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Staking Out the Firm’s Market

Price and the Geometry of Competition
  • John R. Miron
Chapter

Abstract

A firm typically has a market area: a geographic area wherein the firm dominates and does much of its sales. Within its market area, a firm is able to affect the price received to the extent that the effective prices from other suppliers make them uncompetitive. What determines the size and shape of a market area? How might the presence of a market area affect firm behavior? Model 8A considers market area when a firm and its competitor sell at the same f.o.b. price. Model 8B looks at market area when firms have different f.o.b. prices. In 8C, the firm sets a price that maximizes its profit assuming that competitors do not react. Model 8D studies how the firm’s market area boundary adjusts to capacity constraints. Model 8E shows how the firm’s market area boundary varies when it sells a different but perfectly substitutable good. Models 8F, 8G, and 8H introduce differences among consumers as well as imperfectly substitutable goods.  Chapter 1 argues that there are important linkages between prices and localization. I illustrated that idea in Model 2D wherein price and localization were joint outcomes for the monopolist. However, since that chapter, the models in this book have been concerned only with how prices affect localization. This chapter considers how a firm sets its price in response to the proximity of competitors and the prices they set. This chapter explores how, as a consequence, localization and price are jointly determined.

Keywords

Shipping Cost Demand Curve Market Area Budget Share Destination Choice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A firm typically has a market area: a geographic area wherein the firm dominates and does much of its sales. Within its market area, a firm is able to affect the price received to the extent that the effective prices from other suppliers make them uncompetitive. What determines the size and shape of a market area? How might the presence of a market area affect firm behavior? Model 8A considers market area when a firm and its competitor sell at the same f.o.b. price. Model 8B looks at market area when firms have different f.o.b. prices. In 8C, the firm sets a price that maximizes its profit assuming that competitors do not react. Model 8D studies how the firm’s market area boundary adjusts to capacity constraints. Model 8E shows how the firm’s market area boundary varies when it sells a different but perfectly substitutable good. Models 8F, 8G, and 8H introduce differences among consumers as well as imperfectly substitutable goods.  Chapter 1 argues that there are important linkages between prices and localization. I illustrated that idea in Model 2D wherein price and localization were joint outcomes for the monopolist. However, since that chapter, the models in this book have been concerned only with how prices affect localization. This chapter considers how a firm sets its price in response to the proximity of competitors and the prices they set. This chapter explores how, as a consequence, localization and price are jointly determined.

8.1 The Market Area Problem

My focus in this chapter is on the geometry of market areas in the presence of competition and its relationship to the linkage between price and localization.1 There is a long history of research in competitive location theory from the classical work of Fetter through contemporary work on destination choice modeling.2 As we have already discussed, in Economics, a market is a locus where buyers and sellers intersect. Economists normally—and perhaps wisely—stop here. Those from other disciplines might prefer to augment this definition with additional concepts: e.g., including institutions and mechanisms operating at a site or portal through which offerings of a commodity can be viewed and information about prices is readily available. As has been shown in the past few chapters, a market may also have a geographic area insofar as buyers and/or sellers come from different places to participate in this market, or as a commodity gets shipped for sale elsewhere.

The concept of a market area has been widely used: e.g., in Development Studies, Economics, Education Studies, Geography, Health Studies, Housing Studies, Management Science, Marketing and Retailing, Public Finance, Planning, Real Estate, Regional Science, Transportation, and Urban Studies.3 To exemplify, this includes studies that focus on (1) the customer area served by a firm, school, hospital, or other institution or facility; (2) the area from within which an employer draws its workers; and (3) the area within which dwellings or properties are thought to form a housing or property market or submarket.4 In keeping with this book’s focus on the firm, I emphasize here models that look at the customer area served by a firm.

I start with the notion of a market area for a commodity produced by a monopolist and sold at an f.o.b. price. The firm is limited only by its range, that is, the market area as geographic area outside of which a prohibitive unit shipping cost means that the consumer would sooner go without the commodity altogether (whatever the reasons). Here, I retain the assumption of a linear individual inverse demand curve with its implication of a price above which consumers no longer demand the good. I then consider the monopolist who uses discriminatory pricing to extract more profit and show that here too market area is limited; it is not profitable to sell the commodity beyond some maximum distance.

I then consider the market area for a firm—be it producer, supplier, or retailer—when there are other vendors nearby selling a good that is the same or similar (substitutable). In such cases, the market area of the firm may be further limited by the presence of competitors or other alternative establishments. Looking at where the firm locates and the price that it sets in response to the prices and locations of competitors gives us new insights into the linkage between the price of a commodity and localization. The literature in this part of location theory has a history that traces back to Hotelling (1929) and Lösch (1954) and that has benefited from advances in game theory.5

Models of the market area for a firm—in the presence of competitors—come in two varieties. In one variety (as in Models 8A through 8C below), the model assumes we know precisely which customers a firm will serve; in a simple case, the firm captures all potential customers within its market area and none outside. Where, for example, competitors price f.o.b. and lowest effective price is the sole determinant of customer choice, the firm’s market area is the geometric shape within which no other competitor is able to sell their product. In the other variety (as in Models 8D through 8G), the customer chooses a firm (supplier) based at least in part on considerations outside the realm of the model. Taking a model in which the only explanatory variable is the firm’s f.o.b. price, Hotelling (1929, p. 41) argues that (1) in spite of moderate differences of price, some purchasers of a commodity buy from one seller, some from another, and (2) If the supplier of a good gradually increases his price while his rivals keep theirs fixed, the diminution in his volume of sales will take place continuously rather than abruptly.

In either variety, a range of models of competitive behavior is possible here. In a simple model, we might assume a firm and its competitors—with geographic locations given—each offering a given good (or variety of goods) at a given f.o.b. price and a given level of service. Customers who bear the unit shipping cost then choose from among the firms. In more sophisticated models, we might imagine the firm and its competitors adjusting their prices, geographic locations, the kinds of goods they offer for sale, and the quality of service they provide. Hotelling (1929, p. 44) argues that when a seller increases price he will only gradually lose business to his rivals. Some customers will still prefer to trade with him because he is more convenient, provides better service, or sells goods they desire. In Hotelling’s view, such customers make every vendor a monopolist within a limited geographic area, and there is no monopoly that is not confined to a limited geographic area. In any of these models, imagine a spatial equilibrium exists in which no firm has an incentive to change its behavior. In the presence of unit shipping costs, effective price might vary across the landscape; so too might localization (the density of competitors nearby) as well as the kind of goods and level of service locally. In all of these models, the firm is competing in a world of differentiated goods, even if the differentiation is solely on the basis of unit shipping cost. Such ideas help us think about what might be causing Hotelling’s stability of competition.

To exemplify stability, Hotelling uses a bounded linear market along which customers are spread at a given uniform density. Two competitors at given locations along this line compete for customers by each setting a f.o.b. price. In Hotelling’s model, there was no inherent reason for customers to patronize one particular store other than a lower effective price. However, in a linear market each competitor has a protected flank: the customers between him and the end of the line away from his competitor. Unless the competitor’s f.o.b. price is so low as to eliminate our firm’s market completely, the firm can always count on its protected flank. Critics might say that, in practice, bounded linear markets are uncommon in an unbounded rectangular plane—again assuming a uniform density of customers everywhere—the notion of a protected flank vanishes. However, I think that Hotelling would simply argue here that a linear market was just a representation that illustrates a source of stability.

The purpose of this chapter is to give readers the flavor of this line of research through a sequence of models that emphasize aspects of the problem. Sometimes, these models are best cast in the context of a firm purchasing inputs from a supplier. Other times, these models are best cast in terms of a customer choosing a retail outlet (store) from which to purchase a commodity. Models 8A through 8H are presented in sequence.
8A

Market areas when competitors charge the same f.o.b. price, shipping cost is everywhere proportional to distance, customers are identical, uniformly spread across geographic space, and purchase where shipping cost is lowest (Thiessen Polygons).

8B

Market areas when competitors charge different f.o.b. prices for their commodity. Other assumptions remain the same as Model 8A.

8C

Market areas when there is price competition among firms selling the same product.

8D

Market areas when each establishment has a different capacity to supply the commodity.

8E

Market areas when competitors supply different but perfectly substitutable commodities. Other assumptions remain the same as in Model 8B.

8F

Market areas when customers are of two different types. This builds on Model 8E.

8G

Market areas when competitors supply unrelated commodities (zero cross price elasticity).

