Capital Accumulation and the Intensity of Market Competition: a Differential Game

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1 Capital Accumulation and the Intensity of Market Competition: a Differential Game Luca Colombo Trinity College Dublin and University of Bologna Paola Labrecciosa Trinity College Dublin and University of Bologna January 8, 2005 Abstract We adopt a differential oligopoly model to study the relationship between firms capacity investments and the intensity of market competition, captured by the number of active firms in the industry. We characterize a Markov perfect equilibrium and proceed with a comparative static analysis around the steady state (saddle path) as the number of firms increases. We determine how individual investments, industry capacity and welfare vary according to the curvature of the inverse market demand and the intertemporal parameters. JEL Classification: C73, D43, L11, O31. Key Words: investment analysis, capital accumulation, market competition, differential games. We would like to thank the audience at the II International Industrial Organization Conference, Chicago, April 04, and the audience at the II World Congress of the Game Theory Society, Marseille, July 04. Corresponding author. IIIS, Trinity College Dublin, Dublin 2, Ireland; colombol@tcd.ie; tel ; fax Dept. of Economics, University of Bologna, Strada Maggiore 45, Bologna, Italy; colombo@spbo.unibo.it; tel ; fax

2 1 Introduction The aim of this paper is to examine the equilibrium relationship between capacity investments and the intensity of market competition, captured by the number of active firms in the industry. We consider a differential oligopoly game where, at any point in time, each firm may invest in order to increase its productive capacity. We characterize a Markov perfect equilibrium and proceed with a comparative static analysis around the steady state (saddle path) as the number of firms increases, looking for the long run response of individual investment, industry capacity and welfare. At any point in time, a market for a homogeneous product exists. There is a generic number of firms supplying the good which is treated as a continuous variable. Competition takes place in quantities and the object of each firm is to maximize the discounted value of its profits stream subject to the dynamics of capital. For each firm, the strategic variable (control variable) is the level of investments in productive capacity, and the state variable is the level of individual capacity. We consider a one to one production function, meaning that one unit of physical capital is required for the production of one unit of the final good, assuming away inventories. Finally, we assume that, at any point in time, each firm has to bear an istantaneous cost for adjusting its capital stock. Our main results can be summarized as follows. The sign of the derivative of the individual equilibrium investments w.r.t. the number of firms depends on the order relationship between the curvature of the inverse market demand and the number of firms, while, as to industry capacity, whether it increases with the number of firms depends on the elasticity of the investment locus (the relationship between individual capacity and investments such that the investments level remains constant over time). Social welfare increases as industry capacity does likewise; when, instead, industry capacity decreases, the effect on social welfare is ambiguous. We show that, in Schumpeter s vein (1942), if the industry is not sufficiently capitalized, in the sense that the aggregate level of capacity has not reached a certain threshold, an increase in the number of competitors may be socially harmful. On the contrary, if the industry has reached a mature level of capitalization, an increase in the number of competitors surely improves market performance, as predicted by Arrow (1962). Therefore, with regard to the policy implications of the model, more competition is socially desirable only in those countries characterized by mature industries, while, in those countries whose industries are not sufficiently developed, the policy maker should promote investments by means of subsides or other forms of incentives, rather than trying to foster competition. The remainder of the paper is structured as follows. Section 2 relates our paper to the relevant existing literature. The dynamic setup is described in section 3. The analysis is carried out in section 4. Concluding remarks are in section 5. 2

