R&D for Quality Improvement and Network Externalities

Lambertini, Luca; Orsini, Raimondello Working Paper R&D for Quality Improvement and Network Externalities Quaderni - Working Paper DSE, No. 516 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Lambertini, Luca; Orsini, Raimondello (2004) : R&D for Quality Improvement and Network Externalities, Quaderni - Working Paper DSE, No. 516, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4776 This Version is available at: https://hdl.handle.net/10419/159357 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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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/3.0/ R&D for Quality Improvement and Network Externalities Luca Lambertini§and Raimondello Orsini# §,# Department of Economics, University of Bologna Strada Maggiore 45, 40125 Bologna, Italy [email protected], [email protected] fax: +39-051-2092664 §ENCORE, Faculty of Economics & Econometrics University of Amsterdam Roetersstraat 11, 1018 WB Amsterdam August 27, 2004 Abstract We investigate the bearings of network externalities on product quality improvements requiring costly R&D investments. The model considers the dynamic behaviour of a monopolist alternatively maximising profits or social welfare. On the one hand, we confirm much of the acquired wisdom from the static literature on the same topic, about the arising of quality undersupply at the private optimum. On the other, we identify the initial conditions that must be met for R&D activity to be observed under profit-seeking behaviour. We also show that the presence of network externalities affects the optimal behaviour of the profit-seeking firm but not that of a benevolent planner, who serves all consumers and smooths the R&D costs leading to a steady state quality which is independent of network concerns. JEL Classification: D62, D92, L12 Keywords: monopoly, network externality, product quality 1 Introduction The analysis of dynamic monopoly is a long standing issue, dating back to Evans (1924) and Tintner (1937), who investigated the pricing behaviour of a firm with convex costs. The analysis of intertemporal capital accumulation appeared later on (Eisner and Strotz, 1963). However, several other aspects of monopoly behaviour have never been looked upon with the tools of optimal control theory. One such aspect is the provision of product quality, which has been debated in static models to highlight the monopolist’s incentive to undersupply quality as compared to the social optimum (Spence, 1975; Mussa and Rosen, 1978; Itoh, 1983; Gabszewicz et al. 1986; Besanko, Donnenfeld and White, 1987; Champsaur and Rochet, 1989). We develop a monopoly model where the firm may invest to increase quality over time, and consumers enjoy both the utility attached to intrinsic quality and a network effect, whereby the satisfaction of a generic consumer is increasing in the number of individuals purchasing the same good or service (see Cabral, Salant and Woroch, 1999; Shy, 2000).1In a static model with the same ingredients, it is shown that the monopolist trades off quality for quantity as the network effect becomes more relevant (Lambertini and Orsini, 2001, 2003a). Here, the dynamic formulation of the problem permits to single out some additional features of such a market. There exist a parameter region where the monopolist does not find it convenient to improve product quality because the overall willingness to pay of consumers is too low. This must be contrasted with the behaviour of a benevolent social planner, who always improves quality irrespective of how affluent consumers are. As far as the 1Our model is close in spirit to a stream of literature where product quality interacts with the formation of goodwill through advertising (see Feichtinger, Hartl and Sethi, 1994). 