Bilateral oligopoly in pollution permit markets: experimental evidence.
Schnier, Kurt ; Doyle, Martin ; Rigby, James R. 等
Bilateral oligopoly in pollution permit markets: experimental evidence.
We experimentally investigate behavior in a bilateral oligopoly
using a supply function equilibria model discussed by Klemperer and
Meyer (1989), Hendricks and McAfee (2010), and Malueg and Yates (2009).
We focus on the role that market size and the degree of firm
heterogeneity have on the market equilibrium. Our results indicate that
subjects within the experiment recognize the strategic incentives in a
bilateral oligopoly, but they do not exploit these incentives to the
exact magnitude predicted by theory. We find weaker support for
predicted market outcomes, as market efficiency does not depend on
market size, and in some cases buyers or sellers are more successful at
extracting the rents from the market. (JEL L13, Q5, C9)
I. INTRODUCTION
Pollution permit markets have been applied to a wide variety of
pollution problems. Recently, there has been quite a bit of interest in
optimizing the scale of these markets (Wiliams 2003; Krysiak and
Schweitzer 2010; Yates et al. 2013). When a market is optimized with
respect to scale, the polluting firms are divided up into zones and
firms may trade emissions only with other firms in their zone. In a
given zone, there may be a small number of buyers and sellers, and if
so, then all market participants in the zone, not just the buyers or
just the sellers, exert market power. This market structure is called a
bilateral oligopoly. In general, the strategic incentives inherent in
bilateral oligopoly lead to a reduction in trade between firms relative
to a competitive benchmark. For pollution permit markets, this reduction
in trade has a cost (firms must expend more resources to reduce
emissions) as well as a benefit (there are fewer localized damage hot
spots). These effects tend to cancel out, implying that the bilateral
oligopoly performs about as well overall as a competitive market (Yates
et al. 2013). So a regulator need not be overly concerned if it turns
out that a zone contains only a small number of firms, at least
according to theory.
Optimizing permit markets with respect to scale offers promise for
the design of future permit markets. But regulators may be reluctant to
put these insights into practice and create a zone with a small number
of buyers and sellers without assurances that the resulting market will
behave according to theoretical predictions. Toward that end, we conduct
the first experimental test of a bilateral oligopoly utilizing a supply
function equilibria market structure. Using varying market sizes and
cost heterogeneity, we find that subjects within the experiment
recognize the strategic incentives in a bilateral oligopoly. However, we
find weaker support for predicted market outcomes as the overall market
efficiency does not appear to depend on market size. Our cautious
conclusion is that permit markets that are optimized for scale should
generally perform as predicted.
In comparison to other market structures (e.g., monopoly,
monopsony, bilateral monopoly, oligopoly, or oligopsony), bilateral
oligopoly has received surprisingly little attention. Virtually all of
the extent literature is theoretical. Hendricks and McAfee (2010) use
extensions of Klemperer and Meyer's (1989) supply function
equilibria models to characterize bilateral oligopoly equilibria in a
single market. Weretka (2011) uses similar techniques to analyze a
general equilibrium setting. Bjornerstedt and Stennek (2001) do not use
supply function equilibria, but rather analyze sequential bilateral
bargaining between market participants in a network structure. Lange
(2012), Malueg and Yates (2009), and Wirl (2009) have all studied
bilateral oligopoly in the specific context of pollution permit markets.
These papers are conceptually similar in that they all use techniques
related to supply function equilibria. There are some differences in
their exact structure, however, and here we focus on the model in Malueg
and Yates (2009).
The key feature of the Malueg and Yates (2009) model is a net-trade
function, which specifies how many permits the firms are willing to buy
and sell at various prices. Firms select a net-trade function and then
submit it to a market maker. The market maker selects the market price
such that overall net trades are zero. Firms take the behavior of the
market maker into account, of course, when they select their net-trade
functions. Malueg and Yates (2009) describe the resulting equilibrium.
Compared to the competitive outcome, a given firm in the bilateral
oligopoly equilibrium makes an adjustment that depends on both the
number of firms in the market and the variance of firms' abatement
costs. This adjustment increases as the number of firms decreases and it
increases as the cost variance increases. These theoretical predictions
provide the foundation for our experimental analysis.
Although the literature on bilateral oligopoly is relatively small,
the underlying supply function equilibria technique has been the subject
of considerable theoretical and empirical analysis. Interestingly, there
has been little in the way of experimental testing of supply function
equilibria. (1) Perhaps, one reason for this is that supply function
equilibria markets have been typically applied to electricity markets.
There are two salient features of these markets that may help to explain
why these markets have not been experimentally investigated. First,
market participants have a strong incentive to learn about the market
process because they are buying or selling the primary commodity for
their firm. Second, there is rapid feedback due to the fact that the
market operates on a daily time scale. Given these features, it has been
argued that traders will quickly learn and react to the economic
incentives in the market and as a result there should likely be good
agreement between theory and actual behavior (Green and Newbery 1992).
Therefore, the value of an experimental investigation of a similar
market may be low.
These features, however, are unlikely to be found in pollution
permit markets. In most of these markets, the commodity being traded is
not the primary output of the firm. For example, a waste water treatment
plant manager is likely to be more concerned about the successful
operation of their plant rather than the subtleties of the market for
nitrogen emissions. In addition, and perhaps more importantly, the
feedback is slow due to the long time scales of the market. Many
pollution permit markets operate on a yearly basis, if not longer
(Holland and Moore 2013). Thus traders may take a long time to develop
expertise within the market. This is often the case for many
environmental compliance markets (e.g., carbon offsets, Clean Water Act
compensatory mitigation), as these markets typically involve infrequent
market transactions (e.g., land developer needing wetland credits once
every few years, or possibly just once). Given this, it is natural to
question whether or not actual permit markets with a small number of
buyers and sellers will function as predicted by theory. Economic
experiments in a laboratory are ideally suited to answer questions of
this type. (2) If subjects do not behave according to theory, then this
suggests that similar behavior may be exhibited in actual permit markets
and may persist for some time.
Our results indicate that both buyers and sellers understand and
exploit the opportunity to exert market power by manipulating their
net-trade functions, although the evidence is stronger for buyers than
for sellers, and the magnitude of the effects are different than
predicted by theory. We generally find good agreement with the
predictions for comparative statics with respect to the number of firms
and the cost variance. As the cost variance increases, the high cost
(low cost) firms increase (decrease) their indifference price between
being a buyer or a seller of permits and, more importantly, their
behavior increasingly deviates from the competitive equilibrium.
Similarly, as the number of firms decreases, their behavior increasingly
deviates from the competitive equilibrium. We also analyze the overall
market outcomes. Counter to theoretical predictions, we find that market
efficiency does not depend on market size, and there are circumstances
where either buyers or sellers are successfully able to generate more
profits than the other. However, overall the benefits to buyers and
sellers are for the most part balanced.
II. THEORETICAL MODEL
There are n firms that trade pollution permits. The total endowment
of permits is fixed at W. Each firm i is given an endowment of permits
[w.sub.i] with [summation][w.sub.i] = W. Firm i has a quadratic
abatement cost as a function of emissions of pollution [e.sub.i] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
For interior levels of emissions, Firm i's marginal abatement
cost is
=[C'.sub.i] = [[theta].sub.i] - [e.sub.i].
The minus sign is introduced because the abatement cost function is
defined with respect to emissions rather than abatement. The
business-as-usual level of emissions is [[theta].sub.i]. This
corresponds to what the firm would emit if there were no regulations.
The market price is defined through net-trade functions. A
net-trade function for a given firm is piecewise linear in the market
price and contains one parameter a, that is selected by the firm. The
net-trade function v, for firm i is
(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Equation (2) specifies how many permits firm i is willing to buy
(sell) as a function of the market price p. Firms select the intercept
of the net-trade function, but the slope is fixed at one. Firms are not
allowed to sell a quantity of permits that is greater than their initial
endowment. The net-trade function is shown in Figure 1. As illustrated
in this figure, a, has a useful interpretation as the price at which
subjects are indifferent between buying and selling permits. Let [??] =
l/n [summatin][a.sub.i] denote the average of the firms' choices of
[a.sub.i].
