Cross&ownership, R&D Spillovers, and Antitrust Policy

52 Pages·2016·0.5 MB·English

Cross-ownership, R&D Spillovers, and Antitrust
Policy
(cid:3)
`ngel L. L(cid:243)pez Xavier Vives
y z
May 2016
Abstract
Thispaperconsiderscost-reducingR&DinvestmentwithspilloversinaCournot
oligopoly with minority shareholdings. We (cid:133)nd that, with high market concen-
tration and su¢ ciently convex demand, there is no scope for cross-ownership to
improve welfare regardless of spillover levels. Otherwise, there is scope for cross-
ownershipprovidedthatspilloversaresu¢ cientlylarge.Thesociallyoptimaldegree
ofcross-ownershipincreaseswiththenumberof(cid:133)rms,withtheelasticityofdemand
and of the innovation function, and with the extent of spillover e⁄ects. In terms of
consumer surplus standard, the scope for cross-ownership is greatly reduced even
under low market concentration.
JEL classi(cid:133)cation numbers: D43, L13, O32
Keywords: competition policy; partial merger; collusion; innovation; minority sharehold-
ings; modi(cid:133)ed HHI
We thank Ramon Faul(cid:237)-Oller for his insightful initial contributions to this paper. For helpful com-
(cid:3)
ments we thank Isabel Busom, Luis Cabral, Guillermo Caruana, Ricardo Flores-Fillol, Richard Gilbert,
GerardLlobet,PeterNeary,PatrickRey,YossiSpiegel,andJavierSuarez.Wehavereceivedconstructive
suggestionsfromseminarparticipantsatIESE(PPSRC),Universit(cid:228)tMannheim, Universit(cid:224)diBologna,
Universitat de Barcelona, CEMFI, Paris School of Economics, and Universitat Rovira i Virgili as well as
from conference participants at CRESSE 2012, (cid:147)Industrial Organization and Spatial Economics(cid:148)(HSE,
Saint Petersburg), 5th Barcelona IO Workshop (Universitat Pompeu Fabra), EEA2013 (University of
Gothenburg), Barcelona GSE Winter Workshop 2013 (IAE-CSIC), CEPR Applied IO Workshop 2014,
Searle Antitrust Conference 2014, EARIE 2014, XXIX Jornadas de Econom(cid:237)a Industrial (Barcelona
GSE),theESBostonMeeting2015,theBarcelonaGSESummerForum2015,theWorldCongressofthe
Econometric Society 2015, and SAEe 2015. Financial support from the Spanish Ministry of Science and
Innovation under ECO2011-29533, from the Spanish Ministry of Economy and Competitiveness under
ECO2015-63711-P, and from AGAUR under SGR 1326 (GRC), is gratefully acknowledged.
L(cid:243)pez: Departamentd(cid:146)EconomiaAplicada,UniversitatAut(cid:242)nomadeBarcelona,andPublic-Private
y
Sector Research Center, IESE Business School. E-mail address: [email protected]
Vives: IESE Business School.
z
1
1 Introduction
Inmanyindustries, minorityshareholdingsareprevalentintheformofcross-shareholding
agreementsamong(cid:133)rmsorcommonownershipbyinvestmentfunds.Thetendencyofsuch
arrangementstoreducepricecompetitionhasbeendocumentedintheairlineandbanking
industries (Azar et al. 2015, 2016), and it has raised antitrust concerns. However, cross-
ownership arrangements (COAs) may have a bene(cid:133)cial e⁄ect on investment provided
there are positive spillovers across (cid:133)rms. The reason is that COAs help to internalize the
spillover externality, which is especially important for highly innovative industries. To
what extent, and by what means, should antitrust authorities limit the (cid:147)partial(cid:148)mergers
that result from cross-ownership in innovative industries? In this paper we provide a wel-
fare analysis of COAs(cid:151)when (cid:133)rms compete in quantities and invest in cost reduction(cid:151)in
the presence of spillovers; we also derive some implications for competition policy. The
analysis may help elucidate whether the documented increase in cross-ownership arrange-
ments has outrun its social value.
