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Even if large commercial margins are unavoidable in the absence of lump-sum taxation, their

existence is important for policy design. For example, large commercial margins have a profound

eﬀect on the welfare cost of sales taxes, because a zero tax rate on sales is already a large departure

from the Þrst best, so the standard result that small taxes cause negligible dead-weight-losses does

not apply. Moreover, because of standard tax equivalence results, the welfare costs of income taxes

must also be much larger when commercial margins are taken into account.

For simplicity, we have abstracted from the existence of money by assuming that payments can

be made through a central clearing-house. In the absence of this centralized system of payments,

money is a useful device that facilitates trade as in Kiyotaki and Wright (1989 and 1993). Faig

(2001) introduces money in a simpler version of the present model where sellers are constrained

by the set-up to make oﬀers that consist of a single quantity-payment pair (q, z). The main com-

plication of introducing money in the present set-up is that in equilibrium when buyers are lucky

to Þnd a good for which they have a high valuation they would like to spend more money than

they carry. Because this is not possible, they are liquidity constrained. Moreover, these liquidity

constraints aﬀect equilibrium price schedules. Despite this complication, the model remains an-

alytically tractable, but the algebraic expressions are longer and harder to interpret than in the

present contribution. For this reason, we plan to study the monetary version of the present model

in a separate paper.

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APPENDIX

Derivation of (5)

Let q

be the quantity purchased and ε

be the preference shock in eachmatch j betweena buyer

of the household and a seller, where j = 1,2,... Assume the following Dixit-Stiglitz aggregator:





lim

J0→∞

j=1

1−σ





1−σ

(68)

Here, J and J

denote the number of matches and the number of household members respectively.

The goods acquired have diﬀerent valuations for the household. For a given quantity purchased

, the household gets maximum utility when the realization of the shock in a match is ε

=1,

and this utility declines linearly to 0 as ε

decreases to 0. Because there is a countable number of

potential purchases and a continuum of goods, with probability one an additional purchase brings

anew good to the set of goods consumed by the household.

Since the number of household members is large, the measure of matches between the buyers

of the household and a potential seller is lim

J0→∞

J/J

=bm

(θ). Hence,





(θ) lim

J→∞

j=1

1−σ





1−σ

(69)

Moreover, the stochastic variables ε

are uniformly distributed on the interval [0,1] and are inde-

pendent across meetings. Let q

denote the quantity purchased by a buyer when the realization of

the preference shock is ε. The Law of Large Numbers implies the formula in (5).

Proof of Proposition 1

Let the indirect utility of a type-ε buyer be deÞned as

≡U

;ε).

(70)

Using v

,the incentive compatibility constraint (18) can be restated with the help of the following

standard result (see Mas-Colell, Winston and Green, 1995, Proposition 23.D.2):

Lemma A direct revelation mechanism satisÞes the incentive compatibility constraint (18) if and

only if q

is non-decreasing in ε and the indirect utility function satisÞes

∂

∂x

;x)dx =

ψq

1−σ

dx, for all ε ∈ [0,1].

(71)

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Using this lemma and constraints (19) and (20) together with the deÞnitions (14), (16), and

(70), the incentive eﬃcient direct revelation mechanisms can be characterized as the solution to the

following program:

max

}

ε∈[0,1]

(1− ω)v

+ω

εψq

1−σ

−µq

−v

¶¸

dε

(72)

subject to

úv

ψq

1−σ

(73)

≥0,

(74)

≥0,

(75)

≡

εψq

1−σ

1− σ

−v

−µq

≥0,

(76)

is non-decreasing,

(77)

=0, and v

=0.

(78)

Here,π

denotes the seller’s surplus uponmeeting a buyer of type ε. The payment inthe transaction

satisÞes:

λz

εψq

1−σ

−v

=π

+µq

(79)

This program can be solved with a standard application of the Pontryagin’s Maximum Principle.

We organize the solution to the program in three steps:

(a) Constraints (73) to (78) imply that there is a γ ∈ [0,1) such that the solution {q

}

ε∈[0,1]

program (72) obeys: q

=0 and v

=0 for ε ∈ [0,γ], and q

>0 and v

>0 for ε > γ.

Constraints (77) and (78) immediately imply that there is γ ∈ [0,1] such that q

= 0 for

ε∈ [0,γ], and q

>0 for ε > γ. When q

=0, constraints (75) and (76) imply that v

=0. With

these results, constraint (73) implies that q

>0 if and only if v

>0. Finally, if γ were 1, the

optimized value of (72) would be zero, which cannot be a solution to the maximization program

because there are many feasible direct revelation mechanisms that achieve a positive value. For

example, q

=0 if ε < 0.5, and q

ψ(2ε−1)

and v

1−σ

[ψ(2ε − 1)]

σ−1

otherwise.

39úv

denotes the derivative of v with respect to ε evaluated at ε.

40Thisistheoptimaldirectmechanism

for ω = 1.

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(b) For ω ∈ (0.5,1], the solution to program (72) subject to (73) to (78), is the following:











for 0 ≤ ε < γ

ε−γ

1−γ

for γ ≤ ε ≤ 1

and

(80)











for 0 ≤ ε < γ

σµ(1−γ)

1−σ

ε−γ

1−γ

for γ ≤ ε ≤ 1

(81)

where

γ=

2ω − 1

3ω − 1

(82)

Let ζ

denote the co-state variable associated with the diﬀerential equation (73). The current-

value Hamiltonian of the program is:

H(q

,ζ

,ε) = (1− ω)v

+ω

εψq

1−σ

1− σ

−µq

−v

+ζ

ψq

1−σ

1− σ

(83)

The Þrst-order necessary condition with respect to the control variable q

is (H

=0) :

ψεq

−σ

−µ

+ζ

ψq

−σ

=0.

(84)

The co-state variable must obey

=−H

=−(1 −2ω).

(85)

Finally, the transversality implies

=0.

(86)

The value of the co-state variable ζ

can be solved for explicitly using conditions (85) and (86):

=(2ω− 1)(ε − 1).

(87)

Substituting (87) into (84), solving for q

,and using (82), we obtain

ε−γ

1− γ

(88)

Integrating (73) from γ to ε and noting that v

=0, we obtain

1− σ

(1 −γ)µq

(89)

41Thetransversalityconditionisζ

=0. However, v

>0 because of step (a).

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Finally, using the deÞnition of π

in (76), we obtain

=γ

µq

1−σ

σ+

1−ε

ε−γ

(90)

So far, we have disregarded constraints (74) to (78). However, for ε ≥ γ, the values of q

and π

in (88), (89), and (90) are well deÞned and satisfy these constraints. Moreover, at ε = γ,

=0. Hence, (a) implies q

=π

=0 for ε ≤ γ. The Hamiltonian is strictly concave

with respect to q

,and when the Hamiltonian is evaluated at the optimal choice of q

it is concave

with respect to v

.Therefore, (80) to (82) characterize the unique maximum of the program. ¤

and

(91)

1− σ

µq

(92)

When ω = 0.5, the variable v

cancels in the objective (72) so the problem becomes separable

across types. Ignoring constraints (76) and (77), the Þrst-order conditions of the problem yield

(91). Integrating (73) from 0 to ε and noting v

=0, we obtain (92). Using the deÞnition in (76),

we obtain π

=0 for all ε ∈ [0,1]. The values of q

and v

in (91) and (92) satify constraints (76)

and (77), so they solve program (72) subject to (73) to (78). Furthermore, the solution for ω = 0.5

maximizes the expected payoﬀ of the buyer subject to a zero expected payoﬀ for the seller. A

fortiori, given constraint (76), the same solution must apply for ω ∈ [0,0.5), which assigns a lower

weight to the seller in the maximized welfare function (72). ¤

Finally, Proposition 1 results from combining steps (b) and (c) together with (79). ¥

Proof of Proposition 2

Abuyer facing a price schedule γ purchases a quantity that satisÞes the following specialization

of (10)

εψq

−σ

=γψq

−σ

+(1 −γ) µ.

