Convergence to symmetry in dynamic strategic models of R&D: The undiscounted case

*Corresponding author.

Journal of Economic Dynamics & Control25 (2001) 1881}1897

Convergence to symmetry in dynamic strategicmodels of R&D: The undiscounted case

Sumit Joshi��*, Nicholas S. Vonortas��

�Department of Economics, The George Washington University, 2201 G Street NW, Washington,DC 20052, USA

�Department of Economics, The George Washington University, USA�Center for International Science and Technology Policy, The George Washington University, USA

Received 17 April 1997; accepted 16 February 2000

Abstract

Will R&D increase or decrease the asymmetry between "rms over time? We examinethis issue in the context of a dynamic, alternate-move duopoly model of non-cooperativeR&D. The asymmetry we consider is with respect to the initial stocks of technologicalknowledge which provides one "rm a greater potential for current and future pro"ts inthe product market. Utilizing a value loss process constructed from the OptimalityEquation of dynamic programming, we show that for the undiscounted case the asym-metry between the "rms disappears over time. We obtain this convergence-to-symmetryresult by imparting a temporal character to R&D through the notion that R&Dinvestment cannot be changed instantaneously, by allowing research externalities, comp-lementarity or substitutability between own and appropriated R&D, and either increas-ing, decreasing or constant returns to scale in the production of technical knowledgefrom own and rival R&D. � 2001 Elsevier Science B.V. All rights reserved.

JEL classixcation: C72; C73; L13

Keywords: R&D; Dynamic duopoly; Alternate-move; Asymmetry; Convergence

0165-1889/01/$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 1 5 - 4

�See, for example, D'Aspremont and Jacquemin (1988), De Bondt and Veugelers (1991), De Bondtet al. (1992), Kamien et al. (1992), Katz and Ordover (1990), Simpson and Vonortas (1994), andSuzumura (1992).

�The issue of R&D with asymmetric "rm sizes has been explored in the static framework byRosen (1991) and Poyago-Theotoky (1996). These papers do not address the issue which is ofconcern to us * temporal catching up by a lagging "rm in the presence of R&D.

1. Introduction

In the literature on the strategic incentive to invest in R&D, cast in theframework of atemporal two-stage games, the typical approach has been onewhere R&D investment in the "rst stage reduces the production costs of a "rmthereby conferring a strategic advantage in the second stage product marketcompetition.� This strategic role of R&D continues to be important in a dy-namic context providing a channel through which R&D a!ects current pro"ts.However, adopting and extending a formulation "rst developed by Spence(1984), an additional strategic role of R&D in a dynamic framework is toaugment the stock of technological knowledge of a "rm thereby a!ecting futurepro"t possibilities. Within an explicitly temporal paradigm, it is this latter aspectof R&D which is truly vital because it carriers the potential to in#uence theasymmetry between "rms. It is this issue of whether R&D will obviate orexacerbate an asymmetry between "rms which is explored in the paper.�We consider an asymmetry between "rms with respect to di!erent initial

conditions } the competing "rms start with di!erent initial stocks of technicalknowledge. Adopting the linear-demand and constant-returns-to-scale-cost-function speci"cation of the atemporal literature, the asymmetry in the initialstock of technical knowledge has implications for both current and future pro"tsvia the unit cost function. In e!ect then, the "rms start with asymmetric currentand future pro"t possibilities. We demonstrate that, under the same standardassumptions as those employed in the atemporal literature to comparecooperative and non-cooperative R&D, the asymmetry in the unit costs (engen-dered by the asymmetry in the stocks of technical knowledge) disappears overtime and the lagging "rm is able to catch up with the leader. That is, the "rmsgravitate towards an outcome where their current and future pro"t potential} the determinants which in#uence their level of R&D } are identical. The simpleparametrization of demand and cost functions is useful because it enables us topresent the convergence result in a transparent way while staying close to thefunctional forms and the assumptions of the atemporal model.The de"nitive work on temporal catching-up or lagging behind in the context

of the evolution of a duopoly is that of Budd et al. (1993). To maintaintractability in a highly complex stylized model, they kept the role of R&D

