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Federal Reserve Bank of Chicago
More on Middlemen: Equilibrium Entry
and Efficiency in Intermediated Markets
Ed Nosal, Yuet-Yee Wong, and Randall Wright
November 2014
WP 2014-18
More on Middlemen: Equilibrium Entry and Efficiency in
Intermediated Markets*
Ed Nosal, Federal Reserve Bank of Chicago
Yuet-Yee Wong, Binghamton University
Randall Wright, University of Wisconsin-Madison, Federal Rerserve Bank of
Minneapolis, FRB Chicago and NBER
JEL Codes: G24, D83
Keywords: Middlemen, Intermediation, Search, Bargaining, Entry
*
For their input on work related to this paper we thank Cyril Monnet, Guido Menzio,
Benoit Julien, Adrian Masters, Dale Mortensen, Asher Wolinsky, Guillaume Rocheteau,
Yiting Li and participants at several presentations. We espcially thank Luigi Paciello for an
excellent discussion. Wright thanks the National Science Foundation and the Ray Zemon
Chair in Liquid Assets in the Wisconsin School of Business. The usual disclaimers apply.
1
Abstract
This paper generalizes Rubinstein and Wolinsky’s model of middlemen (intermediation) by incorporating production and search costs, plus more general
matching and bargaining. This allows us to study many new issues, including
entry, efficiency and dynamics. In the benchmark model, equilibrium exists
uniquely, and involves production and intermediation for some parameters
but not others. Sometimes intermediation is essential: the market operates
iff middlemen are active. If bargaining powers are set correctly equilibrium is
efficient; if not there can be too much or too little economic activity. This is
novel, compared to the original Rubinstein-Wolinsky model, where equilibrium
is always efficient.
2
1
Introduction
This paper continues the development of theories of middlemen, or intermediaries, going back to Rubinstein and Wolinsky (1987) — hereafter RW. As
they said at the time, “Despite the important role played by intermediation
in most markets, it is largely ignored by the standard theoretical literature.
This is because a study of intermediation requires a basic model that describes
explicitly the trade frictions that give rise to the function of intermediation.
But this is missing from the standard market models, where the actual process
of trading is left unmodeled.” Since then, many people have helped to rectify
the situation, contributing to the discussion with various models, but often
using search-and-bargaining theory.1 What makes the models more than a
relabeling of, say, textbook search models of labor, goods, or marriage is that
they involve three-sided markets — in addition to firms and workers, buyers and
sellers, or men and women, they have third parties potentially intermediating
between the other two.2
We extend RW on several dimensions, not merely for the sake of generality, but because this allows us to address new substantive issues, including
efficiency. Our extensions consist of the following: RW have an endowment
economy, while we have production; they do not have search costs, while we do;
they assume a special matching process with equal numbers of buyers and sellers, while we use a more general population and matching process; they only
1
We do not review the literature, since that was recently done in Wright and Wong
(2014). Surveys by Williamson and Wright (2010) and Nosal and Rocheteau (2011) provide
more discussion of work on financial intermediation, in particular, with an emphasis on
search. We mention below other papers when they are directly related.
2
For labor-market models, see Mortensen and Pissarides (1994) or Pissarides (2000); for
goods-market models, see Osborne and Rubinstein (1990), Shi (1995) or Trejos and Wright
(1995); for marriage-market models see Burdett and Coles (1997,1999) or Shimer and Smith
(2000).
3
consider the case where buyers and sellers exit the market after they trade,
while we allow them to potentially stay in; and they only consider symmetric bargaining, while we allow general bargaining powers, which is especially
important for understanding efficiency. Moreover, by taking advantage of advances in search theory over the past 25 years, we provide a more parsimonious
presentation of the generalized model, and of RW as a special case.
In terms of results, we first characterize the set of steady-state equilibria for
a benchmark model, verifying existence and generic uniqueness.3 Although in
principle there are many candidate equilibria, there are basically three distinct
outcomes: (i) degenerate equilibria where the market does not open (producers are inactive, middlemen are irrelevant); (ii) equilibria with direct trade
between producers and consumers but no intermediation (producers are active, middlemen are not); and (iii) equilibria with direct and intermediated
trade (both are active). For some parameters, only a fraction of producers enter the market, as in macro-labor models along the lines of Pissarides (2000).
In RW, all producers are always on the market. Whether middlemen, as well
as producers, are active depends on parameters, including production and
search costs, bargaining power, the matching process, and the probabilities
that agents exit the market after trade. In RW, middlemen are active iff they
meet consumers faster than producers meet consumers.
We then solve for efficient outcomes. Equilibria are not efficient in general, due to holdup problems, since some costs are sunk when trades occur.
Suppose, e.g., all the bargaining power goes to the agent that passes the good
on to the next agent — i.e., the producer when he trades with a consumer or
3
While uniquenss obtains in our benchmark model, the extension that has producers and
consumers exiting the market after trade with some probability can have multiple equilibria
for certain parameters, although not for those in the original RW specification.
4
a middleman, and the middleman when he trades with a consumer. Then
there is generally too much entry by producers for reasons discussed below.
For arbitrary bargaining powers, there may be too much or too little entry
by producers, and there can be too little but not too much intermediation.4
However, we show that if bargaining powers are set appropriately, related to
Mortensen (1982) and Hosios (1990), equilibrium is efficient. These results
are novel compared to the RW model, with no production or search costs and
symmetric bargaining, where equilibrium is always efficient.
The rest of the paper involves laying out the details and proving the claims.
Section 2 presents the baseline model. Section 3 describes equilibria. Section
4 contains a discussion of extensions and alternative formulations. Section 5
compares efficient and equilibrium outcomes. Section 6 generalizes the model
by allowing producers and consumers to exit the market probabilistically. Section 7 concludes.
2
The Model
There are three types of agents, labeled , and for producers, middlemen
and consumers, who live forever in continuous time. The measure of type is
with + + = 1. There is an indivisible good that only values,
4
Unlike the caveat about uniquness in fn. 3, the result that there can be too little but not
too much intermediation survives in the extension where agents probabilistically exit the
market after trade. Still, the result is somewhat model dependent. Li (1998) can get too
much or too little intermediation depending on bargaining powers. Her model is different in
that agents choose to be either middlemen or producers, so too few of the latter necessarily
means too many of the former. Shevchenko (2004) can also get too much or too little
intermediation. His model is different in that he allows middlemen to hold multiple units
in inventory to study efficiency on the intensive as well as the extensive margin. See also
Johri and Leach (2002). One more example is given by Masters (2007,2008). He generally
gets too much intermediation, because middlemen in his setup perform no socially useful
function — they simply buy low and sell high to consume without producing.
5
enjoying utility from consuming one unit. Good is storable, but only one
unit at a time. It is produced by , who has an entry cost to participate in
the market and another cost to generate a unit of for a trading partner.
One interpretation is that is a cost of raw materials and is a cost to finalize
the output. We assume + , as otherwise the market shuts down. If
pays , we say that he is in the market, looking to trade, in which case he also
has a search or storage cost . While cannot produce , and has no desire
to consume it, he may acquire it from with the intent of retrading it to ,
in which case is in the market, with a search or storage cost . Notice
there are no costs for agents looking to acquire , only for those holding .
Also, there are no production or improvement costs for , although one can
add these, as may be appropriate in applications such as flipping real estate.
There is another good that is perfectly divisible but nonstorable. Any
agent can produce at unit utility cost. All agents derive utility () from
consuming , in general, although here we assume () = (how this matters
is discussed in Wright and Wong 2014). Therefore, in this exercise, as in
RW, one can say there is transferrable utility. Where we generalize their
model is that they have = = = = 0. Adding production and
search costs seems obviously relevant for many substantive applications. It is
also interesting to have these costs, so that there are nontrivial decisions by
producers and middlemen to participate in the market, because this allows us
to analyze when there is too little or too much activity in equilibrium. Also,
while the only role for intermediation in RW is that might be able to meet
faster than can meet , in this model differences in search costs and other
parameters also matter.
The timing is important. For , costs and are sunk when he meets
6
a potential trading partner, while is paid only when delivers the goods.
