Observable versus Unobservable R&D Investments in Duopolies By Kyung Hwan Baik and Sang-Kee Kim * Journal of Economics, June 2020, 130 (1): 37-66. Abstract We study a quantity-setting duopoly with homogeneous products in which two firms first make their cost-reducing R&D investments, and then compete in quantities. When making its R&D investment, each firm is uncertain about its R&D outcome. Its new marginal cost is probabilistically determined later, but before the firm chooses its output level. When choosing its output level, each firm has private information regarding its own new marginal cost. We develop the observable-investments and the unobservable-investments models. We compare the outcomes of these two main models, and perform comparative statics of them with respect to each of the parameters, respectively. As variations, we consider the observable-investments and the unobservable-investments model based on a price-setting duopoly with product differentiation. JEL classification: D43, L13, C72 Keywords: Observable R&D investment; Unobservable R&D investment; Uncertain R&D outcome; Private information regarding R&D outcomes; Cost-reducing R&D investment; Information sharing Baik: Department of Economics, Sungkyunkwan University, Seoul 03063, South Korea (e- * mail: [email protected]); Kim (corresponding author): Department of International Business, Chungbuk National University, Cheongju, Chungcheongbuk-do 28644, South Korea (e-mail: [email protected]). We are grateful to Chris Baik, Anne-Christine Barthel, Jerry Carlino, Hsueh-Jen Hsu, Jin-Hyuk Kim, Jong Hwa Lee, Wooyoung Lim, Daehong Min, seminar participants at Korea University and Yeungnam University, two anonymous referees, and the editor-in-chief of the journal, Giacomo Corneo, for their helpful comments and suggestions. We are also grateful to Hyok Jung Kim and Gwihwan Seol for their excellent research assistance. Earlier versions of this paper were presented at the 87th Annual Conference of the Western Economic Association International, San Francisco, CA, July 2012; and the 78th Annual Meeting of the Midwest Economics Association, Evanston, IL, March 2014. Part of this research was conducted while the first author was Visiting Professor at the University of Illinois at Urbana-Champaign.
Transcript
Observable versus Unobservable R&D Investments in Duopolies
By Kyung Hwan Baik and Sang-Kee Kim*
Journal of Economics, June 2020, 130 (1): 37-66.
Abstract We study a quantity-setting duopoly with homogeneous products in which two firms firstmake their cost-reducing R&D investments, and then compete in quantities. When making itsR&D investment, each firm is uncertain about its R&D outcome. Its new marginal cost isprobabilistically determined later, but before the firm chooses its output level. When choosingits output level, each firm has private information regarding its own new marginal cost. Wedevelop the observable-investments and the unobservable-investments models. We compare theoutcomes of these two main models, and perform comparative statics of them with respect toeach of the parameters, respectively. As variations, we consider the observable-investments andthe unobservable-investments model based on a price-setting duopoly with productdifferentiation.
JEL classification: D43, L13, C72
Keywords: Observable R&D investment; Unobservable R&D investment; Uncertain R&D outcome; Private information regarding R&D outcomes; Cost-reducing R&D investment; Information sharing
Baik: Department of Economics, Sungkyunkwan University, Seoul 03063, South Korea (e-*
mail: [email protected]); Kim (corresponding author): Department of International Business,Chungbuk National University, Cheongju, Chungcheongbuk-do 28644, South Korea (e-mail:[email protected]). We are grateful to Chris Baik, Anne-Christine Barthel, Jerry Carlino,Hsueh-Jen Hsu, Jin-Hyuk Kim, Jong Hwa Lee, Wooyoung Lim, Daehong Min, seminarparticipants at Korea University and Yeungnam University, two anonymous referees, and theeditor-in-chief of the journal, Giacomo Corneo, for their helpful comments and suggestions. Weare also grateful to Hyok Jung Kim and Gwihwan Seol for their excellent research assistance.Earlier versions of this paper were presented at the 87th Annual Conference of the WesternEconomic Association International, San Francisco, CA, July 2012; and the 78th AnnualMeeting of the Midwest Economics Association, Evanston, IL, March 2014. Part of thisresearch was conducted while the first author was Visiting Professor at the University of Illinoisat Urbana-Champaign.
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1. Introduction
In some countries, such as the United States and the United Kingdom, firms are required
to disclose information regarding their research and development (R&D) investments, while in
others, such as France and Germany, there are no such disclosure requirements (see Hall and
Oriani 2006). Even in countries with compulsory disclosure requirements, firms may still have
substantial discretion on whether to disclose true information regarding their R&D investments.
Koh and Reeb (2015) report that 2,110 of the 3,000 NYSE-listed firms in their sample fail to
provide any information regarding their corporate R&D efforts. All this then implies that firms
may or may not disclose true information regarding their R&D investments, and at the same time
that firms may or may not observe the R&D investments of their rival firms.
An interesting question in such environments is how the observability or unobservability
of R&D investments affects firms' R&D investments, their profits, consumer surplus, and social
welfare. Another interesting question is whether firms have an incentive to disclose true
information regarding their R&D investments. Yet another interesting question is whether
compulsory disclosure requirements benefit either consumers or firms. Still yet another
interesting question is whether compulsory disclosure requirements increase social welfare.
To address these questions, we study specific situations where firms first make R&D
investments to reduce their production costs, and then produce their products at the resulting
reduced costs. When making its R&D investment, each firm is uncertain about its R&D
outcome or, precisely, its resulting production cost. After obtaining its R&D outcome, each firm
has private information regarding its R&D outcome.
Such situations with both ex ante uncertainty and ex post private information regarding
R&D outcomes may well be readily observed. For example, when making their R&D
investments in semiconductor technology, Samsung Electronics Co., Ltd., Intel Corp., and SK
Hynix Inc. may well be uncertain about their R&D outcomes. When competing in the
semiconductor market, they may not know the other firms' new production costs. Indeed, ex
ante uncertainty regarding R&D outcomes may well occur naturally. Ex post private
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information regarding R&D outcomes may well occur because firms may not disclose
information regarding their R&D outcomes, because such information disclosed or signaled by
other firms may be unverifiable, and/or because it may take a long time to obtain true
information regarding other firms' R&D outcomes.
Accordingly, we study those situations with both ex ante uncertainty and ex post private
information regarding R&D outcomes in two separate cases. One is the case where firms
observe the R&D investments of their rival firms before they produce their products. The other
is the case where firms do not observe.
Formally, we first study a quantity-setting duopoly with homogeneous products in which
two firms first make their cost-reducing R&D investments, and then compete in quantities. In
this duopoly, when choosing its R&D investment, each firm is uncertain about how much
marginal cost it can reduce by undertaking the R&D investment. Each firm's new marginal cost
is probabilistically determined later, but before the firm chooses its output level. When choosing
its output level, each firm has private information regarding its own new marginal cost.
We develop two models: the observable-investments model and the unobservable-
investments model. These two models differ depending on whether or not the firms know the
R&D investment of their rival firm when choosing their output levels. The first model assumes
public information regarding R&D investments, whereas the second model assumes private
information regarding R&D investments. By considering the two models and further comparing
their outcomes, we can better understand the above-mentioned situations with both ex ante
uncertainty and ex post private information regarding R&D outcomes, and may argue for or
against, for example, policies or regulations or institutions which require the firms to announce
and commit to their R&D investments.
In the observable-investments model, we set up the following two-stage game. In the
first stage, each firm independently makes its cost-reducing R&D investment, and then the two
firms simultaneously announce and commit to their investments. In the second stage, after
observing their R&D investments, the firms choose their output levels simultaneously and
3
independently. When choosing its output level, each firm knows exactly its own new marginal
cost, but knows only the probability distribution of the rival firm's new marginal cost.
In the unobservable-investments model, we set up the following game. First, the firms
independently make their cost-reducing R&D investments. Naturally, each firm observes its
own R&D investment. But it cannot observe the rival firm's R&D investment. Next, the firms
choose their output levels simultaneously and independently. When choosing its output level,
each firm knows exactly its own new marginal cost, but knows only the probability distribution
of the rival firm's new marginal cost given its belief about the rival firm's R&D investment.
