Online Appendices to “Trading Networks with Frictions” Tam´as Fleiner
Ravi Jagadeesan Zsuzsanna Jank´o Alexander Teytelboym October 2, 2017
Contents C Proof of Theorem B.1 C.1 Passing to demand language . . . . . . . . . . . . . . . . . . . . . . . C.2 Theorem B.1 in demand-language . . . . . . . . . . . . . . . . . . . . C.3 Proof of Theorem B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 2 4 6
D Proof of Theorem 1 D.1 The modified economy . . . . . . . . . D.2 Outcomes in the modified economy . . D.3 Completion of the proof of Theorem 1 D.4 Proof of Theorem D.1 . . . . . . . . .
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6 6 9 10 11
E Other proofs omitted from the text E.1 Proof of Theorem 2 . . . . . . . . . E.2 Proof of Proposition 1 . . . . . . . E.3 Proof of Theorem 3 . . . . . . . . . E.4 Proof of Theorem 4 . . . . . . . . . E.5 Proof of Corollary 2 . . . . . . . . . E.6 Proof of Theorem 5 . . . . . . . . . E.7 Proof of Corollary 3 . . . . . . . . . E.8 Proof of Corollary 4 . . . . . . . . . E.9 Proof of Lemma A.1 . . . . . . . .
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F Examples omitted from the text
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References
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1
C
Proof of Theorem B.1
Fix a firm f ∈ F . We first translate the statement of Theorem B.1 to demandlanguage, taking care to account for the possibility that a trade is not available at any finite price. We then apply a perturbation argument similar to the proof of Theorem B.1 in Hatfield et al. (2015) to prove a demand language version of Theorem B.1, which is equivalent to Theorem B.1. We note that the notation and the lemmata discussed in this section are also used in the proof of Theorem 1.
C.1
Passing to demand language
We use infinite prices to denote unavailable trades for the sake of notational convenience. Formally, define a set of prices by P = (R ∪ {−∞})Ωf → × (R ∪ {∞})Ω→f , where R ∪ {−∞} and R ∪ {∞} are topologized with the disjoint union topologies. Given p ∈ P and Ξ ⊆ Ωf , let U f (Ξ|p) = uf Ξ, pΞf → , (−p)Ξ→f , 0Ωf rΞ
denote f ’s utility of trading set Ξ of contracts at price vector p, where we write uf (Ξ, t) = −∞ if tω = −∞ for some ω ∈ Ωf . Define the extended demand correspondence Df : P ⇒ P(Ωf ) by Df (p) = arg max U f (Ξ|p) . Ξ⊆Ωf
Note that the restriction of the extended demand correspondence to RΩf is precisely the demand correspondence Df . We write full substitutability in demand language similarly to Hatfield et al. (2015). Definition C.1 (Hatfield et al., 2015). Df is (demand-language) fully substitutable if for all p ≤ p0 ∈ P with |Df (p)| = |Df (p0 )| = 1, we have Ξ0 ∩ {ω ∈ Ωf → | pω = p0ω } ⊆ Ξ Ξ ∩ {ω ∈ Ω→f | pω = p0ω } ⊆ Ξ0 , 2
where Df (p) = {Ξ} and Df (p0 ) = {Ξ0 }. We now write the constitutent conditions of strong full substitutability in demand language similarly to Hatfield et al. (2015). Definition C.2. Df is (demand-language) increasing-price fully substitutable for sales if for all p ≤ p0 ∈ P and Ξ ∈ Df (p), there exists Ξ0 ∈ Df (p0 ) with Ξ0 ∩ {ω ∈ Ωf → | pω = p0ω } ⊆ Ξ. Definition C.3. Df is (demand-language) decreasing-price fully substitutable for sales if for all p ≥ p0 ∈ P and ψ ∈ Ξ ∈ Df (p) with pψ = p0ψ , there exists Ξ0 ∈ Df (p0 ) with ψ ∈ Ξ0 . It is straightforward to verify that the original substitutability conditions are equivalent to their demand-language analogues, as the following lemma shows formally. Lemma C.1. C f is fully substitutable (resp. increasing-price fully substitutable for sales, decreasing-price fully substitutable for sales) if and only if Df is. Proof. Given a finite set of contracts Y ⊆ X, define a price vector pf (Y ) ∈ RΩf by sup (ω,q)∈Y q pf (Y )ω = inf q (ω,q)∈Y
for ω ∈ Ωf →
,
for ω ∈ Ω→f
so that pf (Y )ω is the most favorable price at which ω is available in Y . Due to the definitions of C f and Df , we have C f (Y ) = {(ω, pf (Y )ω ) | ω ∈ Ψ} | Ψ ∈ Df (pf (Y )) for all finite subsets Y ⊆ X. It follows that C f is fully substitutable (resp. increasing-price fully substitutable for sales, decreasing-price fully substitutable for sales) whenever Df is. Note also that Df (p) = τ (Y ) | Y ∈ C f ({(ω, pω ) | pω ∈ R}) for all p ∈ P. It follows that Df is fully substitutable (resp. increasing-price fully 3
substitutable for sales, decreasing-price fully substitutable for sales) whenever C f is. Note that Df is upper hemi-continuous by Berge’s Maximum Theorem. Considering perturbations shows that extended demand is generically single-valued on P. Claim C.1. The set {p ∈ P | |Df (p)| = 1} is open and dense in P. Proof. Let S = {p ∈ P | |Df (p)| = 1}. The set S is open because Df is upper hemi-continuous and P(Ωf ) is discrete. To see that S is dense, note that for all Ξ 6= Ξ0 ⊆ Ω, the set {p ∈ P | U f (Ξ|p) = U f (Ξ0 |p) 6= −∞} is nowhere dense. Indeed, if U f (Ξ|p) = U f (Ξ0 |p) 6= −∞, we have U f (Ξ|p0 ) 6= U f (Ξ0 |p0 ) for any p0 = pΩr{ω} , pω + and ω ∈ (Ξ r Ξ0 ) ∪ (Ξ0 r Ξ).