The firms report the net-trade functions to the market maker. The
market maker selects the equilibrium price such that the aggregate net
trade function ([summation][v.sub.i]) is equal to zero. This equilibrium
price can be found by applying a simple algorithm. The first step is to
determine a candidate market price p = [??]. If, at this price, all
firms are on the negatively sloped part of their net-trade function,
then the candidate price is in fact the market price. If not, then
collect all firms that are not on the negatively sloped part and group
them into the exceptions group. The next candidate market price is
determined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Then test to see if there are any new exceptions with this
candidate market price. If not, then the candidate price is the market
price. If yes, then add the new exceptions to the exceptions group and
repeat until there are no new exceptions. (3)
Once the market price is specified, a firm's net-trade
function combined with their permit endowment determines their pollution
emissions:
[e.sub.i] = [v.sub.i] + [w.sub.i]
For example, if the net-trade function, evaluated at the market
price, is positive then the firm buys permits in the market and thus
they are able to emit more pollution than their initial endowment. In
the case that the net-trade function is negative the firm becomes a
seller of permits and their emission levels are lower than their permit
endowment.
Firms exert market power through their choice of [a.sub.i]. In
particular, firms realize that the value of [a.sub.i] they select will
influence the market price. In our experiment, each firm receives a
fixed payment [[pi].sub.i] for participating in the market. (4) The
payoff to firm i is equal to this payment minus the abatement cost and
permit expenditures:
[V.sub.i] = [[pi].sub.i] - ([C.sub.i] ([v.sub.i] + [w.sub.i]) +
p[v.sub.i]).
Firm i selects [a.sub.i] to maximize this expression.
In our model, the abatement cost function is truncated at business
as usual emissions and the net-trade function is truncated at the price
[??] = [w.sub.i] + [a.sub.i] (the price at which the firm is just
willing to sell all of their permit endowment). Malueg and Yates (2009)
consider a slightly different model in which these functions are not
truncated. They show that there is a unique interior Nash equilibrium
for the choice of [a.sub.i]. In this interior Nash equilibrium, a given
firm's emissions of pollution are not greater than
business-as-usual and the market price is not more than [??]. This
equilibrium will still be an equilibrium in our model with truncated
functions, but its uniqueness is now an open question. In the Appendix,
we show there cannot actually be any equilibria on the truncated
portions of the abatement cost functions and net-trade functions, so
that the interior Nash equilibrium described by Malueg and Yates (2009)
is indeed the unique equilibrium in our model.
The equilibrium value for [a.sub.i] is given by
(3) [a.sup.*.sub.i] = [a.sup.c.sub.i] + [[DELTA].sub.i],
where
[a.sup.c.sub.i] = [[theta].sub.i] - [w.sub.i]
is the competitive outcome, (5) and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is an adjustment term (again, we use the tilde to represent
averages across all firms). According to Equation (3), firms make an
adjustment to the competitive outcome that depends on the difference
between their own marginal abatement costs at the permit endowment
([[theta].sub.i] - [w.sub.i]) and the average marginal abatement cost at
the permit endowment ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN
ASCII]). Firms that have higher-than-average marginal costs expect to
purchase permits, so they want to reduce their net-trade function to put
downward pressure on prices. Hence they adjust [a.sub.i] below the
competitive outcome. The magnitude of this adjustment decreases in n so
that as n approaches infinity, the equilibrium specified by Equation (3)
approaches the competitive outcome.
In a traditional model of market power, the market price is often a
good indicator of the degree of market power. For example, in an
oligopoly, when firms have greater market power the price is further
from marginal production cost. This will not be the case in bilateral
oligopoly. Buyers are trying to put downward pressure on prices, and
sellers are trying to put upward pressure on prices. Under the
assumptions of the model, the net effect of these efforts cancel. The
market price in equilibrium is:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This is the same as the competitive price.
To characterize the degree of market power in bilateral oligopoly,
we turn to another measure: the variance in marginal abatement costs at
the permit endowment. Let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If the endowment is such that marginal abatement costs are equal
across all firms, then var([theta] - w) = 0. In this case, the
adjustment from the competitive outcome in Equation (3) is zero. In
other words, firms do not exert any market power. If the endowment is
such that there is great variation in marginal abatement costs, then
var([theta] - w) is large. In this case, on average, the adjustment from
the competitive solution in Equation (3) is large and firms exert
significant market power.
The welfare loss in an oligopoly model is due to the increase in
price and the decrease in the quantity of production relative to the
competitive outcome. In this bilateral oligopoly model, there is no
price distortion, but there remains an output distortion. Malueg and
Yates (2009) show that the quantity of trade is lower in bilateral
oligopoly than in competition.
III. EXPERIMENTAL ENVIRONMENT
In our experiment, subjects play the role of firms in the theory
described above. We now describe the experimental environment. Section
III.A focuses on the experimental design that we use to test six
research hypotheses that arise from the aforementioned theory. Section
III.B discusses the experimental procedures used to execute our design
and the testing of our research hypotheses.
A. Experimental Design and Hypothesis
We employ two treatment variables and a three-by-three design. The
first treatment variable is n, the number of subjects in a given market.
This variable takes on three values: 4, 8, and 16. The second treatment
variable is var([theta] - w), the variance of the marginal abatement
costs at the permit endowment. This variable also takes on three values:
0, [sigma].sup.2], and 4[[simgma].sup.2]. Under the endowment rule
discussed below, var([theta] - w) simplifies to become a function of
[theta] only. To further simplify, we allow for only two values for
[theta], and assign these values to an equal number of subjects. This
effectively segregates the subjects into two equal size groups: high
cost subjects and low cost subjects. Although it is possible for a high
cost subject to be a seller of permits (if, for example, the price of
permits is quite high), we would ordinarily expect high cost subjects to
be buyers and low cost subjects to be sellers. So we refer to them using
these terms, denoted by the subscript b ands respectively. The values
for [[theta].sub.b] and [[theta].sub.s] vary according to the
var([theta] - w) treatment, as specified in Table 1. In the first row of
this table, all subjects have identical costs at the permit endowment
and we would expect to get the competitive outcome. In the second and
third rows, the values for [[theta].sub.h] and [[theta].sub.s] are equal
deviations above and below the value in the first row.
We also have several nontreatment parameters. Each of these are a
function of [[theta].sub.i]. Subjects are given an initial permit
endowment equal to half of their business-as-usual emissions: [w.sub.i]
- [[theta].sub.i]/2. The constant [A.sub.i] in each abatement cost
function is selected such that the abatement cost function is zero at
the business-as-usual emissions (6): [A.sub.i] = (1/2)
([[theta].sup.2.sub.i]). Finally, the constant it,? is selected so that
each subject receives the same no-trade payoff, k, that we set equal to
five within the experiment. (7) So we have [[pi].sub.i] = [kappa] +
([[theta].sup.2.sub.i]/8).
Our first four research hypotheses focus on the individual
decisions of subjects within the experiment. The theoretical predictions
that underlie these hypotheses are based on Equation (3). In particular,
this equation identifies the value of [a.sub.i] selected by the players
as a function of the competitive outcome and the adjustment from the
competitive outcome. The predicted values for buyers are given in Table
2 and for sellers in Table 3. Our first hypothesis directly investigates
these values.
Experimental Hypothesis 1. The [a.sub.i] selected by buyers and
sellers are given by Tables 2 and 3.
The next three hypotheses are based on comparative statics across
treatments. As can be seen from Tables 2 and 3, the var([theta] - w)
treatment variable affects both the competitive outcome and the
adjustment term, but the number of subjects only affects the adjustment
term. As a consequence, we utilize two hypotheses for the var([theta] -
w) treatment and one for the number of subjects treatment.
Experimental Hypothesis 2. As var([theta] - w) increases, the
[a.sub.i] selected by buyers (sellers) increases (decreases).
This hypothesis addresses whether subjects understand the structure
of the market, but it does not discriminate between strategic and
competitive behavior. For example, a buyer may simply select a high
indifference price because they have high costs, not because they want
to manipulate the market. As seen from Table 2, a buyer that does behave
strategically will shade their indifference price below the competitive
outcome to put downward pressure on the market price.
The third and fourth research hypotheses discriminate between
strategic and competitive behavior.
Experimental Hypothesis 3. As var([theta] - w) increases, the
magnitude of the adjustments from the competitive outcome ([DELTA] =
[a.sub.i] - [a.sup.c.sub.i]) selected by buyers and sellers increases.
Experimental Hypothesis 4. For the nonzero variance cases, as the
number of subjects increases, the magnitude of the adjustments from the
competitive outcome ([[DELTA].sub.i] = [a.sub.i] - [a.sup.c.sub.i])
decreases for both buyers and sellers. For the zero-variance cases, as
the number of subjects increases, the adjustments stay fixed at zero.
Our final two hypotheses focus on the overall market outcomes. When
the cost variance is nonzero, theory indicates that as the market size
increases the market should converge toward the competitive equilibrium.
This implies that the market should become more efficient as the market
size increases, relative to the perfectly competitive market. When the
cost variance is zero, the market size has no impact on market
efficiency. Combining these theoretical predictions generates our fifth
research hypothesis.
