Bargaining Frictions in Trading Networks

Arnold Polanski; Fernando Vega-Redondo

doi:10.1515/bejte-2016-0089

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Bargaining Frictions in Trading Networks

Arnold Polanski und Fernando Vega-Redondo

Veröffentlicht/Copyright: 25. Juli 2017

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Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 18 Heft 1

Abstract

In the canonical model of frictionless markets, arbitrage is usually taken to force all trades of homogeneous goods to occur at essentially the same price. In the real world, however, arbitrage possibilities are often severely restricted and this may lead to substantial price heterogeneity. Here we focus on frictions that can be modeled as the bargaining constraints induced by an incomplete trading network. In this context, the interplay among the architecture of the trading network, the buyers’ valuations, and the sellers’ costs shapes the effective arbitrage possibilities of the economy. We characterize the configurations that, at an intertemporal bargaining equilibrium, lead to a uniform price. Conceptually, this characterization involves studying how the network positions and valuations/costs of any given set of buyers and sellers affect their collective bargaining power relative to a notional or benchmark situation in which the connectivity is complete. Mathematically, the characterizing conditions can be understood as price-based counterparts of those identified by the celebrated Marriage Theorem in matching theory.

Keywords: bargaining; frictions; networks; markets; arbitrage

JEL Classification: D41; D61; D85; C78

Appendix

In this Appendix, we provide the formal proof of our main result, Proposition 1. The proof relies on two separate Lemmas, which are stated and proven first.

Lemma 1

Let the bipartite network G={B∪S,L} and type profile θ=((cs)s∈S,(vb)b∈B) define an UPC (G,θ) with a trading price p. Consider the network G′={B∪S,L∪{s′b′}}, with s′∈S, b′∈B, and s′b′∉L. Then, (G′,θ) defines an UPC with the same trading price p.

Proof:

Let x∗(G,θ) be the LBO induced by the UPC (G,θ) where all trades occur at the price p. By Theorem 2 in Nguyen (2015), this outcome is the unique solution to the following (quadratic) optimization problem:

(11)minx(∑b∈Bxb2+∑s∈Sxs2),s.t.∀sb∈L,xs+xb≥max{vb−cs,0}.

Consider any seller s and buyer b such that cs≤p≤vb. Their equilibrium payoffs must then be, respectively, given by xs∗=p−cs and xb∗=vb−p. To see this, consider the case of the seller s. Either some of her neighboring buyers in G trade at price p with other buyers or they do not trade at all. In the former case, seller s can also trade at price p. In the latter case, instead, by Assumption PL, there is some price at which s can trade profitably with some of his neighboring buyers. Thus, seller s must be active at equilibrium and his trading price must also be p, since the configuration (G,θ) is an UPC.

Note that the preceding argument applies to any seller-buyer pair (s,b), independently of whether they are connected in G or not. Thus suppose that the link s′b′ added to G to obtain G′ indeed satisfies cs′≤p≤vb′. Then, the solution to the optimization problem eq. (11) must satisfy:

(12)xs′∗+xb′∗=p−cs′+vb′−p=vb′−cs′,

which implies that adding the constraint xs′∗+xb′∗≥vb′−cs′ is redundant, and therefore x∗(G,θ) is still a solution to the optimization problem obtained after adding this constraint. This means that x∗(G′,θ)=x∗(G,θ).

Consider now the alternative case in which the link s′b′ added to G does not satisfy cs′≤p≤vb′. For concreteness, suppose that cs′≤vb′≤p. Then seller s′ trades in G but the buyer b′ does not. Hence the corresponding payoffs satisfy xs′∗=p−cs′, xb′∗=0 and, therefore,

xs′∗+xb′∗=p−cs′>vb′−cs′.

Hence, again, if the link s′b′ is added to G and the constraint xs′∗+xb′∗≥vb′−cs′ is added to eq. (11) the solution remains unchanged. Thus, as before, we find that x∗(G′,θ)=x∗(G,θ), which completes the proof of the Lemma. □

Lemma 2

For any disjoint non-empty subsets S′,S′′⊆S and B′,B′′⊆B in the bipartite network G={B∪S,L} with the type profile θ=((cs)s∈S,(vb)b∈B) it holds that,

P(S′,B′)<P(S′′,B′′)⇒P(S′,B′)<P(S′∪S′′,B′∪B′′)<P(S′′,B′′),

where P(.) is defined in eq. (6).

Proof:

By the definition eq. (6) and by the fact that S′∩S′′=∅ and B′∩B′′=∅, it follows for p=P(S′∪S′′,B′∪B′′) that,

∑b∈B(p)∩B′(vb−p)+∑b∈B(p)∩B′′(vb−p)=∑s∈S(p)∩S′(p−cs)+∑s∈S(p)∩S′′(p−cs).

