FS IV 95 - 24
Simultaneous Choice o f Process and Product Innovation
Stephanie Rosenkranz
September 1995
ISSN Nr. 0722 - 6748
discussion papers
Forschungsschwerpunkt Marktprozeß und Unter nehmensentwicklung Research Unit
Market Processes and
Zitierweise/ Citation:
Stephanie Rosenkranz, Sim ultaneous Choice o f Process and P ro du ct Innovation, Discussion Paper FS TV 95 - 24,
Wisenschaftszentrum Berlin, 1995.
W issenschaftszentrum Berlin fur Sozialforschung gGmbH,
ABSTRACT
Simultaneous Choice of Process and Product Innovation
This paper investigates the strategic decisions o f two identical duopolists, who choose production technology as well as product differentiation through their R&D investment. The product market is characterized by heterogeneous Cournot competition. Firms have an incentive to invest into both, process innovation and product innovation. The optimal division between these two kinds o f R&D activities changes with the market size. The higher consumers' willingness to pay, the more firms' investment is driven to product differentiation. If firms coordinate their R&D activities and share R&D costs but remain rivals in the product market, they will reduce costs and differentiate their products more than under competition. The optimal proportion o f R&D investment is driven more to product innovation than under R&D competition. Further it can be shown that welfare is increased if firms coordinate their research activities and share R&D costs. When firms cooperate but do not share their R&D costs, welfare is only enhanced if product innovations are not too expensive.
ZUSAMMENFASSUNG
Simultane Wahl von Prozeß- und Produktinnovation
Dieses Papier beschäftigt sich mit den strategischen Entscheidungen von identischen Duopolisten, die über ihre Forschungs- und Entwicklungsinvestitionen den Umfang ihrer Prozeß- und Produktinnovationen bestimmen. Auf dem Produktmarkt konkurrieren die Unternehmen über die anzubietenden Mengen. Es kann gezeigt werden, daß die Unternehmen einen Anreiz haben in beide Arten von Innovationen zu investieren. Die optimale Aufteilung ihres Forschungsbudgets verändert sich mit dem Marktpotential. Je größer die Zahlungsbereitschaft der Konsumenten, desto mehr lohnt es sich für die Unternehmen, in die Differenzierung ihrer Produkte zu investieren. Wenn die Unternehmen ihre Forschungsstrategien koordinieren und gleichzeitig ihre Forschungskosten teilen, aber trotzdem weiterhin bei der Vermarktung ihrer Produkte miteinander konkurrieren, dann erhöhen sie ihre Investitionen in Verfahrensinnovationen und Produktdifferenzierung. Die optimale Aufteilung des Forschungsbudgets verschiebt sich jedoch im Vergleich zur Wettbewerbssituation zugunsten von Produktinnovationen. Weiterhin kann gezeigt werden, daß sich die Wohlfahrt erhöht, wenn die Unternehmen bei gleichzeitiger Teilung der Forschungskosten ihre Strategien koordinieren. Wenn die Unternehmen im Joint Venture ihre Forschungskosten nicht teilen, dann ist diese Form der Kooperation nur dann wohlfahrtssteigemd, wenn Produktinnovationen nicht zu teuer sind.
1 Introduction
Em pirical evidence indicates th a t firms usually have a portfolio of R&D projects, some targ eted m ore a t process innovations, some at pro d u ct in
novations. The m anagem ent literatu re for exam ple m entions ratios like 60:40 of to tal R&D budget for process relative to product innovation in Japanese firms.1 In fact, it is frequently discussed th a t Japanese firms invest more in process innovations while w estern firms concentrate m ore on pro d u ct in
novations. As possible reasons for differing innovation strategies, cultural distinctions as well as differences in firm size are specified.2 In th e m odel pre
sented here, a th ird possible explanation will be presented. It tu rn s out th a t th e m agnitude of po ten tial m arket dem and also influences firm s’ innovation strategies.
The em pirical observation allows at least for th e vague conclusion th a t firms choose strategically am ong th e two alternative kinds of innovation, usually w ithout a com plete specialization in one. This fact was widely ignored in the microeconomic theory of innovation. Most models suggest th a t firms invest either in process innovation or product innovation and determ ine optim al innovation strategies under this assum ption. In this paper th e analysis is extended by allowing firms to choose optim ally between b o th kinds of inno
vation. T he first intention of this paper is to show th a t an optim al division of R&D investm ent can be established. The second focus of analysis is on w hether this optim al division is altered when th e size of th e m arket in the sense of m arket po ten tial changes. At first sight, this question is related to
1See Imai (1992) and Kuemmerle (1993).
2See Albach (1994). Cultural reasons for this difference are discussed by distinguish
ing between the process-orientation in Japan originating from the samurai tradition and the result-orientation in America and Europe originating from Calvinistic moral values.
Scherer (1989) suggested the effects of different firm sizes as possible explanations. See also Gilbert & Strebel (1987).
th e “technological life-cycle” m odel of A bernathy & U tterback (1982) based on em pirical evidence and intuitive reasoning. It pictures th e absolute fre
quency of product and process innovations in a p roduct line and its associated production process. Initially, when m arket needs are ill-defined b u t m arket p otential is large, product innovations will tend to be predom inant. T he em phasis will change from product to process innovation when perform ance criteria have been standardized and prices become th e new critical factor of success.
The gam e theoretic model presented here is a static approach and product innovation is understood as product differentiation or m arket segm entation.
