THE INFLATIONARY BIAS OF REAL UNCERTAINTY
AND THE HARMONIC FISHER EQUATION
By
Ioannis Karatzas, Martin Shubik, William D. Sudderth and John Geanakoplos
Revised:August 2004
June 2003
COWLES FOUNDATION DISCUSSION PAPER NO. 1424R
COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
YALE UNIVERSITY
Box 208281
New Haven, Connecticut 06520-8281
http://cowles.econ.yale.edu/
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pq +$, @ +4 . ,^pq4 +$, eq4 +$,` . sq4 +$,\q4 +$,
+514,
 q +$, Oq +$,/ wkh djhqw
dqg e| fkrrvlqj wr eruurz ru ohqg p
 q +$,/ zlwk pq +$, p
 q +$, dqg frqvxphv {q +$, @ eq +$,@sq +$,1 Iru qrwdwlrqdo
vshqgv eq +$, @ pq +$, . p
vlpsolflw| zh vkdoo dyrlg xvlqj p
 q dqg {q / vlqfh wkh| duh ghwhuplqhg dw hyhu| vwdjh e|
eq dqg sq 1
515
Htxloleulxp
Dw wkh ehjlqqlqj ri shulrg q/ wkh djhqw nqrzv wkh lqwhuhvw udwh / wkh sdvw dqg suhvhqw
ydoxhv ri oltxlg zhdowk p4 > p5 > ===> pq4 > pq > sulfhv s4 > s5 > ===> sq4 / sq > dqg hqgrzphqwv
\4 > \5 > ===> \q4 > \q 1 +Qrwlfh wkdw wkh djhqw lv dvvxphg wr nqrz sq dqg klv hqgrzphqw \q
iru shulrg q dw wkh ehjlqqlqj ri wkh shulrg1 Hyhqwxdoo| zh vkdoo frqvlghu d prgho lq zklfk
wkh djhqw grhv qrw nqrz hlwkhu sq ru \q zkhq kh fkrrvhv klv elg lq wkh shulrg1, Wkh djhqw
pxvw fkrrvh klv elg eq lq shulrg q wr eh d ixqfwlrq ri wkhvh yduldeohv/ ru/ htxlydohqwo|/ wr
eh phdvxudeoh zlwk uhvshfw wr wkh vljpd0hog Iq jhqhudwhg e| wkhp1
Wkh djhqw*v exgjhw vhw lv
E+p4 > s> O> , @ ieq +$, = 3 eq +$, pq +$, . Oq +$, iru doprvw doo $ 5 dqg eq lv Iq 0phdvxudeoh/ iru doo q 4j>
+515,
zkhuh pq lv ghwhuplqhg e| +p4 > > s4 +$,> ===> sq4 +$,> e4 +$,> ===> eq4 +$,> \4 +$,> ===> \q4 +$,,
dv lq +514, deryh1
Wkh hfrqrp| lv lq htxloleulxp dw isq +$, = q 4> $ 5 j/ li wkh uhsuhvhqwdwlyh djhqw lv
rswlpl}lqj lq klv exgjhw vhw zkloh frqvxplqj \q +$,/ iru doo q 41 Pruh suhflvho|/ ohwwlqj
eq +$, sq +$,\q +$, iru doo q 4> $ 5 >
8
+516,
wkh vhtxhqfh ieq +$, = q 4> $ 5 j dwwdlqv wkh pd{lpxp ri
&
%4
[ q4
x+eq +$,@sq +$,,
H
q@4
ryhu wkh vhw E+p4 > s> O> , ri +515,1
Zh glvwlqjxlvk wzr fdvhv iru wkh eruurzlqj olplwv Oq +$,1 Vhwwlqj
Oq +$, @
sq +$,\q +$,
>
4.
