Joint pricing and location decisions in a heterogeneous market

Accepted Manuscript

Joint Pricing and Location Decisions in a Heterogeneous Market Nafiseh Sedghi, Hassan Shavandi, Hossein Abouee-Mehrizi PII: DOI: Reference:

S0377-2217(17)30105-4 10.1016/j.ejor.2017.01.055 EOR 14236

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

20 April 2016 30 January 2017 31 January 2017

Please cite this article as: Nafiseh Sedghi, Hassan Shavandi, Hossein Abouee-Mehrizi, Joint Pricing and Location Decisions in a Heterogeneous Market, European Journal of Operational Research (2017), doi: 10.1016/j.ejor.2017.01.055

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Highlights • We consider the problem of joint location and pricing optimization in a heterogeneous market. • Optimal price and location are closely related, and there is a need for a simultaneous optimization.

CR IP T

• When customers are non-uniformly dispersed, uniform assumption leads to non-optimal decisions.

• Transportation cost and level of heterogeneity have great impact on firm’s strategic decisions.

AC

CE

PT

ED

M

AN US

• Ignoring the heterogeneity can significantly reduce the firm’s profit.

1

ACCEPTED MANUSCRIPT

Joint Pricing and Location Decisions in a Heterogeneous Market

February 6, 2017

AN US

Abstract

CR IP T

Nafiseh Sedghi, Hassan Shavandi Department of Industrial Engineering, Sharif University of Technology n [email protected], [email protected] Hossein Abouee-Mehrizi (Corresponding Author) Department of Management Sciences, University of Waterloo [email protected]

In this paper we consider the problem of joint location and pricing optimization for a firm in a heterogeneous market producing a single product. We assume that customers have a different willingness to pay for the product. We consider two classes of customers who are not uniformly distributed in the market and develop an analytical framework to determine the relationship between the optimal price and location of the firm. We demonstrate that the optimal price and

M

location are closely related to each other, and thus there is a need for simultaneous optimization of the price and location. We provide both analytical and numerical results to illustrate the

ED

impact of transportation cost and the level of heterogeneity on the firm’s strategic decisions. Our results show that simplifying the analysis of such markets with a uniform demand assumption and a homogeneity of customers may reduce the firm’s profit significantly.

Introduction

CE

1

PT

Keywords: Location, heterogeneous customers, non-uniform market, pricing

AC

In the 2011 recession, the US market witnessed a sharp drop-off in the sale of green products. However, the market share of independent brands, such as Method and Seventh Generation, did not decrease as the share of big consumer-product companies like Clorox did (Clifford and Martin, 2011). The reason behind this trend was that Method and Seventh Generation offer products for “dark green” customers who have a high reservation price for eco-friendly products, while Clorox products are targeted more to “light green” customers who do not or cannot pay high prices for these products (Szaky, 2011). In this paper, we examine the effects of the heterogeneity of customers on the strategic decisions of a firm entering a new market. Specifically, we consider a market with 2

ACCEPTED MANUSCRIPT

two classes of customers that are not uniformly distributed in the market and have a different willingness to pay for the same product. We investigate how the heterogeneity of customers affects location and price decisions of the firm in the market. Strategic decisions on location and price when entering a new market (with only one type of customer) have been addressed in the operations management and economics literature. However,

CR IP T

the common practice is often hierarchical, i.e., they first determine the location of the firm and then the price of the product. Simultaneous optimization of price and location decisions is encouraged in Hanjoul et al. (1990) and Aboolian et al. (2008). They both indicate that although price setting may be considered a short term decision as opposed to the location decision, deciding on location without considering the pricing strategy results in a non-optimal location strategy. In fact,

AN US

the proper time for deciding a pricing strategy is when the firm is setting its location since the flexibility to change the price after locating in the market is often limited (Aboolian et al., 2008). Therefore, the simultaneous decision on price and location can shed light on the profitability of the new market for the firm and as a result help the firm avoid entry in a market with very low profits. Relatively few papers have considered the simultaneous decision on price and location in mo-

M

nopolistic and static competitive settings. Dobson and Stavrulaki (2007) consider the simultaneous decision on price, location, and capacity in a monopoly market with time sensitive customers. In

ED

the spatial pricing literature, Hwang and Mai (1990) and Cheung and Wang (1995) consider discriminatory pricing with non-uniform demand and show that jointly deciding on price and location results in a different strategy than the hierarchical model. They conclude that the decision of

PT

pricing and location can affect each other. Tang and Cheng (2005) consider this approach for the optimal location and price of web services with intermediaries.

CE

Inspired by Hotelling (1929), we consider a stylized linear market to model the customer’s behavior. The Hotelling linear market is well used in the economics literature for location analysis

AC

whether in a geographical space or in a virtual space, like product characteristics (e.g., Lacourbe et al. (2009) and Lauga and Ofek (2011)). In the latter case (also known as product positioning), it is assumed that customers have ideal points in the feature space and if the product characteristics deviate from that point, they incur a cost proportional to the magnitude of the deviation. In the context of virtual space, our model considers the concept of customer’s heterogeneity in product valuation to examine the effects of heterogeneity on positioning and pricing of a product. In the context of geographical space, our work investigates the effect of customer heterogeneity on the

3

ACCEPTED MANUSCRIPT

location and pricing decisions of a firm. To capture the heterogeneity in the market, we consider two types of customers, namely H-type and L-type, who have high and low willingness to pay for the product, respectively. Considering two types of customers to segment the market has been used by Lacourbe (2012), Zhou et al. (2014), and Adner and Zemsky (2006) among others. We assume that the firm sells one type of product (or a class of products) in the market and show that it is

CR IP T

critical for the firm to address the right type of customers. Although there is a vast body of literature that considers linear location and product positioning, few papers study multiple types of customers. Desai (2001) is the first paper that considers a location model with two classes of customers to combine the idea of vertical differentiation and horizontal differentiation. He considers a firm that produces two versions of a product with high and

AN US

low quality aiming at the H- and L-type customers, respectively. Unlike our model, he assumes that customers are uniformly distributed in the market. Iyer and Seetharaman (2008) and Caldieraro (2016) use the same model as Desai (2001) to find the equilibrium price and quality in such markets. However, they assume exogenous locations for the firms and full coverage of the market. Unlike the papers in the literature, we relax the uniform dispersion assumption, i.e., in our

M

model H- and L-type customers have a non-uniform distribution on the Hotelling line. We assume that H-type customer distribution linearly increases towards the right of the market whereas L-

ED

type customer distribution linearly increases towards the left side of the market.This assumption makes the model more realistic in some markets; however, it introduces more complications into the analysis of the problem. To the best of our knowledge, this has not been considered in the pricing-

PT

location models with multiple types of customers. In the competitive models of location on a line, there are several papers that relax the uniformity assumption, including Neven (1986), Tabuchi

CE

and Thisse (1995), Gupta et al. (1997), Anderson et al. (1997), Calv´o-Armengol and Zenou (2002), Liu and Weinberg (2006) and Meagher et al. (2008) (See Kress and Pesch (2012) for other papers

AC

in this area and their underlying assumptions). These papers investigate how density choice affects the equilibrium outcome in different versions of Hotelling model. However, they all assume the full coverage framework. In our model, we assume that customers may not buy the firm’s product since they are able to find a substitute product. In fact as noted in Aboolian et al. (2007) the full market coverage is a common assumption in competitive location models. Previous papers in the literature assume that willingness to pay of customers is high enough to ensure full coverage of the market. Among sparse papers, Dasci and Laporte (2005) and Kwasnica and Stavrulaki (2008) consider elastic demand in the linear context. 4

ACCEPTED MANUSCRIPT

Note that there are many evidences that show customers separated geographically may have different willingness to pay. For example, residents of different neighborhoods may have different range of income. Booza et al. (2006) point out that the trend of increased sorting of high- and low-income families into neighborhoods is one of the challenges in the big cities that has been more visible recently. Other studies also find a gradual spatial shift of lower-income families from central to sub-

CR IP T

urbs in several US cities, resulting in neighborhood income polarization (Berube and Frey (2002), Cooke and Marchant (2006)). Fry and Taylor (2012) also empirically show that this pattern has increased over the past several decades.

In summary, this paper examines the simultaneous decision of price and location with more realistic assumptions on the types of customers and their distributions in the market. To analyze

AN US

the problem, we first determine the relation between the optimal price and location decisions and then derive the optimal solution. We then analyze the sensitivity of strategic behavior of the firm with respect to the market conditions such as transportation cost, type of customer’s reservation price, and the percentage of each type of customer in the market. Our results demonstrate that transportation cost can play a significant role in determining the strategic decision of a firm. For

M

a high transportation cost, the optimal price and location are not sensitive to the slight changes in the number of H-type customers. Moreover, we highlight the importance of analyzing such

ED

markets prior to offering promotions to each type of customer. We also investigate the value of simultaneous versus sequential decision making on price and location and show that simultaneous decision making could increase the firm’s profit significantly. Furthermore, we observe the effect

PT

of non-uniformity of customers’ distribution on the strategic decision of the firm. Finally, we show that when there is a substantial difference between customers’ willingness to pay, it is critical for

CE

the firm to consider the heterogeneity in its marketing analysis. We demonstrate that under some conditions ignoring this heterogeneity can dramatically reduce the firm’s profit.

