
R;xj
c
w
_
f
1Rx
¤c
w_
1Rx
c
w_
0Rx
c
w_
x
¤N
w
|
1x
N
w_
1x
¤N
w_
0x
N
w_0x
¤N
w_
xj
N
w
_
f
xj
¤N
w
_
f
N
w_;
¤N
w_
c
w_;
¤c
w_
1Rx
c
w_
0Rx
c
w_0Rx
¤c
w_1Rx
¤c
w_
{
{
1x
N
w_
1x
¤N
w_
{
0x
N
w_0x
¤N
w_
{
N
w_;
¤N
w_
0
1
"_1
"_
d
'd
C
'C
'c
c
1
"_
1
"_
o
45
A
'b
'a
a
b
B
'A'B
f˙wN,˙wc|R,x
R;xj
c
w
_
f
R;xj
¤c
w
_
f
c
w_;
¤c
w_
1Rx
¤c
w_
0Rx
¤c
w_
0Rx
c
w_1Rx
c
w_
ABB0A0WN c
Rx
acbb0c0a0
f˙wc|R,x
f˙wN|x
ρR
x(·,·)
˙wN,˙wcρR∗
x(·,·)
˙wN∗,˙wc∗x
R R
˙wN,˙wcρR
x˙wN,˙wc>0
˙wN,˙wc˙wN<˙wc
CRx ≡˙wN,˙wc∈ WNc
Rx : ˙wN<˙wc
f˙wN,˙wc|R,x
x R
f˙wN,˙wc|R,x 45
acc0a0cc0
add0c0a0dd0
ρR
x˙wN,˙wc%R
xCRx
45 ˙wN∗,˙wc∗
˙wN∗−˙ε1,˙wc∗−˙ε1 ˙wN∗+ ˙ε1,˙wc∗+ ˙ε1A=˙wN∗
x0,˙wc∗
Rx1
˙wN,˙wcaa0
AB
abb0a0WNc
Rx
C∗
Rx
˙wN∗<˙wc∗˙wN∗+ ˙ε < ˙wc∗+ ˙ε˙ε
˙wN<˙wcPr ˙wN<˙wc|R, x= Pr ˙wN∗<˙wc∗|R, x%R
x
R x
%R
x= Pr δ= 1|˙wN<˙wc, R, x
=Pr (δ= 1|R, x)
Pr ( ˙wN<˙wc|R, x)
( ˙w, x)x∈ X ˙w
x
%R
xR∈ Rxf˙w|x
f˙wN∗,˙wc∗|R,x
%R
x˙wN= ˙wN∗
˙wc= ˙wc∗
f˙w|xf˙wN|x
f˙w∗|xf˙wN∗|x
f˙w|x( ˙w|x) = X
ϑ∈R
Pr (R=ϑ|x)f˙w|R,x ( ˙w|ϑ, x)
f˙w|R,x
R∈ R f˙w|R,x
f˙w|R,x ( ˙w|R, x) = f˙wN|R,x ( ˙w|R, x) + LR
x( ˙w) + GR
x( ˙w)
LR
x( ˙w)≡ −f˙wN,δ|R,x ( ˙w, 1|R, x)≤0
GR
x( ˙w)≡f˙wc,δ|R,x ( ˙w, 1|R, x)≥0
LR
x( ˙w)GR
x( ˙w)
f˙w|R,x f˙wN|R,x ˙w
x R
LR
x(·)GR
x(·)CRx f˙wN,˙wc,δ|R,x
LRx GRx WN
Rx Wc
Rx
LRx ≡˙wN∈ WN
Rx :˙wN,˙wc∈ CRx
GRx ≡˙wc∈ Wc
Rx :˙wN,˙wc∈ CRx
LRx CRx WN
Rx GRx CRx Wc
Rx
LRx ( ˙w)GRx ( ˙w)LRx GRx
LRx ( ˙w)≡˙wN∈ LRx :˙wN,˙w∈ CRx
GRx ( ˙w)≡ { ˙wc∈ GRx : ( ˙w, ˙wc)∈ CRx}
LR
x(·)GR
x(·)
LR
x( ˙w) =
−´v∈GRx( ˙w)ρR
x( ˙w, v)f˙wN,˙wc|R,x ( ˙w, v|R, x)dv ≤0,˙w∈ LRx
0, o/w
GR
x( ˙w) =
´$∈LRx( ˙w)ρR
x($, ˙w)f˙wN,˙wc|R,x ($, ˙w|R, x)d$ ≤0,˙w∈ GRx
0, o/w
LR
x( ˙w)6= 0 ˙w∈ LRx GR
x( ˙w)6= 0 ˙w∈ GRx
CRx =∅ LRx =GRx =∅LR
x( ˙w) = GR
x( ˙w) = 0
Wc
Rx WN
Rx R /∈ RxLR
x( ˙w) = GR
x( ˙w) = 0
Wc
Rx ⊂ WN
Rx R∈ RxLR
x( ˙w)≥0GR
x( ˙w)≥0
LRx WN
Rx
f˙w|R,x GRx
f˙wN∗,˙wc∗,δ |R,x
˙wc
˙wN
f˙w|x( ˙w|x) = f˙wN|x( ˙w|x) + X
ϑ∈Rx
Pr (R=ϑ|x)Lϑ
x( ˙w) + Gϑ
x( ˙w)
f˙w|x( ˙w|x)−f˙wN|x( ˙w|x)
˙w
f˙w|x( ˙w|x)−f˙wN|x( ˙w|x) =
=X
ϑ∈Rx
Pr (R=ϑ|x)f˙w|R,x ( ˙w|R, x)−f˙wN|R,x ( ˙w|R, x)
=X
ϑ∈Rx
Pr (R=ϑ|x)Lϑ
x( ˙w) + Gϑ
x( ˙w)
f˙w|R,x ( ˙w|R, x)−f˙wN|R,x ( ˙w|R, x)f˙w|R,x
˙w f ˙w|x˙w
R∈ R f˙w|R,x
=LR
x( ˙w)
=GR
x( ˙w)f˙w|R,x R∈ Rx
R∈ R x
Pr (R|x)
LR
x(·)GR
x(·)
LR
x(·)GR
x(·)
R∈ R \ Rxf˙w|R,x
Wc
Rx WN
Rx LRx =GRx =∅LR
x( ˙w) = GR
x( ˙w) = 0
Wc
Rx ⊂ WN
Rx ˙wc˙wc∈ Wc
Rx = [ ˙wc
Rx0,˙wc
Rx1]
LR
x( ˙w) =
−%R
x´v∈GRx( ˙w)f˙wN,˙wc|R,x ( ˙w, v|R, x)dv , ˙w∈ LRx
0, o/w
GR
x( ˙w) =
%R
x´$∈LRx( ˙w)f˙wN,˙wc|R,x ($, ˙w|R, x)d$ , ˙w∈ GRx
0, o/w
LRx =˙wN
x0,˙wc
Rx1GRx = [ ˙wc
Rx0,˙wc
Rx1]LRx ∩GRx = [ ˙wc
Rx0,˙wc
Rx1)
Wc
Rx ⊂ WN
Rx ˙wc˙wc∈ Wc
Rx ={˙wc
Rx1}
LR
x( ˙w) =
−%R
xf˙wN|x( ˙w|x),˙w∈ LRx
0, o/w
GR
x( ˙w) =
%R
xF˙wN|x( ˙wRx1|x),˙w∈ GRx
0, o/w
LRx =˙wN
x0,˙wc
Rx1GRx ={˙wc
Rx1} LRx ∩ GRx =∅
Wc
Rx ⊂ WN
Rx %R
x>0LR
x( ˙w)WN
Rx
˙wc
Rx1
˙wN= ˙w
GR
x( ˙w)Wc
Rx
˙wc= ˙w
LR
x( ˙w)GR
x( ˙w)
˙wc
c
w_;
N
w_
