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A production function analysis of English maternity hospitals
A PRODUCTION FUNCTION ANALYSIS OF ENGLISH MATERNITY HOSPITALS ROBERT J. LAVERSf and DAVID K. WHYNESS University of Nottingham.
Umversuy
Park. Nottingham NG72R. England
(Received 16 August 1977; receiued
for publication 17 November 1977)
Abstract--In recent years, a number of attempts have been made to explain the output of hospitals by means of production function analysis and, in this particular study, the authors estimate Cobb-Douglas and log-quadratic functions from data for the 193 English maternity hospitals. Of the various inputs into maternity care, the numbers of beds and nurses appear as the most significant determinants of throughput, although the relative quantities actually employed differ from the “technical” optimum. Returns to scale in the maternity service appear to be, at best, constant. The effects of hospital location and type are also analysed and they suggest the existence of significant disparities in levels of efficiency between different hospitals.
The important work of Feldstein[l] has paved the way for a number of analyses of the application of neoclassical production functions to the operation of hospitals. Feldstein’s original approach involved the isolation of the appropriate functional form to explain changes in the inputs into hospitals, such as doctors, nurses, drugs and dressings, which result in variations in hospital output, measured on the number of cases treated in each time period. While the present analysis is very much in the Feldstein tradition, it does make two significant departures. First, by analyzing acute hospitals in aggregate, Feldstein’s production functions become open to criticism on the basis of output mix. Is it reasonable, it can be argued, to conceive of the output of one treatment of a broken limb as commensurate with a case of neuro-surgery? Is it not also likely that, in a large acute hospital, we in fact observe a number of different production functions in each of the different treatment areas? By restricting ourselves to maternity hospitals, we believe that these objections may be overcome; the output of such a process is likely to be much more homogenous and can therefore be measured much more realistically by hospital throughput. In addition, it is more likely that, in such hospitals, there will be less variation in the types of productive process. Second, we explore a wider variety of functional forms than has been undertaken previously; this we believe to be important owing to the restrictive nature of assumptions which are implicit in any one particular type of function. In the same way as Feldstein, however, and, indeed, in common with other researchers in the area, we have followed the convention of using costs as a proxy for real inputs. This is necessary owing to lack of any usable data for this latter variable. In addition to advancing the developments of production function work, we also attempt to provide some insights into the operations of maternity hospitals. In particular, we seek answers to the following questions: (1) what is the direction and magnitude of returns to scale in maternity hospitals, e.g. if inputs are increased by lo%, do outputs increase by the same amount, by more, or by less? (2) is the mix of factors appropriate, e.g. are there too many or too few nurses per bed? (3) how responsive are outputs to changes in partitlnstitute of Social and Economic SDepartment of Economics.
cular factor inputs, e.g. will a 10% increase in doctors have a greater or lesser impact than a similar increase in nurses? (4) what difference does the type and location of the hospital make to output? PRELIMINARYANALYSIS
The data used were obtained from the summaries of hospital costs published annually by the fourteen English Regional Boards, and relate to the financial year ended 31st March, 1972. The total of 193 hospitals ranged in size from a tiny unit with 4 available beds to a relatively large one with 216, and almost two-thirds of them (122) had between 10 and 30 available beds. With a mean available bed size of just under 40, which was also the value of the standard deviation, the size distribution has a fairly pronounced positive skew. Occupied beds have a mean and standard deviation of around 24, implying an average occupancy figure of just over 60%, and values of 50% were not uncommon. Summary statistics of the inpatient data used are shown in Table 1. Many of the items in the conventional classification of costs are of relatively little importance, such as expenditure on patients’ applicances and on some of the treatment departments. The more important items which reflect activities conducted in all hospitals. such as catering and laundry, exhibit little variation from hospital to hospital, while such items as operating theatres and X-ray, with a high coefficient of variation, reflect activities undertaken solely in certain types of hospital (i.e. consultant units in the case of the treatment departments). Of the 193 hospitals, 118 contained only GP beds and the remaining 75 were consultant units containing beds for obstetrics or special care baby units. The analysis that follows makes use of data on annual expenditure on a few selected inputs, notably nurses, doctors and drugs and dressings in addition to average daily occupied beds as a measure of the services yielded by the capital of each hospital. Four possible reasons may account for the fact that many categories of inputs have been omitted from the analysis: first, as noted above, many items make a very small contribution of total expenditure; second, even where items do form a sizeable part of all inputs, as with catering and power, heat and light for instance, their variance is low; third, expenditure on many items is highly correlated with the included inputs; and fourth, a number of inputs are
Research.
85
R.J.
86
LAVERS and
D. K. WHYNES
Table 1. Beds, in-patients and in-patient costs in the 193 English maternity hospitals during the year 1971-72
standard Available beds Occupied beds In-patient days Discharges Total.Expenditure (&) Costs per in-patient week (S): Pay, iredical Pay, nursing Pay, dmstic Pay, other IhYF Pre-packed dressings Other dressings Patients' appliances Equiprent, major Equipmat, traditional Equipment, disposable Cont-mct services ODerattie theat-Rs
&diothe;$W Diagnostic X-ray PdthOlOgy Physiotherapy pharmacy Ancillary departnents Nmesintraining catering Staff residences
timdry Pajer, lipbt and heat Building maintenance Medical records General administration General portering Geneml. cleaning Maintenance of gmmds nwlsport Other services Direct credits costs per case, ward costs percase,treaw?nt
sourres :
39.4
94.1
9,881.O
1,422.0 110,600.O 2.0
38.3 5.9
0.2 1.8
0.3 0.4
0.006 0.1 1.0 0.6 O.O? 0.4 o.cco7 0.04 0.9 0.09
0.3 0.06 0.6 8.0 1.3 2.5 3.0 3.0 0.4 3.4 1.2 1.8 1.1 0.6 5.1 1.5 46.0 47.8
39.1 23.9
11,280.O 1,483.O 123,900.o 1.9 11.6 3.8 0.7 0.4 0.4 0.4 0.05 0.2 0.7 0.6 0.5 1.1 0.003 0.1 1.0 0.2 0.3 0.2 1.4 2.2 1.8 0.9 1.2 2.0 0.6 1.2 1.3 1.8 1.2 0.7 1.8 1.0 13.2 13.6
Summries of hospital.costs for the year ended 31st March 1972, published by the 14 Regional Hospital Beads.
endogenous, that is to say, affected by the level of discharges so that the relationship between them and output is approximately one of fixed proportions. The presence of inputs which are relatively unimportant, in respect both of their magnitude and variability, has already been remarked upon and may be observed in Table 1. As evidence of the existence of some fairly highly correlated inputs, the zero-order coefficients of correlation between each of the three variables medical pay, nursing pay and expenditure on drugs on the one hand, and all categories of input used in in-patient departments on the other hand, are presented below in Table 2. The 5% critical values of r are kO.16. As evidence for the existence of fixed proportions for many inputs, the results obtained from fitting the model X, = a,yp’, where X,, denotes the input of category i to hospital j and y, denoted discharges from hospital j, are presented in Table 3. Such a function has been successfully used in the analysis of plants generating electricity, for example (Nerlove[2]) and provided the y, are appropriately regarded as exogenous (which would seem to be in order in view of the low occupancy levels referred to earlier) the parameters may legitimately be estimated by the least squares regression of the logarithm of each input on the logarithm of discharges. It is clear from the t-values shown in brackets that for almost all inputs both the scale and proportionality coefficient are significantly different from zero, but some of the low r2 values in the
final column indicate that for many inputs the model gives a poor fit. The penultimate column indicates whether the proportionality coefficient is below or above unity, or whether the null hypothesis that B = I was accepted. In the event of this last happening (as, for example, with the laundry input) the additional input required to produce an additional discharged patient may plausibly be regarded as equal to the average ratio of input to output. It is clear from Table 3 that for most inputs the marginal requirement exceeds the average. while for the inputs beds, nurses, appliances, catering and power, heat and light a less than proportionate increment is required. In the case of a small number of inputs (domestic staff, other dressings, laundry, building and engineering maintenance, maintenance of grounds and transport) an intercept value of zero and a slope value of unity are indicated, implying a constancy of input per unit of output at all levels. Such inputs are unlikely to be closely substitutable for other inputs. THECOBB-DOUGLASFUNCTION
In order to shed some light on the matters of returns to scale and the responsiveness of output to different inputs, the parameters of the Cobb-Doublas function (e.g. Y, + AB,“N,“, where y,, B, and N, denote respectively cases discharged from and inputs of beds and nurses used by hospital i), were first estimated by the ordinary least-squares regression of logarithm of costs discharged
A production
function analysis of English maternity hospitals
87
Table 2. Zero order correlation coefficients between input categories for the 193 English maternity hospitals 1971-72 Category
Medical cay
Nursinj? pay
Bugs
Available beds Occupied beds In-patient days C&es Total.expaditm k!&icalpay Nwsing pay Domesticpay Other pay ws Dressin@, packed Dressings,other Appliances Equipnmr, major Equipmat, traditional Equipnt, disposable Cmtmct services Operatingtheatres Radiotherapy DiagnosticX-my P&lOlO~ Physiotherapy Pb.==cy Ancillarydeparhnents Nurses in training Catering Staff residences Lamdry Pmer, heat, light Maintenance t?edical records Generaladministraticm Generalportering
0.4110 0.47 0.42 0.39 0.49 1.00 0.12 0.16 0.11 0.42 -0.01 0.15 0.04 0.19 0.36 0.13 0.00 0.22 0.02 0.32 0.43 0.27 0.06 0.23 0.38 0.01 0.31 0.01 0.12 0.04 0.15 0.22 0.23 -0.06 -0.10 0.19 0.41
-0.45 -0.50 -0.50 -0.53 -0.42 -0.12 1.00 0.37 -0.07 -0.11 -0.09 0.10 -0.06 -0.13 -0.19 -0.18 -0.09 -0.28 -0.09 -0.25 -0.34 -0.02 -0.04 -0.03 -0.24 0.41 -0.19 0.05 0.40 0.24 0.06 0.44 -0.13 0.20 0.36 0.16 0.08
0.40 0.42 0.40 0.40 0.44 0.42 -0.11 -0.01 0.03 1.00 0.08 0.11 -0.02 0.15 0.41 0.18 0.03 0.11 0.13 0.29 0.42 0.13 0.20 0.31 0.21 -0.07 0.20 0.00 0.00 0.05 0.05 0.19 0.26 -0.01 -0.08 0.04 0.35
General cleaning
Grounds Transport Other services
Table 3. Esttmated coefficients and f-values when the fixed proportions model with scale differences a, + b, In _v,is fitted to the input data for the 193 English maternity hospitals 1971-72
Nurses
pity.