Why these seven models? These seven are not exhaustive but do illustrate factors that shape market area, a kind of spatial price equilibrium, and localization.6

Following these seven models, I briefly consider market areas when destination choice can be thought to be subject to uncertainty (discrete choice modeling). Each place can be thought of as a store selling the commodity at an f.o.b. price. In simple versions of this model, each store sells the same commodity and is similar in other respects such as store attractiveness, purchase warranty, and level of service. The stores are each just a place from which the commodity can be obtained. The only relevant consideration is the cost of purchasing a unit of the commodity: f.o.b. price plus any shipping cost. More realistic models can be envisaged in which some of these assumptions are relaxed. Ignored here are shopping safaris where a customer purchases commodities while traveling from place to place.7 Ignored also is the practice of bulk buying where the customer acquires a substantial inventory (i.e., stocks up on purchases) so as to minimize the effect of trip cost. Instead, the customer is presumed to incur a fixed shipping cost for each unit of the commodity consumed. Finally, the consumer (or analyst) may well have imperfect information, that is, does not know the price and availability of the commodity at all places.

In this chapter, as elsewhere so far in this book, I take the locations of consumers as given. I do not ask why consumers have come to be where they are. Nor do I ask why they might not respond to differences in price by changing their location. Such considerations are left to later chapters in this book. I also assume that the firm, its customers, and competitors share a common fiat market economy. As in  Chapter 2 and  Chapter 3, price differences among places open up the question of the happiness of customers. Other things being equal, we might expect customers at places where the effective price is higher to want to relocate to places where the effective price is lower. In this chapter, I continue to ignore such inclinations.

8.2 Range and Geographic Size of Market

To start here, suppose a monopolist is located on a rectangular plane with a uniform density of customers (g customers per unit area). Assume the monopolist sells its commodities at an f.o.b. price (P), that customers bear the shipping cost, that the shipping rate per kilometer is a constant (s), and that all distances are shortest paths (i.e., as the crow flies). This is different from  Chapter 2 where I had assumed that the firm used discriminatory pricing; by setting a single f.o.b. price, the firm earns less profit than is possible with discriminatory pricing. As in earlier chapters, I assume each customer has a linear inverse demand curve. For a customer at distance x from the monopolist, the effective price is \(P+sx\), and the individual demand curve is given by (8.1.1): see Table 8.1. Here total trip cost for a customer is, therefore, proportional to consumption. There is no notion here of the consumer having an inventory of the commodity at home and making a tradeoff between trip frequency (trip cost) and household inventory (inventory cost).8
Table 8.1

The monopolist’s market in two-dimensional space using f.o.b. pricing

Individual linear inverse demand curve for customer at x km

 

\(P+sx=\alpha-\beta q\)

(8.1.1)

Range of commodity (since \(q \ge 0\))

 

\(X=(\alpha-P)/s\)

(8.1.2)

Aggregate demand for the firm’s product

 

\(Q=\int_0^x {2\pi gxdx}\)

(8.1.3)

Aggregate demand after accounting for (8.1.1) and integrating

 

\(Q=2\pi g[ {( {( {\alpha-P} )/( {2\beta } )} )X^2-( {1/( {3\beta } )} )sX^3 } ]\)

(8.1.4)

Aggregate demand after accounting for (8.1.2)

 

\(Q=( {1/3} )\pi g( {\alpha-P} )^3 / (\beta s^2)\)

(8.1.5)

Marginal revenue

 

\(\alpha\,{\rm{-}}\,( {4/3} )( {3Q\beta s^2/\pi g} )^{1/3}\)

(8.1.6)

Profit of firm

 

\(Z = ( {P-C} )Q-F\)

(8.1.7)

Profit-maximizing f.o.b. price, assuming \(Z \ge 0\)

 

\(P=( {1/4} )\alpha+( {3/4} )C\)

(8.1.8)

Range of commodity at profit-maximizing price, assuming \(Z \ge 0\)

 

\(( {3/4} )( {\alpha-C} )/s\)

(8.1.9)

Maximized profit

 

\(( {9/326} )\pi g( {\alpha-C} )^4/( {\beta s} )^2-F\)

(8.1.10)

Consumer benefit (CB)

 

\(\alpha Q-( {3\beta s^2/( {\pi g} )} )^{1/3} ( {3/4} )Q^{4/3}\)

(8.1.11)

Producer cost including F (PC)

 

\(CQ+F\)

(8.1.12)

Consumer surplus (CS)

 

\(CB-PQ\)

(8.1.13)

Producer surplus (PS)

 

\(PQ-PC\)

(8.1.14)

Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Unit cost of production; F—Fixed cost; g—Density of consumers everywhere; P—F.o.b. price set by firm; s—Unit shipping rate; x—Distance from customer to firm; α—Intercept of individual inverse demand curve; β—Slope of individual inverse demand curve. Outcomes (endogenous): Q—Aggregate amount demanded; q—Quantity demanded by a consumer; X—Range of commodity.

Lösch characterized models like (8.1.1) as giving rise to a demand cone9 over the rectangular plane.10 This implies that, given an f.o.b. price P, there is a maximum distance, range (X), beyond which customers will demand zero: see (8.1.2). Of course, this is also the radius (geographic size) of the market for this monopolist. The higher the firm sets its f.o.b. price P, the smaller the geographic area of the market. Aggregate quantity demanded is given by (8.1.3). After rearranging (8.1.1), substituting, and integrating, we get (8.1.4); I show this intermediate step to help the reader with the derivation. Finally, after substitution from (8.1.2), the aggregate quantity demanded reduces to an expression for the demand cone (8.1.5). As one might expect, quantity demanded is an increasing function of α and g, and a decreasing function of β, P, and s. This is the so-called free spatial demand curve; “free” in the sense that the firm does not here consider competitors.

The aggregate demand equation (8.1.5) is not linear in price even though we have assumed each customer individually has a linear demand. This is a consequence of the fact that the firm’s demand curve is aggregated over customers: some close by and paying a low effective price, others further away and paying a high effective price. This is illustrated in Fig. 8.1 where I assume the market contains just 3 customers, each with the same linear inverse demand curve. Customer 1, nearest the firm, has the lowest effective price; Customer 2 is further away, and Customer 3 still more so. At a high f.o.b. price, the firm would see demand only from Customer 1. As it lowers its f.o.b. price, it eventually attracts demand from Customer 2 and at a still lower price from Customer 3. That the aggregate demand curve here is kinked, a polyline, is a result of assuming just three customers. However, (8.1.5) assumes customers are spread evenly across the market; thus, (8.1.5)—cubic function of P that it is—can be thought of as the continuous equivalent of a kinked aggregate demand curve. Put differently, when geographic space is continuous, so too is the aggregate demand curve; when geographic space composed of discrete places (punctuated), the aggregate demand curve is kinked.
Fig. 8.1

Aggregate demand curve in market with three customers, each with same linear inverse demand curve but paying different effective prices

Now, let us look at the behavior of a firm that maximizes profit. Assume the firm has a fixed cost (F) and a marginal cost (C); profit is given by (8.1.7). After substituting from (8.1.5), we can find the first-order condition for profit maximization: marginal revenue equals marginal cost. See (8.1.6). This yields the f.o.b. price: (8.1.8). In  Chapter 2, I argue it is helpful to think of C (marginal cost of production) as a minimum price to complement the idea of α as the maximum price. The profit-maximizing price is now one-quarter of the way from C to α. This is a lower price than is charged by the firm in the simplest version of the two-market model in  Chapter 2; there, the firm sets its price halfway between C and α. This difference arises because I here assume f.o.b. pricing; in  Chapter 2, I had assumed that the firm discriminates in pricing between customers at Place 1 and customers at Place 2.

How large is the geographic area of the firm’s market here? After substituting (8.1.8) into (8.1.2), we get (8.1.9). From (8.1.2), we see that, were the firm to price at C, the range for its commodity would be \((\alpha - C)/s\). This is the maximum possible range; therefore, the maximum possible geographic area is \(\pi ((\alpha - C)/s)^2\). However, because the firm maximizes profit, it sets the higher f.o.b. price in (8.1.8) that generates a range 3/4 the size of the maximum possible range. Put differently, the firm foregoes serving customers at radiuses from beyond \((3/4)(\alpha - C)/s\) to \((\alpha - C)/s\) because these remote customers are not profitable enough.11 I presume here that the market radius given by (8.1.9) is sufficiently large to make the firm profitable. As shown in (8.1.10), α and g must be sufficiently large and/or β, C, F, and s sufficiently small for this to happen.

Finally, even though this is the market for the monopolist’s product and even though a single f.o.b. price is determined there, the effective price is different for customers depending on how far they are from the firm. Up until now, we have thought of a market in geographic terms as consisting of one or more places. At each place, customers pay the same price for a commodity. While price may vary from one place to the next, this is still consistent with the notion of a market. Places might here be thought to be submarkets in that the price of the commodity is systemically higher in some of them compared to others. In this chapter, we imagine a market in which potentially every customer faces a different effective price. A firm may sell at the same f.o.b. price to everyone, but customers face different effective prices; each must pay a unique shipping cost to get the commodity home. It is perhaps easiest in this chapter to imagine a continuum of submarkets arrayed by distance ring from the firm, with a price premium (in terms of effective price) for rings further away from the firm. Here, we might be able to retain the notion of a market because, after all, firms are selling at the same f.o.b. price. However, we then turn to a model in which firms each might take advantage of a local monopoly to sell at different f.o.b. prices. In such a model, what does the notion of a market now mean? We return to this question below.