We write the last equality as f(p;B′,S′)+f(p;B′′,S′′)=0, where,

f(p;B~,S~)≡∑b∈B(p)∩B~(vb−p)−∑s∈S(p)∩S~(p−cs),∀B~⊆B,S~⊆S.

Clearly, the function f(p;.) is strictly decreasing in p∈R. Then, we have for p′=P(S′,B′)<p′′=P(S′′,B′′) the following inequalities,

f(p′;B′,S′)=0⇒f(p′;B′,S′)+f(p′;B′′,S′′)=f(p′;B′′,S′′)>0,f(p′′;B′′,S′′)=0⇒f(p′′;B′,S′)+f(p′′;B′′,S′′)=f(p′′;B′,S′)<0.

As the function f(p;.) is also continuous, it follows that there is a unique p∈R solving f(p;B′,S′)+f(p;B′′,S′′)=0 and p∈(p′,p′′). □

Proof of Proposition 1

We establish the desired equivalence by proving in turn a sufficient set of different implications.

(8)⇒(7):

For the sake of contradiction, assume that the condition in eq. (8) holds but (G,θ) is not an UPC. Then, there are at least two connected and disjoint subnetworks G′ and G′′ of G, where trade takes place at the uniform prices p′ and p′′, respectively, such that p′≠p′′. We will call each such subnetwork a trading component (TC). In each TC of G, we add all missing links until all buyers and sellers in this component are connected by a complete subnetwork. By Lemma 1, this will not affect the price in this component. Each node that does not belong to any TC (i.e., does not trade in equilibrium) is connected to at least one trading node due to our Assumption PL. For any such non-trading player v, we select one of her trading neighbors in some TC and connect v to all players from the opposite side in this TC. Again, this operation will not change the price in G′ as v will be still inactive. Thus, we obtain a collection of completely connected TCs that cover disjoint sets of nodes, whose union is S∪B, and each TC displays a uniform price.

If we now add all missing links between two completely connected TCs with the respective prices p′ and p′′, then Lemma 2 implies that the price in the merged component lies in the interval (p′,p′′). If we proceed in this way iteratively merging components, we will arrive at the completely connected bipartite network with the set of nodes S∪B, where all trade takes place at the price p∗=P(B,S). By the iterative application of Lemma 2, this price must lie strictly between the minimum pL and the maximum pH price of the initial completely connected TCs,

pL<p∗=P(B,S)<pH.

Now, denote by H a trading component with the price pH. Furthermore, let HB and HS be, respectively, the (non-empty) set of active buyers and sellers in H. Note that any active seller s∈NG(HB) (who must of course have her cost cs≤pH) will sell at equilibrium only at the highest price pH, since this price is available to her. Hence, we can write HS=NG(HB) and, therefore,

P(HB,HS)=P(HB,NG(HB))=pH>p∗,

which contradicts eq. (8).

(7)⇒(8):

For the sake of contradiction assume that (G,θ) is UPC but eq. (8) does not hold. Then, there exists a non-empty set B′⊆B such that,

(13)P(B′,NG(B′))=p1>p∗≡P(B,S).

We add all missing links between B′ and NG(B′) until these two sets are connected by a complete subnetwork G1 and we do the same for the sets S′=S∖NG(B′) and NG(S′) obtaining the complete subnetwork G2 (G1 and G2 are only connected by links between NG(S′) and NG(B′)). We denote by G~ the entire network that resulted from this link addition to G and note that (G~,θ) is an UPC by the Lemma 1.

Considering now G1 and G2 separately (i.e., ignoring all links between them), they cover disjoint sets of nodes, whose union is S∪B, and each subnetwork displays a uniform price due to their completeness. Then, Lemma 2 and p1>p∗ imply that p2 must verify,

p2=P(NG(S′),S′)<p∗.

Considering now the entire network G~, trading at pk in its subnetwork Gk, k=1,2, and disagreement (no trade) for any connected pair (s,b)∈G1×G2 forms a limit SSPE with the (unique) expected payoff vector x∗. In particular, it is optimal not to trade for each pair (s,b)∈NG(B′)×NG(S′), i.e., s∈G1, b∈G2, as the sum of their expected payoffs is higher than their joint surplus,

xs∗+xb∗=max{p1−cs,0}+max{vb−p2,0}≥max{p1−cs+vb−p2,0}>vb−cs.

As the equilibrium trade in G~ occurs at two different prices, p1>p2, the configuration (G~,θ) cannot be an UPC.

Given the formal symmetry between buyers and sellers in the model, it is clear that it readily follows that both (7)⇒(9) and (9)⇒(7). This establishes the equivalence among eqs (7), (8), and (9), thus completing the proof. □

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Published Online: 2017-7-25

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https://doi.org/10.1515/bejte-2016-0089

Schlagwörter für diesen Artikel

bargaining; frictions; networks; markets; arbitrage