C.2
Theorem B.1 in demand-language
The following technical result exploits the upper hemi-continuity of extended demand and uses perturbations to perform certain selections from the extended demand correspondence. Claim C.2. Let p ∈ RΩ and let V ⊆ RΩf be open and dense in some neighborhood of 0. (a) For all Ψ ∈ Df (p), there exists ∈ V such that Df (p + ) = {Ψ0 } ⊆ Df (p) with Ψ0 ⊆ Ψ. (b) If there exists Ψ ⊆ Ωf with ψ ∈ Ψ, then there exists ∈ V such that Df (p+) = {Ψ0 } ⊆ Df (p) with ψ ∈ Ψ0 . Proof. By shrinking V if necessary, we can assume that Df (p + ) ⊆ Df (p) for all ∈ V (by upper hemi-continuity). We begin the proof of Part (a). First, we show that there exists ∈ V such that 0 Ψ ⊆ Ψ for all Ψ0 ∈ Df (p + ). Take = 0Ψ , δΩ→f rΨ , −δΩf → rΨ , where δ > 0 is such that ∈ V. Note that U f (Ξ|p + ) ≤ U f (Ξ|p) for all Ξ ⊆ Ω with equality if and only if Ξ ⊆ Ψ. It follows that Ψ0 ⊆ Ψ for all Ψ0 ∈ Df (p + ). 4
To complete the proof of Part (a), we perturb . More precisely, let V0 be an open neighborhood of 0 ∈ RΩf such that Df (p + + 0 ) ⊆ Df (p + ) for all 0 ∈ V0 —such a V0 exists by upper hemi-continuity. By Claim C.1, there exists 0 ∈ V0 such that + 0 ∈ V and |Df (p + + 0 )| = 1. The proof of Part (b) is similar. The assertion follows from Claim C.1 and upper hemi-continuity if |pψ | = ∞. Thus, we can assume that pψ ∈ R. First, we show that there exists ∈ V such that ψ ∈ Ψ0 for all Ψ0 ∈ Df (p + ). Without loss of generality, assume that ψ ∈ Ωf → . Take = 0Ωf r{ψ} , δψ , where δ > 0 is such that ∈ V. Note that U f (Ξ|p + ) ≥ U f (Ξ|p) for all Ξ ⊆ Ω with equality if and only if ψ ∈ / Ξ. It 0 0 f follows that ψ ∈ Ψ for all Ψ ∈ D (p + ). To complete the proof of Part (b), we perturb as in the proof of Part (a). Using suitable selections, Claim C.2 implies a demand-language version of Theorem B.1. Claim C.3. If Df is fully substitutable, then Df is increasing-price fully substitutable for sales. Proof. Let p ≤ p0 ∈ P, and let Ξ ∈ Df (p). Let V = ∈ RΩf | Df (p0 + ) ⊆ Df (p0 ) and |Df (p0 + )| = 1 , which is non-empty and dense in a neighborhood of 0 by Claim C.1 and upper hemicontinuity. By Claim C.2(a), there exists ∈ V such that Df (p + ) = {Ψ} with Ψ ⊆ Ξ. Note that Df (p0 + ) = {Ξ0 } for some Ξ0 ∈ Df (p0 ) by construction. Because Df is fully substitutable, we have Ξ0 ∩ {ω ∈ Ωf → | pω = p0ω } ⊆ Ψ ⊆ Ξ. It follows that Df is increasing-price fully substitutable for sales. Claim C.4. If Df is fully substitutable, then Df is decreasing-price fully substitutable for sales. Proof. Let p ≥ p0 ∈ P, let Ξ ∈ Df (p), and suppose that ψ ∈ Ξ satisfies pψ = p0ψ . Let V = ∈ RΩf | Df (p0 + ) ⊆ Df (p0 ) and |Df (p0 + )| = 1 , 5
which is non-empty and dense in a neighborhood of 0 by Claim C.1 and upper hemicontinuity. By Claim C.2(b), there exists ∈ V such that Df (p + ) = {Ψ} with ψ ∈ Ψ. Note that Df (p0 + ) = {Ξ0 } for some Ξ0 ∈ Df (p0 ) by construction. Since Df is fully substitutable, we must have ψ ∈ Ξ0 . Thus, Df is decreasing-price fully substitutable for sales.
C.3
Proof of Theorem B.1
Clearly SFS implies FS. It remains to prove the converse. Suppose that C f is fully substitutable. Lemma C.1 and Claim C.3 imply that C f is increasing-price fully substitutable for sales. Lemma C.1 and Claim C.4 imply that C f is decreasing-price fully substitutable for sales. Similarly, C f must be decreasing-price and increasingprice fully substitutable for purchases. Thus, C f is strongly fully substitutable.
D
Proof of Theorem 1
The strategy of the proof is to reduce Theorem 1 to a different existence result, Theorem D.1. Theorem D.1. Under FS and BWP, competitive equilibria exist. We first modify utility functions so that BWP is satisfied (Lemma D.1), ensuring that our modification preserves FS (Lemma D.2). We then show that any competitive equilibrium in the modified economy yields a competitive equilibrium in the original economy (Lemma D.4). We conclude the proof of Theorem 1 by applying Theorem D.1, which guarantees the existence of competitive equilibria in the modified economy. We then prove Theorem D.1. We note that the modification and the lemmata discussed in this section are also used in the proof of Theorem 3.
D.1
The modified economy
For f ∈ F, let Kf = −
inf
uf (Ξ,t)≥uf (∅,0)
which is finite by BCV. Let Π ≥ 1 +
P
f ∈F
6
X
tω ,
ω∈Ωf
Kf be arbitrary.
We modify the economy by allowing agents to trade ω at a cost of Π.1 Formally, for f ∈ F, define u bf : P(Ωf ) × RΩf → R by u bf (Ξ, t) = max uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞ
Ξ⊆Ψ⊆Ωf
.
