# Product Line Pricing under Marginal Moment Model with Network Effect

Authors: Wei Qi; Xinggang Luo; Xuwang Liu; Yang Yu; Zhongliang Zhang
Journal: Complexity
 (2019)

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## Abstract

The complex network effect of the product exhibits has a significant impact on the product line optimization design. The multinomial logit (MNL) model which is used to simulate consumer choice behavior is applied in most of product line optimization problems. However, its assumptions, independence of irrelevant alternatives (IIA) and the same Gumbel distribution of random error terms, are usually difficult to be met in practice. The marginal moment model (MMM) can be used when the mean and variance of consumer’s utility error are known. The MMM not only has weak assumption conditions but also overcomes the IIA problem of MNL model. In this paper, we study the product pricing problem based on MMM with endogenous negative network effect. Firstly, we construct a variant of MMM considering network effect in product line optimization design. Secondly, we prove that the revenue function is concave in market share. We propose the solving methods of the model to obtain the optimal price, the corresponding market share, and the revenue under three different scenarios, i.e., developing single product, homogeneous products, and heterogeneous products. Finally, numerical experiments show that the proposed model can better simulate consumer choice behavior and potentially increase revenue.


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## Body

## 1. Introduction

With the rapid development of Internet economy, the traditional consumption structure has been changed greatly, and the research on the influence of complex network effect for development and pricing of new products has become an important academic issue. Many products will exhibit network effect or network externality when the utility a consumer obtains from buying the products depends on the number of sales of the same or similar products [1]. In traditional process of product line pricing, it is assumed that utility is only determined by customer’s personal features. However, if a product line has the characteristic of network effect, firms need to analyze how a customer’s purchasing behavior is influenced by other customers and reassess the estimated market size. Since product price is directly related to utility and market size affects the cost structure of the product line, network effect also influences the process of pricing and quality design of a product line. For these firms that develop new products with network effect, a significant challenge is to simulate consumer choice behavior that incorporates network effect and use this model to design a product line to maximize revenue.The network effect usually includes two aspects: positive network effect and negative network effect. For example, there are many products that display positive network effect, e.g., online games, social software (QQ, WeChat, Facebook, and microblog), online movies, and group purchase products. The more people that buy them or use them, the more utilities the consumers can get from buying these items. There are also some products that exhibit negative network effect, for example, the choice of traffic modes or the selection of personalized customers for products. If a firm produces a product line of women’s bags or clothes, customers with personalized requirements may be reluctant to buy the products when more people buy the same or similar products in the product line. If there are too many people buying cars in a small city and resulting in traffic congestion, the utility of owning a car will decrease. If the customers of some luxury goods feel that these goods are possessed by too many consumers, the value or attractiveness of these goods will decline. The more people choose these products or services, the less utility the consumers who choose the same or similar items will obtain.Many firms provide a product line with a variety of products that display network effect to meet different consumers’ needs and obtain maximum profit or revenue. To design a product line, a firm needs to consider a lot of factors, such as pricing decision, quality of product, development cost, price sensitivity of consumer, and network effect.In the process of product line design, consumer choice model [2] is vital to solve the product line optimization problem. There are two kinds of consumer choice models: one is deterministic choice model and the other is probabilistic choice model. Because probabilistic choice model can simulate consumer choice behavior more practically, it has been widely applied in marketing, operation management, transportation, and other fields.Because of the simple formula and good simulation results, the multinomial logit (MNL) model [3] is one of the most widely used probabilistic choice models. However, its assumptions, independence of irrelevant alternatives (IIA) and the same Gumbel distribution of random error terms, are difficult to be met in practice. The IIA property shows that the ratio of choice probabilities of two alternatives does not depend on other alternatives, regardless of their similarities. This is usually exemplified by the famous “red bus/blue bus" paradox [1]. Some other probabilistic choice models, such as nested logit (NL) model, multinomial probit (MNP) model, or mixed logit (MIXL) model, are also limited in application due to the complex calculation process or pregiven specific distributions of measuring errors.To better simulate consumer choice behavior, Natarajan et al. [4] propose a semiparametric choice model which is called as marginal moment model (MMM). The MMM can be applied when both the mean and the variance of utility error are known. The model does not need to know the specific distribution of error in advance and can also overcome the shortcoming of IIA in the MNL model; hence the MMM has better applicability in practice [5].In the research of product pricing problem, many scholars study the influence of positive network effect on consumer choice behavior based on MNL model [6–9], and it has been found that the optimal prices may be different even for homogeneous products case. However, the issue of product optimization with considering negative network effect cannot be ignored and needs to be further solved. In many product line design optimization problems, there exist a lot of products with negative network effect [1] and the problem of IIA may not be met. In this case, consumer choice behavior simulation and product pricing would be very interesting practical issues. The network effect is usually endogenized in the utility of consumer choice model, and the MMM that overcomes IIA problem can better simulate consumer choice model; therefore, the MMM with network effect can solve the product line pricing problem better. This paper focuses on the influence of the consumer choice behavior based on MMM with endogenous negative network effect on product line design problem. The contributions of this research are given as follows:(1) We apply the MMM with negative network effect to simulate consumer choice behavior in product line optimization problem. For the utility of consumer, we add a network effect term consisting of a coefficient of network effect and the sale quantity of the product.(2) Under the scenarios of single product, homogeneous products, or heterogeneous products, we prove the concavity of revenue function in market share with considering the network effect and present the solving equations for the optimal price, the optimal market share, and the optimal revenue, respectively.(3) By performing numerical experiments, we first demonstrate the variation of the optimal solutions with different parameters including network effect parameter, quality of product, price sensitivity parameter, and variance of utility error term. We then analyze the importance of considering network effect when a product line is designed. Finally, to illustrate the applicability of the proposed model, we test the robustness of solutions when the estimation of network effect parameters has some errors.The remainder of this paper is organized as follows. In Section2, we review the related literature papers about product line pricing and network effect. In Section 3, we propose a marginal moment model with negative network effect to simulate consumer choice behavior in product line design and present the solving methods of the optimal solutions. In Section 4, we present the results of some numerical experiments. In Section 5, we conclude the paper.


## 2. Literature Review

In this section, we review the literature papers related to this paper, including the product line pricing considering consumer choice behavior and the study of products with network effect.


### 2.1. Product Line Pricing

There are a lot of literature papers studying product pricing in the product line optimization design [10, 11]. The optimization objective is mainly maximizing profit [12, 13]. There are also some literatures which aim at obtaining maximin utility or minimax regret [14, 15]. And price is usually set as endogenous variable [16, 17].Product line pricing problem is designed in different competitive market. For a monopoly company, Heese and Swaminathan [18] analyzed a stylized model of product line design with component commonality to reduce production costs. They showed the strategy could produce the products of higher quality and acquire higher revenue. In a bilateral monopoly market, Dong et al. [19] investigated the product line pricing with two-stage game. Different from the previous study, they extended to the complementary products and endogenized the price for both the retailer and the manufacturer. Under oligopoly, Yayla-Küllü et al. [20] analyzed how the capacity constraint and competition affected the product-mix decisions. They found that when the resources were limited, the firm should only offer the product that had the largest profit per unit of capacity.Moreover, the simulation of consumer choice behavior is also vital for product line optimization design. Since McFadden [3] proposed multinomial logit (MNL) model, the model has been widely applied in many fields, such as marketing, economics, operation management, and transportation. But its restrictive assumptions, the independence of irrelevant alternatives (IIA) and the same Gumbel distribution of random error terms, are hard to be met in reality. Subsequently, some other probabilistic choice models are presented, for example, the nested logit (NL) model [21], multinomial probit (MNP) model [22], generalized extreme value (GEV) model [23], and mixed logit (MIXL) model [24].Recently, some researchers propose new probabilistic choice models. Kim et al. [25] developed a probit choice model under sequential search, which avoided the computation of high-dimensional integrations for partial analytic model. To address the problem of assortment optimization, Blanchet et al. [26] introduced a Markov chain choice model, which was a good approximation for general choice model under mild assumptions. To alleviate the IIA property of MNL model, Natarajan et al. [4] proposed two kinds of semiparametric choice models—marginal distribution model (MDM) and marginal moment model (MMM). They could be used when only the distribution or mean and variance of the utility error were known. Mishra et al. [27] further studied the theoretical characteristic of MDM. They proved that MDM generalized the MNL, MMM, and GEV models.


