Intra-sector and inter-sector competition in a model of growth

Di Cintio, Marco; Grassi, Emanuele Working Paper Intra-sector and inter-sector competition in a model of growth Economics Discussion Papers, No. 2015-49 Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Di Cintio, Marco; Grassi, Emanuele (2015) : Intra-sector and inter-sector competition in a model of growth, Economics Discussion Papers, No. 2015-49, Kiel Institute for the World Economy (IfW), Kiel This Version is available at: https://hdl.handle.net/10419/111913 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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Licensed under the Creative Commons License - Attribution 3.0 Discussion Paper No. 2015-49 | July 10, 2015 | http://www.economics-ejournal.org/economics/discussionpapers/2015-49 Intra-Sector and Inter-Sector Competition in a Model of Growth Marco Di Cintio and Emanuele Grassi Abstract The role of patents is threefold: first, they are important to state the property rights of an invention; second, they are necessary to secure financing for starting a new venture; third, they are fundamental to recoup R&D investments. The main difficulty in preventing unauthorized use of an innovation is in the establishment of ranges and contexts of patents applicability. Noting the imperfections of the patent legal system, the authors are in a position to consider an economy with two levels of competition under different market structures: the inter-sector monopolistic competition and the intra-sector Cournot oligopoly. The explicit consideration of strategic interactions in a model of endogenous growth produces interesting results. Considering the sectorial market share as the indicator of patent system enforcement, the authors find that growth takes place, if and only if, there are some property rights of private knowledge produced by R&D activities. In turn, the patent system translates into a low degree of competition among firms. Its influence on the growth rate goes in a single unambiguous direction. As competition rises, few resources are available for R&D, so the growth rate goes down. JEL E10 L13 L16 O31 O40 Keywords Product differentiation; endogenous growth; market structure; R&D Authors Marco Di Cintio, Department of Management, Economics, Mathematics and Statistics, University of Salento – via per Monteroni – 73100 – Lecce, Italy, [email protected] Emanuele Grassi, Department of Management, Economics, Mathematics and Statistics, University of Salento, Lecce, Italy Citation Marco Di Cintio and Emanuele Grassi (2015). Intra-Sector and Inter-Sector Competition in a Model of Growth. Economics Discussion Papers, No 2015-49, Kiel Institute for the World Economy. http://www.economicsejournal.org/economics/discussionpapers/2015-49 conomics Discussion Paper Introduction The aim of this paper is to investigate the relationship between the growth rate and the intensity of market competition when monopolistic and oligopolistic competition coexist in a model with an expanding variety of products. The inter-sector monopolistic competition is more or less intense on the basis of the degree of substitutability among differentiated goods, while the degree of intra-sector competition depends on the number of active firms in each sector. Remarkable contributions on the endogenous growth theory are focused either on oligopoly or monopolistic competition. On the one hand, Romer (1990), Grossman-Helpman (1991) and Aghion-Howitt (1992) propose different approaches based on monopolistic competition to generate an endogenous process of knowledge acquisition, where they rely on the assumption that a large number of firms results in a negligible effect of individual choices on the aggregate price index. On the other hand, the difficulty of defining a balanced growth rate under differentiated oligopoly limits the scope of the literature under this market structure. However, the frequent adoption of the Dixit-Stiglitz (1977) aggregation method in models of growth (under monopolistic competition) may be well explained through its many attractive proprieties. First, the CES formulation of the utility function implies fair properties of the aggregate demand functions, i.e. a tractable analytical form. Second, a single (constant) parameter characterizes the degree of product differentiation (which is itself related to the “love for variety”, the degree of substitutability and the market power), facilitating the analysis between the market power of firms and the growth rate. The last property is the symmetry between old and new varieties, which removes product obsolescence and, as a consequence, excludes improvements in quality. However, many economists have abandoned the hypothesis of monopolistic competition in order to introduce oligopolistic markets and to study the effects of strategic interaction on the growth rate. Remarkable