Historic note
 Title Page

Group Title: Circular - University of Florida Institute of Food and Agricultural Sciences ; 474
Title: Marginal analysis
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00072493/00001
 Material Information
Title: Marginal analysis a farm management technique
Series Title: Circular Florida Cooperative Extension Service
Physical Description: ii, 18 p. : ill. ; 28 cm.
Language: English
Creator: Alvarez, Jose, 1940-
Publisher: Florida Cooperative Extension Service, Institute of Food and Agricultural Sciences, University of Florida
Place of Publication: Gainesville
Publication Date: 1980?
Subject: Farm management   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
non-fiction   ( marcgt )
Bibliography: Bibliography: p. 18.
Statement of Responsibility: Jose Alavrez.
General Note: Cover title.
Funding: Circular (Florida Cooperative Extension Service)
 Record Information
Bibliographic ID: UF00072493
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: oclc - 08851531

Table of Contents
    Historic note
        Unnumbered ( 1 )
    Title Page
        Title Page
        Page i
        Page ii
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
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        Page 13
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        Page 15
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        Page 18
        Page 19
Full Text


The publications in this collection do
not reflect current scientific knowledge
or recommendations. These texts
represent the historic publishing
record of the Institute for Food and
Agricultural Sciences and should be
used only to trace the historic work of
the Institute and its staff. Current IFAS
research may be found on the
Electronic Data Information Source

site maintained by the Florida
Cooperative Extension Service.

Copyright 2005, Board of Trustees, University
of Florida

Circular 474



Jose Alvarez

Florida Cooperative Extension Service
Institute of Food and Agricultural Sciences
University of Florida, Gainesville
John T. Woeste, Dean for Extension

B.- -

.,- -o

-2 /
rC~ V6
~r Clb


LIST OF TABLES . . . .... . ii

LIST OF FIGURES. . . . . . ii


How Much Fertilizer to Apply on One Crop; or The Single
Variable Input-Single 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 Inputs-Single Output Production Process 9


Production Possibilities. . . ....... .. ..10
Profit Maximization Criterion . . . .. 14

REFERENCES . . . . . . 18


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


Figure Page

1 Production function for the fertilizer-sorghum relation-
ship described in Table 1 . . . . 5



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



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 Input-Single 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


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 problems--the equi-marginal 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 equi-marginal principle can be stated now as

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-


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 half-between 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-


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

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

conducted at West Florida Experiment Station, Jay,

bEach value is the average of four replications.

Source: [1].


E 5,300

o 4,950



0 100 200 300 400
Input (kg of nitrogen/ha)

Figure 1.--Production function for the fertilizer-sorghum relationship described
in Table 1

Table 2.--A method to determine the most profitable level of fertilizer

application on the grain sorghum

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---------
1 0.077 0.44 364.98 352.66 1,241.92
2 0.077 0.44 332.25 307.61 1,244.44
3 0.077 0.44 279.28 242.32 1,183.44
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 (VMP-the 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 output--5,595 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 Inputs-Single 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.


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 input-two output production


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,

Source: [2].

Table 4.--Effects of applied nitrogen on the yields of grain sorghum and

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)
















aData for 336 and 448 kg/ha have been estimated for clarity of pres-

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


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 equi-marginal 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:

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 two-output 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.



11] Lutrick, M. C. "Preliminary Report on the Response of Grain Sorghum
to Applied Nitrogen," Soil and Crop Science Society of Florida
30(1970): 46-50.

[2] Snyder, G. H., et al. "Field Response of Four Tropical Legumes to
Lime and Superphosphate," Agronomy Journal 70(1978): 269-273.

[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): 1-5.

This public document was printed at a cost of $98.07, or 14 cents per copy, to help the public with farm management.
2 700-80

SCIENCES, K. R. Tefertlller, director, In cooperation with the United States Department of Agriculture, publishes this Infor-
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