Though government intervention is prevalent in the market for research and development (R&D), most literature has focused on the use of subsidies, patents or joint research ventures to obtain the efficient R&D investment. By using a two-stage duopoly model in which firms first choose the level of investment and then output, our paper shows that the introduction of a price ceiling by the regulator will result in the optimal level of R&D. This interesting but counterintuitive result contrasts with the existing literature and advances our understanding about price ceilings.Research and development; Subsidy; Price ceiling

Cristina M. Reiser

Youping Li

Zhou Yang

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 2 
  
Inducing R&D investment with price ceilings 
  
 
 
Youping Li  
Department of Economics, University of Tennessee 
Cristina M. Reiser  
Department of Economics, University of Tennessee 
Zhou Yang  
Department of Economics, University of Tennessee
Abstract 
Though government intervention is prevalent in the market for research and development (R&D), most literature has 
focused on the use of subsidies, patents or joint research ventures to obtain the efficient R&D investment. By using a 
two-stage duopoly model in which firms first choose the level of investment and then output, our paper shows that the 
introduction of a price ceiling by the regulator will result in the optimal level of R&D. This interesting but 
counterintuitive result contrasts with the existing literature and advances our understanding about price ceilings.
We are very grateful to Bill Neilson for the support and comments throughout our development of this paper. 
Citation: Youping Li and Cristina M. Reiser and Zhou Yang, (2010) ''Inducing R&D investment with price ceilings'', Economics Bulletin, 
Vol. 30 no.2 pp. 1548-1553. 
Submitted: Apr 12 2010.   Published: May 26, 2010. 
 
