Delivered Pricing, FOB Pricing, and Collusion in Spatial ... · pricing systems (in particular, UDP and basing-point pricing) are used in highly monopolistic and in highly competitive

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Delivered Pricing, FOB Pricing, and Collusion in Spatial MarketsAuthor(s): Maria Paz EspinosaSource: The RAND Journal of Economics, Vol. 23, No. 1, (Spring, 1992), pp. 64-85Published by: Blackwell Publishing on behalf of The RAND CorporationStable URL: http://www.jstor.org/stable/2555433Accessed: 24/04/2008 13:48

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RAND Journal of Economics Vol. 23, No. 1, Spring 1992

Delivered pricing, FOB pricing, and collusion in spatial markets

Maria Paz Espinosa*

The article examines price discrimination and collusion in spatial markets. The problem is analyzed in the context of a repeated duopoly game. I conclude that the prevailing pricing systems depend on the structural elements of the market. Delivered pricing systems emerge in equilibrium in highly monopolistic and highly competitive industries, while FOB is used in intermediate market structures. Thefact driving this result is that delivered pricing policies allow spatial price discrimination that facilitates collusion, but at the same time they have a very competitive feature: they are the only pricing rules that could be sustained in a very competitive market structure.

1. Introduction

* In some markets, sellers have the ability to discriminate among consumers by making price vary according to some characteristic of the buyer. Here I analyze a particular type of price discrimination common in spatial markets where a consumer's location is observable. In this situation the choice variable for the firm is a price system that specifies a price per unit of product at each location: each firm must choose a function p(x), where x is the distance between the location of the consumer and the location of the firm.

In practice, different industries use different types of pricing systems, p(x), and this suggests that the prevailing pricing system depends on the structural elements of the market. When the pricing policy is FOB,' consumers can pick up the product at the mill, paying the mill price p and incurring the transportation cost from the producer's to the consumer's location, i.e., p(x) = p + t(x), or the seller may deliver the good to the buyer's location, as long as it charges mill price plus transportation costs. Delivered pricing policies are pricing rules p(x) that are not based on consumers picking up the product at the mill; the firm delivers the product at the consumer's location. In a perfectly competitive world with a continuum of firms, an FOB pricing system would be expected: p(x) = c + t(x), where c is the marginal cost of production. In the case of a market with two firms located at the

* Universidad del Pais Vasco, Bilbao, Spain. This article is based on Chapter 1 of my Ph.D. thesis. I would like to thank Ramon Caminal, Richard Caves,

Andreu Mas-Colell, Garth Saloner, and two anonymous referees for their comments. I am especially grateful to Michael Whinston for many helpful suggestions and advice. Needless to say, any remaining errors and shortcomings are mine. Financial support from Harvard University Social Science Dissertation Fellowship is gratefully acknowl- edged.

' Free on board.

64

ESPINOSA / 65

same point, assuming there is no collusion, we would also expect p(x) = c + t(x) (the Bertrand solution). A monopolist, however, could profit from price discrimination and therefore would not use FOB. This association between FOB and competition is part of the reason why other pricing systems have been labelled as "monopolistic" practices and devices that facilitate collusion.2

Several arguments have been given that associate delivered pricing3 with collusive practices.4 Some of them relate to incomplete information and the problem of detection. Under an FOB pricing system firms could disguise price cuts as lower transportation costs, while in a delivered pricing system this problem could not arise, since the implicit agreement is on final prices including freight charges. This argument does not seem to be a strong support for the idea of delivered pricing systems as collusive devices, since it relies on the unlikely assumption that firms observe the price to the consumer in a delivered pricing system but not when FOB is used.5

Stigler (1964) holds that identical delivered pricing systems would make collusion easier in markets with geographically unstable demand, by allowing firms in a territory with low demand to invade territories with high demand. This invasion would eliminate the need for frequent changes in the price schedule and make any collusive price more stable.

I shall argue that in an oligopolistic market, when firms are spatially dispersed a delivered pricing system allows each seller to discriminate among consumers, charging a higher net price (price minus cost) to consumers with higher net reservation value (reservation value minus cost) for the product of that particular seller.6 This increases profits for the seller in the same way standard price discrimination increases the profits of a monopolist. On the contrary, an FOB pricing system entails the same net price for all consumers, no matter their location, and therefore it eliminates the possibility of price discrimination. Thus, there is a sense in which delivered pricing systems can be considered "more collusive" than FOB. Since they increase the profits to be made by colluding, they ease collusion.

The view that delivered pricing policies are monopolistic practices has been widely maintained (by Fetter (1937), Machlup (1949), and Scherer (1980), among others), and it is also supported by casual empirical evidence indicating that many industries characterized by high concentration and a spatially differentiated product (for example, cement, steel, and corn in the United States) did use delivered pricing systems.7 Stigler (1949) provides examples of industries where cartelization coincided with the introduction of delivered pricing systems. The German cement industry was on an FOB price system until a cartel was formed, after which it used delivered prices. The bituminous coal industry in Great Britain was using FOB prices before the compulsory cartelization of 1930, and moved to a delivered price system after that date.

2 "[TI]he delivered-price system as here used provides an effective instrument which, if left free for use of the respondents, would result in complete destruction of competition and the establishment of monopoly in the cement industry" (Mr. Justice Black, Federal Trade Commission v. Cement Institute et al., 333 U.S. 683 (1948)).

3There is a variety of delivered pricing policies. In an identical delivered pricing system, all the firms quote the same price to any one buyer. The most common identical delivered pricing rules are basing-point pricing and uniform delivered pricing. In a uniform delivered pricing system, each firm quotes the same price to all the consumers. Under a basing-point pricing rule, firms decide on the location of a base point and a price at that location; the price at any other location is calculated as the base price plus a transportation charge from the base point.

4 Scherer ( 1980) argues that delivered pricing systems lessen competition because they reduce a "complicated price quotation problem, if executed independently, to a relatively simple matter of applying the right formula." FOB mill pricing would make the "avoidance of independent pricing more difficult." See also Carlton ( 1983).

5 Carlton ( 1983) argues that when firms do not observe prices and use shifts in business to detect deviations, FOB would be a better collusive device.

6 Cost includes transportation cost, so that for a consumer, the net reservation value is different depending on which seller he is considering buying from.

'See Greenhut, Greenhut, and Li ( 1980).

66 / THE RAND JOURNAL OF ECONOMICS

The view of the Federal Trade Commission is that delivered pricing systems lessen competition. In Boise Cascade v. FTC ( 1980), the FTC challenged the practice of a basing- point pricing system in which transportation charges were always calculated from the west coast, regardless of the product's origin. In another case, Ethyl et al. ( 1981), the FTC challenged the use of uniform delivered prices in which the price charged is independent of the buyer's location.

However, identical delivered pricing systems have a very competitive feature. Given that in an identical delivered pricing system all firms are charging the same price at a given location, if a firm slightly decreased its price it would get the whole market, or at least that part of the market that it is profitable to sell to; whereas in an FOB pricing system, a small decrease on the price at the mill gets only the marginal consumers for the firm that lowered its price, and to increase its market share significantly a firm needs to decrease its price by a considerable amount. In this sense, identical delivered pricing systems are a method of increasing the degree of competition through interpenetration of regional markets.8 This argument seems to indicate that a cartel trying to enforce collusion will try to avoid identical delivered pricing systems, and that these pricing systems cannot be considered as collusive practices. And in fact there must be other reasons that explain the use of delivered pricing, since in many highly competitive industries, such as retail drugstores, pizza deliveries, food retailing, furniture stores, and mail-order retailing, uniform delivered pricing (UDP), also called the postage stamp system, is quite prevalent.

