A. Muren J. of Economic Behavior Org. 41 2000 147–157 151
or to expect the other sellers to undercut. The latter strategy implies setting a price at least as high as all other sellers, the former setting a lower price than at least some other seller.
Consider the high-price strategy first. With efficient rationing, the lower priced firms will sell to consumers with high willingness to pay for the good. The high-price firm, i, will face
a residual demand curve of the form P Q = 38 − Q
− i
− q
i
, where Q
− i
is the combined capacity of the two other sellers and q
i
is firm i’s sales. The high-price firm will set its price to maximize revenue, i.e. at p
i
= 38 − Q
− i
2 if its capacity is large enough for this. If not, that is if q
i
is smaller than 38 − Q
− i
2, the firm will set its price to clear the market. If a firm’s capacity is well above the revenue-maximizing sales level for the high-price
strategy, a low-price strategy might yield higher profits. Not all firms can successfully apply low-price strategies since they would then have no other firm to undercut, so under-
cutting would involve mixed strategies. For a detailed determination of equilibria including mixed-strategy equilibria we refer to Kreps and Scheinkman 1983. We will here limit our-
selves to describing the pure strategy equilibrium, i.e. with all firms applying the high-price strategy.
Consider situations where all firms are capacity constrained in the price-setting stage in the sense defined above, i.e. q
i
≤ [38 − q
1
+ q
2
+ q
3
− q
i
]2 for all firms i. All firms will then set the market-clearing price, and since all firms sell to capacity there will be no
incentive to undercut by lowering the price. To get a sense of the values of q
1
, q
2
and q
3
that we are considering, note that e.g. values at or below q
1
= 10, q
2
= q
3
= 9 are consistent
with a high-price strategy on the part of all firms. Under the condition that all firms are capacity-constrained in the price-setting stage, we
now know that firms will set the market clearing price. The capacity-setting stage is then identical to the Cournot quantity-setting game and the prediction for that stage is that firms’
capacity choices will be equal to the Cournot equilibrium quantities. In this experiment the Cournot market output with three sellers is equal to 21 units and the Cournot equilibrium
prediction is thus that each seller would produce 7 units. Replacing the Cournot output volumes with capacities, the prediction is that sellers will install 7 units of capacity each,
and sell these at price SEK 17, which is also the market clearing price.
5. Experimental outcome
5.1. Capacities The data on market capacities for the inexperienced markets are shown in Table 1.
5
In the inexperienced sessions most of the markets produce aggregate capacities above the Kreps and Scheinkman prediction of 21 units. In some of the inexperienced markets there
is a difference between decisions taken at the beginning of a session — in the first four periods — and decisions in the latter part of the session, in that market capacities are closer
to the prediction in the later periods. This difference appears in the A1, A2, B2, C2, A3 and C3-markets and is reflected in lower mean and median market capacities from period five
onwards.
5
Individual seller data on capacity choices are available from the author.
152 A. Muren J. of Economic Behavior Org. 41 2000 147–157
Table 1 Aggregate market capacities in inexperienced rounds. The predicted market capacity is 21 units. Markets A1 to
C2 are from 1998 and markets A3 to D3 from 1999 Period:
1 2
3 4
5 6
7 8
9 10
A1-market 23
28 29
43 24
26 28
25 26
26 B1-market
22 43
48 29
45 66
55 40
36 58
C1-market 38
27 37
33 25
55 49
25 26
30 A2-market
34 50
40 26
27 26
27 25
19 21
B2-market 39
35 32
34 17
17 18
18 19
19 C2-market
40 28
30 50
22 20
23 24
25 30
A3-market 23
30 42
33 36
20 24
45 22
26 B3-market
39 60
59 58
51 41
45 27
29 49
C3-market 37
45 50
49 32
26 26
37 18
18 D3-market
48 74
75 84
65 47
43 27
21 31
Mean 34.3
42.0 44.2
43.9 34.4
34.4 33.8
29.3 24.1
30.8 Median
37.5 39
41 38.5
29.5 26
27.5 26
23.5 28
Table 2 Aggregate market capacities in experienced rounds A8 to C8 from 1998 and A9 to C9 from 1999
Period: 1
2 3
4 5
6 7
8 9
10 A8-market
30 29
37 21
21 27
21 22
32 22
B8-market 16
21 20
23 25
25 29
20 22
27 C8-market
26 30
17 23
21 23
28 23
21 23
A9-market 23
26 25
26 27
27 29
31 34
32 B9-market
20 24
24 22
22 22
22 22
22 37
C9-market 19
18 19
21 20
20 21
19 20
19 Mean
22.3 24.7
23.7 22.7
22.7 24
25 22.8
25.2 26.7
Median 21.5
22.5 22
22.5 21.5
24 25
22 22
25
The median capacities for the inexperienced treatment are above the Kreps and Scheinkman prediction of 21 market units and 7 individual units, particularly in the first four periods. In
fact, several markets have capacity levels well above the competitive equilibrium level of 28 units, which would be the Bertrand competition outcome. Table 2 shows market capacities
for the experienced markets. The markets with experienced subjects are closer to the Kreps and Scheinkman prediction from the beginning and throughout.
