A. Mantovani

G. Rossini

L. Lambertini

English

Research Papers in Economics

R&D in transport and communication in a
Cournot duopoly
Luca Lambertini1, Andrea Mantovani2, Gianpaolo Rossini3
Dipartimento di Scienze Economiche
University of Bologna
Strada Maggiore, 45
I-40125 Bologna, Italy
February 13, 2001
1lamberti@spbo.unibo.it
2mantovan@spbo.unibo.it
3rossini@spbo.unibo.itAbstract
WeanalyseR&D activity in transport and communication technology (TCRD),
in a Cournot duopoly. Transport and communication costs are of the iceberg
type, i.e., using up some portion of the product along its path to the …nal
buyer. Firms invest in TCRD to increase the net amount of the product that
reaches consumers. A variety of equilibria arise as a result of the di¤erent
levels of TCRD e¢ciency. If TCRD’s productivity is high, the game is a
prisoner’s dilemma where both …rms invest in TCRD.
As the e¢ciency of the TCRD progressively fades we come across …rst a
chicken game and, then, at lower e¢ciency, a game with an equilibrium in
dominant strategies where the pro…ts are at the highest. Social welfare is
maximised by market strategies only when TCRD is very e¢cient.
JEL classi…cation: D43,L13, O31.
Keywords: R&D, transport and communication costs.1 Introduction
By and large, R&D expenditure can be devoted either to process or to prod-
uct innovation. In most of the cases product innovation decreases the degree
of substitutability between rival products in oligopolies. No matter which
…rm engages in product innovation there is a bene…cial e¤ect also on ri-
vals that …nd competing products less close. Literature has emphasized the
di¤erent degree of e¢ciency of process innovating R&D in a Cournot mar-
ket setting vis à vis a Bertrand setting (Brander and Spencer, 1983; Dixon,
1985). Recently Lambertini and Rossini (1998, 1999) and Lambertini et al.
(1998) have shown that R&D in product innovation may give rise to the
choice of no heterogeneity as a result of a prisoner’s dilemma, no matter
whether Bertrand or Cournot competition is assumed. This appears to be
quite consistent with the externality brought about by product innovation
through its e¤ect on substitutability.
To be precise, other kinds of R&D activities may be considered. Casual
observation suggests that …rms invest in R&D that is neither devoted to
product innovation nor to process innovation, yet it is a kind of R&D that
allows …rms to reach markets in a more e¢cient way and be more competi-
tive just in their serving customers. The activities involved concern mainly
transport and communication needed to let the product reach the …nal con-
sumer. The related R&D may be …gured out as an expenditure that is going
to improve the technology of the last stage of the production process. Belong
to this category the investment in the Internet, in more advanced logistics,
or in faster transport technology. We de…ne this sort of activity transport
and communication R&D (TCRD). Most of the times transport and com-
munication services are modeled as if a portion of the output is used up to
1produce them, while only a fraction of the …nal product is …nalised to the
consumer. In such a framework, the purpose of investing in TCRD is just to
reduce this chunk of product lost while approaching the …nal buyer.
We borrow from trade theory (see, e.g., Helpman and Krugman, 1985)
the modeling of transport and communication costs, assuming that they are
of the iceberg type: a quantity qi of product i is produced, yet only a fraction
t 2]0;1] of the product reaches the consumer. This fraction depends on the
investment policy of the …rm, since, by committing to TCRD a …rm may
increase it. In doing so the …rm indirectly reduces production costs while
making rival products virtually come closer, even though they remain homo-
