PAUL D. BERGER*
This article discusses how parameters of a cooperative advertising arrangement are currently decided upon and presents decision models which demonstrate
that quantitative analysis can lead to a better decision when profits are to be
maximized. A real-world application of the methodology is presented.
Vertical Cooperative Advertising Ventures
Many articles have described the state of cooperative
advertising [2, 3, 4, 5, 8]. However, the application of
mathematical methodology to decisions on levels of
cooperative advertising and the resulting dollar split has
not been investigated. This article will show how quantitative analysis can be used to determine the optimal
parameters for vertical cooperative advertising ventures.
COOPERATIVE
ADVERTISING
IN
to maximize:
X)S(A
by choice of X, assuming that the retailer, once told -Y,
will choose A to maximize:
c -\- X)S{A) -
(2)
A.
Using calculus of variations, one can solve for the optimal values of X and A:
PRACTICE
More than two billion dollars is expended annually
for vertical cooperative advertising, and 27% of all
manufacturers use this type of advertising [4]. In practice, cooperative advertising ventures frequently are
specified on a 50-50 basis, with each participant paying
half of the expense. Historically this 50 50 basis seems
to be derived from a sense of fairness rather than systematic profit maximization. This sense of fairness is
misguided because other arrangements based on profit
maximization could benefit both parties.
A DECISION
- r -
(1)
(3J
(k -
r)[S'{A)f
-
[S'{A)Y + S(A)S"(A)
^ 0
and once determining A from {?'), X is such that:
(4)
X = \/S'{A) - ( k - c).
Optimal conditions (3) and (4) arc derived in Appendix
A. However, if (3) yields a negative value for A, the
optimal A = 0, and if (4) yields a negative value for X,
the optimal X ^ 0.
The above conditions hold when SiA) is concave,
otherwise these conditions are necessary but not sufficient. Here one must evaluate PR for each A satisfying
(3) and pick the optimal A as that yielding the largest
P,i . It has been proven that the optimal A must either
be zero or a value on a concave portion of the S{A)
curve [1]; thus any convex portion of the curve can he
ignored as a possible source for the optimal A.
Appendix A also deals with solving (3) for A and
demonstrates that there may be a remarkably simple
analytical solution. In general, (3) can be solved via
many standard computer search routines.
Now the assumption of a deterministic S(A) curve
is eliminated. In the more realistic situation, S{A) is a
random variable with expected sales as a function of ^ ,
ii(A). If one redefines the criterion as maximization of
expected profit, the optimal value of A is found in (3)
by replacing the terms S{A), S'lA], and S"{A), by
fi(A}, ix'{A) and M"(^), respectively.
In the present situation, P,M and Pti represent the
profits now earned by the manufacturer and the re-
MODEL
A manufacturer sells a product to a retailer at a price
fixed by competition, and an advertising allowance is
given to the retailer as a fixed discount per item. Suppose
that the variable cost to the manufacturer is Sl'/unit,
the selling price from manufacturer to retailer is Sc/unit,
and the gross margin of the retailer is S/:/unit. Suppose
that the manufacturer will give SA" unit discount to the
retailer as an advertising allowance. The retailer chooses
A, his advertising level. The manufacturer wishes to
determine the best level of X.
One first assumes that S(A), the sales resulting as a
function of ^, is a deterministic function of ^. When the
sales function is deterministic, the best level of X is that
which maximizes profit. Thus the manufacturer wishes
* Paul D. Berger is Assistant Professor of Management, Boston
University.
309
Journal of Murki'iing Research,
Vol. IX (August I972t, 309-12
310
JOURNAL OP MARKETING RESEARCH, AUGUST 1972
Figure 1
BENEFIT FROM COOPERATION
variance of sales also a function of A, a'^iA). By looking
at past data and using standard techniques of assessment of subjective judgment, management's assessments
of tj.{A) and a"{A) were fitted to the functions:
= IO" (4 - 2e-•*'«') standard cases'
= 100
This assessment was the weak link in the procedure.
Because variance of sales increases as expected sales
increase, it was easier to assess variability related to
expected sales, as opposed to directly measuring variability of sales. Management's uncertainty about ti{A)
is reflected in a large assessed variance. One can graph
n{A) as in Figure 2. Not only did the variance of sales
vary with advertising level, but also the reasonable
range of sales. Further assessment led to a range of
sales as a function of advertising, and the following
values were examples:
RETAILER'S PROFIT
tailer, who each choose one parameter (the manufacturer choosing X, the retailer, A). If the two parties
cooperate (move away from their individual optimal
values), -dwd jointly set X and A, they can attain a total
profit in the system, T, higher than the sum of PM and
Pff . As shown in Figure 1, parties must now choose a
point along the total profit line. Clearly, the agreedupon point must be between points a and b for any
point between these represents a situation in which both
parties are better off by cooperating. At point d, the
retailer will refuse to cooperate, essentially forcing the
situation back to point c. At point e, the manufacturer
will refuse to cooperate, and thus the result will be
point c. Details of this analysis are in Appendix B.