AC

The rest of the paper is organized as follows. In Section 2, we define the problem and present the location-pricing model with two types of customers. In Section 3, we derive the relation between the optimal location and price and use it in Section 4 to simultaneously optimize the price and location. In Section 5, we discuss some managerial insights using numerical examples. All proofs appear in the Appendix.

5

ACCEPTED MANUSCRIPT

2

Modeling a Heterogeneous Market

We consider a firm that needs to decide on the location and price strategies for a new product in a heterogeneous market. There are two types (classes) of customers in the market, namely H and L. H-type customers care more about having the product and therefore are willing to pay a higher price to get the product. On the other hand, L-type customers’ willingness to pay for the

customers for the product, respectively, where vL < vH .

CR IP T

product is lower than H-type customers. Let vH and vL denote the valuation of H-type and L-type

The market is linear and represented by a [0, 1] Hotelling line. The total population of the market is normalized to 1, where γ percent of customers are type H and (1 − γ) percent are type L. Customers of type j (j ∈ {L, H}) are distributed on the line according to the function fj (.).

AN US

We assume fH (y) is linearly increasing in y and fL (y) is linearly decreasing in y, where y is the location of a customer. Note that the distribution of customers on the line can also be interpreted as the probability of buying one unit of the product by a customer at every location. Let x be the location of the firm on the line. Customers decide to buy one unit of the product from the firm if their utility is positive and not to buy otherwise. Specifically, the utility that a

M

customer of type j in location y obtains by buying the product is Uj (y|p, x) = vj − p − t|x − y|, where p is the price of the product and t is the transportation cost per unit of travel distance. Note

ED

that since vH > vL , for any given location y, we have UH (y|p, x) > UL (y|p, x). The objective of the firm is to maximize its profit by considering the heterogeneity in the market.

PT

Without loss of generality, we set the production cost to zero. Therefore, the profit that the firm P gains in the market is π = pDj , where Dj denotes the demand of type j customers. j=H,L

CE

Let rj =

v −p max{ j t , 0}

denote the coverage radius of the firm for customers of type j (j ∈ {L, H}).

This means that a type j customer is willing to travel at most the distance rj to purchase the prod-

AC

uct at price p. Problem (P1) represents the nonlinear optimization problem that the firm should consider to obtain the optimal location and price for the new market. The demand of each type of customer is a nonlinear function of x and p; hence, depending on the location of the firm and price of the product, the demand function can have different slopes. Demand of type j customers Rx is Dj = x j γj f j (y) dy , where γH = γ, γL = 1 − γ, xj = min{x + rj , 1} and xj = max{x − rj , 0} j

(see Constraints 2 to 4). xj and xj assure that there is no demand beyond the line [0, 1] to be

captured by the firm.

6

ACCEPTED MANUSCRIPT

(P1)

max π =

X

pDj

(1)

j=H,L

subject to: Dj =

Z

xj

xj

j ∈ {L, H},

(2)

j ∈ {L, H},

(3)

xj = max{0, x − rj }, vj − p rj = max{ , 0}, t

j ∈ {L, H},

(4)

j ∈ {L, H}

(5)

(6)

AN US

0≤x≤1

CR IP T

xj = min{1, x + rj },

p ≥ 0,

3

γj f j (y) dy,

Optimal Location as a Function of Price

Considering that the demand is not uniformly distributed on the line, the effects of different parameters on deriving the total demand captured by the firm should be studied carefully. This

M

makes the analysis of the optimal decisions in the market more complicated than in a market in which customers are uniformly distributed. To reduce the complexity of the analysis, we derive the

ED

optimal location of the firm as a function of the price. Let Q = {p ≥ 0, t ≥ 0} denote the joint state space of the product’s price and transportation cost of the customers. In order to analyze the optimal location strategy, the state space Q is divided

PT

into seven regions (subsets) on the basis of price and transportation cost, as illustrated in Figure 1. The regions are as follows:

vH + vL − t t , p ≤ vL , p ≤ } 2 2 t vH + vL − t = {(p, t) ∈ Q|p ≤ vH − , p ≤ vL , p ≥ } 2 2 t = {(p, t) ∈ Q|p ≥ vH − , p ≤ vL } 2 t = {(p, t) ∈ Q|p ≥ vH − , p ≥ vL , p < vH } 2 t = {(p, t) ∈ Q|p < vL − } 2 t = {(p, t) ∈ Q|p ≥ vL , p < vH − } 2

CE

S1 = {(p, t) ∈ Q|p ≥ vL − S2

AC

S3

S4

S5 S6

S7 = {(p, t) ∈ Q|p ≥ vH }. Note that the regions are defined based on the coverage radius of L-type and H-type customers, rL 7

ACCEPTED MANUSCRIPT

and rH , so that the structure of the demand function remains the same in each region. This means that these regions determine if customers of each class would be fully covered, partially covered, or not covered at all as discussed below. When rL = 0 (no coverage of L-type customers), based on the value of rH , the coverage of H-type

• rH = 0: no coverage of H-type customers (region S7 )

CR IP T

customers can have one of the following structures:

• 0 < rH ≤ 12 : partial coverage of H-type customers (region S4 ) • rH > 21 : full coverage of H-type customers (region S6 ) 1 2

(partial coverage of L-type customers), based on the value of rH , the coverage

AN US

When 0 < rL ≤

of H-type customers can have one of the following structures:

• 0 < rH ≤ 21 : partial coverage of H-type customers (region S3 ) 1 2

and rH + rL ≤ 1: partial coverage of the market area (Hoteling line)(region S2 )

• rH >

1 2

and rH + rL > 1: full coverage of the market area (Hoteling line) (region S1 ) 1 2

since rH > rL (full coverage of all customers):

ED

When rL > 12 , then rH >

M

• rH >

• rH > 21 : full coverage of H-type customers (region S5 )

PT

We next analyze the optimal location of the firm as a function of price. It is straightforward to show that the regions S5 , S6 , and S7 are dominated by other regions and cannot include the

CE

optimal price. Region S7 refers to the case that price is greater than willingness to pay of H-type customers. Thus, in this region the firm generates no demand and no profit. Region S5 can be

AC

dominated by setting the price equal to vL − 2t , which enable the firm to capture all the demand

in the market. Similarly, region S6 is dominated by setting the price equal to p = vH − 2t , where all the H-type demand is captured. We next discuss the remaining regions S1 to S4 , which can potentially include the optimal price

of the firm. Observe that setting a price in regions S1 , S2 , or S3 indicates that the firm is targeting both types of customers. However, any price picked from region S4 results in serving only H-type customers in the market. Moreover, in regions S1 and S2 it is possible to set a price such that all the H-type customers in the market are covered. More specifically, in region S1 the firm can 8

ACCEPTED MANUSCRIPT

࢖

‫ݒ‬ு

ܵସ

ܵ଺ ܵହ

ܵଵ

ܵଶ

ܵଷ

AN US

‫ݒ‬௅

CR IP T

ܵ଻

2‫ݒ‬௅

‫ݒ‬ு + ‫ݒ‬௅

2‫ݒ‬ு

࢚

Figure 1: Different regions in space Q.

M

cover all the H-type customers and maximize capturing of L-type customers. However, in region S2 there is a trade-off between covering all the H-type customers and capturing a greater fraction

ED

of the L-type customers. If the transportation cost is high and the firm sets its price high enough to be in region S3 , the firm cannot cover all the customers of any type in the market. Hence, it targets a fraction of each type of customer to find the best location on the line. In region S4 , the

PT

firm decides to set the price so high to target only H-type customers. In the following lemmas, we determine the optimal location as a function of price in regions

CE

S1 to S4 . We then demonstrate in Proposition 1 that there is a linear relationship between the optimal location and price.

AC

Lemma 1. Given that (p, t) ∈ S1 , the optimal location of the firm is x∗ = rL . Lemma 1 demonstrates that if the firm sets its price so low that the combination of the price

and transportation costs falls in region S1 , the firm will cover all H-type customers even if it locates on the left side of the market where the majority of customers are L-type. Therefore, the optimal location is x∗ = rL , where not only all H-type customers are captured but also the coverage of the L-type customers is maximized.

9

ACCEPTED MANUSCRIPT

Lemma 2. Given that (p, t) ∈ S2 , the optimal location of the firm is x∗ = 1 − rH if and only if γ 1−γ

≥

2rL (fL (rL )−fL (1−rH )) . r +r 1−fH ( L 2 H )(rL +rH )

Otherwise x∗ = rL .

Lemma 2 shows that the optimal location of the firm in region S2 is either 1−rH or rL depending on how large the market of the H-type customers is compared to the L-type market.

or 1 − rH . Specifically,    rL     ∗ x = rH       1 − rH

γ 1−γ ,

the optimal location can only be rL , rH ,

rL (fL (rL )−fL (rH )) r +r r +r fH (rH )rH −fH ( L 2 H )( L 2 H )

if

γ 1−γ

if

rL (fL (rL )−fL (rH )) r +r r +r fH (rH )rH −fH ( L 2 H )( L 2 H )

if

γ 1−γ

≤

≥

CR IP T

Lemma 3. For (p, t) ∈ S3 , depending on the amount

0 (x) −rL fL 0 (x) r H fH

<

γ 1−γ

<

0 (x) −rL fL 0 (x) r H fH

AN US

Lemma 3 indicates that the optimal location of the market depends on how large the market

of the H-type customers is compared to the one of L-type customers. When

γ 1−γ

is large, the firm

locates at the right of the market where the H-type customers are the majority. When the fraction of H-type customers decreases, the firm first moves to somewhere in the middle at rH and then by

M

further decreases in γ moves to the left of the market to concentrate more on the L-type customers. Lemma 4. In the case of (p, t) ∈ S4 , the optimal location of the firm is x∗ = 1 − rH .

