Combining semi-endogenous and fully endogenous growth: A generalization ^☆

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JEL classification

• O30;
• O40

Keywords

• Strong scale effect;
• Semi-endogenous growth;
• Fully endogenous growth

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1. Introduction

In a recent paper, Cozzi (2017) combined the semi-endogenous growth model by Jones (1995), Kortum (1997), and Segerstrom (1998) with the fully endogenous growth models without scale effect by Smulders and van de Klundert (1995), Dinopoulos and Thompson (1998), Peretto (1998), Young (1998), and Howitt (1999). Each of these schools of thought captures an important element of the growth process. The semi-endogenous approach, while inheriting from Romer (1990) the idea that the aggregate flow of innovations increases in the size of the R&D employment, stresses the increase in the difficulty of generating a constant TFP growth¹. The scale free fully endogenous school of thought stresses that the number of innovations per sector increases with the fraction of the labour force which each sector employs in R&D, without assuming an increasing difficulty of R&D². They have different long-term predictions, though similar transitional predictions.

If both growth mechanisms have some empirical confirmation, Cozzi (2017) claims that they may both be present in the growth process, which would result from a weighted average of the semi-endogenous and the fully endogenous growth engines. Unlike one could expect, though, regardless of the weight of each growth engine in the aggregate growth process, Cozzi (2017) proves that only one would dictate the balanced growth path prediction: with high enough population growth rates the semi-endogenous solution will prevail in the long run, while with slowly increasing, constant, or shrinking population the fully endogenous solution will eventually dominate in the long run.

These results are striking, but they may depend on the simple averaging adopted by Cozzi (2017), while most likely the overall growth process could combine the two growth mechanisms in much more general ways. A natural way to generalize the process is by assuming a general constant elasticity of substitution (CES) aggregator of the two growth mechanisms. This generalization is meaningful because, as I will show in this paper, depending on whether the two growth mechanisms are gross complements or gross substitutes in the overall growth process opposite conclusions are obtained: if they are gross complement the semi-endogenous steady state growth rate prevails at all values of the population growth rate; if they are gross substitutes, the long run implications of the fully endogenous growth mechanism will dominate the long run provided population growth does not exceed an endogenous threshold level.³

This paper also differs from Cozzi (2017) also in minor aspects: it allows for heterogeneous innovation technology parameters; and it sets the model in discrete time, potentially useful for embedding the model in a business cycle framework.⁴

The paper is organized as follows: Section 2 extends Cozzi’s (2017) main result and characterize long-run growth in terms of growth engines complementarity and substitutability. The final section concludes.

2. Growth mechanics

As in Cozzi (2017), let us assume the following aggregate production function:

where Y_t$Y_t$ is output at time t$t$, A_t$A_t$ is total factor productivity, and L_Yt$L_{Yt}$ is labor employed in manufacturing. In a balanced growth path L_Yt$L_{Yt}$ is a constant fraction 0<s_Y<1$0<s_Y<1$ of the total labour force L_t$L_t$ , which in turn grows at the - possibly negative - constant net rate n$n$.