1882 S. Joshi, N.S. Vonortas / Journal of Economic Dynamics & Control 25 (2001) 1881}1897

especially simple: a "rm's pro"ts were simply a function of the di!erencebetween its level of R&D and the rival's R&D. This speci"cation, however,abstracted from at least four important features of R&D which we accommod-ate in our analysis.(i) Temporal aspect of R&D: To the extent that investment in R&D involves

creation of research facilities and the acquisition of highly specialized physicaland human capital, there is a degree of "xity in this investment in the sense thatit cannot be changed instantaneously. It is this short run inelasticity whichimparts a temporal character to R&D. In dynamic analysis, where time is ofessence, we need to address this short run in#exibility in R&D investment.In this paper we bring into sharp focus the temporal nature of R&D by

utilizing the notion put forward inMaskin and Tirole (1988a, b) in the context ofquantity and price competition of alternating moves and short-run commit-ment. Extending their framework to accommodate competition in R&D, in ouranalysis a "rm cannot change its investment in R&D for a "nite number of timeperiods. In particular, there are two "rms, i and j, and the competition betweenthem takes place in discrete time, indexed by t51. In any odd-numbered timeperiod t, Firm i determines its R&D level after observing the R&D level of Firmj in period t!1. The R&D level of Firm i is then "xed till period t#2. Identicalconsiderations apply for Firm j which decides on its R&D levels only ineven-numbered time periods. Therefore, in time period 2t#1 (respectively,2t#2), the R&D levels which are observed consist of the current R&D decisionof Firm i (respectively, Firm j) and the previous period 2t (respectively, 2t#1)R&D decision of Firm j (respectively, Firm i). Such an alternative-move frame-work, while retaining a temporal role for R&D, also has the desirable featurethat dynamic programming arguments permit a study of perfect equilibriagenerated by Markov strategies.(ii) Spillovers from R&D: Investment in R&D also generates externalities as the

gains in technical knowledge ensuing from R&D of a "rm cannot in general befully appropriated by the "rm and there are leakages to the rival "rms. This facthas been stressed in the atemporal literature (De Bondt, 1997). In the temporalcontext, its importance is heightened by the fact that if we ignore externalities inR&D, then the asymmetry between "rms resulting from di!erential level ofR&D investments (and the consequent move towards a monopoly outcome) willbe overstated. If research externalities are included in the description of themodel, then the actual asymmetry or gap between the leader and the follower"rm will be smaller given that some fraction of the R&D e!ort of the former willleak to the latter.(iii) Substitutability of complementarity between own and appropriated R&D:

Typically, own and appropriated R&D have been assumed in the literature to besubstitutes. This creates an incentive in terms of ex post cost reduction but alsoa serious ex ante disincentive to invest in R&D that is subject to imitation byrivals. The resulting expectation of low rates of technological progress in

S. Joshi, N.S. Vonortas / Journal of Economic Dynamics & Control 25 (2001) 1881}1897 1883

�Recent evidence suggests that, in certain industries at least, economies of scope may beresponsible for a positive relationship between the size of the R&D e!ort and its productivity toa larger extent than economies of scale (Henderson and Cockburn, 1996).

industries with widespread spillovers, however, is often contradicted by empiri-cal studies (Levin, 1988). The observed phenomenon of clustering of high-technology "rms, for example, has been attributed in part to complementaritiesgenerated by research externalities combined with increasing returns from ownR&D and the fraction of rival R&Dwhich spills over. In such cases* which aretotally compatible with &cumulative' technological advance (see below)* it canbe argued that a "rm invests in R&D not only to advance its own technologybut also to increase its capacity to assimilate knowledge generated elsewhere.This e!ect has been captured formally by endogenizing the absorptive capacityof a "rm (Cohen and Levinthal, 1989). It can also be captured by allowing theR&D expenditure of the "rm to be complementary to that of its rival in the sensethat the knowledge appropriated from the rival increases the marginal produc-tivity of own R&D. Our analysis is #exible enough to permit own R&D andappropriated R&D to be either substitutes or complements.(iv) Returns to scale in the production of technical knowledge: For decades

economists have debated two &Schumpeterian' hypotheses suggesting a positiverelationship between innovation and monopoly power, on the one hand, andbetween innovation and "rm size, on the other. Both hypotheses imply a posit-ive relationship between the size of the R&D undertaking and the ability toinnovate which, since Galbraith (1952), has been supported by many analysts onthe grounds that innovation * a sophisticated, costly, and highly uncertainactivity * has become institutionalized. However, economies of scale in theproduction of technological knowledge have been hard to substantiate empiric-ally (Cohen, 1995). In some industries scale economies may well be presentwhereas in others constant or diminishing returns are more likely (Kamien andSchwartz, 1982; Dasgupta, 1986).� More recently, there has been considerablediscussion over the existence of dynamic economies of scale in the production oftechnological knowledge, primarily as a result of &learning-by-doing', &learning-by-using' and learning-to-learn' which are embedded in the innovation process(Dosi, 1988; Rosenberg, 1982). The implication here is that people and organiza-tions can learn how to improve/create wholly new things by the very processof being engaged in the activity. Dynamic economies of scale are said tobe important when technological advance is `cumulativea (Willinger andZuscovitch, 1988) which, in turn, ties in well with the dynamic model elaboratedin this paper where "rms invest period after period to improve their technology.Our framework is malleable enough to accommodate increasing, constant ordecreasing returns to scale in the production of technical knowledge, and our


