A. Mantovani

L. Lambertini

R. Cellini

English

Research Papers in Economics

Persuasive Advertising in Oligopoly:
A Linear State Diﬀerential Game
Roberto Cellini,# Luca Lambertini§ and Andrea Mantovani∗
# Department of Economics, University of Catania
cellini@unict.it
§ Department of Economics, University of Bologna;
ENCORE, University of Amsterdam
lamberti@spbo.unibo.it
* Department of Economics, University of Bologna
mantovan@spbo.unibo.it
April 5, 2006Abstract
We investigate a linear state diﬀerential oligopoly game with advertising,
under either Cournot or Bertrand competition. We show that a unique sad-
dlepoint equilibrium exists in both cases if the marginal cost of advertising
is suﬃciently low. Then, we prove that Bertrand competition entails more
intense advertising than Cournot competition. This is due to the fact that
enhancing reservation prices is more relevant to ﬁrms when market com-
petition is tougher. Ultimately, this may entail that Cournot outperforms
Bertrand when it comes to social welfare.
JEL Classiﬁcation: C73, D43, D92, L13, M37
Keywords: Advertising, Diﬀerential games, Oligopoly1 Introduction
After the pioneering contributions of [20] and [16], a wide debate on the
nature and eﬀects of advertising has developed. The existing literature on
dynamic models of advertising can be broadly partitioned into two classes:
the ﬁrst establishes a direct relationship between the rate of change in sales
(or market shares) and the advertising eﬀorts of ﬁrms (see, e.g., [7], [9] and
[15]); the second considers advertisement as an instrument to increase the
stock of goodwill or reputation and establishes a link between the advertising
eﬀorts of a ﬁrm and her market demand ([11], [13], [17] and [4]).1
We present a dynamic oligopoly model with diﬀerentiated products, where
ﬁrms compete either àl aCournot or àl aBertrand in the market phase, and
each ﬁrm invests in advertising activities aimed at increasing its market size,
or equivalently the consumers’ reservation price for its product. The adver-
tising activity which we consider in the present dynamic framework is of a
persuasive nature, has long-run eﬀects and presents positive external eﬀects
that spill over to the rival ﬁrms. In modelling this issue, we elaborate upon
[3].
As to the market decision, we consider an oligopoly with linear demands
for substitute goods.2
Our main results can be summarised as follows. First, we show that our
model is a linear state game (see [6]), i.e., the state variables enter both
the state equations and the objective functions linearly. Therefore, the solu-
tion of the diﬀerential oligopoly game beneﬁts both from the easy analytical
procedure of the open-loop concept and from the strong time consistency
1See [6, ch. 11], [8], [10], [12] and [14] for exhaustive surveys.
2As is well known, this approach encompasses monopolistic competition and homoge-
nous oligopoly as special subcases (see [1], [19] and [21]).
1property of the closed-loop concept. Second, we characterise the conditions
ensuring that the dynamic system converges to a unique and economically
meaningful steady state. We also prove that, under the same conditions, such
a steady state is stable in the saddlepoint sense. The third result concerns
the comparison of advertising incentives under price-setting and quantity-
setting behaviour, respectively. We ﬁnd that the parameter region where an
internal solution does exist is wider under Cournot competition; moreover, in
the subset of the parameter space where the steady state exists irrespective
of the nature of market competition, there appears that advertising eﬀorts
are more intense under price competition than under quantity competition.
This is clearly due the fact that, with substitute goods, Bertrand behaviour
entails a harsher competition which drives ﬁr m st oi n v e s tal a r g e ra m o u n to f
resources to increase reservation prices as compared to what they do under
a milder type of competition such as Cournot behaviour. As to the social
welfare at the market level, we ﬁnd - consistently - that Cournot competi-