geneous. Investing in TCRD is then somehow similar to investing in product
innovation R&D, but with an opposite e¤ect, as far as substitutability is con-
cerned. TCRD has a further e¤ect similar to that of process innovation R&D.
Investing in TCRD is then a sort of combination of process and, reversed,
product innovating R&D.
Our aim is to analyse in a Cournot setting various scenarios in which
…rms behave symmetrically or asymmetrically as to TCRD. Our …ndings
can be summarised as follows. At the subgame perfect equilibrium, …rms
invest in TCRD only if the resulting increase in e¢ciency is large enough.
This game is a prisoner’s dilemma. When the equilibrium e¢ciency level of
the transportation technology is lower, …rms do not invest, which con‡icts
with social incentives. From a policy standpoint, a remedy could consist in
providing …rms with a subsidy to TCRD. As the e¢ciency of TCRD becomes
negligible, investment in TCRD appears undesirable both from a private and
from a social standpoint.
The paper is organized as follows. In the next section we analyse the
choice between investing in TCRD and not investing. In section 3 we go
2through the reduced form of the game played by …rms. In section 4 we
provide the welfare evaluation of the market solutions. Final remarks are in
section 5.
2 The model
We analyse a duopoly where …rms i and j compete in a two stage framework
in a Cournot setting. In the …rst stage they decide whether to invest either
in TCRD or not to invest. The second stage is the market stage. We resort
to backward induction to solve the game and get subgame perfection. The
R&D strategy spaceisgiven by the binary choice between undertakingTCRD
or doing nothing f0;kg; with capital expenditure in TCRD represented by
k > 0: We assume, for the sake of simplicity, that, if the …rm invests in TCRD
she will be able to ship the entire product to her customers and no portion
will be lost in the way (t = 1). Otherwise, if she doesn’t invest in TCRD
she will be able to ship only a fraction t 2]0;1] of the product. Marginal
production cost is assumed constant and equal to c:
In a Cournot duopoly setting we consider 3 cases.
2.1 Only one …rm invests in TCRD (case a)
Firm i invests in TCRD while …rm j does not. Firm i is able to deliver the
entire product to her customers, while …rm j a¤ords only a portion t 2]0;1[ of
the product to reach the consumer after its production, since 1¡t is used up
in transport and communication due to an inferior technology. We assume
linear market demand for the two homogeneous products with a unitary
3reservation price. Then we have:
p = 1 ¡ qi ¡ tqj (1)
Operative pro…ts are respectively:
a¼i = qi(1 ¡ qi ¡ qjt) ¡ cqi (2)
a¼j = tqj(1 ¡ qi ¡ qjt) ¡ cqj: (3)
From market stage …rst order conditions (FOCs),1 we get the following quan-
tities:
aq
¤
i =
c + t ¡2ct
3t
aq
¤
j =
¡2c + t + ct
3t2 :
Equilibrium total pro…ts are:
a¼
¤
i =
(2ct ¡ c ¡t)2
9t2 ¡ k (4)
a¼
¤
j =
(ct ¡ 2c + t)2
9t2 : (5)
2.2 Both …rms invest in TCRD (case b)
Assume …rms i and j invest in TCRD. Operative pro…ts are
b¼i = qi(1 ¡ qi ¡ qj) ¡ cqi (6)
1Second order conditions are always satis…ed, as it may be easily checked in this and
subsequent cases.
4b¼j = qj(1 ¡ qi ¡ qj) ¡ cqj: (7)
Equilibrium quantities are:
bq
¤
i =b q
¤
j =
1 ¡c
3
and equilibrium pro…ts are:
b¼
¤
i =b ¼
¤
j =
(1 ¡ c)2
9
¡k: (8)
2.3 None invests in TCRD (case c)
Assume that neither …rm i nor …rm j invests in TCRD. Operative pro…ts
become:
c¼i = tqi(1 ¡ tqi ¡tqj) ¡ cqi (9)
c¼j = tqj(1 ¡ tqi ¡ tqj) ¡ cqj: (10)
Equilibrium quantities are:
cq
¤
i =c q
¤
j =
t¡ c
3t2 (11)
while equilibrium total pro…ts are
c¼
¤
i =c ¼
¤
j =
(c ¡ t)2
9t2 : (12)
In case d, …rm i does not invest in TCRD, while …rm j does. Therefore we
just obtain the reversed payo¤s of case a, i.e.: a¼¤
