DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES
CALIFORNIA INSTITUTE OF TECHNOLOGY
PASADENA, CALIFORNIA 91125
Price Formation in Multiple, S
imultaneous
Continuous
Double A
uctions
, with Implications for Asset Pricing
John Ledyard
California Institute of Technology
Elena A
sparouhova
University of Ut
ah
Pe
ter B
ossaerts
University of Melbourne
SOCIAL
SCIENCE
WORKING
PAPER
1450
July 20 20
Draft
Price Formation in Multiple, Simultaneous
Continuous Double Auctions, with
Implications for Asset Pricing
Elena Asparouhova, Peter Bossaerts and John Ledyard
July 26, 2020
Draft
PRICE FORMATION IN MULTIPLE, SIMULTANEOUS CONTINUOUS DOUBLE
AUCTIONS, WITH IMPLICATIONS FOR ASSET PRICING
1
Elena Asparouhova
a
, Peter Bossaerts
b
,
2
and John Ledyard
c
We propose a Marshallian model for price and allocation adjustments in par-
allel continuous double auctions. Agents quote prices that they expect will max-
imize local utility improvements. The process generates Pareto optimal alloca-
tions in the limit. In experiments designed to induce CAPM equilibrium, price
and allocation dynamics are in line with the model’s predictions. Walrasian ag-
gregate excess demands do not provide additional predictive power. We identify,
theoretically and empirically, a portfolio that is closer to mean-variance optimal
throughout equilibration. This portfolio can serve as a benchmark for asset re-
turns even if markets are not in equilibrium, unlike the market portfolio, which
only works at equilibrium. The theory also has implications for momentum, vol-
ume and liquidity.
Keywords:
Continuous Double Auction, Walrasian Equilibrium, Marshallian
Equilibration, Experimental Economics, Asset Pricing.
1.
INTRODUCTION
General equilibrium has become the widely accepted theoretical model for competitive
markets and the benchmark against which those markets are empirically evaluated. A
compelling reason to be interested in equilibrium is the “argument, familiar from Marshall,
... that there are forces at work in any actual economy that tend to drive an economy
1
We would like to thank Bernard Cornet for pointing out a mistake in an earlier draft, and to Sean
Crockett, Dan Friedman and Sophie Moinas for their comments and suggestions. We gratefully acknowl-
edge comments on prior versions from participants in seminars (Bocconi, UBC, Copenhagen Business
School, Columbia University, GATE (Lyon, France), U of Geneva, Hebrew University, U of Lausanne,
U of Michigan, NYSE, New University in Lisbon, Norwegian Business School, Norwegian School of Eco-
nomics and Business Administration, Ohio State University, SEC, Stanford, SIFR, Tel Aviv University,
UC Berkeley, UC Irvine, UC San Diego, UC Santa Cruz, U of Kansas, U of Paris–Dauphine; U of Vienna,
U of Zurich) and conferences (2003 ESA meetings, 2003 WFAs, 2003 and 2009 SAET, 2004 Kyoto Confer-
ence on Experiments in Economic Sciences, 2005 Princeton Conference on Econometrics and Experimental
Economics, 2005 Purdue Conference in honor of Roko Aliprantis, 2006 Decentralization Conference in
Paris, 2010 Experimental Society Conference in Gothenburg, 2016 Experimental Finance Society Confer-
ence, Tucson), and the 2018 ESAM meetings in Auckland, New Zealand. We acknowledge the support
from NSF grant SES-0527491 (Bossaerts, Ledyard), SES-0616431 (Bossaerts), SES-106184 (Asparouhova,
Bossaerts), and the Swiss Finance Institute (Bossaerts).
2
Corresponding Author
a
David Eccles School of Business, University of Utah.
b
Brain, Mind and Markets Laboratory, University of Melbourne.
c
Division of The Humanities and Social Sciences, California Institute of Technology.
1
2
toward an equilibrium if it is not in equilibrium already.”
1
While there is wide consensus as to the appropriate equilibrium model, there is little
consensus as to the “forces at work.” Many models have been proposed, but none have been
accepted as the appropriate canonical model. How the equilibrium prices and allocations
are attained, and how, if at all, trading occurs out of equilibrium, remains to be discovered.
The lack of a consensus model of the forces that drive an economy towards equilibrium is
a problem for applied economics, including policy analyses. If an inappropriate model is
used in the design of economic policy, outcomes will not be as intended.
Until recently, attempts to settle this question have been mostly theoretical in nature
2
with no real evidence or philosophical foundation available to help sort the sensible from
the inane. Traditional empirical analyses of markets shed no light on the processes because
they do not have access to the fundamentals. But, with the advent and development of
experimental economics, it is now possible to explore the forces that drive equilibrium.
The market organization we focus on in this paper is the continuous double auction
(CDA) where individuals can submit bids (to buy) or asks (to sell) at any price, and
whenever the highest bid is at a price at or above the lowest ask, a trade takes place
immediately. In modern instances of the double auction, called the open-book system,
bids and asks that are surpassed by more competitive orders (bids at a higher price or
asks at a lower price) remain available, unless cancelled. The open book system is the
preferred exchange mechanism of financial markets around the world, and in particular,
of stock exchanges (NYSE, Euronext, LSE, NASDAQ, etc.). Recent advancements have
been proposed where instead of immediate execution, there is a small interval over which
orders accumulate in the book, called Frequent Batch Auctions (Budish, Cramton, and
Shim, 2015). The model we propose also applies to those mechanisms.
