Price-Quantity Competitions and Formationof Buyer-Seller Networks∗
Ryoji Okatani†
Keywords: Price competition; Network formation;
Journal of Economic Literature Classiﬁcation Numbers: C72; D85

1Introduction

During 2006-2007, Intel and AMD, which are suppliers of CPU, were said to
be experienced a price competition. In 2008, the price competition calmed
down, and then the both companies improved their performances. Customersof such suppliers are manufacturers of computer. For example, Dell, HP, Apple, and so on. In such a market, once a manufacturer adopts a componentof its product, it usually uses the component over a long period because it
is diﬃcult to change the component immediately. Therefore, for a manufacturer, the decision that which components it adopts is important. For asupplier, it is important that how many customers they can get.
We model such a market as a network formation game. Buyers choose
components (i.e. sellers) they adopt. Buyers’ choices are described by anetwork, in which a link between a buyer and a seller indicates the buyer
adopts the seller’s good, and they can trade. Sellers are engaged in pricecompetition with quantity precommitment. Each seller decides a quantityand a price without knowing buyers’ exact valuations.
This paper is closely related to the industrial organization literature oncapacity competition. For example, by Kreps and Sheinkman (1983), the∗The author is indebted to Jun Iritani (Kobe University), Eiichi Miyagawa (KobeUniversity), and Tomoyuki Kamo (Kyoto Sangyo University) for helpful comments. Theauthor is responsible for all errors.
†Graduate school of Economics, Kobe University, 2-1 Rokkodaicho, Nada, Kobe 6578501, E-mail: okatani0820@gmail.com

1result that quantity precommitment and Bertrand competition yield Cournot
outcomes is well known.1
Furthermore, this paper is related to the model of buyer-seller networks.
In a series of papers, Kranton and Minehart (2000a, b, and 2001) analyze
the formation of buyer-seller networks. Kranton and Minehart (2001) characterized the equilibrium of the English auction in buyer-seller networks. In
their model, the price at which markets clear is competitive, and then, anallocation as a result of trading at the price is eﬃcient. Corominas-Bosh(2004) studies a bargaining between buyers and sellers who are connected bya network. Besides them, the present paper focuses on a competition amongsellers.

2Model2.1

Players and goods

Let K = {1, . . . , K} be the set of indivisible goods. Let B = {b1 , . . . , bm } bethe set of buyers who each demand only one indivisible unit of good. Let
S = {s1 , . . . , sn } be the set of sellers who each sj sell good kj ∈ K. Eachbuyer bi has valuations vki for good k ∈ K. Let Vk = [0, 1] denote the set ofivaluations for good k. Let V = V1 × · · · × VK . Let v i = (v1i , . . . , vK) be a
1m
proﬁle of buyer i’s valuations. Let v = (v , . . . , v ) denote a proﬁle of buyers’valuation proﬁles.

2.2

NetworksBuyers and sellers can exchange only if they are linked. Let gij ∈ {0, 1}
denote a relationship between buyer bi and seller sj . If bi and sj are linked,then gij = 1, and 0 otherwise. Let G be a m × n matrix, where the (i, j)thelement is gij . We denote a network by G. Let G be the set of all networks.
For a given network G and a set of buyers B 0 ⊆ B, let L(B0 ) = {j ∈ S : i ∈B 0 and gij = 1} denote the buyers’ linked set of sellers. Similarly, for a setof sellers S 0 ⊆ S, let L(S 0 ) = {i ∈ B : j ∈ S 0 and gij = 1} denote the sellers’linked set of buyers.

1Levitan and Shubik (1980), and Davidson and Deneckere (1990) also study such amarket.