3 2 Related literature There exists a huge operations research literature which studies capacity expansion, dating back to Manne (1961, 1967). 1 The class of capital accumulation game was originally proposed by Spence (1979). He studied an infinite horizon game with irreversible investments, highlighting the preemptive role of capital (see also Fudenberg and Tirole, 1983). The case of reversible investment was formulated by Fershtman and Muller (1984), Reynolds (1987) and Dockner (1992). Reynolds (1987), in particular, considered a two player game with a concave revenue function, a convex instantaneous investment costs and a dynamics of capital à la Solow (1956) to study the behavior of investment in the industry when capital depreciates. Reynolds (1991) extended the analysis to the N player case. As in Reynolds (1991), we analyze a differential oligopoly game with capital accumulation, assuming capital depreciates over time, and convex instantaneous costs for adjusting the capital stock (see Abel, 1979 and Hayashi, 1982). These adjustment costs can be due to intrinsic factors (i.e. costs of installation) or extrinsic factors (rising supply price, as in Foley and Sidrauski, 1970). We take the conventional approach in assuming that the investment costs do not depend on the firm s size. 2 Following the above literature, we focus on Markovian (state dependent) strategies and Markov perfect equilibrium. A Markov perfect equilibrium is such that, starting from any point in the game tree, a player (firm) selects the action that maximizes the discounted value of its profits stream, given the subsequent moves of itself and its rivals. In Markovian games, the current action of a player affects his future profits in two ways: first, each player has the ability to change the game he is playing by adjusting structural variables like production scale, and, second, he has the ability to influence the behavior of its rivals. As Dutta and Sundaram (1998) argue, Markovian games thus extend dynamic programming problems (which include only the first effect) and repeated games (which include only the second), by including both effects. Throughoutthepaper,weassumethatplayersfollowopenloopstrategies. Under this solution concept, players decide upon the path of their control variable at the beginning of the game and stick to it forever. 3 This is equivalent to say that, in general, open loop solutions are not Markov perfect (subgame perfect). However, there exists a class of games, which our game belongs to, where the open loop and the closed loop memoryless solutions coincide (see Cellini and Lambertini, 2001 and the references therein). The term memoryless refers to the fact that, in each point in time, players take their decisions by looking only at the level of the current state variable, which summarizes all the relevant history of the game, and the initial conditions. 1 An exhaustive survey is in Luss (1982). More recent research has focused on expansion under uncertainty (Paraskevopoulos et al., 1991; Bean et al., 1992; Eberly and Van Mieghem, 1997). 2 For an alternative way of modelling adjustment costs see Uzawa (1969), who uses a broad definition of capital including also human capital. 3 For the concepts of open loop and closed loop strategies see Başar and Oldser (1982, )and Dockner et al. (2000). 3

4 In our paper, we address the issue of how the steady state levels of individual investments, industry capacity and welfare vary as the intensity of market competition increases. There exists a lot of literature investigating what happens to the static Cournot Nash equilibrium when market competition becomes harder. By and large, it is possible to partition this literature into two subgroups, according to the way stability analysis is treated. On one side, some possible equilibria are disregarded in that they do not possess desired stability criteria (see Hahn, 1962; Novshek, 1980; Seade, 1980). On the other side, it is believed that it is a logical contradiction to deal with stability within static models, the reason being that ad hoc assumptions on the adjustment process are required to make the model dynamic (see Friedman, 1983). By considering an appropriate subset of the parameter space, our dynamic game is able to mimic the static Cournot equilibrium. Therefore, it can be used to study the robustness of some comparative static results we are familiar with from the static literature. Before turning to the setup, it remains to point out that there are a lot of links between the literature on capital accumulation games and the literature on dynamic advertising: instead of considering a stock of physical capital, the marketing literature considers a stock of goodwill or reputation (Nerlove and Arrow, 1962) or explicitly deals with the evolution of sales in response to advertising campaigns (Vidale and Wolfe, 1957). 4 Hence, the results provided in our model can be easily referred to the literature on differential games in marketing. The new relevant question becomes: in the long run, how do firms react in terms of advertising investments to an increase in the number of competitors? 3 The setup Time is continuous and denoted with t [0, ). Consider a set of firms P = {1, 2,..., n} and a market for a homogeneous product. The inverse market demand is given by a time invariant function p : R + R +. This function is strictly positive on some bounded interval (0,X) on which it is twice continuously differentiable and strictly decreasing. For every industry output Q (t) R + this function specifies the market clearing price p(q (t)),withp(q(t)) = 0 Q (t) X< and p (0) = p<. Every firm i P is assumed to posses the same product technology described by a time invariant one to one production function f (k i (t)) = k i (t) i P, meaning that one unit of input (physical capital) is required for the production of one unit of output (homogeneous good). Industry output is Q (t) = P n i=1 q i (t), whereq i (t) represents the quantity produced by each firm at time t. We assume that, at any point in time, each firm may invest in order to increase its productive capacity. The 4 Foranexhaustivesurveyondifferential games in marketing see Feichtinger et al. (1994). 4

