“Parametric analysis of the Bass model”

In this research, the authors explore the influence of the Bass model p , q parameters values on diffusion patterns and map p , q Euclidean space regions accordingly. The boundaries of four different sub-regions are classified and defined, in the region where both p , q are positive, according to the number of inflection point and peak of the non-cumulative sales curve. The researchers extend the p , q range beyond the common positive value restriction to regions where either p or q is negative. The case of negative p , which represents barriers to initial adoption, leads us to redefine the motivation for seeding, where seeding is essential to start the market rather than just for accelerating the diffusion. The case of negative q , caused by a declining motivation to adopt as the number of adopters increases, leads us to cases where the saturation of the market is at partial coverage rather than the usual full coverage at the long run. The authors develop a solution to the special case of p + q = 0, where the Bass solution cannot be used. Some differences are highlighted be-tween the discrete time and continuous time flavors of the Bass model and the implication on the mapping. The distortion is presented, caused by the transition between continuous and discrete time forms, as a function of p , q values in the various regions.


Introduction
The theory of diffusion of innovation, proposed by Bass (1969), has been explored, implemented and extended by numerous researches. Some researchers extend the model to explicitly consider market or industry characteristics or behavior. Others examine implementation of the model on various cases and show that adding more factors improves the capability of the model to capture complex behavior in details. Another direction taken by some researchers was to explore the factors that influence parameters values estimation and forecasts accuracy. Some researches explore empirically how the Bass model parameters vary across products and markets, usually at the ranges when p << q and 0.1< q < 0.7. Bass (1969) notes that there are two different categories of diffusion curve patterns. When q > p, the periodic sales grow until they reach a peak and, then, decline asymptotically to zero. When q < p, periodic sales decline asymptotically to zero starting from launch. Little attention has been directed since then to a further comprehensive exploration of the constraints and classification of the Bass model parameters and how they affect the adoption curve patterns.
Another issue that has attracted little attention in diffusion theory is the transition between the continuous time form and the discrete time form of the Bass model. While many researchers switch between these two flavors without further notice, attention needs to be paid to verifying, for each implementation, that this transition has little impact on parameters' values and on forecasts' accuracy.
In this paper, we perform a comprehensive exploration and classify and map the different diffusion patterns over the p, q space. The exploration is theoretic and is not limited to specific empiric cases. We refer to the different flavors (continuous vs. discrete) and highlight the differences in mapping, constraints and classification. We also define the conditions that need to be checked when switching between them.

