A Kind of Neither Keynesian Nor Neoclassical Model (6): The Ending State of Economic Growth

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1 Open Access Libay Jounal 2017, Volume 4, e3589 ISSN Online: ISSN Pint: A ind of Neithe eynesian No Neoclassical Model (6): The Ending State of Economic Gowth Ming an Zhan 1, Zhan Zhan 2 1 Yunnan Univesity, unming, China 2 Westa College, Southwest Univesity, Chongqing, China How to cite this pape: Zhan, M.A. and Zhan, Z. (2017) A ind of Neithe eynesian No Neoclassical Model (6): The Ending State of Economic Gowth. Open Access Libay Jounal, 4: e Received: Apil 10, 2017 Accepted: May 14, 2017 Published: May 18, 2017 Copyight 2017 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 In taditional macoeconomics agues that the decision and fluctuation of output level is shot-tem theoy, and the gowth of output is a long-tem theoy. The fome is detemined by the demand; the latte is detemined by the poduction. No one has questioned why the fome is detemined by poduction and the latte is detemined by demand. This pape agues that the factos that affect the output ae the same in the shot and long tem, but thee is no need to analyze the poblem of fluctuation in the long tem. Based on the analysis of the gowth path in ou pevious pape, this pape fist examines whethe ou model is applicable to the Pontyagin maximum pinciple, and then analyzes the diffeence of the Cobb-Douglas function in the exogenous gowth model and the endogenous gowth model. Reveal the special ole of paamete A in Cobb-Douglas function: as long as A is consideed as output-elated vaiables, thee is no substantial diffeence between the so-called exogenous model and endogenous mode. Finally, accoding to the tend of the gowth path and the natue of A, the pape deives the final state of Cobb-Douglas function. Subject Aeas Economics eywods Cobb-Douglas Function, Dynamic Optimization, Exogenous Gowth, Endogenous Gowth 1. Peface Lucas has been attacted by the questions on economic gowth: Is thee some DOI: /oalib May 18, 2017

2 action a govenment of India could take that would lead the Indian economy to gow like Indonesia s o Egypt s? Once one stats to think about them, it is had to think about anything else. [1] While it may be necessay to swap out the ole, Lucas s question still has impotant theoetical and pactical significance befoe figuing out whethe economic gowth is endogenous o exogenous. Unlike the economics of the Adam Smith ea, moden economics, which is close to vaious labels, is not content with the liteay logic of explaining economic phenomena. Howeve, moe use of the mathematical logic does not guaantee that the analysis esults ae bette than the liteay logic. The advantage of mathematical logic is that it is easie to find bugs in the pocess of easoning, and that poblems such as peconditions, vaiable settings, and implicit conditions ae even wose than liteay logic. Dynamic Stochastic Geneal Equilibium (DSGE) has been used fo economic poblems extensively since the 1960s, and the eynesian and Neoclassical schools have used it to demonstate thei theoy. Of couse, we can also use it in ou model. What will happen? 2. Dynamic Optimization Analysis on Condition d = o d Y = Y In the Solow model, output gowth depends on capital gowth, capital gowth is a function of net investment, and net investment is pat of the output, which can use the mathematical tool of DSGE to analyze the time path and the ending state of the output changes. In the dynamic analysis of the utility function of consumption, we do not know the specific fom of the utility function, but as long as the utility function is convegent and satisfies some constaints, we can get some impotant esults accoding to the standad solution of the dynamic maximum pinciple. The study of Cass shows that assume the utility function of pe capita consumption U( c ) at a cetain time satisfies the condition du( c) dc > 0, 2 2 d d 0 lim du c dc lim du c dc = 0, and the Solow U( c) c <, ( ) =, ( ) c 0 c model k t I ( δ n) k d d = g + is used as the tansfe equation (among them Ig = y c) in the dynamic optimization, then the shadow pice λ of k, the time path of the pe capita consumption c and the capital k can be solved accoding to the Pontyagin maximum pinciple. The esults have no diffeence fom the oiginal Solow model [2]. Howeve, ou study shows I. If the constaint condition d = I in Solow model is eplaced by d = (whee = Y = Y ) [3], then we need to e-veify the following thee conditions of the Hamilton function in the dynamic maximum pinciple: lim H = 0 (1) H = 0 c (2) H dλ = k dt (3) Let = Nk (whee N is the labo population), substituting it into the iden- 2/18