Experimental Hypothesis 5. For the nonzero variance cases, the
market efficiency (relative to perfect competition) increases with the
size of the market. For the zero variances cases, the market efficiency
is constant.
Owing to the symmetric nature of the [theta]'s used in the
experiment (see Table 1), neither the buyers or sellers should be able
to exhibit more market power relative to each other. This implies that
the split in total profit between the buyers and sellers should be equal
regardless of the market size or the variance in cost, generating our
sixth research hypothesis.
Experimental Hypothesis 6. The share of profits is evenly split
between buyers and sellers independent of variance and market size.
The following section outlines the experimental design utilized to
test our six research hypotheses.
B. Experimental Procedures
The experiments were conducted at the Experimental Economics Center
(ExCEN) in the Andrew Young School of Policy Studies at Georgia State
University. A total of 380 students participated in the experiment
spread across the nine treatments. In the market size of 4 and 8
treatments there were 32 students enrolled, except for the zero variance
treatment when the market size was four. In this case only 28 students
were enrolled (seven groups of four). For the market size of 16
treatments there were 64 subjects enrolled, split across two
experimental sessions. A higher number of subjects were recruited for
the market size of 16 to increase the number of market-level
observations for our investigation of hypotheses 5 and 6. All of the
subjects were recruited using an email database that randomly selects
eligible students for recruiting. All experiments were conducted using
computer terminals with each student randomly assigned to a treatment.
Upon entering the ExCEN laboratory subjects were instructed to sit
at a specific computer terminal within the laboratory, where a copy of
the instructions was waiting for them to review. (8) Prior to beginning
the experiment the instructions were read orally to each group in detail
and they completed five rounds of practice followed by a survey to
determine how well the subjects understood the experimental treatments.
During the five practice rounds the subjects were instructed that they
were participating in a market with computer-generated decisions and the
purpose of the practice was to familiarize themselves with the
experiment software. In the case that the subjects did not answer one of
the survey questions correctly, the question was highlighted and the
subjects were informed of the correct answer. Following completion of
the practice rounds the subjects were asked whether or not they had any
additional questions regarding the experiment. After answering any
additional questions they entered the experimental treatment.
In the treatment portion of the experiment the subjects were
randomly paired in each of the 20 periods into the specified market
size. In addition, there were an equal number of buyers and sellers
within each group and one's type remained constant throughout the
entire experiment. The setting was context neutral in that subjects were
told that they needed a permit for each unit of production, but this
production was not tied to pollution.
At the beginning of each period, subjects were informed of their
personal cost function as well as those for the other subjects in the
experiment and the total number of subjects within their market.
Conditional on this information they were asked to submit their
individual supply functions (determined by their selection of a,). An
example of the decision screen presented to the subjects is illustrated
in Figure 2. The software dynamically illustrated how their selection of
a, and the market price would affect their expected net permit
purchases, fixed payment from operating, costs of operating, permit
expenditures, and total profits. In particular, the selection of
[a.sub.i] was done by manipulating a sliding tab near the bottom left of
the screen. For any given position of the sliding tab, a table on the
right of the screen displayed the monetary information as a function of
various market prices. In addition, a graph showed the subject's
net-trade function and indicated price regions in which the subject
would be a buyer or seller of permits. Once subjects selected a final
value for [a.sub.i], the market equilibrium was calculated and they were
informed of the market clearing price and their experimental earnings.
Earnings within the experiment were based on their cumulative
returns within the experiment over the 20 periods. Subjects could also
look at the history of all their previous actions and the resulting
market outcomes at any time during the experiment. Each experimental
dollar earned in the experiment was converted assuming that 1
experimental dollar equaled 0.25 US Dollars. In addition to their
experimental earnings, the subjects were paid a five dollar show-up fee
for their participation. The average subject payment within the
experiment was 32.68 US Dollars, including the show-up fee.
IV. RESULTS
Prior to investigating our primary research hypotheses it is
important to illustrate that subjects varied their behavior depending on
their role as a buyer or seller as well as the treatment environment.
Figure 3 illustrates buyer and seller choices of [a.sub.i] for each
experimental treatment. The buyer and seller decisions for the zero
variance cases, far left column of the figure, are nearly identical and
focused close to the value of five, the equilibrium prediction. As the
variance increases, moving from the left column to the right column, the
divergence in the buyer and seller selection of a, becomes more evident.
The buyers' distribution shifts upward on the price axis, while the
sellers' distribution shifts downward. By the time the variance
reaches its largest value of 4[[sigma].sup.2] there is a clear
distinction between buyer and seller decisions.
Table 4 illustrates the average and standard deviation of [a.sub.i]
within each treatment broken down by whether or not the subject was a
buyer, b, or a seller, s. We have elected to illustrate the descriptive
statistics for all 20 periods, the last 15 periods and the last ten
periods to illustrate the stability of subject decisions within the
experiment. Although the standard deviation does decrease as we shorten
the time window, the average values of a, are not statistically
different, assuming the same treatment and subject type.
The data in Table 4, in conjunction with the visual evidence from
Figure 3, suggest that buyers behave different than sellers, and these
differences depend on the various treatments. Formal statistical tests
verify this observation. We first test for differences between buyer and
seller behavior within the same treatment. Table A1 shows the results of
Mann-Whitney, t, and Komolgorov-Smirnov tests of equivalent [a.sub.i]
selections for buyers and sellers. For the nonzero variance treatments,
we would expect the behavior of the buyers and sellers to be different,
which suggests that the tests should give statistically significant
results. As shown in Table Al, this does indeed occur. For the zero
variance treatments, we would expect the behavior of the buyers and
sellers to be the same, which suggests that the tests should give
statistically insignificant results. This generally occurs, although the
test results for the market size of 8 are mixed.
Next we investigate whether or not subject behavior differs across
the experimental treatments. Here we conduct three paired analyses of
distributional equivalence. The first compares the buyers across
treatments, the second compares the sellers across treatments and the
last one compares the buyers and sellers across treatments. For each
analysis, we conducted Mann-Whitney, t, and Komolgorov-Smirnov tests.
The results are shown in Tables A2, A3, and A4. For the zero variance
cases, the tests are almost in complete agreement with theory, with only
a few cases of mixed evidence. The results are similarly strong for the
nonzero variance cases. There are only few instances in which the
evidence is mixed, and one instance in which all three tests indicate a
statistically insignificant difference (sellers, variance of
4[[sigma].sup.2], market size of 4 vs. 8). Having validated differences
between buyers and sellers across treatments, we now turn to examining
our research hypotheses.
A. Hypotheses for Individual Behavior
To test our first research hypothesis that the subjects'
selection of at matches the predicted value, we use a series of Wilcoxon
rank-sum tests (see Tables A5 and A6). First we consider buyers. For
buyers there is only one case where their selection of a, accords with
the theoretical predictions. This occurs when variance of zero is paired
with market size of 16 (z-statistic is -0.369). However, when we conduct
a parametric t-test for this treatment the test statistic indicates that
the decisions are statistically different from our predictions
(t-statistic is -3.586). (9) Therefore, these results indicate that the
buyers' selection of at does not accord with our theoretical
predictions. The selection of [a.sub.i], for the sellers accords with
our theoretical predictions of at more often than with the buyers. The
cases in which the sellers' selection of [a.sub.i] matches the
theory are variance of zero paired with market size of 4, 8, and 16, as
well as variance of 4[[sigma].sup.2] paired with market size of 8. (10)
Given that neither the buyers' or sellers' behavior
perfectly accords with the theoretical predictions of [a.sub.i], an
important question is whether or not their selection of [a.sub.i]
reflects strategic behavior. Alternatively, subjects might not recognize
their ability to influence prices. If this is indeed the case, then the
actual [a.sub.i] would be closer to the competitive prediction
[a.sup.c.sub.i]. To distinguish between strategic and competitive
behavior, we conducted an additional test for the six nonzero variance
cases. (11) We computed the midpoint between the theoretically predicted
at and the competitive alternative [a.sup.c.sub.i] and then created a
difference variable, [d.sub.i] by subtracting the midpoint from the
subject's selection of [a.sub.i]
For buyers, the predicted value of a, is smaller than
[a.sup.c.sub.i]. If they follow a strategy closer to the strategic
prediction, then we would expect [d.sub.i] to be statistically negative.
If they follow a strategy closer to the competitive prediction, then we
would expect [d.sub.i] to be statistically positive. This pattern
reverses for the sellers because their predicted [a.sub.i] is greater
than [a.sup.c.sub.i].
