C PROOF OF THEOREM??
Fix a firm𝑓 ∈𝐹. We first translate the statement of Theorem B.1 to demand-language, 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. [2019] to prove a demand language version of Theorem B.1, which is equivalent to Theorem B.1. We note that the notation and the lemma (Lemma C.1) 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 ,
whereR∪ {−∞}andR∪ {∞}are topologized with the disjoint union topologies. Given𝑝∈PandΞ⊆Ω𝑓,let 𝑈𝑓 (Ξ|𝑝)=𝑢𝑓
Ξ,
𝑝Ξ
𝑓→,(−𝑝)Ξ→𝑓,0Ω
𝑓rΞ
denote𝑓’s utility of trading setΞof contracts at price vector𝑝,where we write𝑢𝑓(Ξ, 𝑡)=−∞if𝑡𝜔 =−∞for some 𝜔∈Ω𝑓.Define theextended demand correspondenceD𝑓 :P⇒P (Ω𝑓)by
D𝑓(𝑝)=arg max
Ξ⊆Ω𝑓
𝑈𝑓 (Ξ|𝑝).
Note that the restriction of the extended demand correspondence toRΩ𝑓 is precisely the demand correspondence𝐷𝑓. We write full substitutability in demand language similarly to Hatfield et al. [2019].
Definition C.1(Hatfield et al., 2019). D𝑓 is(demand-language) fully substitutableif for all𝑝≤𝑝0∈Pwith|D𝑓(𝑝) |=
|D𝑓(𝑝0) |=1,we have
Ξ0∩ {𝜔 ∈Ω𝑓→|𝑝𝜔=𝑝𝜔0} ⊆Ξ Ξ∩ {𝜔 ∈Ω→𝑓 |𝑝𝜔=𝑝0
𝜔} ⊆Ξ0, whereD𝑓(𝑝)={Ξ}andD𝑓(𝑝0)={Ξ0}.
We now write the constitutent conditions of strong full substitutability in demand language similarly to Hatfield et al. [2019].
Definition C.2. D𝑓 is(demand-language) increasing-price fully substitutable for sales if for all𝑝 ≤ 𝑝0 ∈ Pand Ξ∈D𝑓(𝑝),there existsΞ0∈D𝑓(𝑝0)with
Ξ0∩ {𝜔 ∈Ω𝑓→ |𝑝𝜔=𝑝𝜔0} ⊆Ξ.
Definition C.3. D𝑓 is(demand-language) decreasing-price fully substitutable for salesif for all𝑝 ≥ 𝑝0 ∈ Pand 𝜓∈Ξ∈D𝑓(𝑝)with𝜓∈Ω𝑓→and𝑝𝜓 =𝑝0
𝜓
,there existsΞ0∈D𝑓(𝑝0)with𝜓 ∈Ξ0.
The original substitutability conditions are equivalent to their demand-language analogues, as the following lemma shows formally.
Lemma C.1. 𝐶𝑓 is fully substitutable (resp. increasing-price fully substitutable for sales, decreasing-price fully substitutable for sales) if and only ifD𝑓 is.
Proof. Given a finite set of contracts𝑌 ⊆𝑋 ,define a price vector𝑝𝑓(𝑌) ∈RΩ𝑓 by 𝑝𝑓(𝑌)𝜔=
sup(𝜔 ,𝑞) ∈𝑌𝑞 for𝜔 ∈Ω𝑓→
inf(𝜔 ,𝑞) ∈𝑌𝑞 for𝜔 ∈Ω→𝑓
,
so that𝑝
𝑓(𝑌)𝜔is the most favorable price at which𝜔is available in𝑌. Due to the definitions of𝐶𝑓 andD𝑓,we have 𝐶𝑓(𝑌)=
n n
𝜔 , 𝑝𝑓(𝑌)𝜔
|𝜔 ∈Ψo
|Ψ∈D𝑓 𝑝𝑓(𝑌) o
for all finite sets𝑌 ⊆𝑋. It follows that𝐶𝑓 is fully substitutable (resp. increasing-price fully substitutable for sales, decreasing-price fully substitutable for sales) wheneverD𝑓 is. Note also that
D𝑓(𝑝)= n
𝜏(𝑌) |𝑌 ∈𝐶𝑓 ({(𝜔 , 𝑝𝜔) |𝑝𝜔∈R})o
for all𝑝∈P.It follows thatD𝑓 is fully substitutable (resp. increasing-price fully substitutable for sales,
decreasing-price fully substitutable for sales) whenever𝐶𝑓 is.
