Katugampola Generalized Conformal Derivative Approach to Inada Conditions and Solow-Swan Economic Growth Model
G. Fernández-Anaya, L. A. Quezada-Téllez, B. Nuñez-Zavala, D. Brun-Battistini
aa r X i v : . [ ec on . T H ] J un Katugampola Generalized Conformal DerivativeApproach to Inada Conditions and Solow-SwanEconomic Growth Model
G. Fern´andez-Anaya a , L. A. Quezada-T´ellez b, ∗ , B. Nu˜nez-Zavala a , D.Brun-Battistini a a Universidad Iberoamericana, Ciudad de Mexico, Mexico. b Universidad Autonoma Metropolitana-Unidad Cuajimalpa, Ciudad de Mexico, Mexico.
Abstract
This article shows a new focus of mathematic analysis for the Solow-Swan eco-nomic growth model, using the generalized conformal derivative Katugampola(KGCD). For this, under the same Solow-Swan model assumptions, the Inadaconditions are extended, which, for the new model shown here, depend on theorder of the KGCD. This order plays an important role in the speed of conver-gence of the closed solutions obtained with this derivative for capital ( k ) andfor per-capita production ( y ) in the cases without migration and with negativemigration. Our approach to the model with the KGCD adds a new parameterto the Solow-Swan model, the order of the KGCD and not a new state vari-able. In addition, we propose several possible economic interpretations for thatparameter. Keywords:
Solow-Swan Economic Growth Model, Katugampola GeneralizedConformal Derivative, Inada Conditios.
1. Introduction
The economic growth models of [1] and [2], that henceforth, because of thesimilarities between them, we will call the Solow-Swan model (SSM), are models ∗ Corresponding author
Email address: [email protected] (L. A. Quezada-T´ellez)
Preprint submitted to Physic A July 2, 2019 hat try to explain income and its growth in an economy through the amountsof resources involved in production (capital and labor) and technological changeor technical progress. The model assumes, among other things, perfect compe-tition, full employment, decreasing marginal returns in the use of capital andlabor, constant returns to scale for a production function that is homogeneousand is considered exogenous because it includes variables or parameters whosevalue is determined outside the model or is an external data.In the SSM, the technical progress, denoted by the constant A , is an ex-ogenous parameter defined as a factor of increasing scale that multiplies theproduction function [1] and includes in that concept the improvements in hu-man factors through time [3]. Recently, the SSM continues to be studied, forexample in [4] it is done using a Kadiyala production function.Two are the main criticisms of SSM from the perspective of endogenousgrowth models: 1) its theoretical inability to explain long-term economic growth,which can only be achieved if it is imposed exogenously through technicalprogress (a strong technological change) to increase income and per-capita cap-ital levels, as well as welfare of families over time, either before reaching theirsteady state (in which economy growth stops) or once this has been reached ([5]and [6]), and 2) the impossibility of empirically verifying in the long term, theconvergence that the model predicts for all the economies of the planet, thosedeveloped and those that are not, to the same steady state [6].This second criticism has also been addressed by those [7] who, based onthe theories of endogenous economic growth, model the differences between theeconomies of the world, adding human capital (the knowledge, skills and com-petencies of workers as individuals) to the two state variables of the (SSM).When this variable was included, the convergence rates of the different types ofeconomies to the same steady state improved, however, they remained insuffi-cient.Nonetheless, in all the mentioned cases and in others as case [8] that ana-lyze the closed solution for the economic growth in relation to the migration,the method to obtain the solutions for the variables of the proposed models2endogenous and exogenous) was typical of derivatives of an integer order.Other works using models based on fractional order derivatives have also ad-dressed the problem of economic growth. For example, in [9], a generalizationof the economic model of natural growth is suggested, which takes into accountthe memory effect of the power law type. The memory effect implies the de-pendence of the process, not only on its current state but also on its history ofchanges in the past. For the mathematical description of the economic processwith power law memory, the theory of non-integer derivatives and fractionaldifferential equations is used. They conclude that the memory effect can leadto a decrease in the output and not to its growth, which is typical of a modelwithout memory.In [10], economic models