Imperfect competition and basing-point pricing.

Gilligan, Thomas W.

I. INTRODUCTION

Basing-point pricing occurs when firms that are geographically differentiated use a common location to formulate the transportation costs, and thus delivered prices, charged to consumers. Consumers are not generally permitted to purchase the commodity at free-on-board mill prices since such prices exist only for firms located at the basing-point. Basing-point pricing is sometimes associated with other seemingly unusual practices such as cross-hauling, the systematic transshipment of apparently identical commodities, freight absorption, the shipment of identical commodities longer distances at delivered prices that do not reflect additional transportation charges, and phantom freight, the assessment of fictitious transportation charges. Basing-point pricing has a rich history in the United States and more recently was widely practiced by firms in the softwood plywood industry.(1)

Economists have long been interested in the determinants of basing-point pricing.(2) Their inquires have generally led them to conclude that basing-point pricing is adopted to facilitate collusion.(3) This view is articulated by Stigler |1949~ who combined two important branches of the literature to formulate a theory of how basing-point pricing and, more generally, any delivered pricing system can solve the problems of cartel instability confronting firms that are geographically differentiated and face stochastic demand.(4) Examining the case of multiple production centers with limited numbers of firms serving geographically uncertain demand, Stigler |1949, 1148~ argued that the "unpremeditable flexibility of mill prices would make it extremely difficult to maintain collusion among firms at the production center, and even more difficult to maintain collusion between firms at different production centers." The systematic freight absorption associated with any delivered pricing system provides a solution to these problems.

|With a delivered pricing system~, there is a single price at each point in the market (if transportation charges are agreed upon), so price rivalry is eliminated. One production center can sell in the "natural" territories of other production centers when this is necessary to obtain its share of the industry's sales; these distant sales involve freight absorption, moreover, and are therefore partly self-limiting. The various prices need not change often, so collusion is possible. Given the unstable geographical pattern of demand and the antitrust laws, systematic freight absorption permitted efficient collusion.

Stigler also provided some evidence, such as the general magnitude of demand variability in those industries characterized by basing-point pricing, to support his hypothesis that delivered prices and basing-point pricing facilitate collusion. Building upon Stigler's theory, McGee |1954~ argued that cross-hauling is a symptom of the collusion facilitated by basing-point pricing.

In a recent paper, Haddock |1982~ challenged this view and argued that basing-point pricing can arise if a firm at other than the base site takes the delivered price of the basing-point product as given when formulating its production decision. Given this behavior, a non-base site firm faces a kinked marginal revenue schedule that supports basing-point pricing for a wide range of marginal production and transportation cost schedules. The non-base firm earns a markup on its product in a particular market equal to the difference between the delivered price charged by base site firms and the non-base site firm's costs of production and delivery. The behavior and markups earned by firms located at the base point are not restricted in Haddock's model and, presumably, can include Bertrand behavior and competitive markups. The behavior of non-base site firms is explicitly monopolistic. Haddock also provided some non-collusive rationales for cross-hauling. In contrast to the view that basing-point pricing is adopted to facilitate collusion among all firms in the industry, Haddock's argument is that basing-point pricing can have little to do with pervasive collusion and may at most be associated with positive markups and phantom freight charges by firms on the periphery of production.(5)

The purpose of this article is to develop a theoretical model to explore the relationship between imperfect competition and delivered prices in a spatially differentiated industry. The model consists of two production sites with multiple firms at both sites employing linear homogeneous technologies to produce for a fixed demand that is uniformly distributed along a line connecting the two sites. The equilibrium of the model is assessed under various assumptions regarding the behavior of the firms in the industry. It is shown that mill pricing emerges in a delivered pricing model if firms at all production sites are Bertrand competitors; no firm believes that it can affect the total sales made in any given market. Single basing-point pricing emerges if firms at the base site are Bertrand competitors, firms at the non-base site are less than perfectly competitive with each other and are von Stackleberg leaders with respect to base site production, and the products produced at alternative sites are perfectly homogeneous. Firms at the base site must be Bertrand competitors for basing-point pricing. Cooperation among non-base site firms is sufficient (given von Stackleberg behavior) but not necessary for basing-point pricing. Cross-hauling results from product heterogeneity under mill pricing and basing-point pricing and is not sufficient for collusion. In general, the theoretical analysis integrates the two views and indicates that single basing-point pricing can be unrelated to collusion among firms located at both the base and non-base sites and occurs when the only deviation from perfect competition is that non-base site firms take the delivered price of the base point product as given when formulating their production decisions. Thus, existing theoretical analyses, even Haddock's |1982~, tend to overstate the amount of cooperation necessary to sustain basing-point pricing.

The current analysis leaves many important issues unresolved. For instance, scholars have long been interested in the effect of alternative spatial pricing policies on the locational decisions of firms in the economy. What are the consequences of basing-point pricing for the geographic concentration of production? This is the question addressed in Greenhut's |1956~ seminal analysis and the focus of the more recent work of Soper, Norman, Greenhut, and Benson |1991~. In contrast to the current analysis, these analyses fix the spatial pricing policy and examine its effects on the distribution of firms in the economy. An analysis that blends both concerns would look at the emergence of pricing policies and locations of firms as a function of firm behavior. Additionally, in a recent paper Thisse and Vives |1988~ examine the effects of precommitment on the formation of spatial pricing policies. In a simple two site, duopoly model they show that basing-point pricing would never arise as an equilibrium policy under non-cooperative behavior and conclude that, therefore, it may be best to think of basing-point pricing as a coordinating or collusive device, although clearly they provide no analysis of this type. They also fail to examine the consequences of multiple firms located at identical sites, a feature that in a non-cooperative setting is likely to undermine discriminatory pricing of the type they find.