Note thatD𝑓 is upper hemi-continuous by Berge’s Maximum Theorem. Considering perturbations shows that extended demand is generically single-valued onP.
Claim C.1. The set{𝑝∈P| |D𝑓(𝑝) |=1}is open and dense inP.
Proof. Let𝔖 ={𝑝 ∈P| |D𝑓(𝑝) | =1}. The set𝔖is open becauseD𝑓 is upper hemi-continuous andP (Ω𝑓)is discrete. To see that𝔖is dense, note that for allΞ≠Ξ0⊆Ω,the set
{𝑝∈P|𝑈𝑓(Ξ|𝑝)=𝑈𝑓 Ξ0|𝑝
≠−∞}
is nowhere dense. Indeed, if𝑈𝑓 (Ξ|𝑝)=𝑈𝑓 (Ξ0|𝑝)≠−∞,we have𝑈𝑓(Ξ|𝑝0)≠𝑈𝑓 (Ξ0|𝑝0)for any𝑝0=
𝑝Ωr{𝜔}, 𝑝𝜔+𝜖
and𝜔 ∈ (ΞrΞ0) ∪ (Ξ0rΞ).
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. Let𝑝∈RΩand let𝔙⊆RΩ𝑓 be open and dense in some neighborhood of 0.
(a) For allΨ∈D𝑓(𝑝),there exists𝜖∈𝔙such thatD𝑓(𝑝+𝜖)={Ψ0} ⊆D𝑓(𝑝)withΨ0⊆Ψ. (b) If𝜓 ∈Ψ∈D𝑓(𝑝),then there exists𝜖∈𝔙such thatD𝑓(𝑝+𝜖)={Ψ0} ⊆D𝑓(𝑝)with𝜓∈Ψ0.
Proof. By shrinking𝔙if necessary, we can assume thatD𝑓(𝑝+𝜖) ⊆D𝑓(𝑝)for all𝜖∈𝔙(by upper hemi-continuity).
We begin by proving Part (a). First, we show that there exists𝜖∈𝔙such thatΨ0⊆Ψfor allΨ0∈D𝑓(𝑝+𝜖). Take 𝜖=
0Ψ, 𝛿Ω
→𝑓rΨ,−𝛿Ω
𝑓→rΨ
,where𝛿 >0 is such that𝜖∈𝔙. Note that𝑈𝑓 (Ξ|𝑝+𝜖) ≤𝑈𝑓 (Ξ|𝑝)for allΞ⊆Ωwith equality if and only ifΞ⊆Ψ.It follows thatΨ0⊆Ψfor allΨ0∈D𝑓(𝑝+𝜖).
To complete the proof of Part (a), we perturb𝜖. More precisely, let𝔙0be an open neighborhood of 0∈RΩ𝑓 such thatD𝑓(𝑝+𝜖+𝜖0) ⊆D𝑓(𝑝+𝜖)for all𝜖0∈𝔙0—such a𝔙0exists by upper hemi-continuity. By Claim C.1, there exists 𝜖0∈𝔙0such that𝜖+𝜖0∈𝔙and|D𝑓(𝑝+𝜖+𝜖0) |=1.