It is well known from the experimental analyses of CDA markets (summarized in Crock-
ett, 2013) that, in the first period of these experiments, (1) competitive equilibrium is not
reached immediately – there is a process of adjustment – and (2) prices follow neither
the Walrasian tatonnement (whereby prices react to aggregate excess demands, but allo-
1
Arrow and Hurwicz (1958), p. 263.
2
Exceptions that study multiple simultaneous markets include the works of Plott (2001), Anderson, e.a.
(2004) and Gillen e.a. (2020). These works report (price) dynamics that are in line with those reported
here, as discussed later.
3
cations are not adjusted) nor any of the various extant non-tatonnement theories (where
allocations can also change). If the fundamentals and markets are repeated for additional
periods, then (3) prices and allocations converge to their general equilibrium values and
(4) between-period price changes follow the Walrasian tatonnement.
In this paper, we present a theory that explains the paths of prices and allocations
within the first few periods of market experiments, before beliefs of likely paths could
reasonably have been formed, and hence, where bets on their nature are pure speculation.
It deserves emphasis that we model the paths
of allocations
as well as the paths of prices.
The extant literature tends to focus only on price dynamics (Crockett, 2013).
There are three main assumptions underlying the theory. First, in the spirit of Marshall
(1890), quantity moves to those offering the highest surplus to the market. Second, indi-
viduals quote prices that maximize their local utility gains taking the rules of engagement
as given. Third, agents do not speculate, which means that they do not perceive drift in
terms of trade that could improve their eventual allocations by postponing or accelerating
transactions. Under these assumptions, the resulting offers are a convex combination of
agents’ marginal valuations and the prices.
The analysis is
not
on each bilateral trade separately as traditional CDA would require.
Instead it invokes local market clearing,
3
defined as the transaction prices that cause
net trades to sum to zero. In this sense, our model is more appropriate for the recently
suggested frequent batch market mechanism (Budish, Cramton, and Shim, 2015).
Our theory is related to that of Friedman (1979), which itself follows up on the work
of Smale (1976). Friedman identifies a process where allocations move in a Marshallian
fashion: throughout, prices are a weighted average of individuals’ willingness-to-pay. Fried-
man (1979) focuses on stability and shows that the process converges to a Pareto-optimal
allocation. However, the model misses detail on how offers are generated and how offers
lead to trade. That is what our theory delivers.
Our theory is also related to that in Smith (1965) (see also Inoua and Smith, 2020).
Smith shows that bids of many agents have an impact on prices and trades, not just
those of the marginal agents, as in neoclassical accounts of Marshallian price adjustment
(Samuelson, 1947). Our theory shares this prediction. In contrast to Smith’s analysis,
3
The local clearing prices are equal to the average of all offers.
4
however, bids in our theory do not derive from Walrasian demand (or supply) functions.
Instead, they result from agents’ attempts to maximize local utility gains from trade.
To show the theory’s power, we apply it to asset markets. It has a particularly intuitive
appeal in the case of quasi-linear utility functions like mean-variance utility functions.
Quasi-linear preferences naturally apply to the finance application of general equilibrium:
the Capital Asset Pricing Model (CAPM) and its multi-factor generalizations (Roll, 1977).
We confront the finance application with data from nine experimental sessions, each with
6 to 8 replications (“periods”) with varying parametrizations. The results provide strong
support for the predictions regarding price and allocation dynamics. We test whether tra-
ditional Walrasian aggregate excess demands explain the remainder. We find that they do
not. That is, Walrasian adjustment theory predicts neither price nor allocation dynamics.
The theory has important implications for empirical asset pricing, where for decades the
concern has been to identify one mean-variance efficient portfolio, or a number of “factor
portfolios” that add up to this efficient portfolio.
4
We find that price dynamics push one
particular portfolio towards mean-variance efficiency throughout equilibration. Unlike in
CAPM (equilibrium), it is not the market portfolio, but a risk-aversion weighted endow-
ment portfolio. We refer to it as the Risk-Aversion Scaled Endowment Portfolio (RASE).
In the experiments, we demonstrate that the RASE portfolio generates significantly higher
average reward-to-risk ratios (Sharpe ratios) than the market portfolio.
The rest of the paper is organized as follows. The model setup and the theoretical results
are presented in Section 2. Experimental methods are discussed in Section 3. Results are
reported in Section 4. Implications for empirical asset pricing are in 5. Section 6 concludes.
2.
TWO MODELS OF MARKET DYNAMICS
2.1.
Preliminaries
2.1.1.
The Economic Exchange Environment
Our analysis is done within the context of the standard model of pure exchange. There
are
I
consumers, indexed by
i
= 1
,...,I
. There are
K
= 1 +
R
commodities, where the
last
R
commodities are indexed by
k
= 1
,...,R
, and the first is indexed by
0
. We reserve
4
See (Fama and French, 2004). Since the set of mean-variance optimal portfolios is spanned by two
portfolios, one of which necessarily is the risk-free security, it suffices to identify one additional mean-
variance optimal portfolio to describe the entire set. See Roll (1977).
5
this first commodity as a special one, and will designate it as the numeraire when needed.
Each individual
i
owns initial endowments
ω
i
= (
ω
i
0
,...,ω
i
R
)
,
ω
i
k
>
0
for all
i
and
k
.
x
i
= (
s
i
,r
i
1
...,r
i
R
)
is the allocation of
i
.
s
i
is
i
’s quantity of the numeraire commodity.