     1. Introduction
It is well known that the public good nature of research and development (R&D) results in sub-
optimal levels of R&D chosen by proﬁt-maximizing ﬁrms. Because a single ﬁrm cannot appropri-
ate rents from the positive spillovers created from their own R&D investment, it chooses a level
that is below what a social planner would choose. In the existing literature, the role of the social
planner to address this market failure has traditionally been to subsidize ﬁrm R&D, to encourage
joint research cooperatives or to grant property rights over the outcomes of R&D (e.g. patents).1
Using a two-stage duopoly game, this paper shows that a price ceiling can be used to induce the op-
timal level of R&D investment. However, because of the information burden it imposes, it is not to
be interpreted as a policy prescription, but rather as a surprisingly interesting (and counterintuitive)
theoretical result.
The two-stage duopoly model with positive R&D spillovers introduced by D’Aspremont and
Jacquemin (1988) provides the theoretical framework for the model employed here. The ﬁrst
stage is characterized by ﬁrms choosing R&D levels which result in process innovation (lowering
of marginal production costs). The second stage is characterized by Cournot competition. It is
the case that ﬁrms fail to recognize the marginal social beneﬁt created from their investment in
R&D when spillovers exist. A price ceiling can be used to raise the marginal private beneﬁt of
R&D.2 Simply, when the ﬁrm is faced with a ﬁxed price and quantity, its choice variable simpliﬁes
to choosing R&D levels that lower marginal costs. The more a ﬁrm produces, the more it is
motivated to lower its marginal costs by investing in R&D. Therefore, by making the market more
“competitive”, ﬁrms want to innovate more.
The remainder of the paper is as follows. In Section 2, we employ the standard two-stage
duopoly model to show that a proﬁt-maximizing ﬁrm chooses a level of R&D that is below what
is optimal. Since the use of a R&D subsidy is most prevalent in existing literature, we derive the
relevant subsidy that would correct the market failure. In Section 3, we show how a price ceiling
on output can be used to induce the optimal R&D investment. In Section 4, we conclude with a
brief comparison of mechanisms used to correct the market failure and the limitation of using a
price ceiling.
2. The Modeling Framework
Consider a model with two ﬁrms producing a homogeneous good.3 Firms are identical and indexed
by i, i=1;2. Firms face an inverse demand function P=P(Q), where Q = q1+q2, qi denotes ﬁrm
i’s output, and P0(Q) < 0. Each ﬁrm has an initial constant marginal cost c0, and can reduce c0 by
investing in research and development. Firms independently invest in R&D, but R&D is a public
good in that one ﬁrm’s R&D spending also beneﬁts the other ﬁrm. We denote the marginal cost
function for each ﬁrm as ci = c(R), where R = r1+r2, ri is ﬁrm i’s investment in R&D, c(0) = c0;
c0(R) < 0, and c00(R) > 0. The convexity of the cost function reﬂects diminishing returns to R&D
spending: each ﬁrm beneﬁts from its own R&D investment and the other ﬁrm’s R&D investment,
1For example, see Tirole (Ch. 10, 1988) for a detailed discussion.
2This is in contrast to a subsidy which effectively lowers the marginal cost of R&D.
3Our results derived from a two-ﬁrm model can be easily extended to an n-ﬁrm case.
1but at a decreasing rate. The timing of ﬁrms’ decisions is as follows. In the ﬁrst stage, each ﬁrm
makes a decision on how much to invest in R&D independently, and in the second stage they
compete on quantity with their costs of production determined in stage one.
Using backward induction, given the R&D investment levels chosen in the ﬁrst stage, ﬁrms
engage in Cournot competition in the second stage. Firm i solves the following problem:
Max
qi
[P(q1+q2) c(r1+r2)]qi.
The ﬁrst-order condition, P0(q1+q2)qi+P(q1+q2) c(r1+r2) = 0, implicitly deﬁnes the best
response function for ﬁrm i given ri determined in the ﬁrst stage. Assuming a symmetric equilib-
rium exists, we can express the equilibrium solution as a function of (r1;r2) : q
i = q(r1+r2). It
is easy to show ¶q
i =¶ri >0.4 That is, a greater R&D investment leads to a greater output in the
second stage.
In the ﬁrst stage, ﬁrm i chooses R&D investment by solving the following:
Max
ri
[P(2q(r1+r2)) c(r1+r2)]q(r1+r2) ri.
From the ﬁrst order condition, we obtain:
[2P0(2q(RNC))q0(RNC) c0(RNC)]q(RNC)+[P(2q(RNC)) c(RNC)]q0(RNC) = 1; (1)
where RNC = rNC
1 +rNC
2 denotes the non-cooperative industry R&D investment level.
Assume the demand and cost functions are well behaved such that the second order condition
is strictly negative at all ranges we consider. Equation (1) implicitly determines the industry R&D
level, RNC, and the best response R&D investment for each ﬁrm at the symmetric equilibrium.5
However, it is not optimal. Firm i only takes into account of its own beneﬁt and ignores the
spillover effect of its R&D expenditures on the other ﬁrm.
Suppose an industry regulator exists and chooses R&D levels in the ﬁrst stage that maximizes
the combined proﬁts of two ﬁrms without changing the nature of competition in stage two. This
is analogous to the setting of cooperation in R&D but not in output market as in D’Aspremont
and Jacquemin (1988). Thus, the problem in stage two is unchanged, and the regulator solves the
following maximization problem in stage one:
Max
r1;r2
å
i
f[P(2q(r1+r2)) c(r1+r2)]q(r1+r2) rig.
The ﬁrst order condition shows:
[2P0(2q(R))q0(R) c0(R)]q(R)+[P(2q(R)) c(R)]q0(R) =
1
2
; (2)
where R = r
1 +r
2 is the optimal industry R&D level and r
i is the desired R&D for each ﬁrm from
the perspective of the industry regulator.
Comparing equation (2) with equation (1) allows us to infer whether underinvestment in R&D
occurs in the non-cooperative setting. Let us denote the left hand side of the ﬁrst-order conditions
4 ¶q
i
¶ri =
c0()
SOC > 0 , where SOC denotes the second order condition that is assumed to hold.
5There are potentially many equilibrium outcomes, but we only look at the symmetric one.
2in stage one as f(R), then equations (1) and (2) can be rewritten as f(RNC) = 1 and f(R) = 1=2,
respectively. From our earlier assumption, with well behaved demand and cost functions, f0(R) <
0. Recognize that f is a decreasing function implying RNC
i < R
i , that is, the non-cooperative
industry R&D investment is below the optimal industry level and both ﬁrms produce less than
their optimal.6 This is consistent with the literature on R&D investment with spillovers. Without
