Optimized Pricing Model of New Product Based on the Probability of Demand XIA Yiqun School of Economy and Management, Shanghai Institute of Technology, P.R. China, 200235 [email protected] Abstract: Pricing the new product was one of the most important strategic factors in its market diffusion process. Probability of demand function described the inherent relationship in the price of new product, the perceived price of customers and the purchase behaviors. The optimal price from the profit function modified by the probability of demand reflected and effected exactly the market diffusion. The result of analysis showed that appropriately low price motivated the imitative consumers’ innovative purchase behaviors, and extended the potential market scale. However, the extremely low price damaged the innovative consumers’ purchase utility, furthermore, damaged the market diffusion. Moreover, the perceived price of consumers increased if the public utility of new product increased while the price was stable, which result in the accelerative process of market diffusion. In conclusion, this probability of demand function was significant for the pricing and diffusion process of new product. Keywords: Technology Innovation, Market Diffusion, New Product, Pricing

1 Introduction The price of new product and its market diffusion process effect for each other. Bass had described the diffusion process of new product in the given potential consumers who would not re-purchase this product according to a simple time-variable mathematic model, and predicted the marketing volume of new product in the variable time[1,2]. Based on the Bass’ prediction, and in order to estimating the probability of new product in revenue, Chinese researchers discussed the time-distribution of the potential revenue[3]. More literatures agreed that pricing the new product is the main marketing strategic factors, and effected the demand in market and the profit of enterprises[4,5]. The research on consumers behaviors showed that price had been accepted as the symbolization or mouthpiece of quality by the consumers[6]. However, the price of new product always is higher than the cost so far. The dominant factor is just the perceived value of consumers that means how much money the consumers would like to pay for it. The famous Rogers model for new product diffusion explained vividly that different consumers had various recognitions on the same new product, and then, had different effect on market[7]. Consumers decided theirs perceived value according to their incomes, status, risk preference, product information in learn, [7] and their recognition to the value of product . While the perceived value effected the purchase [8,9] behaviors adversely . Most literatures (including Bass model) considered the influence on price from demand and cost. They divided consumers into buyers and non-buyers, while ignored the influence on the marketing diffusion from the consumers who had demand but was drifting, which resulted in the over-estimation for the future and latent market. In order to amending this limitation, this paper will found the “probability of demand” to connect all the kinds of consumers and to represent the interactive effect among the price, perceived value and market demand. Moreover, it will be used to optimize the profit function model, and then help to research how the price of new product influences its information diffusion and the profit of enterprises.

2 The process of new product diffusion and the demand probability Defining “potential consumers” as general ones who probably learn this product and S is its total number. Suppose the accumulative total number of consumers who had bought this product at time t is f (t); the accumulative total number of consumers who had learnt this product (had demand if price is 0) at time t 13

is g(t); f ′(t) and g′(t) denote respectively the increasing number of consumers who purchase and learn this product at time period [t,t+dt]. In the Bass model and present literatures, g(t) and f (t) is the same. They suppose f ′(t) = g′(t), which will result in the over-estimation for the future and latent market. In fact, the more actual situation is that g′(t) ≥ f ′(t), g(t) ≥ f (t), g(0)=0, f(0)=0. As the market demand is influenced by the price p, this paper model their relationship with the probability of demand D(p): f ′(t)= g′(t)D(p) (1) because of g(0)=0, f(0)=0, yields f (t)= g(t)D(p) (2) The profit of enterprises was influenced directly by f (t), the number of consumers who had bought actually the product other than g(t). Moreover, the consumers who purchase new product can be divided into two classes[2,3]: one is innovative consumers, namely, they will purchase new product according to their demand entirely; the other is imitative consumers, namely, they will decide to buy or not buy this new product according to present or future information of product and sales. Suppose innovative rate (coefficient) at every t time, namely, the number of buyer at unit time is constant a, while, imitative rate (coefficient) is b f (t) which is various as time being different. The relationship of g(t) and g′(t) is g′(t)=(a+ bf(t))(S-g(t)) (3) here, (S-g(t)) other than (S- f (t)) represents potential consumers, which avoid the over-estimation of value in Bass model. Probability of demand D(p) describes the probability that consumers who have learnt the new product would like to purchase it with given price p. This desire of purchase is associated directly with the perceived value ( the moreover perceived utility), which explains that the consumers who have learnt the product and demand it, will not always purchase it, namely, at any time t, g(t)≥f(t), unless the price of product is lower than perceived value. Suppose price of new product is p in any time period, U is the public utility of new product, namely the public information of new product, θ [0,1] is the private information of new product from consumers, namely, experience of product value. More experience means more acceptances, and the experience depends on private taste, incomes, and so on. So, Uθ is perceived price, and u=Uθ-p is real utility of purchase behaviors of consumers. According to u>0, u=0, u0, and f ′δ(t)>0). And integrating fp(0)=fδ(0)=0, limt→∞bfp(t) limt→∞fδ(t) S(1-p/ p′), Eq. (6) can be transformed T  a(δ − a )  e δt − 1 e δT − 1 δ −a  Max V (δ ) = ) p ′ − c  e − rt ( δT ) + r ∫ ( δt )e −rt dt   (1 − 0 δ b bS ae − a + δ ae − a + δ    s.t. (8) a ≤ δ ≤ bS (1 − c / Uθ ) + a in Eq.(8), V(δ) is continuous in closed interval [a, bS(1-c/ p′)+a], V(a) 0 and V(bS(1-c/ p′)+a) 0, so, inequality a 0 a bS   Supposing a simulating h(y)=yey-ey+1 with h(0)=0, and for arbitrary y > 0, yields h′(y)=yey>0, moreover, arbitrary y>0, yields h(y)>0, so

＝

＝

＝

＝

＝

＝

；

h(δ *t ) = δ *t e δ *t − e δ *t + 1 > 0

(12) According to Eq.(11) and Eq.(12), on the condition of optimized solution, first term in the right of Eq.(10) is always positive, then, negative second term can be deduced, namely, 2(δ * −a ) p ′ + c − p ′ > 0 , with the following attributes: bS p′ + c δ * −a (13) ) p′ < p* = (1 − bS 2

4.