ED

Region S4 refers to the case that the firm only targets the H-type customers. Therefore, it is optimal for the firm to locate at the right side of the market close to H-type customers to maximize

PT

the demand.

Following Lemmas 1-4, we prove in Proposition 1 that there is a linear relationship between the

CE

optimal location and price.

Proposition 1. For any given price, the optimal location of the firm is either rL , rH or 1 − rH .

AC

Note that the optimal location is always at the center of the market (line) when customers are uniformly distributed. However, under the non-uniform heterogeneous market assumption, the optimal location is a linear function of price. Lemmas 2 and 3 show that a threshold for the proportion of the H-type population to the L-type’s plays an important role in determining which side of the market is better for the firm to position its product. Proposition 1 plays the main role in deriving the joint optimal decision of the firm: The optimal solution of problem (P1) can be obtained by investigating the optimal location and price just on the lines x = 1 − rH = 1 − x = rH =

vH −p t ,

and x = rL =

vL −p t .

10

vH −p t ,

ACCEPTED MANUSCRIPT

4

Joint Optimal Decision on Price and Location

In this section, we investigate the joint optimal price and location by considering regions S1 , S2 , S3 , and S4 . To keep the analysis simple and illustrate the market better, we consider the case in which fH (y) = 2y, fL (y) = −2y + 2 in the following sections. This assumption, while not changing the structure of the results, reduces the complexity of the model and helps us focus on the important

CR IP T

model inputs such as vL , vH , γ, and t.

In region S1 , the optimal location is rL ( as shown in Lemma 1), and the firm sets its price such that all H-type customers are captured, which results in a price lower than

vH +vL −t . 2

Moreover,

since the location of the firm is close to L-type customers, it is beneficial for the firm to cover a fraction of these customers which leads to a price lower than vL . Considering these properties of

AN US

region S1 , we get the optimal price as stated in the following lemma.

Lemma 5. Suppose that (p, t) ∈ S1 . Then, the optimal location of the firm is x∗ = rL , and the optimal price of the firm is:

vH + vL − t , p1 }, 2

M

p∗1 = min{vL , where

√

∆1

,

ED

2(1 − γ)vL − 2t + p1 = 3(1 − γ)

PT

t 2 3 ∆1 = (1 − γ)2 (vL − ) + (1 − γ)t2 . 2 4

The boundaries of region S2 , namely p ≤ min{vH − 2t , vL }, allow the firm to attract a fraction

CE

of both customer classes. However, unlike region S1 , the firm cannot cover all H-type customers if it locates close to L-type customers at rL . Therefore, the firm should analyze the trade-off between

AC

the H- and L-type customers to determine the optimal decisions. The following lemma provides the optimal price considering the potential optimal locations of the firm in region S2 characterized in Lemma 2.

Lemma 6. Suppose that (p, t) ∈ S2 . Then, the optimal decision of the firm is one of the following

cases: • The optimal location is x∗ = rL , and the optimal price is p∗2.1 = max{

t vH + vL − t , min{p2.1 , vH − , vL }}, 2 2 11

ACCEPTED MANUSCRIPT

where p2.1 =

  p 1 (4 − 6γ)vL − 2γvH − 2(1 − γ)t + ∆2.1 , 6(1 − 2γ)

∆2.1 = (2(1 − γ)t + 2γvH − vL + (6γ − 3)vL )2 − (6γ − 3)(4(1 − γ)(t − vH ) + γ(vH + vL )). • The optimal location is x∗ = 1 − rH , and the optimal price is t vH + vL − t , min{p2.2 , vH − , vL }}, 2 2

where p2.2

1 = (vH + vL − 3

CR IP T

p∗2.2 = max{

s

2 − vL2 − vL vH + vH

3γ t2 ). 4(1 − γ)

To obtain the optimal decision in region S2 , the firm should compare the profit gained by

AN US

considering two cases provided in Lemma 6 and choosing the one that results in a higher profit. In region S3 , similar to region S2 , the optimal decision of the firm on location and price depends on whether or not the firm gains a higher profit by capturing a higher proportion of H-type customers. The following lemma provides the optimal decision of the firm in region S3 . Lemma 7. Suppose that (p, t) ∈ S3 . Then, the optimal decision of the firm is one of the following

M

two cases:

ED

• The optimal location is x∗ = rL , and the optimal price is p∗3.1 = max{vH −

t , min{p2.1 , vL }}. 2

CE

PT

• The optimal location is x∗ = 1 − rH , and the optimal price is

where

AC

p3.2 =

p∗3.2 = max{vH −

t , min{p3.2 , vL }}, 2

 p  1 γ(3vH + vL ) − (vH + vL ) − γt + ∆3 , 3(2γ − 1)


2 ∆3 = (γ(3vH + vL ) − (vH + vL ) − γt)2 −3(1 − 2γ) (1 − γ)vH vL − γvH + γtvH .

Similar to region S2 , the firm should compare the total profit gained by considering either of

the two cases discussed in Lemma 7 to determine the optimal decision on location and price in region S3 . In region S4 , the price is above the willingness to pay of L-type customers, p > vL . This means that the focus of the firm is to capture H-type customers, which results in locating on the right side 12

ACCEPTED MANUSCRIPT

of the market at 1 − rH . The following lemma determines the optimal price of the firm in region S4 . Lemma 8. Suppose that (p, t) ∈ S4 . Then, the optimal location of the firm is x∗ = 1 − rH and the optimal price of the firm is

where p4 =

1 (2vH − t + 3

CR IP T

p∗4 = max{vL , p4 }, q 2 − tv + t2 ). vH H

Table 1 summarizes the joint optimal location and price of the firm for all regions. These results show that there are six potential candidates for the joint optimal location and price. The optimal

AN US

solution of problem (P1) can be obtained by comparing the profit of these six candidates. The optimal solution is the candidate with the highest profit.

Table 1: Optimal solution in different regions of Q

Region x∗

p∗

2(1−γ)vL − 2t +

√

∆1

S1

rL

min{vL , vH +v2 L −t ,

S2

rL

1 max{ vH +v2 L −t , min{ 6(1−2γ) (2(2 − 3γ)vL − 2γvH − √ 2(1 − γ)t + ∆2.1 ), vH − 2t , vL }} √ max{ vH +v2 L −t , min{ 31 (vH + vL − ∆2.2 ), vH − 2t , vL }}

rL

S4

M

1 max{vH − 2t , min{ 6(1−2γ) (4 − 6γ)vL − 2γvH − 2(1 − √ γ)t + ∆2.1 , vL }}

1 max{vH − 2t , min{ 3(2γ−1) (γ(3vH + vL ) − (vH + vL ) − √ γt + ∆3 ), vL }} q 2 − tv + t2 )} max{vL , 31 (2vH − t + vH H

CE

1 − rH 1 − rH

}

ED

S3

PT

1 − rH

3(1−γ)

Profit function(π) p(4(1−γ)rL (1−rL )+γ) p(4(1 − γ)rL (1 − rL ) +

γ(rL + rH )2 )

p(4(1 − γ)rL rH + γ) p(4(1 − γ)rL (1 − rL ) + γ(rH + rL )2 ) p(4(1

−

γ)rL rH

+

4γrH (1 − rH )) p(4γrH (1 − rH ))

AC

We next discuss an interesting property of problem (P1) in Proposition 2.

Proposition 2. In a market where the ratio of valuation of H-type to L-type customers (i.e.,

vH vL )

is less than 2, there is a threshold such that if the transportation cost t is greater than this threshold,

it is optimal to target both classes of customers. Proposition 2 demonstrates that if the valuation of H-type customers is not as high as twice of the L-type valuations, the firm serves both types of customers in the case of high transportation 13

ACCEPTED MANUSCRIPT

cost. Since the high transportation cost restricts the demand of H-type customers, the firm should set the price less than vL to attract enough demand by considering both classes of customers. On the other hand, if H-type customers are willing to pay a high price for the product, the firm can charge a very high price to compensate for the low demand. We can see this behavior in some branded products where the firm decides to serve only a special class of customer. Note that

CR IP T

Proposition 2 emphasizes that even if the firm has no idea about γ in the market, it can determine its optimal strategy about targeting customers by just estimating the relative valuations and the transportation cost.

Numerical Results

AN US

5

In the previous section, we derived the optimal location and price of the firm. In this section, we discuss several managerial insights based on some numerical examples. To evaluate the effects of different parameters on the optimal decision, we set vH = 15, vL = 10 and investigate the optimal decision of the firm for different values of γ, the percentage of the H-type customers in the market.