3 tity d= Nk d + kn d, and the following tansfe equation is obtained fom d = [3]: d kdn dn dk = = k = k nk = ( n) k (4) N N N If the objective function is assumed to be the maximum utility of pe capita consumption as the dynamic model of Cass, its dynamic optimization poblem is as follows: ( nt ) Max U( c) e dt 0 dk s.t. = ( n) k, k( 0) = k dt The coesponding Hamilton function is: e nt 0 (5) H = U c + λk n (6) Among them, c = C N is the pe capita consumption, the discount ate in e nt is assumed to be the maket inteest ate, λ is the costate vaiable, k = N is the pe capita capital, n = dn N is the gowth ate of labo foce. The Hamilton function is diffeentiated fo c and k espectively: H U( c) e = nt (7) c c H = λ ( n) (8) k Accoding to the necessay condition shown by Equation (2) H c = 0, ( nt ) U( c) e = 0 c (9) In the above equation, the discount facto e nt is not 0 at time t, so we can only make U( c) c = 0. This means that the utility U( c ) is a constant geate than 0. Theefoe, the fist pat of the Hamilton function U( c) e nt is not 0 when t. Fom the necessay condition shown by Equation (3) H k = dλ dt, dλ = λ ( n) dt (10) whee and n ae exogenous vaiables and assume that the ate of population gowth n is less than the maket inteest ate, that is, ( n) > 0, the geneal solution of Equation (10): λ0 λ = (11) e nt Fom the tansfe equation dk dt k( n) =, the geneal solution of k is: ( ) k = k e nt 0 (12) λ in the Hamilton function, 0 lim k ( n λ nt ) lim 0e lim t t nt e k n λ λ k n = = Then, the Equation (11) (12) is bought into the second pat k( n) 3/18

4 It is shown that unless lim tion e nt = λ0 0 in the Hamilton func- U c k n e H = U c nt + λk n, othewise lim lim e H = U c nt + λk( n) 0. It is not possible to satisfy the neces- say conditions shown in Equation (1). Equation (9) shows that the consumption utility U( c ) is a constant and λ0k0( n) > 0, so limu c e nt = λ k n cannot be established. 0 0 The eason fo this is that dk k is deceasing in the tansfe equation dk dt = y( k) c ( δ + n) k of the Solow model, but dk k is not diminish dk dt = n k. when the incease of k in ou tansfe equation We can also constuct dynamic models in anothe way. Since dy and Y ae aleady pesent in ou basic equation dy Y = [3], thee is no need to intoduce the labo population N in ode to convet Y and C to pe capita output y and consumption c as the dynamic model of Cass. We diectly use the total consumption C in the utility function, the dynamic optimization poblem can be simplified as: t Max U( C) e dt 0 dy s.t. = Y, Y ( 0) = Y dt 0 (13) The coesponding Hamilton function is: H = U ( C)e t + λy (14) H U C e t Y = + λ C C C H = λ Y Accoding to the necessay conditions H C = 0 and the Equation (15), ( ) 1 U C λ = Y C C e t Since we do not know the specific fom of the utility function U( C ), the limit of the costate vaiable λ can not be detemined by Equation (17). Fom the necessay conditions shown in Equations (16) and (3), (15) (16) (17) dλ = λ (18) dt Since in Equation (18) is an exogenous vaiable geate than 0, we can get the geneal solution of λ : λ λ = (19) 0 e t λ λ = = (20) 0 lim lim 0 e t lim = 0, because the function H = U ( C)e t + λy also contains the vaiable Y, we need to examine the changes in Y. Accoding to the tansfe equation dy dt = Y, the geneal solu- Equation (20) can not guaantee H( t) 4/18

5 tion of Y is: Fom the Equation (19) and (21), Y = Y 0 et (21) λ0 λy = Y e t e = λ Y lim λy = λ Y 0 Theefoe, unless limu( C) e t λ0y0 t λ t =, othewise t H U( C) Y U( C) λ0y0 sense, the utility t sumption is meaningless), it is, limu( C) e 0 lim = lim e + = lim e + 0. In the economic U C should not be less than 0 (if utility less than 0 the con-, so thee is no e t λ0 0 limu C = y. In the Solow model, since tion of ( U( c) c) ( nt ) lim e 0 =, U c is bounded by the assump- lim d d = 0. Theefoe, the limit of the fist tem in the Hamilton function is always 0; the second tem consists of the costate vaiable λ and c the tansfe equation dk dt = y c ( δ + n) k. The effect of c and δ on the dk in the tansfe equation is negative and the magin of y is deceasing, so the limit of the second tem of the Hamilton function is also 0. When the tansfe equation is changed to ou d = o dy = Y, the consumption and the depeciation do not estict the gowth of and Y because dy has aleady includes inceased consumption and depeciation. If the fluctuation of output in shot-tem ae ignoed, the output gowth ate dy Y is independent of the size of and Y at any time (including t ), only by the maginal state vaiable. As long as > 0, d o dy will not be 0, so the limit of Hamilton function can not be 0 and the tansvesality condition lim H = 0 can not be satisfied. The model of pe capita vaiables, which consides the change in the labo foce as in the Equation (5), is no exception. Fom the Cobb-Douglas function Y = A L, Y = Y, let Y =, then = Y. Assuming that and ae independent of and Y changes (we will see that the assumption is easonable fom the equation Y = 2 ), then d = ( ) dy, and then by the statistical identity dy = dc + dd+ I 1, 1 Thee ae thee types of vaiables in the macoeconomic analysis: incemental, stock, and mixed. In a closed economic system that ignoes foeign tade, the statistical identity of the output is expessed as Y = C+ I = C+ I + D, whee C is consumption, I g g is the goss investment, I is the net investment, and D is depeciation. Exactly, thee is a theoetical bug in this identity. Assuming that the output of this peiod (t) is Y and the output of the pevious peiod (t 1) is Y t t, the exact identity 1 should be expessed as follows: Y = Y t t + Y = ( Ct + Dt ) + ( C+ D+ It) = ( Ct + C) + ( Dt + D) + I = C + D + I t t. Theefoe, t t in the identity, Y, C, D and I ae incemental vaiables, Y t t, C 1 t and D 1 t ae stock 1 vaiables, Y, t C and t D should be called the amount of mixing because t Y both incemental t and stock vaiable. Since I is the incement athe than the stock, theoetically it can only appea in the incemental expession: Y = C+ D+ I. 5/18