contributions are those by Vencatachellum (1998), Peretto (1999) and Cellini (2000). Anyway also in the presence of strategic interaction, many papers usually rely on the assumption that a large number of firms results in a negligible effect (of individual choices) on the aggregate price index, even though this is acceptable only in a world of monopolistic competition1. The literature typically conceives the two market structures as separate or unconnected and, sometimes, the distinction between oligopoly with differentiated goods and monopolistic competition is also unclear. Often, the two terms are used with a vague sense of imperfect competition: while the oligopoly describes few firms competing with or without free entry, the monopolistic competition refers to numerous firms and free entry 2 . By contrast, we study a framework where monopolistic and oligopolistic competition coexist at different levels. In particular, 1See Yang-Heijdra (1993) and D’Aspremont et al. (1996). 2 Following as example, Hart (1985) or Wolinsky (1986), the four standard properties of monopolistic competition are: (1) there are many firms producing differentiated commodities; (2) each firm is www.economics-ejournal.org 2 conomics Discussion Paper our aim is twofold: on the one hand, we propose a different approach where two market structures simultaneously coexist in a growth model; on the other hand, we study the influence of the degree of competition on the growth rate when strategic interaction really plays a role. Our model is based on three simple ingredients. The first is related to the two dimensions of competition: the inter-sector monopolistic competition between differentiated products, and the Cournot oligopoly at the intra-sector level. The second is the traditional R&D technology à la Grossman-Helpman. The third is the assumption that the R&D output is of public domain. Because of the imperfections in the patent system, property rights may be difficult to define, so inventors are unable to exclude others from freely using their innovative ideas. The model explains clearly the relationship between the degree of market competition and the endogenous growth path. Sustained innovations are possible if, and only if, some intellectual property rights prevent the free use of an invention; otherwise, the market tends to be highly competitive. In this case, few resources are available for R&D activity and the growth rate falls. By contrast when no firm has direct competitors, the state of knowledge moves forward because the private incentives for further research are maintained. The discussion is organized as follows. The description of preferences is presented in section 1, while in section 2 we analyze the production side. Sections 3 and 4 describe the structure of R&D activities and the dynamic equilibrium. The last section concludes. 1 Preferences Consider an economy with ¯ L identical households and differentiated goods produced in Nm varieties, [xi]Nm i=1 . Preferences are identical for all consumers. Households maximize the lifetime utility: U(t0) = Z∞ t0 e−ρ(t−t0)lnu(t)dt (1) subject to the intertemporal budget constraint, such that the present discounted value of expenditure cannot be greater than the present discounted value of lifetime labour income, plus initial wealth: Z∞ t0 R(t)Y(t)dt ≤A(t0)+Z∞ t0 R(t)w(t)dt (2) where ρ>0 is the individual discount rate, R(t) = e−Rt t0r(s)ds is the cumulative discount factor, Y is nominal per capita expenditure, and A is the initial wealth. The household takes the path of wages and the interest rate as given. Throughout the analysis, the wage is the numéraire. negligible; (3) free entry results in zero-profit of active firms; (4) the equilibrium price exceeds the marginal cost. www.economics-ejournal.org 3 conomics Discussion Paper We assume that there is a large number of varieties, all of which enter symmetrically into the instantaneous utility function u(t) , which we assume to be of the Dixit-Stiglitz type3: u= Nm ∑ i=1 xβ i!1 β (3) where xi is the consumption of each variety and 0<β<1 . As it is well known, this specification has proved to be the most tractable when product differentiation is the main concern 4 . Over time, innovation can expand this subset, and Nm(t) is the number of varieties at time t . This utility function implies constant elasticity of substitution between any couple of varieties: σ=1 1−β>1 (4) The solution of this problem can be derived in two stages. From the Euler equation, we first obtain the optimal dynamic expenditure path: ˙ Y Y=r−ρ(5) which also defines optimal saving behavior. Then, by taking the time-path of expenditure as given, we solve the static household maximization problem for any t,i.e. the maximization of usubject to Y= Nm ∑ i=1 pixi. The h - th household’s demand function for the i - th variety (where i∈[1,Nm] ) is xh i(pi) = Y qpi q−σ (6) where piis the price of the i-th brand, and qis the ’dual’ price index: q="Nm ∑ i=1 p1−σ i#1 1−σ (7) Aggregating over ¯ L identical consumers, we obtain the demand schedule faced by firms producing the i-th brand: xi(pi) = ¯ LY qpi q−σ (8) 3In the rest of the paper the time variable, t, is suppressed. 