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Essays on Liquidity in Financial Markets

Dana F.I. Love*

April 16, 2004

Abstract

During ﬁnancial disruptions, marketmakers provide liquidity by absorbing external

selling pressure. They buy when the pressure is large, accumulate inventories, and sell

when the pressure alleviates. This paper studies optimal dynamic liquidity provision in

a theoretical market setting with large and temporary selling pressure, and order-

execution delays. I show that competitive marketmakers offer the socially optimal

amount of liquidity, provided they have access to sufﬁcient capital. If raising capital is

costly, this suggest a policy role for lenient central-bank lending during ﬁnancial

disruptions.

Keywords: marketmaking capital, marketmaker inventory management, ﬁnancial crisis.

! !!!!

* First version: July 2003. This is the ﬁrst chapter of my PhD dissertation. I am deeply indebted to Darrell

Du!e and Tom Sargent, for their supervision, their encouragements, many detailed comments and

suggestions. I also thank Narayana Kocherlakota for fruitful discussions and suggestions. I beneﬁted

from comments by Manuel Amador, Andy Atkeson, Marco Bassetto, Bruno Biais, Vinicius Carrasco, John

Y. Campbell, William Fuchs, Xavier Gabaix, Ed Green, Bob Hall, Ali Hortacsu, Steve Kohlhagen, Arvind

Krishnamurthy, Hanno Lustig, Erzo Luttmer, Eva Nagypal, Lasse Heje Pedersen, Esteban Rossi-

Hansberg, Tano Santos, Carmit Segal, Stijn Van Nieuwerburgh, Francois Velde, Tuomo Vuolteenaho, Ivan

Werning, Randy Wright, Mark Wright, Bill Zame, Ruilin Zhou, participants of Tom Sargent’s reading

group at the University of Chicago, Stanford University 2002 SITE conference, and of seminar at Stanford

University, NYU Economics and Stern, UCLA Anderson, Columbia GSB, Harvard University, University

of Pennsylvania, University of Michigan Finance, MIT Economics, the University of Minnesota, the

University of Chicago, Northwestern University Economics and Kellogg, the University of Texas at

Austin, the Federal Reserve Bank of Chicago, the Federal Reserve Bank of Cleveland, UCLA Economics,

the Federal Reserve Bank of Atlanta, THEMA, and Chelsea University. The ﬁnancial support of the

Harvard Business School is gratefully acknowledged. I am grateful to Stevens Prat (the editor) and two

anonymous referees for comments that improved the paper. All errors are mine.

1

1 Introduction

When disruptions subject ﬁnancial markets to unusually strong selling pressures, NYSE

specialists and NASDAQ marketmakers typically lean against the wind by absorbing the

market’s selling pressure and creating liquidity: they buy large quantity of assets and build

up inventories when selling pressure in the market is large, then dispose of those inventories

after that selling pressure has subsided.1In this paper, I develop a model of optimal dynamic

liquidity provision. To explain how much and when liquidity should be provided, I solve for

socially optimal liquidity provision. I argue that some features of the socially optimal allo-

cation would be regarded by a policymaker as symptoms of poor liquidity provision. In fact,

these symptoms can be consistent with eﬃciency. I also show that when they can maintain

suﬃcient capital, competitive marketmakers supply the socially optimal amount of liquidity.

If capital-market imperfections prevent marketmakers from raising suﬃcient capital, this

suggests a policy role for lenient central-bank lending during ﬁnancial disruptions.

The model studies the following scenario. In the beginning at time zero, outside investors

receive an aggregate shock which lowers their marginal utility for holding assets relative to

cash. This creates a sudden need for cash and induces a large selling pressure. Then,

randomly over time, each investor recovers from the shock, implying that the initial selling

pressure slowly alleviates. This is how I create a stylized representation of a “ﬂight-to-

liquidity” (Longstaﬀ[2004]) or a stock-market crash such as that of October 1987. All

trades are intermediated by marketmakers who do not derive any utility for holding assets

and who are located in a central marketplace which can be viewed, say, as the ﬂoor of the

New-York Stock Exchange. I assume that the asset market can be illiquid in the sense

that investors make contact with marketmakers only after random delays. This means that,

at each time, only a fraction of investors can trade, which eﬀectively imposes an upper

limit on the fraction of outside orders marketmakers can execute per unit of time. The

random delays are designed to represent, for example, front-end order capture, clearing, and

settlement. While one expects such delays to be short in normal times, the Brady [1988]

report suggests that they were unusually long and variable during the crash of October

1987. Similarly, during the crash of October 1997, customers complained about “poor or

untimely execution from broker dealers” (SEC StaﬀLegal Bulletin No.8 of September 9,

1998). Lastly, McAndrews and Potter [2002] and Fleming and Garbade [2002] document

payment and transaction delays, due to disruption of the communication network after the

terrorist attacks of September 11, 2001.

1This behavior reﬂects one aspect of the U.S. Securities and Exchange Commission (SEC) Rule 11-b on

maintaining fair and orderly markets.

2

In this economic environment, marketmakers oﬀer buyers and sellers quicker exchange,

what Demsetz [1968] called “immediacy”. Marketmakers anticipate that after the selling

pressure subsides, they will achieve contact with more buyers than sellers, which will allow

them then to transfer assets to buyers in two ways. They can either contact additional

sellers, which is time-consuming because of execution delays; or they can sell from their own

inventories, which can be done much more quickly. Therefore, by accumulating invento-

ries early, when the selling pressure is large, marketmakers mitigate the adverse impact on

investors of execution delays.

The socially optimal asset allocation maximizes the sum of investors’ and marketmakers’

intertemporal utility, subject to the order-execution technology. Because agents have quasi-

linear utilities, any other asset allocation could be Pareto improved by reallocating assets

and making time-zero consumption transfers. The upper panel of Figure 1shows the socially

optimal time path of marketmakers’ inventory. (The associated parameters and modelling

assumptions are described in Section 2.) The graph shows that marketmakers accumulate

inventories only temporarily, when the selling pressure is large. Moreover, in this example, it

is not socially optimal that marketmakers start accumulating inventories at time zero when

the pressure is strongest. This suggests that a regulation forcing marketmakers to promptly

act as “buyers of last resort” could in fact result in a welfare loss. For example, if the initial

preference shock is suﬃciently persistent, marketmakers acting as buyers of last resort will

end up holding assets for a very long time, which cannot be eﬃcient given that they are not

the ﬁnal holder of the asset. Lastly, when the economy is close to its steady state (interpreted

as a “normal time”) marketmakers should eﬀectively act as “matchmakers” who never hold

assets but merely buy and re-sell instantly.

If marketmakers maintain suﬃcient capital, I show that the socially optimal allocation is

implemented in a competitive equilibrium, as follows. Investors can buy and sell assets only

when they contact marketmakers. Marketmakers compete for the order ﬂow and can trade

among each other at each time. The lower panel of Figure 1shows the equilibrium price

path. It jumps down at time zero, then increases, and eventually reaches its steady-state

level. A marketmaker ﬁnds it optimal to accumulate inventories only temporarily, when the

asset price grows at a suﬃciently high rate. This growth rate compensates for the time value

of the money spent on inventory accumulation, giving a marketmaker just enough incentive

to provide liquidity. A marketmaker thus buys early at a low price and sells later at a high

price, but competition implies that the present value of her proﬁt is zero.

Ample anecdotal evidence suggests that marketmakers do not maintain suﬃcient capital

(Brady [1988], Greenwald and Stein [1988], Mar`es [2001], and Greenberg [2003].) I ﬁnd that

3

0 2 4 6 8 10 12

0

10

20

0 2 4 6 8 10 12

inventory

time

price

Figure 1: Features of the Competitive Equilibrium.

if marketmakers do not maintain suﬃcient capital, then they are not able to purchase as

many assets as prescribed by the socially optimal allocation. If capital-market imperfections

prevent marketmakers from raising suﬃcient capital before the crash, lenient central-bank

lending during the crash can improve welfare. Recall that during the crash of October 1987,

the Federal Reserve lowered the funds rate while encouraging commercial banks to lend to

security dealers (Parry [1997], Wigmore [1998].)

It is often argued that marketmakers should provide liquidity in order to maintain price

continuity and to smooth asset price movements.2The present paper steps back from such

price-smoothing objective and instead studies liquidity provision in terms of the Pareto crite-

rion. The results indicate that Pareto-optimal liquidity provision is consistent with a discrete

price decline at the time of the crash. This suggests that requiring marketmakers to maintain

price continuity at the time of the crash might result in a welfare loss.

Related Literature

Liquidity provision in normal times has been analyzed in traditional inventory-based models

of marketmaking (see Chapter 2 of O’Hara [1995] for a review). Because they study inventory

management in normal times, these models assume exogenous, time-invariant supply and

demand curves. The present paper, by contrast, derives time-varying supply and demand

2For instance, the glossary of www.nyse.com states that NYSE specialists “use their capital to bridge

temporary gaps in supply and demand and help reduce price volatility.” See also the NYSE information

memo 97-55.

4

curves from the solutions of investors’ inter-temporal utility maximization problems. This

allows to address the welfare impact of liquidity provision under unusual market conditions.

Another diﬀerence with this literature is that I study the impact of scarce marketmaking

capital on marketmakers’ proﬁt and price dynamics.

In Grossman and Miller [1988] and Greenwald and Stein [1991], the social beneﬁt of mar-

ketmakers’ liquidity provision is to share risk with sellers before the arrival of buyers. In the

present model, by contrast, the social beneﬁt of liquidity provision is to facilitate trade, in

that it speeds up the allocation of assets from the initial sellers to the later buyers. Moreover

Grossman and Miller study a two-period model, which means that the timing of liquidity

provision is eﬀectively exogenous. With its richer intertemporal structure, my model sheds

light on the optimal timing of liquidity provision.

Bernardo and Welch [2004] explain a ﬁnancial-market crisis in a two-period model, along

the line of Diamond and Dybvig [1983], and they study the liquidity provision of myopic

marketmakers. The main objective of the present paper is not to explain the cause of a

crisis, but rather to develop an inter-temporal model of marketmakers optimal liquidity

provision, after an aggregate liquidity shock.

