# On the Effect of Labour Productivity on Growth: Endogenous Fluctuations and Complex Dynamics

Authors: F. Grassetti; C. Mammana; E. Michetti
Journal: Discrete Dynamics in Nature and Society
 (2018)

---

## Abstract

This paper introduces a sigmoidal production function that considers production possible even when the only input is labour. The long-run behaviour of an economy described by the neoclassical Solow-type growth model with differential savings is investigated considering the technology presented. It is found that labour productivity influences the existence of boom and bust periods as well as the level of capital per capita in equilibrium.


---

## Body

## 1. Introduction

Economic growth models explain a country’s capital accumulation over time. One of the most used neoclassical models is the model developed by Solow and Swan (see Solow [1] and Swan [2]). The Solow-Swan model considers the growth rate of labour, the depreciation rate of capital, the saving rate, and the technology of production as determinants of the long-run behaviour of an economy. An extension of the Solow model considers two different saving propensities for workers and shareholders (see Böhm and Kaas [3]).In literature, different technologies have been studied in the Solow framework to analyse the evolution of capital per capita over time. Production technologies with Constant Elasticity of Substitution between production factors (see Brianzoni [4, 5], Masanjala and Papageorgiou [6], and Papageorgiou and Saam [7]) and Variable Elasticity of Substitution (see Brianzoni et al. [8], Cheban et al. [9], and Karagiannis et al. [10]) have been considered in order to verify which parameters influence the generation of cycles and of more complex dynamics for an economy. It has been found that nonsimple dynamics arise when the elasticity of substitution is sufficiently low and shareholders save more than workers. Moreover, attention has been paid to sigmoidal production functions (see Brianzoni et al. [11, 12]), since they are able to describe growth dynamics of nondeveloped and developing countries.In all the above-mentioned works the technology considered,f(k), satisfies f(0)=0, where k is the capital per capita. Therefore, all those works provide a production rule for which no capital can be generated without capital. Thus, they do not take into consideration a possibility of deriving from one of the assumptions of the Solow model: in his work, Solow [1] defined the capital stock as the amount of already-produced output. This implies that an initial production of capital should be possible when the only production factor is labour. Moreover, it is known that nondeveloped countries may base their early stage development only on labour (agriculture and handicraft manufacture), as Azariadis and Stachurski [13]) discussed.In this paper we present a sigmoidal technology that satisfiesf(0)>0, providing the condition for an initial amount of production possible without capital. We then investigate the Solow-type model with differential savings considering the technology introduced, in order to verify how the long-run behaviour of an economy is influenced by labour productivity (a measure of the quantity of output that can be produced from labour).The rest of the paper is organised as follows. In Section2 we describe the production function, in Section 3 we study the existence and stability of the steady states, while in Section 4 we investigate the influence of labour productivity on long-run dynamics. Section 5 concludes the paper.


## 2. Model Setup

Nonconcave production functions are used in literature to describe countries with a low level of economic development, since they present increasing returns to scale at an early stage of development and diminishing ones at a later stage (see, among all, Skiba [14] who introduced them). Capasso et al. [15] presented the sigmoidal production function mapping capital per worker into output per worker, as given by(1)fkt=αktρ1+βktρwhere α and β are nonnegative and ρ≥2 is the output elasticity of capital. Notice that an inflection point kf exists such that f(kt) is convex for kt∈(0,kf) and it is concave for k∈(kf,+∞) with kf=ρ-1/β(ρ+1)1/ρ, so it is a convex-concave production function. Such a function can be considered an extension of the Cobb–Douglas (CD) production function, since it approximates the CD case when α=1 and β=0. Notice that f(k) does not fulfil the Inada Conditions (see Barro and Sala-i Martin [16]), being limk→0f′(k)=0. The S-shape behaviour of the function allows us to consider less-developed countries, but it does not take into account those economies driven by craftsmanship, i.e., in which low levels of output are possible even without capital, that is, with f(0)>0. In order to consider these economies, we add a positive constant c to the sigmoidal production function. The resulting shifted sigmoidal (SS) production function is given by(2)fskt=αktρ1+βktρ+cand its graph is in Figure 1(a).Figure 1
(a) Production functionfs(kt) for α=.5, β=.2, ρ=3, and c=1. (b) Map F(kt) and its fixed points for n=sw=.2, δ=sr=.7, α=1, β=.5, and c=α(ρ-1)2/4βρ; one fixed point for ρ=2.5 (green map), two fixed points for ρ=2.8 (purple map), and three fixed points for ρ=3.2 (blue map).
(a) (b)Notice that when the SS production function is considered, production is possible without capital, beingfs(0)=c>0. This kind of technology has been already considered in literature when studying the evolution of capital per capita over time (see, among all, the Nobel Prize Solow [1]).We now consider the wage of a worker. When the per capita wage of a worker equals the marginal product of labour, that is,w(kt)=f(kt)-ktf′(kt), and the SS production function is considered, then the wage is given by(3)wkt=αktρβktρ-ρ+1βktρ+12+cbeing(4)fs′kt=αρktρ-11+βktρ2.In order for the wage to be nonnegative, we assume c≥α/β(ρ-1)2/4ρ=cmin, so that the minimum point of w(k) is either positive or equal to zero. This consideration establishes an economically meaningful lower bound of c for the setup, as stated in the following remark.Remark 1.
The shifted sigmoidal production function (SS) is given by(5)fskt=αktρ1+βktρ+cwhere α≥0, β≥0, ρ≥2, and c≥α/β(ρ-1)2/4ρ for the wage being nonnegative.As it can be observed in Figure1(a), the SS production function describes economies in which output is generated even without capital. Recall that the elasticity of substitution (ES) between production factors measures the ease with which capital and labour can be substituted for one another and, for twice differentiable production functions, it is mathematically defined by (see Sato and Hoffman [17]): (6)σk=-f′kfk-kf′kkfkf′′k.Being(7)fs′′kt=-αρktρ-2βρ+1ktρ-ρ+1βktρ+13and the SS production function as given by (2), the resulting ES is(8)σkt=cβktρ+12+αktρβktρ-ρ+1βρ+1ktρ-ρ+1ktρα+βc+cTherefore, this production function belongs to the class of VES (Variable Elasticity of Substitution) production functions, since σ depends on the capital per capita level. Notice that the ES may be negative. Production functions with negative ES between inputs can be found in literature (see, among all, Prywas [18], Thompson and Taylor [19], Jurgen [20], and Grassetti et al. [21]). The work of Paterson [22] suggests that such eventuality occurs when production inputs are complementary.Consider now the Solow-Swan-type growth model in which workers and shareholders have different but constant saving rates (respectively,sw∈(0,1) and sr∈(0,1)) and assume that shareholders receive the marginal product of capital π(k), while the wage rate w(k) equals the marginal product of labour. Following Böhm and Kaas [3], the map describing the capital accumulation over time t∈N is then given by(9)kt+1=Fkt=11+n1-δkt+swwkt+srπkt,kt≥0where the labour force grows at a rate n≥0 and δ∈(0,1] is the depreciation rate of capital.In order to consider less-developed economies, in which production is possible using only labour, we assume that the technology is described by the SS production function as defined in (5). Consequently, the per capita wage of workers is given in (3), while the income per worker of a shareholder is given by(10)πkt=αρktρ1+βktρ2.Taking into account (9), (3), and (10), the final continuous and smooth one-dimensional map describing the capital per capita evolution is given by(11)kt+1=Fkt=11+n1-δkt+swαktρ1+βktρ+sr-swαρktρ1+βktρ2+swc,where F(kt)≥0∀kt≥0 for all combinations of parameter values. Notice that in Brianzoni et al. [11], where a sigmoidal production function is considered, the condition sr≥sw is required in order to assure that F(kt) is nonnegative for all kt. Differently, if the SS production function (as defined in Remark 1) is considered, then the final map F(kt) has always economic meaning, thus allowing us to consider both cases sr≥sw and sr<sw and, therefore, this can be considered a generalisation of the previous work. This is possible because capital is not an essential input for production and hence even an economy where shareholders are not saving can be considered.