We believe that this paper makes at least the following contributions to the literature on
R&D competition. First, we study R&D competition with both ex ante uncertainty and ex post
private information regarding R&D outcomes. Second, we analyze the unobservable-
investments model, treating it as a between thesimultaneous-move game with sequential moves
two firms. Third, we compare the outcomes of the observable-investments model with those of1
the unobservable-investments model. Finally, we examine how the outcomes of the two models
respond when each of the parameters changes, .ceteris paribus
Comparing the outcomes of the two models, we find that, in equilibrium, each firm
makes a greater cost-reducing R&D investment in the observable-investments model than in the
unobservable-investments model. We can explain this result as follows. In the observable-
investments model, each firm can make a strategic commitment for the second-stage output
competition by choosing and announcing its R&D investment in the first stage. Given this
possibility, each firm chooses and announces a "large" R&D investment in an attempt to gain its
competitive advantage in the second-stage output competition. Such strategic behavior of the
firms, which is absent from the unobservable-investments model, leads to the result.
1Baik and Lee (2007) define a between twosimultaneous-move game with sequential movesparties, and come up with a solution technique for it. They define it as a game in which eachparty has two sequential moves, the first action chosen by each party is hidden from the otherparty, and the parties choose their second actions simultaneously.
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Next, we find that, in equilibrium, each type of each firm chooses a smaller output level,
but each firm's expected output level is greater, in the observable-investments model than in the
unobservable-investments model. The former statement comes from the fact that each type of2
each firm faces strong types low marginal costs of the rival firm with higher probabilities in
the observable-investments model. The latter comes from the fact that each firm has low
marginal costs with higher probabilities in the observable-investments model.
Next, , in equilibrium, each type of each firm earns less expected grosswe find that
profits in the observable-investments model than in the unobservable-investments model. We
may explain this result using the fact that each type of each firm faces strong types of the rival
firm with higher probabilities in the observable-investments model.
Finally, but most importantly, we find that the firms' equilibrium expected profits, the
equilibrium expected consumer surplus, and the equilibrium expected social welfare are less in
the observable-investments model than in the unobservable-investments model. On the basis of
this result, we may well argue that socially desirable enact policies or regulationsit is not to
which require firms to disclose information regarding their R&D investments. Indeed, we may
well argue against such mandatory disclosure policies or regulations.
Performing comparative statics of the outcomes of the two models with respect to each of
the parameters, respectively, we find several interesting results. For example, we find in each of
the models that, as the size of the market increases, the equilibrium R&D investment and the
equilibrium expected profits of each firm increase. The intuition behind this is that an increase
in the market size enables the firms to capture more profits, and increases the returns from their
R&D investments, which leads to the firms increasing their R&D investments. We also find in
each of the models that, as the initial marginal cost of production decreases, the equilibrium
R&D investment and the equilibrium expected profits of each firm increase.
2As will be clear shortly, each firm's type is determined by its realized marginal cost.
5
As variations, we consider the observable-investments and the unobservable-investments
model based on a price-setting duopoly with product differentiation. Comparing the outcomes of
these two models, we find, for example, that each firm makes a smaller cost-reducing R&D
investment in the observable-investments model than in the unobservable-investments model.
Performing comparative statics of the outcomes of the two models with respect to the product
differentiation parameter, we find that the outcomes of each model almost all increase as the
firms' products become more differentiated.
The paper proceeds as follows. In Section 2, we introduce a quantity-setting duopoly
with R&D. Section 3 develops and analyzes the observable-investments model, and obtain its
outcomes. Section 4 develops and analyzes the unobservable-investments model, and obtains its
outcomes. In Section 5, we compare the outcomes of the observable-investments model with
those of the unobservable-investments model. In Section 6, we perform comparative statics of
the outcomes of the two models with respect to the parameters. Section 7 discusses variations of
the two main models. Finally, Section 8 offers our conclusions.
1.1. Related literature
This paper is closely related to Brander and Spencer (1983). They study a duopoly in
which two firms undertake their cost-reducing R&D investments, and compete in quantities.
They consider two models: the strategic model and the nonstrategic model. In the strategic
model, when choosing its output level, each firm knows the rival firm's R&D investment (and
new production cost). On the other hand, in the nonstrategic model, each firm chooses its R&D
investment and output level without observing those chosen by the rival firm.simultaneously
They show that, under perfect symmetry, each firm undertakes a greater R&D investment,
chooses a greater output level, and earns less profits, in the strategic model than in the
nonstrategic model. They show also that, in some cases, strategic R&D is welfare improving,
compared with nonstrategic R&D.
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This paper differs from Brander and Spencer (1983), notably in several respects. Firstly,
unlike their paper, this paper assumes in both main models that each firm is uncertain about how
much production cost it can reduce when choosing its R&D investment. Secondly, unlike their
paper, this paper assumes in both main models that the precise effect of each firm's R&D
investment on cost reduction is not known but the probability distribution of the firm's new
marginal cost is "known" to the rival firm when the firms choose their output levels. Thirdly, 3
in the unobservable-investments models of this paper, each firm chooses its R&D investment
and output level without observing those chosen by the rival firm, whereas in thesequentially
nonstrategic model of their paper, each firm chooses its R&D investment and output level
simultaneously. Fourthly, this paper obtains that, in equilibrium, each type of each firm chooses4
a smaller output level in the observable-investments model than in the unobservable-investments
model, which is in contrast with their result that each firm chooses a greater output level in the
strategic model than in the nonstrategic model. Fifthly, this paper obtains that the equilibrium
expected social welfare is less in the observable-investments model than in the unobservable-
investments model, which is in contrast with their result that strategic R&D is welfare improving
in some cases. Finally, unlike their paper, this paper considers also a price-setting duopoly with
R&D.
Many economists have studied cost-reducing R&D in different contexts. Examples
include d'Aspremont and Jacquemin (1988), Delbono and Denicolo (1991), Thomas (1997), Lin
and Saggi (2002), Haaland and Kind (2008), Tishler and Milstein (2009), Bourreau and Dogan
(2010), Goel and Haruna (2011), Ishida et al. (2011), Lin and Zhou (2013), Milliou and Pavlou
(2013), Kawasaki et al. (2014), Tesoriere (2015), Sengupta (2016), and Chatterjee et al. (2019).
3Brander and Spencer (1983) assume in the strategic model that the precise effect of each firm'sR&D investment on cost reduction is known to the rival firm when the firms choose their outputlevels.4One may classify the unobservable-investments model of this paper as a simultaneous-movegame with sequential moves, following Baik and Lee (2007), and the nonstrategic model inBrander and Spencer (1983) as a standard simultaneous-move game.
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In particular, Tishler and Milstein (2009) study a Cournot oligopoly with differentiated products
in which each firm first chooses its R&D program, and then chooses its output. In their two-
stage game, the outcomes of each firm's R&D program are random, and their realizations are
known to all the firms when the firms choose their outputs. Ishida et al. (2011) study a quantity-
setting oligopoly with different initial production costs in which asymmetric firms first make
their cost-reducing R&D investments, and then compete in quantities. In their model, each
firm's R&D investment and its new marginal cost are known to the rival firms when the firms
choose their output levels. Milliou and Pavlou (2013) study a vertically related industry in
which two upstream firms and two downstream firms play a five-stage game. In their model, the
R&D process is deterministic, and all the actions including each firm's R&D investment are
observable. Tesoriere (2015) studies a model in which firms in R&D cartels first make their
cost-reducing R&D investments, and then compete in quantities. In his model, each firm's R&D
investment and its new marginal cost are known to the rival firms when the firms choose their
output levels. Sengupta (2016) studies a duopoly with homogeneous products in which firms
first make their cost-reducing R&D investments, and then compete in prices. She considers two
cases: the case with observable R&D investments and the case with unobservable R&D
investments. In each case, when making its R&D investment, each firm is uncertain about its
R&D outcome, and when choosing its price, each firm has private information regarding its own
R&D outcome.
