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WORKING PAPERS
Population, Health, and Nutrition
Population and Human Resources
Department
The World Bank
November 1992
WPS 1044
Hospital Cost Functions
for Developing Countries
Adam Wagstaff
and
Howard Barnum
A critical survey of the techniques available for analyzing
hospital costs and a review of the few hospital cost-function
studies undertaken for developing countries.
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Policy Research
Populallon, Health, and NLitrtion
WPS 104
Thispaper-a productofthe Health andNutrition Division, Population and Human Resources Department
- is part of a larger effort in the departnent to examine the efficiency of resource allocation for human
services. Copies of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC
20433. Please contact Otilia Nadora, room S6-065, extension 31091 (November 1992, 32 pages).
There is an extensive literature on hospital cost Although their paper is intended primarily
functions for industrial countries, and a small for those working in developing countries, the
literature for developing countries. Yet the issues discussion of cost function methodology has
facing policymakers in all countries are much the broad implications for interpreting econometric
same: Are hospitals overcapitalized, as is often cost functions and for examining economies of
claimed of U.S. hospitals? Are hospitals ineffi- scale and scope in both developing and industrial
cient in other respects? Do hospitals vary in countries.
efficiency? Are private hospitals more efficient
than their public counterparts? Should hospitals Their survey of econometric techniques is
specialize or provide a broad range of services? not uncritical. They question, for example, the
Should costs be reduced by concentrating cases validity of recent tests of overcapitalization
in fewer hospitals? undertaken on American hospitals. They also
make general observations about the methods
Wagstaff and Barnum critically survey the used to investigate economies of scope and
techniques available for analyzing hospital costs economies of scale.
and review the few hospital cost-function stu4ies
undertaken for developing countries.
The Policy Research Working Paper Series disseminates the findings of work under way in the Bank. An objective of the series
is to get these findings out quickly, even if presentations are less than fully polished. The findings, interpretations, and
conclusions in these papers do not necessarily represent official Bank policy.
Produced by the Policy Research Dissemination Center
Hospital Cost Functions For Developing Countries
by
Adam Wagstaff
School of Social Sciences
University of Sussex
Brighton BN1 9QN
UK
and
Howard Barnum
Population and Human Resources Department
The World Bank
Washington DC 20433
USA
Table of Contents
1. Introduction .......................................,, 1
2. Some econo nics and econometrics of hcspitals .1
2.1 Are hospitals over-capitalized?. 1
2.2 Are hospitals ineffcient?. 4
2.3 Should hospitals specialze?. 8
2.4 Toc many hospitals? .......................... 10
3. Hospital costs in Kenya .................................. 15
3.1 Are Kenyan hospitals over-capitaHzed? .16
3.2 Inefficiency in Kenyan hospitals. 16
3.3 Economies of scope in Kenyan hospitals. 16
3.4 Too many hospitals in Kenya? .17
4. Hospital costs in Peru .17
4.1 Are Peruvian hospitals over-capitalized? .............. 18
4.2 Inefficiency in Peruvian hospitals .18
4.3 Economies of scope in Peruvian hospitals .19
4.4 Too many hospitals in Peru? .19
5. Hospital costs in Ethiopia .20
5.1 Are Ethiopian hospitals over-capitaized? .20
5.2 Inefficiency in Ethiopian hospitals ............... 21
5.3 Economies of scope in Ethiopian hospitals .21
5.4 Too many hospital in Ethiopia? .22
6. Health care costs in Nigeia .22
6.1 Are Nigerian health care institutions over-capitalized? .24
6.2 Inefficiency in Nigeria health care institutions .24
6.3 Economies of scope in Nigerian health care institutions .25
6.4 Too many health care institutions in Nigeria? .25
7. Conclusions .26
References .30
1
1. Introduction
There is an extensive literature on hospital cost functions for
industrialized countries, but a much smaller literature for developing
co: 'tries.1 Yet the issues facing policy-makers in both sets of
cot .tries are much the same: Are hospitals over-capitalized, as is
often claimed of US hospitals? Are hospitals inefficient in other
respects? Do hospitals vary in their degree of inefficiency? Are
private hospitals more efficient than their public counterparts? Should
hospitals specialize or provide a broad range of services? Could costs
be reduced by concentrating cases in fewer hospitals? Such issues are
as relevant to policy-makers in developing countries as to their
counterparts in the industrialized world.
Moreover, it seems likely that the econometric techniques that
have been used to such effect in the industrialized world would be
equally capable of informing policy debate in the developing world. Our
purpose in this paper is to provide a survey of these techniques and to
summarize the work to date on developing countries. Though the paper is
written primarily with developing countries in mind, it is hoped that it
may be of some value to researchers in industrialized countries. Our
survey of econometric techniques is not uncritical. We question, for
example, the validity of recent tests of over-capitalization undertaken
on American hospitals. We also make various general observations about
the methods used to investigate economies of scope and economies of
scale.
Section 2 provides a survey of the issues facing policy-makers in
the hospital sector and the relevant econometric techniques. sections 3
to 6 contain surveys of four recent studies of hospital costs in Kenya,
Peru, Ethiopia and Nigeria. The final section contains our conclusions
concerning both the studies undertaken to date and the methods for
analyzing hospital costs.
2. Some economics and econometrics of hospitals
As indicated above, there are, broadly-speaking, four sets of
policy issues that can be addressed using cost function analysis: (i)
are hospitals over-capitalized?, (ii) are hospitals inefficient?, (iii)
should hospitals specialize or provide a broad range of services?, and
(iv) are there too many hospitals?
2.1 Are hosoitals over-canLtalized?
The issue is whether hospitals have a capital stock that is too
large given their output level. In developed countries - especially the
United States - it is often argued that this is the case, the
'. For reviews of the American and British literature see Cowing, Noltmann and Peters (1983) and Wagstaff
(1989a). Several of the more recent studies for industriaLized countries are referred to in the present paper.
To our knowledge there has been no survey of developing country studies.
2
implication being that hospitals ought to reduce their capital stock.
It is important to realize that the issue of over-capitalization is
different from the issue of economies of scale. The latter concerns the
effect on costs of further expansion of output and therefore bears on
the question of whether output levels are too high or too low. Authors
of many early cost function studies confused the two issues. Writers
such as Feldstein (1967) and Anderson (1980), for example, claim to
address the issue of economies of scale, but in the event analyzed
over-capitalization .2
The currently favored approach to over-capitalization dates back
to Cowing and Holtmann (1983), who proposed analyzing the issue by
reference to the parameters of the variable cost function. In the
short-run, when the capital stock is fixed, short-run costs, CS, can be
written as the sum of total fixed costs and total variable costs
(1) Cs = F + Cv(Y WV,K) ,
where F=wKK is total outlay on the fixed input, y is output, wv is the
price(s) of the variable input(s) and K is the stock of capital. It is
worth noting that K enters CS twice: once as a determinant of fixed
costs and once as a determinant of variable costs - an issue to which we
return below. The long-run cost-minimizing capital stock, K*, is that
which minimizes costs at each level of output. Differentiating (1) with
respect to K and setting the derivative equal to zero yields
(2) -wK = 6Cv/6K.