We consider a general symmetric model of cross-ownership; this model allows for
a range of corporate control and for distinguishing between stock acquisitions made by
investorsandthosemadebyother(cid:133)rms. Inourbenchmarkmodel, weconsidersimultane-
ous R&D and output decisions. That approach aids tractability while helping to capture
the imperfect observability of (cid:133)rms(cid:146)R&D investment levels.1 We test the robustness of
results by way of a two-stage speci(cid:133)cation. The model subsumes earlier contributions to
the literature that were based on linear or constant elasticity of demand and on spe-
ci(cid:133)c innovation functions (Dasgupta and Stiglitz 1980; Spence 1984; d(cid:146)Aspremont and
Jacquemin 1988; Kamien et al. 1992). Perhaps the work closest to ours in spirit is the
paper by Leahy and Neary (1997).
Our paper seeks to answer the following questions: How do R&D and output levels
vary with minority shareholdings? What are the key determinants of the socially optimal
extent of cross-ownership? How is that optimal level a⁄ected by structural parameters
(demandandcostconditions, industrytechnologicalopportunity,extentofspillovers)and
by the competition authority(cid:146)s objective (to maximize total or rather consumer surplus)?
The main results can be summarized as follows. If demand is not too convex, then
1Simultaneousmodels(cid:151)inthepresenceorabsenceofR&Dspillovers(cid:151)areanalyzedby,amongothers,
Dasgupta and Stiglitz (1980), Hartwick (1984), Levin and Reiss (1988), Ziss (1994), Leahy and Neary
(1997), Cabral (2000), and Vives (2008).
2
increasing the partial ownership interest in rivals will increase (resp. decrease) both R&D
and output when spillovers are high (resp. low); for intermediate levels of spillovers,
an increase in such ownership interest will increase R&D but reduce output. These are
testable predictions. We identify the key determinants of a welfare-optimal degree of
cross-ownership: the curvature of demand, the degree of market concentration, and the
extent of spillovers. A su¢ cient (but not necessary) condition for the absence of minority
shareholdingstobeoptimal(cid:151)fromeitherthetotalsurplus(TS)orconsumersurplus(CS)
perspective, and for any extent of spillover(cid:151)is that the relative degree of convexity of
demandbegreaterthantheinverseoftheHer(cid:133)ndahl(cid:150)Hirschmanindex(HHI).Otherwise,
the range of spillovers is typically partitioned into three regions: one optimally with no
cross-ownership for low levels of spillovers; one optimally with positive cross-ownership
(by TS and CS standards) for high levels of spillovers; and one optimally with positive
cross-ownership (by the TS standard only) in an intermediate region. We remark that the
consumer surplus standard is always more stringent than the total surplus standard. We
also (cid:133)nd that, if the e⁄ectiveness of R&D is independent of the degree of cross-ownership,
then under the CS standard there is a (cid:147)bang bang(cid:148)solution: either independent own-
ership or cartelization is optimal. Numerical results reveal that the (TS-based) socially
optimal extent of cross-ownership is increasing in the number of (cid:133)rms, in the elasticity of
demand and of the innovation function, and in the level of spillover e⁄ects. Qualitatively
similar results hold for the CS-based optimal extent, except that the scope for minority
shareholdings is much reduced.