The purpose of the article is to reconcile the two arguments and explain why delivered pricing systems (in particular, UDP and basing-point pricing) are used in highly monopolistic and in highly competitive industries. I find that, under some restrictions on the pricing rules, UDP appears as the equilibrium pricing policy in collusive and competitive industries, while FOB is the equilibrium policy for intermediate market structures. With no restrictions on the pricing rules, I obtain that basing-point pricing emerges in collusive equilibria but is also the only pricing system that could be sustained in a very competitive market structure; moreover, basing-point pricing appears as the pricing rule in the optimal punishment path sustaining the collusive equilibria.

The article is organized as follows. In Section 2, a repeated duopoly game is used to compare UDP and FOB pricing systems. Markets are characterized by two parameter values: the discount factor 6 and the transportation cost per unit value of the commodities.9 For a given discount factor, a higher transportation cost implies higher market power for each seller in its local markets, and this makes possible more collusive outcomes. Note that a lower transportation cost means a more elastic demand function, and this we associate with a more competitive industry. On the other hand, for a given transportation cost, a higher discount factor also implies that more collusive outcomes are possible in equilibrium. I shall refer to a market as "competitive" if it has low transportation costs and a low discount factor, and a market will be said to be "monopolistic" when it has a high discount factor and/or high transportation costs. For 6 = 0, t(x) = 0, we get the most competitive result: price is equal to marginal cost and profits are zero (Bertrand competition). For 6 = 1 we can get the monopoly solution, and for t(x) high enough we also get the monopoly solution. The idea of competitiveness is also usually related to the number of firms in the market; however, our interest is in industries where, due to fixed costs for example, there is room for only a few firms. When the number of firms is not a variable, the degree of competitiveness is best measured by the values of 6 and t. I start by looking at the simplest case in which only UDP and FOB are feasible (since there are costs associated with the implementation

8 Scherer ( 1980) expresses the belief that this greater interfirm contact leads mainly to more intense nonprice rivalry, but not to price cuts as a competitive weapon.

9 The transportation cost is assumed to be linear: t(x) = tx.

ESPINOSA / 67

of a complicated pricing rule, firms may choose pricing systems with very few parameters). The result is that UDP will be used in very competitive and also in very monopolistic markets. For market structures that are neither very monopolistic nor very competitive, the model predicts FOB.

In Section 3 I generalize the result to linear pricing schedules. The main conclusion of Section 2 still holds: UDP will be used in very monopolistic and in very competitive market structures. For market structures that are neither very competitive nor very monopolistic, the model predicts pricing systems with slope in the interval [0, tI, and the value of the slope is nonincreasing in (. This result is consistent with the empirical findings of Greenhut, Greenhut, and Li ( 1980), who find that the more competitive the market is, the steeper is the delivered price schedule. I also obtain the result that when firms are able to implement market share agreements, the pricing system has slope in the interval [-t, 0 ], and the slope is nondecreasing in (. I find this result to be consistent with existing empirical evidence: in Japan and West Germany, countries where coordination on market share is not as difficult as in the United States, delivered pricing policies tend to have negative slopes (see Greenhut (198 1 )), while for American firms the slope is positive.

Finally, Section 4 considers the case of unrestricted pricing policies. It is shown that to sustain any FOB price, the market must be monopolistic. If the possibilities for collusion are limited, then FOB will not emerge as the equilibrium of the game. This result strongly contradicts the idea of FOB as a competitive pricing system. It shows that, even though FOB is the outcome of a perfectly competitive market with a continuum of firms, or a market where the products are not geographically differentiated, this idea cannot be carried over to markets where only a few firms are present and where products are spatially differentiated. In this type of market, and for values of 6 such that the monopoly solution is not sustainable, basing-point pricing and, in general, identical delivered pricing rules appear as the equilibrium policies.

2. FOB and UDP: competitive versus collusive theories

* The debate on the competitive or collusive nature of delivered pricing has centered on the comparison of very simple pricing rules and always has FOB as the reference point. A reason why firms may be restricted to simple pricing rules (rules with only a few parameters to optimize over) is that there are costs associated with the implementation of a complicated pricing rule. Since explicit communication between firms for price setting is illegal and the agreements are implicit, the greater the number of parameters the firms have to agree on, the more difficult it will be to sustain collusion; even in the absence of collusion, it may be costly for a firm to implement a complex pricing rule. A different explanation would be that simple rules may be optimal. I start by comparing UDP and FOB pricing rules. Later on I shall analyze the case of more general pricing systems.

o A repeated duopoly game. Since our main concern is to determine the relationship between collusion among firms and the spatial pricing systems that they use, we need a model that allows for repeated interaction and in which products are spatially differentiated. In this section I develop a simple model with these features.

There are two firms that produce a homogeneous good. They are located at the extreme points of the interval [0, 1]. There are no costs of production, and the transportation cost is equal to t per unit of distance. I assume that the firms interact repeatedly with an infinite horizon and maximize discounted profits. The discount factor is denoted 6, 6 E [0, 1).

Consumers are uniformly distributed with a unit density along the interval [0, 1]. Their preferences are as follows: each consumer has a reservation value R for the good and con- sumes precisely one unit of the good per period of time, buying from the firm that has the lowest final (delivered) price, as long as the total payment does not exceed the reservation


value, and buys nothing otherwise. When the two firms quote the same delivered price at a given location, the consumer chooses the nearest supplier. The good cannot be stored, but in a given period consumers may buy the product and resell it for a profit to other consumers.

In each period, firms simultaneously announce a price function. The delivered price that corresponds to FOB is p(x) = PC + tx for all x E [0, 1]. Analogously, for UDP, p(x) = Pu for all x E [0, 1]. G( UDP, FOB) denotes the repeated game in which firms are allowed to use either a UDP or an FOB pricing system. A UDP strategy is a function that selects, for any history of play, a pair (Pu, a) E Go X [0, 1]. In a UDP pricing system a firm may not want to sell to consumers located too far away, since the firm has to pay the transportation costs and the price Pu may not be high enough to cover costs. Therefore, I assume that firms may refuse to sell; ai denotes the fraction of the market firm i is willing to sell to (whenever I do not specify the value of a it is supposed to be 1 ). An FOB strategy is a function that selects, for any history of play, an element Pc E P. In an FOB pricing system firms are always willing to serve the entire market, since the transportation costs are added on to the price.1

The payoff functions of the firms in each period are given below. When both firms use UDP strategies,

ll[(pui, ai), (puj, aj)] = (Pui - tx)dx,

where

0 if pui > R

min [1, ai] if Puj > Pui, Pui ? R Z=

min [a, 1 - aj] if Puj <Pui, Pui ? R

min {ai, max [1/2, 1 - a]} if Puj = Pui, Pui ? R.

When both firms use FOB strategies,

[0 if Pci > PC1 + t

1 (pc, Pc) Pci min (

1 P tc

if Pci < Pcjt (1)

Pci min Pcj pci+t (R pci)] if IPci -Pcj <?t.

When firm i uses a UDP strategy and firm j uses an FOB strategy,

ji ((Pui, ?ai ) , Pcj) = (Pui - tx)dx, (2)

where

A max [min [ai 1 _ (PuiP)] 0] if p R

t 0 if pui > R

I1j((Pui ai) Pc) j)= min{ I- max [min [i 1 - t P ] ?] t Pcj

This is true as long as pc is greater than or equal to marginal cost. With regard to equilibrium analysis, there is no loss of generality in restricting FOB prices to be greater than marginal cost.