tion may lead to a larger social welfare as compared to Bertrand competition.
This happens precisely because the tougher competition characterising price-
setting behaviour leads ﬁrms to higher investment eﬀorts and hence to lower
proﬁts that outweigh the larger consumer surplus.
The remainder of the paper is organised as follows. Section 2 illustrates
the basic setup. Section 3 carries out the equilibrium analysis and the com-
parative assessment of optimal advertising eﬀorts. Section 4 presents the
results concerning social welfare. Section 5 contains some concluding re-
marks.
22T h e m o d e l
As in [3], we consider an oligopoly game played over continuous time t ∈
[0,∞). The set of ﬁrms is P ≡ {1,2,3,...N}. Each ﬁrm i produces one diﬀer-
entiated variety of the same good. Let pi (t) denote the price of good i,a n d
qi (t) the quantity of good i at time t. Firm i faces the following instantaneous
demand function, as in [18]:
qi(t)=
Ai(t)
1+S(N − 1)
−
pi(t)[1+S (N − 2)]
(1 − S)[1+S(N − 1)]
+
S
P
j6=i pj(t)
(1 − S)[1+S(N − 1)]
(1)
Variable Ai describes the market size or the reservation price for good i.
Parameter S ∈ [0,1) measures the degree of substitutability between any
pair of diﬀerentiated goods. If S =0 , goods are independent and each ﬁrm
becomes a monopolist. In the limit case where S → 1, goods are perfect
substitutes and the model collapses into the homogeneous oligopoly model.3
Market size may be increased by ﬁrms through advertising. The dynamics
of ﬁrm i’s market size is:
dAi(t)
dt
≡
.
Ai(t)=ki(t)+γ
X
j6=i
kj(t) − δAi(t) (2)
where kh is the eﬀort in advertising made by ﬁrm h, γ ∈ [0,1] is a parameter
capturing the external eﬀect of the advertising of a ﬁr mo nt h em a r k e ts i z e
of diﬀerent ﬁrms and δ ≥ 0 is a depreciation parameter. Given that invest-
ment increases market size or reservation price, this is a form of persuasive
advertising.
Advertising entails a quadratic cost Γi(ki(t)) = α(ki(t))2/2 with α>0;
production entails a linear costs, ci(qi(t)) = cqi(t) with c>0. Prices and
3For diﬀerential games where S is a state variable that changes through ﬁrms’ R&D
eﬀorts for product diﬀerentiation, see [2] and [5].
3advertising eﬀorts are controls, while market sizes are states. Each player
chooses the path of her control variables over time, in order to maximize
the present value of her proﬁt ﬂow, subject to (i) the motion laws regarding
the state variables, and (ii) the initial conditions. Formally, the problem of
player i m a yb ew r i t t e na sf o l l o w s .T h eo b j e c t i v ef u n c t i o ni s :
max
pi(t),ki(t)
Ji ≡
Z ∞
0
πi(t)e
−ρtdt (3)
where the factor e−ρt discounts future gains, and the discount rate ρ is as-
sumed to be constant and common to all players. Instantaneous proﬁts
are πi(t)=[ pi(t) − c]qi(t) − Γi(ki(t)). Function (3) is subject to the set
of N dynamic constraints of type (2), and to the set of initial conditions
A(0) = A0. The Hamiltonian function of ﬁrm i is
Hi(t) ≡ e
−ρt
(
πi(t)+λii(t) ·
·
Ai +
X
j6=i
λij(t) ·
·
Aj
)
(4)
where λij(t)=µij(t)eρt is the costate variable (evaluated at time t) associated
by player i with state variable Aj.
3 Equilibrium analysis
With reference to the labels used by [6, ch. 7], the problem at hand is a linear
state diﬀerential game whose open-loop solution is strongly time consistent.
To prove this fact, it suﬃces to examine the ﬁrst order conditions (FOCs)
4and costate equations taken on the Hamiltonian function of ﬁrm i:
Hi(t) ≡ e
−ρt
∙
(pi(t) − c)
µ
Ai(t)
1+S(N − 1)
−
pi(t)[1+S (N − 2)]
(1 − S)[1+S(N − 1)]
+
+
S
P
j6=i pj(t)
(1 − S)[1+S(N − 1)]
¶
−
α
2
(ki(t))
2 +
+λii(t) ·
Ã
ki(t)+γ
X
j6=i
kj(t) − δAi(t)
!
+
+
X
j6=i
λij(t) ·
Ã
kj(t)+γ
X
h6=j
kh(t) − δAj(t)
!#
(5)
T h eF O C sf o rt h eo p e n - l o o ps o l u t i o na r e : 4
∂Hi(t)
∂pi(t)
=
Ai(t)
1+S(N − 1)
−
[2pi(t) − c][1+S (N − 2)] − S
P
j6=i pj(t)
(1 − S)[1+S(N − 1)]
=0
(6)
∂Hi(t)
∂ki(t)
= −αki(t)+λii(t)+γ
X
j6=i
λij(t)=0 (7)
∂λii(t)
∂t
= λii(t)(δ + ρ) −
pi(t) − c
1+S(N − 1)
(8)
∂λij(t)
∂t
= λij(t)(δ + ρ) (9)
N o w ,f r o m( 6 )w eh a v et h a t∂2Hi(t)/∂pi(t)∂Aj(t)=0for all j 6= i. Likewise,
