Volume 7, Number 3, May 2003
CONTENTS
WHAT DIFFERENCE DOES THE CHOICE OF SES MAKE IN HEALTH
INEQUALITY MEASUREMENT?
Adam Wagstaff and Naoko Watanabe .. .......... .......... ......... .......... ..........3
SIMULATION OF A HIRSCHMAN-HERFINDAHL INDEX WITHOUT
COMPLETE MARKET SHARE INFORMATION
Eric Nauenberg, Mahdi Alkhamisi and Yuri Andrijuk .... ......... .......... ..........9
MANAGED CARE AND SHADOW PRICE
Ching-to Albert Ma . .......... .................... .......... .......... ......... .......... ..........17
Professor W. David Bradford
Medical University of South
Carolina
Department of Health
Administration and Policy
19 Hagood Ave., Suite 408
P.O. Box 250807
Charleston, SC 29425
USA
E-mail: [email protected]
Dr. James F. Burgess, Jr.
Department of Veterans Affairs
Management Science Group
200 Springs Road, Bldg. 12
Bedford, MA 01730
USA
E-mail: [email protected]
Professor Andrew Jones
Director of the Graduate Programme
in Health Economics
Department of Economics and
Related Studies
University of York
Y010 5DD
UK
E-mail: [email protected]
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HEALTH ECONOMICS
HEALTH ECONOMICS LETTERS
Health Econ. (in press)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/hec.805
What di¡erence does the choice of SES make in health
inequality measurement?
Adam Wagstaffa,b,* and Naoko Watanabec
a
Development Research Group and Human Development Network, The World Bank, Washington DC, USA
School of Social Sciences, The University of Sussex, Brighton, UK
c
Development Data Group, The World Bank, Washington DC, USA
b
Summary
This note explores the implications for measuring socioeconomic inequality in health of choosing one measure of
SES rather than another. Three points emerge. First, whilst similar rankings in the two the SES measures will result
in similar inequalities, this is a sufficient condition not a necessary one. What matters is whether rank differences are
correlated with health – if they are not, the measured degree of inequality will be the same. Second, the statistical
importance of choosing one SES measure rather than another can be assessed simply by estimating an artificial
regression. Third, in the 19 countries examined here, it seems for the most part to make little difference to the
measured degree of socioeconomic inequalities in malnutrition among under-five children whether one measures
SES by consumption or by an asset-based wealth index. Copyright # 2003 John Wiley & Sons, Ltd.
Keywords
health inequality; socioeconomic inequality in health; socioeconomic health differentials
Introduction
The literature on socioeconomic health inequalities examines the distribution of health by some
measure of socioeconomic status (SES), the question of interest being the degree to which persons
with lower SES are more likely to suffer from ill
health and die early. A variety of different
measures of SES have been used, including social
(or occupational) class [1], educational attainment
[2], income [3], dwelling size [4], consumption [5]
and ownership of certain household assets as
reflected in a ‘wealth’ index [6]. For some
purposes, it may be of interest to know whether
the choice of SES indicator makes a difference to
the measured degree of socioeconomic inequality
in health. For example, given the relative ease with
which asset data can be collected compared to
consumption data [7], one might ask whether using
assets rather than consumption makes much of a
difference to the degree of measured socioeconomic inequalities in health.
This paper sets out a framework for comparing
health inequalities measured using different measures of SES, and develops a simply implemented
statistical test that enables the analyst to determine
whether the difference is statistically important.
The approach is illustrated using anthropometric
data on child malnutrition in 19 developing
countries. Two measures of SES are employed –
household consumption and an asset-based wealth
index.
Some theory
Suppose we have a scalar measure of health that is
decreasing in good health. The measure might be
*Correspondence to: The World Bank, 1818 H Street NW, Washington, DC 20433, USA. E-mail: awagstaff@worldbank.org
Copyright # 2003 John Wiley & Sons, Ltd.
Received 29 August 2002
Accepted 27 November 2002
A.Wagsta¡ and N.Watanabe
the presence or absence of chronic illness, beddays, malnutrition or even death. Suppose we have
two alternative scalar measures of SES, SES1
and SES2, both increasing in SES. If we rank
individuals by, say, SES1, beginning with the most
disadvantaged, and graph on the x-axis the
cumulative proportion of individuals ranked by
SES1 and on the y-axis the cumulative proportion
of ill health, we obtain the concentration curve for
ill health [8,9]. This will lie above the diagonal or
‘line of equality’ if the disadvantaged suffer from
higher levels of ill health. If the concentration
curve for SES1 lies further from the diagonal than
the curve for SES2, then there is more health
inequality by SES1 than there is by SES2.
Twice the area between the line of equality and
the concentration curve equals the concentration
index, C [8,9], which is our measure of socioeconomic health inequality. This is negative in the
case where ill health is more common among the
more disadvantaged, zero if the concentration
curve coincidences with the diagonal, and positive
if ill health is more common among the better off.
In the case of SES1, C can be written [9]:
C1 ¼
n
2 X
hi r1i 1;
nm i¼1
ð1Þ
where C1 is the concentration index for SES1, n is the
sample size, m is the mean of the ill health variable, h,
and r1i is person i’s fractional rank in the SES1
distribution (r1i=1 being the fractional rank of the
person with the highest SES). A similar expression
can be written down for C2 (the concentration index
for the SES2 measure) by replacing r1 by r2, the
fractional rank in the SES2 distribution. By comparing C1 and C2 we can see how much more (or less)
socioeconomic health inequality there is when we
rank by SES1 rather than by SES2. If the difference
between C1 and C2 is small, the choice of SES
measure makes little difference to the measured
degree of socioeconomic health inequality.
Under what circumstances will C1 and C2 be the
same? Using Equation (1) and the analogue for
SES2, we can write:
2 X
2 X
hi r1i 1 hi r2i þ 1
C1 C2 ¼
nm i
nm i
2 X
¼
hi Dri
nm i
2
¼ covðh; DrÞ
m
Copyright # 2003 John Wiley & Sons, Ltd.
ð2Þ
where Dri=r1ir2i is the difference between the two
fractional rank variables, which has a zero mean.
So, C1 and C2 will be equal if the rankings in the
two distributions coincide (i.e. Dri=0 for all i). But
this is a sufficient condition, not a necessary one.
C1 and C2 will be equal if health and the rank
difference do not covary – in other words, people
may not occupy the same position in the two SES
distributions, and yet if the rank differences are not
correlated with health, socioeconomic inequalities
in health will be the same.
