Optimal Advertising with Persistent Demand

International Journal of Business Marketing and Management (IJBMM)
Volume 6 Issue 10 October 2021, P.P. 25-32
ISSN: 2456-4559
www.ijbmm.com
International Journal of Business Marketing and Management (IJBMM) Page 25
Optimal Advertising with Persistent Demand
Daniel Son
Academy of Culinary Arts and Hospitality Administration, Bergen County Academies, United States
Abstract: How much of a firm’s budget should it devote to advertising? If it devotes too little money, the
marginal cost of an additional advertising dollar will be below the gain from additional customers, causing it to
miss out on a potentially substantial increase in revenue. If it devotes too much money, the firm ceases to reach
new customers, and wastes money as a result. We study a model with stochastic demand shocks, where a firm
aims to equalize marginal expenditures on advertising with expected gains from demand. We show that a firm
that tracks previous patterns in demand significantly outperforms a firm that fails to do so. We also develop a
model of persistent shocks and menu costs that can be generalized to further fields for future research.
Keywords: Advertising, Decision Theory, Economics, Marketing, Optimization JEL Classification Marketing
and Advertising.
I. Introduction
Advertising is an enormous industry, comprising 19% of US GDP last year [1]. In the modern era,
rather than simply posting a newspaper advertisement, advertising campaigns are multifaceted. Techniques for
advertising range from Google Ads to directed mailers. Firms utilize a myriad of communication sources in
order to reach their target audience in the most effective manner. However, just as important as the question of
how to advertise, is the question of how much to spend on advertising. The US Small Business Association
recommends that companies allocate
“7 to 8 percent of your [their] gross revenue for marketing and advertising if you’re [they’re] doing less
than $5 million a year in sales and your [their] net profit margin – after all expenses – is in the 10 percent to 12
percent range” [2]. However, what this advice fails to take into account is the firm’s expected future revenue,
using current revenue instead as a myopic prediction for the future. Indeed, a firm seeing rapid growth in sales
would under spend should it use the above guideline with accounting for demand growth. Tailoring the level of
advertising to demand is quite important. In this paper, we examine the real-world problem of a firm that aims to
calibrate its level of advertising optimally. Overshooting involves excess expenditures that may fail to reach
new convertible audiences, while undershooting leaves money on the table in the form of missed sales.
Achieving the optimal level of advertising is essential to firms, as it allows them to achieve maximum efficiency
when choosing how much to invest in advertising. We aim to devise a strategy that can be applied consistently
over time to maximize firm profit from advertising. In order to do so, we develop a model of a firm facing
stochastic demand. The firm seeks to optimize how often and to what extent it should change its level of
advertising. However, changing its advertising scheme involves incurring a logistical cost each time the firm
chooses to do so. Demand is modeled through a Markovian process and follows a persistent pattern. That is, if
demand went up in the previous period, it is more likely to go up again the following period. Similarly, if
demand went down in the previous period, it is more likely to go down again the following period. The firm
responds to the level of demand by changing the level of advertising to match its expectation of future demand
trends. We characterize the difference between the firm’s profit maximizing strategy and a na¨ıve myopic
strategy. We show that it is optimal for a firm to overshoot or undershoot when setting advertising levels at the
end of a given period, conditional on the previous shift in demand. This follows due to the increased alignment
between advertising and expected demand, given the persistence present. Thus, the optimal mechanism tracks
patterns in demand and use those predictions to selectively set the level of advertising.
II. Literature Review
The study of environments with costly adjustment began with [3]. In their paper, Diamond examines
the behavior of consumers when firms are aware of consumer demand. We differ here by considering the
problem where the firm faces incomplete information and must commit ahead of time to changing the level of

