Estimation of Hospital Cost Functions
and Efficiency Measurement by Stochastic Frontier Method and
DEA
Daniela Serban, PhD
Simona Vasilache, Student, SELS English, Second Year
1
1.
Introduction
When a person is sick and decides to seek treatment from a hospital, few questions run
in his mind which we call them ‘the customers’ needs and expectations’. He will be thinking
whether the treatment he needs is readily available in the hospital. Are the people there
ffriendly? Will he be safe and not lost in the compound of the hospital? How long he will wait
to see the doctor and to take medications? How much is the hospital fees? Will his sickness
improve with treatment? His perception of the hospital services sometime differs from what
we understand. We produce a list of what we promise to do in the form of a patient’s charter,
but sometimes this is not what the patient needs or expects. The gap between the services the
hospital can offer and the needs and expectations of the patient will determine the level of
satisfaction. Patient will be satisfied if he gets all that he needs. He will be more satisfied if
the hospitality rendered to him is beyond his expectation. Hence hospital managers have to
know the customers’ needs and expectations before they can plan effective hospital
management practice to meet these needs and expectations.
This paper will deliberate the following issues:
·
The needs and expectations of the customers
·
The objectives of the hospital
·
The social, ethical and ability-to-pay issues
·
Providing personalized attention and quality assurance to the customers
·
The strategies for hospitals to improve customers’ satisfaction
·
The future look for public and private hospital management practice.
It provides a study of the development in the econometric estimation of structural cost
functions, efficiency and productivity. In the early seventies, hospital cost estimations were
realistically curve fitting exercises intended to forecast costs. Since then, the emphasis has
been on microeconomic theory, as a basis for the specification of cost functioning. Several
excellent reviews are available on estimating cost functions, e.g., Cowing, Holtmann, and
Power (1983), Wagstaff and Barnum (1992), and Barnum and Kutzin (1993). In the context of
efficiency measurement, there are some intensive contributions by Farrall (1957), Fare,
Grosskopf and Lovell (1985, 1994), Coelli, Rao and Battese (1998), Emrouznejad, A (1995).
The aim of this paper is to cull the issues as presented by literature and to provide a sense of
what other issues have to be considered to improve future efficacy and efficiency.
Recent econometric research on hospital cost functions has tried to address many
issues that are helpful to review the basic intent of such investigations. The estimation and
interpretation of hospital cost functions constitute an attempt to study, under a set of hospital
behavior assumptions1, the structure of costs and production of a set of hospitals. This study
postulates,
a) average cost behavior;
b) cost per unit of output rise or decline as a hospital produces more output; and
c) the level of output at which cost per unit of output is at its lowest level and if the hospitals
are producing at this level of output.
As costs are affected by the technology of the production process, hospital cost
functions affords another angle from which production-related issues can be examined.
Postulation of enquires are critical towards,
1
By applying microeconomic theory, models of hospital behavior include standard cost-minimisation, profit-,
output- or utility-maximisation.
2
a)
b)
c)
d)
optimal size of a hospital;
number of beds a hospital maintains;
current output levels and hospitals capital equipment;
technical efficiency; (i.e., do hospitals obtain the maximum levels of output from their
inputs?); and
e) allocative efficiency? (i.e., do hospitals choose optimal combinations and levels of inputs,
given their outputs?); and
f) economies of scope.
There is even greater variation in the choice of model specification in the cost function
literature than in the production function literature. Significant issues raised in cost function
literature are,
a) effects of case-mix on average cost;
b) short-run average and marginal costs;
c) economies of scale;
d) economies of scope;
e) extent of factor substitution and the existence of complements; and
f) technical and scale efficiency of health care institutions
Hospitals may not fit economists’ standard notion of a firm, and this gives rise to a
host of challenges in properly estimating the cost function of hospitals. It is arguable to
presume a definition and recognition of hospital output. If it is allowed (extending Grossman,
1972) that people utilise hospital services because their health stocks have fallen below some
critical levels, then, the restoration of the health stock of its patients, ought to be regarded as
the outputs of hospitalisation (Breyer, 1987 and Ellis, 1992). These measures may lessen for
those terminally-ill, for whom the object of hospitalisation may be the management of pain
before death and of patients, who undergo elective cosmetic enhancements, with nose
reconstructions and breast implants. Ellis (1992) elucidates that measuring the improvement
in health status is unrealistic, as health status is a multifaceted concept. It is a nebulous
concept that is not easy to define and measure, in an operationally feasible manner, much less
to compare and aggregate across patients
Researchers have used measures of throughputs or intermediate outputs, in the number
of cases treated, of patient-days served per hospital department, and of outpatient visitors.
