1. health and fitness
2. exercise

What is Brought by R&D Cooperation?
A Global, Dynamic Analysis
Jeroen Hinloopen∗
Grega Smrkolj∗†
Florian Wagener∗
August, 2011
Preliminary and Incomplete
Abstract
We provide a detailed dynamic, global analysis for a strategic model of
R&D, which allows us to consider the R&D process prior to the production
stage. Contrary to the received literature, we do not have to assume that
initial marginal costs are below the choke price. Our global analysis yields
four distinct scenarios, depending on initial marginal cost to be below or
above the choke price and whether or not the R&D process leads to the
steady state; if not, the market is either not formed at all or R&D efforts
are progressively scaled down, implying that the technology will die out
over time while production is eventually terminated. We then compare
two different possibilities across these scenarios: firms cooperate in R&D
only or they extend this cooperation also to the product market. This
yields two main results. First, the set of initial marginal costs above the
choke price that will induce a stable equilibrium is larger if firms also
collude in the product market. Second, the set of initial marginal costs
that induces firms to abandon the technology in time is larger if firms
cannot collude in the product market. We also discuss the consequences
of these results for the rigour with which antitrust policies should be
assessed.
Keywords: Bifurcations, Cartel, Cooperation, R&D, Spillovers
JEL: D43, D92, L13, L41, O31, O38
1
Introduction
The possible trade-off between static and dynamic efficiency remains a subject of
debate. One reason for a consensus view not to have surfaced is the ambiguous
relation between competition intensity on product markets and the incentives
to invest in Research and Development (R&D) (see for a recent overview of this
debate Schmutzler, 2009). The another reason is the wide array of models that
are used to analyze the problem. There is a fundamental distinction between
static and dynamic models. Early contributions focussed on patent races (Reinganum, 1981) or on the strategic aspects of R&D investments (d’Aspremont
∗ University
of Amsterdam and Tinbergen Institute
author. University of Amsterdam, Amsterdam School of Economics,
Roetersstraat 11, 1018 WB Amsterdam, The Netherlands. E-mail address: g.smrkolj@uva.nl
† Corresponding
1
and Jacquemin, 1988). Several recent contributions analyze dynamic versions
of well-known static models (Cellini and Lambertini, 2009, Lambertini and Mantovani, 2009, Kov´
aˇc et al., 2010). In this paper we also present a dynamic model
of R&D. The distinguishing feature of our approach is that we provide a truly
global analysis.
A widely implemented policy response to the possible conflict between static
and dynamic efficiency is to allow independent firms to cooperate in R&D while
remaining competitors on the ensuing product market.1 This preserves the
benefits of competition while at the same time removes obstacles for R&D investments. Because of the public goods character of R&D, the market is believed
to typically provide too little incentives for investing in R&D. Allowing firms to
form R&D cooperatives internalizes, at least to some extent, the concomitant
positive externalities. However, as Scherer (1980) observes: “the most egregious
price fixing schemes in American history were brought about by R&D cooperatives.” Martin (1995) shows that, indeed, the formation of R&D cooperatives
makes it easier to sustain tacit collusion in the product market: “common assets create common interests, and common interests make it more likely that
firms will non-cooperatively refrain from rivalrous behavior.”2 Supported by a
prevalently declinatory position of economic theory towards market collusion,
the authorities of both the USA and EU have made market cartels explicitly
illegal.3
The competition authorities are therefore faced with some kind of a trade-off
between static and dynamic efficiency. Whereas the extension of cooperation in
R&D to that in the product market may reduce output, without it, incentives
to pursue R&D may be lower. In this paper we examine to what extent the
fear that firms cooperating in R&D may also seek to optimize their production
plans is justified.4
Recently, Hinloopen, Smrkolj, and Wagener (2011) have warned that received models, which require that marginal costs are below the choke price at
any point in time and thereby impose that investments at all time coexist with
1 National Cooperation Act passed by Congress in the USA in 1984 allows firms to cooperate
in R&D (research joint ventures are explicitly evaluated on a rule of reason basis) provided this
cooperation is not extended to product markets. The EU competition law has been even more
sympathetic to joint research projects as joint ventures are exempt from antitrust restrictions
and have often even been initiated and subsidized by different EU programs and government
bodies.
2 That cooperation in R&D may to an extent blur competition can also be gathered from
occasional press releases by corporative managers. For instance, when asked with whom they
compete in the upstream market, Andrew N. Liveris, the CEO of Dow Chemical, answered
how “...Frankly it is not very clear anymore, because I’m joint-venturing with all of them.”
(The Focus (2007)).
3 In the USA, price-fixing conspiracies are banned by the Sherman Act. Under EU competition law, market collusion is banned by Articles 85 and 86 of the Treaty of Rome. Some
exceptions like those for agreements that facilitate technological progress are provided. However, these should not completely restrict competition and very few such exemptions were
granted in practice by the European Commission.
4 Geroski (1992) argues that it is the feedback from product markets that directs research
towards profitable tracks and that, therefore, for an innovation to be commercially successful
there must be strong ties between marketing and development of the product. Jacquemin
(1988) observes that R&D cooperatives are fragile and unstable. He reasons that when there
is no cooperation in the product market, there exist a continuous fear that one partner in
R&D may be strengthened in such a way that he will become a dangerous competitor in the
product market. Preventing firms from collaborating in the product market may therefore
destabilize or prevent many valuable R&D agreements.
2
production, cannot by definition capture the very nature of industry dynamics
as this assumption is obviously violated in the early stages of the innovation
process. In this paper we extend their dynamic framework describing the R&D
investments of a monopolist to allow for two potentially competing firms. A
distinguishing feature of our analysis, compared to the received literature, then
is that we consider a much larger parameter space. In particular, we do not
rule out a priori all those cases where the initial marginal costs are above the
choke price. Static models are forced to impose the assumption of ‘low enough’
marginal costs; dynamic versions of these models maintain this assumption without further justification. However, research has started long before a prototype
sees the light; development begins long before the launch of a new product. It
is then only natural to take this period into account when setting up a dynamic
model of R&D. Indeed, the use of proper bifurcation theory allows us to model
the R&D process prior to the production stage for all possible initial technologies. In particular, it yields a bifurcation diagram that tells us for every possible
parameter combination the qualitative features of any equilibrium. Our global
analysis yields four distinct scenarios: (i) initial marginal cost are above the
choke price and the R&D process is initiated, after some time yielding production for which marginal cost continue to fall with subsequent R&D investments;
(ii) initial marginal cost are above the choke price and the R&D process is not
initiated, yielding no production at all; (iii) initial marginal cost are below the
choke price and the R&D process is initiated, directly leading to production
whereby marginal cost continues to fall over time, and (iv) initial marginal cost
are below the choke price and the initiated R&D process is progressively scaled
down, implying that the technology will die out over time while production is
eventually terminated.
We then compare two different possibilities across these scenarios; one in
which firms form a full cartel, and one in which firms collude in R&D only.
We find that the range of initial marginal costs that leads to production is
larger when firms collude in both R&D and production. Furthermore, the set
of initial marginal costs that induces firms to abandon the technology in time is
larger if firms cannot collude in the product market. The speed at which firms
abandon non-prospective technologies is larger as well if firms cannot collude in
the product market.
Our results are not without policy implications. For designing an optimal
antitrust policy it is important to understand that these policies not only affect
current markets, but also markets that are not yet visible. In particular, preventing firms from colluding in the product market reduces the number of R&D
trajectories that successfully lead to new markets. In itself this constitutes a
welfare loss.
The remainder of the paper is organized as follows. Section 2 presents the
model. Section 3 describes all possible equilibria in case firms collude in both
R&D and on the product market, and analyzes the properties of the global
equilibrium dynamics. Section 4 derives all equilibria in case firms collude in
R&D only, which are confronted with the full collusion case in Section 5. Section
6 concludes.
3
2
The model
Time t is continuous: (t ∈ [0, ∞)). There are two a priori fully symmetric firms
which both produce a homogenous good at constant marginal costs. In every
instant, market demand is:
p(t) = A − qi (t) − qj (t),
(1)
where p(t) is the market price, qi (t) is the quantity produced by firm i = {1, 2},
qj (t) is the quantity produced by its rival (i 6= j), and A is the choke price.
Each firm can reduce its marginal cost by investing in R&D. That is, we
consider process innovations which bring about reductions in production costs.
The R&D process is subject to technological spillovers.5 Firm i exerts R&D
effort ki (t) and as a consequence of these investments, its marginal cost evolves
over time as follows:
dci (t)
≡ c˙i (t) = ci (t) (−ki (t) − βkj (t) + δ) ,
dt
(2)
where kj (t) is the R&D effort exerted by its rival and where β ∈ [0, 1] measures
the degree of spillover. The parameter δ > 0 is the constant rate of decrease
in efficiency due to the ageing of technology. Both firms start with the same
technology: ci (0) = cj (0) = c(0). The cost of R&D efforts per unit of time is
for firm i given by:
Γi (ki (t)) = b(ki (t))2 ,
(3)
where b > 0 is inversely related to the cost-efficiency of the R&D process. Hence,
the R&D process exhibits decreasing returns to scale.6 The firms discount the
future with the same constant rate ρ > 0. The instantaneous profit of firm i is:
πi (qi , qj , ki , ci ) = (A − qi − qj − ci )qi − bki2 ,
(4)
yilding total discounted profit:
Z
Πi (qi , qj , ki , ci ) =
∞
πi (qi , qj , ki , ci )e−ρt dt.
(5)
0
The model has five parameters: A, β, b, δ, and ρ. The analysis can be
simplified by considering a rescaled version of the model which, as defined in
Lemma 1, carries only three parameters: β, φ, and ρ˜. That is, rescaling the
model reduces the five-dimensional parameter space into a three dimensional
one.
Lemma 1. By choosing the units of t, qi , qj , ci , cj , ki , and kj appropriately,
we can assume A = 1, b = 1, and δ = 1. This yields the following rescaled
version of the model:
π
˜i (˜
qi , q˜j , k˜i , c˜i ) = (1 − q˜i − q˜j − c˜i )˜
qi − k˜i2 ,
(6)
5 These spillovers are the result of industrial espionage, common sources of basic research
(e.g, universities), and other various ways a firm can learn about its competitor’s research
experience (e.g, by hiring competitor’s employees). Their extent depends on the strength of
intellectual property protection and on the ability of firms to protect their knowledge.
6 In assuming decreasing returns to scale, we follow the bulk of the literature. See Hinloopen, Smrkolj, and Wagener (2011) for a discussion.
4
˜ i (˜
Π
qi , q˜j , k˜i , c˜i ) =
Z
∞
˜
π
˜i (˜
qi , q˜j , k˜i , c˜i )e−ρ˜t dt˜
(7)
0
c˜˙i = c˜i 1 − k˜i + β k˜j φ ,
with
ci =
c˜i (0) = c˜0 ,
q˜i ≥ 0,
k˜i ≥ 0
ρ˜ > 0,
φ>0
c˜i ∈ [0, ∞) ∀ t˜ ∈ [0, ∞)
conversion rules defined as: qi = A˜
qi , qj = A˜
qj , ki = √Ab k˜i ,
˜
A
A˜
ci , cj = A˜
cj , πi = A2 π
˜ i , π j = A2 π
˜j , φ = δ √
, t = δt , ρ˜ = ρδ .
b
(8)
(9)
(10)
kj =
A ˜
√
k ,
b j
The rescaled version of the model introduces a new parameter φ, which
captures the efficiency of the R&D process: a higher b implies that each unit
of R&D effort costs a firm more, whereas a higher δ implies that each unit of
R&D effort decreases the marginal cost by less. Therefore, a higher (lower) φ
corresponds to a more (less) efficient R&D process. For notational convenience
we henceforth omit tildes in the notation.
3
R&D and market cooperation (Full Cartel)
In this section we assume that each firm operates its own R&D laboratory
and production facility, however, firms adopt a strictly cooperative behavior
in determining both their R&D efforts and output levels. These assumptions
amount to imposing a priori the symmetry conditions ki (t) = kj (t) = k(t) and
qi (t) = qj (t) = q(t). Equation (8) thus reads as7 :
c˙ = c(1 − (1 + β)φk).
(11)
The profit of each firm in every instant is:
π(q, k, c) = (1 − 2q − c)q − k 2 ,
yielding its total discounted profit over time:
Z ∞
Π(q, k, c) =
π(q, k, c)e−ρt dt.
(12)
(13)
0
The optimal control problem of the two firms is to find controls q ∗ and k ∗ that
maximize the profit functional Π subject to the state equation (11), the initial
condition c(0) = c0 , and two boundary conditions which must hold at all times:
q ≥ 0 and k ≥ 0.8 Note that according to (11), if c0 > 0, then c(t) > 0 for all t.
The set of all possible states at each time t is given by c ∈ [0, ∞).
7 It may seem reasonable to assume that when cooperating in this way, firms fully share
their information, that is, that the level of spillover is at its maximum (β = 1). For the sake
of the generality of a solution, we do not a priori fix the value of β. Intuitively, this way
we allow for incomplete complementarity between the results of firms’ R&D activities that
may occur despite otherwise complete information sharing (e.g., there might still be some
ex post duplication or substitutability in R&D outputs if firms maintain their laboratories
separated). This way, we more properly define β as the level of complementarity between the
R&D outputs, and not solely as the level of bare information sharing.
8 Note that due to symmetry maximizing the per-firm total profit is the same as maximizing
the two firms’ combined total profit.
5
To solve the dynamic optimization problem, we introduce the current-value
Pontryagin function9
P (c, q, k, λ) = (1 − 2q − c) q − k 2 + λc (1 − (1 + β)φk) ,
(14)
where λ is the current-value co-state variable. It measures the marginal worth
of the increment in the state c for a firm in the cartel at time t when moving
along the optimal path. As marginal cost is a ‘bad’, we expect λ(t) ≤ 0 along
optimal trajectories.
Pontryagin’s Maximum Principle states that if the pair (c∗ , q ∗ , k ∗ ) is an
optimal solution, then there exists a function λ(t) such that c∗ , q ∗ , k ∗ , and λ
satisfy the following conditions:
1.) q ∗ and k ∗ maximize the function P for each t:
P (c∗ , q ∗ , k ∗ , λ) = max 2 P (c∗ , q, k, λ),
(q,k)∈R+
(15)
2.) λ is a solution to the following co-state equation:
−
∂P
= λ˙ − ρλ
∂c
⇔
λ˙ = q + (ρ − 1 + (1 + β)φk) λ,
(16)
which is evaluated along with the equation for the marginal cost
c˙ = c(1 − (1 + β)φk)
(17)
and the initial condition c(0) = c0 .
Let Q(c, λ) and K(c, λ) solve the problem max(q,k) P (c, q, k, λ) for every
(c, λ). We define the current-value Hamilton function H(c, λ) = P (c, Q(c, λ), K(c, λ), λ).
The above necessary conditions (15)-(16) for an optimal solution are complemented by the transversality conditions:10
lim e−ρt H(c, λ) = 0
(18)
lim e−ρt λc = 0.
(19)
t→∞
and
t→∞
If problem (15) has a solution, this solution necessarily satisfies the following
Karush-Kuhn-Tucker conditions:
∂P
= 1 − 4q − c ≤ 0,
∂q
q
∂P
= 0,
∂q
(20)
∂P
∂P
= −2k − (1 + β)φλc ≤ 0,
k
= 0.
(21)
∂k
∂k
Conditions (20) imply that either i) q = 0 and c ≥ 1 or ii) q > 0 and c < 1.11
In particular,
(
(1 − c)/4 if c < 1
q∗ =
(22)
0
if c ≥ 1
9 Also
called pre-Hamilton or un-maximized Hamilton function.
necessity of (18), which allows exclusion of non-optimal trajectories, was proven by
Michel (1982). For the necessity of transversality condition (19), see Kamihigashi (2001).
11 Observe that in the original (non-rescaled) model, the analogous condition for positive
production is c(t) < A.
10 The
6
Conditions (21) imply that either i) k = 0 and
k > 0 and ∂P
∂k = 0. In particular,
(
λc
if
− (1+β)φ
∗
2
k =
0
if
∂P
∂k
≤ 0 (implying λ ≥ 0) or ii)
λ≤0
λ>0
.
(23)
The above conditions for the optimal production yields two regimes of the
state-control systems. The first one is characterized by positive production, in
the second one there is no production.
1. The system with positive production. In the region where c < 1, using
(16), (17), (22), and (23), we obtain the following optimal state and co-state
dynamics:
(
c˙ = c 1 + 21 (1 + β)2 φ2 λc
(24)
λ˙ = 1 (1 − c) + ρ − 1 − 1 (1 + β)2 φ2 λc λ,
4
2
12
where λ ≤ 0. Equation (23) depends on the co-state and state variable. Note
that the correspondence between k and λ is one-to-one if, and only if, λ ≤ 0,
and that therefore the system can be rewritten in the state-control form.
Differentiate equation (23) with respect to time to obtain the dynamic equation
(1 + β)φ ˙
dk
≡ k˙ = −
λc + λc˙ .
(25)
dt
2
Substitute λ˙ from (16) and c˙ from (17). Note that equation (23) also implies
that
2k
λ=−
.
(26)
(1 + β)φc
Substitute this expression into (25), together with the expression for the optimal
output level (q = (1 − c)/4), to obtain:
(1 + β)φ
k˙ = ρk −
c(1 − c).
8
(27)
Hence, the system with positive production (c < 1) in the state-control form
consists of the following two differential equations:
(
k˙ = ρk − (1+β)φ
c(1 − c)
8
(28)
c˙ = c (1 − (1 + β)φk)
2. The system with zero production. In the region c ≥ 1, substituting (16),
(17), and (26) into (25) and imposing q = 0, gives the following expression:
k˙ = ρk.
(29)
Hence, the state-control system with zero production consists of the following
two differential equations:
(
k˙ = ρk
(30)
c˙ = c (1 − (1 + β)φk)
12 The
differential equations in the main text are all valid for λ ≤ 0. For a complete
specification of optimal state and co-state dynamics (including the case when λ > 0), we refer
the reader to Appendix E.
7
The state-co-state analogue is:
(
c˙ = c 1 + 12 (1 + β)2 φ2 λc
λ˙ = ρ − 1 − 1 (1 + β)2 φ2 λc λ
(31)
2
The Hamilton function, obtained by substituting (23) and appropriate expressions for q from (22) into the Pontryagin function (14), is given by
(
H(c, λ) =
1
8 (1
− c)2 + 14 (1 + β)2 φ2 λ2 c2 + λc
1
4 (1
2 2 2 2
+ β) φ λ c + λc
if
c ∈ [0, 1)
if
c ∈ [1, ∞),
(32)
where λ ≤ 0. The state-control analogue is given by
H(c, k) =