ESPINOSA / 69

Our focus will be on symmetric equilibria of G( UDP, FOB). It is worth noting that when Pu ? R is charged as the UDP price by both firms, profits for each firm are

rmin[ a,'/2 ]

ll((pu, a), (Pu, a)) = J (Pu - tx)dx,

while when an FOB price of Pc is charged by both firms, profits for each firm are

{ Pc . 2R-t 2 2f Pc? 2

1c(PcPC)=l R-pC 2R-t Pc t if PC > 2

The explicit consideration of the repeated nature of the game expands considerably the range of possible outcomes. In particular, outcomes that are more cooperative than the static solution are attainable. Since our main concern is collusion, I focus on the optimal collusive equilibria of the game for given values of the market parameters; i.e., among the symmetric equilibria, our interest is in the maximal amount of profits sustainable and, most importantly, the form that pricing strategies take in such equilibria.

0 Characterization of equilibria. In order to understand the importance of spatial differentiation among firms, let us start with the case in which both firms are located at the same point in the interval [0, 11. 11

Abreu ( 1988 ) has shown that any given path is sustainable in a perfect equilibrium if and only if it can be sustained by reversion, in case of a deviation from that path, to a punishment that is the deviator's worst possible perfect equilibrium. Therefore, knowing the worst perfect equilibrium for each player allows the characterization of all perfect equilibrium paths. When there is no geographical differentiation among the products of the firms, setting an FOB mill price of zero repeatedly is a perfect equilibrium path that yields zero profits to both players. Since in this model firms always have the option to obtain zero profits by not producing, the repeated FOB solution p(x) = t(x) is the lowest-payoff perfect equilibrium path. Any other perfect equilibrium path is sustainable if and only if it can be sustained by reversion to this FOB path: if a player deviates from the specified actions, the FOB policy p(x) = t(x) starts the following period. Since the firms are located at the same point, a deviation consisting of undercutting the opponent by some small amount would yield twice as much profit as conforming to the action that gives the specified payoffs: lI(p(x), p(x)). It follows that to sustain profits higher than the static Nash equilibrium profits, 6 should be greater than '/2: 12

1- _ H(p(x), p(x)) > 2I1(p(x), p(x)) + 1 (30 ?6 ? V2.

Moreover, for 6 2 1/2 the monopoly solution is sustainable. It is clear that for a monopolist, the best pricing policy would be a UDP p(x) = R.13 Since UDP extracts all the consumer surplus, no other pricing system could give higher profits. Thus, when the products are not spatially differentiated, FOB appears as the equilibrium policy for competitive markets, while UDP would require 6 2 1/2.

Consider now the opposite case. Firms are located at the endpoints of the interval [0, 1], and the transportation costs are very high. As a consequence, spatial differentiation

I Or the transportation cost is zero. 12 See Abreu ( 1988). 13 Note that this result comes from our assumptions about demand; when reservation values are not identical

for all consumers, a monopolist who owns the two plants may prefer a different pricing system.


is also high. In particular, when t > 2R, the transportation costs are so high that both firms

are monopolists in their respective local markets. Each firm has a fraction R

of the market t

and does not find it profitable to sell in the other firm's local market. There is no competitive interaction between the firms. In this case, a UDP pricing system p(x) = R and any a E [1/2, 1] is the profit-maximizing strategy for both firms in G(UDP, FOB) for any value of a.

Define w = . In the remainder of this section the symmetric equilibria of t

G( UDP, FOB) are characterized for the case w ? 1; that is, given that firms are located at the endpoints of the interval [0, 1], any of the firms could find it profitable to sell in the other firm's local market. There is competitive interaction in any segment of the market.

We start by calculating the values of 6 such that the equilibria of G( UDP, FOB) coincide with the monopolist solution. For high enough values of the discount factor, firms will be able to implement the monopoly solution, and therefore a UDP price p(x) = R will also be the solution in this case. We now calculate the values of 6 that allow the duopolists to implement a UDP pricing policy p(x) = R.

For a UDP p(x) = R to be sustainable in a perfect equilibrium of G( UDP, FOB), 6 should satisfy

1 1/2 e1 a Zj g (R-tx)dx > (R-tx)dx+1 a K(6, t, R), (3)

where K(a, t, R) denotes the lowest payoff attainable in a perfect equilibrium of G(UDP, FOB): its value for the set of 6 that sustain p(x) = R is zero (see Appen- dix A). This inequality states that the profits from conforming to a UDP pricing policy p(x) = R are greater than the profits from deviating optimally and reverting to an optimal punishment path forever after. Note that when a firm is considering whether to deviate from the path [(R, 1 ), (R, 1 )] , the best deviation is to undercut the other firm slightly and sell to the entire market (given our assumption that the transportation cost is not too high, w ? 1).

From (3) we obtain 6 a 4? This result can be stated as the following: 8w - 4

4w - 3 Lemma 1. For every w ? 1 and 6 2 8 ,4 a UDP policy Pu = R is sustainable as a perfect

equilibrium of G( UDP, FOB).

This result gives the values of 6 for which it is possible to sustain full collusion. Now let us turn to the case in which only imperfect collusion is attainable.

UDP policies. The optimal deviation from a UDP policy is to undercut the opponent by some small amount.14 Two cases must be distinguished depending on whether the optimal deviation entails selling to the whole market or to only a fraction of it.

Case a: Pu ? t. The set of Pu sustainable in a perfect equilibrium, conditioned on Pu 2 t, is given by the inequality

1 C1/2 a 1- aI (Pu-tx)dx >

J (Pu-tx)dx + i K(6, t, R). (4)

14 To deviate from a UDP pricing rule, firms will never use an FOB policy. The reason is that FOB implies an unnecessarily low price in the local market for the deviating firm, while UDP is a more aggressive type of

ESPINOSA / 71

In this case, given that the price is higher than the cost of delivering the good to the consumers located at the other end of the interval, when a firm deviates it finds it profitable to sell to the whole market. From (4), and using the fact that K( (, t, R) 2 0,

3 - 46 PU < 4 - 86 3-

Since Pu is constrained to be greater than t, 4 8-4 must be at least 1, which implies

that 6 2 1/4 is a necessary condition for Pu ? t. We also must check what the value of K((6, t, R) is; in Appendix A I show that for 6 2 1/4, K((6, t, R) = 0. Thus, 6 1/4 is sufficient as well as necessary for sustaining Pu ? t.

t Case b: Pu < t. Obviously, P5 = - is always sustainable since it is a Nash equilibrium

2 t

of the one-shot game. We are interested in the set of Pu greater than - that are sustainable 2

in a perfect equilibrium. The set of sustainable Pu in the interval [2 t) is given by the

inequality

1 l1/2 Pu/lt 6

1 6(J (Putx)dx 2 J (Pu-tx)dx + 1 K(65 t, R). (5)

In this case, when a firm deviates it does not find it profitable to sell to the entire

market. The marginal consumer is at a distance Pu from the firm. From (5), t

1+ 1/i

PU<2( 1 - 6)

Since we are conditioning Pu to be smaller than t, this corresponds to the case 6 < 1/4.

