Export Restraints in a Model of Trade with Capital Accumulation

Transcription

1 Export Restraints in a Model of Trade with Capital Accumulation Giacomo Calzolari Luca Lambertini December 30, 2003 Abstract This paper examines the impact of voluntary export restraints (VERs) in an international duopoly modelled as a differential game. With a Ramsey capital accumulation dynamics, the game admits multiple steady states, and a VER cannot be voluntarily employed by the foreign firm in case of Cournot behaviour in demand substitutes. Hence, the dynamic framework confirms the results of the VERs literature with static interaction in output levels. In the case of price behaviour, the adoption of an export restraint may increase the profits of both firms if products are substitutes and the steady state is market-driven. However, contrary to the acquired wisdom based upon the static approach, the dynamic analysis also admits an equilibrium outcome, identified by the Ramsey golden rule, where the incentive to adopt a VER is ruled out, irrespective of whether firms are quantity- or price-setters. This is confirmed by our analysis with the alternative capital accumulation scheme due to Solow, Nerlove and Arrow. JEL Classification: D43, D92, F12, F13, L13 Keywords: intra-industry trade, trade policy, differential games, capital accumulation We thank Ennio Cavazzuti, Roberto Cellini and the audience at the European University Institute, EARIE 2001 (Trinity College, Dublin) and ISDG 2002 (University of StPetersburg) for helpful suggestions. The usual disclaimer applies. Department of Economics, University of Bologna, Piazza Scaravilli 2, Bologna, Italy. calzolari@economia.unibo.it Department of Economics, University of Bologna, Strada Maggiore 45, Bologna, Italy, fax: , lamberti@spbo.unibo.it. 1

2 1 Introduction Strategic trade policy literature is, with few exceptions, essentially based on a static framework (see Brander 1995). Nevertheless, long-term interactions characterizing international oligopolistic markets are at odds with the one-shot static games generally employed. 1 Voluntary export restraints (VERs) are often considered as coordinating or quasi-collusive devices (see, e.g., Berry, Levinsohn and Pakes, 1999). Indeed, most of the existing theoretical literature justifies this view only insofar as firms are price setters. Harris (1985) first analyzed VERs in a static duopoly model. He showed that when firms compete àlabertrand on differentiated products, then a VER at the free trade level of imports increases profits of both domestic and foreign firms. This result is consistent with what has been found by Mai and Hwang (1988) in a more general analysis based on a conjectural variation static approach. However, their paper shows that with Cournot competition VER is ineffective, but in more collusive settings (i.e., those with positive quantity conjectures) it hurts the foreign firm and fails to be a voluntary strategic trade policy. In general, the consequences of quantity restrictions are known to depend on whether imports are strategic and/or demand substitutes or complements for domestic products. This, in turn, depends on whether market interaction takes place in outputs or prices. When firms set quantities (prices) and goods are demand substitutes (complements), output restrictions impede the ability of the foreign firm to compete in the domestic market, thereby acting to facilitate collusion and raise prices and profits (see Krishna, 1989). 2 This view is reinforced by Suzumura and Ishikawa (1997), who explore the implications of a voluntary export restraint agreement on profits and welfare in a duopoly model with product differentiation and conjectural variations. They assume that the imposition of a VER makes the domestic firm into a Stackelberg leader, and show that a VER introduced at the free-trade equilibrium level of export is welfare-improving for the importing country if and only if the foreign exporter is forced to comply with the restraint involuntarily. This paper analyzes VERs in a dynamic setting where oligopolistic firms interact participating in a differential game. To our knowledge, the only existing contribution in this vein is due to Dockner and Haug (1991), who analyse VERs in a dynamic oligopoly game with Cournot competition, adopting a sticky-price model of dynamic oligopoly. 3 Restricting, for simplicity, the analysis to a speed of price adjustment which goes to infinity, they showed that indeed VER is voluntary as it increases the profits 1 Some of these exceptions are Cheng (1987), Driskill and McCafferty (1989a, 1996), Dockner and Haug (1990, 1991) and Calzolari and Lambertini (2003). Herguera, Kujal and Petrakis (2000) study the effects of quantity restrictions (such as VERs) on the long run choice of quality in a vertical product differentiation model with Cournot competition. 2 See also Pomfret (1989) for a detailed survey on VERs. 3 This model is due to Simaan and Takayama (1978). It has been extended by Fershtman and Kamien (1987, 1990) and Tsutsui and Mino (1990). 2