To test for over-capitalization Cowing and Holtmann suggest estimating
the variable cost equation Cv(y,wv,K) and testing the hypothesis that
6Cv/6K=-wK.
The intuition here can be illustrated with fig 1, which shows
three possible input mixes to produce a single output y. Without loss
of generality set wK equal to 1. Then the vertical intercept of each
isocost line is equal to Cs, the value of K is equal to F and the
vertical distance between Cs and F is equal to Cv. The long-run optimal
stock of capital is K. At point '1', the capital stock is too small
and a move from point '1' towards point '2' would cause total cost to
fall. (Fixed costs would rise, but this rise would be more than offset
by a fall in variable costs.) Hence at point '1' 6Cv/6K<-wK. At point
'3', by contrast, the capital stock is too large and a reduction in
capital towards K would reduce total cost. (Variable costs would rise,
but this would be more than offset by the reduction in fixed costs.)
Hence at point '3' 6Cv/6K>-wK. Only at point '2', where the cap:tal
stock is at its long-run equilibrium value, is 6Cv/5K equal to -wK.
2. That the approach used by Feldstein is itt-suited to the analysis of economies of scale was argued long
ago by Davis (1968) and Mann and Yett (1968). That his approach is better suited to the analysis of
over-capitaLization has not, to our knowledge, been argued before.
3
In Cowing and Holtmann's
study, as in most cost function K
studies in which a proxy for ')\
capital stock is included among 3
the regressors, the derivative F
of cost with respect to capital
is positive. Cowing and
Holtmann interpret this as
evidence of
over-capitalization. However, 2 2
this interpretation is, in K .\ ....... .
fact, inconsistent with F' I c . -
economic theory. An increase Yo
in capital ought always to v
reduce variable costs, because
usage of variable inputs must Figure I
decline if output is to remain
unchanged. Hence 6Cv/6K ought
always to be negative. That
the derivative of cost with respect to capital is positive in such
studies suggests that the dependent variable may not actually be
variable costs, but may instead include some fixed costs. If, for
example, all fixed costs are included in the cost variable, a positive
6Cs/6K makes perfect sense; indeed this is precisely the condition for
over-capitalization.
This suggests an
alternative testing strategy,
which addresses the fact that
it will typically be difficult cs
to purge cost data of fixed
costs: estimate a short-run C CY, WV K)
total cost equation and test
the hypothesis that 6Cs/6K=O. ......
This is similar to the approach Cs
adopted long ago by Feldstein '
(1967), though he confusingly c2 ............
argued that he was
investigating economies of K' K3 K
scale rather than
over-capitalization. From fig _
1 it is evident that the Figure 2
partial relationship between
short-run total (and average)
cost and capital stock is
U-shaped (fig 2). Moving along this curve is equivalent to moving from
one short-run average cost curve to another (fig 3). At point '1'
6C5/6KO
(and, by implication, 6Cv/6K>-wK). Hence if one estimates a short-run
total (or average) cost function and finds that a hospital is operating
to the right of the minimum point of the partial relationship between
4
cost and capital stock, one can conclude that the hospital is
over-capitalized.
Theory suggests various
restrictions that need to be c
placed on a short-run cost
function and it is worth*
spelling these out. First, the C5
equation could be additive in cL5
fixed and variable costs. If, \
C
as is often the case, the
shadow price of capital is c
unknown, the fixed cost
component of the equation ought
not to contain an intercept and
instead capital ought to be
entered as a regressor; its
coefficient would then be Q -
interpreted as the shadow price Fiure3
of capital. Second, assuming
convexity of isoquants, total
variable costs ought to be
decreasing in capital stock with the derivative approaching zero
asymptotically. In other words, as substitution of labor for capital
gets harder and harder, an ever increasing share of the extra cost
associated with increases in capital stock will be taken up by extra
fixed costs. The partial relationship between short-run total cost and
capital stock ought, as indicated above, to be U-shaped.
2.2 Are hbositals inefficient?
A distinction is normally made between technical inefficiency
(failing to produce the maximum possible output from a given bundle of
inputs) and allocative inefficiency (employing inputs in the wrong
proportions given their prices and productivity at the margin). The
issue of over-capitalization examined above is, of course, one aspect of
allocative efficiency. In principle both types of inefficiency might be
present in the hospital sector and it is useful for policy-makers to
know the extent of any such inefficiency in the hospital sector as a
whole, as well as any variation across hospitals. It is also of
interest, of course, to know whether there is any variation between one
sub-sector (eg the private sector) and another (eg the public sector).
Technical efficiency can be analyzed using either non-statistical
approaches3, such as data envelopment analysis, or statistical methods,
3. For an example of the non-statistical approach in the context of the hospitaL sector see Grosskopf and
Valdmanis (1988).
4
such as the frontier production function4. We focus here on the latter.
Suppose for simplicity that the production function is
Cobb-Douglas:
(3) lny, = 00 + Ejpjlnxu + ui
where y is output, the xj are inputs, the P's are output elasticities
and u is an error term. Feldstein (1967) suggested that the residuals
of eq (3) might be used as estimates of technical inefficiency, so that
a hospital with a zero residual is said to be of average technical
inefficiency, while a hospital with a positive (negative) residual is
said to be of above-average (below-average) technical efficiency.5 A
disadvantage of this approach is that it provides no information on the
level of efficiency. Clearly, it is important to know whether
inefficient hospitals are very inefficient or only marginally so. This
defect can be overcome using the deterministic frontier model (DFM) of
Aigner and Chu (1968), which differs from eq (3) in that it constrains
the error term to be non-positive. Hospitals can thus operate on or
below the production frontier but not above it, and the extent of
technical inefficiency is indicated by the estimated residuals, u1. The
DFM can be estimated using a variety of methods, the simplest of which
is Corrected Ordinary Least Squares: this involves shifting up the OLS
estimate of the intercept until one residual is zero and all the rest
are negative.6
There is, however, a second problem with Feldstein's approach,
which is not overcome by the DFM, namely that it implicitly assumes that
all cross-sample variation in the error term of the estimating equation
is due to variation in efficiency. In reality the residuals are also
likely to reflect random influences outaide the hospital's control, as
well as statistical 'noise'. A better tool is the stochastic frontier
model (SFM) [cf eg Aigner et al. (1977)]:
(4) lny; = 00+ EjAjlnx1j + vj+ ui uiS0,
so that the error term vj+u1 is composed of two parts, v; being two-sided
and capturing random shocks and statistical noise, and ui being
one-sided and reflecting inefficiency, which is constrained to be
4. For surveys of parametric and non-parametric approaches to efficiency measurement see Schmidt (1986)
and Barrow and Wagstaff (1989).
5. The rationale behind this is that the output of a hospital with a residual equal to zero is exactly the
output that wouLd be expected of It given its input utilization and the average estimated productivity of the
inputs. By contrast, a hospitaL with a positive (negative) residual produces more (Less) than it would have been
expected to produce on the basis of its input usage and the estimated parameters of the production function.
