noterbc (w/ matlab exercise)
TRANSCRIPT
1 b 1 09
~ 1V1e (ec(,.tYYin~ fluc':\uo.tioYls tltlo.t occ~( [Vl reed output & ptvlploYtvleVl.t
1t o~ qV<Q'IrefS ,6 fY-o /VI peav... -."
Pectl< in veal GDP (Anti \ tVO\,f4k +0 trou~lt\
1CJL/-g:4 2 -1.1- 0/0
1C15?J ~ 2. .3 - 2.1"0'/0
1Q5'7: '6 1- - 3.1- 11/0
1'1bo: 1 .3 - 1. b fI/p
1Q-=J-O: 3 1 -1.1 ·t
,)Cl1-'> ~ 4 5 - 3.'+ ./, If you. wav,t cfo.+o. '''loY€,
f(e((v~lI\..tll.( '200t}_
ill.l. vfS4b
4' 't1J"dev-sY-t; =: Ct -+ It + Got + (X t - 1fv1~) ~ dtttt...
(i) g~Ii2d ~+s
A) Ou.tplA.t Gtow~
~ IS distviblAted vou~\t\\y gylVlMetvicodl'l o.vouvlOL_ its MeCw) ,
IolA+ 'd- ~eerv\S to be CV\cn'~acte"izect loy (elat;ve1v IOVlg peviods Vt-I.t1~Vl it £li~ltttL~
£Above Hs usucJ.. po.t~, 'lntevupted by bvief peYiOcl5 WAeVl it is Yelariv€-l~
few 'oe-lov\J,
gton~(ol
MeaVl oeviatioV\
y 0.0003\ 0.034-5
C o.001tJ. b 0.0201
r 0.00163 O.O(,&'O
G -O.OODbS O. Oqb~
X-1M -0 .oo·,r~~ o.085:t
2
g) wah'as ittr'\ t-AOdlll
1) &ot1tt outplAt ttll\cL input (Vl~d'\ets C\ft. competihve.. MO!'kets.
it,) No extevvlOJities
~ \AJ\tW>ut ~\t\ocks G) ti1e eco v\O(V\'y' in tkt. R01V1fey (VI0cle,l. eo'flver~eg TO
(). 'BoJaVlcecl Gtowt!A. Po.th, a\l1ci t~eV1 grv'WS $rvtooth''f' ( COYlsta Vlt growth ro-\'e)
t:xteV1ct the, CVloV\-~roclAastjc) f2..OtMsey Mode\ to \YlCOV-POyoJe- S,~oCKS to
prvcl~ teGVlrJ olo9 'I
-+----------'"1> t
c) ShockS
1) Tec.hVlo109Y Snoc,ks
to<',c....) -::: /A)K-oc LaO \ ./
'}A ~ -r€GIA. g~o~
€t· &lYlplAtevg I t<o~ots altev procP' process ~ f PfoclutHviftt·
2.) Wea~eV" SnoCkS L NattAyc:U. Oisasi"er'"oS
€.-f Agvicu.\h)Ye c... tvUvlSM '\rletu'sttle.S dGpevui on weatV1e~
~;'. ~ weo.thev IS 0. potential £o(,{ret of f'tuciuQ hoV).
1) Mone-tdr\1 g\1o~ks
~. MOVleV gupply g,. :rVlr~vest (2(Ate, . ~-,·effec,tinfltthOV1
4) Po\iticoJ. ~hockS
e.l. FaSnioVl ~ Fads
-ItJ sp€tilol info 0) Pcopa9a.+iOYl Ma.(,V\at1i~M
(JorJ. ~ M~ch().YlI~Mg thClt otvlp\if'l S'~oCk$ g propa9ate tV1(OlAg1A tiMe.
1) IVl+eV'"'tempot'aA ~ub~ti-hAtio n
~lnovk in t blAt affec+ at- t+1 )
Q.) stiCk'f P..1ce.t
~ Ne~. fuO,UCt1\1Ii"l1 gvlOU<- g, :;:) MPL l =) W~
16lAt) 'In recJ'l"L we" cannot ~w 1n tW- ~lttOY+ rlAVl.
-:;)
/~
/ ~) F:ic'iioYlS "In FiVltll'lG-iaJ. ML\fkets
€.t. AsI().V\ C"is".;> Ht:tMbevgeV" III U.S. ~
4
@ Choices Unf.ieV" Uncev-ft\iY\ty
A) Modelli"9 Ut1c'Y'i"tlintreS (sCtJli'cks)
~ To see "13IASif1€SS Cl(cAe" -;) j lAst adol \Ar1c€'fmin+y to tVle PiVet"' ~ _
~
we Ye~o.'1et'\Procl~.fi.l'· GlS the ecoY'lOr\1'IC ~ysteVV\.
'" "BusiVl€SS CLfcte II as 0-. f'ai\uve. of ~ eCO(1OM;C Gv~te'N\.
ASS()rVlp+iOV\~ :
-",) Theve aye.. 2 peYio~s; t ~cl t+ 1
1<-) Let s == 1, '2, ... , 8'
=- possl~le evevrrs I gt-ates of t1t<rufe thCH (V1Ot"( occlt\y iv) ~Je, t .... 1
4 eiD" 6 x mt-e,. J "-
'l\;~) ;;;'iTs probCl~irlty thert GtCtte g LNiI\ OCCtJY"
~ AH possibilih'eS of 0.11 eveVlts. S:I
S c 1 £(1):1 n(1)::') U [.c(1), 1- )'\(1)] >< 1i'~
s :. 2. .-C ('2.), n (2.) ==) lA Ge(1), 1- (\(2)J >< iT'2,
s = S"
~v) ProO" FV\: I~ = ~. FCK, ~~ 'lCtVlc1oM \/. lal?"(
lIv\Aev€t A = tec[.,V\olo9Y
At+,:::: S pOSS'\~ v~lue.s of tVtt- t~~~IYJOlogy In t+ 1.
3" 1'3 1 0'/
v) A
.j, J {$ iY
vt~t, 1-Ylt) + (3 t, ITs U[.ct-+l(,s), 1-Ylt~I(.s~ ~ J I
\/ 0" v
- A V'ep('e5e~w.\ivt HH ~uppl ie-S V\. unitS of tll\t1e evto(O(,AJfv1elAt
as. lo.boY
\--nc; \.N~ il\cof\llt = LNt V1t
- 111e HH ctivi elLS ii1e pevio et i InCOMe \: ('0 ~,S""" pliO",: -Ct
$'Cl,vl ~: k-t+ I
- ~~e. t Con~tat\V\.t of 16t Puioril :
l-<!~ + kHI - k~ {, W. Vtt + V"t kt] L; Cwoou Vtov-J (\I1(,{ch tiMe. tWy'lI C1VOVI.<?
2nd Peviool: - LNo~k lAMe!. CoV75Utvle
- me. HH veVlts e.-o.pital to -t;ytvlS.
- He suppli~s vvovk e.ffov+ to t1(}VJg.
- He, ConSl,WAeS all tne. ·lhc.O(Vle· eaYned ",V) t-hi$ peviool. --} ...Ct;+1
- P.JlAd.~et(~OVlstmJ(\t of .z,t\ol Peno~
not a f'n of £
% 'rh; clitlJ~· eV\ ·froM F€/ioct t
v~~) \'10 Dep'lecianoVl
b
Max i-l0., 1- 0'4) +: ~ l'lfs· U f'''CS), l-"'~+Ils)J /~ At[c-t -r kt-tl - k t - WtV\t - Y'e k tl ~
f At..-,(S) [..c-t+1 (s) - W:t+ICS)- ~~TI(S) - [1.,. '(t.. I(S)]. kt +11
"i,L~) The Ufiliht fuVlCtiOV\ is 1°9" virl1M1C .
\ LA (.c, 1- rt) = ku: -+ cfi.h (1-1'1.) I )
.~
;;t :: c McAx k.c~ + epkC1-Ylt) +~ t,lfs: · kv~~I(S) + et> kt [1-nt-tI(~)J
At[s + kt-t-I - k-t- WtV\.-b - Vtk~ : )
~ A,...,(.S) t.c<tt(S) - W",(S)·Vlt..(s) - D+ ....<sij-k..,1
foe: d;t = _1 At =.0-d];.t .£t
~ = - cP--L + A-tWt; = 0 ~Yl-t 1-l'l.t
-:::~ot rb 1r~ 1 - At..., (S) '= 0 d..c.-t-t,(S) Ctt-,(S)
'd;t = - (bTls G .-JJL -t- "\t+,(S) wt,-tiS) = 0 d Y\t-t,(S) 1- n-t+,(s)
;)£. =- -Ai ,+ 2. A..t-tI(S) [1 + vt+,CS)] - 0 . ~kt"-l S
~
.ct : 1 (1 )= , t7; 'A;)
T (~)ht :, 'P _1_, :: l6;W-t;
1- t'4
.ct+1 (5) : (bTls _1_ -::: At+I(S) (3) 4i" CS)
'
Vlt+t(S) :
Ai; = S' (5)_ L.At~I(G)[1+l't+I(s)J
~ OVl\"( i'ni<; vavia~ (!.aVl eonnect 1 Beg inro the O!?Jec'+lve.. ~.
A t<ep~evttanve. FirM M.~X'Mi ze~ pmfit (II) :
~ by 0(.,101 se"g of .f-ctc"tOVg of prvet t\ ervtploy(?o(.I ;-aki C19 p~/ic.t.S aG c;]iveV\.
"I / \
MeAX' 11 = ~'FO( f'J») - w·N - r·K K..N ~~-- ' ,
Let ~_(K_J_N_) _=__K~_N_"_-00-J)
1
Foe; = 0 -==) ('to = eO-'\ 1-00
oOAtK t Nt MP",
-=) ~ (Nt
~ /.. .!...
= ( I» -co1-00) AtKt Nt :: MPL
tk"
~ :::::. C1-cc") Yt N-f:;
toMpe.titive ~q.fA.i \,btiUM :
~ eOVlsists of cdlo cotiOVlS
- qIAClVltHie.s \ HH tkooSl 5 [ -c" 1'1., [!:.... t,), VI..,(SB:., ' FioVl c(,."ooSot (K K N· N )t, t+l, -t, tot,
- pn(c..e.S tY'ot, (,.J-t, [Y'tTl (s), Wt-t,(S)l~1 1 ~
/. ·treat ~ (k ~ pC! ~ Mete' ~
.-, Given PD'c..e-s,
tVu. a-UOw.tion f-Ct , Y\t, [c-tTICS), Vtt+,(S)]s:" kt ... ,1solves HH's prolo\eM . ,&
fu..t CdloCtttlOV'\ (K-t:l K-t-tp Ni;, Nt.... ) ~olves the ~f M'S problEUVI.
Mo.rkets cleClV" '1V1 eve~ dOtte .&. state.
DO. :. Sy
-" FOCs 3.. eu~t CoY1stro.iV\ts (.~xe usee{ to Ctl'ltl\"(ze. t~e. (eSOurCe alloco.hoVls."
os VJ0l\ a~ the ol'(()Qtv\icg of -rhiS econoMy.
.~
This ~tt ucdio\'1 gives TYle COVlcfltioV\ toV ~\lo ca.+iy1~
c.OYlSlAMpnoY\ g Le\su.ie optiMCl,ll y.
~
This is, r_r<.-S-'f:,-{1--n-)-=--w-'t-----cP-o-1-_.£_Y\_,]
~
4 (vtoX0 ~NtL 8~'1 \i hvl'utvl
~ tZttlv'\1'c., ~biel\A.
1- 1-Vl
K..t\fV\t. eVlQ.O'i"lyvlQV\-\'".
q
One- Pevio(i Model
AS$().(III e, 1,) "c = Y
LL) w = :t.. N
~i,~) V\, (Of N) will be C\. tOVlstc<V\+ g be- 'IVlcUpellldeVl +of wCt-}eo
- Ft:JM HH,
oc 1-1l4 {;Jt - (1,-- oa) ~A ~ Kt N~
== (1-oC).1. y.#; Not
= ( 1-00) .1. ,e,~ ____- __ "'-t 11;Wt _1~~]
Two - Pevio~ tv\odel
4 Lobov- leisure, cltloice 01Vl oiffeveVlt peviocls aye lin ked
)lAb.stitt.tteR (2) ,Clt) 'Wl1"O (6)
At :: "2 At....,(s) L1 +Yt + l(&)} s
ep '1 ::::.'~ I- lis ¢ [1 +Yt-tI(S)]
(1-V\~) ""~ [1-Y\t-t1(~)}(,.Jt-+ts)
~ ¢ (1+V'tt,)= ( 1 - Vltt,)· Wt-t-I
. (2J(1-V\~) , (1- V't+l)
10
As (b. (1- Ylt) -(1- Y)t-t.)
If W-e-tl '/ W-t ~/ '(b (1-Ylt) ~ 1 1+Vt+1 (1-Vlt;....)
-0 -!!I =) )
wt +\ l' -:=.) ~ (1- tlt) (1- nt-t') Wt,(1+V-t-t-\)
1-Vlt '> 1- Vl t +1
not; <- t'l-t+1
.~
LeiS"Ufe. IV) ~ eurreVlt p€viocl is ~lgltle( Yela,tive, -to the, \ejgCAVe 'If) ~ thtLt(fJ
pvloci,. 1- Vlt i 1- Ylt..-I
~
repre,j,8vrreo. OS Opf"i'hAVI"j-h-f CO's'r
==
::::.=
U' v
= ~ L lis [~ [1 + Vt+1 (,)] ( (1-) s ..c-tof-I(S) J I
4
I
11
~Lt~S+l-hfi,-ti()9 tV1e, bud.5e-t Cliv1strtlliY1f~ iVlto t&u.. ~L,(.leY t7q~atioVl ;
- (.3 uct~ t COY\st filt,-t Ot 1s-l- P~ 0c;l ~ {p,5-)
(.ttr ::. w-t; Vlt" + Y-t; k-t- kt +, .of kt\.. //
'::. WtV\.t + (1 + 'ft-) kt; - kt +\
- S\A~t CoYlstril(\"t of flnIA Pe-ria{)\,: (P,s-)
'cHI (s) ~ . Wt-tl' n-t+-l + (1 .... Vt-t,) kt-t'
~
1 := ~ E (1 + X't.-t,) ConsvMpfi€M- S~ [wt V\~ ... (1-T~t) K - kt...~ ~-t+, ~I + (1 +vu ,) k"bt.Jt 'DuicS 1M
~\·n(t. 1W Proci~ m i'.; Col?~~Vo~\()S,
&. f{-()"" C31.{~t" Cott-s +(~ i {~ t :
Y-t; == t-t, -+ kt,+,
( 1+ V{..,)
1 Yt - kt-t,
-
~
=
DC Yt+l
k'tH
~EPC~J ~,
:: ~Et~J
2"cot -::. ~
kt,ot-' -::--) Lk~TI -== ot1~e-1; {
12
\<t+l
'It - kt +,
k't+,
kt +,
(1 + (OEoe) kt +,
kit-,
£It
~ub~tHvth~ tVtls
.t. (N~
..ct
-l- (1- 00) X-h t1~~
(1 + 00 {b) (1-oC)
- ~Eof)
~e oC ('It - kt-t-.)==
:::= ~E rxJ 'It - ~Eoo kt +,
'::: ~EocYt
= &~oC Yt 1+f;toa
:::
~ Yt: - ~+I
-:::. y~ - EJ2- '1-t; 1+~(?
:::. C-~ ) y~ 1+04~
::::
=
into ~ -mV\~eVJcl( cono.ih'oYl 'fl oM p. q
- ¢ .-L 1- Y1-t
<p_1_ (;et ..n. ~ (1-oC)(1+oc/b)-1- r'lt l'
(1- Y\~)Jl. '= '4> V\-l.
= 4>/~ .J1. - .1L Vlt =. cP Vlt-S1-V\.t
JL -= (<\> + SL.) nt"
( 1-\1\,,) (1 + 00 ~') (1-~) :. = '" V\.'t ~t :;~J
13
~) {("ponse- -to sv.ockg
~ use" elaStiCity" to Measure C\ vo.yio.~le 's yeSpOVlS ;veVless to -rYie,
COYlteY\l\poroV\eous tecnoloqy shock.
1
ocA. A koo 1-ac k
~ t t Y\~ - 1;
1+ ocp
::: :1>
\4
e.14-,~ COYlsicieY" an econorvly cOYlsistin9 of a COVlstaVltp'opCdcxtlc)'V' of. '1\1 fi V1itely:
"rved IVldivicLuaAS .
The fepre&eV\-tative '\V\ctivictuwl (Vt().)(IMiUt; tkt eKpecteol value o~
It
o-JAere, e '/ 0
,) Sttictll( Corl CtlV e..
.' OLAtput j; \ineCtv '\Vl tCtpHoJ. + OVl odditive dLsmv0o.V1ce: / ,/p..)::~/,v.."'~~t r g
Yt - AtK.-e + ~ ~ ;; VlOtk.
kt" ¢~1~\~'
1vu~(e. is no cte-p (ec iatiaVI -:::=) ~lK_'.:,..t..:..l_::='_'Y._-t_-_Ct... __+_K.......:J
A£~Urv1e.1 fA = P1::: iVltevest.
FiYlQll'l, thQ dist\,AY"baYlce. followS 'a -tlrst- or-dey aLAtO(e~fessloV\ process,
ret == CP et - 1 + £.."t r """"""N- ~ lVleCtVl- zero i,~.d. 9ViOC,KS / i ""
_ . If ., " ' , F> 0 <.+)111Q:,~V1W,V\"t Idl!,Vll1eat eks·htJ blA.\ioV'
~
~e~~l = ~et" + 6~~;) ',···H
o '-9l1'v 0i;;$ (tfe f\..tec..\Vl-:zero
15" (5 1 0'1
(jt'f\U urH/ht is G~Clolyt(hC fV((V1
~ Foe blCOMes lineav:
"1 I
Conh0\ V. S pc,.;si\?le vall,lt of ~'V1()(ik.