X
i
=
{
(
s
i
,r
i
)
∈ <
K
|
r
i
≥
0
}
is the admissible consumption set for
i
.
5
Each
i
has a
quasi-concave utility function,
u
i
(
x
)
. We assume that
u
i
∈
C
2
(that is,
u
i
has continuous
second derivatives) although many of our results would hold under weaker conditions. We
also assume that
{
x
|
u
i
(
x
)
≥
u
i
(
ω
i
)
}⊂
Interior
(
X
i
)
and
u
i
0
=
∂u
i
(
x
i
)
∂x
i
0
>
0
,
∀
x
i
∈
X
i
,
∀
i.
2.1.2.
Time and the Continuous Double Auction
In a CDA experiment, traders begin with an endowment of commodities,
ω
i
. They
proceed to make bids and offers over time. Often these are retained in a public book
unless the trader decides to withdraw their bid or offer. A bid or offer in the book can be
accepted by anyone. If accepted, trade occurs at that price. This goes on until a stopping
rule is implemented. Although the CDA operates in continuous time, the intuition behind
the theory is easier to understand in discrete time. Time is divided into discrete intervals
of length
∆
. With slight abuse of notation, the interval
t
is
[
t,t
+ ∆)
.
x
i
t
= (
s
i
t
,r
i
t
)
denotes
i
’s holdings at the beginning of interval
t
. Trade takes place and the change in
i
’s holdings
during interval
t
is
∆
x
i
t
= (∆
s
i
t
,
∆
r
i
t
) = (
s
i
t
+∆
−
s
i
t
,r
i
t
+∆
−
r
i
t
)
.
p
t
= (1
,q
t
)
∈ <
K
+
is the
vector of
K
prices at which trades take place in interval
t
.
2.2.
The Walrasian Model
Here we describe the standard Walrasian model of market dynamics as well as the
variants known as non-tatonnement processes. There is nothing new here. We include this
only as a reminder to the reader.
Given a price vector
p
∈ <
K
+
, the individual excess demand function of
i
is
e
i
(
p,ω
i
) =
arg max
d
i
u
i
(
ω
i
+
d
i
)
subject to
p
·
d
i
= 0
and
ω
i
+
d
i
∈
X
i
. The aggregate excess demand
of the economy is
E
(
p,ω
) =
∑
i
e
i
(
p,ω
i
)
, where
ω
= (
ω
1
,ω
2
,...,ω
I
)
.
Definition
1
A price
p
∗
and an allocation
x
∗
= (
x
∗
1
,...,x
∗
I
)
constitute a competitive
equilibrium at
ω
= (
ω
1
,...,ω
I
)
if and only if
1. Given prices
p
∗
,
the allocations
x
∗
i
are optimal:
x
∗
i
=
e
i
(
p
∗
,ω
i
) +
ω
i
,
∀
i
, and
5
There is no lower bound on the numeraire.
6
2. Markets clear; that is,
E
(
p
∗
,ω
) = 0
.
By Walras’ law, we can limit our attention to the excess demands of all but the nu-
meraire commodity, denoted
e
i
(
p,ω
i
)
and
E
(
p,ω
)
, respectively. Also, since the price of
the numeraire is fixed at 1, individual and aggregate excess demands can be written as
e
i
(
q,ω
i
)
and
E
(
q,ω
)
, respectively, where
p
= (1
,q
)
.
In Walrasian adjustment models, the main force driving price changes is the
taton-
nement
. Prices of goods in excess demand (supply) go up (down). Let
B
be an
R
×
R
diagonal matrix with positive diagonal elements. The Walrasian tatonnement is:
q
t
+∆
−
q
t
∆
=
BE
(
q
t
,ω
)
(2.1)
x
i
t
=
ω
i
if
E
(
q
t
,ω
)
6
= 0
e
i
(
q
t
,ω
i
) +
ω
i
if
E
(
q
t
,ω
) = 0
(2.2)
The tatonnement is really only a model of prices since trades do not occur until prices
have converged to their equilibrium values. (2.2) is not what is going on in most continuous
markets where trading occurs as prices are changing.
6
Recognizing that, researchers have
proposed many alternatives under the heading of
Non-Tatonnement
(NT) processes.
7
An NT process works as follows. At the beginning of each time interval, agents know their
individual holdings,
x
i
t
. Trade takes place during the interval at prices
p
t
. The holdings at
the end of the interval are
x
i
t
+∆
. A new price is computed based on the excess demands
at the price
p
t
and the holdings
x
t
. The Walrasian non-tatonnement dynamics are:
q
t
+∆
−
q
t
∆
=
BE
(
q
t
,x
t
)
(2.3)
x
i
t
+∆
−
x
i
t
∆
=
g
i
(
q
t
,x
i
t
)
,
(2.4)
where
g
i
is a vector function,
∑
i
g
i
(
q
t
,x
i
t
) = 0
, that also satisfies the Lipshitz condition.
Different NT models impose different additional assumptions on the functions
g
i
, see
6
The tatonnement might describe, for example, the “book building” process in a call market if orders
can be withdrawn (Plott and Pogorelskiy, 2017).
7
See e.g. Negishi (1962), Uzawa (1962), Hahn and Negishi (1962).