. Cf. Forsund, Lovelt and Schmidt (1980) and Schmidt (1986).
6
non-positive. Inefficiency is measured relative to the stochastic
frontier (po+Ejpjx+vi). There are, broadly-speaking, two approaches to
estimation of the SFM. One involves m&king an assumption about the
distribution of the ul in a cross-section, the most common assumption
being that the u; are half-normal. The model can then be estimated by
supplementing the information normally used in the estimation of the
regression model with information on the extent of skewness in the
residuals (see Schmidt and Lovell (1979)].7 One then ends up with a
residual for each hospital, an estimate of the mean of the u1, but not
an estimate of ul. What one can estimate, however, is E(u1jv1+u,) - the
expected value of u,, given the value of the composite error (see
Jondrow et al. (1982)). The alternative estimation approach involves
the use of panel data and assumes that inefficiency remains constant
over time (cf Schmidt and Sickles (1984)]. By working with data in
terms of deviations from temporal means, one can eliminate the
unobservable inefficiency term, which can then be recovered once the
parameters of the production function have been estimated.
The detection of allocative inefficiency is, in principle at
least, relatively straightforward. Allocative efficiency requires that
for each pair of inputs j and m
(5) ZP/MP/ = wi/wl
where MPj is the marginal product of the jth input and wj is its price.
In the case of the Cobb-Douglas production function this condition
becomes
(6) pj/l. = wjxj/w,xm,
i.e., the ratio of expenditures on the two inputs equals the ratio of
their output elasticities. Because the , are invariant with respect to
the amount of each of the inputs employed, the left-hand side of eq (6)
can be treated as a datum. Any discrepancy between the ratio of output
elasticities and the expenditure ratio is therefore to be attributed to
incorrect usage of one or both of the inputs. Once one has estimates of
the P,, one can determine whether one input is over- or under-employed
relative to another. If, for example, the right-hand side of (6) is
larger than the left-hand side, the kth input is being over-employed
relative to the jth. For other functional forms - such as the translog
- allocative efficiency can be assessed by comparing the ratio of
marginal products with the ratio of input prices.8
7. An alternative to this so-called moments estimator is a maximum Likelihood (ML) estimator (see eg Greene
(1980, 1982)1.
8, The standard shares equation approach in this context is cLearly inappropriate, since it assumes that
the residuals of the shares equations have zero mean [cf Wagstaff (1989b)1.
7
One possible way of measuring allocative inefficiency is to
calculate by how much a hospital's output would increase if it optimally
reallocated its budget and use the ratio of actual output to feasible
output ae the measure of allocative inefficiency (cf Feldstein (1967)).
Alternatively, in the two-input case, one might employ the index
proposed by Goldman and Grossman (1983):
(7) Ali =(MPH/MP ) (w2/w,)- |,
which in the case of the Cobb-Douglas becomes
(7') AI = I(P1/P2).(w2x2i/w1x1)-l-
Allocative efficiency gives a zero value of AI, while allocative
inefficiency results in AI being positive.
The effect of both technical and allocative inefficiency is, of
course, to raise a hospital's costs above their feasible minimum. It is
instructive, therefore, to consider the relationship between a
hospital's production function and its cost function in the presence of
such inefficiency, not least because doing so ought to help establish
how the estimation of cost functions might shed light on the issue of
efficiency. Consider the case of the Cobb-Douglas production function
(4). If hospitals are technically and allocatively efficient, so that
the u1 in eq (4) are zero for all hospitals, the associated cost
function takes the form
(8) lnC; = M + (1/r)lny1 + (1/r)Ejpjlnpji - (1/r)vj,
where r=E1Pjdenotes returns to scale (RTS) and M is a function of the
parameters of the cost function.9 Suppose now that hospitals are both
technically inefficient and allocatively inefficient. Thus uiO. With diseconomies
of scope, Sc1
implies diseconomies of scale and a rising average cost. It is
customary to measure economies of scale using the reciprocal of e: thus
(11) S = AC/MC = C / (6C/6y)-y = 1/(6lnC/5lny),
which is positive if economies of scale exist and negative if
diseconomies of scale exist.
In the light of the result above concerning the optimal number of
producers, and the definition of S, it is understandable that authors
whose results point towards the existence of economies of scale conclude
that too many hospitals exist, while those whose results point towards
diseconomies of scale conclude that too few hospitals exist."5 There
is, however, a complication that needs to be borne in mind that is often
overlooked, namely that the issue of whether there are too few or too
many hospitals is necessarily a long-run problem and hence economies of
scale should be evaluated in the context of the long-run, as is
envisaged in the traditional textbook definition of economies of scale.
Thus in eq (11) AC should be interpreted as long-run average cost and MC
as long-run marginal cost.
Because of this, it is tempting to try to estimate a long-run cost
curve directly. It is often assumed that this can be achieved simply by
omitting capital from the cost function (cf eg Granneman et al. (1986)].
The problem with this approach is obvious: only in the rather unlikely
event that hospitals have adjusted their capital stock to its long-run
equilibrium value will a hospital be operating on its long-run cost
'r. Although this conclusion is sound when 9 is large reLative to Yi, it is evident that when y' is small
relative to ym, it may welt be optimal for a small number of producers to be producing to the left or right of
Ym. The presence of economies or diseconomies of scale is not necessarily indicative therefore of there being
too many or too few hospitals.
12
curve. If hospitals have not adjusted their capital stocks to the
long-run optimal level, their actual average costs will exceed LAC and
their actual marginal costs will differ from LMC. Attempting to
estimate a long-run equation in this context will yield unreliable
estimates of LMC and LAC and hence of S.
Some authors have instead sought to infer economies of scale from
the variable cost function Cv(y,wv,K). Cowing and Holtmann (1983), for
example, compute S as the reciprocal of 6lnCv/6lny. As Vita (1990)
notes, this does not in fact capture economies of scale. Eq (2) above
can be solved to obtain the long-run equilibrium level of capital
(12) K - K*(y,wK,wv),
which can then be substituted into (1) to obtain the long-run cost curve
(13) C(y,wK,wv) = WKK (Y,WK,Wv) + Cv(Y,Wv,kX(y,WK,Wv)).
The partial derivative 6Cv/6y is equal to short-run marginal cost, while
what is required is the total derivative of total cost with respect to
y. The latter, unlike the former, takes into account the effects of
output changes on the optimal capital stock and the effects of these
changes in capital on fixed and variable costs. What the derivative
6lnCv/6lny does show is whether a hospital has exploited all possible
returns to the variable input(s) and therefore whether the hospital is
operating to the left or right of the minimum point of its short-run
average cost curve. This may be of some interest in its own right, but
it does not bear on the issue of economies of scale.
It is possible, however, to draw inferences about economies of
scale from a variable cost curve by invoking the envelope condition. As
is reported in, for example, Braeutigam and Daughety (1983), S can be
computed as
(11') Sv = (l-6lnCv/6lnK)/(6lnCv/6lny).