¯
°
®
0Rx
c
w_1Rx
c
w_
'
w_
R;xjw
_
f
1Rx
c
w_N
w_
R;xj
N
w
_
f
0x
N
w_1x
N
w_
1x
N
w_
0x
N
w_0Rx
c
w_1Rx
c
w_
'C
'A'B
C
1x
N
w_
0x
N
w_
BA
N
w_
o
45
0
c
w_
'
w_
)
'
w_(
x
R
L
)
'
w_(
x
R
G
0Rx
c
w_
LR
x(·)GR
x(·)f˙w|R,x ˙wc
1x
N
w_
0x
N
w_
BA
N
w_
o
45
0
w_;
N
w_
C
c
w_
1x
N
w_
0x
N
w_N
w_
R;xj
N
w
_
f
0x
N
w_1x
N
w_
R;xjw
_
f
1Rx
c
w_
1Rx
c
w_
1Rx
c
w_
)
'
w_(
x
R
L
)
1Rx
c
w_(
x
R
G
'
w_
'
w_
®
c
w_;
N
w_
LR
x(·)GR
x(·)f˙w|R,x ˙wc
WN
Rx =˙wN
x0,˙wN
x1Wc
Rx = [ ˙wc
Rx0,˙wc
Rx1] ˙wN
x0<˙wc
Rx0<˙wc
Rx1<˙wN
x1
WNc
Rx =WN
Rx × Wc
Rx AA0B0B
LR
x(·) ˙w∈ LRx =˙wN
x0,˙wc
Rx1
f˙wN|R,x GR
x(·)
˙w∈ GRx = [ ˙wc
Rx0,˙wc
Rx1]
f˙wN|R,x
Pr (δ= 1|R, x)
f˙w|R,x
WN
Rx =˙wN
x0,˙wN
x1Wc
Rx ={˙wc
Rx1}˙wN
x0<˙wc
Rx1<˙wN
x1
WNc
Rx =WN
Rx × Wc
Rx AB
LR
x(·) ˙w∈ LRx =˙wN
x0,˙wc
Rx1
f˙wN|R,x GR
x( ˙wc
Rx1)
f˙wN|R,x ˙wc= ˙wc
Rx1
f˙w|R,x
˙wc∗Wc∗
Rx ⊂ WN∗
Rx
˙wc
Wc
Rx ⊂ WN
Rx ˙wc∗Wc∗
Rx ⊂ WN∗
Rx
˙wc∗Wc∗
Rx ⊂ WN∗
Rx
˙w0GRx LRx GRx
GRx ˙w0=˙w0,˙wc
Rx1iβγ β
LR
x˙w0f˙wN,˙wc|R,x ˙w0,·|R, x
−%R
x
˙w0LRx ˙w0=h˙wN
x0,˙w0
αβ β GR
x˙w0
f˙wN,˙wc|R,x ·,˙w0|R, x%R
x
˙wc∗Wc∗
Rx ⊂ WN∗
Rx
˙w0LRx GRx ˙w0={˙wc
Rx1}
α LR
x˙w0f˙wN|R,x ˙w0|R, x−%R
x
LRx ( ˙wc
Rx1) = ˙wN
x0,˙wc
Rx1AC C
GR
x( ˙wc
Rx1)
f˙wN|R,x (·|R, x)%R
x
f˙wN|R,x
˙wcf˙w|R,x
˙wc
Rx0
LRx\GRx
GRx = [ ˙wc
Rx0,˙wc
Rx1]
f˙w|R,x ˙wc
f˙w|R,x
˙wc
Rx0= ˙wc
Rx1LRx
˙wc
Rx1
f˙w|R,x
WRx f˙w|R,x
Wc
Rx WN
Rx WRx WN
x=˙wN
x0,˙wN
x1
Wc
Rx ⊂ WN
Rx ˙wcWRx
WN
x=˙wN
x0,˙wN
x1%R
x<1WN
x\˙wN
x0,˙wc
Rx1=˙wN
Rx0,˙wN
x1%R
x= 1
f˙w|x
f˙w|R,x R∈ R Pr (R|x)
f˙wN|x
f˙wN|R,x R∈ R
R
f˙w|x
Wxf˙w|x
LRx ∪
GRx LRx GRx
f˙wN,˙wc|R,x
f˙wN|R,x f˙wN|x
Rx⊂ R WxWN
x
Rx=R WxWN
x%R
x<1R∈ R
%R
xLϑ
x(·)
Gϑ
x(·)%R
x
xRxRx
RxR∈ Rx˙wc∗
˙wc∗
Rx1˙wc∗
Rx16= ˙wc∗
R0x1R0∈ RxR06=R fε
f˙w|x˙wc∗
Rx1
R∈ Rx
Pr ( ˙w= ˙wc∗
Rx1|x) = %RR
xF˙wN|x( ˙wc∗
Rx1|x)
%RR
x≡Pr (R|x)%R
x
R
˙wc∗
Rx1
f˙wN∗|x
R
%R
x%RR
x
x
f˙w|R,x
%R
x
RPr (R|x)
˙wc∗
Rx1Pr ( ˙w= ˙wc∗
Rx1|x)
F˙wN|x( ˙wc∗
Rx1|x)
%R
x%RR
x
xRx
˙w∈ Wx
F˙w|x( ˙w|x)−F˙wN|x( ˙w|x) = −X
ϑ∈Rx
%ϑϑ
xcϑ
x( ˙w)
cR
x( ˙w)≡¨
($,v)∈CRx :$< ˙w,v> ˙w
f˙wN,˙wc|R,x ($, v|R, x)dvd$ ≥0
f˙wN,˙wc|R,x CRx
˙w
Wc
Rx ⊂ WN
Rx ˙wccR
x( ˙w)
cR
x( ˙w) =
F˙wN|x( ˙w|x),˙wN
x0≤˙w < ˙wc
Rx0
´˙w
˙wN
x0´max GRx($)
˙wf˙wN,˙wc|R,x ($, v|R, x)dvd$ , ˙wc
Rx0≤˙w < ˙wc
Rx1
0,˙wc
Rx1≤˙w≤˙wN
x1
min LRx ( ˙w)LRx ( ˙w) max GRx ( ˙w)
GRx ( ˙w)
Wc
Rx ⊂ WN
Rx ˙wccR
x( ˙w)
cR
x( ˙w) =
F˙wN|x( ˙w|x),˙wN
x0≤˙w < ˙wc
Rx1
0,˙wc
Rx1≤˙w≤˙wN
x1
˙wc
Rx1R∈ Rx
%RR
xR∈ Rx
cR
x˙w0
R˙w0<˙wc
Rx1
˙w0Wx
%RR
x
Kx≥0RxKx>1
Rx={Rk:k= 1, . . . , Kx} {Rk−1, Rk}⊆Rxk= 2, . . . , Kx
˙wc
Rk−1x1≤˙wc
Rkx1Kx= 1 Rx={R1}
x Kx≥1Rx={R1, . . . , RKx}
R∈ Rx⇔ Wc
Rx ⊂ WN
Rx Kx>1k= 2, . . . , Kx
{Rk−1, Rk}⊆Rx˙wc
Rkx16= ˙wc
Rk−1x1
Wx=WN
x˙w0
1,..., ˙w0
Kx˙wx0<˙w0
1<˙wc
R1x1˙wc
Rk−1x1<
˙w0
k<˙wc
Rkx12≤k≤Kxf˙w|x
kWc
Rx R∈ Rx
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−
Kx
X
j=1
%RjRj
xcRj
x˙w0
1
F˙w|x˙w0
2|x−F˙wN|x˙w0
2|x=−
Kx
X
j=2
%RjRj
xcRj
x˙w0
2
. . .