The proof of Part (b) is similar. Note that𝜓is never demanded if|𝑝𝜓|=∞.Hence, we must have𝑝𝜓 ∈R. First, we show that there exists𝜖 ∈𝔙such that𝜓 ∈ Ψ0for allΨ0 ∈D𝑓(𝑝+𝜖). Without loss of generality, assume that 𝜓∈Ω𝑓→. Take𝜖=
0Ω
𝑓r{𝜓}, 𝛿𝜓
,where𝛿 >0 is such that𝜖∈𝔙. Note that𝑈𝑓 (Ξ|𝑝+𝜖) ≥𝑈𝑓 (Ξ|𝑝)for allΞ⊆Ω with equality if and only if𝜓∉Ξ.It follows that𝜓∈Ψ0for allΨ0∈D𝑓(𝑝+𝜖).
To complete the proof of Part (b), we perturb𝜖as in the proof of Part (a).
Using suitable selections, Claim C.2 implies a demand-language version of Theorem B.1.
Claim C.3. IfD𝑓 is fully substitutable, thenD𝑓 is increasing-price fully substitutable for sales.
Proof. Let𝑝≤𝑝0∈P,and letΞ∈D𝑓(𝑝). Let 𝔙 =n
𝜖∈RΩ𝑓 |D𝑓(𝑝0+𝜖) ⊆D𝑓(𝑝0)and|D𝑓(𝑝0+𝜖) |=1 o
,
which is non-empty and dense in a neighborhood of 0 by Claim C.1 and upper hemi-continuity. By Claim C.2(a), there exists𝜖∈𝔙such thatD𝑓(𝑝+𝜖)={Ψ}withΨ⊆Ξ.Note thatD𝑓(𝑝0+𝜖)={Ξ0}for someΞ0∈D𝑓(𝑝0)by construction.
BecauseD𝑓 is fully substitutable, we have
Ξ0∩ {𝜔∈Ω𝑓→|𝑝𝜔=𝑝0
𝜔} ⊆Ψ⊆Ξ.
It follows thatD𝑓 is increasing-price fully substitutable for sales.
Claim C.4. IfD𝑓 is fully substitutable, thenD𝑓 is decreasing-price fully substitutable for sales.
Proof. Let𝑝≥𝑝0∈P,letΞ∈D𝑓(𝑝),and suppose that𝜓 ∈Ξsatisfies𝑝𝜓 =𝑝0
𝜓
.Let 𝔙 =n
𝜖∈RΩ𝑓 |D𝑓(𝑝0+𝜖) ⊆D𝑓(𝑝0)and|D𝑓(𝑝0+𝜖) |=1 o
,
which is non-empty and dense in a neighborhood of 0 by Claim C.1 and upper hemi-continuity. By Claim C.2(b), there exists𝜖∈𝔙such thatD𝑓(𝑝+𝜖)={Ψ}with𝜓∈Ψ.Note thatD𝑓(𝑝0+𝜖)={Ξ0}for someΞ0∈D𝑓(𝑝0)by construction.
SinceD𝑓 is fully substitutable, we must have𝜓 ∈Ξ0∈D𝑓(𝑝).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 that𝐶𝑓is fully substitutable. Lemma C.1 and Claim C.3 imply that𝐶𝑓 is increasing-price fully substitutable for sales. Lemma C.1 and Claim C.4 imply that𝐶𝑓 is decreasing-price fully substitutable for sales. Similarly,𝐶𝑓 must be decreasing-price and increasing-price fully substitutable for purchases. Thus,𝐶𝑓 is strongly fully substitutable.
D PROOF OF THEOREM??
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 that competitive equilibria exist 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.
We modify the economy by giving agents to option to make any trade for a cost ofΠ.41Formally, for𝑓 ∈𝐹 ,define 𝑢b𝑓 :P (Ω𝑓) ×RΩ𝑓 →Rby
𝑢𝑓 is clearly continuous and strictly increasing in theRΩ𝑓 factor. Consider a modified economy in which utility functions are given by
𝑢b𝑓 for𝑓 ∈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 make any trade for a cost ofΠachieves.
Lemma D.1. Under BCV, the modified economy satisfies BWP.