The num ber of firms is given exogenously. In this setting it can be shown th a t an increase in consum ers’ willingness to pay causes firms to increase R&D investm ent b u t also to shift th eir R&D investm ent m ore to product innovation if the R&D efficiencies of th e two kinds of innovation are similar.
The sam e is tru e if firms cooperate in th eir R&D decisions. Obviously, pro
cess innovations get less “tough” when th e m arket size increases. This result relates th e m odel to the work of Fudenberg & Tirole (1984) and De Bondt
& Veugelers (1991).
A nother question addressed in this paper concerns th e effects of cooperative R&D activities. Two different forms of cooperative agreem ents are analyzed.
The case of R& D -cartel is com pared to an R JV -cartel, assum ing th a t the la tte r allows firms not only to coordinate R&D strategies as to maximize joint profits b u t also to share R&D efforts. It can be shown, th a t firms in
vest m ore in an R JV -cartel th a n under com petition and th a t investm ent is shifted tow ards product innovation. Furtherm ore, cooperative agreem ents have positive welfare im plications if they include th e sharing of efforts. In this setting, th e increase in efficiency (no wasteful duplication of efforts) will
always overcom pensate the im pact of negative externalities on firm s’ invest
m ent incentive. If th e cooperative agreem ent is lim ited to pure jo in t profit m axim ization, welfare im plications depend strongly on the assum ptions con
cerning th e underlying R&D-cost functions. Still, it is most probable th a t firms will shift th eir investm ent towards product innovation. We find th a t product innovations inhibit positive externalities if consum ers’ dem and is characterized by “love of variety” . In relation to the findings of D ’Asprem ont
& Jacquem in (1988) it can be shown th a t this positive com petitive spillover on th e product m arket can (under certain conditions concerning th e R&D cost functions) outweigh negative externalities from process innovation. It can therefore take th e role of technological spillovers and cause welfare im provem ents in case of cooperative decision m aking w ithout cost sharing.
B ester & P etrakis (1993) analyze th e incentives of a firm to invest in process innovation or cost reduction given the m arginal costs of its rival and given some degree of product differentiation. They analyze com parative static prop
erties w ith respect to product su b stitu tab ility and analyze these properties under b o th C ournot and B ertrand com petition. W hen goods are im perfect su b stitu tes, b o th forms of com petition induce the firm to underinvest com
pared to th e social optim um . Overinvestm ent can occur when th e goods are sufficiently close substitutes. Cournot com petition provides a stronger or weaker incentive to innovate depending on w hether th e degree of sub
stitu ta b ility is low or high. Their assum ptions concerning th e dem and side are sim ilar to those of the model presented here. B ut Bester & Petrakis re
duce th eir analysis only to th e process innovation decision of one single firm.
Here th e analysis is extended to both firms, choosing two variables sim ul
taneously, m arginal costs and product substitutability. The com parison of different forms of m arket com petition is om itted.
The following section describes the dem and stru ctu re of a simple differen
tiated duopoly. Section 3 derives firm s’ quan tity decisions for any level of m arginal costs and product substitutability. C oncentrating on sym m etric equilibria only, we characterize optim al investm ent behavior under R&D com
p etition in section 4 and com pare this to investm ent under R&D cooperation in sections 5.1 and 5.2. C om parative statics in th e size of th e m arket and a welfare analysis are presented in section 6 and in section 7. The last section provides some concluding rem arks.
2 The model
T he dem and stru ctu re is adopted from Dixit (1979). The representative con
sum er’s utility is a function of consum ption of th e two goods and th e nu
m eraire good m . It is given by U f a , Xj, d) + m w ith
U (xi, Xj, d) ~ a(xi + Xj) — (x 3 + 2dxiXj + ^ j ) / 2
where a > max[c,-, Cj] w ith c,, Cj representing firm s’ m arginal production costs and 0 < d < 1. The assum ption th a t preferences are linear in th e num eraire good elim inates income effects and allows to perform a p artial equilibrium analysis. The specification of {/(.) generates a linear dem and stru ctu re which simplifies the analysis. The param eter d m easures th e degree of product sub
stitutability. The higher d, th e higher is the degree of substitutability. W hen d tends to zero, th e two firms effectively becom e m onopolists, when d = 1 the two goods are perfect substitutes.
T he inverse dem and function of firm i is the linear function P;(a, £;, Xj, d) = a — (xi + d x j).3
3In the following consumers’ prohibitive price a is only given explicitly when it is discussed in comparative statics.
On the supply side we consider an oligopolistic industry, consisting of two firms i , j th a t produce a differentiated good. The two firms operate under constant returns to scale. Firm s unit costs of production are given by c,- and Cj wdth c,-,Cj £ [0, o], which can be chosen through R&D investm ent before th e m arket opens. T he degree of product differentiation given by d :=
d — di — dj can also be influenced by the firms i , j through R&D investm ent in di, and dj respectively, w ith d i,d j £ [0, |] . An im p o rtan t assum ption is th a t any technological process innovation in one firm has no value for the com petitor. T h a t is, no technological spillovers exist. This could be th e case for duopolists producing different goods by employing a different technology.
For investm ent into product differentiation, th e opposite is true. Investm ent by one firm has a direct effect on the profit of its rival due to com petitive spillovers. Spillovers in this kind of innovation are com plete. Firm s do not com pete in a p aten t race where only one firm can win and adopt th e more efficient technology exclusively. The cost function for R&D is th e sam e for bo th firms and is described by /<(c!) + G'(d,) w ith K ' < 0, G' > 0 and K " > 0, G" > 0.4 The higher th e m arginal costs a firm chooses, th e lower is th e needed research investm ent whereas a higher level of product differentiation requires a higher R&D investm ent. Further, it is assum ed th a t lim ^ ^ o /< (c ,) = oo and limdt_ j / 2 G (di) = co to guarantee interior solutions.