Sensitivity analysis

We discuss the influences among the parameters a, b, S, p′ and p* though sensitivity analysis. 2 2 2 2 2 Inference 1: if ( p′ − c)bS + ( p′ − c) b S + 4a p′ < δ * < 2 a + bS ( p ′ − c) , 2 2 p′ * * then, ∂δ / ∂a < 0 , thereby, ∂p / ∂a < 0 Proof: According to Eq.(8), if δ approach smoothly to δ* (namely, limδ→δ*), then, 15

( p ′ − c)bS + ( p′ − c) 2 b 2 S 2 + 4a 2 p′ 2 2

is ∂V ′(δ ) >0, then, ∂a

< δ < 2a + bS ( p ′ − c) . Calculating the derivative a in Eq. (10), if there 2 p′

V ′(⋅) is increasing function in the parameter a, moreover, according the last section,

V′(δ*) is decreasing in neighborhood field δ*, thereby, ∂δ * / ∂a < 0 and ∂p * / ∂a < 0 . So, if limδ→δ*, yields 2 2  ∂V ′(δ ) = 1  (δ − a ) p ′  + δ (c − p ′)  2  ∂a bS ba   +

δT − rT T  e δt − 1 − 1  2(δ − 2a) p ′   e (e − 1) e −rt dt  + r∫ + c − p′   δT  δ 0 ae t − a + δ b  bS ae a − + δ   − rT

δT

δt

 e −1 a  2(δ − 2a ) p ′   e (e − 1) + r T e − rt dt  + c − p ′    2 2 δT δt 0 b bS (ae − a + δ )   (ae − a + δ )  In this equality, the first term is positive, the sum of second and third term is positive. Therefore, ∂V ′(δ ) >0 is set up in the neighborhood filed where δ approaching closely to δ*. +

∫

∂a

This inference explain that, if innovative consumers increase, price should be decreased to some extent with the condition p*< (p′+c)/2, which in turn motivates the imitative consumers to purchase the new product actively. However, if p* is low excessively, either the purchasing desire of innovative consumers or the profit of enterprise is impacted. Therefore, there should be a lower limit of for p*. Based on the relationship between p* and δ* discuss in section 3, we extend inference 1 as 2 2 2 2 p ′ + c ap ′ U b S (1 − c / p ′) + 4a + − 2 bS 2bS

> p* > p ′ + c − ap ′ . It show us that optimized p* is satisfied to not 2

bS

only p*< (p′+c)/2, but also p*> (p′+c)/2-ap′/bS. In this paper, (p′+c)/2-ap′/bS is defined as “threshold price”, which means that purchasing utility of innovative consumers will be impacted if price of product is lower than it. Inference 2: If ∂δ * / ∂b > (δ * − a ) / b , then, ∂p * / ∂b < 0 . Methodology of proof is the same as that of inference 1, which is solving the inequality ∂V ′(δ ) >0 ∂b

integrating Eq.(9). Inference 2 explains that if “lower price” strategy is executed by enterprise, the number of imitative consumers will increase. Inference 3: If ∂δ * / ∂S > (δ * − a ) / S , then, ∂p * / ∂S < 0 . Methodology of proof is the same as that of inference 1, which is solving the inequality ∂V ′(δ ) >0 ∂m

integrating Eq.(9). Inference 2 explains that if “lower price” strategy is executed by enterprise, the rate of purchase will increases, the high perceived price of consumers will extend the potential market. Inference 4: If δ * < bS / 2 + a , then, ∂δ * / ∂p ′ > 0 thereby, ∂p * / ∂p ′ < 0 . Methodology of proof is the same as that of inference 1. Supposing δ * < bS / 2 + a , in the neighborhood filed where δ approaching closely to δ*, inequality δ < bS / 2 + a is exact. So, solving the inequality ∂V ′(δ ) >0 integrating Eq.(9). Be similar to the inference 1, inference 4 can be extend,

，

∂p ′

namely, if p*>Uθ/2, then, ∂p * / ∂p ′ < 0 . Here Uθ is used to show that if the public utility U is increased because of the quality, service, brand while the price is stand, the perceived price will increase, consequently, probability of demand increase, and f ′(t) will approach to g′(t), which result in the accelerative process of market diffusion.

5 Conclusion

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Market diffusion of new product is the process actualizing the value of S&T innovation. Price is the most sensitive factors in this process that impact the consumers’ behavior, but in turn, is impacted by consumers’ behaviors, moreover, influences the information diffusion of new product and profit of enterprise. In the pricing process, deciders should consider not only the scale of market and cost factors, but also the perceive price of consumers which is more important. Therefore, in order to evaluating market and pricing new product exactly, it is necessary to differentiate the various utility and behaviors of consumers. This paper proposes that price of new product, perceived price of consumer, purchase behaviors of consumers can be connected by a novel probability of demand. We modify the profit function with this probability of demand and give the optimized price solution. The conclusions and inferences are significant for the related management behaviors.

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