Theoreical Ecoomics Leers, 0,, 33-40 h://dx.doi.org/0.436/el.0.05 Published Olie May 0 (h://www.scirp.org/joural/el) The Ecoomic Dyamics of Iflaio ad Uemloyme Tamara Todorova Dearme of Ecoomics, America Uiversiy i Bulgaria, Blagoevgrad, Bulgaria Email: odorova@aubg.bg Received November 8, 0; revised December, 0; acceed Jauary 3, 0 ABSTRACT We sudy he ime ah of iflaio ad uemloyme usig he Blachard reame of he relaioshi bewee he wo ad akig he moeary olicy codiio io accou. We solve he model boh i coiuous ad discree ime ad comare he resuls. The ecoomic dyamics of iflaio ad uemloyme shows ha hey flucuae aroud heir ieremoral equilibria, iflaio aroud he growh rae of omial moey suly, resecively, ad uemloyme aroud he aural rae of uemloyme. However, while he coiuous-ime case shows uiform ad smooh flucuaio for boh ecoomic variables, i discree ime heir ime ah is exlosive ad ooscillaory. The hyseresis case shows dyamic sabiliy ad covergece for iflaio ad uemloyme o heir ieremoral equilibria boh i discree ad coiuous ime. Whe iflaio affecs uemloyme adversely he ime ahs of he wo, boh i discree ad coiuous ime, are dyamically usable. Keywords: Ecoomic Dyamics; Secod-Order Differeial Equaios; Secod-Order Differece Equaios; Phillis Curve; Iflaio; Uemloyme. Iroducio The relaioshi bewee iflaio ad uemloyme illusraed by he so called Phillis curve was firs discussed by Phillis [] i a ah-breakig aer iled The Relaioshi bewee Uemloyme ad he Rae of Chage of Moey Wage Raes i he Uied Kigdom, 86-957. The sadard reame of he relaioshi bewee iflaio ad uemloyme i dyamics ivolves he execaios-augmeed Philis curve, he adaive execaios hyohesis ad he moeary olicy codiio. Solvig he model allows sudyig he ecoomic dyamics of he variables reaed as fucios of ime. Thus, for examle, we are able o fid he ime ah ad codiios for dyamic sabiliy of acual iflaio as well as of real uemloyme. I sudyig he relaioshi bewee iflaio ad uemloyme ecoomiss such as Phels [,3] have foud o log-ru radeoff bewee hese wo, oosie o wha he Phillis curve imlies. I a iflueial 968 aer iled Moey- Wage Dyamics ad Labor Marke Equilibrium Phels [4] sudies he role of adaive execaios i seig wages ad rices. There he iroduces he coce of he aural rae of uemloyme ad argues ha labor marke equilibrium is ideede of he rae of iflaio. This fidig reders Keyesia heory of corollig he log-ru rae of uemloyme i he ecoomy ieffecive. I his book Macroecoomics Blachard [5] offers a aleraive reame of he relaioshi bewee iflaio ad uemloyme. He icororaes i he model he aural rae of uemloyme U a which he acual ad he execed iflaio raes are equal. The rae of chage of he iflaio rae is roorioal o he differece bewee he acual uemloyme rae U ad he aural rae of uemloyme U. The urose of our aer is o sudy he ecoomic dyamics ad ime ah of iflaio ad uemloyme from he ersecive of Blachard s equaio of he relaioshi bewee iflaio ad uemloyme. We solve he model boh i coiuous ad discree ime ad comare he resuls. We discuss hree cases, a simle model of Blachard s equaio wih he moeary olicy codiio ake io accou. The we exed he model o he hyseresis case, where iflaio is adversely affeced o oly by uemloyme bu by is rae of chage also. Fially, we solve he model whe here is he oosie effec, ha of iflaio o uemloyme. I sudyig he ime ah of iflaio ad uemloyme we fid ha hey flucuae aroud heir ieremoral equilibria, iflaio aroud he growh rae of omial moey suly, resecively, ad uemloyme aroud he aural rae of uemloyme. However, while he coiuous-ime case shows uiform ad smooh flucuaio for boh ecoomic variables, i discree ime heir ime ah is exlosive ad ooscillaory. Furhermore, i he secial case whe rese, o revious, iflaio is cosidered, he discree-ime soluio shows a o-flucua- Coyrigh 0 SciRes.
