Socially Optimal Subsidy for Entry: The Case of
Medicare Payments to HMOs
Shiko Maruyama1
University of New South Wales
October 19, 2008
Abstract
Should governments subsidize entry to promote competition? The US Medicare has
increased its spending on private Medicare plans, based on the idea that they provide
consumers more alternatives and enhance the program’s e¢ ciency. In this paper, I develop an empirical framework with endogenous entry to evaluate welfare consequences
of such policy interventions with the data from 2003 and 2004. Counterfactual simulation reveals the following: (1) in Medicare, subsidizing entry can be justi…ed to enhance
national welfare (no excessive entry); (2) the level of payments in 2008, however, is far
beyond the optimal level; and (3) the geographic inconsistency between the payment
rate and the Medicare fee-for-service (FFS) cost is another source of ine¢ ciency. Aligning payments with FFS costs raises the national welfare gain of contracting with private
plans by an estimated 3 billion dollars, or 33 percent, compared to the simulated 2008
1
I am thankful to Leemore Dafny, David Dranove, Michael Mazzeo, Aviv Nevo, and Robert Porter for
guidance. Helpful comments from two anonymous referees, Raquel Bernal, Rosa Matzkin, and Hodaka
Morita are gratefully acknowledged. I thank the Centers for Medicare and Medicaid Services for kindly
providing the data …les and answering my inquiries. E-mail: [email protected].
1
benchmark, with lower Medicare spending on private plans. This signi…cant gain stems
from the restructuring of entry, not from lower payments.
1
Introduction
Should governments subsidize entry to promote competition? Advocates of market competition argue for more deregulation and liberalization of markets. On the other hand, few argue
that competition is so welfare-enhancing that governments should intervene further to bring
about more entry and intensify competition. The theoretical literature has con…rmed that,
from the social welfare point of view, whether additional entry has a positive or negative
impact on welfare is ambiguous, due to the trade-o¤ between the variety of products and
ine¢ cient replication of …xed costs.
The US Medicare program has struggled with this trade-o¤ for two decades. US Medicare
is the federal entitlement program that provides comprehensive health insurance coverage to
individuals aged 65 and older and certain disabled people (hereafter "Medicare eligibles"). In
addition to a government-administered fee-for-service plan (hereafter "traditional Medicare
plan"), Medicare features private health insurance plans, which are currently called Medicare
Advantage plans (formerly Medicare+Choice plans). The underlying idea is that private
plans give Medicare eligibles more alternatives and make the system more e¢ cient through
private plans’incentives and ‡exibility to take advantage of managed care techniques such
as utilization review and price negotiation with providers.
Private plans make entry-exit and premium decisions in each county every year, taking
into consideration the responses of both the demand side and competing …rms. Medicare
eligibles make their choice between the government plan and the private plan(s) available
in their county, taking into account each plan’s monthly premium, if any, and the insurance
coverage, e.g. whether it covers prescription drugs. The key government instrument here
2
is the rate of per-enrollee government payments to private plans, since payments can be
regarded as a price subsidy. Payments are the private plans’main source of revenue. Payment
rates vary across counties, but the same rate applies to every plan in a county. Thus,
the payment rate directly a¤ects the government spending and indirectly a¤ects consumer
surplus by altering private plans’coverage, premiums, and entry decision. Consequently, the
appropriateness of the payment rate has been an intense policy debate.
In this paper, I propose an empirical framework to evaluate the welfare consequences
of a policy intervention that a¤ects …rms’entry decisions, and to explore socially desirable
payment rates in Medicare, using national data of the Medicare market for the years 2003
and 2004. By allowing the …rms’entry-exit decision to be endogenous, the framework enables
evaluation of the welfare impacts of such policy interventions. To make policy simulations
even more relevant, the model incorporates product di¤erentiation and …rm heterogeneity.
This study advances the empirical literature of entry and provides unique, relevant policy
implications for the Medicare payment debate.
The literature on industrial organization theory …nds that free entry does not guarantee the socially optimal number of entrants. As discussed in Spence (1976) and Mankiw
and Whinston (1986), in the homogeneous product setting, free entry can lead to socially
excessive entry. Additional entry would expand consumer surplus with lower prices but, at
the same time, would reduce the market shares of other …rms ("business stealing") and lead
to an ine¢ cient replication of …xed costs. In di¤erentiated product markets, however, the
direction of bias is unclear, since entry brings more variety and consumers’appreciation of
variety causes a trade-o¤. In Medicare, for example, higher payments attract more entrants
and promote competition. New entrants enhance consumer surplus through lower prices,
improved product quality, and more variety. On the other hand, new entry lowers social welfare through duplicative set-up costs and "business stealing" from incumbents. In addition,
price subsidies cause market distortions. Therefore, whether the government can achieve
3
higher social welfare by encouraging entry is an empirical question.
Despite its policy importance, this empirical question has not been well explored. In an
attempt to answer this question, my empirical framework adopts three key features: endogenous entry, social welfare analysis, and fully incorporated product di¤erentiation and …rm
heterogeneity. As discussed in the next section, no previous work has combined these three
characteristics at once, mainly due to the computational complexity. A technical novelty
in my empirical framework is adoption of a recursive conditioning simulator in a maximum
simulated likelihood (MSL) estimator, which facilitates estimation of a very general class of
discrete choice sequential-move games. This estimator outperforms the methods standard in
this literature, such as the method of simulated moments, in several ways. The estimator’s
performance and ‡exibility help (1) obtain more e¢ cient estimates, (2) explicitly incorporate …rm heterogeneity into the framework, (3) estimate structural payo¤ functions that are
much more complicated than those used in the previous literature, and (4) analyze markets
with signi…cantly di¤erent structures in the same framework. As a result, I can exploit
detailed …rm/product level data and draw relevant policy implications by simulations with
endogenous entry, which is beyond the scope of previous studies.
The Medicare market is one of the most fruitful applications, because the products are
highly di¤erentiated and the payment policy is directly related to entry. The changes in
payment policy over the years re‡ect the di¢ culty of determining the socially optimal level
of subsidy. Before 1997, the base payment rate in each area was set at 95 percent of the
average fee-for-service (FFS) costs of the government plan. As a natural consequence, highcost markets, which are typically large, urban markets, attracted more entrants. After 1997,
Medicare started shifting payment rates toward uniform national rates, by increasing rates
in low-cost counties and decreasing them in high-cost counties in order to equalize potential
pro…ts of entrants and the level of entry across counties. This led to a signi…cant reduction
in the average payment and massive exodus of private plans around 2000. Since then,
4
Medicare has gradually implemented several adjustments to boost payments, such as "‡oor"
rates. MedPAC (2008) reports that average payments to private plans are 13% higher than
the FFS costs in 2008 and that payments amounted to $75 billion in 2006, representing 20%
of Medicare spending. To remove the inconsistency between the payment rate and the FFS
cost and to contain the growing expenditure, the government currently plans to set the base
payment rates equal to the FFS costs by 2012 (…nancially neutral payment rates).2
I use data from 2003 and 2004. Based on the estimated parameters, I perform counterfactual simulations to draw policy implications for the 2008 payment rates. Since this
approach does not utilize the information from recent years and incorporates neither the recent institutional changes nor the heathcare cost increases, the results need to be interpreted
with caution. On the other hand, this approach has several advantages. First, the system
was institutionally simpler in 2003 and 2004. The legislation in 2006 introduced Medicare
Part D and changed the way payment rates were determined.3 In addition, various new
types of private plans became popular, such as private fee-for-service plans. By avoiding
these complications, my approach helps develop an appropriate framework relatively easily
and concentrate on the essential questions on entry and payments. In addition, my sample
years are relatively stable because they occur long after the huge exodus of private plans
and su¢ ciently before the implementation of the 2006 legislation. Since I use a static entry
game framework, the stable environment is desirable for providing reliable estimates. Along
the same lines, I concentrate this research on Health Maintenance Organizations (HMOs),
since in 2003 and 2004, the vast majority of the private plans were HMOs.
My empirical strategy is as follows. All estimations are at the plan-county-year level.
I …rst estimate a discrete choice demand model of the Medicare health insurance market.
2
MedPAC (2008) provides detailed information and discussion about the payment rate.
The legislation is the Medicare Prescription Drug Improvement and Modernization Act. Currently "plan
bids" partially determine the payment rates, though the previous payment rates are used as "benchmarks"
and still largely determine current payment rates.
3
5
The consumer surplus is calculated using these results. Since private plans are allowed
to charge monthly premiums, I use premiums as a price. Second, by using the estimated
demand parameters and assumptions on …rms’price-setting behavior, I recover the health
plans’ marginal costs. Third, the pool of potentially entering HMOs is created. This is
necessary because the entry game is played not only by the observed …rms but also by …rms
that choose not to enter and so do not appear in the data. I then estimate HMO pro…t
functions by using a sequential-move entry game, which guarantees a unique pure-strategy
equilibrium. The observed market structure and inclusion of the demand model into the
framework allow me to identify the level of …xed and variable pro…ts, despite the lack of
cost information. Finally, I perform counterfactual simulations, where all the consumers
and HMOs re-optimize their decisions on plan choice, price, and entry-exit in response to
an exogenous change, which generates a new market equilibrium. I follow the approach
that is customary in the empirical entry literature and employ no scope of dynamics, since
my dataset has only a few time points and fully incorporating …rm heterogeneity remains a
considerable challenge in dynamic modeling.
The results are as follows. First, the national welfare gain of having private health
insurance plans in Medicare was approximately 10 billion dollars per year in 2003 and 2004,
which is consistent with the study by Town and Liu (2003). Second, the Medicare HMO
market exhibits no evidence of socially excessive entry. Consumers appreciate increase in
variety of products so much that entry subsidy beyond the …nancially neutral level can be
a justi…able option for greater national welfare. Third, the positive e¤ect of such subsidy,
however, is small, and the 2008 level of payments is far beyond the optimal level. Fourth, the
geographic inconsistency between the payment rate and the FFS cost is another source of
ine¢ ciency. Aligning the payment with the FFS cost raises national welfare by an estimated
3 billion dollars, or 33 percent more than the simulated 2008 benchmark case. This gain
occurs together with a healthier government budget and stems from the restructuring of
6
entry, not from the adjustment of the average of payments. These results illustrates how
extensively the current payment distorts the incentives for the Medicare private plans and
harms social welfare.
2
Empirical Entry Literature
The study I follow most closely is Berry and Waldfogel (1999), one of the few to apply the
empirical entry literature to welfare analysis. They model the broadcasting industry as a
homogeneous product market4 and quantify social ine¢ ciency from free entry. Dutta (2005)
extends their work to the case of di¤erentiated products. Her study of the pharmaceutical
industry in India performs counterfactual simulations of entry and …nds no evidence of
excessive entry. Her empirical framework is similar to mine, except that her estimation
relies heavily on the assumption of certain …rm homogeneity. I extend these studies by
incorporating full …rm heterogeneity.