Equation (2) also provides the basis for a simple
statistical test to see whether the difference between
C1 and C2 is important statistically. In much the
same way as the concentration index itself can be
computed easily by means of a convenient regression of health on the fractional rank [9], so too can
the concentration index difference be computed by
means of a simple artificial regression:
2varðDrÞ
hi ¼ a þ bDri þ ei :
ð3Þ
m
The left-hand side is individual i’s ill-health score,
multiplied by twice the variance of the rank
difference variable and divided through by the
mean of the ill-health variable. The coefficient b is
equal to C1C2, and the standard error of b allows
one to test the significance of the difference
between the two concentration indices. This is
valid for small and large samples. Strictly speaking, the expression for Cj (j=1,2) has an additional
term on the right-hand side, equal to 1/n, where n
is the sample size [10]. This tends to zero as n
increases and in any case cancels out in the
difference in Equation (2). The testing procedure
lends itself to a comparison of two alternative
measures of SES. Where more than one measure
of SES is being explored, one could use a
sequential testing procedure, but it is possible that
there may be no significant differences between any
of the measures, and it seems possible that the
results may not even be transitive.
Empirical illustration
The literature to date on socioeconomic inequalities in health in the developing world has focused
for the most part on maternal and child health, in
part because of the wealth of data in this area, but
also because of the objectiveness of data on
child mortality and anthropometrics (malnutrition
Health Econ. (in press)
SES and Health Inequality Measurement
measures based on weight, height and age measurements). A key source of data has been
USAID’s Demographic and Health Survey
(DHS), which has been fielded in over 50 countries.
The DHS, unlike the World Bank’s Living
Standards Measurement Survey (LSMS), does
not have a measure of household consumption
[11], and in its absence an asset-based wealth index
has been developed [12,13] and has been used by
the World Bank to date to generate data on health
inequalities in 42 countries from the DHS [6]. One
issue that has arisen – and which is explored below
– is whether it makes a difference whether child
health inequalities are measured across consumption groups or across wealth groups.
Data
Our data are from the 19 multipurpose LSMS-type
household surveys listed in Table 1 – for further
details see [14]. Not all are nationally representa-
tive. We included only children under the age of
five years. The sample size is after deletion of cases
with missing values for any of the variables used in
the analysis. We measure child health by two
binary variables indicating whether the child is
underweight (low weight for age) or stunted (low
height for age). These are obtained, as is common
practice in anthropometry [15], by comparing the
child’s weight-for-age and height-for-age with a
hypothetical population of well-nourished children
assembled by the US National Center for Health
Statistics (NCHS). Children with a z-score below
2 (using the NCHS mean and standard deviation
as references) are classified as underweight and
stunted.
Our two measures of SES are equivalent household consumption and an asset-based wealth
index. Consumption is a better measure of living
standards than income or expenditure, since it
captures what households consume whether or not
they purchase it or produce it themselves, and
whether they finance it through current, future or
Table 1. Survey details
Country
Survey name
Survey year
Bangladesh
Matlab Health and
Socioeconomic Survey
Presquisa sobre Padro* es
de Vida
China Health and Nutrition
Survey
LSMS
Egypt Integrated
Household Survey
LSMS
Guatemalan Survey of
Family Health
LSMS
Indonesian Family Life
Survey
LSMS
LSMS
LSMS
LSMS
LSMS
Cebu Longitudinal Health
and Nutrition Survey
1996
1506
1995–1996
1693
1991
861
1988
1997
1090
1426
1987–1988
1995
2349
2794
1992–1993
1993
589
1236
1990–1991
1996
1993
1991
1994
1991
2121
1596
3284
3773
2093
2031
1996
1993
1992–1993
1996
3740
3961
2622
4483
Brazil
China
Co# te d’Ivoire
Egypt
Ghana
Guatemala
Guyana
Indonesia
Morocco
Nepal
Nicaragua
Pakistan
Peru
Philippines
Romania
South Africa
Vietnam
Zambia
LSMS
LSMS
LSMS
Living Conditions Monitoring
Survey I
Copyright # 2003 John Wiley & Sons, Ltd.
N
Comments on survey
Covers only a rural region of Matlab,
located to south of Dakha.
Covers only south-east and north-east.
Eight provinces covered including urban
and rural areas.
Covers 4 departments (out of 22).
Covers 13 provinces, representing 83 %
of the population.
Survey area is city of Cebu and
surrounding area – the regional center
of Central Visayas region.
Health Econ. (in press)
Copyright # 2003 John Wiley & Sons, Ltd.
4.91
4.58
3.41
2.52
0.78
5.19
6.78
4.26
3.14
8.39
8.29
8.27
6.00
8.72
4.31
2.43
7.19
5.24
9.44
0.067
0.245
0.151
0.099
0.034
0.105
0.106
0.201
0.062
0.251
0.121
0.245
0.066
0.308
0.107
0.088
0.141
0.068
0.155
Bangladesh
Brazil
China
Côte d’Ivoire
Egypt
Ghana
Guatemala
Guyana
Indonesia
Morocco
Nepal
Nicaragua
Pakistan
Peru
Philippines
Romania
South Africa
Vietnam
Zambia
0.037
0.218
0.043
0.061
0.101
0.054
0.050
0.055
0.071
0.259
0.107
0.255
0.066
0.299
0.158
0.067
0.139
0.067
0.168
C2
2.72
3.52
1.07
1.65
2.17
2.63
3.18
1.04
3.58
8.82
7.17
9.08
5.96
8.40
6.35
1.94
7.14
5.09
10.24
t-value
Wealth index
0.031
0.027
0.107
0.043
0.069
0.051
0.053
0.145
0.009
0.008
0.015
0.010
0.001
0.008
0.052
0.021
0.002
0.002
0.012
C1C2
1.68
0.48
2.03
1.05
1.42
1.94
2.39
2.48
0.42
0.27
0.84
0.41
0.13
0.27
2.39
0.46
0.09
0.11
0.86
t-value
Difference
0.049
0.193
0.140
0.106
0.039
0.094
0.079
0.146
0.076
0.185
0.065
0.227
0.077
0.281
0.191
0.051
0.199
0.088
0.101
C1
3.40
5.80
5.05
2.89
1.10
5.24
8.84
2.20
4.32
9.23
4.45
12.83
6.74
16.10
6.07
3.01
12.97
8.23
11.42
t-value
Equivalent
consumption
0.035
0.150
0.035
0.027
0.102
0.082
0.041
0.125
0.071
0.211
0.062
0.267
0.084
0.254
0.181
0.038
0.170
0.073
0.103
C2
2.44
4.05
1.29
0.81
2.91
4.70
4.70
1.85
4.03
10.88
4.13
15.80
7.42
14.30
5.98
2.26
11.22
6.68
11.60
t-value
Wealth index
Stunting
0.014
0.043
0.106
0.090
0.066
0.013
0.036
0.022
0.005
0.027
0.004
0.039
0.007
0.025
0.005
0.013
0.028
0.015
0.002
C1C2
0.72
1.27
3.03
2.36
1.79
0.56
2.95
0.27
0.28
1.41
0.22
2.59
0.61
1.50
0.18
0.62
2.18
1.24
0.27
t-value
Difference
Notes: C1 and C2 are concentration indices for the consumption and wealth SES measures respectively. The t-values for the indices are relevant to testing the hypothesis
that the indices are zero and are derived from Newey–West standard errors that correct for the serial correlation induced by the fractional rank variable in the artificial
regressions. The difference C1C2 is estimated using (3), the t-value in this cases relevant to testing the hypothesis that C1 and C2 are the equal to one another.
