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1 econsor Make Your Publicaions Visible. A Service of Wirschaf Cenre zbwleibniz-informaionszenrum Economics Todorova, Tamara Aricle The Economic Dynamics of Inflaion and Unemloymen Theoreical Economics Leers Suggesed Ciaion: Todorova, Tamara (0) : The Economic Dynamics of Inflaion and Unemloymen, Theoreical Economics Leers, ISSN 6-086, Vol., Iss., , h://dx.doi.org/0.436/el.0.05 This Version is available a: h://hdl.handle.ne/049/48369 Sandard-Nuzungsbedingungen: Die Dokumene auf EconSor dürfen zu eigenen wissenschaflichen Zwecken und zum Privagebrauch geseicher und koier werden. Sie dürfen die Dokumene nich für öffenliche oder kommerzielle Zwecke vervielfäligen, öffenlich aussellen, öffenlich zugänglich machen, verreiben oder anderweiig nuzen. Sofern die Verfasser die Dokumene uner Oen-Conen-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gesell haben sollen, gelen abweichend von diesen Nuzungsbedingungen die in der dor genannen Lizenz gewähren Nuzungsreche. Terms of use: Documens in EconSor may be saved and coied for your ersonal and scholarly uroses. You are no o coy documens for ublic or commercial uroses, o exhibi he documens ublicly, o make hem ublicly available on he inerne, or o disribue or oherwise use he documens in ublic. If he documens have been made available under an Oen Conen Licence (esecially Creaive Commons Licences), you may exercise furher usage righs as secified in he indicaed licence.

2 Theoreical Economics Leers, 0,, doi:0.436/el.0.05 Published Online May 0 (h:// The Economic Dynamics of Inflaion and Unemloymen Tamara Todorova Dearmen of Economics, American Universiy in Bulgaria, Blagoevgrad, Bulgaria odorova@aubg.bg Received November 8, 0; revised December, 0; acceed January 3, 0 ABSTRACT We sudy he ime ah of inflaion and unemloymen using he Blanchard reamen of he relaionshi beween he wo and aking he moneary olicy condiion ino accoun. We solve he model boh in coninuous and discree ime and comare he resuls. The economic dynamics of inflaion and unemloymen shows ha hey flucuae around heir ineremoral equilibria, inflaion around he growh rae of nominal money suly, resecively, and unemloymen around he naural rae of unemloymen. However, while he coninuous-ime case shows uniform and smooh flucuaion for boh economic variables, in discree ime heir ime ah is exlosive and nonoscillaory. The hyseresis case shows dynamic sabiliy and convergence for inflaion and unemloymen o heir ineremoral equilibria boh in discree and coninuous ime. When inflaion affecs unemloymen adversely he ime ahs of he wo, boh in discree and coninuous ime, are dynamically unsable. Keywords: Economic Dynamics; Second-Order Differenial Equaions; Second-Order Difference equaions; Phillis Curve; Inflaion; Unemloymen. Inroducion The relaionshi beween inflaion and unemloymen illusraed by he so called Phillis curve was firs discussed by Phillis [] in a ah-breaking aer iled The Relaionshi beween Unemloymen and he Rae of Change of Money Wage Raes in he Unied Kingdom, The sandard reamen of he relaionshi beween inflaion and unemloymen in dynamics involves he execaions-augmened Philis curve, he adaive execaions hyohesis and he moneary olicy condiion. Solving he model allows sudying he economic dynamics of he variables reaed as funcions of ime. Thus, for examle, we are able o find he ime ah and condiions for dynamic sabiliy of acual inflaion as well as of real unemloymen. In sudying he relaionshi beween inflaion and unemloymen economiss such as Phels [,3] have found no long-run radeoff beween hese wo, oosie o wha he Phillis curve imlies. In an influenial 968 aer iled Money- Wage Dynamics and Labor Marke Equilibrium Phels [4] sudies he role of adaive execaions in seing wages and rices. There he inroduces he conce of he naural rae of unemloymen and argues ha labor marke equilibrium is