Firm heterogeneity has not been fully incorporated into the empirical entry literature
for several reasons. The di¢ culty of dealing with multiple equilibria is one of the largest issues. To guarantee certain equilibrium properties, researchers have speci…ed the econometric
setting in a very restrictive way, such as homogeneous product markets, symmetric games,
simple reduced-form pro…t functions, and an in…nite pool of homogeneous potential entrants.
The computational burden is another di¢ culty, since the introduction of …rm heterogeneity
requires the estimation of an asymmetric game.
Various approaches have been proposed to address these di¢ culties. Berry (1992) avoids
this problem by introducing …rm heterogeneity into the …xed cost component, thereby keeping his game symmetric. Berry and Waldfogel (1999) and Town and Liu (2003) follow the
4
In their framework, while listeners’ demand for broadcasting is modeled as a discrete choice demand,
the industry’s primary output –advertisement –is treated as a homogeneous product.
7
same approach. Dutta (2005) incorporates …rm heterogeneity by categorizing …rms into three
types and, within each type, …rms are symmetric. Mazzeo (2002) models …rm heterogeneity
through endogenous choices of location and product types. Seim (2001, 2004) proposes the
use of an incomplete information setting to alleviate the computational burden. Ciliberto
and Tamer (2004) pursue a more ‡exible model of …rm heterogeneity and identities, by not
making point identifying assumptions on equilibrium selection. All of these studies, however, still use a reduced-form pro…t function and/or certain symmetry assumptions in entry
games, which is a restriction, especially when a researcher’s goal is a welfare simulation.
Berry (1992) and Mazzeo (2002) employ the idea that the use of a sequential-move game can
avoid the multiple equilibrium issue by the uniqueness of the subgame perfect equilibrium.
Schmidt-Dengler (2006) uses this approach to analyze the timing of MRI adoption of US
hospitals. His framework incorporates …rm heterogeneity, but its tractability relies on the
simplicity in his game setting, such as the irreversibility of adoption and the focus on only
small oligopolistic markets. I introduce a new maximum simulated likelihood estimator,
which allows me to fully incorporate …rm and market heterogeneity in a more general entry
game setting and to perform nationwide welfare simulations.
Another key feature of my model is the structural combination of the demand and supply
sides, which o¤ers a signi…cant advantage: counterfactual simulations by extrapolation. It
enables analysis of how a policy change a¤ects entry and exit, market competition and, in
turn, social welfare. I am not the …rst to embed demand into an entry model. Reiss and
Spiller (1989) estimate the demand and supply of airline services simultaneously, but only
for carefully selected, small homogeneous product markets. Berry and Waldfogel (1999)
explicitly model pro…t functions with price and quantity, which allow …rms to have di¤erent
…xed costs, but their focus is on homogeneous product markets. Another study along these
lines is Town and Liu (2003). They quantify the welfare impact of Medicare HMOs by
combining structural models of demand and entry. Though they model the market as a
8
di¤erentiated product market, their entry game assumes in…nitely many plans with the
same …xed costs and does not fully incorporate …rm heterogeneity. By adding more supply
side structure to their framework, my model can evaluate each plan’s …xed costs, estimate
the potential social ine¢ ciency due to excessive or insu¢ cient entry and perform various
counterfactual simulations.
3
Data
Medicare is administered by the Centers for Medicare and Medicaid Services (hereafter
CMS). The main body of the dataset comes from the CMS. I use data from 2003 and 2004.5
The three main data sources are the following: (1) the Enrollment data at the county-plan
level, (2) the Monthly Report data, and (3) the Plan Bene…t Package data (PBP data).6
3.1
The Unit of Observation
The unit of observation is a plan-county-year. In the CMS data …les, there is a clear distinction between organization, plan (or contract, as the CMS sometimes calls it), and product (or
plan, as the CMS sometimes calls it). The organization is the governing body: for example
Paci…care, Aetna, and Humana. In the Medicare program, each organization may enter into
one or more plans (or contracts). Each of these plans is assigned a unique contract number by the CMS. In addition, each plan can o¤er more than one product, i.e. a particular
bene…t package with unique bene…ts, copayments, premiums, and counties covered. I focus
this study on the plan level analysis, because many important variables — such as county
enrollment — are reported at the plan level.7 I aggregate product level information to the
5
Information from 2002 is also used to identify incumbents in the entry estimation.
The following description of institutional backgrounds is based on the Medicare program in 2003 and
2004. For the detailed description of the data, see Maruyama (2007).
7
The plan level analysis is also e¤ective at reconciling the demand and supply side models. While HMOs
are likely to maximize their pro…ts at the plan or organization level, for the demand side, the product level
6
9
Table 1: The Number of HMO Plans and their Market Share in 2004
Number of plans Enrollment Share
HMO (Health Maintenance Organizations) 145
4,684,304
85%
PPO (Preferred Provider Organizations)
41
119,110
2%
Other Medicare+Choice types
70
266,247
5%
Other prepaid plan types
44
428,833
8%
Total prepaid plans
300
5,498,494 100%
Entire Medicare eligibles
42,992,077
Source: Medicare Managed Care Contract Report and the Enrollment data
plan level in each county by choosing the product with the largest enrollment as the representative product. For ambiguous cases, I average the variables across products, but such
cases are rare.
3.2
The Sample Population
Under the Medicare program, several types of private health plans are eligible to participate,
but HMOs are dominant players. An HMO forms its network of health providers to enhance
its purchasing power and to restrict and manage the ‡ow of its patients. A Medicare HMO
receives a monthly payment for each enrollee from the CMS and, in return, is responsible
for providing all covered services and takes full …nancial responsibility for the actual costs
generated. Due to these patient management devices and …nancial incentives, Medicare
HMOs are assumed to be more e¢ cient than the government plan.
Table 1 shows that HMOs’ enrollment share in private health plans was 85% in 2004.
While I use most plan types in the demand estimation, I focus my supply side model on
HMOs because the shares of non-HMO plans are small and because their pro…t structure may
di¤er from that of HMOs. I estimate pro…ts only for HMOs and the presence of non-HMO
types is treated as exogenous throughout this research.
analysis may be more reasonable because consumers are more likely to choose a product than a plan or an
organization.
10
I choose the observations used in this research as follows.
Omitted Plan Types With various types of private Medicare plans, I exclude the plan
types regarded as not open to regular Medicare bene…ciaries, such as plans in the Program of
All-Inclusive Care for the Elderly (PACE) and long-term care Demonstration plans. Ignoring
these plans excludes only 3% of the entire private plan enrollment.
Choosing Operating Plans
Next, at the plan-county-year level, I select operating plans.
In the CMS Enrollment data …les, individual members are assigned to counties according
to the reported residence. This implies that many counties contain an unrealistically high
number of health plans with very low enrollment. I include a plan-county observation in the
dataset as a valid observation if it satis…es both of the following: (1) it is in a plan’s o¢ cial
service counties under its contract to the government, and (2) it has at least 50 members.8
A very small number of observations with missing values are also excluded. Then, I recalculate the market size by subtracting the enrollment in the excluded observations from the
original number of Medicare eligibles. This completes the dataset for the demand estimation.
Creating Potential Entrants In my supply side model, all the potential operating plans
play a one-shot entry game each year. The pool of potential entrants consists of the operating
HMOs observed in the data and hypothetical potential entrants that I add to the dataset.
I limit my entry model to entrants that are operating in other areas in the same state
or the same metropolitan statistical area (MSA).9 This means that I exclude: (a) entrants
from outside the state or MSA and (b) brand-new entrants from the commercial sector.
8
I set this threshold higher than the previous studies to make the data stabler. Town and Liu (2003),
for example, de…ne the HMOs operating in a county as the HMOs with at least 12 members if the county
has fewer than 5,000 eligibles, and at least 35 members otherwise. Lowering my threshold to 35 does not
signi…cantly a¤ect the estimates, although it makes the standard errors slightly larger and the estimation
procedure less stable and more time-consuming.
9
Multi-state MSAs exist. Entry from outside the state but in the same MSA is considered in the analysis.
11
The potential bias due to this exclusion, however, seems to be minimal to a …rst order
approximation. First, no entry from outside the state or MSA is observed in my sample
periods. Second, although each Medicare HMO plan has its main entity in the commercial
sector, only a few HMOs newly started Medicare business during my sample periods. In
2004, for example, only …ve plans were new in Medicare.
The pool of potential entrants consists of the following six types: (1) plans that actually
operate in the county in the year, (2) plans that operate in the same MSA in the year, (3)
plans that operate in the same state in the year, (4) plans that operate in the county in
the previous year, (5) plans that operate in the same MSA in the previous year, and (6)
plans that operate in the same state in the previous year. Thus, in the supply estimation,
observations that satisfy one of (2) to (6) but not (1) are created and included as hypothetical
entrants.10 As a result, the supply dataset has many counties with one or more hypothetical
entrants but no observed entrants. These counties are retained in the sample in order to
avoid unwanted selection bias and make use of the information value from the fact that no
entry occurs. The plan level characteristics of the newly created potential entrants are the
same as the original plan. For the product level characteristics, I pick the product with the
largest enrollment under the plan as the representative product.11
3.3
Variables, Sample Size, and Summary Statistics
The PBP dataset contains a signi…cant amount of detailed information about the additional
bene…ts that a product o¤ers. In Maruyama (2007) I used factor analysis to summarize this
10
The concept of hypothetical potential entrants di¤ers from that of incumbency. A hypothetical potential
entrant that operates in the previous year but not the current year plays the entry game as an incumbent.
11
Another approach is to construct predicted characteristics of hypothetical entrants by running a reducedform regression. This approach can take into account the possibility that a plan adjusts its characteristics
to suit the markets it enters, although it ignores the possibility that product type choice depends on the
decision of other …rms. I test this approach as a robustness check by estimating a simple probit model that
predicts the drug bene…t availability based on market and plan types. This generates slight di¤erences,
especially in the estimates of producer surplus, but all in all the results are not signi…cantly a¤ected.
12
Table 2: The Sample Size
Number
Number
Number
Number
Number
of
of
of
of
of
Number of
Number of
Number of
2003
Demand dataset
plans
196
plans (HMO)
133
counties
829
plan-counties
1,478
plan-counties (HMO)
997
Entry estimation dataset
HMO plans
133
counties
2,574
plan-counties
9,617
2004
Total
207
138
938
1,811
1,094
207
138
948
3,289
2,091
138
138
2,501 2,630
9,927 19,544
information. Based on the results, I aggregate relevant bene…t variables into six composite
bene…t variables: drug, education, physicals, peripheral1, peripheral2, and screenings. The
payment rate data in the original CMS datasets is the standardized, payment base in each
county. The actual payment for a particular enrollee is determined by certain formulae,
which take demographic and risk factors into consideration. In this study, I use the average
of actual payments to private plans in the county calculated by using demographic and risk
factor information, rather than using the payment base.
Table 2 shows the sample size for the demand and supply estimation. Table 3 shows descriptive statistics of selected variables. The minimum of payment rates is negative because,
since 2003, the premium can be "negative" in the form of a premium rebate. A comparison
of the average and standard deviation of monthly premiums and payment rates indicates
that payments from the government are the primary source of revenue, though private plans
adopt various strategies in setting their premiums.