6 d = dy = dc + dd+ I In the Cobb-Douglas function, C and I ae not the cause of the change in output, but the esult. Since the investment and depeciation incement is only pat of the output incement, I + dd is always less than dy, and because of Y <, dy < d, the investment is much smalle than the capital incease. Theefoe, it is impossible to obtain the coect conclusion fo all dynamic optimal analysis based on the hypothesis of d = I. The statistics show that about 80% of the newly inceased output Y is used fo the newly inceased consumption C. In the above equation because > 1, so d > I, I is only a small pat of d. d > I is not uneasonable in eality. Because net investment I in addition to geneating equal assets with I, but also bing the poduction technology upgades, impove poduction efficiency and othe non-physical assets changes, it is these changes make geedy investos willing to take possible isks. 3. Exogenous and Endogenous Gowth Assuming L in Y = A L as a labo population can give us a benefit that simplifies Y = A L to a fomal univaiate function y = Ak, whee y = Y N, k = N, N is the labo population. Full diffeential of y = Ak : dy da dk d = + + ln k (22) y A k Assuming that and A ae constant, dy dk = (23) y k If d = I is futhe assumed, then d = I = Ig δ, whee I is the goss g investment and δ is the depeciation ate. By the identity d = d( kn ) = kdn + Ndk, Ig δ = kn d + Nk d, the gowth ate of population n = dn N, then dk I g sy sa = ( δ + n) = ( δ + n) = 1 ( δ + n ) (24) k k Among them, Y = C+ I. Assuming that the consumption ate g c= CY, then I g = Y cy = ( 1 c) Y = sy. Obviously, the savings ate s is detemined by the consumption ate c. Substituting Equation (24) into Equation (23): dy I g sa = ( δ + n) = 1 ( δ + n ) y k The Equation (23) and (24) show that since the maginal output of k deceases, the contibution of k to dy is getting smalle and smalle, and the depeciation δ k gows linealy with k inceases. Theefoe, egadless of the cuent state of gowth of the economic system, even if the population gowth ate n is 0, the economic system can not avoid the end state, that is, dy y = 0. Fo a finite 1 1 ( 1 ) time, when k gows to sa k = δ + n, o k = sa ( δ + n), the net capi- (25) 6/18

7 tal gowth will be lost to the depeciation of capital stock. Thus, even if people do not consume at all, output all fo investment ( s = 1 ), it can only extend the time that the output gowth ate is educed to 0. To make dy y in the Solow model not diminish, a simple impovement is to think of the A in the Cobb-Douglas function as a vaiable that gows with k. 1 The Equation (23) shows that thee is dy y > 0 as long as sa k > δ + n, o A ( n) k 1 > δ + s. people geneally believed that A is the technical condition in the poduction pocess, is exogenous facto and does not adjust automatically with k changes. Theefoe, the Solow model is consideed a kind of exogenous gowth theoy. In the subsequent endogenous gowth theoy, someone diect assumptions that the poduction function is the maginal output does not the diminish functions [4] [5] [6], anothe one deived such equations by e-assuming the meaning of the poduction facto L. In fact, the A as a constant o that it is an exogenous vaiable, but people s subjective assumptions. By Y = A L, Y Y Y Y =, = (26) L L Let Y =, Y L = w, when > 0, w > 0, Assume L = N, then Since y Y w w A = = = = = L L L = Ak, so we can have: Y w N A= = = y (27) (28) A= k (29) This shows that A is closely elated to the maginal state of the poduction and the distibution state ( ) of the poduction factos and should be a compehensive vaiable that is affected by vaious factos in the Cobb-Douglas function. The Equation (28) is obtained at the hypothesis L = N, and L = N is not necessay to analyze the pe capita output level y. Without assuming L = N, the pe capita output should be expessed as: Among them, Y A L L L y = = = A = Ak N N N N N w N A= = L When L N, the size of A is elated to the size of N and L, except fo, and k. Theefoe, suppose L = N,we can tansfom the Cobb-Douglas function k (30) (31) 7/18

