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Circular 787
Economic Analysis of Fertilizer
Response for Crop Rotations with
Quadratic Functions
Central Science
Library
I COMPUTER SERIESI
Jose Alvarez
JAI 3 0 1990
University of Florida
1 __.... .
I Ejtamlem Sf W y I Ill/ mtu of PeF mad AgriMutural Scnces. / Univritvof Plorid / Jan T Woust, Dean
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TABLE OF CONTENTS -
Pa.e
INTRODUCTION . . . .. . . . . . . . . 4
HARDWARE REQUIREMENTS . . . . . . ... . . 5
GETTING STARTED . . . . . . . . . . . 5
RUNNING THE PROGRAM. . . . . . . . . . 6
ORDERING INFORMATION . . . . . . . . . 12
REFERENCES . . . . . . . . . . . . 12
APPENDIX A: METHODOLOGY. . . . . . . . . ... 13
APPENDIX B: EXAMPLES OF DATA SOURCES . . . . . 17
ABSTRACT
This manual explains how to use the "Economic Analysis of
Fertilizer Response for Crop Rotations with Quadratic Functions"
microcomputer program. The program provides a framework for
performing economic analyses of fertilizer responses in rotations
where the application is done before the first crop to take
advantage of residual effects and for which quadratic response
functions are known. After completing the analysis, input
variables can be changed to determine their effect on the optimal
rate of fertilizer to be applied and on the extra net returns
generated by both crops due to the fertilizer application. The
program also calculates a sensitivity index for each of the
variables to show their relative impact on fertilizer rates and
net returns.
Keywords: Crop rotation, discounting, fertilizer, microcomputer.
ECONOMIC ANALYSIS OF FERTILIZER RESPONSE FOR
CROP ROTATIONS WITH QUADRATIC FUNCTIONS
Jose Alvarez
S1988, IFAS, UNIVERSITY OF FLORIDA
*
Jose Alvarez is Area Economist, Food and Resource Economics
Department, University of Florida, Everglades Research and
Education Center, Belle Glade, Florida 33430.
UfErsitEIRy OF F
X IflWMOIt
DISCLAIMER
The Board of Regents of the State of Florida, the State of
Florida, the University of Florida, the Institute of Food and
Agricultural Sciences, and the Florida Cooperative Extension
Service, hereinafter collectively referred to as "UF-IFAS," will
not be liable under any circumstances for direct or indirect
damages incurred by any individual or entity due to this software
or use thereof, including damages resulting from loss of data,
lost profits, loss of use, interruption of business, indirect,
special, incidental or consequential damages, even if advised of
the possibility of such damage. This limitation of liability will
apply regardless of the form of action, whether in contract or
tort, including negligence.
UF-IFAS does not provide warranties of any kind, expressed or
implied, including but not limited to any warranty of
merchantability or fitness for a particular purpose or use, or
warranty against copyright or patent infringement.
The entire risk as to the quality and performance of the program
is with you. Should the program prove defective, you assume the
entire cost of all necessary servicing, repair, or correction.
The mention of a tradename is solely for illustrative purposes.
UF-IFAS does not hereby endorse any tradename, warrant that a
tradename is registered, or approve a tradename to the exclusion
of other tradenames. UF-IFAS does not give, nor does it imply,
permission or license for the use of any tradename.
IF USER" DOES NOT AGREE WITH THE TERMS OF THIS LIMITATION OF
LIABILITY, USER SHOULD CEASE USING THIS SOFTWARE IMMEDIATELY AND
RETURN IT TO UF-IFAS. OTHERWISE, USER AGREES BY THE USE OF THIS
SOFTWARE THAT USER IS IN AGREEMENT WITH THE TERMS OF THIS
LIMITATION OF LIABILITY.
*
IBM PC is a trademark of International Business Machines Corp.
