Stochastic diffusion models with advertising and word-of-mouth effects

This paper presents a stochastic diffusion model which incorporates advertising word-of-mouth effects. The model defines a 3 variate stochastic process based on explicit assumptions regarding consumer behavior and consumers' response to advertising. The model generalizes a wide variety of advertising models and in addition includes several classes of consumers, interaction effects between these consumers and, of course, a stochastic framework that may be used for assessing the risk implications of advertising policies and for empirical analyses purposes.     

Advertising Response Functions and Media Planning

In the first part of this paper the definition and use of advertising response functions are examined critically. It is shown that advertising response functions cannot be regarded as models of the advertising process whose parameters are known, that only a small section of the complex functions often postulated is relevant to media planning, that measures of advertising impacts or exposures are relative rather than absolute, and consequently that the “response function” only shows a notional relationship between advertising response and exposure, which normally needs to be calibrated to the advertising appropriation.In the second part a media planning system which utilizes a “control parameter” rather than a “response function” is described. The main feature of the “control parameter” is that it adapts itself so that an appropriate relationship between response and exposure is used, whatever the appropriation. Other features of the system are described.     

Hybrid subgames and copycat games in a pulsing model of advertising competition

Two recent papers,^6,7 introduced the game of pulsing competition (PC) in advertising together with its related subgames of alternating pulsing competition (APC) and matching pulsing competition (MPC) for a duopoly. Following a game theoretic approach in conjunction with a continuous Lanchester model, the above authors basically concluded that when at least one of the response functions is convex, generalising monopolistic advertising pulsation results to a competitive setting might not be adequate. This paper expands the scope of the PC game by incorporating in its structure for the first time in the literature, two versions of a hybrid pulsing competition (HPC) subgame. The article compares the payoffs of the four alternative subgames and provides an analytical solution of a special case of the PC game. In addition, the article also introduces for the first time a variant of the PC game designated by ‘the copycat advertising game’ and shows analytically that for such a game the policy of constant advertising spending over time is optimal for both firms irrespective of the shape of their advertising response functions. The paper illustrates at its end how to solve numerically the expanded PC game in its general form using linear programming and how to derive a solution for a copycat advertising game.     

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.     

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.     

Stochastic optimal budget decision for advertising considering uncertain sales responses

Firms are faced with uncertain sales responses even though they advertise appropriately. To help marketing managers make optimal budget decisions in this situation, we develop a stochastic model, depicting the problem of advertising budget decision as a special Markov decision process where a new objective, maximizing expected market utility, is proposed. In the model we introduce a two-dimension state variable including accumulative sales, which vary randomly with advertising budget, and the predicted probability that an advertising campaign obtains a full sales response. We make an analysis of the model on the premise of growing infinite market potential, deriving the property of optimal policies and that of optimal value function. These results are successfully used to make advertising budget decisions for a private university in Xi’an, China.     

Forecasting call frequency at a financial services call centre

A forecasting model is developed for the number of daily applications for loans at a financial services telephone call centre. The purpose of the forecasts and the associated prediction intervals is to provide effective staffing policies within the call centre. The model building process is constrained by the availability of only 2 years and 7 months of data. The distinctive feature of the data is that demand is driven in the main by advertising. The analysis given focuses on applications stimulated by press advertising. Unlike previous analyses of broadly similar data, where ARIMA models were used, a model with a dynamic level, multiplicative calendar effects and a multiplicative advertising response is developed and shown to be effective.     

A Competitive Advertising Model: Some Theoretical and Empirical Results

Using a modified Lanchester model, this article demonstrates that the policy of constant advertising spending is superior to a cyclic counterpart, provided that the advertising response functions of competing firms are concave. The competitive model is estimated using filter and non-filter cigarette data. The empirical results suggest the presence of over-advertising and that the relative effectiveness and elasticity of filter cigarette advertising are both higher than those of non-filter cigarette advertising.     