34 T. TODOROVA ig exlosive ime ah. I he hyseresis case he resuls are ideical ad show dyamic sabiliy ad covergece for iflaio ad uemloyme o heir ierermoral equilibria boh i discree ad coiuous ime. I he case whe iflaio affecs uemloyme adversely he ime ahs of he wo boh i discree ad coiuous ime are dyamically usable. The aer is orgaized as follows: Secio reveals he sadard reame of he ieremoral relaioshi bewee iflaio ad uemloyme. I Secio 3 we solve a iovaive model of his relaioshi usig Blachard s equaio. Secios 4 ad 5 exed his model o he hyseresis case ad reverse ifluece case, resecively. Secio 6 rasforms hese coiuous-ime soluios io discree-ime resuls. The aer eds wih cocludig remarks.. Iflaio ad Uemloyme: The Sadard Treame The sadard reame of he relaioshi bewee iflaio ad uemloyme has well bee sudied by mahemaical ecoomiss such as Chiag [6], Pembero ad Rau [7] ad Todorova [8]. The origial Phillis relaio shows ha he rae of iflaio is egaively relaed o he level of uemloyme ad osiively o he execed rae of iflaio such ha U hπ, 0,0h where is he rae of growh of he rice level, i.e., he iflaio rae, U is he rae of uemloyme ad π deoes he execed rae of iflaio. Thus he execaio of higher iflaio shaes he behavior of firms ad idividuals i a way ha simulaes iflaio, ideed (execig rices o rise, hey migh decide o buy more resely). As eole exec iflaio o go dow (as a resul of aroriae goverme olicies, for examle), his, ideed, brigs acual iflaio dow. This versio of he Phillis relaio ha accous for he execed rae of iflaio is called he execaios-augmeed Phillis relaio. The adaive execaios hyohesis furher shows how iflaioary execaios are formed. The equaio The exaded versio of he Phillis relaio icororaes he growh rae of moey wage w where he rae of iflaio is he differece bewee he icrease i wage ad he icrease i labor roduciviy T, ha is, w T. Thus iflaio would resul oly whe wage icreases faser ha roduciviy. Furhermore, wage growh is egaively relaed o uemloyme ad osiively o he execed rae of iflaio or w U hπ where U is he rae of uemloyme ad π is he execed rae of iflaio. If iflaioary reds ersis log eough, eole sar formig furher iflaioary execaios which shae heir moey-wage demads. j π 0 j illusraes ha whe he acual rae of iflaio exceeds he execed oe, his urures eole s execaios so 0. I he oosie case, if he acual iflaio is below he execed oe, his makes eole believe ha iflaio would go dow so π is reduced. If he rojeced ad he real iflaio ur ou o be equal, eole do o exec a chage i he level of iflaio. There is also he reverse effec, ha of iflaio o uemloyme. Whe iflaio is high for oo log, his may discourage eole from savig, cosequely reduce aggregae ivesme ad icrease he rae of uemloyme. We ca wrie km k 0 or uemloyme icreases roorioally wih real moey where m is he rae of growh of omial moey. The exressio m gives he rae of growh of real moey, or he differece bewee he growh rae of omial moey ad he rae of iflaio m m rm m where real moey is omial moey divided by he average rice level i he ecoomy. The model he becomes U hπ,, 0,0h (execaios-augmeed Philis relaio) j π 0 j (adaive execaios) km k 0 (moeary olicy) We solve his model by subsiuig he firs equaio io he secod which gives j U jh π Differeiaig furher wih resec o ime, U d π d j j h ad subsiuig for d U we obai d π jk m jh d d where he secod equaio of he model imlies Coyrigh 0 SciRes.
