Double Auction Dynamics:
Structural Consequences of Non-Binding Price Controls*
Dhananjay (Dan) K. Gode
Leonard N. Stern School of Business
New York University
[email protected]
212-998-0021
and
Shyam Sunder
Graduate School of Industrial Administration
Carnegie Mellon University
[email protected]
412-268-2103
Current Revison: February 1999
We are grateful for comments received at workshop presentations at California Institute
of Technology, Carnegie Mellon University, Econometric Society, Economic Science
Association, European University Institute, Renmin University, University of Amsterdam,
University of Bonn and Universitat Pompeu Fabra.
*Available for download in PDF format from http://www.gsia.cmu.edu/andrew/sunder/limit.pdf
Double Auction Dynamics:
Structural Consequences of Non-Binding Price Controls
In an astonishing variety of contexts, apparently complex structures or behaviors emerge from
systems characterized by very simple rules.
Murray Gell-Mann (The Quark and the Jaguar p. 99-100)
Abstract
In competitive equilibrium, non-binding price controls (that is, price floors below and ceilings
above the equilibrium) should have no effect on market outcomes, but in laboratory experiments
they do. We build a simple dynamic model of double auction markets with “zero-intelligence”
(ZI) computer traders that accounts for many, though not all, of the discrepancies between the
data and the Walrasian tatonnement predictions. Success of the model in organizing the data,
and decomposing various consequences of price controls, suggests that the simple ZI model can
be a powerful tool to gain insights into the dynamics of market institutions.
JEL Classification: D44
1. Introduction
In competitive equilibrium, price floors below or ceilings above the equilibrium should
not affect market outcomes, but in laboratory experiments they do [Isaac and Plott (1981), Smith
and Williams (1981), and Coursey and Smith (1983)]. We present a simple dynamic model of
markets with "zero-intelligence" (ZI) traders that organizes the data better than competitive
equilibrium and allows us to decompose the effects of price controls into their components.
ZI traders submit random bids and asks while avoiding losses, have no memory, or
knowledge of market rules [see Becker (1962) and Gode and Sunder (1993a, 1993b, and 1997)].
They are hardly a descriptive model of human behavior; a glaring discrepancy between the two
is that ZI traders do not learn. The simplicity of the ZI model makes the analysis of markets in
which a small number of traders act sequentially tractable when an analysis with strategic traders
is difficult. The ZI model also establishes a useful benchmark for measuring the effects of
varying market rules by allowing us to fix individual behavior.
Rules of markets and other social institutions are open to observation; individual
motivations, information and strategies are inherently private1. Using observable market rules as
explanatory variables helps build testable theories of market phenomenon. It also explains how
important experimental outcomes can be robust to variations in subject backgrounds when the
effect of market rules dominates the effects of individual behavior. The traditional strategic
model in which traders respond fully to changes in market rules is another benchmark. The two
benchmarks can be seen as complements to the extent the actual observations lie somewhere
between the two.
We show that some important consequences of price controls can be qualitatively
replicated using ZI traders. This highlights the powerful role of rules in markets. We provide
several examples of how the simplicity of ZI traders enables us to comprehend interactions
among them and yields simple structural explanations for complex market-level phenomena. For
example, the extramarginal demand and supply, largely ignored earlier, are shown to play a
critical role in market dynamics and convergence to equilibrium.
1
Testing the effect of conjectured trading strategies by coding these conjectures into computer programs designed to
mimic human behavior can be very complex [Rust, Palmer, and Miller (1993)]. Since these conjectures specify only
a few aspects of behavior and trader behavior varies widely, the possible set of programs is bewilderingly large.
Even if market outcomes with these traders are similar to those with human traders, it is often difficult to understand
1
Price controls such as minimum wage laws and agricultural price maintenance are an
important and common form of regulatory intervention in markets. Walrasian tatonnement
model predicts that in competitive equilibrium (CE) non-binding price controls should have no
effect on transaction prices and quantity traded.2 In contrast, Scherer (1970, p. 352] suggests that
non-binding price ceilings may have an impact if they serve as a “focal point” for collusion
among sellers to keep the prices near the ceiling, i.e., above the CE price. It is difficult to resolve
such conflicting predictions from the field data because the demand and supply conditions are
rarely known with sufficient precision. Laboratory markets allow testing of these predictions
under controlled conditions.
The laboratory experiments of Isaac and Plott (1981), Smith and Williams (1981), and
Coursey and Smith (1983) reject the focal point hypothesis. They show that once human markets
converge to competitive equilibrium (typically after repeated trading with the same demand and
supply), non-binding price controls have no effect. However, such controls have the following
effects on convergence to equilibrium: (1) Transaction prices are initially skewed away from the
ceiling, and the skew diminishes as the ceiling is raised (the "buffer effect"). (2) Intra-period
transaction price series has a positive slope. (3) Variability of transaction prices declines for later
transactions within a period. (4) Allocative efficiency remains close to 100 percent, i.e., total
realized profits are marginally affected by non-binding ceilings. (5) The lowering of transaction
prices transfers some profits from sellers to buyers. (6) The difference in bargaining ability of
buyers and sellers persists if it was present before the imposition of ceiling. (7) There is transient
overshooting of transaction prices above the price ceiling after the ceiling is removed.
Isaac and Plott (1981) and Smith and Williams (1981) conjecture that non-binding price
ceilings just above the CE price depress transaction prices by forcing sellers to lower their initial
asks to or below the ceiling, which induces buyers to lower their own initial bids. Isaac and Plott
(1981) further conjecture that removal of non-binding ceilings affects prices by making available
additional strategies to the traders or because mere announcement of a change has a
how exactly trader behavior translates into which characteristics of market outcomes, and why these outcomes are
robust to variations in trader behaviors.
2
Price floor refers to the lower limit and price ceiling refers to the upper limit on bids, asks, and transaction prices.
Since treatment and effect of price ceilings and floors is symmetrical, we will refer only to price ceilings in this
paper. Non-binding ceilings are price ceilings above the competitive equilibrium (CE) price.
2
disequilibrating effect on experimental markets. It is difficult to use the traditional tools to
analyze initial phases of trading when traders are learning about market conditions.
We show that the first five effects of non-binding price controls reported by Isaac and
Plott, and Smith and Williams can be replicated with the ZI traders.3 This shows that even if
traders do not change their behavior in response to rules, market outcomes can resemble those
that are usually attributed to changes in behavior. Of course, not all of the important effects of
price controls are due to market rules. But the existence of effects of price controls with ZI
traders who ignore price controls suggests that market rules, even if they do not directly affect
behavior, may strongly affect market outcomes by restricting the feasible set of bids, asks, and
transaction prices.
Our analysis shows that the consequences of price ceilings can be classified into two
groups. The first set of consequences originate in the bid prices above the ceiling being reset to
the ceiling or zero. This set includes four observed phenomena: (1) lowering of the average
transaction price; (2) an upward trend in the transaction prices within a period towards the
equilibrium price; (3) transaction prices are skewed away from the equilibrium price; (4)
variance of transaction prices declines within a period. This effect by itself has no consequences
for allocative efficiency, but explains the bulk of the price effects of price ceilings. This effect
would exist even if there were no extramarginal traders.