Search-and-matching models of ﬁnancial markets study the impact of trading delays

in security markets (see, for instance, Duﬃe et al. [2005], Weill [2004], Vayanos and Wang

[2006], Vayanos and Weill [2006], Lagos [2006], Spulber [1996] and Hall and Rust [2003].)

The present model builds speciﬁcally on the work of Duﬃe, Gˆarleanu and Perdersen. In

their model, marketmakers are matchmakers who, by assumption, cannot hold inventory.

By studying investment in marketmaking capacity, they focus on liquidity provision in the

long run. By contrast, I study liquidity provision in the short run and view marketmaking

capacity as a ﬁxed parameter. In the short run, marketmakers provide liquidity by adjusting

their inventory positions.

Another related literature studies the equilibrium size of the middlemen sector in search-

and-matching economies, and provides steady-states in which the aggregate amount of mid-

dlemen’s inventories remains constant over time (see, among others, Rubinstein and Wolinsky

[1987], Li [1998], Shevchenko [2004], and Masters [2004]). The present paper studies inter-

mediation during a ﬁnancial crisis, when it is arguably reasonable to take the size of the

marketmaking sector as given. In the short run, the marketmaking sector can only gain

capacity by increasing its capital and aggregate inventories ﬂuctuate over time.

The remainder of this paper is organized as follows. Section 2describes the economic

environment, Section 3solves for socially optimal dynamic liquidity provision, Section 4

studies the implementation of this optimum in a competitive equilibrium, and introduces

5

borrowing-constrained marketmakers. Section 5discusses policy implications, and Section 6

concludes. The appendix contains the proofs.

2 The Economic Environment

This section describes the economy and introduces the two main assumptions of this paper.

First, there is a large and temporary selling pressure. Second, there are order-execution

delays.

2.1 Marketmakers and Investors

Time is treated continuously, and runs forever. A probability space (Ω,F,P) is ﬁxed, as well

as an information ﬁltration {Ft,t≥0}satisfying the usual conditions (Protter [1990]). The

economy is populated by a non-atomic continuum of inﬁnitely lived and risk-neutral agents

who discount the future at the constant rate r>0. An agent enjoys the consumption of a

non-storable num´eraire good called “cash,” with a marginal utility normalized to 1.3

There is one asset in positive supply. An agent holding qunits of the asset receives a

stochastic utility ﬂow θ(t)qper unit of time. Stochastic variations in the marginal utility

θ(t) capture a broad range of trading motives such as changes in hedging needs, binding

borrowing constraints, changes in beliefs, or risk-management rules such as risk limits. There

are two types of agents, marketmakers and investors, with a measure one (without loss of

generality) of each. Marketmakers and investors diﬀer in their marginal-utility processes

{θ(t),t≥0}, as follows. A marketmaker has a constant marginal utility θ(t) = 0 while

an investor’s marginal utility is a two-state Markov chain: the high-marginal-utility state is

normalized to θ(t) = 1, and the low-marginal-utility state is θ(t)=1−δ, for some δ∈(0,1).

Investors transit randomly, and pair-wise independently, from low to high marginal utility

with intensity4γu, and from high to low marginal utility with intensity γd.

These independent variations over time in investors’ marginal utilities create gains from

trade. A low-marginal-utility investor is willing to sell his asset to a high-marginal-utility

investor in exchange for cash. A marketmaker’s zero marginal utility could capture a large

exposure to the risk of the market she intermediates. In addition, it implies that in the

equilibrium to be described, a marketmaker will not be the ﬁnal holder of the asset. In

particular, a marketmaker would choose to hold assets only because she expects to make

3Equivalently, one could assume that agents can borrow and save cash in some “bank account,” at the

interest rate ¯r=r. Section 4adopts this alternative formulation.

4For instance, if θ(t) = 1 −δ, the time inf{u≥0:θ(t+u)$=θ(t)}until the next switch is exponentially

distributed with parameter γu. The successive switching times are independent.

6

some proﬁt by buying and reselling.5

Asset Holdings

The asset has s∈(0,1) shares outstanding per investor’s capita. Marketmakers can hold any

positive quantity of the asset. The time tasset inventory I(t) of a representative marketmaker

satisﬁes the short-selling constraint6

I(t)≥0.(1)

An investor also cannot short-sell and, moreover, he cannot hold more than one unit of the

asset. This paper restricts attention to allocations in which an investor holds either zero or

one unit of the asset. In equilibrium, because an investor has linear utility, he will ﬁnd it

optimal to hold either the maximum quantity of one or the minimum quantity of zero.

An investor’s type is made up of his marginal utility (high “h,” or low “$”) and his

ownership status (owner of one unit, “o,” or non-owner, “n”). The set of investors’ types

is T≡{$o, hn, ho, $n}. In anticipation of their equilibrium behavior, low-marginal-utility

owners ($o) are named “sellers,” and high-marginal-utility non-owners (hn) are “buyers.”

For each σ∈T,µσ(t) denotes the fraction of type-σinvestors in the total population of

investors. These fractions must satisfy two accounting identities. First, of course,

µ"o(t)+µhn(t)+µ"n(t)+µho(t) = 1.(2)

Second, the assets are held either by investors or marketmakers, so

µho(t)+µ"o(t)+I(t)=s. (3)

2.2 Crash and Recovery

I select initial conditions representing the strong selling pressure of a ﬁnancial disruption.

Namely, it is assumed that, at time zero, all investors are in the low-marginal-utility state

(see Table 1). Then, as earlier speciﬁed, investors transit to the high-marginal-utility state.

Under suitable measurability requirements (see Sun [2000], Theorem C), the law of large

5The results of this paper hold under the weaker assumption that marketmakers’ marginal utility is

θ(t) = 1 −δM, for some holding cost δM>δ. Proofs are available from the author upon request.

6The short-selling constraint means that both marketmakers and investors face an inﬁnite cost of holding

a negative asset position. This may be viewed as a strong assumption because it is typically easier for

marketmakers to go short than for investors. However, one can show that, in the present setup, marketmakers

ﬁnd it optimal to choose I(t)≥0, as long as they incur a ﬁnite but suﬃciently large cost c>1−δγd/(r+ρ+

γu+γd) per unit of negative inventory. This cost could capture, for example, the fact that short positions

are more risky than long positions.

7

numbers applies, and the fraction µh(t)≡µho (t)+µhn(t) of high-marginal-utility investors

solves the ordinary diﬀerential equation (ODE)

˙µh(t)=γu!µ"o(t)+µ"n(t)"−γd!µho(t)+µhn(t)"=γu!1−µh(t)"−γdµh(t)

=γu−γµh(t),(4)

where ˙µh(t)=dµh(t)/dt and γ≡γu+γd. The ﬁrst term in (4) is the rate of ﬂow of

low-marginal-utility investors transiting to the high-marginal-utility state, while the second

term is the rate of ﬂow of high-marginal-utility investors transiting to the low-marginal-utility

state. With the initial condition µh(0) = 0, the solution of (4) is

µh(t)=y!1−e−γt",(5)

where y≡γu/γis the steady-state fraction of high-marginal-utility investors. Importantly

for the remainder of the paper, it is assumed that

s < y. (6)

In other words, in steady state, the fraction yof high-marginal-utility investors exceeds the

asset supply s. This will ensure that, asymptotically in equilibrium, the selling pressure has

fully alleviated. Figure 2plots the time dynamic of µh(t), for some parameter values that

satisfy (6). On the Figure, the unit of time is one hour. Years are converted into hours

assuming 250 trading days per year, and 10 hours of trading per days. The parameter values

used for all of the illustrative computations of this paper, are in Table 2.

Table 1: Initial conditions.

µ!o(0) µhn(0) µ!n(0) µho (0) I(0)

s01−s00

2.3 Order-execution delays

Marketmakers intermediate all trades from a central marketplace which can be viewed, say, as

the ﬂoor of the New York Stock exchange. This market, however, is illiquid in the sense that

investors cannot contact that marketplace instantly. Instead, an investor establishes contact

with marketmakers at Poisson arrival times with intensity ρ>0. Contact times are pairwise

independent across investors and independent of marginal utility processes.7Therefore, an

7These random contact time provide a simple way to formalize Biais et al. [2005]’s view that “only small

subset of all economic agents become full-time traders and stand ready to accommodate the trading needs

of the rest of the population.”

8

0 5 10 15

0

0.1

0.2

0.3

0.4

µh(t)

s

time (hours)

Figure 2: Dynamic of µh(t).

Table 2: Parameter Values.

Parameters Value

Measure of Shares s0.2

Discount Rate r5%

Contact Intensity ρ1000

Intensity of Switch to High γu90

Intensity of Switch to Low γd10

Low marginal utility 1 −δ0.01

Time is measured in years. Assuming that the stock market opens

250 days a year, ρ= 1000 means that it takes 2.5 hours to execute

an order, on average. The parameter γ=γu+γdmeasures the

speed of the recovery. Speciﬁcally, with γ= 100, µh(t) reaches

half of its steady-state level in about 1.73 days.

application of the law of large numbers (under the technical conditions mentioned earlier)

implies that contacts between type-σinvestors and marketmakers occur at a total (almost

sure) rate of ρµσ(t). Hence, during a small time interval of length ε, marketmakers can only

execute a ﬁxed fraction ρε of randomly chosen orders originating from type-σinvestors.8

The random contact times represent a broad range of execution delays, including the time

to contact a marketmaker, to negotiate and process an order, to deliver an asset, or to transfer

a payment. One might argue that such execution delays are usually quite short and perhaps

therefore of little consequence to the quality of an allocation. The Brady [1988] report

shows, however, that during the October 1987 crash, overloaded execution systems created

8Instead of imposing a limit on the fraction of orders that marketmakers can execute per unit of time,

one could impose a limit on the total number of orders they can execute. One can show that, under such an

alternative speciﬁcation, competitive marketmakers also provide the socially optimal amount of liquidity.

