Strategic location decisions in a growing market

Regional Science and Urban Economics 31 (2001) 523–533 www.elsevier.nl / locate / econbase

Strategic location decisions in a growing market Dongsheng Zhou a , *, Ilan Vertinsky b a

City University of Hong Kong, Faculty of Business, 83 Tat Chee Avenue, Kowloon, Hong Kong b The University of British Columbia, Faculty of Commerce and Business Administration, 2053 Main Mall, Vancouver, B.C., Canada, V6 T 1 Z2 Received 31 July 1998; accepted 10 September 2000

Abstract The optimal location and time of entry are examined in a horizontally differentiated market using a simple duopoly model with sequential entry. Assuming high entry and relocation costs, we find that (1) the late entrant always locates in the periphery (maximum differentiation), (2) the first entrant will choose either a peripheral location or a central location (minimum differentiation) but not an intermediate location, (3) the choice of a central location is optimal when the effectiveness of entry deterrence is enhanced by low transportation costs, high interest rates, high fixed costs and a low market growth rate.  2001 Elsevier Science B.V. All rights reserved. Keywords: Spatial competition; Deterrence; Sequential entry; Fixed costs JEL classification: D43; L13; R3

1. Introduction Due to differences in information, technology and capital endowments, firms seldom enter the same regional markets simultaneously. Opening a new plant or relocation of an existing plant incurs large fixed costs. In making location and time of entry decisions, firms must consider not only immediate profits but also take account of future market growth and the impacts their location may have on rivals’ *Corresponding author. Tel.: 1852-2788-7981; fax: 1852-2788-9146. E-mail address: [email protected] (D. Zhou). 0166-0462 / 01 / $ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S0166-0462( 00 )00070-3

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entry in the future. Early entrants can choose their locations strategically so as to deter late entrants. While the importance of entry deterrence, in general, is well documented in the literature (see, e.g. Rao and Rutenberg (1979), Dixit (1980) and Brander and Eaton (1984)), there are only a few studies on entry deterrence in spatial competition. Aguirre et al. (1998) have explored spatial price discrimination as an entry deterrence strategy. Johnson and Parkman (1988) have examined empirically spatial monopoly and entry deterrence in the cement industry. The dynamic context of spatial competition adds new variables to the location choice problem. In addition to transportation costs and fixed cost structures, the impact of the dynamic elements, including interest rates and market growth rates and their interactions with other variables, must be considered. In this paper we investigate the effects that interest and market growth rates, transportation costs and fixed cost structures may have on spatial competition in general, and the use of entry deterrence strategies in particular. Spatial competition has attracted many researchers since Hotelling (1929) first published his most celebrated paper. The various modifications and extensions of the original model have introduced alternative cost structures (D’Aspremont et al., 1979; Economides, 1986), examined circular and two dimensional markets (Salop, 1979; Tabuchi, 1994; Veendorp and Majee, 1995), markets with non-uniform customer distributions (Tabuchi and Thisse, 1995) and markets with multiple firms (Eaton and Lipsey, 1975; De Palma et al., 1987). Except for a few papers (e.g. Prescott and Visscher (1977), Gupta (1992) and Neven (1987)), most studies on spatial competition assumed that firms choose locations simultaneously, that locations can be changed instantly and with little cost and that market demand is fixed, thus ignoring some important strategic dimensions of the location decision. In this paper, we consider firms entering a growing market sequentially, incurring significant fixed costs. We find that with the introduction of sequential entry and a growing market, an asymmetry between firms emerges resulting in behaviours which are different from those prescribed by the models of simultaneous entry to a fixed market. Using a quadratic transportation cost function as most models in this area do (e.g. Tabuchi and Thisse (1995) and Aguirre et al. (1998)), we find that the late entrant always chooses peripheral location (i.e. locating at the end point which is furthest from the first firm’s location). The first entrant chooses either a peripheral location (locating at the other end point) or a central location, but it will never choose any location in between. The choice of location depends on a simple function of the following factors: its rival’s fixed cost, the interest rate, the market growth rate and the transportation cost. The paper is organized as follows. In the next section, we describe the model. Section 3 gives the solution to the model using backward induction. There are two sub-sections in this section. In Section 3.1, we look at firm 2’s optimal location and time of entry decisions while in Section 3.2 we investigate the first firm’s optimal location and time of entry decisions. In Section 4 we present conclusions.