Table 4 shows the distribution of observations and the average market size, payment
rates, and monthly premiums by the number of observed entrants in a county. The majority
of observations are from concentrated markets. The table suggests that counties with larger
demands and/or higher payment rates attract more private plans. In addition, the more
13
Table 3: Descriptive Statistics of Selected Variables
Average
Plan enrollees
Plan market share
Monthly premium
Payment rate
bene…t: drug
3,060
0.063
51.3
549.4
0.61
(2003) (2004)
3,339 2,833
0.066 0.061
58.9
45.0
530.7 564.7
0.55
0.65
Std.dev.
Min
Max
7,715
0.070
47.5
91.1
0.48
50
0.0002
62.6
296.7
0
142,050
0.447
189.0
1067.5
1
The number of observations: 3289 plan-county-years. Monthly premium and payment rate are in dollars.
competitive a county is, the lower premiums they charge. Payment rates positively correlate
with FFS costs, but with smaller variance, due to the government payment policy described
above.
Table 4: Sample Distribution, Average Market Size, Average Premiums, and Average Payment Rates by the Number of Plans in Each County: 2004
Number
Number of
Mean:
County mean:
plan-counties:
premium eligibles Std FFS per base payof plans in
a county demand data supply data
capita cost ment rates
0
5,386
1
497
1,860
51.6
15,215
543.7
585.3
2
456
1,167
45.8
35,606
561.6
600.7
3
339
635
44.0
50,835
615.6
644.2
4
184
367
54.1
59,641
629.3
649.2
5
130
180
51.6
65,552
611.5
645.9
6
84
146
30.0 130,593
732.5
694.9
7
21
40
14.6 234,969
806.0
785.5
8
24
41
3.2 157,267
686.7
828.0
9
36
56
4.8 261,881
903.0
812.5
10
40
49
8.1 513,123
926.2
780.1
Total
1,811
9,927
45.0
40,535
570.7
605.8
Non-HMO plans included. Mean values are simple averages with no weights.
Table 5 explains the variables used in this study. The names and de…nitions of the
variables are given in the …rst two columns, and the next three columns indicate whether a
given variable is used to estimate demand, supply, or marginal cost. Summary statistics are
14
provided in the following columns.
4
Econometric Speci…cations
4.1
The Demand
The demand model follows the approach of Berry (1994) and Town and Liu (2003); it uses
market share data with the nested Logit model. A Medicare bene…ciary chooses a Medicare
plan every year by comparing the utility from each plan available in the bene…ciary’s county
and picking the plan with the greatest utility. Utility is derived from health plan characteristics such as premiums and bene…ts. The notation is as follows:
M : Year-markets (year-counties), m = 1; :::; M
J : Private Medicare plans, j = 1; :::; Jm
If j = 0, it means the traditional Medicare plan (the outside option).
Im : Medicare bene…ciaries, i = 1; : : : ; Im :
Bene…ciary i’s utility from plan j in year-market m is denoted as
0
uijm = wjm
(1)
jm
pjm +
+
jm
j
+
M SA
+
jm
+ "ijm
+ "ijm ;
where wjm is a vector of observed plan characteristics, pjm is the premium, "ijm is a nested
Logit error, and
j
+
M SA
+
jm
is a scalar contribution of the plan characteristics and
demand shocks unobserved by the econometrician.
jm
is the mean utility. Without loss of
generality, the error term is normalized as E["ijm ] = 0. Thus a bene…ciary’s expected utility,
E[uim ], increases with better bene…ts, lower premiums, and a greater number of choices. To
alleviate the endogeneity problem discussed below, I apply …xed e¤ects to plans and selected
15
Table 5: Definitions of Variables, Estimations Used (Demand, Supply, and Marginal Cost), and Summary Statistics
Variable Name
Definition
D S MC Obs
Mean Std. dev. Min
Max
(i) Plan or plan-year level variables (the numbers of plan and plan-year observations: 207 and 403)
no experience
experience year
nonprofit
national chain
group model
staff model
IPA model
entire enrollees
DEMO plan
cost plan
cost missing
PPO plan
PFFS plan
PSO plan
POS
=1 if a plan has no experience in Medicare before
Years in business since the first Medicare enrollee
Dummy variable: not-for-profit organizations
Dummy variable: national chains
=1 if the plan is a group model HMO (The omitted category is ”network,
mixed, and other models”.)
=1 if the plan is a staff model HMO
=1 if the plan is an IPA model HMO
Total enrollees of the plan over its entire service areas (in 1,000)
Dummy variable: DEMO plans
Dummy variable: cost contract HMOs
Dummy variable: cost contract HMOs with missing data
Dummy variable: PPO plans
Dummy variable: PFFS plans
Dummy variable: PSO plans
Dummy variable: HMOPOS
X X X
X X X
X X
X X
X X
403
403
207
207
207
0.11
8.18
0.34
0.29
0.47
0.31
6.44
0.47
0.46
0.50
0
0
0
0
0
1
27
1
1
1
X X
X X
X
X
X
X
X
X
X
X
207
207
403
207
207
207
207
207
207
207
0.07
0.43
25,313
0.15
0.13
0.05
0.17
0.02
0.01
0.03
0.25
0.50
57,033
0.36
0.33
0.22
0.38
0.15
0.12
0.17
0
0
52
0
0
0
0
0
0
0
1
1
648,281
1
1
1
1
1
1
1
(ii) Plan-county level variables (the number of plan-county-year observations: 3,289)
s jm
ln s j | private, m
monthly premium
# product in plan
benefit: drug
benefit: education
benefit: physicals
benefit: peripheral1
benefit: peripheral2
benefit: screenings
avg # competitor
competitor: npo
competitor: chain
competitor: IPA
competitor: group
competitor: staff
copayments
Plan j ’s market share in market m
”Within-group share” in log
Monthly premium in $ (Medicare premium not included)
The number of products the plan offers in the market
Dummy variable: supplemental benefits in outpatient prescription drugs
Composite score: supplemental benefits in health education/welness
Composite score: supplemental benefits in routine physicals
Composite score: supplemental benefits in preventive and comprehensive
dental, chiropractic, and acupuncture
Composite score: supplemental benefits in eye exams, eye wear, hearing
exams, and hearing aids
Composite score: supplemental benefits in screenings (mammography,
colorectal, prostate cancer, pap smears, and pelvic exams)
Average number of competitors in the other markets served by the plan
The number of competing NPO plans in the market
The number of competing national-chain plans in the market
The number of competing IPA model plans
The number of competing group model plans
The number of competing staff model plans
Out-of-pocket costs (averaged over demographic groups, net of premiums)
X
X
X
X
X
X
X
X
3,289
3,289
3,289
3,289
3,289
3,289
3,289
3,289
0.06
-1.21
51.3
2.16
0.61
0.36
0.51
0.10
0.07
1.44
47.5
1.28
0.48
0.27
0.21
0.18
0.00
-7.55
-62.6
1
0
0
0
0
0.45
0.00
189.0
10
1
0.9
1
1
X
3,289
0.48
0.29
0
1
X
3,289
0.20
0.26
0
1
IV
IV
IV
IV
IV
IV
3,289
3,289
3,289
3,289
3,289
3,289
X 3,289
1.75
0.70
0.71
0.97
0.74
0.11
335.4
1.69
1.13
1.08
1.30
0.99
0.34
85.9
0
0
0
0
0
0
0
9
5
5
6
4
2
589.9
0.88
527.5
519.7
552.8
0.27
3.18
0.33
0.22
32,939
0.16
0.55
0.19
0.28
0.25
0.14
0.04
0.53
0.95
0.11
80.5
118.8
131.0
0.36
4.58
0.19
0.17
64,364
0.36
0.50
0.39
0.45
0.44
0.35
0.19
0.50
0.04
X 1,767
X 1,767
0.93
0.86
0.09
0.12
0.73
0.52
1.66
1.53
X 1,767
X 1,767
137.4
1.06
38.4
2.06
70.1
0.00
231.8
51.97
X 1,767
67,939 157,012
(iii) County or county-year level variables (the numbers of county and county-year observations: 948 and 1,767)
s 0m
payment rate
FFS per capita cost
Std FFS per capita cost
per capita # hospital
per capita # hospital bed
per capita # medical doctor
HMO penetration rate 98
eligibles
county: no hospital
MSA
county: small
county: medium
county: large
county: extra large
county: huge
year: 2004
demo factor
demo factor HMO
risk factor HMO
medigap premium
hospital expenditure
inpatient days
The market share of the traditional Medicare plan
CMS monthly payment rate per enrollee (case-mix adjusted for each plan)
Average fee-for-service monthly cost per enrollee
FFS per capita cost standardized by demographic factor
The number of hospitals per 10,000 residents in 2001
The number of hospital beds per 1,000 residents in 2001
The number of general practice M.D.s per 1,000 residents in 2001
Estimated (HMO enrollment / total population) in 1998
Medicare eligibles (in 1,000)
Dummy variable for counties with no hospital
Dummy variable for counties in an MSA
County dummy: 3,000 < number of eligibles < 6,000
County dummy: 6,000 < number of eligibles < 15,000
County dummy: 15,000 < number of eligibles < 50,000
County dummy: 50,000 < number of eligibles < 150,000
County dummy: 150,000 < number of eligibles
Year dummy variable for 2004
Demographic factors (e.g. age, sex, institutional status, and Medicaid
status) to adjust the payment rate (the higher, the more costly)
Demographic factors for HMO enrollees
Risk factors for HMO enrollee (e.g. disease group, community factors, and
institutional factors) to adjust the payment rate
Monthly Medigap premium in $
Total reported facility expenditures (short term general hospitals) (not
only Medicare) per 1,000 residents in $1,000 in the county
Total Medicare inpatient days (short term hospitals) in 1,000 days
X
X
X X
IV X X
IV X X
IV X X
X
X
X
IV X X
X
X
X
X
X
X
X X X
X
1,767
1,767
1,767
1,767
948
948
948
948
1,767
1,767
948
1,767
1,767
1,767
1,767
1,767
1,767
1,767
0.50
1.00
296.7 1,067.5
214.9 1,217.3
203.6 1,262.2
0.00
5.12
0.00
87.50
0.00
1.54
0.00
0.96
370 1,046,829
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0.82
1.11
0 2,351,152
”IV” means the variable used in the estimation as an instrument. For the summary statistics, Non-HMO observations are included. The ”hypothetical potenti
entrants” used in the supply estimation are not included. The six benefit variables from ”benefit: drug” to ”benefit: screenings” are the quality composite
variables made by aggregating the detailed characteristics information in the PBP data set, based on the results of the factor analysis (Maruyama (2007)).
15
MSAs, which are represented by
j
+
12
M SA .
Thus,
jm
is the …rst-di¤erenced demand
shock. I normalize the utility from the traditional Medicare plan as: ui0m = "i0m . This
normalization does not a¤ect the welfare calculation, because consumer surplus in my study
measures the utility gain from enrolling in private Medicare plans relative to traditional
Medicare.
