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Original Research ARTICLE

  1. Article excerpt
  2. Upcoming Events
  3. A Simple Mathematical Model of War Events - Persée
  4. Comparison and Analysis of Diffusion Models
  5. Modelling Innovation Diffusion

Learn more. The article presents theoretical and empirical research findings which incorporate price and replacement purchases in new product diffusion models. On the theoretical side, this paper characterizes, qualitatively, optimum pricing policies for new products. Possible entry of rivals is not considered, but repeat sales, cost learning dynamics and discounting of future profit streams are allowed. Theoretical research findings suggest that the inclusion of repeat purchases in the diffusion model significantly changes the derived optimal pricing policy even if replacements were not price dependent.

On the empirical side, alternative first purchase and repeat purchase models have been estimated and compared using nonlinear procedures. The diffusion data analyzed is related to nine consumer durables. Empirical research findings suggest that, for the considered product categories, diffusion is basically an imitative process, price can affect first and replacement purchases, and unit production cost is a decreasing function of cumulative first purchases.

Managerial implications of the research findings are also discussed. Volume 26 , Issue 4. If you do not receive an email within 10 minutes, your email address may not be registered, and you may need to create a new Wiley Online Library account. If the address matches an existing account you will receive an email with instructions to retrieve your username. Decision Sciences Volume 26, Issue 4. Tools Request permission Export citation Add to favorites Track citation.

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Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract The article presents theoretical and empirical research findings which incorporate price and replacement purchases in new product diffusion models.

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Citing Literature. Volume 26 , Issue 4 July Pages Related Information. Appendix B. The main finding of the last evaluated case, with 5. The need for stochastic analysis observably from the actual future values. This is because the population size of the market that slows down the Despite the fact that diffusion models often succeed in diffusion rate does not participate as a parameter in any describing a diffusion process, stochastic considerations of the other evaluated benchmark models.

Nevertheless, they are of major im- In Table 2, the estimated market potentials for all of portance, due to the existence of rapidly changing environ- the evaluated models and the participating countries are mental socioeconomic factors, which in turn affect the dif- presented. Moreover, quite accurately, especially when the conventional models the deterministic realizations of diffusion models are able provide observably diverging estimates.

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Since many of the external parameters are random the uncertainties that accompany a diffusion process. The in nature, they cannot be estimated accurately and used for first is based on the assumption that the parameters of an forecasting purposes. Table 2 Comparison of the estimated market potential for all models. Logistic Gompertz Bass Population Austria 1. The stochastic differential equation described by Eq. There- 5. Model development fore, in order to estimate its parameters, a suitable trans- formation should be applied so that the sde can be approx- Stochastic differential equations SDEs are used quite imated by an equivalent linear sde, based on the available frequently, and in a wide range of fields of application.

As was mentioned in the introductory section, the work presented by Eliashberg and Chatterjee was one of 5. Local linearization the earliest studies to propose the need to employ SDEs in the estimation and forecasting of diffusion processes. The analysis is based on the addition of Nikolau, ; Ozaki, ; Singer, For the needs of a noise term to the initially developed model, which is the present work, the local linearization method proposed represented by a Wiener process.

Wiener processes play by Shoji and Ozaki is adopted, as it provides a a vital role in stochastic calculus and diffusion processes. According This analysis is important as a measure of the volatility to this method, the original stochastic differential equation of the deterministic process, since diffusion is frequently is locally approximated by a linear stochastic differential characterized by uncertainty.

Thus, the discretized process can Wiener processes Gardiner, ; Oksendal, , in ad- be obtained by the discretization of the sample path. Again, as in the deterministic formulation of volatility, or the width of the noise of the process. Linearization The linearization of Eq. After substituting Eqs. This can diffusion model, based on the linearized approach of Eq. However, the stochastic Explicit formulas for the calculation of the parameter analysis revealed that possible values vary from about 1.

This stochastic analysis, together with the corre- exist, as is the case for the function in Eq. Genetic algorithms, as posited by Goldberg , are search algorithms based on the mechanisms of natural 7. Conclusions and future work selection and natural genetics. The key points of the process are reproduction, crossover and mutation, which The aim of this research was to contribute to the ex- are performed according to a given probability, just isting knowledge regarding diffusion estimation and fore- as happens in the real world.

Reproduction involves casting, for both research and practice. This was attempted copying reproducing solution vectors, crossover involves by developing a diffusion model which explicitly incor- swapping partial solution vectors, and mutation is the porates the size of the market, as expressed by the cor- process of randomly changing a cell in the string of responding population.

The two formulations of the pro- the solution vector, thus preventing the possibility of posed PDM, both deterministic and stochastic, provide the algorithm being trapped. The process continues until quite accurate results in terms of diffusion estimation and it reaches the optimal solution of the fitness function. In the context of the present work, the fitness function In such cases, the widely used diffusion models cannot to be optimized is the log-likelihood function, described by predict the saturation level accurately, mainly because the Eq.

The proposed model historical data. After , diffusion slowed down, as the mar- early stages of the corresponding diffusion processes. More- on initial values equal to the ones estimated in the over, the estimation of the underlying level of uncertainty deterministic case of the model. In addition, other, will stimulate further research, in order to produce more randomly chosen, initial values were also used, in order to accurate forecasts.

