The Welfare Effects of Third Degree Price

Discrimination In Intermediate Good Markets: The

Case of Bargaining

Daniel P. O’Brien1

U.S. Federal Trade Commission

601 Pennsylvania Avenue, N.W.

Washington, D.C. 20580

February 5, 2002

Abstract

This paper examines the welfare effects of third degree price discrimination by an intermediate good monopolist selling to downstream

firms with bargaining power. One of the downstream firms (the “chain

store”) may have a greater ability than rivals to integrate backward

into the supply of the input. In addition to this outside option, the

firms’ relative bargaining powers depend on their disagreement profits, bargaining weights, and concession costs. If the chain’s integration

threat is not a credible outside option, and if downstream firms cannot coordinate their bargaining strategies, then price discrimination

reduces input prices to all downstream firms.

1Economist, U.S. Federal Trade Commission. The views expressed herein are my own and do not purport

to represent the views of the Federal Trade Commission or any Commissioner. I thank Ian Gale, Dan

Gaynor, Steve Matthews, John Panzar, Greg Shaffer, Abraham Wickelgren, and seminar participants at

Northwestern University and the University of Michigan for helpful comments. I take full responsibility for

any errors. The author can be reached at dobrien@ftc.gov or danobrien@cox.rr.com.
“When a degree of non-transferability...sufficient to make [price] discrimination profitable is present, the relation between the monopolistic seller and each

buyer is, strictly, one of bilateral monopoly. The terms of the contract that will

emerge between them is, therefore,...subject to the play of that ‘bargaining’...”

(A.C. Pigou, 1932, 278).

Price discrimination policy in the U.S. focuses mainly on intermediate good markets in

which buyers have bargaining power. In fact, the primary U.S. law governing price discrimination, the Robinson-Patman Act (1936), arose from concerns that large downstream

firms (e.g., chain stores) were harming smaller rivals by negotiating larger discounts with

suppliers.2In the first formal analysis of buyer-specific price discrimination policy in intermediate good markets, Michael L. Katz (1987) examined the effects of forbidding third

degree price discrimination when the bargaining power of chain stores comes from their ability to threaten credibly to integrate backward into the supply of the intermediate good. For

downstream markets characterized by Cournot oligopoly, he showed that “if there is no integration under either regime [i.e., whether price discrimination is allowed or forbidden], then

total output and welfare are lower when price discrimination is practiced than when it is

forbidden.” (Katz, Proposition 1). This is an important result for public policy toward price

discrimination. Until this result, the Robinson-Patman Act had received little support3in

the economics literature.4

2Although most of the Robinson-Patman claims brought by the FTC have been against sellers, it was well

understood when the law was passed that the discriminating seller was often the “innocent victim” of the

buyer’s bargaining power. See, for example, Phillip Areeda and Louis Kaplow (1988), pp. 979-80. Congress

made this recognition explicit with Section 2(f) of the Act, which makes it unlawful for a buyer “knowingly

to induce or receive a discrimination in price which is prohibited by this section.”

3

It should be noted that Katz does not argue that his analysis provides support for the enforcement of the

Robinson-Patman Act. However, prior to Katz’s article, the Robinson-Patman Act was almost universally

criticized by economists as an anticompetitive law. See, for example, Marius Schwartz (1986).

4Until Katz’s article, the formal analysis of buyer-specific price discrimination had focused on the case

of independent demands (e.g., Joan Robinson (1934), Richard Schmalensee (1981), Hal Varian (1985)).

Subsequent theoretical work on third degree price discrimination has focused on discrimination by a take-it

or leave-it monopolist (e.g., Marius Schwartz (1990) and David A. Malueg (1993) obtain additional results

for final good markets, and Patrick DeGraba (1990) and Yoshihiro Yoshida (2000) do so for input markets.)

There has been work on the effects of price discrimination in input markets under buyer-specific nonlinear

contracts. See Daniel P. O’Brien and Greg Shaffer (1994). Best-customer clauses have effects that are similar

to a policy against price discrimination. These clauses have been examined by Thomas Cooper and Theodore

Fries (1991) for the case of linear pricing and independent demands, and by Patrick DeGraba and Andrew

1
The ability to pursue an outside option, such as backward integration, is an important

potential source of a buyer’s bargaining power, but it is not the only source. The modern literature on bargaining identifies three additional factors that may affect the relative

bargaining powers of a buyer and a seller: the costs of making price concessions, the loses

inflicted on each other by delaying agreement, and bargaining costs. Other factors equal, a

buyer’s bargaining power is greater the higher the cost it bears from granting a small price

concession to the seller, the greater the loss it inflicts on the seller by delaying agreement,

and the less costly it is to hold out for a better deal. Once these additional sources of bargaining power are recognized, the analysis of price discrimination is more complex than it is

in the take-it or leave-it environment studied by Katz. Even if an explicit threat to integrate

backward is not credible, the chain may receive a discount if it has greater bargaining power

than the independent. Moreover, a policy forbidding price discrimination may do more than

simply constrain the prices the seller can set; it may alter relative bargaining powers.

In this paper, I extend Katz’s take-it or leave-it model to a Nash bargaining framework

that incorporates four sources of bargaining power: outside options, concession costs, inflicted losses, and bargaining costs. Intermediate prices are negotiated in pair-wise meetings

between the supplier and individual downstream firms, one of which (the chain store) has

lower costs of integrating backward into the supply of the input. As in Katz’s model, the

chain’s integration advantage may allow it to negotiate lower prices than its rival. However,

the bargaining model provides three plausible explanations for chain discounts. First, the

chain may be able to threaten credibly to integrate backward, as in Katz’s model. Second,

the chain may earn higher profits than its rival if it fails to reach an agreement with the

supplier. In bargaining language, the chain may have a higher disagreement profit than the

Postlewaite (1992), R. Preston McAfee and Marius Schwartz (1994), and DeGraba (1996) for the case of

nonlinear pricing in intermediate good markets.

2
rival, which gives it greater bargaining power. Third, the chain may have lower bargaining

costs than the rival. For example, the chain may have a lower discount rate than the rival,

which makes it less costly for the chain to hold out for a better deal.5

I find that the effects of forbidding price discrimination depend crucially on the credibility of chain’s threat to integrate backward. If the integration threat is a binding outside

option, then Katz’s results are unchanged; forbidding price discrimination reduces the average wholesale price if the chain chooses not to integrate in both regimes. However, if the

integration threat is not a binding outside option, then forbidding price discrimination raises

the average wholesale price for a wide range of parameters that determine relative bargaining

powers. Thus, when bargaining power comes from sources other than outside options, the

results of the take-it or leave-it model are often reversed.

An important difference between the bargaining and take-it or leave-it environments is

that in the former, forbidding price discrimination alters the firms’ relative costs of making

price concessions, which affects their relative bargaining powers. When price discrimination

is allowed, a buyer’s price concession (an agreement to pay a higher price) weakens its

competitive position in the downstream market relative to rivals that do not make the same

concession. When discrimination is forbidden, however, a buyer’s concession does not weaken

its competitive position (though it reduces its profits), because the higher price must be paid

by every firm that purchases the product from the supplier. Therefore, the cost of a price

concession is lower for a buyer when price discrimination is forbidden than when it is allowed.

