FARM ORGANIZATION AND RESOURCE FIXITY: MODIFICATIONS
OF THE LINEAR PROGRAMMING MODEL
Peter E. Hildebrand
Agricultural Economics Department
Michigan State University
East Lansing, Michigan
November 10, 1959
TABLE OF CONTENTS
I INTRODUCTION,........ ......................................... ..
The Nature of Fixed Resources *.5.o..............,............
The Effect of Predetermined Resource Fixities Without Regard
to MVP ,**......6..........-.........-..................-0., 6
Endogenous Determination of Resource Fixity ................... 7
Some Previous Linear Programming Models Incorporating Various
Aspects of the Problem ..*.................................0 8
The Farm Situation and Credit Supply Function ................. 9
Thesis Organization ******.......**..* ..****.********..o****** 11
II THE ANALYTICAL MODEL *...........................0.............0 12
The Specialized Equations *..o.........................0....... 13
The Double Purpose Acquisition Activities .................... 17
The Credit Activities ..............*............,...........o 17
The Cash Coefficients *.........................0.............. 18
Specialization and Diversification and the Effect of a Single
Fixed Resource on the Solution ......................... 20
Discrete Investment Levels *2........................*......... 22
III APPLICATION OF THE MODEL ....,....,0...0......0...0 .............o 26
Crop Production .............................................. 26
Milk Production *................. *........................... 27
Derivation of the Technical Matrix and Restrictions ........... 28
Acquisition and Salvage ,......*,*...... .................... 28
The Range of Possible Solutions .2............................ 29
IV SOLUTION OF THE FARM MODEL .*......*........................... .0 30
The Initial Optimum Solution *.............. .... ..... ..... 30
The Discrete Investment Series ................................ 37
The Final Farm Organization *..,.o............oo..........0... 46
V SUM1ARY AND CONCLUSIONS ...........,..o....... ...*............... 49
Application of the Model 49................................. 9
The Empirical Results .**..*......,............. ........** ...* 52
Further Study Indicated **..................................... 03
BIBLIOGRAPHY .......................*............,.. .............o 55
LIST OF TABLES
2.1 Cash Coefficients for the Various Groups of Activities ,.........
4.1 Original and Optimum Inventories *.....*..., ....,...............
U42 Profit and Organization for 320 Acres and 360 Acres ,..........
4,3 Profit and Organization for 320 Acres with Four and Five Tractors,
4.4 Profit and Organization for 320 Acres, Four Tractors, With and
Without a Forage Chopper *................* ....,...*..........**
4,5 Profit and Organization for 320 Acres, Four Tractors, One Chopper
and Two and Three Corn Pickers **...,.... ..e.e,............C.....*
4.6 Complete Inventory Change: Original Organization to Final Farm
L.7 Comparison of Profit: Optimum Solution and Final Farm Plan ......
4.8 Disposable Income, Final Farm Plan ..**..,.,.,...............
B,1 Crop Activity Titles and Profit Coefficients ...............
B.2 Dairy Activity Titles and Profit Coefficients, Per Cow ........
B.3 Acquisition, Credit and Salvage Activity Titles and Profit
Coefficients .,....,** ...........** *............**** ...... ***
B,4 Initial Optimum Solution *.....................................
B.5 Optimum Solution--320 Acres ............... ....................
B.6 Optimum Solution-320 Acres, I Tractors .........................
B,7 Optimum Solution--320 Acres, U Tractors, 1 Chopper *..............
B,8 Optimum Solutionr-320 Acres, 4 Tractors, 1 Chopper, 3 Corn Pickers
B.9 Optimum Solution--Final Farm Plan................
B.10 Purchasable Assets: Price, Credit Terms and Depreciation ........
B.ll Cost of Machinery Repair .....................ee ...o....,...*..
B.12 Fertilizer Application and Crop Yield Estimates ...........*......
LIST OF TABIES--continued
B.13 Time Requirements for Field Operations .........................
B.14 Number of Field Working Days Per Month *.........................
B.1S Dairy and Crop Cash Costs ...................................
B.16 Dairy Labor Requirements ..................... .............. C
B.17 Rations and Production for the Milking Herd, Per Cow ,...........
B.18 Rations in Hay Equivalents and Corn Equivalents Per Cow Per Year,
Includes Replacements .......................
B.19 An Example of the Computation of Machine and Power Restrictions .
When the conventional linear programming problem is formulated with fixed
restraints, the level at which the resources of the/firm are fixed are of
primary concern because these restrictions indiqcte the boundaries of the
solution to the organizational problem of the firm. The linear nature of the
profit function of the linear programming problem would indicate infinite
production in the absence of these resource limitations.
To predetermine a set of fixed resources for any firm usually builds into
the optimal solution a certain amount of unrealism. Many of the assets of a
firm are not fixed in an economic sense, i.e., when the marginal value product
lies between acquisition and salvage values. A farm firm is constantly adjusting
many of the factors of production which are normally considered fixed in the
usual formulation of the linear programming model for analyzing the resource
allocation problems of the firm. Land is one of the most commonly fixed resources
in programming an optimal operation of a farm. Many farmers, however, rent,
buy and sell parts of farms or whole farms and recombine their land holdings.
An important consideration in determining the optimum organization of a farm is
to" find the right amount of land to combine with the other factors. Similarly,
all other factors are subject to acquisition and salvage and should be considered
so in determining an optimum farm organization. In addition to the ability of
the manager, important limits to farm size and organization involve the amount
of funds over which the manager can gain control and some reasonable limit to
the area in which land can be purchased.
A procedure allowing for variations in the initial asset structure of the
firm, therefore, is the principal goal of this thesis--i.e., to determine a
process whereby the resource restrictions in a linear program become endogenously
determined. The procedure involves the use of increasing factor supply functions--
primarily, that of the supply of credit-- and a differential between acquisition
and salvage prices of the factors. The approach involves essentially an
increasing cost function for credit.
A problem which always exists in the interpretation of the results of a
linear program involves the assumption of infinite divisibility of factors
and products. Infinite divisibility is particularly a problem when considering
investments in non-divisible assets such as tractors, silos, milking parlors
and buildings. Some non-divisible assets such as tractors can be rented by
time period and using such a method is satisfactory in certain problems.
However, when investment in buildings and silos, etc. is being considered,
renting in small units is undesirable or even impossible as a solution. An
arbitrary rule for dealing with indivisibility in investments is developed and
used in the thesis.
The Nature of Fixed Resources
In the most simple sense, fixed resources are those which cannot be or are
not varied in quantity. In an economic sense, fixed resources are those which
it does not pay to vary, i.e., those resources for which acquisition price is
greater than or equal to marginal value product which is, in turn, greater than
or equal to salvage value. In some cases, resources appear to be physically
fixed. This could be the case for an old building, possibly constructed of
stone or blocks or even of wood. It would appear that regardless of the MVP
of such a building, assuming it to be very low, it would never pay to salvage
it. This is an indication of a negative salvage value where a cost, greater
than sale value, is involved in removing the building from the farm. Since it
is not rational to produce where an MVP is negative, the building is, indeed,
a fixed factor, even if it is not used at all. If the returns from the use of
the land on which the building stands plus the sale value of the materials is
greater than the cost of salvaging plus the MWP of the building, it would, of
course, be salvaged. A factor is not fixed, then, if (1) the costs of removing
it are exceeded',.by ,thei a sm "bf expected revenues occurring as a result of its-
salvage, or (2) the costs of acquiring it are exceeded by the sum of expected
revenues occurring as a result of obtaining it. It is this principle which
is used in constructing the model for this thesis.
Another form of fixity which may be effective are institutional restrictions*
Acreage allotments may limit production of a given crop even though the MVP's
of the factors in producing the crop exceed their marginal factor cost. Using
wheat as an example, a combine may be fixed because no more than one is needed,
even though its MVP may be greater than its MFC. The amount of credit which
any firm can extend to an individual may also be limited by institutional
restrictions. It is this type of restriction which partially determines the
supply of credit available to a farmer.
The Effect of Predetermined Resource Fixities Without Regard to MVP
If a specific farm or "typical" farm is used as a basis for a linear
programming problem, and the given resources are fixed at the initial levels,
two types of error are likely to exist. A resource fixed in abundant amounts
can be utilized to the point where its MVP drops to zero, indicating that
salvage price is considered to be zero when it actually is greater than zero.
The other extreme is a resource fixed in short supply. In this case, the MVP
of the resource may be much higher than the MFC of another unit.
Both bases lead to a less than optimum allocation of resources. A factor
fixed in abundance will cause the program to select inefficient technologies
with respect to that factor. For example if labor is fixed in large amounts,
labor saving technology becomes unimportant. Similarly, highly restricted
factors will impose artificial requirements for technology favoring efficient
use of this factor. If adjustment in factor quantity cannot be based upon the
productivity of the factor, when, in fact no real barriers to adjustment exist,
less desirable solutions will result.
The solution of a linear programming problem nmputes values to the fixed
resources. These values are the MVP of the resource to the firm -- the amount
of income which the firm would gain or lose by buying or selling, respectively,
one unit of the resource. If the resources are artificially fixed, the imputed
value would be unreasonable if that value were greater than acquisition price
or less than salvage value. The true value of a factor to a firm is never less
than its salvage value since the firm could realize at least this amount if it
disposed of the factor in the market. Similarly, if the productivity is greater
than cost of acquisition (MFC) the firm would gain by purchasing and using more
of the asset.
A further undesirable characteristic of using fixed quantities of resources
in optimizing a farm organization is that the stock of capital and credit is not
converted into resources, but is used only for cash expenses for the completely
variable or non-durable factors (factors for which cost of acquisition equals
salvage value). In actuality, the stock of funds available to the firm is
convertible into stock resources as well as factors comprising the list of
Endogenous Determination of Resource Fixity
A linear programming model incorporating the endogenous determination of
resource fixity requires acquisition and salvage activities for all durable
resources. The acquisition and salvage of durable assets presents a stock-
flow problem since the use value of the asset during a time period is derived
from the flow of services available from the stock of the resource on hand.
Short term profit maximization would undoubtedly involve the sales of all owned
resources during the first time period. Therefore, it is essential that the
stock price be appropriately distributed over the series of time periods during
which its services would be available so that the costs from buying, and
returns from selling, correspond to the time period involved in the flow of
The costs of acquiring an additional unit of a durable asset for a one
year period are the annual depreciation, interest, repairs and taxes. The sum
of these four items rather than the market price is the annual marginal factor
cost to the firm of acquiring the asset.1 The corresponding annual salvage
value to the firm of selling the asset is the sum of the depreciation, interest,
repairs and taxes based on the salvage price of the asset at time of sale.
The MFC of a factor produced on the farm is the marginal cost of production
to the firm, or the market price of the last unit delivered to the farm whichever
is lower. So long as the MC is lower than the cost of purchasing the marginal
unit, it will pay the firm to produce the factor if more is desired. When MC
exceeds the cost of the marginal unit in the market, it will pay the firm to
purchase the factor.
The imputed value of resources given in a model incorporating endogenous
fixities will equal (1) annual cost of acquisition for all resources increased
in quantity, (2) annual salvage value for all resources decreased in quantity,
or (3) the manual value in use for all resources fixed at the original quantity
and neither purchased nor sold. Thus, all durable assets in this model receive
an imputed value based on the annual flow of services from it.
Some Previous Linear Programming Models Incorporating Various Aspe'ots of the
Many programming projects have been reported in the various Journals.
Most of them follow the standard pattern with but slight variation. Two models
which have been reported in the Journal of Farm Economics, while not closely
related to the model developed here, incorporate some of the aspects of the
problem under consideration.
1 For a fuller discussion of the pricing problem see footnote 1 on page 18.
Victor E. Smith has constructed a model which incorporates a price
differential between acquisition and salvage values for some factors and
products. He incorporates cash and credit into a lump sum to which is added,
in one model, the proceeds from hay sales. These funds are used to purchase
feeder stock, protein supplement and corn but not labor nor shelter which, in
addition to funds, are considered as fixed resources. In his second model the
buying and selling prices of hay and corn are differentiated.