T. TODOROVA 35 π. Subsiuig his las exressio for j we obai j d π j k π m jh This is a secod-order differeial equaio i π rasforms io d π k j h jkπ jkm, or aleraively π k j h π jkπ jkm which Give he roeries of secod-order differeial equaios, we have he followig arameers a k j h a j k b jkm The coefficies a ad a are boh osiive i view of he sigs of he arameers. We fid he equilibrium rae of execed iflaio o be he aricular iegral b π a m Hece, he ieremoral equilibrium of he execed rae of iflaio is exacly he rae of growh of omial moey. I order o esablish he ime ah of π we eed o fid he characerisic roos of he differeial equaio which we ca do usig he formula r, a a 4a The ime ah of π would deed o he aricular values of he arameers. Oce we fid his ime ah we migh be able o deermie ha of uemloyme U or he rae of iflaio.. 3. Iflaio ad Uemloyme: A Exeded Model I his book Macroecoomics Blachard [5] offers a aleraive reame of he relaioshi bewee iflaio ad uemloyme. He iroduces i he model he aural rae of uemloyme U a which he acual ad he execed iflaio raes are equal. The rae of chage of he iflaio rae is roorioal o he differece bewee he acual uemloyme rae U ad he aural rae of uemloyme U such ha d U U 0 Therefore, whe U U, ha is, he acual rae of uemloyme exceeds he aural rae, he iflaio rae decreases ad whe U U, he iflaio rae icreases. The iuiive logic behid his is ha i bad ecoomic imes whe may eole are laid off, rices ed o fall. A his oi he acual uemloyme would exceed he ormal levels. I imes of a boom i he busiess cycle he rae of acual uemloyme would be raher low bu high aggregae demad would ush rices u. Blachard s equaio reveals a imora relaio as i gives aoher way of hikig abou he Phillis curve i erms of he acual ad he aural uemloyme raes ad he chage i he iflaio rae. Furhermore, i iroduces he aural rae of uemloyme as i relaes o he oacceleraig-iflaio rae of uemloyme (or NAIRU), he rae of uemloyme required o kee he iflaio rae cosa. We solve his aleraive model of he relaioshi bewee iflaio ad uemloyme by assumig ha U is cosa ad ha a ay give ime he acual uemloyme rae U is deermied by aggregae demad which, o is ow, deeds o he real value of moey suly give by omial moey suly M divided by he average rice level. Thus uem- loyme is egaively relaed o real moey suly M accordig o he relaioshi U l M, 0 We solve by differeiaig he firs equaio d ad he secod equaio o obai d U d U d dl l M M dl m We assume ha he growh rae of omial moey suly m is cosa which could be i accordace wih sysemaic goverme laig or moeary olicy. The equaio ha obais is ideical o he moeary-olicy equaio iroduced i he sadard reame of he Phillis curve. Combiig he wo resuls yields d m d m which is a secod-order differeial equaio i iflaio rae. Solvig he differeial equaio, we have Professor Blachard [5] formulaes his origial equaio i discree U U. ime as Coyrigh 0 SciRes.