Following previous studies, I assume a nested Logit error. The group structure is the
private plan group against the traditional Medicare plan. Thus, my assumption allows
substitution among private plans to di¤er from substitution between private plans and the
traditional Medicare plan. The market share of plan j is derived as:
sjm = sjjprivate;m sprivatejm
exp 1 1 jm
h
i;
=
(1 )
Dm 1 + Dm
(2)
where
represents the similarity within the private plan group, and
Dm
X
exp
kjprivate;m
1
1
km
The correlation within a group is possible through . As
correlation becomes one and, when
:
goes to one, the within-group
= 0, the model is reduced to the multinomial Logit
model. When the market share data is used, this model can be estimated using the following
linear form (see Berry 1994):
(3)
ln sjm
0
ln s0m = wjm
pjm +
j
12
+
M SA
+ ln sjjprivate;m +
jm
County …xed e¤ect dummies are not used because many counties appear in the dataset with only one
observation. By the same token, I apply MSA …xed e¤ects only to MSAs for which at least 10 observations
exist in the dataset.
17
for j = 1; :::; Jm : For the estimation, I use the instrumental variable (IV) method to deal
with the potentially endogenous variables in expression (3): the monthly premium, pjm , and
the within-group share, ln(sjjprivate;m ). These two variables may be correlated with the plan
demand shock,
jm .
Since I use the …xed-e¤ect approach, the identi…cation of parameters
comes from within-plan and within-MSA changes in the instruments.
I construct instruments using three strategies. The …rst set of instruments comprises
a county dummy variable for counties with no hospital and the per-resident numbers of
hospitals, hospital beds, and general practice medical doctors. These variables are valid
instruments because they a¤ect the plans’ relative bargaining power with providers, their
cost structure, and the number of competitors. Second, I use the characteristics of competing
plans in a county. This approach is traditional in the literature on product-di¤erentiated
market demand (e.g. Bresnahan 1987 and Berry 1994).13 This is valid if competitors’entryexit decisions and changes in product characteristics are uncorrelated with changes in
jm .
Speci…cally, I choose not-for-pro…t ownership, chain a¢ liation, and HMO network types
(IPA, Group, and Sta¤ models in Table 5), which are expected to be relatively …xed and
predetermined. I use the sums of these indicator variables of rival plans as instruments.
Third, I use the average number of competitors in the other markets in which the plan
operates. This use of the panel structure of data is a strategy similar to that of Hausman
(1997) and Nevo (2001). Private Medicare plans typically set premiums for each product
rather than for individual counties. Thus, the plan premium in a county is likely to correlate
with the competitive environment in the plan’s other service counties.
jm
is speci…c to
the market, so it is likely to be uncorrelated with this instrument.14 In the end, I have 10
13
Dafny and Dranove (2005) also use this approach in the demand estimation of Medicare HMOs.
The argument here does not hold if each plan chooses its premium at the county level, as I assume below.
However, due to set-up costs of products, the reality is in between. Without set-up costs, a plan could set
up a product in each county so that it could charge an optimal price in each county. In reality, however, a
product usually covers several markets. A plan carefully chooses its products’service area boundaries, which
allows a plan to charge roughly optimal prices with lower set-up costs. On this basis my instrument is valid,
14
18
instruments for two endogenous variables. The …rst stage regression is reported in Appendix
A. For both pjm and ln(sjjprivate;m ), the F -statistics reject the joint hypothesis that the
coe¢ cients on the instruments are all zero.
4.2
4.2.1
HMO’s Behavioral Model
The Game Speci…cation
pot
potentially operating HMO plans in each market. Each county can accomThere are Jm
modate zero, one, or more than one HMO plan. At the beginning of each year, all the
potential entrants in each market play the following game. The game consists of two parts, a
sequential-move entry game and a simultaneous-move price-setting game. Only HMOs that
choose "enter" play the price setting game. If a plan chooses "not enter", it receives zero
pro…t from the county. The plan characteristics are exogenously …xed in this study.
I employ a complete information sequential-move entry game. This guarantees the existence of a unique subgame perfect Nash equilibrium, even for large asymmetric games.15
In the sequential game, I assume the following decision order. Among all the potential entrants in a market, the incumbents move …rst. Incumbents make decisions in the order of
the following incumbency classes: (1) presence in the county during the previous year, (2)
presence in the same MSA during the previous year, and (3) presence in the same state
during the previous year. Within the same incumbency class, HMOs make decisions based
on the size of their entire enrollment. This assumption is more likely if larger incumbents
must announce their changes in service area earlier. I also perform a robustness test of this
while the county-level pro…t-maximization assumption still holds as an approximation. Exclusion of this IV
does not change the results signi…cantly.
15
There is no guarantee that this assumption will be realistic, but a simultaneous game is no less a priori
than a sequential game. The order of decisions is studied by Berry (1992), Mazzeo (2002), Einav (2003), and
Schmidt-Dengler (2006). In general, these papers suggest that di¤erences resulting from di¤erent decision
orders or di¤erent game settings are small.
19
order assumption by randomizing the order. As reported in the result section, this does not
signi…cantly a¤ect the results.
The game is assumed to be a complete information game, because private (or incomplete)
information models with publicly observable heterogeneity are relatively di¢ cult to estimate.
The number of potential entrants, plan characteristics, random shocks in pro…ts, and other
information are common knowledge for all players. As the only exception, I assume that
no player observes the random shock in the demand,
jm ,
when players make decisions.
This technical assumption is required to keep the estimation consistent and well-behaved.
Since the value of
jm
can only be calculated as the residual term in the demand esti-
mation, it cannot be obtained for hypothetical entrants whose price and quantity are not
observed in the dataset. However, to calculate more precise national estimates of welfare
components, this residual term should be included in the calculation. Thus I need the assumption that allows me to incorporate
game. By de…nition,
jm
jm
in the welfare calculation but not in the entry
is mean-zero and independent of the decision of plans, so the
assumption that this is unknown ex-ante does not a¤ect the desirable equilibrium properties
of the sequential-move game as long as plans are risk-neutral. At the same time, I believe
that this simplifying assumption is reasonably realistic. First, though there certainly exists
information on quality and characteristics that is known to plans but not to econometricians,
most of such information is likely to be captured by the plan and MSA …xed e¤ects. Second,
the residual from the demand estimation consists of not only variation in characteristics
but also demand shocks at the plan-market level, which are often di¢ cult even for plans to
perfectly predict when they make entry and price decisions. As shown later, my demand
model explains approximately 80% of market share variation. If the accuracy of the plan’s
demand predictions is somewhat similar, this assumption of
jm
is more or less realistic.
I use the concept of the subgame perfect pure strategy Nash equilibrium (SPNE). An
SPNE in a county is obtained when: (1) all entering …rms are pro…table with their optimized
20
prices, and (2) all …rms that do not enter expect non-positive pro…ts from entry. Each HMO’s
entry-exit strategy in county m is represented by a binary variable, yjm , which takes the value
"0" if plan j does not enter and "1" if it enters. A market con…guration is denoted as ym ,
which is a vector that stacks (y1m ; :::; yJ m m ). The equilibrium solution, ym , can be calculated
by the backward induction algorithm.
4.2.2
The Pro…t Function
I assume that each …rm maximizes pro…ts independently at the plan-county level, by assuming the pro…t to be additively separable across counties, and across plans if an organization
has multiple plans. Relaxing this assumption to incorporate the economy of scale or strategic
cooperation within a county is beyond the scope of this paper, since it requires substantial
modi…cation of the estimation algorithm. I assume the following local pro…t function for
plan j in county m:
(4)
jm (x m ; zjm ; "jm ; ym ;
) = [pjm (x m ; ym ) + Payment Rate m
M Cjm ]
qjm (p m (x m ; ym ); x m ; ym ) + zjm + "jm
V Pjm (x m ; ym ) + zjm + "jm ,
where the expression in square brackets is marginal pro…ts and qjm () is the demand. p
is (p1m ; :::; pJ m m )0 , x
m
m
is (x1m ; :::; xJ m m )0 , xjm is the set of predetermined observables and
…xed e¤ects in the demand function, namely (wjm ;
j ; M SA ),
and Payment Rate means
the expected government payment rate per enrollee. zjm is the vector of the independent
variables that explain the …xed part of the pro…ts,
is a parameter vector to be estimated,
and "jm is an idiosyncratic shock to plan j in market m that is observed by all the …rms
ex-ante, but unobserved by the econometrician. V Pjm () denotes variable pro…ts. HMOs are
assumed to incur …xed costs at the county level, which is the sum of the last two terms. For
21
the independent variables, zjm , I choose market-speci…c variables and relatively exogenous or
predetermined plan-speci…c variables, such as non-pro…t status, chain a¢ liation, and years
of experience. Since I regard this …xed cost part as a reduced form, I exclude variables that
are likely to correlate with "jm , such as bene…t coverage.
Before estimating the entry model, I estimate marginal costs. The estimation of marginal
costs for the …rms that actually enter relies on the …rst order condition of the price-setting
game. The …rst order condition in market m can be written as
(5)
sjm (p m ; x m ; ym ) + (pjm + Payment Rate m
M Cjm )
@sjm
= 0;
@pjm
enter
enter
for j = 1; :::; Jm
, where Jm
is the number of actual entrants in county m. This equation
can be solved for M Cjm by writing16
@sjm
@pjm
(6) pjm +Payment Rate m M Cjm =
1
sjm =
1
1
sjjprivate;m
(1
)sjm
:
To calculate M Cjm using (6), I use …tted shares for sjm and sjjprivate;m , instead of observed
shares, under the assumption that
jm
is unknown to plans when they make decisions.17
For hypothetical entrants, the lack of observed pjm and sjm requires M Cjm to be extrapolated. I obtain M Cjm by a reduced-form linear regression, in which I choose independent
variables that are likely to be exogenous or predetermined, such as non-pro…t status, chain
a¢ liation, and market characteristics variables, as listed in Table 5. The result is reported
in Appendix A.
16
While plans have been allowed to charge "negative" premiums in the form of Medicare premium rebates
since 2003, only 0.5% of the plan-county observations charge negative premiums in my data and 30.0% charge
zero premiums. The mass point at zero premiums may re‡ect the administrative costs of charging non-zero
premiums. As long as this censoring occurs symmetrically around zero, the bias is considered to be modest,
if any.
17
Including
jm changes the mean of M Cjm by only 0:2%, so the potential bias of this behavioral
assumption is negligible.
22
4.3
The Estimation Algorithm
I assume the components unobserved to the econometrician to correlate across plans within
each market and to exhibit heteroskedasticity based on the market size, Im . Speci…cally,
(7)
"jm = Im
where !; ; and
!
jm
are parameters to be estimated.
+
jm
m
and
;
m
are distributed i.i.d. standard
normal across plans and markets and assumed to be independent of z m . The correlation of
the unobservable "jm across plans in a given market is then Im
.18 I use Im to control the
variance of the error, since the market size varies signi…cantly.