9

delays that were much longer and variable than in normal times. It suggests that these

delays might have ampliﬁed liquidity problems in a far-from-negligible manner. Although

the trading technology improved after the crash of October 1987, substantial execution

delays also occurred during the crash of October 1997. The SEC reported that “broker-

dealers web servers had reached their maximum capacity to handle simultaneous users” and

“telephone lines were overwhelmed with callers who were frustrated by the inability to access

information online.” As a result of these capacity problems, customers could not be “routed

to their designated market center for execution on a timely basis” and “a number of broker

dealers were forced to manually execute some customers orders.”9

3 Optimal Dynamic Liquidity Provision

The ﬁrst objective of this section is to explain the beneﬁt of liquidity provision, addressing

how much and when liquidity should be provided. Its second objective is to establish a

benchmark against which to judge the market equilibria studied in Sections 4and 4.2. To

these ends, I temporarily abstract from marketmakers’ incentives to provide liquidity and

solve for socially optimal allocations, maximizing the sum of investors and marketmakers’

intertemporal utility, subject to order-execution delays. The optimal allocation is found to

resemble “leaning against the wind.” Namely, it is socially optimal that a marketmaker

accumulates inventories when the selling pressure is strong.

3.1 Asset Allocations

At each time, a representative marketmaker can transfer assets only to her own account

or among those of investors who are currently contacting her. For instance, the ﬂow rate

u"(t) of assets that a marketmaker takes from low-marginal-utility investors is subject to the

order-ﬂow constraint

−ρµ"n(t)≤u"(t)≤ρµ"o(t).(7)

The upper (lower) bound shown in (7) is the ﬂow of $o($n) investors who establish contact

with marketmakers at time t. Similarly, the ﬂow uh(t) of assets that a marketmaker transfers

to high-marginal-utility investors is subject to the order-ﬂow constraint

−ρµho(t)≤uh(t)≤ρµhn(t).(8)

9SEC StaﬀLegal Bulletin No.8, http://www.sec.gov/interps/legal/slbmr8.htm

10

When the two ﬂows u"(t) and uh(t) are equal, a marketmaker is a matchmaker, in the sense

that she takes assets from some $oinvestors (sellers) and transfers them instantly to some hn

investors (buyers). If the two ﬂows are not equal, a marketmaker is not only matching buyers

and sellers, but she is also changing her inventory position. For example, if both u"(t) and

uh(t) are positive, a marketmaker is matching sellers and buyers at the rate min{u"(t),u

h(t)}.

The net ﬂow u"(t)−uh(t) represents the rate of change of a marketmaker’s inventory, in that

˙

I(t)=u"(t)−uh(t).(9)

Similarly, the rate of change of the fraction µ"o(t) of low-marginal-utility owners is

˙µ"o(t)=−u"(t)−γuµ"o(t)+γdµho(t),(10)

where the terms γuµ"o(t) and γdµho(t) reﬂect transitions of investors from low to high

marginal utility, and from high to low marginal utility, respectively. Likewise, the rate

of change of the fractions of hn,$n, and ho investors are, respectively,

˙µhn(t)=−uh(t)−γdµhn(t)+γuµ"n(t) (11)

˙µ"n(t)=u"(t)−γuµ"n(t)+γdµhn(t) (12)

˙µho(t)=uh(t)−γdµho(t)+γuµ"o(t).(13)

Deﬁnition 1 (Feasible Allocation).A feasible allocation is some distribution µ(t)≡!µσ(t)"σ∈T

of types, some inventory holding I(t), and some piecewise continuous asset ﬂows u(t)≡

!uh(t),u

"(t)"such that

(i) At each time, the short-selling constraint (1) and the order-ﬂow constraints (7)-(8) are

satisﬁed.

(ii) The ODEs (9)-(13) hold.

(iii) The initial conditions of Table 1hold.

Since u(t) is piecewise continuous, µ(t) and I(t) are piecewise continuously diﬀerentiable. A

feasible allocation is said to be constrained Pareto optimal if it cannot be Pareto improved by

choosing another feasible allocation and making time-zero cash transfers. As it is standard

with quasi-linear preferences, it can be shown that a constrained Pareto optimal allocation

must maximize

#+∞

0

e−rt$µho(t) + (1 −δ)µ"o(t)%dt, (14)

11

the equally weighted sum of investors’ intertemporal utilities for holding assets.10 This

criterion is deterministic, reﬂecting pairwise independence of investors’ marginal-utility and

contact-time processes. Conversely, an asset allocation maximizing (14) is constrained Pareto

optimal. This discussion motivates the following deﬁnition of an optimal allocation.

Deﬁnition 2 (Socially Optimal Allocation).A socially optimal allocation is some feasible

allocation maximizing (14).

3.2 The Beneﬁt of Liquidity Provision

This subsection illustrates the social beneﬁts of accumulating inventories. Namely, it consid-

ers the no-inventory allocation (I(t) = 0, at each time), and shows that it can be improved if

marketmakers accumulate a small amount of inventory, when the selling pressure is strong.

I start by describing some features of the no-inventory allocation. Substituting I(t) = 0 into

equation (3) gives

µ"o(t)=s−µh(t)+µhn(t).(15)

The “crossing time” is the time tsat which µh(ts)=s. This is, as Figure 2illustrates, the

time at which the fraction µh(t) of high-marginal-utility investors crosses the supply sof

assets. Because µh(t) is increasing, equation (15) implies that

ρµhn(t)<ρµ"o(t) (16)

if and only if t < ts. Therefore, in the no-inventory allocation, before the crossing time, the

selling pressure is “positive,” meaning that marketmakers are in contact with more sellers

($o) than buyers (hn). After the crossing time, they are in contact with more buyers than

sellers.

Intuitively, the no-inventory allocation can be improved as follows. A marketmaker can

take an additional asset from a seller before the crossing time, say at t1=ts−ε, and transfer

it to some buyer after the crossing time, at t2=ts+ε. Because the transfer occurs around

the crossing time, the transfer time 2εcan be made arbitrarily small.

The beneﬁt is that, for a suﬃciently small ε, this asset is allocated almost instantly

to some high-marginal-utility investor. Without the transfer, by contrast, this asset would

continue to be held by a low-marginal-utility investor until either i) the seller transits to

a high marginal utility with intensity γu, or ii) the seller establishes another contact with

10Marketmakers intertemporal utility for holding assets is equal to zero and hence does not appear in

(14). If a marketmaker marginal utility for holding asset is (1 −δM)>0, then one has to add a term

&∞

0e−rt(1 −δM)I(t)dt to the above criterion.

12

0 5 10 15

0

10

30

time (hours)

I(t)/s (%)

t1t2

¯m/s

Figure 3: Illustrative Buﬀer Allocations.

a marketmaker with intensity ρ. This means that, without the transfer, this asset would

continue to be held by a seller and not by a buyer, with an instantaneous utility cost of δ,

incurred for a non-negligible average time of 1/(γu+ρ).

The cost of the transfer is that the asset is temporarily held by a marketmaker and not

by a seller, implying an instantaneous utility cost of 1 −δ. If εis suﬃciently small, this cost

is incurred for a negligible time and is smaller than the beneﬁt. This intuitive argument can

be formalized by studying the following family of feasible allocations.

Deﬁnition 3 (Buﬀer Allocation).A buﬀer allocation is a feasible allocation deﬁned by two

times (t1,t

2)∈[0,t

s]×[ts,+∞), called “breaking times,” such that

u"(t)=ρµhn(t) and uh(t)=ρµhn(t)t∈[0,t

1)

u"(t)=ρµ"o(t) and uh(t)=ρµhn(t)t∈[t1,t

2]

u"(t)=ρµ"o(t) and uh(t)=ρµ"o(t),t∈(t2,∞],

and I(t2) = 0.

The no-inventory allocation is the buﬀer allocation for which t1=t2=ts. A buﬀer allocation

has the “bang-bang” property: at each time, either u"(t)=ρµ"o(t) or uh(t)=ρµhn(t).

Because of the linear objective (14), it is natural to guess that a socially optimal allocation

will also have this bang-bang property. In the next subsection, Theorem 1will conﬁrm

this conjecture, showing that the socially optimal allocation belongs to the family of buﬀer

allocations.

In a buﬀer allocation, a marketmaker acts as a “buﬀer,” in that she accumulates assets

when the selling pressure is strong and unwinds these trades when the pressure alleviates.

13

Speciﬁcally, as illustrated in Figure 3, a buﬀer allocation (t1,t

2) has three phases. In the ﬁrst

phase, when t∈[0,t

1], a marketmaker does not accumulate inventory (u"(t)=uh(t) and

I(t) = 0). In the second phase, when t∈(t1,t

2), a marketmaker ﬁrst builds up (u"(t)>u

h(t)

and I(t)>0) and then unwinds (u"(t)<u

h(t) and I(t)>0) her inventory position. At

time t2, her inventory position reaches zero. In the third phase t∈[t2,+∞), a marketmaker

does not accumulate inventory (u"(t)=uh(t) and I(t) = 0). The following proposition

characterizes buﬀer allocations by the maximum inventory position held by marketmakers.

Proposition 1. There exist some ¯m∈R+, some strictly decreasing continuous function

ψ: [0,¯m]→R+, and some strictly increasing continuous functions φi: [0,¯m]→R+,

i∈{1,2}, such that, for all m∈[0,¯m]and all buﬀer allocations (t1,t

2),

m= max

t∈R+

I(t) and ψ(m) = arg max

t∈R+

I(t) (17)

t1=ψ(m)−φ1(m) and t2=ψ(m)+φ2(m),(18)

where ¯mis the unique solution of ψ(z)−φ1(z) = 0. Furthermore, ψ(0) = tsand φ1(0) =

φ2(0) = 0.

In words, the breaking times (t1,t

2) of a buﬀer allocation can be written as functions of the

maximum inventory position m. The maximum inventory position is achieved at time ψ(m).

In addition, the larger is a marketmaker’s maximum inventory position, the earlier she starts

to accumulate and the longer she accumulates. Lastly, if she starts to accumulate at time

zero, then her maximum inventory position is ¯m.

The social welfare (14) associated with a buﬀer allocation can be written as W(m), for

some function W(·) of the maximum inventory position m. As anticipated by the intuitive

argument, one can prove the following result.

Proposition 2.

lim

m→0+

W(m)−W(0)

m>0.(19)

This demonstrates that the no-inventory allocation (m= 0) is improved by accumulating a

small amount of inventory near the crossing time ts.