The function u bf is clearly continuous and strictly increasing in the RΩf factor. Consider a modified economy in which utility functions are given by u bf for f ∈ F . The remainder of this subsection verifies that the modified economy satisfies BWP and FS. We first show that the modified economy satisfies BWP. Intuitively, note that this property is precisely what giving firms the option to trade extra contracts at price Π achieves. Lemma D.1. Under BCV, the modified economy satisfies BWP. Proof. We claim that BWP is satisfied with M = Π + 1. Let f ∈ F, let ω ∈ Ωf r Ξ, let Ξ ⊆ Ωf , and let t ∈ RΩf be such that tω = 0. Note that, for all ω ∈ Ψ ⊆ Ωf , we have u
f
Ψ, tΩf rΨ∪Ξ , −Mω , (t − Π)ΨrΞr{ω}
whenever uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞ have f
Ξ ∪ {ω}, tΩf r{ω} , −Mω
u b
< uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞ
∈ R, because M > Π = Π − tω . Hence, we
= max u Ψ, tΩf rΨ∪Ξ , −Mω , (t − Π)ΨrΞr{ω} Ξ∪{ω}⊆Ψ⊆Ωf < max uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞr{ω} Ξ∪{ω}⊆Ψ⊆Ωf ≤ max uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞr{ω} f
Ξ⊆Ψ⊆Ωf
=u bf (Ξ, t). Therefore, firm f will never choose a contract (ω, pω ) with pω > M (resp. pω < −M ) if ω ∈ Ω→f (resp. ω ∈ Ω→f ). Since f, ω, Ξ, and t were arbitrary, the claim follows. The following claim, which asserts that giving a firm the option to trade a given 1
Hatfield et al. (2015) show that such trade endowments preserve full substitutability when preferences are quasilinear (see Theorem 2 in Hatfield et al., 2015).
7
contract at a cost of Π preserves full substitutability, will be used to prove that FS holds in the modified economy. Claim D.1. Let Π be a positive real number. Given a utility function uf and ϕ ∈ Ωf , define u bfϕ : P(Ωf ) → RΩf → R by o n . u bfϕ (Ξ, t) = max uf (Ξ, t), uf Ξ ∪ {ϕ}, tΩf r{ϕ} , (t − Π)ϕ If uf is fully substitutable, then so is u bfϕ . Proof. The proof of this claim is similar to the proof of Lemma A.2 in Hatfield et al. (2013) and uses the notation of Appendix C.1. Lemma C.1 guarantees that Df b f denote the extended demand correspondence for the is fully substitutable. Let D utility function u bfϕ . Without loss of generality, assume that ϕ ∈ Ω→f . bf f 0 b We first show that D is fully substitutable. Let p ≤ p ∈ P be such that D (p) = bf 0 b f (p) = {Ξ} and let D b f (p0 ) = {Ξ0 }. Define q ∈ P by D (p ) = 1. Let D q = pΩf r{ω} , min {Π, pω }ω . and define q 0 ∈ P similarly. Note that q ≤ q 0 always holds. We divide into cases based on the order between pϕ , p0ϕ , and Π to show that Ξ0 ∩ {ω ∈ Ωf → | pω = p0ω } ⊆ Ξ Ξ ∩ {ω ∈ Ω→f | pω = p0ω } ⊆ Ξ0 .
(D.1)
b f (p) = Df (q) and Case 1: pϕ ≤ p0ϕ ≤ Π. In this case, we have p = q, p0 = q 0 , D b f (p0 ) = Df (q 0 ), and so (D.1) follows from the full substitutability of Df . D b f (p) = Df (q). Let Case 2: pϕ ≤ Π < p0ϕ . In this case, we have p = q and D ω ∈ Ωf → r Ξ satisfy pω = p0ω —note that ω 6= ϕ by construction. By IFSS, there exists Ψ0 ∈ Df (q) such that ω ∈ / Ψ0 . Since Ξ0 = Ψ0 r {ϕ}, we have ω ∈ / Ξ0 . Similarly, if ω ∈ Ξ→f satisfies pω = p0ω , IFSP implies that there exists Ψ0 ∈ Df (q) such that ω ∈ Ψ0 . Since Ξ0 = Ψ0 r {ϕ}, we have ω ∈ Ξ0 . (D.1) follows. Case 3: Π < pϕ ≤ p0ϕ . Let Ψ ∈ Df (q) be arbitrary, and note that Ξ = Ψ r {ϕ}. Let ω ∈ Ωf → r Ψ satisfy pω = p0ω . By IFSS, there exists Ψ0 ∈ Df (q) such that ω ∈ / Ψ0 . Since Ξ0 = Ψ0 r {ϕ}, we have ω ∈ / Ξ0 . Similarly, if ω ∈ Ψ→f satisfies 8
pω = p0ω , IFSP implies that there exists Ψ0 ∈ Df (q) such that ω ∈ Ψ0 . Since Ξ0 = Ψ0 r {ϕ}, we have ω ∈ Ξ0 . (D.1) follows. b f is fully substitutable. The cases exhaust all possibilities, completing the proof that D By Lemma C.1, u bfϕ must be fully substitutable as well. Claim D.1 and a straightforward inductive argument imply that FS holds in the modified economy. Lemma D.2. Under FS, the modified economy satisfies FS.
D.2
Outcomes in the modified economy
This subsection shows that competitive equilibria in the modified economy give rise to competitive equilibria in the original economy (Lemma D.4). The following lemma, which is also used in the proof of Theorem 3, shows that agent f can only produce Kf units of surplus in the modified economy and that trade endowments can only be used at social cost Π. As will be seen in the proof of Lemma D.4, it follows that trade endowments cannot be used in any competitive equilibrium. Lemma D.3. Let Ξ ⊆ Ωf and let t ∈ RΩf . Suppose u bf (Ξ, t) ≥ u bf (∅, 0). Under BCV: P (a) We have ω∈Ωf tω ≥ −Kf . (b) If we have uf (Ξ, t) < u bf (Ξ, t), then we have
P
ω∈Ωf tω
≥ Π − Kf .
Proof. Note that u bf (∅, 0) ≥ uf (∅, 0) and thus we have u bf (Ξ, t) ≥ uf (∅, 0). Let Ξ ⊆ Ψ ⊆ Ωf be such that u bf (Ξ, t) = uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞ
.
The definition of Kf implies that −Kf ≤
X
tω +
ω∈Ωf rΨ∪Ξ
X
(tω − Π) = −Π · |Ψ r Ξ| +
ω∈ΨrΞ
X
tω
ω∈Ωf
so that |Π| · |Ψ r Ξ| − Kf ≤
X ω∈Ωf
9
tω .