F˙w|x˙w0
Kx|x−F˙wN|x˙w0
Kx|x=−%RKxRKx
xcRKx
x˙w0
Kx
%RR
xR∈ Rx
cR
x(·)F˙w|x(·)F˙wN|x(·)
˙w0
1,..., ˙w0
K
%RR
xR∈ Rx
Wc
Rx R∈ Rx˙w0
1,..., ˙w0
Kx
%RR
xR∈ Rx
F˙w|x(·)F˙wN|x(·) ˙w0
1,..., ˙w0
K
x Kx≥1Rx={R1, . . . , RKx}
R∈ Rx⇔ Wc
Rx ⊂ WN
Rx Kx>1k= 2, . . . , Kx{Rk−1, Rk} ⊆
Rx˙wc
Rk−1x1<˙wc
Rkx0Wc
Rk−1x∩ Wc
Rkx=∅
˙w0
1,..., ˙w0
Kx˙wx0<˙w0
1<˙wc
R1x0˙wc
Rk−1x1<˙w0
k<˙wc
Rkx0
2≤k≤Kxf˙w|x
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=− Kx
X
j=1
%RjRj
x!F˙wN|x˙w0
1|x
F˙w|x˙w0
2|x−F˙wN|x˙w0
2|x=− Kx
X
j=2
%RjRj
x!F˙wN|x˙w0
2|x
. . .
F˙w|x˙w0
Kx|x−F˙wN|x˙w0
Kx|x=−%RKxRKx
xF˙wN|x˙w0
Kx|x
F˙w|xF˙wN|x
˙w0
1,..., ˙w0
K%RR
xR∈ Rx
˙ε1
˙ε1
˙ε1
˙ε1
( ˙w, x)Wc
Rx
˙w0
1,..., ˙w0
Kx
Wc
Rx
˙wc
Rx0
˙wc
Rx1
Wc
Rx
˙wc
Rx0˙wc
Rx1
Wc
Rx
f˙wN|x
f˙wN∗|xf˙wN|x
f˙w|x
f˙wN∗|xf˙wN|x
F˙wN|x( ˙w, x)
xRx˙wc
RKxx1< mxmx
f˙w|x
˙w < ˙wc
RKxx1
F˙wN|x( ˙w|x) = 1 −F˙w|x(2mx−˙w|x)
f˙wN∗|xx
%R
x
Pr (R|x)%RR
x
R x
R
R %R
xR x
Pr (R|x)
%R
x
Pr (R|x)%RR
x
xRx%RR
x
R∈ Rx
%R
x∈"%RR
x
1−Pϑ∈Rx\{R}%ϑϑ
x
,1#, R ∈ Rx
Pr (R|x)∈
h%RR
x,1−Pϑ∈Rx\{R}%ϑϑ
xi, R ∈ Rx
h0,1−Pϑ∈Rx\{R}%ϑϑ
xi, R ∈ R \ Rx
%RR
xR∈ Rx
%R
xRRx
Pr (R|x)RRx
RRx
x
R=n R =r
R=fR={n, r, f}
R=n˙wc∗f˙wc∗|˙wN∗,R=n,x
Wc∗
Rx =Wc∗
nx ={0}
R=r˙wc∗˙
Pe∗
f˙wc∗|˙wN∗,R=r,x f˙
Pe∗|˙wN∗,R=r,x
Wc∗
Rx =Wc∗
rx =h˙
Pe∗
x0,˙
Pe∗
x1i
R=f˙wc∗−∞
f˙wc∗|˙wN∗,R=f,x
x n r
Rx
Rx
X
˙wN∗=˙
Pe∗+ ˙τ˙τ
x∈ X H
0<˙wN∗
x0<˙
Pe∗
x0≤˙
Pe∗
x1<˙wN∗
x1
Rx={r}
x∈ X M
˙wN∗
x0<0<˙
Pe∗
x0≤˙
Pe∗
x1<˙wN∗
x1
Rx={n, r}
x∈ X L
˙wN∗
x0<˙
Pe∗
x0<0<˙
Pe∗
x1<˙wN∗
x1
Rx={n, r}
r∈ Rx
x∈ X H∪ X M∪ X L
%r
x≡Pr (δ= 1|R=r, x)
Pr ˙wN∗<˙
Pe∗|R=r, x
˙wN∗
h˙
Pe∗
x0,˙
Pe∗
x1i⊂˙wN∗
x0,˙wN∗
x1˙
Pe∗˙τ
Wc∗
nx ={0} WN∗
nx =˙wN∗
x0,˙wN∗
x1Wc∗
rx =h˙
Pe∗
x0,˙
Pe∗
x1i⊂ WN∗
nx
Wc∗
nx ⊂ WN∗
nx
Wc∗
rx ⊂ WN∗
rx
x
x∈ X M∪ X L
%n
x≡Pr (δ= 1|R=n, x)
Pr ( ˙wN∗<0|R=n, x)
x
x
¤N
w
|
1
'
w_0 0x
¤N
w_0x
¤e
P
_
1x
¤e
P
_
1x
¤N
w_
xj
N
w
_
f
xjw
_
f
¤N
w_=
N
w_
¤
w_=w_
1
'
w_
xj
N
w
_
f
xjw
_
f
w_;
N
w_
0x
N
w_0x
e
P
_
1x
e
P
_
1x
N
w_
x
¤N
w
|
x∈ X Hf˙ε˙w0
1∈
( ˙wx0,˙wc
rx0) = ˙wN∗
x0,˙
Pe∗
x0
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−%rr
xF˙wN|x˙w0
1|x
x∈ X Hf˙ε
˙w0
1∈( ˙wx0,˙wc
rx0) = ˙wN
x0,˙
Pe
x0Wc
rx =h˙
Pe
x0,˙
Pe
x1i=h˙
Pe∗
x0−˙ε1,˙
Pe∗
x1+ ˙ε1i
f˙
Pe∗|˙wN∗,R=r,x f˙
Pe∗|R=r,x
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−%rr
xF˙wN|x˙w0
1|x
x
¤N
w
|
¤
w_=w_
¤N
w_=
N
w_
2
'
w_
1
'
w_1x
¤e
P
_
1x
¤N
w_
0x
¤e
P
_
0x
¤N
w_ 0
xj
N
w
_
f
xjw
_
f
x
¤N
w
|
1x
N
w_
0x
e
P
_
1x
e
P
_
0x
N
w_1
"_{ 1
"_0
1
'
w_2
'
w_
w_;
N
w_
xjw
_
f
xj
N
w
_
f
x∈ X Mf˙ε˙w0
1∈
( ˙wx0,˙wc
nx0) = ˙wN∗
x0,0˙w0
2∈( ˙wc
nx1,˙wc
rx0) = 0,˙
Pe∗
x0
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−(%nn
x+%rr
x)F˙wN|x˙w0
1|x
F˙w|x˙w0
2|x−F˙wN|x˙w0
2|x=−%rr
xF˙wN|x˙w0
2|x
Pr ( ˙w= 0|x) = %nn
xF˙wN|x(0|x)
x∈ X Mf˙ε
˙w0
1∈( ˙wx0,˙wc
nx0) = ˙wN
x0,−˙ε1˙w0
2∈( ˙wc
nx1,˙wc
rx0) = ˙ε1,˙
Pe
x0
˙
Pe
x0>˙ε1⇔˙
Pe∗
x0>2 ˙ε1
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−(%nn
x+%rr
x)F˙wN|x˙w0
1|x
F˙w|x˙w0
2|x−F˙wN|x˙w0
2|x=−%rr
xF˙wN|x˙w0
2|x
x∈ X Lf˙ε˙w0
1∈
( ˙wx0,˙wc
rx0) = ˙wN∗
x0,˙
Pe∗
x0
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−(%nn
x+%rr
x)F˙wN|x˙w0
1|x
Pr ( ˙w= 0|x) = %nn
xF˙wN|x(0|x)
x
¤N
w
|
¤
w_=w_
¤N
w_=
N
w_
0
0x
¤e
P
_
1x
¤N
w_
0x
¤N
w_1x
¤e
P
_
xj
N
w_
f
xjw_
f
1
'
w_
0
0x
e
P
_
xjw_
f
xj
N
w_
f
1
"_{ 1
"_
0x
N
w_1x
e
P
_
1x
N
w_w_;
N
w_
1
'
w_x
¤N
w
|
x∈ X Lf˙ε
˙w0
1∈( ˙wx0,˙wc
rx0) = ˙wN
x0,˙
Pe
x0
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−(%nn
x+%rr
x)F˙wN|x˙w0
1|x
˙
Pe∗
x0
˙
Pe∗
x1
˙