~Foe: c: 1- '2.ec = A
K: /\ = _1_ 2.f:
'\Is YK [K', e(5)] 1-t'P $::.1
\/V
~
1-2ec = _1 2.. "s 'YK[~" ecs)]1-H' ".. , \
/ /
~velop~ ~ ~: v~ tK,e,l = (1tA)A
:::;;;. (1+Aj (1-2 ec) /
/~
r-~c C =
~
[c~
lb
FOC: Cob : o (~)
o
~l"a A~ P -t, -/
~) (1- 26C-t) ::: ~A) Et(At+l)A-:-/
V'-~Ct) :: Et (%~Je Ctot;)
(10): ~tC~, 1~t ~(e..'s ~vadoM Wal£'
./ j;l'1euV' ·fo{lv\
b) Gue-ss tVlat C.oY1~U1V\P\iO(l +o.kes title foVlVl Ct, = oC -t-~Kl; -+ ret.
Give V1 -this ~ve,s J Wvtcd IS Kt +, Q~ 0 fUrlChOYl of Kt ttVld, tt ? c) hL1~t v~lue.s Must thL paaOMe'tM 00) (b, Y VtCtvt fby iC-u.. FOe. to be batlSfie6A. 1by all
volue.S of! Kt ~ et ~ h-oM e\,t~t &V1Sh'7Jjn.-t-=
(!t + Kt-tl ::: (1+A)Kt + et;
~Ku, (1+A) Kt + e-t - Ct
Kt +-, =- (1+ A)K~ of- et - [00 + (b K-t'" Ye~]
= (1 +A- ~ ) Kt + (1 - 0) et - ocl~t~l)
Ct ~ G.t; C-t+,
oC + ~Kt of-yef:;::: ~t [00 + ~Kt+V'" ¥ etot, ]
-::. ~ 4- fb [(1+A - f.»Kt + (1--r)e-t - oe1 -+ r Gt et+t
Ct = oc + t1"-A ~f>; K -""~I ... ~~~.... ,f>(1-Y),et -- 0
gh1e..t et ::; oC + jb~t -+- If4 in ,il1eCl V to fVtll ~ ~
QIJv~ CiT = CO ..... \(b(1+A-~~ K~ + f(1--r? 4 -~ " V\ SlAlel'S
~
-::: .-::'y~Kt (!J (1 +A-fJ) \<t A -=
let : (!; (1- '() et. "='J '( - " ~ 1+~
~cIJ = oc-oo(b . ") oIJ 0
t
n1~ \JCtlue.g: cr thi.. PCllOtjVI~h~{S (Vll-{st be- ·fbv 1UL Fo C ·w b~ s·o.h SAe d
,Pl1\" all voJlAes of Kt a..,.J et
c
6 0
Kt+\
-!K..,
~
4 ~(W\~ Btte:h'oYl of '\itt,' S
:::::.K't-t' Kt + .L e't (b
'Iou Irtttve., K t lAMa e-t; ~ f'roclllcL
i\JeKt p~jiod Ou.tp~d : You VtClIlf.. \.<'t-tl:; K't + _1_ ei; 1+A
C1+A) Kt + ,• = (1+A-) [K + j.. 1.1
1+A (f •
(1+A)K+e
d) IMA.cAt ore tVte- etfectr of a oVle--tteVle slttovK. tv ~ on itJe pllth~ Ot Y;1 K, av,.c{ c ? W~ QSSlMVlt '\'t.v-. -fbl\ow,"j AR(1) FWce·5S
,.
for tkt- tecVJVlo lo~V ~V1oCV"""
et :=: <p et - 1 4- ~t l'
<!on~\c£uS as <fYll~-·\1(Vl.l., ~\Ao(Y-
C'rI~ ~ 1: 1~O) ... , 0, . .. j &7ns~t 0. ~ y~t~ of S'ntte VO\Ylttvle~~
kt~' -= k-t + .-1- tot;1+A
/q
~tttvt ftol'V\ tW. i n",tia,l pevioc! vJ.1e Ye. ~t;:::: 1
l~:l ~ [~ ;ll~l ~ [~l! 1V\eY\} ~e, follGw\ng pevioct,
0r~:1 ~ [~ ~1[;:1 ~ [;l= [~. ~Ar[~~ ~ [: .:Al [~}1
l10 :A11~:1 + [~l· 0I~:l ::
= l~ ~r[::1 + [~ ; l~ [~1- 1
- l~ ~13
[~:l + [~;:~)J
-:=.
[~l [~ :lf~:l + [~].o
= r: ~Ar[~l + [: ;A]3 [~1· 1
If:::: [~ ;] t:l + [f+A(1;3~+ ~~)J
/
" 2/
/5 09
C) Plat1Vling Pm b\eM
~ tJSe. 2'f1d. we,\mvt 1Vleov-efVl tv (ilaiw. title pflll?lefVl +0 -t1'()d out me PY1'CL$
"In the S'Ibter\ll.
\1H (5 BentMIioY':
;t = Mo.-x k.,c-t + <P ~(1- V\t) + ~ L i1s l k..c (s) -+ q,k [1- v\CS)] ~
- A[.cot -t kt ... , - Wi; V\~ - (1 +Vi;) ~J - '2 A.L5) f.ccS) - wCS) I'll!.) - [ 1HlS) k H.l i
tJlvp
rO.C; "ct;; @ =::
@ I"V,-". = A,w't
fvI gS..c1 1-"'~ '= wt -::.
~
<pet 1- (lot
1- V'.t; W(S) :: d> £C$)
1-r1(S)
~'ils ::: A(5) .cCS)
~ 'ils cP = .A (s) trJ CS) \ S 1,2.."",$
1 - r\lS)
~A L A(S) [1 + yes)]
~
.!.. ~ 2. &7i$ L1 -+ yes)] 'ct ..ccs)
1 =: I.1- ~ L lis J.- [1 -+YCS)]..c{; I...c(')
/
2'2
FiYf\ll ~ 6eho.vioY:
v-t ::
.c 1-oC ::Mo.xi\ Al t Nt - 1Nt.V\t - (1-t Yt-)K t
rOc..~ Wt : (1-oe) A",K; N~oC = (1-00) Yt
N-eoC-1 1-oC
:::. 'C:('1+ Yt) cO At, Kt Nt oeJi K.t
-=..1- ~ P2.l\s 1 .ct. kt +,
fu(Vl p, '\1 :
.f!; :: wV\+ (1-+y)k - Y ~
o(J 1-00 'lot ::: ..ct + kt-t' = A.d<t; N~
oC 1-OC £\:; + ot) (b,t;t - At; Kt Nt
00 1-oe ::::- 1 At K't Not. Eql,t\li ltN-tuMIJ2t
1 ....oef> I LottSVI"1ptiOr1 Ceo"'~r't -w p,12)
OC 1-00. -:::== Y--t+1 + ~"'I AtK-t Nt
oO~ ~ 'I-oC
::(~(b + 1) ~, At"Kt Ni;
Eq,ui \iItlviVM &0l,,1~
I
2'3.
(1-00) (1+oa~) 'it (1-y'\~) ': ¢' Vi; Yl.t
Let ...n..::: (1~oC)(1-toe(b)
-Y;
-:;:...JL ~ (1-n~) <\> ~ V'\t
.sL. - .Jl. V\ i:;; :- <PVt-t (cp+.n.) not; = ..n.
~ .il- lAm SupplC( '= ~biN' D~.tt l4>+51 (At -\'fA'§" [;((,e-ti ~ i ~Y1~
'2422 1 09
~ WMCUS Pa.P€vs en f2,eeJ. 6i4s;ille&S C'IcAe-s.
A) EGfwt;\('a c. Presc.ot+ &. FiVlt1 E. Kydla.VIC)l
~ sueks tv elCp1a.iV\ yeCLA(re Vlt .p [ue-tLAat10V1S IVl eco flO (\11 jc ac+ivHy ~
Mo-l<IM\ zi~ gtrtlte~ ie,~ r() ttu.. Ivtovkets.
-'> Key pcuarVle te~.r ()re, QSS i'iJrleet volue-s bo~ec{ OYl o\o~evvohoY\ S VlO+ di{e~tl~ (elo.ted
tv \)vtSlness cycles
f?Jo. - HH'5 b~iora1 responses
.., fhJot~ Input ~VtavU
~ tlVlalt{-ze bv.Si(le-~s evCAes in Z. stepS:
1'tot ~tep ; lbvti \0. a Mode-(
4 inc{ucltd OVll'1 yeat qp,uVltief. i )
t.e. olAtp(..olt J re-Itdive. pdUJ
t'~ 5te.p : 6X+eV1et t(,.\J.. IVlOck!
w "ncLucU- ViOMi () aL q()'QVltihes
~.e. (Vlot1e~ J.. abroLLtte P(lCU.
~
to()cwSiOYl~ -~ ~econcJ. MC4-~ be uVlVleCeSsQr'Lf
- C3uSillesS C~vle g ea V\ 't;)e expla.iVleet alMost eV\ti(e,l~ by just real QU1Vltifus.
e:tQY1O /Vll.( is gf-a;te. (kt 7 &+;)
rvl,(~e,lI\old,ts Bekaviov: 'tVl,'r eKpectafuV\ is Ott +1tv1e 0 bosevl 01""1 dO+Cl Cl+ -\l(LV\t 0 ?
W1C-"ll-tt\V"I~ "\1f.vducut
- ()V1ceytaiV\t~ IYlvolvec{
25
e\a{;.livilvt of gub&iitu\iot\ %et 1 \e,I>!.t(e" J 1-« ~
( C -¢1 .l ct» - 1 . Glce, 1-n.) = ' 1-0
-t e\a\;n(il~ of ~~'jA.l-?~·hh.A.nl)Y') ~vv (Jetl'oc1.
~ f'.' j E \ t:'. 1~IS(,{(e., (:fl.,u.. lhvt€.- e,tIlJov.;:vtRvvt Is iWfMcJizeJ. to :1) nl\fI\. t.t ef ~q(,('U\Ot1: . ~
DC t r '-t 4> 1-~ J ~{lltei'S:v1,(t g..el"'\f'lV'1{\"~4NtMo..x Eo L ~ (Ct.l ) - 1 iY!J 11 171!- \({,l.y'i+~ ",(1\'\
t~o 1-If ' «WhWIUM.
~~
(1-4» (C~~lt~)-Y c~~,t~ - At t-h 1-cP 4l -¥ 1-cfl 04'-1)'i' (Ct ..it) (Ct At '::; Atw-t
l-<.t+\ : At == E1;At-t, (1+V-t-~)
:::. Et ~"':.')C1-+Y'-to - £) .
= &t ~(c:~; )t~l)o ~~!~~,J (1+ Yt-c\) "--'=<t---¥' ~
~ 5-b [ (C:+ l It'1-' f 7f
c~~ It+. (1 + Yt - ~ )J t l'
.f-t
=
0t'f1(,t. (kt ? 'l./;) ,s e.[Ov1()(I11.t'J r·t-zt;re.. .~) FMc{
CD F(,'~r5. ?OliVi fV1 II (k i7 'l~) (2) i.fr..we,~id<~ tbltCi( h'l: )if"t )(t. J 7 t- )
4rv4.<st I_~ ",p-nOAaA.~v
gNen 4W.. p-rici~ ~
If) (k ._', -= n't:)~t)
. , k lk"7'Zt;) ~ kt;
4 ~~~t b~ opn (,-'It'\£.,._ . qlvl"1 'It.... D,ICl,'~ f,\ .1:,: I:~(.+ &f
. ~oH~
(]) A 1(;\\(11 ~F Mf)l1r."Yl k1.' : 1'"A:-,\
C (tt-l:-? V-t, lL) ~
qrk. "2.. \ " "'t; J z;;;.!;J
® ~;';\d/'1 hr.·~ (I'l Ck-b~ ~I; )
r(.~,tb )
2b
vr ::;. "lot - Yt k t - WtVlt
Zt' f (Kt , not) Oar-put Ct:.,nsJ'fZullt-- O(ifp~l.r fuVl 0e {,t,/locuted. .ft1 uuveVlt Ct:'()( \Ylv'e,~hV\eVlt-~
C\Z.S PioJ"~,
tUVtVlOLCXj'i paritJVle:r-tr gltwlks ~ n:u,\o.OM V
: B:h. , V\t;
::.k-t ; =- (1-c(»Yt kt
! Nc:t'hovuJ JvQ;M e
I,e U~· IVll:\1UY1CJ. Jl1W'M{J./::: O.J.;i+
:::. yw Iflt. (l(,te,., + d€pf€c-ia.:Hot\ rtlte (RHS)
t ., ~ us. = t+cJ~ :;. O,Oi..f tY-OIYl K'fcJ.luvu;.{ ::: 0.0-1
~ ~ifelo{ ii1; r (o1a.t/Wh\J2- ::: r0 l{ ~'S
if ~ -= O·b4
C1 .... cP)~ C'/~b
~ "::- 'l.b 'I
L.o.w of' Motion: J......k_t-t-_\_=--:-_~_').(-t_)_-+_(_~ -_6_)_K_-t~1 ~
6 ~ ·fj-v.6!i'OYt of Ov.lp,..tr-.?!:i + (1-b JIk:) )~ lV1I/~teli\,
Y1; f' ~ 1-0 =.lYOiAiCiVl of O~{\r",tlf ' 0,1 i
't<; WI)~i.tMecl,? 2' . II) ~\~ f\A.V1 .~
Mafke+e. (UtI;
OUtplilt Mark£tg c,[eCtv -:;') Cob of" )..lot = y~
~TVlput MCtvLdts Clettf -:) C4 kr
s S =Vlt Vlt
l(.lvvt l09 :
-)"kc~ + ~ ('t-1)hvt Wi:; -:::: - P -O~Ct1"' -+ cPC¥-1) kw-t-t, + Yt+1 - b
;~~:+~ + cf (~-1) kLWt
1.5
- p -t- Yt-tl - b + cP C~-1) P \---y--./
O.IJ"i... o~o If
~ (,'"'t+1
13/a'iVl v'2e-' I
~ vJrl,tVl p::: 0,02. ~>
=; CoIVeSPOYlJ·tv LCj
I
P ::: O.ott (6=_1_ = tJ_q'f 1+~
t+
G-hez & BeckeV' (lq~5) :
<:p == 23
2 NoVl- Mavket Activi ties. 3
;
1-~ 1 :. )4V1
T -~
1-cP - n ..£T
~
.f -::=- '2Vl '1"
Ma7Y\i ./uJ£ (l.p L\
•
f iC neVl0o,L.<M, t-tCtVl~V\ Owtct S/Y)glewVl (,qgq..)
~ ::: £5 b
.J/;
1 :: 5Vl
Te(ivlVlolO~lf ~~ock.: C~olovJ IS)
Y :::: -z K1-~ VlcP
~
Not o.CCOIAV1t (tw qrolN'~ In Input:
Prescott ~
6kv bh~ - ep~kV\-= .'\" .
\CV\O VIi) ~ 0,1: vd\'! Ct I
o.iJ&.t 1\'\ v,s
ealmllLte, V(XY'(AMAl) b.-r YUnYl~ Ve~(e5SI'oYl ;
L1 ~ ~-t - D.tvt, i!t-1 4- ~~
l-t; = P "21;-·1 -+- 0 ~ 0~ 1.
V(J.y (~.fM l.t) ~ VOV' C~ivt y) + e 2-Va'/" (A Jvt YL-) - 1. e Cf;V (D~Y) D~v\..-)
J. Wt-. ~ ~ I'\-z.
J'tteaSlAve,MeVlt BvV"ov (4~ V\.) ~ Use, iJ surve'1 fo
4 A su'(ve~ of eMploVevS ccttr~aVloL ~ioLe.) H-H_
4 A ~uyve'1 of HH (~upP"1 siclL)
To f1Ylc! VQv(~.tu Vt) =; l;(Se. eov(bAt-Vl", ~Jvt. Vl.-z) rY1stectol of VDV (;::,1M, rt) ''f ,1'
VI + V1 Vl-f- Vz..
v
33 2'7- 1 oq
GH2. 1,
Llo/u.lV' 1)'ffeV'"9V1Ce fq,(,ttltio lAS
fiaM : ~e'" FoYMev-; "The MaC'(l) econof'll ics of Se,lf~ fulf; lilt; fl9 Pmphecles ~ 6!1t,2
--S) -this ~ecHoV' foe-uses OYl a ctepevdence Df itJe pfeseV\t On The, pa~t the CtrfteS 1Y-om
-\11e ClCc,uMulttnoV\ of C-Ctp·,ta,,\,
A) l.intari-z.in9 NOr1\intAV" Mod.glg
0.-) Nontit1eaV" Models to ftpfeteV\t elofloMief;
;",ppost, [.Yt '" fcYt-1; ,,;,) 1 / .
e:'tDC;£ Vl eo u SeVlclo:JeVi e-o us vCtYi(;lbili.
VCtriCt~ 1\l'
G!eteo"'i Ylt ct ~ '\W. cl.e,terrvltlle.et by wc+oyS it-lCtt aye. Q(.,(~iclL of econofVlic 0e-k Ot ~l'\.ts ~ cRoMaiV\ O-P iY'lC/.;uify
~ GNP ~. weetthev; polih'cc.l p'ocess.
lh(s etLlA.O.;lioY\ is 0. c;(e,~c,(Jption o~ ()l,,{y be..li e,f a~olllt tCtu- wCt~ tYt~t Yt- d~P~Vlot$ OVl
- its OwYl past h,stoY\f; Yt-1
- e/<.o~Vleo~S vatl(i~; ),.(t
refllc,f NtfN ~~. f\Aotltl ? Th"S qropk Lt~lt<dll( I)seet toY' DVYlo.MIC MOr1J)...
f (Y't,-1 , i(.) 1'V1;ev€. (Xlt (VtO-lt- tVt.aVl 1. tteAA4. gmte.-.
111i~ IS still (). t<"Ch'OVl
-bY
r U-V1 (}taI? \e. If Yr:J be,~I\Y) 1/\ ~ intef\lU, CY~ yc.) =) ~ ~ yc
'SrUt£.t'{ t;;~{e"
If Yo V' (Y~ Vb) :;) Y6 4\A y~
b) L.tnear i2Jf19 AtAtot'\OMOI,(~ Eq,UAb"Or'l
~ FUVt6fioYl f(,) does Vlof deptvtd e~pllcHl~ on -HMo(.-;=-.
'! == f(y,x)
This fu,Vlc.HoY\ fee) has ~ fixed.. pO'If\tS' a,t y.tl, <, 9'b~ 'Ie .'
4.o.re ~'lnte(~eC+ioVl~ lY\ .,1tv,. ~q).p\t\.·l.AMt{e,
l:t. ~ y!]
