Optimal dynamic advertising policies in the presence of continuously distributed time lags

The Nerlove-Arrow model of optimal dynamic advertising policies is generalized by incorporating a continuously distributed lag between advertising expenditures and increases in the stock of goodwill. This leads to a control problem where the equation of motion is given by an integro-differential equation. The transitory and steady-state properties of the optimal policies are examined, both for a general lag function and for a gamma distributed lag. The dependence of the steady-state solution on the parameters of the gamma distribution is also investigated. An example is given using specific demand and cost functions.     

Delayed effects of cooperative advertising in goodwill dynamics

This paper proposes a tool to control cooperative advertising which increases the goodwill of companies operating in a competitive market. We introduce the lag between advertising exposure and customer reaction in the goodwill dynamics evolved à la Nerlove–Arrow. As a result, we obtain a cooperative differential game with immediate and delayed effects of control variables for which we investigate the optimal solution. We examine the role the pre-coalition programmes and the length of delayed response in generating goodwill.     

Pure-strategy Nash equilibria in an advertising game with interference

Two manufacturers produce substitutable goods for a homogeneous market. The advertising efforts of the two manufacturers determine the demand for the goods and interfere negatively with each other. The demand of each good is a piecewise linear function of the product goodwill, and the latter is a linear function of advertising efforts. In a game with two competing profit-maximizing manufacturers who have access to a set of several advertising media, the pure-strategy Nash equilibria are characterized and their existence is shown.     

Optimal advertising strategies with age-structured goodwill

The problem of a firm willing to optimally promote and sell a single product on the market is here undertaken. The awareness of such product is modeled by means of a Nerlove–Arrow goodwill as a state variable, differentiated jointly by means of time and of age of the segments in which the consumers are clustered. The problem falls into the class of infinite horizon optimal control problems of PDEs with age structure that have been studied in various papers either in cases when explicit solutions can be found or using Maximum Principle techniques. Here, assuming an infinite time horizon, we use some dynamic programming techniques in infinite dimension to characterize both the optimal advertising effort and the optimal goodwill path in the long run. An interesting feature of the optimal advertising effort is an anticipation effect with respect to the segments considered in the target market, due to time evolution of the segmentation. We analyze this effect in two different scenarios: in the first, the decision-maker can choose the advertising flow directed to different age segments at different times, while in the second she/he can only decide the activation level of an advertising medium with a given age-spectrum.     

An advertising game with multiplicative interference

Two competing manufacturers provide a homogeneous market with substitutable products and want to maximize their profits. Each firm may advertise its brand, with a positive effect on its own brand and negative on the competitor's one. Moreover, each firm may choose an advertising medium to use among several available media. We assume that the advertising effect on demand is mediated by the goodwill variable and that a competitor's interference may be represented as a proportional reduction of the virtual goodwill. We model the manufacturers' problem as a noncooperative game under complete information and discuss the existence and features of its Nash equilibria.     

A joint dynamic pricing and advertising model of perishable products

Advertising and dynamic pricing play key roles in maximizing profit of a firm. In this paper a joint dynamic pricing and advertising problem for perishable products is investigated, where the time-varying demand rate is decreasing in sales price and increasing in goodwill. A dynamic optimization model is proposed to maximize total profit by setting a joint pricing and advertising policy under the constraint of a limited advertising capacity. By solving the dynamic optimization problem on the basis of Pontryagin’s maximum principle, the analytical solutions of the optimal joint dynamic pricing and advertising policy are obtained. Additionally, to highlight the advantage of the joint dynamic strategy, the case of the optimal advertising with static pricing policy is considered. Numerical examples are presented to illustrate the validness of the theoretical results, and some managerial implications for the pricing and advertising of the perishable products are provided.     