1
8 (1
2
− c)2 + k k − (1+β)φ
2
k k − (1+β)φ
if
c ∈ [0, 1)
if
c ∈ [1, ∞),
(33)
where k ≥ 0.
3.1
Steady-state solutions
The steady-state solutions to (28) and (30) are obtained by imposing the stationarity conditions k˙ = 0 and c˙ = 0.
Lemma 2. The ‘no-production’ state-control system (30) has no steady-state
in the region where it is defined.
Proof. See Appendix B.
Consider now system (28). Imposing the stationarity condition k˙ = 0, we obtain
kF C =
(1 + β)φ
c(1 − c) ≥ 0
8ρ
∀c ∈ [0, 1],
(34)
where superscript FC stands for Full Cartel.13 Steady-state marginal cost follows from inserting (34) into the state dynamics (17) and imposing the stationarity condition c˙ = 0:
(1 + β)2 φ2
c˙ = c 1 −
c(1 − c) = 0.
(35)
8ρ
This yields:
cF C = 0,
where
1
V =
2
cF C =
s
1
± V,
2
(36)
32ρ
.
(37)
(1 + β)2 φ2
√
Observe that V is real if, and only if, φ ≥ 32ρ/(1+β), in which case V ∈ [0, 21 )
and cF C ∈ [0, 1). The solution to the system is summarized in Proposition 1
and is depicted in Figure 1.
13 All
1−
steady-state values have superscript FC.
8
√
Proposition 1. If φ > 32ρ/(1 + β), the state-control system with positive
production (28) has three steady states:
i) (cF C , k F C ) = (0, 0) is an unstable node,
1
is either an unstable node or an unstable foii) (cF C , k F C ) = 21 + V, (1+β)φ
cus, and
1
iii) (cF C , k F C ) = 21 − V, (1+β)φ
is a saddle-point steady state.
√
At φ = 32ρ/(1 + β), a so-called “saddle-node bifurcation” occurs as the last
two steady states coincide and form a so-called “semi-stable” steady state.
√
If φ < 32ρ/(1 + β), the state-control system with positive production has one
single steady state: the origin (cF C , k F C ) = (0, 0), which is unstable.
Proof. See Appendix C.
SN
12
3 steady
states
10
Β=0
8
Φ
Β = 0.5
Β=1
6
4
Hc, kL = H0, 0L is the only steady state
2
0
0
1
2
3
4
5
Ρ
Figure 1: Steady states of the state-control system. The
√ drawn curves are the
saddle-node bifurcation (SN) curves given by φ = 32ρ/(1 + β). They are
drawn for three fixed values of β. Passing through them from above, the two
steady states other than origin coincide, form a ”semi-stable” steady state, and
then disappear.
In the saddle-point steady state, instantaneous marginal cost, output, and profits of each firm are, respectively:
p
(1 + β)φ − (1 + β)2 φ2 − 32ρ
FC
(38)
c
=
2(1 + β)φ
p
(1 + β)φ + (1 + β)2 φ2 − 32ρ
qF C =
,
(39)
8(1 + β)φ
9
π
3.2
FC
p
(1 + β)2 φ2 − 16(1 + ρ) + (1 + β)φ (1 + β)2 φ2 − 32ρ
.
=
16(1 + β)2 φ2
(40)
Global Optimality & Industry Dynamics
(Bifurcations of optimal vector fields)
The stable manifold of the saddle-point steady state is one of the candidates
for an optimal solution. However, as the following remark shows, it may not
(always) be the true optimum.
Remark 1. The Pontryagin function of the optimization problem as given in
(14) is not jointly concave in the state and control variables. Hence, necessary
conditions for an optimum are not necessarily sufficient.
Proof. See Appendix D.
Therefore, we have to carry out a detailed bifurcation analysis14 to assess
the dependence of the solution structure on the model parameters. This yields
a complete overview of how optimal R&D investment and concomitant production respond to changes in the initial level of marginal cost and to varying
characteristics of the R&D process. To analyze global optimality, we have to
inspect the complete state-control space, which is the set of solutions (c(t), k(t))
of the state-control system considered as parameterized curves (trajectories) in
a plane. Its general form is sketched in Figure 2.
The dotted vertical line c = 1 separates the region with zero production
from the region with a positive level of production. In the left-side region, the
parabola which achieves its maximum at c = 1/2 is the locus k˙ = 0. The
1
loci representing points where c˙ = 0 are the horizontal line with height (1+β)φ
and the k-axis. Loci c˙ = 0 and k˙ = 0 intersect in the saddle-point steady
state (S2 ) and the two unstable steady states (S1 and S3 ). The black arrows
in squares indicate the direction of trajectories in each respective region. The
horizontal arrows indicate an increase or a decrease in c, whereas the vertical
ones refer to changes in k. A number of trajectories is indicated by grey arrows.
Together, the trajectories cover the entire space. However, there are only two
trajectories leading to the saddle-point steady state; these are called “stable
paths” or “stable manifolds”. They are indicated by the connected black arrows
pointing towards S2 . The lighter arrows pointing away from S2 indicate the two
unstable paths. Trajectories spiral out from S3 in a counter-clockwise motion.
The solution to the state-control system (30), which gives trajectories in the
region with zero production, is given by:
k(t)
= C1 eρt
c(t)
= C2 e
eρt (1+β)φC1
t−
ρ
(41)
,
(42)
where C1 and C2 are positive real constants.
14 The variations in the values of parameters typically lead to qualitative changes in the
solution structure (e.g., some steady states lose their stability, indifference points appear).
Such qualitative changes in the solution structure due to smooth variations in parameters are
called bifurcations. For a good introduction to bifurcation theory and precise definitions of
terms used in this section, see Grass et al. (2008). For a recent discussion, see, for instance,
Kiseleva and Wagener (2009).
10
q>0
q=0
k˙ = 0
k
L1
L2
S2
S
1
(1+β)φ
3
c˙ = 0
L3
S1
0
0.5
1
1.5
c
Figure 2: Illustrative sketch of the state-control space.
A solution to the state-control system (28), which yields trajectories in the
region with positive production, cannot be obtained analytically; we therefore
make use of geometrical-numerical techniques.15
A particular trajectory is a candidate for an optimal solution if it satisfies all
necessary conditions: those given by Pontryagin’s Maximum Principle together
with transversality conditions (18) and (19). As Figure 2 shows, there exists a
whole range of solutions (c(t), k(t)) to the state-control systems. These trajectories all satisfy the conditions of Pontryagin’s Maximum Principle. By ruling
out those trajectories that do not satisfy the transversality conditions, we are
left with the candidates for an optimal solution.
Lemma 3. The points along the c-axis in Figure 2 in the set
{(c, k) : c ∈ (0, 1), k = 0}
cannot be a part of any optimal trajectory.
Proof. See Appendix E.
Lemma 4. All trajectories along which k → ∞ and c → 0 as t → ∞ can be
ruled out as optimal solutions.
Proof. See Appendix F.
For instance, the trajectory denoted by L1 in Figure 2 is in the set of
Lemma 4.
Lemma 5. The trajectory through the point (c, k) = (1, 0) satisfies the transversality conditions (18) and (19).
15 See
also [12, 28]. All numerical calculations were done in Matlab.
11
Proof. See Appendix G.
In Figure 2, the trajectory of Lemma 5 is labelled L3 . We call this the “exit
trajectory” as it implies that the firms (eventually) exit the market. Indeed, at
some stage the exit trajectory enters the region where c ≥ 1, at which point
both R&D investment and production come to a halt.16
Notice that the stable path (denoted by L2 in Figure 2) to the saddle-point
steady state (S2 ) also satisfies the transversality conditions. Notably, along this
trajectory, both c and k converge to finite limits and, therefore, so does the
co-state variable (26).
As the following corollary to the above lemmas shows, we are left with two
candidates for an optimal solution.
Corollary 1. The set of candidates for an optimal solution consist of the stable path of the saddle-point steady state and the trajectory through the point
{(c, k) = (1, 0)}.
Proof. See Appendix H.
To find an optimal solution from the candidates, we make use of the following
lemma.
Lemma 6. If the pair (c(·), q(·), k(·)) satisfies the necessary conditions (15)(19), then
Z ∞
Π(c, q, k) =
π(q, k, c)e−ρt dt
0
1
= P (c(0), q(0), k(0), λ(0))
ρ
1
= H (c(0), k(0)) ,
ρ
(43)
where H is the Hamilton function in state-control form.
Proof. See, e.g., Grass et al. (2008), ch. 3, pp. 161-162.
Lemma 6 is useful whenever several paths satisfy the necessary conditions
and we have to compare the corresponding values of the objective (total discounted profit) function in order to determine which one of them is the optimal
solution. As the lemma states, these values are simply equal to the values of
the Hamiltonian (33) at the corresponding initial points.
Figure 3 provides an example in which the optimal solution is the stable
path leading to the saddle-point steady state. Note that the initial marginal
costs can be higher than c = 1. The optimal decision leads the firms to first
produce nothing but to invest increasingly in the reduction of their marginal
costs. When marginal costs are below one, they start producing. The level of
16 Strictly speaking, firms’ marginal costs continue to increase along the exit trajectory due
to the positive depreciation rate δ. However, we interpret any situation in which a firm stays
inactive as if it has left the market.
12
R&D investments then gradually decreases to its long-run steady-state level.17
The value of the total discounted profit function is increasing with decreasing
marginal cost. Clearly, starting with lower initial marginal costs allows each
firm in the cartel to obtain a higher present discounted value of its profit flow
(notice that the profit function is just a rescaled Hamilton function, as stated
in Lemma 6).
Note that having a phase of zero production as a part of the stable path is
possible only if the range of initial marginal costs is not bounded to be below
the choke price. In this phase instantaneous profits are negative, but total
discounted profits are positive; the firms invest in R&D because once marginal
costs are below the choke price, production becomes profitable. Any model
imposing marginal costs to be below the choke price at all times cannot capture
these dynamics.18
0.32
0.25
0.3
0.28
0.2
0.26
0.15
q
k
0.24
0.22
0.2
0.1
0.18
0.16
0.05
0.14
S2
0.12
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0
1.6
0
0.5
1
1.5
2
c
t
(a)
(b)
1.6
1.4
0.32
0.1
0.3
0.09
0.28
0.08
0.26
0.07
0.24
0.06
2.5
3
3.5
4
k
c
1
0.8
0.6
Π
1.2
0.22
0.05
0.2
0.04
0.18
0.03
0.4
0.16
0.2
0
0.02
0.14
0
1
2
3
4
5
t
(c)
6
7
8
9
10
0.12
0.01
0
1
2
3
4
5
t
(d)
6
7
8
9
10
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
c
(e)
Figure 3: State-control space (a), time paths for quantity (b), marginal cost (c),
and R&D effort (d), and the correlation diagram between the total discounted
profit and marginal cost (e), respectively. All plots show curves for parameter
values (β, ρ, φ) = (1, 0.5, 4).
17 The shape of the stable path complies well to the empirical evidence of the level of
R&D activities over time. For instance, in his study of the petrochemical industry, Stobaugh
(1988) finds that the level of process innovation is rising over the first years after the product
innovation, whereas it is decreasing over time. Furthermore, he provides evidence that the
probability of the next process innovation being major innovation also decreases over time.
This relates to the concave shape of our stable path which implies that it takes increasingly
more effort over time to reduce marginal cost.
18 Related, the literature on optimal technology adoption predicts a substantial time lag
between the discovery of a new technology and its adoption (see e.g. Hoppe (2002) for an
overview). Delaying the adoption provides additional information as to the profitability of a
new technology; it is also rational if a better technology is expected to become available in the
near future (see also Doraszelski (2004)). In this literature, however, technological progress is
exogenous.
13
In the example of Figure 3, the firms optimally select an appropriate k such
that (c0 , k0 ) is located on the stable path. However, as the Arrow-Mangasarian
