A Kind of Neither Keynesian Nor Neoclassical Model (2): The Business Cycle

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1 Open Access Libay Jounal 2016, Volume 3, e3215 ISSN Online: ISSN Pint: A Kind of Neithe Keynesian No Neoclassical Model (2): The Business Cycle Ming an Zhan 1, Zhan Zhan 2 1 Yunnan nivesity, Kunming, China 2 Westa College, Southwest nivesity, Chongqing, China How to cite this pape: Zhan, M.A. and Zhan, Z. (2016) A Kind of Neithe Keynesian No Neoclassical Model (2): The Business Cycle. Open Access Libay Jounal, 3: e Received: Novembe 9, 2016 Accepted: Decembe 19, 2016 Published: Decembe 23, 2016 Copyight 2016 by authos and Open Access Libay Inc. This wok is licensed unde the Ceative Commons Attibution Intenational License (CC BY 4.0). Open Access Abstact The Cobb-Douglas function not only leads to a long-tem elationship between the ate of output change and the inteest ate, but also analyzes why they fluctuate in the shot-tem. This pape fist divides the fluctuation cycle of the inteest ate in the statistical data of the past 45 yeas by using the mathematical phase diagam method, and daws the phase diagam of the ate of output change on the inteest ate accoding to the cycle equation of output. Fom this phase diagam, we explain the eason that the phase diffeence between the inteest ate and the ate of output change in the fluctuation. Then, accoding to the optimal elation between L and K in the Cobb-Douglas function, we futhe deive the employment equation and its elation to the eal inteest ate and the ate of eal output change, and veify the theoetical speculation with statistical data. Finally, it is concluded that the business cycle is a kind of endogenous poduction phenomenon. Subject Aeas conomics Keywods Cobb-Douglas Function, Business Cycles, Phase Diagam, nemployment Rate 1. Intoduction Neoclassic theoies believe that the suplus of output and unemployment would be cleaned out duing competition and equilibium is the nomality of the economic system, theefoe the fluctuation of macoeconomic vaiables is geneated by extenal factos. The Real Business Cycle Model (RBC) aisen duing 1980s consides the total facto poductivity o the andom petubations of technology detemine fluctuation of DOI: /oalib Decembe 23, 2016

2 othe vaiables [1] [2] [3]. Howeve, othe economists [4] [5] gave this theoy some questions: accoding to RBC, if the economic boom was geneated by the technological pogess, then the economic ecession should blame on technological setbacks. Nevetheless, what ae easons of technological setbacks? Moeove, if we use othe methods to measue the technological impact, it can be a facto that eliminates business cycles [6] [7] [8] [9] [10]. Hayek said: the incopoation of cyclical phenomena into the system of economic equilibium theoy, which they ae on appaent contadiction. [11]. Since the inteaction of the total supply and total demand leads to convegence athe than divegence, the convegence would flatten the fluctuation eve if thee wee extenal impacts assumed by the RBC theoy. Pehaps the biggest poblem with macoeconomics is explaining business cycles. 2. The Business Cycle quation and Divisions Fom the Cobb-Douglas function Y = AK α L β, we deduce the maginal evenue of the capital Kin the pape A kind of neithe Keynesian no neoclassical model (1): fundamental equation [12]: Y αy MPK = = K K In a competitive maket, assuming that the maginal cost of using K is detemined by the maket inteest ate, the optimal allocation condition fo K in poduction is MPK =, then αy K = o Y =, (2) K α diffeential on both sides of the equation, so: dy dk d dα = +, (3) Y K α accoding to the basic equation, K = K so quation (3) can be ewitten as: dk K K = lim = lim =, (4) K K 0 K K 0 K (1) dy d d Y = + α α (5) When and α ae constant, d= 0, dαα= 0, dy Y=. This is the macoeconomic basic equation. quation (5) contains moe infomation than the fundamental equation dy Y=. It can be used to analyze the elationship of shot-tem fluctuations between dy Y and. In Y = AK α L β, the income distibution paamete α mainly affects the long-tem gowth (We will analyze this poblem in anothe pape The economic gowth ), assuming dαα= 0, then quation (5) can be simplified as: OALib Jounal 2/14