t-value
C1
Country
Equivalent
consumption
Underweight
Table 2. Inequalities in malnutrition by consumption and wealth
A.Wagsta¡ and N.Watanabe
Health Econ. (in press)
SES and Health Inequality Measurement
past income [11]. We used pre-computed consumption aggregates except in the cases of
Guatemala, the Philippines and Zambia, where
we computed our own using as, far as possible,
standard LSMS methodology [11,16]. In the case
of Guatemala, the consumption data were somewhat limited, and in the case of China we had to
make do with income data. We took into account
household size using an equivalence scale equal to
the square root of household size. This is
equivalent to raising household size to an elasticity
power (e) equal to 0.5, this being an intermediate
position between the assumption that there are no
economies of scale in household consumption (it
costs two people twice as much to live as one, or
e=1) and the assumption that two can live as
cheaply as one (e=0) [16,17]. Our asset-based
wealth index is a linear combination of a variety of
indicators of household living standards, such as
ownership of various household durables (e.g.
radio, refrigerator, TV, and motorcycle), whether
the household has electricity, the number of rooms
per person, whether the floor is finished, the type
of drinking water and sanitation, and so on. The
weights used are the first component from a
principal components analysis of the wealth
indicator data [12,13], this being the linear
combination that maximizes the variance in the
observed indicators.
Methods
The concentration indices C1 and C2 were computed by means of an artificial regression of the
malnutrition variable (multiplied by twice the
variance of the fractional rank variable divided
by mean malnutrition) on the fractional rank
variable [9]. The Newey–West [18] estimator was
used to correct standard errors for the autocorrelation induced by the fractional rank variable [9].
Estimates of C1C2 and their standard errors were
obtained directly by using OLS to estimate
Equation (3).
Results
The concentration indices and their t-statistics in
Table 2 indicate that however SES is measured
inequalities in both underweight and stunting
significantly disfavor poor children in almost all
countries (the indices are negative and are mostly
Copyright # 2003 John Wiley & Sons, Ltd.
significantly different from zero). The exceptions
are Egypt in the case where children are ranked by
equivalent consumption and China in the case
where they are ranked by the wealth index. With
the exception of Morocco, it is in the Latin
American countries where socioeconomic inequalities in malnutrition are most pronounced.
Of more interest in the present context are the
differences between the consumption-based and
wealth-based concentration indices. On average,
inequalities in malnutrition are larger (in absolute
size) by equivalent consumption than by wealth,
but the difference between C1 and C2 is, on
average, reasonably small – 12–14% of the average
concentration index. Furthermore, of the 38
differences between C1 and C2, fewer than one
quarter are significant at the 95% level. Thus in
this particular application, and for this particular
set of countries (or at least surveys), the balance of
probability is that it does not make a significant
difference to the estimated magnitude of socioeconomic inequalities in health whether one uses
one measure of SES (consumption) or the other
(wealth).
Conclusions
The aim of this note has been to explore the
implications for measured socioeconomic inequalities in health of choosing one measure of SES
rather than another. Three points seem worth
emphasizing. First, whilst similar rankings in the
two the SES measures will result in similar
inequalities, this is a sufficient condition not a
necessary one. What matters is whether rank
differences are correlated with health – if they are
not, the measured degree of inequality will be the
same. Second, the statistical importance of choosing one SES measure rather than another can be
assessed simply by an artificial regression along the
lines discussed in the paper. Third, in the 19
countries examined here, it seems for the most part
to make little difference to the measured degree of
socioeconomic inequalities in malnutrition among
under-five children whether one measures SES by
consumption or by an asset-based wealth index.
Acknowledgements
Without wishing to incriminate them in any way, we are
grateful to Eddy van Doorslaer for helpful discussions in
Health Econ. (in press)
A.Wagsta¡ and N.Watanabe
the course of this work, and to an anonymous referee for
helpful comments on an earlier version of the paper. The
findings, interpretations and conclusions expressed in
the paper are entirely those of the authors, and do not
necessarily represent the views of the World Bank, its
Executive Directors, or the countries they represent.
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Health Econ. (in press)
HEALTH ECONOMICS
HEALTH ECONOMICS LETTERS
Health Econ. (in press)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/hec.814
Simulation of a Hirschman^Her¢ndahl index without complete
market share information
Eric Nauenberga,c*, Mahdi Alkhamisib and Yuri Andrijukd
a
Department of Economics, State University of New York at Buffalo, USA
Department of Statistics, University of Toronto, Canada
c
Department of Health Policy, Management, and Evaluation, University of Toronto, Canada
d
Statistical Research and Consulting, East Amherst, NY 14051, USA
b
Summary
This paper utilizes maximum likelihood methods to simulate a Hirschman–Herfindahl index (HHI) for markets in
which complete market share information is unavailable or delayed. Many jurisdictions either may be unable to
administratively collect data or experience delays in collection that make data regarding turbulent markets of limited
use. With the development of this method, regulatory authorities monitoring health-care competition or health-care
firms can now use market surveys – in which reliable recall is often limited to the largest three or four firms – to
produce an on-the-spot measure of market concentration. Copyright # 2003 John Wiley & Sons, Ltd.
Keywords
industrial organization; statistical methods; economics of information; Hirschman–Herfindahl index
Introduction
The Hirschman–Herfindahl index (HHI) is a
widely used measure of market concentration
[1,2]. As the sum of the squares of each firm’s
market share, the HHI incorporates information
regarding both the number of firms in the market
and the distribution of market share within the
industry. This index provides a concise and
informative summary of market concentration,
and as such, the US Department of Justice has
adopted this index for reviewing merger cases [3].
There are also regulatory authorities recently set
up in many jurisdictions to monitor health-care
competition and health-care firms that potentially
could make use of this index in measuring market
concentration. However until now, the utility of
this index has been limited to instances in which
complete market share information has been
available. We extend the utility of this index by
presenting a method for calculating a limited
information HHI that could be applied to
incomplete data such as those obtained from
market surveys.
