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Copyright 2005, Board of Trustees, University
of Florida
Circular 474
MARGINAL ANALYSIS:
A FARM MANAGEMENT TECHNIQUE
Jose Alvarez
Florida Cooperative Extension Service
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
John T. Woeste, Dean for Extension
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TABLE OF CONTENTS
Page
LIST OF TABLES . . . .... . ii
LIST OF FIGURES. . . . . . ii
MAXIMIZATION OF PROFITS FOR THE SINGLE ENTERPRISE. . .. 1
How Much Fertilizer to Apply on One Crop; or The Single
Variable InputSingle Output Production Process . 2
Profit Maximization Criterion. . . . 7
Change in Input Prices . . . . 8
Change in Product Prices . . . .. 9
What Fertilizer Combination to Apply on One Crop; or The Two
and More Variable InputsSingle Output Production Process 9
MAXIMIZATION OF FARM VS. ENTERPRISE PROFITS. .... .... 10
Production Possibilities. . . ....... .. ..10
Profit Maximization Criterion . . . .. 14
REFERENCES . . . . . . 18
LIST OF TABLES
Table Page
1 Effects of applied nitrogen on the yield of grain sorghum
in west Florida . . . ... . 4
2 A method to determine the most profitable level of fertil
izer application on the grain sorghum example . 6
3 Combinations of lime and superphosphate that will produce
five metric tons of desmodium per hectare . .. 11
4 Effects of applied nitrogen on the yields of grain sorghum
and eggplant. . . . . ... . 12
5 Combinations of grain sorghum and eggplants that can be
produced with 448 kg of elemental nitrogen per hectare. 13
6 Effects of applied lime on the yields of centro and
desmodium in south Florida. . . . .... 15
7 Combinations of centro and desmodium that can be produced
with 3,360 kgs of lime per hectare. . . ... 16
LIST OF FIGURES
Figure Page
1 Production function for the fertilizersorghum relation
ship described in Table 1 . . . . 5
MARGINAL ANALYSIS:
A FARM MANAGEMENT TECHNIQUE
Jose Alvarez
The relationship between inputs and output is expressed in terms of
a production function. For each set of inputs there is a corresponding
maximum level of output associated with it. This Circular focuses on
the theoretical basis for determining the economic level of inputs to use
in an enterprise assuming the farmer attempts to maximize profits.
The technique used is called marginal analysis. It is a valuable
tool for determining the most profitable combination of resources and
products. Since the concern is with the last added or marginal unit of
input and product, profits are maximized when marginal returns and
marginal costs are equated. Producers can use marginal analysis in many
of their management decisions. For example, how much fertilizer should
be applied to one crop and how the answer varies when fertilizer cost or
product price change. When confronted with two crops, and a limited
amount of fertilizer, how much should be applied to each one. These,
and other examples, will be developed to demonstrate how to use this
technique.
MAXIMIZATION OF PROFITS FOR THE SINGLE ENTERPRISE
A common decision a farmer has to make is the selection and level of
inputs to use in producing a crop. A wide variety of inputs is usually
JOSE ALVAREZ is Area Economist, Food and Resource Economics Depart
ment, University of Florida, Agricultural Research and Education Center,
Belle Glade.
required in any production process. There are many qualitatively different
types of labor, capital and materials that can be used to produce any out
put. Some remain fixed while others may be varied. The problem the farmer
faces is to choose the most economic combination of inputs based on the
physical quantities and prices of inputs and output.
How Much Fertilizer to Apply on One Crop;
or The Single Variable InputSingle Output Production Process
Let us consider a farmer who wants to devote his farm to grain
sorghum production. Once the decision to grow sorghum is made, the cost
of seed, chemicals, machinery, labor, interest, land, etc., are fixed.
One of the variable costs is fertilizer. Adding more fertilizer will
increase yields up to a point. Beyond this point, additional fertilizer
can damage the plants and reduce yield as the result of the law of
diminishing returns. (This law states that, as increasing amounts of
a variable input are applied to a fixed quantity of the other inputs,
the amount added to the total product by each additional unit of the
variable input will eventually decrease.) The question is how much to
apply.
A farmer deciding whether to add a little more fertilizer to his
sorghum crop needs two pieces of information. First, he needs to know
how much the extra fertilizer will cost; and secondly, he needs to know
how much more valuable his sorghum crop will be as a result of the added
fertilizer. If the value of the added sorghum is greater than the added
cost, a farmer interested in making money will add the fertilizer. The
example seems like straightforward common sense, and it is. It also
demonstrates the basic principle of marginal analysis applied to farm
management problemsthe equimarginal principle, that is: a decision
maker should keep using additional units of a productive input as long
as the added input earns or saves more money than it costs.