Why start by assuming f.o.b. pricing? After all, as was seen in  Chapter 2, a delivered pricing scheme is more profitable. Following the findings of  Chapter 7, assume the firm sets its price so as to cover half freight; we then need find only the price (P 0 ) set for a customer adjacent to the firm. Given the firm has a fixed cost (F), a marginal cost (C) of production and pays shipping cost, its marginal cost is \(C+sx\) for the customer at distance x. As seen in  Chapter 2, marginal revenue for that customer is \(\alpha-2\beta q\). Equate these: see (8.2.5) in Table 8.2. Remembering \(q=( {{\alpha / \beta }} )-( {{1 / \beta }} )P\), where P is now delivered price, (8.2.5) yields the profit maximizing price: see (8.2.6). This is different from the f.o.b. pricing solution in the following respects.
  • Compare the delivered price in (8.2.6) with the f.o.b. price in (8.1.8). For customers nearby (small x), delivered price is higher than f.o.b. price; for customers further away, it is smaller. Were a firm to switch from f.o.b. pricing to delivered pricing, customers nearby would become worse off while more remote customers would become better off.

  • In  Chapter 2, we saw the half-freight rule. Here, (8.2.6) instances it again.

  • Profit under delivered pricing, now given by (8.2.11), is higher than it was for f.o.b. pricing: compare (8.2.11) with (8.1.10).

  • The geographic size of market under delivered pricing—see (8.2.7)—is the largest possible; it is larger than it would be, for instance, under f.o.b. pricing.

Table 8.2

The monopolist’s market using delivered pricing assuming \(P=P_0+0.{\rm{5}}sx\), hence \(X=( {\alpha-P_0 } )/( {0.5s} )\)

Aggregate demand

 

\(( {4/3} )\pi g( {\alpha-P_0 } )^3/( {\beta s^2 } )\)

(8.2.1)

Total revenue

 

\(Q( {\alpha+P_0 } )/2\)

(8.2.2)

Marginal Revenue

 

\(0.5( {a+P_0 } )-( {1/6\,} )\{ {( {3/4} )\beta s^2 Q/( {\pi g} )} \}^{1/3}\)

(8.2.3)

Total cost

 

\(F+Q_d [ {C+\alpha-P_0 } ]\)

(8.2.4)

Marginal cost equals marginal revenue for customer at distance x

 

\(C+sx=\alpha-2\beta q\)

(8.2.5)

Profit maximizing price for customer at distance x

 

\(P=0.5\alpha+0.5( {C+sx} )\)

(8.2.6)

Radius of market (where P = α)

 

\(X=( {\alpha-C} )/s\)

(8.2.7)

Quantity at profit-maximizing price

 

\(( {1/6} )\pi g( {\alpha-C} )^3/( {\beta s^2 } )\)

(8.2.8)

Revenue of firm

 

\(\int_0^x {2\pi gxqPdx}\)

(8.2.9)

Profit of firm defined

 

\(\int_0^x {2\pi gx( {P-C-sx} )qdx-F}\)

(8.2.10)

Maximized profit

 

\(( {1/24} )( {\pi g} )( {\alpha-C} )^4/( {\beta s^2 } )-F\)

(8.2.11)

Consumer benefit (CB)

 

\(\alpha Q-( {3\beta s^2/( {\pi g} )} )^{1/3} ( {3/4} )Q^{4/3}\)

(8.2.12)

Producer cost including F (PC)

 

\(CQ+F\)

(8.2.13)

Consumer surplus (CS)

 

\(CB-PQ\)

(8.2.14)

Producer surplus (PS)

 

\(PQ-PC\)

(8.2.15)

Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Unit cost of production; F—Fixed cost; g—Density of consumers everywhere; P—Delivered price set by firm; s—Unit shipping rate; x—Distance from customer to firm; α—Intercept of individual inverse demand curve; β—Slope of individual inverse demand curve. Outcomes (endogenous): P0—Price set by firm for an adjacent customer; P0—Price set by firm for an adjacent customer; Q—Aggregate amount demanded; q—Quantity demanded by a consumer; X—Range of commodity.

Figure 8.2 illustrates the two pricing schemes where \(\alpha=10,\,\beta=1,\break s=0.10,\,g=20,\,C=2,\ {\rm{ and }}\ F=0\). There, we see the effective price (under f.o.b. pricing) and the delivered price increase steadily the further away the customer, until we reach a distance at which demand drops to zero (i.e., effective or delivered price to the customer is α). That defines the range of the market under each pricing scheme.
Fig. 8.2

f.o.b. and delivered pricing compared. Notes: \(a=10; \, \, b=1; \, \, s=0.10; \,\, g=20; \,\, C=2, \,\, f=100,000\). With f.o.b. pricing: \(P=4.00; \,\, R=60; \,\, Q=452,389; \,\, Z=804,779\). With delivered pricing: \(P( {r=0} )=6.00; \,\break R=80, \,\, Q=536,165, \,\, Z=972,330\). Horizontal axis scaled from 0 to 90; vertical axis scaled from 0 to 12

8.3 Trade Area and Market Area in Retailing

Retail analysts use two distinct concepts in thinking about a store’s sales potential; trade area and market area.

The trade area is the (larger) geographic area from within which most of the customers of a store and its competitors come from. The trade area represents most of the customers that one store might attract potentially were it to be highly successful against its competitors. If the trade area lies entirely inside a circle of radius X (the range), then potentially, the store could attract all of them. On the other hand, if you and your competitors are spread out so that some of the competitors’ customers are more than X kms away from you, you will not attract them even if customers might otherwise prefer your store.

In contrast, the market area is the (smaller) geographic area within which you currently dominate and draw a substantial proportion of your customers. Put differently, the market area represents the core of your current customers. There are two critical numbers involved in this definition: (i) the firm’s share of customer purchases locally and (ii) sales within the market area as a percentage of the firm’s total sales. We can imagine that, in the retail sector containing a set of stores, there is a market. In a simple case, the trade area would be partitioned into a set of market areas, one for each store. In that sense, any one firm’s market area is a geographic subset of the market (i.e., the trade area) in which it and its competitors compete.

We can characterize a market area of a firm as (u, v) where u is proportion of sales (or, alternatively, customers) by all firms in the area that accrue to the firm, and v is the proportion of the firm’s total sales (or customers) that originate in that area. For example, in a (60, 80) market area the firm captures 60% of purchases made by customers from there, and the market area accounts for 80% of the firm’s total sales. If a trade area contains just four firms, all clustered at a central location, and doing equal sales, each would share the same (25, 100) market area. If the absence of competitors means a firm’s market area is limited only by range, we could find the (100, 100) market area for this firm. In practice, there are, in general, many ways of drawing a (u, v) market area for any one firm. Usually, analysts choose a compact area: e.g., a circle around the firm’s site with a minimum radius that satisfies u and v.

8.4 Model 8A: Two Firms Selling Commodity at Same f.o.b. Price

Assume the world can be represented as a rectangular plane. Assume only two firms (labeled 1 and 2): each firm sells the same product and at the same f.o.b. price (P). The two firms are at distinct places. Without loss of generality, assign Places 1 and 2 the Cartesian coordinates (0, 0) and (d, 0), respectively: Place 2 is, therefore, d kilometers due east of Place 1. See Table 8.3. Now, assume a customer firm purchases this product for use as an input in its own production. The customer pays a fixed unit shipping rate, s, for each unit of the input purchased. Suppose a firm is located x1 kilometers from Place 1 and x2 kilometers from Place 2. For the firm, the cost of a unit of the input is, therefore, \(P+sx_1\) purchased from Place 1, or \(P+sx_2\) purchased from Place 2. Assume the customer has a linear inverse demand function and \(P+sx_1\) and \(P+sx_2\) are each less than the maximum price, α. Assume also that each firm has sufficient capacity to meet all demand.
Table 8.3

Model 8A: two suppliers on rectangular plane selling commodity at same f.o.b. price (P)

Location of supplier 1

 

(0, 0)

(8.3.1)

Location of supplier 2

 

(d, 0)

(8.3.2)

Location of customer on boundary

 

(X, Y)

(8.3.3)

Distance from supplier 1 to customer

 

\(x_1=( {X^2+Y^2 } )^{0.5}\)

(8.3.4)

Distance from supplier 2 to customer

 

\(x_2=( {( {X-d} )^2+Y^2 } )^{0.5}\)

(8.3.5)

Equal effective prices on boundary

 

\(P+sx_1=P+sx_2\)

(8.3.6)

Boundary condition

 

\(X=d/2{\hbox{ for any }}Y\)

(8.3.7)

Notes: Rationale for localization (see Appendix A): Z3—Implicit unit price advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): d—Distance (in kilometers) between firm and competitor; P—F.o.b. price set each by firm and by competitor; s—Unit shipping rate. Outcomes (endogenous): XX-coordinate of customer on boundary between the two firms; x 1 —Distance from customer to firm; x 2 —Distance from customer to competitor; YY-coordinate of customer on boundary between the two firms.

