Stage 1: The location game
3.3 Distribution cost is paid by firm
In the previous case, consumers have to purchase the products at the firms’ places after receiving informative advertisement. Consumers pay both the product price and the transportation cost. In this section, we discuss the condition from the perspectives of the firms and modify our model into a different scenario. Consider the industries like furniture or pizza, consumers order the products and ask for the delivery service after receiving advertisement information. The firms have to deliver the products to consumers’ locations and cover the distribution costs. The other assumptions of this case are identical with those of the previous case. The two firms locate in a line segment of unit distance,
[ ]
0,1 . Consumers are uniformly distributed along the segment of unit distance. Each consumer is characterized by the location xˆ in the segment. Every consumer purchases one unit of the product.24
Assume that there is a headquarter who delegates two firms (subsidiaries) to operate in the market. Though most of the advertisement of the two firms is made by the headquarter directly, the two firms still utilize other ways to advertise their product quality.
The lowest price a firm expects to obtain from the consumer is called desired value, v . If Firm i does not provide advertisement information but distributes the product to the consumer’s location, a consumer xˆ purchases one unit of the product from Firm i , Firm i ’s surplus is given by:
)2
ˆ (ˆ
ˆi i
i p d x y
PS = −ν − − (33)
where d >0 is the parameter of the rate of distribution fee. The distance between Firm i and consumer is xˆ− . The quadratic form of distance, yˆi (xˆ−yˆi)2, shows that the distribution costs become greater while the distance increases. Hence, Firm i has to spend d(xˆ−yˆi)2 when delivering one unit of the product.
When the two firms compete with each other, they provide consumers informative advertisement for promoting their product quality. The aggregated advertisement effects received by consumers about Firm i’s product quality is given by Eˆi
j i
i e e
Eˆ = ˆ +βˆˆ (34)
After advertising the product quality, the firms expect to obtain more revenues.
The firm’s desired value will increase by Eˆi. When the consumer xˆ purchases from Firm i , the surplus of Firm i is
)2
ˆ (ˆ ˆ )
ˆi ( i i
i p E d x y
PS = − ν + − − (35)
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We setup a three-stage game to discuss the advertisement spillover effect on firms’
location choices. In the first stage, the two firms choose their locations. In the second stage, the two firms decide their advertisement expenditures. In the third stage, the two firms engage in price competition. We adopt backward induction to solve the subgame-perfect equilibrium. The procedure of this case is the same as Fig. 3.
Stage 3: The price subgame
In stage 3, the two firms engage in price competition. Suppose that there is a critical consumer locates at k∈[0,1]. No matter who sell and deliver the product to the critical consumer, the two firms receive the identical surplus. As Firm i delivers its product to the critical consumer, the surplus of Firm i is given as:
)2
( ˆ ˆ )
ˆi ( i i
i p E d k y
PS = − ν + − − (36)
Therefore, we have
2 2 2
2 2 1 1
1 ( ˆ ) ( ˆ ) ˆ ( ˆ ) ( ˆ )
ˆ E d k y p E d k y
p − ν + − − = − ν + − − (37)
From Eq. (37), we can derive k
ˆ ) (ˆ 2
ˆ ) (ˆ ˆ) )(ˆ 1 ˆ ( 2
ˆ ) (ˆ
1 2
1 2 1 2 1
2
y y d
p p e e y
k y
−
−
−
− + −
= + β
(38)
Delivering the product to consumer who locates between 0 and k , Firm 1 can receive greater surplus than Firm 2; delivering the product to the consumer between k and 1, Firm 2 can receive greater surplus than Firm 1. Therefore, Firm 1 dominates in the segment of
[ ]
0,k and Firm 2 dominates in the segment of[ ]
k,1 . As the assumption of every consumer purchases one unit of the product, the market demand of Firm 1 is k and that of Firm 2 is (1−k).26
The profit functions of Firm 1 and Firm 2 are then given by6
2 1 1
1 1
1 ˆ
2 ˆ ˆ ˆ ˆ
ˆ pk a pk ρe
π = − = − (39)
2 2 2
2 2
2 ˆ
2 ) ˆ 1 ˆ ( ) ˆ
1 ˆ (
ˆ p k a p k ρe
π = − − = − − (40)
where ρˆ represents the parameter of conveying informative message. If ρˆ is great, there are difficulties in conveying product information. Therefore, the cost that the firm spends on promoting the product quality gets higher. If ρˆ is trivial, the informative advertisement is easier to transmit. The cost that the firm spends on transmitting the product quality becomes lower. We consider the cost spent on advertising the product quality is the advertisement expenditure, aˆ . The quadratic i form, ˆ2
2 ˆ
ei
ρ , is used to illustrate that the advertisement cost increases with the
difficulty in conveying informative message. Without loss of generality, the operation costs are omitted in this model.
From the first-order-condition of Eqs. (39) and (40), we solve the equilibrium prices as:
ˆ)]
)(ˆ 1 ˆ ( ˆ ) 2 ˆ
ˆ )(
(ˆ 3[
ˆ1 1 d y1 y2 y2 y1 e2 e1
p∗ = − + + − −β − (41)
ˆ)]
)(ˆ 1 ˆ ( ˆ ) 4 ˆ
ˆ )(
(ˆ 3[
ˆ2 1 d y1 y2 y2 y1 e2 e1
p ∗= − − − + −β − (42)
We then check whether the advertisement effects will influence the equilibrium prices.