On the other hand, it is more costly for the seller to make a price concession (an agreement to

accept a lower price) when price discrimination is forbidden, because a lower price must then

be given to every downstream firm. Under both concession cost effects, a policy forbidding

5A fourth source of countervailing power identified in the literature is the ability of buyers to accumulate

a backlog of unfilled orders that, similar to a boom in demand, forces sellers to collude on low prices to

prevent undercutting (Christopher M. Snyder, 1996). This source of bargaining power does not arise in my

model, which focuses on the behavior of a single upstream firm.

3
price discrimination reduces the bargaining power of downstream firms relative to the seller.

The implications of the concession cost effect of price discrimination policy are clearest

when downstream firms have equal bargaining power and the chain cannot credibly threaten

to integrate backward when price discrimination is allowed. In this case, a policy forbidding

price discrimination raises the wholesale price charged to both downstream firms, reducing

total output and welfare. Critics of price discrimination policy have often argued that the

effects of forbidding price discrimination reach beyond markets exhibiting persistent asymmetries in input prices. A common criticism is that forbidding price discrimination prevents

sporadic and selective discounts by cartel members that might break down cartel discipline

and lead to lower prices.6 My results show that upstream competition is not necessary for a

policy forbidding price discrimination to raise prices in markets where systematic discrimination is not observed. All that is required is buyer bargaining power.

If the chain has greater bargaining power than the rival, then the analysis is less clear

cut. The key complication is that when price discrimination is forbidden, the supplier prefers

to have the weaker buyer negotiate the common price, while the buyers both prefer to have

the stronger buyer negotiate the price. I show that if the supplier negotiates with the weaker

buyer, and if the chain’s integration threat does not bind when discrimination is allowed,

then a policy forbidding price discrimination raises the average wholesale price. On the other

hand, if the buyers can arrange to have the stronger buyer negotiate price, forbidding price

discrimination can reduce the average price if the discounts received by the stronger buyer

when discrimination is allowed are large enough.

The remainder of this paper is organized as follows. The bargaining model is introduced

in Section I. Section II presents the implications of the bargaining model when price discrimination is allowed. Section III examines the effects of forbidding price discrimination.

6See, e.g., Dennis W. Carlton and Jeffrey M. Perloff (1994), p. 416.

4
Section IV examines two special cases of the model: independent demands, and “almost

competitive” conduct in the downstream market. Section V concludes the paper. Technical

proofs are presented in the Appendix.

I. A Bargaining Model of Third Degree Price Discrimination

To facilitate comparison with Katz, I adopt a similar framework with some additional notation required to account for bargaining. There is a monopoly supplier of an input used

by two downstream firms to produce a final product. The downstream firms are engaged in

rivalry summarized by their reduced form profit functions. Let π1(w1, w2) and π2(w1, w2) be

the equilibrium profits of downstream firms 1 and 2, respectively, when their wholesale prices

are w1 and w2 (the negotiations determining w1 and w2 are detailed below). The supplier’s

reduced form profit is U(w1, w2). I assume that the profit functions have the usual properties, i.e., ∂πi/∂wi < 0, 0 ≤ ∂πi/∂wj < |∂πi/wi

|, and ∂U/∂wi > 0 up to some maximum

for i ∈ {1, 2}, i 6= j. I also assume that downstream rivalry (but not bargaining power) is

symmetric in the sense that π1(w, w

0

) = π2(w

0

, w) and U(w, w

0

) = U(w

0

, w) for all w, w

0

.

Downstream firm 1 (the “chain store”) has the option of integrating backward into the

supply of the input. If the chain integrates backward, it incurs a fixed cost F and pays

a wholesale price v. Its profits are then π

I

(v, w2) = π1(v, w2) − F, firm 2’s profits are

π2(v, w2), and the upstream firm’s profits are U(v, w2) if price discrimination is allowed

and U

F

(v, w2) ≤ U(v, w2) if price discrimination is forbidden.7 Downstream firm 2 (the

“independent”) cannot integrate backward.8

7

If the chain integrates backward and price discrimination is allowed, the supplier can win the right to

supply it by offering to sell at (or just below) v. If price discrimination is forbidden, the supplier’s profit U

F

is the maximum of the profit earned from selling to both firms at v or only to the independent at w2.

8The chain’s integration profits πI

can also be interpreted as its profits after spending the fixed cost F

to purchase from an alternative supplier at a wholesale price of v. One motivation for the chain’s superior

outside opportunities is that it operates in more markets than the independent and spreads the fixed costs

5
Following Ken Binmore et al. (1986), I model negotiations using an asymmetric Nash

bargaining framework and motivate the role of outside options, disagreement payoffs, and

bargaining weights from an underlying noncooperative bargaining game.9 When price discrimination is allowed, the supplier attempts to negotiate a separate wholesale price with

each downstream firm. Suppose the supplier and firm 2 have agreed (or are expected to

agree) to the price w2 (the negotiations determining w2 will be described shortly). In an

equilibrium in which the chain chooses not to integrate backward, the asymmetric Nash

bargaining solution between the supplier and the chain solves

max

w1

φ1(w1, w2) = [U(w1, w2) − du1]

1−γ1

[π(w1, w2) − d1]

γ1 s.t. π1(w1, w2) ≥ πI

(v, w

I

2

(1) )

where du1 and d1 are the disagreement payoffs of the supplier and the chain, respectively;

w

I

2

is the wholesale price that the independent would pay if the chain were to integrate

backward; and γ1 is the chain’s bargaining weight in negotiations with the supplier.10 These

parameters are explained as follows.

The bargaining weight γ1 can be motivated from an alternating-offer bargaining model

(Rubinstein, 1982) that underlies the Nash bargaining solution, as demonstrated in Binmore

et al. (1986). In this class of models, firms are motivated to reach agreement by the cost

of bargaining delays. In one variant (the time preference Nash solution), delays are costly

because firms discount the future at positive rates. In the other variant (the standard Nash

solution), delays are costly because there is an exogenous risk that negotiations might break

down after each period in which the firms fail to reach an agreement. Both motivations may

play a role in negotiations over intermediate good prices, since firms generally discount the

future at positive rates and there is often some risk that a profitable opportunity will be

of obtaining alternative supplies over greater sales.

9This approach to modelling negotiations is consistent with what Binmore and Partha Dasgupta (1987)

call the “Nash program,” which seeks to motivate cooperative approaches to the bargaining problem like the

Nash bargaining solution from an underlying noncooperative game.

10The individual rationality constraints, U ≥ du1 and π1 ≥ d1, are omitted for brevity.

6
exploited by a third party. Binmore et al. show that the unique subgame perfect equilibrium

to the Rubinstein bargaining game converges to an asymmetric Nash bargaining solution as

the time between offers becomes small. Each firm’s bargaining weight in this solution is a

decreasing function of its bargaining cost, as measured by its discount rate.11 Intuitively,

the more costly it is for a firm to reject an offer, the less bargaining power the firm has. This

is reflected by a lower bargaining weight in the asymmetric Nash bargaining solution.

The disagreement payoffs can also be motivated from the same Rubinstein-style bargaining model that yields the bargaining weights. Binmore et al. show that in the time

preference solution, the disagreement payoffs are the profits earned by firms while they are

negotiating prices. In the standard solution, the disagreement payoffs are the profits received

if negotiations break down.12 I assume that the disagreement profits are small enough that

there exists a set A of wholesale prices such that U(w1, w2) > dui and πi(w1, w2) > difor all

(w1, w2) ∈ A, i ∈ {1, 2}.

