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Co-op Advertising and Pricing Models
In a Manufacturer-Retailer Supply Chain*
Alexandre Neyret, Jinxing Xie
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, CHINA
Abstract: This paper is concerned with co-op advertising
strategies and equilibrium pricing in a two-members channel
of distribution. Four different models are discussed which
are based on three non-cooperative game (like the leaderfollower game and the Nash game) and one cooperative
game. We identify optimal pricing and optimal advertising
strategies (with co-op advertising policies) for both firms
mostly analytically but we have to resort to numerical
simulation in one case. Comparisons are made about the
various outcomes in every case and especially profits. This
leads to consider more specifically the cooperation case in
which profits are the highest for both retailer and
manufacturer and how they share the extra join profit
achieved by moving to cooperation.
Keywords: co-op advertising; pricing; supply chains; game
theory; cooperation; bargaining problem
I. Introduction
Vertical co-op advertising is an interactive relationship
between a manufacturer and a retailer in which the retailer
initiates and implements a local advertisement and the
manufacturer pays part of the cost. It is often used in
consumer goods industry and plays a significant role in
market strategy for many companies. The main reason for a
manufacturer to use co-op advertising is to strengthen the
image of the brand and to motivate immediate sales at the
retail level. The manufacturer’s national advertising is
intended to influence potential consumers to consider its
brand and to help develop brand knowledge and preference
whereas retailer’s local advertising is to stimulate
consumer’s buying behavior. Most studies to date on vertical
co-op have focused on a relationship where the manufacturer
is the leader and the retailer is a follower, which implies that
the manufacturer dominates the retailer. But today’s retail
market for most consumer goods in dominated by large
retail chains who retain equal or more power than most
manufacturers. Hence, recently Jorgensen, Sigue and
Zaccour [10] used differential game theory to study a two
member channel in which a manufacturer and an exclusive
retailer can make advertising expenditures that have both
short and long term impact on the retailer’s sales and the
manufacturer can also support retailer’s advertising efforts
through a co-op advertising program. Two scenarios were
considered: the manufacturer leader versus the retailer
*
follower game and the non-cooperative Nash game. Their
results state that supporting both types of retailers’ ad
provides more profit for both channel members. Lately,
Huang and Li [3] and [4] explored the role of vertical co-op
advertising efficiency with respects to transactions between
a manufacturer and a retailer through brand name
investments, local advertising expenditures and sharing rules
of advertising expenses. Three co-op advertising models are
discussed which are based on two-non cooperative games
(the manufacturer is the leader and Nash) and one
cooperative game, which can provide the highest profits for
both manufacturer and retailer.
Nevertheless, though a fundamental task for supply-chain
managers is to determine wholesale and retail prices (see for
example Choi [2], [3] and Ingene and Parry [5], [6], [7]),
both studies assumed that the market demand is only
influenced by the advertising level and not in any way by the
retail price. In fact, the literature dealing with both pricing
and advertising strategies at the same time is very sparse.
See for example, Jorgensen and Zaccour [9] who only
studied conflict and coordination case in a two-member
channel of distribution through pricing and advertising but
without any co-op advertising policy. That’s why this paper
focuses on pricing and (co-op) advertising strategies in a
manufacturer/retailer supply chain through all the possible
(four) different cases: the manufacturer is the leader, the
retailer is the leader, the conflict Nash case and the
cooperation case.
The paper proceeds as follows. Section II presents our
model and our assumptions. Section III identifies analytical
equilibrium solutions for all the cases except the one where
the manufacturer is the leader. That’s why we have to resort
to numerical simulation in Section IV, which also compares
the outcomes under the different scenarios. Section V
summarizes the results.