The results of the analysis suggest that single basing-point pricing is indicative of imperfectly competitive, von Stackleberg leadership by non-base site firms and Bertrand behavior among base site firms. These findings raise some interesting theoretical issues that have yet to be addressed. For instance, how does this configuration of behavior emerge? Haddock |1982, 296-98~ illustrates that it is frequently optimal for an isolated firm at the periphery of production to take the delivered price of base site production as given when formulating its production plans for distant markets.(6) Is the sequential development of geographically differentiated productive capacity necessary for the emergence of basing-point pricing, as a casual study of its most famous uses would seem to suggest? And once having emerged, how is basing-point pricing sustained in the presence of a growing number of non-base site producers? Such growth should undermine the von Stackleberg behavior of firms located at other than the base site. Does the delivered price gradient generated from Bertrand behavior by base site firms provide a focal price coordinating the repeated interactions of non-base site producers? The importance of inquiries on causes and consequences of basing-point pricing and delivered pricing more generally seems to rest on the answers to such questions.

And lastly, the current analysis is silent on the proposition that basing-point pricing is used to support cooperation among geographically differentiated competitors, as many past and contemporary scholars appear to believe. The analysis does, however, identify some inconsistencies in this argument. For one, it is easy to show within the context of the current model that perfectly colluding firms would not choose basing-point pricing, but instead would select a more complex system of non-linear discriminatory prices. In a recent insightful paper, Espinosa |1992~ shows that in an infinite-horizon, repeated play duopoly model, basing-point pricing can be consistent with a wide range of behavior, including perfect competition. Clearly, additional research on this issue is warranted.

This article is organized as follows. The next section presents a theoretical framework to explore the relationship between the nature of competition among geographically differentiated firms and the characteristics of a delivered pricing system. Section III identifies the properties of spatial prices under Bertrand competition while section IV uncovers the behavioral determinants of single basing-point pricing. Section V contains a comparative statics analysis of the imposition of mill pricing on an industry practising basing-point pricing. Section VI is the conclusion.

II. ASSUMPTIONS AND THE MODEL

The purpose of this section is to construct a theoretical model to identify the behavioral determinants of alternative spatial pricing policies. The chief assumption of the model is that firms in the industry choose the amount of output to deliver to each market in the economy in order to maximize profits subject to the constraint that equilibrium delivered prices provide no opportunities for intermarket arbitrage among consumers. The model places no restrictions on the type of pricing policy (e.g., mill pricing, basing-point pricing, or more complicated nonlinear delivered price schedules) that can emerge in equilibrium, but instead derives equilibrium delivered pricing characteristics from assumptions regarding the competitive interactions among and equilibrium behavior of firms in the industry.(7) The model is also used to examine the consequences of imposing a mill pricing constraint on firms whose pricing policies resemble basing-point pricing.

To begin, consider an economy with a continuum of identical consumers distributed uniformly along a line of unit length. Production of two goods, |q.sub.0~ and |q.sub.1~, occurs at the end points of the line.(8) There is a competitive numeraire sector. Table I provides a convenient summary of the variables used in the model. Table I also contains brief definitions of these variables.

A representative consumer located at some h |is an element of~ |0, 1~ has a utility function that is linear and separable in the numeraire good. The consumer maximizes u(|q.sub.0~, |q.sub.1~) - |p.sub.0~(h)|q.sub.0~ - |p.sub.1~(h)|q.sub.1~ where |p.sub.0~(h) and |p.sub.1~(h) are the per unit delivered prices paid by the consumer located at h for goods |q.sub.0~ and |q.sub.1~, respectively. The function u(|center dot~) is assumed quadratic, (strictly) concave and symmetric in |q.sub.0~ and |q.sub.1~. Thus, |Mathematical Expression Omitted~ where |Alpha~ |is greater than~ 0, |Beta~ |is greater than or equal to~ |Gamma~ |is greater than~ 0. Goods |q.sub.0~ and |q.sub.1~, which are substitutes given |Gamma~ |is greater than~ 0, are perfect substitutes if and only if |Beta~ = |Gamma~. Inverse demands at any point h are given by |p.sub.0~(h) = |Alpha~ - |Beta~|Q.sub.0~(h) - |Gamma~|Q.sub.1~(h) and |p.sub.1~(h) = |Alpha~ - |Gamma~|Q.sub.0~(h) - |Beta~|Q.sub.1~(h) where |Q.sub.0~(h) and |Q.sub.1~(h) are the aggregate levels of sales for goods |q.sub.0~ and |q.sub.1~, respectively, in the hth market.

Production of |q.sub.0~ and |q.sub.1~ occur at h = 0 and h = 1, respectively. There are |n.sub.0~ and |n.sub.1~, firms, indexed by i and j, at these respective locations. Firms at either production site face constant and equal marginal costs which, for convenience, are set equal to zero. Transportation costs vary according to the location of the market. For a market located at some arbitrary h, the per unit transport costs are th|q.sup.i~(h) and t(1 - h)|q.sup.j~(h) for the ith firm located at h = 0 and for the jth firm located at h = 1, respectively, where t is the per unit distance transport costs and |q.sup.i~(h) and |q.sup.j~(h) are the sales of the ith and jth firms in the hth market. The (non-negative) profits to the ith and jth firms derived from the hth market are ||Pi~.sup.i~(h) = |p.sub.0~(h)|q.sup.i~(h) - th|q.sup.i~(h) and ||Pi~.sup.j~(h) = |p.sub.1~(h)|q.sup.j~(h) -t(1 - h)|q.sup.j~(h), respectively. Total profits for these firms are given by

|Mathematical Expression Omitted~.