The principle can be stated more precisely by using symbols. Let X
stand for the productive input (fertilizer in our example), Y stand for
the physical output (sorghum in our example), and AX and AY stand for
"the change in X" and "the change in Y" respectively, and P stand for
price. The equimarginal principle can be stated now as
P AY P
Y X x (1)
This criterion tells that the price of sorghum times the change in sorghum
output over the change in fertilizer must be equal to the price of fertil
izer.
In order to estimate the value of the additional sorghum we need
data showing the relationship between fertilizer applications and levels
of output.1 Data in Table 1 portray such production function. When
plotted, the data illustrate the law of diminishing returns (Figure 1).
We now will show the steps to follow in determining the level of fertil
izer use that maximizes the farmer profits.
Knowing the quantities of input and output, their respective changes
can be written halfbetween the interval for which they are computed.
Assuming the price of sorghum to be $0.077 per kg ($0.035/lb), and the
price of fertilizer $0.44 per kg ($0.20/lb), Table 2 can easily be com
puted.
The Florida Agricultural Experiment Stations Annual Reports contain
a good number of references for many crops in different areas of the state.
Caution should be exercised when determining if the data found apply to
the situation being analyzed.
Table 1.Effects of applied nitrogen on the yield of grain
west Florida
sorghum in
X Yb AX AY AY/AX
Input,
nitrogen Output, Additional Additional
Fertilizer applied grain sorghum input output (kg sorghum/
level (kg/ha) (kg/ha) (kg of N) (kg/ha) kg of N)
0 0 4639
112 741 6.62
1 112 5380
112 215 1.92
2 224 5595
112 48 0.43
3 336 5547
112 358 3.20
4 448 5189
aExperiments were
Florida.
conducted at West Florida Experiment Station, Jay,
bEach value is the average of four replications.
Source: [1].
5,650
E 5,300
S.
o
1
o 4,950
I,,
4
4,600
0 100 200 300 400
Input (kg of nitrogen/ha)
Figure 1.Production function for the fertilizersorghum relationship described
in Table 1
Table 2.A method to determine the most profitable level of fertilizer
example
application on the grain sorghum
P P P AY P YP X
y x yX y x Fertilizer Sorghum
Fertilizer (Price of (Cost of cost increases price increases
level grain sorghum) fertilizer) (VMP) (Budgeting) (Budgeting) (Budgeting)
 $/ha  /ha $/ha
0 0.077 0.44 357.20  357.20  1,113.36
,_____________________________________________________________________, L
0.51
1 0.077 0.44 364.98 352.66 1,241.92
0.15
2 0.077 0.44 332.25 307.61 1,244.44
iJ
0.03
3 0.077 0.44 279.28 242.32 1,183.44
0.25
4 0.077 0.44 202.43 153.15 1,048.24
Profit Maximization Criterion
Determining the fertilizer level at which profits are maximized is
now easy. Tables 1 and 2 show that, as more units of fertilizer are
applied, the total product and the total revenue, and, therefore, the
value of the marginal product (VMPthe change in output times the price
of the output) and net revenue, increase and then decrease as a result
of the law of diminishing returns. At $0.44/kg ($0.20/lb), total costs
increase by a constant amount (assuming that the fertilizer cost includes
application cost and that the price does not change as additional units
are used). It is around fertilizer level one where the value of the
marginal product (VMP) equals the price of fertilizer. It is at that
point where returns above fertilizer costs are maximized. The budgeting
method of the last column of Table 2 can be used to check that conclusion.
The highest return above fertilizer cost ($364.98) is obtained when 112 kg
of fertilizer are used. This is a turning point: If fewer kg are used,
the VMP of fertilizer is greater than the price of fertilizer and thus
it pays to keep adding more units. If more units are used, the VMP is
less than the price of fertilizer and therefore added costs are greater
than added returns. At 112 kg, the additional cost of fertilizer equals
the additional revenue that it yields. Interesting, it does not pay to
fertilize for a maximum output5,595 kg as opposed to 5,380 kg. Indeed
it will only pay to produce at that level when fertilizer is free.
There are different ways of solving this problem, all yielding
the same result. We need, of course, to use only one method to arrive
at the answer. We have used the VMP= Px (equation #1) and the
"budgeting method" (total returns minus fertilizer costs) to check the
result. As long as prices remain constant, the answer will be the same.
But input and product prices are constantly changing and it is important
to consider how these price changes affect the most profitable level of
input use.
Price changes cause no problems. Simply use the new or expected
prices and rework the analysis. Since the turning point is identified,
we likely only need to calculate new values for input levels close to
the turning point.
Change in Input Prices
As additional units of inputs are added, the marginal product de
creases under conditions of diminishing returns. The value of the marginal
product thus dictates how input usage should be varied in response to
price changes.