There are many other interesting papers on R&D. For example, Vives (1989) studies an
oligopoly in which firms first choose their technological positions cost-reducing investments
in the cost reduction model and plant designs in the plant design model and then compete in
quantities. In the oligopoly, each firm is uncertain about demand when making its technological
choice, but receives a private signal about demand before choosing its output level. Each firm's
technological choice and its effect on production cost are observable to the rival firms. Bagwell
and Staiger (1994) evaluate the case for R&D subsidies in export sectors, considering a model in
which each firm is uncertain about its R&D outcome when making its R&D investment, its new
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production cost is randomly determined later, but before the firms compete in the product
market, and each firm observes the realizations of the firms' new production costs before
competing in the product market.
Slivko and Theilen (2014) study a Cournot duopoly model in which each firm chooses
between two R&D strategies, innovation and imitation. They consider a two-stage game in
which the firms first choose and publicly announce their R&D strategies, and then compete in
quantities. In their model, each firm's R&D outcome is deterministically determined before the
firm chooses its output. Chang and Ho (2014) study a duopoly model with product R&D in
which firms first make and publicly announce their R&D investments, and then compete in
quantities or prices. In their model, each firm's R&D outcome is deterministically determined
before the firm competes in the product market. Steinmetz (2015) studies a duopoly model in
which firms engage in step-by-step innovation to reduce their production costs, and experience
learning-by-doing and organizational forgetting in R&D. In his model, innovations are
uncertain. Ishii (2017) studies an international duopoly model in which a firm in one country
undertakes process R&D and a firm in the other country undertakes product R&D. He considers
a three-stage game in which the governments choose and publicly announce their R&D subsidy
rates, then the firms choose and publicly announce their R&D investments, and finally the firms
choose their outputs (or exports). In his model, each firm's R&D outcome is deterministically
determined before the firm chooses its output.
This paper is related to Baik and Lee (2007), Nitzan and Ueda (2011), Baik and Kim
(2014), and Baik (2016). These papers study contests, and consider models in which the
decision-makers in each party choose two actions. In some of these models, the firstsequential
action chosen by each party is observed by the other parties before the parties choose their
second actions. In others, the first action chosen by each party is hidden from the other parties.
For example, Baik and Kim (2014) study two-player contests with delegation in which
each player first hires a delegate and independently writes a contract with her delegate, and then
the two delegates expend their effort simultaneously and independently to win a prize. They find
9
that, if the players are symmetric, then the observable-contracts and the unobservable-contracts
models each player offers her delegate theyield the same outcomes. That is, in both models,
same contingent compensation, the equilibriumeach delegate chooses the same effort level,
expected payoffs for each player are the same he equilibrium expected payoffs for each, and t
delegate are the same.
Baik (2016) studies contests in which the players in each group first decide jointly how to
share a prize among themselves if one of them wins it, and then the players in all the groups
choose their effort levels simultaneously and independently to win the prize. In the case where
there is more than one group, he finds that, in the model of observable sharing rules, each group's
equilibrium winner's fractional share is greater, the equilibrium total effort level is greater, and
(unless the number of players is sufficiently large) the equilibrium expected payoff for each
player in each group is , than in the model of unobservable sharing rules. less
This paper is related also to the literature on information sharing. Papers in this literature
focus on examining, in different contexts, whether firms have incentives to share their private
information with their rivals. Examples include Vives (1984), Gal-Or (1985), Gill (2008), Baik
and Lee (2012), and Kovenock et al. (2015).
2. A quantity-setting duopoly with R&D
Consider a quantity-setting duopoly with homogeneous products in which two profit-
maximizing risk-neutral firms, 1 and 2, first make R&D investments to reduce their production
costs, and then produce their products at the resulting reduced costs. When making its R&D
investment, each firm is uncertain about its R&D outcome. Each firm's new production cost is
probabilistically determined later, but before the firm chooses its output level. We assume in
Section 3 that the firms observe their rival firm's R&D investment, and assume the opposite in
Section 4. The precise effect of each firm's R&D investment on cost reduction is hidden from
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the rival firm when the firms choose their output levels. However, if given an R&D investment5
of the rival firm, each firm knows the probability distribution from which the rival firm's new
cost is drawn.
Let , for 1, 2, represent the R&D investment that firm makes, where 0. Thex i i xi iœ
cost of firm 's R&D investment is assumed to be quadratic, and more specifically, it is given byi
# #x R q2i + iÎ2, where 1. Let denote the set of all nonnegative real numbers. Let represent 6
the output level of firm , where . The cost function of firm is given by ( ) fori q R i T q c qi + i i i i− œ
all , where ( ) represents the cost to firm of producing units of its product, and isq R T q i q c i + i i i i−
a positive constant, which represents a constant marginal cost to firm .i
The products of the firms are sold at a single price, which is determined by the inverse
market demand function together with the output levels of the firms:
for 0P a Q Q aœ Ÿ Ÿ
0 for ,Q a
where represents the market price, is a positive constant, and .P a Q q qœ 1 2
The firms initially have the same marginal cost of production, which is publiclycM
known. We assume at this point that 0 or, equivalently, 0 1. The firms Î c a c aM M7
know that their marginal costs of production can be reduced by undertaking R&D investments,
5Thomas (1997), Sengupta (2016), and Chatterjee et al. (2019) consider duopoly models inwhich each firm's new marginal cost resulting from R&D investment is hidden from the rivalfirm when the firms compete in the product market. Vives (2002) considers a Cournot market inwhich firms first decide whether to enter the market or not, and then compete in quantities. Inhis model, when making its entry decision, each firm is uncertain about its production cost; whenchoosing its output level, each firm has private information regarding its production cost.Bagnoli and Watts (2010) consider a Cournot duopoly in which each firm has privateinformation regarding its production cost when the firms compete in the product market.6Quadratic R&D cost functions are used in, for example, d'Aspremont and Jacquemin (1988),Aghion et al. (2001), Lin and Saggi (2002), Haaland and Kind (2008), Tishler and Milstein(2009), Bourreau and Dogan (2010), Ishida et al. (2011), Lin and Zhou (2013), Milliou andPavlou (2013), and Chang and Ho (2014).7We will make further restrictive assumptions on the parameters later, during the analysis (seefootnote 20).
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and further know the probability distributions from which their new marginal costs resulting
from R&D investments will be drawn. Let ( ; ) represent the cumulative distributionF c xi i i
function of firm 's marginal cost which results from its R&D investment, and let ( ; )i C f c xi i i i
represent the corresponding probability density function. We assume8
( ; ) [ ( ) ] ( ) ( )F c x c c x I c I ci i i M i c x i c x ik kiœ Î $ (0, ] ( , )M i M i $ $ ∞
and
( ; ) [ ( ) ] ( ), (1)f c x kc c x I ci i i M i c x ik kiœ Î1
(0, ] $ M i$
where 0 1, 1, and ( ) is the indicator function of a set . We also assume that the †$ k I ZZ9
two random variables, and , are independent. Note that, as the parameter or firm 's R&DC C i1 2 $
investment increases, , the upper limit of the support decreases and ( ; )x ceteris paribus f c xi i i i
increases at all values of in the resulting support. Note also that is uniformly distributedc Ci i10
over the interval (0, ] if 1, and that, as the parameter increases, ,c x k k ceteris paribusM i $ œ
f c x c ci i i i i( ; ) increases at "high" values of in the support while it decreases at "low" values of
(see Figure 1). The latter implies that, given firm 's R&D investment , the greater thei xi
parameter is, the greater is the probability that firm has a high realized marginal cost.k i
In fact, those cumulative distribution and probability density functions are derived, as in
Melitz and Ottaviano (2008), by assuming that 1 has a Pareto distribution. RecentÎCi11
empirical research has commonly concluded that the distribution of firm productivity across
8We use a capital letter to denote a random variable, and the corresponding small letter to denotea value of the random variable.9We assume that is sufficiently large that ( ) 0 in a relevant range of firm 'sc c x iM M iÎ $ $ R&D investments.10The greater firm 's R&D investment, the better the probability distribution of its new marginalicost. Here "better" could be taken to mean that one distribution first-order stochasticallydominates the other.11Let be a random variable with a Pareto distribution with the lower bound parameter andY ym
the tail length shape parameter , where 0 and 0. Then its cumulative distributionα αym function is ( ) [1 ( ) ] ( ). This paper assumes that 1 has a ParetoF y y y I y Cœ Î Îm y i
α[ , )m ∞
distribution with the lower bound parameter 1 ( ) and the tail length shape parameter .Î c x kM i $
12
sectors and countries fits well a Pareto distribution, and that the distribution of returns on
patented inventions fits a Pareto distribution (see Bessen and Maskin 2009). Other papers in
which a Pareto distribution is assumed for the distribution of productivity across firms include,
for example, Chaney (2008) and Bustos (2011).