Firms in the industry choose the amount of output to deliver to each market in the economy in order to maximize profits subject to two important conditions. First, the equilibrium delivered prices must preclude all opportunities for inter-market arbitrage among consumers. A delivered price schedule that permits a consumer to purchase and profitably resell a commodity at some other site is unsustainable in equilibrium absent some additional assumptions, such as informational or other transactional difficulties, that limit arbitrage among consumers. Second, the output choices must depend on the conjectural industry output variations held by firms in the industry. A firm located at a given production site must postulate how total sales in a given market are affected by changes in that firm's sales choice in that market. The equilibrium sales choices of a firm must recognize its guesses about the nature of competition among firms in the industry.(9) The vector |Theta~ = (||Theta~.sub.00~, ||Theta~.sub.01~, ||Theta~.sub.10~, ||Theta~.sub.11~) summarizes the relevant information about conjectural variations held by firms.(10) The term ||Theta~.sub.00~ = |Delta~|Q.sub.0~(h)/|Delta~|q.sup.i~(h) represents firm i's conjecture of how total sales in a given market by all firms also located at h = 0 respond to changes in its sales level. If ||Theta~.sub.00~ = 0, for example, firm i believes that total sales of |q.sub.0~ in a given market are unaffected by its level of sales in that market since its competitors at site h = 0 will change their sales to counter its alterations. Since changes in its sales do not affect total sales and therefore TABULAR DATA OMITTED price, ||Theta~.sub.00~ = 0 implies competitive or Bertrand behavior among firms at site h = 0. The term ||Theta~.sub.01~ = |Delta~|Q.sub.1~(h)/|Delta~|q.sup.i~(h) is firm i's guess about how total sales in market h by firms located at the other end of the economy, h = 1, respond to changes in its sales level. If ||Theta~.sub.01~ = 0 for example, a firm at h = 0 believes that sales of |q.sub.1~ will not be affected by changes in its sales. The term ||Theta~.sub.10~ = |Delta~|Q.sub.0~(h)/|Delta~|q.sup.j~(h) represents firm j's conjecture of how total sales in a given market by firms located at h = 0 respond to changes in its sales. If ||Theta~.sub.10~ equals the true reaction of site h = 0 firms to firm j's sales choice, for example, firm j is avon Stackleberg leader with respect to site h = 0 production. And ||Theta~.sub.11~ = |Delta~|Q.sub.1~(h)/|Delta~|q.sup.j~(h) is firm j's guess of how total sales in a market by firms also located at h = 1 respond to changes in its level of sales. If, for example, ||Theta~.sub.11~ = |Q.sub.1~(h)/|q.sup.j~(h) so that each firm is matching exactly the behavior of all other firms at h = 1, these firms are behaving cooperatively. Conjectural variations are a convenient way of summarizing firm behavior in alternative competitive environments.

Nash behavior is used to characterize the stationary points of the model. These points have the property that equilibrium in a market exists when no firm can increase its profits by changing its sales in that market given the sales of all other firms in that market. Equilibrium in the economy obtains when each market is in equilibrium. Inter-market arbitrage by consumers must also be precluded. Formally,

DEFINITION: An equilibrium is the set {|q.sup.i~(h)*, |q.sup.j~(h)*}, |q.sup.i~(h)* |is greater than or equal to~ 0 and |q.sup.j~(h)* |is greater than or equal to~ 0, i = 1,...,|n.sub.0~, and j = 1,...,|n.sub.1~, such that |for every~ h |is an element of~ |0,1~ given |Theta~, (1) ||Pi~.sup.i~(h)(|q.sup.i~(h)*, |q.sup.j~(h)*) |is greater than or equal to~ ||Pi~.sup.i~(h)(|q.sup.i~(h), |q.sup.j~(h)*); (2) ||Pi~.sup.j~(h)(|q.sup.i~(h)*, |q.sup.j~(h)*) |is greater than or equal to~ ||Pi~.sup.j~(h)(|q.sup.i~(h)*, |q.sup.j~(h)); (3) |p.sub.0~(h)* |is less than or equal to~ |p.sub.0~(h|prime~)* + t/h - h|prime~/ |for every~ h, h|prime~ |is an element of~ |0,1~; and (4) |p.sub.1~(h)* |is less than or equal to~ |p.sub.1~(h|prime~)* + t/h - h|prime~/ |for every~ h, h|prime~ |is an element of~ |0,1~.

Conditions (1) and (2) comprise the standard definition of a Nash equilibrium when the strategy space of firms is output. These conditions imply that for |q.sup.i~(h)* |is greater than~ 0 and |q.sup.j~(h)* |is greater than~ 0,

(1) |Alpha~ - |q.sup.i~(h)*||Beta~(|n.sub.0~ + ||Theta~.sub.00~) + |Gamma~||Theta~.sub.01~ - |Gamma~|n.sub.1~|q.sup.j~(h)* - th = 0

and

(2) |Alpha~ - |q.sup.j~(h)*||Beta~(|n.sub.1~ + ||Theta~.sub.11~) + |Gamma~||Theta~.sub.10~

- |Gamma~|n.sub.0~|q.sup.i~(h)* - t(1-h) = 0.

Conditions (3) and (4) preclude arbitrage by requiring that a consumer who purchases a unit of the good at one location cannot profitably offer this good for sale at some alternative location utilizing the same transportation technology as the firms in the industry. The equilibrium delivered price of, say, |q.sub.0~ at some particular location (|p.sub.0~(h)* = |Alpha~ - |Beta~|Q.sub.0~(h)* - |Gamma~|Q.sup.1~(h)*) must be no greater than its delivered price at some other locations plus the costs of transshipment (|p.sub.0~(h|prime~)* + t/h - h|prime~/).

The equilibrium sales of firms can be used to derive the delivered price schedules of goods |q.sub.0~ and |q.sub.1~, the ultimate objects of the current inquiry. This is done by substituting |Q.sub.0~(h)* and |Q.sub.1~(h)* into the inverse demand equations yielding |p.sub.0~(h)* = |Alpha~ - |Beta~|Q.sub.0~(h)* - |Gamma~|Q.sub.1~(h)* and |p.sub.1~(h)* = |Alpha~ - |Gamma~|Q.sub.0~(h)* - |Beta~|Q.sub.1~(h)*. Differences in the delivered prices of the goods as markets become more or less distant from the site of production can be derived through total differentiation. These derivatives have implications for the spatial pricing policies observed in equilibrium. Mill pricing obtains if and only if d|p.sub.0~(h)*/dh = (t and d|p.sub.1~(h)*/dh = -t; the delivered price of both commodities increases by t per unit of output for each unit of distance the market is farther from the production site. Basing-point pricing occurs if and only if (d|p.sub.0~(h)*/dh) = (d|p*.sub.1~(h)/dh) = t or (d|p.sub.0~(h)*/dh) = (d|p*.sub.1~(h)/dh) = -t; the delivered price of both commodities increases by t per unit of output for each unit of distance the market is farther from one, but not both, production sites. Below we examine the relationship between the competitive interaction of firms in the industry, as summarized by |Theta~, and the gradient of the delivered price schedules.

