1. science
2. physics
3. thermodynamics

Output Decisions and Price-Matching: Theory and
Experiment∗
Mongoljin Batsaikhan†
Norovsambuu Tumennasan
‡
March 19, 2015
Abstract
We study the effects of price-matching in a duopoly setting in which each firm selects
both its price and output, simultaneously. We show that the availability of a pricematching option leads to the Cournot outcome in this setting. This result is a stark
contrast to the one obtained in the standard price competition that the most likely
market price in the presence of a price-matching option is the monopolistic price. We
further confirm the validity of our theoretical result in an experimental study. Thus,
both theoretical and experimental studies suggest that the effect of price-matching
depends on whether the output is a choice variable for the firms.
Keywords: Price matching, output decision, the Cournot outcome
JEL Classifications: L00, L01, L02, D4
1
Introduction
Retail businesses often use price-matching guarantees: if a product the seller carries is sold
for a cheaper price by some other seller, then the seller will offer the product for the same low
price. The retail giant Walmart, for example, offers price-matching guarantees frequently,
especially during the Christmas shopping seasons.
Since Salop (1986) price-matching has largely been considered an anti-competitive practice: a price-matching firm warns its competitors that it will not be underpriced; it thus
eliminates the rivals’ incentive to underprice their product. For this reason, the market
∗
This version supersedes the previously circulated version “Price-Matching leads to the Cournot Outcome.”
†
School of Foreign Service in Qatar, Georgetown University. E-mail: [email protected]
‡
Department of Economics and Business, Aarhus University, Denmark. Email: [email protected]
1
price ranges from the monopolistic to the Bertrand price when the firms have options to
price-match (Salop, 1986). Despite the multiplicity of equilibria, the literature considers the
monopolistic price as the most likely outcome for two theoretical reasons: (i) the monopolistic price is the most profitable equilibrium price for the firms, and (ii) the monopolistic price
alone survives the process of iterative elimination of weakly dominated strategies (Doyle,
1988). This result is further confirmed in a laboratory setting (Dugar, 2007). However, all
the above-mentioned studies overlook the fact that the output decision is strategic.1 Particularly, these studies assume implicitly that the firms’ output adjusts automatically to the
market demand. This assumption is perhaps realistic if the production takes place after
firms receive pre-orders. In all other cases, because the production is costly, the firms must
choose their output carefully. Therefore, in this paper we assume that the output decision
is strategic for the firms. Specifically, we study a duopoly model in which each firm selects
its output, price, and price-matching option, simultaneously.
We first show that the Cournot outcome is the unique equilibrium outcome in our model.
The key reason for our result is that at any equilibrium the total supply equals the market
demand corresponding to the higher price in the market. As a result, it is possible for a firm to
earn a positive profit even when it is underpriced – the reason why the Bertrand equilibrium
is not an equilibrium in our setting. In addition, the availability of price-matching options
allows the firms to protect their market share from possible underpricing. Subsequently, each
firm responds to the other’s strategy in the same way as it would in the Cournot competition.
Thus, the Cournot outcome emerges as the unique equilibrium in our model.
The influential paper of Kreps and Scheinkman (1983) shows that the Cournot outcome
is the only subgame perfect eqiulibrium (SPE) in the setting in which the firms first select
their output and then set their prices.2 The timing of their setting therefore is different than
ours: the firms make their output and pricing decisions simultaneously in our model. In
fact, for the purpose of welfare comparison, the model in which the firms select their output
and price at the same time in the Kreps-Scheinkman setting is a benchmark, but this model
has not been explored to the best of our knowledge.
We thus study the benchmark model and show that there is no pure equilibrium as long as
1
To the best of our knowledge almost the entire literature on price-matching does not model the firms’
output decisions explicitly. One exception is Tumennasan (2013), which we will discuss in greater detail
later.
2
Davis (1999), Muren (2000), Anderhub et al. (2003) and Goodwin and Mestelman (2010) test the KrepsScheinkman predictions in a laboratory setting. The first three studies report that participants in general
set their output higher than the Cournot output. Goodwin and Mestelman (2010) confirm this result for
the case of inexperienced participants. However, experienced participants make similar choices in both the
Kreps-Scheinkman and the standard Cournot setting. In the static setting with price-matching options, the
subjects’ decisions in our main treatment are consistent with the theoretical prediction of our paper, the
Cournot outcome.
2
the cost of production is positive.3 We then identify a mixed equilibrium in which the firms
turn out to make an 0 profit in expectation. Therefore, the firms’ profit is unambiguously
higher in the presence of a price-matching option. However, this result does not necessarily
imply the negative impact of price matching on consumers because both firms’ price exceeds
the Cournot price with a positive probability under this mixed equilibrium. In fact, for the
linear demand case, we show that the effect of price matching on consumers is ambiguous.
We next test our theoretical result in a laboratory setting. In our baseline treatment,
we consider Salop’s standard model; subjects only selected their price and price-matching
option. Our results for this treatment are highly consistent with the results of Dugar (2007)4 :
the monopolistic outcome is the most likely equilibrium outcome in this setting. In our main
treatment, subjects selected their price, price-matching option and output, simultaneously.
The experimental results at the market level, i.e., the average market price and the subjects’
output, are highly consistent with the theoretical predictions of the paper. At the individual
level, however, the results are not as consistent, but there is a clear trend indicating a
convergence to the theoretical predictions.