In Appendix A I show that K((, t, R) > 0 for 6 < 1/4, so that the inequality is strict. The following table gives, as a function of the discount factor, the maximum values of

UDP prices and profits sustainable in a symmetric perfect equilibrium.15

6 Pu llu

E4w - 3 14R- t

[8Jw 4 R 8

I4w-i3) 3-46t 1 -6 l (6)

[0, 1/4) < l+1/ 1 + 6 +

2V t

______________ 2(1 -(5) 8(1 -(5)

FOB policies. The above has assumed that firms follow UDP rules. I now show the most collusive FOB policies.

deviation given that it allows a greater penetration of the rival's market, without having to decrease prices in the home market.

15 Note that for low values of the discount factor, in particular for 6 < 1/4, I give only the upper bounds for the sustainable UDP price and profits (see Appendix A).


Lemma 2. For w > 1, in G( UDP, FOB) a UDP deviation is always an optimal deviation from an FOB pricing system. The profits from using the optimal deviation are

(PC + t)2 if PC ' 2t, PC'R--

I t 2 lI[pu(PC) PC] = (7)

tPc if PC>2t, pC R--. PC2 2

Proof. Available on request.

Figure 1 illustrates the fact that in G( UDP, FOB) the optimal deviation is always a UDP deviation. From Lemma 2 and assuming K(a, t, R) = 0, the optimal collusive FOB prices are given by

(2-PC (6tp )2( ) for PC?2t p2?R-- (8)

PC (PC - - (I - ) for PC>2t, pC R- - (9) 2 2 ~~~~~~~~~~~~~2

t Note that the constraint Pc < R - 2 is always satisfied for w > 1. From (8),

2

PC +

2(6 + 1 t, and since we are constraining PC to be less than 2t, we must have

1 + 26 + 126 - 3 less than 2, which implies 6 a

1/3. 2( 1 - 6) - From (9), Pc = t which corresponds to the case of Pc greater than 2t, i.e.,

(1 - 2a) 6 > '/3.

Note that for 6 < 1/3, the expression for Pc has no meaning unless 6 1/4 . As Lemma 4 in Appendix B shows, this is due to the fact that for 6 < 1/4, FOB prices cannot be sustained in equilibrium.

FIGURE 1

Pu (Pc)

Pc

P (Pc)

0 1

ESPINOSA / 73

Therefore, the profits from using the optimal collusive FOB pricing system in G(UDP, FOB) are16

[min + 26 + f12a-3 2R-t] if ?4a?1/3

1iC= - L

2R -6 t i 1/C 13 (10)

min [2(1 _ 26) tR 4 ] if a> V3.

For 6 < 1/4 there is no FOB price sustainable in G( UDP, FOB). The reason is that for low values of the discount factor, deviations from the system are very profitable. The lack of effective penetration into the opponent's market (since each firm's price increases with distance at a rate t) leaves the opponent with the possibility of very profitable deviations in its own local market. Therefore, FOB is not likely to be observed in markets with a spatially differentiated product where the possibilities for cooperation among the firms are limited.

The profits from using the optimal collusive FOB pricing system in G( UDP, FOB) are given by (10), and the profits from using the optimal collusive UDP policy have been calculated earlier. The main result of this section is summarized in Figure 2 (see also Prop- osition 4 in Appendix B).

This result indicates that UDP is likely to be observed in very monopolistic industries (defined as industries where the transportation cost is high, w c 5/4, and/or the discount factor is high, 6 2 6*( w)), but also in very competitive industries (industries with a low discount factor, 6 <5 k+ and a low transportation cost, w > 5h). For intermediate market structures the model predicts FOB. This is consistent with the observation that UDP is utilized in industries with the presumption of collusive behavior but also in very competitive markets, and it may provide an explanation for this apparent paradox.

The result derives from the fact that, unless prices are low, the incentive to deviate is lower when the rival is using FOB rather than UDP. In a UDP system the two firms are charging the same price at a given location, so that if a firm slightly decreases its UDP price it gets the whole market; this greater interpenetration of markets makes deviations very

FIGURE 2

UDP

1/2

1/4 a+

UDP

0 5/4

16 See Appendix A for the value of K(B, t, R).


profitable. When prices are low, however, UDP is less profitable to deviate from, since for low prices a firm is not interested in serving the entire market (it would make losses in the consumers located far away), while FOB leaves the opponent with the possibility of very profitable deviations in its own local market.

Thus, for high values of 6, a discriminatory pricing policy is preferable for a cartel, since the enforcement of collusion is not a pressing question, and price discrimination allows the duopolists to extract a greater proportion of the consumer surplus. However, as 6 goes below 0*( w), the enforcement of collusion becomes harder and FOB looks more attractive as a way of softening the incentive to cheat and hence obtaining higher prices. This is the case until 6 reaches (+k. For lower values of the discount factor, no FOB price can survive in equilibrium; UDP deviations are too profitable compared to the profits to be made along the FOB path.

Nevertheless, below (+ UDP is still sustainable because for 6 < (+, UDP deviations do not involve selling to the entire market. Take for instance the case 6 = 0. For 6 = 0 the

t UDP equilibrium price is Pu =2 in a UDP pricing system the firm pays the transportation

cost, so that in that equilibrium none of the firms has any incentive to gain additional business (any sale further than the midpoint of the interval would generate a loss). Thus, for 6 = 0 the profits from deviating from UDP are zero. However, for 6 = 0 the deviation

1 ) profits for FOB are - (Pc + t)2 (see (7)), which is strictly positive for any PC 2 0; this is 6t

due to the fact that for any FOB price, there is always an incentive for a firm to deviate from the system by charging a higher price to the local customers, which makes it impossible to sustain any FOB rule.

The result does not depend on the form of the punishments used. If, for example, after t

a deviation the firms revert to the static Nash equilibrium, Pu = 2 the same conclusion 2'

obtains, although the value of (*( w) and 6+ will be different, and for each 6 equilibrium profits will be lower. The assumption that when consumers are indifferent they go to the nearest firm is crucial. Without it, we could obtain nonexistence of equilibrium for some values of (; in particular, for 6 = 0, if consumers randomize when indifferent, there is no equilibrium in pure strategies.

The assumption that firms can change the pricing rule (from FOB to UDP or vice versa) without incurring any cost is not crucial either. As long as the switching cost is not too high, UDP will still be the best deviation from FOB, and therefore UDP will emerge as the equilibrium policy for low values of the discount factor.

Nevertheless, if the cost of switching pricing systems is prohibitive, and if once in FOB firms cannot use deviations out of the pricing system, then we would observe FOB for low values of the discount factor. Actually, if firms can credibly commit to a pricing policy at the beginning of the game, this amounts to restricting deviations to follow the same pricing rule. In the case of UDP this is not a real constraint, since the best deviations from UDP are also UDP prices, but in the case of FOB the constraint excludes the most profitable deviations. In Appendix B I show that for w 2 w, with an infinite cost of switching pricing

2w -2 2w - 2 systems, for 6 > 4 firms will choose UDP pricing policies, while for 6 < FOB

4w - 3 4w - 3 pricing rules are optimal.

3. Linear pricing systems

* The previous section has tried to clarify the main arguments relevant in the discussion on the collusive nature of delivered pricing rules, which is usually presented as a comparison

ESPINOSA / 75

between UDP (or another identical delivered pricing policy) and FOB. However, firms may have other policies available, even if they are restricted to simple pricing rules; and the presence of these policies could affect the optimality of UDP or FOB. In this section I generalize the results to the class of linear pricing schedules. The main conclusion of Sec- tion 2 still holds: UDP pricing policies will be used in very competitive and in very monopolistic market structures. For market structures that are neither very competitive nor very monopolistic, the model predicts pricing systems with slope that is in the interval [0, t] and is nonincreasing in 3. The range of 6 for which FOB is optimal is reduced. When market share agreements are allowed, the pricing system has slope in the interval [-t, 0), and the slope is nondecreasing in 3.