The context analyzed here is of more than theoretical interest. Minority sharehold-
ings are widespread in many industries (e.g., automobiles, airlines, (cid:133)nancial, energy, and
steel) and have attracted increasing antitrust attention (see EC 2013). There is grow-
ing interest among competition authorities in assessing the competitive e⁄ects of partial
stock acquisitions. This increased attention stems mainly from two factors: (i) the rapid
growth of private equity investment (cid:133)rms, which often hold partial ownership interests
in competing (cid:133)rms (Wilkinson and White 2007); and (ii) some notorious cases, such as
Ryanair(cid:146)s acquisition of Aer Lingus(cid:146)s stock and the Renault(cid:150)Nissan alliance (under which
Renault owns 44.3% of Nissan even as Nissan owns 15% of Renault).2 These factors have
2As Gino (2000) delineates, four other cases that have attracted considerable interest are as follows:
(i) a $150 million investment by Microsoft in the nonvoting stock of Apple in 1997; (ii) the purchase by
Northwest Airlines (fourth-largest US airline) of 14% of the common stock of Continental Airlines (the
3
triggered a debate in Europe over the possibly anticompetitive e⁄ects of partial owner-
ship. Yet the European Commission (EC) is not authorized to examine the acquisition of
minorityshareholdings,3 andithasproposedextendingthescopeofitsmergerregulations
so that it can intervene in cases involving minority shareholdings among competitors or
in a vertical relationship (EC 2014).4 In Canada and the United States, cross-ownership
is scrutinized under prevailing merger control rules. More speci(cid:133)cally, minority share-
holdings in the latter country are often examined with reference to the Clayton Act and
the Hart(cid:150)Scott(cid:150)Rodino Act.5 However, there is an exception to antitrust scrutiny if the
participation is (cid:147)solely for investment(cid:148)purposes, although it is subject to interpreta-
tion whether institutional investors can hold as much as 15% without needing to notify.
These US provisions are important because common ownership by investment funds is
widespread in many sectors.6
The extant literature, most of which focuses on the potential bene(cid:133)ts of cooperative
R&D or on how innovation is a⁄ected by mergers, has largely ignored the topic of how
innovation is a⁄ected by minority shareholdings(cid:151)despite clear evidence that antitrust
policy attends closely to innovation. During the period 2008(cid:150)2014, 36% of the mergers
challenged by the US Department of Justice or the US Federal Trade Commission were
characterized as harmful to innovation; of the challenged mergers, 76% were in high(cid:150)
R&D intensity industries. The anticompetitive e⁄ects of minority shareholdings tend to
be weaker than those of a merger; at the same time, minority shareholdings seldom
yield the e¢ ciencies (e.g., rationalization, fewer duplicated costs) that may arise from a
(cid:133)fth-largest) while agreeing to limit its voting power (however, an antitrust suit forced Northwest to sell
back nearly half of its purchased stake); (iii) the purchase by TCI (largest US cable operator) of a 9%
stake in Time Warner (the second-largest); (iv) the acquisition by Gillette of more than a (cid:133)fth of the
nonvoting stock (and more than a tenth of the debt) of Wilkinson Sword, one of its main competitors.
3In some European countries (e.g., Austria, Germany, the United Kingdom), national merger control
rules give competition authorities the scope to examine minority shareholdings. Currently, the EC can
consider the e⁄ects on competition only of (pre-existing) minority shareholdings in the context of an
anticipatedOK? merger (and in which the merging (cid:133)rms each have stakes in a third (cid:133)rm).
4The European Commission (EC 2014) has proposed a (cid:147)targeted transparency(cid:148)system under which
the EC and its member states must be noti(cid:133)ed of potentially harmful acquisitions. Included in this cat-
egorywouldbeacquisitionsofaminorityshareholding(cid:151)inacompetitororverticallyrelatedcompany(cid:151)
when either the acquired shareholding amounts to 20% or ranges between 5% and 20% but allows the
acquirer(cid:147)ade-factoblockingminority,aseatontheboardofdirectors,oraccesstocommerciallysensitive
information of the target" (p. 13).
5Section 7 of the Clayton Act prohibits acquisitions (of any part) of a company(cid:146)s stock that (cid:147)may(cid:148)
substantially lessen competition either by (a) enabling the acquirer to manipulate, directly or indirectly,
pricesoroutputorby(b)reducingitsownincentivestocompete. Althoughthereisnoclearlyestablished
threshold, acquisitions of less than 25%(cid:151)but of at least 15%(cid:151)have been adjudged to be in violation
(Salop and O(cid:146)Brien 2000).
6See Azar et al. (2015, 2016).