indeenden of he rae of inflaion. This finding renders Keynesian heory of conrolling he long-run rae of unemloymen in he economy ineffecive. In his book Macroeconomics Blanchard [5] offers an alernaive reamen of he relaionshi beween inflaion and unemloymen. He incororaes in he model he naural rae of unemloymen U n a which he acual and he execed inflaion raes are equal. The rae of change of he inflaion rae is roorional o he difference beween he acual unemloymen rae U and he naural rae of unemloymen Un. The urose of our aer is o sudy he economic dynamics and ime ah of inflaion and unemloymen from he ersecive of Blanchard s equaion of he relaionshi beween inflaion and unemloymen. We solve he model boh in coninuous and discree ime and comare he resuls. We discuss hree cases, a simle model of Blanchard s equaion wih he moneary olicy condiion aken ino accoun. Then we exend he model o he hyseresis case, where inflaion is adversely affeced no only by unemloymen bu by is rae of change also. Finally, we solve he model when here is he oosie effec, ha of inflaion on unemloymen. In sudying he ime ah of inflaion and unemloymen we find ha hey flucuae around heir ineremoral equilibria, inflaion around he growh rae of nominal money suly, resecively, and unemloymen around he naural rae of unemloymen. However, while he coninuous-ime case shows uniform and smooh flucuaion for boh economic variables, in discree ime heir ime ah is exlosive and nonoscillaory. Furhermore, in he secial case when resen, no revious, inflaion is considered, he discree-ime soluion shows a non-flucua-

3 34 T. TODOROVA ing exlosive ime ah. In he hyseresis case he resuls are idenical and show dynamic sabiliy and convergence for inflaion and unemloymen o heir inerermoral equilibria boh in discree and coninuous ime. In he case when inflaion affecs unemloymen adversely he ime ahs of he wo boh in discree and coninuous ime are dynamically unsable. The aer is organized as follows: Secion reveals he sandard reamen of he ineremoral relaionshi beween inflaion and unemloymen. In Secion 3 we solve an innovaive model of his relaionshi using Blanchard s equaion. Secions 4 and 5 exend his model o he hyseresis case and reverse influence case, resecively. Secion 6 ransforms hese coninuous-ime soluions ino discree-ime resuls. The aer ends wih concluding remarks.. Inflaion and Unemloymen: The Sandard Treamen The sandard reamen of he relaionshi beween inflaion and unemloymen has well been sudied by mahemaical economiss such as Chiang [6], Pemberon and Rau [7] and Todorova [8]. The original Phillis relaion shows ha he rae of inflaion is negaively relaed o he level of unemloymen and osiively o he execed rae of inflaion such ha U hπ, 0,0h where is he rae of growh of he rice level, i.e., he inflaion rae, U is he rae of unemloymen and π denoes he execed rae of inflaion. Thus he execaion of higher inflaion shaes he behavior of firms and individuals in a way ha simulaes inflaion, indeed (execing rices o rise, hey migh decide o buy more resenly). As eole exec inflaion o go down (as a resul of aroriae governmen olicies, for examle), his, indeed, brings acual inflaion down. This version of he Phillis relaion ha accouns for he execed rae of inflaion is called he execaions-augmened Phillis relaion. The adaive execaions hyohesis furher shows how inflaionary execaions are formed. The equaion The exanded version of he Phillis relaion incororaes he growh rae of money wage w where he rae of inflaion is he difference beween he increase in wage and he increase in labor roduciviy T, ha is, w T. Thus inflaion would resul only when wage increases faser han roduciviy. Furhermore, wage growh is negaively relaed o unemloymen and osiively o he execed rae of inflaion or w U hπ where U is he rae of unemloymen and π is he execed rae of inflaion. If inflaionary rends ersis long enough, eole sar forming furher