Under these assumptions, the customer will be in the market area of the firm at Place 1 if \(P+sx_1 < P+sx_2\) (i.e., if \(x_1 < x_2\)). and Place 2 if \(x_2 < x_1\). The customer will be on the boundary if \(x_1=x_2\). Now, assume a customer on the boundary has Cartesian coordinates (X, Y). Application of the Pythagorean Formula implies the following: \(( {X-0} )^2+( {Y-0} )^2=( {X-d} )^2+( {Y-0} )^2\). Upon simplification, the boundary is given by (8.3.7). See Table 8.3 wherein I summarize equations, assumptions, notation, and rationale for localization in Model 8A. In other words, for any given Y, X is halfway between the x-coordinates of the two places. Therefore, viewed geometrically, the boundary between the two market areas is the perpendicular bisector between the two firms.12 See the line CE on the map in Fig. 8.3.
Fig. 8.3

Model 8B: map of boundary between market areas of two suppliers

Because of competition among firms, we might expect that the firm will not be able to realize the monopoly profit given in (8.1.10). After all, if the firm were to earn substantial profits, other firms would be attracted into the industry. Where they choose locations with a market radius that overlaps that of our firm, they cut into the profit and the market area of our firm. Imagine the following thought experiment. Two competitors—each pricing f.o.b. and producing the same product—locate d kilometers apart on a rectangular plane. Assume initially the competitors are too far apart to impinge on each other’s market: i.e., \(d>2X\) where X is defined in (8.1.9). They, therefore, each earn the profit given by (8.1.10) and have a circular market whose radius is given by (8.1.9). Now, imagine we push the two competitors closer together until \(d < 2X\). At this stage, the two markets intersect, and the boundary between the firms’ market areas is a perpendicular bisector. If customers purchase from the nearest firm, the market area for each firm shrinks the more the markets intersect. Continuing this thought experiment, as we push the two firms closer together, the revenue and profit earned by each firm shrinks.

Thiessen13polygons14—also known as a Voronoi15 or Dirichlet Diagram—are a common tool in retail analysis.16 In this approach, a map of a trade area is drawn with each store represented as a point (dot). Thiessen polygons are created using an algorithm. For each store, a subset of the perpendicular bisectors with all other stores (as well as the map boundary) form the set of line segments that constitute a Thiessen polygon. The corresponding map showing the lines joining places sufficiently near to have a boundary in common is called a Delaunay triangulation.

Under the assumptions made and ignoring knife-edge cases (customer straddles the boundary), a Thiessen polygon is a (100, 100) market area. That is because there would be no incentive for a customer inside a polygon to purchase the commodity from a firm elsewhere, or for a customer outside the polygon to purchase from this firm.

So far, we have assumed the firms each charge the same f.o.b. price for their input. Is that reasonable? Presumably, to the extent each firm can affect the price they receive, they would want to set a price where marginal revenue equals marginal cost as we saw earlier in (8.1.6). At the same time, one might imagine competition between the two firms should drive their prices to be similar if not identical. However, the two firms each have their own market area. This implies the firms are not perfectly competitive; is there any reason, therefore, why competition should lead to the same price for each firm?

8.5 Model 8B: Market Area Boundary Between Two Firms Selling Same Commodity at Different f.o.b. Prices

Retain all of the assumptions above, except now assume the firm at Place 1 on a rectangular plane sells the commodity at one f.o.b. price (P1), while the firm at Place 2, located d kilometers from Place 1, sells for another, P2.17 For a purchaser located x1 kilometers from Place 1 and x2 kilometers from Place 2, the effective price of a unit is, therefore, \(P_1+sx_1\) purchased from Place 1 or \(P_2+sx_2\) purchased from Place 2. See Table 8.4. For the moment, let us not ask why the two firms have different f.o.b. prices.
Table 8.4

Model 8B: two suppliers on rectangular plane selling commodity at different f.o.b. prices (P1 and P2)

Location of supplier 1

 

(0, 0)

(8.4.1)

Location of supplier 2

 

(d, 0)

(8.4.2)

Location of customer on boundary

 

(X, Y)

(8.4.3)

Distance from supplier 1 to customer

 

\(x_1=( {X^2+Y^2 } )^{0.5}\)

(8.4.4)

Distance from supplier 2 to customer

 

\(x_2=( {( {X-d} )^2+Y^2 } )^{0.5}\)

(8.4.5)

Equal effective prices on boundary

 

\(P_1+sx_1=P_2+sx_2\)

(8.4.6)

Boundary condition

 

\(x_1-x_2=( {P_2-P_1 } )/s\)

(8.4.7)

(8.4.7) assumes shipping sufficiently costly to sustain differences in f.o.b. prices: i.e., \(P_1 <\,P_2+sd\) and \(P_2 <\,P_1+sd\) or \(s>| {P_1-P} |/d\)

(8.4.8)

Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): d—Distance (in kilometers) between firm and competitor; P1—F.o.b. price set by firm; P2—F.o.b. price set by competitor; s—Unit shipping rate. Outcomes (endogenous): XX-coordinate of customer on boundary between the two firms; x1—Distance from customer to firm; x2—Distance from customer to competitor; YY-coordinate of customer on boundary between the two firms.

Under these assumptions, a purchaser will be on the boundary if \(x_1-x_2=( P_2-P_1)/s\). The implication here is that the boundary is the set of all places such that the difference in distance to two fixed places is a fixed amount (i.e., \({{( {P_2-P_1 } )} / s}\). However, this is also the definition of a hyperbola. In the special case where the difference in distance is zero (i.e., \(P_1=P_2\)), the hyperbola reduces to a straight line (the perpendicular bisector) as we have already seen.

Under the assumptions made and again ignoring knife-edge cases where customers straddle a boundary, a market area constructed in this way is (100, 100). As in the case of the Thiessen polygon, there would be no incentive for a customer inside the market area to purchase the commodity from a firm elsewhere or for a customer outside the polygon to purchase from this firm.

Imagine now a thought experiment in which the two competitors initially charge the same f.o.b. price. Assume the firms are close enough that the boundary between them incorporates a perpendicular bisector. Now, suppose the firm at Place 2 were to lower its price. The relevant portion of the market boundary would now become a hyperbola. It would generally lie closer to Place 1 than does the perpendicular bisector. See the boundaries FH (when \(P_1=P_2+1\)) and IK (when \(P_1=P_2+2\)) in Fig. 8.3. As hyperbolae, these each bend back toward Place 1. Put intuitively, a firm holds onto to customers best when they are close by or behind it. There is an upper limit to the price the firm at Place 1 can set if it wants to have any market area at all: \(P_1 < P_2+sd\). Of course, if positions were reversed and firm 1 had the lower price, there would be a minimum price charged by the firm at Place 1 below which the firm at Place 2 would lose its markets: \(P_1+sd < P_2\).

I am not saying anything about the amount any one purchaser might demand; I am saying only that the purchaser will or will not patronize a particular firm. Nonetheless, there is an interesting insight here into the demand curve faced by a firm. To see this, suppose each purchaser has a price-inelastic demand for just one unit of the commodity per time period. Suppose further N purchasers in total. If firm 1 sets its price too high \(( {P_1 > P_2+sd} )\), the demand for its product is zero. If firm 1 sets it price sufficiently low \(( {P_1 < P_2-{sd} }) \) to undercut its competitor, its demand is N. If firm 1 sets a price anywhere in between these two, it gets a point on a negatively sloped demand curve. In other words, firm 1 has a demand curve kinked at both \(P_1=P_2-sd\) and \(P_1=P_2+sd\). Imagine a thought experiment in which we push firms 1 and 2 closer together. As a polar case, \(d=0\) when Places 1 and 2 are adjacent. Now, the demand curve for either firm reduces to the familiar perfect competition case: a horizontal line. In that sense, the kink is not inconsistent with neoclassical theory. However, we again have to ask why the kink arises. The answer once again is tied up inextricably with the discreteness of geography. It is a consequence of the idea that firm 2 is where it is, neither closer nor further away.

8.6 Model 8C: Why Do Prices Differ Among Firms?

The purpose of this model is to better understand when and why competitors might charge different f.o.b. prices for the same product. Introducing the possibility that firms sell at different prices raises a question. What determines price in such instances? Making the assumptions that firms are price takers and everyone faces the same price is consistent with perfect competition. However, the notion implicit in a market area is that with f.o.b. pricing customers face different prices depending on location. Add to this the possibility that different firms set different prices, and the notion of market equilibrium is called further into question.