(93)

Solving this equation for q

,we obtain (21). Using (24), the implied payments are given by (22).

Finally, γ taking values in the interval [0,0.5] spans all values of γ attained in (23) for ω ∈ [0,1].

Thus, Proposition 2 follows. ¥

Proof of Proposition 3

Giventhat the equilibriumvariables are recursively determinedby the set of equations described

in the main text. Existence and uniqueness is implied if equations (33) and (38) have a unique

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admissible solution. The right-hand side of (33) is a continuous and decreasing function of θ,

which image spans the interval (0,2]. The left-hand side of (33) is a continuous and non-decreasing

function of θ, which is bounded away from 0 and 1. Therefore, equation (33) has a unique solution.

The left-hand side of (38) is a decreasing function of n that maps [0,1] onto [0,∞), while the

right-hand side is positive and independent of n. Hence, equation (38) has a unique solution for n

in the interval (0,1). ¥

Proof of Proposition 4

Equation (47) together with the deÞnitions of θ, m

,and η in (1), (2), and (31), implies (48)

and (51). The labor allocation constraint together with (48) and (51) yields (50). Using (44) to

solve the integrals in (46) and (43), we obtain

(n) = M

(b,s)

1− σ2

, and

(94)

f(n) = M(b,s)

1+ σ

(95)

The deÞnitions (1), (2), and (31) imply

(b,s)

M(b,s)

1− η(θ)

(96)

Equation (49) results from combining (50) with (94) to (96). Finally, combining (44) and (95), we

obtain (52). ¥

Proof of Proposition 5

The equations that determine θ and n in a directed search equilibrium, (33) and (38)-(30), and

the Þrst best allocation, (48) and (49), can be written as follows:

+θ

= 1− η(θ), and

(97)

(1− n)f

(n)

f(n)

1−σ

(98)

where a

=1 in the Þrst best, and a

=0.5, a

=(1+ θ)/(0.5 + θ) > 1 in equilibrium. The

Implicit Function Theorem applied to (97) and (98) yields dθ/da

<0 and dn/da

>0. Hence, we

obtain (54) and (53).

If f has constant elasticity α, equations (37)-(38) and (49)-(50) imply

σ[1 −η (θ)]

α(1− σ)

(99)

for both the Þrst best and the equilibrium. Therefore, (55) follows from (99), (53), and η being a

non-increasing function of θ.

The resource constraint (7) together with f being constant returns to scale implies that

∗

n∗

(θ

∗

)

ms(θ)

(100)

Therefore, (56) follows from (55), (53), and m

being increasing.

Proof of Proposition 6

(a) When buyer types are public information the directed search equilibrium and the Þrst best allo-

cations coincide.

As in Section 3, all bilateral trades must be pairwise eﬃcient in equilibrium and the marginal

rates of substitution between θ and ξ of buyers and sellers must coincide. With full information,

the eﬃcient quantities q

are calculated by maximizing the trade surplus, U

=argmax

εψq

1−σ

1− σ

−µq

(101)

The solution to (101) is:

ψε

(102)

The expected trade surplus is:

εψq

1−σ

1− σ

−µq

dε =

1− σ2

σ−1

(103)

When the market tightness is θ and buyers receive a fraction ξ of the trade surplus, the expected

utilities of buyers and sellers are:

(θ,ξ) = ξm

(θ)

1− σ2

σ−1

, and

(104)

(θ,ξ) = (1− ξ)m

(θ)

1− σ

σ−1

(105)

In a directed search equilibrium, the marginal rates of substitution between θ and ξ of buyers and

sellers must coincide. Diﬀerentiating (104) and (105), this implies:

ξ= η(θ).

(106)

Households allocate b and s so V

(θ,ξ) = V

(θ,ξ). Using (104) and (105), this equality yields

(48). Also, households allocate s and n so V

(θ,α) = µf

(n). Using (105) and (106) this equality

implies:

µf

(n) = m

(θ)[1 −η(θ)]

1−σ2

σ−1

(107)

Using (102) to calculate the average sales, the resource constraint becomes

f(n) = sm

(θ)

1+ σ

(108)

Combining (107) and (108), and using (1), (48), and (9), we obtain (49) and (50). From (107),

f(n)

sms(θ)

1+ σ

(109)

which combined with (2) and (102) gives (52). This completes the proof of (a).

