Eddleman, Bobby R.

English

Research Papers in Economics

SOUTHERN  JOURNAL  OF AGRICULTURAL  ECONOMICS  JULY,  1972
AREAWIDE  PLANNING FOR OPTIMUM LOCATION  OF
HOSPITAL FACILITIES FOR RURAL PEOPLE
B. R. Eddleman
The  agricultural economic  literature  is  abundant  In  addition  to  health  service  production  costs,
with  least-cost  location  studies  of  agricultural  recipients  must  bear  any travel  costs associated  with
processing  industries.  Yet  very  little  effort has  been  obtaining  the  health  service.  Average  travel  cost per
directed  toward  similar  planning  for  public facilities  patient  depends  upon  several  factors  such  as  the
in rural  areas. The  purpose  of this paper is to present  existing  geographic  distribution  of health  facilities,
a  conceptual  model  and  its  application  to  the  the distribution  and characteristics  of the population,
planning  of  locations  for  additional multiple  service  and  the  configuration  of  travel  routes.  In  general,
hospital  facilities  in  a  predominantly  rural  area  for  however,  if the sizes of hospitals were increased (with
1980.  The  approach  is  based  on  the  premise  that  constant  utilization  rates),  their  geographic  service
areawide  planning, as opposed  to substantial reliance  areas would expand,  and, consequently,  average travel
upon  market  mechanisms,  is  required  to  allocate  cost would rise.
health  care  resources  among  communities.  In  this  These  relationships  are  presented  graphically  in
approach  it is assumed that all health care  service cost  Figure  1.  Line  AA  represents  the  relationship
functions  are  long  run  and  that the  time  period  of  between  average  travel cost and hospital size,  and line
adjustment  is  such  that  fixed  capital  goods  do  not  BB  represents  average  cost  per  patient  day incurred
restrict  adjustments in  the service  output  of hospital  by the hospital as the  size of the institution increases.
facilities.  Since  these  two costs  must be borne by recipients, it
is the sum of these  costs that is relevant. If these costs
THE MINIMUM  TOTAL ACCESS COST MODEL  do  not  depend  upon  one  another  in  any  way,  it  is
permissable  to add  these  curves  vertically  in Figure  1
The  areawide  planning  problem  may be  divided e areawde  panning  probem maye d  ed  to  obtain  line  CC  which  represents  average  total
into  two  parts:  (1)  the  demand  for  health  care access  cost as a function of hospital size.
services  in a given area,  and (2) the distribution of the
facilities  and  personnel  that  will  most  efficiently  Williamson  [11]  has  shown  that  if we  assume
provide the care to be utilized  [1].  uniformity  of terrain,  uniformity  and  continuity  in
Given  knowledge  of  the  level  of  demand  or  transportation  facilities,  and  constant  over-the-road
health  care  needs  in  a  region,  the problem  is one  of  costs per  patient  mile  of travel,  the  size of hospital
determining  the  number,  sizes  and  locations  of  facility that  results in minimum total access cost will
various  types  of  multiple  service  hospital  facilities  increase  or  decrease  as  patient  population  density
that will minimize the combined cost of patient travel  increases  or  decreases.  Thus,  there  is  a  positive
and  health  service  output,  i.e.,  total  regional  access  relationship between  density of population  in an area
cost.  If  we  assume  there  are  both  economies  and  and  the  optimum  size  of  hospital  facility.  This
diseconomies  to  size  in  the  production  of  health  relationship  can  be  taken into  account  in estimating
services  in hospitals,  it might be generalized  that over  patient travel cost functions for an area.
the  usual  range  of operation,  average  health  service  From  the foregoing  discussion,  it is  evident that
production  cost  first  decreases  at  a  decreasing  rate  both the  cost of producing  services  in a  hospital and
and then increases as output is increased.1 the  cost  of  patient  travel  affect  the  optimum  size
B. R. Eddleman  is associate professor of the Department of Food and Resource Economics  at the University  of Florida.
Empirical evidence exists from numerous studies to indicate that unit output costs initially decline and then rise as the
size of hospitals  is increased. For examples, see  [1,  2, 3].
