## Predicting the diffusion of LCD TVs by incorporating price in the extended

## Gompertz model

### Bi-Huei Tsai

### ⁎

Department of Management Science, National Chiao Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan, ROC

### a r t i c l e i n f o

### a b s t r a c t

Article history:

Received 22 February 2011

Received in revised form 30 March 2012 Accepted 12 July 2012

Available online 12 August 2012

In addition to reflecting how price decline stimulates consumption behavior, an effective diffusion model of liquid crystal display televisions (LCD TVs) featured by coexistence of multi-generational LCD TVs could also forecast accurately LCD TV sales under a market segmentation framework. As technological advances have led to a substantial decline in the price of LCD TVs, the diffusive prediction of LCD TVs based on prior conventional models, which neglect how price affects the LCD diffusions, must be biased. Based on anchoring and adjustment theory, this work develops extended Gompertz models that incorporate the consumers’ comparison of the initial prices with the later reduced prices to analyze the purchasing decisions of consumers for various sized LCD TVs from psychological perspectives. The effective ranges of market penetration rate are located under which the estimated parameters conform to the theoretical assumptions of product diffusion. Their forecasting accuracy is examined by further comparing prediction errors of the conventional Gompertz model and the extended Gompertz model. Empirical results indicate a significantly positive correlation between price reduction and LCD TV sales. Additionally, the market penetration rate is higher for smaller-sized LCD TVs than for larger-sized ones, implying that smaller-sized LCD TVs have reached market saturation, while larger-sized LCD TVs still have remaining market potential. Furthermore, the comparison results demonstrate that the effectiveness of the extended Gompertz model in predicting future shipment orbits of LCD TVs is superior to that of the conventional Gompertz model, since the extended model incorporates price factors. © 2012 Elsevier Inc. All rights reserved. Keywords:

Gompertz equation Price elasticity Nonlinear least square Market penetration Anchoring

1. Introduction

Determining the diffusion of hi-tech products has become increasingly essential to marketing, production, and operational strategies. As hi-tech products, liquid crystal display televisions (LCD TVs) hold numerous advantages over conventional TVs. For instance, LCD TVs are slimmer and lighter, as well as have a higher resolution and a larger screen size than conventional TVs. Moreover, LCD TVs are brighter, have a higher contrast, and consume less power as well[1]. Notably, LCD TVs are rapidly replacing conventional TVs in households[2], explaining why consumers increasingly view LCD TVs as durable products. LCD TVs are both durable products and technological commodities whose prices decline as the technology matures and production expands. Smaller and larger-sized LCD TVs are commercially available. The capital intensive nature of the thin film transistor (TFT)-LCD industry explains why manufacturers must decide when and how they can recover R&D and production costs. Therefore, forecasting the market potential, namely, the total number of potential LCD TV adopters, and future shipments of multi-generational LCD TVs is of worthwhile interest when making investments and operational decisions.

Studies involving multi-generational high-tech products have suggested that next-generation products may replace dynamic random access memory (DRAM)[3]. Previous studies viewed the product characteristics and functions of DRAM and flash memory in various generations as homogeneous, explaining the rapidly diminishing market for previous product generations. In

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0040-1625/$– see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.techfore.2012.07.006

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contrast to such homogenous products, the attributes and functions of LCD TVs vary across generations. LCD TV purchasers use these TVs in different locations (e.g., public spaces, living rooms and bedrooms) have different size preferences. Consumer demand for various sizes of LCD TVs is typically heterogeneous. Larger-sized LCD TVs are not as commercially available as smaller ones, explaining why the larger ones in successive generations fail to fully replace smaller ones in previous generations. Distinctive consumer groups in favor of different sizes of LCD TVs behave according to the market segmentation hypothesis of Smith[4]and Kotler[5]. The commercial availability of various sizes of LCD TVs simultaneously differs from the generational substitutions of technological products containing a homogeneous property among multiple generations. Forecasting the sales of LCD TVs is biased if a previous substitution model constructed for homogenous technological products. Because academic literature still lacks the research concerning sales forecasting for heterogeneous products, this work presents a novel model to forecast the sales of LCD TVs, which incorporates the qualities of durable products and high-technological commodities, under the market segmentation framework.

Most of the above studies applied the conventional diffusion theory to elucidate market dynamics[6,7]. Bengisu & Nekhili[8] forecasted the trends of 20 emerging technologies by using the conventional Gompertz model. Lee and Lee[9]analyzed the growth patterns of telecommunication services by using the typical logistical S-curve. In the setting of the original growth model, product diffusion is assumed to remain unaffected by prices[10–12]. To our knowledge, above studies fail to explore the market dynamics of specific products characterized as both “durables” and “high-technological commodities” under the market segmentation framework, while considering price effect as well.

However, retail prices of LCD TV panels have decreased as their production has expanded, leading to a decline in the unit cost owing to economies of scale. Given the continuous reduction in retail prices of LCD TV panels, the successive consumers are more encouraged by the favorable responses of consumers having already purchased LCD TVs positively influence potential customers. An immediate reduction in retail prices is likely to cause the diffusive growth of different sizes of LCD TV panels. In particular, the extent of price reductions is greater for larger-sized LCD TV than for smaller ones, explaining why price declines generally accelerate the pace of market saturation for larger-sized LCD TVs at a higher speed than for smaller ones. Therefore, estimating LCD TV diffusion is likely to be biased when using the restrictive Gompertz model because the model disregards price effect. Owing to the lack of empirical evidence on how prices affect sales with different generations coexisting commercially, this work is the first to develop and apply an extended Gompertz model incorporating price factors to address the diffusion patterns of a product with dual property- durable and high-tech products. While relaxing the restrictive assumptions that prices do not impact the LCD TV diffusion to purchase LCD TVs, this work hypothesizes that the diffusion of each generation of LCD TVs is a function of its price reduction. Under this hypothesis, each generation of LCD TVs occupies its own market share independently under market segmentation framework. In particular, the proposed model follows the anchoring and adjustment heuristic findings of Tversky and Kahneman[13]. The anchoring and adjustment heuristic theory of Tversky and Kahneman[13]assumes that consumers regard the initial price of a new style LCD TV, upon its market launch, as the“reference point”. When the price drops, consumers compare the new price with the“reference point”. A more price declines imply a greater inclination of successive adopters to purchase LCD TVs. In other words, deciding to purchase a LCD TV depends on consumers’ comparison of the initial price with the later reduced price around the purchasing time. From such psychological viewpoints of consumers' comparison of the initial price with the later reduced price, this work applies anchoring and adjustment heuristic theory to explore how the price declines of LCD TV increase consumer purchasing inclinations.

In addition to the price effect, this work examines the range of market penetration rate for multi-generational LCD TVs. Calculated through dividing cumulative shipments by market potentials, market penetration rate refers to the number of active LCD TV adoptions within the upper demand bound for LCD TVs. This work estimates the model for various market penetration rates and, then, selects the rate that most accurately forecasts the future shipment orbits for three reasons. First, parameters estimated under various levels of market penetration rates can respond to environmental changes. The economic environment and technology production skills of the hi-tech industry fluctuate continuously. Hi-tech products have a short life cycle, and a breakthrough in product technology affects the market potential of existing products. As mentioned earlier, market penetration rate is calculated by dividing cumulative shipments with market potential, explaining why the market penetration rate fluctuates. That is, a sudden technological breakthrough shrinks the market potential of products originally circulating in the market, subsequently inducing the market penetration rate to rise. Second, the demand and receptiveness of consumers for hi-tech products fluctuate as well, which also increases the market potential and market penetration rate of technological products. For instance, LCD panels have quickly replaced cathode ray tubes (CRTs), subsequently curtailing the potential market size of CRTs and shrinking the market for CRTs. Market prospects for the CRT industry are pessimistic, owing to the development of LCD panel technology. Third, Trappy and Wu[12]also assessed the diffusion of products of a short life cycle with a fluctuating penetration rate. As is widely suggested in academia, a forecasting model should consider the uncertainty of market potential, a feature of short life-cycle products. When the sales volume of LCD TVs is forecasted using the Gompertz curve, the upper limit of the curve represents the market potential for a product. Caution must be taken to include the actual situation when setting the market potential in both optimist and pessimist circumstances in order to forecast the shipments of short life cycle LCD TV[8]. Therefore, estimating the parameters and evaluating the forecasting accuracy under various levels of market penetration rates can accurately forecast the market potential of hi-tech technological products in both a optimist and a pessimist environment, thus providing managerial implication for the TFT-LCD industry, as well as for academia. However, if the proposed model assumes that the economic environments and manufacturing technology skills remain constant and set the potential market size of the LCD TV product as a fixed value, parameters estimated by the proposed nonlinear least squares (NLS) model fail to respond to environmental changes. Therefore, based on statistical simulation, this work estimates parameters based on various levels of

market penetration rates and then identifies the effective ranges of the market penetration rate. The parameter subsequently generated conforms to the theoretical assumptions of product diffusion. Effective levels of market penetration rate allow LCD TV manufacturers not only to evaluate total profits gained from investing capital and resources in LCD TVs, but also to anticipate the time needed to recover R&D and production costs.

