Markets with Intermediaries

Transcription

1 Markets with Intermediaries Episode Baochun Li Professor Department of Electrical and Computer Engineering University of Toronto

2 Network Models of Markets with Intermediaries (Chapter )

3 Who sets the prices? 3

4 Who sets the prices? Matching markets embody a number of basic principles People naturally have different preferences for different kinds of goods Prices can decentralize the allocation of goods to people Such prices can in fact lead to allocations that are socially optimal 3

5 Who sets the prices? Matching markets embody a number of basic principles People naturally have different preferences for different kinds of goods Prices can decentralize the allocation of goods to people Such prices can in fact lead to allocations that are socially optimal But who sets the prices, who trades with whom, if there are many buyers and many sellers? 3

6 Trade with intermediaries In a wide range of markets, buyers and sellers do not interact directly with each other, but instead trade through intermediaries Think real estate brokers, stock markets There are many kinds of markets, each serving a purpose A particular good can be traded in multiple markets How do buyers and sellers interact in a stock market? 4

7 $5.5 $5.5 ASK $5. ASK $4. BID $4. BID $3.5 $3.5 (a) (b) The order book

8 A network structure with multiple traders With multiple buyers, sellers, and traders (intermediaries) connected in a network structure, what is the trading behaviour at equilibrium? B T S S B T S B 6

9 Assumptions in our simple network model Do not consider multiple goods for sale, or multiple quantities Each seller i holds one unit of the good, willing to sell it at any price that s at least vi Each buyer j values one unit of the good at vj, willing to pay no more than this value all buyers have the same valuation No one wants more than one unit All buyers, sellers and traders are assumed to know all the valuations Network structure is fied not affected by valuations 7

10 vi sellers traders buyers vj S B T S2 B2 T2 S3 B3

11 Model it as a game 9

12 Model it as a game Two stages The traders simultaneously announce their ask and bid prices to the sellers and buyers connected Each seller and buyer chooses at most one trader (but doesn t have to) all sellers and buyers simultaneously make the choice Let s model it as a dynamic game Maimizing the payoffs: a buyer j maimizes vj - atj; a trader maimizes its profit; a seller i maimizes bti - vi 9

13 sellers traders buyers S.2.8 B T.2.8 S2.3.7 B2 T2 S3 First stage B3

14 sellers traders buyers S.2.8 B T.2.8 S2.3.7 B2 T2 S3 Second stage B3

15 When a seller or buyer is indifferent between accepting or rejecting, then we (as the modellers) have a choice.

16 Best responses and equilibrium T is making several bad decisions He is losing the opportunity of dealing with S2 and B2 to T2 If he were to raise his bid to seller S2 to.4, and lower his ask to.6 sellers traders buyers S B T.4.6 S2.3.7 B2 T2 S3 B3 3

17 What are bid and ask prices at equilibrium? 4

18 What are bid and ask prices at equilibrium? First think about the sellers and buyers they choose optimally given whatever prices that traders have posted and the traders know this 4

19 What are bid and ask prices at equilibrium? First think about the sellers and buyers they choose optimally given whatever prices that traders have posted and the traders know this Then think about the problem faced by the traders, in deciding what prices to post in the first stage Each trader chooses a strategy that is a best response to both the strategies the sellers and buyers will use and the strategies the other traders will use 4

20 What are bid and ask prices at equilibrium? First think about the sellers and buyers they choose optimally given whatever prices that traders have posted and the traders know this Then think about the problem faced by the traders, in deciding what prices to post in the first stage Each trader chooses a strategy that is a best response to both the strategies the sellers and buyers will use and the strategies the other traders will use At equilibrium (called a subgame perfect Nash equilibrium), what are the bid and ask prices? Let s start with something simple 4

21 Monopoly sellers traders buyers S T B For any other bid and ask, the trader can slightly lower the bid or raise the ask to get a higher profit 5

22 Perfect competition sellers traders buyers T S B T2 T at equilibrium has a common bid and ask of If T performs the trade, T2 must have a bid and ask of since b <= and a >= But if a > b, then T could lower the bid or raise the ask and make a positive profit 6

23 Back to our eample sellers traders buyers S B T S2 B2 T2 S3 B3 7

24 Implicit perfect competition S T B T2 T3 B2 T4 S2 8

25 Modelling single-item second-price auctions T B w T2 B2 S T3 B3 y w > > y > z T4 B4 z 9

26 Modelling single-item second-price auctions T w B w T2 B2 S z y T3 y B3 y T4 z B4 z The second-price rule is not built into the formulation of the auction; it emerges naturally as an equilibrium. 2

27 Social welfare in trading networks Are the solutions at equilibrium socially optimal? maimizing the sum of payoffs of all the players Each good that moves from seller i to buyer j contribute vj - vi to the social welfare (bti - vi) + (atj - bti) + (vj - atj) = vj - vi As a result, more richly connected networks can potentially allow a flow of goods achieving a higher social welfare 2

28 Equilibria and social welfare: eample <= <= 2 <= y <= 2 <= z <= 3 S B S B T z T y S2 B2 2 S2 z y B2 2 T2 3 B3 3 T2 3 B3 3 S3 4 S3 4 social welfare (a) B4 4 social welfare +2+4 = = 9 (b) B4 4 22

29 In every trader network, there is always at least one equilibrium, and every equilibrium produces a flow of goods that is socially optimal.

30 Chapter..5