1 extent of market coverage is concerned, we show that (i) the optimal (private) monopoly output is always increasing in the amount of externalities; yet (ii) the profit-seeking firm never covers the entire market, whatever the network effect is, while the planner serves all consumers from the outset to the steady state. The remainder of the paper is structured as follows. The basic model is in section 2. Section 3 contains the analysis of the profit-seeking monopoly equilibrium, while the comparison with the social planner’s behaviour is investigated in section 4. Concluding remarks are in section 5. 2 The setup Consider a monopoly market over an infinite (continuous) time horizon, t∈[0,∞).Consumers are indexed by their marginal willingness to pay for quality, measured by parameter θ, uniformly distributed with density 1 over [0, θ].2Accordingly, the size of the market is θ. The generic consumer at θ∈£0, θ¤buys one unit of the good iff: U(t) = θq (t) + αy (t)−p(t)≥0 (1) where p(t) and q(t) are the price and the quality of the good supplied by the monopolist at time t;αy (t) is the network externality which is assumed to be linear in market demand y(t).When inequality (1) is reversed, the consumer located at θdoes not buy and his utility is U= 0. Under partial market coverage, there will be a marginal consumer at b θ(t),who is indifferent 2Parameter θcan be thought of as the reciprocal of the marginal utility of income, so that high-income consumers are indexed by high levels of θ, and conversely for low-income consumers (see Tirole, 1988, ch. 2). 2 between buying or not and identifies the lower bound of demand: y(t)≡ θ−b θ(t).By definition, the indifference condition writes: b θ(t)q(t) + α³θ−b θ(t)´−p(t) = 0 ⇒(2) b θ(t) = αθ −p(t) α−q(t). Then, plugging b θ(t) into y(t)−θ+b θ(t) = 0 and solving w.r.t. the price, we obtain the inverse demand function: p(t) = θq (t)+(α−q(t)) y(t).(3) Quality improvement involves an R&D investment process summarised by the following differential equation: · q=bk (t)−δq (t), b > 0 (4) where k(t) is the instantaneous investment and δ∈[0,1] is a constant depreciation rate. The instantaneous cost involved by investing k(t) is C(k(t)) = c[k(t)]2.For simplicity, we normalise the marginal production cost of output to zero. Hence, instantaneous monopoly profits are: π(t)≡p(t)y(t)−c[k(t)]2(5) and, given a constant discount rate ρ, the monopolist must choose y(t) and k(t) so as to maximise: Π≡R∞ 0©p(t)y(t)−c[k(t)]2ªe−ρtdt s.t. :· q=bk (t)−δq (t).(6) If instead the firm is run by a benevolent social planner, the scale of production and the intensity of R&D efforts are chosen to maximise the discounted 3 flow of social welfare, defined as the sum of profits and consumer surplus. The latter, at any time t, corresponds to: cs (t)≡Zθ b θ U(t)dθ. (7) Therefore, the discounted stream of consumer surplus is: CS ≡Z∞ 0 cs (t)e−ρtdt. (8) Accordingly, the planner’s problem is max y(t),k(t)SW ≡Π + CS (9) under (4). 3 Monopoly optimum The Hamiltonian of the firm is: HM=e−ρt ©£θq (t) + (α−q(t)) y(t)¤y(t) −c[k(t)]2+λ(t) [bk (t)−δq (t)]ª(10) where λ(t) = µ(t)eρt, µ(t) being the co-state variable associated to quality. The initial and transversality conditions are q(0) = q0and lim t→∞ µ(t)q(t) = 0.(11) 4 The FOCs are (henceforth we omit the indication of time and discounting):3 ∂HM ∂k =−2ck +bλ = 0 (12) ∂HM ∂y =θq + 2y(α−q) = 0 (13) −∂HM ∂q = · λ−ρλ ⇒ · λ=λ(ρ+δ)−y¡θ−y¢.(14) FOC (12) yields: λ=2ck b; · k=b · λ 2c.