RE-INTERPRETATION OF THE DIFFUSION MODEL - KOASAS

RE-INTERPRETATION OF THE DIFFUSION MODEL:

A MICROECONOMIC APPROACH

Sang-June Park¹ Minhi Hahn²

1 School of Business & Information, Dongseo University, San 69-1, Churye-2 Dong, Sasang-Gu, Pusan 617-716 Korea.

2Graduate School of Management, Korea Advanced Institute for Science and Technology (KAIST), 207-43 Cheongryangri-Dong, Dongdaemun-Gu, Seoul 130-012 Korea.

ABSTRACT

A potential adopter of an innovation must decide when to adopt. Authors assume that the consumer actively chooses the adoption time that maximizes his/her utility. The utility of newness of the innovation decreases while that of certainty of the innovation increases over time. Also, they assume that each potential adopter perceives greater certainty of the innovation by observing others’ adoption. Under such microeconomic assumptions, they show that the adoption probability in diffusion models can be expressed as a Nash equilibrium.

KEYWORDS

Diffusion; Nash Equilibrium; Utility

1. Introduction

The diffusion is the process by which an innovation is communicated through certain channels over time among the members of a social system (Rogers [13] p.6). Fourt and Woodlock [4] propose that the diffusion process is primarily driven by mass-media communication, whereas Mansfield [10] suggests that the process is primarily driven by word- of-mouth communication. Bass model [1], which has been a main impetus for variety of diffusion research in marketing, assumes that the diffusion is driven by both the mass-media and word of mouth channels. The Bass model

fits the empirical adoption curve quite well for a variety of new products and technological innovations. But, Russell [14] and other economists insist that the Bass Model is incomplete. The Bass model does not incorporate the contagion effects of traditional economic variables such as price. Furthermore, it is not based on typical microeconomic assumptions.

Bass, Krishnan, and Jain [2] show that the Bass model fits the adoption curve very well without such economic variables when the changes of the variables are regular. Also, various extended-Bass models have been developed to incorporate the effects of marketing variables and other economic variables into the Bass model.

Several researchers (i.e., Hiebert [5], Jensen [6], Stoneman [15], Oren and Schwarz [11], Roberts and Urban [12]) attempted to develop diffusion models under microeconomic assumptions. Such diffusion models are built upon aggregating the demand of individual consumers who behave in a neoclassical microeconomic way. The models assume that potential adopters are smart and are not just carriers of information. They assume that the potential adopters maximize some objective functions such as expected utility or benefit coming from the new product adoption. The models also take the uncertainty of the new products into account (Mahajan, Muller, and Bass [9]).

However, current diffusion models do not explain how the Bass model can be described under neoclassical microeconomic assumptions explicitly. Also, these models do not consider a fact that a consumer’s utility of each alternative (available adoption time) depends on the other consumers' adoption behavior. A consumer is feels more certainty about the new product when they observe the adoption of other consumers. In general, certainty has utility.

In this article, we assume that the consumer strategically chooses the adoption time. Under such a microeconomic assumption, we show that the adoption probability of diffusion models can be expressed as a Nash equilibrium. Thus, we re-interpret the diffusion models under a microeconomic perspective.

2. Model Development

Let us consider a situation where consumers must choose the adoption time after an innovation is introduced in the market. Following the diffusion literature, the number of consumers in the market determines the market potential.

We assume that there are two key factors that each consumer considers when he or her determines the adoption time.

The first is perceived newness of an innovation. The newness inherently involves some degree of uncertainty (Rogers [13], p.6). The second is perceived certainty of the innovation that a consumer gets from external communication sources. The level of perceived newness is decreased over time because it is negatively correlated to the number of adopters. Thus, the earlier a consumer adopts an innovation, the higher utility of newness he or she feels from adopting it. On the other hand, the level of perceived certainty is increased over time because it is positively correlated to the number of adopters.

Thus, we assume that a consumer determines the possible adoption time by considering the gain of utility due to increased certainty and the loss of utility due to the decreased newness over time. Separating the two factors, i.e., newness and certainty, will give us better insights into understanding consumers’ new product adoption behavior.

