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TABLES
State of the Market
TABLE 1 SUMMARY STATISTICS OF SAMPLE DATA Number Average Average of Quantity Price Markets per Firm American 14 13,586.61 92.74 (8,055.24) (62.93) United Monopoly 29 19,191.53 138.13 (only one Firm enters) (8,699.98) (39.90) Total 43 17,366.67 123.35 (8,809.43) (52.44) American 40 31,302.64 127.91 (20,481.25) (65.19) United Duopoly 40 36,635.66 128.71 (two Firms enter) (26,628.18) (63.44) Total 40 33,969.15 128.31 (22,066.86) (64.13) American 54 26,709.59 118.79 (19,646.95) (65.88) United Active 69 29,304.07 132.67 (at least one Firm enters) (22,650.92) (54.64) Total 83 25,367.86 125.74 (18,466.34) (58.06) Overall 100 . . (100 Markets)
Standard errors in parantheses.
26
Average Mileage 441.14 (463.91) 754.13 (385.78) 652.23 (433.42) . . 698.17 (556.02) 631.53 (541.49) 721.69 (489.25) 674.37 (668.51) 715.28 (493.74)
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TABLE 3 INCOMPLETE INFORMATION SIMULATION RESULTS (Average across all Markets)
Probability that the Market is Expected Quantity per Firm Expected Price Expected Profit per Firm Expected Consumer Surplus
Monopoly
State of the Market Duopoly
Active
Overall
(only one Firm enters)
(two Firms enter)
(at least one Firm enters)
(100 Markets)
0.32 (0.19) 27,429.00 (20,657.04) 133.56 (61.46) 804,538.52 (1,184,272,21) 302,607.07 (448,104.03)
0.54 (0.36) 27,130.17 (20,772.13) 121.00 (57.07) 392,865.68 (584,678.93) 2,014,144.56 (3,485,116.57)
0.86 (0.28) 27,199.00 (20,386.48) 123.89 (57.84) 486,968.57 (699,237.09) 1,127,444.41 (2,572,921.30)
. 18,992.84 (14,282.98) . 340,725.41 (488,335.02) 981,921.02 (2,367,420.58)
Standard errors in parantheses.
TABLE 4 COMPARISON BETWEEN OBSERVATIONS AND SIMULATIONS
American
Sample Markets in United
Monopoly Total American
Sample Markets in United
Duopoly Total American
Active United
Sample Markets Total
Overall
Per firm
Relative Difference between Actual and Simulated Expected Quantity (in %) -0.51 (14.31) 0.03 (16.14) -0.15 (15.40) -5.40 (28.81) 5.44 (25.46) 0.02 (10.36) -4.13 (25.80) 3.17 (22.05) -3.65 (12.63) -0.07 (13.13) Standard errors in parantheses.
28
Relative Difference between Actual and Simulated Expected Price (in %) -2.46 (14.72) 0.30 (9.29) -0.60 (11.24) 0.41 (9.97) 1.79 (11.13) 1.27 (9.74) -0.34 (11.31) 1.17 (10.35) 0.30 (10.52) 0.30 (10.52)
TABLE 5 COMPLETE INFORMATION SIMULATION RESULTS (Average across all Markets)
Probability that the Market is Expected Quantity per Firm Expected Price Expected Profit per Firm Expected Consumer Surplus
Monopoly
State of the Market Duopoly
Active
Overall
(only one Firm enters)
(two Firms enter)
(at least one Firm enters)
(100 Markets)
0.52 (0.25) 41,170.47 (25,847.47) 130.16 (59.55) 1,176,839.39 (1,626,334.05) 333,348.38 (450,119.43)
0.46 (0.30) 24,577.39 (19,314.73) 127.04 (59.04) 516,927.94 (751,433.30) 1,893,240.42 (3,160,894.11)
0.98 (0.21) 30,564.647 (21,474.42) 128.08 (58.96) 754,974.62 (1,030,123.59) 1,085,103.58 (2,338,940.18)
.
