Strategic R and D investments with uncertainty
Toshihiro Matsumura
Institute of Social Science, University of Tokyo

Abstract
I introduce uncertainty into the model of strategic cost−reducing R and D investments andreexamine welfare implications. I discuss two models. In one model an increase in
expenditure decreases production costs when R\Dsucceeds, and in the other model it
increases probability of success. I show that two models yield completely different
implications for tax−subsidy policies on R and D investments. In the former model
equilibrium investment level is always too low from the viewpoint of social welfare, while in
the latter model it can be either too low or too high and relatively risky (safe) investments
should be subsidized (taxed).

I am grateful to Takanori Ida, Kazuhiko Kato, Noriaki Matsushima, Tatsuhiko Nariu, Tadashi Sekiguchi,Daisuke Shimizu,
Yoshiaki Tojyo and participants of the seminars at University of Tokyo and Kyoto University for their helpful comments andsuggestions. I am also indebted to an anonymous referee and an associate editor, Professor John Wooders, for their valuable andconstructive suggestions. Needless to say, I am responsible for any remaining errors. I gratefully acknowledge financial supports
from Grant−in−Aid of the Japanese Ministry of Education, Science and Culture.
Citation: Matsumura, Toshihiro, (2003) "Strategic R and D investments with uncertainty." Economics Bulletin, Vol. 12, No. 1
pp. 1−7
Submitted: April 1, 2003. Accepted: June 13, 2003.
URL: http://www.economicsbulletin.com/2003/volume12/EB−03L10004A.pdf

1

Introduction

I reexamine welfare eﬀect of strategic cost-reducing R&D investments in oligopoly markets by introducing uncertainty of success. In contrast to a huge body of literature on
R&D competition models of patent race, quite a smaller body of literature on strategic
cost-reducing R&D deals with uncertainty. The literature with strategic R&D competition is fairly large.1 Most studies assume that R&D succeeds with probability one.
Without any doubt, most R&D investments fail with a positive probability. Thus, investigating the eﬀect of uncertainty is important.
In this paper I introduce into the model of Brander and Spencer (1983) and Lahiri andOno (1999) uncertainty, where R&D investments fail with a positive probability, and discuss whether or not the equilibrium investment level exceeds the eﬃcient one. I consider
two models. One is the model where an increase in investments decreases production
cost when R&D succeeds, and another is the model where an increase in investments
increases the probability of success. I ﬁnd that two models yield quite diﬀerent welfare
implications. In the former model the equilibrium investments are insuﬃcient from theviewpoint of social welfare. In the latter model the equilibrium investments are insuﬃcient (excessive) if the equilibrium probability of success is low (high). In other words,
the private incentive of R&D investments for relatively risky (safe) projects is insuﬃcient(excessive). These results indicate that whether or not subsidies for R&D investments
improve welfare depends on whether ﬁrms spend money on increasing innovation size or
the probability of success, and on the property of the project (risky or safe).
I use a standard strategic commitment games. In the ﬁrst stage ﬁrms engage in costreducing R&D investments, and in the second stage they compete à la Cournot. A lot has
been written on such two-stage models. Recently Lahiri and Ono (1999) point out that,very few studies analyze a question of R&D subsidies, while they are widely introduced
in many countries. They also emphasize the importance of ex ante asymmetries between
ﬁrms. In this paper I consider ex ante symmetric games. Introducing uncertainty, asymmetries between two ﬁrms appear in the second stage with a positive probability. I showthat this ex post asymmetries are quite important for analyzing optimal R&D subsidies.
The remainder of this paper is organized as follows. Section 2 presents the model.
Section 3 investigates the equilibrium investment level. Section 4 discusses the welfare
implication by comparing the equilibrium investment level with the socially eﬃcient one.
1

See, among others, Brander and Spencer (1983), Spence (1984), d’Aspremont and Jacquemin (1988),
Kamien et al. (1992), Suzumura (1992), Okuno-Fujiwara and Suzumura (1993), Matsumura (1995), andLahiri and Ono (1999). See also Martin (2000), which is a unique work discussing strategic cost-reducing
investments in the context of a patent race.

1Section 5 concludes the paper.