To see how prices might be set, consider a simple geography wherein customers are spread uniformly along an east−west line of infinite length and that firms are also evenly spread out (one firm each d kilometers).18 Consider one firm with a competitor to the west (labeled w) and another to the east (labeled e). Assume all firms price f.o.b. Let \(0 < x < d\) be the location (kilometers) of a customer to the west of the firm. For that customer, effective price of a unit purchased from the firm is \(P+sx\) and the effective price from competitor w is \(P_w+s( {d-x} )\). Assume \(P_w \le P_e\) and that w does not price e out of the market: i.e., \(P_w+2sd > P_e\) (remembering w and e are 2d kilometers apart). Further, assume here \(P < P_w+sd\) and \(P+sd > P_e\) so the three firms coexist. At the boundary, X w , between competitor w and the firm, we get (8.5.1). See Table 8.5. This yields a formula for the western boundary—see (8.5.2)—wherein our firm gets half the market to its west (d/2) adjusted for any difference in f.o.b. price with respect to its competitor there. Similar results apply in the market to the east of our firm: see (8.5.3). Therefore, the total length of market, \(X=X_w+X_e\), for our firm is given by (8.5.4).
Table 8.5

Model 8C: customers spread uniformly along a line market (zero price elasticity) with supplier every d kilometers and f.o.b. pricing

Effective price at western boundary, assuming \(s>| {P_w-P} |/d\)

 

\(P+sX_w=P_w+s( {d-X_w } )\)

(8.5.1)

Western boundary, assuming \(s>| {P_w-P} |/d\)

 

\(X_w=( {1/2} )( {P_w-P} )/s+( {1/2} )d\)

(8.5.2)

Eastern boundary, assuming \(s>| {P_e-P} |/d\)

 

\(X_e=( {1/2} )( {P_e-P} )/s+( {1/2} )d\)

(8.5.3)

Total length of supplier’s market

 

\(X=( {( {1/2} )( {P_w+P_e } )-P} )/s+d\)

(8.5.4)

Aggregate quantity demanded

 

\(Q=gqX\)

(8.5.5)

Profit

 

\(( {P-C} )gqX-F\)

(8.5.6)

First-order condition for myopic profit maximization

 

\(X-( {P-C} )/s=0\)

(8.5.7)

Profit-maximizing price

 

\(P=( {1/2} )( {1/2} )( {P_w+P_e } )+( {1/2} )( {C+sd} )\)

(8.5.8)

Notes: Rationale for localization (see Appendix A): Z2—Implicit unit cost advantage at some locales; Z8—Limitation of shipping cost. Givens (parameter or exogenous): C—Unit cost of production; d—Distance (in kilometers) between firm and competitor to East or West; F—Fixed cost; g—Density of consumers (assumed constant along line); Pe—Price of competitor to east; Pw—Price of competitor to west; q—Quantity demanded by a consumer; s—Unit shipping rate. Outcomes (endogenous): P—F.o.b. price set by firm; Q—Aggregate amount demanded; X—Total length of firm’s market; Xe—Distance to customer on boundary to east of firm; Xw—Distance to customer on boundary to west of firm.

Suppose further customers are spread uniformly along this line at density g and that each customer demands q units of the commodity regardless of price. In this case, aggregate demand for the firm’s product is given by (8.5.5). If the firm has a marginal cost of production, C, and a fixed cost of production, F, then its profit is given by (8.5.6). The firm then sets a price so as to maximize profit: see (8.5.7). Assume here the firm is myopic; it does not expect either firm w or firm e to change their price in response to the price chosen. This yields the profit-maximizing price shown in (8.5.8): halfway between the average of P w and P e on the one hand and a cost \(( {C+sd} )\) on the other hand. Note \(C+sd\) is greater than the actual cost \(( {{{C+sd} / 2}} )\) of serving someone at a place midway between any pair of firms.

There is something special about \(C+sd\) here. If all three firms were to each have an f.o.b. price of \(C+sd\), there would be no incentive for our firm or, under symmetric conditions, for either competitor to change price. If the two competitors each set a price lower than \(C+sd\), there would be an incentive for our firm to set its price higher than them and closer to \(C+sd\). On the other hand, if the two competitors each set a price higher than \(C+sd\), there would be an incentive for our firm to set its price lower and closer to \(C+sd\). Therefore, \(C+sd\) is a kind of equilibrium price for all firms. I do not want to make too much of this idea, however, because of the following conundrum. If all firms in the market, not just the three examined here, were to conspire, they would want to set the price as high as possible since we have assumed demand is insensitive to price. However, in (8.5.8), there is no mechanism that would drive prices up in this way. Of course, you might say, this is a competitive market so why would we expect prices to be driven up?  If it is indeed competitive, why isn’t profit being driven to zero? At price equal to \(C+sd\) each firm has the potential to some excess profit unless fixed cost is prohibitive. Why? Presumably, this is because the market is rich enough to support firms at a density at least equal to one firm per d kilometers. This is the so-called Löschian equilibrium problem in which we imagine free entry of new firms, and the rearrangement of these firms in geographic space, and a shrinking of d until the excess profit earned by the marginal firm is effectively zero.19

As in  Chapters 3 and  http://6, this chapter assumes that customers each demand a fixed amount of output, q units, regardless of price. Can we release this assumption? Is it possible, for instance, to imagine a linear inverse demand curve for each customer here? The firm’s aggregate demand Q on this linear market would then be defined by (8.6.1): see Table 8.6. Profit would then be given by the identity (8.6.3). Assuming again the firm is myopic in that it sets a price for its product ignoring the possibility that the two competitors might react by changing their prices implies (8.6.4). To solve this equation, we need to differentiate (8.6.2): see (8.6.5). As well, we need to remember X w , X e , and X here are determined by (8.5.2), (8.5.3), and (8.5.4). Unfortunately, substituting these equations back into (8.6.4) gives a quadratic equation (8.6.6) from which it is difficult to obtain insights into the determinants of P. That is why I presented the zero price elasticity of demand solution in Table 8.5; it is easier to solve and interpret.
Table 8.6

Model 8C: customers spread uniformly along a line market (nonzero price elasticity) with supplier every d kilometers

Demand for firm’s product

 

\(Q=\int_o^{x_w} {( {\alpha-( {P+sx} )} )( {g/b} )dx+\int_o^{x_e} {( {\alpha-( {P+sa} )} )( {g/b} )dx} }\)

(8.6.1)

Which yields

 

\(Q=( {g/{\beta}} )( {( {\alpha-P} )X-( {s/2} )( {X_w ^2+X_e ^2 } )} )\)

(8.6.2)

Profit

 

\(Z=( {P-C} )Q-F\)

(8.6.3)

Myopic first-order condition

 

\(\partial Z/\partial P=Q+( {P-C} )\partial Q/\partial P=0\)

(8.6.4)

where

 

\(\partial Q/\partial P=( {g/\beta} )( {-X/2-( {\alpha-P} )/s} )\)

(8.6.5)

Yields a quadratic in P

 

\(a_1 P^2+b_1 P+c_1=0\)

(8.6.6)

where

 

\(a_1\,{\rm{=}}\,6-{3s}\)

(8.6.7)

\(b_1= 2s(\alpha+C)-2(2\alpha+C)-3({P_w} + {P_e}+2ds)\)

(8.6.8)

\(c_1={( {2\alpha+C} )\,( {P_w+P_e+2ds} )}+ s(P_w+ds)\,(P_e+2ds)-\)

(8.6.9)

\((s/2)\,(P_w+P_e+2ds)^2-2saC\)

 

Note: See also Table 8.5.

8.7 Model 8D: Market Area Boundary Between Two Firms with Different Capacities

In the previous chapter, I presented the Hitchcock–Koopmans problem in which a monopolist minimizes the cost of production and shipment in meeting a fixed schedule of demand. We saw there the economic insight that a firm will use a particular factory (establishment) to serve customers at a given place if the opportunity cost of using that factory is the lowest of all potential sites. In a related vein, we came to understand that the geographic area supplied by a factory—its market area—could be outlined from the pattern of shipments in the least cost solution. Where demand at a place is met entirely by shipments from one firm, that customer place is part of the firm’s market area. Suppose that where demand at a place is met by shipments from two or more firms we assign that place to the market area of the firm with the largest shipment. Constructed in this way, the market area of a firm may not necessarily include all of its customers but will include all those customer places where the firm is the exclusive or principal source of shipments. However, what the Hitchcock–Koopmans problem does not do is to give us an idea about the geography of the areas served by a given factory under the Hitchcock–Koopmans model. Inspired by Hall (1989), I show when and how these areas can be delimited.

Suppose now firm 1, unlike firm 2, has insufficient capacity to meet the needs of all the purchasers who might otherwise want to purchase there. As in  Chapter 7, I also assume demand for the commodity everywhere has a price elasticity of zero. There are two possible scenarios here.