Firm s play a non-cooperative two-stage game under com plete inform ation. In the first stage, they decide on th eir m arginal costs by investing in a research project generating a process innovation. Simultaneously, they decide on the optim al degree of product differentiation by investing in another research project generating a (non essential) product innovation. On th e second stage, firms choose quantities. Since these two-stage games are solved by backward 4In section (4) it is shown, th at the slope of both R<hD cost functions has to be suffi
ciently high to guarantee the existence of a unique symmetric pure strategy equilibrium.
induction, we first analyze firm s’ quantity decision in th e second stage.
3 Quantity decisions
The q uantity gam e is a p artial equilibrium model w ith heterogeneous prod
ucts and C ournot com petition. Since we consider a m arket in which th ere are two firms offering quantities X{ and xy, firm z’s profit function is given by:
7Tj(xi,Xj) = X i P i f a i , X j , d ) — C iX i (1)
Each firm maximizes its profit given the quan tity chosen by the other firm.
The first-order condition of profit m axim ization for firm i is:
— P i ( x i j X j , d ) — a — X i — 0. (2)
Substituting and solving gives th e equilibrium solution of the C ournot sub
game:
* _ 2(a — Ci) — d(a — Cj)
The game obviously has a unique equilibrium in pure strategies in which the firms choose x*(ci, Cj, d) and
X
j(
cj,
c,-, d).5 For fu rth er analysis it is useful to analyze com parative static properties of th e optim al quantities w ith respect to changes in m arginal costs and in th e degree of product substitutability.D ifferentiation of (3) w ith respect to Ci and Cj yields:
- 2 (4 - d2)
d (4 - d2)
< 0 ,
> 0.
(4) (5) 5Second-order conditions are satisfied and c,- = constant and P " = 0 and also
D ifferentiating bo th first-order conditions w ith respect to th e degree of prod
u ct differentiation d, su b stitu tin g and solving gives:
2x* - dx-
3 > 0 for x* < and i j , or d2 4
-x*
(6)
< 0 for x* = x*- and i j . x
2 + d
As expected in a Cournot setting, each firm ’s optim al quan tity decreases w ith an increase of its own m arginal costs and increases w ith th e m arginal costs of its rival. A firm ’s reaction to a change in th e level of product differentiation is twofold. If firms offer th e same quantities, an increase in product differentia
tion (which corresponds to a decrease in d) leads to a lower optim al quantity.
If firms are sufficiently different, one firm ’s quan tity increases while the other firm ’s q u an tity decreases in d.
The reduced form profit function of firm i is given by:
(a(d — 2) + 2c,- — d cj)2
(ci ; cj ) 5 ) — (7)
(d2 - 4)2
In th e next subsection th e first stage of th e gam e is analyzed, in which firms choose th e optim al level of m arginal costs and product differentiation through R&D investm ent.
4 The innovation decisions under R & D competition
A nticipating th e outcom e of th e stage-two game, firms choose optim al R&D projects. Possible R&D projects are targ eted at process innovation and at product innovation. Through th e form er they choose m arginal costs of pro
duction and through the la tte r they choose a degree of product differentia
tion. F irm s’ strategies are (cp,d p) G R 2, w ith c„ G [0, a] and d„ G [0, d] w ith v = i, j and a being consum ers’ prohibitive price.
The N ash-equilibrium strategies of b o th firms (c,-,d,) and (c j,d j) are defined by th e m utual best-response property. T h a t is for firm i:
(q, di) G argm ax{IR - Tr*(ci,Cj,di,dfi) - K (c i) — Glfffi)}, (8) Ci ,di
P r o p o s itio n 1 Under the given assumptions there exists an equilibrium in sym m etric pure strategies, which is the unique sym m etric equilibrium, in which firm s invest in both process innovation and product differentiation.
P roof: F irm z’s profit is given by:
I R = 7 T * ( c ,- , Cj, d i , d j ) - K ^C i) - G(dfi), (9)
and firm J ’s profit is defined analogously. Maximizing (9) w ith respect to c,-, su bstituting (5) and rearranging leads to the following first-order condition:
=><’ w
This condition can be satisfied since both sides of th e equation are negative.
The optim al level of m arginal costs is given by th e equality of m arginal revenues of cost reduction and m arginal R&D costs. If R&D for process innovation is inefficient or th e initial level of m arginal costs is already very low, (10) should lead to a solution in [0, a]. Due to th e assum ption concerning the R&D cost function, th e only corner solution which can arise is c* = a meaning th a t th e firm does not invest into process innovation.
M aximizing firm ’s profit with respect to di, we restrict atten tio n to sym m etric equilibria. Substituting (6), using x* — x* and rearranging yields:
Since firms are identical at the outset, we find analogous conditions for firm j . Condition (11) can be satisfied since b o th sides of the equation are positive.
Due to B row er’s fixed point theorem , we can verify th a t in this strategic-form game under condition (12), there exists a pure-strategy N ash equilibrium:
Strategy spaces 7) are nonem pty com pact convex subsets of an Euclidean space and th e payoff functions II, are continuous in t and quasi-concave in ti, w ith ti — (c,-, di}, if th e following condition holds:6
K « , ) 2 <
- G"').