must take into account the memory effects causedby the fact that economic agents, in their decisions, remember the history ofchanges in the exogenous and endogenous variables characterizing economic pro-cesses. The continuous-time description of the economic processes by decreasingmemory of the power law type can be described using fractional calculus andfractional differential equations. The inclusion of memory effects in economicmodels can lead to new results with the same parameters.In [11] with sufficient conditions to analyze the Mittag Leffler stability type,a Solow type model of fractional order is introduced as a new tool in mathemat-ical finance. The main advantage of the proposed approach is the non-localityproperty of these fractional derivatives that are convenient for the modeling ofreal financial situations and macroeconomic systems.Models based on other types of derivatives have also been studied, for exam-ple in [12], axioms for the system of differential equations (time scales) of Solowtype are formulated under the assumption that a certain function is constant,proving stability and balance results in positive coordinates. A Cobb-Douglastype production function is also considered.In [13], a general Solow model on time scales is introduced from which a first-order nonlinear dynamic equation that describes the model is deduced. Basedon a Cobb-Douglas type production function, several cases are considered when3here is no technological development or changes in the population. Subse-quently, they consider the case with technological development and populationgrowth. These models include, as particular cases, different types of derivativessuch as quantum calculus and some of conformal type.The case of slow growth when the order of the fractional derivative ρ ∈ (0 ,
1) is similar to the situation that appears in physics with the descriptionof anomalous diffusion models [14], which is related in form of sub-diffusionfor 0 < ρ <
1. The anomalous diffusion equations have been used to describefinancial processes [15, 16, 17, 18, 19, 20, 21].We propose derivatives of conformal type [22, 23] that could, without in-creasing state variables, offer an alternative to solve the two criticisms raisedpreviously against the SSM for the case with negative migration. The modelobtained with the KGCD is simpler than the models presented with fractionalderivatives, or with derivatives of the time-scales type. In addition, we presentclosed solutions of the SSM with the KGCD, very similar to the model of the in-teger SSM order, and therefore it is possible to give an analogous interpretationto those of the classical model.Applying the KGCD of order ρ [23] instead of the usual derivative of the in-teger order in the SSM, we show two results that preserve the Inada conditions,imposing restrictions on the ρ order of the KGCD related to the marginal con-tribution of capital ( α ) to the production obtained ( Y ). Later, we obtain closedsolutions for capital and per-capita production on time for zero and negativemigration cases. These closed solutions with the KGCD introduce a new pa-rameter ρ , and not a new state variable to the solutions obtained for the SSM asother authors do. The analysis of the SSM model with the KGCD allows us toshow that the ρ parameter plays a fundamental role in maintaining consistencywith Inada’s well-known conditions. As expected, when the parameter ρ takesthe value of 1, we recover the derivative of integer order, and consequently theclassic SSM.This article is organized as follows. In the second section the mathematicalpreliminaries are presented. In the third section the congruence of the KGCD4ith the Inada conditions of integer order is shown. In the fourth section, closedsolutions are obtained for the SSM model without migration and with negativemigration. In the negative migration case, the analysis on the restrictions im-posed for capital and per-capita production is presented, and also the timesfor which the aforementioned variables increase to infinity. In the sixth sectionsome representative graphs of the results obtained in the previous sections arepresented. Finally, the general conclusions are presented.
2. Preliminaries of KGCD
This section provides the main definitions of the KGCD, as well as thefoundation of the SSM.
This subsection discusses the definition of the conformal derivative, as wellas its properties of order ρ ∈ [0 , Definition 2.1. [23] Given a function f : [0 , ∞ ) → R . Then the KGCD oforder ρ , m is defined by: D ρm f ( t ) = lim ǫ → f ( te ǫt − ρ k ) − f ( t ) ǫ , (1) where e tm = m X i =0 t i i ! , (2) for all t > , ρ ∈ [0 , . As a consequence of the previous definition, the following Lemma is obtained.
Lemma 2.1. If f is differentiable, then D ρm f ( t ) = t − ρ dfdt . Proof 1.