lkmstic pay Other pay mgs PP Ikessings Other dressings Patient'sappliances Major equipnt Traditionalequipment Disposableequipment Operatingtheatres Radiotherapy DiagnosticX-ray Pathology Physiotherapy PhmCy Ancillarymedical service depa%wnts Nursesi.ntMinine . Catering Staff residence LamdTy Paer, heat and light Btildin~mdEn@neering maintenance Medicalrecords @nerd1 administmtion
(t-values) +1 I,...\ ' IL1.l.l
-3.6 -13.8 4.6 0.6 -13.7 -2.1 -10.0 -1.9 -8.2 -21.6 -3.4 -6.4 -25.1 -6.9 -25.0 -14.8 -18.9 -10.0
(t-values) bi ^ ^^ (-15.7) “.YL 2.86 -(-6.5) 0.84 (30.6) 1.06 (0.4) 1.75 (-4.9) 1.29 (-7.2) 1.75 (-3.4) 1.03 (-1.5) 0.62 (-6.3) 2.95 (-8.2) 1.43 (-5.3) 1.67 (-3.6) 3.24 (-11.1) 0.40 (-6.3) 3.26 (-12.8) 2.85 (-6.9) 2.57 (-7.2) 2.04 (-5.0)
-19.6 -31.1 2.3 -24.2 0.4 1.4
(-8.1) (-11.4) (11.3) (-7.3) (1.7) (6.1)
2.59 4.50 0.94 3.73 1.05 0.93
(7.5) (11.4) (31.5) (7.8) (28.9) (27.4)
0.3 -13.2
(0.7) (-5.2) (3.6) (-7.1) (-2.7) (1.1) (-0.8) (1.7)
1.08 2.38 1.03 3.33 1.70 0.60 0.93 1.16
(19.5)
? Occupiedbeds Med?cal Da"
0.9
-18.6 -6.1 1.7 -1.5 0.4
‘
(9.3)
(38.4) (4.6) (4.3) (30.7) (4.0) (5.9) (3.3) (7.8) (15.4) (6.5) (9.9) (2.5) (11.6) (9.2) (6.8) (6.9)
y
-z = .c = < > > > > < > > > > > > < > :
(6.5)
>
(32.7)
>
r2 0.81 0.31 0.89 0.10 0.09 0.83 0.08 0.15 0.05 0.24 0.55 0.18 0.34 0.03 0.41 0.31 0.19 0.20 0.23 0.41 0.84 0.24 0.81 0.80 0.66 0.18 0.82 0.29 0.12 0.03 0.55 0.85
in X,, =
R. J. LAVERS and D. K. W~IYE~SS
88
on logarithms of occupied beds, and other inputs. The assumption here is that inputs are exogenous and for a given level of inputs output will vary depending on a random term which reflects managerial skill, luck, location, etc. (Wallis[3]). Such a function has been previously used to analyse the characteristics of the large acute non-teaching hospitals (Feldstein [ 11). Apart from the ease with which its parameters may be estimated, the Cobb-Douglas function has the advantage that the parameters themselves are readily identifiable, and it is therefore particularly useful for illus~~tive purposes: the parameters a and #? in the above expression measure the percentage change in output resulting from a one per cent change in beds and nurses respectively, while the magnitude of their sum indicates whether returns to scale are constant (sum = 1.) decreasing or increasing with the size of hospital as measured by the number of cases discharged. In view of the fact that the parameters, CX,8, etc., are measures of the elasticity of output with respect to the appropriate input, it follows that a v/f? and @. F/R are estimates of the margins products of beds and nurses respectively in the average hospital. From this it follows that comparisons may also be made between actual ratios of inputs in the average hospital, N/L?,and the implied optimal ratios, a/& which would maintain if all inputs could be adjusted so that their marginal products were equal. As an example, the results of fitting the function using the four inputs beds (B), doctors (M), nurses (N) and drugs and dressings (II) were as follows (togged values):
doctors, this works out at f1380; for nurses the figure is f889 and for drugs &137.The true annual value of the capital services of a bed in 1971-2 was probably of the order of El000 (Feldsteintl], formed a rough upper estimate of f750 for the value of an acute bed in 1960-61) so the above ratios imply that inputs of doctors are rather too high and those of nurses somewhat too low in relation to beds. The average of expenditure per bed on drugs and dressings is so far below the implied optimum that the latter figure can hardly be taken seriously: even the most extravagent user of drugs and dressings recorded a value of just over f500 per bed. In addition to such off implications as this lastmentioned, the properties of the Cobb-Douglas function render it inappropriate as a means to the estimation of the extent to which each input may be substituted for the others: by its very nature, the function allows the possibility that a given output may be produced by any unit of inputs, as long as each input is positive, and the conventional measure of the elasticity of substitution entailed by the function takes the value of unity for all pairs of inputs at al1 levels. If, therefore, one wishes to estimate the elasticity of substitution for pairs of inputs, the Cobb-Douglas is a particularly inappropriate function to use. For this reason, we went on to fit the log-quadratic function, which includes the squares and cross-products of the logged input variables (Sargan[4]), and which has been used to test behavioural hypotheses about 6Q non-teaching, short-term hospitals in Ohio (Hellinger[S]). A typical equation of the log-quadratic form is as follows flogged variables):
Cases = - 0.97 4 0.221ili 0.~2~ + 0.057N f 0.19f) ( - 1.97 (4.4)
(O-4)
0.9)
(3.9)
The percentage of total variance explained by the regression was 91, and the figures in parentheses are t values, from which it may be seen that only the parameter for doctors failed to have a value significantly different from zero. The elasticity of output is greatest for inputs of nurses, while the input of one per cent more beds has siightIy more impact than an addi~onal one per cent spent on drugs and dressings. The sum of the coefficients amounts to 0.97, indicating that returns to scale are slightly decreasing, but if it is assumed that the input variables are statistically independent the standard error of the sum of the coefficients is 0.099, and a t-test of the null hypothesis that the true value in unity is not rejected (t = 0.31). It thus seems that a maternity hospital using twice as much of each input as another will have approximately twice as many discharges. The implied ratios of inputs derived from the c~~cients in the equation differ considerably from those observed in the average maternity hospital: MB
Implied ratio ( + 0) Observed mean (f) Observed Standard deviation
NIB
0.088 2616 121.5 2326 193.5 2812
D/B
0.861 118.1 188.1
Since the implied ratios of inputs to beds are calculated from elasticity values, it is necessary to multiply them by the annual capital cost of a bed(k) in order to make them comparable with the observed data. ~ternatively, one may calculate the implied capital cost of a bed if the observed ratios were equal to the implied optima: for
Cases = 36.8 + 5.9B - 7.8N t O.lL?‘t 0.9NZ- 1.1BN (8.3) (11.1) (8.5) (1.4) (9.8) (-8.7). The figures in parentheses are t-values, and the value of RZ for this particular equation was 0.94. It is clear that in addition to the linear terms for beds and nurses, the coefficients of the quadratic term for nurses and of the interaction term are also significantIy different from zero. The elasticity of cases with respect to nurses now depends on the level of inputs of beds and nurses themselves, and the implication is that hospitals in which expenditure on nurses is below a certain level in relation to beds (N < 12.2B -t8.67) will have a negative elasticity: among such hospitals, that is to say. those with relatively more nurses discharged on average relatively fewer cases. Similarly, the elasticity of cases with respect to beds is such that it takes negative values if B < 11N - 5.9, and although this result is somewhat uncertain in view of the lack of significance of the coefficient of B2, it is what one might expect in the light of the low observed percentage occupancy in the maternity hospitals. The average values of B and N were 24 and f54,063 respectively, so we would expect positive elasticity with respect to nurses and negative elasticity with respect to beds to prevail. Detai&d interpretation of the estimated coefficients of the log-quadratic function, however, is more satisfactorily done when the data used has been appropriately scaled. The main purpose served by the last equation, therefore, is to indicate that the simple log-linear form, as well as being unsatisfactory on logical grounds, does not perform as well empi~caily as the log-quadratic function, and attention is now turned to the use of the latter function on suitably scaled data.