3 dy d = + (6) Y This is the business cycle equation. The change of dy Y in shot tem depends on changes of and d. The inteaction between and d might leads to endless fluctuation of dy Y. By = MPK we can see that eflects the poduction efficiency in the use of K. The peiodic fluctuation in the statistical data can be descibed by the phase diagam in the mathematical model. sing as the hoizontal axis and d as the vetical axis, we can plot the phase diagam of ~d. As shown in Figue 1(a), since d= 0, d = 0 at points A and C, so should not continue to change. Howeve, since d = 0, d( d) d 0, the point A o C is not a stable point on the Phase diagam ~d. At point B o D, d 0, so will continue to change. Theefoe, once moves up, the phase diagam ~d will endogenously change and make dy Y in equation (7) peiodically fluctuate. Moeove, we will see that the fluctuations of othe vaiables ae diectly o indiectly elated to the fluctuation of. Accoding to the phase diagam ~d, we can define the chaacteistic of peiodic change of, o be called the thee elements of the phase diagam: fistly, point in Figue 1(a) is the coe of one cycle, and the inteest ate that coesponds to point is an aveage value of inteest ate of a business cycle (, namely the coe inteest ate). Secondly, A o C is the adius of fluctuation of. Thidly, the otation of the phase diagam ~d is clockwise. Theoetically, we can call fou phases of A B, B C, C D and D A in Figue 1 as peiod of ecovey, pospeity, ecession and depession. Based on the phase diagam ~d, we can easily daw out the time path of as Figue 1(b) shows. In eality, since the value of coe inteest ate might be affected by extenal factos, such as the technological impact, theefoe the location and adius of ~d can be affected also. Theefoe the time path of cannot be a egula path as tigonometic functions. Figue 2 is a phase diagam ~d based on statistical data of the nited States duing Although we can distinguish diffeent cycles by these phase diagams (as indicated by the dashed line in Figue 2 fo two peiods of and , the cycle stating point is always on -axis o the neaest it below), most of them have only two phases of ise and fall, without theoetical fou phases of ecovey, pospeity, ecession and depession. Some of them even otate counteclockwise, fo example, duing This is caused by the collection of data has longe inteval time than the eal fluctuation. Take semi-annual data to edaw this diagam, it is the dotted cuve athe than the solid cuve duing Theefoe, and ae two diffeent cycles. In the same way, and ae also two cycles in Accoding to annual statistical data and the dividing ule showed in Figue 1(a), thee ae 9 cycles of fluctuation of inteest ate duing of the nited States: , , , , , , , , Accoding to the pesent statistical data (10/2016), in 2016 may not be OALib Jounal 3/14

4 Figue 1. Mathematical phase diagam about peiodic fluctuation of. (a) Phase diagam ~ d/. (b) Time path of. highe than 2015, so we guess 2016 is also in the cycle since Figue 3 shows the coesponding time path. OALib Jounal 4/14

5 Figue 2. Phase diagam ~d/ based on statistical data. Souces: 1) d. 2) In this figue, is the nominal, calculated by the simple aveage of the annualized of vaious teasuy bonds in the maket. The vaieties of teasuy bonds we collect ae 3-months, 6-months, 1-yea, 2-yea, 5-yea, 7-yea, 10-yea, 20-yea and 30-yea (Resouces: Thei aveage is in the gaph. Although statistics of in some yeas ae incomplete, but we still calculate the aveage of based on othe vaieties. Fom 1969, thee ae six vaieties of moe statistical data, so we will be 1969 as the stating yea. 3) Solid cuve is geneated by Office-xcel automatically. Dotted cuve is dawn atificially to show the path when and d ae semi-annual values. Figue 3. Business cycles divided by the change of. Souces: Light and dak aeas show diffeent cycles. Figue 2 shows base of the phase diagam ~d. Data of ae the same as Figue The Relationship between the Peiodicity of dy/y and Accoding to quation (6), the change ate of output dy Y is not only elated to but also to the change ate of inteest ate d. Based on the phase diagam ~d, we can daw a phase diagam ~dyy as Figue 4 shows. In ode to deduce the phase diagam ~dyy fom ~d, we fist daw an auxiliay line which is dy Y=, then fix points a, b, c, d and e that coespond to points A, B, C, D and on ~d based on dyy= + d. OALib Jounal 5/14