II. Model and Assumptions
We consider a single manufacturer-single retailer channel in
which the retailer sells only the manufacturer brand within
the product class. Decision variables of the channel
members are their advertising efforts, their prices
(manufacturer price and retail price) and the co-op
advertising reimbursement policy. Denote by a and q
respectively the retailer’s local advertising level and the
manufacturer’s national brand name investments. The
consumer demand function V depends on the retail price pr
Supported by China 863 Plan Project No. 2001AA414230 and NSFC Project No. 69904007.
and the advertising levels a and q in a multiplicatively
separable way (see [9]), V (a, q, pr )  g ( pr )S (a, q ) where
g ( pr ) is linearly decreasing and S (a, q) is the same
function as in [3] to model in a static way advertising effects
on sales: V (a, q, p r )  g ( p r ) S (a, q)  (  p r )( A 
where
 ,  , B,  , 
B (1)
)
a q

are positive constants, A>0 is the sales
saturate asymptote. We denote by
t , pm respectively: the
fraction of total local advertising expenditures, which the
manufacturer agrees to share with the retailer (that is to say
the manufacturer’s co-op advertising reimbursement policy)
and the manufacturer transfer price to the retailer. Then if
we agree to denote by c=const>0 the manufacturer unit
production cost and by d=const>0 the retailer unit
production cost then the manufacturer’s, retailer’s and
system’s profits are as follows:
B
 m  ( pm  c)(   pr )( A    )  ta  q
a q
B
)  (1  t )a
a  q
B
 m  r  ( pr  c  d )(   pr )( A    )  a  q
a q
Now to handle the problem in a more convenient way, it
could be shown that with an appropriate change of variables
we can have:
1
 m  pm (1  pr )( A*    )  ta  q
a q
(2)
1
 r  ( pr  pm )(1  pr )( A*    )  (1  t )a
a q
1
 m  r  pr (1  pr )( A*    )  a  q
a q
 r  ( pr  pm  d )(   pr )( A 
obvious that the manufacturer’s profit is increasing with p m.
But pm can’t be equal to 1 otherwise there is no profit at all
for both manufacturer and retailer… We have to add one
more hypothesis: if the manufacturer and the retailer are in a
symmetric relationship then we could assume that their
respective margins are equal that is: ( pr  pm )  pm (5)
A Nash symmetric equilibrium is obtained by
simultaneously solving the following three first-order
conditions for the manufacturer and the retailer plus
condition (5) (for the details of the calculus see [12]):
pmN  1/ 3, prN  2 / 3
a N  [( /  )  / 9]1/(  1) , q N   /  a N
tN  0
3.2 Stackelberg Retailer Equilibrium
We now model the relationship between the manufacturer
and the retailer as a sequential no cooperative game with the
retailer as the leader and the manufacturer as the follower.
The solution of this game is called Stackelberg (Retailer)
equilibrium. The retailer, as the leader, first declares the
level of local advertising expenditures that he is willing to
pay and set the retail price for the product. The manufacturer,
as the follower, then set its own brand name investments
level and its manufacturer price.
In order to determine Stackelberg retailer equilibrium, we
first solve the manufacturer optimal problem (3) by finding
optimal t, q and pm that is t  t r , pm  pmr , q  q
S
S
Sr
.
Once again we get t=0. As above, the manufacturer’s profit
is increasing with pm but, especially in a follower position,
it’s hard to set the manufacturer’s price as big as desired and
the natural constraint is that the manufacturer margin should
not be bigger than the retailer’s one so we add (5) again.
Next, the optimal values of a  a r , pr  pr r (6) are
S
S
determined by maximizing the retailer’s profit subject to the
constraints imposed by equations (6). That is to say:
III. The four scenarios
Max  r with t  0, pm  pmSr , q  q sr
3.1 Nash Equilibrium
0 a ,0 pr 1
In this section, we assume a symmetric relationship between
the manufacturer and the retailer who simultaneously and no
cooperatively try to maximize their own profits. It is called a
simultaneous move game and the solution provided by this
structure is called Nash equilibrium.
Hence the manufacturer’s optimal problem and the
retailer’s optimal problem are respectively:
(3)
Max
m
0t 1,0 q ,0 pm 1
Max  r
pmSr  1/ 4, prSr  1/ 2
a Sr  [(


 (  1) S
)
]1/(  1) , q S 
a
 (  1) 8(  1)

r
r
t Sr  0
3.3 Stackeleberg Manufacturer equilibrium
and
0 a ,0 pr 1
We finally find (for details of the calculus see [12]):
(4)
Now, it’s obvious that the optimal value of t is zero
because of its negative coefficient in the objective. It’s also
In this part, we model the relationship between the
manufacturer and the retailer as a sequential no cooperative
game in the same way than before but with the manufacturer
as the leader and the retailer as the follower. The solution of
this game is hence also called Stackelberg (manufacturer)
equilibrium. We shall proceed exactly as above.