i =d ¼¤
j and d¼¤
i =a ¼¤
j:
53 The reduced form of the game
The reduced form of the game is represented in normal form in matrix 1.
firm j
firm i
0 k
0 c¼¤
i =c ¼¤
j d¼¤
i; d¼¤
j
k a¼¤
i; a¼¤
j b¼¤
i =b ¼¤
j
Matrix 1
The above game has di¤erent solutions according to the productivity of
TCRD. By partitioning the admissible set of k into three regions, we can
derive the following:
Proposition 1 When TCRD is very e¢cient, i.e., at the lower bound of
the feasible set of k 2
"
0;
4c(c ¡ t)(t¡ 1)
9t2 = k1
#
; the game is a prisoner’s
dilemma with a unique solution in dominant strategies and both …rms invest
in TCRD.
As TCRD becomes less e¢cient, i.e., for k 2
"
k1;
4ct(1 ¡c)(1 ¡ t)
9t2 = k2
#
;
the game becomes a chicken game and there exist two asymmetric equilibria
in which only one …rm invests in TCRD.
For all k 2 [k2;1); the game has an equilibrium in dominant strate-
gies where the aggregate payo¤ of the …rms is maximized by not investing in
TCRD. Such an equilibrium is Pareto-e¢cient from the …rms’ standpoint.
Proof. First consider non-negativity constraints on quantities. In case a)
we have that q¤
i ¸ 0 if t ¸
c
2c ¡1
and q¤
j ¸ 0 if t ¸
2c
1 + c
: If we compare the
6two threshold levels of t we …nd that, if 1=2 < c < 1; then
2c
1 + c
<
c
2c ¡ 1
:
Therefore, in orderto have qi;j > 0; it must be t >
c
2c ¡ 1
: While if c 2 [0;1=2)
the condition turns out to be t >
2c
1 + c
.
In case b) the same requirement boils down simply to c · 1, while in case
c) it becomes t ¸ c:
Taking into account the above constraints on the parameters, we compare
the payo¤s appearing in matrix 1.
First, we can see that c¼¤
i;j ¸b ¼¤
i;j, since
(1 ¡ c)2
9t2 ¸
(1 ¡ c)2
9
¡ k:
Then compare b¼¤
i;j with a¼¤
j. We see that
b¼
¤
i;j ¸a ¼
¤
j
if
k ·
4c(c ¡t)(t ¡1)
9t2 = k1: (13)
Third, compare a¼¤
i with c¼¤
i;j: It appears that a¼¤
i ¸ c¼¤
i;j if
k ·
4ct(1 ¡ c)(1 ¡t)
9t2 = k2: (14)
Eventually, if we compare k1 with k2 we see that:
k1 ¡ k2 < 0
in the admissible region of parameters. Therefore:
i) if 0 < k · k1 the sequence of payo¤s becomes
a¼
¤
i ¸ c¼
¤
i;j ¸ b¼
¤
i;j ¸ a¼
¤
j
7and the game is a prisoner’s dilemma with a unique equilibrium where both
…rms invest in TCRD.
ii) if k1 · k · k2 the sequence of payo¤s is
a¼
¤
j ¸ c¼
¤
i;j ¸ a¼
¤
i ¸ b¼
¤
i;j
and the game is a chicken game with two equilibria o¤ the principal diagonal,
where only one …rm alternatively invests in TCRD.
iii) if k ¸ k2 the sequence of payo¤s becomes
b¼
¤
i;j ¸ a¼
¤
j ¸ a¼
¤
i ¸ c¼
¤
i;j
and the game has a unique equilibrium in which none of the …rms invests in
TCRD, and the aggregate payo¤s of the …rms is maximized. It can be easily
checked that the sequence of payo¤s presented is invariant as the value of the
parameter c varies within its admissible range.
4 Welfare analysis
If we now go to the welfare assessment, we can state the following:
Proposition 2 The solution of the TCRD game is also the outcome pre-
ferred by the social planner when the e¢ciency of TCRD is high and both
…rms invest, i.e., case b: For lower levels of TCRD e¢ciency, i.e., for larger
k; the social planner may prefer …rms not to invest. At intermediate levels
of k …rms do not invest while the social planner would like them all to do so.
Proof. We start calculating the social welfare in the three cases a; b; c:
In case a) the consumer surplus is:
csa =
(c ¡ 2t+ ct)2
18t2
8while social welfare, de…ned as the sum of consumer surplus and pro…ts is:
swa =
c2(11 ¡ 14t+ 11t2) ¡8t(c ¡ t + ct)
18t2 ¡k
In case b) both …rms invest in TCRD and then we get:
csb = 2
µ1 ¡c
3
¶2
while social welfare is
swb =
2(2 ¡4c + 2c2)
9
¡ k:
In case c) we have:
csc = 2
µc ¡t
3t
¶2
while social welfare is
swc =
4(c ¡t)2
9t2 :
Compare …rst swb with swc and substitute k1 to k: It then appears that
swb > swc if (t ¡ 1)(1 ¡ c) < 0, that is always true. If we substitute k2 for