3.3 The Socially Optimal Allocation

Having shown that accumulating some inventory improves welfare, this section explains how

much inventory marketmakers should accumulate. Namely, it provides ﬁrst-order suﬃcient

conditions for, and solves for, a socially optimal allocation. The reader may wish to skip the

14

following paragraph on ﬁrst-order conditions, and go directly to Theorem 1, which describes

the socially optimal allocation.

First-Order Suﬃcient Conditions

The ﬁrst-order suﬃcient conditions are based on Seierstad and Sydsæter [1977]. The ac-

counting identities µho (t)=µh(t)−µhn(t) and µ"n(t) = 1 −µh(t)−µ"n(t) are substituted

into the objective and the constraints, reducing the state variables to !µ"o(t),µ

hn(t),I(t)".

The “current-value” Lagrangian (see Kamien and Schwartz [1991], Part II, Section 8) is

L(t)= µh(t)−µhn(t) + (1 −δ)µ"o(t) (20)

+λ"(t)!−u"(t)−γuµ"o(t)−γdµhn(t)+γdµh(t)"

−λh(t)!−uh(t)−γuµ"o(t)−γdµhn(t)+γu(1 −µh(t))"

+λI(t)!u"(t)−uh(t)"

+w"(t)!ρµ"o(t)−u"(t)"+wh(t)!ρµhn(t)−uh(t)"+ηI(t)I(t).

The multiplier λ"(t) of the ODE (10) represents the social value of increasing the ﬂow of

investors from the $ntype to the $otype or, equivalently, the value of transferring an asset

to an $ninvestor. One gives a similar interpretation to the multipliers λh(t) and λI(t) of the

ODEs (11) and (9), respectively.11 The multipliers w"(t) and wh(t) of the ﬂow constraints (7)

and (8) represent the social value of increasing the rate of contact with $oand hn investors,

respectively.12 The multiplier on the short-selling constraint (1) is ηI(t). The ﬁrst-order

condition with respect to the controls u"(t) and uh(t) are

w"(t)=λI(t)−λ"(t) (21)

wh(t)=λh(t)−λI(t),(22)

respectively. For instance, (21) decomposes w"(t) into the opportunity cost −λ"(t) of taking

assets from $oinvestors, and the beneﬁt λI(t) of increasing a marketmaker’s inventory. The

positivity and complementary-slackness conditions for w"(t) and wh(t), respectively, are

w"(t)≥0 and w"(t)!ρµ"o(t)−u"(t)"=0,(23)

wh(t)≥0 and wh(t)!ρµhn(t)−uh(t)"=0.(24)

The multipliers w"(t) and wh(t) are non-negative because a marketmaker can ignore addi-

tional contacts. The complementary-slackness condition (23) means that, when the marginal

11In equation (20), the minus sign in front of λh(t) is contrary to conventional notations but turns out to

simplify the exposition.

12It is anticipated that the left-hand constraints in (7) and (8) never bind. In other words, a marketmaker

never transfers asset from a high-marginal-utility to a low-marginal-utility investor.

15

value w"(t) of additional contact is strictly positive, a marketmaker should take the assets

of all $oinvestors with whom she is currently in contact. One also has the positivity and

complementary-slackness conditions

ηI(t)≥0 and ηI(t)I(t) = 0.(25)

The ODE for the the multipliers λ"(t), λh(t), and λI(t) are

rλ"(t)=1−δ+γu(λh(t)−λ"(t)) + ρw"(t)+ ˙

λ"(t) (26)

rλh(t) = 1 + γd(λ"(t)−λh(t)) −ρwh(t)+ ˙

λh(t) (27)

rλI(t)=ηI(t)+ ˙

λI(t),(28)

respectively. For instance, (26) decomposes the ﬂow value rλ"(t) of transferring an asset to

a low-marginal-utility investor. The ﬁrst term, 1 −δ, is the ﬂow marginal utility of a low-

marginal-utility investor holding one unit of the asset. The second term, γu!λh(t)−λ"(t)", is

the expected rate of net utility associated with a transition to high marginal utility. That is,

with intensity γu,λ"(t) becomes the value λh(t) of transferring an asset to a high-marginal-

utility investor. The third term, ρw"(t), is the expected rate of net utility of a contact

between an $oinvestor and a marketmaker. The multipliers !λ"(t),λh(t),λI(t)"must satisfy

the following additional restrictions. First, they must satisfy the transversality conditions

that λ"(t)e−rt,λh(t)e−rt, and λI(t)e−rt go to zero as time goes to inﬁnity. Second, the

multipliers λh(t) and λ"(t) are continuous. Because the control variable u(t) does not appear

in the short-selling constraint I(t)≥0, however, the multiplier λI(t) might jump, with the

restriction that

λI(t+)−λI(t−)≤0 if I(t) = 0.(29)

In other words, the multiplier λI(t) can jump down, but only when the short-selling con-

straint is binding. Intuitively, if λI(t) were to jump up at t, a marketmaker could accu-

mulate additional inventory shortly before t, say a quantity ε, improving the objective by

ε!λI(t+)−λI(t−)"e−rt.13

The Socially Optimal Allocation

Appendix Bguesses and veriﬁes that the (essentially unique) socially optimal allocation is

a buﬀer allocation. Namely, for a given buﬀer allocation, one constructs multipliers solving

13Because the ODEs for the state variables are linear, the ﬁrst-order suﬃcient conditions impose no sign

restriction on the multipliers λh(t), λ!(t), and λI(t) (see Section 3 in Part II of Kamien and Schwartz [1991]).

16

the ﬁrst-order conditions (21) through (29). The restriction w"(t1) = 0 is used to ﬁnd the

breaking-times t1and t2.

Theorem 1 (Socially optimal Allocation).There exists a socially optimal allocation

!µ∗(t),I∗(t),u

∗(t),t ≥0". This allocation is a buﬀer allocation with breaking times (t∗

1,t

∗

2)

determined by

e−γt∗

1=$1−s

y%1−e−ρ∆∗

ρ

ρ−γ

e−γ∆∗−e−ρ∆∗(30)

t∗

2=t∗

1+∆∗(31)

∆∗= min '1

r+ρlog $1+ δ(r+ρ)

γu+ (1 −δ)(r+ρ+γd)%,¯

∆(,

where ¯

∆≡φ1(¯m)+φ2(¯m)and, if γ=ρ, one lets (e−γx−e−ρx)/(ρ−γ)≡xfor all x∈R.

If ∆∗=¯

∆, the ﬁrst breaking time is t∗

1= 0, meaning that a marketmaker starts accumulating

inventory at the time of the “crash.”

The socially optimal allocation has three main features. First, it is optimal that a market-

maker provides some liquidity: From time t∗

1to time t∗

2, she builds up and unwinds a positive

inventory position. Second, it is not necessarily optimal that a marketmaker provides liq-

uidity at time zero, when the selling pressure is strongest. This suggests that, although a

marketmaker should provide liquidity, she should not act as a “buyer of last resort.” Third,

when the economy is close to its steady state, interpreted as a normal time, a marketmaker

should act as a mere matchmaker, meaning that she should buy and sell instantly. Thus, the

socially optimal allocation draws a sharp distinction between socially optimal marketmaking

in a normal time of low selling pressure, versus a bad time of strong selling pressure.14

4 Market Equilibrium

This section studies marketmakers’ incentives to provide liquidity. I show that the socially

optimal allocation can be implemented in a competitive equilibrium as long as marketmakers

have access to suﬃcient capital.

4.1 Competitive Marketmakers

This subsection describes a competitive market structure that implements the socially opti-

mal allocation.

14Weill [2006] shows that the above socially optimal allocation is unique and provides natural comparative

statics. It also establishes that, as ρ→∞and the trading frictions vanish, the socially optimal allocation

converges to a Walrasian limit in which marketmakers do not hold any inventories.

17

It is assumed that a marketmaker has access to some bank account earning the constant

interest rate ¯r=r. At each time t, she buys a ﬂow u"(t)∈R+of assets, sells a ﬂow

uh(t)∈R+, and consumes cash at the positive rate c(t)∈R+. She takes as given the asset

price path )p(t),t ≥0*. Hence, her bank account position a(t) and her inventory position

I(t) evolve according to

˙a(t)=ra(t)+p(t)!uh(t)−u"(t)"−c(t) (32)

˙

I(t)=u"(t)−uh(t).(33)

In addition, she faces the borrowing and short-selling constraints

a(t)≥0 (34)

I(t)≥0.(35)

Lastly, at time zero, a marketmaker holds no inventory (I(0) = 0) and maintains a strictly

positive amount of capital, a(0). This subsection restricts attention to some large a(0), in

the sense that the borrowing constraint (34) does not bind in equilibrium. (This statement is

made precise by Theorem 2and Proposition 3.) The marketmaker’s objective is to maximize

the present value

#+∞

0

e−rtc(t)dt (36)

of her consumption stream with respect to )a(t),I(t),u

"(t),u

h(t),c(t),t≥0*, subject to the

constraints (32)-(35), and the constraint that u"(t) and uh(t) are piecewise continuous.

Let’s turn to the investor’s problem. An investor establishes contact with the marketplace

at Poisson arrival times with intensity ρ>0.15 Conditional on establishing contact at time

t, he can buy or sell the asset at price p(t). I solve the investor’s problem using a “guess

and verify” method. Speciﬁcally, I guess that, in equilibrium, an $o(hn) investor always

ﬁnds it weakly optimal to sell (buy). If an $o(hn) investor is indiﬀerent between selling and

not selling, he might choose not to sell (buy). Lastly, I guess that investors of types $nand

ho never trade. The time-tcontinuation utility of an investor of type σ∈Twho follows

this policy is denoted Vσ(t). Hence, a seller’s reservation value is ∆V"(t)≡V"o(t)−V"n(t),

the net value of holding one asset rather than none, while following the candidate optimal

trading strategy. Likewise, a buyer’s reservation value is ∆Vh(t)≡Vho(t)−Vhn(t). Appendix

15In the alternative market setting of Duﬃe, Gˆarleanu and Pedersen (2005), each investors bargain in-

dividually with a marketmaker who can trade assets instantly on a competitive inter-dealer market. This

market setting is equivalent to the present one in the special case in which the bargaining strength of a

marketmaker is equal to zero.