ECONOMIC ANALYSIS OF FERTILIZER .RESPONSE FOR
CROP ROTATIONS WITH QUADRATIC FUNCTIONS
Jose Alvarez
(C) 1988, IFAS, UF
INTRODUCTION
Crop rotations are common practices in many areas of the
world, including the United States, for controlling pests and
increasing farm income. In many cases, a single application of
fertilizer is done before the first crop to take advantage of
residual effects. In most instances, farmer recommendations
derived from experiments consider only the agronomic results.
However, maximum physical yields do not make sense in a world
where fertilizers are not free. Farmers must be informed about
the economic optimal amount of fertilizer that maximizes net
revenues in a given rotation.
The purpose of this publication is to present and describe a
microcomputer program that provides farmers as well as research
and extension scientists with a framework for performing economic
analyses of fertilizer responses in crop rotations for which
quadratic response functions are known.
Although the numbers used in the.example pertain to a rice-
sugarcane rotation practiced in the Everglades, the program
should be useful in other areas. A few popular rotations include
corn-wheat in some areas of the United States; corn-soybean,
rice-soybean, soybean-rice and cotton-soybean in parts of the
South; and, more recently, rice-wheat in Louisiana. Sources of
data to run the program for these or other rotations for.which
quadratic response functions have been estimated include the
Agricultural Experiment Stations reports and numerous
professional journals. A few examples are included in Appendix B.
HARDWARE REQUIREMENTS
The computer program "Economic Analysis of Fertilizer
Response for Crop Rotations with Quadratic Functions" consists of
this user's manual and a distribution disk. The program works on
IBM PC microcomputers, or compatibles, with a minimum
configuration of 64K memory and one disk drive. A printer is
optional.
GETTING STARTED
Before running the program for the first time, the user
should make a back-up copy of the distribution disk and store the
original in a safe location. The instructions for duplicating
diskettes can be found *in your microcomputer operating system
manual.
The distribution disk contains one BASIC file called
ROTATION.BAS. To start a session with one disk drive, you must:
-First, load BASIC by booting the DOS disk and, after the A>
prompt is displayed, type BASIC and then press the
key.
-After the Ok prompt, remove the DOS disk and replace it
with the distribution disk.
-Type LOAD"ROTATION and then press the key.
-Then type RUN or simply press the {F2} key and the title
screen will be displayed on the screen.
In the case of two disk drives:
-Boot the DOS disk in drive A with the distribution disk in
drive B.
-Type BASIC after the A> prompt and then press the
key.
-After the Ok prompt, type LOAD"B:ROTATION and press the
key.
-Then type RUN or press the (F2} key.
If your DOS disk does not have BASIC on it, consult your
BASIC manual to load it and then follow the instructions to load
ROTATION after the Ok prompt.
In the case of a hard disk, type BASIC after the C> prompt
and then press . Place the distribution disk in drive B
and follow the instructions given for the case with two disk
drives.
Any screen of information displayed on the monitor can
easily be copied to a printer by holding down the SHIFT <1> key
and pressing the key. Since there are no hardcopy
functions in the program, using this feature is useful for
generating a printed copy of the analyses. If at any point in the
program you want to return to the beginning, hold down the
key and press the key and then type RUN"ROTATION or press
the (F2} key.
RUNNING THE PROGRAM
The first screen of the program displays the title and
version of the program. It also gives the user two options. The
first option shows the credit and disclaimer screens of the
program and then allows you to proceed to the input section.
Users are encouraged to select this option when running the
program for the first time. To read this information, answer the
MAKE SELECTION question with the number 1 and then press .
The credit screen will be displayed. By pressing at the
end of the screen, the first part of the Disclaimer will be
shown. Press again to read the last part of the
Disclaimer. Pressing again will display some instructions
about the program. Pressing the key again will take the
user to the input section. If number 2 is selected from the title
screen, the program will go directly to the screen containing the
instructions and then to the input section.