### 2.2. Network Effect

There are two streams of researches on the products with network effects. One is global network effect, where the utility consumers get from buying a product depends on the total sales of that product; the other is local network effect, where the consumer’s utility only depends on the number of purchases made by his “neighbors”. Our work only considers global network effect. For local network effect, please refer to Candogan et al. [8], Bloch and Quérou [28], and references therein for recent researches.The research on network effect mainly focuses on the global network effect [29, 30]. For new product pricing problem, Du et al. [9] established the MNL model with network effect. The authors analyzed the properties of the optimal market share and the optimal price for homogeneous coefficients and heterogeneous coefficients case. They found that the optimal prices might be different when network effect existed even for homogeneous case. Different from the literature Du et al. [9], Wang and Wang [1] investigated the assortment optimization problem under MNL with endogenous network effects. They found that the quasi-revenue-ordered assortment was optimal under certain conditions. In addition, they showed that it was very necessary to consider the network effect if such effects did exist.


## 2.1. Product Line Pricing

There are a lot of literature papers studying product pricing in the product line optimization design [10, 11]. The optimization objective is mainly maximizing profit [12, 13]. There are also some literatures which aim at obtaining maximin utility or minimax regret [14, 15]. And price is usually set as endogenous variable [16, 17].Product line pricing problem is designed in different competitive market. For a monopoly company, Heese and Swaminathan [18] analyzed a stylized model of product line design with component commonality to reduce production costs. They showed the strategy could produce the products of higher quality and acquire higher revenue. In a bilateral monopoly market, Dong et al. [19] investigated the product line pricing with two-stage game. Different from the previous study, they extended to the complementary products and endogenized the price for both the retailer and the manufacturer. Under oligopoly, Yayla-Küllü et al. [20] analyzed how the capacity constraint and competition affected the product-mix decisions. They found that when the resources were limited, the firm should only offer the product that had the largest profit per unit of capacity.Moreover, the simulation of consumer choice behavior is also vital for product line optimization design. Since McFadden [3] proposed multinomial logit (MNL) model, the model has been widely applied in many fields, such as marketing, economics, operation management, and transportation. But its restrictive assumptions, the independence of irrelevant alternatives (IIA) and the same Gumbel distribution of random error terms, are hard to be met in reality. Subsequently, some other probabilistic choice models are presented, for example, the nested logit (NL) model [21], multinomial probit (MNP) model [22], generalized extreme value (GEV) model [23], and mixed logit (MIXL) model [24].Recently, some researchers propose new probabilistic choice models. Kim et al. [25] developed a probit choice model under sequential search, which avoided the computation of high-dimensional integrations for partial analytic model. To address the problem of assortment optimization, Blanchet et al. [26] introduced a Markov chain choice model, which was a good approximation for general choice model under mild assumptions. To alleviate the IIA property of MNL model, Natarajan et al. [4] proposed two kinds of semiparametric choice models—marginal distribution model (MDM) and marginal moment model (MMM). They could be used when only the distribution or mean and variance of the utility error were known. Mishra et al. [27] further studied the theoretical characteristic of MDM. They proved that MDM generalized the MNL, MMM, and GEV models.


## 2.2. Network Effect

There are two streams of researches on the products with network effects. One is global network effect, where the utility consumers get from buying a product depends on the total sales of that product; the other is local network effect, where the consumer’s utility only depends on the number of purchases made by his “neighbors”. Our work only considers global network effect. For local network effect, please refer to Candogan et al. [8], Bloch and Quérou [28], and references therein for recent researches.The research on network effect mainly focuses on the global network effect [29, 30]. For new product pricing problem, Du et al. [9] established the MNL model with network effect. The authors analyzed the properties of the optimal market share and the optimal price for homogeneous coefficients and heterogeneous coefficients case. They found that the optimal prices might be different when network effect existed even for homogeneous case. Different from the literature Du et al. [9], Wang and Wang [1] investigated the assortment optimization problem under MNL with endogenous network effects. They found that the quasi-revenue-ordered assortment was optimal under certain conditions. In addition, they showed that it was very necessary to consider the network effect if such effects did exist.


## 3. Product Line Design Based on MMM with Network Effect

Natarajan et al. [4] proposed a marginal moment model (MMM). Assume that there are n (i=1,…,n) products to be developed to maximize the revenue. We assume that the competitors, in the short run, do not respond to the firm’s new products [31, 32]. The market has m (j=1,…,m) potential consumers. If consumer j purchases product i, the consumer has a utility uij and(1)uij=vij+εij,where vij is the deterministic component of the utility from the observed product and consumer attributes and εij is a random variable that represents consumer-specific idiosyncrasies. Assume that we know the mean 0 and variance σij2 of the random error; that is, the utility uij has the mean vij and variance σij2 and the choice probability (market share) is given as [4, 33](2)qij=121+vij-λjvij-λj2+σij2,where the Lagrange multiple λj is found by solving(3)∑i=1n121+vij-λjvij-λj2+σij2=1.In our model, we consider a variant of MMM that incorporates network effects in consumer’s utilities. The consumer’s utility that is similar to the literature of Du et al. [9] is decided by the quality of product yi, its price pi, its overall consumption xi, price sensitivity parameter bij (bij≥0), and the network effect sensitivity parameter αij, and(4)vij=yi-bijpi+αijxi.The parameterαij represents the strength that consumer j’s utility is affected by the overall consumption of product i. The larger the parameter αij, the more sensitive the network effect of consumer purchasing the product. The network effect on vij only depends on the total consumption level of product i. Throughout, we assume that αij≤0, and we normalize the total market size to 1 (m=1). Thus the choice probability qi is equal to the total market consumption of product i, i.e., xi=qi. (For a general m, xi=mqi, we can redefine αi~=mαi and the two problems are equivalent [9].) And we have vi=yi-bipi+αiqi and then the choice probability is defined as(5)qi=121+yi-bipi+αiqi-λyi-bipi+αiqi-λ2+σi2.In model (5), network effect is endogenized in the utility of MMM in the form of network strength parameter. Each consumer buys at most one product. Product 0 indicates a no-purchase option in the product line or an external option. For the no-purchase option, the utility is v0 and the variance of error term is σ02. Meanwhile, both parameters v0 and σ0 are known. The choice probability of no-purchase option is(6)q0=121+v0-λv0-λ2+σ02.By (6), we have λ=v0-1/2σ0(2q0-1)·(q0-q02)-1/2. And by (5), the price of product i is defined by(7)pi=1biyi+αiqi-v0+12σ02q0-1·q0-q02-1/2-12σi2qi-1qi-qi2-1/2,where σi>0 and σ0>0. The development costs of all products are assumed to be 0 [9]. Defining q=(q0,q1,…,qn), the firm’s total revenue is (8)πq=∑i=1nqi·piq=∑i=1nαibiqi2+∑i=1nqibi·yi-v0+σ022q0-1·q0-q02-1/2-∑i=1nσi2bi2qi2-qiqi-qi2-1/2. The product line optimization problem is stated as(9)maxπq0,q1,q2,…,qns.t.∑i=1nqi+q0=1qi,q0≥0i=1,2,…,n.In the following sections, we study that how network effects impact the optimal prices, the corresponding market shares, and the optimal revenues in different scenarios, including developing one new product, homogeneous scenarios of multiple products with the same coefficients, and heterogeneous cases of multiple products with different parameters.


