Trading Networks and Equilibrium Intermediation Maciej H. Kotowski1 1 John
C. Matthew Leister2
F. Kennedy School of Government Harvard University 2 Department
of Economics Monash University
December 11, 2015
Intermediation
“Intermediation” is 25% of the U.S. Economy (Spulber 1996, JEP) ◮
Retail & Wholesale Trade
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Finance
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Other (Real Estate Brokers, Transport, . . . )
Trading Networks
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Trading Networks
Seller ↔ Intermediary ↔ · · · ↔ Intermediary ↔ Buyer {z } | We Study This Part of the Market
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Intermediaries have a network of relationships
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Intermediaries have different (private) costs of trade
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Intermediaries bid competitively to provide “intermediation services” that move goods from the seller to the buyer
Some Related Work
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Networks and exchange ◮ ◮ ◮
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Middlemen ◮
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Rubinstein & Wolinsky (1987)
Experiments ◮
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Kranton & Minehart (2001) Manea (2015) Condorelli, Galeotti, Renou (2015)
Gale & Kariv (2009)
Many others cited in the paper.
Outline A tractable network structure “Multipartite Networks” 1. Model Second Price Auctions A tractable trading protocol A tractable cost structure Binary
Outline A tractable network structure “Multipartite Networks” 1. Model A tractable trading protocol Second Price Auctions A tractable cost structure Binary
2. Analysis
Stability
Network Persistence / “No Mergers”
Equilibrium Network Formation / “Free Entry”
Outline A tractable network structure “Multipartite Networks” 1. Model A tractable trading protocol Second Price Auctions A tractable cost structure Binary
2. Analysis
Stability
3. Conclusion
Network Persistence / “No Mergers”
Equilibrium Network Formation / “Free Entry”
Stability + Equilibrium
Final Remarks
Just an Example
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Traders
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Degree of Intermediation: R Example: R = 3 b
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Degree of Intermediation: R Example: R = 3 b
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Configuration of Traders: n = (n1 , . . . , nR ) Example: n = (4, 2, 3) b
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Model: Odds and Ends
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Network structure common knowledge.
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Buyers’ valuations are henceforth normalized to 1 and are common knowledge.
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Ties are broken at random.
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Trade “breaks down” if all bidders/traders bid “ℓ.”
Model: Trading Costs Each trader has a private trading (inventory cost) that he must incur when he receives the item. ◮
p — probability trading cost is 0.
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1 − p — probability trading cost is c¯ > 1.
Distribution of trading costs is common knowledge. Realized trading costs are private information.
Model: Trading Costs Each trader has a private trading (inventory cost) that he must incur when he receives the item. ◮
p — probability trading cost is 0.
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1 − p — probability trading cost is c¯ > 1.
Distribution of trading costs is common knowledge. Realized trading costs are private information.
Trader’s Payoffs (Re)sale Revenue - Purchase Costs - Trading Cost
Exchange in a Fixed Network Theorem There exists a perfect Bayesian equilibrium of the trading game where each agent i (in row r ) adopts the following strategy: 1. If the agent’s costs are low and the asset is being sold by an agent in row r + 1, the agent places a bid equal to the asset’s expected resale value conditional on all available information and on others’ strategies. 2. Otherwise, the agent bids ℓ. Buyers bid their value for the asset.
NB. Multiple second price auctions =⇒ Many other equilibria.
Exchange in a Fixed Network ◮
Asset does not backtrack or stall.
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Inductive structure. Given n = (n1 , . . . , nR ), the equilibrium path bid of a low-cost trader in row r :
Exchange in a Fixed Network ◮
Asset does not backtrack or stall.
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Inductive structure. Given n = (n1 , . . . , nR ), the equilibrium path bid of a low-cost trader in row r :
ν1 = 1 ν0 = 1
(Buyers’ value)
Exchange in a Fixed Network ◮
Asset does not backtrack or stall.
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Inductive structure. Given n = (n1 , . . . , nR ), the equilibrium path bid of a low-cost trader in row r :
ν2 = δ(n1 ) ν1 = 1 ν0 = 1
(Buyers’ value)
δ(n) := 1 − (1 − p)n − np(1 − p)n−1
Exchange in a Fixed Network ◮
Asset does not backtrack or stall.
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Inductive structure. Given n = (n1 , . . . , nR ), the equilibrium path bid of a low-cost trader in row r :
ν3 = δ(n2 )δ(n1 ) ν2 = δ(n1 ) ν1 = 1 ν0 = 1
(Buyers’ value)
δ(n) := 1 − (1 − p)n − np(1 − p)n−1
Exchange in a Fixed Network ◮
Asset does not backtrack or stall.
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Inductive structure. Given n = (n1 , . . . , nR ), the equilibrium path bid of a low-cost trader in row r :
νr =
rY −1
δ(nk ) = δ(nr −1 )νr −1
k=1
.. . ν3 = δ(n2 )δ(n1 ) ν2 = δ(n1 ) ν1 = 1 ν0 = 1
(Buyers’ value)
δ(n) := 1 − (1 − p)n − np(1 − p)n−1
Expected Payoffs Ex ante expected trading profit of a row r trader given n = (n1 , . . . , nR ):
πr (n) =
rY −1
δ(nk ) ×
|
{z
k=1
[1]
}
p × (1 − p)nr −1 × |{z} {z } | [2]
[3]
µ(n) := 1 − (1 − p)n δ(n) := 1 − (1 − p)n − np(1 − p)n−1
R Y
µ(nk )
k=r +1
|
{z [4]
}
Expected Payoffs Ex ante expected trading profit of a row r trader given n = (n1 , . . . , nR ):
πr (n) =
rY −1
δ(nk ) ×
|
{z
k=1
[1]
}
p × (1 − p)nr −1 × |{z} {z } | [2]
[3]
R Y
µ(nk )
k=r +1
|
{z [4]
µ(n) := 1 − (1 − p)n δ(n) := 1 − (1 − p)n − np(1 − p)n−1
Fact: πr (nr , n−r ) is decreasing in nr and increasing in n−r . ◮ Traders in the same row are substitutes. ◮ Traders in others rows are complements.