government intervention, positive R&D externalities are not considered by ﬁrms when making an
innovation decision.
The most common tool to correct the market failure is R&D subsidies. Effectively, a subsidy
lowers the marginal cost of R&D spending (the RHS of the ﬁrst-order condition in equation (1)).
For example, suppose the industry regulator sets a subsidy of s dollars per dollar of R&D invest-
ment spent by the ﬁrm. The objective function in stage one is altered such that ﬁrm i chooses ri to
solve:
Max
ri
[P(2q(r1+r2)) c(r1+r2)]q(r1+r2) (1 s)ri.
Using the same notation as before, the ﬁrst order condition becomes f(R) = 1 s. Equating this
ﬁrst order condition with the optimal condition (equation (2)) gives the level of optimal subsidy,
s = 1=2. To induce the optimal level of R&D, the government must cover half of each ﬁrm’s
expenditure.
3. Price Ceiling
Equation (1) indicates that a ﬁrm will choose the R&D spending level which makes the marginal
beneﬁt to innovate (cost savings on production) equal to the marginal cost of R&D, given the other
ﬁrm’s choice. Note that the cost savings arising from each dollar of R&D spending is associated
with the production level the ﬁrm chooses. Thus, in this stage of the game, a higher level of output
implies a greater marginal beneﬁt of innovation. The more each ﬁrm produces, the more each
ﬁrm wants its production cost to be lower. To induce the two ﬁrms to spend more on innovation
activities, a price ceiling can be employed to force the ﬁrms to produce more.
Let the price ceiling be Pc. As long as Pc is higher than the marginal cost after innovation,
the two ﬁrms will produce according to the demand curve. In the symmetric equilibrium, these
two ﬁrms evenly split the market, and each ﬁrm produces qi = P 1(Pc)=2. Equating the marginal
beneﬁt from one dollar of spending on R&D to the marginal cost, we have
 c0(r1+r2)qi = 1, or, P 1(Pc) =  
2
c0(R)
,
which deﬁnes the relationship between the level of R&D spending the ﬁrms will choose and the
price ceiling the government imposes: R = R(Pc). It is easy to show that the market R&D in-
vestment level is decreasing in the price ceiling: R0(Pc) < 0 for Pc  c(R). This means imposing
a lower price ceiling causes ﬁrms to spend more on R&D as long as they will make a proﬁt.
To induce ﬁrms to choose the optimal level of R&D investment, the industry regulator simply sets
R(Pc
) = R. Solving the equation, we get Pc
=P( 2=c0(R)), which proves the following result.
6This is usually the case even with incomplete spillovers.
3Proposition 1. Firms invest in cost-reducing R&D activities at the optimal level under the price
ceiling Pc
= P( 2=c0(R)) in the symmetric equilibrium, as long as the following holds:
P(  2
c0(R))  c(R) R
c0(R)
2 : (3)
The condition we need (expression (3)) for a price ceiling to work is a participation constraint
which simply says a ﬁrm’s proﬁt is nonnegative. The revenue from selling products should be at
least as large as the sum of production cost and R&D spending. As long as the market is large
enough to satisfy this condition, the optimal level of R&D, as we outlined in section 2, can be
achieved.
Generally, priceregulationisoftenoverlookedasatooltoachievesociallydesiredoutcomes. In
that light, our result that an appropriate price ceiling induces the optimal level of R&D investment
is quite surprising. In a setting of positive R&D spillovers, the effective role of price ceiling arises
from the particular assumption that R&D reduces marginal costs. Under these circumstances,
the beneﬁt from investing in R&D is associated with the ﬁrm’s production level. Consequently,
making a ﬁrm invest more in R&D can be accomplished by making a ﬁrm produce more. The
price ceiling forces ﬁrms to choose higher production levels, and thus beneﬁt more from additional
R&D spending.
4. Conclusive Discussion
In a market where R&D investment can spillover to other ﬁrms, the standard free-riding problem
existsandunderinvestmentinR&Doccurs. R&Dsubsidies, jointresearchcooperativesandpatents
have been used in economic theory (and in public policy) to allow ﬁrms to internalize the positive
externality. In the case of subsidies, the marginal cost of R&D is reduced; for cooperatives and
patents, the marginal private beneﬁt of R&D is raised. In the preceding analysis, we showed that,
in theory, a price ceiling for Cournot duopolists could be used to raise the marginal private beneﬁt
of R&D as well. Although the use of a price ceiling is less practical from a policy standpoint, the
analysis still provides a surprising mechanism that achieves the optimal level of R&D.
Still, even these aforementioned mechanisms have their issues worth mentioning. Subsidies
place ﬁnancial burdens on the government which may result in large welfare losses when lump-
sum taxation is not used. And if a subsidy is given retroactively to the ﬁrm, moral hazard problems
exist if monitoring of R&D is too costly (Romano 1991). Assigning patents to the outputs of R&D
(such as a new production technology) can have signiﬁcant welfare effects as well if they create
excess market power (Romano 1991). Research joint ventures may increase the probability that
the participating ﬁrms collude on the output market (Leahy and Neary 1997).
Theoretically, a price ceiling would not have these particular issues. However, ﬁnding a proper
price ceiling requires the regulator having perfect information about the ﬁrms’ cost structure re-
lating to R&D spending and information on market demand as well. This places extra burdens on
the regulator. As a result, the policy implication of this paper is rather limited. With that said,
our ﬁnding that a price ceiling induces Cournot competitors to innovate contributes to the existing
R&D literature. While price ceilings are often seen as distorting the market and creating efﬁciency
losses, we ﬁnd a circumstance where it can enhance efﬁciency.
4References
D’Aspremont, C. and A. Jacquemin (1988) “Cooperative and Noncooperative R&D in Duopoly
with Spillovers” American Economic Review 78(5), 1133-1137.
Leahy, D.andJ.P.Neary(1997)“PublicPolicytowardsR&DinOligopolisticIndustries”American
Economic Review 87(4), 642-662.
Romano, R.E. (1991) “The Optimal R&D Policy: Patents, Public Funding, or Both” Southern
Economic Journal 57(3), 703-718.
Tirole, J. (1988) The Theory Of Industrial Organization, The MIT Press.
5

Inducing R&D investment with price ceilings

http://www.accessecon.com/Pubs/EB/2010/Volume30/EB-10-V30-I2-P142.pdf

Inducing R&D investment with price ceilings

Abstract

Similar works

Full text

Available Versions

Research Papers in Economics