This strategy summons a new set of problems related to the homogeneity of hospital output.
Two aspects that have received widespread attention are,
a) case-mix; and
b) quality of care
Hospital case-mix refers to the variety of illness, that are treated in a hospital environ.
The case-mix of cost function estimation presents two problems,
a) if hospitals do not administer to the same kinds of ailments (or if they follow radically
different treatment protocols), then their production and cost are bound to be different, and
they ought not to be regarded as belonging to the same class of firms; and
b) the correct specification of the cost function requires the inclusion of all outputs of
hospitals in the set of regressors. Otherwise, the regression equation runs the risk of being
misspecified.
Given the sheer number of diseases and conditions for which patients seek treatment in
a hospital, aggregation of throughputs is necessary to avoid degrees of freedom problem (i.e.,
where the number of parameters to be estimated is greater than, or equal to, the number of
observations in the data set) in the estimation of cost function. Unfortunately, the appropriate
method of aggregation is still unsettled in literature, although there is no shortage of
proposals.
3
Different sources of hospital funding present methodological difficulties. Diverse
sources and levels of medical payments for public and private patients in public hospitals are
of specific relevance:
a) identification of marginal, as distinct from average, costs;
b) incorporation of outpatient services;
c) adjustments for teaching and research; and
d) further refinement of DRGs for certain specialist services (eg, peadiatrics).
Schemes employed in actual hospital cost function studies, range from simple
breakdowns of medical cases into the number of outpatient visits and of in-patient admissions
(Wouters, 1993), to elaborate stratification. This includes, number of in-patient days by
hospital department and emergency room visiting (Cowing and Holtmann, 1983), frequencies
of in-patient days by age group (child vs. non-child), and mode of payment (Medicare vs.
non-Medicare) (Conrad and Strauss,1983).
Though the case-mix issue can be resolved through a variety of suggested and
implemented methods of aggregation, the proposed methods will be not able to accommodate
the same breadth of disease categories as a specification, based on microeconomic theory
(Caragay, 1989)
The quality of care provided by hospitals has not satisfactorily dealt with in empirical
literature. Ellis (1992) notes that hospitals, with high mortality or re-admission rates, ought to
not be regarded as having the same outputs as hospitals with lower rates. Yet, this
misspecification is indicative of when quality measures are not included in the set of
regressors of cost function. As throughputs are used as the measures of hospital outputs, vital
information is lost on the effectiveness of treatments.
Appropriate measures of quality care predicate,
a) the inherent uncertainty about the outcomes of medical treatments, which makes mortality
and re-admission rates indicative but noisy measures of quality at best;
b) the bundling of (medical and non medical or hostel) services in a hospital stay, each of
which may have a qualitative aspect; and
c) the perception of patients regarding quality of care.
Measures of quality proposed or used in literature highlight,
the teaching status of hospitals;
the number or proportion of specialists on the medical staff;
the location and accessibility of the hospital;
the attributes of amenities (e.g., cleanliness of facilities, hospitality of the staff, quality of
food); and
e) the occupancy rate of hospitals2.
a)
b)
c)
d)
Physicians play a significant role in the provision of hospital care. Cowing, Holtmann,
and Powers(1983) stress that although doctors supply what may be considered indispensable
inputs in the treatment of patients in a hospital setting, they are often paid separately, either by
patients or health insurance companies. Costs of their services are usually not reflective of
production factors in hospital cost figures. Physicians enjoy privileged relationships with their
patients, which allow a wide degree of latitude in treatment procedure choice. Special bonds
may cause physicians to induce demand for particular procedures, for there is a notion, that
qualification and reputation of doctors in the medical staff of hospitals, increase both the cost
2
The occupancy rate of hospitals was used by Friedman and Pauly (1981) as a measure of quality. This is based
on the argument that as admissions approach hospital capacity, resulting in lower overall quality of services
provided.
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of and the demand for care in hospitals. All these aspects pose important questions for the
correct analysis of hospital costs.
