The Role of Social Norms for the Diffusion of Eco-Innovations: Tipping Point, and Lock-in Effects

Abstract

In the innovation literature, paradigm changes in supply have been elaborated during the last three decades, while interdependencies between technology competition and social norm changes on the demand-side have received less attention. This paper investigates the concept of the social norm to model green product diffusion. It offers a social perspective on the systemic phenomena of tipping point and lock-in effects in relation to green product diffusion; this is our first contribution. Social interaction effects of distinct technology adoption patterns are conceptualized as social norm competition. We apply the method of simulation based theory building, to test the system behavioral implications of the postulated nonlinear socio-technical norm effect. We show that this conception provides an endogenous explanation of tipping behavior in s-shaped diffusion m1odels. This complements pure probabilistic technology diffusion models that neglect both endogenous and social influences on adoption decisions. We perform simulations for two and three competing technologies, using the example of vehicle fleet penetration with alternative drivetrain technologies. We show that the critical mass and the transition pathway is path dependent. Our second contribution is the specification of the critical mass within distinct socio-technical norm regimes. We apply a mathematical analysis of the technological landscape potential to visualize the characteristics of the tipping point. The tipping point is explained by the built up of a critical mass of users that signal a new socio-technical norm fostering transition to irreversible substitution. The offered approach and perspective is intended to be useful for effective long term policy making and to enhance the intuition about feedback rich sustainability transitions.

Appendix

1.1 Derivation of the Potential for the Three Competing Technologies

In Fig. 5.9, the stock and flow structure for drivetrain technology i is shown twice: the picture to the right introduces short names for the model variables which are used in the equations below. The index i represents one of the drivetrain technologies considered in the model.

Fig. 5.9

Stock and flow structure as it is applied to every technology platform with abbreviated variable names.

In Subsection 2, we have introduced Eq. 5.2 of the rate for X _i , which reads

$$ \frac{d{X}_i}{ dt}=\frac{C{A}_i*\left(S{N}_i+I{A}_i-S{N}_iI{A}_i\right)}{{\displaystyle {\sum}_kC{A}_i*\left(S{N}_k+I{A}_k-S{N}_kI{A}_k\right)}}\Big({\displaystyle \sum_j\frac{X_j}{\tau_j}+ NS\Big)-\frac{X_i}{\tau_i}} $$

(5.3)

For simplification we

• Set the net new sales ns equal to 0, keeping the total number of vehicles in the fleet constant.

• Normalize Eq. 5.3 to the total number of vehicles N in the whole fleet: ${x}_i=\frac{X_i}{N}.$ x _i represents the fleet share of technology i.

• Use the same lifetime τ for all drivetrain technologies.

• Assume that all involved technologies have the same comparative attractiveness. Therefore the term CA _i is cancelled out.

These simplifications yield

$$ \frac{d{x}_i}{ dt}=\frac{S{N}_i\left(1-a{n}_i\right)+I{A}_i}{{\displaystyle {\sum}_k\Big(S{N}_k\left(1-I{A}_k\right)+I{A}_{k\Big)}}}{\displaystyle \sum_j\frac{x_j}{\tau }}-\frac{x_i}{\tau }. $$

(5.4)

The social norm SN _i is given as a function of the fleet share x _i . As Ulli-Beer et al. (2010) point out, that the behavioral norm should be represented by a nonlinear function. The easiest nonlinear case is a pure quadratic function (see the Results section and Appendix):

$$ S{N}_i=f\left({x}_i\right)={x}_i^2 $$

This yields

$$ \frac{d{x}_i}{ dt}=\frac{x_i^2\left(1-a{n}_i\right)+I{A}_i}{{\displaystyle {\sum}_k\left({x}_k^2\left(1-I{A}_k\right)+I{A}_k\right)}}{\displaystyle \sum_j\frac{x_j}{\tau }}-\frac{x_i}{\tau }. $$