4
merger. The commonly held view is that, overall, minority shareholdings tend to lessen
competition. Nonetheless, the evidence of spillover-induced underinvestment in R&D
suggests that cross-ownership could be bene(cid:133)cial.7
The paper proceeds as follows. We review the literature in Section 2. In Section 3, we
describe the di⁄erent types of minorityshareholdings that canbe analyzedviaourmodel,
which is presented in Section 4. That section characterizes the equilibrium responses of
outputandR&Dinresponsetoachangeinthedegreeofcross-ownership. InSection5,we
examinethesociallyoptimaldegreeofcross-ownershipandthenillustratetheresultswith
threeleadingspeci(cid:133)cationsfromtheliterature:thed(cid:146)Aspremont(cid:150)JacqueminandKamien(cid:150)
Muller(cid:150)Zang models, and a constant elasticity model as in Dasgupta and Stiglitz (1980).
Section 6 extends our model to allow for strategic R&D commitments in a two-stage
game. Section 7 explores an alternative interpretation of our model when cooperation in
R&D extends to the product market; with regard to this case there is empirical evidence,
antitrust case study evidence, and also experimental evidence. We conclude in Section 8.
Unless noted otherwise, all proofs are given in Appendix A. Appendix B provides details
onouranalysisofthethreemodelspeci(cid:133)cationsconsidered. Notethatweo⁄erapplication
software (available on the Web), which the reader can use to conduct simulations with
the models.
2 Review of the literature
Previous literature has analyzed the anticompetitive e⁄ects of cross-ownership (Bresna-
han and Salop 1986; Reynolds and Snapp 1986). These researchers show that the pres-
ence of partial ownership interests in a Cournot industry may result in less output and
higher prices (even if those interests are relatively small). This is because the compet-
itive decisions of one (cid:133)rm(cid:151)with stakes in a competitor(cid:146)s pro(cid:133)t(cid:151)will take those stakes
into account by reducing output (or raising the price) so as to increase that competitor(cid:146)s
pro(cid:133)t and hence its own (cid:133)nancial pro(cid:133)t. Azar et al. (2015) document the substantial
common ownership interests of institutional investors (e.g., BlackRock, Vanguard, State
Street, Fidelity) in competing technology (cid:133)rms (Apple, Microsoft), pharmacies (CVS,
7For example, Bloom et al. (2013) report underinvestment in R&D (because of spillovers) in a panel
of US (cid:133)rms from 1981 to 2001. These authors (cid:133)nd that (i) the e⁄ects of technology spillovers are much
greater than those of product market spillovers and (ii) the socially optimal level of R&D is 2(cid:150)3 times
as high as the observed level of R&D.
5
Walgreens), and banks (JP Morgan Chase, Bank of America, Citigroup). In that study
of how passive investments by institutional investors a⁄ect market outcomes in the US
airline industry, the authors (cid:133)nd that ticket prices are about 10% higher on the average
route than they would be with no cross-ownership (or if strategy decisions were made
without regard to the investors(cid:146)minority shareholdings). Similar results are obtained for
the banking industry (Azar et al. 2016).
There is an extensive literature on the e⁄ects of cooperation and competition in R&D
with spillovers, starting from the seminal articles of Brander and Spencer (1984), Spence
(1984), Katz (1986), and d(cid:146)Aspremont and Jacquemin (1988, 1990). Leahy and Neary
(1997) present a general analysis of the e⁄ects of strategic behavior and cooperative R&D
in the presence of price and output competition; they also study optimal public policy
toward R&D in the form of subsidies.8 One of this literature(cid:146)s primary objectives is to
examine underprovision of R&D and the welfare e⁄ects of moving from a noncooperative
to a cooperative regime in R&D. For example, d(cid:146)Aspremont and Jacquemin (1988, 1990)
show that, when spillovers are high enough, R&D cooperation (with subsequent competi-
tion at the output stage) leads to increased output, innovation, and welfare. Cooperative
R&D enables (cid:133)rms to internalize their externalities and thus preserves the incentives to
invest in R&D.