Research on hospital cost function estimation (roughly since 1983) has proceeded under
the assumption, that hospitals are a class of multi-product firm, with an objective to minimise
cost subject to an output constraint. In this paper, cost-minimising behavior is rationalized on
the strength of the following arguments,
a) many hospitals constitute as non-profit organization. As such these hospitals may have
objectives other than profit-maximisation;
b) cost-minimisation is a necessary condition for profit- and budget-constrained outputmaximization and thus is a legitimate objective under a wide variety of circumstances;
c) hospitals do not control their output level, but merely responds to the demand for medical
care in their catchment areas; and
d) it is contended that hospitals do not exercise monopsonistic powers over inputs.
In production theory, the iso-revenue or equal-income curve may be represented as a
convex-to-the origin curve, since there are many possible combinations of products that yield
the same level of income. But given product-price relationships, there is only one
economically efficient combination. Convexity of the iso-revenue curve is due to the short-run
market condition that prices need to be lowered to sell more of the product. This implies that
there are combinations of the product, which yield the same level of income. Due to current
production costs, these combinations are costly to produce compared to those that are the most
efficient and of least cost. Iso-revenue curves apply to firms competing in the marketplace
under competitive market condition. Hospital in the US may face such conditions and
experience convex-to the origin iso-revenue curves. In Australia, where 'market conditions'
are quite different and distinct, the convexity of the iso-revenue curve may be of doubtful
validity. Revenue generation is not a goal of many Australian hospitals in terms of typical
profit-maximisation hypothesis. Any critical analysis could be redirected to ponder 'value' that
the hospital might be trying to maximise. The hospital might seek to maximise the amount of
each product it can produce, maximising the collection of characteristics embodied in each
product given the available resources.
2. Cost Functions
2.1
Basic Concept and Interpretation of Fix Factors and Variable Factors
Assessing the variable cost structure is complex. This section develops a framework of
basic concepts to assess hospital cost functions. Salient features of this framework are,
a) cost function is the minimum cost of producing a given output level, during a given time
period, expressed as a function of input prices and output;
b) variable factors of production refer to factors that vary continuously with output and can
be varied on short notice; and
c) fixed factors refer to factors of production that cannot be changed in a short run.
The importance of fixed factors in the theory of the multi-product firm arises from the
possibility of transferring units of a fixed factor from use in producing one product, to usage
in producing a different product. A second source of the special importance of fixed factors in
the multi-product firm is the available quantity of a fixed factor may not be used entirely
during any operating period. Such a condition can not exist with variable factors, as the total
quantity used by the firm can be adjusted in the short run.
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2.2
Ad Hoc Cost Function
Multivariate analysis and associated techniques are most used in deriving the ad hoc
functional form cost models. This approach assumes, that the impact on unit costs of the
various cost determinants, are both linear and separable additively. Lave, Lave and Silverman
(1974) used a linear regression to estimate the cost per patient, using 65 Western
Pennsylvanian hospitals to analyse 32 parameters. Several case types were classified into two
sets,
a) characteristic variables; and
b) diagnosis variables.
Grannemann, Brown and Pauly (1986) used a hybrid functional form which
incorporates desirable features, both of some ad hoc functions and of commonly used forms
for structural cost functions, in estimating a hospital cost function, based on data from 867
hospitals in the US. In contrast to the "pure" structural cost functions, which incorporate only
output quantities and input prices (as used in the Conrad-Strauss and Cowing-Holtmann
studies), the hybrid form includes potentially relevant variables on an ad hoc basis, while
allowing economies of scope and scale to be incorporated as testable hypotheses. While the
large sample size in this study eased the trade-off between flexibility in functional form and
parameter parsimony, the use of interactive terms between a range of output categories still
result in a relatively high degree of output aggregation3. In total, 63 regressors were included
in the estimated equation. These hospital cost studies confirm the above argument that
adoption of such a form involves a trade-off between flexibility of functional form and
parameter parsimony.
2.3
The Behavioural Cost Function
Evans (1971) argued that the true cost curves (based on either the profit maximisation or
cost minimisation process) can not be derived for the hospital sector, as hospital management
and/or medical staff committees influence rates of hospital utilisation, through scheduling of
laboratory and operating times, bed availability and regulatory constraints. Hospital managers
endeavour to choose both inputs and outputs to reach an optimum position than maximum
profit. Deviation from this optimum would be a function of random ‘disturbance’ factors, and
could not be relied upon, to trace out a technological cost curve.