APPLICATION
To apply these basic ideas, a soft drink manufacturer's
optimal strategy will be determined and compared with
current practice. This manufacturer deals with the only
independent bottler in a given area distributing the
manufacturer's product to retail outlets. The bottler
signs a contract specifying the price at which the concentrate is supplied and the fraction of the advertising
expense that the manufacturer will pay (currently,
50%). The bottler is then free to choose the advertising
level; the amount o^ advertising done by the manufacturer is assumed to be already pledged and thus unaffected by this contract. The manufacturer controls
only the price and fraction of the bottler's advertising
expenses that he pays.
The mathematical model has sales as a random variable, with expected sales a function of A, p.{A), and
Expected
.sales
One standard
deviation
2 X 10"
3 X 10"
4 X lOi^
.53 X 10'-'
.72 X 10''
.90 X 10"
Range of sales
.7 X 10^ -^ 3.,30 X
1.24 X 10^ -> 4.,76 X
1.81 A iU -^ 6,,19 X
Next, k was specified at .25 per standard case and V
at .03 per standard case, where k and r were the gross
margin of the bottler and the variable cost of the
manufacturer.
Further, the manufacturer was assumed to be linear
in utility (an expected value decision maker), while the
utility function of the bottler was assessed. Using
standard techniques for quantification of preference
[7], the representative utility function for the bottler
was fitted to the curve U{X) = —l/X for positive
dollar amounts X. The bottler is averse to risk and would
pay a risk premium of I % of assets to avoid a gamble of
50% chance of gaining \0V<, of its assets and 50% chance
of losing 10% of its assets.
The structure of the actual situation suggested two
constraints:
1. Negotiating constraint: the bottler's expected utility
must be at least some value U{SP).
2. Operating constraint: if the bottler's actual profit is
below some value of SL, the manufacturer must make
up the difference.
It was decided initially to have P = S200,000 and L =
S50,000.
The optimal solution was found to be: c = S.178,
/ ^ .75, and A = S160,000. That is, the manufacturer
charges S.178 per standard case, pays 75% of the bottler's advertising expense, and with these input parameters, the bottler will choose an advertising level of
S160,000. At this solution, the manufacturer has an
expected profit of S412,000 and the bottler has a cash
equivalent of $200,000. For a decision maker averse to
' A standard case is equivalent to 24 X 8 = 192 oz. of soft drink.
VERTICAL COOPERATIVE ADVERTISING VENTURES
311
risk, the cash equivalent is less than the expected value
of the gamble. Details of the analysis are in [1].
Figure 2
EXPECTED SALES AS A FUNCTION OF ADVERTISING
A COMPARISON
OF
SOLUTIONS
The arbitrary practice of setting/at .50 can be quite
costly to the manufacturer. For example, using no
quantitative analysis, l e t / = .50 and c = $A4- (halfway
between r and k). The manufacturer would receive an
expected profit of S316,000. If partial quantitative
analysis is used, let / = .50, but c be set to maximize
expected profit. This would yield c = S.166 and an
expected profit to the manufacturer of S398,OOO. Thus
the manufacturer loses an expected profit of at least
814,000 and perhaps up to vS96,000. The higher profit
could be realized at no cost to the bottler. With complete
cooperation and an overall system maximization at
A — SI50,000, the manufacturer would receive an
expected profit of S433,OOO if the bottler remains as well
off as before. (There is no actual c or / here, as the
total dollars in the system are simply divided appropriately after the fact.) In practice, however, complete
cooperation may not be operational, and partial cooperation may be more fruitful. This takes the form of
increasing L up from ^S50,000. That is, the manufacturer
guarantees the bottler a minimum actual profit of L.
As L increases from S50,000 to a maximum of S200,000,
the latter essentially being full cooperation, the following results are obtained. For all cases but manufacturer's expected profit of 8412,000, the bottler would
probably receive some of the profit for cooperation:
S[A)
4 X 10'
2x 10
parameters (c, / , A) are singularly or simultaneously
varied from optimality.
APPENDIX
A
To maximize:
(5)
p^ = {c - r ~
X)S(A)
given that once told X, the retailer will choose A to
maximize:
(6)
p^ ^ (k ^ c -\- X)S{A) -
A.
Then from (6), the retailer will choose A such that:
(7)
d P ^ / d A ^ (k ~ c + X ) S ' { A )
- 1 = 0 ,
or:
f
$ 50,000
100,000
125,000
150,000
175,000
198,000
2a). 000
Manufacturer's
expected profit
$412,000
412,000
416,000
420,000
425,000
432,000
433,000
Bottler's cash
equivalent
$200,000
2a), 000
2{X),000
200,000
200,000
200,000
200,000
In summary, this solution yields an expected profit
to the manufacturer significantly higher than current
practice. Increasing the degree of cooperation will
yield even a higher expected profit, while complete
cooperation yields the highest expected profit.