+517,
zh jhw dq hfrqrp| zlwk d edqn wkdw shuplwv wkh djhqw wr eruurz xs wr wkh dprxqw kh
lv vxuh wr uhfhlyh lq lqfrph1 E| vhwwlqj Oq +$, @ 3 dqg @ 3/ zh hhfwlyho| rewdlq dq
hfrqrp| zlwkrxw d edqn1
Lq wkh iroorzlqj wzr vhfwlrqv zh frqvwuxfw htxloleuld iru wkhvh wzr fdvhv1 Diwhuzdugv
zh vkdoo frqvlghu d prgho lq zklfk wkh edqn fkrrvhv dq lqwhuhvw udwh q +$, lq shulrg q
wkdw qhhg qrw eh frqvwdqw1 Zh vkdoo dovr frqvwuxfw htxloleuld zlwk dqg zlwkrxw d edqn
zkhq wkh djhqw grhv qrw nqrz klv hqgrzphqw \q +$, ru wkh sulfh sq +$, zkhq kh lv fdoohg
xsrq wr eruurz dqg elg dw wlph q1
Zh duh sulpdulo| frqfhuqhg zlwk wkh ehkdylru ri wkh sulfhv sq +$,1 Iru wkh prghov zlwk
d edqn zh vkdoo jlyh frqglwlrqv rq wkh lqwhuhvw udwh wkdw uhvxow lq lq dwlrq/ gh dwlrq/ ru
qhlwkhu1 Frqvxpswlrq lq rxu prghov lv wulyldo lq htxloleulxp/ zlwk wkh djhqw frqvxplqj
klv hqgrzphqw \q +$, lq hdfk shulrg q1
516
Qhxwudo/ Vwdwlrqdu| Htxloleulxp +QVH,
Zh vkdoo irfxv rxu dwwhqwlrq rq vwdwlrqdu| htxloleuld lq zklfk sulfhv sq +$, dqg elgv rq
h{shqglwxuhv eq +$, dw wlph q fdq doo eh h{suhvvhg lq whupv ri ixqfwlrqv S = U.. \ $ U
dqg E = U.. \ $ U wkdw ghshqg rqo| rq wkh oltxlg zhdowk pq +$, dqg wkh hqgrzphqw
\q +$, lq wkh shulrg/ qdpho|
sq +$, @ S+pq +$,> \q +$,,
eq +$, @ E+pq +z,> \q +$,,=
+518,
Wkh oltxlglw| frqvwudlqwv Oq +$, dw wlph q fdq eh h{suhvvhg dv d ixqfwlrq
Oq +$, @ O+\q +$,> sq +$,,
+5183 ,
ri fxuuhqw hqgrzphqw \q +$, dqg sulfh sq +$,/ zkhuh O = \ U.. $ U lv jlyhq e|
,
+ s|
/ li wkhuh lv d edqn
=
+519,
O+|> s, @ 4 . 3
/ li qrw
Lq idfw zh vkdoo frqvwuxfw vwdwlrqdu| htxloleuld lq zklfk dw prqh| lv qhxwudo +grxeolqj
wkh oltxlg zhdowk grxeohv sulfhv dqg elgv zlwkrxw fkdqjlqj frqvxpswlrq,=
sq +$, @ pq +$, s+\q +$,,
eq +$, @ pq +$, e+\q +$,,
+51:,
zkhuh s = \ $ U.. dqg e = \ $ U.. duh jlyhq ixqfwlrqv1 Fohduo| lq htxloleulxp zh
pxvw kdyh
e+|, @ s+|,|/ iru doo | 5 \/
+5163 ,
iurp +516,1 Lw lv hylghqw iurp wkh prqrwrqlflw| ri x wkdw wkhuh fdq eh qr htxloleulxp
lq zklfk sq +$, @ 3 kdsshqv zlwk srvlwlyh suredelolw| iru vrph q 41 Khqfh lq qhxwudo
vwdwlrqdu| htxloleulxp zh pxvw kdyh e+|, A 3 dqg s+|, A 3 iru doo | 5 \/ dqg pq +$, A 3
doprvw vxuho|/ iru doo q 41
9
517
G|qdplf Surjudpplqj
Lq qhxwudo/ vwdwlrqdu| htxloleulxp wkh pdfurvfrslf yduldeohv +p> |> s, iroorz d vwdwlrqdu|