8 into a simple function y = Ak fo easy analysis, but also confuse the diffeence between pe capita output and hypothesis L = N. ( 0) If L is human esouce and let L = EN, E is the coefficient of woke quality E >, then the poduction function is y = AE k. If E is a constant, this assumption is not substantially diffeent fom the Solow model and the diminishing of output gowth ate can not be changed. Like the Leaning by doing model [7], if the quality of human esouces is consideed to gow with the capital, that is, E =, then L = EN = N, the coesponding poduction function is y = AN k. This is fee fom the impact of diminishing maginal output of capital with output gowth, and the gowth of population N not only does not impede pe capita output gowth as the Solow model, but also pomote the gowth of pe capita output. If E = N, then L = EN =, so Y = A L = A = A, o y = Ak. This is called the A model and is the most concise endogenous gowth model [5] [6]. Note that thee is no A as an endogenous vaiable in the above-mentioned poduction function, which deived fom the vaious assumptions about human esouce L above. If A is egaded as an endogenous vaiable and is substituted into the Leaning by doing model and the A model, it is found that thee is no essential diffeence between the two models. Fom Equation (31), in the Leaning by doing model and the A model, the expession of A should be: Leaning by doing model : A model : N N 1 A= k = k = L N N N N A= k = k = L They ae substituted into the Leaning by doing model y = AN k and the A model y = Ak, espectively, the, Leaning by doing model : A model : 1 y = AN k = N k = k N y = Ak = k It like a game in mathematics. As long as A is egaded as an endogenous vaiable, all models ae nothing but an expession of the equation y = ( ) k. The poblem is that when > 0, Y = A L and Y = ( ) can be used to expess the elationship between Y and, but A contains vaiables associated with in Y = A L. Lucas discoveed the gap between ich and poo counties in capital maginal 8/18

9 output in his analysis of Indian data in 1990 [8]. This is called Lucas s Paadox. In fact, if a clea endogenous A, we will undestand the so-called Lucas s Paadox. The equation k= y (fom = yk ) is bought into the Cobb- Douglas function y Ak y= A y, this is, =, then 1 1 = A y (32) whee, is Y. The tap of Equation (32) is that if A is seen as a constant, we can pedict the maginal output y k( = ) by the pe capita output of a county. Fo example, the pe capita output of the United States is 15 times that of India, assuming that = 0.4 both counties in Equation (32), then India s maginal output Y ( = ) will be 58 times that of the United States. In such a big gap, Ameican capital should go to India to get amazing benefits, so Lucas said that neoclassical theoy simply can not easonably explain the eal wold. Howeve, if the Equation (29) A= ( ) k is bought into Equation (32), we find that Equation (32) is anothe expession of = yk. The equation = yk indicates that the diffeence in inteest ate between the two counties is detemined by the diffeence in output-capital atio yk between the two counties, not detemined by the diffeence in the level of pe capita output y. Although the pe capita output of the United States is 15 times that of India, if the US pe capita capital k is also 15 times that of India, the inteest ate (capital maginal output) of the two counties will be vey simila (assuming the diffeence of between the two counties is not significant). Theefoe, although the pe capita income of counties may have a big gap, but the inteest ate gap is not lage (especially the eal inteest ate). Lucas said thee wee seveal ways to make the economy of India apidly gow, the fist one is to impove the oiginal poduction efficiency, the second one is to make full use of domestic esouces though intenational tade, the thid one is to ean othe county s tansfe payments though diplomatic access that have significant effect on its own economy. The essence of tade to be ich is to tun the idle esouces of a county into a eal income that can impove people s living standads. If use some evenues fom tading to impot technologies and to impove the domestic poduction efficiency, the pocess of getting ich will be faste and moe sustainable. The expot of cheap oil in the Middle East, and expot of cheap labo in China with the help of two outsides (poducts and aw mateials), two lage (expot and impot) ae the example of tade to be ich in economic globalization. In theoy, as long as the clea A= w, no matte how the assumption that L can be, it will only cause the fomal changes of the poduction function. Fo example, assuming that the monetay value of human esouces is the income it obtains in the maket: L = Y, then Y = A L = A ( Y ), that is, Y ( A ) 1 =. In the philosophical sense, if the contibution of mateial esouce to output is also deived fom human wisdom, all income is the contibution of human 9/18