f r o m( 7 )w eh a v e∂2Hi(t)/∂ki(t)∂Aj(t)=0 , since costate equations (8-9) are
indeed independent of the state variables. Accordingly, we may state:
Lemma 1 The game is a linear state one. Hence, the open-loop solution is
strongly time consistent, or subgame perfect.
We can now proceed to characterise the equilibrium. From (7),
ki(t)=
λii(t)+γ
P
j6=i λij(t)
α
(10)
4We omit the indication of exponential discounting for the sake of brevity.
5which can be diﬀerentiated w.r.t. time to obtain the dynamics of the adver-
tising eﬀort:5
·
ki =
·
λii + γ
P
j6=i
·
λij
α
(11)
The next step consists in noting that (9) is a separable diﬀerential equation
admitting the solution λij =0at all t. Therefore,
·
ki =
·
λii/α. Moreover, the
expression of λii can be obtained from (7), as follows:
λii = αki (12)
which, together with (8), can be plugged into the advertising dynamics:
·
ki =
1
α
∙
(ρ + δ)αki −
pi − c
1+S(N − 1)
¸
. (13)
Having rewritten the dynamics of ki in terms of controls and parameters only,
we can introduce the symmetry condition whereby pj = pi = p, kj = ki = k
and Aj = Ai = A for all i and j. By doing so, we may solve (6) to yield the
optimal pricing rule:
p
∗ =
A(1 − S)+c[1 + S (N − 2)]
2+S (N − 3)
. (14)
Note that p∗ =0in the limit case where S → 1, as is well known from the
static oligopoly literature (see, e.g., [18]). Using (14), (13) rewrites as follows:
·
k =( ρ + δ)k −
(A − c)(1− S)
α[1 + S(N − 1)][2 + S (N − 3)]
(15)
and imposing stationarity,
·
k =0yields:
k
∗ =
(A − c)(1− S)
α(ρ + δ)[1+S(N − 1)][2 + S (N − 3)]
. (16)
5Henceforth we will omit the indication of time for the sake of brevity.
6As a last step, we may plug (16) into (2) and impose
·
A =0to obtain the
steady state level of the reservations price:
AB =
c(1 − S)[1+γ(N − 1)]
(1 − S)[1+γ(N − 1)] − αδ(ρ + δ)[1+S(N − 1)][2 + S (N − 3)]
.
(17)
This goes along with the steady state advertising investment:
kB =
c(1 − S)δ
(1 − S)[1+γ(N − 1)] − αδ(ρ + δ)[1+S(N − 1)][2 + S (N − 3)]
.
(18)
It is easily checked that AB and kB are both positive iﬀ α ∈ [0,α B), where:
αB =
(1 − S)[1+γ(N − 1)]
δ(ρ + δ)[1+S(N − 1)][2 + S (N − 3)]
. (19)
The stability properties of the steady state characterised above can be eval-
uated on the basis of the trace and determinant of the following Jacobian
matrix:
JB ≡
⎡
⎣
−δ 1+( N − 1)γ
−
1 − S
α[1 + S(N − 1)][2 + S (N − 3)]
ρ + δ
⎤
⎦ (20)
The trace is T (JB)=ρ, while the determinant is:
∆(JB)=
(1 − S)[1+γ(N − 1)]
α[1 + S(N − 1)][2 + S (N − 3)]
− δ(ρ + δ) (21)
with ∆(JB) < 0 for all α ∈ [0,α B). Consequently, it appears that the steady
state is stable in the saddlepoint sense within the same parameter range
where the equilibrium values of states and controls are indeed economically
acceptable.
We may summarise the foregoing discussion by stating:
Proposition 2 Provided α ∈ [0,α B), the advertising game with Bertrand
competition admits a unique saddlepoint equilibrium (AB,p B,k B).
7From [3], we know that the corresponding Cournot game produces a
unique saddlepoint equilibrium at (AC,q C,k C) with
kC =
δc
1+( N − 1)γ − αδ(ρ + δ)(2 + S(N − 1))
> 0 (22)
AC =
c(1 + (N − 1)γ)
1+( N − 1)γ − αδ(ρ + δ)(2 + S(N − 1))
> 0 (23)
for all α ∈ [0,α C), where
αC =
1+( N − 1)γ
δ(ρ + δ)(2 + S(N − 1))
(24)
and αB <α C always. This entails the following result:
Lemma 3 The parameter region where the saddlepoint equilibrium exists is
wider under Cournot competition than under Bertrand competition.
Keeping in mind that α measures the marginal cost of advertising, the
reason for this fact appears to be that Cournot competition is softer than
Bertrand. Thus, it is easier for ﬁrms to ﬁnance advertising campaigns when
market behaviour follows the Cournot rule.
Accordingly, the two games can be compared only for α ∈ [0,α B). In this
range, we may compare kB against kC and AB against AC. Take ﬁrst the
optimal investment levels:
kB − kC ∝ αδ(ρ + δ)[1+S(N − 1)][2 + S (N − 3)] · Ψ +
−(1 − S)[1+(N − 1)γ] (25)
Ψ ≡ αδ(ρ + δ)[2+S(N − 1)] − 1 − (N − 1)γ
with the r.h.s. of (25) being equal to zero in correspondence of αB and αC.
This suﬃces to prove the following result:
8Proposition 4 kB >k C for all α ∈ [0,α B), irrespective of the value of γ
