Brokken, Ray F.

English

Research Papers in Economics

The  Case  of a  Queer  Isoquant:
Increasing  Marginal  Rates of Substitution  of
Grain for  Roughage  in  Cattle  Finishing
Ray F. Brokken
The  purpose  of this paper is to argue the reality
of  increasing  marginal  rates  of  substitution  of
grain  for  roughage  in  beef production.  Most  eco-
nomists  and  animal  scientists  have  the  idea  that
these  marginal  rates  of technical  substitution  are
decreasing.  Brokken  et  al.,  Heady  et  al.,  and
Goodrich  et  al.  have  shown  increasing  marginal
rates  of substitution of grain for roughage in cattle
feeding  response  functions.1 The  practical  im-
plications  of  such  substitution  relations  are  enor-
mously  important  for  both  the  producers  and
consumers  of  beef  as  well  as  for  economic  and
technical  research in beef production.
Three Empirical Examples
In  the  three  examples  cited,  grain-roughage
isoquants  for  cattle  finishing  are  found  to  be
concave  to  the  origin  over  at  least  part  of  the
physical  region  of  substitution.  All  three  use
different  methodology  in  deriving the substitution
relations.
Goodrich et al.
Data  from  17  midwestern  university  experi-
ments  involving 878 steer calves on rations varying
in  the  proportions  of corn  silage  and  corn  grain
were used. Rate  of gain and total dry matter intake
per  pound  of gain  were  each  related  to  the  pro-
portion  of  corn  silage  in  the  ration.  From  these
relationships,  the  expected  quantities  of  the
separate  ingredients  required  to  obtain  a  given
Ray  F.  Brokken  is  an  agricultural  economist,  Economic
Research  Service,  U.S.  Department  of  Agriculture,  sta-
tioned at Oregon  State University,  Corvallis.
Space  prevents  discussion of the distinction between
the  neoclassical  concept  of the  production  function  and
the  concept  of  a  response  function  for  livestock  pro-
duction  [Dillon].
gain and the expected time required were calculated
[table  1,  p.  19].  A  plot  of  the  data  shows  a
concave  (to  the  origin)  isoquant  for  corn  silage-
corn grain substitution.
Heady et al.
Heady  et  al.  analyzed  response  functions  for
finishing  steers  on  various  proportions  of  corn
and  soilage.  Among  the  alternative  regression
equations was a function allowing increasing and/or
decreasing  marginal  rates  of substitution  of corn
for  soilage  in  the  range  of  diminishing  marginal
physical  product  of both  soilage  and corn. 2 Even
though  this  function  had the  best  statistical  fit  of
the  alternatives  tried,  it  was  rejected  because  it
".  ..  gave  sigmoid  isoquant  contours  denoting
first  increasing  marginal  rates  of substitution  and
then  decreasing  marginal  rates  of  substitution
. "  [p.883].  They  state  [p.918],  "Even  though
the  coefficients  of  determination  for  this model
of  the  beef-cattle  production  function  were  quite
high, the model  was rejected on the basis of logic."
The logic used was not explained.
Brokken et al.
In this study,  an appetite function which  relates
daily  voluntary  feed  intake  to  the  energy  con-
centration  (calories/kilogram)  of  the  ration  is
combined  with  a  function  which  relates  daily
feed  requirements  for  maintenance  and  gain  to
body  weight,  rate  of  gain  and  ration  energy
2  Response function was of the form G  = aFbecR + dR2
where  G  is cumulative  weight  added,  F  is  soilage, e is the
base  for natural  logarithms,  R  is the  soilage to corn ratios
and  a,  b,  c  and  d  are  parameters  to  be  estimated.  This
function  was  fit  as  a  part  of a  special  form  of recursive
system  which  consisted  of  the  response  function,  the
ration  relation,  the  gain  relation  and  the  consumption
function.
