Exact solutions for a Solow-Swan model with non-constant returns to scale
aa r X i v : . [ ec on . T H ] A ug Exact solutions for a Solow-Swan model with non-constantreturns to scale
Nicolò Cangiotti & Mattia Sensi University of Pavia, Department of Mathematics, via Ferrata 5,27100 Pavia (PV), Italy. Email: [email protected] University of Trento, Department of Mathematics, via Sommarive 14,38123 Trento (TN), Italy. Email: [email protected]
Abstract
The Solow-Swan model is shortly reviewed from a mathematical point of view. Byconsidering non-constant returns to scale, we obtain a general solution strategy.We then compute the exact solution for the Cobb-Douglas production function, forboth the classical model and the von Bertalanffy model. Numerical simulations areprovided.
Keywords : Solow-Swan model, Cobb-Douglas production function, returns to scale, vonBertalanffy model.
JEL Classification Codes : C60, C65, C67.
1. Introduction
The Solow-Swan model plays an important role in neoclassical economics. Eventhough more than 60 years have passed since it was developed, independently, byRobert Solow [19] and Trevor Swan [20] in 1956, the model is still being analyzed andgeneralized, as evidenced by a large literature, which involves many fields of studies[1, 6, 7, 9, 10, 12, 15, 17].This work is devoted to the deepening of the mathematical point of view ofthe model. In particular, we are interested in investigating a model with weakerconditions on the returns to scale than the usual ones (see, e.g., [10]). In fact, weare going to relax the hypothesis of constant returns to scale, which in the classicalmodel allows to rewrite the production function as a function of the output pereffective unit of labour; instead, we let the production function to have increasing
Preprint submitted to Elsevier r decreasing returns to scale. We obtain a non-autonomous first order differentialequation, for which we provide the exact solution.The paper is organized as follows. In Section 2, we present the Solow-Swanmodel, with a focus on the Cobb-Douglas production function; moreover, we studythe non-constant returns to scale case, obtaining the exact solution for the model.In Section 3, we explore a different model, namely the von Bertalanffy model, byusing the same techniques. Section 4 is devoted to numerical analysis, to betterunderstand the behaviour of such solutions. Finally, in Section 5 we shall suggestsome perspectives for future research.
2. The classical model
As highlighted in the introduction, Solow-Swan models have a key role in neo-classical growth theory. Let us denote by C ( R ) the class of twice continuouslydifferentiable functions F : R → R . We actually can restrict the study to R , i.e.the first quadrant, the only economically relevant subset in this setting. The mathe-matics of the model is based on the hypothesis that a production function F ( x , x ) satisfies the following conditions: ∂F∂x i > , ∂ F∂x i ∂x j < , lim x i → + ∂F∂x i = + ∞ , lim x i →∞ ∂F∂x i = 0 , (1)for i, j = 1 , . In literature, conditions (1), which are aimed at ensuring the existenceof an unique stable steady state in a neoclassical growth model, are called Inadacondtions . For further details and properties about the Inada conditions, we refer to[2, 13, 16, 21, 22].Classically, the variables x and x are denoted with K and L , respectively. Weswitch to this notation for the remainder of the article.Moreover, the quite stringent assumption that F has constant returns to scale israther frequent in many studies. In fact, thanks to such a hypothesis, it is not hardto obtain an exact solution for the ODE describing the model, at least in its mostfamous autonomous form.Since it is a useful step towards our more general construction, we briefly recall thisstrategy. Let us suppose that the rate of change of K is proportional to F , andthe labor force grows exponentially; such a setting can be described by the following2ystem: d K d t = sF ( K, L ) , d L d t = γL, with s, λ > constants. Thus, since the equation for L is autonomous and easilysolved, we focus our attention on the ODE describing the evolution in time of K ,which is, explicitly: d K d t = sF ( K, L ) . (2)We notice that the constant return to scale hypothesis implies that F ( λK, λL ) = λF ( K, L ) . (3)Dividing both sides of (2) by L , the equation becomes L d K d t = sF (cid:18) KL , (cid:19) . (4)Let us now consider the following derivative:dd t (cid:18) KL (cid:19) = 1 L d K d t − K d L d t L = 1 L d K d t − γ KL . (5)Combining (4) and (5), and introducing