The constraint in the maximization problem reflects the “outside option principle” (Avner

Shaked and John Sutton, 1984). Binmore et al. showed that when the Nash bargaining

solution is motivated from an alternating-offer bargaining game, an outside option should be

modelled as a constraint on the equilibrium prices, separately from the disagreement payoffs.

The outside option affects the bargaining solution only if the solution to the bargaining

11Specifically, suppose the one-period discount rates of firm i and the upstream supplier are ri and ru,

respectively. In the time preference solution, where there is no probability that negotiations will terminate

in a breakdown, Binmore et al. show that the subgame perfect equilibrium to the Rubinstein game converges

to an asymmetric Nash bargaining solution in which firm i’s bargaining weight is γi = ri/(ri + ru). Suppose

instead that firms do not discount the future, but that there is an exogenous probability αi that negotiations

between the supplier and firm i will break down after any period that one of them rejects the other’s offer.

In this case, Binmore et al. show that the alternating-offer bargaining game between the supplier and firm

i converges to the (symmetric) Nash bargaining solution. Abhinay Muthoo (1999) considers an alternating

offer bargaining model in which both types of delay costs (discounting and the risk of a breakdown) are

present. His results imply that firm i’s bargaining weight in this case is γi = (ri + αi)/(2αi + ri + ru).

12Let Ui and πi be the profits earned by the supplier and firm i, respectively, each period during their

negotiations; let bui and bi be their profits (per period) in the event negotiations break down. When both

motivations for reaching agreement are present, the results of Muthoo (1999) imply that the disagreement

profits are dui = (Ui + αibu)/(ru + αi) and di = (πi + αibi)/(ri + αi).

7
game ignoring the outside option yields a lower payoff to the chain than it would receive by

exercising its option. If the chain chooses integration, I assume that the supplier and firm

2 negotiate (or renegotiate) their price. This leads to a wholesale price for firm 2 of w

I

2

. I

assume that there exists a set A1 of wholesale prices for firm 1 such that U(w1, w

I

2

) > dui

for all w1 ∈ A1, i ∈ {1, 2}.

The wholesale price negotiated by the supplier and firm 2 solves a similar Nash bargaining

problem without the integration constraint:

max

w2

φ2(w1, w2) = [U(w1, w2) − du2]

1−γ2

[π2(w1, w2) − d2]

γ2

(2)

where du2 and d2 are the disagreement payoffs of the supplier and firm 2, and γ2 is firm

2’s bargaining weight.13 These parameters are interpreted the same way as the analogous

parameters in negotiations with the chain.

The first order conditions for (1) and (2) are

(1 − γ1)

∂U

∂w1

[π1 − d1] + γ1

∂π1

∂w1

[U − du1] + λ

∂π1

∂w1

(3) = 0,

λ ≥ 0, λ[π1 − π

I

(4) ] = 0;

(1 − γ2)

∂U

∂w2

[π2 − d2] + γ2

∂π2

∂w2

(5) [U − du2] = 0

where λ is a Lagrangian multiplier. The solution to conditions (3) and (4) defines a “bargaining reaction function,” R1(w2), which expresses the wholesale price negotiated by the

manufacturer and firm 1 as a function of the wholesale price negotiated with firm 2. The bargaining reaction function for firm 2, R2(w1), is defined symmetrically as the solution to (5).

A bargaining equilibrium when price discrimination is allowed is a pair of wholesale prices

(w

A

1

, w

A

2

) (the superscript ‘A’ for “allowed”) such that w

A

1 = R1(w

A

2

) and w

A

2 = R2(w

A

1

).

I make the following additional assumptions:

13Note that I am ignoring the possibility that firm 2 might want to choose a wholesale price that would

induce chain integration. This possibility is ruled out by Assumption 4 below.

8
Assumption 1 U(w1, w2) is strictly quasi-concave.

Assumption 2 φi(w1, w2) is strictly quasi-concave in wi, i ∈ {1, 2}.

Assumption 3 φi(w, w) is strictly quasi-concave in w, i ∈ {1, 2}.

Assumption 4 −1 < R0

i

(wj ) ≤ (∂πi/∂wj )/(∂πi/∂wi), i ∈ {1, 2}, i 6= j.

Assumptions 1–3 imply that the supplier’s profit function and the Nash products with and

without price discrimination are single-peaked. Assumption 4 embodies two assumptions.

The second inequality is true if and only if firm i’s profits from its negotiations with the

supplier are increasing in firm j’s wholesale price. This assumption is quite natural, although

it is not implied by the others. Combining this assumption with the first inequality in

Assumption 4 ensures that the bargaining equilibrium is (locally) strictly stable. These

assumptions are satisfied in a variety of environments, e.g., under Cournot or differentiated

Bertrand competition with linear demand and constant marginal cost.

II. Price Discrimination in the Bargaining Model

A. Equilibrium When Price Discrimination is Allowed

Intuition about the bargaining solution can be gained by rewriting the first order condition

for the negotiations between the supplier and firm i when the integration constraint is slack:

γi[−∂πi(w

A

1

, w

A

2

)/∂wi]

πi(wA

1

, wA

2

) − di

=

(1 − γi)[∂U(w

A

1

, w

A

2

)/∂wi]

U(wA

1

, wA

2

) − dui

(6)

or

Firm i’s weighted concession cost

Firm i’s net profits =

Supplier’s weighted concession cost

Supplier’s net profits (7) .

That is, in a bargaining equilibrium, the wholesale price negotiated by the supplier and firm

i equalizes their weighted concession costs as a percentage of their gains from trade, where

9
the weights are the firms’ bargaining weights. The intuitive interpretation of this condition

is that the firm with the lower percentage concession cost loses less when improving its

offer and thus should do so to facilitate reaching agreement.14 Note that πiis decreasing

and U is increasing in wi over the range of conflict relevant for bargaining. Using this

fact, it is straightforward to see the effects of different sources of firm i’s bargaining power

from condition (6). Other factors equal, an increase in firm i’s (absolute) concession costs

(−∂πi/∂wi) requires an increase in firm i’s relative profits for condition (6) to continue to

hold. This requires a reduction in firm i’s wholesale price. Similarly, an increase in firm

i’s bargaining weight, an increase in its disagreement profit, or a decrease in the supplier’s

disagreement profit will result in a decrease in firm i’s wholesale price for any given wj.

w1

w2

U(w ,w ) = U(I(w ),w )

1 2 2 2

R (w )

1 2

1

R (w )

1 2

0

I(w )

2

T T

T argmax U(w ,w )

w1,w

1

2

2

R (w )

A A 2 1

1

0

I(w )

2

T

w2

T

Figure 1: Bargaining equilibrium when price discrimination is allowed.

The equilibrium prices are illustrated in Figure 1, which is analogous to Figure 1 in

Katz. The curve I(w2) represents the value of w1 such that the chain is indifferent be14This heuristic is due to Frederik Zeuthen (1930), who described this solution for the case of equal

bargaining weights. John Harsanyi (1956) demonstrates that Zeuthen’s heuristic solution is equivalent to

the symmetric Nash bargaining solution.