convergence-to-symmetry result is robust with respect to any of the threespeci"cations (or, any combination of them).Dynamic programming techniques provide us with an additional dimension

of #exibility. Our result on convergence to symmetry starting from di!erentinitial conditions is reminiscent of a classical turnpike result from the theory ofeconomic growth that optimal paths of capital accumulation from di!erentinitial stocks converge (McKenzie, 1976). Exploiting an idea from turnpiketheory, we use the Optimality Equation of dynamic programming to constructa &value loss process' which records a strictly positive value loss whenever thereis a divergence greater than some critical amount between the stock of technicalknowledge of the "rms. This sensitivity of the value loss process, coupled with itsboundedness, enables us to prove the temporal catching-up property.In this paper we abstract from any uncertainty and restrict attention to an

undiscounted framework. The value loss technique, however, is malleable toextensions of the basic model to a stochastic environment. Uncertainty can beeasily included via the Spence (1984) production function which speci"es howR&D augments the stock of technical knowledge (yielding as a byproduct thevalidity of the convergence result in a stochastic framework). The value lossmethod used in this paper cannot handle discounting (we discuss in the con-clusion to the paper the reason for this by drawing upon our construction of thevalue loss process). Recall that the undiscounted and discounted cases requirevery di!erent treatments even in the in"nitely repeated game literature. Ina related paper (Joshi and Vonortas, 1997), we consider the tools required tohandle the discounted case.

2. A dynamic alternate-move model of non-cooperative R&D

The model is based on Joshi and Vonortas (1996) who extend the alternate-move Cournot duopoly of Maskin and Tirole (1988a) to dynamic R&D games.It is described for Firm i and is identical for Firm j. There are two stages in eachperiod t51. In accordance with the principle of backward induction, we startour description with the second stage.

2.1. Stage 2: The product market

In Stage 1, the tuple (x��, x�

�) of R&D levels of the two "rms is determined

non-cooperatively (in a manner to be elaborated in a later subsection). Thistuple (x�

�, x�

�), via some technical production function, increases the stock of

technological knowledge available to both "rms in period t, (K��, K�

�). In the

second stage, the "rms play a Cournot game in a homogenous product marketwhere they face the linear inverse demand function

P"a!b(q��#q�

�), a, b'0. (1)


�While unit costs lie in the interval (0, a] for any "niteK�, we close this interval by adding the limitpoint �0� so that the range of unit costs is compact. The importance of compactness is discussed inRemarks 2 and 5.

The stock K��in#uences the second stage cost function of Firm i which is

assumed to be of the constant returns to scale form

C(K��, q�

�)"F(K�

�)q�

�. (2)

The unit cost function, F: ��

P��

is assumed to satisfy:(A.1) F is twice continuously di!erentiable on �

�.

(A.2) F�(0 and F�'0 on ��. Further, F(K�)P0 as K�P#R.

(A.3) F(0)"a.

Remark 1. Assumption (A.1) is a standard continuity requirement. Assumption(A.2) delineates the mechanism through which R&D, by e!ecting the stock oftechnical knowledge, in#uences the current pro"ts of the pro"t. This is anextension of a formulation commonly seen in the atemporal literature: the stockof technical knowledge reduces unit costs of production. The Inada condition isdesigned to ensure that unit costs do not go to zero in "nite time. Assumption(A.3), along with (A.2), ensures that the range of the unit cost function isrestricted to the interval [0, a].� Such a restriction on the intercept of the inversedemand and the (constant) marginal cost parameter is common in static sym-metric or asymmetric cost Cournot games in order to ensure positive levels ofoutput in the Nash equilibrium.

Solving for the second stage Cournot}Nash equilibrium in period t, andsubstituting in the objective function, yields the reduced form (or, "rst stage)pro"t function ��: ��

�P�

�:

��(K��, K�

�),

1

9b[a!2F(K�

�)#F(K�

�)]�. (3)

Remark 2. Since the range of the unit cost function is the compact interval[0, a], the reduced form pro"ts will also lie in a compact interval which does notdepend on the tuple (K�, K� ). This is a consequence of the fact that reduced formpro"ts is a quadratic (and, therefore, continuous) function of unit costs, and theimage of a compact set under a continuous mapping is also compact. Thisuniform bound will permit us to derive the Optimality Equation of dynamicprogramming for the undiscounted case by using a result in Ross (1983, The-orems 2.1 and 2.2). A sharper characterization of the bounds on the pro"tfunction by restricting the initial stocks of technical knowledge is provided inLemma 1.