Assume the price of fertilizer has increased from $0.44 to $0.55/kg
(or $0.25/lb). As a result, the equality does not hold anymore since the
VMP is now less than the price of fertilizer. Since the producer cannot
change any of the prices, he has to increase the marginal product (get
more sorghum out of the last unit of fertilizer) and make it equal to the
price of fertilizer. He can only do that by reducing the use of fertilizer;
down to around zero in this case (Table 2).
After the adjustment, returns above fertilizer cost peak at $357.20
when zero fertilizer is applied. Therefore, when the price of an input
increases relative to the price of the product, a reduction in input use
(and therefore in output) is necessary in order to maximize returns above
input costs. The result is obvious when we look at equation (1): If
P increases, AY has to decrease in order to balance the equation again
x A
since the farmer generally can not affect the level of P The reverse
holds true for a decrease in the price of the input.
Change in Product Prices
A similar situation exists when P increases relative to P Assume
y x
the price of sorghum increase to $0.24/kg (or $0.11/lb), an unlikely
event but it makes our point in this case. The VMP is now greater than
the price of fertilizer. It is necessary to increase the level of
fertilizer use in order to increase the value of MP. $1,244.44 is now
the highest return above fertilizer cost, which can be obtained by in
creasing the use of fertilizer up to 224 kg/ha (Table 2). A decrease in
the price of grain sorghum would require a reduction in the level of
fertilizer used.
What Fertilizer Combination to Apply on One Crop;
or The Two and More Variable InputsSingle Output Production Process
Many farmers face the problem of producing one crop by varying more
than one single input. Let us consider the situation of a legume pro
ducer who needs five metric tons of desmodium per hectare to feed his
cattle. He wants to determine the lowest cost fertilizer program of
lime and superphosphate for that level of yield. The research data in
Table 3 illustrate this situation.
As the farmermoves from combination one to combination two, he
must add 15 kg of superphosphate per ha. However, he saves 30 kg of
lime per ha. A farmer interested in lowering cost will be willing to
switch to combination two if the 15 kg of superphosphate are worth less
than the 30 kg of lime, or if PxAX1 < Px2 JAX21, where the vertical bars
x1 2^ c
on AX2 mean "the absolute value of AX2," or "ignore the minus sign."
Assume P = $.11 per kg and P = $.012 per kg. Then the 15 extra
kg of superphosphate cost him $1.65 while he saves $0.36 by giving up
30 kg of lime. Since he would spend more money by switching to com
bination two, he is willing to stay at combination one.
The budgeting procedure shows this fact: By minimizing the last
column of Table 3, the budgeting equation (P Y P X) is maximized.
This procedure offers an easy method of solving small problems.
MAXIMIZATION OF FARM VS. ENTERPRISE PROFITS
In most cases the farm manager's problem is maximizing profits from
two or more enterprises with limited resources. Some examples are: A
rancher has $5,000 to buy winter feed for his herd; should he buy corn,
hay or molasses? Should he buy a combination? If so, how much of what?
Another example is a rancher with three types of pasture. How much
fertilizer should he put on each?
Let us consider a farmer who has 448 kg of elemental nitrogen per ha
(400 Ib/A). He wants to know how much of it he should apply to his
sorghum and eggplant crops (single variable inputtwo output production
process).
Production Possibilities
The sorghum and eggplant production functions are given in Table 4.
The 448 kg of nitrogen can be put on one hectare of each crop in the
combinations shown in the first and second columns of Table 5. The third
and fourth columns show corresponding yields of each crop that will be
produced with each allocation.
Table 3.Combinations of lime and superphosphate that will produce five
metric tons of desmodium per hectarea
Superphos Nutrient
Combi phate Lime costs
nation Yield X1 X2 AX1 AX2 P xAX1 Px2 AX 2 Pxl X+P x2X
kg/ha $
1 5000 30 880 13.86
15 30 1.65 0.36
2 5000 45 850 15.15
aExperiments were conducted at the Agricultural Research Center in
Ft. Pierce, Florida. The estimated equation from the experiments was:
Y = 485.5 + 3080.1 L 922.7L + 2660.9 P 580P2 + 401.5LP, where Y
is output in terms of kg/ha, L and P are kg/ha of lime and superphosphate,
respectively.
Source: [2].
Table 4.Effects of applied nitrogen on the yields of grain sorghum and
eggplant
Xl Y1 Y2
(kg of elemental (kg of sorghum/ha (kg of eggplants/ha
nitrogen/ha) for given level of N) for given level of N)
0
112
224
336
448
4,639
5,380
5,595
5,547
5,189
4,620
22,546
23,063
15,893
8,612
aData for 336 and 448 kg/ha have been estimated for clarity of pres
entation.
Source: [1, 3].