To test for significance, we utilize a series of Wilcoxon rank-sum
tests. For the buyers, all test statistics for [d.sub.i] are
statistically negative (z-statistics range from -4.826 to -10.073). This
indicates that buyers behave closer to the strategic prediction than the
competitive alternative. For the sellers, in three of the six cases the
test statistics for d; are not significant at the 95th percentile. These
cases occur when the variance of [[sigma].sup.2] is paired with a market
size of 4 (z-statistic is -0.659), when the variance of 4[[sigma].sup.2]
is paired with a market size of 8 (z-statistic of 1.693), and when the
variance of 4[[sigma].sup.2] is paired with a market size of 16
(z-statistic of -1.827). In these cases, the sellers' behavior is
not statistically distinguishable from either the strategic prediction
or the competitive alternative. The test statistics in the remaining
cases indicate that sellers are closer to the strategic prediction when
variance of [[sigma].sup.2] is paired with market size of 16
(z-statistic is 6.295), (12) but sellers behave closer to the perfectly
competitive predictions when the variance of 4[[sigma].sup.2] is paired
with a market size of 4 (z-statistic of -2.308) and when the variance
[[sigma].sup.2] is paired with market size of 8 (z-statistic is -3.560).
Taken together, the tests of our first hypothesis provide strong
evidence that buyers internalize the strategic incentives present and
manipulate the price via their selection of [[sigma].sup.i] but not to
the same extent as predicted by theory. The evidence for sellers is
somewhat mixed. For three of the cases the sellers' behavior is not
distinguishable from either the strategic or competitive predictions,
and in one case they behave closer to the strategic predictions.
However, in the remaining two cases they behave closer to the
competitive predictions. Overall, this suggests that sellers are less
able to internalize the strategic incentives present within the decision
environment. Additional insight into strategic behavior and the
differences between buyers and sellers will be provided below in the
analysis of our third and fourth hypotheses.
Turning now to comparative statics, our second research hypothesis
is that as var([theta] - w) increases, the [[sigma].sub.i] selected by
buyers (sellers) increases (decreases). Evidence in favor of this
hypothesis can be easily seen in both the descriptive statistics (Table
4) and the visual representation of the data (Figure 3). More rigorous
evidence is based on a series of two-sample Wilcoxon rank-sum tests
comparing the selections of [[sigma].sub.i], across the different
variance levels holding the market size constant. The results from these
tests are contained in Table 5. All of the test statistics confirm our
second hypothesis. The negative and statistically significant
z-statistics for the buyers indicate that as the variance increases
(holding the market size constant) the subject's selection of
[[sigma].sub.i], increases. The positive and statistically significant
z-statistic for the sellers indicates the opposite; as the variance
increases the sellers selection of at decreases. The combined evidence
offers strong support for our second research hypothesis and confirms
that the subjects realize that when they have higher (lower) costs, they
should increase (decrease) their indifference price.
Our third and fourth research hypotheses shed additional light on
the issue of whether or not subjects act strategically to manipulate the
market price. They are based on changes in [[DELTA].sub.i] across
treatments. To test these hypotheses, we calculated [[DELTA].sub.i] for
each subject's decision within the experiment, as illustrated in
Table 6. The third research hypothesis states that as var([theta] - w)
increases, the magnitude of the adjustments from the competitive outcome
([[DELTA].sub.i] = [a.sub.i] - [a.sup.c.sup.i]) selected by buyers and
sellers increases. The fourth research hypothesis states that, for
nonzero variance cases, as the number of subjects increases, the
magnitude of the adjustments from the competitive outcome selected by
buyers and sellers decreases. Evidence in favor of both hypotheses is
illustrated in Table 6, although it appears to be stronger for buyers
than sellers.
To more formally investigate the third hypothesis, a series of
two-sample Wilcoxon rank-sum tests were conducted comparing all
[[DELTA].sub.i]'s holding the subject market size constant. The
z-statistic for each of these tests is illustrated in Table 7. Our prior
for all of the comparisons is that the z-statistics for the buyers
should all be positive and statistically significant, whereas they
should be negative and statistically significant for the sellers. For
all of the comparisons made for the buyers the z-statistics match up
with our priors. For the sellers this is not always true. When the
market size is 4 the comparison between a variance of [[sigma].sup.2]
and 4[[sigma].sup.2] indicates that the [[DELTA].sub.i]'s are not
statistically significant from each other, but the sign on the
difference is as expected. This is also true when the market size is 8
and we compare the zero variance case with [[sigma].sup.2]. Finally, the
test statistic for the comparison between [[sigma].sup.2] and
4[[sigma].sup.2] when the market size is 16 indicates that differences
are statistically significant but the differences are of the opposite
expected sign. These results indicate the behavior of the buyers
perfectly accords with our theoretical predictions, whereas for sellers
there are a few exceptions.
The formal comparison of the fourth hypothesis utilizes a series of
two-sample Wilcoxon ranksum tests comparing the magnitude of the
[[DELTA].sub.i]'s across the different market sizes holding the
variance constant. For the nonzero variance cases, a negative and
statistically significant z-statistic supports our fourth research
hypothesis. The test statistics are illustrated in Table 8. For the zero
variance cases, the results for the sellers perfectly agree with the
theory as none of the test statistics are statistically significant. For
the buyers this is also true when comparing the market size of 4 and 8,
but the difference is marginally significant (90th percent level) when
comparing the market size of 4 and 16 and is statistically significant
and negative when comparing the market size of 4 and 16. Therefore,
buyers do tend to lower their [[DELTA].sub.i] as the market size
increases but sellers do not. For the nonzero variance cases, theory
suggests that the magnitude of [[DELTA].sub.i] decreases as market size
increases. When the variance is [[sigma].sup.2] the z-statistics
indicate that buyers do tend to decrease the magnitude of their
[[DELTA].sub.i]'s as the market size increases, except when
comparing the market size of 4 and 8. For sellers, the statistically
significant and positive z-statistic when comparing the market size of 4
and 8 is the opposite of our prediction, but the other comparisons are
consistent with our predictions. When the variance is 4[[sigma].sup.2]
the test statistics perfectly accord with our theoretical predictions
for buyers, but do not for any of the comparisons conducted for the
sellers. Once again we see good agreement with the theory for buyers and
weaker agreement for sellers.
The results in Tables 7 and 8 show some asymmetries in the behavior
of buyers and sellers in selecting [[DELTA].sub.i], even though the
experiment was constructed so that they would have symmetric strategic
incentives. One possible explanation for these differences is that the
buyers in the experiment tend to "overshoot" the predicted
[[DELTA].sub.i] whereas the sellers tend to "undershoot" it.
This can be seen by comparing the predicted [[DELTA].sub.i]'s in
Tables 2 and 3 with the actual [[DELTA].sub.i]'s in Table 6. In a
number of cases the buyers nearly double the predicted [[DELTA].sub.i]
whereas the sellers rarely meet the predicted [[DELTA].sub.i] and in
some cases the magnitude of [[DELTA].sub.i] is very small and not of the
expected sign. However, buyers also possess a higher standard deviation
for [[DELTA].sub.i]. Therefore, it is not entirely clear that this
asymmetry implies an advantage for the buyers in the market. This will
be more formally investigated in our sixth research hypothesis.
In summary, the evidence from testing hypotheses 3 and 4 confirms
and reinforces the evidence from testing hypothesis 1. So we conclude,
with the caveat that the evidence is stronger for buyers than sellers,
that subjects generally understood and acted upon the strategic
incentives in the experiment, but not to the degree predicted by theory.
B. Hypotheses for Overall Market Outcomes
Our final two research hypotheses focus on the market outcomes of
efficiency and the distribution of profits. The latter is critically
determined by the observed market price. Figure 4 illustrates the market
price by variance and market size. The average market price by treatment
over all time periods and the last ten periods is shown in Table 9.
Recall that the predicted market price is 5 for all cases. The price in
the zero-variance cases is relatively stable across time but it is
generally increasing in the nonzero variance cases as the market size
increases.
Figure 5 shows the observed market efficiency. Market efficiency
was calculated for each treatment by estimating the ratio of total
profits earned to the profits that would be earned if everyone behaved
according to the perfectly competitive outcome. (13) The temporal
patterns of market efficiency do not necessarily have to follow those
found for the market price. Efficiency is determined by the trades that
occur within the market. So, for example, it is possible that some
subjects end up with too many permits and others with too few, all the
while preserving the predicted equilibrium price. Indeed, we do find a
difference in these temporal patterns; market efficiency increases as
the experiment progresses for all variance cases, not just the nonzero
cases (see Table 10).
Our fifth research hypothesis is that as the market size increases
so too does the level of market efficiency. This can be seen by changes
in the predicted [[DELTA].sub.i] as the number of subjects increases
(see Tables 2 and 3). This results in an increased number of permit
trades which enhance market efficiency. The graphical results and
descriptive statistics for all the periods, as well as the last ten
periods, do not consistently support our research hypothesis. To more
rigorously investigate our hypothesis we conducted a series of
two-sample Wilcoxon rank-sum tests under the hypothesis that as the
market size increases so too will the market efficiency, holding the
variance constant. Given the changes in market efficiency as the
experiment progresses we have elected to use just the last ten periods
of each treatment. The results from these tests are contained in Table
11.