10
tween integration and non-integration when the wholesale prices are I(w2) and w2, i.e.,

π1(I(w2), w2) = π

I

(v, w2) for all w2. The chain prefers integration over non-integration for

all (w1, w2) to the right of I(w2). The integration constraint was the only source of chain

bargaining power in the take-it or leave-it model studied by Katz. He showed that if the

supplier finds it profitable to sell to the chain, it maximizes profits by choosing the wholesale prices represented by point T, where the supplier’s iso-profit contour is tangent to the

integration constraint. The chain receives a discount relative to the independent because it

has a credible threat to integrate backward.

In the bargaining model, the integration constraint will not bind if the chain has enough

bargaining power from other sources. For example, point A0represents a bargaining equilibrium when downstream firms have symmetric bargaining power (d1 = d2, du1 = du2, γ1 = γ2)

high enough that the integration constraint is slack. An increase in the chain’s bargaining

power through any of the mechanisms described above is represented as a leftward shift

of its bargaining reaction function, e.g., from R0

1

(w2) to R1

1

(w2). This changes equilibrium

wholesale prices from point A0to point A1. The chain’s wholesale price falls unambiguously.

The independent’s price may rise or fall, but Assumption 4 implies that it cannot fall by

more than the chain’s price. Thus, an increase in the chain’s bargaining power allows it

to negotiate a discount relative to the independent. These results are summarized in the

following Lemma.

Lemma 1 Suppose the integration constraint is slack. Then in a bargaining equilibrium

when price discrimination is allowed, firm i’s wholesale price w

A

i

is strictly decreasing in γi,

di, and −dui. Moreover, firm i’s equilibrium discount, w

A

j − w

A

i

, is strictly increasing in γi,

di, and −dui.

11
B. How Bargaining Can Lead to Chain Discounts

When Robinson-Patman was passed, the common perception was that chain stores tended to

pay lower wholesale prices than independents. In this model, four types of factors affect the

chain’s ability to negotiate discounts: the chain’s integration threat, the buyers’ disagreement

profits, the supplier’s disagreement profits, and the firms’ bargaining weights. I discuss briefly

the role each factor might play in generating chain discounts.

The potential for backward integration (or, more generally, the ability to seek alternative

supplies) affects the chain’s bargaining power by giving it an outside option. The effect of

this option is similar to its role in the take-it or leave-it model studied by Katz. In both

cases, the potential for backward integration may constrain the wholesale prices the supplier

can charge without inducing chain integration. The main difference between the models is

that the threat to integrate backward is not a binding constraint in the bargaining model if

the chain has enough bargaining power from other sources.

From Lemma 1, a sufficient condition for the chain to receive a discount is that d1 ≥ d2,

du1 ≤ du2, and γ1 ≥ γ2, with at least one of these inequalities being strict. Consider first the

disagreement profits of the downstream firms. The chain’s disagreement profit will exceed the

independent’s if the chain has a more profitable “inside option,” i.e., if it earns greater profits

than the independent during its negotiations with the supplier.15 For example, the chain may

be able to substitute another product during negotiations that achieves greater sales or a

higher margin than the independent could achieve with its next best alternative. The chain’s

disagreement profit will also exceed the independent’s if it earns greater profit in the event

negotiations break down. For example, suppose the exogenous risk of a breakdown arises

from the possibility of new entry that would make it unprofitable for the downstream firms

15Muthoo (1999) refers to the profit earned by a player during negotiations as its “inside option.”

12
to carry the supplier’s product. If the chain has better alternatives than the independent,

then d1 > d2.

The chain can also receive a discount if, other factors equal, the supplier’s disagreement

profit in negotiations with the chain is lower than its disagreement profit in negotiations with

the independent, i.e., du1 < du2. Suppose that if negotiations break down with the chain, but

not the independent, the chain will be a stronger competitor against the independent than

the independent would be against the chain in the opposite situation where negotiations

broke down with only the independent.16 If the chain integration constraint is slack in the

event negotiations break down with the independent, we would expect du1 < du2. The logic

is that if the supplier is unconstrained in selling to only one of the downstream firms, it is

better off selling unconstrained to the firm that faces less vigourous competition.17

Finally, the chain will also receive a discount if, other factors equal, it has a greater

bargaining weight than the independent. This occurs if the chain has a lower discount rate,

which might be the case if it has lower capital costs than the independent.

III. The Effects of Forbidding Price Discrimination

When price discrimination is forbidden, the two buyers will pay a single price.18 It is not

obvious what role each firm will play in determining that price. One possibility is that the

supplier can select one of the downstream firms to negotiate a common price. At an intuitive

16A breakdown with only one of the downstream firms might occur if an entrant comes in and displaces

only that firm.

17Plausible reasons can also be given for why du1 might exceed du2. For example, if the chain integration

constraint binds in the event negotiations with the independent break down, then the supplier might earn

more selling through the independent than the chain if the chain has better outside opportunities than the

independent. As another example, suppose the supplier’s inside option in negotiations with firm i is the

profit it earns from sales to firm j while it negotiates with firm i. If the price charged to independent

during negotiations with the chain is higher than the price charged the chain during negotiations with the

independent in this event, then it is also possible to have du1 > du2.

18In this paper I am abstracting from enforcement costs that might permit some price discrimination to

go unchallenged even when discrimination is illegal.

13
level, one might argue that even if the other downstream firm would like to participate in the

bargaining, the supplier does not have to listen to the other firm while it negotiates with the

firm it selects. On the other hand, the downstream firms have a joint incentive to have the

stronger firm negotiate price because both benefit from having a lower, common marginal

cost. If the downstream firms can coordinate their strategies, the weaker firm may simply

refuse to listen to the supplier, thinking that its rival may be able to negotiate a lower price.

Another issue when price discrimination is forbidden is whether the first buyer to reach

agreement determines the wholesale price for the second, or whether the second buyer can

negotiate a new price that must then be given to first. The answer to this question is likely

to depend on the nature of supplier and buyer liability under the rule that prohibits price

discrimination. For example, suppose the supplier first negotiates a “high” wholesale price

with the independent. The chain might then try to negotiate a lower price. However, if the

supplier keeps the high price in place with the independent, then the chain may risk being

sued by the independent and the authority that enforces the rule against price discrimination.

As noted in footnote 2, the chain would be liable under Section 2(f) of the Robinson-Patman

Act. The supplier would also be liable under the other Sections of the Act, but if a lawsuit

would result in a cease and desist order that would allow the supplier to charge both buyers

the high price, then price discrimination may be worth the risk. If this were true, the chain

would be better off simply accepting the high price or pursuing its outside option. On the

other hand, if the supplier’s liability is sufficiently high, or if a cease and desist order would

lead to the low price for both buyers, then the supplier may accede to the chain’s demands. A

third possibility is that the supplier would simply refuse to sell to the chain unless it agreed

to the same price as the independent. If this were credible, the chain would presumably

agree to the higher price or exercise its outside option.

In this paper, I will not attempt to resolve the buyer coordination and legal liability

14
questions that affect the influence each buyer is likely to have in negotiating a common price.

Instead, I will consider two cases distinguished by which buyer negotiates the wholesale price.

These two cases represent endpoints of the set of agreements that are likely to emerge from

bargaining when price discrimination is forbidden.

