ANALYSIS OF ONFARM RESEARCH
AEB 5167
Economic Analysis in Small Farm Livelihood Systems
Fall Semester, 1995
PRODUCTION FUNCTION
EXERCISES
Peter E. Hildebrand
Food and Resource Economics Department
University of Florida
Gainesville, FL 326110240
AEB 5167
Fall 1995
ANALYSIS OF ONFARM RESEARCH
PRODUCTION FUNCTION EXERCISES
In order to reflect realistic responses, many kinds of research whose purpose is to generate technology, must be
conducted on the kinds of farms where the technology is expected to be useful. This can create much more
variability in the data than does research conducted on experiment stations where many factors not included as
treatments are controlled. Often, nontreatment factors onstation are also controlled at levels high enough so
that they do not limit the potential of the variables being tested. The result is to create environments that are
much less variable and much more productive than those found on most farms.
For researchers trained only in onstation research, the lack of control over nonexperimental variables and the
resulting high CVs can be exasperating. Many believe that "good" research cannot be done on farms for these
reasons. On the other hand, it is becoming well recognized that farmers seldom can use, directly, research
results from experiment stations. One kind of research results that seldom can be extrapolated successfully from
experiment stations to farms is fertility research. Because experiment stations have been used for research, the
soils have been modified by amendments to the point they no longer resemble the soils on most farms.
Following are a series of exercises to familiarize you with the nature of onfarm research results. The data are
taken from a real onfarm maize fertility trial conducted by CIMMYT in Mexico. In the trial, there were four
farms each with three replications of all 12 treatments and the trials were conducted two different years.
Unfortunately, only the individual treatment averages by farm and by year are available. Furthermore, the data
on the characteristics of the individual fields where the trials were located are not available nor are climatic data
for each year. Nevertheless, this is an excellent data set to work with. Therefore, these data will be used to
show a number of ways the results can be analyzed and interpreted. The first series of exercises will look at
the data set as a whole. The second series will consider groups of farms so that more specific recommendations
can be made.
516795/FAEIEXER
AEB 5167
Fall 1995
Exercise FA1
Below are data generated from a fertilizer trial on maize conducted on 4 farms over two years by personnel from
CIMMYT in Mexico. Notice that in this trial, there are four levels of N and three of P25O. It is a 3 x 4 factorial (with
12 plots per block). There were three blocks or replications per farm, but only the average of the three blocks for each
treatment are available and shown in the table.
Table 1. Maize yields (kg ha' of 14 percent moisture grain) by fertilizer treatment, 4 farms, 2 years
Fertilizer treatment (kg ha'1)
N: 0 50 100 150 0 50 100 150 0 50 100 150
P205 0 0 0 0 25 25 25 25 50 50 50 50 Avg.
Fm/Yr
1/A 400 1240 3630 3760 0790 2580 4230 4720 1670 2510 3280 3660 2710
2/A 1530 2600 5140 5320 1670 3790 5100 6830 1410 4130 5890 6270 4140
3/A 4150 4860 4800 4870 4440 5000 4970 5280 5120 5660 6360 6620 5180
4/A 2420 3820 5230 4480 2360 4540 6260 7170 1610 4410 5380 6580 4520
1/B 1640 1920 2080 2190 2040 3210 3120 2930 1440 3440 3320 3620 2580
2/B 1610 2940 4140 4340 1810 3920 3610 3810 1180 3890 5380 4920 3460
3/B 4740 5410 4290 4920 4910 5220 5380 5140 5100 4880 4540 5280 4980
4/B 1210 2330 1970 2230 1530 2780 2490 2800 1370 3510 3750 4350 2530
A04 40/f ;3 4 431'" 4W 13(2 4ps 4jt31 3;* 4
Avg. 12210 3140 3910 4010 2440 3880 4400 4840 2360 4050 4740 5160 3760
Source: Perrin, Richard K., et al. 1976. From Agronomic data to farmer recommendations. An economics
training manual. CIMMYT. Information bulletin 27.
1. First look at the data. Are the farms and years quite similar or are they different? If they are quite different, we
should probably look at them as if there were going to be more than one recommendation domain.