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2. The model We assume that there is a ‘linear market’ of length 1 along which consumers are uniformly distributed with density f(t) 5 a t (here the total population is also a t), where a is a positive constant and t is the time variable. Two profit-maximizing firms, indexed by 1 and 2, enter this market. The firms sell the same physical good and firm 1 enters the market earlier than firm 2.1 Each firm incurs a fixed cost Fi , i 5 1, 2, when it enters the market, and the marginal production costs of both firms are assumed to be constant (without loss of generality, let them be zero). The interest rate is r. We further assume, as most models in this area do, that each consumer has a perfectly inelastic demand for one unit of the good, and that consumers incur transportation costs which are a quadratic function of distance with a cost parameter c. Consumers are willing to buy at the smallest total cost to them so long as the latter does not exceed the gross surplus (S) they obtain from the goods. We assume S is sufficiently large (S . 2c) so that all consumers are always covered. We want to find the optimal location and time of entry decisions for both firms. The evolution of market structure is characterized by two periods. The first period starts at the entry of firm 1. Firm 1 enjoys monopoly profit during this period as the sole seller in the market. The entry of firm 2 marks the beginning of the second period where both firms compete for customers. To facilitate computation, the decision process is modelled as a three-stage game: in the first stage, firm 1 chooses time t 1 and location a to enter the market taking into account the effect of its decisions on the entry decisions of firm 2 and on the price competition that will take place once firm 2 enters. Firm 1 will choose its location and time of entry decisions so as to achieve the highest profits possible. Without loss of generality, we can assume that a # 1 / 2. In the second stage, after observing firm 1’s time of entry and location decisions, firm 2 chooses a time of entry t 2 and a location b taking into account the prices that will result as a consequence of competition with firm 1. In the third stage (the beginning of the second period), having observed the time of entry and location decisions, firm 1 and firm 2 announce simultaneously their prices. This three-stage game can also be understood in the following way. In the first two stages, firms play a Stackbelberg location / time of entry game with firm 1 as the leader, while in the last stage firms play a Nash price game. We want to find the subgame perfect equilibrium in this three-stage game. From time t 1 to t 2 , firm 1 is the monopoly and can charge a high monopoly price Pm . Because the market is covered, the optimal monopoly price should be given by the reservation price (S) minus the largest transportation cost to firm 1. 1

To focus on the strategic location decisions, here we treat the order of entry as exogenous (firm 1 may have some advantages in capital or technology over firm 2 so it enters the market earlier). If the order of entry is endogenous, then we have a much more complicated game involving predation (see Tirole (1988, Chapters 8–10)).

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Starting from time t 2 , firm 2 is also in the market and both firms compete for consumers. If we denote p1 , D1 as the price and demand for firm 1 while p2 , D2 as the price and demand for firm 2, then firm 1’s profit function can be obtained as follows: t2

`

E

p1 5 Pm a t e

2rt

dt 2 F1 e

2rt 1

Ep De

1

t1

1

1

2rt

dt.

(1)

t2

Firm 2’s profit function is: `

Ep D e

p2 5

2

2

2rt

dt 2 F2 e 2rt 2 .

(2)

t2

3. The solution We obtain the solution to the model by backward induction. We start with the third-stage price game. Following the steps in D’Aspremont et al. (1979), we obtain the optimal prices and demand for both firms as follows: p1 5 c(b 2 a)(2 1 b 1 a) / 3, p2 5 c(b 2 a)(4 2 b 2 a) / 3; D1 5 a t(2 1 b 1 a) / 6, D2 5 a t(4 2 b 2 a) / 6.

3.1. Firm 2’ s location and time of entry decisions Now we turn to the second-stage game. In this stage, firm 2 optimally chooses its time t 2 and location b to enter the market, given its observation of firm 1’s location and time of entry decisions in stage 1, and its expectation of the Nash price game in stage 3. The first-order conditions for firm 2’s maximization problem are dp2 / dt 2 5 0, dp2 / db 5 0. The optimal time of entry decision is t 2 5 (F2 r) /(A 2 ca ), where A 2 5 (b 2 a)(4 2 b 2 a)2 / 18. As for the optimal location decision, since dA 2 / db 5 (4 2 3b 1 a)(4 2 b 2 a) / 18 . 0, for b # 1 and a # 1 / 2, we have dp2 / db 5 et`2 ≠A 2 / ≠b c a t e 2rt dt . 0. So, the optimal location choice for firm 2 is to locate at the furthest point from firm 1. Since b # 1, the optimal location is b 5 1. Substituting b 5 1 into the price and demand functions, we get (1 2 a)(3 1 a) (1 2 a)(3 2 a) 31a 32a p1 5 ]]]]c, p2 5 ]]]]c; D1 5 ]]a t, D2 5 ]]a t 3 3 6 6 2 F2 r (1 2 a)(3 2 a) t 2 5 ]], A 2 5 ]]]]] (3) A 2 ca 18