## 3. Equilibria and Local Dynamics

In this section we consider the question of the existence and stability of steady states for an economy described by the Solow-Swan growth model with differential savings as proposed in Böhm and Kaas [3], while considering the SS production function as defined in Remark 1.


### 3.1. Existence of Steady States

We now analyse the number of steady states mapF may own, considering different conditions on parameters values.The first difference from most of the models that consider the same initial framework (see, among all, Brianzoni et al. [12] and Klump and de La Grandville [23]) concerns the existence of a steady state characterised by zero capital per capita. While previous works established the existence of this equilibrium, when the SS production function is considered, the origin is not a fixed point, being F(0)>0. This happens when production is possible without capital (i.e., only labour is necessary), as the works of Solow [1] and Miyagiwa and Papageorgiou [24] demonstrate. Notice that this model better describes the behaviour of nondeveloped economies: even in those countries in which only labour is available (no infrastructures, machineries, or investments are given), a minimum level of production is possible, i.e., agriculture and handicraft manufacture (with regard to the early stages of economic development see Azariadis and Stachurski [13]). While Brianzoni et al. [11] proved that, when the sigmoidal production function given by (5) is considered, a poverty trap exists (the origin is a stable fixed point); this work found that, if a technology in which the only necessary input factor is labour is considered, the existence of the ’vicious circle of poverty’ (see Azariadis and Stachurski [13]) can be avoided. Notice that, even though an economy in which only labour is given will not converge to a zero growth rate, attention should be paid to the existence of multiple positive steady states: different equilibrium levels of capital per capita would explain the coexistence of nondeveloped, developing, and developed countries.To this aim we now consider the following functions:(12)Gkt=ktρ-11+βktρsw+ρΔs1+βktρand(13)Hkt=n+δα-swcαkt,with Δs=sr-sw∈(-sw,1-sw). Then the fixed points of (11) are the solutions of equation G(kt)=H(kt), kt≥0.As far as the properties of functionG(kt) are concerned, the following proposition holds.Proposition 2.
FunctionG(kt) as given by (12) is defined ∀kt≥0, it is continuous and differentiable such that G(0)=limkt→+∞G(kt)=0. Furthermore, there exist 0<km<kM such that (i)
ifρ-1/ρsw≤sr<1, then G(kt) admits a unique maximum point kM;(ii)
if0<sr<ρ-1/ρsw, then G(kt) admits a minimum point km and a maximum point kM, 0<km<kM.Proof.
The first part is trivial. As far as the number of critical points ofG(kt) is concerned, it is possible to write (14)G′kt=fktα1+M2kt2a1M2+a2M+a3where f(kt) is given by (1), M=βktρ, a1=-sw, a2=sw(ρ-2)-(ρ2+ρ)Δs, and a3=sw(ρ-1)+(ρ2+ρ)Δs.(i)
ConsiderΔs>0, then G(kt) admits a unique maximum point given by kM=[(-a2-a22-4a1a3)/(2a1β)]1/ρ as it was proved in Brianzoni et al. [11].(ii)
ConsiderΔs<0, then a1<0, a2>0 while a3 can be positive or negative. Define Z(M)=a1M2+a2M+a3 and notice that Z′(0)>0. Then
(a)
ifa3≥0, i.e., sr≥ρ-1/ρsw, the equation Z(M)=0 admits a unique positive solution; hence G(kt) admits a unique maximum point kM;(b)
ifa3<0, Z(M)=0 may admit two, one, or zero solutions, depending on the sign of Z-a2/2a1=-a22-4a1a3/4a1. Notice that Z-a2/2a1>0 iff a22>4a1a3. After some computations, it can be observed that this inequality is always verified; hence, if sr<ρ-1/ρsw, then Z(M)=0 admits two solutions. The minimum and maximum points are given, respectively, by km=[(-a2+a22-4a1a3)/(2a1β)]1/ρ and kM, where 0<km<kM.Taking into account Proposition2 and considering that H(kt) given by (13) is a hyperbola with H(kt)<n+δ/α∀kt>0, limkt→0+H(kt)=-∞, limkt→+∞H(kt)=n+δ/α>0, H′>0, and H′′<0∀kt>0, then the following proposition holds.Proposition 3.
MapF(kt) given by (11) admits at least one positive fixed point and at most three positive fixed points.Proof.
Recall thatH(kt) is concave, limkt→+∞H(kt)=n+δ/α and limkt→+∞G(kt)=0. (i)
Assume0<sr<ρ-1/ρsw. Then G(kt) is bimodal with km<kM. Then
(a)
ifG(km)>H(km) no intersection exists in (0,km). Moreover, if G(kM)≥H(kM), one intersection exists in [kM,+∞) and at most two intersections may exist in [km,kM), whereas if G(kM)<H(kM), no intersection exists in [kM,+∞) and from one up to three intersections exist in (km,kM);(b)
ifG(km)≤H(km), one intersection exists in (0,km]. Moreover, if G(kM)≥H(kM), one intersection exists in (km,kM) and one in [kM,+∞), whereas if G(kM)<H(kM), at most two intersections exist in (km,kM) and no intersection exists in [kM,+∞).(ii)
Assumeρ-1/ρsw≤sr<1. Then G(kt) is unimodal with maximum point kM. Then
(a)
ifG(kM)<H(kM), from one up to three intersections exist in (0,kM) and no intersection exists in [kM,+∞);(b)
ifG(kM)≥H(kM), one intersection exists in [kM,+∞). Moreover at most two intersections exist in (0,kM).Notice that pointsG(kM) and H(kM) determine the position of the higher equilibrium: whatever the saving propensity of shareholders, when G(kM)>H(kM), one fixed point exists in the interval [km,+∞), while no fixed points exist in that interval otherwise.Differently from previous models using this initial framework, when production is possible even without capital, the origin is not a fixed point. Moreover, up to three fixed points may exist (see Figure1(b)). Notice that the number of fixed points is in agreement with that obtained by Brianzoni et al. [8] and Grassetti et al. [21] considering different types of VES production functions, but, whereas in previous works the condition sr≥sw was required, in this case multiple equilibria may exist even if workers save more than shareholders.


