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5/08 KzlProleocaPros/CIRCIDaIeDue.indd

ECONOMIC ANALYSIS or *mumm
INPUT-OUTPUT DATA FROM THE
CAUGA VALLEY, COLOIBIA

13!
HERNAE‘I BERTOLDTTO

AH ABSTRACT

Subnltted to the College or Agrloulturo of
Michigan State University or Agriculturo
and Applied Science in partial fulfillment
of the requirements for the degree of

MASTER OF SCIENCE

Dam-hunt. of Agricultural Economic.
1909

APPROVED

Bartolotto
1

ABSTRACT

Current fertilizer recommendations generally reflect
inadequate attention to economic considerations.

Farmers are being eupplied with fertilizer information
free which, implicitly or eXplicitly, the conclusion is
being drewn that the moat adequate level of fertilization
ie the one at which maximm yield: per acre are attained,
m ie eeldom consistent with the more important concern
of maximizing profite. Profits are increased only so long
no the cost of adding fertilizer inpute ie leee tinn the
added return derived. from their uee.

The experimental work to determine fertilizer input-
ercp output reletionehipe and to provide information for
making more dependable recommendation to farmers eat
ecndncted emperetivelr by the Colombia Project of Michigan
Btete Univereity, the Feeulted de Agronomic of the Univerei-
ded-flecicnel de colon'bie. at Pnlnire. (colonize) and e
dole-mien farmer. Bencr Edgardo Patina. during 1957.

the We studied ere corn and beene. The variable
nutriente etudied ere nitrogen, phosphoroue and poteeeiun.

‘i'he enelyeie of the date produced by theee experimente
penite more adequate analysis of fertilization retee and

of the recommendation which are given to farmere.

Bertolottc
2

The analyeie of theee'data are based on the. concept
of a continuous mathematical production function. Yield
reepcneee to different fertilizer nutriente are dependent
upon the levele of the variable nutrient inpute. The
economic Optima are determined where the marginal value
productiv1ty of a nutrient input ie equal to the coat of
adding another unit of such input of fertilizer.

m two-variable functicne were fitted to the
experimental data for corn Patino Lower and corn Patina
Upper Field. After applying varioue etatietical teete
it vae decided that a eroee product production function
of the fonu ’

Ye=a§bN§cP§dN3§ePz+fNP

where to in yield and H and P are per acre applicatione of
nitrogen and phoephoric acid nae coneidered a better
repreeentation of the functional relationahipe involved
tmn a square root equation which was the alternative
pruduetion function fitted to the data.

For eornPatino Lower Field data, nitrogen wee found
to exert the predominant influence on yield, even though
reeponee wae aleo obtained from the applicatione of
phoephoric acid.

For corn Patino Upper Field data , only phosphoric acid

Bertolbttc
3

applications were found to influence yield.
three three-variable functions were fitted to the
emerinental data for bean Patina 1957/58. After applying

statistical teete a Cobb-Douglas function of the fern:

Ivan considered a more appropriate fit for thie eat of
experimental data, firm the square root and oroee product
production functions the alternative equations fitted to
the data.

The economic Optima conditione are based both in the
phyeical functional relationshipe and in the price con-
ditions for fertilizer inputs and product output existim
at a given moment.

If the price relationships involved change, a new
optimun amount of fertilizer inputs and nutrient
conbinatione to apply become profitable as determined by
the new nutrient-crop price ratio existing after the
change.

Ii‘hie study shows that further emperinental work in
corn, beans and other crepe ie needed in the Canon Valley.
the eXperieental design used in this study has proved
ueeful to obtain the kind of eXperinental data needed
to make sound recommendatione to farmers.

In view of present agricultural development projects

Bertolette
4

under way in Colombia,-thie kind of reeearch.work may be
the beat any to promote an efficient reallocation of
resources and can make an important contribution to

increased.productive capacity or the Colombian agriculture.

Em;fibf‘ilc AERLYGIZB 1!»? F 3563.11.11.23qu
INPUTUCUTPU? DATA F335 THE
CAUCA VAZLEY, CCLOEHSIA

8:!
me ammw no

A THESIS

Submitted to the (bums of Agricultur- of
Hicnignn State! University of Agriculture
and Applied Bcianoc in partial fulfillment
of the requirmcmtc for the degree of

HASTER U? HGIPNGE
Departnmt of Agricultural Economics

1959

11

ACKNOWLEDGMENTS

The author wishee to eXpreee his appreciation to
Dr. Glenn L. Johnson, major professor, for his counsel,
guidance and for constant encouragement throughout thie _
etudy and during the whole or the author's graduate work.

The author in particularly indebted to International
Cooperation Adminietreticn (1.0.A.) or waehington, D. 0.,
and to ite Mieeicn to Chile. for providing the echolarehdp
which enabled graduate work to be undertaken at Michigan
State University.

thanks areleleo due to the Catholic‘Univereity
(Chile) tcr’granting the necessary leave to study in the
United Statee.

The author ie aleo indebted to Dr. Leonard Kyle who
reed the manuecript and.provided valuable euggeeticne and
conetructive criticiee.

Thanke are given to Bernard Hoffnar and to the
pereonnel or the etatietical pool for carrying out a major

portion or the computatione.

111

TABLE 0? CONT}: T3

Page
ACKHUELEDGWEflTS e e e e e e e e e e e e e e e e e e e 11
LIST OF TABLES e e e e e e e e e e e e e e e e e e e 7

LIST 0? FIGURES e e e e e e e e o e e e e e e e e e e V11

can-mm
1 THE IETUWZEZATIOUSFM3 OF Aflfifflififilc AN?)
ECUHUHIC C(JI-Scm’TS In r‘mTILIzm RZSJEMECH . . 1
The Type of Information Hooded by
Farmers 000 e e e e e e e e e e e e e e e 1
Economic-Agronomic Integration in
Fertilizer Research e e e e e e e e e e e 4
2 THE TE‘QTX‘JHE Ami) LOGICAL FC‘UHIFLTION OF
PfiODUCTIUB FUNCTION ANALYQIS e e e e e e e e 7
The Production Function in Fertilizer
RBBBRPOD e e e e e e e e O D e e e e e e 7
The Concept of Eaximizetion . . . . . . . 7
3 BfiflTIBTXCAL REASUEEHEflTB e e e e e e e e e e 10
Introduction I e e e e e e e e e e e e e 10
Graphic Analysis of Functional
Relationships e e e e e e e e e e e e e‘e 11
THE Estimating Equation e e e e e e e o e 11
The Standard Error of Estimate. . . . . . 13
heaeurement 0: Correlation. e e e e e e e 15
Variation EXpleinod e The Coefficient
of Determination. e I e e e I e e e e e e 16
The Coefficient of correlation. e . . . . 17
Tests of Correlation Results. . . . . . . 19
(a) the Validity teste e e e e e e e 19
(b) thfi ”Elifiblllty tQSt e e e e e e B
4 KC3I'LTKIFEI‘IIiTlsL WCRK AND WHJRCE CF DfsTA . e e e 23
Th3 Corn Data 9 e e e e e e e e e e e e e 23

Th6 BB£H Data 0 e e e e e e e e e e e e e 24

com-m

5 DERIVnW 10% :F PRODUCTIUH IUHLiIOWS FROM
XIELD DATA e e e e e e e e e e e e e e e e e

Production Functions Fitted to fiata . . .
Previous hyperinontal lleeulte U.' =.ing
these Nationfie e e e e e e e e e e e e
(a) nouare hoot Production Function.
('0) {moss Product Prozluotion
Function e e e o e e e e e e e e
(o) Cobtrbouglae Pmrluotion
Function e e e e n e e e e e e e
Chooeing the "3.331;" Fitting Function. 9 e
Analysis Of the Data. e e e e e e e e 0
Analysis of the Patina Lower Field
Corn Data e e e e e e e e e e e e e e e
(a aquaro float Production ?unotion.
(13 Cross Product Production
» Function e e e I e e e e e
Frofiuotion Surface Estimates. 1 e e
IEoonomio Ohtima o e e e e e e e e
Analysis of the Patino Upper Field
Corn note e e e e e o e e e e e c e e e e
(a) Square hoot Production Function.
(b) Cross Product Production
Function e e e e e e o e e
Production Surface EStlmatfifle e o e
Economic option 9 e e e e e e e e e
Analysis or the Patino Boone Date ‘ .
1957/58 e e e e e e e e e e e e o e e e e
(a) Square Root.Produotion Funotion.
(b) Cross Product Production
Function e e e e e 9 e e o e e e
(o) Cobh~noug1ae Production
Function e e e e o e e e e e e e
P’DdUGtian Surface ES‘lmatflae 9 e e e e e
Eoonomic Optima o e e e I e e e e e e e e

I b e
a o I
I I O

C 0.
CI 0
O. C

6 EVALUATIGH MED CUflCLUSIOHB‘ . . .in g g . . . _.

arr own A

?nrinnoe e e e e e e e e e e u e e e e e
Validity of Experimental Reeulto

Over Time e e e e e e e e e e e e e e
COHOIUS1038 o q n e e e e e e e e e e
Implicatione. e e e e e e e o e e i e t

I C. O C i U l O 0' I g I O I O D I I O

O

~§
.0.

BIBLIQGW’EY!Q.DICIQQ6......IIOGI

iv

TABLE

LIST OF TAELHS

Levels of Fertilization for the Corn
ExPorimonts, Patina Lower and Patina
UFDBP Fifilde 1957e I e e e e e e e e e e e e

Experimentnl Design of the Corn
Experimente Patina Lower and retina
Uppfir FIB-Id, 1957e e e e e e e e e e e e e e

Levels of fertilization for the Beans
Experiment, Patino, 1957/53e e e o e n e e e

Experimentnl Design of the Boone
Experiment, Patina, 1957/58e e e e e e e e e

nines or “n“ and “213* for Two Variablo
nutrients and voluee or “t“ for Individual
Regression Coefficients, tor‘Petino Lower
PieldCornlmtn.u...e........

Total Calculated Corn Kielde from opeoified

Rates of Application of Nitrogen and
Phoephoric Acid, Predioted from the Cross
Product Equation, ror’Patino Loner Field
Corn Data, 1957 y e e e u e e e e e e e e e

Total Predicted and Cbeerted‘Yieldo
’0? Corn Patina LOWS? Flfild. 1957e e e e e e

iarginal rrodnctivitiee of Nitrogen and
Phosphoric Acid in the Production of Corn
for Inputs Indicated (Nitrogen at TOP of
Enoh.Pnir and Phosphoric Acid at Bottom}
tor'Patino Lower Field Corn Date, 1957 . . e

Veluoe of “h“ and “R3' tor*Two Variable
Hotriente and Values of “t“ for

Individual Regression Coefficients, for
Pntino Upper Field Corn Date, 1957 . . . . .

Page

49

64

11

12

13

14

15

16

17

Total Calculated Corn Yields from

opeoified Katee of Arplicntion of

nitrogen and Phosphoric Acid,

Predicted from the Croce Froouot Function
for Patino Upper Field Corn Date, 195? . . .

Total Predicted and Observed X1olds for
GOP“ 9ntino Upper F1615 Data, 195? c e e e e

Harginal Productivitiee or nitrogen and
Phosphoric Acid in the Production of

Corn for Inputs Indicated (nitrogen at

TOp of Each Fair and Phoepmric Acid at
Bottom) for Patina Upper Field Yield

Data, 1957 I e e e e e e e e e e e e e e e 9

Values of “R“ and ‘32., for Three

Variable nutrients and Values of 't‘

for Indiriduol Regression coefficients,
tOr'PfitinO Bean Data, 1957/53e e e e e e e e

Total Calculated Bean Yield from Specified
Rates of Application of Nitrogen and
Phoephorio Acid, nnd.Potach, Predicted

from the Cobb-Douglas Equation, for.Petino
Been Data. 1957/58: e e e e e e e d e e e e

Total Predicted and ObeerVeJ'Yielde for
BERN P&tln° Field Data, 1957/58. D e e e e e

Calculated finrginal Productivitieo of a
Pound of Nitrogen in the Production ot_
Beans at Various Katee of Application of

- Nitrogen and fieleoted Bean Prices, for

Petino Fifild Data, 1957/58 e e e e e e e e e

Calculated Harginal Productivities of n
Pound of Phosphoric Acid in the Production
of Boone at Various Rates of Application

of Phosphoric Acid and Selected Bean Fricee.
for'Pntino Field Data, 1957/58 e e e e e e e

60

61

64

66

71

72

73

74

FIGURE

:0

a:

I.

b.

0.

'0.

vii

Lie? 0? FIGURE*

Total Yield with fiitrogen .

Variable end.POU Fixea at Three

Levels, PotincuLower Field Yield

Data for Corn, 1957. a o I Q a o o I a o 46

Total Yield with 221106 Variable

end flitrogen Fixed at Three
Levels, for Petino Lower Field
Yield Date for Corn, 195?- o 0 o o I a o 46

Scale Line for Nitrogen and P905
Increased in : fixed 1:1 PrOpcrticn

for’Patino Lower Field Corn Yield
beta, 1957 n o u a o I o n o o o u n o a 48

Total Ileld with Nitrogen

Variable and.P¢0 Fixed at Three

Levels, PutinOWUpper Field corn

Eleld Data, 1957 o v o o c o o o o o a u 58

Total Xiald'with.P005 Iariable and

Hitrogen Fixed at lures Levels, ficr
Patino Upper Field Corn Yiela Data,

195?’OOIII‘IQQO§OIO‘OO. 5’8

ficalo Line for Nitrogen and P005

Increased in a Fixed 1:1 PrepSrtion,
for Patina Upper.Pield Corn Yield
Data. 1957 Q I I I O O I O C. I U C I O O 58

03 m3" T173331 1

T2133: Ifi’i‘zifiinfdlixfliilfi.zsliIP'.236 up Adfiuflmfit} Maj.) :.:1Cu:-::;;:-::IG
L1=454C2F£3 13:5 9351;111:313 ii;3‘.3?7;"«3i{:}£
:81; Mg Qfiglnfargrfitggn fine-flew; 1);: ’r‘f'gmgrmjw
Una. aspect of
rartllizer resaarcn deals with.tha presence or absence or
respanao 1n crap yields to fertilizer aypllcatlans.