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2.3

Strategies

Buyers and seller play the following two-stage game.
Stage 1. Each buyer chooses links that connect to sellers.
Each seller chooses the quantity of its product.
Stage 2. Each Seller names the prices for the buyers.
The buyer bi ’s valuation for good k is distributed over the interval Vk =[0, 1] according to uniform distribution function F : Vk → [0, 1] with associated density function f . All distributions are common knowledge. Thevaluations are private information for buyers. Sellers do not know the exactvaluations.
In Stage 1, buyers and sellers act simultaneously. Let λi : V → {0, 1}n belinking of buyer bi , denoted by()λi (v i ) = λi1 (v i ), . . . , λin (v i ) ∈ {0, 1}n ,where λij (v i ) = 1 indicates that bi can adopt seller sj ’s good as input after bi
receives his valuation v i . The network that is formed in this stage is denotedby a m × n matrix G = [λij (v i )]. Let λ = (λ1 , . . . , λm ) denote a proﬁle ofbuyers’ linkings, and let Λ be the set of linkings. We ∑suppose that a linkcosts d for the buyer. Therefore, buyer bi pays the cost nj=1 λij (v i )d in thisstage.
Each seller sj chooses a quantity of production. Let qj ∈ Z+ denote thequantity of sj ’s production. Let q = (q1 , . . . , qn ) denote a proﬁle of sellers’production. We assume that producing one unit costs c for the seller.
In Stage 2, only sellers act. Sellers know the network and quantities thatare ditermined in Stage 1. Let pj : G → Rn+ be seller sj ’s pricing, which isdenoted by pj (G) = (p1j (G), . . . , pnj (G)), where pij (g) is the price named byseller sj for bi . Let p = (p1 , . . . , pn ) denote a proﬁle of all sellers’ pricing.

2.4

PayoﬀsWe now deﬁne buyers’ and sellers’ payoﬀ function. First, we deﬁne a buyer’s
payoﬀ function. If buyer bi purchases a good kj , then the net beneﬁt fromgood kj is vki j − pij . Furthermore, buyer bi pays a linking cost xd, where x is
the number of links bi formed. Then, bi ’s payoﬀ is vki j − pij − xd.
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The buyer’s payoﬀ function is Vi : Λ × Zn+ × Rn++ × V → R, deﬁned by∫iVi (λ, q | v , p) =δij (λ, q, p, v −i | v i )(vki j − pij )f K−1 (v −i )dv −i
V (K−1)−n∑λij (v i )d,j=1

where δij (λ, q, p, p, v −i | v i ) is the probability that buyer bi buys from seller
sj . The probability δij is deﬁned in the next section.
The seller’s expected proﬁt function is Πj : Λ × Zn+ × Rn++ → R, deﬁnedby∫θij (λ, q, v, p)pij ([λij (v i )])f K (v)dv − cqj ,Πj (λ, q; p) =V (n)where θij (λ, q, v, p) is a probability that seller sj sells a good to buyer bi when
prices are pij . The probability θij is deﬁned in the next section.
This two-stage game is solved backwardly. In Stage 2, given quantities
and the network, each seller names prices to maximize its expected proﬁt. At
the end of Stage 1, each seller updates information of the buyers’ valuations.
For example, if buyer bi forms a link to seller sj , then buyers consider thatvki j > d. In this case, we suppose that sellers consider that bi ’s valuation is
uniformly ditrbuted over (d, 1]. Each seller maximizes its expected proﬁt byusing the newly obtained distribution function f˜K . The expected revenue inStage 2 is∫θij (λ, q, p, v)pij (G)f˜K (v)dv.
V (n)Therefore, seller sj names prices pij , i = 1, . . . , n such that∫θij (λ, q, p, v)pij (G)f˜K (v)dv(n)V∫θij (λ, q, p0i , p−i , v)p0ij (G)f˜K (v)dv.
≥V (n)In Stage 1, each buyer chooses its links to maximize the expected payoﬀ
Vi subject to the expected prices that will be named in Stage 2. Each sellerchooses quantities to maximize the expected payoﬀ Πj .
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3A small market: two buyers and two sellers

We consider a small market that both numbers of buyers and sellers are
two. We suppose that sellers s1 and s2 produce diﬀerent goods 1 and 2,respectively.
For simplicity, we suppose that costs of producing a unit and forming alink are two types: cL = 1/16, cH = 1/4, dL = 1/16, and dH = 1/4.