10 esouces, that is, assuming L 1 Y A L A Y A = Y, then = = =. Obviously, this hypothesis deived poduction function has also the chaacteistics of Leaning by doing model and the A model that the maginal output does not diminish. Table 1 summaizes the changes in the Cobb-Douglas function unde vaious assumptions of L. In addition to these assumptions shown in Table 1, we can also think of moe easons to assume moe. In shot, by assuming a diffeent L and subjective intepetation of L, any desied endogenous gowth o exogenous gowth model can be obtained. Howeve, as long as the coesponding change of = Y and A is intoduced, all models ae not diffeent, and ae the defomation of the maginal state equation = Y o Y = of the Cobb-Douglas function. Equation Y = is not only the stating point fo the analysis of the cyclical changes in output [9], but also implies the output gowth path and the Table 1. The influence in the vaious hypotheses of L on the Cobb-Douglas function and the paamete A. Hypothesis The poduction function expessed by total output Y The poduction function expessed by the pe capita output y ( y = Y N, N is the labo population) The vaiable A coesponding to the hypothesis of L Note L= L Y = A L Y L = A N N N L y = Ak N w A = Y A = L Assuming = Y, w= Y L, then Y =, Y L= w, so A= ( w ) ( ) L= whee = Y Y = A L Y = A N N A y = N 1 2 k 2 2 Y A = A= is calculated by the definition = Y (whee,y and ae fom the statistical data), athe than calculated by the fixed assets. L= N Y = A N L = N L= L= Y L= Y Y = A N Y = Y AN = A Y = A ( ) Y = A Y 1 Y = A Y = A Y 1 Y = A Y N = A N N N y = Ak Y N = A N N N Y N y = Ak Y = A N N y = Ak A 1 = ( ) 1 N y = A k Y N 1 = y = A N 1 A k Y A= = y N A= k Y A = = N N A = N Y A = = A = Y 1 A = = Y A = Y A = Y A = A is diffeent in the Cobb-Douglas function with the assumption of L. In A ( w ) ( ) =, assuming L= N, then The esult of A is not only elated to the calculation of w, but also to the dimensions of N and othe vaiables. If the Equation (28) ( ) A= y is calculated (whee y = Y N ), then Y and N using the diffeent dimensions (such as Y and N dimensions wee billion and thousands of people, o million and people), size A will be diffeent. If the Equation (29) A= ( ) k is calculated (whee k = N ), when is calculated fom = Y, The esult is not diffeent fom Equation (28); If the statistics fixed assets and consume goods as, then the Equation (29) A will be diffeent fom the esults of Equation (28). Only the A calculated by assuming L= will not show diffeent esults, and is a dimensionless numbe. 10/18

11 final state of the infomation. Figue 1 shows the diffeence in A assuming L = and L = N. Unde the assumption of L = ), the change of A is obviously deceasing, and in the othe way, A shows a slight upwad tend. Theefoe, A is the vaiable associated with the hypothesis L, not a constant. Gollop and othes on the statistical analysis think that the contibution of capital is the fist, the labo foce is second, and the contibution of the poductivity is less than one quate in the economic gowth of the United States [10]. Afte the Cobb-Douglas function Y = A L is fully diffeentiated, the da A in Equation (33) is called Total Facto Poductivity (TFP). dy da d dl = + + (33) Y A L Since the sum of ( d ) and ( dn N) is less than dy Y in the statistical data, It is believed that the esidual da A is the contibution of technological pogess. If L is detemined by L =, the aveage of da A will be negative (duing , the aveage value of da A is ). Because A deceases with output (see Figue 1). This diffeence is due to the fact that Gollop s estimates do not take into account the maginal elationship between the vaiables in the Cobb-Douglas function. If the maginal equation Y = is fully diffeent, dy d d d = + (34) Y Substituting Equation (34) into Equation (33), accoding to + = 1 and d = ( d ) + ( d ), da d dl d d = + (35) A L In Equation (35), the fist tem shows that da A is also affected by d and dl L, the second tem d is the fluctuating vaiable that causes peiodic change, and the thid tem d is inceased and fluctuates peiodically 11/18