(12)For a unique sym m etric equilibrium there has to exist only one sym m etric pair of (c*,d*) w ith v = i , j , th a t satisfies all four first-order conditions.
Imposing sym m etry, th a t is c,- = Cj and di = dj, on th e expressions (10) and (11), th e system reduces to two equations only. Now consider th e (unique) solution to equation (10) which is given by c^(d„).7 The system has a unique sym m etric solution, if after substitution of c ^ d y) into (11) this equation has a unique solution in d„. This in tu rn is (at least) the case, if 7r^ (c,(d,), di) — G' is m onotonically decreasing in d,:8
+ - G" <
0Substituting c,d. and rearranging yields th e following condition on the slope of m arginal R&D cost functions, which guarantees the existence of a unique sym m etric equilibrium in pure strategies:
- G")- (13)
Applying th e contraction m apping theorem , th a t is D et Cicj Cidj > 0 ,
and imposing sym m etry leads to th e sam e condition. As expected, this con
dition is stronger th a n condition (12). □
6Firms’ second-stage profits net of R&D costs are represented by 7r while gross profits are denoted as II.
7It is easily checked, th at (10) is linear in c„.
sThe superscript s denotes symmetry.
5 The innovation decisions under R & D co
operation
Now suppose th a t firms form a Research Joint Venture in which they coordi
n ate th eir R&D activities while they rem ain com petitors in th e second stage of th e game. If firms coordinate R&D activities in th e first stage, they choose R&D investm ent as to maxim ize th eir joint profit, 7r, + 7rj. Concerning the costs of R&D, two cases can be considered: a) Firm s coordinate th eir research strategies b u t conduct research in two separate labs, or b) they coordinate th eir research strategies and share th e costs of R&D by building up only one research unit. The first case can be characterized as R& D -cartelization, the second as R JV -cartelization.9 We will begin w ith the analysis of the first case.
5.1 Firm s m axim ize jo in t profit but do not share R&;D costs
Since th e second stage is unaffected by any cooperative agreem ent, only op
tim al quantities are presented here. Due to th e fact th a t firms are com m itted to choose a common level of m arginal costs and product differentiation, they will definitely offer sym m etric quantities:10
* o Cfc
Xk ~ T + d '
C om parative statics with respect to m arginal costs yield:
* _ - 1 x kCk ~ 2 + <N
9See Kamien, Muller V Zang (1992) for this terminology.
10This assumption is critical, since Salant V Shaffer (1992) show that instead of investing identical amounts on cost reduction and then slugging it out as equally matched rivals, the firms can often earn strictly higher joint profits by investing at the first stage to induce monopoly or at least asymmetric duopoly at the second stage. Still, here symmetry is assumed.
which is different to th e com petitive case since firms ignore strategic effects.
Turning to th e first stage, now assum e th a t firms coordinate th eir strategies as to m axim ize joint profit and b u t do not share th eir R&D costs. This m ore unrealistic scenario is included into analysis because it allows us to isolate the strategic effects of cooperative R&D investm ent and th e influence of cost sharing agreem ents. F irm s’ joint profit is
IR = 27r*(Q , 4 ) - 2 Ä ( c, ) - 2 G '( d fc).
Using (15) and rearranging leads to th e following im plicit function which gives o ptim al investm ent into process innovation c£:
= K '- (16)
M axim izing joint profit w ith respect to th e level of product differentiation dk yields th e im plicit function for d*k.
= (17)
Obviously also under R&D cooperation th ere is an optim al division of inno
vative activity between th e two alternatives, leading to a positive investm ent for b o th kinds of innovation.
Let us now exam ine the relationship betw een the first-order necessary con
ditions if firms choose their research strategies so as to maximize only their individual profits, expressions (10) and (11), and com pare these to th e con
ditions when th ey maximize combined profits, expressions (16) and (17).
Com paring (10) and (16) shows, th a t only if d = d = 0, w ith d denoting th e level of pro d u ct differentiation under this kind of cooperation, b o th functions can lead to th e sam e c,- = q
.
For d = d = 1 we find c,- < q,
since — jz £ (., 1) >— |a :* (.,l) and K 1 < 0 and K " > 0. Therefore, under cooperation without
R&D costs sharing, firms choose for each d a higher level of m arginal costs c. A com parison of (11) and (17) reveals th a t firms choose for any level of m arginal costs c, a lower level of product su b stitu tab ility d.
B ut w hat happens in equilibrium ? Note th a t in this case of cooperation, th e two problem s are com parable in th e sense th a t e.g. in case of process innovation
d iR _ d ih a n , dck dci + dci ’
and IIj is th e externality conferred by firm z’s cost reduction on th e profit of its rival j . Analogously, also th e externality induced through product inno
vation by firm i on th e profit of firm j is added to th e com petitive advantage externality th e firm ’s R.&D effort has on its own profit through increasing the am ount of differentiation of its com petitor. Those externalities, positive or negative, are ignored when each firm chooses its R&D expenditure so as to m axim ize its own profit. They are internalized when th e firms coordi
n ate th eir R&D strategies. Each firm ’s internalization of these externality is w hat causes th e individual m axim ization problems to be equivalent to the joint m axim ization problem th a t would be solved by a single director of the
R & D -cartel.11
To determ ine th e effect of those strategic term s, th e first-order conditions given by (16) and (17) can be w ritten as
= 0, and (18)
Kidi ~ G ' + ßKjd> = 0, (19)
respectively, w ith ß = 1. By applying com parative statics w ith respect to ß the effects of adding these strategic term s to firm s’ first-order conditions of
11 See Kamien, Muller & Zang (1992) for a similar argument in the comparison of R&D- cartels to competitive R&D. They apply the same formal approach but in their model firms choose only one strategic variable.
profit m axim ization can be analyzed.12 Taking all variables as functions of ß , differentiation of (18) and (19) w ith respect to ß yields:
(’T.e.c, - K " + ) d c i - (ffic.d + ß ffjc.d)d di + irje.d ß = 0, (7r«d,.ci + ß i ^ c i )dCi - (fftd.d + G" + di + 7TJd.d ß = 0.
