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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.

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 [

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 [

In all the above-mentioned works the technology considered,

In this paper we present a sigmoidal technology that satisfies

The rest of the paper is organised as follows. In Section

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 [

(a) Production function

Notice that when the SS production function is considered, production is possible without capital, being

We now consider the wage of a worker. When the per capita wage of a worker equals the marginal product of labour, that is,

The shifted sigmoidal production function (SS) is given by

As it can be observed in Figure

Consider now the Solow-Swan-type growth model in which workers and shareholders have different but constant saving rates (respectively,

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 (

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 [

We now analyse the number of steady states map

The first difference from most of the models that consider the same initial framework (see, among all, Brianzoni et al. [

To this aim we now consider the following functions:

As far as the properties of function

Function

if

if

The first part is trivial. As far as the number of critical points of

Consider

Consider

if

if

Taking into account Proposition

Map

Recall that

Assume

if

if

Assume

if

if

Notice that points

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 Figure

Function

Consider

If

Proposition

Assume

From Proposition

Recall from Proposition

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 [

Consider two economies described by SS technology and differing only in the constant

As the proof of Proposition

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 Figure

Labour productivity, annual growth rate (%),

Notice that the condition of Proposition

Consider

Map

Map

Recall (

if

if

If neither conditions of Proposition

Let

When one of the conditions of Proposition

As demonstrated by Proposition

In order to observe fluctuations and more complex dynamics, map

Let

Recall from Proposition

Proposition

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 or

Notice that cycles and chaotic dynamics arise when a fixed point

Parameter values

The previous considerations are summarised in the following proposition.

Assume

no other fixed point exists;

a nonhyperbolic fixed point

two fixed points

Therefore, a multistability phenomenon may occur (see Figure

(a) Cases (a) in blue, (b) in green, and (c) in purple of Proposition

Bifurcation diagrams. Parameter values

Given the form of

Figure

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.

The data used to support the findings of this study are included within the article.

This paper did not receive specific funding and was performed as part of the employment of the authors to University of Macerata.

The authors declare that there are no conflicts of interest regarding the publication of this paper.