The figures in the following example run are for
demonstration purposes and are always underlined to show that
they are provided by the user and not the computer. The equations
pertain to a rice-sugarcane rotation practiced in the Everglades
Agricultural Area of Florida and were taken from Alvarez et al.
who made the conversions to English units from Anderson et al.
They show the response of rice and sugar to a calcium silicate
slag application before the rice crop. The equations and a brief
description of the methodology used are included in Appendix A.
There is only one input section:
ENTER THE FOLLOWING DATA:
LINEAR TERM IN THE FIRST-CROP EQUATION ? 722.1
QUADRATIC TERM IN THE FIRST-CROP EQUATION ? 39.5
LINEAR TERM IN THE SECOND-CROP EQUATION ? 738.7
QUADRATIC TERM IN THE SECOND-CROP EQUATION ? 53.5
PRICE OF THE FIRST CROP ($/UNIT) ? 0.10
PRICE OF THE SECOND CROP ($/UNIT) ? 0.0813
COST OF FERTILIZER INCLUDING APPLICATION ($/UNIT) ? 38
ADDITIONAL COST OF THE FIRST CROP ($/UNIT) ? 0.022
ADDITIONAL COST OF THE SECOND CROP ($/UNIT) ? 0.045
INTEREST RATE FOR DISCOUNTING INCOME (%; EX: 10, 9.5, ETC.) ? 10
YEARS TO USE IN DISCOUNTING INCOME (#; EX: 1, 1.5, ETC.) ? 1.5
ARE THESE ENTRIES OK (Y/N) ? Y
The linear and quadratic terms of the response equations are
entered first. These values pertain to the example used to
demonstrate the program and are incorrect for other crop
rotations. The price of the first crop (rice) was set at
$0.10/lb. The price of the second crop (sugar) received by the
cane producer was estimated to be $0.0813/lb, equivalent to an
average market price for sugar of $0.20/lb received by the mill.
The current cost of the slag was $38/ton which includes
application costs. The additional cost of harvesting, hauling and
drying the extra rice produced was assumed at $0.022/lb. The
additional cost of harvesting, loading and hauling the extra
sugar produced was set at $0.045/lb, equivalent to $10.75/gross
ton of sugarcane. Revenues from sugar were discounted at 10% (to
reflect the opportunity cost of money) for 1.5 years since these
are not received within the year of the investment.
The last line in the input screen allows correcting the
information entered when necessary. Typing the letter N takes the
user to the beginning of that section. If the entries are
correct, typing a Y and then pressing the key will show
the output screen:
i /. i ii i i i i ] ,
FIRST-CROP EQUATION
SECOND-CROP EQUATION
LINEAR TERM
722.1
738.7
QUADRATIC TERM
39.5
53.5
A. PRICE OF THE FIRST.CROP ($/UNIT) .1
B. PRICE OF THE SECOND CROP ($/UNIT) .0813
C. COST OF FERTILIZER INCLUDING APPLICATION ($/UNIT) 38
D. ADDITIONAL COST OF THE FIRST CROP ($/UNIT) .022
E. ADDITIONAL COST OF THE SECOND CROP ($/UNIT) .045
F. INTEREST RATE FOR DISCOUNTING INCOME (%) 10
G. YEARS TO USE IN DISCOUNTING INCOME (#) 1.5
F* (AMOUNT OF FERTILIZER TO APPLY)= 4.36
NR (EXTRA NET RETURNS FROM BOTH CROPS)= $
***************
* SENSITIVITY *
* INDEX *
* 2.62 *
* .1.71 *
* -1.83 *
* -.58 *
* -.95 *
* .77 *
* .77 *
***************
90.66
LETTER TO CHANGE (A-G) OR Z TO END ?
This screen shows the values previously entered along with
the results. The economic optimal rate of calcium silicate slag,
with the price and cost structure assumed, is 4.36 tons/acre
which yields an extra net return of $90.66/acre for both crops as
the result of the material application.