12 in the statistical data. It would be easie to undestand the natue of A by ewiting the Equation (27)as follows: w A = whee and w ae the maginal states of and L, and the function of and is the effect-popotion of and w in the poduction. In this way, A is an endogenous vaiable that contibutes to the maginal contibution of factos and L with the distibution paametes and. It plays an impotant ole as bidge in the Cobb-Douglas function Y = A L and the maginal state equation = Y. Ignoing the endogeneity of A ignoes the elation between Y = A L and = Y. 4. The Final State of Economic Gowth Accoding to A ind of Neithe eynesian No Neoclassical Model (5): The Path of Economic Gowth [11] we know that egadless the poduction state of the system is Y < Y L (on the left side of the Cobb-Douglas function bisecto function bisecto = L), o Y > Y L (on the ight side of the Cobb-Douglas = L), the state will close to the bisecto = L diven by maket competition. And the state of limit is Y = Y L, namely = w, at that point = = 12. Theefoe the final state of vaiable A is: w lim A= lim = 2 (36) 1 1 In Cobb-Douglas function Y = A L, because L = [9], when 1, L, theefoe, lim Y = lim A L = lim A lim L = 2 (37) This is the esult of the A model o the Leaning by doing model : the contibution of to the gowth of output is not maginally diminishing. The A model and Leaning by doing model ae deived fom the subjective intepetation of L, and ou endogenous model is deived fom the final state of A and Y. Equation (37) shows that the final state of output Y is still a function but not a constant. In the final state function Y = 2, the elation between Y and evolves fom an exponential elationship of Y = A L to a linea elationship, and the output gowth does not teminate due to the gowth of. Diffeentiate A= 2, then daa= d. In the final state, da A only has the second item in Equation (35). This shows that A will fluctuate due to the fluctuation of ( fluctuation easons see efeence [9]). It is simple to deive the final state of the output Y fom the maginal state equation Y = of the Cobb-Douglas function. Since = = 12 when Y = Y L, lim Y = lim = 2 (38) /18

13 Compaing Equations (37) and (38), Y = A L and Y = ae the equivalent expession of Cobb-Douglas function. Fom the equation Y = 2 we can see that when is an exogenous vaiable, dy d = 2. This suggests that the maginal output of poduction factos is deceasing is a elative definition. When the maginal output of all factos in poduction is equal to each othe, the output incement dy is not necessaily diminishing compaed to the facto incement d, since the incement of anothe facto dl( = d) also contibutes the same to dy. Micoeconomics divides poduction into thee states. In the fist, the maginal output of is geate than the aveage output ( MP > AP ). In the second, MP < AP, the thid is the inefficient state, because MP < 0. In the macoeconomic field, since the maginal output MP = Y = Y, the aveage output of AP = Y, < 1, so MP < AP. This suggests that, as long as Y > 0, the poduction in the macoscopic sense is always in the second state of efficiency. We have defined Y ( + L) as unit capital output [11], and now its economic significance is moe obvious. Since = Y, L = Y w, when + = 1, = 1, = w, then, Y Y lim = lim = 1 + L 1Y Y + w Unit Resouce Output Y ( + L) is actually Total Facto Aveage Poduct (TFAP). At Y Y L, TFAP is always geate than the maginal output Y o Y L. As the output Y gows and 1, the TFAP will become moe optimized [11] and will be close to the maginal output Y o Y L. Accoding to statistical data of the United States, as shown in Figue 2, thee ae always A 2 > Y ( + L) >. And based on Equation (36) and (39), we can get the following inequality, whee the equation is satisfied when = 1. A Y (40) 2 + L (39) 13/18

14 As in the case of taditional theoy, it is impossible to have the above inequality if L is the labo foce N. The most staightfowad eason is that thee is no compaability between A, Y ( + L) and, because thei dimensions ae inconsistent. Fo example, when L is the employment population, the dimension of thousands of people, What is the dimension of A and Y ( + L)? The final state of the poduction function does not mean that the output gowth will eventually stop. Diffeentiated the final state equation Y = 2, then: dy d d = + (41) Y This is not contadictible to the fully diffeential equation of Y =, since Y = is not the final state, so d 0. In the final state = 12, so d= 0. Fom the basic equation d = [3], then dy d = + (42) Y This is the cycle equation of the gowth of output that we have peviously deived [9], but at that time we assumed that in the shot tem d= 0. Theefoe, although the final state poduction function Y = 2 is diffeent fom the poduction function Y = A L o Y =, the factos that detemine the coe output gowth ate in shot-, medium- and long-tem ae the same, which is the coe inteest ate. And as long as the coe inteest ate is geate than 0, Even if the poduction function evolves to Y = 2, the business cycle will not disappea. If we neglect the peiodic fluctuations in output gowth, then dy Y=. Re- dy dt Y =, the geneal wite it as a diffeential equation vaies along time solution is: Y = Y 0 et (43) Although this time path function of Y does not have abundant poduction state infomation as the Cobb-Douglas function, it is not like Y = A L, when = Y = Y L = 0, the esult is not undestandable: = + =. Fo the time path function Y = Y0 et, when Y ( Y ) ( Y L) L 0 = Y = Y L = 0, then Y = Y0, the system is in a simple epoduction state. The Haod-Doma model assumes that the poduction function is Y = a [12] [13]. This can be seen as a simplified fom of the maginal state equation Y = expessed. Although the subsidiay hypothesis = I is not coect, when intoduce coefficient b in = I, then = bi. And with the poduction function Y = a, then Y = a = abi. Accoding to the statis- Y = Y + Y, assuming I = sy, whee s is tical data, I is a pat of the output anothe coefficient that is diffeent fom b (note: we do not call s fo the savings ate), so Y = abi = absy. Rewite it as the diffeential equation consideing time dy dt = abi = absy, then its solution is Y = Y e abs t 0. As long as the coefficient (abs) is egads as the inteest ate, thee is no diffeence with Equation t 1 14/18