Let’s take h ( t, ǫ ) = ǫt − ρ (cid:16) ǫt − ρ + ǫ t − ρ ) + ... + ǫ m t m (1 − ρ ) m ! (cid:17) = ǫt − ρ (1 + O m ( ǫ )) = ǫt − ρ + ˆ O m ( ǫ ) , (3)5 here ˆ O m ( ǫ ) = ǫt − ρ O m ( ǫ ) . Then, the definition of the KGCD of order ρ , m is as follows: D ρm f ( t ) = lim ǫ → f ( te ǫt − ρ m ) − f ( t ) ǫ = (4) lim ǫ → f ( t + ǫt − ρ + ˆ O m ( ǫ )) − f ( t ) ǫ = (5) lim ǫ → f ( t + h ( t, ǫ )) − f ( t ) ǫ = (6) lim ǫ → f ( t + h ( t, ǫ )) − f ( t ) h ( t,ǫ ) t ρ − O m ( ǫ ) = (7) lim ǫ → t − ρ df ( t ) dt . (8) If the KGCD of f exists, then we will say that the function f is ρ − differentiablein some interval (0 , a ) with a > , the KGCD of the fuction f exits and the lim t → + D ρm f ( t ) exists as well, hence we define D ρm f (0) = lim t → D ρm f ( t ) . (9) Note that D ρ f ( t ) = lim ǫ → f ( t + ǫt − ρ ) − f ( t ) ǫ (10) is the KGCD defined for m = 1 and D ρ ∞ f ( t ) = lim ǫ → f ( te ǫt − ρ ) − f ( t ) ǫ (11) is the KGCD defined for m = ∞ . (cid:3) Let it be noted that the derivatives (10) and (11) are particular cases ofthe derivative KGCD (1). Which correspond to the conformal derivative in [22]and the derivative of Katugampola in [23]. Therefore, the following results aresimilar due to Lemma (2.1). 6 heorem 2.1. [22, 23] If a function f : [0 , ∞ ) → R is ρ − differentiable in t > , ρ ∈ (0 , , so f is continuous on t . Theorem 2.2. [22, 23] Let it be that ρ ∈ (0 , and f, g are ρ − differentiable for t > . Therefore D ρm ( af + bg ) = aT ρ ( f ) + bT ρ ( g ) for every a, b ∈ R . D ρm ( t q ) = qt q − ρ for all q ∈ R . D ρm ( λ ) = 0 , for all constant functions f ( t ) = λ . D ρm ( f g ) = f T ρ ( g ) + gT ρ ( f ) . D ρm (cid:16) fg (cid:17) = gT ρ ( f ) − fT ρ ( g ) g . In the following section, the Inada conditions are presented applying theKGCD.
3. Inada conditions for the Solow-Swan model with the KGCD
The SSM is a benchmark for most economic growth analysis. In time, themodel is represented as in Eq.(12) where Y is the national production, K and L (state variables) are the quantities of capital and labor factors used in produc-tion, both measured with the appropriate units and A represents a technologicalconstant that is usually interpreted as the total productivity of all factors. Fora period of time t , national production is the result of combining capital andlabor, given a certain technological constant, represented mathematically as: Y ( t ) = F ( A, K ( t ) , L ( t )) (12)where t indicates time [8]. Definition 3.1.
The production Y ( t ) : R + → R is a function and must fulfillthe following properties [8], known in Economic Science as the Inada conditions.Replacing the derivative of integer order with the KGCD, we obtain the followingmodified Inada conditions: The function Y ( t ) is increasing for both state variables, capital ( D ρK,m Y > and labour force ( D ρL,m Y > . ii) The function Y ( t ) will have constant returns to scale, Y ( λK, λL ) = λY ( K, L ) , ∀ λ > . iii) The function Y ( t ) satisfies the conditions: lim K → D ρK,m Y = lim L → D ρL,m Y = + ∞ and lim K →∞ D ρK,m Y = lim L →∞ D ρL,m Y = 0iv) The function Y ( t ) also satisfies that ( D ρK,m Y < and that ( D ρL,m Y < Where: D ρK,m Y y D ρL,m Y are the partial derivatives of Y of order ρ , m ofKGCD with respect to the variables K and L respectively. Let it be noted that when ρ is 1 we recover the usual Inada conditions ofinteger order. Thereafter, we present the following propositions, where we willconsider A, K, L >
0. Let’s observe the following: D ρK,m Y ( K, L ) = K − ρ D K,m Y ( K, L ) (13) D ρL,m Y ( K, L ) = L − ρ D L,m Y ( K, L ) (14) D ρK,m Y ( K, L ) = D ρK,m ( D ρK,m Y ( K, L )) D ρK,m Y ( K, L ) = K − ρ [(1 − ρ ) D K,m Y + KD K,m Y ] (15)and D ρL,m Y ( K, L ) = D ρL,m ( D ρL,m Y ( K, L )) D ρL,m Y ( K, L ) = L − ρ [(1 − ρ ) D L,m Y + LD L,m Y ] (16) Theorem 3.1.
If the Inada conditions for the KGCD are satisfied for some ρ ∈ (0 , , then the Inada conditions of integer order are satisfied. Remark 1.