5 dynamics of physical capital is described as follows: 5 k i (t) t = ki (t) =I i (t) δk i (t), i P (1) where I i (t) and k i (t) stand for firm i s investment and capital stock at time t, respectively, and δ 0 is the constant physical decay rate, common to all firms. Two cases can be distinguished (see Dockner et al., 2000): when δ =0and I i (t) is constrained to be non negative, the investment is irreversible; when δ > 0 or I i (t) is unconstrained, the investment is reversible. Since we do not impose any restriction on the set of admissible investments, in this paper investment is perfectly reversible. The constant unit input price is p I 0, then the instantaneous cost of investing I i (t) units of capital is equal to p I I i (t). We also assume that, in any point in time, each firm has to bear a convex instantaneous cost for adjusting its capital stock, exhibiting decreasing returns to scale. Formally, c : R + R +,withc(0) = 0, c p (I i (t)) > 0, c p (0) = 0 and c pp (I i (t)) > 0, i P and t. For the sake of simplicity, production entails no cost. Hence, the instantaneous firm i s profits write: π i (.) =p(q i (t)+q i (t))q i (t) p I I i (t) c(i i (t)) (2) where Q i (t) = P n 1 j6=i q j (t). To keep things as simple as possible, as in Kreps and Scheinkman (1983), we assume that each firm always operates at full capacity, i.e. q i (t) =k i (t) and Q i (t) =K i (t) i P and t. Furthermore, we assume that firm i s profits function is concave w.r.t. k i (t) to be sure that second order conditions are satisfied (see the Appendix). Competition takes place in a Cournot fashion, and the object of each firm consists in determining the investment path that maximizes the discounted value of its profits stream: Π i = Z 0 e ρt π i (t) dt (3) where π i denotes the firm i s instantaneous profits level, ρ > 0 being the rate at which future profits are discounted. We define the social welfare function we will use throughout the paper: W = π + CS (4) where π denotes industry profits and CS is the consumers surplus given by: CS = Z Q 0 p(q)dq Qp(Q) (5) 5 This formulation dates back to Solow (1956). Nerlove and Arrow (1962) apply this kinematic to a dynamic advertising model. For a differential game with advertising à la Nerlove and Arrow see, among others, Colombo and Lambertini (2003) and the references therein. 5

6 3.1 Degenerate Markov Perfect Equilibrium In the remainder of this section, in order to save space, let us omit the indication of time. Firm i s current value Hamiltonian function writes: ) Xn 1 H i = e (p(k ρt i + K i )k i p I I i c(i i )+λ ii [I i δk i ]+ λ ij [I j δk j ] (6) where λ ii = e ρt µ ii, µ ii being the co-state variable associated to k i and λ ij = e ρt µ ij, µ ij being the co-state variable associated to k j,withj 6= i. The necessary condition on the control variable is: H i =0 λ ii = p I + c p (I i ) (7) I i which can be differentiated w.r.t. time to get: j6=i λ ii = c pp Ii (8) It is worth noting that (7) does not contain any rivals capital stock, which implies that feedback effects are nil. Accordingly, the open loop solution is strongly time consistent, i.e. Markov perfect. 6 Moreover, (7) does not contain λ ij,sincewehave assumed separated dynamics. This allows us to set λ ij =0and specify only one co-state equation per firm: H i k i = rλ ii λ ii λ ii = λ ii (ρ + δ) π i k i (9) The transversality and initial condition are lim t µ ii k i =0and k i (0) = k 0 > 0. By using (9), (8) and (7), we obtain the dynamics of firm i s investment: Ii= 1 pi + c p (I c pp i ) (ρ + δ) π i (10) (I i ) k i We are interested in characterizing the steady state property, i.e. the long run equilibrium of the industry under consideration. Implicitly, the steady state, defined by {ki,ii }, is the solution to the system { Ii= 0, ki= 0}. 7 It is easy to verify that the investment locus ( Ii= 0)is monotonically decreasing in k i : I i (k i ) = 1 2 π i 1 k i c pp (I i ) 2 k i ρ + δ < 0 (11) 6 For the coincidence between open loop and closed loop solutions see Clemhout and Wan (1974), Reinganum (1982), Mehlmann and Willing (1983), Dockner et al. (1985), Fershtman (1987) and Fershtman et al. (1992). 7 By imposing p I =0and δ =0,itresultsthatki corresponds to the static Cournot solution. 6

