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Search in Asset Markets: Market Structure, Liquidity, and Welfare
By Ricardo Lagos and Guillaume Rocheteau*
In many markets, trade is facilitated by intermediaries, e.g., dealers, market makers, and specialists. The degree of market power that these
intermediaries have is commonly viewed as a
key determinant of the liquidity of the market
in which they operate. Recently, several regulations have been introduced to foster competition
in financial markets. The available evidence suggests that these reforms have had an impact on
trading costs and have also affected the incentives
of financial intermediaries to make markets.
In order to understand how the organization
of the market in which an asset is traded affects
the standard financial measures of liquidity, we
generalize the search-theoretic model of financial intermediation of Darrell Duffie, Nicolae
Gârleanu, and Lasse H. Pedersen (2005, DGP
hereafter) by introducing entry of dealers and a
nontrivial choice of asset holdings. (Asset holdings are restricted to lie in {0, 1} in DGP.) These
extensions allow us to discuss several dimensions of market liquidity, such as trading costs,
trade volume, and execution delays. Motivated
by the recent regulatory changes, we center our
analysis around the effects of changes in the
dealers’ market power.
We find that a reform that reduces the dealers’
market power can lead to lower trading costs,
higher trade volume, and a net entry of dealers,
in line with the evidence provided by Weston
(2000). Our model can also generate multiple
steady-state equilibria, suggesting that markets
with similar structures may differ considerably
in terms of their liquidity outcomes.
I. The Environment
Time is continuous and infinite. There are
two types of infinitely-lived agents: a unit measure of investors and a large measure of dealers.
There is one asset, one perishable good called
special good, and a general consumption good
defined as numéraire. The asset is durable, perfectly divisible, and in fixed supply A [ R1. Each
unit of the asset produces a unit flow of special
good. There is no market for the special good.
The numéraire good is produced and consumed
by all agents. The instantaneous utility function
of an investor is eiu(a) 1 c, where a [ R1 is the
consumption of special goods (which coincides
with the investor’s asset holdings), c [ R is the
net consumption of the numéraire good (c , 0
if the investor produces more of the numéraire
good than he consumes), and i [ {L, H} indexes
an idiosyncratic component, with eL , eH. The
function u(a) is continuous and twice differentiable, with u9 . 0 and u0 , 0. Each investor
receives an idiosyncratic preference shock with
Poisson arrival rate d. Conditional on the preference shock, the probability the investor draws
preference type i [ {L, H} is pi, with pL 1 pH
5 1. Dealers’ instantaneous utility is simply
c, their consumption of the numéraire good.
Dealers who choose to participate in the market
also incur a flow cost g . 0 which represents
the ongoing costs of running the dealership. All
agents discount at rate r . 0.
Participating dealers have continuous access
to a competitive asset market. An investor can
adjust his asset holdings only through a dealer
whom he contacts at random and bilaterally with
Poisson rate a. Once they have made contact, the
dealer and the investor negotiate over the quantity of assets that the dealer will acquire on behalf
of the investor and an intermediation fee. Then,
they execute the transaction and part ways.
The rate at which investors contact dealers,
a, is a continuously differentiable function of
* Lagos: Economics Department, New York University,
19 W. 4th Street, 6th Floor, New York, NY 10012, and
Federal Reserve Bank of Minneapolis (e-mail: ricardo.
[email protected]); Rocheteau: Research Department, Federal
Reserve Bank of Cleveland, PO Box 6387, Cleveland, OH
44101-1387 (e-mail: [email protected]). We
thank Joe Haubrich, Ed Nosal, Peter Rupert, Neil Wallace,
and Pierre-Olivier Weill for useful comments. Financial support from the C. V. Starr Center for Applied Economics at
New York University is gratefully acknowledged. The views
expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Cleveland,
the Federal Reserve Bank of Minneapolis, or the Federal
Reserve System.
See Michael Barclay et al. (1999) and James P. Weston
(2000) for accounts of these regulatory reforms.
In Table 6, Weston (2000) documents that the 1997
reform in the NASDAQ was accompanied by a net entry of
dealers, higher turnover of stocks, and lower spreads.
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VOL. 97 NO. 2
Search in Asset Markets: Market Structure, Liquidity, and Welfare
the measure of active dealers in the market, y.
Investors contact a dealer faster when the measure of active dealers is larger, i.e., ar 1 # 2 . 0.
Furthermore, a(0) 5 0, a(`) 5 `, a(`)/` 5 0,
and a9(0) 5 `. We capture the notion of competition for order flow by assuming that the rate
at which dealers contact investors, a 1 y 2 /y, is
decreasing in y.
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that eiur 1 a 2 is a weighted average of current and
future expected marginal utilities. The weight
on the current marginal utility decreases with
trading delays, 1/a, and with dealers’ bargaining power, h.
The free entry of dealers implies
(3) G(y, h) K
a1y2
y
II. Equilibrium
3 {fL[aH( y, h)]nHL 1 fH[aL(y, h)]nLH}
We restrict our attention to steady-state equilibria where the price of the asset in terms of the
numéraire good, p [ R1, is constant over time.