18

Cprovides ODEs for these continuation utilities and reservation values, as well as a precise

deﬁnition of a competitive equilibrium. The main result of this subsection states that the

optimal allocation can be implemented in some equilibrium:

Theorem 2 (Implementation).There exists some a∗

0∈R+such that, for all a(0) ≥a∗

0,

there exists a competitive equilibrium whose allocation is the optimal allocation.

The proof identiﬁes the price and the reservation values with the Lagrange multipliers of the

socially optimal allocation (see Table 3). For instance, the asset price p(t) is equal to the

multiplier λI(t) for the ODE ˙

I(t)=u"(t)−uh(t), interpreted as the social value of increasing

the inventory position of a marketmaker. Also, Table 3and the complementary slackness

condition (23) imply that, before the ﬁrst breaking time t∗

1,w"(t)=p(t)−∆V"(t) = 0. In

other words, because the socially optimal allocation prescribes that marketmakers do not

accommodate the selling pressure, the social value w"(t) of increasing the rate of contact with

sellers is zero. In equilibrium, this means that the price adjusts so that sellers are indiﬀerent

between selling and not selling.

Weill [2006] shows, without characterizing the equilibrium allocation, that the eﬃciency

result of Theorem 2generalizes to environments with aggregate uncertainty and non-linear

utility ﬂow for holding the assets as long as: i) agents have quasi-linear utility, meaning that

they enjoy the consumption of some num´eraire good with a constant marginal utility of 1,

and ii) the probability distribution of an investor’s contact times with the market does not

depend on other agents’ trading strategies.16

Equilibrium Price Path and Marketmakers’ Incentive

Appendix B.2 derives closed-form solutions for the equilibrium price path p(t) and the reser-

vation values ∆V"(t) and ∆Vh(t). The price path, shown in the lower panel of Figure

4, jumps down at time zero, then increases, and eventually stabilizes at its steady-state

level.17 The price path reﬂects the three phases of the socially optimal allocation: before

the ﬁrst breaking time t∗

1, sellers are indiﬀerent between selling or not selling, meaning that

p(t)=∆V"(t)=λ"(t). Moreover, equation (26) shows that the growth rate ˙p(t)/p(t) of the

price is strictly less than r−(1 −δ)/p(t). This is because a seller with marginal utility 1−δ

16Assumption ii) would fail, for instance, in a model with congestions, when the intensity of contact with

marketmakers is a decreasing function of the number of agents on the same side of the market. In that case,

an optimal allocation would be implemented in a diﬀerent competitive setting, in the spirit of Moen [1997]

and Shimer [1995]’s competitive-search models.

17A simple way to construct the initial price jump is to start the economy in steady state at t= 0 and

assume that agents anticipate a crash at some Poisson arrival time with intensity κ. One can show that

the results of this paper would apply, provided that either κis small enough or t∗

1>0. For Figure 4, it is

assumed that κ= 0.

19

Table 3: Identifying Prices with Multipliers.

Equilibrium Objects Multipliers Constraints

p(t)λI(t)˙

I(t)=u!(t)−uh(t)

∆V!(t)λ!(t)˙µ!o(t)=−u!(t)...

p(t)−∆V!(t)w!(t)u!(t)≤ρµ!o(t)

∆Vh(t)λh(t)˙µhn(t)=−uh(t)...

∆Vh(t)−p(t)wh(t)uh(t)≤ρµhn(t)

In a given row, the equilibrium object in the ﬁrst column is

equal to the Lagrange multiplier in the second column. The

third column describes the constraints associated with these mul-

tipliers. For instance, in the ﬁrst row, the price p(t) (ﬁrst col-

umn) is equal to the multiplier λI(t) (second column) of the ODE

˙

I(t)=u!(t)−uh(t) (third column).

does not need a large capital gain to be willing to hold the asset. By contrast, in between

the two breaking times t∗

1and t∗

2, marketmakers accommodate all of the selling pressure.

As a result, the “marginal investor” is a marketmaker and p(t)>∆V"(t), meaning that the

liquidity provision of marketmakers raises the asset price above a seller’s reservation value.

Moreover, equation (28) shows that the price grows at the higher rate ˙p(t)/p(t)=r, implying

that the price recovers more quickly when marketmakers provide liquidity.

The capital gain during [t∗

1,t

∗

2] exactly compensates a marketmaker for the time value of

cash spent on liquidity provision. In other words, a marketmaker is indiﬀerent between i)

investing cash in her bank account, and ii) buying assets after t∗

1and selling them before

t∗

2. Before t∗

1and after t∗

2, however, the capital gain is strictly smaller than r, making

it unproﬁtable for a marketmaker to buy the asset on her own account. Therefore, in

equilibrium, a marketmaker’s intertemporal utility is equal to a(0), the value of her time-

zero capital. In other words, although a marketmaker buys low and sells high, competition

drives the present value of her proﬁt to zero.

The equilibrium of Theorem 2implements a socially optimal allocation with a bid-ask

spread of zero. Conversely, Weill [2006] shows that, if marketmakers could bargain individ-

ually with investors and charge a strictly positive bid-ask spread, then they would provide

more liquidity than is socially optimal. However, this ﬁnding that eﬃciency requires a zero

bid-ask spread crucially depends on the speciﬁcation of the trading friction. For example,

Weill [2006] shows that if marketmakers face a ﬁxed upper limit on the total number of orders

they can execute per unit of times then the equilibrium liquidity provision would continue

20

0 4 6 10 12

t∗

1t∗

2

p(t)

∆Vh(t)

∆V"(t)

I(t)

Figure 4: The Equilibrium Price Path.

to be socially optimal, but the equilibrium bid-ask spread would be strictly positive.

One might wonder whether the present results survive the introduction of a limit-order

book. Indeed, although investors are not continuously in contact with the market, their limit

orders could be continuously available for trading, and might substitute for marketmakers

inventory accumulation. Weill [2006] addresses this issue in an extension of the model where

the arrival of new information exposes limit orders to picking-oﬀrisk (see, among others,

Copeland and Galai [1983]). It is shown that, if the picking-oﬀrisk is suﬃciently large, then

the limit-order book is empty, liquidity is only provided by marketmakers, and the asset

allocation is the same as the one of Theorem 2.

4.2 Borrowing-Constrained Marketmakers

The implementation result of Theorem 2relies on the assumption that the time-zero capital

a(0) is suﬃciently large. This ensures that, in equilibrium, a marketmaker’s borrowing con-

straint (34) never binds. There is, however, much anecdotal evidence suggesting that, during

the October 1987 crash, specialists’ and marketmakers’ borrowing constraints were binding.

Some market commentators have suggested that insuﬃcient capital might have ampliﬁed

the disruptions (see, among others, Brady [1988] and Bernanke [1990]). This subsection

describes an ampliﬁcation mechanism associated with insuﬃcient capital and binding bor-

rowing constraints. Speciﬁcally, the following proposition shows that if marketmakers are

borrowing constrained during the crash and if their time-zero capital is small enough, then

21

they do not have enough purchasing power to absorb the selling pressure, and therefore fail

to provide the optimal amount of liquidity.

Proposition 3 (Equilibrium with small capital).There exists a∗

0≤a∗

0such that:

(i) If marketmakers’ aggregate capital is a(0) ∈[0,a

∗

0), there exists an equilibrium whose

allocation is a buﬀer allocation with maximum inventory position m∈[0,m

∗).

(ii) If marketmakers’ aggregate capital is a(0) ∈[a∗

0, a∗

0), there exists an equilibrium whose

allocation is a buﬀer allocation with maximum inventory position m∗.

If t∗

1>0, then a∗

0=a∗

0and the interval [a∗

0, a∗

0)is empty. Lastly, in all of the above, the

equilibrium price path has a strictly positive jump at the time tmsuch that I(tm)=m(that

is, tm=ψ(m)for the function ψ(·)of Proposition 1).

The equilibrium price and allocation are shown in Figures 5and 6. The price jumps up at

time tm. It grows at a low rate ˙p(t)/p(t)<rfor t∈[0,t

1), at a high rate ˙p(t)/p(t)=rfor

t∈(t1,t

m)∪(tm,t

2), and at a zero rate after t2. Because of the price jump, a marketmaker

can make positive proﬁt: for instance, he can buy assets the last instant before the jump

at a low price p(t−

m) and re-sell these assets the next instant after the jump at the strictly

higher price p(t+

m). An optimal trading strategy maximizes the proﬁt that a marketmaker

extracts from the price jump, as follows: i) a marketmaker invests all of her capital at the

risk-free rate during t∈[0,t

1) in order to increase her buying power, ii) spends all her capital

in order to buy assets before the jump, during t∈(t1,t

m), iii) re-sells all her assets after

the jump, during (tm,t

2). A marketmaker does not hold any assets during t∈[0,t

1) and

t∈(t2,∞) because the price grows at a rate strictly less than r. Because the price grows at

rate rduring (t1,t

m) and (tm,t

2), a marketmaker is indiﬀerent regarding the timing of her

purchases and sales, as long as all assets are purchased during (t1,t

m), sold during (tm,t

2),

and all capital is used up at time tm.

The price jump at time tmseems to suggest the following arbitrage: a utility-maximizing

marketmaker would buy more assets shortly before tmand sell them shortly after. This

does not, in fact, truly represent an arbitrage, because a marketmaker runs out of capital

precisely at the jump time tm, so she cannot purchase more assets.

Perhaps the most surprising result is that, if t∗

1= 0, then there is a non-empty interval

[a∗

0, a∗

0) of time-zero capital such that the price has a positive jump and marketmakers ac-

cumulate the optimal amount m∗of inventories. If t∗

1>0, then a∗

0=a∗

0and the interval is

empty. Intuitively, if the interval were not empty, then a small increase in time-zero capital

combined with a positive price jump would give marketmakers incentive to provide more

22

0 3 9

0

5

15

20

tm

m

time

I(t)/s(%)

small capital

large capital

Figure 5: Equilibrium Allocations.

