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Horizontal differentiation and economic growth under non-CES aggregate production function

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Abstract

We present a model of economic growth driven by horizontal innovation in which, unlike the existing literature, the final output sector employs a non-specified, non-CES, additive production function. Our motivation in conducting such analysis is based on the recognition that the use of a CES aggregate production function in the final output sector leads to the unrealistic conclusion that the gross markup of price over marginal costs set in the monopolistically-competitive intermediate sector is constant. We derive necessary and sufficient conditions for an equilibrium with perfect competition in the final output market to exist even in the presence of a non-CES technology. These conditions generalize the usual properties of the CES case. We also analyze the long-run relation between economic growth and variable markups.

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Notes

  1. See, among many others, (Barro and Sala-i-Martin (2004), Ch. 6); (Aghion and Howitt (2009), Ch. 3).

  2. See Zhelobodko et al. (2012) for a more general critique towards the CES model of monopolistic competition.

  3. It is possible to show that in Romer (1990), under the usual assumption of CES aggregate production function, the total factor productivity of the final output sector does not affect at all the markup rate of the monopolistically-competitive intermediate sector (see, for example, Barro and Sala-i-Martin (2004)).

  4. This dichotomy has recently been introduced by Zhelobodko et al. (2012). In our setting it is meant to denote the differential impact that changes in the production of a generic monopolistically-competitive, intermediate firm, induced for example by a diverse degree of competition in the intermediate sector, may have on this firm’s price and mark-up choices.

  5. Zhelobodko et al. (2012) paper analyzes in complete isolation the monopolistic competition market. So, these authors do not consider at all the dynamic, general equilibrium relations among R&D activity, final output manufacturing and intermediate-inputs production, which are instead the focus of our contribution. All in all, we believe that the relation of the Zhelobodko et al. (2012) paper to ours is the same as the relation of the Dixit-Stiglitz (1977)’s seminal contribution to Romer (1990).

  6. See (Barro and Sala-i-Martin (2004), Ch. 6).

  7. In an extended version of our model, presented in Sect. 7, the number of firms producing final output, n, and their size, h, are simultaneously endogenized.

  8. See (Aghion and Howitt (2009), Ch. 3, p. 70, Eq. 3.2).

  9. Under symmetry, \(i.e.\,\, x_j =x, j\in [0, N ]\), \(1/r_f (x)\) is the elasticity of substitution between two factor-inputs in the final output sector’s production function.

  10. One recent example of such extensions is represented by Zhelobodko et al. (2012), where the so-called relative love for variety is a function similar to our \(r_f \text{( }\cdot \text{) }\).

  11. If, additionally to (2), we assume that \(f{'''}>0\), then (3) follows from Lemma 1 when the latter is applied to function \(\left[ {-f'(\cdot )} \right] \).

  12. Under condition (3), inequality \(e_d >1\) takes place. This corresponds to the familiar property that any monopolist tends to produce along the elastic branch of her inverse demand curve.

  13. In the benchmark case, where \(f( x)=x^\alpha \) and \(0<\alpha <1\), we have: \(e_f ( x)\equiv \alpha \text{; } r_f ( x)\equiv 1-\alpha \text{; } r_{{f}'} ( x)\equiv 2-\alpha \). Thus, in the benchmark case, (3) and (13) are met. Instead, (10), (12) and (14) are satisfied as equalities.

  14. Similar pairs of strict inequalities provide necessary and sufficient conditions for strict convexity downwards (strict convexity upwards).

  15. Notice that conditions (13), (19) and (20) may straightforwardly be interpreted in the light of the properties of an intermediate firm’s revenue function, \(R( x)=px=g( h)f'( x)x=\delta f'( x)x\), where \(\delta \) is taken as an exogenous constant by any generic producer of intermediate inputs. Given this function, inequality (13) can be understood as a condition of concavity of R( x), while (19) and (20) are Inada conditions applied to R( x).

  16. See, for example, the version of the Romer (1990) model in (Aghion and Howitt (2009), Section 3.2.1, p. 72).

  17. We assume that in this economy individuals offer inelastically one unit of labor per unit of time. Hence, there is no difference between per-capita and per-worker variables. We also normalize the size of the representative household to one. This implies that there is no difference between aggregate and per-capita variables, as well.

  18. Indeed, prima facie, this conclusion is consistent with the analysis of (Benassy (1998), Eq. 17, p. 66), who was among the first to show that in ‘Romerian’-type growth models the rate of economic growth is an increasing function of the monopolistic markup, as long as the measures of market power and returns to specialization are disentangled. In Benassy’s words (Benassy (1998), p. 63), the returns to specialization measure “...the degree to which society benefits from specializing production between a larger number of intermediates”.