### 3.2. Local Dynamics

FunctionF(kt) as given by (11) can be written in terms of function G(kt) as follows:(15)Fkt=11+n1-δkt+αktGkt+swc1+nHence(16)F′kt=11+n1-δ+αSkt,where(17)Skt=Gkt+ktG′kt=ρktρ-1sw-ρΔsβktρ+ρΔs+sw1+βktρ3.Notice that G(0)=limkt→0+ktG′(kt)=0; hence F′(0)∈(0,1), that is, ∃I+=(0,r1) s.t. F(kt) is strictly increasing ∀kt∈(0,r1). Since limkt→+∞F′(kt)=1-δ/1+n∈(0,1), then ∃I∞=(r2,+∞) s.t. F(kt) is strictly increasing ∀kt∈(r2,+∞). Hence, taking into account also the properties of function G(kt), two cases may occur: F(kt) is strictly increasing or F(kt) is bimodal; i.e., it admits a minimum point kmin and a maximum point kmax. The following proposition holds.Proposition 4.
ConsiderF(kt) as given by (11). If sr∈ρ-1/ρsw,ρ+1/ρsw, then F(kt) is strictly increasing.Proof.
Ifsw-ρΔs>0 and sw+ρΔs≥0, then F′(kt)>0∀kt>0.Proposition4 establishes a sufficient condition for the strict monotonicity of F(kt): if this condition is not fulfilled, map F can be strictly increasing or bimodal. Notice that, when the condition of Proposition 4 holds and only one equilibrium exists, the long-run behaviour of an economy can be easily understood, as the following proposition states.Proposition 5.
Assumesr∈ρ-1/ρsw,ρ+1/ρsw, and let k∗ be the unique fixed point of map F. Then k∗ is globally stable.Proof.
From Proposition4 it is known that, when sr∈ρ-1/ρsw,ρ+1/ρsw, F′(kt)>0∀kt>0. Moreover, F(0)>0 and limkt→+∞F(kt)-kt<0. Being k∗ the only fixed point of F, i.e., F(k∗)=k∗, then F(kt)>(<)k∗∀0≥kt<k∗(kt>k∗) and hence k∗ is globally stable.Recall from Proposition3 that map F may have from one up to three fixed points. When the condition of Proposition 4 holds and F has one fixed point, then it isglobally stable. Modifying the parameter values, afold bifurcation occurs (when parameters reach the bifurcation values, one nonhyperbolic fixed point is created) and a pair of fixed points emerges so map F has three fixed points k1<k2<k3 (see Figure 1(b)). Being F(kt) increasing, the fixed points are alternatively stable and unstable, with the unstable equilibrium k2 that separates the basins of attraction of k1 and k3. Notice that, when 3 fixed points exist the model simultaneously explains the behaviour of economies that, starting from different initial conditions, reach different equilibria level of capita per capita in the long-run.We now focus on the role of labour productivity on growth. Notice that the productivity of labour is the measure of the amount of output produced with one hour’s labour. Moreover, the existence of a positive relation between labour productivity and wage has been demonstrated (see Meager and Speckesser [25]). For the SS production function it has F(0)=c; therefore parameter c influences the production without capital: when the only input of production is labour, the higher the value of c, the higher the value of output for the economy. Moreover, considering the wage equation as given by (3), it follows that ∂w(kt)/∂c>0, so a positive correlation between c and w(kt) is evident. Given these considerations, c can be seen as an indirect measure of labour productivity. The following proposition highlights the role played by parameter c on the existence and position of the equilibrium level of an economy.Proposition 6.
Consider two economies described by SS technology and differing only in the constantc, both with only one equilibrium level for capital per capita. Then the one with higher production without capital (i.e., higher value of c) will experience a higher level of capital per capita in the steady states.Proof.
As the proof of Proposition3 showed, the locations of the steady states depend on the intersection between G(kt) and H(kt). Recall that G(kt) has a maximum point kM and can have a minimum point km<kM. Moreover, limkt→0+G(kt)=limkt→+∞G(kt)=0 while H(kt) is strictly increasing with limkt→0+H(kt)=-∞ and limkt→+∞H(kt)=n+δ/α. Notice that parameter c has no effect on the shape of function G(kt), while ∂H(kt)/∂c<0, so that H(kt) shifts downward as c increases. Therefore, when only one intersection between G and H exists, a higher value of c determines a higher value of the intersection point.The previous proposition demonstrates a positive correlation between labour productivity and the value of the capital stock for the higher steady state. This result is in agreement with the Organisation for Economic Cooperation and Development (OECD) data: as it can be seen in Figure2, the years of economic crisis are related to lower levels of labour productivity.Figure 2
Labour productivity, annual growth rate (%),2000-2016. Source: OECD.Notice that the condition of Proposition4 is fulfilled when sr∈I(sw,ϵ). Therefore, when the saving behaviours of the two income groups are similar, the economy does not experience boom and bust periods (since F is strictly increasing, no fluctuation can be generated). To investigate the appearance of fluctuations, we now prove that function F may be unimodal, depending on parameter values.Proposition 7.
