Cleveland, O.A.

Salkin, Michael S.

English

Research Papers in Economics

SOUTHERN  JOURNAL  OF  AGRICULTURAL  ECONOMICS  JULY,  1975
PROJECTION OF COTTON  WAREHOUSE  FACILITIES  FOR
OKLAHOMA:  A  MARKOV  PROCESS
0.  A. Cleveland  and Michael S. Salkin
The  cotton  industry's  decline  in  competitive  movement  of  firms  between  states  over  time  be
strength  is  a  matter  of  increasing  concern  to  in-  regarded as a stochastic  process.l  Once states have
dustry  members  and  public  policymakers.  There  been defined,  it is possible to estimate probabilities
are  many  complex  economic  adjustments  arising  (Pij)  of firms  moving  from  any  state  i  (Si)  to  any
from changes  in government policies,  technological  state  j  (Sj).  These  are  expressed  as  a  transition
advancement,  and  changing  market  conditions  at  matrix,  P, where:
home  and  abroad.  The  impact  of  these  factors,
coupled  with  increased  competition  of  synthetic  n
fibers  and  foreign-grown  cotton,  has  been  felt  by  [1]  X  Pij - 1, for all i,
j-1 all  segments  of  the  industry.  Industry  experience  [2  0  P  1 for all i and j.
and  recent  research  results  [1,  3]  indicate  that
serious overcapacity has  developed in warehousing  Each Pij represents  that fraction of firms start-
facilities  and,  as  a  consequence,  producers  and  ing in period t at position  Si and moving to Sj  the
consumers  have  suffered  through  increased  per-  following  period,  t+  1. The  movement  interval
bale  storage and  assembly costs. bale  storage  and  assembly  costs.  in the study,  t to t +  1, is five years.
In view  of  these factors,  this paper  addresses
itself  to projecting  Oklahoma  warehouse  viability  If  all  states  are  accessible,  indicating  a  non-
and  determining  sizes  (bale  capacity)  of  active  zero probability  of  moving from  state  i to  state  j
firms.  Firms surviving  in the long run are assumed  in a finite  number of time periods,  then P defines
to  take  advantage  of  economies  of  scale.  In  sup-  a regular Markov  chain.  If  any Pii element  in the
port  of  the  projections,  cost  of  storage  and  as-  transition  matrix  takes  on  a  value  1.0,  a  special
sembly  data  are  employed  to construct  an  econo-  case  of  the  Markov  chain  exists.  This  is  known
mic planning curve.  It is hypothesized that the bale  as  an  "absorbing  chain."  A Markov  chain  is  ab-
capacity  at  the  minimum  point  on  an  envelope  sorbing if it has  at least one absorbing  state  which
curve  will  represent  capacity  level  of  warehouses  can be entered from all non-absorbing  states. Once
predicted  to survive.  a  firm  goes  out  of  business  it  is  mathematically
absorbed,  since the "out of business"  size category
MARKOV  PROCESS  is an  absorbing  state.
Markov  chains  are  employed  as  a  predictive  When  a  Markov  chain  is  absorbing,  some
tool,  assuming  firm  growth  (changing  bale  capa-  measures  characterizing  the  system  can be  calcu-
city)  is a first-order  Markov  process.  The Markov  lated.  These  aie:  (1)  mean  length  of time  in each
tool provides long-run projections  of firms  operat-  transient  state,  (2)  probability  of  absorption  and
ing in various  size  categories.  (3)  length  of  time to  absorption  [2].  A more  ex-
This  process  requires  that  any  population  of  haustive  discussion  of the  Markov  technique  can
firms  be  classified  into  n  different  states,  and  that  be found  in Kemeny  and Snell  [5].
Agricultural  Economist,  Economic  Research  Service,  USDA  and  Assistant  Professor,  Department  of  Agricultural  Economics,
Oklahoma  State  University.
