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Profit taxation and capital accumulation in a dynamic oligopoly model
Japan and the World Economy 23 (2011) 13–18
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Profit taxation and capital accumulation in a dynamic oligopoly model§ Massimo Baldini a,1, Luca Lambertini b,c,* a b c
Dipartimento di Economia Politica, Universita` di Modena e Reggio Emilia, Viale Berengario, 41100 Modena, Italy Dipartimento di Scienze Economiche, Universita` di Bologna, Strada Maggiore 45, 40125 Bologna, Italy ENCORE, Faculty of Economics & Econometrics, University of Amsterdam, WB1018 Amsterdam, The Netherlands
A R T I C L E I N F O
A B S T R A C T
Article history: Received 11 December 2009 Received in revised form 21 April 2010 Accepted 27 May 2010 Available online 4 June 2010
We illustrate a differential oligopoly game using the capital accumulation dynamics a` la Ramsey. We evaluate the effects of profit taxation, proving that there exists a tax rate yielding the same steady state social welfare as under social planning. Contrary to the static approach, our dynamic analysis shows that, in general, profit taxation affects firms’ decisions concerning capital accumulation and sales. In particular, it has pro-competitive effects provided that the extent of delegation is large enough (and conversely). ß 2010 Elsevier B.V. All rights reserved.
JEL classification: D43 D92 H20 L13 Keywords: Differential games Capital accumulation Open-loop equilibria Closed-loop equilibria Profit taxation
1. Introduction There exists a relatively large literature on profit taxation in static models of imperfect competition (Levin, 1985; Besley, 1989; Delipalla and Keen, 1992; Dung, 1993; Denicolo` and Matteuzzi, 2000; Ushio, 2000, inter alia). A well established result of this literature is that the taxation of operative profits (defined as the profits gross of fixed costs) is neutral, in that it does not affect first order conditions on market variables. The dynamic interaction between capital accumulation and taxation has been analysed by Hall and Jorgenson (1967).2 However, their analysis, as well as the debate stemming from what is now conventionally labelled as Jorgenson’s model, is carried out focussing upon monopoly.3 § We would like to devote this paper to Massimo Matteuzzi, in memoriam. We thank Yasushi Hamao (Editor), an anonymous referee, Vincenzo Denicol o`, Davide Dragone, Massimo Matteuzzi and the seminar audience at the University of Helsinki, the University of Padua and Universidad Carlos III, Madrid, for helpful comments and discussion. The usual disclaimer applies. * Corresponding author. Tel.: +39 0512092623; fax: +39 0512092664. E-mail addresses: [email protected] (M. Baldini), [email protected], [email protected] (L. Lambertini). 1 Tel.: +39 0592056922/23; fax: +39 0592056927. 2 For an exhaustive overview on the effects of uncertainty on investment decisions, with and without taxation, see Dixit and Pindyck (1994). 3 For the analysis of optimal taxation of polluting emissions in a monopoly model, see Benchekroun and Van Long (2002).
0922-1425/$ – see front matter ß 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.japwor.2010.05.002
In the light of the above mentioned streams of literature, one would like to characterise the influence of taxation on the behaviour of firms in a dynamic setting where strategic interaction is duly accounted for. To this aim, we propose a dynamic capital accumulation game in a Cournot oligopoly a` la Ramsey (1928), i.e., a ‘‘corn–corn’’ growth model, where accumulation is based upon unsold output and coincides with consumption postponement. In both settings, our aim consists in characterising the effects of profit taxation on the steady state behaviour of firms and the associated performance of profits and social welfare. In order to account for the (more realistic) possibility for firms not to be strict profit-seeking agents, we assume, throughout our analysis, that firms may delegate control over their strategic decisions to managers who are interested in expanding sales a` la Vickers (1985; see also Fershtman and Judd, 1987). This, to the best of our knowledge, is a new perspective on the effects of taxation in the Ramsey model, which so far has been extensively analysed under the classical assumption of a representative consumer/producer embedded in a perfectly competitive environment, as in Xie (1997); Karp and Lee (2003) and Cellini and Lambertini (2007a), inter alia. In these contributions, and in many related ones, the discussion focusses upon the time consistency of optimal taxation and the possibility of delegating the design of such a policy to a benevolent planner as a remedy to the myopic behaviour of the (atomistic) representative
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
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agent. Here, we explicitly assume the market be characterised by a non-negligible degree of market power, so that crucial production decisions are taken by oligopolistic firms interacting strategically. We focus on the open-loop solution of the dynamic game. Our main results are as follows. First, profit taxation distorts capital accumulation and the associated market performance of firms in steady state, as long as firms are managerial. The distortion disappears if all firms are strictly entrepreneurial units, i.e., pure profit-seekers. This sharply contrasts with the conventional wisdom generated by the static approach to taxation in oligopoly; such a difference comes from the fact that, if one takes the more realistic view that capacity accumulation is a dynamic process, then one can verify that indeed the presence of a tax rate on profits enters firms’ optimality conditions in a non-neutral way, contrary to what happens in a static model where taxation has only a scale effect on profits. Second, we prove that, even if all firms were pure profitseeking units, then taxation would affect the optimal sales decision at any time during the game, except in steady state. This is tantamount to saying that, while the neutrality result drawn from the static analysis indeed portrays the long run outcome of a dynamic model, it cannot grasp the ad interim distortionary bearings of taxation on firms’ decision (and therefore also on market price and consumer surplus). It is worth emphasising that this is not just a qualitative nuance of the model, but a substantive feature of it, to the extent that the game may last for long (in fact, in our approach, forever), with the distortionary effects of taxation disappearing only on doomsday. These properties are replicable under the feedback information structure, although the feedback solution cannot be fully outlined analytically. However, based on the concavity of the value function, there clearly emerges that the more intense strategic interaction generated by feedback information makes the feedback equilibrium output more sensitive to the presence of profit taxation, as compared to the market-driven solution of the open-loop game. We also characterise the optimal tax rate, from the standpoint of a policy maker aiming at the maximization of social welfare in steady state. We show that there exists a tax rate driving the price to marginal cost in steady state, and that such a tax rate is monotonically increasing in the size of the market and monotonically decreasing in the number of firms. The remainder of the paper is structured as follows. The setting is laid out in Section 2. Section 3 contains the equilibrium analysis and examines the effects of taxation at the steady state equilibrium. The feedback game is briefly dealt with in Section 4. Section 5 contains concluding remarks. 2. The basic setup The market exists over t 2 ½0; 1Þ, and is served by N firms producing a homogeneous good. Let qi ðtÞ 0 define the quantity sold by firm i at time t. The marginal production cost is constant and equal to c for all firms. Firms compete a` la Cournot, the demand function at time t being: N X pðtÞ ¼ A BQ ðtÞ; Q ðtÞ qi ðtÞ:
(1)
i¼1
In order to produce, firms must accumulate capacity or physical capital ki ðtÞ over time, according to the following dynamic equation (Ramsey, 1928): dki ðtÞ k˙ i ¼ f ðki ðtÞÞ qi ðtÞ dki ðtÞ; dt
(2)
