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1 Chape 3: Adjusmen Coss n he abou Make I Movaonal Quesons and Execses: Execse 3 (p 6): Illusae he devaon of equaon (35) of he exbook Soluon: The neempoal magnal poduc of labou s epesened by (3) = = E λ hee ( ) s he mmedae magnal poduc of labou s he eal nees ae denoes he poduc demand s he level of employmen [ ] E s he expecaon opeao based on he nfomaon se avalable a and s he age Usng dynamc pogammng e can ee (3) as follos: (3) = = E λ If e e-ndex ( = ) fom = nsead of = e have = and heefoe (33) = = E λ By he la of eave expecaons he above equaon can be en as (34) = = E E λ By (3) e can defne λ as follos (35) = = E λ Subsung (35) no (34) gves

2 (36) λ ( ) E [ λ ] = hch s equaon (35) n he exbook Execse 3 (p 5): Consde he o-sae Makov pocess (36) x a = b h pob p f x = a; h pob h pob q f x = b; h pob ( p) ( q) f x = b f x = a Wha s he expeced duaon of he Makov pocess n sae a? Soluon: Gven ha he pocess s cuenly n sae a p s he anson pobably fo he Makov pocess o sa and say n sae a and D s he duaon of sae a e have (37) D = f x = a and x a; pob D = f x = x = a and x D = 3 f x = x = x = a and x M ( D = ) = ( p) a; pob( D = ) = p( p) a; pob( D = 3) = p ( 3 p ) The expeced duaon of sae a can hen be deved as (38) E = ( D) = a pob( D a) M = a= pob pob 3 pob ( x a x = a) ( x = a x a x = a) ( = a = a a = a) ( p) p( p) 3 p ( p) ( p) = x x x 3 x Execse 33 [Blanchad and Fshe (989) Poblem 3 p 49]: Consde a fm and a unon ho bagan h each ohe ove ages and employmen Wages ae chosen so as o maxmse ( - A)[R() ] subjec o he gh-o-manage consan R () = s employmen R() s he evenue funcon and A s he expeced level of ncome fo hose ho ae no employed by he fm (a) Deve he age level esulng fom he ash bagan as a funcon of he labou ncome o pofs ao θ he age elascy of labou demand η and A (b) e A be gven by ( u) Bu hee u s he ae of unemploymen he ousde age f employed elsehee and B he ncome f unemployed Deve he ae of unemploymen conssen h a symmec equlbum assumng fo convenence ha B s consan and denoed by ρ

3 Soluon: (a) Consde he gh-o-manage model n hch he age s deemned n a non-coopeave ash bagan and n hch he fm subsequenly chooses opmal labou demand fom he labou demand cuve The objecve funcon of he unon s (-A) hle he objecve funcon of he fm s [R() - ] Consequenly he ash baganng soluon s deemned by (39) ( A)[ R( ) ] [ R ( ) ] max λ We ge he follong o fs-ode condons: (3) [ ( A) ] [ R( ) ] ( A) [ R' ( ) ] λ [ R' ] = and (3) R ( ) = Equaon ( ) mples ( ) = We heefoe ge R (3) [ ( A) ] [ R( ) ] ( A) [ R' ( ) ] = Snce (33) ( R' ( ) = [ R' ( ) ] = R ) = e have [ ( A) ] [ R( ) ] = ( A) η θ η A ( A) = θ A η = θ θ A = η θ ( A) [ R( ) ] = ( A) ( A) = ( A) Thus hee e have used he defnons of he age elascy of labou demand labou ncome o pof shae θ = ( R( ) ) η = and he When η s nceasng he unons ancpae he lage mpac of hghe ages upon employmen and heefoe he age level ll be loe In he lm fo η e ge = A (b) Inseng A = ( u) Bu = [ ( u) ρ] = [ ( ρ) u] hee ρ = B no he eal age equaon and ecognsng R ( ) = e ave a 3

