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Hospital scale economies due to stochastic demand and service
Economics Letters North-Holland
HOSPITAL
193
23 (1987) 193-197
SCALE ECONOMIES
DUE TO STOCHASTIC
DEMAND
AND SERVICE
James G. MULLIGAN University of Delaware, Newark, DE 19716, USA Received
18 September
1986
This paper uses queuing theory to approximate empirically the potential for hospital scale economies due to the stochastic nature of hospital demand and service. These results suggest that scale economies due to stochastic demand and service are likely to be important only for the consolidation of small, specialized hospital units.
1. Introduction The hospital market has two important features: (1) the good produced is a service, and (2) consumers are not affected for the most part by the marginal cost associated with their hospital stay. Anywhere from 85 to 90% of Americans using hospitals have third party insurance policies [Bothwell and Cooley (1982)]. When a firm produces a service and cannot ration its service with price, excess demand must be rationed by other means. The most likely mechanism is a queue. The cost to a potential customer is a wait in the queue. The customer will make a market decision based on how much time he or she expects to wait, the expected quality of the service and the quality and availability of alternatives. This rationing process can be made formal with a queuing model. While several researchers [e.g., Joskow (1980), Coyte (1983), DeVany (1976), and MacStrevic (1981)] have recognized that fluctuations in the demand for hospital services pose a queuing problem, no one has exploited the potential of queuing-theoretic models in a way that could determine the expected economies associated with hospital operations. This paper uses a queuing model to assess the potential gains and demonstrates that scale economies due to stochastic demand and service are likely to be important only for the consolidation of small, specialized hospital units.
2. Economies from pooling hospital beds DeVany (1976) offered one of the strongest statements that stochastic demand and service may lead to economies for large scale operations. DeVany used a queuing model to describe a natural-monopoly market for a non-storable good or service and related this model to the issue of the economies of large hospitals. DeVany suggested that his model implies that large hospitals would operate at significantly higher occupancy rates than smaller hospitals, leading to substantial scale economies. Despite these claims, DeVany interpreted his model incorrectly. 1 ’ Mulligan (1986) provides a detailed criticism of the DeVany model and shows that what DeVany scale would be a change in technology for the stochastic model used in his paper.
01651765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
defined
as a change
in
194
J.G. Mulligan
/ Hospital scale economies
In the literature the stochastic models used for the hospital market are normally based on the assumption that both the demand (i.e., arrival) rate and service rate are Poisson random variables. While there is some support for this assumption for demand [e.g., Thompson et al. (1960)], it is unlikely to be the case for the service rate. This assumption would imply that the standard deviation of the length of hospital stays equals the mean (e.g., approximately eight days in the late 1970s and early 1980s). For consistency with the literature, I will maintain the Poisson assumption initially. Later I will relax this assumption and consider the Poisson distribution as a special case. As a starting point, potential arrivals (i.e., prospective patients) are assumed to have only one hospital from which to choose. They refer themselves or are referred by their doctors. The individual’s decision to seek admittance to the hospital (i.e., to become ill) is a random variable whose associated probability distribution is exogenously determined. Price is assumed not to be a factor in the consumer’s decision. The i th hospital faces a population of potential arrivals K,. Each potential arrival is homogeneous in the sense that he or she faces the same probability of requiring hospitalization. Assume that the arrival rate of each potential customer is a Poisson random variable with mean X. If the probability of becoming ill is not correlated across individuals, the arrival rate faced by the i th hospital will be Poisson with mean XK,, which will be referred to as X,. The hospital consists of c,, perfectly substitutable beds. Length of stay will vary randomly. Although the expected length of stay is assumed to be known, there is no knowledge prior to admission how long a particular patient will stay. The length of stay will be assumed to be distributed subject to a negative exponential distribution with mean l/p. p is the mean service rate per bed (i.e., patients that can be treated per day). The service rate is a Poisson random variable, if l/p is distributed subject to a negative exponential distribution. Given these arrival and service distributions, the hospital will face a positive probability that prospective patients will seek admittance when the hospital is full. This forces the hospital to ask them to wait for a bed. * The hospital takes requests on a first-come, first-serve basis regardless of the nature of the illness. 3 As currently specified, this process is a direct application of a M/M/c queuing model. 4 If one knew the expected daily census and the number of beds, one could use this relationship to calculate expected queue length, probability of being asked to wait, expected idle-bed time and marginal cost. Since A,/p is equal to the expected daily census in this model, Xi can be determined once the average length of stay (l/p) is known.