There is an extensive literature on price matching.5 Salop (1986) and Doyle (1988) argue
that price-matching leads to the monopolistic price. Corts (1995) and Hviid and Shaffer
(1999), on the other hand, counter that the Bertrand outcome is the more likely outcome.
Specifically, Corts (1995) extends the price-matching policy to the price beating policy.6
Hviid and Shaffer (1999) introduce hassle costs, i.e., consumers have to bear certain costs
to convince a price-matching firm that there is a lower price in the market. In their model,
a firm can steal the other’s market share by underpricing because customers save the hassle
costs by buying from the price cutter, thus, restoring the Bertrand price.7 We show that the
Cournot price, a third price, emerges as the unique equilibrium outcome if the output is a
choice variable for the firms.
The closest study to ours is Tumennasan (2013). The main difference between Tumennasan (2013) and this paper is the timing of price and output decisions. He considers Kreps
and Scheinkman (1983)’s two-stage setting in which the firms select their output in the first
3
Without price matching, the firms have an incentive to underprice the other slightly as long as the
market price exceeds the marginal cost as in the Bertrand model. If both firms price their product at the
marginal cost, each firm can earn a positive profit by increasing its price slightly and setting its production
so that it sells its whole production at the new price. Thus, there is no pure equilibrium if the firms select
their output and price at the same time. For more information see Proposition 2.8.
4
We discuss Dugar (2007) in more detail below.
5
See Arbatskaya et al. (2004) for more information.
6
If firms can beat the effective price of its competitors, then Salop (1986)’s results are restored (Kaplan,
2000).
7
Dugar and Sorensen (2006) take the model of Hviid and Shaffer (1999) to an experimental lab and find
a significantly different price than the Bertrand price.
3
stage and make their pricing decisions in the second stage. In contrast, the firms make price
and output decisions simultaneously in our setting. This difference in timing leads to a very
different set of results: Tumennasan (2013) shows that a multiplicity of equilibria exists
while we show that a unique equilibrium exists.8
Dugar (2007) studies the effects of price matching in a laboratory setting for the standard
model in which firms do not select their output. He considers triopoly markets where firms
select their price-matching options first, and then having observed its competitors’ pricematching options, each firm sets its price. He finds that the monopolistic outcome emerges
in a laboratory setting. The results from our baseline treatment (i.e., the one without output
choice) are consistent with Dugar (2007)’s results despite the differences between the two
models.
The paper is organized as follows: Section 2 lays out the model. Section 3 proves our main
result that price-matching leads to the Cournot outcome. Section 4 reports our experimental
results, and Section 5 concludes. We collect longer proofs in Appendix A and tables in
Appendix B.
2
Theoretical Study
2.1
Model
Two identical firms produce a homogeneous product; the market demand for this product is
P (x) or D(p) = P −1 (p) where x and p stand for quantity and price, respectively.
Each firm i selects (i) its output ki ∈ R+ , (ii) announced price pi ∈ R+ , and (iii) pricematching option oi ∈ {0, 1} where 1 means “match” and 0 means “do not match.” The buyers
are aware of the firms’ choices. As a result, the price-matching options allow the firms to
alter the actual price of its product. In particular, if a firm does price match, then it sells
its product for the lowest price on the market. On the other hand, if the firm does not price
match, then it sells its product for its announced price. The effective price of firm i is the price
the firm sells its product for, i.e., pei (p1 , o1 , p2 , o2 ) := (1 − oi )pi + oi min{p1 , p2 }. The effective
prices are instrumental for our analysis as they, along with the outputs, determine the sales
quantity of the firms. To simplify the notation, we often write pei instead of pei (p1 , o1 , p2 , o2 ).
We now formulate the sales quantity of the firms. Let pei and pej be the effective prices
8
Specifically, Tumennasan (2013) shows that price-matching weakly lowers the market price if the cost of
production is “high”. If the cost of production is “low,” the effects of price-matching is ambiguous because
the set of SPE prices includes the Cournot price in its interior.
4
for firms i and j (6= i), respectively. Then firm i sells

e


 min {k
oo
n i , D(pi )}
n
xi (pe1 , pe2 , k1 , k2 ) =
min ki , max D(p) − kj , D(p)
2


 min {k , max{0, D(pe ) − k }}
i
i
j
if pei < pej
if pei = pej = p
(1)
if pei > pej .
The above formulation implicity assumes that the firms split the market if they announce
the same price as long as each firm’s output is sufficiently large. In addition, the efficient
rationing rule is used, i.e., the consumers with a higher valuation buy from the firm with the
lower effective price.
Let the strategy profile be ((k1 , p1 , o1 ), (k2 , p2 , o2 )). Then the profit of firm i = 1, 2 is
πi (k1 , p1 , o1 , k2 , p2 , o2 ) = pei xi (pe1 , k1 , pe2 , k2 ) − cki
where c > 0.
The cost function is assumed to be linear in the formulation of the profit. We emphasize
here that this assumption is not important in our analysis: the main result of the paper is
valid as long as the cost function is strictly increasing and convex. We also maintain the
following assumption throughout the paper.