The impact of product recall on advertising decisions and firm profit while envisioning crisis or being hazard myopic

The occurrence of a product recall can have a disastrous effect on the firm responsible for the recall. Any major recall by a firm can negatively affect the goodwill of the firm. Consequently, the firm incurs a substantial indirect cost due to decline in sales and loss in profit. Moreover, a competitor’s opportunistic reaction can intensify the recalling firm’s damages. Strategic use of advertising recovers lost goodwill and mitigates the damages made by a product recall. In this paper, using a goodwill based model under a differential game framework, we analyze the equilibrium strategies of two competing manufacturers when either one firm or both can issue a product recall at a random time, and investigate (i) the firms’ equilibrium advertising strategies (ii) analyze the impact of the recall on a firm’s profit (iii) introduce and investigate the effect of “hazard myopia” (a firm’s inability to foresee the crisis likelihood) on a firm’s advertising decisions and profit. Our study finds that the equilibrium advertising strategies of competing firms depend on the impact and likelihood of the recall. Notably, we find that when both the firms are focal firms without the prior knowledge of who will recall first in a planning horizon, adjusting optimal advertising at an appropriate time is essential. Surprisingly, a product-recall with a minor impact can increase the focal firm’s long-term expected profit. On the other hand, hazard myopia can be profitable if the long-term effect of the recall is small. Our findings suggest that advertising levels of firms should differ in pre-recall and post-recall regimes depending on the impact and likelihood of the recall.     

Advertising and quality improving strategies in a supply chain when facing potential crises

In this paper, we consider a supply chain that faces a potential brand crisis, with one manufacturer deciding quality improvement and global advertising levels, and one retailer determining local advertising effort. The goodwill model proposed by Nerlove and Arrow (1962) is adopted here under the assumption that when the crisis happens, the companies suffer a sharp decrease in the goodwill. We characterize the feedback Nash equilibrium, and then we compare the corresponding quality and advertising strategies and outcomes with those of the case where the potential crises are absent, and where the companies do not invest in quality. The effects of the instantaneous crisis rate and the short-term and long-term damages are also evaluated. Our results reveal that the pre-crisis quality improvement accelerates the goodwill build-up before the crisis, and also helps the recovery in post-crisis regime. Its twofold function suggests that one of the pre- and post-crisis regimes/instants ought to be matched with more intense investment in both quality and global advertising, depending on the overall effect of instantaneous crisis rate, short-term damage and long-term damage. This carryover effect also brings a non-monotonicity of quality improvement effort and value functions with respect to the instantaneous crisis rate. These properties leave the chance to mitigate the loss by anticipating crisis for both members under certain circumstances.     

Modelling impact of advertising and optimizing advertising policy: an application in recreation

This paper deals with the problem of the desirable level of advertising expenditure, the optimal distribution of this expenditure in time and the allocation over the media: TV, radio and newspaper for a recreation park in the Netherlands.Although the model id developed for the specific situation of this park, in principle it can be applied in all situations where the interest is in short-term (day-by-day) effects of promotional activities on sales. Examples are: other situations in the recreation and leisure business, cultural events (theatre, cinema) and sales promotions (e.g. weekend offerings) for products in supermarkets.First a model was specified and estimated that relates number of visitors to advertising effort. It also takes into account non-advertising variables that effect the number of visitors.Then this model was used in a heuristic advertising planning procedure, which by means of incremental analysis, for a given budget level searches for the optimal allocation of the advertising budget over media and time.With this procedure, ways to readjust the advertising policy were found: by allocating the budget differently over media and time and by changing the overall budget level.Several recommendations were made to the management of the park, a number of which have already been implemented.     

Dynamics of competing with quality- and advertising-based goodwill

Goodwill formation is a complex process and many factors influence the formation of goodwill of a firm. The implications of advertising enabled goodwill formation are reported in several articles in the research literature. In this paper, we extend this stream of research by including quality in the goodwill formation process. We adopt a dynamic model of competition utilizing a differential game approach and derive expressions for open-loop Markovian Nash equilibrium investments in advertising and quality. The insights gained from the analysis of our model and from the equilibrium solutions are presented in the form of research propositions.     