Pe∗
x0˙
Pe∗
x1
%R
x= 1 %RR
x
Pr (R|x)
Pr (R=n|x)%nn
x
Pr (R=r|x)
%rr
x
F˙wN|xPe∗
rx|x−F˙w|xPe∗
rx|x=%r r
x
F˙wN|xPe∗
rx|x
2
˙w0=Pe∗
rx ≡E˙
Pe∗|R=r, x
˙wN∗˙
Pe∗R x
f˙
Pe∗|R=r,x
˙ε1<˙
Pe
x0⇔˙ε1>˙
Pe∗
x0
2
%nn
x
%rr
x˙ε1< P e∗
rx
Pr ( ˙w= 0|x)f˙wN|xPr ( ˙w∗= 0|x)f˙wN∗|x
%nn
x=Pr( ˙w=0|x)
F˙wN|x(0|x)= Pr (R=n|x)
f˙w|xf˙wN|xf˙w∗|xf˙wN∗|x
%rr
x=2[F˙wN|x(Pe∗
rx|x)−F˙w|x(Pe∗
rx|x)]
F˙wN|x(Pe∗
rx|x)= Pr (R=r|x)
Pe∗
rx R=r x
Pe∗
rx =E˙
Pe|R=r, x
˙ε1˙ε1>˙
Pe
x0⇔˙ε1>˙
Pe∗
x0
2
%R
x
%R
x=Pr δ= 1,˙wN∗<˙wc∗|R, x
Pr ( ˙wN∗<˙wc∗|R, x)
Pr δ= 1,˙wN∗<˙wc∗|R, x=¨
($,v):$<v
f˙wN∗,˙wc∗,δ|R,x ($, v, 1|R, x)dvd$
=¨
($,v):$<v
ρR
x($, v)×
×f˙wN∗,˙wc∗|R,x ($, v|R, x)dvd$
Pr δ= 1,˙wN∗<˙wc∗|R, x=¨
($,v):$<v
%R
xf˙wN∗,˙wc∗|R,x ($, v|R, x)dvd$
=%R
xPr ˙wN∗<˙wc∗|R, x
˙ε
˙w∗+ ˙ε=δ·˙wc∗+ (1 −δ) ˙wN∗+ ˙ε=δ·( ˙wc∗+ ˙ε) + (1 −δ)˙wN∗+ ˙ε
˙w=δ·˙wc+ (1 −δ) ˙wN
˙wc∗6= ˙wN∗˙wc∗+ ˙ε6= ˙wN∗+ ˙ε⇔˙wc6= ˙wN
˙wNR x ˙wN
˙ε f ˙wN|R,x ( ˙w|R, x) =
=´˙ε1
−˙ε1f˙wN∗|R,x ( ˙w−|R, x)f˙ε()d
f˙wN|R,x ( ˙w|R, x) = ˆ˙ε1
−˙ε1
f˙wN∗|x( ˙w−|x)f˙ε()d
=f˙wN|x( ˙w|x)
f˙wN|xf˙wN∗|xf˙ε
f˙wN|x
f˙wN|xf˙wN|xwN∗
x−c|x=f˙wN|xwN∗
x+c|x
wN∗
x≡E˙wN∗|xwN∗
x≡E˙wN|x
f˙wN|xwN∗
x−c|x=ˆ˙ε1
−˙ε1
f˙wN∗|xwN∗
x−c−|xf˙ε()d
f˙wN∗|xwN∗
x−c−|x=f˙wN∗|xwN∗
x+c+|x
f˙wN∗|xwN∗
xf˙ε() = f˙ε(−)f˙ε0
f˙wN|xwN∗
x−c|x=ˆ˙ε1
−˙ε1
f˙wN∗|xwN∗
x+c+|xf˙ε(−)d
=f˙wN|xwN∗
x+c|x
f˙wN|x˙wN≡˙wN∗+ ˙ε˙wN∗∈˙wN∗
x0,˙wN∗
x1
˙wN
x0≡min ˙wN|x= min ˙wN∗+ ˙ε|x= min ˙wN∗|x+ min ( ˙ε|x) =
˙wN∗
x0−˙ε1˙wN
x1≡max ˙wN|x= max ˙wN∗+ ˙ε|x= max ˙wN∗|x+ max ( ˙ε|x) =
˙wN∗
x1+ ˙ε1
f˙wN|xE˙wN|x=E˙wN∗+ ˙ε|x=E˙wN∗|x+E( ˙ε|x) = E˙wN∗|x
f˙ε
f˙wN|xV ar ˙wN|x=V ar ˙wN∗+ ˙ε|x=V ar ˙wN∗|x+V ar ( ˙ε|x)+
2Cov ˙wN∗,˙ε|x=V ar ˙wN∗|x+V ar ( ˙ε) ˙ε
˙wc= ˙wc∗
Rx1+ ˙ε
˙ε
f˙wc|R,x f˙ε
f˙wc|R,x ˙wc≡˙wc∗+ ˙ε˙wc∗∈[ ˙wc∗
Rx0,˙wc∗
Rx1]
˙wc
Rx0≡min ( ˙wc|R, x) = min ( ˙wc∗
Rx1+ ˙ε|R, x) = min ( ˙wc∗
Rx1|R, x) +
min ( ˙ε|R, x) = ˙wc∗
Rx1−˙ε1˙wc
x1≡max ( ˙wc|R, x) = max ( ˙wc∗
Rx1+ ˙ε|R, x) =
max ( ˙wc∗
Rx1|R, x) + max ( ˙ε|R, x) = ˙wc∗
Rx1+ ˙ε1
f˙wc|R,x E( ˙wc|R, x) = E( ˙wc∗
Rx1+ ˙ε|R, x) = ˙wc∗
Rx1+E( ˙ε|R, x) =
˙wc∗
Rx1+E( ˙ε) = ˙wc∗
Rx1f˙ε
f˙wc|R,x V ar ( ˙wc|R, x) = V ar ( ˙wc∗
Rx1+ ˙ε|R, x) = V ar ( ˙ε)
˙wc˙wN
˙ε
f˙wc|˙wN,R,x f˙wc∗|˙wN∗,R,x f˙ε
f˙wc|˙wN,R,x
f˙wc|˙wN,R,x ˙wc≡˙wc∗+ ˙ε˙wc∗∈[ ˙wc∗
Rx0,˙wc∗
Rx1]
˙wc
Rx0≡min ( ˙wc|R, x) = min ( ˙wc∗+ ˙ε|R, x) = min ( ˙wc∗|R, x) +
min ( ˙ε|R, x) = ˙wc∗
Rx0−˙ε1˙wc
x1≡max ( ˙wc|R, x) = max ( ˙wc∗+ ˙ε|R, x) =
max ( ˙wc∗|R, x) + max ( ˙ε|R, x) = ˙wc∗
Rx1+ ˙ε1
f˙wc|˙wN,R,x E˙wc|˙wN, R, x=E˙wc∗+ ˙ε|˙wN, R, x=
=E˙wc∗|˙wN, R, x+E˙ε|˙wN, R, x=E˙wc∗|˙wN, R, x+E( ˙ε) = E˙wc∗|˙wN, R, x
f˙ε
f˙wc|˙wN,R,x V ar ˙wc|˙wN, R, x=V ar ˙wc∗+ ˙ε|˙wN, R, x=
=V ar ˙wc∗|˙wN, R, x+V ar ˙ε|˙wN, R, x+2Cov ˙wc∗,˙ε|˙wN, R, x=V ar ˙wc∗|˙wN, R, x+
V ar ( ˙ε) ˙ε
ρR
x˙wN,˙wc
˙wN˙wc˙ε
ρR
x˙wN,˙wc= Pr δ= 1|˙wN,˙wc, R, x
=ˆ˙ε1
−˙ε1
Pr δ= 1|˙wN∗= ˙wN−, ˙wc∗= ˙wc−, R, xf˙ε()d
=ˆ˙ε1
−˙ε1
ρR∗
x˙wN−, ˙wc−f˙ε()d
ρR∗
x˙wN−, ˙wc−=
%R
x,˙wN− < ˙wc−
0, o/w
ρR∗
x˙wN−, ˙wc−=
%R
x,˙wN<˙wc
0, o/w
ρR∗
x˙wN−, ˙wc−˙wN
˙wc
R x
˙wN∗<˙wc∗R
˙wN∗+ ˙ε < ˙wc∗+ ˙ε⇔˙wN<˙wc
%R
x=Pr(δ=1,˙wN∗<˙wc∗|R,x)
Pr( ˙wN∗<˙wc∗|R,x)
Pr δ= 1,˙wN∗<˙wc∗|R, x= Pr δ= 1,˙wN∗+ ˙ε < ˙wc∗+ ˙ε|R, x=
= Pr δ= 1,˙wN<˙wc|R, xPr ˙wN∗<˙wc∗|R, x= Pr ˙wN∗+ ˙ε < ˙wc∗+ ˙ε|R, x=
= Pr ˙wN<˙wc|R, x%R
x=Pr(δ=1,˙wN<˙wc|R,x)
Pr( ˙wN<˙wc|R,x)= Pr δ= 1|˙wN<˙wc, R, x
Pr δ= 1,˙wN<˙wc|R, x= Pr (δ= 1|R, x)R x
δ= 1
˙wN<˙wc
x R ˙w
˙w
˙wN= ˙w δ = 0 ˙w
˙wc= ˙w δ = 1 f˙w|R,x
f˙w|R,x ( ˙w|R, x) = f˙wN,δ|R,x ( ˙w, 0|R, x) + f˙wc,δ|R,x ( ˙w, 1|R, x)
=f˙wN|R,x ( ˙w|R, x)−f˙wN,δ|R,x ( ˙w, 1|R, x)+