A production function analysis of English maternity hospitals THELOG-QUADRATIC FUNCTION
In view of the fact that any function of several variables may be approximated as closely as one wishes by a Taylor expansion about some particular vector of values, if the logs of the mean input values form the elements of that vector, then the log-quadratic may be regarded as a close approximation to any general production function of whatever form. Instead of using Bi, Ni, _etc.,as input variables, the transformed values In(BJB), In(N,/N), etc., were used. The coefficients in expressions like yi = uO+ alxl, + tizX2i+ fbllx:t +fbz~~ZZi + blZXlrXZr (where y, = In CJC, x,, = In B,/& etc.) could then be estimated by the usual method, and the parameters ai, a2 are the first derivatives and the parameters b,,, bz2, blz the second derivatives at the “typical” values x1 = 0 (corresponding to the mean value of B) and x2 = 0 (where N = N). The constant term a0 also has a straightforward interpretation, since the predicted mean value of y is a0 and therefore the corresponding value of cases discharged is exp(a,JC. Marginal products may also be calculated in natural units; for beds, for example, the marginal_product is a,exp(ao) c/l?, and for nurses, a2 exp(ao) C/N. While marginal products are appropriate quantities to consider when small changes in the composition of inputs are under consideration, the elasticity of substitution between pairs of inputs is a more relevant measure where larger changes are concerned. The usual definition of elasticity of substitution is the relative change in ratio of inputs in response to a unit relative change in the marginal rate of substitution of one input for the other. Following the notation of Sargan[4], if p = dC/dN + dC/dB is the marginal rate of substitution of nurses for beds when cases discharged and all other inputs are held constant, and r = N/B the ratio of nurses to beds, then elasticity of substitution e = (dr/r)/(dp/p). Where different types of input have to be used in fixed proportions, r does not change as p varies and e is therefore zero, implying a Leontief-type production function; where inputs are very close substitutes, p varies little and r will change a lot in response, so that e will take numerically large values. For the Cobb-Douglas function mentioned earlier, the value of e is by defiqition -1, while for the log-quadratic function it is a rather complicated function of the parameters taking the following form: -(a,+ak)
e,k = a,tak)-
b+i2 - 2b,&ak + bkka, wk
where j and k denote inputs and a and b denote coefficients of the linear and quadratic terms respectively. One further general point worth mentioning in connection with the log quadratic function is that by imposing constraints of various kinds on the values that can be taken by its parameters one can derive as special cases the homogeneous production function, the constant returns to scale (i.e. homogeneous of degree one) function and the Cobb-Douglas function. The homogeneous function is obtained by constraining the coefficients of quadratic terms in each input variable to sum to zero, constant returns to scale additionally require that the coefficients of all linear terms sum to unity, and the Cobb-Douglas function requires each coefficient of quadratic terms to be zero. To illustrate the
89
kind of results obtained both with and without the imposition of such constraints, Table 4 gives the results of estimating the parameters when cases discharged are regressed on beds (B), medical staff (M), nurses (N) and drugs and dressings (D) (all variables scaled so that at the mean they take the value zero). It is clear that the explanatory power of the equation (as measured by RZ or 8’) is greatest with no constraints imposed (GLQ) and falls as the function is forced to be homogeneous (HLQ) and Cobb-Douglas (CD). The constant term is not significantly different from zero, which is as it should be if the predicted mean number of cases discharged is to be equal to the observed average, and the sum of the coefficients of linear terms is slightly below unity, indicating slightly decreasing returns to scale (if the hypothesis Z%(Y = 1 is tested, f = - 3.2 and the hypothesis is rejected in favour of the alternative Pa, < 1). Almost all of the quadratic terms in the unconstrained equation fail to have coefficient values significantly different from zero (except where drugs interact with doctors and nurses), although this changes somewhat when homogeneity is enforced. The insignificant or zero coefficient values for medical staff reinforce the observation by Barr[6], that compared with nurses, doctors are relatively unimportant in the maternity hospitals. The results are therefore also presented in Table 4 of the same analysis with medical staff omitted, and it is clear that the resultant reduction in R2 (or d’ adjusted for degrees of freedom) is negligible. One particularly noteworthy feature of the results shown in Table 4 is that as further restrictions are imposed on the coefficients, the linear effect for beds falls while that for nurses rises, which means that estimates of marginal products will differ considerably when derived from the general log-quadratic function from those obtained from the Cobb-Douglas function. Marginal products derived from the various equations are shown below in Table 5: the marginal _ product of a bed on average, for example, is a, C/B discharged cases, where a, is the coefficient of the linear term in beds in the fitted function. The average number of cases discharged in 1971-72 was 1422,and apart from beds, inputs are measured in hundreds of pounds of expenditure. The results suggest that as far as small changes are concerned beds are by far the most important input with a marginal product lying somewhere between 15 and 30 cases and that drugs have a somewhat surprisingly higher marginal product of between 5 and 8 cases per flO0 of expenditure than nurses, with 1 or 2 cases. However, it might be objected that in view of the fact that only about f2 per patient was spent on drugs and dressings, working in El00 units gives a somewhat distorted view of the impact of marginal changes in inputs of drugs. Finally, we present in Table 6 some results of attempts to estimate elasticities of substitution between paris of inputs. As explained above when results of the fixed proportions model presented in Table 3 were being discussed, it seems likely that such inputs as domestic staff, laundry, maintenance and transport are not substitutes for other inputs. Beds, nurses and drugs. however, might be to some extent substitutes for each other and it was thought that the parameters estimated by the logquadratic regression of cases discharged on these three inputs (shown in Table 4) might enable the extent of substitutability to be measured. Apart from the estimated elasticities for beds and nurses, the results are disappointing: most of the values are positive, which arises
11
M
N
D
0.47
(7.2)
-
-
-O.CDi 0.25 t-0.25) (6.41)
CD
-
-
-
0.63 0.12 u.l.4f (3.7)
0.46 0.09 (7.4) (2.01
0.36 0.15 (4.6) (2.8)
0.35 0.10 (4.5) (0.6)
0.m. 0.63 0.11 (0.3)(11.5) (3.6)
0.45 0.012 (0.76) (7.2)
0.43 O.DO6 (0,361 CS.9)
C-O.81
-0.01
-0.005 0.25 C-0.3) (6.51
0.42 -0.02 0.40 0.16 5.x7 (0.00) (6.6) C-1.3) (4.961 (2.9)
-0.Q4 0.33 0.13 (7.3) C-2.0) (3.8) (2,4)
R
0.50
CW
w
CL9
-0.52
(-0.9)
3 varinhler
CD
w
w
variablesckm!st.
M2
N2
0.30 (4.9)
0.31 (5.1)
0.06 (0.6)
"
-
-
0.29 -o.em 14.8) f-1.2)
D2
BM
EIN
BD
rpf
MLI
IID
R2
;i?
0.15 0.03 -0.28 -0.19 -0.05 -0.006 0.08 0.947 0.943 (1.4) (1.6) (-1.7) C-1.5) (-1.8) (-5.8) (6.1)
-0.49 0.29 (-2.7) C2.9)
-0.50 0.22 (-2.81 (3.3)
0.50 0.14 (1.51 (1.3)
-
-
-
o.l.9 -0.49 (1.71 C-6.1?
0.21 -0.52 (X.83 C-6.3)
-0.28 10.15 f-3.7) (-1.2)
-
*
-
-
-
-
0.908 0.903
0.30 0.936 0.932 (3.3)
0.29 0.937 0.934 (3.2)
o.@l 0.944 0.941 (0.2)
0.908 0.901
0.03 -o.cm 0.26 0.933 0.934 -0.49 0.24 -0.02 0.21 -0.49 f-2.7) (3.3) C-O.91 Cl,91 l-.5.5> (l.4f C-0.61 12.7)
0.04 -0.003 0.61 (0.5, C-1.1) (1.7)
B2
Table 4. Estimated coetkients and f-values when the log-quadraticproduction function is fitted to data on cases,beds. medical staff, nursing staff and drugs and dressings for the 193English Maternity hospitals, 1971-72
8
A production Table 5. Estimated
marginal
function
analysis
of English
maternity
hospitals
91
products of beds (units), nurses, doctors and drugs and dressings for the 193 English maternity hospitals, 1971-72
(f100 of Expenditure)
bugs,
3
Table
variable function:
6. Estimated
elasticities
of substitution
between
59.0
2.63
40.29
50.37
29.5
24.8
0.87
-1.6
6.55
1.05
-0.8
8.06
14.8
1.66
27.7
0.92
5.04
25.4
0.95
7.56
26.6
1.21
4.53
14.8
1.66
6.04
0.04
etc
5.54
pairs of inputs used by the 193 English
maternity
hospitals,
1971-72
Function
ESaislNurses
GLQ w CRLQ
Beds/Drugs
Nurses/Dru@
t1.13
+4.3
-22.6
+0.49
-0.86
-45.5
t1.01
dO.40
-1.2
because the coefficients of quadratic terms in drugs (D) are relatively large and which implies that the production function itself is not well behaved in the sense of being concave. Even for the inputs beds and nurses the results are puzzling, for while of the correct sign the degree of responsiveness appears to increase as further restrictions are imposed, and yet in the most constrained case we know elasticity to be by definition -1.0. The most that can be concluded is that there appears to be some scope for substitution between nurses and beds, but not between nurses and drugs or beds and drugs. It also seems in view of these results that the estimation of elasticity of substitution is unlikely to be very successful using data as coarse and highly aggregated as those employed here. AOSPITALTYPEANDlOCATION
One feature of maternity hospitals which springs readily to mind when factors affecting the relationship between inputs and outputs are being considered is that just under 40% of them are consultant units, which possibly treat a rather more complex mix of cases with somewhat different methods from those used by the G.P. units which form the majority. The simplest way in which this aspect of maternity hospitals might affect the production function is by the existence of a scale factor, and test for the presence of such a factor was made by the inclusion in the log-quadratic regressions of a dummy variable taking a value one for consultant units and zero for G.P. units. The results of this analysis are displayed in Table 7 from which is may be inferred that the effect of hospital type is only significantly different from zero at the 10% level, and this is evident only in the general and constant returns forms of the log-quadratic fitted with the three input variables beds, nurses and drugs. Taking the estimate of 0.067 to be a measure of the effect of
consultant units, the implied scale effect is exp (O&70.012) = 1.07, so that cases discharged and marginal products of factors are on average about 7% higher than in G.P. units. It seems reasonable to conclude therefore that if there is a difference in the relationship between input and outputs in the two types of maternity hospital, it is not very sizeable. In other respects also the differences between consultant and G.P. units appear to be small or statistically insignificant. It has been mentioned earlier that many of the input categories listed in the cost returns are closely related to those inputs upon which attention has been focussed in the present study. As part of the analysis of the relation between cases discharged and inputs as a recursive system in which cases are a function of all inputs, but only a few key inputs (beds, nurses, etc.) are exogenous, the other inputs depending on the values of these inputs, expenditures on drugs alone and on drugs and dressings were regressed on the key inputs, with a dummy variable included in respect of consultant units. The rest&s appearing in Table 8 indicate that only when drugs are considered alone is there any evidence of a difference between the two types of hospital, and in this case the estimated additional expenditure of consultant units is of the order of f400 per annum. In other respects the results are what one might expect: approx. 90% of the variation in expenditures on drugs and dressings is accounted for by the three key inputs, although in the case of drugs alone beds appear to be unimportant. While additional expenditure on doctors has the same impact on inputs of drugs as additional expenditure on nurses, nurses have a far greater effect on drugs and dressings combined. Although the obstetric and special baby units clearly do not use fewer drugs and dressings than G.P. units, the evidence is not very clear that they use a great deal more.