While information concerning the number of
firms in the market is normally available through
such sources as Standard Industry Classification
(SIC) data, recollection of market share information by survey respondents may only be accurate
for a subset of the market – usually the largest
three or four firms. Complete information may be
available through administrative data sets but has
an associated time lag that renders it useless in
measuring market concentration in a turbulent
marketplace. Most often, the solution is to rely on
such market surveys to produce a four-firm
concentration ratio (C4) or even an HHI based
*Correspondence to: Health Economics, 8th Floor, 80 Grosvenor Street, Toronto, ON M7A 1R3, Canada.
E-mail: [email protected]
Copyright # 2003 John Wiley & Sons, Ltd.
Received 27 June 2002
Accepted 30 January 2003
E. Nauenberg et al.
on these known firms (i.e. quasi-HHI); however,
these measures discard potentially useful information like the total number of firms in the market.a
This paper improves upon the precision of a
previously published simulation technique by
employing maximum likelihood methods [4].
Accordingly, we calculate the maximum likelihood
estimator of a distribution shape parameter, a,
using the joint probability function of the largest
cumulative market shares [5]. That is, individual
market shares from the three or four firms with the
largest shares are sorted by magnitude and
sequentially subtracted from 100% to produce
cumulative market shares used to calculate the
estimator. This estimator is then used to produce a
fitted distribution function (FDF) for the remaining unknown part of the cumulative market share
distribution. And from this distribution, individual
market shares can be extracted to estimate an HHI
(i.e. simulated HHI).
Although this method is to be employed in
instances where data is incomplete, it was necessary to use complete data from which an actual
HHI could be calculated in order to test the
accuracy of the proposed method. Since it
provides a wide assortment of market sizes and
configurations, we chose to test the method using
hospital discharge data from the State of New
York. The analysis proceeds by first simulating an
HHI based on the largest three or four firms –
temporarily blinding the researcher to the remaining market shares – and then comparing this value
to the actual HHI and various alternative measures of market concentration.
modeling the distribution of market share or
cumulative market share. While the binomial
distribution is suited for modeling proportions, it
is functionally awkward for modeling cumulative
proportions since it is not easily integrated. For
this purpose, the Bradford distribution is a
reasonable alternative [6,9]. The functional form
is ideal since its domain is in the interval [0,1], and
there is a single shape parameter with domain
(1, +1). A single unbounded parameter, a,
both facilitates maximum likelihood estimation
and ensures the existence of a unique solution.
We develop an algorithm for estimating the nj
smallest market shares from simulating cumulative
market shares. We work with the latter because a
plot of such data against the proportion less than a
particular value produces a curve that is strictly
concave and bounded by the points (0,0) and
(1,1) – conditions that are necessary for fitting a
Bradford distribution to the data. This is a trivial
task if complete information is available; however,
when information is incomplete, the curvature of
such a distribution is estimated based upon known
information using the Bradford distribution as a
parametric model. The remaining unknown values
are then simulated. Specifically, the model takes
the form
ln 1 ð1 SÞ=1 þ expðaÞ
;
FðS; aÞ ¼ 1 ln 1 1=ð1 þ expðaÞ
05S51; 15a51
ð1Þ
with the probability density function (pdf) defined
as
f ðS; aÞ ¼ F 0 ðS; aÞ
Statistical model
Let X1, X2,.P
. ., Xn designate the market shares of n
n
firms
Pnj with i¼1 X ðiÞ ¼ 1. In addition, let SðnjÞ ¼
i¼1 XðiÞ denote the cumulative value of the
smallest observations and assume that only the j
largest shares X(n), X(n1), . . ., X(nj+1) are known.
Note that the distribution of cumulative values is
by construction strictly concave with S(n)=1 – an
additional known data point (i.e. there are j+1
known cumulative shares). Historically, economic
analyses involving firm size have been modeled
using Pareto-type distributions – classified as
Pearson type VI [6,7]. The Zipf distribution has
also been shown to be appropriate [8]. However,
these functional forms are not appropriate for
Copyright # 2003 John Wiley & Sons, Ltd.
¼ 1=ð1 þ expðaÞÞ
1ð1 SÞ=1þexpðaÞ ln 1 1=1 þ expðaÞ
ð2Þ
The joint likelihood function (L) of the ordered
variates SðnjÞ 4 4SðnÞ is given by:
L ¼ Lða; SðnjÞ ; :::; Sðn1Þ ; SðnÞ Þ
¼ ðn jÞ:::ðn 2Þðn 1ÞðnÞFðSðnjÞ Þðnj1Þ
f ðSðnjÞ Þ:::f ðSðn1Þ Þf ðSðnÞ Þ
where F(S) designates the distribution function of
the underlying population and f(S)=F0 (S) represents its pdf [5].
The maximum likelihood estimator (MLE) of a
is the root of @L=@a ¼ 0 with @2 L=@a2 50; that
can
be solved numerically for any underlying
´
Health Econ. (in press)
Simulation of a Hirschman^Her¢ndahl Index
functional form that meets the stated restrictions
for the cumulative distribution F(S) [10].
Denoting the MLE of a as a# ; the entire fitted
distribution FðS; a# Þ is generated. Note that, for a
given S, the ordinate is the maximum proportion
of observations with cumulative value 4S: The
following relation segments the abscissa into n
quantiles (i.e. cumulative market shares) of which
the nj smallest are added to the j+1 known
cumulative values
F 1 ðni ; a#Þ ¼ xi=n
for i ¼ 1; . . . ; n
ð4Þ
A simple algorithm to generate the unknown
values using the maximum likelihood function in
(3) is as follows:
(a) Extract the unknown cumulative values from
the relation
ni
SðniÞ ¼ F 1 ðnj
FðSðnjÞ ; a#Þ; a#Þ
for i ¼ n 1; . . . ; j þ 1
ð5Þ
(b) Add the known values ðSðnjÞ ; . . . ; SðnÞ Þ to those
in step (a) to obtain a complete set of
cumulative market shares for each hospital
market.
catchment area and calculating the percent associated with each hospital. Similar techniques have
been utilized in other studies [11–14]. For the
simulation, the research team remained blinded to
all but the three or four hospitals with the largest
market share until testing for accuracy against the
actual HHI was performed.
During the simulation phase, hospital markets
were also classified according to whether they were
located Downstate (New York City and immediate vicinity) or Upstate. Closer geographical
proximity between institutions in the more generally urban Downstate locations presumably
leads to larger markets than in more generally
rural Upstate locations; therefore, these two
regions were useful for comparing the accuracy
of the simulation methodology in different environments. The simulations were conducted for all
markets in which there were at least six hospitals
(177 markets).b
To determine the marginal value of conducting
the simulation, scatter plots are produced comparing patterns of association between the actual HHI
and both the simulated HHI and the quasi-HHI,
and between the actual HHI and the C4.
All analyses were conducted using S-Plus 2000.