+f˙wc,δ|R,x ( ˙w, 1|R, x)
LR
x(·)GR
x(·)
LR
x(·)
LR
x( ˙w) = −ˆv∈Wc
Rx
f˙wN,˙wc,δ|R,x ( ˙w, v, 1|R, x)dv
=−ˆv∈Wc
Rx
ρR
x( ˙w, v)f˙wN,˙wc|R,x ( ˙w, v|R, x)dv
ρR
x˙wN,˙wc≡Pr δ= 1|˙wN,˙wc, R, x
CRx ρR
x( ˙w, v) ( ˙w, v)∈ CRx LRx
GRx ( ˙w) ( ˙w, v)∈ CRx ˙w
LRx vGRx ( ˙w)
GR
x(·)
GR
x( ˙w) = ˆ$∈WN
Rx
f˙wN,˙wc,δ|R,x ($, ˙w, 1|R, x)d$
=ˆ$∈WN
Rx
ρR
x($, ˙w)f˙wN,˙wc|R,x ($, ˙w|R, x)d$
CRx ρR
x($, ˙w) ($, ˙w)∈ CRx
GRx LRx ( ˙w) ($, ˙w)∈
CRx ˙wGRx $LRx ( ˙w)
RxR /∈ RxCRx =∅
LRx =GRx =∅
Lϑ
x(·)Gϑ
x(·)
f˙wN|x( ˙w|x) = X
ϑ∈R
Pr (R=ϑ|x)f˙wN|R,x ( ˙w|ϑ, x)
f˙w|x( ˙w|x) = f˙wN|x( ˙w|x) + X
ϑ∈R
Pr (R=ϑ|x)Lϑ
x( ˙w) + Gϑ
x( ˙w)
LR
x(·)GR
x(·)R /∈ Rx
f˙w|xf˙wN|xf˙w|R,x
f˙wN|R,x
Wc
Rx WN
Rx CRx =∅
ρR
x( ˙w, v)
( ˙w, v)∈ CRx ˙w < v
ρR
x( ˙w, v)%R
x
ρR
x($, ˙w)
($, ˙w)∈ CRx $ < ˙w ρR
x($, ˙w)
%R
x
LRx GRx
CRx add0c0a0
dd0CRx WN
Rx =˙wN
x0,˙wN
x1LRx
˙wN
x0,˙wc
Rx1CRx Wc
Rx = [ ˙wc
Rx0,˙wc
Rx1]GRx
[ ˙wc
Rx0,˙wc
Rx1]
˙wcf˙wN,˙wc|R,x
f˙wN|R,x f˙wN|x
GR
x( ˙w) =
%R
x´$∈LRx( ˙w)f˙wN|x($|R, x)d$ , ˙w∈ GRx
0, o/w
GRx ={˙wc
Rx1} LRx =˙wN
x0,˙wc
Rx1˙w∈ GRx ⇔
˙w= ˙wc
Rx1LRx ( ˙w) = LRx ( ˙wc
Rx1) = ˙wN
x0,˙wc
Rx1
f˙wN|x˙w
F˙wN|x( ˙wRx1|x)
LRx GRx
CRx AC
CCRx WN
Rx =˙wN
x0,˙wN
x1LRx
˙wN
x0,˙wc
Rx1CRx Wc
Rx ={˙wc
Rx1} GRx {˙wc
Rx1}
˙wc∗fε
f˙w|R,x f˙wN|R,x
WRx =WN
Rx
f˙wN|R,x f˙wN|xWN
xWRx =WN
x
˙wc∗fε
˙wcfε˙wcCRx
ACC 0A0C C0˙wc∗
fε˙wcCRx
acc0a0cc0
LRx GRx
f˙w|R,x ( ˙w|R, x) =
f˙wN|R,x ( ˙w|R, x) + LR
x( ˙w),˙w∈ LRx\GRx
f˙wN|R,x ( ˙w|R, x) + GR
x( ˙w),˙w∈ GRx\LRx
f˙wN|R,x ( ˙w|R, x) + LR
x( ˙w) + GR
x( ˙w),˙w∈ LRx ∩ GRx
f˙wN|R,x ( ˙w|R, x), o/w
f˙wN|R,x
LRx GRx
˙wc˙wc
Rx0= ˙wc
Rx1
f˙w|R,x ( ˙w|R, x) =
f˙wN|x( ˙w|x) + LR
x( ˙w),˙wN
x0≤˙w < ˙wc
Rx0
f˙wN|x( ˙w|x) + LR
x( ˙w) + GR
x( ˙w),˙wc
Rx0≤˙w < ˙wc
Rx1
f˙wN|x( ˙w|x) + GR
x( ˙w),˙w={˙wc
Rx1}
f˙wN|x( ˙w|x),˙wc
Rx1<˙w≤˙wN
x1
WRx ⊆ WN
x=˙wN
x0,˙wN
x1˙wN
x0,˙wc
Rx0⊂ WRx 0≤%R
x<1
˙wN
x0,˙wc
Rx0∩ WRx =∅%R
x= 1 LR
x( ˙w) = −f˙wN|x( ˙w|x) ˙w∈
˙wN
x0,˙wc
Rx0˙wc[ ˙wc
Rx0,˙wc
Rx1)⊂ WN
x
˙w∈[ ˙wc
Rx0,˙wc
Rx1)LR
x( ˙w)< f ˙wN|x( ˙w|x)GR
x( ˙w)≥0
˙wc
Rx1,˙wN
x1⊂ WRx GR
x( ˙w)≥0
f˙wN|x( ˙w|x)>0˙wc
Rx0,˙wN
x1⊂ WN
x
%R
x= 1 WRx =WN
x\˙wN
x0,˙wc
Rx0=˙wN
Rx0,˙wN
x1
WRx =WN
x=˙wN
x0,˙wN
x1
˙wc
Wx=SR∈R WRx Pr (R|x)6= 0 R∈ R
WRx ⊆ WN
xR∈ R
Wx⊆ WN
xWx=WN
xWRx ⊂ WN
xR∈ R
Rx⊂ R f˙w|R,x R∈ R\RxWRx =WN
x
R∈ R\RxWx=WN
x
R∈ Rx⇔ Wc
Rx ⊂ WN
Rx WRx =WN
Rx
%R
x<1Wx=SR∈R WRx WxWN
x%R
x<1
R∈ R
fε˙w= ˙wc∗
Rx1R∈ Rx
LR
x( ˙w)GR
x( ˙w)f˙w|R,x
R∈ Rx˙w= ˙wc∗
Rx1
Pr ( ˙w= ˙wc∗
Rx1|R, x) = GR
x( ˙wc∗
Rx1) = %R
xF˙wN|x( ˙wc∗
Rx1|x)%R
x=%R
x
Pr ˙wN= ˙wc∗
Rx1|R, x= Pr ˙wN= ˙wc∗
Rx1|x= 0 f˙wN|xLR
x( ˙wc∗
Rx1) =
0 Pr ( ˙w= ˙wc∗
Rx1|R, x) = 0 R∈ R\ RxLR
x(·)
GR
x(·) ˙w
f˙w|x
˙wc∗
Rx1R∈ Rx
Pr ( ˙w= ˙wc∗
Rx1|x) = Pr (R|x)%R
xF˙wN|x( ˙wc∗
Rx1|x)
=%RR
xF˙wN|x( ˙wc∗
Rx1|x)
%RR
x≡Pr (R|x)%R
x
F˙w|x( ˙w|x)−F˙wN|x( ˙w|x) =
=ˆ˙w
wN
x0(X
ϑ∈Rx
Pr (R=ϑ|x)Lϑ
x($) + Gϑ
x($))d$
=X
ϑ∈Rx
Pr (R=ϑ|x)"ˆ˙w
wN
x0
Lϑ
x($)d$ +ˆ˙w
wN
x0
Gϑ
x(v)dv#
wN
x0WN
x
wx0Wx
ˆ˙w
wN
x0
Lϑ
x($)d$ =−%R
xAR( ˙w)
AR
x( ˙w) = ˆmin(˙w, ˙wc
R1x1)
wN
x0ˆ
v∈GRx($)
f˙wN,˙wc|R,x ($, v|R, x)dvd$
f˙wN,˙wc|R,x CRx
˙w Lϑ
x($) = 0 $ > max LRx = ˙wc
R1x1
ˆ˙w
wN
x0
Gϑ
x(v)dv =%R
xBR( ˙w)
BR
x( ˙w) =
0,˙w < min GRx
´min(˙w, ˙wc
R1x1)
wN
x0´$∈LRx(v)f˙wN,˙wc|R,x ($, v|R, x)d$dv , ˙w≥min GRx
f˙wN,˙wc|R,x CRx
˙w Gϑ
x(v)=0 v < min GRx min GRx = ˙wc
R1x0
˙wcmin GRx = ˙wc
R1x1˙wc
Gϑ
x(v) = 0 v > max GRx max GRx = ˙wc
R1x1˙wc
cR
x˙w0≡AR
x( ˙w)−BR
x( ˙w)
%R
x=%R
x
F˙w|x( ˙w|x)−F˙wN|x( ˙w|x) = X
ϑ∈Rx
Pr (R=ϑ|x)−%R
xcR
x( ˙w)
=−X
ϑ∈Rx