A sensitivity index is also computed for each variable. This
value represents the percentage change in net returns caused by a
percentage change in the variable under consideration, calculated
at 20% intervals above and below the value entered in the input
section. The sign indicates the direction of that change. For
example, the 2.62 sensitivity index for the price of rice means
that, for each percentage increase in the price of rice, net
returns increase by 2.62%. Although fully explained in Appendix
A, the user must be aware of the fact that these values may
change every time one variable is changed.
The last line on the screen allows the user to answer "what
if" questions or end the program. Since costs and prices are
constantly changing, the LETTER TO CHANGE question can be used to
perform a sensitivity analysis by changing, one at a time, any
variable value to assess the effect on the optimal fertilizer
rate and on net returns. The difference between the new result
and the sensitivity index computed from the new value entered is
that the latter takes -into account all other variables when
calculating the index at the 20% intervals described above.
Let us consider the effect of an increase in the cost of the
slag. To do that, one has to answer C to the LETTER TO CHANGE
question and then enter the new value:
(LETTER TO CHANGE (A-G) OR Z TO END ? C
NEW COST OF FERTILIZER INCLUDING APPLICATION ($/UNIT) ? 40
An increase from $38 to $40/ton lowers the optimal rate of
slag from 4.36 to 4.15 tons/acre and the net returns from $90.66
to $82.15 while the sensitivity index values changed by small
amounts:
FIRST-CROP EQUATION
SECOND-CROP EQUATION
LINEAR TERM
722.1
738.7
QUADRATIC TERM
39.5
53.5
PRICE OF THE FIRST CROP ($/UNIT) .1
PRICE OF THE SECOND CROP ($/UNIT) .0813
COST OF FERTILIZER INCLUDING APPLICATION ($/UNIT) 40
ADDITIONAL COST OF THE FIRST CROP ($/UNIT) .022
ADDITIONAL COST OF THE SECOND CROP ($/UNIT) .045
INTEREST RATE FOR DISCOUNTING INCOME (%) 10
YEARS TO USE IN DISCOUNTING INCOME (#) 1.5
* SENSITIVITY *
* INDEX *
* 2.79 *
* 1.83 *
* -2.02 *
* -.62 *
* -1.02 *
* .82 *
* .82 *
***************
F* *(AMOUNT OF FERTILIZER TO APPLY)= 4.15
NR (EXTRA NET RETURNS FROM BOTH CROPS)= $ -82.15
LETTER TO CHANGE (A-G) OR Z TO END ?
I i I I I I I __ I I I III |_II I I_--0_0
Now consider a change in the price of rice with the cost of
slag set at the original figure. First, let us change the cost of
slag back:
(LETTER TO CHANGE (A-G) OR Z TO END ? _C.
NEW COST OF FERTILIZER INCLUDING APPLICATION ($/UNIT) ? 38
Then enter the change in the price of rice:
( LETTER TO CHANGE (A-G) OR Z TO END ? A
NEW PRICE OF THE FIRST CROP ($/UNIT) ? 0.13
The new result shows that the optimal amount of slag to be
applied increased to 5.31 tons/acre and net returns to $168.00:
FIRST-CROP EQUATION
SECOND-CROP EQUATION
LINEAR TERM
722.1
738.7
QUADRATIC TERM
39.5
53.5
A. PRICE OF THE FIRST CROP ($/UNIT) .13
B. PRICE OF THE SECOND CROP ($/UNIT) .0813
C. COST OF FERTILIZER INCLUDING APPLICATION ($/UNIT) 38
D. ADDITIONAL COST OF THE FIRST CROP ($/UNIT) .022
E. ADDITIONAL COST OF THE SECOND CROP ($/UNIT) .045
F. INTEREST RATE FOR DISCOUNTING INCOME (%) 10
G. YEARS TO USE IN DISCOUNTING INCOME (#) 1.5
F* (AMOUNT OF FERTILIZER TO APPLY)- 5.31
NR (EXTRA NET RETURNS FROM BOTH CROPS)- $ 168.00
**************
* SENSITIVITY *
* INDEX *
* 2.09 *
* 1.01 *
* -1.2 *
* -.36' *
* -.56 *
* .45 *
* .45 *
***************
LETTER TO CHANGE (A-G) OR Z TO END ?