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Stability
Persistence of a trading network is a puzzel. Why? Adjacent traders have an incentive to merge or collude. We call such deviations “partnerships.” In a stable market, traders should not deviate in this manner, i.e. the network is valuable.
A partnership is any group of adjacent traders that function as a single entity. A bc
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A partnership is any group of adjacent traders that function as a single entity. A
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Partnerships
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Timing: A partnership forms conditional on n but before trading costs are realized.
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Once present, a partnership can trade just like any trader.
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Denote partnership membership by m = (m1 , . . . , mR ). Example: m = (0, 2, 1, 0) ◮ ◮
m ¯ — highest row with a partnership member. m — lowest row with a partnership member.
Partnerships: Benefits and Costs ◮
Probability that partnership m has low trading cost: pm =
m ¯ Y
k=m
µ(mk )
Partnerships: Benefits and Costs ◮
Probability that partnership m has low trading cost: pm =
m ¯ Y
µ(mk )
k=m
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Costs of partnership formation ζ(m) = ch
m ¯ X
r =m
|
(mr − 1) + cv · (m ¯ − m) | {z } [2] {z } [1]
Partnerships: Benefits and Costs ◮
Probability that partnership m has low trading cost: pm =
m ¯ Y
µ(mk )
k=m
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Costs of partnership formation ζ(m) = ch
m ¯ X
r =m
| ◮
(mr − 1) + cv · (m ¯ − m) | {z } [2] {z } [1]
Costs of partnership formation
Exchange The trading game can be analyzed as before, but a partnership enjoy direct and indirect advantages. A
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A partnership is any group of adjacent traders that function as a single entity. A
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Stability
A trading network n is stable if for all feasible partnerships m ≤ n, X mr πr (n) ≥ πm (n) − ζ(m). r
Stability
A trading network n is stable if for all feasible partnerships m ≤ n, X mr πr (n) ≥ πm (n) − ζ(m). r
Theorem If ch > 0 and cv ≥ 0, then there exists a pˆ > 0 such that for all p < p, ˆ the trading network is stable.
Equilibrium Networks
Our model of network formation. 1. R is fixed. 2. There is a large pool of potential traders. 3. A trader can enter any row at an entry cost of κ > 0. 4. Traders make entry decision before learning their cost-type. 5. Traders enter until expected profits are zero.*
Equilibrium
The network configuration n∗ = (n1∗ , . . . , nR∗ ) is an equilibrium configuration if for all r , πr (n∗ ) − κ ≥ 0 and πr (n1∗ , . . . , nr∗−1 , nr∗ + 1, nr∗+1 , . . . , nR∗ ) − κ < 0.
See also Gary-Bobo (1990).
Existence and Example
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There exists a nontrivial equilibrium n∗ iff there exists n such that for all r , πr (n) − κ ≥ 0.
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If n∗ is an equilibrium, nr∗ ≥ nr∗+1 .
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Multiple equilibria may exist.
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Equilibria form a directed set. (n∗ ≥ n∗∗ ⇐⇒ nr∗ ≥ nr∗∗ for all r .)
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There exists a unique “maximal” equilibrium.
An example: R = 6, p = 0.5, κ = 0.01 bc
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An example: R = 6, p = 0.5, κ = 0.01 7 bc
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Welfare Aggregate Welfare
Ω(n) =
n0 π0 (n) + | {z }
Buyers’ Payoffs
R X r =1
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nr (πr (n) − κ) + πR+1 (n) . | {z } {z } Seller’s Payoff
Traders’ Payoffs
Welfare Aggregate Welfare
Ω(n) =
n0 π0 (n) + | {z }
Buyers’ Payoffs
R X r =1
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nr (πr (n) − κ) + πR+1 (n) . | {z } {z } Seller’s Payoff
Traders’ Payoffs
Theorem ˆ maximizes Ω(n), then nˆr = nˆr ′ for all r and r ′ . If n ˆ ≥ n∗ . Moreover, if n∗ is an equilibrium configuration, then n (cf. Mankiw & Whinston 1986)
Stability and Equilibrium: An Example If R = 5, p = 1/2, and κ = 0.015, there are two equilibrium configurations: n∗ = (4, 3, 3, 2, 1) and n∗∗ = (5, 5, 5, 5, 4).
.1
.03 .01 0
No Stable Equilibrium
cv
{n∗ } Stable
.07 .1
{n∗ , n∗∗ } Stable
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Concluding Remarks
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Developed a tractable model of exchange in a network.
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Proposed definition of stability (no mergers) and equilibrium configurations (free entry). ◮
(Network) Externalities =⇒ Multiple Equilibria.
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A stability-efficiency tradeoff: A trading network may be stable, but improving efficiency may lead to instabilities.