221Western J. Agr. Econ.
concentration.  From  the  combined  functions,  a
relationship  showing  daily  rate  of  gain  as  a func-
tain of energy concentration (calories per kilogram)
of the ration  is derived.  Total feed intake for each
energy  level  is obtained by aggregating daily intake
over  time  (by  either  discrete  or  continuous  sum-
mation).  The proportion  of grain and roughage  for
each  energy  level  are,  respectively,  multiplied  by
total feed required to produce a given output to ob-
tain coordinates for the  grain-roughage isoquants.
The  appetite  function  has two  phases.  In phase
I, gut  fill limits intake, but daily  dry matter intake
increases  as  ration  energy  concentration  increases
because the rate  of digestion increases  as the ration
is  enriched.  This  phase extends  from  rations  with
all  hay  up  to about  35 to 40%  grain.  In phase  II,
further enrichment  of the ration causes daily intake
to decline  and gut  fill does  not limit intake. How-
ever,  the  rise  in  energy  concentration  more  than
offsets  the  decline  in  dry  matter  intake  so  that
the net energy intake is also increasing in phase II.
This  study was concerned  only with diets in phase
II  of  the  appetite  function.  The  system  is  sum-
marized as follows.
Notation.  Let  Y  represent  daily  voluntary  dry
matter  intake  per  unit  metabolic  weight,  i.e.,
Y  - F/W.  75  where  F  is  daily  feed  intake,  W is
body  weight  (W.75 is  metabolic  weight);  i
represents  the  ith  animal;  X is  net energy  for gain
per kilogram  of feed i.e.,  the energy concentration
of  the  ration; g is rate  of gain  in Kg/day; t  repre-
sents  the  tth  day  of the  feeding  period;  Wo  is
weight  on  day  zero; n is  the  total number of days
in the  feeding period; Wn  is  the weight on the nth
day of the  feeding period.
System.  The  appetite functions are:
1) Daily dry matter intake,  Yi = fi (X) = Ai + BiX,
2)  Daily energy intake, XYi = Xfi(X) = AiX + BiX2.
Daily energy requirements  for gain, g(X, gi):
3)  XYi = .08089X-  .03185X2 + .05272gi + .00684gi.
Equating  2 and 3 and solving for gi obtains
gi (X) = G  [Xfi(X), g(X, gi)]:
4)  gi = -3.8538  +
r4.8  8  Xfi(X  08089X+.03185X
2/2
4.8517  +'  .00684
Daily feed intake  is Fit = YiWt 75 = Yi(Wo + gt)' 75.
Hence total feed intake  over n  days (TF) is:
5) TF  Yi  (Wo+ gt) 75 dt
r(W  F  )175  W17s
Yi  (Wo  +ng)  -WQ
gi  1.75
Set  Wo  + ng = Wn  and consider  TF  for a given
span of weight from  Wo  to Wn  (n  being variable).
The  right hand expression  of equation  5 becomes
a  constant  (r)  and total feed  becomes  a  function
of ration energy concentration:
6) TF= Yi  fi(X)  p. 6)TF=-F=  r.
gi  gi(X)
Multiplying  TF  by  Px  and  (1 - Px),  respectively,
the  proportions  of  roughage  and  concentrates  in
the  ration,  obtains  the  coordinates  for  plotting
the concentrate-roughage  isoquant.
The  concentrate-roughage isoquant. The  shape
of the isoquant  depends  on  the  appetite function,
fi(X),  and  on  the  gain  function,  gi  (X),  both  of
which  appear  in  equation  for  total  feed  intake.
The  gain  function  depends,  in  turn,  on  the  ap-
petite  function,  fi  (X),  and  on  the  requirements
function,  g(X,  gi).  The  general  shapes  of  these
functions  are  well  established  in  the  literature.
For  purposes  of  this  argument,  the  requirements
function  of  the  California  net  energy  system  is
used with alternative  appetite functions.
Four  appetite  functions  are  shown  in  figure  1.