Weshallidentifytheconditionsunderwhichminorityshareholdingsmayincreasetotal
surplus, and even consumer surplus, in industries where R&D investment is important
and spillovers are signi(cid:133)cant. Farrell and Shapiro (1990) show that passive (cid:133)nancial
stakes may be welfare increasing in asymmetric oligopolies; here we demonstrate the
possibility in a symmetric oligopoly. There is some evidence that common ownership
improves e¢ ciency. He and Huang (2014), using data on US public (cid:133)rms from 1980 to
2010, estimate the e⁄ect of common ownership on market performance and report that
(cid:133)rms increase their market share (up to 3.2%) through common ownership.9
The results that we derive complement those in the extant literature. For instance,
Leahy and Neary (1997) (cid:133)nd that R&D cooperation leads both to more output and
8Suzumura (1992) extends the analysis to multiple (cid:133)rms and general demand and cost functions
in Cournot competition. Ziss (1994) does likewise but also considers product di⁄erentiation and price
competition. Kamien et al. (1992) analyze the e⁄ects of R&D cartelization and joint research ventures.
For a survey, see Gilbert (2006).
9This result is consistent with Giannetti and Laeven (2009), who show that stock acquisitions by
pension funds enhance (cid:133)rm valuation.
6
to more R&D when spillovers are positive. Yet we show that, when there are minority
shareholdings, this result holds only when spillovers are high enough. We also identify
conditionsunderwhichacartelizedResearchJointVenture(RJV)isoptimal,generalizing
Kamien et al. (1992) and (cid:133)nding that this result depends on the innovation function
having little curvature.
3 Minority shareholdings
We may consider two types of acquisitions: when investors acquire (cid:133)rms(cid:146)shares, called
common ownership or (partial) cross-ownership by investors; and when (cid:133)rms acquire
other (cid:133)rms(cid:146)shares, called cross-ownership by (cid:133)rms. We discuss two cases of cross-
ownership by investors (common ownership) and one case of cross-ownership by (cid:133)rms. In
each case we show that, when the stakes are symmetric, the (cid:133)rm-i manager(cid:146)s problem is
to maximize
(cid:30) = (cid:25) +(cid:21) (cid:25) ; (1)
i i k
k=i
X6
where the value of (cid:21) depends on the type of ownership and corresponds to what Edge-
worth (1881) termed the coe¢ cient of (cid:147)e⁄ective sympathy(cid:148)among (cid:133)rms. The analysis is
developed in Appendix A.
3.1 Cross-ownership by investors (common ownership)
In this situation, (cid:133)rms(cid:146)stakes are held only by investors(cid:151)for example, large institutional
investors such as pension or mutual funds, which now have stakes in nearly three fourths
of all publicly traded US (cid:133)rms. Consider an industry with n (cid:133)rms and n investors;10
we let i and j index (respectively) investors and (cid:133)rms. The share of (cid:133)rm j owned by
investor i is (cid:29) , and the parameter (cid:16) corresponds captures the extent of i(cid:146)s control over
ij ij
(cid:133)rm j (Salop and O(cid:146)Brien 2000). The total (portfolio) pro(cid:133)t of owner i is (cid:25)i = (cid:29) (cid:25) ,
k ik k
where (cid:25)k are the pro(cid:133)ts of portfolio (cid:133)rm k. The manager of (cid:133)rm j takes intoPaccount
10We make this assumption solely to simplify notation. In fact, we need only that the total number
of investors, say I, be no less than the total number of (cid:133)rms (n). Then the expressions that follow for
(cid:21)SFI and (cid:21)PC hold if we replace n with I. However, this assumption also has the bene(cid:133)t of facilitating
comparisons with the case of cross-ownership by (cid:133)rms.