In order to determine Stackelberg manufacturer
equilibrium, we first solve the retailer optimal problem (4)
by finding optimal values a
values of
Sm
, prSm . Next, the optimal
q, pm and t are determined by maximizing the
manufacturer’s profit subject to a  a
We can get expressions of a
Sm
Sm
Sm
r
, p
, pr  prSm .
,q
Sm
, t
Sm
function
does also implicitly define the constant join extra-profit 
such as:
   comr   mmaxr  ( com   mmax )  ( cor   rmax )   m   r  0
that manufacturer and retailer achieved by moving to a
cooperation game equilibrium and that they will have to
share. Of course the more the manufacturer gets, the less the
retailer and vice versa. So they will bargain over ( pm , t )
with boundaries defined by inequalities (7), (8),
and 0  pm  p
co
r
0  t 1
(see Fig. 1) .
S
of pmm but the problem is that in that case equation doesn’t
S
lead to a closed-form result for pmm so we have to use
t
numerical simulation: that’s the point of the fourth part (for
details about equations see [12]).
  m
  r
 
3.4 Cooperation
In the previous three sections, we analyzed two sequential
move and one simultaneous move no cooperative game
structures. In this part, we focus on a cooperative game
structure that is to say manufacturer and retailer both agree
to take decisions in order to maximize the total system profit
(joint payoff maximization).
We hence have the following optimization problem:
Max
0 pr 1,0 a ,0 q
pr , a and q . If
pr , a, q are respectively equal to prco , a co , q co that is (see
calculation details in [12])
p  1/ 2
co
r
a co  [( /  )  / 4]1/(  1)
q co   /  a co
Then the system profit is maximized with t
and pm free
to take any value between 0 and 1 (provided of course that
pm  prco ). But obviously, the manufacturer’s profit and
the retailer’s profit are not independent of
t and pm . More
precisely, neither the manufacturer nor the retailer would be
willing to maximize the system profit and accept fewer
profits with cooperation than with no cooperation. We call a
co
 r   rmax
0
qco  max
m
*
A
prco 
aco  max
r
A*
pm
Fig. 1 Feasible solutions and the bargain problem
 m r
We notice that it depends only on
co
Iso-profit line
 m   max
m
co
solution ( pr , a , q , pm , t ) feasible if and only if it
satisfies both inequalities (7) and (8):
 com   m ( prco , a co , q co , pm , t )  max( mSm ,  mSr ,  mN )   mmax (7)
 cor   r ( prco , aco , qco , pm , t )  max( rSm ,  rSr ,  rN )   rmax (8)
It could be shown (see [12]) that feasible solutions do
exist and both manufacturer and retailer are willing to
cooperate. But we should notice that inequalities (7) and (8)
IV. Comparison between the different cases
4.1 Numerical Assumptions
In the previous part, we failed to analytically solve the
Stackelberg Manufacturer case. In order to solve it
numerically we need an estimation of the following
parameters:  ,  and A*. Our four assumptions are:

In most cases, studies found out that the average
advertising expenditures level represent five percent of the
net company sales so q  a  5 /100  m  r
net sales

If advertising effects are modeled by equation (1)
then we can reasonably argue that good estimations for A
 
 
and ( B / a q ) are A  2 and B / a q  0.5 that is to
say the maximum advertising policy can’t increase the sales
by more than 100% and the average usual advertising level
already makes an 50% up.

According to our previous results a good guess for
pr is pr  1/ 2 .
Those assumptions allow us to have a general idea about
the value of A*. With regards to  ,  , a good range of value
could be from 0,1 to 3 to keep as much generality as
possible.
We now can calculate the manufacturer price for different
values of  ,  in the manufacturer Stackeleberg case.
Simulation with Mat lab 6.1 show (see details in [12]) that
the manufacturer price is quite stable with typical values
ranging from 0.45 to 0.5 whereas the manufacturer
participates in the local advertising expenditures of its
retailer (that is to say t m  0 ) only if we approximately
have   1 . This result is compatible with Proposition 1 of
Huang and Li [3]. More exactly, as  increases from 0 to
S
1,the participation t
Sm
strictly decreases from 0.5 to 0.