Howavar, once reapunaes have bean round to 9113:, the
farmer needs to ounslficr fartillzsr alang with other
reaouroon and practices in his farm managemant decisions.

These decisions can be made moat efficiently 1f
fertilizer information 13 prcviflad 1n the tern.or 1ncra~
mental rngponsa data.

Ineromental response data show tha'auocsaalva ada1t10na
ta yield rasulting tram suecnsaive fartillzar appliontldpn..
Regardingly, onaa research has shown that crap yields no
reapond to tartlllzer, the max: stays in rasekroh.mre to
estimata:

(a) the lnaranental yielda fortncauing from airfarent

rates or tartllizer appliaations unfier apaolried crap an&
2011 conditions, and

(b) tha eoanamla Optimum quantity of fertilizer, conngerlng
crap and fertilizer'prioaa and production costs.

Farmara can be divlaed luto two groups! thaao who

have ample capital and those who have limited capital.

they are seldom interested in maximizing yields per acre,
and not even the farmer with unlimited eapitai ie interested
in unnimunwper acre yields: he is interested in hhgher acre
yields only to the extent thigreeter production adds more
to reuurne than to caste.

the extent to which.higher yields increase profits
depend on: _

(a) the rate at which inputs are transformed into crepe,
and
(b) the price ratio.

Maximum profits come when the crop/fertilizer trans-
formation ratio is equal to the fertilizer/crap price
ratio: the transformation ratio declines with heavier
fertilization rates under diminishing returns. The
elape or the response function represents the incremental
er'marginal yield due to small increases in fertilizer
use. The farmer with.iinited capital needs this information
in determining how mach fertilizer to apply.

For inetenoel. suppose that a farmer with.limited
capital can earn 82-60 return on tunde spent eieewhero
in his business (each as motor fuel, crap need, or beg

supplement).

_. . .A..- A
W ‘ v . ,_v , V .7 ‘— 7 _'

11161117, B. 0., "Methodological Problems in Fertilizer
Use“, (in) 321211, E. In. Heady, 17:. 0., Blackmore, J.,
‘Hethodolo§geal Prgge-rmren in the Economic Amalgam 0:;

t 11 er Us Dar 3, Men, Iowa Estate College—“Frees,
6, 9113}; or , pp. 3.

    
 

lie in given information showing that; one discrete
level of fertilization, 30 pounds of nitrogen, will more-nee
oat yield by 17 bushels. Eith oatsat 70 cents per bushel
and nitrogen application coating 18 cents per pound, the
tote]. return in $11.90 and the total cost in £35.40, a net
of 2.16.50.

However, the return per dollar spent on fertilizer
(11.90336MO) in only $2.20 and the farmer will allocate
hie scarce rundn where he can get £32.60.

suppose, however, that the farmer ie given even three
points from a response function moving: in. first 10
pounds of nitrogen has a marginal yield or 10 bushels;
the second 10 pounds hoe n marginal yield of 6 bushels,
and the third 10 pounds has a marginal yield of 2 bushels.

with a unit caning $1.80, the first 10 pounds returns
$5.89 per dollar invented in fertilizer, and the second
return: $1.95.

Hence, since the tamer can realize only 32.50
elsewhere in his business, he new is encouraged to
invest in at least lo pound: of nitrogen.

Clarion-1y, than, knowledge or the response function,
coupled with information on the economics of fertilizer
woman encourage a greater investment in this resource

on the great majority of fame with limited capital.

4

The netted If research and form of presentation, when
the findings and recommendations are in tome of "one
discrete level. can lead the tamer to uee no fertilizer
when fertilization actually represente a profitable
investment within hie situation of limited capital.

Knowledge of the responee function is equally
important for the farmer with unlimited eopital.

It ie known that the optimum or most profitable level
of fertilization for these farmer-e ie defined in equation
(1)

21:21
a? par

where the term to the left of the equality ie the marginal
yield or response and the term to the right ie the price
ratio (price per unit of fertilizer divided by the price
per unit of yield).

The marginal yield ie the derivative of yield with
respect to nutrient; it ie the 310]” of the response
function for any particular input level. Thi- ie the type
of information haeio for making recommendations to fax-mere
who no): to maximize profite when unlimited capital in
available to than.

iwmniorozlronomie Intezrotion

in Fer-til; 3:13P Hesterqgw
Ae
Johnson1 pointe out , ”fertilization research should be

‘— -— M A A 4+... 4‘

1Johan-on. 6.1... "Interdisciplinary Considerations in
Designing merimente to {study the Profitability of Fertilizer
030'. in Baton, Eel-we. 8.88.”, 3.0., Blackmore, Je, gge gi‘e.pe 22o

r

looked at from the point of vim: or an agriculturist

rather than I'm tha confined viewpoints of ths farm
management specialist, the soil Specialist, the marketing
01:00:41.1“. the mathematical statistician, or the specialln
1n leguminous nitrogen fixation”.

Agronomna and omnomlata recognize that fertilizer
recommendations should be! based on data and principle:
dram In: both salmon.

first, it 1. necessary that 33902103110 findings be
avntlablo for.applioazion of tho relevant oaonomio
principles. ntatlng that. find how much tortilla“- would
I” unad- ‘I'ho 01001103110 13148131910 15. or course, quit.
uterus without tho raaponao data to go with it.

Rayner. agronomic data alono do not yrs-wide the
bill! for efficient fertilizer use.

803:. of the reasons for the lack of integration of
economic and agronomic principlea 1n tartlluer raseamh
in tho past and at tho present time in many cases and.
muntriea, as Haadyl points cu: are the: following:

(a) lack at training of agricultural ecnnomata 1n
matkwmatlcal economics and statistical techniques to

davalop the kind of estimates for economic analyais.

____ ‘4. . “A

w .— vv— w w v ———-—v

18mm, 3.32... Heady, 3.0. “Over-all Economio Gonna-ru-
tions in Fertilizer Use", (in Baum. Ed... Heady, E.0.,
Peach, J.‘I‘., Hildra‘bh, 3.0., “figg‘tiliaer Innovationg ang
gigsqurogflsg', Am”, Iowa Stat? Collega Proms, 7195?, 1313.128.

6

(b) overspeaialization in landvgrant collagen and othsr
rodenroh institutions has not always encouraged sufficient
unaporativo work.

(0) improved statistical techniques far handling multifi
variable tortillas: eXperimants have been emphasized only
racontly.

(d) in many areas of tbs United atatea, fertilizer became
an important factor of production only recently.

(0) the raluotanca or agronomiats to consider economic
Optima studio. as a part or their resaaroh program duo,

in part, to a lack of understanding of the mathamatioal
procedural; used by the economists.

(f) agronomiatll
typo studies to establish.rosponso and relate it to soil

characteristion rathmr than to dotcrmino the marginal

had been interested largely in Variance-

quantitioa And the aptimum use of fartilizor.

# __._‘. AA “A; .4 ____. A“ A; .4-‘4 _‘ —J-

lfiaady, E.O. Peaek, J.T., “A Fertilizor*Production
surfaoo with specifications of Economic Optima for Corn
Grown on Calaaraoua 153 311% Loam”, ggurnal of Fang
Economggo, Volume 36, August 1954, pp; 466. "fl"

 

momma 2

Ti ItUJdbfiod FJHG” 3% Ana EHEL 0.NCETI
O, i i lulu}: EA: 102‘
Thgbggoduction Function in Pa; :3; 11m 2~3"'~fi;~

 

As has been
pointed out by Johnsonl, tho typical ex;oriment deoign
inroatigatos a production function or the form of equation (2)
(2) i 3 ft”,:, K / L4......2{n) § u
where Y 18 yield in buahels per aoro; N, ? ani E are the
uaoally investigated independent variable: or levolo of
fertilization; xé......xn include Variables "fixed” at a
cortain level (oultivations, insect control, rotations,

Ph Invola. varieties, drainago, soil uniformity, oto.).
U is the variation in yield not explainod by the

experimental variations in the indepandont variables.

T3 n or Kari 'atio .~

Produotion functions express
the funokional relationship between ronouroa inputs and
product output .

Th. contentional procedure in a production function

__ “4‘ .L A __
v—w ~—

lJohnaon, G.L.,!Planning Agronomic-Economic Research
in View of Roodlta to Iato' (in) Sana, E.L., l=¢dr, 3.0.,
Pesok, J.?., Hildroth 0.6., “bertiliLer =Innovations and

Rooouro , Tie 10% a State ouiiabo EToaa, Ar.es, glowa,
I§57, fihapfor 19,p pp. 219.

.A...

‘——,—~=' ‘— —r v v w—v

study is to predict the total output curve or surface as
an estimating (or regression) equation.

Maximization concepts help locate cuch.economic
optimum as the quantity or‘r to produce maximum profit
and the least cost combination of fertilizer F1 and F2
to use in producing that amount ofjproduct output Y, and

also how these Optima shift with price change..1
In a function such as equation (:5),

in‘whidh

3c 3 yield of crap

P0 3 price of crap

’1 3 fertilizer input

F2 3 fertilizer inpnt

P11 3' pricc ct fertiliser’input 31

‘Pm 3- price of fertilizer input 1'2

Xd...?..x, ‘ input. fixed at upcoificd conditions.
" 3 profit

when 921; constant

(4 an...
) m-c

A

W .7. ._ —.- fir.— v 7w ’7

130mg”, 3. L. , “Interdisciplinary consideration: in

‘ Designing Emerinentl to Study the Profitability of
Fertiliser Use“, (in) Baum, E. I... Heady, E. 0.,
Bllchflfl‘ JO. fl. a;t., Chaptflr 2' p. 27.

defines the most profitable amount of F1 to use with the

constant. amount 0! P2. Under ordinary campetitlva con-
ditione, the condition for maximizing profits is defined

as in equation (5)

(5) Mr : ch . rye-13:1 n O
3'5").

I: the most profitable combination of F1 and F2 in
producing: a given amount or 20 is desired, equation (6)
defines the least cost combination of F1 and F2 to use in
producingx the amount of Yc under consideration:

(5)
2.1.3
1”}.

(123
d? 2

:9
‘6‘5?
rt .

A: the condition defined in equation (5) ie, for F1
-and :3 respectively, the slap. or the;production function
defined in equation (5). these oonditionc permit deter-
mination of tho moat profitable (least-coat) combination of
F1 and F2 to use in obtaining a given yield (where‘ic in
hold constant).

When it is desired to determine the most profitable

amount: of F1 and F to use and at Ya to produce, the

2
derivatives for profit with respect to F1 and F2 are set

equal to zero, and solved simultaneously for F1 and FP‘
Having eecurcd r and F in this way, the values are

l 2
substituted in equation (3) and solved for‘Yc.

io
omega :5
STAT I 3T1 CM, M 3‘13. Siflfiifi i-‘J‘ETfi

Int rgduct 19g . -

when the values assumed by one variable (If)
dependcn (i.c., are a function of) the values taken by
one or more other‘variablee (X1,X3.....Xn) a functional
relationship is defined-

! is called the I'dcpendent variable' and the variables
on which‘! depends are referred to no “independent variables".

correlation analysis given e measure of how the
dependent variable changes with e given change in the
variable or variablee on which it depende.

This measure in an 'eetimating equation' which malice
possible estimates of the dependent variable from the
independent variable or variables.

Correlation.cnelyeie cleo providee a measure of the
eccurecy of Inch estimates - “the standard error of estimate“:
and finally, 'the coefficient of correlation“ tells the
degree of correlation.

When the enelyeie is limited to two variables the
method ie celled lsimple correlation“ but it is often
necessary to include the influence of several independent
veriablee to explain the variation in the dependent variable

and this is known on "multiple correlation“.

11

Gganhlc Annlyein of Functional Relationsh Ja.~

A method of
plotting the ante and a method peculiarly suited to the
ennlyeie of functional relationships is that or the scatter
diagram.

Here one variable ie scaled on the x axis, the eecond
on the‘! axis, and the paired values or the two variables
are plotted on these eoaiee.

Because of the characteristic of the data, the tendency
of the point: ie to eeetter diagonally across the diagram
from the lower left hand to the upper right.

When two variiblee are plotted in the scatter diagram,
one ordinarily should be characterized ne indepennent
(in the present study it ie fertilizer input) and the
other an dependent variable (being output or crop yield).

the independent variable is the one upon which.the
variation in the second scene to depend. The independent
variable is ordinarily'plotted on the}{exie and the flependent

variable on the‘! axia-

Th. . inn Benn o .*

gathematioally computed line! may
be penned.through.the date, which.ere celled I'2Linee of
average relationship“, or 'regreeeion lines", because they

reveal the typical change in the dependent variable'! which

has accompanlad a given changa 1n the independent variable
or variablaa.

This average relationship may he datermined math-
ematically by tha mathod of least squares.

In computing tha equation for an estimating equation,
an.‘a‘ value must be accurad'whioh'will be the value of the
regression lino at its origin, and a “b" value, which will
desorlbfi tha average changg ln‘Y with,a 31Ven change in x.

When thus: two values are obtainad, a complete
description or & regression line 13 secured, the mathematical
characteristle'cf which.may‘bo described in equatlan (7)
for functional relationships with.ona indapandent variable.

(’7) _{
Eu 3 a § bx

To obtain the ”a“ value, the follawlng equation 19
used:

(8) av? 1x32: -;_x gig
:3 1x5 - (xx)?