Suppose first that the supplier negotiates a common price with the independent. If both

firms prefer to have the chain remain non-integrated, their Nash bargaining solution solves

max

w

φ2(w, w) = [U(w, w) − du2]

1−γ2

[π2(w, w) − d2]

γ2 s.t. π1(w, w) ≥ πI

(v, w

I

2

).

The first order condition is

0 = (1 − γ2)

X

i

∂U

∂wi

[π2 − d2] + γ2

X

i

∂π2

∂wi

[U − du2] + η

X

i

∂π1

∂wi

=

∂φ2

∂w2

+

(

(1 − γ2)

∂U

∂w1

[π2 − d2] + γ2

∂π2

∂w1

[U − du2]

)

+ η

X

i

∂π1

∂wi

(8) ,

η ≥ 0, η[π2(w, w) − π

I

(v, w

I

2

(9) )] = 0

where η is a Lagrangian multiplier. Recall that ∂φ2/∂w2 is the derivative of the Nash product

for negotiations between the supplier and firm 2 when discrimination is allowed. Suppose

first that firms 1 and 2 are symmetric except for their abilities to integrate backward (i.e.,

d1 = d2, du1 = du2, and γ1 = γ2). Then w

A

1 = w

A

2

, and ∂φ2(w

A

2

, w

A

2

)/∂w2 = 0 by the first

order condition for the optimal choice of w

A

2

. The term in curly braces in (8) is positive

at (w

A

1

, w

A

2

) because firm 2’s profit is increasing in w1, ∂U/∂w1 is positive over the range

of conflict, and net profits are positive in a bargaining equilibrium when discrimination is

allowed. Therefore, if the integration constraint is slack at w = w

A

2

, the price that solves

(8), say w

F

(the superscript ‘F’ for “forbidden”), must exceed w

A

2

.

19 Assuming that welfare

(measured as the sum of consumer and producer surplus) is decreasing in the wholesale price,

we have the following proposition.

19This follows from the strict quasi-concavity of the Nash product φ2, Assumption 2.

15
Proposition 1 Suppose that when price discrimination is allowed, the integration constraint

is slack. If downstream firms are symmetric, and if there is no integration under either

regime, the wholesale price is lower and welfare is higher when price discrimination is practiced than when it is forbidden.

The intuition for this result can be seen by rewriting condition (8) when the integration

constraint is slack as

γ2[(−∂π2/∂w2) + (−∂π2/∂w1)]

π2 − d2

=

(1 − γ2)[(∂U/∂w2) + (∂U/∂w1)]

U − du2

(10) .

Notice that this condition is the same as condition (6) except that firm 2’s concession cost is

lower by ∂π2/∂w1 and the supplier’s concession cost is higher by ∂U/∂w1. A policy forbidding

price discrimination reduces firm 2’s concession cost because an agreement to pay a higher

price requires its rival to pay a higher price too. On the other hand, the policy increases

the supplier’s concession cost because an agreement to charge a lower price must be granted

to firm 1 as well as firm 2. Both concession cost effects strengthen the supplier’s relative

bargaining position, allowing it to negotiate a higher wholesale price.

Next, suppose that the chain has greater bargaining power than the independent and

that the supplier still negotiates the common price with the independent. The bargaining

equilibrium when discrimination is allowed is represented by point A1in Figure 2, where

wholesale prices are (w

A1

1

, w

A1

2

). The effects of forbidding price discrimination can be seen by

evaluating condition (8) at the wholesale prices that would be chosen when discrimination is

allowed if the chain’s bargaining power were the same as the independent’s. This is point A0

in Figure 2, where wholesale prices are (w

A0

, w

A0

). Since this price lies on the independent’s

bargaining reaction function, it must be true that ∂φ2(w

A0

, w

A0

)/∂w2 = 0. Since the terms

in curly braces in (8) are positive and the Nash product is strictly quasi-concave, this implies

that the wholesale price that solves (8), w

F

, exceeds w

A0

.

16
w1

w2

R (w )

1 2

1

R (w )

1 2

0

I(w )

2

T

R (w )

A A 2 1

1

0

(w ,w )

A A

1 1

(w ,w )

A A

0 0

1 2

(w ,w )

F F

wA

1

T'

A

2

(w ,w )

A A

1 1

1 2

(w ,w )

(w ,w )

A A

2 2

Figure 2: Wholesale price effects of forbidding price discrimination.

Consider the line through point A1 with a slope of negative one. The average wholesale

price is constant along this line and is given by w

A1

= (w

A1

1 + w

A2

2

)/2. By Lemma 1,

A0lies above and to the right of this line. Since w

F > wA0

, it follows that forbidding

price discrimination raises the average wholesale price. Note that the supplier’s profits are

increasing in the average wholesale price by the assumption that U is strictly quasi-concave.

Note further that if the chain were the weaker firm, and if the supplier negotiated the

common price with the chain, the same argument would hold with the subscripts reversed.

Therefore, if the supplier can select the downstream firm that will negotiate the wholesale

price, the average wholesale price will be higher when discrimination is forbidden than when

it is practiced.

Proposition 2 Suppose that when price discrimination is allowed, the integration constraint

is slack, and that the supplier can select which downstream firm will negotiate the common

price. Then if there is no integration under either regime, the average wholesale price is

higher when discrimination is practiced than when it is forbidden.

17
Augustin Cournot (1838) has shown that the Cournot equilibrium output is decreasing

in the average marginal cost of the Cournot competitors. If downstream firms produce under

the same fixed proportions technology, welfare is an increasing function of the total output.

These observations yield the following corollary to Proposition 2 for the case of Cournot

competition in the downstream market.

Corollary 1 Suppose that when price discrimination is allowed, the integration constraint

is slack, and that the supplier can select which downstream firm will negotiate the common

price. Suppose further that downstream firms are Cournot competitors that employ a fixed

proportions technology. Then if there is no integration under either regime, total output and

welfare are higher when price discrimination is practiced than when it is forbidden.

Katz’s Proposition 1 shows that in the take-it or leave-it environment, if integration

occurs in neither regime, output and welfare are lower when price discrimination is practiced

than when it is forbidden. Propositions 1 and 2 (and Corollary 1) above show that this result

is reversed if three conditions hold: i) downstream firms have bargaining power from sources

other than outside options; ii) the chain integration threat is not credible (i.e., the constraint

is slack) when price discrimination is allowed; and iii) downstream firms are symmetric, or

the supplier can select which downstream firm will negotiate a common price. The results of

the take-it or leave-it and bargaining environments can be compared using Figure 2. In the

take-it or leave-it environment, a policy forbidding price discrimination causes the supplier to

reduce wholesale prices from T to T

0

. Intuitively, a reduction in the independent’s wholesale

price to bring it in line with the chain’s price reduces the chain’s profits, requiring a reduction

in the chain’s wholesale price to prevent it from integrating backward. Thus, both wholesale

prices fall when discrimination is forbidden.20 This result relies on the chain having a credible

20Katz also considered a case in which the chain’s integration incentives are increasing in the price w2

18
threat to integrate backward at the prices offered by the supplier in both regimes, so that

a prohibition on price discrimination causes the supplier to adjust prices along the chain

integration constraint I(w2). If the integration constraint is slack when discrimination is

allowed, this effect is absent. The policy still affects concession costs, however, in a way that

increases the supplier’s relative bargaining power with either downstream firm. Under the

conditions of Propositions 1 and 2, this effect causes both wholesale prices to rise.