R. J. LAVERSand D. K. WHYNES
92
Table 7. Estimated coefficients and r-values for the log-quadratic function. including a dummy variable for consultant units: 193 English maternity hospttals, 1971-72 3 variables Const. GUI
N
B
-0.012 0.46
C-0.8)
(7.0)
B2
D
3.34 0.15 (4.3) (2.K)
N2
I?
BN
'!3D
0.04 0.39 0.15 -0.24 -0.17 (0.5) (1.1) (1.4) (-1.4) (-1.4)
NT!
Ihmy
0.07
R2
0.067 0.945
K7
0.942
(0.4) (1.9)
Iw)
0.006 0.43 0.36 0.17 0.29 -0.53 0.23 0.23 -0.53 (0.4) (6.8) (4.5) (3.1) (4.7)(-7.9) (3.4) (2.0) (-6.5)
0.30 0.038 0.938 0.934 (3.3) (1.1)
c%Q
0.011 0.43 0.42 0.15 0.28 -0.54 0.21 0.23 -0.52 (0.7) (7.0) (6.3) (2.8) (4.6)(-3.0) (7.2) (7.0) (-6.4)
0.30 0.057 0.937 0.933 (3.4) (1.9)
CD
-0.003 0.25 0.61 0.14 (-0.2) (6.6) (10.9) (3.9)
0.046 (1.4)
Grn
-rl.OlS0.53 0.43 (-0.9) (5.4) (6.7)
-
0.03 0.87 (0.3) (9.6)
-
-0.48 (-8.2)
-
HLO
0.014 0.61 0.39 (0.9)(10.9) (6.1)
-
0.45 0.45 (7.7) (7.7)
-
-0.45 (-7.7)
-
-
-0.029 0.923 0.920 (-1.01
c&Q
0.014 0.61 0.19 (0.9)(11.0) (7.0)
-
0.45 0.45 (8.0) (8.0)
-
-0.45 (-8.0)
-
-
-0.029 0.923 0.920 (-1.5)
CD
0.76 0.003 0.29 (0.17)(7.7) (17.6)
-
_
_
_
_
-0.019 0.904 (-0.7)
_
_
0.031 0.943 (1.0)
0.941
0.901
Table 8. Results of the regression of expenditures on drugs alone and drugs and dressings on other Inputs. including a dummy variable for consultant units, 193 English maternity hospitals. 1971-72
const.
DruRs blY
LW@ and DreSSiIlgS
B
M
N
R2
-532.1 (-5.1)
324 .8 (1.7)
-5.1 (-1.0)
0.05 (2.3)
0.05
0.88
-645.7 (-6.9)
418.9 (2.3)
-4.5 (-0.9)
-546.4 (-5.3)
263.9 (1.5)
lxlnw
-1008.5 (-6.8)
198.4
(0.7)
21.4 (3.0)
-1032.3 (-7.9)
218.1 (0.8)
21.6 (3.0)
-947.8 (-6.3)
456.8 (1.8)
A slightly more complex way in which G.P. units and consultant units might differ in respect of their use of drugs and dressings would be for the marginal effect of one of the key inputs to be higher in one type of hospital than in the other. If forces of this kind are at work, they may be detected by including a dummy variable for the slope coefficient of the key inputs, as illustrated in the following equations: D - - 673.3 +0.059( + 0.017) N - O.O03M+20.2B (-3.0) (6.1) (1.9) (-0.1) (2.9) (B” = 0.90) D = - 447.3 + 0.039+ 0.038 N - 0.002~4 (-2.9) (1.9) (1.8) (-0.06) + 59.5( - 41.O)B
(I?* = 0.90)
(1.7)( - 1.1)
The additional coefficient values for N and B shown in parentheses represent the additional effect due to consultant units compared with G.P. units, and while the
(12.5) 0.05 (19.3)
0.87 0.88
0.05 (2.2)
(13.6)
0.01 (0.3)
0.07 (13.9)
0.90
0.08 (19.6)
0.90
0.08 (17.0)
0.90
0.01 (0.5)
0.05
slope value seems to be unaltered for beds, that for nurses is significantly different from zero at the 10% level and indicates that an additional $1000 expenditure on nursing inputs in consultant units could result in additional expenditure on drugs and dressings which was almost 30% higher in these hospitals than in G.P. units. The presence of differences in the relationship between inputs and outputs which are attributable to locational factors was tested for by including a dummy variable for every region except the Leeds region in the log-quadratic regression of cases discharged on inputs. The results when the two inputs beds and nurses were included are shown in Table 9. The value of 8’ for this equation was 0.87 compared with the value of 0.94 for the unrestricted log-quadratic regression containing the same two inputs without regional dummies, and the values of the coefficients of input terms were approximately the same as those shown above. The only difference introduced by the inclusion of regional variables, therefore, is that a regional scale factor is discernible which serves to reduce the average
A production function analysis of English maternity hospitals
93
Table 9. Results of the log-quadratic regression of cases discharged on beds, nurses and 13regional dummy varrables. 193English maternity hospitals 1971-72 Variable
Ccefficient value
t-value
0.15 0.80 -0.51
1.8 3.2 -8.9
Newcastle
-0.04
Sheffield E. hglia N.W. Met N.E. Met. S.E. Wt. S.W. tit. oxford s. western Binningham Manchester LiVerpool wessex
0.03 -0.07 -0.25 -0.15 -0.25 -0.11 0.12 -0.02 0.12 0.W -0.05 0.14
-0.6 0.4 -0.9 -3.3 -2.1 -3.4 -1.1 1.7 -0.3 1.8 0.7 -0.6 1.8
number of cases discharged in certain areas (the London regions) and possibly to raise it in others (Birmingham, Wessex and possibly Oxford). The extent of reduction or increase varies from 11% to 28%, and, as with the effect of type of hospital, estimated marginal products would also be affected by the same order of magnitude. The regional scale factor, it should be stressed, had been calculated with the Leeds region as the standard or norm; if one of the regions in the Northern or Eastern part of the London area had been selected as the standard, for example, many more coefficients with a value significantly in excess of zero would have been recorded. When other inputs were included in the log-quadratic regression, the same sort of results were obtained except that Liverpool sometimes appeared alongside Birmingham among the “over-producers”.
scale are slightly decreasing or at best constant. Less success has been achieved in estimating the values of conventional measures of the extent to which some inputs may be substituted for others, but we have been able to establish that assumptions that have been widely adopted in the past (e.g. that valves of elasticity of substitution are the same for all pairs of inputs, or that they assume certain fixed values) are unlikely to be tenable. On the question of the effect of the location of hospitals on the relationship between outputs and inputs, we have discovered that hospitals in some regions appear to be significantly less efficient than those in others, and differences in the relationship between inputs and outputs may also be attributed to such other features of hospitals as whether they are G.P. or consultant units, although the evidence here is less clear cut.
CONCLUSION
Acknowledgements-We gratefully acknowledgethe assistance of the D.H.S.S. for the funding of a research project, of which
Our production function analysis of English maternity hospitals goes some way towards answering the questions posed at the beginning of this paper. With regard to the degree of responsiveness of output to different inputs, it has been shown that, on the average, hospital beds are marginally more important than nurses whilst drugs and dressings are of considerably less importance, and medical staff have a negligible impact in the maternity hospitals. From such measures of responsiveness, moreover, we are able to conclude that the relative amounts of different inputs which are actually employed in hospitals are rather different from the “best” (in some relevant sense) relative amounts which could be employed. The estimated elasticity values obtained have also enabled the conclusion to be drawn that returns to
this material forms a part. REFERENCES
1. M. S. Feldstein, Economic Analysis for Health Service Ejiciency, Amsterdam (1967). 2. M. Nerlove, Returns to scale in electricity supply. In Measurement in Economics. (Edited by C. F. Christ). Stanford University Press, California (1963). 3. K. F. Wallis, Topics in Applied Econometrics, London (1973). 4. J. D. Sargan. ProductIon functions. In Qualified Manpower and Economic Performance, (Edited by P. R. G. Layard, et al.), London (1971). 5. Fred J. Hellinger, Specification of a hospital production function. Appl. Econ. 7, 149-160(1975). 6. A. Barr. Maternity Hospital Costs. Oxford R.H.A.. Occasional Paper No I.. 1974.