 

The farmula for “b“. nhowing ~ha average change in'Y
for n givnn change in x 1: given 1n (9)

(9)
b: a ZXYg-HZX g};

a 1x3 - (81:02
If two or'mora independent variables are used.to naplain
tho change: in a depandent variable. 1: becomes possible to
measuro the influence or each or these (X2 and X5, for
example) when the Influenos of the ether (x2 and X5) 13

considered.

13

ThQ:Standard_Errorflpf Estimgt§.~

Error must be expected in
all estimates made tram regression equations. If there are
variables which have been ignored in computing the regression
equation and.they are important, the estimates mafia may be
very poor and the actual observations may scatter widely
about the regroaslon lino.

Since the daparturo of the observations from the line
of regression is due to such "other” factors a3 have been
suggested, these deviations are knnun as ”rcaifluals".

Thay are rsaidualu 1n the sauna, than, that after X has
been used to QXplain the variation of Y there may remain
a residual variation whioh.1s due to a large number of
forces which has been ignorua in the correlation.

If, on the other hand, the raaiauals in simple
oorrolstion are not due to a mass or other influences,
but can be explained by the introduction of one or morn
sadltlonal inaspsndant variables. the analysis should bs
converted to a problem in multiple correlation.

The new function with several indepafidont variables
may providc s‘bsttar explanation of the variation in Y,
and one svidenoo of this will be a reductisn 1n the
residuals.

Thu next step in the analysis will be to measure the

14

residual variation. If it proves to be small, the forces
ignored in the function as stated have little influence on
I.

In oraer to gain some idea or the ndqquacy of the
rogroaalon equation as an eXpIanntion of tho vr‘iationo
In I, it 13 accessory to have a mathematical devioo'whloh
V111 measure the scatter of the points around the regression
lino. If the regression line 13 a good fit and the aotual
data plot dloso to it, thorn 19 indioation that the values
of I are related to those of x in the manner described
by the regression equation.

In thin noon, the derived.mathenat1oal measure or the
residual variatlon ohould give a low value.

Should the points scatter widoly from the regression
line. tho us. of that estimating equation as an explanation
of the variations in I must be questioned and any estimate
or‘! baaod.on its functional relationship to X must be
expootod to be Inaccurate.

In this tune. the mathematical measure of thezrooldnnlo
would have I rulativoly large value.

The neanuro of the scatter of the pout: 13 known on
the 'atandordforror of ootimate'. It In 13 used.to symbolize
tho computed Values of I, nnd.¥ to symbollzo the actual

value. of I. this pronoun of calculating tho standard crror

15

of estimate is as follows in equation (10), where By
Mixes the standard error of estimate:

10
‘ ’ ay=\l 11W"???

 

 

H
The interpretation of the ctnnainrd error of estimate
is eimilnr to that of the standard deviation. It: may be
said that approximatelr'fiafi of the points in the scatter
diagram will be within the range or the regression line

plus and minus one standard error.

nonsurongng cg Correlg}: on.-

f The estimating equation-reveals
the change in the dependent Variable which typically

 

accompanies a given elmngo in x.

The scatter of points around a regression line gives
a fir-t visual impression of the extent to which the
independent variable, or nrinblcc, actually cucceed in
explaining the Variation in the dependent variable, and
whether useful estimates of it can be made from those
relations. The standard error of estimate gives a measure
of this scatter.

How, it in desired to obtain one summary figure which
will indicate the 'extent' to which two or more variable:
are correlated.

This should be A pure number, so that the units in
which the values are quoted will. not affect it.

It should have known linitc no that it may be readily

16

interpreted. The “coefficient of correlation“ (symbolized
by 'r’ for simple correlation and by "a" for multiple

correlation) is such a measure.

Eiatiom}mvlained v The Coefwnt of Detenninfitiogow
'1lm
procedure in oorrelation analysis ie to nonpute the'per can!
of the variation of Y‘whieh is exnlained by the independent
variablei.
iroditionelly, thin result is obtained by first competing
the per cent variation of I'whdoh.ie not explained'oy the
independent variable or Striablee.
(a) The garietion of rhino Value of 6:12:
oinoe the variation
of I can be defined no the etendcrd deviation squared, the
calculation in mode substituting the Yaluee tor Y, 1‘3 and
H in the formula (11):

 

(11) o 2

-? _o 9:

 

Just no the variation of Y is the etandard deviation
squared, the variation of I not explained by x,ie the
etanderd error or eetimate squared (Byg). It the variation
of I were completely explained by X, all observations of I
would tell on the regression line and the value or By and
Eye would be zero.

17

The larger the value of 6:; end oyg the lean perfect the
exylanation of the variation o! I‘by the independent variable
or variables. and the more important in determining Y are
other factors, which were not included in the function.

Elvira‘fore, the larger {Bye the gnome:- the per cent of
”the variation of 2 not- oxylnined by 2!.

(12)

 

eye 8 2x3.- Egg) 4 Vb tug].
n

(c) We?ACr-zlnfi"ain‘thew’e’qriotion‘pf Y}? Jalainguuvv
The
variation not explained by X is divided by the variation of
I, that in, Byz/yg and by subtraction then from one resulte
the per cent of the variation of X emlained by the inde-
pendent variable or variables, which is one or the neat
useful resulte in correlation analysis, and is known ee
the 'ooetrioient of determination“, as crown in (1:5)

(13)
coefficient of denomination 3: 1 «- 3:72

"'8"
6‘1"
The per cent, Byalfl‘ya ie :cnmm no the “coefficient of
non detemination‘ because it represente the per cent of

the variation of if not explained. by the inciepondont variable.

W 00. ff 1m 01‘ Germaine-c

 

i-fhon the etamiard error of

18

estimate is zero there is perfect correlation. It tho
standard error of estimate is zero, actual ano estimated
values of‘! are identical.

That is, tho regression equation prOViaoa a perfect
fit to the actual values or I} and the variation 01'! is
completely exploinod by tho indeyandsnt variable and
complet aly fieynmflent u; on: it.

In that oaoe one value or Sy' 0 and the ratio
aye/Cr? will be zero imlioating zero per cent of the
variation explained by‘tho indop ndont Variable or variables.

I! the standard error is equal to, or nearly equal to
tho standard oeviation, there io no oorrolation. Unoor this
condition the value or the ratio on/C‘yg would be one,
or approximately one, tho yer cont of the variation
unexplainod.wou1d be near 100 per cont and.tho variation
explained would be near zero. In this case, the cooffioiont
or actormination will approximate zero.

Tho 'oooffioiont of correlation” is bogod on the
coefficient of determination, that 19, the par cont of

La variation explained by tho inaopondant variable. Thus,

when tho standard error or estimate-is zero an& the explained
variation is thoroforo 100 per cent, tho ooefficiont of
oorrolation has a value of one.

If tn. Variation Lunoiplainofi (SW3) 33100113. be £3 largo

19

shoulLi be as large as too total variation 5'3“? than the
unexplained variation is 100 per cent, the oxylained
variation is zero, and the coefficient of correlation is
also zero.

The ooofrioient of correlation is a pure number not
influenced.by the unito in Ufilafl tho data are quoted and
it is computed as in oiuation (1%)

(14) ~

 

W
6'!“

when.more than one independent Variable is used to
oXplain the variation of the dependent variable, ”R“ in
need to eymbolizo the ooefrioiont of multiple correlation,

being conceptually the some as the oootfioiont of correlation.

1‘2th of: @331: tion 3.0311211: gu-

A

H5

Simply beoauoo a high value
or P. is obtained in a correlation armiysio. it cannot be
assumed that a valid, reliable, oni suitable funotion h&8
been oetobiiexoo.

There are acme further tests that sheila be node to
establish the reliability of the roanlto from correlation
analysis.

(a) Validity Tog§.v

The volifiity toot consists of a critical

appraisal of rooulto to assure tho reoeorohor‘that the
relations aaoumed in the function are corrootg‘that the
rolntionohipo do not violate :oanona of reasonableness:
that tho ohoervod results are consistent among thomselvos
and that the terms included actually reflect the Variables
they aro intondofl to rnyrooonto

The voliaity toot rests partly on theory, partly on
ozporimontution w to otuor alternati?o (auctions, and partly
on the comparison 01‘ the attained result. with those aohiond
in other similar studies.

his toot in important booauoo whom the relations

revealed or. not valid, in tho oonoo tho torn is used horo.
tho: likely will not be stool. ovor a poriod or timo'fiithor,
no vary poor estimates may result. Fhroharmoro, wrong
unavora may be suggested for analytical and Operational
problems and easy satisfaction with.high 8': values may
discourage further roooaroh.whionuvould product muoh.b¢$t¢r

re B‘dlt 3.

(b) _Boiiabiiiix;§gg§v-

Another reason to take a aocond
critical look at correlation rooulta is that they may look
roliabiiity.

Tho most obvious case is when random aam;laa are used.

If nany ouch.aamp1es were taken from the some pepulation,
can can be sure that the regression coefficient. for
example, woulfl differ from sample to sample: that a sampling
distribution could be constructed and tho standarfl error

of the regression coefficients estimated.

or course, 5y (tho standard error of actinato) and 6‘?
must also b9 expected to differ somewhat, from sample to
sample, so thot “R“ which.éepondc on than, will also have
a nonpling distribution and ito own standard error.

Tonto or significance or the rngraacion coefficient
can be made under a null hypothesis. The hypothesis in
that tho pepuiation voice of the regression coefficients
are zoro and that the cczimatod value is one to sampling
error or other chance «lemontc in the cchrimcnt.

is Tintnerl points out, the logic of tho teats or
cigaitioanoo consists in determining what is the probability
that certain deviations from a postuintod iypothocia (called
the null hypothosic) could have orioon by chanoc. If this
pmifrability 1:: email, then, the chinoao are that the null
hypothesis does not hold.

What is necdod to test this hypothcoio is a critical

ratio subject to probability distribution. For thio test,

A“ A Aw _._ iii _ A. A A m

‘ ‘ .T —V —v "'-

1Tintnor, G.,'Significanccriccta in Production Function
Roooarohfi, (in) Hoody, Er0., Johnson, G.L., Hardin, L.8.,
£0 21 0. Chapter 14. pp. 1380

g:
i

the standard errcr of the sampling distribution or the
regroaaion coeffieiants is as follows:

(115)

we
regression ooaffioien : _A *

.L

(T'3 ~x~P
x (1 d)

The Gritiaal ratio is tha diffwrenee bfituusn the
hypothetical regressiua ifisfilcient zero and the unnerved
vain. of regraaaion coefficient over the standard error of
tha rsgronaion coefficient as ahawn below in (15)

(is)

t t .ggyreggiqn cqetficient 7 #
Standard arrar of regreasion cceffioiani

{‘3
08

C EN’T HR 4

ILYJ’JJ. ffflTAi. HUME M43) :30 ”123-105 (:2? DATA

who experimental work to actornina fertilizer input-
crop output relationships was conducted coOperotively by
tho Colombia Project of Michigan state University, the
Facultcd dc.A3rcnomia.ot the Universidod fiaoional dc
Colombia at Palmira (Colombia) and a Colombian former,
Honor Edgardo Patino.

Doctor Leonard Kyle and Hr. Gerald Trant from the
Agricultural Economic Department; 3r. Kirk Lawton from the
Soil Department, at Michigan ototc University, and members
of the staff from the Michigan Agricultural Experiment
Station, participated in the design of tho cchrincnt and

colloction cf the experimental data.

Th D‘-.t' .-
Thc corn experiments were developed at
SencrtPatino‘c farm near Florida (Colombia) on a well
drained clay loam coil.
The Ochrimcntc incluaod all three of tho primary plant
nutricnts, nitrogen, phosphoric acid and potash, the first
two in varying combinations and the last generally constant.

The corn experiments were conducted in two fields which

24

will be called Patino Loner Field and Potino Upper Field
respectively.

Both experiments had the some design and in both the
some trentmcnt levels of fertilizer were used.

Eight treatment lcvclc were included for nitrogen,
ccvcn treatment levels for phcaphoric acid, with,potach
gonorcliy conctnnt.

Those trcatmcnt levels measured in pounds per core are:

H — O 2 40 60 80 100 180 140

P - O :3 4.0 60 BO 100 120

Except for the zero treatment lcvclc, potash was hold
constant at 60 pounds per acre.

The levels of fertilization for corn arc shown in detail

in Table i, and the design of the experiment in Table 2.

Thc_8ccn_Datn.-

The objective of this exocrimcnt conducted
also in Potino'c farm rrcn.00tch P loo? to January 1958, was
to evaluate the rccponcc of beans to different combinations
of fertilizer inputs.

Five treatment levels were included for nitrogen, five
treatment levels forjphoopnoric ncid, with.potosh generally
constant.

Tomcc treatment levels measured in poundc per core arc:

n - O 2 40 60 80

P ~ 0 £6 50 75 100

Except for the zero trontmont levels, potash was held

constant at 40 pounds per core.

TABLE 1

szst c? FamTILIaATIOH Pea THE away :xrgmlxxnws, PATINO
1.0mm AND “mm mama mum, 1957.