Note, th a t firms are sym m etric in th e sense th a t 7rJc. = 7r,-Cj. 7Tjc d = 7Tic d =
—fficd. and 7Tjc.c. = 7r»CjCj • Further, since product innovation by one firm com
pletely spills over to th e other firm, also ffid.d. = ffjd.d. = ~ffdtd- Substituting these term s and applying C ram er’s rule leads to:
C? Ci if
Sign d / ^ = + G" + + ))
Sign = signH ’r«cici “ K " + + ))•
Since the sign of th e right-hand sides of b oth expressions is ambiguous, ob
viously the slopes of th e m arginal R&D cost functions determ ine w hether R&D investm ent is increased or decreased. R equiring decreased investm ent in process innovation, e.g. c8/3 > 0, m eans th a t the right-hand side of the first expression has to be positive, which is tru e in case:
G" >
ß^idfd^icjd 7r«d,d7r’c_, ) + 7r’did7r«cJ'
:= G ".
7T,-
Cooperation increases investm ent in product innovation, e.g. d ^ > 0, when
ever th e right-hand side of the second expression is positive, or if:
\ f »5 f f f Ii d j C j J f f t C j C jf f l j M ,/ J f f t r ~ ' t c j C j f f 1 Jt d j I 7T. 7T. , t c j t d j C j „
7T.1d i
These conditions on th e R&D cost functions can only be m et, if th ey allow for the existence of an equilibrium . To ensure th a t there exists an equilib
rium for all values of ß , th e second-order condition, given by (12), has to be 12If ß = 0 we are in the competitive case.
transform ed to:
G" > TTid . + ßTTj d . -
,d' ^ , c, + ^ c , - K "
Substituting K " into this new second-order condition of profit m axim ization, reveals th a t whenever K " < K " it is necessary for th e equilibrium to exist th a t G" > G ". Similarly, G" < G" requires th a t K " > K " . Hence, it can definitely be excluded th a t firms invest more in process innovation, cJ/3 < 0, and less into product differentiation, diß < 0 when they coordinate their research strategies in an R& D-cartel. On th e other hand, in case m arginal R&D costs increase slowly for process innovation, th a t is K " < K " , firms will invest less in process and less in product differentiation com pared to th e non- cooperative equilibrium . If th e slope of m arginal R&D costs is low for product innovation, th a t is G" < G", firms invest more in process innovation and m ore in product innovation. If m arginal R&D cost functions are sufficiently steep bu t do not differ significantly, coordination of strategies induces firms to invest m ore in product differentiation b u t to reduce investm ent in process innovation. This result corresponds to th e trad itio n al conjecture, th a t the internalization of positive externalities should increase incentives to conduct R&D while the opposite should hold for negative externalities. B ut contrary to th e trad itio n al result, in this scenario th e following proposition holds:
P r o p o s itio n 2 A coordination o f strategies m ay induce firm s to increase their investm ent in product differentiation, whenever product innovation is not too expensive, while at the same tim e they increase their investm ent in process innovations.
P roof: See th e argum ent above for G" < G" and K " > K " .
So if firms can easily differentiate their products, th e internalization of posi
tive externalities overcom pensates negative externalities.
5.2 F irm s m axim ize jo in t profit and share R & D costs
By changing th e assum ption concerning R&D costs again only th e first stage of th e gam e is affected. Since all analytical steps are corresponding to those discussed previously, form alities will be confined to th e second-stage first- order conditions of joint profit m axim ization and th e m axim ization problem itself, which is given by:
IIS = 27T*(cs, ds) - K (c s) - G(ds).
O ptim al investm ent into process innovation is d eterm ined by:
(20) For pro d u ct innovation th e first-order condition is:
= G'. (21)
s V2 + <T v 1
Obviously also under this kind of R&D cooperation there is an optim al divi
sion of innovative activity between the two alternatives, leading to a positive investm ent for both kinds of innovation.
The optim ization problems under the two regimes, R&D com petition and R JV -cartelization, now differ in two aspects: F irst, under cooperation firms consider th e effects of th eir investm ent on th eir rival’s profit, and second, R&D cost functions change due to effort sharing. A comparison of optimal investm ent under th e two regimes leads to th e following proposition:
P r o p o s itio n 3 Firm s reduce costs and differentiate their products more if they cooperate in their R& D activities and share R& D costs. Under com
petition firm s proportionally invest more in process innovation than under RJV-cartelization.
P roof: For a com parison of firm s’ optim al RfeD investm ent under th e two regimes we characterize the resulting equilibrium points in the (c, d) space:
Each pair of first-order conditions gives us an im plicit function c(.) = c(d).