This work investigates the dynamic diffusion of 15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 40-inch, 42-inch, and 46-inch LCD TVs by incorporating time-varying price decline factors into the extended Gompertz model. Effects of price elasticity, and market penetration on LCD TV shipments are systematically analyzed based on statistical examinations. This work also forecasts the future orbits of global LCD TV shipments and, then, compares the accuracy of the conventional Gompertz model with that of the proposed extended Gompertz models. In doing so, whether the proposed model, which identifies price reduction, performs better than the conventional Gompertz model is determined. Analysis results indicate that the Gompertz equation explains cumulative sales increases exponentially over time for multiple generations of LCD TVs. Our results further demonstrate that larger-sized LCD TVs continue to have many potential consumers, while sales for smaller-sized LCD TVs are approaching market saturation. In particular, this model illustrates how reducing the retail price of LCD TVs stimulates the diffusion of LCD TVs. Owing to the statistical significance of the modeled price effect in LCD TV diffusions, the proposed model more accurately forecasts the price impact than the conventional one does.

This work significantly contributes to technology management in several ways. First, this work adopts a market segmentation framework to forecast the sales volume of products as Tsai et al.[14]. This work emphasized the critical role of segmentation in devising marketing strategies aimed at potential adopters, leading to the general consensus that enterprises should coordinate technological, forecasting and marketing strategies concerning consumer segmentation simultaneously. Second, this work attempts to explain the product life cycle of hi-tech products by using the Gompertz curve, because Gompertz curve corresponds to the shipment distribution of hi-tech products. The Gompertz curve can generally be divided into two non-symmetrical parts based on the inflection point; the left-hand section where the curvature is positive before the inflection point and the right-hand section where the curvature is negative after the inflection point. This pattern of Gompertz curve represents a situation in which consumer appeal to hi-tech products creates a tremendous demand for such products during the early stage of the product life cycle. The demand quantity of such products decrease over time as the related hype has diminished. In particular, the left-hand section accounts for a relatively smaller portion of the entire whole graph than the right-hand section does. This phenomenon reflects the fact that the early stage of a product life cycle in which sales grow dramatically persists shorter than the later stage when sales decline. Third, the anchoring and adjustment heuristic theory adopted here illustrates how price affects the purchasing behaviors of LCD TV consumers. Based on psychological viewpoints, the proposed model accentuates the role of consumers in comparing the current price with the initial price at the first launching of LCD TVs. LCD TV diffusions are thus specified as the function of such price comparison in our extended Gompertz model to forecast the global adoptions of multi-generational LCD TVs, which is in contrast to the assumptions in previous studies of no price effect[15–17]. Although previous studies highlight the role of price in determining market dynamics[18–20]in the growth model, products combining the attributes of durables and hi-technological products based on anchoring and adjustment heuristic theory have received little attention. To our knowledge, this work applies for the first time an extended Gompertz model that combines the role of price in the purchasing behaviors of potential consumers to explain the diffusion of a specific commodity with durable and high-tech attributes. Fourth, in addition to simulating the effective ranges of market penetration rates for multi-generational LCD TVs, this work analyzes the adoptive features of LCD TVs under such effective market penetration levels. Unexpected technological advances and fashion trends affect the total adoption number of LCD TVs, thus inducing the market penetration rate variant. Parameter simulation under both optimistic and pessimistic market scenarios allows LCD TV manufacturers to make precise decisions under uncertain economic circumstances, highlight the diffusion characteristics of LCD TVs and review the feasibility of their marketing strategies.

The remainder of this paper is organized as follows.Section 2provides a background of LCD TV industry.Section 3summarizes the conventional Gompertz model and extended Gompertz models with the price effects in our research.Section 4gives the empirical results of coefficient estimations, market potential evaluations, and forecast accuracy.Section 5provides the additional testing of alternative forecasting models. Finally, conclusions are given inSection 6.

2. Background

Fig. 1depicts the coexistence of cumulative shipment for 15-inch, 20-inch, 26-inch, 37-inch, 40-inch, 42-inch, and 46-inch LCD TV panels. It is clearly observed fromFig. 1that the curve of 15-inch and 20-inch LCD TVs changes from being concave upwards (positive curvature) prior to inflection point to concave downwards (negative curvature) subsequent to inflection point. 15-inch and 20-inch LCD TVs nearly approach market saturation, while the curvatures of 40-inch, 42-inch and 46-inch LCD TVs appear positive; briefly, their quarterly shipment volumes still grow increasingly. The slope of shipment curve for 40-inch and 42-inch LCD TVs are still getting steeper with time even after 46-inch LCD TVs have been entered into markets. Since varying household space requirements are associated with different sizes of LCD TVs, the attributes and functions LCD TVs is heterogeneous among various generations. Hence, larger-sized LCD TVs in successive generations hardly substitute for the smaller-sized LCD TVs in previous generations. Distinctive consumer groups prefer different sizes of LCD TVs under market segmentation framework. This coexistence of various sizes of LCD TVs in the market is different from the generation substitutions of technological products that contain homogeneous characteristics among multiple generations.

More specially, technology progress and industrial competition of LCD TV manufacturers enhance the cost down of LCD TV production, reducing the price of LCD TVs. The price of various size levels of LCD TVs has been standardized in terms of average prices per square meters and denoted the inFig. 2. Prices of LCD TV panels decrease with time as their degrees of productivity expands, leading unit cost to drop due to economy of scale. In particular,Fig. 2exhibits the differential patterns of price reduction among multiple generations of LCD TVs. The accumulated price reduction was 4,805 US dollars (per square meters) for a 26-inch LCD TV from its debut in the fourth quarter of 2004 to the fourth quarter of 2009, while the accumulated price reduction was only 3,088 US dollars (per square meters) for a 15-inch LCD TV from its launch in the first quarter of 2002 to the fourth quarter in 2009. The price reduction for 26-inch LCD TVs is almost one and half times of that for 15-inch LCD TVs. Also, the price of 42-inch and 46-inch LCD TVs keep dropping to the lowest point due to the technical maturity after their appearance in the market. The unit price appeared greater for larger-sized LCD TV panels than smaller ones as the LCD TV panels newly entered the markets. Briefly, only when the productivity of each size level finally reaches the economy of scale, does the unit price of various size levels of LCD TV panels converge to be approximately the same. Combining the shipment pattern inFig. 1and price trend inFig. 2, the diffusive growth of different sizes of LCD TV panels accompany the immediate advent of lower prices. Particularly, the extent of price reductions is more immense for larger-sized LCD TV than for smaller ones, so price declines accelerate the pace of market saturation for larger-sized LCD TVs at higher speed than small ones.

0 1000 2000 3000 4000 5000 6000 7000 8000

Average Area Price (unit: thousand US dollar/m

2) 15-inch 20-inch 26-inch 32-inch 37-inch 40-inch 42-inch 46-inch 2002Q12002Q32003Q12003Q32004Q12004Q32005Q12005Q32006Q12006Q32007Q12007Q 3 2008Q12008Q3

Fig. 2. The distribution of average price per square meters for 15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 40-inch, 42-inch, and 46-inch LCD TV panels (Data source: DisplaySearch databases).

0 5000 10000 15000 20000 25000 30000 35000 2002Q12002Q32003Q12003Q32004Q12004Q32005Q12005Q32006Q12006Q32007Q12007 Q3 2008Q12008Q3

cumulative shipment (thousand units)

15-inch 20-inch 26-inch 37-inch 40-inch 42-inch 46-inch

Fig. 1. The distribution of cumulative shipment for 15-inch, 20-inch, 26-inch, 37-inch, 40-inch, 42-inch, and 46-inch LCD TV panels (Data source: DisplaySearch databases). The cumulative shipment of 32-inch LCD TVs is approximately four times of the cumulative shipments of other sizes of LCD TVs by the fourth quarter of 2009. To report the curvature clearly for other sizes of LCD TVs, 32-inch LCD TVs is not included in Fig. 1.