For the zero-variance case we predict that the market efficiency
should not vary with market size. The z-statistics support this
hypothesis for two cases but indicate that the market efficiency is
actually greater when the market size is 8 versus 16.14 When the
variance increases to [[sigma].sup.2] the results indicate that market
efficiency does increase when the market size increases from 4 to 14 8,
but the efficiency when the market size is 16 is not greater than 4 and
the differences between the market size of 8 and 16 indicate that the
efficiency is greater when the market size is 8. Finally, when the
variance is 4[[sigma].sup.2] the test results indicate that the market
efficiency does not increase between the market size of 4 and 8 and 4
and 16, but it does between 8 and 16. On the whole, these results
provide little support for our fifth research hypothesis. A possible
explanation for these results is that the expected change in market
efficiency as the market size increases is relatively small. Within the
experiment the expected change in efficiency as the market size
increases within the experiment is at most 2.5 percentage points (going
from a market size of 4 to perfect competition with a variance of
4[[sigma].sup.2]) with many of the expected differences being much
lower. As a result of this the expected differences are too small to be
observed in the data given the high variation in the intra-treatment
efficiency scores. This variation generates the inconsistent pattern
observed in our test statistics. In any event, the test statistics do
not support our fifth research hypothesis.
Our final hypothesis focuses on how the profits are split between
the buyers and sellers. By construction, buyers and sellers are
symmetrically placed on either side of the market. Therefore, the
profits should be identical whether subjects behave strategically or
competitively. To investigate this we determined the total profits
earned for each period of the experiment and then determined the average
percentage of those earnings that were earned by the buyers. The results
are illustrated in Figure 6. There is considerably more variation in the
percentage of profits earned by buyers when the market size is 4 versus
the other market sizes. As the market size increases the percentage
becomes more stable and tightens around the even split. This is a result
of the "thinness" of the market. In a market size of 4 there
are only two buyers and two sellers and a poor decision by either of the
two (buyers or sellers) can generate a substantial swing in the profits.
When the market size increases, a poor decision, although damaging at
the individual level, is dampened at the market level.
Given the observed average market prices reported earlier, we would
expect that the buyers earned a larger split of the profits except when
the market size is 16 and the variance is 4[[sigma].sup.2]. However,
given the substantial variation that exists in the observed equilibrium
price and the reported fractions observed in Figure 6, a more rigorous
investigation is required. Due to the sizable differences in market
efficiency as each experiment progressed, discussed earlier (see Figure
5), we have elected to focus our test of hypothesis six on the data
observed in the last ten periods of each experimental treatment. To do
so we conducted a series of two-sample Wilcoxon rank-sum tests within
each treatment to determine whether or not the buyers or sellers earned
a larger portion of the profits using period specific information versus
the percentages reported in Figure 6. The results from these tests are
illustrated in Table 12.
The z-statistics indicate that five of the nine test statistics
support our sixth hypothesis that there is no statistical difference in
the split between the buyers' and sellers' profits. The
remaining test statistics are split where two indicate that buyers
exceed sellers and another two indicate the opposite. The two
environments where the sellers outperformed the buyers were when the
variance was 4[[sigma].sup.2] and the market size was 8 and when the
variance was zero and the market size was 16. The two markets where the
buyers outperformed the sellers occurred when the variance was
4[[sigma].sup.2] and the market size was 4 and when the variance was
[[sigma].sup.2] and the market size was 8. With the exception of the two
instances when the sellers outperformed the buyers, the signs of these
results are in agreement with the observations about market prices
discussed earlier. (15) In total, the results indicate that the split on
profits was predominantly balanced, with a few situations in which
buyers outperformed sellers and vice versa. However, the differences
that did arise were not systematic.
V. CONCLUSION
This research represents the first detailed study of a supply
function equilibria bilateral oligopoly market using experimental
economic methods. By systematically varying the number of firms as well
as the variance in cost, we determined that subjects respond to the
strategic incentives inherent in this market structure. The magnitude of
the strategic behavior exhibited by subjects, however, does not
perfectly accord with theory. In general, buyers tend to overshoot their
strategic incentives whereas sellers tend to undershoot them. Perhaps,
as a consequence, our results tend to be stronger for buyers than
sellers. The results for the overall market outcomes did not agree with
theory. The observed market efficiency did not increase with the market
size and there were cases where buyers outperformed the sellers and vice
versa. However, for the most part the revenues earned by buyers and
sellers in the market were balanced.
From a policy perspective, these results represent a necessary
first step in investigating behavior within a bilateral oligopoly. Given
the increasing interest in applying pollution markets to localized
pollution problems like nitrogen emissions from waste water treatment
plants, it is imperative that we develop a better understanding of the
bilateral oligopoly market structure. Should the supply function
equilibria market be utilized in pollution markets, the long temporal
delays in market feedback necessitate that we study the market's
comparative statistics with respect to market characteristics before
incurring the social, political, and pecuniary costs of their
utilization. By investigating how market size and firm heterogeneity
affect the market equilibrium we have provided policymakers with
information on the primary factors that may impact the market
equilibrium.
More formally, our research has indicated that although participant
behavior does not perfectly accord with our strategic predictions,
participant behavior does accord with their expected responses with
respect to the market's comparative statistics. This provides a
degree of confidence in the potential utilization of this market
structure to manage environmental pollution. Furthermore, given the
prevalence of market mechanisms in environmental compliance, markets
adding this mechanism to the suite of options may expand a
regulator's capacity for market-based environmental management.
Perhaps, a productive next step would be to further investigate the
apparent differences in the behavior of buyers and sellers documented in
our study to determine if these results are robust to alternative market
environments. However, our findings do provide important policy-relevant
findings that may be informative for future pollution markets.
ABBREVIATION
ExCEN: Experimental Economics Center
doi: 10.1111/ecin.12087
APPENDIX
Proof that there cannot be an equilibrium on the truncated portions
of abatement costs and net-trade functions.
From Malueg and Yates (2009), we know there exists a unique
interior Nash Equilibrium ([a.sup.*.sub.i], [a.sup.*.sub.2], ...,
[a.sup.*.sub.n]) such that pollution levels are less than or equal to
business-as-usual and the price is less than or equal to [??]. We now
show that there cannot be another equilibrium on the truncated portions
of the abatement costs and net-trade functions. First consider abatement
cost functions. Suppose there exists another equilibrium
([a.sup.t.sub.1], [a.sup.t.sub.2], ..., [a.sup.t.sub.n]) and at this
equilibrium at least one subject j has emissions of pollution that
exceed business as usual. Then this subject could lower their [a.sub.j],
buy less permits, and reduce pollution at no extra cost. Clearly, they
would be better off, which contradicts the definition of an equilibrium.
Now consider the net-trade function. Suppose there exists another
equilibrium ([a.sup.t.sub.1], [a.sup.t.sub.2], ..., [a.sup.t.sub.n]) and
at this equilibrium at least one subject j is on the truncated portion
of their net-trade function. A small change in [a.sub.j] would have no
effect on the market price and net-trades (because this subject would
still be in the exceptions group). Hence any value for [a.sub.j] such
that at the equilibrium price subject j is on the truncated portion of
the net-trade function would be an equilibrium choice for player j
(given, of course, the choices of the other players). In particular,
consider the choice [a.sup.s.sub.j] such that [v.sub.j] =
[a.sup.s.sub.j] - p = -[w.sub.j]. The subject is just on the boundary of
the truncated portion of the net-trade function. So we still have
([a.sup.t.sub.1], [a.sup.t.sub.2], ..., [a.sup.s.sub.j], ...,
[a.sup.t.sub.n]) as an equilibrium. But, this equilibrium is now
interior, contradicting the uniqueness of (a.sup.*.sub.1],
[a.sup.2.sub.2], ..., [a.sup.*.sub.n]).
SUPPORTING INFORMATION
Additional Supporting Information may be found in the online
version of this article:
Appendix S1. Experiment Instructions.