Assumption 2.1. P (x) is strictly positive on some bounded interval (0, x¯) on which it is
twice continuously differentiable, strictly decreasing, and concave. For x ≥ x¯, P (x) = 0.
We now turn our attention to the standard Cournot competition. Thanks to Assumption
2.1, one can easily show that the profit function P (x+y)x−cx is concave on [0, x¯ −y]. Let r(y)
be the Cournot best response to the rival’s production y, i.e., r(y) = arg max0≤x≤¯x−y P (x +
y)x − cx.
The following lemma, which is instrumental in our analysis, is from Kreps and Scheinkman
(1983).
Lemma 2.2. (a) The Cournot best response rb is nonincreasing in y. In addition, rb is
continuously differentiable and strictly decreasing over the range where it is strictly positive
(b) r0 (y) ≥ 1, with strict inequality for y with r(y) > 0, so that y + r(y) is nondecreasing in
y.
Proof. See Kreps and Scheinkman (1983).
Due to Assumption 2.1, there is a unique Cournot duopoly equilibrium with each firm
supplying xc . Let pc := P (2xc ) and π c := (pc − c)xc . Furthermore, we find it useful to have
5
notations for the monopolistic price (pm ) and half of the monopolistic quantity (xm ). Let
pm := arg maxp (p − c)D(p), xm := D(pm )/2 = arg maxx (P (2x) − c)x, and π m := (pm − c)xm .
2.2
Main Result
In this subsection, we prove our main result: the only equilibrium outcome in our model is
the one that results in the Cournot outcome. We will first study some properties of equilibria,
which in turn will lead to our main conclusion.
The first key property is that both firms set the same effective price at any equilibrium.
If this is not the case, then the firm with the higher price must not be price-matching. Then
it is optimal for the firm with the lower price to supply enough output to meet the market
demand at its price because its price strictly exceeds c (see the formal proof). As a result,
the firm with the higher price nets a non-positive profit because there is no demand beyond
what the lower price firm supplies. However, the firm with the higher price can net half of
the lower price firm by setting its price to that of the lower price firm and its output to half
of the lower price firm’s output. Consequently, both firms set the same effective price at
any equilibrium. In addition, the range of the equilibrium effective price must be in the set
(c, P (0)). We collect these results below.
Lemma 2.3. If (pe1 , pe2 ) is an equilibrium effective price pair, then pe1 = pe2 ∈ (c, P (0)).
Proof. See Appendix A.
We use the terminology market price at an equilibrium to refer to the common effective
price the firms offer at this equilibrium. The following lemma asserts that, at an equilibrium,
each firm supplies half of the market demand corresponding to the market price at this
equilibrium. The key observation for this result is that at any equilibrium the total market
supply equals the market demand corresponding to the market price at this equilibrium.
Indeed if there is an excess supply (or excess demand), then one of the firms has an incentive
to decrease (increase) its output without altering the market price. Now if the two firms do
not supply the same output, then the one with the lower output has an incentive to increase
its output while leaving the market price unaltered.
Lemma 2.4. If the market price at an equilibrium is p ∈ (c, P (0)), then each firm’s output
is D(p)/2 at this equilibrium.
Proof. See Appendix A.
6
Lemmas 2.3 and 2.4 imply that each firm nets the same profit at a given equilibrium.
In other words, for a given market price at an equilibrium, each firm nets half of the total
market profit corresponding to this market price.
In the lemma below, we first show that both firms price-match at each equilibrium.
Indeed if a firm does not price-match, then the other firm, by slightly undercutting the
equilibrium market price, can steal the market share of the non-price-matcher. The lemma
also establishes a new and a tighter upper bound on the equilibrium market prices at the
monopolistic price. If the market price strictly exceeds pm , then each firm has an incentive
to set its price to the monopolistic price and its output to half of the monopolistic quantity.
The deviating firm nets half of the monopolistic profit, which is superior to half of the total
market profit corresponding to any price. Lastly, the lemma shows that, in the case that the
market price at an equilibrium is strictly below the monopolistic price, both firms’ announced
prices equal the market price. If the firms set different prices in this case, then the firm with
the lower price has a profitable deviation. Specifically, this firm increases its price slightly
and decreases its output to half of the market demand corresponding to its new price. At
the deviation, the firm nets half of the total market profit corresponding to the altered price.
But this profit increases as the price approaches the monopolistic price. Thus, both firms
must have the same announced prices at a given equilibrium if the market price is strictly
below the monopolistic price.
Lemma 2.5. Consider any equilibrium strategy profile ((k1 , p1 , o1 ), (k2 , p2 , o2 )).
(i) Both firms must price match, i.e., o1 = o2 = 1
(ii) The equilibrium market price p := pe1 = pe2 does not exceed pm , i.e., p ≤ pm .
(iii) If the equilibrium market price p := pe1 = pe2 is strictly less than pm , i.e., p < pm , then
p1 = p2 = p.
Proof. See Appendix A.
We are now ready to prove our main result that the only equilibrium in our setting is the
one which results in the Cournot outcome. To see this, first recall that each firm’s output
at an equilibrium is half of the market demand that corresponds to the market price at this
equilibrium. Let this quantity be x∗ . Then each firm’s profit is (P (2x∗ ) − c)x∗ . If a firm sets
its price to P (x∗ + r(x∗ )) and its output to r(x∗ ) and does not price-match, it sells r(x∗ )
for P (x∗ + r(x∗ )). The reason for this result is that the other firm cannot sell more than its
output, x∗ . Clearly, (P (2x∗ ) − c)x∗ < (P (x∗ + r(x∗ )) − c)r(x∗ ) by the definition of r(·) as
long as x∗ 6= xc . Thus, the market price must be the Cournot price.