ConsiderF and S as given, respectively, by (11) and (17). Define r=ρ[(3ρ2+1)Δs+4sw]Δs+sw2 and assume kS1=2ρ2Δs+sw+r/β(ρ+1)(ρΔs-sw)1/ρ and kS2=2ρ2Δs+sw-r/β(ρ+1)(ρΔs-sw)1/ρ.
MapF is bimodal with kmax<kmin if one of the following conditions holds: (i)
sr≥ρ+1/ρsw∧S(kS2)<δ-1/α;(ii)
sr<ρ-1/ρsw∧S(kS1)<δ-1/α.
MapF is strictly increasing otherwise.Proof.
Recall (16). The critical points of F are solutions of S(kt)=δ-1/α. Notice that S(0)=limkt→+∞S(kt)=0. Moreover, S′(kt)=ρktρ-2Q(kt)/βktρ+14, where (18)Qkt=ρ+1ρΔs-swβ2kt2ρ-22ρ2Δs+swβktρ+ρ-1ρΔs+sw,and therefore S′(kt)=0 iff Q(kt)=0 that is for kt=kS1 and kt=kS2. Assume kS=ρΔs+sw/β(ρΔs-sw)1/ρ and notice that kS1<kS<kS2. Then (i)
ifsr≥ρ+1/ρsw, function S(kt) is negative ∀kt>kS and it has one minimum point given by kS2. Therefore S(kt) intersects the negative and constant function δ-1/α iff S(kS2)<δ-1/α;(ii)
ifsr<ρ-1/ρsw, function S(kt) is negative ∀kt<kS and it has one minimum point given by kS1. Therefore S(kt) intersects the negative and constant function δ-1/α iff S(kS1)<δ-1/α.If neither conditions of Proposition7, (i) or (ii) are fulfilled, map F is strictly increasing and the same considerations derived from Proposition 4 hold: only convergence to a positive equilibrium is possible and no complex dynamics may emerge. Notice that when the system has three fixed points, a policy increasing the capital per capita would push the economy to the higher equilibrium level. On the other hand, when F is bimodal, parameter c influences the existence of a globally stable fixed point, as the following proposition states.Proposition 8.
LetF as given by (11) be bimodal with minimum point kmin. Then ∃c¯ s.t., if c>c¯, map F has one fixed point k∗>kmin and it is globally stable.Proof.
When one of the conditions of Proposition7 holds, map F is bimodal with kmax<kmin. Notice that ∂F(kt)/∂c>0, therefore ∃c¯ s.c. F(kt)>kt∀kt<kmin if c>c¯. Being F(0)>0 and limkt→+∞F(kt)-kt<0, it follows that one fixed point k∗>kmin exists and it is globally stable.As demonstrated by Proposition8, when the output productivity is sufficiently high, the convergence to the equilibrium level k∗>kmin is guaranteed. This result is in line with that obtained by Tugcu [26]: considering a binary dependent variable model, he found that a higher expenditure in secondary education, health, and R&D (i.e., policies that increase labour productivity) pushes the economy out of the middle trap (the phenomenon for which an economy gets stuck when it achieves the middle-income status) and towards higher levels of income.In order to observe fluctuations and more complex dynamics, mapF has to be nonmonotonic. Notice that several works (see, among all, Brianzoni et al. [8] and Tramontana et al. [27]) demonstrate that cycles and chaotic dynamics appear when the ES between production factors is sufficiently low. Therefore, we now investigate the role of the output elasticity of capital ρ on the shape of F.Proposition 9.
LetF be as given by (11) and assume sr≥ρ+1/ρsw or sr<ρ-1/ρsw. Then ∃ρ~ s.t. map F is bimodal ∀ρ≥ρ~.Proof.
Recall from Proposition7 that F is bimodal if sr≥ρ+1/ρsw∧S(kS2)<δ-1/α or sr<ρ-1/ρsw∧S(kS1)<δ-1/α, where δ-1/α<0. Moreover ∂S(kS1)/∂ρ<0, ∂S(kS2)/∂ρ<0, and limρ→+∞S(kS1)=limρ→+∞S(kS2)=-∞; therefore ∃ρ~ s.t. the minimum point of S(kt) is below the constant function q for all ρ≥ρ~.Proposition9 establishes a necessary condition for the arising of fluctuations, cycles, and complex dynamics, that is, a sufficiently high ρ and sufficiently different saving behaviours between workers and shareholders. In this section we showed that, when the SS production function is considered in the Solow-type growth model with different saving propensities between workers and shareholder, from one up to three equilibria exist for the capital per capita level. Differently from previous works, in which production is not possible without capital, the origin is not a fixed point: thanks to labour productivity, even a country in which only labour is available for production can experience economic growth. Notice that in neoclassical growth models the stock of capital K (from which the capital per capita k derives as a ration between capital and labour) is defined as the stock of the already-produced commodities (see Solow [1]). Given this definition, an initial condition in which the first unit of capital was produced must exist. In contrast to this requirement, previous literature considers the condition kt=0 as a fixed point, implying that if no "already-produced" goods exist, no goods can be produced. Our result overcomes the lack of previous works. Furthermore, we found that the productivity of labour positively influences the existence and stability of a high equilibrium level for the economy. In the next section, thanks to numerical simulations, we analyse the influence of the elasticity of substitution between production factors and labour productivity on the existence of fluctuations for an economy.