sufficiency conditions are not satisfied (see Remark 1), an investment policy that
follows the stable path is not necessarily optimal at all times. Indeed, it could
not even be a possible solution as the saddle-point steady state does not exist
for all parameter values (recall Proposition 1). That is, the solution structure
depends on the system’s parameters.
Figure 4 presents the bifurcation diagram, which shows the optimal R&D
investments for varying parameter values.19 Every combination of the exogenous
parameters yields a point on this diagram. It consists of five distinct regions,
each representing a particular structure of the set of optimal solutions. In
general, either there exists an upper bound of initial marginal cost below which
the firms selects the stable path towards the saddle-point steady state, or the
(new) technology is always phased out by firms (eventually) exiting the market.
Hence, if it exists, convergence to the saddle-point steady state along the stable
path is locally optimal only; it is never optimal globally.
Figure 5 differs from Figure 4 in that in the former β is fixed at 0.1 instead
of at 1. As the two figures indicate, changing β does not qualitatively change
our conclusions, yet it changes them quantitatively. Decreasing β (the degree of
spillovers), pushes the curves upward. This was expected as larger beta implies
larger complementarity of firms’ R&D efforts, which enhances the ability of the
firms to reduce investment costs in circumstances with a convex investment cost
function and so yields a higher return to their R&D.
Solutions to the optimization problem for parameters outside the shaded
region in Figure 4 are characterized by the presence of threshold values; the
stable path of the saddle-point steady state is optimal if the initial marginal
cost is below some threshold level, otherwise the exit trajectory is optimal. The
threshold point can be either an indifference point20 or a repeller. Indifference
points are initial states c = c0 for which the firm is indifferent between several
possible investment policies (i.e., the stable path and the exit trajectory). A
repeller differs from an indifference point in that at a repelling point, it is optimal
to stay at that point forever after.21
In Region I, the point of indifference is above one. Therefore, the system will
converge to the saddle-point steady state for all initial technologies c0 ∈ (0, 1].22
A typical example is depicted in Figure 6. The arrows indicate the direction of
motion along optimal trajectories. The two regions (production, no production)
are split by the dotted vertical line c = 1. For a given ρ and β, Region I is
characterized by a relatively high φ, that is, by relatively large demand and/or
high R&D efficiency. In such a favorable environment, the firms can compensate
for their early stage losses if, intitally, production is postponed as that would
yield negative mark-ups. There exists, however, a finite upper bound cˆ > 1
for the initial marginal costs c0 beyond which future profits are not enough to
19 The
bifurcation diagram in Figure 4 is drawn for the parameters of the non-rescaled model.
called Skiba points or DNSS points (see Grass et al., 2008).
21 More formally, a threshold point separates two basins of attraction: the initial values of
the marginal cost below these points constitute the basin of attraction of the saddle-point
steady-state; the initial values of the marginal cost above these points constitute the basin
of attraction to exit the market. An indifference point lies in both basins of attractions, a
repeller in neither.
22 Note that c = 0 corresponds to an unstable steady state; for this trivial value of the
0
marginal cost, the firms stay at that point forever.
20 Also
14
β=1
12
10
I
II
8
A
√
δ b6
IR
III
SN
ISN
4
IA
2
IV
V SN’
0
0
0.5
1
1.5
2
2.5
ρ/δ
3
3.5
4
4.5
5
Figure 4: Bifurcation diagram for β = 1. The uppermost curve represents
parameter values for which the indifference point is exactly at c = 1. SN
stands for the “saddle-node bifurcation curve”, SN’ for the “inessential saddlenode bifurcation curve”, IA for the “indifference-attractor bifurcation curve”,
and IR for the “indifference-repeller bifurcation curve”. ISN indicates the
“indifference-saddle-node point”, which is the point from which indifferenceattractor, indifference-repeller, saddle-node and inessential saddle-node curves
emanate. It’s coordinates are ρ˜ ≈ 2.14 and φ ≈ 4.1376. Roman numerals (IV) indicate the corresponding parameter regions discussed in the main text.
The shaded region indicates the parameter region for which firms always exit
the market. Notice that the axes are excluded from the admissible parameter
space. The curve representing indifference points at c = 1 obtains a value of
φ ≈ 1.41 for ρ˜ = 1e − 5.
compensate for short-run losses (ˆ
c ≈ 1.75 in the case considered). In that case,
the technology will never enter the market. In Figure 6, the point of indifference
(indicated by a dashed vertical line) occurs where the total profit function for
the stable path (Π1 ) in the right-hand picture obtains a value of zero (which
is the value of the total profit function for the exit trajectory Π2 in the region
c ≥ 1). As the total profit beyond this point is negative (the firms make a net
loss), the firms prefer no investment at all (giving them total profits of zero).
Starting with any initial value of the marginal cost below the indifference point,
the firms’ optimal decision is to follow the stable path. The exact position of
the indifference point is defined in Lemma 7.
Lemma 7. If an indifference point between the stable path and the exit trajectory
does exist in the region with zero production (c ≥ 1), it is given by the value
2
of the marginal cost cˆ ≥ 1 for which the point (c, k) = cˆ, (1+β)φ
lies on the
stable path. For c0 = cˆ, total discounted profits are zero for both trajectories.
15
β=0.1
12
III
IR
I
II
SN
10
8
ISN
A
√
δ b
6
IA
IV
4
SN’
V
2
0
0
0.5
1
1.5
2
2.5
3
ρ/δ
3.5
4
4.5
5
Figure 5: Bifurcation diagram for β = 0.1. The curve representing indifference
points at c = 1 obtains a value of φ ≈ 2.56 for ρ˜ = 1e − 5. The coordinates of
the ISN point are ρ˜ ≈ 2.14 and φ ≈ 7.523.
0.02
0.26
0.24
0.015
0.22
Π1
0.01
Π
k
0.2
0.18
0.005
0.16
Π2
0
0.14
0.12
−0.005
0.1
0.08
S2
0
0,5
1
1,5
1,75
2
−0.01
2,5
c
1
1,5
1.75
2
2,5
c
Figure 6: State-control space and the total profit function when the indifference
point is in the region with zero production. The plots are drawn for parameter
values (β, ρ, φ) = (1, 1, 6).
Proof. See Appendix I.
Region I is bounded from below by the curve representing those points for
which the point of indifference is exactly at c = 1. This particular situation is
depicted in Figure 7. For all initial values of the marginal cost c0 ∈ (0, 1), the
optimal-solution trajectory is the stable path of the saddle-point steady state,
while for c0 ∈ (1, ∞), it is optimal not to produce and invest anything. At c0 =
cˆ = 1, either strategy is optimal. The right-hand picture indicates that the total
profit Π1 for the stable path is zero at exactly c = 1. For comparison, the total
profit Π2 on the exit trajectory, which passes through the point (c, k) = (1, 0),
is plotted up to its conjugate point (defined later). We see that Π2 is smaller
16
than Π1 for c < 1, and it is equal to Π1 at the indifference point c = 1.
0.25
0.09
0.08
Π1
0.07
0.06
0.2
k
Π
0.05
0.04
0.03
0.15
0.02
0.01
Π2
0
S2
0.1
0.2
0.4
0.6
0.8
1
1.2
−0.01
1.4
c
0
0.2
0.4
0.6
0.8
1
1.2
c
Figure 7: State-control space and the total profit functions when the indifference
point is exactly on the boundary between the two regions (ˆ
c = 1). The plots are
drawn for parameter values (β, ρ, φ) = (1, 1, 4.7).
The stable path in the region with zero production (c ≥ 1) is characterized by
a decreasing k and an increasing c as t → −∞ (see (41)-(42) and Figure 6). From
Lemma 7 follows that the indifference point between the stable path and the exit
trajectory is exactly at c = 1 when the stable path reaches k = 2/ ((1 + β)φ)
exactly at c = 1 (see Figure 7). As investment is increasing in φ (for a given ρ and
β), we can find for every ρ and β a high enough φ such that the indifference point
is either at c = 1 or in the region with c ≥ 1. A higher φ for a given ρ and β in
Region I just moves the indifference point further beyond c = 1. Consequently,
Region I is not bounded from the right. Moreover, as k decreases along the
stable path as t → −∞, at some instant the stable path, having a value of k
at c = 1 above k = 2/ ((1 + β)φ)), reaches the point with k = 2/ ((1 + β)φ) for
some c > 1 as t → −∞. Therefore, for parameters in Region I, we always have
an indifference point at some c = cˆ ≥ 1 and, in particular, the convergence to
the saddle-point steady-state is for these parameters guaranteed if c0 ∈ (0, 1],
but not for all c0 ∈ (1, ∞).
In Region II the indifference point cˆ is below c = 1. This implies in particular that at the indifference point the monopolist produces a positive quantity.
Figure 8 illustrates this case. The indifference point (indicated by the dashed
vertical line) occurs where the two respective total profit functions (Π1 for the
stable path, Π2 for the exit trajectory) intersect, as indicated in the right-hand
picture. For the particular parameter values considered in Figure 8, the indifference point occurs at cˆ ≈ 0.59. Although it is also profitable to invest in R&D for
some initial technology level c0 ∈ (ˆ
c, 1), the technology is not promising enough
for the firms to select the stable path towards the saddle-point steady-state.
Rather, they invests in R&D at some smaller rate that retards the decay of the
technology level optimally; but eventually the technology will leave the market,
and the firms with it.23
For an even higher initial level of marginal cost (c0 ≥ 1), the firms do not
initiate any activity; the new technology is not developed at all and the firms
do not enter the market.24
23 Note that if c ∈ (ˆ
c, 1), the alternative strategy of producing but not investing anything
0
in R&D, which corresponds to ‘following’ the c-axis below the exit trajectory in Figure 8,
yields lower total discounted profits than following the exit trajectory (recall Lemma 3).
24 Conjugate points P and P indicate the last element on the respective trajectory (when
1
2
17
0.25
0.06
0.05
Π1
0.2
0.04
0.03
0.15
P1
P2
S3
k
Π
S2
0.1
0.02
0.01
P2
Π2
0
0.05
−0.01
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
−0.02
0.2
1.2
c
P1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
c
Figure 8: State-control space and the total profit functions when the indifference
point is within the region with positive production. The plots are drawn for
parameter values (β, ρ, φ) = (1, 1, 3.5). The optimal-solution trajectories are
marked by thick lines. The two candidates for the optimal solution, which both
spiral out of the unstable steady state S3 , are the stable path and the exit
trajectory.
Incidently, it is not uncommon to observe continuous incremental improvements of an existing technology while it has been decided that it will be taken
off the market at some future instant.
An interesting example of a dying industry is ice harvesting, which was
destined to disappear after the emergence of machine-made ice. According
to Utterback (1994), the ice harvesters were however still making numerous
improvements in harvesting, storing and delivery processes (e.g., steam-powered
saws and conveyors were introduced), which resulted in greater volume and
lower unit cost of ice. Similarly, even as the door was closing for producers of
gas lamps after the electric light bulb had been invented by Edison, this did
not stop their continuous process innovation. Utterback reports how among
other improvements the Welsbach mantle, which increased the efficiency of gas
lighting by five times, was successfully introduced. Yet another example is the