3. Methodology 3.1. Gompertz Model

The Gompertz model was first used to calculate mortality rates in 1825 and has been widely applied to the technology forecasting[21]. Different from the simple logistic curve, Gompertz curve is not symmetric about the inflection point. The Gompertz model reaches the point which occurs early in the growth trend and is expressed as:

yt¼ Le−ae −bt

; ð1Þ

where ytis the cumulative shipments of LCD TVs. L is the upper bound which means the potential market size and should be set before estimating the parameters a and b. Natural logarithms are used to transform the original Gompertz model to linear equation:

Yt¼ ln ln L=yð ð tÞÞ ¼ ln að Þ−bt: ð2Þ

The distribution of Gompertz model follows s-shape curve. The Gompertz curve is not symmetrical about the inflection point which occurs at t = (ln(a)/b), where yt= L/e. There are two examples for Gompertz function tracking inFigs. 3 and 4and both figures set upper bound L at 12,392, which means the potential market size for technological products is 12,392 ifFigs. 3 and 4are applied to describe the sale for technological products. The inflection point occurs at 4,757.344 thousand units, at which the curvature changes sign. The curve changes from being concave upwards prior to inflection point to concave downwards subsequent to inflection point. In brief, the curvature is positive before the inflection point, while negative after the inflection point. The features of Gompertz curve properly explains the life cycle of high-technology products. At the beginning stage when high-technology products go on shelves, consumers flock in to purchase them. Demand quantity for LCD TVs surges, so the marginal growth rate of shipment is thus positive in the early stage of product life cycle. However, as the hype fades, high-technology products have less appeal to consumers, so the demand quantity decreases over time. Thus, the marginal growth rate of shipments turns to be negative, and the cumulative shipments curve tends smooth, whose pattern is the same as the long and smooth tail of the Gompertz curve. In addition, the right-hand asymptote of the Gompertz function is approached much more gradually by the curve than the left-hand asymptote, in contrast to the logistic function in which both asymptotes are approached by the curve symmetrically. The characteristics of Gompertz curve can account for the shipment concentration of high-technology products in the short period at the early stage of product life cycle. Because LCD TVs contain the property of durables and technology products, Gompertz model is employed to forecast the diffusion of LCD TVs in our studies.

Fig. 3clearly indicates that the change in coefficient b affects the shape yet does not affect the location. In the Gompertz equation, ytrefers to the positive function of coefficient b when both upper bound L and coefficient a are fixed as constants. Also, parameter b denotes the growth rate of the logarithm of the reciprocal of penetration rateyt

L, which indicates the speed by which

the cumulative shipment ytgradually approach market potential L. In Eq.(2)Yt= ln(ln(L/yt)) = ln(a)−bt,because the operator before parameter b is a minus. This finding suggests that if parameter b is positive, the speed by which the cumulative shipments ytapproach market saturation decreases progressively. According toFig. 3, when parameter b is positive, the closer the Gompertz curve approaches market potential L implies a slower product diffusion. When the Gompertz curve is used to explain a product shipment, a larger parameter b implies an earlier period that the market reaches saturation; in addition, the curve becomes flatter and closes in to market potential L in later stages. This finding suggests that the speed at which a shipment is made slows down at the end of the product cycle.

Conversely, the change in coefficient a affects the location only and does not affect the shape. Parameter a is the logarithm of the reciprocal of the penetration rate,yt

L. A smaller parameter a implies a larger penetration rate when the product is launched,

implying that potential consumers immediately purchase a product upon its market entry.Fig. 4clearly reveals that ytrefers to the negative function of coefficient a when upper bound L and coefficient b are fixed as a constant in the Gompertz equation. If the market evolutions of LCD TVs are stated using the Gompertz equation, cumulative shipment ytdecreases with parameter a.

0 2,000 4,000 6,000 8,000 10,000 12,000 14,000 0 5 10 15 20 25 30 Time cumulative shipment b=0.05 b=0.1 b=0.2 b=0.4

Fig. 3. The incremental effect of the parameter b on the Gompertz curves. The four Gompertz curves set the same value of the upper bound L and the parameter a (L = 12,392 and a = 6). The parameter b affects mainly the magnitude of adoptions per period.

From Eq.(1), the Gompertz model has definite limitations (upper bound L), when used to forecast high-technological products with shorter product lifecycles than durable commodities. Because it is almost impossible to estimate the correct upper limit for a new product when it is first introduced to the market. This paper statistically simulates parameters based on different levels of market penetration rates. The market penetration rate of LCD TV is a term generally used to describe the number of active LCD TV adoptions within the upper demand bound for LCD TV products, so it can be calculated by dividing the cumulative shipment volumes by the potential market size. In our study, the parameters are simulated based on the market penetration rate datasets of 10%, 20%, 30%, 50%, 70%, 80%, 90%, and 95%. Then, this investigation determines under what level of market penetration rate the parameters can be reasonably estimated. In theory, the coefficients a and coefficients b are presumed to be positive in Gompertz model concerning the dynamic diffusion of LCD TV panels. If the parameters estimated are negative under a certain market penetration level, the chosen market penetration rate could be inferred to be invalid. Hence, this investigation locates the effective ranges of market penetration rates, under which the coefficients are estimated positive, to be the predictive level of market penetration rate. Next, we calculate the potential market size, which means the upper bound of LCD TV product demand, by dividing the current cumulative shipments by the predictive market penetration rate.

3.2. The Extended Gompertz Model that Incorporates Price Effect

Aforementioned conventional Gompertz models are assumed unaffected by price reductions. However, as LCD TV prices drop, more and more potential consumers are likely to shop for LCD TVs in response to the persuasions of previous adopters. Prior articles emphasized the impact of price on market potential dynamics of products[22–24]. It is inferred that the LCD TV diffusion should be sensitive to price reduction. The assumptions of no price effect in conventional Gompertz model violate the real situations. In practice, the decreasing prices of LCD TVs can strongly encourage the successive consumers to buy LCD TVs. The unit price of LCD TVs declines in high speed after LCD TVs are first to be promoted on sale. InFig. 2, unit prices of various types of LCD TVs in 2005 decrease to half of that in 2003. Price reduction is the key factor to strengthen the purchasing behaviors of potential LCD TV adopters, so this study relaxes the restrictive assumption of ignoring internal price elasticity.

Both the absolute change in the level of prices and the ratio of price changes are specified as the price decline indicators in our proposed models because this work attempts to explain the consumer receptiveness towards the initial price of a new style of LCD TV by using anchoring and adjustment heuristic theory of Tversky and Kahneman[13]. The extended Gompertz model which incorporates the absolute price changes and the change ratio of prices are defined here as“extended Gompertz models with the absolute price change value” and “extended Gompertz models with the change ratios of prices”. LCD TV diffusion is specified as the function of price decline expressed as Eqs.(3) and (4):

b1t¼ b1exp½γ1 Pð 0−Pt−1Þ: ð3Þ b2t¼ b2exp γ2 P0−Pt−1 ð Þ P0 : ð4Þ

In Eqs.(3) and (4), Ptis the price at time t. We set up the time 0 as the time point when a new style of LCD TV initially launch in the market, so P0is the initial price. The styles for LCD TVs are classified by their size, including 15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 42-inch and 46-inch LCD TVs. Obviously, the term (P0−Pt−1) andðP0−Pt−1Þ

P0 in Eqs.(3) and (4)represents the price gap in

the interval [0, t-1], from the time point when a new style of LCD TV is first to be promoted on sale (t = 0) to time (t-1). Because price reductions enhance the growth rate of LCD TV consumptions, the coefficient b1tand b2tvaries with LCD TV panel prices over time and is stated as a function of price reduction during period (t-1). The parameters of price elasticityγ1andγ2denote the

0 2,000 4,000 6,000 8,000 10,000 12,000 a=3 a=6 a=12 a=24 0 5 10 15 20 25 30 Time cumulative shipment

Fig. 4. The incremental effect of the parameter a on the extended Gompertz curves. The four Gompertz curves set the same value of the upper bound L and the parameter b (L = 12,392 and b = 0.1). The coefficient a affects mainly the intercept of the diffusive curve, and the necessary time to reach the maximum number of adoptions L. The smaller parameter a is, the sooner the diffusive curve reaches L.

marginal effect of price reduction on the sales growth of LCD TVs. For this reason, we put the Eqs.(3) and (4)into Eq.(2)and it can be rewritten as Eqs.(5) and (6):

yt¼ Le−a1 e−b1exp½γ1 P0−Pt−1ð Þt : ð5Þ yt¼ Le−a2 e−b2exp γ2P0−Pt−1 ð Þ P0 h i t : ð6Þ

Natural logarithms are used to transform the conventional Gompertz model to linear equation:

Yt¼ ln ln L=yð ð tÞÞ ¼ ln að Þ−b1 1exp½γ1 Pð 0−Pt−1Þt: ð7Þ
Yt¼ ln ln L=yð ð tÞÞ ¼ ln að Þ−b2 2exp γ2
P0−Pt−1
ð Þ
P0
t_{:} _{ð8Þ}

Eqs.(5) to (8)explain the LCD TV shipments during time period t by using the product's price gap in the interval [0, t-1]. Restated, during period t, consumers have already observed the price decline from period 0 to period (t-1), explaining why they decide to purchase a LCD TV during period t. With such an adjustment, shipment volumes of LCD TV during period t do not affect the margin of the price decline during period t-1, thus excluding the endogeneity problem with the model.