Let it be noted that we do not assume an explicit form for thefunction Y ( K, L ) . roof 2. Let’s suppose that ρ ∈ (0 , exists so that the Inada conditions aresatisfied with the KGCD (3.1). Therefore i) Since D ρ K,m Y = K − ρ D K,m
Y > , then D K,m
Y > and similarly D ρ L,m Y = L − ρ D L,m
Y > , then D L,m
Y > . ii) Is the same statement as for ρ = 1 . iii) If lim K → D ρ K,m Y = lim K → K − ρ D K,m Y = ∞ , then D K,m Y it is of or-der O ( K − (1 − ρ + ǫ ) ) with ǫ > . Therefore lim K → D K,m Y = ∞ . Similarly, lim L → D ρ L,m Y = ∞ implies lim L → D L,m Y = ∞ .If lim K →∞ D ρ K,m Y = lim K →∞ K − ρ D K,m Y = 0 , then D K,m Y is of order O ( K − (1 − ρ + ǫ ) ) with ǫ > . Therefore lim K →∞ D K,m Y = 0 . The case of lim L →∞ D L,m Y = 0 is similar. iv) D ρ K,m
Y < if and only if K − ρ [(1 − ρ ) D K,m Y + KD K,m Y ] < , whereby (1 − ρ ) D K,m Y + KD K,m
Y < , but (1 − ρ ) , K and D K,m Y are positivebecause of item i), therefore D K,m
Y < . Similarly D ρ L,m
Y < , then D L,m
Y < . (cid:3) Theorem 3.2.
Let’s consider the SSM, with production function Y = AK α L − α .The Inada conditions for the KGCD are satisfied if and only if ρ > max [ α, − α ] . Proof 3.
First let’s note that Y = AK α L − α satisfies the Inada condition ofinteger order. i) D ρK,m ( Y ) = K − ρ D K,m ( Y ) > and D ρL,m ( Y ) = L − ρ D L,m ( Y ) > because K, L, D
K,m ( Y ) y D L,m ( Y ) are positive for Y = AK α L − α . ii) It is the same statement for integer order and KGCD. iii) lim K → D ρK,m ( Y ) = lim K → K − ρ D K,m ( AK α L − α ) = αAL − α lim K → K α − ρ = ∞ , if ρ > α . lim L → D ρL,m ( Y ) = lim L → L − ρ D L,m ( Y ) = AK α (1 − α ) lim L → L − ( ρ + α ) = ∞ , if ρ > − α . Therefore ρ > max [ α, − α ] . Analogously lim K →∞ D ρK,m ( Y ) = αAL − ρ lim K →∞ K α − ρ = 0 , if ρ > α .Similarly lim K →∞ D ρK,m ( Y ) = 0 , if ρ > − α . Therefore ρ > max [ α, − α ] . D ρK,m ( Y ) = K − ρ [(1 − ρ ) D K,m ( Y ) + KD K ( Y )] < if and only if | D K ( Y ) | > − ρK | D K,m ( Y ) | (17) because D K,m ( Y ) > y D K,m ( Y ) < .Substituting Y = AK α L (1 − α ) in Eq. (17) and considering the absolutevalues, we obtain Aα (1 − α ) K α − L − α > (1 − ρ ) AαK α − L − α if and onlyif − α > − ρ equivalently ρ > α . Similarly D ρL,m ( Y ) < if | D L,m ( Y ) | > (1 − ρ ) L | D L,m ( Y ) | ⇐⇒ AK α (1 − α ) L − α − (1 − ρ ) < AK α (1 − α ) αL − α − giventhat − ρ < α ⇐⇒ ρ > − α , therefore both conditions are fulfilled ⇐⇒ ρ > max [ α, − α ] . Notice that α ∈ [0 , implies that ρ ∈ (1 / , since ρ cannot be greater than by the propositions that consider the Cobb-Douglasform Y = AK α ( t ) L − α ( t ) , α ∈ (0 , . (cid:3) These results show that the Inada conditions of integer order are preservedwith the new derivative KGCD. In the next section the SSM is described usingthe KGCD.