inflaionary execaions which shae heir money-wage demands. j π 0 j illusraes ha when he acual rae of inflaion exceeds he execed one, his nurures eole s execaions so 0. In he oosie case, if he acual inflaion is below he execed one, his makes eole believe ha inflaion would go down so π is reduced. If he rojeced and he real inflaion urn ou o be equal, eole do no exec a change in he level of inflaion. There is also he reverse effec, ha of inflaion on unemloymen. When inflaion is high for oo long, his may discourage eole from saving, consequenly reduce aggregae invesmen and increase he rae of unemloymen. We can wrie k k 0 or unemloymen increases roorionally wih real money where is he rae of growh of nominal money. The exression gives he rae of growh of real money, or he difference beween he growh rae of nominal money and he rae of inflaion rm m where real money is nominal money divided by he average rice level in he economy. The model hen becomes U hπ,, 0,0h (execaions-augmened Philis relaion) j π 0 j (adaive execaions) k k 0 (moneary olicy) We solve his model by subsiuing he firs equaion ino he second which gives j U jh π Differeniaing furher wih resec o ime, U d π d j j h and subsiuing for d U we obain d π jk jh d d where he second equaion of he model imlies

4 T. TODOROVA 35 π. Subsiuing his las exression for j we obain j d π j k π jh This is a second-order differenial equaion in π ransforms ino d π or alernaively k jh jkπ jk, π k j h π jkπ jk which Given he roeries of second-order differenial equaions, we have he following arameers a k j h a j k b jk The coefficiens a and a are boh osiive in view of he signs of he arameers. We find he equilibrium rae of execed inflaion o be he aricular inegral b π a Hence, he ineremoral equilibrium of he execed rae of inflaion is exacly he rae of growh of nominal money. In order o esablish he ime ah of π we need o find he characerisic roos of he differenial equaion which we can do using he formula r, a a 4a The ime ah of π would deend on he aricular values of he arameers. Once we find his ime ah we migh be able o deermine ha of unemloymen U or he rae of inflaion.. 3. Inflaion and Unemloymen: An Exended Model In his book Macroeconomics Blanchard [5] offers an alernaive reamen of he relaionshi beween inflaion and unemloymen. He inroduces in he model he naural rae of unemloymen U n a which he acual and he execed inflaion raes are equal. The rae of change of he inflaion rae is roorional o he difference beween he acual unemloymen rae U and he naural rae of unemloymen U n such ha d U Un 0 Therefore, when U Un, ha is, he acual rae of unemloymen exceeds he naural rae, he inflaion rae decreases and when U Un, he inflaion rae increases. The inuiive logic behind his is ha in bad economic imes when many eole are laid off, rices end o fall. A his oin he acual unemloymen would exceed he normal levels. In imes of a boom in he business cycle he rae of acual unemloymen would be raher low bu high aggregae demand would ush rices u. Blanchard s equaion reveals an imoran relaion as i gives anoher way of hinking abou he Phillis curve in erms of he acual and he naural unemloymen raes and he change in he inflaion rae. Furhermore, i inroduces he naural rae of unemloymen as i relaes o he nonacceleraing-inflaion rae of unemloymen (or NAIRU), he rae of unemloymen required o kee he inflaion rae consan. We solve his alernaive model of he relaionshi beween inflaion and unemloymen by assuming ha U n is consan and ha a any given ime he acual unemloymen rae U is deermined by aggregae demand which, on is own, deends on he real value of money suly given by nominal money suly M divided by he average rice level. Thus unem- loymen is negaively relaed o real money suly M according o he relaionshi U ln M, 0 We solve by differeniaing he firs equaion d and he second equaion o obain d U d U d dlnm ln M dln We assume ha he growh rae of nominal money suly is consan which could be in accordance wih sysemaic governmen lanning or moneary olicy. The equaion ha obains is idenical o he moneary-olicy equaion inroduced in he sandard reamen of he Phillis curve. Combining he wo resuls yields d d which is a second-order differenial equaion in inflaion rae. Solving he differenial equaion, we have Professor Blanchard [5] formulaes his original equaion in discree U U. ime as n