### 3.1. Pricing for One Product

In this section, we will investigate the product pricing problem when one product is developed. The quality of the product isy, the price sensitivity parameter is b, the network effect parameter is α, the variance of utility error term is σ2, and the purchasing probability of the product is q. By (7), the price of the product is(10)p=1by+αq-v0+12σ02q0-1q0-q02-1/2-12σ2q-1q-q2-1/2.We assume σ=σ0>0 and q+q0=1, so the price is changed as(11)p=1by+αq-v0+12σ01-2qq-q2-1/2-12σ2q-1q-q2-1/2=1by-v0+αq+σb1-2qq-q2-1/2.The firm’s revenue is(12)π1q=p·q=αbq2+y-v0bq+σbq1-2qq-q2-1/2.The optimal product pricing optimization problem is(13)maxπ1qs.t.q≥0.The following proposition proves thatπ1(q) is concave in q when only one product is developed. When network effect is considered, the optimal price and the optimal market share can be derived by the implicit equations; then the maximum revenue can be obtained by formula (12) according to the corresponding optimal price and the optimal market share.Proposition 1.
When only one product is developed,π1(q) is concave in q, and the optimal price and the optimal market share can be obtained by the following implicit equations:(14)4αq+2y-v0+σq-q2-3/24q3-6q2+q=0p=1by-v0+αq+σb1-2qq-q2-1/2.Proof.
Because(15)∂π1q∂q=2αbq+y-v0b+σb1-4qq-q2-1/2+q-2q2-12q-q2-3/21-2q=2αbq+y-v0b+σ2bq-q2-3/24q3-6q2+q,∂2π1q∂q2=2αb+σ2b-32q-q2-5/21-2q·4q3-6q2+q+q-q2-3/212q2-12q+1=2αb+σ2bq-q2-5/2-q2-12q,and b≥0,σ>0,α≤0, so ∂2π1(q)/∂q2≤0, π1(q) is concave in q.
By∂π1(q)/∂q=0, 2α/bq+y-v0/b+σ/2b(q-q2)-3/2(4q3-6q2+q)=0. Then the optimal q∗, p∗ can be obtained by (14):Even if the network effect is taken into account, Proposition1 proves that the revenue function is concave in choice probability when only one product is developed. Meanwhile, the optimal price and the corresponding choice probability can be solved, and the optimal revenue can also be obtained. Therefore, when the firm plans to develop one new product to obtain maximum revenue, we find that the optimal choice probability (market share) is related to the network effect, the quality, the utility of no-purchase option, and the variance of utility error term; in addition to the above factors, the optimal price is also related to the price sensitivity parameter.


### 3.2. Pricing for the Homogeneous Products

In this section, we consider a special case that alln products have the same coefficients; i.e., all products have the same quality, price sensitivity parameter, network effect parameter, and the variance of utility error term, i.e., yi=y, αi=α, bi=b, and σi=σ. Without loss of generality, we assume that the price sensitivity parameter is b=1 in the remainder of this section.By (7), the price can be rewritten as(16)pi=y+αqi-v0+12σ02q0-1q0-q02-1/2-12σ2qi-1qi-qi2-1/2,and the firm’s total revenue (8) is expressed as(17)π2q=∑i=1ny-v0+αqiqi+∑i=1nqi·σ022q0-1·q0-q02-1/2-∑i=1nqiσ22qi-1qi-qi2-1/2. Thus the product line optimization problem can be formulated as follows:(18)maxπ2q0,q1,q2,…,qns.t.∑i=1nqi+q0=1qi,q0≥0i=1,2,…,n.The following proposition shows that when the network effect exists, the revenue functionπ2(q0,q1,q2,…,qn) is still concave in choice probability qi and the optimal price and the optimal market share can still be obtained by implicit equations. Meanwhile, the optimal revenue can be obtained by formula (17).Proposition 2.
The revenue functionπ2(q0,q1,q2,…,qn) is concave in qi, and the optimal price and the optimal market share can be obtained by the following implicit equations:(19)2αqi+y-v0+12σ0q0-q02-3/2-2q03+3q02-12q0-12-σ2qi-qi2-3/2-2qi3+3qi2-12qi=0pi=y-v0+αqi+12σ02q0-1q0-q02-1/2-12σ2qi-1qi-qi2-1/2q0=1-∑i=1nqi.Proof.
Forπ2(q), let us divide it into three sections. Let t1=∑i=1n(y-v0+αqi)qi, and then(20)∂t1∂qi=αqi+αqi+y-v0=2αqi+y-v0,∂2t1∂qi2=2α≤0.
Lett2=∑i=1nqi·σ0/2(2q0-1)(q0-q02)-1/2, and then(21)∂t2∂qi=12σ02q0-1q0-q02-1/2+∑i=1nqi·12σ0-2·q0-q02-1/2+2q0-1-12q0-q02-3/2-1+2q0=12σ0q0-q02-3/2-2q03+3q02-q0-12∑i=1nqi,∂2t2∂qi2=12σ0-32q0-q02-5/2-1+2q0-2q03+3q02-q0-12∑i=1nqi+q0-q02-3/26q02-6q0+12=12σ0q0-q02-5/2-12q0-542+132≤0.
Lett3=-∑i=1nqiσ/2(2qi-1)(qi-qi2)-1/2=-σ/2∑i=1n(2qi2-qi)(qi-qi2)-1/2, and then(22)∂t3∂qi=-σ2qi-qi2-3/2-2qi3+3qi2-12qi,∂2t3∂qi2=-σ2qi-qi2-5/212qi2+14qi≤0.
So∂2π2/∂qi2=∂2t1/∂qi2+∂2t2/∂qi2+∂2t3/∂qi2≤0. And ∂2π2/∂qi∂qj=∂2t2/∂qi2≤0 (i≠j). For the Hessian matrix of revenue function π2(q), let ∂2π2/∂qi2=Bi≤0, ∂2π2/∂qi∂qj=A≤0, and B=mini⁡Bi, B≥A. Because(23)BA⋯AAB⋯A⋮⋮⋱⋮AA⋯B=B+n-1AA⋯AB+n-1AB⋯A⋮⋮⋱⋮B+n-1AA⋯B=B+n-1A1A⋯A1B⋯A⋮⋮⋱⋮1A⋯B=B+n-1A·B-An-1≤0,ifnisanoddnumber;>0,ifnisanevennumber.Assuming that Bi=B+Δi(Δi≤0,Bi≤0), then(24)BA⋯A⋯AAB⋯A⋯A⋮⋮⋱⋮⋮⋮AA⋯Bi⋯A⋮⋮⋮⋮⋱⋮AA⋯A⋯B=BA⋯A⋯AAB⋯A⋯A⋮⋮⋱⋮⋮⋮AA⋯B⋯A⋮⋮⋮⋮⋱⋮AA⋯A⋯B+BA⋯A⋯AAB⋯A⋯A⋮⋮⋱⋮⋮⋮00⋯Δi⋯0⋮⋮⋮⋮⋱⋮AA⋯A⋯B=B+n-1A·B-An-1+Δi·-1i+i·BA⋯AAB⋯A⋮⋮⋱⋮AA⋯Bn-1×n-1=B+n-1A·B-An-1+Δi·B+n-2A·B-An-2≤0,ifnisanoddnumber;>0,ifnisanevennumber.
We can obtain the similar conclusion when there are multipleBi in the above matrix. So π2(q0,q1,q2,…,qn) is concave in qi. By ∂π2(q)/∂qi=0, we have(25)2αqi+y-v0+12σ0q0-q02-3/2-2q03+3q02-12q0-12-σ2qi-qi2-3/2-2qi3+3qi2-12qi=0.
Therefore, the optimal price and the market share can be solved by the implicit equations (19).According to Proposition2, we find that when multiple homogeneous products are developed, the revenue function is still concave and the optimal price and the optimal choice probability are also obtained. Note that the optimal prices with network effect are different from the optimal prices without network effect. In literature [9], when network effect is positive in homogeneous scenario, the optimal solution takes one of two different forms. However, when network effect is negative, the optimal solution is unique.