The next case to consider arises when firms are asymmetric and the supplier negotiates

a common price with the stronger buyer, assumed to be the chain. This might occur if

downstream firms could coordinate on the order in which they bargain, or if the supplier

is better off accepting the chain’s terms than either risking a lawsuit for engaging in price

discrimination or selling only to the independent. A diagrammatic argument similar to the

one above suggests that the effects of forbidding price discrimination are generally ambiguous

in this case. To see this, suppose that point A1in Figure 2 represents the equilibrium

wholesale prices when discrimination is allowed. If the independent had bargaining power as

high as the chain’s, the equilibrium price would be point A2. By arguments similar to those

made above, the supplier and the chain will negotiate a common price greater than w

A2

,

but this price may or may not exceed w

A1

. If the chain has only a little more bargaining

power than the independent (i.e., if R1

1

is close to R0

1

), then w

F > wA1

.

21 But if the chain’s

bargaining power exceeds the independent’s by a large amount, (R1

1 well to the left of R01

),

then it is possible that w

F < w

A1

.

22

offered by the supplier to the independent. This can arise in his model if a higher value of w2 signals that

the supplier has higher costs and would charge a higher price to the independent after chain integration. In

this case, the integration constraint is downward sloping. A policy forbidding price discrimination can result

in a higher wholesale price for the chain, but it still causes the average wholesale price to fall in his model

if the chain chooses non-integration in both regimes.

21This follows from the continuity of the equilibrium prices in the parameters that affect bargaining power.

22The effects of forbidding price discrimination in my model have an analogy with “pattern bargaining”

in union-labor negotiations. Under pattern bargaining, the labor union negotiates wage rates with one

firm and then offers them on a take-it or leave-it basis to the other. Paul W. Dobson (1994) examined the

circumstances under which unions prefer pattern bargaining over simultaneous negotiations with downstream

Cournot competitors who face linear demand. He shows that the union gains by negotiating with the firm

19
To get an idea of the degree of asymmetry required for a policy against price discrimination to be beneficial when the stronger buyer conducts the bargaining, I have solved

an example with Cournot competition in the downstream market. The inverse demand is

P(X) = 1 − X where X is aggregate output, the supplier’s marginal cost is c = 0, and

γ1 = γ2 = 1/2.23 In the example, I varied d1, d2, du1, and du2 over a four dimensional grid

such that Assumptions 1–4 are satisfied and profitable equilibria without integration exist.24

The results are presented in Figure 3. In this example, the smallest chain discount such

0 0.2 0.4 0.6 0.8 1

Chain Discount

-0.1

-0.075

-0.05

-0.025

0

0.025

0.05

0.075

% Change in Welfare

Figure 3: Welfare effects of forbidding price discrimination for different values of the discount

negotiated by the chain when discrimination is allowed.

that a policy forbidding price discrimination increases welfare is 7.7 percent. The welfare

that is in the weaker bargaining position. My analysis differs from his in three main respects. First, I

consider how the potential for backward integration and inside and outside options affect different sources

of bargaining power, and how each source is affected by a policy forbidding price discrimination. Second, I

focus on the welfare effects of forbidding price discrimination. Third, I allow for more general rivalry in the

downstream market, and consider how the effects of forbidding price discrimination vary with the nature of

downstream rivalry (section IV below).

23The Cournot equilibrium output of firm i is xi(w1, w2) = (1 − 2wi + w2)/3, i ∈ {1, 2}, and total output

is X(w1, w2) = x1(w1, w2) + x2(w1, w2). The firms’ profits are πi(w1, w2) = (1 − 2wi + wj )

2/9, i ∈ {1, 2},

and U(w1, w2) = (w1 − c)x1(w1, w2) + (w2 − c)x2(w1, w2).

24In this example the joint monopoly profit is 0.25. Varying the supplier’s disagreement profits from 0 to

0.14 and the buyers’ disagreement profits from 0 to 0.15 was sufficient to cover the relevant part of the grid.

I varied du1 and du2 in increments of 0.02 and d1 and d2 in increments of 0.01.

20
benefits of forbidding price discrimination, when they exist, tend to be small (less than 3

percent if the chain discount is less than 30 percent.) The welfare cost of forbidding price

discrimination can be as high as about 10 percent even when the chain negotiates discounts

as high as 25 percent.

IV. Two Special Cases

Equations (6) and (10) are useful for motivating how the results are affected by different

assumptions about downstream rivalry. Suppose that firms 1 and 2 are monopolists in

different markets. In this case ∂π2/∂w1 = 0 because the downstream firms do not compete.

A policy forbidding price discrimination therefore does not affect the downstream firms’

concession costs, but it still increases the supplier’s concession costs (e.g., by ∂U/∂w1 in

negotiations with firm 2), leading to a higher wholesale price.25

The presence of downstream competition reinforces the concession cost effect of forbidding

price discrimination because the policy then affects downstream firms’ concession costs. A

natural question is whether this bargaining effect persists as downstream rivalry becomes

more intense. Intuitively, one might think that as the downstream market becomes more

competitive, each downstream firm’s bargaining power would fall. If the supplier had all

the bargaining power in both regimes, the bargaining model would collapse to the take-it or

leave-it model. This intuition turns out to be incorrect. In fact, the wholesale price effects of

forbidding price discrimination can get worse as downstream rivalry becomes more intense.

To demonstrate this point, I consider an example in which downstream firms sell homogeneous products and engage in rivalry that can be summarized by the following pricing

25This result is similar to the finding of Cooper and Fries (1993) that most-favored-nation pricing by a

monopolist to independent buyers can lead to higher prices when prices are negotiated. Cooper and Fries

did not consider the case of competing downstream firms, asymmetric bargaining powers, or the potential

for backward integration.

21
rule:

P(X) = wi − θP

0

(X)xi (11) , i ∈ {1, 2}

where X is industry output, P(X) is the inverse demand, xiis firm i’s output, and θ ∈ [0, 2] is

a parameter that measures industry conduct.26 I assume that the industry revenue function,

R(X) = P(X)X, is concave. The solution to the two equations in (11) yields downstream

equilibrium quantities x

θ

1

and x

θ

2

. As θ falls, rivalry becomes more intense in the sense that

price falls toward marginal cost. The Cournot outcome occurs when θ = 1. If w1 = w2, the

monopoly outcome occurs when θ = 2, and the competitive outcome arises when θ = 0.

Denote the reduced form profits the same way as in previous sections, but with the

superscript θ to indicate their dependence on downstream conduct. Firm i’s equilibrium

profits are π

θ

i = (P − wi)x

θ

i

, i ∈ {1, 2}, and the supplier’s profits are U

θ = (w1 − c)xθ

1 +

(w2 −c)x

θ

2

. If the integration constraint is slack, the equilibrium wholesale prices when price

discrimination is allowed solve

γi[−∂π

θ

i

/∂wi]

π

θ

i − di

=

(1 − γi)[∂U

θ/∂wi

]

Uθ − dui

(12) , i ∈ {1, 2}.

Assuming symmetry, the equilibrium wholesale price when price discrimination is forbidden

solves

γi[(−∂π

θ

i

/∂wi) + (−∂π

θ

i

/∂wj )]

π

θ

i − di

=

(1 − γi)[∂U

θ/∂wi + ∂Uθ/∂wj

]

Uθ − dui

(13) , i ∈ {1, 2}, i 6= j.

