Inventory costs and the optimal spacing of retail stores

Economics Letters 58 (1998) 127–131

Inventory costs and the optimal spacing of retail stores Ralph M. Braid* Department of Economics, Wayne State University, Detroit, MI 48202, USA Received 3 June 1997; accepted 3 July 1997

Abstract This paper examines the socially optimal spacing of stores, assuming an explicit inventory model of costs. In the basic model, there is equality between the capacity costs of stores, the fixed costs of deliveries, and the travel costs of consumers.  1998 Elsevier Science S.A. Keywords: Inventory costs; Spatial model; Retail stores JEL classification: R32; L81

1. Introduction In one standard model of the socially optimal spacing of retail stores, consumers are spread evenly at linear density D along an infinite roadway (or circular roadway). Each consumer has a completely inelastic demand for a retail good, and has travel costs of t per unit of one-way travel distance. Each store has fixed costs, F, and constant marginal costs, c. Stores are spaced evenly at separation R along the infinite roadway. Under these assumptions, the socially optimal spacing between stores is given by the simple square root formula, R* 5 2(F /tD)1 / 2 , a result which is common in the literature.1 This note derives simple formulae for the optimal spacing of retail stores when an explicit inventory model is used in place of the assumption (made above) that each store has fixed costs F. Section 2 presents the assumptions and analysis of the basic model. Sections 3 and 4 vary the model by considering two dimensions and quadratic transportation costs, respectively. 2. Assumptions of the model I make the following assumptions. Store i is located at position Ri along an infinite roadway, where *Tel.: (1-313) 577-2540; fax: (1-313) 577-0149. 1 See for example (34) of Salop (1979), page 1052 of Capozza and Van Order (1980), and page 335 of Vickrey (1964). My variables R, F, t, and D are equivalent to Salop’s L /n, F, c, and 1 (where n is the number of brands around a circle of circumference L); Capozza and Van Order’s 2R, k, t, and 1 (my R* corresponds to 2R 0 in their paper); and Vickrey’s L /N, C, t, and Q /L. 0165-1765 / 98 / $19.00  1998 Elsevier Science S.A. All rights reserved. PII S0165-1765( 97 )00230-9

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R is endogenously chosen by a government planner, and i assumes all integer values.2 Consumer density along the infinite roadway is uniformly equal to D. Each consumer has a completely inelastic demand to buy 1 unit of a desired good per month.3 Each store serves the consumers closest to it, so each store sells a quantity DR of the retail good each month. The trips of different consumers are distributed uniformly over the month. I assume the following costs. First, the capacity cost of each store is bS (after amortization of capital costs on a monthly basis), where S is the size (capacity) of the store. Second, each delivery of goods to a store has fixed costs, f, and constant marginal costs, c. Third, each consumer has travel costs equal to t multiplied by one-way travel distance.4 The delivery costs associated with each store (per month) can easily be calculated. Let n be the number of uniformly spaced deliveries (per month). The amount delivered in each shipment is clearly (1 /n)DR. Units of measurement may be chosen such that S 5 (1 /n)DR,

(1)

so that the inventory of the retail good varies from S (just after a shipment) to 0 (just before the next shipment). The delivery costs (per month) for a store are clearly fn1cDR. Similarly, the total travel costs of consumers (per month) are easily calculated. The maximum one-way travel distance is R / 2, and the average one-way travel distance is R / 4. Since travel costs are t per unit of one-way travel distance, the total travel costs of all DR consumers going to a store each month are (1 / 4)tDR 2 . Combining the various terms above, the total social costs (TSC) associated with each store (per month) are given by TSC 5 bS 1 fn 1 cDR 1 (1 / 4)tDR 2 .

(2)

If R is regarded as fixed, then the third and fourth terms on the right hand side are fixed, so the minimization of Eq. (2) subject to Eq. (1) represents a very simple inventory model of the Baumol (1952); Tobin (1956) type which can be solved for the optimal value of n. However, R is variable here, so the appropriate objective function is the minimization of per-capita social costs (PCSC). Each store serves DR consumers. Dividing Eq. (2) by DR, and substituting Eq. (1), it is seen that PCSC(n, R) 5 (1 /n)b 1 fn(DR)21 1 c 1 (1 / 4)tR,