The impact of transportation cost 16

12

PT

Optimal price (p*)

14

ED

5.1

M

We continue the numerical analysis by investigating the value of our modeling assumptions.

AC

CE

10 8

=0.3

6

=0.5

4

=0.7

2 0 0

10

20

30

40

50

Transportation cost (t)

Figure 2: Optimal price with respect to transportation cost.

Figure 2 illustrates how the optimal price of the firm changes by varying the transportation cost. Note that when t is very high, the optimal price is less than the valuation of L-type customers, vL , independent of the value of γ as we demonstrated in Proposition 2. As shown in Figure 2, the behavior of the optimal price with respect to transportation cost is not always monotonically 14

ACCEPTED MANUSCRIPT

0.9

Optimal location (x*)

0.8 0.7 0.6 0.5 =0.3 0.4 =0.5 0.3 =0.7 0.2

0 0

10

20

30

Transportaion cost (t)

40

CR IP T

0.1 50

Figure 3: Optimal location with respect to transportation cost.

decreasing. We also observe that when the number of H-type customers increases in the market,

AN US

depending on the amount of the transportation cost, the optimal price might change considerably or just slightly. As can be seen from Figure 3, for γ = 0.3, the firm locates on the left side of the market, closer to the L-type customers, while for γ = 0.7, the firm locates on the right side of the market, concentrating more on the H-type customers. However, for γ = 0.5, the location strategy

M

changes as travel cost increases.

In Figure 3, for γ = 0.5, the optimal location has a non-smooth behavior, and for some t it is even less than the optimal location for γ = 0.3. This shows that when the number of H-type

ED

customers increases in the market, it is not always the best strategy to move to the right side of the market where the density of H-type customers is high; on the contrary, it might be better to move

PT

to the left side where the majority of customers are L-type. This result is shown in Lemmas 2 and 3 where we proved that when vL −p t .

is less than a threshold, the optimal location of the firm is at

As the percentage of the H-type customers in the market, γ, increases from 0.3 to 0.5,

CE

rL =

γ 1−γ

the firm sets a higher price for the product, which results in a lower rL and the firm locates closer

AC

to the left side of the market. Figure 4 shows the relation between the optimal profit and transportation cost. This figure

illustrates that although the transportation cost has a negative effect on the profit of the firm, it is not the same for all amounts of γ. This is because of the non-uniform distribution of customers in

the market.

15

ACCEPTED MANUSCRIPT

12

Optimal profit ( *)

10 8 =0.3

6

=0.5 4 =0.7

0 0

10

20

30

Transportation cost (t)

40

CR IP T

2

50

Figure 4: Optimal profit with respect to transportation cost.

The Impact of Promotion Strategy

AN US

5.2

We next consider the case for which the population of any type of customer is fixed and the firm targets to increase the H-type customers’ utility through some promotion or price discount. We assume that promotion will increase the willingness to pay of the targeted customers and investigate the effect of the promotion. Providing discount to a group of customers can be modeled similarly.

M

The relation between optimal profit and vH is illustrated by Figure 5 for different amounts of the transportation cost. vL and γ parameters are set to 10 and 0.3, respectively. As Figure 5

ED

shows, it is not a good strategy to target H-type customers, especially when t is low. The reason is that in the case of a low transportation cost, the H-type market is fully covered and making the

PT

product more attractive for H-type customers does not increase demand. However, increasing vL while keeping vH constant seems a better strategy, especially in the case of low transportation cost,

AC

CE

as illustrated in Figure 6.

16

ACCEPTED MANUSCRIPT

'

!"#$%&'!()*#"'+,-.

& % $

()!

#

()!% ()"%

"

()$

!!

CR IP T

!

!& "! /%&0%"#)1')*'23"4!56'+72.'

"&

Figure 5: Optimal profit versus H-type customer’s reservation price for the product.

AN US

%!

!"#$%&'!()*#"'+,-.

% $ # "

'

%% %( /%&0%"#)1')*'23"4!56'+72.

*+%) *+!) *+"

%)

ED

&

M

!

*+%

5.3

PT

Figure 6: Optimal profit versus L-type customer’s reservation price for the product.

Simultaneous vs. Hierarchical Optimization

CE

The exact analysis of simultaneous price and location optimization might be cumbersome for a small firm. A simpler alternative approach for entering in the market is to set the location first and

AC

then adjust the price to the market conditions. While this approach leads to a suboptimal profit for the firm (see Hanjoul et al., 1990), the amount that the firm loses is not clear. Note that the optimal location in a heterogeneous market where customers of both classes are

uniformly distributed is at the center. Therefore, a simple analysis would show that both the hierarchical and simultaneous approaches yield the same location-price set. However, this is not the case in a market with a non-uniform dispersion of customers and the joint location and price decision is crucial to maximize the profit.

17

ACCEPTED MANUSCRIPT

Table 2: Percentage of profit loss associated with location choices on the line, for vL = 10, vH = 15.

γ = 0.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

γ = 0.7

t = 10

t = 20

t = 30

t = 10

t = 20

t = 30

t = 10

t = 20

t = 30

20.42 7.22 0.14 6.31 17.09 27.69 37.79 46.54 53.93

22.82 1.32 2.10 4.36 5.39 8.66 19.85 32.20 45.67

11.97 1.32 3.50 4.50 5.28 5.80 6.00 17.15 36.28

20.00 6.67 1.87 8.25 14.28 19.52 23.19 28.66 34.82

42.14 24.46 19.78 14.49 7.41 0.45 4.00 15.72 29.49

44.51 34.26 29.47 23.21 16.67 9.87 2.80 4.00 24.62

24.94 12.43 6.09 6.03 5.33 0.03 4.57 12.70 20.27

59.99 46.51 38.39 28.42 16.72 4.81 0.31 9.92 25.28

65.80 56.18 47.40 37.27 26.96 16.50 5.94 1.31 21.74

CR IP T

x

γ = 0.5

AN US

To highlight the value of the simultaneous decision of the price and location, consider a market with non-uniform distributions of customers as in Section 4. For numerical analysis, we set vL = 10 and vH = 15 and consider low, medium and high amounts of γ and t. The percentage of profit loss when the firm optimizes its price after locating in the market is shown in Table 2. The results in this table indicate that the profit loss could be significant if the firm is located far from the optimal

5.4

ED

their non-uniform distributions.

M

location, on the wrong side of the market. The loss is due to the heterogeneity of customers and

The Impact of Heterogeneous Assumption

PT

In this section, we investigate the importance of distinguishing between different classes of customers when a firm is entering a new market. We study the case in which the firm assumes homogeneity

CE

of customers and considers the average customer behavior to obtain its optimal location and price. More specifically, the firm considers a market with only one type of customer with the reservation

AC

price of v = γvH + (1 − γ)vL and a distribution of f (.) on the [0, 1] line, where f (y) = γfH (y) + (1 − γ)fL (y). The process of obtaining optimal location and price is similar to what we offer for the market with two classes of customers in sections 3 and 4. A summary of optimal location and price is provided in Table 3. To demonstrate the importance of distinguishing the customers in the market, we numerically compare the profit gained by considering the average customers’ behavior versus the two types of customers. We consider two examples. In the first, the difference between the willingness to

18

ACCEPTED MANUSCRIPT

Table 3: Optimal solution considering the average behavior of customer.

γ

x∗

p∗

γ < 0.5

r

1 max{ 3−6γ ((2 − 4γ)v + (1 − γ)t +

√

∆0 ), v − 2t }

∆0 = (1 − 2γ)2 v 2 − (1 − γ)(1 − 2γ)vt + (1 − γ)2 t2

[r, 1 − r]

max{ v2 , v − 2t }

γ > 0.5

1−r

1 max{ 6γ−3 4(γ − 12 )v − γt +

q 2 2 3(γ − 12 ) v 2 + ((γ − 12 )v − γt) , v − 2t }

CR IP T

γ = 0.5

pay of the two types of customers is moderate, namely vL = 10 and vH = 15. Figure 7 shows the percentage of profit loss for three different values of the transportation cost. We observe the amount of the profit that the firm loses by considering that a single class of customer depends on

AN US

both t and γ. For example, when t = 40 and γ = 0.4, the firm loses 25% of its profit by considering average customer behavior instead of the heterogeneous market.

% ! $ !

M

# ! " ! !

ED

!"#!$%&'!()*(+")*,%(-)..

& !

'"

'#

'$

'%

,-"& ,-#& ,-%

'&

'(

')

'*

'+

)"%,)$()*(/0%1+!.(,$(%2!(3&"4!%(5 6

CE

vH = 15, vL = 10.

PT

Figure 7: Percentage of profit loss when the firm applies average customer behavior in a market with

Figure 8 illustrates the result of an example in which the degree of market heterogeneity is

AC

high, with vL = 10, vH = 20. Here, when t = 40 and γ = 0.4, there is a 45% profit loss for the firm if it considers no customer segmentation in the market, which is indeed a great loss in profit. In general, the profit loss is higher if the average valuations result in locating on the non-optimal side of the market. In these situations, considering a heterogeneous market is vital for the firm. However, there are several situations where considering the heterogeneity analysis in the market does not add much to the profit of the firm. These cases are mostly the ones in which the firm estimates that γ is very high or very low, customers show little difference in their willingness to pay, or the transportation cost is low. 19

ACCEPTED MANUSCRIPT

!"#!$%&'!()*(+")*,%(-)..