(15) From (13), we have y∗ M=θq/ [2 (q−α)] >0 for all q > α, which entails ∂y∗ M/∂q ≤0 for all α≥0.4On this basis, we can claim: Lemma 1 The monopolist trades off quantity and quality along the equilibrium path, provided any positive network effect operates. The above Lemma illustrates what is by now a well known result in the static models on the interplay between network effects and product quality, according to which the presence of the externality, while inducing the monopolist to expand output, brings also about an otherwise undesirable reduction of the quality level (see, e.g., Lambertini and Orsini (2001, 2003a). Here, we extend this conclusion to a dynamic setting. Now we are in a position to characterise the steady state equilibrium. Using y∗ M,we may write the dynamics of the R&D investment as follows: · k=8c(ρ+δ) (α−q)2k−θ2bq (q−2α) 8c(q−α)(16) 3Throughout the paper, we also omit the analysis of second order (concavity) condition, which are always satisfied at saddle point equilibria. 4Throughout the paper, we use stars to indicate optimal controls and states along the path to the steady state, and superscript ss to identify steady state levels. 5 and imposing · k= 0,we get k∗ M=θ2bq (q−2α) 8c(ρ+δ) (α−q)2>0∀q > 2α. (17) Note that the positivity of k∗ Malso involves a requirement on the initial condition, i.e., q0>2α. Should this condition not be met, the monopolist would not start R&D activities for quality improvement. From (4), · q= 0 in q∗ M=bk/δ. Plugging it into (17), we have three steady state levels of the R&D effort: kss M1= 0,which is economically meaningless, and kss M2,3=16αcδ (δ+ρ) + b2θ2∓bθ√Ψ 16bc (δ+ρ)(18) where Ψ≡b2θ2−32αcδ (δ+ρ)≥0 (19) for all θ≥p32αcδ (δ+ρ)/b. On the basis of (4) and (16), we can write the Jacobian matrix: JM≡     ∂· q ∂q ∂· q ∂k ∂ · k ∂q ∂ · k ∂k     (20) where: ∂· q ∂q =−δ;∂· q ∂k =b(21) ∂ · k ∂q =−α2θ2b 4c(q−α)3;∂ · k ∂k =ρ+δ. (22) Hence, the trace and determinant of the Jacobian matrix JMare: T(JM) = ρ > 0 (23) ∆ (JM) = α2θ2b2 4c(q−α)3−δ(ρ+δ)<0. 6 References [1] Besanko, D., S. Donnenfeld and L. White (1987). Monopoly and quality distortion: effects and remedies. Quarterly Journal of Economics,102, 743-768. [2] Cabral, L., D. Salant and G. Woroch (1999). Monopoly pricing with network externalities. International Journal of Industrial Organization, 17, 199-214. [3] Champsaur, P. and J.-C. Rochet (1989). Multiproduct duopolists. Econometrica,57, 533-557. [4] Eisner, R., and Strotz, R.H. (1963). Deteminants of business investment. In Impacts of Monetary Policy. Research Studies Prepared for the Commission on Money and Credit, Englewood Cliffs, NJ, Prentice-Hall. [5] Evans, G.C. (1924). The dynamics of monopoly. American Mathematical Monthly,31, 75-83. [6] Feichtinger, G., R.F. Hartl and P.S. Sethi (1994). Dynamic optimal control models in advertising: recent developments. Management Science, 40, 195-226. [7] Gabszewicz, J.J., A. Shaked, J. Sutton and J.-F. Thisse (1986). Segmenting the market: the monopolist optimal product mix. Journal of Economic Theory,39, 273-289. [8] Itoh, M. (1983). Monopoly, product differentiation and economic welfare. Journal of Economic Theory,31, 88-104. 13 [9] Lambertini, L. and R. Orsini (2001). Network externalities and the overprovision of quality by a monopolist. Southern Economic Journal,67, 969-982. [10] Lambertini, L. and R. Orsini (2003a). Monopoly, quality, and network externalities. Keio Economic Studies,40(2), 1-16. [11] Lambertini, L. and R. Orsini (2003b). Network externalities in a dynamic monopoly. in R. Bars and E. Gyurkovics (eds), Proceedings of the IFAC Symposium on Control Applications of Optimization (CAO 2003). Oxford, Pergamon Press, 99-104. [12] Mussa, M. and S. Rosen (1978). Monopoly and product quality. Journal of Economic Theory,18, 301-317. [13] Rohlfs, J. (1974). A theory of interdependent demand for a communications service. Bell Journal of Economics,5, 16-37. [14] Shy, O. (2000). The Economics of Network Industries. Cambridge, Cambridge University Press. [15] Spence, A.M. (1975). Monopoly, quality and regulation. Bell Journal of Economics,6, 417-429. [16] Tintner, G. (1937). Monopoly over time. Econometrica,5, 160-170. [17] Tirole, J. (1988). The Theory of Industrial Organization. Cambridge, MA, MIT Press. 14