  19. (Aghion and Howitt (1998), p. 407) were among the first to introduce the idea that there exist conflicting specialization and complexity effects in relation to an increase in scope, N (rather than in scale, x). For a more recent analysis of these two effects within a horizontal differentiation-driven growth model, see Bucci (2013). We believe that, from the point of view of the internal organization of a firm, an increase in scope or in scale produce comparable (positive and negative) effects.

  20. Indeed, \(r_f ( x)=\text{ const }\) leads to the differential equation \(x{z}'=const\cdot z\), where \(z={f}'\). By solving this equation, we find \(z={f}'=ax^b\), where a and b are constants. Hence, by virtue of the condition \(f( 0)=0\), the function f has the form \(f( x)=kx^l\), where k and l are constants. The latter case is precisely what one may find in Romer (1990) and in Barro and Sala-i-Martin (2004), where CES functions are explicitly used.

  21. Li (2000) and Peretto and Smulders (2002) were among the first to propose theoretical solutions to the removal of scale effects in R&D-based growth models. See also Laincz and Peretto (2006) for a survey.

  22. Compare this growth rate with the one obtained by (Aghion and Howitt (2009), Ch. 3, p. 76).

  23. In versions of the Romer (1990) model with labor/human capital as an R&D input and in which the markup ratio equals the reciprocal of the factor-share for intermediates (Barro and Sala-i-Martin (2004), Eq. 6.58; Aghion and Howitt (2009), Ch. 3, p. 76), the correlation between markups and economic growth is always negative. In other versions of the Romer (1990) model in which the measure of market power is disentangled from the degree of returns to specialization (Benassy (1998), p. 66, Eq. 17), it is found that economic growth is always an increasing function of the monopolistic markup. According to Schumpeter (1942), the intuition for the latter result is straightforward: product market competition reduces the monopoly rents that reward successful innovators and thereby discourages R&D investments and long-term economic growth.

  24. An excellent review of these two branches of the literature on product market competition, incentives to innovate, and economic growth can be found in Aghion and Griffith (2005). In particular, the third chapter of this book extends the endogenous growth model from the first chapter. At the end of their analysis, Aghion and Griffith (2005) claim that, for some parameter values, the steady-state aggregate innovation intensity of their model is correlated in an inverted-U-shaped manner with the degree of competition. Funk (2007), however, demonstrates that this claim is incorrect, as innovation and growth are, instead, always falling with product market competition. According to (Funk (2007), p. 114), the mistake is probably due to the wish of Aghion and Griffith (2005) of (over-)simplifying the analysis conducted in full in Aghion et al. (2005), which is correctly able to generate an inverted-U-shaped relation between the degree of competition and the economy’s rate of growth.

  25. While Nickell (1996) and (Blundell et al. (1995, (1999)), for instance, find that competitive pressures encourage innovation and, therefore, have a positive effect on productivity growth in a long-run perspective, other papers (notably, Aghion et al. (2005)) show that the relationship between competition and growth is inverted U-shaped in the data.

  26. See also Aghion et al. (2009) and Acemoglu and Akcigit (2012) for comparable examples of Schumpeterian growth-frameworks with step-by-step innovation.

  27. Aghion et al. (2005) explain theoretically such a relation through the interplay between two opposing effects, the “escape-competition” and the “schumpeterian” effect, respectively.

  28. The possibility that competition across numerous firms might not only decrease but also increase prices is now not new to the literature. Simple economic intuitions for this phenomenon (i.e., price-increasing competition) can be found in Weyl and Fabinger (2013), Quint (2014), and Gabaix et al. (2016). For previous papers focusing on the idea that prices may rise with more intense competition, see (Gabaix et al. (2016), footnote 7, p. 6). These authors also list a number of works documenting the presence of industries (especially those producing financial products in a broad sense) where high markups, and hence high prices, do coexist with homogeneous goods and many competing firms.

  29. See Cette et al. (2013) for a recent discussion on the link between competitive pressures, innovative activity, and productivity growth.

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Acknowledgments

Vladimir Matveenko acknowledges financial support from the Russian Foundation for Basic Research (Grants 14-01-00448 and 14-06-00253). This work was initiated during a visit of Alberto Bucci at the National Research University-Higher School of Economics (Center for Market Studies and Spatial Economics, MSSE), Saint Petersburg, Russian Federation. Alberto Bucci thanks this institution for the warm hospitality. We would like to thank an Editor (prof. G. Corneo) and two anonymous reviewers of this Journal for providing constructive remarks and suggestions. Responsibility for any remaining errors and/or omissions lies exclusively with the authors.

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  1. Department of Economics, Management and Quantitative Methods (DEMM), University of Milan, Milan, Italy

    Alberto Bucci

  2. National Research University Higher School of Economics, St. Petersburg, Russia

    Vladimir Matveenko

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  1. Alberto Bucci
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Correspondence to Alberto Bucci.

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Bucci, A., Matveenko, V. Horizontal differentiation and economic growth under non-CES aggregate production function. J Econ 120, 1–29 (2017). https://doi.org/10.1007/s00712-016-0497-1

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