3 of both domestic and foreign firms. However, this result stems from the fact that, since the price is the state variable, interaction among output levels takes place only through co-state equations, as each firm s first order condition w.r.t. own quantity is independent of the other firms. Instead, one could consider that capital accumulation is one of the most important strategic decision firms are confronted with. Hence, following the literature initiated by Spence (1979), we explicitly model firm s dynamic capital accumulation decisions and study the effects of a VER on firms profits and equilibrium prices. When dealing with differential games, different strategies and solution concepts may be applied. The existing literature mainly concentrates on two kind of strategies: 4 the open-loop and the closedloop ones. In the former case, firms precommit to an investment path over the whole time horizon of the game, and the relevant equilibrium concept is the open-loop Nash equilibrium. In the latter, firms do not precommit on investment paths and their strategies at any instant depend upon all the preceding history of the game, as described by the evolution of state variables and their influence upon the evolution of control variables. Dockner and Haug (1991) restrict the analysis to the Markov Perfect Nash equilibrium and, as in most of the literature, they adopt a refinement of the closed-loop Nash equilibrium, which is known as the feedback Nash equilibrium. 5 In the present paper, relying on a result which is proved by Cellini and Lambertini (2001), we will not restrict to this refinement and deal with both the open-loop and closed loop no-memory solutions. Under these two solution concepts, we will study whether a VER leads to more or less cooperative equilibria and then higher or smaller profits for firms, as compared to the free trade equilibrium. We explicitly deal with the effectsofversoverprofits and equilibrium quantities, using the Ramsey (1928) model of capital accumulation (i.e., the well known corn-corn growth model) and considering both Cournot and Bertrand competition. We show that when firms supply substitute goods and compete on quantities any VER benefits the domestic firmbutithurtstheforeignfirm which imposes it. It follows that a VER cannot be observed in equilibrium. Although this confirms the analysis in the mentioned static literature, this result is here obtained as one outcome out of the richer set of equilibria that our dynamic setting provides. Similarly, under Bertrand competition, the VER may increase the profits of both firms, as we are used to see in static models of price competition, and exactly for the same reason, namely that provided products are substitutes, a quantity commitment on the part of the foreign firm yields a quasi-stackelberg price equilibrium, with the foreign firm in the follower s position. 6 However, contrary 4 See Kamien and Schwartz (1981); Başar and Olsder (1982); Mehlmann (1988). 5 For a clear exposition of the difference among these equilibrium solutions see Başar and Olsder (1982, pp , and chapter 6, in particular Proposition 6.1). 6 We also show that, if goods are complements, exactly the opposite considerations apply. 3

4 to what a static setting delivers, we also show that with substitute products and price-setting firms there are equilibria that are specific to the dynamic framework and where VER cannot arise. We also extend our analysis to an alternative capital accumulation scheme, the Solow (1956) and Nerlove and Arrow (1962) model of reversible investment (i.e., capital accumulation with depreciation). In this setting, capacity and sales coincide in each period so that price and quantity competition are observationally equivalent (in steady state). This means that the differential game with reversible investment in capacity encompasses the well known result of Kreps and Scheinkman (1983), who show in a static two-stage game that capacity-constrained price competition gives rise to a Cournot equilibrium. Under the Solow-Nerlove-Arrow accumulation rule, a free trade equivalent export restraint induces the same equilibrium price which would prevail with free trade and, as a consequence, firm s profits are unaffected. However, this does not hold if the VER further reduces imports with respect to the free trade level. In this case, since the game with substitute goods also features substitutability between strategic variables (i.e. firms investment efforts), the firm reducing its production will be negatively affected and a VER cannot be observed at equilibrium. 7 These results allows us to draw the following implication, which extends to a dynamic setting a conclusion reached by most of the aforementioned static literature on this topic. As VERs are usually observed in several markets, and their adoption is not justified when firms set output levels, then the viability of VERs as coordinating or quasi-collusive instruments is confined to those cases where firms supply substitute (complement) goods and compete in prices (quantities), and the decision on sales is not influenced by investment plans concerning capacity. The paper is organized as follows. The general setting is laid out in section 2. Section 3 is devoted to the analysis of Cournot competition under the two alternative capital accumulation rules. Bertrand behavior in the two models is investigated in section 4. Concluding remarks are in section 5. 2 The Ramsey model As in the previous literature on this topic, we consider a duopoly market supplied by a domestic producer (firm D) and a foreign rival (firm F ). The model is built in continuous time. The market exists over t [0, ). Let q i (t) define the quantity sold by firm i, i = D, F, at time t. The inverse demand function of firm i at time t is: p i (t) =a q i (t) sq j (t), (1) with i = D, F, i 6= j. When s (0, 1], products are substitutes, while they are complements when s [ 1, 0), and independent with s =0. The substitutability parameter s [ 1, 1] ensures that 7 Again, the opposite holds when goods are demand complements. 4

5 quantities are never negative. 8 In order to produce, firms must accumulate capacity or physical capital k i (t) over time. Following Ramsey (1928), we assume that the capital accumulation process takes place according to the following dynamic equation: dk i (t) = f(k i (t)) q i (t) δk i (t), (2) dt where f(k i (t)) = y i (t) denotes the output produced by firm i at time t. That is, capital accumulates as a result of intertemporal relocation of unsold output y i (t) q i (t). This can be interpreted in two ways. The first consists in viewing this setup as a corn-corn model, where unsold output is reintroduced in the production process. The second consists in thinking of a two-sector economy where there exists an industry producing the capital input which can be traded against the final good at a price equal to one (see Cellini and Lambertini, 1998, 2000). The control variable is q i (t), whilethestatevariable is k i (t). Concerning instantaneous variable costs, we assume that unit production cost is constant and equal across firms. For the sake of simplicity, and without further loss of generality, we also assume it to be nil. Accordingly, firm i s instantaneous profits are π i (t) =p (t) q i (t), and output q i (t) must be chosen so as to maximise the discounted flow of profits: J i (t) = Z 0 π i (t) e ρt dt (3) under the dynamic constraint (2). The discount rate ρ 0 is constant and equal across firms. 2.1 Cournot competition Here, we examine quantity competition. Under the capital accumulation rule (2), the Hamiltonian function of firm i is the following: H i (t) = e ρt {q i (t)[a q i (t) sq j (t)] + +λ ii (t)[f(k i (t)) q i (t) δk i (t)] + +λ ij (t)[f(k j (t)) q j (t) δk j (t)]}, (4) where λ ij (t) =µ ij (t)e ρt, and µ ij (t) istheco-statevariableassociatedbyfirm i to state k j (t); k i (0) k i0 defines the initial condition for firm i. As a preliminary step, we can prove the following: Lemma 1 The open-loop Nash equilibrium is subgame perfect, and admits λ ij (t) =0,j6= i, at any t [0, ). 8 This formulation of market demand functions with product differentiation dates back to Bowley (1924) and is commonly used in the industrial organization literature since Dixit (1979) and Singh and Vives (1984). 5