(D.2)
As |Ψ r Ξ| ≥ 0 always holds, Part (a) follows from (D.2). If uf (Ξ, t) < u bf (Ξ, t), then we must have Ψ 6= Ξ. As |Ψ r Ξ| ≥ 1 in this case, Part (b) follows from (D.2) as well. We now show that competitive equilibria in the modified economy give rise to competitive equilibria in the original economy. Lemma D.4. Under BCV, any competitive equilibrium in the modified economy is a competitive equilibrium in the original economy. Proof. For f ∈ F, let tf = pΞf → , (−p)Ξ→f , 0Ωf rΞf . Since [Ξ; p] is a competitive equilibrium in the modified economy, we have u bf (Ξf , tf ) ≥ u bf (∅, 0) for all f ∈ F . Note that XX f ∈F ω∈Ωf
tfω =
XX
tfω =
f ∈F ω∈Ξf
X (pω − pω ) = 0. ω∈Ξ
P In light of the fact that Π > f ∈F Kf , it follows from Lemma D.3 that uf (Ξ, t) ≥ u bf (Ξ, t) for all f ∈ F . Let f ∈ F be arbitrary. For any Ψ ⊆ Ωf , we have uf (Ξf , tf ) ≥ u bf (Ξf , t) ≥ u bf Ψ, pΨf → , (−p)Ψ→f , 0Ωf rΨ
≥ uf Ψ, pΨf → , (−p)Ψ→f , 0Ωf rΨ
,
where the second inequality is because [Ξ; p] is a competitive equilibrium in the modified economy and the third inequality follows from the definition of u bf . It follows that Ξ ∈ Df (p). Since f was arbitrary, [Ξ; p] is a competitive equilibrium in the original economy.
D.3
Completion of the proof of Theorem 1
Theorem D.1 and Lemmata D.1 and D.2 imply the modified economy has a competitive equilibrium [Ξ; p], which is a competitive equilibrium in the original economy by Lemma D.4.
10
D.4
Proof of Theorem D.1
Let M be as in BWP. Intuitively, we consider a grid of size in [−2M, 2M ]Ω , chosen so that there are no indifferences. We then use the Gale-Shapley operator of Hatfield and Kominers (2012) and Fleiner et al. (2016) to produce an -equilibrium. Sending → 0, we obtain a competitive equilibrium. Formally, a vector δ ∈ (−, )Ω is -regular if Df is single-valued on [−2M, 2M ]Ωf ∩ ZΩf + δΩf for all f ∈ F . The following claim asserts that there are many regular vectors. Claim D.2. For any > 0, the set of -regular vectors is dense in (−, )Ω . Proof. For a firm f ∈ F, let Sf = p ∈ RΩ | Df pΩf = 1 , which is open and dense in RΩf by Claim C.1. Let n = 2M + 1 and let T = Ω ([−n, n] ∩ Z) . Note that δ is -regular if δ + T ⊆ Sf . For any t ∈ T , the set of vectors δ such that δ + t ∈ Sf is open and dense in (−, )Ω since Sf is open and dense in RΩ . As T is finite, it follows that the set of -regular vectors contains an open and dense subset of (−, )Ω . An arrangement [Ξ; p] is an -equilibrium if every agent f demands Ξf when given access to sales at prices p and buys at prices p + . Definition D.1. An arrangement [Ξ; p] is an -equilibrium if p ∈ [−2M, 2M ]Ωf and Ξ ∈ Df pˆf,Ξ, for all f, where
pˆf,Ξ, = ω
p
ω
if ω ∈ Ξ or f = s(ω)
.
p + if ω ∈ / Ξ and f = b(ω) ω
The following claim shows that -equilibria exist. We will conclude by sending to 0. Claim D.3. Under FS and BWP, there exists an -equilibrium for any 0 < < M.
11
Proof. Let δ be an -regular vector, which exists by Claim D.2. Let Pω = [−2M, 2M ]∩ (Z + δω ) , and let [ b= X ({ω} × Pω ) ⊆ X. ω∈Ω
f
bf Note that C is single-valued on P X bf . {Cf (Y )} for Y ⊆ X
by -regularity, and so write C f (Y ) =
2 2 b b by Following Hatfield and Kominers (2012), define Φ : P X → P X Φ(X B , X S ) = ΦB (X B , X S ), ΦS (X B , X S ) [ B b r XS) ∪ ∪ XfS→ f → ΦB (X B , X S ) = (X Cf X→f f ∈F S
B
S
B
b rX )∪ Φ (X , X ) = (X
[
B Cf X→f ∪ XfS→
→f
.
f ∈F
As in Fleiner (2003), Hatfield and Milgrom (2005), Hatfield and Kominers (2012), and 2 b by letting (X B , X S ) v (X ¯ B, X ¯ S ) if X B ⊇ X ¯ B and Fleiner et al. (2016), order P X ¯ S . As Hatfield and Kominers (2012) and Fleiner et al. (2016) have shown, XS ⊆ X Φ is isotone (with respect to v) under FS. The Tarski (1955) fixed point theorem guarantees that Φ has a fixed point (X B , X S ). Given f ∈ F, since (X B , X S ) is a fixed-point of Φ, we have B bf → r XfS→ ) ∪ Cf X→f XfB→ = (X ∪ XfS→ B Since Cf X→f ∪ XfS→ over f, we have
f→
b= X
f→
.
(D.3)
bf → . Taking unions ⊆ XfS→ , it follows that XfB→ ∪ XfS→ = X
[ f ∈F
Xf → =
[
XfB→ ∪ XfS→ = X B ∪ X S .
f ∈F
(D.3) also implies that B XfB→ ∩ XfS→ = Cf X→f ∪ XfS→
S B B X→f ∩ X→f = Cf X→f ∪ XfS→
f→
.
Similarly, we have
12
→f
,
(D.4)
and it follows that B (X B ∩ X S )f = Cf X→f ∪ XfS→ .
