Post on 23-Jan-2023
Outline
A mathematical model for the diffusion ofphotovoltaics
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
Outline
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
Outline
1. The Innovation Diffusion Theory
2. A model for the diffusion of photovoltaics
3. 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
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
The diffusion of an innovation is 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
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
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
Marketing problems
The Bass model, 1969
dA(t)
dt= p(m − A(t)) +
q
mA(t)(m − A(t)) (1)
p is 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
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
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 actually
the costs of photovoltaics are still so high and the incentives aresubstantially 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
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
THE MODEL
Aim
We will propose a time delay model to simulate the adoption processwhen 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 stability of 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
THE MODEL
The model
X The total population C consists of the adopters of thenew technology, A, and of the potential adopters, P.
X i is a government incentive and c represents theproduction costs, eη(i−c) = a is an external factor ofinfluence.
X τ 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 − τ and in theinterval [t − τ, 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
THE MODEL
The model
X ρ is the percentage of individuals that don’t adopt thetechnology after the test period, e−ρτ = k is the fractionof individuals that are still interested in the adoption afterτ .
X α is the valid contact rate of adopters with potentialadopters, δ the discontinuance rate of adopters and γ avalid contact rate between the adopters at time t andthose at time t − τ .
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
THE MODEL
The model
A(t) + P(t) = C⇓
dA(t)
dt= [a + αA(t − τ)] (C−A(t−τ))k−δA(t)+γA(t)A(t−τ), ∀τ > 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
THE MODEL
The equilibrium
A∗ =(αCk − ak − δ)−
√(αCk − ak − δ)2 − 4(γ − αk)akC
2(γ − αk)
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
The model
THE MODEL
Sufficient condition for a positive equilibrium
TheoremIf (γ − αk) < 0, that is if the growth factor due to the interpersonalcommunication inside the adopters class contributes in increasing thenumber of adopters less than the factor of imitation, then the time delaydifferential equation admits an unique positive equilibrium.
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
Stability
Stability
Letx(t) = A(t)− A∗.
Then
dx(t)
dt= a0x(t) + b0x(t − τ) + f (x(t − τ), x(t))
where
a0 = (γA∗ − δ),
b0 = (αkC − ak − 2αkA∗ + γA∗),
f (x(t − τ), x(t)) = γx(t)x(t − τ)− αkx2(t − τ).
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
Stability
Stability
Poincare-Liapunov Theorem⇓
The characteristic equation of the delayed differential equation linear partis
− λτeλτ + a0τeλτ + b0τ = 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
Stability
Stability
TheoremThe equilibrium is asymptotically stable if
τ 2 <θ2
[φ(α, δ, γ, a, k, C)]2 + 2(γA∗ − δ)φ(α, δ, γ, a, k, C)
where φ(α, δ, γ, a, k, C) =p
(αkC − δ − ak)2 − 4akC(γ − αk) and θ is theroot of θ = (γA∗ − δ)τ tan θ, such that(
0 < θ < π2
when A∗ > δγ,
π2
< θ < π when δγ− φ(α,δ,γ,a,k,C)
2γ< A∗ < δ
γ.
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
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
Stability switch
Procedure
Consider the following first order characteristic equation
D(λ, τ) = a(τ)λ + b(τ) + c(τ)e−(λτ) = 0, ∀τ > 0 (3)
wherea(τ) = −τ , b(τ) = τa0(τ) and c(τ) = τb0(τ)and the condition b(τ) + c(τ) 6= 0 is fulfill because b0(τ) 6= a0(τ).
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
Stability switch
Procedure
If λ = iω with ω > 0 is a root of our characteristic equation, we musthave
F (ω, τ) := τω2 + (τa0(τ))2 − (τb0(τ))2 = 0, ∀τ > 0 (4)
ω(τ) =√
b0(τ)2 − a0(τ)2
which is true only if
αk√
(αCk − ak − δ)2 − 4(γ − ak)akC + γ(αCk − ak + δ)
αkδ> 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
Stability switch
Procedure
Consider
sinϑ(τ) =ω(τ)a(τ)
c(τ)=
√b0(τ)2 − a0(τ)2
b0(τ)
and
cosϑ(τ) = −b(τ)
c(τ)= −a0(τ)
b0(τ)
where ϑ(τ) ∈ (0, 2π)
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
Stability switch
Switches
Set
τn(τ) =ϑ(τ) + 2nπ
ω(τ)for n ∈ {0, 1, ...}
Then STABILITY SWITCHES occur at the zeros of the followingfunction
Sn(τ) = τ − τn(τ), n ∈ {0, 1, ...}. (5)
Since it is difficult to obtain the zeros of Sn(τ) analytically, we fix theparameters and use numerical calculations.If n = 0, C = 20, δ = 0.2, α = 0.1, a = 0.1, k = e−0.2τ and γ = 0.05,then τ = 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
Part IV
Conclusions
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
Conclusions
Conclusions
Conclusions I
Conclusions
I We have proposed a mathematical model with stage structure tosimulate adoption processes. The model includes the awarenessstage, the evaluation stage and the decision-making stage.
I 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).
I 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 Foggia
A mathematical model for the diffusion of photovoltaics
Conclusions
Conclusions
Conclusions I
Conclusions
I 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 Foggia
A mathematical model for the diffusion of photovoltaics
Future Research
Part V
Future Research
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
Future Research
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 Foggia
A mathematical model for the diffusion of photovoltaics
References
References I
F.M. Bass.A new product growth model for consumer durables.Management Sci., 15:215–227, 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 withdelay-dependent parameters.SIAM J. Math.Anal., 33:1144–1165, 2001.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics
References
References II
D. de Solla Price.The science/technology relationship, the craft of experimentalscience, and policy for the improvement of high technologyinnovation.Research Policy, 13:3–20, 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 Foggia
A mathematical model for the diffusion of photovoltaics
References
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, 30:129–146, 2006.
Viviana Fanelli, Lucia Maddalena Dipartimento di Scienze Economiche Matematiche e Statistiche Universita di Foggia
A mathematical model for the diffusion of photovoltaics