6 Proof. See the appendix. The game we are investigating belongs to the class of the so-called linear state games, asdefined in Dockner et al. (2000, section 7.2). As it can be verified in this reference, when the first order conditions on control variables do not contain the state variables but only the co-state ones, then the open-loop equilibrium is Markov-perfect (Dockner et al., 2000, pp ), where Markov-perfect is an alternative but equivalent way of labelling subgame perfection and strong time consistency. Moreover, the resulting equilibrium does not depend on initial conditions, and the admissibility of λ ij (t) =0 reveals that the rival s dynamic is immaterial to firm i. Accordingly, we move on to the solution of the open-loop problem by reformulating the Hamiltonian of firm i as follows: H i (t) =e ρt {q i (t)[a q i (t) sq j (t)] + λ i (t)[f(k i (t)) q i (t) δk i (t)]}. (5) From the first order condition on q i (t), we obtain the best reply function of firm i : qi br (t) = a sq j(t) λ i (t). (6) 2 The co-state equation of firm i writes as follows: H i(t) k i (t) = µ i(t) λ i(t) = ρ + δ f 0 (k i (t)) λ i (t). (7) The best reply function (6) can be differentiated w.r.t. time to yield: Then, using dq i (t) dt = s dq j(t)/dt + dλ i (t)/dt 2. (8) λ i (t) =a 2q i (t) sq j (t) (9) and (7), we obtain: dq i (t) dt = 1 2 s dqj (t)/dt +[a 2q i (t) sq j (t)] ρ + δ f 0 (k i (t)) ª (10) which, invoking symmetry, can be rearranged to yield: dq(t) dt = 1 2+s [a c (2 + s) q(t)] ρ + δ f 0 (k i (t)) Imposing dq(t)/dt =0and solving, we obtain the following set of solutions: f 0 (k) =ρ + δ (11) and q ss = a 2+s, (12) 6

7 where q ss is the demand-driven equilibrium (observationally equivalent to the static Cournot equilibrium), while f 0 (k) =ρ+δ is the Ramsey golden rule equilibrium dictated by intertemporal capital accumulation alone. This entails a capital endowment equal to k R and quantity q R = f k R, with R standing for Ramsey. The phase diagram illustrating the dynamics of the system is in figure 1, where the locus dk/dt =0 as well as the behaviour of k, depicted by horizontal arrows, derive from (2). Steady states are identified by the intersections between loci. Figure 1: Steady state equilibrium under free trade q 6 ¾ q ss? L 6 - P -? ¾ 6 ¾6 - -6? k R f 0 1 (ρ + δ) f 0 1 (δ) ¾? - k It is worth noting that the situation illustrated in figure 1 is only one out of several possible configurations, due to the fact that the position of the vertical line f 0 (k) =ρ + δ is independent of demand parameters, while the horizontal loci q ss shifts upwards (downwards) as a increases (decreases). Here, we confinetothecasewherethehorizontallocusq ss intersects the locus dk/dt =0in the region where the latter is increasing in k, to the left of the Ramsey equilibrium f 0 (k) =ρ + δ. Steady state points are identified as L and P. Intersections to the right of k = f 0 1 (δ) are clearly inefficient and therefore can be disregarded. Stability analysis reveals that {L, P} are, alternatively, a saddle point and an unstable focus. In particular, in the case depicted in figure 1, L is a saddle point while P is an unstable focus. If, e.g., parameter a is large enough to drive L to the right of P, then P becomes a saddle point while L becomes an unstable focus. 9 9 The other intersection between the horizontal locus and dk/dt =0can be disregarded, as it is clearly inefficient. The stability analysis is omitted for the sake of brevity. See Cellini and Lambertini (1998) for details. 7

8 The foregoing discussion can be summarised as follows: Lemma 2 Under free trade, for all {a, s} such that a/ (2 + s) f k R, the system reaches a steady state at q ss = a/ (2 + s), which is a saddle. In correspondence of either q ss = a/ (2 + s) or q R = f k R, firms steady state profits under free trade are, respectively: a 2 π ss = (2 + s) 2 ; πr = a 2f k R f k R. (13) Now we shall take into consideration the alternative setting where firm F adopts an export restraint q F (which, for instance but not necessarily, can be fixed at the free trade level). 10 The issue can be quickly dealt with by observing how the best reply of firm D modifies in the presence of an export restraint. Suppose the Cournot equilibrium prevails (i.e., it is a saddle point) under free trade. Then, from (6), we can write: qd br (t) = a sq F λ D (t), (14) 2 where q F a/ (2 + s). It is immediate to verify that dq D (t) dt = dλ D(t) dt = ρ + δ f 0 (k D (t)) λ D (t), (15) where, from (14), λ D (t) =a 2q D (t) sq F. This entails that the optimal quantity offered by the domestic firm in steady state, as a reaction to a VER, may be either the Cournot best reply to the VER, or the Ramsey output, depending on which one identifies a saddle point. While in the free trade setting the imposition of symmetry entails that both firms converge either to the demand-driven or to the Ramsey equilibrium, here the adoption of a VER amounts to abandoning symmetry, with the domestic firm being in steady state at either f 0 (k R )=ρ + δ or q D =(a sq F ) /2. Define the optimal domestic choice as qd VER. Then, qver D =(a sq F ) /2 if (a sq F ) /2 <f(k R ), and qd VER = f k R otherwise. 11 An interesting limit case may arise, where q F is sufficiently lower than a/ (2 + s) and consequently q D becomes sufficiently large to coincide with the Ramsey equilibrium. This situation is illustrated in figure 2 (the horizontal and vertical arrows describing the dynamics of {k, q} are omitted). 10 Since we focus upon the steady state analysis, we assume that q F = q F. In general, during the adjustment process towards the steady state, it can be the case that q F < q F, but excess capacity cannot be observed in the subgame perfect steady state equilibrium. 11 In fact, recall that the smaller output level always defines a saddle point, while the larger always defines an unstable focus. 8

