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.     

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.     

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.     

The effects of advertising,prices and distribution on market share volatility

Although the influence of marketing decision variables on market share has received ample attention in the literature, less is known about their effects on volatility. This study attempts to shed light on this issue by empirically examining the effects of advertising, pricing and distribution, not only on market share but also on its volatility, using an EGARCH model. We argue that establishing a link between management-controlled actions, such as advertising, pricing and distribution, and market share volatility may benefit firms seeking to reduce uncertainty in their market share performance. Application of the proposed model to data on two markets (SUVs and Minivans), suggested that advertising, pricing and distribution significantly influence market share volatility.     

Consumption-Investment Problem with Subsistence Consumption,Bankruptcy, and Random Market Coefficients

We consider a general continuous-time finite-horizon single-agent consumption and portfolio decision problem with subsistence consumption and value of bankruptcy. Our analysis allows for random market coefficients and general continuously differentiable concave utility functions. We study the time of bankruptcy as a problem of optimal stopping, and succeed in obtaining explicit formulas for the optimal consumption and wealth processes in terms of the optimal bankruptcy time. This paper extends the results of Karatzas, Lehoczky, and Shreve (Ref. 1) on the maximization of expected utility from consumption in a financial market with random coefficients by incorporating subsistence consumption and bankruptcy. It also addresses the random coefficients and finite-horizon version of the problem treated by Sethi, Taksar, and Presman (Ref. 2). The mathematical tools used in our analysis are optimal stopping, stochastic control, martingale theory, and Girsanov change of measure.     

Optimal Multi-Period Advertising Budget Allocation within a Competitive Environment

A mathematical model approach is developed for the purpose of aiding advertising and marketing executives in advertising budget allocation decision-making in the face of a competitive environment. Two alternative model formulations are examined to study the dynamic market response to advertising expenditures. These embody numerous realistic characteristics of the advertising phenomenon including carry-over of past expenditures, diminishing returns and saturation effects, response decay in the absence of advertising and product diffusion effects. Through mathematical programming, the model determines the optimal advertising expenditures over a predetermined planning horizon under alternative constraint options (including competitive advertising assumptions). Illustrations of model applications are also presented.     

Signaling advertising by multiproduct firms

We consider the use of advertising expenses as quality signals in multiproduct firms, extending previous results on single product firms. In our model, a firm introduces sequentially two products whose qualities are positively correlated. We investigate whether there exist information spillovers from the first to the second market. We show that, when correlation is high, the equilibrium in market 2 depends on the quality reputation the firm has gained in market 1. Moreover, if a firm with a high-quality product 1 wants to separate from its low-quality counterpart, it needs to advertise more in this market than if the qualities of the two products are unrelated. This advertising level signals not only high quality in the first market, but also the likely quality of the second product. Thus, advertising in the first market has information spillovers in the second market.     

Feedback Competitive Advertising Strategies with a General Objective Function

We introduce a general objective function, which incorporates competitive situations, such as conservative, punitive, and predatory advertising. Linking together the particular situations into a two-parameter family of max–min problems, and using the Lanchester model to describe the dynamics of the market, a bilinear-quadratic differential game is obtained. For this game, we find saddle-point feedback time-invariant advertising strategies and show when these strategies are Nash equilibrium strategies. In an empirical application involving duopolistic competition in the cola market, we find evidence of a punitive motivation for the advertising strategies.     

Advertising in a Differential Oligopoly Game

We illustrate a differential oligopoly game where firms compete à la Cournot in homogeneous goods in the market phase and invest in advertising activities aimed at increasing the consumers reservation price. Such investments produce external effects, characterizing the advertising activity as a public good. We derive the open-loop and closed-loop Nash equilibria, and show that the properties of the equilibria depend on the curvature of the market demand function. The comparative assessment of these equilibria shows that the firms advertising efforts are larger in the open-loop equilibrium than in the closed-loop equilibrium. We also show that a cartel involving all the firms, setting both output levels and advertising efforts, may produce a steady state where the social welfare level is higher than the social welfare levels associated with both open-loop and closed-loop noncooperative settings.     

New product advertising in dynamic oligopolies

The success of the introduction of a new product in a market is very sensitive to the marketing decision variables adopted by the firm. In the present paper we are concerned with the question of new product advertising in a heterogeneous oligopoly market consisting of N firms. A dynamic game is formulated to model strategic as well as sales interactions in such a market. Optimal advertising strategies are identified as open-loop Nash solutions.The comments of two anonymous referees are appreciated. The first author wishes to acknowledge support from NSERC (Grant No. OGP0037342).     