(c) Extract the individual market share values from
the following formulae:
Results
XðiÞ ¼ SðiÞ Sði1Þ
for i ¼ 2; . . . ; n
Xð1Þ ¼ Sð1Þ
(d) The corresponding simulated HHI is
Pn
i¼1
ð6Þ
2
XðiÞ
:
Data and methods
To construct an HHI estimate for the hospital
industry, we used hospital discharge data for 1997
obtained from the New York State Department of
Health Systemwide Policy and Research Cooperative System (SPARCS) for over 250 hospitals
statewide. From these data, we constructed a
market area for each hospital consisting of those
zip codes from which 85% of the patients
indicated their residence. Other hospitals were
included in the market as competitors if they had
at least a 3% share of the total hospital discharges
from any one of these zip codes. Market share was
then calculated for each competitor by adding up
the discharges across zip codes in the 85%
Copyright # 2003 John Wiley & Sons, Ltd.
We simulated an HHI for markets with at least six
hospitals in 1997. Utilizing either three or four
hospitals with the largest market share to produce
estimates for both the other market shares and a
simulated HHI, we test our technique’s accuracy
against the actual HHI. Figure 1 is a graphical
example of the simulations produced for a
particular market. The first panel in the figure
contains the distribution of cumulative market
shares used to develop a parametric fit to the data.
The second panel contains the distribution of
marginal market shares both actual and simulated
(i.e. obtained from the fitted distribution function).
This technique is accurate in both the more
urbanized Downstate areas as well as in the
generally more rural Upstate areas. For simulations involving three known market shares, the
simulated HHI values were on average within 5%
of the actual HHI in Upstate New York and 8%
Downstate. The maximum error among 86 Upstate hospital markets was 22.3% (based on
HHIactual=842.1 and HHIsimulated=654.2) and
Health Econ. (in press)
Proportion of Market Shares <= X
proportion of cumulative market shares <= S
E. Nauenberg et al.
Distribution of Cumulative Market Share
1.0
0.8
0.6
0.4
0.2
actual
simulated
0.0
0.0
0.2
0.4
0.6
0.8
S - cumulative market share
1.0
1.0
actual HHI – particularly when the value is less
than 1500. The simulated HHI maintains a high
degree of accuracy throughout the range of the
index with average error less than 10% throughout.
Comparing C4 to the actual HHI, the presence
of large market shares in highly concentrated
markets will have a much more substantial impact
on the simulated HHI than on the C4. This results
in a major departure from linearity between the
measures at high values of the HHI rather than at
low values as was true in the comparison with the
quasi-HHI. Since it is felt that changes in
market shares among larger firms have a disproportionate impact upon market behavior, the extra
weighting given to larger firms in the HHI is
therefore warranted and suggests that it is a more
accurate measure of market concentration [3]
(Figure 3).
0.8
0.6
Discussion
0.4
0.2
actual
simulated
0.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
X - market share
Figure 1. Sample market: simulated vs. actual values. Distributions for both cumulative (S) and marginal (X) market
shares, n=23
among 91 Downstate markets 19.5% (based on
HHIactual=1,087.7 and HHIsimulated=875.5). If an
improvement upon these error margins is considered economically significant, then the marginal
utility of information at this level is potentially
great since adding an extra data point – i.e. from 3
to 4 known market shares – reduced mean errors
to within 3% Upstate and 5% Downstate.
Maximum errors also decreased to 16.2% Upstate
and 14.7% Downstate.
Contained in Figure 2 is a scatter plot and
combination bar chart comparing the performance
of the quasi-HHI to the simulated HHI based on
four known market shares. The scatter plots of
both measures against the actual HHI suggests
that the quasi-HHI consistently underestimates the
Copyright # 2003 John Wiley & Sons, Ltd.
There are a variety of methods to impute data in
cases where it is either censored, truncated, or
missing. These include regression methods, hot
decking (based on neural networks), and mean
substitution [15]. While these methods have their
advantages and disadvantages, few of them are
applicable to situations where the available information is so sparse as is the case in these
simulations. In some of the samples, the top four
firms comprised less than 15% of the total market
share from which we had to predict the rest of the
distribution; yet, this method was highly accurate
consistently producing differences of under 10%
and often under 2%. Although not a closed-form
solution, this method is practical now that
computers are both powerful and widely available.
There are also alternative functional forms for
the fitted distribution than the one selected.
Previous discussion outlined the problems
with candidate distributions such as Pareto-type
and binomial distributions; however, there are
other flexible functional forms, such as the
Weibull, that might be appropriate. Beyond
problems with a domain exceeding [0,1], the
Weibull has two parameters to estimate
making maximum likelihood estimation computationally difficult and often precluding the possibility of a unique solution. Other alternative forms
either have more than a single parameter to
Health Econ. (in press)
Simulation of a Hirschman^Her¢ndahl Index
4000
HHI (estimated)
3000
2000
1000
simulated HHI vs. actual HHI
quasi-HHI vs. actual HHI
0
0
1000
2000
3000
4000
HHI (actual)
Error by HHI Magnitude and Method of Computation
simulated HHI (error magnitude)
quasi HHI(error magnitude))
simulated HHI (%)
quasi HHI (%)
300.0
80.0%
70.0%
250.0
60.0%
50.0%
150.0
40.0%
Error %
Error Magnitude
200.0
30.0%
100.0
20.0%
50.0
10.0%
0.0
0.0%
(0-500)
(n = 32)
(501-1000) (1001-1500) (1501-2000) (2001-2500) (2501-5500)
(n = 34)
(n = 36)
(n = 25)
(n = 21)
(n = 29)
HHI magnitude
Figure 2. Marginal value of simulation: actual HHI vs quasi (four firm) HHI and actual HHI vs simulated HHI). Sample includes
all markets with at least six hospitals (n=177)
estimate, a domain outside of [0,1], or restrictions
on the domains of the parameters. Although
the Bradford is an obscure functional form,
it meets the specifications needed for this
simulation.
Compared to the quasi-HHI, the simulated
HHI is more sensitive to variations in market
concentration particularly when markets are
highly competitive (i.e. when the HHI is low in
value). This is because the lower values
of the index are associated with markets that
Copyright # 2003 John Wiley & Sons, Ltd.
have larger numbers of firms that would not be
included in the quasi-HHI measure whereas
higher values are often associated with smaller
markets. When markets become highly concentrated, the linear relationship between C4 and
the HHI increasingly breaks down since the
former weights the market shares of all firms
equally while the latter gives added weight to
larger firms; therefore, neither the quasi-HHI
nor the C4 are adequate proxy measures for the
HHI.
Health Econ. (in press)
E. Nauenberg et al.
standardized value for C4
1
0
-1
-2
-1
0
1
2
standardized value for actual HHI
3
Figure 3. Comparison between actual HHI vs C4 (All measures transformed to standardized values). Sample includes all markets
with at least six hospitals (n=177).