production of dry plate photography, which was put on the road to extinction
by Kodak roll film cameras. The old technology still found ways of improving
itself (e.g., by introducing self-setting shutters, small plate cameras, celluloid
substitutes for glass). See Utterback (1994) for numerous other examples of
industry dynamics. These examples in which producers in a dying industry
still invest in their products give much support for exit trajectories that are
characterized by positive R&D efforts. As already noted in the discussion of
Region II, our model shows that the ‘exit trajectory investment schedule’ leading
up to eventually discontinuing the product gives a higher total profit than the
investment schedule that is characterized by zero investment along the way of
exiting the market. Hence, theoretical predictions conform well to real world
practice.
Region III is similar to Region II in that for all c ≥ 1 the firm does not
initiate any activity and only for some small enough c ∈ (0, 1) there is converstarting from its end) where the marginal cost is still increasing/decreasing. Considering a
particular trajectory beyond this point would give us two possible k-points for each c, of which
the one beyond the P point is suboptimal. Observe that in the plot of the profits, the value
of the total profit for the exit trajectory (starting from its end) increases all along the way to
P2 (where it obtains a cusp point), beyond which it starts decreasing.
18
gence to the saddle-point steady state. There is, however, a small qualitative
difference between the two regions. Namely, in Region III, the threshold point
is not an indifference point but a repeller. Region III is separated from Region
II by a type-2 indifference-repeller bifurcation curve (IR1 (2)), at which a repeller changes into an indifference point (see Hinloopen, Smrkolj, and Wagener
(2011)).
Region IV√is separated from Region III by a saddle-node bifurcation curve,
given as φ = 32ρ/(1 + β). This curve represents parameter combinations for
which the saddle and the unstable node S3 coincide and form a semi-stable
steady state. After the bifurcation, both the saddle and the unstable node
disappear. In such a case (see Figure 9), the only solution left is the exit
trajectory: for c0 ≥ 1 the firms never initiate any activity, while for c0 ∈ (0, 1)
the firms do invest in R&D but they will leave the market at some future instant.
This is again intuitive as in Region IV we have that φ is relatively small (for a
given ρ and β). That is, R&D is relatively inefficient and/or market demand is
relatively small. Note that in Figure 9 R&D investments are initially increasing
over time for very low values of marginal costs. These investments reduce the
speed at which the firms will leave the market; for low initial marginal cost it
turns out to be profitable to slow down this process in order to profit optimally
from low marginal costs.
0.12
0.12
0.1
0.08
k
Π
0.1
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
0.2
0.4
0.6
0.8
1
1.2
0
1.4
c
0
0.2
0.4
0.6
0.8
1
1.2
1.4
c
Figure 9: State-control space and the total profit function for parameters of the
shaded region. Parameter values are (β, ρ, φ) = (1, 1, 2.5). The exit trajectory
covers the whole space and it is the only solution. Observe that the total
discounted profit is positive for c < 1.
Finally, Region V resembles Region IV in that only the exit trajectory is
optimal. This is in spite of the fact that points in Region V have higher φ
values than points in Region IV with the same value of ρ and β, and that the
saddle-point steady state (S2 ) together with its stable path exists. The exit
trajectory always yields a higher discounted profit flow than the stable path.
Consequently, the saddle-point steady state is not even locally optimal. The
IA1 curve that bounds Region V from above is a so-called indifference-attractor
bifurcation curve (see Hinloopen, Smrkolj, and Wagener (2011) for details).
The curve bounding Region V from below is an inessential saddle-node bifurcation curve (SN01 ). It is similar to the SN1 curve in that two new steady
states appear when crossing it. However, it is inessential for the optimization
problem in the following sense. The bifurcation does not entail a qualitative
change of the optimal strategy, as the steady states involved in the SN01 bifurca19
tion are not associated with the optimal strategy; as above, the exit trajectory
remains the optimal trajectory. Consequently, while the state-control system
does bifurcate, the structure of the optimal solutions does not.
Together, Region IV and Region V (the shaded region) represent the parameter region for which there is never convergence to the saddle-point steady
state because it is either not optimal to approach it (Region V ) or because there
is no saddle-point steady state at all (Region IV ). Moving from the shaded region, where the firms always exit the market, by increasing φ while keeping ρ
and β fixed, we arrive at Region II, where staying in the market is possibly
an optimal solution depending on the initial level c0 of the marginal cost. The
interpretation is straightforward: staying in the market becomes an option if
R&D becomes more efficient and/or if market demand increases.
4
The R&D Cartel
We now turn to analyze the R&D cartel. We still assume that each firm
operates its own R&D laboratory and production facility. However, while
firms select their output levels non-cooperatively, they adopt a strictly cooperative behavior in determining their R&D efforts so as to maximize their joint
profits. These assumptions amount to imposing a priori the symmetry condition ki (t) = kj (t) = k(t).25 As ci (0) = cj (0) = c(0), this implies that
ci (t) = cj (t) = c(t). Equation (8) thus reads as:
c˙ = c(1 − (1 + β)φk).
(44)
The instantaneous profit of firm i is:
πi (qi , qj , k, c) = (1 − qi − qj − c)qi − k 2 ,
(45)
yielding its total discounted profit over time:
Z ∞
Πi (qi , qj , k, c) =
πi (qi , qj , k, c)e−ρt dt.
(46)
0
We look for feedback (also called closed-loop) Nash equilibria. As firms
cooperatively decide on their R&D efforts, the only competitive decisions are
that of production levels. However, as quantity variables do not appear in
the equation for the state variable (44), production feedback strategies of a
dynamic game are simply static Cournot-Nash strategies of each corresponding
one-period game.
Solving maxqi ∈R+ πi gives us standard Cournot best-response functions for
the product market:
(
1
qj < 1 − c,
2 (1 − c − qj ) if
(47)
qi (qj ) =
0
if qj ≥ 1 − c,
25 In this paper we are looking only for symmetric equilibria. However, it is perhaps possible
that joint profits could be increased if firms made unequal investments and produce unequal
amounts. Such an asymmetric strategy would raise investment costs (the R&D cost function
is convex) but might create more than offsetting benefits in the production stage. Salant and
Shaffer (1998) consider this possibility within the framework of a static model of strategic
R&D.
20
where the second part comes from the boundary condition for the non-negative
production levels qi,j ≥ 0. Solving for the Cournot-Nash production levels, we
obtain
(
1
c < 1,
N
3 (1 − c) if
(48)
q =
0
if c ≥ 1.
Consequently, the instantaneous profit of each firm is
(
1
2
2
if c < 1,
9 (1 − c) − k
π(c, k) =
2
−k
if c ≥ 1.
(49)
Then, the optimal control problem of the R&D cartel is to find control k ∗ (R&D
effort of each firm) that maximizes the total discounted joint profit of the two
firms taking into account the state equation (44), initial condition c(0) = 0, and
the boundary condition k ≥ 0 which must hold at all times. Note that according
to (44), if c0 > 0, then c(t) > 0 for all t. The set of all possible states at each
time t is given by c ∈ [0, ∞).26
To solve the dynamic optimization problem, we introduce the current-value
Pontryagin function27
(
1
2
2
c < 1,
9 (1 − c) − k + λc(1 − (1 + β)φk) if
(50)
P (c, k, λ) =
2
−k + λc(1 − (1 + β)φk)
if c ≥ 1.
where λ is the current-value co-state variable of a firm in the R&D cartel. It
measures the marginal worth of the increment in the state c for each firm in
the cartel at time t when moving along the optimal path. As marginal cost is a
‘bad’, we expect λ(t) ≤ 0 along optimal trajectories.
Pontryagin’s Maximum Principle states that if the pair (c∗ , k ∗ ) is an optimal
solution, then there exists a function λ(t) such that c∗ , k ∗ , and λ satisfy the
following conditions:
1.) k ∗ maximizes the function P for each t:
P (c∗ , k ∗ , λ) = max P (c∗ , k, λ),
k∈R+
(51)
2.) λ is a solution to the following co-state equation:
−
∂P
= λ˙ − ρλ
∂c
(52)
which is evaluated along with the equation for the marginal cost
c˙ = c(1 − (1 + β)φk)
(53)
and the initial condition c(0) = c0 .
26 In other words, firms that jointly decide on their R&D efforts know that as the result
of this cooperation, the marginal cost of both firms will be the same and that their product
market profits directly depend on it (recall (49)). Knowing this, they will select such a time
path of R&D efforts, and through them the path of their marginal cost over time, that their
joint profits, which in our symmetric case they share equally, will be the highest possible.
27 We again omit a factor of 2 for combined profits to obtain the solution expressed in
per-firm values.
21
Let k = K(c, λ) solve the problem maxk P (c, k, λ) for every (c, λ). We define the current-value Hamilton function H(c, λ) = P (c, K(c, λ), λ). The above
necessary conditions (51)-(52) for an optimal solution are complemented by the
transversality conditions:
lim e−ρt H(c, λ) = 0
(54)
lim e−ρt λc = 0.
(55)
t→∞
and
t→∞
If problem (51) has a solution, this solution necessarily satisfies the following
Karush-Kuhn-Tucker condition:
∂P
= −2k − (1 + β)φλc ≤ 0,
∂k
Conditions (56) imply that either i) k = 0 and
k > 0 and ∂P
∂k = 0. In particular,
(
− (1+β)φ
λc
if
2
k∗ =
0
if
k
∂P
∂k
∂P
= 0.
∂k
(56)
≤ 0 (implying λ ≥ 0) or ii)
λ ≤ 0,
λ > 0.
(57)
The two-part composition of the Pontryagin function leads us to differentiate
between two regimes of the state-control system. The first one is characterized
by positive production, whereas the second one corresponds to no production.
1. The system with positive production. In the region c < 1, using (52), (53),
and (57), we obtain the following optimal state and co-state dynamics:
(
c˙ = c 1 + 21 (1 + β)2 φ2 λc ,
(58)
λ˙ = 2 (1 − c) + ρ − 1 − 1 (1 + β)2 φ2 λc λ,
9
2
where λ ≤ 0. Equation (57) depends on the co-state and state variable. Note
that the correspondence between k and λ is one-to-one if, and only if, λ ≤ 0,
and that therefore the system can be rewritten in the state-control form. Differentiate equation (57) with respect to time to obtain the dynamic equation
dk
(1 + β)φ ˙
≡ k˙ = −
λc + λc˙ .
dt
2
(59)
Substitute λ˙ from (52) and c˙ from (53). Note that equation (57) also implies
that
2k
λ=−
.
(60)
(1 + β)φc
Substitute this expression into (59) to obtain:
(1 + β)φ
k˙ = ρk −
c(1 − c).
9
(61)
Hence, the system with positive production (c < 1) in the state-control form
consists of the following two differential equations:
(
k˙ = ρk − (1+β)φ
c(1 − c),
9
(62)
c˙ = c (1 − (1 + β)φk) .
22
2. The system with zero production. In the region c ≥ 1, substituting (52),
(53), and (60) into (59) gives us the following expression:
k˙ = ρk.
(63)
Hence, the state-control system with zero production consists of the following
two differential equations:
(
k˙ = ρk,
(64)
c˙ = c (1 − (1 + β)φk) .
The state-costate analogue is:
(
c˙ = c 1 + 21 (1 + β)2 φ2 λc ,
λ˙ = ρ − 1 − 1 (1 + β)2 φ2 λc λ.
(65)
2
The Hamilton function, obtained by substituting (57) into the Pontryagin function (50), is given by
(
H(c, λ) =
1
9 (1
− c)2 + 14 (1 + β)2 φ2 λ2 c2 + λc
1
4 (1
2 2 2 2
+ β) φ λ c + λc
if
c ∈ [0, 1),
if
c ∈ [1, ∞),
(66)
where λ ≤ 0. The state-control analogue is then
H(c, k) =