III. SPATIAL PRICES UNDER BERTRAND COMPETITION

Suppose that |Theta~ = ||Theta~.sub.1~ = (0,0,0,0) for all i and j. No firm in the industry believes that it can change the total quantity of sales made in any market by altering its level of sales into that market. Since these conjectures yield price-taking behavior, we refer to ||Theta~.sub.1~ as Bertrand competition. The following result is well-known.(11)

PROPOSITION 1: d|p.sub.0~(h)*/dh = t and d|p.sub.1~(h)*/dh = -t if |Theta~ = ||Theta~.sub.1~.

Proposition 1 shows that under Bertrand competition, mill pricing obtains. The delivered price of any good, say |q.sub.0~, under ||Theta~.sub.1~ must equal the marginal costs of production plus the marginal costs of delivery. Thus, |p.sub.0~(h)* = th and |p.sub.1~(h)* = t(1 - h). It is thus clear that the delivered prices of the goods increase by t for each unit of distance the market is further from the production site. Figure 1 illustrates the delivered pricing equilibrium for |Theta~ = ||Theta~.sub.1~.

Special competition under Bertrand behavior illustrates one non-collusive determinant of cross-hauling. Cross-hauling occurs when products with similar attributes are hauled in opposite directions past one another. This so-called "trans-shipping" of commodities is troublesome because of the implied inefficiencies of duplicative transport costs. Define |h.sub.0~ and |h.sub.1~ as the first location into which |q.sub.0~ is not shipped and |q.sub.1~ is shipped, respectively. That is, |q.sub.0~ is sold in h |is an element of~ |0, |h.sub.0~~ while |q.sub.1~ is sold in h |is an element of~ ||h.sub.1~, 1~ It is easy to calculate(12) that for |Theta~ = ||Theta~.sub.1~,

(3) |h.sub.0~ = ||Alpha~(|Beta~ - |Gamma~) + t|Gamma~~/t(|Gamma~ + |Beta~)

(4) |h.sub.1~ = |a(|Gamma~ - |Beta~) + t|Beta~~/t(|Gamma~ + |Beta~).

Cross-hauling occurs if and only if |h.sub.0~ |is greater than~ |h.sub.1~. Inspection of (3) and (4) indicates that cross-hauling is absent only if |Beta~ = |Gamma~; the two goods are perfect substitutes. So long as there is some product heterogeneity, cross-hauling in equilibrium is present, as illustrated in Figure 1. Haddock |1982, 293-94~ identifies some potential sources of product heterogeneity between seemingly identical products, such as the diversification of sources by buyers to insure against supply interruptions. This result indicates that the existence and magnitude of cross-hauling are functions of product heterogeneity and are not sufficient to infer collusive behavior among firms in a spatially differentiated economy.

IV. BEHAVIORAL DETERMINANTS OF SINGLE BASING-POINT PRICING

Suppose that |Theta~ = ||Theta~.sub.2~ = (0,0, |Delta~|Q.sub.0~(h)/|Delta~|q.sup.j~(h), |Q.sub.i~(h)/|q.sup.j~(h)). Firms at site h = 0 are Bertrand competitors and take site h = 1 production as given when formulating their production decisions. Firms at site h = 1 are von Stackleberg leaders with respect to site h = 0 production and act cooperatively in determining site h = 1 production. Provided demand is large relative to unit transport costs (|Alpha~ |is greater than or equal to~ 2t) and the products produced at the ends of the economy are perfect substitutes (|Beta~ = |Gamma~), the following result, which is illustrated in Figure 2, holds.

PROPOSITION 2: (d|p*.sub.0~(h)/dh) = *(d|p*.sub.1~(h)/dh) = t if |Theta~ = ||Theta~.sub.2~, |Beta~ = |Gamma~, and |Alpha~ |is greater than or equal to~ 2t.

Proposition 2 shows that under ||Theta~.sub.2~, sufficiently large markets, and product homogeneity, single basing-point pricing obtains. Since site h = 0 firms behave competitively, the delivered price of |q.sub.0~ to any market must equal the sum of marginal production and marginal transportation costs; |p.sub.0~(h)* = th. Site h = 0 firms adjust their production to market conditions and site h = 1 production to insure that |p.sub.0~(h)* = th holds. Under von Stackleberg leadership, site h = 1 firms profitably substitute sales of good |q.sub.1~ for |q.sub.0~ at the delivered price of |p.sub.1~(h)* = |p.sub.0~(h)* = th. This substitution occurs only in markets that are closer to site |h.sub.1~ than to site |h.sub.0~.(13) Cooperative behavior among site h = 1 firms supports the delivered price gradient established by site h = 0 firms. The assumption that all markets are large relative to unit transport costs (|Alpha~ |is greater than or equal to~ 2t) insures that site h = 1 firms, even under perfectly cooperative behavior, have no incentive to reduce the delivered price of good |q.sub.1~ below |p.sub.0~(h)* = |p.sub.1~(h)* = th for any market in the economy. That is, |Alpha~ |is greater than or equal to~ 2t insures that the monopoly delivered price of good |q.sub.1~ is greater than |p.sub.0~(h)* = th for all h |is an element of~ |0,1~.

Proposition 2 is the classic illustration of single basing-point pricing. The equilibrium delivered price of good |q.sub.1~ is increasing in direct proportion to the true transportation costs (t) relative to h = 0, the basing point, and not the actual production site (h = 1). That is, the consumer is assessed a freight charge that depends on the actual transportation technology but uses the base point, not the production point, as the reference for calculating total freight charges. This case also illustrates freight-absorption where some producers are shipping goods for lower delivered prices to more distant locations. Producers at site h = 1 are shipping |q.sub.1~ to more proximate consumers at higher delivered prices than are charged more distant consumers.