(3)

where PCSC is a function of both n and R, as indicated. Note that b, f, D, c, and t are constants. Minimizing Eq. (3) with respect to n and R, and then using Eq. (1), it is possible to establish the following results. Proposition 1. The optimal values of the endogenous variables are 2

Alternatively, the roadway is long and circular. Instead of 1 unit per month, the consumer could consume any other fixed amount, q, per month, but instead of including q as another exogenous parameter, it is sufficient to interpret D as the product of q and consumer density along the line. 4 The number of shopping trips by each consumer is fixed. If it is fixed at more than one trip (per month), this is factored into the definition of t. 3

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n* 5 (4b 2 Dt 21 f 21 )1 / 3 ,

(4)

R* 5 2(2bfD 21 t 22 )1 / 3 ,

(5)

S* 5 (4f 2 Db 21 t 21 )1 / 3 .

(6)

To establish Eqs. (4) and (5), simply note that these values of (n, R) represent the only solution to the first-order conditions for (n, R) in the positive quadrant, and that the second-order conditions for a local minimum are satisfied at this point.5 Substituting Eqs. (4) and (5) into Eq. (1) yields Eq. (6). Based on Eqs. (4)–(6), we have simple cube root formulae for the optimal values of the number of monthly deliveries to each store (n*), the market length served by each store (R*), and the size (maximum inventory capacity) of each store (S*). From Eqs. (4)–(6), it is possible to examine the comparative statics of changes in b, t, D, or f on the optimal values of n, R, and S. The results are not particularly surprising. It is worth noting that the elasticities of R* with respect to D and t are 2(1 / 3) and 2(2 / 3), respectively, in contrast to the standard model mentioned in Section 1, in which both elasticities are 2(1 / 2). It is interesting to compare the first, second, and fourth terms on the right-hand side of Eq. (2), when evaluated using Eqs. (4)–(6). It is found that bS* 5 fn* 5 (1 / 4)tD(R*)2 5 (4f 2 b 2 Dt 21 )1 / 3 .

(7)

This leads to the following proposition. Proposition 2. When the endogenous variables are chosen optimally, the capacity cost of each store, the fixed component of the delivery costs for each store, and the travel costs of the consumers going to each store are all equal to each other.

3. Two dimensions In the real world, consumers tend to be spread over a two-dimensional plane rather than along a one-dimensional line (or around a circle). One possible set of assumptions is as follows. Consumers are spread uniformly at population density D over an infinite plane. Stores are located in a square grid pattern, such that store (i, j) is located at (Ri, Rj), where R is endogenous, and i and j assume all integer values. A dense square grid of roadways allows travel only in the northeast–southwest and northwest–southeast directions (a block metric, as described by Eaton and Lipsey (1980); Economides (1986), and other authors). The travel costs of a consumer are t multiplied by one-way travel distance (along the roadway network) to the nearest store. Braid (1991) uses these assumptions in parts of his paper, and points out (in agreement with earlier statements in the literature) that the arrangement of stores described above, in which the market area of each store is a square which is rotated at a 45-degree angle relative to the roadway network (a 5

The function defined by Eq. (3) is not convex for all values of (n, R) in the positive quadrant. Thus, the argument must be completed by noting that Eq. (3) is infinite if n50 or R50, or in the limit that n or R is infinite, so that the global minimum of Eq. (3) in the positive quadrant must be in the interior of this quadrant.

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‘‘diamond pattern’’), represents the optimal set of store locations, if R is chosen optimally. Sections 6 and 7 of Braid show that if each store has fixed costs, F, then the optimal value of R is ] R*5(3Œ2)1 / 3 (F /tD)1 / 3 . In the remainder of this section, I will make the assumptions of the inventory-based model of Section 2, except that the travel cost assumptions are the same as described two paragraphs above. Section 6 of Braid (1991) shows that the average travel distance of a consumer to the nearest store is ] (Œ2 / 3)R. The number of consumers going to each store is clearly DR 2 . Thus the modified versions of Eqs. (1)–(3) that are relevant here are S 5 (1 /n)DR 2 ,

(8)

] TSC 5 bS 1 fn 1 cDR 2 1 (Œ2 / 3)tDR 3 ,

(9)

] PCSC(n, R) 5 (1 /n)b 1 fn(DR 2 )21 1 c 1 (Œ2 / 3)tR.