(D.5)
Let ω ∈ Ω be arbitrary. Since < M , we have max Pω > M and min Pω < −M. Thus, we have (ω, max Pω ), (ω, min Pω ) ∈ / X B ∩ X S by BWP and (D.5). If (ω, max Pω ) ∈ / X B , then adding (ω, max Pω ) to X B and removing it from X S preserves (D.4) and (D.5) by BWP for f = s(ω). Thus, we can assume that (ω, max Pω ) ∈ X B r X S . Similarly, we can assume that (ω, min Pω ) ∈ X S r X B . Define pω = max p0ω | (ω, p0ω ) ∈ X S , which exists as Pω is finite and (ω, min Pω ) ∈ X S . We claim that [Ξ; p] is an -equilibrium, where Ξ = τ (X B ∩ X S ). Note that since f,Ξ, ∈ Pω forall ω ∈ Ωand f ∈ F. The (ω, max Pω ) ∈ / X S for all ω ∈ Ω,we have pˆ ω b(ω),Ξ, s(ω),Ξ, definition of pω also ensures that ω, pˆω ∈ X B and ω, pˆω = (ω, pω ) ∈ X S B ∪ XfS→ for all for all ω ∈ Ω. It follows that X B ∩ X S f ⊆ κ Ωf ; pˆf,Ξ, ⊆ X→f f ∈ F . Thus, (D.5) implies that Ξf ∈ Df pˆf,Ξ, for all f ∈ F, so that [Ξ; p] is an -equilibrium. Since [−2M, 2M ] is sequentially compact, Claim D.3 implies that there exists an arrangement [Ξ; p], a sequence n1 < n2 < · · · of positive integers, and a sequence p1 , p2 , . . . ∈ [−2M, 2M ]Ω such that [Ξ; pk ] is a n1k -equilibrium for all k and pk → p. f,Ξ, n1 f,Ξ, n1 1 f k k → pΩf for all f ∈ F because nk → 0. Because Ξf ∈ D pˆk Note that pˆk for all k and Df is upper hemi-continuous, it follows that Ξf ∈ Df pΩf for all f ∈ F . Thus, [Ξ; p] is a competitive equilibrium.
E E.1
Other proofs omitted from the text Proof of Theorem 2
Competitive equilibrium outcomes are clearly individually rational. It remains to show that no trail locally blocks a competitive equilibrium outcome. Let [Ξ; p] be a competitive equilibrium and let A = κ([Ξ; p]). Suppose for the sake of deriving a contradiction that there is a locally blocking trail (z1 , . . . , zn ). Let zi = (ωi , p0i ). Let fi = s(xi ) and let fn+1 = b(xn ). Since Af1 ∈ / C f1 (Af1 ∪ {x1 }) 13
and [Ξ; p] is a competitive equilibrium, we must have p01 > pω1 . Similarly, we must have p02 > pω2 . A simple inductive argument shows that p0n > pωn . But we must have / C fn+1 (Afn+1 ∪ {xn }). Thus, there are no locally blocking trails. p0n < pωn since Afn+1 ∈
E.2
Proof of Proposition 1
We adapt the proof of Lemma 5(ii) in Fleiner et al. (2016) to our setting. Inspired by Fleiner et al. (2016), we say that a circuit (z1 , . . . , zn ) is locally blocking if every pair of adjacent contracts is demanded by their common agent in every choice set. Definition E.1. Let Y be an outcome. A sequence of contracts (z1 , . . . , zn ) is a locally blocking circuit if: • for all 1 ≤ i ≤ n, we have {zi , zi+1 } ⊆ W for all W ∈ C fi+1 (Yfi ∪ {zi , zi+1 }) , where fi+1 = s(zi+1 ) = b(zi ). Here, we write zn+1 = z1 . To prove Proposition 1, we show (as in Fleiner et al., 2016) that every shortest locally blocking circuit or locally blocking trail gives rise to a blocking set. Claim E.1. Let Y be an individually rational outcome. Under FS, if (z1 , . . . , zn ) is the shortest among all locally blocking circuits and locally blocking trails for Y, then the set {z1 , . . . , zn } blocks Y . Proof. We prove the contrapositive of the claim. Suppose that (z1 , . . . , zn ) is a locally blocking circuit or locally blocking trail. If (z1 , . . . , zn ) is a locally blocking trail / W, then (zi+1 , . . . , zn ) is a locally and there exists W ∈ C fi+1 ({zi , zi+1 }) with zi ∈ blocking trail. Similarly, if (z1 , . . . , zn ) is a locally blocking trail and there exists W ∈ C fi+1 ({zi , zi+1 }) with zi+1 ∈ / W, then (z1 , . . . , zi ) is a locally blocking trail. Now, suppose that Z = {z1 , . . . , zn } does not block Y . Then, there is a firm f, a contract zj ∈ Zf , and a set W ∈ C f (Yf ∪ Zf ) with zj ∈ / W . Without loss of generality, we can assume that f = s(zj ), so that f = fj . We show that there is a locally blocking circuit or locally blocking trail that is shorter than (z1 , . . . , zn ). By the logic of the previous paragraph, we can assume that {zi , zi+1 } ⊆ W for all W ∈ C fi+1 ({zi , zi+1 }) if (z1 , . . . , zn ) is a locally blocking trail, as otherwise there is a shorter locally blocking trail. By Theorem B.1, SFS must be satisfied. We divide into cases based on whether j = 1 and whether we have a trail or a circuit to complete the proof of the claim. 14
Case 1: j = 1 and (z1 , . . . , zn ) is a locally blocking trail. By IFSS, there exists W 0 ∈ C f (Yf ∪ Zf → ) with z1 ∈ / W 0 . Among all such W 0 , take W to minimize |W 0 r Yf |. Since Yf ∈ / C f (Yf ∪ {z1 }), we have Yf ∈ / C f (Yf ∪ Zf → ), and thus W 0 6⊆ Yf . Let zk ∈ W r Yf be arbitrary. By IFSS, we must have Yf ∈ / C f (Yf ∪ {zk }). Thus, (zk , . . . , zn ) is a shorter locally blocking trail. Case 2: j 6= 1 or (z1 , . . . , zn ) is a locally blocking circuit. In either case, zj−1 is well-defined. By IFSS, there exists W 0 ∈ C f (Yf ∪ {zj−1 } ∪ Zf → ) with zj ∈ / W 0. Among all such W 0 , take W to minimize |W 0 r Yf |. Since {zj−1 , zj } ⊆ B for all B ∈ C f (Yf ∪ {zj−1 , zj }), we have zj−1 ∈ W by DFSP. Let zk ∈ W r Yf be arbitrary. By IFSS, we must have zk ∈ B for all B ∈ C (Yf ∪ {zj−1 , zk }). If k < j, then (zk , . . . , zj−1 ) is a shorter locally blocking circuit. If k > j, then (z1 , . . . , zj−1 , zk , . . . , zn ) is a shorter locally blocking circuit or locally blocking trail. f
The cases exhaust all possibilities, completing the proof of the claim. Claim E.1 clearly implies Proposition 1.