Pr (R=ϑ|x)%R
xcR
x( ˙w)
=−X
ϑ∈Rx
%ϑϑ
xcϑ
x( ˙w)
%ϑϑ
x≡Pr (R=ϑ|x)%ϑ
xcR
x( ˙w)
f˙wN,˙wc|R,x
CRx ˙w
˙wc∈ Wc
Rx = [ ˙wc
Rx0,˙wc
Rx1] ˙wN
x0<˙wc
Rx0<˙wc
Rx1<˙wN
x1
˙w≥˙wc
Rx1˙wN,˙wc∈ CRx : ˙wN<˙w, ˙wc>˙w
˙wc≤˙wc
Rx1
˙w˙wN
x0≤˙w < ˙wc
Rx1
%R
x>0
cR
x( ˙w) = ˆ˙w
min LRx ˆmax GRx($)
max( ˙w,min GRx($))
f˙wN,˙wc|R,x ($, v|R, x)dvd$
=ˆ˙w
˙wN
x0ˆmax GRx($)
max( ˙w,min GRx($))
f˙wN,˙wc|R,x ($, v|R, x)dvd$
min LRx = ˙wN
x0
˙w˙wN
x0≤˙w < ˙wc
Rx0
max ( ˙w, min GRx ($)) = min GRx ($) min GRx ($)≥min GRx =
˙wc
Rxo
cR
x( ˙w) = ˆ˙w
˙wN
x0ˆmax GRx($)
min GRx($)
f˙wN,˙wc|R,x ($, v|R, x)dvd$
=ˆ˙w
˙wN
x0ˆv∈GRx($)
f˙wN,˙wc|R,x ($, v|R, x)dvd$
CRx ˙w
CRx ˙w
CRx ˙wc
˙wc∗˙wc∗
WNc
Rx ˙w
˙w˙wN
x0≤˙w < ˙wc
Rx0
cR
x( ˙w) = ´$< ˙wf˙wN|R,x ($|R, x)d$ =F˙wN|R,x ( ˙w|R, x)
F˙wN|R,x ( ˙w|R, x) = F˙wN|x( ˙w|x)
˙wc
Rx0≤˙w < ˙wc
Rx1
max ( ˙w, min GRx ($)) = ˙wmin GRx ($) = ˙wc
Rxo
˙wc∈ Wc
Rx ={˙wc
Rx1}˙wN
x0<˙wc
Rx1<˙wN
x1
˙w≥˙wc
Rx1˙wN,˙wc∈ CRx : ˙wN<˙w, ˙wc>˙w
˙wc= ˙wc
Rx1
˙w˙wN
x0≤˙w < ˙wc
Rx1
%R
x>0
cR
x( ˙w) = ˆ˙w
min LRx
f˙wN|R,x ($|R, x)d$
=ˆ˙w
˙wN
x0
f˙wN|R,x ($|R, x)d$ =
=F˙wN|x( ˙w|x)
R1∈ RxWc
R1x⊂ WN
R1xWN
R1x=WN
x=˙wN
x0,˙wN
x1
˙wN
x0<˙wc
R1x1Wx=WN
x
[ ˙wx0,˙wx1] = ˙wN
x0,˙wN
x1˙wx0<˙wc
R1x1˙w0
1
˙wx0<˙w0
1<˙wc
R1x1˙wc
R1x1<˙wc
Rkx1k > 1
˙w0
1<˙wc
Rkx1k≥1cRk
x˙w0
16= 0 k≥1
F˙w|x˙w0
1|x−F˙wN|x˙w0
1|x=−
Kx
X
j=1
%RjRj
xcRj
x˙w0
1
k= 2, . . . , Kx˙w0
k˙wc
Rk−1x1<˙w0
k<˙wc
Rkx1
˙wc
Rkx1<˙wc
Rk−1x1k˙w0
k<˙wc
Rκx1
κ≥k cRκ
x˙w0
k6= 0 κ≥k
F˙w|x˙w0
k|x−F˙wN|x˙w0
k|x=−
Kx
X
j=k
%RjRj
xcRj
x˙w0
k,1< k ≤Kx
k= 2, . . . , Kx{Rk−1, Rk} ⊆
Rx˙wc
Rk−1x1<˙wc
Rkx0˙wc
Rkx16= ˙wc
Rk−1x1˙wc
Rkx0≤˙wc
Rkx1
˙wx0<˙w0
1<˙wc
R1x0˙wx0<˙w0
1<˙wc
R1x1˙wc
Rk−1x1<˙w0
k<˙wc
Rkx0
˙wc
Rk−1x1<˙w0
k<˙wc
Rkx12≤k≤Kx
˙w0
1,..., ˙w0
Kxk=
1, . . . , Kx
F˙w|x˙w0
k|x−F˙wN|x˙w0
k|x=−
Kx
X
j=k
%RjRj
xcRj
x˙w0
k
cR
x˙w0=F˙wN|x˙w0|xR˙w0<˙wc
Rx0
˙wc
Rkx1≤˙wc
Rk−1x1k˙wc
Rkx16= ˙wc
Rk−1x1
˙w0= ˙w0
kcRj
x˙w0
k=F˙wN|x˙w0
k|xRjj≥k˙w0
k<˙wc
Rkx0≤
˙wc
Rkx1<˙wc
Rjx0j > k
F˙w|x˙w0
k|x−F˙wN|x˙w0
k|x=−
Kx
X
j=k
%RjRj
xF˙wN|x˙w0
k|x
=− Kx
X
j=k
%RjRj
x!F˙wN|x˙w0
k|x
f˙wN|x˙w2mN
x−˙w
mN
xf˙wN|x
F˙wN|x( ˙w|x) = 1 −F˙wN|x2mN
x−˙w|x
f˙w|x
˙wc
RKxx1LR
x( ˙w)GR
x( ˙w)R=Rk∈ Rx
˙w < ˙wc
Rx1˙wc
RKxx1≥˙wc
Rkx1Rk∈ Rx
F˙w|x( ˙w|x) = F˙wN|x( ˙w|x),˙w≥˙wc
RKxx1
f˙w|x
˙wc
RKxx1
F˙w|x( ˙w|x)≤F˙wN|x( ˙w|x),˙w < ˙wc
RKxx1
qαx qN
αx α f ˙w|xf˙wN|xmN
x=
qN
50x
qN
αx < qαx , qαx <˙wc
RKxx1
qN
αx =qαx , qαx ≥˙wc
RKxx1
˙wc
RKxx1< mxmx≡q50xf˙w|x
α qαx >˙wc
RKxx1qαx =qN
αx
α= 50 mx=mN
x˙w
˙w < ˙wc
RKxx1< mx˙w f ˙w|x
2mN
x−˙w
F˙wN|x( ˙w|x)=1−F˙w|x2mN
x−˙w|xmx=mN
x
F˙wN|x( ˙w|x) = 1 −F˙w|x(2mx−˙w|x)
%R
x= Pr δ= 1|˙wN<˙wc∗, R, xR∈ Rx
%R
x≥0
%R
x≤1
πR
x≡Pr (R|x)R∈ R
πR
x≥0
πR
x≤1
X
ϑ∈R
πϑ
x= 1
%RR
x%R
xR∈ Rx
%R
x≥%RR
x≥0, R ∈ Rx
%RR
xπR
x
R∈ Rx
πR
x≥%RR
x≥0, R ∈ Rx
πR
x∈[0,1] R∈ R\Rx
πR
x
πR
x≥
%RR
x≥0, R ∈ Rx
0, R ∈ R\Rx
πR
x= 1 −X
ϑ∈R\{R}
πϑ
x
πϑ
xϑ∈ R\ {R}
πR
xR∈ R
πR
x≤1−X
ϑ∈Rx\{R}
%ϑϑ
x≤1, R ∈ R
%R
x≥%RR
x
1−Pϑ∈Rx\{R}%ϑϑ
x
≥%RR
x≥0, R ∈ Rx
%R
x
Kx= 1 Rx={r}
%r
x
Rx⊂ R
Wx=WN
x˙wx0= ˙wN
x0fε˙wx0= ˙wN∗
x0
R=r˙wc∗
rx0=˙
Pe∗
x0fε
˙wc
rx0=˙
Pe∗
x0
0<˙wN∗
x0<˙
Pe∗
x0Rx={R1}
R1=r˙wx0<˙wc
rx0⇔˙wx0<˙wc
R1x0
˙w0
1∈˙wx0,˙wc
R1x0= ( ˙wx0,˙wc
rx0) = ˙wN∗
x0,˙
Pe∗
x0
Rx⊂ R Wx=WN
x˙wx0= ˙wN
x0
fε˙wx0= ˙wN∗
x0−˙ε1
˙wc
rx0= ˙wc∗
rx0−˙ε1=˙
Pe∗
x0−˙ε1
0<˙wN∗
x0<˙
Pe∗
x0Rx={R1}
R1=r˙wx0<˙wc
rx0⇔˙wx0<˙wc
R1x0
˙w0
1∈( ˙wx0,˙wc
rx0) = ˙wN
x0,˙
Pe
x0
Kx= 2 Rx={n, r}
%n
x%r
x
Rx⊂ R Wx=WN
x˙wx0= ˙wN
x0
fε˙wx0= ˙wN∗
x0
R=n˙wc∗
nx1= ˙wc∗
nx0= 0 R=r
˙wc∗
rx0=˙
Pe∗
x0˙wc∗
rx1=˙
Pe∗
x1fε˙wc