Equilibrium Pricing and Advertising Strategies in a Marketing Channel

This paper is concerned with conflict and coordination in a two-member channel of distribution. We propose a differential game model that includes carryover effects of advertising, expressed by a retailer-specific stock of advertising goodwill. Pricing and advertising strategies for both firms are identified under channel conflict as well as coordination. Dynamic advertising policies are designed as stationary Markov perfect strategies. In a symmetric case, these strategies can be determined in closed form, taking into consideration explicitly nonnegativity constraints on advertising rates. We establish a global result for the relationship between the advertising strategies of the two firms under conflict and coordination.     

Intervention analysis to identify significant exposures in pulsing advertising campaigns: an operative procedure

The purpose of this paper is to develop an operational method to detect the most effective exposures in the context of a given pulsing advertising campaign. For most effective, are intended those exposures that produce a statistically significant increase in the level of a response variable, either temporarily or permanently. The method consists in specifying an intervention model for the response variable, where the significant exposures are selected on the basis of a probabilistic criterion, and is empirically evaluated by using brandlevel data from five advertising tracking studies that also include the actual spending schedules. Given a pulsing advertising campaign, the proposed method serves both as an a-posteriori improvement of the campaign itself and as an a-priori additional information for programming future scheduling. Mathematics Subject Classification: 62M10.     

Using mixture‐amount modeling to optimize the advertising media mix and quantify cross‐media synergy for specific target groups

One of the critical decisions in media planning is how to allocate advertising efforts across different media. While studies indicate that marketers can create positive synergy effects by spreading their effort across several media, there is little understanding of how much should be invested in each specific medium to optimize advertising results. In this study, we apply a novel methodology, mixture‐amount modeling, which allows advertisers to determine the optimal allocation of advertising effort across media as a function of the total advertising effort. Moreover, we test how the optimal allocation and the resulting response change for consumers with distinctive media usage patterns and varying degrees of product category experience. Based on these results, we quantify the potential synergy between media and calculate the synergistic capacity for specific target groups. We apply the model to data from 52 beauty care advertising campaigns that ran on TV and in magazines in the Netherlands and Belgium. We determine the optimal allocation of advertising investments (measured through Gross Rating Points) to maximize campaign recognition. Our findings support the existence of positive synergistic effects between magazine and TV advertising and illustrate that these effects depend on consumers' media usage and product category experience.     

Advertising decisions for a segmented market

We consider the problem of choosing the levels of a set of advertising media in order to maximize the firm profit when the market is heterogeneous. Advertising efforts affect the demand of the different segments variably and we assume that the advertising effects on demand over time are mediated by a vector goodwill variable. A first general advertising decision problem is stated and solved in the non-linear programming framework. A preference index is then obtained for the medium selection problem when each segment demand function is linear in goodwill and each medium advertising cost function is quadratic in its level. Finally the theoretical case of disjoint advertising media is discussed.     

Recent developments in dynamic advertising research

A variety of continuous-time differential functions have been developed to investigate dynamic advertising problems in business and economics fields. Since major dynamic models appearing before 1995 have been reviewed by a few survey papers, we provide a comprehensive review of the dynamic advertising models published after 1995, which are classified into six categories: (i) Nerlove–Arrow model and its extensions, (ii) Vidale–Wolfe model and its extensions, (iii) Lanchester model and its extensions, (iv) the diffusion models, (v) dynamic advertising-competition models with other attributes, and (vi) empirical studies for dynamic advertising problems. For each category, we first briefly summarize major relevant before-1995 models, and then discuss major after-1995 models in details. We find that the dynamic models reviewed in this paper have been extensively used to analyze various advertising problems in the monopoly, duopoly, oligopoly, and supply chain systems. Our review reveals that the diffusion models have not been used to analyze advertising problems in supply chain operations, which may be a research direction in the future. Moreover, we learn from our review that very few publications regarding dynamic advertising problems have considered the supply chain competition. We also find that very few researchers have used the diffusion model to investigate the dynamic advertising problems with product quality as a decision variable; and, the pricing decision has not been incorporated into any extant Lanchester model. The paper ends with a summary of our review and suggestions on possible research directions in the future.     