-" At \Stable. steadl( etofe-; JY. 0= Cl + byt-,l ltv~eve b:- f'l CV,)<:) IS ~LL stope. of f( .. )
- - e<.otultteol at nI(eof polf\t y:= y-by
SoLliLrlOY/: ~ ~eC{ueV1c..e. tYo, Y1, ,.,} OpprOCll/iAf y
~
1lAe c(lffoveVl(L ~ft" W V10V111{\ea.V ~ t(·) ~ ttu ctfl/rOXi MttiiOVl b€co~ 'S Y"lttl Iev,
c) L.ifleavizifl9 NOVl,(l.c.(tDnol'4ou~ ettuationt
-'J ASSUMe." Yt::::o mnc{OM vc{Yia~ vy t!: S(v1ttll bOUVlded. ~t CoYltaiYlS ).(
~
-tc> cttHl1cl y ~ fey, li)
g Op(xll'XIMate fe-) ~ First-order' Ta1'[ov S~v1V; ~ y...
~ ~~
[flo 0.. + bY, + C~tl='
WrJe(e" C- ~ fit (.v, );t)
(A =" Y- by - c )i
-» \hL o.s5\Al"lpnOV\~ -y IS swbQ ( eVlsuv:e that tvre etevioJi oV1 of t)(t) 'h) fYl}V\I1
" v. is $'MGtll \ l):{, 'f ~ CUVI be kept C!vbitrovi~ SMUll "It by p,ckl~ 4W.. bOlAV\&S On ).(, o<ppr-opvitAt(!"Ll(,
.J~9(e-CULnOY\
'"
f(k~) 1'.5 strJ'Ctllf tOYlCttue, g;
o ~ S <:: 1.
kt
~I~TI C1- ~ ) K~ + Yt - Ct, ,
K k*1 tf; ::: (1-S)Y-c
Lf{)eo,yjLI~ fA Non lmwY" Dit'f'€y0Y1c.Q. Bq,LW;h'vv\ 1Lt- ~
:: otAAt o
Gross Ivwes-tvvIell\t :
$ eoostaVl-t gavi~ mte..
Lt ~ 1
0' -:::: q(7)wt'tA mte.. ot teul/1VJo lo~ j ccJ.. p(7)~feSS
cgi f\ c;., P(o) IS hDMO~(1 eous of d~(et i
~
1"kt Solow MoclU has Ot -0trst- orc;(ei'" VlO (l- \ineaY' dlfferenC£.- ettutlh~ ~
'0 kt.j.\ :: s· f(kt;) ..... (1- 6)~-t
(;J;Ie (e. k~ ~ Kt :;. K~
Ati,Ctl At - ~ ~
1
At ~ Pmclutn vH'1 qrovvt~ furtJlA.,lA fi·~
-::; f(k ) :- Yf; Y.,Yt t ~
Atl-t At
=�
~i'()u.. rCo) IS l"ilonoronie-cJLy '{)CyeD~jV19 Ir'l V(t�
{f.;�
The, rvtO£:h.;l VtCl5 o.t MOSt 1 ~tyictl( pos.,tive- gt-eo..ctlf srate tho.t satisfies ~
qoC'~ lVll.tvKe+ elearlllB equo..+ioV\�
S·fCk) ~ (o+b-1)k�
~ u~r~ r-/
k t ~ k t - k�
k�--W ~
I'
k t +, cA. kt
tiMeve 0.. == s: .tk (k) + (1- ~) o�
1V\a golow Mode) te-lls that kt LM11 eov1Ver~ to k .(l'
S·/r'l(.t kt is tkt fa+io' of capltaA stoelt.. to Cl '3WW\Il~ P(-octucJ1vity treYlc!.
y :=:~, "-" fCk) k -= K-\:
Ao 5 t�
fCk) VI;== Aao"t
-l/) CapitaL &. OlAtplAt qrow tv1mugk nM~.
~l,ttl K OMcA Y d.o Vlot qiVw D..t 0 Sre-tl d~ ({A tt.�
~O ,II< (»..cJ.. '( HuchAcrte, OroUVltt ().. 0IOlIv'''j +YeVlcJ. (iVl ttu.- U.S,)�
4 the fluc-hAa,tcOYlS Ctve- r'tt vtclo(Vl vov1CH~ft,s. ~
t()1Y"ottua ~ ,die<. of 0 SfocVJosh'c.,., c£ tfFefeVlve,. et(uation.
- -=
( 1y(;;
f,) ~lvi~ Rn;t- ()'f(}ev UneaV' Mod.tJ..' ~ 10 l,tt'ldefstaVld Vlow (). StOCV1tl~tiC lineav rvIocte-l genemtes De ~totioncj(1 pwb oUst,
\'Va, (Vtu£,r first \<.VlOW f,th o.bout -r~e Vlonsto.tiOYlGl 'f"( belt\. of tIrl.e. VlOVlgtoC!rlObh'c, (\IJoci£,l •
(}..) First- OrtJ.ev VeteV'Ministic eq,(,t(ltlOr'\$
NeW GIOWttA fVloettl:
A~SUfV1e, At stuble s~e{(dl( gt-(;t1E~,:
Yt ::=. fCYt-1) c.::',
::: (fcy); + f'CY)(YH - y..) "~ .
~ 'e:;).J ~~ Y- fey) ~/';-l-(f~g)'}yt_1 s\ov'·
! ,f('i> -:- --~~ ~ .:-j-=-...::.-:=-,-=-....--".----~r ./" ' .
I
IYt =' 0. -\- b Yt-1 I I i I
Yt :: fcy,):{) + ty(v,x). (Yt -,- Y) + fl.«(v,i~J().(t-~)
Ye = fo,~) - fy(Y,,~,,)y- f"CY7):i»){ + fy(Y,;()Yt-, + \. "----..r---/" •
---v--- -~---• CbPi ~ed. COYlst-a.,l,II,T
"
0-
Y ==' Yt;-1 ~
y -= Y, -'/ : a+bY
/
If 1'01 ~ 1
'It tv(lver~!' +l> a 1-b
o.s 1,'-'00 'i y"
1.f
1.r -1 '
'0
'0 4
L.. -1
0 Yt
Y-l;
eonve~$
e;tive~S
• ....,
gc;
b) R~t, Or(j.e~ Stoc~~ttic Eq,l.latiC'yi~
).( € [A 131 -;) IV1trocllAce Iike a SV10ck lV1tv tVu. c;;:¢efVl .. ~.
iVle 1){~tri~\AtiO(\ in -thiS loouVlol Vlevev CVtaV1~. ~ f (). Markov j?me-efseye. ()
If Yo IS ou~ldt. tW. rnte-rv",1 C ,/\/\/,
[ ().+CA , tl+CBJ 1-1:> 1-1::>
~ "0 YH l y" ~ -') tW. i{lt-evvcJ IH 10Y1~ t\~ 0,( b.e: 1.
" l\A1A~~ \?e. ;(\~~ "",is \oo ..d.. If A~ 0
A<.SUMpnonS:
'It) "l-tt i~ o.rali"VI .(}OM 0. pf?Jb. c(1s-t. tV1Dt is' jnvavioV\t t\t1rouBVt tilVlt.
i~) ~ parCHVI@,tey 0 L b L. 1
~
If (i.) 0'( (iD ,\; violate~ ~) it1...t MOcW.. will flot qeVl£>,ate. ex smtyoy)Of~ dtst.
toy' tVle va Ilit e.s o-f ~ eVlcl09 eVlOUS va v1a~S.
~oll,ttiOYl~ To Cho.rz;lcJeviu ~ se.qul?nCL tY,~ in Terrl.1S of f).(1;~
U(1ole"v COYld.ifiOYlib on I'7U1c/O/VI'l/').{t =) tftu- prol? dl'st 0f 'it (!oYlVer¥S to en1 invariavlt
pr-ob c{cH vy tW- IllteiVoJ..
4-0
XC) £ettLAenug ()f ProIo~loility Dlsm~1i£>V1S
[lot :0 (). + b Yt -] - NO(1&-+velttt<Sh'c PI fferen c..e eQuC{nOY1
c;olu-h'oYl ~ A $eq,uene.t Ot' iWb, O'St - tlttCtt obeys, the fZ.~cl.«(sive. Y'elctlio~SVli p cleBnecA Of +i1e ei(uCI.\lOJl\ Itrel'(
1/
rep(('SeVli! tW. Prob. tC1at- Vt; wll lie in iVle s~t S CIt t
qivetI\ 00 info +VJo.t Y "Yo cd elate 0
... , -'. t
."'.
, ... ~ ..
Att-:;.'2
1Yie~e.. caVl I?e ('Oi1S1'tucJecA. y-eeuy'give.l~ ~(;yVl t!A.£ know\') cUsts of' l ~t ~
If Ibl.( i =1 tG-t (SI'to)l:=l c.onvero/'~ to aVl if'lVo.viaVlt cli$t.
l'ilYi GtC· I Y,,) = G(·) V Yo t .. oe
Gt(Yt "' TI tY5~:=O ) As T gets lar¥y =) ; nf about tht.. vollAt of Yt bl2wiVle.s Iess useful y ~olo
li.:oe G-t('i4THY.~:=) = GC·) 9
••
Lt./
eJ SoLving Hi~\1eY-Ot'c;teY lAnea" MDd~l~: .
~ SecDYld - Ou).e'f ,Avttv (e0 (e.)s i,,€' Prvgn:trvl
//~
'f'he, n()+tAnJ. geneadiwlioV\ of tW... Y!fS-f-Orclev G't()aI1OV\ Cp· 3'1) is
(). lb1vcho.stic vectoy oiffeYG'nce eqVlatioV1.~ ( \ [Yt :::: b + AYH + C XJ ~~:: [, ::: :'~1 V:::,":ee:;:~: ~ndOM v
b ~ n >< 1 vectoy of eDV1St~V\tS
A J. C = MatyiceS of toetfic;evtts
SecoVlcJ.-o(ciev ve.c,tvy ~'1ste(V\ :
-IZt -::; b1 + C1 Vt-Ai It-1 + Az 'l.t-:z. -+ ] j
1Vle~e SeCoYltl- Ordef vec+ov S'vste(\l/g ave eabill( Vltinoled aG -Arot-ordey- vecto(~ysft-s
by inuea~in9 iW. olifV1eVloioYl of itu. S'tate vector avd (Qdetlilf'''& varjOt~eu ...
lY\ ttu.- Ma-h1ces s. vectOY al?ove. . •
1VtfV1, Yb -::: ~i +~'1t-, -I- CVi;
kev tv DyIWIV\',G q\{~ref'l' " va(uR.t- "}is tll~O (~VI e,¥Vl /.
II-'s ,'lot je,st ,1. S\()~, \:M \
~ The \?ev,o.viov- of tin;t-ordev- vectov- mode-Is is tOUl'1ct by deco(V)pO~in9 tC1.L
(V\o.tvi X £ytteM '\VIto 0. set of fi(st - orctev- equatioVl5'
Ei~tV\ vfA.l\,tes -:::: ·~V1orac.teYistic roots"
~ poYaMetey~ tVlC\t gove\'Yl the. b'tav'ility of' '!tie. tyste(Vl of ~ql.AC\tiOVlS
A :: 0.. SColo.'Y
, i
'2.1"'2- tAtI.n1X A
e1efV1ell\'I' of A
A tollection at' 2 vec.tvv £ "") Mtth1>< A -= [~ ~1
1 1..
~2
1- ---------
~-i-
1/-'3 Zq 0/ 09
AX - ,A T" .x = 0
(A - >.. I",) X -= 0
- Ar""X A [~ ~J [::1
\0- 11 o-)"
L Ct Z\
.. ~ ..
1$ \!lOMOIJlVlet'US eq~Cl.J11lVl qyrte.tVI.
=) ~1(l91,(IOt~ MCdYl)(.
iI' V
~ ....
=) (A -AI",)-1
eJ<1~tS
rvOYl q, i110 v{ Ial"" MOt..fyl)C
~
-1 (A-Ar",) (A-AI,,) x :::c 0
X -::- 0 (C<N1+<~diCtVM)
I.A I qeVll?1Pctef> '1V~ LllJ.Cll;{rzl. hC et(,uoJiOVl Z-'('2,.
C\A-"'\I~1 =
JA' - A({;t11 + an') + (A11 Q Zl - o.1Zjl2.l) ~ o f l/ \"
la., \1>thtVl, we, qo t A /\. ('l vllues of' ~)
-Htle((!., ore also Z Ii flea'lll.( cttp~t
et(ua\i0Vl5 ill tW- 'eAeC't~~ o~ ~t -U- ~
o.eterl'mYie,S ~divechDYI.~ ~~
VA;! ylA. ,
c' (Ay~ AY1
1· , ~ Yt-1
YL = b + 0.. Y"-l
I
6' b yb '.I.
A Move~ ev€cr pO;Ylt to ().. cHffeyent poif)t blf flu. ~Y1sfvrf\llc:1.tiOY\ 'I ~ AY. z.x2
Poi i')fS thoJ line.. 0Y1 tht eiqeVl vectors Y Gcre MOVect in a spec1cJ. waif,
~
b) Hi~heV'-Oretev Det€Y'f\I\irtistic f''l.tA.O.titmg.
gOMe- properties of ~q,()O:(e.. fV\O.mCQ,s t ti1eiv- oppliCCdioYl h? dif'feYeVlC£.- equatioYls
f) It A is syrvtrvtetvic =) tt-1L (Vots of A ave., (ecJ..
(Otv,eV'wl}.{.;> tVt.tl{ MCn.r be, eOl1l1plex)
2) If %u. 'fOot~ we c-olVlple)( ;:;.) ~I.( MCll( 't/e- "iVl~ic£L" 0'1' "oLAtsicU" tW uVlit evelt,
1{ M roots ore re 01. =::) "ls > 1
(J( As .( 1
3) H '\he. roots are..- coMpleX" => titLcAI COMe. in C,Orvlple)( paiY"'s
S<. boiiA roots of Ct poiV' are eitv,ev "iV15"ic;U" Of 'o/Atrid£.
+tv (A l1it C4cl.c.
1+) Ii all (ocn$ ore "iVlSicU" 'tCu l)V\'lt Cl.(c1e,
~
5) If fVu. roots aye "ou.tsicitt" '+W. uvlit Clfc1e.,
~
b) If 'ttv. svsteNl Vto.s at leCt~t 1 rvot t!-1.at IS' ''outsid.£-'' ~ uVlit tL/CIe-,
:j
4h
Y-t = b + A Yt-1 .+ :cx-j;;
,
A
IA1 [ .c 1 AI.\ ~ 1
1 00
o ).,,2,: o· ~ o 0
> 1.
1V1 (). ~ecoVlcl.- Order SysteM -:;;) VlClS 2 solutioYls', ,A'" awl Ab
~) itO ~ ,As < 1 ~') tC-u 8Vsh?1\I\ eonllef¥S fV\oflotoV\i cCtllL( to
iW- steCAett.( gtate-,
2.) it tW- rvois CAve ('j)(\I\~)( ~ lie, "
iVl~icV.,II tW. UVj·lt eI...-cle.
~
3) If at lQCt$t 1- root IS ne-~a\-1\Je" bAt both IAs I '" 1
~
theYl} tf,\,C.. systeVVI w\1\ flip troM 1. sieLL Ot itu steaC:<lf ~mte to fC..e.. ot0~v
Oes It to(\\}e(~s.
A source
B) Sf ftI.L roof) Ode, ComplQ,)( & lie "outrioll' "tt1e- uVlit Clvcle-9
~
to) If at least 1 root (~ negCtl1ve-J !:Mt both IAsl > 1 .g,
~Vl/ M sy1;-teM wl\ -P\lp {YOJY1 :1- slott of thL gteAAtf ~tate to tf;u ot/llev
as It dive~s to 'rntiYl"I+Y.
@ H IA1 1 > 1 =::) 4iv £teMl.( gmte.. I~ cCt(led a CaiJ.d.Ie. PO'llt
1/\.z.\ <. 1- \ -Y; ..'\01 )..2. > 0 fVu. Sys+-efll1 IS UI'1~+ctb& foy- Cllrvtost- aLL Ivlfh'c'£ COYdil)e;r(
1
(unless *u- Ce-t of iYlinM eondi11OY1 s +Wet be9i(\ on tW.. ei /1.U1V(2(:!VV" a.,soc.iat~ol lIY t!N.. b~h.et.. et'~V\value-)
e~", Xt ~ 0. + £Xt.-1
4 \b \ L 1 => COnve(~eV\.t seq,()e~1CL tv SOMe, [0Y7SWV1+
l Ia\ :> 1 ==> Oive(~V\,t ='> outsiok tW w1it circle
(not (;\ wMer$iVlt sequ en(..£. uvJ\es s Jo =- 0 )
.,)1
.... tI-.
•
c) V~ona,Ii'Z.ir1g SvsteMg of Nonswchasn'c e'luahoYlS
-) W Metl10ct of COMpl.khYl9 solutioV1 S by +1ndiYl~ linear COfV1vino,t1ov\£, of
-rmVlSttl'(Mea VO-v1tA~S,
NOl'\stvChttS+JC fl{[,(OetlOYl ~
Yi; = 'q--±~Y~-1 71D', '"'JJ
~ Oi~oV1Q.li2i~ A - by sto.ck,'~ 2. ~rqeV1VCtlue equah'OYlS.
'l l? • - Y a-t Yare ,n!hpetAdt.rl·
of e<>lO(" otLow',
~ .by t,\ncl ine )ineeIY &Jt>!VlflCthO'n" r
of tG1V1~f'vn"ed. \ioYiow.sLet Q. = [Y~ Vb] I I
is Ctcl-1ieved ~ cfiogo>1cJ'zAJ A , ~
\ "--..
0 l -1-~ :::: exA Q[A"() A'o
~
IA -= QI\Q-1{ . ~e(e /\ :; olia00Y1cJ.. Matrix of ei¥V1\1al\A£.-S,
1- WiI<cV\ derenli\iYie,g 'iIlSictt/OLA.fS'W.·~ l,{Vl',r circl~.
'1VI.e,Vl, d.e~\)~ G\. V1~VJ vect()v of truV1Sto((V]eet voriQ.?U s; "Zt
t< 0.. trClVlS forMe.d.. Vettov o.p c-oeffieieVl tr ;.. (1;, . ., ",
frOM Yt = '0 + AYt-1'"
~ -tyz:,,,sforMi'L5 •i
(~1_~ =: lQ1~)+ A\Q1 Yt~~/' '"
"..