nx0= ˙wc
nx1= 0
˙wc
rx0=˙
Pe∗
x0˙wc
rx1=˙
Pe∗
x1
˙wN∗
x0<0<˙
Pe∗
x0<˙
Pe∗
x1
˙wc
nx1<˙wc
rx1Rx={R1, R2}R1=n
R2=r
˙wx0<˙wc
nx0⇔˙wx0<˙wc
R1x0˙wc
nx1<˙wc
rx0⇔˙wc
R1x1<˙wc
R2x0
˙w0
1∈˙wx0,˙wc
R1x0=
( ˙wx0,˙wc
nx0) = ˙wN∗
x0,0˙w0
2∈˙wc
R1x1,˙wc
R2x0= ( ˙wc
nx1,˙wc
rx0) = 0,˙
Pe∗
x0
R=n˙wc∗= 0 R=r
˙wc∗=˙
Pe∗Rx={n}R=n
˙wc∗
Rx1= ˙wc∗
nx1= 0
Rx⊂ R Wx=WN
x˙wx0= ˙wN
x0
fε˙wx0= ˙wN∗
x0−˙ε1
˙wc
nx0= ˙wc∗
nx0−˙ε1=−˙ε1˙wc
nx1= ˙wc∗
nx1+ ˙ε1= ˙ε1˙wc
rx0= ˙wc∗
rx0−˙ε1=
˙
Pe∗
x0−˙ε1˙wc
rx1= ˙wc∗
rx1+ ˙ε1=˙
Pe∗
x1+ ˙ε1
˙wN∗
x0<0<˙
Pe∗
x0<˙
Pe∗
x1
˙wc
nx1<˙wc
rx1Rx={R1, R2}R1=n
R2=r
˙wx0<˙wc
nx0⇔˙wx0<˙wc
R1x0˙wc
nx1<˙wc
rx0⇔˙wc
R1x1<˙wc
R2x0˙
Pe
x0>
˙ε1⇔˙
Pe∗
x0>2 ˙ε1
˙w0
1∈˙wx0,˙wc
R1x0= ( ˙wx0,˙wc
nx0) = ˙wN
x0,−˙ε1˙w0
2∈˙wc
R1x1,˙wc
R2x0= ( ˙wc
nx1,˙wc
rx0) =
˙ε1,˙
Pe
x0
Kx= 2 Rx={n, r}
%n
x%r
x
Rx⊂ R Wx=WN
x˙wx0= ˙wN
x0
fε˙wx0= ˙wN∗
x0
R=n˙wc∗
nx1= ˙wc∗
nx0= 0 R=r
˙wc∗
rx0=˙
Pe∗
x0˙wc∗
rx1=˙
Pe∗
x1fε˙wc
nx0= ˙wc
nx1= 0
˙wc
rx0=˙
Pe∗
x0˙wc
rx1=˙
Pe∗
x1
˙wN∗
x0<˙
Pe∗
x0<0<˙
Pe∗
x1
˙wc
nx1<˙wc
rx1Rx={R1, R2}R1=n
R2=r
˙wx0<˙wc
nx0⇔˙wx0<˙wc
R1x0˙wc
nx1≮˙wc
rx0⇔˙wc
R1x1≮˙wc
R2x0
˙w0
1∈˙wN∗
x0,˙
Pe∗
x0˙w0
1<˙wc
nx0⇔˙w0
1<˙wc
R1x0˙w0
1<˙wc
rx0⇔
˙w0
1<˙wc
R2x0
R=n˙wc∗= 0 R=r
˙wc∗=˙
Pe∗Rx={n}R=n
˙wc∗
Rx1= ˙wc∗
nx1= 0
Rx⊂ R Wx=WN
x˙wx0= ˙wN
x0
fε˙wx0= ˙wN∗
x0−˙ε1
˙wc
nx0= ˙wc∗
nx0−˙ε1=−˙ε1˙wc
nx1= ˙wc∗
nx1+ ˙ε1= ˙ε1˙wc
rx0= ˙wc∗
rx0−˙ε1=
˙
Pe∗
x0−˙ε1˙wc
rx1= ˙wc∗
rx1+ ˙ε1=˙
Pe∗
x1+ ˙ε1
˙wN∗
x0<˙
Pe∗
x0<0<˙
Pe∗
x1
˙ε1<˙
Pe∗
x1+ ˙ε1⇔˙wc
nx1<˙wc
rx1Rx={R1, R2}R1=n
R2=r
˙wx0<˙wc
nx0⇔˙wx0<˙wc
R1x0˙wc
nx1≮˙wc
rx0⇔˙wc
R1x1≮˙wc
R2x0
˙w0
1∈˙wN
x0,˙
Pe
x0˙w0
1<˙wc
nx0⇔˙w0
1<˙wc
R1x0˙w0
1<˙wc
rx0⇔
˙w0
1<˙wc
R2x0
ResearchGate has not been able to resolve any citations for this publication.
How Wages Change: Micro Evidence from the International Wage Flexibility Project
Article
Full-text available
  • Oct 2007
  • J ECON PERSPECT
Workers' wages are not set in a spot market. Instead, the wages of most workers-at least those who do not switch jobs-typically change only annually and are mediated by a complex set of institutions and factors such as contracts, unions, standards of fairness, minimum wage policy, transfers of risk, and incomplete information. The goal of the International Wage Flexibility Project (IWFP)-a consortium of over 40 researchers with access to individual workers' earnings data for 16 countries-is to provide new microeconomic evidence on how wages change for continuing workers. Wage changes due to worker mobility are governed by different processes and are beyond the scope of this study. A key question in the theoretical and empirical literature, as reviewed in
View
Rigid Wages: What Have We Learnt from Microeconometric Studies?
Chapter
Full-text available
  • Jan 2001
In this survey, I intend to describe the latest efforts of labour economists to analyse wage rigidity both in its existence and in its consequences in various countries, in particular the United States and France. There has been recently a renewed interest in questions surrounding wage rigidity, a central concern of many macroeconomists. In addition, the existence of wage rigidities has been viewed by some analysts as the main reason for the high level of European unemployment, in contrast to the North American situation. The stakes are clear enough so that I do not need to spend much time in this introduction on justifying why we should, as economists, be interested in this topic.
View
View
View
View
View