36 T. TODOROVA a 0, a ad b m. Hece, he aricular iegral is m ad he characerisic equaio is e r r, 0 i where h 0 ad v Thus he geeral soluio ivolves comlex roos ad akes he form o () m e Bcos Bsi m Bcos Bsi Similar o he sadard model we ca sudy he dyamic sabiliy of acual iflaio. Sice h 0, he fucio of iflaio rae dislays uiform flucuaios aroud he rae of growh of moey suly which gives he equilibrium level of iflaio. 3 Sice he growh rae of omial moey suly deeds o goverme olicies ad chages wih hose, i is a movig equilibrium. Such flucuaig ime ah aroud he ieremoral equilibrium ca be grahed as i Figure. Alhough he ime ah is o coverge, moeary olicy ca somewha seer iflaio ad limi i wihi a uel as i flucuaes aroud m. Give he remises of he model ad he values of he arameers, a diverge ime ah ad, herefore, a ucorollable level of iflaio are imossible. To fid he ime ah of uemloyme U as he d ex se we exress as d Bsi Bcos ad subsiue i io d U U U Bsi Bcos U B si B cos where he cosas B ad B have o bee defiiized. I follows ha, similar o he iflaio rae, he uemloyme rae dislays regular flucuaios bu is ieremoral equilibrium is he aural rae of uemloyme. Sice his is he rae a which execed ad acual iflaio are equal, we ca view ieremoral equilibrium as he sae i which execaios coicide 3 The ime ah of a geeral comlemeary fucio of he ye h y e B cosvb si v c deeds o he sie ad cosie fucios h as well as o he erm e. Sice he eriod of he rigoomeric fucios is π ad heir amliude is, heir grahs reea heir shaes every ime he exressio v icreases by π. () 0 h 0 Figure. The ime ah of acual iflaio. wih realiy. Sice agai we have h 0, he ime ah of uemloyme is eiher coverge, or diverge. I follows, herefore, ha wih he assage of ime acual uemloyme cao subsaially deviae from he aural rae of uemloyme. 4. The Blachard Model: A Hyseresis Sysem The equaio formulaed by Professor Blachard ca be exeded furher o he so called hyseresis sysem. This versio of he model assumes ha he rae of chage of he iflaio rae is a decreasig fucio o oly of he level of uemloyme, bu also of is rae of chage. Thus eve he seed wih which uemloyme icreases will have a favourable effec o rice hikes. For examle, very low uemloyme ha icreases raidly would affec he iflaio rae egaively. The iflaio-uemloyme model he becomes d U U,, 0 U l M, 0 Subsiuig for U, d M d l U l ad differeiaig wih resec o order differeial equaio i d d m m M gives a secod- Agai, omial moey suly m is a saioary value for iflaio rae. Here we have a, a ad b m. Hece, he aricular iegral is m ad he characerisic roos are e r, a a 4a 4 Thus he geeral soluio for iflaio deeds o he values of he characerisic roos where if 4, we have real roos such ha Coyrigh 0 SciRes.
T. TODOROVA 37 r r m Ae Ae Sice he cosas ad are osiive, he roos (or heir real ar) ur ou o be egaive ad he equilibrium is dyamically sable. For he uemloyme rae from he firs equaio of he model we have d U U which is a firs-order differeial equaio i uemloyme wih a cosa coefficie ad a variable erm. For differeial equaios wih a variable erm ad a variable coefficie of he ye dy u y v where v 0, he geeral soluio is give by he formula u d u d d y e A ve. Subsiuig i his formula i order o solve he equaio, d U e A U e d where u ad v U ad rasformig furher, d U U Ae e where by differeiaio of he iflaio rae we have d r r A re Are ad, hece, r r U U Ae Are Are e r r U Ae Are Are r r Ar Ar U Ae e e r r The resuls are cosise wih our revious fidigs. The aural rae of uemloyme agai gives he ieremoral equilibrium rae for U. Furhermore, a dyamically sable ime ah for uemloyme is ossible, sice all exoeial erms could ed o zero. The firs exoeial erm disaears wih he assage of ime, while he secod ad he hird disaear whe r, r. 5. The Effec of Iflaio o Uemloyme Le us ow cosider a versio of he exeded iflaio-uemloyme model where here is o hyseresis, ha is, iflaio is uaffeced by he rae of chage of he uemloyme level bu, raher, here is he oosie effec, ha of iflaio o uemloyme. I fac, may socially orieed ecoomiss roose maiaiig some healhy levels of iflaio so ha o kee uemloyme low. Le us assume ha he rae of chage of he iflaio rae is a decreasig fucio of he level of uemloyme bu he uemloyme rae iself is a decreasig fucio of boh real moey suly M ad he ifla io rae. A icrease i, icreases aggregae demad ad, herefore, lowers uemloyme. Now he iflaio-uemloyme model akes he form d U U 0 U l M,, 0 We ca agai aalyze he ime ahs of Subsiuig for U, d M l U ad differeiaig wih resec o d m d ad U. Agai, omial moey suly m is a saioary value for iflaio rae. Here he arameers are a, a ad b m. Hece, he aricular iegral is m ad he characerisic roos are r, a a 4a 4 Thus he geeral soluio for iflaio would deed o he values of he characerisic roos. If i haes ha 4, we have real roos. If 4, he we obai com lex roos for he ime ah of iflaio. I all cases, hough, we kow ha his ime ah is usable sice he arameers ad are osiive ad he real ar of he characerisic roos is also osiive. r r () m Ae Ae From he exressio for he uemloyme rae we obai U U d which agai gives he aural rae of uemloyme as he equilibrium rae for U. The geeral soluio for uemloyme by differeiaio of he iflaio rae is r r U U Are A re ad shows a dyamically usable ime ah for uemloyme. Coyrigh 0 SciRes.