2

The Model

I formulate two-stage duopoly games. Two ﬁrms are symmetric before the game. In theﬁrst stage each ﬁrm engages in cost-reducing R&D activities. Firm i (i = 1, 2) succeeds
in reducing its cost with probability qi , independently of whether or not ﬁrm j succeeds(j = 1, 2, j = i). Let ci denote the (constant) marginal production costs. If ﬁrm i
succeeds in cost-reducing, its marginal production cost becomes ci = c − ∆i . Otherwise,
its marginal production cost is ci = c.
I consider two games. One is the game where each ﬁrm i chooses ∆i given q1 = q2 = q.
In this game an increase in R&D expenditure decreases the production costs when it
succeeds. The other is the game where each ﬁrm i chooses qi given ∆1 = ∆2 = ∆(> 0).
In the second game, an increase in R&D expenditure increases the probability of thesuccess of R&D. Let I(∆i , qi ) denote the investment costs. I assume that I(∆i , qi ) is
twice continuously diﬀerentiable, increasing in qi and ∆i , and suﬃciently convex.2 I also
assume that
∂I
∂I
∂I
∂I
= 0, lim= 0, lim= ∞, lim=∞
lim
qi →0 ∂qi
qi →1 ∂qi
∆i →0 ∂∆i∆i →c ∂∆iso as to ensure the interior solution.
At the beginning of the second stage each ﬁrm observes its rival’s cost. Then the
duopolists produce perfectly substitutable commodities for which the market demand
function is given by p = a−Y (price as a function of quantity), where Y is the total outputof duopolists. Let yi denote the output of ﬁrm i. Each ﬁrm i chooses yi independently.

3

Equilibrium

I use subgame perfect Nash equilibrium as equilibrium concept. Let superscript ‘E’denote the equilibrium outcome. In the Cournot game at the second stage, given c1 andc2 , the equilibrium output yiE and the proﬁt πiE of ﬁrm i are given by(a − 2ci + cj )yiE =,(a − 2ci + cj )
πiE =

2

− I(∆i , qi ) (i, j = 1, 2, i = j).
3
9In the ﬁrst stage the expected proﬁt of ﬁrm i is given by
E[πi ] =2

1[qi qj (a − c + 2∆i − ∆j )2 + qi (1 − qj )(a − c + 2∆i )2
9

For the suﬃcient convexity of investment cost functions, see footnotes (5) and (9).

2(1)

+ (1 − qi )qj (a − c − ∆j )2 + (1 − qi )(1 − qj )(a − c)2 ] − I(∆i , qi ).

(2)First, consider the game where each ﬁrm chooses ∆i , given q1 = q2 = q. The ﬁrst
order condition is given by
∂I
4q
= 0.
(−q∆j + 2∆i + a − c) −
9
∂∆i

(3)

Henceforth, I restrict attentions to the symmetric equilibrium.3 Substituting ∆1 = ∆2 =∆ into (3) yields∂I4q(−∆q + 2∆ + a − c) =
.
(4)
9
∂∆
Let ∆E (q) denote the equilibrium innovation size. It is derived from (4).
Next, consider the game where each ﬁrm chooses qi , given ∆1 = ∆2 = ∆. The ﬁrst
order condition is given by
4∆∂I(−qj ∆ + ∆ + a − c) −
= 0.
9
∂qi

(5)

Substituting q1 = q2 = q into (5) yields∂I4∆(−q∆ + ∆ + a − c) =
.
9
∂q

(6)

Let q E (∆) denote the equilibrium q. It is derived from (6).44

Results

In this section I discuss the following second best problem so as to compare the equilibrium outcomes discussed in the previous section to the eﬃcient one. The welfaremaximizing social planner chooses the action of each ﬁrm in the ﬁrst stage, given the
Cournot competition in the next stage.
3

Since the reaction curve of the ﬁrst stage game has a negative slope (i.e., strategic substitutes) and iscontinuous, I can show that the symmetric equilibrium is unique. However, it is possible that asymmetricequilibria also exist. A suﬃcient condition for the uniqueness of equilibrium is ∂ 2 I/∂∆2 > 4q(2 + q)/9,
which is a more restrictive condition than the second order condition. For the discussion of asymmetricequilibria, see Amir and Wooders (1998).
4

Since the reaction curve of the ﬁrst stage game has a negative slope (i.e., strategic substitutes) andis continuous, the symmetric equilibrium is unique. However, it is possible that asymmetric equilibria
also exist. A suﬃcient condition for the uniqueness of equilibrium is ∂ 2 I/∂q 2 > 4∆2 /9, which is a more
restrictive condition than the second order condition. See also footnote (3).

3

First, I consider the ﬁrst game where innovation size of each ﬁrm is determined.
Suppose that the social planner chooses ∆1 = ∆2 = ∆, given q1 = q2 = q.5 Then eachﬁrm faces Cournot competition discussed in the previous sections. Consumer surplus isgiven by1
(2a − c1 − c2 )2,(7)
CS = Y 2 =218
and social welfare (total surplus) is given by
W = CS + π1 + π2 .