Evans (1971) used the 1967 discharge data of acute care hospitals in Ontario, Canada, to
relate measures of cost per day and per case to a variety of hospital characteristics including
hospital size, levels of activity, and patient diagnostic and age-sex mix. The diagnoses were
grouped into 40-category male/female in each five-year age span from 0 to 100. Sets of four
vectors of proportions were derived for the diagnostic groups, age-sex characteristics and total
days of care supplied.
The vectors of proportion were condensed, using the factor analytical technique, with
the diagnostic variable vector reduced to 10 factors and each age-sex vector to 6. The reduced
factors for the diagnostic proportions accounted for 70 per cent of those vectors based on days
of care provided. Results showed the importance of the diagnostic and age-sex factors in
3
The in-patient discharges were classified as being acute, intensive care, or sub-acute and other, and five
categories were used to differentiate the case mix of specialist admissions, e.g., pediatrics, surgery, psychiatry,
obstetrics and gynecology, and others.
6
explaining inter-hospital cost variation. Adjustment of these factors accounted for 85 per cent
of inter-hospital variance in cost per case.
Given the importance of the effects on economic efficiency of different hospital
objectives, Feldstein (1988) undertook an examination of the behavior objectives of hospitals.
Feldstein (1988) stipulate three groups important in the determination of hospital
objectiveness. These groups include, a) the trustees; b) the administrators; and c) the medical
staff. The behavior models assume that one group is dominant in the decision-making process.
He concluded, that in an era of extensive third-party reimbursement and cost-based payment
of hospitals, the decision makers recognise that the output mix of the hospital industry is
unlikely to be optimal.
2.4
Neoclassical Cost Function
In the neoclassical theory of the firm, cost function is defined to describe minimum cost
of providing a given volume of output, as a function of the exogenous vector of input prices.
The cost functions were developed from theoretically consistent properties and propositions
about short-run and long-run cost structures
An example of the use of a multiproduct translog cost function is provided in the case
of US railroads by, Brown, Caves and Christensen (1979). Using two outputs (freight and
passenger services) and three inputs (capital, labour and fuel), an unrestricted (except for
linear homogeneity in factor prices), translog cost function together with four restricted
versions were estimated. The four versions were based upon,
a) constant returns to scale or linear homogeneity in outputs;
b) input/output separability;
c) homogeneity plus separability; and
d) a separable form with Cobb-Douglas aggregator functions for inputs and outputs.
Breyer (1987) provides a list of hospital and patient-related characteristic used in
hospital cost analysis,
 Capacity (bed size);
 Global indicators of hospital activity such as case flow, average occupancy rate or average
length of stay;
 Case-mix, measured by the proportion of patients in various diagnostic categories, defined
by a detailed classification code;
 The wage level of hospital employees;
 Dummy variables for teaching status;
 Indicators of hospital facilities and services;
 Characteristics of the market for inpatient services with regional income level, physician
density or hospital bed density.
The concept of returns to scale and returns to size are closely related. Elasticity of scale,
measures how output responds as one moves out along a scale line from the origin in output
space. Elasticity of size measures cost response associated with movements along the locus of
cost-minimizing points in input space being the expansion path. The two measures are based
on different input combinations generally.
Recently, studies on economies of scale and scope have been undertaken, using flexible
functional forms, allowing for a general specification of cost structure. This is known as the ‘
7
production-theoretic’ model which employs the translog variable cost function for the
exploration of production within multi-product firms, because of their ability to represent any
underlying arbitrary structure of production at the chosen point of approximation. (Schuffham,
Devlin and Jaforullah, 1996 and Vita, 1990).
3.Implementation of the method
Breyer (1987) suggests that the case-mix issue be handled by grouping patients,
according to an arbitrary (manageable) number of diagnostic categories and by specifying that
each diagnostic group raises total costs only by a constant. Given N diagnostic groups and y1,
y2, ……, yN cases per group, the effect on total costs of these groups is given by,
Wagstaff and Barnum (1992) noted, however, that this type of specification assumes
away the possibility of economies of scope. That is, the cost of jointly producing various
classes of outputs cannot be lower than the costs of producing each output category
separately, if total costs are merely the sum of all outputs
After recognising that the approach used for traditional markets4 is inappropriate in the
case of hospitals because of price and non-price distortions caused by different insurance
schemes, Ellis (1992) claimed that the usual technique in aggregating hospital throughputs
included a case-mix index in the specification of the hospital cost function5. This case-mix
index is generated by dividing the severity-weighted sum of hospital admissions, in which
diagnostic resource group (DRG) costs are used as measures of illness severity weighted by
the total number of admissions. Ellis cautions, though, that a drawback in using such a casemix index is that the index infects output variables with measures of inputs.