6 Figue 4. Phase diagam ~dyy. Notice: In Figue 4, find a, e, c based on the auxiliay line, and find othe points based on phase diagam ~d and equation dyy= + d. Since point h on phase diagam ~d is consistent with that on ~dyy, theefoe ( dy Y ) max exists ealie than max. Point a and c on ~dyy ae easy to be fixed, since on these two points dy Y=, d= 0 (on the phase diagam d= 0, is the minimum value min o the maximum value max ), theefoe point a and c ae located on the auxiliay line dy Y=. Points b, d and e coespond to the maximum, minimum and aveage values of d. dyy d dyy = + d. Similaly, we can de- In that case, ( ) = + ( ), ( ) ( ) b max temine the othe points, connect these points to get the phase diagam ~dyy. Figue 5 shows the time path of dy Y and accoding to ~dyy. Figue 5(b) and Figue 5(c) ae time paths of dy Y and, espectively. In ode to compae, we exchange the hoizontal and vetical coodinates of Figue 5(b) to get Figue 5(d). By compaing Figue 5(c) and Figue 5(d), thee is diffeent of fluctuation of dy Y and. We call it phase diffeence. As shown in Figue 5(c) and Figue 5(d), the time path of dy Y and ae elated as: when inceases, dy Y >. Fom point b to c, when dy Y deceases, keeps incease. Similaly, when deceases, dy Y d min <. And fom point d to a, when dy Y stats to incease, keeps decease. This shows the peak and tough of dy Y exist ealie than that of. Although the change of might exist late that of dy Y, it does not mean dy Y is the cause of the change of. Contaily, the change of dy Y is the esult of inteaction of and d. OALib Jounal 6/14

7 Figue 5. Phase diffeence of time paths of dy Y and. In Figue 6, statistical data duing veified the phase diffeence between time paths of dy Y and. Figue 4 is geneated unde the assumption of dαα= 0. If dαα 0, then the auxiliay line dy Y= should be dyy= dαα. Theefoe when dαα> 0, the phase diagam ~dyy would move downwads o be deflected, vice vesa. So the flunctuation of dy Y in eal wold is much moe complex than that of, the peak o tough of dy Y would move fowad o backwad in the business cycle unde the effect of dα α. The foundation equation dy Y= deived assuming that and α ae constant. =. Although the coe ate of Accoding to Figue 4, it should be ewitten as ( dy Y) output gowth ( dy Y ) is equal to the coe inteest ate, the value of dy Y and ae not always equal, since them have diffeent phases. Compaed to the analysis of total demand and supply, the analysis of phase diagam could give bette explanation of elations between shot-tems values of macoeconomic vaiables. 4. Peiodicity of the nemployment Rate Based on algebaic ules, no matte what ae the oiginal state of α and β, thee is a dynamic uppe limit of Cobb-Douglas function Y = AK α L β : α β ( K ) + ( L ) 2 2 α β Y = AK L A (7) 2 Appaently, the condition of AK α L β to each its uppe limit is: L β α = K (8) OALib Jounal 7/14

