Can a Natural Economy Operate in Macroeconomy? A Caution for Deviation from Natural Economy

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1 Ital Econ J 206 2:23 42 DOI 0.007/ x RESEARCH PAPER Can a Natural Economy Oerate in Macroeconomy? A Caution for Deviation from Natural Economy Koji Akimoto Received: 27 March 205 / Acceted: 5 January 206 / Pulihed online: 5 Feruary 206 Società Italiana degli Economiti Italian Economic Aociation 206 Atract The fundamental rolem iue of thi aer i that a fundamental caue of economic crie, uch a the dot-com ule, the Lehman crah, or PIIGS, from hich the current caitalit economy i uffering, lie in a large deviation from a natural economy. Any large deviation from the natural economy entail high rik. Akimoto Int J Econ Sci III4: 37, 204 roved thi rooition. Thi lead to a quetion of hether a mechanim,in hich a natural economy oerate to egin ith, exit in the macroeconomy. Firt, e adot the definition y Painetti Structural change and economic groth; a theoretical eay on the dynamic of the ealth of nation. Camridge Univerity Pre, 98 from a macroeconomic vieoint, ecaue it deend on the tructure of the roduction roce. Hoever, there i no guarantee that the natural economy defined y Painetti Structural change and economic groth; a theoretical eay on the dynamic of the ealth of nation. Camridge Univerity Pre, 98 oerate in the macroeconomy. If Akimoto Int J Econ Sci III4: 37, 204 analyi i correct, e need to rove that a mechanim through hich the natural economy oerate doe in fact exit in the macroeconomy. We contruct a macroeconomic game ith caitalit and orker a layer. The macroeconomy involve circulating layer aving to invetment via caital market. Thi roce contruct Kaldor fundamental equation. We rove that a Nah equilirium exit, hich carrie out a natural economy and alanced economic groth. In addition, the theoretical analyi demontrate that Kaldor fundamental equation, hich i Keyneian, ecome an identity, i.e., =. Thi imlie that the Keyneian equation ring aout a claical reult. Akimoto 204. B Koji Akimoto akimoto_koji@kurume-u.ac.j Faculty of Economic, Kurume Univerity, 635 Mii-Machi, Kurume, Fukuoka , Jaan 23

2 24 K. Akimoto Keyord Non-cororative macroeconomic game Natural economy Balanced economic groth Kaldor fundamental equation Laor theory JEL Claification B2 B6 E0 E2 C72 P0 Introduction Thi tudy examine hether a natural economy can oerate in the macroeconomy. The fundamental rolem iue of thi aer i that a fundamental caue of economic crie, from hich the current caitalit economy i uffering, lie in a large deviation from a natural economy. Thi lead u to ak if a mechanim exit in the macroeconomy through hich the natural economy oerate. An overvie of the core iue needed to dicu a natural economy ill firt e offered. To remarkale trend in the caitalit economy ince the 980 have een the raid exanion of the orld financial aet and the correonding raid GDP increae in high aet area. The exanion of money hould e conidered a detructive to the economy alance ee Akimoto 204. From an economic theory vieoint, thi detruction i conidered to e a deviation from macroeconomic equilirium and therefore a deviation from a natural economy. Hoever the natural economy i not a ynonym for macroeconomic equilirium. Nonethele, a natural economy, deite difference in definition eteen variou vieoint, involve an economic equilirium. Therefore, taking a contraoition, deviation from uch an equilirium can e conidered a deviation from a natural economy. Akimoto 204 dicue the high rik entailed in the deviation from natural economy and concluded that human cannot control the economy artificially ignoring natural economy Thu, one of the imortant rolem remaining to e invetigated i hether a mechanim y hich the natural economy oerate doe in fact exit in the macroeconomy. At thi tage, e hould exlore ho the natural economy function. A the economit Smith 776 decried, a market rice fluctuate around the natural rice. Thi fluctuation ha to imortant imlication. Firt, a natural economy contitute the economy aic tructure. Therefore, tudying the natural economy mean analyzing the aic la that underin and control the economy. Second, any large deviation from the natural economy entail high rik. Such rik therefore rovide an imortant alarm for controlling the economy ithout rigorou rule. We cannot artificially control the economy y ignoring the natural economy. When e tudy the natural economy y Smith 776, e hould alo analyze natural interet rate. After Smith, many economit tried to define a natural interet rate from macroeconomic vieoint. Firt, e hould highlight the definition y Wickell 898. Wickell defined a natural interet rate a eing neutral for a rice level etalihed in the real market. More reciely, it i the rate at hich demand i equal to uly in the real market, making a caital market unneceary. A i ell knon, thi definition affected Keyne 930. In hi ook, A Treatie on Money 930, Keyne For intance, ee htt://.meti.go.j/reort/tuhaku2008/2008honun/html/i20000.html. 23