Downward Wage Rigidity in Europe: A New Flexible Parametric Approach and Empirical Results
Article
  • May 2010
  • GER ECON REV
We suggest a new parametric approach to estimate the extent of downward nominal wage rigidity in ten European countries between 1995 and 2001. The database used throughout is the User Data Base of the European Community Household Panel (ECHP). The proposed approach is based on the generalized hyperbolic distribution, which allows to model wage change distributions characterized by thick tales, skewness and leptokurtosis. Significant downward nominal wage rigidity is found in all countries under analysis, but the extent varies considerably across countries. Yearly estimates reveal increasing rigidity in Italy, Greece and Portugal, while rigidity is declining in Denmark and Belgium. The results imply that the costs of price stability differ substantially across Europe. Copyright 2009 The Authors. Journal Compilation Verein für Socialpolitik and Blackwell Publishing Ltd. 2009.
View
How Strong Is the Macroeconomic Case for Downward Real Wage Rigidity?
Article
  • May 2009
  • J MONETARY ECON
We explore the existence of downward real wage rigidity (DRWR) at the industry level, based on data from 19 OECD countries for the period 1973–1999. The results show that DRWR compresses the distributions of industry wage changes overall, as well as for specific geographical regions and time periods, but there are not many real wage cuts that are prevented. More important, however, DRWR attenuates larger real wage cuts, thus leading to higher real wages. There is stronger evidence for downward nominal wage rigidity than for DRWR. Real wage cuts are less prevalent in countries with strict employment protection legislation and high union density.
View
Robustness and Real Consequences of Nominal Wage Rigidity
Article
  • Mar 2003
  • J MONETARY ECON
Nominal wage rigidity has been shown to exist in periods of high inflation, while reduction in nominal pay has been hypothesized to occur in times of low inflation. Nominal wage rigidity would therefore become irrelevant because there is little need to cut nominal pay under high inflation, while the necessary cuts would occur under low inflation. We test this hypothesis by examining Swiss data in the 1990s, where wage inflation was low. Nominal wage rigidity proves robust in a low inflation environment, constituting a considerable obstacle to real wage adjustments. Real wages would indeed respond to unemployment without downward nominal rigidity. Moreover, wage sweep-ups caused by nominal rigidity correlate strongly to unemployment, suggesting downward nominal wage rigidity fuels unemployment.
View
Real and Nominal Wage Rigidities in Collective Bargaining Agreements
Article
  • May 2007
  • LABOUR ECON
An earlier study of wage agreements, reached in the Canadian unionized sector between 1976–99, found that wage adjustment is characterized by downward nominal rigidity and significant spikes at zero. We extend this earlier approach to encompass the possibility of real as well as nominal wage rigidity. The addition of real wage rigidity variables enhances earlier results and suggests that real rigidity increases significantly the mass in the histogram bin containing the mean anticipated rate of inflation, as well as in adjacent bins. Downward nominal wage rigidities and spikes at zero remain important.
View