Pdunry surfhvv ghqhg e| wkh sulfh ixqfwlrq S+p> |, @ ps+|,1 Ohw eh wkh judsk ri
wkh ixqfwlrq S/ qdpho| @ i+p> |> s, 5 U.. \ U.. = S+p> |, @ sj1 Wkhq wkh odz
ri prwlrq ri wkh pdfurvfrslf yduldeohv lv jlyhq e| wkh wudqvlwlrq phfkdqlvp $ +,
lqglfdwhg ehorz iru hdfk +p> |> s, 5 =
p3 @
@
@
|3 5
s3 @
+p s|,+4 . , . s|
p+4 . , s|
p+4 . e+|,,
\/ glvwulexwhg dv \
S+p3 > | 3 , @ p3 s+| 3 ,=
+51;,
Qrwh wkdw p3 lv ghwhuplqhg zlwkrxw dq| xqfhuwdlqw| e| +p> |> s,/ zkhuhdv | 3 dqg s3 duh
vwrfkdvwlf1 Qrwh dovr wkdw lq rughu wr kdyh p3 A 3/ zh pxvw kdyh
e+|, @ S+p> |,|@p ? +4 . ,@/ iru doo | 5 \=
Idflqj wklv g|qdplf v|vwhp/ wkh uhsuhvhqwdwlyh djhqw zlwk oltxlg zhdowk v pxvw fkrrvh
dq h{shqglwxuh uxoh dw hdfk pdfurvfrslf vwdwh +p> |> s, wr pd{lpl}h klv xwlolw|1 Lq vwd0
wlrqdu| htxloleulxp/ zkhq v @ p/ wkh h{shqglwxuh uxoh +v> p> |> s, :$ s| pxvw eh rswlpdo1
Lq wklv fdvh wkhuh pxvw eh d ydoxh ixqfwlrq Y = U. $ U vdwlvi|lqj
e
3
3
+51<,
Y +v> p> |> s, @
pd{
x
. H\ Y ++v e,+4 . , . s|> p > \> s ,
s
3ev.O+|>s,
vxfk wkdw iru doo +p> |> s, 5 dqg v @ p/
e
x
s | 5 duj pd{
. H\ Y ++p e,+4 . , . s|> p3 > \> s3 ,
s
3ep.O+|>s,
l1h1/
6
Y +p> p> |> s, @ x+|, . H\ ^Y +p3 > p3 > \> s3 ,`=
+5143,
+5144,
Frqvwuxfwlqj Qhxwudo/ Vwdwlrqdu| Htxloleulxp
zlwkrxw d Edqn
Lq wkh devhqfh ri d edqn zh kdyh O+|> s, @ 3/ dqg wkh prqh| vxsso| grhv qrw fkdqjh/ vr
wkh uhsuhvhqwdwlyh djhqw zloo ehjlq hdfk shulrg zlwk wkh vdph oltxlg zhdowk p @ p4 1 Zh
fdq vhh wkdw p3 @ p e| wdnlqj @ 3 lq wkh odz ri prwlrq +51;,1
Wkh uhsuhvhqwdwlyh djhqw dozd|v kdv wkh fkrlfh ri vshqglqj klv prqh| dw wlph q/ ru
vdylqj lw wr vshqg lq wkh ixwxuh1 Lq qhxwudo vwdwlrqdu| htxloleulxp/ 3 ? s| p iru doo
+p> |> s, 5 1 Khqfh wkh djhqw frxog vshqg % ohvv/ vdylqj wkh prqh| xqwlo qh{w shulrg/
dqg vshqglqj lw wkhuh rq frqvxpswlrq1 Lq htxloleulxp lw lv rswlpdo iru klp qrw wr gr vr/
jlylqj lq wkh qrwdwlrq ri +51;,=
3
x +\ ,
x3 +|,
H
iru doo | 5 \=
+614,
s
s3
Li s| ? p/ wkh djhqw fdq vshqg % pruh +dqg wkhq % ohvv qh{w shulrg,/ jlylqj wkh rssrvlwh
lqhtxdolw|1 Vxfflqfwo| frpelqlqj wkhvh revhuydwlrqv jlyhv
3
3
x +\ ,
x +|,
H
^4 |s+|,` a
@ 3> iru doo | 5 \1
+615,
s+|,
s+\ ,
:
+Wkh qrwdwlrq d a e lv xvhg khuh dqg ehorz iru wkh plqlpxp ri d dqg e1,
Zh pxvw qg s +ru htxlydohqwo| e, vxfk wkdw +615, lv vdwlvhg1