Proof. We claim that BWP is satisfied with𝑀=Π+1.Let𝑓 ∈𝐹 ,let𝜔∈Ω𝑓 rΞ, letΞ⊆Ω𝑓,and let𝑡∈RΩ𝑓 be
The following claim, which asserts that giving a firm the option to make one trade for a cost ofΠpreserves full substitutability, will be used to prove that FS holds in the modified economy.
Claim D.1. LetΠbe a positive real number. Given a utility function𝑢𝑓 and𝜑 ∈Ω𝑓,defineb𝑢
41Hatfield et al. [2019] show that suchtrade endowmentspreserve full substitutability when preferences are quasilinear (see Theorem 2 in Hatfield et al., 2019).
If𝑢𝑓 is fully substitutable, then so isb𝑢
𝑓 𝜑.
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 thatD𝑓 is fully substitutable. LetbD𝑓 denote the extended demand correspondence for the utility function
follows from the full substitutability ofD𝑓.
Case 2: 𝑝𝜑≤Π<𝑝𝜑0.In this case, we have𝑝=𝑞andDb𝑓(𝑝)=D𝑓(𝑞). Let𝜔∈Ω𝑓→rΞsatisfy𝑝𝜔=𝑝𝜔0—note
The cases exhaust all possibilities, completing the proof thatbD𝑓 is fully substitutable. By Lemma C.1, b𝑢
𝑓
𝜑must 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 agent𝑓 can only produceK𝑓 units of surplus in the modified economy and that trade endowments can only be used at social costΠ. As will be seen in the proof of Lemma D.4, it follows that trade endowments cannot be used in any competitive equilibrium.
Proof. Note that The definition ofK𝑓 implies that
−K𝑓 ≤ Õ
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.
Since[Ξ;𝑝]is a competitive equilibrium in the modified economy, we have
𝑢b𝑓(Ξ𝑓, 𝑡𝑓) ≥b𝑢𝑓(∅,0)for all𝑓 ∈𝐹. Note
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
b𝑢𝑓.It follows thatΞ𝑓 ∈𝐷𝑓(𝑝).Since𝑓 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.
D.4 Proof of Theorem D.1
Let𝑀be as in BWP. Intuitively, we consider a grid of size𝜖in[−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. [2018b] to produce an 𝜖-equilibrium. Sending𝜖→0,we obtain a competitive equilibrium.
Formally, a vector𝛿 ∈ (−𝜖, 𝜖)Ωis𝜖-regularif𝐷𝑓 is single-valued on[−2𝑀 ,2𝑀]Ω𝑓 ∩
𝜖ZΩ𝑓 +𝛿Ω
𝑓
for all𝑓 ∈𝐹. The following claim asserts that there are many regular vectors.
Claim D.2. For any𝜖>0,the set of𝜖-regular vectors is dense in(−𝜖, 𝜖)Ω.
An arrangement[Ξ;𝑝]is an𝜖-equilibrium if every agent𝑓 demandsΞ𝑓 when given access to all sales, as well as purchases inΞ,at prices𝑝, and other purchases at prices𝑝+𝜖.
The following claim shows that𝜖-equilibria exist.
Claim D.3. For all0<𝜖<𝑀 ,under FS and BWP, there exists an𝜖-equilibrium. Following Hatfield and Kominers [2012], defineΦ:P
𝑋b
As in Fleiner [2003], Hatfield and Milgrom [2005], Hatfield and Kominers [2012], and Fleiner et al. [2018b], orderP 𝑋b
2
by letting(𝑋𝐵, 𝑋𝑆) v (𝑋¯𝐵,𝑋¯𝑆)if𝑋𝐵⊇𝑋¯𝐵and𝑋𝑆⊆𝑋¯𝑆. As Hatfield and Kominers [2012] and Fleiner et al. [2018b]
have shown,Φis isotone (with respect tov) under FS. The Tarski [1955] fixed point theorem guarantees thatΦhas a fixed point(𝑋𝐵, 𝑋𝑆).