Loftsgard and Heady2 develop a model to obtain a solution over a series of
years, "... with the optimum for any one year depending on the optimum in other
years, on the availability of and returns on capital in other years, on the need
for household consumption at different points in time, etc."3 This model is of
more interest as a suggested extension of the model developed in this thesis
than as an explicit aspect of it and will be discussed in this respect in a
later section. In their model, however, account is taken of investments added
to the initial inventory of durable goods and includes expenditures for
depreciation, taxes and insurance. They do not, however, include the problem
of endogenous determination of resource fixity.
The Farm Situation and Credit Supply Functions
The farm to be programmed is a "typical" central Michigan dairy farm
located on moderately productive soils (with Miami as the major soil series)
containing 160 acres of which 132 are tillable. Included are a fall line of
equipment with a PTO forage chopper, two field tractors and one "chore" tractor,
and a one row corn picker plus a 180 ton upright silo, a 32 stanchion barn which
meets Grade A market requirements and 32 cows and their replacements. The silo
1 Victor E. Smith, "Perfect vs. Discontinuous Input Markets," Journal of Farm
Economics. Vol. 37 (August, 1955), p. 538.
Laurel D. Loftsgard and Earl 0. Heady, "Application of Dynamic Programming
Models for Optimum Farm and Home Plans," Journal of Farm Economics, Vol. 41
(February, 1959), p. 51.
3 Ibid., p. 51.
is equipped with an unloader, but feeding is not automatic. It is considered
that the milking routine is set up for average efficiency but the farmer is
capable of managing a highly efficient organization including automatic silage
feeders and either a walk-through or a herringbone parlor.
Possible investments include new machinery of the same type already on the
farm, additional upright silos or bunter silos, either a walk-through or herring-
bone milking parlor, additional bulk tanks, more cows and replacements, and
automatic silage feeding bunks in the case of upright silos. Feeding from a
bunker silo is on a self-feeding basis for efficient operation and the investment
includes movable feeding gates for this purpose. In order to keep the farm an
entity, that is, not spread over too wide an area, 480 acres is the madxmum
amount of land considered available for purchase. No limit is placed on the
amount of the other resources which can be purchased except that imposed by
the availability of spendable funds.
The debt-asset structure of the farm includes a total asset value of
$45,090 with an estimated net worth of $36,000 and a debt of $9,090. The assets
are $7,545 in machinery, $10,545 in cattle plus $3,000 in a bulk tank and
$24,000 in land valued at $150 per acre. All initial debt was considered as
land mortgage at 5.5 percent interest. The total amount of land mortgage
available is 45 percent of current market value, $250 per acre, or $18,000.
Deducting the mortgage outstanding leaves $8,910 of land mortgage available.
In addition to the land mortgage available, credit is available for pur-
chasing the additional 480 acres of land. A 5.5 percent land mortgage is
available for up to 160 acres, requiring a down payment of 55 percent. Two
land contracts are considered available. One contract-requires 6 percent
interest, the other 7 percent; both require only 10 percent down payment. Each
contract can be used for as much as 160 acres purchased in 40, 80 and 120 acre
units. A chattel mortgage is available for $10,545 which is half the value of
the chattels and carries a 6.5 percent interest charge. The credit supply
function also includes $20,000 at 13 percent from machinery dealers and $14,000
at 9.4 percent from a silo dealer. Real estate credit is payable over a 20 year
period and all other sources of credit must be repaid in 3 years. Interest is
First (Chapter II) the analytical model is presented and discussed. In
Chapter III the problems of applying the model to the fazm situation are
discussed. The initial optimal solution, the succeeding solutions of the
discrete investment series and the final farm organization are presented in
Chapter IV. To simplify the material presented in the text, most of the
technical data and results are listed in the Tables of Appendix B beginning
on page 6).
THE ANALYTICAL MODEL
Many equations and activities in the model are of standard form, i.e., the
type usually used in a resource allocation model as applied to a farm firms and
should require no clarification other than description. Labor from April to
October inclusive is divided into monthly periods. November through March
labor is considered one resource. Tractor services, measured in hours, are
divided into the same monthly periods as labor. Machinery services are on a
monthly basis and their availability is specifically taken into account only
for those months in which they are required--there are no equations for equipment
service during months when that service is not required. The unit for measuring
machinery service is the number of acres which can be covered by that machine
in an eight-hour day, accounting for the number of days each month the land can
Since the unit of measure for the capacity of milking parlors is commonly
time per cow, the services from the parlors are measured in 100 hour units. All
other dairy equipment is measured on a per cow and replacement basis. Land is
measured in tillable acres and all monetary equations are in $100 units. The
crops produced on the farm are transferred into crop equations so that they
either can be sold or fed to the dairy stock. In contrast, the milk production
activities account for the sale of milk, since milk is not an input for other
In constructing the model it was found necessary to include several
specialized equations to handle satisfactorily, the investment activities. The
asset acquisition and credit activities also require explanation since they
contain some aspects peculiar to the model. Table 2.1 on page 19a should help
clarify the following narrative.
The Specialized Equations
Some difficulty is encountered in explaining the three sets of specialized
equations individually since there is a degree of relationship between them.
However, as explaining them jointly would probably create confusion, they are
explained individually, with some of the coefficients being more fully interpreted
later in the chapter.
1. The "Sum" Equation. The name of this equation is unimpor ant and is
not closely related to its function. The equation essentially states that the
sum of the annual net revenue of the firm must be at least as great as the sum
of all annual commitments which must be met if the farm is to remain solvent.
Symbolically, omitting the variables: }NR _..AnC. The annual commitments of
the firm include those which accrue within the solution as well as any previous
commitments the farmer has made or must pay such as taxes, debt repayment,
depreciation and family living expenses. For convenience, let the sum of the
initial annual commitments be called K. The equation then reads: NR ~ AnC + K.
The K becomes the restriction or bi value: ,NR An3C g K.
To remove the inequality from the last equation a slack activity with the
appropriate coefficient must be added.
NR -AC S S K
NR-m AC S + S =K
In these equations, S is the regular slack coefficient. A positive slack
coefficient, S must be added to complete the identity matrix used as the
first solution in solving the problem by the simples procedure. However, the
positive slack activity, corresponding to the coefficient Sp, is artificial
and should be prevented from entering the final solution. This artificial
activity, therefore, requires an appropriate penalty coefficient in the
2. The Credit Source Restrictions (CSR). The model contains four of
these equations, one for machinery dealer credit (CSRMD), one for silo dealer
credit (CSRSD), and one each for land mortgage (CSRIM) and land contracts
(CSRLC).1 These equations are related to the acquisition of machinery, silos
and land respectively, and state that no more of the particular source of
credit is available than is generated by the purchase of that particular asset.
For example, machinery dealer credit is not available unless, in fact, a piece
of machinery has been purchased. The CSR for machinery dealers will serve as
an example to explain the formulation of the equations.
It is necessary to pay 25 percent of the price (P) of a piece of machinery
as down payment (Dp). Thus, machinery dealer credit cannot exceed 75 percent
of the value of machinery purchased. It is important to note that purchase of
machinery does not force the use of dealer credit. The purchase can be made
wholly with cash. The equation, then, is:
Dealer credit available (DCA) e P- Dp
or DCA (P-Dp) 0
and removing the inequality:
DCA (P-Dp) + S = 0
The DCA coefficient is part of the dealer credit acquisition activity and
the P-Dp coefficients are in the machinery acquisition activities. The negative
sign preceding P-Dp indicates that machinery acquisition increases the amount of
credit available from this source by the amount of the coefficient Since the
initial restriction or bi value is zero, no credit from this source is available
unless machinery is purchased. The other CSR equations are exact duplicates of
the CSRMD equation explained above except that the value of the down payment
varies for each.
1 The abbreviations are used in Table 2.1 on page 19a.
3, The Cash Equations. The model contains two cash equations. The first
(Cash 1) is similar to the standard capital equation found in most programming
models, with one exception. All funds acquired through the credit transactions
are transferred into this restriction. Every credit acquisition activity
increases the available supply of capital as expressed in the Cash 1 equation.
In addition, all transactions and activities requiring cash draw the full
amount involved from this equation. The cash expenses for the production
activities are drawn from this equation as well as the full purchase price of
all assets acquired.
The sales of assets increase the supply of funds since they will be sold
at the beginning of the year, but the sale of products does not increase the
amount of funds in Cash 1. Crop sales revenue is received after most of the
expense for the farm has accrued. It would be unrealistic to add this income to
cash to be used in its own production. An exception would be milk income which
is generally received in monthly checks. In order to be realistic in adding
this income to cash, it would be necessary to consider the capital restrictions
by months. To consider monthly capital restrictions would involve a large
amount of complication in the cash transfer and utilization activities. For
this reason, income from milk production is not added to cash available for
operation and asset purchase.
The second cash equation (Cash 2) concerns the minimum down payment required
for any transaction. The acquisition activities involving all items which can
be purchased with direct credit (machinery, silos, land) contain the down
payment required, as a coefficient in this equation. The firm must have at
least this much cash available before the purchase can be made. Since this
equation involves only the actual cash available and not the total amount of
funds, as does Cash 1, the credit activities such as land contracts which do
not transfer cash, do not transfer funds into Cash 2. This is the major
difference between the two cash equations. Cash 1 involves the total amount
of funds the farmer has to work with including the full amount of credit
acquired from machinery dealers, silo dealers, from land mortgages and contracts,
The Cash 2 equation considers only the actual cash the farmer has to work with
This amount of cash includes cash on hand and cash received from land and chattel
mortgages only. In effect, the Cash 2 equation states that the money balance
or cash on hand must be at least as great as the minimum amount necessary for
purchase of the asset.
The minimum amount necessary for purchase of an asset is not always a
down payment. Consider a bunker silo for example. The materials come from
various sources, most of which do not offer credit plans. The usual procedure
would be for a farmer to acquire a loan from some source either on his land or
chattels and make cash purchases of the necessary material and labor. In this
case, the coefficient in the Cash 2 equation is equal to that in the Cash 1
equation. And repeating, funds for purchases of this type are available from
land mortgage and chattel mortgage acquisition activities.
One further aspect of the cash equations should be mentioned. Depreciation
accrues to the firm as the products are sold. Since storable crops frequently
are sold during the year following production, depreciation can accumulate at
any time during the year. As an arbitrary choice, half the drpreciation is
added to the cash account at the outset. This makes it necessary to add half the
apnual- depreciation of an asset to cash at the time of purchase, which is assumed
to be at the first of the year. Similarly, half the depreciation must be
removed from cash if the asset is sold. Therefore, for all depreciable assets,
the full coefficient for the Cash 1 equation in the acquisition activities is
price minus one-half the depreciation (P-1/2D). The corresponding coefficient
for salvage activities is one-half of the depreciation minus the salvage value
The Double Purpose Acquisition Activities
Two methods exist for incorporating the asset acquisition activities into
the model, The first, and less desirable involves, for each asset, one
activity for cash purchases and one for purchases with direct credit. This
would be necessary if only one cash equation were used since the two types of
purchases require different amounts of cash.
The addition of the second cash equation reduces the number of activities
needed by requiring only one acquisition activity for each asset. A single
acquisition activity contains the coefficients for both cash equations and
simultaneously handles both types of purchase. A direct credit purchase enters
the solution only if one of the direct credit acquisition activities enters. If
no direct credit acquisition activity has entered, all funds in both cash
equations are derived only from cash on hand plus cash loan activities. Therefore,
if none of the direct credit acquisition activities has entered, all purchases
are on a cash basis and only the Cash 1 equation would be effectively limiting.