3. The model with observable R&D investments
Consider a quantity-setting duopoly with R&D introduced in Section 2. In the current
model, we add to it the assumption that each firm's R&D investment is observable to the rival
firm.
We consider the following two-stage game. In the first stage, each firm independently
makes its cost-reducing R&D investment, and then the two firms simultaneously announce and
commit to their investments that is, firm 1 announces publicly the value of , and firm 2 x1
announces publicly the value of . Thus each firm observes both firms' R&D investments. Inx2
the second stage, the firms choose their output levels simultaneously and independently. When
choosing its output level, each firm knows exactly the realization of its own new marginal cost,
but not that of the rival firm's new marginal cost. However, firm knows the probabilityi
distribution specifically, ( ; ) and ( ; ) of possible realizations of the rival firm's new F c x f c xj j j j j j
marginal cost .Cj12
Let ( ) represent firm 's second-stage expected profits of producing ( ), given1i i i ic i gross q c
its realized marginal cost of that is, firm 's second-stage expected profits withoutc ii
subtracting its cost of R&D investment. Then the gross profit function for firm whichi
associates a value of ( ) with each possible output level of firm is1i ic i
( ) ( ( ) ( ) ) ( ) ( ; ) , (2)1i i i i j j i i i j j j jc xc a q c q c c q c f c x dcœ
0M j$
12Throughout the paper, when we use and at the same time, we mean that .i j i jÁ
13
where ( ) represents the output level of firm , which is firm 's belief, when is firm 'sq c j i c jj j j
realized marginal cost.
Next, consider the firms' expected profits computed in the first stage of the game or, more
precisely, at the time when they choose their R&D investments. At this time, given its belief
about the rival firm's R&D investment, each firm seeks to maximize its expected profits over its
R&D investment, having perfect foresight about the equilibrium of every subgame that starts at
the beginning of the second stage. Let ( , ) represent firm 's expected profits when itH x x ii 1 2
chooses an R&D investment of , forming its belief that firm chooses an R&D investment ofx ji
x i'j. Then firm s profit function in the first stage is
( , ) ( ) ( ; ) 2, (3)H x x c f c x dc xi i i i i ic x N
i i1 2 02œ Î M i$
1 #
where ( ) represents the second-stage equilibrium expected gross profits for firm , given the1Ni ic i
R&D investments and , resulting when is firm 's realized marginal cost in the secondx x c i1 2 i
stage.
We end the description of the model by assuming that all of the above is common
knowledge between the firms.
Now, to obtain the equilibrium R&D investments and the equilibrium output levels of the
firms, we work backward, considering first the second stage of the two-stage game. In the
second stage, the firms choose their output levels simultaneously and independently, each
knowing the R&D investments, and , chosen in the first stage, the realization of its ownx x c 1 2 i
marginal cost, and the probability density function ( ; ) of the rival firm's marginal cost .f c x Cj j j j
We analyze this second stage by formulating it as a Bayesian game.13
In this Bayesian game, since firm announced publicly its R&D investment in the firstj xj
stage, type of firm knows the probability density function ( ; ) of firm 's marginal costc i f c x ji j j j
C c x f c xj M j j j j, and the upper limit ( ) of the support of ( ; ). Thus when choosing its output $
13For the notion and definition of a Bayesian game, see, for example, Osborne (2004).
14
level, it uses ( ; ) to assign probabilities to the events that it competes against different typesf c xj j j
of firm .j
Given the R&D investments, and , type of firm seeks to maximize its expectedx x c i1 2 i
gross profits (2) over its output level ( ), taking the output levels of all types of firm as given,q c ji i
where (0, ]. From the first-order condition for maximizing function (2), we obtainc c xi M i− $
the reaction function of type of firm :c ii
q c a c q c f c x dci i i j j j j j jc x( ) 0.5( ) 0.5 ( ) ( ; ) , (4)œ
0M j$
for , 1, 2 with . It is straightforward to check that the second-order condition fori j i jœ Á 14
maximizing (2) is satisfied. Then, using those reaction functions of all the types of firms 1 and15
2, one for each type of each firm, we obtain the Nash equilibrium of the second-stage subgame
(see Appendix A):16
q c a c k c x x kNM1 1 1 2 1( ) (2 3 ) 6 ( 2 ) 6( 1)œ Î Î $ $
for every type of firm 1 with (0, ], and (5)c c c x1 1 1− M $
q c a c k c x x kNM2 2 2 1 2( ) (2 3 ) 6 ( 2 ) 6( 1)œ Î Î $ $
for every type of firm 2 with (0, ]. Note that this Nash equilibrium consists ofc c c x2 2 2− M $
the output levels of all the types of firms 1 and 2, one for each type of each firm.17
14The reaction function of type of firm shows its best response to every possible "vector" ofc ii
output levels, one for each type of firm , that all types of firm might choose.j j15Note that the second-order condition is satisfied for every maximization problem in this paper;however, for concise exposition, we do not state it explicitly in each case.16We assume at this point that is sufficiently larger than , in order to have that ( ) 0 fora c q cM i
Ni
all (0, ] (see footnote 20).c c xi M i− $17At the Nash equilibrium of the Bayesian game, the output level of each type of firm is the bestiresponse to the output levels of all the types of firm , one for each type.j
15
Let ( ) represent the expected gross profits for type of firm at the Nash1Ni i ic c i
equilibrium of the second-stage subgame. Then, substituting the second-stage equilibrium
output levels in (5) into function (2), we obtain
( ) ( ( ) ( ) ) ( ) ( ; ) 1N N N Ni i j ii i j i i j j j j
c xc a q c q c c q c f c x dcœ 0
M j$
{ ( )} . (6)œ q cNi i
2
Next, consider the first stage in which the firms choose their R&D investments
simultaneously and independently. Each firm knows what will be the probability density
function of its own marginal cost if choosing a certain level of its R&D investment, and also
knows what will be the probability density function of the rival firm's marginal cost if the rival
firm chooses a certain level of its R&D investment. In this stage, taking firm 's R&Dj
investment as given, firm seeks to maximize its expected profits, ( , ) in (3), over itsx i H x xj i 1 2
R&D investment , having perfect foresight about the Nash equilibrium of every second-stagexi
subgame or, equivalently, having perfect foresight about the second-stage equilibrium output
levels in (5) for any values of and . Note that ( ) in (3) is specified by function (6).x x c1 2 1Ni i
From the first-order condition for maximizing function (3), we obtain firm 's best response ( )i r xi j
to . Firm 's reaction function, ( ), which shows its best response to every possiblex i x r xj i i jœ
value of that firm might choose, is thenx jj
x A k k x Bi j { 8 ( 2) } ,œ Î$2 2
for , 1, 2 with , where {8( 1)( 2) (8 16 9) } andi j i j A k k k a k k cœ ´ Á $ 2M
B k k k k k´ 18 ( 1) ( 2) (16 32 9). By solving the system of two simultaneous# $2 2 2
equations, ( ) and ( ), we obtain the firms' R&D investments, and , whichx r x x r x x x1 1 2 2 2 1* *1 2œ œ
are specified in the equilibrium of the two-stage game.