I employ maximum likelihood estimation. I denote the observed market con…guration as
o
, and the maximum likelihood problem can be written as
ym
bM L = arg max
(8)
where
(
)
M
1 X
o
ln Pr [ym
= ym (x m ; z m ; )] ,
M m=1
is the vector of model parameters, ( ; !; ; ), and ym is the predicted market
con…guration given (x m ; z m ; ; "jm ), which can be solved uniquely by backward induction.
The probability in the likelihood, however, does not have an analytical solution due to
the multi-dimensional integrals, so I use the maximum simulated likelihood method.19 To
make the estimation of the large game computationally feasible, I propose a modi…cation of
the Geweke-Hajivassiliou-Keane (GHK) simulator, one of the smooth recursive conditioning
simulators.20 This simulator provides a di¤erentiable, unbiased estimator of likelihood with
18
Unlike usual discrete choice entry models, normalization for the variance of the error is not required.
The variance is identi…ed because variable pro…ts are unparameterized.
19
In the literature, method of simulated moments (MSM) estimators have been used (Berry 1992, SchmidtDengler 2006). However, this approach is less e¢ cient, especially when the number of playes in a game varies
across markets.
20
For the GHK simulator and related methods, see Geweke and Keane (2001).
23
smaller variance than the crude frequency simulator. The original GHK simulator allows
interactions across j through the disturbance structure, but not strategic interactions across
j. In Appendix B, I modify the simulator to …t my model, by claiming that the simulator can
be dovetailed with sequential-move games by exploiting its recursive conditioning structure.
To solve the game and obtain ym in each likelihood evaluation, the (partial) equilibrium
prices and quantities, p
m
and q m , need to be available for any market con…guration, ym ,
in order to determine the payo¤s. First I calculate p
m
by solving the system of the …rst
order conditions in the price-setting game, (5). Given the estimated demand parameters,
enter
equations
( ; ; ), and the values of M C m ; x m , Payment Rate m , and ym , there are Jm
enter
and Jm
unknowns. Due to the numerical property of the discrete choice model, however,
the price function, pjm (x m ; ym ), does not have a closed form solution. I solve the system
of equations for pjm using the following numerical algorithm. The …rst order condition, (5),
can be written as:
pjm =
sjm (p m ; x m ) (
@sjm (p m ; x m )
)
@pjm
1
Payment Rate m + M Cjm ;
enter
for j = 1; :::; Jm
: The following numerical iteration gives the numerical solution for pjm .
(9)
pt+1
jm =
sjm pt m ; x
m
(
@sjm (pt m ; x m )
)
@pjm
1
Payment Rate m + M Cjm ;
enter
for j = 1; :::; Jm
and t = 0; 1; 2; :::. The observed values of pjm are used for the initial
values of ptjm . It turns out that this numerical iteration converges in an acceptable amount
of time.21 After p
m
is calculated, I can use the estimated demand equation to calculate q m .
21
The demand shock,
jm , is consistently excluded from these calculations. If I use M Cjm calculated
with
,
the
performance
of this numerical iteration is sometimes poor.
jm
24
4.4
Welfare Measures and Simulations
The net welfare gain of having private plans in Medicare is given by
(10)
W = (CSw=PrivatePlans
(Gw=PrivatePlans
CSw=oPrivatePlans )
Gw=oPrivatePlans ) + private plan pro…ts;
where CSx denotes the aggregated consumer surplus attributable to program x, and Gx
denotes government expenditures. I use HMO pro…ts instead of the pro…ts of all private
plans, as discussed above. Following McFadden (1981), annual expected consumer surplus
is derived as:
(11)
CSm
(CSw=PrivatePlans
= Im
12
(1
b
CSw=oPrivatePlans )
2 ent
Jm
bjm +
X
b) ln 4
exp
1 b
j=0
jm
!3
5:
The wealth e¤ect is assumed to be zero. In the dataset, all premiums and payment rates are
de…ned on a monthly basis, so the calculated surplus is multiplied by 12.
Counterfactual simulations use the same framework as above. The only di¤erence is that
I exclude
jm
from all calculations in the simulations, because
jm
can be calculated only
for observed entrants. Due to the convexity of the expression in square brackets in (11),
consumer surplus is understated without
jm ,
even though E
jm
= 0. By the same
token, the predicted number of HMO enrollees, government gain, and HMO pro…ts are also
understated. In the simulations, readers should focus on changes in numbers, not on the
level.
Not only bene…ciaries and suppliers but also the government may enjoy the welfare gain
from the program. This gain comes from per-enrollee cost di¤erence between per-enrollee
25
payments to private plans and expected per capita costs in the traditional Medicare plan that
the government would have to pay without private Medicare plans. Thus, if the government
payment rate is lower than the expected fee-for-service cost in a county, the government expects more gain when more members transfer from traditional Medicare to Medicare HMOs.
I calculate Gw=oPrivatePlans by using demographic data and the average fee-for-service cost of
traditional Medicare. I discount the average cost data by 8.0% to take favorable selection into
account; favorable selection refers to the widely acknowledged phenomenon that Medicare
bene…ciaries who choose to join an HMO are systematically healthier than bene…ciaries who
choose to remain in the fee-for-service sector even after adjusting for the demographic characteristics included in Medicare’s HMO payment methodology (e.g. Mello et al. 2003). GAO
(2000) concludes that Medicare HMO enrollees are an average of 11.7% less costly than other
Medicare eligibles. My choice of 8.0% is fairly arbitrary, but I choose 8.0% instead of 11.7%
for the following reasons: (1) the Medicare HMO market had been shrinking since the years
analyzed by GAO (2000), while the main source of the cost di¤erence is new HMO joiners;
(2) the cost di¤erence for new enrollees has been shrinking over the years (GAO 2000); and
(3) after 1998, the government gradually introduced a payment methodology with more detailed risk adjustments. If 8.0% is misspeci…ed by 1%, for example, the net social welfare
gain is a¤ected by only 2 to 5%. This implies that the following welfare simulation results
and policy implications are robust overall with respect to favorable selection.22
26
Table 6: Nested Logit Demand with Fixed E¤ects
Dependent variable:
ln(sjm ) ln(s0m )
ln sjjprivate;m
monthly premium
bene…t: drug
bene…t: education
bene…t: physicals
bene…t: peripheral1
bene…t: peripheral2
bene…t: screenings
# products in plan
no experience
ln experience year
HMO penet rate 98
year: 2004
constant
Adjusted R2
Nested
without IV
0.639
(.014)
0.0022 (.0008)
0.203
(.062)
0.139 (.150)
0.457
(.149)
0.351
(.195)
0.069
(.113)
0.370
(.196)
0.112
(.021)
0.364 (.077)
0.720
(.195)
1.802
(.137)
0.118
(.040)
4.277 (.288)
0.831
Logit
with IV
0.348
(.055)
0.0105 (.0057)
0.338
(.068)
0.296
(.182)
0.419
(.167)
0.551
(.236)
0.096
(.124)
0.517
(.232)
0.163
(.027)
0.389
(.107)
0.794
(.236)
1.255
(.194)
0.253
(.075)
4.955
(.411)
0.797
The number of observations: 3,289. Standard errors are in parentheses.
5
5.1
Results
Parameter Estimation
Table 6 shows the results of the demand estimation. The signi…cant coe¢ cient on ln sjjprivate;m
implies that the imposed grouping structure is relevant. The results indicate that consumers
are attracted by plans with lower premiums, more bene…ts (except for education bene…ts), more options ("# products in plan"), and longer experience of business in Medicare.
Medicare HMOs have more popularity in the counties where HMOs are more common in the
commercial sector. Table 7 reports price elasticities, marginal costs, and per-capita consumer
surplus. The conventional price elasticity is not de…ned, because charging a non-positive pre22
Another possible concern is that the degree of favorable selection may depend on market share. However,
in the counterfactual simulations below, the simulated changes in market share are generally local. The
aggregate market share of private plans ranges from 8% to 20%, with most shares around 13%, so assuming
a constant discount factor is a reasonable approximation.
27
mium is a common practice. Table 7 shows two alternative measures of price sensitivity:
the semi-elasticity,
jm
point of view,
(pjm + Payment Ratem ).23 The estimated semi-elasticity is somewhat
jm
(@sjm =@pjm ) (1=sjm ) and the price elasticity from the producers’
higher than the estimate of Town and Liu (2004)’s,
0.009. A possible explanation of this
di¤erence may be the huge decline in HMO enrollment between their sample periods and
mine. In this market, those who choose private plans tend to be with less risk and more
price sensitive. It may be the case that those who are more price sensitive remained in the
private plans in my sample years. The latter elasticity is well-de…ned because premiums are
much smaller than payment rates. With values less than
1:0, the estimated elasticity does
not contradict the assumption that …rms maximize pro…t. The table also shows that elderly
people in counties with at least one Medicare HMO enjoy an average consumer surplus of $50
per month, regardless of whether the person joins a private plan or the traditional Medicare
plan. As discussed in the model section, I calculate these values in two ways, depending on
whether I include
jm
or not, i.e. depending on whether I use observed market share or
…tted market share. When
jm
is in the calculation, the values obtained are more accu-
rate, but the values calculated without
jm
are necessary for counterfactual simulations.
The table shows that this does not make a di¤erence in the estimated elasticities, but does
produce some di¤erence in the consumer surplus.
Table 8 shows the results of the entry estimation, in which the dependent variable is
each plan’s entry decision and the estimated equation is the latent pro…t function, (4). Since
variable pro…ts are all computed numerically in the entry game, the estimated coe¢ cients
represent each variable’s contribution to …xed pro…ts. The results suggest that, while a
newcomer in Medicare incurs signi…cant setup costs in each county, the length of business in
Medicare is irrelevant. The results that non-pro…t and local plans face lower …xed costs can be
explained by the view that those plans tend to stay in the market for lower pro…t thresholds,
23
If the semi-elasticity is
0.0124, a $1 increase in the premium is expected to reduce enrollment by 1.24%.
28
Table 7: Price Semi-Elasticity, Marginal Costs, and Consumer Surplus
# obs
Mean
Std. dev.
Own-price semi-elasticity (in 100%)
with
3,289
0.0124
0.0026
jm
w/o
3,289
0.0125
0.0024
jm
Own-price elasticity for private plans (in %)
with
3,289
7.529
2.246
jm
w/o
3,289
7.575
2.167
jm
Marginal costs (monthly in $)
w/o
3,289
517.6
103.0
jm
Consumer surplus (per capita, monthly in $)
with
1,767
53.2
80.3
jm
w/o
1,767
41.1
64.4
jm
Min
Max
0.0161
0.0161
0.0058
0.0075
17.452
16.583
2.841
2.952
196.2
1146.3
0.02
0.09
512.8
396.3
because their optimization behavior involves not only pro…ts but also other factors, such
as quality and quantity of services. This view is consistent with the arguments in Gaynor
and Vogt (2003) that nonpro…t …rms act like for-pro…t …rms with di¤erent cost functions.