3 Can a Natural Economy Oerate in Macroeconomy? 25 contructed the fundamental equation herey a natural interet rate a defined a a rate that made invetment equal to aving. At the natural interet rate, the rice level i equal to the monetary income er outut aid for the roduction factor. Hoever, a ituation herein invetment equal aving imlie deendence on the hae of oth invetment and aving function. For intance, if an innovation i exected to occur, the exectation of thi innovation ill change the invetment function and the natural interet rate ill increae. In thi cae, no information i given ith reect to develoment in the roduction roce. Hoever, it i aolutely certain that the natural interet rate deend uon the tructure of the roduction roce. Painetti 98 rooed a different aroach to that of Wickell. Uing an aroach aed on the theory of laour, Painetti concentrated on the tructure of the roduction roce and innovation. Firt, he egan ith the condition of full emloyment of laour and caital tock, uing a vertically integrated analyi defined y the multi-ector model. Second, he introduced the concet of natural rate of rofit and contructed a natural economy. Finally, he introduced financial aet to thenatural economy and roved the emergence of a rate of interet. After thee rearation, Painetti defined the on-rate of interet for each commodity and rigorouly analyzed the relation eteen the nominal rate of interet and real rate of interet tandard real rate of interet. After defining the concet of interet rate at each tage, he finally reached the definition of the natural rate of interet. 2 From the aove dicuion, e ee that the natural interet rate defined y Wickell and Painetti are oth rereentative and aed on a macroeconomic equilirium. Among thee, e adot the definition y Painetti ecaue it deend on the tructure of the roduction roce. Therefore, the concet of natural economy in thi tudy deend on definition of oth Smith 776 and Painetti 98. Although Painetti roved the exitence of a natural economy, he did not demontrate a mechanim through hich a natural economy oerate. Therefore, the aim of thi aer i to ho that thi mechanim exit in a macroeconomy and to demontrate the reaon that the mechanim hold. That i, if only one condition i not atified, the hole natural economy ould collae. To kee the macroeconomic vieoint, e adot the fundamental equation of Kaldor 3 ho i one of the mot imortant decendent of Keye. In addition, e introduce Riccardian rice ytem. Riccardo i a decendent of Adam Smith. The main reult e ould like to rooe are folloing to remark. There exit a Nah equilirium. 2 The Nah equilirium correond to the natural economy defined y Painetti 98 and Smith 776 contruct an alanced economic groth. 3 Therefore, there exit a mechanim y hich the natural economy can oerate and caitalit economy can urvive. The aove concluion demontrate that e hould take off the factor hich otruct the condition under hich the Nah equilirium exit. 2 Painetti 98, Kaldor

4 26 K. Akimoto 2Model In thi ection, e contruct a game theoretic model ith caitalit, orker and a caital ditriutor a layer. The caital ditriutor i a eronification of money and aet market knon a a caital market here and decide the ditriution of caital good eteen roduction ector. 2. Model contruction The model contain roduction ector, a rice ytem and the fundamental equation of Kaldor model. Let u no contruct our model. Production ector We aume that the economy ha to roduction ector, a ector roducing conumtion good and a ector roducing caital good. We call the former ector 0 and the latter ector and allocate uffixe 0 and, reectively. The technology of each ector i denoted y laour coefficient n i and the caital-outut ratio i i = 0,. The coefficient n i and i are oitive contant. Furthermore, let X i and N i rereent the outut of ector i and the laour emloyed in ector i, reectively. The relationhi eteen ector i outut, X i and the caital tock in ector i, K i, i rereented y X i = i K i. i = 0, We denote the amount of caital tock in the economy yk. We further denote the ratio of the ditriution of caital tock to ector 0 y v and the ratio to ector y v. Then, e otain K 0 = vk, 2 K = vk, 3 X 0 = 0 vk, 4 In addition, e otain X = vk. 5 N 0 = n 0 X 0, 6 N = n X, 7 N = N 0 + N, 8 here N i the total amount of laour emloyed. We no can tate our aumtion aout caital tock. Aumtion Caital tock i never deleted. Aumtion 2 Caital tock can move freely eteen ector 0 and ector. 23