The  assumption  of  a  constant  daily  feed  intake,
as  in  appetite  function  II,  is  frequently  used  in
economic  analysis.  This  assumption  means  that
daily  energy  intake  (i.e.,  X times  Y)  is increasing
linearly  as  energy  concentration  is  increased  and
represents  a  rather  extreme  bias  in  favor  of high
concentrate  feeds.  Another  extreme,  represented
by  appetite  function  III,  is  the  assumption  that
the  rate  of gain  would  be  the  same  regardless  of
energy  concentration  of  the  ration.  In  between
these  two  assumptions  lies  appetite  function  I
which  represents  the  function  for  an  animal  of
higher  than average  performance  from  the  experi-
mental  sample  used  in the  study  by Brokken  and
Dinius.  The fourth  function represents  the unlikely
case  in  which  rate  of gain  first increases  then de-
creases  as ration energy concentration is increased.
The  isoquants  corresponding  to  these  four ap-
petite functions  are  shown in figure  2. Proportions
222
June 1977Increasing  Substitution of Grain for Roughage
Fig.  1.  Appetite  functions  used  for  illustrating
shape  of concentrate-roughage  isoquants.
Dry Matter  Intake  (Kg/W: 7/Day)
Fig. 2.  Isoquants  corresponding to appetite  func-
tions in  figure 1.
Kilograms of Roughage
.15
.10
.8 1.3
Ration  Energy  Concentration  (Mcal  NEg/Kg)
of  roughage  and  concentrates,  corresponding  to
each  energy  concentration  are  shown  in  table  1
and  the  rates  of gain  and total feed requirements
for each  ration for the  four appetite functions are
shown  in  table  2.  Isoquants  corresponding  to
appetite  functions  I,  II,  and  III are  all  concave  to
Table  1.  Roughage-concentrate  proportions by diet
Ration Concentration,  Mcal  NEg/Kg
Item  .8  .9  1.0  1.1  1.2  1.3
Roughage %  63  53  42  32  23  14
Concentrates %  37  47  58  68  77  86
800
600
400
200
200  400 600  800
the  origin.  To  obtain  an  isoquant  convex  to  the
origin,  it  was  necessary  to  have  rate  of gain  first
increasing  then  decreasing  as  the  proportion  of
concentrates  in  the  diet  are  increased  (appetite
function  IV  and  the  corresponding  isoquant  IV).
This is not always a sufficient condition; a minimal
change  in  gain  would  not  obtain  this  behavior.
Another  behavior  resulting  in  a  convex  (to  the
origin)  isoquant  is  to  have  an  appetite  function
such  that  the  rate  of  gain  is  continuously
Table  2.  Rate of gain and  total feed  requirements by diet and  by appetite  functiona
Ration  Concentration,  MCal  NEg/Kg
Appetite.
Functions  Item  .8  .9  1.0  1.1  1.2  1.3
Gain  (Kg/day)  1.08  1.16  1.23  1.29  1.34  1.38
Total Feed  (Kg)  1265  1125  1011  915  834  764
II  Gain  (Kg/day)  1.08  1.24  1.40  1.57  1.74  1.92
Total Feed  (Kg)  1265  1101  973  869  784  712
III  Gain  (Kg/day)  1.08  1.08  1.08  1.08  1.08  1.08
Total  Feed  (Kg)  1265  1152  1055  971  896  828
IV  Gain  (Kg/day)  1.08  1.09  1.08  1.05  .99  .91
Total Feed  (Kg)  1265  1147  1054  982  927  889
aTotal  feed required  for a Fof 104  (eg. for growth from 320 Kg  to 437  Kg).
223
I  Y= .18607-.062X
II  Y =  .13647X
.06485
Ill Y=  +  .08089 -. 03185X X
IV  Y=.22587-.11175X
II
III
IV
I  I
63:37
53.47
42;58
32:68
23;77
__  I  __  L  ￿_￿  ￿_ ...  - ._  ,  _  _  _  _
.Vi ---  L  --  - --- ---  IO-  -~~~~~~~~~~~~~~~
I,  I  I  I  II,  ,,
Brokken
m
IWestern J. Agr. Econ.
decreasing.  This  also  is  not a  sufficient  condition
unless  the  decline  in  gain  is  large,  which  is
very unlikely.