7
Negishi (1962). In the CDA, there is no Walrasian auctioneer to set prices. There, (2.3)
is interpreted as a predictive theory of prices: it predicts the price changes at
t
+ ∆
based
on prices and allocations at
t
.
2.2.1.
A Problem
In most multi-market CDA experiments, competitive equilibrium does not occur instan-
taneously except, perhaps, with replication in later periods. In addition, neither taton-
nement, nor non-tatonnement dynamics fit the data.
8
A better theory is needed.
2.3.
ABL Market Dynamics
Here, we describe a model based on Marshall’s intuition but with a consistent micro-
foundation. The model rests on four key hypotheses. The first captures the Marshallian
intuition that
quantity moves to those individuals offering higher surplus to the
market
. Let
b
i
t
= (
b
i
1
,t
,...,b
i
R,t
)
be
i
’s bid during the interval
t
.
b
i
k,t
is
i
’s stated willingness
to pay (accept) to buy (sell) a unit of
k
in terms of the numeraire commodity 0.
Hypothesis
1
Marshallian Trading
∆
r
i
t
(=
r
i
t
+∆
−
r
i
t
) =
A
(
b
i
t
−
q
t
)
, i
= 1
,...,I
(2.5)
where
A
is an
R
×
R
positive diagonal matrix and
A
kk
=
α
k
,
k
= 1
,...,R.
In some markets, aggressive bidding attracts larger volume than in others. In this sense,
α
k
is a
liquidity parameter.
It is assumed that it does not vary over time.
The next two hypotheses are almost always requirements of a CDA system.
Hypothesis
2
Instant Settlement (Payment with numeraire occurs at each trade)
∆
s
i
t
=
−
q
t
·
∆
r
i
t
i
= 1
,...,I.
(2.6)
8
See Asparouhova, Bossaerts and Plott (2003), Anderson, e.a. (2004), Asparouhova and Bossaerts
(2009), Gillen e.a. (2020), and Crockett (2013).
8
Hypothesis
3
Feasible Trading (Whatever is bought, is sold)
I
∑
i
=1
∆
r
i
t
= 0
.
(2.7)
The last hypothesis, Hypothesis 4, specifies how individual traders determine their bids
in a continuous double auction. It captures the idea that agents only consider small trades
and do not speculate. Faced with the fact that large orders will move prices unfavorably,
intractable strategic uncertainty, and a lack of futures markets and rational expectations,
agents make only small (local) adjustments to their holdings
. This can be mo-
tivated using game theory, but it is also a
fact
in field markets.
9
Faced with uncertainty
about where prices will go next,
agents do not speculate
. They take current prices as
given.
To motivate Hypothesis 4, assume traders only consider small local adjustments that
maximize their gain in local utility
∆
u
i
t
. For very small
∆
,
∆
u
i
t
≈ ∇
u
i
(
x
i
t
)
·
(∆
s
i
t
,
∆
r
i
t
)
where
∇
u
i
(
x
i
t
)
is the gradient of
u
i
at
x
i
t
. Under Hypotheses 1 and 2, the change in
i
’s
utility that results from a bid
b
i
t
at time
t
will be:
∆
u
i
t
≈
u
i
0
(
x
i
t
)(
ρ
i
(
x
i
t
)
−
q
t
)
·
∆
r
i
t
=
u
i
0
(
x
i
t
)(
ρ
i
(
x
i
t
)
−
q
t
)
·
A
(
b
i
t
−
q
t
)
,
where
ρ
i
k
(
x
i
)
denotes the marginal rate of substitution between commodities 0 and
k
for
k
= 1
,...,R
if
i
’s holdings are
x
i
.
10
ρ
i
k
represents
i
’s marginal willingness to pay
(or be paid) for units of
k
in units of commodity
0
.
ρ
i
(
x
i
) = (
ρ
i
1
(
x
i
)
,...,ρ
i
R
(
x
i
))
and
∇
u
i
(
x
i
t
) =
u
i
0
(
x
i
t
)(1
,ρ
i
(
x
i
t
))
.
To locally optimize,
i
wants to choose
b
i
t
so that the direction of change of
x
i
t
= (
s
i
t
,r
i
t
)
is proportional to the gradient. This means they want
A
(
b
i
t
−
q
t
) =
c
i
∆(
ρ
i
(
x
i
t
)
−
q
t
)
, where
the parameter
c
i
is a characteristic of
i
. It determines the step size and rate of trading.
Larger
c
i
imply a greater urgency to trade. We call this
i
’s
impatience parameter
and
9
Financial markets have become more competitive, and trade sizes have decreased dramatically. “Split-
ting orders” has become an important concern in algorithmic trading. See Avellaneda, Reed and Stoikov
(2011). Further empirical evidence that trade takes place “in smalls” can be found in O’Hara, Yao and
Ye (2014). In a market with continuous order submission and trading, the small-orders assumption can
easily be justified theoretically; see Rostek and Weretka (2015).
10
ρ
i
k
(
x
i
) =
∂u
i
(
x
i
)
/∂x
i
k
∂u
i
(
x
i
)
/∂x
i
0
.
9
assume it does not change over time.
Remark
1
This behavior is incentive compatible in the following sense. If both the quan-
tity adjustment rule, Hypothesis 1, and the price setting rule, Hypothesis 3, are known and
taken as given, and
α
k
=
α
, for
k
= 1
,...,R
, then there are
(
c
1
,...,c
I
)
such that the bids
derived above are a local Nash equilibrium.