9 Figure 2: Steady state equilibrium under a VER q 6 q ss D P q ss F L k P f 0 1 (ρ + δ) f 0 1 (δ) - k The foregoing discussion proves the following result, which holds if the (static) Cournot equilibrium obtains with free trade: Lemma 3 Suppose s > 0 and the demand-driven equilibrium q ss prevails under free trade. Any export restraint may drive the domestic firm either to the demand-driven equilibrium with VER qd ss = (a sq F ) /2, or to the Ramsey equilibrium where f 0 (k R )=ρ + δ and q R = f(k R ) > q F. In the first case, the export restraint benefits firm D and hurts firm F for all q F <q ss. In the second case the same applies for all q F q ss. Therefore, in general: π D (q F ) π ss π F (q F ). Yet, under free trade, the Ramsey equilibrium may be reached, whereby firms choose capacity and sales in relation to intertemporal parameters only. Here, the following holds: Lemma 4 Suppose s>0 and the Ramsey equilibrium q R prevails at the free trade equilibrium. If so, then two situations may arise, where the export restraint is either q F = f(k R ) or any q F <f(k R ). In the first case, q D = q F = f(k R ). Therefore, the profits associated with the VER are observationally the same as under free trade. In the second case, firm D s best reply is either qd ss =(a sq F ) /2 or q D = q F = f(k R ). Therefore, in general: Lemmata 2-4 produce the following result: π D (q F ) π ss π F (q F ). 9

10 Proposition 1 A voluntary export restraint cannot be observed at equilibrium in the Ramsey model, for all s (0, 1]. Of course, the VER becomes desirable if s [ 1, 0) and q F 6= f(k R ). In such a case, the Cournot game with complement goods exhibits increasing best replies, and therefore the VER is a profitable restriction for both firms. Otherwise, if q F = f(k R ) because the free trade equilibrium is the Ramsey golden rule and firm F sticks to it, the best reply to the VER is q D = q F = f(k R ) and the VER is ineffective. Summing up, notwithstanding the fact that the dynamic game yields a richer set of equilibria as compared to the static game, the Cournot-Ramsey game leads to the same qualitative conclusions we are accustomed with in static settings. 2.2 Bertrand competition With price competition, from (1) the demand function firm i faces at time t is: q i (t) = a 1+s p i(t) 1 s 2 + sp j(t) 1 s 2 (16) where p i (t) and p j (t) are respectively the price set by firms i and j, respectively. The problem of firm i is the following: ½ a H i (t) = e ρt p i (t)+ (17) λ ii (t) λ ij (t) 1+s p i(t) 1 s 2 + sp j(t) 1 s 2 f(k i (t)) f(k j (t)) a 1+s + p i(t) 1 s 2 sp j(t) 1 s 2 δk i(t) a 1+s + p j(t) 1 s 2 sp i(t) 1 s 2 δk j(t) + ¾, where λ ij (t) =µ ij (t)e ρt, and µ ij (t) is the co-state variable associated by firm i to state k j (t). Proceeding as with Lemma 1, one can simply show that the open-loop Nash equilibrium is subgame perfect also when firms compete on prices and λ ij (t) =0is admissible (the proof is omitted for brevity). Moving on to the solution of the open-loop problem, from the first order condition on p i (t), we obtain the best reply function of firm i : p br i (t) = a(1 s)+sp j(t)+λ i (t). (18) 2 Function (18) can be differentiated w.r.t. time to yield: dp i (t) = 1 dpj (t) s + dλ i(t). (19) dt 2 dt dt Then, using λ i (t) =2p i (t) a(1 s) sp j (t), (20) 10