15 (43). Theefoe, egadless of whethe the analysis of Haod-Doma model is coect, the time path of its output is oughly the same as ou conclusion. The Solow model uses Y = A L as the poduction function and egads A as a constant. Although the auxiliay assumptions = I and I = sy ae the same as those of the Haod-Doma model, the output will finally stop incease in a limited time since the gowth of output is constained by dual factos of the diminishing capital maginal contibution pe capita and stock capital depeciation [14]. Compaing Equations (41) and (34), it can be seen that the diffeence between the gowth ate of the final state and the shot-tem is that thee is less ( d ) in Equation (41). The aveage numbe of d in the United States fom 1970 to 2016 is , so befoe the Cobb-Douglas function evolves to the final state equation Y = 2, the eal output gowth ate dy Y fo the emoval of the inflation ate should be slightly educed. of the United States inceased fom in 1970 to in 2016, and expeienced an annual gowth of about fo 47 yeas. Assuming a linea incease at this ate, afte anothe 71 yeas, in the United States will each 1/2. As the theoetical gowth ate of is deceasing, so in 2087 we may not see the actual value of equal to 1/2. Since is not a vaiable that detemines whethe the maginal output is geate than zeo o not, the gowth of output is detemined by the inteest ate in the poduction function Y = o Y = 2. The speed of output gowth is not the poduction function itself, but on the poduction efficiency. Adam Smith said in An Inquiy into the Natue and Causes of the Wealth of Nations that thee was an invisible hand that led self-inteest people pomoted public inteest [15], but Smith had neve evealed the metapho in the book. Some people say invisible hand is the elationship between supply and demand, some said that it is the maket pice, and some said that it is maket competition, and so on. We believe that invisible hand means maket competition. Maket competition aises the poduction efficiency and makes it necessay fo poduces to etun the suplus geneated by the incease of efficiency to the society. Duing , the aveage of eal inteest ate in the United States is Assuming that the numbe can last 200 yeas, the actual output Y will also incease at the aveage annual ate of Then afte anothe 200 yeas in yea 2216, the actual GDP will be times the cuent GDP. Accoding to the annual population gowth ate of the United States in , by 2216 the total population will be 7.32 times the cuent. At that time, the eal pe capita GDP in the United States would be times that of 2016, o pe capita GDP of 651,684 USD (eal GDP in 2016 is 16,660 billion USD, 323,391 thousands population, 51,517 USD/peson). The question is with the income level of up to eal 650,000 USD/peson, will people be involved in the exhausting competition fo highe income? Pehaps when it has not yet eached this figue, many people have aleady voted fo the 15/18

16 paty who advocates fo the univesal welfae policy. Theefoe, the teminato of economic gowth is not the Cobb-Douglas poduction function, but at what level of income, those who believe in maket competition will become social minoities. 5. Conclusions Table 2. Summay of macoeconomic model based on Cobb-Douglas function. The issues we ae concened with today ae not much diffeent fom what Adam Smith was concened about in his geat wok of two hunded yeas ago, but at that time most of the eseach methods wee empiical and conjectue, and we now have to use stict analytical methods and data validation. Ou model is deived fom the Cobb-Douglas function, and only > 0 will have dyy> 0, > 0 is detemined by the maginal state of eal poduction and inflation, so we conclude that the equation of gowth is endogenous, the cause of gowth is exogenous. No matte the gowth of output is geneated by the economic system evolution, the impovement of poduction management o the pogess of science and technology, ultimately the esult is the incease of poduction efficiency. These ae in the model that the maginal output is geate than 0. So, egadless of Name of model Hypothesis o cause Infeence o conclusion Note Pecondition Y = A L Y Y = = Y Y w = = L L Cobb-Douglas function Fundamental equation, Investment decision [3] = 0 = 0 Y = C+ I + D Y =, = Y I ( 1 ) Y = > I If sepaating coe vaiables and fluctuating vaiables, then dy dy dy Y = = + = Y + Y Y Y Y then Y =, =. Exactly, all equations that ignoe fluctuation of vaiable should be expessed as such. Business cycle equation [9] d 0 = L = de dw = E w dy d = + Y dr d dn E = 1 + R N E The simplified symbol can be expessed as: dy d Y = = + = + Y R = 1 N E + Inflation equation [16] [17] M = ay Y = PY, Y = P + Y V = Y M V = θ P P θ P = M Y M = M + M = + P = P + P Y = + (when P = 0 and dp P = P dp P = P P = ) The path and the ending state of economic gowth [11] t Y Y = 1 w L 1 w A = Y 1, A 2, L + L Y = 2, Y = Y0 et A Y 2 + L 16/18