* Mulligan (1984) showed that the hospital will choose to place patients in a queue rather than turn them away. 3 One could consider alternative queue disciplines. Although the effect on capacity utilization will not be altered if a priority system or randomly selected discipline is used, the expected wait of each priority category will be affected. 4 M/M signifies that the arrival and service processes are Poisson. c indicates that there are c servers (i.e., hospital beds). Once one knows X, (the mean arrival rate), ~1 (the mean service rate of each bed) and c, (the number of beds), the steady-state conditions of this model are specified fully. For this model the expected daily census (i.e., expected number in the hospital at any point in time) is equal to h,/p. The number in the queue, N4, is given by
where c, is the number Pa is equal to
of hospital
beds (i.e., servers) and Pa is the probability
-1
c,-1
n~o(l/n!)(h,/P)n +(l/c,!)(h,/~)~‘c,/(c, - Xl/P) [
1
of no patients
in the hospital
or in the queue.
J. G. Mulligan Table 1 Scale effects on queue waits (assuming ADC
occupancy
195
rate)
Bed
Expected queue length
Expected wait (l/p
5 10 20 40 80 100 150
2.217 1.637 1.024 0.485 0.139 0.079 0.020
0.5541 0.2046 0.0640 0.0151 0.0022 0.0010 0.0002
(X,/P) 4 8 16 32 64 80 120
an 80% expected
/ Hospital scale economies
queue = 1)
Expected wait (l/p
queue = 7.7)
4.267 1.576 0.493 0.116 0.017 0.008 0.001
The American Hospital Association reports the average daily census (ADC), average length of stay and number of beds for each separately identifiable unit of a hospital. In this paper the ADC is used as an approximation for the expected daily census (i.e., X,/p in this model). Dividing ADC by the average length of stay (l/p) yields X, for each hospital unit. Once h, is known, one can calculate expected waiting time in the queue. Expected waiting time in the queue equals N,/X, (i.e., expected queue length divided by expected arrival rate). As shown earlier N4 is a function of X,/p and c,. A hospital can reduce expected waiting time either by adding additional hospital beds or by merging two existing hospitals. Since a merger or coordination effort among two existing equal-sized hospitals will consolidate both hospital beds and potential customer arrivals, one can simulate the scale-economy potential of a merger by increasing the number of beds and the mean arrival rate (i.e., population) by the same percentage and by observing the change in expected waiting time and expected idle time of a hospital bed. Alternatively, one could hold expected waiting time constant, increase the arrival rate, and note the number of beds needed to keep the waiting time constant. In this case, a less than proportional increase in beds needed would be the indication of the economies experienced. 5 Local regulatory agencies make decisions concerning hospital expansion based on expected occupancy rates for the hospitals under their jurisdiction. Typically, these agencies will not approve hospital bed expansions if expected occupancy will fall below 80%. In table 1 the expected occupancy rate is held constant at 80% by holding A,/p and c, in the ratio of 4 to 5. Since X,/p is equal to the expected daily census and c, is the capacity of the hospital, (X,/p)/ c, is the expected occupancy rate. As long as h,/p and c, are kept in the ratio of 4 to 5, the expected occupancy rate will always equal 80% regardless of the scale. Since the occupancy rate is held constant, the effect of scale will be a reduction in the expected wait in the queue. As scale increases the expected wait will approach zero. Table 1 shows the queue wait when l/p (i.e., the average length of stay) equals 1.0 and 7.7. The average length of stay varies considerably depending on the patient’s illness. In 1976 it was 7.7 days for short-term hospitals. As an example, if A,/p equals 4 and l/p equals 7.7 days, h, is 0.5195 arrivals per day, and the expected wait would be 4.267 days (i.e., 2.217/0.5195). At XI/p equal to 8 and the number of beds equal to 10, the expected wait drops to 1.576 days with expected idle time of a bed equal to 29.73%. Note that the expected wait is less than a day at a capacity of 20 beds and only 0.008 days at a capacity of 100 beds.