1 In  this  study  the  states  are  the  various  bale  capacities  of warehouses.
179EMPIRICAL  RESULTS  So  S1  S2 53  S4 S5
So  1  0  0  0  0  0
Micro-data  on movement  (changes  in bale  ca-  S  .4  .6  0  0  0  0
pacity)  of all firms under  consideration  are essen-  P = S2 .2  0  .8  0  0  0
tial  for  calculating  transition  probabilities.  Okla-  S  .2  0  .8  0  0
homa  warehouse  movement  from  1964  to  1973  S4 .333  0  0  0  .333  .333
was  constructed  using  data  abtained  from  the  S5 0  0  0  0  1 
Fibers  Group,  Economic  Research  Service,  U.S.
Department  of  Agriculture  [4].  A  flow  chart  of  Similar calculations,  using the 1969-73  base,  were
firm movement between bale capacities from  1964  also made  and used in later  analysis.
to  1973  was  constructed.  To allow sufficient  time  Elements  of  the  transition  probability  matrix
for  firms  to  change  size,  two  five-year  intervals,  provide  meaningful  information  not  easily  ob-
1964-1968  and 1969-1973,  were selected.  tained from other types of projections. The princi-
pal  diagonal  elements  of  the transition  matrix  are
The Markov  states (S  . S)  are:  generally  large,  indicating  industry  stability.  The
WAREHOUSE  first  element  row  So  indicates  that  a firm  out  of
STATE  CAPACITY  (BALES)  business  in  1964  stays  out  of  business  in  1968.
So  0  (out of business)  The S55 element indicates a firm in state S5 in 1964
S1  1-15,000  will  remain  in  S5 through  1968;  thus,  So  and  S5
S2 15,001-25,000  are  absorbing  states.  Row Si  proportions  indicate
S 3 25,001-40,000  forty percent of warehouses  having  1-15,000  bale
S4 40,001-75,000  capacity  will  cease  operation  during  the  1964-
S5 > 75,000  1968  period.  Sixty percent will remain.
Given  the  number  of  firms  in  1964  in  six
Given this  information  and warehouse  flows,  the  categorie  s  the  number  of  firms  in  in
size  categories,  So,  .... , S5;  the  number  of  firms
following transition  matrix was constructed  for the  in these  categories in  1968  may  be  obtained.  The
~~period  1964 to  1968.  ^new  industry configuration-by-size  category repre-
Table  1.  PROJECTED AND  ACTUAL  NUMBER  OF OKLAHOMA  COTTON  WAREHOUSES,
1973
Years
State  Size 
1973  Actual  1973  Projected
S0  9  10.24
S  1  1.08
S2 4  2.56
S3 3  2.56
S4 0  .11
S5 1  1.45
Total  18  18.00
2 The transition  matrix  is available  upon  request.
180sents  total  warehouses  in  each  category  in  1964  To analyze  the  absorption  process,  additional
times  the  transition  probabilities  of  moving  be-  projections  were  made  for each five-year  interval
tween size categories.  Projections for  1973 through  to  1998  (Table  2).  Warehouse  capacity  greater
1998  are  calculated,  using  successive  powers  of  than  75,000  bales  (S5)  stablized  in  1983  at  1.5
the transition matrix,  warehouses.3
Using this technique  to test  the Markov mod-
el's  projection  ability,  projections  for  1973  were
movement were developed and are shown in Table made.  They  compared  favorably  to  the  actual  movement were developed  and are shown in Table made.  They  compared  favorably  to  the  actual  .2.  As  a  basis  for  comparison,  projections  were
values,  Table  1. Analysis  indicated  that  by  1973,  m  ae  t  o  1  . T  omparison,  projections  were
ten firms  would go  out of business.  Actually,  nine  reasonay  wel  wth  the  1  projection  compared
firms ceased  operation.  One  was  operating  in the  reasonably  well  with  the  previous  one,  with  one
1-15,000  size  category  in  1973,  thus  supporting  exception.  The  projection  for  15,001-25,0  ca-
the  projection.  Results  obtained  for  states  S2 pacity (S2)  for the first base period was 2.05, com-
through S5  supported the Markov tool's projection  pared to 4.0 for the second.  This resulted  from S2
ability.  being an absorbing state in the second base period.