5 Can a Natural Economy Oerate in Macroeconomy? 27 Aumtion 3 Caital tock i emloyed comletely. By thee aumtion, e otain X = K = K 0 + K, 9 here dot denote dt d and t denote time. Equation 9 define the differential equation of caital tock K. Price ytem Let i denote the rice of the roduct of ector ii = 0,. We aume that the rice are determined a the um of age aid to the laour emloyed and the rofit required to ue the caital. Then, y Aumtion, e otain { 0 = n 0 + π 0, = n + π, 0 here and π denote a age rate and a rofit rate, reectively. We can olve Eq. 0 to otain 4 0 = n 0 + πn 0, π = n π. 2 The term n π in the right hand ide of 2 rereent the laor equivalent. 5 Becaue rice mut e oitive, e aume π<. 3 Equation and 2 contain four variale, 0,,π and. A exlained later, the rofit rate π i determined in the roce of ditriuting caital tock eteen to ector. Therefore, the rice ytem ha one degree of freedom. Thu, e chooe 0 a the numeraire to cloe the rice ytem. That i, 0 =. 4 Equilirium condition: fundamental equation of Kaldor model When 0 and are determined under ome rofit rate π or ome age rate, and X 0, X, N 0 and N are determined in the roduction ector, then GDP Y hich i equal to the um of total rofit and total age W hould e determined in the economy at the ame time. Thi roce can e exlained a follo. 4 n The exanion of the right-hand ide of 2 ecome π = 5 Painetti 98,. 43. { } n + π n + π 2 n +. 23

6 28 K. Akimoto From Aumtion, e otain Sutituting 5, and 2 for5, e otain Y = { πn K 0 + π π It i eay to verify that Y = 0 X 0 + X. 5 n K } n0 + K 0 + n K. 6 0 W = n 0 X 0 + n X = n0 0 K 0 + n K, 7 = π K 0 + X N = πn K 0 + π π Therefore, from Eq. 5, e otain n K. 8 Y = + W. 9 Equation 9 ho that GDP Y i ditriuted to caitalit and orker. Therefore, from equilirium condition, e otain X = + W 20 here and denote the aving rate of caitalit and orker reectively and require 0 and 0. A exlained in the folloing ection, and are the trategie ued y caitalit and orker. Player in the macroeconomic game No, e contruct a macroeconomic model uing game theory. Player are caitalit, orker and the caital ditriutor. A mentioned aove, the caital ditriutor i a eronification of caital market. Caitalit and orker have the ojective of maximizing their rofit rate and the age rate, reectively. Therefore, e can define their rolem a here and are layer trategie. 6 max π.t. 0 2 max.t In ordinary cae, the layer rolem are defined y max 0 e ρt π dt and max 0 e ρt dt here ρ denote the dicount rate of layer time. Hoever, a hon later, the rofit rate π and the age rate 23