11
The final intuition behind Hypothesis 4 concerns the timing of information and actions.
When an agent computes their bid at the start of interval t, they do not know
q
t
. They
only know the prices and allocations at the end of the
t
−
∆
interval. Because
∆
is assumed
to be very small, it is likely that bids at
t
are based on the prices and allocations arrived
at in the interval
t
−
∆
.
Hypothesis
4
Local Optimization and Lagged Prices
b
i
t
=
q
t
−
∆
+
c
i
∆
A
−
1
(
ρ
i
(
x
i
t
)
−
q
t
−
∆
)
,
∀
i,
∀
t >
0
.
For the curious, Section B.1 of the Appendix contains a discussion of the model and its
implications when
q
t
is used in place of
q
t
−
∆
in Hypothesis 4. That model implies that
bids and prices are
simultaneously
determined in the time
∆
. The model is not consistent
with the data, as explained in Appendix B.2.
This leaves the initial price,
q
0
, to be specified. The initial price is likely context-
dependent and can plausibly equal the vector of mean payoffs in an asset pricing setup, be
related to prices in the previous period when applied to replications of the same situation,
or be equal to the average of the values of the initial endowments.
Hypothesis
5
The initial price
q
0
is some arbitrary positive vector.
In our empirical analysis, the focus will be on price
changes
, so Hypothesis 5 is never in
play.
Hypotheses 1-5 are the ABL model.
Remark
2
We have assumed that agents do not speculate. The beginning of an analysis
under speculation can be found in Appendix C. Speculation becomes an important concern
11
This is similar to a result of Roberts (1979). A proof is provided in section A.1 of the Appendix.
10
in later replications in an experiment, when these replications are identical, meaning par-
ticipants have the opportunity to form beliefs about likely price dynamics. Here, we focus
on early replications, or replications with varying parametrizations.
The dynamics of the ABL model are straightforward. Entering interval
t
, consumer
i
has an allocation
x
i
t
= (
s
i
t
,r
i
t
)
and knows the price from the previous interval
q
t
−
∆
. In the
interval, bids are formed based on Hypothesis 4 and trade occurs at new prices based on
Hypotheses 1-3. Prices adjust rapidly to ensure that trading, according to Hypothesis 1
and 2, adds up to zero (Hypothesis 3). Leaving the interval, trader
i
now owns
x
i
t
+∆
=
(
s
i
t
+∆
,r
i
t
+∆
)
and knows the prices
q
t
. This process, given the initial price
q
0
, is formalized
in equations (2.8)-(2.10).
12
r
i
t
+∆
=
r
i
t
+ ∆
(
−
̄
c
( ̄
ρ
t
−
q
t
−
∆
) +
c
i
(
ρ
i
t
−
q
t
−
∆
)
)
(2.8)
s
i
t
+∆
=
s
t
−
q
t
·
(
r
i
t
+∆
−
r
i
t
)
(2.9)
q
t
=
q
t
−
∆
+ ̄
c
∆
A
−
1
( ̄
ρ
t
−
q
t
−
∆
)
(2.10)
where
̄
c
=
∑
i
c
i
I
and
̄
ρ
(
x
t
) =
∑
i
c
i
ρ
i
(
x
i
t
)
∑
i
c
i
.
The limiting behavior of the dynamics is most easily seen in continuous time.
13
Dividing
(2.8) and (2.10) by
∆
and letting
∆
→
0
,
we get the continuous time version, for
t >
0
:
14
dr
i
t
dt
=
c
i
(
ρ
i
t
−
q
t
)
−
̄
c
( ̄
ρ
t
−
q
t
)
,
∀
t >
0
(2.11)
ds
i
t
dt
=
−
q
t
·
(
(
c
i
(
ρ
i
t
−
q
t
)
−
̄
c
( ̄
ρ
t
−
q
t
)
)
,
∀
t >
0
(2.12)
dq
t
dt
=
−
̄
cA
−
1
(
q
t
−
̄
ρ
t
)
,
∀
t >
0
(2.13)
Remark
3
When taking limits, one important subtlety of the ABL model is lost. The
discrete-time equations specify dynamics over
two
intervals:
[
t
−
∆
,t
)
and
[
t,t
+ ∆)
. In
continuous-time, everything collapses effectively to one interval. E.g., in discrete time,
price changes over
[
t
−
∆
,t
)
depend on marginal rates of substitution at the
end
of the
12
See Appendix A.2 for details.
13
Convergence in continuous time implies that if step sizes,
c
i
, are not too large, then there will also
be convergence in discrete time.
14
See Appendix A.3 for details.
11
interval (i.e., at
t
); see (2.10). In continuous time, it does not matter whether marginal
rates of substitution are based on holdings at the beginning or end of an interval, because
adjustment is smooth. To preserve discrete-time subtleties, one could add random shocks
to the adjustment, and appeal to Itô calculus. Limit (Itô) processes are
not
smooth (time
series are nowhere differentiable with respect to time). Consequently, timing subtleties
from discrete time are retained in continuous time. As reported in Section 4 below, the
discrete-time subtleties matter empirically. Price changes within observation intervals in
our trading sessions are driven by holdings at the
end
of each such interval, as predicted
by the ABL model. The Walrasian model, in contrast, predicts that price changes are based
on (excess demands computed from)
lagged
holdings. The Walrasian model fails if only
because of timing issues. Timing is an under-appreciated dimension in which Marshallian
and Walrasian dynamics differ. In Marshallian dynamics, prices are determined by current
willingness to pay; in Walrasian dynamics, prices are determined by past excess demands.
