Abstract-From the demand theorem, sales model theory has fully discussed how price affects the sales process and manufacturer’s competitive strategy under the backgroundof supply surplus,but is little concerned with the effects ofbrand credit degree and choice cost on the sales process. Building sales model based on brand economics, using matlab to model and case studies of the variables such as brand sales cycle, brand purchase proportion of population, the paper provides theoretical support on how to optimize sales process and finally the study should become the theory guidance for corporations using brand as the core to establish business strategy and in order to obtain higher profits and market share .

Keywords-Sales model,brand credit degree,choice cost, case study

I. INTRODUCTION

The sales model is about a mathematical relationship between variables affecting product sales and the sales cycle in the research of brand sale process, which could effectively guide the corporations to establish the optimal marketing strategy and sales promotion mode, and to increase market share and profits rate. Hal R. Varian established the optimal sales model based on price and information to explain the retail behavior of real-world and provided a theoretical support for the corporation product sales[1]. The Peng SY established the sales model in whichprice is a decreasing function of demand rate, and proved the existence and uniqueness of the optimal policy[2]. Taking into account of the uncertainties,Han Xin-she and Ye Heng-kui established the product sales forecast model on product sales in a period of time whichprovided an important reference to forecast durable goods production and sales[3]. Leng Ke-pingdiscussed a simple sales process in the market from the game theory, established corresponding mathematical model and suggested the optimal strategy of the corporation product sales[4]. Mo Jiang-tao, Pan Hong-juan, Mi Fang-fang and Zhou Fang-ming studied the advance sales of the seasonal demand products using a two-stage pricing strategy[5].

On the law of demand, past researches have discussed the mechanism of price impacting on sales and discussed corporations’ competitive strategy in the condition of supply surplus, but lacked research on the influence of indicators in the sales process based on brand credit degree and choice cost. Taking the theory of SIS model for reference, this paper builds repeat purchase brand sales model based on the brand economics[6], and aims to introduce brand credit degree andchoice cost into the sales model, and make digital simulation on data models variables such as brand sales cycle, population proportion, extreme point of brand purchase, and brand coverage etc. It provides a theoretical support for corporationsapplying brand economics theory to brand building and decision-making. It can effectively guide the practice of corporations brand sales.

II.THE DEFINITION OF SALES MODEL VARIABLES BASED ON THE BRAND ECONOMICS

Brand category degree is the mental awareness degree of how consumers specify a certain brand as a category psychologically, and it can generally be obtained by testing and analysis[7].Through the analysis of the audience and means in brand promotion, we can get the correlation coefficient of brand and the promotion means, thus the data of valid daily brand contact rate can beobtained which determined by brand category degree, the correlation coefficient between brand and promotion means, brand investment, which are all proportional tovalid daily brand contact rate. The valid daily brand contact rate represents the degree of the promotion effectiveness.

We first define the valid daily brand contact rateλ which embodied brand credit degreeis the function of Bc, such as,

λ = λ(Bc) (1)

According to the definition of brand economics, Bc=b*s is brand credit degree, brand category degree b∈[0,1], brand strategy s∈[1 0], when b=1 and s=1, thus Bc=1. In order to simplify the analysis, we set s=1, thus Bc∈[0,1][7].Based on the aboveassumptions, we can obtainλby the following equation,

(2)

Where ci is brand promotion investment proportional of the category i of the brand, mki is correlation coefficient between brand category i and promotion means k, and bk is brand category degree when the brand uses the promotion means k.

Equation(2) can be explained economically as follows: assuming that the brand needs k kinds of promotion means, valid daily brand contact rate is multiplied by the investment proportion of the brand in all promotion means, correlation coefficients of the brand and promotion means as well as brand category degree. Valid daily brand contact rate is not only a function of brand investment amount but also a function of the correlation coefficient between brand and promotion means.