Umversuy
Park. Nottingham NG72R. England
(Received 16 August 1977; receiued
for publication 17 November 1977)
Abstract--In recent years, a number of attempts have been made to explain the output of hospitals by means of production function analysis and, in this particular study, the authors estimate Cobb-Douglas and log-quadratic functions from data for the 193 English maternity hospitals. Of the various inputs into maternity care, the numbers of beds and nurses appear as the most significant determinants of throughput, although the relative quantities actually employed differ from the “technical” optimum. Returns to scale in the maternity service appear to be, at best, constant. The effects of hospital location and type are also analysed and they suggest the existence of significant disparities in levels of efficiency between different hospitals.
The important work of Feldstein[l] has paved the way for a number of analyses of the application of neoclassical production functions to the operation of hospitals. Feldstein’s original approach involved the isolation of the appropriate functional form to explain changes in the inputs into hospitals, such as doctors, nurses, drugs and dressings, which result in variations in hospital output, measured on the number of cases treated in each time period. While the present analysis is very much in the Feldstein tradition, it does make two significant departures. First, by analyzing acute hospitals in aggregate, Feldstein’s production functions become open to criticism on the basis of output mix. Is it reasonable, it can be argued, to conceive of the output of one treatment of a broken limb as commensurate with a case of neuro-surgery? Is it not also likely that, in a large acute hospital, we in fact observe a number of different production functions in each of the different treatment areas? By restricting ourselves to maternity hospitals, we believe that these objections may be overcome; the output of such a process is likely to be much more homogenous and can therefore be measured much more realistically by hospital throughput. In addition, it is more likely that, in such hospitals, there will be less variation in the types of productive process. Second, we explore a wider variety of functional forms than has been undertaken previously; this we believe to be important owing to the restrictive nature of assumptions which are implicit in any one particular type of function. In the same way as Feldstein, however, and, indeed, in common with other researchers in the area, we have followed the convention of using costs as a proxy for real inputs. This is necessary owing to lack of any usable data for this latter variable. In addition to advancing the developments of production function work, we also attempt to provide some insights into the operations of maternity hospitals. In particular, we seek answers to the following questions: (1) what is the direction and magnitude of returns to scale in maternity hospitals, e.g. if inputs are increased by lo%, do outputs increase by the same amount, by more, or by less? (2) is the mix of factors appropriate, e.g. are there too many or too few nurses per bed? (3) how responsive are outputs to changes in partitlnstitute of Social and Economic SDepartment of Economics.
cular factor inputs, e.g. will a 10% increase in doctors have a greater or lesser impact than a similar increase in nurses? (4) what difference does the type and location of the hospital make to output? PRELIMINARYANALYSIS
The data used were obtained from the summaries of hospital costs published annually by the fourteen English Regional Boards, and relate to the financial year ended 31st March, 1972. The total of 193 hospitals ranged in size from a tiny unit with 4 available beds to a relatively large one with 216, and almost two-thirds of them (122) had between 10 and 30 available beds. With a mean available bed size of just under 40, which was also the value of the standard deviation, the size distribution has a fairly pronounced positive skew. Occupied beds have a mean and standard deviation of around 24, implying an average occupancy figure of just over 60%, and values of 50% were not uncommon. Summary statistics of the inpatient data used are shown in Table 1. Many of the items in the conventional classification of costs are of relatively little importance, such as expenditure on patients’ applicances and on some of the treatment departments. The more important items which reflect activities conducted in all hospitals. such as catering and laundry, exhibit little variation from hospital to hospital, while such items as operating theatres and X-ray, with a high coefficient of variation, reflect activities undertaken solely in certain types of hospital (i.e. consultant units in the case of the treatment departments). Of the 193 hospitals, 118 contained only GP beds and the remaining 75 were consultant units containing beds for obstetrics or special care baby units. The analysis that follows makes use of data on annual expenditure on a few selected inputs, notably nurses, doctors and drugs and dressings in addition to average daily occupied beds as a measure of the services yielded by the capital of each hospital. Four possible reasons may account for the fact that many categories of inputs have been omitted from the analysis: first, as noted above, many items make a very small contribution of total expenditure; second, even where items do form a sizeable part of all inputs, as with catering and power, heat and light for instance, their variance is low; third, expenditure on many items is highly correlated with the included inputs; and fourth, a number of inputs are
Research.
85
R.J.
86
LAVERS and
D. K. WHYNES
Table 1. Beds, in-patients and in-patient costs in the 193 English maternity hospitals during the year 1971-72
standard Available beds Occupied beds In-patient days Discharges Total.Expenditure (&) Costs per in-patient week (S): Pay, iredical Pay, nursing Pay, dmstic Pay, other IhYF Pre-packed dressings Other dressings Patients' appliances Equiprent, major Equipmat, traditional Equipment, disposable Cont-mct services ODerattie theat-Rs
&diothe;$W Diagnostic X-ray PdthOlOgy Physiotherapy pharmacy Ancillary departnents Nmesintraining catering Staff residences
timdry Pajer, lipbt and heat Building maintenance Medical records General administration General portering Geneml. cleaning Maintenance of gmmds nwlsport Other services Direct credits costs per case, ward costs percase,treaw?nt
sourres :
39.4
94.1
9,881.O
1,422.0 110,600.O 2.0
38.3 5.9
0.2 1.8
0.3 0.4
0.006 0.1 1.0 0.6 O.O? 0.4 o.cco7 0.04 0.9 0.09
0.3 0.06 0.6 8.0 1.3 2.5 3.0 3.0 0.4 3.4 1.2 1.8 1.1 0.6 5.1 1.5 46.0 47.8
39.1 23.9
11,280.O 1,483.O 123,900.o 1.9 11.6 3.8 0.7 0.4 0.4 0.4 0.05 0.2 0.7 0.6 0.5 1.1 0.003 0.1 1.0 0.2 0.3 0.2 1.4 2.2 1.8 0.9 1.2 2.0 0.6 1.2 1.3 1.8 1.2 0.7 1.8 1.0 13.2 13.6
Summries of hospital.costs for the year ended 31st March 1972, published by the 14 Regional Hospital Beads.
endogenous, that is to say, affected by the level of discharges so that the relationship between them and output is approximately one of fixed proportions. The presence of inputs which are relatively unimportant, in respect both of their magnitude and variability, has already been remarked upon and may be observed in Table 1. As evidence of the existence of some fairly highly correlated inputs, the zero-order coefficients of correlation between each of the three variables medical pay, nursing pay and expenditure on drugs on the one hand, and all categories of input used in in-patient departments on the other hand, are presented below in Table 2. The 5% critical values of r are kO.16. As evidence for the existence of fixed proportions for many inputs, the results obtained from fitting the model X, = a,yp’, where X,, denotes the input of category i to hospital j and y, denoted discharges from hospital j, are presented in Table 3. Such a function has been successfully used in the analysis of plants generating electricity, for example (Nerlove[2]) and provided the y, are appropriately regarded as exogenous (which would seem to be in order in view of the low occupancy levels referred to earlier) the parameters may legitimately be estimated by the least squares regression of the logarithm of each input on the logarithm of discharges. It is clear from the t-values shown in brackets that for almost all inputs both the scale and proportionality coefficient are significantly different from zero, but some of the low r2 values in the
final column indicate that for many inputs the model gives a poor fit. The penultimate column indicates whether the proportionality coefficient is below or above unity, or whether the null hypothesis that B = I was accepted. In the event of this last happening (as, for example, with the laundry input) the additional input required to produce an additional discharged patient may plausibly be regarded as equal to the average ratio of input to output. It is clear from Table 3 that for most inputs the marginal requirement exceeds the average. while for the inputs beds, nurses, appliances, catering and power, heat and light a less than proportionate increment is required. In the case of a small number of inputs (domestic staff, other dressings, laundry, building and engineering maintenance, maintenance of grounds and transport) an intercept value of zero and a slope value of unity are indicated, implying a constancy of input per unit of output at all levels. Such inputs are unlikely to be closely substitutable for other inputs. THECOBB-DOUGLASFUNCTION
In order to shed some light on the matters of returns to scale and the responsiveness of output to different inputs, the parameters of the Cobb-Doublas function (e.g. Y, + AB,“N,“, where y,, B, and N, denote respectively cases discharged from and inputs of beds and nurses used by hospital i), were first estimated by the ordinary least-squares regression of logarithm of costs discharged
A production
function analysis of English maternity hospitals
87
Table 2. Zero order correlation coefficients between input categories for the 193 English maternity hospitals 1971-72 Category
Medical cay
Nursinj? pay
Bugs
Available beds Occupied beds In-patient days C&es Total.expaditm k!&icalpay Nwsing pay Domesticpay Other pay ws Dressin@, packed Dressings,other Appliances Equipnmr, major Equipmat, traditional Equipnt, disposable Cmtmct services Operatingtheatres Radiotherapy DiagnosticX-my P&lOlO~ Physiotherapy Pb.==cy Ancillarydeparhnents Nurses in training Catering Staff residences Lamdry Pmer, heat, light Maintenance t?edical records Generaladministraticm Generalportering
0.4110 0.47 0.42 0.39 0.49 1.00 0.12 0.16 0.11 0.42 -0.01 0.15 0.04 0.19 0.36 0.13 0.00 0.22 0.02 0.32 0.43 0.27 0.06 0.23 0.38 0.01 0.31 0.01 0.12 0.04 0.15 0.22 0.23 -0.06 -0.10 0.19 0.41
-0.45 -0.50 -0.50 -0.53 -0.42 -0.12 1.00 0.37 -0.07 -0.11 -0.09 0.10 -0.06 -0.13 -0.19 -0.18 -0.09 -0.28 -0.09 -0.25 -0.34 -0.02 -0.04 -0.03 -0.24 0.41 -0.19 0.05 0.40 0.24 0.06 0.44 -0.13 0.20 0.36 0.16 0.08
0.40 0.42 0.40 0.40 0.44 0.42 -0.11 -0.01 0.03 1.00 0.08 0.11 -0.02 0.15 0.41 0.18 0.03 0.11 0.13 0.29 0.42 0.13 0.20 0.31 0.21 -0.07 0.20 0.00 0.00 0.05 0.05 0.19 0.26 -0.01 -0.08 0.04 0.35
General cleaning
Grounds Transport Other services
Table 3. Esttmated coefficients and f-values when the fixed proportions model with scale differences a, + b, In _v,is fitted to the input data for the 193 English maternity hospitals 1971-72
Nurses
pity.