89Cost per
patient
day
C
|  ~~~~~~~~~~~A
Hospital Size
FIGURE  1.  PRODUCTION  AND  TRAVEL  COST
PER  PATIENT  DAY  IN  RELATION
TO HOSPITAL SIZE.
(and,  thus,  number  and  location)  of  hospitals  and  travel cost  function for  each type  of health service  is
ought  to be  taken into  account in areawide planning.  developed  from  a  matrix  of  total  travel  costs  per
As  the  number  of  hospital  facilities  in  an  area  is  patient  from  each  patient  supply  area  to  each
increased,  total  travel  cost  for  the  area  would  be  potential  facility  location and  a vector of the number
expected  to  decrease  as  a  result  of  smaller  average  of patients  in  each  supply  area  requiring  the health
travel  requirements  per  patient.  Additionally,  total  service.
cost  of  producing  health  services  within  hospitals
might  initially decrease  for  the area as the number of  The  second  step  involves  the  addition  of  the
facilities  is  increased  due  to  diseconomies  of  size  minimized  travel  cost  functions  for  each  type  of
associated  with  excessively  large  facilities.  However,  health  service  to obtain  a  minimized total combined
as  the number of facilities  is  continuously expanded,  patient  travel cost  function  for the various locational
a  number  would  be  reached  beyond  which  total  patterns  and  number  of  hospital  facilities.
health  service  production  cost  for the area would rise  Computational  procedures  required  to  accomplish
as  a result  of both the  fixed  investment  requirement  these  two  steps  are  discussed  in  detail  in  previous
of  hospitals  and  higher  variable  cost  of producing  studies  [7,  8].  From  the  above  computations,  the
health services in  smaller hospital  facilities.  In Figure  optimum allocation of patients requiring each type of
2,  line  AA  represents  the  relationship between  total  health  service  is  obtained  for  the  various  locational
patient  travel  cost  for  an  area  and  the  number  of  patterns and number of locations.
hospital facilities,  line BB represents total production
cost  of  health  services  for  the  area  expressed  as  a  The  third  step  involves  the  calculation  of  the
function  of the number  of hospital facilities, and line  total  cost  of producing  the health  services  (for  each
CC  indicates  their  sum  (i.e.,  total  regional  access  number,  locational  pattern,  and  the  various  sizes of
cost).  hospital  facilities,  indicated  by  the  optimum
The  problem  of minimizing  total regional  access  allocation  of  patients  from  supply  areas  to  facility
cost  with respect  to the number  of hospital  facilities  locations  by  the  above  two  computational  steps)
and  locational  pattern  can  be  accomplished  in  four  from  a  size  economies  equation  estimated  for
steps.  The  first  step is  to obtain  a total patient travel  multiple service hospitals.
cost  function  for each  type of health service  that has  The  final  step  involves  addition  of  minimized
been minimized  with respect to facility locations with  total combined  travel  cost  and  areawide  total cost of
varying  numbers  of hospital facilities.  The minimum  producing  the health  services  to obtain total regional
90Total  C
area  B
cost
/B
L  A
Number of  facilities
FIGURE  2.  TOTAL PRODUCTION  AND TRAVEL
COST IN RELATION  TO NUMBER OF
HOSPITALS.
access  cost as a function of locational pattern and the  [10]  final counts  and  the Kiplinger  estimates of each
number  of hospital  facilities.  The  areawide  optimum  county's  population  in  1970. The  proportion of the
solution  is  determined  where  total  regional  access  projected  population  that  would  require  health
cost is minimum.  services  of various  types  was  then  determined  from
EMPIRICAL  USE  OF THE MODEL  information  obtained  from  a  1967  survey  of health EMPIRICAL USE OF THE MODEL
care  practices  and  needs for  counties in  Florida  [5].