One scenario is where the firms are competitors. Assume the two firms initially intend to charge the same price. In this case, firm 1 finds too many customers to accommodate given its capacity. However, firm 1 may raise its price above that of its competitor in order to ration its output most profitably. As it does, some customers now find it less costly to purchase instead from the more distant firm 2. As firm 1 raises its price, the market boundary becomes a hyperbola bending back toward it as in Fig. 8.3. At whatever price firm 1 eventually settles, the market boundary will be a hyperbola.

The second scenario, as used in the Hitchcock–Koopmans model in the preceding chapter, is where the suppliers are branch factories under common ownership. In that case, the firm faces a shadow price on capacity at each factory. The shadow price here is zero for the factory at Place 2 (excess capacity) and positive for the factory at Place 1 (no excess capacity). In this case, the firm can be thought to allocate customers to factories taking this opportunity cost into account. So, the boundary between the two market areas will therefore again follow a hyperbola. The primal and dual in the Hitchcock–Koopmans problem show us that quantity allocations (shipments) and prices (shadow prices) are just two ways of looking at the same problem. Therefore, we should not be surprised to see that a capacity limitation has the same effect on a market boundary as does a higher unit cost.

This is a powerful conclusion because it extends our understanding of the primal in the Hitchcock–Koopmans problem. In  Chapter 7, we saw a method (linear programming) for finding a solution to the factory allocation problem. However, that solution did not give us much of an idea as to the geography of market areas. Now, we know that the boundary between two suppliers on a rectangular plane will be either a perpendicular bisector or a hyperbola bending back toward the supplier with the higher unit cost or the capacity constraint. So, the conclusion here is that a capacity constraint affects the market boundary exactly the same manner as does a price difference. It causes the market boundary to become hyperbolic and to bend back in the direction of the factory with the binding capacity constraint.

Under the assumptions made (see Table 8.7) and once again ignoring knife-edge cases of customers straddling a boundary, a market area constructed in this way is (100, 100). As in the case of Models 8A and 8B, there would be no incentive for a customer inside the market area to purchase the commodity from a supplier elsewhere or for a customer outside the polygon to purchase from this supplier.
Table 8.7

Model 8D: assumptions and rationale for localization

Assumptions (see Appendix A)

Rationale for localization (see Appendix A)

A1

Closed regional market economy

Z2

Implicit unit cost advantage at some locales

A3

Punctiform landscape

Z5

Capacity constraints

A4

Rectangular plane

Z8

Limitation of shipping cost

B1

Exchange of commodity for money

  

C1

Fixed demand locations

  

D2

Firm minimizes cost of production and shipping

  

D3

No capacity exceeded

  

D5

I factories

  

D8

Reverse-L marginal cost curve

  

E2

Fixed unit shipping rate

  

H2

Locations of the firm is given

  

M1

Firm uses f.o.b. pricing

  

8.8 Model 8E: Market Area Boundary Between Two Firms with Different, but Perfectly Substitutable, Commodities

Now assume the two firms each use f.o.b. pricing. See Table 8.8 for the list of assumptions and rationale for localization. Suppose that customers see the two firms as different but fully substitutable; each customer would purchase a quantity of only one of these two commodities. If the two firms are otherwise identical, the customer’s choice then depends strictly on the price they have to have to pay, inclusive of shipping costs. If customers see no inherent difference between the two commodities, they would choose the less-expensive commodity; Model 8E reduces to Model 8A if the commodities have the same f.o.b. price or Model 8B if their f.o.b. prices differ. On the other hand, suppose the commodities are inherently different in a way that customers prefer commodity from supplier 2 when effective prices are the same. Presumably, there is a difference in effective price at which the customer is indifferent between the two commodities. That difference therefore defines the boundary between the two firms. That difference also gives rise to a hyperbolic boundary as we have already seen in Model 8B. Under the assumptions made (see Table 8.8) and again ignoring straddling consumers, a market area constructed in this way is (100, 100). As in the case of Models 8A, 8B, and 8D, there would be no incentive for a customer inside the market area to purchase the commodity from a firm elsewhere.
Table 8.8

Model 8E: assumptions and rationale for localization

Assumptions (see Appendix A)

Rationale for localization (see Appendix A)

A1

Closed regional market economy

Z7

Variation in goods

A3

Punctiform landscape

Z8

Limitation of shipping cost

A4

Rectangular plane

  

B1

Exchange of two commodities each for money

  

C4

Identical customers

  

E2

Fixed unit shipping rate

  

F7

Location of competitor is given

  

H2

Location of the firm is given

  

M1

Firm uses f.o.b. pricing

  

8.9 Model 8F: Market Area Boundary Between Two Firms with Different, but Perfectly Substitutable, Commodities When Customers Are of Two Types

Until now, we have assumed that customers are everywhere identical. How does customer heterogeneity add complexity to the determination of market areas? Assume two different kinds of customers: types 1 and 2. See the list of assumptions and rationale for localization in Table 8.9. Every customer of type 1 is identical. So too is every customer of type 2. Assume here that the two firms sell different but perfectly substitutable commodities and price f.o.b. See Table 8.9. It is then possible to imagine a market area for any given firm among customers of type 1 based on Model 8E above. Similarly, we can imagine a market area for the same firm among customers of type 2. These two market areas may well differ; for example, customers of type 1 may prefer the product of the first firm (where effective price is the same), whereas customers of type 2 prefer the second. In such cases, it would not be possible in general to construct a single (100, 100) market area that covers both types of customers. Here then is a situation where, no matter how we draw the market boundary we might expect some customers inside the market area for firm 1 to purchase from firm 2, and some customers outside the market area of firm 1 to purchase from that firm nonetheless.
Table 8.9

Model 8F: assumptions and rationale for localization

Assumptions (see Appendix A)

Rationale for localization (see Appendix A)

A1

Closed regional market economy

Z6

Differences among consumers

A3

Punctiform landscape

Z8

Limitation of shipping cost

A4

Rectangular plane

  

B1

Exchange of two commodities, each for money

  

C10

Two kinds of customers

  

E2

Fixed unit shipping rate

  

F7

Location of competitor is given

  

H2

Location of the firm is given

  

M1

Firm uses f.o.b. pricing

  

8.10 Model 8G: Market Area Boundary Between Two Firms Supplying Different Commodities

This analysis generalizes from Model 8E. At this point in the book, we need something more general than a demand curve for a single product; we need to model how a consumer substitutes between two or more commodities. The economist does this by means of a utility function, \(U=f( {q_1 ,q_2 } )\) that shows how the well-being (U) of the consumer varies depending on the amounts of commodities 1 and 2 consumed: q1 and q2, respectively. An indifference curve traces combinations of quantities of commodity 1 and commodity 2 that generate the same level of well-being for the consumer. Maximizing utility subject to a budget constraint in turn yields the individual demand curves that we have used to this point in the book.

In a world of differentiated commodities, it may seem strange to be focused here on the case where two commodities are perfectly substitutable. Is it not reasonable to expect that a given customer would purchase some amount of the commodity from firm 1 and some amount from supplier 2? Assume two suppliers on the map: d kilometers apart. They each sell one product. Each uses f.o.b. pricing. The prices set by the two firms are p1 and p2, respectively. Consumers are identical. They each consume only the commodities provided by the two suppliers. Where q1 is the amount purchased by a customer from the first store monthly, and q2 is the amount purchased from the second store, each is rational; he or she maximizes the log-linearutility function given in (8.10.1).20 See Table 8.10. For ease of exposition, these are the only goods consumed and therefore exhaust consumer income. Assume also each consumer incurs a shipping cost of s dollars per kilometer for each unit of a commodity consumed. Consumers each have the same monthly income (Y). For a customer at distance x1 from one store and distance x2 from the other store, the budget constraint is (8.10.2).
Table 8.10

Model 8G: two stores selling different commodities; consumers with log-linear or CES utility functions

Log-linear utility

 

\(U=q_1 ^\nu q_2 ^{1-\nu }\)

(8.10.1)

Budget constraint

 

\(Y=( {p_1+sx_1 } )q_1+( {p_2+sx_2 } )q_2\)

(8.10.2)

First-order conditions for log-linear utility maximization

 

\(( {p_1+sx_1 } )q_1=\nu Y\)

(8.10.3)

\(( {p_2+sx_2 } )q_2=( {1-\nu } )Y\)

(8.10.4)

Relative expenditures at store 1

 

\(( {p_1 q_1 } )/( {p_2 q_2 } )=( {\nu /( {1-\nu } )} )( {p_1/p_2 } )( {p_2+sx_2 } )/( {p_1+sx_1 } )\)

(8.10.5)

On market area boundary, where \(( {p_1 q_1 } )/( {p_2 q_2 } )=1\)

 

\(x_2=( {1-2\nu } )p_2/( {\nu s} )+( {1-\nu } )p_2 x_1/( {\nu p_1 } )\)

(8.10.6)

CES (Constant elasticity of substitution) utility

 

\(U=( \delta q_1 ^{-\rho }+( 1-d)q_2 ^{-\rho })^{-\,1/\rho }\)

(8.10.7)

First-order condition for CES utility maximization

 

\(q_1/q_2=( \delta/( 1-\delta ))^{1\,/(1+\rho )} ((p_1+sx_1)/( p_2+sx_2 )^{-\,1/(1+\rho )}\)

(8.10.8)

Boundary (fraction of distance) along line from Supplier 1 to Supplier 2 where budget shares are invariant

 

\(X=( {( {1-\delta } )P_1-\delta P_2+( {1-\delta } )s} )/s\)

(8.10.9)

Notes: Rationale for localization (see Appendix A): Z7—Variation in goods; Z8—Limitation of shipping cost. Givens (parameter or exogenous): p i —F.o.b. price charged by firm i; s—Unit shipping rate; Y—Income of customer; ν—Relative preference for commodity 1: log-linear; d—Relative preference for commodity 1: CES; r—Substitutability of. Outcomes (endogenous): q i —Individual consumption of commodity 1; x i —Distance from firm i; X—Fraction of distance from Supplier 1 to Supplier 2; U—Utility of individual.