8 Figue 6. Phase diffeences between statistical data dy Y and. Souces: 1) dy Y YY, Y = GDP. Data of nominal GDP ae fom 2) Shadowed aeas show business cycles as in Figue 3 (the same as heeafte). Data of ae the same as Figue 2. quation (8) L β α = K not only expesses the endogenous optimization of L and K, but also defines the elation between L and K. Accoding to economic pinciples, the amount of poduction factos used in the system is detemined by its maginal evenue. Theefoe the amount of L that be used should be closely elated to the maginal output of L ( Y L). Based on the maginal equation of Y = AK α L β : Y L = βy L, assume w= Y L, then Y βy w = = (9) L L βy d β α βy dw α β K dβ dy α dk α α since L = K, w =, = = + ln Kd α w βy + β Y β K β β β K α β K among them, α + β = 1, dk K = ( A kind of neithe Keynesian no neoclassical model (1): The fundamental equation [12]), Assuming α is a constant, so dβ β = 0 and d( α β ) = 0, then dw dy α = (10) w Y β In Y = AK α L β, L is the monetized human esouces, we cannot diectly use amount of the employment population as amount of L. Thee might be elation between d (the change ate of employment) and dw w (the change ate of maginal output of the monetized human esouces), since we notice that the lage of the maginal contibution of L the moe willingness of hiing employees of companies. Figue 7 shows, the fluctuation of dw w and d ae move to the same diection in shottem, even though the ange of fluctuation ae diffeent especially when the inflation is moe seious. The statistical data Y and ae the nominal values with money when calculate dw w. is the employed population. The natue of is the non-monetay eal vaiable. OALib Jounal 8/14

9 When dpp> 0, dw w which is calculated based on nominal values is geate than d which is calculated based on eal values. If quation (10) is calculated based on eal values Y and, then dw dy α = (11) w Y β Since dy Y = dyy dpp, and = dpp,in which dp P is the ate of inflation, then dw dy α dy dp α dp dy α α dp = = = 1 w Y β Y P β P Y β β P Since d d ( α β) ww= YY, the eal vaiable dw w and the nominal vaiable dw w is expessed as dw dw α dp = 1 w w β P As Figue 8 shows, the fluctuation of dw w and d ae appoximately simila. The eason fo they ae not completely consistent is because we assume α keep unchanged and theefoe the change of α is not included in the impact on dw w. Above statistical data show that the change ate of employment d is detemined by the change ate of maginal eal output of human esouces dw w. Theefoe we assume d= dw w and eplaced dw w in quation (11) with d, then d dy α = Y β (12) Accoding to quation (6) assume dy Y + d, then d dy α d d α 1 α = = + = + Y β β β (13) Figue 7. Relations between dw w and d. Calculation and Resouces: 1) dww= dyy ( α β), in which α = 1 β, β is the compensation of employees in pesonal income. Data of wage income ae fom Data of Y and see Figue 6. 2) d, data of employment population ae fom OALib Jounal 9/14

10 Figue 8. Relations between dw w and d. Calculation and Resouces: 1) dw w = dww 1 α β dpp. Data of dw w, α β, and d see Figue 7. 2) ( ) dp P PP, P is calculated based on the deflato of the nominal GDP and the eal GDP. Data ae fom The stuctue of quation (13) is simila to that of dyy= + d, theefoe we can daw the phase diagam ~d accoding to the phase diagam ~d as Figue 4 shows. Accoding to the ate of change of employed population d and R = N (labo population as N, employed population as ), we can convet d dr d ( N ) d dn into dr R : = =, accoding to quation (13) R N N dr α d dn = 1 + R β N As Figue 9 shows, the diection of otation and shape of the phase diagam ~dr R is the same as that of ~d. Coe vaiables d and dr R locate diffeently on the vetical coodinate, and the diffeence is detemined by the change ate of labo population dn N. In othe wods, we get the phase diagam of ~dr R by moving the phase diagam of ~d downwad a distance of dn N. Theefoe, we can pedict that the fluctuation of d and dr R ae consistent. Accoding to quation (13), the eason of shot-tem fluctuation of d is the peiodicity facto ( + d ), which is simila to the eason of the fluctuation of dy Y. If thee is no phase o only tiny diffeence between the fluctuation of dy Y and dy Y, and the phase diffeence between and is small, then the fluctuating paths of d, dr R and dy Y would be consistent. The statistical data of Figue 10 can veify this conjectue. The phase diagam ~dr R in Figue 9 shows that the change ate of employment ate dr R maybe 0 unde the effect of sometimes. Mathematically, fist ode diffeential dr = 0 indicates thee is exteme value o inflection point on R. Theefoe, thee is a phase diagam between R and dr R pobably. The phase diagam R ~dr R in Figue 11 is fom the statistical data. (14) OALib Jounal 10/14