11 and the co-state equation of firm i whichwritesasin(7),weobtain: dp i (t) = 1 ½ dpj (t) s +[2p i (t) a(1 s) sp j (t)] ρ + δ f 0 (k i (t)) ¾ (21) dt 2 dt Invoking symmetry, this can be rearranged to yield: dp(t) dt = 1 2 s [(2 s) p(t) a(1 s)] ρ + δ f 0 (k(t)) Imposing dp(t)/dt =0and solving, we obtain the Ramsey equilibrium: f 0 (k R )=ρ + δ (22) with quantity q R = f(k R ) and, substituting p = a (1 s) / (2 s) into (1), the demand-driven solution: q ss a = (1 + s)(2 s). (23) The phase diagram illustrating the dynamics of the system is as in figure 1, and Lemma 2 applies qualitatively unmodified, although of course Bertrand behaviour entails a larger output and a lower price in steady state, as compared to Cournot, for all positive s (and conversely). With free trade at q ss, the instantaneous profit eachfirm obtains then is π ss = a 2 (1 s) (2 s) 2 (1 + s). (24) Now let us turn to the case where firm F adopts an export restraint q F. Suppose the demanddriven equilibrium prevails under free trade. Then, replace q F = q F into the inverse demand for firm F and substitute back into the Hamiltonian of firm D. The best reply function of firm D now becomes p br D(t) = a sq F + λ D (t). (25) 2 Differentiating this best reply w.r.t. time yields: Then, using dp D (t) dt = 1 2 dλ D (t). (26) dt λ D (t) =2p D (t) a + sq F (27) and (7), we obtain: dp D (t) = 1 dt 2 [2p D(t) a + sq F ] ρ + δ f 0 (k D (t)) (28) Imposing dp D (t)/dt =0and solving, we obtain the Ramsey and the demand-driven equilibria: f 0 (k R )=ρ + δ q VER D = a sq F 2. (29) 11

12 Note that qd VER coincides with the expression obtained in the Cournot model. This is due to the fact that, once firm F sets her quantity restriction, firm D becomes a monopolist on the residual demand function. Therefore, D must be indifferent between maximising profits w.r.t. a price or a quantity. What determines the desirability of the VER is the slope of reaction functions and the sign of parameter s. When f 0 (k) =ρ + δ holds at equilibrium, then firm D chooses q R = f(k R ). On this basis, we can prove the following: Proposition 2 Under the Ramsey capital accumulation regime, 1. there exists an export restraint q F q ss, that benefits both firms as long as the steady state is driven by demand conditions only, q ss f(k R ) and s (0, 1]. Firm F benefits more than firm D for any s. 2. If instead q ss >f(k R ) or s [ 1, 0), then the steady state profits of firm F under any VER are lower than under free trade, and therefore the VER cannot be part of a subgame perfect equilibrium. Proof. See the Appendix Proposition 2 shows that when firms compete in prices, then VERs may indeed be voluntary and serve as coordinating or quasi-collusive instruments. Ceteris paribus, this depends on the slope of technology. To see this, consider as given the set of demand and intertemporal parameters {a, s, δ, ρ}. If so, the capital level associated with the Ramsey equilibrium increases as f 0 (k) increases. Accordingly, the same holds for the corresponding output level f(k R ). Hence, the production possibility set wherein the VER is adopted in equilibrium is directly related to the marginal productivity of capital. This phenomenon is illustrated in figure 3, where we consider a technical progress increasing the marginal productivity of capital from f0 0(k) to f 1 0(k). 12

13 Figure 3 : The effect of a change in the productivity of capital f 0 (k) 6 f 0 0 (k) f 0 1 (k) δ + ρ - k As a general appraisal of the above analysis, one can say that, when the saddlepoint equilibrium is the one driven by demand parameters, then the steady state of the Ramsey model closely replicates the subgame perfect equilibrium of the corresponding static versions of Cournot and Bertrand games. In such cases, the conclusions drawn from either the static literature or the dynamic reinterpretation of it indeed coincide as to the desirability of a VER at equilibrium. However, considering a dynamic accumulation process also produces an additional equilibrium consisting in the Ramsey golden rule, which, by definition, cannot arise in the static setup. In particular, the golden rule whereby f 0 (k R )= ρ + δ is independent of demand conditions, which implies that this equilibrium appears unmodified under both Bertrand and Cournot competition. Now focus upon the case of demand substitutes, i.e., s (0, 1], where the Cournot output, a/ (2 + s), is lower than the Bertrand output, a/ [(1 + s)(2 s)]. Suppose a is sufficiently low to ensure that the quasi-static (or demand-driven) Cournot and Bertrand outcomes prevail at the free trade equilibrium in the two alternative settings. Then examine the effects of an increase in a. For intermediate values of a, we obtain: a (1 + s)(2 s) f(kr ) > a (30) 2+s which entails that there exists a parameter region where the demand-driven equilibrium obtains only in the Cournot game. Hence, contrary to the acquired wisdom based upon the static approach, the dynamic analysis admits equilibrium outcomes where the incentive to adopt a VER is ruled out, 13

14 function rewrites as: 12 p i (t) =a k i (t) sk j (t). (32) irrespective of whether firms are quantity- or price-setters. This produces the following Corollary to Propositions 1-2: Corollary 1 Contrary to the static models, the dynamic analysis admits equilibrium outcomes where the incentive to adopt a VER is ruled out, irrespective of whether firms are quantity- or price-setters. In the remainder, we further explore the richness of the dynamic analysis and show that the accumulation dynamics may be reformulated so as to imply that, indeed, the incentive to introduce a VER never exists. 3 The Solow-Nerlove-Arrow model The alternative accumulation scheme that we propose here is that due to Solow (1956) and Nerlove and Arrow (1962). In this case the relevant dynamic equation is: dk i (t) dt = I i (t) δk i (t), (31) where I i (t) is the investment carried out by firm i at time t, andδ is the constant depreciation rate. The instantaneous cost of investment is C i [I i (t)] = b [I i (t)] 2, with b>0. To solve this model explicitly, we also assume that firms operate with a constant returns technology q i (t) =k i (t), so that the demand Here, the control variable is the instantaneous investment I i (t), while the state variable is obviously k i (t). As in the Ramsey setup, we assume that firms operate at the same (constant) marginal cost, and we normalise it to zero. Since strategies are defined over the investment space and demand functions are defined in terms of state variables only, in this model we cannot properly distinguish between Cournot and Bertrand competition. We investigate the model by using, alternatively, inverse and direct demand functions, in order to show that (i) the open-loop equilibrium is subgame perfect in both cases, and (ii) the two alternative approaches coincide. 13 The latter conclusion comes from the fact that the present model 12 This assumption entails that firms always operate at full capacity. This, in turn, amounts to saying that this model encompasses the case of Bertrand behaviour under capacity constraints, as in Kreps and Scheinkman (1983). The openloop solution of the Nerlove-Arrow differential duopoly game in a model without trade is in Fershtman and Muller (1984) and Reynolds (1987). In Dockner et al. (2000, Example 7.3, p. 191) one can find a fixed-price version of our Nerlove- Arrow-Solow model, which was originally introduced by Leitmann and Schmitendorf (1978) and Feichtinger (1983) as a model of advertising. 13 As we will discuss in the following, inverting demand functions involves a reformulation of the dynamics of state variables as well as the co-state equations, and this may in principle affect the strategic interaction between firms. 14