7
Theorem 2.6. There exists only one equilibrium. Particularly, at the equilibrium it must
be that (ki , pi , oi ) = (xc , pc , 1) for both i = 1, 2.
Proof. See Appendix A.
Remark 2.7. For our result the assumption that c > 0 is crucial. Indeed if c = 0, then all
the prices from the Bertrand to the monopolistic price can be supported as an equilibrium
price in our setting. To see this, suppose each firm sets its output to its D(0). Because the
cost of production is 0, such a high output is not costly for the firms. Now each firm has
enough output to satisfy the market demand at any price. Thus, we are back in the Salop
world.
2.3
Benchmark Model
In this subsection, we consider our model without price matching option. In other words, the
firms choose their price and output simultaneously. Given that the option of price matching
is not available for the firms, the announced price of a firm is the effective price of the firm.
Thus, by substituting the announced prices for effective prices in expression (1) we determine
the the sales quantity of the firms.
We first point out that in the benchmark model, there is no pure equilibrium. In this
setting, Lemmas 2.3 and 2.4 are still valid. But if the firms’ prices are the same and each
firm’s output is half of the market demand corresponding to their prices, each firm has an
incentive to underprice the other slightly while setting its output to the market demand
corresponding to its new price. Thus, there is no pure equilibrium if the option to price
match is not available to the firms. We state this result in the following proposition.
Proposition 2.8. If a price matching option is not available to the firms, then there exists
no pure equilibrium.
We now focus our attention to identifying a mixed equilibrium in the benchmark model.
Proposition 2.9. The symmetric, mixed strategy profile satisfying the two conditions below
is an equilibrium.9
• An output and price combination, (ki , pi ), is in the support of firm i’s mixed strategy
if (ki , pi ) = (D(pi ), pi ) and (ki , pi ) ∈ [c, P (0)] × [0, D(c)]
9
We are not sure whether this equilibrium is unique or not. Secondly, if we relax the assumption that
the cost function linear, then the mixed strategy considered here is not necessarily an equilibrium.
8
• Each firm’s price is distributed according to the cumulative distribution function


 0
F (p) =
1−


1
if p < c
if p ∈ [c, P (0))
c
p
if p > P (0).
Furthermore, at this equilibrium each firm’s profit is 0.
Proof. See Appendix A.
Proposition 2.9 and Theorem 2.6 lead to the conclusion that price matching affects the
firms positively. When the firms do not have options to price match, at equilibrium each
firm faces a risk of announcing a higher price than the other. In such cases, a firm sells none
of its output and incurs a loss. On the other hand, such a risk is absent at the equilibrium
when there is an option to price match. This elimination of the risk explains partly why price
matching affects the firms positively. Therefore, the firms’ profit gain due to the availability
of a price-matching option does not automatically lead to the loss in consumer welfare. In
fact, the effect of price matching on consumers is ambiguous which we illustrate by the
following example.
Example 2.10. Suppose that the demand function is P (x) = a − bx where a > c. In
, and pc = a+2c
. The consumer surplus when the firms have an
this case, P (0) = a, xc = a−c
3b
3
option to price match is the consumer surplus at the Cournot price which is found as follows:
a
Z
a
Z
D(t)dt =
CS(p) =
pc
pc
a−t
2(a − c)2
dt =
.
b
9b
Now we find the expected consumer surplus at the mixed equilibrium strategy profile
identified in Proposition 2.9. Recall that at the equilibrium strategy, the firm with the lowest
price produces the output which equals the market demand corresponding to the price of
that firm. Thus, the firm with the lowest price sells all its output at its own announced price
while the other sells nothing. Subsequently, if the minimal price on the market is p, then
the consumer surplus, CS(p), is
Z
CS(p) =
a
Z
D(t)dt =
p
p
a
a−t
(a − p)2
dt =
.
b
2b
Now let us calculate the cumulative distribution function, G(·), for the minimal market
price at the mixed equilibrium. Because there are only two firms, the minimum market price
is the second order statistic which is known to be distributed according to the following
9
distribution:
2
2
c
c
c
− 1−
= 1−
, for all p ∈ [c, P (0)] ≡ [c, a].
G(p) = 2F (p) − (F (p)) = 2 1 −
p
p
p
2
Observe here that the distribution for the minimal market price is discontinuous only
at P (0) = a. However, this does not cause any issue in our calculation of the expected
consumer surplus because the consumer surplus at a is 0. The expected consumer surplus
in the absence of a price-matching option is:
Z
E[CS] =
a
Z
a
CS(p)dG(p) =
c
p
(a − p)2 c2
c2
dp =
bp3
b
3 a2 − 4ac
a
+
+ ln
2
2
2c
c
.
Observe here that the consumer surplus in the presence of price matching exceeds the
2
2
one in the absence of price matching when a = 2c (E[CS] = cb (ln(2) − 0.5) < 2c9b = CS(pc )).
However, this inequality is reversed once a = 4c.