The user may make as many changes as desired, always one at
a time, until satisfied with the result. The last line of the
output screen also gives the option finishing the session. When
the latter is done, an END OF PROGRAM message is displayed.
ORDERING INFORMATION
For more information on this and other IFAS software,
contact the local county extension office or write to:
IFAS Software Communication and Distribution
Building 120, Room 203
University of Florida
Gainesville, FL 32611
REFERENCES
Alvarez, J., G.H. Snyder, D.L. Anderson and D.B. Jones.
"Economics of Calcium Silicate Slag Application in a Rice-
Sugarcane Rotation in the Everglades," Agricultural Systems (In
Press).
Anderson, D.L., D.B. Jones and G.H. Snyder. "Response of a
Rice-Sugarcane Rotation to Calcium Silicate Slag on Everglades
Histosols," Agronomy Journal 79 (1987):531-535.
APPENDIX A: METHODOLOGY
The structural form of the equations for both the first and
second crop in the rotation is a quadratic response function:
Y = a + blF + b2F2 + e
where Y = yield per unit of land;
F = rate of fertilizer application; and
e = random error.
Economic optimum conditions are derived from the following
profit function:
S- [Pl*Y1(F)] + [(P2*Y2(F))*(l+i)-n] (CF*F) [Yl(F)*AC1]
[(Y2(F)*AC2)*(l+i)-n] ;
where PI = price of the first crop;
Y1(F) = response function of the first crop;
P2 price of the second crop;
Y2(F) = response function of the second crop when the
fertilizer application occurred prior to the
planting of the first crop;
(l+i)-n = discount factor, where i = interest rate and n =
,number of years;
CF = cost of fertilizer including application;
F rate of fertilizer application;
AC1 additional cost of the extra production of the first
crop;
AC2 = additional cost of the extra production of the second
crop.
Taking the first partial derivative with respect to F and
solving for F provides the equation for the optimal rate of
fertilizer:
F*=[CF + (Ll*AC1)] + [(L2*AC2)*DF] (Ll*Pl) [(L2*P2)*DF]/
(MQ1*AC1) + [(MQ2*AC2)*DF] (MQ1*Pl) [(MQ2*P2)*DF];
where LI = linear term in the first-crop equation;
AC1 = additional per unit cost of the first crop;
L2 = linear term in the second-crop equation;
AC2 = additional per unit cost of the second crop;
DF = discount factor from the formula (l+i)-n;
MQ1 = quadratic term in the first-crop equation multiplied
by 2 when taking the first derivative;
MQ2 = quadratic term in the second-crop equation multiplied
by 2 when taking the first derivative;
.and the rest of the terms are as defined above.
Profits are maximized when marginal cost equals marginal
revenue, provided the second-order condition of profit
maximization holds. After taking the first partial derivative
with respect to the fertilizer variable, the two response
functions plugged into the profit equation do not include the
intercepts. Since the intercepts represent the yield with no
fertilizer, the result obtained is the profit figure from the
extra production generated by the fertilizer application:
p [Pl*(L1*F-Ql*F2)] + [(P2*(L2*F-Q2*F2))*DF] (CF*F) -
[(Ll*F-Ql*F2)*ACl] [((L2*F-Q2*F2)*AC2)*DF];
where Q1 = quadratic term in the first-crop equation;
Q2 quadratic term in the second-crop equation; and the
rest of the terms are as defined above.