0 8

time

p(t)

price jump

tm

Figure 6: Equilibrium Price Path with a Small Marketmaking Capital.

liquidity. Hence, they would start buying assets at some time t1<t

∗

1and would end up

accumulating more inventories than m∗, which would be a contradiction. If t∗

1= 0, this

reasoning does not apply: indeed, an increase in time-zero capital cannot increase inventory

accumulation because marketmakers cannot start accumulating inventories earlier than time

zero.

Price Resilience

Proposition 3illustrates the impact of marketmakers’ liquidity provision on what Black

[1971] called “price resiliency” – the speed with which an asset price recovers from a random

shock. This speed can be measured by the time t2reaches its steady-state “fundamental

value.” Note that this is also the time at which marketmakers are done unwinding their

inventories. The proposition reveals that, when marketmakers provide more liquidity, the

23

market can appear less resilient. Indeed, increasing a0∈[0,a

∗

0) means that marketmakers

purchase more assets and, as a result, take longer to unwind their larger inventory position.

This, in turn, increases the time t2at which the price recovers and reaches its steady state

value.

Numerical calculations reported in Weill [2006] suggest that more liquidity can lower

the price. To understand this somewhat counterintuitive ﬁnding, recall that the liquid-

ity provision of marketmakers reduces the inter-temporal holding cost of the average in-

vestor18 by making marketmakers hold inventories for a longer time. Moreover, when they

hold inventories, marketmakers become marginal investors. Therefore, the reduction in the

inter-temporal holding cost of the average investor is achieved through an increase in the

inter-temporal holding cost of the marginal investor. Because the asset liquidity discount

capitalizes the inter-temporal holding of the marginal investor, more liquidity can lower the

asset price.

5 Policy Implications

This section discusses some policy implications of this model of optimal liquidity provision.

Marketmaking Capital

The model suggests that, with perfect capital markets, competitive marketmakers would

have enough incentive to raise suﬃcient capital. The intuition is that marketmakers will

raise capital until their net proﬁt is equal to zero, which precisely occurs when they provide

optimal liquidity. For example, suppose that, at t= 0, wealthless marketmakers can borrow

capital instantly on a competitive capital market. Then, for t>0, the economic environment

remains the one described in the present paper. If, at t= 0, a marketmaker borrows a

quantity a>0, then she has to repay a×erT at some time T≥t∗

2. One can show that, with

an optimal trading strategy (described at the end of Section 4.2), the net present value of

her proﬁt is

$p(t+

m)

p(t−

m)−1%a, (37)

where the jump-size !p(t+

m)/p(t−

m)−1"depends implicitly on the time-zero aggregate mar-

18Indeed, equation (14) reveals that the planner’s objective is to maximize the inter-temporal utility for

the asset of the average investor or, equivalently to minimize his inter-temporal holding cost.

24

ketmaking capital.19 As long as the jump size is strictly positive, a marketmaker wants to

borrow an inﬁnite amount of capital. Therefore, in a capital-market equilibrium, a market-

maker’s net proﬁt (37) must be zero, implying that p(t+

m)/p(t−

m) = 1 and m=m∗. This

means that marketmakers borrow a suﬃciently large amount of capital and provide the

socially optimal amount of liquidity.

Lending capital to marketmakers, however, might be costly because of capital-market

imperfections associated for example with moral hazard or adverse selection problems. In

order to compensate for such lending costs, the net return !p(t+

m)/p(t−

m)−1"on marketmaking

capital must be greater than zero. This would imply that, in an equilibrium, marketmakers

do not raise suﬃcient capital. As a result, subsidizing loans to marketmakers may improve

welfare.20

During disruptions, some policy actions can be interpreted as bank-loan subsidization.

For instance, during the October 1987 crash, the Federal Reserve lowered the fund rate, while

encouraging commercial banks to lend generously to security dealers (Wigmore [1998]).

Price Continuity

It is often argued that marketmakers should provide liquidity in order to maintain price

continuity and to smooth asset price movements.21 The present paper studies liquidity

provision in terms of the Pareto criterion rather than in terms of some price smoothing

objective. The results are evidence that Pareto optimality is consistent with a discrete price

decline at the time of the crash. This suggests that requiring marketmakers to maintain

price continuity at the time of the crash may result in a welfare loss.

A comparative static exercise suggests, however, that liquidity provision promotes some

degree of price continuity. Namely, in an economy with no capital at time zero (a(0) = 0),

no liquidity is provided and the price jumps up at time ts>0. In an economy with large

time-zero capital, however, the price path is continuous at each time t>0.

Marketmakers as Buyers of Last Resort

19The proﬁt (37) is not discounted by e−rT because a marketmaker can invest her capital at the risk free

rate.

20Weill [2006] provides an explicit model of marketmakers’ borrowing limits based on moral hazard. A

diﬀerent model of limited access to capital is due to Shleifer and Vishny [1997]. They show that capital con-

straints might be tighter when prices drop, due to a backward-looking, performance-based rule for allocating

capital to arbitrage funds.

21For instance, Investor Relations, an advertising document for the specialist ﬁrm Fleet Meehan Specialist,

argues that specialists “use their capital to ﬁll temporary gaps in supply and demand. This can actually

help to reduce short-term volatility by cushioning the intra-day price movements.”

25

A commonly held view is that marketmakers should not merely provide liquidity, they should

also provide it promptly. In contrast with that view, the present model illustrates that

prompt action is not necessarily consistent with eﬃciency. Namely, it is not always optimal

that marketmakers start providing liquidity immediately at the time of the crash, when the

selling pressure is strongest. For example, if the initial preference shock is very persistent,

then marketmakers who buy asset immediately end up holding assets for a very long time.

This cannot be eﬃcient given that marketmakers are not the ﬁnal holders of the asset. This

suggests that requiring marketmakers to always buy assets immediately at the time of the

crash can result in a welfare loss.

6 Conclusion

Although it focuses on liquidity provision during a crisis, the present paper may extend to less

extreme situations. For instance, a similar model may help study the price impact of a large

trader, such as a pension funds, selling its assets. Similarly, the model can help understand

specialists’ liquidity provision on individual stocks, in normal times (see the recent evidence

of Hendershott and Seasholes [2007]).

One question that the paper leaves open is whether the welfare gains of marketmakers’

liquidity provision can be quantitatively signiﬁcant. Clearly, since marketmakers provide

liquidity over relatively short time intervals, the welfare gains can only be signiﬁcant if

sellers ﬁnd it very costly to hold assets during the crisis. Although the magnitude of price

concessions sellers are willing to make during an actual crisis (e.g., the 23% price drop of the

Dow Jones Industrial Average on October 19th, 1987) suggests that their holding costs are

indeed very large, it can be quite diﬃcult to rationalize these costs with standard models.

Taking a step back from the precise setup of the paper, one may also recall the commonly

held view that liquidity provision creates large welfare gains, because it helps mitigate the

risk of a meltdown of the ﬁnancial system and may reduce the adverse impact of a disruption

on the macro-economy. This view goes back, at least, to the famous Bagehot (1873) rec-

ommendation that the central bank provide liquidity on the money market during a crisis.

Of course, the present paper does not explain the mechanism through which a failure to

accommodate investors’ liquidity needs may have an adverse impact on aggregate economic

conditions. Instead, it takes these liquidity needs as given, and provides the optimal liquidity

provision of marketmakers. Understanding precisely how ﬁnancial market disruptions may

propagate to the macro-economy remains an important open question for future research.

26

A Proof of Proposition 1

Hump Shape. Consider a buﬀer allocation (t1,t

2). One ﬁrst shows that I(t) is hump-shaped

and that, given the ﬁrst breaking time t1, the second breaking time t2is uniquely character-

ized. For t∈[t1,t

2), the inventory position I(t) evolves according to ˙

I(t)=u"(t)−uh(t)=

ρ!µ"o(t)−µhn(t)". With equation (3), this ODE can be written ˙

I(t)=−ρI(t)+ρ!s−µh(t)".

Together with the initial condition I(t1) = 0, this implies that, for t∈[t1,t

2], I(t)=H(t1,t),

where H(t1,t)≡ρ&t

t1!s−µh(z)"eρ(z−t)dz. One has ∂H/∂t=−ρH+ρ(s−µh(t)), implying that

∂H/∂t(t1,t

1)=ρ(s−µh(t1)) ≥0 and ∂H/∂t(t1,t

s)=−ρ2&ts

t1(s−µh(z))eρ(z−ts)dz ≤0. There-

fore, there exists tm∈[t1,t

s] such that ∂H/∂t(t1,t

m) = 0. Moreover, ∂H/∂t= 0 implies that

∂2H/∂t2=−ρ˙µh(t)−ρ∂H/∂t=−ρ˙µh(t)<0. This implies that tmis unique, that H(t1,t) is

strictly increasing for t∈[t1,t

m), and strictly decreasing for t∈(tm,∞). Now, because µh(t)>s

for tlarge enough, it follows that H(t1,t) is negative for tlarge enough. Therefore, given some

t1∈[0,t

s], there exists a unique t2∈[ts,+∞) such that H(t1,t

2) = 0.