For , if he acquires from , he generally must transfer some to , and
that as well as are sunk when meets . Sunk costs are interesting in
search-and-bargaining models, generally, because they lead to holdup problems
that can cause market failures. These problems are often described as the
result of imperfect contracting or commitment, and that is accurate here, too.
However, compared to models where such imperfections are imposed in an ad
hoc fashion, in search theory it is obviously natural to say that it is hard to
contract with someone, or commit something to someone, before you meet
them. Therefore these are natural models in which to study the efficiency of
entry or participation.5
Agents meet according to a bilateral random-matching process, where is
the Poisson arrival rate at which type meets . This implies three identities:
= , = and =
The first says the measure of type meeting type is the same as the
measure of type meeting type , and similarly for the others. The vector
= ( ) has 6 elements, but the above identities imply =
, which means that one can choose only 5 independently. In the
background one can imagine a population n = ( ) determining the
arrival rates, but we follow RW and take to be exogenous, since there exists
an n consistent with any such that = . However,
5
RW also have a holdup problem, since whatever gives to is sunk when meets
. This does not affect efficiency in RW, but can affect the distrbution of payoffs. They
discuss a “consignment” arrangement, whereby makes a transfer to only after trading
with , so it is not sunk when bargaining with . Of course, this may or not be feasible,
depending on the physical environment — e.g., it will not work if and cannot reconvene
after trading, or if cannot commit to transfers.
7
we depart from RW by not focusing exclusively on markets where = ,
= and = , the first of which implies = (i.e., the rather
special case of equal measures of producers and consumers).
As in RW, middlemen are recycled after each trade: any type agent
that chooses to enter the market at = 0 stays in the market forever. For
and , RW assume that they exit after one trade, to be cloned by replicas of
themselves to maintain a stationary environment. While it does not matter
much whether and are cloned or recycled in the simple baseline RW model,
it turns out that in our generalized version the latter is much more tractable.
To nest both cases, where they are cloned and where they are recycled, we
assume that after each trade type = agents continue in the market with
probability and exit with probability 1 − . However, for now, as a more
tractable benchmark, we assume = = 1, and revisit the general case in
Section 6.
Denote the surplus in an match by Σ . In any match, there can
be trade if Σ ≥ 0, and must be trade if Σ 0 (see Lemma 2 below). In
matches, when they trade, gives to for some amount of , say .
Similarly, in matches, if has gives he it to for . In matches,
cannot give to if already has , since it is only storable one unit at
a time. If does not have , may or may not give it to him; if he does
then gives in return. Our convention for notation is that in Σ the
subscripts indicate that goes from to , and in they indicate that goes
from to . For future reference let y = ( ). Bargaining determines
the terms of trade: splits the surplus, where is the share (bargaining
power) of , and = 1 − . This outcome follows from generalized of Nash
(1950) bargaining, Kalai (1977) bargaining, and various other solutions when
8
() = , although they can give different outcomes when 00 () 0.6
We now analyze behavior. For it is trivial, since he pays no cost to
participate and trades whenever he can (see Lemma 3). The choices for are
, the probability he enters the market, and , the probability he searches
conditional on holding ; the choices for are , the probability he tries
to enter the market by trading for , and , the probability he searches
conditional on having .7 Let p = ( ) and, since the environment is
stationary, assume agents make once-and-for-all decisions at = 0. Thus,
enters with probability , and if he does then after trading he pays the cost
to remain in the market, while if he does not then he is out forever. Similarly,
decides at = 0 with probability to trade for , and if he does he stays
forever, while if he does not then he is out forever. In other words, agents
randomize once at = 0, and not in each meeting.
As defined below, an equilibrium determines p, and hence determines when
the market is open (some producers are active), and whether there is intermediation (some middlemen are active). We begin with a few preliminary results.
The first says that and would only pay to acquire if they strictly prefer
6
Again see Wright and Wong (2014). That paper also shows that there are belief-based
(bubble) equilibria in a related model with 00 0, something that cannot happen with
00 = 0 (see Section 4). Another point in that paper is that one ought to resist the temptation to call the price and say buys from , since it makes at least as much sense
to call 1 the price and say sells to . The argument is that we can only really
say who is the buyer and seller in monetary exchange, and one should not call money,
even though people often do in related models — i.e., they use the word money as a sloppy
synonym for transferable utility. On reflection, we think that it makes at least as much if
not more sense to call a commodity money: it is a storable asset that and use as a
medium of exchange to acquire . Under this interpretation, one can say that Rubinstein
and Wolinsky (1987) provide a model of commodity money as a by-product of their analysis
of middlemen, the same way that Kiyotaki and Wright (1989) provide a model of middlemen
as a by-product of commodity money.
7
One might anticipate that there are no equilibria where 0 and = 0, or 0 and
= 0. That is true, but it is still important to have and in the strategy profile,
since an agent who is not willing to pay to get may or may not try to trade it if, off the
equilibrium path, he happened to have it.
9
to search. This should be obvious for , since it is costly to enter the market,
and hence he will not do so unless he plans to search. Similarly, would
never acquire unless he plans to search for a trading partner, unless he can
get for free, but will not give it away for free. This constitutes a proof
of Lemma 1. The second result says that trade is mutually agreeable in an
match whenever the total surplus is positive. The third result says that
always trades with anyone who has .
Lemma 1 If 0 then = 1. If 0 and 0 then = 1.
Lemma 2 If 0 then (strictly) wants to trade with iff (strictly)
wants to trade with iff Σ is (strictly) positive.
Lemma 3 If 0 then always trades with , and always trades with
when has .
Given that wants to trade with whenever wants to trade with
, we can delegate the decision to And always trades with . Hence,
once is in the market he trades with anyone that is willing and able. To
determine who is willing and able, let be the fraction of holding . Then
in any match the probability of trade is − , since a fraction of type
decided at = 0 to accept , but a fraction already have it. The law of
motion for is
̇ = ( − ) −
In the first term, there are − type that accept but do not currently
have , they contact at rate , and the probability is that is on
the market and looking to trade, assuming random matching in the sense that
can meet even if the latter is not actively on the market.
10
One way to motivate this is to imagine calling random agents on the
phone at rate . He may call one that is not in the market or not searching,
whence the call goes unanswered. With probability he reaches a who is
active. Note this is not inconsistent with assuming agents with pay a cost
or while those looking to acquire it do not — it simply means phone calls
are free while storage is costly.8 In any case, for the second term in ̇, there
are type agents with , and they trade whenever they contact . The
SS (steady state) condition ̇ = 0 implies
=
+
(1)
Let be ’s payoff or value function. Let be ’s value function, given
that he has decided to enter the market and search (otherwise his payoff is
0). Let 0 be ’s value function when he does not have , given that he has
decided to enter the market and trade when he can (otherwise his payoff is
0). Let 1 be ’s value function when he has , given that he has decided to
search (otherwise his payoff is 0). To develop some intuition, consider first the
flow payoff for ,
= Σ + Σ
The first term says meets at rate , and the probability is that
is on the market with goods to trade, in which case gets a share of the
surplus total Σ . The second term is similar.
As in Lagos and Rocheteau (2009), we simplify notation by letting =
8
Unlike many search and matching models, our specifcation does not admit congestion
effects. At the suggestion of a refee we note that this simplifcation can be justifed by
saying that it allows some new interpretations and additional decisions. So, while slightly
nonstandard, the setup is logically consistent, interesting and especially tractable.
11
combine arrival rates and bargaining powers to get
= Σ + ( − ) Σ −
(2)
0 = Σ
(3)
1 = Σ −
(4)
= Σ + Σ
(5)
the standard DP (dynamic programming) equations. The surpluses are9
Σ = − + max { − 0} − max { 0}
(6)
Σ = max {1 0} − max {0 0} − + max { − 0} − max { 0}
(7)
Σ = + max{0 0} − max {1 0}
(8)
For future reference, let V = ( 0 1 ) and Σ = (Σ Σ Σ ).