A utility-maximizing consumer can be shown to have demands for the two commodities as described by (8.10.3) and (8.10.4). These do not give the linear inverse demand curve that I have used elsewhere in the book so far. The log-linear utility function (8.10.1) used here has the implication that the consumer always spends fixed fractions (ν and \(1-\nu\)) of income on the two goods.21 The demand for either commodity does not depend on the price of the other commodity. This zero cross-price elasticity is another feature of the log-linear utility function (8.10.1) used here. While consumers exhaust their income on spending for the two commodities, \(( {p_1+sx_1 } )q_1\) and \(( {p_2+sx_2 } )q_2\), respectively, the revenues received by each store from the consumer are only \(p_1 q_1\) and \(p_2 q_2\), respectively. Therefore, the expenditure ratio for a customer at any given location is given by (8.10.5). I define the boundary of the market area to be the locations where \({{( {p_1 q_1 } )} / {( {p_2 q_2 } )}}=1\). Under the boundary condition (8.10.6), x2 is a linear function of x1. Note here restrictions on (8.10.6): \(x_1 \ge 0,\ x_2 \ge 0\), and \(x_1+x_2 \ge d\).

What does the market boundary look like? In the special case where \(\nu=0.5\) and \(p_1=p_2\), (8.10.5) reduces to \(x_2=x_1\) which means that the boundary is the perpendicular bisector. For other parameter values, the boundary is of an elliptical form. Under the assumptions made, a market area constructed in this way is never (100, 100). No matter how the market area for supplier 1 is drawn, customers inside it will still purchase some amount from supplier 2, and customers outside it will purchase some amount from supplier 1.

As noted elsewhere in this book, a log-linear utility function implicitly assumes that the cross-price elasticity—e.g., \({{dln[ {q_1 } ]} / {dln[ {p_2 } ]}}\)—is always zero and the elasticity of substitution

$${{dln[ {{{q_1 } / {q_2 }}} ]} / {dln[ {{{p_1 } / {p_2 }}} ]}}$$

is always −1. The CES utility function shown in (8.10.7) is a more general utility function; it has two parameters (δ and ρ) where δ (like ν), constrained to lie between 0 and 1, is the relative preference for good 1 and ρ,constrained to lie between −1 and 0 is tied to the elasticity of substitution. From (8.10.8),

$${{dln[ {{{q_1 } / {q_2 }}} ]} / {dln[ {{{p_1 } / {p_2 }}} ]}}={{-1} / {( {1+\rho } )}}.$$
To illustrate ideas, consider now a customer located somewhere along the straight line joining Suppliers 1 and 2 on a rectangular plane. See Fig. 8.4. Let x denote the fraction of the way from Suppliers 1 to 2: x therefore ranges from 0 (customer located adjacent to Supplier 1) to 1 (customer located adjacent to supplier 2). When x is near 0, \(p_1+sx_1\) is relatively small and \(p_2+sx_2\) is relatively large; the opposite holds when x is near 1. In the case of a log-linear utility function, the budget share for a commodity is the same for customers everywhere along this line; however, for customers closer to Supplier 1, this in turn implies more quantity of that commodity. In the case of a CES utility function, both the budget share and quantity of commodity 1 increase as x approaches 0. As we increase x—move closer to Supplier 2—we find customers with a CES utility function reallocate budget share in favor of commodity 2. In fact, there is a distance (a fraction X of the length from Suppliers 1 to 2) where the budget share is the same regardless of substitutability (ρ). See (8.10.9) and distance OK in Fig. 8.4. We also see that as the elasticity of substitution becomes larger (in absolute value) the budget share schedule transitions from AB to CD to FG to HI. The logical implication is that as the two goods approach perfect substitutes, the budget share approaches a Z shape (HJKI) where budget share is 1.0 when \(x < X\), 0.0 for \(x > X\), and in effect a vertical line at \(x=X\). In such a case, Model 8H reduces to Model 8A (if \(p_1=p_2\)) or 8B (if \(p_1 \ne p_2.\))
Fig. 8.4

Model 8G: budget share schedule along line from Suppliers 1 to 2 using log-linear and CES utility functions. Notes: Horizontal axis scaled from 0 to 1; vertical from 0 to 1.2. Givens also include \(p_1=11,\break p_2=12,\; s=10\)

8.11 Model 8H: Destination Choice Under Uncertainty

In recent decades, the discrete choice model22 has become popular as a means of explaining consumer shopping choices. Modeling of store choice is widely thought to have begun with Reilly (1931).23 In its simplest form, discrete choice modeling imagines that from among all possible stores the consumer formulates a choice set. The consumer is then assumed to collect information about stores in the choice set and then choose one store to make the purchase.24 In so doing, there may be differences among consumers that lead them to formulate different choice sets and/or make different choices. The analyst is assumed to know only about some (not all) of the differences among consumers and some (again not all) of the differences among stores. Because of this imperfect information, we can never be sure that we have properly identified the consumer’s choice set nor the full set of factors that shape choice from within the choice set.

Analysts typically use a variant of a multinomial logit model to predict shopping behavior. In a logit model, the log odds (reflecting the analyst’s uncertainty) of choosing a particular alternative are seen to gradually rise (fall) as conditions for that choice become more (less) favorable. By varying the magnitudes of the slope coefficients in the model, we can make consumers more sensitive (or less sensitive) to a particular factor. Earlier in this chapter, I presented several models in which there is a critical distance at which the customer suddenly and completely shifts from one supplier to another. In principle, it is possible to make the multinomial logit model reproduce a sudden and complete shift, but the model is most appropriate when change is gradual. What is not clear here, however, is just when and why change ought to be gradual. This brings us back to Hotelling’s thoughts on stability in competition.

8.12 Final Comments

In this chapter, I have explored ways of defining the market area boundary for a firm. In Table 8.11, I summarize the assumptions that underlie selected models from 8A through 8G. Many assumptions are in common to all these models: see the list in panel (a) of Table 8.11. In 8A, the firm and its competitor sell the same commodity at the same given f.o.b. price. In 8B, the competitor sets an f.o.b. price that is different but given. In 8C, the firm gets to set a price that maximizes its profit assuming that competitors do not react. In 8D, I study how the firm’s market area boundary adjusts to capacity constraints. In 8E, I show how the firm’s market area boundary varies when it sells a different, but perfect substitutable good. In terms of the geometric shape of market areas, Models 8B, 8D, and 8E give similar outcomes. Models 8F, 8G, and 8H introduce differences among consumers, as well as imperfectly substitutable goods.
Table 8.11

Assumptions in Models 8A through 8G

 

8A

8B

8C

8D

8E

8F

8G

 

Assumptions

 

[1]

[2]

[3]

[4]

[5]

[6]

[7]

(a) Assumptions in common

       

A1

Closed regional market economy

x

x

x

x

x

x

x

A3

Punctiform landscape

x

x

x

x

x

x

x

B1

Exchange of commodity (or commodities) for money

x

x

x

x

x

x

x

E2

Fixed unit shipping rate

x

x

x

x

x

x

x

H2

Location(s) of firm given

x

x

x

x

x

x

x

M1

Firm uses f.o.b. pricing

x

x

x

x

x

x

x

(b) Assumptions specific to particular models

       

F3

Competitor sells at same f.o.b. price

x

      

A4

Rectangular plane

x

x

 

x

x

x

x

F7

Location of competitor is given

x

x

x

 

x

x

x

C4

Identical customers

x

x

x

 

x

 

x

F1

Competitor sells same product

x

x

x

    

F5

Competitor sells at a different f.o.b. price

 

x

x

   

x

A5

Linear landscape

  

x

    

D6

Fixed cost

  

x

    

D7

Horizontal marginal cost curve

  

x

    

F6

Firm is myopic

  

x

    

C1

Fixed demand locations

  

x

x

   

D2

Firm minimizes cost of production and shipping

   

x

   

D3

No capacity exceeded

   

x

   

D5

I factories

   

x

   

D8

Reverse-L marginal cost curve

   

x

   

C10

Two kinds of customers

     

x

 

C7

Maximize same utility function

      

x

F2

Competitor sells a different product

      

x

In  Chapter 1, I argue that there are important linkages between prices and localization. I illustrated that idea in Model 2D wherein price and localization were joint outcomes for the monopolist. In this chapter, I consider how a firm sets its price in response to the prices set by its competitors nearby and hence is affected by the geographic density of firms. The idea here has been to explore how, as a consequence, localization and price are jointly determined. Much more could be done here. There is an extensive game-theoretic literature in which the locations of firms and their prices are reactions to the actions of their competitors. By itself, that is the subject of another book. My modest aim in this chapter has been to provide pointers in that direction.