Analysis of ZI markets reveals a second set of two other subtle effects that have not been
previously identified. Price ceilings reduce the relative bidding ability of intramarginal buyers
and exclude extramarginal sellers above the ceiling. The first effect unambiguously lowers
allocative efficiency by reducing the probability that the intramarginal buyers will keep out
extramarginal buyers by outbidding them. The second effect has a mixed impact. It can increase
efficiency because exclusion of extramarginal sellers above the ceiling reduces the probability
extramarginal sellers may displace intramarginal sellers. It can reduce efficiency because in case
the extramarginal buyers displace intramarginal buyers initially, the intramarginal buyers may
not have anyone to trade with. Our simulations with ZI traders using the demand and supply used
in previously reported experiments indicates that price ceilings have an efficiency increasing
3
Since all ZI traders have identical strategies, we do not expect to see differences in bargaining ability (effect 6).
Since ZI traders lack memory, we do not expect to find the transient overshooting (effect 7), which can only be
based on traders’ memory of price controls imposed earlier. Note that this effect diminishes significantly in human
markets as traders learn to ignore past information that is no longer relevant.
3
effect overall. In the absence of the second set of effects, price ceilings would affect prices in a
simple way; they would have no affect on quantity traded or the allocative efficiency.
The ZI model does well in predicting market outcomes with human traders who have
experience with double auction rules but have little information about the demand and supply
conditions [Cason and Friedman (1996)], which is typically the case in the early stages of
markets. As human traders learn about demand and supply, market outcomes may diverge from
those predicted by the ZI model because ZI traders have no memory. ZI traders are not an
attempt to model human behavior accurately; they are a simple tool to explore the dynamics of
double auctions. They help us identify the powerful effect of market institutions.
Section 2 presents the ZI model, section 3 presents the simulation results, section 4
analyzes and explains the simulation results and section 5 concludes the paper.
2. Model
Our markets have three elements: demand and supply, trader behavior, and market rules.
We describe these elements in order.
2.1 Demand and supply
Consider a market consisting of N buyers and N sellers. Each buyer is endowed with the
right to buy one or more units and each seller is endowed with the right to sell one or more units.
Let vij denote the value of the ith buyer’s jth unit and cij denote the cost of the ith seller’s jth unit.
The costs and values are private information. The vector v representing values sorted in a
descending order constitutes the market demand while vector c representing costs sorted in an
ascending order constitutes the market supply. Units to the left of the point of intersection of
demand and supply are the intramarginal units and units to the right of the intersection of
demand and supply are the extramarginal units.
Similar to experimental auctions and game-theoretic models of auctions, we disallow
retrading; a trader can be either a buyer or a seller but not both. Retrading is often not feasible in
markets for services, perishable products, and products that have high transactions costs.
Multiple rounds of bids and asks for a given trade are permitted.
If traders are endowed with only one unit, then the maximum possible number of trades is
limited by the number of human subjects available. To increase possible number of trades while
keeping the trading simple, experimental subjects are often endowed with multiple units which
they can trade only one at a time. For example, consider the demand and supply shown in Table
4
1 and Figure 1 which is reproduced from Smith and Williams’ (1981) Table 1. Each trader is
endowed with five or six units but can trade only one unit at a time. In these markets it is
assumed that a buyer trades his units in the descending order of their values and a seller sells her
units in the ascending order of their costs. This order maximizes the profits earned by a trader.
The Walrasian tatonnement model predicts the equilibrium price of 46.5 and volume of 12 units.
To facilitate comparison, we report the results of market simulations with this supply and
demand.
When each trader is endowed with multiple units but can only trade one unit at a time, the
market outcomes at any instant are influenced by only the highest valued unit remaining with
each buyer and the lowest cost unit remaining with each seller. Thus, very few, if any,
extramarginal units are “in the market” at the beginning of a period and the chance for
extramarginal units to displace the intramarginal units in trading is small.
When each trader can trade all units he is endowed with in a single transaction, all units
not yet traded affect market outcomes at any given instant. We repeat our market simulations for
a second configuration (shown in Table 2) in which the number of buyers and sellers is increased
so that the aggregate demand and supply remain the same (Figure 1), but each trader is endowed
with the right to trade only one unit. This allows us to test the robustness of our results.
Finally, Smith and Williams’ demand and supply had only six extramarginal units, all
relatively close to the margin. Even if the maximum possible number of units (intra- as well as
extramarginal) were traded, the allocative efficiency of this market would still be 88 percent.4
Therefore, one should expect to see at most small effects of price controls on efficiency. We
therefore use a third demand-supply schedule shown in Table 3, where the efficiency is only 56
percent if all units are traded to increase the possibility of observing the efficiency effect of price
controls.
2.2 Individual Behavior
When bidding for its jth unit the ith ZI buyer, submits a bid that is independently and
uniformly distributed between 0 and vij (unit’s value to the buyer). When asking for its jth unit the
4
Allocative efficiency, an important measure of market performance, is the ratio of the actual to the potential gains
from trade (the sum of Marshallian consumer and producer surplus). Allocative efficiency may fall short of 100
percent is because: (1) traders fail to conclude feasible trades, and (2) extramarginal traders enter into trades [for
further details see Gode and Sunder (1997)]. If retrading is permitted, then an extramarginal trader who enters the
5
ith ZI seller, submits an ask that is independently and uniformly distributed between a market
determined upper bound M and cij (unit’s cost to the seller).
5
In other words, ZI traders avoid
losses. Note that in our simulations ZI traders do not change their behavior in response to other
bids, asks, prior transactions, or price ceilings described below.
2.3 Market Rules
Our market simulations were implemented on a single computer as continuous double
auctions without replacement (see Appendix for details). Each market was run for one hundred
periods. Sufficient time was allowed for each period so that all feasible trades could occur with a
high probability. Thus, the only source of loss of allocative efficiency with ZI traders was
displacement of extra-marginal traders by intramarginal traders.
Price Controls:
In a simulation model, price controls can be defined as a characteristic of the market
institution, or of trader behavior. From an institutional perspective there are two ways of
implementing price controls.6 We shall label them as reject and reset modes respectively. In
reject mode, both bids as well as asks that violate the price controls (e.g., exceed ceiling c) are
simply rejected by the market.
In reset mode, any asks that exceed the ceiling are rejected, but any bids that exceed the
ceiling are reset to the ceiling. In other words, bid = minimum (bid submitted by the trader, the
ceiling), which creates a point mass of probability of bids at the ceiling. The rationale for the
reset mode is as follows. If a seller submits an ask above the ceiling, we cannot assume that he
would also be willing to sell at the ceiling, which is below his ask. Therefore, in this mode, any
asks that exceed the ceiling are simply ignored. In contrast, if a buyer is willing to buy a unit at a
price above the ceiling, then he would certainly be willing to buy the unit at the ceiling.
market will eventually find an intramarginal trader with whom he can retrade profitably, thus eliminating the second
cause of inefficiency.
5
For the sake of symmetry with respect to the demand and supply we used (see Tables 1, 2 and 3), we set M=93.
Raising M shifts the prices slightly upward and vice versa. However, the allocative efficiency does not change.
Maintaining the symmetry isolates the effect of price ceiling on price distribution by abstracting away from the
effect of parameter M.