An equilibrium p = ( ) must satisfy what we call the BR (best
response) conditions. For , these are:
⎧
⎪
⎨
1
if
=
[0 1] if =
⎪
⎩
0
if
For , they are:
⎧
⎪
⎨
1
if 0
and =
[0 1] if = 0
⎪
⎩
0
if 0
⎧
⎪
if 1 0
1
if Σ 0
⎨ 1
=
[0 1] if Σ = 0 and =
[0 1] if 1 = 0
⎪
⎪
⎩
⎩
0
if Σ 0
0
if 1 0
⎧
⎪
⎨
9
(9)
(10)
Heuristically, the max operators embody the notion of subgame perfection. Consider
Σ . If and trade, the instantaneous surplus is − , then decides whether to pay
to remain in the market, so his continuation value is max{ − 0}; if they do not trade,
decides whether to continue search, so his outside option is max{ 0}. In equilibrium, once
decides to enter he is in the market forever, but this way of writing the surplus indicates
this is a best response in every subgame. For the continuation value and outside options
are both , so they cancel, which is one reason the analysis is easier when we recycle .
12
Definition 1 A (steady-state) equilibrium is a list h V pi such that: satisfies the SS condition (1); V satisfies the DP equations (2)-(5); and p satisfies
the BR conditions (9)-(10).
Given an equilibrium the terms of trade are easily recovered. Assuming
0, ’s surplus when trading directly with is − = ( − − ),
and so
= + ( + )
(11)
This is a weighted average of ’s gain and ’s cost, including even though
it is sunk, because has to pay it again to continue in the market. Similarly,
assuming 0 and 0, the transfers in wholesale and retail trade are
= ∆ + ( + ) and = + ∆
(12)
where ∆ ≡ 1 − 0 is ’s gain from getting (cost to giving) , which is
easily computed from (3)-(4). Finally, the wholesale-retail markup, or spread,
≡ − , is given by10
= + ( − ) ∆ − ( + )
3
(13)
Equilibrium
We now characterize equilibria. There are in principle many candidate equilibrium profiles, but one can rule out those with 0 and 1, plus those with
0 and 1, since it cannot be a BR to pay for and not search. There
10
Although the terms of trade are interesting, we do not dwell on them since we are more
concerned with existence and efficiency. However, if one solves for , it clearly does not
vanish as → 0, contrary to RW. In RW, profits exclusively from the impatience of
others when , and as → 0 that advantage vanishes. Here may have other
advantages, including costs and bargaining power.
13
are also candidates with = 0, which are relegated to Appendix A, so we can
concentrate on nondegenerate cases here. The next result further reduces the
set of candidates by establishing that never randomizes, and while may
randomize, he does so only when is in the market with probability 1.
Lemma 4 For generic parameters, in any equilibrium: (i) ∈ (0 1) implies
= 0; and (ii) ∈ (0 1) implies = 1.
Proof: For (i), suppose by way of contradiction 0 and ∈ (0 1). Then
Σ = 0, or 1 − 0 = + . The value functions for are then given by
0 = [1 − 0 − ( + )] −
1 = [ − (1 − 0 )]
Solving for ∆ = 1 −0 = +, Σ = 0 implies = ( + ) ( + )− ,
which is nongeneric. For (ii), suppose 0 and 1. By (i), = 0; then
= Σ − = ( − − ) − . For ∈ (0 1) we need = , which
is nongeneric. ¥
Our quest for nondegenerate equilibria is thus reduced to four candidates.
There are three where enters with probability 1: p = (1 1 0 0), where
does not trade for and would not search if he had it; p = (1 1 0 1), where
does not trade for but would search if he had it; and p = (1 1 1 1), where
trades for and searches when he gets it. There is also one candidate
where enters with probability ∈ (0 1) and = 1. To understand the logic
of ∈ (0 1), note that for to be indifferent to entry we need = . As
varies, the probability that can take off ’s hands when they meet,
1 − , adjusts endogenously to make = . We now consider each of these
candidates in turn.
14
1. Equilibrium p = (1 1 0 0): In this equilibrium, enters with probability 1, while neither accepts nor searches if (off the equilibrium path)
he happens to have . This implies
= ( − − ) −
0 = (1 − 0 − − )
1 = ( − 1 + 0 ) −
= ( − − )
where one should interpret 0 and 1 as the payoffs to if he were active,
even though he is not active in equilibrium. For , = 1 is a BR iff ≥ ,
which reduces after routine algebra to
≤ ̃ ≡ ( − − ) −
(14)
Given = 1, = 1 is automatic (Lemma 1). For , consider a deviation
where he searches when he has . The deviation payoff is
¡
¢
1 = − 1 −
For = 0 to be a BR we need 1 ≤ 0, or ≥ . Given = 0,
= 0 is automatic. Hence p = (1 1 0 0) is an equilibrium iff 0 ≤ ≤ ̃ and
≥ .
2. Equilibrium p = (1 1 0 1): For , the BR condition is again ≤ ̃.
For , it is easy to check = 0 is a BR iff Σ ≤ 0, or
≥ ̃ ≡ ( − − ) − ( + )
15
(15)
Also, = 1 is a BR iff ≤ . Hence p = (1 1 0 1) is an equilibrium iff
≤ ̃ and ≥ ≥ ̃ .
Before moving to other cases, consider Figure 1, where the two equilibria
discussed above exist in ( ) space in the regions to the northwest labeled
(1 1 0 0) and (1 1 0 1). Naturally, is active while is not when is
small and is big. If if very big, would not search for even if he
had ; if is only moderately big would search for if he had , but it
is not worth making the transfer to acquire it. To describe what happens for
lower , it is convenient to consider the lines 0 ( ) and 1 ( ) in Figure 1.
Both are special cases of = ( ), for any ∈ [0 1], given by
( ) ≡ ̃ + (̃ − )
+ +
[ − ()]
(16)
where ̃ is defined in (15) and = () is now written as a function of .
e ) from 0 ( ) to 1 ( ).
As goes from 0 to 1, ( ) rotates around (e
insert fig 1 about here
3. Equilibrium p = (1 1 1 1): For = 1 we need Σ ≥ 0. This reduces
to ≤ ̃ , the reverse of (15). For = 1 we need ≥ , which reduces
to ≤ 1 ( ). Hence this equilibrium exists iff ≤ min {̃ 1 ( )}, as
shown in Figure 1.
4. Equilibrium = ( 1 1 1) with ∈ (0 1): One can check ’s
BR condition holds iff ≤
e , so it remains only to check = ∈ (0 1).
Substituting from (1) into and solving the quadratic equation = for
, we get
− [ ( + ) + ] +
=
2
16
√
(17)
where = [ ( + ) + ]2 − 4 . Algebra implies ∈ (0 1) in the region between 1 ( ) and 0 ( ) in Figure 1.
5. Degenerate equilibria: Appendix A shows that equilibria with = 0
exist in the shaded region of Figure 1. However, there are different equilibria
with = 0, e.g., where is either 0 or 1. Appendix A shows that where are
in the shaded region determines which degenerate equilibrium exists.
In terms of economics, it is no surprise that for or to be active we
cannot have or too high; the preceding analysis worked out the exact
cutoffs. For some parameters, enters with probability ∈ (0 1), with
adjusting endogenously to make = . This is related to discussions
of “search externalities” throughout the literature, although in this model,
by design, entry does not affect meeting rates, it rather affects and hence
the probability of trade when meets . We also emphasize this: Suppose
̃ , as is the case when has a poor storage technology, a low chance of
finding , or low bargaining power when he does find . Then intermediation
is essential in the sense the market operates iff middlemen are active.11
These results are novel relative to RW, where costs are 0, so is always
active, and is active iff exceeds . While intermediation can improve
welfare in RW, the impact here is more dramatic — sometimes the market
opens iff intermediation smooths the way. We summarize as follows:
Proposition 1 Given = 1 (everyone recycles), for all values of the other
parameters, equilibrium exists and is generically unique, as shown in ( )
space by Figure 1. For some parameters intermediation is essential.
11
In monetary theory, money is said to be essential if the set of outcomes that can be supported as equilibria is bigger or better with money than without it (e.g., Wallace 2001,2010).
For money this is nontrivial because, obviously, it is not essential in standard Arrow-Debreu
models. The same is true of intermediation.