(8)The expected welfare is given by
1 2[4q (a − c + ∆)2 + 4(1 − q)2 (a − c)29+ q(1 − q)(8a2 + 8c2 + 11∆2 − 16ac + 8a∆ − 8c∆)] − 2I(∆, q).

E[W ] =

(9)

The ﬁrst order condition isq∂I(−7q∆ + 11∆ + 4a − 4c) =
.
9
∂∆

(10)

Let ∆∗ (q) denote this eﬃcient innovation size ∆. It is derived from (10).
Proposition 1: ∆∗ (q) ≥ ∆E (q) ∀q ∈ [0, 1] and the equality is satisfied if and only if
q = 0 or q = 1.
Proof: The left-hand side of (10) – the left-hand side of (4) = ∆(1 − q)q/3 ≥ 0, andthe equality is satisﬁed if and only if q = 0 or q = 1. Note that 0 ≤ q ≤ 1. From theconvexity of I(∆), I obtain Proposition 1. Q.E.D.
The result that ∆∗ (1) = ∆E (1) has already been shown by Brander and Spencer (1983).6
Proposition 1 implies that introducing uncertainty induces the deviation of the equilibrium innovation size from the eﬃcient one. I now discuss the intuition behind this result.
5This assumption is not innocuous. Suppose that cost is given exogenously. Then the total proﬁts
of two ﬁrms is jointly convex in costs. Thus it is possible that the social planner prefers the asymmetricoutcome to the symmetric one. A suﬃcient condition under which the symmetric outcome is eﬃcientis ∂ 2 I/∂∆2 > q(11 + 7q)/9, which is a more strict condition than what appears in footnote (3). Thisis a reason why we assume that ∂ 2 I/∂∆2 is suﬃciently large. For the related works pointing out thisconvexity in costs and possible eﬃcient asymmetric equilibria, see Amir and Wooders (2000) and Salantand Shaﬀer (1998).
6

They also show that ∆∗ (1) > (<)∆E (1) if p > (<)0. I can show that uncertainty decreases ∆E
more signiﬁcantly than ∆∗ , so ∆∗ can be larger than ∆E even if p < 0, and ∆∗ is always larger than∆E if p > 0.

4

An increase in ∆ increases ﬁrm 1’s proﬁt and aﬀects welfare when ﬁrm 1 is successful inthe project. It induces the production substitution from ﬁrm 2 to ﬁrm 1. With probability 1 − q, the cost of ﬁrm 2 is c, which is higher than c − ∆i . Thus, the above production
substitution economizes the total production costs because it reduces the production of
the less eﬃcient ﬁrm and increases that of the more eﬃcient ﬁrm. Firm i chooses ∆iwithout considering this welfare-improving production substitution eﬀect, so the incentive for increasing the innovation size becomes insuﬃcient. This eﬀect disappears only if
1 − q = 0 (i.e., ﬁrm 2 is always as eﬃcient as ﬁrm 1).7
This result is closely related to that of Lahiri and Ono (1988). They show that a
reduction of the output of the ﬁrm with the higher marginal cost improves welfare.8 In
my model an increase in ∆ increases this welfare-improving production substitution when
ex post asymmetries appear, so it should be promoted by subsidies. My result shows thatthe principle of Lahiri and Ono is important even when ﬁrms are symmetric ex ante.
Next, I consider the second game where the probability of success is determined.
Suppose that the social planner chooses q1 = q2 = q, given ∆1 = ∆2 = ∆.9 Then eachﬁrm faces Cournot competition discussed in the previous sections. The social plannermaximizes (9) with respect to q. The ﬁrst order condition is∂I∆(11∆ + 8a − 8c − 14∆q) =
.
18
∂q

(11)

Let q ∗ (∆) denote this eﬃcient probability of the success. It is derived from (11).
Proposition 2: q ∗ (∆) ≤ q E (∆) if and only if q E (∆) ≥ 1/2.
Proof: The left-hand side of (11) – the left-hand side of (6) = ∆(1 − 2q)/6. Since ∆ > 0,
it is non-positive if and only if q ≥ 1/2. From the convexity of I(q), I obtain Proposition
7

The essential point in this paper is ex post asymmetries rather than uncertainty. Suppose that I
drop the assumption of no-correlation and consider the following model: The probability that both ﬁrms
succeed or fail is (1 + r)/4, and the probability that only ﬁrm 1 succeeds or fails is (1 − r)/4, where r isthe correlation coeﬃcient. Then I can show that ∆∗ ≥ ∆E and that the equality is satisﬁed if and only if
r = 1. Note that, if the projects of two ﬁrms are perfectly correlated, the welfare-improving production
substitution does not occur in the symmetric equilibrium. For the discussion of correlation in the context
of patent race, see Cardon and Sasaki (1998).
8