1
9 (1
2
− c)2 + k k − (1+β)φ
2
k k − (1+β)φ
if
c ∈ [0, 1),
if
c ∈ [1, ∞),
(67)
where k ≥ 0.28
4.1
Steady-state solutions
The steady-state solutions to (62) and (64) are obtained by imposing the stationarity conditions k˙ = 0 and c˙ = 0.
28 Our closed-loop solution differs from that of Cellini and Lambertini (2009), who consider
the case when marginal cost is always lower than the choke price. This is so because their
proof that the open-loop and closed-loop solutions coincide is flawed by the fact that in
their derivation of the closed-loop solution, players’ output choices are not properly treated
as functions of the state variable. The derivations of the authors implicitly assume that
if marginal cost within the R&D cartel changes, the opponent’s quantity does not change,
which is the violation of the feedback principle underlying the closed-loop solution. It is also
counterintuitive as firms in the R&D cartel are supposed to jointly decide on their R&D effort
taking into account that marginal cost in any period affects the Nash-equilibrium profit in the
product market. This profit is a function of the cartel’s marginal cost and reflects changes
in outputs of both players. Our calculations also show that the authors’ solution leads to
situations in which the profit of the R&D cartel can be higher than that of the full cartel.
We shall question any such a result as the full cartel can in principle always do what a lower
form of cooperation (R&D cartel) does if this brings about a higher profit. We therefore stick
to our closed-loop solution, which does not suffer from this oddity, for all future comparisons
between regimes.
23
Lemma 8. The ‘no-production’ state-control system (64) has no steady state
in the region where it is defined.
Proof. See Appendix B.
Consider now system (62). Imposing the stationarity condition k˙ = 0, we
obtain
(1 + β)φ
k RC =
c(1 − c) ≥ 0
∀c ∈ [0, 1],
(68)
9ρ
where superscript RC stands for R&D Cartel.29 Steady-state marginal cost
follows from inserting (68) into the state dynamics (53) and imposing the stationarity condition c˙ = 0:
(1 + β)2 φ2
c˙ = c 1 −
c(1 − c) = 0.
(69)
9ρ
This yields:
cRC = 0,
where
cRC =
1
± V,
2
(70)
s
36ρ
.
(71)
(1 + β)2 φ2
√
Observe that V is real if and only if φ ≥ 6 ρ/(1 + β), in which case V ∈ [0, 12 )
RC
and c
∈ [0, 1). The solution to the system is summarized in Proposition 2.
√
Proposition 2. If φ > 6 ρ/(1 + β), the state-control system with positive production (28) has three steady states:
1
V =
2
1−
i) (cRC , k RC ) = (0, 0) is an unstable node,
1
is either an unstable node or an unstable foii) (cRC , k RC ) = 21 + V, (1+β)φ
cus, and
1
iii) (cRC , k RC ) = 12 − V, (1+β)φ
is a saddle-point steady state.
√
At φ = 6 ρ/(1 + β), a so-called “saddle-node bifurcation” occurs as the last two
steady states collide and form a so-called “semi-stable” steady state.
√
If φ < 6 ρ/(1 + β), the state-control system with positive production has one
single steady state: the origin (cRC , k RC ) = (0, 0), which is unstable.
In the saddle-point steady state, instantaneous marginal cost, output, and
profit of each firm are, respectively:
p
(1 + β)φ − (1 + β)2 φ2 − 36ρ
RC
c
=
,
(72)
2(1 + β)φ
p
(1 + β)φ + (1 + β)2 φ2 − 36ρ
q RC =
,
(73)
6(1 + β)φ
p
(1 + β)2 φ2 − 18(1 + ρ) + (1 + β)φ (1 + β)2 φ2 − 36ρ
RC
π
=
.
(74)
18(1 + β)2 φ2
29 All
steady-state values have superscript RC.
24
5
Market Collusion and Incentives to Innovate:
Full Cartel and R&D cartel confronted
The graphs showing different kinds of industry dynamics, each corresponding
to one of the five distinct regions of the bifurcation diagram, are for the R&D
cartel qualitatively the same as the ones for the full cartel.30 The bifurcation
diagram for the R&D cartel is together with its full cartel counterpart depicted
in Figure 10. Observe that the bifurcation curves for the full cartel lie below
those for the R&D cartel, and this so over the entire parameter space. The
relatively lower position of the shaded region (bounded above by the saddlenode and indifference-attractor bifurcation curves) for the full cartel leads to
our first main result of the global comparisons of the two regimes.
Result 1. For a given discount rate, it takes a larger market size and/or
efficiency or R&D to make staying in a market a possibility for firms in the
R&D cartel than it does for firms in the full cartel.
This result stems from the fact that in the shaded region, the exit trajectory
is the only solution. What is especially interesting to notice is the relative
position of the curves indicating the point of indifference being exactly at c = 1.
The curve for the full cartel lying below its R&D cartel counterpart implies the
existence of the situation in which, for a given parameter combination, the full
cartel regime has an indifference point at c > 1, while in the R&D cartel regime
the indifference point is at some c < 1. This relates to our second main result.
Result 2. Whenever there is a threshold value of initial marginal cost in
both regimes, this value is for the full cartel strictly larger than for the R&D
cartel, for every point in the parameter space.
The above result implies that, for a given market market size and efficiency
of R&D, the full cartel is ex ante more likely to bring a particular technology
to a production phase and less likely to exit the market. This fact is further
discerned by Result 3 and Result 4 below.
Result 3. The set of initial marginal costs that will induce the formation
of a new market is larger if besides cooperating in R&D, firms also cooperate in
the product market.
We already noted that the point of indifference in the region with c ≥ 1 is an
important concept. It determines the highest value of the initial marginal cost
for which the firms still initiate investment. If the firms find out that the initial
marginal cost (reflecting the starting level of the technology) is higher than this
upper bound, they do not go into development and the particular technological
idea is left unrealized - the market is not formed.
Figure 11 shows that this indifference point is for the full cartel at the higher
value of the initial marginal cost (c20 ) than for the R&D cartel (c10 ). Hence,
technologies corresponding to the marginal costs between the two indifference
30 Remark
1, Lemmas 3 - 7, and Corollary 1 apply also to the R&D cartel. Corresponding
proofs are analogous to their full cartel counterparts. Any differences between the proofs are
indicated within appendices corresponding to the full cartel.
25
25
I
20
II
15
IR
A(1+β)
√
δ b
III
SN
10
ISN
IA
IV
5
V
0
0
SN’
0.5
1
1.5
2
2.5
ρ/δ
3
3.5
4
4.5
5
Figure 10: Bifurcation diagram. The bifurcation curves for the R&D cartel
(blue) are drawn together with the bifurcation curves for the full cartel (red).
points (in the interval of a nontrivial size (c10 , c20 )) are the ones that are developed
only if besides cooperating in R&D, firms also cooperate in the product market.
The intuition is as follows. The firms are prepared to incur initial losses only
if they know they will make large enough profits to compensate for these losses
once the production has been initiated. The higher the value of the initial
marginal cost, the more investments it takes to reduce the marginal cost to the
level where the production can start (below the choke price) and so the larger
the initial losses that have to be compensated for. As the firms in the full cartel
are able to extract larger profits from the product market than the firms in
the R&D cartel (which earn only competitive profits), the firms engaged in the
full cartel can compensate for larger initial losses than the firms cooperating in
R&D only.31
As Figure 11 shows, the total discounted profit of the R&D cartel is for
c0 ∈ (c10 , c20 ) zero as firms don’t optimally do anything.32 However, the total
profit of the full cartel is positive. Hence, the full collusion appears endogenously
as a voluntary agreement through which both firms expect to benefit. Moreover,
as the plots in the figure show, this agreement is beneficial also from the social
standpoint as without it, consumers, as well as firms, are left empty-handed
(total discounted profit, consumer surplus, and total surplus are all zero for the
R&D cartel when c0 ≥ c10 ).
31 The assumption that firms know the market demand before the production stage has even
started may seem strong. We could relax it by postulating that firms form expectations about
future demand. However, while adding to the complexity of the model, this would not change
our conclusions as long as these expectations are assumed to be identical between the two
regimes; and we can not think of any good reason for not assuming so.
32 If the firms started developing the product, they would in total make a loss.
26
1
0.5
0.9
0.45
0.8
0.7
0.4
Π
k
0.6
0.35
0.5
0.4
0.3
0.3
0.2
0.25
0.1
0.2
0
0.5
1
c
c10
1.5
2
c20
0
2.5
0
0.5
1
(a)
c
c1
1.5
2
1.5
2
0
2
c0
2.5
(b)
4
2.2
3.5
2
3
1.8
2.5
TS
CS
1.6
1.4
2
1.5
1.2
1
1
0.8
0.5
0.6
0
0
0.5
1
c
c10
1.5
2
c2
2.5
0
0.5
1
c
0
(c)
c10
c20
2.5
(d)
Figure 11: State-control space (a), total discounted profit (b), consumer surplus
(c), and total surplus (d), when the indifference point is in the region with zero
production. The plots are drawn for parameter values (β, ρ˜, φ) = (1, 0.1, 2.25).
Blue curves correspond to the R&D cartel, whereas the red ones correspond to
the full cartel.
Figure 12 illustrates Result 3 further by comparing the position of the indifference point for different parameter combinations. The indifference point is for
the R&D cartel always below the one for the full cartel. For a given discount
rate, increasing the market size and/or the efficiency of R&D, increases the value
of the indifference point and so broadens the range of initial marginal costs that
lead firms to select the stable path. It also increases the discrepancy between
the two regimes (∆2 cˆ > ∆1 cˆ); the more so, the lower the discount rate (and so
the higher the slope of the curves). This is intuitive. The surplus created by
R&D that can be appropriated by the firms in the future (once the production
has started) is for the full cartel larger than for the R&D cartel. And the lower
the discount rate, the higher the value of this future surplus for firms.
Figure 12 also shows that increasing the discount rate, for a given market
size and efficiency of R&D, pushes the indifference point downwards (from C1
to C2 ) as it reduces the value of the future profits. This increase in the discount
rate can in principle be offset by sufficiently increasing the market size and/or
the efficiency of R&D, as indicated in the figure.
At Figure 11 we showed that for c0 ∈ (c10 , c20 ), the technology is developed
only by the full cartel. Assuming c0 = 2 (observe that 2 ∈ (c10 , c20 )), Figure 13
shows how marginal cost and price are continuously decreasing over time to their
27
10
9
ρ/δ = 0.1
ρ/δ = 0.5
8
7
cˆ
6
5
ρ/δ = 1
∆2 cˆ
C1
4
3
2
1
2
∆1 cˆ
4
C2
6
8
10
12
14
16
A(1+β)
√
δ b
Figure 12: Dependence of the indifference point cˆ on model parameters. Curves
are drawn for three fixed values of ρ˜. Curves for the R&D cartel (dotted) lie
below the curves for the full cartel (full).
long-run steady-state levels (0.0412 and 0.5206, respectively). We also plot the
so-called Lerner Index which measures a divergence of price from marginal cost
and is traditionally used as a short-cut estimator of the monopoly power and, as
prevalently claimed, its corresponding welfare loss. Its value is increasing over
time to the ‘high’ long-run level of 0.92. This increasing discrepancy between
price and marginal cost is a consequence of the cartel increasing its mark-up
over time to compensate for investments made before the production stage as
well as to cover its current R&D expenses. This is clear from the plot of the
instantaneous profit against the total discounted profit in Figure 13.
Observe that it is only after a while that the instantaneous profit becomes
positive. Its potential high size combined with a high Lerner Index may raise
eyebrows of antitrust economists worrying about resource misallocation. We
should note two things. First, while the instantaneous profit could well be
judged high, focusing on it, one overlooks huge R&D expenditures made over
years (let alone all the mental effort). Notably, the total discounted profit is
much lower: 0.0131 in our case (see Figure 13). In fact, the latter will be closer
to zero, the closer to the indifference point the initial marginal cost is. Second, at
Figure 12 we saw how market characteristics that affect future production (and
so profitability) affect the position of the indifference point. Hence, if market
regulations reduce mark-ups (for instance, by dismantling the product market
cartel), the range of initial technologies that are developed further shrinks (c20 ↓).
It can then well happen that in our case 2 > c20 and the firms are not able to
cover the costs already incurred in R&D. One should note that in our example,
the full cartel created the market and did not just take over the preexisting
28
2
0.05
1.8
Π
1.6
0
1.4
−0.05
1.2
LI
1
π
−0.1
0.8
p
0.6
−0.15
0.4
−0.2
0.2
c
0
0
2
4
6
8
10
12
14
16
18
−0.25
20
0
2
4
6
8
10
t
t
(a)
(b)
12
14
16
18
20
Figure 13: Marginal cost, price, and Lerner Index (a); total discounted profit and
instantaneous profit (b), for the full cartel. The plots are drawn for parameter
values (β, ρ˜, φ) = (1, 0.1, 2.25) and starting point c0 = 2.
one. Had firms not been able to cooperate also in the product market, the
market would not even have been formed (recall that 2 > c10 ). Hence, the
Lerner Index appears misleading from a correct dynamic perspective as it clearly
cannot indicate ‘social inefficiencies’ in any meaningful way. As shown, (larger)
deviations of price from marginal cost are not a source of inefficiencies but a
requirement for technology to be developed and a market to be formed at all.33
Result 4. The set of initial marginal costs that induce firms to abandon the
technology in time is larger if firms cannot collude in the product market.
Result 4, corresponding to the indifference point in the region with positive
production, is illustrated in Figure 14. For all initial marginal costs in interval
(c10 , c20 ), the firms engaged in the R&D cartel gradually exit the market, whereas
the firms engaged in the full cartel continue to reduce marginal costs and stay in
the market. The revenue the firms in the R&D cartel can potentially earn in the
product market is lower than that of the full cartel. Consequently, the relatively
large expenses needed to stay in the market (to countervail depreciation) become
for the R&D cartel ‘too high’ sooner than for the full cartel. That is, the full
cartel is more resistant to strained investment circumstances.34
33 In the literature, the Lerner Index has been subject to several critiques. Pindyck (1985)
claims that characteristics of dynamic markets (e.g., technological change, innovation, learning
by doing) cause the Lerner Index to give a distorted picture of a firm’s monopoly power. The
idea that deviations of price from marginal cost may arise from the need to cover incurred
investment costs can be traced back to Lindenberg and Ross (1981):“...[The Lerner Index]
does not recognize that some of the deviation of P from MC comes from ... the need to
cover fixed costs and does not contribute to market value in excess of replacement cost” (p.
28). The argument that marginal pricing may be neither feasible nor desirable in technologydriven industries has been occasionally recognized by courts, e. g., in US v. Eastman Kodak