4.2 Comparison
What’s the influence of the scenario on the retail and
manufacturer price, the advertising expenditures and the
profits?
If we agree to consider the manufacturer price in the
Stackelberg manufacturer case approximately equal to 0.47,
we have, according to our previous analytical results and our
numerical simulation, the following interesting table (Fig 2.):
pr
Co-op
Sr
Nash
Sm
0.5
0.5
0.66
0.73
?
0.125
0.111
0.07
?
0.25
0.33
0.47
?
0.125
0.111
0.1245
Retailer margin
( pr  pm )(1  pr )
pm
Manufacturer margin
pm (1  pr )
Fig 2. Influence of the four scenarios on prices and margins
We notice that the highest retail prices occur at the
Stackelberg Manufacturer (Sm) and Nash equilibrium
whereas the lowest retail prices occur when both retailer and
manufacturer cooperate or when the retailer is the leader (S r).
The high value of the retail price in the conflict case versus
the low value of the retail price in the coordination case is
well known in the literature. Now, if the manufacturer is the
leader, he will abuse of its position to impose a very high
price to its retailer who have consequently no choice but to
mark a high retail price and yet to obtain a miserable margin.
On the other hand, if the retailer is the leader, he marks a
low retail price to get a high margin and hence induce a
small manufacturer price; the high manufacturer’s margin is
due to our hypothesis on the equality of retailer and
manufacturer in this case.
In the same way, we could get a comparison between the
different levels of advertising expenditures in the different
models. Concerning the local advertising expenditure of the
retailer, it is the highest in the cooperation case and the
smallest in the Stackelberg retailer case. On the other hand,
the manufacturer brand name investment is the highest in the
Stackelberg retailer and cooperation case and the smallest in
the Stackelberg manufacturer case. Indeed, if the retailer is
the leader, he will manage to make the manufacturer pay the
larger amount of the total advertising expense with a high
brand name investment and a comparatively small level of
local advertising to increase its own benefit. The same
explanation stands for the manufacturer being the leader. In
the same way, in a conflict situation, we have a free-ride
problem where both manufacturer and retailer has the
temptation to invest less in advertising and to benefit from
the investment of the other whereas this problem is solved in
the cooperation case.
But of course, the most interesting results deal with profits.
There are two striking results. The first is that the
manufacturer always prefers to be the follower of the retailer
than to be in a conflict situation with it!! The second
surprising result is: for low values of  , (  0.6) , the
manufacturer even prefers to be the retailer’s follower rather
than to be the leader (though numerical simulation prove the
difference of profits to be very small)!! There are mainly
two explanations to those facts. First, when the retailer is the
leader, we have assumed (5) that leads to a quite a fair
bargain for the manufacturer since though being the
retailer’s follower it gains the same relatively high margin
(see Fig 2.). The second reason is closely linked to the
structure of the stackelberg manufacturer equilibrium itself.
Indeed, we have a manufacturer “margin” equal to 0.125 in
the stackelbeg retailer case, which drops to 0.1245 in the
manufacturer stackelberg case. So we have to reach a certain
value of delta for which the loss of profit due to expensive
brand name investment in the stackelberg retailer case
compensates the slightly higher manufacturer margin.
Anyway, for relatively high  , we have the expected result
that the manufacturer always prefers to be the leader.
With regards to the retailer’s profits we have the
following very simple result: for every couple ( ,  ) , the
retailer always prefers to be the leader and if it can’t, it
always prefers to be in conflict with the manufacturer rather
than to be his follower. Indeed, in Fig. 2 we see that the
retailer’s margin in the Stackelberg manufacturer
equilibrium is so small compared to the retailer’s margin in
the Nash case, that it could never be compensated by the
manufacturer’s advertising allowance. The only way for the
retailer to accept would be to have a relatively low margin
compared to the manufacturer’s one as showed in Huang
and Li [3], but our model assumes equality between both
margins.