In this formula the derivatives ought in principle to be evaluated at K'
(cf Friedlaender and Spady (1981)]. In his analysis of American
hospitals Vita (1990) does not have access to the price of the fixed
input and hence cannot compute K*; he therefore evaluates the
derivatives at the actual value of K, an approach suggested by Caves et
al. (1981). This, like the approach of Cowing and Holtmann, hinges on
the assumption that the dependent variable in the cost equation is
indeed Cv. If, as suggested above, the reported costs include some
element of fixed costs, one's inferences concerning economies of scale
will be unreliable. An alternative would be to estimate a short-run
total cost equation, which is additive in fixed and variable costs, and
then compute Sv using eq (11') and differentiating that part of the cost
function that captures variable costs, or compute S using eq (11) and
bearing in mind that K* depends on y (cf eq (13)].
13
Thus far we have assumed
that hospitals produce just one c
product. Yet as the discussion
above of economies of scope
makes clear, such an assumption
is unwarranted. This raises
the issue of what determines PAC
the cost-minimizing number of
hospitals in a multi-product
context. The appropriate cost R
concept in such a situation is
ray average costs (RAC). Along
the ray R in fig 5 the output
mix is unchanged. The RAC at
each point on this ray is equal
to the slope of the chord drawn Figure 5
from the origin to the total
cost surface above the ray. In
the case illustrated, RAC
reaches its minimum at ym on the ray R. Evidently for each such ray
there exists a different RAC curve and hence, assuming each is U-shaped
as in fig 5, a different output combination at which the RAC reaches its
minimum.
The locus of such output
combinations is shown in fig 6
and is termed the M locus y locu
(Baumol et al. (1982)]. Let y'
be the current output vector on
ray R and let t y' be the
output level at which the RAC
for ray R reaches its minimum tyi Y R
point. Given the logic of the \
single-product case, one might
expect the cost-minimizing
number of producers in the
multi-product case to be equal
to l/t if this an integer, or y
to the integer just below or
just above l/t. Thus if RPP Figure 6
reaches its minimum at one
tenth of industry output, one
might reasonably expect the
cost-minimizing number of producers to be equal to 10. As Baumol et al.
(op cit) show, however, this is not necessarily the case. The lower and
upper bounds that can be derived for the cost-minimizing number of
producers are, it turns out, relatively wide. Moreover, their
calculation requires information on the location of ym for output mixes
other than the current mix.
In these calculations the shape of the RAC surface is crucial,
though the relationship between this and the optimal number of producers
14
is far from straightforward. Empirically the output mixes at which the
RACs reach a minimum can be investigated using the multi-product
analogue of economies of scale, ray economies of scale, defined as
(14) SN = C / 2;(6C/6y1).y1 = 1/E;#
where El is the elasticity of cost with respect to output i. Ray
economies (diseconomies) are said to exist if SN is greater (less) than
unity. Since SN can be shown to be equal to the reciprocal of one plus
the elasticity of RAC(t-y) with respect to t, it follows that ray
economies (diseconomies) of scale imply that RAC is decreasing
(increasing) [cf Baumol et al. (1982)]. As in the single product case,
the formula for economies of scale needs to be modified if a variable
cost equation or a short-run total cost equation has been estimated. In
the case of the former the appropriate formula for SN is
(14') SN = (1-6lnCv/6lnK))/SiE1,
which ought to be evaluated at K*, while in the case of the latter one
can proceed using either of the approaches suggested above for the
single product case.
The extent of any ray economies of scale can be shown to depend in
part on economies of scope and in part on Product-specific economies of
scale. The latter indicate what happens to cost when one alters the
level of production of one product, holding the other output levels
constant. The incremental cost of product 1 in the two-product case is
defined as
(15) ICI = (C(y,,y2)-C(O,y2)],
which indicates the addition to the producer's costs resulting from the
current level of output of product 1. The averace incremental cost of
producing y, is then defined as
(16) AIC, = IC,/y,
and indicates the extra cost associated with producing product 1
averaged over the amount of y, produced. Product-specific economies of
scale in the production of product 1 are then measured as
(17) S, = AIC,/MC,,
where, as before, an index value that is greater (less) than one
indicates economies (diseconomies) of snale. It can be shown [cf eg
Baumol et al. (op cit)] that
(18) SN = [WSI + (1-W)S2j / (-SC),
where w=y,MC,/[y,MC,+y2MC2J. Thus if economies of scope are sufficiently
strong, ray economies can exist even if there are no product-specific
economies. Indeed, sufficiently strong economies of scope might
15
generate ray economies of scale even in the presence of product-specific
diseconomies. Conversely, sufficiently strong diseconomies of scope
might result in ray diseconomies of scale even if product-specific
economies of scale exist. Evidently, since product-specific economies
of scale, like ray economies of scale, are a long-run concept, AIC and
MC ought to be evaluated using a long-run cost function.
3. Hospital costs in Kenya
Anderson (1980) reports the results of a cost function estimated
on data for 51 provincial and district public hospitals in Kenya in
1975/76. His estimating equation takes the form
(19) ln(C/I) = ao + a,lnB + a2lnR + a3lnS + a4ln(OUTP/I) .+ a5lnSAT +
a6PROV + v,
where C is total cost, B is the stock of beds, R is the occupancy rate,
I is inpatient days, S is mean length of stay, OUTP is outpatient
visits, SAT is the number of associated sub-hospital facilities, PROV is
a dummy taking a value of one if the hospital is a provincial hospital
and zero if it is a district hospital and v is an error term. If we
substitute R=I/(365-B) and S=I/A, where A is admissions, eq (19) can be
rearranged as a total cost function of the form
(19') lnC = aO + (a,-a3)lnB + (l+a2+a3-a4)lnI - a3lnA + a4lnOUTP + a5lnSAT +
a6PROV + v.
The equation estimated (eq 19) Table 1: Parameter Estimates of
implies an elasticity of cost with Anderson's Equation
respect to inpatient days of one A
minus the elasticity of cost with Variable Parameter Value
respect to outpatient visits if
occupancy rate and length of stay Constant a0 6.57
are not allowed to change. The inB a, -0.20
total cost form allows R and S to
change, but the interrelationship InR a2 -0.44
of the coefficients in the total InS a. -0.07
cost form remains, of course,
entirely the artifact of the ln(OUTP/1) 04 0.29
equation estimated. Note too that InSAT a. 0.19
this Cobb-Douglas-type cost
function implies that if either PROV a° 0.28
output is zero, cost is
automatically zero, which is
consist ' with the equation being Adjusted R2 0.75
interpreted as a variable cost Note: Dependent variable is ln(C/I).
equation. Anderson's estimates of Parameters taken from column labelled R-
the parameters of eq (19) are 4 of Anderson's table 1.
shown in table 1.
16
3.1 Are Kenvan hospitals over-capitaiized?
The negative coefficient on beds in Anderson's study, coupled with
the lack of fixed costs in the equation, suggests that the estimated
equation can indeed be interpreted as a variable cost equation. Hence
the Cowing-Holtmann test for over-capitalization would be applicable.