4. The Solow-Swan model with the KGCD
In this section we analyse two cases: the case without migration with aMalthusian law considering the population as labour force and the case withnegative constant migration. We will follow a similar procedure to the onedeveloped in [8]. If the production function is of the Cobb-Douglas form [1]: Y = AK α ( t ) L ( t ) − α , α ∈ (0 ,
1) (18)when α is close to 0 it is said that the economy is work intensive and for theopposite case, if α is almost 1, it is capital intensive. According to (18), thecapital stock dynamics is governed by the ordinary differential equation:˙ K = sY − δK = sAK α L − α − δK (19)10here s and δ are the savings constants and the rate of depreciation of capitalrespectively, hence, neoclassically, sY can be taken as the gross investmentand δK is the capital depreciation of the entire economy [8]. In the followingsubsections we present both cases, with and without migration. Deriving out of the SSM of integer order, we present in this subsection anew model, applying the KGCD. In it, the KGCD retrieves the properties ofinteger order, but introduces a parameter in the order of the derivative thatallows greater flexibility to the model and is compatible with the classic SSMwhen ρ takes the unit value.The rest of the work will use the following notations indistinctly: L ( ρ ) m = D ρm ( L ) L ( ρ ) m = γL ⇒ L ( t ) = L e γ tρρ (20)where L > γ the inter-temporal rateof growth or Malthusian parameter. If we replace (20) in (18) we obtain Y = AK α (cid:16) L e γ tρρ (cid:17) − α (21)If we define the per-capita capital, as k ( t ) = K ( t ) L ( t ) (22)and the labour growth rate as, n ( t ) = L ( ρ ) m ( t ) L ( t ) (23)Therefore, deriving (22) with respect to the time and taking n ( t ) = γ , weobtain that k ( ρ ) m = K ( ρ ) m L − γk ( t ). From (19) and replacing now the capital stockusing the KGCD is given by 11 ( ρ ) m = sY − δK = sAK α L − α − δKk ( ρ ) m + ( δ + γ ) k = sAk α (24)where (24) is a Bernoulli equation using the KGCD. Through Bernoulli’s well-known variable change, to obtain the linear equation, we take w = k − α , w ( ρ ) m = (1 − α ) k − α k ( ρ ) w ( ρ ) m + (1 − α )( γ + δ ) w = (1 − α ) sA (25)From (24) and (25) the solution for k ( t ) is given by k ( t ) = (cid:20) c e − (1 − α )( γ + δ ) tρρ + sAγ + δ (cid:21) − α (26)where c is a constant of appropriate units.It is important to note that the steady state of per-capita capital k ∞ , isgiven by k ∞ = lim t → + ∞ k ( t ) = (cid:18) sAγ + δ (cid:19) − α (27)Note that the limit, when t tends to infinity, coincides with the limit ofinteger order. On the other hand, we now define the per-capita product as thetotal production ratio in respect to work, that is, y ( t ) = Y ( t ) L ( t ) = Ak α ( t ) (28)where using the expression (18), in the long term the per-capita productiontends to y ∞ = lim t → + ∞ Ak α ( t ) = A (cid:18) sAγ + δ (cid:19) − α . (29)The solutions obtained here with KGCD have the same mathematical behavioras those of classical SSM. Therefore in the case without migration, the conver-gence to the same steady state of capital and per-capita production will exist12egardless of the value of ρ . However, in this case the convergence speed de-creases as the value of ρ decreases. Under the same assumptions, this subsection presents another KGCD modeladding a constant migration rate ( I ) to the differential equation (20), whichdetermines the labour force of the economy, therefore we have: L ( ρ ) m = γL + I. (30)Calculating the solution of (30) L ( t ) = ce γ ( t − t ρρ + Z t e γ ( t − t ρρ e − γ ( s − t ρρ I ( s − t ) ρ − ds (31)where t is an initial time which we take with value 0 in (31), we obtain L ( t ) = (cid:20)(cid:18) c + Iγ (cid:19) e γ tρρ − Iγ (cid:21) , c = L (32)Note that if ρ = 1, we retrieve the solution of the integer migration case.Therefore, if a labour growth rate is taken as (23) and the KGCD is applied, itresults in ¯ n ( t ) = L ( ρ ) m L = γ ( γL + I )( γL + I ) e γ tρρ − I (33)Again, with (19) and now replacing the stock capital with migration usingthe KGCD, the per-capita capital satisfies the following equation¯ k ( ρ ) m + (¯ n ( t ) + δ )¯ k = SA ¯ k α (34)where (34) was obtained analogously to Eq.