The extent of direct credit purchases which are made depends on the level
at which the pertinent credit acquisition activities enter the solution. To
this extent, funds are added to Cash 1 and not to Cash 2, and both equations
can then become effectively limiting. Thus, the type of purchase made, with
cash or with direct credit, is independent of the acquisition activity and one
activity serves a double purpose.
The Credit Activities
There are three types of credit acquisition activities in the model:
mortgages, dealer credit and land contracts. Land mortgages are divided into
two categories depending on use. A mortgage is available on the land owned by
the farmer at 5.5 percent interest. This is one of the credit activities which
transfers funds to both of the cash equations described above. The other land
mortgage is available for purchase of up to 160 acres, the purchased land being
the collateral. Since this latter activity does not transfer the actual funds
to the farmer, only the Cash 1 equation is credited with the amount of the
mortgage when the activity enters the solution. The land contract acquisition
activities have the same effect on the cash equations as the second type of
land mortgage activity since funds are not transferred directly to the farmer.
A chattel mortgage acquisition activity is available and transfers funds into
both cash equations. The two dealer credit acquisition activities, machinery
dealer and silo dealer, affect the restriction of only the Cash 1 equation.
One additional credit activity should be described. This is the land
mortgage repayment activity. The activity enters the solution only if the
firm goes out of business, sells its assets and repays its debts. Since no
other debts exist at the outset, no other debt repayment activity need be
considered. Funds are drawn from both cash equations if debt repayment is
included in the solution.
The Cash Coefficients
The annual MVP of an asset must exceed the annual cost of ownership of
one more unit of that asset (MFC) in order for the purchase of another unit to
be profitable. The annual cost of ownership includes depreciation (D), interest
(i), repair (R) and taxes (T). These items are, in effect, the cost of the
annual flow of services from the asset. The sum of these items must be charged
against the acquisition of an asset as the MFC of obtaining another unit.
SThe MWP and MFC can be in units of either a stock or a flow so long as both
are in the same unit. To convert the MVP of a flow unit to the MVP of a stock
unit, multiply the MVP by the number of flow units per unit of stock. The
consequences of this relationship are explored in a later chapter.
The annual MFC of a stock unit is not the total market price of the resource
divided by the number of years' use. A durable asset which has a life greater
than one year, need not return its full market price in one year to be profitable
to acquire. In contrast, the MFC of a unit of a non-durable item, which is
expended within the year, is its market price--the equivalent of the annual cost
of ownership of a durable asset. Since the annual cost of ownership of a
durable asset is composed of depreciation, interest, taxes and repairs, these
items comprise the annual MFC of a durable.
In this model, only the depreciation and taxes are charged directly
against the acquisition activity for crop machinery; that is, appear in the
profit equation as a cost coefficient. Repairs are charged as expenses in the
crop producing activities since they are primarily a function of use. This
cost is then reflected in the profit equation as crop producing cost. Charging
repairs in this manner has the effect of reducing the direct annual MFC of the
machine, but simultaneously, it increases the indirect cost by increasing the
cost of producing the crop. Thus, indirectly, the MFC is unchanged. The annual
expenses or "repair" charge on the livestock, i.e., veterinarian fees, breeding
fees, etc., is similarly charged against the milk producing activities.
Repairs on silos, buildings and dairy equipment are included with depreciation
and taxes in the profit equation for acquisition activities.
All interest costs are handled through the credit acquisition activities.
The initial cash on hand has an opportunity cost of four percent through the
cash salvage activity. Capital used for production or asset purchase must
bear a return greater than four percent before cash w ill be so used. When the
initial cash on hand is exhausted, more can be acquired at 5.5 percent through
the land mortgage acquisition activity. Therefore, the MVP of the asset
purchased must be at least as large as the total of repairs, depreciation,
taxes, and the interest charge, the latter being a cost coefficient in the
profit equation for the credit acquisition activity. The profit equation
coefficient for machinery sales activities reflects the savings to the firm
of not owning the asset. That is, the depreciation plus taxes which are saved
by not owning the machine.
The coefficients in the profit equation for the crop producing activities
are cost figures equal to the cash expenses (CE) for non-durable items plus
repairs on the durable assets. This same coefficient is in both cash equations
CASH COEFFICIENTS FOR THE VARIOUS GROUPS OF ACTIVITIES
Activities Land Credit Credit Credit
acq. Credit Credit Credit acq. acq. repay-
cash Land acq. acq. acq. land banks ment Crop Milk
Equations or acq. Machy, land land machy. and (Chattel Machy. land pro- pro-
mort. contract acq. mort. cont, dealer mort. mort.) sales mort. duction duction
Profit -T -T -(D+T) -i -i -i -i -i (D+T) i -(CE+R) NR
Cash 1 P P P-/2D-100 -100 -100 -100 -100 1/2D-Vs 100 CE + R CE
Cash 2 .55P .10P .25-PD-100 -100 /2D-Vs 100 CE + R GE
Sum -T -T -(D+T) -(i+CR) -(i+CR) -(i+CR) -(i+CR) -(i+CR) D+T i+C GCE + R NR
avail, 100 -100
CSRIM -45P 100
CSRMD -,75P 100
CSRIC -.90P 100
Land and mort.
for these activities. The profit coefficients for the milk producing activities
are gross revenue minus cash expense. The cash expenses appear in the cash
equations. The profit coefficients for the crop sales activities are the gross
revenues received from the sales since all costs have been deducted elsewhere
in the program,
The sum equation accounts for changes in net revenue and annual commitments.
The revenue increasing activities--milk production, crop sales, debt repayment
and asset salvage--have the same coefficient in the sum equation as in the
profit equation with a positive sign.1 Asset acquisition activities increase
annual commitments and thus bear a negative coefficient in the sum equation.
Here, again, the coefficient is the same as in the profit equation as is the
case for the coefficients in the crop producing activities which also have a
negative sign. The annual commitment acquired upon the acquisition of credit
includes not only the interest, but also the annual repayment of capital (CR).
The coefficient in the sum equation for the credit acquisition activities,
therefore, is the sum of interest plus capital repayment and bears a negative
sign since it is an annual commitment. The coefficients for the cash equations
have been explained elsewhere.
Specialization and Diversification and the Effect of a Single Fixed Resource on
.. most farmers choose as their principal or main enterprise-- around
which to develop farming programs--an enterprise which has high and
sustained marginal returns; they then produce this product with their
fixed investment as long as marginal returns to the variable inputs exceed
those obtainable from other enterprises. They add to such a crop (or
livestock) other enterprises which will employ unused resources equally
advantageously at the margin. If they are interested only in monetary
returns, this process of expansion is continued until marginal returns are
equal for all enterprises. . it is obvious that the existence of
complementary (and, hence, multified farms) depends upon the production
Debt repayment actually is a cost decreasing activity, but the net effect
is the same as a revenue increasing activity.
relationships existing for the variable factors of production, given
the fixed investments in each enterprise. . if a high proportion
of the inputs used in the production of the various products, is fixed,
complementarity is likely to exist. If a small proportion of the inputs
used in the production of the various products is fixed, then complementarity
is less likely to exist.1
The basic assumption of this model, concerning initial resource fixity,
is that the supply schedule for spendable funds is the only fixed resource.
All other resources, except land to some degree, are variable and thus present
no limit to production. The program, therefore, emphasizes, much as the farmer
described in the above passage, the single most profitable activity relative
to the use of spendable funds. The magnitude of this activity will expand to
the point where the cost of obtaining additional factors of production, a
function of the increasing cost of credit, exceeds the marginal value
productivity of the factors in this one activity or to the limit of a resource,
the MVP for which lies between acquisition and salvage values and is, therefore,
fixed. This process of enterprise expansion can create idle services from
some of the resources during the months in which they are not used. Such idle
services might be used profitably in other enterprises or activities. In effect,
these idle services have become fixed for the firm as a by-product of the ex-
pansion in resources to produce the most profitable product (activity).
An increase in the proportion of services which are thus fixed, tends to
create seasonal complementarity (sometimes called supplementarity) between
enterprises as expressed in the quoted passage above. Therefore, the program,
as would the farmer, selects the next most profitable activity (enterprise) to
make fuller use of the endogenously fixed stock of resources. Thus, it can be
seen that specialization is not a by product of a single, fixed resource if
provision is made for determining fixity endogenously. Unused services from
endogenously determined fixed levels can make diversification a profitable
alternative just as can unused services from predetermined resource fixities,
1 Lawrence A. Bradford and Glenn L. Johnson, Farm Management Analysis (New York:
John Wiley and Sons, Inc., 1953), pp. 171-172. .. .
In a mechanical sense, it would appear that with only one resource
initially fixed, only one production activity could enter the solution since in
a standard linear programming model, when a resource becomes limiting, the
slack activity becomes zero and a production activity enters the solution to
the limit of the scarce resource. In the model presented in this thesis, a
production activity can, but need not enter the solution when a non-money
resource becomes limiting. If the productivity of the factor is such that more
of the asset should be purchased, an acquisition activity will replace the
slack activity. Therefore, a production process or activity may not be
obtained in the solution to replace a slack resource activity unless one of the
resources is just exactly used up and no more acquired, i.e., the resource
has been endogenously fixed at the initial level. However, since spendable
funds are limited in amount, at least one production activity will enter so
long as the solution indicates any production at all (the other possibility
would be to sell out). Other possibilities for production processes to enter
into the solution would be when any of the specialized equations (cash 2, sum
or one of the CSR's) is an exact equality and the slack activity drops out.
Thus, if it is profitable for the firm to diversify, the program, mechanically,
is capable of arriving at such a solution.
Discrete Investment Levels
An ever present problem of linear programming evolves from the assumption
of infinite divisibility. This problem is particularly difficult when con-
sidering investments in expensive durable items, since the purchase of a
complete unit is essential. In this model an arbitrary method has been
incorporated as one possible way of handling the problem.
The problem is to find the most profitable discrete level of investment
for the important investment items. This is equivalent to the most profitable
discrete level at which an asset shoulI e. fixed. Thus the method evoUe4
depends upon the concept of resource fixity.
An asset is fixed to the firm if it# WP Ule between, or is equal to, its
acquisition and salvage values. The greater the differential between the
acquisition and salvage values, the more subject the asset is to fixity because
the VWP will have to change by a greater magnitude before it lies outside these
boundaries. It is also trne that the WP of a fixed asset will vary as the
quantities of the vaaLabk factors used with it vary.
It is a reasonable approach to determine the level of fixity for assets
individually, beginning with the cne most sub3ect to fixity. The variations of
the other assets EIl be 2ws lktely to eauae the MWP of the fixed asset to shift
beyond the bounds of Lrity t ,the one vdth the greatest differential between
acquisition w4 salvage vaees s9 t h Arst to be filed in the solution.
The method, thea for detefirtg discrete investment levels is first to
obtain an optimal solUtionz wit 4a1 assae4 asetue m to be infinitely divisible.
Choose frcm among the as~eta in *iM4 iaveiatst occurred, the one most subject to
fixity. This particular asset is the fraed for the farm at the next higher
and next lower discrete level ty eWaling t ia~tial restrictions by the amount
of the coefficients in the acq isitlon activity waltiplied by the level of the
activity for each case and reag5ra acqEM )a s4tlaga activities for
the asset from the matrix. t~iS process, however, Mt retSlt in negative values
for the restrictions la siore euatios, particularlyl the cas equations, so that
manipulation of some other i atiity levels &ay be neceaeary to increase the
negative values to sors Oaygive or sero le1wV WhSn this process is coart
pleted, the program is rer~a twdts, once for each investment level. After adjust-
ing the profit values for *aeh sobntiO to account fo the different investment
levels, the solution fo iwhbih 3 l eagesa profit was obtained indicates the
most profitable discrete investment level of the asset in question. The process
is then repeated as often as desired, each time using the new set of restrictions
derived from the previous trial solution.1
Figure 1 should help explain the procedure desori~ed ahve. In Figure 1,
ACGK is a portion of the NF c ure for spendable funds and the point E represents
the MVP of dollars invested for the optimum solution. To the left of E, the
MVP of cash would be no lower than IE, and to the right, no greater than BF.
