Marketmaking Middleman Pieter Gautier, Bo Hu and Makoto Watanabe VU - - PowerPoint PPT Presentation

Marketmaking Middleman

Pieter Gautier, Bo Hu and Makoto Watanabe

VU University Amsterdam, Tinbergen Institute

December 2014

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What is a marketmaking middleman?

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What is a marketmaking middleman?

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What is a marketmaking middleman?

Figure: the spectrum, intermediaries de…ned by functioning

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literature

Two-sided market literature: cross-group externality.

Rochet and Tirole (2003, 2006), Caillaud and Jullien (2003), Armstrong (2006).

Middlemen literature: high meeting rate, demand variety and the high inventory capacity.

Rubinstein and Wolinsky (1987), Shevichenko (2004) and Watanabe (2010, 2013).

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the model

a mass B of buyers, a mass S of sellers homogenous goods: unit demand/unit sell, common consumption value normalized to 1, marginal cost is c 2 [0, 1)

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the model

D(ecentralized) market

I random meeting, meeting rate λb, λs 2 (0, 1), Bλb = Sλs I bilateral bargaining, θ for buyers, 1 θ for sellers 7 / 44

the model

C(entralized) market

I marketmaker: pro…ts from the transaction and registration fees. I middleman: stock a mass inventory in a competitive wholesale market,

resells to buyers

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centralized market

a two-sided market

I cross-group externality I subsidize one type, the other type will come soon

price observed by all, buyers can’t coordinate their actions

I more buyers than the seller’s inventory, then some buyers will be

• rationed. (Coordination frictions)

suppose buyers visit the middleman xm, the rest visit the marketplace B xm

I on average each seller gets xs = Bx m

S

I middleman optimally adjusts inventory equal to xm, Pr [served] = 1 I sellers have limited 1 unit inventory, Pr [served] = ηs (xs) 1 (high

coordination frictions).

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centralized market

suppose buyers visit the middleman xm, the rest visit the marketplace B xm

I on average each seller gets xs = Bx m

S

I middleman optimally adjusts inventory equal to xm, Pr [served] = 1 I sellers have limited 1 unit inventory, Pr [served] = ηs (xs) 1 (high

coordination frictions).

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centralized market

suppose buyers visit the middleman xm, the rest visit the marketplace B xm

I on average each seller gets xs = Bx m

S

I middleman optimally adjusts inventory equal to xm, Pr [served] = 1 I sellers have limited 1 unit inventory, Pr [served] = ηs (xs) 1 (high

coordination frictions).

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urn-ball matching process

given fees F, suppose B0 buyers and S0 sellers participate in C market

1

all participating sellers post a price simultaneously

2

all participating buyers choose a seller to visit simultaneously

directed search friction

I in a large market, buyers can’t coordinate their visits I if more than one buyers visit a seller, then allocate the good randomly I the number of buyers visits a seller N follows a Poisson distribution

Pr (N = n) = ex s (xs)n n! where xs = B 0

S 0

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urn-ball matching process

for …nite integers B0 and S0, the number of buyers visits a seller, denoted as N, follows a Binomial distribution Pr (N = n) = B0 n 1 S0 n 1 1 S0 B 0n take B0, S0 ! ∞ and B0/S0 = xs, N follows a Poisson distribution lim

b,s!∞ b/s=x s

Pr (N = n) = ex s (xs)n n! lim

b,s!∞ b/s=x s

Pr (N > 0) = 1 ex s a deviating seller with an expected queue of buyers, denoted as x

I the seller receive at least one buyer with 1 ex I a visiting buyer is served with ηs (x) 1ex

x

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timing

a competitive wholesale market opens. The intermediary purchases a mass k inventory at wholesale price c two retail markets open to buyers, C market and D market the intermediary announces the fee structure for the marketplace

• bserving these fees and inventory, buyers and sellers simultaneously

decide which market to participate in

I singlehoming, agents visit only one market I multihoming, agents can visit both markets C & D market 14 / 44

single-homing

agents can participate in only one market. intermediary essentially becomes a monopolist

I once agents are in the centralized market, switching to the

decentralized market is not feasible,

all trade surplus can be extracted πSingle = (B xm) ηs (xs) (1 c) + xm (1 c) the middleman mode creates more transactions, thus is preferred xm = B

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multi-homing

participating all markets is allowed compete against the outside market, the intermediary has to lower price and fees, give buyers their outside option value V b

(D)

the pro…ts of intermediary πMulti = (B xm) ηs (xs) h 1 c V b

(D)

i + xm h 1 c V b

(D)

i , where V b

(D) = λbex s θ (1 c), xs = Bx m S

.