17 whethe the ending state of Y = A L is Y = 2, as long as the maginal state of poduction is geate than 0, Adam Smith said The Wealth of Nations will continue to gow. The 2008 financial cisis made the fabulous fome Fedeal Reseve Chaiman Alan Geenspan into a nude swimme and humiliated the once billiant taditional macoeconomic theoy, but the textbooks still popagate those misleading theoies. We abandon the taditional analytical famewok, tying to use axiomatic methods to deive the vaious models of macoeconomic poblems, thus foming a new system of macoeconomics with an inteest ate as the coe vaiable. Table 2 summaizes these models and helps us undestand the link between economic gowth and othe macoeconomic poblems. Compaed with the taditional theoy, these models ae not only vey igoous in theoy, but also have the impotant pactical significance: Based on these models, we will identify the chaacteistics and causes of the financial cisis, and make a contibution to the pediction o avoiding the next financial cisis. Refeences [1] Lucas J., R.E. (1988) On the Mechanics of Economic Development. Jounal of Monetay Economics, 22, [2] Cass, D. (1965) Optimum Gowth in an Aggegate Model of Capital Accumulation. The Review of Economic Studies, 32, [3] Zhan, M.A. and Zhan, Z. (2016) A ind of Neithe eynesian no Neoclassical Model (1): The Fundamental Equation. Open Access Libay Jounal, 3, e [4] Rome, P.M. (1986) Inceasing Retuns and Long-Run Gowth. Jounal of Political Economy, 94, [5] Bao, R.J. (1990) Govenment Spending in a Simple Model of Endogenous Gowth. Jounal of Political Economy, 98, [6] Rebelo, S.T. (1991) Long-Run Policy Analysis and Long-Run Gowth. Jounal of Political Economy, 99, [7] Aow,.J. (1962) The Economic Implications of Leaning by Doing. The Review of Economic Studies, 29, [8] Lucas J., R.E. (1990) Why Doesn t Capital Flow Fom Rich to Poo Counties, Ameican Economic Review (Papes and Poceedings), 80, [9] Zhan, M.A. and Zhan Z. (2016) A ind of Neithe eynesian no Neoclassical Model (2): The Business Cycle. Open Access Libay Jounal, 3, e [10] Gollop, F.M., Faumeni, B.M. and Jogenson, D.W. (1987) Poductivity and U.S. Economic Gowth. Havad Univesity Pess, Cambidge, MA. (Repinted, 1999) [11] Zhan, M.A. and Zhan, Z. (2017) A ind of Neithe eynesian no Neoclassical Model (5): The Path of Economic Gowth. Open Access Libay Jounal, 4, e [12] Haod, R.F. (1939) An Essay in Dynamic Theoy. The Economic Jounal, 49, [13] Doma, E.D. (1946) Capital Expansion, Rate of Gowth, and Employment. Econometica, 14, /18

18 [14] Solow, R.M. (1956) A Contibution to the Theoy of Economic Gowth. The Quately Jounal of Economics, 70, [15] Smith, A. (1776) An Inquiy into the Natue and Causes of the Wealth of Nations. 5th Edition, Methuen Co. Ltd., London. [16] Zhan, M.A. and Zhan, Z. (2017) A ind of Neithe eynesian no Neoclassical Model (3): The Inflation Equation. Open Access Libay Jounal, 4, e [17] Zhan, M.A. and Zhan, Z. (2017) A ind of Neithe eynesian no Neoclassical Model (4): The Natue of Philips Cuve. Open Access Libay Jounal, 4, e Submit o ecommend next manuscipt to and we will povide best sevice fo you: Publication fequency: Monthly 9 subject aeas of science, technology and medicine Fai and igoous pee-eview system Fast publication pocess Aticle pomotion in vaious social netwoking sites (LinkedIn, Facebook, Twitte, etc.) Maximum dissemination of you eseach wok Submit You Pape Online: Click Hee to Submit O Contact sevice@oalib.com 18/18