6
Interaction of social institutions with individual behavior makes it difficult to separate the two. For example, one
can say that sellers choose the highest bid (behavioral perspective) or that market rules give priority to the highest
bid (institutional perspective). This duality is reflected in the alternative ways in which price controls can be
implemented within the context of ZI model. Price controls can be implemented by changing individual behavior or
by changing market rules. Since we are interested in the institutional perspective, we assume that ZI traders do not
alter their behavior in response to price controls.
6
3. Results of ZI Simulations with Smith and Williams Demand and Supply
For the sake of comparison, Figures 2 and 3 show the representative results from human
subject experiments (reproduced from Figures 1 and 3 respectively from Smith and Williams,
1981). Three panels of Figure 2 show the demand and supply functions, equilibrium price
(horizontal dashed lines), and transaction price paths (dots) for three markets of four periods
each. In the first market, there is a price floor five points below the equilibrium price; in the
second market, there is a price ceiling five points above the equilibrium price; and in the third
market there are no price controls. The upper panel of Figure 3 corresponds to the middle panel
of Figure 2 for a market with price ceiling at five points above the equilibrium price of 645. On
the left hand side, it shows the frequency polygon of offers (dashed line with hollow circle
markers) and bids (continuous line with dark circle markers). The frequency polygon of
transaction prices is drawn on the right hand side. The bottom panel of Figure 3 corresponds to
the bottom panel of Figure 2 for a market without price controls. Smith and Williams focus on
the following effects of non-binding price ceilings:
(1) Transaction prices are skewed away from the ceiling and the magnitude of skew
diminishes as the ceiling is raised. (2) Intra-period transaction price series has a positive slope.
(3) Variability of transaction prices declines for later transactions within a period. (4) Under nonbinding price ceilings, allocative efficiency remains close to 100 percent, i.e., total realized
profits are unaffected. (5) Transaction prices are lowered transferring some profits from sellers to
buyers. (6) The difference in bargaining ability of buyers and sellers persists if it was present
before the imposition of ceiling. (7) There is transient overshooting of transaction prices above
the price ceiling after the ceiling is removed. They do not imply that all of these effects will be
present with all variations of demand and supply.
We examine the simulation data to see whether the first five effects are present with ZI
traders (in markets with Smith and Williams (1981) demand and supply.) The results of the
examination are presented in sections 3.1 to 3.5. We defer the explanation of ZI outcomes until
section 4, which also discusses the general insights provided by ZI traders about the effect of
price controls.
3.1 Distribution of Prices
Figure 4 shows the histogram of prices for 100 periods of ZI markets whereas Table 4
presents the summary statistics.
Panel 1 of figure 4, which shows market prices when there
7
is no price ceiling, serves as the benchmark The average transaction price 46.5 is equal to the CE
price and price distribution is symmetric.
Panel 2 of Figure 4 and Column 2 of Table 4 show the effect of imposing a price ceiling
at 47. The bids and asks above 47 are simply ignored by the market. Mode of transaction prices
is between the price ceiling of 47 and equilibrium price of 46.5, and the distribution is skewed
away from the ceiling. Average transaction price declines from 46.5 (without price control) to
45.01 (with price control), a statistically significant downward shift (α = .000). This downward
shift parallels the downward shift in prices when price ceiling is imposed on markets with human
traders as can be verified from Figures 2 and 3.
Panel 3 of Figure 4 and Column 3 of Table 4 show the behavior of prices when bids
above the ceiling are reset to the ceiling but the asks above the ceiling are ignored. (see
discussion “ reset mode” in Section 2.3). Now the modal density at the price control is greater
and the mean price rises to 45.67; otherwise the qualitative effect of the price ceiling on
transaction prices is essentially similar to Panel 2.
Panel 4 of figure 4 and Column 5 in Table 4 show the results from Smith and Williams’
market demand and supply functions in which each trader has only one unit to trade (see Table 2
and Section 2.1). Again, in comparison with Panel 2, we see a higher peak at the modal value of
ceiling and a mean price of 45.63; otherwise the effect of ceiling on the distribution of
transaction prices is essentially the same.
Finally, we repeated the simulations after adding extramarginal traders (see Table 3 and
Section 2.3 above). Histogram of prices from this simulation with single unit holdings is shown
in Panel 5 of Figure 4 (and Column 8 of Table 4). Compared to Panel 2, the results are
essentially similar except for a higher modal peak and a mean price of 45.71.
3.2 Path of Transaction Prices
The five panels of Figure 5 (parallel to the five panels of Figure 4) show the paths of
transaction prices averaged across all periods (in ZI environment, there is no learning and all
periods are identical). As expected, the mean price path for the no-price ceiling control in Panel 1
is horizontal at the CE price of 46.5. Panel 2 shows the effect of imposing a non-binding price
ceiling on the mean price path. The greatest impact of the ceiling is on the earliest transactions.
The price of these transactions drops by the largest amount because the sellers of these units, on
average, can afford to sell at a lower price. Prices of the later transactions (usually involving
8
units whose costs are closer to the margin) are hardly affected. This means that the effect of the
price ceiling on profits of the low-cost producers is far greater than its effect on the marginal
producers.
Panel 3 shows the effect of an alternative implementation where the invalid bids are reset
to the price ceiling instead of being ignored. As one would expect, this implementation results in
increased number of bids at the price ceiling, and therefore, higher transaction prices on average
(as compared to Panel 2). Panel 4 shows the transaction price path in the single unit
implementation of demand and supply (see Table 2). Transaction prices are somewhat higher
than panel 2. Panel 5 shows that adding extramarginal traders causes little change in the effect of
price controls on the price path. Thus, the effect of non-binding price ceiling on mean price path
remains robust.
3.3 Dispersion of Transaction Prices
Figure 6 shows the dispersion associated with the mean transaction price paths shown in
Figure 6. The transaction price median (dark square) and the interquartile range (vertical line) in
this figure highlight three things. First, the dispersion of transaction prices declines from early to
later transactions in a period, independent of whether or not price controls are imposed. Second,
the bias of the mean price path can be compared to the interquartile range to get a sense of
significance of the bias. It is clear that the imposition of the ceiling causes an overwhelming
majority of transactions in the first half of the period to occur at a lower price. Third, as one
would expect, the imposition of price ceiling causes the distribution of prices to be skewed away
from the ceiling. This skewness is reflected in the interquartile range.
3.4 Allocative Efficiency
The lower part of Table 4 shows the effect of price ceiling on average efficiency of these
double auctions. ZI markets exhibit high allocative efficiency as shown by columns 1, 2, and 3.
The difference among these three columns is negligible, i.e., similar to Smith and Williams
(1981), non-binding price ceilings seem to have little effect on allocative efficiency.
When the number of traders is increased to endow each trader with only one unit (without
changing the aggregate demand and supply), efficiency declines (Table 4, Column 5). When
more extramarginal traders are added there is a further decline in efficiency. In both these cases,
ceiling does cause a drop of one or two percent in allocative efficiency if bids and asks above the
ceiling are ignored but only a smaller drop in efficiency if bids above the ceiling are reset to the
9
ceiling. This issue is explored further in Section 4. Thus, modification of demand and supply
provide weak evidence that effects of ceilings on allocative efficiency are more difficult to detect
when traders have multiple units but can trade only one unit at a time.