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multi-homing

πMulti = (B xm) ηs (xs) h 1 c V b

(D)

i + xm h 1 c V b

(D)

i , middleman mode is e¢cient in creating more transactions = ) a higher xm; buyer’s outside option value V b

(D) increases in xm =

) a lower xm;

I a higher xm leads to less transactions in the marketplace, more sellers

available in D market, buyers are easier to match in D market

the trade-o¤ between more transactions and more pro…ts per-trade pins down xm < B

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conclusion

non-exclusive intermediation service leads to a hybrid business mode

I an active marketplace to lower buyers’ outside option value

competition for intermediation can deteriorate the e¢ciency and welfare

I social planner’s viewpoint, pure middleman mode since it minimizes

market frictions

I in the presence of outside competition, hybrid mode is more pro…table

for the intermediary

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agenda remainder of talk

the marketmaking middleman under singlehoming (section 3)

I participate in only one market

the marketmaking middleman under multihoming (section 4)

I allowed to participate in both markets 19 / 44

singlehoming, the model

a one-period economy agents: a mass B of buyers, a mass S of sellers, one intermediary homogenous goods:

I unit demand/unit sell/a continuum of inventory I common consumption value normalized to 1; I marginal cost is c 2 [0, 1).

D(ecentralized) market

I random meeting, meeting rate λb, λs 2 (0, 1), Bλb = Sλs I bilateral bargaining, θ for buyers, 1 θ for sellers

C(entralized) market

I marketmaker: transaction fees f b, f s 2 [0, 1] , participation fees

gb, gs 2 [1, 1]

I middleman: pro…ts from inventory 20 / 44

singlehoming, timing of events

1

a competitive wholesale market opens, the intermediary stocks a continuum of inventory k 2 [0, B] from producers/sellers

2

retail markets open

1

the intermediary announces a set of fees F n f b, f s, gb, gso

2

• bserving F, buyers and sellers simultaneously decide which market to

attend (singlehoming)

3

trade occurs in each market (standard directed search game in C market)

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singlehoming, timing of events

1

a competitive wholesale market opens, the intermediary stocks a continuum of inventory k 2 [0, B] from producers/sellers

2

retail markets open

1

the intermediary announces a set of fees F n f b, f s, gb, gso

2

• bserving F, buyers and sellers simultaneously decide which market to

attend (singlehoming)

3

trade occurs in each market (standard directed search game in C market)

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singlehoming, the directed search equilibrium sellers

a deviating seller with an expected queue of buyers, denoted as x

I the seller receive at least one buyer with xηs (x) = 1 ex I a visiting buyer is served with ηs (x) 1ex

x

I the problem

max

p

V s = xηs (x) (p f s c) s.t.V b = ηs (x)

• 1 p f b

equilibrium values V b = ex s (1 f c) V s =

• 1 ex s xsex s

(1 f c) where xs = B 0

S 0 , f f b + f s

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singlehoming, the directed search equilibrium middleman

expected queue of buyers visiting middleman xm, S0xs + xm = B0 the serving probability of a buyer visiting the middleman ηm (xm) = min fk, xmg xm the middleman’s problem max

pm V m

= min fk, xmg (pm c) , s.t.V b (xm) = ηm (xm) (1 pm) . note the large market assumption does not hold V b (xm)

• ptimal xm k

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singlehoming, timing of events

1

a competitive wholesale market opens, the intermediary stocks a continuum of inventory k 2 [0, B] from producers/sellers

2

retail markets open

1

the intermediary announces a set of fees F n f b, f s, gb, gso

2

• bserving F, buyers and sellers simultaneously decide which

market to attend (singlehoming)

3

trade occurs in each market (standard directed search game in C market)

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singlehoming, buyers’ and sellers’ decision

agents hold a pessimistic expectation

I they would rather believe no one of the other type is in C market

resort to divide and conquer strategies

I DbCs

Db : gb λbθ (1 c) , Cs : V s gs 0.