.__ .4‘ 4— A. __. A L ~__._.‘_-

v' .— fil wfi— v. w“

‘v v—V v W i"

Plant Rutrienta Plant fiutrianta

 

(Pounds per Acre) (Pounds per Acre)
N P206 K20 H P205 1.20
O O O 60 60 60
O 0 ' 60 60 100 60
O 20 60 60 1:20 60
0 4O 60
o 60 60 80 o 60
O 80 60 80 2:0 60
o 100 so so 40 so
0 1,, 60 80 80 60

80 120 60
20 O 60 -
20 20 60 100 0 60
20 4O 60 100 2’0 60
20 60 60 100 60 60
2.0 80 60 100 100 60
2.0 100 60 100 3... 60
20 120 60
120 O 60
40 0 60 120 40 60
40 20 60, 1:30 80 60
40 4O 60 1% 15-30 60
40 60 60
40 BO 60 140 O 60
40 120 60 140 20 60
140 40 60
60 O 60 140 60 60
60 20 60 ‘ 140 80 60
60 4O 60 140 100 60

140 120 so

T 2"; Plug 22

Ewmuzm-z’z‘m. DESIGN 02" mix; can zm"1;;‘21x:2-3<T3, P1831330 1.1mm

 

 

Arm m‘rmo urn-521 man, 1957. '
Pounds of Pounds of Pounds of nitrogen
P 0 par K 0 per . per Aura
m Ann 0 20 4.0 so 30 mo 1:20 140
o o
0 60 x x _ x. x x x x x
20 60 x x x x x x x
40 60 x x x x x x x
60 60 x x x x x x
80 60 x x x x x x
100 an x x x x x
1520 60 x x x x x x x x

‘ Each '10 represent: an experimuntal plot.

27

The fertilizer oarriars used Hero, nitrogen as ammonium
sulfate (BO-0‘0); phosphorous no concentrate superphOSphnte
(0-46-0): and potash as potassium chloride.-

lndiviflual plots were two rows by 40 feet in size,
with.tho rows 24 inches apart. Thoro were two replicates
of each treatment anfl the total area covered was approxi-
mately 0.3 sore or 62 x 160 foot.

Tho fertilizer-was to be applied an a bonded side“
dressing to the side and below the seedling rows in furrows,
whan the beans were 2 to 3 inchea tall. The Deana waro
planted approximately Ootdbor 213% and harvested January 14,
1958.

The levels of fertilization for the bean experiment
are proaonted in detail in Table 3 and the design of the

bean experizaent in Table 4.

MES

Lzms or FEfi‘iTILIZA’I‘ION FOP. mo: 372m afo-nnxzao-fiacr, “Two 1957/58.

Plant Hutrionta

(Pounds per Aura)

N r205 x20 :4 P305 1:20
o o o 40 o o
o 40 40 o 40
o 26 o 40 :35 4o
0 25 4:0 40 50 40
o oo 40 40 75 4o
0 75 4:0 4:0 100 40
o 100 40
210 o 0 60 o o
20 o 40 so 0 40
20 :36 4o 60 :25 4o
20 50 4O 60 50 40
20 76 am so 75 40
20 100 40 80 100 40

1-23? m 2.13sz 1353:3101": FOR THE: mow 3:921:212-43-31', PM? 13:0 1957/58. '

A

Pounds of

P205 3101'

I"

I:

76

100

A ._ A”
fl.

TABLE 4

Pound. of
K20 par
nor! .0
O X
40 X
0 x
40 x
40 X
40 X
60 X

L ._AA

Younda of Nitrogen

per Acre
20 4:0 60 80
X X X
X X X X
X X X
X X X
X X X
X X X

fi‘

“ anh.'X' represents an oxporimontol plot.

mum-rm s
Dmxvmmn or mec'i‘xon museum-m mom new 1mm

After collection of yield observations, the next stop
in the analysis 19 that of estimating production functions,

inpuonoutput or response coefficients.

§:§:01UO3;O!! Fig 951033 21$:{3QTt0w—thLDth.“
The production
functional fitted to the data included:

(a) a square root equation of the form of equation (17.)
(17) Yo=aow§np§ar§¢ 9J5§gffi
(b) a cross product production function or the form of
oquatlon (18)
(18) yo = no bu o o? o c1243 6 3123 & mm

(o) a mWDouglaa production function of the form of
equation (19)

a: .b1 b2 b3

- ;t1“'==;1..1*310 -~

 

Thin function has been

A. - _. :.‘. ..‘ 4 ._ A ..
—r *‘I—w . v , v— v. ‘_ w v.

‘rm antiwar is indebted both to Bernard Hotfnar of tho
Agricultural Economics Department and to the personnel or
the Statistical pool of tho same Department, at Michigan
State University, for fitting the alternative production
functions.

31

ulod by Ebodyl in fitting yield data.for corn on calcareous
Ida allt loam I011 In western Iowa in 1952, and the results
proved to‘bo oatlafaotory witthI per cent or the yield
various. explained by thfi varying quantities of the two
nutrients used. nitrogen and.phoaphorio acid.

This tunotion was oolaoted as being the most efficient
for prodioting tho production surface, yield lacquants, and
marginal quantitlos for corn.

Heady, working on red clover2 and alfalfaa on Wobato!
and Hloollot loan in north'oontral Iowa, BXplainod.64 par
cent and 77 par cont respectivoly or the various. preoont
1n the data using the square root produatlon function as
tho predicting equation.

4

Inotaoh working on corn oxperlmonto on a Kalamazoo

sandy loam .011 during 1955 in Kalamazoo and Calhoun
counties and using all three of tho nutrients :4, P3 O5 and
:20, fitted this function to the data. It proved to fit

.— ; _LL 1.. A .__._ L ... 4.; J—h-

w .— V v v—wvvv’ vT—Y w—w—

130a”. EoOo. Poaok, 110%., Brown 3.0., “Crap 33890313.
Surfaces! and :zoonomm Optima 1n Fartihzer Use“, Amioultu
Experimqgt Bta- '; ’ R ear05_3u;;e ‘ ,

H

fimign. pl 312.
3:51:10. I). 31".

‘Knetsoh J.L., 'fiethodologloal Proooduroe and.Applloattonl
for Incorporaéing Economic Considerations into Fertiliser
Recommendations“ Unpublished Master of Balance, Thoala, Dept.
of Agricultural iconomioo, %1ohigan State University, pg 47.

 
 
 

the datasnequately for low applications of nitrogon but
failed to fit at high levels, 41 per oont of tho varianoo
being explalnod.by thin equation.

Ehio function in used by none roscnrohorol

to fit yield
data when extremely largo marginal profluotn over small inputs
are followed by n long range of small and fairly constant
marginal produota. 1.o., in onocs where n atoep oorvo is
expootod at tho autaot. followed by a Slot in tho middle.
The square root equation allows intorootion of tho
nutrienta 1n the production proooaa and also allows the
inputs:
(a) to be outstitutoo only for amnll tnputs.
(b) to be both substitutes and oomplononta for higher
lovolo or output, and

(o) to bo only oompllnnnts at maximum yield lovolsog

(b) _groog:?roonot_Pnpinotiqn Funotion.-
This produotlon
runotion has been taod 1n grovioua atudioo with.?nrinblo

results.

Heady fitted a aroma product equation to corn data

from calcareous Ida silt loam 3011 in western Iowa from

__A A
) A“... .4 A A4. A h M
‘1 P.— V —v ‘v _ W V.—

lfionay, 3.0., Johnoon, 6.1», Eart‘un, L.3., 33. on...
Chapter 1, pp. }O~12.

3mm. , pp. 10-12.
afiendy, 3.0., Poaok, J.T., Brown, w.0.,‘gg. olt., p. 304.

33

.xpdriucnts conductod in 1952, and it was considered a caconc
boot fitting uith.83 per cent or the variance in yield ex~
plainod.h7‘vcrinblo quantities of the two nutrientn nitrogen
and. phosphoric acid.

Hood: rittoc this function to red clover1 yield. data
.f!0l;oxpcr1mcntc conducted.on wcbater and Nicollet loan in
northroontrnl Iowa in 1952, and it was conoidorcd c nooond
best fitting with.68 per cont of tho yield variance explained.

Ready, working on alfalfnz‘fittcd this function to
31011 data from Webster and.nicollct loan in northpoentrnl
Iowa in 1953 and, no in the previous cacao it was considered
tho second best fitting equation with.66 per cent of the
varicnoo explained.

tnotach.rittod tho cross product production function
to corn8 act; from Kalamazoo and Calhoun counties for
cxporinonts conductcd on a Kalamaxco candy loan coil during
1955. It gave a poor fit as indicated.by only 29 per cont
or the variance oxplninorl by the fitting aquation.

Sundquiat, working on oat4 yield data from exocrinontc

‘- . , _
A

_v_ w. , v— ' ——-~~-- -— ..

1£b1go‘ Do 5120

2131c. , p. 31?.
snags-oh, J.L.,lgg. cit., p. 47.

48md<wict v.3... ‘An Economic Analysis of Some Contmllod
Fertilizer Inpu -0utput a‘rpcrimonto in Michigan", Unpublished
Ph.D.'Thocic Dcpt. of Agricultural Economics, Michigan Stat.
University, i957, p. 51.

conducted in Kalamazoo an; Calhoun GOufltiEE on a Kalamazoo
enndy 10am coil flaring 1355, explained only 48 per cent of
the variance in yicla using the cross product production
function.

Bundquict, working on wheat; yield data from experiments
conductea in Kalamazoo an; Galhoun countica on n Enlumazoo
sandy loam 6011 during 1955. explninoa only 4% per cent of
the variance prosont in tho data using his function.

sundquiot, working on corn2 data from experiments
ccnfiuctcd in Kalamazoo and Calhoun counties on.a fialamazoo
sandy loam soil during 1955, fitted this equation: it provofl
to be highly unsatisfactory with only 3 per cent of tho

ariancc eXpIaincd‘by the cross product production function.

Sundquiat fittcé this function to continuous corna
yield data from Tuscola county on a Wiener clay loam soil
for cxporimonta conducted during 1666 and it rcculted in
only 13 per cent of tho variance cxPlaincd. .

SunQQuict, working on beans4 fitted this equation to
act: from Gratict county for experiments conducted during
1955 on a 31mm: loam coil; 42 per cent or the varianco

precont in the data was explaincd.
Sundquist, workingonpotatoa data titted this equation
1333., p. as.
9%., p. '73.
qlgigfb p. 80.
4m” p. as.
5M” p. 92.

WWW

35

for oxparimontal data from a Houghion muck soil on the
Experiment station much farm of the uichigan Agricultural
Experiment Station near East Lansing; tho rogrcasion equation
was capable of orplnining only 25 per cont of tho variance
present in tho onto.

Tho cross product production function allowo interaction
or the nutrients in tho proiuction process and being poly»
nominl with first and :ooond degree torms can show both
dininioning marginal and diminishing total yields.

Tho cross product production function has greater
flexibility than the Cobbunouglno function nni also allows,
like the aquoro root production function, tho inputs:

(1) to bc substitutes only for small inputs

(b) to bo‘both substitutes and oomplcnonts for higher
loyal. of output, and

(o) to be only oowoleaonts at maximum yields.

Fur marginal producto of 'nofiiun' magnitudo for small
resource inputs, followe&.hy on oarly maximum in total

product, it in desirable to try this c1untion.

(a) .gggpnnougloa Proigggionmtunotion.~

This function aasunoo
tout the peroontago inoroaso in yield is constant and equal
to 'b' for all inn-wants of fertilizer.

This function allows the yield to increase at either
a diminishing, constant, or increasing rate, although the
response curve can be represented by only onc of those and
never by a combination.

Therefore, if noro than one stage of production seems
to be prcocnt in tho data, a Cobb~Douglac function is not
adequate, sinoo it can approximate only onc otngo at a time.

Another disadvantage of thin function is that it takes
on a value of zero wholovor any input is zero. This nio-
advnntago has been solved in this analysis by' he odaition
of ans-tonth; of a unit to all zero fertilizer levels.
This introduces an upward bias in tho predicted yield but
overcomes the problem of having Yo 8 0 when any of the
treatments is zero.

0n tho other hand, the Cobb-Douglas function is easy
to fit and worn with. However. in View of the asymptotic
nature of tho function, considerable care must be used
whcn drawing inferences from the extreme ranges of the
data.3

This function has been used in fitting yield data
in previous rcaenrch.wcrk. Trout fittod this function to

to corn3 yield data in the Canon Valley (Colombia). using

__..-n._i___ A _L..i_.l

1that in, 2 pounds for nitrogcn; 2.5 pounds for
phosphorous; and 4 pounds for potash.

gnu eXplanation of the computation of the predicted
yields, high profit point, and marginal proiuotivitieo
using the Cobb—Douglno function is presented in Appeniix A.

afrant, 0.1., ”Implication! of Calculated Economic
Optima in the Canon Valley, Colombia, a.n.~, Journal‘of

 

 

nitrogen and phosphoroue and potassium combined in e 131
propaltion and obtained be per cent of the variance axe
plained in the first experiment; 60 per cent in a first
replication of the name eXperinent; and 45 per cent in e
second replication. In a second experiment on oorn, using
phoepnoroue and potaeeiun no the two independent variables,
no statistical evidence of correlation between the
independent variables and the dependent variable was
found; the coefficient of nultiple determination had a value
of zero.

front, working on auger cone; in the Canon Valley
(Colombia), and using nitrogen as one independent variable
and phosphorous and potassium combined in a 181 preportion
as the other independent variable fitted the Cobb-Douglas
function to date on noel: planted cane. He found no
statistical evidence of correlation between the independent
and dependent variable“ the eoeffioimt of multiple
determination had a value or zero. In fourth cutting none,
as a eecond experiment, the Cobb-Douglas function was
capchle of eXplnining only 19 per cent of the Variance
present in the data.

Knetech fitted this function to corn2 yield data from

M-

21111633011, JeLe' or). 0 to. p. 4.80

unfit

58

coxpcrimcntc conducted during 1955 in Kalamazoo and Calhoun
counties on a Kalamazoo candy loam soil but it was not
considered the boot roprcoentation of the data and it was
chcltcd because of undesirable characteristics when the
function and the data were plotted in natural numbers, although
61 per cent or the Variance was explained by this production
function.

Choocigg_tho "Bost' fitting Function.-
Thc coleoticn of tho

 

boot functional torn in a problcm of considerable importance.
A: Moon’- states, tho problem of choosing the ”best"
function is not colonic from a single set or rules.