(1995), where the court concludes that “Kodak’s film business is subject to enormous expenses
that are not reflected in its short-run marginal costs”. See Elzinga and Mills (2011) for a
comprehensive survey of limitations and qualifications of the index. For a critical discussion
of numerous antitrust cases in business history, see Armentano (1999).
34 Strained in the sense of a relatively high depreciation rate in comparison with the market
size and efficiency of R&D.
29
0.5
1.2
0.45
0.4
1
0.35
0.8
Π
k
0.3
0.25
0.6
0.2
0.4
0.15
0.1
0.2
0.05
0
0
0.1
0.2
0.3
0.4
0.5
c10
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
c20
c
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
c
(a)
(b)
2.5
4.5
4
2
3.5
3
TS
CS
1.5
2.5
2
1
1.5
1
0.5
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
0.5
c
c
(c)
(d)
Figure 14: State-control space (a), total discounted profit (b), consumer surplus (c), and total surplus (d), when the indifference point is within the region with positive production. The plots are drawn for parameter values
(β, ρ˜, φ) = (1, 0.1, 2). Blue curves correspond to the R&D cartel, whereas the
red ones correspond to the full cartel. Full curves correspond to the stable path,
whereas the dotted ones to the exit trajectory. Dots indicate the saddle-point
steady state.
Result 5. When the stable path to the saddle-point steady state is optimal
in both regimes, the steady-state value of marginal cost is for the full cartel lower
than for the R&D cartel. The full cartel always invests more in R&D, be it along
the stable path or the exit trajectory.
From the fact that for every given marginal cost, the trajectory for the full
cartel is characterized by a higher R&D effort (see Figure 11, 14, and 15), the
following result follows.
Result 6. For a given initial value of marginal cost, firms in the full cartel
enter production phase earlier than firms in the R&D cartel. On the contrary,
the speed at which firms abandon non-prospective technologies is lower when
firms also collude in the product market than it is when they do not.
As Result 5 states, the full cartel always achieves a higher production efficiency than does the R&D cartel when they both follow the stable path.35
35 This
follows directly from comparing the steady-state value of the marginal cost of the
30
However, when both regimes follow the stable path, the partial cartel is socially preferred to the full cartel as the higher efficiency brought about by the
full cartel is not relatively higher enough to compensate for the inefficiency
arising by its relatively lower output, and it therefore does not translate into
either higher consumer surplus or total surplus. Likewise, when both regimes
are about to exit the market, socially the R&D cartel is preferred. While the
full cartel indeed abandon production relatively more slowly36 (Result 6 ), this
does not outweigh its relatively lower instantaneous production. Consequently,
both consumer surplus and total surplus are lower for the full cartel (see also
Figure 15).
0.1
0.14
0.09
0.12
0.08
0.1
0.07
0.08
Π
k
0.06
0.05
0.06
0.04
0.03
0.04
0.02
0.02
0.01
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
0.5
c
c
(a)
(b)
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.45
0.25
0.4
0.2
0.35
0.3
TS
CS
0.15
0.25
0.2
0.1
0.15
0.1
0.05
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
0.4
0.5
c
c
(c)
(d)
Figure 15: State-control space (a), total discounted profit (b), consumer surplus
(c), and total surplus (d), when the exit trajectory is an optimal solution. The
plots are drawn for parameter values (β, ρ˜, φ) = (1, 1, 2). Blue curves correspond
to the R&D cartel, whereas the red ones correspond to the full cartel.
However, as Figure 14 shows, the full cartel brings about a higher total
full cartel with that of the R&D cartel (equations (38) and (72), respectively). The reason for
this difference is that while both regimes indeed exert the same R&D effort in the steady state
(the level needed to countervail depreciation), the investment paths of regimes are different.
The full cartel invests more in R&D as it is able to appropriate a higher surplus resulting
from it. This result stands in stark contrast to a strand of literature which claims that a lack
of competition in the product market will through a lack of investment motivation lead firms
to rest content with old methods and be engaged in x-inefficiency (a term first introduced by
Leibenstein (1966).
36 The intuition for this is that, having a monopoly power, the full cartel has still more to
extract from the market and therefore has incentives to leave slower.
31
surplus and also consumer surplus for initial marginal costs in interval (c10 , c20 ).
Recall that for these costs, the R&D cartel eventually exits the market, while
the full cartel follows the stable path and so stays in the market. Continuous
reductions in marginal costs along the stable path by the full cartel eventually
translate into a relatively higher output and in the end also into a higher total
discounted consumer surplus and total surplus. These benefits of the full cartel,
accruing from increasing efficiency over time, diminish with increases in the
discount rate. As Figure 16 indicates, for a relatively higher discount rate, the
R&D cartel can (for some subset of (c10 , c20 )) lead to a higher total surplus when
market size and/or R&D efficiency are sufficiently small even though the firms
engaged in it are destined to exit the market. With a lower market size and
efficiency of R&D, the full cartel itself invests less along the stable path and
thereby brings about lower gains in efficiency; and this future gains are of a
lower value, the higher the discount rate.
0.1
0.2
0.08
0.06
TS
TS
0.15
0.1
0.04
0.05
0.02
0
0.6
0.65
0.7
0.75
0
0.8
0.2
0.25
0.3
0.35
0.4
0.45
c
c
(a)
(b)
0.5
0.55
0.6
0.65
Figure 16: Total surplus for the R&D cartel (blue curves) and full cartel (red
curves) for two different combinations of parameter values: (β, ρ˜, φ) = (1, 1, 4)
in the left figure and (β, ρ˜, φ) = (1, 1, 3.14) in the right figure. Full curves
correspond to the stable path, whereas the dotted ones to the exit trajectory.
Dots indicate the saddle-point steady state.
6
Discussion and concluding remarks
Great benefits of R&D cooperations proven both in theory and practice have led
policy makers not to treat such initiatives as per se violations of the antitrust
law (as it holds for price-fixing agreements) but to consider them on a rule-ofreason basis. A possible extension of cooperation in R&D to that in the product
market has been considered as something that must be prevented at all costs.
In their seminal paper, d’Aspremont and Jacquemin (1988) conclude, within
their static model of R&D, that while the amount of R&D in the case of collusion
in both R&D and product market is higher than in the case of R&D cooperation
only, the quantity produced is still lower. Petit and Tolwinski (1999), who
consider a dynamic model of R&D, conclude the same. Moreover, a higher price
under full collusion leads in their model to a reduction in a consumer surplus
and to the lowest total surplus among regimes considered. Consequently, the
32
authors conclude that “...[full collusion] is socially inferior to other forms of
industrial structures” and “...thus, antitrust legislation is important.” (p. 206).
Our global analysis shows that the received literature suffers from neglecting
the very possibility that the initial marginal cost is higher than the choke price.
By assuming that a particular product is already in the production stage and
this so irrespective of the market regime under consideration, this literature
disregards an important part of the development phase. Indeed, it assumes an
existence of a situation whose formation the true theory of R&D should explain.
The practice shows that firms do not pursue further all conceived technological
ideas at all, but rest their decision upon their (expected) profitability.37 Yet,
this profitability is affected not only by the (expected) demand for a product
but also by the current and (expected) future market structure. The more
competitive the latter, the lower the price, the lower the mark-up, and hence
the lower the profitability of the idea.
We have shown that collusion in the product market through a larger profit
enlarges the set of initial marginal costs for which the firms pursue further development of corresponding technologies. Preventing collaboration in the product
market reduces this set.38 As Hinloopen, Smrkolj, and Wagener (2011) points
out “the loss of total surplus this reduction brings about is a hidden cost of competition policies if the indifference point is above c = 1, as in that case there is
initially no production. Not pursuing the development of the technology does
consequently not surface as an immediate reduction in total surplus. Yet, these
indirect costs should be taken into account when assessing properly the trade-off
between static and dynamic efficiency.” We have also shown that the popular
Lerner Index gives a distorted picture of welfare losses as market concentration
with its ensuing future high product market prices might in fact be needed for
the market to be formed at all. The practice of antitrust prosecutions however
shows that antitrust authorities widely neglect the exposed trade-off. As reported by Armentano (1999) in his comprehensive exposition of past antitrust
cases, “...price-fixing agreements were always illegal and there was to be no rule
of reason with regard to such conspiracies” (p. 143) and “...whether prices have
actually been fixed, or whether they have been fixed at unreasonable levels, has
been immaterial [at a federal court]” (p. 147).
In this paper we have focused on two market regimes (full cartel and R&D
cartel), leaving aside, in particular, a regime where firms compete both on the
product market and in their R&D endeavors. The technical demands of its
global analysis prevent us from including it here. We think it is also interesting
to try to include shochasticity, endogenous choice of spillovers, separate investments into innovative R&D on one hand and imitative R&D on the other hand,
asymmetric R&D abilities and differentiated products in the global framework
we developed here. Furthermore, we abstracted from potential new entry into
the market. These are in our opinion some of the fruitful areas we leave for
37 Elmer Bolton, a scientist-manager at the DuPont company, one of the most innovative
corporations in American business history, was famous for saying to company’s chemists who
in his opinion lacked the awareness that the success of the company depends on its products
being commercially exploitable: “This is very interesting chemistry, but somehow I don’t hear
the tinkle of the cash register.” (Hounshell and Smith, 1988).
38 Notice that the probability of being detected by the antitrust authorities when engaged
in collusion can be accounted for as an increase in a firm’s discount rate (as it is standard in
the literature). We have shown in Figure 12 that a higher discount rate reduces the value of
marginal cost at the indifference point.
33
future research.
34
Appendices
A
Proof of Lemma 1
A rescaled variable or parameter is distinguished by a tilde: for instance, if π
denotes profit, then π
˜ denotes profit in rescaled variables. The profit function
in the original (non-rescaled) model is:
πi = (A − qi − qj − ci ) qi − bki2
Using the conversion rules given in Lemma 1, we obtain:
πi
=
(A − qi − qj − ci ) qi − bki2
=
(A − A˜
qi − A˜
qj − A˜
ci ) A˜
qi − b
= A2 (1 − q˜i − q˜j − c˜i ) q˜i − k˜i2
A
√ k˜i
b
2
= A2 π
˜i
The equation for the evolution of marginal cost over time is in original variables
given by:
c˙i (t) = ci (t) (−ki (t) − βkj (t) + δ) .
Write ci (t) = ci 1δ t˜ . Then,
dci
dt˜
=
=
dci dt
dt dt˜
c˙i
δ
= ci 1 −
kj
ki
−β
δ
δ
.
Setting ki = √Ab k˜i and kj = √Ab k˜j , and substituting them in the previous equation, we obtain:
A dci
= ci 1 − k˜i + β k˜j √ .
dt˜
δ b
A
It is now natural to introduce φ = δ√
.
b
Note that if c˜i = ci /A, then c˜˙i = c˙i /A and
c˜˙i = c˜i 1 − k˜i + β k˜j φ
˜
Observe finally that if t = δt , then e−ρ˜t˜ = e−ρt if and only if ρ˜ = ρδ .
B
Proof of Lemma 2 and Lemma 8
As ρ > 0, the equilibrium condition k˙ = ρk = 0 implies that k = 0. But then
c˙ = c (1 − (1 + β)φk) = c. This is equal to 0 if and only if c = 0. But this
cannot be as in the system with zero production we have c ≥ 1.
35
C
Proof of Proposition 1
√
Assume φ > 32ρ/(1 + β). The stability of the steady states can be analyzed
by evaluating the trace and determinant of the following Jacobian matrix:39
" ∂ c˙ ∂ c˙ # "
#
1 − (1 + β)φk −(1 + β)φc
∂c
∂k
=
JFC =
.
(75)
∂ k˙
∂ k˙
− (1+β)φ(1−2c)
ρ
8
∂c
∂k
At c = k = 0, the trace τ of the matrix J F C is given as
def
τ = tr J F C = 1 + ρ > 0,
its determinant ∆ is
def
∆ = det J F C = ρ > 0,
and its discriminant D is
def
D = τ 2 − 4∆ = (1 − ρ)2 > 0.
Hence, this steady state is an unstable node.
Evaluating the Jacobian matrix at
(1 + β)φ
c(1 − c),
k=
8ρ
1
1 1
c= +V = +
2
2 2
s
1−
32ρ
,
(1 + β)2 φ2
(76)
we obtain τ = ρ > 0 and
p
p
(1 + β)2 φ2 − 32ρ (1 + β)φ + (1 + β)2 φ2 − 32ρ
∆
=
(77)
16
1 1
= (1 + β)2 φ2 V
+ V ,
(78)
8 4
√
which is clearly positive if φ > 32ρ/(1 + β). Hence, this steady state is also
unstable. The discriminant takes the value
p
1
D = ρ(8 + ρ) − (1 + β)φ (1 + β)φ + (1 + β)2 φ2 − 32ρ ,
(79)
4
q
2ρ(8+ρ)2
which is zero for φ = φ0 = (1+β)
2 (4+ρ) , negative for φ > φ0 , and positive for
√
32ρ/(1 + β) ≤ φ < φ0 . The steady state is an unstable node if D > 0 and an
unstable focus if D < 0. In the latter case, the eigenvalues of J F C are complex
conjugates with positive real parts.40
39 Note that the trace of the respective Jacobian matrix is equal to the sum of eigenvalues,
while its determinant is equal to their product. If the real part of each eigenvalue is negative,
then the steady state is asymptotically stable. If the real part of at least one of the eigenvalues
is positive, then the steady state is unstable. More particular, if one eigenvalue is real and
positive and the other one real and negative, the steady state is said to be a saddle. In this
last situation, there are four special trajectories, the separatrices, two of which are forward
asymptotic to the saddle, while the other two are backward asymptotic to it. The union the
former separatrices with the saddle form the stable manifold of the saddle; analogously, the
union of the latter with the saddle form the unstable
manifold.
√
1
40 Note that the eigenvalues of J F C are r
2
1,2 = 2 τ ± τ − 4∆ .
36
Finally, evaluating the Jacobian matrix at
(1 + β)φ
k=
c(1 − c),
8ρ
1
1 1
c= −V = −
2
2 2
s
1−
32ρ
,
(1 + β)2 φ2
(80)
we obtain τ = ρ > 0 and
(1 + β)2 φ2 − 32ρ − (1 + β)φ
∆=
16
p
(1 + β)2 φ2 − 32ρ
.
(81)
√
If φ > 32ρ/(1+β), then ∆ < 0, and the eigenvalues are real and have opposite
sign. Therefore,
(80) is a saddle-point steady state of the system. Observe that
√
for φ = 32ρ/(1 + β), the steady states (76) and (80) coincide at c = 1/2 and
1
k = (1+β)φ
. A saddle-node bifurcation occurs at these parameter values, where
the two equilibria coincide and disappear.41
Substituting the expression for the steady-state marginal cost of the steady
states other than the origin into (34), the expression for the optimal investment
1
.
in the steady state simplifies to k F C = (1+β)φ
D
Proof of Remark 1
The Hessian matrix of the Pontryagin function with respect to control and state
variables is