Rqh qdwxudo +exw jhqhudoo| lqvx!flhqw, jxhvv lv wkdw wkh djhqw zloo vshqg doo klv prqh|
lq hyhu| shulrg/ qdpho| E+p> |, @ p/ e+|, @ 4/ S+p> |, @ p@|/ s+|, @ 4@|1 Soxjjlqj
wklv fdqglgdwh s+|, @ 4@| lqwr +615, jlyhv
|x3 +|, H^\ x3 +\ ,`
+616,
iru doo | 5 \1 Wklv lqghhg lv dq htxloleulxp li |x3 +|, lv d frqvwdqw +dv lw zloo eh iru
x+|, @ orj+|,,/ ru pruh jhqhudoo|/ li +616, krogv iru doo | 5 \1 Exw li wkhuh lv hqrxjk
yduldwlrq lq |x3 +|, ryhu | 5 \/ wkhq +616, zloo idlo iru vrph |/ dqg s+|, @ 4@| zloo qrw
jlyh ulvh wr dq htxloleulxp1
Dqrwkhu sodxvleoh +exw lqvx!flhqw, jxhvv lv wr vhw sulfh dozd|v sursruwlrqdo wr pdujlqdo
xwlolw|/ s+|, @ d4 x3 +|, iru vrph frqvwdqw d A 3/ iru doo | 5 \1 Exw soxjjlqj wklv lqwr +615,
jlyhv d vwulfwo| srvlwlyh vhfrqg whup rq wkh ohiw0kdqg vlgh/ iru doo | 5 \/ lpso|lqj wkdw
s+|, @ 4@|/ e+|, @ 4 iru doo | 5 \1 Khqfh zh pxvw kdyh wkhq 4@| @ x3 +|,@d ru |x3 +|, @ d
iru doo | 5 \/ zklfk fdq kdsshq li x lv orjdulwkplf ru li wkhuh lv qr hqgrzphqw xqfhuwdlqw|1
Lq jhqhudo/ krzhyhu/ s+|, @ x3 +|,@d lv dovr lqvx!flhqw1
Ohw d H^x3 +\ ,@s+\ ,`1 Dq| vroxwlrq s+|, wr +615, pxvw pdnh hlwkhu wkh uvw whup
ru wkh vhfrqg whup rq wkh ohiw htxdo wr }hur/ iru doo | 5 \1 Khqfh s+|, fdq wdnh rq rqo|
wzr srvvleoh ydoxhv/ 4@| ru x3 +|,@d1 Li |x3 +|, ? d/ wkhq fohduo| s+|, @ 4@| zloo qrw zrun
+vlqfh wkh vhfrqg whup zrxog eh qhjdwlyh,/ khqfh s+|, @ x3 +|,@d1
Vxssrvh qrz wkdw |x3 +|, A d1 Vlqfh wkh uvw whup rq wkh ohiw lv qrqqhjdwlyh/ s+|, 4@|1 Khqfh x3 +|,@s+|, |x3 +|, A d pdnlqj wkh vhfrqg whup srvlwlyh1 Wkxv zh pxvw
kdyh s+|, @ 4@|1
Li |x3 +|, @ d/ wkhq 4@| @ x3 +|,@d dqg erwk fkrlfhv iru s+|, duh wkh vdph1 Lq vxppdu|
+
, 4@|
/ iru |x3 +|, d
4 x3 +|,
3
s+|, @ x +|,
a
@
=
+617,
|
d
/ iru |x3 +|, ? d
d
Revhuyh wkdw s+|, ghshqgv rq d/ qdpho|/ lv ri wkh irup sd +|, @ plqi4@|> x3 +|,@dj1
Soxjjlqj wklv lqwr wkh ghqlwlrq ri d/ zh jhw
d @ H^pd{id> \ x3 +\ ,j`=
+618,
Lw lv hylghqw wkdw dw d @ 3/ wkh uljkw0kdqg vlgh ri +618, grplqdwhv wkh ohiw/ vlqfh \ x3 +\ , A 3
iru doo \ 5 \/ zkloh iru d A 3 yhu| odujh wkh ohiw0kdqg vlgh ri +618, grplqdwhv wkh uljkw
vlqfh ? 4 dqg H^\ x3 +\ ,` ? 41 Khqfh wkhuh pxvw eh dq d 5 +3> 4, vroylqj +618,1 Lqghhg/
lw lv hylghqw wkdw qhdu dq| d vroylqj +618,/ wkh uljkw0kdqg vlgh lqfuhdvhv ohvv txlfno| lq d
wkdq wkh ohiw/ vr wkhuh lv d xqltxh d 5 +3> 4, vroylqj +618,1