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527

The second-order conditions for this maximization are satisfied, dp 22 / dt 22 5 2 A 2 c a e 2rt 2 , 0, dp 22 / db 2 5 (28 1 3b 1 a) / 9 , 0. From (3), we have the following lemma: Lemma 1. (A) Regardless of the location and time of entry decisions of firm 1, firm 2 always chooses a location furthest away from firm 1. Firm 1’s market share falls into the interval [1 / 2, 7 / 12] while firm 2’s market share belongs to [5 / 12, 1 / 2]. Firm 1’s price falls into the range of [7c / 12, c] while firm 2’s price belongs to [5c / 12, c]. Firm 1 (the first entrant) has advantages in both market share and price. (B) Firm 2’s optimal time of entry increases with F2 , r and a. It decreases with a and c. In other words, the larger firm 2’s fixed cost, the higher the interest rate and the closer firm 1 locates to the middle of the market (closer to firm 2’s eventual location), the later firm 2 will enter the market. The larger the market growth rate, the larger the transportation cost, the earlier firm 2 will enter the market. The intuition behind this is clear. Large fixed costs and interest rate serve as natural barriers to the entry of firm 2 and thus delay firm 2’s time of entry. A large a means firm 1 locates closer to the eventual location of firm 2, this will certainly increase the competition and reduce the profit of firm 2. As a result, firm 2 will not enter the market until the demand is large enough (thus the profit is large enough) to cover its interest on the fixed cost. On the other hand, a high market growth rate leads to large demands in a short period of time, while high transportation costs allows firms to charge higher prices. Both factors push firm 2 to enter the market earlier. Although firm 2’s optimal time of entry depends on its fixed cost, the interest rate, the transportation cost, the market growth rate as well as firm 1’s location, its optimal location decision is independent of any of these factors and is always at the furthest position b 5 1. Substituting the optimal solutions into (2), we have the maximal profit function of firm 2: ca A 2 2F r 2 / c a A 2 P2 5 ]] e 2 . r2

(4)

Note that firm 2’s profit is an increasing function of the transportation cost c. This is because: (1) a high transportation cost induces firm 2 to enter the market earlier (smaller t 2 ); (2) a high transportation cost leads to high duopoly prices (from results in (3)). On the other hand, since A 2 is a decreasing function of a, from (4) we conclude that firm 2’s profit decreases with a. The intuition is clear. Since a # 1 / 2, the larger the a, the closer firm 1 locates to the middle of the market. As a result, the competition will increase and firm 2’s profit will be reduced.

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3.2. Firm 1’ s location and time of entry decisions Firm 1 can correctly anticipate the effects of its decisions on the entry decisions of firm 2 and the third-stage price competition. It will take into account the consequences of the second and third stage game, i.e. substituting the results in (3) into its profit function (1). Firm 1’s optimization problem is: P1 5max p1 . Its t 1 ,a corresponding first-order conditions are: dp1 / dt 1 5 0, dp1 / da 5 0. Solving the first-order condition for the time variable, we have: F1 r F1 r t 1 5 ]] 5 ]]]. Pm a (S 2 c)a

(5)

We note from (5) that (i) the higher the consumer reservation price (S), the earlier firm 1 will enter the market (smaller t 1 ); (ii) firm 1’s optimal time of entry is an increasing function of its fixed cost, the interest rate, and the transportation cost. It is a decreasing function of the market growth rate. The first-order condition for the location variable a is: dp ≠p ≠t ≠p ≠A ]1 5 ]1 ]2 1 ]]1 ]]1 da ≠t 2 ≠a ≠A 1 ≠a

F

G

(6)

D

(7)

dt 2 (rt 2 1 1)c dA 1 2rt 2 ]] e 5 a sS 2 c 2 A 1 cdt 2 ] 1 ]]] 50 da da r2 where A 1 c 5 p1 d 1 . Using (3), Eq. (6) becomes:

S

2 F 22 r 4 dA 2 dA 1 2 2 F2 r ]] ]] 2 ]] S 2 c 2 A c 1 A c 1 A 2 c ]] 5 0. s d 2 1 2 da a da a

If we define k 2 5 (F2 r 2 ) /(a c), that is k 2 is a function of the fixed cost, the interest rate, the growth rate and the transportation cost, then Eq. (7) can be simplified as follows: dA 2 dA 1 2 2 2 k (S 2 c 2 A 1 c)]] 1 A 2 c(k 1 A 2 )]] 5 0. da da

(8)

Lemma 2. Firm 1’s profit function is a convex function of its location variable a, where 0 # a # 0.5. Proof. Please see Appendix A. The above lemma suggests that, for any a satisfying the first-order condition (8), a is always a local minimum rather than a local maximum. In other words, the optimal location for firm 1 is either a 5 0, which means locating at the other extreme point, or at a 5 0.5, which means locating at the middle of the market.

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Proposition 1. Firm 1’s optimal location strategy is either to locate at the periphery (a 5 0) or locate at the centre (a 5 0.5). It will never choose any location in between. Substituting previous results into (1), we can derive the profit function of firm 1 as follows:

F

S

D

k2 a P1 5 ]2 (S 2 c)e 2k 1 c / (S2c) 2 (S 2 c 2 A 1 c) ] 1 1 e 2k 2 / A 2 A2 r

G

(9)

2

where k 1 is defined in a similar fashion to k 2 as k 1 5 (F1 r ) /(a c). Obviously, firm 1’s profit will depend upon its location and its fixed cost. The reason why firm 1’s profit also depends on firm 2’s fixed cost can be explained as follows: firm 2’s fixed cost affects its time of entry as it will enter the market only when its revenue can cover the interest paid on its fixed cost. The entry of firm 2 will in turn affect the market structure (firms’ prices and demands) and thus firm 1’s profit. Since it is optimal for firm 1 to locate only at one of the two points (either at point a 5 0 or at a 5 0.5), we need only check firm 1’s profits at these two points to find the optimal one. At location a 5 0, we have A 1 5 A 2 5 0.5. Firm 1’s corresponding profit function is: a P 01 5 ]2 f(S 2 c)e 2k 1 c / (S2c) 2 (2k 2 1 1)(S 2 1.5c)e 22k 2g. r At location a 5 0.5, A 1 5 49 / 144, A 2 5 25 / 144. Firm 1’s profit function becomes:

F

S

a 144 P 11 / 2 5 ]2 (S 2 c)e 2k 1 c / (S2c) 2 ]k 2 1 1 25 r

193 DSS 2 ] cDe 144

G.

2144 / 25 k 2

The difference between locating at a 5 0 and a 5 0.5, in terms of firm 1’s profit, is: 144 193 FS] k 1 1DSS 2 ]cDe 25 144 G. 2s2k 1 1dsS 2 1.5cde

a P 01 2 P 11 / 2 5 ]2 r

2144 / 25 k 2

2

22k 2

2

(10)

From (10), it is obvious that firm 1’s optimal location only depends on consumers’ reservation price S and the level of ‘natural’ entry barriers k 2 (where k 2 5 (F2 r 2 ) /(a c)), and not on its own fixed cost. We can show that, for any given S, there is a critical level of ‘natural’ entry barriers k 2* above which central location is most profitable. Clearly, the higher the reservation price S, the lower this critical level k 2* will be. Similarly, if the reservation price S exceeds some critical value S*, given a level of ‘natural’ entry barriers k 2 , firm 1 will locate in

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D. Zhou, I. Vertinsky / Regional Science and Urban Economics 31 (2001) 523 – 533

the centre. The higher the ‘natural’ entry barriers k 2 , the lower this critical reservation price S* will be.

4. Conclusions The introduction of dynamic aspects to spatial competition has highlighted the central role that entry deterrence may play in the location decision of the first entrant (the second entrant always chooses a peripheral location to minimize competition). We have shown that the first entrant will choose either to locate so as to maximize deterrence (central location) or to minimize future competition (locating in the periphery so as to reduce overlap with its future rival). The particular choice of location depends on several factors. Clearly a higher consumer reservation price means that the delay in rival entry obtained from central location is more profitable. A higher interest rate reduces the present value of future losses resulting from a more intense competition, making deterrence less expensive. It also delays the time of entry of the rival. Lower transportation costs enhances monopoly profits thus making deterrence more attractive. It also means that location is less material in the future (competition will be high anyway) while at present it magnifies the power of the deterrence of the first entrant. Lower market growth means lower profits in the future delaying entry of the rival and thus reducing the present value of the costs of deterrence.