This subtle but important difference in the models will be crucial for our empirical work.
There is an analogy to the First Welfare Theorem of General Equilibrium Theory: the
allocation at any rest point is a Pareto-optimal allocation. By the Second Welfare Theorem
the rest point is also a competitive equilibrium at that allocation. If there are no income
effects, the continuous process (2.11)-(2.13) will converge to a rest point from any initial
price and allocation. This may not be true for the discrete process (2.8)-(2.10) if step sizes
are too large.
Theorem
1
Convergence to Pareto Optimal Allocations
15
If (i) there are no income effects, i.e.,
u
i
0
(
x
i
) = 1
for all
i
and all
x
i
∈
X,
and (ii)
r
i
t
>
0
for all
t
, then for the dynamics in (2.11) - (2.13),
(
x
t
,p
t
)
→
(
x
∗
,p
∗
)
where
x
∗
is
Pareto-optimal and
(
p
∗
,x
∗
)
is a competitive equilibrium at
x
∗
.
Remark
4
Along the path generated by (2.11) - (2.13), it is possible that
du
i
t
/dt <
0
.
With the bidding lag,
du
i
t
/dt
=
u
i
0
,t
((
ρ
i
(
x
i
t
)
−
q
t
)
·
c
i
(
ρ
i
(
x
i
t
)
−
q
t
)
−
∑
k
(
ρ
i
k
(
x
i
t
)
−
q
k,t
)
α
k
(
dq
k,t
/dt
))
While the first term is non-negative, the second term is not necessarily so. Traders basing
their bids on lagged prices do not anticipate and cannot protect themselves from “ex post”
15
The proof of this theorem is relegated to Section A.4 of the Appendix.
12
adverse trades. For example, if prices are rising fast, slow agents may trade into increasing
prices when they want to buy.
Remark
5
The possibility that
du
i
t
/dt <
0
(among other differences) distinguishes the
ABL theory from Friedman (1979) and Smale (1976). Specifically, our allocation dynamics
do not satisfy Friedman’s condition (P).
2.4.
Comparing Walrasian vs. ABL Dynamics
The Walrasian and ABL models can imply significantly different paths of price adjust-
ment. This can be seen in the simple example in Figure 1. There
R
= 1
and
I
= 2
, utility
functions are quasi-linear (the inverse demand functions therefore equal the marginal rates
of substitution
ρ
), and the aggregate endowment is
W
=
r
1
t
−
∆
+
r
2
t
−
∆
=
r
1
t
+
r
2
t
.
We mea-
sure the holding of trader 2 from right to left starting at
W
. The competitive equilibrium
allocation and the resting point of the ABL model occur where
ρ
1
and
ρ
2
cross, with
q
e
denoting the equilibrium price.
In Figure 1,
r
1
t
−
∆
denotes 1’s holdings at (
t
−
∆
), while 2 holds
r
2
t
−
∆
=
W
−
r
1
t
−
∆
. The
most recent price,
q
t
−
∆
, is below the equilibrium price. At the given holdings, and given
the most recent price, there is excess demand for the good (at
q
t
−
∆
, individual 2 demands
3 units, and 1 demands more than
W
units) so the Walrasian model requires the price
to
increase
, i.e.,
q
t
−
q
t
−
∆
>
0
. To determine the sign of
q
t
−
q
t
−
∆
, the ABL model uses
the allocations at
t
,
r
1
t
and
r
2
t
. Given small changes in quantities, these allocations will be
close to
r
1
t
−
∆
and
r
2
t
−
∆
, as depicted by the vertical band. As a result, the average weighted
marginal rate of substitution,
ρ
∗
t
= ̄
ρ
(
r
t
)
,
falling in the corresponding horizontal band, is
lower than the price
q
t
−
∆
meaning the ABL model predicts that the price would
fall
, i.e.,
q
t
−
q
t
−
∆
<
0
.
The difference in the implications of the two models when
R >
1
is also very stark
if we restrict attention to a very special environment: the Capital Asset Pricing Model
(CAPM). The CAPM is theoretically simple and is of its own interest since it serves as
the foundation of both asset market experiments and empirical analyses on historical data
from the field. In the CAPM, all utility functions are of the form:
u
i
(
x
i
) =
s
i
+
μ
·
r
i
−
(
a
i
/
2)(
r
i
)
·
(Ω
r
i
)
,
(2.14)
13
0
2
4
6
8
10
1 2
3
4 5 6 7
8
9 W
r
1
t
−Δ
q
t-
Δ
2
4
6
8
10
q
e
9 8
7
6 5 4 3
2
1
r
2
t-
Δ
ρ
t
*
ρ
1
Holdings of person 1 = W - holdings of person 2
MRS
ρ
2
Figure 1:
MRS (Marginal Rate of Substitution
ρ
i
) in a 2-commodity, 2-person economy, as a
function of holdings of agent 1. Equilibrium price equals
q
e
. Last traded price equals
q
t
−
∆
. The
Walrasian equilibration model predicts that the price will increase because, at
q
t
−
∆
, there is
excess demand: agent 2 demands three units and agent 1 demands more than
W
units, while
total supply equals only
W
units. In contrast, ABL predicts that the price will decrease, to
ρ
∗
t
,
which equals the average of the
ρ
i
s at current holdings.
where
μ
is an
R
-dimensional vector of positive constants,
Ω
is an
R
×
R
positive-definite
matrix of constants, and
a
i
is a positive scalar constant. In asset pricing models,
μ
is
interpreted as the expected payoff of an asset,
Ω
is the payoff covariance matrix across
the assets, and
a
i
is a measure of risk aversion. For these utility functions,
ρ
i
(
x
i
) =
μ
−
a
i
Ω
r
i
and
e
i
(
q,x
i
) =
1
a
i
Ω
−
1
(
μ
−
q
)
−
r
i
.