The delivered pricing model with |Theta~ = ||Theta~.sub.2~, |Beta~ = |Gamma~, and |Alpha~ |is greater than or equal to~ 2t shows that cross-hauling of homogeneous products should not occur under basing-point pricing.14 Similar to the case of |Theta~ = ||Theta~.sub.1~ and |Beta~ = |Gamma~, |h.sub.0~ = |h.sub.1~ = 1/2. Site h = 1 firms sell more of their output to markets closer to h = 0 but do not sell in markets more that half the distance to the basing-point, h = 0. The base site product, |q.sub.0~, is sold primarily in markets closer to h = 0 and is not sold at all in markets more than half the distance to site h = 1. Basing-point pricing and cross-hauling should not occur simultaneously, since the former is implied only for homogeneous products and the latter is possible only for |Gamma~ |is less than~ |Beta~.

The equilibrium for |Theta~ = ||Theta~.sub.2~ and |Beta~ = |Gamma~ also illustrates the markups of firms to consumers at alternative locations. Under |Theta~ = ||Theta~.sub.2~, |Beta~ = |Gamma~ |Alpha~ |is greater than or equal to~ 2t, |p.sub.0~(h)* = |p.sub.1~(h)* = th, the sum of the marginal production and transportation costs for site h = 0 firms. The markup for firms at site h = 0 is obviously the delivered price minus actual costs, or |p.sub.0~(h)* - th = 0. The markup on |q.sub.0~ is independent of the consumer's location. The markup charged by firms at site h = 1 is |p.sub.1~(h)* - t(1 - h) = t(2h - 1). This markup is always non-negative since, given product homogeneity (|Beta~ = |Gamma~), site h = 1 firms do not ship their products to markets h |is less than~ 1/2. Moreover, the markup is higher for consumers located further from site h = 0, the base point, and closer to h = 1, the production point. Note that this markup is simply the phantom freight assessed to the shipment, since t(2k - 1) = tk - t(1 - k), the freight charges associated with the delivery of site h = 0 product minus the actual transportation costs incurred from site h = 1. Figure 2 illustrates the markup charged for |q.sub.1~ as a function of the consumer's location.

The base site product, |q.sub.0~, earns competitive profits. The total profits on |q.sub.1~ derived from any market equal the markup per unit of sales (t(2h - 1)) in a given market multiplied by the total sales of |q.sub.1~ in that market ((|Alpha~ - th)/|Beta~).(15) The total profits from |q.sub.1~ equal the integral of this product over site h = 1 firms' market area (|1/2, 1~). The average markup per unit of |q.sub.1~, denoted MU(||Theta~.sub.2~), is simply the total profits on |q.sub.1~ divided by total sales of |q.sub.1~, or

|Mathematical Expression Omitted~.

The average markup on |q.sub.1~, MU(||Theta~.sub.2~), is increasing in transport costs (t) since the markup on any unit of |q.sub.1~ equals the phantom freight charges. MU(||Theta~.sub.2~) is decreasing in |Alpha~ since |Alpha~ increases sales but does not affect delivered prices, which are th for both |q.sub.0~ and |q.sub.1~. It can further be shown that t/2 |is less than~ MU(||Theta~.sub.2~) |is less than~ t. The average markup on |q.sub.1~ is less than the per unit distance transportation fee but greater than half that amount.

Proposition 2 identifies a set of conditions sufficient for basing-point pricing. The current model can also be used to highlight an important necessary condition for basing-point pricing.

PROPOSITION 3: d|p.sub.0~(h)* = d|p.sub.1~(h)* = t only if ||Theta~.sub.00~ = 0.

Firms at the base point must behave as Bertrand competitors to yield a delivered price gradient that reflects only transportation costs. When firms at the base point are other than Bertrand competitors, the delivered price gradient embodies spatial price discrimination leading to delivered prices that are more than the simple combination of a quoted mill price plus the actual shipping charges from the base site to the consumer's location. Basing-point pricing is inconsistent with non-competitive behavior by base site firms.(16)

Proposition 2 generalizes Haddock's result by allowing for more than one non-base site producer. Any number of non-base site producers is theoretically consistent with single basing-point pricing. Proposition 4 also refines Haddock's result by showing that while cooperative behavior among non-base site firms is sufficient to sustain single basing-point pricing in the economy, as Haddock's model suggests, perfect collusion among these firms is not necessary for basing-point pricing.

PROPOSITION 4: Given |Theta~ = (0,0, |Delta~|Q.sub.0~(h) / |Delta~|q.sup.i~(h), ||Theta~.sub.11~), d|p.sub.0~(h)* = d|p.sub.1~(h)* = t only if ||Theta~.sub.11~ |is greater than or equal to~ |n.sub.1~t/(|Alpha~ - t).

Deviations from cooperative behavior are admissible if the size of the markets in the economy (|Alpha~) are large relative to unit transport costs (t) and the number of firms at the non-base location (|n.sub.1~). Deviations from cooperative behavior by non-base firms can be consistent with basing-point pricing. Cooperation is, under the other values of ||Theta~.sub.2~, always sufficient for basing-point pricing. Notice, however, that Bertrand behavior by non-base site firms is never consistent with basing-point pricing. Proposition 4 quantifies the degree of imperfect competition among non-base site firms necessary to sustain basing-point pricing.