Equation(2) also provides a method to measure the effectiveness of brand promotion. Generally, when weput the limited business promotion investment expenses into the maximum correlation coefficient between the brand and promotion means, its valid daily brand contact rate is maximum too. The correlation coefficient of brand and promotion means can be obtained by investigation and testing, we will focus on the λ value influenced by brand category degree in the following analysis.

We secondly define valid daily brand purchase rate embodied choice cost μ, is the function of Cc, such as,

μ = μ(Cc) (3)

According to the definition of brand economics, Cc is choice cost, and Cc=f(Bc), Cc is inversely proportional to Bc[7]. In order to simplify the analysis, we set s=1, thus Bc∈[0,1], μ=μ(f(Bc)), and μis inversely proportional to Bc.Equation (3) can be explained economically as follows, the smaller choice cost of the brand is, the higherdaily brand purchase rate of the brand is.

To establish the repeat purchasebrand sales model, we define variables as follows: 1, i is the purchase population proportion of a certain brand; 2, s is the potential purchase population proportion of a certain brand, and s is determined by brand position,target market size, population income level and consumption propensity; 3, λ is valid daily brand contact rate of a certain brand, or the population proportion contact rate of a certain brand, and is proportional to promotion effectiveness and brand category degree. 4, μ is daily brand purchase rate of a certain brand, or the population proportion purchase a certain brand, and it is inversely proportional to choice cost and is concerned with the product type. 5, б = λ / μ, is the effective consumer contact number added by brand credit degree and choice cost during the brand sales.6, t is the time of sales duration of a certain brand.

III.REPEAT PURCHASE BRAND SALES MODEL ANALYSIS AND SIMULATION RESULTS

Repeat purchase brand sales model mainly analyzes the brands in the field of daily consumption[8]. While brands are repeatedly purchased or products are relatively unimportant, consumers would not be excited in a lot of decision-making activities in the brain[9].The brands will be purchased by consumers without hesitation and maintain a stable brand purchase rate.

We define s0(s0>0) as potential purchase population proportion, and i0(i0>0) as purchasing population proportion, and s0 + i0 = 1.

We can build the Repeat Purchase Brand Sales Model,

, (4)

Usingequation,then we get,

(5)

The solution of (5)is,

(6)

As t→∞,

(7)

Drawn from the above model, for repeat purchase brand, we have:

Proposition I: When б> 1 and λ> μ, the valid dailybrand contact rate is higher than the daily brand purchase rate, regardless of the initial i0 is large or small, as time passes i (∞) converges to a stable value (1-1/б), by the definition ofб, we know that λ and μ determine the size of the i(∞), i(∞) is proportional to μ and i(∞) is inversely proportional to λ. The largerλand the smaller μis , the largerбis, so is the larger i(∞). Thus the corporation has infinite life cycle, and can continue operating forever. At the same time μ determines the duration of each stage of the brand life cycle.The smallerμis, the longer time i(t)convergence to the stable valueneeds.Thus increasing the brand category degree and lowering the choice cost are available to consumers to purchase a certain brand without hesitation. Ultimatelyincreasing μ can effectively improve the stability of brand production and sales as soon as possible.

Using the software matlab[10] to simulate the model,we get the simulation as Fig.1.. The fig.1. shows i(∞) respectively maintaining the same convergence (1-1/8) = 0.875 on 50th and 80th days whenб=8, λ=0.8, μ=0.1, i0 respectively is 0.8 and 0.09. Fig.2.shows i(∞) respectively maintaining the same convergence (1-1/4)=0.75 on 100th and 180th days whenб=4, λ=0.8, μ=0.05, i0 is 0.8 and 0.09respectively.Through the simulation, we get the empirical formula, when i(t) reaches a stable value, t*Bc=100t0(the unit is day), where t0 is a constant determined by product type.The larger Bcand the smaller Cc are,the smaller t is, then corporations can significantly shorten the brand promotion period and cut costs for promotion, meanwhile the production scale can keep stable at a reasonable level as soon as possible.