7 since c pp (I i ) > 0 and 2 π i / 2 k i < 0. 8 We look at the dynamic properties of the system (1-10) in the space {k i,i i } by means of the following phase diagram: I i 6 ki= 0-6 -? ¾? 0 k i ¾ 6 Ii= 0 - k i Fig. 1 : Phase Diagram (12) Proposition 1 The steady state defined by {k i,i i } is stable along a saddle path. Proof. See the Appendix. From the above phase diagram, it should be clear that the steady state can be approached only from north-west or from south-east. We are now in a position to perturb the steady state by exogenously increasing the number of firms in the industry. Provided that the capital locus ( ki= 0)is monotonically increasing and independent from n and that the investment locus is monotonically decreasing and dependent from n, wecanwritethefollowing: Lemma 1 I i (k i ) / I i / k i /. Proposition 2 When the curvature of the inverse market demand is greater than the number of firms in the industry, per firm equilibrium investments increase as the number of firms increases and viceversa. Formally: I i / T 0 if γ T n. Proof. See the Appendix. 8 When k i =0, in order for an admissible steady state to exist, it has to be true that Ii I Given the assumptions made, this is always the case. (0) > 0. 7

8 In what follows, we depict the case γ >n: I i 6 ki= 0 6 Ii= k i k i - k i Fig. 2 : Comparative Statics Once understood the equilibrium relationship between per firm level of investment (so per firm level of capital) and the intensity of market competition, captured by the number of firms in the industry, we are now interested in the sign of K /. Proposition 3 Industry capacity increases if the elasticity of the investment locus w.r.t. the number of firms is lower than unity. Formally: K /dn > 0 if ε I,n < 1. Proof. See the Appendix. The intuition behind the above proposition is qualitatively the same as in the static Cournot game: the reduction in capital (quantity) by each incumbent firm has to be less than proportional w.r.t. the increase in the number of firms. It may be worthwhile to point out that, when capital depreciation rate is nil, the condition stated in Proposition 3 becomes K /dn > 0 if γ < 1+n and otherwise, mimicking the static Cournot case (Svizzero, 1997). The proof is omitted for brevity. Now, we examine the welfare implications of strengthening market competition. Proposition 4 (i) Consider the case in which capital depreciates over time. If K /dn > 0 then W /dn > 0, otherwisetheeffectofanincreaseinthenumber of firms on social welfare is ambiguous. (ii) When, instead, capital depreciation rate is nil, dw /dn dk /dn. Proof. See the Appendix. From the definition of γ: γ K = γ (1 + γ) K 8 (13)

9 implying that the curvature of the inverse market demand is monotonically increasing in industry capacity. As a consequence, referring to Proposition 2, γ >nif K> γ 1 (n) and otherwise. Hence, when the industry has reached a mature level of capitalization, given by γ 1 (n), an increase in the number of competitors surely improves market performance, as predicted by Arrow (1962). On the contrary, in Schumpeter s vein (1942), when the industry is not sufficiently capitalized, in the sense that the aggregate level of capacity has not reached the above threshold, an increase in the number of competitors, leading to a decrease in per firm level of investments, may be socially harmful. 4 Concluding remarks We have analyzed a dynamic oligopoly model to study how an increase in the number of active firms in the industry affects the steady state (saddle path) level of firm s capacity investments, industry capacity and social welfare. We have shown that the derivative of per firm investments w.r.t. the number of firms has the same sign as the difference between the curvature of the inverse market demand, γ, andthe number of firms, n. Thisimpliesthat, withasufficiently convex market demand, the equilibrium relationship between capacity investments and the intensity of market competition is positive, otherwise it is negative. The sign of the derivative of the industry capacity w.r.t. n depends on the elasticity of the investment locus w.r.t. n. Exactly as in the static Cournot game, in order for industry capacity to increase, the incumbent firms have to decrease their investments less than proportionally w.r.t. the increase in the number of firms. Social welfare moves along with industry capacity, in case of nil depreciation rate. When, instead, capital depreciates over time, social welfare increases if industry capacity does likewise, otherwise the effect is ambiguous. As to the policy implications of the model, more competition is socially desirable only in those countries characterized by mature industries, while, in those countries whose industries are not sufficiently developed, in the sense that a certain threshold level of industry capacity has not been reached, the policy maker should promote investments by means of subsides or other forms of incentives, rather than trying to foster competition. 9