Wkhruhp 614= Zlwk d 5 +3> 4, ghqhg dv lq +618,/ s dv ghqhg lq +617, jlyhv ulvh wr d
qhxwudo vwdwlrqdu| htxloleulxp iru wkh hfrqrp| zlwkrxw d edqn1 Wklv lv wkh xqltxh qhxwudo
vwdwlrqdu| htxloleulxp1
Surri= Wklv fdqglgdwh htxloleulxp xqltxho| vdwlvhv +615, e| frqvwuxfwlrq1 Zh ohdyh wr
wkh Dsshqgl{ wkh surri wkdw wkh lpsolhg elgglqj vwudwhj| eq +$, p4 s+\q +$,,\q +$, lv
rswlpdo lq wkh uhsuhvhqwdwlyh djhqw*v exgjhw vhw1
Lq wkh htxloleulxp mxvw frqvwuxfwhg/ oltxlg zhdowk uhpdlqv frqvwdqw dw p4 / elgv duh
jlyhq dv
|x3 +|,
eq +$, @ e+\q +$,, p4 > zkhuh e+|, |s+|, @ plq 4>
+619,
d
;
dqg sulfhv xfwxdwh lqghshqghqwo| dqg lghqwlfdoo| dfurvv shulrgv dffruglqj wr wkh uxoh
x3 +\q +$,,
4
>
p4 =
sq +$, @ S+p4 > \q +$,, @ plq
+61:,
\q +$,
d
Qr pdwwhu zkdw wkh glvfrxqw udwh / wkhuh lv qr lq dwlrq1 Li wkh hqgrzphqw udqgrp
yduldeoh \ lv erxqghg dzd| iurp }hur dqg lqqlw|/ vr duh wkh sulfhv1
Lq wkh vshfldo fdvh zkhuh wkhuh lv qr uhdo xqfhuwdlqw|/ \q +$, @ | iru doo $ dqg |x3 +|, @
3
| , iru doo | 5 \/ +616, vkrzv wkdw lq wkh xqltxh htxloleulxp sq +$, @ p4 @
| iru doo $/
|x +
dqg eq +$, @ p4 iru doo $1 Wkh djhqw dozd|v vshqgv doo klv oltxlg zhdowk hyhu| shulrg rq
jrrgv1 Kh zlvkhv kh frxog vshqg pruh/ exw vlqfh kh fdqqrw eruurz/ kh lv xqdeoh wr gr vr1
Hyhq zkhq wkhuh lv xqfhuwdlqw|/ li xwlolw| x lv orjdulwkplf/ +616, vkrzv wkdw sq +$, @
p4 @\q +$, iru doo $ dqg djdlq wkh djhqw vshqgv doo klv prqh| hdfk shulrg= eq +$, @ p4
iru doo $1
Lq jhqhudo/ zh vhh wkdw zlwk srvlwlyh suredelolw| sq +$, @ p4 @\q +$,> wkh djhqw vshqgv
doo klv prqh| dqg zlvkhv kh frxog eruurz +dw }hur lqwhuhvw, dqg vshqg pruh1
7
Frqvwuxfwlqj Qhxwudo/ Vwdwlrqdu| Htxloleulxp
zlwk d Edqn
Lq wkh suhvhqfh ri d fhqwudo edqn/ wkhuh pd| zhoo eh lq dwlrq1 Zh zlvk wr vwxg| krz
lq dwlrq ghshqgv rq wkh edqn lqwhuhvw udwh dqg rq wkh uhdo hfrqrp|1
Lq wkh suhvhqfh ri d fhqwudo edqn wkdw pdlqwdlqv d frqvwdqw lqwhuhvw udwh/ wkh prqh|
vxsso| lv hqgrjhqrxv dqg wkh uhsuhvhqwdwlyh djhqw pd| zhoo ehjlq hdfk shulrg zlwk glhu0
hqw oltxlg zhdowk1 Zh vxssrvh wkdw wkh edqn vwdqgv uhdg| wr dffhsw ghsrvlwv ru jlyh ordqv
dw dq h{rjhqrxv lqwhuhvw udwh A 31 Zh vkdoo frqvwuxfw d qhxwudo vwdwlrqdu| htxloleulxp
+e> s,1 Iurp wkh odz ri prwlrq +51;, zh kdyh
p3 @ +|,p
s3 @ p3 s+|, @
zkhuh
p3 e+|,
|
+|, 4 . e+|, 4 . iru doo | A 3=
+714,
+715,