17
4
Alternative Assumptions
Here we mention some extensions, including a different way to describe the
results.12 To begin, note that in addition to preferences ( ), arrival rates
( ) and bargaining powers ( ), the model parameters are given by the
vector of costs = ( ). In general, we need all elements of to
characterize the equilibrium set, but sometimes different equilibria generate
the same outcome — e.g., for any ( ), both ( ) = (0 0) and ( ) =
(0 1) entail no intermediation. If one cares only about outcomes, we claim
all that matters is the expected net gain for from trying to trade with
directly, denoted , and the expected net gain for from trying to trade
with , denoted , where
≡ ̃ − = − − ( + ) −
(18)
≡ ̃ − = − ( + ) ( + ) −
(19)
Appendix B translates the results in Section 3 from ( ) space into
( ) space, as illustrated in Figure 2, which is isomorphic to Figure 1,
but is still useful due to the interpretation. First notice that and
are bounded above by ̃ and ̃ . Now, since is the net benefit to of
searching for without using , 0 implies = 1 regardless of ’s
decision. Similarly, is the net benefit to of searching for , so = 1
if 0 and = 0 if 0. Hence, outcomes are obvious in three of the
four quadrants in Figure 2: (i) 0 and 0 imply = 1 and = 1;
12
This is not critical for what follows, and one can move directy to the discussion of
efficiency with little loss of continuity, but one message here is that = = 0 is in a
sense without loss of generality. We also show how to extend the analysis to describe what
happens out of steady state.
18
(ii) 0 and 0 imply = 1 and = 0; (iii) 0 and 0 imply
= 0 and is irrelevant. In the fourth (northwest) quadrant, as we make
a bigger negative number for fixed 0, goes from 1 to ∈ (0 1) to 0.13
insert fig 2 about here
An implication is that there is little loss of generality in setting = = 0
if we care only about outcomes. By analogy, in labor models firms can have
a fixed or flow cost to entering the market, but we do not need both, since
all that matters is the total expected discounted cost. To see how this works
here, consider two economies with ̄ = (̄ ̄1 ̄ ̄ ) and ̂ = (̂ ̂ ̂ ̂ ).
The outcome depends only on
¡
¢
̄ = − ( + ) ̄ + ̄ − ̄
̄ = − ̄ − ( + ) ̄ − ̄
in the ̄ economy, and similarly in the ̂ economy. If we set ̂ = ̂ = 0, then
¡
¢
set ̂ = ( + ) ̄ + ̄ + ̄ and ̂ = ̄ + ( + ) ̄ + ̄ , the outcomes
in the ̂ and ̄ economies are the same. Hence, we can always set = = 0
and not change outcomes, as long as we adjust the ’s.14
Next, consider dynamics. Setting = = 0 to reduce notation, the DP
equation for type without imposing steady state is
̇ = − − ( − )(1 − 0 ) +
(20)
and similarly for the others. These plus the law of motion for define a dy13
It is easy to check = occurs between the ray 0 defined by = − ( + )
and the ray 1 defined by = − ( + + ) ( + ) .
14
At least, this is true if we care only about outcomes in terms of and ; the above
argument does not say that the two economies will have the same terms of trade y.
19
namical system. Given an initial condition 0 , an equilibrium is a nonnegative
solution to this system that is bounded (more accurately, that does not grow
faster than ). In fact, the system is quite simple. First, because type
makes no decisions, ignore them. Second, for type , at any point in time
they choose = 1 if and = 0 if , independent of
anything else that is going on.
insert fig 3 about here
Thus, the relevant decisions are made by , although of course these depend on what is doing. If is inactive, assuming free disposal, jumps
to = 0 and stays there. Then ̇ is linear with slope 0, and the only
bounded solution is = ∀, where = ( − ) . Hence, when is
inactive: =⇒ 0, = 0 ∀; and =⇒ 0, = 1
∀. The outcome is somewhat more interesting when is active. Since it is
obvious the unique steady state is a saddle point, once is fixed, there can be
transitional dynamics as → but no belief-based (bubble) equilibria. So
all we have to do is describe how and evolve over time. We break the
analysis into cases, depending on parameters and initial conditions.
Suppose first parameters are such that in steady state = 1. Figure 3
depicts three subcases differing in 0 . The top panel is subcase a, defined by
0 (1), which means 0 (1) 0 where we now write () and
() for the steady state given . Then type enter at = 0, and stay
in, while → (1) and → (1) as shown. Subcase b has the opposite
initial condition, 0 (1), which means 0 (1). Then two situations
can occur. The middle panel is subcase b1 where 0 0, so enter at = 0,
while → (1) and → (1). The bottom panel is subcase b2, where
20
0 0, which means do not enter at = 0. Since we are supposing = 1,
in this situation ∃1 ∈ (0 ∞) such that = 0 ∀ 1 and = 1 ∀ 1 .
At first, with = 0, inventories fall rapidly, making rise until 1 , at which
point finds it profitable to enter, and then → (1) and → (1). The
other cases can be analyzed similarly.
Finally, consider an extension, suggested by a referee, to incorporate “occupational choice.” Given a fixed measure , the baseline model takes and
as also fixed, but because and can choose to be inactive, the ratio of
active producers to middlemen is endogenous. But another approach is to let
everyone in the 1 − set of nonconsumers choose to be either a producer or
a middlemen, or be neither and sit out. When we worked out this alternative
way to endogenize the ratio, the results were similar: there were still three
possible outcomes — inactive; active but inactive; or both active — and
the analog to Figure 1 looks roughly the same. One interesting difference is
this: given 0, the baseline model always has intermediation when 0 0,
which must be true for small because ’s only other option is to sit out;
the alternative setup may have = 0 even when = 0, because for 0
we now not only need 0 0, we need 0 . Since this and some other
technical features are rather different, we do not include details.
5
Efficiency
We now consider the planner’s problem, in discrete time, which we find easier
and more intuitive, although of course the SS is the same as in continuous
time. The state vector is ( ), where is the stock of ’s in the market
with at the end of a period, and as always is the stock of ’s holding .
21
The control vector is ( ), where is the stock of ’s in the market with
at start of the next period, and denotes the fraction of the 1 − type
’s that are currently without that we instruct to try to trade for next
period. Obviously, ( ) and ( ) are related to the equilibrium variables,
as described below. The planner’s problem is15
̃ ( ) = − ( − ) + [ ( − ) − (1 − ) − ]
(21)
+ ( − ) + ̃ (0 0 )
The first term on the RHS is the current cost to activating ( − )
type agents that are currently not in the market. The second term is the
net benefit from having type ’s active next period, which includes the
following instantaneous payoffs: the net gain − to trading with , which
happens with probability ; the cost − from producing for , which
occurs with probability (1 − ) ; and the search cost − . Similarly, the
third term is net benefit from having type ’s with . The final term is
the continuation value, given the laws of motion
0 = [1 − − (1 − ) ]
(22)
0 = (1 − ) + (1 − )
(23)
Letting ( ) = ̃ ( ) , and using = , rewrite (21) as
( ) = − ( − ) + [ ( − ) − (1 − ) − ]
(24)
+ ( ) ( − ) + (0 0 )
15
Importantly, we are not imposing steady state and then maximizing welfare; we are
solving the dynamic planner’s problem and then imposing steady state. The other problem
only gives the correct answer in limit as → 0.
22
Let denote the RHS of (24). Since ∈ [0 1], their optimal values depend
on
− )
w ( + + )(̃ − ) + (1 − ) (̃
−
w ̃
(25)
(26)
where w means that and take the same sign, and we define
≡ − ( + )( + )
̃
(27)
̃ ≡ ( − ) − ( + )
(28)
are closely related to ̃ and ̃ Section 3: for all values
Note that ̃ and ̃
iff = 1, and ̃ = ̃
iff = 1.
of the other parameters, ̃ = ̃
Given these results, we arrive at the final simplified versions of (25)-(26),
and hence the final answer to the planner’s problem,
⎧
⎪
⎨
1
=
[0 1]
⎪
⎩
0
⎧
⎪
⎨
if ̃ =
⎪
⎩
⎧
⎪
⎨
1
and =
[0 1]
⎪
⎩
0
where to conserve space we introduce
− +
≡ ̃
⎧
⎪
⎨ 0
if
=0
⎪
⎩
0
( + + ) ( + )
(̃ − )
(29)
(30)
It is now straightforward to characterize efficient outcomes, as shown in ( )
space by Figure 4, similar to the equilibrium characterization in Figure 1, except now with ̃ and rather than ̃ and .