Appendix A In this appendix, we prove Lemma 2, i.e. firm 1’s profit function is convex in a. Developing the second-order condition of the location variable (using (8)), we have: ≠ 2 p1 6a 2 14 2 2 (3 2 a)(5 2 3a)(3 1 a)(3a 1 1) ]] 1 k 2 c ]]]]]]]]] 2 5 k 2 (S 2 c 2 A 1 c)]]] 18 ≠a 18 2 8a(3a 2 1 2a 2 13)(3 2 a)3 (1 2 a)c 2 ]]]]]]]]]] (k 2 1 A 2 ) 18 3 (3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4

F

1 6a 2 14 8a(3a 2 1 2a 2 13) ]] ]]] ]]]]]]] 5 m2 n 3 2 a 5 2 3a (3 1 a)(3a 1 1)(1 2 a) (3 2 a)(5 2 3a)(3 1 a)(3a 1 1)c 1 k 22 ]]]]]]]]] 18 2

G

D. Zhou, I. Vertinsky / Regional Science and Urban Economics 31 (2001) 523 – 533

(3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4

531

(A.1)

where (3 2 a) (5 2 3a) (3 1 a) (1 1 3a) m 5 k 22 (S 2 c 2 A 1 c) ]]]]], n 5 A 22 c(k 2 1 A 2 ) ]]]]]. 18 18 From (8), we get m 5 n. Since, for a # 0.5, 5 2 3a $ 3 1 a, 3a 2 1 2a 2 13 , 0, we have,

F

2 ≠ 2 p1 m 6a 2 14 8a(3a 1 2a 2 13) ]] ]] ]]] ]]]]]]] 5 2 3 2 a 5 2 3a (3 1 a)(3a 1 1)(1 2 a) ≠a 2 (3 2 a)(5 2 3a)(3 1 a)(3a 1 1)c 1 k 22 ]]]]]]]]] 18 2

G

(3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4 m(6a 2 14) . ]]]]][1 2 z(a)] (3 2 a)(5 2 3a) (3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4

(A.2)

where 8a (3a 2 1 2a 2 13) z(a) 5 ]]]]]]]] . 2 18a 2 1 54a 2 2 22a 2 14 We separate the proof into two cases: case 1: 0.161 # a # 0.5; case 2: 0 # a , 0.161. Case 1. In this case, we can easily show that 1 2 z(a) , 0. Since a , 1, we have ≠ 2 p1 / ≠a 2 . 0. Case 2. 0 # a , 0.161. From (8), we have k 22 (S 2 c 2 A 1 c) 2 g(a)c(k 2 1 A 2 ) 5 0, where (3 2 a)3 (1 2 a)2 (3 1 a) (3a 1 1) g(a) 5 ]]]]]]]]] . 18 2 (5 2 3a) Since gmax (a) 5 g(0.1050) 5 0.0523, S . 2c, A 1 # 0.5, A 2 # 0.5, we have 2

k 2 (S 2 c 2 0.5c) 2 gmax (a)c(k 2 1 0.5) , 0. Solving the above inequality, we have k 2 , 0.2869. So,

(A.3)

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≠ 2 p1 m(6a 2 14) ]] 2 . ]]]]][1 2 z(a)] (3 2 a)(5 2 3a) ≠a (3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4 k 22 (S 2 c 2 A 1 c)(6a 2 14) ]]]]]]] 5 [1 2 z(a)] 18 (3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4 g(a)(k 2 1 A 2 )(6a 2 14) 5 ]]]]]]][1 2 z(a)] 18 (3 2 a)5 (1 2 a)2 (3 1 a)(3a 1 1)(5 2 3a)c 1 ]]]]]]]]]]]] 18 4 (3 2 a)3 (1 2 a)2 (1 1 a)(3a 1 1)c ]]]]]]]]] 18 3 3 [18(0.2869 1 A 2 )(6a 2 14)(1 2 z(a)) 1 (3 2 a)2 (5 2 3a)2 ] . 0 (A.4) The last inequality is obtained by straightforward calculation. Combining the results in both cases, we conclude that for any a (0 # a # 0.5) that satisfies the first order condition (8), a is always a local minimum rather than a local maximum.

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