Abstract
This paper proposes definitions of implementation and adoption delays, arising from firm and client behaviors, in the context of market share dynamics. Information delay refers to the existence of lags in the information used for the advertising policy, while adoption delay refers to the existence of a lag in the effect of the advertising policy. These are natural lags in the flow of information and have not been considered in several models proposed in the literature. In this paper, these delays are introduced into recently proposed extensions of the Vidale-Wolfe-Deal and Lanchester models of market share dynamics subjected to affine advertising control policies. Conditions for stability of the equilibrium market share are derived. In addition, it is shown that Hopf bifurcations leading to oscillatory behavior exist for certain parameter values, and corresponding conditions for these are given. The main results are: the equilibrium market shares of the extended Vidale-Wolfe-Deal and Lanchester models are both robust to implementation delays, but, in the case of adoption delays, for both models, numerical results show that there is a critical value such that if the sum of the adoption delays exceeds this value, there is an onset of oscillations of market shares, through a Hopf bifurcation.
Highlights
Market share dynamics under advertising in duopolies have a long history, starting with the classical Vidale-Wolfe [1] and Arrow-Nerlove [2] models for monopolies, which have been generalized and studied intensively over the last sixty years, as can be seen in the survey [3]
Inspired by the discussion in [9], we make the following definition, assuming that the advertising effort is defined as a function of the market share: Definition 1 (Implementation Delay): is said to occur when the market share information utilized to define advertising policy is lagged or delayed with respect to the instant when the latter is applied
The timeline diagram (Figure 1) clarifies the definition: From the definition, denoting u (x(t − τ )) as uτ, it follows that the following modification to (1): x = f (x, uτ ) − g(x) represents a market share dynamics model with uniform implementation delay, meaning that information on market share of each firm becomes available with the same delay to both firms
Summary
Market share dynamics under advertising in duopolies have a long history, starting with the classical Vidale-Wolfe [1] and Arrow-Nerlove [2] models for monopolies, which have been generalized and studied intensively over the last sixty years, as can be seen in the survey [3]. A salient feature of all existing market share dynamics models with controls is that they assume instant access to market information as well as an immediate effect of advertising These are not realistic assumptions and consideration of. The contribution of this paper is to define two types of delays, named implementation and adoption delays, and introduce them into the Vidale-Wolfe-Deal and extended Lanchester models. Once this is done, as with all delay models, one of the central questions is to examine the effect of the delays on the asymptotic behavior of the system. The results in this paper were obtained in the PhD thesis of the first author and are unpublished but, under university policy, available on the university thesis archive server [25]
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