The value function of an investor with a preference type i [ {L, H} who holds a quantity of
assets a, Vi(a), satisfies
2 g 5 0,
(4) rVi(a) 5 eiu(a) 1 dpj 3 Vj 1 a 2 2 Vi 1 a 2 4
1 a(y) 3 Vi 1 ai 2 2 Vi 1 a 2 2 p 1 ai 2 a 2 2 fi 1 a 2 4 ,
where { j} 5 {L, H}\{i}. The investor enjoys a
utility flow eiu(a) from holding portfolio a. He
receives a new preference type with instantaneous probability dpj and enjoys a capital gain
Vj(a) 2 Vi(a). Upon contacting a dealer, with
instantaneous probability a 1 y 2 , the investor
buys ai 2 a (sells if negative) and pays the dealer
an intermediation fee, fi(a) [ R1. The quantity
traded, ai, and the fee, fi(a), correspond to the
Nash solution of the bargaining problem,
(ai, fi ) 5
arg max 3 Vi 1 ar 2 2Vi 1 a 2 2p 1 ar2a 2 2f 4 12h fh ,
(a9, f)
where h [ 3 0, 1 4 is the dealer’s bargaining
power. After some calculations,
(1) (2)
fi(a) 5
ēi u91ai 2 # rp, if ai 5 0,
ēi u91ai 2 5 rp, if ai . 0,
h 5 e# i 3 u 1 ai 2 2 u 1 a 2 4 2 rp 1 ai 2 a 2 6
r 1 a1y2 11 2 h2
where nji denotes the measure of investors with
preference type i who hold the quantity of assets
aj, given by
,
where ēi 5 {[r 1 a(y)(1 2 h)]ei 1 dē}/{r 1 d
1 a(y)(1 2 h)} and e 5 pL eL 1 pHeH. Note
(5) nHL 5 nLH 5
nii 5
dpLpH
a1y2 1 d
a 1 y 2 1 dpi
a1y2 1 d
,
pi ,
where aH 1 y,h 2 and aL 1 y,h 2 are implicitly defined
by (1). According to (3), the expected net profit
of dealers, G 1 y,h 2 , must be 0 in equilibrium.
The expected flow revenue of a dealer equals the
expected intermediation fee he earns when he
trades with a random investor, an event that occurs
with Poisson rate a 1 y 2 /y.
Finally, p is determined by the market-clearing condition g i,jnijai 5 A, which, using (4), can
be written as:
(6) pHaH 1 pLaL 5 A.
Condition (6) equates the aggregate demand for
the asset by investors (the left-hand side) to the
asset supply.
A steady-state equilibrium is a list {(nij), (ai, fi( · )), y, p} satisfying (1)–(6). It is easy to
show that limyS0 G(y, h) 5 ` and limyS` G(y, h)
5 2g for all h . 0. So an equilibrium with y .
0 exists, provided that h . 0.
III. Liquidity and Welfare
In order to isolate the direct effects of changes
in the bargaining power, h, on the dealers’ incentives to participate in the market from the general
200
equilibrium effects that operate through the implied changes in the investors’ asset holdings,
we first assume u(a) 5 a12s/(1 2 s) with s S
1/s
0. In this limiting case, ai 5 Ae1/s
i / g j pj ej
is independent of h, specifically, a L S 0 and
a H S A/pH. From (6), the asset price is p 5
1/s s
1 pHe1/s
H 1 pLeL 2 /rAs S eH /r . Thus, from
(2), fLH K fH(aL) S 0 (dealers do not charge a
fee when they sell) and
fHL ; fL 1 aH 2 S
h 1 eH 2 eL 2 A
3 r 1 d 1 a 1 y 2 1 1 2 h 2 4 pH
.
So, for given y, the trading cost fHL increases
with h. From (3), the net expected profit of a
dealer is
G 1 y,h 2 5
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a1y2
y
hAdpL 1 eH 2 eL 2
3r 1 d 1 a 1 y 2 1 1 2 h 2 4 3a 1 y 2 1 d 4
2 g.
The function G 1 y,h 2 is strictly decreasing in y,
so the equilibrium is unique. G is strictly increasing in h so that dy/dh . 0. This last result
implies that fHL increases with h. Finally, the
total volume of trade, V K a(y)nHL(aH 2 aL) 5
A{a(y)dpL/[a(y) 1 d]}, increases with y. The
following proposition summarizes the effects of
h on the different dimensions of liquidity.
Proposition 1: Assume s S 0. An increase
in dealers’ bargaining power raises trading
costs ( fHL ), reduces trading delays (1/a(y)) and
increases trade volume (V).
Assuming r < 0 , steady-state welfare is
oj[ 5L, H6 nH ejaH 2 yg. If a planner could choose
the measure of dealers in the market, he
would pick the unique y* that solves ar 1 y 2
{dpL(eH 2 eL)/[a(y) 1 d]2}A 5 g. (It is easy to
check that the planner would choose the same
portfolio allocations implied by the equilibrium.) By using G 1 y*, h 2 5 0 to solve for h, the
following proposition shows that the optimal
j
allocation can be implemented provided that
dealers enjoy a degree of market power that
gives them enough incentives to participate in
market making.