SinceC𝑓
As[−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 sequence𝑝1, 𝑝2, . . .∈ [−2𝑀 ,2𝑀]Ωsuch that[Ξ;𝑝𝑘]is a 1
E OTHER PROOFS OMITTED FROM THE TEXT E.1 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 let𝐴=𝜅( [Ξ;𝑝]). Suppose for the sake
E.2 Proof of Proposition 1
We adapt the proof of Lemma 5(ii) in Fleiner et al. [2018b] to our setting. Inspired by Fleiner et al. [2018b], we say that a circuit(𝑧1, . . . , 𝑧𝑛)islocally blockingif every pair of adjacent contracts is demanded by their common agent in every choice set.
Definition E.1. Let𝑌 be an outcome. A sequence of contracts(𝑧1, . . . , 𝑧𝑛)is alocally blocking circuitif:
• for all 1≤𝑖≤𝑛,we have{𝑧𝑖−1, 𝑧𝑖} ⊆𝑊 for all𝑊 ∈𝐶𝑓𝑖
𝑌𝑓
𝑖∪ {𝑧𝑖−1, 𝑧𝑖}
,where𝑓𝑖=s(𝑧𝑖)=b(𝑧𝑖−1). Here, we write𝑧0=𝑧𝑛.
To prove Proposition 1, we show (as in Fleiner et al., 2018b) that every shortest locally blocking circuit or locally blocking trail gives rise to a blocking set.
Claim E.1. Let𝑌be an individually rational outcome. Under FS, if(𝑧1, . . . , 𝑧𝑛)is shortest among all locally blocking circuits and locally blocking trails for𝑌 ,then the set{𝑧1, . . . , 𝑧𝑛}blocks𝑌.
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 and there exists𝑊 ∈ 𝐶𝑓𝑖+1({𝑧𝑖, 𝑧𝑖+1}) with𝑧𝑖 ∉ 𝑊 ,then (𝑧𝑖+1, . . . , 𝑧𝑛) is a locally blocking trail. Similarly, if (𝑧1, . . . , 𝑧𝑛) is a locally blocking trail and there exists𝑊 ∈ 𝐶𝑓𝑖+1({𝑧𝑖, 𝑧𝑖+1})with𝑧𝑖+1∉𝑊 ,then(𝑧1, . . . , 𝑧𝑖)is a locally blocking trail.
Now, suppose that𝑍 = {𝑧1, . . . , 𝑧𝑛}does not block𝑌. Then, there is a firm 𝑓 ,a contract𝑧𝑗 ∈ 𝑍
𝑓,and a set 𝑊 ∈𝐶𝑓
𝑌𝑓 ∪𝑍𝑓
with𝑧𝑗 ∉𝑊. Without loss of generality, we can assume that𝑓 =s(𝑧𝑗),so that𝑓 =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} ⊆𝑊for all𝑊 ∈𝐶𝑓𝑖+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 whether𝑗 =1 and whether we have a trail or a circuit to complete the proof of the claim.
Case 1: 𝑗=1 and(𝑧1, . . . , 𝑧𝑛)is a locally blocking trail. By IFSS, there exists𝑊0∈𝐶𝑓(𝑌𝑓∪𝑍𝑓→)with𝑧1∉𝑊0. Among all such𝑊0,take𝑊 to minimize|𝑊0r𝑌𝑓|. As𝑌𝑓 ∉𝐶𝑓(𝑌𝑓 ∪ {𝑧1}),we have𝑌𝑓 ∉𝐶𝑓(𝑌𝑓 ∪𝑍𝑓→),and hence𝑊0*𝑌𝑓.
Let𝑧𝑘 ∈𝑊r𝑌𝑓 be arbitrary. By IFSS, we must have𝑌𝑓 ∉𝐶𝑓(𝑌𝑓 ∪ {𝑧𝑘}),so that(𝑧𝑘, . . . , 𝑧𝑛)is a shorter locally blocking trail.