3
3.1
Experimental Study
Experimental Setup
We conducted a laboratory experiment to test the results predicted by theory. The experiment involved two treatments: T1, where subjects chose a price and decided whether
to price-match, and T2, where subjects chose a price, an amount of output to commit to
produce, and decided whether to price-match or not. We recruited 36 subjects (18 for each
treatment) from the School of Economics and Business at the Mongolian National University
and conducted the experiment in Ulaanbaatar, Mongolia on May 23rd, 2014. Subjects were
randomly matched and played for 10 rounds with the same opponent. Decisions were logged
via computer and the identities of opponents were kept anonymous.
After the subjects granted their consent, subjects were provided with oral and written
instructions in Mongolian. The instructions carefully explained the structure of the experiment and specified the demand function. Subjects were given an opportunity to ask
questions about the experiment. The experiment was conducted via computer, with two
practice rounds to ensure that the subjects were familiar with the computer program. In
each round of T1, subjects chose their price and decided whether to price match or not; in T2
they also chose their output. In addition, each subject was provided an on-screen calculator
which one could use to calculate the profits by entering her and her opponent’s hypothetical decisions. There is a long tradition of providing a profit table or a profit calculator in
10
experiments involving the Cournot duopolies (Fouraker and Siegel, 1963; Holt, 1985; Huck
et al., 1999).10
In both treatments, the marginal cost of production is $40 for each firm, and the market
demand function is
D(p) = 160 − p.
In each round, the market cleared after both subjects in the same group made decisions.
Each subject’s profit was calculated and the decisions and profits of both subjects were
revealed to both subjects. These results were accumulated into a result table for each round.
Subjects were paid 3000 Mongolian Tugrik (MNT) as a show-up fee and 1 MNT for each
unit of profit they earned throughout the actual 10 rounds of the experiment.
3.2
Results
Treatment 1. For this treatment, Salop (1986) shows that the equilibrium price can range
between the Bertrand price, 40, and the monopolistic price, 100. The firms must price
match in all equilibria except the one in which both firms set their prices to the Bertrand
price. There are strong theoretical arguments that the monopolistic price is the most likely
equilibrium outcome: (i) the monopolistic price is the only equilibrium price that survives
the process of iterative elimination of weakly dominated strategies (Doyle, 1988), and (ii) out
of the all equilibria prices the monopolistic one results in the highest profit to the firms.11
Our experimental results strongly support that the monopolistic price is the most likely
outcome. First of all, the subjects use the price matching option overwhelmingly: there were
only 4 cases in all 10 rounds (that is, out of 180 decisions) where a subject chose not to price
match. In addition, each non-price-matching subject in these cases selected a lower price
than its opponent. Secondly, the subjects set their prices to the monopolistic price in 73.3%
of all decisions (132 times out of all 180 cases). We report these results in Table 1.
The results are even stronger if we concentrate on the last five periods: the subjects set
their prices to the monopolistic price in 83.3% of all decisions. There were six instances in
which a subject set a price greater than 100, but in all these instances the opponent always
set her price to 100. In fact, all but one game in the last 5 rounds resulted in the prices of
100 or 99.
10
Requate and Waichman (2011) find that the experimental results in Cournot duopolies do not depend
on whether a profit table or a profit calculator has been provided to the participants.
11
It is straightforward to see that at each equilibrium the two firms earn equal profit. In addition, because
the monopolistic price brings the highest total profit to the firms, this price is Pareto optimal from the firms’
perspective.
11
Dugar (2007) conducts an experimental study that investigates the effects of price matching in the standard model in which the firms do not make output decisions. Even though
our Treatment 1 differs from Dugar (2007)’s in some aspects, our results are consistent with
Dugar (2007)’s12 : the monopolistic price is the most likely equilibrium price in the standard
model of price matching where output decisions are not considered.
Treatment 2. In Section 2.2 we showed that the only equilibrium for this treatment is the
Cournot outcome. Specifically, theory predicts that the subjects price-match and set their
price and output to the Cournot price and output, respectively.
The experimental results of this treatment did not converge to the theoretical predictions
as easily as the results of T1 did. However, the trend shows a clear convergence to the
theoretical predictions.
We first analyze the results at the market level. In our experimental setup, the Cournot
price is $80, and the Cournot production of each firm is 40. Although the Cournot outcome
did not prevail in any market in the first period, it did prevail in 5 out of 9 markets (55.56%)
in the last two periods (see Table 2). In these markets, the minimal price selected by the
participants is the Cournot price, and the two opponents set their output to the Cournot
output.
We next analyze the results at the individual level by three variables – the price matching
option, price and output. The subjects price matched in 177 out of 180 total decisions. The
prices selected by the subjects are more dispersed, but the mode is at 80 – the Cournot price.
This price is selected 55 times which is 20 more than the second highest selected price, 100.
These results are shown in Table 3. In Figure 1 we show the frequency of the chosen prices,
period by period. The number of subjects who set their price to the Cournot price reaches
the highest (9 subjects or 50%) in the last two rounds. It is also worth noting that the
results from this treatment are significantly different from the ones in Treatment 1: the
average price of subjects’ price choice is significantly lower than the monopoly price of 100 –
the predicted price in the baseline treatment. The Mann-Whitney test for the difference in
prices between the two treatments is significant at 1% level, and the result stays the same
in linear regression models with period trend and subject fixed effects.