11 Figue 9. Relations between phase diagams ~d and ~dr R. Notice: Assume a phase diagam ~d, daw the auxiliay line d= ( 1 α β). Find points a, e, c coesponding to points A,, C on ~d, fix othe points accoding to the equation d= 1 α β + d. Then get the phase diagam ~d by connecting these points. ( ) Figue 10. Relations between d, dr R and dy Y. Calculation and esouces: dy Y= dyy dpp, date of Y is same Figue 6. Date of P is same Figue 8. d = d dpp, d is same Figue 2. α β is same Figue 7. Date of N fom In ode to discuss the elationship between the unemployment ate R and othe macoeconomic vaiables based on the expession of Phillips cuve, we often need to convet the employment ate R to the unemployment ate R. Since the labo population N is the sum of employed population and unemployed population, namely N = +, then the elation between R and R, and the elation between dr R and dr R ae: N R = = = 1 = 1 R N N N OALib Jounal 11/14

12 Figue 11. The phase diagam R ~dr R and R ~dr R in Souces: Date of R is same Figue 11. R = 1 R. ( R ) dr d1 dr R dr 1 dr = = = = 1 R 1 R 1 R 1 R R R R (15) Based on quation (15) we can convet the phase diagam R ~dr R to R R R. The phase diagam of R ~dr R and R ~dr R is almost ~d mio images. The dr R is not the R. As thee is a phase diffeence between and d, thee is also a phase diffeence between R and dr R, although thei phase diffeences may not be significant in yealy statistical data. Figue 12 shows the fluctuation of R and dr R in the statistical data. Befoe 1986, the peaks of R and dr R appeaed at the same time (which is the eason fo inflation, which we will descibe in late papes). Afte 1986, the peak of R appeaed late than dr R. The peaks of dr R and dy Y ae the same in Figue 10, because dr R and dr R ae the opposite of the peak, theefoe the peak of R will lag behind the peak of dy Y afte This helps us undestand the elationship and causes of fluctuations of unemployment and output. While the above analysis explains the easons fo the cycle in the unemployment ate, thee is one fundamental poblem that emains unsolved: We cannot detemine the value of R o R accoding to quations (17) and (14) in a business cycle. Because in the diffeential equation dr R = ( 1 1 R ) ( 1 α β) + d dn N, we need to know some initial conditions to detemine R o R. As Y and dy Y, we cannot detemine Y fom Y without knowing the initial condition of Y. This means that even if the dr R of the two economic entities ae exactly the same, thei R ae not necessaily equal, because the maketization degee of employment and the cultue of employment is diffeent in diffeent counties o egions. Fiedman consideed although the unemployment state would be affected by maket impefections, stochastic vaiability in demands and supplies, the cost of gatheing infomation about job vacancies and labo availabilities, the costs of mobility, and so OALib Jounal 12/14