’ Economies from pooling service facilities are known as massed-reserves economies See Mulligan (1983) for a general discussion of the economies of massed reserves.
in the industrial
organization
literature.
196
J. G. Mulligan
/ Hospital scale economies
These waits are expected waits. It is possible that some patients may be asked to wait for much longer times. For example, one could use the model to determine the probability of having to wait longer than n days. If the hospital were required to maintain higher occupancy rates, the hospital would have to be larger to maintain the same expected queue wait. The expected wait will also depend on the expected length of stay. Note in table 1 that the expected wait in the queue is lower when the mean service rate (p) increases. As noted earlier the service rate is likely to be more deterministic than Poisson. The economies identified in the table of this paper are overestimates of what is likely to occur in practice. Since the potential economies identified by this model are nearly non-existent, once a homogeneous hospital unit has 150 beds, it is likely that actual potential economies will be much less. In the extreme and unlikely case that service is deterministic, the corresponding economies are much lower. For example, the queue lengths and expected waiting times shown in table 1 would be twice as much as those for the deterministic service case. As a result, the potential for economies is not as great as in the M/M/c case. In the following section a further relaxation of assumptions results in even less potential for scale economies.
3. Search costs, inter-hospital coordination and hospital-bed homogeneity Until now I have assumed that consumers will wait in the queue until served. This assumption is important because these potential scale economies are a factor, only if prospective patients face significant search costs (e.g., information and transportation costs) in finding an available hospital bed. In the absence of search costs, prospective patients would go to the hospital with the shortest queue. The full benefit of these scale economies could be captured without need for a hospital merger. The main search cost is the restriction that limits the number of hospitals to which a doctor can admit a patient. If doctors had equal access to all hospitals, the search for a hospital bed might involve only a few telephone calls to area hospitals or a call to a central referral service. A patient would be placed in a queue only if all beds were filled at all hospitals in the area. The scale-economies issue becomes more important when the bed-homogeneity assumption is relaxed. Although the majority of beds are for medical-surgical patients, hospitals do maintain beds and related equipment that are not substitutes for one another (e.g., intensive care, pediatric, psychiatric, and obstetric). As Cowing and Holtmann (1983) note, the number of these specialized beds and services may be small even for a large general hospital. In this case the hospital must be considered a collection of separate units. Added to the relatively small number of beds may be a large cost to a wait in the queue for these special cases (e.g., obstetric, pediatric, psychiatric, intensive care) which will lead the hospital to maintain much higher capacity than needed in the absence of increased coordination among hospitals. Thus, while the general medical-surgical component of a hospital may be large enough to capture most of the scale economies because of the substitutability of beds, a consolidation of the smaller units would benefit from scale effects. As noted earlier, most of this benefit could be captured if doctors could refer their patients to any hospital having these specialized services. Specialization will consolidate potential arrivals of a certain type in one facility and will reduce customers’ search costs and waiting time. The benefits from this specialization may be further enhanced if a critical mass of specialized talent and equipment is achieved that improves the service rate per bed.
J.G. Mulligan / Hospital scale economies
197
4. Concluding comments This paper characterized the hospital production unit as a multi-server queuing process. Simulated scale-economies values based on U.S. averages are provided to indicate the potential magnitude for economies due to stochastic demand and service over the relevant range of hospital sizes and occupancy rates. Although several researchers have acknowledged the potential of these economies, no one has made a formal effort to approximate them. This paper provides the formal method. From this discussion it is clear that the potential for these economies for hospital units is limited. This result is fully consistent with the existing empirical literature indicating limited scale economies for hospitals. This paper has focused exclusively on the economies inherent to the stochastic nature of the hospital service function. Although economies and diseconomies may be associated with other functions, such as purchase of supplies and management, the previous literature has focused on the economies associated with the stochastic environment identified here. While this paper does not provide a complete explanation for every possible source of scale economies, it does provide a general theoretically-based account of economies from operating in a stochastic environment.