4 (34) A = A = A = [( u) ub] ( u) u [ ( u) uρ] ( u) u B A hghe ρ = B and heefoe a moe geneous unemploymen benef sysem ll ncease equlbum unemploymen In a symmec equlbum e have = and heefoe equaon (34) smplfes o A (35) = u( ρ) The coespondng equlbum unemploymen ae s A (36) u = ρ Addonal Refeence: Blanchad OJ and S Fshe (989) ecues on Macoeconomcs Cambdge (MIT Pess) II Sofae Tools Sofae Execse 3: Exploe he cyclcal behavou of and neempoal magnal value of pofs n execse 3 on p 3 and pp Soluon: The sang pons of hng and fng decsons ae llusaed by he equaons (*) and (**) on p 44: [ ] β τ (37) h = e dτ τ τ [ ] β τ (38) f = e dτ τ τ hee s he numbe of employees s dscoun ae β > epesens he concavy of poducon funcon h s he magnal cos of hng f s he magnal cos of fng and denoes he π deemnscally flucuang level of demand ( τ ) = K K sn τ hee K > K > The p values of K K and p deemne he magnude and he lengh of he busness cycle I s assumed ha employees do no qu and ha hee ae no quadac coss of hng and fng he fm hes o β fes mmedaely so ha [ ] ( τ ) τ τ e dτ ll neve be geae han h o loe han f The nacon aea s defned by he egme of β ( τ ) [ ] e f < dτ < h τ τ 4

5 = τ e τ dτ can be β ( τ ) Fo he nacon aea hee no hng/fng happens he negal [ ] en as follos: v (39) v = K β K β π sn τ e p ( τ ) dτ hee v solve β ( τ ) [ ] e = τ τ π τ p sn τ e dτ : λx λe γ sn γx e dτ = sn γx cos γx γ λ λ λx (3) dτ The negal fomula on p 45 of he exbook can be used o Subsung he above equaon no (39) yelds (3) v = K β K β λx e γ π π π sn x cos x p p p Fo ( ) h v > he fm mmedaely hes h so ha ( ) h v h = The nely hed okes h lead o a loe magnal poduc of labou so ha v = h agan Smlaly fo v( ) < f he fm mmedaely lays off f so ha v( f ) = f The fng of f employees leads o a hghe magnal poduc of labou so ha v = f agan The Malab fle man_execse_3-m allos o analyse such a sysem of flucuaons due o hng and fng ove he deemnsc busness cycles hch s denoed by he sn funcon n % begnnng of npu daa fo (76) and (77) = ; = ; = ; %nal value of bea = 3; h = ; f = ; K = ; K = 5; p = 4; % end of npu daa 5

6 8 Sofae execse 3: he cyclcal movemenns of and v 6 4 and v v Tme oe ha hee ae some dffeences beeen he me sees of and v snce v s he neempoal value The peaks and oughs of do no concde h he maxmum pons of v The eason s ha v ll only flucuae n he aeas of f v h When v s beyond he peak he fm sops hng Convesely he fm sops fng afe v eaches s ough If hng and fng coss ae vey small say h = and f = hen he flucuaons of ll mosly lead o vaaons n no n v Sofae execse 3: he cyclcal movemenns of and v and v v h= f= Tme On he conay f h = and hee exs subsanal fng coss say f = 6 hen he flucuaons of ll mosly lead o bgge vaaons n v bu smalle flucuaons n 6

7 Sofae execse 3: he cyclcal movemenns of and v 5 and v 5 v h= f= Tme Sofae-Execse 3: Demonsae he mpacs of changes n he qu ae fng coss and unceany upon he hng and fng hesholds n execse 4 on p 4 and pp Soluons: The soluons fo v n he dffeenal equaon on p 46 ae denoed by (3) v η θ δ δ = α α Kη K η ( β ) β hee η = s he numbe of employees s he nees ae δ s he qu ae θ s he df paamee of geomecal Bonan moons of demand σ s he sandad devaon s he age level and K and K ae paamees elaed o dffuson pocesses and had o be solved by he valuemachng and smooh-pasng condons oe ha hee s a ypo n he equaon fo he pacula soluon n he exbook! e and epesen he hng and fng hesholds The value-machng condons deved usng he dynamc pogammng appoach ae denoed by (33) θ β ( β ) K δ δ β α β α ( ) K ( ) = H and (34) θ β ( β ) K δ δ β α β α ( ) K ( ) = F hee H s he magnal hng cos and F s he magnal fng cos The coespondng smooh-pasng condons ae (35) θ β ( β ) α δ α βα α βα K α K = 7