Notwithstanding Anderson's claims to the contrary, the negative
coefficient does not of itself imply under-capitalization. Only if
6Cv/6K is less than -wK can one conclude that "cost savings could be
obtained by expanding existing facilities" (Anderson (op cit, p233)].
Since Anderson does not report an estimate of wK, or the mean values of
Cv and K, one cannot establish from his paper whether Kenyan hospitals
are indeed under-capitalized.
3.2 Inefficiency in Kenvan hospitals
Although Anderson does not explicitly address the issue of
efficiency, it might be argued that he does so implicitly by including
several variables that are not required by economic theory. One problem
with this line of argument is, of course, that it is difficult in the
context of the hospital sector to determine whether a variable is an
'additional' variable, or whether it is included in attempt to capture
better inter-hospital variations in output. Anoth-r problem is that, as
emphasized above, no account is taken in this non-frontier approach of
the fact that inefficiency is cost-increasing.
In the Kenyan study four variables are potentially 'additional'
variables: the occupancy rate, length of stay, the number of associated
hospital sub-facilities, and the province/district dummy. Inclusion of
occupancy rate cannot be rationalized in terms of its being a proxy for
output. The implication of the results in table 1 are that hospitals
with low occupancy rates are inefficient. It is also hard to justify
viewing length of stay as an output proxy, given that inpatient days are
used as the output measure for inpatient care and these already reflect
length of stay. The relevant coefficient is not, however, significant.
Whether variations in the number of associated hospital sub-facilities
reflects output variations is unclear. The province/district dummy
probably does reflect - at least in part - differences in output, since
it is likely that provincial hospitals end up treating more complicated
cases. This may explain at least partly the positive and significant
coefficient on the province/district dummy.
3.3 Economies of scope in Kenyan hospitals
The Cobb-Douglas functional form adopted by Anderson implies that
costs fall to zero whenever the production level of either output falls
to zero. As Baumol et al. (1982:449) note, this implies that industry
costs can (ostensibly) be driven to zero by dividing outputs among
specialized producers. The Cobb-Douglas specification automatically
gives rise to cost anti-complementarities if the cost elasticities are
positive (as they are in the present case) and hence (in the absence of
fixed costs) results in diseconomies of scope. The Cobb-Douglas cost
17
function is thus insufficiently flexible to test for economies of scope
in a multiproduct environment.
3.4 Too many hospitals in Kenya?
Ignoring for the moment the long-run nature of economies of scale,
in the case of eq (19') we have
S= 1 + a2 + C3 - C4 0.2
and
SOUM = a4 = 0.3
so that both outputs have product-specific diseconomies of scale. Given
the result in eq (18) above, we would expect ray economies of scale to
exist if these product-specific economies are sufficiently strong to
offset the assumed diseconomies of scope. In fact
SN = 1 / (1 + a2 + a3) = 2.0.
This does not takes into account that the estimated equation is
not a long-run equation, but rather a variable cost function. Because
of this it is more appropriate to compute SN using eq (14') rather than
eq (14). Ideally 6lnCv/6lnK would be evaluated at K, but since there
is insufficient information in Anderson's paper to calculate KX, we
evaluate SN at the actual value of K. This gives
SN = (1-a, + a2)/(a4 + (1 + a2 + a3 - a4) = 1.55,
implying mild ray economies of scale.''6 The implication is that the ray
average cost curve is downward sloping and that product-specific
economies of scale are also larger than the short-run estimates above
suggest. However, as indicated above, the fact that ray economies exist
in a multiproduct setting does not necessarily mean that there ought to
be fewer hospitals. Moreover, given the restrictions in the estimated
equation, the fact that SN points towards ray economies of scale ought
to be treated with some caution.
4. Hospital costs in Peru
Dor (1987) reports the results of a cost function estimated on
data for 19 urban public hospitals in Peru in 1984. His equation bears
a close resemblance to the equations estimated by Feldstein (1967) and
takes the form
(20) C/A = ao + a,F + a2F2 + a3OUTP + a4%DEL + a5%SURG + adMIN + v,
l. Anderson also concluded that hospitals in his sample exhibited economies of scale. However, his
conclusions were based on the value of the coefficient on the stock of beds, which, as we argued above, bears
on the Issue of over-capitalization rather than economies of scale.
18
where A is the number of admissions, %DEL is the proportion of
admissions taken up by deliveries, %SURG is the proportion of cases
receiving surgery and MIN is a dummy taking value of one if the hospital
is under the control of the ministry. Eq (20) implies a total cost
function of the form
(20') C = aoA + axA-F + a2gA-F2 + a3A-OUTP + a4DEL + asSURG + aX6AMIN + v,
where DEL is the number of
deliveries, SURG is the number of Table 2: Parameter Estimates of Dor's
deliveries, SURG is the number of Equation
inpatients admitted for surgery
and v=A*v. It is evident that if Variable Parameter Value
admissions are zero, total cost is
zero, irrespective of whether any Constant aO 12076.94
outpatients are being seen or not. F a -6168.90
This, coupled with the fact that
the stock of beds does not appear F2 a2 961.16
as a regressor in the estimating OUTP as 0.003
equation, suggests that eq (20')
is probably best interpreted as a %DELIV 04 802.30
long-run total cost function. %SURG a- 562.45
Dor's parameter estimates are
shown in table 2. MIN as -1635.38
4.1 Are Peruvian hospitals
over-capitalized? Adjusted R2 0.98
Note: Dependent variable is cost per
Since Dor' s esti.mating admission. Weighted Least Squares
equation is a long-run equation, estimates taken from column (2.9) of
it is assumed implicitly that the Dor's table 2.
stock of capital is at its
long-run optimal value. No test
for over-capitalization is therefore possible.
4.2 Inefficiency in Peruvian hospitals
Like Anderson, Dor does not analyze efficiency explicitly but
might be said to do so implicitly by including various 'non-traditional'
variables in his cost function. Clearly one would not include among
these the outpatient variable and the two casemix variables, which
reflect output. This leaves three variables that might be argued to be
'additional': the affiliation dummy, caseflow and its square. Whether
the affiliation dummy (which takes a value of one if the hospital is
operated by the Ministry of Health and zero if operated by the social
security system) might capture output variations is unclear. If it does
not, the results in table 2 suggest that hospitals operated by the
Ministry of Health are more efficient. Turning to the effects of
caseflow, it is evident that the partial relationship between average
cost and caseflow is U-shaped, reaching a minimum at 3.2 cases per bed
per month, which is above the sample average of 2.8. The inference
drawn by Dor is that most hospitals have unnecessarily high costs
19
because their caseflows are too low. Much the same conclusion was
reached by Feldstein (1967), who found that the 'inefficient' hospitals
with the below-average caseflows tended to be the larger hospitals and
tended to have low caseflows because of above-averag& lengths of stay.
Several writers have, however, questioned Feldstein's conclusion: it may
well be that larger hospitals may have an above-average mean length of
stay and hence a below-average caseflow because they treat the more
severe cases (cf Barlow (1968), Fuchs (1969), Lave and Lave (1970)3.