7 Can a Natural Economy Oerate in Macroeconomy? 29 Next, e introduce Eq. 20 hich i an equilirium condition that require the ditriution of caital tock. Therefore, the economy hould contain a roce y hich caital tock are ditriuted eteen to roduction ector. Ho i thi ditriution determined? In our game model, aving are inveted in ector a caital. Therefore, e et a layer ho decide the ditriution of caital tock. We call thi layer caital ditriutor CD. 7 The CD rolem i to maximize the rofit rate. In other ord, the CD act a a roxy for all invetor in the caital market. The CD decide a ditriution of caital tock eteen to roduction ector ith invetment ratio v a a trategy. We define the CD rolem a max π,.t. 0 v. 23 v To comlete the model We have no contructed the model. We have 5 unknon, namely X, X 2,, 2, K, K, K 2, N, N, N 2,,π,, v. For thee unknon, e have 5 equation, namely 2 9,, 2, 4 and three condition for Thee condition comlete the equation ytem. 2.2 Structure of the Model Let u determine the macroeconomic tructure of the model hich i hon in Fig.. GDP i roduced in roduction ector, namely ector 0 and ector. It i ditriuted among caitalit and orker. They conume ome art of their income. The remaining amount, i.e. aving, i inveted in ector. Caital good roduced in ector are handed to the CD hoe rolem i to decide the ditriution rate v to maximize the rofit rate. Becaue our model i contructed a a game, information tructure i imortant. The information aout the model reented in Fig. i aumed to e common knoledge among the layer. Namely, the game i contructed aed on comlete and erfect Footnote 6 continued are indeendent from time t. In addition, if e contruct layer rolem y max 0 te ρt dt, max 0 v te ρt dt and max 0 W te ρt dt,here t and W t denote total rofit and total age at time t, then the model ecome a differential game and e hould olve otimal control rolem. Hoever, in uch a game, it i imoile to rove the exitence of alanced economic groth and it taility. 7 From a macroeconomic vieoint, three route exit here aving circulate a invetment: i aving aet market invetment, ii aving financial comanie ank invetment, and iii aving financial comanie aet market invetment. Among thee, route i i imortant in a caitalit economy and i decried a the CD. In addition, for examle, an individual invetor invet hi caital in an individual firm. If thi firm roduce conumtion good and decide to ue thi invetment to contruct additional roductive facilitie, the invetment hould in ector from a macroeconomic vieoint. Therefore, hen e contruct our model ith Sector 0 and Secotor, aving hould e aumed to circulate firt a caital to Sector. And then, a caital good roduced, they are ditriuted y ome mechanim. We trut thi mechanim herein aet market rimarily decide the ditriution of caital to the CD. 23

8 30 K. Akimoto Conumtion of Worker Saving of Worker Worker Caitalit Conumtion of Caitalit GDP Flo of Money Caital Ditriution Fig. Flo of money and caital good in the game Saving of Caitalit Production Sector Sector roducing conumtion good Sector roducing caital good Caital Good Caital Ditriutor 23

9 Can a Natural Economy Oerate in Macroeconomy? 3 information. Deciion making i achieved imultaneouly. Hence, the game olution, if it doe exit, i a Nah equilirium. 2.3 Profit function Let u analyze Eq. 20. We hould conider the folloing three cae. I n 0 0 < n, II n 0 0 > n, III n 0 0 = n. Firt, e analye cae I and II. Sutitute Eq. 2 and 3for7 and 8. Next, utitute Eq. 5, 2, 7 and 8for20 and rearrange them to otain here π = A = { } n n n n 0 0 v n + n n 0 0 v n n C = n, n n 0 0, B = D = n n 0 0 = A + B C + Dv, 24 [ ] n n n 0 0. n n 0 0 Equation 24 exree a hyerola of the variale v. Although A,B,C and D contain the trategie of layer,,, e may deict the grah of 24 a hon in Fig. 2, conidering, a arameter. The lu or minu ign ona, B, C and D deend on the condition of the technical coefficient. We ho the grah in the cae of I. Becaue e can deict the grah of 24 in the cae of II in the ame ay, e do not ho it here. Finally, e analyze the cae of III. Sutitute Eq. 5, 2, 7 and 8 for20 and rearrange them to otain π = We deict the grah in the cae III in Fig. 3. v Analyi We no reented all the information neceary to analyze the game. We reent the folloing rooition. 23

10 32 K. Akimoto π 0 0 < + n n n v The range v can take π 0 0 < + n n n 0 v The range v can take > n n n π v The range v can take a The cae of 0 > B > The cae of 0 > B < c The cae of 0 < B < Fig. 2 Profit function in the cae of I 23

11 Can a Natural Economy Oerate in Macroeconomy? 33 π 0 The range v can take v Fig. 3 Profit function in the cae of III Prooition Nah Equilirium i Equation 26 i a Nah equilirium.,,v = n,, n 0 + n ii The rofit rate and age rate under 26 are { π = 0 +, = n 0 +n. 27 Proof i We ho that each layer trategy i the et reone to the other layer trategie in Eq. 26. Suoe that the trategie and v are given. Sutituting v = for Eq. 24, e otain π = ee Fig. 2. On the other hand, e otain the age-rofit curve n 0 π = 28 + n 0 n 0 from Eq. 4. Then, the age rate i equal to 0. Worker intend to avoid = 0. Hoever, only one trategy enale orker to avoid it. In other ord, orker only need to take the trategy that ring aout B = 0inEq.24. Thi condition i denoted y n = n Therefore, the orker hould take the trategy in reone to caitalit trategy. Hoever, the orker hould analyze hat haen in the game and calculate their ayoff for Eq. 26. Next, e analyze thee oint [ee Eq. 24].ForEq.26, 23