## 3.1. Existence of Steady States

We now analyse the number of steady states mapF may own, considering different conditions on parameters values.The first difference from most of the models that consider the same initial framework (see, among all, Brianzoni et al. [12] and Klump and de La Grandville [23]) concerns the existence of a steady state characterised by zero capital per capita. While previous works established the existence of this equilibrium, when the SS production function is considered, the origin is not a fixed point, being F(0)>0. This happens when production is possible without capital (i.e., only labour is necessary), as the works of Solow [1] and Miyagiwa and Papageorgiou [24] demonstrate. Notice that this model better describes the behaviour of nondeveloped economies: even in those countries in which only labour is available (no infrastructures, machineries, or investments are given), a minimum level of production is possible, i.e., agriculture and handicraft manufacture (with regard to the early stages of economic development see Azariadis and Stachurski [13]). While Brianzoni et al. [11] proved that, when the sigmoidal production function given by (5) is considered, a poverty trap exists (the origin is a stable fixed point); this work found that, if a technology in which the only necessary input factor is labour is considered, the existence of the ’vicious circle of poverty’ (see Azariadis and Stachurski [13]) can be avoided. Notice that, even though an economy in which only labour is given will not converge to a zero growth rate, attention should be paid to the existence of multiple positive steady states: different equilibrium levels of capital per capita would explain the coexistence of nondeveloped, developing, and developed countries.To this aim we now consider the following functions:(12)Gkt=ktρ-11+βktρsw+ρΔs1+βktρand(13)Hkt=n+δα-swcαkt,with Δs=sr-sw∈(-sw,1-sw). Then the fixed points of (11) are the solutions of equation G(kt)=H(kt), kt≥0.As far as the properties of functionG(kt) are concerned, the following proposition holds.Proposition 2.
FunctionG(kt) as given by (12) is defined ∀kt≥0, it is continuous and differentiable such that G(0)=limkt→+∞G(kt)=0. Furthermore, there exist 0<km<kM such that (i)
ifρ-1/ρsw≤sr<1, then G(kt) admits a unique maximum point kM;(ii)
if0<sr<ρ-1/ρsw, then G(kt) admits a minimum point km and a maximum point kM, 0<km<kM.Proof.
The first part is trivial. As far as the number of critical points ofG(kt) is concerned, it is possible to write (14)G′kt=fktα1+M2kt2a1M2+a2M+a3where f(kt) is given by (1), M=βktρ, a1=-sw, a2=sw(ρ-2)-(ρ2+ρ)Δs, and a3=sw(ρ-1)+(ρ2+ρ)Δs.(i)
ConsiderΔs>0, then G(kt) admits a unique maximum point given by kM=[(-a2-a22-4a1a3)/(2a1β)]1/ρ as it was proved in Brianzoni et al. [11].(ii)
ConsiderΔs<0, then a1<0, a2>0 while a3 can be positive or negative. Define Z(M)=a1M2+a2M+a3 and notice that Z′(0)>0. Then
(a)
ifa3≥0, i.e., sr≥ρ-1/ρsw, the equation Z(M)=0 admits a unique positive solution; hence G(kt) admits a unique maximum point kM;(b)
ifa3<0, Z(M)=0 may admit two, one, or zero solutions, depending on the sign of Z-a2/2a1=-a22-4a1a3/4a1. Notice that Z-a2/2a1>0 iff a22>4a1a3. After some computations, it can be observed that this inequality is always verified; hence, if sr<ρ-1/ρsw, then Z(M)=0 admits two solutions. The minimum and maximum points are given, respectively, by km=[(-a2+a22-4a1a3)/(2a1β)]1/ρ and kM, where 0<km<kM.Taking into account Proposition2 and considering that H(kt) given by (13) is a hyperbola with H(kt)<n+δ/α∀kt>0, limkt→0+H(kt)=-∞, limkt→+∞H(kt)=n+δ/α>0, H′>0, and H′′<0∀kt>0, then the following proposition holds.Proposition 3.
MapF(kt) given by (11) admits at least one positive fixed point and at most three positive fixed points.Proof.
Recall thatH(kt) is concave, limkt→+∞H(kt)=n+δ/α and limkt→+∞G(kt)=0. (i)
Assume0<sr<ρ-1/ρsw. Then G(kt) is bimodal with km<kM. Then
(a)
ifG(km)>H(km) no intersection exists in (0,km). Moreover, if G(kM)≥H(kM), one intersection exists in [kM,+∞) and at most two intersections may exist in [km,kM), whereas if G(kM)<H(kM), no intersection exists in [kM,+∞) and from one up to three intersections exist in (km,kM);(b)
ifG(km)≤H(km), one intersection exists in (0,km]. Moreover, if G(kM)≥H(kM), one intersection exists in (km,kM) and one in [kM,+∞), whereas if G(kM)<H(kM), at most two intersections exist in (km,kM) and no intersection exists in [kM,+∞).(ii)
Assumeρ-1/ρsw≤sr<1. Then G(kt) is unimodal with maximum point kM. Then
(a)
ifG(kM)<H(kM), from one up to three intersections exist in (0,kM) and no intersection exists in [kM,+∞);(b)
ifG(kM)≥H(kM), one intersection exists in [kM,+∞). Moreover at most two intersections exist in (0,kM).Notice that pointsG(kM) and H(kM) determine the position of the higher equilibrium: whatever the saving propensity of shareholders, when G(kM)>H(kM), one fixed point exists in the interval [km,+∞), while no fixed points exist in that interval otherwise.Differently from previous models using this initial framework, when production is possible even without capital, the origin is not a fixed point. Moreover, up to three fixed points may exist (see Figure1(b)). Notice that the number of fixed points is in agreement with that obtained by Brianzoni et al. [8] and Grassetti et al. [21] considering different types of VES production functions, but, whereas in previous works the condition sr≥sw was required, in this case multiple equilibria may exist even if workers save more than shareholders.