the variable k := KL , i.e. the capital-laborratio, and the notation f ( k ) := F ( k, , we are finally ready to write the classicSolow-Swan model: d k d t = sf ( k ) − γk. (6)However, in this paper we shall present a different approach to the Solow-Swanmodel compared to the one given in [10], which is ˙ k = sf ( k ) − ( δ + γ ( t )) k, (7)where k is the capital-labor ratio, s is the fraction of output which is saved, δ is thedepreciation rate, f is a production function and γ ( t ) is the ratio ˙ L/L ; ˙ k indicatesthe derivative of k with respect to the time variable t , i.e. dd t k ( t ) . In fact, in [10],the author assumed f to have constant return to scale (as in the original model),3nd γ to be variable in time. Conversely, we assume γ to be constant, from whichwe obtain ˙ LL = γ = ⇒ L ( t ) = L e γt . (8)However, we do not assume our production function f to have constant return toscale. Instead, we choose a generic homogeneous production function, namely F ( λK, λL ) = λ n F ( K, L ) . (9)This means that, if n = 1 , the function has constant return to scale, if n < ( n > )the function has decreasing (increasing) returns to scale. In particular, we noticethat F ( K/L,
1) = F ( L − K, L − L ) = L − n F ( K, L ) . (10)Starting from the usual equations ˙ K = sF ( K, L ) , (11a) ˙ L = γL, (11b)we can derive a non-autonomous equation for the capital-labor ( K/L ) ratio k , asstated in the following proposition. Proposition 1.
The ratio k evolves in time obeying the ODE ˙ k = sL n − ( t ) f ( k ) − γk, (12) where f ( k ) := F ( k, and L ( t ) = L e γt ; recall (8) and (11b) .Proof. By direct computation, we notice thatdd t KL = 1 L d K d t − KL d L d t = 1 L d K d t − γ KL . Now, combining (11a) and (10), we notice that L d K d t = sL n − F ( K/L, . Recalling the definitions of k = K/L and f ( k ) := F ( k, , we conclude the proof. Remark 1.
There are many standard properties of the following Cauchy problem: ( ˙ k = sL n − ( t ) f ( k ) − γk,k (0) = k , (13) that one can easily obtain by simple observations or by using to use the so-called Comparison theorems (for results in that direction see [10, Sec. 3] and [8]). emark 2. The results obtained so far are valid for a wide class of productionfunctions; however, in order to proceed with the analysis of the Cauchy problem (13) one needs to specify a production function.
Our investigation now proceeds with a very natural choice for the productionfunction f ( k ) , i.e. the Cobb-Douglas production function [5]. For the standard Cobb-Douglas production function (in which we fixed, without loss of generality, the total-factor productivity coefficient equal to ) F ( K, L ) = K α L β , < α ≤ , < β ≤ , α + β = n, it is easy to compute the law of the capital-labor ratio k ( t ) : ˙ k = sL n − e ( n − γt k α − γk. (14)The following theorem provides the exact solution for (14). Theorem 1.
Let k ( t ) be a solution of (14) . Then if n = 1 and α = 1 k ( t ) = (cid:18) e ( α − γt (cid:20) s (1 − α ) L n − ( e γβt − γβ + k − α (cid:21)(cid:19) − α , (15) where we denote k := k (0) .Proof. Consider (14), which is clearly a Bernoulli differential equation [11]. We divideboth sides by k α , and apply the substitution v = k − α . Then, after some algebraicsteps, (14) becomes − α ˙ v + γv = sL n − e ( n − γt , We multiply both sides by (1 − α ) , which brings the equation to a standard form ˙ v + (1 − α ) γv = sL n − (1 − α ) e ( n − γt . Recalling β = n − α , we apply the well-known formula to solve this first order linearODE, obtaining (15). Remark 3. If n = α + β = 1 , i.e. if the Cobb-Douglas function has constant returnto scale, we recover Thm. 10 of [10]. Remark 4.
It is clear that for α = 1 Eq. (14) is a linear differential equation. Thesolution of such equation is computable by standard methods and, for k (0) = k , wehave k ( t ) = k exp sL n − (cid:0) e ( n − t − (cid:1) n − − γt ! . . The von Bertalanffy model In this section, we propose a different model, in which the labor force follows avon Bertalanffy law [23]: ( ˙ L = r ( L ∞ − L ) ,L (0) = 0 , (16)where L ∞ = lim t →∞ L ( t ) , (17)is a theoretical maximum asymptote size of the labor force, and r > determines thespeed at which the labor force approaches the asymptote. The model was exhaus-tively studied by Brida and Limas in [3], where the authors present many importantresults for the constant returns to scale case. As in Sect. 2, we are going to relaxthis hypothesis, considering also increasing (and decreasing) returns to scale, and wepresent the exact solution for the model. Remark 5.