where f ðki ðtÞÞ ¼ yi ðtÞ denotes the output produced by firm i at 0 time t. As in setting [A], we assume f @ f ðki ðtÞÞ=@ki ðtÞ > 0 and 2 00 f @ f ðki ðtÞÞ=@ki ðtÞ2 < 0. In this case, capital accumulates whenever yi ðtÞ qi ðtÞ > 0, and conversely. 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 (for further discussion, see Cellini and Lambertini, 1998, 2007b).4 The control variable is qi ðtÞ, while the state variable is ki ðtÞ. In the remainder of the paper, we will consider an oligopoly where the control of firms’ behaviour is delegated to managers characterised by a preference for output expansion. As in Fershtman (1985), Vickers (1985)Fershtman and Judd (1987), Lambertini (2000) and many others,5 we assume that delegation contracts are observable and establish that the manager of firm i maximises a combination of profits and output, so that his instantaneous objective function is: M i ðtÞ ¼ pi ðtÞ þ ui qi ðtÞ
(3)
where parameter ui measures the extent of delegation. If ui ¼ 0, the firm is entrepreneurial, i.e., it is run by stockholders so as to strictly maximise profits. Moreover, we assume that firms’ profits (gross of investment costs) are taxed at a constant rate t . Firm i’s objective is to maximise the discounted payoff flow J i ðtÞ ¼
Z 0
1
Mi ðtÞert dt
where r 0 is a constant discount rate common to all firms, s.t. the set of dynamic constraints (2) and the vector of initial conditions concerning the state variables, which we assume to be symmetric across firms for the sake of simplicity, ki ð0Þ ¼ ki0 for all i ¼ 1; 2; 3; . . . ; N. 3. Equilibrium analysis Given that, in view of the functional form of technology, the problem at hand is not defined in linear-quadratic form, we will focus on the open-loop solution.6 Under the dynamic constraint (2), the Hamiltonian of firm i is: Hi ðtÞ ¼ ert ½A Bqi ðtÞ BQ i ðtÞ cð1 t Þqi ðtÞ þ ui qi ðtÞ X þ lii ðtÞ½ f ðki ðtÞÞ qi ðtÞ dki ðtÞ þ li j ðtÞ½ f ðk j ðtÞÞ j 6¼ i
q j ðtÞ dk j ðtÞ;
(4)
P where t is the constant tax rate, Q i ðtÞ ¼ j 6¼ i q j ðtÞ is the amount of sales of all firms other than i at any time t and li j ðtÞ ¼ mi j ðtÞert is the co-state variable (evaluated at time t) that firm i associates with the state variable k j ðt Þ. The first order condition (FOC) concerning the control variable is:
@Hi ðtÞ ¼ ½A 2Bqi ðtÞ BQ i ðtÞ cð1 t Þ þ ui lii ðtÞ ¼ 0: @qi ðtÞ
(5)
Inspecting the above FOC immediately reveals that, at a generic instant t during the game, profit taxation is non-neutral unless ui and lii ðtÞ are both simultaneously equal to zero. Now note that, while it is admissible that ui ¼ 0 (i.e., the firm has not hired a manager or, if it has, the delegation contract imposes strict profit maximisation forever), the condition lii ðtÞ ¼ 0 at all t’s would clearly imply that the shadow value of an additional unit of capital 4 See also Calzolari and Lambertini (2006, 2007), where the same dynamic structure is used to investigate the viability of tariffs, quotas and export restraints in an open economy with capital accumulation and intraindustry trade. 5 For a recent overview of this literature, see Jansen et al. (2007). 6 Note, however, that if the capital endowment of every firm at time zero is sufficiently large, then the open-loop solution is indeed strongly time consistent and therefore subgame perfect. The proof is in Cellini and Lambertini (2008).
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is nil at all times, something that we cannot innocently expect to hold in general.7 This entails a key result:
Corollary 4. For all u < lðtÞ, profit taxation increases the rate of capital accumulation at any t during the game, and conversely.
Lemma 1. lii ðtÞ 6¼ 0 8 t 2 ½0; 1Þ suffices to ensure that profit taxation will be distortionary at all times during the game.
The latter result can be easily verified by considering the consequences of a decrease in the individual firm’s sales level on the state dynamics (2), all else equal. This implies that taxing instantaneous profits exerts (although indirectly) an influence on the capital accumulation process of the entire industry. The next step consists in characterising the optimal control’s dynamics and the steady state equilibrium. To this aim, we can differentiate (8) w.r.t. time, to write: 2 3 X @q j ðtÞ @li ðtÞ dqi ðtÞ 1 4 5 ð1 t ÞB (11) ¼ 2Bð1 t Þ @t @t dt j 6¼ i
Note that the above Lemma is independent of whether firms are managerial or entrepreneurial units. We can now proceed to examine at the co-state equation of firm i:
@Hi ðtÞ dlii ðtÞ rlii ðtÞ , ¼ dt @ki ðtÞ dlii ðtÞ ¼ lii ðtÞ½r þ d f ðki ðtÞÞ
(6)
dt
whereby
lii ðtÞ ¼ aexp
Z
t 0
0
½d þ r f ðki ðsÞÞds > 0; witha > 0:
(7)
Then, solving (5), we obtain the instantaneous best reply function of firm i: qi ðtÞ ¼
½A BQ i ðtÞ cð1 t Þ þ ui lii ðtÞ ; 2Bð1 t Þ
(8)
(9)
Imposing symmetry across firms, the above derivative yields:
@q ðtÞ u lðtÞ ¼ ; @t 2BðN þ 1Þð1 t Þ2
li ðtÞ ¼ ð1 t Þ½A 2Bqi ðtÞ BQ j ðtÞ c þ ui :
(12)
Now we can impose the symmetry condition qi ðtÞ ¼ qðtÞ and
ui ¼ u for all i, and using (6) and (12) we can rewrite (11) as follows:
which can be differentiated w.r.t. t , taking ui and lii ðtÞ as given, to obtain:
@qi ðtÞ ui lii ðtÞ Bð1 t Þ2 @Q i ðtÞ=@t ¼ : @t 2Bð1 t Þ2
and solve the FOC (5) to obtain the expression of the co-state variable at any t:
(10)
whose sign coincides with the sign of u lðtÞ. This allows us to claim: Proposition 2. At any t during the game, @q ðtÞ=@t < 0 (resp., @q ðtÞ=@t > 0) for all u < lðtÞ (resp., u > lðtÞ). Note that the above Proposition implies that pure profitseeking firms will surely restrict their sales as a response to any degree of profit taxation, since they avoid delegating control to managers. This, in turn, entails that delegation can be seen here as a remedy to the distortionary effects of taxation. Indeed, the effect exerted by profit taxation on the strategic choice of each individual firm can be interpreted on the basis of the balance between the marginal value u associated to an additional unit of sales and the marginal value lðtÞ of an additional unit of capacity, keeping in mind that unsold output contributes to capacity accumulation at every instant. Whenever the latter is higher than the former, @q ðtÞ=@t < 0 as any increase in fiscal pressure induces the firm to postpone a portion of sales in order to accumulate additional capacity along the path to the long-run equilibrium (and conversely). A firm may become neutral to profit taxation by setting u ¼ lðtÞ, i.e., the extent of delegation must equal the shadow price attached to an additional unit of installed capacity. Without further proof, Proposition 2 produces two relevant Corollaries. The first is:
0 dq ðtÞ / ½ð1 t Þð A c qðtÞðN þ 1ÞBÞ þ u f ðkÞ r d dt
which is equal to zero in correspondence of the following steady state solutions: qSS ¼
ð A cÞð1 t Þ þ u ; BðN þ 1Þð1 t Þ
0
f ðkÞ ¼ r þ d:
To ease the exposition, define: n o 0 kˆ k : f kˆ ¼ r þ d :
(14)
(15)
That is, kˆ is the level of capacity associated with the Ramsey steady state equilibrium where the marginal productivity of capital is equal to the sum of depreciation and discount rates. The phase diagram of the present model can be drawn in the space fk; qg. The locus q˙ dq=dt ¼ 0 is given by the solutions in (14). The two loci partition the space fk; qg into four regions, where the dynamics of q is summarised by the vertical arrows. The locus k˙ dk=dt ¼ 0 as well as the dynamics of k, depicted by horizontal arrows, derive from (2). Steady state equilibria, denoted by E1, E2 along the horizontal arm, and E3 along the vertical one, are identified by the intersections between loci. Fig. 1 describes only one out of five possible configurations, due 0 to the fact that the position of the vertical line f ðkÞ ¼ r þ d is independent of demand parameters, while the locus q ¼ ½ð A cÞð1 t Þ þ u =½BðN þ 1Þð1 t Þ shifts upwards (downwards) as A c and/or u (B and N) increases. Therefore, we obtain one out of five possible regimes:
Corollary 3. For all u < lðtÞ, profit taxation induces an increase in market price and a decrease in consumer surplus at any t during the game, and conversely. The second Corollary is: 7 Again, the only possible exception would obtain if every ki0 were large enough to allow each firm to produce the optimal steady state output from t ¼ 0 onwards and let capacity depreciate at the rate d at every instant (as in Cellini and Lambertini, 2008).