The same note of caution would seem appropriate in the context of Dor's
study, if low caseflows are due to long lengths of stay rather than to
low occupancy rates.
4.3 Economies of scope in Peruvian hospitals
Dor has, in effect, four outputs in his equation: deliveries,
surgical procedures, other inpatient care, and outpatient visits.
Denoting by OTH non-surgery and non-delivery admissions, eq (20')
becomes
(21) C = (aO+a4)DEL + a1F-DEL + a2F2DEL + a3DEL-OUTP + a6DEL-MIN
+ (aO+a5)SURG + a,F-SURG + a2F2,SURG + a3SYJRG-OUTP + a6SURG-MIN +
+ aoOTH + a,F'OTH + a2F2.OTH + a3OTH'OUTP + a6OTH*MIN + v-,
Overall economies of scope can then be calculated as
SC = [C(DEL,O,O,0)+C(O,SURG,0,0)+C(O,O,OTH,O)+C(O,O,O,OUTP)
-C(DEL,SURG,OTH,OUTP)]/C(DEL,SURG,OTH,OUTP),
which, in this case, turns out to be equal to
Sc = -a3(DEL+SURG+OTH)-OUTP / C(DEL,SURG,OTH,OUTP).
Since the estimate of a3 is positive, Sc0. The implication is that there are
ray diseconomies of scale in the Peruvian sample. This result is, of
course, prejudged by the model specification. Only in the implausible
case where the marginal cost of an outpatient visit is negative can SN
be larger than one. Since the weights in the multiproduct analogue of
eq (18) sum to one, the relationship between ray economies of scale and
economies of scope in this case is
SN I1 / (l-SC) I
which, given the expression for Sc derived above, confirms the earlier
expression for SN. Thus, because product-specific economies are ruled
out by assumption, it is the diseconomies of scope that give rise to the
ray diseconomies of scale. But since the specification is compatible
only with diseconomies of scope, this finding is uninteresting.
5. Hospital costs in Ethiopia
Bitran and Dunlop (1989) report estimates of a cost function
estimated from an unbalanced panel of 38 observations on 15 public
hospitals in Ethiopia in the mid-1980s. Their estimating equation is
similar to that of Granneman et al. (1986), except that the stock of
beds is included and there are no cubic terms. Their equation is of the
form
(22) lnC = o + a,B + PII + pI2+ +I30UTP + P4OUTP2 + P5I-OUTP +
j6DELIV + 07SURG + P8LAB + v,
where LAB is the number of lab tests and the other variables are as
defined above. The parameter estimates of eq (22) are shown in table 3.
5.1 Are Ethiopian hospitals over-ca2italized?
The positive estimate of al suggests that the equation estimated
by Bitran and Dunlop is not a variable cost equation but rather a
short-run total cost function. The Cowing-Holtmann test for
over-capitalization would therefore seem to be inappropriate and points
towards the use of the alternative Feldstein-type test proposed above.
The fact that 6Cs/6K>O in this study implies that hospitals in the
sample are too large, given their current output levels. One ought,
however, to be wary about taking this conclusion at face value. The
estimating equation does not satisfy the restrictions required of a
21
short-run total cost function:a it
ishnort-run additiveln £ d fction: i Table 3: Parameter Estimates of Bitran and
is not additive in fixed costs and Dunlop's Equation
variable costs, and variable costs
are not a decreasing function of Variable Parameter Value
the stock of capital. Constant 5.45
5.2 Inefficiency i-n B 01 4.71E-3
Ethiocian hosoitals
I Rl ~~~~~2.18E-5
Bitran and Dunlop do not 12 a2 -1.65E-12
analyze efficiency explicitly.
Nor do they include any OUTP 3 1.91E-6
'non-traditional' variables in OUTP2 a4 1.42E-10
their cost function which might be
said to capture inefficiency I*OUTP 5 -7.50E-10
implicitly. DELIV 6 1.68E-4
5.3 Economies of scope in SURG 67 3.21 E-6
EthioDian hospitals LAB a8 7.63E-6
Bitran and Dunlop have, in
effect, five different outputs in Adjusted R2 0.96
their cost function: inpatient
days, outpatient visits, Note: Dependent variable is InC.
deliveries, surgical procedures
and lab tests. In calculating the economies of scope associated with eq
(22) it is important to be clear about which fixed costs would be
incurred and which would avoided in each scenario. An extreme and
clearly implausible assumption would be that all outputs would, if
produced alone, require the same level of beds as at present. In this
case projected stand-alone production costs are those indicated in
column 1 of table 4. The projected costs incurred by the average
hospital at present are 1006231 Birr (Bitran and Dunlop (op cit, table
A.3)). The overall degree of economies of scope are therefore
Sc = (3155873-10062313/1006231 = 2.136,
which implies economies of scope. If, instead, one assumes that
producing outpatient visits and lab tests alone would not necessitate
any beds, the stand-alone production costs are those indicated in column
2 of table 4 and the implied degree of economies of scope is equal to
Sc = (2527777-10062313/1006231 = 1.512,
which, unsurprisingly, is smaller than in the previous case. But even
this probably overstates the true degree of economies of scope, the
reason being that the production of each of the inpatient outputs alone
is unlikely to require the full amount of beds currently being used.
This highlights a weakness of the specification used by Bitran and
Dunlop, namely that unlike the flexible fixed cost functional form
proposed by Baumol et al. (op cit), it does not allow for the
22
possibility that each production 'line' has its own quasi-fixed costs
that are avoided if the production of that output ceases.
5.4 Too many hospitals in Ethiopia?
Suppose we ignore for the moment the long-run nature of economies
of scale and calculate ray economies of scale using eq (14). The cost
elasticity of output i is equal to (6lnC/6y1)y1. If these are evaluated
at the means given in appendix 1 of Bitran and Dunlop, the cost
elasticities obtained are those indicated in table 4. The large and
negative cost elasticity of outpatient visits suggests a serious model
misspecification and therefore the results below ought to be treated
with caution. The elasticities imply that
SN = 1 / 0.296 = 3.38,
and hence imply substantial ray economies of scale. This is despite the
fact that none of the outputs with positive marginal costs are
characterized by product-specific economies of scale (cf table 4]. The
implication is that the ray economies of scale exist despite these
product-specific diseconomies and are to be attributed to the economies
of scope noted earlier.
The ray economies uncovered above ignore the fact that the
equation estimated is not a long-run equation. Nor apparently is it a
variable cost function. One cannot therefore treat the derivative of
the cost function with respect to beds as reflecting 6lnCv/6lnK and use
eq (14), as was done above in the discussion of Anderson's results.
Moreover, since the estimated equation is not additive in fixed and
variable costs, one cannot recover 6lnCv/6lnK from the estimated
equation. Nor, given the lack of additivity, would it make sense to try
to solve for K', even if information on the shadow price of beds were
reported in the study, which it is not. The upshot of this is the
results in table 3 cannot be used to infer the true extent of ray
economies of scale.