The von Bertalanffy equation was widely studied by many authors fromdifferent fields. See, for instance, [4, 14, 18].
The first step is to compute the law of the ratio k . Proposition 2.
The ratio k evolves in time obeying the ODE ˙ k = sL n − ( t ) f ( k ) − rk ( L ∞ − L ( t )) , (18) where f ( k ) := F ( k, and, from (16) , L ( t ) = L ∞ − ( L ∞ − L ) e − rt .Proof. The proof is analogous to the proof of Prop. 1.We consider the Cobb-Douglas production function to proceed with our investi-gation, thus obtaining the following Cauchy problem: ( ˙ k = s ( L ∞ − ( L ∞ − L ) e − rt ) n − k α − r ( L ∞ − L ) e − rt k,k (0) = k . (19)The solution of the Cauchy problem (19) is given by the following theorem. Theorem 2.
Let k ( t ) be a solution of (19) . Then if n = 1 and α = 1 k ( t ) = e − ( α − L ∞ − L ) e − rt k − α e ( α − L ∞ − L ) − ( α − Z t L ( τ ) dτ !! − α , (20)6 here L ( τ ) := s ( L ∞ − ( L ∞ − L ) e − rτ ) n · exp [ rτ + ( α − L ∞ − L ) e − rτ ] L ∞ ( e rτ −
1) + L . Proof.
The proof is analogous to the proof of Thm. 1. In fact, by the same substi-tution v = k − α , we get the following linear differential equation: ˙ v = (1 − α ) s (cid:0) L ∞ − ( L ∞ − L ) e − rt (cid:1) n − + (1 − α ) r ( L ∞ − L ) e − rt v. Thus, applying the classical formula, we compute the solution.
Remark 6.
A comment analogous to Rmk. 4 can be made. For α = 1 , we have thefollowing solution (which involves hypergeometric functions F ), where we introduce,for ease of notation, L ∗ := L ∞ − L : k ( t ) = k · exp (cid:0) e − rt − (cid:1) L ∗ + sL n − (cid:16) − L L ∗ (cid:17) − n F (cid:16) − n, − n ; 2 − n ; L ∞ L ∗ (cid:17) ( n − r − s (cid:16) L ∞ e rt − L ∗ + 1 (cid:17) − n ( − L ∗ e − rt + L ∞ ) n − F (cid:16) − n, − n ; 2 − n ; e rt L ∞ L ∗ (cid:17) ( n − r . For further details on the use of the hypergeometric function in this context see, forinstance, [18].
4. Numerical simulations
In this section we propose some numerical simulations for both the classical andthe von Bertalanffy model, for the specific choice of Cobb-Douglas for our produc-tion function f ( k ) . The results of the classical case agree with the expectations,consistently with neoclassical growth theory with a convergence toward the initialconditions for the decreasing returns to scale and an exponential growth for increas-ing returns to scale (Figure 1). A very interesting output comes from the study of thevon Bertalanffy model, as dysplayed in Figure 2. The latter seems to level out thedifferences between the two cases, namely increasing returns to scale and decreasingreturns to scale. The following graphs show the behaviour of the capital-labor ratio.7 a) For n < , we observe a rapid growth, fol-lowed by a convergence towards the initial con-ditions k = 1 , . , . (b) For n > , we observe an exponentialgrowth, independent on the initial conditions k = 1 , , .Figure 1: Numerical simulations of (15) for (a) n < (b) n > . The value of α is displayed in thetitles of each figure. The other values of the parameters are β = n − α , γ = 0 . , s = 0 . , L = 1 .(a) For n < , we observe a rapid and shortdecrease, dependent on initial conditions, fol-lowed by a exponential growth starting for allinitial conditions k = 1 , , , . (b) For n > , we observe a dependence on ini-tial condition for the first part of the dynam-ics, followed by exponential growth, with initialconditions k = 1 , , , .Figure 2: Numerical simulations of (20) for (a) n < (b) n > . The value of α is displayed in thetitles of each figure. The other values of the parameters are L = 1 , L ∞ = 5 , s = 0 . , r = 0 . . . Conclusions The analysis of the Solow-Swan type models presents several stimulating mathe-matical challenges, which might be explored. In this work, we dwell on the case ofnon-constant returns to scale, providing an exact solution for the model that arisefor the Cobb-Douglas production function. A more complicated case, namely thevon Bertalanffy model, is also studied with similar results. Numerical simulationssupport the economical idea under the behaviour of the capital-labor ratio. Manyissues remain open. One of this is, for instance, the study of the model for the CESproduction function (which does not satisfy the Inada conditions) with non-constantreturns to scale or trying other, more exotic, production functions. We plan toexplore these possibilities in the near future.