Both the absolute change in the level of prices and the ratio of price changes are specified as the price decline indicators in our proposed models because this work attempts to explain the consumer receptiveness towards the initial price of a new style of LCD TV by using anchoring and adjustment heuristic theory of Tversky and Kahneman[13].Fig. 5denotes that the coefficients of price affect the shape. The positive price elasticity coefficient illustrates how the price decline accelerates the diffusions of LCD TV panels. Greater price elasticity coefficient represents larger impact of price decline on consumers’ purchasing decisions, so it is clearly observed that price elasticity coefficients are increasing with LCD TV diffusion.

3.3. Forecasting

The forecasting accuracy of“extended Gompertz models with the absolute price change value” , “extended Gompertz models with the ratios of price changes” and conventional Gompertz models are compared in this section. Our sampled data is divided into two periods; one is the training period and the other is the test period. All the prediction models are developed by using the training sample, which ranges from the initial periods for various sizes of LCD TVs in our sample to the fourth quarter in 2008, and their levels of accuracy are compared with the test sample from the first quarter of 2009 to the fourth quarter of 2009. We apply the estimated parameters calculated from training sample to evaluate shipment orbits in test sample.

To verify whether the extended Gompertz model performs better than conventional Gompertz model, the accuracy comparison of the forecasting orbit in test sample is conducted between Gompertz model and our extended Gompertz models incorporating price factors. As for the prices used in the extended Gompertz models to predict cumulative shipments during the test period, three price indicators are adopted: 1. the last known real price during the training period, i.e. the price in the fourth quarter of 2008; 2 the actual prices from the first to the fourth quarter of 2009; and 3. the estimated price computed by the price

0 2000 4000 6000 8000 10000 12000 14000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Time cumulative shipment γ=0 γ=0.00005 γ=0.0001 γ=0.0002 γ=0.0004

Fig. 5. The incremental effect of the price elasticity coefficientsγ on the extended Gompertz curves. The five extended Gompertz curves set the same values of the upper bound L, the parameter a and the parameter b (L = 12,392, a = 6 and b1= 0.1). In the situation of price declines, (P0−Pt) > 0, the positive price elasticity

coefficientγ enlarges the coefficient bt. Thus, the slope of the Gompertz model considering price reductions (γ>0) is higher than that of the conventional

prediction model (Eq.(9)) which is constructed with data in the training period from the first quarter of 2002 to the fourth quarter of 2008. The price reduction prediction model can be expressed as Eq.(9):

P0−Pt

ð Þ ¼ α þ βt ð9Þ

In Eq.(9), price decline is time-variant. After the accuracy between conventional and extended Gompertz models as well as the collected data is compared and validated in the test sample, this research verifies whether the extended Gompertz model, which specifies the price factors, performs better than the conventional one. We choose mean absolute percentage error (MAPE), mean absolute deviation (MAD), and the root mean squared error (RMSE) of the cumulative quarterly shipments in the test sample to compare the forecasting capability of the modified model with the conventional one. The calculated errors could manifest the results which method is more efficient. The MAPEs are computed by MAPE¼1

n

Pn t¼1

Yt−^Yt

j j

Yt . The MADs are computed by

MAD¼1 n

Pn t¼1 Yt−^Yt

. The RMSEs are computed by RMSE ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Pn t¼1 Yt−^Yt ð Þ2 n s . Table 1

The parameter estimation for the extended Gompertz models with the absolute price change value under the market penetration rate levels of 70%, 80%, 90%, and 95% for various size of LCD TVs.

Size Penetration Rate 70% 80% 90% 95%

Coefficient t-value Coefficient t-value Coefficient t-value Coefficient t-value

15 a1 4.9936 15.18*** 5.0290 10.59***
b1 0.1038 9.68*** 0.0953 6.97***
γ1 X X 0.0001 3.69*** 0.0002 4.28***
L 13,650 12,932
R2
0.9912 0.9855
Adj R2 _{0.9904} _{0.9843}
20 a1 6.4206 28.82*** 6.2287 19.35*** 5.8719 10.36*** 5.4478 7.05***
b1 0.0741 11.22*** 0.0651 7.44*** 0.0475 3.67*** 0.0307 2.19**
γ1 0.0001 5.02*** 0.0002 5.25*** 0.0004 4.32*** 0.0005 3.74***
L 44,063 38,555 34,271 32,467
R2
0.9969 0.9945 0.9846 0.9698
Adj R2
0.9966 0.994 0.9834 0.9673
26 a1 7.8838 8.15*** 6.2452 7.19*** 5.2715 6.93***
b1 0.0758 2.66*** 0.0229 1.52 0.0052 1.05
γ1 X 0.0001 1.73* 0.0004 2.93*** 0.0008 3.64***
L 42,526 37,801 35,812
R2
0.9801 0.9608 0.9427
Adj R2
0.9781 0.9569 0.9370
32 a1 9.6457 11.34*** 9.1330 1.1342 8.1342 5.97*** 7.1369 5.28***
b1 0.0802 4.27*** 0.0649 0.0232 0.0376 1.67* 0.0164 1.12
γ1 0.0001 2.31** 0.0002 0.000092 0.0004 2.46** 0.0006 2.62***
L 143,145 125,252 111,335 105,475
R2
0.9871 0.9756 0.9519 0.9149
Adj R2
0.9858 0.9732 0.9471 0.9064
37 a1 7.0139 14.37*** 6.4754 12.51*** 5.8733 10.39*** 5.3623 9.07***
b1 0.0225 2.93*** 0.0130 2.16** 0.0048 1.43 0.0011 0.97
γ1 0.0004 4.94*** 0.0006 4.97*** 0.0009 4.86*** 0.0013 4.74***
L 37,465 32,781 29,139 27,605
R2
0.983 0.9753 0.9611 0.9467
Adj R2
0.9814 0.9729 0.9574 0.9416
40 a1 8.4939 19.12*** 7.980949 17.3*** 7.4600 13.92*** 7.0802 11.28***
b1 0.0115 3.13*** 0.006368 2.48** 0.0024 1.73* 0.0008 1.23
γ1 0.0005 7.18*** 0.000649 7.34*** 0.0009 6.97*** 0.0011 6.39***
L 32,783 28,685 25,498 24,156
R2
0.9857 0.9812 0.9713 0.9586
Adj R2
0.9844 0.9796 0.9688 0.955

Notes: 1.⁎Significant at the 10% level. ⁎⁎Significant at the 5% level. ⁎⁎⁎Significant at the 1% level.

2.“X” illustrates that the signs of the parameters in extended Gompertz model do not conform to the theoretical assumptions of product diffusion under such market penetration rate level.

3. The measurement unit of potential market size L is thousand units.

4. Because market penetration rate in the fourth quarter of 2008 is equal to the proportion of cumulative shipment in the fourth quarter of 2008 to the potential market size, market potential L can be calculated by dividing the cumulative shipments with the market penetration rate.

3.4. Sample and Data

The units of global quarterly shipment of LCD TV panels and the prices of global LCD TV panels in this study are obtained from DisplaySearch databases. According to the definition of the DisplaySearch database, the global shipments and shipment price refer to shipments from all LCD TV panel manufacturers in the world and the average price of their shipment price. The global LCD TV panel shipments data include all countries. The“price” is defined as the average price of the quoted price from all LCD TV panel manufacturers in the world, including taxes and transportation costs. The period, which contains 32 quarters in our sample, ranges from the first quarter in 2002 to the fourth quarter in 2009. Because various sizes (15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 40-inch, 42-inch and 46-inch) of LCD TV begin to be available to the market in different periods, the actual units of shipment are collected from the first quarter in 2002, the first quarter in 2002, the first quarter in 2003, the first quarter in 2003, the fourth quarter in 2002, the second quarter in 2002, the fourth quarter in 2003, and the first quarter in 2004 for 15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 40-inch, 42-inch and 46-inch LCD TVs, respectively. Therefore, we contain 32, 32, 28, 28, 29, 31, 25 and 24 observations for 15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 40-inch, 42-inch and 46-inch LCD TVs, respectively. 4. Results and Discussion

4.1. The Results of Parameter Estimation

We use the actual market and price data of 15-inch, 20-inch, 26-inch, 32-inch, 37-inch, 40-inch, 42-inch and 46-inch LCD TV panels to fit the extended Gompertz model (Eqs.(5) and (6)) to optimize the model parameters of each size of LCD TV panels (Tables 1 to 4). The effective ranges of market penetration rate are located under which the estimated parameters conform to the theoretical assumptions of product diffusion. Theoretically and practically, the diffusion of new product contains non-negative intercept and growth rates of LCD TV diffusion in extended Gompertz model (a1,a2> 0 and b1,b2> 0). According to the statistically results of parameter estimation, the coefficients are estimated positive under 95% market penetration rate for 15-inch LCD TVs. The coefficient signs of intercept and growth rates are also positive as the market penetration rate is higher than 70% for 20-inch and 32-inch LCD TVs, implying the sale of smaller-sized LCD TV is almost close to the upper limit of the market size. On the other hand, the coefficients are theoretically valid under the low level of market penetration rates for larger-sized LCD TV panels. The effective market penetration rate is higher for smaller-sized LCD TVs than for larger-sized ones, implying that smaller-sized LCD

Table 2

The parameter estimation for the extended Gompertz models with the absolute price change value under the market penetration rate levels of 10%, 20%, 30% and 50% for 37-inch, 40-inch, 42-inch, and 46-inch LCD TVs.