3.5 Division of Surplus between Buyers and Sellers
The bottom rows of Table 4 show the effect of price ceiling on the profits of buyers and
sellers. The data confirm that price ceilings increase profits of buyers relative to sellers by
depressing prices. The magnitude of the shift of the surplus to the buyers is highest in the Smith
and Williams (1981) setting where traders have multiple units to trade (Columns 2 and 3), and is
the least when each trader can trade only one unit in a period and there are additional extramarginal units. The magnitude of the shift depends on the implementation of demand and supply
and of price controls for ZI traders. Magnitude of shift decreases when bids above the ceiling are
reset to the ceiling instead of being ignored.
4. Using ZI Traders as an Exploratory Tool for Markets
The above simulation results for markets populated by ZI traders replicate many of the
important consequences of imposing price controls on double auctions. These results, along with
some prior work (Gode and Sunder, 1993b and 1997) encourage us to think that the simple ZI
model, with its minimalist approach to rationality, is a useful analytical tool to explore the
behavior of market institutions. In this section we supplement the simulation results of the
preceding section by statistical reasoning in simplified environments to enhance our
understanding of the effects of price controls on market outcomes.
Simplicity of the model helps us (1) identify ways in which changes in rules can affect
market outcomes, and (2) appreciate how even seemingly simple changes in rules can cause
complex changes in markets. We discuss only statistical effects of market rules assuming that
trader behavior does not change in response to market rules. We assume throughout that each
trader has only one unit to trade each period. Non-binding price ceilings exclude extramarginal
sellers whose costs equal or exceed the ceiling are excluded from the market. We start by
assuming that all extramarginal sellers are excluded, and relax this assumption later.
If the price limits are imposed after the market has settled to the equilibrium, imposing a
price ceiling above the equilibrium price should have no effect on behavior of the market.
However, when traders are searching for equilibrium, these ceilings affect the convergence
process. We identify three components of this effect. The first of these is obvious; exploration
10
of the dynamics of double auctions with ZI traders reveals the other two. These are (1) the limit
effect; (2) reduction of competitive bidding ability of the intra-marginal buyers (relative to extramarginal buyers); (3) exclusion of extramarginal sellers above the ceiling.
The first, and the obvious, effect of price ceilings is to make it impossible to conduct
transactions above the specified limit. In addition to (a) a reduction in the average transaction
price; it also results in (b) an upward slope of the intra-period transaction price path toward the
equilibrium price; and (c) a skew of transaction prices away from the equilibrium price. The limit
effect, by itself, has no consequences for trading volume or allocative efficiency, but it includes
the bulk of the price consequences of the controls imposed. The statistical intuition behind the
reduction in average price and upward slope of price path is explained in Cases 1 and 2
respectively.
Case 1 (Price level and relative profits of buyers and sellers):
Assume that there is just one seller with a cost of 0 and one buyer with a value of 1 as
shown in Figure 7A. The bids and asks of ZI traders are uniformly distributed from 0 to 1.
Without any price ceilings the expected transaction price is 0.5.
Buyer (value =1)
Ceiling
c
Seller (cost = 0)
Figure 7A
If a price ceiling is imposed at c (0 < c < 1), and bids and asks above the ceiling are
ignored by the market (ignore mode), the expected transaction price is c/2 which is less than or
equal to 0.5. Proportion (1-c) of all bids/asks submitted will be rejected by the market. The
expected number of rounds of bids/asks required for a transaction to occur will increase by a
factor of 1/c.
In an alternative implementation of the market, if bids above the ceiling are reset to c
(reset mode), the transactions that might have occurred between c and 1 without the ceiling will
now occur at c. Thus, imposition of a ceiling lowers the price on average. However, the average
price is higher compared to the ignore mode because in that case the transactions between c and
1 are redistributed over the range 0 to c instead of being concentrated at c. Thus, both
11
implementations of price ceiling (ignore and reset modes), lower the average transaction price
and shift profits from sellers to buyers. The simulation results for more complex demand and
supply environments presented in Section 3 correspond well with these results.
The structural model is able to predict the price shift or “buffer” effect without
assumptions about learning behavior and cognitive processes of the participants. “Overreaction”
to the removal of ceilings requires some sort of “memory” which ZI traders lack. Explaining the
overreaction requires a more sophisticated modeling of individual expectation formation and
adjustment than our simple model permits.
Case 2 (Price Slope and Variance)
We shall use the demand and supply functions shown in Figure 7B, which allow two
trades, to explore the effect of price ceiling on the slope of transaction price path.
Buyer (value =1)
Buyer (value=0.7)
Ceiling =0.5
Seller (cost=0.3)
Seller (cost = 0)
Figure 7B
Without a price ceiling, there will be two transactions, both transactions have expected
price of 0.5. Suppose a ceiling at 0.5 is imposed, and all bids and asks above the ceiling are
ignored (ignore mode). The seller with the cost of 0 is more likely to trade first because it will,
on average, have a lower ask. Since the bids above the ceiling are being ignored, a valid bid is
more likely (10 to 7 odds) to come from the buyer at 0.7 than from the buyer at 1. This is
because a relatively larger proportion of bids from the buyer at value 1 will be above the ceiling.
Thus, the buyer at value 0.7 is more likely to trade earlier than the buyer at 1, and this trade will
occur at a price in the range (0, 0.5). The second transaction is more likely to occur between the
seller at cost 0.3 and the buyer at value 1. The feasible range for such a transaction is (0.3, 0.5)
and therefore the average price of the second transaction will be higher than the average price of
the first. Thus, the transaction price series will have a positive slope.
If, instead of ignoring the invalid bids (which is equivalent to resetting them to 0) we
reset them to the ceiling (reset mode), the average value of bids will increase. Resetting bids to
12
the ceiling would thus raise average transaction prices compared to the ignore mode. Resetting to
the ceiling not only raises the average transaction price, it also reduces the slope of the
transaction price relative to the setting where bids exceeding the ceiling are ignored.
When bids above the ceiling are reset to the ceiling, the seller at 0 is still likely to trade
earlier than the seller at 0.3, but the buyer at 1 is likely to trade earlier than the buyer at 0.7
because the buyer at 1 is likely to have a bid at the ceiling earlier than buyer at value 0.7. This
switch in the order in which buyers trade will increase the first transaction price and lower the
second transaction price relative to the case where the bids above the ceiling were ignored. Since
the low cost seller still trades earlier, the slope of the transaction price remains positive albeit
less steep. Since lower cost units are traded early over a larger range of prices, the earlier
transaction prices have a greater variance than the later prices. The simulation results presented
in Section 3 for more complex demand and supply conditions are consistent with these analytical
predictions.
The other three consequences of price controls are more subtle, and have to do with the
role of extra-marginal traders in the equilibrating process. This role is entirely absent in the
static Walrasian tatonnement, and the analysis of markets populated by ZI traders makes it easier
for us to understand and explain its consequences.
Price ceilings constrain the intra-marginal buyers’ competitive ability to outbid the extramarginal buyers, and thus increase the chances that the extramarginal buyers will be able to
trade. Trades that involve extra-marginal traders lower efficiency and increase the trading
volume. The increase in volume comes about because a trade between the intra-marginal sellers
and extra-marginal buyers eventually leads to an additional trade between extramarginal sellers
and intramarginal buyers.