(2.15)
Combining (2.15) with (2.10) yields:
16
q
t
−
q
t
−
∆
∆
=
A
−
1
Ω
∑
c
i
a
i
e
i
(
q
t
−
∆
,x
i
t
)
I
(2.16)
Comparing (2.16) with (2.3), we can see three fundamental differences between the
16
See section D of the Appendix for the details of the derivation.
14
price dynamics
of the ABL model and those of the Walrasian model in the CAPM envi-
ronment.
17
1.
Cross-Security Effects Emerge.
In the ABL model, changes in the price of com-
modity
k
depend not only on the excess demand for
k
(as in the Walrasian model)
but also on the excess demand of the other commodities. For example, if the off-
diagonal entries of
Ω
are negative (indicating the commodities are complements),
18
the excess demand for
j
6
=
k
puts upward pressure on the price of
k
. This means
that the price of
k
could increase, even though there is an excess supply of it. This
cannot happen under Walrasian price dynamics.
2.
Heterogeneity in Risk Aversion, Impatience and Liquidity Matters.
In the
ABL model the excess demand functions of traders with higher
a
i
c
i
are weighted
more heavily in how they affect the changes in prices. The desires of the more risk
averse and the more impatient thus have a larger impact on price changes. In the
Walrasian model it is the less risk averse who have a larger impact on price changes.
3.
Timing Is Different.
See Remark 3. In the Walrasian model, prices in interval t
are determined by prices and allocations in period
t
−
∆
.
In the ABL model, prices
in period t are determined by prices in period
t
−
∆
and by
allocations in period t
.
The three differences are testable in the lab and motivate the design of our experiment.
As to
allocation dynamics
, using (2.15), the following system of difference equations
describes agent-level changes in allocations:
r
i
t
+∆
−
r
i
t
∆
=
−
Ω
(
c
i
a
i
r
i
t
−
∑
c
i
a
i
r
i
t
I
)
+
(
c
i
−
̄
c
)
(
μ
−
q
t
−
∆
)
(2.17)
In ABL, the changes in an agent’s allocations depend on (i) how far impatience and
risk-aversion scaled holdings are from the average impatience and risk-aversion scaled
holdings, plus (ii) the differences between expected payoffs and lagged market prices,
provided the agent’s impatience is different from the average. The second term disappears
if impatience is the same across agents; the first term remains under equal impatience, as
long as risk aversion is heterogeneous. The covariance matrix pre-multiplies the first term.
17
The premultiplication by
Ω
of the excess demands might remind some of the Newton-Raphson algo-
rithm. We discuss this in section E of the Appendix.
18
A similar analysis applies when the commodities are substitutes or when there is a mix of both.
15
As a consequence, ABL predicts cross-security effects in allocation dynamics in the same
way it predicts them in price dynamics. The effects are opposite for prices and allocations
however, because of the negative sign in front of the first term of (2.17).
Equations (2.16) and (2.17) will form the basis of our empirical analysis.
3.
EXPERIMENTAL METHODS
3.1.
Framework
Our experimental design builds on the CAPM. Agents have mean-variance preferences
with fixed risk-to-reward trade-offs, and hence, no wealth effects. Prior experiments have
shown robust convergence to equilibrium; see Asparouhova, Bossaerts and Plott (2003);
Bossaerts and Plott (2004) and Bossaerts, Plott and Zame (2007).
CAPM predicts that, in equilibrium, one particular portfolio is mean-variance optimal.
This portfolio is the market portfolio. In CAPM, agents’
total
demands (holdings plus
excess demands) are the same for all agents, up to a constant of proportionality equal to
the inverse of risk aversion. This is obtained by rewriting (2.15):
r
i
t
+
e
i
(
q
t
,x
i
t
) =
1
a
i
Ω
−
1
(
μ
−
q
t
)
.
(3.1)
The property is known as “portfolio separation.” As a result, in the Walrasian equilibrium,
the right-hand-side must equal to the total supply of assets, i.e., the “market portfolio.”
The market portfolio is defined as the per-capita endowment portfolio of risky assets, with
holdings equal to
r
=
1
I
∑
I
i
=1
r
i
.
Consequently this means that, in equilibrium, the market
portfolio must be mean-variance optimal, for otherwise it would not be proportional to
agents’ demands. See Roll (1977).
Equilibrium prices are as follows.
19
(3.2)
q
∗
=
μ
−
1
1
I
∑
i
1
a
i
Ω
r.
In the laboratory, CAPM works well; see, e.g., Bossaerts and Plott (2004); Bossaerts,
19
It is straightforward to check that, at these prices, the sum of the individual excess demands (3.1)
equals zero, and hence, markets equilibrate. When converted to restrictions on returns (payoffs divided by
prices), the equation becomes the well-known requirement that expected returns in excess of the risk-free
rate be proportional to the covariance of returns with those on the market portfolio.