2.2. The production function for technical knowledge

Before describing Stage 1, we "rst turn to how the stock of technologicalknowledge evolves. Given an initial stock K�

�3[0, R) in period 0, technical

knowledge cumulates according to

K��"K�

��#k�

�, t"1, 2,2 . (4)

Technical knowledge in period t, K��, is made up of the technical knowledge

accumulated in the previous (t!1) periods, K��

, and the increment to thisstock of technical knowledge in period t, k�

�. The latter is determined by the

production function

k��"g(x�

�, �x�

�), t"1, 2,2, (5)

where g: ��

P��

satis"es the following:(A.4) g is a continuously di!erentiable function on ��

�.

(A.5) g is strictly increasing in x� given x�; g is strictly increasing in �x�

given x�.

Remark 3. In this formulation, the technical knowledge of Firm i increases inperiod t through the R&D expenditures of both "rms: its own research, x�

�, and

the fraction �x��(0(�(1) of the research of Firm j which it can appropriate via

spillovers. Further, the production function for the increment to technicalknowledge is speci"ed generally enough to accommodate either increasing,decreasing or constant returns to scale in the production of the stocks oftechnical knowledge with respect to own R&D and appropriated R&D. It alsoallows substitutability or complementarity between own and appropriatedR&D. An important special case in the literature is the well-known productionfunction of Spence (1984) which embodied constant returns to scale in additionto requiring perfect substitutability between own and appropriated R&D,k��"x�

�#�x�

�.

Using the fact that the stocks of technological knowledge are an increasingsequence, we will obtain an important uniform boundedness property of thepro"t function. From (A.1)}(A.3) and the intermediate value theorem, thereexists K

�3(0, R) such that F(K

�)"a/3. Note that for all stocks of technical

knowledge greater than K�, unit cost is less than a/3.

¸emma 1. Under Assumptions (A.1)}(A.3), for any tuple (K��, K�

�)5(K

�, K

�), there

exists constants 0(�(�M (R such that

�4��(K��, K�

�)4�M .


� It is also in keeping with the atemporal literature where R&D investment is "nanced through theproduct market pro"ts. Permitting otherwise would raise di$cult questions relating to the modes of"nancing investment, the timing of payments, the possibility of default, etc. Note that this does notnecessarily mean that "rms cannot save; the second-order properties of the reduced form pro"tfunction can dictate an interior maximum.

Proof. From the restriction on the initial stocks, 2F(K��)42a/3. Hence,

a!2F(K��)5a/3. Since F(K�

�)50, the lower bound follows by letting

�"a�/81b. Similarly, noting that a!2F(K��)(a and F(K�

�)4a/3, the upper

bound follows by letting �M "16a�/81b. �

Remark 4. Lemma 1 shows that if we pick the initial stocks of technicalknowledge, (K�

�, K�

�), to be larger than (K

�, K

�), then the pro"ts of both "rms

are uniformly bounded away from zero in each subsequent time period. Thisresult is important for two reasons:

(i) In our model, "rms lack external sources of "nancing R&D and funnel theproduct market pro"ts from the previous period to "nance current invest-ment in R&D. This is formalized below in the inequality given by (6).� If thepro"ts of the lagging "rm could be driven to zero in "nite time, then thedivergence in the stocks of technological knowledge between the leader andthe follower would be heightened thereby endowing one "rm with a monop-oly position in the product market; the high-cost follower would never beable to catch up with the leader.

(ii) The uniform bound prevents the low-cost "rm from making strategic R&Ddecisions to force the high-cost "rm out of the market. Therefore, we modelthe optimal R&D decision as the solution to a "rm maximizing its ownintertemporal pro"ts. This is an extension of the multistage atemporal entrydeterrence analysis of Fudenberg and Tirole (1984). In their analysis, if the"rm wishes to keep a rival "rm out of the market, it chooses its R&D not tomaximize its own pro"ts but to drive its potential rival's pro"ts down tozero; if it cannot force the other "rm out, then it accommodates bymaximizing its own pro"ts and softening the post-entry equilibrium. Thepositive lower bound on pro"ts implies that no "rm can force the other out;therefore, each &accommodates' by maximizing its own payo!.

2.3. Stage 1: The alternate move structure of R&D

We now tackle the alternating-move structure of the "rst stage game. Both"rms invest in R&D in all time periods. The time between two consecutivediscrete time periods is T. It is for this length of time that a "rm cannot change itsinvestment in R&D. If t is odd, Firm i chooses its R&D level x�

�which is then


Because of (6), all &feasible' R&D sequences lie in the set ��[�, �M ] which is compact by

Tychono! 's theorem. Further, from the continuity of the reduced-form pro"t functions (Remark 2),it is easily established that the set of R&D sequences satisfying (6) is closed. The conclusion nowfollows from the result that closed subsets of compact sets are also compact.