& ! % ! $ ! ,-"& # ! ,-#& ,-%

" ! ! '#

'$

'%

'&

'(

')

'*

)"%,)$()*(/0%1+!.(,$(%2!(3&"4!%(5 6

'+

CR IP T

'"

Figure 8: Percentage of profit loss when the firm applies average customer behavior in a market with vH = 20, vL = 10.

Linear versus Uniform Demand

AN US

5.5

Recall that in Section 2 we assume that fH (y) is linearly increasing in y and fL (y) is linearly decreasing in y, where y is the location of a customer. In this section, we numerically investigate the accuracy and robustness of this linear assumption. Moreover, we compare the results of our model with the ones obtained based on the uniform demand on the line as is commonly assumed in

M

the literature to examine the accuracy of these assumptions under various market conditions. For the comparison, we obtain the percentage of profit loss when customers are distributed according

ED

to a convex, concave, and S-shaped function.

assume that

PT

To model a market in which customers are distributed according to a convex function, we

fH (y) = cy c−1 ,

fL (y) = c(1 − y)c−1 ,

c > 2,

CE

where fH (y) is an increasing convex function and fL (y) is a decreasing convex function in y. Note that c = 1 and c = 2 represent the uniform and linear functions, respectively. By increasing c the

AC

distribution of customers will be more concentrated at the two extremes of the line. To model a market where customers are distributed according to a concave function, we assume

that

fH (y) = (2 − c)y 1−c ,

fL (y) = (2 − c)(1 − y)1−c ,

0 < c < 1,

where fH (y) is an increasing concave function and fL (y) is a decreasing concave function in y. Note that c = 0 and c = 1 represent the uniform and linear functions, respectively. We also consider a market in which customers distributed according to a S-shaped function 20

ACCEPTED MANUSCRIPT

using fH (y) =

2 1+

e−k(y−0.5)

,

fL (y) =

2 1+

e−k(−y+0.5)

.

In the following examples whenever not explicitly mentioned, we consider vL = 10 , γ = 0.5, vH = 15 and t = 20 as default values.

γ = 0.1

Concave (c = 0.5) S-Shaped (k = 10)

t = 10 t = 20 t = 30 t = 10 t = 20 t = 30 t = 10 t = 20 t = 30

γ = 0.9

Uniform

Linear

Uniform

Linear

Uniform

Linear

26.59 47.44 56.32 8.81 11.77 13.31 24.51 34.82 37.14

2.24 1.51 0.46 1.27 0.17 0.27 1.56 0.00 0.46

25.00 20.28 40.53 11.35 13.63 11.14 29.21 21.34 16.14

0.00 6.87 4.56 0.00 0.68 1.22 0.00 4.77 1.44

11.11 40.36 55.26 5.24 11.13 6.98 11.71 17.97 8.60

2.38 2.59 2.11 2.10 1.47 0.37 2.76 1.12 0.04

AN US

Convex (c = 3)

γ = 0.5

CR IP T

Table 4: Percentage of profit loss due to the uniform and linear assumption when vL = 10, vH = 15.

M

Tables 4 and 5 show the profit loss due to the linear and uniform assumptions on the customers’ distribution in the market when they are distributed according to a convex, concave, or S-shaped

ED

function.

Table 5: Percentage of profit loss due to the uniform and linear assumption when vL = 10, vH = 25.

t = 10 t = 20 t = 30 t = 10 t = 20 t = 30 t = 10 t = 20 t = 30

CE

Convex (c = 3)

PT

γ = 0.1

AC

Concave (c = 0.5)

S-Shaped (k = 10)

γ = 0.5

γ = 0.9

Uniform

Linear

Uniform

Linear

Uniform

Linear

26.59 52.34 51.92 8.81 17.22 13.07 24.51 39.47 36.09

2.24 1.20 0.37 1.27 0.11 0.00 1.56 0.00 0.28

4.44 15.65 37.17 0.22 3.48 11.97 4.69 15.90 31.42

1.75 2.77 2.20 1.49 1.76 1.39 2.13 2.90 1.12

4.44 15.65 37.17 0.22 3.48 11.97 4.69 15.90 31.42

1.75 2.77 2.20 1.49 1.76 1.39 2.13 2.90 1.12

The results of Tables 4 and 5 demonstrate that the optimal location and price obtained based on the model proposed in this paper is robust for the other three distributions in a wide range of model parameters. However, the simple assumption of uniform distribution of customers results in 21

ACCEPTED MANUSCRIPT

substantial profit loss which varies depending on parameters of the market, such as number of Htype customers and transportation cost. When customers in the market are distributed according to a concave function, the model with the uniform assumption results in a low profit. ,-./01234556278.0-3

% ! $ !

CR IP T

!"#!$%&'!()*(+")*,%(-)..

& !

90-:;<

90-=4:;

# !

>?5@47;A

" ! ! '"

'#

'$

'%

'&

'(

')

'*

'+

AN US

)"%,)$()*(/0%1+!(#2.%)3!".4 5

Figure 9: Percentage of profit loss associated with uniform assumption for different values of γ.

,-./0120334567-8.2

M

% ! $ ! # ! " ! !

ED

!"#!$%&'!()*(+")*,%(-)..

& !

'"

'#

'$

98.:/; 98.<0:/ =>3?06/@

'%

'&

'(

')

'*

'+

PT

)"%,)$()*(/0%1+!(#2.%)3!".4 5

CE

Figure 10: Percentage of profit loss associated with linear assumption for different values of γ.

When the number of H-type customers varies in the market, Figures 9 and 10 show the profit

AC

loss of models based on the uniform and linear assumptions, respectively. The profit loss associated with linear assumption is almost negligible for different values of γ while the profit loss due to the

uniform assumption depends severely on the value of γ.

5.6

Effect of Customers’ Valuations

In this section, we investigate the effect of extreme cases of customers’ valuations on the optimal decisions of the firm. There are two possible extreme cases: (a) the 22

ACCEPTED MANUSCRIPT

120

Optimal Price (p*)

100 80 =0.05 60

=0.01

40

=0.1

20

0

100

200 Transportation cost (t)

CR IP T

0 300

400

Figure 11: Optimal price with respect to transportation cost when vH /vL = 10. 0.9 0.8

0.6 0.5 0.4 0.3 0.2 0.1 0 0

50

100

150

AN US

Optimal location( x*)

0.7

200

250

300

=0.01 =0.05 =0.1

350

M

Transportation cost (t)

ED

Figure 12: Optimal location with respect to transportation cost when vH /vL = 10.

customers’ valuations may be almost the same; and (b) there may be a considerable difference between the customers’ valuations. The first case indicates a homogeneous

PT

market analyzed in Section 5.4. Therefore, the second case is examined in this section. Consider a market in which the value of the product for the H-type customers is 10

CE

times more than L-type customers, i.e., vH = 10vL . The optimal decisions and the profit of the firm in this market are depicted in Figures 11, 12, and 13. As illustrated in these

AC

figures, the firm targets L-type customers when γ is 0.01 and H-type customers when γ = 0.1. However, for γ = 0.05, the firm may target either class of customers depending

on the transportation cost. When the transportation cost is low, it is optimal for the firm to target L-type customers. However, when the transportation cost is high, the firm loses a portion of L-type customers, and therefore, it is optimal to target the H-type customers.

23

ACCEPTED MANUSCRIPT

12

Optimal profit ( *)

10 8 =0.01

6

=0.05 4 =0.1

0 0

100

200

300

Transportation cost (t)

CR IP T

2

400

Figure 13: Optimal profit with respect to transportation cost when vH /vL = 10.

Conclusions

AN US

6

In this paper we considered the problem of simultaneous location and price decisions of a firm in the presence of heterogeneous customers. Specifically, we considered a market comprised of two types of customers with high and low willingness to pay for the firm’s product. We analyzed the market more realistically and relaxed the full market coverage and inelastic demand assumptions, a

M

common practice in the literature. We also allowed customers to have non-uniform distribution in the market. We first proved that there is a linear relation between the optimal location and price.

ED

Using this result, we obtained the optimal joint price and location of the firm. We demonstrated that for the markets where two types of customers are not very different

PT

in their willingness to pay, there is a threshold such that if the transportation cost is above the threshold, the firm should target both classes of customers. In our numerical analysis, we highlighted the importance of considering the H-type population along with the transportation cost

CE

since a change in the population results in different strategies for the firm when faced with low and high transportation costs. We also found that for targeted promotions the firm should have a

AC

good estimation of the transportation cost. Our results show that the effect of raising the utility of any type of customer (through price discount, for example) on profit depends on the percentage of H-type customers in the market as well as the transportation cost. Ignoring one of these factors in the analysis results in targeting the wrong type of customer. Moreover, when there is a substantial difference between customers’ willingness to pay, it is necessary for the firm to consider the heterogeneity in its marketing analysis. We also highlighted the importance of the non-uniformity assumption in analyzing heterogeneous markets.

24

ACCEPTED MANUSCRIPT

Future research could consider the competition between two firms in such markets to examine how the optimal firms’ strategies in location and pricing vary when two types of customers with non-uniform distribution exist in the market. Considering that quality is exogenous in our model, another avenue for the future research would be to jointly decide on quality, location and price in

References

CR IP T

such markets in both monopoly setting and competition.