(24) but considering migration. Thisis also a Bernoulli equation with the KGCD. If we now take Z = ¯ k − α and Z (0) = Z = ¯ k − α with Z ( ρ ) m = (1 − α )¯ k − α ¯ k ( ρ ) m , for a similar argument we cometo: Z ( ρ ) m − ( α − δ + ¯ n ( t )) Z = (1 − α ) sA. (35)13ow using Eq.(33), we obtain that: Z ( δ + ¯ n ( t )) t ρ − dt = δ t ρ ρ + Z dγL ( t ) γL ( t ) = δ t ρ ρ + ln h ( γL + I ) e γ tρρ − I i − ln ( γL )(36)where γL ( t ) = ( γL + I ) e γtρρ − I .On the other hand, substituting the integral of the Eq.(36) in the exponentof the exponential function, we get: e ± R y (1 − α )( δ +¯ n ( t )) t ρ − dt = e ± (1 − α ) δ yρρ (cid:20)(cid:18) γL + IγL (cid:19) e γyρρ − IγL (cid:21) ± (1 − α ) . (37)Therefore the solution of Eq.(35) is: Z ( t ) = Z e − R (1 − α )( δ +¯ n ( t )) t ρ − dt + e − R (1 − α )( δ +¯ n ( t )) t ρ − dt Z e (1 − α ) δ yρρ (cid:20)(cid:18) γL + IγL (cid:19) e γyρρ − IγL (cid:21) (1 − α ) (1 − α ) sAt ρ − dt (38)Hence, from the following equation it is possible to obtain the capital andper-capita production: Z ( t ) = Z e ( α − δ tρρ [ (cid:18) γL + IγL (cid:19) e γtρρ − IγL ] ( α − +(1 − α ) sAe ( α − δ tρρ [ (cid:18) γL + IγL (cid:19) e γtρρ − IγL ] ( α − Z t e − ( α − δ yρρ (cid:20)(cid:18) γL + IγL (cid:19) e γyρρ − IγL (cid:21) (1 − α ) y ρ − dy !! (39)where it is obtained that the per-capita capital (¯ k ) is given by¯ k ( t ) = Z ( t ) − α (40)and per-capita production (¯ y ) by¯ y ( t ) = AZ ( t ) α − α . (41)The last two equations (40), (41), give us closed solutions for capital andper-capita production with migration. In the next section we will give the closed14olution for the case of negative migration by explicitly solving the integral of(39) in terms of hypergeometric functions. In this section we applied a similar procedure to used in [8] for to solve theintegral of the second term of the equation (39). J = Z t e (1 − α ) δτρρ (cid:20)(cid:18) γL + IγL (cid:19) e γτρρ − IγL (cid:21) (1 − α ) τ ( ρ − dτ (42)Applying the change of variable u = e γtρρ we have that, du = γρ e γτρρ (cid:0) ρτ ( ρ − dτ (cid:1) = γuτ ( ρ − dτ from where duγu = τ ( ρ − dτJ = 1 γ Z e γτρρ u (1 − α ) δγ − (cid:20) − IγL + (cid:18) γL + IγL (cid:19) u (cid:21) (1 − α ) du (43) J = 1 γ (cid:18) − IγL (cid:19) (1 − α ) Z e γtρρ u (1 − α ) δγ − (cid:20) − (cid:18) γL I (cid:19) u (cid:21) (1 − α ) du (44)The last integral is related to Euler’s integral representation of the GaussianHypergeometric Function F : F a, bc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = ∞ X n =0 ( a ) n ( b ) n z n ( c ) n n != Γ( c )Γ( c − b )Γ( b ) Z t b − (1 − t ) c − b − (1 − zt ) − a dt (45)where ( . ) n = Γ( . + n ) / Γ( . ) is the Pochhammer symbol [24, 25]. The series isconvergent for any a, b, c if | z | <
1, and for Re ( a + b − c ) < | z | = 1. For theintegral representation Re ( c ) > Re ( b ) > z ) denotesthe Gamma Function. Thus, J = 1 γ (cid:18) − IγL (cid:19) (1 − α ) ( J t − J ) (46)15here J = γ (1 − α ) δ F a, bc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z (47)where J t = γe (1 − α ) δ tρρ (1 − α ) δ F a, bc (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ( t ) (48)with a = α − b = (1 − α ) δγ , c = (1 − α ) δγ + 1, z = (cid:16) γL I (cid:17) y z ( t ) = (cid:16) γL I (cid:17) e γ tρρ . And therefore, we obtain formulas that are explicitly closed forcapital and per-capita production with negative migration Z ( t ) = Z e ( α − δ tρρ [ (cid:18) γL + IγL (cid:19) e γtρρ − IγL ] ( α − +(1 − α ) sAe ( α − δ tρρ [ (cid:18) γL + IγL (cid:19) e γtρρ − IγL ] ( α − γ (cid:18) − IγL (cid:19) (1 − α ) ( J t − J ) ! (49)¯ k ( t ) = e − δ tρρ [ (cid:18) γL + IγL (cid:19) e γtρρ − IγL ] − " ¯ k − α + (1 − α ) sAγ (cid:18) − IγL (cid:19) (1 − α ) ( J t − J ) ! − α (50)¯ y ( t ) = A ¯ k ( t ) α (51)The (cid:16) − IγL (cid:17) (1 − α ) factor implies that: if I ≤