E.3
Proof of Theorem 3
Let A be any stable outcome, and let Ξ = τ (A). For ω ∈ τ (A), let pω be the unique price such that (ω, pω ) ∈ A. For f ∈ F, let X Kf = − inf tω , uf (Ξ,t)≥uf (∅,0)
ω∈Ωf
which is finite by BCV. Let Π=1+
X
Kf + 2
f ∈F
X
|pω |.
ω∈Ξ
Recall the definition of u bf : P(Ωf ) × RΩf → R from the proof of Theorem 1, which is u bf (Ξ, t) = max uf Ψ, tΩf rΨ∪Ξ , (t − Π)ΨrΞ Ξ⊆Ψ⊆Ωf
.
Consider a modified economy in which utility functions are given by u bf for f ∈ F . 15
Claim E.2. A is a stable outcome in the modified economy. Proof. The outcome A is clearly individually rational in the modified economy. It remains to prove that A is not blocked in the modified economy. Suppose for the sake of deriving a contradiction that there is a blocking set Z in the modified economy. bf denote the choice function of f in the modified economy. For f ∈ F and Let C bf (Af ∪ Zf ), note that U f Y f ≥ U f (∅) and thus Yf ∈C X
−Kf ≤
X
p0ω −
(ω,p0ω )∈Yff→
X
p0ω ≤
p0ω +
(ω,p0ω )∈Z→f
(ω,p0ω )∈Zf →
f (ω,p0ω )∈Y→f
X
p0ω −
X
|pω |,
ω∈Ξf
where the first inequality is due to Lemma D.3(a), so that X (ω,p0ω )∈Zf →
But note that X X f ∈F
X
p0ω −
p0ω +
(ω,p0ω )∈Z→f
X
|pω | + Kf ≥ 0.
ω∈Ξf
X
p0ω −
p0ω +
(ω,p0ω )∈Z→f
(ω,p0ω )∈Zf →
X
|pω | + Kf = 2
ω∈Ξf
X ω∈Ξ
|pω | +
X
Kf = Π − 1.
f ∈F
It follows that X
X
p0ω −
(ω,p0ω )∈Zf →
(ω,p0ω )∈Z→f
X
p0ω +
|pω | + Kf ≤ Π − 1 < Π
ω∈Ξf
for all f ∈ F, so that X (ω,p0ω )∈Yff→
p0ω −
X
p0ω ≤ −Kf + Π − 1 < −Kf + Π.
f (ω,p0ω )∈Y→f
b f Y f = U f Y f for all f ∈ F . Lemma D.3(b) implies that U Let W ∈ C f (Af ∪Zf ) be arbitrary. In light of the previous paragraph and the fact b f (W ), we must have W ∈ C bf (Af ∪ Zf ). Since Z blocks A in the that U f (W ) ≤ U modified economy, we must have Zf ⊆ W . Hence, Z blocks A in the original economy, which contradicts the hypothesis that A is stable in the original economy. Claim E.2 guarantees that A is a stable outcome in the modified economy. By 16
Proposition 1, A is trail-stable in the modified economy. Lemmata D.1 and D.2 ensure that FS and BCV are satisfied in the modified economy. Thus, there exists a competitive equilibrium [Ξ; p] in the modified economy with κ([Ξ; p]) = A by Theorem 4 (which is proved independently). Lemma D.4 guarantees that [Ξ; p] is a competitive equilibrium in the modified economy.
E.4
Proof of Theorem 4
We set prices that are as high as possible while remaining (weakly) undesirable to sellers. Call a trail (z1 , . . . , zn ) locally semi-blocking if the seller of zi wants to propose zi when given access to zi−1 for all i > 1, and the seller of z1 wants to propose z1 . We consider a contract desirable to a seller if it appears in a locally semi-blocking trail. Definition E.2. A trail (z1 , . . . , zn ) locally semi-blocks an outcome A if: • Af1 ∈ / C f1 (Af1 ∪ {z1 }), where f1 = s(z1 ); • {zi , zi+1 } ⊆ Y for all Y ∈ C fi+1 (Afi+1 ∪ {zi , zi+1 }) for 1 ≤ i ≤ n − 1, where fi+1 = b(zi ) = s(zi+1 ). Formally, let A be an outcome and let Ξ = τ (A). Let M be as in BWP. Let X B be the set of contracts that appear in some locally semi-blocking trail. Thus, X B consists of all contracts that are strictly desirable to their sellers.2 For ω ∈ Ω, define pω = min M,
inf
(ω,p0ω )∈X B
p0ω
,
(E.1)
so that pω is the minimum of M and the highest price at which ω is weakly undesirable to its seller. We will prove that κ([Ξ; p]) = A and that [Ξ; p] is a competitive equilibrium. Claim E.3. Under BWP, if A is individually rational, then we have κ([Ξ; p]) = A. Proof. Suppose that (ω, p0ω ) ∈ A. BWP implies that p0ω < M . As us(ω) is strictly increasing in transfers and A is individually rational, we have (ω, p00ω ) ∈ X B if and only if p00ω > p0ω . It follows that pω = p0ω . Since ω ∈ Ξ was arbitrary, the claim follows. 2
X
B
In the fixed-point interpretation of trail-stable outcomes (Fleiner et al., 2016; Adachi, 2017), is the set of contracts that are available to their buyers.