### 3.3. Pricing for the Heterogeneous Products

Different from the homogeneous case, in this section, we consider the general case (9) when the parameters of all products are distinct; i.e., the quality levels of products (yi), the price sensitivity parameters (bi), network effect parameters (αi), or the variances of utility error terms (σi2) are different for various products.For the revenue function (8), the product line pricing problem is described by (9). Through the following proposition, we find that the revenue function π(q0,q1,q2,…,qn) is concave in choice probability qi when bi=b (i=1,…,n) even though the parameters of all products are heterogeneous and network effect exists. Meanwhile, the optimal price and the optimal market share can also be derived by the implicit equations; and according to the corresponding optimal price and the optimal market share, the maximum revenue can be obtained by formula (8).Proposition 3.
For each producti(i=1,…,m), when bi=b, the revenue function π(q0,q1,q2,…,qn) is concave in qi, the optimal price and the optimal market share can be obtained by the following implicit equations:(26)42αiqi+yi-v0+σ0q0-q02-3/2-4q03+6q02-q0-1+σiqi-qi2-3/24qi3-6qi2+qi=0pi=1biyi+αiqi-v0+12σ02q0-1q0-q02-1/2-12σi2qi-1qi-qi2-1/2q0=1-∑i=1nqi.Proof.
Let us, respectively, consider each part of the revenue function (8).
Letw1=∑i=1nαi/biqi2, ∂w1/∂qi=2αi/biqi, and ∂2w1/∂qi2=2αi/bi≤0. And ∑i=1nqi/bi·(yi-v0) is linear in qi.
Letw2=∑i=1nqi/bi·σ0/2(2q0-1)(q0-q02)-1/2, and then(27)∂w2∂qi=q0-q02-3/2-σ04∑i=1nqibi+σ02bi-q0+3q02-2q03,∂2w2∂qi2=q0-q02-5/23σ082q0-1∑i=1nqibi-σ02biq0-q02.When bi=b,(28)∂2w2∂qi2=q0-q02-5/2·σ08b-2q02+5q0-3=q0-q02-5/2·σ08b-2q0-542+18≤0.
Letw3=-∑i=1nσi/2bi(2qi2-qi)(qi-qi2)-1/2, and then(29)∂w3∂qi=σi2bi1-4qiqi-qi2-1/2-σi4biqi-4qi2+4qi3qi-qi2-3/2=-σi2biqi-qi2-3/2-2qi3+3qi2-12qi,∂2w3∂qi2=σi8biqi-qi2-5/2-2qi2-qi≤0.
So whenbi=b, ∂2π(q)/∂qi2≤0. And ∂2π(q)/∂qi∂qj=∂2w2/∂qi2≤0 (i≠j). The rest of the proof is similar to the proof of Proposition 2, so π(q0,q1,q2,…,qn) is concave in qi. By ∂π(q)/∂qi=0, we have(30)∂π∂qi=1b·2αiqi+yi-v0+σ04bq0-q02-3/2-4q03+6q02-q0-1+σi4bqi-qi2-3/24qi3-6qi2+qi=0.The preceding equation is reduced to(31)42αiqi+yi-v0+σ0q0-q02-3/2-4q03+6q02-q0-1+σiqi-qi2-3/24qi3-6qi2+qi=0.Then the optimal market share and the optimal price can be obtained by the implicit equations (26).In heterogeneous scenario, when the price sensitivity parameters of different products are the same, the revenue function is also concave in choice probability. Meanwhile, we find that the optimal prices of all products are generally distinct. In addition, in literature [9], the optimal choice probability may be one of two different structures; however, when the products exhibit negative network effect, the optimal choice probability is unique.


## 3.1. Pricing for One Product

In this section, we will investigate the product pricing problem when one product is developed. The quality of the product isy, the price sensitivity parameter is b, the network effect parameter is α, the variance of utility error term is σ2, and the purchasing probability of the product is q. By (7), the price of the product is(10)p=1by+αq-v0+12σ02q0-1q0-q02-1/2-12σ2q-1q-q2-1/2.We assume σ=σ0>0 and q+q0=1, so the price is changed as(11)p=1by+αq-v0+12σ01-2qq-q2-1/2-12σ2q-1q-q2-1/2=1by-v0+αq+σb1-2qq-q2-1/2.The firm’s revenue is(12)π1q=p·q=αbq2+y-v0bq+σbq1-2qq-q2-1/2.The optimal product pricing optimization problem is(13)maxπ1qs.t.q≥0.The following proposition proves thatπ1(q) is concave in q when only one product is developed. When network effect is considered, the optimal price and the optimal market share can be derived by the implicit equations; then the maximum revenue can be obtained by formula (12) according to the corresponding optimal price and the optimal market share.Proposition 1.
When only one product is developed,π1(q) is concave in q, and the optimal price and the optimal market share can be obtained by the following implicit equations:(14)4αq+2y-v0+σq-q2-3/24q3-6q2+q=0p=1by-v0+αq+σb1-2qq-q2-1/2.Proof.
Because(15)∂π1q∂q=2αbq+y-v0b+σb1-4qq-q2-1/2+q-2q2-12q-q2-3/21-2q=2αbq+y-v0b+σ2bq-q2-3/24q3-6q2+q,∂2π1q∂q2=2αb+σ2b-32q-q2-5/21-2q·4q3-6q2+q+q-q2-3/212q2-12q+1=2αb+σ2bq-q2-5/2-q2-12q,and b≥0,σ>0,α≤0, so ∂2π1(q)/∂q2≤0, π1(q) is concave in q.
By∂π1(q)/∂q=0, 2α/bq+y-v0/b+σ/2b(q-q2)-3/2(4q3-6q2+q)=0. Then the optimal q∗, p∗ can be obtained by (14):Even if the network effect is taken into account, Proposition1 proves that the revenue function is concave in choice probability when only one product is developed. Meanwhile, the optimal price and the corresponding choice probability can be solved, and the optimal revenue can also be obtained. Therefore, when the firm plans to develop one new product to obtain maximum revenue, we find that the optimal choice probability (market share) is related to the network effect, the quality, the utility of no-purchase option, and the variance of utility error term; in addition to the above factors, the optimal price is also related to the price sensitivity parameter.


## 3.2. Pricing for the Homogeneous Products

In this section, we consider a special case that alln products have the same coefficients; i.e., all products have the same quality, price sensitivity parameter, network effect parameter, and the variance of utility error term, i.e., yi=y, αi=α, bi=b, and σi=σ. Without loss of generality, we assume that the price sensitivity parameter is b=1 in the remainder of this section.By (7), the price can be rewritten as(16)pi=y+αqi-v0+12σ02q0-1q0-q02-1/2-12σ2qi-1qi-qi2-1/2,and the firm’s total revenue (8) is expressed as(17)π2q=∑i=1ny-v0+αqiqi+∑i=1nqi·σ022q0-1·q0-q02-1/2-∑i=1nqiσ22qi-1qi-qi2-1/2. Thus the product line optimization problem can be formulated as follows:(18)maxπ2q0,q1,q2,…,qns.t.∑i=1nqi+q0=1qi,q0≥0i=1,2,…,n.The following proposition shows that when the network effect exists, the revenue functionπ2(q0,q1,q2,…,qn) is still concave in choice probability qi and the optimal price and the optimal market share can still be obtained by implicit equations. Meanwhile, the optimal revenue can be obtained by formula (17).Proposition 2.
The revenue functionπ2(q0,q1,q2,…,qn) is concave in qi, and the optimal price and the optimal market share can be obtained by the following implicit equations:(19)2αqi+y-v0+12σ0q0-q02-3/2-2q03+3q02-12q0-12-σ2qi-qi2-3/2-2qi3+3qi2-12qi=0pi=y-v0+αqi+12σ02q0-1q0-q02-1/2-12σ2qi-1qi-qi2-1/2q0=1-∑i=1nqi.Proof.
Forπ2(q), let us divide it into three sections. Let t1=∑i=1n(y-v0+αqi)qi, and then(20)∂t1∂qi=αqi+αqi+y-v0=2αqi+y-v0,∂2t1∂qi2=2α≤0.
Lett2=∑i=1nqi·σ0/2(2q0-1)(q0-q02)-1/2, and then(21)∂t2∂qi=12σ02q0-1q0-q02-1/2+∑i=1nqi·12σ0-2·q0-q02-1/2+2q0-1-12q0-q02-3/2-1+2q0=12σ0q0-q02-3/2-2q03+3q02-q0-12∑i=1nqi,∂2t2∂qi2=12σ0-32q0-q02-5/2-1+2q0-2q03+3q02-q0-12∑i=1nqi+q0-q02-3/26q02-6q0+12=12σ0q0-q02-5/2-12q0-542+132≤0.
Lett3=-∑i=1nqiσ/2(2qi-1)(qi-qi2)-1/2=-σ/2∑i=1n(2qi2-qi)(qi-qi2)-1/2, and then(22)∂t3∂qi=-σ2qi-qi2-3/2-2qi3+3qi2-12qi,∂2t3∂qi2=-σ2qi-qi2-5/212qi2+14qi≤0.
So∂2π2/∂qi2=∂2t1/∂qi2+∂2t2/∂qi2+∂2t3/∂qi2≤0. And ∂2π2/∂qi∂qj=∂2t2/∂qi2≤0 (i≠j). For the Hessian matrix of revenue function π2(q), let ∂2π2/∂qi2=Bi≤0, ∂2π2/∂qi∂qj=A≤0, and B=mini⁡Bi, B≥A. Because(23)BA⋯AAB⋯A⋮⋮⋱⋮AA⋯B=B+n-1AA⋯AB+n-1AB⋯A⋮⋮⋱⋮B+n-1AA⋯B=B+n-1A1A⋯A1B⋯A⋮⋮⋱⋮1A⋯B=B+n-1A·B-An-1≤0,ifnisanoddnumber;>0,ifnisanevennumber.Assuming that Bi=B+Δi(Δi≤0,Bi≤0), then(24)BA⋯A⋯AAB⋯A⋯A⋮⋮⋱⋮⋮⋮AA⋯Bi⋯A⋮⋮⋮⋮⋱⋮AA⋯A⋯B=BA⋯A⋯AAB⋯A⋯A⋮⋮⋱⋮⋮⋮AA⋯B⋯A⋮⋮⋮⋮⋱⋮AA⋯A⋯B+BA⋯A⋯AAB⋯A⋯A⋮⋮⋱⋮⋮⋮00⋯Δi⋯0⋮⋮⋮⋮⋱⋮AA⋯A⋯B=B+n-1A·B-An-1+Δi·-1i+i·BA⋯AAB⋯A⋮⋮⋱⋮AA⋯Bn-1×n-1=B+n-1A·B-An-1+Δi·B+n-2A·B-An-2≤0,ifnisanoddnumber;>0,ifnisanevennumber.
We can obtain the similar conclusion when there are multipleBi in the above matrix. So π2(q0,q1,q2,…,qn) is concave in qi. By ∂π2(q)/∂qi=0, we have(25)2αqi+y-v0+12σ0q0-q02-3/2-2q03+3q02-12q0-12-σ2qi-qi2-3/2-2qi3+3qi2-12qi=0.
Therefore, the optimal price and the market share can be solved by the implicit equations (19).According to Proposition2, we find that when multiple homogeneous products are developed, the revenue function is still concave and the optimal price and the optimal choice probability are also obtained. Note that the optimal prices with network effect are different from the optimal prices without network effect. In literature [9], when network effect is positive in homogeneous scenario, the optimal solution takes one of two different forms. However, when network effect is negative, the optimal solution is unique.