38 T. TODOROVA 6. Iflaio ad Uemloyme i Discree Time U U formu- by Professor Blachard i discree ime. I is laed equivale o he firs equaio i our coiuous-ime iflaio-uemloyme model Cosider he equaio d U U 0 m 0 We ow cover he model i a discree-ime form ad solve for he ime ah of iflaio. From he firs eq uaio of he model by furher differeiaio we ob- d aied. I discree ime his ivolves a secod differece of rice o he lef side, ha is, The equaio i is discree form becomes U U where from he secod equaio of he model we have i discree ime U U m Thus he ew model becomes U U m U U m Subsiuig he differece er for uemloyme gives a secod-order differece equaio i : m The equilibrium value for is m m. This resul is cosise wih our revious fidigs. The comlemeary fucio of he secod-order differece equaio obaied is of he ye y y y Ab Ab c where for he characerisic roos we have b, a a 4a 44( ) i i which ur ou o be comlex umbers so he ime ah of he iflaio rae mus ivolve seed flucuaio. Sice R a where boh ad are osiive cosas, i mus be ha R. He ce, he flucuaig ah of iflaio, give he assumios of he model, mus be exlosive, as show i Figure. If we assume ha he differece for uemloyme is give b y U U m, ha is, he icrease i uemloyme deeds o iflaio i he rese, o i he revious eriod, he model becomes U U U U m Subsiuig agai he differece erm for uemloyme resuls i m The equilibrium value for is m m. Agai, he ieremoral equilibrium of iflaio is he growh rae of omial moey suly. The characerisic roos are a a 4a b, 4 4 By aalyzig he roos furher we fid b b a a ad b b ( ) 0 Sice boh ad are osiive cosas, oe ossibiliy is for boh roos o be egaive where oe is a fracio. From he secod equaio we also see ha oe () 0 Figure. The discree ime ah of acual iflaio. m Coyrigh 0 SciRes.
T. TODOROVA 39 roo is recirocal of he oher. Therefore, we coclude ha b, b 0 b ad b Sice he absolue value of oe of he roos urs ou o be greaer ha, he ime ah of iflaio is diverge ad ooscillaory. Such ime ah is illusraed by Figure 3. I he secial case of hyseresis he coiuous-ime form of he model was d U U, 0 m 0 We cover he model i a discree-ime form ad solve for he ime ah of iflaio. From he firs equaio of he model by furher differeiaio we have d d U I discree ime his ivolves a secod differece of rice o he lef side ad a secod differece of he rae of uemloyme o he righ side such ha U U U U U U U U U U U The equaio i is discree form becomes U U U U U where from he secod equaio of he model we have i discree ime U U m ad also () 0 Figure 3. The discree ime ah of acual iflaio: rese eriod. m U U U Therefore, he equaio for iflaio becomes m The equilibrium value for is m m which we have obaied reviously. Aalyzig he characerisic roos, b b a ad a b b 0 The las resul imlies ha he characerisic roos ca boh be bigger ha or smaller ha. This meas ha a coverge ime ah for iflaio is o imossible. The codiio 0 esures he dyamic sabiliy of iflaio. If we assume he differece for uemloyme o be U U m, he chage i uemloy me deeds o curre, o o revious, iflaio. The equaio of iflaio is sill U U U U U where U U m ad U U U ( ) Subsiuig i he firs equaio, m The equilibrium value for is m m. For he characerisic roos we have b b a a b b b b 0 The las resul agai shows ha a coverge ime ah for iflaio is o imossible. However, his deeds o he exac values of he arameers. Furhermore, we see ha could be less ha, give he osiive val- Coyrigh 0 SciRes.