(10)

Minimizing Eq. (10) with respect to n and R, and then using Eq. (8), it is possible to establish the following results. Proposition 3. The optimal values of the endogenous variables are n* 5 (18b 3 Dt 22 f 21 )1 / 4 ,

(11)

R* 5 (18bfD 21 t 22 )1 / 4 ,

(12)

S* 5 (18f 3 Db 21 t 22 )1 / 4 .

(13)

The argument is exactly the same as the one following Proposition 1. Based on Eqs. (11)–(13), we have simple fourth root formulae for the optimal values of n, R, and S. As in Section 2, the comparative static results that can be derived from Eqs. (11)–(13) are not particularly surprising. The elasticities of R* with respect to D and t are 2(1 / 4) and 2(1 / 2), respectively, in contrast to the model mentioned above with simple fixed costs, F, in which both elasticities are 2(1 / 3). Based on Eqs. (11)–(13), it is possible to compare different kinds of costs. It is found that ] bS* 5 fn* 5 (1 / 2)(Œ2 / 3)tD(R*)3 5 (18b 3 f 3 Dt 22 )1 / 4 .

(14)

This leads to the following proposition. Proposition 4. When the endogenous variables are chosen optimally, the capacity cost of each store and the fixed component of the delivery costs for each store are equal to each other, and each of these is equal to half of the travel costs of the consumers going to each store.

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4. Quadratic transportation costs Quadratic transportation costs have been used in an infinite line model by Eaton and Wooders (1985), and in a circular model by Economides (1989). Quadratic transportation costs make more sense for spatial product differentiation (goods which are at different locations in product characteristics space) than for spatial competition between retail stores (stores which are at different locations in real space, selling a homogeneous product). Nevertheless, it is interesting to consider the latter interpretation, to see how quadratic transportation costs affect the results of Section 2. Suppose first that each store has fixed costs, F, and that the travel costs of a consumer are a multiplied by the square of one-way travel distance to the nearest store. Then, based on results in Eaton and Wooders (1985) and Economides (1989),6 the optimal spacing between stores is given by the simple cube root formula, R*5(6F /a D)1 / 3 . By contrast, I will now make the same assumptions as in the inventory-based model of Section 2, except that the travel costs of a consumer are as defined immediately above. The maximum value of the square of the one-way travel distance for any consumer is (1 / 4)R 2 , and the average value is (1 / 12)R 2 . Thus the total travel costs of all DR consumers going to a store per month are (1 / 12)a DR 3 . The final terms on the right-hand sides of Eqs. (2) and (3) thus change from (1 / 4)tDR 2 and (1 / 4)tR to (1 / 12)a DR 3 and (1 / 12)a R 2 , respectively. The first three terms on the right-hand side of Eq. (3) are unchanged. Maximizing the modified version of Eq. (3), it is seen that the optimal value of R is R* 5 (36bfD 21 a 22 )1 / 5 .

(15)

Thus we get a fifth root formula instead of the cube root formula of standard quadratic transportation cost models. The elasticities of R* with respect to D and a are 2(1 / 5) and 2(2 / 5), respectively, in contrast to the standard model, in which both elasticities are 2(1 / 3).

References Baumol, W., 1952. The transactions demand for cash. Quarterly Journal of Economics 67, 545–556. Braid, R.M., 1991. Two-dimensional Bertrand competition: Block metric, Euclidean metric, and waves of entry. Journal of Regional Science 31, 35–48. Capozza, D.R., Van Order, R., 1980. Unique equilibria, pure profits, and efficiency in location models. American Economic Review 70, 1046–1053. Eaton, B.C., Lipsey, R.G., 1980. The block metric and the law of market areas. Journal of Urban Economics 7, 337–347. Eaton, B.C., Wooders, M.H., 1985. Sophisticated entry in a model of spatial competition. Rand Journal of Economics 16, 282–297. Economides, N.S., 1986. Nash equilibrium in duopoly with products defined by two characteristics. Rand Journal of Economics 17, 431–439. Economides, N.S., 1989. Symmetric equilibrium existence and optimality in differentiated product markets. Journal of Economic Theory 47, 178–194. Salop, S.C., 1979. Monopolistic competition with outside goods. Bell Journal of Economics 10, 141–156. Tobin, J., 1956. The interest elasticity of the transactions demand for cash. Review of Economics and Statistics 38, 241–247. Vickrey, W.S., 1964. Microstatics. Harcourt, Brace and World, New York. 6

See (22) of Eaton and Wooders (1985) and Theorem 7 of Economides (1989).