lkmstic pay Other pay mgs PP Ikessings Other dressings Patient'sappliances Major equipnt Traditionalequipment Disposableequipment Operatingtheatres Radiotherapy DiagnosticX-ray Pathology Physiotherapy PhmCy Ancillarymedical service depa%wnts Nursesi.ntMinine . Catering Staff residence LamdTy Paer, heat and light Btildin~mdEn@neering maintenance Medicalrecords @nerd1 administmtion
(t-values) +1 I,...\ ' IL1.l.l
-3.6 -13.8 4.6 0.6 -13.7 -2.1 -10.0 -1.9 -8.2 -21.6 -3.4 -6.4 -25.1 -6.9 -25.0 -14.8 -18.9 -10.0
(t-values) bi ^ ^^ (-15.7) “.YL 2.86 -(-6.5) 0.84 (30.6) 1.06 (0.4) 1.75 (-4.9) 1.29 (-7.2) 1.75 (-3.4) 1.03 (-1.5) 0.62 (-6.3) 2.95 (-8.2) 1.43 (-5.3) 1.67 (-3.6) 3.24 (-11.1) 0.40 (-6.3) 3.26 (-12.8) 2.85 (-6.9) 2.57 (-7.2) 2.04 (-5.0)
-19.6 -31.1 2.3 -24.2 0.4 1.4
(-8.1) (-11.4) (11.3) (-7.3) (1.7) (6.1)
2.59 4.50 0.94 3.73 1.05 0.93
(7.5) (11.4) (31.5) (7.8) (28.9) (27.4)
0.3 -13.2
(0.7) (-5.2) (3.6) (-7.1) (-2.7) (1.1) (-0.8) (1.7)
1.08 2.38 1.03 3.33 1.70 0.60 0.93 1.16
(19.5)
? Occupiedbeds Med?cal Da"
0.9
-18.6 -6.1 1.7 -1.5 0.4
‘
(9.3)
(38.4) (4.6) (4.3) (30.7) (4.0) (5.9) (3.3) (7.8) (15.4) (6.5) (9.9) (2.5) (11.6) (9.2) (6.8) (6.9)
y
-z = .c = < > > > > < > > > > > > < > :
(6.5)
>
(32.7)
>
r2 0.81 0.31 0.89 0.10 0.09 0.83 0.08 0.15 0.05 0.24 0.55 0.18 0.34 0.03 0.41 0.31 0.19 0.20 0.23 0.41 0.84 0.24 0.81 0.80 0.66 0.18 0.82 0.29 0.12 0.03 0.55 0.85
in X,, =
R. J. LAVERS and D. K. W~IYE~SS
88
on logarithms of occupied beds, and other inputs. The assumption here is that inputs are exogenous and for a given level of inputs output will vary depending on a random term which reflects managerial skill, luck, location, etc. (Wallis[3]). Such a function has been previously used to analyse the characteristics of the large acute non-teaching hospitals (Feldstein [ 11). Apart from the ease with which its parameters may be estimated, the Cobb-Douglas function has the advantage that the parameters themselves are readily identifiable, and it is therefore particularly useful for illus~~tive purposes: the parameters a and #? in the above expression measure the percentage change in output resulting from a one per cent change in beds and nurses respectively, while the magnitude of their sum indicates whether returns to scale are constant (sum = 1.) decreasing or increasing with the size of hospital as measured by the number of cases discharged. In view of the fact that the parameters, CX,8, etc., are measures of the elasticity of output with respect to the appropriate input, it follows that a v/f? and @. F/R are estimates of the margins products of beds and nurses respectively in the average hospital. From this it follows that comparisons may also be made between actual ratios of inputs in the average hospital, N/L?,and the implied optimal ratios, a/& which would maintain if all inputs could be adjusted so that their marginal products were equal. As an example, the results of fitting the function using the four inputs beds (B), doctors (M), nurses (N) and drugs and dressings (II) were as follows (togged values):
doctors, this works out at f1380; for nurses the figure is f889 and for drugs &137.The true annual value of the capital services of a bed in 1971-2 was probably of the order of El000 (Feldsteintl], formed a rough upper estimate of f750 for the value of an acute bed in 1960-61) so the above ratios imply that inputs of doctors are rather too high and those of nurses somewhat too low in relation to beds. The average of expenditure per bed on drugs and dressings is so far below the implied optimum that the latter figure can hardly be taken seriously: even the most extravagent user of drugs and dressings recorded a value of just over f500 per bed. In addition to such off implications as this lastmentioned, the properties of the Cobb-Douglas function render it inappropriate as a means to the estimation of the extent to which each input may be substituted for the others: by its very nature, the function allows the possibility that a given output may be produced by any unit of inputs, as long as each input is positive, and the conventional measure of the elasticity of substitution entailed by the function takes the value of unity for all pairs of inputs at al1 levels. If, therefore, one wishes to estimate the elasticity of substitution for pairs of inputs, the Cobb-Douglas is a particularly inappropriate function to use. For this reason, we went on to fit the log-quadratic function, which includes the squares and cross-products of the logged input variables (Sargan[4]), and which has been used to test behavioural hypotheses about 6Q non-teaching, short-term hospitals in Ohio (Hellinger[S]). A typical equation of the log-quadratic form is as follows flogged variables):
Cases = - 0.97 4 0.221ili 0.~2~ + 0.057N f 0.19f) ( - 1.97 (4.4)
(O-4)
0.9)
(3.9)
The percentage of total variance explained by the regression was 91, and the figures in parentheses are t values, from which it may be seen that only the parameter for doctors failed to have a value significantly different from zero. The elasticity of output is greatest for inputs of nurses, while the input of one per cent more beds has siightIy more impact than an addi~onal one per cent spent on drugs and dressings. The sum of the coefficients amounts to 0.97, indicating that returns to scale are slightly decreasing, but if it is assumed that the input variables are statistically independent the standard error of the sum of the coefficients is 0.099, and a t-test of the null hypothesis that the true value in unity is not rejected (t = 0.31). It thus seems that a maternity hospital using twice as much of each input as another will have approximately twice as many discharges. The implied ratios of inputs derived from the c~~cients in the equation differ considerably from those observed in the average maternity hospital: MB
Implied ratio ( + 0) Observed mean (f) Observed Standard deviation
NIB
0.088 2616 121.5 2326 193.5 2812
D/B
0.861 118.1 188.1
Since the implied ratios of inputs to beds are calculated from elasticity values, it is necessary to multiply them by the annual capital cost of a bed(k) in order to make them comparable with the observed data. ~ternatively, one may calculate the implied capital cost of a bed if the observed ratios were equal to the implied optima: for
Cases = 36.8 + 5.9B - 7.8N t O.lL?‘t 0.9NZ- 1.1BN (8.3) (11.1) (8.5) (1.4) (9.8) (-8.7). The figures in parentheses are t-values, and the value of RZ for this particular equation was 0.94. It is clear that in addition to the linear terms for beds and nurses, the coefficients of the quadratic term for nurses and of the interaction term are also significantIy different from zero. The elasticity of cases with respect to nurses now depends on the level of inputs of beds and nurses themselves, and the implication is that hospitals in which expenditure on nurses is below a certain level in relation to beds (N < 12.2B -t8.67) will have a negative elasticity: among such hospitals, that is to say. those with relatively more nurses discharged on average relatively fewer cases. Similarly, the elasticity of cases with respect to beds is such that it takes negative values if B < 11N - 5.9, and although this result is somewhat uncertain in view of the lack of significance of the coefficient of B2, it is what one might expect in the light of the low observed percentage occupancy in the maternity hospitals. The average values of B and N were 24 and f54,063 respectively, so we would expect positive elasticity with respect to nurses and negative elasticity with respect to beds to prevail. Detai&d interpretation of the estimated coefficients of the log-quadratic function, however, is more satisfactorily done when the data used has been appropriately scaled. The main purpose served by the last equation, therefore, is to indicate that the simple log-linear form, as well as being unsatisfactory on logical grounds, does not perform as well empi~caily as the log-quadratic function, and attention is now turned to the use of the latter function on suitably scaled data.