V. IMPOSITION OF MILL PRICING

The most frequently used antitrust remedy for basing-point pricing is the imposition of mill pricing. The imposition of a mill pricing constraint obviously has no effect on the prices charged by firms located at h = 0 and at least one certain and one possible effect on the other firms in the industry. First, the imposition of mill pricing affects the delivered prices charged by site h = 1 firms independent of any impact the constraint has on the competitive conjectures held by firms in the industry. Under a mill pricing constraint and the conditions of Proposition 2, site h = 1 firms cooperate in selecting a mill price, |m.sub.1~, to maximize their joint profits given that |q.sub.1~ replaces |q.sub.0~ under von Stackleberg leadership in any market for which the delivered price of |q.sub.1~ is less than or equal to the delivered price of |q.sub.0~, which is th. Given |Theta~ = ||Theta~.sub.2~ and |Beta~ = |Gamma~, the joint profits of site h = 1 firms are

|Mathematical Expression Omitted~

where h(|m.sub.1~), the market in which the delivered prices of |q.sub.0~ and |q.sub.1~ are equal under mill pricing, is given by th(|m.sub.1~) = |m.sub.1~ + t(1 - h(|m.sub.1~)). The profit-maximizing mill price, |m*.sub.1~, equates the additional revenues derived from a small increase in |m*.sub.1~ with the lost sales occurring from this increase. The lost sales result from the reduction in quantity demanded in those markets for which the delivered price of |q.sub.1~ is less than that of |q.sub.0~ plus the loss of market area that results from a higher |m*.sub.1~. Figure 3 illustrates the delivered pricing equilibrium for |Theta~ = ||Theta~.sub.2~ and |Beta~ = |Gamma~ under a mill pricing constraint.

The profits earned by site h = 1 firms are obviously lower under the mill pricing constraint when the competitive conjectures remain unchanged. Mill pricing is an option for these firms absent any constraint and is not chosen because basing-point pricing results in higher profits. The markup per unit of |q.sub.1~ under the mill pricing constraint is constant and equals |m*.sub.1~. Whether |m*.sub.1~ is larger than MU(||Theta~.sub.2~) is subject to inquiry since, under the mill pricing constraint, sales made in markets closer to h(|m*.sub.1~) have larger markups while those made closer to h = 1 have lower markups than those obtained under basing-point pricing. The following proposition illuminates the relationship between |m*.sub.1~ and MU(||Theta~.sub.2~).

PROPOSITION 5: Given a mill pricing constraint, |Beta~ = |Gamma~, and |Theta~ = ||Theta~.sub.2~, 0 |is less than~ |m*.sub.1~ |is less than~ MU(||Theta~.sub.2~).

The average markup on |q.sub.1~ is strictly positive but lower under the mill pricing constraint.

The imposition of a mill pricing constraint may also affect the competitive conjectures of site h = 1 firms. The delivered price gradient of th established by site h = 0 firms may serve as a focal point coordinating the behavior of site h = 1 firms. If the ability of site h = 1 firms to cooperate and lead site h = 0 production is somehow enhanced by the practice of basing-point pricing, the imposition of mill pricing may result in the loss of this coordination, more competitive conjectures, and even lower mill prices than those obtained under the conditions of Proposition 5.

PROPOSITION 6: Given a mill pricing constraint and |Theta~ = ||Theta~.sub.1~, 0 = |m*.sub.1~.

Whether the imposition of mill pricing affects the competitive conjectures of firms in the industry is an empirical issue. If mill pricing induces more competitive conjectures, specifically |Theta~ = ||Theta~.sub.1~, the reduction in the site h = 1 markup should equal the average phantom freight charged under basing-point pricing; MU(||Theta~.sub.2~). If mill pricing leaves the conjectural variations unchanged, the reduction in the site h = 1 markup should be less than average phantom freight charges under basing-point pricing.

VI. CONCLUSION

Basing-point pricing emerges in a delivered pricing system when firms at the base site are Bertrand competitors and firms at non-base sites are imperfectly competitive with each other and are von Stackleberg leaders vis-a-vis base site production. Basing-point pricing is associated with competitive pricing and zero markups for base site firms and positive markups for non-base firms equal to the magnitude of phantom freight collected in a given market. Basing-point pricing is not consistent with collusive conjectures for base site firms since collusion results in a non-linear delivered price schedule. Cooperation among non-base site firms is sufficient but not necessary for basing-point pricing. The imposition of mill pricing on firms practicing basing-point pricing lowers the markups charged by non-base site firms but does not affect base site pricing. If the imposition of mill pricing induces competitive conjectures, the reduction in non-base site markups equals the average phantom freight charges occurring under basing-point pricing. Cross-hauling can result from product heterogeneity and has little to do with the nature of competition among firms in an industry. More generally, cross-hauling, freight-absorption, and phantom freight can all emerge when a preponderance of the industry's production is conducted under maximally competitive conditions.

One implication of the above model is that the imposition of mill pricing on firms in an industry that practises basing-point pricing should not affect the markups of firms located at the basing-point but can reduce the markups of firms at distant sites by as much as the their total phantom freight collections. In a companion empirical piece, we found that the imposition of mill pricing by antitrust authorities in the softwood plywood industry had no effect on the markups earned by firms at the base site (Portland) but reduced the markups of firms located in the southeastern U.S. (Gilligan |1992~). Indeed, consumers located the greatest distance from the basing-point found that basing-point pricing increased the difference between the price of the non-base and base site products by nearly 65 percent of the transcontinental rail costs of shipping plywood. We argue that these results are consistent with Haddock's |1982~ model and the one presented above and somewhat inconsistent with the notion that basing-point pricing supports industry-wide collusive pricing.

APPENDIX

Proof of Proposition 1: Given the assumption that goods |q.sub.0~ and |q.sub.1~ are not perfectly substitutable, three cases must be considered. These cases correspond to the availability of the two goods in a particular market. Let |h.sub.0~ be the last market into which good |q.sub.0~ is sold and |h.sub.1~ the first market into which good |q.sub.1~ is sold. That is, |q.sub.0~ is sold in markets h |is an element of~ |0,|h.sub.0~~ while |q.sub.1~ is sold in h |is an element of~ ||h.sub.1~,1~ Solving for these three cases and substituting ||Theta~.sub.1~ into (1) and (2), the equilibrium sales for a firm located at h = 0 and h = 1 into market h are given by