2.1 Modeling Certainty over Time

As a theory of communications, the diffusion theory mainly focuses on the effects of two communication channels, i.e., means by which information about innovation is transmitted to within a social system. Those means include both the mass media and the interpersonal channel (Mahajan, Muller, and Bass [9]). Let us define the amount of certainty from the communication channels that consumer i gets at time t as Ii(t). We assume that Ii(t) consists of two parts.

Ii (t)=p+qF-i (t), (1) where F-i (t) represents the other consumers’ adoption.

We assume that the amount of information from mass-media, p, is constant over the time horizon. Also, we assume that the amount of information the consumer receives from communication with previous adopters at time t is proportional to the normalized number of the previous adopters. The parameter, q, is the relative importance or sensitivity of the source of information. Then, the cumulative amount of certainty that a consumer i gets at time t is:

CIi(t)=pt+q

òòòò

₀^{t F-i} (u)du. (2) It is similar to the concept of goodwill in advertising studies. However, we do not consider a discount factor in the model for the purpose of parsimony. Equation (2) implies that the level of the certainty the consumer i feels depends on the other consumers’ adoption behavior.

2.2 Modeling Newness over Time

We assume that the level of newness is negatively related to the number of the previous adopters. That is, the level of newness consumer i perceives at time t, denoted as CNi(t), is proportional to the number of the remaining potential adopters. Let the marginal newness a consumer perceives is proportional to the percent change of the size of the remaining market potential that is expressed as d/dt ln [1-F-i(t)].

d/dt CNi (t) =r d/dt ln [1-F-i(t)], (3) where r is a positive constant. From equation (3), we can derive the following equation.

CNi (t)=C+r ln [1-F-i(t)], (4) where C is the integration constant. Equation (4) means that level of the newness is exponentially decreasing over time.

2.3. Optimal Adoption Strategy of Each Consumer

Let us assume that both the perceived newness and certainty are directly related to the utility a consumer perceives.

Then, we know that the consumer i's utility (or pay-off) is determined by other consumers’ adoption from equation (2) and equation (4). The consumer i chooses the timing of adoption such that his or her payoff is maximized. The decision-making problem can be written as:

Max t Ui (t) = CIi(t) + CNi (t). (5) Then, the optimal adoption time of consumer i satisfies the following conditions.

CIi ´(t) +CNi ´(t) = 0 and (6) CIi (t) +CNi (t) ≤ 0. (7) Equation (6) and (7) are the first-order condition (F.O.C) and the second-order condition (S.O.C), respectively. From the F.O.C, we get the following relation:

- CNi ´(t) = r [F-i´(t)/{1-F-i(t)}] = p+qF-i(t) = CIi´(t). (8) Equation (8) describes that the optimal decision is to choose the adoption time such that the marginal newness is equal to the marginal certainty the consumer perceives from communication sources. From Appendix, we know that

CIi (t) +CNi (t) = 0. (9) Equation (9) suggests that the consumer i’s utility (payoff) is constant regardless of the chosen adoption time, if other consumers choose the adoption strategy that satisfies equation (8). However, there does not exist Nash equilibrium unless the consumer i’s strategy is identical to the others’ strategies. In addition, he or she does not have any incentive to deviate from the equilibrium if he or she chooses such a strategy. This argument is verified by Equation (9). Thus, in the market, the equilibrium can be computed from a following nonlinear differential equation.

F´(t)/{1-F(t)} = p/r +q/r F(t), (10) where F(t) is an adoption strategy that all the potential consumers choose. Equation (10) is the same as the hazard rate of the Bass model when r=1. It is the same as that of the Mansfield's model when p=0 and r=1. Solving the differential equation, we can get an optimal adoption strategy as follows:

F(t) = [ 1- exp{-(α+β)t}]/[1+(β/α)exp{-(α+β)t}], (11) where α=p/r and β=q/r. Equation (11) implies that a pure strategy cannot be Nash equilibrium. That is, only a mixed strategy can be Nash equilibrium in the market.

4. Discussion

In this article, we re-interpreted the adoption probabilities of Bass-type diffusion models under a microeconomic perspective. We assumed that potential adopters are not just carriers of information but active decision-makers who maximize some objective functions regarding an innovation adoption. We also assumed that the adopters strategically choose the best adoption strategy, which is the adoption probability over time, because a potential adopter’s payoff depends on the others’ strategies. Under the assumption, we showed that the adoption probability in the Bass model can be interpreted as Nash equilibrium in the market. We believe that our effort provides another theoretical base for the diffusion models.