Price ceilings exclude from trading those extra-marginal sellers whose costs exceed the
price ceiling. This exclusion enhances allocative efficiency, especially in the early part of
trading, by eliminating the possibility that such an extra-marginal seller will displace an intramarginal trader. On the other hand, if an extra-marginal buyer does displace an intra-marginal
buyer in early trading, the only way the displaced intra-marginal buyer can get back into the
market is by trading with one of the extra-marginal sellers. By excluding the high cost extramarginal sellers from trading, price ceilings precludes such trades and thus lower allocative
13
efficiency. We cannot yet tell whether the sum of these two opposite effects on efficiency is
always positive or always negative. These effects are explored in Case 3.
Case 3: (Effect of extramarginal traders on allocative efficiency)
Let us consider the demand and supply shown in Figure 7C. The CE price range is
(β, (1−α)) and the CE quantity is 1. With ZI traders the only source of inefficiency is the entry of
the extramarginal traders. Gode and Sunder (1993b) and (1997) derive the allocative efficiency
of similar markets in the absence of price controls.
Buyer (value =1)
Seller (cost=1-α)
Ceiling=c
Buyer (value=β)
Seller (cost = 0)
Figure 7C
Case 3a: Consider the above demand and supply when α>0 and β=0, i.e., effectively there are
no extramarginal buyers. This setting highlights the role of price ceiling in keeping out the
extramarginal sellers because the only source of inefficiency is the probability that the
extramarginal seller will trade with the intramarginal buyer.
1. c < (1−α): A ceiling below the extramarginal seller’s cost locks it out, thereby raising
efficiency to 100 percent. This effect is independent of whether the bids above the ceiling are
ignored or reset.
2. c > (1−α): The effect of the ceiling depends on whether bids above the ceiling are
ignored (i.e., reset to 0) or reset to the ceiling. Ignoring the bids above the ceiling raises
efficiency by lowering the probability of trade with the extramarginal seller. Resetting the abovethe-ceiling bids to the ceiling does not affect this probability and therefore the ceiling then has no
effect on efficiency.
Thus, an efficiency increasing effect of a price ceiling is that it may lower the chances of
extramarginal sellers entering the market.
Case 3b: Consider the above demand and supply when α=0 and β>0, i.e., there are no
extramarginal sellers.
14
1. c < β: The competitive equilibrium theory predicts no efficiency loss. Isaac and Plott
(1981) show, and the ZI model predicts, a higher efficiency loss. If bids above the ceiling are
ignored, then the intramarginal buyer has a lower chance of submitting a valid bid than the
extramarginal buyer, raising the probability that extramarginal buyer would enter the market and
lower efficiency. If the bids above the ceiling are reset to the ceiling, then the probability of entry
by the extramarginal buyer diminishes but is still higher than that without any ceiling. This can
be explained as follows. Suppose the extramarginal buyer first generates a bid at or above the
ceiling, which is then reset to the ceiling. Suppose the intramarginal buyer then generates a
higher bid. Without the ceiling, the intramarginal buyer’s bid would supersede the extramarginal
buyer’s bid. With the ceiling this is not possible. This setting shows that binding price ceiling
lowers the efficiency more than that predicted by competitive equilibrium theory because it
lowers the probability that the intramarginal buyers will outbid extramarginal buyers.
2. c > β: If bids above the ceiling are ignored, and if the ceiling is relatively low, then the
intramarginal buyer will still have a lower chance of submitting a valid bid as compared to the
extramarginal buyer raising the probability of entry by the extramarginal buyer and lowering
efficiency. This effect would diminish as the ceiling is raised. If the bids above the ceiling are
reset to the ceiling, then the ceiling will have no effect.
The above discussion points to two effect of ceilings on the buyers. (1) frequency effect:
if the bids above the ceiling are ignored, then the intramarginal buyers may have a lower chance
of submitting a valid bid, thereby increasing the chance of entry by the extramarginal sellers,
which lowers efficiency. The frequency effect is absent if bids above the ceiling are reset to the
ceiling (2) truncation effect: Truncation lowers the probability that the intramarginal buyers
would outbid the extramarginal buyers by submitting a bid above the ceiling. This effect is
present even if bids exceeding the ceiling are reset to the ceiling.
Case 3c: Consider the above demand and supply when α>0 and β>0 (1−α > β)
The presence of extramarginal traders on both sides of the market has two opposing
effects on efficiency.
(1) Without extramarginal sellers, the inefficient trade could be caused only by the entry
of extramarginal buyers. Addition of extramarginal sellers now raises the probability of an
inefficient trade because now it can also be triggered by the extramarginal sellers.
15
(2) Some of the efficiency loss due to an initial inefficient trade is recovered due to
subsequent trade between the remaining intramarginal and extramarginal traders. Without
extramarginal sellers the efficiency loss due to initial trade between the extramarginal buyer and
the intramarginal seller is 1−β because the intramarginal buyer is left out of the market. With
extramarginal seller, there will be a subsequent trade between the intramarginal buyer and the
extramarginal seller, which recovers a surplus of α and reduces the efficiency loss from 1−β to
1−β−α.
Let us see how price ceilings affect the above two factors.
1. c < β : The efficiency enhancing effect is that the ceiling precludes initial entry by
extramarginal sellers. However, it also has efficiency lowering effects discussed in case 3b and
also due to the fact that partial recovery of surplus lost due to initial entry of extramarginal
buyers is no longer feasible.
2. β < c < 1-α: The only difference now is that as discussed in case 3b the influence of
extramarginal buyers is weakened as compared to the above setting.
3. c > 1-α: This does not preclude the efficiency recovering trades with extramarginal
sellers following initial efficiency reducing trades with extramarginal buyers. However, the
effects discussed in case 3a remain.
These cases illustrate how ZI analysis helps us tease out the structural effects of price
controls. The richness of the analysis of ZI-populated markets points to the extreme difficulty of
doing the same for strategic human traders. The ZI approach seems powerful enough to helps us
make predictions about market dynamics when it is difficult to derive analytical solutions using
game theory.
5. Concluding Remarks
We use the ZI model of trader behavior to gain insights into the effect of non-binding price
limits in double auctions. The ZI model has two major advantages. First, it provides a useful
benchmark to study the effect of market rules on market outcomes. Since ZI traders do not modify
their behavior in response to market rules, any changes in market outcomes with ZI traders can be
attributed to changes in market rules, not individual behavior. Consequences of intelligent, rational,
or other individual behavior have long been the focus of economics. In contrast, using a simple,
minimally rational, random model for trader behavior allows us to explore the structural
16
consequences instead. Exploration of institutions and their economic properties is an important part
of social sciences (North, 1990).
Second, the simplicity of the ZI model allows us to analyze the effect of market rules in a
tractable way. Markets with a finite number of traders who act sequentially are difficult to analyze
using a strategic model of trader behavior. Conjectures that link market outcomes to traders’ specific
strategies are difficult to verify because traders’ strategies are not observable and it is difficult to
implement conjectures of traders’ strategies through computer programs. The simplicity of the ZI
traders helps us appreciate how even seemingly simple changes in rules can cause complex changes
in markets. With cheaper technology, computational methods have become an increasingly
attractive tool for economics research (Leijonhufvud, 1993). Computational work has helped
distinguish between the individual behavior and aggregate outcomes. 7
Using ZI traders allows us to explore why non-binding price limits affect the outcome of
laboratory markets (Isaac and Plott 1981, Smith and Williams 1981) when competitive equilibrium
theory predicts no such effects. We can qualitatively replicate some features of the experimental
results with ZI traders. These features are: (1) transaction prices are skewed away from the ceiling,
and the magnitude of skew diminishes as the ceiling is raised; (2) intra-period transaction price
series has a positive slope; (3) variability of transaction prices declines for later transactions within a
period; (4) allocative efficiency remains close to 100 percent, i.e., total realized profits are
unaffected; (5) transaction prices are lowered transferring some profits from sellers to buyers. These
results hold under alternative implementations of price controls and for alternative specification of
demand and supply conditions.