## 3.3. Pricing for the Heterogeneous Products

Different from the homogeneous case, in this section, we consider the general case (9) when the parameters of all products are distinct; i.e., the quality levels of products (yi), the price sensitivity parameters (bi), network effect parameters (αi), or the variances of utility error terms (σi2) are different for various products.For the revenue function (8), the product line pricing problem is described by (9). Through the following proposition, we find that the revenue function π(q0,q1,q2,…,qn) is concave in choice probability qi when bi=b (i=1,…,n) even though the parameters of all products are heterogeneous and network effect exists. Meanwhile, the optimal price and the optimal market share can also be derived by the implicit equations; and according to the corresponding optimal price and the optimal market share, the maximum revenue can be obtained by formula (8).Proposition 3.
For each producti(i=1,…,m), when bi=b, the revenue function π(q0,q1,q2,…,qn) is concave in qi, the optimal price and the optimal market share can be obtained by the following implicit equations:(26)42αiqi+yi-v0+σ0q0-q02-3/2-4q03+6q02-q0-1+σiqi-qi2-3/24qi3-6qi2+qi=0pi=1biyi+αiqi-v0+12σ02q0-1q0-q02-1/2-12σi2qi-1qi-qi2-1/2q0=1-∑i=1nqi.Proof.
Let us, respectively, consider each part of the revenue function (8).
Letw1=∑i=1nαi/biqi2, ∂w1/∂qi=2αi/biqi, and ∂2w1/∂qi2=2αi/bi≤0. And ∑i=1nqi/bi·(yi-v0) is linear in qi.
Letw2=∑i=1nqi/bi·σ0/2(2q0-1)(q0-q02)-1/2, and then(27)∂w2∂qi=q0-q02-3/2-σ04∑i=1nqibi+σ02bi-q0+3q02-2q03,∂2w2∂qi2=q0-q02-5/23σ082q0-1∑i=1nqibi-σ02biq0-q02.When bi=b,(28)∂2w2∂qi2=q0-q02-5/2·σ08b-2q02+5q0-3=q0-q02-5/2·σ08b-2q0-542+18≤0.
Letw3=-∑i=1nσi/2bi(2qi2-qi)(qi-qi2)-1/2, and then(29)∂w3∂qi=σi2bi1-4qiqi-qi2-1/2-σi4biqi-4qi2+4qi3qi-qi2-3/2=-σi2biqi-qi2-3/2-2qi3+3qi2-12qi,∂2w3∂qi2=σi8biqi-qi2-5/2-2qi2-qi≤0.
So whenbi=b, ∂2π(q)/∂qi2≤0. And ∂2π(q)/∂qi∂qj=∂2w2/∂qi2≤0 (i≠j). The rest of the proof is similar to the proof of Proposition 2, so π(q0,q1,q2,…,qn) is concave in qi. By ∂π(q)/∂qi=0, we have(30)∂π∂qi=1b·2αiqi+yi-v0+σ04bq0-q02-3/2-4q03+6q02-q0-1+σi4bqi-qi2-3/24qi3-6qi2+qi=0.The preceding equation is reduced to(31)42αiqi+yi-v0+σ0q0-q02-3/2-4q03+6q02-q0-1+σiqi-qi2-3/24qi3-6qi2+qi=0.Then the optimal market share and the optimal price can be obtained by the implicit equations (26).In heterogeneous scenario, when the price sensitivity parameters of different products are the same, the revenue function is also concave in choice probability. Meanwhile, we find that the optimal prices of all products are generally distinct. In addition, in literature [9], the optimal choice probability may be one of two different structures; however, when the products exhibit negative network effect, the optimal choice probability is unique.


## 4. Numerical Experiments

In this section, we present some numerical experiments to show the influence of network effect parameters on the results. First, we study the variation of optimal solutions with different parameters for developing one product, homogeneous coefficients, and heterogeneous coefficients case. Second, we analyze the importance of considering network effect if it does exist. Finally, we test the robustness of solutions when the estimation of network effect parameter has some error.


### 4.1. Variation of the Optimal Solutions with Different Parameters

We consider the influence of different parameters on the optimal price, the optimal market share, and the optimal revenue when different products are developed.


#### 4.1.1. Develop One Product

We first present the variation of the optimal results with different parameters when only one product is developed. For ease of analysis, we consider the case where only one parameter changes and the other parameters remain the same.To analyze the influence of network effect parameter on the optimal solutions, let the variation range ofα be set as [-10,0] and the other parameters be set as y=2, b=1, σ=1, and v0=0.5. The variation of optimal solutions with network effect parameter α is showed in Figure 1.Figure 1
The variation of optimal solutions with network effect parameter.As shown in Figure1, with the network effect parameter reducing, the optimal price gradually is enhanced and the optimal market share or the revenue is going down. That means the network effect is a key factor for the product development if it exists. If the network effect is almost negligible, the optimal price is relatively low, the corresponding market share is large, and the earning revenue is high. Because we only consider the negative network effect, when the absolute value of network effect goes up, the optimal price may be set higher, but both the corresponding market share and the revenue reduce gradually.We next analyze the influence of the product’s quality on the optimal solutions when the network effect exists. We assume the product’s qualityy lies in [1,10]; let α=-2, b=1, σ=1, and v0=0.5. Figure 2 shows the variation of optimal solutions with the quality.Figure 2
The variation of optimal solutions with the quality.From Figure2 we find that the optimal price, the corresponding market share, and the revenue all go up with the quality of product increasing. That is to say, even when network effect exists, improving quality is good for the seller when development costs are not taken into account. Because of the better quality, the price of product may be set higher, the corresponding market share is large, and the earning revenue also increases.We then show the influence of price sensitivity parameter on the optimal solutions when the network effect exists. We assumeb∈[0.1,5], α=-2, y=2, σ=1, and v0=0.5. Figure 3 shows the variation of optimal solutions with the price sensitivity parameter b.Figure 3
The variation of optimal solutions with the price sensitivity parameter.As shown in Figure3, when the network effect is considered, the optimal price, the corresponding market share, and the optimal revenue all go down with the price sensitivity parameter becoming large. When the sensitivity of consumer to the price is relatively low, the price could be high and the earning revenue is also large. However, when the price sensitivity parameter of consumer to the product is large, the optimal price should be set lower and the corresponding revenue also decreases. When the price sensitivity parameter is large enough, the optimal price, the corresponding market share, and the optimal revenue are almost 0.To analyze the influence of the variance of the utility error term on the optimal solutions when the network effect is considered, we setσ∈[0.1,5], α=-2, y=2, b=1, and v0=0.5. Figure 4 shows the variation of optimal solutions with the variance of utility error term σ.Figure 4
The variation of the optimal solutions with the variance of utility error term.From Figure4 we find that as the variance of utility error term increases, the optimal price and the revenue go up and the market share decreases. That means that if the estimated utility has a large error, the price of product may be increased and the earning revenue also goes up, but the corresponding market share reduces.When different products have the homogeneous coefficients, even though the network effect exists, the influences of different parameters on the optimal solutions (the optimal price, the optimal market share, or the optimal revenue) are similar to that when one product is developed. Next, we will study the variation of the optimal solutions with different parameters when all products have the heterogeneous coefficients and the network effect exists.