TABLE A1
Test of Equality in Distribution within Treatments: Buyers Versus
Sellers
Exp 1 0.510
Group = 4, Var = 0 0.997
0.982
Exp 2 0.033
Group = 8, Var = 0 0.223
0.022
Exp 3 0.438
Group = 16, Var = 0 0.619
0.713
Exp 4 0.000
Group = 4, Var = [[sigma].sup.2] 0.000
0.000
Exp 5 0.000
Group = 8, Var = [[sigma].sup.2] 0.000
0.000
Exp 6 0.000
Group = 16, Var = [[sigma].sup.2] 0.000
0.000
Exp 7 0.000
Group = 4, Var = 4[[sigma].sup.2] 0.000
0.000
Exp 8 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000
0.000
Exp 9 0.000
Group = 16, Var = 4[[sigma].sup.2] 0.000
0.000
Notes: Cells represent p values for tests of equality of distribution
between buyers and sellers within experiment. The first line of each
cell are p values resulting from a Mann-Whitney test; the second line
of the cell are p values from a t-test; and the final line of each
cell is the p value from a Kolmogorov-Smirnov test.
TABLE A2
Test of Equality in Distribution Across Experiment: Buyers
Exp 1 Exp 2 Exp 3
Exp 1 ... 0.781 0.875
Group = 4, Var = 0 ... 0.500 0.202
... 0.065 0.045
Exp 2 0.781 ... 0.037
Group = 8, Var = 0 0.520 ... 0.985
0.065 ... 0.016
Exp 3 0.875 0.037 ...
Group =16, Var = 0 0.500 0.985 ...
0.045 0.016 ...
Exp 4 0.000 0.000 0.000
Group = 4, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 5 0.000 0.000 0.000
Group = 8, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 6 0.000 0.000 0.000
Group = 16, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 0.000 0.000 0.000
Group = 4, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 8 0.000 0.000 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 9 0.000 0.000 0.000
Group =16, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 Exp 5 Exp 6
Exp 1 0.000 0.000 0.000
Group = 4, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 2 0.000 0.000 0.000
Group = 8, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 3 0.000 0.000 0.000
Group =16, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 ... 0.133 0.000
Group = 4, Var = [[sigma].sup.2] ... 0.003 0.000
... 0.000 0.000
Exp 5 0.133 ... 0.001
Group = 8, Var = [[sigma].sup.2] 0.003 ... 0.630
0.000 ... 0.000
Exp 6 0.000 0.001 ...
Group = 16, Var = [[sigma].sup.2] 0.000 0.630 ...
0.000 0.000 ...
Exp 7 0.001 0.049 0.136
Group = 4, Var = 4[[sigma].sup.2] 0.001 0.336 0.475
0.000 0.000 0.000
Exp 8 0.000 0.000 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 0.011 0.006
0.000 0.000 0.000
Exp 9 0.000 0.000 0.000
Group =16, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 Exp 8 Exp 9
Exp 1 0.000 0.000 0.000
Group = 4, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 2 0.000 0.000 0.000
Group = 8, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 3 0.000 0.000 0.000
Group =16, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 0.001 0.000 0.000
Group = 4, Var = [[sigma].sup.2] 0.001 0.000 0.000
0.000 0.000 0.000
Exp 5 0.049 0.000 0.000
Group = 8, Var = [[sigma].sup.2] 0.336 0.011 0.000
0.000 0.000 0.000
Exp 6 0.136 0.000 0.000
Group = 16, Var = [[sigma].sup.2] 0.475 0.006 0.000
0.000 0.000 0.000
Exp 7 ... 0.003 0.000
Group = 4, Var = 4[[sigma].sup.2] ... 0.131 0.000
... 0.003 0.000
Exp 8 0.003 ... 0.022
Group = 8, Var = 4[[sigma].sup.2] 0.131 ... 0.001
0.003 ... 0.021
Exp 9 0.000 0.022 ...
Group =16, Var = 4[[sigma].sup.2] 0.000 0.001 ...
0.000 0.021 ...
Notes: Cells are p-values for tests of equality of distribution
between experiments conditional on being a buyer. The first line of
each cell are p-values resulting from a Mann-Whitney test; the second
line of the cell are p-values from a t-test; the final line of each
cell is the p-value from a Kolmogorov-Smimov test.
TABLE A3
Test of Equality in Distribution Across Experiment: Sellers
Exp 1 Exp 2 Exp 3
Exp 1 ... 0.220 0.684
Group = 4, Var = 0 ... 0.672 0.735
... 0.056 0.250
Exp 2 0.220 ... 0.331
Group = 8, Var = 0 0.672 ... 0.344
0.056 ... 0.622
Exp 3 0.684 0.331 ...
Group = 16, Var = 0 0.735 0.344 ...
0.250 0.622 ...
Exp 4 0.000 0.000 0.000
Group = 4, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 5 0.000 0.000 0.000
Group = 8, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 6 0.000 0.000 0.000
Group = 16, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 0.000 0.000 0.000
Group = 4, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 8 0.000 0.000 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 9 0.000 0.000 0.000
Group = 16, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 Exp 5 Exp 6
Exp 1 0.000 0.000 0.000
Group = 4, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 2 0.000 0.000 0.000
Group = 8, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 3 0.000 0.000 0.000
Group = 16, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 ... 0.001 0.469
Group = 4, Var = [[sigma].sup.2] ... 0.329 0.012
... 0.000 0.000
Exp 5 0.001 ... 0.000
Group = 8, Var = [[sigma].sup.2] 0.329 ... 0.000
0.000 ... 0.000
Exp 6 0.469 0.000 ...
Group = 16, Var = [[sigma].sup.2] 0.012 0.000 ...
0.000 0.000 ...
Exp 7 0.000 0.000 0.000
Group = 4, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 8 0.000 0.000 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 9 0.000 0.000 0.000
Group = 16, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 Exp 8 Exp 9
Exp 1 0.000 0.000 0.000
Group = 4, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 2 0.000 0.000 0.000
Group = 8, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 3 0.000 0.000 0.000
Group = 16, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 0.000 0.000 0.000
Group = 4, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 5 0.000 0.000 0.000
Group = 8, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 6 0.000 0.000 0.000
Group = 16, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 ... 0.910 0.005
Group = 4, Var = 4[[sigma].sup.2] ... 0.730 0.007
... 0.172 0.025
Exp 8 0.910 ... 0.010
Group = 8, Var = 4[[sigma].sup.2] 0.730 ... 0.012
0.172 ... 0.000
Exp 9 0.005 0.010 ...
Group = 16, Var = 4[[sigma].sup.2] 0.007 0.012 ...
0.025 0.000 ...
Notes: Cells are p values for tests of equality of distribution
between experiments conditional on being a seller. The first line of
each cell are p values resulting from a Mann-Whitney test; the second
line of the cell are p values from a t-test; the final line of each
cell is the p value from a Kolmogorov-Smirnov test.
TABLE A4
Test of Equality in Distribution Across Experiment Between Buyers and
Sellers
Exp 1 Exp 2 Exp 3
Exp 1 ... 0.610 0.211
Group = 4, Var = 0 ... 0.534 0.733
... 0.060 0.057
Exp 2 0.053 ... 0.161
Group = 8, Var = 0 0.667 ... 0.068
0.011 ... 0.200
Exp 3 0.357 0.775 ...
Group = 16, Var = 0 0.506 0.210 ...
0.129 0.660 ...
Exp4 0.000 0.000 0.000
Group = 4, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 5 0.000 0.000 0.000
Group = 8, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 6 0.000 0.000 0.000
Group = 16, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 0.000 0.000 0.000
Group = 4, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 8 0.000 0.000 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 9 0.000 0.000 0.000
Group = 16, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 4 Exp 5 Exp 6
Exp 1 0.000 0.000 0.000
Group = 4, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 2 0.000 0.000 0.000
Group = 8, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 3 0.000 0.000 0.000
Group = 16, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp4 ... 0.000 0.000
Group = 4, Var = [[sigma].sup.2] ... 0.000 0.000
... 0.000 0.000
Exp 5 0.000 ... 0.000
Group = 8, Var = [[sigma].sup.2] 0.000 ... 0.000
0.000 ... 0.000
Exp 6 0.000 0.000 ...
Group = 16, Var = [[sigma].sup.2] 0.000 0.000 ...
0.000 0.000 ...
Exp 7 0.000 0.000 0.000
Group = 4, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 8 0.000 0.000 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 9 0.000 0.000 0.000
Group = 16, Var = 4[[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 Exp 8 Exp 9
Exp 1 0.000 0.000 0.000
Group = 4, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 2 0.000 0.000 0.000
Group = 8, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp 3 0.000 0.000 0.000
Group = 16, Var = 0 0.000 0.000 0.000
0.000 0.000 0.000
Exp4 0.000 0.000 0.000
Group = 4, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 5 0.000 0.000 0.000
Group = 8, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 6 0.000 0.000 0.000
Group = 16, Var = [[sigma].sup.2] 0.000 0.000 0.000
0.000 0.000 0.000
Exp 7 ... 0.000 0.000
Group = 4, Var = 4[[sigma].sup.2] ... 0.000 0.000
... 0.000 0.000
Exp 8 0.000 ... 0.000
Group = 8, Var = 4[[sigma].sup.2] 0.000 ... 0.000
0.000 ... 0.000
Exp 9 0.000 0.000 ...