The negative coe¢ cient on standardized per capita fee-for-service costs indicates that a
plan incurs additional …xed costs in a county where utilization of medical care is relatively
high. Variables of per capita medical care infrastructure are included to control the costs
of organizing and maintaining the provider network. The results indicate that a plan incurs
the highest cost when there is only one hospital in the county — the case where the plan’s
bargaining power is weakest. While the marginal cost regression suggests that marginal costs
are lower in larger counties, a plan incurs higher …xed costs in larger cities. On the other hand,
being in an MSA is favorable to HMO plans, probably because health providers’bargaining
power is weaker compared to isolated cities, or maybe because the coe¢ cient captures some
economy of scale, which is beyond the scope of my supply side model. The estimated
parameters, ! and , indicate that plan-county-speci…c shocks,
the variance of …xed cost errors and county-speci…c shocks,
estimated value of
m,
jm ,
for 40%. The signi…cantly
rejects the null hypothesis of homoskedasticity.
29
account for 60% of
To con…rm how the order assumption may bias the results, I also estimate the model
using randomized order sequences. The di¤erences in estimated values are nearly as small as
the changes caused by di¤erent simulation draws, while the …t of the model always becomes
worse.
Table 8: Entry Estimation
E¤ects on county level …xed pro…ts in $1,000,000
no experience
1.252
(.241)
experience year
0.006
(.005)
nonpro…t
0.159
(.045)
national chain
0.505
(.064)
group model
0.568
(.196)
IPA model
0.166
(.191)
sta¤ model
0.069
(.209)
Std FFS per capita cost
0.002
(.0002)
per capita # hospital
0.223
(.035)
per capita # hospital bed
0.001
(.004)
per capita # medical doctor
0.114
(.062)
county: no hospital
0.937
(.077)
MSA
0.321
(.078)
county: small
0.554
(.069)
county: medium
1.462
(.095)
county: large
4.430
(.216)
county: extra large
7.334
(.771)
county: huge
12.124
(4.85)
year: 2004
0.091
(.030)
constant
1.158
(.229)
! (plan-county-speci…c error)
0.267
(.017)
(county-speci…c error)
0.216
(.014)
(heteroskedasticity)
0.859
(.014)
The number of markets: 5,075. The number of potential entrants: 19,544.
Standard errors are in parentheses. The number of simulation draws: 40.
30
5.2
Welfare Estimates
Table 9 summarizes the calculated net social welfare gain of the Medicare HMO program.
Adding the three welfare components above results in net welfare gains of 9.04 billion dollars
in 2003 and 10.27 billion in 2004. The majority of the gain comes from producer surplus.
Consumer surplus is much smaller than producer surplus and the government is losing money.
The table also shows the numbers calculated without
jm .
Whether I use …tted market
share or predicted market share a¤ects not only consumer surplus but also HMO pro…ts and
government gain through the change in enrollment.
Table 9: Welfare Results
2003
2004
Consumer surplus
2000 (Town&Liu)
2,806
2,862
(1,362)
(1,366)
HMO pro…ts
7,751
8,027
(210)
(243)
Net government gain
CMS payment to private plans
1,119
371
N/A
34,147 37,825
Expected FFS payment without private plans
33,028 37,454
41,726
Net social welfare gain
9,044
10,267
Consumer surplus w/o
jm
HMO pro…ts w/o
jm
Net government gain w/o
Net social welfare gain w/o
2,262
7,357
1,133
8,486
2,476
7,777
305
9,948
jm
jm
4,061
8,757
N/A
N/A
Annual, in millions of dollars. Standard errors shown in parentheses are
calculated by Monte Carlo simulation over simulation draws and parameters.
My estimates are all smaller than the …gures from 2000 estimated by Town and Liu (2003),
shown in the last column. A drastic decrease in the number of participating HMOs and
enrollees between these years may explain the di¤erences. The decrease in producer surplus
31
is smaller than the decreases in the other two components, possibly because average actual
payments have gradually increased since 1998 (GAO 2000). Overstatement of producer
surplus may be another explanation. Whereas …rms may make their entry decision based
on long-run payo¤, my entry model, as well as Town and Liu (2003), assumes that …rms
consider only the current year’s payo¤s. Under a certain stationarity assumption of the
market, this simpli…cation should be reasonable. However, after the exodus of HMOs around
2001, it is likely that remaining …rms are systematically optimistic about their future pro…ts,
which leads to an upward bias in current-year producer surplus. The assumption of additive
separability across counties may be another source of bias.
5.3
Counterfactual Simulations
The structural estimation strategy with endogenous entry-exit decisions of …rms enables me
to evaluate the impact of payment rate changes on the national social welfare. At the same
time, readers should note the simpli…ed setting used in this study, especially in the discussion
of policy implications in relation to the current payment rate policy. Non-HMO private plans
and Medicare Part D are not included in the simulation framework. I endogenize the entryexit decision but not the decisions on plan characteristics, which implies that …rms make all
the pass-through of additional payments simply through premium changes. The results for
2003 are not shown but are generally consistent with the 2004 results.
Payment Rate Simulations First I perform welfare simulations with four di¤erent payment rates. In the simulations, a change in the payment rate …rst a¤ects the cost structure
of HMOs. They then re-optimize their entry-exit and price decisions by taking the demand
and the rivals’response into consideration, which leads to a new equilibrium. Based on the
recalculated market shares, the welfare components are recalculated.
Table 10 shows the results with four di¤erent payment rates in 2004. A uniform rise of 25
32
Table 10: Payment Rate Simulation: 2004
Payment rate:
$25
$0
Consumer surplus
HMO pro…ts
Net government gain
CMS payments
Expected FFS payment
1,671
7,067
950
27,064
28,013
2,476
7,777
307
35,401
35,094
3,589
8,689
2,237
45,817
43,581
5,121
9,786
5,022
58,388
53,366
8,163
11,683
10,747
73,344
68,994
9,688
9,947
10,041
9,885
9,100
3,813,610
3,270,073
1,712
995
60.6
591.4
4,795,493
4,309,146
1,811
1,094
36.5
615.2
5,952,749
5,516,147
1,895
1,178
14.6
641.4
7,304,955
6,918,159
1,990
1,273
7.0
666.1
9,497,744
9,178,289
2,154
1,437
37.1
699.7
Net social welfare gain
Enrollment (total)
Enrollment (HMO)
# of plan-mkt (total)
# of plan-mkt (HMO)
Avg premium (HMO)
Avg payment rate
+$25
All welfare measures are in millions of dollars, calculated without
+$50
+$85:6
jm :
or 50 dollars in the payment rate leads to a decrease in the average premium and increases
in the enrollment and entrants. This, in turn, increases consumer surplus, producer surplus,
and the government de…cit. Among the …ve di¤erent payment rates, the maximum social
welfare gain is achieved with +$25.
I include the last column as an approximation of the 2008 payment rate. MedPAC
(2008) projects that the 2008 payment to Medicare HMOs is, on average, 112 percent of
the FFS costs. Taking into consideration the fact that MedPAC’s FFS cost estimates are
approximately 2 percentage points lower than CMS’s estimates24 and comparing the average
base payment and standardized FFS cost in my 2004 data (684.8 and 700.4 dollars per
month, respectively), I can roughly approximate the 2008 payment level by increasing the
payment by $85.6.25 According to the CMS data, enrollment in all Medicare private plans
24
This is because MedPAC factors out certain spending related to graduate medical education in teaching
hospitals (Merlis 2007).
25
What I simulate here is only payment rates. The level of health care costs, the number of Medicare
33
was 4.7 million in 2004 and 9.4 million in 2008, which is quite close to the …gures in the table
and indicates that the simulated numbers are consistent with reality. The results show that,
although consumers and …rms bene…t from the 2008 level of subsidy, this level is nevertheless
not justi…able from the social welfare perspective.
Decomposing Welfare Changes Further counterfactual simulations give additional insight into how a change in payment rate a¤ects social welfare. I decompose net social welfare
changes into three components: entry e¤ect, market power e¤ect, and subsidy e¤ect. Entry
e¤ect, the welfare e¤ect due to HMOs’entry-exit response, can be con…rmed by turning the
entry-exit response on and o¤ in a payment simulation. HMOs’ market power may be a
source of social ine¢ ciency. To clarify this point, the changes in entry occurrence and average premium when the payment rate increases by 50 dollars are shown according to type of
county in Table 11. Of the last three columns, the …rst and second show the results with and
without simulated entry, respectively. Each plan sets its premium based on the …rst order
condition, (6), i.e. based on its market power. The table shows that, while monopolists
"bank" approximately 10% of the change in payment rate through increased price-cost margins, when there are nine entrants, most of the payment rate increase is passed to consumers
in the form of premium reductions.26 The last column of the table shows the average payment rate changes made by incumbents accommodating an entrant. The more concentrated
the market is, the more premiums fall as a result of entry.
Subsidizing too much is another possible source of dead weight loss. The per-enrollee
payment from the government to HMOs can be seen as a price subsidy. After the welfare
impact of entry is quanti…ed, I decompose the residual into a market power e¤ect and a
subsidy e¤ect using the following experimental simulation: (1) there is no additional entry,
and (2) each …rm passes all the incremental payment along to its enrollees through premium
eligibles, and other institutional details remain unchanged from 2004.
26
Concerning pass-through behavior of a …rm with a cost advantage, see Besanko et al. (2001).
34
Table 11: Counties Where New Entry Occurs and Average Incumbent Premium Change
When Payments Increase by $50: 2004
Initial number
of plans in
a county
0
1
2
3
4
5 to 7
8 to 9
Total
Number
of
markets
1,918
362
181
49
27
8
8
2,553
Number
of entry
occurrence
110.2
34.9
19.1
7.5
3.0
2.4
2.0
178.9
Entry
occurrence
ratio
5.7%
9.6%
10.5%
15.2%
10.9%
29.6%
24.6%
7.0%
Average change in
premium over all plans
w/ entry w/o entry
46.4
47.6
48.2
48.4
49.4
49.6
44.1
46.8
47.5
48.2
48.8
49.2
Average change
over plans facing
new entry
69.9
55.5
52.4
51.1
51.8
51.1
Average over simulation draws. Non-HMO plans are not included.
reduction.2728
The results of the decomposition are shown in Table 12. The second column shows the net
welfare changes as a result of the payment rate increase. The next three columns show the
three e¤ects, which add up to the total e¤ect. For both +$25 and +$50 cases, subsidy and
entry e¤ects signi…cantly increase consumer surplus by lowering premiums and providing
more choice. The market power e¤ect, in contrast, is relatively small.29 The net welfare
impacts of these three e¤ects di¤er between the two cases. The subsidy e¤ect is positive in
both cases, but larger in the $25 case. On the contrary, the entry e¤ect is negative in both
cases, but larger in the $50 case. As a result, increasing payment rates by 25 dollars has a
positive net welfare impact because of the large subsidy e¤ect, while increasing payment rates
27
I do not assume HMO’s price cost margins to be zero; I assume there is no room for them to exploit
their market power for the $50 payment rate increase.