The line DEF, therefore, represents the extreme range of the MVf of cash on
either side of the optimum value, E. The initial optimal solution indicates the
use of OP dollars of inputs including an investment in 2.4 tractors with a
revenue of OREP or greater, The problem is to determine whether an investment in
two or in three tractors is more profitable.
If investment is fixed at two tractors, revenue will be no less than the
area ORDN, the area lying under the MWP curve. The net cost of moving from 2.4
to 2 tractors is BDEC, the loss in net revenue. Net revenue, of course, is
MVP- -MFC or BDEC between 2 and 2.4 tractors. In moving from 2.4 to 3 tractors,
the net cost is EGHF, the amount by which the change in cost exceeds the change
in revenue. The alternative chosen is the one having the lower net cost-two
tractors would be chosen if the relationships were as in Figure 1.
The difficulty is in determine~ tgthe magnitude of the net cost areas BDEC
and EGHF.. The effect of forcing an investment in either two or three tractors
can change the proportions in whichh the enterprises as well as the inputs
are combined. This can cause a shift in either or both the MVP and MFC such that
it is impossible to predetermine, without computing the two programs, the most
profitable level of investment for the asset under consideration
1 It should be emphasized that this method of determining discreteness leaves:
much to be desired. See Appendix A for a more complete discussion of the
effect on resource fixity from using this method.
.." .. .....-MVP2
APPLICATION OF THE MODEL
The model was applied to a "typical" central Michigan dairy farm situation
for which several alternative organizations were considered. The "typical"
aspects of the farm refer to the initial resource base including type of land,
amount and kind of machinery, size of herd and livestock facilities. The
manager was considered to be above average in capabilities for obtaining higher
than average crop and milk yields and able to use the most efficient type dairy
facilities in use at the present time. The dairy farms of Michigan are presently
undergoing a technological change, increasing labor efficiency particularly for
the milking chores and herd management. Therefore, it is not unreasonable to
consider such possibilities for a man on an average dairy fam.
In all, 33 crop producing activities are included in the model, involving
three crops--corn, oats and alfalfa. The oats and alfalfa are considered as
one crop with one-fourth of each acre devoted to oats, for a nurse crop, and
three-fourths to alfalfa. The proportion of corn in the rotation is independent
at all levels. The solution could involve continuous corn, no corn, or any
amount in between. For each crop, three fertilizer levels are included, the
lowest level being about equivalent to the general level of application currently
in practice. Consequently, the higher fertilizer levels are concurrent with
above average management practices.*
1 The low and medium fertilizer application levels are taken (with slight
modification) from: C.R. Hoglund and R.L. Cook, Higher Profits From
Fertilizer aid Improved Practices, Agricultural Economics Mimeo 545, Michigan
tate University Agricultural Experiment Station and Soil Science Department,
Revised October, 1956. The high application levels are a current revision
of the same publication by Hoglund, Cook, John Guttay and L.S. Robertson.
Silage is an important component of the rations for dairy cattle. It is
desirable, therefore, to include in a dairy farm program, various amounts of
both hay and corn which can be cut for silage. The amount of corn cut for
silage varies by 20 percent intervals from zero to one-fifth, to two-fifths, up
to 100 percent. The oats are all cut for silage in each oat-hay activity, with
the combined oat-hay crop being cut for silage at the rate of one-fourth (oats
only) two-fifths, three-fifths, four-fifths and 100 percent. Thus, there are
six corn production activities and five oat-hay activities each having three
levels of fertilizer application, or a total of 33 crop production activities.
Initially, the farm is equipped with a 32 stall grade A stanchion milking
barn and a 500 gallon bulk tank. Milking is done by machine, but the milk is
carried to the bulk tank. Grain is fed on an individual basis from a cart.
Silage feeding is accomplished with an automatic silo unloader in the upright
silo but without automatic auger feed bunks. The labor efficient operation of
the stanchion system includes an automatic feeder for silage and a pipe line
milking system. If the herd were expanded or more silage fed, additional
investments could include another upright silo or bunker silo. Three stanchion
systems are considered as alternatives in the model: the present system with
"average" labor efficiency; a labor efficient system with upright silos; and a
labor efficient system with additional investment in one or more bunker silos.
Two milking parlors are included--a double three walk-through parlor and a
double six herringbone system. For each type, combinations for (1) "average"
efficiency with upright silos, (2) efficient operations with upright silo and
(3) efficient operations with bunker silos are included. In addition, for each
of the nine different systems, nine rations with varying proportions of hay and
silage and varying levels of grain are used. There are three proportions of
hay and silage with three grain levels for each. Milk production increases
the same piece of machinery. Land, too, can be bought or sold. If the
original acreage is sold, the mortgage on it must be repaid. The acquisition
of a new milking parlor includes the disposal of the old stanchion barn.
Buildings and facilities which are not included in the initial resource base
cannot be sold so no salvage activity exists for these items. In addition, more
cows and replacements, jointly, can be purchased, or any proportion of the herd
Hired labor can be acquired by the month far cropping operations and summer
milking for the months of April through October. Any labor acquired during the
off season would be for milking, so the months of November through March are
grouped together, In case no dairy is included in the solution, the farmer
has the opportunity of off-farm employment of his labor during the slack months
of November through March. The opportunity cost of the farmer's own labor
during the summer months is the possibility of employment a specified number of
days every month up to full time off-farm employment.
The Range of Possible Solutions
First, it is possible for the farmer to sell out completely, invest the
resulting cash at 4 percent, and obtain full time off-farm employment. The
earnings from off-farm employment will satisfy the family living requirements
and thus, the sum equation, since all other annual commitments will be cancelled.
It is also possible to have a complete milk factory with all inputs acquired.
It is possible to purchase all labor, feed and equipment necessary to run this
type of operation. The third possible extreme is to keep the farm but sell
the dairy equipment and herd and end up with a cash crop farm. It is not
necessary for the crops to be sold through the dairy herd.
Given these extremes and the assumptions of linear programming, it is
evident that any combination of the limited number of alternatives considered,
represents a possible solution.
SOLUTION OF THE FARMU MODEL
The initial optimal solution obtained from this model is unique to linear
programing in that the quantity of all resources can be varied should it be
profitable to do so. Consequently, the model allows the determination not
only of the optimum combination of enterprises, but also the optimum combination
of the factors of production subject to the limitation on funds, the initial
asset structure, the acquisition and salvage values of the assets, product
prices and the input-output relationships. Since the principal limit to enter-
prise organization and size results from the increasing cost of obtaining
funds, the solution is optimal with respect primarily, to spendable funds. In
addition, the imputed values of the resources are a function of their acquisition
and salvage values, their use opportunities and their initial level on the farm,
rather than being a function of an arbitrarily set and rigidly fixed limitation
on the amount available to the farm.
The Initial Optimal Solution
The initial assumptions made in formulating the model result in an optimum
organization consisting of a 337.2 acre cash crop farm containing 282.7 acres
of continuous corn, of which 12 acres are cut for silage and sold out of the
field, with the remainder sold as grain. This organization involves the purchase
of 177.2 acres of land, of which 85 percent is assumed to be tillable, and the
complete disposal of the dairy enterprise. Although somewhat unrealistic, it
is more profitable for the farmer to take advantage of full-time off-farm
employment and hire the necessary farm labor.1 Table 4.i shows the change
1 In at least one case, this has actually occurred on a Michigan farm. In
general, however, this is an undesirable course of action since it leaves the
farm without an active manager when only monthly labor is hired. Were the
hired labor on a full time or tenant basis, of course, the organization would
not be unrealistic nor necessarily undesirable. Obtaining such a result in the
solution is a consequence of the static nature of the analysis. The opportunity
cost of full time off farm employment is sufficiently high that, since management
is not considered a necessary resource, the services of the manager are sold off
in inventory between the initial farm assets and those of the optimum organization.
ORIGINAL AND OPTIMUM INVENTORIES
Item Initial Purchased Sold Optimum
Land, total acres 160 177.2 337.2
Land, tillable acres 132 150.7 282.7
Dairy cows 32 32 0
Dairy heifers 11 11 0
Dairy calves 13 13 0
Field tractors 2 3.0 5
Plows 1 1.6 2.6
Disc, drill 1 1 0
Disc, planter 1 1.1 2.1
Cultivator, sprayer 1 0,7 1.7
Mower, rake 1 1 0
Wagons 2 4i8 6.8
Chopper 1 0.8 0.2
Fertilizer spreader 1 1.0 0
Corn pickers 1 2.4 3.4
Bulk tank 1 1 0
After deducting cash expenses, taxes, depreciation and interest for new
debt, but excluding interest on the owned assets and capital repayments to
retire the debt, profit for the optimum solution is $8810.1 Deducting the off
farm income of $4500 leaves a farm profit of $4310. Farm profit includes a
return to owned assets. If the owned capital is charged a 6.5 percent interest
rate, which is the highest rate paid for credit, the remaining amount is $2412.
Adding the off farm income to the $2412 above gives the labor income for
the farm. Labor income is $6912. If the family spends only the minimum amount
for consumption, $3200, then $3712 is available from labor income to retire the
debt. The annual capital repayment contracted upon the acquisition of the debt
is $3635. By paying this amount in full, the family has available for consumption,
1 This is the value which is maximized in the objective function. For purposes
of comparing profit from the various solutions, only those items stated
above are deducted from gross income. This figure could be called return
for family labor and owned capital.
in addition to the minimum $3200, the amount of $77.
To organize the optimum farm requires a full mortgage on the owned land
and a chattel mortgage on all equipment. In addition, 160 acres is purchased
with a 6 percent land contract and an additional 17.2 acres with a mortgage
after meeting the down payment requirements. The total annual interest and
capital repayment commitment which the farm must meet is $7320, In addition
to the credit acquired, cash was increased $8837 by the sale of assets.
The values imputed to the resources and farm produced crops are of major
interest from both an empirical and a theoretical point of view. As one would
expect on a cash crop farm where the crops are not sold through livestock, the
value of the crops are the prices received by the farmer--90 cents per bushel of
corn and $17.50 per ton hay equivalent of silage. Similarly, the imputed
value of assets sold should be equal to their salvage value.
The salvage value of a unit of service from a durable asset is equal to
the savings in depreciation, taxes and interest all based upon the salvage
value of the durable stock. For example, the depreciation and taxes per cow
and replacements as a unit are $42.39. The interest charged at the highest
rate (6.5 percent) on the net salvage price of $139.22 is $9.05. The Delta J
value or imputed value of the cow and replacements unit is $52.07 which is very
near the total of taxes, depreciation and interests, $51.44.2 The imputed value
1 It is, of course, possible for the family to spend for consumption the
interest on owned assets and depreciation, in addition to labor income.
2 The term Delta J stands for the imputed value of the activities. The values
imputed to the slack activities are the MVP's of the resources. An accumulated
round-off error, accounting for the difference between the Delta J and
salvage value is to be expected when working with a large number of equations
and activities, especially with the high degree of interaction expressed in
of one unit of service from the disc, drill asset is 80 cents. The salvage
value of the service unit is 79 cents. The corresponding values for the
forage chopper, which was only partially sold, are $3.52 and $3.47 respectively*1
The acquisition cost of a unit of flow service from an asset is the sum
of the annual cost of depreciation, interest and taxes of the stock divided by
the number of flow units. This cost is computed in the same way as was the
salvage value for assets sold. The imputed value and cost of acquisition
respectively for three acquired resources are: for May plow services, $,34 and
$*29; the services for the disc and corn planter, $.67 and $.63; and an hour of
June labor, $1.14 and $l.13.. A listing of the imputed values of the resources
for each solution appears in Appendix B and are further discussed in a later
The values imputed to non basis or excluded activities indicate the decrease
which would occur in profit if that activity were forced into the solution.2
This information makes possible the determination of the relative profitability
(in a more strict sense, unprofitability) of those activities not in the
solution. Several aspects of the excluded activities are worthy of note.