10 Appendix ProofofProposition1.We consider the system: ki (t) =I i (t) δk i (t) Ii= 1 ½ pi c pp i (I + c p (I i ) (ρ + δ) π ¾ i i) k i We form the Jacobian matrix: Θ = ki k i Ii k i ki I i I i Ii which possesses the following determinant: = δ 1 2 π i 1 2 k i c pp (I i ) ρ + δ (A1) (A2) (A3) Θ = δ (ρ + δ)+ 2 π i 2 k i 1 c pp (I i ) < 0 (A4) Since 2 π i < 0 and c pp (I 2 i ) > 0, the determinant is always negative. Therefore, we k i have saddle path stability. ProofofProposition2.From (10), the investment locus writes: I i (k i )= c p µ 1 1 π i p I ρ + δ k i (A5) By differentiating (A5) w.r.t. n: I i (k i ) = 1 2 π i 1 c pp (I i ) k i r + δ II i (k i ) 2 π i k i (A6) where, by using symmetry: implying that: 2 π i k i = k i p pp k i + p p (A7) 2 π i > 0 if γ >n k i dn (A8) where γ =( p pp Q) /p p is the curvature of the inverse market demand. From Lemma 1: I i (k i ) I i k i T 0 if γ T n 10 (A9)

11 Proof of Proposition 3. We aim to study the dynamic properties of the industry at stake in the space {AI, K}, where AI = ni is the aggregate investment level. From (1) and (10), by multiplying each side for n and by using symmetry: AI= n c pp (I i ) K= AI δk (A10) pi + c p (I i ) (ρ + δ) π i k i (A11) The locus K= 0is an upward sloping straight-line starting from the origin, while the locus AI= 0writes: AI= 0 AI (K) =ni i (k i ) (A12) In order to understand the shape of the locus AI= 0we take the following derivative: AI (K) K = n µ µ 1 1 p pp K + p p + p p (A13) c pp (I i ) ρ + δ n which implies that: AI (K) K ppp K + p p + np p (A14) After rearranging: AI (K) > 0 if γ >n+1 (A15) K Now, we study how the locus AI= 0reacts to an increase in n: AI (K) = I i (k i )+n I i (k i ) By using the definition of elasticity, we can write: (A16) AI (K) > 0 if ε I,n < 1 (A17) where ε I,n is the elasticity of the investment locus w.r.t. n. ProofofProposition4. From (3), we first compute the aggregate steady state profits: Π = p (K ) K nc(δ K n ) (A18) which can be differentiated w.r.t. n: Π = p p K + p δc p K c + δcp ki (A19) 11

12 Notice that the term between brackets is always negative. To check this, it suffices to divide each term of the equation for p and use the definition ε (and the fact that ε > 1). Since we have assumed that the average cost for adjusting the capital stock is increasing, then δc p (I i ) k i >c(i i ). This amounts to saying that when K / <0, then Π / >0, otherwise when K / >0 the overall effect on industry profits is ambiguous. As to consumers surplus, from (5) we have: CS = K K p p CS K (A20) implying that: W = K p δc p c + δc p ki (A21) The term between brackets is clearly always positive, in that p>c p and δ (0, 1). We can conclude that, when K / >0 then W / >0: consumers are better off while the effect on firms profits is ambiguous, with the effect on consumers surplus prevailing. When, instead, K / <0, theeffect on social welfare is ambiguous. It is immediate to verify that when δ =0, c =0,so W / K /. Second order conditions. We apply Mangasarian sufficiency theorem and obtain the following matrix: p pp k i +2p p 0 0 c pp (A22) (I i ) Giventheassumptions ontheconcavity ofprofits and the convexity of the adjustment costs, the determinant of the above matrix is positive, implying that second order conditions are fulfilled. 12

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