In terms of the output choice of the subjects, they set the output to the Cournot output
104 times (57%) and to the monopolistic quantity 21 times (12%). These results are shown
in Table 4.
12
In Dugar (2007)’s study, subjects are divided into groups of three, and each subject chooses her pricematching option first and then sets her price after observing the price-matching decisions of her group
members.
12
Now we consider all the price, output and price-matching decisions jointly. Panel A of
Table 5 shows the frequencies and percentages of subjects who chose the exact Cournot
price and quantity with price matching over 10 periods. Panel B exhibits the frequencies
and percentages of subjects who chose a price higher than Cournot but the Cournot quantity
with price matching. This table suggests that the subjects are converging to the Cournot
outcome.
The Cournot competition has received a significant amount of attention from experimental economists. Holt (1985) finds that with fixed pairs of participants the results of the
Cournot experiments lie somewhere between the Cournot and the collusive outcomes.13 Our
results are consistent with Holt’s results. Given that the experimental setups for the standard Cournot competition is much simpler than ours, it perhaps is unreasonable to expect
that our experimental results converge perfectly to the theoretical prediction.
4
Conclusion
In this paper we study the effects of price-matching in a duopoly setting in which each
firm selects both its price and output, simultaneously. Our theoretical result shows that
the unique equilibrium outcome in this setting is the Cournot outcome. In addition, our
experimental results support our theoretical result.
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Appendix A: Proofs
Proof of Lemma 2.3. Let ((k1 , p1 , o1 ), (k2 , p2 , o2 )) be an equilibrium strategy pair.
First we show that ki > 0 for i = 1, 2. On the contrary, suppose ki = 0 for some
i ∈ {1, 2}. Clearly, i nets 0 profit. In this case, j is essentially a monopolist. It thus must
be that kj = 2xm , pj = pm and πj (k1 , p1 , o1 , k2 , p2 , o2 ) = 2π m . If i sets its output to xm and
price to pm , then i nets π m , which is positive. Thus, ki > 0 for i = 1, 2.
If pei < c for some i, then the firm with the lowest announced price must net a strictly
negative profit because both firms produce a strictly positive output. Thus, pei ≥ c for
i = 1, 2. We now dispose of the pei = pi = c case. Suppose that pei = pi = c. In this case, firm
i nets a non-positive profit. In addition, i must not net a negative profit because i nets 0
profit by producing 0 output. Thus, i nets 0 profit. Observe now that firm i has a profitable
deviation unless pj = c, where j 6= i. Specifically, if pj > c, then by setting its price to c + ,
where > 0 satisfies that c + < pj , and its output to D(c + )/2, firm i nets D(c + )/2,
a positive profit. Thus, pj = pi = c which implies that both firms make 0 profit (recall that
the firms can always net 0 profit by setting their output to 0). Furthermore, it must be that
ki + kj ≤ D(c). Otherwise, one of the firms cannot sell its full capacity, which implies that
the firm earns a negative profit, a contradiction. Without loss of generality, assume that the
firm with the higher output is firm i. Then by setting its price to c + δ, where δ > 0 satisfies
D(c + δ) − kj > 0, and its output to D(c + δ) − kj , i nets δ (D(c + δ) − kj ), a positive profit.
This is a contradiction. Thus, each firm i must set its effective price strictly higher than c,
i.e., pei > c.
Now let us show that pe1 = pe2 . On the contrary, suppose that c < pei < pej . This means
that pi < pj and oj = 0. As a result, firm i nets pi min{ki , D(pi )} − cki . First observe that
pi < P (0). Otherwise, firm i’s profit is not positive. But if i sets its price to pm (which is
strictly lower than P (0), by definition) and its output to xm , then it earns a positive profit,
which is a contradiction. Thus, pi < P (0) or equivalently, D(pi ) > 0. Firm i’s profit is
pi min{ki , D(pi )} − cki . Because pi ∈ (c, P (0)) and D(pi ) > 0, observe that firm i’s profit for
a given price pi is maximized at output D(pi ). In other words, it must be that ki = D(pi ).
Subsequently, firm j’s profit is non-positive because D(pj ) − ki < 0, for all pj > pi and
ki = D(pi ). However, if firm j sets its price to pi and its output to D(pi )/2, then it nets
(pi − c)D(pi )/2, a positive profit. This is a contradiction.
15
Finally, let us show that pe1 = pe2 < P (0). On the contrary, suppose that pei = pej ≥ P (0).
Then neither firm’s profit is positive. But if a firm sets its price to pm (which is strictly lower
than P (0) by definition) and its output to xm , then the firm nets a positive profit. This is
a contradiction.
Proof of Lemma 2.4. We first show that k1 + k2 = D(p). Recall that p ∈ (c, P (0)), due
to Lemma 2.3. If k1 + k2 < D(p), then each firm i nets (p − c)ki . But firm i can net
(p−c)(D(p)−kj ) by changing only its output to D(p)−kj . This is a profitable deviation since
D(p) − kj > ki . If k1 + k2 > D(p), then there is some excess output in the market. Let j be
the firm with the higher capacity. Then j earns p (min{kj , max{D(p)/2, D(p) − ki }}) − ckj .
