This paper reports a new theorem and proof for optimizing the advertising budget. The theorem is that the optimal rate of advertising is equal to gross profit multiplied by advertising elasticity. This does not involve a ratio of elasticities, and so is an advance on the Dorfman-Steiner theorem that has dominated this topic for the last 50 years. The elegant nature of the proof makes it especially suitable for managerial economics textbooks. The simple nature of the theorem means that it is easily adopted, by both large and small businesses, in place of heuristics such as industry advertising to sales ratios. From meta-analysis, the mean advertising elasticity is .11. Therefore, in the absence of any other information, companies should spend 11% of gross profit on advertising

Wright, Malcolm

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MPRA
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A new theorem for optimizing the
advertising budget
Malcolm Wright
Ehrenberg-Bass Institute, University of South Australia
28. April 2008
Online at http://mpra.ub.uni-muenchen.de/10565/
MPRA Paper No. 10565, posted 19. September 2008 10:40 UTC
 1 
A New Theorem for Optimizing the Advertising Budget 
 
 
Malcolm Wright 
Ehrenberg-Bass Institute for Marketing Science 
malcolm.wright@marketingscience.info 
 
 
 
GPO Box 2471, Adelaide 
SA 5001, Australia 
Ph +61-8-8302 7640 
Fax +61-8-8302 0442 
 
Version History 
15 September 2008 
9 May 2008 
28 April 2008 
 
 
Acknowledgements 
 
Thanks to Scott Armstrong, Nick Danenberg, John Dawes, Gerald Goodhardt, Rob Lawson, Cam Rungie, 
Deborah Russell and Byron Sharp for helpful comments on earlier, longer, drafts of this paper. 
 
 
This is a working paper. Feedback is solicited. 
 2 
 A New Theorem for Optimizing the Advertising Budget 
 
Abstract 
This paper reports a new theorem and proof for optimizing the advertising budget. The theorem is 
that the optimal rate of advertising is equal to gross profit multiplied by advertising elasticity. This does 
not involve a ratio of elasticities, and so is an advance on the Dorfman-Steiner theorem that has 
dominated this topic for the last 50 years. The elegant nature of the proof makes it especially suitable for 
managerial economics textbooks. The simple nature of the theorem means that it is easily adopted, by 
both large and small businesses, in place of heuristics such as industry advertising to sales ratios. From 
meta-analysis, the mean advertising elasticity is .11. Therefore, in the absence of any other information, 
companies should spend 11% of gross profit on advertising. 
 
Introduction 
This paper reports a new theorem and proof for optimizing the advertising budget. The theorem is 
extremely simple, and represents an advance over previous advertising optimization methods in both 
marketing and economics. The Dorfman-Steiner theorem (1954) of advertising optimisation uses a linear 
functional form and relies on the advertising to sales ratio being equivalent to the ratio of advertising 
elasticity to price elasticity. Brook (2005) recently demonstrated that, when using an independent linear 
demand specification for advertising and price (or quantity), the Dorfman-Steiner equation reduced to an 
identity, implying that it is of little practical use. The new theorem does not assume linearity, offers a 
result that is more easily interpretable by managers and relaxes the Dorfman-Steiner requirement that 
advertising and price be dependent. Thus, it overcomes Brook’s (2005) objections to the Dorfman-Steiner 
theorem. 
 
Assumptions 
 
(1)  Profit  = G - A 
(1a) G  = (P – C ) Q 
  
Where  P is unit price. 
   C is unit variable cost. 
   Q is quantity demanded, defined below in (2). 
   A is advertising cost. 
 
 3 
(2) Q  = k A e 
 
 Where  k = constant (1 for the purposes of this analysis). 
 e = the exponent for the marketing mix element, in this case advertising (known 
through a simple algebraic proof to be the elasticity of Q to changes in that 
element). 
 
Note the assumption of a multiplicative function in equation (2): while empirical studies of supermarket 
checkout scanner data do not generally identify one functional form as being superior to others in 
marketing mix modelling (e.g. Bolton 1989), multiplicative functions are easily interpretable by managers 
as they have a constant elasticity. They are also widely used: see Danaher, Bonfrer and Dhar (2008) for a 
recent example, in which the multiplicative functional form is justified to avoid aggregation bias. The use 
of this functional form is the major assumption in this work; however the elegance of the proof and 
avoidance of problems arising from linearity lend it some support. 
 
Theorem: Profit is maximised when advertising expenditure equals gross profit times advertising 
elasticity 
 
(Note: This theorem was discovered through simulation. The full simulation method is reported in early 
versions of this working paper.) 
 
Proof 
 
Profit  = G – A    from (1) 
= (P – C) Q – A   by substitution of (1a) 
= (P – C) k A e  – A  by substitution of (2) 
 
d Profit  = e (P – C) k A e - 1 – 1 
d A 
 
To maximise profit we set this differential equal to zero, add one to both sides, and then multiply both 
sides by A. This yields. 
 
A = e (P – C) k A e-1 * A 
 4 
Giving  A = e (P – C) k A e 
That is also  A = e (P – C) Q  from (2) 
Or  A = e G 
Or  A / G = e   to optimise profit.  Q.E.D. 
 