 2
∂ P (·)
∂ 2 P (·)
∂ 2 P (·)


2
−4
0
−1
∂q∂k
∂q∂c

 2∂q

 P (·) ∂ 2 P (·) ∂ 2 P (·)  
2
−2
−(1 + β)φλ 
D(q,k,c)
P =  ∂∂k∂q
= 0
∂k2
∂k∂c 

∂ 2 P (·)
∂ 2 P (·)
∂ 2 P (·)
−1 −(1 + β)φλ
0
2
∂c∂q
∂c∂k
∂c
(82)
The leading principal minor of the first order is −4. The leading principal
minor of the third order, which is also the determinant, of the above matrix is
equal to 2 + 4(1 + β)2 φ2 λ2 . As the leading principal minors of odd order do
not have the same sign, the matrix is indefinite. Consequently, the Pontryagin
function is nowhere jointly concave in state and control variables. Hence, the
Arrow-Mangasarian sufficiency conditions are not satisfied.42
For the R&D cartel, the appropriate Hessian matrix of the Pontryagin function is
 2
 "
#
∂ P (·)
∂ 2 P (·)
−2
−(1 + β)φλ
2
∂k
∂k∂c
2
=
D(k,c) P =  2
(83)
2
∂ P (·)
∂ 2 P (·)
−(1 + β)φλ
9
2
∂c∂k
∂c
The determinant of the above matrix is −4/9 − (1 + β)2 φ2 λ2 , which is clearly
negative. Notice that the determinant is the leading principal minor of the
second order. As the leading principal minor of the even order is negative,
the Hessian is by definition indefinite and, therefore, the Pontryagin function
is nowhere jointly concave in state and control variables. Hence, the ArrowMangasarian sufficiency conditions are not satisfied.
41 In
Figure 2, this corresponds to the locus c˙ = 0, which is the horizontal line k =
becoming tangent to the locus k˙ = 0 at its peak.
42 For details of sufficiency conditions, see, for instance, Grass et al. (2008).
37
1
,
(1+β)φ
E
Proof of Lemma 3
To prove the lemma, we consider the state-co-state form of the solutions, given
by:

0 ≤ c < 1, λ ≤ 0
c 1 + 12 (1 + β)2 φ2 λc ,





c,
0 ≤ c < 1, λ > 0
c˙ =
(84)

c 1 + 21 (1 + β)2 φ2 λc ,
c ≥ 1, λ ≤ 0




c,
c ≥ 1, λ > 0,
λ˙ =






2
τ (1
− c) + (ρ − 1 − 12 (1 + β)2 φ2 λc)λ,
0 ≤ c < 1, λ ≤ 0
2
τ (1 − c) + (ρ − 1)λ,
1 − 21 (1 + β)2 φ2 λc)λ,
0 ≤ c < 1, λ > 0
(ρ − 1)λ,
c ≥ 1, λ > 0,
(ρ −