Straightforward comparative statics show that for all θ ∈ (0, 2], ∂U

θ/∂wi > 0, ∂πθ

i

/∂wi < 0,

and ∂πi/∂wj > 0 over the range of conflict. Thus, if Assumptions 1–4 hold, and if net profits

are positive, Lemma 1 and Proposition 1 go through. The effect of more intense rivalry on

bargaining equilibria in the two regimes depends on how the firms’ reduced-form profits and

concession costs vary with θ. The derivative of these expressions with respect to θ depends

26This rule is familiar from empirical applications in the “New Empirical IO” (NEIO) tradition, which

interpret θ as a measure of industry conduct. See, for example, Timothy F. Bresnahan (1989), David

Genesove and Wallace P. Mullin (1998) and Catherine Wolfram (1999).

22
on the third derivative of the inverse demand function and hence cannot be signed without

further assumptions. However, it is possible to determine the values of profits and concession

costs under symmetry in the limit as θ → 0.

Lemma 2 Suppose w1 = w2 = w. Then the following conditions hold in the limit as the

downstream market becomes competitive:

lim

θ→0

Ã

∂U

θ

∂wi

!

= x

θ

i +

w − c

2P0(Xθ)

(14) ,

lim

θ→0

Ã

−

∂π

θ

∂wi

!

=

x

θ

i

2

(15) ,

lim

θ→0

π

θ

(16) i = 0.

lim

θ→0

Ã

−

∂π

θ

i

∂wi

−

∂π

θ

i

∂wj

!

= −

P

00(Xθ

)Xθ + P

0

(Xθ)

(P0(Xθ))2 Xθ

(17)

Conditions (14) and (15) show that when price discrimination is allowed, the concession costs of the supplier and downstream firm i are bounded and have the expected signs

(∂U

θ/∂wi > 0 for small enough w and −∂πθ

i

/∂wi > 0) in the limit as the downstream market

becomes perfectly competitive. Condition (16) shows that downstream firms’ profits go to

zero as the market becomes competitive, as expected. Thus, downstream firm i’s concession

costs as a percentage of its gains from trade rise to infinity as the downstream market becomes competitive. Since the percentage concession costs of firm i and the supplier must be

equalized in a bargaining equilibrium, the supplier’s percentage concession costs must also

rise to infinity, which requires its net profits to fall to zero too. This establishes the following

proposition.

Proposition 3 Suppose that downstream firms are symmetric. In a bargaining equilibrium

when price discrimination is allowed, the supplier’s net profits fall to zero as the downstream

market becomes perfectly competitive.

23
This result may seem counter-intuitive at first, but it has a natural interpretation in a

bargaining environment. A firm’s bargaining power comes partly from its ability to inflict

a loss on the firm it is negotiating with by delaying an agreement. When the downstream

market is highly competitive, downstream firms earn little profit, so the supplier inflicts

only a small loss on each firm by delaying an agreement. The “inflicted loss” source of

the supplier’s bargaining power falls to zero as the downstream market becomes perfectly

competitive, so that in the limit as θ → 0 the supplier does no better than earning its

disagreement profits.

Next, consider the effects of forbidding price discrimination as the downstream market

becomes competitive. Condition (17) in Lemma 2 implies that when price discrimination is

forbidden, firm i’s concession costs as a percentage of its net profits are positive and finite

(using the assumption that R(X) is concave). From condition (13), the supplier’s percentage

concession costs must also be positive and finite, which requires U

θ − dui > 0. Thus, as the

downstream market becomes more competitive, wholesale prices remain higher when price

discrimination is forbidden than when it is allowed.

Proposition 4 Suppose that downstream firms are symmetric, and that downstream rivalry

is described by the conduct parameter θ in condition (11). For all θ ∈ [0, 2], total output and

welfare are lower when price discrimination is forbidden than when it is practiced.

Figure 4 plots the equilibrium wholesale price and welfare as a function of θ for an example

in which P(X) = 1 − X, c = 0, λ1 = λ2 = 1/2, d1 = d2 = 0, and du1 = du2 = 3/32.27 In this

example, a policy forbidding price discrimination has a larger effect on wholesale prices the

more competitive the market, as measured by the conduct parameter. Welfare rises as the

market becomes more competitive in both regimes, but the percentage reduction in welfare

27The supplier’s profit under successive monopoly is 3/32.

24
0.5 1 1.5 2

Conduct Parameter

0.2

0.3

0.4

0.5

$

wF

wA

Welfare - Disc. Forbidden

Welfare - Disc. Allowed

Figure 4: Wholesale prices and welfare as functions of the conduct parameter.

from a policy forbidding price discrimination remains roughly constant at about 10 percent.

V. Conclusion

The main implication of this paper is that the welfare effects of forbidding price discrimination in intermediate good markets turn partly on the source of the bargaining power used by

downstream firms to extract discounts. Katz’s results imply that if discounts are extracted

by explicit threats to integrate backward, then forbidding price discrimination can be socially beneficial. My results show that if the discounts are derived from some other source of

bargaining power, e.g., a greater disagreement profit for the chain, or a greater bargaining

weight, then forbidding price discrimination is often socially harmful.

The passage of the Robinson-Patman Act provides an experiment that in principle might

allow one to distinguish between the bargaining and take-it or leave-it models in different

industries. Where discounts arose from credible threats to integrate backward, both models

predict that the average wholesale price would have fallen after Robinson-Patman was passed

25
if the supplier adjusted wholesale prices to prevent chain integration. The law would have

made the supplier worse off in such markets. Where discounts arose from other sources of

bargaining power, the model predicts that the average wholesale price could have risen after

the law was passed. The supplier could have benefited from the law in such markets.

A formal study of the effects of the Robinson-Patman Act on prices has not been conducted, to my knowledge. However, the effects of Robinson-Patman on profits have been

explored both informally and using econometric methods. Morris Adelman (1959) conducted

an informal study of the wholesale pricing and purchasing practices of A&P, the dominant

grocery chain of the 1920s and 1930s. He argues that that suppliers with strong brand

positions that sold to A&P benefited from the Robinson-Patman Act.28 This conclusion is

consistent with chain bargaining power arising from concession costs, disagreement profits,

and bargaining weights. It is not consistent with the predictions of the take-it or leave-it

model.29

The most comprehensive formal study is that of Thomas W. Ross (1984). He used event

study methodology to evaluate the abnormal returns of grocery chains and manufacturers

after the Robinson-Patman Act was passed and after various enforcement actions were undertaken by the U.S. Federal Trade Commission (FTC). He found that the stock market

values of the major grocery chains in the U.S. fell significantly after the Act was passed.

This conclusion is consistent with the predictions of the bargaining model. It is not consistent with the take-it or leave-it model, which predicts that the chain firm’s profits should

equal the expected value of its outside option whether price discrimination is allowed or

forbidden.30

28See Adelman, p. 190.

29Adelman’s conclusion would also be consistent with the view that the Robinson-Patman Act facilitates

collusion among suppliers. However, collusion among differentiated manufacturers with strong brand positions is notoriously difficult. Thus, the bargaining model probably provides a more plausible explanation for

Adelman’s conclusion.