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29
22,004.37 (15,452.03) . 544,199.725 (744,184.14) 978,405.64 (2,149,882.41)
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66
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69
APPENDIX 1 #
Market with flights from 1 airline
Airline with flights
Miles
#
Markets with flights from AA and UA
Miles
#
Markets with no flights
Miles
1
Albuquerque, NM
AA
1118
44
Albany, NY
723
84
Wilkes-Barre/Scranton, PA
631
2
Bloomington, IL
AA
116
45
Austin, TX
972
85
Fresno, CA
1730
3
Champaign, IL
AA
135
46
Kalamazoo, MI
122
86
Greensville, SC
577
4
Dubuque, IA
AA
147
47
Hartford, CT
783
87
Bakersfield, CA
1732
5
El Paso, TX
AA
1236
48
Buffalo, NY
473
88
Little Rock, AR
552
6
Evansville, IN
AA
273
49
Iowa City, IA
196
89
Mobile, AL
779
7
Fargo, ND
AA
557
50
Columbus, OH
296
90
Tri-City Airport, TN
481
8
Flint, MI
AA
223
51
Wausau, WI
213
91
Chattanooga, TN
501
9
Lafayette, IN
AA
119
52
Dayton, OH
240
92
Bridgeport, CT
767
10
La Crosse, WI
AA
215
53
Washington National, DC
612
93
Baton Rouge, LA
810
11
Muskegon, MI
AA
118
54
Des Moines, IA
299
94
Melbourne, FL.
1040
12
Rochester, MN
AA
268
55
Sioux Falls, SD
462
95
Augusta, GA
677
13
Toledo, OH
AA
214
56
Fort Wayne, IN
157
96
Beaumont/Pt. Arthur, TX
897
14
Tucson, AZ
AA
1437
57
Green Bay, WI
174
97
Mc. Allen, TX
1238
15
Allentown, PA
UA
654
58
Grand Rapids, MI
137
98
Daytona Beach, FL
962
16
Appletown, WI
UA
160
59
Westchester County, NY
738
99
Santa Barbara, CA
1803
17
Bangor, ME
UA
978
60
Indianapolis, IN
177
100
Youngstown, OH
378
18
Birmingham, AL
UA
584
61
New-York Laguardia, NY
733
19
Boise, ID
UA
1437
62
Kansas City, MO
403
20
Burlington, VT
UA
763
63
Harrisburg, PA
594
21
Columbus, SC
UA
666
64
Moline, IL
139
22
Akron/Canton, OH
UA
344
65
Madison, WI
109
23
Charleston, SC
UA
760
66
New Orleans, LA
837
24
Colorado Springs, CO
UA
911
67
Oklahoma City, OK
693
25
Ft. Lauderdale, FL
UA
1182
68
Omaha, NE
416
26
Spokane, WA
UA
1498
69
Ontario, CA
1700
27
Greensboro, NC
UA
590
70
Portland, OR
1739
28
Huntsville/Decatur, AL
UA
510
71
Peoria, IL
130
29
New Haven, CT
UA
778
72
Providence, RI
849
30
Wichita, KS
UA
588
73
Rochester, NY
528
31
Jacksonville, FL.
UA
865
74
San Diego, CA
1723
32
Lexington, KY
UA
323
75
San Antonio, TX
1041
33
Lincoln, NE
UA
466
76
Seattle/Tacoma, WA
1721
34
Saginaw, MI
UA
222
77
San Jose, CA
1829
35
Manchester, NH
UA
843
78
Sacramento, CA
1781
36
Oakland, CA
UA
1835
79
Orange County, CA
1726
37
Norfolk/ VA Beach, VA
UA
717
80
St. Louis, MO
258
38
Portland, ME
UA
900
81
Syracuse, NY
607
39
Richmond/Wmbg. VA
UA
642
82
Tampa/St. Petersburg, FL
1012
40
Fort Myers, FL
UA
1120
83
Tulsa, OK
585
41
Savannah, GA
UA
773
42
Louisville, KY
UA
286
43
Knoxville, TN
UA
475
37
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6;
TABLE A1 COMPLETE INFORMATION SIMULATION RESULTS State of the Market Monopoly Duopoly Active (only 1 Firm enters)
Probability that the Market is Expected Cost per Firm Expected Quantity per Firm Expected Price Expected Profit per Firm Expected Consumer Surplus Per Market
(At least 1 Firm enters)
0.599
0.401
1.000
.