Several methodological problems remain with DRG funding. These include,
a) the adoption of an acceptable and an appropriate Australian costing model;
b) identification of marginal, as distinct from average costs;
c) incorporation of outpatient services;
d) adjustments for teaching and research; and
e) further refinement of DRGs for certain specialist services (eg, paediatrics).
Feldstein (1967) adopted surrogate variables, as the output mix measure and examined
the relationship between operating expenses per medical case and the proportion of cases in
each of 28 diagnostic proportional composition of case-mix. Various specialities were
grouped into eight mutually exclusive categories,
a) general medicine;
b) paediatrics;
c) general surgery;
d) ETN;
e) traumatic and orthopaedic surgery;
f) other surgery;
g) gynaecology and obstetrics.
4
Which is to deflate the price weighted sum of a subset of products by some price index in order to generate
quantity indices of outputs.
5
However, Ellis (1992) neglects to cite studies on cost function estimation that use this approach.
8
Watts and Klastorin (1980) used a cross-section sample of 315 short-term general
hospitals in the US, to compare the ability of various measures of case-mix to explain the
variation in average cost per admission per hospital. The study uses 10 case-mix variables and
proxy variables, including the number of beds in the institution. Four-element variables
counted the number of facilities and services in each of the four service categories (basic
service, quality-enhancing services and complex services) and the weighted sum of a number
of facilities and services reported by the hospital. The study was able to explain 70% of the
inter-hospital variation, due to four service categories. Given dimensionality, noted direct
case-mix variables performed better than the proxy variables.
Criteria guiding the selection of an appropriate case-mix measure relates to issues of
output homogeneity and the extent to which case-mix complexity can be controlled and
validated adequately, to enable investigations of inter-hospital cost variation.
In this study, we used two methods of analysing the technical efficiency of a hospital:
-the Stochastic Frontier Analysis and the DEA method
3.1. Considered production functions
a) Cobb-Douglas function:
The main property of this function is the constant marginal substitution rate between inputs,
since the partial derivatives are constant.
Then the natural logarithm Ln is:
LPUBLi = A + B1*LRESTIMEi + B2*LRESFUNDi + Ei
(2)
With LRESTIMEi = Ln(RESTIMEi) and LRESFUNDi = Ln(RESFUNDi), while A, B1 and
B2 are parameters to estimate.
b) Translogarithmic function:
It is a generalization of Cobb-Douglas´ function,
regressor. We consider the function to be estimated:
considering the squared variables as
LPUBLi=A+B1*LRESTIME+B2*LRESFUND+
B11*( LRESTIME2 )+B22*( LRESFUND2 )+ B12*LRESTIME*LRESFUND
(3)
With LRESTIMEi = Ln(RESTIMEi) and LRESFUNDi = Ln(RESFUNDi), while A, B1, B2,
B11, B22, and B12 are parameters to estimate.
This function is assumed to satisfy monotonicity and convexity conditions. Nevertheless, the
partial derivatives are not fixed. This feature makes the translog function more flexible than
the Cobb-Douglas function.
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c) C.E.S. function:
The third alternative is to use the constant elasticity of substitution (C.E.S.) function,
because it allows for the possibility for some firms to produce zero levels of a subset of
outputs (thus reducing estimated costs to zero), thus rendering this unappealing, as Baumol,
Panzar and Willig (1982) suggest. The proposed C.E.S. function is:
LPUBLi=A+(B1*LRESTIMEG1 +B2*LRESFUNDG2 )RO
With LRESTIMEi = Ln(RESTIMEi) and LRESFUNDi = Ln(RESFUNDi), where A, B1, B2,
G1, G2, and RO are parameters to estimate.
3.2. Dynamic analysis: Treatment of panel data
To use the average of the considered variables in the four available years to estimate the
proposed model implies a misuse of the available information, being more appropriate to use
the panel of data to carry out the estimate.