Com paring (10) and (20) shows, th a t only if d = d = 1, w ith d denoting the level of p roduct differentiation under cooperation, b oth functions can lead to th e sam e Ci = cs. For d = d = 0 we find cs < c,-, since —2x*(.,0) < —£*(.,0) and K ' < 0 and K " > 0 for v = s ,i. Applying th e im plicit function theorem , th e slopes of th e functions Cid and cSd can be determ ined.13 Since cSd > 0.
and Cid < 0 for d > 2/3, (and vice versa) in the (c, d) space the function cs(d) lies always und ern eath th e function c,-(d). Therefore, under R JV -cartelization firms choose for each d a lower level of m arginal costs c.
Now consider (11) for th e com petitive case and (21) for th e cooperative case respectively. For any given c, = cs we find d < d, w ith d = d — 2d,- and d = d — 2ds, since for any given c we have ailj Q> > q anj G" > 0. A pplying again th e im plicit function theorem , from (21) we find th a t (dCs) -1 > 0 and (dCi) _1 > 0. (See A ppendix B I.) Therefore, under cooperation firms choose for any given c a lower level of d. In th e following graphic all four functions are sketched.
13See Appendix BI.
A com prehension of argum ents allows to conclude th a t under R JV carteliza
tion firms invest more in both, process innovation and product differentia
tion. Still, it is worthwhile to note th a t firms do not only reduce m arginal costs and increase product differentiation more under R JV -cartelization th an under R&D com petition, bu t also change th e proportion of optim al R&D in
vestm ent. Consider th e ratios given by:14
Gi ( 2 - d ) x f 1 )
in th e com petitive case, and
Gs - 1
2a:* (23)
under cooperation. Due to th e fact th a t cs < Ci and d < d for equilibrium quantities th e following holds: x* > a:*, since x*d < 0 and x* < 0. Hence, 14Althoug R&D cost functions are assumed to be the same under all regimes, here we lable them with subscripts i and s in order to be able to distinguish the competitive and the cooperative case.
1^1 > IzM > m eaning th a t under com petition firms proportionally invest more
in process innovation th a n under cooperation. □
Digression: Since most models consider only process innovation, it is interest
ing to analyze how firms R&D investment strategies are affected by the level of product differentiation. Suppose for the moment that the degree of product sub
stitutability d is given exogenously. It is then interesting to note th at firms under R&D cooperation reduce their R&D investment in process innovations, the less the products are differentiated. Under R&D competition, firms invest less into process innovation for d < 2/3 and more for d > 2/3. Obviously, if the products are close substitutes, the strategic importance of marginal costs is stronger than if products are less substitutable. Under cooperation, these strategic effects of marginal costs are absent.
6 Changes in the market size
As analyzed in th e previous section, firms do not invest only in one kind of innovation b u t rath er distrib u te their research effort optim ally among process and product innovation. The analysis concerning R&D cooperation has revealed th a t firm s’ investm ent decisions are strongly influenced through strategic externalities and cost sharing agreem ents. How does this optim al in
vestm ent decision change w ith th e m arket size? Does th e size of th e m arket change th e stren g th of externalities? C om parative statics concerning con
sum ers’ prohibitive price helps to answer these questions if th e prohibitive price is in terp reted as th e m arket potential. Up to now, a was considered to be a constant and therefore not given explicitly. To be m ore precise, the sym m etric equilibrium q uantity in case of R&D com petition is in th e first stage given by x* = z*(a, Cj(o), Cj(a), d — d,(a) — d ;(o )). Hence, it is straight forward to analyze com parative static properties. If a changes, th e first-order
conditions have to rem ain satisfied:15
c . a f ’Tic.e, - K " ) - dia TCic.d + 7T,-C.O = 0 , d ia i^ id d ~ G } — Cia7TiCid + 7T,-d;o = 0 .
Applying C ram er’s rule, th e following relations hold:
sisn ( - J ^ ) = sign ( - » i..,( ! r iaJ - d Ci G") - jt
. ^
tt,;,..)
sign = sign ( - ( ’ ’i,,, -
(24) (25) Obviously, th e signs of b o th expressions are ambiguous and therefore depend on the slopes of th e m arginal R&D cost functions.
Assume first, firms would increase their investm ent in pro d u ct differentiation, which means th a t dia > 0. For this to be true, expression (25) has to have a positive sign, which is th e case if:
K " >
TV\ , TVi j TV v TV i , ~
lc i ci ______ Zcxa Idtj . __ j £ t t
TV. (26)
Now assume th a t investm ent in process innovation would be increased as well, e.g. Cia < 0. This would be th e case if
^ C ta ^ d d T ^ c i d ^ d i a __qh
7T„ (27)
S ubstituting K " into the second-order condition of profit m axim ization, given by (12), shows th a t whenever K " < K " th e existence of a sym m etric pure strategy equilibrium requires G" > G ". Similarly, G" < G" requires K " >
K ". Therefore, an increase in consum ers’ willingness-to-pay m ay never lead to less process innovation, c8a > 0, and less product innovation, dia < 0. In case the slope of m arginal R&D costs is low for process innovation, th a t is
» c , d
15Note th at ^ id.d. — Kidi and due to the additive relation of d and d,-.
Further, note th at 7r,c.o = < 0 and -irid a > 0.
K " < K " , this leads to m ore process innovation b u t also to less product innovation because then for an equilibrium to exist, G" > G" is required.