纵向差异化下的双寡头说服性广告竞争

构建了一个纵向差异化下的双寡头说服行广告竞争模型,考查了均衡时双寡头企业的相对市场份额、均衡价格、企业利润及其决定因素,并对其进行比较静态分析,得出了若干有价值的结论,并构造了一个梯形分布的算例来对结论进行了验证.     

Models of a Consumer Product Market

Aggregate models of the United Kingdom razor blade market are discussed. During one prolonged period when the market was dominated by two major brands, approximately equivalent in terms of quality, price and distribution, a model based on advertising expenditures alone was remarkably accurate at predicting short-term movements of a brand's market share in periods of both growth and decline. An optimum decay rate of advertising effectiveness was identified. The value of consumer promotions could be estimated from the deviations they produced from the model. To cover longer periods and to encompass brands of widely differing qualities and prices a more complex model was necessary. This has been successful in following long-term trends of market shares of all brands and has contributed towards understanding the market structure. It has shed light on the value of a multi-brand policy and has enabled a retrospective evaluation to be made of the effect of a period of production capacity limitation on market share growth.     

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.     

BSDE approach to utility maximization with square-root factor processes

We consider the utility-based portfolio selection problem in a continuous-time setting. We assume the market price of risk depends on a stochastic factor that satisfies an affine-form, square-root, Markovian model. This financial market framework includes the classical geometric Brownian motion, CEV model, and Heston’s model as special cases. Adopting the BSDE approach, we obtain closed-form solutions for the optimal portfolio strategies and value functions for the logarithmic, power, and exponential utility functions.     

Optimal advertising and pricing in a class of general new-product adoption models

In [21], Sethi et al. introduced a particular new-product adoption model. They determine optimal advertising and pricing policies of an associated deterministic infinite horizon discounted control problem. Their analysis is based on the fact that the corresponding Hamilton–Jacobi–Bellman (HJB) equation is an ordinary non-linear differential equation which has an analytical solution. In this paper, generalizations of their model are considered. We take arbitrary adoption and saturation effects into account, and solve finite and infinite horizon discounted variations of associated control problems. If the horizon is finite, the HJB-equation is a 1st order non-linear partial differential equation with specific boundary conditions. For a fairly general class of models we show that these partial differential equations have analytical solutions. Explicit formulas of the value function and the optimal policies are derived. The controlled Bass model with isoelastic demand is a special example of the class of controlled adoption models to be examined and will be analyzed in some detail.     

Advertising Events in a Competitive Framework

We study the problem of how to minimize the advertising costs for a one-time entertainment event taking competition into account. Using the concept of Nerlove-Arrow goodwill, we describe a market where two single players organize two different events. A single-player instance of the same problem has been analyzed already in the literature using both the Nerlove-Arrow dynamics and the Bass dynamics, but without considering a competitive framework. In this work, starting from a linear-quadratic model, we consider competition in the market and we study the problem in the settings of linear-quadratic deterministic and stochastic differential games. The analytical tractability of the problem allows us to characterize the Nash equilibria and to study the new features of the competitive scenario. In the deterministic instance of the model, we show that competition decreases efficiency. On the other hand, when the goodwill evolution is stochastic, competition may even increase efficiency by a diversification effect.     

On Controlled Linear Diffusions with Delay in a Model of Optimal Advertising under Uncertainty with Memory Effects

We consider a class of dynamic advertising problems under uncertainty in the presence of carryover and distributed forgetting effects, generalizing the classical model of Nerlove and Arrow (Economica 29:129–142, 1962). In particular, we allow the dynamics of the product goodwill to depend on its past values, as well as previous advertising levels. Building on previous work (Gozzi and Marinelli in Lect. Notes Pure Appl. Math., vol. 245, pp. 133–148, 2006), the optimal advertising model is formulated as an infinite-dimensional stochastic control problem. We obtain (partial) regularity as well as approximation results for the corresponding value function. Under specific structural assumptions, we study the effects of delays on the value function and optimal strategy. In the absence of carryover effects, since the value function and the optimal advertising policy can be characterized in terms of the solution of the associated HJB equation, we obtain sharper characterizations of the optimal policy.     

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.     

基于博弈论的促销商广告微分方程决策分析

将供应链看作为一个整体,他们需要经手一种新型时髦的商品或季节性商品,促销商作为供应链中一个独立的部门,专门负责季节性商品的促销工作,在供应链管理体系中处于一个核心地位,制造商与零售商则专门按照促销商提供的信息,专职从事制造与销售工作.以此假设为前提,分别建立了三类广告促销的微分方程对策模型,研究广告对季节性商品销售量的影响,以及在促进供应链整体绩效中的作用,进而确定合理的广告强度和时间.最后,在假设广告收益函数与成本函数服从均匀分布的情况下,由建立的广告决策模型得出了恰当的广告使用费用和时间.并由此得到了一些对决策有重要参考价值的结论.     

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.