Greenhut, Greenhut, and Li ( 1980) analyze empirically the influence of several factors on the slope of the delivered price schedules of the firms in their sample and, in particular, the effect of the degree of competitiveness in the market. Two measures of competitiveness are used in their study: the rank (or extent) of competition assigned by the seller as applying to his market (they call this a subjective measure of the intensity of competition) and the number of competitors (an objective measure). They obtain the result that the more competitive the market, measured by the extent of competition assigned by the seller as applying to his market, the steeper the delivered price schedule, and this relationship has a high level of statistical significance. Their study provides some empirical support for our result that the slope of the pricing system is nonincreasing in 6 given that market share agreements are difficult to implement.17 Greenhut ( 1981 ) also observes that in Japan and West Germany, delivered pricing systems tended to have negative slopes, while for American firms the slope was positive. In our model this might be explained by a stricter antitrust legislation in the United States making it harder for American firms to coordinate on market sharing agreements, or by the greater difficulty in using pricing rules with negative slope (due to legislation against price discrimination) .18

o Description of the game. The description of the game is identical with that in Sec- tion 2, except that now firms' pricing policies must be of the form p(x) = po + cx, po E R, c E OR, where x is the distance between the location of the consumer and the location of the firm.

The pricing systems analyzed in the previous section are particular cases of these pricing policies: when c = 0, we have UDP, and c = t corresponds to FOB. Note that a linear pricing rule involves two parameters, po and c. We do not allow in this section piecewise-linear pricing; piecewise-linear would involve at least three parameters and be, in this sense, more complex.

We are interested in the optimal collusive symmetric perfect equilibria of this game, which will be referred to as collusive equilibria.

cl Collusive equilibria. First I characterize in Proposition 1 the optimal collusive equilibria from the set of subgame perfect equilibria for which a = 1. In some collusive equilibria firms do not need to specify a < 1, since in equilibrium they are selling in only half the market. We are interested in these equilibria because a = 1 in equilibrium implies there is no market-sharing agreement. It is worth noting that firms can still refuse to sell to that

17 In our model the degree of competitiveness is measured by the value of 6, which can be interpreted as the extent of competition assigned by the seller as applying to his market.

18 Greenhut ( 1981 ) attributes the differences in the sign of the slope of delivered price patterns to the Robinson- Patman Act and points out that "one spokesman interviewed . . . also admitted that his company had followed this kind of pricing policy [charging a lower delivered price in a distant market than that charged in its home market] but upon the advice of lawyers eliminated the lower (delivered) at distant market points in favour of a more moderate freight absorption practice."


portion of its demand that is not profitable to supply; the restriction a = 1 is not a restriction on the strategy sets of the firms, but rather a criterion to select equilibria where no market- sharing agreement is being used. Later on I shall characterize the optimal collusive equilibria of the game (not only the equilibria with a = 1) and show that firms find it profitable to have market sharing agreements, i.e., in equilibrium they will refuse to sell to customers located further than the midpoint of the interval even though it is profitable (in the short run) to serve them at the announced pricing rule.

We shall assume w 2 2 (transportation cost is not too high). Let (pb(X), ab) be the best response to (p0 + cx, a) and y the fraction of the market captured by a firm when using the strategy (pb(X), ab) against (p0 + cx, a). Lemmas 5 and 6 in Appendix B give the minimum requirements for a collusive symmetric equilibrium. The'slope of the price schedule cannot be greater than t, given the possibility of arbitrage among consumers, and it cannot be negative (if it were negative, each firm would be selling in the other firm's local market). The mill price cannot be negative, and all the consumers have to be served. If any of these requirements is not met, then it is possible to increase the equilibrium profits, contradicting that firms were in a collusive equilibrium. Lemma 6 characterizes the optimal deviation (pb(X), ab) from the symmetric policies [(p0 + cx, a), (p0 + cx, a)] when c 2 0. The optimal deviation involves undercutting the opponent by a small amount at every location: Pb(X) = p0 + c - cx, whenever this is compatible with serving the entire market, i.e., whenever t c p < R - c (see Figure 3, top). If pb(X) = p0 + c - cx, this would imply prices higher than R in some areas, i.e., if p0 ? R - c, then the slope is decreased until the entire market can be served: Pb(X) = R - (R - p0)x (see Figure 3, bottom). Lemma 7 gives conditions that the optimal pricing rule must verify.

Now we can establish

Proposition 1. Assume w 2 2. In the optimal collusive equilibria in which a = 1, UDP pricing policies will be used for very high and also for low values of the discount factor 3.

FIGURE 3

R R

pO+ C Pb(x) = Po + c - cx - R-c

--____---------- Po

0 1/2 1

R R

P + cx

Pb (X) R-(RP )X Po

- R-c

0 1/2 1

ESPINOSA / 77

For intermediate values of 6, the linear pricing system used in the optimal collusive equilibrium has slope in the interval [0, t], and when c > 0, it is nonincreasing in 3.

Proof. See Appendix B.

For very high values of 6, a UDP price Pu = R prevails because it allows perfect discrimination. As 6 decreases, Pu = R cannot be sustained, and then it is optimal to have a pricing system with a positive slope. This pricing policy is equivalent (same profits and same profits from the optimal deviation) to the optimal collusive UDP price Pu < R (see Figure 4). However, as 6 decreases further, deviations from a policy with a positive slope become too profitable: a deviating firm would undercut slightly the rival's price line, and this implies a high price in the deviating firm's local market; then, UDP is strictly better.

We have characterized the optimal collusive equilibria from the set of subgame perfect equilibria for which a = 1. We also may consider the possibility of market-sharing agreements requiring a firm not to supply all of its demand that is profitable to supply (in the short- run sense). The case where firms are allowed to use market-share agreements is considered in the Appendix. The result is

Proposition 2. For w> 2, when firms are allowed to use market-share agreements, UDP pricing policies will be used in the optimal collusive equilibrium for high values of the discount factor 3. For lower values of 6, the linear pricing system has slope in the interval [-t, 0) and the slope is nondecreasing in 3. A market-share agreement a = 1/2 will be used whenever the slope is strictly negative.

Proof. Available on request.

When market-share agreements are possible, firms have pricing rules with a negative slope in an optimal collusive equilibrium. The reason is that this type of policy makes deviations not as profitable as policies with a positive slope. Negatively sloped rules need a market-share agreement, however; otherwise, each firm would be selling in the rival's local market, which is an inefficient arrangement.

4. Unrestricted pricing policies

* Until now we have been assuming that the set of pricing policies available to a firm was somehow constrained to a family of simple pricing rules (UDP or FOB in Section 2 and linear pricing policies in Section 3). This may be a reasonable assumption, since there are costs associated with the implementation of a complicated pricing rule (we may think that pricing rules that involve the choice of only one parameter, like FOB or UDP, are less complex than pricing policies that involve more parameters). However, it is of interest to

FIGURE 4

R R

0p0

0 1~~~~~~~~~


examine what happens in the case where arbitrary pricing systems p(x) can be implemented without cost. In this section we impose no particular restriction on the pricing rule p(x).