6 In Eq. (38), Firm i’s average distribution distance per unit product isk 2. And we will prove that k is a constant of 12 later in p.29. The distribution cost is also a constant of d(k 2)2=d 16. Because the constant distribution cost will not affect the response function of Firm i’s profit function, the distribution costs are omitted for simplifying the following calculations.
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Eq. (43) reveals that the more advertisement effects consumers received from Firm i ’s product, the higher price of Firm i’s product is. In contrast, Eq. (44) shows that more advertisement from Firm i will lower Firm j’s price.
Stage 2: The advertisement subgame
In Stage 2, the two firms decide their advertisement expenditures respectively supposing locations as given. Substituting the equilibrium prices, Eqs. (41) and (42) into Eqs. (39) and (40), we obtain
From the first-order-condition of Eqs. (45) and (46), we get the response functions of advertisement effect:
2
and the equilibrium advertisement effects
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From the comparative static analysis, we find that the two firms’ distance will not influence the advertisement effects.
ˆ 0
Then, solve the equilibrium advertisement expenditures as:
2
Stage 1: The location game
The two firms choose their locations, yˆ in Stage 1. Substituting Eqs. (49) and i
Taking first-order-condition of Eqs. (54) and (55), we solve the equilibrium locations:
29
From Eqs. (56) and (57), we derive the equations below.
ˆ 1
Applying the equilibrium locations, the equilibrium advertisement expenditures and the equilibrium prices are as follows:
ρ firms cover the whole market. Moreover, the firms have symmetric advertisement expenditures and product prices. Substitute Eqs. (58), (59) and (60) into Eq. (38), we can know that the second term of Eq. (38) equals to zero. Hence, the critical consumer kˆ is the person who locates at 1/2. That means, each firm owns a half of the market demand and the two firms sit symmetrically on the two halves of [0,1]. Let
From Eq. (61), we further find that the two firms will choose the same location at 2
1 if the advertisement spillover effect is perfect. Therefore, the headquarter has no
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necessary to delegate two subsidiaries or two exclusive agents to operate in the district. This finding is described as proposition 4.
Proposition 4. When the degree of advertisement spillover effect approaches to one, the headquarter will delegate just one firm.
From the comparative static analysis, the effect of the rate of distribution fee and the advertisement spillover effect on the distance of the two firms are as follows:
0
Eq. (62) shows that the higher the rate of distribution fee, the closer the firms will locate and vice versa. That means as the rate of distribution fee increase, the two firms will more cluster for eliminating the cost difference and increasing their surplus. In contrast, the two firms will locate apart from each other while the rate of distribution fee decreases because the distribution cost is trivial. Eq. (63) reveals that the two firms will choose their locations approximately while the degree of advertisement spillover effect increases. Additionally, from Eq. (63) we can know that the degree of advertisement spillover effect will negatively impact the two firms’ distance results in the two firms getting more cluster. In the real world, we take the exclusive agent or geographic division management for example. The headquarter delegate the two firms to operate within a responsibility district by choosing their locations and advertisement expenditures. If the advertisement effect perfectly spill over to each other’s product quality, the two firms will choose the same location. The headquarter should create only one agent or subsidiary instead of two. Naturally, this result is
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concluded in the situation that the distribution cost is the only consideration. The other factors such as delivery time and service quality are beyond our discussion.
The results are summarized in Proposition 5.
Proposition 5. In the case of distribution cost is paid by firms, the following
variable(s) will positively impact the two firms’ clustering considerations respectively when we consider (1) the rate of distribution fee alone (β is fixed), (2) the advertisement spillover effect alone ( d is fixed), and (3) the rate of distribution fee and the advertisement spillover effect simultaneously.
From the comparative static analysis, the spillover effect on the equilibrium advertisement expenditure is given:
ˆ 0 9
ˆ) 1 ( ˆ
ˆ =− − ≤
∂
∂
ρ β β
ai
(64)
We observe Eq. (59), the advertisement expenditure is not the function of the rate of distribution fee. But in Eq. (64), the advertisement spillover effect will negatively impact the advertisement expenditure. The firm exerting advertisement expects to increase consumers’ awareness of its own product quality and then attracts consumers to purchase its product. If the degree of spillover effect is great, consumers can understand the rival’s product quality, thereby, the firm is hesitating to spend on advertisement in advance. On the other hand, when the degree of advertisement spillover effect is trivial, consumers can hardly understand the rival firm’s product quality. Hence, the firm is willing to spend on advertising its product quality for raising consumers’ awareness. The conclusion of the advertisement spillover effect on advertisement expenditure is shown in Proposition 6.
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Proposition 6. The rate of distribution fee does not impact the firm’s advertisement
expenditure. The advertisement spillover effect will negatively influence the firm’s advertisement expenditure.
3.4 Conclusion
In our model, we discuss the advertisement spillover effect on firms’ location choices from the perspectives of consumers and firms individually. In the first case, we consider consumers’ surplus while consumers need to spend the transportation costs for purchasing the product after receiving the informative advertisement. In the second case, we consider firms’ surplus while the firms have to expend distribution costs delivering the products to consumers’ places after consumers receive the advertisement and order the products.
We finds that no matter the transportation cost is paid by consumer or the distribution cost is paid by firm, the results of both cases show that while the firms compete with each other, each firm owns a half of market demand. And the two firms will locate symmetrically on the two halves of the line segment of unit distance.
Finally, the advertisement spillover effect will positively influence the clustering decisions of the two firms.
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