15 is an investment game rather than a market game, because firms do not choose market variables. Therefore, the incentive to adopt a VER is univocally determined by the strategic interaction between investment plans rather prices or quantities. Using inverse demand functions as in (32), the closed-loop formulation of the Hamiltonian of firm i writes as follows: n H i (t) = e ρt [a k i (t) sk j (t)] k i (t) b [I i (t)] 2 + (33) +λ ii (t)[i i (t) δk i (t)] + λ ij (t)[i j (t) δk j (t)]} where λ ij (t) =µ ij (t)e ρt, and µ ij (t) is the co-state variable associated to k j (t), i,j= D, F. Moreover, let k i (0) k i0 define the initial condition for firm i. On the basis of (33), we can prove the following: Lemma 5 Using the inverse demand functions, the open-loop Nash equilibrium of the Solow-Nerlove- Arrow capital accumulation is a degenerate closed-loop memoryless equilibrium. Therefore, the openloop equilibrium is subgame perfect. Moreover, λ ij (t) =0,j6= i, is admissible at any t [0, ). Proof. See the appendix. Accordingly, in the remainder we use the following Hamiltonian: n o H i (t) =e ρt [a k i (t) sk j (t)] k i (t) b [I i (t)] 2 + λ i (t)[i i (t) δk i (t)]. (34) Firm i s first order conditions, under the open-loop solution, are (the transversality condition is omitted for brevity): H i (t) I i (t) =0 2bI i(t)+λ i (t) =0 (35) H i(t) k i (t) λ i (t) = λ i(t) ρλ i (t) (36) = (ρ + δ) λ i (t) [a 2k i (t) sk j (t)] Now we can explicitly look for steady state points under free trade. From the first order condition w.r.t. I i (t), we obtain: I i (t) = 1 λ i (t) = I i (t)(ρ + δ) a 2k i(t) sk j (t). (37) 2b 2b As an intermediate step, observe that (37) can be rewritten so as to show that investment levels are strategic substitutes (resp., complements) along the equilibrium path, for all s (0, 1] (resp., s 15

16 [ 1, 0)). To this aim, one has to impose the stationarity condition on capacity stocks, dk i (t) /dt =0, yielding k i (t) =I i (t) /δ, which can be plugged into (37) to obtain: I i (t) Now, imposing I i (t)/ =0, we have: = 2[1+bδ(ρ + δ)] I i(t) aδ + si j (t) 2bδ I i (t) = aδ si j (t) 2[1+bδ(ρ + δ)] ; I i (t) I j (t) = si j (t) 2[1+bδ(ρ + δ)] which entails the following:. (38) Lemma 6 Along the equilibrium path, investment levels are strategic substitutes (resp., complements) whenever goods are demand substitutes (resp., complements). The above Lemma suggests that the flavour of the present game is somewhat similar to that associated with the static Cournot game, although the setup under examination cannot be defined at all as a Cournot game. To see this, note that the Solow-Nerlove-Arrow game never produces increasing best replies when s is positive. Solving the system: I i (t) =0; k i(t) =0,i= D, F, (40) we calculate the steady state levels of states and controls: I ss δa = 2+s +2b (ρ + δ) δ ; kss = Iss δ. (41) The pair {I ss,k ss } is a saddle point. 14 Now we revert to direct demand functions: q i (t) = a 1+s p i(t) 1 s 2 + sp j(t) 1 s 2 (42) so that the Hamiltonian of firm i is: ½ a H i (t) = e ρt 1+s p i(t) 1 s 2 + sp j(t) 1 s 2 p i (t) b [I i (t)] 2 + µ a λ ii (t) I i (t) δ 1+s p i(t) 1 s 2 + sp j(t) 1 s 2 + (43) µ a λ ij (t) I j (t) δ 1+s p j(t) 1 s 2 + sp ¾ i(t) 1 s 2 TheequivalentofLemma5iseasytoprove. Thatis,first order conditions on controls do not contain the state variables, and therefore the open-loop equilibrium is subgame perfect. The details are omitted for brevity. Accordingly, we proceed by solving the open-loop formulation of the game, which obtains from (43) by setting λ ij (t) =0and λ ii (t) =λ i (t). The outcome is summarised by the following: 14 The proof is omitted for brevity. See Cellini and Lambertini (2001). (39) 16

