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

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

explaining the probability of adoption
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

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