Case 2: 𝑗≠1 or(𝑧1, . . . , 𝑧𝑛)is a locally blocking circuit. In either case,𝑧𝑗−1is well-defined. By IFSS, there exists 𝑊0∈𝐶𝑓(𝑌𝑓∪ {𝑧𝑗−1} ∪𝑍𝑓→)with𝑧𝑗∉𝑊0.Among all such𝑊0,take𝑊to minimize|𝑊0r𝑌𝑓|. As{𝑧𝑗−1, 𝑧𝑗} ⊆𝐵
for all𝐵∈𝐶𝑓(𝑌𝑓 ∪ {𝑧𝑗−1, 𝑧𝑗}),we have𝑧𝑗−1∈𝑊 by DFSP.
Let𝑧
𝑘 ∈𝑊r𝑌
𝑓 be arbitrary. By IFSS, we must have𝑧
𝑘 ∈𝐵for all𝐵 ∈𝐶𝑓(𝑌
𝑓 ∪ {𝑧𝑗−1, 𝑧
𝑘}). If𝑘<𝑗 ,then (𝑧𝑘, . . . , 𝑧𝑗−1)is a shorter locally blocking circuit. If𝑘>𝑗 ,then(𝑧1, . . . , 𝑧𝑗−1, 𝑧𝑘, . . . , 𝑧𝑛)is a shorter locally blocking circuit or locally blocking trail.
The cases exhaust all possibilities, completing the proof of the claim.
Claim E.1 implies Proposition 1.
E.3 Proof of Theorem 3
which is finite by BCV. Let
Π=1+Õ Consider a modified economy in which utility functions are given by
𝑢b𝑓 for𝑓 ∈𝐹. Claim E.2. The outcome𝐴is stable in the modified economy.
Proof. The outcome𝐴is clearly individually rational in the modified economy. It remains to prove that𝐴is not blocked in the modified economy. Suppose for the sake of deriving a contradiction that there is a blocking set𝑍in the modified economy.
where the first inequality is due to Lemma D.3(a), so that Õ
Hence, Lemma D.3(b) implies that𝑈b
𝑓 original economy, which contradicts the hypothesis that𝐴is stable in the original economy.
Claim E.2 guarantees that𝐴is stable in the modified economy. By Proposition 1,𝐴is trail-stable in the modified economy. Lemmata D.1 and D.2 ensure that FS and BCV are satisfied in the modified economy. Hence, there exists a com-petitive equilibrium[Ξ;𝑝]in the modified economy with𝜅( [Ξ;𝑝])=𝐴by 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 for unrealized trades that are as high as possible while remaining (weakly) undesirable to sellers. Call a trail(𝑧1, . . . , 𝑧𝑛)locally semi-blockingif the seller of𝑧𝑖wants to propose𝑧𝑖when given access to𝑧𝑖−1for all𝑖>1,and the seller of𝑧1wants to propose𝑧1. We consider a contract desirable to a seller if it appears in a locally semi-blocking trail. in some locally semi-blocking trail. Thus,𝑋𝐵consists of all contracts that are strictly desirable to their sellers.42For 𝜔∈Ω,define
so that𝑝𝜔is the minimum of𝑀and the highest price at which𝜔is weakly undesirable to its seller. We prove that 𝜅( [Ξ;𝑝])=𝐴and that[Ξ;𝑝]is a competitive equilibrium.
Claim E.3. Under BWP, if𝐴is individually rational, then we have𝜅( [Ξ;𝑝])=𝐴.
Proof. Suppose that(𝜔 , 𝑝𝜔0) ∈𝐴. BWP implies that𝑝𝜔0 <𝑀. As𝑢s(𝜔)is strictly increasing in transfers and𝐴is individually rational, we have(𝜔 , 𝑝00
𝜔) ∈𝑋𝐵if and only if𝑝00
𝜔>𝑝0
𝜔.It follows that𝑝𝜔=𝑝0
𝜔.Since𝜔∈Ξwas arbitrary,
the claim follows.