Considering a production function:
LPUBLit = f(RESTIMEit; LRESFUNDi; b) + Eit
Where PUBLit represents the “remarkable publications” for the ith firm at the tth observation
period; f(RESTIMEit; LRESFUNDit; b) is a suitable function of the two explanatory
variables representing the factor inputs associated with the “remarkable publications” of the
ith department in the observation period tth; b is a vector of unknown parameters; Eit is the
error term for the ith observation in the tth period. The error term Eit is made up of two
independent components:
Eit = Vit - hit Uit
Where Vit is N(0; ) distributed, that is a two-sided error term representing the usual
statistical noise found in any relationship; Uit is assumed to be independent and identically
distributed non-negative truncations of N(MU, s2), following Battese and Coelli (1992),
where MU, s and sv are parameters to be estimated, and hit is a behaviour specification of the
department effects over time. The density function of Ei is:
,
where F is the standard normal distribution function, S the number of time periods;
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Ei = LPUBLi - f(RESTIMEit; LRESFUNDi; b), is a (Sx1) vector and represents the error
term for the ith department in each of the S considered periods ;is a (Sx1) vector and
represents the functional specification of firm efficiency evolution over time
Therefore, the likelihood function for a sample of n departments is:
Ln(L(X;E)),
the
second
likelihood
function
Next we describe the necessary modifications to analyse efficiency evolution considering
cases of fixed and variable effects. In the last place, we distinguish different formulations
for inefficiency variability over time.
11
3.3.
Fixed effects
When hit =
over time are fixed. In this case, the Ei density function is:
, efficiency effects
And the second likelihood function is the next:
In this case, hit =
is a lineal
specification of efficiency evolution, in which technical efficiency must either increase at a
positive rate (h>0), or decrease at an negative rate (h>0) or remain constant at a zero rate
(h=0):
3.4. Static analysis
The O.L.S. regression analysis (chart I) shows a high level of confidence in the LRESTIME
variable as an explanatory factor of the “remarkable publications” number, while
LRESFUND is significant at 94,8%. The regression analysis is carried out in order to
estimate the parameters as initial conditions of the posterior maximum likelihood estimates.
12
With these results, we proceeded to a maximum likelihood estimate of a Cobb-Douglas´
function adjusted specification, but splitting the residuals in two, a normal and a half-normal
component following Aigner, Lovell and Schmidt (1977). In this estimate it can be observed
(chart II) that the two variables introduced are clearly explanatory, although the variance of
the normal component is not significant, meaning that the residuals follow a half-normal
distribution truncated in the zero value. Problems of collineality are not detected among the
regressors, since the correlations among the parameters are not excessively large.
As a generalization of the Cobb-Douglas based specification, a translogarithmic function is
proposed, under the same assumptions of residual decomposition, and estimated in the same
way through maximum likelihood (chart III), so as to obtain clearly significant parameters,
and validating the fact that the variances of both components of the residuals are significantly
different from zero, confirming the assumption carried out by its distribution. It was be
observed that the correlations among the estimates are larger that in the previous estimate,
causing multicollineality problems; that is, imprecision in the estimates of each of the
parameters, although the precision of the combined adjustment does maintain its validity,
and can be used to carry out predictions concerning the endogenous variable.
The last estimate that is carried out in relation to the average of the four year variables (chart
IV) is a C.E.S. function based specification obtaining, in the same way, that all the
parameters are significant, including the two error terms standard deviations in which the
perturbations are disaggregated. This estimate should be taken with good judgment, since the
correlations among parameters are quite large.
3.5. Dynamic analysis
The Panel data estimate allows for the introduction of more information in the model, to test
the static model results, and, simultaneously, to establish hypothesis concerning the
evolution of technical efficiency.
Using the assumptions of residual desaggregation described in epigraph 2.2, we proceed to
estimate panel data maximum likelihood, using a specification based on the Cobb-Douglas
function, but assuming fixed effects; that is, the residuals do not suffer an evolution caused
by the course of the time. In this estimate (chart V), it is observed that LRESTIME is
influential upon the value of the endogenous variable, but LRESFUND is not. It can be also
be observed that the average as well as the typical deviation (MU and SIG) of the residual
component distributed as a truncated normal significantly equals zero. Thus we can deduce
that in this specification an error term division is unnecessary and the usual residual
normality assumption is valid. Moreover, the correlations between the parameters are not
excessively high, therefore severe multicollineality problems should not exist.
The following analysis consists of relaxing the assumption of invariability of the residuals
over time, considering that these evolve linearly (chart VI). In this case LRESTIME is
explanatory of LPUBL, but non LRESTIME. On the other hand, by means of the functional
specification that represents the evolution of residuals over time, the typical deviation of the
residual component, distributed as a truncated half-normal, is significant although its
average again equals zero.
