Fanelli Maddalena

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Outline

A mathematical model for the diffusion of

photovoltaics

Viviana Fanelli, Lucia Maddalena

Dipartimento di Scienze Economiche Matematiche e StatisticheUniversita di Foggia

Rome, June, 30th 2008ISS on Risk Measurement and Control 2008

Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia

A mathematical model for the diffusion of photovoltaics
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Outline

Outline

1. The Innovation Diffusion Theory

2. A model for the diffusion of photovoltaics3. The stability of the equilibrium

4. Conclusions

5. Future Research

6. References


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Outline



4. Conclusions

5. Future Research

6. References


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4. Conclusions

5. Future Research

6. References


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Outline



4. Conclusions

5. Future Research

6. References



O li
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4. Conclusions

5. Future Research

6. References



O tli
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4. Conclusions

5. Future Research

6. References



Outline
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Outline

Outline



4. Conclusions

5. Future Research

6. References



The Innovation Diffusion Theory
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Part I




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Thediffusion of an innovationis the process by which an innovation iscommunicated through certain channels over time among the membersof a social system (Rogers, 2005)



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u y

The adoption process

The adoption process

Knowledge;Persuasion;Decision or evaluation;Implementation;Confirmation or adoption.

Communication channels

Historical distinctionbetween

technological break thoughts (innovations)and

engineering refinements (improvements)(De Solla Price, 1985)





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y

Marketing problems

Marketing problems

1960:To develop marketing strategy in order to promote the adoption of a newproduct and penetrate the market.



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Marketing problems

The Bass model, 1969

dA(t)

dt =p(m A(t)) +

q

m A(t)(m A(t)) (1)pis a parameter that takes into account as the new adopters joint themarket as a result of external influences: activities of firms in the market,advertising, attractiveness of the innovation,q refers to the magnitude of influence of another single adopter



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Time delay model with stage structure

Stage structure model

Time delay models with stage structure for describing the newtechnology adoption process

W. Wang, and P. Fergola, and S. Lombardo, and G. Mulone, 2006

Beretta (2001)





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Time delay model with stage structure

Photovoltaics diffusion

Photovoltaics technology: critical nature its diffusion must besupported by the government policy of incentives

The use of public subsides accelerate market penetration, but actuallythe costs of photovoltaics are still so high and the incentives are

substantially very low

The process of diffusion is stopped at the first stage and it cannot

develop.



The model
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Part II

A model for the diffusion of photovoltaics



The model
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THE MODEL

Aim

We will propose a time delay modelto simulate the adoption process

when individuals in the social system are influenced by external factors(government incentives and production costs) and internal factors(interpersonal communication). We will find a final level of adopters andwe will carry out a qualitative analysis to study the stabilityof modelequilibrium solution.



The model
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THE MODEL

The model

The total population Cconsists of the adopters of thenew technology, A, and of the potential adopters, P.

i is a government incentive and crepresents the

production costs, e(ic) =a is an external factor ofinfluence.

is the average time for an individual to evaluate if toadopt or not the technology. Then, the knowledge and theawareness of technology occur at time t and in the

interval [t , t] the individual decides whether to adoptor not the technology.



The model


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THE MODEL

The model

is the percentage of individuals that dont adopt thetechnology after the test period, e =k is the fraction

of individuals that are still interested in the adoption after.

is the valid contact rate of adopters with potentialadopters, the discontinuance rate of adopters and avalid contact rate between the adopters at time tand

those at time t .



The model
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THE MODEL

The model

A(t) +P(t) =C

dA(t)

dt = [a+ A(t )] (CA(t))kA(t)+A(t)A(t), >0

(2)



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THE MODEL

The equilibrium

A =(Ck ak )

(Ck ak )2 4( k)akC

2( k)



The model


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THE MODEL

Sufficient condition for a positive equilibrium

TheoremIf( k)


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Stability

Stability

Letx(t) =A(t) A.

Then

dx(t)dt

=a0x(t) +b0x(t ) +f(x(t ), x(t))

where

a0 = (A ),

b0 = (kC ak 2kA + A),

f(x(t ), x(t)) = x(t)x(t ) kx2(t ).