Literature review
The diffusion of innovation model, known as the Bass model, has several flavors and numerous extensions. In this paper we focus on the basic model of Bass (1969), which has a simple analytic solution and not to extensions that are usually solved numerically. We brief the different basic models and highlight the differences between them. We use these models for mapping the patterns and constraints categories over the p, q space and show that each flavor has a slightly different map.
The Bass (1969) presents the diffusion of innovation dynamic equation: (1) This equation represents the innovation and imitation influence on the remaining potential market. The periodic sales, or the rate of cumulative sales change, which is the derivative of the cumulative adoption, are proportional to the multiplication of the remaining market by the sum of innovation and imitation influence. The analytic continuous time solution that Bass presents to his equation is: The cumulative sales function, as well as its derivative or periodic sales function, can be outlined as a graph where time is the X axis and sales, or sales rate, are the Y axis. The shapes of both curves, the cumulative and periodic sales, are determined by the values of the model parameters p and q. Srinivasan and Mason (1986) note that both p and q must be non-negative. Acemoglu and Ozdaglar (2009) note that the levels of p and q scale time, while the ratio q/p determines the overall shape of the curve. Bass (1969) refers to the effect of the q/p ratio or q, p phase and differentiates between two categories. He notes that when q/p > 1, i.e., p, q phase = 45°, the product is successful and sales experience growth and then decline due to saturation. When q/p < 1, which represents an unsuccessful product, sales will start at a certain level and keep declining. Bass (1969) also presents the influence of (p + q) values, between 0.3 and 0.9, on the growth rate. Sultan et al.
(1990) performed a meta-analysis of 213 applications of diffusion models from 15 articles published between 1950 and 1980. They compare several parameters estimation methods (OLS, MLE, Bayesian and non-linear least square) and also how the number of sampling points influences accuracy. They found that the average p value is 0.03, while the average q value is 0.38. Van den Bulte (2002) explored how p and q vary across products and countries, based on a database containing 1586 sets of p and q parameters, from 113 papers published between January 1969 and May 2000. He explains that the parameters p and q provide information about the speed of diffusion. A high value for p indicates that the diffusion has a quick start, but also tapers off quickly. A high value of q indicates that the diffusion is slow at first, but accelerates after a while. He also notes that, when q is larger than p, the cumulative number of adopters follows the type of Scurve often observed for radically innovative product categories. When q is smaller than p, the cumulative number of adopters follows an inverse J-curve often observed for less risky innovations such as new grocery items, movies, and music CDs.
where F d (n) is the cumulative adoption and f d (n) is the periodic sales at period n.
We use the notation p d and q d for the discrete time model parameters to distinguish them from the p, q parameters of the continuous time form. The X(n) function can capture many factors the original Bass model ignores such as advertisement of price changes. When X(n) = 1, it converges back to the original Bass model. For example, Bass et al. (1994) propose including the influence of advertisement of price changes by using: M a x nA n (4) When coefficient captures the percentage increase in diffusion speed resulting from a 1% decrease in price, Pr(n) is the price in period n, coefficient captures the percentage increase in diffusion speed resulting from a 1% increase in advertising and A(n) is advertising in period n. The main advantage of the discrete model is that it can be solved numerically, in cases where there is no analytic solution. While Bass (1969) presents an analytic general solution for the continuous time differential equation (1), neither he nor others propose an analytic solution for the discrete time difference equation (3). Bass (1969) does refer implicitly to the discrete model by providing an insight that the likelihood of a purchase at time t, P(n) is calculated by: While many researchers switch between the discrete model, based on a difference equation (3), and the continuous time model, based on the differential equation (1), the transition is not trivial. Van den Bulte and Lilien (1997) show that OLS or NLS estimations of the continuous Bass model parameters using discrete time data are biased and that they change systematically as one extends the number of observations. An analytic discrete time solution for the Bass continuous time differential equation (1), developed by Satoh (2001), is:  (7) to p and q, a solution of the discrete Bass model (6) provides identical values to the solution of the continuous model (2). For two cases (out of nine) where the estimated value of the p parameter is negative, he notes that the wrong sign indicates that the data are not appropriate for the Bass model.

Mapping the Bass model parameters
We perform the mapping separately for each flavor of the Bass model. First, we develop a formula of the periodic sales inflection points' times, for the continuous time form, and classify the diffusion patterns according to the number of inflection point and whether there is peak. Second, we map each category classified to a sub-region of the positive p and q values of the Euclidean space. Then, we extend the map to include regions where either p or q is negative, which add some insight about seeding and about market saturation. For the discrete time model, equation (3), which is more restricted, we remap the p, q space. The map is similar but has additional constraints about the absolute values of p, q and their sum. We redo the mapping for the Satoh model which is a discrete time solution (6, 7) for the continuous time mode of equation (1, 3).

Mapping the Bass model parameters ratio
space. Bass (1969) and Van den Bulte (2002) distinguish between cases where q > p, and periodic sales have a peak after launch, to cases where q < p and periodic sales keep declining since launch. The periodic sales peak time, as calculated by Bass (1969), is: From (8), we see that when q > p, the peak time is positive. When q < p, peak time is negative, thus, sales keep declining since launch time (t = 0). There are no sales before product launch.
The inflection points' times (see Appendix A) are calculated as: Result 1: the time between the inflection points and between them to peak depends only on (p + q) and not on (p/q).      there is sing ies also for that, for reg alues of bot n 1, while th

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have opposite lute values of h parameters aint. Figure 6 ) k and s p From (7)  vior. In region E, we define a new motivation for seeding which, unlike previous research, is not used only for accelerating the diffusion, but, in certain conditions is essential for starting the market. Another contribution of this paper is defining the conditions to saturation below 100% (unlike previous concept that any product will finally cover the entire market). We also highlight some differences between discrete time and continuous time flavors of diffusion models and the map of the regions, where a switch between them is appropriate. Future research may propose an intuition for the regions that are still white in the pq space maps and explore their properties. Another direction may be developing an analytic solution for the discrete time diffusion difference equation.

Appendix A: Inflection points calculations
Bass solution for the non-cumulative adoption rate is: For finding the inflection points, we need to find where the derivative equals 0.
The derivative is calculated using the formula: