Trading Networks and Equilibrium Intermediation

Transcription

1 Trading Networks and Equilibrium Intermediation Maciej H. Kotowski 1 C. Matthew Leister 2 1 John F. Kennedy School of Government Harvard University 2 Department of Economics Monash University December 11, 2015

2 Intermediation Intermediation is 25% of the U.S. Economy (Spulber 1996, JEP) Retail & Wholesale Trade Finance Other (Real Estate Brokers, Transport,...)

3 Trading Networks F D E Beth Sam C B G A

4 Trading Networks Seller Intermediary Intermediary Buyer }{{} We Study This Part of the Market Intermediaries have a network of relationships Intermediaries have different (private) costs of trade Intermediaries bid competitively to provide intermediation services that move goods from the seller to the buyer

5 Some Related Work Networks and exchange Kranton & Minehart (2001) Manea (2015) Condorelli, Galeotti, Renou (2015) Middlemen Rubinstein & Wolinsky (1987) Experiments Gale & Kariv (2009) Many others cited in the paper.

6 Outline 1. Model A tractable network structure A tractable trading protocol A tractable cost structure Multipartite Networks Second Price Auctions Binary

7 Outline 1. Model A tractable network structure A tractable trading protocol A tractable cost structure Multipartite Networks Second Price Auctions Binary 2. Analysis Stability Network Persistence / No Mergers Equilibrium Network Formation / Free Entry

8 Outline 1. Model A tractable network structure A tractable trading protocol A tractable cost structure Multipartite Networks Second Price Auctions Binary 2. Analysis Stability Network Persistence / No Mergers Equilibrium Network Formation / Free Entry 3. Conclusion Stability + Equilibrium Final Remarks Just an Example

9

10 Seller 4 3 Traders 2 1 Buyers 0

11 4 Degree of Intermediation: R Example: R =

12 4 Degree of Intermediation: R Example: R = Configuration of Traders: n = (n 1,...,n R ) Example: n = (4,2,3) 0

13 T T = 10 T = 10 0

14 T = 10 T = 10 0

15 3 4 T T = 10 T = 10 0

16 0 3 4 T 2 l T = 10 T = 10 0

17 T 2 1 T = 10 T = 10 0

18 3 4 2 l T T = 10 T = 10 0

19 T 1 T = 10 T = 10 0

20 T T = 10 T = 10 0

21 T T = 10 T = 10 0

22 T 0 0 0

23 Model: Odds and Ends Network structure common knowledge. Buyers valuations are henceforth normalized to 1 and are common knowledge. Ties are broken at random. Trade breaks down if all bidders/traders bid l.

24 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. 1 p probability trading cost is c > 1. Distribution of trading costs is common knowledge. Realized trading costs are private information.

25 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. 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

26 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.

27 Exchange in a Fixed Network Asset does not backtrack or stall. Inductive structure. Given n = (n 1,...,n R ), the equilibrium path bid of a low-cost trader in row r:

28 Exchange in a Fixed Network Asset does not backtrack or stall. Inductive structure. Given n = (n 1,...,n R ), the equilibrium path bid of a low-cost trader in row r: ν 1 = 1 ν 0 = 1 (Buyers value)

29 Exchange in a Fixed Network Asset does not backtrack or stall. Inductive structure. Given n = (n 1,...,n R ), the equilibrium path bid of a low-cost trader in row r: ν 2 = δ(n 1 ) ν 1 = 1 ν 0 = 1 (Buyers value) δ(n) := 1 (1 p) n np(1 p) n 1

30 Exchange in a Fixed Network Asset does not backtrack or stall. Inductive structure. Given n = (n 1,...,n R ), the equilibrium path bid of a low-cost trader in row r: ν 3 = δ(n 2 )δ(n 1 ) ν 2 = δ(n 1 ) ν 1 = 1 ν 0 = 1 (Buyers value) δ(n) := 1 (1 p) n np(1 p) n 1

31 Exchange in a Fixed Network Asset does not backtrack or stall. Inductive structure. Given n = (n 1,...,n R ), the equilibrium path bid of a low-cost trader in row r: ν r =. r 1 k=1 ν 3 = δ(n 2 )δ(n 1 ) ν 2 = δ(n 1 ) ν 1 = 1 ν 0 = 1 δ(n k ) = δ(n r 1 )ν r 1 (Buyers value) δ(n) := 1 (1 p) n np(1 p) n 1