7
shareholders(cid:146)incentives (through the control weights (cid:16) ) and maximizes
ij
n
(cid:30) = (cid:25) + (cid:21) (cid:25) ;
j j jk k
k=j
X6
where
n (cid:16) (cid:29)
(cid:21) i=1 ij ik:
jk (cid:17) n (cid:16) (cid:29)
Pi=1 ij ij
We next discuss two important cases:Psilent (cid:133)nancial interests (a.k.a. passive invest-
ments) and proportional control.11
Silent Financial Interest (SFI). In this case, each owner (i.e., the majority or domi-
nant shareholder) i retains full control of the acquiring (cid:133)rm and is entitled to a share of
the acquired (cid:133)rm(cid:146)s pro(cid:133)ts(cid:151)but exerts no in(cid:135)uence over the latter(cid:146)s decisions. If i owns j,
then (i) (cid:16) = 1 and (cid:16) = 0 for k = j and (ii) (cid:133)rm i maximizes n (cid:29) (cid:25) . We consider
ij ik 6 k=1 ik k
the symmetric case, in which each investor i receives a share P(cid:11) in the acquired (cid:133)rms;
hence (cid:29) = 1 (n 1)(cid:11) and (cid:29) = (cid:11) for k = j. Then (cid:21)SFI (cid:11)=(1 (n 1)(cid:11)). The upper
ij ik
(cid:0) (cid:0) 6 (cid:17) (cid:0) (cid:0)
bound of cross-ownership is (cid:11) = 1=n, in which case (cid:21)SFI = 1 and n identical (cid:133)rms will
maximize total joint pro(cid:133)t.
Proportional Control (PC). Under proportional control, the (cid:133)rm(cid:146)s manager accounts
forshareholders(cid:146)own-(cid:133)rminterests inother (cid:133)rms inproportionto theirrespective stakes.
Suppose that each investor acquires a share (cid:11) of those other (cid:133)rms. Then, to compute
(cid:21) for a given k = j, we (cid:133)rst note that if i is the majority shareholder of j then
jk
6
(cid:16) = 1 (n 1)(cid:11) and (cid:29) = (cid:11). Yet if instead i+1 were the majority shareholder of k,
ij ik
(cid:0) (cid:0)
then that investor has control over j equal to (cid:16) = (cid:11) and receives an own-(cid:133)rm pro(cid:133)t
(i+1)j
share of (cid:29) = 1 (n 1)(cid:11). Finally, there are n 2 investors who are minority
(i+1)k
(cid:0) (cid:0) (cid:0)
shareholders of j and k; for these investors, the combination of their pro(cid:133)t shares (and
control) is equal to (cid:11). Thus we obtain
2[1 (n 1)(cid:11)](cid:11)+(n 2)(cid:11)2
(cid:21)PC (cid:0) (cid:0) (cid:0) :
(cid:17) [1 (n 1)(cid:11)]2 +(n 1)(cid:11)2
(cid:0) (cid:0) (cid:0)
As with any silent (cid:133)nancial interest, here (cid:21)PC = 1 when (cid:11) = 1=n. If (cid:11) < 1=n, then (cid:21) is
increasing in both n and (cid:11).
11Other governance structures are total control, partial control, (cid:133)duciary obligation, Coasian joint
control, and one-way control. For a discussion of these structures, see Salop and O(cid:146)Brien (2000).
8
3.2 Cross-ownership by (cid:133)rms
In this situation, (cid:133)rms may acquire their rivals(cid:146)stock in the form of passive investments
with no control rights (e.g., nonvoting shares; see Gilo et al. 2006). This setting features
a complex, chain-e⁄ect interaction between the pro(cid:133)ts of (cid:133)rms. Here (cid:11) denotes (cid:133)rm j(cid:146)s
jk
ownership stake in (cid:133)rm k, and the strategy decisions are made by the controlling share-
holder; thus the pro(cid:133)t of (cid:133)rm j is given by (cid:30) = (cid:25) + (cid:11) (cid:30) . We can derive the
j j k=j jk k
6
pro(cid:133)t of each (cid:133)rm by solving for a (cid:133)xed point of a matrPix equation. In the symmetric
case, (cid:11) = (cid:11) = (cid:11) for all j = k and (cid:11) = 0 for all j. It can be shown that (cid:133)rm j will
jk kj jj
6
maximize (cid:25) +(cid:21) (cid:25) , where (cid:21)PCO (cid:11)=[1 (n 2)(cid:11)] (we use PCO to index partial
j k=j k (cid:17) (cid:0) (cid:0)
6
cross-ownership).PIt follows that the upper bound of cross-ownership is (cid:11) = 1=(n 1), in
(cid:0)
which case (cid:21)PCO tends to 1 as (cid:11) approaches 1=(n 1). Just as in the two previous cases,
(cid:0)
(cid:21) is increasing in the number of (cid:133)rms and in the (cid:133)rms(cid:146)stakes.