Considering first, the corn activities in which all corn is picked for
grain, the activity having the heaviest level of fertilization entered the
solution. The reduction in profit from using the medium level of fertilizer
1 The MVPrs of the resources are expressed in terms of the units in which they
are measured. In the cow and replacements example above, both the salvage
activity and the resource are measured in the same unit. In contrast,
machinery salvage is measured in terms of a stock but the resource in terms
of a flow of services. (As a consequence, such resources are varied in
terms of a flow rather than in terms of a stock.) Therefore, it is necessary
to divide the salvage value of such an asset by the number of units of the
flow service derived from it to put it in the units in which the imputed value
2Non basis activities are the activities which do not enter into the final
would have been $19,60 an acre, determined from the Delta J value of the
activity. Using the lightest application of fertilizer considered in the program
would have reduced profit by $44.12 per acre. The same relationship is true for all
all the corn activities. That is, the heavier the application of fertilizer, the
less would be the reduction in profit or, stated alternatively, the greater the
increase in profit, from incorporating that activity in the solution. Within
the corn activities using a high level of fertilization, the reduction in profit
from increasing the amount of silage would be $3.24 per acre if 40 percent were
so harvested, $.478 if 60 percent, and $6.4O and $8.04, respectively, for 80
and 100 percent silage per acre.
No hay producing activities entered the solution. The least reduction in
profit from forcing hay into the solution (,1.88 per acre of hay) would have
resulted from the most highly fertilized hay of which only the oat nurse crop
was chopped for silage. Here, again, the increasing reduction in profit from
decreasing the level of fertilization is evident as well as from increasing the
amount of silage per acre.
By varying the level of fertilizer within a crop activity series with the
proportion of silage held constant, the change in profit due to changing the
fertilizer can be determined. For example, consider the corn activities in
which 40 percent of the acreage was chopped and 60 percent picked. The imputed
values of the activities at the three fertilizer levels were: low, $21.20;
medium, $10.88 and high, $3.26. Since these figures indicate the loss in
profit, the difference between low and medium, and between medium and high
indicate the increase in profit from heavier applications of fertilizer. The
gain in profit from low to medium is $10.32 and from medium to high is $7.62.
Plotting these values on a graph with dollars of fertilizer on the horizontal
axis, illustrates the decreasing returns as more fertilizer is applied.
0 1 -
8 9 10 11 12
Dollars of Fertilizer per acre
Figure 4.1 (Corn)
Since in the imputed values, all costs are accounted for, the values
plotted in Figure 4.1 are changes in net revenue or profit and can be defined as
gain. Maximum profit is equivalent to zero gain. Therefore, it appears that
even though higher level fertilizer applications were used in this study than
currently in common practice, even higher applications would be profitable.
The total cost of fertilizer applied to corn at the three levels was: $8.32,
$10.42 and $12.92.
The total cost of fertilizer applied to hay at the three levels was:
$3.40, $6.l5i and $9.32. The corresponding imputed values for hay with only
the oats cut for silage are $21.94, $11.94, $11.28 and $1.88. In Figure 4.2
the gain obtained from increased fertilizer application is plotted. Here,
again, it appears that heavier rates of fertilizer would be profitable.
Gain $ 12
4i 6 7 8 9
Dollars of Fertilizer per acre
With only two points on the gain function, it is not possible to determine
the most profitable level of fertilizer to use. However, the closer gain is
to zero, the closer the rate of application to the maximum profit point. Given
the information available, it appears that the rate of fertilizer application
on corn is nearer to the optimum than the rate on hay.
The magnitude of the Delta J values for the milk producing activities
indicate that dairying, under the conditions set forth in the assumptions, is
a poor alternative compared to cash cropping if continuous corn is possible.
S The least unprofitable type of dairy enterprise, a highly labor efficient
herringbone system, would have reduced profit by $2224 if one cow were milked.
Throughout all three milking systems, efficiency in labor utilization has a
market effect on profitability, but there is only a slight profit differential
between upright silos and bunker silos. In choosing between the types of milking
parlors, the double six herringbone has a slight advantage over the double
three walk-through, but investing in either would be considerably more :- *
profitable (less unprofitable) than using the stanchion arrangement already on
The Discrete Investment Series
Had a dairy enterprise been included in the solution, the important assets
for which to determine discrete investment levels would have been the milking
parlor and the silos. These are items with a high acquisition cost and, because
of their permanent nature, a relatively low salvage value.
On a cash crop farm, it is important to determine the size of the farm,
the number of tractors, and the amounts of other expensive machinery. In
addition to determining the level of land and tractor investments, solutions
were obtained to determine whether or not to sell the forage chopper and to find
the most profitable number of corn pickers for the farm.
Fixing the level of any asset for the farm, will decrease the value of the
objective function from the previous solution in which the asset level was not
fixed. Each successive solution, therefore, for which more and more of the
assets are fixed will have a lower profit than the previous solution. That is,
the solutions for a resource fixed at the next higher and next lower discrete
levels, will both exhibit less profit than did the previous solution.1 The
choice of which discrete level of investment to use depends on the relative
profitability between the two levels being programmed.
Forty acres was considered as the most reasonable discrete level of land
investment. Forty acre plots are generally available while an area as small
as 20 acres is not. To restrict purchase to 80 acres puts an unreasonable
demand on farm size. The initial solution indicated an investment in an
additional 177.2 acres or 337.2 total farm acres. Land investment programs were
computed, therefore, for 320 and 360 total acres or for an additional investment
1 It should be pointed out that the higher profits received from prior
solutions are based on infinite divisibility of factors and product and
as such, are only illusionary.
in 160 and 200 acres. The optimum organization and profit for both conditions
is given in Table .2.".
The 320 acre farm incorporates 268 acres of continuous corn and no hay.
Twelve acres of corn is chopped for silage using the services of 0.17 chopper.
An additional 2.2 corn pickers are acquired to pick 256 acres of corn, and the
acquisition of 2,7 tractors increases the stock of tractor services to 4.7
tractors. The addition of 40 acres, giving rise to the 360 acre farm, makes hay
production a profitable alternative by decreasing the necessary investment in
specialized corn equipment. The production of 74.8 acres of hay restricts corn
PROFIT AND ORGANIZATION FOR 320 ACRES AND 360 ACRES
Tractors, beginning inventory
Tractors, ending inventory
Choppers, beginning inventory
Choppers, ending inventory
Corn pickers, beginning inventory
Corn pickers acquired
Corn pickers, ending inventory
Acres in hay, high fert., oats for silage
Acres in corn, high fert., 1/5 for silage
Acres in corn, high fert., all picker
Total acres, picked corn
Profit, nearest dollar
production to 227.2 acres. With fewer acres in corn, a smaller tractor invest-
ment is required since the use of tractor services is spread more evenly
throughout the year, Since all the hay is chopped as well as the oat silage,
more chopper services are retained on the larger farm. Profit comparison between
the alternative organizations, however, favors the smaller farm. Consequently,
succeeding programs are based on 320 acres.
The MVP's of most resources were reduced only slightly, comparing the 320
acre farm with the initial optimal solution. As would be expected, however,
the MVP of land increased (from $20.72 per acre to $22.11 per acre) when it was
fixed at the 320 acre level. Cash, which in the initial optimum was worth
$7.42 per $100, is worth $6.50 per $100 on the 320 acre farm. This occurs as a
result of the limitation on land, which causes some 6.5 percent credit not to be
In Figure l.3, the segmented curve labeled ABOCD*EF represents, again, a
portion of the MFC of dollars to the firm and the line MVP indicates the MVP of
spendable funds in the initial optimum solution. Spendable funds, here, includes
6, . D% i
0 $14,759 $19,289 Spendable Funds
cash, owned land mortgage and chattel mortgage, but not land contract funds nor
land mortgage on purchased land. In the optimal solution, a total of $19,289
of spendable funds was used. This amount includes all the 6.5 percent credit
available. In the 320 acre organization, the land limitation forced down the MVP
of spendable funds so that not all the 6.5 percent credit is exhausted. Were
the MVP of spendable funds greater, an additional iL$430 of 6.5 credit could be
The 320 acre organization indicated an optimum of U.7 tractors. Programs
were computed to determine the most profitable alternative between 4 and 5
tractors. The results appear in Table 4.3
It is of interest to note the effect on organization from fixing the number
of tractors at levels higher and lower than the optimum number in the previous
320 acre solution. Restricting the number of tractors to four has the expected
effect of placing a premium on their services, and as a result, more intensive
use of these services through time is required. Although hay is a less pro-
fitable crop than is corn, it is profitable to more fully utilize these tractor
services than to specialize in the production of corn. Specialized corn pro-
duction makes less efficient use of the relatively scarce tractor services than
does the more diversified previous solution.
The farm organized around five tractors is a sharp contrast to the one
for which tractors are a more limiting resource. On the five tractor farm,
tractor services are relatively abundant. As a consequence, intensification of
their use is not a prerequisite to a profitable farm organization, as is the
case where tractor services are relatively scarce. Because tractor resources
are fixed at a high level on the second (five tractor) farm, specialization is
a profitable alternative.1
1 The comparison of these two programs with reference to the effect of tractor
limitation on organization is a good example of the effect on the ultimate
outcome from predeterming the level of resource fixity,
Although the second farm specializes in a relatively more profitable crops
the added expense of the additional tractor is sufficient to reduce profit
below that for the four tractor farm organization. Since the four tractor farm
is more profitable, it is this organization which was chosen for further
investigation in accordance with the rule developed for this purpose. In an
actual planning situation, however, the profit differential is sufficiently
small that other alternatives should be considered.
PROFIT AND ORGANIZATION FOR 320 ACRES WITH 4 AND 5 TRACTORS
Description Four Tractors Five Tractors
Tillable acres 268 268
Tractors 4 5
Choppers, beginning inventory 1 1
Choppers sold 0.8 1
Choppers, ending inventory 0.2 0
Corn pickers, beginning inventory 1 1
Corn pickers acquired 1.82 2.35
Corn pickers, ending inventory 2.82 3.35
Acres in hay, high fert., oats for silage 2.80 0
Acres in corn, high fert., 1/5 for silage 71.1 0
Acres in corn, high fert,, all picked 168.9 268
Total acres, picked corn 225.8 268
Profit, nearest dollar $8228.00 $8088.00
Profit differential +i.40
The question of whether or not to sell the forage chopper is the next to
be determined. The 4 tractor optimum indicated salvage of 0.8 of the chopper,
using only 0.2 to harvest 28 acres of hay and 14.2 acres of corn silage. If
the chopper is completely sold, only ear corn can be raised since both the
hay and corn silage activities require the services from the chopper and no
provision is made for hiring custom work on the farm. If the chopper is not
sold, one would expect a more diversified farm plan to make fuller utilization
of this fixed, specialized piece of equipment. In some respects, therefore, the
effects of selling or keeping the chopper are more important to the farm
organization than determining the level of fixity for the resources with a more
PROFIT AND ORGANIZATION FOR 320 ACRES, 4 TRACTORS,
WITH AND WITHOUT A FORAGE CHOPPER
Description Without With
Tillable acres 268 268
Tractors 4 4
Chopper 0 1
Corn picker, beginning inventory 1 1
Corn pickers acquired 1.8 1.4
Corn pickers, ending inventory 2.8 2.4
Acres in hay, high fert., oats for silage 0 28.0
Acres in corn, high fert., l/$ for silage 0 240.0
Acres in corn, high fert., all picked 22.56 0
Total acres, picked corn 225.6 192.0
Profit, nearest dollar $6270,00 $8047.00
Profit differential +1777
As expected, the forage chopper has amaaled effect on farm organization
With no chopper, all the corn must be picked. The limitation on October tractor
services prevents more than 22.56 acres of corn from being harvested as ear
corn. Consequently, 42.4 tillable acres on the farm must remain idle--an
unprofitable alternatively On the other hand, having the chopper available on
the farm leads to a diversified organization which fully utilizes all available
tillable acres. With the price restriction still holding for corn pickers, it
becomes profitable to more fully utilize the chopper and reduce the investment
in the corn pickers, so more hay and corn silage is produced relative to the
amount of ear corn than was the case in all previous solution.