17
Claim E.4. Under FS and BWP, if A is trail-stable, then [Ξ; p] is a competitive equilibrium. Proof. Suppose for the sake of deriving a contradiction that Ξf ∈ / Df pΩf . As A is individually rational, it follows from Claim E.3 that Ξ0 ∈ / Df pΩf for all Ξ0 ⊆ Ξf . We perturb prices slightly to ensure that sellers have strict incentives to propose contracts. Due to the upper hemi-continuity of demand, we can ensure that a sufficiently small perturbation does not affect the property that f demands no subset of Ξf . Formally, define O = {p0 ∈ RΩf | Df (p0 ) ∩ P(Ξf ) = ∅}. Since Df is upper hemi-continuous, the set O contains an open neighborhood of pΩf . By (E.1), there exists q ∈ O such that qΞf ∪Ωf → = pΞf ∪Ωf → and (ω, pω ) ∈ X B whenever ω ∈ Ω→f r Ξ and pω < M. We consider the prices q instead of the prices p. By Theorem B.1, SFS must be satisfied. To produce a contradiction, we consider the set of trades that f could demand at price vector q that contains fewest trades outside Ξf . Formally, let Ψ ∈ Df (q) minimize |Ψ0 r Ξ| over all Ψ0 ∈ Df pΩf . Consider the corresponding set of contracts W = κ([Ψ; q]). Note that W 6⊆ A and W→f r A ⊆ X B by construction and BWP. We divide into cases based on whether W r A contains any contracts that are sold by f to produce a contradiction. Case 1: W r A 6⊆ X→f . In this case, we either produce a locally blocking trail or show that any sale in W r A must be appear in some locally semi-blocking trail. Formally, let z ∈ Wf → r Af → be arbitrary and let ω = τ (z). By IFSS, we have z ∈ W 0 for all W 0 ∈ C f (A ∪ {z} ∪ W→f ). Let W 0 ∈ C f (A ∪ {z} ∪ W→f ) minimize |W 0 r A| over all W 0 ∈ C f (A ∪ {z} ∪ W→f ). As qω = pω , the trail ((ω, p0ω )) cannot be locally semi-blocking for any p0ω < qω by (E.1). Hence, we have that Af ∈ C f (Af ∪ {z}) by the upper hemi-continuity 0 of demand. It follows that W→f r A→f 6= ∅. Since W→f r A ⊆ X B , there must 0 exist a locally semi-blocking trail (z1 , . . . , zn ) with zn ∈ W→f . By DFSP, we have zn ∈ W 0 for all W 0 ∈ C f (Af ∪ {zn , z}). We divide into cases based on whether there exists W 00 ∈ C f (Af ∪ {zn , z}) with z ∈ / W 00 to derive contradictions. Subcase 1.1: There exists W 00 ∈ C f (Af ∪{zn , z}) with z ∈ / W 00 . Then, (z1 , . . . , zn )
18
is a locally blocking trail, contradicting the assumption that A is trailstable. Subcase 1.2: z ∈ W 00 for all W 00 ∈ C f (Af ∪ {zn , z}). Then, (z1 , . . . , zn , z) is a locally semi-blocking trail. Since uf is continuous, there exists p0ω < pω such that (z1 , . . . , zn , (ω, p0ω )) is a locally semi-blocking trail, contradicting (E.1). Case 2: W r A ⊆ X→f . Let z ∈ W r A be arbitrary, and let (z1 , . . . , zn ) be a locally semi-blocking trail with zn = z. By DFSP, we have Af ∈ / C f (Af ∪ {z}). Thus, (z1 , . . . , zn ) is a locally blocking trail, contradicting the assumption that A is trail-stable. The cases exhaust all possibilities. We have produced contradictions in all cases, completing the proof of the claim. Claims E.3 and E.4 together imply the theorem.
E.5
Proof of Corollary 2
Competitive equilibria exist by Theorem D.1. Competitive equilibrium outcomes are trail-stable by Theorem 2. Trail-stable outcomes lift to competitive equilibria by Theorem 4.
E.6
Proof of Theorem 5
We prove the contrapositive. Let [Ξ; p] be an arrangement and suppose that A = κ([Ξ; p]) is not strongly group stable. If A is not individually rational, then clearly [Ξ; p] is not a competitive equilibrium. Thus, we can assume that A is not strongly unblocked—that is, that there exists a non-empty set of contracts Z ⊆ X r A and, for each f ∈ F with Zf 6= ∅, a set of contracts Y f ⊆ Zf ∪ Af with Y f ⊇ Zf and U f Y f > U f (Af ) (see Definition 6). Let F 0 = {f ∈ F | Zf 6= ∅}. For each f ∈ F 0 , let X X Mf = sup q uf τ (Y f ), pω − pω − q ≥ U f (Af ) ω∈τ (Y f )f → ω∈τ (Y f )→f
19
denote the compensating variation for f from the change from τ (Af ) to τ (Y f ) at price vector p. For ω ∈ τ (Z), let p˜ω be the unique price such that (ω, p˜ω ) ∈ Z. Define p˜ω = pω for ω ∈ Ω r τ (Z). The definition of Y f ensures that
X
uf τ (Y f ),
X
p˜ω −
ω∈τ (Y f )f →
p˜ω > U f (Af )
ω∈τ (Y f )→f
for all f ∈ F 0 . It follows that Mf >
X ω∈τ (Y
=
X
pω −
f) f→
X
ω∈τ (Y
X
pω −
f) →f
ω∈τ (Y
X
(pω − p˜ω ) +
ω∈τ (Y f )f →
p˜ω +
f) f→
X ω∈τ (Y
p˜ω
f) →f
(˜ pω − pω ).
ω∈τ (Y f )→f
Because pω = p˜ω for ω ∈ / Z and Zf ⊆ Y f , we have Mf >
X
X
(pω − p˜ω ) +
ω∈τ (Zf )f →
Summing over f ∈ F 0 , we have Mf > 0. For such f, we have
(˜ pω − pω ).
ω∈τ (Zf )→f
P
f ∈F 0
Mf > 0. Thus, there exists f ∈ F 0 with
uf τ (Y f ), pτ (Y f )f → , (−p)τ (Y f )→f , 0Ωf rτ (Y f )
> U f (Af ) = uf Ξf , pΞf → , (−p)Ξ→f , 0Ωf rΞ
,
so that Ξf ∈ / Df pΩf . Therefore, [Ξ; p] is not a competitive equilibrium.
E.7
Proof of Corollary 3
Competitive equilibria exist by Theorem D.1. Competitive equilibrium outcomes are strongly group stable by Theorem 5. Strongly group stable outcomes are always stable. Stable outcomes are trail-stable by Proposition 1. Trail-stable outcomes lift to competitive equilibria by Theorem 4.
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E.8
Proof of Corollary 4
Competitive equilibria exist by Theorem 1. Competitive equilibrium outcomes are strongly group stable by Theorem 5. Strongly group stable outcomes are always stable. Stable outcomes lift to competitive equilibria by Theorem 3.