:::
4 ~1Y1(..[ A is oliCl5onai ::::) eeculA of equa.1iOVlg is' inole-petAdeV1t tOt ttu. oit1e-rs
-') fue eql,{QlioYlS Ore 'ScalflY" dfffeveYlu et(~lo. noV)
-!> ECtllh, Z~t is ().. lintaf tDMbi(\a.l'ion. o-F Yt ~e(e" the we{~VJt-G of tke...
~orv1bino.tioY1 aie ttu. eleNleli\t,g Ot oYie of 'lOWJ of fuL I{werse of tkt. fV1Mvi)( [if
r(?t'0eV\ vec+-ofS of A
1J
(;fY)CL Q1 Yo/;'l,,- ~
It = [~" 1"][Y J~21 c( y~: •
- tL Y,t Ii Yzt ( )([l,'l "l
'lzt ct'2I Y,t t(l.Y2tI" If IAbl > 1 :::.') Zzt ::: 0 ~") q,21 Y 4- 17..7.. y =01t zt
11..1~Yz-/; - Y1't
, q}~ . '/
50
Z2.t := \
If =; 'ttu.. S\.{~teNl (;VJ1\ tonve(tp +u C-OYlSraht
~) ... "'" .. JIVM'jC MCfY\OroV\iCo,H1 'tn-Rn"itv,
0If t=l =::-) t.11 == ~1 + A "l1O
t=2. -;;') ~1l ==- fbI + AI).. C~1 -t- )...CA Z10 )
:::: ~1 -I- A~1 + ()..o. /-c10
t:: 3 .,;:0) 1.1~ :::;:c fb1 + A0..[~1 + A~ ~1 -t- (A0.. )~
l10J r
i I
tl. '1 Q.)3 0
- ~1 + A~1 + (At:<) ~1 + ()\ l,o
0
1,-:1. . t
Z1t -::: ~1 ~ (AfAi + (;..0.) 11o J"O
•
5.1
ertj Plcm Y1ev- eq(,tl libriutvl AIIDW.tiOr1
~iep 1.: Oescviption of fi.U1daM eVlttt:ls
~ The eCO(\O~\f IS' C!escYibe-d loy ~ follovvivzg fUVlctCliVleVlmls: ""'" .;:<tl'416\
t) PrefefeYl ce-,g (',ngmVlr-ClV\eous &. tntevtef1t1ponv uti llhf FL{VlCtiC'YI)
L~) tec,{rjt10 l09Y
iii,) (). tOflGtmint S'et.
~ f\!l(;:lvkets +VV' exc!tlul1gifl9 C01YJM odihes5' l'oYIrumeY'$' OY Ar{1llf O!/e no-t
expl'citL~ gpeci fied
-") The, pltlYlV\ ey CVtoDSes iiu. OpiitV1M tOrY/modity o.lIocanoV1 ')
\N,;& velyifl9 On trocU !< pn'cu.
~
1Vu. p lCll1 l'lev c-L.toos.e-S - ClBWe9Clte. e..onS1ANlphOVl
- 00gfegecfe, cop~e- stoCk
1;\ LOboI'
PrvclV\.
IV\~to.V\m.n€ous Utility FUr1ction
IU(Ct)Nt;) - L09Ct + e,Jtl ,
Ct; :: o.qqregate... Con~uC\1ptJ'OVi
N-t; ~ V'- Laboy ~upply
.f..f L-e[svre ?
Fov CaSe Z Cl2o""ev 'S )l
Tect1Vlo logy: [ f O(~)Vi» At ~ K~ N~-oc At ]
(;bnstrtti nt ~e,t:
1) F~s;i l? ility CO()SfntiVlt
[C i + It - K: N~-c<J At I
i,i) Cttpita,\ Ac,curYIulat1OYl eo~stmi(\t
~t-t-I ::=. (1- ~) Kt + :r t {
{;vV1eVe.- 2J (; [OJ 1J ::=. qUlAderle.r depreciafioVl rate
U,~) Leis-CAfe. - Hue..cr~ Ct>(lS+m;V\t c-nr~ CoYlst{(;lI/lt)
r~t+.1c = 11 ~v) AGSL{Me- tV10t ikL In"d-1M CCtpi-rM g-rock 15 po~Ih\le.- &- if IS g/Ve-Vl
Ko > 0
S+OC.VU;\Snc Prod,uctivity Shock
1At-t-I -::=- PAt + ~HI (
-vJ;,ere. it-t-, -- NCO, 6;)
rn~M~tlon : (<7..- It)
~ Ct , Nt, KHI ove c.Vic?Sen b().S"ed on ti-u. info L\vQila.ble., at tlfVl€.- t_ ~
corrlfol 'Ia.rial7les
53
~te.p Q. : Objective, rc.trlC:liOVl & Constmit'\t ~et
~ It's c-onvenieVlt tv reduc.-e IYllAn-'pulate. +i-ct Co'rJstmin,t get
D EliMi nette ),t In tkt Utili-h-t fu
- by QubStitutill9 -f'tOrvl +-kt L-e-isurt- Hour!" Cms-rrnlYlt into IV1SWVlTUVlOUS Uti/~ F>,
!lA ( eli, Nt) = l09 C-t; + 13 C1- Nt-) (
<Xl 1-00 .c/:; + K-t+1 - K t Nt At + C1-~) DC E Co, 1)
At +, = PAt + "1;.1;+1 St+1 A.- N Co) 6~)
Ko > 0
Step?> ~ PlanVJev OptiMization ProbleM
Pletl1n-ey-'s task ~ To jVl"'r1i(\t1ize tiu..lnteYtf'(\t1pOyAA LJt7I'n.r Df Cl r<epreseVltcH7J£
CoY1$ufV1eV ~ubj eL-t tv tf,u. CoY1.st'ru IYl t- J'e-t.
-!:Y
PICmVley ClttooteS equilihqv(V\ a~q(etJate. q,tdintities fV.- tW. e-J,zole econoMy
£t· Ct + Kt-tl = K: N~-oeAt + C1- 6) Kt
At +, ~ PAt + £'t+l
K. > 0
=
gte.p '+ FOC&
Ct :
Nt ;
K-t.tl :
1 Ct
.1l (At'),_/
-==
=
=
~] CAt) (1- oe) Kt
oC
N;CX)At; ',,_/~
~ Et [A t+1 [<X) <:-,1 N:~~ AH1 + (1-bJ] ~
TVC
. ,
IJ
111eVlJ eliMj YlCttlY\9 At" ='> PlclYlYl.ev
1tt.t follolr<ing eqIACltioV\S;
ettui liWvJV1 I'~ ,chafGtctevi'Z.e~ '0'1 f I J
.1C-t"'
=
TVc o
Not~ There eire.
f".
S
8
eq,.LLtdioY7S
vaviables: Kt , Kt +1 , Ct , N-I;, At;
-
58
~tep 5 : "J:Vl+erpremtiDYl
LllboV'"- Lei'llft Cl1oic..e. 00 -00
B - ( 1- DC) Kt Nt At(Jt )
f ~ t
o~ v.\i llR-( M PI.. . MU~~ U1i\flv1
\DSS ~ of
Labor '-
Vt11f.h-t Go.i(\ sn\ib\\t'l
-) OptiMoJ. proviSiOYl of Labov is 6V1oseYl bll eq.uC\.tiIl9 tht COS<Atilikt of supplyitt8 i Move- \.Arf,t of labDY (0) w/. tf,u (VICtV'-oinAt 1 pro{}..ucAivThj [C1-oC) J.<;N-oCAtl
\tJe:l~ntQd tNi'th ~ /VtuCt
-7 This eq,uanoY\ hold.s on 0. peviocl by period. ioClds-,
gi I)C e. It ctoe~ not i nvol ve Clny expectetnOVl to" fV1e, N+I,A Ve. ,
t MUL ct-J
lJtil~ loSS
-7 -1l1is c-onc;fltion eq,lACttes, at t\tle M~Y~ iV), The. ufr\Hy loss at -nf\lle t
(c{ue to postpoVlifl9 tv next peviod 1 tAVlit of con.l'Uf\Ilph"OVl) :J
~ itJe v+it"d\.( ~~·'n 0+ tllVli- tot 1? -r-o.ki(\9 ·Into account fA c{iscoUYlt
-to.c,wv ~ IX. tVLL (e.-tu(V1~ -f}orVJ fywestin9 thClt conl:'UiVlphoYl un"/t.c; I~
~ Tvle elJt!eY'" Eq,uatioVl dnves rVu.. plaVlV'lel'"'~ opn(V1cJ. CoYlWlVlpHOYl &. ,'fIVesi {\'IBVlr
(OVlet itte<ewy/l. itu.. copitM ~iCu(\ltlAlQtiOY\)
5b
I~lAPPDU;,- we" cue 0'" pOlioci to·'.11:; T
- CO()~WV1eY.l 9;v/2- up 0.. un"! t of ConsuMption
~
Vtili~ Loss = (1-\ \Y
II . V
00-1 1-OCvWllvV\ has MP = CO Kt 1"' Nt +, At + 1 - ~
~
o 0 TOMoy-yOW'S expected. retu(t') = Et(ooK~~; N~: At +1 + 1-~) Wvlev€, Et :: expecto.hoV1 opemtvr} COYlcfdioYlcJ at time, t info:
o ! '
'The, pr~seV]t d.i~couV1t vcdu'e of UtilHy GC1in
~
C f (Kt ) Kt + l ):::0 (t
C-t+1 ::: f (Kt+l, Kt +2.) r ~
1}]1$ \s ().. Second. on;ler- £tov~asiic Viffe(erJCL e"tL<&ttlOy);1 IAlt1t'GVt 15
oHeVl Callelt Eulev fq,ucd;OYl,
Fea~i b'i Ii1'( Covtstmi nt
[ G~ + Kt +, == K: N;:,'" A"t + (1- ~) Kb ]
-miS equcdion tells tltlat tt1e ?1C\V1Y\ev- ()~es e:tll yeSOl,ore-es C<.vai lable. IVl
~ eCOYIOM y .
?Jut If -tYJis FeaSi'oi1i'1lf COY1~taiiV\.t l;Je(e VlOt ~atisfiecl ~ et(uaITft-t,
+hen £Ol'Vlt feSOUYCes WOlA,lci Vlot 'oe wed. > .
ConsuMers e;tve Vlot Ovl tltle'lv- bVlct0e.t Ii V\e.-,
-Vi
•", A-, lOI1~ Oes titleve.. VlD btAcl ~ DOOI..S (i.e. poIlunoV! ) :;
COVlIU Mevs wo(,(.let lot ~t1eY' off by MOV', r19 oYl tltu. btAct9M line. -' i Ci;
Iji1uCltjOVl Cl ,'os.! stlA.
"Tntflsversalit ConditiOVl (ne) - Musr be f(lilpoucl ?,I) orct..ev-tv (iAle 0lA, Ice vmiV1type.J' Of etl'.i'lib1o
o~~~ ~T (~T) KT
1 /1fI %a.d.ow ~ (,t. of K. ~ 0 ':71Yut-- !;..Ie. ass~ u'/ili'ht if. Srvictly
/ en cre.£!'1'Y11j . // ---') we <AVe. gOlvi~ QVI 'Infinite VlOV"ZOVl pfVblefVI,
! ~ .' ... I f Kt ~ oc ~ \ WQ, ~V1oulcl p(eveVlt titlcxt 1tle. CtApircJ. stoc.k -follows an Q;<plosive ptAttevVl"
~ ~ ~ TVC E?V1c;u(e.~ *tat explo~ive poHeY'S <Ave ex:cLvtded 1YO(Vl bei (l~ aV)
eq,udibriulV\ sollAhCiYl.
It 's V\O+ worth occurVIvdcding a. so lox~ OrvtO(,(.V1t of ~apita,(
( ~i(jU- its' p-tJ'et. = 0 -:;:;) iW- stock Vlt!S litO eCOYioMic vcdvu.)
1h1. deYivettion of t(,u... TVC is preseVlt-eet In LA S'e.PCHPte. se.t of V10tes .
e~t'f~ Competitive Equi\,bviwVl
~t€-p 1: DescviptioVl f>t the f3CJJrlOtll\y
-'J In -rhis eCOYlOMy;; ft1ere, ('AYe
{i(rv1S
VlO(Vl0gevlOU$ gooel,
-> CoVlsUlVlerS" g, {1'('(Y1S tro.c:lt- comfVIoc:fdies 'IV) tke- vaYiou$ t\tlavkgts fbv goods
~ ".""pro", Inputs,
OvNl itu, eap', tcJ.. stvvk
l""eVlt "1+ tv t'w.. PivlVlS
guppllf laboy tv th.t +7r'lV1S
- ctervlClVlcl COY1SUf\I1pti"oV\. 8< jvwestMeVlt tj0ocl.5
, !
deMand.. ot{pi-r-cJ. Gtvck t. Iol?OY gevvius \ I
gupp\y t~e vlOMOBeVlO()S good (c-t; g. It)
~te,p Q: CDnslAYV\er-s
~ COr1S!A(V\er,s o.ve lAV\ifoYrVlecl.l~ c1istvibvtted. Dvev- tvJe lAVl\t '!Vltev-VtM.,
~
!vIsmV1taneo~s Uti Ii n.tJ1oti 0 r\
Y
l). Cc:, .-e;) = L09 .c; + B "it
/rJIIeve, B > 0
IV\ter-teMporr:t1. Uti Iity furlCfi'an Y
lA ( t.c~ t:o ' [~tf:D) - Eo !o ('t (Log.c~ + B l t )
hVl€l"e, (3 e (OJ 1) ::: of$" C~,lV1t .friIc;n:w As> v.M e., edl C0>1sutVIey£ V1tA.,Ve, +W SCtrVIe. <;ubjeeJ7ve. cHSCOUVlf" factov
Ccrlstro.iV\t Set:
j,) FeaSibility Co~stmiV\t
[.c: + L: = w~ V1~ + Y"t k~ I v===::>
t()YlS"'lIV1eV~ InCOfYIe-.
~ Wt n: WO:JL toil I
r~ k~ == YeV1t7~ e.apiTovt ~toCk W 0rMs.
~ &nS(,((1I1el.Y tttke. P'J'CQ., VVt ()Mol ft; OS given.
~ -e; Cl;v\cl C~ tlye.GlttoSfM:OU.tof'+w"S'ClMe.. q,uaVttiht.:'·
CfVleC{V\s/ Output y IS nOfrvtevliz.eot -tv Vl1ili~.; y;:.c.; + ~:
ii.) CapiW ACCUMUltt-noVl CoV1.Str"{,liVlt
f kt:, = (1-6) k; + ~~ I t,J,1e re 2J £. (0/1)== o{),{cdterll-( eopfu<1 cleprecianoYl mte.
k: = tVu capiW ~tvc.k. oW\'led by 'Y-tV1. COYlSUMey
~~) Leisufe - Hot.trl ~t'lgim;(It
[Vl~+/C =1! lV) AssuMe/ m itl,tia.1 r:apTW gj-OCK. ;~ poSitive.
[(¥ > 10
\J) ~Wcl-l4Stic Pmduc.tivitv 'lttoc[.(
.rAt 1-' - PAt + ~1-1 1 Wvie(e St+1 """NCO)6~)
rnforMLl.nOVI : (oy ~;)
~ .c::; Yl~ ) ttVld ~k:+, ctve,." CVioseVlloo~ect OYi itJe info Met-do.ble, !At tJ'l\I1e t \ -v-..I
Conrro\ vavlo.\:7les
=
Ai; ==
eV1(;{0Bevwus q,+ttte. variable.
exogeVlDu5 bmte. vL\viabU
£tep 2A: eonSIAMey'S CotllStmiVlt !'et
~ It's coYlVeYJieVlt to reduu rYItH'lipulCtte ~ cOYlstrz;tint £'et
, ,
,,
Y y-ct + kt +,
A-t+\
k:
=
:=
'>
Wt V1: +Y-t;kJ + (1-b)k;
P A.., -+ ~t-tl
0
S.t. y
L t y
+ kt-t,
At-t,
kY
"
-=
>
IAI..,ltl: + 1'".., k; -+-
PAt + ft+,
0
(1- 6) k;
b1
~t~p 2C:
Foe: (1)
Ylt y tzJ
= ('f)
T\!C : o
el;MiV10Jjf1.~ ,At :» [oYlSi.A.fvlef'S l?t{u[ik"'1V Nl is- CLttamctel1zed 'o.t
tkt tolloLN1Vlj W1_.~;.::.et.:-i --.+i-.::.'0Y1--=.~_:
e,
TVC
'-.----~-~-------------\
f\Jot~ 'TVtere. CAfe 5" equanonr ,&
5 VClY1Cd::;de.S
Step ~ : Fi (tVI S
~ FIrMS O:r(!, I).Ylifol'rY1ecl\~ c\iSTv1'outed ovey the tAVlit iVlteyvcvl
L.e. f € [0,11
- lol?ov c1e(V1{).Y1a
&\ Investf\1eVlt clef\l\ClVlct
"Foe:
('-t; =
~~ l}; CoMp~t1tive. l;q,lAiiilt'YilAlV\ \)efinitioYl
fs 0. Get Ot tontinl)OUS pnC£.- fiAVlC,tioVl /Nt, Yt suc.L,-, tf;,aT
1') tonslAlVlel1: ave. MCt)(it'V\"\zin9 [)ti li~ - (b) holds
z) 0rrViS elVe., MCI.l<iVVlizinB prvfits - (7) VtoJdS
3) All MCiYke+- e"lear-s ;
UJ Cupi1d-~tvGk MClyket ; ::i k: ctr Sk! d-f f
~LLJ Ivwesfl'VleV\t MCi.vket: J~; eX- y == ), If et t y :J
Lv) COrl s\)(VJ PtiOYl j\/\()yket : L.c; Gty ) .cf elf := f
b'3
Co (V1 pe:ti h\le., l7q,ui li ltMUIV\ IS elttCtt'ttc,telizu;l Ic.t +t1i -to llDWiVL~ . ~clinO)/l ~
16 = .i. //VI;;.£-[
1 - ~ ~t [(~tl) (YH1 + 1 - 2J )] ;:..;
.c~ -+ k;r, - tA!tV1; -I- Ytk; -+ (1-b)k;
~ /' TVC 0~.r:; ~T (j;) k;..l
// At +, - PAt -I- ~-t+ I
!