A  study  was  carried  out  for a  nine county  area  The  services  considered  for  this  study  included
surrounding  the  City  of  Gainesville,  Florida.  Each  inpatient  treatment  (confinement,  diagnoses  and
county  in  the study  area was  designated  as a  patient  treatment)  and  general  outpatient  treatment.  For
supply  area  and  four cities with a  1970 population of  inpatient  treatment,  average  length of stay in hospital
9,000  or  more were considered  as potential sites for a  facilities in the study area was 8.9 days in 1970 which
new  multiple  service hospital  facility  or expansion of  is  slightly  higher  than  the  8.2  days  of stay  for  the
existing hospital capacity.  State  [4].  An  estimate  of  10  days of stay was used
Data  for  the  1980  projections  and  reflects  the gradually
Population and Patient Supply.  Estimates of the  increasing trend  in  length of stay  since  Medicare  and
1980  population  for  each  county  were  made  by  Medicaid  programs  were  initiated.  Estimated
adjusting  the  1970 Kiplinger  [9]  estimates  for  1980  additional  quantities  of each  health  service  required
by  the differences  in  the  1970 Census of Population  by residents in each county are shown in Table  1.
2The  study area  is a subregion of a 19 county area in Northeast  Florida for which a comprehensive  analysis  is currently
underway  by  Louis  Murray,  Graduate  Research  Assistant.  In  this  latter  study  15  potential  sites  for various  types  of multiple
service hospitals are being considered.
91Travel  Distances and Costs.  Patient  travel  was  Pki  =  number  of  persons  residing  within
considered  to  be  by  private  automobile  over  hard  Enumeration District  k of county i in  1970,
surface  roads  from  patient  supply areas to potential  Mkij=  one-way  hard  surface  road  mileage  from  an
facility  locations.  The hard  surface road mileage  from  approximate  midpoint  location  in
an  approximate  midpoint  location  in  each  census  Enumeration  District  k  of  county  i  to
defined  enumeration  district  in each  county to each  facility j located at Lj.
potential  facility  location  was  estimated  from  road  These  computations  resulted  in  a  matrix  of average
maps.  Average  length of travel for persons residing  in  travel  distances per  patient  from each countyto each
each county was computed  as follows:  potential  facility  location  that  was weighted  by the
K  K  1970 population  density of the enumeration districts
(1)  Mij = 2  3  PkiMkij/  Pki  comprising the county.3
k=l  k=1
where  Travel  by  private  automobile  was considered  to
Mij  =  average  two-way  hard  surface  road  mileage  cost  20 cents  per mile.  Thus, the  total cost of travel
from  patient  supply  county  i  to  hospital  per  patient  from  each  county  to  each  potential
facility j located at  Lj,  facility location was estimated as follows:
Table  1. --  ESTIMATED  ADDITIONAL  QUANTITIES  OF HEALTH SERVICES REQUIRED IN
THE STUDY AREA BY 1980
Patient supply  Additional amounts of health service needed by 1980
areas  Inpatient hospital treatment  Outpatient
(counties)  Patient  days  Patient trips  visits
Alachua  52,500  5,250  26,000
Bradford  7,500  750  3,600
Clay  16,000  1,600  8,000
Columbia  13,000  1,300  6,300
Gilchrist  1,800  180  900
Levy  6,500  650  3,000
Marion  34,500  3,450  17,000
Putnam  18,000  1,800  9,000
Union  4,000  400  2,000
Totals  153,800  15,380  75,800
31  am  indebted  to  Louis  Murray  for  the computation  of the average  length  of  travel distances from  each  county  of
patient  origin to each potential  facility location.