E.9
Proof of Lemma A.1
The proof is similar to the proof of Theorem 7 in Hatfield and Kominers (2012). By Theorem B.1 in Appendix B, we can assume that SFS is satisfied. We prove the contrapositive. Let A be outcome that is not stable. If A is not individually rational, then clearly A is not trail-stable. Thus, we can assume that A is blocked by a non-empty blocking set Z. Since Z is non-empty and the network is assumed to be acyclic, there is a firm f1 with Z→f1 = ∅ and Zf1 → 6= ∅. Let z1 ∈ Zf1 → be arbitrary. By IFSS, we have Af1 ∈ / C f1 (Af1 ∪ {z1 }). Let f2 = b(z1 ). If Af2 ∈ / C f2 (Af2 ∪ {z1 }), then (z1 ) is a locally blocking trail. Thus, we can assume that Af2 ∈ C f2 (Af2 ∪ {z1 }) . DFSP implies that z1 ∈ W 0 for all W 0 ∈ C f2 (Af2 ∪ {z1 } ∪ Zf2 → ). Let W ∈ C f2 (Af2 ∪ {z1 } ∪ Zf2 → ) minimize |W 0 r A| among all W 0 ∈ C f2 (Af2 ∪ {z1 } ∪ Zf2 → ). By IFSS, we must have W = {z1 , z2 } for some z2 ∈ Zf2 → . Note that Af2 ∈ / C f2 (Af2 ∪ {z1 , z2 }) by construction. A similar argument to the previous paragraph shows that (z1 , z2 ) is a locally blocking trail or there exists z3 ∈ Z with s(z3 ) = b(z2 ) such that Af2 ∈ / C f2 (Af2 ∪ {z2 , z3 }). By induction and due to acyclicity, we obtain a locally blocking trail. Thus, A is not trail-stable.
F
Examples omitted from the text
The following two examples remove the frictions from Examples 1 and 2, respectively, showing that competitive equilibrium cannot be Pareto-comparable and that adding an outside option that is not used cannot shut down trade. Thus, distortionary frictions are crucial to the conclusions of Examples 1 and 2. Example 3 continued (Cyclic economy with transferable utility). In Example 3, the competitive equilibria are [{ζ, ψ}; p] , where |pζ − pψ | ≤ 10. All competitive equilibria
21
are Pareto-efficient, as guaranteed by the First Welfare Theorem (see, e.g., Hatfield et al., 2013), and trade occurs in every competitive equilibrium. Example F.1 (Cyclic economy with transferable utility and an outside trade—Hatfield and Kominers, 2012). As depicted in Figure 2(b), consider the economy of Example 3 with an additional firm f3 , which interacts with f1 via trade ω 0 . Firm fi has utility function X ufi (Ξ, t) = v fi (Ξ) + tω , ω∈Ωf
where valuations v f1 , v f2 , and v f3 are as in Example 2. Trade ζ 0 cannot be realized in equilibrium due to the technological constraints of f1 and f2 . Hence, we must have pζ 0 ≥ 300 in any competitive equilibrium, since f3 must weakly prefer ∅ over {ζ 0 } in equilibrium. In order for trade to occur, f1 must prefer ζ over ζ 0 , and so we must have pζ ≥ pζ 0 . Hence, the competitive equilibria are [{ζ, ψ}; p] , where |pζ − pψ | ≤ 10 and pζ ≥ pζ 0 ≥ 300. Essentially, adding an outside option simply forces pζ to be at least $300 without shutting down trade between f1 and f2 . The next example shows that a regularity condition, such as BCV, is needed in addition to FS to ensure that competitive equilibria exist. Example F.2 (Competitive equilibria need not exist under FS alone). Consider two firms, b and s, and one trade ω between them with s(ω) = s and b(ω) = b. Suppose that s is not willing to sell ω at any (finite) price, but b would buy ω at any (finite) price. Note that the market does not clear at any price—b always demands ω and s never demands ω. The issue is that the variation needed to exactly compensate b for going from autarky to trade is −∞. If b’s compensating variation were −p, then autarky could be sustained in equilibrium at any price above p. The last example shows that FS needed for stable outcomes to be trail-stable. Example F.3 (Stable outcomes may not be trail-stable without FS). As depicted in Figure 2(a), there are two firms, f1 and f2 , which interact via two trades, ζ and ψ. Firm fi has utility function ufi (Ξ, t) = v fi (Ξ) +
X ω∈Ωf
22
tω ,
where v f1 (∅) = v f2 (∅) = 0 v f1 ({ζ}) = v f1 ({ψ}) = 1 v f1 ({ζ, ψ}) = −∞ v f2 ({ζ}) = v f2 ({ψ}) = −∞ v f2 ({ζ, ψ}) = 1. Note that trades ζ and ψ are not complementary for firm f1 , which implies that f1 ’s preferences are not fully substitutable. The no-trade outcome ∅ is stable, as there is no non-empty set of contracts that is individually rational for both f1 and f2 . However, the trail (ζ, ψ) locally blocks the outcome ∅. Thus, the no-trade outcome is stable but not trail-stable.
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References Adachi, H. (2017). Stable matchings and fixed points in trading networks: A note. Economics Letters 156, 65–67. Fleiner, T. (2003). A fixed-point approach to stable matchings and some applications. Mathematics of Operations Research 28 (1), 103–126. Fleiner, T., Z. Jank´o, A. Tamura, and A. Teytelboym (2016). Trading networks with bilateral contracts. Working paper. Hatfield, J. W. and S. D. Kominers (2012). Matching in networks with bilateral contracts. American Economic Journal: Microeconomics 4 (1), 176–208. Hatfield, J. W., S. D. Kominers, A. Nichifor, M. Ostrovsky, and A. Westkamp (2013). Stability and competitive equilibrium in trading networks. Journal of Political Economy 121 (5), 966–1005. Hatfield, J. W., S. D. Kominers, A. Nichifor, M. Ostrovsky, and A. Westkamp (2015). Full substitutability. Working paper. Hatfield, J. W. and P. Milgrom (2005). Matching with contracts. American Economic Review 95 (4), 913–935. Tarski, A. (1955). A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics 5 (2), 285–309.
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