Optimal Advertising and Pricing in a Dynamic Durable Goods Supply Chain

Cooperative advertising is an incentive offered by a manufacturer to influence retailers’ promotional decisions. We study a dynamic durable goods duopoly with a manufacturer and two independent and competing retailers. The manufacturer, as a Stackelberg leader, announces his wholesale prices and his shares of retailers’ advertising costs, and the retailers in response play a Nash differential game in choosing their optimal retail prices and advertising efforts over time. We obtain the feedback equilibrium policies for the manufacturer and the retailers in explicit form for a linear demand formulation. We investigate issues, like channel coordination and antidiscriminatory legislation, and also study a case, when the manufacturer sells through only one retailer and the second retailer sells a competing brand.     

Modelling advertising budgeting and allocation decisions using modified multinomial logit market share models

Previous research suggests that a multinomial logit model of market share (MNL) is inappropriate for equilibrium analyses of advertising competition. This article shows that when employing simple transformations of the advertising effort, the modified MNL model becomes useful in representing situations of diminishing returns to advertising and appropriate for advertising equilibrium analyses without additional difficulties in its empirical estimation. Using the modified MNL model, optimal advertising budgets together with their allocation over time are derived for both the cases of concave and S-shaped attraction (response) functions in a symmetric oligopoly.     

Optimal Goodwill Path to Introduce a New Product

We consider the problem of determining an optimal goodwill path for the introduction of a new product in a market, while looking for the maximum foreseen profit. The foreseen revenue depends on the product introduction time and on the goodwill level at the same time. We focus on the advertising costs associated with the goodwill evolution and assume that the cost function possesses some rather general features which are shared by the cost functions of the Nerlove-Arrow type models. The dynamic optimization problem is discussed in the calculus of variations framework. A few examples associated with special cost functions are discussed in detail.     

Signaling Quality: Dynamic Price-Advertising Model

This paper extends the existing quality-signaling literature by investigating the roles of price and advertising levels as quality indicators in a dynamic framework. Considering perceived quality as a form of goodwill, we modify the well-known Nerlove-Arrow dynamic model to include price effects. In our model, price is used both as a monetary constraint and as a signal of quality, while advertising spending is used only as a signaling device, and thus purely as a dissipative expense. Utilizing optimal control, we determine optimal decision rules for a firm regarding both price and advertising over time as functions of perceived quality. The results indicate that, when prices act as monetary constraints and are reduced to increase demand, the firm should use the signaling role of advertising by increasing spending to accelerate perceived quality increases. In cases when the value of the perceived quality goes up together with the increase in the perceived quality by more than the demand, in percentage terms, the firm should increase the price (use its signaling role). At steady-state, we find that the level of optimal profit margin relative to price decreases with the elasticity of demand with respect to the brand price. However, higher elasticity of demand with respect to the firm’s perceived quality and/or a higher impact of price (advertising) lead/leads to a higher optimal profit margin (advertising spending) relative to price (revenue).     

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.     

Advertising for a new product introduction: A stochastic approach

We formulate a stochastic extension of the Nerlove and Arrow’s advertising model in order to analyze the problem of a new product introduction. The main idea is to introduce some uncertainty aspects in connection both with the advertising action and the goodwill decay, in order to represent the random consequences of the advertising messages and of the word-of-mouth publicity, respectively. The model is stated in terms of the stochastic optimal control theory and a general study is attempted using the stochastic Maximum Principle. Closed form solutions are obtained under linear quadratic assumptions for the cost and the reward functions. Such optimal policies suggest that the decision-maker considers both the above mentioned phenomena as opportunities to increase her/his final reward. After stating some general features of the optimal solutions, we analyze in detail three extreme cases, namely the deterministic model and the stochastic models with either the word-of-mouth effect only, or the lure/repulsion effect only. The optimal policies provide us with some insight on the general effects of the advertising action. Supported by MIUR and University of Padua.     

Cooperative advertising models in supply chain management: A review

This paper reviews articles on cooperative advertising, a topic which has gained substantial interest in the recent years. Thereby, we first briefly distinguish five different definitions of cooperative advertising which can be found in operations research literature. After that, we concentrate on vertical cooperative advertising, which is the most common object of investigation and is understood as a financial agreement where a manufacturer offers to pay a certain share of his retailer’s advertising expenditures. In total, we identified 58 scientific papers considering mathematical modeling of vertical cooperative advertising. These articles are then analyzed with regard to their general model setting (e.g., the underlying supply chain structure and design of the cooperative advertising program). After that, we explain the different demand and cost functions that are employed, whereupon we distinguish between static and dynamic models. The last dimension of our review is dedicated to the game-theoretic concepts which are mostly used to reflect different forms of distribution of power within the channel.     