A production function analysis of English maternity hospitals THELOG-QUADRATIC FUNCTION
In view of the fact that any function of several variables may be approximated as closely as one wishes by a Taylor expansion about some particular vector of values, if the logs of the mean input values form the elements of that vector, then the log-quadratic may be regarded as a close approximation to any general production function of whatever form. Instead of using Bi, Ni, _etc.,as input variables, the transformed values In(BJB), In(N,/N), etc., were used. The coefficients in expressions like yi = uO+ alxl, + tizX2i+ fbllx:t +fbz~~ZZi + blZXlrXZr (where y, = In CJC, x,, = In B,/& etc.) could then be estimated by the usual method, and the parameters ai, a2 are the first derivatives and the parameters b,,, bz2, blz the second derivatives at the “typical” values x1 = 0 (corresponding to the mean value of B) and x2 = 0 (where N = N). The constant term a0 also has a straightforward interpretation, since the predicted mean value of y is a0 and therefore the corresponding value of cases discharged is exp(a,JC. Marginal products may also be calculated in natural units; for beds, for example, the marginal_product is a,exp(ao) c/l?, and for nurses, a2 exp(ao) C/N. While marginal products are appropriate quantities to consider when small changes in the composition of inputs are under consideration, the elasticity of substitution between pairs of inputs is a more relevant measure where larger changes are concerned. The usual definition of elasticity of substitution is the relative change in ratio of inputs in response to a unit relative change in the marginal rate of substitution of one input for the other. Following the notation of Sargan[4], if p = dC/dN + dC/dB is the marginal rate of substitution of nurses for beds when cases discharged and all other inputs are held constant, and r = N/B the ratio of nurses to beds, then elasticity of substitution e = (dr/r)/(dp/p). Where different types of input have to be used in fixed proportions, r does not change as p varies and e is therefore zero, implying a Leontief-type production function; where inputs are very close substitutes, p varies little and r will change a lot in response, so that e will take numerically large values. For the Cobb-Douglas function mentioned earlier, the value of e is by defiqition -1, while for the log-quadratic function it is a rather complicated function of the parameters taking the following form: -(a,+ak)
e,k = a,tak)-
b+i2 - 2b,&ak + bkka, wk
where j and k denote inputs and a and b denote coefficients of the linear and quadratic terms respectively. One further general point worth mentioning in connection with the log quadratic function is that by imposing constraints of various kinds on the values that can be taken by its parameters one can derive as special cases the homogeneous production function, the constant returns to scale (i.e. homogeneous of degree one) function and the Cobb-Douglas function. The homogeneous function is obtained by constraining the coefficients of quadratic terms in each input variable to sum to zero, constant returns to scale additionally require that the coefficients of all linear terms sum to unity, and the Cobb-Douglas function requires each coefficient of quadratic terms to be zero. To illustrate the
89
kind of results obtained both with and without the imposition of such constraints, Table 4 gives the results of estimating the parameters when cases discharged are regressed on beds (B), medical staff (M), nurses (N) and drugs and dressings (D) (all variables scaled so that at the mean they take the value zero). It is clear that the explanatory power of the equation (as measured by RZ or 8’) is greatest with no constraints imposed (GLQ) and falls as the function is forced to be homogeneous (HLQ) and Cobb-Douglas (CD). The constant term is not significantly different from zero, which is as it should be if the predicted mean number of cases discharged is to be equal to the observed average, and the sum of the coefficients of linear terms is slightly below unity, indicating slightly decreasing returns to scale (if the hypothesis Z%(Y = 1 is tested, f = - 3.2 and the hypothesis is rejected in favour of the alternative Pa, < 1). Almost all of the quadratic terms in the unconstrained equation fail to have coefficient values significantly different from zero (except where drugs interact with doctors and nurses), although this changes somewhat when homogeneity is enforced. The insignificant or zero coefficient values for medical staff reinforce the observation by Barr[6], that compared with nurses, doctors are relatively unimportant in the maternity hospitals. The results are therefore also presented in Table 4 of the same analysis with medical staff omitted, and it is clear that the resultant reduction in R2 (or d’ adjusted for degrees of freedom) is negligible. One particularly noteworthy feature of the results shown in Table 4 is that as further restrictions are imposed on the coefficients, the linear effect for beds falls while that for nurses rises, which means that estimates of marginal products will differ considerably when derived from the general log-quadratic function from those obtained from the Cobb-Douglas function. Marginal products derived from the various equations are shown below in Table 5: the marginal _ product of a bed on average, for example, is a, C/B discharged cases, where a, is the coefficient of the linear term in beds in the fitted function. The average number of cases discharged in 1971-72 was 1422,and apart from beds, inputs are measured in hundreds of pounds of expenditure. The results suggest that as far as small changes are concerned beds are by far the most important input with a marginal product lying somewhere between 15 and 30 cases and that drugs have a somewhat surprisingly higher marginal product of between 5 and 8 cases per flO0 of expenditure than nurses, with 1 or 2 cases. However, it might be objected that in view of the fact that only about f2 per patient was spent on drugs and dressings, working in El00 units gives a somewhat distorted view of the impact of marginal changes in inputs of drugs. Finally, we present in Table 6 some results of attempts to estimate elasticities of substitution between paris of inputs. As explained above when results of the fixed proportions model presented in Table 3 were being discussed, it seems likely that such inputs as domestic staff, laundry, maintenance and transport are not substitutes for other inputs. Beds, nurses and drugs. however, might be to some extent substitutes for each other and it was thought that the parameters estimated by the logquadratic regression of cases discharged on these three inputs (shown in Table 4) might enable the extent of substitutability to be measured. Apart from the estimated elasticities for beds and nurses, the results are disappointing: most of the values are positive, which arises
11
M
N
D
0.47
(7.2)
-
-
-O.CDi 0.25 t-0.25) (6.41)
CD
-
-
-
0.63 0.12 u.l.4f (3.7)
0.46 0.09 (7.4) (2.01
0.36 0.15 (4.6) (2.8)
0.35 0.10 (4.5) (0.6)
0.m. 0.63 0.11 (0.3)(11.5) (3.6)
0.45 0.012 (0.76) (7.2)
0.43 O.DO6 (0,361 CS.9)
C-O.81
-0.01
-0.005 0.25 C-0.3) (6.51
0.42 -0.02 0.40 0.16 5.x7 (0.00) (6.6) C-1.3) (4.961 (2.9)
-0.Q4 0.33 0.13 (7.3) C-2.0) (3.8) (2,4)
R
0.50
CW
w
CL9
-0.52
(-0.9)
3 varinhler
CD
w
w
variablesckm!st.
M2
N2
0.30 (4.9)
0.31 (5.1)
0.06 (0.6)
"
-
-
0.29 -o.em 14.8) f-1.2)
D2
BM
EIN
BD
rpf
MLI
IID
R2
;i?
0.15 0.03 -0.28 -0.19 -0.05 -0.006 0.08 0.947 0.943 (1.4) (1.6) (-1.7) C-1.5) (-1.8) (-5.8) (6.1)
-0.49 0.29 (-2.7) C2.9)
-0.50 0.22 (-2.81 (3.3)
0.50 0.14 (1.51 (1.3)
-
-
-
o.l.9 -0.49 (1.71 C-6.1?
0.21 -0.52 (X.83 C-6.3)
-0.28 10.15 f-3.7) (-1.2)
-
*
-
-
-
-
0.908 0.903
0.30 0.936 0.932 (3.3)
0.29 0.937 0.934 (3.2)
o.@l 0.944 0.941 (0.2)
0.908 0.901
0.03 -o.cm 0.26 0.933 0.934 -0.49 0.24 -0.02 0.21 -0.49 f-2.7) (3.3) C-O.91 Cl,91 l-.5.5> (l.4f C-0.61 12.7)
0.04 -0.003 0.61 (0.5, C-1.1) (1.7)
B2
Table 4. Estimated coetkients and f-values when the log-quadraticproduction function is fitted to data on cases,beds. medical staff, nursing staff and drugs and dressings for the 193English Maternity hospitals, 1971-72
8
A production Table 5. Estimated
marginal
function
analysis
of English
maternity
hospitals
91
products of beds (units), nurses, doctors and drugs and dressings for the 193 English maternity hospitals, 1971-72
(f100 of Expenditure)
bugs,
3
Table
variable function:
6. Estimated
elasticities
of substitution
between
59.0
2.63
40.29
50.37
29.5
24.8
0.87
-1.6
6.55
1.05
-0.8
8.06
14.8
1.66
27.7
0.92
5.04
25.4
0.95
7.56
26.6
1.21
4.53
14.8
1.66
6.04
0.04
etc
5.54
pairs of inputs used by the 193 English
maternity
hospitals,
1971-72
Function
ESaislNurses
GLQ w CRLQ
Beds/Drugs
Nurses/Dru@
t1.13
+4.3
-22.6
+0.49
-0.86
-45.5
t1.01
dO.40
-1.2
because the coefficients of quadratic terms in drugs (D) are relatively large and which implies that the production function itself is not well behaved in the sense of being concave. Even for the inputs beds and nurses the results are puzzling, for while of the correct sign the degree of responsiveness appears to increase as further restrictions are imposed, and yet in the most constrained case we know elasticity to be by definition -1.0. The most that can be concluded is that there appears to be some scope for substitution between nurses and beds, but not between nurses and drugs or beds and drugs. It also seems in view of these results that the estimation of elasticity of substitution is unlikely to be very successful using data as coarse and highly aggregated as those employed here. AOSPITALTYPEANDlOCATION
One feature of maternity hospitals which springs readily to mind when factors affecting the relationship between inputs and outputs are being considered is that just under 40% of them are consultant units, which possibly treat a rather more complex mix of cases with somewhat different methods from those used by the G.P. units which form the majority. The simplest way in which this aspect of maternity hospitals might affect the production function is by the existence of a scale factor, and test for the presence of such a factor was made by the inclusion in the log-quadratic regressions of a dummy variable taking a value one for consultant units and zero for G.P. units. The results of this analysis are displayed in Table 7 from which is may be inferred that the effect of hospital type is only significantly different from zero at the 10% level, and this is evident only in the general and constant returns forms of the log-quadratic fitted with the three input variables beds, nurses and drugs. Taking the estimate of 0.067 to be a measure of the effect of
consultant units, the implied scale effect is exp (O&70.012) = 1.07, so that cases discharged and marginal products of factors are on average about 7% higher than in G.P. units. It seems reasonable to conclude therefore that if there is a difference in the relationship between input and outputs in the two types of maternity hospital, it is not very sizeable. In other respects also the differences between consultant and G.P. units appear to be small or statistically insignificant. It has been mentioned earlier that many of the input categories listed in the cost returns are closely related to those inputs upon which attention has been focussed in the present study. As part of the analysis of the relation between cases discharged and inputs as a recursive system in which cases are a function of all inputs, but only a few key inputs (beds, nurses, etc.) are exogenous, the other inputs depending on the values of these inputs, expenditures on drugs alone and on drugs and dressings were regressed on the key inputs, with a dummy variable included in respect of consultant units. The rest&s appearing in Table 8 indicate that only when drugs are considered alone is there any evidence of a difference between the two types of hospital, and in this case the estimated additional expenditure of consultant units is of the order of f400 per annum. In other respects the results are what one might expect: approx. 90% of the variation in expenditures on drugs and dressings is accounted for by the three key inputs, although in the case of drugs alone beds appear to be unimportant. While additional expenditure on doctors has the same impact on inputs of drugs as additional expenditure on nurses, nurses have a far greater effect on drugs and dressings combined. Although the obstetric and special baby units clearly do not use fewer drugs and dressings than G.P. units, the evidence is not very clear that they use a great deal more.