Size Penetration rate 10% 20% 30% 50%

Coefficient t-value Coefficient t-value Coefficient t-value Coefficient t-value

37 a1 8.9227 33.52*** 8.0421 20.21***
b1 0.0510 9.44*** 0.0409 5.24***
γ1 X X 0.0001 5.17*** 0.0002 4.97***
L 87,418 52,451
R2
0.9963 0.9918
Adj R2
0.9959 0.991
40 a1 12.7049 32.69*** 11.8623 29.15*** 11.2133 25.47*** 9.8232 20.61***
b1 0.0449 9.76*** 0.0442 8.81*** 0.0410 7.41*** 0.0271 4.72***
γ1 0.0001 4.52*** 0.0001 5.49*** 0.0002 5.89*** 0.0002 6.32***
L 229,485 114742 76,495 45,897
R2
0.9939 0.9942 0.9935 0.9905
Adj R2
0.9934 0.9937 0.9929 0.9897
42 a1 9.6664 56.94*** 8.8711 53.36***
b1 0.0179 5.61*** 0.0143 5.34***
γ1 0.0003 8.32*** 0.0003 9.92*** X X
L 211,010 105505
R2 _{0.998} _{0.998}
Adj R2
0.9977 0.9977
46 a1 10.8302 42.83*** 9.9480 39.07***
b1 0.0459 8.12*** 0.0312 6.89***
γ1 X X 0.0002 8.07*** 0.0002 10.63***
L 35,285 21,171
R2 _{0.9988} _{0.9987}
Adj R2
0.9986 0.9985

Notes: 1. *Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.

2.“X” illustrates that the signs of the parameters in extended Gompertz model do not conform to the theoretical assumptions of product diffusion under such market penetration rate level.

3. The measurement unit of potential market size L is thousand units.

4.Regarding LCD TVs whose size is equal to or smaller than 32-inch, the signs of the parameters in extended Gompertz model do not conform to the theoretical assumptions of product diffusion under such low market penetration rate level (10%, 20%, 30% and 50%). Thus, only the parameter estimation of larger-sized LCD TVs simulated under low level of penetration rate are disclosed in this table.

TVs have reached market saturation, while larger-sized LCD TVs still have remaining market potentials. Remaining market potential is defined as market potential minus cumulative shipment units. Regarding the t-statistic test results, all the coefficients, including price elasticity coefficients are statistically significantly at the 10% level, suggesting the enhancement of price decline on LCD TV diffusions.

Referring to the goodness of fit analysis, the adjusted R2_{is higher for smaller-sized LCD TV panels at the high level of market}
penetration rate while it is higher for larger-sized LCD TV panels at the low level of market penetration rate for both conventional
Gompertz and extended Gompertz models. This concludes that smaller-sized LCD TVs reach the market saturation, while the
larger-sized LCD TVs contain a remaining market potential. Comparing the adjusted R2_{of conventional Gompertz model (Eq.}_{(1)}_{)}
with that of the extended Gompertz models, which incorporate the impact of price reductions (Eq.(5) and (6)), the latter is
higher than the former. The extended Gompertz model performs superior to the conventional Gompertz model in predicting LCD
TV diffusions.

In regards to the extended Gompertz model incorporating price factors, price elasticity coefficients (γ1andγ2 are all positive for multi-generational LCD TVs. To our expectation, price decline of LCD TV stimulates consumption caused by the price declines will accelerate the diffusions of LCD TV panels. The empirical results suggest that previous consumers who have adopted LCD TVs Table 3

The parameter estimation for extended Gompertz models with the change ratios of prices under the market penetration rate levels of 70%, 80%, 90%, and 95% for various size of LCD TVs.

Size Penetration rate 70% 80% 90% 95%

Coefficient t-value Coefficient t-value Coefficient t-value Coefficient t-value

15 a2 5.1760 19.99*** 5.2893 13.80*** 5.3281 10.10***
b2 0.1195 12.89*** 0.1167 9.76*** 0.1072 7.44***
γ2 X 0.0413 0.41 0.3492 2.58*** 0.6654 3.72***
L 15356 13650 12931
R2
0.9923 0.9886 0.9831
Adj R2 _{0.9917} _{0.9877} _{0.9817}
20 a2 6.6304 23.79*** 6.4659 16.54*** 6.0809 9.92*** 5.5354 7.48***
b2 0.0799 9.22*** 0.0705 6.39*** 0.0499 3.55*** 0.0294 2.33**
γ2 0.4754 3.42*** 0.8162 4.02*** 1.4977 4.02*** 2.3751 4.10***
L 44,063 38,555 34,271 32,467
R2
0.9955 0.9925 0.9831 0.9711
Adj R2 _{0.9951} _{0.9919} _{0.9816} _{0.9687}
32 a2 9.0457 14.49*** 8.2774 11.56*** 7.2053 9.82*** 6.4158 9.22***
b2 0.0604 4.53*** 0.0397 3.19*** 0.0164 2.20** 0.0057 1.74*
γ2 0.9486 3.71*** 1.5602 4.21*** 2.7740 5.04*** 4.1698 5.96***
L 143,145 125,252 111,335 105,475
R2
0.9902 0.9832 0.9720 0.9637
Adj R2
0.9892 0.9815 0.9692 0.9600
37 a2 7.0046 16.53*** 6.5374 15.03*** 5.9933 13.34*** 5.5841 12.35***
b2 0.0183 3.14*** 0.0107 2.57** 0.0041 2.00** 0.0013 1.66*
γ2 2.2921 5.95*** 3.0719 6.46*** 4.4318 7.19*** 5.9294 7.97***
L 37,465 32,781 29,139 27,605
R2
0.9854 0.9813 0.9755 0.9716
Adj R2
0.984 0.9795 0.9732 0.9688
40 a2 8.6167 22.11*** 8.1871 21.08*** 7.7015 18.67*** 7.3322 16.44***
b2 0.0098 3.61*** 0.0058 3.22*** 0.0024 2.64*** 0.0009 2.17**
γ2 2.9753 8.92*** 3.7227 9.89*** 4.9536 10.74*** 6.2369 11.11***
L 32,783 28,685 25,498 24,156
R2
0.9885 0.9872 0.9844 0.9811
Adj R2
0.9875 0.9861 0.9830 0.9794
42 a2 8.1718 20.49*** 7.6019 14.67*** 6.6696 10.53*** 5.9648 8.74***
b2 0.0602 8.85*** 0.0500 5.72*** 0.0308 3.21*** 0.0163 2.09**
γ2 1.1059 8.55*** 1.4434 7.03*** 2.1823 5.77*** 3.0769 5.20***
L 30,144 26,376 23,445 22,211
R2
0.9942 0.9894 0.9778 0.9626
Adj R2
0.9935 0.9881 0.9752 0.9582
46 a2 11.2691 21.24*** 10.8211 14.15*** 10.0373 8.26*** 9.2320 5.78***
b2 0.1105 15.53*** 0.1048 10.13*** 0.0913 5.50*** 0.0755 3.46***
γ2 0.5937 7.85*** 0.7753 6.58*** 1.0954 4.92*** 1.4461 3.99***
L 15,122 13,231 11,761 11,142
R2
0.9957 0.9916 0.9788 0.9593
Adj R2
0.9952 0.9906 0.9761 0.9542

Notes: 1. *Significant at the 10% level. **Significant at the 5% level. ***Significant at the 1% level.

2.“X” illustrates that the signs of the parameters in extended Gompertz model do not conform to the theoretical assumptions of product diffusion under such market penetration rate level.

3. The measurement unit of potential market size L is thousand units.

4. Because market penetration rate in the fourth quarter of 2008 is equal to the proportion of cumulative shipment in the fourth quarter of 2008 to the potential market size, market potential L can be calculated by dividing the cumulative shipments with the market penetration rate.

are more inclined to recommend potential consumers to successively buy LCD TVs as the prices of LCD TV panels drop. The comparison of the initial prices with the later reduced prices during period (t-1) may induce successive consumers to purchase the LCD TVs. Purchasing a LCD TV based on the absolute change in the level of prices or the ratios of price declines verify that consumers’ purchasing decisions follows Tversky and Kahneman's[13]anchoring and adjustment theory.