When б gets smaller and gradually closes to a value 1,i(∞) becomes smaller, then a brand with low brand category degree will converge to a low brand purchase population proportion. Corporations will fall into "muddy middle zone[11]" or the brand will die out of the market owing to the condition that the product scale fail to expand, then we get proposition II.

PropositionII: When б1,then λμ,and the brand purchase population proportioni(t) will decrease with the time t prolongeduntil it is 0, which means the brand could not be sustainable and has a life cycle.

From λ=λ(Bc), μ=μ(Cc),with the other conditions remain unchanged, brand credit degree Bc and choice cost Ccinfluence б throughλ and μrespectively, then i(t) has different changing curve, the smaller Bc and Cc are, thesmallerб is, thus the faster i(t) decreases. When б=0, λ=0,

.

Fig.1. Simulation graph when б=8, μ=0.1.


Fig.2. Simulation graph when б=4, μ=0.05.

Using the software matlab to simulate the model,we get the simulation as Fig.3., i(∞)converges to 0on about 160th daywhen б=0.8,λ=0.08, μ=0.1,and i0is 0.8 and 0.09 respectively, which means when brand category degree is low, regardless of the initial purchase population proportion is large or small, the brand category will die out of the market,but the brand duration may varies, as shown in Fig.4., i(∞) will converge to about 0.07 and 0.02on 600th daywhen б=0.8, λ=0.008, μ=0.01, i0is 0.8 and 0.09 respectively.Although the brand category can also exists, but will gradually withdraw the marketbecause of lackingprofit margins. When compared Fig.3. and Fig.4., we can infer that the lower the dailybrand purchase rate is, the slower the brand dies, which would confuse the corporation’s decision-makersto mistakenly believe that the brand can developsustainably, thereby to increase the promotion investment.But when brand categorydegree is low,even ifpromotioninvestment increased, the result is still futile and even may be counterproductive to accelerate the death of the brand category.


Fig.3.Simulation graph when б=0.8, μ=0.1.

Fig.4.Simulation graph when б=0.8, μ=0.01.

Proposition I and proposition II illustrate what a key role brand category degree and choice cost play in brand sales model. Brand category degree can be gotten through testing and analysis generally.Valid dailybrand contact rate can be gottenfrom analyzing the promotion investment and the correlation coefficient between brand and promotion means. At the same time daily brand purchase rate can be observed in actual sales process. Combined both rates, we can calculate the brand effective contact number, which can be great help for the certain brand to predict the changerule of i(t) and t, and can guide corporations’ sales and production in the selling process.

IV.REPEAT PURCHASE BRAND CASE STUDIES

A.Chinese toothpaste market

Brand credit degreeequals to the product of the brand category degree and brand strategy.If brand category degree is high and brand strategy is correct, the brand market share will maintain in a high level in long-term. We will research on the brandin Chinese toothpaste marketfor example.According to the survey data of ZhongJing-XianLue Investment Advisory Center,research on the Chinese toothpastes market shows that Crest's shares more than 20% in the Chinese toothpaste market in the first half of 2010, indicating that the implementation of Procter & Gamble Crest "no cavities" brand promotion maintains its high brand credit degree and high brand repeat purchase rate. For repeat purchase brand, market share is the stable everyday brand purchase rate, so i(∞)=(1-1/б)=20%, we can calculate б=1.25, λ=1.25μ, which indicates that Crest toothpaste has high brand category degree by a large marketing promotion investment, making the valid daily brand contact rate higher than daily brand purchase rate to ensure its popularity in the market.

Chinese toothpastes have suffered much for a long time after foreign brands entered the Chinese market - the correct promotion strategy and heavy investment of foreign brands have already made them sidelined in the competition, together with the fact that they have a weak single interest point which impressed consumers that toothpastes are just cleaning supplies. In this reshuffle process, brand category degree and brand credit degree of Chinese toothpastes are low, thus Chinese toothpastes are vulnerable in the competition and even some brands died out.