Next, substituting the firms' equilibrium R&D investments into (5), we obtain the output
levels of all the types of firms 1 and 2, which are specified in the equilibrium of the two-stage
game. Finally, substituting the firms' equilibrium R&D investments and the equilibrium output
16
levels of all the types of the firms into functions (3) and (6), we obtain the firms' expected profits
and the expected gross profits of all the types of the firms, respectively, in the equilibrium of the
two-stage game.
Let denote the equilibrium expected consumer surplus, and let denote theCS SW* *
equilibrium expected social welfare, where is defined as the simple sum of and theSW CS* *
firms' equilibrium expected profits. Lemma 1 reports the outcomes of the observable-
investments model.
Lemma 1. i The firms' equilibrium R&D investments are x x A B k k( ) { 8 ( 2)},* *1 2
2 2œ œ Î $
where A k k k a k k c and B k k k k {8( 1)( 2) (8 16 9) } 18 ( 1) ( 2) (16´ ´ $ # $2 2 2 2M
32 9). 1, 2. ( ) , k Let x x for i ii The equilibrium output level of type c of firm i for* *i i´ œ
i is q c a c k c x k for c c x iii Theœ − 1, 2, ( ) (2 3 ) 6 ( ) 6( 1) (0, ]. ( ) * * *i i i M i Mœ Î Î $ $
equilibrium expected gross profits of type c of firm i for i are c q c fori i ii i*, 1, 2, ( ) { ( )} œ œ1* 2
c c x iv Firm i's equilibrium expected profits are H x x ci M i i* * * c x
i− œ(0, ]. ( ) ( , ) ( )$ 1 2 0* M
*$1
f c x dc x for i where f c x kc c x I c* * * k * ki i ii i i i i M ic x( ; ) ( ) 2, 1, 2, ( ; ) [ ( ) ] ( ), œ Î# 2 1
(0, ]Î œ $M
*$
and H x x H x x holds v The equilibrium expected consumer surplus is1 21 2 1 2( , ) ( , ) . ( )* * * *œ
CS c x a k c x a k k k k* * *M Mœ ( ) [32 ( 1) ( ){16 ( 1)( 3) (5 18 21)$ $2 2 2 2
( )}] 144( 1) . ( ) ( ,c x k vi The equilibrium expected social welfare is SW CS H xM* * * * œ$ Î 2
1 1
x H x x* * *2 1 22) ( , ).
We assume that , where 4 9 ( 1), which leads to ( ) 0.S c a S k k c x_ _ Î œ Î M M*$ #2 $
We assume also that min{ , 1}, which, together with , leads to ( ) 0 forc a S S c a q c_
_M M i*iÎ Î
all (0, ], where {6 ( 1) ( 2) (4 5)} 3 ( 1)( 2)(2 3).c c x S k k k k k k k_
i M*− œ$ # $ # Î 2 2
Note that ( , ) in p positive under the given assumptions on the parameters.H x xi* *1 2 art ( ) isiv
An observation from Lemma 1 is that, in equilibrium, the firms make the same cost-
reducing R&D investment, and earn the same expected profits; they have the same second-stage
"type-output function," ( ), and the same second-stage "type-profits function," ( ). Notq c c*i ii i1*
17
surprisingly, this arises because the firms are symmetric at the start of the game. Actually,
however, the firms may be asymmetric from the second stage on, due to their different realized
marginal costs. If so, then their output levels differ, and so do their profits.
4. The model with unobservable R&D investments
Consider a quantity-setting duopoly with R&D introduced in Section 2. In the current
model, we add to it the assumption that, when choosing their output levels, the firms do not
know their rival firm's R&D investment.
We consider the following game. First, the firms make their cost-reducing R&D
investments simultaneously and independently that is, firm 1 chooses a value of , and firm 2 x1
chooses a value of . Naturally, each firm observes its own R&D investment. But it cannotx2
observe the rival firm's R&D investment. Next, the firms choose their output levels
simultaneously and independently. When choosing its output level, each firm knows exactly the
realization of its own new marginal cost, but not that of the rival firm's new marginal cost.
However, under its belief about the rival firm's R&D investment, the firm "knows" the
cumulative distribution function and the probability density function of the rival firm's new
marginal cost.
Let ( ) represent firm 's expected gross profits of producing ( ) computed at the<i i i ic i q c
time when it chooses its output level given its realized marginal cost of and given its belief ci
that firm chooses an R&D investment of . Then the gross profit function for firm isj x ij18
( ) ( ( ) ( ) ) ( ) ( ; ) , (7)<i i i i j j i i i j j j jc xc a q c q c c q c f c x dcœ
0M j$
where ( ) represents firm 's belief about firm 's output level when is firm 's realizedq c i j c jj j j
marginal cost.
18Note that each firm forms its belief about the rival firm's R&D investment when it chooses itsR&D investment.
18
Let ( , ) represent firm 's expected profits from choosing computed at the timeG x x i xi i1 2
when it chooses its R&D investment given its belief that firm chooses an R&D investment of j
x i'j. Then firm s profit function is
( , ) ( ) ( ; ) 2, (8)G x x c f c x dc xi i i i i ic x m
i i1 2 02œ Î M i$
< #
where ( ) represents the expected gross profits for firm , given its belief that <mi i ic maximum i c
will be its realized marginal cost and given its belief about firm 's output level as a function ofj
firm 's realized marginal cost.j
We end the description of the model by assuming that all of the above is common
knowledge between the firms.
Now, to obtain the equilibrium R&D investments and the equilibrium output levels of the
firms, we begin by obtaining firm 's best response to a pair ( , ) consisting of firm 's R&Di x jj q cj j( )
investment and the output levels of all types of firm (see Baik and Lee 2007). Nowx jj q cj j( )
that firm 's best response to ( , ) consists of firm 's R&D investment and its output level,i x ij q cj j( )
working backward, we consider first firm 's decision on its output level, and then consider firmi
i's decision on its R&D investment.