28
In this decomposition, cross-term e¤ects are included in the entry e¤ect. In other words, the entry e¤ect
here includes not only the pure entry e¤ect but also market power and subsidy e¤ects for new entrants. I
have con…rmed the fact that the e¤ect of this treatment on the results is negligibly small.
29
The reason the market power e¤ect on government gain is positive is that, at this level of payment rates,
it is bene…cial for the government to have fewer HMO enrollees even if it is due to HMOs’price-cost margins.
For the same reason, entry e¤ect increases government de…cits. The entry e¤ect on premiums is positive
because some entry occurs in counties without incumbents, resulting in high, monopoly premiums.
35
Table 12: Welfare Change Decomposition: 2004
Total
e¤ect
2004
+$25
2004
+$50
Total welfare
Cons surplus
Prod surplus
Gov savings
Enrollment
Premium
Total welfare
Cons surplus
Prod surplus
Gov savings
Enrollment
Premium
Subsidy
e¤ect
+94:1
+132:7
+1; 112:4
+949:9
+954:0
+911:8
1; 771:3
1; 930:1
+1; 207; 001
+942; 134
25:0
21:9
61:6
+77:5
+2; 223:6
+2; 644:5
+2; 009:1
+2; 031:6
4; 177:8
4; 715:2
+2; 609; 013 +2; 004; 772
50:0
43:5
Market
power
Entry
e¤ect
13:0
25:5
85:6
+248:1
+27:0
69:2
+45:5
204:4
81; 704 +346; 571
+1:7
+1:3
+8:6
147:6
233:5
+654:4
+78:2
100:8
+163:8
701:2
191; 360 +795; 601
+3:6
+2:8
All welfare …gures are in millions of dollars.
by 50 dollars has a negative net welfare impact because the negative entry e¤ect dominates.
Beyond this level of payment rates, per-enrollee government loss is huge and entry of HMOs
harms social welfare.
Budget Neutral Payment Rate As discussed earlier, the payment rate is intended to
slowly converge toward the FFS cost. I evaluate this policy by setting the base payment
equal to the standardized FFS per capita cost in each county. The county-level correlation
coe¢ cient between the original and the new rates is 0.75. Overall, many urban counties
experience a payment increase, while many rural counties experience a reduction. If I calculate the average of the new payment rate based on the same 2004 enrollment and market
con…guration, it is 2.3 percent higher than the average of the original rate.
Table 13 reports the simulation results. As shown in the …rst and second columns, the
new rate leads to a signi…cant increase in social welfare from 9.9 billion dollars to 12.1
billion, a 21 percent increase compared to the 2004 benchmark case.30 Although the new
30
Conceptually, the FFS rate is a budget-neutral payment rate and should make the net government gain
36
payment rate is only 2.3 percent higher if calculated under the 2004 enrollment and market
con…guration, the simulation results show a remarkable increase in average payment rate
from 615.2 to 684.6 dollars, an increase by more than 10 percent. This is due to the changes
in enrollment and entry decisions of …rms. As a result, consumer surplus and enrollment
increase, which signi…cantly increases the net social welfare gain.
Table 13: Payment Rate Simulation with Fee-For-Service Payment Rate: 2004
Payment rate:
$0
Consumer surplus
HMO pro…ts
Net government gain
CMS payments
Expected FFS payment
Net social welfare gain
Enrollment (HMO)
# of plan-mkt (HMO)
Avg premium (HMO)
Avg Payment Rate
FFS rate
FFS+$20
2,476
7,777
307
35,401
35,094
4,725
8,495
1,153
46,401
45,248
5,886
9,198
2,944
54,862
51,917
8,163
11,683
10,747
73,344
68,994
9,947
12,067
12,140
9,100
4,309,146
1,094
36.5
615.2
5,167,844
1,018
45.5
684.6
6,094,323
1,082
27.59
699.3
9,178,289
1,437
37.1
699.7
All welfare measures are in millions of dollars, calculated without
+$85:6
jm :
This welfare gain is not due to adjustments in the level of payments but rather to restructuring of the payment rate distribution. To clarify this point, I perform payment rate
simulations in which I increase or decrease the FFS rate uniformly across counties. I …nd
that social welfare can be maximized by uniformly adding approximately 20 dollars to the
FFS rate, the results of which are reported in the third column of Table 13. Coincidentally,
the average payment rate in this case is commensurate with the case of the 2008 rate simulation, which is discussed above and shown again in the last column of the table. Despite
zero. However, the results still exhibit some di¤erences between the actual payment rate and the FFS cost
due to technical reasons, in particular the fact that in 2004 the CMS was still undergoing a lengthy transition
to full risk adjustment and favorable selection by HMOs was substantial.
37
Table 14: Counties Where New Entry Occurs When FFS Payment Rate Is Employed: 2004
Initial number of HMO
plans in a county
0
1
2
3
4
5 to 7
8 to 9
Total
Number of
markets
1,918
362
181
49
27
8
8
2,553
Number of
entry occurrence
26.1
7.4
3.9
3.0
0.8
1.3
3.5
45.9
Number of
exit occurrence
71.0
32.2
9.2
9.6
0.3
0.1
122.4
Entry
ratio
1.4%
2.0%
2.2%
6.1%
2.8%
16.4%
44.0%
1.8%
Exit
ratio
Premium
change
19.6%
17.8%
18.8%
35.4%
4.1%
0.6%
4.8%
25.1
22.0
0.3
14.4
27.8
110.1
Average over simulation draws. Non-HMO plans are not included.
the same level of the average payment rates, the optimal case generates signi…cantly greater
social welfare gain with much lower Medicare spending. This …nding illustrates how extensively the current subsidy distorts the incentives for Medicare private plans and harms social
welfare.
How does this payment adjustment bring about such a considerable welfare improvement?
In comparing the FFS rate case with the 2004 benchmark case, the critical di¤erence from
the previous payment rate simulations is the signi…cant decrease in the number of entrants
— from 1,094 to 1,018 — despite the signi…cantly higher average payment. Table 14 details
this point by county type. In large markets, the increase in payments attracts entrants,
while many HMOs exit small markets. In this way, Medicare can exploit the potential
welfare-enhancing ability of HMOs in large markets and avoid unnecessary …xed costs in
small markets. At the same time, as long as an entrant can make positive pro…ts under the
FFS rate, entry occurs even in small markets, which generates additional social welfare.
Naturally this may prompt an equity concern: most of the increase in welfare is generated
in large markets at the expense of small markets. To address this point, I calculate per capita
welfare change by county type. The results, which are shown in Table 15, do not exhibit
such inequality. While the larger counties bene…t greater welfare gain, the change in payment
38
Table 15: Per Capita Welfare Components by Market Type: 2004
Number
of plans in
a county
1
2
3
4
5 to 7
8 to 9
Total
Welfare
85:4
216:9
342:5
475:0
891:9
1; 377:9
2004 rate
Cons
Prod
Surplus Surplus
26:7
77:4
109:8
168:8
232:3
178:6
108:0
218:3
288:7
429:7
633:9
781:2
Total
Welfare
FFS payment rate
Cons
Prod
Surplus Surplus
Gov
Savings
49:2
122:2
78:8
239:0
55:9
368:7
118:5
509:7
25:7
942:7
418:1 1; 853:3
35:5
108:2
71:9
212:8
120:2
298:8
128:8
422:8
334:5
684:1
911:5 1; 007:5
21:5
45:7
50:3
41:8
75:9
65:6
Gov
Savings
Number of plans is based on numbers before simulation.
rates enhances per capita consumer surplus in small markets as well.
Simulating Entry Is the entry level excessive? To answer this question, I perform entry
simulations using the FFS rate case as a benchmark. Changing the level of entry is not
straightforward because the simulation results may depend on which plan is included. I
determine several di¤erent lists of plans using the results of the payment rate simulations
with di¤erent payment rates uniformly changed from the FFS rate. The results, presented in
Table 16, suggest that there is no evidence of excessive entry in this market. The benchmark
FFS entry level lies near, or marginally below, the optimum.
Table 16: Entry Simulation: 2004
HMO entry:
55
Consumer surplus
HMO pro…ts
Net government gain
CMS payments
Expected FFS payment
4,549
8,559
1,091
44,649
43,557
Net social welfare gain
12,017
+64
+132
+207
4,725
8,495
1,153
46,401
45,248
4,889
8,378
1,216
48,036
46,820
5,042
8,212
1,274
49,609
48,334
5,162
8,023
1,321
50,843
49,521
12,067
12,051
11,980
11,864
FFS Benchmark
All welfare measures are in millions of dollars, calculated without
39
jm :
6
Conclusion
In this paper, I develop an econometric model to examine welfare consequences of policy
changes through entry, and I take as a case study the US Medicare HMO market. The
innovation of this paper is the explicit treatment of …rm heterogeneity in the welfare analysis
with endogenous entry. It helps exploit detailed …rm level data and allows relevant policy
simulations, which is beyond the scope of previous studies but important in the analysis of
the Medicare HMO market. I …nd no evidence of excessive entry in terms of social welfare
and I show that aligning the payment rate with the FFS cost may lead to a signi…cant gain
in national welfare.
Appendix
A.1 Additional Tables
Table 17 shows the results of the …rst stage regression in the demand estimation. In general,
the instruments have reasonable coe¢ cients and signi…cances. Table 18 shows the results
of the linear regression for marginal costs. The estimated coe¢ cients should be interpreted
with caution, since the regression is a reduced form and serves only for extrapolation.
A.2 Applying the GHK Simulator
In this appendix, I follow the customary notation and change the subscript for a market from
m to i = 1; :::; N , since each market is the unit for which the individual likelihood is de…ned.
For a vector of indices (1; :::; J), the notation "< j" denotes the subvector (1; :::; j
1), "
j"
denotes the subvector (1; :::; j), and " j" denotes the subvector that excludes component j.
In the sequential-move game, the order of subscripts for …rms (1; 2; :::; Ji ) comprises the
reverse of the decision order in market i — …rm Ji makes a decision …rst, …rm 1 makes a
40
Table 17: The First Stage IV Regression
ln sjjprivate;m
Dependent variable:
avg # competitor
competitor: npo
competitor: chain
competitor: IPA
competitor: group
competitor: sta¤
per capita # hospital
per capita # hospital bed
county: no hospital
per capita # medical doctor
R2
F -test (distributed F (10; 2; 987))
0.063
(.068)
0.207
(.056)
0.022
(.057)
0.547
(.058)
0.269
(.033)
0.596
(.118)
2.067
(.617)
0.003
(.004)
0.012
(.042)
0.156
(.094)
0.770
43.98
***
***
***
***
***
*
***
premium
0.446
(1.189)
5.404 ***
(1.03)
6.423 ***
(1.18)
0.989
(1.03)
0.110
(.601)
3.127
(1.98)
2.783
(10.56)
0.007
(.072)
0.305
(.841)
0.825
(1.50)
0.921
7.68
***
The number of observations: 3,289. *, **, and *** denote p<.1, p<.05, and
p<.01, respectively. Heteroskedasticity consistent standard errors are used.