advertising. [4] study a scenario where inflation continually increases, causing posted firm prices to decrease
automatically over time. For contrast, we study a problem where demand is imperfectly persistent,
symmetrically positive or negative.
The importance of sticky pricing was shown in [5], which uses sticky prices to generate a model of the
business cycle.
A great deal of empirical work has also studied the problem of price adjustment. [6] develop a model
that numerically fits several empirical regularities. For instance, they find that price reductions are larger but
less frequent than price increases. In a related paper, [7] test their asymmetrical predictions on manufacturing
data, finding support for a menu cost model. [8] examine the practical costs behind price adjustment,
importantly the practical costs don’t simply equal physical menu costs, but rather include costs managerial costs.
This motivates our decision to model advertising adjustment as a price adjustment problem in this paper.
III. Discrete Model
We study the decision problem of a firm that aims to invest in the correct level of advertising, in order
to maximize profits. Time is discrete, t ∈ {1, 2, . . . , T}. The firm aims to match the level of advertising, xt , to
demand, rt . Demand follows a random Markov process, wherein each period it may increase or decrease. Each
period, demand moves up or down by k. If it increased last period, the probability of increase this period is p ≥
1/2, otherwise the probability of increase is 1 − p. That is, the demand process features binary persistence. The
initial level of demand is given by r0.
The firm observes the current level of demand in each period. At the beginning of time, the firm
chooses x0, the initial level of advertising. At the end of any period t, the firm may choose xt ∈ R+ and pay a
cost c > 0 to change the level of advertising to that xt . In an arbitrary period t, the firm’s current level of
advertising x(t) is given by the largest τ < t such that xτ 6= xτ−1. Namely, τ was the last period that the
advertising was changed in, not including the current period. The firm’s per period profit is given by u(t) =
f(x(t), rt), where f is decreasing in the distance of its two arguments. It may be helpful to think of f as the losses
from incorrect advertising, rather than the actual level of profit. Note, the firm earns profit before it can change
the level of advertising. Let ti denote the ith time that the firm has changed its level of advertising, and let I be
the collection of all such times. Then, firm’s overall profit at time t is the discounted sum of per period utility:
We specialize to the quadratic loss case, but conjecture that our results generalize to the case where f is any
concave function:
Assumption 1.
IV. Results
IV.I. Two Period Setup
To illustrate our main point, we focus on the case wherein a firm changes its advertising over two
periods, that is, T = 2. For the firm to maximize profit by investing in the most optimal amount of advertising,
portrayed by x ∗ 0 , it must initially find the most optimal level of advertisement at its starting point, or t = 0.
The firm must use an arbitrary number and plug it into the expected utility function, noted by u0. Then, the firm
must solve for x ∗ 0 by taking the derivative of said function and setting it equal to zero, as this would find the
maximum of the function. This would give the firm the most optimal amount of advertising to start off with.
The optimal strategy ensures that the firm will choose a level of advertising that will respond in the best way
possible to the given circumstances. That is, the choice of the level of advertising is not restricted by demand
and advertising levels. In particular, if it is optimal to increase advertising by a certain amount under a fixed
demand, then it is optimal to increase it to the same relative level compared to a different demand. Thus, the
firm is able to make the optimal choice without having to take into account their current levels. Accordingly,

since the same relative decisions are made, we can compare the optimal utilities from two different original
states of demand and advertising and find that the utilities are ultimately equal.
Lemma 1. When the firm opts to change its level of advertising at the end of period 1, the expected utility from
the next period under the optimal strategy is independent of the current levels of demand and advertising.
Proof. To prove this, we compare the optimal utilities from two different initial states, (D, A) and (D0 , A0 )
where D denotes current demand and A current advertising and D0 , A0 alternate values for current demand and
advertising. The utility from choosing x 0 under D, A, u(C, D, A, ↑), is the following
The values from the two equations are rendered equal. Since the utilities hold the same value and this is true for
any x 0 it must also hold for the optimal value of x 0 . Then, the optimal utilities under the two different initial
states are the same.
Whether demand increased or decreased in the previous period, ultimately conditional on changing the
level of advertising, the firm will receive the same level of utility in the optimal mechanism. More specifically,
the expected utility remains the same if demand goes up twice in a row and the firm changes the level of
advertising. To see why, suppose it were optimal to increase advertising relative to demand, conditional on
demand having increased in the previous period. Then, had demand decreased instead, it would have been
optimal to decrease advertising by an equivalent amount. Doing so, ensures the firm receives the same expected
utility, independent of the direction of last period’s demand shift.
Lemma 2. When the firm opts to change its level of advertising at the end of period 1, the expected utility from
the next period under the optimal strategy is independent of whether demand goes or down in period 1.