In this case, the consequences of fixing the farm size at 320 acres, when
42.4 tillable acres remain idle, are plainly evident.
These two solutions, again, provide a good example of the effect of
predetermined resource fixity. With no chopper available to the farm, land was
used to the point where its MVP dropped to zero*. ere land not fixed in this
particular problem, some would be sold--the amount sold stopping at the point
where its MVP reaches salvage value. In this example, the value of land in use
is less than its value in salvage. Since land has a positive salvage value, it
is unrealistic to value it at zero.
The final factor of production to be set at a discrete level in the invest-
ment series are corn pickers. Table L.5 shows the organization and profit for
the two levels of investment.
Varying the amount of corn picker services available has less effect on
organization than when the chopper was varied. The limitation of corn pickers
in the first, 2 pickers, solution restricts the amount of corn which can be
harvested by this method and as a consequence, more hay is produced. It is
PROFIT AND ORGANIZATION OF 320 ACRES, 4 TRACTORS, 1 CHOPPER
AND 2 AND 3 CORN PICKERS
Description Two Three
Corn Pickers Corn Pickers'
Tillable acres 268 268
Tractors 4 4
Chopper 1 1
Corn pickers 2 3
Acres in hay, high fert., oats for silage 68 28
Acres in corn high fert., 1/5 far silage 200 240
Acres in corn, high fert., all picked 0 0
Total acres, picked corn 160 192
Profit, nearest dollar $7337 $7708
Profit differential +$371
interesting to note that although the organization for the 3 picker solution is
the same as for the previous solution with a chopper fixed, the investment in
the additional corn picker reduces profit by $339.
The pattern of MVPls of the various resources throughout the investment
series helps explain the effect of fixing resources arbitrarily at various
levels. In the two land investment problems, when land was fixed at 320 acres,
tillable acres had a value in use of $22.11, but for the 360 acre farm where
land was more abundant, the MVP of tillable acres dropped to $10.28 which is
$8.39 below salvage value. Because the other resources were combined with a
greater amount of land on the 360 acre farm, their MVP's increased relative to
those for the 320 acre organization.
The MVP of tillable acres decreases to $16.0 when the number of tractors is
fixed at four, but increases to $31.78 when five tractors are available. Thus,
it can be seen that in linear programming, as in other computational procedures,
the MVP of one fixed resource increases as the amount of another resource is
The value of the services from the forage chopper and the corn picker
remains constant as tractors are varied from four to five This is to be expected
because in both cases, some of the chopper is sold and some corn pickers acquired.
The value of the flow unit of the chopper is its salvage value and the MVP of
the corn picker is equal to its annual acquisition cost. The MVP of tractor
services for any given month, however, varies, depending upon the proportions of
crops produced. A change in the proportions of crops changes the tractor
requirements and thus their MVP.
The application of the model and the discrete investment rule to the
original farm situation has resulted in a farm organization consisting of a 320
acre cash crop farm with h field tractors, 1 chore tractor, a forage chopper and
3 corn pickers. In addition, for the final farm organization, the remaining
factors were fixed at the following levels: 2 plows, 2 discs, 1 drill, 2 corn
planters, 2 cultivators and sprayers, 1 mower and rake, 5 wagons and 1 fertilizer
spreader. Table 4.6 shows the complete change in farm inventory from the
original organization to the final farm plan, including discrete investment
levels for all assets.
COMPLETE INVENTORY CHANGE
ORIGINAL ORGANIZATION TO FINAL FARM PLAN
Original Inventory Final Inventory Change In
Description Amount Value Amount -Value Value
Land, total acres2
320 $60,000 $36,000
Machinery and Equipment
Total farm investment
0 410.,5 5
Original inventory value plus additional investment
minus depreciation on all units.
2 Includes improvements.
(price x number of units)
-e -- ~- --------- --
The Final Farm Organization
The initial solution derived from the model is an optimum solution under
the assumption of complete divisibility. Succeeding solutions derived from
the investment series are not optimum in the strict sense. The 320 acre farm
organization with other factors variable is optimum only in the sense that it
is more profitable than the 360 acre alternative. (Of course, given the 320
acres, the remaining factors and products are optimum.) A major weakness of
the rule for determining discrete investments is that the previously fixed
resources may actually be fixed at the wrong level as more resource fixation
occurs. That is, additional resource fixation may have a sufficient effect
upon the MVP of previously fixed resources, that the excluded alternative, or
even an alternative not tested, may lead to higher profits. If the MVP of land
drops so low for the last solution that at least 40 acres could be sold before
the MVP increased to the salvage value, it would indicate that given the resource
fixation of succeeding solutions, too much land was acquired in the original
The change in acreage accounts for the greatest amount of change in
inventory value. Although the additional acreage was priced at $250 per acre,
the inventory value is $225, the net price the farmer would receive were he
to sell it. For inventory purposes, the original land is valued at $150 per
acre. Placing a value of $225 per acre on this land would have the effect of
increasing the original net worth of the farmer. Net worth, when the original
farm is valued at the lower price is $36,000. Increasing the value of the land
would increase net worth to $48,000. Either valuation will have no effect on
1 Refer to pages 42-43 for such a solution.
the change in the value of inventory nor in the change in net worth.
COMPARISON OF PROFIT:
OPTIMUM SOLUTION AND FINAL FARM PLAN
Optimum Final Farm Loss Involved
Description Solution Plan in Obtaining
Profit $8,810 $6,796 $2,014
Labor income 6,912 828 2,084
Available for capital repayment 3,712 1,628 2,084
Needed for full capital repayment 3,635 2,548 -
1 For a definition of the income categories, see page 31.
In Table 4.7, a comparison is made between comparable profit figures for
the initial optimal solution and the solution derived from the investment series-
the final farm plan. The third column in Table 4.7 shows the loss in profit
due to fixing the assets at discrete levels.
In the final farm solution, labor income is $4828, In addition to this
amount, the family also has available for consumption or investment (disposable
income) the interest on owned assets and asset depreciation. Final asset value
is $77,324 and the total debt is $59,242. Interest, at 6.5 percent, on the
difference is $1175, and depreciation on the assets is $2729. However, half
the depreciation has already been added to cash (see page 16). The disposable
income obtained by adding interest and half the depreciation to labor income
is $7,367. These figures are summarized in Tab3e 4.8.
It remains to examine the capital accumulation side of the business. The
difference between final total asset value and total debt is $18,082. This is
the net worth of the farmer at the end of the year if none of the debt is
retired. Should the family so choose, a maximum of $4,167 of the debt could be
retired from disposable income if only the minimum $3,200 was used for family
consumption. If this course of action were followed, net worth, at the end of
the year would be $22,249. Therefcae, depending upon the use of disposable
income, net worth at the end of the year would be between 1l8,082 and $22,249.
DISPOSABLE INCOME, FINAL FARM PLAN
Labor income $4,828
Interest on owned assets 1,175
One-half depreciation l3j6&4
Disposable income $7,367
SUMMARY AND C(CCLUSI ONS
Application of the Model
The model developed in this thesis actually is composed of two parts. The
first part, which is the principal development of the thesis is the mathematical
model dealing with the endogenous determination of fixed resources* The second
deals with the discrete investment levels and is more a rule than a model.
The range of application of the mathematical model is as wide as the use of
linear programming for solving maximization and minimization problems involving
resources which, in fact, are subject to variation. The modifications in the
linear programming model made in this thesis would not be necessary nor
especially useful where resources are rigidly fixed.
The model is particularly useful in a business which has resources as
variable as does farming. It is capable of handling the very important resource
allocation problems facing farmers today--such problems as diversification,
specialization and vertical integration. An asset structure fixed at the
initial levels and proportions, predetermines the outcome of an optimizing
problem in a very real sense. The importance of scarce resources is unrealistically
emphasized where the opportunity for further investment actually exists. A
model with predetermined resource levels also has more of a tendency toward a
more diversified solution than will this more general model. A model in which
resources are variable, is not forced to search for employment for factors of
production having a very low or zero productivity. It is much more realistic
to dispose of such resources which in turn will free funds for the expansion of
the more productive enterprises. At the same time, this model does not over-
emphasize specialization which would be an equally undesirable result.
The alternative enterprises considered in the standard programming model
are, by necessity, restricted by the group of resources considered fixed. In
the more general model developed here, this is not the case. The entire initial
set of assets can be disposed of and an entirely new type of business brought
into being if such alternatives are specified in the model. However, the
initial set of resources in this general model, does influence the outcome of the
* program. This is the case because the initial assets will not be sold so long
as their value in use is greater than their salvage value. Therefore, their
value in use, when combined with the other initial resources, or additional
acquired resources, must have an MVP less than their salvage value before the
initial resources would be exchanged for another set of resources--a new type of
business being organized--or sold and the capital invested outside the
It should not be inferred from the above statements that all the analysis
problems of a firm have been solved with the conception of this model. The
model still contains many of the problems organic to linear programming and as
such has many of its shortcomings. An attempt to alleviate one of these short-
comings resulted in the rule creating the discrete investment series described
in the text.
The results obtained from any linear program are limited to the particular
alternatives and activities included in the model. The determination of the
combination of factors within each production activity is exogenous to the model
itself and as such, must be dealt with independently. Erroneous factor com-
binations within the activities result in erroneous conclusions from the model.
In addition to the regular problems encountered in linear programming,
this model is oversimplified and lacks realism concerning the budgeting and
and accounting techniques used. Depreciation and income (particularly dairy
income) accruing through the year are not adequately handled nor are problems
concerning the stock and flow characteristics of resources. The stodk-flow
problem is of major concern. The acquisition and salvage of resources involve
units of stock such as tractors, buildings and machinery. The productivity of
the stock, however, is measured in terms of the flow of services from that
resource. As a result, the differential between acquisition and salvage values
is, operationally, a function of the unit of service, and as a consequence, the
buying and selling of resources, due to the nature of linear programming, is a
function of the flow unit of the resource rather than of the unit of stock.
This characteristic reduces the fixation restrictions for resources and thus
creates a tendency toward more variability than actually exists. In the absence
of a fully discrete programming model, where activities enter only in discrete
units, the infinite divisibility assumption of programming will continue to be
Further, the model is constructed under static economic assumptions. In
the static framework, reference is not made to the management function nor to
the interrelationships between the firm and household. The model assumes
profit maximization as the only motivation for production. At the same time,
enterprises which are distasteful or undesirable to the manager may simply be
excluded as a possibility in the problem. The only management decisions beyond
profit maximization considered in the model are the alternative enterprises
acceptable to the manager, including minimum and maximum size restrictions.
The lack of risk and uncertainty considerations is another characteristic
of the static economic assumptions under which the model is constructed. The
input-output relationships are considered to be single valued. The effect of
diminished crop yields or prices on the liquidity of the firm and status of the
family are not taken into consideration. Its static nature precludes risk
discounts and informal insurance schemes.
A major inconvenience of the model concerns the complex nature of it,
which tends to create great size. To completely analyze a diversified farm
organization, requires at best a large and unwieldly program matrix. Adding
the complex of asset buying and selling activities and capital transfer
activities as well as the specialized equations, compounds the size of the
matrix involved. A complete programming analysis including the features of
this model, will invariably require the services of a large electronic
computer, i.e. one with a large memory system.
The Empirical Results
The optimal solution to the model indicates that, under the conditions
set forth in the problem, a cash crop farm is more profitable in the Central
Michigan area than is a dairy farm even if the dairy utilizes the most labor
efficient type of operation now in practice. It would be unwise to make
recommendations from these results without further study, for several reasons.