As in earlier chapters, I continue to assume that the prices of inputs (including shipping services) used by the firm are exogenous; input prices are all determined in markets outside the scope of the models in this chapter. As well, the models in this chapter are silent on prices in output markets. Once again, I think Walras would have argued that the analysis in this chapter has been only partial in the sense that we have not looked explicitly at the simultaneity among prices in these markets. Once again, we must wait until a later chapter for the opportunity to do that.

The models in this chapter do not take into account the labor, capital, and land inputs used in production. As such, the models are of no use in looking at wages, interest and profit, and land rents. However, the models do provide some insight into the determination of commodity prices and thereby the well-being of consumers.

Footnotes

  1. 1.

    As this book does not deal with optimal location theory, I exclude optimal market areas here. See, for example, Hsu (1997).

  2. 2.

    On contributions to market areas by economists, see Bacon (1992), Eswaran and Ware (1986), Fetter (1924), Greenhut (1952a, 1952b), Greenhut and Ohta (1975), Hartwick (1973a), Hyson and Hyson (1950), and Stern (1972a). On the modeling of destination choice, see Ghosh and McLafferty (1987), Lo (1990, 1991a, 1991b, 1992), and Miron and Lo (1997).

  3. 3.

    Geographers and other regional scientists who have written about the analysis of market area include Batty (1978), Boots (1980), Daly and Webber (1973). Epping (1982), Erickson (1989), Fotheringham (1988), Gillen and Guccione (1993), Golledge and Amedeo (1968), Golledge (1967, 1970, 1996), Huff and Jenks (1968), Lentnek, Harwitz, and Narula (1988), Lo (1990, 1991a, 1991b, 1992), McLafferty and Ghosh (1986), Miron and Lo (1997), Mu (2004), Mulligan (1982), O’Kelly and Miller (1989), Parr (1995a, 1995b, 1997a, 1997b), Pitts and Boardman (1998), Pred (1964), Rushton (1971a), Sheppard, Haining, and Plummer (1992), Solomon and Pyrdol (1986), Stimson (1981), and Wilson (1967).

  4. 4.

    See, for example, Pate and Loomis (1997).

  5. 5.

    Important early work in the field also includes Lerner and Singer (1937) and Smithies (1941). Other work that largely predates the application of game theory include Capozza and Van Order (1978), Devletoglou (1965), Eaton and Lipsey (1975, 1979), Gannon (1972, 1973), Harker (1986), Mills and Lav (1964), Salop (1979), and Webber (1974). For game theory applications, see Anderson, de Palma, and Hong (1992), Anderson, Goeree, and Ramer (1997), Basu (1993,  Chapter 8), Benassi and Chirco (2008), Bester, de Palma, Leininger, Thomas, and von Thadden (1996), Boyer, Laffont, Mahene, and Moreaux (1995), Braid (2008), Damania (1994), De Frutos, Hamoudi, and Jarque (2002), Dorta-González, Santos-Peñate, and Suarez-Vega (2005), Economides (1993a, 1993b), Gabszewicz and Thisse (1986b, 1992), Goettler and Shachar (2001), Gupta, Pal, and Sarkar (1997), Gupta, Lai, Pal, Sarkar, and Yu (2004), Hamilton and Thisse (1992), Hay (1976), Huang (2009), Huck, Müller, and Vriend (2002), Irmen and Thisse (1998), Isard and Smith (1967), Iyer and Seetharaman (2008), Lambertini (1997), Lederer (1994), Lederer and Hurter (1986), Matsumura, Ohkawa, and Shimizu (2005), Osborne and Pitchik (1987), Pires (2005, 2009), Prescott and Visscher (1977), Seim (2006), Selten and Apesteguia (2005), Shaked and Sutton (1982), Stuart (2004), Tabuchi (1994), Tabuchi and Thisse (1995), and Zhu and Singh (2009).

  6. 6.

    I do not, for example, include models based on Marxist perspectives. See Plummer, Sheppard, and Haining (1998).

  7. 7.

    See, for example, McLafferty and Ghosh (1986).

  8. 8.

    See, for example, Bacon (1992). Here, inventory models of the kind used in  Chapter 7 allow us to look at the economics of buying in bulk and such phenomena as a buying safari or buying spree.

  9. 9.

    A term characterizing the tendency for individual quantity demanded to fall off with increasing distance from a supplier pricing f.o.b. because of the rise in effective price.

  10. 10.

    See Long (1971) for further discussion of the properties of a spatial demand curve.

  11. 11.

    See also Alderighi and Piga (2008).

  12. 12.

    To draw a boundary between the two suppliers, simply draw a straight line from Place s1 to 2 and then find a line that is perpendicular and bisects it. One method, familiar from high school mathematics, is to use a compass to draw a circle of radius r (where r > d/2) centered at Place 1, repeat at Place 2, then draw a straight line (the perpendicular bisector through the points of intersection of these two circles.

  13. 13.

    Alfred Henry Thiessen (born 1872), an American climatologist, defined polygons around individual rainfall stations to estimate total rainfall across a region.

  14. 14.

    A map polygon formed on a rectangular plane by constructing perpendicular bisectors to the straight lines joining a point (typically a store) to similar points nearby. The partitioning of a map in this way is called a Voronoi Diagram.

  15. 15.

    Georgy Fedoseevich Voronoi (born 1868), a Russian mathematician worked on polygonal partitioning of a two-dimensional plane. This follows on earlier work by Rene Descartes (writing around 1644) and Johann Peter Gustav Lejeune Dirchlet (writing around 1850).

  16. 16.

    See Boots (1980), Byers (1996), Graham and Yao (1990), Miles and Maillardet (1982), and Sibson (1980).

  17. 17.

    A similar model is studied in Parr (1995b).

  18. 18.

    Other geographies have also been explored. See, for example, Sarkar, Gupta, and Pal (1997).

  19. 19.

    In an industry where firms produce the same commodity and compete by choosing geographic locations, a long-run equilibrium wherein firms no longer have the incentive to enter or leave the industry. Every firm earns normal profit only. See, for example, Mai and Hwang (1994).

  20. 20.

    A ranking of alternative bundles that allows us to predict consumer choice. As a ranking, a given utility function is said to be unique up to a monotonic transformation. For example, the utility functions \(f( {x,y} )=x^b y^{1-b}\) where \(1 < a < 0\) and \(g( {x,y} )=ax^b y^c\) where \(b > 0,c > 0\), and \(b+c < 1\), calculated at consumption of x units of good 1 and y units of good 2, generate the same rank ordering: i.e., \(g( {x,y} )\) is a monotonic transformation of \(f( {x,y} )\). At the same time, consumers are usually imagined to have diminishing marginal utility in that each additional unit of good consumed increases utility by a smaller amount. The two utility functions above exhibit diminishing marginal utility.

  21. 21.

    The literature includes some experimentation with alternative utility functions. See, for example, Lo (1990, 1991a, 1991b, and 1992).

  22. 22.

    A statistical model used to predict consumer choice from among a set of discrete and denumerable alternatives. This includes the Multinomial Logit Model and Nested Logit Model.

  23. 23.

    Other early work in the area includes Carrothers (1956), Brown (1957), and Huff (1963, 1964). Important contributions include Davis (2006), de Palma, Lindsey, von Hohenbalken, and West (1994), Iyer and Seetharaman (2008), Lee and Pace (2005), Miron and Lo (1997), Sheppard, Haining, and Plummer (1992), and Slade (2005). O’Kelly and Miller (1989) and Parr (1997b) are good summaries of work in this area. See also Berry, Parr, Epstein, Ghosh, and Smith (1988,  Chap 7).

  24. 24.

    Dudey (1990) considers the impact of consumer search behavior on the location choices of retailers. Also see Schulz and Stahl (1996).

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department Social SciencesUniversity of Toronto ScarboroughTorontoCanada

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