Writing {t1,t

m,t

2}as a function the the maximum inventory position. The maximum inventory

position of Proposition 1is deﬁned as m≡I(tm). One let tm≡ψ(m), for some function ψ(·)

which can be written in closed form by substituting ˙

I(tm) = 0 and I(tm)=min ˙

I(t)=−ρI(t)+

ρ!s−µh(t)":

ψ(m)=−1

γlog $1−s−m

y%.(38)

Now, solving the ODE ˙

I(t)=−ρI(t)+ρ!s−µh(t)"with the initial condition I(tm)=m, one ﬁnds

I(t)=me−ρ(t−tm)+(s−y)!1−e−ρ(t−tm)"+ρye−γtme−ρ(t−tm)#t

tm

e(ρ−γ)(u−tm)du. (39)

Plugging (38) into (39), and making some algebraic manipulations, show that I(t) = 0 if and only

if t=tm+z, for some zsolution of G(m, z) = 1, where

G(m, z)=$1+ m

y−s%e−ρz+1+ρ#z

0

e(ρ−γ)udu,.(40)

Let’s deﬁne, for x∈[0,+∞), the two functions gi(m, x)=G!m, (−1)i√x",i∈{1,2}. For x>0,

the partial derivatives of giwith respect to xis

∂gi

∂x=$1+ m

y−s%(−1)iρe−ρ(−1)i√x

2√x

×-−1−ρ#(−1)i√x

0

e(ρ−γ)udu +e(ρ−γ)(−1)i√x..(41)

One easily shows that this derivative can be extended by continuity at x= 0, with ∂gi/∂x(m, 0) =

−(1 + m/(y−s)) ργ/2. The term in bracket in (41) is zero at x= 0, and is easily shown to be

strictly increasing (decreasing) for i= 1 (i= 2). This shows that gi(m, ·) is strictly decreasing

over [0,+∞). Moreover, for m= 0, gi(0,0) = 1. For m>0, gi(m, 0) >1, g1(m, x)→ −∞

and g2(m, x)→0 when x→+∞. This implies that, for any m≥0, there exists only one

solution xi=Φi(m) of gi(m, x) = 1. An application of the Implicit Function Theorem (see

27

Taylor and Mann [1983], Chapter 12) shows that the function Φi(·) is strictly increasing and

continuously diﬀerentiable, and satisﬁes Φi(0) = 0, Φ'

i(0) = 2/(ργ(y−s)). Clearly, G(m, z ) = 0 if

and only if z∈{φ1(m),φ2(m)}, with φi(m) = (−1)i/Φi(m). Lastly, the restriction t1≥0 deﬁnes

the domain [0,¯m] of the functions ψ(·), φ1(·), and φ2(·). Indeed t1=ψ(m)−φ1(m), where

the function ψ(m)−φ1(m) is strictly decreasing, strictly positive at m= 0 and strictly negative

for m=s. Hence, there exists a unique ¯msuch that ψ(¯m)−φ1(¯m) = 0. By construction, the

maximum inventory of a buﬀer allocation is less than ¯m.

B Socially Optimal Allocations

This appendix solves for socially optimal allocations. In order to prove the various results of Section

3and 4, it is convenient to assume that, in addition to the trading technology and the short-selling

constraint, the planner is also constrained by an inventory bound I(t)≤M, for some M∈[0,+∞].

B.1 First-Order Suﬃcient Conditions

The current-value Lagrangian and the ﬁrst-order conditions are the one of subsection 3.3, with

an additional multiplier ηM(t) for the inventory bound, that ends up being equal to zero. It is

important to note that, because of the inventory bound, the multiplier λI(t) can also jump up,

with the restrictions λI(t+)−λI(t−)≥0 if I(t)=M. In what follows, it is convenient to eliminate

λh(t) and λ"(t) from the ﬁrst-order conditions using (21) and (22). One obtains the reduced system

rw"(t)=δ−1−γu!wh(t)+w"(t)"−ρw"(t)+ηI(t) + ˙w"(t) (42)

rwh(t) = 1 −γd!wh(t)+w"(t)"−ρwh(t)−ηI(t) + ˙wh(t),(43)

together with the ODE (28), the jump conditions (29),

λI(t+)−λI(t−)≥0 if I(t)=M(44)

λI(t+)−λI(t−)=w"(t+)−w"(t−)=−wh(t+)+wh(t−),(45)

and the transversality conditions that e−rtλI(t), e−rtw"(t), and e−rtwh(t) go to zero as time goes

to inﬁnity. As before, the positivity restrictions and complementary slackness conditions are (23),

(24), and (25). Note that, because the optimization problem is linear, there is no sign restrictions

on the multiplier λh(t) and λ"(t) (see section 3 in Part II of Kamien and Schwartz [1991]). As a

result, the present reduced system of ﬁrst-order condition is equivalent to the original system.

B.2 Multipliers for Buﬀer Allocations

Consider some feasible buﬀer allocation with breaking times(t1,t

2) and a maximum inventory

position m∈[0,min{¯m, M }] reached at time tm. This paragraph ﬁrst constructs a collection

!wh(t),w

"(t),λI(t),ηI(t)"of multipliers solving the ﬁrst-order suﬃcient conditions of Section B.1,

but ignoring some of the positivity restrictions. These restrictions are imposed afterwards, when

discussing the optimality of this allocation. First, summing equations (42) and (43), and using the

transversality conditions shows that w"(t)+wh(t)=δ/(r+ρ+γ), for all t≥0. Then, one guesses

that there are no jumps at t1and t2. With (45), this shows that λI(t+

i)−λI(t−

i)=w"(t+

i)−w"(t−

i)=

−wh(t+

i)+wh(t−

i) = 0, for i∈{1,2}. Now, one can solve for the multipliers, going backwards in

time.

28

Time Interval t∈[t2,+∞).Complementary slackness (24) implies that wh(t2) = 0. Plugging this

into wh(t)+w"(t)=δ/(r+ρ+γ) gives that w"(t)=δ/(r+ρ+γ). With (43), this also implies

that ηI(t)=−γdδ/(r+ρ+γ). Lastly, together with the transversality condition, (28) implies that

rλI(t) = 1 −δγd/(r+ρ+γ).

Time Interval t∈[tm,t

2).First, because I(t)>0, the complementary slackness condition (25)

implies that ηI(t) = 0. Then, one solves the ODE (43) with the terminal condition wh(t2) = 0, and

one ﬁnds that

wh(t)= 1

r+ρ$1−δγd

r+ρ+γ%!1−e(r+ρ)(t−t2)",(46)

for t∈[tm,t

2). With wh(t)+w"(t)=δ/(r+ρ+γ), w"(t)=δ/(r+ρ+γ)−wh(t). Similarly, one

can solve the ODE (28) with the terminal condition rλI(t−

2) = 1 −γdδ/(r+ρ+γ), ﬁnding that

rλI(t) = (1 −δγd/(r+ρ+γ)) er(t−t2).

Time Interval t∈[t1,t

m).In this time interval, ηI(t) = 0. One needs to consider two cases.

Case 1: m< ¯m. Complementary slackness at t=t1shows that w"(t1) = 0, implying that

wh(t1)=δ/(r+ρ+γ). With this and (43), one ﬁnds

wh(t)= 1

r+ρ$1−δγd

r+ρ+γ−$1−δr+ρ+γd

r+ρ+γ%e(r+ρ)(t−t1)%,(47)

for t∈[t1,t

m). Given (46) and (47), the multiplier wh(t) is not necessarily continuous at time

tm. The size wh(t−

m)−wh(t+

m) of the jump can be written as some function b(·) of the maximum

inventory level m, where

b(m)≡1

r+ρ+$1−δγd

r+ρ+γ%e−(r+ρ)φ2(m)−$1−δr+ρ+γd

r+ρ+γ%e(r+ρ)φ1(m),.(48)

Equation (45) implies that λI(t+

m)−λI(t−

m)=b(m). This and the ODE (28) show that rλI(t)=

(rλI(t+

m)−rb(m)) er(t−tm).

Case 2: m=¯m. Then, by construction of ¯m,t1= 0. If b(¯m)<0, the multipliers are con-

structed as in Case 1. If, on the other hand, b(¯m)≥0, then one constructs a set of multipliers

by solving the ODEs (43) and (28) with terminal conditions wh(t−

m)=wh(t+

m) + (1 −α)b(¯m) and

λI(t−

m)=λI(t+

m)−(1 −α)b(¯m), for all α∈[0,1]. That b(¯m)≥0 implies that, for all α∈[0,1],

wh(t1)=wh(0) ∈[0,δ/(r+ρ+γ)]. By construction, the α= 1 multipliers do not jump at t=tm.

Time Interval t∈[0,t

1],m< ¯mComplementary slackness shows that w"(t) = 0, implying that

wh(t)=δ/(r+ρ+γ). With equation (42), this also implies that ηI(t) = 1−δ(r+ρ+γd)/(r+ρ+γ)≥

0, and rλI(t)=ηI(t)+er(t−t1)!rλI(t1)−ηI(t)".

B.3 Proof of Proposition 2

Let us consider the multipliers w0

h(t),w

0

"(t),η0

I(t),λ0

I(t) associated with the no-inventory allocation,

as constructed in Section B.2. Recall that λ0

I(t+

s)−λ0

I(t+

s)=b(0) >0. It can be shown that, for

29

any buﬀer allocation (µm,Im,u

m),

W(m)−W(0) = −#+∞

0

e−rtw0

h(t)!ρµm

hn(t)−um

h(t)"dt −#+∞

0

e−rtw0

"(t)!ρµm

"o(t)−um

"(t)"dt

−#+∞

0

e−rtη0

I(t)Im(t)dt +!λ0

I(t+

s)−λ0

I(t−

s)"Im(ts)e−rts.(49)

Formula (49) follows from the standard comparison argument of Optimal Control (see, for example,

Section 3 of Part II in Kamien and Schwartz [1991]), in the special case of a linear objective

and linear constraints (see Corollary 2in Weill [2006] for a derivation). The ﬁrst two terms in

equation (49) are zero because, for all t≤ts,um

h(t)=ρµm

hn(t) and w0

"(t) = 0, and, for all t≥ts,

um

"(t)=ρµm

"o(t) and w0

h(t) = 0. The third term can be bounded by

0≤#ψ(m)+φ2(m)

ψ(m)−φ1(m)

e−rtη0

I(t)Im(t)dt ≤$1−γd

δ

r+ρ+γ%m!φ2(m)+φ1(m)",

because η0

I(t)≤1−γdδ/(r+ρ+γ). Since limm→0+φi(m) = 0, for i∈{1,2}, this implies that,

as m→0+,1/m &ψ(m)+φ2(m)

ψ(m)−φ1(m)e−rtηI(t)Im(t)dt goes to zero. In order to study the last term of (49)

note that equation (39) shows that Im(ts) = 0 at m= 0. Using the facts that ts=tm+ψ(0)−ψ(m),

and that ye−γtm=y−s+m, diﬀerentiating (39) with respect to mshows that the derivative of

Im(ts) at m= 0 is equal to one. Therefore, Im(ts)/m goes to 1, as mgoes to zero, establishing

Proposition 2.