(13)
Fig. 1. The phase diagram.
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
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1. 2. 3. 4. 5.
There exist three steady state points, with kE1 < kE2 < kE3 (Fig. 1). There exist two steady state points, with kE1 ¼ kE2 < kE3 . There exist three steady state points, with kE2 < kE1 < kE3 . There exist two steady state points, with kE2 < kE1 ¼ kE3 . There exists a unique steady state equilibrium point, corresponding to E2. 0
The vertical locus f ðkÞ ¼ r þ d is a constraint on optimal capital, determined by firms’ intertemporal preferences, i.e., their common discount rate, and depreciation. Accordingly, maximum output level in steady state would be that corresponding to (i) r ¼ 0, and (ii) a capacity such that f 0 ðkÞ ¼ d. Yet, a positive discounting (i.e., impatience) leads firms to install a smaller capacity at the long-run equilibrium. This is the optimal capital ˆ When the market size A c and the extent of constraint k. delegation u are very large (or B and N are low), points E1 and E3 either do not exist (regime 5) or fall to the right of E2 (regimes 2, 3 and 4). In such a case, the capital constraint is operative and firms choose the capital accumulation corresponding to E2. Notice that, as E1 and E3 entail the same levels of sales, point E3 is surely inefficient in that it requires a higher amount of capital. E1 corresponds to the optimal quantity emerging from the static version of the game. It is hardly the case of emphasising that this solution encompasses both monopoly and perfect competition (as, in the limit, N ! 1). The stability analysis of the above system reveals that8: Regime 1. E1 is a saddle point, while E2 is an unstable focus. E3 is again a saddle point, with the horizontal line as the stable arm. Regime 2. E1 coincides with E2, so that we have only two steady states which are both are saddle points. In E1 ¼ E2, the saddle path approaches the saddle point from the left only, while in E3 the stable arm is again the horizontal line. Regime 3. E2 is a saddle, E1 is an unstable focus. E3 is a saddle point, as in regimes 1 and 2. Regime 4. Here, E1 and E3 coincide. E3 remains a saddle, while E1 ¼ E3 is a saddle whose converging arm proceeds from the right along the horizontal line. Regime 5. Here, there exists a unique steady state point, E2, which is a saddle point. We can sum up the above discussion as follows. The unique efficient and non-unstable steady state point is E2 if kE2 < kE1 , while it is E1 if the opposite inequality holds. Such a point is always a saddle. Individual equilibrium output is qSS if the equilibrium is in E1, or the level corresponding to the optimal capital constraint kˆ if the equilibrium is point E2. The reason is that, if the capacity at which marginal instantaneous profit is nil is larger than the optimal capital constraint, the latter becomes binding. Otherwise, the capital constraint is irrelevant, and firms’ decisions in each period are driven by the unconstrained maximisation of singleperiod profits only. Now, examining the steady state solutions in (14) allows us to state, without further proof, the following result:
Proposition 5. In the market-driven steady state, any profit tax rate t > 0 distorts qSS and kE1 if firms are managerial, i.e., iff u > 0. If instead firms are strictly entrepreneurial, i.e., u ¼ 0, profit taxation is neutral. 0 In the Ramsey equilibrium where f ðkÞ ¼ r þ d, profit taxation is always neutral. 8 See Cellini and Lambertini (1998, 2003) for a detailed stability analysis carried out on the Jacobian matrix of the dynamic system.
Consider the market-driven solution. The effects of profit taxation on the equilibrium level of sales are summarised by:
@qSS u ¼ > 0 8 u > 0: @t BðN þ 1Þð1 t Þ2 Since
the
level
@kE1 =@t / @qSS =@t .9
(16)
of sales is proportional to capacity, This proves a relevant Corollary to
Proposition 5:
Corollary 6. If firms are managerial, the optimal levels of sales and capacity are monotonically increasing in t . In a Ramsey model, capacity accumulation involves the intertemporal relocation of production only. Therefore, being absent any instantaneous costs, the introduction of a profit tax rate has an expansionary effect on capital accumulation whenever firms are managerial, the reason being that firms expand sales so as to try to recover through larger market shares some of the profits extracted by the policy maker. Since this incentive exists for all firms, the effect of taxation is definitely pro-competitive in that it translates into an overall expansion of the industry sales and a reduction in the price level. Given the Ramsey accumulation mechanism, by which the unsold output increases capacity, firms do not bear any fixed cost and the social optimum involves firms producing an aggregate output that must be sufficiently large to drive the market price to marginal cost, with zero profits.10 This can be obtained by setting:
t¯ ¼ 1
Nu : Ac
(17)
The above expression allows us to state our last result, whose interpretation is intuitive:
Proposition 7. Provided Nu < A c, the tax rate Nu Ac drives the industry equilibrium price and sales to their perfectly competitive levels. The rate t¯ is (i) monotonically increasing in the size of the market, A c, and (ii) monotonically decreasing in the number of firms, N, as well as in the extent of delegation, u.
t¯ ¼ 1
4. The feedback solution: sketch As stated above, the Bellman equation that would yield the feedback equilibrium of the game cannot be solved analytically since the game at hand is not a linear-quadratic one. However, a relevant implication of the first order condition can be easily drawn.11 The Bellman equation for firm i is: @V ðkðtÞÞ ½ f ðki ðtÞÞ rV i ðkðtÞÞ ¼ max ½ pðtÞ cð1 t Þqi ðtÞ þ ui qi ðtÞ þ i @ki ðtÞ qi ðtÞ X @V i ðkðtÞÞ qi ðtÞ dki ðtÞþ ½ f ðk j ðtÞÞq j ðtÞ dk j ðtÞ @k j ðtÞ j 6¼ i (18) 9 This can be verified intuitively by observing that, whenever the horizontal arm in Fig. 1 identifies the efficient saddle point steady state, then any increase in qSS SS goes along with an increase in k ¼ kE1 along the positively sloped part of the locus k˙ ¼ 0. 10 This of course is admissible under the proviso that the quasi-static perfect competition outcome with marginal cost pricing is socially optimal in this version of the Ramsey model. This is shown to hold true in Appendix A. 11 Here we exclude the trivial case in which the technology f ðki ðtÞÞ is linear, as in this case the open-loop solution is obviously a degenerate feedback one.