These effects are usually attributed to modification of human traders’ strategies in
response to price limits. The persistence of these effects when we replace human traders by ZI
traders who do not respond to market rules suggests that it is not modification of trading
strategies per se that causes non-binding price limits to affect market outcomes. In fact, these
effects arise because human traders are unable to modify their behavior sufficiently to counter the
effect of changes in market rules. The evidence that non-binding price ceilings have no effect if
they are imposed after the market has settled to the equilibrium supports our hypothesis. Once
traders have learned about the underlying market conditions they are able to change their
7
See Duh and Sunder, 1986; Gode and Sunder, 1993, 1997; Jamal and Sunder, 1996; Bosch and Sunder, 1994;
Hardle and Kirman, 1995; Evans, 1997; Luo, 1995; and Cason and Friedman, 1996, 1997.
17
behavior to exactly offset the effect of changes in market rules. However, an analysis of price
limits imposed after a market has converged is not as intriguing as the analysis of price limits
imposed when traders are searching for equilibrium. In the real world, the demand and supply
are constantly changing and price limits are often imposed in response to a crisis.
The ZI model provides insights about the market in its initial chaotic stages when it is in
the process of converging to equilibrium. It shows how market rules affect the dynamics of
double auctions by modifying the opportunity set. A statistical analysis reveals three effects of
market rules: (1) the limit effect; (2) reduction of competitive bidding ability of the intramarginal buyers (relative to extra-marginal buyers); (3) exclusion of extramarginal sellers above
the ceiling. These effects become apparent only because we use a simple non-strategic model of
trader behavior.
Individuals do not act alone; “they stand on the scaffolding of larger scale constraining
structures [that are] sometimes the strongest carrier of maximizing force” (Clark, 1997). If
significant aspects of substantive rationality is achievable through simple procedural rationality
in market mechanisms (Simon, 1996), we may be well on our way to the integration of complex
and weak human models of cognitive sciences with the simplicity and power of exchange
institutions in classical economics.
18
List of References
Becker, Gary S. Irrational Behavior and Economic Theory. Journal of Political Economy 70
(1962): 1-13.
Bosch, Antoni, and Shyam Sunder. Tracking the Invisible Hand: Convergence of Double
Auctions to Competitive Equilibrium. Carnegie Mellon University Working Paper,
Revised February 1999.
Cason, Timothy and Daniel Friedman. 1996. Dynamics of Double Auctions, Journal of Economic
Dynamics and Control.
Cason, Timothy and Daniel Friedman. 1997. Price Formation in Single Call Markets.
Econometrica 65:2 (March 1997), pp. 311-3345.
Clark, Andy. 1997. Being There: Putting Brain, Body, and World Together Again. Cambridge:
MIT Press.
Coursey, Don, and Vernon L. Smith. 1983. Price Controls in a Posted Offer Market. The
American Economic Review 73:1 (March), 218-21.
Duh, Rong-Ruey and Shyam Sunder, Base Rate Fallacy: Incentives and Learning in a Market
Environment, in Shane Moriarty, editor, Laboratory Market Studies, University of
Oklahoma (1986), pp. 50-79.
Evans, Dorla A. 1997. The Role of Markets in Reducing Expected Utility Violations. Journal of
Political Economy 105 (3) pp. 622-636.
Gell-Mann, Murray, (1994). The Quark and the Jaguar: Adventures in the Simple and the
Complex. New York: W. H. Freeman and Company.
Gode, Dhananjay K., and Shyam Sunder. (1993a). Allocative Efficiency of Markets with Zero
Intelligence (ZI) Traders: Market as a Partial Substitute for Individual Rationality.
Journal of Political Economy, 101, (1993): No.1. 119-137.
Gode, Dhananjay K. and Shyam Sunder, (1993b). “Lower Bounds for Efficiency of Surplus
Extraction in Double Auctions,” in D. Friedman and J. Rust, eds., The Double Auction
Market, Santa Fe Institute Studies in Sciences of Complexity, Proceedings Volume XIV.
New York: Addison-Wesley.
Gode, Dhananjay K. and Shyam Sunder, (1997). What Makes Markets Allocationally Efficient?”
The Quarterly Journal of Economics 112 (2) pp. 603-630.
19
Hardle, W. and A. Kirman. 1995. Neoclassical Demand: A Model-Free Examination of PriceQuantity Relations in the Marseilles Fish Market. Journal of Econometrics 67 (1) pp.
227-257.
Hayek, Friederich A. 1945. The Uses of Knowledge in Society, The American Economic Review,
35 (September): 519-30.
Isaac, R. Mark, and Charles R. Plott, (1981). Price Controls and the Behavior of Auction
Markets: An Experimental Evaluation. American Economic Review 71:3 (June), pp. 448459.
Jamal, Karim and Shyam Sunder, “Bayesian Equilibrium in Double Auctions Populated by
Biased Heuristic Traders,” Journal of Economic Behavior and Organization Vol. 31 No.
2 (November 1996), pp. 273-291.
Kagel, John H. 1995. Auctions: A Survey of Experimental Research. In J. H. Kagel and A. E.
Roth, eds., The Handbook of Experimental Economics, pp. 501-85. Princeton, NJ:
Princeton University Press.
Leijonhufvud, A. 1993. Towards a Not-Too-Rational Macroeconomics. Southern Economic
Journal 60 (1), pp. 1-13.
Luo, G. Y. 1995. Evolution and Market Competition. Journal of Economic Theory 67 (1) pp.
223-50.
North, Douglass C. 1990. Institutions, Institutional Change and Economic Performance. New
York; Cambridge University Press.
Rust J., Miller, J.H., and Palmer, R (1993). Behavior of Trading Automata in a Computerized
Double Auction Market. In The Double Auction Market: Institutions, Theories, and
Evidence, edited by D. Friedman and J. Rust, Santa Fe Institute Series in the Sciences of
the Complexity, Proceedings Volume XV. New York, NY, Addison-Wesley.
F. M. Scherer. Industrial Pricing: Theory and Evidence. Chicago: Rand McNally College Pub.
Co. 1970.
Simon, Herbert A. 1996. The Sciences of the Artificial. Third Edition. Cambridge, Mass.: MIT
Press.
Smith, Vernon L. And Arlington W. Williams. 1981. On Nonbinding Price Controls in a
Competitive Market. The American Economic Review 71: (June), 467-74.
20
Appendix: Double Auction Market Rules
Double auctions can be run under a variety of rules. This section describes what we mean by the
term “continuous double auction without replacement.”
Definitions:
Market, periods, transactions and rounds of bidding constitute a nested hierarchy in time.
A market is divided into periods, usually of fixed duration. There could be several transactions in
a period. Each transaction consists of one or more rounds of solicitation of bids and asks from
traders.