The following estimate is identical to the previous one, but assuming that the expression
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modelling the temporal evolution of residuals is exponential instead of lineal, obtaining a
very similar result to the previous analysis, but with the ET parameter positive, a logical
result, since, in the exponential specification, the parameter interpretation is of opposite sign
to the lineal one (chart VII). The likelihood function value corresponding to the lineal
specification is slightly higher than that of the exponential one, suggesting an adjustment
better adapted to the first assumption.
As a generalization of this last specification, based on the Cobb-Douglas function, we
propose a translogarithmic function, with the assumptions of decomposition as in the
previous analysis, considering again that these evolve over time according to an exponential
function. The results (chart VIII) of this estimate shows that all the parameters of the
proposed function, except those related to the interaction of the two explanatory variables,
are 95% significant. It is also observed that, as in previous estimates, the average of the
residual component with truncated normal distribution significantly equals zero, although
the typical deviation is significantly different to zero, so as to affirm that this component is
half-normal adjusted. Given the high correlations between parameters, the values of the
individual estimates of parameters should be taken with caution.
3.6.
Efficiencies
To conclude, chart IX shows the static model efficiencies and the dynamic model yearly and
firm estimates. We use the translog function as a reference, lsince it is the most flexible
functional form of those analysed in the dynamic model. To test the homogeneity of both
samples we have carried out the Kruskal-Wallis test on the five independent samples of the
efficiencies estimates, for each one of the four years panel and the efficiencies
corresponding to the static estimate. As chart X shows, the test allows us to accept the null
hypothesis of homogeneity of the five studied samples with a level significance of 0,8%.
If two units of output are heterogeneous, they belong to different output categories. In the
context of hospitals, the problem is that no two patients may receive exactly homogeneous
treatment, for " the theory is as extensive as the number of patients it serves" (Fetter, et al.
1980,p.1). Given this situation, once cases are aggregated into groups, some within-group
heterogeneity is inevitable and will increase with the level of aggregation. It is because of this
outcome, that the use of restricted functional form for estimating a hospital’s form is not
parsimonious in parameters. With any given number of observations, the number of output
categories which can be used is restricted. This leads to an increase in within-group
heterogeneity.
4.
Teaching Status and Non-teaching Status
An important issue for teaching hospitals has been the cost of medical education.
Researchers have attempted to determine, whether the higher costs of teaching hospitals are
due to the costs of medical education or other factors with severity of the hospital case-load.
There were mixed results in the empirical studies conducted in the US. Some studies
suggested that medical education increased the cost of hospital care for similar groups of
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patients, through increased numbers of diagnostic tests (Frick, Martin, and Schwartz, 1986).
Other studies found that teaching and non-teaching hospital cost differences were reduced
significantly, when hospital case-mix measures were included (Watts and Klastorin, 1980).
A study by Goldfarb and Coffery (1987) examined the differences between teaching and
non-teaching hospitals in admission policies, treatment decisions, and discharges outcomes.
The study attempted to isolate inherent distortions in the use of illness classification systems
to favour hospitals that choose costly treatment. The conclusion is that, in a world in which
allowable classification schemes were restricted to DRGs and where the weighting schemes
may or may not be resource based, teaching hospitals have a more serious case- mix than nonteaching hospitals. However, this is true only when DRGs are the basis of the classification
system and the uses of resources are prominent features of the weighting system.
5.
Treated Cases and Length of Stay
A unit of output measurement commonly adopted in studies of hospital costs is the
patient day, with total output over a given period, then being taken as the total number of
patient days provided over that time period. This is considered as an alternative to measure
output, by using the number of cases treated. Feldstein (1967, p.24) states that "we must
choose between two basic units of output: the case and the patient week". Lave and Lave
(1970) employ the patient day as the unit of output measurement, arguing that although "A
more relevant measure is the number of cases treated. The question of whether patients or
patient-days is the better measure can not be settled a priori"
Feldstein (1967) estimated the various types of marginal costs from total cost equations.
Total cost was expressed as a function of case-mix, the number of case treated and the number
of beds. The first type of marginal cost was derived when only the stock of beds is fixed. The
second type of marginal cost was derived when the number of occupied beds fixed. Linear
version of these equations give estimates of marginal cost on a per case basis equivalent to
21% of the average cost per case (in the case when only the number of stock of bed is fixed)
and 12% of the average cost per case when the occupation rate is fixed. Marginal costs on a
per-patient-day basis can be inferred from the parameter estimates of these equations. These
marginal costs are equivalent to 54% of the average cost figure (in the case where both length
of stay and the occupancy rate can change) and 745 of the average cost figure (in the case
when both length of stay can change). All these results are found to be robust in changes in
functional form and estimation method.