I DsCb

Ds : gs λs(1 θ) (1 c) , Cb : V b gb 0.

I these induce all agents to join the C market, B0 = B and S0 = S. 26 / 44

singlehoming, timing of events

1

a competitive wholesale market opens, the intermediary stocks a continuum of inventory k 2 [0, B] from producers/sellers

2

retail markets open

1

the intermediary accounces a set of fees F n f b, f s, gb, gso

2

• bserving F, buyers and sellers simultaneously decide which market to

attend (singlehoming)

3

trade occurs in each market (standard directed search game in C market)

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singlehoming, the intermediary’s problem

max

f ,k Π

= Bgb + Sgs + Tf + xmpm kc, s.t. DbCs or DsCb where T S(1 ex s ) is the expected trade of sellers. gb and gs are the determined by DC restrictions f is set to be maximum 1 c to extract all surplus from participating agents the trade-o¤ on inventory

I recall the optimal xm k, but k > xm only incurs more cost, k = xm I set f = 1 c and ignore the Bgb + Sgs (constant)

max

k

S

• 1 e Bk

S

• (1 c)

| {z }

marketmaker (k)

+ k (1 c) | {z }

middleman (+k)

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singlehoming, the intermediary’s problem

Proposition

The pro…t-maximizing intermediary divides (subsidies) buyers when θ < 1 θ and divides sellers when θ > 1 θ, the optimal transaction fee is f = f b + f s = 1 c, optimal inventory is k = B. The corresponding pro…ts are ΠDbCs = B

• 1 λbθ
• (1 c) ,

ΠDsCb = B

• 1 λb (1 θ)
• (1 c) .

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agenda remainder of talk

the marketmaking middleman under singlehoming (section 3)

I participate in only one market

the marketmaking middleman under multihoming (section 4)

I allowed to participate in both markets 30 / 44

multihoming, timing of events

1

a competitive wholesale market opens, the intermediary stocks k 2 [0, B] from producers/sellers as inventory

2

retail markets open

1

the intermediary announces a set of fees F n f , gb, gso

2

• bserving F, buyers and sellers simultaneously decide which one or two

markets to attend (multihoming), make the decision on where to trade after outcomes of two markets are revealed.

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multihoming

matching outcomes

I if 1 match from C or D market, chooses the market that is matched I if 2 matches from both markets, chooses the preferred one by

comparing prices.

F price in D market is …xed, θ (1 c) or (1 θ) (1 c) F price in C market depends on f , as will detailed later F this means the intermediary can set f to change agents’ choice

depending on the value of f , there are four possible cases

I BCSC: all types choose C market I BCSD: buyers choose C market, sellers choose D market I BDSC: buyers choose D market and sellers choose C market I BDSD: all agents prefer D market 32 / 44

multihoming, BCSC incentive constraints

buyers prefer middleman to D market 1 pm (1 xsηs (xs)) θ (1 c) buyers prefer sellers in C market to D market 1 ps f (1 xsηs (xs)) θ (1 c) sellers prefer C market to D market ps c ρ (xs, xm) (1 θ) (1 c) where ρ (xs, xm) 1 x m

B 0 ηm (xm) S 0x s B 0 ηs (xs) .

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multihoming, BCSC sellers

Consider a deviating seller characterized by an expected queue x max

p

V s = xηs (x) (p c) + (1 xηs (x)) λsρ (xs, xm) (1 θ) (1 c) subject to V b = ηs (x) (1 p f ) + (1 ηs (x)) λb (1 xsηs(xs)) θ (1 c) where ρ (xs, xm) 1 x m

B 0 ηm (xm) S 0x s B 0 ηs (xs) .

Solution

Solve this problem gives ps, V b and V s.