At the beginning of a period, all market variables are reset and traders receive fresh
endowments. Except for changes in traders’ memory and wealth, all periods are identical. Since
ZI traders lack memory and ignore wealth, each period of a market with ZI traders is statistically
identical.
Each period may have one or more transactions. When bidding starts at the beginning of
a transaction, the current bid is reset to 0 and the current ask is reset to M (an upper bound
specified by the market rules and known to all traders).
Bids and asks are solicited in a series of rounds, and submitted to a central clearinghouse.
If a new bid is higher than the current bid, then it becomes the current bid. Similarly if the new
ask is less than the current ask, then it becomes the current ask. As soon as the current bid equals
or exceeds the current ask, a trade occurs. The transaction price can be (1) the current bid or
current ask depending on which was submitted earlier (or later), or (2) prespecified weighted
average of the current bid and current ask (in human markets current bid mostly equals current
ask rather than exceed it). If a round does not result in a transaction, more rounds follow until
there is a transaction, or the period ends.
Sequencing
Since the computerized clearinghouses are much faster than human traders, the unlikely
event that two or more traders act at the same time is resolved by sequencing the ties in random
order. When human traders are replaced by ZI traders, which are identical computer programs,
all traders theoretically submit their bids and asks at the same time. Therefore, a precise
description of the sequence in which bids and asks are solicited and matched is important.
21
In a synchronized double auction, both the solicitation of bids/asks as well as their
processing for the purpose of matching within each round is simultaneous or batched. All traders
are solicited for their bids and asks. While submitting a bid or ask, a trader does not know the
bids and asks submitted by other traders in that round. The highest bid and the lowest ask from
the round are then used to update the current bid and current ask. If current bid exceeds or equals
current ask, a transaction is completed, the current bid and current ask are reset, and the first
round of bids and asks for the next transaction are solicited. If current bid is less than current ask,
then they are carried over to the next round of bids and asks for the same transaction. Since bids
and asks of all traders in a given round are considered all at once, the order in which they are
solicited does not matter. Since only the highest of all bids and lowest of all asks submitted in a
given round matter, the order in which they are picked for comparison is not relevant.
In a continuous double auction a trader is randomly solicited and his bid/ask is used to
update the current bid/current ask. In solicitation without replacement the next trader is randomly
picked from traders that have not been picked in a round. The process continues until the current
bid exceeds or equals current ask. If all traders have been picked, the next round of random order
solicitation of bids/asks for the transaction starts afresh. Solicitation without replacement ensures
that every trader has a chance to submit a proposal before someone has the chance to submit a
second proposal. Alternatively, random sampling in a continuous double auction could be carried
out with replacement. Such sampling still gives equal chance to all traders, but allows for the
possibility that one trader could be solicited many times while others may not be solicited at all.
Under this sampling scheme, all rounds of a transaction merge into a continuum until the
transaction is completed.
A third possibility is to create a semi-continuous or hybrid mechanism by combining the
batched bid/ask solicitation feature of synchronized double auction with sequential matching
feature of continuous double auction. Similar to a synchronous auction first all bids/asks are
gathered at the clearing house. Then each of them is randomly selected without replacement for
updating the current bid/current ask. If the current bid at any stage of this process matches or
exceeds the current ask, a transaction is completed, the remaining bids/asks are discarded, and
the auction moves to the next transaction; if no match occurs, the auction moves to the next
round.
22
In summary, bids and asks are solicited from traders and matched in a sequence of
rounds. One or more rounds may be needed for a transaction, and all transactions until
endowments are reset constitute a period. A market may consist of one or more periods.
Depending on sequencing of solicitation and matching of bids and asks, we can carry out a
synchronized, continuous (with or without replacement) and hybrid double auction.
In this paper we report simulation results from continuous double auctions without
replacement assuming that the transaction occurs at the earlier of current bid and current ask. We
feel that this market mechanism mimics laboratory auctions. Since in this paper ZI traders ignore
current bid and current ask, the results would be the same if we used semi-continuous or hybrid
auction.
23
Table 1*
Unit Values and Costs
Unit
Theoretical
Profit**
Trader
1
2
3
4
5
6
Buyer 1
53.5
51.0
47.0
46.0
45.0
12.0
Buyer 2
56.0
49
48.0
46.5
45.5
13.5
Buyer 3
56.0
49.0
48.0
46.5
46.0
45.0
13.5
Buyer 4
53.5
51.0
47.0
46.5
45.5
12.0
Seller 5
39.5
42.0
46.0
46.5
47.5
12.0
Seller 6
37.0
44.0
45.0
46.5
47.5
13.5
Seller 7
39.5
42.0
46.0
47.0
48.0
12.0
Seller 8
37.0
44.0
45.0
46.5
47.0
48.0
13.5
*
Reproduced from Table 1 from Smith and Williams (1981) after scaling all values and costs by
a factor of 10.
** Profit if the units were traded at the competitive equilibrium price
Competitive equilibrium price = 46.5, quantity = 12 to 15
Maximum extractable surplus (or maximum total profit of all traders) = 102
Surplus extracted (or total profit of all traders) if all units are traded = 90
24
Table 2*
Unit Values and Costs
Buyers
Value
Theoretical
Sellers
Cost
Profit**