These studies argued that the treated case was, a priori, a more defensible unit of output
measurement than the patient day. Indeed, the latter is more like an input measure relating to
the time dimension of the production of a treated case. The number of patient days, which a
hospital uses to produce a treated case, indicates the time period over which production of one
unit of output takes place, for it does not measure the output itself. Feldstein (1969) disagreed
and showed that there was no clear line to divide inputs and outputs, in the context of efforts
to improve the level of community’s health.
The treatment conception of hospital output accords with the view of hospital output, as
an intermediate product used as an input into the production function for health. This view is
summarised by Berki (1972, p.42):
" If we consider that the final product of the medical care process is the provision of the
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highest attainable of health, given the state of the arts, it is clear that the output of hospitals is
more precisely an intermediate input into this process".
Whatever position is adopted on these conceptual issues, empirical reality is the way
hospital output forms the basis of most applied work on hospital cost analysis. Discussion
showed stress consideration of the unit of measurement to be employed in empirical
applications, based on patients treated or discharged.
Patients receiving treatment face degrees of uncertainty of the effect of that treatment on
their health. If patients pay for treatment and gain nothing in term of their health status, that
amounts to financial loss. Financial loss may include loss of income, due to reduced earning
ability arising from the prolongation of the illness. Under such a circumstance, insurance
could be provided by a third party, who pays for the treatment in the event of being
unsuccessful. It could also be provided by the hospital or the medical practitioner, if they
agreed to accept full payment, only if the treatment improves health. Under these
circumstances, the risk of the financial loss following unsuccessful treatment is shifted to the
supplier of the treatment. The supplier can now be thought of as producing not one but two
outputs: a) the treatment itself; and b) insurance, against the financial loss that may arise
because of the uncertain impact of that treatment on health status.
It should be noted that the treated-case has been argued to be a defensible unit of output,
than patient-day for treatment provided in hospitals. The argument does not apply necessarily
to other institutions which may be providing a different type of output. In-patients in nursing
homes do not generally receive treatment for a specific illness or illnesses, but are assisted or
cared in matters of everyday living (cooking, bathing, and so on). For such institutions a
strong case can be made, that a day of care is the unit of care, not a treatment. This can give
rise to problems in measuring hospital output, if nursing home type patients are being cared in
hospitals (along with the usual patients requiring acute care). They should, therefore, be
recognised as receiving a conceptually distinct type of output. This distinction, however, is
often drawn in theory than in practice, giving rise to empirical problems. In discussing the
concept and measurement of hospital output, this paper has briefly alluded to the fact that
treatments may differ for different illnesses. Under these circumstances, a treated case may
not be a homogeneous unit of output even within a hospital. This is another important
dimension of the multi-product nature of the hospital, to be addressed.
6.Technical and Allocative Efficiency
These issues are providers being:
a) allocatively efficient;
b) technically efficient;
c) producing the mix of outputs society values.
The issue addressed by the cost and production function literature is the question of
technical efficiency. There is attention to levels of technical efficiency of hospital type and if
type A is higher than that of hospital type B? The issue of technical efficiency has received
less attention than the issue of allocative efficiency. Feldstein (1967) was the first to
investigate this issue and suggested the residual of the production function as a measure of
technical efficiency. A hospital with a residuals equal to zero were of average technical
efficiency, whilst, hospitals with residuals greater (smaller) than zero were of above-average
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(below-average) technical efficiency (Feldstein, 1967). The rational being that the output of a
hospital with a residual equal to zero, is exactly the output that would be expected on the basis
of average estimated output elasticities. A hospital with a positive (negative) residual, by
contrast, produces more (less) output than expected on the basis of the estimated parameters
of the production function.
7.
Conclusions
This overview cannot be considered complete, without a brief discussion on issues
pertinent to hospital cost function estimation that will have to be addressed, if empirical work
on the subject is to relevant. These are the effects on hospital costs of,
a) doctors fee and the unique relationship that exists between doctors and their patients;
b) various health insurance schemes;
c) uncertainty in terms of diagnoses and treatment outcomes; and
d) type of hospital structure, ownership, and conduct.
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