While hospital discharge data does not provide
exact information concerning market share, it is
often used to construct an HHI for the hospital
industry [11–14,16]. This industry was a good
laboratory to study the accuracy of this methodology since the complete discharge data was available from the New York State Department of
Health. In other words, the remainder of the
market share distribution could be revealed to the
researcher after the simulations based on the
largest firms was produced; therefore, it was
possible to calculate an actual HHI for this
industry to determine the method’s accuracy.
These simulations are not designed for hospital
markets in which discharge data is both complete
and timely. In practice, though, both conditions
are seldom met. Many countries still do not have
accurate hospital information systems, and for
those that do, there is often a time lag of up to 2
years to obtain accurate discharge data making
current marketplace assessment difficult. Particularly in countries in which internal markets are
operating (e.g. Great Britain, Sweden, New
Zealand), there may be a need to continuously
monitor market competition in a number of
health-care sectors. This methodology helps to
make the HHI measure available to those who
must rely on on-the-spot survey data in the
hospital as well as other health-care sectors. Often
Copyright # 2003 John Wiley & Sons, Ltd.
such surveys ask respondents about the identity
and market shares of their top three or four
competitors. In such instances, the proposed
methodology is of value to those who wish to
employ the HHI in assessing market concentration
rather than inferior measures such as the quasiHHI and the C4.
There is also some concern that there are a few
outlier market configurations that vary strongly
from expectation based on the largest four market
shares. In such cases, the concern is that the
simulated HHI might be substantially below the
threshold of 1800 – the US Department of Justice’s
criteria for concentrated markets in which mergers
presumptively raise antitrust concerns – while the
actual HHI would be above this threshold or vice
versa [3]. The proposed methods, however, alleviate this concern somewhat since the simulated
HHI values based on four known firms were off by
a maximum of 16.2% from the actual value.
While the method used to calculate market
shares and obtain a variety of market sizes and
configurations was sufficient for testing the simulated HHI’s accuracy, there are at least three
additional steps necessary for those wishing to use
hospital discharge data to actually measure market
concentration in the hospital industry. First, since
there is product-line diversification across institutions, hospital markets need to be constructed for
Health Econ. (in press)
Simulation of a Hirschman^Her¢ndahl Index
each product line. For example, if one wanted to
construct a market for a particular hospital’s
obstetrics unit, only hospitals with such units
would be included in the market rather than all
hospitals that fit the inclusion criteria outlined in
the methods section. For those interested, Bamezai
et al. (1999) include such adjustments in their HHI
measure [13].
Second, there were many hospital mergers in
New York State in the early part of the 1990s;
however, it appears that the state continues to
collect discharge information from individual
facilities regardless of changes in ownership due
to mergers or acquisitions [17]. Recently published
research on New York State discharge data up to
1998 shows growing discrepancies between the
number of units from which discharge data is
collected and the number of units based on
common ownership. The implications for the
HHI are often large leading to error margins as
high as 44% [18]. This problem, if it persists in
more recent data, will produce ever greater
inaccuracies in calculating an accurate discharge
HHI. Whether this problem exists in other hospital
discharge databases is an important area of
inquiry.
Finally, there is the possibility that a hospital’s
geographic service area may transcend jurisdictional boundaries so that discharge data
from all competitors may not be available
from a single jurisdiction. For example, some
hospitals on the southern border of New York
State may both serve populations in Pennsylvania/
New Jersey and compete with institutions in those
two states; as a result, it may be necessary to
obtain discharge data from neighboring jurisdictions to complete the market configuration for
some hospitals.
Conclusion
One long-standing issue in both economics and
antitrust practice concerns the correct and workable measurement of market concentration. The
HHI serves as a benchmark measure of market
concentration if all market shares are known; but
unfortunately, the market shares of the smallest
firms are usually unknown. Therefore, in order to
extend the utility of the HHI beyond instances of
perfect information, a method of estimating the
unknown portion of the index is needed. This
Copyright # 2003 John Wiley & Sons, Ltd.
paper fills this void by presenting a formulation to
estimate such an index.
This study will inform public policy in
many jurisdictions and health care sectors.
Given either lack of or delays in the availability
of administrative data and the need for up-to-date
monitoring of market competition (e.g. in
internal markets), a simulated HHI is an
important addition to the toolbox of those
assigned the task of measuring market
concentration. As well, the pace of mergers and
consolidations have been accelerating in many
industries over the last few years, and
the US Department of Justice and other regulatory
bodies have been interested in predicting how these
changes will impinge upon the forces of competition at the local level. Since many markets contain
many local submarkets, for which information is
less readily available than at the level of legally
defined jurisdictions, there is a need to measure
concentration in these markets as well. While a
structural analysis must be complemented by an
investigation of firm conduct and factors such as
the likelihood of entry, the techniques presented
here extend the utility of an important measure for
examining market consolidation and related antitrust considerations.
Acknowledgements
The authors wish to thank Jacek Dmochowski, Manavala Desu, David Andrews, and John Bell for their
helpful discussions.
Notes
a. See Curry B, and KD George. Industrial Concentration: A Survey. J Ind Econ 1983; 31: 203–255 for
discussion of other potential indices. While there are
many full-information indices, limited information
ones are few in number. Besides, the C4 and C8,
there are the I50 and I90 indices which consist of the
number of firms comprising 50% and 90% market
share, respectively. However, in markets consisting
of many small firms, surveys may not reveal
adequate information for producing reliable measures for the I50 and I90.
b. In a five-firm market, the market share of the fifth
firm in any simulation involving four firms is fixed
due to the constraint that all market shares must
add to 100%. A simulation only has marginal value
when simulated values are not constrained to equal
actual ones (n>5).
Health Econ. (in press)
E. Nauenberg et al.
References
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4. Nauenberg E, Basu K, Chand H. Hirschman–
Herfindahl index determination under incomplete
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14. Keeler EB, Melnick G, Zwanziger J. The changing
effects of competition on non-profit and for-profit
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69–86.
15. Little RJ, Rubin DB. Statistical Analysis with
Missing Data. Wiley-Interscience: New York, 2002.
16. White SL, Chirikos TN. Measuring hospital competition. Med Care 1988; 26: 256–262.
17. American Hospital Association. Annual Survey of
Hospitals. 1990–1997 (supplemented by merger data
for 1995–1999 from Irving Levin and Associates,
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Health Econ. (in press)
HEALTH ECONOMICS
HEALTH ECONOMICS LETTERS
Health Econ. (in press)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/hec.817
Managed care and shadow price
Ching-to A. Ma
Department of Economics, Boston University, Boston, USA
Department of Economics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
Summary
A managed-care company must decide on allocating resources of many services to many groups of enrollees. The
profit-maximizing allocation rule is characterized. For each group, the marginal utilities across all services are
equalized. The equilibrium has an enrollee group shadow price interpretation. The equilibrium spending allocation
can be implemented by letting utilitarian physicians decide on service spending on an enrollee group subject to a
budget for the group. Copyright # 2003 John Wiley & Sons, Ltd.