2. Either way, let's begin to examine the data by looking at the overall average for all farms. First, let's look at the
response of the maize to P205. Notice that we have three levels of 1? 9 for each of the four levels of N. Begin by
summarizing the average data for different levels of N as follows:
Maize response for:
P105
0
25
50
N=0 N=50 N=100 N=150
Yield (Mg ha')
2.21
2.44
2.36
3.14 3.91 4.01 \
3.88 4.40 4.84
4.05 4.74 5.16
3. Now, plot the P205 response data for N = 0 on a graph and calculate a quadratic the response equation (production
function) using the visiographic procedure.
4. Repeat this process for the other three levels of N.
,A) J /2
AEB 5167
Fall 1995
Exercise FA2
1. Load the data from the table in Exercise No. 1 in a spreadsheet. Use the same orientation as in Table 1, i.e., farms
and years are the rows and treatments are the columns. You can calculate the averages as a means of verifying accuracy
of the data entered. This data set will be the basis for a number of analyses we will be doing. Be sure to save it in this
form.
2. Now we will set up the data to facilitate estimation of the production functions from Exercise No. 1 on the computer
using the data from all eight farms. By copying and moving your data, set up another working table as below in order
to calculate the response to P205.
Farm No. P205 (P205)2 N = 0 N = 50 etc.
I/A 0 0 400 1240
2/A 0 0 1530
1/A 25 625 790
2/A 25 625 1670
1/A 50 2500 1670
2/A 50 2500 1410
4/B 50 2500 1370 3510 cc) pI
You should have 24 rows of data (4 farms x 2 years x 3 levels of P205). Be careful not to include the averages!
3. Using markers in an xy graph format, look at the data for N = 0 (compare yield only with the levels of P2Os, not
the squared values). Discuss.
4. You can now estimate the equation by regression using QuattroPro with the two columns for PzO5 and (P20)2 as the
independent variables and the column for yield as the dependent variable. The equation should be the same as the first
you estimated in Exercise No. 1 by the visiographic method.
5. Repeat for the other three levels of N. Are all the equations the same as you got from the visiographic method?
They should be.
6. Using the spreadsheet, show the four equations on a single graph. Can you figure out how to do this and get a
smooth curve, not just two straight lines going through three points?
7. Interpret the results.
AEB 5167
Fall 1995
Exercise FA3
1. For each of the production functions in Exercise FA2, find mathematically where production is maximum. Do these
correspond with your graphs?
2. For prices of maize of $lM00OpurMg ($ is no particular currency), N of $8"per kg and P205 of $10 per kg, find for
what level of phosphorus profit is maximized. Repeat for each level of N. Discuss and interpret the results.
3. If you have time, you should repeat exercises FA1 to FA3 varying nitrogen for each level of phosphorus.
'[p
0
AEB 6933
Spring 1995
Exercise FA4
Factor x Factor
So far we have analyzed this data set by finding responses to phosphorus for set levels of nitrogen (FA1, FA2 and FA3)
or by finding responses to nitrogen for set levels of phosphorus (FA33). By doing this, it is possible to answer
questions such as, "If I apply 25 kg ha1 of P205, how much nitrogen should I apply?" But it does not answer the
question of how much of each amendment is best. To do this requires the analysis of nitrogen and phosphorus
simultaneously.
1. To obtain a first estimate of the nature of the 3dimensional surface (shown in two dimensions) construct a graph
with nitrogen on the vertical (Y) axis and phosphorus on the horizontal (X) axis. Use the values from the experiment as
shown in the table in Exercise FA1. Then using the averages from each treatment, write the corresponding yields for
each NP combination at the intersection in the graph. For example, the value for 00 (NP) is 2210 and for 5025 is
3880. Follow the visiographic procedure from Hildebrand and Poey, pp. 108113.
2. Draw in isoquant contours for 2500, 3000, 3500, etc.
3. Using the prices from Exercise FA3, draw in the isocost contours and the expansion path.
4. Interpret your work.
A<
AEB 6933
/ Spring 1995
Exercise FA5
Factor x Factor
In the procedure used in Exercis FA4, we could determine the best combination of N and P to use, but would need to
make successive approximations o determine the most profitable combination of the two for any price combination. In
order to be more "precise" a ma ematical production function incorporating both amendments can be calculated by
regression. The form we will us here is quadratic (with an NP interaction term).