REFERNCES

[1] Bass, Frank M. (1969), “A New Product Growth Model for Consumer Durables,” Management Science, 15 (January), 215-27.

[2] Bass, Frank M., T. V. Krishnan, and Dipak C. Jain (1994), “Why the Bass Model Fits Without Decision Variables,” Marketing Science, 13 (Summer), 203-23.

[3] Ben-Akiva, Moshe and Steven R. Lerman (1985), Discrete Choice Analysis: Theory and Application to Travel Demand, M.I.T Press.

[4] Fourt, L. A. and J. W. Woodlock (1960), “Early Prediction of Market Success for Grocery Products,” Journal of Marketing, 25 (October), 31-8.

[5] Hiebert, L. Dean (1974), "Risk, Learning and the Adoption of Fertilizer Responsive Seed Varieties," American Journal of Agricultural Economics, 56 (November), 764-8.

[6] Jensen, Richard (1982), "Adoption and Diffusion of an Innovation of Uncertain Profitability," Journal of Economic Theory, 27, 182-93.

[7] Kalish, Shlomo (1985), "A New Product Adoption Model with Price, Advertising, and Uncertainty," Management Science, 31 (December), 1156-1584.

[8] McFadden, Daniel (1974), "A Conditional Logit Analysis of Qualitative Choice Behavior," in Frontiers of Econometrics, P. Zarembka, ed. New York: Academic Press, Inc. , 105-142.

[9] Mahajan, V., Eitan Muller, and Frank M. Bass (1990), “New Product Diffusion Models in Marketing: A Review and Directions for Research,” Journal of Marketing, 54 (January), 1-26.

[10] Mansfield, E. (1961),"Technical Change and the Rate of Imitation," Econometrica, 29 (October), 741-766.

[11] Oren, Shumel S. and Risk G. Swartz (1988), "Diffusion of New Products in Risk-Sensitive Markets," Journal of Forecasting, 7 (October-December), 273-87.

[12] Roberts, John H and Glen L. Urban (1988), "Modeling Multiattribute Utility, Ristk, and Belief Dynamics for New Consumer Durable Brand Choice," Management Science, Vol. 34, No. 2, 167-185.

[13] Rogers, Everett M. (1983), Diffusion of Innovation, 3rd ed. New York: The Free Press.

[14] Russell, Thomas (1980), "Comments on "The Relationship Between Diffusion Rates, Experience Curves, and Demand Elasticities for Consumer Durable Technological Innovations," Journal of Business, 53, 69-78.

[15] Stoneman, P. (1981), "Intra-Firm Diffusion, Bayesian Learning and Profitability," Economic Journal, 91 (June), 375-88.

APPENDIX: THE SECOND ORDER CONDITION

CIi ´(t) = p + q F-i(t) and (A1) CNi ´(t)= r [F-i´(t)/{1- F-i´(t)}]. (A2) Thus,

CIi (t) = q F-i ´(t) and (A3) CNi (t) = r [F-i (t){1- F-i(t)}+{ F-i´(t)} ²] / [1- F-i(t)] ² (A4) From the F.O.C, we know that

F-i´(t) = [1- F-i(t)] [p/r + (q/r) F-i (t) ]. (A5) From equation (A5), we can derive equation (A6).

F-i (t) = - F-i´(t) [p/r + (q/r) F-i (t) ] + [1- F-i(t)](q/r) F-i´(t) (A6)

= [ (q-p)/r – (2q/r) F-i (t) ] [1- F-i(t)] [(p/r)+(q/r) F-i(t)].

Then, we know that

CIi (t) = q [1- F-i(t)] [(p/r)+(q/r) F-i(t)] and (A7) CNi (t) = r [ (q-p)/r – (2q/r) F-i(t) ] [(p/r)+(q/r) F-i(t)] + r [(p/r)+(q/r) F-i(t)] ². (A8) Finally, we can derive a following equation.

CIi (t) + CNi (t) = 0. (A9)