Fanelli Maddalena
Transcript of Fanelli Maddalena
-
8/13/2019 Fanelli Maddalena
1/38
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
http://find/ -
8/13/2019 Fanelli Maddalena
2/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
http://find/ -
8/13/2019 Fanelli Maddalena
3/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
http://find/ -
8/13/2019 Fanelli Maddalena
4/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
5/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
O li
http://find/ -
8/13/2019 Fanelli Maddalena
6/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
O tli
http://find/ -
8/13/2019 Fanelli Maddalena
7/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
Outline
http://find/ -
8/13/2019 Fanelli Maddalena
8/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/ -
8/13/2019 Fanelli Maddalena
9/38
The Innovation Diffusion Theory
Part I
The Innovation Diffusion Theory
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/ -
8/13/2019 Fanelli Maddalena
10/38
The Innovation Diffusion Theory
Thediffusion of an innovationis the process by which an innovation iscommunicated through certain channels over time among the membersof a social system (Rogers, 2005)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/ -
8/13/2019 Fanelli Maddalena
11/38
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)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
12/38
y
Marketing problems
Marketing problems
1960:To develop marketing strategy in order to promote the adoption of a newproduct and penetrate the market.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/ -
8/13/2019 Fanelli Maddalena
13/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/ -
8/13/2019 Fanelli Maddalena
14/38
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)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The Innovation Diffusion Theory
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
15/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/ -
8/13/2019 Fanelli Maddalena
16/38
Part II
A model for the diffusion of photovoltaics
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/ -
8/13/2019 Fanelli Maddalena
17/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/ -
8/13/2019 Fanelli Maddalena
18/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
19/38
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 .
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/ -
8/13/2019 Fanelli Maddalena
20/38
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)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/ -
8/13/2019 Fanelli Maddalena
21/38
THE MODEL
The equilibrium
A =(Ck ak )
(Ck ak )2 4( k)akC
2( k)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
22/38
THE MODEL
Sufficient condition for a positive equilibrium
TheoremIf( k)
-
8/13/2019 Fanelli Maddalena
23/38
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 ).
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The stability of the equilibrium
http://find/ -
8/13/2019 Fanelli Maddalena
24/38
Stability
Stability
Poincare-Liapunov Theorem
The characteristic equation of the delayed differential equation linear part
is
e +a0e +b0= 0
Hayes Theorem
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The stability of the equilibrium
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
25/38
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
-
8/13/2019 Fanelli Maddalena
26/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The stability of the equilibrium
http://find/ -
8/13/2019 Fanelli Maddalena
27/38
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().
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The stability of the equilibrium
http://find/ -
8/13/2019 Fanelli Maddalena
28/38
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
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The stability of the equilibrium
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
29/38
Stability switch
Procedure
Consider
sin() = ()a()
c()
=
b0()2 a0()2
b0()and
cos() =b()
c() =
a0()
b0()
where () (0, 2)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The stability of the equilibrium
http://find/ -
8/13/2019 Fanelli Maddalena
30/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
Conclusions
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
31/38
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
http://find/http://goback/ -
8/13/2019 Fanelli Maddalena
32/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
Conclusions
http://find/ -
8/13/2019 Fanelli Maddalena
33/38
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).
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
Future Research
http://find/ -
8/13/2019 Fanelli Maddalena
34/38
Part V
Future Research
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
Future Research
http://find/ -
8/13/2019 Fanelli Maddalena
35/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
References
http://find/ -
8/13/2019 Fanelli Maddalena
36/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
References
http://find/ -
8/13/2019 Fanelli Maddalena
37/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
References
http://find/ -
8/13/2019 Fanelli Maddalena
38/38
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.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di FoggiaA mathematical model for the diffusion of photovoltaics
http://find/