Acknowledgements:
NC and MS would like to thank the University of Pavia and theUniversity of Trento, respectively, for supporting their research.This research did not receive any specific grant from funding agencies in the public, com-mercial, or not-for-profit sectors.
References [1] O. Bajo-Rubio. A further generalization of the solow growth model: the role of the publicsector.
Economics Letters , 68(1):79–84, 2000.[2] P. Barelli and S. de Abreu Pessôa. Inada conditions imply that production function must beasymptotically cobb–douglas.
Economics Letters , 81(3):361 – 363, 2003.[3] J. G. Brida and E. Limas. Closed form solutions to a generalization of the solow growth model.
Applied Mathematical Sciences , 1(40):1991–2000, 2007.[4] J. Cloern and F. Nichols. A von bertalanffy growth model with a seasonally varying coefficient.
Journal of the Fisheries Research Board of Canada , 35:1479–1482, 1978.[5] C. W. Cobb and P. H. Douglas. A theory of production.
Am. Econ. Rev. , 18(1):139–165, 1928.[6] A. Dohtani. A growth-cycle model of solow–swan type, i.
Journal of Economic Behavior &Organization , 76(2):428–444, 2010.[7] R. Farmer.
Macroeconomics [3 rd Ed.] . South-Western College Pub, 2008.[8] G.-C. Rota G. Birkhoff.
Ordinary Differential Equations [4 th Ed.] . John Wiley & Sons, 1989.[9] G. Gandolfo.
Economic Dynamics: Study Edition . Economic Dynamics. Springer, 1997.[10] L. Guerrini. The solow–swan model with a bounded population growth rate.
Journal ofMathematical Economics , 42(1):14–21, 2006.
11] E. Hairer, S. P. Nørsett, and G. Wanner.
Solving Ordinary Differential Equations I - NonstiffProblems . Springer Series in Computational Mathematics. Springer-Verlag Berlin Heidelberg,1993.[12] V. Halsmayer. From exploratory modeling to technical expertise: Solow’s growth model as amultipurpose design.
History of Political Economy , 46(Supplement 1):229–251, 2014.[13] K.-I. Inada. On a two-sector model of economic growth: Comments and a generalization.
TheReview of Economic Studies , 30(2):119–127, 1963.[14] D. Salas Jurado-Molina, J. and R. Villasenor-Talavera. Solution of the von bertalanffy s weightgrowth differential equation (1938) by two distinct methods.
Anales del Instituto del Mar yLimnologìa , 19(2), 1992.[15] D. A. Kulikov. The generalized solow model.
Journal of Physics: Conference Series ,1205:012033, 2019.[16] A. Litina and T. Palivos. Do inada conditions imply that production function must be asymp-totically cobb–douglas? a comment.
Economics Letters , 99(3):498 – 499, 2008.[17] N. Lundström. How to find simple nonlocal stability and resilience measures.
NonlinearDynamics , 93:887 – 908, 2018.[18] G. Mingari Scarpello and D. Ritelli. The solow model improved through the logistic manpowergrowth law.
Annali dell’Università di Ferrara. Sezione 7: Scienze matematiche , 49:73–83, 2003.[19] R. M. Solow. A Contribution to the Theory of Economic Growth.
The Quarterly Journal ofEconomics , 70(1):65–94, 02 1956.[20] T. W. Swan. Economic growth and capital accumulation.
Economic Record , 32(2):334–361,1956.[21] A. Takayama and T. Akira.
Mathematical Economics . Cambridge University Press, 1985.[22] H. Uzawa. On a two-sector model of economic growth ii.
The Review of Economic Studies ,30(2):105–118, 1963.[23] L. von Bertalanffy. A quantitative theory of organic growth (inquiries on growth laws ii).
Human Biology , 10(2):181–213, 1938., 10(2):181–213, 1938.