k we end up with (t ¡ c)(t ¡ 1) < 0; that is always true, since feasibility
requires 1 > t > c: This establishes that, for all k 2 [0;k2]; swb > swc:
Now compare swa with swc: It appears that swa > swc if
c
2(11 ¡ 14t + 11t
2) ¡ 8t(c ¡t + ct) ¡ 18t
2k > 8(c ¡ t)
2
If we substitute k1 in the above expression it appears that it is always
true in the feasible set of parameters. The same happens if we substitute k2:
Then compare swb with swa: It appears that swb > swa regardless of the
value of k: To prove it just use the feasibility condition that t > c:
9Therefore the outcome preferred by the social planner is that associated
with both …rms investing in TCRD. This coincides with the equilibrium of
the game played by …rms, if TCRD is very e¢cient, i.e. for k 2 [0;k1]: The
coincidence disappears for k1 > k > k2 since …rms face a chicken game, while
the social planner would like them all to invest.
When we consider values of k larger than k2; the evaluations of the social
planner change. While swb > swa regardless of the value of k; we …nd that
swc > swb if k > k4 =
ct2 ¡ 2t2 ¡ c + 2t
9t2 and swc > swa if k > k3 =
c2(11 ¡ 14t + 11t2) ¡ 8t(c ¡t + ct) ¡ 8(c ¡ t)2
18t2 : By comparing k4 and k3 we
see that k4 > k3 if t >
11c
8 + 3c
: Mind that
11c
8 + 3c
<
c
2c ¡ 1
<
2c
1 + c
for all
feasible c < t 2]0;1]: Then for k2 > k > k3 …rms do not invest in TCRD
while the planner would like them to invest. For k3 > k > k4 the same
applies, while beyond k4 the planner agrees with the …rms and prefers them
not to invest since it is socially too expensive.
We have seen that for intermediate levels of TCRD commitment the social
planner would like …rms to invest. Therefore, there is a case for public
subsidies to TCRD whenever the cost of TCRD is not too high.
5 Concluding remarks
We have analysed in a simple Cournot duopoly setting the choice of …rms to
undertakea particularkind of R&D, that hasnot been considered sofarin the
literature and that is devoted to improve the transport and communication
(TC) technology that …rms adopt to reach the market.
Firms competing in quantities and producing homogeneous goods have an
incentive to undertake TCRD if the advantage they get is fairly high, that is,
10if the e¢ciency boost associated with the resulting TC technology is large.
This outcome is the result of a prisoner’s dilemma situation where social
welfare is maximised, while …rms do not maximise their aggregate payo¤.
For lower levels of e¢ciency, …rms shun TCRD investment, while the
social planner would like them to undertake it. In such a case, it appears that
a subsidy to TCRD could be introduced to obtain a second best result. As
the e¢ciency of TCRD fades, the stance of …rms and social planner converge
since investment in TCRD is undesirable both from a private and from a
social standpoint.
11References
[1] Brander, J. and Spencer, B. (1983) Strategic commitment with R&D: the
symmetric case. Bell Journal of Economics, 14, 225-35.
[2] Dixon, H.D. (1985) Strategic investment in a competitive industry. Jour-
nal of Industrial Economics, 33, 205-12.
[3] Helpman, E. and Krugman, P. (1985) Market structure and foreign trade.
Cambridge, MA, MIT Press.
[4] Lambertini, L. and Rossini, G. (1998) Product homogeneity as a prison-
er’s dilemma in a duopoly with R&D. Economic Letters, 58, 297-301.
[5] Lambertini, L. and Rossini, G. (1999) Endogenous choice of capacity
and product innovation in a di¤erentiated duopoly. Brown University,
Department of Economics, Working Paper No. 99-30.
[6] Lambertini, L., Poddar, S. and Sasaki, D. (1998) Standardization and
the stability of collusion. Economics Letters, 58, 303-10.
12

R&D in transport and comunication in a Cournot duopoly.

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