12 34 K. Akimoto Conumtion of Conumtion of Caitalit C Income of Worker W WorkerC Saving of Income of Caitalit Π Saving of Caitalit S Worker S Fig. 4 Player conumtion and aving B = 0 and C + Dv = 0. Therefore, the layer cannot determine the rofit rate in the game. Then, hat haen in the game? The fact that the rofit rate cannot determined in the game imlie that it determination deend on condition of the money market. Figure 4 ho the alance eteen the demand and uly of money. Firtly, the caitalit hould end art of their income on conumtion. In Fig. 4, the caitalit conumtion i denoted y C hich i conidered a a hortage of caital. In our model, thi hortage hould e ulied in the money market here orker uly caital. 8 Thu, orker aving hich are denoted ys in Fig. 4 are ulied a caital in the money market. Therefore, e otain C = S W. 30 By definition, C =, S = W. Thu, e otain π = 0 +, 3 y uing Eq. 7, 8 and Sutituting Eq. 3 for28, e otain in Eq. 26. Therefore, e can conclude that i the et reone to,v. 8 From a macroeconomic vieoint, there are three route here aving circulate a invetment. See Footnote 7. In thi cae, money flo through either route ii or route iii. 9 Equation 30 denote a alanced condition eteen the uly of and the demand for money. Therefore, Eq. 3 mean interet rate and not rofit rate. Player trategie yield Eq. 3 and they hould follo thi rate in the game. Hoever, a a reult, the interet rate hich layer follo coincide ith the rofit rate that guarantee alanced economic groth and a natural economy ee Prooition 2 and 3. Regarding natural rate interet, eepainetti 98 VIII. 23

13 Can a Natural Economy Oerate in Macroeconomy? 35 2 Suoe that the trategie and v are given. Then, the rofit rate i π = 0, ecaue the numerator of Eq. 24 ecome zero. Caitalit intend to avoid π = 0. For thi uroe, caitalit hould chooe the trategy that atifie C + Dv = 0 for and v. Accordingly, the caitalit trategy i given y in Eq. 26. Hoever, the caitalit hould analyze hat haen in the game and calculate their ayoff for Eq. 26. In thi cae, layer are alo unale to determine the rofit rate in the game. Therefore, caitalit hould calculate their ayoff uing Eq. 30 and recognize that they can otain Eq. 3. Thu, e can conclude that i the et reone to,v. 3 Suoe that the trategie and are given. In thi cae, e otain B = 0.In the cae of I, e otain π = A < 0inEq.24. Therefore, the CD intend to avoid a ituation here the rofit rate i determined y Eq. 24. For thi uroe, the CD hould chooe the trategy v that atifie C + Dv = 0inEq.24; thi trategy i given y v in Eq. 26. In thi cae, layer are alo unale to determine the rofit rate in the game. Therefore, the CD ayoff can e calculated uing Eq. 30 and layer hould recognize that they can get Eq. 3. Thu, e can conclude that the trategy v i the et reone to,. In the cae of II, if the CD doe not chooe trategyv, then the rofit rate ecome π = A = + n >. 2 n 0 0 n Thi condition i not alloed in the game. See inequality 3. The CD can recognize thi ituation a a layer of the game. Therefore, the CD hould chooe trategy v. In the cae of III, condition 29 ecome =. We can analyze it in the ame manner a cae I and otain the ame concluion. The Nah equilirium alo contruct alanced economic groth. We offer a rooition regarding thi oint. Prooition 2 Balanced Economic Groth The Nah equilirium in Eq. 26 contruct alanced economic groth. The rate of thi alanced economic groth g i g = and the alanced economic groth vector i. Proof Firt, let u conider the utainaility of the game to roduction ector. Note that ector rovide ector 0 ith caital good. Therefore, if a alanced economic groth ath exit, the folloing condition i required, Equation 32 i uorted y v in 26, ecaue e otain Ẋ 0 = Ẋ. 32 K 0 = v K v =