2

Evidencc presented.by Macon seems to giro no indica-

tion of preference or one function over the other from c
strict ctatiatioal point of View. For using as a {prediction
equation“ on; would coon to work as well as another and tho
criteria tor choosing among thou would rest on slopiicity
or couputation considerations.

By the usc of lcast squares procedure: the valuc of
the constants for the equation may be computed. The“
procedural giro the best {it for the partiouiar Iona of

w —‘ v r—vr m

1Macon, David 9., 'Functional Manolo and Echrimcntal
Designs for Characterizing Response Curvcc and Burtoccc'.
(in) Baum, Eel-u. 5686.3. 3000‘ BlflOker. J0. amt.
ChflptBI‘ 5. p. 80.

zfiaaon, David 0., “Statistical Problems of Joint
Hocoarchfi,JggurnaifloguForm Econonigg, Vol. 39, Hay 1957, p.376.

&9

functional moficl, in the canoe of describing a curve from
Ihich.thc mean of tho squares of the deviations of tho
indiViduol point! from that curve is a minimum.

flochcr, the estimating equation is uficd as a.
“production function” to catiootc Optimum rates or level:
or tortilizaticn under various price relationohipa, which
is the ultimate and in estimating these functional relation-
ships, and u it is shown by Manor} considerably more
variation exists among tho cctinatc optima for the various
functions than'bctwoan the estimated values or yield.

It cannot be claimed that any of the functions represent
fundamental biological laws of growth, although.onc may
rationalize tho form or a particular function in a particular
situation.

It in likely that thc‘bost fitting function or the
fertilizcr'prcduction function varioc'by crop, year, soil,
or other variabloc.2

Direct statistical tests (analysis of variance quan-
titicc and F tests) ore available for determining whothcr
a significant reduction in varianoc ic Obtained by including

one norc or loco terms in equations auchbcc the cross product

._ A _ . A . M

,f

111313. , p. 376.

. afieady, 3.0., 'Mothodologicnl Problems in Fertiliser
"36'. (in) Baum, Rollo, Ready, E000. Blnokmorfi, Jo. 2-2- 01“.
Chaptal' 1. pp. 60

40

or oqucrc root functions.

Howarcr, cc Hoodyl points out, "airoct" tootc arc not
available for choosing between ouch.wiaoly oiffcront functions
as tho Cobb-Douglas, Epillnan, cross product or square root
equations.

"Testing tho significancc of coefficients for indiviaual
variable: in an equation which.containc more than one vuriablc
for c givcn plant nutriunt in n procticc of limited useful-
noon.

the related varidbicc in an equation such,ac H, N2,

P, P3, ctc.. arc obviously highly ccrrclatcdi Estimates
of individual parameters may be subjected to large
standard crrorc reflecting those high.intcrccrrolationc
and it might bc concluacd that since individual paramotcrc
arc not ctaticticnlly significant, no significant effects
arc prcocnt.

This line or analysis may‘bc niclccding. If tho
aggrcgatc ctrcct cf'nll variables could to tested for
significance, the toot might indicate a significant
aggregate ctrcct for the nutrient.

One way to approach.tho problem or selecting the

.._. . . A -5. ‘_ _‘ ‘_. A“

,fi V“ f ‘.—_.. ,w

1

Ready, E.P.. "Technical Considerations in Estimating
Production Functions“, (in) Handy, E.0., Johnson, 0.L..
mrdln. L030. w. 21“. Chapter 1' p. 13.

boot fanation has boon ouggootod by Johnaon‘.

It consists in testing the degree to wh.on tho altorna~
tivo functions individually meet the uguol aosumptions with
roapoot to the distribution 0: unoXplainoi residuals on
which objective statistical tests are based, and rejooting
:11 functions that fail to pass this toot.

Ir n 0120100 botwoon than runotions whioh do moot
tho assumptions is necessary, objootivo tests are used'as
a second atop in tho selection of the boat function.

I! ouoh'toatn also fail to reveal statistically
dirroronoon between the alternative functions, loan
objectivo criteria, cocoons tho oxporimontor'n Judgment

and oxpcrt Opinion could be applied, but only it noooeoory.

oi , Da .-
V Tao results from fitting the above

equations to the yiold data are presented in this section.

Tho basic ototistioa relating to the functions fitted
am given in Table 8 for Patina Lower Field oom data;
in.Tablo 9 for'Potino Upper Field corn data. and in
Table 13 for Patino boon data.

. Tho meaning and significance or these stitiatios are

L... ___._ ‘__ ._. A

1Johnson, Glenn L., ”Discussion: Eoonomio Implications

of Agricultural FXporimonto',.ggqgngigtwggrm qunomioo,
Volume 39. Hay 1957, pp. 395”596Q

4___ MM “
.7 ‘_ . 7—7

oxylainod in detail in Chapter 5.
finelyeig_of the Patinoflower Field Corn‘finte»1957.-

(a) ignore Root; Enod‘uoticngiftinotvlrvww
The first formulation
of the functional relationship which was attempted for the
date was a square root function or tno form of equation
(17) above.
This formulation oontalning the eetimated parameters
is shown in equation (20). Values lietoa.below the
estimated.peramotero and included in parentheses are
stenaerd errors of‘the respectivo parameters. fl and.?
represent per acre applicationa or nitrogen and phoe-
phorio acid respectively and yield in maneured in bushels
per aore.
(20) Yo 3 42.754948 4 .095'72'7036 N .. .376305107 P -
(.064078601) (.073615241)
- .313685120 (:7 o 5.095714910 W - .oeso'zelee W
(.933415823) (.980684688) (.066703825)
Tho coefficient of multiple oorrelotion for this
equation was .656. The ooerfioient of multiple determination
indicated that 45 per oent or the variance present in the

yield data wee explained by this regression equation.

(b) _groeaArroduot Proiuction_§unotion.-

Because of the large
amount of variance not associated with the estimating
equation, a second formulation of the functional relation-
ship was attempted of the type or equation (18) above.

The equation with the estimataa.parametera is shown
in equation {21):
(21)

20 = 42.925175 e .065559729 N t .846996724 p e .000102522 a9 -
(.063513311) (.075582381) (.030422563)

.. .06405752569 9‘3 - .0004’7068339 up
(.000559148) (.00041177279)

The coefficient of multiple correlation for this equation
was .747. The coefficient of multiple determination indicated
that about 56 per cent or the variance in the yield data was
aseoeiated‘vith the regression equation.

Based on the statistical measurements derived, the
arose product equation.van considered a more apprOpriato
formulation than the equare root equation to represent the
functional relationship between varied quantities of input
nutrients and orOp yield output.

Eggdggtggn Euggage gatimat§§.-

One set or production surface
esthmatee is presented in Table 6. These quantities ehowing

the total per acre yield of corn for various rates of H and

TABLE 5

VALUES 9? 'R' AND “R2” FOR T‘s-f0 VARIAE‘SME §€U"i"RI?IéiT:'3 AND VALUES
0? 't‘ F011 12-1 DIVIDUAL RESRISSSIQE? (302.??I CIE-‘li‘i'fé-S, FOR COIN PATIRO

'“tER FIELD 1957'

Equation Value Valuer) Value of “t“ for
of “FL" of “Rh“ coefficients in order
listed in equation.

5— A- ._ 4 L‘— - A_.____-

y.— f w fl —‘ W '

square .656 .430 N a 1.49
r°°‘ P s b.ll
m . 034

{5'3 5.20

VEF : .65

‘Crcee .747 .558 H’: .96
5’°d“°t P s 7.24

fig! .24
P3: 7.27
NP . 1.14

45

P205 are the counterpart of a production surfece; they
represent specific points on the surface for specific
points in the nutrient input plane.

While not all or the treatments in every cell in
Table 3 were included in the experiment , their yiolz’le can
now be predicted. Each column in Table 6 ie the counterpart
of a vertical slice through the production surface parallel
with the 9205 axis and yields oorreepond to point: on a
single-Variable input output our". F205 is the Variablo
while nitrogen is fixed in the amount shown in the tap at
the column. '

The rows represent the some thing with nitrogen
variable am} P205 fixed. It is obvious that interaction
exists, the productivity of any input of one nutrient
depends on the quantity or the other with which it in
combined.

Figures La and 1.?) illustrate the effect of one
nutrient on the pronuotivity of the other. They are
response curves for one variable nutrient with the other
one fixed in the quantity indicated. 11: can be seen from
these figures tint: nitrogen exert: the predominant effect
on yields.

In Figure 1.9. with P205 fixed and nitrogen variable.

IdsuELs [Aug

FIGURE 1

CORN PATINO LOWER FIELD 1957

 

 

      

 

 

 

7: (A. 0‘ (B)
O O
. A
. l
0 w
95
“r o <
O
o . 3,
0 an
O
SOL-v". .
I
0;- M' 0 us .
' $5M” .... 5:: 2; q. :22:
Mr M’ m as a! or mm
a 40 so no I60 «)0 I20 too '

”Pb-Jugs (\p 92.09

900”” OF- \“TIO‘EU
YEW-N ”vi-KS:

V 1-; K #CKE

SKA“ Ll“: WV?“ N
“’9 505‘ N H I title
(‘4

 

 

 

 

 

70
m
1
gm
\
Q
J
u
3
Q.
'3.
a
50
I» Q inuioiumg
oesemnfions
40. I .
5 4‘5 to no (60

“WW5 0" u. Mus no:
".9. ACRE.

 

4?

increasing yields for nitrogen applications are obtained,
being the yields higher and higher until P205 is fixed at
the 60 pounds per acre level (not shown in Flame). If
P205 is fixed at a higher than 60 pounds per acre level,
increasing yields: for nitrogen are obtained, but the
yields are lower than those obtained with n 60 pounds per
nor. level. In other words, too much P305 is being combined
with nitrogen and diminiehing returns for P20; are present.
In Figure l.'b with N fixed and 9205 Variable, it can
be soon that when nitrogen in being fixed at higher and
higher levela, higher yieldl are encountered. until P205,
the nutrimt variable in this case recon the 60 pounds per
acre level. Afterwards. diminishing returns and diminishing
total yields are present.

Hgogomig Ogtimgw
As was .ohown in detail in Chapter 2. the

purpose of deriving functional relationships between
fertilizer nutrient input and crap yield output was to
provide the beais for making more efficient recommendations
in the coononio use of fertilizer specifying:

(a) the combination of nutrients to give least cost for the
particular yield, and.

(b) the amount of the nutrients to apply to maximize profits.

43

‘E'ABLE 8

'1’?th CAMUIATED 0:31:64 31:11.33 FFK'JEQI 31933633131) RATES OF APPLICATION
C? EITROGE'M A339 3?Ii(}.§"§if}?xIC ACID, PHEDIC‘i‘Z-EZD F3061 Tiifiél 13305.23 PRODUCT

EQUATIGR FOR GDP“ PAI‘I§*6O LUMZR ir‘IrLD 1957.

__ _A__ _ __

,r ,w .T‘, W --v- ‘r— wwq

M
M “M ‘Ah‘ A

v, V 7y..— . , F— w 7‘ v. —-——~ W

 

 

Pounds of Pounds of Nitrogen
r905 per per.Aore
Acre _ t : M f , f.“ 4 :A W i Y
O 20 40 60 60 100 120 140
0 42.9 44.2 46.6 47.1 46.7 60.3 52.0 65.6
20 52.03 5304 54.6 55.9 57.9. 58.7 50.3 61.8
40 58.3 69.2 60.3 61.6 62.6 63.8 65.1 66.6
60 61.1 61.8 62.7 63.6 64.8 66.7 66.8 66.1
80 60.? 61.2 61.9 62.6 63.4 64.3 66.2 66.3
100 66.9 67.3 67.8 68.5 68.9 69.6. 60.4 61.3
130 60.0 60.2 60.4~ 50.8 61.2 51.7 68.6 63.0

v. “ fiw—v—V i w . v.7 . ~— _—.._

* Yields are given in bushels por*eore¢

TABLE '7

TOTALiPREDIGTED.AND OBSERVED'IIELDS FOR CORN PATINO LOWER.FIELD,
1957.

 

 

 

Treatment Prod: cted Observed
(Pounds per Acre) Yiola Yield.
__ W , A M. {fininorgz (Sm/norgl

H P205 {1’20
0 o ~o 42.9 47.1
0 20 60 62.?! 44.9
0 40 60 66.5 65.4
0 60 60 6193. 599‘}.
0 80 60 60.? 2.9
0 106 60 66.9 65.6
0 120 60 50.0 43.5
20 O 60 44.8 42.
20 20 60 53.6 66.2
20 40 60 69.2 66.6
30 60 6O $1.6 65.6
20 80 60 61.2 69.5
20 100 6‘0 57.3 4.3
20 120 60 50.2 50.?
40 0 60 45.6 48.1
40 2O 60 54.6 4?.8
40 40 60 60.3 63.2
40 60 60 62.7 , 64.2
4:0 80 60 61.9 59.6
40 100 60 57.8 m
40 120 60 50-4 49.3
60 0 60 47.1 46.3
60 20 60 56.9 68.7
60 40 60 61.4 72.3
60 60 60 6.5.6 6.1.6
60 100 60 68.3 61.1
60 120 60 60.6 63.8

TABLE

80

80
80

Eh
us:

80
80

100
100

100
100
100

120

120
120
120
190
130

140
140
140
140
140
140
140

7 ' mntmuedo

ruw
330 8833380

60

8

100
1r_:.

20
60

100
120

60
80

190

68.?
57.3
63.6
6 ‘73 o 3
63.4
{3:3 o 9
51.2

50.3
it")
w 0

63.8
65.7
64.3
69.6
61.?

52.0
60.3
65.1
635.8
65.2
$3004
52.3

53.8
51.8
63.6
68.1
66.3
81.3
63-0

47.2
6Q.8
59.2

54.5

--"

67.9

49.7
55.4

73.?