92Table  2.  --  TOTAL  COST  OF TRAVEL PER PATIENT  FROM  PATIENT  SUPPLY  AREAS  TO
POTENTIAL HOSPITAL FACILITY  LOCATIONS
Patient supply  Potential facility locations (cities)
areas
(counties)  Gainesville  Lake City  Ocala  Palatka
Alachua  5.96  17.88  15.28  18.12
Bradford  10.92  12.56  23.40  16.44
Clay  21.12  25.76  25.20  17.20
Columbia  18.80  1.96  31.60  31.00
Gilchrist  13.12  16.28  24.00  31.08
Levy  14.28  25.36  16.40  30.08
Marion  15.92  31.64  5.08  21.96
Putnam  18.68  32.04  22.00  4.72
Union  14.36  10.04  27.20  22.24
(2)  TTCmij  = .20 Mij  Results
where  TTCmij  is the  total cost  per patient  for  travel
from  county  i to  facility  j  located  at  Lj  to  obtain  Total  regional  access  costs  were  estimated  for
health  service  m.  Total  costs  of  travel  per  patient  each  of the possible  locational patterns with varying
from  supply  areas  to  potential  facility  locations  are  numbers  of  hospital  facilities.  Cost  components  for
given in Table 2.  the  optimum  solutions  are  summarized  in Table  3.
Total  health  service  production  cost  relates  to  The  additional  quantities  of  health  services  can  be
the  combined  cost  of  producing  both  types  of  provided with minimum total access costs to residents
services  in  hospitals  of  various  sizes.  For this paper,  in the region by expansion of hospital capacity in two
cost  of  producing  health  services  in  hospitals,  as  cities,  Gainesville and Ocala. With one facility the size
estimated  by Carr  and  Feldstein  [1], are  adjusted  to  required  would be  an  approximate  450 bedhospital.
correspond  with  1970 hospital daily service charges.' correspond  with  1970  hospital daily service charges.4  Sizable  external  costs  of  patient  travel  as  well  as
The  resulting  relationship  of  adjusted  costs  to  diseconomies  to  size  in the production  of the health
hospital size is as follows:  services  are  encountered  when  only  one  facility  is
(3)  TPC  =  325,062 + 57.59  (PD.)  considered.  As  the  number  of facilities  is  increased (3)  TPCj  325,062 +57.59  (PDj) beyond  two,  the  increased  cost  of  health  service
+ .0000667 (PDj)2 + 9.14 (OPV-)  production  more  than  offsets  the  cost  savings
associated  with  shorter  patient  travel  distances.  The
where  estimated  cost  savings  obtained  from  the  areawide
TPCj  =long-run  total  production  costs  for  optimum  location  for  two  facilities  (Gainesville  and
facility j at location  Lj,  Ocala)  is about  $75,000  annually when  compared  to
PDJ  =size  of hospital  facility  j  located  at  Lj  the optimum  location for  three facilities (Gainesville,
measured in patient  days,  Ocala  and  Palatka),  and  about  $588,000  annually
OPVj  =number  of  outpatient  visits  at facility j  when  compared  to  the  optimum  location  for  one
located at Lj.  facility (Gainesville).
4The function estimated by Carr and Feldstein was:
TC = 171,085 + 30.31 (PD) + .0000351  (PD)2 + 4.81 (OPV)
The data were for 1963.  Estimates of TC were adjusted upward by multiplying each of the terms in the equation by  1.9, the 1970
index  of hospital  daily  charges for the  South with  the years  1963-64  as the base.  In  the study by Murray  a cost function using
data from 67 hospitals  in North Florida and South Georgia is being developed  for use in the 19 county study area.
93TABLE3.--  OPTIMUM  HOSPITAL  FACILITY  LOCATIONS  AND  MINIMIZED  TOTAL
REGIONAL  ACCESS COST FOR VARIOUS  NUMBERS  OF FACILITIES
Number  Optimum  Minimized patient  travel costs  Joint  Minimized
of  loca-  pro-  total
facil-  tional  Inpatient  Outpatient  Combined  duction  regional
ities  pattern  treatment  visits  travel  costs  access costs
cost
1  Gainesville  203,648  1,003,800  1,207,448  11,270,167  12,477,615
2  Gainesville,  166,250  819,520  985,770  10,903,913  11,889,683
Ocala
3  Gainesville,  134,850  662,520  797,370  11,167,150  11,964,520
Ocala,
Palatka
4  Gainsville,  111,230  547,788  659,018  11,337,321  11,996,339
Lake City,
Ocala, Palatka
It should  be emphasized  that  all residents in the  possibilities  for aiding in areawide  planning of present
area  would  not  pay  the  same  for  inpatient  services  and  future  developments  in  the  health  services
due  to economies  and  diseconomies of size in service  industry.