We further apply Eqs.(3) and (4)to calculate the time-variant coefficient btover time for multi-generational LCD TV panels. At 95% market penetration rate level, the pattern of time-variant growth coefficients of the extended Gompertz model with absolute price change values and the extended Gompertz with ratios of price changes (b1t and b2t) are denoted in Figs. 6 and 7, respectively. The result shows that the coefficients increase gradually for multi-generational LCD TV panels. Combining the price trend depicted inFig. 2and the growth fluctuations denoted inFigs. 6 and 7, the price reductions of LCD TV panels increased dramatically for multiple generations of LCD TVs surrounding 2004, so the diffusive growth rate is computed to drop. After 2004, the adoptions of LCD TV expand due to continuous price reduction. The decrease in price will promote the LCD TV sales. We then take the diffusion of 32-inch LCD TVs as an example, the price of the 32-inch of LCD TV panel has declined from 4,606 US dollars (per square meters) in the first quarter in 2003, the time when 32-inch LCD TV was newly available to the market, to 694 dollars (per square meters) in the fourth quarter in 2008. Meanwhile, the quarterly shipments of LCD TV panels have grown from 4,000 to more than 10,000,000 according to the statistics in DisplaySearch database. Consistent with our expectation, causal relations are also observed between LCD TV panel shipments and price reductions.

FromTables 1 to 2, we can also discover that the smaller-sized LCD TV panel's coefficient of growth rate (b1and b2) is greater than the larger ones. Besides, the price elasticity coefficient (γ1andγ2) of the larger-sized LCD TV panel has been found greater than that of the smaller ones. The diffusion of larger-sized LCD TV panels is more sensitive to the price reduction than smaller-sized ones. InFig. 2, the price reduction is 4,805 US dollars (per square meters) from the first quarter in 2003, the time when a 26-inch LCD TV begins to be promoted on sale, to the fourth quarter in 2009, while the price reduction is only 3,088 US dollars (per square meters) from the first quarter in 2002, the time when a 15-inch LCD TV begins to be promoted on sale, to the fourth quarter in 2009. The price reduction to 26-inch LCD TVs is almost one and half times of that to 15-inch LCD TVs. The higher price elasticity coefficientγ and larger price reduction (P0−Pt) enable the diffusion of larger-sized LCD TVs to become more obvious than smaller-sized ones. At 95% market penetration rate, the diffusive growth coefficient b1tincrease from 0.1 to almost 0.18 for 15-inch LCD TVs, while increase from 0 to almost 0.18 for 26-inch, 32-inch, 37-inch and 40-inch LCD TVs in the extended Gompertz model with absolute values of price changes. The diffusive growth increases smoothly for 15-inch LCD TVs, while Table 4

The parameter estimation for extended Gompertz models with the change ratios of prices based on the market penetration rate levels of 10%, 20%, 30% and 50% for 37-inch, 40-inch, 42-inch, and 46-inch LCD TVs.

Size Penetration Rate 10% 20% 30% 50%

Coefficient t-value Coefficient t-value Coefficient t-value Coefficient t-value
37 a2 10.2867 41.85*** 9.5155 34.43*** 8.9649 27.98*** 7.9659 20.19***
b2 0.0539 10.40*** 0.0532 8.92*** 0.0494 7.19*** 0.0353 4.62***
γ2 0.2188 2.09** 0.4272 3.43*** 0.6525 4.15*** 1.2785 5.04***
L 262,255 131,127 87,418 52,451
R2
0.9964 0.9960 0.9963 0.9912
Adj R2 _{0.9961} _{0.9957} _{0.9959} _{0.9903}
40 a2 12.7962 26.07*** 11.8240 23.32*** 11.0242 21.83*** 9.6202 21.9***
b2 0.0446 7.18*** 0.0413 6.26*** 0.0352 5.39*** 0.0205 4.26***
γ2 0.5008 3.36*** 0.7540 4.29*** 1.0577 5.05*** 1.8948 6.86***
L 229,485 114,742 76,495 45,897
R2
0.9914 0.9914 0.9909 0.9898
Adj R2 _{0.9906} _{0.9907} _{0.9902} _{0.9889}
42 a2 10.7208 40.36*** 10.0279 41.84*** 9.5920 43.27*** 8.9209 36.18***
b2 0.0621 15.73*** 0.0667 17.75*** 0.0688 19.21*** 0.0684 16.57***
γ2 0.3068 4.61*** 0.4168 6.96*** 0.5182 9.24*** 0.7466 11.17***
L 211,010 105,505 70,336 42,202
R2
0.9953 0.9968 0.9976 0.9976
Adj R2
0.9948 0.9965 0.9974 0.9973
46 a2 13.3959 26.93*** 12.7892 26.58*** 12.4197 27.4*** 11.8677 29.37***
b2 0.0956 15.7*** 0.1037 16.99*** 0.1084 18.51*** 0.1128 21.14***
γ2 0.0080 0.12 0.1025 1.56 0.1851 3.06*** 0.3589 6.63***
L 105,855 52,927 35,285 21,171
R2
0.9917 0.9937 0.9952 0.9970
Adj R2
0.9906 0.9929 0.9946 0.9966

Notes: 1. ⁎Significant at the 10% level. ⁎⁎Significant at the 5% level. ⁎⁎⁎Significant at the 1% level. 2. The measurement unit of potential market size L is thousand units.

3. Regarding LCD TVs whose size is equal to or smaller than 32-inch, the signs of the parameters in extended Gompertz model do not conform to the theoretical assumptions of product diffusion under such low market penetration rate level (10%, 20%, 30% and 50%). Thus, only the parameter estimation of larger-sized LCD TVs simulated under low level of penetration rate are disclosed in this table.

increases dramatically for LCD TV panels whose size is equal to or more than 26-inch. This suggests that the impact of price reduction on the sale is more profound for the larger-sized LCD TV panels than for the smaller ones. When the price of larger-sized LCD TVs drops, a great number of consumers are able to afford the larger-sized LCD TVs. A greater price reduction accelerates consumers to purchase larger-sized LCD TVs in a greater magnitude. Thus, the time-varying growth rate expands monotonously with the size of LCD TVs. Price reduction enhances the substantial diffusion of larger-sized LCD TV panels, curtailing the time of larger-sized LCD TV panels for reaching the inflection point or upper limit.

4.2. The Results of Forecasting

In this section, the ability of the conventional Gompertz model and our extended Gompertz model to predict LCD TV shipments is assessed. The parameters of both models are estimated by using quarterly LCD TV shipments in the training sample ranging from the initial period to the fourth quarter in 2008. This work chooses the market penetration rates that consistently have the lowest prediction errors among the three price indicators in the extended Gompertz model to depict their cumulative shipments in the test sample. The prediction error of 15-inch, 20-inch, 32-inch, and 46-inch TVs are consistently the lowest under 95%, 95%, 70% and 50% penetration rate among all the market penetration rates. Therefore, the shipments of these four sizes of LCD TVs under these four penetration rate were selected to be simulated by both the extended and conventional Gomprtz models. Figs. 8 to 11 depict the actual shipments and the simulated shipments, as estimated by the extended and conventional Gomprtz models under 95%, 95%, 70% and 50% market penetration rate for 15-inch, 20-inch, 32-inch, and 46-inch LCD TVs, respectively. Forecasting results by the extended Gompertz models have a trend similar to that of actual time-series LCD TV diffusions. Conversely, the forecasted trend of LCD shipments by the conventional Gompertz model differs from actual shipments. Cumulative shipment units simulated by the extended Gompertz model are closer to actual shipments than those simulated by the conventional Gompertz model.

Forecasted quarterly LCD TV shipments from the first quarter in 2009 to the fourth quarter in 2009 are then compared with actual quarterly shipments. Forecasting errors for each model is then measured by MAPE, MAD, and RMSE.Tables 5 to 10lists the forecast errors of extended Gompertz model which selects the three price indicators to predict cumulative shipments in the test period: 1. the last known real price, i.e. the price in the fourth quarter of 2008; 2. the estimated prices computed by the extended Gompertz model, and 3. the actual prices in the test period.