Proposition 2 The solution to the planners problem has the following prop
erties in steady state: (i) ̃ and ̃
implies = = 1; (ii) ̃
23
and ̃
implies = 1 and = 0; (iii) ̃ and ̃
implies
implies three subcases: if
= 0, so is irrelevant; (iv) ̃ and ̃
0 then = 0 and is irrelevant, if 0 then = 1 and = 1, and if
= 0 then = 1 and = ∈ (0 1) is given by
=
− ( + 2 ) +
q
− )(̃ − )
2 − 4 (̃
2
(31)
insert fig 4 about here
While equilibrium outcomes and efficient outcomes may not coincide, in
general, the next result demonstrates that there are conditions on bargaining
powers, related to Mortensen (1982) and Hosios (1990), that imply they do
coincide.
Proposition 3 There exists a unique = (
) that implies the
equilibrium outcome is efficient for all values of the other parameters, given
=
= 1 and
by
=
+ +
+ + 2
(32)
where is given in (31).
Proof: The efficient outcome ( ) = (1 1) requires that ̃
and
̃ ; the equilibrium outcome ( ) = (1 1) requires that ̃ and
implies ̃ for all values of the other parameters
̃ . Now ̃
iff = 1, and ̃ implies ̃ for all values of the other parameters
and ̃ = ̃ . Then from
parameters iff = 1. When = = 1, ̃ = ̃
(30) efficient outcome ( ) = ( 1) requires
̃
¢
( + + ) ( + ) ¡
− = −
̃ −
24
(33)
The equilibrium outcome ( 1), with = = 1, requires
̃
¢
( + + (1 − ) ) ( + ) ¡
− = −
̃ −
(34)
Setting (33) to equal (34) and = , we obtain (32). ¥
Although efficiency obtains when the ’s are set just right, for arbitrary
parameters, the equilibrium outcomes can be inefficient. In particular,
Proposition 4 Depending upon parameters: (i) can be too high or too low;
(ii) can be too low but not too high.
Proof. (i) Suppose 0 1 and = = 1. It is not hard to check
implies and
implies .
from (17) that
Hence, can be too high or too low.
. Then = 1. If 0 1, then
(ii) Suppose ̃ and ̃
̃ . For sufficiently small, ̃ , which implies = 0. Hence,
̃
can be too low. Equilibrium requires that = 1 iff ̃ ; otherwise = 0.
; otherwise = 0. Since ̃
≥ ̃ ,
Efficiency requires that = 1 iff ̃
we have ≥ . Hence, we would be able to conclude that the equilibrium
cannot be too high, if we could verify that = 0 implies = 0. We need
the latter condition because otherwise can be too high when 0 1 and
= 0. The result we need, that = 0 implies = 0, is true as long as
0 ( ) lies everywhere above 0 ( ) for all ̃ where 0 ( ) ≥ 0, as shown
in Figure 5 (the graph is drawn assuming ̃ ̃ and ̃
̃ but ̃ = ̃
and ̃
= ̃ has the same qualitative features).
insert fig 5 about here
25
Since 0 ( ) and 0 ( ) are linear, it suffices to show the intercept of
0 ( ) exceeds the intercept of 0 ( ). This is equivalent to
(1 − ) ( − − ) ≥ [ ( − − ) − ( + )]
−[ ( − − ) − ( + )]
+
+
(35)
We can set = = 1 here, without sacrifice, since if (35) holds for these
values it holds for all ∈ [0 1). When = = 1, (35) simplifies to
≤ 1, which is true. Hence, we have established that = 0 implies = 0.
As remarked above, this allows us to conclude that cannot be too high. ¥
Heuristically, the intuition for the above results is as follows. We can
, because this means is not being
make too low by setting
sufficiently compensated for his sunk costs when he meets . We cannot set
= 1, however, so we cannot make too high. By a similar logic,
, but in this case we can make
we can make too low by setting
, because
∈ (0 1). The reason for
1
too high by setting
is similar to results in other search-and-bargaining models. When decides
to enter the market, he considers his own costs and benefits, but not those of
others. Thus, he ignores the fact that when there are more ’s in the market,
increases, and this makes it harder for all ’s to trade. The bargaining
1, is determined so that the socially optimal
power that gives efficiency,
measure of producers enter.
6
Random Recycling
In this section we reintroduce , the probability that type recycles (stays in
the market) after trade, where = 1 in our baseline specification and = 0
26
in the original RW specification. This is relevant for several reasons. First, the
analysis presented above is not really a generalization of RW because, although
we added general costs, bargaining weights and so on, we also changed from
0 to 1. Consideration of = 0, and a fortiori ∈ [0 1], delivers a strict
generalization of RW. Also, it turns out to be interesting in its own right to
understand what happens for different . In particular, we want to know how
our results on existence, uniqueness and efficiency are affected when we allow
1, which means agents randomly continue or exit.
To begin, note that affects agents’ outside options, and therefore affects
the surpluses, as follows:
Σ = − + max{ − 0} − max{ 0} − (1 − ) max{ 0}
(36)
Σ = − + max{ − 0} − max{ 0} + max {1 0} − max {0 0}
(37)
Σ = − (1 − ) max{ 0} − max {1 0} + max {0 0}
(38)
However, other than using the Σ’s in (36)-(38) instead of the special case
in (6)-(8), the DP, BR and SS equations are unchanged, as is the definition
of equilibrium. In what follows, we characterize equilibrium outcomes with
∈ [0 1], where to reduce notation we set = = 0, but one can say that
this is without loss in generality given the results in Section 4. Interestingly,
there is now a greater variety of outcomes, including more possibilities for
mixed-strategies, and sometimes multiplicity.
Before going through each case, to develop some intuition, suppose that
is big enough that we can be sure = 0. Then there are effectively two types,
and , which allows us to illustrate some results easily. The DP equations
27
are given by
= [ − (1 − ) − (1 − ) ] −
= [ − (1 − ) − (1 − ) ]
Now = 1 is an equilibrium iff ≥ 0, which reduces to
≤
b ≡
+ (1 − )
Similarly, = 0 is an equilibrium when ≥ . Then solve = 0 for
, and notice ∈ (0 1) iff
b . Hence, conditional on = 0,
we immediately get existence, generic uniqueness, and for some parameters an
interior solution for , as we found in Section 3.
However, the logic behind ∈ (0 1) here is different from the logic with
= 1. With = 1, we found that ∈ (0 1) was only possible when = 1,
and the equilibrating mechanism was that adjusted to make = . Now
we can get ∈ (0 1) without intermediation, with the terms of trade rather
than the probability of trade equilibrating entry. To see this, solve for:
=
Υ − (1 − ) [ + (1 − ) ]
[ + (1 − ) + (1 − )]
where Υ ≡ + (1 − ) + (1 − )(1 − ) . With 1, depends
on ; with = 1, it does not. Figure 6, drawn for 1, shows the
following: Starting from a low value, with = 1, as increases initially
falls while initially rises, because higher makes more keen to trade
and this decreases . At =
b , hits 0, at which point there emerges a
mixed-strategy equilibrium. In this mixed equilibrium, as rises further,
falls and rises to keep = 0.
28
insert fig 6 about here
For the rest of the candidate equilibria, it turns out there are two scenarios
to be considered, shown in Figures 7 and 8. The two scenarios correspond to
(1 − ), as in Figure 7, and (1 − ), as in Figure 8. Let
us begin with the former case (details are in Appendix C). As one can see,
in Figure 7 we still have existence and generic uniqueness, but now there can
be three distinct types of mixed-strategy equilibria, ( 1), ( 0) and (1 ).