"xed till time period t#2. If t is even, then Firm j chooses its R&D levelx��which then remains "xed till time period t#2:

x��"x�

��, t odd; x�

�"x�

��, t even.

In choosing their R&D levels, each "rm observes the R&D level of the other"rm in the previous time period and the accumulated stock of technical know-ledge of both "rms. Further, in any odd (respectively, even) time period when the"rm has the discretion to vary its level of investment, Firm i's (respectively, Firmj's) choice of R&D expenditure is constrained by the product market pro"tsavailable from the previous period. In particular, letting ��

�(respectively, ��

�) be

the initial investment funds for Firm i (respectively, Firm j), we impose thefollowing feasibility constraint:

x��4��(K�

��, K�

��), t odd; x�

�, (K�

��, K�

��), t even. (6)

Remark 5. As noted earlier, the feasibility constraint is in keeping with theatemporal literature where investment in R&D is funded through productmarket pro"ts. In the dynamic case, given the uniform bounds on the reduced-form pro"t functions by Lemma 1, the feasibility constraint facilitates theexistence of an equilibrium by ensuring that the set of &feasible' R&D sequencesis compact.

Given the time-variance of the pro"t and production functions, we restrict ourattention to equilibria where the "rms employ stationary Markov R&D strat-egies which take the form

x��

"��(K��, K�

��, x�

��; ��

�, �), x�

��"x�

��, t"0, 1, 2,2,

x��

"��(K��

, K��

, x��

: ��, �), x�

��"x�

��, t"1, 2,2 . (7)

There may, of course, be other equilibria generated by non-stationary non-Markov R&D strategies. Following Du$e et al. (1994), we con"ne attention tostationary Markov strategies for two reasons. The "rst is analytical tractability.Second, given their relative simplicity in relation to more complex time-varyingrules, they are likely to be the focal points of the dynamic game. Our result onconvergence to symmetry is shown only for the equilibrium consisting ofMarkov strategies. The Markov strategies of each "rm, in addition to the statevariables, is in general also parametrized by the initial investment funds and thespillover term. From now on, to save on space and notation, we only indicatethe pertinent argument of the Markov strategies.


In this paper we assume that "rms are su$ciently patient and do not discountthe future. With no discounting, however, the in"nite sum of one-period pro"tsof a "rm will invariably diverge. This requires a de"nition of optimality alterna-tive to one of maximizing the in"nite stream of one-period pro"ts. Conse-quently, in this paper, we use an average reward criterion. Given � �, theintertemporal pro"t of Firm i generated from time period s onwards from anystate (K�

�, K�

�) by the Markov strategy � � is given by

��(s, K��, K�

�; ��, �� ),liminf

��

(1/N)��

��(K��, K�

�). (8)

A pair of Markov R&D strategies, (��H, ��

H), constitute a Markov perfect equilib-

rium (MPE) if for any s"1, 2,2, and any state (K��, K�

�)

��H

3argmax��(s, K��, K�

�; ��, ��

H)�,

��H

3argmax��(s, K��, K�

�; ��

H, �� )�. (9)

Remark 6. The reward functions �� and �� have been introduced to formallyde"ne aMPE in the undiscounted case. In the next section, we will show that theMPE satis"es the functional equation of dynamic programming. In our sub-sequent analysis, we will draw upon this functional equation and will not refer tothe reward functions explicitly. Also note that in this paper, we do not digressinto the technical issue of the existence of a MPE because such results based on"xed point theorems are by now well established (for instance, Dutta andSundaram, 1992).

2.4. The optimality equation for the undiscounted model

The Optimality Equation of dynamic programming for the undiscountedmodel can be determined by considering the discounted model. We providea heuristic discussion of the derivation; a rigorous presentation is available inRoss (1983, Chapter V). Consider a discount factor 3(0, 1). Note from Remark2 that the in"nite sum of discounted one-period pro"ts lies in a compactinterval. Now, from dynamic programming arguments, (�� , ��) is a -MPE ifand only if there exists -value functions (<�� , J��) such that �� is the solution to

<�� (K��, K�

�, x�)"Sup[��(K�

�#g(x�, �x� ), K�

�#g(x�, �x�))