6. Health care costs in Nigeria
Wouters (1990) reports the results of a cost function estimated on
24 Nigerian health care institutions, of which eight are health centers,
seven are maternity units and nine are dispensaries. Her estimating
equation is of the form
(23) lnC = ao + aolnA + a2lnOUTP + a3ln(%DRUGS) + a4lnwHw + a5lnwNHW +
a6DBms + a7DBE-lnB + a8lnAI + v,
where C is total cost, %DRUGS is the percentage of drugs available, wHw
and wNHW are the wages of health workers and non-health workers
respectively, DBED is a dummy taking a value of one if the facility has
beds and AI is an index of allocative inefficiency calculated from
Table 4: Economics of scope and scale-Bitran-Dunlop study
Output Cost of each output alone MC Cost AIC S y yOMC
Elasticity
With AN Beds Only IP beds
Inpatient days 883532 883532 2.582 0.073 2.534 0.981 28410 73361
Outpatient visits 548476 268063 -12.226 40.310 -19.542 n.a. 25520 -311995
Deliveries 564877 231544 169.047 0.171 155.407 0.919 1016 171752
Surgical procedures 478934 478934 3.230 0.006 3.221 0.997 1758 5678
Lab tests 680054 332371 7.678 0.356 6.459 0.841 46691 358472
Sum 3155873 2527777 0.295 297268
Projected cobt of avg. beds 1006231
Economies of scope (all beds) 2.136
Econs. of scope (only IP beds) 1.512
Ray econs of scale (SR) 3.385
Elasticity of cost wrt beds 0.716
Ray economies of scale (LR) 0.962
24
estimates of a production function (see section 2.2 above).'7 Like
Anderson's equation, this Cobb-Douglas type equatior implies that cost
is zero if either output is zero.
6.1 Are Nigerian health care institutions over-capitalized?
The positive estimates of a6 and a7 suggests that Wouters'
equation, notwithstanding her claims to the contrary, is best
interpreted as a short-run total cost function rather than as a variable
cost equation. As in the Ethiopia study, these positive coefficients
imply over-capitalization. But once again one ought to be wary about
taking this conclusion at face value, since the estimating equation does
not satisfy the restrictions required of a short-run total cost
function.
6.2 Inefficiency in Nigerian health care institutions
Of the four studies included in this survey, Wouters' is the only
one which analyses efficiency explicitly. She explores both technical
and allocative inefficiency. Her method differs slightly from that
outlined in section 2.2 above. Rather than estimating a SFM, she first
divides her sample into efficient
and inefficient facilities on the Table 5: Parameter Estimates of Wouters's
basis of the number of visits per EquatiOn
health worker per year and then Variable Parameter Value
estimates a conventional
Cobb-Douglas production function Constant a0 1.63
on the 'efficient' subsample. The InA al 0.01
shortcomings of this approach are
obvious: it is arbitrary and fails InOUTP a2 0.60
to provide any evidence on In(%DRUGS) a3 -1.36
variation in technical
inefficiency within each group. InwHw 4 0.59
Nonetheless, it does have an InwNHw a5 0.39
advantage over the frontier
production function approach in DBEDS ae 0.22
that whereas the frontier approach D,05.inB a7 0.09
models technical efficiency simply
as a shift factor, Wouters' InAl a8 -0.16
approach allows the shape of the
production function to differ
between 'efficient' and Adjusted R2 0.91
'inefficient' facilities. Note: Dependent variable is InC.
Moreover, it is interesting to
note that almost all of the
private facilities in Wcouters' sample fall below the relatively generous
cutoff point for inclusion in the 'efficient' subsample.
". Because of zero values admissions and beds are, in fact, not entered in logarithms but are instead
transformed using the Box-Cox metric (L=0.10). The interpretation of the coefficients is much the same as for
variables expressed in logarithms.
25
Wouters' analysis of allocative inefficiency is more conventional.
From her estimates of the Cobb-Douglas production function estimated on
the 'efficient' subsample, she finds that the ratio of the marginal
product of non-health workers to that of health workers is less than the
relevant wage ratio, implying that non-health workeri are, on average,
over-employed relative to health workers. Allocative inefficiency is
then measured by AI in eq (7). Interestingly, the index is smaller in
value in the private sector than in the public sector, suggesting that
at least among the technically efficient facilities, allocative
inefficiency is lower in the private sector.'8 Finally, Wouters
includes the log of her inefficiency index in her cost function as a
regressor. Surprisingly, its coefficient is negative (though not
significant), implying literally that allocative inefficiency reduces
costs.
6.3 Economies of scope in Nigerian health care institutions
Like Anderson's estimating equation, eq (23) implies that costs
fall to zero whenever the production level of either output falls to
zero. The specification therefore automatically implies diseconomies of
scope and an Sc value of -1.
6.4 Too many health care institutions in Nigeria?
Ignoring for the moment the long-run nature of economies of scale,
eq (14) gives a value of SN equal to
SN = 1 / a,+a2 = 1. 645,
which implies ray economies exist. This is despite the fact that, as
noted above, Wouters' cost function automatically results in
diseconomies of scope. The implication is that these diseconomies of
scope are offset by sufficiently strong product-specific economies of
scale. This is indeed the case. From Wouters' estimating equation and
table 5 we obtain
SI a15, 90.909
and
SOUrP= 1 / a2 = 1.675,
so that both outputs enjoy product-specific economies of scale. The
marginal costs of admissions and outpatient visits are a,(C/A) and
a2(C/OUTP) respectively, and therefore the weights in eq (18) are
a,/(a,+a2) and a2/(a,+a2) respectively. Eq (18) thus becomes
SN = (0.018-90.91 + 0.982-1.645] / 2 = 1.630,
l. The sample size is, however, very smaLl, the number of private facilities being onLy 3.
26
which is close to the value of SN obtained above and confirms that the
product-specific economies of scale more than offset the (assumed)
diseconomies of scope.
These estimated ray economies ignore, however, the fact that the
equation estimated is not a long-run equation but rather a short-run
total cost equation. As in the case of Bitran-Dunlop study above, this
means that one cannot obtain a true measure of ray economies of scale
from the results reported in Wouters' paper.
7. Conclusions
It is evident from the above, and not surprising perhaps, that
none of the studies of hospital costs in developing countries would
appear to provide reliable evidence on all of the issues of interest to
policy-makers.
Although the issue of over-capitalization is clearly important in
its own right, it also has implications for the investigation of other
issues. Thus even if one's interest lies with, say, economies of scale,
the issue of over-capitalization cannot be ignored, since if hospitals
are not in a long-run equilibrium, the estimated parameters will not
provide reliable evidence on economies of scale. Estimating a short-run
cost function, irrespective of whether or not one is interested in
over-capitalization, which is the practice advocated by Cowing, Holtmann
and Powers (1983), seems eminently sensible. It is therefore
unfortunate that Dor, in his study of hospital costs in Peru, elected to
estimate a long-run cost function. With regard to testing for
over-capitalization, we have argued that since in practice it is
difficult to separate fixed costs from variable costs, the test proposed
by Cowing and Holtmann (1983) is unlikely in general to be appropriate,
since it relies on the estimated equation being a variable cost
function. Of the three short-run cost functions included in this
survey, only in Anderson's was the sign of 6C/6K consistent with the
equation being interpreted as a variable cost equation. But although
the Cowing-Holtmann test seems valid in this case, the negative value of
6C/6K and the absence of information on the shadow price of capital make
testing for over-capitalization impossible. In the case of the other
two studies, where the positive value of 6C/6K suggests that the
equation estimated is a short-run total cost equation, we argued that a
more appropriate test would be to test whether 6C/6K=0. In both studies
6C/6K was positive, which interpreted literally implies
over-capitalization. We argued, however, that this conclusion ought not
to be taken at face value, since the estimated equations are not, as
theory requires, additive in fixed and variable costs.