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

The market boundaries |h.sub.0~ and |h.sub.1~ are arrived at by finding those parameter values such that |q.sup.i~(h)* and |q.sup.j~(h)* equal zero, respectively. Calculation yields |h.sub.0~ = ||Alpha~(|Beta~ - |Gamma~) + t|Gamma~~/t(|Gamma~ + |Beta~) and |h.sub.1~ = ||Alpha~(|Gamma~ - |Beta~) + t|Beta~~/t(|Gamma~ + |Beta~); both |h.sub.0~ and |h.sub.1~ approach 1/2 as |Beta~ |approaches~ |Gamma~. The equilibrium delivered prices are obtained through simple substitution of the equilibrium sales levels into the inverse demand equations; |p.sub.0~(h)* = |Alpha~ - |Beta~|Q.sub.0~(h)* - |Gamma~|Q.sub.1~(h)* and |p.sub.1~(h)* = |Alpha~ - |Gamma~|Q.sub.0~(h)* - |Beta~|Q.sub.1~(h)*. The change in the delivered price of |q.sub.0~ as h increases is obtained by total differentiation; d|p.sub.0~(h)*/dh = -|Beta~d|Q.sub.0~(h)* / dh - |Gamma~d|Q.sub.1~(h)* / dh = t. The change in the delivered price of |q.sub.1~ as h increases is given by d|p.sub.0~(h)* / dh = -t. It is easily verified that consumer arbitrage opportunities are absent in equilibrium.

Proof of Proposition 2: Given ||Theta~.sub.00~ = ||Theta~.sub.01~ = 0, (1) reduces to |Alpha~ - |Beta~|Q.sub.0~(h) - |Gamma~|n.sub.1~|q.sup.j~(h) = 0 and the true reaction of site h = 0 production to increases in |q.sup.j~(h) is |Delta~|Q.sub.0~(h) / |Delta~|q.sup.j~(h) = -|Gamma~|n.sub.1~ / |Beta~. Since |Beta~ = |Gamma~ (goods |q.sub.0~ and |q.sub.1~ are perfect substitutes) and site h = 1 firms are von Stackleberg leaders with respect to production, good |q.sub.1~ replaces |q.sub.0~ in all markets in which the delivered price is as least as low as the delivered price of good |q.sub.0~. Two case, thus, must be considered; markets in which only |q.sub.0~ is sold and markets in which only |q.sub.1~ is sold. Substituting |Theta~ = ||Theta~.sub.2~ and |Beta~ = |Gamma~ into (1) and (2), the equilibrium level of sales by firms located at h = 0 and h = 1 into market h is given by

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

|p.sub.1~(h)* = th is the highest price site h = 1 firms can charge and secure all sales in market h. It must be confirmed that site h = 1 firms do not wish to charge less than th in all markets in which |q.sub.1~ is sold. For h |is greater than or equal to~ 1/2, the unconstrained optimal level of output for a firm located at site h = 1 is ||Alpha~ - t(1 - h)~/2|Beta~|n.sub.1~; the monopoly optimum. However, this level of sales results in a delivered price for |q.sub.1~ greater than the price at which site h = 0 firms are willing to supply all of that particular market's demand. |q.sub.i~(h)* = (|Alpha~ - th)/|Beta~|n.sub.1~ is the lowest output level, and thus the highest delivered price, at which firms at site h = 1 secure all of the sales in the h-th market. The assumption |Alpha~ |is greater than or equal to~ 2t guarantees that ||Alpha~-t(1-h)~/2|Beta~|n.sub.1~ |is less than or equal to~ (|Alpha~ - th)/|Beta~|n.sub.0~. The equilibrium delivered prices are |p.sub.0~(h)* = |Alpha~ - |Beta~|Q.sub.0~(h)* - |Gamma~|Q.sub.1~(h)* and |p.sub.1~(h)* = |Alpha~ - |Gamma~|Q.sub.0~(h)* - |Beta~|Q.sub.1~(h)*. The change in the delivered price of |q.sub.0~ as h increases is given by d|p.sub.0~(h)*/dh = -|Beta~d|Q.sub.0~(h)*/dh - |Gamma~d|Q.sub.1~(h)*/dh = t. The change in the delivered price of |q.sub.1~ as h increases is given by d|p.sub.0~(h)*/dh = t.

Proof of Proposition 3: Under perfect substitutability (|Beta~ = |Gamma~), the two goods are never sold in the same markets. Thus, |p.sub.0~(h)* = |Alpha~ - |n.sub.0~|(|Alpha~ - th)/(|n.sub.0~ + ||Theta~.sub.00~ + ||Theta~.sub.01~)~, and (d|p.sub.0~(h)*/dh) = |n.sub.0~t/(|n.sub.0~ + ||Theta~.sub.00~ + ||Theta~.sub.01~). The delivered price gradient, which is linear, differs from the gradient of the true transport cost by those factors (the number and behavior of firms) that affect the perceived marginal revenue of a firm at site h = 0.

Proof of Proposition 4: At the equilibrium delivered price of |p.sub.1~(h)* = th, the marginal revenue of any firm at site h = 1 equals |MR.sup.j~(h) = th - |q.sup.j~(h)*|Beta~||Theta~.sub.11~ - t(1 - h) = th - (|Alpha~ - th)||Theta~.sub.11~/|n.sub.1~ - t(1 - h). For |p.sub.1~(h)* = th, |MR.sup.j~(h) |is less than or equal to~ 0; no firm at site h = 1 must have an incentive to increase its sales in any market. |MR.sup.j~(h) can be strictly less than zero since, at |p.sub.1~(h)* = th, any reduction in |q.sup.j~(h) will increase the delivered price of |q.sub.1~ resulting in the loss of all sales to site h = 0 firms. Rearranging the expression for |MR.sup.j~(h) implies that ||Theta~.sub.11~ |is greater than or equal to~ |n.sub.1~ |th - t(1 - h)~/(|Alpha~ - th). The right-hand side of this expression takes its greatest value for h = 1; the market at the production site of |q.sub.1~. Substituting h = 1 into this expression implies that ||Theta~.sub.11~ |is greater than or equal to~ |n.sub.1~t/(|Alpha~ - t). Since the smallest value the denominator of this expression can take is |Alpha~ = 2t, the largest value the left-hand side can attain is |n.sub.1~. Thus, ||Theta~.sub.11~ = |n.sub.1~ is sufficient behavior among non-base site firms to sustain the basing-point pricing equilibrium. Notice, however, that for |Alpha~ |is greater than~ 2t, the conjectures held by firms at h = 1 need not necessarily be perfectly collusive. If |Alpha~ is large relative to t and |n.sub.1~, imperfect collusion among non-base site firms can be sufficient for basing-point pricing.