Table  2.  PROJECTIONS  OF  COTTON  WAREHOUSES  IN OKLAHOMA  USING  1964-1968  AND
1969-1973  BASE  PERIODS FOR 1978,  1983,  1988,  1993  AND  1998
1964-1968  Base  Period  1969-1973  Base  Period
Year  S0  S1 S2 S3 S4 S5 Total  S0  S1  S2 S3 S4 S5 Total
1978  11.74  .65  2.05  2.05  .03  1.48  18.00  12.14  .11  4.00  .75  .00  1.00  18.00
1983  13.62  .24  1.32  1.32  .00  1.50  18.00  12.59  .04  4.00  .37  .00  1.00  18.00
1988  14.73  .09  .84  .84  .00  1.50  18.00  12.81  .01  4.00  .18  .00  1.00  18.00
1993  15.43  .03  .52  .52  .00  1.50  18.00  12.91  .00  4.00  .09  .00  1.00  18.00
1998  15.86  .00  .32  .32  .00  1.50  18.00  13.00  .00  4.00  .00  .00  1.00  18.00
Since  the  second  set  of  projections  indicate  the matrices  must be multiplied  times the scaler 5.
four  firms  at  absorption  in  S2,  and  the  first  set  The  sum  of  the  elements  of  each  row  represents
indicate  no  active  firms at  absorption,  a  question  mean  length  of  time  all  firms  remain  in  non-
arises  as to which is more accurate.  While the first  absorbing  states.  The  probability  of  firms  being
set were  accurate  for actual  1973  firm  configura-  absorbed  was  found  by  multiplying  the  funda-
tion,  the  second  may  be  more  accurate  for  1978  mental  matrix  times  probabilities  of  absorption
through  1998,  because  they  were  based  on  more  found  in  the  southwest  corner  of  the  canonical
recent  industry  movements. 4 Studies  [1,  3]  indi-  matrix.  The fundamental  matrix  for the first time
cate  firms  of  size  15,001-25,000  operate  eco-  interval under study is:
nomically,  thus  supporting  the  latter  projections.  r  S  S 2 S 3 S4
Calculations  regarding  warehouse  movement  Si  2.5  0  0  0
between  categories  were  made.  Specifically,  these  S2 0  5  0  0 
included  mean number  of  years  a firm  spends  in  S3 0  5  0
each  non-absorbing  size  category,  time  before  S 4 0  0  1.5
absorption,  and probability  of being  absorbed.  To  The  matrix  containing  long  run  probabilities
estimate these values,  it was necessary to calculate  of absorption  is:
fundamental  matrices  from  canonical  forms  of  So  S5
1964-1968  and  1969-1973  transition  matrices.  S1  1  0
These  are  a collection  of mean  stay  times in each  S2  1  0
non-absorbing size category; however,  since a five-  Sa  1  0
year  movement  interval  was  selected,  elements  in  S4  .5  .5
3  Integer  values  are appropriate  for industry configuration;  however, actual  calculations  are  presented.
4 A chi-square  test for  stationarity was preformed  and the  null hypothesis  of stationarity  could not be  rejected  at the 95%  level. The calculated  test  statistic was  4.45 with  30  degrees  of freedom.
181The  probability  is  1.0  that  Si,  S2, and  S3  will  go  and
out  of business  in  the  long  run,  and  0.5  that  S4 X3 = Bales received in thousand of bales.