Of the studies included in this survey, only Wouters' analyses the
issue of efficiency explicitly. Although her method for investigating
technical inefficiency is somewhat ad hoc and falls short of the
frontier production function approach, her approach to allocative
inefficiency is fairly conventional. She finds evidence in her sample
of under-employment of health workers relative to non-health workers,
27
and that the degree of allocative inefficiency is greater in the private
sector. Her finding that allocative inefficiency reduces costs must,
however, cast some doubt on these findings. Since two of the other
studies include 'non-traditional' variables in their cost function, it
might be argued that they too address the issue of inefficiency, albeit
implicitly. The results of these studies suggest, for example, that low
occupancy rates are a sign of inefficiency, and that hospitals operated
by the Peruvian Ministry of Health are more efficient than those
operated by the Peruvian Social Security. It is apparent from the above
that the large and growing econometric literature on efficiency
measurement has remained virtually untapped by the authors of hospital
cost functions in developing countries, as indeed is true of researchers
in industrialized countries. Much more research could usefully be done
on this topic.
Of the four studies covered in the present survey only one employs
a specification that is sufficiently general not to prejudge the issue
of economies of scope. Both Anderson and Wouters employ a multiproduct
Cobb-Douglas production, which, as Baumol et al. (1982) note, implicitly
assumes cost anti-complementarities and hence assumes diseconomies of
scope unless there are sufficiently strong offsetting fixed costs.
Dor's specification is less rigid but is consistent with economies of
scope only in the implausible case where the marginal cost of an
outpatient visit is negative. Only in the Bitran-Dunlop study is the
model specification sufficiently general not to prejudge the issue of
economies of scope. In the event the authors' results imply overall
economies of scope. The extent of these economies is reduced, but are
still positive, if it is assumed, not unreasonably, that beds are
required only in the provision of inpatient services. However, although
the Bitran-Dunlop specification is much less restrictive than the
specifications adopted in the other studies, it still has the
disadvantage that it does not take into account that each output may
have its own quasi-fixed costs. Both of the aforementioned estimates of
the economies of scope index assume that if a hospital were to provide
no outpatient services and only one line of inpatient services (eg
deliveries), it would still require all the beds it currently uses, even
if the number of cases treated in the retained line were no greater than
the current number. This is clearly unrealistic. A more sensible
functional form is the flexible fixed cost function proposed by Baumol
et al. (1982), in which the quasi-fixed costs of each product line are
distinguished from one another.19
Turning finally to economies of scale, we have seen that of the
three studies that prejudge the issue of economies of scope, two also
prejudge the issue of economies of scale. In Anderson's specification
the cost elasticities sum to unity by construction. Thus if one
calculates SN using eq (14), the specification guarantees a value of SN
'O. A disadvantage of this functional form is that the separate quasi-fixed costs are identifiable only
in samptes where not atl producers produce all outputs.
28
of one. If instead the equation is treated as a variable cost function
(which is consistent with the value of 6C/6K), the value of SN depends
solely on the value of 6lnC/6lnK. Dor's specification also prejudges
the issue of ray economies of scale by implicitly assuming that the
product-specific economies-of-scale index Si is equal to one for all
outputs. Given this, the link between ray economies of scale, economies
of scope and product-specific economies of scale, and that the
specification in effect rules out economies of scope, it follows that
the specification forces ray diseconomies of scale. Neither of the
remaining two studies prejudge the issue of ray economies of scale.
Bitran and Dunlop find slight product-specific diseconomies of scale
(although the negative marginal cost of one output suggests a model
misspecification) but find ray economies of scale. The implication is
that these stem from the economies of scope noted above. Wouters also
finds ray economies of scale but in contrast to Bitran and Dunlop finds
product-specific economies. The economies-of-scale estimates of both
studies ought, however, to be treated with some caution, since these
results are calculated from equations which are probably best
interpreted as short-run total cost equations, albeit equations that do
not meet the theoretical requirement of being additive in fixed and
variable costs. At best the results indicate the effects of moving
along short-run cost curves.
We feel that it would be unwise to try to draw firm policy
conclusions from the four studies included in this survey. In several
studies the issues being investigated are in effett prejudged by the
model specifications. Where this is not the case, the model
specifications are often inconsistent with economic theory. Some
authors claim, for example, to estimate a variable cost equation and yet
the parameter estimates are inconsistent with this interpretation. Nor
are the specifications consistent with being interpreted as short-run
total cost functions, since the equations are not additive in fixed and
variable costs.
We believe, however, that certain firm conclusions can be drawn
about the methodology of hospital cost function estimation. These
conclusions would appear to be relevant to the estimation of hospital
costs functions in industrialized countries. First, since hospitals may
well not be employing their long-run equilibrium quantities of capital,
it seems sensible to follow the line taken by Cowing et al. (1983) and
estimate a short-run cost function. If capital is not wholly exogenous,
the obvious answer is to employ simultaneous equation techniques. One
cannot, we believe, convincingly argue, as Granneman et al. (1986) seek
to, that the possible endogeneity of capital justifies the estimation of
a long-run equation. Second, given the difficulty of purging fixed
costs from variable costs, it may be best to estimate a short-run total
cost equation, rather than a variable cost equation. This ought to be
additive in fixed and variable costs, as required by economic theory,
and variable costs ought to be a decreasing function of the stock of
29
capital.0 One can then test for over-capitalization by testing to see
whether 6Cs/8K is positive. Third, since hospitals may well be
technically and allocatively inefficient, and since cost frontier models
are relatively straightforward to estimate, it seems desirable that
greater use should be made of these models in future work in this area.
Disentangling technical and allocative inefficiency via a frontier
production function analysia also seems desirable. Fourth, as the
studies in this survey indicate it is all too easy to specify a cost
function in a way that prejudges the issues that are of interest to
policy-makers. Equations must be specified sufficiently flexibly to
allow the data rather than the model specification to indicate the
extent of any economies of scope, product-specific economies of scale,
and hence ray economies of scale. Ideally these specifications would
allow one to determine the extent of any quasi-fixed costs associated
with each product line. Finally, since economies of scale are a
long-run phenomenon, this needs to be borne in mind when calculating
their extent. If a short-run cost function is estimated, this will
necessitate the calculation of the optimal capital stock, which will in
turn require data or estimates of the shadow price of capital.
. Barer's (1982) specification meets the first of these requirements but not the second.
30
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