5 36 T. TODOROVA a 0, a and b. Hence, he aricular inegral is and he characerisic equaion is e r 0 r, i where h 0 and v Thus he general soluion involves comlex roos and akes he form o () e Bcos Bsin Bcos Bsin Similar o he sandard model we can sudy he dynamic sabiliy of acual inflaion. Since h 0, he funcion of inflaion rae dislays uniform flucuaions around he rae of growh of money suly which gives he equilibrium level of inflaion. 3 Since he growh rae of nominal money suly deends on governmen olicies and changes wih hose, i is a moving equilibrium. Such flucuaing ime ah around he ineremoral equilibrium can be grahed as in Figure. Alhough he ime ah is no convergen, moneary olicy can somewha seer inflaion and limi i wihin a unnel as i flucuaes around. Given he remises of he model and he values of he arameers, a divergen ime ah and, herefore, an unconrollable level of inflaion are imossible. To find he ime ah of unemloymen U as he d nex se we exress as d Bsin Bcos and subsiue i ino d U Un Un Bsin Bcos Un B sin B cos where he consans B and B have no been definiized. I follows ha, similar o he inflaion rae, he unemloymen rae dislays regular flucuaions bu is ineremoral equilibrium is he naural rae of unemloymen. Since his is he rae a which execed and acual inflaion are equal, we can view ineremoral equilibrium as he sae in which execaions coincide 3 The ime ah of a general comlemenary funcion of he ye h y e B cosvb sin v c deends on he sine and cosine funcions h as well as on he erm e. Since he eriod of he rigonomeric funcions is π and heir amliude is, heir grahs reea heir shaes every ime he exression v increases by π. () 0 h 0 Figure. The ime ah of acual inflaion. wih realiy. Since again we have h 0, he ime ah of unemloymen is neiher convergen, nor divergen. I follows, herefore, ha wih he assage of ime acual unemloymen canno subsanially deviae from he naural rae of unemloymen. 4. The Blanchard Model: A Hyseresis Sysem The equaion formulaed by Professor Blanchard can be exended furher o he so called hyseresis sysem. This version of he model assumes ha he rae of change of he inflaion rae is a decreasing funcion no only of he level of unemloymen, bu also of is rae of change. Thus even he seed wih which unemloymen increases will have a favourable effec on rice hikes. For examle, very low unemloymen ha increases raidly would affec he inflaion rae negaively. The inflaion-unemloymen model hen becomes d U Un,, 0 U ln M, 0 Subsiuing for U, d M d ln Un ln M and differeniaing wih resec o gives a secondorder differenial equaion in d d Again, nominal money suly is a saionary value for inflaion rae. Here we have a, a and b. Hence, he aricular inegral is and he characerisic roos are e r, a a 4a 4 Thus he general soluion for inflaion deends on he values of he characerisic roos where if 4, we have real roos such ha

6 T. TODOROVA 37 r r Ae Ae Since he consans and are osiive, he roos (or heir real ar) urn ou o be negaive and he equilibrium is dynamically sable. For he unemloymen rae from he firs equaion of he model we have d U Un which is a firs-order differenial equaion in unemloymen wih a consan coefficien and a variable erm. For differenial equaions wih a variable erm and a variable coefficien of he ye dy u y v where v 0, he general soluion is given by he formula u d u d y e A ve. Subsiuing in his formula in order o solve he equaion, d U e A Un e d where u and v Un and ransforming furher, d U Un Ae e where