Summary and Conclusions
The  examples  offered  do not constitute  a proof
that  grain-roughage  isoquants  will  be  concave  to
the  origin  for all  types  of cattle under  all growing
conditions  and  all  types  of  grain  and  roughage.
However,  they  do  constitute  a  strong  argument
that  isoquants  are  concave  in  many, if not most,
cattle  finishing  programs.  Only  two  kinds of  ap-
petite  functions  will  give  isoquants  that  are  con-
vex  to the  origin.  One  is  an appetite function that
obtains  a  substantially  decreasing  rate  of  gain  as
the  proportion  of  grain  in  the  diet  is  increased.
The  other  is  an  appetite  function  that  obtains
first an increasing rate of gain and then a decreasing
rate  of  gain  as  the  proportion  of  grain  in  the
diet is increased.  The author has never encountered
either of  these patterns  of performance  relative  to
varying proportions of hay and grain  in the diet.
This idea  of concavity  also has intuitive appeal.
Total  energy  required  for  a  given  gain  decreases
as  rate  of  gain  increases,  because  fewer  days  of
maintenance  are  required.  As  the  proportion  of
grain  in the  ration is increased, replacing roughage,
the  rate  of  gain  is  expected  to  increase.  Hence,
grain  substitutes  for  roughage  with  increasing
efficiency.  This  means  the  isoquant  curve  is  con-
cave  to the origin.
Is  this  a  queer isoquant?  It would seem  so from
the  point  of  view  of our  economic  training  and
practice.  However,  the  reality  has  always  been
independent  of  our  assumptions  about  it.  On
careful  examination,  the  concave  isoquant should
not have  seemed  so unusual.  It is  not inconsistent
with  plain  intuition  and  has  been  observed  when
the  assumed  functional  forms  did  not  preclude
its  observation.  What  does  this  tell  us  about  our
economic  training and  practice?  It tells us that we
can  easily  become  entrapped  in  the  confines  of
our  paradigms.  This  disconcerting  condition
challenges  us  to open up and examine  our assump-
tions,  even  those  related  to  the  most  elementary
concepts.  Where  else  can  increasing marginal  rates
of technical  substitution  between  factors  of  pro-
duction  be found?
References
Brokken,  Ray  F.,  and  D. A.  Dinius.  "Effect  of Dietary
Energy  Concentration  on  Feed  Intake  of Beef Steers
and  Implications  for  Performance  and  Heat  Pro-
duction."  CED  Working  Paper,  Economic  Research
Service, U.S. Dept. of Agriculture,  1976.
Brokken,  Ray  F., T.  H.  Hammonds,  D.  A.  Dinius,  and  J.
Valpey.  "Framework  for Economic  Analysis  of Grain
Vs.  Harvested  Roughage  for  Feedlot  Cattle."  Amer.
Jour. Agr. Econ. 58:245-258,1976.
Dillon, John L.  The Analysisof Response in Crop and Live-
stock Production. New York:  Pergamon  Press,  1968.
Goodrich,  R.  D.,  D.  W.  Crawford,  M.  L.  Thonney, and J.
C.  Meiske.  "Influence  of  Corn  Silage  Level  on  the
Performance  and  Economic  Returns  of Steer  Calves."
1974  Minnesota  Cattle  Feeders' Report,  Dept.  of
Animal  Sci.,  University  of Minnesota,  St.  Paul,  1974:
14-19.
Heady,  Earl  0., Glenn  P.  Roehrkasse,  Walter Woods,  and
J.  M.  Scholl.  Beef  Cattle Production Functions in
Forage  Ulilization.  Iowa  State  University  of Science
and Technology  Research  Bulletin 517, July  1963.
224
June 1977

THE CASE OF A QUEER ISOQUANT: INCREASING MARGINAL RATES OF SUBSTITUTION OF GRAIN FOR ROUGHAGE IN CATTLE FINISHING

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