But we have already shown the most classical result that
is co-operation guarantees the highest profits for both
retailer and manufacturer with regards to every other
situation. We know from above that if the retailer has the
possibility to be the leader, it will and the manufacturer will
accept to be his follower. On the other hand, if the
manufacturer has the possibility to be the leader, it will but
the retailer won’t accept and we have a conflict situation
(Nash). In the first case, numerical simulation reveals a join
extra-profit varying from 1% to 3% (it depends on  ,  ) of
the total system profit. And in the second case we have a
join extra-profit varying from 14% to 16%. If by any chance
the manufacturer manage to make a follower of the retailer
then moving to cooperation generates a join extra-profit of
34% to 39% of the total system profit.
But how to share that extra-profit? We have a well-known
bargain problem (see above the end of Section 3.4). The
approach that could be used to solve this issue is the Nash
bargaining model (see Nash in [11]) where the bargaining
outcome is obtained by maximizing the product of
individual marginal utilities (here  m (t , pm ) and
 r (t, pm ) ) over the feasible solution area (see Fig. 1).
Huang and Li [3] used that model to determine the
advertising allowance t and find out that the more riskaverse a member is, the lower his share of the profit is. Our
bargaining problem is also solved using that method in [12].
V. Conclusion
This paper attempts to identify the optimal pricing and
advertising (included co-op advertising) strategies in four
classical types of relationship between a manufacturer and a
retailer using the game theory modeling.
We find again some classical results like the advertising
expenditures of both members is generally higher in a
coordinated situation that in a non-coordinated situation but
this doesn’t come at the expense of consumers, since the
retail price is the lowest in the coordinated case (and the
highest in the case where the manufacturer is the leader…).
In the same way, the leader will always manage to make the
follower invest more in advertising that it usually does in
other case while the leader itself pays a smaller amount.
With regards to profits, the retailer always prefers to be the
leader whereas the manufacturer needs a certain influence of
its brand name investment on sales to choose to be the leader
otherwise it always gains to be the retailer’s follower...
Furthermore, in our model, the retailer always prefers to be
in conflict with the manufacturer rather than to be its
follower though it may get advertising allowance. This may
sounds also surprising but our model doesn’t take into
account others factors than profit that may lead the retailer to
accept to be the manufacturer’s follower like the competition
between retailers for example or the manufacturer’s brand
fame…Anyway, we prove that coordination always
guarantees higher profits for both manufacturer and retailer
than in any other case. But they have to bargain over the
manufacturer price and the advertising allowance to share
the system profit gain achieved by moving to cooperation.
The approach that could be used to solve this issue this
bargaining problem is the Nash bargaining model.
Other interesting issues would be to relax the classical
two channel members situation to a three channel members
situation (either two manufacturers and one retailer or two
retailers and one manufacturer) to move one step towards
the understanding of the role of competition and cooperation.
This kind of study has only been done in the field of pricing
(see Choi [2], [3] and Ingene and Parry [5], [6], [7]). Next,
our model suffers from some limitations due to the choice of
its demand function. Changing it may yield some interesting
results.
References
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pricing when retailers compete, Marketing Letters, Vol. 9, No.1, 6577, 1998.
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up to be? , Journal of Retailing, Vol. 76, No.4, 511-547, 2000.
[9] Jorgensen S. and Zaccour G., Equilibrium pricing and advertising
strategies in a marketing channel, Journal of Optimization theory and
applications, Vol. 102, No.1, 111-125, 1999.
[10] Jorgensen S., Zaccour G. and Sigue S.P. Dynamic cooperative
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[12] Neyret A., Xie J., Co-op advertising models and pricing models in a
manufacturer-retailer supply chain, working paper.
Alexandre Neyret received the B.Sci. degree in Mathematics and Physics
from Ecole Centrale Paris in 2000. He is currently an M.Sci. student in
Operations Research Group at the Department of Mathematical Sciences,
Tsinghua University.
Jinxing Xie is a professor of operations research at Tsinghua University.
He received a B.S. in applied mathematics in 1988 and a Ph.D. in
computational mathematics in 1995 from Tsinghua University. He has
published in European Journal of Operational Research, Operations
Research Letters, International Journal of Production Research,
International Journal of Production Economics, Production and Operations
Management, Production Planning and Control, Decision Sciences, Supply
Chain Management, Computers and Mathematics with Applications, and
other journals. His current research interests include Supply Chain
Management, Production and Inventory Control Systems, Machine
Scheduling and Sequencing, Mathematical Modeling and Optimization.