To compare the outcome of our capacity precommitment experiment with Fouraker and Siegel’s results we computed the support for the same three equilibrium solutions in our
data for both the inexperienced and the experienced sessions.
6
These are displayed in Table 3.
For the inexperienced sessions these values have the same tendency as Fouraker and Siegel’s results in that there are more rivalistic markets than CournotKreps and Scheinkman
together with cooperative. For the experienced sessions the tendency is reversed and there
6
The monopoly equilibrium market capacity is 14 units and the competitive equilibrium capacity is 28 units. Market capacities in the interval [0, 17] were counted as cooperative, capacities in the interval [18, 24] as Cournot
and capacities ≥ 25 as rivalistic.
A. Muren J. of Economic Behavior Org. 41 2000 147–157 153
Table 3 Number of market outcomes supporting each of three equilibrium solutions
Period: 4
5 6
7 8
9 10
Fouraker and Siegel period 21 Inexperienced
Cooperative 1
1 CournotK-S
2 2
3 2
5 3
5 Competitive
10 7
7 7
8 5
7 6
Experienced Cooperative
CournotK-S 5
4 3
3 5
4 3
Competitive 1
2 3
3 1
2 3
are now more CournotKreps and Scheinkman outcomes than competitive ones there are no cooperative outcomes in periods 4–10. This indicates that experience makes a difference.
Table 3 does not take information about numerical capacity values within each of the three ranges into account but looking at means and medians there is a considerable difference
between inexperienced and experienced markets. A period by period comparison shows that the market capacities are significantly greater in the inexperienced markets for periods 1, 2,
3 and 4.
7
For the last six periods means and medians are still greater in the inexperienced markets. In the experienced treatment 7 out of 10 means and 8 out of 10 medians are
within the CournotKreps and Scheinkman range. To summarize the evidence on capacity levels we conclude that in the inexperienced
sessions capacity levels are higher than predicted by the Kreps and Scheinkman model and in line with the results reached in Cournot triopolies by Fouraker and Siegel. With experience,
within the experiment and more noticeably between experiments, capacity choices approach the Kreps and Scheinkman prediction quite well.
5.2. Prices So far we have compared the capacity precommitment experiment with its theoretical
prediction and with a Cournot experiment, which means that we have concentrated on the quantitycapacity dimension. To check if the capacity precommitment model yields the
Kreps and Scheinkman outcome we also need to see to what extent the installed capacity is sold. This is interesting also because the efficient rationing mechanism that was used is
complicated to understand.
In Section 4 we determined conditions under which firms will set the market-clearing price in the price-setting stage. This will be the case if capacities are not too high, specifically
if q
i
≤ [38−q
1
+ q
2
+ q
3
− q
i
]2 for all firms i. To test whether the Kreps and Scheinkman price-setting prediction is confirmed by the experimental data or not, we inquire if it is the
case that sellers set the market-clearing price when the capacity constraints given above are satisfied.
Table 4 shows, for the inexperienced treatment, on the one hand the markets and periods where the capacity constraint is satisfied and those where it is not, marked with a C or a –,
7
A Kolmogorov–Smirnov two-sample, one-tailed test rejects the null hypothesis that the two samples are from the same distribution at the 0.05 significance level for periods 1–4.