Claim E.4. Under FS and BWP, if𝐴is trail-stable, then[Ξ;𝑝]is a competitive equilibrium.
Proof. Suppose for the sake of deriving a contradiction thatΞ𝑓 ∉𝐷𝑓
𝑝Ω
𝑓
. As𝐴is individually rational, it follows from Claim E.3 thatΞ0∉𝐷𝑓
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 that𝑓 demands no subset ofΞ𝑓.Formally, define
𝔒 ={𝑝0∈RΩ𝑓 |𝐷𝑓(𝑝0) ∩ P (Ξ𝑓)=∅}. Since𝐷𝑓 is upper hemi-continuous, the set𝔒contains an open neighborhood of𝑝Ω
𝑓. By (E.1), there exists𝑞∈𝔒 such that𝑞Ξ
𝑓∪Ω𝑓→ =𝑝Ξ
𝑓∪Ω𝑓→,we have𝑞𝜔=𝑝𝜔whenever𝑝𝜔=𝑀 ,and(𝜔 , 𝑝𝜔) ∈𝑋𝐵whenever𝜔∈Ω→𝑓 rΞand 𝑝𝜔<𝑀 .We consider the prices𝑞instead of the prices𝑝.
By Theorem B.1, SFS must be satisfied. To produce a contradiction, we consider the set of trades that𝑓 could demand at price vector𝑞that contains fewest trades outsideΞ𝑓. Formally, letΨ∈𝐷𝑓(𝑞)minimize|Ψ0rΞ|over allΨ0∈𝐷𝑓(𝑞).
42In the fixed-point interpretation of trail-stable outcomes [Adachi, 2017, Fleiner et al., 2018b],𝑋𝐵is the set of contracts that are available to their buyers.
Consider the corresponding set of contracts𝑊=𝜅( [Ψ;𝑞]). Note that𝑊 *𝐴and𝑊→𝑓 r𝐴⊆𝑋𝐵by construction and BWP. We divide into cases based on whether𝑊r𝐴contains any contracts that are sold by𝑓 to produce a contradiction.
Case 1: 𝑊r𝐴*𝑋→𝑓. In this case, we either produce a locally blocking trail or show that any sale in𝑊r𝐴 with𝑧∉𝑊00to derive contradictions.
Subcase 1.1: There exists𝑊00∈𝐶𝑓(𝐴𝑓∪{𝑧𝑛, 𝑧})with𝑧∉𝑊00.Then, the trail(𝑧1, . . . , 𝑧𝑛)is locally blocking, contradicting the assumption that𝐴is trail-stable.
Subcase 1.2: 𝑧 ∈𝑊00for all𝑊00∈𝐶𝑓(𝐴𝑓 ∪ {𝑧𝑛, 𝑧}).Then,(𝑧1, . . . , 𝑧𝑛, 𝑧)is a locally semi-blocking trail.
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 that𝐴=𝜅( [Ξ;𝑝])is not strongly group stable.
If𝐴is not individually rational, then clearly[Ξ;𝑝]is not a competitive equilibrium. Thus, we can assume that𝐴is not strongly unblocked—that is, that there exists a non-empty set of contracts𝑍⊆𝑋r𝐴and, for each𝑓 ∈𝐹with𝑍𝑓 ≠∅,
ensures that
. Therefore,[Ξ;𝑝]is not a competitive equilibrium.
E.7 Proof of Corollary 3
Competitive equilibrium outcomes exist and coincide with trail-stable outcomes by Corollary 2, and are strongly group stable by Theorem 5. Strongly group stable outcomes are always stable. Stable outcomes are trail-stable by Proposition 1.
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. Let𝐴be outcome that is not stable. If𝐴is not individually rational, then clearly𝐴is not trail-stable. Thus, we can assume that𝐴is blocked by a non-empty blocking set𝑍.