B.4 Proof of Theorem 1

This paragraph veriﬁes that some buﬀer allocation is constrained-optimal with inventory bound

M. First, if some buﬀer allocation is constrained-optimal, it must satisfy the jump condition (44),

meaning that b(m)≥0 and b(m)(M−m) = 0. In particular, if there is no inventory constraint,

then the jump must be zero. One deﬁnes the maximum msuch that the jump b(m) is positive,

m∗= sup{m∈[0,¯m]:b(m)≥0}. Furthermore, since b(·) is decreasing, b(m)≥0 for all m≤m∗.

Proposition 4.. For all M∈[0,∞], the buﬀer allocation with maximum inventory position

m= min{m∗,M}is socially optimal with inventory bound M.

In order to prove this Proposition, let’s consider this allocation and its associated multipliers,

constructed as in the previous subsection. Two optimality conditions remain to be veriﬁed: the

jump conditions (44) and the positivity restrictions of (23) and (24) . Because m≤m∗, the

jump condition (44) is satisﬁed. Also, because wh(t) is a decreasing function of time, wh(0) ∈

[0,δ/(r+ρ+γ)] and wh(t2) = 0, it follows that, at each time, wh(t)∈[0,δ/(r+ρ+γ)], and

therefore that w"(t)≥0. The inventory-accumulation period ∆∗and the breaking times (t∗

1,t

∗

2) of

Theorem 1are found as follows. First, ∆∗=φ1(m∗)+φ2(m∗). Then, simple algebraic manipulations

show that b(m)≥0 if and only if

e(r+ρ)!φ1(m)+φ2(m)"≤1+ δ(r+ρ)

γu+ (1 −δ)(r+ρ+γd).(50)

If m∗<¯m, then (50) holds with equality at m∗, and if m∗=¯m, it holds with inequality. This is

equivalent to the formula of Theorem 1. Then, given ∆∗, the ﬁrst breaking time t∗

1is a solution of

H(t∗

1,t

∗

1+∆∗) = 0, where H(t1,t) was deﬁned in Section A. Direct integration shows that

H(t1,t) = (s−y)01−e−ρ(t−t1)1+ρye−γt1e−γ(t−t1)−e−ρ(t−t1)

ρ−γ,(51)

30

where we let (e−γx−e−ρx)/(ρ−γ)=xif ρ=γ. Simple algebraic manipulation of (51) give the

analytical solution of Theorem 1.

C Proofs of Theorem 2and Proposition 3

Deﬁnition. We ﬁrst provide a precise deﬁnition of a competitive equilibrium. First, investors’

continuation utilities solve the ODE

rV"n(t)=γu!Vhn(t)−V"n(t)"+˙

V"n(t) (52)

rV"o(t) = 1 −δ+γu!Vho(t)−V"o(t)"+ρ!V"n(t)−V"o(t)+p(t)"+˙

V"o(t) (53)

rVhn(t)=γd!V"n(t)−Vhn(t)"+ρ!Vho(t)−Vhn (t)−p(t)"+˙

Vhn(t) (54)

rVho(t) = 1 + γd!V"o(t)−Vho(t)"+˙

Vho(t),(55)

where ˙

Vσ(t)≡dVσ(t)/dt. Hence, the reservation value ∆V"(t) of a seller and of a buyer solve

r∆V"(t) = 1 −δ+γu!∆Vh(t)−∆V"(t)"+ρ!p(t)−∆V"(t)"+∆˙

V"(t) (56)

r∆Vh(t) = 1 + γd!∆V"(t)−∆Vh(t)"−ρ!∆Vh(t)−p(t)"+∆˙

Vh(t).(57)

Lastly, in order to complete the standard optimality veriﬁcation argument, I impose the transversal-

ity conditions that both ∆Vh(t)e−rt and ∆V"(t) go to zero as time goes to inﬁnity. Conversely, given

the reservation values, one ﬁnds the continuation utilities by solving the ODE (52) and (54) for V"n

and Vhn, and by letting Vjo =Vjn +∆Vjfor j∈{h, ,}. Indeed, subtracting (52) from (54), inte-

grating, and assuming transversality, one ﬁnds that Vhn (t)−V"n(t)=&+∞

te−(r+γ)(z−t)ρ!∆Vh(z)−

p(z)"dz. And, replacing this expression in (52), that V"n(t)=&+∞

te−r(z−t)γu!Vhn(z)−V"n(z)"dz.

Acompetitive equilibrium is made up of a feasible allocation !µ(t),I(t),u(t)", a price p(t), a collec-

tion !∆V"(t),∆Vh(t)"of reservation values, a consumption stream c(t), and a bank account position

a(t) such that: i) given the price p(t), !I(t),u(t),c(t),a(t)"solves the marketmaker’s problem, and

ii) given the price p(t), the reservation values !∆V"(t),∆Vh(t)"solve equations (56)-(57), satisfy

the transersality conditions, and satisfy at each time,

p(t)−∆V"(t)≥0 (58)

∆Vh(t)−p(t)≥0 (59)

!p(t)−∆V"(t)"!ρµ"o(t)−u"(t)"=0 (60)

!∆Vh(t)−p(t)"!ρµhn(t)−uh(t)"=0.(61)

Equations (58) through (61) verify the optimality of investors’ policies. For instance, equation (58)

means that the net utility of selling is positive, which veriﬁes that a seller ,oﬁnds it weakly optimal

to sell.22 Equation (60), on the other hand, veriﬁes that a seller’s trading decision is optimal.

Namely, if the net utility p(t)−∆V"(t) of selling is strictly positive, then u"(t)=ρµ"o(t), meaning

that all ,oinvestors in contact with marketmakers choose to sell. If, on the other hand, the net

utility of selling is zero, then ,oinvestors are indiﬀerent between selling and not selling. As a result,

u"(t)≤ρµ"o(t), meaning that some ,oinvestors might choose not to sell.

Equilibrium: Solution method. The idea is to identify equilibrium objects with the multipliers of

Appendix B.4, as in Table 3. For each m∈[0,m

∗], let Λ(m) be the (set of) multipliers associated

22Because of linear utility, it also shows that selling one share is always weakly preferred to selling a smaller

quantity q∈[0,1].

31

with the buﬀer allocation m. If m∈[0,m

∗), or if m=m∗and m∗<¯m, then Λ(m) is a singleton.

If m=m∗and m∗=¯m, then Λ(m∗) is a set (see Case 2 of Appendix B.4). Let λ(respectively

λ) be the element of Λ(m∗) with largest (smallest) λI(t−

m∗). By construction λhas no jump at

t=tm. Lastly, one lets a∗

0≡λI(t−

m∗)e−rtm∗m∗and a∗

0≡λI(t−

m∗)e−rtm∗m∗. By construction,

a∗

0= limm→m∗λI(tm)e−rtmm. Moreover, a∗

0≤a∗

0, with an equality if m∗<¯m. Now, one can

construct a competitive equilibrium using the following “backsolving” method. First, one picks

some buﬀer allocation m∈[0,m

∗] and multipliers λ∈Λ(m). Then, given λ, one guesses that price

and values are given as in Table 3. If m < m∗, or if m=m∗and λ$=λ, one takes time-zero capital

to be a(0) = λI(tm)e−rtmm. If m=m∗and λ=λ, one can take any a(0) ∈[a∗

0,∞). In the next

paragraph, I verify that, given this time-zero capital, the buﬀer allocation, the price, and the values

are the basis of a competitive equilibrium.

Conversely, let’s consider any a(0) ∈[0,a

∗

0). Given that m,→ λI(t−

m)e−rtmmis continuous

and is zero at m= 0, the construction of the previous paragraph implies that there exists a

competitive equilibrium implementing a buﬀer allocation with some inventory bound m∈[0,m

∗).

For any a(0) ∈[a∗

0,∞), the previous paragraph implies that there exists a competitive equilibrium

implementing the buﬀer allocation with maximum inventory m∗. In particular, if a(0) ≥a∗

0, the

multipliers do not jump at time tm∗. This establishes Proposition 3and Theorem 2.

Equilibrium: Veriﬁcation. The current value Lagrangian for the representative marketmaker’s

problem is

L(t)=c(t)+ˆ

λI(t)!u"(t)−uh(t)"+ˆ

λa(t)!ra(t)+p(t)(uh(t)−u"(t)) −c(t)"

+ˆηI(t)I(t) + ˆηa(t)a(t) + ˆwc(t)c(t).

The ﬁrst-order suﬃcient conditions are

1 + ˆwc(t)=ˆ

λa(t) (62)

ˆ

λI(t)=ˆ

λa(t)p(t) (63)

rˆ

λI(t) = ˆηI(t)+ ˙

ˆ

λI(t) (64)

˙

ˆ

λa(t)=−ˆηa(t) (65)

ˆwc(t)≥0 and ˆwc(t)c(t) = 0 (66)

ˆηI(t)≥0 and ˆηI(t)I(t) = 0 (67)

ˆηa(t)≥0 and ˆηa(t)a(t) = 0 (68)

ˆ

λa(t+)−ˆ

λa(t−)≤0 if a(t) = 0,(69)

together with the transversality conditions that λx(t)x(t)e−rt go to zero as tgoes to inﬁnity, for

for x∈{I, a}. The Bellman equations and optimality conditions for the investors are (56)-(61).

Direct comparison shows that a solution of the system (62)-(68), (56)-(61) of equilibrium equations

(with transversality) is p(t)=λI(t), ˆ

λa(t) = 1 + (λI(t+

m)/λI(t−

m)−1) for t < tm,ˆ

λa(t) = 1, for

t≥tm,ˆηa(t) = 0, ˆ

λI(t)=ˆ

λa(t)λI(t), ˆηI(t)=ˆ

λa(t)ηI(t), ˆwc(t)=ˆ

λa(t)−1, ∆V"(t)=λ"(t),

and ∆Vh(t)=λh(t), together with the corresponding inventory-constrained allocation, and some

consumption process c(t) such that c(t) = 0 for t≤t2and c(t)=ra(t2), for t > t2. To conclude the

optimality veriﬁcation argument for a marketmaker, one needs to check the jump condition (69), and

that a(t)≥0 for all t≥0. To that end, one notes that, for t∈[t1,t

2], d/dt!a(t)e−rt"=−p(t)˙

I(t)

and ˙

I(tm) = 0. This implies that that a(t)e−rt is continuously diﬀerentiable and achieves its

minimum at t=tm. By construction a(tm) = 0.

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