by differeniaion of he inflaion rae we have d r r A re Are and, hence, r r U Un Ae Are Are e r r Un Ae Are Are r r Ar Ar Un Ae e e r r The resuls are consisen wih our revious findings. The naural rae of unemloymen again gives he ineremoral equilibrium rae for U. Furhermore, a dynamically sable ime ah for unemloymen is ossible, since all exonenial erms could end o zero. The firs exonenial erm disaears wih he assage of ime, while he second and he hird disaear when r, r. 5. The Effec of Inflaion on Unemloymen Le us now consider a version of he exended inflaion-unemloymen model where here is no hyseresis, ha is, inflaion is unaffeced by he rae of change of he unemloymen level bu, raher, here is he oosie effec, ha of inflaion on unemloymen. In fac, many socially oriened economiss roose mainaining some healhy levels of inflaion so ha o kee unemloymen low. Le us assume ha he rae of change of he inflaion rae is a decreasing funcion of he level of unemloymen bu he unemloymen rae iself is a decreasing funcion of boh real money suly M and he infla ion rae. An increase in, increases aggregae demand and, herefore, lowers unemloymen. Now he inflaion-unemloymen model akes he form d U Un 0 U ln M,, 0 We can again analyze he ime ahs of and U. Subsiuing for U, d M ln Un and differeniaing wih resec o d d Again, nominal money suly is a saionary value for inflaion rae. Here he arameers are a, a and b. Hence, he aricular inegral is and he characerisic roos are r, a a 4a 4 Thus he general soluion for inflaion would deend on he values of he characerisic roos. If i haens ha 4, we have real roos. If 4, hen we obain comlex roos for he ime ah of inflaion. In all cases, hough, we know ha his ime ah is unsable since he arameers and are osiive and he real ar of he characerisic roos is also osiive. r r () Ae Ae From he exression for he unemloymen rae we obain d U Un which again gives he naural rae of unemloymen as he equilibrium rae for U. The general soluion for unemloymen by differeniaion of he inflaion rae is r r U Un Are A re and shows a dynamically unsable ime ah for unemloymen.

7 38 T. TODOROVA 6. Inflaion and Unemloymen in Discree Time Consider he equaion U Un formulaed by Professor Blanchard in discree ime. I is equivalen o he firs equaion in our coninuous-ime inflaion-unemloymen model d U Un 0 0 We now conver he model in a discree-ime form and solve for he ime ah of inflaion. From he firs equaion of he model by furher differeniaion we ob- d ained. In discree ime his involves a second difference of rice on he lef side, ha is, The equaion in is discree form becomes U U where from he second equaion of he model we have in discree ime U U m Thus he new model becomes U U U U m Subsiuing he difference erm for unemloymen gives a second-order difference equaion in : The equilibrium value for is. This resul is consisen wih our revious findings. The comlemenary funcion of he second-order difference equaion obained 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 urn ou o be comlex numbers so he ime ah of he inflaion rae mus involve seed flucuaion. Since R a where boh and are osiive consans, i mus be ha R. Hence, he flucuaing ah of inflaion, given he assumions of he model, mus be exlosive, as shown in Figure. If we assume ha he difference for unemloymen is given by U U m, ha is, he increase in unemloymen deends on inflaion in he resen, no in he revious eriod, he model becomes U U U U Subsiuing again he difference erm for unemloymen resuls in The equilibrium value for is. Again, he ineremoral equilibrium of inflaion is he growh rae of nominal money suly. The characerisic roos are a a 4a b, 4 4 By analyzing he roos furher we find b b a a and b b ( ) 0 Since boh and are osiive consans, one ossibiliy is for boh roos o be negaive where one is a fracion. From he second equaion we also see ha one () 0 Figure. The discree ime ah of acual inflaion.