16
Plott and Zame (2007). Here is an example, from a classroom session in an advanced
investments class at the University of Melbourne. Forty-eight students were asked to
trade to maximize their payoffs given by (2.14), with
a
i
= 0
.
01
, for all
i
,
μ
=
[
5 5 5
]
and
Ω =
16
−
5
−
14
−
5
16
9
−
14
9
16
.
Notice that mean-variance preferences are
induced
by asking students to directly optimize
the CAPM payoff function. In the sequel, we will nevertheless refer to
μ
as the vector of
expected payoffs, and
Ω
as the covariance matrix.
The three securities had equal expected payoffs and equal variances. But in equilibrium
prices differ because (i) supplies were unequal, with the third security being in the shortest
supply and (ii) the first security had negatively correlated payoffs with the others, while
the other two had positively correlated payoffs. Equilibrium prices were:
q
∗
=
[
5
.
125 1
.
5 3
.
5
]
.
The equilibrium price of the third security is not the highest even if it is in the shortest
supply. The intuition is simple: the first security, with the highest equilibrium price, is more
valuable because its payoff is negatively correlated with that of the others. Participants
were not told the per-capita supplies. Hence, even if they knew CAPM, they could not
possibly compute equilibrium prices.
20
Trade in this sample laboratory market took place in an online continuous open-book
trading platform (called Flex-E-Markets
21
). Participants could submit limit orders for any
of the securities for the duration of the class exercise (about 35 minutes). Participants were
provided with a tool that evaluated the performance of their current portfolios as well as
the net performance of any trades they wished to make.
Figure 2 displays the evolution of trade prices, during the first replication, of the three
risky securities (referred to as Stock A, Stock B and Stock C). Prices convincingly evolved
20
The results of a quick poll before trading confirmed that most participants expected prices to be
equal).
21
See http://www.adhocmarkets.com.
17
from expected values to equilibrium levels.
22
!
"!!
#!!
$!!
%!!
&!!
'!!
!(!!(!!
!(!)("#
!("%(#%
!(#"($'
!(#*(%*
!($'(!!
!"#$%
&#'%
!"#$%&'"()%*
+,-./01
+,-./02
+,-./03
Figure 2:
Transaction prices (in cents) during a class experiment. Forty-eight participants traded
three risky securities (“Stocks” A, B and C) with known, equal payoff distributions but different,
unknown total supplies. Predicted equilibrium prices, in cents: 513 (A; blue), 150 (B; orange)
and 350 (C; grey).
Participants were divided into three groups based on their initial portfolio allocations.
They only knew their own allocation. The first group started with 15 of the first security
and none of the other securities; the second group started with allocations of 9, 20 and 0,
and the third group started with 0, 10, and 18. In equilibrium, they should all end up with
the same allocation, since they all faced the same risk aversion parameter. Final allocations
necessarily equal the market portfolio. Figure 3 plots the evolution of the
difference
of the
per-capita holdings of Group 1 and the market portfolio, over intervals of 5 trades each.
The figure shows how per-capita holdings gradually move towards the equilibrium level.
Notice that the evolution is far more gradual than the price evolution.
In the class experiment, we induced mean-variance preferences, by tying performance
directly to the CAPM utility function in (2.14). There was no explicit uncertainty in
the experiment; performance (payoffs) were immediate once allocations were known. We
could also have drawn payoffs from distributions with mean
μ
and covariance matrix
Ω
,
but then we would not have controlled the risk aversion parameter, so we could not have
unambiguously derived equilibrium price levels. In addition, we would have to make the
22
We were agnostic as to the price levels markets would start from; see Hypothesis 5. In the experiment,
prices started from expected value. That is,
q
0
=
μ
.
18
0
10
20
30
40
50
60
70
80
90
100
110
Time (5-trade intervals)
-8
-6
-4
-2
0
2
4
6
8
Per capita holdings minus market
Stock A
Stock B
Stock C
Figure 3:
Evolution of differences between (i) per-capita holdings of A (blue), B (orange) and
C (grey) of the first group of participants, and (ii) the market portfolio. Initial holdings are 15
units of A each and 0 of B and C. The market portfolio consisted of (per capita) 8 units of A,
10 of B and 6 of C. Differences converge to zero, implying that per-capita holdings converged to
CAPM predictions. Time is measured in intervals of 5 transactions.
auxiliary assumption that mean-variance preferences explain choices in the experiment.
23
As with the classroom experiment presented above, the experimental sessions we ran
to test the theory of this paper also relied on induction of mean-variance preferences. To
simplify the setup, the experiments had two, not three, risky securities.
24
Also, since the
theory has predictions for economies with heterogeneous risk aversion, we varied the risk
aversion coefficient across subjects.
3.2.
Hypotheses
The theory makes precise predictions about the evolution of
prices
as well as
allocations
.
Allocation changes depend on risk aversion and are therefore analyzed as average changes
in holdings across subjects who belong to homogeneous groups. Groups are defined by
initial allocations and/or risk aversion coefficients. The parameters
μ
and
Ω
in the payoff
23
When introducing uncertainty explicitly, Bossaerts and Plott (2004) and Bossaerts, Plott and Zame
(2007) show that mean-variance preferences provide only a crude approximation of individual choices,
even if CAPM accurately predicts prices.
24
Appendix F.2 reports results from earlier sessions with three risky securities, but where mean-variance
preferences were not induced.