#### 3.1. Basic Bass Model

Bass Model argues that when a new product is released to a market, the diffusion rate of the product is mainly determined by two factors: innovation and imitation [

14,

16]. Innovation influences the product diffusion process mainly through external sources, such as mass-media advertising. It reflects the explicit characteristics of a product, which are usually easy to be identified by consumers. Imitation, on the contrary, influences product diffusion through internal sources, e.g., word-of-mouth. It reflects some product features that are not easily identifiable by consumers, namely implicit characteristics.

According to the basic Bass model, first, let

f(

t) be the probability function of the ratio of the number of new buyers to the number of total potential buyers at time

t;

F(

t) is the ratio of the number of accumulative total buyers to the number of total potential buyers at time

t.

p is the coefficient of innovation influence,

q represents the influence of imitation, and it is assumed that

p,

q ∈ [0, 1]. Then the basic Bass Model can be specified as follows:

Next, let

m represent the maximal market potential.

n(

t)indicates the increased number of buyers at time

t, and

N(

t) indicates the accumulated number of buyers at time

t. Then the model is specified as follows:

#### 3.3. The Proposed New Model

Erkut [

21,

22] take the perspective of evolutionary economics and proposes a conceptual model that analyzes the process of how a new idea is formulated, turned into technological knowledge and finally casted a new market segment. The conceptual model includes four continuous dimensions that combine the neuroscience, cognition science, psychology, marketing knowledge, and product innovation.

The model provides foundations for our new model. Specifically, as Myers [

23], Ram [

24] and Cherrier, Black, and Lee [

25] suggests, there are always a group of consumers who are not willing to adopt the new product or new innovations. As a result, they tend to be resistant to the change of new product generations. In this study, we consider such factors by including a new coefficient,

$\alpha $, in the classic NB model. We then have the number of new adopters as follows:

The ratio of the number of accumulated buyers to the number of all potential buyers can be predicted using the following equation:

Product consistency also influences consumers’ perceived value of the new product introduction and thus influences the product diffusion process [

7]. Specifically, in this study, we define product consistency as the extent to which the new green product offering will be connected to the existing product offering in terms of market targeting, production, and channel of distribution. In our model, we assume that the product consistency between the existing product offering and the new green product is a continuous variable,

$u\in \left[0,1\right]$.

Due to different new product development strategies, the value of

u could fall into three categories (see

Figure 1). Specifically, first,

u = 0. Under this condition, the new green product is completely different from the existing product offering, which suggests that the new green product will have a different target market, new production requirement, or new channel of distribution. Thus, there is no consistency between the two product offerings. As a result, the two products will have separate diffusion process. Second, 0 <

u < 1. This condition suggests that some characteristicsof the new green product are inherited from the existing product. For example, the new green product may have similar function, or the same channel and OEM manufacture. Meanwhile, the two products also show some differences, such as sustainability of the materials used or slightly different target market. Under this condition, the performance of the new green product will influence the performance of existing product because they both use similar resources in the product diffusion process. Third,

u = 1. Under this condition, there is no significant difference between the new green product and the existing product. In other words, there is no change in the target market, production, and channel of distribution. Thus, the diffusion of the new green product is also the substitution of the existing product, which is a typical research set in the classic NB model.

Given researchers have provided solutions for scenarios one and three (i.e., u = 0 and u = 1) in previous research, this study focuses on examining scenario two (i.e., 0 < u < 1), in which there is some consistency between the new green product and the existing product.

We specify the product consistency between the two products as follows:

where

x_{1} is the product end usage consistency. Larger value of

x_{1} indicates that the two products show more similar functionalities, which suggests that the target market for the two products is more likely to be the same.

x_{2} reflects the product consistency in production, which involves the requirements for manufacture, technology, and materials. Thus, the larger the value of

x_{2} is, the more possible it is for the new green product to replace the existing one because no further resources are required to produce the new green product. Finally,

x_{3} indicates the consistency in channel of distributions. Larger

x_{3} suggests that the new green product is more likely to choose the same channel of distribution as the old product does.

To model the product diffusion process considering both consumer resistance and product consistency, we make the following assumptions:

A company sells a product on the market, then, it will introduce a green product offering to the market after a certain time of period (t > t_{0}).

Because the new green product is not completely replacing the existing product offering, thus, different from the classic NB model, our model does not require the influences of innovation, imitation, and consumer resistance are persistent. Specifically, p_{1} at time t_{0} is set not always equal to p_{1}′ at time t. Similarly, q_{1} at time t_{0} is not always equal to q_{1}′ at time t; α_{1} is not always equal to α_{2}.

According to assumption 1, the substitution process of the new green product on the old product is a two-stage process: in the first stage, a company sells the old product. The performance of the old product can be predicted using the original Bass model. In the second stage, the company introduces the new green product to the market while they still offer old product. The performance of both the green product and the old product can be predicted using the NB model. Then, we suggest that there is some consistency between the old product and the new green product, thus, we add the product consistency,

u, to the NB model. According to assumption 2, the ratio of the accumulated buyers to the total market potential,

F_{1}’(

t), is the same as the value at t

_{0}, which is

F_{1}(

t). Given

F_{2}(

t) = 0 if

t <

t_{2}, the performance of the two products can be predicted using the following model:

where,

u ∈ [0,1]. In particular, when

u = 0, the model examines the performance of two independent products while when

u = 1, the model investigates the product performance of two similar products.

Based on the proposed model (Equations (8) and (9)), we present the following propositions:

**Proposition** **1.** As the product consistency between the existing product and the new green product increases, the remaining market potential for the existing product after the introduction of green product decreases.

**Proposition** **2.** As the consumer resistance to new technology/product increases, the growth rate of the green product will decrease. In particular, when the resistance to green product is greater than one of the existing product, the peak sales potential for the existing product will increase and will reach the peak point sooner, vice-versa.