will be  either  absorbed by  So or  S5. Neither t-values nor standard  errors  of the  regres-
Similar calculations were made using the  1968-  sion  coefficient  were  reported  in  [4];  however,  it
73  base  period.  Results  indicate  all  firms  in  size  was  reported  that  all  coefficient  were  significant
classes Si  and S 3 will,  in the long run, be absorbed  at the 99-percent level.6 Summing the total cost  of
by So.  Length  of time prior to  absorption was  7.5  each warehouse  service yields a function represent-
years for Si and  10 for S 3. As of  1973,  there  were  ing total warehouse cost. This can be expressed  as:
no  firms  in  S4. Firm  survival,  then,  could  not  be 
determined.  In the  long  run,  four  firms  are  in S.  Y = -0.139 +  1.467X1 +  1.652X2 +  2.763X 3
and one  in S,.5 Economic  theory hypothesizes  that  where:  Y = Total  storage  and  assembly  cost  in
firms surviving  in the long run  operate on the low  thousands  of dollars  and  independent
point  of  industry long  run  average-cost  curve.  In  variables  are  as earlier defined.
view  of  this,  Markov  projections  are  interesting  As  shown,  the  above  function,  representing
because firms  surviving  in the long run are  not  of  total  warehouse  cost,  is  composed  of  six  com-
unique  size.  Markov  results  show  firms  in  size  ponents,  each  expressed  in  functional  form.  In-
classes S2  and S5 will survive in the long  run,  sug-  creased  cotton  production  density  results  in  in-
gesting  that firms  of  either  size  have  cost  curves  creased  total  assembly  costs  and  in  decreased
tangent to the falt portion of the long run average-  average  costs,  as  reflected  by coefficients  in  each
cost curve.  of the six  subequations.  For  example,  the  greater
To test this hypothesis,  average  cost functions  the density  of  production,  the  greater  the  receipt
for all warehouse  sizes had to be  developed.  These  level  by  a warehouse.  When  number  of  bales  re-
were derived  from total cost functions  determined  ceived  (X 3)  increases,  a  greater  percentage  occu-
by  combining  costs  of  all  services  performed  by  pancy  (X2)  will  result.  Increases  in  either  X2 or
warehouses.  The  functions utilized  were  those  re-  will result in a total cost  increase
ported by USDA.  [4]. ported by USDA.  [4]  p  d  ad  te  a  The model was  used to construct average  cost
The  services  performed  and  the  associated  warehouses.  Individual
total  cost  functions  are:  curves  for  various  size  warehouses.  Individual
warehouse  curves enveloped by the planning curve
Receiving:  Storing-variable cost:  are presented in Figure  1. SAC  1 represents  a firm
Y =  10.367 +  0.785X.  Y=-24.339 +  0.529X 1 of  bale  capacity  19,000,  SAC's  2,  3,  4  and  5
R2  = 98.4, N = 18  +  1.652X 2 representing firms of bale capacity 32,000,  50,000,
R2 =  97.3, N =  18  75,000 and  95,000. These  were  derived for ware-
Storing-fixed  cost:  Breakout:  houses  corresponding  to  the  Markov  states  pre-
Storeng-ftxed cost:  Breakout:
Y =  -16.207 +  0.938X,  9  =  1.770 +  0.511X 3 sented earlier.
R2 = 98.6, N =  18  R2= 82.4, N = 18  Points A  and  B  represent  points  of tangency
between  the relatively  flat portion  of the  envelope
Shipping:  ompression:  curve  and points  on  the  32,000  and 95,000  bale
Y =  9.577 +  0.308X 3 Y =  18,693  +  1.159X 3 SAC  curves,  respectively.  At  these  points,  aver-
R2  =  94.1, N =  18  R2 - 97.2,  N'  18  age  cost of handling  and storing per bale is lower
where:  Y =  Total cost of the associated  service  than  that  resulting  from  other  warehouse  sizes.
in thousands  of dollars,  Warehouses represented by SAC1, SAC 3 and SAC 4
Xi = Plant capacity  in thousands  of bales,  have  higher  per  bale  costs  than  the  32,000  bale
X2  =  Percent  occupancy  of warehouse,  plant,  SAC2,  has  at  15,001-25,000  bales.  SAC2
5 The long  run  projection  is  1998  since  convergence  occurred.
6The  total  cost models  presented  imply downward  sloping  average  costs  and  constant  marginal  costs.  It  should  be  noted  these
models  were the  ones selected over  several  alternative  models.