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
where V i ðkðt ÞÞ is firm i’s value function and kðt Þ fk1 ðt Þ; k2 ðt Þ; . . . kN ðt Þg is the vector of states. Now, taking the first order condition, we have: ½A 2Bqi ðtÞ BQ i ðtÞ cð1 t Þ þ ui
@V i ðkðtÞÞ ¼0 @ki ðtÞ
(19)
which coincides with (5) except for the presence of the partial derivative of the value function in place of the co-state variable. Hence, provided @V i ðkðt ÞÞ=@ki ðt Þ 6¼ 0 (which will hold in general, with the exception of the case where also lii ðt Þ ¼ 0, as already mentioned), the same considerations emerging under the openloop rule apply also here, in particular the equivalent of Lemma 1, Proposition 2 and Corollaries 3 and 4. Notwithstanding the fact that we cannot determine the exact form of V i ðkðt ÞÞ, we may nonetheless formulate some additional qualitative considerations on the linkage between the optimal feedback output and the level of profits taxation, by noting that the imposition of symmetry across firms on (19) yields (superscript F stands for feedback): qF ¼
ðA cÞð1 t Þ þ u @VðkðtÞÞ=@kðtÞ BðN þ 1Þð1 t Þ
@qF u @VðkðtÞÞ=@kðtÞ ¼ @t BðN þ 1Þð1 t Þ2
(20)
(21)
Unlike what happens in the open-loop game, where lii ¼ 0 in steady state, here @V ðkðt ÞÞ=@kðt Þ will differ from zero even at the steady state, precisely because firms take into account the feedback effects generated through the vector of states at any time and these will carry over to the long-run equilibrium allocation. Hence, even if an explicit solution remains out of reach, we are able to say that Lemma 1, Proposition 2 and Corollaries 3 and 4 will apply also to the equilibrium and not only to the transition towards it. As a last remark, note that @V ðkðt ÞÞ=@kðt Þ < 0 will be required in order for the value function V ðkðt ÞÞ to be concave. Therefore, the comparison between (16) and (21) reveals that
@qF @qSS @V ðkðt ÞÞ=@kðtÞ ¼ > 0; @t @t BðN þ 1Þð1 t Þ2
17
sales, provided that firms are managerial. If instead firms are under the direct control of stockholders, then taxation is neutral at the long-run equilibrium, but not along the path leading to the steady state. Analogous properties can be shown to hold in the feedback game as well (and they appear to be strengthened by feedback information), although the analytical solution of the associated Bellman equation remains out of reach. Finally, we have also investigated the welfare-maximising tax rate, finding that the Ramsey accumulation rule allows for the attainment of the perfectly competitive outcome through an appropriate tax rate, for any given extent of managerial incentives. Appendix A. The social optimum Here we briefly characterise the socially efficient solution, under the same demand and technological conditions specified in Section 2. A benevolent social planner aims at maximising the discounted flow of social welfare yielded by an industry made up by N firms. Of course, the command optimum is indeed insensitive to the specific market variable being chosen, either a single price or N output levels. To keep the exposition consistent with the foregoing analysis, we shall assume the planner controls the vector of quantities. The instantaneous social welfare function is SW ðt Þ ¼
N X
pi ðtÞ þ CSðtÞ
(23)
i¼1
where ½A pðtÞ CSðtÞ ¼
N X qi ðtÞ i¼1
(24)
2
is the instantaneous consumer surplus level. Accordingly, the planner’s problem is Z 1 max q1 ðtÞ;...qN ðtÞ SWðtÞert dt (25) 0
(22)
which can be spelled out as follows:
Proposition 8. The feedback equilibrium control is more sensitive to the tax rate than the open-loop (market-driven) equilibrium control, all else equal. A possible explanation of this result is to be found in the fact that feedback information intensifies the degree of strategic interaction among firms, thereby leading them to increase production through a well known incentive towards preemption.12 5. Conclusions We have investigated a dynamic oligopoly game embedded in a Ramsey growth model, in order to characterise the effects of gross profit taxation on the performance of firms in steady state. In particular, we have established that, if capacity accumulation is modelled as a dynamic investment process, then taxation exerts distortionary effects on the amount of capital and the level of sales at equilibrium. In the Ramsey-type model we have adopted here, taxation exerts a pro-competitive effect on optimal capacity and 12 See, e.g., Reynolds (1987) for an analogous phenomenon in the partial equilibrium version a` la Solow-Swan of the dynamic oligopoly game with capacity accumulation (Solow, 1956; Swan, 1956).
s.t. the set of dynamic constraints (2) and the vector of initial conditions ki ð0Þ ¼ ki0 for all i ¼ 1; 2; 3; . . . N. Let hi ðtÞ ¼ ni ðtÞert be the co-state variable (evaluated at time t) that the social planner (SP) associates with the state variable ki ðt Þ. The planner’s Hamiltonian function: ( " # ) N N X X HSP ðtÞ ¼ ert pi ðtÞþ CSðtÞ þ hi ðtÞ½ f ðki ðtÞÞ qi ðtÞ dki ðtÞ i¼1
i¼1
(26) yields the following set of necessary conditions:
@HSP ðtÞ ¼ A Bqi ðtÞ BQ i ðtÞ c hi ðtÞ ¼ 0 @qi ðtÞ
(27)
@HSP ðtÞ dhi ðtÞ rhi ðtÞ , ¼ dt @ki ðtÞ dhi ðtÞ ¼ hi ðtÞ½r þ d f ðki ðtÞÞ:
dt
Imposing symmetry across controls, states and co-states, and using (27), we have: 0 dqSP ðtÞ 1 dhðtÞ ½ A c BNqðtÞ f ðkÞ r d ¼ ¼ (29) dt BN dt BN
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with 8 Ac < dqSP ðtÞ qSP ¼ ¼ 0in BN : 0 dt f ðkÞ ¼ r þ d
(30)
The demand-drive steady state solution qSP ¼ ð A cÞ=ðBN Þ corresponds to marginal cost pricing, while the other steady state solution is the Ramsey golden rule we are accustomed with from the macroeconomic version of the model (Ramsey, 1928). The stability analysis follows the same line as in the oligopoly model, so that the horizontal arm qSP ¼ ð A cÞ=ðBN Þ identifies a saddle point equilibrium if the associated capacity endowment is lower than than the Ramsey capacity kˆ (and conversely). In such a case, adopting a profit tax rate may drive the oligopolistic industry to the (quasi-static) perfectly competitive outcome, as from Proposition 7, although of course not necessarily at the same capital accumulation rate.13 Note that, in general, we may assume u < ðA cÞð1 t Þ=N for all t 2 ½0; t¯ Þ, so that qSP > qSS in such a range. All this of course applies whenever the market-driven optimal output is smaller than the quantity corresponding to the Ramsey golden rule under both regimes (oligopoly and planning). Alternatively, if both qSP and qSS are larger than the output level associated to ˆ the market will converge to the Ramsey the Ramsey capacity k, equilibrium irrespective of whether firms are managerial Cournot oligopolist or are driven by a benevolent planner, and therefore designing taxation to reach the socially optimal steady state is not an issue; if the socially optimal solution is the Ramsey golden rule while managers lead their firms to the market-driven steady state for all t 2 ½0; t¯ Þ, then the adoption of t ¼ t¯ drives them into the 0 Ramsey equilibrium where f ðkÞ ¼ r þ d.
References Benchekroun, H., Van Long, N., 2002. On the multiplicity of efficiency-inducing tax rules. Economics Letters 76, 331–336. Besley, T., 1989. Commodity taxation and imperfect competition: a note on the effects of entry. Journal of Public Economics 40, 359–367. Calzolari, G., Lambertini, L., 2006. Tariffs vs quotas in a trade model with capital accumulation. Review of International Economics 14, 632–644.
13 This is due to the fact that, in general, the adoption of the constant tax rate t¯ will not ensure that lii ðtÞ ¼ hi ðtÞ at all times.