References Bothwell, James L. and Thomas F. Cooley, 1982, Efficiency in the provision of health care, Southern Economic Journal, April, 911-972. Cowing, Thomas G. and Alphonse G. Holtmann, 1983, Multiproduct short-run hospital cost functions: Empirical evidence and policy implications from cross-section data, Southern Economic Journal 49, no. 3, Jan., 637-653. Coyte, Peter, 1983, The economics of Medicare: Equilibrium within the medical community, Journal of Labor Economics 1, no. 3, July, 254-285. DeVany, Arthur S., 1976, Uncertainty, waiting time and capacity utilization: A stochastic theory of product quality, Journal of Political Economy 84, June, 523-542. Joskow, Paul, 1980, The effects of competition and regulation on hospital bed supply and the reservation quality of the hospital, Bell Journal of Economics, Autumn, 421-447. MacStrevic, R.E.S., 1981, Admissions scheduling and capacity pooling: Minimizing hospital bed requirements, Inquiry 18, no. 4, Winter, 345-350. Mulligan, James G., 1983, The economies of massed reserves, American Economic Review 73, Sept., 725-734. Mulligan, James G., 1984, Hospital scale economies, Paper presented at the Annual Meeting of the European Association for Research in Industrial Economics, Fontainebleau, Aug. Mulligan, James G., 1985, The stochastic determinants of hospital-bed supply, Journal of Health Economics 4, no. 2. Mulligan, James G., 1986, Technical change and scale economies given stochastic demand and supply, International Journal of Industrial Organization 4, no. 2, 189-201. Thompson, John B. et al., 1960, How queuing theory works for the hospital, Modem Hospital, March, 75.
HOSPITAL
193
23 (1987) 193-197
SCALE ECONOMIES
DUE TO STOCHASTIC
DEMAND
AND SERVICE
James G. MULLIGAN University of Delaware, Newark, DE 19716, USA Received
18 September
1986
This paper uses queuing theory to approximate empirically the potential for hospital scale economies due to the stochastic nature of hospital demand and service. These results suggest that scale economies due to stochastic demand and service are likely to be important only for the consolidation of small, specialized hospital units.
1. Introduction The hospital market has two important features: (1) the good produced is a service, and (2) consumers are not affected for the most part by the marginal cost associated with their hospital stay. Anywhere from 85 to 90% of Americans using hospitals have third party insurance policies [Bothwell and Cooley (1982)]. When a firm produces a service and cannot ration its service with price, excess demand must be rationed by other means. The most likely mechanism is a queue. The cost to a potential customer is a wait in the queue. The customer will make a market decision based on how much time he or she expects to wait, the expected quality of the service and the quality and availability of alternatives. This rationing process can be made formal with a queuing model. While several researchers [e.g., Joskow (1980), Coyte (1983), DeVany (1976), and MacStrevic (1981)] have recognized that fluctuations in the demand for hospital services pose a queuing problem, no one has exploited the potential of queuing-theoretic models in a way that could determine the expected economies associated with hospital operations. This paper uses a queuing model to assess the potential gains and demonstrates that scale economies due to stochastic demand and service are likely to be important only for the consolidation of small, specialized hospital units.
2. Economies from pooling hospital beds DeVany (1976) offered one of the strongest statements that stochastic demand and service may lead to economies for large scale operations. DeVany used a queuing model to describe a natural-monopoly market for a non-storable good or service and related this model to the issue of the economies of large hospitals. DeVany suggested that his model implies that large hospitals would operate at significantly higher occupancy rates than smaller hospitals, leading to substantial scale economies. Despite these claims, DeVany interpreted his model incorrectly. 1 ’ Mulligan (1986) provides a detailed criticism of the DeVany model and shows that what DeVany scale would be a change in technology for the stochastic model used in his paper.