0.342
0.421
0.374
0.499
0.572
0.285
0.457
0.286
0.673 0.207
0.579 0.067
0.635 0.151
. 0.089
0.177
0.178
0.178
0.178
TABLE A2 INCOMPLETE INFORMATION SIMULATION RESULTS State of the Market Monopoly Duopoly Active Probability that the Market is Expected Cost per Firm Expected Quantity per Firm Expected Price Expected Profit per Firm Expected Consumer Surplus Per Market
Overall
(Firms 1 and 2 enter)
Overall
(only 1 Firm enters)
(Firms 1 and 2 enter)
(At least 1 Firm enters)
0.213
0.770
0.983
.
0.443
0.445
0.444
0.499
0.295
0.291
0.292
0.255
0.841 0.137
0.575 0.057
0.633 0.075
. 0.059
0.057
0.197
0.167
0.164
39
Addendum 1: Estimation Results for Alternative Specifications of the Demand Function 1. Regression with Rival’s Complementarities The estimated model is the demand function in (4.1) including rivals’ complementarities: Pi,m,t = α0 + α1INCm + α2P OPm + α3 ln(P OPm) + α4 MILESm + α5 DP OPm +α6 QT Rt + β
M qi,m ,t + λ M N qj,m ,t − γ N qj,m,t + εi,m,t .
m =m
m =m j =i
j=1
The estimation results are: TABLE FGLS ESTIMATES Variable Parameter Estimate Std Errors Constant α0 -34.42 26.73 INC α1 1.72E10-3* 2.19E10-4 POP α2 1.03E10-5* 1.78E10-6 ln(POP) α3 9.65* 1.76 MILES α4 0.11* 1.94E10-3 DPOP α5 37.22* 8.12 QTR α6 -2.24E10-3* 7.90E10-4 β 2.71E10-6 5.89E10-6 λ -3.45E10-7 5.90E10-6 γ 6.47E10-4* 5.28E10-5 R2∗ = 0.852
Standard Errors in Parentheses. * indicates that the parameter is significant at a 5% Level
We fail to reject H0 : λ = 0 at a 5% significance level. The log-likelihood of the model, computed at the FGLS estimates, equals -3618.9644. For reference, the log-likelihood of the model with no rivals’ complementarities (i.e. the model chosen for the paper) equals -3618.9691. The two models may be compared with a Likelihood Ratio test for nested models (see e.g. Vuong (1989)). The likelihood ratio statistics is equal to 0.0093 and the null hypothesis that the models are equivalent cannot be rejected at the 5% level. Reference: VUONG, Q. “Likelihood Ratio Tests for Model Selection and Non-Nested Hypotheses.” Econometrica, Vol. 57 (1989), pp. 307-333.
2. Regression with Different Levels of Own Complementarity
The estimated model is the demand function in (4.1) including an interaction term for own complementarities and AA (i.e. variable AAi,m,t M m =m qi,m ,t), but not including rivals’ complementarities: The estimated model is:
Pi,m,t = α0 + α1INCm + α2P OPm + α3 ln(P OPm) + α4 MILESm + α5 DP OPm +α6 QT Rt + β
M qi,m ,t − γ N qj,m,t + εi,m,t + δ AAi,m,t M qi,m ,t
m =m
j=1
m =m
where AAi,m,t is a dummy variable equal to 1 if firm i on market m at time t is AA, equal to 0 otherwise. The null hypothesis is: H0 : δ = 0 . The estimation results are: TABLE FGLS ESTIMATES Variable Parameter Estimate Std Errors Constant α0 -39.57 26.59 INC α1 1.71E10-3* 2.19E10-4 POP α2 1.02E10-5* 1.78E10-6 ln(POP) α3 9.52* 1.76 MILES α4 0.11* 1.94E10-3 DPOP α5 36.66* 8.12 QTR α6 -1.28E10-3* 6.48E10-4 β -1.13E10-5 7.71E10-6 γ 6.45E10-4* 5.28E10-5 δ -6.35E10-6 3.38E10-6 2 R∗ = 0.851
Standard Errors in Parentheses. * indicates that the parameter is significant at a 5% Level
The parameter δ is not significantly different from 0 at a 5% significance level. In addition, the log-likelihood of the model, computed at the FGLS estimates, equals -3618.8097. The Likelihood Ratio statistics (see Section 1 in this addendum) is equal to 0.316 and the null hypothesis that this model is equivalent to the model estimated in our paper cannot be rejected at the 5% level.