17 Lemma 7 The steady state of the Solow-Nerlove-Arrow game with direct demand functions is observationally equivalent to the steady state of the same game with inverse demand functions. Proof. See the Appendix. The above result has the following intuitive explanation. The usual interpretation of the difference between Cournot and Bertrand in static games is that firms optimise w.r.t. either quantities or prices. However, in this differential game, using direct demand functions for the Bertrand case does not modify the strategy space for control variables, which are investment efforts. Therefore, in the Solow model, firms are not choosing prices or quantities and consequently the specific formulation of instantaneous profits is immaterial to the subgame perfect equilibrium emerging in steady state. Nevertheless, this conclusion was not obvious at the outset, in that inverting demand functions involves a reformulation of the dynamics of state variables as well as the co-state equations, as one can check in the Appendix. Since the specification of the demand system is irrelevant, we may examine the foreign firm s incentive to adopt a VER by using the inverse demand functions. If an export restraint (equivalent to the free trade level of kf ss ) is adopted by firm F, the domestic firm s optimization problem becomes max H D(t) =e ρt I D (t) = Iss F where k F = kf ss δ firm D coincide with (35-36). n a kd (t) sk F kd (t) b [I D (t)] 2 + λ D (t)[i D (t) δk D (t)]o (44). It is immediate to verify that the first order conditions for the optimum of The above discussion proves the following result: Lemma 8 Under the Solow-Nerlove-Arrow capital accumulation dynamics, with a free trade equivalent export restraint, the steady state equilibrium price in the domestic market is the same under both free trade and VER. As a corollary, notice that both firms steady state profits are also the same as under free trade. Essentially, the above result is driven by the fact that, in the Solow-Nerlove-Arrow model, there is no direct strategic interaction in the choice of optimal investment on the part of firms, i.e., firm i s first order condition on investment (35) only contains the own control, and not the rival s. Hence, the behaviour of firm D is unaffected by an export restraint set by firm F atthefreetradelevel. Obviously, this does not hold if k F (0,kF ss ). The intuition for this comes directly from Lemma 6 whereweshowthatfirms investment and capital levels are strategic substitutes so that any reduction in the capacity of firm F entails an increase in the capacity of firm D and the same applies to profits. Therefore, as for the desirability of a VER, one obtains similar conclusions as with static quantity competition: 17

18 Proposition 3 Consider s>0. A voluntary export restraint cannot be observed at equilibrium in the Solow-Nerlove-Arrow model. Of course, as in the Ramsey-Cournot model, exactly the opposite holds in the negative range of s, since best replies are everywhere increasing and therefore the VER becomes profitable. 4 Conclusions In this paper, we have analyzed the effects of voluntary export restraint in a continuous time differential game where we have explicitly introduced the firms accumulation dynamics. In both models, open-loop and closed-loop (no-memory) Nash equilibria always coincide, so that firms play subgame perfect and strongly time consistent strategies. Under the Ramsey (1928) accumulation dynamics with substitute goods and quantity-setting firms, any VER hurts the firm employing this policy. A similar conclusion holds also under the Solow (1956) or Nerlove and Arrow (1962) accumulation dynamics. Hence, contrary to the conclusions reached by Dockner and Haug (1991), VERs are not voluntarily employed by Cournot firms. The opposite holds when Cournot firms supply complements. Therefore, the above analysis suggests that the empirical observation of VERs corresponds to their use as either coordinating or quasi-collusive devices in markets for where firms have increasing reaction functions, which applies to either price competition with substitutes or quantity competition with complements. This is confirmed by the analysis of price competition in the Ramsey accumulation model, where the adoption of an export restraint increases the profits of the foreign firm, provided that the marketdriven equilibrium prevails and goods are substitutes. If instead the domestic firm is at the Ramsey equilibrium or goods are complements, the VER will not be adopted by the foreign rival. This outcome contrasts with the result in the static literature where VERs emerge where firms compete on prices. 18

19 Appendix ProofofLemma1.Firm i s first order condition concerning the control variable is: H i (t) q i (t) = a 2q i(t) sq j (t) λ ii (t) =0. (45) Now look at the co-state equation of firm i, for the closed-loop solution of the game: H i(t) k i (t) H i(t) q j (t) q j (t) k i (t) = µ ii(t) (46) where q j (t) =0 (47) k i (t) as it appears from a quick inspection of best replies obtained from (45): qi br (t) = a sq j(t) λ ii (t). (48) 2 Moreover, (48) suffices to establish that the co-state equation: H i(t) k j (t) H i(t) q j (t) q j (t) k j (t) = µ ij(t) (49) is indeed redundant since µ ij (t) = λ ij (t)e ρt does not appear in the first order condition on the control variable. That is, the Ramsey game yields that the open-loop solution is a degenerate closedloop solution because the best reply function of firm i does not contain the state variable pertaining to the same firm. Additionally, from (48) it also appears that λ ij (t) =0is admissible as qi br (t) is independent of λ ij (t). Proof of Proposition 2. To begin with, we have to compare the profits firms F and D obtain when they respectively use the VER q F and the implied market-driven equilibrium quantity qd VER, against the profits they obtain with no quantity restrictions adopted by firm F. Calculating firm F s profit withthever Π VER F = q F a(2 s) qf (2 s 2 ), (50) 2 the comparison of instantaneous profits for firm F reveals that Π VER F Π ss =0at q a (1 + s)(2 s) 2 ± (1 + s) 2 (2 s) 4 8(2 s 2 )(1 s 2 ) q F 1,2 = 2(2 s 2 (51) )(1 + s)(2 s) Similarly, for firm D, we have Π VER D = (a sq F ) (52)