3.3 Comparative statics on the degree of cross-ownership ((cid:21))
Table 1 summarizes the value of (cid:21) according to the type of cross-ownership. We can see
that more (cid:133)rms and higher investment stakes are both positively associated with (cid:21). In
addition, it is straightforward to show that (cid:21)PC > (cid:21)SFI > (cid:21)PCO. The implication is that,
in order to attain the same degree of cross-ownership (and for a given number of (cid:133)rms),
the symmetric investment stake with proportional control must be lower than with silent
(cid:133)nancial interests, which in turn must be lower than with partial cross-ownership by
(cid:133)rms: (cid:11)PC < (cid:11)SFI < (cid:11)PCO.
Table 1: Comparative Statics on (cid:21)
Common Ownership, Common Ownership, Cross-ownership
Silent Financial Interests Proportional Control (by (cid:133)rms)
(cid:21) (cid:11) 2(cid:11)[1 (n 1)(cid:11)]+(n 2)(cid:11)2 (cid:11)
(cid:0) (cid:0) (cid:0)
1 (n 1)(cid:11) [1 (n 1)(cid:11)]2+(n 1)(cid:11)2 1 (n 2)(cid:11)
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
@(cid:21)=@n + + +
@(cid:21)=@(cid:11) + + +
9
4 Framework and equilibrium
We consider an industry consisting of n 2 identical (cid:133)rms, where each (cid:133)rm i = 1;:::;n
(cid:21)
chooses simultaneously their R&D intensity (x ) and production quantity (q ). Firms
i i
produce a homogeneous good characterized by a smooth inverse demand function f(Q),
where Q = q . We make the following three assumptions.
i i
P
A.1. f(Q) is twice continuously di⁄erentiable, where (i) f (Q) < 0 for all Q 0 such
0
(cid:21)
that f(Q) > 0 and (ii) the elasticity of the slope of the inverse demand function,
Qf (Q)
00
(cid:14)(Q) ;
(cid:17) f (Q)
0
is constant and equal to (cid:14).
The parameter (cid:14) is the curvature (relative degree of concavity) of the inverse demand
function, so demand is concave for (cid:14) > 0 and is convex for (cid:14) < 0. Furthermore, demand
is log-concave for 1+(cid:14) > 0 and is log-convex for 1+(cid:14) < 0. If 1+(cid:14) = 0, then demand is
bothlog-concaveand log-convex.12 AssumptionA.1isalwayssatis(cid:133)edbyinversedemand
functions that are linear or constantly elastic. In particular, the family of inverse demand
functions for which (cid:14)(Q) is constant can be represented as
a bQ(cid:14)+1 if (cid:14) = 1;
f(Q) = (cid:0) 6 (cid:0)
8
>a blogQ if (cid:14) = 1;
<
(cid:0) (cid:0)
>
:
here a is a nonnegative constant and b > 0 (resp., b < 0) if (cid:14) 1 (resp., (cid:14) < 1).
(cid:21) (cid:0) (cid:0)
A.2. The marginal production cost or innovation function of (cid:133)rm i, or c , is independent
i
ofoutputandisdecreasinginbothownandrivals(cid:146)R&Dasfollows:c = c(x +(cid:12) x ),
i i j=i j
6
where c0 < 0, c00 0, and 0 (cid:12) 1 for i = j. P
(cid:21) (cid:20) (cid:20) 6
A.3. The cost of investment is given by the function (cid:0)(x ), where (cid:0)(0) = 0, (cid:0) > 0, and
i 0
(cid:0) 0.
00
(cid:21)
12We remark that (cid:14) is also related to the marginal consumer surplus from increasing output(cid:151)that is,
toMS= f (Q)Q. Aftersetting(cid:15) astheelasticityoftheinverse marginalconsumersurplusfunction
0 MS
(cid:0)
(so that (cid:15) = MS=(MSQ)), Weyl and Fabinger (2013) argue that (cid:15) measures the curvature of the
MS 0 MS
logarithm of demand. Under A.1, we can write 1=(cid:15) =1+(cid:14).
MS
10