A pricing policy that has been commonly used in practice and will play an important role in this section is the basing-point pricing rule. When using this pricing policy, firms decide on the location of a base point, Xb, and a price at that location, Pb. The price at any other location is calculated as the base price plus the transportation cost from the base point. Formally, a basing-point pricing rule is a function p(x) such that p(x) = Pb + tXb(X),

where xb(x) expresses the distance from the consumer to the base point as a function of the distance between the consumer and the producer.

Examples of this pricing policy are the Pittsburgh Plus system used in the steel industry and the Portland Plus system used for plywood. It was also implemented as a punishment path after deviations in the cement industry during the Great Depression. The Supreme Court described the punishment as

simple but successful. Other producers made the recalcitrant's plant an involuntary base point. The base price was driven down with relatively insignificant losses to the producers who imposed the punitive basing-point but with heavy losses to the recalcitrant who had to make all its sales on this basis. In one instance, where a producer had made a low public bid, a punitive base point price was put on its plant and cement was reduced ten cents per barrel; further reductions quickly followed until the base price at which this recalcitrant had to sell its cement dropped to 75 cents per barrel, scarcely one half of its former base price of $1.45. Within six weeks after the base price hit 75 cents, capitulation occurred and the recalcitrant joined a Portland cement association.19

Basing-point pricing can be a very severe form of punishment after a deviation, and therefore firms could sustain higher profits through the threat of making the deviator an involuntary base-point. Actually, we can prove that making the deviator an involuntary base-point is the deviator's worst possible perfect equilibrium, and then any given path is sustainable in a perfect equilibrium if and only if it can be sustained by reversion, in case of a deviation from that path, to a basing-point punishment.

Lemma 3. For any 6 E [0, 1], there is a subgame perfect equilibrium in basing-point strategies that yields zero payoff to one of the players.

Proof Consider the following pair of basing-point strategies:

[(Pb, Xb)1, (Pb, Xb)210 = [(O, location of firm 2), (0, location of firm 2)].

Figure 5 illustrates this pair of identical delivered pricing policies. Since consumers always buy from the nearest supplier in an identical delivered pricing

t system, the corresponding payoffs per period are - for firm 1 and zero for firm 2.

4

FIGURE 5

ten

Firm 1 1/2 Firm 2

19 As quoted in Machlup ( 1949).

ESPINOSA / 79

This is a subgame perfect equilibrium. Note that given firm 2's strategy, firm 1 cannot profitably deviate in any period (undercutting firm 2 only lowers total revenue, by decreasing revenue from old customers and/or attracting customers for which the transportation cost is higher than the price, while raising price only makes firm 1 lose customers). The same is true for firm 2.

This is the subgame perfect equilibrium that yields the lowest payoff to firm 2. Similarly,

[(Pb, Xb)I, (Pb, Xb)2 180 = [(0, location of firm 1), (0, location of firm 1 )] is the subgame perfect equilibrium that yields the lowest payoff to firm 1.20 Q.E.D.

The following proposition characterizes the collusive equilibria of the game. Let us define a basing-point type policy as any p(x) = Pb + axb(x), i.e., firms can use freight rates different from the actual ones. Then, basing-point appears as an optimal rule whenever the

monopoly solution is not sustainable, i.e., for values of 6 less than 8w -3

Proposition 3. In optimal collusive equilibria, when there is no restriction on the pricing

policies, for 3 2 8w 4 X firms will use a UDP pricing rule. For values of 6 in the interval

(?' 8 ) , a variety of identical delivered pricing policies could be observed (including

UDP, basing-point pricing, etc.). Finally, for 6 = 0, the model predicts basing-point pricing. FOB strategies are never used in an optimal collusive equilibrium.

Proof. For 3 2 8 - 3

the monopoly price p(x) = R extracts all the consumer surplus 8w-4'

and it is sustainable (see Section 2). For 6 = 0, the best sustainable pricing system is basing- point pricing:

p(x) = p112 + tX112(X) and P1/2 = 2

i.e., the base point is at the middle point of the interval and the base price is t/2, as indicated t

in Figure 6. Profits are - for each firm. 4

To see that the indicated basing-point rule is an equilibrium for 6 = 0, it is sufficient to prove that none of the firms has any incentive to deviate. A profitable deviation for firm 1 would involve either an increase in market share from [0, 1/2] to [0, a], a e (1/2, 1 ] and/or an increase in the price to the existing customers in [0, 1/2]. An increase in market share for firm 1 would require selling in [1/2, a ] at a price not higher than the transportation cost, given that the prices of firm 2 in [1/2, 1 ] are p(x) = tx, where x is the distance from the consumers to firm 1. Thus, an increase in market share cannot increase profits for firm 1. On the other hand, an increase in the price to some of the consumers in [0, 1/2] is not profitable either: it would imply losing those consumers to the competition, given that in [0, 1/2] firm 2 is pricing at p(x) = t - tx, where x is the distance from the consumers to firm 1.

Moreover, the indicated basing-point rule p(x) is the collusive equilibrium of the game for 6 = 0. Assume there is another pricing system p'(x) that yields higher profits and is

20 This punishment path is stationary. If we introduced production costs, then we could have a "stick and carrot" basing-point optimal punishment path, which would resemble more closely the punishments implemented in the cement industry.


FIGURE 6

t t

0 1/2 1

sustainable for 6 = 0. Then, in some interval z, p'(x) > p(x) for x E z. The firm that is not serving customers in z (firm j) would deviate selling in z since p'(x) > p(x) 2 tx, where xis the distance from a consumer in z to firmj, hence contradicting that p'(x) was sustainable for 6 = 0.

For intermediate values of the discount factor, identical delivered pricing systems will be optimal. To see that FOB (or any other price rule different from identical delivered pricing) cannot be observed in a collusive equilibrium, one need only note that when at a given point the two firms have different prices, this only increases the deviating profits for the firm that has the lower price (compared to the situation in which both firms have the lower price). This implies that given a pricing system with nonidentical prices at a given point, it would be possible to increase profits, contradicting that the pricing policy is used in a collusive equilibrium.

Define B as the highest level of profits in excess of t ( are the maximum profits when

6 = 0, with a basing-point pricing rule) that can be made in equilibrium. Then, for each

value of 6, the level of profits B is given by

t t~~~~~~ 1 4+ B = 4+ 2B5 ( 1 )

where - + B are the profits from conforming to the equilibrium price policies and - + 2B 4 4

are the profits from deviating, given that an identical delivered price rule is optimal (see Figure 7) and the punishment path has a payoff of zero (by Lemma 3). B represents the profits due to the fact that 6 > 0. From (11),

B 4(1 -2) t

B is increasing in 3 when 3 E [0, 1/2]; since we are focusing upon the case when

8w< - -4 and given that 8 -3 < - it follows that B is increasing in 3 over the

relevant range of discount factors. The maximum level of profits that can be sustained is t

determined by 6, and any identical delivered pricing system that yields B(b) + - is indif- 4

ferent. Q.E.D.

ESPINOSA / 81

FIGURE 7

t . _ t

0 1/2 1

Note that unless the discount factor is very high (at least /2 in this model), FOB cannot be sustained in equilibrium. The reason is that when pricing policies are not restricted, deviations can be very profitable, resulting in the impossibility of sustaining any FOB price. This may explain the opinion often expressed that delivered pricing systems serve the purpose of stabilizing the industry (if FOB cannot be sustained in equilibrium, any FOB price will be necessarily unstable).