2
3. Regression with Different Demand Slopes
The estimated model is the demand function in (4.1) including an interaction term for total quantity and AA (i.e. variable δ AAi,m,t N j=1 qj,m,t ), but not including rivals’ complementarities: The estimated model is: Pi,m,t = α0 + α1INCm + α2P OPm + α3 ln(P OPm) + α4 MILESm + α5 DP OPm +α6 QT Rt + β
M qi,m ,t − (γ + δ AAi,m,t) N qj,m,t + εi,m,t
m =m
j=1
where AAi,m,t is a dummy variable equal to 1 if firm i on market m at time t is AA, equal to 0 otherwise. The null hypothesis is: H0 : δ = 0 . The estimation results are: TABLE FGLS ESTIMATES Variable Parameter Estimate Std Errors Constant α0 -34.66 26.52 INC α1 1.72E10-3* 2.19E10-4 POP α2 1.04E10-5* 1.78E10-6 ln(POP) α3 9.66* 1.76 MILES α4 0.11* 1.94E10-3 DPOP α5 37.13* 8.11 QTR α6 -2.24E10-3* 3.81E10-4 β 2.51E10-6 2.11E10-6 γ 6.46E10-4* 5.29E10-5 δ 5.34E10-6 1.64E10-5 2 R∗ = 0.852
Standard Errors in Parentheses. * indicates that the parameter is significant at a 5% Level
The null hypothesis H0 : δ = 0 cannot be rejected at a 5% significance level. In addition, the log-likelihood of the model, computed at the FGLS estimates, equals -3618.8999. The Likelihood Ratio statistics (see Section 1) is equal to 0.122 and the null hypothesis that this model is equivalent to the model estimated in our paper cannot be rejected at the 5% level.
3
4. Test of Endogeneity In the augmented regression approach as outlined in Davidson and MacKinnon (1993), we first estimate this model:
N qj,m,t
j =1
= θ0 + θ1 IN Cm + θ2P OPm + θ 3 ln(P OPm ) + θ4 MILESm + θ 5DP OPm +θ6 QT Rt + θ7
M qi,m ,t + θ AM Im,t + θ AP Im,t + εi,m,t 8
m =m
9
where AM Im,t and AP Im,t areour two instruments. We compute the residuals for this N regression, denoted by residual j=1 qj,m,t . We then add these residuals to the demand function in (4.1): Pi,m,t = α0 + α1 IN Cm + α2 P OPm + α3 ln(P OPm ) + α4MILESm + α5DP OPm M N N +α6QT Rt + β qi,m ,t − γ qj,m,t + εi,m,t + δ residual qj,m,t .
m =m
j =1
j=1
The null hypothesis is: H0 : δ = 0 . The estimation results are: TABLE FGLS ESTIMATES Variable Parameter Estimate Std Errors Constant α0 -59.37 41.84 INC α1 1.87E10-3* 2.86E10-4 POP α2 1.12E10-5* 2.04E10-6 ln(POP) α3 10.96* 2.45 MILES α4 0.11* 1.94E10-3 DPOP α5 44.76* 12.75 QTR α6 -2.03E10-3* 4.82E10-4 β 2.13E10-6 1.73E10-6 γ 7.54E10-4* 1.45E10-4 δ 1.23E10-4 1.57E10-4 R2∗ = 0.852
Standard Errors in Parentheses. * indicates that the parameter is significant at a 5% Level
The null hypothesis H0 : δ = 0 cannot be rejected at a 5% significance level. In addition, the log-likelihood of the model, computed at the FGLS estimates, equals -3618.7237. The Likelihood Ratio statistics (see Section 1) is equal to 0.488 and the null hypothesis that this model is equivalent to the model estimated in our paper cannot be rejected at the 5% level. 4
Addendum 2 Alternative Estimation method We now propose a two-step semi-parametric technique based upon the first order conditions of the problem, in order to estimate the unknown parameter of the distribution of airlines’ private costs θ = (µ0, µ1, σ) ∈ Θ where Θ = R×]0, ∞[2. The first step consists of a non-parametric estimation of the expected quantities. Consider the function: g (zm) = E [qm |θ0] , where zm = MILESm is the exogenous variable, qm is a random variable representing the quantity produced by a firm on market m, and θ0 is the true value of the unknown parameter. We can then write the regression model q m = g (zm) + εm , where q m = 12 (q1,m + q2,m), qi,m is the quantity produced by firm i = 1, 2 on market m, and εm is a traditional error term with mean zero. For a given value of θ, the function g (.) may be consistently estimated by the non-parametric estimator q l k zmh−g zl l=m g (zm ) = M , z − z m l k hg