S'iA) ^ l/(k -
(8)
Forming the Lagrangian function from (5) and the
constraint (7):
(9)
L=
{c~r
The major implication of this study is that quantitative analysis may yield significantly better solutions to
cooperative advertising decisions than those solutions
currently in practice. Other types of cooperative arrangements, for example one in which the advertising
subsidy is a fixed dollar amount per item purchased,
may be analyzed by similar methods.
The sensitivity of the solution to the assessment of
the n{A), (T^(A) functions can be examined in detail
[1]. Also, the profit figures can be examined as various
X)S{A)
+ \((k
~ c -h X)S'(A)
-
I).
Taking OL/dA and dL/dX, and setting them equal to
zero:
00)
(c - r - X)S'{A) -\-\{k
- c + X)S''(A)
= 0
and
(11)
CONCLUSIONS
~
-S{A)
+ \S'(A) - 0.
Solving for X in (11) and plugging it into (10), and
solving together with (7) yields:
(12) (k - T)[S\A)f
- [S'{A)]' + S(A)S"(A) ^ 0,
as the equation that the optimal value of A must
satisfy. Then X is found from (8).
Solving (U) may be analytically formidable, but
under certain circumstances the solution may be remarkably simple. Suppose S{A) is a concave exponential function:
(13)
S(A) =
Ni~
Ni>
N,>
0, m > 0
JOURNAL OF MARKETING RESEARCH, AUGUST 1972
312
^. = ((c - r -
pictured below:
X)S{A))
(16)
S(A)
which equals:
+ ((/c - c + X)S{A) -
A),
(17)
Taking the first and second derivative of S(A), let
R = f-'"^, and one can solve (11) analytically and find
that R is the positive root of the quadratic equation
{(k - r)Nrm + N-,)R' - N^R - A'I = 0, and from
the definition of R, A ^ -\og R/m. If i? > 1, the
optima! ^ - 0. Then find X = e'^^/mNi - {k - c)
or 0, whichever is larger.
Now suppose that S{A) is a random variable with
expected sales as a function of advertising, t^i^)- To
maximize expected profit, (5) is replaced by:
= E((c - r (14)
= {c-T
X)S{A))
-
and (6) is replaced by:
E(P,) = E{{k - c + X)SiA) - A)
(15)
^ {k-
c + X)ii{A) -
A.
Here p.{A) has replaced S{A). It follows that the
optimal conditions (12) and (8) for determining A and
X are the same here except n{A), fi'{A), and fi"{A) replace S{A), S'{A), and S"{A), respectively. Expected
sales as a function of advertising form the basic input.
APPENDIX
B
The choice of A can maximize the sum of the profits
and thus total dollars in the system:
{k - Y)S{A)
- that
A. S'{A) = \J{k —
The above is maximized
at A such
r ) , a value different from that chosen when total profit
is not of concern; but each party essentially performs a
suboptimization. If the parties cooperate and set A
at the amount that maximizes total dollars, extra
revenue is generated. How this extra revenue is shared
must be worked out between the two parties involved.
With sales as a random variable, the total expected
profit is E{PM + PR) which equals (fc - V)\i.{A) - A,
and total expected profit is larger with A such that
\i{A) ^ l/(fc — r ) than the sum of the previously
expected profits. Similar results are obtained by examining E(PM) and E(PR); the extra expected profit is
shared (after negotiation) by jointly choosing an X
such that each party has higher expected profit than
without cooperation.
REFERENCES
1. Berger, I'aul D. A Decision Theoretic Approach to Cooperative
Adverlising," unpublished doctoral dissertation, Massachusetts
Instituie of Technology, 1969.
2. Burton, Philip Ward. Retail Advertising for the Small Store.
Englewood Clilfs, NJ.: Prentice-Hall, 1951.
3. Hutciiins. Musher. Cooperative Advertising The Way to Make
it Pay. New York: Ronald Press, 1953.
4. Lockley, Lawrence C. Vertical Cooperative Advertising. New
York: McGraw-Hill, 1957.
5. Lyon, Leveretl. Advertising Allowances. Washington, D.C:
The Brookings Institute, 1932.
6. Luce, Robert and Howard Raiffa. Games and Decisions. New
York: John Wiley & Sons, 1957.
7. PraU, John, Howard Raiffa, and Robert Schlaifer. Jntroduciion
to Slalislieal Deeision Theory. New York: McGraw-Hill, 1965.
8. Salisbury, Philip. Cooperative Adverlising—An Effeetive Merehanciisiitg Tool or a Disguised Racket? Sales T'raining File
Number 4, S-52G, Newspaper Advertising Executives Association, 1952.