Gft Gfreew-\:; =- (1-oC) V"t ,Vl t At-
r{, =: (J/r- I GfcJ] t; n-t)HC At
rhl :: at! L1t-t1
kH1
Ct-i.5: NoV"! lJ,tvchostic £tettcA\f gtate. (NSSS)
AJ OeYivanoYl f)f fA NSSS
~ WrJeV1 we, <;olve 0. dl(f\D fV1i C MOcW !At (ASing Cl Lo~-~ln€aV" Appn"ciMtl+ioYl
~
~ NSSS -:::: tVJeve.. rs (lO uncevtCtIVlty
r At ~ PCA) =) A]
~) NSSS fDr- ttu. PlaVlnev- Cq,CA.i li~Urv1
-) EClUililv¥iLArv"I (<; GVto.~ctevlU'd 10vr tke- ti?1l0wI equcdiOYl S ;
g =: _1 (1-00) K: N~oe At (1") Ct
(2..)..1- :: ~ Ei; [..1.- (oe K::,' f-J:7 Atl-I-+- '1 - 6)1Ct CHI
Ct + Kt +1 = K~ N:-oo At-+- (1- S) Kt
109 At+ , - (1-p) l00 A + P L0C) At -+- \vt . '-..J .
) 0 :=: liM E{i>T~) KTJ ,/'f""oc CT
/
//
N!1ere, ::: 1Ai\t]lte vlOis- e..- Ar"Vi
b-5
,i1".rV1
iheV1] Ctbstrv.c.;!l'ng frOrYl ~ 1VC, tV1ere CUe. -:5 etlucdiOY1~
- '3 UnLMO~Y1S: CT N- K I , " TT
I~S [ ]l~ = C, N, K] NSSS Ve(;l-ov
c) TVVlPOsifl9 (). tJSSS
1) E?11(vi'IYlIA+-e- e,Kpec,t"uhOYl open;d"OY1
2.) \)<l)p 0<\\ bU~Sc,vipts,
o se,t?tll voYiuWg - S!; value.£
2; <:SiGVlp\ify t solve ttu UYlk,vtowYl (c) iV, K)
os (}v -BAVtchen of p0((il(Y1 ete<j g p(edeterrvlInec{ quuntities
4-") Ar;suCYle; A:= 1
~
- oC --rXJ1 (1-oe) K N AE
i
c
t;.t€tP i: frOM tke- euler- &quatioYl
CSlil(J., A ~ 1-)
f1 - oC ~ )-1 + 1 2>
F 1
ConsiCUx tht- capittJ. A-cC!Arv]lA\e<.noYl CQfl.stIetiYl't": - h-oM C~)~tep '2. =
of:
C ==- K N1-"" -Sl-Z KoO 1-01:C - f'I - ~ 1Z - ,KK
00-1 1- 011
f. - K N - S K C ::: (~)OO-1 _ ~ V....
-C - K(F-a) Cq)
Step'?> : COYisickv Cb)
-::: 1e C1-00)~)cO C
-~~ !3 :::: .1 C1-00) F 1-"" (IOJ
C
~+ep q. ~ £\A bS ti -Ivt e.- een Into (10) - ~olve- tDy K~
F ,:.00G -:::. .-1- (1-oO) .- -$
K(f-~)
-.5!2K,-
~ -=- 1 ( '1- OC) F 1-"" (r-6) e
~o~ve, .fay N· ~ (1)~
=1 c£t~~1-o&' + 1 ~ (:>
~ ~)-oe - .1.+1-cS (!I
G+ 1- a) ~1-~
j....
~ C+~;~·r K
C = KoO 1-IlCl C
N - oK.
Ctep 5: HSSS
K
N
VeCTOY
1 C~)= F-,b -B
..1-:1-00
::: ~+~; ~d)
- -;2?F 1-.0
K
C
-I
y
:::
:=
:::
00 1-aC
KN -c\K
~K
-00 -1-OC K N
4
Ol-t b =
Lc>~ - LlneClvizanoY'
-"r DfteV1 c1YlL1CY1J/ eCCiVlOMiC fVlod£,ls 6 Ve" tvo oiHic(..d+ to $'olvC,,; % tf"u.'f involve,
0.. SysteM of VlOfl-lit'leav et(UlA.tiOYl.r ::0) S'ollAh'OYlq does not et(i.rts
~
IVII (/ V\JO-lj S tv line~,rju fi.u. MOetd..;
T,,) L~tmi£jl.tt) UneCtyiz.cdioY1
it) Lo~ - l\neCtyj ZOJiOVl
--!?
mVi/ 1) vv~ e.OMpute, 1'W-- SoluhoYl (lulVley;c.cdll( Ciet ttu.. C-O/)l\putev .folve
\'W.. ~vste{V\ Df eqUtltiOYl g -BY" giVe-Vi pan;:UVleteV" .1',
:2) We CtlVl Ctpprz.>)(;~te. tVu. MOeuA Ot( well CU; pOSJi/o1e- by c< fVlocid
thClVl we. CCtVl &,Dlve,
qel'le@te~ e><p(essiOYlg 'In term\? of e.{a$ticiti~.
er. P.-od'" e{oS he-iti e..s
'i,ubStitutiOVl elasticihe-s
~te.p 1: lOW d<zy;va.tive-~ 1L TGl'llcw f:x avzsioV\
(Jy = afcu"u .• ).-t'I1) d).(1 +.~.+ ofc).('"" .• ).-(.,) O}(no)€, d~n
PI (,4:J ffoM (2.)
1 =Ct
~At gt-eaetlf stuJe.. : Ct C-t-t-I -== 1
K-t -:::. Kt;+1 K.
-:;> Fa.,..- ~,'Mp"'Ci1tt:J 9et (\It :;; .1 VtNt == = N"'t+ I !Nvu", lo&-II>7eclVi2-tiH~
~At At +1 - i
1~I 1
i (b~) [oOKOC-~~J&41-bJ Y
~ Ge K oC
-1 + 1- ~1
i
O-lj
.0 0-2 ::. - ( )
00-\ ~ oC oo-,'K
0..1. ::. - ~oC KoC -
1
01.\: -=
o.s ~ KoC - 1
{).." -;::. - C
I-<
c
~tep'Z ; gfo-ndQ,rd MltfflX ForM ;) ,Ei¥nv~lULS'$,., Ei$t'vweofo,rs '
1 0 Ct; 0-2- Cl.3 Ct'l-I 0 <i-t'l-\1 a~ a,J
/I /\
CAb o.q. K,. 0 1 0 + f) o 0 0KH1 WtT/
0 0 At 0 0 1 AH1 -1 o 0 0 w"t-t,
I lrJA
~1 1\ "
WI
Ao Yt; /
1 y~
rr.Mo€vt
1Vl(~ ghows tVtClt 1tu Modt,.,1 ckp.eVlQt,!' OYl MD.1Yi X' A.
~
It's COYlveVlieV\t to eulculo.te., E'¥Vl val(,{£-S of' Matrix A
(A - I).J 'I = 0
we V\lj.ol \ A - I A\ - 0
Ai, L i :=:) 'lV1sic(;.. tltu- ('{Yl'lt cive-le,
Ai, '> 1 ...;') ou:tsiclL ttu.. unit civc,te..
92
g-tlf~: . rordan OecoMpoSHidYl
-') l)ec..otvlpo.sitVj MlA1Y1 x A
Frvrtl (13J
Yo\; ::. A'Y~+I + .B wt"~J ...~
~ ~V,,\~ e)(pec..m:\0n<; o·~ c.oncti11on on tl'Mt 1:; lnfo
[A L
.~.A
~_j§I\~-')E{;'i~1 -1
:: 1\ IS Q. Yt +1
le..+ Z{; = Q-'y{; = Q Kt-f'lA~ ~
~\ Z{; 1\ (;:t, l-t-+l 0·b)
r1 . N 6_1
A, E", &.HTI
~Z{; = ).2. E~ l:z.t+1
z.>~ A3 Eb "Z!Ml
I
qu" ~
/ -1L = I\"l -t :::. 1\ G! '{{; [l.., 0
2!t+1 = ~, A2 ~lrt -~ -~~ (~:J 3 q,~1 1.3'Z. l{.n \t:y
C\-:rjo 0 t{,11 Ct +'l1~·Kb + c/"3 At =. A 0 A A A
~ A ttZI Ct; + C/,tz V\t .+ 'l..·nAtB D
]f' , '1o 3 " A I't{,l,\c t + 'l.37,. K t+ q,J3 At
~te.p q. ~ L.AW of Itamted - ~>cpecmtiov.t S -
l~ = 1\ Et;~~ - ......,./.
i CHI = 1\ E-I:;+I 'l-t-tl.
~
"It; = f\ Et; (/\ Et-t-I Lt;.;2,.) 2
== 1\ Et 1: t-t'l
~
7 "T
t = 1\ f-t It-t''
£-te.p 5: TrvlpoS'e ~strl:tint
'RrJ(Vl (11") :
I
2.) If A1 > 1
~ I
li-tl = A 'lot -1 AIQ-1 Y::& Y-t~1 t
\I --"7...i. .·---v--·-·-----------~--~./
( '111 Ct-tl + 112 Kt-t-I + q,1'!> Avl)
L ---y-- ~
::: 0 . >M C£ ).1 '> 1 =-) ;.\.1 i-? ovttri04 -ltu. tArI"lt 'Cifc:,\ e. ~
z,t..,::: 0
I
g~ b: Vec-\oy A"';to(e~(e$sioVl ,g PO\\~\f (1.((1CtiOVlS
Gu.bSl1M.1e,. (\~) '\(\,To (12.);
~ [a. + a'~)J K, +
~ 17, .
~ A A
KH \ -::. \o1 1,1(t -\- bz At
10I~...l ~
_1, bJI~~ -\- 1°1 ~H'ClAu, ] lo rJ lA.l l1 1
iZem? At +l ~ eA", + (".+1
() 0 The L.lwJ ot Mo-noYl of +kt. gte<:t"e" vt~xjClbU--5 In f!tlis (V10~·
~
'ihe,Vl/ tW. evolu;tioVl of tkc. yeiVIo.iYl"Vl.B vonaWt D.ye vttte1w\IVleet -I1rJyoVt~l.-\
+W. tollow\~ '" Poll elf funcJiOVl S "
ct., Yo, ,,) =' If [t1 ~ttp ~: -rWlp~lse, (<~r0rtS~ fur'\C-nOfl
~l.ApPO>t-, 1;Je. o.tt \n"re(e~teo. '1(\ tVv... d~no.(Y\;Ct "MpOl.GJ ot ,-tiM€, t inY\ovaliOY\ tv
/+~(;\t)ilO\O~;CcJ. 0V\o.~:~t =) Impact of £,~ OVl K"i"1 g. ;Zt-t-I
g:l -- [: :·1 [~1.~ l~2l
-
PrJ Prescott (l'lQ;5) ~ ~ Alru.ac{ () f- Busineg Cf{(//e, f\-1eaSuteM£V1.t,
Mo-x. Eo[~o (-;-1 (1-ep) k Ct· + ep ~L-(1-V\t)J (Ct,I<-t-tI.Y\tr.'
t-;.o
Foe:
\ e l k1 e - 6-1At t t V't
Kt-t-I :
\./v
--(1)
- ~OVl-L.ineav EquafioYJ 1-Vli;
l /(
TI'CV1e (tllO co..hoV\ ('lJYlo(inOYl -4 tM[g It!ovv iv itUDca-t-e, ~-w t.oVlslAfV1pnOYl g.. £t.1viflj
fO"to MO«O (rI,
111M) L09-line&\Yiz\~ orounc;t +tA.L ~eacLl{ state:
At steCtcl~ gh:de ; r::::: 1
ki;:::: k-t-t, :::: k
L.HS :
+ @~41 ek'-·(1$-(~~t Q~~)Bk'-'n'-~ - ~)ek'-'n'-® ~ ~' ~
A
K =- dk",t
k-t;
A Yl t :: etVi t
Yl t
" :::.D t d.c t
..ct
= ~ [1 -Q:L\Vltl)1=V1 l1-Vli·J
1\
'c't =
/I /I
kt+\ V-t -1 A /\
Q .ct-tr - A ..et-A II 1\
Vlttl Vie AQ-1
A
"It-tr Zt
A A
=) fJ:. e(/Mj VlUre.- ..c.t GlM.~ .-CHI
~ /I
11\", - ()..() k.t + Cl 1 Zt
~ V ~tt1.te. V•
.. .
x0) gwckastic Vtc.tt'V" Viff~re~CL E'etuatiovt,
1Y. ~. b+ AY,-, + c;J ~Vldo"", y,
At t=-1 , '11 = b+A'Io + GX 1
t-=2 , '1 2 '== b + A (b + A Yo + C)( 1) + G)(l.
~ -t-l
~Yf, 2. AS (b + CXt-_S) + At,yo 8=0
1AS -=
~
[QI\Q-1Y = Q AS Q-J
If V lA-I ~ 1. :::) eU'c\tl elelVH?lIltr of AS conve,e;es tv 0 os S ~ 00
OM,! IAJ> 1 =) /\ \IIJI1\ eoYl1-ttifi 0. dictVOYlcJ. -e1e(VleVlT tV!Ci-t qrows ~ bOlAnd,
[, (o,v\~ J2,i~Vt \letl(,\)., ~t Cs- o~ds{eu ~ cmH- civcle-)
1Vu. ~tabiliht ~j nOV' foil' CI. vec,f()('-valw..ct <zquah'oYl y-eqwve.g
0,1\ et'seVl~va.l~ of A q~ou.lO< be iflsiciL ~ un'ltcircle-, Iy I~T~ 1--1 ~ ------'0
111;S eq,uah'oY\ Yep(e~eV1t&" a tiMe £'E'viM -:;;.) ~ Movin~ Ave~ Rep;-eseVltation
4 sinu. Y-t; is Movi,'"e! averu'j-t- <lfc:dL
PC\$t ...-ea1iz.£Hions of oljstuv~aYlc..£ tevM
Xt.
g-o,
e'f.. Beho,vioY of tW.. Solow l<e£idutNl . ,.
[Y-t = b+ Ct Yt-1 + ct + dVt-]
= vt. + c3 vt - 1
t procluc;.-!ivih.t_ pmductivlh.( ctiShAt17aVlU- - ~'ltJD(.,V,
.. FroM uS. olaffi., K-u. (e~id.ua,(. I()f -tW... Y~r-i?5siOY1 of 'It on Yt - t-l;v1o{ t gtiH S/;'00'SI
<;ubstaVlhal OtuwCOr-veta.tiOV1 .
~
yVioc;U.,li'''-j Vt a~ &tv\' tlt.,d-oeoYfe,.(a:teol proce5~.
o ~'Solow Residual" rMI
~CVMVIO'j;CoJpa'~
(c.aV1 M€.(lSVJe. tW.. eOlV1b'lfle.o.. .effec.i1 of At; ~vtol V-t; by 9\AQrm(..-h\~
Q.f' the growth In "0P\..{1'S ft'OfVI tiIu.. qrowth of ()utput:) oC 1-ocY r= Vt> (K1) (At L t ) I' .t
IhIS eC(,iJCl.hoY1 vvlll V10t wor-k if labo"'- ~L.Appl~ flue-hAutes Over- tiMe,
Gina i Lt} \iVlll be cor-vela-teet vy oIisfuv bane.£. -\-e-nvl. g
L..eCt~t gq;uaf€-S e"'titvtute,~ of 00 ClMol [1-00) \iV11l be bias.
o P If vt is 'oi~ vall.t£ ::0) L'" i =) recJ wCl.J-€ r
f\~~iAMpt1OV\S: Prod'" m ,'s Co blo- Do~let'S
- Co fV1 peti hve, fac+ov rvtCl y tats
~
Foe: . {1-oC)Yt- = Wf;
I.-t
/' ~.~ ~ $WOl/K .Vt/a4v,-, + ;<t + e,
t ~,","'" i«L k"'''' .;~ p",'''Uiv,''t • .
MeaSU(e.~ -+W. perbistevJ(/~: Of c,{evitHioY'S ·f.(-I?NltreVl (1.
[ -ct.] Iy+-] =1
o 1 s;r't
" " ~ outplAt yesr ctuoJ ;~ -hvrY\ ttu. Y(2.B ress iOVl 0 f Vi; on Yt-1 QMo( S 1't-1
CltMeye Vtod de Vlotes c(evietJioVl ~CiyY) tr-eVlol) A /I
'11-1e SR yesictuo..t IS trom Cl. yeg(e.ssioVl of' r;('t on SI"t-1
~
lVlQSe. (esictu.oJ. gkou.lct be. i oUVlt1c.cJ. uVldev- tVu rV100W wI!t,'Gh rs o(Vl'Ve.V1 by
0.. £ in91e. \::Y1ou<) et;.
~2 3 2. 01
Pvescott (l'teb) - U51'n~ fv1o.tlAb
A vepreseV\totive o.~eV\t fItll1xi 1\I1·IZe.-S·;
-~t + Ct
L. -
1-eI ~ StKt nt
I CaIlil-lJ. ACCUrvllAloJiOYl:
e..W'Y€Vlt Procl '" TOW Yeso"n:.,{. t<vo.ilab-Q
- (1)
- Cz.)
- (Lf)
e.tiU1i !t>rJ'Ur'l1 IS iVLe(e·ft,fe, Q Vlonhnecw (e;<pectatiOr'UtL 'llysr-€NI of it d,i'ffevenu- ettl,lClIiWlS &. TW LAw of MohOYlfbr i:'-t> )V1 it
vootic<~~
Cz, "1<,;, Ct> V1~)
1 :::. (5)
-it (~) l-n
= C~)'::i 1-~ &k k Yl + [1-6) k - C Lg)
\IFa.... ~1"'1pHcihr ~ v
'" '" (tl ) ~e-t V\. = 17
C = Et CA
t +1 + ell. EO k-t+, + C<3 Et i H, A A A
tltu,Vl ('2) J. (i?) Ofe _. '" (10)k t +, 0-4- k t + o.s Zt + o.(,,-C-t> \'3 r10{~1;t
" (II)==It-t, pit, + cc,t+1
j,,-~r ~ \;~) ~ Jt~-' 1i ~~- } I 'L_"; = O-!J Yt + (Al2: t -+ ~;t _
CAz ::=. ~e (1- e) k-e
-:::: -(:>(1-f)) k- e 3° k-e
Ott = C1-b)+(1-e) ,
k-eas -=
=- -~0." k
=
~0- Unear'izcdion : ! j
t - ~ (Ct~ c) == f.> 6[1-0+ (H) k-e@)]- f> aG-&+ (I-e) k--rtJ ~ &',,;C)
I .