For other discussions of welfare-enhancing production substitution eﬀects, see also Brander (1981),
Ono (1990), Riordan (1998), Matsumura (1998, 2003), Lahiri and Ono (1998), Ushio (2000), and Matsushima and Matsumura (2003).
9

A suﬃcient condition under which the symmetric outcome is eﬃcient is ∂ 2 I/∂q 2 > 7∆2 /9, which isa more strict condition than what appears in footnote (4). This is a reason why we assume that ∂ 2 I/∂q 2
is suﬃciently large.

5

2. Q.E.D.
Proposition 2 states that the equilibrium R&D expenditure is too small from the viewpoint of social welfare if the equilibrium probability of success is less than 1/2. In otherwords, each ﬁrm has an insuﬃcient incentive for increasing the probability of success if
the project is relatively hard to result in a success.
I explain the intuition behind the result. Consider the following situation. The initialprobability of success is 0.1 and it increases up to 0.2 by additional investments. Supposethat ﬁrm 2 has already failed in the project and its cost is c. Suppose that the cost ofﬁrm 1 becomes c − ∆ from c. The reduction of the cost increases the proﬁt of ﬁrm 1,
and it also induces the production substitution from ﬁrm 2 to ﬁrm 1. Since ﬁrm 1 ismore eﬃcient than ﬁrm 2, this production substitution improve welfare. Since ﬁrm 1
does not fully care about this welfare-improving production substitution eﬀect, ﬁrm 1’s
incentive for investments becomes insuﬃcient. Suppose that ﬁrm 2 has already succeededin the project and its cost is c − ∆. Suppose that the cost of ﬁrm 1 becomes c − ∆ fromc. The reduction of the cost induces the production substitution from ﬁrm 2 to ﬁrm
1. Since ﬁrm 2 is eﬃcient whether or not ﬁrm 1’s project is successful, this production
substitution from ﬁrm 2 to ﬁrm 1 is not desirable. Thus, in contrast to the case whereﬁrm 2 fails, ﬁrm 1’s incentive for investments becomes excessive when ﬁrm 2 succeeds.
In short, given that ﬁrm 2 fails (succeeds), ﬁrm 1’s incentive for investments becomes
insuﬃcient (excessive). Therefore, if equilibrium q2 is low (high), ﬁrm 1’s incentive forinvestments becomes insuﬃcient (excessive).
This result makes a sharp contrast with Proposition 1. For example, as is shown in
the proof of Proposition 1, in the ﬁrst game the diﬀerence between marginal social andmarginal private beneﬁts of R&D expenditure is proportional to q(1 −q). It is maximized
when q = 0.5. On the other hand, Proposition 2 states that, in the second game social
beneﬁts of R&D expenditure is equal to private one when q = 0.5, so neither taxes nor
subsidies are required in this case. These results imply that the optimal tax-subsidy for
R&D investments crucially depends on what kind of investments ﬁrms practice. Anotherexample is the welfare eﬀect of R&D subsidies for relatively safe project. For example,
if q = 0.6, subsidies on R&D expenditure improve welfare if they expand the innovation
size, while they are harmful if they raise the probability of success. The desirable policy
crucially depends on the purpose of R&D investment.

6

5Concluding Remarks

In this paper I investigate the optimal tax-subsidy policies on R&D investments. I introduce uncertainty into the standard strategic cost-reducing R&D investments. I discuss
two duopoly models. In model 1 (2) an increase in expenditure decreases production
costs when R&D succeeds (probability of failure).
I show that two models yield completely diﬀerent implications for tax-subsidy policies
on R&D investments. In model 1 the equilibrium investment level is always too low fromthe viewpoint of social welfare. On the other hand, in model 2 it can be either too low
or too high. I also ﬁnd that risky (safe) investments should be highly subsidized (taxed),
which is never derived in model 1. These results indicate that the optimal tax-subsidy
for R&D investments depends on what kind of investment the ﬁrms practice. In many
countries both direct and indirect subsidies for R&D investments are widely adopted.
This paper indicates that the government should carefully choose what kind of R&D
should be prompted.
In this paper spillover eﬀects of R&D and the associated free-rider problem are ignored. In order to derive richer policy implications, considering spillover eﬀects areindispensable. Without them, the policy implication becomes limited. This important
extension remains for future research.

7

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