#J��(K��#g(x�, �x� ), K�

�#g(x�, �x�))], (10)

where the supremum in (10) is taken over all x�4��. <�� is the -value function

when in any (odd) time period t, Firm i chooses its current R&D optimallyconditional on the R&D of Firm j from the previous period, and in the futureboth "rms react optimally according to (�� , �� ) when they have the discretion tochange their R&D levels. Note that <�� will also in general depend on the


parameters (��, ��

�). We have not explicitly referred to this in order to simplify

the notation. J�� is the -value function when Firm i cannot vary its R&D whileFirm j can choose its R&D level optimally contingent on Firm i's R&D levelfrom the previous period; subsequently both "rms, when they have the discre-tion to change R&D, react optimally according to (�� , �� ). By recursivity orstationarity of the dynamic programming problem

J��(K��, K�

�)"�

�(K�

�#g(x�, �� (x� )), K�

�#g(��(x� ), �x� ))

#<�� (K��#g(x�, ��(x�)), K�

�#g(��(x�), �x�), ��(x�)), (11)

Parallel arguments apply for Firm j. Now "x an arbitrary state,(KI �

�, KI �

�, x �

�, � �

�, � �

�), and de"ne

��(K��, K�

�, x� ),<��(K�

�, K�

�, x� )!<��(KI �� , KI �� , x �), (12)

��(K��, K�

�),J��(K�

�, K�

�)!<��(KI �� , KI �� , x �). (13)

Note once again that for notational ease we have not shown the dependence ofthe value functions on the tuple (� �

�, � �

�), Then (10) can be rewritten as

(1!)<�� (KI �� , KI �� , x �)#�� (KI �� , KI �� , x �)

"Sup[��(K��#g(x�, �x� ), K�

�#g(x�, �x�))

#��(K��#g(x�, �x�), K�

�#g(x�, �x�))],

where the supremum is taken over all x�4��. The next result shows that the

Optimality Equation for the undiscountedmodel can be derived by letting P1in the above expression.

¸emma 2. Assume (A.1)}(A.5) hold. Then:

(i) There exists bounded functions (��, ��) and a constant g� such that:

g�#��(K��, K�

�, x�)"Sup[��(K�

�#g(x�, �x� ), K�

�#g(x�, �x�))

#��(K��#g(x�, �x�), K�

�#g(x�, �x�))], (14)

where the supremum in (14) is taken over all x�4��

and where along somesequence �

; n"1, 2,2�:

��(K�, K�, x�)" lim�

�� (K�, K�, x�), ��(K�, K�)" lim�

�� (K�, K�),

g�"lim��

(1!)<��(KI �� , KI �� , x��).

(ii) Given a MPE (��H, ��

H) dexned by (9), the Markov R&D strategy ��

Hattains the

supremum in (14) for any given state (K�, K�, x�).


Proof. The "rst part of Lemma 2 follows from Ross (1983, Theorem 2.2) if themodi"ed value functions given by (12) and (13) lie in a compact interval for any3(0, 1) and any given state. But this follows from Remark 2 since the reducedform pro"t function lies in a compact interval which does not depend onthe tuple (K�, K�). The second part of Lemma 2 follows from Ross(1983, Theorem 2.1). �

We will now rewrite (14) in an alternative form which will simplify the valueloss theorem of the next section. For this purpose, we will assume the existenceof a &worst' initial state, the one-period pro"ts from which are Pareto-dominatedby the one-period pro"ts from a given initial state.(A.6) There exists an initial state, (KK �

�, KK �

�, x( �

�, �( �

�, �( �

�), such that the reduced

form pro"ts generated by theMPE (��H, ��

H) is dominated coordinatewise by the

reduced form pro"ts generated by (��H, ��

H) from the given initial state

(K��, K�

�, x�

�, ��

�, ��

�):

��(K��, K�

�)5��(KK �

�, KI �

�), t"1, 2,2 .

Remark 7. This assumption does not involve any loss of generality. For in-stance, we can let the &worst' state be (KK �

�, KK �

�, x( �

�, �( �

�, �( �

�),(0, 0, 0, 0, 0). It then

follows from Assumption (A.3), Eq. (3) and inequality (6) that reduced formpro"ts are zero in every period starting from the &zero state'. Consequently, thesezero pro"ts are dominated by pro"ts from any given initial state satisfying therestrictions of Lemma 1 and with strictly positive level of initial investmentfunds.