The crop yields considered in the application correspond to a very high degree
of management skill--it would require a very good manager to obtain the results
indicated by the most productive crop activities. Secondly, under exceptional
management, milk yields may be greater than the maximum of 11,000 pounds
considered in the model. An individual iho was a very good dairy farmer, but
lacked this ability in producing crops, may well find the profit situation
reversed from the optimal solution.
The assumptions made, relative to labor, have an important effect on the
outcome of the problem. The problem assumes off farm employment is available
only a specified number of days every month for each of the two time periods
During the cropping season, this assumption makes it profitable to hire all
necessary labor, so that the farmer's labor is fully employed throughout the
period. Were monthly off farm labor employment available, it would have been
profitable to accept off farm employment only during slack months, hiring labor
only in excess of that supplied by the farmer during the rush seasons.
The fact that alternative employment is considered available off the farm
during the winter months, has an influence upon the profitability of dairying.
If the farmer's labor were not utilized off the farm during these months, the
opportunity cost of dairying may be sufficiently great that this enterprise
would enter the optimum solution.
The method of handling income from the dairy enterprise quite probably has
an important influence on the outcome. If the monthly milk checks were reflected
in the cash account,less cash would need to be borrowed outside the firm. Since
cash in the initial optimum solution has a marginal value product of $7.42 per
$100, the addition of the milk income to the cash account each month may have
been sufficient to cause the dairy enterprise to enter the solution.
Price considerations should also be taken into account before making
recommendations on the basis of the results of the program. While both the
crops and the milk were conservatively priced, the relationship between the
two has an important bearing on the outcome of the problem.
The optimum cropping program, itself, should receive special scrutiny.
Since the initial assumptions were organized around a dairy farm, the possibilities
of a larger variety of crops was not considered. This is perhaps, the most
serious restriction of the results. In making the initial assumptions, the
possibility of forming a cash crop farm as a solution was desirable, but since
the farm was a dairy farm, more emphasis was put on dairy organization than on
the organization of a cash crop farm.
Further Study Indicated
The model as applied in this thesis, considers investment, organization
and operation only for a one year period. Obviously, the optimum program the
following year could not be a duplicate of the first year's solution. It would
be highly desirable to incorporate the features of this model with the model
developed by Loftsgard and Heady and referred to earlier in the introduction.
Their model makes use of dated variables and arrives at an optimum solution
through time, but does not consider the investment alternatives made possible
by the incorporation of a model considering endogenously determined resource
fixities. The combination of the two models should produce a much more realistic
answer than either is able to produce alone.
Further work is required on the stock-flow problem, which, as indicated
previously, is not sufficiently handled by this model. Two problem areas exist
with respect to this problem. One concerns the use of assets over time and the
corresponding investment plan through time. The other concerns the effect on
the fixity restrictions caused by imputing productivity values to flows rather
than to stocks.
The application of linear programming to dynamic economics is worthy of
further study. Price and resource mapping are examples of previous work in this
area. The mapping technique, sometimes called parametric programming, considers
the effects of changes in prices and resources on farm organization. An important
problem, which has as yet not been solved, is programming in terms of risk and
uncertainty using distributed coefficients*
Allen, R. G. D., Mathematical Economics, Macmillan and Co. Ltd., London, 1957.
American Society of Agricultural Engineers, Agricultural Engineers Yearbook,
2nd edition, 1955.
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The Macmillan Co., New York, 1947.
Botts, Ralph R., Amortization of Loans, Its Application to Farm Problems,
United States Department of Agriculture, Agricultural Research Service,
Washington, D. C., May, 1954.
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at Particular Locations, Iowa State College Research Bulletin 426,
Agricultural Experiment Station, Ames, Iowa, April, 1955.
Bradford, Lawrence A. and Johnson, Glenn L., Farm Management Analysis, John Wiley
and Sons, Inc., New York, 1953.
Brown, B. A., Snyder, W. W., Hoglund, C. R. and Boyd, J. S., "Comparing Efficiency
of Herringbone with Other Type Milking Parlor," unpublished article,
Michigan Agricultural Experiment Station, Michigan State University,
Brown, Lauren H., Farming Today, Michigan State University Cooperative Extension
Service Department of Agricultural A, Ec. 751 (Area 5), 1959.
Candler, Wilfred, "A Modified Simplex Solution for Linear Programming with
Variable Capital Restrictions," Journal of Farm Economics, Vol. 38
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Variable Prices," Journal of Farm Economicsp Vol. 39 (May, 1957), p. 409.
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Analysis, McGraw-Hill Book Co., Inc., New York, 1958.
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unpublished Ph. D. Thesis, Michigan State University, 1958.
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Resource Fixity and Discrete Investment Levels
(Text reference pp. 25-26)
First, consider the case of a resource which is acquired (positive investment).
In the optimum solution, the MVP of the resource will equal its acquisition
value. Assuming some fractional acquisition level in the optimum solution, the
rule for obtaining discreteness will be applied,
For the discrete level in which the fraction is dropped (next lower discrete
level), one would expect the MVP to be greater than in the optimal solution
since a smaller amount of the resource is combined with at least as great an
amount of the other resources. Immediately, then, the resource in question is
no longer economically fixed (PMVP>Ca). But at this lower discrete level, the
second asset to be fixed in discrete units will, in all probability, itself be
at a fractional amount. Fixing the second asset at a lower level will generally
decrease the MVP of the first resource, and, conversely, fixing the second asset
at the next higher level will further increase the MVP of the first asset con-
sidered, Thus, if all succeeding assets are fixed at the next higher discrete
level, the WVP of the first will, in general, continue to increase, diverging
more and more from its acquisition cost. At some point, it may become profitable
to acquire an additional full unit of the first resource.
In the case when the first resource is initially fixed at the next higher
discrete level, its WMP will decrease relative to that in the initial optimum
solution. If, in the new solution, the MVP>Vs, there is no problem--the
resource remains flxed. Should the MWP become less than the value of salvage
and continue to decrease as more assets are fixed at discrete levels, it may
become profitable, at some point, to salvage one full unit of the first resource,
The argument in favor of using this method to deal with indivisibility could
be based on attaching equal probabilities to all values taken by the MVP of
one resource as others are fixed at discrete levels* Given this assumption,
the greater the probability of the MVP of the resources fixed at discrete
levels, falling between these values and the resource actually being economically
fixed in the final solution.
It is quite evident, however, that the distribution of the values of the MVP
of a resource when fixing other resources at discrete levels is not a uniform
distribution. It seems much less likely that either of the extreme cases
discussed above will occur than that some intermediate point will be reached.
Thus, one would expect a distribution more like the normal distribution with a
mean near or equal to the acquisition price. If the MVP values are normally
distributed about the acquisition price, then it is equally likely that the
final MVP will be greater than the acquisition price as below acquisition price.
In this case, too, however, the greater the differential between acquisition and
salvage values, the more likely the asset will be economically fixed in the
CROP ACTIVITY TITLES AND PROFIT COEFFICIENTS
Description Profit or CJ
Number Crop Unit % Cut for Silage Fertiliser Level Coefficient
_ ~ _~_ __
DAIRY ACTIVITY TITLES AND PROFIT COEFFICIENTS, PER COW
.. ........... ... .. ......... ...- . .. ..o. .,o,
Description t or
Labor CJ Coefficient
Number Ration No. Parlor Type Efficiency level Silo Type Dollars
Description Profit or
Labor C Coefficient
Number Ration No# Parlor Type Efficiency Level Silo Type Dollars
____ __. ~__ _~~_~___ _,____~_ ___ .. ,,_ __ __.~_~_ _
ACQUISITION, CREDIT AND SALVAGE ACTIVITY TITLES AND
Profit or Cj
Number Description Coefficients
Acquisition, Upright silo
Acquisition, Bunker silo
Acquisition, Herringbone parlor
Acquisition, Walkthrough parlor
Acquisition, Automatic feed bunk
Acquisition, Disc, drill
Acquisition, Disc, planter
Acquisition, Cultivator, sprayer
Acquisition, Mower, rake
acquisition, Fertilizer spreader
Acquisition, Corn picker
Acquisition, Bulk tank
Acquisition, Loafing area, per cow
Acquisition, Cow and replacements
Acquisition, Non-auto silage feed bunk, per cow
Acquisition, Hay storage and feeding, per cow
Acquisition, Corn, 100 bushels
Acquisition, Hay, 10 tons
Acquisition, April labor, 260 hours
Acquisition, May labor, 260 hours
Acquisition, June labor, 260 hours
Acquisition, July labor, 260 hours
Acquisition, August labor, 260 hours
Acquisition, September labor, 260 hours
Acquisition, October labor, 260 hours
Acquisition, November to March labor, 1300 hours
Land acquisition, cash and mortgage, 10 acres
Land acquisition, contract, 10 acres
Credit acquisition, land and mortgage, $100
Credit acquisition, 6% land contract, $100
Credit acquisition, 7% land contract, $100
Credit acquisition, land mortgage, $100
Credit acquisition, chattel mortgage, $100
Credit acquisition, silo dealer, $100
Profit or Cj
Number Description Coefficients
153 Credit acquisition, machinery dealer, $100 13.00
154 Salvage, tractor 356.70
155 Salvage, plow 18.63
156 Salvage, disc, drill 88.37
157 Salvage, disc, planter 48.22
158 Salvage, cultivator, sprayer 55.21
159 Salvage, chopper 279.31
160 Salvage, wagon 33.82
1161 Salvage, mower, rake 98.81
162 Salvage, fertilizer spreader 26.10
163 Salvage, corn picker 156.31
164 Salvage, cow and replacements 42.39
165 Salvage, corn, 100 bushels 90.00
166 Salvage, hay, 10 tons 175.00
167 Salvage, land, 10 acres 12.50
168 Salvage, summer labor, 14 days 175.00
169 Salvage, winter labor, 10 days 125.00
170 Salvage, cash, l$000 40O00
171 Salvage, bulk tank 173,25
172 Credit repayment, $1000 55.00
173 Positive unit vector, sum equation, penalty 444I.00
174s Negative unit vector, sum equation 0.00
175 Salvage hay equipment
1 All hay equipment was combined for the final computations. The set includes
the disc and drill, mower and rake, and the fertilizer spreader.
INITIAL OPTIMUM SOLUTION
Organization and Imputed Values
Organization T Selected Imputed Values
Activity No. Unit Amount Resource Unit MVP Activity No. Delta J
Unused 5.5% credit
Unused 6.5% credit
Unused 7% credit
1 Acres coverable.
Auto bunk capacity
6% land contract
Bulk tank capacity
2 Tillable acres.
w WLIWW . .
P I .
OPTIMUM SOLUTION, 320 ACRES
Organization and Imputed Values
Organization Selected Imputed Values
Activity No, Unit -Amount source Unit MVP Activity No. Delta J
5 Acres 60,00 Disc, drill AC1 $ 0.79 I $ 3.23
Unused 5.5% c
Unused 6.5% c
Auto bunk capacity
Cow and replacement
6% land contract
Bulk tank capacity
1 Acres co -able. 2 Tillable acres.
OPTIMUM SOLUTION, 320 ACRES, 4 TRACTORS
Organization and Imputed Values
Activity Organization uSelected Impted Vales
Number Unit Amount Resource Unit MVP
with this activity in this and succeeding solutions.