However, if m arginal R&D costs of product innovation increase slowly, th a t is G" < G", product innovation will be increased and process innovation decreased, since again equilibrium requires K " to be sufficiently high. In case the slopes of bo th m arginal cost functions do not differ significantly, an increase in the m arket size will induce firms to invest m ore in product innovation and m ore in process innovation. Clearly, th e higher the outp u t th e larger is th e to tal gain from a given reduction in u n it costs of production.
ft can fu rth er be shown th a t this result still holds if firms cooperate in their R&D decisions, independent from th e specific form of agreem ent.16 B ut does the optim al division between th e two kinds of innovation rem ain unchanged?
Differentiation of (22) w ith respect to a yields:
K "c iaG' - G"diaK ' 4 < 2 « - 2 « d + ciax * J G'2 (2 - cP)2x* + ((2 - d ) x f >
It is easily checked th a t the ratio on the right hand side is positive. Since the ratio of m arginal R&D costs is negative, an increase in this case represents a reduction of th e value of th e ratio. Again, this result does not change under R&D cooperation. We can conclude th a t in a larger m arket firms invest pro
portionally m ore in product differentiation th an in process innovation. Given some initial distribution of investm ent, firms will shift th eir R&D investm ent from process to product innovation if they are confronted w ith a larger m ar
ket if bo th kinds of innovation are characterized by rapidly increasing R&D cost functions. And we observe fu rth er th a t in a sm all m arket firms invest less in R&D th an in a large m arket.
This analysis reveals th a t firm s’ optim al choice between process and product 16See Appendix B2 for a formal proof of this statement.
innovation depends on th e one hand on th e m arket size and on th e other on th e relative slopes of m arginal R&D cost functions.
C ontrary to this result, it can be shown th a t firms always increase their investm ent in both kinds of innovation if m arket size increases, if they coor
dinate th eir R&D strategies and share RfeD costs.
Com paring (in the sym m etric setting) th e relative m agnitude of m arginal rev
enues and externalities for b o th kinds of innovation, one finds th a t for product innovation this relation does not change w ith th e m arket size, while for pro
cess innovation th e m arginal revenue increases m ore th a n th e externality.17
9 7T'
T h at is 13%'c'~ 1 > 0. Obviously, w ith an increase in th e m arket size process innovation gets less “tough” in th e sense defined by Fudenberg &; Tirole (1984).
7 Welfare
W hen analyzing R&D cooperation in this setting it is also interesting to consider welfare im plications. Welfare is given by:
W = 2(a - c,)x, - (1 + d - 2di)x? - ß K (c ,) - ß G (di) (28) w ith ß = 1 and I = s for R&D cooperation in th e sense of R JV -cartelization, and ß = 2 and I = i for com petition. Here th e firms always have an incentive to cooperate in th eir R&D activities because they could have mimicked the non-cooperative strategies, if this were optim al :
d*,c* = argm ax{7r;(ds ,c s) + 7r;(ds,c s) - /<(cs) - G (ds)},
17 Actually, one finds iry > 0 and Hjd ,o > 0. However, a reduction of c,- reduces firm j ’s profit since 7rJc. > 0, so the negative externality is actually captured by —irJc. < 0.
Therefore, one finds also < 0, meaning that the negative externality is reinforced through an increase in market size.
and
d*,c* / di , ci = argmax{7ri(dl-,ct) - K (c t ) - G(dt )|dJ, c*}.
Hence, it m ust be th e case th a t th e following holds:
- | ( A ( < ) + < ? (« )) > » ( < ? ,« ) - ( A ( < ) + G « ) ) . (29)
P r o p o s itio n 4 I f firm s have an incentive to cooperate in their R& D activi
ties and share their R& D costs, this cooperation increases welfare.
P roof: N ote th a t (29) can be rew ritten as:
( » - < ) < - ( i +
+
g w»
> (« - < ) < - (1 + d ) x ? - (A '(< ) + G « ) ) ,
w ith d = d — 2d* for th e com petitive case and d = d — 2d* under cooperation.
W hereas welfare is increased whenever:
(« - - | ( i + < j > : 2 - |( a « ) + g « »
> (« - - ~(1 +
d)x’2 -
(/<(e?) + G « ) ) .Given th a t (29) is satisfied, welfare is increased if:
j ( l + < W - j ( l + < W 2 > 0
O ( l + d X 2 > ( l + d)x*2.
Substituting x* and x* and rearranging leads to:
> (1 + d ) ( 2 + rip (« - c*)2 (1 + d)(2 + d fi
This inequality holds, because due to c* > c* the right hand side of (30) is larger th an one and due to d < d th e left hand side is sm aller th a n one.
Obviously, if (29) is satisfied, and cooperation leads to m ore investm ent into bo th kinds of R&D, also welfare will be increased. □ It is easily seen th a t welfare is increasing in a reduction of (c, d) as long as the increase in R&D costs does not overcom pensate th e gains of producer and consum er surplus or, in other words, as long as firms have an incentive to cooperate. This is not surprising since consumers as well as producers benefit from an increase in product differentiation and from reducing m arginal costs.
The only lim iting factor is th e increase of R&D costs.
If firms do not share th eir R&D costs when they cooperate, welfare im pli
cations are ambiguous. If th e slopes of m arginal R&D cost functions do not differ substantially, cooperative firms differentiate th eir products m ore which benefits th e consumers. B ut m arginal costs are higher under this kind of co
operation th a n under com petition. Therefore th e effect on consum er surplus depends strongly on th e relative strength of th e two effects. W ith o u t assum
ing concrete functions for R&D costs no general conclusion can be drawn.