We have assumed that whenever the firms quote the same delivered price at a given location, the consumers choose the nearest supplier. This assumption is crucial to get ex- istence of equilibrium. Consider 6 = 0 and assume, for example, that consumers randomize among suppliers when the same delivered price is quoted; then the game has no equilibrium in pure strategies.

The fact that basing-point pricing is used for low 6 and that identical delivered pricing policies are optimal for values of the discount factor such that the monopoly solution cannot be implemented does not depend on the assumed form of the demand function at each point, the space considered (the Hotelling line), or the number of firms (as long as it is finite and firms are spatially differentiated). However, the prediction of UDP for high 6 does depend on the demand function. Let D(p) be the demand function at each point and assume that pD(p) is concave. Then, for 6 high enough, the delivered pricing policy p(x), for x E [0, ?/2], is implicitly defined by D'(p)(p - tx) + D(p) = 0, which corresponds to a

t price schedule p(x) increasing in x. When demand is linear, the slope of p(x) is - 2

5. Concluding remarks

* This article has examined the question of whether identical delivered pricing policies should always be considered practices that impair competition. The results indicate that the answer to this question is negative. Although delivered pricing systems allow spatial price discrimination, which facilitates collusion, they are also a very aggressive form of competition, so that for low values of the discount factor FOB is not sustainable in a subgame perfect equilibrium, and identical delivered pricing policies will be observed.

The opinion expressed by Scherer ( 1980) and others that in order to increase the degree of competitiveness in markets, firms should not be allowed to price differently from FOB, except perhaps to undercut another firm, is not supported by the results obtained here. To impose FOB allowing deviations out of the pricing system would be a solution to the welfare loss problem only in monopolistic markets. If the market structure is competitive (low 6 or low t), this policy would lead to nonexistence of equilibrium and presumably to great instability in the market.


Another option is to enforce FOB strictly, i.e., only FOB is allowed, even for deviations. The welfare effects of this measure are ambiguous and can go in the wrong direction, since banning the most effective deviations from the pricing system means that the level of delivered prices sustainable in equilibrium would go up. This seems to be what happened in the cement and steel industries when they were ordered to discontinue the practice of basing- point pricing. In 1948 the Supreme Court declared the basing-point system unlawful, sustaining the decision of the Federal Trade Commission. After the forced abandonment of the system, cement producers were able to raise delivered prices and revenues with FOB prices. At the same time, the steel industry started selling steel on an FOB basis, expecting a similar decision from the Federal Trade Commission, and again there was an increase in delivered prices and revenues for the industry.21 This seems to be contradictory, for if FOB was more profitable than basing-point, why was the industry not using FOB before? The answer is that when basing-point is unlawful, FOB is more profitable than when firms are also allowed to use basing-point (the optimal deviation from an FOB price is to make the opponent a base point, undercutting its mill price by some small amount). When FOB is made compulsory, deviations are less profitable, and thus higher FOB prices can be sustained in equilibrium.

The fact that in a perfectly competitive world with an infinite number of firms FOB would prevail does not imply that FOB is the most competitive pricing policy in spatial markets where products are spatially differentiated and, due to fixed costs for example, there is room for only a few firms. We have seen that in these markets, delivered pricing may be more competitive than FOB. The conclusion is that delivered pricing policies cannot be labelled as facilitating collusive practices in all instances; the structure of the market should be taken into consideration.

An interesting question is how uncertainty of detection of deviations from a specified pricing policy can affect the incentives to use a given price rule. The presence of uncertainty would also bring arguments like the one given by Stigler ( 1964), who rationalizes the uti- lization of basing-point pricing as a stabilizing device in a context of geographically unstable demand. The introduction of uncertainty would bring new elements to the discussion and deserves further research.

Appendix A

* Optimal punishment path. Let A (6, t, R) be the optimal collusive outcome (per period) that could be sustained by reversion to a zero-payoff punishment path, and PA(6, t, R) the price that implements that outcome.

Consider the following pair of UDP pricing policies:

[(Pu, a)I, (pu, )2] = [(0, 1 ), (0, 1 )] (Al)

The corresponding payoff is 8 per period for each firm. 8

The optimal punishment path has two phases. For Tperiods, firms follow the pricing policy (A l ). Tis chosen such that

(t T-1I <x j j=Tb1+A 2 b6=0. (A2)

8j=O j= T

After that first phase of T periods, they follow the policies (PA(6, t, R), 1) that yield A(6, t, R) per period forever after.

We have to check whether this "stick and carrot" punishment path,

sr,r I ), {as I\T, r[ (p \A n , Il ), (pA I

21 "Irate consumers wrote letters to editors, Congressmen, and Senators, complaining about the new system [FOB] which made them pay prices higher than before," Machlup ( 1949).

ESPINOSA / 83

is a subgame perfect equilibrium. By deviating from the first phase of the path, a firm can increase its profits at

most by 8per period and make zero profits. This deviation is not worthwhile, since the payoff of the punishment

path is also zero. And given the way PA(6, t, R) has been defined, neither firm can find it profitable to deviate from the second phase of the punishment path. This shows that the punishment path (A3) is a perfect equilibrium. It is also the lowest-payoff perfect equilibrium, since zero is the lowest payoff that is individually rational.

We have implicitly assumed that the solution for T given in (A2) is an integer. If it is not an integer, then the first phase would last for a number of periods equal to the highest integer smaller than T; in the next period a random device would decide whether they continue the first phase one more period or go on to the second phase (with the right probabilities for each event). This would require that firms be able to use correlated strategies; if they are unable to use such strategies, in the last period of the "stick" phase, firms would have a positive price (properly chosen). A more serious problem is the possibility that the solution for T given in (A2) is a number less than 1 (in particular, this happens for very low values of the discount factor). In this case, the lowest-payoff perfect equilibrium would yield a strictly positive payoff Here we do not need to calculate the lowest-payoff perfect equilibrium for all possible values of 6; everything that is interesting for us happens in the set of 6 for which T 2 1 and there is a punishment path that yields zero payoff.

* To see that the value of K(6, t, R) for the set of 6 that sustain p, = R is zero, we have to check that

(i8 )+ ( > 4 t 0 (i.e., T 2 1). This inequality is satisfied for 6 2 The set of 6 that sustain \8 \1 - 6 -0 4w

5 4w -31\r i Pu = R 6: 6 2 8w- 4) is included in the set 6: 6 2 4J , so that the value of K(6, t, R) for the set of 6 that

sustain Pu = R is zero.

3 -46 * To see that the value of K(6, t, R) for the set of 6 that sustain Pu = 4-86 t is zero, we have to check that

-t 6 1- 8 + -6 4(1 -26) t 2 0 (i.e., T 2 1 ). This inequality is satisfied for 6 2 ?/4. The set of 6 that sustain

P, {6: 6 E [I4w 3] is included in the set { 6: 6 2 I/4}, so that the value of K(6, t, R) for the set of 6 that

3 - 46 sustain p,, = 8 t is zero. Pa4 - 86

-t 6 1+6+2V-b For 6 < I/4, T < 1, since -8 +

- 6 (1 -) t < 0 and K(6, t, R) will be strictly positive. In this case the

UDP price Pu = 2I ; t is an upper bound for the sustainable UDP price. 2(1 -6)

* To calculate the value of K(6, t, R) for the set of 6 that sustain pc = 2(1 I 6) t, we check whether

+ 6 4(1-6) t 0 O. This inequality holds for 6 2 d', where d' is the solution to

1 = 6(4 + 36 + 2V126 - 3). Therefore, K(6, t, R) is zero for 6 2 d' and strictly positive for 6 < d'. Since d' < ?/4,

1 +26+V/126-3 K(6, t, R) is zero for 6 E [ I/4, 1/3], which is the set of 6 that sustain PC = 2(1 1 6) tI

Appendix B

* With commitment, for 6> 2 - 2

firms will choose UDP pricing policies, while for 6 < 2 -

2, FOB 4w -3' w-3

pricing rules are optimal.