.. 1
=- [(1 ... elk~& -+ C~-~)l K(kt-k\ -+ k1-8riJ/l~-"l.) - ~~.rtt-C) ~~ It. "). k \: f k\: C
-:: L(1-a)l.<~e-t- (1-0J E: t k~ -I
-------------
A
a:z.. 1 (42- 'Yt-tl
C
o 0 ~t+\1 KH1 +[~: [: :J ~..] [: :,] tv l
/'
0 wt"' lZt 0 'i!t'l-I 0
~"'I
I ( f '/1; - Ai YH1 -\- A2 Wt"'r'
'4V t -tW,(e, is jIl<? e;<peufah£vvl
~ bu r odd. e,Yt( +e(I'\A D'~n;<,,~
~
l Yt = AYt +, + 13 wti-\ 1 (12) •
-1B -:::. Ao AZ
= [' ooT~I> 0't~. Az. o 0 (oJ
...--.-.--~
JOV"MVl Oecof\1posHioVl
~ c\(!(OmpOS\V1g Y"ltltnx A
[A =: QAQ-1]
f\ "" 0. cJ. i050fl oJ. Mt\-\l1)( of eigeVl vt\Iu.e-.r
~
I
\ A L~ K 2J =
iheV/} etefJ'ne 0 flew vectvv- TYZAVlStooll1eet VO r-IQWg, ; /<.1 .'''\ :-1 -1/ -1 ::(Q Yt +1 I\~ !~ - A 13 Q Wt-tl
!1
'::'/ lnsicLL'tW- undo C·'I''(.,le.. :::::) this !'WI" ,'~ ·foyward.A1 .c. i ~~ble. root!
~o o o 1.02... 0
o o .8
A A A
o o 'Vi1 .t-+;, + ¥,t.. kt + V'3 2t;
,"" "" '" o ,V21 '..ct + Vu. k t + VZI l'l;
1\ " /Io o 0·8 V~\L+;, + ,vn k'l; + V:?3'2t
'3,,3 '31'1
- prvo'(.{ChVe,h\v..L 'j-v YlO.') - rv1CtY r..e r ach'viH-&J Cal i~tiO)1: -,- oUS-Count- (Me uf 4-h/v peVal'l{)tAM.
= O.01Z[ _ depredah'tN\ Q·F eapilnt I) uU,.d. -#tv 10 l(r.r
l1- CCtPj~ C-QVl be. us.ed Ion¥Y -=) ~ is YMal/er-:e :::: O,'f =c' 0, bLf - (abC!>"s !;inav€. i()f r'ylC~
6 -= o.76'S .- <;,E (}ft-e{)'!V/olo51 l:itxk 7
{frY'" rob"", veriotu,q;J!.s.
y/<- =
1° =
.L!k = II<. - b
:::. ~ b (1- e) k-8 c:1.:l.
:= -~(1-&) k-e Cl'3
::=:: ( 1 - ~ ) + (1 - 8) k.-~Ott
L,-ecAs- ==- r, ~ y-k
= !:... = c_k°b r
,
"
~6Lu+l0Y1 toM MatLo.b:
EI<~V1-Vtll~ : LO.'H?q
J\ ~ c)
o
rvtcdyj X of 81§£V\veGtof'S;
- O. b't 08 -O.Oq]-1
Q = -O.'T:j. -O.qz -o.H [ o o O.c1~
II
'-c 1\
t - -o.:r:rkt -o.12'1t;
~
~Ul:Jstiluh~ -li1i~ Intv ~ law of MOh'OV\ :
o /' A A II
I Kt +\ = -0. '33 Gt;' + 1.01 kt;' + 0. 0 (, Zt; 1\ 1\ A II
- -():~'3 (- O.11kt-- O.1gZt;) + 1.D1kt: -+ 0.010 Zt;
~
; OyrEll SVS~"l:" o~o;.J [~:l 1 1
exo~eneous
'w Mo,i'O v\.
.
I
~
1-e Zt; kt + (1 -;- 6) k t
lOt) It'neClV"
AT steaoll{ St-at-e, :
..c + k
/C
e k
=
=
=
kH~ + (1-h)k
-(7k - bY-..
Y - 2> v....
~+td-e.
/I
kt /I
Zt;
V, / ~ "'
"..,\~k.., ---->
it ~ "-
Yt
Yt"+1 II
CHI
A
kHZ
FroM C'l):
Let S-t
T~kl\~ i2xpectrdioY\~ of COYlct',tiOYlo1.
Yt -;::: (A)E:t;Yt+,
'it ::: (~ Et Yt+1
Q-1 Y.L 1 "t; = 1\ ~ Q Yt+1 ,
- Q-1 y-\;,7
- St - f\ S SiTI
()Yl t;~ t Info J
,., (13)
~
[~t15z !;
S3t
=
A1 Et s1 t-tl >
.Az £t.Sz;t-tl
.AJ Et S~,tTI
(Itt)
La.w of Itarated.. expectations
gt == AE£Sty
(~~(-5tH ~
£t = AEt (A Et +, s,tt-z.) 2
::: 1\ E:t St+2.
0 T
~t 1\ Et 5 tH
Foy all )..~ ~ 1. :
Let lilV) E S-t+T does not elCploh -Iva ~t i~oc
Fov Ai <. 1.
t;rl\ (.l. 'St -= Q-l Yt
- ~ee p.5tj
Vec-!oy AlAtor~(es~joV\ s.. fbliCtf J:Ul1ctitms
~(,dZ)b-liIvt+e. (IS) 'I/ltv (10)
vecfov hctt"re9 "'eSSi0'Y1
(VAfl.) Fot'IV1.
1"V1.eVl, i1tu. evolution of w refVlCliVlinfl voviable--s DVe ol£teftVl'Vled t0rvu9 h. 'ftv...
tollOlN"~ "pollC!.f FLWlc:nOVls"
( c" G" ") Tt n:l
C[a
2) Divisible Laboy
~ All tfldivilituo)s ove eMployed
C"" tJ-t; \') tOYlswV\t)
~ IV1dividutJ~ COntiYllAously C<oIjU5t hot
~ ~ 1<()tQ, e;tll 'InclividucJs o..ve -the. sa IV] e. eve fVJ ageVlt VlDY ks the SCt/Vle Vlumbe( of . LJ 0ou~.
'V
0·0 in ClV1 t<~c tv'lodeA} eve"1 a0enf c-wooses hot
~ witVl {,tV1 1tec~ nolo5Y:J g(eo"tev (efurflg to Wovkl~ wilt pYDviciL
aVl IflceV1+1ve, -Iv wovk (VI()(e. \t1ours.
H Not~ FlucfvtatiOn<5 In ll0qveljate hOlA(J \!Iovked in ttu. eCOYlDMY are d,()e,. tv
LMOJJ¥g IYl botVi \ #" of- lrlb~fs peopet c,woo£..L tv work CinteVlSive. rvIQyq,'n) (1\1) .
:tt of peop-U.. eVlteri~ $. Il?utVl(~ +0 wOt!'<' wrc..e Cex t-e tI15 Iv e, (VIC! v"f); 1\ )
(
Let H t -toM VlOl1 rr workevt Ci e,. ®~ '" V< hti)::0
h,,::. Clve~ hours li'JOvv...eet
Nt': :;t Of incLividua.Qg at vvOy~
<>dlc4tC;h'vi f1( ~ ..Y. \V'V Nt
B) HGlflseV\ (l.:tg,.): JY\divlsible L-abOl'" etVl()l tW- ~v.s;()ess Ctrv1e
AJ Ind.ivisib/e, LaboY' Vs Vivis-ible- ~bt1Y'
1} TV]cfi\/i~ible L.abov
..., l'iH-/o du.e -Iv t> Nt. rotWi tCttCtv\' Ah
~c kW.. MoM IS r'\O()- ~onvex:'11vt ? (;e..) i'vIodJ,). ~ Kyclltl,Vlcl &< Prescott CICl~z.)
~ OH WV\e t!Aat
1) 11\oi \;icWcJ s ('em eitVJef - IN()ik. (). set :If of hours (i,e E 'n~o<fV I'lot ev(!,,"'! HH\ '~_) is e{)1p'/)lfP~
Vlot LNDl"k at all I
en ~\.{ are. UVlCllole. tv tNol"k On inrev(V1eot",tlf e n{.AyVJl:7el" of! VtovuJ
~
~c fY\Ost peoptt- ~ worK hAll -nP/le...
not VV Of If.. 0+ tU I
-~ 1niS pc<pev sltJows ttwt OVl eC{;(IDflt1y V'j/ ·''ldivi';/ble,.. LaboV" exViibits V(}'('-!
. h ~ \.()X~ .flv\cfvtCthDn~ In hours wOY v~e..ol (~ati.;e, to prvc!Ucrivirvr
~
9t 0.( Ylon-eOY\ve.xi+vt (ot(,u,{ P~Pef gGDW SI"\tl.l\ t'(uc.flIl&ln(>V1)CI .
~ JlI\ aVl 12.l2>e" MoiUt, H'/1 CVtoo~ Q. pmb of wDvki"j g a. l~tteV'y
de.h?fMI(1~S vJAet~ev" tCu HH a.cfua,(.(~ (;v'oyK~
~
bfflCL all HH s o.rQ iW- So.Me =) tVtey all cWoo.l'e the Sa(Vle (Xoh of WOYki~ 9
bvtt +W. 0utco f\II e 0 f ttu. lotte'1 Ctvill be oJffe re VI. t. ,
1-1' I P r----~1'flO II~WI~
Ii
I, I
__ ~I
I / j---,1_ p /em I I J-10 II I I
-10 0 +10 0-10 +10
Pm 17 O"st'. h...
T"teLilS"ive. MoV'8it'\ ~J. DiVl,?!ble LctQO(.· ""V\. .
--" Inolivic:{,lAe<AS Me .fre-e. tv actjtAst eonnfllJ{JUS', ttu.. .:J:h0.P, VtuJ.-{rJ vvovv.eci
EJ<tenSi\le. M'U6~1\ =) t"Vldivigible t-tlboY"
-7 No if\ctivicluai~ eV1tev7~ / [ec(Vi~ ervJplolfNv~r
-" h; i'MpDYr~iI\,t fb( o(?Kplain3 rv S af bo~ M,'cro J, MaCrO (eve{
~ b. Ht c;<.lAe -tv 4 Nt; rtttWr *'-alii b. ht"
20'[· < 5S'/'
~ Most peop-{t e'li"tJev -woY-1.<- -tU.1I h'ML NOvi - COYlV€ >eiNeS .
~ Vlot wovk a.t o.ll ..)y c.Mic;(,.. i'M 100Ue" al?;1i l-f fo,l
i,) N()l'1Co()ve'(i~ "}- ?rVctV' htt. -h\M.l~''''''~ M-OotV .
FIYst <fa\.( of week; ~ rL. f� Lote r Cr1 : (vtP.. t�
~
lVlis jVIe(tvls, TVlcti violu aJ. fOtc e.s ('Onve)( Prvcl" m o.t .f;'tst; avtr;/.. tiM. III
'o~O,vle.. CO(l CCi,V€,.,
~
Mal.( be, ~c- of "WIH M lAp" t/lVIJ,., be.fo-{e.--W~l~ f)(vc;Lu.c,tl Ve..
(!;
'" Que.~ 0. te.d1V10lo~~ Cbl),lcJ. Il1etVc.£ Il1d;vicilAOt.{S tv !rIovk a. lot (J'V /IIOt ~t clil
-1 9r-na MOSt at tW. voviaHOYl "IVI \-1 is due hJ I\l-,
~ -te-,;S <;,uppovtq, Moclell'j of RI3C vy fnoi vi sible la0ov:
'" ='; (VlU.ti CoppoY!lAI'lH., eo~t rot WOYIu'''B /oYl& cHswrlU ~ {/?>s li1itl'f to vIOYV- OYJII.( tl l-1 cdf do. '1
(5{ uf£, tf;u.iv' .eVInce -nMR- eVldolrJ(Vl(? V\t ctr lei~l)(e.-
(not Viorl.<-otall),
*- :. \fo.seVl qJW,(V}e.& tW. NOV\;... convexi -h.1 IS 0. prvpefty of vef'eyenc£ q, 'rf) ttu.. te<S-t.
Df "e>deVlsive. lY1M0 1() " C\;~ €><plQin ~~ Rtll-ll'Me.. \.N'orkeY~ aye. po.lcl (VlOve 'lttaYl rost-ti'~ wo\,kerS,
/ lNcvk ~ h... "" /.1 0 > 0
~Ylot- vvoY'V_~ ~ ::0 0
He. ecvVl efttool£ aVl'l :t:t of nou"-S rW1Be. he (ikf,S =='? C'cYlnnu()uS�
6\,Vlo\ it M()S"t be ',vi '1Vu ['ov1vel<ih.j�
rf UVl bOlAVI cJ-.
---I-----+~O'+- __ h t
~ No SollttiOVl
~ -ro Make it bOlAV'lcUet by O-HlAI"1i nB 'rnCOf\o1e I's liMitecl
\-It
-+--t---f---+-+--+--~hoC
~~ \lOIot'11 ~ iYl 'ta!ioll;:>ie ;~ ~is pO\Yl'\" i'SVl't- (11 tf,u. ~e{-.
AS~~CV1P nOYl s ~
1') HOl<SeVIOlcLs ?lye iolevltical
- to HH live, irAl,,,",teL'1 ?) on ~closed iYlteyval [07
11 li.,~) 1}Jeve is 0 ~il'1gle .flY(VJ Cid£.V1h'cuL hrMS>
~v) CObb-DoutjlttU Prod" FVJ (fec-!AVlol"8 lof ('I7YISt(~'I(I+)
~(Zt,kt,ht;) = i\k~h.~el -:=) CR5 (1) ~
1\ ==- 0 (under perte£..t ) CC' lYI P€:litioY1
kt OCCl<MLAICHecl C'c..pit-cvl�
Zt; ::: YOVldoVVl glr-lOCk -;::-) -fDllow~ Q. r+ocV1cl~hc proceSS,�
v) . ~\Ad.ge,t eoY1strvt~n,t:
(2.-)IoCt + ~t ~\f\Itl-\t+ V"t-Kt 1 ,=-7C..J
Ou,-tptA\" (~i0i:b'I!(~O( I,:)e &7flsw\uetl ~nveS1t'd
wheYe (!t ~ eOYl 5l.AMpfiOvl
\;t == ,Y"!VestMent
VOl) wwof Motion fay tkl- i:'a.prt-a,L gt00V
Et-t' = (1-d)kt -r l-tJ O~ ~ ~ :1
c.oY1<;{rhamR. fVlt(h1:
:C"~U\) = YE("Zt;) + En:....,) 1Vu. .;+oc k Df (' apitQ;L 1'5 Olrmed loy HHs, w"o wJI CCl.pIW se~i C£.- tv ttu tl'lVl (PirST c<u-lo-'{eqr<!~SJve p<liWS )
TedttVlO \0'0'( ~Vtocl.<. is CW()IVU d. tv follow tl -Arst -0 rde~ MClvkov pafC(?SS
g"obel('s #U lavV of (\I10HoYl
-Z--j;+-\-::::---O-Z-t-+-~t-t"""'ll C"· (I. b (4-) C ( [_.... 1 ../"t") ~ -nn1re "'ppW Ov\ '1 d. ;:=./ Y '" -t") 0- VJa<f~
tJ.,jer€,. £'"t Or€.- iio{ C"f' PI5..L~. F =') 'V--- iid('l-Y, 6;)
I
erg�
LX) HH rv1Ct)(iMiz,u -tW eiCpec.ted-.Jltl1lAe of otscouVlteGt (,1e-h'vv..L uti1lht:r ;;~ t ~tu(Ct, 1<) ! D.c.(6L:t
fA.t"eye� 0 ;:; diS eOUVlt ~tvy
it ;:; L~isu(e
)<) 1h..e-� eVlc{OW J\I1('Vlt Df -h\(1.!IL IS' (lOffVItJi-zet;{ -tv be OVI e,
Iif; + k" 1 ] -;') At ==' 1- h t=0
)(c) un IHv( FUV\ctl PYl :
[lA LC, -f t ) ::: -T- A\og "iJlora c.t
wlIleye.� A ~ 0
I'n ttu..� interval [0,1 J
C.OVlVllK·I~ .
1'1'<1",,,\ o� hI> 1 L--y---.J�
he cetYl ehoc1(" eiitne.....� h. or 0
Eo.cV1. period; '3 COMy'\Iloetitie-s Dye tn:;toltd - O~+PIA+- eOfl/lMoo/t-y
LCtbov
(lClpil-cJ
t<LL) A~eVlt5' Ove, OtsS(,{C\I\.l~ to tallow rcUl(1'(lcJ. QXP"CttAh"OVl g/ (incw.w. Vi;:7 1tJ~)
1'WL( u~e ttll Mailab-&. Info L(,{P tv pef1Dol t) tv mk.e- tXeo's-{OV1S
in pCl/lod t.
l<;~i~) ~IYt..ee no (?xteo'laN'[ec, ==:) -\W, corvlpetltil1t ~q(,(I;"'(1rJrvl 1<; Peueto optlMtt.£ I
Thg SDGloJ.. planne r'S probleM ,'s
Mo.-x U :: E z: ~i;, lACe, 1- ht;} t,..o
L;, t. ? ke h1 - ef (z\, k t , ltlt) = t 't t
oCt + lit t[ fU\,k t , h.t )
kt+\ = (1-6) kt + ~t
"l t+l == -0 'l.t + ~t"-t-I
ko,"Zo ayt 0IVt-Vl
o ~ 6 f 1
Cv l~ot (0, 61.)r-/
7
1Vlt~ probleNl eoV\ be S'olveet 'oy us"n9 DvnClM;C fUlUqrn/VIy\lf, '1j t-ecvwdqpes
~
13>ellMClVl(S equtlh'orr~
= ere)
~{; Gt +It ~ Zt kf h~-9 + (1-~)kt
2.1.+1 = )' Zt + 'V-I;f-l�
2) Atl eCofloMy wft1..1 IVldivigilole. /.A.~oY"
AS(;UMptiOYl S :
't,) Actdi~ ~ CH~vli"lptioY) ofinc!lvi~i ble.' \aboV' tv iW- above ~tvcV1astic
qrvlNttA (V1ootet ~
/J. Not (eflnts 'odJ'u&t(\'1ev1t tt10Y19 tGe exteVlsille, fV1c<rB in.
ll,) Utilitv ru(lctiov\
LJJ .r<ep((?seVlmtive GtgeVlt LM1\ ICVlply ,'iVl+evtel\1poV''''' elQ$ticitV ()t gC.t kI'fitu tiOVJ
~
AVl 0lasncity of rubShJ-unoV\ ~w \eisuve,. jn otiffNevlt pef)'octJ
.!:!J 1vrted-R 1'1 pOlo!