The stability of the equilibrium
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Stability

Stability

Poincare-Liapunov Theorem

The characteristic equation of the delayed differential equation linear part

is

e +a0e +b0= 0

Hayes Theorem





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Stability

Stability

TheoremThe equilibrium is asymptotically stable if

2 < 2

[(,,, a, k, C)]2 + 2(A )(,,, a, k, C)

where(,,, a, k, C) =p

(kC ak)2 4akC( k) and is theroot of= (A )tan , such that

( 0< < 2 when A > ,2

< < when

(,,,a,k,C)2


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Stability switch

Stability switch

We apply the methods proposed by Beretta (2006) and used by W.Wang,P. Fergola, S. Lombardo, and G. Mulone (2006) to demonstrate the timedelay differential equation admits stability switches.



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Stability switch

Procedure

Consider the following first order characteristic equation

D(, ) =a() +b() +c()e

() = 0, >0 (3)

wherea() =, b() =a0() and c() =b0()and the condition b() +c()= 0 is fulfill because b0()=a0().



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Stability switch

Procedure

If= i with >0 is a root of our characteristic equation, we musthave

F(, ) := 2 + (a0())2 (b0())

2 = 0, >0 (4)

() =

b0()2 a0()2

which is true only if

k

(Ck ak )2

4( ak)akC+ (Ck ak+ )k

>2





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Stability switch

Procedure

Consider

sin() = ()a()

c()

=

b0()2 a0()2

b0()and

cos() =b()

c() =

a0()

b0()

where () (0, 2)



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Stability switch

Switches

Set

n() = () + 2n

() for n {0, 1,...}

Then STABILITY SWITCHES occur at the zeros of the followingfunctionSn() = n(), n {0, 1,...}. (5)

Since it is difficult to obtain the zeros ofSn() analytically, we fix theparameters and use numerical calculations.

Ifn= 0, C= 20, = 0.2, = 0.1, a= 0.1, k=e

0.2 and = 0.05,then = 1.32.



Conclusions


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Part IV

Conclusions

Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics

Conclusions


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Conclusions

Conclusions I

Conclusions

We have proposed a mathematical model with stage structure tosimulate adoption processes. The model includes the awareness

stage, the evaluation stage and the decision-making stage.

We have found the final level of adopters for certain parameters and,from studying the local stability of the equilibrium, we have foundthat the final level of adopters is unchanged under smallperturbations (if the evaluation delay is small).

The model can be used to predict the number of adopters for certainparameters and economic and financial estimations about marketprofit may be made if the equilibrium is stable.


Conclusions
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Conclusions

Conclusions I

Conclusions

We have shown that the model admits stability switches. Thismeans the number of adopters could fluctuate in time. (This is aneffective characteristic of the new technology markets and, inparticular, of the photovoltaics market).


Future Research
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Part V

Future Research


Future Research
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Future Research

Future Research I

First The existence of stability switches suggests to investigatethe existence of periodic solutions and their stability.

Second Since there exist stochastic factors in the environment aswell as in the interior system and the action of thesefactors can cause a random adoption pattern of the newtechnology, we wish to investigate the effect of stochasticperturbations for the model.


References
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References I

F.M. Bass.A new product growth model for consumer durables.Management Sci., 15:215227, January 1969.

R. Bellman and K.L. Cooke.Differential-difference equations.Academic Press, New York, 1963.

E. Beretta and Y. Kuang.Geometric stability switch criteria in delay differential system with

delay-dependent parameters.SIAM J. Math.Anal., 33:11441165, 2001.


References
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References II

D. de Solla Price.The science/technology relationship, the craft of experimentalscience, and policy for the improvement of high technologyinnovation.Research Policy, 13:320, 1984.

J.K. Hale and S.M.V. Lunel.Introduction to Functional Differential Equations.Springer-Verlag, New York, 1993.

V. Mahajan, E. Muller, and F.M. Bass.

New-Product Diffusion Models.in Eliashberg, J, Lilien, G.L (Eds),Handbooks in Operations Researchand Management Science, Elsevier Science Publishers, New York,NY, 1993.


References
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References III

E.M. Rogers.Diffusion of Innovations.

The Free Press, New York, 4th edition, 1995.

W. Wang, P. Fergola, S. Lombardo, and G. Mulone.Mathematical models of innovation diffusion with stage structure.Applied Mathematical Modelling, 30:129146, 2006.

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