Since kj ≥ ki and k1 + k2 > D(p) (by assumption), kj > max{D(p)/2, D(p) − ki }. This
means that firm j does not sell its entire output at p. If j decreases its output slightly but
keeps its effective price at p, then j’s income does not change but its cost of production
decreases. Hence, firm j has a profitable deviation which would be a contradiction. Thus,
it must be that k1 + k2 = D(p). Finally, let us show that k1 = k2 = D(p)/2. Suppose
ki < D(p)/2. Then it must be that kj > D(p)/2 > D(p) − kj = ki . Consequently, firm i
nets (p − c)ki . If firm i increases its output slightly to ki + ( > 0 is small enough so that
D(p)/2 > ki + ) and sets its effective price to p, then it earns (p − c)(ki + ) > (p − c)ki .
This is a contradiction.
Proof of Lemma 2.5. Due to Lemma 2.3, it must be that pe1 = pe2 = min{p1 , p2 } := p > c. In
addition, it must be that k1 = k2 = D(p)/2 (Lemma 2.4). Thus, both firms net (p−c)D(p)/2.
To the contrary of (i), suppose oi = 0. This and the result that pe1 = pe2 = p > c (Lemma
2.3) imply that pi = p. Now firm j, by setting its price to p − , where > 0, and its output
to D(p − ), nets (p − − c)D(p − ), which is a profitable deviation if is sufficiently small.
Therefore, oi = 1.
To the contrary of (ii), suppose p > pm . If firm i sets its price to pm and its output to xm ,
then it nets (pm − c)xm = (pm − c)D(pm )/2. But by the definition of pm , (p − c)D(p)/2 <
(pm − c)D(pm )/2. Hence, i has a profitable deviation which is a contradiction.
To prove (iii), suppose that p < pm and pi = p < pj . From the first part of this lemma
we already know that oj = 1. Thanks to Lemma 2.4, firm i nets (p − c)D(p)/2. If firm i sets
its price to p¯ = min{pj , pm } and its production to D(¯
p)/2, then it nets (¯
p − c)D(¯
p)/2. As
m
the function (y − c)D(y)/2 is concave and maximized at p , it must be that (¯
p − c)D(¯
p)/2 >
(p − c)D(p)/2. Therefore, i has a profitable deviation which is a contradiction. Thus,
p1 = p2 = p if p < pm .
Proof of Theorem 2.6. First let us show that the strategy profile in which both firms price
match and set their output to xc and their price to pc is an equilibrium. At this equilibrium,
16
each firm nets (pc − c)xc . Suppose firm i deviates unilaterally by setting its effective price
to p ∈ [c, pc ] and its production to k. Then it earns
p min {k, D(p) − xc } − ck
because D(p)−xc > D(p)/2, for all p < pc . For any given p, observe that the above expression
is maximized at k = D(p) − xc . Thus, firm i earns at most (p − c)(D(p) − xc ) by setting its
price to p. Now let us find the maxp≤pc (p − c)(D(p) − xc ). Denote x = D(p) − xc . Then our
problem is equivalent to maxx≥xc (P (x + xc ) − c)x. This expression, by the definition of r(·),
is maximized at x = r(xc ) = xc . Thus, maxp≤pc (p − c)(D(p) − xc ) = (pc − c)xc . Therefore,
there is no profitable deviation for firm i in which its price does not exceed pc . Suppose firm
i deviates unilaterally by setting its effective price to p > pc and its production to k. Then
it earns
p min {k, max{D(p) − xc , 0}} − ck.
Similarly to the previous case one can see that firm i does not have any incentive to increase its price above pc . This completes the proof that the proposed strategy profile is an
equilibrium.
We now show that there is no equilibrium in which the resulting effective price is different
from pc . On the contrary, suppose there is an equilibrium at which the effective price is
p∗ 6= pc . Due to Lemma 2.3, p∗ ∈ (c, P (0)). When this result is combined with Lemma
2.4, we obtain that firm i nets (p∗ − c)D(p∗ )/2 or (P (2x∗ ) − c)x∗ where x∗ = D(p∗ )/2.
Now if firm i sets its price to p = P (x∗ + r(x∗ )) and its output to D(p) − x∗ , then it earns
(p − c) max{0, (D(p) − x∗ )} = (P (x∗ + r(x∗ )) − c)r(x∗ ). The last equality is due to the fact
that x∗ < D(0) and that r(x) > 0 for all x ≤ D(0). Now observe that, by the definition
of r(·), we obtain that (P (x∗ + r(x∗ )) − c)r(x∗ ) > (P (2x∗ ) − c)x∗ . Hence, no p∗ 6= pc is an
equilibrium effective price.
The fact that p∗ = pc is the unique equilibrium effective price along with Lemmas 2.4
and 2.5(iii) yields the uniqueness of the equilibria.
Proof of Proposition 2.9. Fix a firm i ∈ {1, 2}, and suppose i plays the mixed strategy
described in the proposition. We now show that all the strategies in the support of firm
j’s strategy described in the proposition maximizes firm j’s profit. Any price outside of the
support of j’s mixed strategy, i.e., any price pj such that pj < c or pj > P (0), does not bring
a nonnegative profit to j. If pj = P (0), then the maximal profit of firm j is 0 which j can
achieve by setting its output to 0. Consider now any pj = p ∈ [c, P (0)). By combining p
with an output kj , firm j nets
17
p min{kj , D(p)}(1 − F (p)) − ckj .