0, therefore γ >
0, and if
I > γ <
0. In the next section we offer restrictions on the values that migration I and time t can take.
5. Analysis for I negative. In this section an analysis is made of the restrictions on capital and per-capita production, for
I < γ > emma 5.1. i) L ( t ) = (cid:16) Iγ + L (cid:17) e γ tρρ − Iγ > with γ > , < Iγ + L and I < , if I ∈ [ − γL , . ii) L ( t ) = (cid:16) Iγ + L (cid:17) e γ tρρ − Iγ > with γ > , Iγ + L < and I < , if I ∈ ( −∞ , − γL ) y t < t f where t f = h ln (cid:16) γL I (cid:17)i − γ . Proof 4. i) If γ > , < Iγ + L and I < ⇔ − γL < I < ⇔ I ∈ [ − γL , ⇒ L ( t ) = (cid:16) Iγ + L (cid:17) e γ tρρ − Iγ > . ii) If γ > , Iγ + L < and I < ⇔ I < − γL ⇔ I ∈ ( −∞ , − γL ) and (cid:16) Iγ + L (cid:17) e γ tρρ > Iγ ⇔ e γ tρρ > IγIγ + L = γL I ⇒ t < t f = h ln (cid:16) γL I (cid:17)i − γ . (cid:3) Lemma 5.2. γ > and | z | < and | z | < ⇔ I ∈ ( −∞ , − γL ) y t < t f = h ln (cid:16) γL I (cid:17)i − γ . Proof 5. | z | < ⇔ − < γL I < ⇔ − < γL I < . | z | < ⇔ − < (cid:16) γL I (cid:17) e γ tρρ < ⇔ − γL I < e γ tρρ < γL I − γL I < e γ tρρ is fulfilled if γL I > ⇔ − < γL I .If γ > and I < or γ < and I > then − γL I < e ⇒− < γL I < . Note that − < γL I < ⇔ I ∈ ( −∞ , − γL ) . On the another hand, e γ tρρ < γL I is fulfilled if t < t f . Therefore, γ > and | z | < and | z | < ⇔ I ∈ ( −∞ , − γL ) and t < t f . (cid:3) Remark 2.
If the conditions of lemma 5.2 are met, so are the conditions ofitem ii) of the lemma 5.1. Hence, we can only consider the case I ∈ ( −∞ , − γL ) and t < t f . As a consequence of the previous results, it is not possible to perform anasymptotic analytic for ¯ k ( t ) when t → ∞ and for any ρ ∈ [0 , J t y J converge only for | z | < | z | < I ∈ ( −∞ , − γL ) and t < t f . However, we can calculate the limitlim t → t f ¯ k ( t ) = ∞ , which coincides with the case of integer order.17ote that L ( t ) > ρ is incorporated, and not a newstate variable to the solutions obtained for the SSM model. This parameter ρ represents the order of the KGCD applied and can vary between 0 and 1 tomeet the Inada conditions if the KGCD is applied. It allows to recover thesolutions of the SSM when ρ adopts its maximum value and is consistent withthe Inada conditions, and therefore with the SSM, if its value is greater than max [ α, − α ].In a simple way, the parameter ρ added to the classic SSM when using theKGCD, can be interpreted as the speed with which an economy approachestowards its steady state in which there will be no further growth. The aforesaidspeed will be slower the smaller the said parameter is. In this trajectory, per-capita capital decreases and, therefore, per-capita production and the economy’sgrowth rate do it as well.In this sense, the possibility of incorporating into the classic SSM a setof differences between the economies of the planet that makes their trajectoriestowards the same steady state do not coincide in speed and/or value. Therefore,there is the possibility of solving the second criticism of said model that wasaforementioned.The ρ parameter could, in addition, allow economic science researchers tomodel endogenously (and without adding a state variable) some data of variablesfrom the economic, socio-political and institutional spheres that, the closer theyare to zero, more they decelerate the falling rate of the trajectories of capitaland the per-capita production over time towards its respective long-term steadystate.Among the useful economic data to represent ρ there could be: 1) The per-centage of available resources not used, that is, the percentage of idle capacity18f available resources. 2) The percentage of profits (payment to capital) that isused for purposes other than the creation of new capital through innovation. 3)The inflation rate of an economy, this is the speed with which prices increase.At a higher rate, per-capita capital and, therefore, also per-capita productionwill fall to lower values and more rapidly than in the case of economies withlower inflation rates. 4) The interest rate. 5) The tax rate applied in directand indirect taxes. 6) The unemployment rate. 7) The natural rate of unem-ployment. 8) The size of the total public deficit with respect to GDP. 9) Thesize of imports regarding GDP. 10) The complement of the ratio of the TradeOpenness Index (1 − T OI ). 11) The standardized risks measured between zeroand one by the rating agencies. 12) The percentage of concentration in the eco-nomic structure (monopolies and oligopolies). 13) The degree to which othermarket failures arise (externalities, incomplete markets, public goods and socialgoods, among others). In the socio-political and institutional spheres, ρ couldrepresent, among other things, the degrees of: 14) Risk perceived by economicagents. 