Calzolari, G., Lambertini, L., 2007. Export restraints in a model of trade with capital accumulation. Journal of Economic Dynamics and Control 31, 3822–3842. Cellini, R., Lambertini, L., 1998. A dynamic model of differentiated oligopoly with capital accumulation. Journal of Economic Theory 83, 145–155. Cellini, R., Lambertini, L., 2003. Differential oligopoly games. In: Bianchi, R., Lambertini, L. (Eds.), Technology, Information and Market Dynamics: Topics in Advanced Industrial Organization. Edward Elgar, Cheltenham, pp. 173–207. Cellini, R., Lambertini, L., 2007a. Time consistent Fiscal policies in a Ramsey economy. Mathematical Social Sciences 53, 296–313. Cellini, R., Lambertini, L., 2007b. Capital accumulation, mergers, and the Ramsey golden rule. In: Quincampoix, M., Vincent, T., Jørgensen, S. (Eds.), Advances in Dynamic Game Theory and Applications, Annals of the International Society of Dynamic Games, vol. 8. Birkha¨user, Boston, pp. 487–505. Cellini, R., Lambertini, L., 2008. Weak and strong time consistency in a differential oligopoly game with capital accumulation. Journal of Optimization Theory and Applications 138, 17–26. Delipalla, S., Keen, M., 1992. The comparison between Ad Valorem and specific taxation under imperfect competition. Journal of Public Economics 49, 351– 367. Denicolo`, V., Matteuzzi, M., 2000. Specific and Ad Valorem taxation in asymmetric Cournot oligopolies. International Tax and Public Finance 7, 335–342. Dixit, A.K., Pindyck, R.S., 1994. Investment under Uncertainty. Princeton University Press, Princeton, NJ. Dung, T.H., 1993. Optimal taxation and heterogeneous oligopoly. Canadian Journal of Economics 26, 933–947. Fershtman, C., 1985. Internal organizations and managerial incentives as strategic variables in a competitive environment. International Journal of Industrial Organization 3, 245–253. Fershtman, C., Judd, K., 1987. Equilibrium incentives in oligopoly. American Economic Review 77, 927–940. Hall, R.E., Jorgenson, D.W., 1967. Tax policy and investment behaviour. American Economic Review 57, 391–414. Jansen, T., van Lier, A., van Witteloostuijn, A., 2007. A note on strategic delegation: the market share case. International Journal of Industrial Organization 25, 531– 539. Karp, L., Lee, I.H., 2003. Time-consistent policies. Journal of Economic Theory 112, 353–364. Lambertini, L., 2000. Extended games played by managerial firms. Japanese Economic Review 51, 274–283. Levin, D., 1985. Taxation within Cournot oligopoly. Journal of Public Economics 27, 281–290. Ramsey, F.P., 1928. A mathematical theory of saving. Economic Journal, 38, 543–59. Reprinted in Stiglitz, J.E. and H. Uzawa (1969, eds.), Readings in the Modern Theory of Economic Growth, Cambridge, MA, MIT Press. Reynolds, S., 1987. Preemption and commitment in an infinite horizon model. International Economic Review 28, 69–88. Solow, R., 1956. A contribution to the theory of economic growth. Quarterly Journal of Economics 70, 65–94. Swan, T.W., 1956. Economic growth and capital accumulation. Economic Record 32, 334–361. Ushio, Y., 2000. Welfare effects of commodity taxation in Cournot oligopoly. Japanese Economic Review 51, 268–273. Vickers, J., 1985. Delegation and the theory of the firm. Economic Journal, 95 (Conference Papers), 138–147. Xie, D., 1997. On time consistency: a technical issue in Stackelberg differential games. Journal of Economic Theory 76, 412–430.
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Japan and the World Economy journal homepage: www.elsevier.com/locate/jwe
Profit taxation and capital accumulation in a dynamic oligopoly model§ Massimo Baldini a,1, Luca Lambertini b,c,* a b c
Dipartimento di Economia Politica, Universita` di Modena e Reggio Emilia, Viale Berengario, 41100 Modena, Italy Dipartimento di Scienze Economiche, Universita` di Bologna, Strada Maggiore 45, 40125 Bologna, Italy ENCORE, Faculty of Economics & Econometrics, University of Amsterdam, WB1018 Amsterdam, The Netherlands
A R T I C L E I N F O
A B S T R A C T
Article history: Received 11 December 2009 Received in revised form 21 April 2010 Accepted 27 May 2010 Available online 4 June 2010
We illustrate a differential oligopoly game using the capital accumulation dynamics a` la Ramsey. We evaluate the effects of profit taxation, proving that there exists a tax rate yielding the same steady state social welfare as under social planning. Contrary to the static approach, our dynamic analysis shows that, in general, profit taxation affects firms’ decisions concerning capital accumulation and sales. In particular, it has pro-competitive effects provided that the extent of delegation is large enough (and conversely). ß 2010 Elsevier B.V. All rights reserved.
JEL classification: D43 D92 H20 L13 Keywords: Differential games Capital accumulation Open-loop equilibria Closed-loop equilibria Profit taxation
1. Introduction There exists a relatively large literature on profit taxation in static models of imperfect competition (Levin, 1985; Besley, 1989; Delipalla and Keen, 1992; Dung, 1993; Denicolo` and Matteuzzi, 2000; Ushio, 2000, inter alia). A well established result of this literature is that the taxation of operative profits (defined as the profits gross of fixed costs) is neutral, in that it does not affect first order conditions on market variables. The dynamic interaction between capital accumulation and taxation has been analysed by Hall and Jorgenson (1967).2 However, their analysis, as well as the debate stemming from what is now conventionally labelled as Jorgenson’s model, is carried out focussing upon monopoly.3 § We would like to devote this paper to Massimo Matteuzzi, in memoriam. We thank Yasushi Hamao (Editor), an anonymous referee, Vincenzo Denicol o`, Davide Dragone, Massimo Matteuzzi and the seminar audience at the University of Helsinki, the University of Padua and Universidad Carlos III, Madrid, for helpful comments and discussion. The usual disclaimer applies. * Corresponding author. Tel.: +39 0512092623; fax: +39 0512092664. E-mail addresses: [email protected] (M. Baldini), [email protected], [email protected] (L. Lambertini). 1 Tel.: +39 0592056922/23; fax: +39 0592056927. 2 For an exhaustive overview on the effects of uncertainty on investment decisions, with and without taxation, see Dixit and Pindyck (1994). 3 For the analysis of optimal taxation of polluting emissions in a monopoly model, see Benchekroun and Van Long (2002).
0922-1425/$ – see front matter ß 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.japwor.2010.05.002
In the light of the above mentioned streams of literature, one would like to characterise the influence of taxation on the behaviour of firms in a dynamic setting where strategic interaction is duly accounted for. To this aim, we propose a dynamic capital accumulation game in a Cournot oligopoly a` la Ramsey (1928), i.e., a ‘‘corn–corn’’ growth model, where accumulation is based upon unsold output and coincides with consumption postponement. In both settings, our aim consists in characterising the effects of profit taxation on the steady state behaviour of firms and the associated performance of profits and social welfare. In order to account for the (more realistic) possibility for firms not to be strict profit-seeking agents, we assume, throughout our analysis, that firms may delegate control over their strategic decisions to managers who are interested in expanding sales a` la Vickers (1985; see also Fershtman and Judd, 1987). This, to the best of our knowledge, is a new perspective on the effects of taxation in the Ramsey model, which so far has been extensively analysed under the classical assumption of a representative consumer/producer embedded in a perfectly competitive environment, as in Xie (1997); Karp and Lee (2003) and Cellini and Lambertini (2007a), inter alia. In these contributions, and in many related ones, the discussion focusses upon the time consistency of optimal taxation and the possibility of delegating the design of such a policy to a benevolent planner as a remedy to the myopic behaviour of the (atomistic) representative
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
14
agent. Here, we explicitly assume the market be characterised by a non-negligible degree of market power, so that crucial production decisions are taken by oligopolistic firms interacting strategically. We focus on the open-loop solution of the dynamic game. Our main results are as follows. First, profit taxation distorts capital accumulation and the associated market performance of firms in steady state, as long as firms are managerial. The distortion disappears if all firms are strictly entrepreneurial units, i.e., pure profit-seekers. This sharply contrasts with the conventional wisdom generated by the static approach to taxation in oligopoly; such a difference comes from the fact that, if one takes the more realistic view that capacity accumulation is a dynamic process, then one can verify that indeed the presence of a tax rate on profits enters firms’ optimality conditions in a non-neutral way, contrary to what happens in a static model where taxation has only a scale effect on profits. Second, we prove that, even if all firms were pure profitseeking units, then taxation would affect the optimal sales decision at any time during the game, except in steady state. This is tantamount to saying that, while the neutrality result drawn from the static analysis indeed portrays the long run outcome of a dynamic model, it cannot grasp the ad interim distortionary bearings of taxation on firms’ decision (and therefore also on market price and consumer surplus). It is worth emphasising that this is not just a qualitative nuance of the model, but a substantive feature of it, to the extent that the game may last for long (in fact, in our approach, forever), with the distortionary effects of taxation disappearing only on doomsday. These properties are replicable under the feedback information structure, although the feedback solution cannot be fully outlined analytically. However, based on the concavity of the value function, there clearly emerges that the more intense strategic interaction generated by feedback information makes the feedback equilibrium output more sensitive to the presence of profit taxation, as compared to the market-driven solution of the open-loop game. We also characterise the optimal tax rate, from the standpoint of a policy maker aiming at the maximization of social welfare in steady state. We show that there exists a tax rate driving the price to marginal cost in steady state, and that such a tax rate is monotonically increasing in the size of the market and monotonically decreasing in the number of firms. The remainder of the paper is structured as follows. The setting is laid out in Section 2. Section 3 contains the equilibrium analysis and examines the effects of taxation at the steady state equilibrium. The feedback game is briefly dealt with in Section 4. Section 5 contains concluding remarks. 2. The basic setup The market exists over t 2 ½0; 1Þ, and is served by N firms producing a homogeneous good. Let qi ðtÞ 0 define the quantity sold by firm i at time t. The marginal production cost is constant and equal to c for all firms. Firms compete a` la Cournot, the demand function at time t being: N X pðtÞ ¼ A BQ ðtÞ; Q ðtÞ qi ðtÞ:
(1)
i¼1
In order to produce, firms must accumulate capacity or physical capital ki ðtÞ over time, according to the following dynamic equation (Ramsey, 1928): dki ðtÞ k˙ i ¼ f ðki ðtÞÞ qi ðtÞ dki ðtÞ; dt
(2)
where f ðki ðtÞÞ ¼ yi ðtÞ denotes the output produced by firm i at 0 time t. As in setting [A], we assume f @ f ðki ðtÞÞ=@ki ðtÞ > 0 and 2 00 f @ f ðki ðtÞÞ=@ki ðtÞ2 < 0. In this case, capital accumulates whenever yi ðtÞ qi ðtÞ > 0, and conversely. 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 (for further discussion, see Cellini and Lambertini, 1998, 2007b).4 The control variable is qi ðtÞ, while the state variable is ki ðtÞ. In the remainder of the paper, we will consider an oligopoly where the control of firms’ behaviour is delegated to managers characterised by a preference for output expansion. As in Fershtman (1985), Vickers (1985)Fershtman and Judd (1987), Lambertini (2000) and many others,5 we assume that delegation contracts are observable and establish that the manager of firm i maximises a combination of profits and output, so that his instantaneous objective function is: M i ðtÞ ¼ pi ðtÞ þ ui qi ðtÞ
(3)
where parameter ui measures the extent of delegation. If ui ¼ 0, the firm is entrepreneurial, i.e., it is run by stockholders so as to strictly maximise profits. Moreover, we assume that firms’ profits (gross of investment costs) are taxed at a constant rate t . Firm i’s objective is to maximise the discounted payoff flow J i ðtÞ ¼
Z 0
1
Mi ðtÞert dt
where r 0 is a constant discount rate common to all firms, s.t. the set of dynamic constraints (2) and the vector of initial conditions concerning the state variables, which we assume to be symmetric across firms for the sake of simplicity, ki ð0Þ ¼ ki0 for all i ¼ 1; 2; 3; . . . ; N. 3. Equilibrium analysis Given that, in view of the functional form of technology, the problem at hand is not defined in linear-quadratic form, we will focus on the open-loop solution.6 Under the dynamic constraint (2), the Hamiltonian of firm i is: Hi ðtÞ ¼ ert ½A Bqi ðtÞ BQ i ðtÞ cð1 t Þqi ðtÞ þ ui qi ðtÞ X þ lii ðtÞ½ f ðki ðtÞÞ qi ðtÞ dki ðtÞ þ li j ðtÞ½ f ðk j ðtÞÞ j 6¼ i
q j ðtÞ dk j ðtÞ;
(4)
P where t is the constant tax rate, Q i ðtÞ ¼ j 6¼ i q j ðtÞ is the amount of sales of all firms other than i at any time t and li j ðtÞ ¼ mi j ðtÞert is the co-state variable (evaluated at time t) that firm i associates with the state variable k j ðt Þ. The first order condition (FOC) concerning the control variable is:
@Hi ðtÞ ¼ ½A 2Bqi ðtÞ BQ i ðtÞ cð1 t Þ þ ui lii ðtÞ ¼ 0: @qi ðtÞ
(5)
Inspecting the above FOC immediately reveals that, at a generic instant t during the game, profit taxation is non-neutral unless ui and lii ðtÞ are both simultaneously equal to zero. Now note that, while it is admissible that ui ¼ 0 (i.e., the firm has not hired a manager or, if it has, the delegation contract imposes strict profit maximisation forever), the condition lii ðtÞ ¼ 0 at all t’s would clearly imply that the shadow value of an additional unit of capital 4 See also Calzolari and Lambertini (2006, 2007), where the same dynamic structure is used to investigate the viability of tariffs, quotas and export restraints in an open economy with capital accumulation and intraindustry trade. 5 For a recent overview of this literature, see Jansen et al. (2007). 6 Note, however, that if the capital endowment of every firm at time zero is sufficiently large, then the open-loop solution is indeed strongly time consistent and therefore subgame perfect. The proof is in Cellini and Lambertini (2008).
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
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is nil at all times, something that we cannot innocently expect to hold in general.7 This entails a key result:
Corollary 4. For all u < lðtÞ, profit taxation increases the rate of capital accumulation at any t during the game, and conversely.
Lemma 1. lii ðtÞ 6¼ 0 8 t 2 ½0; 1Þ suffices to ensure that profit taxation will be distortionary at all times during the game.
The latter result can be easily verified by considering the consequences of a decrease in the individual firm’s sales level on the state dynamics (2), all else equal. This implies that taxing instantaneous profits exerts (although indirectly) an influence on the capital accumulation process of the entire industry. The next step consists in characterising the optimal control’s dynamics and the steady state equilibrium. To this aim, we can differentiate (8) w.r.t. time, to write: 2 3 X @q j ðtÞ @li ðtÞ dqi ðtÞ 1 4 5 ð1 t ÞB (11) ¼ 2Bð1 t Þ @t @t dt j 6¼ i
Note that the above Lemma is independent of whether firms are managerial or entrepreneurial units. We can now proceed to examine at the co-state equation of firm i:
@Hi ðtÞ dlii ðtÞ rlii ðtÞ , ¼ dt @ki ðtÞ dlii ðtÞ ¼ lii ðtÞ½r þ d f ðki ðtÞÞ
(6)
dt
whereby
lii ðtÞ ¼ aexp
Z
t 0
0
½d þ r f ðki ðsÞÞds > 0; witha > 0:
(7)
Then, solving (5), we obtain the instantaneous best reply function of firm i: qi ðtÞ ¼
½A BQ i ðtÞ cð1 t Þ þ ui lii ðtÞ ; 2Bð1 t Þ
(8)
(9)
Imposing symmetry across firms, the above derivative yields:
@q ðtÞ u lðtÞ ¼ ; @t 2BðN þ 1Þð1 t Þ2
li ðtÞ ¼ ð1 t Þ½A 2Bqi ðtÞ BQ j ðtÞ c þ ui :
(12)
Now we can impose the symmetry condition qi ðtÞ ¼ qðtÞ and
ui ¼ u for all i, and using (6) and (12) we can rewrite (11) as follows:
which can be differentiated w.r.t. t , taking ui and lii ðtÞ as given, to obtain:
@qi ðtÞ ui lii ðtÞ Bð1 t Þ2 @Q i ðtÞ=@t ¼ : @t 2Bð1 t Þ2
and solve the FOC (5) to obtain the expression of the co-state variable at any t:
(10)
whose sign coincides with the sign of u lðtÞ. This allows us to claim: Proposition 2. At any t during the game, @q ðtÞ=@t < 0 (resp., @q ðtÞ=@t > 0) for all u < lðtÞ (resp., u > lðtÞ). Note that the above Proposition implies that pure profitseeking firms will surely restrict their sales as a response to any degree of profit taxation, since they avoid delegating control to managers. This, in turn, entails that delegation can be seen here as a remedy to the distortionary effects of taxation. Indeed, the effect exerted by profit taxation on the strategic choice of each individual firm can be interpreted on the basis of the balance between the marginal value u associated to an additional unit of sales and the marginal value lðtÞ of an additional unit of capacity, keeping in mind that unsold output contributes to capacity accumulation at every instant. Whenever the latter is higher than the former, @q ðtÞ=@t < 0 as any increase in fiscal pressure induces the firm to postpone a portion of sales in order to accumulate additional capacity along the path to the long-run equilibrium (and conversely). A firm may become neutral to profit taxation by setting u ¼ lðtÞ, i.e., the extent of delegation must equal the shadow price attached to an additional unit of installed capacity. Without further proof, Proposition 2 produces two relevant Corollaries. The first is:
0 dq ðtÞ / ½ð1 t Þð A c qðtÞðN þ 1ÞBÞ þ u f ðkÞ r d dt
which is equal to zero in correspondence of the following steady state solutions: qSS ¼
ð A cÞð1 t Þ þ u ; BðN þ 1Þð1 t Þ
0
f ðkÞ ¼ r þ d:
To ease the exposition, define: n o 0 kˆ k : f kˆ ¼ r þ d :
(14)
(15)
That is, kˆ is the level of capacity associated with the Ramsey steady state equilibrium where the marginal productivity of capital is equal to the sum of depreciation and discount rates. The phase diagram of the present model can be drawn in the space fk; qg. The locus q˙ dq=dt ¼ 0 is given by the solutions in (14). The two loci partition the space fk; qg into four regions, where the dynamics of q is summarised by the vertical arrows. The locus k˙ dk=dt ¼ 0 as well as the dynamics of k, depicted by horizontal arrows, derive from (2). Steady state equilibria, denoted by E1, E2 along the horizontal arm, and E3 along the vertical one, are identified by the intersections between loci. Fig. 1 describes only one out of five possible configurations, due 0 to the fact that the position of the vertical line f ðkÞ ¼ r þ d is independent of demand parameters, while the locus q ¼ ½ð A cÞð1 t Þ þ u =½BðN þ 1Þð1 t Þ shifts upwards (downwards) as A c and/or u (B and N) increases. Therefore, we obtain one out of five possible regimes:
Corollary 3. For all u < lðtÞ, profit taxation induces an increase in market price and a decrease in consumer surplus at any t during the game, and conversely. The second Corollary is: 7 Again, the only possible exception would obtain if every ki0 were large enough to allow each firm to produce the optimal steady state output from t ¼ 0 onwards and let capacity depreciate at the rate d at every instant (as in Cellini and Lambertini, 2008).
(13)
Fig. 1. The phase diagram.
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
16
1. 2. 3. 4. 5.
There exist three steady state points, with kE1 < kE2 < kE3 (Fig. 1). There exist two steady state points, with kE1 ¼ kE2 < kE3 . There exist three steady state points, with kE2 < kE1 < kE3 . There exist two steady state points, with kE2 < kE1 ¼ kE3 . There exists a unique steady state equilibrium point, corresponding to E2. 0
The vertical locus f ðkÞ ¼ r þ d is a constraint on optimal capital, determined by firms’ intertemporal preferences, i.e., their common discount rate, and depreciation. Accordingly, maximum output level in steady state would be that corresponding to (i) r ¼ 0, and (ii) a capacity such that f 0 ðkÞ ¼ d. Yet, a positive discounting (i.e., impatience) leads firms to install a smaller capacity at the long-run equilibrium. This is the optimal capital ˆ When the market size A c and the extent of constraint k. delegation u are very large (or B and N are low), points E1 and E3 either do not exist (regime 5) or fall to the right of E2 (regimes 2, 3 and 4). In such a case, the capital constraint is operative and firms choose the capital accumulation corresponding to E2. Notice that, as E1 and E3 entail the same levels of sales, point E3 is surely inefficient in that it requires a higher amount of capital. E1 corresponds to the optimal quantity emerging from the static version of the game. It is hardly the case of emphasising that this solution encompasses both monopoly and perfect competition (as, in the limit, N ! 1). The stability analysis of the above system reveals that8: Regime 1. E1 is a saddle point, while E2 is an unstable focus. E3 is again a saddle point, with the horizontal line as the stable arm. Regime 2. E1 coincides with E2, so that we have only two steady states which are both are saddle points. In E1 ¼ E2, the saddle path approaches the saddle point from the left only, while in E3 the stable arm is again the horizontal line. Regime 3. E2 is a saddle, E1 is an unstable focus. E3 is a saddle point, as in regimes 1 and 2. Regime 4. Here, E1 and E3 coincide. E3 remains a saddle, while E1 ¼ E3 is a saddle whose converging arm proceeds from the right along the horizontal line. Regime 5. Here, there exists a unique steady state point, E2, which is a saddle point. We can sum up the above discussion as follows. The unique efficient and non-unstable steady state point is E2 if kE2 < kE1 , while it is E1 if the opposite inequality holds. Such a point is always a saddle. Individual equilibrium output is qSS if the equilibrium is in E1, or the level corresponding to the optimal capital constraint kˆ if the equilibrium is point E2. The reason is that, if the capacity at which marginal instantaneous profit is nil is larger than the optimal capital constraint, the latter becomes binding. Otherwise, the capital constraint is irrelevant, and firms’ decisions in each period are driven by the unconstrained maximisation of singleperiod profits only. Now, examining the steady state solutions in (14) allows us to state, without further proof, the following result:
Proposition 5. In the market-driven steady state, any profit tax rate t > 0 distorts qSS and kE1 if firms are managerial, i.e., iff u > 0. If instead firms are strictly entrepreneurial, i.e., u ¼ 0, profit taxation is neutral. 0 In the Ramsey equilibrium where f ðkÞ ¼ r þ d, profit taxation is always neutral. 8 See Cellini and Lambertini (1998, 2003) for a detailed stability analysis carried out on the Jacobian matrix of the dynamic system.
Consider the market-driven solution. The effects of profit taxation on the equilibrium level of sales are summarised by:
@qSS u ¼ > 0 8 u > 0: @t BðN þ 1Þð1 t Þ2 Since
the
level
@kE1 =@t / @qSS =@t .9
(16)
of sales is proportional to capacity, This proves a relevant Corollary to
Proposition 5:
Corollary 6. If firms are managerial, the optimal levels of sales and capacity are monotonically increasing in t . In a Ramsey model, capacity accumulation involves the intertemporal relocation of production only. Therefore, being absent any instantaneous costs, the introduction of a profit tax rate has an expansionary effect on capital accumulation whenever firms are managerial, the reason being that firms expand sales so as to try to recover through larger market shares some of the profits extracted by the policy maker. Since this incentive exists for all firms, the effect of taxation is definitely pro-competitive in that it translates into an overall expansion of the industry sales and a reduction in the price level. Given the Ramsey accumulation mechanism, by which the unsold output increases capacity, firms do not bear any fixed cost and the social optimum involves firms producing an aggregate output that must be sufficiently large to drive the market price to marginal cost, with zero profits.10 This can be obtained by setting:
t¯ ¼ 1
Nu : Ac
(17)
The above expression allows us to state our last result, whose interpretation is intuitive:
Proposition 7. Provided Nu < A c, the tax rate Nu Ac drives the industry equilibrium price and sales to their perfectly competitive levels. The rate t¯ is (i) monotonically increasing in the size of the market, A c, and (ii) monotonically decreasing in the number of firms, N, as well as in the extent of delegation, u.
t¯ ¼ 1
4. The feedback solution: sketch As stated above, the Bellman equation that would yield the feedback equilibrium of the game cannot be solved analytically since the game at hand is not a linear-quadratic one. However, a relevant implication of the first order condition can be easily drawn.11 The Bellman equation for firm i is: @V ðkðtÞÞ ½ f ðki ðtÞÞ rV i ðkðtÞÞ ¼ max ½ pðtÞ cð1 t Þqi ðtÞ þ ui qi ðtÞ þ i @ki ðtÞ qi ðtÞ X @V i ðkðtÞÞ qi ðtÞ dki ðtÞþ ½ f ðk j ðtÞÞq j ðtÞ dk j ðtÞ @k j ðtÞ j 6¼ i (18) 9 This can be verified intuitively by observing that, whenever the horizontal arm in Fig. 1 identifies the efficient saddle point steady state, then any increase in qSS SS goes along with an increase in k ¼ kE1 along the positively sloped part of the locus k˙ ¼ 0. 10 This of course is admissible under the proviso that the quasi-static perfect competition outcome with marginal cost pricing is socially optimal in this version of the Ramsey model. This is shown to hold true in Appendix A. 11 Here we exclude the trivial case in which the technology f ðki ðtÞÞ is linear, as in this case the open-loop solution is obviously a degenerate feedback one.
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
where V i ðkðt ÞÞ is firm i’s value function and kðt Þ fk1 ðt Þ; k2 ðt Þ; . . . kN ðt Þg is the vector of states. Now, taking the first order condition, we have: ½A 2Bqi ðtÞ BQ i ðtÞ cð1 t Þ þ ui
@V i ðkðtÞÞ ¼0 @ki ðtÞ
(19)
which coincides with (5) except for the presence of the partial derivative of the value function in place of the co-state variable. Hence, provided @V i ðkðt ÞÞ=@ki ðt Þ 6¼ 0 (which will hold in general, with the exception of the case where also lii ðt Þ ¼ 0, as already mentioned), the same considerations emerging under the openloop rule apply also here, in particular the equivalent of Lemma 1, Proposition 2 and Corollaries 3 and 4. Notwithstanding the fact that we cannot determine the exact form of V i ðkðt ÞÞ, we may nonetheless formulate some additional qualitative considerations on the linkage between the optimal feedback output and the level of profits taxation, by noting that the imposition of symmetry across firms on (19) yields (superscript F stands for feedback): qF ¼
ðA cÞð1 t Þ þ u @VðkðtÞÞ=@kðtÞ BðN þ 1Þð1 t Þ
@qF u @VðkðtÞÞ=@kðtÞ ¼ @t BðN þ 1Þð1 t Þ2
(20)
(21)
Unlike what happens in the open-loop game, where lii ¼ 0 in steady state, here @V ðkðt ÞÞ=@kðt Þ will differ from zero even at the steady state, precisely because firms take into account the feedback effects generated through the vector of states at any time and these will carry over to the long-run equilibrium allocation. Hence, even if an explicit solution remains out of reach, we are able to say that Lemma 1, Proposition 2 and Corollaries 3 and 4 will apply also to the equilibrium and not only to the transition towards it. As a last remark, note that @V ðkðt ÞÞ=@kðt Þ < 0 will be required in order for the value function V ðkðt ÞÞ to be concave. Therefore, the comparison between (16) and (21) reveals that
@qF @qSS @V ðkðt ÞÞ=@kðtÞ ¼ > 0; @t @t BðN þ 1Þð1 t Þ2
17
sales, provided that firms are managerial. If instead firms are under the direct control of stockholders, then taxation is neutral at the long-run equilibrium, but not along the path leading to the steady state. Analogous properties can be shown to hold in the feedback game as well (and they appear to be strengthened by feedback information), although the analytical solution of the associated Bellman equation remains out of reach. Finally, we have also investigated the welfare-maximising tax rate, finding that the Ramsey accumulation rule allows for the attainment of the perfectly competitive outcome through an appropriate tax rate, for any given extent of managerial incentives. Appendix A. The social optimum Here we briefly characterise the socially efficient solution, under the same demand and technological conditions specified in Section 2. A benevolent social planner aims at maximising the discounted flow of social welfare yielded by an industry made up by N firms. Of course, the command optimum is indeed insensitive to the specific market variable being chosen, either a single price or N output levels. To keep the exposition consistent with the foregoing analysis, we shall assume the planner controls the vector of quantities. The instantaneous social welfare function is SW ðt Þ ¼
N X
pi ðtÞ þ CSðtÞ
(23)
i¼1
where ½A pðtÞ CSðtÞ ¼
N X qi ðtÞ i¼1
(24)
2
is the instantaneous consumer surplus level. Accordingly, the planner’s problem is Z 1 max q1 ðtÞ;...qN ðtÞ SWðtÞert dt (25) 0
(22)
which can be spelled out as follows:
Proposition 8. The feedback equilibrium control is more sensitive to the tax rate than the open-loop (market-driven) equilibrium control, all else equal. A possible explanation of this result is to be found in the fact that feedback information intensifies the degree of strategic interaction among firms, thereby leading them to increase production through a well known incentive towards preemption.12 5. Conclusions We have investigated a dynamic oligopoly game embedded in a Ramsey growth model, in order to characterise the effects of gross profit taxation on the performance of firms in steady state. In particular, we have established that, if capacity accumulation is modelled as a dynamic investment process, then taxation exerts distortionary effects on the amount of capital and the level of sales at equilibrium. In the Ramsey-type model we have adopted here, taxation exerts a pro-competitive effect on optimal capacity and 12 See, e.g., Reynolds (1987) for an analogous phenomenon in the partial equilibrium version a` la Solow-Swan of the dynamic oligopoly game with capacity accumulation (Solow, 1956; Swan, 1956).
s.t. the set of dynamic constraints (2) and the vector of initial conditions ki ð0Þ ¼ ki0 for all i ¼ 1; 2; 3; . . . N. Let hi ðtÞ ¼ ni ðtÞert be the co-state variable (evaluated at time t) that the social planner (SP) associates with the state variable ki ðt Þ. The planner’s Hamiltonian function: ( " # ) N N X X HSP ðtÞ ¼ ert pi ðtÞþ CSðtÞ þ hi ðtÞ½ f ðki ðtÞÞ qi ðtÞ dki ðtÞ i¼1
i¼1
(26) yields the following set of necessary conditions:
@HSP ðtÞ ¼ A Bqi ðtÞ BQ i ðtÞ c hi ðtÞ ¼ 0 @qi ðtÞ
(27)
@HSP ðtÞ dhi ðtÞ rhi ðtÞ , ¼ dt @ki ðtÞ dhi ðtÞ ¼ hi ðtÞ½r þ d f ðki ðtÞÞ:
dt
Imposing symmetry across controls, states and co-states, and using (27), we have: 0 dqSP ðtÞ 1 dhðtÞ ½ A c BNqðtÞ f ðkÞ r d ¼ ¼ (29) dt BN dt BN
M. Baldini, L. Lambertini / Japan and the World Economy 23 (2011) 13–18
18
with 8 Ac < dqSP ðtÞ qSP ¼ ¼ 0in BN : 0 dt f ðkÞ ¼ r þ d
(30)
The demand-drive steady state solution qSP ¼ ð A cÞ=ðBN Þ corresponds to marginal cost pricing, while the other steady state solution is the Ramsey golden rule we are accustomed with from the macroeconomic version of the model (Ramsey, 1928). The stability analysis follows the same line as in the oligopoly model, so that the horizontal arm qSP ¼ ð A cÞ=ðBN Þ identifies a saddle point equilibrium if the associated capacity endowment is lower than than the Ramsey capacity kˆ (and conversely). In such a case, adopting a profit tax rate may drive the oligopolistic industry to the (quasi-static) perfectly competitive outcome, as from Proposition 7, although of course not necessarily at the same capital accumulation rate.13 Note that, in general, we may assume u < ðA cÞð1 t Þ=N for all t 2 ½0; t¯ Þ, so that qSP > qSS in such a range. All this of course applies whenever the market-driven optimal output is smaller than the quantity corresponding to the Ramsey golden rule under both regimes (oligopoly and planning). Alternatively, if both qSP and qSS are larger than the output level associated to ˆ the market will converge to the Ramsey the Ramsey capacity k, equilibrium irrespective of whether firms are managerial Cournot oligopolist or are driven by a benevolent planner, and therefore designing taxation to reach the socially optimal steady state is not an issue; if the socially optimal solution is the Ramsey golden rule while managers lead their firms to the market-driven steady state for all t 2 ½0; t¯ Þ, then the adoption of t ¼ t¯ drives them into the 0 Ramsey equilibrium where f ðkÞ ¼ r þ d.
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13 This is due to the fact that, in general, the adoption of the constant tax rate t¯ will not ensure that lii ðtÞ ¼ hi ðtÞ at all times.
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