Although this is a proof for advertising, it can equally be applied to any marketing mix element that has a 
positive elasticity, as such an item (for example distribution expenditure) could simply be substituted for 
A in (2). However, the theorem does not address interactions between marketing mix elements. 
 
Discussion  
Rather than use the advertising to sales ratio, and assume a linear dependence between price and 
advertising, this theorem implies that advertising budgets should be set so the advertising/gross profit 
ratio is equal to the advertising elasticity. This is a new budgeting rule for advertising expenditure, simple 
enough to be applied by any business that can calculate a gross profit. From meta-analysis, we know that 
the average advertising elasticity is .11 (Sethuraman and Tellis 1991). Thus, in the absence of any other 
information firms should spend 11% of gross profit on advertising. If gross profit is negative, firms 
should instead close their doors, as they are not covering variable costs, let alone overheads. 
In practice, this is of course overly simplistic. There may be both short and long term advertising 
effects, or advertising effectiveness may vary under different circumstances or with different advertising 
executions. Conceptually, however, many of these issues can be dealt with by adjusting the advertising 
elasticities. Here are four examples. 
First, if we accept the BehaviorScan results that the average advertising elasticities are .26 for 
new products (for three years after launch) and .05 otherwise (Lodish et al. 1995), we can optimize 
advertising under each circumstance. In this case advertising expenditure should be 26% of anticipated 
gross profit for the first three years post-launch, dropping back to a maintenance budget of 5% of gross 
profit thereafter. (While the sample of BehaviorScan experiments is large (n = 389) they are conducted 
using a single method, so their elasticities are less generalizable than the multi-method meta-analysis of 
Sethuraman and Tellis 1991). 
Second, we can apply Danaher et al.’s (2008) work on clutter. They found pure advertising 
elasticities of .16, compared to .076 under conditions of high clutter (note that the average of these values 
is .12, close to that of the meta-analysis). Danaher et al. recommended scheduling advertising to avoid 
competitive interference. It was not clear what would happen everybody followed this advice, as they did 
not calculate a game-theoretic equilibrium. We can extend their recommendations to find a budget 
equilibrium (although not a scheduling equilibrium) without resorting to game theory. The equilibrium 
 5 
advertising budget under conditions of high clutter, in the markets Danaher et al. studied, is simply 7.6% 
of gross profit. 
Third, the theorem can guide the advertising budget during a recession. Businesses are prone to 
slash advertising at such times; the theorem developed in this paper suggests that advertising should be 
cut to the extent that gross profits are expected to be cut, but no further. 
Fourth, what if a creative execution is particularly effective? More effective advertising implies a 
higher elasticity for advertising expenditure. Therefore, effective creative executions should be backed by 
a higher advertising budget – as long as this does not promote earlier wearout of effectiveness. 
While the new theorem is extremely simple, many enhancements are possible to relax 
assumptions and develop more complex models of market behavior. These could include introducing 
game-theoretic competition, splitting out long-term and short-term effects, modelling interactions and 
incorporating tactical elements such as price promotions, media buying decisions and message strategy. 
Actual advertising policies could also be investigated, to see whether theoretically sub-optimal decisions 
lead to lower firm profits. The theorem could be developed to include other advertising measures, such as 
an adstock decay variable, or a competitive share of voice. New dependent variables could be developed 
that are closer to the true objective of most firms: increasing shareholder wealth or market to book ratios. 
The theorem provides an elegant and productive line of enquiry for investigating such issues; they can be 
framed as relaxations of the basic assumptions made in (1) and (2). 
Meanwhile, a small business with limited expertise or resources currently has no sound rule 
available to optimize advertising expenditure. This new theorem provides such a rule. It is so simple that 
the theorem itself can be used as a heuristic by anybody capable of basic arithmetic. In the absence of any 
other information, small businesses should spend 11% of their gross profits on advertising. 
 6 
References 
 
Bolton, Ruth N (1989), "The Robustness of Retail-level Price Elasticity Estimates," Journal of Retailing, 
65 (2), 193-219. 
 
Brook, Stacey, (2005), “Is the Dorfman-Steiner Rule Always Optimal?” The American Economist, 49, 2 
(Fall), 75-77. 
 
Danaher, Peter, Bonfrer, Andre. and Sanjay Dhar, (2008), “The Effect of Competitive Advertising on 
Sales for Packaged Goods,” Journal of Marketing Research, XLV (April), 211-225. 
 
Dorfman, R. & P. Steiner, (1954), "Optimal Advertising and Optimal Quality," American Economic 
Review, 44, 826-836. 
 
Lodish, Leonard M, Magid Abraham, Stuart Kalmenson, Jeanne Livelsberger, Beth Lubetkin, Bruce 
Richardson, and Mary Ellen Stevens (1995), "How T.V. Advertising Works:  A Meta-Analysis of 389 
Real World Split Cable T.V. Advertising Experiments," Journal of Marketing Research, 32 (May), 125-
39. 
 
Sethuraman, Raj and Gerard J Tellis (1991), "An Analysis of the Tradeoff Between Advertising and Price 
Discounting," Journal of Marketing Research, 28 (May), 160-74. 
 
 


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