But when m arginal costs for product innovation do not increase too rapidly, also cooperation in th e sense of R& D -cartelization increases welfare.
8 Conclusion
In this paper we have used a simple linear dem and stru c tu re which enables us to analyze firm s’ R&D decision in a differentiated industry, when they can determ ine m arginal costs and product su b stitu tab ility simultaneously.
In p articu lar, we com pare firm s’ R&D decision under com petition and co
operation and conduct some com parative statics concerning th e m arket size.
The m ain findings are th a t firms do not necessarily specialize in one kind of innovative activity bu t ra th e r allocate their R&D budget optim ally among the two altern ativ e forms of innovation, process and product innovation. Only
if th e R&D costs for one of th e two innovations is such th a t investm ent would be inefficient, we find th e extrem e case of com plete specialization, which is m ostly discussed in th e R&D literatu re. Furtherm ore, the optim al division between the two kinds of innovative activity changes w ith th e m arket size.
U nder R&D com petition th e business stealing effect induces firms to increase (reduce) th eir investm ent in process (product) innovation, com pared to non- strategic decisions. As far as product innovations are concerned, investm ent is fu rth er reduced through th e public good effect. The larger th e m arket, the m ore firms invest in R&D and th e m ore th e investm ent is driven to product innovations provided th a t R&D efficiencies are similar.
We therefore find th a t an increase in th e m arket size affects th e stren g th of com petitive spillovers: “Tough” investm ent becomes less “to u g h ” . (In this scenario th e positive spillover is not affected through an increase in th e m ar
ket size. It rem ains to be shown if in different m arket settings also “soft”
investm ent becomes m ore or less “soft” .)
If firms coordinate their R&D activities and share th e costs for innovation, they will reduce m arginal costs and product su b stitu tab ility m ore th a n un der com petition. They will also proportionally invest more into product in
novation since th e strategic im portance of m arginal costs diminishes if firms cooperate (internalization of com petitive spillovers). In this case of R&D com petition, th e welfare im plications of joint research are positive. T he cost sharing agreem ent together w ith th e internalization of positive com petitive spillovers outweigh the incentive to reduce investm ent in process innovations.
If firms cooperate, b u t for any reason cannot share their R&D costs, they still proportionally invest m ore into product innovation. However, in this case they differentiate their products m ore b u t reduce m arginal costs less th an under com petition. The welfare im plications of an R JV in which firms
only elim inate strategic com ponents b u t do not reduce overall R&D costs are ambiguous. Only if product differentiation is not too expensive, firms increase th eir investm ent for both kinds of innovation, and welfare increases.
In this scenario, positive com petitive spillovers from the product m arket can play a sim ilar role as technological spillovers on th e R&D m arket. If they are sufficiently strong, any cooperative agreem ent is welfare enhancing.
Of course one could th in k of process and pro d u ct innovations as taking place sequentially. The presented m odel could be extended by separating these decisions in two stages. The question of which decision is taken first is not obvious. E ith er th e production technology is considered to be th e m ore com
m ittin g decision or it is assum ed th a t firms first decide on the m arket niche they w ant to position th eir product in. B oth seem to be reasonable assum p
tions and it should be verified by em pirical evidence which of th em is m ore realistic. If rem aining in th e sim ultaneous framework, the assum ption con
cerning th e separability of th e R&D cost function could be altered. Assuming one cost function could lead to unconditioned results concerning th e welfare im plications of cooperative research or concerning com parative statics in the m arket size. A th ird extension of th e m odel could be to assum e price com pe
titio n on th e product m arket. Furtherm ore, th e com parative static analysis could be extended to changes in th e slope of the dem and function. A m ore technical aspect which could be included into th e analysis is th e description of asym m etric equilibria (in case they exist) w ithin th e existing setting, or to analyze equilibria in an a priory asym m etric setting. It would be interesting to analyze these aspects in future research.
Appendix A
For any im plicit function cv = c„(dp) given by th e function F {cv{dv'},d„) = 0, we find c„d>/ = ~ ^ r - The functions c,d and dic. can be derived from (10) and
(11) respectively as:
a-*4(3rf-2)
(cZ2-4)2
TT.
—4xi*
( W and d , = TT.
AA»CjC,-
The denom inators of b oth functions are negative since th e second-order con
ditions for profit m axim ization are assum ed to be satisfied. Therefore the signs of Cid, and d,c. are given by th e signs of th e num erators, which means th a t dic. is negative and Cjd. < 0 for d < 2/3. Using 9ci/U~2^ ) = — 2cPd w ith v = i,s , shows th a t dCi > 0 and Cid > 0 for d < 2/3.
For th e cooperative case, cSd can be derived from (16):
8a?:
_ _ (d + 2 )2
- n
A1Scscs
Again, th e sign of cSd is given by th e sign of th e num erator, which is positive.
Applying again th e im plicit function theorem , from (17) we find th a t -i6z:
U+2)2
< 0 (31)
holds, since the num erator as well as the denom inator are negative. Therefore, also dCs > 0. Inverting dCs and dCi allows us to depict all functions in one graphic.
Appendix B
Under cooperation firms invest more into bo th kinds of innovation since applying C ram er’s rule yields for process innovations, w ith v — s ,k :
sign ( ) = sign (- n c, „ n rfydl/+ IICu^ I I d ,a),dcy
<o <o
< 0. (32) And for product innovations it yields:
. , ddy . Slgn ( d Y
> 0 > 0
> 0. (33)
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