This paper examines costs in the Swiss nursing home industry, an issue of concern to Swiss policy makers because of the explosion of elderly care costs and the aging of the population. The paper considers estimation of a translog cost function employing panel data for a sample of 36 nonprofit nursing homes operating in Ticino, a canton of Switzerland, over the period 1993-1995. Using a set of dummy variables quality differences are considered. The results of this analysis indicate the existence of economies of scale for most output levels.nursing homes, Switzerland, panel data, cost function

Massimo Filippini

English

Research Papers in Economics

1
ECONOMIES OF SCALE  IN THE SWISS NURSING HOME
INDUSTRY§
by
Massimo FILIPPINI
Department of Economics - Università della Svizzera Italiana
Via Ospedale 13
6900 Lugano, Switzerland
E-mail: massimo.filippini@lu.unisi.ch
January 1999
ABSTRACT
This paper examines costs in the Swiss nursing home industry, an issue of concern to
Swiss policy makers because of the explosion of elderly care costs and the aging of the
population. The paper considers estimation of a translog cost function employing panel data
for a sample of  36  nonprofit nursing homes operating in Ticino, a canton of Switzerland,
over the period 1993-1995. Using a set of dummy variables quality differences are
considered.
The results of this analysis indicate the existence of economies of scale for most output
levels.
                                                       
§  Part of this work was done while I was a visiting research fellow at the Center for Health Economics, University
of York (Summer 1998). I am grateful to Antonio Giuffrida and Andrew Jones for their comments on earlier
versions of this paper and the Dipartimento Opere Sociali, Canton Ticino, for general assistance. The views
expressed in this paper are strictly personal. Responsability for any remaining errors lies solely with the author.2
1. Introduction
The cost of residential provision for the elderly has risen steadily in the last 15 years.
This dramatic increase and the negative situation of the Government’s budget deficit have
placed pressure on nursing homes to contain costs. A possible way to reduce costs per unit of
service  could be to promote nursing home mergers. Arguments for consolidation of the
provision of residential services for the elderly are based on the presumed existence of
economies of scale in the production of such service. If economies of scale exist, then a strong
economic case could be made for the consolidation of small nursing homes in larger ones.
Possible sources of such economies  could be, for instance, a better utilization of the
administrative, medical records and laundry services.
This paper estimates scale economies in the nursing home industry, using data from
nursing homes operating in Ticino, a canton of Switzerland.1 For this purpose we estimate a
translog cost function using data for 38 nonprofit nursing homes over the period 1993-1995.2
These nonprofit nursing homes are regulated in terms of quality standards and rates by the
cantonal government, which gives them subsidies  for the operating costs.
The paper is organized as follows. Section 2 discusses the cost model; section 3
presents the data, while section 4 illustrates the  Parameter estimates of the total cost function
and the results in terms of economies of scale. Section 5 concludes the paper.
2 A Total Cost Function for the Nursing Homes
The costs of operating a nursing home are the costs of the building and the equipment
and the  costs of taking care of the residents. A nursing home can, therefore, be represented as
a firm transforming two inputs into patient-days of nursing home care. Moreover, in the cost
model specification we take into account a number of output characteristic variables, which
should capture the heterogeneity dimension of the output of a nursing home.
                                                       
1 Switzerland is a confederation composed of 26 cantons and approximately 3000 municipalities. This federal
state is characterized by a high degree of decentralization in the provision of public services. For instance, each
canton is very independent in the organization and regulation of the provision of long-term care for elderly
people. Moreover, some cantons decentralize this task to the communes (local government), while others tend
to organize the provision of this service at the cantonal level.
2 See  Vitalino & al. (1994), Mobley and Bradford (1997) and McKeay (1990) for previous studies on the cost
structure of Nursing Homes.3
Assuming that output and input prices are exogenous, and that (for a given technology)
firms adjust input levels so as to minimize costs, the firm's total cost of operating a nursing
home can be represented by the cost function
  TC = h ( Y, Q1, Q2, PK, PL, DIF, T) (3)
 