4 The love for variety could alternatively be modeled in a slightly different framework, by extending preferences over a continuous product space and assuming that at any given moment in time only a subset of potential varieties are available (Grossman and Helpman, 1989; Krugman, 1980). www.economics-ejournal.org 4 conomics Discussion Paper Equation (8) is used in the analysis of a firm’s price-setting behavior. Since we are interested in quantity competition between firms, we consider the corresponding inverse demand function, along the lines suggested by Spence (1976): pi(xi) = ¯ LY xβ−1 i Qβ(9) where pi is the price of the i - th variety, xi is the aggregate production of the i - th sector, and Qis the industry quantity index given by: Q="Nm ∑ i=1 xβ i#1 β (10) Notice the immediate interpretation of βin terms of both market structure and preferences. As β→0 , the degree of substitution between any couple of varieties reaches the minimum level (i.e. σ→1 ) and varieties of different sectors become highly differentiated. As β→1 , we obtain a set-up with an homogeneous product, the degree of substitutability becomes infinite (i.e. σ−→ ∞ ) and each brand is perfectly substitutable with the others of the remaining Nm−1 sectors. Clearly, the demand function given in (8) or (9) encompasses both traditional formulations of oligopoly with a homogeneous good, and the standard monopolistic competition. 2 Technology On the production side, firms undertake two activities. First, they produce the existing varieties; second, they can divert resources to investment in R&D in order to create new designs. While it is generally assumed that each variety is produced by a single firm, in what follows we will assume that each variety will be manufactured by N competing firms. This assumption can be justified in different ways. The innovative brand may not be patentable because its inventor has difficulties to prevent unauthorized use of its ideas. Alternatively, one may think at this kind of innovation as a new combination of existing knowledge. In the latter interpretation, the new product may indeed look new to consumers, but, being not really original, it is not patentable. Another way to justify our assumption is that, especially in the case of trade openness, similar varieties could exhibit many overlapping characteristics 5 , and a (nearly) identical brand is produced by many firms. Finally, we recall that Grossman and Helpman (1991) exclude any incentive to imitation on the basis that an intra-sector price competition would immediately lead profits to zero, so that the copier would not be able to recoup the positive cost of imitation. Their argument is clearly based on the idea that firms compete under a Bertrand fashion. But if we imagine Cournot competition, the scope for imitation may indeed arise. If the intra-sector competition 5The case of the automobile sector provides clear examples in this respect. www.economics-ejournal.org 5 conomics Discussion Paper is consistent with a positive mark-up over marginal costs, the imitation costs can be covered and firms could find it profitable to produce the same (homogeneous) good. Since there are Nm varieties, each of them produced by N firms, each firm simultaneously faces two different competitive environments. Horizontally, at the inter-sector level each firm competes with other firms producing an imperfect substitute of its own product. Also, it competes with other firms producing a homogeneous product at the intra-sector level 6 . Therefore, there is an inter-sector competition (i.e. between different varieties) of the standard monopolistic type, and an intra-sector competition (within the same variety). As suggested above, we assume that the latter is in quantities, so that the market for each variety can be thought as a traditional Cournot oligopoly. The j - th firm ( j∈[1,N] ) operating in the i - th sector, is mono-product. Each good can be produced through labour according to the linear technology: zi j(Li j) = Li j (11) where Li j is the amount of labour employed in the i - th sector by the j - th firm, and zi j is the firm’s output. Hence, for the j - th firm, the cost function is C(zi j) = zi j (remember that the wage, w , is the numéraire). Obviously, the aggregate production for the i-th sector is: xi= N ∑ j=1 zi j!(12) Therefore, the number of workers employed in the i-th sector is given by: Li= N ∑ j=1 zi j!=xi(13) while the total amount of workers employed in production is: LX="Nm ∑ i=1 N ∑ j=1 Li j!