Aboolian, R., O. Berman, and D. Krass (2007). Competitive facility location and design problem. European Journal of Operational Research 182 (1), 40–62.

Aboolian, R., O. Berman, and D. Krass (2008). Optimizing pricing and location decisions for competitive

AN US

service facilities charging uniform price. Journal of the Operational Research Society 59 (11), 1506–1519. Adner, R. and P. Zemsky (2006). A demand based perspective on sustainable competitive advantage. Strategic Management Journal 27 (3), 215–239.

Anderson, S. P., J. K. Goeree, and R. Ramer (1997). Location, location, location. Journal of Economic Theory 77 (1), 102–127.

M

Berube, A. and W. H. Frey (2002). A decade of mixed blessings: urban and suburban poverty in census 2000. Brookings Institution, Center on Urban and Metropolitan Policy. Booza, J., J. Cutsinger, and G. Galster (2006). Where did they go?: The decline of middle-income neigh-

ED

borhoods in metropolitan America. Metropolitan Policy Program, Brookings Institution. Caldieraro, F. (2016). The role of brand image and product characteristics on firms entry and oem decisions.

PT

Management Science.

Calv´o-Armengol, A. and Y. Zenou (2002). The importance of the distribution of consumers in horizontal

CE

product differentiation. Journal of Regional Science 42 (4), 793–803. Cheung, F. K. and X. Wang (1995). Spatial price discrimination and location choice with non-uniform demands. Regional Science and Urban Economics 25 (1), 59–73.

AC

Clifford, S. and A. Martin (2011, 21 April). As customers cut spending, green products lose allure. http: //www.nytimes.com/2011/04/22/business/energy-environment/22green.html?

Cooke, T. and S. Marchant (2006). The changing intrametropolitan location of high-poverty neighbourhoods in the us, 1990-2000. Urban Studies 43 (11), 1971–1989.

Dasci, A. and G. Laporte (2005). A continuous model for multistore competitive location. Operations Research 53 (2), 263–280. Desai, P. S. (2001). Quality segmentation in spatial markets: When does cannibalization affect product line design? Marketing Science 20 (3), 265–283.

25

ACCEPTED MANUSCRIPT

Dobson, G. and E. Stavrulaki (2007). Simultaneous price, location, and capacity decisions on a line of time-sensitive customers. Naval Research Logistics (NRL) 54 (1), 1–10. Fry, R. and P. Taylor (2012). The rise of residential segregation by income. Pew Research Center . Gupta, B., D. Pal, and J. Sarkar (1997). Spatial cournot competition and agglomeration in a model of location choice. Regional Science and Urban Economics 27 (3), 261–282.

spatial price policies. Management Science 36 (1), 41–57.

CR IP T

Hanjoul, P., P. Hansen, D. Peeters, and J.-F. Thisse (1990). Uncapacitated plant location under alternative

Hotelling, H. (1929). Stability in competition. The Economic Journal , 41–57.

Hwang, H. and C.-C. Mai (1990). Effects of spatial price discrimination on output, welfare, and location. The American Economic Review 80 (3), 567–575.

gasoline markets. QME 6 (3), 205–234.

AN US

Iyer, G. and P. B. Seetharaman (2008). Too close to be similar: Product and price competition in retail

Kress, D. and E. Pesch (2012). Sequential competitive location on networks. European Journal of Operational Research 217 (3), 483–499.

Kwasnica, A. M. and E. Stavrulaki (2008). Competitive location and capacity decisions for firms serving time-sensitive customers. Naval Research Logistics (NRL) 55 (7), 704–721.

M

Lacourbe, P. (2012). A model of product line design and introduction sequence with reservation utility. European Journal of Operational Research 220 (2), 338–348.

ED

Lacourbe, P., C. H. Loch, and S. Kavadias (2009). Product positioning in a two-dimensional market space. Production and Operations Management 18 (3), 315–332. Lauga, D. O. and E. Ofek (2011). Product positioning in a two-dimensional vertical differentiation model:

PT

The role of quality costs. Marketing Science 30 (5), 903–923. Liu, Y. and C. B. Weinberg (2006). The welfare and equity implications of competition in television broad-

CE

casting: the role of viewer tastes. Journal of Cultural Economics 30 (2), 127–140. Meagher, K. J., E. G. Teo, and W. Wang (2008). A duopoly location toolkit: Consumer densities which yield unique spatial duopoly equilibria. The BE Journal of Theoretical Economics 8 (1).

AC

Neven, D. J. (1986). On hotelling’s competition with non-uniform customer distributions. Economics Letters 21 (2), 121–126.

Szaky, T. (2011, April 28). Why sales of green products are down. http://boss.blogs.nytimes.com/ 2011/04/28/why-sales-of-green-products-are-down/.

Tabuchi, T. and J.-F. Thisse (1995). Asymmetric equilibria in spatial competition. International Journal of Industrial Organization 13 (2), 213–227. Tang, Q. C. and H. K. Cheng (2005). Optimal location and pricing of web services intermediary. Decision Support Systems 40 (1), 129–141.

26

ACCEPTED MANUSCRIPT

Zhou, W., X. Chao, and X. Gong (2014). Optimal uniform pricing strategy of a service firm when facing two classes of customers. Production and Operations Management 23 (4), 676–688.

7

Appendix vH +vL −t }, 2

which is

CR IP T

Proof of Lemma 1. Recall that S1 = {(p, t) ∈ Q|p ≥ vL − 2t , p ≤ vL , p ≤

equivalent to {rL , rH |0 ≤ rL ≤ 21 , rH > 12 , rH ≥ 1 − rL }. Therefore, in S1 we have the following inequalities: 0 ≤ 1 − rH ≤ rL ≤ 1 − rL < rH ≤ 1. Since rL > 1 − rH , by locating at x = rL the firm can cover all the market of H-type customers while maximizing the demand of L-type customers. Thus, in this case x∗ = rL .

AN US

Note that in the case of rH > 1, it is obvious that the optimal location is x∗ = rL . The case that rH > 1, can only be in S1 , because in the regions that rL > 0, the condition rL + rH ≤ 1, prevents rH to be greater than one and when rL = 0, the optimality condition will make rH ≤ 1.

Proof of Lemma 2. In region S2 = {(p, t) ∈ Q|p ≤ vH − 2t , p ≤ vL , p ≥

vH +vL −t }, 2

we have

M

{rL , rH |0 ≤ rL ≤ 12 , rH ≥ 21 , rH ≤ 1 − rL }, which yields to the following inequalities:

(7)

ED

0 ≤ rL ≤ 1 − rH ≤ rH ≤ 1 − rL ≤ 1.

Since the demand of the firm is a function of its location, we consider five cases with boundary

PT

overlaps based on the inequalities given in (7) to obtain the demand. Case 1: x ∈ [0, rL ]

AC

CE

xH = min{x + rH , 1} = x + rH ,

Since

dD(x) dx

xH = max{x − rH , 0} = 0,

xL = min{x + rL , 1} = x + rL , xL = max{x − rL , 0} = 0, Z x+rH Z x+rL fH (y)dy + (1 − γ) fL (y)dy. D(x) = γ 0

0

= γfH (x + rH ) + (1 − γ)fL (x + rL ) > 0, the firm can increase its demand by moving

to the right. Thus, in this case the optimal location is at rL . Case 2: x ∈ [ rL , 1 − rH ] xH = min{x + rH , 1} = x + rH ,

xH = max{x − rH , 0} = 0,

xL = min{x + rL , 1} = x + rL ,

xL = max{x − rL , 0} = x − rL . 27

ACCEPTED MANUSCRIPT

The demand of the firm in this region is Z Z x+rH fH (y)dy + (1 − γ) D(x) = γ d2 D(x) dx2

fL (y)dy.

x−rL

0

Since

x+rL

0 (x + r ) > 0, the demand function is convex in x. Therefore, the optimal = γfH H

location is at one of the two ends of the interval, i.e., x∗ = rL or x∗ = 1 − rH . A simple analysis

CR IP T

shows that D(rL ) ≤ D(1 − rH ) if and only if :

γ 2rL (fL (rL ) − fL (1 − rH )) ≥ . H 1−γ 1 − fH ( rL +r )(rL + rH ) 2

(In deriving (8), since fH (x) and fL (x) are linear functions we used j = H, L).

xL = max{x − rL , 0} = x − rL .

xL = min{x + rL , 1} = x + rL ,

In this case the demand function is: D(x) = γ + (1 − γ)

R x+rL x−rL

fL (y)dy.

Case 4: x ∈ [rH , 1 − rL ]

M

= fL (x + rL ) − fL (x − rL ) < 0 (recall that fL (x) is decreasing in x), then x∗ = 1 − rH .

ED

dD(x) dx

fj (x) = (b − a)fj ( a+b 2 ) for

xH = max{x − rH , 0} = 0,

xH = min{x + rH , 1} = 1,

Since

a

AN US

Case 3: x ∈ [ 1 − rH , rH ]

Rb

(8)

In this case it is clear that the firm can attract more customers by moving to the left, i.e., locating

x∗ 6∈ [rH , 1 − rL ].