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multihoming, BCSC the middleman

The middleman is characterized by an expected queue xm, the optimal price is obtained by solving max

pm V m = min fxm, kg (pm c)

subject to V b = ηm (xm) (1 pm) + (1 ηm (xm)) λb (1 xsηs (xs)) θ (1 c) To rule out increasing demand function, we need λb to be small enough.

Solution

When λb is small, solve this problem gives the optimal queue xm k and the equilibrium pm.

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multihoming, divide and conquer conditions

divide and conquer conditions are modi…ed as

I DbCs

Db: gb 0, Cs: V s gs λsρ (xs, xm) (1 θ) (1 c) .

I DsCb

Ds: gs 0, Cb: V b gb λbex s θ (1 c) .

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multihoming, the intermediary’s problem

The intermediary’s problem max

f ,k Π = Bgb + Sgs + Tf + xmpm kc,

subject to DbCs or DsCb constraint and BCSC constraints. It is optimal to set maximum f , so the restriction on f must be binding.

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multihoming, the intermediary’s problem

illustrate choice of k by DsCb, impose c = 0 (for shorter expression) max

k

Π = k

• 1 λbex s θ
• + S
• 1 ex s

1 ps λbex s θ

• .

∂Π ∂k =

• 1 λbex s θ
• + ∂S
• 1 ex s

∂k

• 1 ps λbex s θ
• |

{z }

transaction e¤ect: positive

• h

k + S

• 1 ex s i ∂ex s λbθ

∂k | {z }

competition e¤ect: negative

+S

• 1 ex s ∂ (1 ps)

∂k | {z }

externality e¤ect: positive

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multihoming, the intermediary’s problem

∂Π ∂k jk=B = λsθ (1 c) < 0 implies k < B. Thus,

Π (k) > Π (k = B) = B

• 1 λbθ
• Same analysis applies to DbCs strategy, but divide buyers means giving up

the surplus of buyers, less pro…table.

Proposition

In the case of BCSC, when λ is small, DsCb strategy is optimal. The

• ptimal transaction fee is its upper bound and the optimal inventory is set

to be strictly smaller than B, 0 k < B. The corresponding pro…t is higher than B

• 1 λbθ
• (1 c).

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multihoming, the intermediary’s problem

Proposition

With λ is small, the DsCb strategy involving BCSC directed search equilibria constitutes the optimal strategy. Strategies involving BDSC and BDSD equilibria, the max amount of transactions B

• 1 λb

, and the max pro…t is B

• 1 λb

(1 c).. Strategies involving BCSD equilibria, k = B, and the max pro…t is B

• 1 λbθ
• (1 c).

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Conclusion

Without outside competition (singlehoming), middleman mode (inventory trade) is optimal. With outside competition (multihoming), using a hybrid mode can be pro…table for the intermediary. The intermediary is a marketmaking middleman. The marketmaking middleman reduces its inventory from the e¢cient

• level. Hence, competition for intermediation can deteriorate the

e¢ciency and welfare.

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Appendix on multihoming

BCSC incentive constraints

BC: 1 pm (1 xsηs (xs)) θ (1 c) , (1) 1 ps f (1 xsηs (xs)) θ (1 c) , (2) SC: ps c ρ (xs, xm) (1 θ) (1 c) , (3) indi¤erent between visiting sellers and the middleman V b

(m)

= V s

(m)

1 pm = ηs (1 ps f ) + (1 ηs) λbexθ (1 c)

• nly (1) and (3) can be binding.

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Appendix on multihoming

the intermediary’s problem

Corollary

When buyers’ D market value is big enough, satis…es

• 4λb + 3
• θ > 5 and
• 3λb 1
• θ > 1, k = 0 is optimal for some small but strictly positive

buyer-seller ratio B/S > 0.

Corollary

An increase in buyer’s bargaining power θ in the decentralized market leads to a lower inventory level k.

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Appendix on multihoming

BCSC incentive constraints

With ps (f ) and pm (f ), BCSC incentive constraints become 1 f c h λsρ (xs, xm) (1 θ) +

• 1 λb

θ + λbexθ i (1 c) , 1 f c 1 ex λsxex 1 ex xex ρ (xs, xm) (1 θ) + λbexθ

• (1 c) .

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