and S.
This is fully consistent with the ex ante intuition which would suggest
that the incentive to increase reservation prices should be higher under price-
setting behaviour, as this is harsher than quantity-setting behaviour, all else
equal. Indeed, the above Proposition tells that such incentive is higher under
Bertrand competition than under Cournot competition everywhere, provided
that the condition for an internal solution to obtain is satisﬁed.
4S o c i a l w e l f a r e
It is immediate to compare the social welfare at the market level, in steady
state, under Bertrand and Cournot behaviour.
As a social welfare index, we consider the sum of consumer surplus and
the proﬁts of ﬁrms (net of the cost of advertising investments). Hence, it is
SW = N
∙
(A − c)q −
µ
1
2
+ S(N − 1)
¶
q
2 −
α
2
k
2
¸
> 0 (26)
Taking the relevant values of variables from [3], one can ﬁnd the social wel-
fare in the steady state under Cournot competition (SWC). Similarly, sim-
ple substitutions for the relevant values under price competition allow to
obtain the steady state social welfare level under for the Bertrand game
(SWB). The diﬀerence between the two indeces is DSW = SWB − SWC,
which is quite a cumbersome expression, depending on the set of parameters
{α,γ,δ,ρ,c,N,s}.6
6The details of mathematical calculations are omitted for brevity. They are available
from the authors upon request.
9From an analytical point of view, this diﬀerence may take both sings.
However, numerical simulations show that, taking severalreasonable param-
eter regions, the diﬀerence is generally negative, indicating that the market
social welfare is larger under Cournot behaviour than under Bertrand be-
haviour. This is due to the fact that, as we know from the foregoing analy-
sis, the harsher competition characterising the Bertrand regime leads ﬁrms
to larger investment eﬀorts in advertising, and hence to lower proﬁts which
outweight the beneﬁts associated with consumer surplus. Figures 1 and 2
show the patterns of DSW under speciﬁc parameter conﬁgurations. In both
cases, we have set α =2 ,γ=3 /10,δ=1 /100,ρ=2 /100,c=1 . Then, in
F i g u r e1w eh a v es e ts =1 /2 with N varying over the interval of integers
[2,50] while in Figure 2, N is set equal to 10 and the diﬀerence is plotted
against s over the unit interval.
s =1 /2,N ∈ [2,50]
10s ∈ [0,1],N =1 0
In the numerical simulations illustrated above, the social welfare advan-
tage under Cournot is monotonically increasing in the number of existing
varieties, while it is non-monotonic in the degree of product substitutability.
This pattern, intuitively, may be reversed in correspondence of other speciﬁ-
cations of parameter values. Nonetheless, a remarkable feature of the present
model is precisely that it allows for non-negligible regions of the parameter
space where a reversal of fortune obtains as to the welfare performance of
the industry, this being fully consistent with the intuition based upon the
steady state equilibrium strategies of ﬁrms.
115C o n c l u d i n g r e m a r k s
We have characterised the diﬀerent incentives to conduct advertising cam-
paigns under either quantity or price competition in a linear state diﬀerential
game where each ﬁrm’s advertising eﬀort exerts a positive externality to any
other ﬁrm in the market. After showing that a unique saddlepoint equilib-
rium exists in both cases if the marginal cost of advertising is suﬃciently
low, we have focussed on the parameter range where both equilibria do ob-
tain, proving that Bertrand competition entails more intense advertising than
Cournot competition. The intuitive reason is that enhancing the reservation
p r i c ei sm o r er e l e v a n tt oﬁrms when market competition is tougher. Con-
sistently, social welfare can be larger when ﬁrms behave àl aCournot, as
compared to the Bertrand behaviour.
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Persuasive Advertising in Oligopoly: A Linear State Differential Game

Persuasive Advertising in Oligopoly: A Linear State Differential Game

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