The first we encountered in Section 3; the second we discussed just above; and
we now discuss the third.
insert fig 7 about here
insert fig 8 about here
There is a region in Figure 7 where the unique equilibrium entails =
∈ (0 1). This region is bounded above by 0 ( ) and below by 1 ( ) and
3 ( ) (Appendix C). The condition (1 − ) makes 1 ( ) lie below
0 ( ) in the relevant range. In particular, for a relatively low , we have
= 1 when ≤ 1 ( ), ∈ (0 1) when 1 ( ) 0 ( ), and = 0
when 0 ( ) . Naturally, when is higher, middlemen are active with a
lower probability, which turns the terms of trade in their favor to compensate
for higher costs. Figure 8 is similar, except the condition (1 − )
makes 1 ( ) lie above 0 ( ) over the relevant range. The equilibrium region
for ∈ (0 1) in this case is bounded below by 0 ( ) and above by 1 ( )
and 3 ( ). The logic is similar, except now the regions overlap, so there are
multiple equilibria. We summarize as follows:
Proposition 5 For any ∈ [0 1], for all values of the other parameters,
equilibrium exists. If (1 − ) it is generically unique, as shown by
29
Figure 7. If (1− ) there are multiple equilibria for some parameters,
as shown by Figure 8.
Multiplicity cannot arise in the baseline model with = 1 because then
we cannot satisfy the condition (1 − ), so we are necessarily in
the scenario depicted in Figure 7. When → 1, the regions with ( 0) and
( ) disappear, and when → 1 the region with ( 0) disappears. Hence,
as → 1 the model of course collapses to the benchmark case. In the special
case with no search/storage costs, the outcome is especially simple:
Proposition 6 If = = 0 then equilibrium exists, is generically unique,
and has = 1 for all parameters. It also has = 1 if (1 − ) and
= 0 if (1 − ).
This is a strict generalization of the original RW result, where depends
only on ≷ , rather than ≷ (1 − ), because they had = 12
and = 0. It is good to know that a version of their main result holds when
= 0, even for general ’s, ’s and ’s. Things are more interesting,
however, when 0, because this allows mixed equilibria with ∈ (0 1) or
∈ (0 1), plus multiple equilibria.16
Some results are the same for any ∈ [0 1], including the efficiency results
in Section 5. This is because the planner’s problem is unaffected by changing
, since our planner regards incumbent traders and their clones as perfect
substitutes. Hence, efficient outcomes are still as shown in Figure 4. We
16
As a special case, if = 0 and = 1, then is active with probability = 1 whenever
0. This is because = 1 implies has no opportunity cost of trading his output
to , as he can always produce again and continue in the market. This of course uses
= = 0, which we said above was without loss of generality; that is based on results in
Section 4, where it was proved that one can always set = = 0 as long as one resets the
’s, which in general would not lead to = 0. In other words, fixing = 0 means we
cannot also set = = 0 without loss of generality.
30
already know can be too high or too low, in general, because we verified
this for = 1. Comparing Figure 4 to Figures 7 and 8, it is still the case that
can be too low but not too high. Before, with 1 = 1, we could get = 0
and = 1; now we can also get ∈ (0 1) and = 1. In any case, can
be too low but not too high, while can be too low or too high.
7
Conclusion
This project has continued the development of intermediation theory by extending the original Rubinstein-Wolinsky (1987) specification on several dimensions. We verified existence and generic uniqueness in a benchmark case
where all agents stay in the market forever. The results are more complicated
when consumers and producers continue in the market probabilistically, but
the framework is still tractable. An interesting feature compared to RW is
that for certain parameters equilibria entail mixed strategies, with some but
not all potential entrants participating in the market. What equilibrates participation can be either the terms of trade or the time it takes to trade, which
is an attractive feature of a search-based approach. Having participation costs
made it interesting to study efficiency. We found that there can be too little
or too much production, and there can be too little but not too much intermediation. However, with bargaining powers set appropriately equilibrium is
efficient.17
17
A direction for future research might be to consider directed search, with the terms
of trade posted rather than negotiated, which one might conjecture could deliver efficiency
endogenously (based on work on labor and other markets by, e.g., Moen 1997, Mortensen
and Wright 2001, Shimer 2005 or Eeckhout and Kircher 2010). However, it is not clear how
to introduce directed search without compromising some features of a three-sided market,
including the feature that can randomly meet and trade with either or . Watanabe
(2010) provides one avenue of exploration for middlemen with directed search.
31
One can use the framework to address many issues in finance, banking, real
estate and other areas where intermediation plays a big role. An example of
results that we did not have the space to discuss concerns the relative terms of
trade in direct, wholesale and retail transactions. Since these obviously depend
on bargaining powers, in general, consider the special case of the baseline
model with = 12. Then one can show (retail exceeds
direct exceeds wholesale), at least if = are not too big. This is not general,
however, and one can show if is small. Another natural
case is = , which means equilibrium is efficient, and implies =
. Hence, the efficient outcome is described by having direct and retail
transfers from the consumer the same, and above the wholesale transfer from
middlemen to producers, reflecting the very real service that intermediation
provides in markets with frictions.
Finally, we mention that we have to this point not emphasized that intermediation per se increases production and consumption in these models. Due
to good being storable only one unit of a time, once produces , he cannot
produce again until he trades. When takes off ’s hands, therefore,
produces more often. Although this depends on the technical assumption that
inventories are in {0 1}, one can also say that it rings true: well-functioning
intermediation allows to tie up fewer resources in marketing and get back
more easily to making stuff, something in which he specializes.
32
Appendix A: Results for Section 3
Here we characterize regions of parameter space where degenerate equilibria
exist. Figure 9 shows the results, along with the regions with nondegenerate
equilibria discussed in the text.
insert fig 9 about here
1. Equilibrium p = (0 0 0 0): The BR condition for = 0 is ≤ 0
which reduces to ≤ ( + ). Given ≤ 0, the BR condition for = 0
is not binding. The BR condition for = 0 is 1 ≤ 0, which reduces to
≤ . Given 1 ≤ 0, the BR condition for = 0 is not binding.
2. Equilibrium p = (0 0 0 1): The BR conditions for = 0 and = 0
are the same as in p = (0 0 0 0). The BR condition for = 1 is 1 ≥ 0,
which reduces to ≥ . Letting
( ) ≡ − ( + ) − [ ( − ) − ]
+
+
the BR condition for = 0 is ( ) ≤ .
3. Equilibrium p = (0 0 1 1): The BR condition for = 0 is ≤ 0
which reduces to ≥ ( ), where
( ) ≡ − ( + ) + [ ( − ) − ]
+
Given ≤ 0, the BR condition for = 0 is not binding. The BR condition for
= 1 is Σ ≤ 0, which can be simplified to ≤ ( ). The BR condition
for = 1 is ≤ .
4. Equilibrium p = (0 1 0 0): The BR condition for = 0 is ≥
( − ) − ( + ) . The BR condition for = 1 is ≤ ( − ). The
BR condition = 0 is not binding. The BR condition for = 0 is ≥ .
33
5. Equilibrium p = (0 1 0 1): The BR condition for = 0 is ≥
( − ) − ( + ) . The BR condition for = 1 is ≤ ( − ).
The BR condition for = 0 is ≥ ( ): The BR condition for = 1 is
≤ .
6. Equilibrium p = (0 1 1 1): The BR condition for = 0 is ≥
0 ( ), where 0 ( ) is defined above. The BR condition for = 1 is ≤
( ). The BR condition = 1 is ≤ ( ). The BR condition for = 1
is ≤ .
Appendix B: Characterization in Figure 2
1. Outcome ( ) = (1 0): In this case all producers enter, but middlemen are inactive. As shown above, this happens in two equilibria, one with
= 0 and another with = 1, although the outcome is the same because
= 0 in both cases. From Figure 1, an equilibrium with ( ) = (1 0) exists
iff ≤ ̃ and ≥ ̃ , which from (18) and (19) are equivalent to ≥ 0
and ≤ 0.
2. Outcome ( ) = (1 1): Now all producers and middlemen are active.
This occurs in the equilibrium that exists when ≤ ̃ and ≤ 1 ( ),
conditions that are equivalent to ≥ 0 and
≥ −
( + + ) ( + )
given that − (1) = ( + ).
3.
Outcome ( ) = ( 1) with ∈ (0 1): Now some producers
and all middlemen are active. This happens in the equilibrium that exists iff
1 ( ) ≤ ≤ 0 ( ), conditions that are equivalent to
≤ −
( + + ) ( + )
+
and ≥ −
34
since in this case − (0) = 0.