Consider firm 's decision on its output level. After knowing its own realized marginali
cost , firm (or rather type of firm ) seeks to maximize its expected gross profits (7) over itsc i c ii i
output level ( ), taking firm 's R&D investment and the output levels of all types ofq c j xi i j q cj j( )
firm as given, where (0, ]. From the first-order condition for maximizing functionj c c xi M i− $
(7), we obtain
q c a c MBRi i i j( ) 0.5( ), (9)œ
for , 1, 2 with , where ( ) ( ; ) . Then, substituting ( ) in (9)i j i j M q c f c x dc q cœ ´Á j j j j j j j ic x BR
i
0M j$
into function (7), we obtain the maximum expected gross profits for type of firm :c ii
( ) ( ( ) ( ) ) ( ) ( ; ) <m BR BRi i ii i j j i i j j j j
c xc a q c q c c q c f c x dcœ 0
M j$
{ ( )} . (10)œ q cBRi i
2
19
Next, consider firm 's decision on its R&D investment. Taking firm 's R&D investmenti j
x j ij and the output levels of all types of firm as given, firm seeks to maximize its expectedq cj j( )
profits, ( , ) in (8), over its R&D investment , having perfect foresight about ( ) forG x x x q ci i iBRi1 2
every type of firm with (0, ], for any value of . Because ( ) in (8) isc i c c x x ci i M i i imi− $ <
specified by function (10), function (8) can be rewritten as
( , ) ( ) 4( 2) ( ) ( ) 2( 1)G x x k c x k a M k c x ki M i j M i1 22œ $ $Î Î
( ) 4 2. (11) Î a M x Î j i2 2#
From the first-order condition for maximizing function (11), we obtain
x k k a M k c k DBRi j M {( 2)( ) ( 1) } ( 1) , (12)œ Î $
for , 1, 2 with , where 2 ( 2) .i j i j D k kœ ´ Á # $2
We now obtain the reaction functions for firm . Firm has two reaction functions, onei i
from firm 's decision on its output level and one from firm 's decision on its R&D investment:i i
q c q c x xi i i iBR BRi i( ) ( ) and .œ œ
Finally, we obtain the firms' equilibrium R&D investments and the equilibrium output
levels of all the types of the firms, ( , , , ). Because , ( ), , andx x x q c x** ** **1 2 1 1 2
** **1q c q c** **
1 21 2( ) ( )
q c**2 2( ) satisfy all the four reaction functions of the firms simultaneously, we obtain them by
solving the following system of four simultaneous equations:
( ) 0.5( ), (13)q c a c M1 1 1 2œ
{( 2)( ) ( 1) } ( 1) , (14) x k k a M k c k D1 2œ Î $ M
( ) 0.5( ), (15)q c a c M2 2 2 1œ
and
{( 2)( ) ( 1) } ( 1) , (16) x k k a M k c k D2 1œ Î $ M
20
where ( ) ( ; ) and ( ) ( ; ) . First, usingM q c f c x dc M q c f c x dc2 2 2 2 2 2 2 1 1 1 1 1 1 10 0´ ´ c x c xM M $ $2 1
function (1) and equations (13) and (15), we obtain ( , ) and ( , ). Next,M x x M x x1 1 2 2 1 219
substituting the expressions for ( , ) and ( , ) into equations (14) and (16), and thenM x x M x x1 1 2 2 1 2
solving the resulting pair of simultaneous equations for and , we obtain the firms'x x1 2
equilibrium R&D investments, and . Then, substituting ( , ) into equation (13),x x M x x** **1 2 1 22
** **
and ( , ) into equation (15), we obtain ( ) and ( ), respectively.M x x q c q c1 1 21 2 1 2** ** ** **
Note that the firms' equilibrium expected profits and the equilibrium expected gross
profits of all the types of the firms are obtained by substituting the firms' equilibrium R&D
investments and the equilibrium output levels of all the types of the firms into functions (11) and
(10), respectively. Let denote the equilibrium expected consumer surplus, and let CS SW** **
denote the equilibrium expected social welfare. Now, using the above results, we report the
outcomes of the unobservable-investments model in Lemma 2.
Lemma 2. i The firms' equilibrium R&D investments are x x( ) ** **1 2œ œ
$ $ #k k k a k k c k D k k where D k{2( 1)( 2) (2 4 3) } {3( 1) ( 2)}, 2 ( 2) Î ´ 2 2 2 2M
$2k Let x x for i ii The equilibrium output level of type c of firm i for i. 1, 2. ( ) , 1,** **i i´ œ œ
2, ( ) (2 3 ) 6 ( ) 6( 1) (0, ]. ( ) is q c a c k c x k for c c x iii The** ** **i i i M i Mœ Î Î − $ $
equilibrium expected gross profits of type c of firm i for i are c q c fori i ii i**, 1, 2, ( ) { ( )} œ œ<** 2
c c x iv Firm i's equilibrium expected profits are G x x ci M i i** ** ** c x
i− œ(0, ]. ( ) ( , ) ( )$ 1 2 0** M
**$<
f c x dc x for i where f c x kc c x I c** ** ** k ** ki i ii i i i i M ic x( ; ) ( ) 2, 1, 2, ( ; ) [ ( ) ] ( ), œ Î# 2 1
(0, ]Î œ $M
**$
and G x x G x x holds v The equilibrium expected consumer surplus is1 21 2 1 2( , ) ( , ) . ( )** ** ** **œ
CS c x a k c x a k k k k** ** **M Mœ ( ) [32 ( 1) ( ){16 ( 1)( 3) (5 18 21)$ $2 2 2 2
( )}] 144( 1) . ( )c x k vi The equilibrium expected social welfare is SW CSM** ** ** œ$ Î 2
G x x G x x1 21 2 1 2( , ) ( , ).** ** ** **
19Using a technique similar to the one for solving for and in Appendix A, we obtainK KN N1 2
M x x a k c x x k i j i ji M i j( , ) 3 ( 2 ) 3( 1) for , 1, 2 with .1 2 œ Î Î œ Á$ $
21
We assume that , where 3 ( 1), which leads to ( ) 0.U c a U k k c x_ _ Î œ Î M M**$ #2 $
We assume also that , which, together with , leads to ( ) 0 for allc a U U c a q c_
_M M i**iÎ Î
c c x U k k k k k_
i M**− œ(0, ], where {2 ( 1)( 2) } ( 2)(2 3) 1. Note that$ # $ # Î 2
G x xi** **( , ) in p positive under the given assumptions on the parameters.1 2 art ( ) isiv
The firms are symmetric at the start of the game. Accordingly, as stated in Lemma 2, the
firms' equilibrium R&D investments are the same, and so are their equilibrium expected profits;
they have the same "type-output function," ( ), and the same "type-profits function," ( ),q c c**i ii i<**
in equilibrium. Actually, however, the firms may be asymmetric after their marginal costs are
realized. If so, then their output levels differ, and so do their profits.
5. Comparison of the two main models
In this section, we compare the outcomes of the observable-investments model with those
of the unobservable-investments model. Using Lemmas 1 and 2, we obtain Proposition 1.20
Proposition 1. ( ) i Firm i's equilibrium R&D investment is greater in the observable-investments
model than in the unobservable-investments model: x x ii The equilibrium output level of* ** . ( )
type c of firm i for i is less in the observable-investments model than in thei , 1, 2, œ
unobservable-investments model: q c q c for c c x iii The equilibriumi ii i i M** **( ) ( ) (0, ]. ( ) − $
expected gross profits of type c of firm i for i are less in the observable-investmentsi , 1, 2, œ
model than in the unobservable-investments model: c c for c c x iv1 <* **i ii i i M
*( ) ( ) (0, ]. ( ) − $
Firm i's equilibrium expected profits are less in the observable-investments model than in the
unobservable-investments model: H x x G x x for i v Firm i's equilibriumi i* * ** **( , ) ( , ) 1, 2. ( ) 1 2 1 2 œ
expected output level is greater in the observable-investments model than in the unobservable-
investments model: E q C E q C for i vi The equilibrium expected consumer[ ( )] [ ( )] 1, 2. ( )i ii i* ** œ
surplus is less in the observable-investments model than in the unobservable-investments model:
20We have assumed in Section 3 that min{ , 1}, and in Section 4 thatS c a S__
Î M
U c a U U S U S S c a U_ _ _ __ _ __
Î Î M M. Because , we assume in this section that .
22
CS CS vii The equilibrium expected social welfare is less in the observable-investments* ** . ( )
model than in the unobservable-investments model: SW SW* ** .
The proof of Proposition 1 is relegated to Appendix B. Figure 2 illustrates the
equilibrium probability density functions of firm 's marginal cost in the two main models,i Ci
which are drawn with, for example, 5. Note that, in parts ( ) and ( ), firm 's type isk ii iii i cœ i
limited to the interval (0, ] because ( ) and ( ) are defined only in that interval.c x q c cM i i*
i i $ * *1
says that, in equilibrium, each firm makes a greater cost-reducing R&DPart ( )i
investment in the observable-investments model than in the unobservable-investments model.
What accounts for the difference between the equilibrium R&D investments of the two models?
We respond to this question as follows. In the observable-investments model unlike in the
unobservable-investments model each firm can make a strategic commitment for the second-
stage output competition by choosing and announcing, in the first stage, its R&D investment.