1. In a spreadsheet, set up e data in th following form:
N P N2 P2 NP Y
0 0 0 0 0 400
50 0 2500 0 0 1240 .
100 0 10000 0 0 3630
150 0 22500 0 0 3760
0 25 0 625 0 790
50 25 2500 625 1250 2580
etc. etc. etc. etc. etc. etc.
By using all the data, you should have 96 (8 X 12) rows. Calculate the production function Y = f(N,P,N2,P2,NP) by
blocking the first five columns as independent variables in the regression menu. In the output, the coefficients are in the
same order as the columns.
2. Determine mathematically, for what quantities of N and P roducti i maximized. How much maize is produced
with these quantities of N and P? Do these values correspond with your graph from Exercise FA4? Explain any
differences.
3. Using the same prices as before, find mathematically the quantities of N and P w ih maximize profit. How much
maize is produced at this level of fertilizer use? Do these quantities fall on the expansion path from your graph from
Exercise FA4? Explain any differences.
GROUPING FARMS INTO
RECOMMENDATION DOMAINS
Introduction
In the previous exercises, we have been looking at all the farms as a single group. Any recommendations made from
these analyses would presumable apply to all the farms in the sample and all other similar farms. However, it is evident
from the data in Exercise FA1 that the farms (or at least the fields) where the trials were conducted are quite different.
The environments in the fields from farms 1A, 1B and 4B, for example, are obviously poorer for producing maize than
are the environments in the fields from farms 3A, 4A and 3B. Unfortunately, as mentioned in the introduction to these
exercises, no information is available on the characteristics of the environments at these locations. We do not know if
the lower yields are associated with less rainfall, poorer fertility, late planting, less effective weeding, or what. But we
do know that for some reason there are distinct environmental differences in the fields and on the farms where these
trials were planted. Even though these differences are real, and form the real world of the farmers, they are the reasons
that many researchers complain about doing research on farms. However, in order for us to have confidence in the
conclusions we make regarding maize response to N and P, for example, it is necessary to conduct such trials on farms
and under real farm conditions. Because these conditions and the growing environments they create are so highly
variable, it is necessary to have a means to separate recommendations into specific domains of similar environments and
farmer needs.
Without minimizing the importance of characterizing each environment where onfarm trials are conducted, there is a
convenient means available to quantify the evident environmental differences. This is to convert the overall farm (field)
average (over all treatments) into an index that reflects quality of the environment in each field and differentiates it from
the environments in other fields. Thus for example, for farm (field) 1A, the average yield is 2710 kg ha"1. This
converts into an index (without units) of 2710. The index for farm (field) 2A is 4140. It is quite obvious that
environment 2A, representing farm or field 2, with an index of 4140 is a better environment for producing maize than
environment 1A with an index of 2710. Even though some statistical purists feel the use of this index in regression is
marred, it has been in use for over 50 years and has been invaluable in the absence of any other available method.
We will use the term El for this "environmental index" as a component of additional variables in the production function
from the previous exercise..
AEB 5167
Fall 1995
EXERCISE EI1
1. To the table in Exercise FA5, add four more columns of data: El, NEI, N2EI and PEI.1 Then blocking all nine
columns as independent variables, calculate the production function:
Y = f(N,P,N2,P2,NP,EI,NEI,N2EI,PEI)
Compare the R2 values for the two equations and the deviations from regression (Std Err of Y Est). How do you
explain the differences?
2. What would you expect this equation to look like if you set the El equal to the average El for all farms (3760 from
the table in Exercise FA1)? Do this and compare the resulting equation with the equation from Exercise FA5. Did you
expect the similarity? Explain why this happens. Solve for maximum production and maximum profit. How do the
quantities of N, P and Y compare with those from Exercise FA5?.
3. Now set El to represent the poorest environments, say El = 2500. Notice how the equation changes. Solve this
equation for maximum production and maximum profit. Compare resulting values of N, P and Y with those from the
previous equation. Explain the results.
4. Do the same for an El representing high environments, say El = 5000. Compare with the results of the other
production functions.
5. Discuss the implications for making specific recommendations.
1 Note that other combinations could be used. These are the ones that are most significant and have
the greatest impact on the surface.