14 36 K. Akimoto from v = v, K 0 = vk, K = vk and alo otain Ẋ 0 = K 0 0 = K = Ẋ from Eq. and 33. Therefore, v guarantee the condition required for alanced economic groth. Next, let u contruct the alanced economic groth ath. From Eq. 3 and 9, e otain K K =. On the other hand, fromk 0 = vk andk = vk, e otain K 0 = v K = v K, K = v K = v K. Thee equation can e denoted y [ ] [ ] K 0 K0 = M, here M = K K [ 0 v 0 v ], Calculating the eigenvalue and eigenvector of the matrix M, e otain λ 0 = 0, λ = v and h 0 = [ 0 0 ], h = [ v v ] here the eigenvector h i correond to the eigenvalue λ i i = 0,. Therefore, the caital vector that denote the alanced economic groth i ] { v = K ex K [ K0 t} [ ] v v here K denote the initial level of caital tock. On the alanced economic groth ath, it i eaily verified that K 0 K 0 = K K = v. 35

15 Can a Natural Economy Oerate in Macroeconomy? 37 Furthermore, the groth rate of the caital tock K K i calculated a follo, K K = K 0 + K K 0 + K = K K + K 0 K = Sutituting v for Eq. 35 and 36, e otain From Eq. and 37, e otain = + K 0 K v. 36 K K = K 0 = K =. 37 K 0 K 0 + Ẋ 0 = Ẋ = = g 38 X 0 X Natural Economy No e analyze a natural economy, firt ith a definition of natural economy uing concet of Smith 776 and Painetti 98. Definition Natural Economy If an economy atifie the folloing condition, e call it a natural economy. i A natural rice, defined y Smith, i etalihed in the rice of a commodity. ii The rice of caital good follo a ure laour theory. 0 iii Total rofit i equal to total aving, and total age i equal to total conumtion. iv The rofit rate i equal to the alanced groth rate of the economy. 2 We rove that the Nah equilirium 26 atifie condition i, ii, and iii. Hoever, for rearation in advance, e conider the natural rice a defined y Adam Smith. In articular, ee the folloing entence, regarding the natural age rate. There i in every ociety or neighorhood an ordinary or average rate oth of age and rofit in every different emloyment of laour and tock. Thi rate i naturally regulated, artly y the general circumtance of the ociety, and artly y the articular nature of each emloyment. 3 Thu far at leat eem certain, that, in order to ring u a family, the laour of huand and ife together 0 Painetti 98, Iid., Iid., Smith 776, Chater VII,

16 38 K. Akimoto mut, even in the loet ecie of common laour, e ale to earn omething more than hat i reciely neceary for their on maintenance; ut in hat roortion, hether in that aove mentioned, or in any other, I hall not take uon me to determine. 4 We hould reciely verify thi oint, uing our model. We adot conumtion good a a numeraire and et Eq. 4. That i, age rate, rofit rate, and caital tock are meaured y conumtion. At thi tage, uoe that X unit of laor, required to e inut, roduce one unit of conumtion good, and that the age rate i. Then, Eq. 4 hould e reciely analyzed. I The technology of the economy e are conidering require X unit of inutted laor to roduce one unit of conumtion good. II Worker maintain their live y conuming. III Therefore, the natural age rate imlie that X unit of laor hould e maintained y conuming one unit of good. That i, a orker and hi or her family hould e maintained y the natural age rate. Thi imlie that the natural age rate i = X. Therefore, to rove that the Nah equilirium 26 etalihe a natural economy,e hould demontrate that the rice of conumtion good i accurately exreed y the quantity of laor inut. On the other hand, Adam Smith took care to not commit himelf regarding the level of the natural rofit rate. We quote him again: Profit i o very fluctuating, that the eron ho carrie on a articular trade cannot alay tell you himelf hat i the average of hi annual rofit. It varie, therefore, not only from year to year, ut from day to day, and almot from hour to hour. 5 Thee entence imly that Adam Smith analyi a concentrated on the mechanim of the market. In contrat, Painetti 98 reciely defined a natural rofit rate aed on macroeconomic. A roduction roce and an economic tructure of economy lie ehind Painetti 6 definition of natural rofit rate. We quote him, firt, regarding the natural rate of rofit: No that a natural rate of rofit i introduced, all relation ecome much more traightforard. The theory of value imlied y the reent theoretical cheme ecome a theory in term of imle laour a ure laour theory of value. 7 In addition, it i imortant to undertand the meaning of the natural rofit rate from a macroeconomic vieoint. 8 Next, e offer a rooition regarding the natural economy. 4 Iid., Chater VIII, Iid., Chater IX, Painetti 98, Iid., Iid., Secially, ee Footnote in. 47 and Footnote in