PT

at rH . However, as we showed in Case 3, x = 1−rH captures more demand than x = rH . Therefore

Case 5: x ∈ [1 − rL , 1]

CE

The argument in this case is the same as in the previous case. The firm can attract more customers if it locates at 1 − rL . However as we showed in case 4, locating at x = rH captures more demand

AC

than x = 1 − rL , which results in x∗ 6∈ [1 − rL , 1] .

By considering the five cases discussed above, we see that the optimal location is x∗ = 1 − rH if and only if

γ 1−γ

≥

2rL (fL (rL )−fL (1−rH )) . r +r 1−fH ( L 2 H )(rL +rH )

Otherwise x∗ = rL .

Proof of Lemma 3. When (p, t) ∈ S3 = {(p, t) ∈ Q|p ≥ vH − 2t , p ≤ vL }, (which is equivalent to {rL , rH |0 ≤ rL ≤ 1/2, 0 < rH ≤ 1/2} we have 0 ≤ rL ≤ rH ≤ 1 − rH ≤ 1 − rL ≤ 1. Similar to the proof of Lemma 2, we consider all the possible locations for the firm in the following cases with boundary overlaps: 28

ACCEPTED MANUSCRIPT

Case 1: x ∈ [0, rL ] xH = min{x + rH , 1} = x + rH ,

xH = max{x − rH , 0} = 0,

xL = min{x + rL , 1} = x + rL ,

xL = max{x − rL , 0} = 0.

Case 2: x ∈ [rL , rH ]

R x+rH 0

+(1 − γ)

R x+rL

is linearly increasing. Therefore, x∗ = rL .

0

CR IP T

In this case, D(x) = γ

xH = min{x + rH , 1} = x + rH ,

xH = max{x − rH , 0} = 0,

xL = min{x + rL , 1} = x + rL ,

xL = max{x − rL , 0} = x − rL .

The demand of the firm is D(x) = γ

R x+rH 0

fH (y)dy +(1−γ)

R x+rL x−rL

fL (y)dy. Since

d2 D(x) dx2

0 (x+ = γfH

rH ) > 0, D(x) is a convex function of x. Therefore, the maximum demand is either happens at rL

AN US

or rH . A simple analysis show that D(rL ) < D(rH ) if and only if

rL (fL (rL ) − fL (rH )) γ > . H H 1−γ fH (rH )rH − fH ( rL +r )( rL +r ) 2 2 Case 3: x ∈ [rH , 1 − rH ]

xH = max{x − rH , 0} = x − rH ,

M

xH = min{x + rH , 1} = x + rH ,

xL = max{x − rL , 0} = x − rL .

xL = min{x + rL , 1} = x + rL ,

ED

In this case the demand function is D(x) = and fL (x) are linear functions of x,

dD(x) dx

R x+rH x−rH

γfH (y)dy+

PT

0 (x) + (1 − γ)2r f 0 (x), the optimal γ2rH fH L L −r f 0 (x) γ at rH if and only if 1−γ > rHLf 0L(x) . H

CE

location is

x−rL

(1 − γ)f L (y)dy. Since fH (x)

= γ(fH (x+rH )−fH (x−rH ))+(1−γ)(fL (x+rH )−fL (x−

0 (x) + (1 − γ)2r f 0 (x) (Note that in this case rH )) = γ2rH fH L L

the sign of

R x+rL

d2 D(x) dx2

= 0). Therefore, depending on

location is either rH or 1 − rH . The optimal

AC

Case 4: x ∈ [1 − rH , 1 − rL ] xH = min{+rH , 1} = 1,

xH = max{x − rH , 0} = x − rH ,

xL = min{x + rL , 1} = x + rL ,

xL = max{x − rL , 0} = x − rL .

The demand of the firm is D(x) = γ

R1

x−rH

fH (y)dy+(1−γ)

−γfH (x − rH ) < 0 .Therefore, x∗ = 1 − rH .

R x+rL x−rL

fL (y)dy. In this case

Case 5: x ∈ [ 1 − rL , 1] xH = min{x + rH , 1} = 1,

xH = max{x − rH , 0} = x − rH ,

xL = min{x + rL , 1} = 1,

xL = max{x − rL , 0} = x − rL . 29

dD(x) dx

=

ACCEPTED MANUSCRIPT

In this case, since

dD(x) dx

< 0, the firm should locate at 1 − rL . However, x = 1 − rL ,

cannot be the optimal location, because the result of Case 4 shows that x = 1 − rH results in a higher demand than 1 − rL . In summary, the only cases that may include the optimal location are Cases 2, 3, and 4. Therefore,

x∗ =

    rL    

rH       1 − r

H

rL (fL (rL )−fL (rH )) r +r r +r fH (rH )rH −fH ( L 2 H )( L 2 H )

if

γ 1−γ

if

rL (fL (rL )−fL (rH )) r +r r +r fH (rH )rH −fH ( L 2 H )( L 2 H )

if

γ 1−γ

≤

≥

0 (x) −rL fL 0 (x) rH fH

<

γ 1−γ

<

0 (x) −rL fL 0 (x) rH fH

(9)

AN US

rH or 1 − rH :

CR IP T

γ depending on the number of H-type to L-type customers, ( 1−γ ) the optimal location can be at rL ,

Proof of Lemma 4. When (p, t) ∈ S4 = {(p, t) ∈ Q|p ≥ vH − 2t , p ≥ vL }, the price is more than vL and the firm serves only H-type customers:

xH = max{x − rH , 0} = x − rH .

Then, the demand of the firm is Z

x+rH

γfH (y)dy=γ

Z

x+rH

2ydy = 4γxrH ⇒ x∗ = 1 − rH .

ED

D(x) =

M

xH = min{x + rH , 1} = x + rH ,

x−rH

PT

x−rH

Proof of Proposition 1. According to Lemma 1 to 4 the optimal location when (p, t) is in regions

CE

S1 , S2 , S3 or S4 is either x = 1 − rH , x = rH or x = rL .

AC

Proof of Lemma 5. Lemma 1 shows that when (p∗ , t) ∈ S1 , then x∗ = rL . Hence the profit is   Z rL +rH Z 2rL π1 = p γ fH (y)dy + (1 − γ) fL (y)dy = p (4(1 − γ)rL (1 − rL ) + γ) 0  0  vL − p t − v L + p = p 4(1 − γ)( )( )+γ , t t

and we have the following optimization problem:

30

ACCEPTED MANUSCRIPT

π1∗ = max π1 p

subject to:

  t vH + vL − t . vL − ≤ p ≤ min vL , 2 2

(10)

Ignoring the constraints, the optimal solution can be obtained by the First Order Condition (FOC)

CR IP T

and checked by the Second Order Condition (SOC): q 2 2(1 − γ)(vL − 2t ) + (1 − γ)2 (vL − 2t ) + 34 (1 − γ)t2 p1 = . 3(1 − γ)

Since π1 is concave in p, considering the constraints of the problem, the optimal price is,

Note that, here p1 > vL −

t 2

since:

2(1 − γ)(vL − 2t ) +

p1 =

vH + v L − t , p1 }. 2

AN US

p∗1 = min{vL ,

q 2 (1 − γ)2 (vL − 2t ) + 43 (1 − γ)t2

3(1 − γ) 2(1 − γ)(vL − + (1 − γ)(vL − 2t ) t > = vL − . 3(1 − γ) 2

M

t 2)

ED

Proof of Lemma 6. Based on Lemma 2, if (p∗ , t) ∈ S2 , then the optimal location is either rL or 1 − rH . Note that we cannot use the threshold obtained for γ in Section 3 since it depends on price.

CE

PT

If x∗ = rL , then the profit function is   Z rL +rH Z 2rL   π2.1 = p γ fH (y)dy + (1 − γ) fL (y)dy = p 4(1 − γ)rL (1 − rL ) + γ(rL + rH )2 , 0 0   vH + vL − 2p 2 vL − p t − v L − p = p 4(1 − γ)( )( )+( ) , t t t

AC

and we have the following optimization problem: ∗ π2.1 = max π2.1 p

subject to:

  vH + vL − t t ≤ p ≤ min vH − , vL . 2 2

Considering the FOC and SOC, we have: p2.1 =

  p 1 (4 − 6γ)vL − 2γvH − 2(1 − γ)t + ∆2.1 , 6(1 − 2γ) 31

ACCEPTED MANUSCRIPT

where ∆2.1 = (2(1 − γ)t + 2γvH − vL + (6γ − 3)vL )2 − (6γ − 3)(4(1 − γ)(t − vH ) + γ(vH + vL )). Since π2.1 is concave in p, the optimal price considering the constraints is: p∗2.1 = max{

vH + vL − t t , min{p2.1 , vH − , vL }}. 2 2

constraints remain the same,  Z 1 Z fH (y)dy + (1 − γ) π2.2 = p γ

fL (y)dy

1−rL −rH

0

vL − p vH − p )( ) + γ). t t

Similar to the previous case, s

1 = (vH + vL − 3



= p(4(1 − γ)rL rH + γ)

AN US

= p(4(1 − γ)(

p2.2

1

CR IP T

But in case of x∗ = 1 − rH , the objective function of the optimization problem changes while the

2 − vL2 − vL vH + vH

3γ t2 ). 4(1 − γ)

M

L −t Therefore, p∗2.2 = max{ vH +v , min{p2.2 , vH − 2t , vL }}. 2

Proof of Lemma 7. In the case of (p∗ , t) ∈ S3 the optimal location is either rL , rH or 1 − rH

ED

depending on other parameters (according to Lemma 3). If x∗ = rL , the profit function is:   R R 2r r +r π3.1 = p γ 0 L H fH (y)dy + (1 − γ) 0 L fL (y)dy = p(4(1 − γ)rL (1 − rL ) + γ(rH + rL )2 ).