15) Informality in the economy. 16) Inequality in income distribution.17) Poverty and Malnutrition. Health and education shortfalls (illiteracy). 18)Corruption. 19) Impunity. 20) Social violence. 21) Public insecurity. 22) Theefficiency with which private economic agents, with their rational expectations,inhibit countercyclical economic policy measures that the government designsand executes to reduce and decelerate the capital and per-capita productiondownfalls.One more possibility is that ρ is an index that results from a combinationof all or some of the aspects of any of the areas indicated previously, providedit reflects in an aggregated and hierarchical manner the differences between theeconomies of the planet in normalized values between zero and one.Finally, ρ could also represent the rate of diminishing returns of capitalthat generates convergence to the same steady state and that could be differentfor each economy of the world, since it would be determined by the economic,socio-political and institutional aspects listed above.19 . Graphics of some representative examples In this section we present the different cases of the solution for capital andfor production, both per-capita, without migration and with negative migration.In Figures 1 and 2 the following values were taken: γ = 0 . α = 0 . δ = 0 . s = 0 . A = 1, k = 200, L = 100, and − γL = −
2. For the Figures 3 and4 γ = 0 . α = 0 . δ = 0 . s = 0 . A = 1, k = 100, L = 100, and H = − . Figure (1) shows the trajectories of per-capita capital for the case withoutmigration and for different values of ρ > max [ α, − α ] compared with the one ofinteger order ( ρ = 1). It can be noted that, without migration ( I = 0) and witha s lower than α , the per-capita capital decreases at a lower speed while theorder of the derivative KGCD is smaller and it does so converging to the samesteady state regardless of the value of ρ . That is, the lower the ρ , the lower andslower will be the decrease in per-capita capital in an economy in which, withoutmigration, the savings rate and its conversion into new capital is insufficient tocounteract the depreciation of capital and growth of the population. Figure 1: The value of per-capita capital k t ) without migration of integer order ρ = 1 . k t ) with fractional order of ρ = 0 . k t ) with fractional order of ρ = 0 .
90 and k t ) withfractional order of ρ = 0 . .Figure (2) graphically describes the trajectories of per-capita productionwithout migration and for different values of ρ > max [ α, − α ] compared with20 igure 2: The value of per-capita product y t ) without integer order migration ρ = 1 . y t ) with fractional order of ρ = 0 . y t ) with fractional order of ρ = 0 .
90 and y t ) withfractional order of ρ = 0 . the ρ of integer order. Figure 2 shows that the per-capita production under theKGCD follows a similar behaviour to that of the per-capita capital. This means,the smaller the parameter ρ is, the lower and slower its trajectory towards thesame steady state will be. This is consistent with what happens in the SSM. Figure (3) shows the trajectories of the per-capita capital with negativemigration of −
19 for different values of ρ > max [ α, − α ] compared to theone of integer order ( ρ = 1). From this we can deduce that negative migrationgenerates two different phases in the trajectory of per-capita capital over time:one descending and another ascending. In both phases it happens that, insideof the aforementioned interval, the smaller the ρ value the slower or less fast thedecrease and the rise of the per-capita capital trajectories.An interesting aspect on this graph is that the lower the value of said para-meter, the corresponding trajectory reaches its vertical asymptote more slowly.This implies that, in the presence of negative migration, if it is possible that per-capita capital, and therefore the per-capita production, will grow indefinitely ina given time ( t ∗ ). The figure (4) that appears below shows this. This forces tointuit the divergence in contrast to the convergence predicted by the classicalSSM.Figures 3 and 4 show the possibility of solving the first criticism for the SSM21 igure 3: The value of the per-capita capital kf t ) with migration I = − . ρ = 1 . kf t ) with fractional order of ρ = 0 . kf t ) with fractional order of ρ = 0 . kf ρ = 0 . yf t ) with migration I = − . ρ = 1 . yf t ) with fractional order of ρ = 0 . yf t ) with fractional order of ρ = 0 . yf ρ = 0 . cited in the introduction of this paper. This is provided that three conditionsare met: a negative migration ( I < I ∈ ( −∞ , − γL ) and t < t f . This isdue to the fact that capital and per capita production grow, approaching theirvertical asymptote, without stagnation.
7. Conclusions