With assumption (A.6) the Optimality Equation characterizing the MPE(��

H, ��

H) becomes

�K �(K��, K�

�, x�)"��(K�

�#g(��

H, �x�), K�

�#g(x�, ��

H))!��(KK �

�, K�

�)

#�K �(K��#g(��

H, �x�), K�

�#g(x�, ��

H)) (15)

any, by recursivity

�K �(K��, K�

�)"��(K�

�#g(x�, ��

H(x�)), K�

�#g(��

H(x�), �x�))!�

�(KK �

�, KK �

�)

#�K �(K��#g(x�, ��

H(x�)), K�

�#g(��

H(x�), �x�)), ��

H(x�)), (16)

where we have de"ned

�K �(K��, K�

�, x�

�),��(K�

�, K�

�, x�

�)!��(KK �

�, K�

�, x�

�)

and similarly for �K ��. Essentially, we have taken the Optimality Equation similar

to (14) for the &worst' initial state and subtracted it from that corresponding tothe given initial state to cancel out the g� term and generate (15) and (16).Identical considerations apply for Firm j.


�We provide the proof in detail because it is applicable to general reduced form pro"t functionsand will, therefore, be helpful in future extensions of the linear model to general parametric forms ofdemand and costs.

3. The value loss process

We now construct a value loss process using (15) and (16) which is instrumen-tal in proving convergence to a symmetric outcome. For any �'0, the objectiveof the value loss process is to display sensitivity to a divergence greater than� between the unit costs of the two "rms. In particular, the main result of thissection is:�

¹heorem 1. Assume (A.1)}(A.6) hold and let (K��, K�

�)5(K

�, K

�). There exists

a bounded process ��; t"1, 2,2� satisfying the following sensitivity property:

for any �'0, there exists a constant �(�)'0 such that for any period t"1, 2,2:

F(K��)!F(K�

�) 5�N�

�!�

��'�(�). (17)

Proof. Consider Firm i and de"ne the process ��; t"0, 1, 2, 2� as follows:

��,�K �(K�

�, K�

�, x�)

and for any t"1, 2,2:

��,�K �(K�

�, K�

�, x�

�), t odd; ��

�,�K �(K�

�, K�

�), t even. (18)

Further, let

FM�"max�F(K�

�), F(K�

�)�, t"0, 1, 2,2.

Then, for any t"1, 2,2, it follows that

��!��

��"

1

9b[a!2F(K�

�)#F(K�

�)]�!

1

9b[a!2F(KK �

�)#F(KK �

�)]�

5

1

9b[a!2F(K�

�)#F(K�

�)]�!

1

9b[a!2FM

�#F(K�

�)]�

"

2

9b[2a!2FM

�!2F(K�

�)#2F(K�

�)][FM

�!F(K�

�)]

5

2

9b[2a!4FM

�#2F(K�

�)][FM

�!F(K�

�)]

5

8a�

81b[FM

�!F(K�

�)]. (19)


The "rst equality follows from (15) or (16), depending on whether t is odd oreven, respectively, and substituting from Eq. (3) for the reduced pro"t function;the subsequent inequality follows by adding and subtracting the expression[a!2FM

�#F(K�

�)]�/9b and recalling that it dominates the corresponding pro"ts

from the worst state; the next equality follows from the fact that an expression ofthe type ��!�� can be written as (�#�)(�!�); the next inequality follows asa consequence of FM

�5F(K�

�); the last inequality follows, as in Lemma 1, from

2a!4FM�52a/3 (since FM

�4a/3 from the choice of the initial stocks of technical

knowledge) coupled with 2F(K��)50.

Now suppose that F(K��)!F(K�

�) 5�. Note that if F(K�

�)!F(K�

�)5�, then

FM�!F(K�

�)"0; if F(K�

�)!F(K�

�)5�, then FM

�"F(K�

�) and FM

�!F(K�

�)5�.

Combining these facts with (19) yields

��!��

��'�

0 if F(K��)!F(K�

�)5�,

�8a�

81b,�(�) if F(K�

�)!F(K�

�)5�.

(20)

Following an identical argument, we can construct a process ��; t"0, 1, 2,2�

for Firm j where

��,��(K�

�, K�

�), ��

�,��(K�

�, K�

�, x�

�), t even, ��

�,��(K�

�, K�

�), t odd

and which satis"es

��!��

��'�

0 if F(K��)!F(K�

�)5�,

�(�) if F(K��)!F(K�

�)5�.

(21)

Combine the value loss processes for Firms i and j by letting ��"��

�#��

�,

t"0, 1, 2,2. Boundedness of the process follows from Lemma 2. The sensitiv-ity property described by (17) follows from inequalities (20) and (21). �

Remark 8. The restriction on the initial stocks of technical knowledge has twoimportant roles in the analysis. First, as seen in Lemma 1, it prevents anasymmetric monopoly outcome by ensuring strictly positive pro"ts for both"rms. Second, as seen in (19), it is instrumental in ensuring that the process��

�; t"0, 1, 2,2� records a strictly positive value loss of �(�) whenever the unit

cost functions of the two "rms are more than �-distance apart.