01651765/87/$3.50
0 1987, Elsevier Science Publishers
B.V. (North-Holland)
defined
as a change
in
194
J.G. Mulligan
/ Hospital scale economies
In the literature the stochastic models used for the hospital market are normally based on the assumption that both the demand (i.e., arrival) rate and service rate are Poisson random variables. While there is some support for this assumption for demand [e.g., Thompson et al. (1960)], it is unlikely to be the case for the service rate. This assumption would imply that the standard deviation of the length of hospital stays equals the mean (e.g., approximately eight days in the late 1970s and early 1980s). For consistency with the literature, I will maintain the Poisson assumption initially. Later I will relax this assumption and consider the Poisson distribution as a special case. As a starting point, potential arrivals (i.e., prospective patients) are assumed to have only one hospital from which to choose. They refer themselves or are referred by their doctors. The individual’s decision to seek admittance to the hospital (i.e., to become ill) is a random variable whose associated probability distribution is exogenously determined. Price is assumed not to be a factor in the consumer’s decision. The i th hospital faces a population of potential arrivals K,. Each potential arrival is homogeneous in the sense that he or she faces the same probability of requiring hospitalization. Assume that the arrival rate of each potential customer is a Poisson random variable with mean X. If the probability of becoming ill is not correlated across individuals, the arrival rate faced by the i th hospital will be Poisson with mean XK,, which will be referred to as X,. The hospital consists of c,, perfectly substitutable beds. Length of stay will vary randomly. Although the expected length of stay is assumed to be known, there is no knowledge prior to admission how long a particular patient will stay. The length of stay will be assumed to be distributed subject to a negative exponential distribution with mean l/p. p is the mean service rate per bed (i.e., patients that can be treated per day). The service rate is a Poisson random variable, if l/p is distributed subject to a negative exponential distribution. Given these arrival and service distributions, the hospital will face a positive probability that prospective patients will seek admittance when the hospital is full. This forces the hospital to ask them to wait for a bed. * The hospital takes requests on a first-come, first-serve basis regardless of the nature of the illness. 3 As currently specified, this process is a direct application of a M/M/c queuing model. 4 If one knew the expected daily census and the number of beds, one could use this relationship to calculate expected queue length, probability of being asked to wait, expected idle-bed time and marginal cost. Since A,/p is equal to the expected daily census in this model, Xi can be determined once the average length of stay (l/p) is known.
* Mulligan (1984) showed that the hospital will choose to place patients in a queue rather than turn them away. 3 One could consider alternative queue disciplines. Although the effect on capacity utilization will not be altered if a priority system or randomly selected discipline is used, the expected wait of each priority category will be affected. 4 M/M signifies that the arrival and service processes are Poisson. c indicates that there are c servers (i.e., hospital beds). Once one knows X, (the mean arrival rate), ~1 (the mean service rate of each bed) and c, (the number of beds), the steady-state conditions of this model are specified fully. For this model the expected daily census (i.e., expected number in the hospital at any point in time) is equal to h,/p. The number in the queue, N4, is given by
where c, is the number Pa is equal to
of hospital
beds (i.e., servers) and Pa is the probability
-1
c,-1
n~o(l/n!)(h,/P)n +(l/c,!)(h,/~)~‘c,/(c, - Xl/P) [
1
of no patients
in the hospital
or in the queue.