where TC represents total cost and Y is the output represented by the total number of
patient-days of nursing home care. PK and  PL are the prices of capital and labor,
respectively. Q1  is an index that measures the average required assistance of a home’s
patients with normal daily activities such as eating, personal care and performing
physiological functions. This index is calculated yearly by the Regional Department of Public
Health. Q2 is the ratio of the number of nurses employed by a nursing home and the number
of nurses that should theoretically be employed according to the guidelines of the Regional
Department of Public Health.3 These two variables, Q1 and Q2, are, therefore, introduced in
the model as output characteristics. In order to distinguish nursing homes which offer the
possibility for a relatively small part of the people to live with a small support in an apartment
located beside the main building, we introduce in the model a dummy variable, DAPP. T is a
time variable which captures the shift in technology representing change in technical
efficiency.
To be consistent with economic theory the  translog cost function (2) should be
concave and linearly homogeneous in input prices and  non-decreasing in input prices and
output.4
To estimate the cost function (1), a  translog functional form is employed. This
flexible functional form is a local, second-order approximation to an arbitrary cost function. It
places no a priori restrictions on the elasticities of substitution and allows the economies of
scale estimate to vary with the output level.5 The translog approximation to (1) is
                                                       
3 The nursing homes are not obliged to exactly follow the guidelines of the regional Department of Public
Health.
4See Cornes (1992), p.106.
5 A translog function requires the approximation of the underlying cost function to be made  at a local point,
which in our case is taken at the median point of all variables. Thus, all  independent variables are
normalized at their median point.4
T DAPP Q
P
P
Q
P
P
Q Q
Q y Q y
P
P
y
P
P
Q
Q y
P
P
Q Q y
P
TC
T DAPP
K
L
LQ
K
L
LQ Q Q
yQ yQ
K
L
yL
K
L
LL Q Q
Q Q yy
K
L
L Q Q y
K
a a a a a
a a a a a
a a a a a a a
+ + + + +
+ + + + +
+ + + + + + =
2 ln ln 1 ln ln 1 ln 2 ln
2 ln ln 1 ln ln ln ln ) (ln
2
1
) 2 (ln
2
1
) 1 (ln
2
1
) (ln
2
1
) ln( 2 ln 1 ln ln ) ln(
2 1 1 2
2 1
2 2
2 2
2
1 1
2
2 1 0
 (2)
Note that by normalizing total cost and input prices by one of the input prices, we
impose the theoretical condition that the cost function is linearly homogeneous in input prices.
In order to improve the efficiency of the estimation of least squares parameter
estimates for the cost function, a cost system is estimated. This system consists of the translog
cost function (2) and the factor share equation for labor. By applying Shephard's lemma, the
resulting share equation takes the familiar form:
2 ln 1 ln ln ) ln( 2 1 Q Q y
P
P
S LQ LQ yL
K
L
LL L L a a a a a + + + + =
(3)
The price by which we normalize is that of the input whose share equation we
dropped from the estimating system.
The  translog cost function permits scale economies to vary with the level of
output, factor prices and the output characteristic variables. Economies of scale (ES) are
defined as the proportional increase in total cost resulting from a proportional increase
in output (Y), holding all input prices and output characteristic variables fixed. This is
equivalent to the inverse of the elasticity of total cost with respect to the output (Caves,
Christensen, and Tretheway 1984) :
2 ln 1 ln ln ln
1
ln
ln
1
2 1 Q Q
P
P
y
Y
TC
ES
yQ yQ
K
L
yL yy y a a a a a + + + +
=
¶
¶
=        (5)
We will talk of economies of scale if ES is greater than 1, and accordingly, identify
diseconomies of scale if ES is below 1. In the case of ES = 1  no economies or diseconomies
of scale exist. Economies of scale exist if the average costs of a nursing home decrease as
output increases.5
3. The data
This study is based on a combined time series and cross-sectional data set for 38
nonprofit nursing homes for elderly people operating over the period 1993-1995 in Canton
Ticino.
For regulatory purposes each year the nonprofit nursing homes are required to send to
the cantonal Department of Public Health information on operating costs and on the
characteristics of the residents.6 Thus, the data set used in this study has been created using
the information contained in these annual reports by the nonprofit nursing homes.
Total cost is taken to be the total expenditures of the nursing homes. Output is measured
in total number of patient-days of nursing home care. Average yearly wage rates are estimated as
the weighted mean of the average wage rates of the different professional categories working in a
nursing homes: doctors, nurses, administrative and technical staffs.
Following Friedlaender and Wang Chiang (1983) and  Filippini and Maggi (1993), the
capital price is calculated from the residual capital costs divided by the capital stock. Residual
cost is total cost minus labor cost. According to  Wagstaffs (1989), the capital stock is
approximated by the number of beds owned and operated by a nursing homes. Unfortunately no
data are available which would allow us to calculate the capital stock using the capital inventory
method. The quality indicators, Q1 and Q2, are calculated yearly by the regional Department of
Public Health. All input prices, total costs and variable costs were deflated to 1996 constant
Swiss francs using the Consumer Price Index.
4. Estimation Results and Economies of Scale
The multivariate system of equations (2) and (3) has been iteratively estimated using
the Zellner (1962) procedure for seemingly unrelated regressions.7
The estimated coefficients of the translog cost model (2) are presented in Table 1.
The results are satisfying in so far as all first order coefficients and most of the second order
coefficients are significant and carry the expected signs. As can be seen, the corrected R2 for
the model is also satisfying with a value of 0.98. Since variable cost as well as the dependent
variables are in natural logarithms and have been normalized, the first order coefficients are
interpretable as cost elasticities evaluated at the sample median.   The output elasticity is
                                                       