According to the basic principles of brand economics, Chinese toothpastes have to establish a mutually exclusive category with the foreign brands, in order to increase brand credit degree and reduce choice cost, namely to implement category confrontation strategy[7]. In recent years, the Chinese herbal medicine efficacy toothpastes gradually rise, for example YUNNANBAIYAO toothpaste[12] got market share around 9% to 10% in China in the first half of 2010. This means that its brand credit degree has been significantly elevated; the core reason for this is that these Chinese toothpaste brands used the unique Chinese herbal formula for oral health care, which is of great significanceto stabilize its brand repeat purchase rate by establishing a single interest point.

B.Coca-Cola and Qinchi wine

Other examples are new-formula Coca-Colaand Qinchi (a wine brand in China). Due to the wrong product positioning, brands or categories above lasted only a very short time. The core reason is that they both have low brand category degree or product quality, leading to brand credit plummeting. In April 1985, Coca-Cola announced it would use the new formula instead of the old one, because the Company believed that“the bold decision was the best decision the Company has made[13]”. However, in July 1985, Coca-Cola Company announced to reuse the previous Coca-Cola formula because of the pressure. Only after 82 days, the new Coca-Cola quit from the market because of the wrong product positioning. Even Qinchi was the advertising pillar Wang in CCTV, unfortunately, its market situation was rapidly deteriorated because of bad product quality[14].

C.Porsche and Apple

Porsche is the world famous car brand, the production of Porsche 911 has been by far the world's best-selling since 1963 when the car was produced. It updated constantly averaging about 1.5 years on technology and shape, launched about 30 kinds of car models for consumers to choose totally. Porsche has a very high brand category degree in the field of luxury sports car, thereby enhanced its brand credit degree, thus it need constantly introduce new products to meet market needs and to ensure sustainable development of corporations.The new product release time is proportional to the validdaily brand contact rate and inversely proportional to daily brand purchase rate. The smaller λ and the larger μ are, the shorter intervals of new product releases is. Luxury brand should arrange the production and stocking to prevent temporary supply shortage and surplus of products based on the extreme point of purchase population proportion.

Apple Inc. takes a completely different category confrontation strategy from the "Wintel" compatibility architecture and continues to invest into innovative research and product development, thus makes Mac, iPad, iPod, iphone and other products maintain a high brand category degree.Apple Inc. establishs unique market position and wins high recognition of the market. Because of its high brand category degree and the correct marketing strategy, Apple brand credit degree increases and accumulates rapidly. The queuing up to purchase and short supply of the recent products (iphone4s) is a good proof.

Corporaions need to innovate or introduce new products to maintain its brand credit degree in the brand salescycle in order to improve their brand coverage. If the brands lack sustained R&D strength and ability to innovate, they will disappear without new category release after a short honeymoon-like time relying on powerful promotion.Different corporations can analyze the problems of their own brand in order to improve the final brand coverage by increasing brand promotion when brand category degree is a certain value, and increasing the valid daily brand contact rate. It also shows the importance of the rule “Have a promotion theme every year, Have promotion activities every month.” in corporation marketing promotion [15].

V. CONCLUSION

The new sales model as proposed in this research paper on the basis of brand economics is an in-depth explorationand extension to traditional arrangement of new product launch and distribution. Through the research, we get functions of the variables and variation disciplines of brand sales cycle, brand purchase population proportion by adding brand credit degree, choice cost and other indicators to the sales model.

Realistic simulation results obtainedby using software matlab to simulate the repeat purchase brand model brings usto a firmconclusion that brand can prolong the brand life cycle, quicken the process ofproduct stability, boost sales, elevate purchase rate and enlarge market share as well by increasing brand category degree and reducing choice cost. In this way, brand economics theory is further developed in this paper, which should be significant in brand building and sales practice because it could provide a theoretical support to guild corporationsto establish their core brand business strategy to obtain higher profits and raise market share through increasing brand awareness and using correct brand strategy.