Optimal advertising pulsation policies: a dynamic programming approach

This study formulates and solves an advertising pulsation problem for a monopolistic firm using dynamic programming (DP). The firm aims at maximising profit through an optimal allocation of the advertising budget in terms of rectangular pulses over a finite planning horizon. Aggregate sales response to the advertising effort is assumed to be governed by a modified version of the Vidale–Wolfe model in continuous time proposed by Little. Using a numerical example in which a planning horizon of one year is divided into one, two through ten equal time periods, computing routines are developed to solve 150 DP problems. Computational results show among other findings that the performance yielded by the DP policy dominates the uniform advertising policy (constant spending) for a concave advertising response function and the advertising pulsing policy (turning advertising on and off) for a linear or convex response function.     

A dynamic model for advertising and pricing competition between national and store brands

We study the relationship between the pricing and advertising decisions in a channel where a national brand is competing with a private label. We consider a differential game that incorporates the carryover effects of brand advertising over time for both the manufacturer and the retailer and we account for the complementary and competitive roles of advertising. Analysis of the obtained equilibrium Markov strategies shows that the relationship between advertising and pricing decisions in the channel depends mainly on the nature of the advertising effects. In particular, the manufacturer reacts to higher competitive retailer’s advertising levels by offering price concessions and limiting his advertising expenditures. The retailer’s optimal reaction to competitive advertising effects in the channel depends on two factors: (1) the price competition level between the store and the national brands and (2) the strength of the competitive advertising effects. For example, in case of intense price competition between the two brands combined with a strong manufacturer’s competitive advertising effect, the retailer should lower both the store and the national brands’ prices as a reaction to higher manufacturer’s advertising levels. For the retailer, the main advantage from boosting his competitive advertising investments seems to be driven by increased revenues from the private label. The retailer should however limit his investments in advertising if the latter generates considerable competitive effects on the national brand’s sales.     

Existence and uniqueness of optimal dynamic pricing and advertising controls without concavity

We consider a pricing and advertising dynamic-optimization problem where the goodwill dynamics evolve à la Nerlove–Arrow. The firm maximizes its profit over a finite-planning horizon corresponding to the product’s lifespan, and it turns out that the Hamiltonian is non-concave. We show the existence and uniqueness of an optimal solution under some mild conditions.     

On the impact of initial performance on the effectiveness of advertising pulsation policies

Using a modified version of a Vidale–Wolfe model, proposed by Little, this paper examines the impact of initial sales rate on the performance of a variety of discrete, piecewise-continuous advertising policies for a finite planning horizon. The deployment of a non-discounted measure of performance reveals, irrespective of the shape of the advertising response function, that when the initial sales rate is different from zero at the beginning of the planning period: (1) a firm would be better off concentrating its advertising effort at the end rather than at the beginning of the planning period for a Blitz Policy (BP), (2) for an Advertising Pulsing/Maintenance Policy (APMP), it is more lucrative for a firm to alternate between a lower level of advertising followed by higher level (low–high) in a cyclic manner rather than to cycle the opposite way (high–low), and (3) in the presence of an initial sales rate, the pattern of the optimal advertising policy determined by dynamic programming can be significantly different from its alternative counterpart in its absence. In addition, it has been demonstrated, among other theoretical findings, that, for any given mean rate of advertising, the mean sales is bounded from below and is a decreasing function of the length of the planning horizon. Numerical examples are introduced to illustrate and reinforce the above research findings.     

Optimal advertising policy with the contagion model

This paper considers an optimal control problem for the dynamics of a contagion model, the optimal control being the rate of advertising expenditure that maximizes the present value of net profit streams over an infinite horizon. By using a Green's theorem approach, it is shown that there are multiple optimal stationary equilibria and that the optimal path from any given initial condition is a nearest feasible path to one of these equilibria.This work was partially supported by the National Research Council of Canada, Grant No. A4619.