Observe here that any kj that strictly exceeds D(p) does not maximize the expression
above. Indeed if kj > D(p), j would improve its profit by setting its output to D(p). On the
other hand, if kj ∈ [0, D(p)], then j’s profit is
pkj (1 − F (p)) − ckj = (p(1 − F (p)) − c) kj .
Now by substituting that F (p) = 1 − pc in the expression above, we obtain that the
expression above equals 0 for any kj ∈ [0, D(p)]. Given that for all p ∈ [c, P (0)), max pkj (1 −
F (p)) − ckj = 0 and D(p) = arg maxkj pkj (1 − F (p)) − ckj , we obtain that all the output and
price combinations in the support of the mixed strategy given in the proposition maximizes
firm j’s profit against firm i’s strategy. In addition, because this maximal profit is always 0,
we have proved that the mixed strategy profile in the proposition is an equilibrium.
18
Appendix B: Tables and Figure
Table 1: Frequency of Price and Price Matching Decisions in Treatment 1 (all rounds)
Price
79
80
89
90
95
97
98
99
100
101
110
119
120
140
Total
All Rounds
Price Matching
No
Yes
Total
1
2
3
0
3
3
1
1
2
1
2
3
0
3
3
0
1
1
0
1
1
1
15
16
0
132
132
0
1
1
0
1
1
0
1
1
0
12
12
0
1
1
4
176
180
Rounds 6-10
Price Matching
No
Yes
Total
1
0
1
1
0
7
75
8
75
0
0
1
5
1
5
2
88
90
Notes: The table shows the subjects’ choices for price and price matching
in Treatment 1. The first half of the table shows the frequencies of prices
chosen along with its price matching choices in all ten periods. The second
half of the table shows the frequencies of prices chosen along with its price
matching choices in the last five periods. The prices ranged from 79 to 140.
The most frequent choice was at 100, which is the monopoly price.
19
Table 2: The Percentage of Games that Result in the Cournot Outcome
Period Freq.
1
0
2
3
3
4
4
2
5
1
6
4
7
3
8
3
9
5
10
5
Total
30
Percent
0.00
33.33
44.44
22.22
11.11
44.44
33.33
33.33
55.56
55.56
33.33
Notes: The table shows the frequency and its percentage of
Cournot outcome out of all possible market equilibria over ten
periods. Each pair of subjects is considered as a market, and
there were 9 markets with 18 subjects in each round.
20
Table 3: Frequency of Price and Price Matching Decisions in Treatment 2
Price
70
75
76
78
79
80
82
84
85
89
90
92
95
98
100
101
110
150
Total
All Rounds
Price Matching
No
Yes
Total
0
2
2
0
8
8
0
3
3
1
2
3
0
5
5
2
53
55
0
1
1
0
4
4
0
13
13
0
4
4
0
30
30
0
3
3
0
9
9
0
2
2
0
35
35
0
1
1
0
1
1
0
1
1
3
177
180
Rounds 6-10
Price Matching
No
Yes
Total
0
1
1
0
2
2
0
3
3
0
1
1
0
1
1
0
32
32
0
0
0
0
0
0
2
6
2
13
1
6
2
6
2
13
1
6
0
0
18
1
18
1
0
90
90
Notes: The table shows the subjects’ choices of price and price matching in Treatment
2. The first half of the table shows the frequencies of prices chosen along with its price
matching choices in all ten periods. The second half of the table shows the frequencies
of prices chosen along with its price matching choices in the last five periods. The prices
ranged from 70 to 150. The most frequent choice was at 80, which is the Cournot price.
Table 4: Frequency of Output Decisions in Treatment 2
Output
Frequency
30
21
35
15
38 40
6 104
41
1
42
2
44 45
4 13
48 50
1 12
60
1
Notes: The table shows the frequency of outputs chosen by subjects in all ten periods.
The output choices ranged from 30 to 60. The most frequent one was at 40, which is the
Cournot output.
21
2
3
4
5
6
7
8
9
10
0
10
5
0
Frequency
5
10
1
60
70
80
90 100
60
70
80
90 100
60
70
80
90 100
60
70
80
90 100
60
70
P
Graphs by Period
Figure 1: Frequency of Price Choices of the Subject over 10 Rounds
22
80
90 100
Table 5: The Percentage of Subjects who set Their Output to the Cournot Output
Panel
Period Frequency
1
3
2
3
3
4
4
4
5
4
6
7
7
4
8
5
9
6
10
8
Total
48
A
Panel B
Percent Frequency
Percent
16.67
5
27.78
16.67
9
50.00
22.22
14
77.78
22.22
10
55.56
22.22
8
44.44
38.89
13
72.22
22.22
10
55.56
27.78
10
55.56
33.33
12
66.67
44.44
12
66.67
26.67
103
57.22
Notes: The table shows the subjects’ choices for price and outcome in Treatment 2.
Panel A of Table 5 shows the frequencies and percentages of subjects who chose the
exact Cournot price and quantity with price matching over 10 periods. Panel B exhibits
the frequencies and percentages of subjects who set their price (weakly) higher than the
Cournot price and selected the Cournot quantity along with the price matching option.
23