55.8
69.9

43.1

’m

72.0
58.6

“m

55.9

65.1
64.8
83.8
65.6
67.0
59.9
61.0

These Optima nrs attained when the partial derivatives
(the marginal product) tor‘both nutrianta (in this oaso N
and.P) are equal to tho nutrient/crop price ratio.

The marginal produotlvltiea or nitrogen and.phoephorlo
aold are presented In Table 8, with the marginal productn
lvity for nitrogen at the tap at each pair and the marginal
preduotlvlty of phosphoric aola at the bottom.

Then. marginal productivitloa are computed from
equation (22) for nitrogen and from equation (23) for
phosphoric acid.

($32 $1? 2 .0655697739 0 BhOflOlOffififimfi 4- .47068339CIE’

(u .

(25) .gg : .543996734 - 2(.004067526900)? - .000470583590N
a:

In computing the Optimum amount of fertilizer nutrient
input to use. the following prlooa were used:
Corn I ,$1.45/bu.
N n 8 .16/1b.
P : 3 .1o/lb.
To find the amount or nutrionta to apply to maximlzo

profits, equations (22) and (35) are solved aimulfianoouslfl

“A w 4‘ u .. A _ “

v—w ,— fi —-r

lrho aquatlona solved were as follows:

(3) F0; throgen:
81 ”1 Q1
(1)) For ngpmroug:

~2(.004oo7526900)n - .030470683390P : .069 ~ .546996724
a. 0
2 2

52
using equations (34) and (£5);

(:24) (01 b2) ~ (03 ‘01)

N a: .7 W-
(at; b2) " (b1 as)

 

(2.5)

 

{a1 32%;.“ (‘2 31!
(31 b2) "' (b1 :2)

The nutrient/crap price ratio for nitrogen was found
to be .1103 and the nutrient/ore}: prion ratio for pmaphorio
acid was found to be .059. Using the above prion ratios,
the optimma amount of nitmgon to use was found to be
$20.34 gonna: per non, which is an extrapolation well
beyond the ram” of recorded oxparinental observation:
which down not indicate, by any moans, an ooonomio Optimum
point from which actual fertilizer recommendations can be
no.6...

However, this ofimaoteriotio indicates that further
orperinontnl work is needed with higher fertilizer levels
than thou studied in this particular experiment and for
which no information in aflllablo.

who Optimum. amount of phosphoric acid to use was found
to be 40.28 pounds per note.

11’ these values with the above qualification are

53

substituted in equation (31) a predicted yield of 89.08
bushels per acre in round but, again, this figure represents
a theoretical solution based on the above value for'nitrogen

whose accuracy cannot be determined.

Anglzole 0g the rating_ggpen_Fi§lfl Corn Dotn:l9§Z.-
(e) _§guegg figct Eggduogioq Fugotlflg.~

The first formulation
of the functional relationship which.wne attempted for the
data was a square root function of the form of equation (17)
above.

This formulation containing the eetimated.paremetere
in chem in equation (2'7). N and. 1’ represent per acre
applications of nitrogen and.phosphoric acid respectively
and yield in measured in buahdsper acre.

(27) ‘10 e 30.107083 - .oseoiioeeon - .0873990461P +
(.oooeoooeloi (.445073575)

i .714140543 W t 1.46594an 5 - .0224733044 43
(.035481598) (.472677396) (.0521503848)
The basic etatietioe for this function are shown in
Table 9. The coefficient of multiple correlation for this
equation‘waa .470 and the coefficient of multiple deter-
mination indicated that only 22 per cent of the variance
present in the yield data was explained by this formulation.

TMHLE 8

HARGIZEAL PEJQDUCTIVITIE'R‘S 0? NIT :‘KL‘GiA NE D THUS}? BURL} ACID I}! TILE

PIODUCTIOH OF CORN FOR INPUTS momma» (1:1

fl‘h‘xr

1.115

10:“; £5? T1213 0? “£01"!

PAIR. A151) 3", '0; WHO? £10 11610 M ”n CT ICE), FOR 0011i; 1'!in *0 LC mat-i

FIZLD 1957. *

 

Pounds of Pounds of Eitrogen
P206 P” pox- Acre
Acre , _ m fl r 4 i _A M m T T w A
O 30 4O 60 80 100 120 140
0 0 q057 0071 .075 .079 0084 .088 .092
O O O 0 O O 0 O
20 0 .053 .062 .066 .070 .074 .076 .08?
.584 «"374 ~366 .356 .154.” .557 .327 .318
40 o .043 .0522 .057 . .061 .065 .089 .073
o 93" 9 23-2 o 303 o 193 .183 017‘ .135 o 155
60 o .039 .043 .04? .051 .066 .059 .0“
.053 .049 004*.) I030 «0'31 ".011 .002 ‘oOO?
80 O .030 .034.- 0038 .0433 004-6 0050 .054
*0 103 ‘0 113 “'9 123 *g 133 “a 141 “I 150 "o 160 “’0 179
100 O 9030 .034 .028 0032 4.037 .041 02045
“a 960 “9275 “9385 -9294 -0 304: “Owl 1?! ”a 523 -0353
1% 0 .011 .015 .019 .093 .0‘3'7 .031 .058
-0429 ‘0408 -0448 “045? -0 406 “.43“; "0485 “0495

' Herginnl proauotivitiee are givem in bushels

 

 

W

w

per acre.

(b) grace Prolucgégrggucticn Kunctiqg.~
Because of the large
amount of variance not eeeooiated‘with.tne functional re-
lotionehdp above explained, a second formulation was
attempted or the type of equation (18) above.
This equation and the cat1mcted.porametcre are
preeented.in equation(afi)s

(28) to :- 31.33.8103 a» .oecmeocgm o .1so4rgogyggp ..
(.0355700598) (.040420165)

- .00044996016252 - .001034584183’2 - .ooooeeleeenm
(.0002259778) (.0003990206) 5000220207)

The coefficient of multiple correlation for thin
equation won .505. The coefficient of multiple determination
indicated that about 26 per cent or the variance present in
the data was explained by this equation.

.Based on the statistical measurements derived neither
of the two equations was considered to fit the data
adequately. Nonetheless, the crane product functional

oxprooeion gave a slightly better fit than the square root

production function.

 

The cross product production
function can used to computo the predicted yields for

specific points on the production surface.

TABLE 9

VALUES 0!" '3‘ AND .32. FOR Tiff.) VARINBLE IIUTIKI'ESLEITS mu) ViaLUEB
G!“ ”t“ FOR IEDIVIDIIAL REG-3133:3023 COEFFIUIEQWS FOR 0031:! PATH?!)
UPPER FIE-31:1) 1957..

Equation Value Valuen Value of “t“ For
of "R“ 0: “HQ“ Caerficiente in Order
Listed in Equation.

Square 9.470. .221 - N 3 1‘73
I‘OOt P I 2046
Fr? 3 1.60

G’- :‘ 3.08

Ea? :. .70

Cross .505 .255 H x 1.60
product P ‘ 5‘73
N2 a 1.99

P2 3; 5:46

HP 3 .31

8']

As was the can for corn Patino Lower Field 195?,
each column in Table 10 represents a vertical slice through
the production surface parallel with the P205 axis, and
«on row represents the same thing with nitrogen variable
and ng find...

Figures 2.3 and 2.!» illuctratc the affect or on.
nutrient on the productivity of the other. They arc
rcsponec curves tor one variable nutrient with the other
fixed in the quantity indicated.

It can be soon from Table 10 and from Figure- 2.c
and 2.1: that both nitrogcn and phosphoric acid have a.
negligible effect on crop yield: and that diminishing
total yieldc are rapidly encountered.

In Figure 2.: with P205 fixed and nitrogen variablc,
the level or the crop yield dopondl to c certain extent
on the amount or P205 present in the coil. Up to tho
80 pounds per car. lovcl or P205 slightly higher ionic
of yields rccult as a. consequencc or the interaction
between nitrogen and phosphoric acid.

Beyond that level, too much P205 is being combincd
with the specified level of nitrogen and yields are at c
lowcr level.

In Figure 2.1) in which nitrogen is being held. constant
and P205 varied, higher levels at which nitrogen in fixed

FIGURE 2
CORE? PATII‘IO UPPER FIELD 1957

 

 

6

  
  

u)

i’
3,. i
3 3
§ é
a A

 

 

 

 

 

 

 

0 40 to up “a 0 4'0 8.0 do Ce
Peon» 9? (50‘, Hosts 0F Mime».
me. “6 PEG. ACRE

sauce Lites. who to.
has he; m [:1
r“ :’

 

 

 

 

LC)
. .
«to
. e
u
3‘
i s:-
e
‘1}
i‘
a
9
a
$0 - O I
e mewieo hL
oueuefiws
I»
3 4‘0 {.0 Illa {59

towns OF a. Mix. Q0;—
?E.l Ange

89

increase the action of P205. However, beyond the level of
60 pounds per acre or nitrogen this element is redundant

and the total yield is deoreaeed.

 

is was 3}an for the Patina Loner Field
195? corn date, the economic Optimum is attained when the
partial derivatives tor'both.nutriente are equal to the
nutrient/crap price ratio.

The marginal productlfltice of nitrogen and phosphoric
acid are presented in table 12, with the marginal product-
ivity of nitragen at the top of ench pair and the marginal
productivity of phosphoric acid at the bottom.

These marginal productivitiee ere computed from
equation (29) for nitrogen and from equation (30) for

phosphoric acid.

(ti-39) gr = .056748099 - R(.000449960160)N - .oemceueeio?

(30) 6% a 450470579200 - 8.(.001034684180)P - .ooeoeeieeeion

In, computing the wmxmt of. nutricn‘be to apply to maxinixe
profits, the following prices were used:
Corn : $1.45/hu.
H z & .16/lb.
P I 3 .io/lb.

 

   

-
D
e
u
‘
I
f
. 0
e
‘
- , I
e
. .
y I

2". BL?) 10

60

TOTAL CA1...CULATEEID OWN 351233.133 FEW“; {"51“ Cl “‘13-3‘9 Riff-3&5 UP Mi’Ii’LICfisTIG‘H

CF lflTI‘KEGE'LIi ASE) 1‘30“} EKZRIG A313, 1? 3253391 5'3." 2'57".) FRCI‘H T333 CFWSS P 2:32.331] CT

mum 1014 me c we: zwrzm 111’? m new, 3.9 5? .

W l. , 7 v. — ‘r . y.—

TW._-——

 

 

Pcunde of Pounds or Nitrogen
P205 per per.Aorc
Acre .i A it .i_ _“i it :e A i_
O 20 40 60 so 109 120 140
O 31.3 .32.3 52.9 33.1 33.0 32.5 31.? 30.5
20 55.9 54.9 35.4 55.6 55.5 35.0 54.1 32.9
40 ' 55.7 36.6 3?.1 5?.5 57.1 35.6 35.? 34.4
60 56.6 57.5 38.0 38.2 38.0 37.4 56.5 65.3
80 38.8 37.6 38.1 38.2 58.0 3?.i 36.4 35.1
100 56.0 36.9 57.5 37.d 37.1 36.5 55.8 34.2
120 34.5 65.3 35.7 58.3 35.5 54.9 53.8 33.6

1... 4 k .__ #

fin... v. --——~——~

* Xielde are given in bushels per acre.

wfi fl r—vfi‘fi

61

I’R , .. 1 £40.
n-i‘B‘dlb 11

TOTAL rahnxcrun awn cesxnvxn YIELDS ran otnm PATIRO unrun FIELD

 

 

1957.
‘L‘reetmcnt Predicted Observed
(Pounds per Acre) Yield Kield
.9

M i __ Sim. (Agrgz (flu .19.ch

 

 

0 O O 51.3 2’7q6

' i0 23 60 33:9 32.5
o 40 60 35.7 55.6
0 80 60 556.6 56.3
0 80 60 38. .3 36.3
0 100 60 33.0 36.1
0 1530 80 54.5 2579 E
3330 0 60 152.3 54.5
20 20 60 54-9 $0.?
20 40 (30 (56.8 35.2
20 6O 60 37.5 3-3. 2
20 80 60 57e6 33.4
20 10 60 55.9 40.6
20 1:720 60 3559.3 32. 4
40 O (’10 3299 9.9.3
40 20 50 5594 32.5
40 40 150 37.1 139.0
40 co 60 38.0 $1.6
4-0 80 60 38.1 35. 2
40 130 50 359'? 354.3
0 60 3391 35.4

‘10 60 3793 4094

80 50 60 I33. 2 4.196
60 100 60 37.4 57.1

60 120 60 325e8 3894

TABLE ll, Continued.

130

120
120
ISO
130

140
140
140
140
140
140
140

90

60
80
100
120

80

60
60
60
60
60

60
30
60
60
60
60
60

60
50
60
60
80
60
60
60
60

60.

60
60
60

36.1
35.3
39.0

34.1

50.7

50.2
55.4

41.9

39.3
55.8

29.1
29.3
34.5

35.8

33.2
34.6
34.3
35.0
34.3
35.0
31.1

65

To find the amount of nutrients to apply to maximize
profits, equations (39) and (50) are solved einultnneouely1
using equations (34) and (35) an before.

fining .1103 es the nutrient/orOp price ratio for
nitrogen and .069 as the nutrient/crap price ratio for
phosphorous, the Optimum amount of nitrogen to use ie a
negative quantity ~62.?0 pounde per-core, and the Optimum
amount of phosphoric acid to uee in 41.46 pounds per acre.

Substituting these quantities in equation (28), thn
predicted yield is round to be 30.65 buehele per acre.

thin solution for the profit maximizing point ie valid
only if the price conditions specified above are true. It
the giten.prioce or fertilizer‘nutrient inputs end/or crap
change, co change the profit maximizing level of fertilizer
inputs end its combinations to use end.the amount of crap

yield to produce.