Naturally, to achieve its competitive advantage in the second-stage output competition, each firm
has an incentive to make itself "strong" or "aggressive," and actually does so by choosing and
announcing a "large" R&D investment. Such strategic behavior of the firms, which is absent
from the unobservable-investments model, leads to the finding that .x x* **
Part ( ) says thatii , in equilibrium, of each firm chooses a smaller output level ineach type
the observable-investments model than in the unobservable-investments model. (Note that here
we do not compare each firm's equilibrium output levels in the two models.) This canexpected
be explained as follows. In the observable-investments model, after observing firm 'sj
equilibrium R&D investment , type of firm chooses its output level to maximize itsx c i*j i
expected gross profits, taking the output levels of all types of firm as given. Likewise, takingj
firm 's equilibrium R&D investment as given, the same type of firm does the same thing inj x i**j
the unobservable-investments model. However, as shown in Figure 2, the equilibrium
probability density functions of firm 's type differ in the two models: ( ; ) over thej C f c xj j j*
j
interval (0, ] in the observable-investments model, and ( ; ) over the interval (0,c x f c xM j jj j* ** $
23
c xM j** $ ] in the unobservable-investments model. Comparing these two functions, we see that
type of firm faces strong types of firm low values of with higher probabilities in thec i j ci j
observable-investments model than in the unobservable-investments model. This leads to the
finding that ( ) ( ).q c q ci ii i* ** 21
Part ( )iii says that, in equilibrium, each type of each firm earns less expected gross profits
in the observable-investments model than in the unobservable-investments model. As in part
( ), we may explain this using the fact that the equilibrium probability density functions of theii
rival firm's marginal cost differ in the two models. Type of firm faces strong types of firmc ii
j c low values of with higher probabilities in the observable-investments model than in thej
unobservable-investments model, so that we obtain that ( ) ( ).1 <* **i ii ic c
Part ( )iv may come from the following two facts. Firstly, in equilibrium, the cost of each
firm's R&D investment is greater in the observable-investments model than in the unobservable-
investments model: ( ) 2 ( ) 2. Secondly, as stated in p the equilibrium# #x x* **2 2Î Î art ( ),iii
expected gross profits of each type of each firm is less in the observable-investments model than
in the unobservable-investments model.
Part ( )v comes from the fact that each firm has low marginal costs with higher
probabilities due to its greater cost-reducing R&D investment in the observable-investments
model than in the unobservable-investments model, as shown in Figure 2.
Part ( )vi can be explained as follows. There are two countervailing facts. On the one
hand, it follows from part ( ) that, given their realized marginal costs, the firms choose smallerii
output levels, and thus the consumer surplus is less, in the observable-investments model than in
the unobservable-investments model. On the other hand, as Figure 2 illustrates, "low" values of
c ii, for 1, 2 which result in "high" output levels and thus high consumer surpluses areœ
realized with higher probabilities in the observable-investments model than in the unobservable-
21Note that, in a standard quantity-setting duopoly with homogeneous products, a firm with ahigher marginal cost chooses a smaller output level in equilibrium than the rival firm.
24
investments model. Understandably, because the first fact dominates the second one, we obtain
part ( ).vi
We can explain part ( ) with those explanations above for parts ( ) because vii vi( ) and theiv
equilibrium expected social welfare is defined as the sum of the firms' equilibrium expected
profits and the equilibrium expected consumer surplus.
Interestingly, Proposition 1 establishes that the firms' equilibrium expected profits, the
equilibrium expected consumer surplus, and the equilibrium expected social welfare are less in
the observable-investments model than in the unobservable-investments model. With this result,
we are tempted to say that firms may have an incentive not to disclose information regarding
their R&D investments. Also, based on we may well argue that it is the result, harmful to firms
and consumers thus, it is not to socially desirable enact or establish policies or regulations
or institutions which require firms to disclose information regarding their R&D investments.
6. Comparative statics
6.1. Observable R&D investments
We perform comparative statics of the outcomes except and of the CS SW* *
observable-investments model with respect to each of the parameters, respectively. Using22
Lemma 1, we obtain Proposition 2, which is also summarized in Table 1.23
Proposition 2. ( ) , , , ( ), ( ), a As the size a of the market increases the outcomes x q c c and* *i ii i1*
H x x all increase b As the initial marginal cost c of production decreases thei M* *( , ), . ( ) , 1 2
investment x and H x x increase while the outcomes q c and c decrease c As the ( , ) , ( ) ( ), . ( ) * * * *i i ii i1 2
*1
22We cannot determine the sign of the comparative statics of or with respect to each ofCS SW * *
the parameters, without assuming specific values for the other parameters.23Because the proofs of the comparative statics results presented in Propositions 2 and 3 involvevery long mathematical derivations, we do not provide them for concise exposition. But they areavailable from the authors upon request.
25
parameter k increases from unity the investment x increases while the outcomes q c and, , ( )* *i i
1*i i
*( ), . ( ) , c first decrease and then increase d As the parameter increases the investment x$
increases while q c c and H x x decrease e As the parameter increases the* * *i ii i i( ), ( ), ( , ) . ( ) , 1*
1 2 #
investment x decreases while q c c and H x x increase ( ), ( ), ( , ) .* * * *i ii i i1*
1 2
Part ( ) is stated in more detail as follows. As the parameter increases, ( ) and ( )c k q c c*i ii i1*
decrease if , and increase if min{ , 1}, where 8 ( 1)S c a S S c a S S k k__
Î Î ´ M M‰ ‰ ‰ $2
{2 ( 2) } {18 ( 1) ( 2) (2 1)(4 5)}.# $ # $ #k k k k k k k Î 2 2 2 2 2 2 2
Part ( ) makes intuitive sense. An increase in the parameter means that the quantitya a
demanded increases at each price or that consumers are willing to pay more for the products of
the firms. Thus it enables the firms to capture more profits, and increases the returns from their
R&D investments, which leads to the firms increasing their R&D investments.
Part ( ) can be explained as follows. A decrease in the parameter , ,b c ceteris paribusM
tends to decrease the upper limit of the support of ( ; ) and to increase ( ; ) at all values off c x f c xj j j j j j
c c ij i in the resulting support. This implies that type of firm now faces strong types of firm
j c low values of with higher probabilities in the second stage. Knowing this, what does, orj
should, each firm do in the first stage? To achieve its competitive advantage in the second-stage
output competition, each firm makes itself stronger by choosing and announcing a larger cost-
reducing R&D investment.
Next, because a decrease in , together with the resulting increase in , decreases thec xM*
upper limit of the support of ( ; ) and increases ( ; ) at all values of in the resultingf c x f c x c* *j jj j j j j
support, type of firm now faces strong types of firm with higher probabilities in the secondc i j i
stage, and thus it chooses a smaller output level and earns less second-stage expected gross
profits than before (see footnote 21).
As decreases, the equilibrium expected gross profits for firm in the firstc iM
stage increase because the upper limit of the support of ( ; ) decreases and ( ; ) f c x f c x* *i ii i i i
26
increases at all values of in the resulting support. On the other hand, as decreases, thec ci M24
equilibrium R&D investment increases, and thus the cost ( ) 2 of each firm's R&Dx x* *# 2Î
investment increases. Now that the first effect dominates the second one, ( , ) increases.H x xi* *1 2
Part ( ) says that, as the parameter increases, the equilibrium R&D investment of eachc k
firm increases. This can be explained as follows. Recall from Figure 1 that, as the parameter k
increases, ( ; ) increases at high values of in the support while it decreases at low values off c x ci i i i
c k ceteris paribusi. This implies that, as the parameter increases, , the probability that each firm
has a "high" realized marginal cost in the second stage increases. Hence, given an increase in
the parameter , each firm has an incentive to decrease the upper limit of the support, andk
actually does so by increasing its cost-reducing R&D investment.
We may explain part ( ) in the same way as we explain part ( ) because an increase in ,d b $
as a decrease in does, tends to decrease the upper limit of the support of ( ; ) and toc f c xM j j j
increase ( ; ) at all values of in the resulting support.f c x cj j j j
Part ( ) says that, as the parameter increases, ( ) and ( ) increase. This can bee q c c# 1*i ii i
*
explained as follows. An increase in leads to a decrease in , which in turn increases the# x*
upper limit of the support of ( ; ) and decreases ( ; ) at all values of in the resultingf c x f c x c* *j jj j j j j
support. Thus, facing weak types of firm high values of with higher probabilities, typej c j
c ii of firm now chooses a greater output level and earns more second-stage expected gross
profits in equilibrium than before.