17 Can a Natural Economy Oerate in Macroeconomy? 39 Prooition 3 Natural Economy The Nah equilirium in Equation 26 contruct a natural economy. Proof e indicate that the Nah equilirium in Eq. 26 atifie the condition of a natural economy. i Sutituting π in 27 for, e otain 0 = n 0 + n. Notice that the term n on the right-hand ide of thi equation i aid to caitalit a rofit. Hoever, thi art i aid to orker to roduce one unit of caital good, hich i required to roduce conumtion good. That i, 0 urely exree the quantity of laor inut in the economy. By 4, the age rate correonding to π i, in 27. The age rate imlie that one unit of conumtion good hould utain n 0 + n unit of laor. ii Next, e analyze π in Eq. 27. A hon in Footnote 9, it i the interet rate determined y the layer trategie and Eq. 30. Player hould follo in thi rate in the game. Hoever, a hon in Prooition 2, π i equal to the alanced economic groth rate carried out y the Nah equilirium. In addition, uoe that one unit of conumtion good and one unit of caital good are roduced. Production require 0 + unit of caital good. In thi roundaout roduction roce, 0 + contitute the additional ocial cot that the economy hould ay. 9 In the groing economy, 0 + imlie the natural rofit rate. 20 Therefore, a a reult, π coincide ith the natural rofit rate defined y Painetti Uing thee rearation, e exand the right-hand ide of Eq. 2, according to it Footnote 4. Sutituting the rofit rateπ for Footnote 4, e otain } = n { + g + g 2 + g Equation 39 doe not exre a laor equivalent ut rather a ure laor theory correonding to the alanced economic groth. 22 iii Next, e rove that the Nah equilirium 26 atifie one of the condition of natural economy defined y Painetti. 23 We quote Painetti 98. In the natural economic ytem, total rofit turn out to e equal to total aving, and total age turn out to e equal to total conumtion. 24 Total rofit emerge a a kind of rior claim to hare in the final national income, hilt total age emerge a a kind of reidual or urlu that remain over and aove hat ha een charged for rofit Iid., Iid., See Equation VII.3. in Painetti 98, Painetti 98, Iid., Iid., Iid.,

18 40 K. Akimoto The Nah equilirium in Eq. 26 ring aout Eq. 27. We indicate that Eq. 27 atifie the Painetti condition. From Eq. 30 here C =, S = W, e otain 0 X 0 = + W = W. 40 From Eq. 5, 9 and 40, e otain X =. 4 In the roce of the aove calculation, the rofit rate i determined rior to the age rate. Therefore, total age are interreted a a urlu that remain over and aove hat ha een charged for rofit. 26 Therefore, the economic reult 27 involve a character of the natural economy defined y Painetti iv It i aarent from aove analyi. Finally, e otain the folloing rooition hich indicate the relationhi eteen the Nah equilirium and the technological coefficient of the to roduction ector. Prooition 4 Production Structure and Saving Rate in a Nah Equilirium In the Nah equilirium in Eq. 26, e otain the folloing condition. If if n 0 < n, then 0 <, n 0 > n, then 0 >, and if n 0 0 = n, then =. Proof We otain Prooition 4 from Eq. 26 directly. Prooition 4 indicate the relationhi eteen roduction tructure and the layer aving rate. Hoever, e hould note the relationhi eteen Prooition 4 and Painetti analye. Painetti 962 roved that the condition < a neceary for the macroeconomy to e tale. Let u conider thi rolem. Rememer that Kaldor fundamental Eq. 20 contruct a central core of Keyneian economy and require < for the economy to e tale. 26 Iid., Iid.,