40 T. TODOROVA ues of he arameers, which also allows for covergece. If he exeded iflaio-uemloyme model i is coiuous-ime form is d U U 0 d m, 0 we modify he model i a discree-ime form U U U U m Subsiuig he differece erm for uemloyme gives a secod-order differece equaio i, m The equilibrium value for m m. For he characerisic roos we have a a is b b b b 0 Here sice cao be bewee 0 ad, he roos cao boh be fracios. Therefore he ime ah of iflaio would o be dyamically sable. If a differe assumio is made abou uemloyme such as U U m he equaio becomes m The ieremoral equilibrium for is m m. For he characerisic roos we have a a b b 0 Here sice cao be bew ee 0 ad, he roos cao boh be fracios. Therefo re he ime ah of iflaio would o be dyamically sable agai. 7. Coclusio Sudyig he ecoomic dyamics of iflaio ad uemloyme we fid ha heir ime ahs show flucuaio boh i coiuous ad discree ime. Boh iflaio ad uemloyme flucuae aroud heir ieremoral equilibria, iflaio aroud he growh rae of omial moey suly, reflecig he moeary olicy of he goverme, ad uemloyme aroud he aural rae of uemloyme. However, while he coiuous-ime case shows uiform ad smooh flucuaio for boh ecoomic variables, i discree ime heir ime ah is exlosive ad ooscillaory. Furhermore, i he secial case whe rese, o revious, iflaio is cosidered, he discree-ime soluio shows a o-flucuaig exlosive ime ah. I sudyig he hyseresis case where iflaio is adversely affeced o oly by uemloyme bu by is rae of chage also, he resuls are ideical i boh discree ad coiuous ime. The hyseresis case shows dyamic sabiliy ad covergece for iflaio ad uemloyme o heir ieremoral equilibria. Fially, i he case whe iflaio affecs uemloyme he ime ahs of he wo boh i discree ad coiuous ime are dyamically usable. I all cases he dyamic sabiliy of iflaio ad acual uemloyme deeds o he secific values of he arameers. REFERENCES [] A. W. Phillis, The Relaioshi bewee Uemloyme ad he Rae of Chage of Moey Wage Raes i he Uied Kigdom, 86-957, Ecoomica, New Series, Vol. 5, No. 00, 958,. 83-99. [] E. S. Phels, e al., Microecoomic Foudaios of Emloyme ad Iflaio Theory, W. W. Noro, New York, 970. [3] E. S. Phels, Iflaio Policy ad Uemloyme Theory, W. W. Noro, New York, 97. [4] E. S. Phels, Moey-Wage Dyamics ad Labor Marke Equilibrium, Joural of Poliical Ecoomy, Vol. 76, No. 4, 968,. 678-7. doi:0.086/594 38 [5] O. J. Blachard, Macroecoomics, d Ediio, Chaers 8-9, Preice Hall Ieraioal, Uer Saddle River, 000. [6] A. Chiag, Fudameal Mehods of Mahemaical Ecoomics, 3rd Ediio, McGraw-Hill, Ic., New York, 984. [7] M. Pembero ad N. Rau, Mahemaics for Ecoomiss: a Iroducory Texbook, Macheser Uiversiy Press, Macheser, 00. [8] T. P. Todorova, Problems Book o Accomay Mahemaics for Ecoomiss, Wiley, Hoboke, 00. Coyrigh 0 SciRes.