30Ross was not able to reject the hypothesis that the Robinson-Patman Act had no effect on the stock

26
The implications of bargaining for antitrust policy are not well understood. However,

bargaining is prevalent in intermediate good markets, where a large share of the antitrust

enforcement in developed countries takes place. This paper shows that bargaining has important implications for understanding the effects of the Robinson-Patman Act. There is

every reason to believe that bargaining could have important implications for understanding

the effects of antitrust laws governing mergers, vertical restraints, and collusion as well.

market values of grocery manufacturers in the U.S.

27
APPENDIX

Proof of Lemma 1

Differentiating the system (3) and (5) when the integration constraint is slack yields

∂w

A

i

∂γi

=

∂

2φi

∂w2

i

3

∂U

∂wi

[πi − di] −

∂πi

∂wi

[U − dui]

 ́

D

(18) ,

∂(w

A

j − w

A

i

)

∂γi

=

−

1⁄2

∂

2φi

∂w2

i

+

∂

2φi

∂wi∂wj

3⁄4 3

∂U

∂wi

[πi − di] −

∂πi

∂wi

[U − dui]

 ́

D

(19) ,

∂w

A

i

∂di

=

∂

2φi

∂w2

i

3

[1 − γi]

∂U

∂wi

 ́

D

(20) ,

∂(w

A

j − w

A

i

)

∂di

=

−

1⁄2

∂

2φi

∂w2

i

+

∂

2φi

∂wi∂wj

3⁄4 3

[1 − γi]

∂U

∂wi

 ́

D

(21) ,

∂w

A

i

∂dui

=

∂

2φi

∂w2

i

3

γi

∂πi

∂wi

 ́

D

(22) ,

∂(w

A

j − w

A

i

)

∂dui

=

−

1⁄2

∂

2φi

∂w2

i

+

∂

2φi

∂wi∂wj

3⁄4 3

γi

∂πi

∂wi

 ́

D

(23) ,

where

D =

Ã

∂

2φ1

∂w2

1

! Ã

∂

2φ2

∂w2

2

!

−

Ã

∂

2φ1

∂w1∂w2

! Ã

∂

2φ2

∂w2∂w1

!

(24) .

Note that ∂U/∂wi > 0 and ∂πi/∂wi < 0 over the range of conflict, and the net profit terms

U − dui and πi = di are positive in equilibrium. The second derivative ∂

2φi/∂w2

i

is negative

by Assumption 2. Thus, the sign of the derivatives in (18)-(23) depends on the signs of the

terms in curly brackets and the sign of D. Differentiating (3) and (5) individually when the

integration constraint is slack yields

∂

2φi

∂wi∂wj

= −R

0

i

(wj )

∂

2φi

∂w2

i

(25) .

Substituting (25) into (24) and using Assumptions 4 and 2 implies that D > 0. Substituting

(24) and (25) into the derivatives (18) through (23) yields the results in the Lemma.

28
Proof of Lemma 2

For future reference we need the derivatives of the equilibrium quantities with respect to

w1. Differentiating the system (11) with respect to w1 gives

Ã

Ω11 Ω12

Ω21 Ω22 ! Ã

∂x

θ

1

/∂w1

∂x

θ

2

/∂w1

!

=

Ã

1

0

!

(26)

where Ωii = θP

00xi + (1 + θ)P0 and Ωij = θP00xi + P0

, i ∈ {1, 2}, i 6= j. Let Dθ =

Ω11Ω22 − Ω12Ω21 = P

0

[θ

2P00Xθ + θ(2 + θ)P0

]. Solving (26) gives

∂x

θ

1

∂w1

=

Ω22

Dθ

=

θP

00xθ

2 + (1 + θ)P

0

Dθ

(27)

∂x

θ

2

∂w1

=

−Ω21

Dθ

=

−[θP

00xθ

2 + (1 + θ)P

0

]

Dθ

(28) .

We can now calculate the derivatives in the Lemma and verify the limits. Differentiating

the supplier’s profits and evaluating at w1 = w2 = w yields

∂U

θ

∂w1

= x

θ

1 + (w − c)

Ã

∂x

θ

1

∂w1

+

∂x

θ

2

∂w1

!

(29)

= x

θ

1 +

(w − c)θ

∆

(30)

where ∆ = Dθ/P0 = θ

2P00Xθ + θ(2 + θ)P0

. Let Xθ

0

, and ∆0 denote derivatives with respect

to θ. By L’Hopital’s rule,

lim

θ→0

θ

∆

= lim

θ→0

μ

1

∆0

¶

(31)

= lim

θ→0

Ã

1

2θP00Xθ + θ

2

[P000Xθ + P00]Xθ

0 + 2(1 + θ)P0 + θ(2 + θ)P00Xθ0

!

(32)

=

1

2P0

(33) .

Substituting (33) into (30) and imposing symmetry yields (14).

The limit of firm 1’s concession cost is

lim

θ→0

Ã

−

∂π

θ

1

∂w1

!

= lim

θ→0

Ã

−P

0

"

∂x

θ

1

∂w1

+

∂x

θ

2

∂w1

#

x

θ

1 + x

θ

1

!

(34)

= lim

θ→0

Ã

−

P

0

θ

∆

x

θ

1 + x

θ

1

!

(35) .

29
Substituting (33) into (35) and imposing symmetry yields (15).

Inspection of condition (11) implies that P → wi as θ → 0. This implies condition (16).

Let w1 = w2 = w. Using symmetry, the limit of firm 1’s concession costs when price

discrimination is forbidden is

lim

θ→0





−

·

∂π

θ

1

∂w1

+

∂π

θ

1

∂w2

 ̧

π

θ

1



 = lim

θ→0

μ

−2P

0

·

∂x

θ

1

∂w1

+

∂x

θ

2

∂w1

 ̧

x

θ

1 + x

θ

1

¶

(P − w)x

θ

1

(36)

= lim

θ→0

Ã

θ

2

[P

00Xθ + P0

]

(P − w)∆ !

(37) .

Both the numerator and denominator of (37) converge to zero as θ → 0. Let A = P

00Xθ +P0

,

and let A0 denote the derivative of A with respect to θ. Applying L’Hopital’s rule to condition

(37) twice, we have

lim

θ→0

Ã

θ

2

[P

00Xθ + P0

]

(P − w)∆ !

= lim

θ→0

Ã

2θA + θ

2A0

P0Xθ

0∆ + (P − w)∆0

!

(38)

= lim

θ→0

Ã

2A + 4θA0 + θ

2A00

[P00(Xθ

0

)

2 + P0Xθ

00]∆ + 2∆0P0Xθ

0 + (P − w)∆00 !

(39)

= lim

θ→0

μ

2A

2∆0P0Xθ

0

¶

(40) .

The derivative of ∆ is

∆

0 = 2θP00Xθ + θ2

[P

000Xθ + P00]Xθ

0

+ (2 + 2θ)P

0 + θ(2 + θ)P00Xθ

0

(41)

Taking the limit gives

lim

θ→0

∆

0 = 2P0

(42) .

The derivative of Xθis found by differentiating (11) with respect to θ and imposing symmetry. This gives

X

θ

0

=

−P

0Xθ

θP00Xθ + (2 + θ)P0

(43)

and

lim

θ→0

X

θ

0

=

−Xθ

2

(44)

Condition (17) results from substituting conditions (42) and (44) into (40).

30
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