M
l=m
where k (.) is a kernel function and hg is a bandwidth controlling the smoothness of the 2 estimates. In practice, we consider a Gaussian kernel of the form k (x) = √12π exp 2x and the bandwidth is approximated by least square cross-validation (see Pagan and Ullah (1999)). In step 2, we use the set of first order conditions of the problem, as well as the function g (zm ), to recover the cost ci,m (when qi,m > 0) and the upper bound ci,m (ci, m ). Indeed, when qi,m > 0, we can estimate ci,m by inverting the Kuhn-Tucker equation (3.2): −
−
ci,m = αm +
M
(β m + β m ) qi,m + λm
m =m
M
m =m
g (zm ) − γ mg (zm) − 2γ mqi,m ,
where qi,m is the observed quantity produced by firm i on market m, and (αm , β m , λm , γ m ) are the known parameters of the demand function. We can also estimate the upper bound ci,m (ci, m ) from equation (3.3) −
ci,m = αm +
M
(β m + β m ) qi,m + λm
m =m
M
g (zm ) − γ mg (zm) .
m =m
Finally, we can estimate the parameter θ = (µ0, µ1, σ) ∈ R×]0, ∞[2 by θ SP =Arg max
θ∈Θ
i,m
1−F
ci,m |θ zm
I I {qi,m =0} ci,m {qi,m >0} |θ f zm
where f (.) and F (.) are, respectively, the log-normal probability density function and c . cumulative distribution for the random variable cpmi,m = zi,m m Note that this estimation method is robust in the sense that it does not rely upon a given equilibrium strategy, since it is constructed around the first order conditions of the problem. In other words, unlike the MSM technique, this method provides consistent estimates, whether or not the model has a unique solution, as long as players actually use Nash equilibrium strategies. Note also that this semi-parametric technique relies upon the same parametric assumptions as the MSM. We opted for the MSM for three main reasons: i) We wanted to use a standard estimation method with well established asymptotic properties; ii) Numerous Monte-Carlo simulations, and the comparison of the estimates obtained with this non-parametric technique and the MSM, strongly suggest that the model has a unique solution; iii) Provided that the equilibrium selected is correct, the MSM is significantly more accurate than the nonparametric technique. Finally, our objective is not only to estimate the parameter θ but also to run some policy simulations requiring the determination of the equilibrium strategy. To compare whether the two methods generate significantly different estimates, we generate by bootstrap some simulated samples, and we estimate parameters
l the and with the two techniques. As a result, we obtain two sequences θSP l=1,...,MC
l where MC = 104 is the size of the Monte Carlo simulation. These seθ MSM
l=1,...,MC
quences have an empirical mean of θSP = (0.189, 5.103E10−5, 1.888E10−2 ) and θMSM = (0.193, 5.059E10−5 , 1.874E10−2) and a standard deviation of (1.103E10−1, 1.344E10−5 , 5.162E10−3) and (6.345E10−2 , 1.651E10−6 , 2.207E10−4 ). Note that these two sequences are almost perfectly distributed around their means. To test whether these sequences have the same mean ( H0 : θi,SP = θi,M SM for i = 1, 2, 3), we apply the non-parametric Wilcoxon signed-rank test for comparing two non-independent samples. The p-values are (0.345, 0.553, 0.604), and we cannot reject the null hypothesis at the usual significance levels. This result suggests that the two methods estimate the same parameters, and that the equilibrium strategy derived by our numerical algorithm in the MSM method is not inconsistent with the data.
PAGAN A. and ULLAH, A. Nonparametric Econometrics. Cambridge University Press, 1999.
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