Anelxeigqgitheangino BeenuggthLQEZ/fig.-
(a) Sounre Root Proiucticgqgu;etiqg.o

The first formulatian

‘— A‘ ’ A ‘ A “‘L .L- .4“ Li. A.— A
"w...— wva ‘—

 

rvv 1- _ w t

like equations solved were as follows:

(a) gbr Hitrogggt
-2(.000449e60160)n - .000089145810? 3 .1105 ~ .056748099
3”a. ‘31. ”1
(b) qur_§hoephorggex

-e(.ooioe§eeeieo)m - .ooooeemeeiop a .039 - 450470979200
2 ”*2 °2

64

Rid-IGINAL PRQZMCTIVITIES O? NITMHHQZ A33) I‘BOSI’S’BEHO ACID IN
THE FfiflDUC?IOH OF CQHH FOR IRPUTS INQICflTED (HITRQGKN AT TOP
OF ERCH PAIE.AED PHOSPEOHIU ACID i? BOTTOM), Ffifi GOES PATIRO
moot mam, 1957... *

 

 

Pounds of Pounds of nitrogen
P203 par par Acre
Marc A __ A g A f «-
MOA L 20 4 4f) 60 _ A ‘80 100 120 A140
0 O 0.038 9020 0-002 “.015 “'00:. “.051 *13089
0 O O 0 0 O O O
20 0 0.037 1:019 0001 ' ”@018 -0034 “c052 “6070
.109 .107 .106 .104 .103 .1o2 .100 .099
40 o .055 .017 .000 «.013 «0:35 «.054 «.072
306? .065 0.064 0063 .053 o p 0059 .053
60 O .034 0016 “.001 -0019 “$05? “.056 -0073
.026 .024 .035 . .022 .030 .019 .013 .016
80 O .033 9015 “c002 «In-09.0 -0038 -0056 ‘0074
-3016 ”9016 ‘001? “.019 “.0533 ’00? -0023 “083‘
100 O 0051 c013 “NOOQ ":03?! “904:0 “.058 ”.076
.0036 “.00? -0059 .i060 “0061 “.063 “0084 “1066
120 0 #030- 0012 ‘HOOB ’0023 “c041 ’0059 .0077
-4397 .1099 ““0 100 “c 103 “0103 *0 104 .0 106 3'0 107

M‘—

" Marginal productiviusc are given in buoholg per oer-3.:

 

 

w

A‘

h—

65

of the functional relationship nhioh.wno fitted to the bean
yield data was a square root function of the form or equatian
(1?) above.

Thin formfilntion containing the ootimated parameters
is shown in equation (31). N 5nd,? roprooont per sore
applicationa or nitrogen and phosphoric acid respectively
an& yield is measured in bushels per acre.

(31) Etc 3 133957525 «- .ooooo'roomm: - .oomsmoomp -
(«0059601QSJ (.oooloooo?)
- .oomoozaau 4! .18528605968 W «1 .0493035974 {'17
(.005029390) (.047112116) (.046056907)

The coefficient of multiple correlation for this
equation was .806 and tho coefficient of multiple determin-
aticn was .650, indicating that 85 per cont or the variance
present in the yield data was OXplainod by tho indopcndont
variabloc-

The basic statistics for this and the following
cquntions fittefl for this not or yield data aro presented
in.€ablc 13.

(b) 02055 Profluqt Proguction Funqtioqin
A scout! formulation
of the functional relationship was ottomytod.uoing a cross

product production function of the form of equation (18) above.

TI‘B LIE 13

VALUES OF ”R." AND “Ii?” FUR T319333 VATLIAEiLE-‘J RUTRIEEWS AWE) VALUES
0? 't' FOR INDIVIDUAL RLfiflfiafilOfi COHFFICIHfiTB FOR.BEAR PATINQ
1957/58;

 

Equation value Value Value of 't'.£or
Coefficients in Order
Li st 66. in Equation.

___ .4. A... . M

lle

aqua. .806 .550 N ‘_
root P i .45
I I 071
(I? ! 30.88
6 '. 1106
Cross .816 .665 n 3- 6.97
. preduot P ' 1.12
K i, .39

8 .

HI 3‘ 40 24
P2 t 0.70
Cobb-Douglas .849 .720 H :11.oe
P l 1.98

Klqéfi

6"

This equation with the estimated.pcremetcre is shown
in equation (52).

(32) to e 1.760349605 e .ooooocooooen o .00452240615? o
(.005303363) (.004119018)

+ .00115231883x ~ .00032175563fi2 - .0000281858P3
(.002976016) (.000075872) (.000040272

The coefficient of multiple correlation for this
equation vue .816 end the coefficient of multiple detec-
minetion vol .665, indicating that about 68 per cent of the

variance was explained by the regression equation.

(0). Qggozpouglne Production Funcsiggpo
A third formulation

of the functional relationship was attempted.ueing e
Cobb~Douglal production function of the form of equation
(19) above.

this formulation with the eetimated.paramctere ie
ehown in equation (35) and in ite logarithmic form to
equation (34).
(33) 'Ic a 1.768 N.106357643 P’017555083 x~.005592104

(34) log to 3 .244929 e 405357543 log N + .017355082 103 p -
(.009260381) (.009027808)

~ ~005593104 log K
(.012126062)

The coefficient of multiple correlation for this
equation was .849 end the coefficient of multiple

63

determination was .720 indicating that 72 per cent of the ,
variance present in the data was explained by the independent
variablee of the regression equation.

Beecd.on the statistical measurements derived, the
Gobb-Douglae production function was considered a more
opprOpriete formulation than the two previously derived
equatione. to represent the functional relationship
between varied quantities of input nutrients and crOp
yield output.

genetic; surfing! estimatggw

A act of production surface

estimates is presented in Table 14. The“ quantitiel

 

chewing the total per acre yield of been: for varioue
retee of R end.P205 repreeent specific pointe on the
eurfece for epecific pointe in the nutrient input plane.

Each column in.feble 14 correeponde to a vertical
elice through.the production curfece parallel with the
P205 exie, and they chew the yielde when nitrogen is
fixed at e given level and P205 is being varied.

The rows repreeent the same thing with nitrogen
variable enszo5 fixed in the amount specified.

Inyfable ld'the total predicted and observed yielde

for been: are compared to chew how accurately thie function

ie capable of predicting then from their functional releticnehip.

69

JEccnomic opting.-
A detailed explanation on how to work with
the Cobb-Douglas function has been presented in fippendix A,
so these concepts will not be repeated here.
In computing the profit maximization point the following
prices have been used:
Bean: 3 $3.?6/bu~

n s 3 .io/ib.
P 3 s eIO/lbe
K I 3 eOG/lbe

The Optimum amount of nitrogen to use was found to be
56.9 pounce per core: the Optimum enount of phosphoric
acid 13.0 pounds per core; and the optimum amount of
poteeh.wee e negative quantity of *8.1 pounds per acre.

The estimated yield was found to be 14.3 bushels per
acre.

The calculated marginal productivity of a pound of
nitrogen in the production of beans at various ratee of
application of nitrogen are preeented in Table 16, for
selected beans prices.

Ae has been explained, profit maximization concepts
must include price relationships nnd.when these price
conditione change, no change the amount and compoeition of

fertilizer nutrients to apply and of product Output to produce.

70

It can be seen, that when beans are priced at $ 5.75
per bushel, with the above indicated fertilizer prices,
the Optimum level of application of nitrogen is at the
level of the 60 pounds per scro level when tho condition

MVP“ * Pn : 9

is ustiatiod.

Th1 can. analysis has boon done in Table 17, in which
tho marginal productivitios or a pound.cr phosphoric acid
in the production of boon: at various rates or application
of phosphoric acid, are presented.

Hith.ths above specified price Qanditicns, tho
Optimum lovol or application of phosphoric acid is at
tho level of tho 15 pounds per acre.

T1123 11E 1 4

TOTAL GXLQILATED BEAR 13131.1) P723321 i‘fi‘iiCIFIEED PoiTES 0F NVPLICJILTIOH
01’ HITRCGK! MID PEWLEI’EKHHC ACID, PI-‘ifiDIC’l’Zi'fiJ FIKOM THIS (IJBBJDUUCLAS

EnonTIcn FOR BEAN PATIno, 1957/58. *

 

 

Pounds of Pounds of Nitrogen
P305 par par Acre
Acre L , fl“? .1 _ h w
0 ~ 20 40 60 80
0 11.4 14.5 15.6 16.3 16.8
26 11.9 15.1 18.8 17.0 17.5
50 12.0 15.3 18.4 17.2 17.7
75 12.1 15.4 16.6 17.5 17.8
100 12.1 10.6 16.6 17.4 17.9

14%

__‘__

M

* Yields are given in bushels per acre.

TABLn 15

72

Tom. 1332:3991ch AND 0mm? ED mama Fun 8.1M: pint-30 1957/53.

 

w

A.

M

7

 

 

 

Treatment Prom ct ed Observed
(Pounds per Acre) Xi old Yield.
# “ (Bu./Aoggl (nut/Acrg;

0 O 0 11.4 10.6

" '0. 25 ‘10 11.9 11. 2
D 50 40 12.0 10.9
0 75 40 12.1 11.2
0 100 40 12.1 11.5
80 O 40 14.5 15.1
5‘50 $3 40 15.1 15.4
20 50 40 15.3 13.5
‘30 '75 4.0 15.4 14. 2
20 100 40 15. 5 14.3
40 O 40 15.6 14.6
40 I: 40 13. 2 18.1
40 50 40 16.4 19.4
40 75 40 16.6 13.?
60 O 40 16.3 15.1
60 25 40 17.0 15.7
60 50 40 17.2 18.2
60 75 40 17.3 17.6
50 10.0 40 17.4 19. 1
80 O 40 16.8 15.1
30 50 ‘10 17.7 m
30 3.00 40 17.9 9-“

73
TABLE 18

cmwmmn 22m: 12m. mooucrlmms on A mono or termrzvom IN
was; moouonon or rooms AT maroon 3:17:33 0? Afi—‘LI mums or
1:117me nun "wanna Emu PRICE-c, FOR PATIHO, low/cc.

A... ___..

‘7‘. .7 _‘

M A A- ._ -A _. _ -..

—'—v “w—V'Y‘ Y— —_, , 7

Price of Beans (Dollars / Buohcl)

 

 

Pounds

or E

par 0 6.25 § 0.50 $ 0.76 g 6.00 3 6.25

Acro _ v fr _~V-i ; w # _ “._tf A 1.

MP? ’ m “V? HY? EV? MVP MY?

1 1.01 7.93 8.31 8.68 9.06 9.44
2 .76 3.94 4.13 4.31 4.50 4.69
3 .50 2.65 2.75 2.88 5.00 5.13
4 .58 2.00 2.09 2.19 2.28 2.56
5 .30 1.58 1.65 1.73 1.80 1.88
6 .35 1.51 1.38 1.44 1.50 1.56
7 .33 1.16 1.21 1.87 1.52 1.38
B .19 1.00 1.05 1.09 1.14 1.19
9 .17 .89 .94 .98 1.02 1.06
10 .15 .79 .83 .86 .90 .94
20 .08 .43 .44 .46 .48 .50
4O .04 .2 .23 .23 .24 .25

60 .03 .18 .17 .17 .18 .19

80 .02 .11 .11 .12 .12 .15

k

' Bushols 5.. acre.

74
TABLE 17

u...“ ‘CULATED 0:001:00. I’mUUU’I'IYI‘i" 1:05 02? A Pom-:0 0F mumvfiuiuc
ACID 10 THE rnovucrxoa 0? EEAHS AT vaaxuua 06066 0? AerlchICH
0? 0500900010 ACID 000 SELECTED BEAR 001000, r03.rarx60, 1057/08.

A A m __ .4. A. _ .

_,__ "— 1‘ w w... ,_. 7—7 . w— — “x.

#‘ ‘44 --—‘ A.____~ A. - A “.1 4.__ A

w. V. V , v r. ‘— _'._ r.— v “ ‘——~' w

Pound: of Price 01' Beans (Dollar-6 / Bushel)

 

 

P005 per _

“Aer. 6 0.25 3 0.00 0 5.75 6 0.00 3 6.20
:03? * MP W mm m m

1 .26 1.01 1.30 1.44 1.60 1.56
2 .12 ’:..63 .06 .69 .72 .75
a .00 .42 .44 .46 .48 .60
4 .00 .02 .50 .00 .36 .30
6 .05 .26 .2? .2 .60 .51
6 .04 .21 .00 .0 .24 .06
7 .04 .21 .26 .23 .24 .20
6 .03 > .16 .17 .17 .10 .19
0 .05 .16 .17 .17 .18 .19
10 .03 .16 .17 .17 .10 .10
25 .01 .05 .06 .00 .06 .06
60 .006 .03 .00 .03 .03 .05
70 .005 .00 .02 .03 .02 .02
100 .002 .01 .01 .01 .01 .01

v—n— q..- w. ‘v—w v .— V ,

‘ 8001101: per acre.

75
mwma 6
WALUATION, cummmzmgs mm In: 1.10;": 10:43

‘ggaluatigg.*
Eagegimentgl_flotg and Gonnaggigl_Far§§.-

Comparing thfi
conditions faced by the researcher varking with experimental
plots and the situation faced by the Operator’on a commercial
farm, it can be seen that, even though, the number of
independent variables usually included in experimental
work in very small nnd.many problems encountered.by the
farmers (rotations, ate.) cannot b0 solved entirely within
the framework or a ninglo cxperiment, the information
obtained from it can be combined and improved by subaoquent
research leading to better and broader knowledge of tha
functional relationship: involvad.

Th0 elements oonnidired “fixed“ in oxperimantai work
ouch.aa the recommended.practiooa, are also controllable
by the farmer.

The trouhlesome element seems to lie on the difference
of levels at which.non¢oontroliablo, non-studied variables
are fixed in the experimantal work and on the farm.