Simulation of the stochastic formulation of the population model. The dashed line corresponds to the mean value of the process, which coincides with the results of the deterministic formulation of the model. Table 3 Percentage diffusion over the population of mobile telephony for countries in the wider European area. Source: ITU. The results are enhanced by forecasting further. In this direction, the development of the estimation of the level of uncertainty, provided by the appropriate price elasticity functions and their incorpo- stochastic analysis.

Thus, the model will enhance our ability to develop effective strategies for introducing and adopting diffusion framework would provide important managerial new technologies. As with any model, this model has certain limitations that need to be investigated in future work, the first of which is the population size, which was considered to be Appendix A. PDM evaluation results constant over the evaluation period. Therefore, incorporat- ing the population rate of change would be expected to im- This appendix presents the detailed evaluation results prove the forecasting and estimation of the market satu- from the PDM.

Thus, a framework accommodating the flex- ibility of the market ceiling, due to a varying population, Moreover, the evaluation results from the Gompertz, should be developed. In addition, sumption that decision variables, such as price, are ex- the historical data for the evaluation process are given in ogenous to the system.

Allowing these variables to be Table 3.


A Simple Mathematical Model of War Events - Persée

PDM forecasting results In this appendix, the forecasting ability of the proposed participating models, namely the Gompertz, linear logistic model is evaluated, together with the results of the other and Bass models. A simple substitution model of technological change. A new product growth model for consumer durables. Management Science, 15, — Gardiner, C. Handbook of stochastic methods for physics, Bemmaor, A.

Comparison and Analysis of Diffusion Models

The impact of heterogeneity and chemistry, and the natural sciences 3rd ed. Berlin: Springer. Marketing Geroski, P. Models of technology diffusion. Research Policy, 29, Science, 21, — Bergstrom, A. Continuous time econometric modelling. Oxford: Giovanis, A. A stochastic logistic innovation Oxford University Press.

Modelling Innovation Diffusion

A flexible logistic growth-model the United States. Technological Forecasting and Social Change, 61, with applications in telecommunications. International Journal of — Forecasting, 4, — Goldberg, D. Genetic algorithms in search, optimization, and Box, G. A note on the generation of random normal machine learning.

Reading, Mass: Addison-Wesley. The Annals of Mathematical Statistics, 29, — Gompertz, B. On the nature of the function expressive of the law of Boyce, W. Elementary differential equations and human mortality, and on a new mode of determining the value of life boundary value problems 8th ed. Hoboken, NJ: Wiley. Determination of the uncertainties in , — S-curve logistic fits. Competition and innovation: the diffusion of mobile — Information Dembo, A.

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On the parameters estimation of Economics and Policy, 13, 19— The diffusion of mobile telecommunica- Transactions on Automatic Control, 32, — European Economic Review, 45, Eliashberg, J. Stochastic issues in innovation — Wind Eds. Diffusion-models for technology-forecasting. Bullinger Publishing Company. Stochastic-evolution of a non-linear Evans, L.

An introduction to stochastic differential equations. Available at: 7, 59— Kumar, V. Cross-national diffusion Fildes, R. Telecommunications demand forecasting—a research: what do we know and how certain are we? Journal of Product review. International Journal of Forecasting, 18, — Innovation Management, 15, — Multinational diffusion models: an Wrede, R. Theory and problems of advanced calculus alternative framework.

Marketing Science, 21, — New York: McGraw-Hill. Linton, J. Forecasting the market diffusion of disruptive and discontinuous innovation. Mahajan, V. Innovation diffusion and new product Christos Michalakelis holds a degree in Mathematics University of growth-models in marketing. Journal of Marketing, 43, 55— Athens, Department of Mathematics , an M.

First-purchase diffusion-models of of Informatics and Telecommunications, Interfaculty course of the new-product acceptance. He also holds a Ph. Technological substitution—framework of stochastic- and with a focus on the demand estimation and forecasting of high models. Technological Forecasting and Social Change, 36, — Michalakelis, C.

Impact He is currently a Lecturer on telecommunications technoeconomics of cross-national diffusion process in telecommunications demand in the Department of Informatics and Telematics at the Harokopio forecasting. Telecommunication Systems, 39, 51— University of Athens. Previously, he was with the Greek Ministry of Michalakelis, C.

Diffusion Education for 7 years, in the Managing Authority of Operational Program models of mobile telephony in Greece. Michalakelis has participated in a number of projects concerning the Milstein, G. Numerical integration of stochastic differential design and implementation of database systems, and now participates equations. Dordrecht: Kluwer. He tial equations in R1. Journal of Statistical Computation and Simulation, has also developed or made a major contribution to the development of a 75 8 , — Oksendal, B.

Stochastic differential equations: an introduction with Mr. Non-linear time series models and dynamical systems. Hannan Ed. He has authored a number of papers Amsterdam: North-Holland. Appropriate models for technology substitution. Journal journals and books. He serves as a reviewer for a number of scientific of Scientific and Industrial Research, 58, 14— Ruiz-Conde, E. Marketing vari- ables in macro-level diffusion models. Seber, G.

Nonlinear regression. Hoboken, New Jersey: Wiley.