#### 4.1.2. Heterogeneous Case

We consider the influence of different parameters on the optimal solutions when all products are heterogeneous and the network effect exists. Without loss of generality, we assume only two products are developed in the market.First, we consider the influence of the network effect parameter on the optimal solutions. For ease of exposition, we assume the network effect parameters of two products are the same andα1=α2∈[-10,0]. Let y1=2, y2=3, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 5 shows the variation of the optimal solutions with the network effect parameter.Figure 5
The influence of the network effect parameter on the optimal solutions.As shown in Figure5, the variations of the optimal prices, the optimal market shares, and the optimal revenues are distinct. With the decrease of the network effect, the optimal prices of two products first reduce and then increase, the market shares of two products and the total market share decrease generally, and the optimal revenues of two products and the total revenue also go down gradually. When the network effect decreases, the optimal prices should be set higher, but the corresponding optimal revenues decrease, which is similar to that one product developed.Second, we study the variation of the optimal solutions with the quality of product when the network effect exists. We assume the qualities of two products are the same andy1=y2∈[1,10]. Let α1=-2, α2=-1, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 6 shows the variation of the optimal solutions with the quality of product.Figure 6
The influence of the quality of product on the optimal solutions.We find from Figure6 that the optimal prices, the corresponding market shares, and the optimal revenues gradually go up with the quality increasing even if the network effect exists. This tendency is the same as the case that one product is developed. For the sellers, if the quality of product can be improved, the sale price can be set higher and the market share and the earning revenue are also increased.Next, we analyze the influence of the price sensitivity parameter on the optimal solutions. We assumeb∈[0.1,5], α1=-2, α2=-1, y1=2, y2=3, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 7 shows the variation of the optimal solutions with the price sensitivity parameter when the network effect exists.Figure 7
The influence of the price sensitivity parameter on the optimal solutions.From Figure7, we can see that the optimal prices and the optimal revenues both become small when the price sensitivity parameter gets big, but the corresponding market share is constant even if the price sensitivity parameter changes. For the market share, we can find from Proposition 3 that the market share has nothing to do with the price sensitivity parameter. In addition, when the price sensitivity of consumer is large, the seller should cut price to obtain the optimal revenue.Finally, we investigate the influence of the variance of utility error term on the optimal solutions when the network effect exists. We assume that the variances of utility error terms are all uniform andσ0=σ1=σ2∈[0.1,10]. Let α1=-2, α2=-1, y1=2, y2=3, v0=0.5, and b=1. Figure 8 shows the variation of the optimal solutions with the variance of utility error term.Figure 8
The influence of the variance of utility error term on the optimal solutions.From Figure8 we can find that the variations are distinct among the optimal prices, the optimal market shares, and the optimal revenues. As the variance of utility error term becomes large, the optimal prices of two products increase, the optimal market shares of two products and the total market share reduce, and the revenue of product 1 gradually increases, but the revenue of product 2 and the total revenue first decrease and then increase. It can be seen that the variance of the estimated utility can affect the price of product, the market share, and the earning revenue.


### 4.2. The Importance of Considering Network Effect

In this section, we demonstrate the importance of considering network effect when making pricing decision if network effect does exist. Otherwise, the seller will suffer a certain loss. The test is presented by three parts of numerical experiments, including developing one product, homogeneous products, and heterogeneous case.For one product, we sety=2, b=1, σ=1, and v0=0.5. For the homogeneous case, we assume that only two products are developed and fix y=2, σ=1, and v0=0.5. For the heterogeneous case, there are also two products in the market, and y1=2, y2=3, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. We assume that the network effect parameter α∈[-10,0].In Table1, the values of the optimal revenues π1, π2, and π are obtained by Propositions 1, 2, and 3. The values of π10, π20, and π0 indicate the revenue without considering the network effect. Where the prices p0 are solved by Propositions 1, 2, and 3 with α=0, the corresponding market shares q0 are derived by (5) with the actual α. In general, q0≠q unless the actual network effect parameter is 0. The revenue without considering the network effect is earned by p0·q0. The values of l1, l2, and l indicate the loss of revenue if the pricing decision does not consider the network effect.Table 1
The comparison of revenues whether considering network effect.
α 0 -1 -2 -3 -4 -5 -10 π 1 0.7666 0.6248 0.5393 0.4822 0.4409 0.4093 0.3182 π 1 0 0.7666 0.6225 0.5320 0.4694 0.4231 0.3872 0.2823 l 1 (%) 0 0.37 1.35 2.65 4.04 5.40 11.28 π 2 1.0469 0.9166 0.8217 0.7507 0.6957 0.6517 0.5171 π 2 0 1.0469 0.9160 0.8188 0.7442 0.6850 0.6368 0.4848 l 2 (%) 0 0.07 0.35 0.87 1.54 2.29 6.25 π 1.5646 1.2679 1.0610 0.9171 0.8140 0.7368 0.5280 π 0 1.5646 1.2583 1.0531 0.9138 0.8132 0.7368 0.5222 l (%) 0 0.76 0.74 0.36 0.10 0 1.10We can see from Table1 that the optimal revenues outweigh the revenues which are obtained when the network effects are not considered but those do exist, whenever there are products, homogeneous products, or heterogeneous cases in the market. From Table 1, when α=-10, the loss is as high as 11%. With the network effect parameter becoming small, the loss could be even higher. Therefore, if there exist network effects in the market, the network effect needs to be considered when making pricing decision; otherwise, the seller may suffer some losses.


### 4.3. Robustness of the Solutions

In this section, we investigate the robustness of the revenue if the network effect parameter has some estimation error. Without loss of generality, we only consider the homogeneous case. Assume that the seller has an estimationα~ to the network effect parameter and the pricing decision p~(q~) is made with α~. But the network effect parameter is actually α. According to p~(q~), the consumers have the corresponding market share q¯ based on α. If α=α~, q¯=q~. Otherwise, q¯ and q~ are usually different. And the revenue is π~=p~(q~)·q¯.For the homogeneous case, we assume that there exist two products to be developed. Lety=2, σ=1, and v0=0.5. Define l=100×π∗-π~/π∗ as the revenue loss, where π∗ defines the optimal revenue when the network effect parameter does not have any estimation error. Table 2 shows the results of revenues and the loss.Table 2
The robustness of revenue for estimation of network effect.
α ~ -2.5 -2.8 -3 -3.2 -3.5 -3.8 α = - 2.8 π ~ 0.7633 0.7634 0.7634 0.7633 0.7629 0.7625 l (%) 0.01 0 <0.01 0.01 0.07 0.12 α = - 3.5 π ~ 0.7207 0.7211 0.7214 0.7215 0.7216 0.7215 l (%) 0.12 0.07 0.03 0.01 0 0.01As shown in Table2, when the network effect parameter has some estimation error, the revenue shows good robustness. From Table 2, we can see that the highest revenue loss percentage is 0.12% and most percentages are below 0.1%. We also can find from Figure 1 that the revenue curve is relatively flat with the network effect changing. When the estimation error of network effect parameter is within a reasonable range, we may obtain the optimal robust revenue. Therefore, our model shows well robustness.


## 4.1. Variation of the Optimal Solutions with Different Parameters

We consider the influence of different parameters on the optimal price, the optimal market share, and the optimal revenue when different products are developed.