Other independent variables and …xed e¤ects are used but not reported.
decision last, and so on. Firm j’s strategy in market i is represented by yj;i ; a binary variable
that takes "1" if the …rm enters and "0" otherwise. To simplify the notation, I omit the
di¤erence between the independent variables in the demand and the …xed pro…ts, xjm and
zjm , and let Xji denote the set of independent variables. The pro…ts of entering …rm j in
market i are:
(12)
ji (X i ; "ji ; y j;i ;
)
V P (yi ; X i ) + Xji + "ji :
41
Table 18: Marginal Cost Regression
Dependent variable:
copayments
entire enrollees
no experience
experience year
nonpro…t
national chain
group model
sta¤ model
IPA model
DEMO plan
cost plan
cost missing
PPO plan
PFFS plan
PSO plan
POS
demo factor HMO
risk factor HMO
Std FFS per capita cost
medigap premium
ln per capita # hospital
ln per capita # hospital bed
ln hospital expenditure
impatient days
per capita # medical doctor
county: no hospital
HMO penetration rate 98
eligibles
year: 2004
constant
R2
M C jm
0.108
0.082
18.30
0.343
18.06
18.51
39.57
29.92
31.52
31.62
35.39
147.8
11.69
24.29
33.40
31.35
346.0
149.0
0.332
0.396
2.228
4.588
6.928
0.033
19.60
22.21
68.82
0.090
33.05
165.3
(.023)
(.011)
(3.71)
(.226)
(2.41)
(2.29)
(4.19)
(5.69)
(4.65)
(6.72)
(4.36)
(8.82)
(7.43)
(4.61)
(12.6)
(4.36)
(21.3)
(9.97)
(.014)
(.046)
(1.84)
(2.16)
(2.24)
(.012)
(6.25)
(5.47)
(8.67)
(.031)
(2.12)
(27.4)
0.830
The number of observations: 3,289. Standard errors are in the
parentheses. Heteroskedasticity consistent standard errors
are used. MSA …xed e¤ects are included in the estimation.
42
A subgame perfect Nash equilibrium (SPNE) is obtained when (1) all entering …rms are
pro…table with their optimal prices, and (2) all …rms that do not enter expect non-positive
pro…ts from entry. Formally, an SPNE strategy in market i, (yie ); is any strategy that satis…es:
(13)
ji
(14)
ji
X i ; "ji ; y e j;i
0; if …rm j enters,
e
e
X i ; "ji ; y>j;i
; y<j;i (y>j;i
; yj;i = 1)
0; if …rm j does not enter,
for all j = 1; :::; Ji ; where y<j (y j ) is the solution to the downstream subgame, i.e. the best
responses of the downstream players given X; "<k ; and the upstream players’strategies. The
unique equilibrium solution always exists. Denote this solution, after dropping index i, as
y (X; "; )
fy is the unique solution in the sequential-move game with X; "; g:
The component unobserved to the econometrician is speci…ed as
(15)
ji
and
"ji = !
i
ji
+
i.
are assumed to be independent of X i , and distributed i.i.d. standard normal across
…rms and markets.31 The heteroskedasticity adjustment in (7) is dropped for simplicity of
explanation. This simpli…cation does not a¤ect the argument in this appendix.
The log likelihood function can be written as
(16)
31
bM L = arg max
(
)
N
1 X
ln Pr [yio = yi (X i ; )] ,
N i
Unlike the previous studies, no normalization such as ! 2 + 2 = 1 is necessary because the level of pro…ts
is identi…ed. Moreover, this speci…cation of the error components is not crucial for the argument below.
43
where yio is the observed market con…guration and
is the vector of parameters, ( ; !; ).
The probability expression in the likelihood does not have an analytical solution because of
the multi-dimensional integrals involved. Thus, unless the dimension of the unobservables is
very small, the numerical approximation is not feasible. Hence, I use the maximum simulated
likelihood (MSL).
The Modi…ed GHK Simulator The most straightforward simulator for MSL is the
crude frequency simulator. However, such simple discontinuous simulators require many
random draws and are not feasible in practice. My dataset has at most 16 players in some
markets, and the use of the backward induction technique makes each likelihood evaluation
computationally expensive.
I propose the use of the GHK simulator. The GHK simulator is a smooth recursive
conditioning simulator and is often useful when the log-likelihood function involves highdimensional integrals with a multivariate normal distribution. The GHK algorithm draws
recursively from truncated univariate normals. It relies on the decomposition,
f (v1 ; :::; vJ ) = f (v1 )f (v2 jv1 ):::f (vJ
1 jvJ 2 ; :::; v1 )f (vJ jvJ 1 ; :::; v1 );
along with the fact that the conditional normal density can be written as a univariate normal.
The GHK simulator produces probability estimates that are bounded away from 0 and 1.
The estimates are continuous and di¤erentiable with respect to parameters, because each
contribution is continuous and di¤erentiable. It is also an unbiased estimator of individual
likelihood, l( ; !; ; yio ; X i ). It has a smaller variance than the crude frequency simulator,
because each element is bounded away from 0 and 1.
The use of the GHK simulator in a game-theoretic situation, however, is not straightforward, because the original GHK simulator can deal with interactions across j through the
44
disturbance structure, but not strategic interactions across j. The use of the GHK simulator in this study relies on the sequential-game assumption. In a sequential-move game,
the prior players’decisions are known to a player. This fact harmonizes the estimation of a
sequential-move game with recursive conditioning simulators, as shown below.
The GHK simulator exploits the Cholesky triangularization to decompose the multivariate normal into a set of univariate normal distributions. The multivariate normal disturbance
vector, "i , can be rewritten as: "i =
standard normal variates,
i
where
N (0; IJi +1 ), and
i
Thus, "i can be rewritten as: "i
i i,
is a (Ji + 1)
i
i
1 vector of independent
(Ji + 1) parametric array,32
is a Ji
2
3
0 7
6!
6
7
...
6
7
6
7
=6
:
.. 7
...
6
7
.7
6
4
5
0
!
N (0;
i ),
where
i
is a positive de…nite matrix,
i
=
0
i i.
It follows that "i can be written by using the Cholesky decomposition as:
(17)
"i = L( i ) vi ;
where L( ) is the lower-triangular Cholesky factor of
multivariate standard normal vector, vi
, or LL0 =
N (0; IJi ).
The individual likelihood can be written, after dropping index i, as
l( ; y o ; X) = Pr [y o = y (X; ; !; )]
Z
=
n("; )d":
y o =y (X;"; )
32
More ‡exible models can be dealt with by changing
45
i
and the size of
i.
, and vi is another
This expression involves multiple integrals, which are hard to compute in a straightforward
way. The general objective here is to obtain random draws from the distribution "i subject
to y o = y (X; ; !; ). The probability expression that explicitly expresses the rectangle in
which the event, y o = y (X; ; !; ), occurs can be rewritten as:
8
3
>
< > 0 if y o = 1
j
7
6
o0
o0
; 1)0 ); )
Pr [y o = y (X; ; !; )] = Pr 4for 8j, j (Xj ; "j ; (y>j
; y<j (y>j
5:
>
: 0 if yjo = 0
2
By de…ning
8
>
< a = 0; b = 1 if y o = 1
j
j
j
>
: a = 1; b = 0 if y o = 0
j
j
j
the probability can be written as:
9
>
=
>
;
;
(18)
Pr [y o = y (X; ; !; )] = Pr[for 8j, aj (yjo )
o0
o0
0
j (Xj ; "j ; (y>j ; y<j (y>j ; 1) );
)
bj (yjo )]:
Using the form of the pro…t function, (12), and de…ning
aj
aj
Xj
o
o
V P ((y>j
; y<j (y>j
; 1)); X)
bj
bj
Xj
o
o
V P ((y>j
; y<j (y>j
; 1)); X);
(18) can be rewritten as
Pr [y o = y (X; ; !; )] = Pr[for 8j, aj (y o ; X; "<j ; )
"j
bj (y o ; X; "<j ; )]:
This expression shows us the rectangle in which the event, y o = y (X; ; !; ), occurs. To
obtain the interval of "j , we only need "<j . This is because the upstream …rms’decisions are
46
given for …rm j. The Cholesky decomposition (17) makes this expression:
(19)
Pr [y o = y (X; ; !; )] = Pr[for 8j, aj (y o ; X; v<j ; ; )
L( ) v
bj (y o ; X; v<j ; ; )]
or
Z
o
Pr [y = y (X; ; !; )] =
for 8j, aj (y o ;X;v<j ; ; ) L( ) v bj (y o ;X;v<j ; ; )]
"
#
J
Y
(vj ) dv;
j=1
where () is the probability density function of standard normal.
Now we are ready to apply the GHK simulator. For each time of simulation draws,
a vector of independent uniform (0; 1) random variates, (u1 ; :::; uJi ), is drawn. De…ne the
following function:
(20) q(u; a; b)
1
( (a) (1
u) + (b) u) , where 0 < u < 1 and
1
a<b
1:
This function, q( ), is a mapping that takes a uniform (0; 1) random variate into a truncated
standard normal random variate on the interval [a; b].
For given y o ; X; u; ; L, it is possible to recursively de…ne for j = 1; :::; J:
ve1
ve2
veJ
a1 b1
;
L11 L11
a2 (e
v1 ) L2;1 ve1 b2 (e
v1 ) L2;1 ve1
q u2 ;
;
L22
L22
q u1 ;
:::
q uJ ;
aJ (e
v<J
1)
LJ;1 ve1 :::
LJJ
LJ;J
eJ 1
1v
47
;
bJ (e
v<J
1)
LJ;1 ve1 :::
LJJ
LJ;J
eJ 1
1v
and
Q1
Pr
Q2
Pr
:::
QJ
Pr
a1
b1
v1
L11
L11
a2 (e
v1 ) L2;1 ve1
L22
aJ (e
v<J
1)
v2
LJ;1 ve1 :::
LJJ
b2 (e
v1 ) L2;1 ve1
L22
LJ;J
eJ 1
1v
vJ
bJ (e
v<J
LJ;1 ve1 :::
LJJ
1)
LJ;J
eJ 1
1v
Given all the a; b; L; and ve, every Qj is truncated univariate standard normal, so Qj can be
calculated by, for example,
a1
L11
b1
L11
Q1 =
:
Based on repeating this simulation R times, the likelihood contribution simulator is de…ned
as
1 XY
Qj (e
v1r ; :::; veJ
R r=1 j=1
R
(21)
e
l( ; ; y o ; X; R; u)
J
1;r ):
The model is estimated by solving the following maximum simulated likelihood problem:
(22)
bM SL = arg max
= arg max
(
(
)
N
1 X e
ln l( ; ; yio ; X i ; R; ui )
N i
N
R
J
1 X 1 XY
ln
Qj (e
v1r ; :::; veJ
N i
R r=1 j=1
)
1;r )
.
In the computation, I use the Quasi-Newton method with BFGS updating algorithm for
the maximization routine. In making the random draws for the simulator, I use antithetics
to reduce simulation variance and bias. For more details, see Maruyama (2007).
48
:
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51