1~,~ ~ elasncity of'" ~u0rlitutlOv1 is IYlfi (\ ire g inciepeVldeVlt )k -+----"'--~~~4 "
""-- loqc i + (1-1j,) A (1-h17) + AC1-!rt.,) _ A(1-ho)..{t;. => logci-l?Jt
~., 1-1 0
i~i) The eonsuNlptioVl possibilities set is vermc+.eol. tv be.- "Conve)("
~
. 'IVlctividua,tg caV\ '\ WDik tu.\\ tiM.(. ~o: lD9 Ct 1- A109 (1-ho)
~ \110+ 8~~:~1t oJI :J.i ", 1 loq C-t + AX(1)o
H'H '5 p'DoleM can ,veach (\. Compen hie equ'd ikmlJrvl.
'W) A LoHeY"( dete(lVIine-6" ~ethe( OV not tW. H H actucJ.lr wov kS.
~ t> lol-terie--s)
iVle (lew COMr11odify bel'rt3 trv.cU.ol IS 0.. Contm:::! ~"" tVu.. h'pl/J t. HH
'thCtt COfVll'V\its tW HH tv wovk h o ~ pcvb, otJ-I:
.~
.. ",� ,-'iHI--t· qe,ts ptl.iol Wtleii1er" it INOY~$ ot' not.
- Firm 1'5 ?1OcluC.i"j f1All er1l1plOl.(Mevlt /nSlAmVlUtD fir1e ~/oYkeyS
--------
";rnu? tf Cile, of ~ ~QmMoc;tj1jei 1l:rz;LctwI... IS .l.,o' Cas tnrtv.-iXoove. (VI()ol.J)
~
tW-v'/, tW.. eofl~l.uIiJPtlOY1 prob ge-t wi'J I be noV!, ep-nllex.
:;y
To tonve"i fy tVJe, COY1SUMphOfl prob. .let:
by CVlOosiVlB lotteYl'e~ yotvlev tltlc<V1 VII' (.-. Lott-9 qtttlrttVlre,e.s -t{1\().t tW tOY1\VfVlpii?Jv1 pro!? S'et- (NitI be,. ~OYlVe.x.) .
. ~ .tW,.. UMJ.,-(;.S""dOte.. alWli\:JtiVIV1 a\\oct\..hJv1S Me. S
v) All HHs D.ve -lOlev'/ticM leX" ?\t'1te,,) tw-Rl(t;tca..~e,.·
cveVi tt1.0~V1 <eve~ OYle e-liiooSe.S *u saMe GCi; ,
'\'W. OUTCOV1t1e. ~ of eavltt loHeyy \.NJ11 b~ dtffeveVlt ; oC.,.. --------� f1~
~ lr _
(J( '~-liDYJ of'~ ••ifO-fOJ 'Vt!1J.'WJ.:J
o 0 !?~(,h HH vvi1\ otlffev el< post.
<:Je-t :::- -fi-ac.HoYl o'r'rw- c.on·t1V11hlt'V! o-P JiH I/vll( wovk. 6" ¥Uve.sl will VlOt
(b"l c(iv j·etittg '\(M COrl ti nUll /VI of t:{~ eVlf$ Inw :2 ~L(b~e.ts"
b.� Lottev"Y ~evves tv Make -!noliviet<AtU5 bett-evoff
~
v/c It \ncY'eu5e~ tW.- vVlOI'W avoHab--&�
~ {}'Ill( (00 __. 0)
If lotte/le..s we-ve. VlOt o.voilable. '=") HH li'IOUlolj\ ,be. able tctCMQ?J~ not Wf/YK
\ *' (00=1) woYk h.
A(\'\oetU or ,. (()l'l.\i 11 IAU 1"1 of A90Vl rs 'f ( 10) ~
[OJ\J dE [011J
V'l} HH'S Utillit.f:
j'-u-(-c-t,':""'oe-i;-)-=-l-Oe-c.-t-+-A-oC'--c-,0-9-(1---ho-'i CC( )
11'� V\\,j E"xp~c.ted.. Utility OVI peviocl t; 0
cX?t, [I09C~ -t- A109 C1-~Q)J + :1-oe t ) [lD~C-t + t)Joqc1)1 ~~ + oCiA tD9C1-~) v
ioqC't ~ ~tv~~i,) Tecvlnolo9Y COYlstmint:�
f (It J k,.)/t,) = 'It k~ i-\:e�
-vX') Tee vJn0 lOB\{ rkock I'r CWW'\.Lol tb fOllow cA -Arst - orclev (vtavkov prOCJ?JJ
L~t-t, '::: "'6 i't + £'t-t"'J f'./ i ~ot ( o} 6(.," )C; C/;f\'\L as o\iviS' m -",
Co.'ooY CeI.'"t .
x"~) Law of MOrlOYl tOy ~pil-t<t ~tvcv...
Ikt +, = (1-cS)kt + Lt 7� ~iA. o{' 11 H :
1t1e- 12eprMevttahve. ~Vlt IS ProbleM;
MCAl< Eo to ~1; [109 e~ + AoC l09 (1- ko)] C/z)
c; ..t. tC'l~,~] ht-) -= t:t k~ h:-e
C-t+ ~-t- ~ f'Cz",kt,ht-J ~+att v, ~ Kt; g Zt;
kt +! C1-b) kt + i~ oecls iOYl v..' cCj; J " Itt ....f; J
[01�
Vt- i ~ oOi ~
C1 - .it)' -; oc,i- h.o
LA(Ct,oC-t;) = logc-t; + AoCt I09(1-Mc )
-;;; log Ct + A C-h~-t) lo~ (1-.no)
Let B::: ~ A 1°9 (1- t-to) ho
(Ilf )
o. \V1IS uti li1L.t fv,. ($ lineal in leisLAve.
~
1Vt1~ IcYlplie-& ,~ B((;(.sh'c~ of IYltefte(lllporcJ.. ~(..{Os-\i-\v.nC}YI (Of l.eisLlv€. ?- of)
4 & '\(\ot~IJeVlc{e,Vltt of +w'willi"1ness of <lflcf/vloWa.1~ -tv
~ubs-nh.tte- leisUie.- across -H(\'I.(,
7
'-.
(02�
a1p~OV1\l, 1'61 DJ ~ol(,(+ioV\ Md~oci & eo.libfllnOYl
~ 'tWYI/ it's v'\Ot posSib-C.e. +0 ~olue aV1lA,L'1ticMt~ .fof decisiOYl Jules.
~
~~ t-\-o.vtUVl lJ.W Ctppro\(:, Ma:t,'oYI tv e~+iMa:te pa(~{(\Ilete(" volues +'or- ~
I·, "h'Me se,rie.-r; MOdV) tollolrli~ KyollaVlcl &. P(e~cott)s (i1eittoololoB~
b().~eO. OYl ~roLN+1A lD'oseNo.tiOYlS g, Micro gtvtolie.s .
. ~
1VI.f,Vl i-trs pDssiw -tv ob+ctin expll'c.it, etecJSIoY1 ruler +&r ~Je.. apprvxiMCtf-t.J
ee(IY/(;Mi€ S..}J
13'f e><rreSSI'~ (-;r) tlMo.. (13) as 'DynaMIC OpnMi Zo.h'£/Yl Probl~M$'
~ - gtate., V,: Zt, Kt
- oltciSIOYl 1/,: tt, lttt
~ ~
At -
SfeaolLi 9~-te, : L.j, -=- 'SV-. ~ v vt
h-t =- II\. vt
'l1tk~ ==- k .(;,... I-Jo~ €,cOYJOtVI,'€J
gl.Atl ~t'e.a.c;tl.f ~taf'e. h, to .... eco,nOM w etivi~i ~ la't:1O'(' :
Itt == (1-e), P-+J>
\ 3( f'-t~) + e( P-t~~)
~~t€- ~~te k toY' ecO'f\Ot'A'1�
h "" C1-e)(p-t~) ]t(f'-tb -eb)
--y -l\,
parameter� definition value
e� Capital's share in 0.36 production
c5� Rate of depreciation 0.025 (annual of capital rate = 0.10) --101.
Discount factor 0.99 (steadyfJ state annual real interest rate = 4)
:> 9-\eo.ot.( Stlittt h-+ 1;A (if) 'I-tu vl1liht m)� From =" 2 (households' ~
i 1'\ ctivic;i "Ie Io.bo.,..u(c, ,I,) =logc, + A logl, take leisure 2/3, oQ.sework 1/3 of ti me)
y\ 'vl()~ F o;,st. Rate of tech 0.95 j" '. .......� 2
'i~ '" III;!. (1-Y, 61,.)� persistence !
,ho� Hours worked 0.53 f.< /� I ,
~ I"
Results •� Hansen reports seasonally adjusted, logged, and detrended�
data in the table. The first two columns are U.s. data, and the� other four are results from his models.�
Quarterly u.s time series Economy wi divisible Economy wi indivisble 55.3-84.1 labor labor
Series S.D.sin %� Correlations S.D.sin % Correlations S.D.sin %; Correlations i� with output with output with output;�
,Output ------"� 1.00 ~.16) 1.00 Cf5) 1.00~ (GNP) Y (.00) ..l\ (.21) (.00)�
Consumption It 1.29 .85 .42 .89 'f' .51 .87�
Co (.06) (.03) (.08) (:04)�
Invesbnent 8.60 .92 4.24 .99 5.71 .99�
~ (.51) (.00) (.70) (.00)�
.05�
(non-res.) Ie. \ (.07) (.07) \ (.10) (.07)�
Capital Stock 0.63 0.04 .36 .06 .47
.' .98 .98Hours� .76h. C~~ ~ .16)(Non-ag.)� ,\.(.DS)"· (.01) ~ (.01)
.87�
(Output/Hrs) (.08) ·(.01) (.07) (.03)�
Productivity 1.18 .42 .68 .98 .50
I.'r -tW. do:\~ WUe,� 10 eel =; S.O :: MQOYl of, olevi~.:I1O'f\g frO(lll h€Vlol.
104
[) The, i'(\d.ivi~i~le Lal?ov eC0'f10fVI Lj c{iSpJaVs lor-gel"" tluc+vtail"OV1g, iftCt Vl�
k ctiviq,ib\e. Jabov elUr\OMy.,�
~L) rV1d.ivi~i,b\e lCAbov' Increase-s- ti1e volatilit~ of' itJe. stvcktlstic. q(OlN~
(V1octe.A.ft,v a 0iveVl <3tocltlostlc process -for' -tkL tect..lVIolo~y "hoe-k,
[SD)� h'v) The 'Ind.ivi~·lb\e ModeA ge()emte~ St~h1ctafo. oevlo"tlovH tltJat ave�
c.losey- tv tL1e. obseyved Vo.lue.6.
l.v) The fllA(/tlAl\:tiOV"\~ in Most vo.v1o.lole£ oye lorBey'" foY'" ifle ac.tlAM ecofJol'1Y
t("OVl w( -me.. ·Inctiv)~ib\e. lo.~o( eC0r10My.
(th'IS is likely oue to (V1easvyeMeV\1" eyyoy)
\/) lnoivi4ible labov' 0enea;\fe-s 0. h'gn\y procyclico..A. Cl\JGt~ rote Cprvcluctiviht;
[)A'de u.~. o.o:ro. <lVtowg 1110.t w~S aye. anl~ MDcU~tellf Pl'Vtyc,{j~
VL) 1ltu. Most Biql'lifico.rl.t d.isc.CJve~ :
fu QMt of'6, h r-elo.t1ve-ro ,1(~) IS vef\j C<iffev(;7nt ~"" oivisible- [abCi ....
.& Illoivit;i/XL lobo'" Mtldl;/;
~ ~ ~ ,__;_'.~_i:_'d:_}~_e_1L_a_'oo_(_I_T_V1-'-;---'-;_~~-'~-'2-.~_bO_t'---,---\--=-~-"-';'-'-:A",-"C_'-~--=1'-'4--1
V"l~) Acco(.(()'ti(l~ tOy'" tht Ia.rg e 6 ( l1ol.ll~. '\ (Zttio VICl5 betVl 0 p.-v~lefVI in equili\.vviuf\I1 ~mducnvlhJ
BV.f>lne~S Cl(cle. i'ht0i'1 ~
\-wwevef, we. WOlA.let expe(..t tiu.. 'WlC::tivic;ib\e lo.wf lVIocLtA 's \'"0.110 to k;1e. hi';} ~eY'
o/c,.. "Inc;l.ivisible. Io.bof 'IMplies tltlo.t all fluCtlACtnOV1$ oewv'" ot- tht l'xteVlsive.. f'VlctYB'Vl.,
l05
F) eellcwsioV\~
-t.,) !hIS IS OV1 eqy.ili\?viuM iVloet~ -\'ho.t exl1ibits L/.VleMp\o'((\1eVlt f/LAc+utl..nOflS
',1'1 f(?SpOVlSe tv oqgveeo.:re. bVJockS o..~ 'I-lHs aye, .ftJ(ceol to ~ntev J< ex'l;
-\ill€" lobof force in (espoYlSe tv tecVJV10 lo~'( g~OCY..s,
GC) 1"'he, el()~tivit'f of Q(,{:b~titvd10Yl o/vJ leisuYe 'In oliHeveVlt pevioo.s toY'"
t\tle Dggfego.te econoMi I~ inf1Vlite, &. "!V1o.epeV\(}eV\t of titL eLa5htiht
of ~ubS+ihttiOV1 iMplied ~y Hrl'S V1i\I~ FlI\s)
Wtlere.aS 0e-toV'e, It \I'i~S c;lt.pe-Vlt:kV\t oY1 +hL~"\l"I .. ' u-l1\1lvt f.ns_
\,1v
l\tlis feothey' enoble,g +{;u. econ(J(\l\Y +-0 e><lrliloi+ IOI¥ flLtchAatiOfl~ 'IVI k t
v£lcdive- 1-v Hu.c-kc~J;oYl~ f'n productlvHl.f~)
t~\,) fVlod.e \$ Ir/lth crlvi~il?le lobDY no.ve be.tVl. LM1Successtui o.,t e"pl(;(IV'iV\~
t\tlis -Peo.tnev of ~5. tiMe !!.evle& do.-m..
~v) A ('/lodel -Mat lAlIolNg toy o.O:jUstrvl€t'\t o.\oVlB both tVle hlteVl£ilie. t
Qdevtc;lve. (Y\or~"lns LNoulol. \tutue () 5000. 6\rio.nce. tof s\ll.cce.:isndl~
COYJ B-oY\1i ns -ttu. oo:w.,
v) No\'\- Cot'\veKiHe-S fV\0-~ Ot "IGYlPOftaVlt -tvy Q)<pla.ini(\9 t\t1e volO-tHin, of h~
'(e lative tb prodtlActiV\~ (~) eveV\ vW\0V\ HHs oJe. (elo-~vel"1 l-tVlv-hlli6lj ~
c;lAl?stihAte teiSLl.(~ OCOSS tiMe ~
lnts MQ.Kes "f ?9ss",l?le ~f o-n eqi,\iWv'iVi\ll (Yloete.l to Qxltl1vit
~ luchAAtloYlg, 'In eMp\t~MeV\.t
c) ChwiSiiarto J. £..~heV\bttIMV1 L.lqQZ)
"CLtl'Yertt l2l3t ~€-S g.. Ag~(e9o.;re. wboY-,.v!ovket FI~ctu£tHOYI~"
-I.tet� Y\.t" ~ nOLO'''S WDyKecl�
N 'Tot-tAl lrtOI).V'~ wDvke.cl�
Q :.
='-/ ~)(is\i~ ~6c Mod-e-Ag pfeo.ict ~ tWtovrelah'OVl ~ cJ,tj
-::;;") Ac~ co((tla:\ioYl ~ 0
~ 1hL. &la~slco1. &. V-e({l'1e,s-iOv\ MoMS" £'VU1ofe. 1W- ~SSL<{VjpriOY1 t£-to,t
w~ {).McJ., vtt; ore srnble
1Vt.L onl~ iYVlpuJSe..s 0e(lemhnB fluc-hAClt1oY1S
IYl o.9qregate emplol.( Ment (he.- ~tocl.1t:t$tic..
Ghift in HP.j".
l~ '\VIe- h'cYl€t <:;;evie.-S t)Y1 V'tt 8< ~t- ().vt YVloc(.().ecl a~ fk.t, ICltertttf'ioV\ .oF /
()c ~"'CVto.l~C lo.boY dJ.""()N<r/.. tu,vt '1' " tixf~ l"bCff ~up('ly
~ '.� iJ"e~ ("edict" s~ 1"'(. co((elah'On ~w "'. IJ 1'1.
[09�
--,) NittA diffefeV\t iMpvdseS SlrJi.ftiY1~: ~ la~ot c.uppl\( t· /;abOf:cf;vvl,~ Fns ;)
ft1~(e, ts VlO pVi0Y'j l'eNOn {Ov Vii; to be co((el~teol. In (J.I1Y Vlia~ w/ WI;
~
CClVldidettes -tot rlJClA ~lt1ott.<-~ (f1Cluctt�
1) fj, w)( rate,�
;2) rVlYlOvo.tiOV\S tv '\i-v. monel{ ~uppll(
'3) demo~f7Aphic .6 labo\'" force
-tOws (JY1 iC1,~ 4) S'VltJC,!;t$ tv qovt gpeVld;~.
~l 0etiro; PD$". gVtDCk.? 'v qovt' ~peV!ti<j{~ ~') Ght:(t lo.bor guppll.( o(..(two.rd.
~
v;,. cOCV1ln~itlivtj MP,- (vfo te['~Y101o~~ glttot-k) lNnl qeVl~t(.,. a
If\e~. conelanoVl ~w Y'lt- J. Wi> 'In RBC.