Because of computational complexity involved, we cannot algebraically determine the
sign of the comparative statics of ( , ) with respect to the parameter . However, on theH x x ki* *1 2
basis of our extensive numerical investigation, we conclude that ( , ) decreases as theH x xi* *1 2
parameter increases from unity. We do not report this result in Proposition 2, but do report itk
in Table 1, using a south east arrow.
24Recall from Lemma 1 that ( ) increases as decreases.1*i i ic c
27
6.2. Unobservable R&D investments
We perform comparative statics of the outcomes except and of the CS SW** **
unobservable-investments model with respect to each of the parameters, respectively. Using
Lemma 2, we obtain Proposition 3, which is also summarized in Table 1.
Proposition 3. ( ) , , , ( ), ( ), a As the size a of the market increases the outcomes x q c c and** **i ii i<**
G x x all increase b As the initial marginal cost c of production decreases thei M** **( , ), . ( ) , 1 2
investment x and G x x increase while the outcomes q c and c decrease. c As ( , ) , ( ) ( ), ( ) ** ** ** **i i ii i1 2
**<
the parameter k increases from unity the investment x increases while the outcomes q c, , ( )** **i i
and c first decrease and then increase d As the parameter increases the investment<**i i( ), . ( ) , $
x increases while the outcomes q c and c decrease e As the parameter ** **i ii i , ( ) ( ), . ( ) <** #
increases the investment x decreases while the outcomes q c and c increase, , ( ) ( ), .** **i ii i<**
Part ( ) is stated in more detail as follows. As the parameter increases, ( ) andc k q c**i i
< $** o o o 2i i M M( ) decrease if , and increase if , where 2 ( 1)c U c a U U c a U U k k_
_ Î Î ´
{2 ( 2) } {6 ( 1) ( 2) (2 2 1)}.# $ # $ #k k k k k k k Î 2 2 2 2 2 2 2 2
The explanations for the comparative statics results in Proposition 3 may be made
similarly to those for the comparative statics results in Proposition 2, and therefore are omitted.
It is algebraically intractable to determine the sign of the comparative statics of ( ,G xi**1
x k**2 ) with respect to the parameter . However, on the basis of our extensive numerical
investigation, we conclude that ( , ) decreases as the parameter increases from unity.G x x ki** **1 2
We do not report this result in Proposition 3, but do report it in Table 1, using a south east arrow.
Finally, we find that the sign of the comparative statics of ( , ) with respect to theG x xi** **1 2
parameter or is ambiguous. We leave blank the corresponding entries in Table 1.$ #
28
7. Discussion: variations of the main models
In this section, we consider variations of the main models presented in Sections 3 and 4.
As the first variations, we consider models in which there is no ex post private information
regarding R&D outcomes. The second variations are models in which there is no ex ante
uncertainty regarding R&D outcomes. The third variations are price-setting duopoly models
with product differentiation.
7.1. Models without ex post private information regarding R&D outcomes
Consider situations in which firms are required to disclose or voluntarily disclose true
information regarding their R&D outcomes. Or consider situations in which firms "quickly"
obtain true information regarding other firms' R&D outcomes, for example, through "signals"
created due to their market interactions. Such situations may be studied with models without ex
post private information regarding R&D outcomes.
We formally consider a quantity-setting duopoly with R&D which is the same as that in
Section 2 with the exception that each firm now knows the realization of the rival firm's new
marginal cost when choosing its output level.
In this case, the model with observable R&D investments and that with unobservable
R&D investments are analytically equivalent. Indeed, each firm's output level decision, and thus
its R&D investment decision, does not depend on whether or not the firm knows the R&D
investment of the rival firm, because the firm knows the rival firm's new marginal cost when
choosing its output level.
On the basis of this result, one may conclude that our main models are useful for
situations in which it takes time for firms to obtain information regarding their rival firm's R&D
outcome.
29
7.2. Models without ex ante uncertainty regarding R&D outcomes
We consider a quantity-setting duopoly with R&D which is the same as that in Section 2
with the exception that each firm is now certain about its R&D outcome when making its R&D
investment that is, each firm knows for sure how much marginal cost it can reduce by
undertaking the R&D investment. Let , for 1, 2, represent firm 's new marginal cost whichc i ii œ
is deterministically determined before the firm chooses its output level. We now assume that
c c xi M iœ $ $, where 0 1.
In this case, the model with observable R&D investments has also no ex post private
information regarding R&D outcomes, while the model with unobservable R&D investments has
ex post private information regarding R&D outcomes.
In the model with observable R&D investments, we obtain for 1, 2: i xœ *i œ
4 ( ) (9 4 ), 3 ( ) (9 4 ), ( , ) (9 8 )( )$ # $ # # $ # # $a c q a c H x x a c Î œ Î œ ÎM M i M*i
2 2 * * 2 21 2
(9 4 ) , 18 ( ) (9 4 ) , and 4 ( ) (9 4 ).# $ # # $ # # $ œ Î œ Î 2 2 * 2 2 2 2 * 2 2CS a c SW a cM M
In the model with unobservable R&D investments, we obtain for 1, 2: i xœ **i œ
$ # # # # #( ) (3 ), ( ) (3 ), ( , ) (2 )( )a c q a c G x x a c Î Î ÎM M i M** ** **i œ œ $ $ $2 2 2 2
1 2
2(3 ) , 2 ( ) (3 ) , and (4 )( ) (3 ) .# # # # # # $ $ $ $2 2 ** 2 2 2 2 ** 2 2 2 2CS a c SW a cœ Î œ ÎM M
Comparing the outcomes of the two models, we obtain for 1, 2: , ,i x x q qœ i i i i* ** * **
H x x G x x CS CS SW SWi i* * ** **( , ) ( , ), , and . As expected, the comparison results1 2 1 2
* ** * **
that and ( , ) ( , ), are the same as those in Proposition 1. Interestingly,x x H x x G x xi i i i* * ** *** **1 2 1 2
however, the comparison results that and , are opposite to those inCS CS SW SW* ** * **
Proposition 1.
7.3. Price-setting duopoly models with product differentiation
Consider a price-setting duopoly with differentiated products in which two firms, 1 and
2, first make R&D investments to reduce their production costs, and then produce and sell their
products at the resulting reduced costs. When making its R&D investment, each firm is certain
30
about its R&D outcome, even though the R&D outcome is determined later. Each firm cannot
directly observe the rival firm's R&D outcome.
In this section, we abuse notation by using the same notation as in Sections 2 to 4. Let ,xi
for 1, 2, represent firm 's R&D investment. As in Section 2, the cost of firm 's R&Di i iœ
investment is given by 2. The cost function of firm is given by ( ) .#x i T q c q2i i i i iÎ œ
The market demand function facing firm , for 1, 2, is given byi i œ
q a p p i ji i jœ ) for ,Á
where represents the quantity of firm 's product demanded, represents firm 's price, q i p i pi i j
represents firm 's price, and and are positive constants. We assume that 0 1.j a ) )
The firms initially have the same marginal cost of production, wherecM
0 (1 ). When making its R&D investment, each firm knows for sure how much Î c aM )
marginal cost it can reduce by undertaking the R&D investment. Let , for 1, 2, representc ii œ
firm 's new marginal cost which is deterministically determined before the firm chooses itsi
price. We assume that , where 0 1.c c xi M iœ $ $
We formally consider the following two games. First, the firms make their cost-reducing
R&D investments simultaneously and independently. In the model with observable R&D
investments, each firm observes the rival firm's R&D investment; in the model with
unobservable R&D investments, each firm cannot observe it. Next, the firms choose their prices
simultaneously and independently.25
In the model with observable R&D investments, we obtain for 1, 2: 2 (2 )i xœ *i œ $ )2
{ (1 ) } { (2 )(2 ) 2 (1 )(2 )}, { (4 ) 2 (2 )a c p a a Î œ ) # ) ) $ ) ) # ) $ )M*i
25In the model with observable R&D investments, each firm knows the rival firm's new marginalcost when choosing its price. However, in the model with unobservable R&D investments, eachfirm does not know the rival firm's new marginal cost when choosing its price.
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