production.  For  the  optimum  locations  shown  in  Some  major  problems  occur  in  the  empirical
Table  3, costs per patient  day range  from $67.00 for  application  of the model. One problem is in acquiring
an  approximate  200  bed  facility  to  $78.00  for  an  accurate  information  on the  health  care  needs  of a
approximate 50 bed facility.  population.  Detailed  information  on  the  effects  of
the distribution  and characteristics  of the population
CONCLUDING REMARKS  CONCLU  G  RS  (including  race,  sex,  age,  income,  residence,  etc.) on
A  model  permitting  minimization  of combined  health  practices and the configuration of travel routes
patient  travel  costs  and  health  service  production  is  required.  A  second  problem  is  acquiring
costs  with  economies  to  size  affecting  the  level  of  information  on  health  service  production  costs  for
production costs  was presented.  This model was used  various service level facilities. However, the possibility
to estimate the optimum number,  size and location of  of making  more productive  use' of health facility cost
hospital  facilities  and  the  total  regional  access  costs  functions, patient travel cost functions and other data
for  projected  additional health  service  requirements  should  encourage  an increased  effort to acquire more
in  a  nine  county  area.  The  model  appears  to  have  accurate information.
94REFERENCES
[1]  Carr, W.  John, and  Paul J.  Feldstein,  "The  Relationship  of Cost to Hospital Size," Inquiry 4: 45-65, June,
1967,
[  2]  Cohen,Harold A.,  "Variations  in Cost  Among Hospitals  of Different  Sizes,"  Southern Economic Journal,
33:  35 5-366, Jan.  1967.
[3]  Feldstein,  Paul  J.,  "An  Empirical  Investigation  of  the  Marginal  Cost  of Hospital  Services,"  Chicago:
Graduate  Program in Hospital Administration, pp.  1-57,  1961.
[  4]  Florida Department of Health and Rehabilitative Services, Bureau of Community Medical  Facilities, Florida
State Plan For Construction of Hospitals. Tallahassee,  1970.
[  5]  Florida Hospital  Association, Statewide Patient  Origin Study. Orlando,  1968.
[  6]  Polopolus, Leo, "Optimum  Plant Numbers and Locationsfor  Multiple Product Processing,"  Journal  of Farm
Economics, 47: 287-295,  May  1965.
[  7]  ,  "An  Analytical  and  Operational  Framework  for  Solving  Problems  of  Plant  Location,"  in
Contemporary Agricultural Marketing (Irving  Dubov,  ed.),  pp.  157-183,  The  University  of Tennessee
Press,  1968.
[  8]  Stollsteimer,  John  F.,  "A Working  Model  for Plant  Numbers and  Locations,"  Journal  ofFarm Economics
45: 631-645, Aug.  1963.
[  9]  The Kiplinger  Washington Editors, Inc.,  "1970 Kiplinger Forecast of  Florida's Growth During the Next Ten
Years - By  Localities," The Kiplinger  Florida  Letter. Washington,  1970. 
[10]  U.  S.  Department  of  Commerce,  Bureau  of  the  Census,,  1970 Census of Population, Final Population
Counts for Florida. PC (V1)-l 1, Dec. 1970.
[11]  Williamson,  J.  C., Jr.,  "The  Equilibrium  Size  of Marketing  Plants  in a Spatial Market,"  Journal of Farm
Economics, 44: 953-967, Nov.  1962.
95

AREAWIDE PLANNING FOR OPTIMUM LOCATION OF HOSPITAL FACILITIES FOR RURAL PEOPLE

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