J. G. Mulligan Table 1 Scale effects on queue waits (assuming ADC
occupancy
195
rate)
Bed
Expected queue length
Expected wait (l/p
5 10 20 40 80 100 150
2.217 1.637 1.024 0.485 0.139 0.079 0.020
0.5541 0.2046 0.0640 0.0151 0.0022 0.0010 0.0002
(X,/P) 4 8 16 32 64 80 120
an 80% expected
/ Hospital scale economies
queue = 1)
Expected wait (l/p
queue = 7.7)
4.267 1.576 0.493 0.116 0.017 0.008 0.001
The American Hospital Association reports the average daily census (ADC), average length of stay and number of beds for each separately identifiable unit of a hospital. In this paper the ADC is used as an approximation for the expected daily census (i.e., X,/p in this model). Dividing ADC by the average length of stay (l/p) yields X, for each hospital unit. Once h, is known, one can calculate expected waiting time in the queue. Expected waiting time in the queue equals N,/X, (i.e., expected queue length divided by expected arrival rate). As shown earlier N4 is a function of X,/p and c,. A hospital can reduce expected waiting time either by adding additional hospital beds or by merging two existing hospitals. Since a merger or coordination effort among two existing equal-sized hospitals will consolidate both hospital beds and potential customer arrivals, one can simulate the scale-economy potential of a merger by increasing the number of beds and the mean arrival rate (i.e., population) by the same percentage and by observing the change in expected waiting time and expected idle time of a hospital bed. Alternatively, one could hold expected waiting time constant, increase the arrival rate, and note the number of beds needed to keep the waiting time constant. In this case, a less than proportional increase in beds needed would be the indication of the economies experienced. 5 Local regulatory agencies make decisions concerning hospital expansion based on expected occupancy rates for the hospitals under their jurisdiction. Typically, these agencies will not approve hospital bed expansions if expected occupancy will fall below 80%. In table 1 the expected occupancy rate is held constant at 80% by holding A,/p and c, in the ratio of 4 to 5. Since X,/p is equal to the expected daily census and c, is the capacity of the hospital, (X,/p)/ c, is the expected occupancy rate. As long as h,/p and c, are kept in the ratio of 4 to 5, the expected occupancy rate will always equal 80% regardless of the scale. Since the occupancy rate is held constant, the effect of scale will be a reduction in the expected wait in the queue. As scale increases the expected wait will approach zero. Table 1 shows the queue wait when l/p (i.e., the average length of stay) equals 1.0 and 7.7. The average length of stay varies considerably depending on the patient’s illness. In 1976 it was 7.7 days for short-term hospitals. As an example, if A,/p equals 4 and l/p equals 7.7 days, h, is 0.5195 arrivals per day, and the expected wait would be 4.267 days (i.e., 2.217/0.5195). At XI/p equal to 8 and the number of beds equal to 10, the expected wait drops to 1.576 days with expected idle time of a bed equal to 29.73%. Note that the expected wait is less than a day at a capacity of 20 beds and only 0.008 days at a capacity of 100 beds.
’ Economies from pooling service facilities are known as massed-reserves economies See Mulligan (1983) for a general discussion of the economies of massed reserves.
in the industrial
organization
literature.
196
J. G. Mulligan
/ Hospital scale economies
These waits are expected waits. It is possible that some patients may be asked to wait for much longer times. For example, one could use the model to determine the probability of having to wait longer than n days. If the hospital were required to maintain higher occupancy rates, the hospital would have to be larger to maintain the same expected queue wait. The expected wait will also depend on the expected length of stay. Note in table 1 that the expected wait in the queue is lower when the mean service rate (p) increases. As noted earlier the service rate is likely to be more deterministic than Poisson. The economies identified in the table of this paper are overestimates of what is likely to occur in practice. Since the potential economies identified by this model are nearly non-existent, once a homogeneous hospital unit has 150 beds, it is likely that actual potential economies will be much less. In the extreme and unlikely case that service is deterministic, the corresponding economies are much lower. For example, the queue lengths and expected waiting times shown in table 1 would be twice as much as those for the deterministic service case. As a result, the potential for economies is not as great as in the M/M/c case. In the following section a further relaxation of assumptions results in even less potential for scale economies.