6 Unfortunately no data are available for the for-profit nursing  homes. These nursing homes, which do not
receive subsidies from the Department, are not obliged to send to the cantonal Department of Public Health
information on operating costs and on the characteristics of the residents.
7  The model has been estimated using the computer program LIMDEP 7.0.6
positive and implies that a 1% increase in the number of patient-days of nursing home care
will increase the total cost by approximately 0.86%.
The cost elasticities with respect to the output chartacteristics variables, Q1 and Q2,
are positive and imply that an increase in the average required assistance of a home’s patients
or an increase in the ratio of the number of nurses employed by a nursing home and the
number of nurses that should be theoretically be employed will increase total cost. The
estimated coefficient for time (proxy for technical change) is negative and significant,
showing the presence of neutral technical change.
Table 1 : Total cost parameter estimates (standard errors in paranthese)
     Parameters
Constant 15.029***
(0.017)
a Y 0.860***
(0.023)
a Q1 0.844***
(0.090)
a Q2 0. 630***
(0.047)
a PL 0.817***
(0.002)
a YY 0.251**
(0.069)
a Q1Q1 0.044
(0.716)
a Q2Q2 0.504**
(0.122)
a PLPL 0.101***
(0.545)
a YQ1 0.816
(0.225)
a YQ2 -0.408**
(0.122)
a YPL 0.007
(0.006)
a Q2PL 0.082***
(0.008)
a Q1PL 0.186***
(0.021)
a Q1Q2 -0.009
(0.244)
a APP 0.297
(0.017)
a T -0.013**
(0.006)
R2 0.986
    *, **, *** significantly different from zero at the 90, 95 and 99 % confidence level.7
Parameter estimates of the cost function satisfy the regularity condition of concavity
in input prices at the median point of approximation, which requires that the own-price
elasticities of inputs be negative and that the Hessian Matrix,  ¶ ¶ ¶
2 C w w i j / , be negative
semi-definite. Because homogeneity in input prices and symmetry of the second order terms
were imposed, the estimated functions satisfy all regularity conditions of a theoretically valid
total cost model.
Economies of scale
The estimation results from Table 1 can be utilized to calculate, using equation
(3), the values of the economies of scale in the production of the Swiss nursing homes.
Table 2 presents in more detail the results for small, medium-sized and large nursing
homes, respectively. We note that all values of the indicator for economies of scale are
greater than 1, which means that the majority of the nursing homes operate at an
inappropriately low scale. Up to a capacity of  about 120 beds,  the economies of scale
are exhausted.
 
  Table 2  Economies of scale 8
                                                       
  8  Equation (5) has been evaluated at the input prices  and output characterisctics variables, Q1 and Q2, of the
median nursing home.
Nursing
home size
small
Patient-days =  15160
 Bed capacity = 41
medium
Patient-days =
20187
 Bed capacity = 57
large
Patient-days =  26733
 Bed capacity = 80
ES  Model 1 1.19 1.18 1.15
5 Conclusion
The paper consider estimation of a translog cost function for a sample of  36 Swiss
nursing homes using panel data for the period 1993-1995.8
The empirical evidence indicates the existence of economies of scale for most output
levels. This result suggests that efficiency gains could result from merging smaller nursing
homes operating in the same area. Of course, mergers between two small nursing homes do
not, as a rule, result in a single larger nursing homes. The merger will generate the sought
after efficiencies only if the two nursing integrate some clinical and administrative functions
so as to act as a single nursing home. Moreover, by planning the size of new nursing homes
the economies of scale effects on the costs should be taken into account. However, the size of
the nursing homes should not reach an overly large size which could determine a deterioration
of the quality in term of human contacts.
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Economies Of Scale In The Swiss Nursing Home Industry

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Economies Of Scale In The Swiss Nursing Home Industry

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