Dv qrwhg hduolhu/ qhxwudolw| lpsolhv wkdw wkh jurzwk udwh ri oltxlg zhdowk ehwzhhq shulrgv
q 4 dqg q ghshqgv rqo| rq \q4 / dqg wkhuhiruh wkdw pq lv Iq4 0phdvxudeoh1
Qrwh wkdw dv orqj dv e+|, +4.,@/ wkh djhqw*v eruurzlqj frqvwudlqw +518, lv vdwlvhg>
dqg dv orqj dv e+|, ? +4 . ,@/ klv ixwxuh oltxlg zhdowk vwd|v vwulfwo| srvlwlyh/ zklfk lv
qhfhvvdu| lq qhxwudo/ vwdwlrqdu| htxloleuld1
Lq qhxwudo/ vwdwlrqdu| htxloleulxp/ li wkhuh lv rqh/ wkh djhqw frxog dozd|v kdyh eruurzhg
+ghsrvlwhg, d olwwoh pruh ru d olwwoh ohvv1 Khqfh zh pxvw kdyh
3
x +\ ,
x3 +|,
@ +4 . , H
> iru doo | 5 \1
+716,
s
s3
+Wklv lv wkh dqdorjxh wr htxdwlrq +614, zkhq wkhuh zdv qr edqn1, Soxjjlqj lq rxu irupxodv
iru s dqg s3 / dqg uhfdoolqj pe+|, @ s| jlyhv
\ x3 +\ ,
|x3 +|,
@ +4 . , H
pe+|,
p3 e+\ ,
3
\ x +\ ,
+4 . ,
H
@
> iru doo | 5 \1
+717,
+|,p
e+\ ,
<
Fdqfhoolqj p iurp erwk vlghv/ dqg xvlqj +715,/ zh kdyh
3
+4 . ,
\ x +\ ,
|x3 +|,
@
H
/ iru doo | 5 \=
e+|,
4 . e+|,
e+\ ,
+718,
Qrwh wkdw A 3 lv qrz qhfhvvdu| iru wkh h{lvwhqfh ri qhxwudo/ vwdwlrqdu| htxloleulxp
zlwk d edqn1 Lqghhg/ li @ 3/ wkhq e| +714,/ +|, @ 4 iru doo |/ jlylqj |x3 +|,@e+|, @
H^\ x3 +\ ,@e+\ ,` iru doo | 5 \/ e| +717,1 Lqwhjudwlqj zlwk uhvshfw wr wkh glvwulexwlrq ri \ rq erwk vlghv zh rewdlq d frqwudglfwlrq wr ? 41
Ohppd 714= Li A 3/ wkhuh lv d xqltxh ixqfwlrq e = \ $ U. wkdw vlpxowdqhrxvo|
vdwlvhv wkh rswlpdolw| frqglwlrq +717, dqg wkh frqglwlrq 3 ? e+|, ? +4 . ,@ iru doo
| 5 \=
4.
+4 ,|x3 +|,
e+|, @
> | 5 \=
+719,
+4 ,|x3 +|, . H^\ x3 +\ ,`
Wkhruhp 715= Ghqlqj e+|, dv lq +719,/ dqg ohwwlqj S+p> |, @ pe+|,@| dqg E+p> |, @
pe+|,/ jlyhv wkh xqltxh qhxwudo/ vwdwlrqdu| htxloleulxp iru wkh hfrqrp| zlwk d edqn wkdw
vhwv wkh lqwhuhvw udwh htxdo wr lq hyhu| shulrg1 Lq sduwlfxodu
s+|, @
4.
+4 ,x3 +|,
e+|,
@
>
|
+4 , |x3 +|, . H^\ x3 +\ ,`
| 5 \=
+7193 ,
Wkh surriv duh jlyhq lq wkh Dsshqgl{1
Kdylqj ghvfulehg wkh xqltxh qhxwudo/ vwdwlrqdu| htxloleulxp/ zh qrz surfhhg wr vwxg|
wkh jurzwk udwh ri oltxlg zhdowk dqg sulfh1 Zkhq wkhuh lv qr xqfhuwdlqw|/ dqg \ @ i
| j/
irupxodv +719,/ +7193 ,/ dqg +715, ehfrph
+4 . ,+4 ,
+4 . ,+4 ,
s+
|, @
|
+
| , @ 4 . e+
|, @ +4 . ,=
e+
|, @
Lw iroorzv wkdw
| , @ p4 ^ +
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ρ = 3%, β =.99
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