In our 3-drivetrain system with constant fleet we have x₃ = 1−x₁−x ₂ and therefore $\frac{d{x}_3}{ dt}=-\frac{d{x}_1}{ dt}-\frac{d{x}_2}{ dt},$ leading to two independent rate equations:

$$ \begin{array}{l}\frac{d{x}_1}{ dt}=\frac{1}{\tau}\Big(-{x}_1+\dots \\ {}\dots +\frac{x_1^2\left(1-a{n}_1\right)+a{n}_1}{x_1^2\left(1-a{n}_1\right)+{x}_2^2\left(1-a{n}_2\right)+{\left(1-{x}_1-{x}_2\right)}^2\left(1-a{n}_3\right)+a{n}_1+a{n}_2+a{n}_3}\end{array} $$

$$ \begin{array}{l}\frac{d{x}_2}{ dt}=\frac{1}{\tau}\Big(-{x}_2+\dots \\ {}\dots +\frac{x_2^2\left(1-a{n}_2\right)+a{n}_{21}}{x_1^2\left(1-a{n}_1\right)+{x}_2^2\left(1-a{n}_2\right)+{\left(1-{x}_1-{x}_2\right)}^2\left(1-a{n}_3\right)+a{n}_1+a{n}_2+a{n}_3}\end{array} $$

Following the lightweight-ball metaphor introduced by Ulli-Beer et al. (2010) and extending it to three competing technologies leads to the elegant form including a potential V(x ₁ ,x ₂ ):

$$ \left(\begin{array}{c}\hfill \frac{d{x}_1}{ dt}\hfill \\ {}\hfill \frac{d{x}_2}{ dt}\hfill \end{array}\right)=-\nabla V\left({x}_1,{x}_2\right)=\left(\begin{array}{c}\hfill -\frac{\partial V\left({x}_1,{x}_2\right)}{\partial {x}_1}\hfill \\ {}\hfill -\frac{\partial V\left({x}_1,{x}_2\right)}{\partial {x}_2}\hfill \end{array}\right) $$

(5.5)

However, it is not possible to define a global potential V(x ₁ ,x ₂ ), satisfying Eq. 5.5. This can easily be shown by calculating the rotation of the left-hand side.

Although no global potential V(x ₁ ,x ₂ ) exists, we can find an approximated potential U(x ₁ ,x ₂ ). For this purpose we use a general fourth order function of x ₁ and x ₂ of the following form:

$$ \begin{array}{l}U\left({x}_1,{x}_2\right)={a}_1{x}_1-{a}_2{x}_1^2+{a}_3{x}_1^3-{a}_4{x}_1^4+{b}_1{x}_2+f{x}_1{x}_2-c{x}_1^2{x}_2\\ {}\kern4.7em -{b}_2{x}_2^2-d{x}_1{x}_2^2+e{x}_1^2{x}_2^2+{b}_3{x}_2^3-{b}_4{x}_2^4\end{array} $$

There is no constant term because this would just cause a translation of the potential function, and can be omitted. We now insert τ = 15 and IA _i  = 0.02, for all i, and set up a homogeneous system of equations using the roots of the dynamic equations. Solving for the parameters a₁, a₂, a₃, a₄, b₁, b₂, b₃, b₄, c, d, e, f leads to the approximated potential

$$ \begin{array}{l}U\left({x}_1,{x}_2\right)=0.4{x}_1-11.9{x}_1^2+23.6{x}_1^3-12.1{x}_1^4+0.4{x}_2+{x}_1{x}_2-1.9{x}_1^2{x}_2\\ {}\kern4.8em -11.9{x}_2^2-1.9{x}_1{x}_2^2+23.9{x}_1^2{x}_2^2+23.6{x}_2^3-12.1{x}_2^4\end{array} $$

1.2 Quadratic Versus S-Shaped Norm Function

To keep the mathematical analysis simple we used a quadratic function for the social behavioral norm. With a logistic function for SN _i , Eq. 5.4 would read

$$ \frac{d{x}_i}{ dt}=\frac{\frac{1-I{A}_i}{1+{e}^{-10\left(-0.5+{x}_i\right)}}+I{A}_i}{{\displaystyle {\sum}_{k=1}^3\left(\frac{1-I{A}_k}{1+{e}^{-10\left(-0.5+{x}_k\right)}}+I{A}_k\right)}}{\displaystyle \sum_{j=1}^3\frac{x_j}{\tau }-}\frac{x_i}{\tau }. $$

Without giving a profound analysis, we illustrate the similarity of the resulting potentials and behavior using the underlying force field as in Fig. 5.7. These are plotted in Fig. 5.10b for the s-shaped and in Fig. 5.10a for the quadratic norm function respectively. Qualitatively the two yield the same results in the area of interest.

Fig. 5.10

Driving force field plot with a quadratic social norm function (a) and its equivalent with an s-shaped social norm function (b), here a logistic function