32 Expected Payoffs Ex ante expected trading profit of a row r trader given n = (n 1,...,n R ): π r (n) = r 1 k=1 δ(n k ) }{{} [1] p }{{} [2] (1 p) nr 1 }{{} [3] R k=r+1 µ(n k ) } {{ } [4] µ(n) := 1 (1 p) n δ(n) := 1 (1 p) n np(1 p) n 1

33 Expected Payoffs Ex ante expected trading profit of a row r trader given n = (n 1,...,n R ): π r (n) = r 1 k=1 δ(n k ) }{{} [1] p }{{} [2] (1 p) nr 1 }{{} [3] R k=r+1 µ(n k ) } {{ } [4] µ(n) := 1 (1 p) n δ(n) := 1 (1 p) n np(1 p) n 1 Fact: π r (n r,n r ) is decreasing in n r and increasing in n r. Traders in the same row are substitutes. Traders in others rows are complements.

34 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.

35 A partnership is any group of adjacent traders that function as a single entity. A m

36 A partnership is any group of adjacent traders that function as a single entity. A B m

37 Partnerships Timing: A partnership forms conditional on n but before trading costs are realized. Once present, a partnership can trade just like any trader. Denote partnership membership by m = (m 1,...,m R ). Example: m = (0,2,1,0) m highest row with a partnership member. m lowest row with a partnership member.

38 Partnerships: Benefits and Costs Probability that partnership m has low trading cost: p m = m k=m µ(m k )

39 Partnerships: Benefits and Costs Probability that partnership m has low trading cost: p m = m k=m µ(m k ) Costs of partnership formation m ζ(m) = c h (m r 1) r=m } {{ } [1] +c v ( m m) }{{} [2]

40 Partnerships: Benefits and Costs Probability that partnership m has low trading cost: p m = m k=m µ(m k ) Costs of partnership formation m ζ(m) = c h (m r 1) r=m } {{ } [1] +c v ( m m) }{{} [2] Costs of partnership formation

41 Exchange The trading game can be analyzed as before, but a partnership enjoy direct and indirect advantages. A B m

42 A partnership is any group of adjacent traders that function as a single entity. A B m

43 Stability A trading network n is stable if for all feasible partnerships m n, m r π r (n) π m (n) ζ(m). r

44 Stability A trading network n is stable if for all feasible partnerships m n, m r π r (n) π m (n) ζ(m). r Theorem If c h > 0 and c v 0, then there exists a ˆp > 0 such that for all p < ˆp, the trading network is stable.

45 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 > Traders make entry decision before learning their cost-type. 5. Traders enter until expected profits are zero.*

46 Equilibrium The network configuration n = (n1,...,n R ) is an equilibrium configuration if for all r, π r (n ) κ 0 and π r (n1,...,n r 1,n r + 1,n r+1,...,n R ) κ < 0. See also Gary-Bobo (1990).

47 Existence and Example There exists a nontrivial equilibrium n iff there exists n such that for all r, π r (n) κ 0. If n is an equilibrium, n r n r+1. Multiple equilibria may exist. Equilibria form a directed set. (n n n r n r for all r.) There exists a unique maximal equilibrium.

48 An example: R = 6, p = 0.5, κ =

49 An example: R = 6, p = 0.5, κ =

50 Welfare Aggregate Welfare Ω(n) = n 0 π 0 (n) }{{} Buyers Payoffs + R n r (π r (n) κ) r=1 } {{ } Traders Payoffs + π R+1 (n). }{{} Seller s Payoff

51 Welfare Aggregate Welfare Ω(n) = n 0 π 0 (n) }{{} Buyers Payoffs + R n r (π r (n) κ) r=1 } {{ } Traders Payoffs + π R+1 (n). }{{} Seller s Payoff Theorem If ˆn maximizes Ω(n), then ˆn r = ˆn r for all r and r. Moreover, if n is an equilibrium configuration, then ˆn n. (cf. Mankiw & Whinston 1986)

52 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). c v.1 No Stable Equilibrium {n } Stable {n,n } Stable c h

53 Concluding Remarks Developed a tractable model of exchange in a network. Proposed definition of stability (no mergers) and equilibrium configurations (free entry). (Network) Externalities = Multiple Equilibria. A stability-efficiency tradeoff: A trading network may be stable, but improving efficiency may lead to instabilities.