The quadratic response functions used in the example run
were taken from Alvarez et al., who adapted them from Anderson
et. al., and pertain to a rice-sugarcane rotation practiced in
the Everglades Agricultural Area of Florida. The rice equation is
Y = 6284 + 722.1S 39.5S2, and the sugar equation is Y = 10490 +
981.6S 75.8S2.
The sensitivity index for each variable is calculated by
dividing the percentage change in profits by the percentage
change in the variable under consideration, holding the remaining
variables at their original value. Since this is done for a 20%
above and below the input figure, the two resulting numbers are
added and then divided by two.
The interpretation of the sensitivity index is similar to
that of an elasticity. When the sensitivity index is greater than
one, the percentage change in net returns is greater than the
percentage change in the variable under consideration. When the
sensitivity index is less than one, the percentage change in net
returns is less than the percentage change in the variable under
consideration. When the sensitivity index is equal to one, the
two percentage changes are equal to each other. The sign
indicates the direction of change. For example, a 2.62 index for
the price of the first crop means that, for each positive
percentage change in that price, net returns increase by 2.62%,
while they decrease by 2.62% when the price decreases by one
percent. A negative sign represents just the opposite.
The user should keep in mind that the purpose of the
sensitivity indexes is to compare the relative importance of each
variable for a particular set of values. The input variables
should be changed within a reasonable range of values in order to
obtain meaningful results. It is also important to notice that
the net returns curve need not have the same sensitivity index
over every part of the curve. For that reason, the sensitivity
index changes-every time variable values are changed.
Finally, the sensitivity indexes for the interest rate and
number of years are not discount factors, although the latter are
used in their calculations. For that reason, these indexes do not
show the expected negative signs that would indicate an inverse
relationship between these variables and net returns. Finally,
the two are almost always identical because the values of the
variables in the equations where they are calculated are always
the same while the differences obtained in the discount formula
are minor ones. The small difference between the interest rate
and years variables disappears due to rounding.
APPENDIX B: EXAMPLES OF DATA SOURCES
Abshahi, A., F.J. Hills and F.E. Broadblut. "Nitrogen
Utilization by Wheat from Residual Sugarbeet Fertilizer and Soil
Incorporated Sugarbeet Tops," Acronomy Journal 76 (1984): 954-
958.
Baldock, Jon D., Roger L. Higgs, William H. Paulson, Joseph
A. Jackobs and William D. Shrader. "Legume and Mineral N Effects
on Crop Yields in Several Crop Sequences in the Upper Mississippi
Valley," Agronomy Journal 73 (1981): 885-890.
Hargrove, W.L., J.T. Touchton and J.W. Johnson. "Previous
Crop Influence on Fertilizer Nitrogen Requirements for Double-
Cropped Wheat," Agronomy Journal 75 (1983): 855-859.
Hesterman, O.B., C.C. Sheaffer, D.K. Barnes, W.E. Lueschen
and J.H. Ford. "Alfalfa Dry Matter and Nitrogen Production, and
Fertilizer Nitrogen Response in Legume-Corn Rotations," Agronomy
Journal 78 (1986): 19-23.
Levin, A., D.B. Beegle and R.H. Fox. "Effect of Tillage on
Residual Nitrogen Availability from Alfalfa to Succeeding Corn
Crops," Agronomv Journal 79 (1987): 34-38.
Sharpe, R.R., J.T. Touchton, F.C. Boswell and W.L. Hargrove.
"Effect of Applied and Residual P on Double-Cropped Wheat and
Soybean under Conservation Tillage Management," Agronomy Journal
76 (1984): 31-35.
Sutherland, W.N., W.D. Shrader and J.T. Pesek. "Efficiency
of Legume Residue Nitrogen and Inorganic Nitrogen in Corn
Production," Agronomy Journal 53 (1961): 339-342.
This publcadon was produced at a cost of $1986.38. or 3.01 cents per copy. to demonstrate a micrcomputer
program for use in performing economic analysis of fertilizer response in crop raltone. 7-66-88
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