19 Can a Natural Economy Oerate in Macroeconomy? A Caution for Deviation from Natural Economy At thi tage, e hould analyze the ackground of Prooition 4. The rolem i the Nah equilirium in Eq. 26. It reject the rooition that the rofit rate i determined y the rofit function in Eq. 24, hich i contructed from Kaldor fundamental Eq. 20. In other ord, if the rofit rate i determined y the rofit function, each layer i laced in a ore ituation. Therefore, the Nah equilirium make Eq. 20 the identity, or =, and avoid the rofit rate eing determined y Eq. 24. A a reult, in the Nah equilirium, the rofit rate i determined at a level here caitalit conumtion i rovided y orker aving. Thi condition ha a common feature ith Wickell natural interet rate. 28 We contruct our model uing Kaldor fundamental equation. Hoever, the Keyneian equation caue a condition here it doe not function. A a reult, it ring aout a claical rofit rate. Prooition 4 i organized on the ai of thi condition. We conclude that the outcome of caitalit or orker aving rate, hichever i larger, deend on the technological tructure of the to roduction ector. Thi analyi imlie that if only one condition i detroyed, the hole economy ould loe it alance. From the aove dicuion, one dituring factor for a natural economy make it aearance: Kaldor fundamental Eq. 20 ould revive to determine rofit and age rate. Rememer the roof of the Nah equilirium: caitalit have a trong trategy that yield π = and = 0. Worker have a defenive trategy that atifie Eq. 29. Thi mean that if the natural economy i detroyed, orker have their ack againt the all. From a macroeconomic vieoint, the alance in income ditriution i comletely detroyed. Thi i the tart of an economic crii Concluion Uing game theory, e roved that a natural economy i oerated in macroeconomy. Thi tudy i aed on the aarene of the iue dicued in Akimoto 204, namely that the fundamental caue of the current economic crii i a large deviation from the natural economy. Hoever, ome limitation to our finding hould e dicued. Firt, e hould oint out that the game-model in thi aer doe not involve innovation. In other ord, a i hon in Eq., 7 and 8, the model i contructed uing the fixed technological coefficient. Hoever, a Schumeter noted, the fundamental imule that et and kee the caitalit engine in motion come from the ne conumer good, the ne method of roduction or tranortation, the ne market, the ne form of indutrial organization that caitalit enterrie create. 30 Thi i the fundamental henomenon of economic develoment. 3 Therefore, the aumtion of fixed technological coefficient hould e removed. Regarding thi oint, r the tudy y Painetti 98 contructed the vertically integrated analyi that enaled 28 Wickell 898, Chater VIII. 29 Akimoto Schumeter 954, Schumeter 934, II. 23

20 42 K. Akimoto innovation to e examined. Therefore, e hould integrate the vertically integrated analyi into our game-model. Second, e hould once again deignate the imortance of the natural economy. Although many imortant tudie regarding the natural economy exit, fe examine the rik of deviating from the natural economy. A Akimoto 204 indicated, the raid increae in monetary aet can caue an imalance eteen the real economic ector and the monetary ector. Thi imalance imlie a deviation from the natural economy and erve a a fundamental caue of economic crie. Thi ource of rik hould e tudied from variou oint of vie. Thi tudy thu illutrate that the rolem of hether the natural economy can e roduced in a macroeconomy that involve innovation and technological change till needed to e examined. The game-model in thi aer excluded thee factor a riori. If the natural economy cannot oerate in a macroeconomy involving innovation, e hould examine the condition and olicie hich uort the natural economy. We ill examine thi rolem in an ucoming aer. Reference Akimoto K 204 A fundamental caue of economic crii; a macroeconomic game eteen the real economic ector and monetary ector. Int J Econ Sci III4: 37 Kaldor N Alternative Theorie of ditriution. Rev Econ Stud 23:83 00 Keyne JM 930 A treatie on money II; the alied theory of money. The Macmillan and Co., Ltd, London Painetti LL 962 Rate of rofit and income ditriution in relation to the rate of economic groth. Rev Econ Stud, vol xxix 3, No 8 Painetti LL 98 Structural change and economic groth; a theoretical eay on the dynamic of the ealth of nation. Camridge Univerity Pre, London Schumeter JA 934 The theory of economic groth. Harvard Univerity Pre, Camridge Schumeter JA 954 Caitalim, ocialim and democracy, fourth edition, London. George Allen & Unin: firt edition, Harer & Ro 942, Ne York Smith A 776 An inquiry into the nature and caue of the ealth of nation, edited, ith an introduction, note, marginal ummary and an enlarged index y Edin Cannan. LL.D. Modern Lirary Edition, Ne York, 937 Wickell JFK 898 Geldzin und Güterreie, Eine Studie üer die den Tauchert de Gelde etimmenden Urachen, Jena Englih verion: Interet and Price, A Study of the Caue Regulating the Value of Money, tranlated y R.F. Kahn, Macmillan, London,