20 and then Π VER D Π ss =0at q F 3,4 = h a (1 + s)(2 s) 2 ± 2(s 2) p i (1 s 2 ) s(1 + s)(2 s) 2 (53) The expression for q Fi,i=1, 2 is real if and only if s (0, 1]. Given the sign of the coefficients of q F in Π VER F Π ss, this reality condition proves that the VER cannot be adopted when goods are complements. Now observe that Π VER F Π ss > 0 for all q F (q F 2, q F 1 ), while Π VER D Π ss > 0 for all q F (q F 4, q F 3 ). It is a matter of tedious algebra to check that q F 3 > q F 1 > q F 4 >q ss > q F 2. Therefore, any q F (q F 2,q ss ) benefits both firms. Moreover, it is also immediate to verify that arg max Π VER F = a (2 s) / 2 2 s 2 q F, with q F (q F 2,q ss ). This entails that Π VER F increases as q F decreases from q ss to q F, which, in turn, suffices to prove that Π VER F > Π VER D for all q F (q F,qss ) and s (0, 1]. Nowexaminethecasewhereq D = f(k R ), because either (i) q ss >f(k R ), or (ii) q F is larger than the value of q F that drives the domestic firm to the Ramsey equilibrium. Case (i) describes a situation where the free trade equilibrium is the symmetric Ramsey golden rule. Case (ii) captures the situation where the output expansion carried out by the domestic firm as a reaction to the VER is sufficiently large to generate the Ramsey output for firm D only. In the first case, firm F has no incentive to adopt a VER because firm D cannot expand output beyond the Ramsey level. In the second case, firm D reaches the Ramsey output starting from the market-driven solution. Any further reduction by firm F is unprofitable. ProofofLemma5.The necessary conditions for firm i = D, F require: (i) H i(t) I i (t) =0 2bI i(t)+λ ii (t) =0 (ii) H i(t) k i (t) H i(t) I j (t) I j (t) k i (t) = λ ii(t) ρλ ii (t) λ ii(t) + ρλ ii (t) =a 2k i (t) sk j (t) δλ ii (t) (iii) H i(t) k j (t) H i(t) I j (t) I j (t) k j (t) = λ ij(t) ρλ ij (t) (iv)lim µ ii (t) k i (t) =0; lim µ ij (t) k j (t) =0, t t where (iv) is the transversality condition. Notice that by (54.i) we have I i(t) =0for all i and j. Moreover, condition (54.iii), which yields k j (t) λ ij (t)/, is redundant in that λ ij (t) does not appear in the first order conditions (54.i) and (54.ii). Therefore, the open-loop solution is indeed a degenerate closed-loop solution Note that, however, the open-loop solution does not coincide with the feedback solution (see Reynolds, 1987). For (54) 20

21 ProofofLemma7.The first order condition on investment is: yielding H i (t) I i (t) =0 2bI i(t)+λ i (t) =0, (55) λ i (t) = 2bI i (t) (56) I i (t) = 1 λ i (t) (57) 2b When one uses direct demand functions, deriving the co-state equation is more involved than in the alternative case where inverse demand functions are adopted. In the system of direct demand functions, capacity k i (t) is expressed as a function of the price vector {p i (t),p j (t)}. Therefore, H i (t) k i (t) = H i(t) p i (t) p i (t) k i (t) + H i(t) p j (t) p j (t) k i (t) (58) where H i (t) p i (t) = a(1 s) 2p i(t)+sp j (t)+δλ i (t) 1 s 2 ; H i (t) p j (t) = s [p i(t) δλ i (t)] 1 s 2 ; p i (t) k i (t) = 1 ; p j(t) k i (t) = s. Using (59), the co-state equation writes as follows: λ i (t) Hi (t) p i (t) p i (t) k i (t) + H i(t) p j (t) p j (t) k i (t) = λ i(t) (59) ρλ i (t) (60) = a (1 s) p i(t) 2 s 2 + sp j (t)+λ i (t)(ρ + δ) 1 s 2 1 s 2. (61) Then, plugging (56) and (61) into (57) and imposing the symmetry condition p j (t) =p i (t), we obtain: I i (t) = a p i(t)(2+s)+2bi i (t)(ρ + δ)(1+s) 2b (1 + s) (62) with I ss i I i (t) =0at I ss i can be substituted into (31), which simplifies as follows: k i (t) = p i(t)(2+s) a 2b (ρ + δ)(1+s). (63) = p i(t)(2+s) a 2b (ρ + δ)(1+s) δ [a p i(t)] 1+s further details, see Cellini and Lambertini (2001), as well as the discussion in Driskill and McCafferty (1989b, pp ). Classes of games where this coincidence arises are illustrated in Clemhout and Wan (1974); Reinganum (1982); Mehlmann and Willing (1983); Dockner, Feichtinger and Jørgensen (1985); Fershtman (1987). For an overview, see Mehlmann (1988); Fershtman, Kamien and Muller (1992). (64) 21

22 with k i (t) =0at p ss i = a [1 + 2bδ (ρ + δ)] 2[1+bδ (ρ + δ)] + s Now, using (65), we can simplify the expression for the steady state levels of investment and capacity: which coincide with (41). I ss i = δa 2+s +2b (ρ + δ) δ ; kss i (65) = Iss i δ, (66) 22

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