(b) If buyer types are private information but sellers charge a lump-sum fee to prospective buyers

prior to the realization of ε, then the directed search equilibrium and the Þrst best allocations

coincide.

Let p be the fee a seller charges to prospective buyers. After the fee p has been paid and ε

is realized, the trading game between a buyer and a seller is identical to the one in Section 3.

Hence, the payment schedule net of the fee p that implements incentive eﬃcient direct revelation

mechanisms has still the functional form (24) and the quantities purchased by the buyer are given

by (21). The expected utilities of the traders in a market with congestion θ, price schedule γ, and

fee p are

(θ,γ,p) = m

(θ)

1− σ2

σ−1

−λp

, and

(110)

(θ,γ,p) = m

(θ)

2σ

1−σ2

σ−1

γ(1− γ) + λp

(111)

In a directed search equilibrium, the marginal rates of substitution of θ for γ, and of γ for p must

be equal for buyers and sellers, so

η(θ) −1

η(θ)

1− 2γ

2σ2

1−σ2

σ−1

γ(1 −γ) +λp

σ2

1−σ2

σ−1

γ2 − λp

;

(112)

γ = 1.

(113)

Substituting (113) in (112) and solving for p,

1−σ2

σ−1

[1− η(θ)].

(114)

Note that this fee implies V

(θ,γ,p) > 0, so buyers are willing to pay the fee.

Households allocate b and s so V

(θ,γ,p) = V

(θ,γ,p). Using (113) and (114), this equality

yields (48). Also, households allocate s and n so V

(θ,α) = µf

(n). Using (113) and (114), this

equality yields (107). To show that (49), (50) and (52) hold, we use the exact same steps used in

the proof of (a). ¥

Proof of Proposition 7

Atype-ε buyer chooses

=argmax

εψq

1−σ

1− σ

−λZ(q)

ψε

µη

(115)

Hence, the return of a seller when matched with a type-ε buyer is

λTZ(q

)− µq

=(Tη − 1)µq

σ(1− η)

1− σ

ψε

µη

µ,

(116)

where η = η(θ) and θ is the Þrst-best level of congestion given by (48). By (48), η ≤ 1. Thus, the

seller’s individual rationality constraint is satisÞed.

When the market tightness is θ, the expected utilities of the traders are:

(θ) = m

(θ)η

1−σ2

µη

µ;

(117)

(θ) = m

(θ)(1 −η)

1− σ2

µη

µ.

(118)

The household chooses its labor allocation so that V

(θ) = V

(θ) = µf

(n). The Þrst equality

implies that θ satisÞes (48) so η = η(θ). The second one implies

(n) = m

(θ)(1 − η)

1−σ2

µη

(119)

Using (115) to calculate the average sales, the resource constraint becomes

f(n) = sm

(θ)

1+σ

µη

(120)

Combining (119) and (120), and using (1), (48), and (9), we obtain (49) and (50). From (120),

f(n)

(θ)

1+ σ

(121)

which combined with (2) and (115) gives (52). ¥

Proof of Proposition 8

The direct revelation mechanism chosen by the planner must be incentive eﬃcient, so we can

restrict our search to price schedules in (24) for some unknown parameter γ. Substituting (21) into

(5), solving for the integral, and using (2) we obtain:

(θ)σ

1+ σ

(1− γ)(1+ σγ)

1−σ

(122)

Denoting f(n) = a+wn, and using (6), (7), (32), (35), (37), and (38), equation (122) simpliÞes to:

2wσ

1+σ

1−σ

(1 −σ)(a +w)

1−σ

[γ(1− γ)m

(θ)]

1−σ

(123)

Since (32) holds, the maximization of c implies:

(2γ − 1)m

(θ) +γ(1 − γ)(m

)

(θ)

2γ

=0.

(124)

Given (31) and (32), (124) implies (30). ¥

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