Group = 16, Var = 4[[sigma].sup.2] 0.000 0.000 ...
0.000 0.000 ...
Notes: Cells are p values for tests of equality of distribution
between experiments. Columns are conditional on being a buyer, rows
are conditional on being a seller. The first line of each cell are p
values resulting from a Mann-Whitney test; the second line of the
cell are p values from a t-test; the final line of each cell is the p
value from a Kolmogorov-Smimov test.
TABLE A5
Wilcoxon Rank-Sum Test and t-Tests of Theoretical Equivalence for
Buyers
[a.sub.i]-- [a.sub.i]--
[a.sub.i] = 0 [a.sup.c.sub.i] = 0
Exp 1 0.0192 0.0192
Group = 4, Var = 0 (0.1219) (0.1219)
Expv2 0.0042 0.0042
Group = 8, Var = 0 (0.0016) (0.0016)
Exp 3 0.7222 0.7222
Group = 16, Var = 0 (0.0004) (0.0004)
Exp 4 0.0008 0.0000
Group = 4, Var = [[sigma].sup.2] (0.0000) (0.0000)
Exp 5 0.0074 0.0000
Group = 8, Var = [[sigma].sup.2] (0.0000) (0.0000)
Exp 6 0.0080 0.0000
Group = 16, Var = [[sigma].sup.2] (0.0000) (0.0000)
Exp 7 0.0000 0.0000
Group = 4, Var = 4[[sigma].sup.2] (0.0000) (0.0000)
Exp 8 0.0000 0.0000
Group = 8, Var = 4[[sigma].sup.2] (0.0000) (0.0000)
Exp 9 0.0000 0.0000
Group = 16, Var = 4[[sigma].sup.2] (0.0000) (0.0000)
Wilcoxon rank-sum test on top and t-test in parentheses below.
TABLE A6
Wilcoxon Rank-Sum Test and t-Tests of Theoretical Equivalence for
Sellers
[a.sub.i]-- [a.sub.i]--
[a.sub.i] = 0 [a.sup.c.sub.i] = 0
Exp 1 0.1473 0.1473
Group = 4, Var = 0 (0.1518) (0.1518)
Exp 2 0.8086 0.8086
Group = 8, Var = 0 (0.2505) (0.2505)
Exp 3 0.1542 0.1542
Group = 16, Var = 0 (0.0008) (0.0008)
Exp 4 0.0001 0.0046
Group = 4, Var = [[sigma].sup.2] (0.0009) (0.4885)
Exp 5 0.0000 0.1840
Group = 8, Var = [[sigma].sup.2] (0.0000) (0.4465)
Exp 6 0.0001 0.0000
Group = 16, Var = [[sigma].sup.2] (0.0000) (0.0000)
Exp 7 0.0000 0.0000
Group = 4, Var = 4[[sigma].sup.2] (0.0000) (0.0001)
Exp 8 0.1495 0.0000
Group = 8, Var = 4[[sigma].sup.2] (0.5281) (0.0000)
Exp 9 0.0009 0.3784
Group = 16, Var = 4[[sigma].sup.2] (0.0481) (0.3079)
Wilcoxon rank-sum test on top and t-test in parentheses below.
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University of Texas at Austin, Working Paper, 2003.
Wirl, F. "Oligopoly Meets Oligopsony: The Case of
Permits." Journal of Environmental Economics and Management, 58,
2009, 329-37.
Yates, A. J., M. Doyle, J. R. Rigby, and K. E. Schnier.
"Market Power, Private Information, and the Optimal Scale of
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(1.) An exception is Brandts, Pezanis-Christou, and Schram (2007)
who include a supply function equilibria case in one of their
experiments. They do not, however, provide a robust investigation of the
properties of the supply function equilibria.
(2.) Alternatively, one might consider a field experiment. But at a
yearly time scale, it would take decades to generate the amount of data
that can be generated in a few laboratory hours. Moreover, over such a
long time period it would be extremely likely that other confounding
factors would arise that would complicate the analysis of the market.
(3.) Within the experiment the exception rule was utilized
approximately 41% of the time, with the largest percentage occurring
when the market size is the largest (16 subjects).
(4.) Using profit maximization, rather than cost minimization,
facilitates the experimental analysis. The fixed payment [[pi].sub.i],
is different from a show up fee, as it is firm specific and depends on
the values of the other experimental parameters.
(5.) The competitive outcome can be found by assuming that the
firms ignore the effect of their choices on the market price.
(6.) [A.sub.i] solves [C.sub.i]([[theta].sub.i]) = 0.
(7.) [[pi].sub.i] solves [[pi].sub.i] - [C..sub.i] ([w.sub.i]) =
[[pi].sub.i] - ([A.sub.i] - [[theta].sub.i][w.sub.i] +
(1/2)([w.sup.2.sub.i])) = [kappa].
(8.) A copy of the instructions can be obtained from K.S.
(9.) The f-tests generate one other counterfactual to the Wilcoxon
rank-sum tests for buyers and that is when a variance of zero is paired
with market size of 4.
(10.) All of these are robust to a parametric test except when
variance of zero is paired with market size of 16.
(11.) The three cases where the sellers followed the strategic
predictions were all for the zero variance case. When the variance is
zero we cannot differentiate between strategic behavior and perfect
competition because the theoretical predictions are identical.
Therefore, we must focus on the nonzero variance cases to determine
whether or not the buyers/sellers behave closer to the strategic
predictions or those under perfect competition.
(12.) If we use the 90th percentile as our threshold of statistical
significance there are two situations where sellers' behavior more
closely follows the strategic predictions. The second treatment is when
the variance is 4[[sigma].sup.2] and the market size is 8.
(13.) The predicted market efficiency varies depending on the
variance and market size. When the variance is zero the predicted market
efficiency is one. When the variance is the predicted market efficiency
is 99.16, 99.79, and 99.95% when the market size is 4, 8, and 16,
respectively. When the variance is 4 of 2 the predicted market
efficiency is 97.60, 99.40, and 99.85% when the market size is 4, 8. and
16, respectively.
(14.) The z-statistics comparing the market sizes of 4 and 8 as
well as 8 and 16 are statistically significant at the 90th percent
level.
(15.) The exception seem to be driven by the high degree of
heterogeneity in the profits earned by buyers in the market. In a number
of cases, buyers actually sold permits and this led to a large loss for
those individual subjects.
KURT SCHNIER, MARTIN DOYLE, JAMES R. RIGBY and ANDREW J. YATES *
* This research was supported by NSF grant numbers 0909275,
0908679, and 0909056.