7 Assuming  an  oligopolistic  industry  setting  and  an increasing  cost  industry,  warehouse  rent  can  be  defined  as  the  difference
between  total  revenue  and  total  cost.  The  charge  per  bale  made  by  warehouses  for  storing  and  handling  cotton  is  relatively
invariant  with  location  in  Oklahoma;  hence,  per  unit  revenue  is equal  for  all  warehouses.  Costs,  however,  are  dependent  on
bales  received.  As  discussed,  average  costs  per  bale  decline  with  increasing  volume;  hence,  warehouses  located  in  the  high
production  density  areas earn  more  rent  than  warehouses  located  in  the  less  dense  areas.  Rent  calculations,  although  possible,
were not in the  objectives  of this study.
8 Warehouse  sizes  used  to  construct  the  SAC  curves  are  twenty  percent  larger  than  warehouse  sizes  employed  in  the  Markov
analysis.  This  adjustment  was  necessary  because  warehouse  receipts  can be  no  greater than  eighty  percent  of capacity  [6].
182Figure 1. AVERAGE  COST  OF  STORING  AND  HANDLING  COTTON  IN  OKLAHOMA
WAREHOUSES
40
36-
32
- 28  s
,24  -!4,,.SACI 1  wSAC 2 0,  24-'
,  20  \\  \ \SAC4 o  '\  \Y>-sAc 4 \  \\
16  \\  o-SAC
12
8-  A
4  (  Planning  Curve
I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I 0  5  10  15  20  25  30  35  40  45  50  55  60  65  70  75  80
Thousands  of  Bales
and  SAC5 correspond  to Markov states  S2 and  S5,  The analysis indicated  five firms, in two  size cate-
and are tangent to the flat portion  of the envelope  gories,  would survive.
curve.  Therefore,  the  economies  of size  hypothe-
sis used  to explain  the existence  of two warehouse  g  eted  a  iteret in  a  unique.  siz  e  ategory  sug-
sizes  could  not  be  rejected,  thus  supporting  long  a  e  aaities  eist  ing  quetion.  i  iffent
run projections.  bale  capacities  exist?  A  function  representing run projections.
warehouse  total  cost  was used  to develop  average
COMMENTS  costs  of  various  size  firms  and  an  economy  of
Markov chains were  employed  to analyze long  size  hypothesis  was  used  to  support  the  projec-
run  viability  of  cotton  warehouses  in  Oklahoma.  tions. The hypothesis  could  not be  rejected.
183REFERENCES
[1]  Chandler,  Whitman M.,  Jr.,  and  Joseph  L.  Ghetti.  "Cost  of  Storing  and  Handling  Cotton  at  Public
Storage  Facilities,"  U.S.  Department  of Agriculture,  ERS-515,  April,  1973.
[2]  Cleveland,  O. A., Jr., Richard  E. Just and Michael  S.  Salkin.  "Application of Markov Chain Analysis,"
Oklahoma  Agricultural  Experiment  Station  Bulletin,  September,  1974.
[3]  Ghetti,  Joseph  L.,  Whitman  M.  Candler,  Jr.,  Roger  P.  Strickland,  Jr.  and  Rodney  C.  Kite.  "Storing
and Handling  Cotton in Public Facilities,"  U.S.  Department  of Agriculture,  ERS-469,  April,  1971.
[4]  Ghetti,  Joseph  L. Working  papers  within  Fibers  Group,  ERS,  U.S.  Department  of  Agriculture,  Sep-
tember,  1973.
[5]  Kemeny,  J.  G.,  and  J. L.  Snell.  Finite Markov Chains, Princeton,  D.  Van Nostrand  Company,  Inc.,
1960.
[6]  Looney,  Zolon M.  Working  papers  within  Fibers Group, ERS, U.S. Department of Agriculture, April,
1972.
184

PROJECTION OF COTTON WAREHOUSE FACILITIES FOR OKLAHOMA: A MARKOV PROCESS

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