H(c, λ) =











1
τ (1
− c)2 + 14 (1 + β)2 φ2 λ2 c2 + λc,
1
4 (1
1
2
τ (1 − c)
2 2 2 2
c ≥ 1, λ ≤ 0
(85)
0 ≤ c < 1, λ ≤ 0
+ λc,
0 ≤ c < 1, λ > 0
+ β) φ λ c + λc,
c ≥ 1, λ ≤ 0
λc,
c ≥ 1, λ > 0,
where τ = 8 for the full cartel and τ = 9 for the R&D cartel.
Introduce the characteristic function χS of a set S by
(
1 if x ∈ S
χS (x) =
0 if x ∈
/ S.
(86)
(87)
Differential equations (84) and (85) can be rewritten more compactly as:
1
def
c˙ = c + χ(−∞,0] (λ) (1 + β)2 φ2 λc2 = F (c, λ)
2
(88)
and
1
2
def
λ˙ = (ρ − 1)λ − χ(−∞,0] (λ) (1 + β)2 φ2 cλ2 + χ[0,1) (c) (1 − c) = G(c, λ). (89)
2
τ
The state-co-state space is sketched in Figure 17. We first consider the trajectories in the region where c ≥ 1 and λ > 0, which are the solution to the following
canonical system:
(
c˙ = c
(90)
λ˙ = (ρ − 1)λ,
given by λ(t) = C1 e(ρ−1)t and c(t) = C2 et , where C1 and C2 are positive constants. As mentioned in the main text, every candidate for an optimal solution
must necessarily satisfy the transversality condition limt→∞ e−ρt λ(t)c(t) = 0.
As we now show, the trajectories in the region considered violate this condition.
Moreover, they also violate the following transversality condition:
lim e−ρt H(c, λ) = 0,
t→∞
38
(91)
q > 0,
0.6
k=0
q = 0,
k=0
q = 0,
k>0
0.4
S1
λ˙ = 0
λ
0.2
0
c˙ = 0
λ˙ = 0
S
3
−0.2
−0.4
q > 0,
0
k>0
0.5
1
1.5
c
Figure 17: Illustrative sketch of the state–co-state space (for the full cartel with
ρ < 1). The dotted vertical line c = 1 separates the region with zero production
from the region with a positive level of production, whereas the horizontal line
λ = 0 separates the region with positive investment from the region with zero
investment. The loci c˙ = 0 and λ˙ = 0 intersect in the two unstable steady states
(S1 and S3 ) and the saddle-point steady state S2 (not indicated). A number
of trajectories is indicated by black curves: the arrows point in the direction of
the flow. The thick curve indicates the stable path leading to the saddle-point
steady state.
which in this case coincides with the first transversality condition, as shown
below. It follows from Michel (1982) that condition (91)43 is also a necessary
condition, and hence that it allows exclusion of trajectories which verify the
other necessary conditions in an infinite horizon optimization problem. The
value of the Hamiltonian function evaluated along the considered trajectories is
H(c, λ) = λ(t)c(t) = C1 C2 eρt ; it follows that
lim e−ρt H(c, λ) = lim e−ρt λ(t)c(t) = C1 C2 6= 0.
t→∞
t→∞
(92)
Hence, no trajectory in the region given by the restriction c ≥ 1 and λ > 0 can
be optimal.
Consider now trajectories in the region with 0 < c < 1 and λ > 0. These
trajectories are the solution to
(
c˙ = c
(93)
λ˙ = τ2 (1 − c) + (ρ − 1)λ.
43 In
words, the above condition means that the present value of the maximum of the
Pontryagin function (the present value of the Hamiltonian) converges to zero when time goes
to infinity.
39
We show that any trajectory in this region sooner or later enters the region with
c ≥ 1 and λ > 0.
Observe that the equation c˙ = c has as a solution c(t) = C3 et , where C3 is
a positive constant. Hence, along any trajectory in this region, c is increasing.
From (85) follows that if λ = 0, then
2
λ˙ = (1 − c) > 0
τ
for all c ∈ (0, 1). Hence, trajectories in the region with 0 < c < 1 and λ > 0
cannot exit this region through the line segment {(c, λ) : c ∈ (0, 1), λ = 0}. We
now show that they also cannot exit through the point (c, λ) = (1, 0).44
Let x = (c, λ) and x0 = (c0 , λ0 ) = (1, 0). Furthermore, let F : D → R2 be a
vector function defined as
F(c, λ) = F (c, λ), G(c, λ) ,
where F is defined in (88) and G in (89).45 Its domain is D = R+ × R ⊂ R2 .
We are then looking for a solution to a 2-dimensional nonlinear autonomous
dynamical system of the form
x˙ = F(x),
x(0) = x0 .
(94)
Take 0 < r < 1. Then the set
D0 = Br (x0 ) = {x ∈ D : kx − x0 k ≤ r},
(95)
is a neighborhood of x0 . First, we show that the restriction of the functions F
and G on D0 (a compact subset of D) is Lipschitz 46 . Consider first the function
(1 + β)2 φ2 2
c .
2
and set h(λ) = χ(λ)λ. Then,
F (c, λ) = c + λχ(−∞,0] (λ)
Write for brevity χ = χ(−∞,0]
|h(λ) − h(0)| = |χ(λ)λ − 0| = |χ(λ)λ| ≤ |λ| = 1 · |λ − 0|.
(96)
Hence, h is Lipschitz on D0 . As on compact sets continuously differentiable
functions are Lipschitz, as well as sums and products of Lipschitz functions, it
follows that F is Lipschitz on D0 .
Consider now on D0 the function
G = (ρ − 1)λ − λ2 χ(−∞,0] (λ)
(1 + β)2 φ2
2(1 − c)
c+
χ[0,1) (c).
2
τ
44 If they exited through this point, they would satisfy the transversality conditions as λ
˙ =0
for λ = 0 and c ≥ 1.
45 Though continuous, functions F and G are not differentiable: F is not differentiable with
respect to λ at λ = 0, whereas G is not differentiable with respect to c at c = 1. Using
Peano’s existence theorem, the continuity of F implies that at least one solution to (94) exists.
However, continuity is not enough to guarantee uniqueness. An additional condition that
needs to be fulfilled to guarantee uniqueness of solutions, at least in some neighborhood of
x0 , is that F is locally Lipschitz in x at x0 .
46 Simply put, a function f : D → Rn is said to be Lipschitz on B ⊆ D if there exists
a constant K > 0 such that kf (x) − f (y)k ≤ Kkx − yk, for all x, y ∈ B. See Kelley and
Peterson (2010), ch. 8, and Sohrab (2003), ch. 4, for the introduction to concepts and precise
definitions of terms used in this section.
40
The first term of this expression is a linear function; the second term is the
function h introduced above times a differentiable function. That the final term
is Lipschitz at c = 1 is demonstrated in the same way as for the function h. It
follows that G is Lipschitz on D0 .
We have proved that the functions F and G are locally Lipschitz at x0 .
Consequently, so is also the vector function F. By the Picard-Lindel¨of Theorem,
the system (94) has a unique solution in a neighborhood of x0 . As this point is
on the exit trajectory, no other trajectory can pass through it, in particular no
trajectory from the region with λ > 0. As c˙ > 0 in the region with 0 < c < 1 and
λ > 0, as shown above, all trajectories in this region exit through {c = 1, λ > 0}
and enter the region with c ≥ 1 and λ > 0 as t → ∞. We have already shown
that none of these can be optimal.
Moreover, trajectories through points in the set {(c, λ) : c ∈ (0, 1), λ = 0}
satisfy λ˙ > 0; they move in the region for which λ > 0 and hence cannot
be a part of any optimal trajectory. Due to a one-to-one correspondence between λ and k as given in the first part of (23), this set corresponds to the set
{(c, k) : c ∈ (0, 1), k = 0} in the state-control representation of the solution. As
we have shown, all trajectories leading to these points violate the transversality
conditions which every optimal trajectory must necessarily satisfy.
F
Proof of Lemma 4
Assume that there is an optimal investment schedule for which c → 0 and
k → ∞ as t → ∞.
The instantaneous profit function of each firm in every instant is:
π = (1 − 2q − c)q − k 2 .
(97)
As c → 0 along the trajectories considered, they sooner or later enter the region with positive production, where we have q = (1 − c)/4. Substituting this
expression into (97), we obtain:
π=
(1 − c)2
− k2 .
8
(98)
As c → 0, (1 − c)2 /8 approaches its upper bound of 1/8. The second term in
the above equation, −k 2 , decreases beyond all bounds as k → ∞. As t → ∞,
there is therefore a time t0 such that π(t) = 0 for t = t0 , and π(t) < 0 for all
t > t0 . Changing the investment schedule to k(t) = 0 for all t ≥ t0 would yield
a higher value of total discounted profits Π, contradicting the assumption.
In the case of the R&D cartel the instantaneous profit is
π(c, k) =
(1 − c)2
− k2 .
9
(99)
As c → 0, (1 − c)2 /9 approaches its upper bound of 1/9. The second term in
the above equation −k 2 however increases beyond all bounds as k → ∞. The
rest of the argument is analogous to the one for the full cartel above.
41
G
Proof of Lemma 5
We know that the state-control system with zero production is:
˙
k(t)
= ρk(t)
c(t)
˙
= c(t) (1 − (1 + β)φk(t)) .
˙
At k = 0, this reduces to: k(t)
= 0, c(t)
˙ = c. Hence, marginal costs increase
to infinity along the exit trajectory as t → ∞. However, as λ = 0 for c ≥ 1 along
the exit trajectory, the transversality condition (19) is satisfied. Observe that
along the exit trajectory k > 0 for 0 < c < 1 and k = 0 for c ≥ 1. Consequently,
it follows from (23) that λ = 0 at c = 1. Then, from (85) we have that indeed
λ = 0 for all c ≥ 1. Increasing marginal costs along the exit trajectory sooner or
later exceed the value of 1, for which the value of H(c, λ) becomes 0 (see (86)).
Hence, the transversality condition (18) is satisfied as well.
H
Proof of Corollary 1
TO BE DONE
The result of the corollary follows directly from lemmata (3)-(5) and the
directions of trajectories in the phase diagram provided in Figure 2. It is easy
to observe that given the trajectories in the phase diagram, if we exclude trajectories defined in Lemma 3 and Lemma 4, we are left exactly with the two
trajectories defined in the corollary. As the latter both satisfy the transversality
conditions, the result of the corollary is obtained. We now prove that the directions of trajectories in different sectors of the phase diagram are indeed those
that are in the phase diagram indicated in squared brackets.
˙ For c < 1,
Consider first the equation for k.
(1 + β)φ
k˙ = ρk −
c(1 − c).
8
(100)
FC
˙
The k˙ = 0 isocline is given by k F C = (1+β)φ
8ρ c(1 − c). Then, k > 0 if k > k
and k˙ < 0 if k < k F C . For c ≥ 1, k˙ = ρk. Hence, in this case, k˙ > 0 for all
k 6= 0.
Consider now equation for c,
˙ given by
c˙ = c(1 − (1 + β)φk).
(101)
1
The c˙ = 0 isoclines are given by c = 0 and k = (1+β)φ
= K. Assume c > 0.
Then, c < 0 if k > K and c > 0 if k < K.
˙ For c < 1,
For the R&D Cartel: Consider first the equation for k.
(1 + β)φ
k˙ = ρk −
c(1 − c).
9
(102)
RC
˙
The k˙ = 0 isocline is given by k RC = (1+β)φ
9ρ c(1 − c). Then, k > 0 if k > k
RC
and k˙ < 0 if k < k . For c ≥ 1, k˙ = ρk. Hence, in this case, k˙ > 0 for all
k 6= 0.
42
Consider now equation for c,
˙ given by
c˙ = c(1 − (1 + β)φk).
The c˙ = 0 isoclines are given by c = 0 and k =
Then, c < 0 if k > K and c > 0 if k < K. Q.E.D.
I
(103)
1
(1+β)φ
= K. Assume c > 0.
Proof of Lemma 7
From (33), we know that the Hamiltonian for c ≥ 1 is given by:
2
H(c, k) = k k −
(1 + β)φ
(104)
and from Lemma 6, we know that the comparison of the total discounted profits
of each two candidate optimal paths amounts to comparing the values of the
respective Hamiltonians in the initial point of each respective path. As the
Hamiltonian in the case of zero investment and zero production is zero, the
indifference point between the stable path and the exit trajectory in the region
with zero production must be the point at which the Hamiltonian evaluated
along the stable path obtains the value of zero.
2
=0
H(c, k) = k k − (1+β)φ
⇒k=0
or k =
2
(1+β)φ
The solution structure (41)-(42) tells us that the stable path in the region
with c ≥ 1 decreases as t → −∞, assuming that the stable path covers the
respective region. Observe that the derivative of the Hamiltonian with respect
2
1
to k is ∂H(c,k)
= 2k − (1+β)φ
, which is positive for k > (1+β)φ
and negative
∂k
1
for k < (1+β)φ .
If the stable path enters the region with zero production at all, it enters this
1
; this follows directly from equation (30).
region at the point where k > (1+β)φ
The same equation implies that k is decreasing along the stable path as
t → −∞, assuming the stable path enters the zero-production region. The above
conclusions lead us to distinguish three cases. First, if the stable path crosses
2
the boundary line c = 1 at k > (1+β)φ
, then the value of the Hamiltonian is
decreasing along the stable path in the region with c ≥ 1 as t → −∞, passes zero
2
at k = (1+β)φ
and is negative afterwards. In this case, the indifference point is
2
the value of the marginal cost cˆ > 1 that corresponds to the point cˆ, (1+β)φ
on
2
the stable path. Second, if the stable path reaches c = 1 at exactly k = (1+β)φ
,
then the indifference point is c = 1. Third, for lower values of k on the stable
path at c = 1, the value of the Hamiltonian evaluated along the stable path in
the zero-production region is negative, such that the point of indifference (if at
all) must be in the region with positive production.
J
Proof of Proposition 2
√
Assume φ > 6 ρ/(1 + β). The stability of the steady-states can be analyzed by
evaluating the trace and determinant of the following Jacobian matrix:
43
"
J
RC
=
∂ c˙
∂c
∂ k˙
∂c
∂ c˙
∂k
∂ k˙
∂k
#
"
=
1 − (1 + β)φk
−(1 + β)φc
− (1+β)φ(1−2c)
9
ρ
#
.
(105)
36ρ
,
(1 + β)2 φ2
(106)
At c = k = 0, the trace τ of the matrix J RC is given as
def
τ = tr J RC = 1 + ρ > 0,
its determinant ∆ is
def
∆ = det J RC = ρ > 0,
and its discriminant D is
def
D = τ 2 − 4∆ = (1 − ρ)2 > 0.
Hence, this steady state is an unstable node.
Evaluating the Jacobian matrix at
(1 + β)φ
c(1 − c),
k=
9ρ
1
1 1
c= +V = +
2
2 2
s
1−
we obtain τ = ρ > 0 and
p
p
(1 + β)2 φ2 − 36ρ (1 + β)φ + (1 + β)2 φ2 − 36ρ
∆ =
(107)
18
1 2
= (1 + β)2 φ2 V
+ V ,
(108)
9 9
√
which is clearly positive if φ > 6 ρ/(1 + β). Hence, this steady state is also
unstable. The discriminant takes the value
p
2
(109)
D = ρ(8 + ρ) − (1 + β)φ (1 + β)φ + (1 + β)2 φ2 − 36ρ ,
9
√
3 ρ(8+ρ)2
which is zero for φ = φ0 = √
, negative for φ > φ0 , and positive for
2
2
√
6 ρ
(1+β)
(1+β) (4+ρ)
≤ φ < φ0 . The steady state is an unstable node if D > 0 and an unstable
focus if D < 0. In the latter case, the eigenvalues of J M are complex conjugates
with positive real parts.47
Finally, evaluating the Jacobian matrix at
s
(1 + β)φ
1
1 1
36ρ
k=
c(1 − c), c = − V = −
1−
,
(110)
9ρ
2
2 2
(1 + β)2 φ2
we obtain τ = ρ > 0 and
(1 + β)2 φ2 − 36ρ − (1 + β)φ
∆=
18
p
(1 + β)2 φ2 − 36ρ
.
(111)
√
If φ > 6 ρ/(1 + β), then ∆ < 0, and the eigenvalues are real and have opposite
sign. Therefore, (110) is the saddle-point steady state of the system. Observe
47 Note
that the eigenvalues of J RC are r1,2 =
44
1
2
√
τ ± τ 2 − 4∆ .
√
that for φ = 6 ρ/(1 + β), the two steady states (106) and (110) coincide at c =
1
. A saddle-node bifurcation occurs at these parameter values,
1/2 and k = (1+β)φ
where the two equilibria collide and disappear. Substituting the expression for
the steady-state marginal cost of the steady states other than the origin into
(68), the expression for the optimal investment in the steady state simplifies to
1
.
k RC = (1+β)φ
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46