Schnier: Professor of Economics, School of Humanities, Social
Sciences and Arts, University of California, Merced, CA 95343. Phone
+l-(209) 205-6461, Fax (209) 228-4007, E-mail kschnier@ucmerced.edu
Doyle: Professor of River Science and Policy, Nicholas School of
the Environment, Duke University, Durham, NC 27708. Phone
+1-919-613-8026, Fax (919) 613-8070, E-mail martin.doyle@duke.edu
Rigby: Research Hydrologist, USDA-ARS National Sedimentation
Laboratory, Oxford, MS 38655. Phone +1-(662) 232-2951, Fax (662)
281-5706, E-mail jr.rigby @ars.usda.gov
Yates: Assistant Professor, Economics/Curriculum for the
Environment and Ecology, University of North Carolina, Chapel Hill, NC
27599. Phone 919-966-2385, Fax 919-966-4986, E-mail
ajyates@email.unc.edu
TABLE 1 Values for Buyers and Sellers
var([theta]--w) [[theta].sub.b] [[theta].sub.s]
0 10 10
[[sigma].sup.2] 12.5 7.5
4[[sigma].sup.2] 15 5
TABLE 2
Experimental Design and Predictions: Buyers' Values for [a.sub.1] =
[a.sup.c.sub.i] + [[DELTA].sub.i]
n: Number of Subjects in Market
4
[a.sub.i] [a.sup.c.sub.i] [[DELTA].sub.i]
var([theta]--w): Variance of costs
0 5 5 0
[[sigma].sup.2] 5.93 6.25 -0.32
4[[sigma].sup.2] 6.87 7.5 -0.63
n: Number of Subjects in Market
8
[a.sub.i] [a.sup.c.sub.i] [[DELTA].sub.i]
var([theta]--w): Variance of costs
0 5 5 0
[[sigma].sup.2] 6.09 6.25 -0.16
4[[sigma].sup.2] 7.19 7.5 -0.31
n: Number of Subjects in Market
16
[a.sub.i] [a.sup.c.sub.i] [[DELTA].sub.i]
var([theta]--w): Variance of costs
0 5 5 0
[[sigma].sup.2] 6.17 6.25 -0.08
4[[sigma].sup.2] 7.34 7.5 -0.16
TABLE 3
Experimental Design and Predictions: Sellers' Values for
[a.sub.i] = [a.sup.c.sub.i] + [[DELTA].sub.i]
n: Number of Subjects in Market
4
[a.sub.i] [a.sup.c.sub.i] [[DELTA].sub.i]
var([theta]--w): Variance of costs
0 5 5 0
[[sigma].sup.2] 4.07 3.75 0.32
4[[sigma].sup.2] 3.13 2.5 0.63
n: Number of Subjects in Market
8
[a.sub.i] [a.sup.c.sub.i] [[DELTA].sub.i]
var([theta]--w): Variance of costs
0 5 5 0
[[sigma].sup.2] 3.91 3.75 0.16
4[[sigma].sup.2] 2.81 2.5 0.31
n: Number of Subjects in Market
16
[a.sub.i] [a.sup.c.sub.i] [[DELTA].sub.i]
var([theta]--w): Variance of costs
0 5 5 0
[[sigma].sup.2] 3.83 3.75 0.08
4[[sigma].sup.2] 2.66 2.5 0.16
TABLE 4
Descriptive Statistics: Subject Choice Variable a, (Broken Down by
All 20 Periods, Last 15 Periods, and Last 10 Periods)
Market Size
4: b, s
M SD M SD
Periods 1 -20
var([theta]--w): 0 4.89 1.21 4.89 1.30
Variance of costs [[sigma].sup.2] 5.38 1.82 3.81 1.42
4[[sigma].sup.2] 5.86 1.87 2.81 1.35
Periods 6-20
var([theta]--w): 0 4.97 1.11 4.91 1.19
Variance of costs [[sigma].sup.2] 5.57 1.70 3.75 1.19
4[[sigma].sup.2] 6.16 1.71 2.69 1.16
Periods 11-20
var([theta]--w): 0 4.89 0.93 4.88 1.15
Variance of costs [[sigma].sup.2] 5.86 1.21 3.80 1.10
4[[sigma].sup.2] 6.23 1.61 2.68 1.15
Market Size
8: b, s
M SD M SD
Periods 1 -20
var([theta]--w): 0 4.83 0.95 4.93 1.09
Variance of costs [[sigma].sup.2] 5.75 1.17 3.72 0.79
4[[sigma].sup.2] 6.11 2.29 2.77 1.08
Periods 6-20
var([theta]--w): 0 4.91 0.73 4.95 0.94
Variance of costs [[sigma].sup.2] 5.88 1.11 3.70 0.61
4[[sigma].sup.2] 6.26 2.16 2.76 0.99
Periods 11-20
var([theta]--w): 0 4.94 0.69 4.98 0.80
Variance of costs [[sigma].sup.2] 5.91 0.91 3.67 0.52
4[[sigma].sup.2] 6.37 2.08 2.80 0.96
Market Size
16: 6,s
M SD M SD
Periods 1 -20
var([theta]--w): 0 4.83 1.20 4.86 1.05
Variance of costs [[sigma].sup.2] 5.79 1.36 4.01 1.13
4[[sigma].sup.2] 6.54 1.76 2.55 1.35
Periods 6-20
var([theta]--w): 0 4.90 1.02 4.96 0.94
Variance of costs [[sigma].sup.2] 6.03 0.98 4.06 0.92
4[[sigma].sup.2] 6.93 1.45 2.36 1.28
Periods 11-20
var([theta]--w): 0 4.89 0.89 4.94 0.76
Variance of costs [[sigma].sup.2] 6.13 0.82 4.03 0.91
4[[sigma].sup.2] 7.06 1.20 2.22 1.27
TABLE 5
Comparison of [[alpha].sub.i], 's Across Treatments Holding Market
Size Constant: z-Statistic
Variance Comparison Buyers Sellers
Market size: 4
0 > [[sigma].sup.2] -7.153 12.677
0 > 4[[sigma].sup.2] -7.563 16.215
[[sigma].sup.2] > 4[[sigma].sup.2] -3.461 11.630
Market size: 8
0 > [[sigma].sup.2] -13.454 16.789
0 > 4[[sigma].sup.2] -12.566 18.509
[[sigma].sup.2] > 4[[sigma].sup.2] -6.562 12.821
Market size: 16
0 > [[sigma].sup.2] -17.933 17.495
0 > 4[[sigma].sup.2] -18.601 25.819
[[sigma].sup.2] > 4[[sigma].sup.2] -11.152 20.608
TABLE 6
Descriptive Statistics: [[DELTA].sub.i] 's (Broken Down by Buyers [b]
and Sellers [5])
Market Size
4: b, s
M SD M SD
Periods 1-20 var([theta]--w): Variance of costs
0 -0.1120 1.28 -0.1116 1.30
[[sigma].sup.2] -0.8692 1.82 0.0551 1.42
4[[sigma].sup.2] -1.6357 1.87 0.3053 1.36
Market Size
8: b, s
M SD M SD
Periods 1-20 var([theta]--w): Variance of costs
0 -0.1688 0.95 -0.0702 1.09
[[sigma].sup.2] -0.5042 1.17 -0.0340 0.80
4[[sigma].sup.2] -1.3858 2.29 0.2718 1.08
Market Size
16: b, s
M SD M SD
Periods 1-20 var([theta]--w): Variance of costs
0 -0.1703 1.20 -0.1389 1.05
[[sigma].sup.2] -0.4614 1.36 0.2672 1.13
4[[sigma].sup.2] -0.9561 1.76 0.0544 1.35
TABLE 7
Comparison of [[DELTA].sub.i] 's Across Treatments Holding Market
Size Constant: z-Statistic
Variance Comparison Buyers Sellers
Market size: 4
0 > [[sigma].sup.2] 4.766 -3.768
0 > 4[[sigma].sup.2] 10.447 -4.464
[[sigma].sup.2] > 4[[sigma].sup.2] 5.804 -1.027
Market size: 8
0 > [[sigma].sup.2] 6.038 -0.882
0 > 4[[sigma].sup.2] 8.233 -2.894
[[sigma].sup.2] > 4[[sigma].sup.2] 4.658 -3.469
Market size: 16
0 > [[sigma].sup.2] 4.684 -8.276
0 > 4[[sigma].sup.2] 9.037 -2.057
[[sigma].sup.2] > 4[[sigma].sup.2] 5.541 4.113
TABLE 8
Comparison of [[DELTA].sub.i] 's Across Treatments Holding Variance
Constant: z-Statistic
Market Size Comparison Buyers Sellers
Variance = 0
4 > 8 -0.278 -1.227
4 > 16 -1.708 -0.408
8 > 16 -2.089 0.971
Variance = [[sigma].sub.2]
4 > 8 -1.501 3.390
4 > 16 -3.634 -0.724
8 > 16 -3.202 -5.086
Variance = 4[[sigma].sub.2]
4 > 8 -2.972 0.113
4 > 16 -5.717 2.794
8 > 16 -2.297 2.580
TABLE 9
Average Market Price by Variance and Market Size
Variance 0 [[sigma].sup.2] 4[[sigma].sup.2]
Market Size 4 8 16 4 8 16 4 8 16
All time periods 4.89 4.88 4.85 4.60 4.73 4.91 4.43 4.53 4.75
Last ten periods 4.89 4.96 4.92 4.84 4.79 5.08 4.55 4.67 4.93
TABLE 10
Average Market Efficiency by Variance and Market Size
Variance 0 [[sigma].sup.2] 4[[sigma].sup.2]
Market Size 4 8 16 4 8 16 4 8 16
All time periods 88.5 90.1 88.4 81.2 91.9 87.6 82.1 79.1 86.1
Last ten periods 92.3 95.3 93.4 91.0 95.7 93.8 88.3 83.1 93.7
TABLE 11
Comparison of Efficiency Across Treatments Holding Variance Constant:
z-Statistic.
Variance
Market Size Comparison 0 [[sigma].sup.2] 4[[sigma].sup.2]
4 > 8 -1.871 -2.550 1.539
4 > 16 1.898 0.582 -1.342
8 > 16 3.878 4.234 -3.002
TABLE 12
Comparison of Buyer and Sellers Profits within Each Treatment:
z-Statistic ([H.sub.0] Is That the Profit for Sellers Is Equal
to the Profit for Buyers)
Variance
Market Size 0 [[sigma].sup.2] 4[[sigma].sup.2]
4 0.767 -0.589 -4.494
8 -0.447 -3.570 3.002
16 1.963 1.771 1.058
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