Each individual experiment field has certain unique

characteristics associated with it, which are the detarminant

factors when the results from the experimental field arc
trying to be generalized for a large number of forms.

Thus, the problem, is to try to reduce the variance
in the experimental results to conform closer to those on
the farm, enabling tho economic optimum conditions to be
defined more accurately.

This problem, on the other hand, is nggrnvatcd'by the
desire of the researcher of minimizing within field
variability choosing the location of the experiment in
auoh,n way, that generally the 167913 at which those
uncontrolled variables are fixed in the experimental work
and tho levels at which they are tixod on tho commercial

farms nro pushcd still fartnor apart.

Vagigngg.-

The unotudied and uncontrolled variables causing
largo amounts of uncxplnined betwecnuplot variance may be
important elements when determining:

(a) tho apprOprinto mathanntionl function to fit, and
(b) the Optima located onasalcctod function.
One cause or this unoxolained variance is believed
to be the small also of tho plots usually used in experimental
work. However, as has been suggcotcd, he variance present
in yield data obtained from small experimental plots migr

bc highor than the amount of variance experienced by tho

?7

Operator cn-a commercial form in which larger areas are
involvcfi.

This is u very important point to oonsiior, because
the accuracy of tho recommendations undo to farmers nopenda
to a great extent on how representative of tnc conditions
on the averngc turn are the estimates secured from experi-
mental work.

A: to ways to handle this problem, the following have
been suggested:

(a) Incrcnoc in the Sizcugt_tho Engarimontnl Platon—

Larger
plots should be used in experimental work. Khan the causes
of the variance are randomly aistributod throwghout the
experimental area. the use of larger plots will be indicated,
but, to tho extent that the causes of variance are not
randomly distributed but arc correlated.bctwccn chnccnt
small plots, replications of plots become relatively more
effective than larger plots in reducing variance.

(1:)

 

Poisoning Young;

The cnuacs of unexplained variance
might be invcstigatod and measured and incluaed in the
study no independent variables.

However, obstacles are encountered at present to

78

study accurately this point because appr0pr1ate methods

of measurement are still not very well developed.

Va11d1tz of Eagerlmgntal Results over Time.-
Yisld data from

one your eXperimental work has been analyzed in previous
chapters only.

As has been pointed out previously, the uncontrolled
and unstudled varlables present in the eXperinental work
influence the results in such.s way that, based on one
year's date, generalizations cannot and must not be made
trying to extend the analysis to future years.

The U element analyzed above In likely to change
year after year and predictions based on such.unstsble
ground V111 have a large percentage of probsbxlitiee to
be wrong.

Preblems of residual fertility accumulation and
depletion and rotation effects become important when lens
run conclusions are to be drsvn teen the experimental data

and long run decisions are to be made on commercial terms.

Conelus10g3.*
Three sets of yield data were analyzed in the

present study.

79

The first one. yield data for corn Patina Lower Field
195? included nitrogen and phosphoric acid as the independent
variables; with potash.gonera11y held constant at 60 pounds
per core.

Two production functions or the type of equations (17)
and (18) above were fitted to the data. Statistical
measurements indicated that'tns cross product type or
production function fitted the dots better than the square
root production function.

As it was seen before, the economic Optima conditions
are based both.in the physical functional relationships
and in the price conditions for fertilizer inputs and
product Output existing at that time.

If the price relationships involvod change, a new
optimum amount of fertilizer inputs and nutrient combine.
ticns to apply become profitable as determined,hy the new
nutrient/crop price ratio existing site: the change.

This point is generally overlooked in the present
fertilizer'recommendations‘whieh.are being given to farmers.

‘ It is thejprinoipnl reason why a new approach inte~
grating agronomic and economic concepts is being used in
the design of fertilizer-experiments Which allows the

location of points at which.mnxinum profits from a given

80

yplioation of fertilizer nutrients are possible. At the
some time, recomncnlaticns to farmers are made in terms
or maximum.profita in which changing price conditions of
fertilizer and/or craps are considered, that is, these
recommendations are mode more realistic approaching
situations which.orc ucuolly {need by the farmers in the
planning of their fertilizer prcgrams.

A significant response to both nitrogen and thSphOP10
acid was found to exist for this set or data.

The economic Optima point «no computod. with respect
to nitrogen it was found to be located outside the range
of experimental observations. Therefore, the figure or
530.24 pounds per acre of nitrogen 13 an extrapolation
and cannot'bo uaod for actual fertilizer racemnondation
purposes. However, it does indicate tnat further research
using higher fertilizer treatment levels would be very
useful to complement the experimental resultcgresented hero.

The Optimum amount of phosphorous to uce was found to
be 40.98 pounds per core which is within the experimental
range observed.

The same two functions above indicated were fitted
to the second set of data for corn Patina Uppcr Field 195?.

This yield data included nitrogen nnd.phospnoric acid as

81

the two independent variables, with.potash generally constant
at 60 pounds per acre.

The two functionc iniicatei gave a very poor fit of
the yield data, with the crooa product proiuotion ruzction
considered.to be a slighzly bottor representation of the
functional relationships involved. The most profitable
amount of fertilizer input to use was JOHpUted using the
cross proiuot equation and was found to be a ncgativa
quantity 01-62.?0 pounds of nitrogen per acre and 41.46
pounds of pncaplsorio acid per core.

The predicted yield computed for the profit maxiniza-
tion point was 30.85 bushels per acre.

A significant response was recorded only for
phocphorio aoifi with no reoponsc on yield recorded for
nitrogen.

The third set of antn for bean Patina 1957/58 was
fitted with the same two production functions previously
cited and'with a Cobb~Douglas function of the type of
equation (13) obovo.

Based on tho statistical nonouroncnts derived, the
Cobb-Douglas function nonconsidorofl n more opprOpriatc fit
for this act of yield iota.

Economic optima quantities of fertilizer inputs to

apply were computed using this equation ani‘tho results

ehcwad that 56.9 pounds of nitrogen per acre ought to be
ussd for profits to be at a maximum, indicafiing a sigh ~
nitloant response to this element.

A slight response to phosphoric acid was recorfied
inflicatad by an Optimum amount of 15.0 pounds per acre to
be used.

The ostluated yield as gutai using this functional
relationship was reuni to ba 14.3 muahels per acra.

The same onnacpta on haw the npfilmum amount at nutrients
to use are dependent on tha fertilizer/crap price relation-
ship existing at tho time an thny were explained above for
the corn efipcrlmsnts are velld in this case and they must

be taken into onnziierwtion.

In"). 11 (33132033.-

 

alts fer tha 3gggggg§gg.~

It has been
rcmarkedl “that the only time an experiment can be
preperly designed in after it has been completed“.

Ono sometimes finds, after a act of experimente have

v -—v— — ‘7 w v—

1fiox, 6.52.?” Hunter, J.S., "Tb-w mlo‘mtion and

Rzplcitatlon of Heaponaa aurraoes‘, (as cited in) Mason,
David 3., “Statistical Frnbloms of Joint Research“,
Jaurnnl of Farm Eceqogiog, Vuluna 69, May 19d7, p. 371.

83

been made, that one or more important variables have been
overlooked or that more could have been learned it the
factor-e could have been varied. over different rangee.

Al has been the. can in the present study, additional
information would have been extremely useful it higher
levels of fertilization would have been included for
Corn Patina Lower Field 1967', in which the economic
optimum point eetmate'wae well beyond the levels within
which experimental observaucne were recorded.

Thus, the results from the above analysis have
indicated the direction in which further research work
with fertilizer may be pretitably carried out in corn
and bean orapl.

“the” reculte "cm to smut the need for additional
experimental work in the Ounce. Valley, Colombia, for corn
and been: and other important crepe which over the yam
would provide useful and dependable information for
making fertilizer recommendations to farmers. As he.
been shown in this study, agronomic and economic concepts
are closely interrelated in fertilizer research.

However. economic maceration. are still neglected
in the current fertilizer recommendations that are given to

I fame". In Colombia, as elsewhere. this in also true and

84

it ie nocoeenry thnt reocarotore planning future fertilizer
reeearch.work be able and willing to recognize these inter-
relationships and to include them in their ctudionl It in
not on coey task to obtain cooperation, and in many oneee
to be willing to cGOperato in research work which seems
outside one's area of specialization or interests. This
is particularly true in an environment in which efforts
toward this and have not been nude yet. Nevertheless,

this OOOperction in neceeenry and prdbnbly bndly needed

in countries like Colombia. in which agricultural developv
ment prejectc are undcr'woy and where capital and trained
technicians are usually in short supply.

A fertilixer reeearohqproject deeigned.to provide
experimental data when plant nutrient inputs are varied
over different ranges end from which economic Optima eati-
nates can be located, certainly represents a considerable
improvement over experimental designo from which.merginnl
productivitiee and economic optimum cannot be determined.

no a matter of foot, more information is provided by
design: of the type considered herein both or agronomic
and economic interact.

The ratio or useful information to expenditures in

probably higher for agronomic-economic 1:an for purely

85
agronomic designs.

_§ignificnngc cg rccnl§o_jor tho Furfior.~

The analysis of
experimental work presented in this ctufly is ratncr limited
in scape with reopoct to coil conditions, crops, and
growing cannons.

For the reasons stated cloonnerc, seldom are any
reconmenfintlons made Upon the basis of n alnglo experiment
such no this onc.

Additional work is nocfied to oupyort or dong the
Optimum plant nutrient treatment octimntcc precentod
hero and before rollahlo rocomcndationa can be none to
farmers for rational planning or their'fcrtilizor*programs.

Historically, this information has not been available
to farmers for the vary simple reason that fertilizer
research has been conducted indcnondcntly from any economic
oonaiderationa.

‘Tha farmers arc being copplicd.nizh fartilizor infoan~
ntion in which, implicitly or oXplicitly, the conclusion
in Doing drawn that the moat conquntc level of fertilization
in the one at which.naximum yields For acre are attained. -

As has been 32mm, maximum yields per com- and maximum

profits from a given application of fertilizcr arc seldom

88

located at the same level. -

Therefore, the dolombian farmer in the Canoe Valley
and, it is suspected, elsewhere elongiend across the main
agricultural regions or South America, ought to be supplied
with fertilizer information in which recommendations are
made to maximize profit instead of yields.

when resources are so tar-out of adjustment as other
similar studies in the Cauce'Valley show they are} hit.
of internation provided by partial etudiee and preliminary
surveys or the fertilizer problems or a given area, are
perhaps the “at way and the most economic oneto promote
a reallocation of those resources even though more refined
and elaborated etudiee may prove to be useful afterwards.

If reliable information could be secured with respect
to the returne the Colombian farmer in the Cauce Valley is
earning an inputs other than fertilizer in his bucineee, e
coupariacn or their marginal productivitiee would provide
an additional tool of decision-making to farmers for whcm
limited capital in an important consideration.

in surveys or the Colombian agriculture show a rate
of increase in the agricultural production higher than
that eccmnpliehed during the past year. will be needed to
keep pace with the increasing pepulation and improving levels

.__._.

1Trent, G. 1., 92. git.

or living.

Colombian farmers will certainly be required to
increase the productivity of their farms.

Fertilizer, as well as other forms of capital and
”know how" representing technical improvements in agri-
culture een make an important contribution toward that

Gflde

88
M‘PifiEDIX A

(A) The Cob‘erouglac equation in of the form of equation (1)

(1)
YO : a o Ilbl 6 Phi; e K

133
in which Io is the predicted yield, the tom ”a“ is a.
constant, and bl, b8, b3 are the regression coefficients
and the elasticitiec of the dependent variable with respect
to each dependent variable, that is, the percentage change
in. the dependent variable associated with one per cent
airings in the dependent vnria‘ole.

the Cobb-Jouglas function becomes easier to manage
in logarithmic form such no equation (2)

(if) 103. Yo a: log. a 4 bl loam ; b3 logd’ <1 b3 log. K
(B) The marginal phyeicel productivity in this function is
defined in the equation (5)

1’

(5’%1¥::ble%rg

in which :4 takes different value: according to the treat-
ment levels specified. In the some way it is possible to
compute the marginal productivitiee for P and X.
from equation (3), equation (4) is derived indicating
he condition of higher profit point:

(4) b1 0 “Y‘s "' Pn C 0
H

in which Pn is the price of (he) nitrogen.

89

The name condition can be derived for P nnd.K. The
exeroalion (4) is need now to derive another equation,
solving for»! (or P or X) such as (5)

(5) N . b1 0 YB
13"”

The expression.§§ is a constant K
I.
n

1 (KO for P, and

K3 for K) and.now the equation for yield can be expressed

an in (6):

b1
(6) Y0 : 8. O (311 I Y0) i (K2 I Y°)b2 c (I: Q :{O)b3

3
and in logarithmic forms as in tsprnosion (7):

(7) log;¥c : iog.n f b1 103.):1 { b1 log. Yo fb? leg. K2 f

f b2 10g.Yo f b5 log. K3 i b3 log.'Yo

or in a more abbreviate form cucn.ae (8):

(8) 1051 Y0: log. a f b1 log.X1 ¥ b3 log'xp ‘ b3 log.K3

1~b1~b9~b3
(C) To work with the Cobb—Douglas equation, constants

K In. and K,, are first computed, being necessary to know
(5, u

1:
the prices of N, P, and K, from equation (5).

Then an estimate or yield: (Yo) can be made using
equation (8) and converting the logarithm into a natural
nuancr.

How, the Optimum qunntitiea of N, P, and K, can be

estimated from equatien (6) substituting the appropriate
values in it.

Finally, estimated. marginal physical. productivitiea
can be computed substituting; the corresponding; values in
equation ((3), and at the Efigh1‘3mf1t point $37.; emmllty
(4) must be true.

91
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F3053”; .‘?SE OELY

é?
'75
- f; 1900

   
 

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