3. Search costs, inter-hospital coordination and hospital-bed homogeneity Until now I have assumed that consumers will wait in the queue until served. This assumption is important because these potential scale economies are a factor, only if prospective patients face significant search costs (e.g., information and transportation costs) in finding an available hospital bed. In the absence of search costs, prospective patients would go to the hospital with the shortest queue. The full benefit of these scale economies could be captured without need for a hospital merger. The main search cost is the restriction that limits the number of hospitals to which a doctor can admit a patient. If doctors had equal access to all hospitals, the search for a hospital bed might involve only a few telephone calls to area hospitals or a call to a central referral service. A patient would be placed in a queue only if all beds were filled at all hospitals in the area. The scale-economies issue becomes more important when the bed-homogeneity assumption is relaxed. Although the majority of beds are for medical-surgical patients, hospitals do maintain beds and related equipment that are not substitutes for one another (e.g., intensive care, pediatric, psychiatric, and obstetric). As Cowing and Holtmann (1983) note, the number of these specialized beds and services may be small even for a large general hospital. In this case the hospital must be considered a collection of separate units. Added to the relatively small number of beds may be a large cost to a wait in the queue for these special cases (e.g., obstetric, pediatric, psychiatric, intensive care) which will lead the hospital to maintain much higher capacity than needed in the absence of increased coordination among hospitals. Thus, while the general medical-surgical component of a hospital may be large enough to capture most of the scale economies because of the substitutability of beds, a consolidation of the smaller units would benefit from scale effects. As noted earlier, most of this benefit could be captured if doctors could refer their patients to any hospital having these specialized services. Specialization will consolidate potential arrivals of a certain type in one facility and will reduce customers’ search costs and waiting time. The benefits from this specialization may be further enhanced if a critical mass of specialized talent and equipment is achieved that improves the service rate per bed.
J.G. Mulligan / Hospital scale economies
197
4. Concluding comments This paper characterized the hospital production unit as a multi-server queuing process. Simulated scale-economies values based on U.S. averages are provided to indicate the potential magnitude for economies due to stochastic demand and service over the relevant range of hospital sizes and occupancy rates. Although several researchers have acknowledged the potential of these economies, no one has made a formal effort to approximate them. This paper provides the formal method. From this discussion it is clear that the potential for these economies for hospital units is limited. This result is fully consistent with the existing empirical literature indicating limited scale economies for hospitals. This paper has focused exclusively on the economies inherent to the stochastic nature of the hospital service function. Although economies and diseconomies may be associated with other functions, such as purchase of supplies and management, the previous literature has focused on the economies associated with the stochastic environment identified here. While this paper does not provide a complete explanation for every possible source of scale economies, it does provide a general theoretically-based account of economies from operating in a stochastic environment.
References Bothwell, James L. and Thomas F. Cooley, 1982, Efficiency in the provision of health care, Southern Economic Journal, April, 911-972. Cowing, Thomas G. and Alphonse G. Holtmann, 1983, Multiproduct short-run hospital cost functions: Empirical evidence and policy implications from cross-section data, Southern Economic Journal 49, no. 3, Jan., 637-653. Coyte, Peter, 1983, The economics of Medicare: Equilibrium within the medical community, Journal of Labor Economics 1, no. 3, July, 254-285. DeVany, Arthur S., 1976, Uncertainty, waiting time and capacity utilization: A stochastic theory of product quality, Journal of Political Economy 84, June, 523-542. Joskow, Paul, 1980, The effects of competition and regulation on hospital bed supply and the reservation quality of the hospital, Bell Journal of Economics, Autumn, 421-447. MacStrevic, R.E.S., 1981, Admissions scheduling and capacity pooling: Minimizing hospital bed requirements, Inquiry 18, no. 4, Winter, 345-350. Mulligan, James G., 1983, The economies of massed reserves, American Economic Review 73, Sept., 725-734. Mulligan, James G., 1984, Hospital scale economies, Paper presented at the Annual Meeting of the European Association for Research in Industrial Economics, Fontainebleau, Aug. Mulligan, James G., 1985, The stochastic determinants of hospital-bed supply, Journal of Health Economics 4, no. 2. Mulligan, James G., 1986, Technical change and scale economies given stochastic demand and supply, International Journal of Industrial Organization 4, no. 2, 189-201. Thompson, John B. et al., 1960, How queuing theory works for the hospital, Modem Hospital, March, 75.