4. Outcome = 0: This occurs in a degenerate equilibrium, which exists
in the complement of set of parameters where the other outcomes occur.
Appendix C: Results for Section 6
We now characterize regions of parameter space where different equilibria
exist for the case of general ∈ [0 1], where to conserve space we focus on
( ), with implicit.
1. Equilibrium ( ) = (0 0): The BR condition for = 0 is ≤ 0, or
≤ . The BR condition for = 0 is 1 ≤ (1 − ) , which reduces to
2 ( ) ≤ , where
2 ( ) ≡
[ − (1 − )] + (1 − )( + )
+ (1 − )
2. Equilibrium ( ) = (0 1): The BR condition for = 0 is 0 ( ) ≤
, where
0 ( ) ≡ [
+
( + ) + ] −
The best response condition for = 1 is 2 ( ) ≥ .
3. Equilibrium ( ) = (1 0): The BR condition for = 1 is ≥ 0
which reduces to
b ≡
≥
+ (1 − )
The best response condition for = 0 can be simplified to ≥ 0 ( ), where
0 ( ) ≡
[ − (1 − )] + (1 − )[ + + (1 − )]
+ (1 − ) + (1 − )
4. Equilibrium ( ) = (1 1): The BR condition for = 1 is 1 ( ) ≥
35
, where
−1
1 ( ) ≡ [ ( + + ) + (1 − )]
−1
− {( + + ) [ + (1 − ) ] + (1 − ) ( + )}
= (1 − ) + (1 − ) [ + (1 − )( + )], and =
( + ). The BR condition for = 1 is 1 ( ) ≥ , where
1 ( ) ≡
[ − (1 − )] + (1 − )[ + + ( + )(1 − )]
+ (1 − ) + ( + )(1 − )
5. Equilibrium ( ) = ( 1): The BR condition for ∈ (0 1) is given
by = 0, which is
[ ( + + ) + (1 − )]
− {( + + ) [ + (1 − ) ] + (1 − ) ( + )}
= (1 − ) + (1 − ) [ + (1 − )( + )]
where = ( + ). When → 0, → 0 ( ), and when
→ 1, → 1 ( ). Thus, ∈ (0 1) when 0 ( ) ≤ ≤ 1 ( ). Given
= 0, the BR condition for = 1 is 1 ≥ 0 , which reduces to
≥
+ (1 − )( + ( ) )
The is complicated because is a complicated function of parameters. However, it is clear from numerical analysis that it generates a positive relation
between and , as shown by 3 ( ) in the diagrams.
6. Equilibrium ( ) = ( 0): The BR condition for ∈ (0 1) is = 0,
which solves for
=
( − )
(1 − )
36
This means ∈ (0 1) iff b
≤ ≤ . Then the BR condition for = 0 is
1 ≤ 0 , which reduces to ≤ .
7. Equilibrium ( ) = (1 ): The best response condition for = (0 1)
is given by 1 = (1 − ) , which can be simplified and solved for
=
= [ − (1 − )]−1 + (1 − )[ + + (1 − )]−1
− [ + (1 − ) + (1 − )] −1
where = (1 − )[ − (1 − )]. Note that the value of depends on
the relative values of and (1− ); hence, so do the BR conditions. Upon
simplification, we have the following: if (1 − ) we need 1 ( )
0 ( ) and 3 ( ) for this equilibrium; and if (1 − )
we need 0 ( ) 1 ( ) and 3 ( ), where
3 ( ) ≡
2
{[ ( + ) −
(1 − )] + (1 − )[ ( + ) + ]}
[( + )( + ) − ]
These two cases, (1 − ) and (1 − ), correspond to the
scenarios in Figures 7 and 8.
37
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40
m
(1,1,0,0)
mcu
(1,1,0,1)
~m
(1,1,1,1)
(πe,1,1,1)
~p
h1 ( p )
h0 ( p )
Figure 1: Equilibria in ( ) Space.
41
p
Am
0
1
~m
e , 1
1
0
(0,0)
1, 0
Figure 2: Equilibria in ( ) Space.
42
~p
Ap
V
p 0
V
pt
(1 )
s ( 1 ) ( 0 ,1 )
t (1 )
0
a.
V ps (1 ) 0
0
t
s (1 )
V
0
pt
(1 )
t (1 )
V ps (1 ) 0
s ( 1 ) ( 0 ,1 )
V p0
t
b1 .
0
s (1 ) , V
p0
0
0
V
t (1 )
0
s (1 ) ( 0 ,1)
t
t1
V p0
pt
V ps (1 ) 0
(1 )
1
b 2.
0
s
(1 ) , V
p0
0
Figure 3: Dynamics and Payoffs for , = 1
43
m
o 1, 0
o 0
~mo
o 1, 1
o (0,1), 1
~ po
h1o ( p )
h0o ( p )
p
Figure 4: Efficient Outcomes in ( ) Space
m
~mo
~m
h 0 ( p )
h0o ( p )
~ p ~ po
p
Figure 5: = 0 Implies = 0
44
Figure 6: Equilibium with = 0 when 1
m
( e ,0 )
(0, 0 )
F2( p )
(1,0 )
mc u
F3 ( p )
( 0 ,1)
( e ,1)
(1, e )
F0 ( p )
( 1 ,1 )
F1 ( p )
H1( p )
ˆ p
pc u
H0 ( p )
p
Figure 7: Equilibrium Outcomes for 1 and (1 − )
45
m
( 0 ,0 )
F2 ( p )
( e , 0 )
mc u
( 0 ,1 )
(1, 0 )
F3 ( p )
( e ,1 )
F1 ( p )
ˆ p
F0 ( p )
pc
H 0 ( p )
u
H 1 ( p )
(1,0),(1,τ),(1,1)
(1,1)
p
(1,0),(1 ,τ),(π,1)
Figure 8: Equilibrium Outcomes for 1 and (1 − )
m
(1,1,0,0)
(0,1,0,0)
mcu
(0,0,0,0)
(0,0,0,1)
(1,1,0,1)
f ( p )
(0,1,0,1)
(0,0,1,1)
~m
(0,1,1,1)
(1,1,1,1)
g( p )
(πe,1,1,1)
~p h ( )
1 p
h0 ( p )
Figure 9: All Equilibria
46
p
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Corporate Average Fuel Economy Standards and the Market for New Vehicles
Thomas Klier and Joshua Linn
WP-11-01
The Role of Securitization in Mortgage Renegotiation
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-02
Market-Based Loss Mitigation Practices for Troubled Mortgages
Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff
WP-11-03
Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay
WP-11-04
The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz
WP-11-05
Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička
WP-11-06
A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy
WP-11-07
Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen
WP-11-08
Fertility Transitions Along the Extensive and Intensive Margins
Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder
WP-11-09
Black-White Differences in Intergenerational Economic Mobility in the US
Bhashkar Mazumder
WP-11-10
Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein
WP-11-11
Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture
Ezra Oberfield
WP-11-12
Equilibrium Bank Runs Revisited
Ed Nosal
WP-11-13
Are Covered Bonds a Substitute for Mortgage-Backed Securities?
Santiago Carbó-Valverde, Richard J. Rosen, and Francisco Rodríguez-Fernández
WP-11-14
The Cost of Banking Panics in an Age before “Too Big to Fail”
Benjamin Chabot
WP-11-15
1
Working Paper Series (continued)
Import Protection, Business Cycles, and Exchange Rates:
Evidence from the Great Recession
Chad P. Bown and Meredith A. Crowley
WP-11-16
Examining Macroeconomic Models through the Lens of Asset Pricing
Jaroslav Borovička and Lars Peter Hansen
WP-12-01
The Chicago Fed DSGE Model
Scott A. Brave, Jeffrey R. Campbell, Jonas D.M. Fisher, and Alejandro Justiniano
WP-12-02
Macroeconomic Effects of Federal Reserve Forward Guidance
Jeffrey R. Campbell, Charles L. Evans, Jonas D.M. Fisher, and Alejandro Justiniano
WP-12-03
Modeling Credit Contagion via the Updating of Fragile Beliefs
Luca Benzoni, Pierre Collin-Dufresne, Robert S. Goldstein, and Jean Helwege
WP-12-04
Signaling Effects of Monetary Policy
Leonardo Melosi
WP-12-05
Empirical Research on Sovereign Debt and Default
Michael Tomz and Mark L. J. Wright
WP-12-06
Credit Risk and Disaster Risk
François Gourio
WP-12-07
From the Horse’s Mouth: How do Investor Expectations of Risk and Return
Vary with Economic Conditions?