JEL classification: D45; H40
Keywords
managed care; shadow price
Introduction
Managed care refers to the set of instruments for
controlling the delivery of health services when
minimal financial incentives are imposed upon
consumers. Managed care increasingly has replaced traditional financial control such as patient
deductibles, and copayments. The formal modeling of the way managed care delivers services to
consumers continues to be an important research
topic.
An early attempt by Baumgardner [1] simply
postulates that a fixed quantity of service will be
supplied under managed care (see also [2]). More
recently, Keeler et al. [3] put forward a shadow
price approach to model the way managed-care
companies allocate resources to patients. Frank
et al. [4] further develop this methodology for
managed care into a model with many patient
groups and many services; they use it to study
selection and empirically estimate the extent of
distortion. More recently, Glazer and McGuire [5]
extend it to study policy implications.
The shadow price approach posits that a
managed-care company will allocate resources of
a service to its consumer groups until each group’s
marginal benefit is equal to the service shadow
price.a The shadow price approach has been a
significant theoretical development. First, it is a
simple theory, and under some assumptions, can
be used for empirical analysis. Second, given the
service shadow prices, resources are allocated
efficiently. Third, it also has been claimed that
managed-care firms setting shadow prices may be
consistent with profit maximization and capture
health plans’ actions in practice.
In this note, I examine the theoretical foundation of service shadow price. Contrary to the
earlier analysis, I do not require that a managedcare company must impose a single shadow price
on each service. Here, setting service shadow price
is a feasible strategy, but a managed-care plan is
free to allocate resources of many services to many
*Correspondence to: Department of Economics, Boston University, 270 Bay State Road, Boston, Massachusetts 02215, USA.
E-mail: [email protected]
Copyright # 2003 John Wiley & Sons, Ltd.
Received 2 July 2002
Accepted 10 February 2003
C. Ma
groups to maximize profit. I show that it is never
profit-maximizing to use service shadow prices.
The optimal strategy is for the managed-care firm
to allocate resources to a group of individuals to
equalize marginal utilities of all services. In other
words, the equilibrium can be described by a set
of group shadow prices, one for each group of
individuals covered by the managed-care company.
Given a fixed revenue rate for each group of
consumers, profit maximization by the managedcare company can be given the following interpretation. First, the managed-care company
chooses a total budget for each group. Given the
budget for a group, the final allocation will
maximize the utility of consumers in that group.
Relative to the total amount of resources set aside
to a group of consumers, these consumers receive
an efficient allocation. Maximizing consumer
utility is profit maximizing because that will attract
more consumers, each of whom yields a fixed
revenue for the firm. To maximize the utility of
consumers in a group, each dollar spent on each
service for the group must yield the same marginal
utility; were this condition not satisfied, the
managed-care company could have reallocated
the same spending amount to increase utility, and
hence profit.
Equilibrium shadow price
A set of consumers consider joining and receiving
services from a managed-care company. There are
I different groups of consumers. For my purpose,
it is unnecessary to consider aggregation issues. So
I regard each group as consisting of a single,
representative individual. The representative preferences of each group are denoted by a strictly
increasing and concave function Ui defined on a
managed-care organization’s spending on S services. The index i ¼ 1; . . . ; I is for the consumer
groups; the index s ¼ 1; . . . ; S, for the services. If
the managed-care company allocates a monetary
amount mis for service s on group i, group i
consumers have a utility Ui ðmi1 ; . . . ; miS Þ þ mi ,
where mi follows a distribution Fi with density fi .
Group i consumers have a reservation utility
U% i . The managed-care organization receives a
capitation rate ri for providing services to group
i consumers. These capitation rates are assumed to
be exogenous; consumers do not directly pay for
services.
Copyright # 2003 John Wiley & Sons, Ltd.
The game proceeds in the following way. The
managed-care company decides on the spending,
mis ; i ¼ 1; . . . ; I; s ¼ 1; . . . ; S. Group i consumers
observe the realization of mi ; i ¼ 1; . . . ; I, and
decide whether to join the managed-care plan to
enjoy the services. Given the spending allocation,
the probability that group i joins the managed-care
firm is PrðUi þ mi > U% i Þ; that is, the demand from
group i is 1 Fi ðU% i Ui ðmi1 ; . . . ; miS ÞÞ. The managed-care firm makes an expected profit:
I
X
"
½1 Fi ðU% i Ui ðmi1 ; . . . ; miS ÞÞ ri i¼1
S
X
#
mis
s¼1
ð1Þ
In an equilibrium, the managed-care firm picks
mis ; i ¼ 1; . . . ; I; s ¼ 1; . . . ; S to maximize its
profits.b
To understand how the equilibrium spendings
mis are chosen, break up the maximization
problem into two steps. First, for group i, let the
managed-care
PS firm commit to a level of total
spending,
s¼1 mis . Then find the allocation of
spending to maximize profit given this preset total.
Second, adjust the total spending to achieve the
global maximum profit.
P
From (1), once Ss¼1 mis is fixed in the first step,
the managed-care firm chooses the spending to
maximize its demand 1 Fi . This is equivalent to
maximizing the consumer utility function Ui
subject to the spending level. For those services
for which the managed-care firm chooses positive
spendings, marginal utilities of these services are
equalized:
@Ui ðmi1 ; . . . ; miS Þ @Ui ðmi1 ; . . . ; miS Þ
¼
;
@mis
@mit
s; t ¼ 1; . . . ; S:
For any given consumer group, each service
generates the same marginal utility. So I call this
value of marginal utility the shadow price for the
consumer group. From the second step, given the
equal-service-marginal-utility
Pproperty, the profitmaximizing total spending Ss¼1 mis is chosen to
satisfy
condition: the price–cost margin
Pthe usual P
ðri Ss¼1 mis Þ= Ss¼1 mis will be inversely related
to the elasticity of demand.
The following derivation confirms the intuition.