R. J. LAVERSand D. K. WHYNES
92
Table 7. Estimated coefficients and r-values for the log-quadratic function. including a dummy variable for consultant units: 193 English maternity hospttals, 1971-72 3 variables Const. GUI
N
B
-0.012 0.46
C-0.8)
(7.0)
B2
D
3.34 0.15 (4.3) (2.K)
N2
I?
BN
'!3D
0.04 0.39 0.15 -0.24 -0.17 (0.5) (1.1) (1.4) (-1.4) (-1.4)
NT!
Ihmy
0.07
R2
0.067 0.945
K7
0.942
(0.4) (1.9)
Iw)
0.006 0.43 0.36 0.17 0.29 -0.53 0.23 0.23 -0.53 (0.4) (6.8) (4.5) (3.1) (4.7)(-7.9) (3.4) (2.0) (-6.5)
0.30 0.038 0.938 0.934 (3.3) (1.1)
c%Q
0.011 0.43 0.42 0.15 0.28 -0.54 0.21 0.23 -0.52 (0.7) (7.0) (6.3) (2.8) (4.6)(-3.0) (7.2) (7.0) (-6.4)
0.30 0.057 0.937 0.933 (3.4) (1.9)
CD
-0.003 0.25 0.61 0.14 (-0.2) (6.6) (10.9) (3.9)
0.046 (1.4)
Grn
-rl.OlS0.53 0.43 (-0.9) (5.4) (6.7)
-
0.03 0.87 (0.3) (9.6)
-
-0.48 (-8.2)
-
HLO
0.014 0.61 0.39 (0.9)(10.9) (6.1)
-
0.45 0.45 (7.7) (7.7)
-
-0.45 (-7.7)
-
-
-0.029 0.923 0.920 (-1.01
c&Q
0.014 0.61 0.19 (0.9)(11.0) (7.0)
-
0.45 0.45 (8.0) (8.0)
-
-0.45 (-8.0)
-
-
-0.029 0.923 0.920 (-1.5)
CD
0.76 0.003 0.29 (0.17)(7.7) (17.6)
-
_
_
_
_
-0.019 0.904 (-0.7)
_
_
0.031 0.943 (1.0)
0.941
0.901
Table 8. Results of the regression of expenditures on drugs alone and drugs and dressings on other Inputs. including a dummy variable for consultant units, 193 English maternity hospitals. 1971-72
const.
DruRs blY
LW@ and DreSSiIlgS
B
M
N
R2
-532.1 (-5.1)
324 .8 (1.7)
-5.1 (-1.0)
0.05 (2.3)
0.05
0.88
-645.7 (-6.9)
418.9 (2.3)
-4.5 (-0.9)
-546.4 (-5.3)
263.9 (1.5)
lxlnw
-1008.5 (-6.8)
198.4
(0.7)
21.4 (3.0)
-1032.3 (-7.9)
218.1 (0.8)
21.6 (3.0)
-947.8 (-6.3)
456.8 (1.8)
A slightly more complex way in which G.P. units and consultant units might differ in respect of their use of drugs and dressings would be for the marginal effect of one of the key inputs to be higher in one type of hospital than in the other. If forces of this kind are at work, they may be detected by including a dummy variable for the slope coefficient of the key inputs, as illustrated in the following equations: D - - 673.3 +0.059( + 0.017) N - O.O03M+20.2B (-3.0) (6.1) (1.9) (-0.1) (2.9) (B” = 0.90) D = - 447.3 + 0.039+ 0.038 N - 0.002~4 (-2.9) (1.9) (1.8) (-0.06) + 59.5( - 41.O)B
(I?* = 0.90)
(1.7)( - 1.1)
The additional coefficient values for N and B shown in parentheses represent the additional effect due to consultant units compared with G.P. units, and while the
(12.5) 0.05 (19.3)
0.87 0.88
0.05 (2.2)
(13.6)
0.01 (0.3)
0.07 (13.9)
0.90
0.08 (19.6)
0.90
0.08 (17.0)
0.90
0.01 (0.5)
0.05
slope value seems to be unaltered for beds, that for nurses is significantly different from zero at the 10% level and indicates that an additional $1000 expenditure on nursing inputs in consultant units could result in additional expenditure on drugs and dressings which was almost 30% higher in these hospitals than in G.P. units. The presence of differences in the relationship between inputs and outputs which are attributable to locational factors was tested for by including a dummy variable for every region except the Leeds region in the log-quadratic regression of cases discharged on inputs. The results when the two inputs beds and nurses were included are shown in Table 9. The value of 8’ for this equation was 0.87 compared with the value of 0.94 for the unrestricted log-quadratic regression containing the same two inputs without regional dummies, and the values of the coefficients of input terms were approximately the same as those shown above. The only difference introduced by the inclusion of regional variables, therefore, is that a regional scale factor is discernible which serves to reduce the average
A production function analysis of English maternity hospitals
93
Table 9. Results of the log-quadratic regression of cases discharged on beds, nurses and 13regional dummy varrables. 193English maternity hospitals 1971-72 Variable
Ccefficient value
t-value
0.15 0.80 -0.51
1.8 3.2 -8.9
Newcastle
-0.04
Sheffield E. hglia N.W. Met N.E. Met. S.E. Wt. S.W. tit. oxford s. western Binningham Manchester LiVerpool wessex
0.03 -0.07 -0.25 -0.15 -0.25 -0.11 0.12 -0.02 0.12 0.W -0.05 0.14
-0.6 0.4 -0.9 -3.3 -2.1 -3.4 -1.1 1.7 -0.3 1.8 0.7 -0.6 1.8
number of cases discharged in certain areas (the London regions) and possibly to raise it in others (Birmingham, Wessex and possibly Oxford). The extent of reduction or increase varies from 11% to 28%, and, as with the effect of type of hospital, estimated marginal products would also be affected by the same order of magnitude. The regional scale factor, it should be stressed, had been calculated with the Leeds region as the standard or norm; if one of the regions in the Northern or Eastern part of the London area had been selected as the standard, for example, many more coefficients with a value significantly in excess of zero would have been recorded. When other inputs were included in the log-quadratic regression, the same sort of results were obtained except that Liverpool sometimes appeared alongside Birmingham among the “over-producers”.
scale are slightly decreasing or at best constant. Less success has been achieved in estimating the values of conventional measures of the extent to which some inputs may be substituted for others, but we have been able to establish that assumptions that have been widely adopted in the past (e.g. that valves of elasticity of substitution are the same for all pairs of inputs, or that they assume certain fixed values) are unlikely to be tenable. On the question of the effect of the location of hospitals on the relationship between outputs and inputs, we have discovered that hospitals in some regions appear to be significantly less efficient than those in others, and differences in the relationship between inputs and outputs may also be attributed to such other features of hospitals as whether they are G.P. or consultant units, although the evidence here is less clear cut.
CONCLUSION
Acknowledgements-We gratefully acknowledgethe assistance of the D.H.S.S. for the funding of a research project, of which
Our production function analysis of English maternity hospitals goes some way towards answering the questions posed at the beginning of this paper. With regard to the degree of responsiveness of output to different inputs, it has been shown that, on the average, hospital beds are marginally more important than nurses whilst drugs and dressings are of considerably less importance, and medical staff have a negligible impact in the maternity hospitals. From such measures of responsiveness, moreover, we are able to conclude that the relative amounts of different inputs which are actually employed in hospitals are rather different from the “best” (in some relevant sense) relative amounts which could be employed. The estimated elasticity values obtained have also enabled the conclusion to be drawn that returns to
this material forms a part. REFERENCES
1. M. S. Feldstein, Economic Analysis for Health Service Ejiciency, Amsterdam (1967). 2. M. Nerlove, Returns to scale in electricity supply. In Measurement in Economics. (Edited by C. F. Christ). Stanford University Press, California (1963). 3. K. F. Wallis, Topics in Applied Econometrics, London (1973). 4. J. D. Sargan. ProductIon functions. In Qualified Manpower and Economic Performance, (Edited by P. R. G. Layard, et al.), London (1971). 5. Fred J. Hellinger, Specification of a hospital production function. Appl. Econ. 7, 149-160(1975). 6. A. Barr. Maternity Hospital Costs. Oxford R.H.A.. Occasional Paper No I.. 1974.