Buyer 1
56.0
9.5
Seller 22
37.0
Buyer 2
56.0
9.5
Seller 23
37.0
Buyer 3
53.5
7.0
Seller 24
39.5
Buyer 4
53.5
7.0
Seller 25
39.5
Buyer 5
51.0
4.5
Seller 26
42.0
Buyer 6
51.0
4.5
Seller 27
42.0
Buyer 7
49.0
2.5
Seller 28
44.0
Buyer 8
49.0
2.5
Seller 29
44.0
Buyer 9
48.0
1.5
Seller 30
45.0
Buyer 10
48.0
1.5
Seller 31
45.0
Buyer 11
47.0
0.5
Seller 32
46.0
Buyer 12
47.0
0.5
Seller 33
46.0
Buyer 13
46.5
0
Seller 34
46.5
Buyer 14
46.5
0
Seller 35
46.5
Buyer 15
46.5
0
Seller 36
46.5
Buyer 16
46.0
0
Seller 37
47.0
Buyer 17
46.0
0
Seller 38
47.0
Buyer 18
45.5
0
Seller 39
47.5
Buyer 19
45.5
0
Seller 40
47.5
Buyer 20
45.0
0
Seller 41
48.0
Buyer 21
45.0
0
Seller 42
48.0
*
Transformation of Table 1 by assigning a single unit to each buyer and seller;
** Profit if the units were traded at the competitive equilibrium price
Competitive equilibrium price = 46.5, quantity = 12 to 15
Maximum extractable surplus (or maximum total profit of all traders) = 102
Surplus extracted (or total profit of all traders) if all units are traded = 90
25
Theoretical
Profit**
9.5
9.5
7.0
7.0
4.5
4.5
2.5
2.5
1.5
1.5
0.5
0.5
0
0
0
0
0
0
0
0
0
Buyers
Value
Buyer 1
Buyer 2
Buyer 3
Buyer 4
Buyer 5
Buyer 6
Buyer 7
Buyer 8
Buyer 9
Buyer 10
Buyer 11
Buyer 12
Buyer 13
Buyer 14
Buyer 15
Buyer 16
Buyer 17
Buyer 18
Buyer 19
Buyer 20
Buyer 21
Buyer 21A
Buyer 21B
Buyer 21C
Buyer 21D
56.0
56.0
53.5
53.5
51.0
51.0
49.0
49.0
48.0
48.0
47.0
47.0
46.5
46.5
46.5
46.0
46.0
45.5
45.5
45.0
45.0
44.0
44.0
42.0
39.5
Table 3*: Unit Values and Costs
Theoretical
Sellers
Profit**
9.5
Seller 22
9.5
Seller 23
7.0
Seller 24
7.0
Seller 25
4.5
Seller 26
4.5
Seller 27
2.5
Seller 28
2.5
Seller 29
1.5
Seller 30
1.5
Seller 31
0.5
Seller 32
0.5
Seller 33
0
Seller 34
0
Seller 35
0
Seller 36
0
Seller 37
0
Seller 38
0
Seller 39
0
Seller 40
0
Seller 41
0
Seller 42
0
Seller 42A
0
Seller 42B
0
Seller 42C
0
Seller 42D
Cost
37.0
37.0
39.5
39.5
42.0
42.0
44.0
44.0
45.0
45.0
46.0
46.0
46.5
46.5
46.5
47.0
47.0
47.5
47.5
48.0
48.0
49.0
49.0
51.0
53.5
*
Theoretical
Profit**
9.5
9.5
7.0
7.0
4.5
4.5
2.5
2.5
1.5
1.5
0.5
0.5
0
0
0
0
0
0
0
0
0
0
0
0
0
Transformation of Table 1 by assigning a single unit to each buyer and seller and addition of
extramarginal units;
** Profit if the units were traded at the competitive equilibrium price
Competitive equilibrium price = 46.5, quantity = 12 to 15
Maximum extractable surplus (or maximum total profit of all traders) = 102
Surplus extracted (or total profit of all traders) if all units are traded = 57
26
Table 4: Summary Statistics
(100 periods for each column)
Smith and Williams market demand and supply
Extended demand and supply (more
extramarginal traders)
Multiple units per trader (Table 1)
Single unit per trader (Table 2)
Single unit per trader (Table 3)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
Panel 2
Panel 3
Panel 4
Panel 5
Panel 1
No Ceiling Ceiling = 47 Ceiling = 47 No Ceiling Ceiling = 47 Ceiling = 47 No Ceiling Ceiling = 47 Ceiling = 47
Ignore
Reset
Ignore
Reset
Ignore
Reset
invalid bids invalid bids
invalid bids invalid bids
invalid bids invalid bids
TRANSACTION
PRICE
Mean
Median
Std. Deviation
Skewness
Kurtosis
Volume
75 percentile
25 percentile
EFFICIENCY
Total
Mean
Std. Error of Mean
Buyers
Mean
Std. Error of Mean
Sellers
Mean
Std. Error of Mean
46.45
46.47
2.75
-0.02
1.55
1298
47.62
45.25
45.01
45.89
2.09
-1.49
1.51
1337
46.51
44.31
45.67
46.49
1.98
-2.01
3.70
1285
47.00
45.25
46.45
46.46
2.30
-0.15
2.35
1488
47.52
45.46
45.63
46.20
1.56
-2.17
5.49
1480
46.65
45.17
46.07
46.66
1.43
-2.44
7.15
1406
47.00
45.74
46.55
46.52
2.35
0.03
2.26
1510
47.58
45.50
45.71
46.25
1.45
-2.18
5.68
1467
46.64
45.25
46.01
46.66
1.59
-2.43
6.50
1402
47.00
45.76
99.90
0.03
99.88
0.03
99.97
0.01
98.14
0.13
96.24
0.44
99.46
0.06
97.36
0.18
96.29
0.49
99.08
0.12
101.26
1.64
138.78
1.07
120.81
1.08
99.72
1.32
117.73
1.26
110.69
1.01
95.93
1.31
115.29
1.15
111.69
0.94
98.55
1.64
60.99
1.07
79.13
1.08
96.56
1.31
74.74
0.94
88.23
1.00
98.78
1.32
77.29
0.84
86.48
0.97
27
Figure 1: Demand, Supply and Price Ceiling
60
55
50
Price
Price Ceiling = 47
Demand
Equil. Price = 46.5
45
Supply
Price Ceiling
Eq. Price
40
35
30
0
5
10
15
Quantity
28
20
25
Figure 2: Results from Laboratory Experiments with Human Subjects
(Figure 1. Experiment 2:26 from Smith and Williams, 1981)
29
Figure 3:Results from Laboratory Experiments with Human Subjects
(Figure 3 from Smith and Williams, 1981)
30
Figure 4: Histograms of Transaction Price Distributions
1: No Price Limits
2: Ceiling = 47
3: Ceiling = 47
Bid = Min(Bid, Ceiling)
4: Ceiling = 47
One unit per trader
5: Ceiling = 47
One unit per trader
More extramarginal
traders
55
55
55
55
55
53
53
53
53
53
51
51
51
51
51
49
49
49
49
49
47
47
47
47
47
45
45
45
45
45
43
43
43
43
43
41
41
41
41
41
39
39
39
39
39
37
37
0
200 400 600 800
37
0
200
400
600
800
37
0
200 400 600 800
31
37
0
200 400 600 800
0
200 400 600 800
Figure 5:Intra-Period Mean of Transaction Price Series
1: No Price Ceiling
2: Ceiling =47
3: Ceiling = 47
Bid = Min(Bid, Ceiling)
4: Ceiling = 47
One unit per trader
56
56
56
56
56
54
54
54
54
52
52
52
52
50
50
50
50
48
48
48
48
5: Ceiling = 47
One unit per trader
More extramarginal
traders
54
52
50
48
46 1
5
9
13
17
46 1
5
9
13
17
46 1
5
9
13
17
46 1
5
9
13
17
46 1
44
44
44
44
44
42
42
42
42
42
40
40
40
40
40
38
38
38
38
38
36
36
36
36
36
32
5
9
13
17
Figure 6:Intra-Period Median and Inter-Quartile Range of Transaction Price Series
1: No Price Ceiling
2: Ceiling =47
3: Ceiling = 47
Bid = Min(Bid, Ceiling)
4: Ceiling = 47
One unit per trader
56
56
56
56
56
54
54
54
54
54
52
52
52
52
52
50
50
50
50
50
48
48
48
48
48
46 1
4
7
10 13 16 19
46 1
4
7
10 13 16 19
46 1
4
7
10 13 16 19
46 1
4
7
10 13 16 19
46 1
44
44
44
44
44
42
42
42
42
42
40
40
40
40
40
38
38
38
38
38
36
36
36
36
36
33
5: Ceiling = 47
One unit per trader
More extramarginal
traders
4
7
10 13 16 19
Figure 7: Demand, Supply, and Price Controls
Buyer (value =1)
Ceiling
c
Seller (cost = 0)
Figure 2A
Buyer (value =1)
Buyer (value=0.7)
Ceiling =0.5
Seller (cost=0.3)
Seller (cost = 0)
Figure 7B
Buyer (value =1)
Seller (cost=1-α)
Ceiling=c
Buyer (value=β)
Seller (cost = 0)
Figure 7C
34