The values from the two equations are rendered equal. Since the utilities hold the same value and this is true for
any x it must also hold for the optimal value of x ∗ . Then, the optimal utilities under the two different initial
states are the same
Lemma 3. When the firm opts to change its level of advertising at the end of period 1, the expected utility next
period is always:
Proof. Because of Lemmas 1 and 2, we can assume demand increased in period 1, and set the current level of
demand to 0. Then, the expected utility conditional on changing advertising to x is −p(k − x) 2
− (1 − p)(−k − x)2
. Taking the first order condition and solving for x generates, x = −k(1 − 2p). Last, plugging x back into the
expected utility generates equation 9.
The firm is able to calculate the overall expected utility starting from period 0 and continuing over the
following two periods by accounting for anticipated changes in demand. This expected utility comprises several
components. First, the utility conditional upon the change in demand in period 1. The firm anticipates it will act
optimally at the end of period 1, making a decision at that point regarding whether to change the level of
advertising. The firm must account for the fact that the current level of demand will impact the probability that
in period 1 it chooses to change the level of advertising.
Then, the firm must calculate the utility conditional on the probability distribution over period 2
demand. Lemma 3 implies that the utility conditional on having changed the level of advertising at the end of
period 1 remains the same whether demand went up or down in period 1. Then, the firm’s overall expected
utility is the sum of four different terms: the utility in beginning of period 1, the utility if demand goes up during
period 2 given advertising levels were not changed at the end of period 1, the period 2 utility if demand
decreases given advertising levels were not changed at the end of period 1, and the period 2 utility when
advertising levels are changed in period 1. We compute each of these terms, then take the derivative with
respect to x0, the initial level of advertising, and use a first-order condition approach to characterize the optimal
initial advertising level.
Theorem 1. The optimal firm welfare when T = 2 is:
Proof. As it turns out, the optimal strategy for the firm in the second period, depends on the following two
thresholds. If the cost to change the level of advertising is above both thresholds, the firm never changes its
advertising in the second period. If the cost is below both, it always changes its level of advertising in the
second period. Last, if it is in between the two, it changes the level of advertising only if demand decreased.

These thresholds are computed by comparing the utility from changing the level of advertising to the utility
from leaving them fixed and solving for c. We can then break the problem down into the above subcases,
conditional upon where c falls on a case by case basis. Then, in each subcase, we proceed by determining the
optimal value for x0. To begin, we calculate V0 when c < c1(x), which implies that the level of advertising will
always be changed in the second period.

V. Naivety
Suppose a firm lacked foresight, and simply set its level of advertising based upon the current level of
demand. That is, the firm ignored the persistence in demand trends. We show that such a firm does worse
relative to a firm that chooses advertising optimally when the persistence in demand increases. Formally, define
the na¨ıve strategy as the one where the firm sets advertising, xt equal to demand, rt in every period.
We proceed by computing the expected payoff from the na¨ıve strategy:
Then, we take the derivative in respect to p.
Note that the optimal strategy improves with respect to p. Therefore, the na¨ıve strategy becomes relative worse
as the persistence in demand increases. That is, a firm that myopically sets advertising equal to current demand
does relatively worse as more information is present with respect to demand.
VI. Conclusion
We created a model in which a firm determines the optimal level of advertising in response to
stochastic demand when aiming to maximize its expected profit over time. Our model shows that setting
advertising levels higher than demand levels is beneficial when demand is persistent and has previously
increased. In particular, a na¨ıve strategy that fails to take persistence into account does especially bad in real-
world settings where stock trends are largely consistent over time.
Notably, our definitions of demand and advertising are broad. The research question we studied, could
have been reframed as the optimal decision of a firm that aims to match one quantity to another stochastic but

persistent quantity. For instance, we touch on the menu costs literature by providing insight when inflation isn’t
deterministic.
Following the optimal strategy allows a firm to maximize their profits and minimize inappropriate
expenditure on advertising. Utilizing the optimal model takes into accounts factors such as demand and trends,
which firms may find helpful. This is especially the case when a firm may find themselves focused on other
elements such as audience, platforms, and more. Utilizing a demand and anticipation-focused model allows
firms to take into account more elements than they may see at a first glance. For example, that firm could take
engagement into account to improve its assessment of future demand. An extension of our model would be able
to incorporate this by generalizing demand from a binary variable to a continuous distribution. We think that
this could be a productive question for future research
References
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states, 2015. URL https://www.ana.net/content/show/id/37679.
[2]. Caron Beesley. How to set a marketing budget that fits your business goals and provides a high return
on investment, 2012. URL https://www.sba.gov/taxonomy/term/15051? page=37.
[3]. Peter A Diamond. A model of price adjustment. Journal of economic theory, 3(2):156–168, 1971.
[4]. Laurence Ball and N Gregory Mankiw. Asymmetric price adjustment and economic fluctuations. The
Economic Journal, 104(423):247–261, 1994.
[5]. George A Akerlof and Janet L Yellen. A near-rational model of the business cycle, with wage and price
inertia. The Quarterly Journal of Economics, 100(Supplement):823–838, 1985.
[6]. Tore Ellingsen, Richard Friberg, and John Hassler. Menu costs and asymmetric price adjustment.
Mimeo, 2006.
[7]. Jakob B Madsen and Bill Z Yang. Asymmetric price adjustment in a menu-cost model. Journal of
economics, 68(3):295–309, 1998.
[8]. Mark J Zbaracki, Mark Ritson, Daniel Levy, Shantanu Dutta, and Mark Bergen. Managerial and
customer costs of price adjustment: direct evidence from industrial markets. Review of Economics and
statistics, 86(2):514–533, 2004.

Optimal Advertising with Persistent Demand

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