3.1

Sellers’ pricing

In Stage 2, sellers name their prices to buyers. Since if a seller produces nounit, then the seller’s expected revenue is zero, We now consider all sellers
produce at least one unit. We have three cases as follows.
• Case 1. A buyer links only one seller.
Let bi and sj be lnked, and bi and sj 0 not be linked. Then, seller sj
consider buyer bi ’s valuation as vji > d. Thus, sj calculates the distributionof bi ’s valuation as the uniform distribution over [d, 1], which is denoted byF̃ : [d, 1] → [0, 1]. If seller sj names the price as pij , then sj ’s expected
revenue is pij (1 − F̃ (pij )). Since sj maximizes its expected revenue without
considering a sj 0 ’s price, sj prices (d + 1)/2. Thus, if d = dL or d = dH , thenpij = or pij =, respectively.
• Case 2. A buyer links two sellers.
Let bi and s1 , and bi and s2 be linked. Then, each seller considers bi ’svaluation as vji > 2d. Sellers update the distribution of bi ’s valuation as
the uniform distribution over [2d, 1], which is denoted by F̃ : [2d, 1] → [0, 1].
Each seller prices considering the price named by the other seller.
If v1i − pi1 > v2i − pi2 , then bi buys from s1 . Thus, the probability that b1
buys from s1 is{()2(1 − 2d)2 − (1 − 2d) − (pi2 − pi1 ) /2 if pi1 ≤ pi2 ,()2
1 − 2d − (pi1 − pi2 ) /2
if pi1 > pi2 .
Thus, seller s1 ’s expected revenue is{ (()2 )pi1 (1 − 2d)2 − (1 − 2d) − (pi2 − pi1 ) /2 if pi1 ≤ pi2 ,()2
if pi1 > pi2 .
pi1 1 − 2d − (pi1 − pi2 ) /2
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Similarly, seller s2 ’s expected proﬁt is{ (()2 )pi2 (1 − 2d)2 − (1 − 2d) − (pi1 − pi2 ) /2 if pi2 ≤ pi1 ,()2pi2 1 − 2d − (pi2 − pi1 ) /2
if pi2 > pi1 .
In equilibrium, they name the price (2d + 1)/2. Thus, if d = dL or d = dH ,then pij = or pij =, respectively.
• Case 3. No buyer forms any link.
Let no link be formed. Then, each seller consider buyer bi ’s valuation asvji < d. In this case, no trade occurs.
In the model, sellers consider that if a buyer forms two links, then they
have the higer valuations than those that when they form only one link.

3.2

Buyers’ linking and sellers’ production

In Stage 1, buyers choose links, and sellers determine quantities. Buyers andsellers decide their actions considering the prices will be set in Stage 2. The
results of the game in Period 1 are summarized as follows.
Proposition 1. There exist three-types symmetric equilibria due to the link
cost and the production cost.
(i) Each buyer forms at most one link, and each seller produces one unit ifa link cost is dH and a production cost is cH .
(ii) Each buyer forms at most one link, and each seller produces two units ifa link cost is dH and a production cost is cL .
(iii) Each buyer forms at most two links, and each seller produces one unit
if a link cost is dL and a production cost is cH .
The proof is in Appendix.

4Concluding remarks

We consider that price-quantity competitons in the networked market. Evenif a buyer can exchange multiple sellers, prices that the buyer faces are higher
6than those that when it can exchange only one buyer. The reason is thatto trade with multiple sellers, a buyer form multiple links spending a large
cost and, then, sellers consider that the buyer’s valuations for goods are
so high. Thus, in the model, price competitions do not work similar to
the Bertrand competition, in which prices ditermined by sellers are set incompetitive levels. Particulary, the Bertrand competition leads the price tothe suﬃciently low level in the sense that the price is equal to the marginal
cost of production.
In our model, competitive prices are as follows. For a network G andvaluations v, a price vector p is competitive if:(i) if a buyer bi and a seller sj exchange good kj , then vki j ≥ pij ≥ 0 and
pij = min{pij 0 : sj 0 ∈ L(bi )} and vki j − pij ≥ vki j0 − min{pij 00 : sj 00 ∈ L(bi )},(ii) if a buyer bi does not buy a good, then vki ≤ min{pij : j ∈ L(bi )}, k ∈ K,and(iii) if a seller sj does not sell a good then pij = 0.
Cleally, if sellers produce one unit, then prices need not satisfy the condition (ii). Furthermore, even if prices set in the game is competitive, there is
no guarantee that the prices are suﬃciently low. This result is derived fromincomplete information.

References[1] Corominas-Bosch, M. (2004), “Bargeining in a network of buyers andsellers,” Journal of Economic Theory 115, 35-77.
[2] Davidson, C. and R. Deneckere (1990), “Excess capacity and collusion,”International Economic Review 31, 521-41.
[3] Kranton, R. and D. Minehart (2000), “Competition for goods in buyerseller networks,” Review of Economic Design 5, 301-332.
[4] Kranton, R. and D. Minehart (2000), “Networks versus Vertical Integration,” RAND Journal of Economics 31, 570-601.
[5] Kranton, R. and D. Minehart (2001), “Buyer-seller networks,” American
Economic Review 91, 485-508.

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[6] Kreps, D. M. and J. A. Sheinkman (1983), “Quantity precommitment
and Bertrand competition yield Cournot outcomes,” Bell Journal ofEconomics 14, 326-337.
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