Proof of Proposition 5: Under |Theta~ = ||Theta~.sub.2~, the joint profits of site h = 1 firms equal

|Mathematical Expression Omitted~

where h is the endogenously determined lower boundary of site h = 1 firms' market. Maximizing with respect to h implies that the equilibrium market boundary must satisfy |h.sup.2~4t - h2(2|Alpha~ + t) + 3|Alpha~ - t = 0, which implies that h |is less than~ 3/4. h |is less than~ 3/4 implies that |m*.sub.1~ |is less than~ t/2, which is lower than MU(||Theta~.sub.2~).

Proof of Proposition 6: From Proposition 1 above, |Theta~ - ||Theta~.sub.1~ implies that |p.sub.1~(h)* = t(1 - h), or |m*.sub.1~ = 0.

1. Machlup's |1949~ important work catalogues many prominent uses of basing-point pricing, including steel, cement and corn products. Wilcox |1966~ also provides an illuminating history of basing-point pricing, including its first recorded use in the steel industry under the "Pittsburgh-plus" plan. The Federal Trade Commission |1978~ provides a comprehensive description of how basing-point pricing was practised in the plywood industry during the early and middle parts of the 1970s.

2. Many inquires have focused on cost-based determinants of basing-point pricing, such as economies in the production or transportation of geographically differentiated goods. This article ignores such cost-based arguments. Also, apart from the determinants of basing-point pricing, economists and policy-makers have also been concerned with its effects. DeCanio |1984~ provides an analysis of the relative consequences of mill and multiple basing-point pricing under varying degrees of competition.

3. This wisdom is reflected in the conduct of contemporary public policy towards basing-point pricing. In Federal Trade Commission |1978~, the government argued that the effects of the use of basing-point pricing in the plywood industry were, among other things, to "reduce and hinder actual and potential competition among |firms~ in the sale and distribution of softwood plywood." Karlson's |1986~ response to Haddock's |1982~ article also takes this view. Wilcox |1966, 229~ shares this view and reports that, "where delivered prices have regularly been matched by all sellers, the practice has usually been facilitated by some sort of an agreement or understanding with respect to many, if not all, of the factors that influence the prices that are charged. The possibility that variations in price may result from differences--whether deliberate or inadvertent--in the methods of selling and pricing employed by different members of the industry is thus removed."

4. Machlup's |1949~ arguments are consistent with this view, too.

5. Benson, Greenhut, and Norman |1990~ also show that basing-point pricing can arise when production at the basing point is competitive while production at the periphery is monopolized. In contrast to Haddock, they focus on the behavior of the non-base site firm to highlight the non-rivalrous foundations of basing-point pricing. In their model, they further show that monopolized periphery production is necessary for basing-point pricing. Below we show that this is not generally true.

6. Karlson |1986~ identifies the conditions necessary for this result and some important exceptions.

7. Other authors that have provided derivations of equilibrium spatial pricing policies under a variety of imperfectly competitive conditions include Greenhut and Greenhut |1975~ and Spulber |1981~. The present model differs from Greenhut and Greenhut in that it entertains a more complex set of competitive relationships among firms in the industry. Spulber examines the spatial pricing policies of a monopolist assuming that consumer arbitrage possibilities are limited.

8. Thus, the locations of firms in the model are assumed fixed and exogenous. To be sure, spatial pricing policies affect the distribution of firms throughout the economy, as the seminal work of Greenhut |1956, chapter II~ and the more contemporary effort of Soper et al. |1991~ show. The spatial policies in these analysis are exogenous. A more complete analysis than the present one would solve for the equilibrium spatial pricing policy and the equilibrium distribution of firms as a function of firm behavior.

9. As always, this conceptualization is not analogous to a dynamic model of intra-industry behavior. Rather, it is a static framework that incorporates differing assumptions about the degree of non-competitiveness among firms in the industry. For a further discussion of these issues, see Tirole |1988, 244-5~.

10. As the notation indicates, it is assumed that (1) firms at a given location hold identical conjectures and (2) the conjectures are not market specific. The later assumption implies that, for instance, if a firm believes that it is a Bertrand competitor with all other firms in one market, it also believes that it is a Bertrand competitor with these same firms in all other markets.

11. Proofs to the propositions are contained in the Appendix.

12. See the proof of Proposition 1 in the Appendix for the derivation.

13. Greenhut |1956, 57-83~ recognized the necessity of this behavior, which he termed "restrained price competition," for basing-point pricing. His focus, however, was on the effects of restrained price competition on firm location.

14. The absence of cross-hauling under basing-point pricing is informally referred to as an illustration of "spontaneous basing-point pricing." Clearly, there are historical uses of basing-point pricing that involve rampant cross-hauling--the "Pittsburgh-plus" system in the U.S. steel industry. The current analysis thus supports McGee's |1954~ contention that cross-hauling of identical products should not generally occur under spontaneous basing-point pricing. Greenhut |1956, Chapter II~ provides an analysis of the locational choices of firms under basing-point pricing with cross-hauling.

15. Recall that the inverse demand for |q.sub.1~ in any market is given by |p.sub.1~(h) = |Alpha~ - |Beta~|Q.sub.1~(h).

16. Capozza and Van Order |1977~ present a similar result and Haddock |1982, 299-300~ exposes its implications for the hypothesis that basing-point pricing is associated with collusion among base-site firms.

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Capozza, Dennis R., and Robert Van Order. "Pricing Under Spatial Competition and Spatial Monopoly." Econometrica, September 1977, 1329-44.

DeCanio, Stephen A. "Delivered Pricing and Multiple Basing Point Equilibria: A Reevaluation." Quarterly Journal of Economics, May 1984, 329-49.

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