Regarding the extended Gompertz model inTable 6andTable 9, the MAPE, MAD and RMSE is the smallest under 95%, 95%, 70%, 50%, and 30% level of market penetration rate for 15-inch, 20-inch, 32-inch, 37-inch, and 40-inch LCD TVs when actual price in the test period is used to predict shipments in the test period. The empirical results reveal that the extended Gompertz model with absolute change in the level of prices and the extended Gompertz model with the change ratio of prices have similar forecast accuracy. The findings suggest that consumers purchase LCD TVs based on the comparison of the initial price with the later reduced prices, a finding consistent with the anchoring and adjustment theory. Combining the aforementioned results of parameter estimation with the findings of forecasting errors in this section, the market penetration rate has been found diminishing with the size level of LCD TVs. Market penetration rate is high for the smaller-sized LCD TVs while the market penetration rates is low for larger-sized LCD TV panels. This reflects that most larger-sized LCD TVs do not reach the market saturation and still have a remaining market share. In addition, the findings show that 32-inch LCD TV panels contain the greatest market share (143,145 thousand units), implying its mainstream position in TFT-LCD industry. When the houses are large enough, consumers tend to purchase larger-sized LCD TVs in living rooms and 32-inch LCD TVs in bedrooms. In addition to the

Time 2002Q1 2003Q1 2004Q1 2005Q1 2006Q1 2007Q1 2008Q1 bt 0.00 0.05 0.10 0.15 0.20 0.25 0.30 15-inch 20-inch 26-inch 32-inch 37-inch 40-inch

Fig. 6. The time-varying coefficient b1tversus time for extended Gompertz models with the absolute price change value at the 95% market penetration rate. The

consumers who live in large houses, 32-inch LCD TVs are also favorable for those who in small houses for putting them in the living rooms. Thus, the potential market size is greatest for 32-inch LCD TVs.

In regard to the prediction ability, MAPE of the conventional Gompertz model is listed inTable 11. The comparison results suggest that the prediction ability is improved dramatically as price reduction is considered for 26-inch and 32-inch LCD TVs. Price elements incrementally validate the predictions of LCD TV panel shipments. We strongly recommend for using a price-related diffusion model, such as the extended Gompertz model, when analyzing LCD TV shipments under a multiple generation structure. Although the predictive ability of the extended Gompertz model is not as good as that of conventional Gompertz model for 15-inch, 37-inch or 42-inch LCD TV panels under the optimal market penetration rate level, the simulated error difference is relatively minor. Martin and Witt[25]explained that the forecasting capacity of a model is“excellent” as MAPE is smaller than 10% (MAPEb10%). Forecasting capacity is “good” as MAPE locates in the interval [10%, 20%] (10%bMAPEb20%). Forecasting capacity is“reasonable” as MAPE is in the interval [20%, 50%] (20%bMAPEb50%). In the view of Martin and Witt[25] criteria, the forecast ability of the extended and conventional Gompertz models is“excellent” in predicting the cumulative shipments of 15-inch and 37-inch LCD TVs. The forecast ability of our extended Gompertz model also maintains to be“good” in predicting the cumulative shipments of 42-inch LCD TVs. The MAPE difference is minor when comparing conventional Gompertz and our extended Gompertz model. Conversely, the forecast ability of the extended Gompertz models with absolute price change values is“good” in predicting the cumulative shipments of 26-inch and 32-inch LCD TVs, while the forecast ability of the conventional Gompertz models is only“reasonable”. Since 32-inch LCD TVs are the mainstreams and have the greatest market share in LCD TV industry, the improvement of the extended Gompertz model is worthwhile for LCD TV manufacturers in making investment and production decisions.

5. Additional Testing

5.1. Multi-generational Gompertz Model Incorporating Price Factors

Previous constructed multi-generational models attempted to identify the substitutions of next generation products for previous generation ones[3, 26]. Owing to that similarly sized LCD TVs tend to replace each other, this work also constructs a multi-generational Gompertz model, which accumulates data of 37-inch, 40-inch, 42-inch and 46-inch LCD TVs to forecast LCD TV shipments. Price reductions are incorporated in specific multi-generational Gompertz models. Multi-generational Gompertz model are expressed as follows:

S1ð Þ ¼ Ft 1ð ÞMt 1½1−F2ðt−τ2Þ;

S2ð Þ ¼ Ft 2ðt−τ2Þ M½ 2þ F1ð ÞMt 1 1−F½ 3ðt−τ3Þ;

S3ð Þ ¼ Ft 3ðt−τ3Þ M½ 3þ F2ðt−τ2Þ M½ 2þ F1ð ÞMt 1 1−F½ 4ðt−τ4Þ;

S4ð Þ ¼ Ft 4ðt−τ4Þ M½ 4þ F3ðt−τ3Þ M½ 3þ F2ðt−τ2Þ M½ 2þ F1ð ÞMt 1;

ð10Þ

where Si(t) denote the cumulative sales at period t. Mirefers to market potentials of the ith generation without considering
market substitution of multiple generations. Generations of LCD TVs are classified by size. The 37-, 40-, 42-, and 46-inch LCD TVs
represent the first, second, third, and fourth generation, respectively. Notably,_{τi}refers to the time of introduction of successive
product generations. Assume thatτi= 0 for the first generation. For comparisons of the aforementioned extended Gompertz

Time 2002Q1 2003Q1 2004Q1 2005Q1 2006Q1 2007Q1 2008Q1 bt 0.00 0.05 0.10 0.15 0.20 0.25 0.30 20-inch 32-inch 37-inch 40-inch 42-inch 46-inch 15-inch

Fig. 7. The time-varying coefficient btversus time for extended Gompertz models with ratios of price changes at 95% market penetration rate. The coefficient btis

calculated by b2t¼ b2expγ2ðP0−PP0tÞ

h i

models, both the absolute price changes and the change ratio of prices are utilized to develop multi-generational models. The extended multi-generational Gompertz model which incorporates the absolute price changes and the change ratio of prices are defined here as“multi-generational Gompertz models with the absolute price change value” and “multi-generational Gompertz models with the change ratios of prices”. Cumulative percentage of adoptions in multi-generational Gompertz models with the absolute price change value” and “multi-generational Gompertz models with the change ratios of prices” can be written as Eqs.(11) and (12), respectively:

Fið Þ ¼ et −a1 e−b1 exp γ1 p i;0−pi;t−1 h i t ; t≥0: ð11Þ Fið Þ ¼ et −a2 e−b2 exp γ2 p i;0−pi;t−1 Pi;0 h i t ; t≥0: ð12Þ

Since such a multi-generational Gompertz model has four more parameter values, the parameter optimization fail to converge when this work utilize the statistical NLS methods to optimize parameters. Our proposed multi-general Gomertz models are estimated using a numerical method. This work iterates genetic algorithm simulations 1,000 times, subsequently generating

2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 12.0x103 12.1x103 12.2x103 12.3x103 12.4x103 12.5x103 12.6x103 12.7x103 12.8x103 actual actual_p last_p estimated_p conventional Time Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 12.0x103 12.1x103 12.2x103 12.3x103 12.4x103 12.5x103 12.6x103 12.7x103 12.8x103 actual actual_p last_p estimated_p conventional

## a)

### Extended Gompertz models with the absolute price change value

## b)

### Extended Gompertz models with the change ratios of prices

Fig. 8. Comparison of actual shipments of 15-inch LCD TVs with simulated values, by the conventional and extended models at 95% penetration rate level. “Actual,” “conventional,” “actual_p,” “last_p” and “estimated_p” in the figure represent the three price indicators used to predict shipments in test periods: actual price in test prices, the last prices in the training period, and estimated price calculated by price prediction model (Eq.(9)).

different initial values each time and ultimately yielding 1,000 sets of parameters. This work selects parameter values located within one standard deviation among these 1,000 sets of parameters and, then, calculates their means, t-statistics and prediction accuracy (Table 12). According to t-statistics results, the coefficient of price is significantly positive, indicating that price can increase sales. This observation confirms that price affects shipment volumes, which is the same if predicted with our extended Gompertz model. Additionally,Table 12lists the outcome of the extended multi-generational Gompertz models. This table reveals that the shipments of 40-inch, 42-inch and 46-inch LCD TVs are overestimated, particularly for the large-sized LCD TVs. Additionally, accuracy analysis of a multi-generational model indicates that the multi-generational model is less accurate than our extended Gompertz model. This difference is largely attributed to the fact that different sizes of LCD TVs are considered as homogeneous products in a multi-generational model; in general, consumers tend to select the optimum size for their living space. According to the market segmentation theory, different sizes of LCD TVs do not tend to substitute one another, explaining why a forecast done with a multi-generational model should not generate as accurate an outcome as our extended Gompertz model.