4. Convergence to symmetry

Utilizing the value loss process we now show that the divergence in the unitcost functions of the two "rms, which is at the source of the asymmetry in cur-rent and future pro"t possibilities between the two "rms, is eradicated over time.


¹heorem 2. Assume (A.1)}(A.6) hold and let (K��, K�

�)5(K

�, K

�), K�

�OK�

�.

Then, F(K��)!F(K�

�) P0 as tPR.

Proof. Consider any �'0 and de"ne the binary variable ��, t50 as

��"�

0 if F(K��)!F(K�

�) (�,

1 if F(K��)!F(K�

�) 5�.

(22)

Then ��

��is the total number of time periods the unit cost functions of the two

"rms are more than � distance apart. Using the sensitivity of the value lossprocess ��

�; t"0, 1, 2,2� given by (17), for any arbitrary ¹"1, 2,2

�(�)��

��4

��

(��!�

��)"�

�!�

��

(��,�K �(K�

�, K�

�, x� )#�K �(K�

�, K�

�).

The extreme term on the right-hand side is bounded independently of ¹ (fromLemma 2i), yielding �

��(R from the Monotone Convergence theorem.

Hence, ��P0. That is, for any 0(�(1, it is possible to choose a t(�, �)(R

such that ��(� for all t5t(�, �). Since ��

�can only take the values 0 or 1, it

follows that ��"0 for all t5t(�, �).

We now claim that F(K��)!F(K�

�) P0. If not, then there exists a ��'0 such

that for each t we can "nd a t�5t with F(K�� )!F(K�

� ) 5��. Constructing t(��, �)

as in the preceding paragraph, this implies using (22) that we can "nda t5t(��, �) such that ��

�"1, a contradiction. �

Remark 9. Theorem 2 shows that the assumptions of the atemporal multistagemodel drive the symmetry result of the dynamic model. It is useful at this pointto highlight the essential features of the model which lead to convergence toa symmetric outcome. This result is critically dependent on twin properties ofthe value loss process: its boundedness and its sensitivity. Note that by construc-tion and the bounded value assumption, R'�

�'�

�'�

�'2. Hence, the

sequence ��; t"0, 1, 2,2� converges. From the triangle inequality, this means

��!�

��can be made arbitrarily small by choosing t large enough. But now

note from the contrapositive of the sensitivity property given by (17) that F(K�

�)!F(K�

�) can then be made arbitrarily small for a su$ciently large

t giving us the desired temporal catching-up property.

5. Conclusion

In this paper we examined the issue of whether R&D will increase or decreasethe asymmetry between "rms in the context of an alternate-move undiscounteddynamic model of non-cooperative R&D with Cournot competition in each


period. Our model had the desirable characteristics of imparting a temporaldimension to R&D (through the notion of short term commitment), ac-commodating externalities in R&D investment, allowing substitutability orcomplementarity between own and appropriated R&D, permitting increasing,decreasing or constant returns to scale in production of technical knowledgefrom own and spillover R&D, and exploiting to the fullest extent the tools ofdynamic programming and turnpike theory. Our primary objective was to showthat the linear paradigm and the assumptions of the atemporal multistage R&Dliterature (which was used to compare cooperative and non-cooperative R&D)could be used in a di!erent context to productively analyze the dynamic issue ofconvergence or divergence between asymmetric "rms over time. We demon-strated that, maintaining the same assumptions as are standard in the multistageliterature, and adopting their linear parametrization, we could show conver-gence to symmetry under suitable restrictions on the initial stock.Our analysis is restricted, however, by the assumption of a homogeneous

product market with linear inverse demand and constant-returns-to-scale costfunction. There are two important dimensions along which the convergence-to-symmetry result needs to be extended. The "rst is to allow general demand andcost functions into the analysis. Since reduced form pro"ts may no longer beuniformly bounded away from zero as in Lemma 1, this may introduce thepossibility of an asymmetric monopoly outcome. Second, it may be desirable toallow discounting in the model. The problem with accommodating discountingin the present framework can be seen from the fact that the value loss process,derived from the Optimality Equations, becomes ��

�; t"0, 1, 22� and re-

cords a value loss of ��(�) whenever the unit cost functions of the "rms are morethan �-distance apart. Since value loss is no longer uniformly bounded awayfrom zero, Theorem 2 is no longer valid. This means that, as in optimal growththeory with discounting (McKenzie, 1976), the value loss process has to beconstructed via a shadow price system (derived from separating hyperplanetheorems) which supports the value functions. These two issues have beenexplored partially in Joshi and Vonortas (1997).

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Convergence to symmetry in dynamic strategic models of R&D: The undiscounted case

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