1 Selling of crops is combined
OPTIMUM SOLUTION, 320 ACRES, 4 TRACTORS
Organization and Imputed Values
Activity Organization Selected Imputed Values
Number Unit Amount Resource Unit MVP
5 Acres 240.00 Hay equipment AC 0.99
6 do 0.0 May tractor 1 hour 7.69
19 do 28,0 May plow AC 0.32
175 Set 0.8 Disc, planter do 0.63
137 260 hours 0.05 Cultivator, sprayer do 0,58
138 do 1.85 Sept. chopper do 0.00
139 do 1.55 Oct. wagon do 1.35
140 do 0.07 Corn picker do 3.13
141 do 0.40 Labor 1 hour 1.43
142 do 1.03 Cash 1 $100 6.50
143 do 1.96 5.5% credit do 1.00
129 One unit 1.4 6.5% credit do 0.00
123 do 0.75 Land Til. acre 20.12
126 do 2.80
124 do 0.48
121 do 1.22
Unused cash $100 0.00
credit do 0.00
credit do 69.18
OPTIIUM SOLUTION--320 ACRES, 4 TRACTORS, 1 CHOPPER, 3 CORN PICKERS
Organization and Iuted Values
Sganization Selected Imputed Values
Activity No. Unit Amount Resource Unit VP
5 Acres 240.00 Hay equipment AC 0.99
6 do 0.00 May tractor 1 hour 8.94
19 do 28,00 May plow AC 0.32
175 Set 0.80 Disc, planter do 0.63
137 260 hours 0.05 Cultivator, sprayer do 0.58
138 do 1,85 Sept. chopper do 0.00
139 do 1.55 Oct. wagon do 1.35
140 do 0.07 Corn picker do 0.00
141 do 0.40 Labor 1 hour 1.43
142 do 1.03 Cash 1 $100 6.50
143 do 1.96 5.5 credit do 1.00
123 One unit 0.75 6.5% credit do 0.00
126 do 2.80 Land 20.12
124 do 0.48
121 do 1.22
Unused cash $100 0.00
credit do 0.00
credit do 60.49
OPTIMUM SOLUTION, FINAL FAPE PLAN
Organization and Imyuted Values
Organization Selected and Iuted Values
Activity No. Uit Amount Resource Unit MV
$ Acres 80.00 Hay equipment AC 0.00
6 do 136,00 May tractor 1 hour 0.00
19 do 52.00 May plow AC 17.84
137 260 hours 0.09 Disc, planter do 0.00
138 do 1.66 Cultivator, sprayer do 0.00
139 do 1.78 Sept. chopper do 0.00
140 do 0.13 October wagon do 0.85
141 do 0.75 Corn picker do 0.00
142 do 0.34 Labor 1 hour 1.43
143 do 2.04 Cash $100 6.50
Unused cash $100 0.00 5.5% credit do 1.00
Unused 5.5 6.5% credit do 0.00
credit do 0.00 Land Til. acre 22.1
credit do 53.03
PRICE, CREDIT TERMS AND DEPRECIATION
Including Credit Annual Annual
Asset Description Pricel Credit Fee Balance Terms Payment Depreciation2
Dollars Dollars Dollars Years $/100 Dollars
Tractor, 2-3 plow 3480 874 2610 3 42,51 348.00
Plow, 2 x 14" 257 68 193 3 do 18.00
Disc, drill 1153 292 865 3 do 84.50
Corn planter, disc, 2 row 613 157 460 3 do 46.69
Cultivator, sprayer, 2 row 625 160 469 3 do 53.65
Chopper, PTO, Silo filler 2725 685 204i 3 do 272.50
Wagon 330 86 248 3 do 33.00
Mower, side rake, 7' 1125 285 844 3 do 96.00
Fertilizer spreader, 10' 360 94 270 3 do 25.20
Corn picker, pull type, 1 row 1525 385 1143 3 do 152.50
Upright silo, 20'x 60', complete 6669 3169 3500 3 39.76 284.823
Bunker silo, 6x30x130, complete 5100 5100 714.003
Bulk tank, 500 gal. 3300 829 2475 3 42.51 165.00,
Double 3 Walkthrough, complete 6175 3775 2400 3 do 688.254
Double 6 Herringbone, complete 9359 5527 3832 3 do 1063.85
Loafing area for 10 cows 800 800 6.0
Hay storage, feeding, 10 cows 400 400 32.004
Feed bunk, non auto, 10 cows 50 50 8
Feed bunk, auto, 10 cows 852 -- -- 161.88
Cow and replacements 332 332 156
Land, cash purchase, 10 acres 2900 2500 -
Land, mortgage, 10 acres 2500 1375 1125 20 8.40 -
Land, 6% contract, 10 acres 2500 250 2250 20 8.70
land, 7% contract, 30 acres 2500 250 2250 20 9.40
- Prices were obtained from various dealers. Dealer prices were then increased
2 Depreciation rates from: Nielson, James M., Application of the Budget Method
Ph.D. thesis, Harvard University, Cambridge, Massachusetts, 1953.
SIncludes repairs, taxes, insurance.
uniform y by 10%.
in Farm Planning, unpublished
COST OF MACHINERY REPAIR
Repairs as Percent of
Machine Machine Cost L
Corn planter 3
Side rake 2
Fertilizer spreader 3
Corn picker 7
Silo filler 7
1 Data from: Nielson, James M., op cit
FERTILIZER APPIaCATION AND CROP YIELD ESTIMATES1
Oat Hay Hay Corn Corn
Item Silage Silage Grain Silage
5-20-10 200 Ibs 210 Ibs. 210 Ibs.
0-20-20 60 Ibs 60 Ibs.
Yield 5.0 tone 2.5 tons 7.5 tons 60 bu. 10.6 tons
5-20-10 300 Ibs 250 250
0-20-20 200 Ibs. 200 Ibs.
Sidedress, N 40 Ibs. 40 lbs.
Yield 8.0 tons 3.4 tons 10.2 tons 76 bu. 12.5 tons
5-20-10 400 lbs 300 Ibs. 300 lbs,
0-20-20 300 Ibs. 300 Ibs.
Sidedress, N 30 Ibs. 80 Ibs. 80 lbs.
Yield 8.5 tons 4.2 tons 12.6 tons 90 bu. I5 tons
1 Data modified from: Hoglund, C. R., and Cook, R. L., Higher Profits from:
Fertilizer and Iproved Practices. Agricultural Economics Eimeo 545, Michigan
State University Agricultural Experiment Station and Soil Science Department,
East Lansing, October, 1956. The high roles and yields are from unpublished
data by the same authors.
TIME REQUIREMENTS FOR FIELD OPERATIONS1
Acres per Hours per Acres per
Operation Hour Acre 8 Hour Day
Plow 0.90 1.11 7.2
Disc 2.80 0.36 22.4
Drill 3.50 0,29 28.0
Plant corn 1.90 0.53 15.2
Cultivate 2.40 0.42 19.2
Spray weeds 2.50 0.40 20.0
Pick corn 0.75 1.33 6.0
Mow hay 2.0 0.50 16,0
Rake hay 1.9 0.53 15.2
Chop hay 1.1 0.91 8.8
Chop corn 0.8 1.25 6.4
Spread fertilizer 1.5 0.67 12,0
1 Primarily from: American Society of Agricultural Engineers, Agricultural
Engineers Yearbook, 2nd Edition, 1955, p. 89.
NMER OF FIELD WORKING DAYS PER MONTH1
1 Data from unpublished sources.
DAIRY AND CROP CASH COSTS1
Item Unit Amount
Fuel and oil Per hour tractor time $ 0.70
Alfalfa seed Bushel 25.00
Oat seed Bushel 1.45
Corn seed Bushel 12.50
5-20-10 Ton 79.20
0-20-20 Ton 47.55
45-0-0 Ton 118.00
Weed spray Per acre 3.00
elec., etc. Per head 20.00
Milk for calves Per head 4.00
Bedding Per head 24.00
1 Data from various unpublished sources.
DAIRY LABOR REQUIREMENTS1
Parlor Level of Type of Minutes per
Type Efficiency Silo Day per Cow2
Stanchion average Upright 17.76
Stanchion efficient Upright 10.56
Stanchion efficient Bunker 10.56
Walkthrough average Upright 12.06
Walkthrough efficient Upright 7.50
Walkthrough efficient Bunker 7.50
Herringbone average Upright 10.92
Herringbone efficient Upright 6.90
Herringbone efficient Bunker 6.90
1 Primarily from: Hoglund, C. R., Boyd, J. S. and Snyder, W. W. "Herringbone
and Other Milking Systems," Quarterly Bulletin, Michigan Agricultural Experiment
Station, Michigan State University, East Lansing, Vol. 41, No. w (February, 1959)
and Hoglund, C. R. and Wright, K. T., Reducing Dairy Costs on Michigan Farms
Michigan State University Agricultura Experiment Station Special Bulletin 376,
East Lansing, May, 1952.
2 Includes care of the entire herd.
RATIONS AND PRODUCTION FOR THE MILKiNG HERD, PER COW1
Silage Hay Corn
Per Per Per Per Per Per Total Production
Ration Day Year TDN Day Year TDN Day Year TDN TDN Per Year
NIumber Pounds ounounds P ds unds Pounds Pounds Pounds Pounds Pounds Pounds Pounds Pounds
1 50 18,300 3,600 10 3,650 1,825 6.8 2,470 1,975 7,400 11,000
2 50 18,300 3,600 10 3,650 1,825 5.7 2,090 1,675 7,100 10,500
3 50 18,300 3,600 10 3.650 1,825 4.7 1,720 1,375 6,800 10,000
4 40 14,600 2,920 12 4,380 2,190 7.8 2,860 2,290 7,400 11,000
5 40 14,600 2,920 12 4,380 2,190 6.8 2,490 1,990 7,100 10,500
6 40 14,600 2,920 21 4,380 2,190 5.8 2,120 1,690 6,800 10,000 0
7 30 10,950 2,190 15 5,470 2,740 8.5 3,090 2,470 7,400 11,000
8 30 10,950 2,190 15 5,470 2,740 7.4 2,720 2,170 7,100 10,500
9 30 10,950 2,190 15 5,470 2,740 6.4 2,340 1,870 6,800 10,000
1 Modified from: Jensen, Einar, et al., Input-Output Relationships in Milk Production,United States
Department of Agriculture Technical Bulletin No. 815, Washington D. C., May, 1942
RATIONS IN HAY EQUIVALENTS AND CORN EQUIVALENTS
PER COW PER YEAR, INCLUDES REPLACEMENTS
Hay, Silage, by Total Grain, Grain, Total
Ration Silage, Cows Cows Replacements Roughage Cows Replacement Grain
Number Tons Tons HE Tons HE Ton Tons HE Bu CE Bu CE
1 9.15 3.05 1.82 2.00 6.87 t4.1 13.3 57.4
2 9.15 3.05 1.82 2.00 6.87 37.4 13.3 50.7
3 9.15 3.05 1.82 2.00 6.87 30.7 13.3 44.0
4 7.30 2.43 2.19 2.00 6.62 51.1 13.3 614.
5 7.30 2.43 2.19 2.00 6.62 44.5 13.3 57.8
6 7.30 2.43 2.19 2.00 6.62 37.9 13.3 51.2
7 5$.7 1.82 2.74 2.00 6.56 55.2 13.3 68.5
8 5.47 1.82 2.74 2,00 6.56 48.6 13.3 61.9
9 5.47 1.82 2.74 2.00 6.56 41.8 13.3 55.1
AN EXAMPLE OF THE COMPUTATION OF MACHINE
AND POWER RESTRICTIONS
1. April power restriction
(a) Plowing is the most limiting restriction
(b) One tractor can plow 7.2 acres per day
(c) Twelve field working days in April
(d) One tractor can plow a maximum of 86.6 acres in April.
2. April disc and drill restriction
(a) One tractor can disc (2.8)(8) = 22.4 acres per day
(b) One tractor can drill (3.5)(8) = 28.0 acres per day
(c) Set up a set of simultaneous equations where:
x number of days to disc
y = number of days to drill
We have x + y = 12
22.4x = 28.0 y
Disc 6.67 days
Drill 5.33 days
(e) Thus, one tractor can disc (6.67)(22.4) = 1h9 acres
and one tractor can drill (5.33)(28.0) = 149 acres.