Since𝑍is non-empty and the network is assumed to be acyclic, there is a firm𝑓1with𝑍→𝑓
1=∅and𝑍𝑓
A similar argument to the previous paragraph shows that(𝑧1, 𝑧2)is a locally blocking trail or there exists𝑧3∈𝑍 withs(𝑧3)=b(𝑧2)such that𝐴𝑓
2∉𝐶𝑓2(𝐴𝑓
2∪ {𝑧2, 𝑧3}).By induction and due to acyclicity, we obtain a locally blocking trail. Thus,𝐴is 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.
Example3continued(Cyclic economy with transferable utility). In Example 3, the competitive equilibria are[{𝜁 , 𝜓};𝑝], where
𝑝𝜁−𝑝𝜓
≤10.All competitive equilibria are Pareto-efficient, as guaranteed by the First Welfare Theorem (see, e.g., Hatfield et al. [2013]), and trade occurs in every competitive equilibrium.
ExampleF.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 firm𝑓3,which interacts with𝑓1via trade𝜔0. Firm 𝑓𝑖has utility function
𝑢𝑓𝑖(Ξ, 𝑡)=𝑣𝑓𝑖(Ξ) + Õ
𝜔∈Ω𝑓
𝑡𝜔,
where valuations𝑣𝑓1, 𝑣𝑓2,and𝑣𝑓3are as in Example 2.
Trade𝜁0cannot be realized in equilibrium due to the technological constraints of𝑓1and𝑓2.Hence, we must have 𝑝𝜁0≥300 in any competitive equilibrium, since𝑓3must weakly prefer∅over{𝜁0}in equilibrium. In order for trade to occur,𝑓1must prefer𝜁 over𝜁0,and so we must have𝑝𝜁 ≥𝑝𝜁0.Hence, the competitive equilibria are[{𝜁 , 𝜓};𝑝],where
|𝑝𝜁−𝑝𝜓| ≤10 and𝑝𝜁 ≥𝑝𝜁0≥300.Essentially, adding an outside option simply forces𝑝𝜁 to be at least $300 without shutting down trade between𝑓1and𝑓2.
The next example shows that a regularity condition, such as BCV, is needed in addition to FS to ensure that competitive equilibria exist.
ExampleF.2 (Competitive equilibria need not exist under FS alone). Consider two firms,𝑏and𝑠,and one trade𝜔 between them withs(𝜔)=𝑠andb(𝜔)=𝑏. Suppose that𝑠is not willing to sell𝜔at any (finite) price, but𝑏would buy𝜔 at any (finite) price. Note that the market does not clear at any price—𝑏always demands𝜔and𝑠never demands𝜔 .The issue is that the variation needed to exactly compensate𝑏for going from autarky to trade is−∞.If𝑏’s compensating variation were−𝑝,then autarky could be sustained in equilibrium at any price above𝑝.
The last example shows that FS needed for stable outcomes to be trail-stable.
ExampleF.3 (Stable outcomes may not be trail-stable without FS). As depicted in Figure 2(a), there are two firms,𝑓1 and𝑓2,which interact via two trades,𝜁 and𝜓. Firm𝑓𝑖has utility function
𝑢𝑓𝑖(Ξ, 𝑡)=𝑣𝑓𝑖(Ξ) + Õ
𝜔∈Ω𝑓
𝑡𝜔,
where
𝑣𝑓1(∅)=𝑣𝑓2(∅)=0 𝑣𝑓1({𝜁})=𝑣𝑓1({𝜓})=1
𝑣𝑓1({𝜁 , 𝜓})=−∞
𝑣𝑓2({𝜁})=𝑣𝑓2({𝜓})=−∞
𝑣𝑓2({𝜁 , 𝜓})=1.
Note that trades𝜁and𝜓are not complementary for firm𝑓1,which implies that𝑓1’s preferences are not fully substitutable.
The no-trade outcome∅is stable, as no non-empty set of contracts is individually rational for both𝑓1and𝑓2. However, the trail( (𝜁 ,0),(𝜓 ,0))locally blocks the outcome∅.Thus, the no-trade outcome is stable but not trail-stable.