13 Figue 12. Relations between R and dr compaison puposes, dr R is only 1/10 of the oiginal data. R. Souces: The data is same Figue 11. Fo on [13], the unemployment ate fluctuated aound the natual ate of unemployment in shot tem. Factos that Fiedman took as examples ae also initial conditions that affect the diffeential equation R = f ( dr R ). Howeve, the natual ate of unemployment is not the coe unemployment ate R, since R move back and fowads in diffeent cycles as Figue 11 shows. 5. Conclusions 5.1. Hypothesis Poduction function in the maket system: Y = AK α L β. A maginal condition: = Y K = αy K. β α dαα= 0, L = K, d= dw w Results The cycle equation about the output: dyy= + d. The cycle equation about the employment: dr R = ( 1 α β) + d dn N, o dr R = ( 1 1 R ) ( 1 α β) + d dn N Discussion Taditional macoeconomics cannot logically explain contadictions of economic poblems in long-tem and shot-tem. Classical theoy seems to be handy in explaining elationships between vaiables in long-tem, but it is difficult to undestand the phenomenon in shot-tem. Keynesian theoy, while able to explain some phenomenon in shot-tem, but thee will be idiculous infeence in long-tem. Fom the model in this pape and A kind of neithe Keynesian no neoclassical model (1): the fundamental equation [12], we can logically and consistently see elationships between the macoeconomic vaiables in the long-tem and the shottem, and use statistical data to veify these elationships. OALib Jounal 13/14

14 The cycle is affected by many factos, but as long as the maginal poduct of the economic system is not zeo, thee is the business cycle even without these extenal stochastic factos. Since and d ae the fist and second ode diffeentials of Y, the fluctuation of output is due the effect of the second ode diffeential on the fist ode diffeential. Theefoe the business cycle is fundamentally an endogenous poducing action. Because both endogenous and exogenous foces have impacts on the cycle, so it is difficult to see two cycles ae the same in tems of fluctuation and time consumption. On the intechangeability between inflation and unemployment in the Phillips Cuve, we need to futhe ou study on the elationship between money and Cobb-Douglas function. Due to the limited data souces and the heavy wokload of pocessing data, this pape is limited to the veification of annual data. It is not known whethe these peiodic equations also apply to quately o monthly data. Refeences [1] Kydland, F.. and Pescott,.C. (1988) The Wokweek of Capital and Its Cyclical Implications. Jounal of Monetay conomics, 21, [2] Kydland, F.. and Pescott,.C. (1990) Business Cycle: Real Facts and a Monetay Myth. Reseve Bank of Minneapolis Quately Review, 14, [3] Kydland, F.. and Pescott,.C. (1991) The conometics of the Geneal quilibium Appoach to Business Cycle. Scandinavian Jounal of conomics, 93, [4] Summes, L.H. (1986) Some Skeptical Obsevations on Real Business Cycle Theoy. Fedeal Reseve Bank of Minneapolis Quately Review, 10, [5] Mankiw, N.G. (1989) Real Business Cycles: A New Keynesian Pespective. Jounal of conomic Pespectives, 3, [6] Buside, C., ichenbaum, M. and Rebelo, S. (1996) Sectoal Solow Residuals. uopean conomic Review, 40, [7] Basu, S. and Kimball, M.S. (1997) Cyclical Poductivity with nobseved Input Vaiation. NBR Woking Pape Seies, numbe [8] Muellbaue, J. (1997) The Assessment: Business Cycles. Oxfod Review of conomic Policy, 13, [9] Gall, J. (1999) Technology, mployment and Business Cycles: Do Technology Shocks xplain Aggegate Fluctuation? Ameican conomic Review, 89, [10] Ramey, V.A. and Neville, F. (2002) Is the Technology-Diven Real Business Cycle Hypothesis Dead? Shocks and Aggegate Fluctuations Revisited. nivesity of Califonia at San Diego, conomics Woking Pape Seies qt6x80k3nx. [11] Hayek, F.A. (1929) Monetay Theoy and the Tade Cycle, New Yok: Augustus M. Kelley, 1966; epint of 1933 nglish dition, London: Jonathan Cape; oiginally published in Geman in [12] Zhan, M.A. and Zhan, Z. (2016) A Kind of Neithe Keynesian No Neoclassical Model (1): The Fundamental quation. Open Access Libay Jounal, 3, e [13] Fiedman, M. (1968) The Role of Monetay Policy. Ameican conomic Review, 58, OALib Jounal 14/14

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