CE

PT

and we have the following optimization problem:

∗ π3.1 = max π3.1 p

(11)

subject to: vH −

t ≤ p ≤ vL . 2

AC

Since the objective function π3.1 is the same as π2.1 , we have p3.1 = p2.1 . The optimal price is: p∗3.1 = max{vH −

t , min{p2.1 , vL }}. 2

If we consider x∗ = 1 − rH , then the profit function changes as follows:  Z 1  Z 1 π3.2 = p γ fH (y)dy + (1 − γ) fL (y)dy = p(4(1 − γ)rL rH + 4γrH (1 − rH ). 1−2rH

Therefore, p3.2 =

1−rH −rL

 p  1 γ(3vH + vL ) − (vH + vL ) − γt + ∆3 , 3(2γ − 1) 32

ACCEPTED MANUSCRIPT

and


2 ∆3 = (γ(3vH + vL ) − (vH + vL ) − γt)2 −3(1 − 2γ) (1 − γ)vH vL − γvH + γtvH , p∗3.2 = max{vH −

In proof of Lemma 2 and 3 we show that if

t , min{p3.2 , vL }}. 2

rL (fL (rL )−fL (rH )) r +r r +r fH (rH )rH −fH ( L 2 H )( L 2 H )

≤

γ 1−γ

≤

0 (x) −rL fL 0 (x) r H fH

then

the optimal location is at rH . However in the special case of fH (x) = 2x and fL (x) = 2 − 2x, this 4rL 3rH +rL

≤

γ 1−γ

≤

rL rH

rL rH

4rL 3rH +rH

4rL 3rH +rL .

CR IP T

condition results in:

which cannot hold since

=

≤

Hence

the optimal location cannot be at x = rH , and we do not consider this location in our results. Proof of Lemma 8. In the case of (p∗ , t) ∈ S4 , the optimal location is x∗ = 1 − rH based on

AN US

Lemma 4. The profit function of the firm is   Z 1 vH − p t − vH + p fH (y)dy = p (4γrH (1 − rH )) = 4γp( π4 = p γ )( ). t t 1−2rH

(12)

and we have the following optimization problem:

π4∗ = max π4 p

M

subject to:

max{vL , vH −

(13)

t } ≤ p. 2

ED

Ignoring the constraints and by FOC and checking the S OC for concavity, we have:

PT

p4 =

1 (2vH − t + 3

q 2 − tv + t2 ). vH H

(14)

CE

Considering the constraints, the optimal price is:

−

t 2)

q 2 2 + (v H − 2t ) + 43 t ) ≥ vH − 2t .

AC

Note that p4 =

1 3 (2(vH

p∗4 = max{vL , p4 }.

Proof of Proposition 2. Before we prove Proposition 2, we provide the following Lemma. Lemma 9. For γ < 1, it is not optimal for the firm to set the price at the willingness to pay of the L-type customers, vL . Proof of Lemma 9. Let πi denote the profit of the firm when (p, t) ∈ Si (Si is defined in section 3 and illustrated in Figure 2). Note that p = vL is a boundary line for regions S1 , S2 , S3 , S4 and S6 . 33

ACCEPTED MANUSCRIPT

By considering conditions p = vL and p < vH − t/2 ,which is a condition in regions S2 and S6 , one could obtain that (p = vL , t) is in regions S1 , S2 and S6 if and only if t < 2(vH − vL ) otherwise (p = vL , t) is in regions S3 and S4 . We first check the optimality of p = vL when t < 2(vH − vL ).

In this case (p, t) ∈ S6 for p > vL . In section 3 we defined S6 = {(p, t) ∈ Q|p > vL , p ≤ vH − 2t }. Thus, considering that the firm only targets H-type customers and covers all the market(because ∂π6 ∂p |vL

= γ > 0 , vL cannot be an optimum because the

firm can increase its profit by raising the price.

CR IP T

of low transportation cost), π6 = pγ. Since

Now, consider the case that t > 2(vH −vL ). This condition implies that (vL , t) be at the intersection of S3 and S4 . Since vL ≤ p for (p, t) ∈ S4 , the necessary condition for optimality of vL in region S4 is

∂π4 ∂p |vL

≤ 0. Recall that the profit function π4 is p(4γrH (1 − rH )) (see Table 1).The slope of π4

AN US

at vL is

4 4 ∂π4 2 |vL = 2 γ(−vH + vH t + 4vH vL − 2vL t − 3vL2 ) = 2 γ(vL 2 − (vH − 2vL )2 + t(vH − 2vL )). ∂p t t ∂π4 ∂p |vL

≤ 0 if either of the following conditions satisfies:   t >  t <

∂πS3 ∂p |vL

2 (vH −2vL )2 −vL vH −2vL

∂π4 ∂p |vL

ED

Note that when vH = 2vL , then

2 −(v −2v )2 vL H L 2vL −vH

M

Therefore,

if vH < 2vL

(15)

if vH > 2vL

= vL 2 > 0. We next show that if

∂π4 ∂p |vL

≤ 0, then

< 0, which implies that pricing lower than vL will increase the profit in region S3 .

PT

When the firm considers the price p = vL − , (the price in which almost H-type customers can

buy the product) the optimal location in region S3 is at x∗ = 1 − rH . (Note that since we assumed

CE

γ < 1, π3 is different from π4 ) and we have π3 =p(4(1 − γ)rL rH + 4γrH (1 − rH )) (see Table 1), and

AC

∂π3 4 2 |vL = 2 ( −γvH − (4γ − 1)vL2 + (5γ − 1)vH vL − 2γvL t + γvH t) ∂p t 4 2 = 2 (−γvH − 4γvL2 + vL2 + 5γvL vH − vH vL + γt(vH − 2vL )) t 4 = 2 (−γ(vH − 2vL )2 + vL2 + γvH vL − vH vL + γt(vH − 2vL )). t

If vH < 2vL , then using (15), we have: v 2 − (vH − 2vL )2 t2 ∂π3 ( |vL ) ≤ −γ(vH − 2vL )2 + vL2 + γvH vL − vH vL + γ( L )(vH − 2vL ) 4 ∂p 2vL − vH = (1 − γ)vL2 − (1 − γ)vL vH = (1 − γ)vL (vL − vH ) < 0.

34

ACCEPTED MANUSCRIPT

If vH > 2vL , then using (15), we have: (vH − 2vL )2 − vL2 t2 ∂π3 )(vH − 2vL ) ( |vL ) ≤ −γ(vH − 2vL )2 + vL2 + γvH vL − vH vL + γ( 4 ∂p vH − 2vL = (1 − γ)vL (vL − vH ) < 0. Therefore, if

∂π4 ∂p |vL

≤ 0, then

∂πS3 ∂p |vL

< 0 . Thus, vL is not a local optimum and hence cannot be

We next prove Proposition 2. Let t¯ =

2 −(v −2v )2 vL H L . 2vL −vH

CR IP T

a global optimum of problem P1.

Note that t¯ > 2(vH − vL ) since

vL (vH − vL ) > 0 ⇒ t¯ > 2(vH − vL ). t¯ − 2(vH − vL ) = 2vL − vH

AN US

Since t¯ > 2(vH − vL ), when the firm targets only H-type customers its price will be in region S4 (Recall that the firm serves only H-type customers in regions S4 and S6 ). We next show that when vH < 2vL , the optimal price cannot be in S4 . We prove it by contradiction. Suppose that (p∗ , t) ∈ S4 . In the proof of Lemma 9, we showed that

∂π4 ∂p

is negative at p = vL when t > t¯ and

−8γ(t − 2vH − 3vL ) d2 π4 (x) |vL = dp2 t2

M

vH < 2vL . Also, considering π4 given in (12) we get,

ED

L −vH ) H −vL )vL −8γ( (vH −v2vLL)(3v −8γ( (v2v + vH − vL ) ) −8γ(t¯ − 2vH − vL − vH ) −vH L −vH ≤ = = <0 2 2 2 t t t

PT

which implies that π4 is concave at p = vL . Moreover, π4 is concave at p4 = 13 (2vH − t + q 2 − tv + t2 ) and ∂π4 | vH H ∂p p4 = 0 (see proof of Lemma 8). Therefore, considering the concavity of π4 at p4 and vL as well as

∂π4 ∂p |vL

< 0 and

∂π4 ∂p |p4

= 0, we conclude that p4 < vL (since π4 is a

CE

cubic function). This results in p∗4 = vL based on Lemma 8. However, this is a contradiction to Lemma 9 which indicates that p∗4 6= vL for γ < 1. Therefore, the optimal price cannot be in the

AC

region S4 . In other words, in this situation the firm targets both type of customers.

35