5.2. Gompertz Models which Adopts Market Potential as the Incremental Parameter

This work also constructs alternative Gompertz models which takes market potential L as the incremental parameter instead of estimating market potential by the market penetration rate. As for the extended Gompertz model which takes market potential

## a)

### Extended Gompertz models with the absolute price change value

Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 30.5x103 31.0x103 31.5x103 32.0x103 32.5x103 33.0x103 33.5x103 34.0x103 actual actual_p last_p estimated_p conventional

## b)

### Extended Gompertz models with the change ratios of prices

Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 30.5x103 31.0x103 31.5x103 32.0x103 32.5x103 33.0x103 33.5x103 34.0x103 actual actual_p last_p estimated_p conventional

Fig. 9. Comparison of actual shipments of 20-inch LCD TVs with simulated values, by the conventional and extended models at 95% penetration rate level. “Actual,” “conventional,” “actual_p,” “last_p” and “estimated_p” in the figure represent the three price indicators used to predict shipments in test periods: actual price in test prices, the last prices in the training period, and estimated price calculated by price prediction model (Eq.(9)).

L as an incremental parameter, the NLS statistical method generally has difficulty in optimizing the parameters of extended Gompertz equations. Statistical NLS optimization methods fail to estimate the model parameters of the extended Gompertz model, except for the 46-inch LCD TV data. Although the parameter optimization can converge for 46-inch LCD TV data, the estimated parameter value is unacceptable. The estimated market potential parameter of the 46-inch LCD TV is 591,401 thousand units. The market penetration rate is only 1.01% in the last quarter of 2008, when this work uses this estimated market potential of 591,401 thousand units to calculate the market penetration rate. These estimated results contradict the actual circumstances in the LCD TV panel industry where the market is nearly saturated and supply has exceeded demand in recent years. As for the conventional Gompertz model, only the 40-inch LCD TV data can statistically optimize the parameters of the extended Gompertz model. Although the parameter optimization can converge to estimate parameters for 40-inch LCD TV data, the estimated parameter value is unreasonable as well. The estimated market potential of the 42-inch LCD TV is 827,926 thousand units. The market penetration rate in the last quarter of 2008 is only 0.58% when this work uses this estimated market potential 827,926 thousand units to calculate the penetration rate. These estimated results also fail to reflect actual circumstances in recent years, in which the supply for LCD TVs has exceeded demand. This phenomenon implies that taking market potential L as an incremental parameter in the Gompertz model to optimize the parameter would be infeasible.

## a)

### Extended Gompertz models with the absolute price change value

Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 40x103 60x103 80x103 100x103 120x103 140x103 160x103 180x103 actual actual_p last_p estimated_p conventional

## b)

### Extended Gompertz models with the change ratios of prices

Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 40x103 60x103 80x103 100x103 120x103 140x103 160x103 180x103 actual actual_p last_p estimated_p conventional

Fig. 10. Comparison of actual shipments of 32-inch LCD TVs with simulated values, by the conventional and extended models at 70% penetration rate level. “Actual,” “conventional,” “actual_p,” “last_p” and “estimated_p” in the figure represent the three price indicators used to predict shipments in test periods: actual price in test prices, the last prices in the training period, and estimated price calculated by price prediction model (Eq.(9)).

6. Conclusions

This work presents a novel model to overcome the limitations of the conventionally adopted Gompertz growth model by incorporating price factors and estimating product demand. To our knowledge, this work explores for the first time the diffusive evolutions for various size levels of LCD TV panels by applying a Gompertz growth model that considers how price declines increase shipments. Feasibility of the proposed model is also demonstrated by comparing the goodness of fit and predictive ability of the extended Gompertz models which incorporate price factors with those of the conventional Gompertz model.

Analytical results indicate that the Gompertz curve reliably features the life cycle of LCD TVs. When LCD TVs first go on the shelves, consumers flock to purchase them. Demand quantity for LCD TVs surges, explaining why the marginal growth rate of LCD TV shipments is positive in the early stage of product life cycle. However, as popularity of the LCD TV diminishes, LCD TVs appeal less to consumers, explaining why the demand quantity decreases over time. Thus, the marginal growth rate of shipments becomes negative, and the cumulative shipments curve tends to level out, whose pattern is the same as the long and smooth tail of the Gompertz curve.

Especially for all LCD TV sizes in this work, consumer purchasing tends to increase with the price reductions in the extended Gompertz model. Comparing the initial price with the later reduced prices reveals that the price reductions induce consumers to purchase a LCD TV. Analytical results again support the anchoring and adjustment heuristic theory of Tversky and Kahneman[13]. Hi-tech products generally depend more on consumers to accept reduced prices than durable products do. This difference is owing to that few individuals are familiar with the functions of hi-tech products upon market entry. Time-varying price declines

## a)

### Extended Gompertz models with the absolute price change value

Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 8x103 10x103 12x103 14x103 16x103 18x103 20x103 22x103 actual actual_p last_p estimated_p conventional

## (b)

### Extended Gompertz models with the change ratios of prices

Time 2008Q4 2009Q1 2009Q2 2009Q3 2009Q4 8x103 10x103 12x103 14x103 16x103 18x103 actual actual_p last_p estimated_p conventional

Fig. 11. Comparison of actual shipments of 46-inch LCD TVs with simulated values, by the conventional and extended models at 50% penetration rate level. “Actual,” “conventional,” “actual_p,” “last_p” and “estimated_p” in the figure represent the three price indicators used to predict shipments in test periods: actual price in test prices, the last prices in the training period, and estimated price calculated by price prediction model (Eq.(9)).

Table 5

Forecast accuracy of extended Gompertz models with the absolute price change value, in which the price in the fourth quarter of 2008 is used to predict cumulative shipments in the test periods.

Penetration rate 70% 80% 90% 95% Size MAPE 0.0492 0.0222 15 MAD X X 610 275 RMSE 615 276 20 MAPE 0.1164 0.0682 0.0200 0.0064 MAD 3689 2160 632 205 RMSE 3737 2174 639 249 26 MAPE 0.2003 0.1978 0.1978 MAD X 9196 9155 9195 RMSE 10142 10363 10546 32 MAPE 0.1973 0.2202 0.2351 0.2342 MAD 27858 31071 33230 33239 RMSE 31150 34662 37244 37673 37 MAPE 0.0982 0.125127 0.1517 0.1653 MAD 3468 4380.323 5280 5742 RMSE 4345 5295.814 6237 6724 40 MAPE 0.1220 0.1536 0.1874 0.2061 MAD 4075 5067 6112 6688 RMSE 5218 6219 7242 7799 Penetration rate 10% 20% 30% 50% Size MAPE 0.0637 0.0471 37 MAD X X 2008 1678 RMSE 2109 2275 40 MAPE 0.2579 0.1158 0.0470 0.0825 MAD 7848 3398 1355 2740 RMSE 7930 3434 1505 3500 MAPE 0.3934 0.2258 42 MAD 11325 6298 X X RMSE 11397 6343 MAPE 0.0898 0.0714 46 MAD X X 1216 1129 RMSE 1277 1461

Notes: 1. The minimum errors among various level of market penetration rate are marked in grey shade for each size of LCD TVs.

3. MAPE ¼1 n Pn t¼1 Yt− ^Yt j j Yt ,MAD¼ 1 n Pn t¼1 Yt−^Yt , RMSE ¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PT t¼1 Yt−^Yt ð Þ2 n s

, where Ytand ^Ytrepresent the actual and predicted shipment at time t and n is the number of

Table 6

Forecast accuracy of extended Gompertz models with the absolute price change value, in which the actual prices in test periods are used to predict shipments in test periods. Penetration rate 70% 80% 90% 95% Size MAPE 0.0490 0.0220 15 MAD X X 608 273 RMSE 611 274 20 MAPE 0.1237 0.0757 0.0263 0.0063 MAD 3923 2398 831 199 RMSE 3987 2428 843 217 26 MAPE 0.1977 0.1940 0.1951 MAD X 9092 9004 9088 RMSE 10082 10284 10493 32 MAPE 0.1957 0.2184 0.2332 0.2328 MAD 27716 30905 33065 33122 RMSE 31319 34831 37419 37849 37 MAPE 0.0759 0.1062 0.1398 0.1595 MAD 2727 3753 4887 5550 RMSE 3764 4770 5890 6550 40 MAPE 0.1007 0.1371 0.1780 0.2013 MAD 3423 4563 5825 6541 RMSE 4715 5800 6991 7668 Penetration rate 10% 20% 30% 50% Size MAPE 0.0855 0.0414 37 MAD X X 2732 1451 RMSE 2874 1914 40 MAPE 0.2839 0.1409 0.0644 0.0646 MAD 8650 4170 1849 2174 RMSE 8773 4247 1979 3018 MAPE 0.4555 0.2793 42 MAD 13120 7840 X X RMSE 13314 8030 MAPE 0.1291 0.0534 46 MAD X X 1811 829 RMSE 1844 1037

Notes: 1. The minimum errors among various level of market penetration rate are marked in grey shade for each size of LCD TVs.

3.MAPE¼1 n Pn t¼1 Yt− ^Yt j j Yt ,MAD¼1n Pn t¼1Yt−^Yt ,RMSE¼ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ PT t¼1 Yt− ^Yt ð Þ2 n s