154 A. Muren J. of Economic Behavior Org. 41 2000 147–157
Table 4 Capacity constraints and market clearing in inexperienced treatment
Period: 1
2 3
4 5
6 7
8 9
10 A1
C– ––
–– ––
C– CP
C– CP
CP CP
B1 C–
–– ––
–– ––
–– ––
–– ––
–– C1
–– C–
–– ––
C– ––
–– CP
CP –P
A2 ––
–– ––
C– C–
CP C–
CP CP
CP B2
–– ––
–– ––
CP CP
CP CP
CP CP
C2 ––
–– ––
–– CP
CP CP
CP CP
–– A3
C– ––
–– ––
–– C–
C– ––
C– C–
B3 ––
–– ––
–– ––
–– ––
–– ––
–– C3
–– ––
–– ––
–– C–
C– ––
CP CP
D3 ––
–– ––
–– ––
–– ––
–– C–
–– Table 5
Capacity constraints and market clearing in experienced treatment Period:
1 2
3 4
5 6
7 8
9 10
A8 ––
–– ––
C– C–
–– C–
C– ––
CP B8
CP CP
C– CP
–– CP
–– CP
CP ––
C8 –P
–– CP
CP CP
CP ––
C– CP
CP A9
C– C–
C– C–
C– C–
–– ––
–– ––
B9 C–
C– C–
C– C–
C– CP
CP CP
–– C9
C– CP
C– C–
CP CP
C– CP
CP CP
and on the other hand whether all units were sold or not for the same markets, marked with a P or a –, respectively. Table 5 shows the same information for the experienced markets.
The tables suggest several conclusions about pricing behaviour. In the first place, although markets clear when the capacity constraint is satisfied in quite a few of the markets, they
do not always. Looking at each market individually, we note that after a market has cleared once, it continues to do so in subsequent periods. One possible explanation for this could
be that the efficient rationing mechanism was difficult for the subjects to understand and that they learned something about how to price when their market cleared.
In general it is interesting to notice that in the experienced markets capacities came close to the Kreps and Scheinkman prediction in spite of the fact that sellers did not seem to
be entirely certain about the way the rationing mechanism worked. This suggests that the model’s predictions may be robust for slight divergences from efficient rationing, which
strengthens the case for the Kreps and Scheinkman somewhat, particularly since efficient rationing is an extreme form which seems unlikely to appear with any frequency in real
markets see Davidson and Deneckere for a discussion of rationing mechanisms.
8
8
Davidson and Deneckere, who show that although the Kreps and Scheinkman result does not emerge under proportional rationing when the cost of capacity is small, note that when capacity costs increase, equilibrium
capacities under proportional rationing approach Kreps and ScheinkmanCournot values.
A. Muren J. of Economic Behavior Org. 41 2000 147–157 155
Table 6 Frequency of losses in the inexperienced treatment
Mean losses per seller sellers receiving SEK 0
A1-market 430
1 B1-market
2130 3
C1-market 1230
3 A2-market
630 1
B2-market 530
1 C2-market
630 1
A3-market 1430
1 B3-market
1930 3
C3-market 1230
2 D3-market
1830 3
Table 7 Frequency of losses in the experienced treatment
Mean losses per seller sellers receiving SEK 0
A8-market 630
B8-market 130
C8-market 230
A9-market 830
1 B9-market
530 C9-market
130
5.3. Incentives The average incentive payments made to subjects after sessions were SEK 39 for the
inexperienced treatment and SEK 136 for the experienced treatment.
9
In several of the sessions of the inexperienced treatment and on occasion in the experienced treatment,
subjects made losses due to not being able to sell all their installed capacity in one or more periods. Since losses were not subtracted from the participation payment but only
from earnings in the subsequent periods, large losses may reduce incentives to act as a profit-maximizer. To investigate whether this appears to have happened we computed the
frequency of losses over all periods in each session and noted the number of sellers in each market who received less than zero in accumulated earnings. This information is displayed
in Tables 6 and 7.
Clearly the frequency of losses is considerably lower in the experienced treatment and it seems reasonable to conclude that the fact that losses were not subtracted was not im-
portant for experienced markets. In the inexperienced markets we note that the frequency of losses is lower in the A1, A2, B2, C2 and A3 markets, which are among the mar-
kets where capacities seem to approach the prediction after period 4. Certainly the lower capacities associated with this contribute to lower losses, but it might also be the case
9
An allowance for income tax dues 30 percent tax rate was added to each subject’s payment and paid directly to the Tax Authority, so that payment to subjects was net of taxes.
156 A. Muren J. of Economic Behavior Org. 41 2000 147–157
that the preserved incentives resulting from lower losses have contributed to the fact that capacities approach the prediction. Looking at pricing behaviour in the inexperienced mar-
kets Table 4 we see that it is only in three out of the 10 markets that neither capac- ities nor prices eventually approach Kreps and Scheinkman results, which suggests that
incentives to act as profit-maximizing sellers have been preserved for a majority of the subjects.
6. Conclusions