# (14) Each firm chooses the level of production in order to maximize profits: πi j =pi(xi)zi j −zi j (15) Notice that, given the large number of existing varieties, each firm perceives the industry quantity index as given. In turn, this implies that the negligibility assumption holds: each firm considers the change in its own level of production, zi j , as irrelevant with respect to the industry aggregate production index, Q. Therefore it is the 6 Notice that also in Grossman and Helpman (1991) there is a schematic discussion of possible forms of intra-sector competition. In particular they suggest that the research labs could be involved in quality improvements of existing varieties, so that intra-sector competition may turn to vertical product differentiation. www.economics-ejournal.org 6 conomics Discussion Paper negligibility assumption that allows for the inter-sector monopolistic competition. On the contrary at the intra-sector perspective, competition is à la Cournot. Substituting (9) into (15), and using (12), we can rewrite profits in terms of individual quantity: πi j =¯ LY "N ∑ j=1 zi j#β−1 zi j Qβ−zi j (16) The first order condition under Cournot conjectures for any given level of zhk , h6=i and k6=j, is ∂πi j ∂zi j =0⇐⇒ ¯ LY Qβ (β−1) N ∑ j=1 zi j!β−2 zi j + N ∑ j=1 zi j!β−1 −1=0 (17) Under symmetry zi j =z∀i,j, the Nash equilibrium is: z∗=¯ LY (β−1+N) N2Nm .(1) From (12), the aggregate production of each sector is: x∗=¯ LY (β−1+N) NNm ,(2) and the related market price is given by (9) p∗=N β−1+N.(3) The resulting level of profits at the equilibrium is: π∗=¯ LY 1−β N2Nm (21) Notice that the optimal quantity produced by any firm is inversely proportional to the number of existing varieties, Nm . The same holds for profits, while the price level is independent of Nm . Notice, also, the influence of the degree of substitutability. For a low level of β , inter-sector competition is less fierce because of the low interdependence among sectors. Table 1 summarizes the equilibrium outcome under these two extreme configurations of the inter-sector competition. At the intra-sector level, a simple indicator of the degree of competition is given by the number of active firms in the sector. In this respect, on the one hand a large number (i.e. N−→ ∞ ) means that no limits to imitation exist; on the other hand, this implies a negligible market share for each firm of the sector (i.e. z x=1 N=s−→ 0 ). In this case, the intra-sector competition resembles perfect competition: prices equal marginal costs and profits are driven to zero. On the contrary, when a strict patent www.economics-ejournal.org 7 conomics Discussion Paper Table 1: Equilibrium outcomes Homogeneous product: β→1 inter-sector perfect competition p x π marginal cost ¯ LY Nm0 Table 2: Equilibrium outcomes Free entry: s→0 intra-sector perfect competition p x π marginal cost ¯ LY 1 Nm0 system prevents imitation and unauthorized entry into the sector, only a single firm supplies the entire sector (s=1), i.e. this firm behaves like a monopolist7. Table 2 summarizes the extreme configurations of intra-sector competition. It must be stressed that the market share s can be interpreted in two different ways. It is an index of the degree of competition of market structure, but it can also be seen as an indicator of the degree of enforcement of patent law. In this respect, the extreme values, s→0 and s=1 , arise under perfect competition (absence of patents) and monopoly power (perfect patents), respectively. For intermediate values of s , we have some degree of strategic interaction: the higher the value of s , the lower the degree of competition and the higher the level of patentability. Finally, we recall that if the intra-sector competition were à la Bertrand, we would have the competitive price (because of homogeneity), independently from the properties of the inter-sector competition. 3 Research & Development Following Grossman and Helpman (1991) and Lucas (1988), we assume that the production of new varieties takes place according to the innovation function: ∂Nm ∂t=˙ Nm=1 aLRk(t)(22) where a is a positive parameter, LR is the number of workers employed in R&D and k(t) is the stock of knowledge at time t . Equation (22) is the most common formulation of R&D technology in the endogenous growth literature: it shows a positive relationship between the development of new varieties and the stock of available knowledge at each moment in time. Since the number of varieties changes 7 This latter situation collapses to that described by the Grossman and Helpman model, where the intra-sector competition is absent. www.economics-ejournal.org 8 Please note: You are most sincerely encouraged to participate in the open assessment of this discussion paper. You can do so by either recommending the paper or by posting your comments. Please go to: http://www.economics-ejournal.org/economics/discussionpapers/2015-49 The Editor © Author(s) 2015. Licensed under the Creative Commons Attribution 3.0.