## 3.2. Local Dynamics

FunctionF(kt) as given by (11) can be written in terms of function G(kt) as follows:(15)Fkt=11+n1-δkt+αktGkt+swc1+nHence(16)F′kt=11+n1-δ+αSkt,where(17)Skt=Gkt+ktG′kt=ρktρ-1sw-ρΔsβktρ+ρΔs+sw1+βktρ3.Notice that G(0)=limkt→0+ktG′(kt)=0; hence F′(0)∈(0,1), that is, ∃I+=(0,r1) s.t. F(kt) is strictly increasing ∀kt∈(0,r1). Since limkt→+∞F′(kt)=1-δ/1+n∈(0,1), then ∃I∞=(r2,+∞) s.t. F(kt) is strictly increasing ∀kt∈(r2,+∞). Hence, taking into account also the properties of function G(kt), two cases may occur: F(kt) is strictly increasing or F(kt) is bimodal; i.e., it admits a minimum point kmin and a maximum point kmax. The following proposition holds.Proposition 4.
ConsiderF(kt) as given by (11). If sr∈ρ-1/ρsw,ρ+1/ρsw, then F(kt) is strictly increasing.Proof.
Ifsw-ρΔs>0 and sw+ρΔs≥0, then F′(kt)>0∀kt>0.Proposition4 establishes a sufficient condition for the strict monotonicity of F(kt): if this condition is not fulfilled, map F can be strictly increasing or bimodal. Notice that, when the condition of Proposition 4 holds and only one equilibrium exists, the long-run behaviour of an economy can be easily understood, as the following proposition states.Proposition 5.
Assumesr∈ρ-1/ρsw,ρ+1/ρsw, and let k∗ be the unique fixed point of map F. Then k∗ is globally stable.Proof.
From Proposition4 it is known that, when sr∈ρ-1/ρsw,ρ+1/ρsw, F′(kt)>0∀kt>0. Moreover, F(0)>0 and limkt→+∞F(kt)-kt<0. Being k∗ the only fixed point of F, i.e., F(k∗)=k∗, then F(kt)>(<)k∗∀0≥kt<k∗(kt>k∗) and hence k∗ is globally stable.Recall from Proposition3 that map F may have from one up to three fixed points. When the condition of Proposition 4 holds and F has one fixed point, then it isglobally stable. Modifying the parameter values, afold bifurcation occurs (when parameters reach the bifurcation values, one nonhyperbolic fixed point is created) and a pair of fixed points emerges so map F has three fixed points k1<k2<k3 (see Figure 1(b)). Being F(kt) increasing, the fixed points are alternatively stable and unstable, with the unstable equilibrium k2 that separates the basins of attraction of k1 and k3. Notice that, when 3 fixed points exist the model simultaneously explains the behaviour of economies that, starting from different initial conditions, reach different equilibria level of capita per capita in the long-run.We now focus on the role of labour productivity on growth. Notice that the productivity of labour is the measure of the amount of output produced with one hour’s labour. Moreover, the existence of a positive relation between labour productivity and wage has been demonstrated (see Meager and Speckesser [25]). For the SS production function it has F(0)=c; therefore parameter c influences the production without capital: when the only input of production is labour, the higher the value of c, the higher the value of output for the economy. Moreover, considering the wage equation as given by (3), it follows that ∂w(kt)/∂c>0, so a positive correlation between c and w(kt) is evident. Given these considerations, c can be seen as an indirect measure of labour productivity. The following proposition highlights the role played by parameter c on the existence and position of the equilibrium level of an economy.Proposition 6.
Consider two economies described by SS technology and differing only in the constantc, both with only one equilibrium level for capital per capita. Then the one with higher production without capital (i.e., higher value of c) will experience a higher level of capital per capita in the steady states.Proof.
As the proof of Proposition3 showed, the locations of the steady states depend on the intersection between G(kt) and H(kt). Recall that G(kt) has a maximum point kM and can have a minimum point km<kM. Moreover, limkt→0+G(kt)=limkt→+∞G(kt)=0 while H(kt) is strictly increasing with limkt→0+H(kt)=-∞ and limkt→+∞H(kt)=n+δ/α. Notice that parameter c has no effect on the shape of function G(kt), while ∂H(kt)/∂c<0, so that H(kt) shifts downward as c increases. Therefore, when only one intersection between G and H exists, a higher value of c determines a higher value of the intersection point.The previous proposition demonstrates a positive correlation between labour productivity and the value of the capital stock for the higher steady state. This result is in agreement with the Organisation for Economic Cooperation and Development (OECD) data: as it can be seen in Figure2, the years of economic crisis are related to lower levels of labour productivity.Figure 2
Labour productivity, annual growth rate (%),2000-2016. Source: OECD.Notice that the condition of Proposition4 is fulfilled when sr∈I(sw,ϵ). Therefore, when the saving behaviours of the two income groups are similar, the economy does not experience boom and bust periods (since F is strictly increasing, no fluctuation can be generated). To investigate the appearance of fluctuations, we now prove that function F may be unimodal, depending on parameter values.Proposition 7.