### 4.1.1. Develop One Product

We first present the variation of the optimal results with different parameters when only one product is developed. For ease of analysis, we consider the case where only one parameter changes and the other parameters remain the same.To analyze the influence of network effect parameter on the optimal solutions, let the variation range ofα be set as [-10,0] and the other parameters be set as y=2, b=1, σ=1, and v0=0.5. The variation of optimal solutions with network effect parameter α is showed in Figure 1.Figure 1
The variation of optimal solutions with network effect parameter.As shown in Figure1, with the network effect parameter reducing, the optimal price gradually is enhanced and the optimal market share or the revenue is going down. That means the network effect is a key factor for the product development if it exists. If the network effect is almost negligible, the optimal price is relatively low, the corresponding market share is large, and the earning revenue is high. Because we only consider the negative network effect, when the absolute value of network effect goes up, the optimal price may be set higher, but both the corresponding market share and the revenue reduce gradually.We next analyze the influence of the product’s quality on the optimal solutions when the network effect exists. We assume the product’s qualityy lies in [1,10]; let α=-2, b=1, σ=1, and v0=0.5. Figure 2 shows the variation of optimal solutions with the quality.Figure 2
The variation of optimal solutions with the quality.From Figure2 we find that the optimal price, the corresponding market share, and the revenue all go up with the quality of product increasing. That is to say, even when network effect exists, improving quality is good for the seller when development costs are not taken into account. Because of the better quality, the price of product may be set higher, the corresponding market share is large, and the earning revenue also increases.We then show the influence of price sensitivity parameter on the optimal solutions when the network effect exists. We assumeb∈[0.1,5], α=-2, y=2, σ=1, and v0=0.5. Figure 3 shows the variation of optimal solutions with the price sensitivity parameter b.Figure 3
The variation of optimal solutions with the price sensitivity parameter.As shown in Figure3, when the network effect is considered, the optimal price, the corresponding market share, and the optimal revenue all go down with the price sensitivity parameter becoming large. When the sensitivity of consumer to the price is relatively low, the price could be high and the earning revenue is also large. However, when the price sensitivity parameter of consumer to the product is large, the optimal price should be set lower and the corresponding revenue also decreases. When the price sensitivity parameter is large enough, the optimal price, the corresponding market share, and the optimal revenue are almost 0.To analyze the influence of the variance of the utility error term on the optimal solutions when the network effect is considered, we setσ∈[0.1,5], α=-2, y=2, b=1, and v0=0.5. Figure 4 shows the variation of optimal solutions with the variance of utility error term σ.Figure 4
The variation of the optimal solutions with the variance of utility error term.From Figure4 we find that as the variance of utility error term increases, the optimal price and the revenue go up and the market share decreases. That means that if the estimated utility has a large error, the price of product may be increased and the earning revenue also goes up, but the corresponding market share reduces.When different products have the homogeneous coefficients, even though the network effect exists, the influences of different parameters on the optimal solutions (the optimal price, the optimal market share, or the optimal revenue) are similar to that when one product is developed. Next, we will study the variation of the optimal solutions with different parameters when all products have the heterogeneous coefficients and the network effect exists.


### 4.1.2. Heterogeneous Case

We consider the influence of different parameters on the optimal solutions when all products are heterogeneous and the network effect exists. Without loss of generality, we assume only two products are developed in the market.First, we consider the influence of the network effect parameter on the optimal solutions. For ease of exposition, we assume the network effect parameters of two products are the same andα1=α2∈[-10,0]. Let y1=2, y2=3, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 5 shows the variation of the optimal solutions with the network effect parameter.Figure 5
The influence of the network effect parameter on the optimal solutions.As shown in Figure5, the variations of the optimal prices, the optimal market shares, and the optimal revenues are distinct. With the decrease of the network effect, the optimal prices of two products first reduce and then increase, the market shares of two products and the total market share decrease generally, and the optimal revenues of two products and the total revenue also go down gradually. When the network effect decreases, the optimal prices should be set higher, but the corresponding optimal revenues decrease, which is similar to that one product developed.Second, we study the variation of the optimal solutions with the quality of product when the network effect exists. We assume the qualities of two products are the same andy1=y2∈[1,10]. Let α1=-2, α2=-1, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 6 shows the variation of the optimal solutions with the quality of product.Figure 6
The influence of the quality of product on the optimal solutions.We find from Figure6 that the optimal prices, the corresponding market shares, and the optimal revenues gradually go up with the quality increasing even if the network effect exists. This tendency is the same as the case that one product is developed. For the sellers, if the quality of product can be improved, the sale price can be set higher and the market share and the earning revenue are also increased.Next, we analyze the influence of the price sensitivity parameter on the optimal solutions. We assumeb∈[0.1,5], α1=-2, α2=-1, y1=2, y2=3, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 7 shows the variation of the optimal solutions with the price sensitivity parameter when the network effect exists.Figure 7
The influence of the price sensitivity parameter on the optimal solutions.From Figure7, we can see that the optimal prices and the optimal revenues both become small when the price sensitivity parameter gets big, but the corresponding market share is constant even if the price sensitivity parameter changes. For the market share, we can find from Proposition 3 that the market share has nothing to do with the price sensitivity parameter. In addition, when the price sensitivity of consumer is large, the seller should cut price to obtain the optimal revenue.Finally, we investigate the influence of the variance of utility error term on the optimal solutions when the network effect exists. We assume that the variances of utility error terms are all uniform andσ0=σ1=σ2∈[0.1,10]. Let α1=-2, α2=-1, y1=2, y2=3, v0=0.5, and b=1. Figure 8 shows the variation of the optimal solutions with the variance of utility error term.Figure 8
The influence of the variance of utility error term on the optimal solutions.From Figure8 we can find that the variations are distinct among the optimal prices, the optimal market shares, and the optimal revenues. As the variance of utility error term becomes large, the optimal prices of two products increase, the optimal market shares of two products and the total market share reduce, and the revenue of product 1 gradually increases, but the revenue of product 2 and the total revenue first decrease and then increase. It can be seen that the variance of the estimated utility can affect the price of product, the market share, and the earning revenue.


## 4.1.1. Develop One Product

We first present the variation of the optimal results with different parameters when only one product is developed. For ease of analysis, we consider the case where only one parameter changes and the other parameters remain the same.To analyze the influence of network effect parameter on the optimal solutions, let the variation range ofα be set as [-10,0] and the other parameters be set as y=2, b=1, σ=1, and v0=0.5. The variation of optimal solutions with network effect parameter α is showed in Figure 1.Figure 1
The variation of optimal solutions with network effect parameter.As shown in Figure1, with the network effect parameter reducing, the optimal price gradually is enhanced and the optimal market share or the revenue is going down. That means the network effect is a key factor for the product development if it exists. If the network effect is almost negligible, the optimal price is relatively low, the corresponding market share is large, and the earning revenue is high. Because we only consider the negative network effect, when the absolute value of network effect goes up, the optimal price may be set higher, but both the corresponding market share and the revenue reduce gradually.We next analyze the influence of the product’s quality on the optimal solutions when the network effect exists. We assume the product’s qualityy lies in [1,10]; let α=-2, b=1, σ=1, and v0=0.5. Figure 2 shows the variation of optimal solutions with the quality.Figure 2
The variation of optimal solutions with the quality.From Figure2 we find that the optimal price, the corresponding market share, and the revenue all go up with the quality of product increasing. That is to say, even when network effect exists, improving quality is good for the seller when development costs are not taken into account. Because of the better quality, the price of product may be set higher, the corresponding market share is large, and the earning revenue also increases.We then show the influence of price sensitivity parameter on the optimal solutions when the network effect exists. We assumeb∈[0.1,5], α=-2, y=2, σ=1, and v0=0.5. Figure 3 shows the variation of optimal solutions with the price sensitivity parameter b.Figure 3
The variation of optimal solutions with the price sensitivity parameter.As shown in Figure3, when the network effect is considered, the optimal price, the corresponding market share, and the optimal revenue all go down with the price sensitivity parameter becoming large. When the sensitivity of consumer to the price is relatively low, the price could be high and the earning revenue is also large. However, when the price sensitivity parameter of consumer to the product is large, the optimal price should be set lower and the corresponding revenue also decreases. When the price sensitivity parameter is large enough, the optimal price, the corresponding market share, and the optimal revenue are almost 0.To analyze the influence of the variance of the utility error term on the optimal solutions when the network effect is considered, we setσ∈[0.1,5], α=-2, y=2, b=1, and v0=0.5. Figure 4 shows the variation of optimal solutions with the variance of utility error term σ.Figure 4
The variation of the optimal solutions with the variance of utility error term.From Figure4 we find that as the variance of utility error term increases, the optimal price and the revenue go up and the market share decreases. That means that if the estimated utility has a large error, the price of product may be increased and the earning revenue also goes up, but the corresponding market share reduces.When different products have the homogeneous coefficients, even though the network effect exists, the influences of different parameters on the optimal solutions (the optimal price, the optimal market share, or the optimal revenue) are similar to that when one product is developed. Next, we will study the variation of the optimal solutions with different parameters when all products have the heterogeneous coefficients and the network effect exists.