Gene Amromin and Steven A. Sharpe
WP-12-08
Using Vehicle Taxes To Reduce Carbon Dioxide Emissions Rates of
New Passenger Vehicles: Evidence from France, Germany, and Sweden
Thomas Klier and Joshua Linn
WP-12-09
Spending Responses to State Sales Tax Holidays
Sumit Agarwal and Leslie McGranahan
WP-12-10
Micro Data and Macro Technology
Ezra Oberfield and Devesh Raval
WP-12-11
The Effect of Disability Insurance Receipt on Labor Supply: A Dynamic Analysis
Eric French and Jae Song
WP-12-12
Medicaid Insurance in Old Age
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-12-13
Fetal Origins and Parental Responses
Douglas Almond and Bhashkar Mazumder
WP-12-14
2
Working Paper Series (continued)
Repos, Fire Sales, and Bankruptcy Policy
Gaetano Antinolfi, Francesca Carapella, Charles Kahn, Antoine Martin,
David Mills, and Ed Nosal
WP-12-15
Speculative Runs on Interest Rate Pegs
The Frictionless Case
Marco Bassetto and Christopher Phelan
WP-12-16
Institutions, the Cost of Capital, and Long-Run Economic Growth:
Evidence from the 19th Century Capital Market
Ron Alquist and Ben Chabot
WP-12-17
Emerging Economies, Trade Policy, and Macroeconomic Shocks
Chad P. Bown and Meredith A. Crowley
WP-12-18
The Urban Density Premium across Establishments
R. Jason Faberman and Matthew Freedman
WP-13-01
Why Do Borrowers Make Mortgage Refinancing Mistakes?
Sumit Agarwal, Richard J. Rosen, and Vincent Yao
WP-13-02
Bank Panics, Government Guarantees, and the Long-Run Size of the Financial Sector:
Evidence from Free-Banking America
Benjamin Chabot and Charles C. Moul
WP-13-03
Fiscal Consequences of Paying Interest on Reserves
Marco Bassetto and Todd Messer
WP-13-04
Properties of the Vacancy Statistic in the Discrete Circle Covering Problem
Gadi Barlevy and H. N. Nagaraja
WP-13-05
Credit Crunches and Credit Allocation in a Model of Entrepreneurship
Marco Bassetto, Marco Cagetti, and Mariacristina De Nardi
WP-13-06
Financial Incentives and Educational Investment:
The Impact of Performance-Based Scholarships on Student Time Use
Lisa Barrow and Cecilia Elena Rouse
WP-13-07
The Global Welfare Impact of China: Trade Integration and Technological Change
Julian di Giovanni, Andrei A. Levchenko, and Jing Zhang
WP-13-08
Structural Change in an Open Economy
Timothy Uy, Kei-Mu Yi, and Jing Zhang
WP-13-09
The Global Labor Market Impact of Emerging Giants: a Quantitative Assessment
Andrei A. Levchenko and Jing Zhang
WP-13-10
3
Working Paper Series (continued)
Size-Dependent Regulations, Firm Size Distribution, and Reallocation
François Gourio and Nicolas Roys
WP-13-11
Modeling the Evolution of Expectations and Uncertainty in General Equilibrium
Francesco Bianchi and Leonardo Melosi
WP-13-12
Rushing into American Dream? House Prices, Timing of Homeownership,
and Adjustment of Consumer Credit
Sumit Agarwal, Luojia Hu, and Xing Huang
WP-13-13
The Earned Income Tax Credit and Food Consumption Patterns
Leslie McGranahan and Diane W. Schanzenbach
WP-13-14
Agglomeration in the European automobile supplier industry
Thomas Klier and Dan McMillen
WP-13-15
Human Capital and Long-Run Labor Income Risk
Luca Benzoni and Olena Chyruk
WP-13-16
The Effects of the Saving and Banking Glut on the U.S. Economy
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti
WP-13-17
A Portfolio-Balance Approach to the Nominal Term Structure
Thomas B. King
WP-13-18
Gross Migration, Housing and Urban Population Dynamics
Morris A. Davis, Jonas D.M. Fisher, and Marcelo Veracierto
WP-13-19
Very Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, Jan Tilly, and Nan Yang
WP-13-20
Bubbles and Leverage: A Simple and Unified Approach
Robert Barsky and Theodore Bogusz
WP-13-21
The scarcity value of Treasury collateral:
Repo market effects of security-specific supply and demand factors
Stefania D'Amico, Roger Fan, and Yuriy Kitsul
Gambling for Dollars: Strategic Hedge Fund Manager Investment
Dan Bernhardt and Ed Nosal
Cash-in-the-Market Pricing in a Model with Money and
Over-the-Counter Financial Markets
Fabrizio Mattesini and Ed Nosal
An Interview with Neil Wallace
David Altig and Ed Nosal
WP-13-22
WP-13-23
WP-13-24
WP-13-25
4
Working Paper Series (continued)
Firm Dynamics and the Minimum Wage: A Putty-Clay Approach
Daniel Aaronson, Eric French, and Isaac Sorkin
Policy Intervention in Debt Renegotiation:
Evidence from the Home Affordable Modification Program
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
Tomasz Piskorski, and Amit Seru
WP-13-26
WP-13-27
The Effects of the Massachusetts Health Reform on Financial Distress
Bhashkar Mazumder and Sarah Miller
WP-14-01
Can Intangible Capital Explain Cyclical Movements in the Labor Wedge?
François Gourio and Leena Rudanko
WP-14-02
Early Public Banks
William Roberds and François R. Velde
WP-14-03
Mandatory Disclosure and Financial Contagion
Fernando Alvarez and Gadi Barlevy
WP-14-04
The Stock of External Sovereign Debt: Can We Take the Data at ‘Face Value’?
Daniel A. Dias, Christine Richmond, and Mark L. J. Wright
WP-14-05
Interpreting the Pari Passu Clause in Sovereign Bond Contracts:
It’s All Hebrew (and Aramaic) to Me
Mark L. J. Wright
WP-14-06
AIG in Hindsight
Robert McDonald and Anna Paulson
WP-14-07
On the Structural Interpretation of the Smets-Wouters “Risk Premium” Shock
Jonas D.M. Fisher
WP-14-08
Human Capital Risk, Contract Enforcement, and the Macroeconomy
Tom Krebs, Moritz Kuhn, and Mark L. J. Wright
WP-14-09
Adverse Selection, Risk Sharing and Business Cycles
Marcelo Veracierto
WP-14-10
Core and ‘Crust’: Consumer Prices and the Term Structure of Interest Rates
Andrea Ajello, Luca Benzoni, and Olena Chyruk
WP-14-11
The Evolution of Comparative Advantage: Measurement and Implications
Andrei A. Levchenko and Jing Zhang
WP-14-12
5
Working Paper Series (continued)
Saving Europe?: The Unpleasant Arithmetic of Fiscal Austerity in Integrated Economies
Enrique G. Mendoza, Linda L. Tesar, and Jing Zhang
WP-14-13
Liquidity Traps and Monetary Policy: Managing a Credit Crunch
Francisco Buera and Juan Pablo Nicolini
WP-14-14
Quantitative Easing in Joseph’s Egypt with Keynesian Producers
Jeffrey R. Campbell
WP-14-15
Constrained Discretion and Central Bank Transparency
Francesco Bianchi and Leonardo Melosi
WP-14-16
Escaping the Great Recession
Francesco Bianchi and Leonardo Melosi
WP-14-17
More on Middlemen: Equilibrium Entry and Efficiency in Intermediated Markets
Ed Nosal, Yuet-Yee Wong, and Randall Wright
WP-14-18
6
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