ConsiderF and S as given, respectively, by (11) and (17). Define r=ρ[(3ρ2+1)Δs+4sw]Δs+sw2 and assume kS1=2ρ2Δs+sw+r/β(ρ+1)(ρΔs-sw)1/ρ and kS2=2ρ2Δs+sw-r/β(ρ+1)(ρΔs-sw)1/ρ.
MapF is bimodal with kmax<kmin if one of the following conditions holds: (i)
sr≥ρ+1/ρsw∧S(kS2)<δ-1/α;(ii)
sr<ρ-1/ρsw∧S(kS1)<δ-1/α.
MapF is strictly increasing otherwise.Proof.
Recall (16). The critical points of F are solutions of S(kt)=δ-1/α. Notice that S(0)=limkt→+∞S(kt)=0. Moreover, S′(kt)=ρktρ-2Q(kt)/βktρ+14, where (18)Qkt=ρ+1ρΔs-swβ2kt2ρ-22ρ2Δs+swβktρ+ρ-1ρΔs+sw,and therefore S′(kt)=0 iff Q(kt)=0 that is for kt=kS1 and kt=kS2. Assume kS=ρΔs+sw/β(ρΔs-sw)1/ρ and notice that kS1<kS<kS2. Then (i)
ifsr≥ρ+1/ρsw, function S(kt) is negative ∀kt>kS and it has one minimum point given by kS2. Therefore S(kt) intersects the negative and constant function δ-1/α iff S(kS2)<δ-1/α;(ii)
ifsr<ρ-1/ρsw, function S(kt) is negative ∀kt<kS and it has one minimum point given by kS1. Therefore S(kt) intersects the negative and constant function δ-1/α iff S(kS1)<δ-1/α.If neither conditions of Proposition7, (i) or (ii) are fulfilled, map F is strictly increasing and the same considerations derived from Proposition 4 hold: only convergence to a positive equilibrium is possible and no complex dynamics may emerge. Notice that when the system has three fixed points, a policy increasing the capital per capita would push the economy to the higher equilibrium level. On the other hand, when F is bimodal, parameter c influences the existence of a globally stable fixed point, as the following proposition states.Proposition 8.
LetF as given by (11) be bimodal with minimum point kmin. Then ∃c¯ s.t., if c>c¯, map F has one fixed point k∗>kmin and it is globally stable.Proof.
When one of the conditions of Proposition7 holds, map F is bimodal with kmax<kmin. Notice that ∂F(kt)/∂c>0, therefore ∃c¯ s.c. F(kt)>kt∀kt<kmin if c>c¯. Being F(0)>0 and limkt→+∞F(kt)-kt<0, it follows that one fixed point k∗>kmin exists and it is globally stable.As demonstrated by Proposition8, when the output productivity is sufficiently high, the convergence to the equilibrium level k∗>kmin is guaranteed. This result is in line with that obtained by Tugcu [26]: considering a binary dependent variable model, he found that a higher expenditure in secondary education, health, and R&D (i.e., policies that increase labour productivity) pushes the economy out of the middle trap (the phenomenon for which an economy gets stuck when it achieves the middle-income status) and towards higher levels of income.In order to observe fluctuations and more complex dynamics, mapF has to be nonmonotonic. Notice that several works (see, among all, Brianzoni et al. [8] and Tramontana et al. [27]) demonstrate that cycles and chaotic dynamics appear when the ES between production factors is sufficiently low. Therefore, we now investigate the role of the output elasticity of capital ρ on the shape of F.Proposition 9.
LetF be as given by (11) and assume sr≥ρ+1/ρsw or sr<ρ-1/ρsw. Then ∃ρ~ s.t. map F is bimodal ∀ρ≥ρ~.Proof.
Recall from Proposition7 that F is bimodal if sr≥ρ+1/ρsw∧S(kS2)<δ-1/α or sr<ρ-1/ρsw∧S(kS1)<δ-1/α, where δ-1/α<0. Moreover ∂S(kS1)/∂ρ<0, ∂S(kS2)/∂ρ<0, and limρ→+∞S(kS1)=limρ→+∞S(kS2)=-∞; therefore ∃ρ~ s.t. the minimum point of S(kt) is below the constant function q for all ρ≥ρ~.Proposition9 establishes a necessary condition for the arising of fluctuations, cycles, and complex dynamics, that is, a sufficiently high ρ and sufficiently different saving behaviours between workers and shareholders. In this section we showed that, when the SS production function is considered in the Solow-type growth model with different saving propensities between workers and shareholder, from one up to three equilibria exist for the capital per capita level. Differently from previous works, in which production is not possible without capital, the origin is not a fixed point: thanks to labour productivity, even a country in which only labour is available for production can experience economic growth. Notice that in neoclassical growth models the stock of capital K (from which the capital per capita k derives as a ration between capital and labour) is defined as the stock of the already-produced commodities (see Solow [1]). Given this definition, an initial condition in which the first unit of capital was produced must exist. In contrast to this requirement, previous literature considers the condition kt=0 as a fixed point, implying that if no "already-produced" goods exist, no goods can be produced. Our result overcomes the lack of previous works. Furthermore, we found that the productivity of labour positively influences the existence and stability of a high equilibrium level for the economy. In the next section, thanks to numerical simulations, we analyse the influence of the elasticity of substitution between production factors and labour productivity on the existence of fluctuations for an economy.