Proof To simplify the exposition we will take a high enough value of w (w 2 5 will suffice). Let G(UDP) be the game where only UDP is feasible, G(FOB) the game where only FOB is feasible. If the firms can commit to a pricing policy before the game starts, the punishments in case of any deviations must follow the pricing rule. We denote K&(6, t, R) and KJ(6, t, R) the lowest payoffs attainable in a perfect equilibrium of G( UDP) and G(FOB) respectively. Then we can write, as a function of the discount factor, the maximum values of UDP prices and profits sustainable in a symmetric perfect equilibrium of G( UDP):


a Pu nu

[4w- 3 1 _-__

[8w-4'] R 4R-t

I _4w-_3 3-46 _-__ _

\4' 8w - 4/ 4 - 86 4(1 - 26)

| 0 'A4 I + V6- ( -6ku t I + 6 + 2V6 8(1 -6)ku t |

[O,?] ~~~~2(1 - ) 8(1-s)

where ku =Ku

The maximum values of FOB prices and profits sustainable in a symmetric perfect equilibrium of G(FOB) are as follows:

6 PC RC

2R-t 2R-t 1,] 2 4

02Y 61) | 2(1 - a)-26kc t - 1 6-__ k l

1 + 6 + 2 Vb-26(1- )kc 1 + 6 + 2 V-26(1 - 6)k l [O,(52] t 2(1-5)

where kc = Kc 62 is the solution to 462 - 56 + 1 + 26(1 - )kc =0, and 61 is the solution to

2w(1 - 26) + ((6 + 4kc) - 5 = 0.

Note that kc E [0, ?/2] and k,, E [0, I/8] (the payoffs in the static Nash equilibrium of G(FOB) and G( UDP)

respectively). Note also that 1/4 < 62 < 61 < 8w - 3

8w - 2w4-

Comparing Hc and II, we conclude that Hc ' ll4 6 2 . Q.E.D.

4w - 3

Lemma 4. If 6 < 1/4, then there is no pair of FOB prices (Pc, Pc) sustainable in equilibrium.

Proof We know from previous discussion that for 6 < 1/3, any pc sustainable in a symmetric equilibrium should be

such that Pc ' 2t. In that case the optimal deviation yields profits II [p"(pc), Pc] = I (Pc + t)2. 6t

Thus, for (ps, Pc) to be sustainable it is necessary that PC > I (Pc + t)2( 1 - 6) + (K((, t, R). 2 6t

For this to hold it has to be the case that PC > I (Pc + t)2( 1 - 6), since 6 2 0 and K((, t, R) 2 O. 2 6t Rewriting the expression we get (1 - 6)p2 - t(l + 26)pc + (1 - 6)t2 < 0, which reaches a minimum at

C= ( 26) t. Substituting this value it yields -( I + 26)2 + 4(1 _- )2 < 0. The left-hand side is nonpositive as YC-2(1 -(5)

long as 6 ( 1/4, so that for 6 < 1/4, the inequality cannot hold. Thus, when 6 < 1/4 it is impossible to sustain FOB prices (Pc, Pc) in equilibrium. Q.E.D.

Proposition 4. For w > 1, there is a function (*( w), nondecreasing in w, 0 < (*( w) < 1, such that for every 6 ( 6*( w) the optimal collusive equilibria of G( UDP, FOB) are UDP equilibria. There is a (+, (*( w) > 6+ > 0, such that for 6 < (+ all the equilibria of G( UDP, FOB) are in UDP strategies. For (*( w) > 6 > (+ the optimal collusive equilibria are FOB equilibria.

5 for 1 < w < -

4 ~~~~~~4 (*(w)=

{

2w-2__ 5 Iw

- 2 for W2-

4w- - 3 4

ESPINOSA / 85

Proof Available on request.

Lemma 5. For w 2 2, in a collusive equilibrium, (i) I cI < t, (ii) if a > 0, then po 2 0, and (iii) if a = 1, then C 20.


Lemma 6. For w ? 2, when c ? 0: (i) ifpo <R - c, then pb(x) =po + c -cx;

ab for po E [t, R-t] and

ab<l for po<t;

(ii) if po ? R -c, then pb(X) = R - (R -p)x;

(iii) if po 2 t, then y = 1.


Lemma 7. For w 2 2, in a collusive equilibrium with a = 1: (i) p('/2) < R; (ii) either c = 0 orp('/2) = R; (iii) if pO < R-t/2, then c = 0; (iv) if c > 0, then pO = R - c/2.


Proof of Proposition 1. For 6 2 8- 3, we know that a UDP price Pu = R is sustainable. For 6 < 8w - , by

Lemma 7 (ii), either p('/2) = R or c = 0. If in a collusive equilibrium p('/2) = R, the slope must be c > 0 (since

4w< 8 -3 Then there is a UDP pricing system Pu = Po + - = R+ P equivalent to p(x): it yields the same 8w -4J 4 2

profits, f/2 (p0 + cx-tx)dx = 1/2 (Pu - tx)dx, and the profits from the optimal deviation are the same (see Figure 4): fo (R - (R - po)x - tx)dx = fX (P -tx)dx (see Lemma 5 for the form of the optimal deviation). Since the maximum sustainable UDP price is nondecreasing in 6, if p('/2) = R the slope of the equivalent pricing policy with c > o must be nonincreasing in 6, and p0 must be nondecreasing in 6. It has been shown that UDP is always optimal and that a p(x) with c > 0 is equivalent to the optimal UDP as long as p('/2) = R. However, there is a 6 such that for 6 < 6 in a collusive equilibrium p('/2) < R and, by Lemma 6 (iii), the optimal policy is necessarily UDP. Q.E.D.

References

ABREU, D. "On the Theory of Infinitely Repeated Games with Discounting: A General Theory." Econometrica, Vol. 56, (1988), pp. 383-396.

CARLTON, D.W. "A Reexamination of Delivered Pricing Systems." Journal ofLaw and Economics, Vol. 26 (1983), pp. 51-70.

FETTER, F.A. "The New Plea for Basing-Point Monopoly." Journal of Political Economy, Vol. 45, (1937), pp. 577-605.

GREENHUT, J., GREENHUT, M.L., AND LI, S.Y. "Spatial Pricing Patterns in the United States." Quarterly Journal of Economics, Vol. 94 (1980), pp. 329-350.

GREENHUT, M.L. "Spatial Pricing in the United States, West Germany and Japan." Economica, Vol. 48 (1981), pp. 79-86.

MACHLUP, F. The Basing-Point System. Philadelphia: Blakiston, 1949. SCHERER, F.M. Industrial Market Structure and Economic Performance. Chicago: Rand McNally College Publishing

Co., 1980. STIGLER, G.J. "A Theory of Delivered Price Systems." American Economic Review, Vol. 39 (1949), pp. 1143-

1159. . "A Theory of Oligopoly." Journal of Political Economy, Vol. 72 (1964), pp. 44-61.

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Delivered Pricing, FOB Pricing, and Collusion in Spatial ... · pricing systems (in particular, UDP and basing-point pricing) are used in highly monopolistic and in highly competitive