8 T. TODOROVA 39 roo is recirocal of he oher. Therefore, we conclude ha b, b 0 b and b Since he absolue value of one of he roos urns ou o be greaer han, he ime ah of inflaion is divergen and nonoscillaory. Such ime ah is illusraed by Figure 3. In he secial case of hyseresis he coninuous-ime form of he model was d U Un, 0 0 We conver he model in a discree-ime form and solve for he ime ah of inflaion. From he firs equaion of he model by furher differeniaion we have d d U In discree ime his involves a second difference of rice on he lef side and a second difference of he rae of unemloymen on he righ side such ha U UU U U U UU U U U The equaion in is discree form becomes U U U U U where from he second equaion of he model we have in discree ime U U m and also () 0 Figure 3. The discree ime ah of acual inflaion: resen eriod. U U U Therefore, he equaion for inflaion becomes The equilibrium value for is which we have obained reviously. Analyzing he characerisic roos, b b a and a b b 0 The las resul imlies ha he characerisic roos can boh be bigger han or smaller han. This means ha a convergen ime ah for inflaion is no imossible. The condiion 0 ensures he dynamic sabiliy of inflaion. If we assume he difference for unemloymen o be U U, he change in unemloymen deends on curren, no on revious, inflaion. The equaion of inflaion is sill U U U U U where U U and U U U ( ) Subsiuing in he firs equaion, The equilibrium value for is. For he characerisic roos we have b b a a b b 0 The las resul again shows ha a convergen ime ah for inflaion is no imossible. However, his deends on he exac values of he arameers. Furhermore, we see ha could be less han, given he osiive val-

9 40 T. TODOROVA ues of he arameers, which also allows for convergence. If he exended inflaion-unemloymen model in is coninuous-ime form is d U U n 0 d, 0 we modify he model in a discree-ime form U U U U m Subsiuing he difference erm for unemloymen gives a second-order difference equaion in, The equilibrium value for. For he characerisic roos we have a a is 7. Conclusions Sudying he economic dynamics of inflaion and unemloymen we find ha heir ime ahs show flucuaion boh in coninuous and discree ime. Boh inflaion and unemloymen flucuae around heir ineremoral equilibria, inflaion around he growh rae of nominal money suly, reflecing he moneary olicy of he governmen, and unemloymen around he naural rae of unemloymen. However, while he coninuous-ime case shows uniform and smooh flucuaion for boh economic variables, in discree ime heir ime ah is exlosive and nonoscillaory. Furhermore, in he secial case when resen, no revious, inflaion is considered, he discree-ime soluion shows a non-flucuaing exlosive ime ah. In sudying he hyseresis case where inflaion is adversely affeced no only by unemloymen bu by is rae of change also, he resuls are idenical in boh discree and coninuous ime. The hyseresis case shows dynamic sabiliy and convergence for inflaion and unemloymen o heir ineremoral equilibria. Finally, in he case when inflaion affecs unemloymen he ime ahs of he wo boh in discree and coninuous ime are dynamically unsable. In all cases he dynamic sabiliy of inflaion and acual unemloymen deends on he secific values of he arameers. 0 REFERENCES Here since canno be beween 0 and, he roos canno boh be fracions. Therefore he ime ah of inflaion would no be dynamically sable. If a differen assumion is made abou unemloymen such as U U he equaion becomes [] A. W. Phillis, The Relaionshi beween Unemloymen and he Rae of Change of Money Wage Raes in he Unied Kingdom, , Economica, New Series, Vol. 5, No. 00, 958, [] E. S. Phels, e al., Microeconomic Foundaions of Emloymen and Inflaion Theory, W. W. Noron, New York, 970. [3] E. S. Phels, Inflaion Policy and Unemloymen Theory, W. W. Noron, New York, 97. [4] E. S. Phels, Money-Wage Dynamics and Labor Marke Equilibrium, Journal of Poliical Economy, Vol. 76, No. 4, 968, doi:0.086/59438 The ineremoral equilibrium for is. [5] O. J. Blanchard, Macroeconomics, nd Ediion, Chaers 8-9, Prenice Hall Inernaional, Uer Saddle River, 000. For he characerisic roos we have a a b b 0 Here since canno be beween 0 and, he roos canno boh be fracions. Therefore he ime ah of inflaion would no be dynamically sable again. [6] A. Chiang, Fundamenal Mehods of Mahemaical Economics, 3rd Ediion, McGraw-Hill, Inc., New York, 984. [7] M. Pemberon and N. Rau, Mahemaics for Economiss: an Inroducory Texbook, Mancheser Universiy Press, Mancheser, 00. [8] T. P. Todorova, Problems Book o Accomany Mahemaics for Economiss, Wiley, Hoboken, 00.