## 4.1.2. Heterogeneous Case

We consider the influence of different parameters on the optimal solutions when all products are heterogeneous and the network effect exists. Without loss of generality, we assume only two products are developed in the market.First, we consider the influence of the network effect parameter on the optimal solutions. For ease of exposition, we assume the network effect parameters of two products are the same andα1=α2∈[-10,0]. Let y1=2, y2=3, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 5 shows the variation of the optimal solutions with the network effect parameter.Figure 5
The influence of the network effect parameter on the optimal solutions.As shown in Figure5, the variations of the optimal prices, the optimal market shares, and the optimal revenues are distinct. With the decrease of the network effect, the optimal prices of two products first reduce and then increase, the market shares of two products and the total market share decrease generally, and the optimal revenues of two products and the total revenue also go down gradually. When the network effect decreases, the optimal prices should be set higher, but the corresponding optimal revenues decrease, which is similar to that one product developed.Second, we study the variation of the optimal solutions with the quality of product when the network effect exists. We assume the qualities of two products are the same andy1=y2∈[1,10]. Let α1=-2, α2=-1, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 6 shows the variation of the optimal solutions with the quality of product.Figure 6
The influence of the quality of product on the optimal solutions.We find from Figure6 that the optimal prices, the corresponding market shares, and the optimal revenues gradually go up with the quality increasing even if the network effect exists. This tendency is the same as the case that one product is developed. For the sellers, if the quality of product can be improved, the sale price can be set higher and the market share and the earning revenue are also increased.Next, we analyze the influence of the price sensitivity parameter on the optimal solutions. We assumeb∈[0.1,5], α1=-2, α2=-1, y1=2, y2=3, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. Figure 7 shows the variation of the optimal solutions with the price sensitivity parameter when the network effect exists.Figure 7
The influence of the price sensitivity parameter on the optimal solutions.From Figure7, we can see that the optimal prices and the optimal revenues both become small when the price sensitivity parameter gets big, but the corresponding market share is constant even if the price sensitivity parameter changes. For the market share, we can find from Proposition 3 that the market share has nothing to do with the price sensitivity parameter. In addition, when the price sensitivity of consumer is large, the seller should cut price to obtain the optimal revenue.Finally, we investigate the influence of the variance of utility error term on the optimal solutions when the network effect exists. We assume that the variances of utility error terms are all uniform andσ0=σ1=σ2∈[0.1,10]. Let α1=-2, α2=-1, y1=2, y2=3, v0=0.5, and b=1. Figure 8 shows the variation of the optimal solutions with the variance of utility error term.Figure 8
The influence of the variance of utility error term on the optimal solutions.From Figure8 we can find that the variations are distinct among the optimal prices, the optimal market shares, and the optimal revenues. As the variance of utility error term becomes large, the optimal prices of two products increase, the optimal market shares of two products and the total market share reduce, and the revenue of product 1 gradually increases, but the revenue of product 2 and the total revenue first decrease and then increase. It can be seen that the variance of the estimated utility can affect the price of product, the market share, and the earning revenue.


## 4.2. The Importance of Considering Network Effect

In this section, we demonstrate the importance of considering network effect when making pricing decision if network effect does exist. Otherwise, the seller will suffer a certain loss. The test is presented by three parts of numerical experiments, including developing one product, homogeneous products, and heterogeneous case.For one product, we sety=2, b=1, σ=1, and v0=0.5. For the homogeneous case, we assume that only two products are developed and fix y=2, σ=1, and v0=0.5. For the heterogeneous case, there are also two products in the market, and y1=2, y2=3, b=1, v0=0.5, σ0=0.5, σ1=1, and σ2=0.8. We assume that the network effect parameter α∈[-10,0].In Table1, the values of the optimal revenues π1, π2, and π are obtained by Propositions 1, 2, and 3. The values of π10, π20, and π0 indicate the revenue without considering the network effect. Where the prices p0 are solved by Propositions 1, 2, and 3 with α=0, the corresponding market shares q0 are derived by (5) with the actual α. In general, q0≠q unless the actual network effect parameter is 0. The revenue without considering the network effect is earned by p0·q0. The values of l1, l2, and l indicate the loss of revenue if the pricing decision does not consider the network effect.Table 1
The comparison of revenues whether considering network effect.
α 0 -1 -2 -3 -4 -5 -10 π 1 0.7666 0.6248 0.5393 0.4822 0.4409 0.4093 0.3182 π 1 0 0.7666 0.6225 0.5320 0.4694 0.4231 0.3872 0.2823 l 1 (%) 0 0.37 1.35 2.65 4.04 5.40 11.28 π 2 1.0469 0.9166 0.8217 0.7507 0.6957 0.6517 0.5171 π 2 0 1.0469 0.9160 0.8188 0.7442 0.6850 0.6368 0.4848 l 2 (%) 0 0.07 0.35 0.87 1.54 2.29 6.25 π 1.5646 1.2679 1.0610 0.9171 0.8140 0.7368 0.5280 π 0 1.5646 1.2583 1.0531 0.9138 0.8132 0.7368 0.5222 l (%) 0 0.76 0.74 0.36 0.10 0 1.10We can see from Table1 that the optimal revenues outweigh the revenues which are obtained when the network effects are not considered but those do exist, whenever there are products, homogeneous products, or heterogeneous cases in the market. From Table 1, when α=-10, the loss is as high as 11%. With the network effect parameter becoming small, the loss could be even higher. Therefore, if there exist network effects in the market, the network effect needs to be considered when making pricing decision; otherwise, the seller may suffer some losses.


## 4.3. Robustness of the Solutions

In this section, we investigate the robustness of the revenue if the network effect parameter has some estimation error. Without loss of generality, we only consider the homogeneous case. Assume that the seller has an estimationα~ to the network effect parameter and the pricing decision p~(q~) is made with α~. But the network effect parameter is actually α. According to p~(q~), the consumers have the corresponding market share q¯ based on α. If α=α~, q¯=q~. Otherwise, q¯ and q~ are usually different. And the revenue is π~=p~(q~)·q¯.For the homogeneous case, we assume that there exist two products to be developed. Lety=2, σ=1, and v0=0.5. Define l=100×π∗-π~/π∗ as the revenue loss, where π∗ defines the optimal revenue when the network effect parameter does not have any estimation error. Table 2 shows the results of revenues and the loss.Table 2
The robustness of revenue for estimation of network effect.
α ~ -2.5 -2.8 -3 -3.2 -3.5 -3.8 α = - 2.8 π ~ 0.7633 0.7634 0.7634 0.7633 0.7629 0.7625 l (%) 0.01 0 <0.01 0.01 0.07 0.12 α = - 3.5 π ~ 0.7207 0.7211 0.7214 0.7215 0.7216 0.7215 l (%) 0.12 0.07 0.03 0.01 0 0.01As shown in Table2, when the network effect parameter has some estimation error, the revenue shows good robustness. From Table 2, we can see that the highest revenue loss percentage is 0.12% and most percentages are below 0.1%. We also can find from Figure 1 that the revenue curve is relatively flat with the network effect changing. When the estimation error of network effect parameter is within a reasonable range, we may obtain the optimal robust revenue. Therefore, our model shows well robustness.


## 5. Conclusions

Nowadays, with the rapid development of e-commerce, the influence of complex network effect on the development and pricing of new products cannot be ignored. Consumer choice behavior not only has positive network effect but also has negative network effect, which is also common in practice. In addition, the assumptions of MNL model, which is one of the most widely used consumer choice models, may not be met in some practical cases. Therefore, in this paper, we study the product line pricing problem considering negative network effect based on the MMM. We establish an improved MMM with endogenous network effect. By proving the concavity of the profit function, we obtain the solving equations of the optimal price, the corresponding market share, and the optimal revenue in three different markets, including developing one product, homogeneous products, and heterogeneous case. Through numerical experiments, we first show the variation of the optimal solutions with different parameters, then analyze the importance of considering network effect if it does exist, and finally test the robustness of solution when the estimation of network effect parameter has a certain error.Our paper also has some limitations that are showed as follows. First, we only study the product pricing with negative network effect. We may also explore the product line optimization problem considering positive network effect. Second, we simulate the consumer choice behavior based on MMM with network effect. We could further introduce another probabilistic choice model to describe consumer choice behavior when the network effect is considered. Finally, we only obtain the solving equations of the optimal solutions. We would continue to explore the analytical expression of the optimal price or the optimal revenue. These issues would be the topics of our future work.



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*Source: 6104805-2019-02-18.xml*