Overview & ScheduleP1: Topic AnalysisP2A: Diffusion ModelsP2B:Literature Review & Model JustificationP3:Preliminary Draft Outline

### Key Words/Interests/Concepts

At this stage, my topic is more highly-centralized with the idea of “Tick-Tock Innovation Strategy”, as described in the beginning part of the analysis below.
This analysis aims to deliver my thinkings, ideas and relevant previous researches.
— Viimeksi päivitetty / last updated: 25/09/2015

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Literature Review in Diffusion Models

### • Bass Diffusion Model (Hazard-function form)

Model #1.1 $$h(t)= \frac{ f(t)}{ 1-F(t)} = p + qF(t)$$ ,

where
▻▻ $$h(t)$$ is the conditional probability that an adoption will occur at time t given that an adoption has not yet occurred;
▻▻ $$q$$ coefficient of innovation (purchase innovator);
▻▻ $$p$$ coefficient of imitation (purchase imitator);
▻▻ $$f(t)$$ is the probability (or fraction of total population) of adoption at time $$t$$ . It is a probability density function (pdf) .
▻▻ $$F(t)$$ is cumulative density function (cdf), which implies the proportion of adopters at time $$t$$ ;

This is a stochastic differentiation equation, a (unique) solution to it, given $$F(0)=0$$ , is:
▻▻ Model #1.2 $$F(t)=\frac{ 1-e^{-(p+q)t}}{ 1+\frac{ q}{ p} e^{-(p+q)t} }$$ ;
▻▻ Model #1.3 $$f(t) = F'(t) = \frac{ \frac{ (p+q)^2}{ p} e^{-(p+q)t}}{(1+\frac{ q}{ p}e^{-(p+q)t} )^2}$$ ;
▻▻ $$t^*=\frac{ \ln q – \ln p }{ p+q}$$ is the inflection point (where S-curve changes direction)
It is obvious that the probability of adoption is sole dependent on innovation and imitation coefficient and, of course, time.

### • Gernerized Bass Diffusion Model(Bass, Krishna & Jain, 1994)

Model #2.1 $$h(t) = \big( p+qF(t) \big)\ x(t)$$ ,
Model #2.2 $$x(t) = 1+ \beta_1 \frac{ \partial P(t)}{ \partial t} +\beta_2 \max \big( 0, \frac{ \partial A(t)}{ \partial t} \big)$$,

where
▻▻ $$x(t)$$ is current marketing efforts;
▻▻ $$P(t)$$ price index;
▻▻ $$A(t)$$ advertising expenditure.

### • Other Substitutional Models with Diffusion (Fisher & Pry, 1971)

Model #3.1 $$\frac{ d}{ dt}s(t) = ks(t) (1-s(t))$$,

where
▻▻ $$s(t)$$ is market share in time $$t$$ .
▻▻ the function is somewhat the same as Bass Diffusion Model, but it is an often used model of technological substitution (Norton & Bass, 1987).
a possible solution to this SDE, given $$s(0)=\frac{ 1}{ 2}$$ , is
Model #3.2 $$s(t) =\frac{ 1}{ 1+e^{-kt}}$$

Conclusions

### • Conclusion: Regression Model Candidates

General conclusion: use hazard function (conditional probability) is an acceptable and feasible way, because:
(1) sufficient previous literature to support authencity;
(2) can simply translated into real word behavior;
(3) is a suitable modeling for my research topics;
(4) valid in both Stochastic and Econometrics, which means it is qualified for both “estimation” and “verification”.

Note: the residual terms is omitted in the following models

Candidate #1 $$h(t) = \underbrace{ \beta_0}_{publicity}+\underbrace{\beta_1}_{WOM}F(t)+ \underbrace{\beta_2 \ln A(t)}_{research\ object}$$ .
• Previous research translate $$A(t)$$ as advertising expenditure;
• The use of $$\ln, \log$$ form is technically optional.

Candidate #2 $$h(t) = \underbrace{ \beta_0}_{publicity}+\underbrace{\beta_1}_{WOM}F(t)+ \underbrace{\beta_2\ \alpha (A_t)F(t)}_{research\ object}$$ .
• Linking imitation $$F(t)$$ and (accumulation of) other independent variable $$\alpha A(t)$$
explaining market share
Candidate #3 $$\ln h^*(t)= \ln \underbrace{\frac{ s}{ 1-s}}_{s:\ market\ share}= \underbrace{\beta_0}_{publicity} + \underbrace {\beta_1\ t}_{profitability\ of \ investment} + \underbrace{\beta_2 A(t)}_{research\ object}$$.
• exponential regression model
Viimeksi päivitetty / last updated: 19/11/2015 01.30pm