Table 5.Combinations of grain sorghum and eggplants that
with 448 kg of elemental nitrogen per hectare
can be produced
Nitrogen Y1 Y P P P Y +P Y
allocation Y2 y1 2 2
Y1 Y2 Sorghum Eggplant Price of Price of
Sorghum Eggplants output output sorghum eggplant Total returns
kg/ha $/kg  $/ha 
448 0 5,189 4,620 0.077 0.26 1,600.75
336 112 5,547 22,546 0.077 0.26 6,289.08
224 224 5,595 23,063 0.077 0.26 6,427.09
112 336 5,380 15,893 0.077 0.26 4,546.44
0 448 4,639 8,612 0.077 0.26 2,596.32
Profit Maximization Criterion
Assume the price of sorghum to be $0.077 per kg ($0.035/lb), and the
price of eggplants to be $0.26/kg ($0.12/Ib). Again the most profitable
way to use the nitrogen resource can be determined by several means. The
budgeting approach indicated in the last column of Table 5 is the easiest
and most common sense approach. Since the same amount of nitrogen (448 kg)
and therefore the same cost are used for all possible combinations of
sorghum and eggplants, all we need to do is maximize the total returns
equation:
Total returns = P Y1 + P Y
y11 y y2 (2)
It becomes clear that the most profitable combination is the third
one (224 kg of N on each crop). That combination brings a total gross
revenue of $6,427.09.
Let us now work on an example when the fertilizer is not equally dis
tributed between two crops. Assume the farmer is a legume producer. He
knows the production functions for centro and desmodium (Table 6). He has
3,360 kg of lime to apply to both crops. The first and second columns
of Table 7 show some of the feasible combinations. The third and fourth
columns show the corresponding yields. The farmer wants to sell his out
put as hay and the price of both centro and desmodium is $55/ton, or $.06/kg.
If he wants to maximize total returns, the last column of Table 7 indicates
$670.14 as the highest attainable gross revenue when 1,120 kg of lime
per ha are applied to his centro crop and 2,240 kg /ha to his desmodium
crop. In this example, he needs to allocate different amounts of his
input between both crops in order to maximize total returns.
Table 6.Effects of applied lime on the yields of centro and desmodium
in south Florida
X1 Y1 Y2
(kgs of centro/ha (kgs of desmodium/ha
(kgs of lime/ha) for a given level of lime) for a given level of lime)
0 3004 2516
1120 4577 5477
2240 5542 6592
3360 5901 5861
Source: [2].
Table 7.Combinations of centro and desmodium that can be produced with 3,360 kgs of lime per hectare
Lime Y Y AY AY P AY P AY P lY+P Y2
aloao 1 2 1 2 y 1 y2 2 y 1 y2 2
allocation 1
Centro Desmodium
Centro Desmodium output output
 kg/ha
3,360 0 5,901 2,516 505.02
359 2,961 21.54 177.66
2,240 1,120 5,542 5,477 661.14
965 1,115 57.90 66.90
1,120 2,240 4,577 6,592 670.14
1,573 731 94.38 43.86
0 3,360 3,004 5,861 531.90
The opportunity cost principle can also be used to obtain the solution.
This principle, anotherform of the equimarginal principle, states that
for profit maximization, the marginal value product of a resource used on
one activity must be equal to the marginal value product of the resource
used in all other alternatives:
P AY P AY
1 1 = 2 2 ; or P AYI = PY AY
AX1 AX1 Y 1 y2 2 (3)
When Y1 and Y2 compete for a fixed amount of a resource, the only way
we can change Y1 is by changing Y2 in the opposite direction. The condition
stated by the last equation is fulfilled at the combination of 1,120 kg
applied to the centro crop and 2,240 kg to the desmodium crop of Table 7.
The twooutput case can now be generalized when dealing with several
products. Using the last equation, the profit maximization conditions for
n number of products, ignoring signs, would be
P AY = P Y2AY = Py3 AY = ..."' = Pn Yn (4)
1 1 y2 2 y3 3 y n (4)
In the problems discussed so far, the farmer is faced with only a few
alternatives. In these cases, the budgeting method provides the easiest
way of finding the answer.
18
REFERENCES
11] Lutrick, M. C. "Preliminary Report on the Response of Grain Sorghum
to Applied Nitrogen," Soil and Crop Science Society of Florida
30(1970): 4650.
[2] Snyder, G. H., et al. "Field Response of Four Tropical Legumes to
Lime and Superphosphate," Agronomy Journal 70(1978): 269273.
[3] Sutton, Paul and E. E. Albregts. "Response of Eggplant to Nitrogen,
Phosphorous, and Potassium Fertilization," Soil and Crop Science
Society of Florida 30(1970): 15.
This public document was printed at a cost of $98.07, or 14 cents per copy, to help the public with farm management.
2 70080
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