## 4. Fluctuations and the Role of Labour Productivity

The aim of this section is to investigate the factors that may influence the generation of boom and bust periods for an economy. As discussed in the previous section, fluctuations and complex dynamics may arise only if the difference between saving propensities of workers and shareholders is sufficiently high orρ is high enough. Therefore, in the following we consider the case in which one of the conditions (i) or (ii) given in Proposition 7 holds.Notice that cycles and chaotic dynamics arise when a fixed pointkn∗ loses stability via flip bifurcation; i.e., F′(kn∗) passes the value -1, since a fixed point placed on the decreasing branch of map F is required. Thus, a necessary condition for fluctuations is F(kmax)>kmax∧F(kmin)<kmin. Assume kn∗∈(kmax,kmin) to be a fixed point, then, considering Proposition 3 and being F(0)>0 and F′(kt)>0∀k<kmax, it follows that up to two more fixed points may exist and, given the monotonicity of F for k<kmax and k>kmin, the number of fixed points determines their stability: if one fixed point knh∗ exists, it is nonhyperbolic, while if two fixed points ks∗ and ku∗ exist, they are, respectively, stable and unstable (in Figure 3 the cases kn∗<ku∗<ks∗ and ks∗<ku∗<kn∗ are depicted, respectively, in panels (a) and (b)).Figure 3
Parameter valuesn=1.72, δ=.46, sr=.44, α=2.65, β=1.73, and ρ=8.82. (a)  kn∗<ku∗<ks∗ for sw=.83 and c=2.95, (b)  ks∗<ku∗<kn∗ for sw=.15 and c=7.5.
(a) (b)The previous considerations are summarised in the following proposition.Proposition 10.
AssumeF bimodal with F(kmax)>kmax∨F(kmin)<kmin. Then map F has one fixed point kmax<kn∗<kmin with F′(kn∗)<0. Moreover three cases may occur: (a)
no other fixed point exists;(b)
a nonhyperbolic fixed pointknh∗ exists;(c)
two fixed pointsks∗ and ku∗ exist, with ks∗ stable and ku∗ unstable.Therefore, a multistability phenomenon may occur (see Figure4 panel (b)) and the model is able to explain why economies starting from different capital per capita levels may experience different long-run behaviours. This situation is depicted in Figure 5: starting from different initial conditions, two different attractors (in black and blue) may be reached.Figure 4
(a) Cases (a) in blue, (b) in green, and (c) in purple of Proposition10. (b) Two coexisting attractors.
(a) (b)Figure 5
Bifurcation diagrams. Parameter valuesα=.2, β=.4, δ=n=.7, sr=.8, and sw=.1, initial conditions kmin (black) and k0=.02 (blue): (a) with respect to ρ for c=10; (b) with respect to c for ρ=18.
(a) (b)Given the form ofF, an analytical study of the influence that parameters have on the loss of stability for the equilibrium kn is not possible. Therefore, we analyse this influence thanks to numerical simulations.Figure5(a) shows that the elasticity of substitution between production factors influences the generation of chaotic dynamics, as previous works demonstrated while considering economies in which capital is necessary for production. A different role is, instead, played by parameter c, which is related to labour productivity. Numerical simulations (an example is given in panel (b) of Figure 5) showed that fluctuations arise for low values of c. Moreover when the productivity of labour is sufficiently high, the attractor is a fixed point, no cycles can be generated, and the value of capital per capita in equilibrium increases if c increases, since if c is sufficiently high and F is bimodal, F(kmin)>kmin. Given these results, an economic policy that increases the productivity of labour would push an economy out of boom and bust periods and increase the level of capital per capita in equilibrium.


## 5. Conclusions

In this work we introduced a technology that considers production possible even if the only input is labour, making criticism of the production functions previously used in economic growth models. We then investigated the long-run dynamics of the Solow-type growth model with differential savings between workers and shareholders, considering the function presented. Differently from previous works, we found that the origin is not a fixed point: when in the early stage of development an economy may produce only with labour, a positive evolution of capital per capita is observed. We showed that fluctuations may arise if the difference between saving propensities is sufficiently high. Moreover, we showed that an economic policy intended to increase labour productivity would push an economy out of boom and bust periods and increase the level of capita per capita in the equilibrium.



---

*Source: 6831508-2018-10-09.xml*