[;to
A) 1. P(l)totypiea-t f<ec MorJ.eJ.S
As,.. I , nAp 1->' I'M' (l '-' v"Of l£,t'1'. .., .0.. ., ...... , II vO '.1 .~ rt11A l T"l l V
~ )i.)" rtle-asulf, hi",/' bV APt. (ra*'e( *,OtVl (ew--I W~
~~) AIIOfrJ\''1 for .noneoflve"iHu 1(1 LoboV' S'/,{ppl~ C'o-t l-ttlrlUVI terns)
~~~) Govt 'S lf1coypom-teol In'ID t(u. aVl(1L~SI~
'LIJ) ONlpliCAtllf) [1Abllc g.. fMVlxte .;;pevl{;:!I'1l9 nave. lcltn.n caJ.. effRctg, 0Yl
HH's (\J\U�
v) HI-t MQ,xi""iu .~ €Kpec;,t-et:l value of cli~co(,A.nted. IlfitJMe�!(vI",~ Eo tt. Ib' [kct + ~ V (N- rl,lJ 11_
~eve., Nt ~ \;i rvle <i'VI~ow(V\eV' t of HH ~ o p~s.rca,lat"
Ct ::: COYlSLAMptiOY'
V (N-'(\Jo) ::;. hAvl cti()Y\ 0 f leis\J(€..
"( ::: C\. pos. rco..\O\V"
v ':::
,.j~) CMsWVlptiCM rs. veLat-ect -tv pt1vate. g.. publiC {!{7YlSLIfYIph'M
[c~ ~ 0: + oC q*- l (2)
e-MU€- oC is ().. pan:l Mete I"'" -ItICtt 00vevns +Cu ti9t1 f>. (\1~~Ylifuctt
of +(,u.. C<~v1vaJ; ve Ot fV\Uci P I). «~spe(.+ tv a ,
J t; .
"3t> is an VtV1COl'1ttO\lt\~ ~tz;C/Vlus+ic.. proc..e..H =') gmt€- v.
VOlli) run(i\ioYl of leisu rf,; VeN - Vl~)
(1)Oivisil-AL Lo.~Oy : [ V CN- Vlb-) :::: ~(N-V1b) \it1
l'Vl{;iiv iSiW Lc< q,r- : \ v eN - V't;) - N-V't 1 'it C~)
·t tti,lMthOn (I.\-) ~VtovvS fuCtt ~ ufili1l.t.fvls' ave liY1ft(v lnle-fslXft,. (S€.L 1'>:1'2.)
&
UUU((l, ~ ~ ;rV1cavl~i ble [0.'001 ~flect.Q -1itIic, a.s~lA,MpHOYl • '-----"~.-.- - -'i'~'-'~- --....._--./
\aYJov suppl~ \~ ii'lc{ivi)ibtt..
lno.i 1I1ctuoJ. elA.~ e;fuev� - lNovk 'JOYVlot- po~, ~ of' hov.(S .
~ Vlot v/ovk. at- ~Il LJ
V, J Ii
letfvYL> Po-re+v-opliM..&J. Alloccdloy\ of ct J.. (N-Vl ...)�
l (Ie,) H'H OWJo~"\otte-vie.5"· rothe..... ttwlV' V1t; -=0) S'uppoit Povl!fv.�
Cobb- D0<1la~
G~ =
~',() T.eV("VlO!O~~ has w +JMt- ~evie.s ,yep(€seVlmtiOYi�
-:: L t -1 el<pc At) {�1~b
WAeft,. At =� ~e"';cJ.l1 uncovvelateol. hc;l..£.,peV1c;ll.Y\t
t ~,;~.cl (A., 6,J ~
f,O.
xj \<esourrL CoYJSt!1AIY1t.
(;-;r) Ie: + qt; + Kt.. ,� Yt- + (1- S) kt;I. c:. ~..\- "---',. 'v-------'
CdY1luM0'v>l ; fl"€Sh..t-"'~ _ OlA~p"d'
!12.�
~c) ~ of f\1OnOYl
kqb :: 11
1OV
(1-()) ~g
(pI L 1.
~ov(t>J. PItMmi~ ProbleM
h'C$t {'u~tHv.te. (2.) a...cl (5") 1nto (-q.)l
c: == 1<': (l:~V1t,)1-6 + C1-~)kt - k t +, - qt ..~
(/0"" - ceq!;" '= k~ (~tVld-e + (1-,Ok t - kt-+I - t3t Ct- = k: ~!; V11;)H~ -I- (1 -6) k t - kt-t-, + (oC-1) q-t
~
~. t. a low Of MOti oYI fo{" Cd..
o}/ cI;vl~i~ LaM v(· ') ......... illdivl'~I~ lC<~OV'"
__-------3�
l,J,Ievt- Y-(Y1t)~)kt+l, g~) A-t") ::= k IV\1;e k~ el<p(-6At) + ellpC-At)(1-b)kt
- kt 1-' + (oC-1) gt1 + '(VCN-Vl..,)
co t @ == Eo L ~ ~ "It;�
\- e"o� is\;>eyoVlet ~ plcH1Yle(~ probleM ;;:.) It tJlltn b~ d's'(09tl rd..eJ,
If v
o 0 1/. ,j/ ......
~)CP (-A+) (1-1) kt - kt +1 + (li-~)~t]
I J
QDlUhOY1 ex-I'sfs WtltV\ ~ -:::. S :::. 1 Ix
V C•) fS CJ1'vtVl by (~ ) \1
oC::. 1
l1 if�
ConcluSion S ; v . ~l
1) He, toUYlet/I ghocA,<3 0 t govt tpevWtr~ t-r'att1e'l 'twa VI nc:Vi no to Vlf J i l7iOCt"J}�
CalML fluc-\vtoJiOn~ M ag9Y~a.tl.- US ouf-pl;{,t. .~
~
Cqn~ctlCk +0 e)C,'Sh~ flee +ktc-n'e..s assWVlI'''j ~t only t.eVV1VlOI05L(
~l..tovV...s ttffeur ~ ~~(e~ Me.. OlA.~vd,
1Vu.-st. S~cN...1j VLC<-Vt. aY\ flVlpetc"t On ~ MOdLA ~1fV1', Let f +0 t-u[m'1oLt>jy
t!M.cI. iiu.'1 L?ontYi wt!.- w tt p(JSih've,. t?oYVt-ltJvh~ ~IN hout''Q vvo-rk.LoL
J..- prod~c..f.vfht
as aLlo~~ .f&.t ,l1oUvlsibil'flrfS 1''1 /I)I!J(J-duft(
z) r;ho0v...~ ot qovt ~p~vtol;~ al~ 0 '~iSflC<\f Cl.59{tcdt IClboy -fVlMk£+- flucAutt~
(elCtsl-i"i 1Z-t3C- MOel.(J.S) f'[ucMc(:h'l/110 if) a~9(eCt,te. :e""plDl{M.L4I\..-t !Aye,
affeMuL Io..t OYl MPt.- )
.g) J"-/Yvctud~ Y'\OY\coflvelCi -\1'e.& ',n laboY" ~uppl~ li1Cls alfVloSt no ,'01 P~(;t OVl
~ MOcUJ.'S prvoli'Ch'OYl wY VlIS frrw0"1 Co«e,[aHOYI.
\:·\olt'evev} intrvetl..0~ qovr mtv ~ i!M£tUjsiS Gc --0) does 'fed-tAU
«Vll~ Df to'fYe-\A.hon.� ~
[')Jevev'~le~s, alLowi"'5 .Q;;y (lDYI(lot1vtxi-h'e,) In la ~(~V(ppll{ J.. IY)'kC'auc-l~
qovt Wltv t("u. QVLCtl\.{sts I'Mpt1>VeS 4W. MOcLd's ptrfz:Jr(YtA.Vl.C,Il Vj/
r-esp.evt- tv M volaH I~ of Vl t velOth\.;e. tv lft-Ylt
It.?
~\c.L '\W. V'e,su\-r .{}O(\l\ ChYiStiCtVlO av.ol EicltJeV1baUM Cltiqz.); }, :;: o.otb O{:t)
~\flU l-tz;tvt~ {Lq815) CtlSUCll\..t-S +w.. teC'lrlVlolD~t{ ~ Iitock'~ a&~V1'J
0- 1l7AVlSieVlt e-ffec.:'f OYl &-4:;; Z" ~ )' 2-1;-1 -+- fhHI ~
V'JAe(e-M 0lrvJ'Stl'Q.V10 ~vtot. elvhtvtbtH~M (tetqZ-) as£wv!L' ~t
<\WIt e£fu.-t ts pel/lll.BW1eV\t, '\lILU~e} +W.. \otNtQ.(,/\P~
~ubShktiov\ effee-+ of 0. SVtOvv- tv tec.V1nol(}s~ t~ COYlSid..tyo.bl1
NlU0 nih eol t'vr Hot vL~v'\ Clq~) (Vl()r;W.,
11 b .5 '3 oq
. ~. ~di A Pr-o+oi'{p" Mod-tl\
-') SIMi\QY -tv ?r-eScott (ICig~)
~ep(~et1~iive A9tVlt
I('1\0-)( Eo ~Q (3t LA (C t ,1-Ylt;)] (1)
Wit.ere nt ;;; Lobo)'"
Inst(AlIltt\rttOI,(S Utiliil.t Functi (}V\
G(Ct , 1- n to-) = oC k Ct + (1-OC) k (1- V\~) 1
(2-)
~wc~asnc T~~~o~~vs~oc~
-? is OSSUM£.c( to follovv tW.. AI«1) pmc..e.ss
[Zt+1 = PLt + ~HJ C'3)
WvlH€- q,HI "" NCO, 6'b"-)
\l V
"rOC: c-(; : 00 ,A.' (4-)-= " yCot )'l-'
A. Zt k& -6n t : ( 1-00) =// t(1-s)e tV\t C8) (1-Ylt) /
/" : :=. f. G Zt 9-' 1-& (b) "kH1 0 ~ 6 t },.t+1 e- e kt+1 Vl t + , + (1-6)J f
111
Ct� ::: C t + 1
::: C
:::- -:;;k t� kt-t, k
Vl t� ; .. ..~. .Vlt+1 = V\ .... ....
'It ::: Zt~1 -: 1
0 e"l :::: 1
.•. ,� 1 ~ • ,.
00(1-1'\) = 1 (be)� C
2.) 41A\::Isti lute,. Cb) /ntv (4) 1
:::: (bt ~) ( e~~e-, nHJ~ 1-$)�Jt 1 (3 e~:9"'1-c'i
~t-l
3) Fro (V\ (2.) ; 1�
1-&�C +� k = (;Je k....e Y\ -+ (1-,nk
c -= (kG' n1-
fY- b k..
'---" ....
(\_1~)f)
~ e~ :::: 1� (:t)
"
1-0<: of) (1-e) (}f'� (~)=1- n. C
~1 :::: e(~~fH + 1-S Ul)
(,\'10\f'1Y] : C - n- 1 &tj@ -b k� (10;C, k, n
nt'M: , .
.-.
roc;
(' :=. e.:i. k
Y, C, k'-)
VI. (;kVe. UVlVvVlOWY]
LNe n.e;.oI. tv VvVlO"" tfA.t. valu,e "r\" f;~t,
-t
~) rtttio (;-OfVl Ot.{y kVlv(,oIled'¥- Obou.i r:
-to\.tVlvt .,..;\\~
GJ K.
~m~ [~e{ 1 . (11)
I*~iil (12)
I~r = Y:.. e } y
~
Clg)
[c = '1-~k I (/'f-)
y
i :: )(b
f"@:CH
==- .1.- (1-8)[f .(b r ~
..g) Lo(,! - Li flea¥" ized. . . ,., " v ,
00 .- A"FroM C/.i-J C
etJ--- = A ~t-A) ~ ~) T~
I'
Ct; == .)t At;-\ "
A
(15)j- 2~ - At ]
- D- (1-e) k&v'1.A e [AC1-e) k!1y1eJ kt1"t
-e[\C1-e)k& vie] ~t -t- [;\C1-e1k&v)"fJ] Z-t
o :;, = ~ - ~r (At'" ek.... it ~
Let l4>1 = ~+ 1~n)-1 j ~_.__..._-_.---------- -
I ~~ ... z.)l= ep, (;\. ... ekt (1.) ..�
1'20�
1
frOM U?); A == ~ A (e~~ke-lvrE/ -t 1 - ~) 6 1 1 e
Ai;,\/\ ~ ~ (e k - n - + 1-d) ~~~:c0 -+ ~;( [e (e-1") ke
-2 A-e
] k' ~t+,-k\ )c k /
+� ~lc[e(,-e) kEHr\"eJ ~(V1~+I-V\) A Y\.
1
+ ~"k [e ke-l~-eJ~0t+;C)
= ~ [e ~~r + 1-.s1;\t+l� + ~ [e (e-1) ~rl Jkt + ,
+ ~[e(1-.1~r] "<+1 + ~ [e(\:.fJ 4T1
e-1 'I ~ FrOM (13)~) e
1./V
/\ "� A A
~ At +1 + (3(e-1Jf kt +1 + ~(1-e)'('nt+l + fJ'('2 t + ,
- --- - --
A A A /I� A
/\" = At;t-I -+ (b� (e -1) r kt-tl -t ~ (1-fj) r <1'1 At+, + (2) (1-e) rep, e ~t-tl A A
+ ~ (1-e) '{' 4>1 'It-tl� + (br It+1 cP. _.. _~ 0
- (1 + ~(1-e) r4>11� ,At:-I-l + ~r[(e-1) + (1 -e~47)l kt +1
+� ~f [1 + (1-€)) <pJ ~+I-
~r(1-e) (cPo-1) kt~, + ~r[1+ (1-e)4'1l Zt-tl ./
v -
ll.b
A
~ lAQ~t1-tute, (I b")� In~ (l<1) to eli,vti ('-(Ate" V\t"
kt,t-t,� ~ k + (r+1-cS1ifr,t -+ C1-G~ (j/, (Atte~,t-tiJ 4- '/?f; - CCt
=- k + [(\r-t1-S) k + (1-17) 'Y~~J kt + LY + (1-e)YcP1Jit 1\ , ['\\
-;- (1-e") yA. At :,_'1� Crt ')Y'1 \ ~ )
\\ A
~ -+Af;
k L,,, = k 4- [c 4- 0-~) y<1>,l).t 4- f,:'~m. + (1 - e)~ ~t + Y, [ 1 + (1-,17,,) 1\).1 Zt /k~ v(';) ._
.� . v / S:kllL y" ~) k.
- - -_.
(V"+1-b)k+(1-e)yeep, == 'y@)Y+ ~)Y-~)YcS -t YcPo-yelPo
"� ~ ev+ ~(~)I<-- (~{~)k6 + '1~,,-)'e<Po '
:::. '/(e -t (\)0 - e4lo) + (1-s)k
== 'I [e -\- (1 - e)¢"J + (1-S) k.
122.
a a1 ..2.. ./ ""
...--I\-----..,.....(::::-=:;:;;>''\,~ r' ~ 1\
fuVl) k kt +, == k + [c+(1-e)VcF1] ~t + Ly[e+(1-e)<po1 +C1-cOkJ kt; C2-0 )
+ Y[1 + (1-e) Q>11 "l.t
a ~
Ihe(em'(e" &t1 :::: v[e+ (1-e)cj)ol + (1-$) k -'" . .. ~ ..
02. =- C + (1-e)'1<P1
Cl:; -= Y [1 + C1- e) ep11
Olf- = ~ I" ( 1- e) ( 4'0 - 1)
0. 5 == 1 + ~r(1-e)cP1
::::CAl. ~ Y' [1 + (1 - e) '1'11
g1z1~m. Matrix forM A 1\
k 'kt 0. 1 /Io 01 ~, ~1 kt~"
/I
0.4- C<~ lAb At 0 1 0 AtTI -r U}~I 0 0 P 0 o ·1'lot; It-l"l
t J\
AD y~ A1 Y-e-t-I-1\
+ [~]~, ~
1\ 1 1\'If; - A~ -1 [01A1 VH1 + Ao ~ q,.'t+1
L.et A :::: A-1
o A1
~~ A~'[n -'\ " 'lot, == AYt-t-I + l3 'Vtrl
1 (21 )r:--= A-
1 Y, ->- A- B ..., 1
12,�
/IA
kt-tl� k-t; A� A -1A-1At-t'� At + A (3 !"{--t"1
:It;Ct+ 1
-1� ' -1 -1 Q Yt +1 /\ Q� Yt + A B'ttHI
A� 1\
Kttl� . ktq" C{12. 'l'3� ct,. CiZl f{13 A I� 1\ -1�
({2.1 Ci-z/.. lz3' ).t-tl /\ qZ\ QZ2. 1..n A~ -t- A 13 'Vt-tl�
lHIq~1 Qj2. q33� Q'31 q,1 Q13 Ct
,.
~') ~u tiIli)'j-z) f1'llc( tW..� re,w,hcmsh,p .At; s.. 2-S'i-ti ta5. 7 1
/r
~ 0 0
rf /\ ::::;. () 0:\20 a /\3
t;\ilt..e- A1 > 1 J ~\tw.Jc,t :::. 0�
~------
( 11 , kt;+, -+ q .AU1 -+ Q 'It"-tl\ '\'11 1L l.z.3 )
VVe. ~()'Yl LL~ -tv e,lif\ltlrl.9--te- A "-
fYOM OLtV d.~nOMico-i. ~VS~e,IV1" A� 1\
'oIf wvd11 A (AS l?l \ineav t?oMbiflCtnOrl ot k Ovv"cl 'l..�
t I\.,� ~O~V1MlAS . e.lCoseVl€Ou.S Gtztte. VOr1~u.e. ~+o-te. v(il""a~
12Lf
Le-t .A-1
..
A ...
cr1 (Ai; + e kt + 'It)
= ~, f[~~::) k, - ~ ltJ ~ ek: +Zt J.
- 4>, [~- tJ kt + 0-t-)Z~J
=[~:l
126�
® ftAAt Lab� P()'fl'4Met'er VaLLte.s -0) Cotil?rafioYl
{\A~+lab
~ = O.q~ bew. == o.q~;
~, \l'1 ~
e DC
2:J
V\
P
cP1
=
:::
--:::
--
-
O.l.\
0.33
0.02-5
0.2g
O.g
1
e-~)1-1"
them :: 0.£+', o.\pho. = 0.33;
delta = D.02.6 • ~
V\ := D 1.8' •• :J
rho :::: o.~ . :J
?V1i 1 == 1/ (them - (vd (1-Vl))) ;
p. 12.\ cPo = e <1'1 ph,o ::. them * ph"li j
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