Proposition 2: Assume s S 0 and r < 0.
The constrained, efficient allocation is achieved
if and only if
h5 c
a 1 y* 2 1 d
y*ar 1 y* 2
[ 1 0, 1 2 .
a 1 y* 2 1 d 1 yar 1 y* 2 a 1 y* 2
d
IV. Portfolio Effects
Proposition 1 predicts a positive correlation
between trading costs and the number of dealers.
This result, however, hinges on the fact that aH
and aL become independent of h as s S 0. For
more general preferences, however, increases in
the dealers’ market power distorts individuals’
asset holdings, which in turn affects the volume
of trade and the profitability of dealers. The following lemma, immediate from (1), formalizes
these portfolio effects.
Lemma 1: (i) 0aH 1y, h 2/0h , 0 and 0aL 1y, h 2/
0h . 0; (ii) If h , 1 then 0aH 1y, h 2/0y . 0 and
0aL 1y, h 2/0y , 0.
According to part (i) of Lemma 1, a reduction
in h (for given y) raises the asset demand from
high marginal utility investors and reduces the
asset demand from low marginal utility investors.
Intuitively, investors put more weight on their current utility relative to their future expected utility
when dealers have less market power. Thus, all
else equal, a reduction in h makes the distribution of portfolios more disperse. But changes in
h also induce changes in y. According to part
(ii) of Lemma 1, the effects of an increase in y
on aH and aL are similar to those of a reduction in
h. As shown in the following propositions, these
portfolio effects can have important positive and
normative implications.
Proposition 3: For some parameter values,
there are multiple equilibria.
Consider the following parametrization: r 5
0.1, d 5 1, eH 5 1, eL 5 0, pL 5 pH 5 0.5,
s 5 1.7, A 5 1, g 5 0.12, and a 1 y 2 5 5y0.9 .
Search in Asset Markets: Market Structure, Liquidity, and Welfare
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VOL. 97 NO. 2
H
Figure 1. Effects of a Change in Dealers’ Market Power
The first panel of Figure 1 plots G 1 y, h 2 as a
function of y. It shows that there are multiple
(three) steady-state equilibria if h 5 0.51.
This multiplicity is removed when h is raised
to 0.55 (only the equilibrium with a high measure of dealers remains) or when h is reduced
to 0.44 (only the low equilibrium survives).
Thus, a small change in dealers’ market power
can have a dramatic effect on market outcomes. The possibility of multiple equilibria
also suggests that markets with identical fundamental structure can exhibit very different liquidity properties. An equilibrium with
slow execution, wide spreads, and low volume
of trade can coexist with another equilibrium
with fast execution, narrow spreads, and a large
volume of trade.
Proposition 4: A reduction in dealers’ market power can generate a reduction of trading
costs, a net entry of dealers, an increase in trade
volume, and an increase in welfare.
Keeping the investors’ contact rate constant,
an increase in h tends to reduce the dispersion of
portfolios. But there is also a general ­equilibrium
202
MAY 2007
AEA PAPERS AND PROCEEDINGS
effect according to which an increase in h can
raise y (and the investors’ contact rate), which
in turn can increase the dispersion of portfolios.
As shown in the second panel of Figure 1, the
direct effect dominates for large values of h. As
a result, the volume of trade (third panel) and the
dealers’ revenue decline, and execution delays
increase (fourth panel) with the dealers’ bargaining power for sufficiently high values of h.
The fifth panel shows that intermediation costs
per unit of asset traded (expressed as a proportion of the asset price) increase with h. Thus,
provided that h is high enough, a reduction in
dealers’ market power improves all dimensions
of liquidity. The volume of trade is larger and
trades are executed faster and at a lower cost.
Furthermore, a reduction in h can raise welfare
according to the last panel.
In the last five panels of Figure 1, we plot the equilibrium with the highest measure of dealers whenever there
are multiple equilibria.
In contrast to the linear case of Proposition 2, there
is no value of h that implements the constrained-efficient
References
Barclay, Michael J., William G. Christie, Jeffrey
H. Harris, Eugene Kandel, and Paul H. Schultz.
1999. “Effects of Market Reform on the Trad­
ing Costs and Depths of NASDAQ Stocks.”
Journal of Finance, 54(1): 1–34.
Duffie, Darrell, Nicolae Gârleanu, and Lasse H.
Pedersen. 2005. “Over-the-Counter Markets.”
Econometrica, 73(6): 1815–47.
Lagos, Ricardo, and Guillaume Rocheteau. 2006.
“Search in Asset Markets.” Federal Reserve
Bank of Cleveland Working Paper 06-07 and
Federal Reserve Bank of Minneapolis Staff
Report 375.
Weston, James P. 2000. “Competition on the
NASDAQ and the Impact of Recent Market Reforms.” Journal of Finance, 55(6): 2565–98.
allocation, since dealers’ market power distorts investors’
asset holdings. See Lagos and Rocheteau (2006) for further
details.