The first-order derivative of (1) with respect
Health Econ. (in press)
Managed Care and Shadow Price
to mis is
½1 Fi ðU% i Ui ðmi1 ; . . . ; miS ÞÞ
"
#
S
X
@Ui
mit fi ðU% i Ui ðmi1 ; . . . ; miS ÞÞ
þ ri @m
is
t¼1
ð2Þ
If there is a corner solution, then the above
expression will be negative and the value of mis is
set at 0. For an interior solution, I set the firstorder derivative to zero. After rearranging, I
obtain the necessary condition for a profit-maximizing choice of an interior mis :
"
#1
S
X
@Ui
1 Fi ðU% i Ui ðmi1 ; . . . ; miS ÞÞ
¼ ri mit
@mis
fi ðU% i Ui ðmi1 ; . . . ; miS ÞÞ
t¼1
i ¼ 1; . . . ; I; s ¼ 1; . . . ; S
ð3Þ
I consider now those spendings that are strictly
positive. The right-hand side expression in (3)
holds for all such spendings for service s ¼ 1; . . . ; S
for group i ¼ 1; . . . ; I. For group i, the profitmaximizing spendings will equalize group i’s
marginal utilities for these services. The equilibrium reveals a shadow price, Pi , for each
consumer group i: for any two services (with
strictly positive spendings) s and t
@Ui ðmi1 ; . . . ; miS Þ @Ui ðmi1 ; . . . ; miS Þ
¼
Pi
@mis
@mit
Rearranging (3), I also obtain
P
1
ri St¼1 mit
fi @Ui
mis
¼
mis
PS
PS
1 Fi @mis
t¼1 mit
t¼1 mit
The term inside the square brackets on the righthand side is simply the elasticity of group i demand
with respect to mis . The price–cost margin for
group i is inversely related to its demand elasticity.
In practice, a managed-care firm has to rely on
its physicians and other health-care professionals
to deliver services. How can the profit-maximizing
allocations be implemented? I now describe this
implementation when physicians decide on the
services on a utilitarian basis. Suppose the
managed-care organization allocates a budget Bi
for consumer group i; i ¼ 1; . . . ; I. Under the
utilitarian assumption, the physicians will choose
service spendings to maximize the group’s utility
given the resources available to this group. That is,
the physicians choose
mis to maximize Ui ðmi1 ; . . . ;
P
miS Þ subject to Ss¼1 mis ¼ Bi .
Copyright # 2003 John Wiley & Sons, Ltd.
To maximize utility for consumer group i
subject to budget Bi , the physicians choose
spending mis characterized by
@Ui
@Ui
¼
s; t ¼ 1; . . . ; S
@mis @mit
To implement the profit-maximizing spending, the
managed-care company simply picks Bi such that
the above marginal utility equals Pi . In other
words, choose the budget for group i to ensure
that physicians implement spending across services
according to the group shadow price Pi . The
implementation of the optimal spending may
look similar to that under service shadow price,
as Frank, Glazer, and McGuire [4, p. 386]
describe: ‘Cost-conscious management allocates a
budget or a physical capacity for a service.
Clinicians working in the service area do the best
they can for patients...management is in effect
setting a shadow price for a service through its
budget allocation.’ In fact, here the equilibrium
can be implemented by management allocating a
budget for a group of enrollees, instead of a
service.
Concluding remarks
I characterize a managed-care company’s profitmaximizing spending by a set of shadow prices,
one for each of its consumer groups. These group
shadow prices depend on the capitation rates and
consumer preferences. My results show that
service shadow prices are suboptimal. Furthermore, equilibrium properties of managed-care
spending are to be recovered from the group
shadow prices, not service shadow prices. Therefore, the relevance of empirical estimation of
service shadow prices is being questioned. Further
empirical work on enrollee group shadow prices
may well shed new light on adverse selection and
managed care.
Arguments for service shadow price in the
literature have centered on the practicality and
perhaps fairness of such an approach. Physicians
allocating resources to different groups of consumers according to service shadow price will not
have to know these consumers’ capitated payments. Each group of consumers also obtains the
same (marginal) value from a given service.
Nevertheless, an enrollee group’s capitation rate
presumably reflects the group’s expected usage
Health Econ. (in press)
C. Ma
cost and premium. In my model, groups that have
higher capitation rates will be allocated a higher
budget. This does not seem to be an unfair
procedure if higher capitation rates correspond
to higher premium rates. In fact, this is the usual
way the market allocates resources: consumers
who have paid more expect to receive more
services.
My model uses the same informational setup as
work in the earlier literature (for example, Frank
et al. [4]) the managed-care company possesses
perfect information, and a contractible state for
resource allocation is service s on group i. In
this setup, the managed-care company will not
lose profit if it sets a shadow price for each
enrollee group, and asks the utilitarian physician to implement the allocation. Conversely,
asking utilitarian physicians to implement an
allocation via service shadow prices leads to lower
profits.
Under perfect information, a managed-care
company can simply compute the optimal spending. The usual justification for delegation revolves
around asymmetric information and expertise.
What is the optimal spending when providers
possess private information on patient characteristics and their own costs? What is the second-best
allocation, and will service or enrollee group
shadow price implement it? There are related
issues, too. For example, what are physicians’ motives? Is the pure utilitarian assumption
a good one? Can physicians dilute or manipulate
budgets for different groups? Do physicians’
actions fully reflect consumer preferences? For a
broader perspective, one must also ask how the
capitation rates are set. If these rates do not reflect
the group premium rates or costs, adverse
selection problems must be addressed. The note
here presents a foundation for studying these
problems.
Copyright # 2003 John Wiley & Sons, Ltd.
Acknowledgements
Work on this study is supported by grant PO1-HS10803
from the Agency for Health Care Research and Quality. I
thank Karen Eggleston, Randy Ellis, Marty Gaynor, Jacob
Glazer, Ann Holmes, Mingshan Lu, and Tom McGuire for
their comments and discussing related issues with me.
Coeditor Jim Burgess and two referees gave me very helpful
advice. Opinions expressed here are the author’s.
Notes
a. Suppose that the utility function of a group i is
Ui ; i ¼ 1; . . . ; I, and that the shadow price of service
s is ps ; s ¼ 1; . . . ; S. According to the service
shadow price approach, the resources allocated to
group i; mis , satisfy @Ui ðmi1 ; . . . ; mis ; . . . ; miS Þ=
@mis ¼ ps , for each group i. For example, a group
of individuals who have a low marginal valuation of
psychiatric services will be given a smaller number of
outpatient visits compared to those who have a high
marginal valuation.
b. The firm must make a nonnegative profit: setting
each of mis to 0 is a feasible allocation.
References
1. Baumgardner J. The interaction between forms of
insurance contract and types of technical change in
medical care. Rand J Econ 1991; 22(1): 36–53.
2. Pauly M, Ramsey SD. Would you like suspenders to
go with that belt? An analysis of optimal combinations of cost sharing and managed care. J Health
Econ 1999; 18(4): 443–458.
3. Keeler EB, Carter G, Newhouse JP. A model of the
impact of reimbursement schemes on health plan
choice. J Health Econ 2002; 17(3): 297–320.
4. Frank RG, Glazer J, McGuire TG. Measuring
adverse selection in managed health care. J Health
Econ 2000; 19: 829–854.
5. Glazer J, McGuire TG. Setting health plan premiums
to ensure efficient quality in health care: minimum
variance optimal risk adjustment. J Public Econ 2002;
84: 153–173.
Health Econ. (in press)
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