On the superiority of pulsing under a concave advertising market potential function

The authors study the superiority of advertising pulsing policy (turning advertising on and off in a cyclic fashion) over its uniform (constant spending) counterpart that costs the same under the assumption that sales dynamics follow a modified Vidale–Wolfe aggregate advertising model. The authors show that pulsing can be superior if the product of the concave market potential function and the linear or concave advertising response function is convex in advertising. Similar to previous studies in the literature, the average undiscounted profit over the infinite planning horizon is considered as a performance measure according to which alternative advertising pulsation policies are compared.     

Innovation diffusion uncertainty,advertising and pricing policies

We develop and analyze a normative and structurally stochastic model of innovation diffusion by depicting the market at an aggregate level. Model dynamics are defined through the flow pattern of individuals that move from the innovation unaware stage, to the innovation aware, and ultimately to the adopter stages. The stochastic evolution of this stage-wise transition unfolds according to tractable stochastic processes and is influenced by such factors as price, word of mouth, and advertisement efforts. In this environment, techniques of contingent claims analysis and stochastic control theory are employed to obtain optimal pricing or advertising policies that maximize the value of the innovation. To account for their optimal adjustment over time, these policies are modeled as positive real-valued adapted processes. Given this setting, policy adjustments over time (i.e. advertising or pricing) are viewed as a value additive sequence of nested real options. We present closed-form analytic results regarding the optimal policies. Simulations provide a numeric insight to the models' behavior.     

Optimal dynamic advertising policies for hereditary processes

In this paper, the Nerlove-Arrow model of optimal dynamic advertising policies is generalized by assuming a general probability distribution of the forgetting time, rather than the exponential one. A control problem with integrodifferential equations of motion is defined for which the transitory and steady-state properties of the optimal advertising policy are examined. The effects of assumptions like IHR-distributions and DHR-distributions, the existence of an upper bound for the forgetting time, etc., are explained. It is shown that there are two (in the case of an exponential distribution even three) different current-value adjoint functions associated with the problem, and relations between the two (three) are established. Also provided is a sensitivity analysis.Thanks are due to G. Feichtinger and S. Jorgensen for useful discussions.     

A goodwill model with predatory advertising

We investigate the dynamic advertising policies of two competing firms in a duopolistic industry, assuming a predatory phenomenon between their advertising campaigns. The resulting model is a differential game which is not linear-quadratic. We show that there exists a Markovian Nash equilibrium, and that it leads to time constant advertising strategies. According to this model, predatory advertising produces a negative externality: the interference between the advertising campaigns decreases the total demand of the market.     

Public Disclosure Programs vs. traditional approaches for environmental regulation: Green goodwill and the policies of the firm

A Public Disclosure Program (PDP) is compared to a traditional environmental regulation (exemplified by a tax/subsidy) in a simple dynamic framework. A PDP aims at revealing the environmental record of firms to the public. This information affects its image (goodwill or brand equity), and ultimately its profit. A firm polluting less than its prescribed target would win consumer’s sympathy and raise its goodwill, whereas it is the other way around when the firm exceeds its emissions quota. The evolution of this goodwill is assumed to depend also on green activities or advertising expenditures. Within this framework, we analyze how a PDP affects the firm’s optimal policies regarding emissions, pricing and advertising as compared to a traditional regulation. We show that advertising acts as a complementary device to pricing and that emissions are increasing in goodwill. The role of a standard or target level for emissions turns out to be totally different under both policy regimes. In the case of a tax/subsidy approach, this target level only acts as constant who increases or decreases profit by a fixed amount, but it does not affect the policy of the firm. On the contrary, if a PDP is implemented, the target value for emissions enters in an important way in the goodwill accumulation mechanism and determines how the firm reacts to the regulation and what is the time path for the economic and environmental variables. Moreover, this value is also crucial to determine the possibility that a PDP is profit improving. A policy implication of this fact is that regulators should be particularly careful in fixing the emission standard when a PDP is applied. The theoretical results are complemented with a numerical illustration.