8 and (36) θ β ( β ) α δ α βα α βα K α K = Equaons (35) and (36) can be solved numecally usng he Malab m fles fun_execse_3_m and man_execse m The npu daa ae as follos: % begnnng of npu daa = ; bea = 3; dela = 5; hea = ; sgma = ; = ; age = ; H = ; F = 6; choceofplo = ; % "" fo plo of F; % "" fo plo of sgma; % "3" fo plo of dela fo execse 3 (c); % end of npu daa Gven he choce of he value of choceofplo o o 3 he Malab m fles ll geneae a plo ha shos he effec of F o σ o δ on he hng hesholds and he fng hesholds The Effec of F on Thesholds As F nceases he hng hesholds fall he fms ae moe elucan o fe magnal employees The effec of F on hng hesholds oks ndecly va he dffuson em K ( β α ) Hghe F leads o α ( β a loe value of K ) a he fng hesholds and also loes K ( β ) a he hng hesholds Theefoe he hng hesholds ae hghe fo sng F α 8

9 4 3 Sofae execse 3: he effec of F on hesholds - and F The Effec of σ on Thesholds The value of σ only affecs he dffuson ems of he soluons As expeced a ske envonmen (hghe σ ) dens he nacon aea Thus a se n σ makes he fm elucan o he o fe magnal employees 4 3 Sofae execse 3: he effec of sgma on hesholds - and sgma The Effec of δ on Thesholds As δ nceases he pacula negal falls and leads o hghe hng and fng hesholds Hoeve a sng δ has a negave mpac on he opon (dffuson) ems and leads o loe hesholds As β α β α K K ) s negave ndcaed by he gaph belo he sum of he dffuson ems ( 9

10 4 3 Sofae execse 3: he effec of dela on hesholds - and dela Sofae-Execse 33: Wokng me and employmen decsons An exenson of sofae execse 3 Assume ha a fm has he follong mmedae pof: β Π = ( g( H )) [ ( H ) x] β hee x s he fxed coss of employmen g s a funcon elaed o okng hous age s also a funcon of okng hous H denoes okng hous and all ohe vaables ae he same as n sofae execse 3 agan follos a geomecal Bonan moon The exsence of fxed coss pe oke x ends o make fms demand longe okng hous n ode o spead hese coss ove moe hous of ok g and have he follong funconal foms: g( H ) H = H γ s γ H / H s δ H H > H H s s ( H ) s = H s H s a s H H s H H > H H s s I s assumed ha < δ < so ha g(h) s scly concave and he poblem of he fm s ell defned An exogenous educon of sandad hous H s may ncease o decease g(h) and dependng on he oveme age pemum ncease o decease employmen and labou sevces The fm pays a consan pemum a > on oveme hous (H H s ) The magnal coss of hng and fng ae denoed by T and F especvely and he employees neve qu The fm smulaneously chooses acual hous and employmen o maxmse s expeced dscouned value of pofs (a) Deve he Bellman equaon of he above se-up and solve he okng me employmen decsons se-up (b) Analyse numecally he mpac of he oveme age pemum a fng coss F and hng coss T on he employmen hesholds and hous oked

11 Soluon: (a) The fm s expeced value of dscouned pofs hou any fng and/o hng coss s β s (37) V max E ( g( H )) [ ( H ) x] e ds = H β hee s he eal ae of nees and E[ ] s he expecaon opeao Accodng o equaon (37) he fm chooses ho many people o employ and he specfc numbe of hous gven he age schedule Usng Iô s emma he Bellman equaon fo he value V a me zeo n he connuaon egon s β (38) V = max ( g( H )) [ ( H ) x] ηv H σ V The fs em on he gh-hand sde s evenue [ ( H ) x] s he employmen-elaed bll ηv s he gan due o a shock and he las em s he change n he value of he fm caused by changes n demand The fs-ode condons fo H ae: β β (39) ( β ) g ( H) g' ( H) ' ( H) = Afe solvng he above equaon he vaable H becomes a funcon of gven he funcons of and g The fs-ode condon h espec o s denoed by σ β β (33) v = g ( H ) [ ( H ) x] η v v hee v = V s he value of employng he magnal oke As n he pevous execse he nacon aea fo hng and fng s deemned by he condon (33) F < v < T The hng hesholds ae deemned hen v = T and he fng hesholds happen hen F = v In he absence of hng and fng coss he pacula negal may be expessed as β β (33) v ( ) = E g ( H ) [ ( H ) z] β β s g [ ] ( H ) ( H ) e ds = P θ The fm s opon value of hng n he fuue and s opon value of fng once he oke s employed ae measued by he homogenous pa of he equaon z (333) v = θ v σ v α α The hng opons ae denoed by A and he fng opons ae epesened by A hee A and A ae paamees o be deemned by he bounday condons and α and α ae he posve and negave oos of he follong chaacesc equaon:

12 σ α β θα = (334) ( ) The value-machng condons of hng and fng follo: (335) β g θ β ( H ) ( H ) x α α A = T A and (336) ( H ) ( H ) x α α β β g θ A = F A The lef-hand sdes of (335) and (336) sho he magnal benef fom hng/fng a oke and he gh-hand sdes gve he coespondng magnal coss The smooh-pasng condons ensue ha hng (fng) s no opmal ehe befoe o afe he hng (fng) heshold s eached: (337) β β g θ ( H ) α α A α T = Aα T and (338) β β g θ ( H ) α α A α = Aα Equaons (335) - (338) fom a non-lnea sysem of equaons h fou unknon paamees A and A and can be solved numecally once he soluons fo α and α ae obaned fom (334) and opmal values fo H ae found fo he values of and va equaon (39): β β (339) ( β ) g ( H) g' ( H) ' ( H) = and β β (34) ( β ) g ( H) g' ( H) ' ( H) = fo hng and hng hesholds especvely T F Equaons (335) - (34) can be solved numecally usng he follong Malab m fles: fun_execse_3_m fo value-machng and smooh-pasng condons fun_execse_3 gm fo funcon g(h) fun_execse_3 g_dm fo he devave of g(h) fun_execse_3 m fo funcon (H) fun_execse_3 dm fo he devave of (H) fun_execse_3 Hhm fo deemnaon of hous n hng hesholds fun_execse_3 Hfm fo deemnaon of hous n fng hesholds man_execse_3_3m fo he man pogam The npu daa ae as follos:

13 % begnnng of npu daa = 8; % nees aes bea = 3; % fo CD funcon hea = ; % df paamee fo geomecal Bonan moons sgma = 3; % sk paamee fo geomecal Bonan moons = ; % he level of employees T = ; % magnal hng cos F = 6; % maganl fng cos a = 4; % pemum ae fo ove me HS = 5; % benchmak value fo hous WS = ; % benchmak value fo ages gama = 8; % fo age funcon dela = 8; % fo age funcon xx = 5; % x he fxed cos elaed ages choceofplo = 3; % "" fo plo of F; % "" fo plo of T; % "3" fo plo of a fo execse 3 (c); % end of npu daa Gven he choce of he value of choceofplo o o 3 he Malab m fles ll geneae a plo ha shos he effec of F o T o a on he hng hesholds and he fng hesholds and coespondng hous The Effec of F on Thesholds The effec of F on he hng and fng hesholds ae smla o he ones n sofae execse 3 excep he nacon aea s de As F nceases he fm loes he okng hous fuhe befoe fng; he se n F also leads o hghe hng hesholds snce he fm ops nsead fo nceasng okng hous Sofae execse 3: he effec of F on hesholds and hous 8 and - and Hh and Hf H H F The Effec of T on he Thesholds The ncease n T has a dec mpac on hng hesholds amplfed by he hghe okng hous noduced o offse sng hng coss The mpac of changes n T on fng hesholds s mnmal due o smalle fng opons (fom hgh fng coss n he benchmak value) 3

14 Sofae execse 3: he effec of sgma on hesholds and hous 4 and - and Hh and Hf H H T The Effec of he Oveme Pemum (a) on he Thesholds Fo loe oveme pemums he fm pefes nceasng okng hous ahe han hng addonal employees The mpac of changes n a on he fng hesholds s smalle snce he oveme pemum only decly affecs he age funcon n hng decsons Sofae execse 3: he effec of dela on hesholds and hous 8 6 and - and Hh and Hf H H a oveme pemum Addonal Refeence: Chen Y-F and Funke M (4) Wokng Tme and Employmen Unde Unceany Sudes n onlnea Dynamcs and Economecs 8 Issue 3 Acle 5 (bepesscom/snde/vol8/ss3/a5) 4

p E p E d ( ) , we have: [ ] [ ] [ ] Using the law of iterated expectations, we have:

p E p E d ( ) , we have: [ ] [ ] [ ] Using the law of iterated expectations, we have: Poblem Se #3 Soluons Couse 4.454 Maco IV TA: Todd Gomley, gomley@m.edu sbued: Novembe 23, 2004 Ths poblem se does no need o be uned n Queson #: Sock Pces, vdends and Bubbles Assume you ae n an economy